Lecture Notes in Applied and Computational Mechanics Volume 52 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. D...

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Lecture Notes in Applied and Computational Mechanics Volume 52 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers

Lecture Notes in Applied and Computational Mechanics Edited by F. Pfeiffer and P. Wriggers Further volumes of this series found on our homepage: springer.com Vol. 52: Pilipchuk, V.N. Nonlinear Dynamics: Between Linear and Impact Limits 364 p. 2010 [978-3-642-12798-4]

Vol. 40: Pfeiffer, F. Mechanical System Dynamics 578 p. 2008 [978-3-540-79435-6]

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Nonlinear Dynamics Between Linear and Impact Limits

Valery N. Pilipchuk

123

Valery N. Pilipchuk Professor (Research) Mechanical Engineering Wayne State University 5050 Anthony Wayne Dr., 2140 Detroit, Michigan 48202 USA E-mail: [email protected]

ISBN: 978-3-642-12798-4

e-ISBN: 978-3-642-12799-1

DOI 10.1007/ 978-3-642-12799-1 Lecture Notes in Applied and Computational Mechanics

ISSN 1613-7736 e-ISSN 1860-0816

Library of Congress Control Number: 2010925555 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

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Preface

The main objective of this book is to introduce a uniﬁed physical basis for analyses of vibrations with essentially unharmonic, non-smooth or may be discontinuous time shapes. It is known that possible transitions to nonsmooth limits can make investigations especially diﬃcult. This is due to the fact that the dynamic methods were originally developed within the paradigm of smooth motions based on the classical theory of diﬀerential equations. From the physical standpoint, these represent low-energy approaches to modeling dynamical systems. Although the impact dynamics has also quite a long pre-history, any kind of non-smooth behavior is often viewed as an exemption rather than a rule. Similarly, the classical theory of diﬀerential equations usually avoids non-diﬀerentiable and discontinuous functions. To-date, however, many theoretical and applied areas cover high-energy phenomena accompanied by strongly non-linear spatio-temporal behaviors making the classical smooth methods ineﬃcient in many cases. For instance, such phenomena occur when dealing with dynamical systems under constraint conditions, friction-induced vibrations, structural damages due to cracks, liquid sloshing impacts, and numerous problems of nonlinear physics. Similarly to the wellknown analogy between mechanical and electrical harmonic oscillators, some electronic instruments include so-called Schmitt trigger circuits generating nonsmooth signals whose temporal shapes resemble mechanical vibro-impact processes. In many such cases, it is still possible to adapt diﬀerent smooth methods of the dynamic analyses through strongly non-linear algebraic manipulations with state vectors or by splitting the phase space into multiple domains based on the system speciﬁcs. As a result, the related formulations are often reduced to discrete mappings in a wide range of the dynamics from periodic to stochastic. Possible alternatives to such approaches can be built on generating models developing essentially nonlinear/unharmonic behaviors as their inherent properties. Such models must be general and simple enough in order to play the role of physical basis. As shown in this book, new generating systems can be found by intentionally imposing the ‘worst case scenario’ on conventional methods in anticipation that failure of one asymptotic may

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point to its complementary counterpart. However, the related mathematical formalizations are seldom straightforward and require new principles. For instance, the tool developed here employs nonsmooth (impact) systems as a basis to describing not only impact but also smooth or even linear dynamics. This is built on the idea of non-smooth time substitutions/transformations (NSTT) proposed originally for strongly nonlinear but still smooth models. On the author’s view, the methodological role of NSTT is to reveal explicit links between impact dynamics and hyperbolic algebras analogously to the link between harmonic vibrations and conventional complex analyses. In particular, this book gives the ﬁrst systematic description for NSTT and related analytical and numerical algorithms. The text focuses on methodologies and discussions of their physical and mathematical basics. Detailed applications are mostly excluded from this book, however, necessary references on journal publications are provided. The material of this book was prepared during several years of work at Dnipropetrovsk National University and Technological University (Ukraine), Wayne State University, and Research and Development of General Motors Corporation (Michigan, USA). The author greatly appreciates discussions on diﬀerent subjects related to this book during diﬀerent periods of time with Professors I.V. Andrianov, R.A. Ibrahim (Wayne State University), L.I. Manevich (Russian Academy of Science), A.A.Martynyuk (National Academy of Science of Ukraine), Yu.V. Mikhlin (Kharkov Polytechnic Institute), A.F. Vakakis (University of Illinois), A.A. Zevin (National Academy of Science of Ukraine), and V.F. Zhuravlev (Russian Academy of Science). The author also would like to thank Dr. Kelly Cooper (ONR) for her support and attitude to fundamental research that was very helpful at the ﬁnal stage of work on this book.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Brief Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Asymptotic Meaning of the Approach . . . . . . . . . . . . . . . . . . . . 1.2.1 Two Simple Limits of Lyapunov Oscillator . . . . . . . . . . 1.2.2 Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis . . . . . . . . . . . . . . . . . . . 1.3 Quick ‘Tutorial’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Remarks on the Basic Functions . . . . . . . . . . . . . . . . . . . 1.3.2 Viscous Dynamics under the Sawtooth Forcing . . . . . . 1.3.3 The Rectangular Cosine Input . . . . . . . . . . . . . . . . . . . . 1.3.4 Oscillatory Pipe Flow Model . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Periodic Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Strongly Nonlinear Oscillator . . . . . . . . . . . . . . . . . . . . . 1.4 Geometrical Views on Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Geometrical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Nonlinear Equations and Nonlinear Phenomena . . . . . 1.4.3 Rigid-Body Motions and Linear Systems . . . . . . . . . . . 1.4.4 Remarks on the Multi-dimensional Case . . . . . . . . . . . . 1.4.5 Elementary Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Example of Simpliﬁcation in Nonsmooth Limit . . . . . . 1.4.7 Non-smooth Time Arguments . . . . . . . . . . . . . . . . . . . . . 1.4.8 Further Examples and Discussion . . . . . . . . . . . . . . . . . . 1.4.9 Diﬀerential Equations of Motion and Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Non-smooth Coordinate Transformations . . . . . . . . . . . . . . . . . 1.5.1 Caratheodory Substitution . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Transformation of Positional Variables . . . . . . . . . . . . . 1.5.3 Transformation of State Variables . . . . . . . . . . . . . . . . .

1 1 4 4 6 9 9 9 11 12 15 15 17 17 19 21 23 24 25 26 28 30 33 33 33 35

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2

3

4

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Smooth Oscillating Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linear and Weakly Non-linear Approaches . . . . . . . . . . . . . . . . 2.2 A Brief Overview of Smooth Methods . . . . . . . . . . . . . . . . . . . . 2.2.1 Periodic Motions of Quasi Linear Systems . . . . . . . . . . 2.2.2 The Idea of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Averaging Algorithm for Essentially Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Averaging in Complex Variables . . . . . . . . . . . . . . . . . . . 2.2.5 Lie Group Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 38 39

Nonsmooth Processes as Asymptotic Limits . . . . . . . . . . . . . 3.1 Lyapunov’ Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlinear Oscillators Solvable in Elementary Functions . . . . . 3.2.1 Hardening Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Localized Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Softening Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Nonsmoothness Hiden in Smooth Processes . . . . . . . . . . . . . . . 3.3.1 Nonlinear Beats Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Asymptotic of Equipartition . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Asymptotic of Dominants . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Necessary Condition of Energy Trapping . . . . . . . . . . . 3.4.4 Suﬃcient Condition of Energy Trapping . . . . . . . . . . . . 3.5 Transition from Normal to Local Modes . . . . . . . . . . . . . . . . . . 3.6 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Normal and Local Mode Coordinates . . . . . . . . . . . . . . . . . . . . 3.8 Local Mode Interaction Dynamics . . . . . . . . . . . . . . . . . . . . . . . 3.9 Auto-localized Modes in Nonlinear Coupled Oscillators . . . . .

51 51 54 56 59 60 61 62

Nonsmooth Temporal Transformations (NSTT) . . . . . . . . . . 4.1 Non-smooth Time Transformations . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Positive Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 ‘Single-Tooth’ Substitution . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 ‘Broken Time’ Substitution . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Sawtooth Sine Transformation . . . . . . . . . . . . . . . . . . . . 4.1.5 Links between NSTT and Matrix Algebras . . . . . . . . . 4.1.6 Diﬀerentiation and Integration Rules . . . . . . . . . . . . . . . 4.1.7 NSTT Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.8 Generalizations on Asymmetrical Sawtooth Wave . . . . 4.1.9 Multiple Frequency Case . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Idempotent Basis Generated by the Triangular Sine-Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Deﬁnitions and Algebraic Rules . . . . . . . . . . . . . . . . . . . 4.2.2 Time Derivatives in the Idempotent Basis . . . . . . . . . .

93 93 94 96 96 97 101 102 103 105 107

41 43 44

64 69 71 73 74 74 74 76 81 85

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4.3 Idempotent Basis Generated by Asymmetric Triangular Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Deﬁnition and Algebraic Properties . . . . . . . . . . . . . . . . 4.3.2 Diﬀerentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Oscillators in the Idempotent Basis . . . . . . . . . . . . . . . . 4.3.4 Integration in the Idempotent Basis . . . . . . . . . . . . . . . 4.4 Discussions, Remarks and Justiﬁcations . . . . . . . . . . . . . . . . . . 4.4.1 Remarks on Nonsmooth Solutions in the Classical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Caratheodory Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Other Versions of Periodic Time Substitutions . . . . . . 4.4.4 General Case of Non-invertible Time and Its Physical Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 NSTT and Cnoidal Waves . . . . . . . . . . . . . . . . . . . . . . . .

IX

112 112 114 115 116 117 118 119 122 125 125

5

Sawtooth Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Manipulations with the Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Smoothing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sawtooth Series for Normal Modes . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Periodic Version of Lie Series . . . . . . . . . . . . . . . . . . . . . 5.3 Lie Series of Transformed Systems . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Second-Order Non-autonomous Systems . . . . . . . . . . . . 5.3.2 NSTT of Lagrangian and Hamiltonian Equations . . . . 5.3.3 Remark on Multiple Argument Cases . . . . . . . . . . . . . .

131 131 131 135 135 138 138 141 144

6

NSTT for Linear and Piecewise-Linear Systems . . . . . . . . . 6.1 Free Harmonic Oscillator: Temporal Quantization of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Non-autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Standard Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Idempotent Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Systems under Periodic Pulsed Excitation . . . . . . . . . . . . . . . . 6.3.1 Regular Periodic Impulses . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Harmonic Oscillator under the Periodic Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Periodic Impulses with a Temporal ‘Dipole’ Shift . . . . 6.4 Parametric Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Piecewise-Constant Excitation . . . . . . . . . . . . . . . . . . . . 6.4.2 Parametric Impulsive Excitation . . . . . . . . . . . . . . . . . . 6.4.3 General Case of Periodic Parametric Excitation . . . . . 6.5 Input-Output Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Amplitude-Phase Equations . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Amplitude Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 147 147 148 149 149 151 155 157 157 159 161 163 165 166 167

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6.6.3 Phase Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Remarks on Generalized Taylor Expansions . . . . . . . . . 6.7 Multiple Degrees-of-Freedom Case . . . . . . . . . . . . . . . . . . . . . . . 6.8 The Amplitude-Phase Problem in the Idempotent Basis . . . . 7

8

9

Periodic and Transient Nonlinear Dynamics under Discontinuous Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Nonsmooth Two Variables Method . . . . . . . . . . . . . . . . . . . . . . 7.2 Resonances in the Duﬃng’s Oscillator under Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Strongly Nonlinear Oscillator under Periodic Pulses . . . . . . . . 7.4 Impact Oscillators under Impulsive Loading . . . . . . . . . . . . . . Strongly Nonlinear Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Periodic Solutions for First Order Dynamical Systems . . . . . . 8.2 Second Order Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 8.3 Periodic Solutions of Conservative Systems . . . . . . . . . . . . . . . 8.3.1 The Vibroimpact Approximation . . . . . . . . . . . . . . . . . . 8.3.2 One Degree-of-Freedom General Conservative Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero . . . . . . . . . . . . . . . . . . . 8.3.5 A Generalized Case of Odd Characteristics . . . . . . . . . 8.4 Periodic Motions Close to Separatrix Loop . . . . . . . . . . . . . . . 8.5 Self-excited Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Strongly Nonlinear Oscillator with Viscous Damping . . . . . . . 8.6.1 Remark on NSTT Combined with Two Variables Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Oscillator with Two Nonsmooth Limits . . . . . . . . . . . . 8.7 Bouncing Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 The Kicked Rotor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 172 173 177 179 179 182 185 189 195 195 196 198 198 202 205 207 209 211 214 218 222 225 230 234 235

Strongly Nonlinear Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.1 Wave Processes in One-Dimensional Systems . . . . . . . . . . . . . . 241 9.2 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

10 Impact Modes and Parameter Variations . . . . . . . . . . . . . . . . 10.1 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Parameter Variation and Averaging . . . . . . . . . . . . . . . . . . . . . . 10.3 A Two-Degrees-of-Freedom Model . . . . . . . . . . . . . . . . . . . . . . . 10.4 Averaging in the 2DOF System . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Impact Modes in Multiple Degrees of Freedom Systems . . . .

245 245 249 252 253 256

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10.5.1 A Double-Pendulum with Amplitude Limiters . . . . . . . 258 10.5.2 A Mass-Spring Chain under Constraint Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 10.6 Systems with Multiple Impacting Particles . . . . . . . . . . . . . . . . 262 11 Principal Trajectories of Forced Vibrations . . . . . . . . . . . . . . 11.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Principal Directions of Linear Forced Systems . . . . . . . . . . . . . 11.3 Deﬁnition for Principal Trajectories of Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Asymptotic Expansions for Principal Trajectories . . . . . . . . . . 11.5 Deﬁnition for Principal Modes of Continuous Systems . . . . . .

265 265 267

12 NSTT and Shooting Method for Periodic Motions . . . . . . . 12.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Sample Problems and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Smooth Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Step-Wise Discontinuous Input . . . . . . . . . . . . . . . . . . . . 12.3.3 Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Periodic Solutions of the Period - n . . . . . . . . . . . . . . . . 12.4.2 Two-Degrees-of-Freedom Systems . . . . . . . . . . . . . . . . . 12.4.3 The Autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 275 277 279 279 286 286 290 290 293 294

13 Essentially Non-periodic Processes . . . . . . . . . . . . . . . . . . . . . . 13.1 Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Impulsively Loaded Dynamical Systems . . . . . . . . . . . . . . . . . . 13.2.1 Harmonic Oscillator under Sequential Impulses . . . . . . 13.2.2 Random Suppression of Chaos . . . . . . . . . . . . . . . . . . . .

295

14 Spatially-Oscillating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Periodic Nonsmooth Structures . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Averaging for One-Dimensional Periodic Structures . . . . . . . . 14.3 Two Variable Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Acoustic Waves from Non-smooth Periodic Boundary Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Spatio-temporal Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Membrane on a Two-Dimensional Periodic Foundation . . . . . 14.8 The Idempotent Basis for Two-Dimensional Structures . . . . .

268 269 271

295 298 301 303 305 305 312 313 315 319 323 326 332

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Chapter 1

Introduction

Abstract. This chapter contains physical and mathematical preliminaries with diﬀerent introductory remarks. Although some of the statements are informal and rather intuitive, they nevertheless provide hints on selecting the generating models and corresponding analytical techniques. The idea is that simplicity of a mathematical formalism is caused by hidden links between the corresponding generating models and subgroups of rigid-body motions. Such motions may be qualiﬁed indeed as elementary macro-dynamic phenomena developed in the physical space. For instance, since rigid-body rotations are associated with sine waves and therefore (smooth) harmonic analyses then translations and mirror-wise reﬂections must reveal adequate tools for strongly unharmonic and nonsmooth approaches. This viewpoint is illustrated by physical examples, problem formulations and solutions.

1.1

Brief Literature Overview

Analytical methods of conventional nonlinear dynamics are based on the classical theory of diﬀerential equations dealing with smooth coordinate transformations, asymptotic integrations and averaging [96], [27], [87], [124], [204], [160], [79], [50], [114], [86], [8], [50], [161], [77], [89], [115], [120], [117], [116], [204], [97], [62]. The corresponding solutions often include quasi harmonic expansions as a generic feature that explicitly points to the physical basis of these methods namely - the harmonic oscillator. Generally speaking, some of the techniques are also applicable to dynamical systems close to integrable but not necessarily linear. However, nonlinear generating solutions are seldom available in closed form [9]. As a result, strongly nonlinear methods usually target speciﬁc situations and are rather diﬃcult to use in other cases. Generating models for strongly nonlinear analytical tools with a wide range of applicability must obviously 1) capture the most common features of oscillating processes regardless their nonlinear speciﬁcs, 2) possess simple enough solutions in order to provide eﬃciency of perturbation schemes, and 3) V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 1–36, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

2

1 Introduction

describe essentially nonlinear phenomena out of the scope of the weakly nonlinear methods. So the key notion of the present work suggests possible recipes for selecting such models among so-called non-smooth systems while keeping the class of smooth motions still within the range of applicability. Note that diﬀerent non-smooth cases have been also considered for several decades by practical and theoretical reasons [81], [39], [17], [42], [127], [72], [196], [82], [204], [172], [45], [65], [204], [169], [29], [14], [93], [30], [180], [51], [206], [94], [44], [173], [25], [162], [55], [185], [85], [130], [129], [128], [1], [68], [69], [37]. On the physical point of view, this kind of modeling essentially employs the idea of perfect spatio-temporal localization of strong nonlinearities or impulsive loadings. For instance, sudden jumps of restoring force characteristics are represented by absolutely stiﬀ constraints under the assumption that the dynamics in between the constraints is smooth and simple enough to describe. As a result, the system dynamics is discretized in terms of mappings and matchings diﬀerent pieces of solutions. The present work however is rather close to another group of methods dealing with the diﬀerential equations of motion on entire time intervals despite of discontinuity and/or nonsmoothness points. Such methods are developed to satisfy the matching conditions automatically by means of speciﬁc coordinate transformations on preliminary stages of study. To some extent, these can be qualiﬁed as adaptations of the diﬀerential equations of motion for further studies by another methods. Among such kind of transformations, the Caratheodory substitution [45] can be mentioned ﬁrst. This linear substitution, which includes the unit-step Heaviside function, eliminates Dirac deltafunctions participating as summands in diﬀerential equations. Much later, non-smooth nonlinear coordinate transformations were suggested in [199], [204] for the class of impact systems. This strongly nonlinear transformation eﬀectively eliminates stiﬀ barriers by unfolding the system conﬁguration space in a mirror-wise manner with respect to the barrier planes. A similar idea was implemented later regarding the system phase space [72] in order to resolve certain problems related to non-elastic impacts. Some technical details and discussions on these methods are included below in Section 5 for comparison reason. In contrast, the present approach employs time histories of impact systems as new time arguments. Originally such a nonsmooth temporal transformation (NSTT), or a sawtooth oscillating time1 , was introduced for strongly nonlinear but smooth periodic motions with certain temporal symmetries [132]. Then, it was shown that such an approach still works for general cases by generating speciﬁc algebraic structures in terms of the coordinates [134], [144]. The occurrence of such algebraic structures seems to be essential 1

Here and further, the term ‘sawtooth’ is used for periodic piecewise-linear functions regardless symmetries of their teeth; in most cases, it is the triangular sine wave.

1.1 Brief Literature Overview

3

feature of the approach since it justiﬁes and simpliﬁes analytical manipulations with non-invertible temporal substitutions such as NSTT. Further, the technique was applied to diﬀerent problems of theoretical and applied mechanics [15], [101], [104], [103], [47], [189], [92], [91], [175], [170], [177], [171], [16], [179], [109], [181], [67]. Basic ideas and some of the tools were adapted earlier for the nonlinear normal mode analyses and included in the monograph [190]. However, much of the material presented in this book is either new or signiﬁcantly updated. The entire text is organized as follows. As mentioned in the Preamble, this Chapter 1 contains physical and mathematical preliminaries and diﬀerent introductory remarks. Chapter 2 gives a brief overview of selected analytical methods for smooth oscillating processes. The description focuses on the ideas and technical details that are used further in combinations with non-smooth approaches. In particular, the method of asymptotic integration of the diﬀerential equations of motion based on the Hausdorf equation for Lie operators is reproduced. Chapter 3 includes diﬀerent examples of smooth vibrating systems that, under some conditions, show close to non-smooth or even non-smooth time histories. Such cases are usually most diﬃcult for conventional analyses. Much of the content focuses on nonlinear beat and localization phenomena. Recent interest to this area is driven by the idea of nonlinear energy absorption. Chapter 4 provides a new description of NSTT with proofs of the basic identities and rules for algebraic, diﬀerential and integral manipulations. In particular, ﬁnal subsections show how to introduce nonsmooth arguments into the diﬀerential equations. Such manipulation imposes two main features on the dynamical systems namely generates speciﬁc algebraic structures for unknown functions and switches formulations to boundary-value problems. Notice that the transformation itself imposes no constraints on dynamical systems and easily applies to both smooth and nonsmooth systems. Any further steps, however, should account for physical properties of the related systems. Chapter 5 preliminary illustrates the NSTT’ advantages by introducing power-series expansions for general periodic processes. This becomes possible because the new temporal argument - the triangular sine wave - is itself periodic in the original time. Therefore, such expansions can be viewed as some alternative to Fourier series for processing periodic signals especially with step-wise discontinuities. Then a periodic version of the Lie series is introduced. As a result, formal analytical solutions for normal mode motions of dynamical systems are obtained. Chapters 6, 7 and 8 describe NSTT based analytical tools for linear, weakly nonlinear and strongly nonlinear vibrating systems, respectively. In particular, applying NSTT to linear and weakly nonlinear systems may be very eﬀective in those cases when nonsmooth loadings are present and thus harmonic or quasi harmonic approaches require multiple term expansions for capturing essential features of the dynamics.

4

1 Introduction

Chapters 9 through 11 deal with the concept of nonlinear normal modes. In particular, it is shown that NSTT leads to adequate formulations of the normal mode problem for impact systems. Also the idea of nonlinear normal modes for the case of forced vibration is formulated in terms of NSTT. Chapter 12 presents a semi-analytical approach combining NSTT with the shooting method that essentially extends the area of applications. Chapter 13 describes a possible physical basis for NSTT in case of essentially non-periodic processes. Finally, Chapter 14 illustrates diﬀerent applications to spatially oscillating structures such as one-dimensional elastic rods with periodic discrete inclusions and two-dimensional media with a periodic nonsmooth boundary source of waves.

1.2

Asymptotic Meaning of the Approach

1.2.1

Two Simple Limits of Lyapunov Oscillator

Let us introduce some preliminaries and remarks based on a one-degree-offreedom model as shown at the top of Fig. 1.1.2 This relatively simple model however depicts the gradual evolution from linear to extremely nonlinear dynamics as the exponent n runs from unity to inﬁnity. Notably, all the temporal mode shapes of the oscillator are described by special functions except two boundaries of the interval 1 ≤ n < ∞. Both boundaries represent simple asymptotic limits within the class of elementary functions. Consider ﬁrst the limit of harmonic oscillator, n = 1, that generates the sine and cosine waves as illustrated on the left column of the diagram in Fig. 1.1. From the very general standpoint, a widely known convenience of using this couple of functions can be explained by their certain link to the group of rigid-body motions namely the subgroup of rotations. The standard complex plane representation and the Euler formula can be mentioned here as related tools. Further, taking the linear combination of harmonic waves with diﬀerent frequencies and keeping in mind the idea of parameter variations invokes the area of harmonic and quasi harmonic analyses for both signal processing and dynamical systems. Such tools therefore represent complicated dynamic processes as a combination of the very simple rigid-body rotations with certain angular speeds. Let us consider now another limit n = ∞, when the restoring force vanishes inside the interval −1 < x < 1 but becomes inﬁnitely growing as the system reaches the potential barriers at x = ±1. The physical meaning of this limit is introduced at the top of the right column of the diagram. Despite of the strong nonlinearity caused by impacts, the limit oscillator is also 2

Note that oscillators with power-form characteristics were considered for quite a long time since possibly Lyapunov who obtained such oscillators while investigating degenerated cases in dynamic stability problems; see Chapter 3 for details.

1.2 Asymptotic Meaning of the Approach

5

..

x x2 n1 0 n

Harmonic oscillator

Impact oscillator

Sine and cosine waves Rotation group

Triangular and rectangular waves Translation and reflection groups

1

1

..

x x2 n1 0 n1

1

sin t cos t

Π

2Π

Τt

t et

1

2

4

Conventional complex numbers

Hyperbolic complex numbers

z x yi i2 1

x X Τ Y Τ e e2 1; e Τ t

Fourier series

Power series

k0 Ak cos kt Bk sin kt Quasi harmonic methods

k0

1 k

X k 0 Τ k

1 k

t

Y k 0 Τ k e

Strongly nonlinear methods

Fig. 1.1 Two asymptotic limits described by elementary functions associated with subgroups of rigid-body motions.

described by quite simple elementary functions such as triangular sine and rectangular cosine, say τ (t) and τ˙ (t). These two are associated with another subgroup of the rigid-body motions namely translation and reﬂection. Therefore, analogously to the above case n = 1, the upper limit n = ∞ can play the same fundamental role by generating an hierarchy of alternative to quasi harmonic tools. On ﬁrst look, such alternative tools can still be developed within the paradigm of Fourier expansions by taking appropriate multiple frequency combinations of τ (t) and τ˙ (t). Although such approaches could work for signal processing, it is unclear how to deal with a large number of singularities if substituting the related expansions into diﬀerential equations of motion. Indeed nonsmoothness of such Fourier basis seems to contradict

6

1 Introduction

Fig. 1.2 Complex and hyperbolic planes are shown on the left and right, respectively; in contrast to the circle, each of the hyperbola branches is covered exactly once as the hyperbolic angle is varying in the inﬁnite interval.

the very language of dynamical systems involving time derivatives.3 In addition, the Fourier expansions are closely linked to the linear superposition principle which is inadequate to nonlinear cases. Finally, the major role of Fig. 1.1 is to convince the reader that any presence of functions τ (t) and τ˙ (t) in further developed analytical algorithms is not a simple match of diﬀerent pieces of solutions4 . On the contrary, it has its real physical basis and invokes speciﬁc mathematical tools.

1.2.2

Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis

So instead let us introduce another formalism dealing with same basic frequency but adjusting the triangular sine shape to the desired one through 3

4

Generally speaking, it is possible to start with an expansion for high-order derivatives and then come backward to coordinates by integrations. However, algebraic complexity of such approaches may overshadow any advantages as compared to the regular Fourier expansions. It is quite clear that harmonic waves can also be interpreted as those combined of the same pieces of curves, but such a viewpoint would eliminate much of the vibration theory and many physical eﬀects.

1.2 Asymptotic Meaning of the Approach

7

polynomials and other simple functions of the argument τ . Such formalism is based on the following statement [133]: Any periodic process x (t) of the period normalized to T = 4 can be expressed through the dynamic state of the impact oscillator, {τ (t), τ˙ (t)}, in the form of ‘hyperbolic complex number’ x = X (τ ) + Y (τ ) τ˙

(1.1)

On the right-hand side, the functions X and Y are easily expressed through the original function x(t), if this function is known; see Chapter 4 for further details. In case x(t) is an unknown periodic motion of some dynamical system, equations for X and Y components are obtained by substituting (1.1) into the corresponding diﬀerential equation of motion. Then either analytical or numerical procedures can be applied. For instance, one may seek solutions in the form of power series with respect to the ‘oscillating time’ τ . Therefore expression (1.1) can be qualiﬁed as non-smooth time transformation, t → τ , on the manifold of periodic motions. Note that the structure of hyperbolic numbers has been known for quite a long time mostly as a formal extension of the regular complex numbers with no relation to either vibrating systems or nonsmooth functions; see the references below. In the mathematical literature, such an extension (which is often regarded to as a simple particular case of Cliﬀord’s algebras5) is introduced as follows. Whereas the algebraic equation p2 = 1 has the real number solutions p = ±1, the existence of a unipotent u is assumed such that u = +1 and u = −1 but u2 = 1. Then by considering the elements {1, u} as a standard basis, any hyperbolic number w ∈ H can be written in the form w = x + uy, where x and y are real numbers [176]. The hyperbolic √ conjugate of w is deﬁned by w− = x−uy so that |w|H = ww− is the norm of w. Fig. 1.2 illustrates the diﬀerence between complex and hyperbolic planes. More details and some, rather abstract, applications of this algebraic theory can be found in references [90], [176], [7], [187]. As to applied areas, the same kind of algebraic structures occurred in hydrodynamics in connection with characteristics of partial diﬀerential equations [90]. In our case, the unipotent is not a number but the discontinuous function of certain physical nature i.e. the rectangular cosine wave e(t). Indeed since t is running then there is no unique choice for the magnitude of e, whereas always6 e2 = 1. Therefore, identity (1.1) generates the hyperbolic structure from the very general properties of periodic processes. Note that both terms on the right-hand side of (1.1) are essential as those responsible for components with diﬀerent temporal symmetries. For instance, 5 6

William Kingdon Cliﬀord (1845-1879), English mathematician who, in particular, developed the idea that space may not be independent of time. More precisely, for almost any t.

8

1 Introduction

Fig. 1.3 Geometrical interpretation of the particular case with a sine-wave temporal symmetry: observing the coordinate x does not allow to conclude which of the two temporal variables, τ or t, is actually ‘running.’

Fig. 1.3 illustrates geometrical meaning of the particular case of temporal symmetry with no Y -component, where x(t) ≡ x[aτ (t/a)] = X(τ ). It is important to note that, under some conditions on X and Y , combination (1.1) can be of any class of smoothness even though the couple {τ, τ˙ } has singularities at such time instances t where τ = ±1. Finally, let us mention that the hyperbolic plane has another natural basis associated with the two isotropic lines separating the hyperbolic quadrants as shown in Fig. 1.2. The transition from one basis to another is given by e± = (1 ± e)/2 or, inversely, 1 = e+ + e− and e = e+ − e− . Therefore, x = X + Y e = X(e+ + e− ) + Y (e+ − e− ) = (X + Y )e+ + (X − Y )e− ≡ X+ (τ )e+ + X− (τ )e− where x = x(t) any periodic function as deﬁned in (1.1). whose period is normalized to T = 4. On one hand, the advantage is that the elements e+ and e− are mutually annihilating (idempotents) so that e+ e− = 0, e2− = e− and e2+ = e+ . It is clear also that ee+ = e+ and ee− = −e− . Due to the annihilation property, the idempotent basis signiﬁcantly eases diﬀerent algebraic manipulations, for instance, 2 2 e+ + X− e− (X+ e+ + X− e− )2 = X+ On the other hand, this basis usually couples the corresponding smoothness (boundary) conditions; see Chapter 4 for further details and examples.

1.3 Quick ‘Tutorial’

1.3

9

Quick ‘Tutorial’

1.3.1

Remarks on the Basic Functions

First note that the basic functions, τ (t) and τ˙ (t) are expressed through the standard elementary functions in the closed form as τ (t) = and τ˙ (t) = cos

πt 2 arcsin sin π 2

−1 πt πt = e(t) cos 2 2

(1.2)

(1.3)

Obviously τ (t) is a triangular sine wave whose amplitude is unity and the period is T = 4, whereas e(t) is a rectangular cosine wave with step-wise discontinuities as shown in Fig. 1.1. Therefore, expression (1.3) holds ‘almost everywhere.’ For calculating purposes, it can be represented in the form e(t) = sgn [cos(πt/2)]. Practically, for given t, one can calculate the functions τ (t) and e(t) based on their piecewise-linear graphs even with no tables nor calculators involved. Necessary mathematical details regarding the discontinuities of e(t) are discussed later; see also [144]. At this stage however, solutions can be compared with exact solutions obtained in a diﬀerent way or numerical ones in order to validate the technique. Moreover, during the derivations, there is no need in keeping in mind expressions (1.2) and (1.3). It is suﬃcient to take into account linear independence of the elements 1, e and e, ˙ as those from diﬀerent classes of smoothness, and the following properties τ˙ = e e˙ = 2

∞

[δ (t + 1 − 4k) − δ (t − 1 − 4k)]

(1.4)

k=−∞

e2 = 1 where, strictly speaking, the equalities should be interpreted in terms of the distribution theory due to the presence of singularities that occur whenever τ = ±1. Let us consider few examples illustrating the technique and giving a hint regarding those cases when the NSTT is reasonable to apply.

1.3.2

Viscous Dynamics under the Sawtooth Forcing

Consider ﬁrst the very simple one-dimensional case. A light particle in a viscous ﬂuid is subjected to the sawtooth force so that the diﬀerential equation of motion is reduced to

10

1 Introduction

x(t) ˙ = qτ (t)

(1.5)

where q represents the force amplitude per unit coeﬃcient of viscosity. The general solution of equation (1.5) is given by

t

x(t) = x(0) + q

τ (ϕ)dϕ

(1.6)

0

where x(0) is an arbitrary initial position, and a part of the problem, which is calculating the integral, still persists. Now let us represent the particular (periodic) solution in the form x = X(τ ) + Y (τ )e

(1.7)

where τ = τ (t) and e = e(t). Substituting (1.7) in (1.5) and taking into account properties (1.4), gives Y (τ ) − qτ + X (τ )e + Y (τ )e˙ = 0

(1.8)

Based on the linear independence of 1, e and e, ˙ equation (1.8) is equivalent to the boundary value problem Y (τ ) = qτ ,

X (τ ) = 0,

Y (±1) = 0

(1.9)

where the boundary condition provides zero factor for all δ−functions of the derivative e. ˙ In contrast to (1.5), the integration of equation (1.9) is strightforward since the variable of integration is τ . So substituting obvious solution of problem (1.9) into (1.7), gives general solution of equation (1.5) q x = C + (τ 2 − 1)e 2

(1.10)

where C is an arbitrary constant. Then taking into account that τ (0) = 0 and e(0) = 1, gives x(0) = C − q/2 and therefore solution (1.10) takes the form x = x(0) +

q q + (τ 2 − 1)e 2 2

(1.11)

As compared to the direct approach (1.6), the NSTT allowed for conducting the integration of the diﬀerential equation of motion with no dealing with the piece-wise structure of the integrand. Moreover, comparing solutions (1.6) and (1.11), gives the result of direct integration (1.6) in the form

t

τ (ϕ)dϕ = 0

1 {1 + [τ 2 (t) − 1]e(t)} 2

(1.12)

1.3 Quick ‘Tutorial’

11

As a simple test, expression (1.12) can be veriﬁed by taking ﬁrst derivative of the both sides. Note that there are two boundary conditions for Y in (1.9). Since the Y − component of the solution appeared to be even with respect to τ , then both of the conditions are satisﬁed even though one arbitrary constant only is available for Y . If the forcing function in (1.5) were of even degree with respect to τ , for instance, qτ 2 , then the corresponding boundary-value problem would have no solution. This fact is explained by the absence of periodic solutions under the loading qτ 2 , which has a non-zero mean value, whereas representation (1.7) imposes periodicity on the function x(t). However, the NSTT is applicable in case of any odd degree polynomial of τ on the right-hand side of equation (1.5), x(t) ˙ =

n

qk τ 2k−1 (t)

(1.13)

k=1

where qk are constant coeﬃcients. Analogously to the above particular case, taking into account the modiﬁed equation, n qk τ 2k−1 Y (τ ) = k=1

gives periodic solution n n qk qk 2k x = x(0) + + (τ − 1) e 2k 2k k=1

(1.14)

k=1

This solution is veriﬁed by the direct substitution of (1.14) in (1.13).

1.3.3

The Rectangular Cosine Input

Consider the case of step-wise discontinuous periodic loading x(t) ˙ = pe(t)

(1.15)

Substituting (1.7) in (1.15), gives Y + (X − p)e + Y e˙ = 0 The boundary value problem therefore takes the form Y (τ ) = 0,

X (τ ) = p,

Y (±1) = 0

(1.16)

In this case, Y ≡ 0 and the solution of equation (1.15) is x = x(0) + pτ (t)

(1.17)

12

1 Introduction

In this simple case, the above solution could be written though based on the deﬁnition of e(t) (1.3). As a generalization of the right-hand side of equation (1.15), let us consider equation m pi τ i (t) e(t) (1.18) x(t) ˙ = i=0

where pi are constant coeﬃcients. In this case, the periodic solution x = x(0) +

m pi i+1 τ i + 1 i=0

(1.19)

does exist for both odd and even exponents of the polynomial in (1.18). Finally, combining the right-hand sides of equations (1.13) and (1.18) and considering the corresponding inﬁnite series, gives ∞ ∞ 2k−1 i qk τ (t) + pi τ (t) e(t) ≡ Q(τ ) + P (τ )e (1.20) f (t) = k=1

i=0

It will be shown later expansion (1.20) represents a very general class of zero mean periodic functions with the period normalized to T = 4. The corresponding periodic solution of the equation x(t) ˙ = f (t) is obtained by combining solutions (1.14) and (1.19).

1.3.4

Oscillatory Pipe Flow Model

As a possible application, consider a simpliﬁed model of pipe ﬂow driven by a regularly repeating rectangular pressure wave [183]. The pipe ﬂow is assumed to have lumped inertance L and quadratic-law resistance with the coeﬃcient K. The ﬂow Q(t) is described by ﬁrst-order non-linear diﬀerential equation LQ˙ = P0 + P1 e(t/a) − KQ2

(1.21)

where P0 and P1 are constants characterizing the pressure drop, and 4a is the period of the rectangular pressure wave. Introducing parameters k = K/L, p0 = P0 /L and p1 = P1 /L, brings equation (1.21) to the form Q˙ + kQ2 = p0 + p1 e(t/a)

(1.22)

Temporal behavior of the ﬂow essentially depends on the model parameters and initial conditions. Let us assume however that the ﬂow becomes eventually periodic with the period of rectangular pressure wave. The objective is to ﬁnd the average steady state ﬂow. The corresponding periodic solution can be represented in the form

1.3 Quick ‘Tutorial’

13

Q(t) = X(τ ) + Y (τ )e

(1.23)

where τ = τ (t/a) and e = e(t/a) are triangular and rectangular waves, respectively, with the period T = 4a. Substituting (1.23) in (1.22), gives a−1 (Y + X e + Y e ) + k(X 2 + Y 2 + 2XY e) = p0 + p1 e

(1.24)

where e = de(t/a)/d(t/a), and therefore Y + ak(X 2 + Y 2 ) = ap0 X + 2akXY = ap1

(1.25)

Y (±1) = 0 Introducing the new unknowns U = X + Y and V = X − Y , brings the boundary value problem (1.25) to the form U + akU 2 = aF V − akV 2 = −aG

(1.26)

U (±1) − V (±1) = 0 where F = p0 + p1 and G = p0 − p1 are constant. Both equations in (1.26) are separable and thus admit general solutions of the form √ 2 exp(−2a kF τ ) F √ U (τ, C1 ) = 1− k C1 + exp(−2a kF τ ) √ 2 exp(−2a kGτ ) G √ (1.27) 1− V (τ, C2 ) = − k C2 + exp(−2a kGτ ) where C1 and C2 are arbitrary constants of integration to be determined from the boundary conditions U (1, C1 ) − V (1, C2 ) = 0 U (−1, C1 ) − V (−1, C2 ) = 0

(1.28)

Equations (1.28) appear to be nonlinear algebraic set of equations that, generally speaking, admits multiple solutions of which some solutions may be real, whereas others are complex. Each real solution for the constants C1 and C2 gives a periodic solution of diﬀerential equation (1.22). However, simple numerical tests show that some of the periodic solutions may appear to be unstable. (Detailed parametric study and stability analysis go beyond of this introductory illustration.) After a set of the arbitrary constants has been

14

1 Introduction

determined, the pipe ﬂow function can be represented in either standard or idempotent basis as follows 1 1 [U + V ] + [U − V ]e 2 2 = U (τ, C1 )e+ + V (τ, C2 )e−

Q(t) = X + Y e =

(1.29)

where τ = τ (t/a), and e± = [1 ± e(t/a)]/2 are elements of the idempotent basis; algebraic properties and further details of transition from one basis to another are discussed in Chapter 4. Also, it will be shown in Chapter 4 that, temporal mean value of a periodic function can be found by averaging its X(τ )-component with respect to τ . Applying this statement to solution (1.29), gives the average steady state ﬂow G F − k k 0 −1 √ √ 1 [1 + C1 exp(2a kF )][1 + C2 exp(−2a kG)] √ √ + ln 4ak [1 + C1 exp(−2a kF ][1 + C2 exp(2a kG)]

1 < Q >≡ T

T

1 Q(t)dt = 2

10

T8

9

T4

1

1 X(τ )dτ = 2

(1.30)

Q 8 7 6 0.0

0.5

1.0 tT

1.5

2.0

Fig. 1.4 Proﬁles of steady state pipe ﬂows obtained for two diﬀerent periods of the rectangular pressure wave.

Fig. 1.4 shows what happens to the steady state ﬂow proﬁle as the period of pressure wave becomes twice longer. The model parameters are k = 0.03, p0 = 2.0, and p1 = 1.5. In cases T = 4 and T = 8, the arbitrary constants are C1 = 8.2248, C2 = −0.3125 and C1 = 11.7200, C2 = −0.2651, respectively. Note that the average ﬂow for the period T = 4 is < Q >= 8.12292, whereas the longer period T = 8 gives somewhat smaller average, < Q >= 8.02996.

1.3 Quick ‘Tutorial’

1.3.5

15

Periodic Impulsive Loading

Let us consider the one-dimensional motion of a material point under the periodic impulsive loading inside the linearly viscous ﬂuid. The corresponding diﬀerential equation of motion can be represented in the form v˙ + λv = 2p

∞

[δ (t + 1 − 4k) − δ (t − 1 − 4k)] = pe˙

(1.31)

k=−∞

where λ and p are a coeﬃcient of viscosity and a half of the pulse amplitude per unit mass, respectively. Equation (1.31) has a simple form involving the distribution. Such kind of problem is usually solved by applying either the generalized Fourier series or Laplace transform, or by considering the equation between the pulses by matching diﬀerent pieces of the solution at the pulse times. Alternatively, the solution can be obtained in few quick steps by using identity (1.1). Indeed, substituting (1.1) in (1.31) and taking into account (1.4), gives (Y + λX) + (X + λY ) e + (Y − p)e˙ = 0 or

Y + λX = 0,

X + λY = 0,

Y |τ =±1 = p

(1.32)

Solving boundary value problem (1.32), gives the closed form particular solution p (sinh λτ − e cosh λτ ) (1.33) v=− cosh λ The entire general solution is obtained by adding the term with an arbitrary constant, C exp(−λt), due to linearity of equation (1.31).

1.3.6

Strongly Nonlinear Oscillator

Now let us consider the nonlinear oscillator shown in Fig. 1.1, x ¨ + x2n−1 = 0

(1.34)

where n is an arbitrary positive integer. Note that this example gives the asymptotic basis for introducing the sawtooth temporal argument τ justiﬁed by the limit n → ∞, in which x(t) takes the triangular sine shape [132]. The root of the mathematical problem here is that the limit is nonsmooth, whereas large but ﬁnite numbers n generate close to the sawtooth but smooth oscillations. Let us show that changing the temporal variable t → τ facilitates a natural transition to the limit n → ∞. Taking into account the temporal symmetry of the oscillation, reduces the substitution to the particular case X(−τ ) ≡ −X(τ ) and Y ≡ 0 so that

16

1 Introduction

x = X(τ ),

τ = τ (t/a)

(1.35)

where a = T /4 is an unknown quarter of the period. Substituting (1.35) in (1.34) and taking into account (1.4), gives 1 (X + X e ) + X 2n−1 = 0 a2

(1.36)

Due to the oddness of X(τ ), equation (1.36) is equivalent to a one-point boundary value problem X = −a2 X 2n−1 ,

X |τ =1 = 0

(1.37)

Now the idea is to take advantage of the fact that the new temporal argument is periodic and bounded, −1 ≤ τ ≤ 1. So, in order to design an analogous of the quasi harmonic approaches, successive iterations can be applied. This tool requires no small parameter to be present explicitly. As a result, one generally must rely on convergence of the procedure. Although the convergence is usually slow7 , the physical basis of the convergence can always be determined whenever the problem has indeed some physical content. Note that the harmonic balance method also uses no explicit small parameter by assuming however that temporal mode shapes are close to harmonic. In other words, high frequency terms just correct but not exceed the principal harmonic. In the present case, solution is approximated by the triangular sine, which is corrected by higher powers of the same triangular sine. On the physical point of view, the model under consideration has to be close to the impact oscillator rather then the harmonic one. In terms of the new time variable τ , such an assumption simply means that the right-hand side of the diﬀerential equation of motion (1.37) is small enough to justify the following generating system X0 = 0

(1.38)

Indeed, this equation describes a family of impact oscillators with the triangular sine wave time histories X0 = Aτ (t/a)

(1.39)

where A is an arbitrary constant and another constant is zero due to the symmetry X(−τ ) ≡ −X(τ ). There are diﬀerent ways to formalizing the entire procedure. For instance, next term of the series can be obtained by substituting the generating solution (1.39) into the right-hand side of equation (1.37) and then integrating twice so that τ 2n+1 (1.40) X1 = Aτ − a2 A2n−1 2n(2n + 1) 7

This is rather a side eﬀect of the generality of successive approximation techniques.

1.4 Geometrical Views on Nonlinearity

17

Note that the linear term Aτ occurred again as a result of integration in (1.37). Keeping the same arbitrary constant A automatically includes generating solution (1.39) into ﬁrst-order approximation. This represents a certain mathematical constraint, which actually enables one of determining the parameter a by satisfying the boundary condition in (1.37). This gives a2 = and therefore

2n A2n−2

τ 2n+1 X1 = A τ − 2n + 1

(1.41)

(1.42)

High-order algorithms and the corresponding error estimates are described in Chapter 8. In particular, expression (1.41) is sequentially improved as follows

2n 2n − 1 2 a = 2n−2 1 + A 4n + 2 Note that analogous successive approximation algorithms can be applied to equation (1.37) with a quite general right-hand side. In the above case, however, the physical meaning of the parameter, n, shown in Fig. 1.1, directly relates to the idea of algorithm. As a result, solution (1.42) performs better as the exponent n increases. Note that, a physically meaningful transition to the asymptotic limit n → ∞ can be implemented by replacing the parameter A in (1.42) with the initial velocity v0 = A/a which in contrast to A is independent of n. Finally, despite of the manipulations with nonsmooth and discontinuous functions, solution (1.42) is twice continuously diﬀerentiable with respect to the argument t; this can be veriﬁed by taking ﬁrst two formal derivatives of (1.42). Generally speaking, the boundary value problem, such as (1.37), may appear to be complicated for any analytical method. In such cases, a combination of NSTT with the shooting method can be eﬀectively used as a semi-analytical approach [153], [144].

1.4

Geometrical Views on Nonlinearity

1.4.1

Geometrical Example

As shown in the previous sections, the asymptotic of linearity has a strongly nonlinear but simple enough counterpart so that both may complement each other as possible approaches to vibration problems. In other words, two couples of basic functions, {sin t, cos t} and {τ (t), e(t)}, naturally associate with opposite boundaries of parameter intervals of certain physical systems. Note that the oscillator (1.34) is not a unique example showing such a property.

18

1 Introduction

Fig. 1.5 Rigid-body rotation that generates both sine and triangular waves within the same class of elementary functions.

Another example is given by a strongly nonlinear oscillator which is exactly solvable in terms of elementary functions [78], [122]. Below, a geometrical interpretation for this case is introduced. Let us consider a geometrical model generating the sine and triangular sawtooth waves as two asymptotic limits of the same family of periodic functions. With reference to Fig. 1.5, the distance between two ﬁxed points C and O is equal to unity. A disc of the radius CP = α is rotating around its geometrical center C with some angular speed ω so that the angle between CO and CP is ϕ = ωt. The edge point P of the rotating disc has its image Q on the co-centered circle of the unit radius. The image is obtained under the condition that, during the rotation, P Q remains parallel to CO. Then position of the image Q is given by either its Cartesian coordinate Y or by the arc length y as follows Y = α sin ϕ (1.43) y = arcsin(α sin ϕ)

(1.44)

Expression (1.44) is obtained by equating projections of CP and CQ on OY and taking into account that the angle QCO is equal to the corresponding arc length y. Obviously, (1.44) becomes equivalent to (1.43) and thus describes the harmonic sine wave as α → 0.

1.4 Geometrical Views on Nonlinearity

19

Suppose now that α → 1 − 0. In this case, (1.44) takes the triangular sine wave shape

2ϕ π y→ τ (1.45) 2 π Interestingly enough, both types of observation of the above rigid-body model are associated with diﬀerent elastic oscillators. Namely, on one hand, as a function of time, expression (1.43) satisﬁes equation Y¨ + ω 2 Y = 0

(1.46)

On the other hand, function y(t) (1.44) satisﬁes the diﬀerential equation of motion of conservative oscillator tan y =0 cos2 y

(1.47)

α2 + ω −2 = 1

(1.48)

y¨ + under condition,

In particular, condition (1.48) implies that the frequency ω depends upon the disc radius monotonically in such a way that 1 ≤ ω < ∞ as 0 ≤ α < 1. Therefore, nonsmooth limit (1.45) is reached under the inﬁnitely large frequency ω, when the total energy of oscillator (1.47) also becomes inﬁnitely large, tan2 y y˙ 2 + →∞ (1.49) H= 2 2 The reasons for considering oscillator (1.47) as a generating model in nonlinear dynamics will be described later. At this stage, let us bring attention to the fact that both equations (1.46) and (1.47) actually describe the same rigid-body rotation but in diﬀerent coordinate systems.

1.4.2

Nonlinear Equations and Nonlinear Phenomena

The purpose of next few subsections (1.4.2 through 1.4.6) is to ﬁnd some basis for the idea of using elementary nonsmooth systems as generating systems in nonlinear dynamics. The discussion below may provide a hint for selecting nonlinear generating systems within the class of nonsmooth systems even though some statements of the discussion may have no rigorous proof. As a starting point, it could be useful to clarify why the harmonic oscillator represents a good generating model. Indeed, deﬁning the nonlinearity regardless mathematical formalizations appears to be a challenging task. Namely it is diﬃcult to ﬁnd appropriate physical principals for such a deﬁnition that would qualify nonlinearity as a natural phenomenon rather than the speciﬁc form of mathematical expressions. Brieﬂy speaking, nonlinear phenomena may occur at high energy levels. The following scenario is described in the book [126]: “...once the power or violence of a system is increased, it leaves

20

1 Introduction

the familiar linear region and enters the more complex world of nonlinear eﬀects: rivers become turbulent, ampliﬁers overload and distort, chemicals explode, machines go into uncontrollable oscillations, plates buckle, metals fracture, and structures collapse.” The author goes even further by saying that “within such an approach, mind may no longer appear as an alien stuﬀ in a mechanical universe; rather the operation of mind will have resonances to the transformations of matter, and indeed, the two will be found to emerge from a deeper ground.” However, mathematical approaches can still be helpful based on the following logical identity Nonlinear systems = All systems - Linear systems

(1.50)

Although this relationship seems to bring no physical contents, nevertheless it shows how physical understanding of nonlinearity can possibly be achieved. In other words, by clarifying the physical basis that provides the well known simplicity of the linear systems and harmonic analyses, one could ﬁnd what actually makes systems nonlinear. The standard mathematical deﬁnition for linear systems based on the superposition principle. Generally, a linear system can be deﬁned in terms of operators. Let L be an operator acting on state or may be coordinate vectors {q}. Then the operator L is linear, if for any two states, say q1 and q2 , and any two constants, C1 and C2 , the following relationship holds L(C1 q1 + C2 q2 ) = C1 Lq1 + C2 Lq2

(1.51)

Now, according to (1.50), a system is nonlinear if it is not linear. This mathematical deﬁnition, however, may appear to be confusing from physical viewpoints. For instance, based on deﬁnition (1.51), system 2

ρ ¨(t) − ρ(t) ϕ(t) ˙ = ρ(t) [sin 2 ϕ(t) − 2] ρ(t)¨ ϕ(t) + 2ρ(t) ˙ ϕ(t) ˙ = ρ(t) cos 2 ϕ(t)

(1.52)

does not satisfy its conditions and therefore is nonlinear, whereas x ¨1 (t) + 2x1 (t) − x2 (t) = 0 x ¨2 (t) + 2x2 (t) − x1 (t) = 0 is a linear system whose operator reads

2 2 d /dt + 2 −1 L= −1 d2 /dt2 + 2

(1.53)

(1.54)

so that Lq = 0, where q = [x1 (t) , x2 (t)]T . The linear superposition principle is therefore applicable to equations (1.53) whereas it does not work for equations in the nonlinear form (1.52).

1.4 Geometrical Views on Nonlinearity

21

However, both sets of equations (1.52) and (1.53) describe the same mechanical system in polar and Cartesian coordinates, respectively. On the plane of conﬁgurations, the corresponding coordinate transformation is given by x1 = ρ cos ϕ,

x2 = ρ sin ϕ

In this example, it is clear that the system is linear from the physical point of view; indeed nonlinearity of equations (1.52) is due to speciﬁc choice for coordinates. Therefore a ‘physical deﬁnition’ for linear systems must specify the type of reference coordinate system, for instance, as follows: A mechanical system is linear if its diﬀerential equations of motions in Cartesian coordinates are linear 8 . This deﬁnition involves already some physical principles since the Cartesian coordinates are uniquely associated with general properties of the physical space. Note that the above deﬁnition still involves the mathematical notion of coordinate systems. As known from observations however, some dynamic phenomena are perceived as nonlinear with no explicit system coordinates. So next two subsections make an attempt to clarify why such a perception of nonlinearity is actually possible.

1.4.3

Rigid-Body Motions and Linear Systems

As mentioned at the beginning of this introduction, sine and cosine waves are associated with the subgroup of rigid-body rotations. On the other hand, same temporal shapes describe vibrations of deformable linearly elastic bodies such as harmonic oscillators. Therefore, one-dimensional dynamics generated by linearly elastic restoring forces can be represented as free rigid-body rotations however in the two-dimensional space. In other words, linearly elastic forces are eﬀectively eliminated by expanding the dimension of space. Therefore, linear dynamics can be viewed as kinematics of freely rotating discs, where the number of discs is the number of modes. Such a viewpoint can be formalized as follows. Let z be a complex vector frozen into a rigid body (disc), whose position is observed in the empty space. Note that the notion of ‘position’ becomes quite vague if there is only one body. Therefore it is assumed that the observer represents another physical body, which is a single point. Then a straight line between this point and the center of the disc gives a reference line for determining the vector z direction. Since the disc is assumed to be rigid then z z¯ = z z¯ 8

(1.55)

This deﬁnition was suggested by V.Ph.Zhuravlev during a private discusion at International Conference “Nonlinear Phenomena,” Moscow, 1989.

22

1 Introduction

where the over bar means complex conjugate, and the prime denotes any new position of the body. Expression (1.55) recalls mathematical objects generated by the Galilean group of rotations. The corresponding operator G is introduced through relationship (1.56) z = Gz where G must depend on some parameter, say ϕ, characterizing the transition z → z. Substituting (1.56) in (1.55), gives 2

¯ = |G| = 1 =⇒ G = exp(iϕ), GG

i2 = −1

(1.57)

Since ϕ is the only parameter of the only one ‘process’ then ϕ must be qualiﬁed as time with possibly some scaling factor. Introducing an arbitrary scaling factor, say ω, gives then G = exp (iωt) = cos ωt + i sin ωt

(1.58)

Now a simple calculation shows that

d d ¯=0 − iω G = 0 and + iω G dt dt ¯ satisfy the diﬀerential equation of harTherefore both operators, G and G, monic oscillator associated with the operator

d d d2 − iω + iω = 2 + ω 2 (1.59) dt dt dt By representing the scaling factor in the form ω = k/m, one can interpret then m and k as, respectively, mass and stiﬀness of a mass-spring model associated with the ‘product’ of clockwise and counter-clockwise rotations of the disc. Let us consider now n rotating discs with the angular frequencies {ω1 , ..., ωn } by generalizing product (1.59) as follows

d − iω1 dt

n 2 d d d d 2 + iω1 ... − iωn + iωn = + ω j dt dt dt dt2 j=1

(1.60) Operator (1.60) represents an arbitrary n-degrees-of-freedom linear elastic system. Replacing d/dt → λ, gives the corresponding characteristic equation in the form n 2 λ + ωj2 = 0 (1.61) j=1

1.4 Geometrical Views on Nonlinearity

23

Finally, let us consider the asymptotic limit n → ∞. In order to calculate the limit, the frequency dependence on its index j must be speciﬁed. If, for instance, ωj = ja with some constant a then equation (1.61) gives sinh

πλ =0 a

(1.62)

as n → ∞. Equation (1.62) associates with a linearly elastic string of the length l = π with two ﬁxed ends, ∂ 2 u (t, x) ∂ 2 u (t, x) − a2 = 0; 2 ∂t ∂x2

u (t, 0) = u (t, π)

(1.63)

Therefore, it is shown that basic linearly elastic dynamic models can be logically obtained from the kinematics of freely rotating discs or even from a more fundamental concept given by expression (1.55). To some extent, this may explain why nonlinearity is perceptible as a physical phenomenon. Indeed the linear dynamics is simply matching with the usual perception (experience) of the space. All other motions give therefore an impression of unusual, in other words, nonlinear phenomena.

1.4.4

Remarks on the Multi-dimensional Case

The multi-dimensional example below gives another (purely geometrical) viewpoint on the link between rigid-body motions and the notion of linearity. So, by deﬁnition, the mapping f : Rn −→ Rn is an isometry if relation

f (v) − f (w) = v − w

(1.64)

holds for all v, w ∈ Rn . Deﬁnition (1.64) requires distances between any two images and their preimages be same; this is a generalization of relationship (1.55). Let us consider such isometries with a ﬁxed point O in Rn , that is symbolically f (O) = O. Let us show that an isometry that ﬁxes the origin is a linear mapping f (av + bw) = af (v) + bf (w)

(1.65)

where a and b are arbitrary scalars. First, note that the relations

f (v) = v = −v = f (−v)

and

f (v) − f (−v) = v− (−v) = 2 v

hold due to (1.64), and thus f (v) and f (−v) are antipodal: f (−v) = −f (v).

24

1 Introduction

Second, taking into account the latter result and deﬁnition (1.64), transforms the polarization identity as follows 1 2 2 2

v + w − v − w

2 1 =

f (v) − f (−w) 2 − f (v) 2 − f (−w) 2 2 1 2 2 2 =

f (v) + f (w) − f (v) − f (w)

2 = f (v) · f (w)

v·w =

Therefore, the inner product is preserved. Now, if {ui }is orthogonal basis in Rn , then {f (ui )} is another orthogonal basis. Taking the dot product of the both sides of identity (1.65) with the arbitrary basis vector f (ui ), gives f (av + bw) · f (ui ) = af (v) · f (ui ) + bf (w) · f (ui ) or (av + bw) · ui ≡ av · ui + bw · ui The above relation proves identity (1.65) in terms of the coordinates associated with the basis {f (ui )}. All linear isometries of Rn are denoted by the symbol O (n). By choosing a basis for Rn , one can represent every element of O (n) as a matrix. It can be shown that O (n) consists of all n × n matrixes such that A−1 = AT : 1 = det I = det AA−1 = det AAT = det A det AT = (det A)2

(1.66)

Relationship (1.66) is therefore a multidimensional matrix analogy of the relationship (1.57).

1.4.5

Elementary Nonlinearities

Let us consider now the temporal coordinate of Galilean spatio-temporal continuum. In particular, the idea of ‘perfectly rigid time’ is expressed by relationship

2 dt =1 (1.67) dt2 = dt2 ⇐⇒ dt By considering (1.67) as a diﬀerential equation with respect to t = s (t), one obtains two possible solutions t = ± (t − a) where a is an arbitrary temporal shift.

(1.68)

1.4 Geometrical Views on Nonlinearity

25

Now, combining diﬀerent branches of (1.68) for t < a and t > a, gives t = s (t) = |t − a|

(1.69)

Function (1.69) also satisﬁes equation (1.67), for all t except may be single point t = a, where the classic derivative of s (t) has no certain value. Namely, equality holds for almost all t except may be t = a. Nevertheless, solution (1.69) admits an obvious physical interpretation since it describes a free material point moving in the space splitted into two subspaces by a perfectly stiﬀ plane, were the normal to the plane velocity is scaled by (1.70) s˙ 2 = 1 Since the perception of empty space rejects any built in stiﬀ planes then a sudden V -turn of the particle will represent unusual, in other words, strongly non-linear event. This is obviously the most elementary nonlinear event whose simplicity nevertheless will be employed further in less trivial cases.

1.4.6

Example of Simpliﬁcation in Nonsmooth Limit

The idea that transition to most severe nonlinearity may actually simplify a system response ﬁnds its support in very diﬀerent examples. Consider, for instance, the diﬀerential equation of motion

2x (1.71) x ¨ = exp − 2 The nonlinear force on the right-hand side gives the ‘rigid body’ limit as the parameter approaches zero. Let us assume that x˙ = −1 and x → ∞ as t → −∞. In this case, equation (1.71) has exact solution

t−a (1.72) x = 2 ln cosh 2 where a is an arbitrary parameter such that x(a) ˙ = 0; see Fig. 1.6, where a = 1. Now let us consider the asymptotic limit → 0. Taking into account evenness of cosh-function, brings solution (1.72) to the form

|t − a| s 2 2 (1.73) x = ln cosh ln cosh = 2 2 This simple manipulation represents a useful preliminary step for asymptotic estimates because the new temporal argument, s, remains always positive as the original time runs in the interval −∞ < t < ∞.

26

1 Introduction x 2.0 Ε0.1 Ε0.5

1.5

Ε1.0 1.0

0.5

1

1

2

3

t

Fig. 1.6 Time histories of the particle when the barrier approaches rigid body limit, → 0.

Further algebraic manipulations bring solution (1.73) to the form

2s 2 x = s + ln (1.74) 1 + exp − 2 2 Taking into account that 0 < exp −2s/2 < 1, gives the ‘rigid-body’ limit x → s = |t − a|

(1.75)

as → 0. The above example shows that (1.69) can be viewed as a natural time associated with the temporal symmetry of the V-turn. Note however that the new temporal argument is nonsmooth and non-invertible with respect to the original time. Therefore, using (1.69) for temporal substitutions in the diﬀerential equations of motion has certain speciﬁcs as discussed below.

1.4.7

Non-smooth Time Arguments

As mentioned above, piece-wise linear function (1.69), s(t), can play the role of natural temporal argument associated with the elementary nonlinear events such as dynamic V-turns caused by potential barriers. Namely the temporal symmetry of V-turns is captured by the function s(t) regardless possible shapes of the potential barriers. Therefore, using the function s(t) as a temporal argument should simplify analyses since the global information about V-turns is a priory built into the new argument. The corresponding time substitution however is not straightforward since its inverse version does not exist in the form t = t(s) but requires the following generalization t = t(s, s). ˙ Below, this case is discussed ﬁrst. Then the periodic version is

1.4 Geometrical Views on Nonlinearity

27

introduced that reveals structural similarity of expressions as far as periodic motions are in fact regular sequences of v-turns. Non-periodic case. Let us start with the inverse version of (1.69), which includes both the image s and its time derivative s˙ as follows t = t(s, s) ˙ = a + ss˙

(1.76)

Now relation (1.70) reveals that combination (1.76) represents speciﬁc algebraic structure with the basis {1, s}. ˙ In other words, time belongs to the algebra of hyperbolic ‘complex numbers’ with the table of products generated by (1.70). For instance, taking (1.76) squared, gives another hyperbolic ‘number’ (1.77) t2 = a2 + s2 + 2ass˙ Moreover, it is easy to prove for any function x(t) that x (t) = X (s) + Y (s) s˙

(1.78)

where 1 [x (a + s) + x (a − s)] 2 1 Y (s) = [x (a + s) − x (a − s)] 2

X (s) =

For instance, setting a = 0 in the case x(t) = exp t, gives exp (ss) ˙ = cosh s + s˙ sinh s

(1.79)

In contrast to the conventional complex algebra, division is not always possible, so the following relationship is meaningless for |a| = s or t = 0: 1 (a − ss) ˙ a 1 s = = = 2 − 2 s˙ t a + ss˙ (a + ss) ˙ (a − ss) ˙ a − s2 a − s2 In particular case of even function x (t), with respect to t = a, one has Y (s) ≡ 0; see (1.73) for example. Periodic case. The periodic version of nonsmooth time transformations as that preliminary introduced in Section 1 involves triangular and rectangular waves described by the periodic piecewise-linear function πt 2 t for − 1 ≤ t ≤ 1 ∀t , τ (t) = τ (4 + t) τ (t) = arcsin sin = −t + 2 for 1 ≤ t ≤ 3 π 2 and its Schwartz derivative e(t) = τ˙ (t); see Fig. 1.1. Remind that from the physical point of view, the functions τ (t) and e (t) describe dynamic states of a particle oscillating between two perfectly rigid

28

1 Introduction

Fig. 1.7 Mechanical model generating the saw-tooth sine and rectangular cosine, τ (t) and τ˙ (t)

planes with no energy loss as shown in Fig. 1.7, where spatio-temporal coordinates are normalized so that dt2 = dτ 2 ⇐⇒ e2 = τ˙ 2 = 1

(1.80)

Note that relationship (1.80) is a periodic version of (1.70) whereas, for any periodic function of the period T = 4, representation (1.1) is a periodic version of (1.78). In other words, any periodic motion uniquely associates with the standard impact vibration, for instance as follows x (t) = A sin

πt πτ πτ πt + B cos = A sin + B cos e 2 2 2 2

(1.81)

where τ = τ (t), e = τ˙ (t), and A and B are arbitrary constants. Exercise 1. By taking sequentially derivatives, show that diﬀerentiation keeps the result within the set of hyperbolic numbers provided that the corresponding process is smooth enough. In particular, π 2n−1 πτ πτ πτ πτ −B sin + B cos e /dt2n−1 = + A cos e d2n−1 A sin 2 2 2 2 2 πτ πτ πτ πτ π 2n A sin + B cos e /dt2n = + B cos e d2n A sin 2 2 2 2 2

1.4.8

Further Examples and Discussion

The Fourier analysis is based on the standard trigonometric pair of functions {sin ωt, cos ωt} or {exp (iωt) , exp (−iωt)} so that periodic processes are described by linear combinations of these functions with appropriate frequencies {ωk }. For instance, the dynamic states of impact oscillators are represented as follows

8 1 1 3πt 5πt πt τ (t) = 2 sin − 2 sin + 2 sin − ... π 2 3 2 5 2

4 3πt 1 5πt πt 1 e (t) = − cos + cos − ... (1.82) cos π 2 3 2 5 2

1.4 Geometrical Views on Nonlinearity

29

The well known convenience of Fourier series for handling partial and ordinary diﬀerential equations is due to the fact that exp(iωt) is an eigen function of the time derivative, in other words, d exp(iωt)/dt = iω exp(iωt). This important advantage is unfortunately missing when using the set {τ (ωk t), e(ωk t)} as a nonsmooth basis for Fourier expansions. The alternative way is based on power series expansions. Indeed, in terms of the oscillating time τ , polynomial approximations can be applied with no periodicity loss. For example, πτ πτ 1 πτ 3 1 πτ 5 πt = sin = − + − ... 2 2 2 3! 2 5! 2

πτ 1 πτ 2 πt 1 πτ 4 = cos e= 1− cos + − ... e 2 2 2! 2 4! 2 sin

(1.83)

While being polynomials, truncated series of (1.83) preserve periodicity at cost of smoothness loss though; see Fig. 1.8 for explanation. Fortunately, nonsmoothness times Λ = {t : τ (t) = ±1} are same for every term of the series and this enables one of smoothing the series by re-ordering their terms as follows

3 πτ π π π3 τ τ5 πt τ3 = sin = + − sin τ− + 2 2 2 3 2 16 3 5

3 5 5 7 π π π τ τ + + − + + ... (1.84) 2 16 768 5 7

πτ π2 2 πt 2 = cos e = [1 − τ + 1 − τ − τ4 cos 2 2 8

π2 π4 4 + 1− + τ − τ 6 + ...]e 8 384 Fig. 1.9 illustrates both convergence and smoothness of the transformed series as compared with the diagrams in Fig.1.8. Moreover, it will be shown that the power series of τ can be re-ordered in such a manner that their particular sums become as smooth as necessary. Still the convenience of such kind of series is that they can accurately approximate non-smooth or close to them processes just by ﬁrst few terms. Finally, there are many physical processes easily described by very simple combinations of the triangular sine and rectangular cosine waves, which otherwise would require quite long Fourier sums; see Fig. 1.10, for examples.

30

1 Introduction

Fig. 1.8 Truncated Maclaurin NSTT expansions with respect to τ for sin(πt/2) and cos(πt/2) including one, two, and three ﬁrst terms of the expansions as indicated on the curves.

1.4.9

Diﬀerential Equations of Motion and Distributions

Any nonsmooth substitution in the diﬀerential equations of motion must be carefully examined since diﬀerentiation is involved. In many physically meaningful cases though suﬃcient justiﬁcations can be achieved by understanding equalities as integral identities. For illustration purposes, let us consider conservative oscillator (1.85) x ¨ + Px (x) = 0 where P (x) is the potential energy.

1.4 Geometrical Views on Nonlinearity

31

Fig. 1.9 First term of the modiﬁed NSTT series with respect to τ for sin(πt/2) and cos(πt/2); 0 - original functions, 1 - ﬁrst term of the series.

Classical pointwise interpretations of diﬀerential equations require no discontinuities to occur on the left-hand side of (1.85). Therefore, ‘real motions’ must be described by at least twice continuously diﬀerentiable functions of time, x (t) ∈ C 2 (R). Practically, however, physical systems cannot be observed at every time instance. In other words, point-wise deﬁnition of equality (1.85) appears to be somewhat restrictive from the physical standpoint. As an extension of classical approaches, the left-hand side of (1.85) can be considered as some force producing zero work on arbitrary path variations δx during the observation interval, t1 < t < t2 , so that

32

1 Introduction

t2

(¨ x + Px (x)) δxdt = 0

(1.86)

1 2 x˙ − P (x) dt = −δS = 0 2

(1.87)

t1

or

t2

−δ t1

where S is the action by Hamilton.

Fig. 1.10 Sample temporal shapes of periodic processes easily described by combinations of the triangular sine and rectangular cosine

1.5 Non-smooth Coordinate Transformations

33

Expression (1.87) represents the Hamiltonian principle, where the integral can be calculated for less smooth functions x (t) ∈ C (R). Therefore, it is possible to essentially extend the class of solutions by ‘keeping in mind’ interpretation (1.86) when considering equation (1.85). Note that equation (1.85) may be strongly nonlinear, however the highest derivative must participate in a linear way as a summand, otherwise transition from (1.86) to (1.87) may become impossible.

1.5

Non-smooth Coordinate Transformations

1.5.1

Caratheodory Substitution

In general, including Dirac’s δ-functions in nonlinear diﬀerential equations complicates mathematical justiﬁcations of the modeling. It is important however how δ-functions participate in equations [46]. For instance, let the differential equation include δ-impulse as a summand x˙ = kx3 + qδ (t − t1 )

(1.88)

where k, q and t1 are constant parameters. In this case, the δ-input generates step-wise discontinuity of the response x(t) at t = t1 , however the nonlinear operation in (1.88) still can be conducted. Moreover, the δ-pulse is easily excluded from equation (1.88) by substitution (1.89) x (t) = y (t) + qH (t − t1 ) where y (t) is a new unknown function and H (t − t1 ) is the unit-step Heaviside function. Indeed, substituting (1.89) in (1.88) and taking into account that H˙ (t − t1 ) = δ (t − t1 ), H 2 (t − t1 ) = H (t − t1 ), and H 3 (t − t1 ) = H (t − t1 ), gives y˙ = k[y 3 + (q 3 + 3yq 2 + 3y 2 q)H (t − t1 )]

(1.90)

where y (t) is now continuous on the interval of consideration including the point t1 .

1.5.2

Transformation of Positional Variables

Now let us discuss the method of nonsmooth coordinate transformation (NSCT) for mechanical systems with perfectly stiﬀ constraints [200]. According to this approach, the new coordinates are introduced in order to automatically satisfy the constraint conditions. Let us reproduce the idea based on one-degree-of-freedom Langrangian system L=

1 2 x˙ − P (x) 2

(1.91)

34

1 Introduction

whose motion is limited by interval −1≤x≤1

(1.92)

It is assumed that the particle collides with the obstacles x = ±1 with no energy loss. Note that relationships (1.91) and (1.92) give no unique diﬀerential equation of motion on the entire time domain since every collision indicates transition from one system to another. Indeed, even though the form of Lagrangian (1.91) remains the same, one must deal with diﬀerent solutions before and after the collision, say x− (t) and x+ (t). Matching such solutions is usually not straightforward since the collision time is a priory unknown. The major reason for applying NSCT is that it gives a single diﬀerential equation of motion for the entire time interval with no constraint conditions. As a result, the mentioned above matching procedure simply becomes unnecessary. In addition, after the transformation, a new system appears to be well suited for averaging since no impact forces are involved any more. In the above case (1.91) and (1.92), the NSCT is introduced as follows9 x = τ (s)

(1.93)

where s(t) is a new positional coordinate. Substituting (1.93) into (1.91), gives L=

1 1 [τ (s)s] ˙ 2 − P (τ (s)) = s˙ 2 − P (τ (s)) 2 2

(1.94)

where the prime indicates diﬀerentiation with respect to s, and the relationship [τ (s)]2 = e2 (s) = 1 has been taken into account. Now, condition (1.92) is satisﬁed automatically since −1 ≤ τ (s) ≤ 1, whereas the Hamiltonian principle gives the diﬀerential equation of motion with no constrains dP e(s) = 0 (1.95) s¨ + dτ On geometrical point of view, transformation (1.93) unfolds the conﬁguration space by switching the coordinate direction on opposite whenever the particle collides with an obstacle. As a result, the new conﬁguration space acquires a cell-wise non-local structure as illustrated in [191] and [148]. This usually makes the diﬀerential equation essentially nonlinear even when the system in between the constraints is linear. Let, for instance, the potential energy function represents the harmonic oscillator, P (x) = ω 2 x2 /2. Then the coordinate transformation (1.93) brings system x¨ + ω 2 x = 0, 9

−1≤x≤1

(1.96)

Note that both notations and normalization for the period for the triangular wave function diﬀer from the original work [200].

1.5 Non-smooth Coordinate Transformations

35

to the form s¨ + ω 2 τ (s)e(s) = 0,

−∞<s<∞

(1.97)

Note that, as a side eﬀect of the elimination of constraints, the diﬀerential equation lost its linearity. In contrast to (1.97) however, the linear diﬀerential equation in (1.96) does not describe the entire system, which is strongly non linear from the very beginning due to impact interactions with constraints. Finally, results of comparison between NSCT and NSTT can be summarized as follows [143]: Coordinate transformation

Time transformation

Unfolds the space with no eﬀect on Folds the time, and generates the the time variable hyperbolic algebraic structures in space Targets rigid barriers (constraints) Barriers do no matter Dynamic regime independent Assumes certain temporal symmetries of motion Applies to a function (image) Transforms an argument (preimage) Essentially changes the ODE struc- Preserves most structural properties ture, for instance, linear on strongly of ODEs nonlinear As follows from the table, NSCT is quite opposite to NSTT from the very diﬀerent viewpoints.

1.5.3

Transformation of State Variables

Under some conditions, the NSCT can still be adapted to the case of nonelastic interactions with constraints in a purely geometrical way [204]. However, generalized approaches to such cases should involve both coordinates and velocities [73], [72]. For illustrating purposes, let us consider the case of harmonic oscillator under the constraint condition x˙ = Ax,

x1 > 0

where x = [x1 (t), x2 (t)]T is the state vector such that x2 = x˙ 1 , and

0 1 A= −ω 2 0

(1.98)

(1.99)

It is assumed that every collision with the constraint x1 = 0 results in a momentary energy loss characterized by the restitution coeﬃcient κ. The idea is to unfold the phase space in such way that the energy loss would automatically occur whenever the system crosses a preimage of the line x1 = 0. The corresponding transformation is

36

1 Introduction

x = Sy where y = [s(t), v(t)]T is a new state vector, and

10 S= sgn(s) 0 1 − ksgn(sv)

(1.100)

(1.101)

where k = (1 − κ)/(1 + κ). Transformation (1.100) is of-course strongly nonlinear due to non-smooth dependence (1.101). Substituting (1.100) in (1.98), gives equation −1

y˙ = (S

AS)y

(1.102)

In the component-wise form, expressions (1.100) and (1.102) are written as, respectively, x1 = x1 (s, v) ≡ ssgn(s) x2 = x2 (s, v) ≡ sgn(s)[1 − ksgn(sv)]v

(1.103)

and s˙ = [1 − ksgn(sv)]v v˙ = −ω 2 s[1 + ksgn(sv)]/(1 − k 2 )

(1.104)

Now both unknown components of the state vector are continuous, whereas speciﬁcs of non-elastic collisions are captured by transformation (1.103). Finally, consider the general case of one-degree-of-freedom nonlinear oscillator x˙ 1 = x2 x˙ 2 = −f (x1 , x2 , t)

(1.105)

with non-elastic constraint x1 = 0 of the restitution coeﬃcient κ. Applying transformation (1.103) to system (1.105), gives s˙ = [1 − ksgn(sv)]v v˙ = −f (x1 (s, v), x2 (s, v), t)sgn(s)[1 + ksgn(sv)]/(1 − k 2 )

(1.106)

Although the above illustrations is one-dimensional, similar coordinate transformations can be introduced also for multiple degree-of-freedom systems in which one of the coordinates is normal to the constraint. The corresponding analytical manipulations can be conducted in terms of Routh descriptive functions such that the normal to the constraint coordinate is Lagrangian whereas other generalized coordinates and associated momenta are Hamiltonian.

Chapter 2

Smooth Oscillating Processes

Abstract. This chapter gives a brief overview of selected analytical methods for smooth oscillating processes. Most of such methods are indeed quasilinear. In other words, the corresponding technical implementations employ harmonic oscillators as generating models. The description focuses only on the ideas and technical details that are further combined with non-smooth methods. As most eﬀective way, procedures of asymptotic integration of the diﬀerential equations of motion bring original systems to such simple form that further solution becomes straightforward. In particular, the method of asymptotic integration of the diﬀerential equations of motion based on the Hausdorﬀ equation for operators Lie is reproduced.

2.1

Linear and Weakly Non-linear Approaches

By both practical and theoretical reasons, the quantitative methods of dynamics were developed ﬁrst for smooth processes. As a rule, smooth oscillations can be directly observed under no special conditions. For instance, projection of any ﬁxed point of a body rotating with constant angular speed, makes a perfect impression about harmonic oscillations. Interestingly, in 1693, Leibniz derived the diﬀerential equation for sine geometrically by considering a circle. Much later, original analytical ideas of nonlinear vibrations emerged from the celestial mechanics considering perturbations of circular orbits of rigid-body motions rather than any mass-spring oscillators. Robert Hooke (1635-1703) was probably ﬁrst who suggested the basic elastic mass-spring model, whereas Galileo and Huygens were investigating the pendulum. Later, d’Alambert, Daniel Bernoulli and Euler considered a one-dimensional continual model of a string. It was found that the vibrating string represents the inﬁnity of harmonic oscillators corresponding to diﬀerent mode shapes of the string. It is well known that a serious discussion arised about whether or not V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 37–49, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

38

2 Smooth Oscillating Processes

the sum of smooth functions, such as sines, can represent a non-smooth shape of the string. These discussions were ﬁnalized by the Fourier theorem. Let us reproduce the result for a periodic function of time f (t) of the period T in the complex form f (t) =

∞

ck exp(iωk t)

(2.1)

k=−∞

1 ck = T

T /2

−T /2

f (t) exp(−iωk t)dt, ωk =

2π k T

This relation generates a one-to-one mapping between function and its Fourier coeﬃcients f (t) ←→ {...c−2 , c−1 , c1 , c2 , ...}

(2.2)

Note that mathematical expressions (2.1) do not necessarily imply that the periodic process f (t) must be produced by linear systems even though the right-hand side of (2.1) combines free vibrations of linear oscillators, in other words - ‘rigid-body rotations’ as discussed in Chapter 1. Therefore, the Fourier analysis and associated analytical tools provides a ‘linear language’ for nonlinear systems regardless speciﬁcs of algorithm implementations. Indeed, most quantitative methods for weakly nonlinear periodic motions, actually estimate Fourier coeﬃcients of the corresponding solutions. As a result, on one hand, such tools possess a high level of generality. On the other hand, even ‘elementary’ strongly nonlinear phenomena (as qualiﬁed in Chapter 1) may become quite diﬃcult to describe in terms of the ‘linear language.’ Nevertheless, the quantitative theory of nonlinear vibration has been advanced by new asymptotic techniques developed originally for solving nonlinear diﬀerential equations. Most traditional methods are essentially based on perturbation or averaging methods [50]. Similar results can be obtained within the theory of Poincare’ normal forms [118], which retains resonance terms, whereas all non-resonance terms are eliminated by means of a coordinate transformation. Such a normal form is qualiﬁed as the simplest possible form of the equations of motion.

2.2

A Brief Overview of Smooth Methods

2.2.1

Periodic Motions of Quasi Linear Systems

Consider a weakly unharmonic oscillator of the form x ¨ + ω02 x = εf (x, x) ˙

(2.3)

2.2 A Brief Overview of Smooth Methods

39

where ε is a small parameter, 0 < ε 1, and f (x, x) ˙ is smooth enough function. Periodic solutions of equation (2.3) can be found by splitting the nonlinear system into the sequence of linear oscillators by means of the power series x = x0 + εx1 + ε2 x2 + ...

(2.4)

The perturbation on the right-hand side of equation (2.3) changes the principal frequency of the oscillator so that ω 2 = ω02 (1 + εγ1 + ε2 γ2 + ...)

(2.5)

The new frequency is introduced explicitly into the diﬀerential equation of motion by re-scaling the temporal argument ϕ = ωt

(2.6)

As a result, series (2.4) appears to be composed of trigonometric functions of multiple phases ϕ, 2ϕ, 3ϕ,... . A similar idea was implemented by Lyapunov for systems of ﬁrst-order equations, for instance x˙ 1 = a11 x1 + a12 x2 + f1 (x1 , x2 )

(2.7)

x˙ 2 = a21 x1 + a22 x2 + f2 (x1 , x2 ) where f1 and f2 are nonlinear functions. It is assumed that system (2.7) admits ﬁrst analytical integral and the corresponding linearized system has only periodic solutions. Then periodic solutions of (2.7) admit power series expansions with respect to the amplitude parameter. There exist at least two extensions of Lyapunov’s theory, such as local and global approaches to nonlinear normal modes, see for instance [100], [190], [119].

2.2.2

The Idea of Averaging

Let us illustrate diﬀerent implementations of the idea of averaging by reproducing some technical details. The following description focuses on such tools that remains applicable to non-hamiltonian systems. Following Van-der-Pol’s approach, let us transform system (2.3) by chang˙ → {a, ϕ}: ing the variables {x, x} x = a cos ϕ,

x˙ = −aω0 sin ϕ

(2.8)

40

2 Smooth Oscillating Processes

As a result, one obtains ε f (a cos ϕ, −aω0 sin ϕ) sin ϕ ω0 ε f (a cos ϕ, −aω0 sin ϕ) cos ϕ ϕ˙ = ω0 − ω0 a

a˙ = −

(2.9)

Despite of a formal complexity, system (2.9) has essential advantage due to diﬀerent time scales of the new variables. This enables one of eliminating the fast phase ϕ on the right-hand side of the system by applying the averaging 2π 1 · · · dϕ < · · · >ϕ ≡ 2π 0 as follows ε < f (a cos ϕ, −aω0 sin ϕ) sin ϕ >ϕ ω0 ε < f (a cos ϕ, −aω0 sin ϕ) cos ϕ >ϕ ϕ˙ = ω0 − ω0 a a˙ = −

(2.10)

Solutions of system (2.10) are considered then as approximate solutions of the original system (2.9). This method was essentially generalized in thirties [28] by incorporating the Lindstedt-Poincare and Van-der-Pol’s ideas as follows. Let us consider the general system with one fast phase x˙ = εX(x, y) y˙ = ω(x) + εY (x, y)

(2.11)

where y and x are scalar and vector variables respectively. In contrast to (2.9), the frequency in (2.11) depends on the slow vectorfunction x. Sometimes, such kind of systems is called essentially nonlinear since the condition ε = 0 does not make the frequency state independent. However, if ε = 0 then system (2.11) has no fast phase on the right-hand side. The problem is to ﬁnd close to identical transformation x = q + εu(q, ψ) + O(ε2 ) y = ψ + εv(q, ψ) + O(ε2 )

(2.12)

which eliminates the fast phase entirely from the system by bringing equation (2.11) to the form q˙ = εA(q) + O(ε2 ) ψ˙ = ω0 (q) + εω1 (q) + O(ε2 )

(2.13)

2.2 A Brief Overview of Smooth Methods

41

This problem is solved by substituting expansions (2.12) into equations (2.11) and enforcing then equations (2.13).

2.2.3

Averaging Algorithm for Essentially Nonlinear Systems

In order to illustrate the corresponding procedure, let us specify system (2.11) as follows l˙ = μR(μl, s, ϕ, θ) s˙ = μ2 S(μl, s, ϕ, θ) θ˙ = μQ(l, s) + μ2 Θ(μl, s, ϕ, θ)

(2.14)

ϕ˙ = Ω(μl, s) + μ2 G(μl, s, ϕ, θ) where all the coordinates and functions are scalars, and μ 1. Such kind of equations may occur when considering ‘essentially nonlinear’ systems under diﬀerent resonance conditions. This is the reason for using another notation for small parameter. √ In resonance cases, original small parameters are often modiﬁed as μ = ε to capture speciﬁcs of the dynamics near resonance surfaces [9], [205]. The basic approximation is obtained from system (2.14) by applying the averaging procedure directly to the terms of order μ on the right hand side. This gives, l˙ = μ < R(0, s, ϕ, θ) >ϕ θ˙ = μQ(l, s)

(2.15)

where s should be considered as a constant. System (2.15) is easily integrated and the result is known to give an error of order μ on time intervals of order 1/μ. In many cases however, ﬁrst approximation gives incomplete characterizations of systems. In order to illustrate the basic stages of second approximation, consider the ﬁrst equation only. It is suﬃcient for illustration of the procedure, which is sequentially applied in the same way to other equations. Let us represent the ﬁrst equation of system (2.14) in the form l˙ = μ < R(0, s, ϕ, θ) >ϕ +μ[R(0, s, ϕ, θ)− < R(0, s, ϕ, θ) >ϕ ] (2.16) +μ2 lRμl (0, s, ϕ, θ) + O(μ3 )

Following the idea of averaging, one eliminates the second term on the righthand side by means of the coordinate transformation l = q + μf (q, s, ϕ, θ)

(2.17)

42

2 Smooth Oscillating Processes

Then, substituting (2.17) into (2.16), gives

∂f ∂f ∂f ∂f ˙ q˙ + μ q˙ + s˙ + ϕ˙ + θ ∂q ∂s ∂ϕ ∂θ = μ < R(0, s, ϕ, θ) > ϕ + μ[R(0, s, ϕ, θ)− < R(0, s, ϕ, θ) > ϕ] (2.18) +μ2 qRμl (0, s, ϕ, θ) + O(μ3 ) Now the fast phase ϕ is eliminated from the equation in ﬁrst order of μ by taking into account (2.14) and imposing condition ∂f Ω(0, s) = R(0, s, ϕ, θ)− < R(0, s, ϕ, θ) >ϕ ∂ϕ

(2.19)

Further, f (q, s, ϕ, θ) is independent of q because the terms of order μ on the right-hand side of equation (2.18) are independent of q. As a result, equation (2.18) takes the form q˙ = μ < R(0, s, ϕ, θ) > ϕ + μ2 qRμl (0, s, ϕ, θ)

∂f ∂f 2 −μ qΩ (0, s) + Q(q, s) + O(μ3 ) ∂ϕ μl ∂θ

(2.20)

Since the fast phase ϕ is eliminated from the terms of order μ then the averaging procedure is applied to the terms of order μ2 analogously to the ﬁrst stage of the method. As follows from (2.19), < ∂f /∂ϕ >ϕ =< ∂f /∂θ >ϕ = 0, therefore after the averaging, equation (2.20) takes the form q˙ = μ < R(0, s, ϕ, θ) >ϕ +μ2 < qRμl (0, s, ϕ, θ) >ϕ +O(μ3 )

or q˙ = μ < R(μq, s, ϕ, θ) >ϕ +O(μ3 )

(2.21)

The second approximation therefore is obtained by applying the operator of averaging to original equation (2.14). Note however that the meaning of the coordinate q is now diﬀerent. Namely, the original coordinate l is expressed through the new coordinate q by relationship (2.17), which due to (2.19) takes the form μ l=q+ Ω(μq, s)

ϕ (R(μq, s, ϕ, θ)− < R(μq, s, ϕ, θ) >ϕ )dϕ + O(μ2 ) (2.22) 0

In this expression, the variable μq was put back into the expression R instead of zero. Although such a manipulation has no eﬀect on the order of approximation, the new form is in a better match with the form of equation (2.21). Expressions (2.21) and (2.22) summarize the averaging procedure in second order of μ.

2.2 A Brief Overview of Smooth Methods

43

Note that the case of multiple fast phases turns out to be more complicated in many respects due to the well known problem of small denominators.

2.2.4

Averaging in Complex Variables

In the physical literature, vibration problems are usually considered in terms of complex variables [89]. The idea of using the complex variables may be suggested by the standard manipulations of the variation of constants for oscillator (2.3) as follows. If ε = 0 then general solution of equation (2.3) is represented in the complex form 1 x = [A exp(iω0 t) + A¯ exp(−iω0 t)] (2.23) 2 where A and A¯ are arbitrary complex conjugate constants. The velocity is x˙ =

iω0 [A exp(iω0 t) − A¯ exp(−iω0 t)] 2

(2.24)

If ε = 0 then the constants are assumed to be time dependent whereas expressions (2.23) and (2.24) are considered as a change of the state variables ¯ {x, x} ˙ → {A, A}

(2.25)

under the compatibility condition dA dA¯ exp(iω0 t) + exp(−iω0 t) = 0 dt dt

(2.26)

By solving equations (2.23) and (2.24) with respect to A one obtains A=

1 exp(−iω0 t)(x˙ + iω0 x) iω0

(2.27)

Similar kind of complex amplitudes is used in both physics [89] and nonlinear mechanics [102]. Now equation (2.3) gives dA ε = exp(−iω0 t)f dt iω0

(2.28)

where f = f (x, x) ˙ is expressed trough (2.23) and (2.24). Equation (2.28) is still exactly equivalent to (2.3). If the parameter ε is small then the amplitude A is slow, and one can apply the averaging

44

2 Smooth Oscillating Processes

ε dA = dt 2πi

2π/ω 0

exp(−iω0 t)f dt

(2.29)

0

On theoretical point of view, complex amplitudes may bring some convenience compared to the traditional Van-der-Pol variables. Firstly, until the certain stage of manipulations, it is usually possible to keep only one equation since another one is its complex conjugate. Secondly, such a symmetry of the equations helps sometimes to reveal interesting features of the dynamics. Note that the above manipulations remain valid in degenerated cases of multiple degrees of freedom systems. For instance, equation (2.3) can be interpreted as a vector equation with the scalar factor ω02 .

2.2.5

Lie Group Approaches

The one-parameter Lie1 group approaches are motivated by the idea of matching the tool and the object of study as explained in works [201] and [204]. Brieﬂy, it is suggested to seek transformation (2.12) among solutions of dynamical systems rather than the class of the arbitrary nonlinear transformations. Original materials and overviews of the mathematical structure of Lie groups, Lie algebras and Lie transforms with applications to nonlinear differential equations can be found in [38], [61], [22], [95], [35]. An essential ingredient of this version is the Hausdorﬀ formula, which relates the Lie group operators of the original and new systems, and the operator of coordinate transformation. According to [202] and [204], most of the averaging techniques just reproduce this formula, each time implicitly, during the transformation process. But, there is no need of doing this, since it is reasonable to start the transformation using Hausdorﬀ’s relationship. The corresponding algorithms therefore enable one of optimizing the number of manipulations for high-order approximations of asymptotic integration. The theory of Lie groups deals with a set of transformations. In other words, some dynamical system y˙ = f (y, ε) is transformed into its simplest form z˙ = g (z, ε) by means of a coordinate transformation y → z produced by solution z = z(y, ε) of the third dynamical system dz = T (z, ε) , dε

z |ε=0 = y

where the choice for vector-function T (z, ε) depends upon desired properties of the transformed system. 1

Marius Sophus Lie ( 1842-1899 ), Norwegian mathematician; diﬀerent mathematical objects are named after him, for instance, groups, operators, algebras, and series.

2.2 A Brief Overview of Smooth Methods

45

As mentioned, one of the advantages of the group formulation is that it speciﬁes a general class of near identical transformations. Speciﬁcally, one should select the expression z = z(y, ε) among solutions of a dynamical system, but not among all classes of the near identical transformations. Another basic advantage is that all manipulations of the scheme can be done in linear terms of the monomial Lie group operators. Moreover, the result of transformation in general terms of operators is well-known and is given by the Hausdorﬀ formula. The description below presents all the stages starting with the traditional Newtonian form of the diﬀerential equations of motion as implemented in [147]. The original system will be reduced to its normal form by Poincare. In terms of the principal coordinates qk , a nonlinear dynamical system of n-degrees of freedom may be described by a set of n + 1 autonomous diﬀerential equations written in the standard form q¨k + ωk2 qk = εFk (q1 , ..., qn+1 , q˙1 , ..., q˙n+1 );

k = 1, ..., n + 1

(2.30)

where an overdot denotes diﬀerentiation with respect to time t, ε is a small parameter, and an external excitation has been replaced by the coordinate qn+1 . The functions Fk include all nonlinear terms and possibly parametric excitation terms, and ωk are the principal mode frequencies. It is assumed that the functions Fk admit Taylor expansions near zero. The Poincare normal form theory deals with sets of ﬁrst-order diﬀerential equations written in terms of normal form coordinates. In this case it is convenient to transform the n + 1 second-order diﬀerential equations (2.30) into n + 1 ﬁrst-order diﬀerential equations plus their conjugate set. This can be done by introducing the complex coordinates yk = q˙k + iωk qk qk =

1 (yk − y¯k ), 2iωk

q˙k =

(2.31) 1 (yk + y¯k ) 2

(2.32)

Introducing the transformation (2.31) into the equations of motion (2.30), gives d (q˙k + iωk qk ) − iωk (q˙k + iωk qk ) = dt dyk − iωk yk = εFk (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 ) = dt

q¨k + ωk2 qk =

or y˙ k = iωk yk + εFk (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 )

(2.33)

and the corresponding complex conjugate (cc) set of equations, where the functions Fk (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 ) are obtained by substituting (2.32) in the right hand side of equation (2.30). These terms can be represented in the polynomial form

46

2 Smooth Oscillating Processes

Fk =

m

l

n+1 l1 n+1 Fkσ y1m1 · · · yn+1 y¯1 · · · y¯n+1

(2.34)

|σ|=2,3,...

where the Taylor coeﬃcients are Fkσ =

∂ |σ| Fk 1 | m ln+1 y=0 n+1 σ! ∂y1m1 · · · ∂yn+1 ∂ y¯1l1 · · · ∂ y¯n+1

and multiple-index notations have been introduced as follows σ = {m1 , ..., mn+1 , l1 , ..., ln+1 } |σ| = m1 + · · · + mn+1 + l1 + · · · + ln+1 σ! = m1 ! · · · mn+1 !l1 · · · ln+1 ! Equations (2.33) correspond to the standard form, which is ready for analysis in terms of Lie group operators. To apply the theory of the Lie groups we rewrite equations (2.33) in the form (2.35) y˙ = Ay, A = A0 + εA1 where y = (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 )T , and A0 =

n+1 k=1

iωk yk

∂ + cc ∂yk

and

A1 =

n+1

Fk

k=1

∂ + cc ∂yk

(2.36)

are operators of linear and nonlinear components of the system, respectively. In order to bring the equations of motion to their simplest (Poincare) form, we introduce the coordinate transformation y → z in the Lie series form y = e−εU z = z−εU z+

ε2 2 U z − ... 2!

(2.37)

T

where z = (z1 , ..., zn+1 ; z¯1 , ..., z¯n+1 ) , and the operator of transformation U is represented in the power series form with respect to the small parameter ε U = U0 + εU1 + · · ·

(2.38)

The coeﬃcients of this series are Uj =

n+1 k=1

Tj,k

∂ + cc ∂zk

(2.39)

where Tj,k = Tj,k (z1 , ..., zn+1 ; z¯1 , ..., z¯n+1 ) are unknown functions to be determined.

(2.40)

2.2 A Brief Overview of Smooth Methods

47

One of the advantages of this process is that the inverse coordinate transformation to the form (2.37) can be easily written as eεU y = z

(2.41)

where one should simply replaces z with y in the operator of transformation, U. If ε = 0, transformation (2.37) becomes identical, y = exp (0) z = z. In this case, equation (2.35) has already the simplest linear form and there is no need to transform the system. For ε = 0 transformation (2.37) converts the system (2.35) into the following one: z˙ = Bz

(2.42)

where the new operator B is given by the Hausdorﬀ formula [22]: B = A + ε [A, U ] +

ε2 [[A, U ] , U ] + ... 2!

(2.43)

where [A, U ] = AU − U A is the commutator of operators A and U . An optimized iterative algorithm for high-order solutions of equation (2.43) was suggested in [202] and [204]. In order to illustrate just the leading order terms of asymptotic expansions, let us follow the direct procedure though. Substituting the power series expansions for A and U given by relations (2.35) and (2.38) into (2.43) gives B = A0 + ε (A1 + [A0 , U0 ])

1 2 +ε [A0 , U1 ] + [A1 , U0 ] + [[A0 , U0 ] , U0 ] + ... 2!

(2.44)

A simple calculation gives B=

n+1

{iωk zk + ε [Fk + (A0 − iωk ) T0,k ]}

k=1

∂ + O ε2 + cc ∂zk

(2.45)

where the terms of order ε2 have been ignored, Fk = Fk |y→z and A0 = A0 |y→z . The above relationships show that a transformation of the system y˙ = Ay → z˙ = Bz can be considered in terms of operators A → B. In order to bring the system into its normal form, one must eliminate as many nonlinear terms as possible from the transformed system such that the system dynamic characteristics are preserved. It follows from (2.45) that all nonlinear terms of order ε could be eliminated under the condition

48

2 Smooth Oscillating Processes

Fk + (A0 − iωk ) T0,k = 0 Representing the unknown functions in the polynomial form mn+1 l1 ln+1 σ m1 T0,k z1 · · · zn+1 z¯1 · · · z¯n+1 T0,k =

(2.46)

|σ|=2,3,...

and taking into account (2.34), gives m1 mn+1 l1 ln+1 σ Fkσ + iΔσk T0,k z1 · · · zn+1 z¯1 · · · z¯n+1 Fk + (A0 − iωk ) T0,k = |σ|=2,3,...

where Δσk = (m1 − l1 − δ1k ) ω1 + ... + (mn+1 − ln+1 − δn+1,k ) ωn+1 m

(2.47) l

n+1 l1 n+1 To reach zero-th coeﬃcient of the monomial z1m1 · · · zn+1 z¯1 · · · z¯n+1 , one must put Fσ σ = i kσ T0,k Δk

under the condition that Δσk = 0. If Δσk = 0 for some k and σ then the corresponding nonlinear term cannot be eliminated from the transformed equation since it is qualiﬁed as a resonance term. Finally, the result of transformation is summarized as follows. The original set: mn+1 l1 ln+1 Fkσ y1m1 · · · yn+1 y¯1 · · · y¯n+1 (2.48) y˙ k = iωk yk + ε |σ|=2,3,...

The transformation of coordinates:

yk = zk − ε

|σ|=2,3,... Δσ k =0

i

Fkσ m1 mn+1 l1 ln+1 z1 · · · zn+1 z¯1 · · · z¯n+1 + O ε2 σ Δk

(2.49)

The transformed set: z˙k = iωk zk + ε

mn+1 l1 ln+1 Fkσ z1m1 · · · zn+1 z¯1 · · · z¯n+1 + O ε2

(2.50)

|σ|=2,3,... Δσ k =0

Equations (2.50) represent the normal form of the system, where the summation is much simpler than that in the original set (2.48). Namely, the summation in (2.50) contains only those terms that give rise to resonance while the ﬁrst term on the right hand side stands for the fast component of the

2.2 A Brief Overview of Smooth Methods

49

motion. The fast component of the motion can be extracted by introducing the complex amplitudes ak (t) as follows zk = ak (t) exp(iωk t)

(2.51)

Substituting (2.51) into (2.50) and taking into account the resonance condition, Δσk = 0, gives mn+1 l1 ln+1 1 (2.52) Fkσ am ¯1 ...¯ an+1 + O ε2 a˙ k = ε 1 ...an+1 a |σ|=2,3,... Δσ k =0

System (2.52) describes the dynamics in terms of slowly varying amplitudes and, as a result, reveals global properties of the dynamics in a much easier way then the original system. After solution of system (2.52) is obtained, the coordinate transformations (2.51) and (2.49) can interpret the result in terms of the original coordinates.

Chapter 3

Nonsmooth Processes as Asymptotic Limits

Abstract. In this chapter, we consider diﬀerent physical models generating both smooth and nonsmooth temporal mode shapes as appropriate conditions occur. The objective is to bring attention to the fact that nonsmooth processes may naturally occur as high-energy asymptotics in diﬀerent oscillatory models with no intentionally introduced stiﬀ constraints or external impacts. In other words, nonsmooth temporal mode shapes may be as natural as sine waves generated by oscillators under low-energy conditions. Essentially nonlinear phenomena, such as nonlinear beats and energy localization are also considered. In particular, it is shown that energy exchange between two oscillators may possess hidden nonsmooth behaviors.

3.1

Lyapunov’ Oscillator

Let us consider a family of oscillators described by the diﬀerential equation x ¨ + x2n−1 = 0

(3.1)

where n is a positive integer. In the particular case n = 1 one has the simplest harmonic oscillator. However, when n > 1 the system becomes essentially nonlinear and cannot be linearized within the class of vibrating systems. Moreover, as the parameter n increases, the temporal mode shape of oscillator (3.1) while remaining smooth is gradually approaching the triangular wave non-smooth limit. In general, such transition represents a challenging problem from both physical and mathematical viewpoints. Therefore, it is important to understand some basic cases, such as oscillator (3.1) and those considered in the next section. These special cases admit exact solutions, so that it is possible to actually see how smooth motions are approaching the non-smooth limit. There are also methodological reasons for considering equation (3.1) as a simple example of oscillators including both linear and strongly nonlinear cases. It is known that, for an arbitrary positive integer n, general solution of equation (3.1) V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 51–91, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

52

3 Nonsmooth Processes as Asymptotic Limits

can be expressed in terms of special Lyapunov’s function [96], [77], [56] such as snθ and csθ as deﬁned by expressions1

θ=

snθ

1 − nz 2

1−2n 2n

dz,

cs2n θ + n sn2 θ = 1

0

These functions possess the properties cs0 = 1,

sn0 = 0,

dsnθ = cs2n−1 θ, dθ

dcsθ = −snθ dθ

The period is given by √ T =4 n

1 0

1 √ dx π Γ 2n √ =2 n Γ n+1 1 − x2n 2n

Now, the general solution of equation (3.1) can be written as x = Acs An−1 t + α

(3.2)

where A and α are arbitrary constants. Note that the scaling factors A and An−1 are easily predictable based on the form of equation (3.1). Indeed, equation (3.1) admits the group of transformations x = A¯ x(t¯), where t¯ = An−1 t. For n = 1 the functions snθ and csθ give the standard pair of trigonometric functions sin θ and cos θ, respectively. Interestingly enough, the strongly nonlinear limit n → ∞ also gives a quite simple pair of periodic functions. Despite some mathematical challenges, this case admits interpretation by means of the total energy x2n 1 x˙ 2 + = (3.3) 2 2n 2 where the number 1/2 on the right-hand side corresponds to the initial conditions x (0) = 0 and x˙ (0) = 1. Taking into account that the coordinate of the oscillator reaches its amplitude value at zero kinetic energy, gives the estimate −n1/(2n) ≤ x (t) ≤ n1/(2n) for any time t. Since n1/(2n) −→ 1 as n −→ ∞ then the limit motion is restricted by the interval −1 ≤ x (t) ≤ 1. Inside of this interval, the second term on the left-hand side of expression (3.3) vanishes and hence, x˙ = ±1 or x = ±t + α± , where α± are constants. By manipulating with the signs and constants one can construct the sawtooth sine τ (t) - triangular wave - since there is no other way to providing the periodicity condition. 1

Another version of special functions for equation (3.1) was considered in [167].

3.1 Lyapunov’ Oscillator

53

So the family of oscillators (3.1) includes the two quite simple cases associated with the boundaries of the interval 1 ≤ n < ∞. Respectively, one has the two couples of periodic functions {x, x} ˙ = {sint, cost},

if n = 1

(3.4)

if n → ∞

(3.5)

and {x, x} ˙ → {τ (t), τ˙ (t)},

where τ˙ (t) is a generalized derivative of the sawtooth sine and will be named as a rectangular cosine. Earlier, the power-form characteristics with integer exponents were employed for phenomenological modeling amplitude limiters of vibrating elastic structures [191] and illustrations of impact asymptotics [132], [137]. It should be noted that such phenomenological approaches to impact modeling are designed to capture the integral eﬀect of interaction with physical constraints bypassing local details of the dynamics near constraints. Such details ﬁrst of all depend upon both the vibrating body and amplitude limiter physical properties. In many cases, Hertz model of interaction may be adequate to describe the local dynamics near constraint surfaces [64]. Note that direct replacement of the characteristic x2n−1 by the Hertzian restoring force kx3/2 in (3.1) gives no oscillator. The equation, x¨ + kx3/2 = 0

(3.6)

which is a particular case of that used in [64], must be obviously accompanied by the condition 0 ≤ x, where x = 0 corresponds to the state at which the moving body and constraint barely touch each other with still zero interaction force. The following modiﬁcation brings system (3.6) into the class of oscillators with odd characteristics x ¨ + ksgn(x)|x|3/2 = 0

(3.7)

However, oscillator (3.7) essentially diﬀers from oscillator (3.1) since equation (3.7) describes no gap (clearance) between the left and right constraint surfaces. In other words models (3.1) and (3.7) still represent physically diﬀerent situations. The gap 2Δ with its center at the origin x = 0 can be introduced in equation (3.7) as follows x ¨ + k[H(x − Δ)|x − Δ|3/2 − H(−x − Δ)|x + Δ|3/2 ] = 0

(3.8)

where H means Heaviside unit-step function. This is a generalization of model (3.7), which is now obtained from (3.8) by setting Δ = 0. Equation (3.8) can be viewed as a physical impact oscillator that accounts for elastic properties of its components. As compared to

54

3 Nonsmooth Processes as Asymptotic Limits

phenomenological model (3.1), equation (3.8) was obtained on certain physical basis given by the Hertz contact theory. Finally, oscillators with power-form characteristics, including their generalizations, can be found in physical literature[26], [121], [58], [60], [98] and diﬀerent areas of applied mathematics and mechanics [112], [11], [3], [4], [43], [107], [5], [34], [6], [63], [193], [48], [108], [164]. In [146], the power-form restoring forces were introduced to simulate the eﬀect of liquid container’ walls on liquid sloshing impacts; see also review article [66].

3.2

Nonlinear Oscillators Solvable in Elementary Functions

A class of strongly nonlinear oscillators admitting surprisingly simple exact general solutions at any level of the total energy is described below. Although the fact of exact solvability of these oscillators has been known for quite a long time [78], it did not attract much attention possibly due to the speciﬁc form of the oscillator characteristics with uncertain physical interpretations. It is clear however that, in a phenomenological way, such characteristics capture suﬃciently general physical situations with hardening and softening behavior of the elastic forces. For instance, these oscillators were recently used as a phenomenological basis for describing diﬀerent practically important physical and mechanical systems [122], [40], [41]. The hardening characteristic is close to linear for relatively small amplitudes but becomes inﬁnity growing as the amplitude approaches certain limits. As a result, the corresponding temporal mode of vibration changes its shape from smooth quasi harmonic to nonsmooth triangular wave. In contrast, the softening characteristic behaves in a non-monotonic way such that the vibration shape is approaching the rectangular wave as larger amplitudes are considered. Earlier, amplitude-phase equations were obtained for a coupled array of the hardening oscillators [157]. It will be shown below that such oscillators admit explicit introduction of the action-angle variables within the class of elementary functions. As a result, conventional averaging procedures become applicable to a wide range of nonlinear motions including transitions from high- to low-energy dynamics. In particular, analytical solutions are obtained under small damping conditions. These solutions show a good match with the corresponding numerical solutions at any energy level even within the ﬁrstorder asymptotic approximation. Hardening and softening cases of these oscillators are, respectively, H=

tan2 x tan x p2 + ⇒x ¨+ =0 2 2 cos2 x

(3.9)

and

tanh2 x tanh x p2 + ⇒x ¨+ =0 2 2 cosh2 x where p = x˙ is the linear momentum of the Hamiltonian H. H=

(3.10)

3.2 Nonlinear Oscillators Solvable in Elementary Functions

55

Further objectives are to investigate the high-energy asymptotics with transitions to nonsmooth temporal mode shapes and to show that both of the above oscillators can play the role of generating systems for regular perturbation procedures within the class of elementary functions. tan x cos2 x 100

50

Π

2

Π

Π

4

4

Π

2

x

50

100 Fig. 3.1 Hardening characteristic.

tanh x cosh2 x 0.5

Π

2

Π

Π

4

4

Π

2

x

0.5 Fig. 3.2 Softening characteristic.

Notice that oscillators (3.9) and (3.10) complement each other as those with stiﬀ and soft characteristics represented in Figs. 3.1 and 3.2, respectively. These oscillators can be represented also in the form x ¨ + tan x + tan3 x = 0

(3.11)

56

3 Nonsmooth Processes as Asymptotic Limits

x ¨ + tanh x − tanh3 x = 0

(3.12)

Further analyses of equations (3.11) and (3.12) can be quite easily conducted by means of substitutions q = tan x and q = tanh x, respectively. Interestingly enough, oscillators (3.11) and (3.12) without the qubic terms, namely x¨ + tan x = 0 and x ¨ + tanh x = 0, were considered by Timoshenko and Yang [182]. But, despite of the simpliﬁed form, the corresponding solutions were found to be special functions.

3.2.1

Hardening Case

Consider ﬁrst stiﬀ oscillator (3.9), whose solution is

t x = arcsin sin A sin cos A

(3.13)

where A is an arbitrary constant, and another constant is introduced as an arbitrary time shift t− > t + const., since the equations admits the group of temporal shifts. Therefore, (3.13) represents a general periodic solution of the period T = 2π cos A, and the total energy is expressed through the amplitude A as E=

1 tan2 A 2

(3.14)

In zero energy limit, when the amplitude A is close to zero, the oscillator linearizes whereas solution (3.13) gives the corresponding sine-wave temporal shape. On the other hand, the energy becomes inﬁnitely large as the parameter A approaches the upper limit π/2. In this case, the period vanishes while the oscillation takes the triangular wave shape, as it is seen from expression (3.13). Fig. 3.3 illustrates the evolution of the vibration shape in the normalized coordinates. Below, the action-angle variables are introduced in terms of elementary functions. This enables one of considering non-periodic motions by using exact solution (3.13) as a starting point of the averaging procedure. For a single degree-of-freedom conservative oscillator, the action coordinate I is known to be the area bounded by the system’ trajectory on the phase plane divided by 2π whereas the angle ϕ coordinate is simply phase angle [8], [124]. In the case of stiﬀ oscillator (3.9), one obtains 1 1 I= pdx = −1 (3.15) 2π cos A and, ϕ= respectively.

t cos A

(3.16)

3.2 Nonlinear Oscillators Solvable in Elementary Functions

57

1.0 Α0.001 0.5 Α0.999 Ns

0.0

0.5

1.0 0

1

2

3

4

5

6

Fig. 3.3 Normalized temporal mode shapes of the stiﬀ oscillator, N s(ϕ, α) = arcsin(α sin ϕ)/ arcsin α.

The original coordinate and the velocity are expressed by the action-angle variables as follows [158] √ √ (1 + I) 2I + I 2 cos ϕ 2I + I 2 (3.17) sin ϕ , p = x = arcsin 1+I 1 + (2I + I 2 ) cos2 ϕ In order to observe the convenience of action-angle coordinates, let us chose the Hamiltonian description of the oscillator. Taking into account expressions (3.14) and (3.15), and eliminating the amplitude A, gives the total energy and thus the Hamiltonian in the form 1 H = I + I2 2

(3.18)

The corresponding diﬀerential equations of motion are derived as follows ∂H =1+I ∂I ∂H I˙ = − =0 ∂ϕ

ϕ˙ =

(3.19)

As it is seen, the diﬀerential equation of the oscillator (3.9) takes the linear form with respect to the action-angle coordinates, and thus possess the exact general solution (3.20) I = I0 , ϕ = (1 + I0 ) t + ϕ0 where I0 > 0 and ϕ0 are arbitrary constants. By substituting (3.20) in (3.17), one can express the solution via the original coordinates. The meaning of the initial action is clear from the energy relationship

58

3 Nonsmooth Processes as Asymptotic Limits

1 1 E = I0 + I02 = tan2 A 2 2

(3.21)

Note that the linearity of the Hamiltonian equations is due to the speciﬁc strongly non-linear form of the coordinate transformation (3.17). In other words, the system nonlinearity has been ‘absorbed’ in a purely geometric way by the nonlinear coordinate transformation. As mentioned at the beginning, simplicity of the transformed system and that of the corresponding solution can be essentially employed for the purpose of perturbation analysis. Let us consider, however, the diﬀerential equation of motion in the Newtonian form x ¨+

tan x = εf (x, x) ˙ cos2 x

(3.22)

were ε is a small parameter. This system is weakly non-hamiltonian. However, it is still possible to consider expressions (3.17) as a change of the coordinates {x, p} −→ {I, ϕ} by imposing the compatibility condition dx/dt = p, where x and p must be taken from (3.17). This gives εf (x, p) sin ϕ (1 + I) (2I + I 2 ) [1 + (2I + I 2 ) cos2 ϕ] √ εf (x, p) 2I + I 2 cos ϕ I˙ = 1 + (2I + I 2 ) cos2 ϕ

ϕ˙ = 1 + I −

(3.23)

where the function f (x, p) must be expressed through the action-angle coordinates by means of (3.17). For instance, in the case of linear damping, f (x, p) ≡ −p, one obtains ε cos ϕ sin ϕ 1 + (2I + I 2 ) cos2 ϕ ε (1 + I) 2I + I 2 cos2 ϕ ˙ I=− 1 + (2I + I 2 ) cos2 ϕ

ϕ˙ = 1 + I +

(3.24)

At this stage, let us implement just one step of the procedure and evaluate its eﬀectiveness. Applying the operator of averaging with respect to the phase variable gives the corresponding ﬁrst-order averaged system in the linear form ϕ˙ = 1 + I,

I˙ = −εI

(3.25)

Substituting the general solution of system (3.25) in (3.17), ﬁnally gives

2I0 exp (−εt) + I02 exp (−2εt) 1 − exp (−εt) sin t + I0 + ϕ0 x = arcsin 1 + I0 exp (−εt) ε (3.26)

3.2 Nonlinear Oscillators Solvable in Elementary Functions

59

where I0 and ϕ0 are arbitrary constants. The corresponding time history records and phase plane diagrams for diﬀerent damping coeﬃcients are shown in Fig. 3.4. Even the ﬁrst order approximation appears to be perfectly matching with numerical solution for all range of amplitudes. The analytical and numerical curves can be distinguished only at relatively large magnitudes of the damping parameter ε. Also, the diagrams show that the temporal mode shape is gradually changing from the triangular to harmonic as time increases and thus the amplitude decays. 1.5 1.0 0.5 x 0.0 0.5 1.0 1.5

0

10

20

30

40

50

1.5 1.0 0.5 x 0.0 0.5 1.0 1.5

0

10

t

v

30

40

50

1.0

1.5

t

10

10

5

5

0

v 0

5

5

10 1.5 1.0 0.5 0.0 x

20

0.5

1.0

1.5

1.5 1.0 0.5 0.0 x

0.5

Fig. 3.4 The dynamics of the linearly damped stiﬀ oscillator under the initial conditions I0 = 10 and ϕ0 = 0, and two diﬀerent damping parameters: ε = 0.2 (on the left,) and ε = 0.8 (on the right.) Analytical and numerical solutions show a slight mismatch only on the top right diagram.

3.2.2

Localized Damping

Let us consider the case of nonlinear damping x ¨+

tan x + 2εx˙ tan2 x = 0 cos2 x

(3.27)

In this case, the perturbation is given by f (x, p) ≡ −2p tan2 x. Such a damping is rapidly growing near the boundaries of the interval −π/2 ≤ x ≤ π/2, but it becomes negligible when the amplitude is small, |x| << 1.

60

3 Nonsmooth Processes as Asymptotic Limits

In the action-angle coordinates, ﬁrst order averaging gives ϕ˙ = 1 + I, and thus ϕ=t+

I˙ = −εI 2

1 ln (1 + εI0 t) + ϕ0 , ε

I=

I0 1 + εI0 t

Using the coordinate transformation (3.17), gives solution

I0 (2 + I0 + 2εI0 t) ln(1 + εI0 t) sin t + + ϕ0 x = arcsin 1 + I0 + εI0 t ε

(3.28)

where I0 and ϕ0 are arbitrary constants. Note that the amplitude decay of solutions (3.26) and (3.28) is qualitatively diﬀerent. For instance, the amplitude of vibration (3.28) originally decays in a fast rate and then becomes very slow. In contrast, the amplitude of vibration (3.26) ﬁrst decays slowly then the decay rate abruptly increases and then slows down again.

3.2.3

Softening Case

Let us consider now softening oscillator (3.10), whose exact solution is

t x = arc sinh sinh A sin (3.29) cosh A As Fig. 3.5 shows, the high-energy vibration shape approaches the rectangular wave and thus essentially diﬀers of that observed in the stiﬀ case. Based on solution (3.29), the action-angle coordinates are introduced by means of expressions √ √ (1 − I) 2I − I 2 cos ϕ 2I − I 2 sin ϕ , p = (3.30) x = arc sinh 1−I 1 − (2I − I 2 ) cos2 ϕ where

1 (3.31) cosh A Nevertheless, all the analytical manipulations are analogous to those in the stiﬀ case. For instance, taking into account (3.31), gives the total energy as a function of the action coordinate I =1−

E=

1 1 tanh2 A = I − I 2 2 2

(3.32)

3.3 Nonsmoothness Hiden in Smooth Processes

61

1.0 Α106 0.5 Α0.001 Nh

0.0

0.5

1.0 0

1

2

3

4

5

6

Fig. 3.5 Normalized temporal mode N h(ϕ, α) =arcsinh(α sin ϕ)/arcsinh α.

shapes

of

the

soft

oscillator,

In the presence of viscous damping, x ¨+

tanh x = −εx˙ cosh2 x

(3.33)

one obtains, compare to (3.25), ϕ˙ = 1 − I,

I˙ = −εI

(3.34)

and general solution of the original equation takes the form

2I0 exp (−εt) − I02 exp (−2εt) 1 − exp (−εt) sin t−I0 +ϕ0 x = arc sinh 1 − I0 exp (−εt) ε (3.35) The corresponding time history records and phase plane diagrams are shown in Fig. 3.6 for diﬀerent damping coeﬃcients. The ﬁrst order approximation appears to perfectly match the corresponding numerical solution for all range of amplitudes, unless the initial action I0 approaches the magnitude 1. As follows from expressions (3.32), this magnitude corresponds to the maximum value of the total energy of the oscillator. Note that the energy of the hardening oscillator has no maximum.

3.3

Nonsmoothness Hiden in Smooth Processes

In this section, we consider nonlinear beats phenomena as another source of nonsmooth behavior that brings certain physical meaning to oscillator (3.22). Note that nonlinear beats became of growing interest just few decades ago

62

3 Nonsmooth Processes as Asymptotic Limits

0.8 0.6 0.4 0.2 x 0.0 0.2 0.4 0.6

0.6 0.4 x 0.2 0.0 0.2 0

10

20

30

40

50

0

10

20

t 1.0

40

50

1.0

0.5 v

30 t

0.5 v

0.0

0.0

0.5 0.60.40.2 0.0 0.2 0.4 0.6 0.8 x

0.5 0.2

0.0

0.2 x

0.4

0.6

Fig. 3.6 The dynamics of the linearly damped softening oscillator under the initial conditions I0 = 0.5, ϕ0 = 0, and two diﬀerent damping parameters: ε = 0.2 (on the left,) and ε = 0.8 (on the right.) Analytical and numerical curves practically coincide.

from diﬀerent viewpoints of physics and nonlinear dynamics [84], [59], [99], [188], [88]. Interestingly enough, phase variables of interacting oscillators with close natural frequencies may show non-smoothness of temporal behavior during the beating [59], for instance similar to that of a vibro-impact process [101], [104].

3.3.1

Nonlinear Beats Model

Consider two linearly coupled Duﬃng oscillators x ¨1 + Ω 2 x1 = −β(x1 − x2 ) − αx31 ≡ f1 x ¨2 + Ω 2 x2 = −β(x2 − x1 ) − αx32 ≡ f2

(3.36)

where α and β are nonlinearity and linear coupling parameters, respectively. Let us introduce complex coordinates Aj (t) and A¯j (t) as follows 1 [Aj exp(iΩt) + A¯j exp(−iΩt)] 2 iΩ x˙ j = [Aj exp(iΩt) − A¯j exp(−iΩt)] 2 xj =

where j = 1, 2.

(3.37)

3.3 Nonsmoothness Hiden in Smooth Processes

63

Convenience of using the complex amplitudes for linear and nonlinear mode analyses have been known for a long time [131], [174]. In physical literature though, complex amplitudes are introduced more often as vectors rotating on complex phase planes of oscillators, but the resultant equations are usually similar to those obtained below. Expressions (3.37) implement indeed a complex version of the parameter variation method based on the solution of the corresponding linear system. The related compatibility condition is imposed in the form dAj dA¯j exp(iΩt) + exp(−iΩt) = 0 dt dt

(3.38)

By substituting (3.37) in (3.36) and taking into account (3.38) gives dAj 1 = exp(−iΩt)fj dt iΩ

(3.39)

Assuming that the coupling and nonlinearity parameters are suﬃciently small and averaging the right-hand side with respect to Ωt, gives

βi 3α 2 ¯ A˙ 1 = A1 A1 A1 − A2 + 2Ω 4β

βi 3α 2 ¯ ˙ A A2 A2 = −A1 + A2 + 2Ω 4β 2

(3.40)

Further, following work [104], a slowly rotating subcomponent on the complex plane is eliminated by means of substitution

Aj = ψj (t) exp

iβ t 2Ω

Substituting (3.41) into (3.40), gives

˙ψ = − βi ψ − 3 α ψ 2 ψ ¯ 1 2 2Ω 4β 1 1

˙ψ = − βi ψ − 3 α ψ 2 ψ ¯ 2 1 2Ω 4β 2 2

(3.41)

(3.42)

This system has two integrals as follows ¯ +ψ ψ ¯ K = ψ1 ψ 1 2 2

(3.43)

64

3 Nonsmooth Processes as Asymptotic Limits

3α 4 4 ¯ +ψ ψ ¯ |ψ 1 | + |ψ 2 | G = ψ1 ψ 2 2 1− 8β Integral (3.43) admits substitution

√ 1 π ψ 1 = K cos θ(t) + exp[i δ1 (t)] 2 4

√ 1 π ψ 2 = K sin θ(t) + exp[i δ 2 (t)] 2 4

(3.44)

(3.45)

Substituting (3.45) in (3.42) and (3.44), gives G = −K cos Δ cos θ −

3 K2 α (3 − cos 2 θ) = const. 32 β

(3.46)

β 3K α Δ˙ = − cos Δ tan θ + sin θ (3.47) Ω 8Ω β sin Δ (3.48) θ˙ = Ω where Δ = δ 2 − δ 1 + π. It will be shown below that the phase variable θ, which determines the process of energy ﬂow between the oscillators, is described by the oscillator (3.22). Equations (3.47)-(3.48) are similar to those obtained in [101], [104], where it was noticed that temporal shapes of the phase variables θ and Δ may resemble the behavior of the state variables of impact oscillator. This observation seems to be important since it is hard to expect any “impact oscillators” in weakly nonlinear systems of type (3.36). Below, the corresponding ‘conservative oscillator’ admitting the impact limit will be explicitly obtained and analyzed by using the methodology described in the previous section. However, the approach below deals with a general class of nonlinear restoring force characteristics admitting powerseries expansions. It will be shown also that equations (3.47)-(3.48) can be derived by introducing the standard set of amplitude-phase variables and applying then the traditional one fast phase averaging technique.

3.4

Nonlinear Beat Dynamics: The Standard Averaging Approach

Let us consider two identical linearly coupled oscillators u ¨1 + b(u1 − u2 ) + p(u1 ) = 0 u ¨2 + b(u2 − u1 ) + p(u2 ) = 0

(3.49)

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

65

where b is the coupling stiﬀness per unit mass, and p(u) is the restoring force characteristic, which is assumed to be an analytic function that admits a power series expansion. Assuming that the system has equilibrium at zero and introducing the notation Ω2 = b + k ε = b/Ω 2 = b/(b + k)

(3.50)

f (u) = [(b + k)/b][p(u) − ku] where f (u) is a nonlinear component of the characteristic, and k=p’(0). Note that the power series expansion for f (u) starts from at least second degree of u. Therefore, the order of magnitude of the function f (u) can be manipulated by making appropriate assumptions as to the magnitude of the total energy of the system. Taking into account the above notations and introducing the velocities v1 (t) and v2 (t) brings the original system to the form u˙ 1 = v1 u˙ 2 = v2 v˙ 1 = −Ω 2 u1 + ε[Ω 2 u2 − f (u1 )]

(3.51)

v˙ 2 = −Ω u2 + ε[Ω u1 − f (u2 )] 2

2

As ε → 0, the system degenerates into two identical harmonic oscillators whose total energies are conserved of-course since neither damping nor external loading are present. At non-zero ε, the oscillators become non-linear and interact with each other in such a way that one of the oscillators is loaded by the force proportional to the displacement of another oscillator. Since system (3.51) is still perfectly symmetric and conservative, it is reasonable to assume a relatively slow periodic energy exchange between the oscillators. In order to describe this process in physically meaningful terms, let us introduce new set of variables as follows {u1 , v1 , u2 , v2 }− > {K, ϕ, δ1 , δ2 }: √ u1 = K cos ϕ cos(Ωt + δ1 ) √ v1 = − KΩ cos ϕ sin(Ωt + δ1 ) √ (3.52) u2 = K sin ϕ cos(Ωt + δ2 ) √ v2 = − KΩ sin ϕ sin(Ωt + δ2 ) In case ε = 0 and constant {K, ϕ, δ1 , δ2 }, expressions (3.52) gives an exact general solution of system (3.51). Therefore, relationships (3.52) simply implement the idea of parameter variations; the corresponding diﬀerential equations will be given below.

66

3 Nonsmooth Processes as Asymptotic Limits

Now, in order to track the oscillator energies during the vibration process, let us use quantities 1 2 (v + Ω 2 u21 ) = 2 1 1 E2 = (v22 + Ω 2 u22 ) = 2

E1 =

1 2 Ω K cos2 ϕ 2 1 2 Ω K sin2 ϕ 2

(3.53)

and

1 2 Ω K (3.54) 2 Besides, expressions (3.53) and (3.54) clarify the physical meaning of the variables K and ϕ participating in transformation (3.52), where other two variables, δ1 and δ2 , are phases of the vibrating oscillators. In particular, K is proportional to the total energy of the decoupled and linearized oscillators, whereas the phase ϕ characterizes the energy split between the oscillators. In case ε = 0, the energy parameter K will have small temporal ﬂuctuations due to coupling and nonlinear terms in (3.51). Nevertheless expressions (3.53) and (3.54) still will be used as the energy related quantities for characterization of the energy exchange process between the oscillators. In order to conduct the transition to the new variables, let us substitute (3.52) in (3.51) and then solve the set of equations with respect to the derivatives as follows √ 2ε K ˙ K = −εKΩ sin 2ϕ sin(2Ωt + δ1 + δ2 ) + Ω √ ×{f [ K cos ϕ cos(Ωt + δ1 )] cos ϕ sin(Ωt + δ1 ) √ +f [ K sin ϕ cos(Ωt + δ2 )] sin ϕ sin(Ωt + δ2 )} ε ε ϕ˙ = Ω[sin(δ1 − δ2 ) − cos 2ϕ sin(2Ωt + δ1 + δ2 )] − √ 2 KΩ √ ×{f [ K cos ϕ cos(Ωt + δ1 )] sin ϕ sin(Ωt + δ1 ) √ (3.55) −f [ K sin ϕ cos(Ωt + δ2 )] cos ϕ sin(Ωt + δ2 )} δ˙ 1 = −εΩ cos(Ωt + δ1 ) cos(Ωt + δ2 ) tan ϕ √ ε cos(Ωt + δ1 ) sec ϕf [ K cos ϕ cos(Ωt + δ1 )] +√ KΩ δ˙ 2 = −εΩ cos(Ωt + δ2 ) cos(Ωt + δ1 ) cot ϕ √ ε cos(Ωt + δ2 ) csc ϕf [ K sin ϕ cos(Ωt + δ2 )] +√ KΩ E0 = E1 + E2 =

System (3.55) is still an exact equivalent to system (3.49) and represents a standard dynamic system with a single fast phase, ψ = Ωt. As a next natural stage, let us apply the direct averaging to the right-hand side of (3.55) with respect to the fast phase ψ:

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

K˙ = 0 ε ϕ˙ = Ω sin(δ1 − δ2 ) 2 ε ε δ˙ 1 = − Ω cos(δ1 − δ2 ) tan ϕ + F1 (K, ϕ) 2 Ω ˙δ 2 = − ε Ω cos(δ1 − δ2 ) cot ϕ + ε F2 (K, ϕ) 2 Ω

67

(3.56)

where the residue theorem has been applied so that 2π √ 1 1 F1 (K, ϕ) = √ f ( K cos ϕ cos ψ) cos ψdψ K cos ϕ 2π 0 √ 1 1 1 1 1 = √ Res{f [ K cos ϕ (z + )] (z + 2 )} 2 z 2 z K cos ϕ 2π √ 1 1 F2 (K, ϕ) = √ f ( K sin ϕ cos ψ) cos ψdψ K sin ϕ 2π 0 √ 1 1 1 1 1 Res{f [ K sin ϕ (z + )] (z + 2 )} = √ 2 z 2 z K sin ϕ First equation in (3.56) shows that the energy parameter K introduced in (3.54) remains averagely constant regardless the magnitude of coupling and nonlinearity parameter ε. This gives a justiﬁcation for using quantities (3.53) and (3.54) for describing the energy exchange between the oscillators: indeed, neither the coupling nor nonlinear stiﬀness in (3.51) can accumulate the energy during one vibration cycle. Further complete description of the dynamics can be conducted in terms of the two phase shift parameters, Δ(t) and θ(t), introduced as follows δ2 = δ1 + Δ − π 1 1 ϕ= θ+ π 2 4

(3.57)

Substituting (3.57) in (3.56) and introducing the slow time parameter t1 = εΩt, gives dθ = sin Δ dt1 dΔ = − cos Δ tan θ + F (θ) dt1

(3.58)

where

1 1 1 1 1 [F2 (K, θ + π) − F1 (K, θ + π)] Ω2 2 4 2 4 It can be shown that system (3.58) has the integral F (θ) =

G = − cos Δ cos θ + h(θ) = const.

(3.59)

68

3 Nonsmooth Processes as Asymptotic Limits

where h(θ) = −

F (θ) cos θdθ

(3.60)

Now let us show that system (3.58) can be reduced to a single strongly nonlinear oscillator with respect to the coordinate θ. Taking time derivative of both sides of the ﬁrst equation in (3.58) and eliminating from the result dΔ/dt1 and cos Δ by means of the second equation in (3.58) and the integral of motion, (3.59), respectively, gives tan θ d2 θ = R(θ) + dp2 cos2 θ

(3.61)

where p = |G|t1 = ε|G|Ωt is a new slow temporal argument, θ = θ(p) and F (θ) tan θ R(θ) = G−2 h(θ)[2G − h(θ)] 2 − [G − h(θ)] (3.62) cos θ cos θ Equation (3.61) represents a principal equation describing the energy exchange in the coupled set of oscillators (3.49) through the phase shift (3.57). Note that the right-hand side in (3.61) is due to only nonlinearity associated with the nonlinear stiﬀness f (u); see relationships (3.50) and (3.51). In case R(θ) = 0, equation (3.61) has exact analytical solution

p (3.63) θ(p) = arcsin sin θ0 sin cos θ0 where θ0 is the amplitude of θ, whereas another constant can be introduced as an arbitrary temporal shift, which is admitted by equation (3.61). Generally speaking, it is still possible to ﬁnd implicit solutions of equation (3.61) in terms of quadratures for nonzero R(θ). In most cases, however, the corresponding expressions appear to be technically complicated for analyses. Therefore secondary asymptotic approaches to oscillator (3.61) may be reasonable for understanding its behaviors. For illustrating purposes, let us consider the example when the nonlinear stiﬀness in (3.50)-(3.51) consists of cubic and ﬁfth-degree terms f (x) = α3 x3 + α5 x5

(3.64)

In this case, integration in (3.60) gives h=

K(6α3 + 5Kα5 ) cos2 θ 32Ω 2

(3.65)

whereas equation (3.61) takes the form tan θ d2 θ = μ sin 2θ + 2 dp cos2 θ

(3.66)

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

69

where

K 2 (6α3 + 5Kα5 )2 (3.67) 2048G2 Ω 4 Substituting (3.57) in (3.53), gives the corresponding energy values versus phase θ: μ=

1 KΩ 2 (1 − sin θ) 4 1 E2 = KΩ 2 (1 + sin θ) 4 E1 =

3.4.1

(3.68)

Asymptotic of Equipartition

For suﬃciently small amplitudes of θ, equation (3.66) can be reduced to the following Duﬃng equation d2 θ 4 + (1 − 2μ)θ + (1 + μ)θ3 = 0 dp2 3

(3.69)

As follows from (3.68) on physical point of view, the equilibrium point θ = 0 corresponds to equal energy distribution between the oscillators (3.49). So when the linear stiﬀness is positive 1 − 2μ > 0, equation (3.69) describes periodic energy exchange between the oscillators (3.49) provided that the initial energy distribution is close to equal and the cubic approximation for the characteristic is justiﬁed. The period of the energy exchange process can be easily estimated based on the corresponding solution of equation (3.69). However, the linear stiﬀness becomes negative when μ>

1 2

(3.70)

Condition (3.70) says that the equal energy distribution may become unstable if the parameter μ is suﬃciently large. As a result, system (3.69) can stay in one of the two new stable equilibrium positions so that a larger portion of the total energy is localized on one of the two identical oscillators (3.49). Note that the equilibrium points of system (3.69) correspond to nonlinear normal mode regimes. The phase-plane diagrams of oscillator (3.66) are shown in Figs. 3.7, 3.8, and 3.9 for diﬀerent magnitudes of μ. It is seen how two more equilibria occur when μ exceeds the critical value (3.70). Note that condition (3.70) is obviously necessary but not suﬃcient to guarantee that the localization will actually occur. Necessary and suﬃcient conditions will be discussed below. In order to determine the initial state of oscillator (3.66) and its parameter μ, let us consider transformation (3.52) at t = 0. With no loss of generality,

70

3 Nonsmooth Processes as Asymptotic Limits 3

2

dΘdp

1

0

1 2 3

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ

Fig. 3.7 Periodic energy exchange case, μ = 0.2. 3

2

dΘdp

1

0

1 2 3

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ

Fig. 3.8 Energy trapping bifurcation, μ = 0.5

one can select δ1 (0) = 0. Then, taking into account expressions (3.57), (3.53) and (3.54), gives

√ π θ(0) u1 (0) = K cos + 4 2 v1 (0) = 0

√ π θ(0) u2 (0) = − K cos Δ(0) sin + (3.71) 4 2

√ θ(0) π + v2 (0) = KΩ sin Δ(0) sin 4 2

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

71

3

2

dΘdp

1

0

1 2 3

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ

Fig. 3.9 Super-critical energy trapping diagrams, μ = 0.9

and

θ(0) = Δ(0) = K= G=

E1 (0) − E2 (0) − arcsin E1 (0) + E2 (0)

v2 (0) − arctan Ωu2 (0) 2 [E1 (0) + E2 (0)] Ω2 K(6α3 + 5Kα5 ) − cos Δ(0) cos θ(0) + cos2 θ(0) 32Ω 2

(3.72)

where 1 2 v1 (0) + Ω 2 u21 (0) 2 1 2 v2 (0) + Ω 2 u22 (0) E2 (0) = 2 E1 (0) =

Since the initial velocity of the ﬁrst oscillator is ﬁxed v1 (0) = 0 then the system initial state is determined by the three quantities K, θ(0), and Δ(0); see (3.71). Alternatively, one can specify u1 (0), u2 (0) and v2 (0) and then ﬁnd K, θ(0), and Δ(0) from (3.72).

3.4.2

Asymptotic of Dominants

As the amplitude of θ is getting closer to π/2 then the phase θ oscillations acquire nonsmooth temporal shapes. Expression (3.63), for instance, shows that, at amplitudes near π/2, the energy exchange phase will be close to the triangular wave shape with a relatively small wave-length. In this case,

72

3 Nonsmooth Processes as Asymptotic Limits

Duﬃng equation (3.69) seems to be not an adequate model. So assuming that μ is suﬃciently small, let us introduce action-angle variables as described earlier in reference [158] √ 2I + I 2 sin φ θ = arcsin 1+I √ (1 + I) 2I + I 2 cos φ θ = (3.73) 1 + (2I + I 2 ) cos2 φ Substituting (3.73) in (3.66) under the compatibility condition θ = dθ/dp, gives still exact equivalent of oscillator (3.66) dI I(2 + I) =μ sin 2φ dp (1 + I)2 dφ 2μ = 1+I − sin2 φ dp (1 + I)3

(3.74)

Note that the coordinate transformation (3.73) still would be valid for general case (3.61) although with diﬀerent to (3.74) result. In contrast to (3.66), system (3.74) is weakly nonlinear with a very simple solution at μ = 0. In [158], the direct averaging was applied to the right-hand side of (3.74) in order to obtain the ﬁrst-order solution. The idea of averaging can be also implemented as asymptotic integration of system (3.74) by means of the coordinate transformation J(2 + J) cos 2ψ + O(μ2 ) 2(1 + J)3 (J 2 + 2J − 2) φ = ψ−μ sin 2ψ + O(μ2 ) 4(1 + J)4 I = J −μ

(3.75)

Transformation (3.75) is obtained from the condition eliminating the fast phase ϕ from the terms of order μ on the right-hand side in such a way that the new system takes the form dJ = O(μ2 ) dp μ dψ = 1+J − + O(μ2 ) dp (1 + J)3 System (3.76) is easily integrated as follows

μ ψ = 1+J − p (1 + J)3 where J = const.

(3.76)

(3.77)

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

73

Then the reversed chain of transformations back to (3.73) is applied. Solution (3.73) through (3.77) appears to have a good match with the corresponding numerical solution when μ << 1/2; see (3.70) for interpretation. The solution can work well even under condition (3.70), but the corresponding initial conditions must keep the oscillator out of the triple equilibria region. The oscillators’s motion within such region is better approximated by Duﬃng’s equation (3.69). On physical point of view, the applicability loss for solution (3.73) through (3.77) is due to the energy localization phenomenon, which is not captured by the above solution despite of its strong nonlinearity. Indeed, the term, which is responsible for occurring the triple equilibria is ignored in the leading-order approximation.

3.4.3

Necessary Condition of Energy Trapping

On physical point of view, necessary condition of localization is the presence of triple equilibrium positions of oscillator (3.66) within the basic interval −π/2 < θ < π/2, which is provided by condition (3.70). However, to guarantee the energy localization, the initial conditions must keep the oscillator within one of the two branches of the separatrix loop. Let us bring both of the above conditions to the explicit form. Necessary condition. For simplicity reason, let us consider the case of cubic nonlinearity and introduce two dimensionless parameters ν= and

3E0 α3 8Ω 4

ζ=

ΔE0 E0

(3.78)

2 (3.79)

where E0 = E1 (0) + E2 (0) is the total initial energy deﬁned by (3.54), and ΔE0 = E1 (0) − E2 (0), so that √ ν estimates the weight of nonlinearity in the system dynamics, whereas ζ is the initial energy disbalance per total initial energy E0 . Calculating the constants K and G from (3.72) and making algebraic manipulations, eventually brings the necessary condition of localization (3.70) to the form (2 − ζ)ν ζν < cos Δ(0) < √ (3.80) −√ 1−ζ 1−ζ Let us recall that condition (3.80) provides the presence of two stable equilibrium position of oscillator (3.66) near the origin (θ, dθ/dp) = (0, 0) which itself becomes unstable. However (3.80) does no guarantee that oscillator (3.66) is in the neighborhood of a stable equilibrium condition.

74

3 Nonsmooth Processes as Asymptotic Limits

3.4.4

Suﬃcient Condition of Energy Trapping

Now let us deﬁne that the energy localization takes place whenever the initial conditions keep oscillator (3.66) within one of the two separatrix loops surrounding the stable equilibrium points. A manifold of such initial conditions is obtained from ﬁrst integral of oscillator (3.66) as follows

dθ dp

2 + tan2 θ + μ cos 2θ < μ

(3.81)

Taking into account equation (3.58), dθ/dp = |G|−1 sin Δ, and calculating the left - hand side of (3.81) at p = 0, gives (3.82) ζ[cos Δ(0) − ν 1 − ζ]2 + sin2 Δ(0) < ζν 2 Note that both conditions (3.80) and (3.82) include the same set of parameters, such as the initial phase shift from the out-of-phase mode, Δ(0), the parameter characterizing the initial energy distinction between the oscillators, ζ, and the parameter characterizing the total energy and thus strength of nonlinearity, ν.

3.5

Transition from Normal to Local Modes

The transient mode localization phenomenon is considered in a mechanical model combined of a simply supported beam and transverse nonlinear springs with hardening characteristics. Two diﬀerent approaches to the model reduction, such as normal and local mode representations for the beam’s center line, are discussed. It is concluded that the local mode discretization brings advantages for the transient localization analysis. Based on the speciﬁc coordinate transformations and the idea of averaging, explicit equations describing the energy exchange between the local modes and the corresponding localization conditions are obtained. It was shown that when the energy is slowly pumped into the system then, at some point, the energy equipartition around the system suddenly breaks and one of the local modes becomes the dominant energy receiver. The phenomenon is interpreted in terms of the related phase-plane diagram which shows qualitative changes near the image of out-phase mode as the total energy of the system has reached its critical level. A simple closed form expression is obtained for the corresponding critical time estimate. The text below is an update of recent publication by the author [159].

3.6

System Description

The model under investigation represents a simply supported elastic beam of length l with two masses attached to the beam and connected to the base

3.6 System Description

75

Fig. 3.10 The mechanical model admitting both normal and local mode motions; all the springs have hardening restoring force characteristics.

by nonlinear springs; see Fig. 3.10. The corresponding diﬀerential equation of motion and boundary conditions are, respectively, ρA

∂4w ∂2w + EI 4 = f1 (t)δ(y − y1 ) + f2 (t)δ(y − y2 ) 2 ∂t ∂y

and w(t, y)|y=0,l = 0,

∂ 2 w(t, y) |y=0,l = 0 ∂y 2

(3.83)

(3.84)

where fi (t) = −f [w(t, yi )] − c

∂w(t, yi ) ∂ 2 w(t, yi ) ; i = 1, 2 −m ∂t ∂t2

(3.85)

are transverse forces applied to the beam from masses attached at the two points y = y1,2 . It will be assumed that the model is perfectly symmetric with respect to y = l/2 so that the springs are attached at points

76

3 Nonsmooth Processes as Asymptotic Limits

y1 = l/3 and y2 = 2l/3

(3.86)

Below we consider the case of the hardening restoring force characteristics of the springs and show that, under appropriate conditions, a slow energy in-ﬂow leads to the localization of vibration modes. As a result, the system energy is spontaneously shifted to either the left or right side of the beam - the symmetry break. The adiabatically ‘slow’ energy increase means that the energy source has a minor or no direct eﬀect on the mode shapes. For simulation purposes, such an energy in-ﬂow is provided by the assumption that the viscous damping coeﬃcient c is suﬃciently small and negative; the physical basis for such an assumption was discussed in [158], [157]. This remark, which is substantiated below by the corresponding numerical values of the parameters, is important to follow, otherwise the phenomenon, which is the focus of this paper, may not be developed. In contrast, the dissipation (c > 0) can lead to a spontaneous dynamic transition from local to normal modes, when the total energy reaches its sub-critical level. Note that the presence of Dirac δ-functions in equation (3.83) requires a generalized interpretation of the diﬀerential equation of motion in terms of distributions [166]. The corresponding compliance is provided by further model reduction based on the Bubnov-Galerkin approach, which actually switches from the point-wise to integral interpretation of equations.

3.7

Normal and Local Mode Coordinates

Normal mode coordinates. Let us evaluate two possible ways to discretizing the model (3.83). In this paper, the reduced-order case of two degrees-offreedom is considered, when the conventional normal mode representation for the boundary value problem (3.83) - (3.84) is w(t, y) = W1 (t) sin

πy 2πy + W2 (t) sin l l

(3.87)

Substituting (3.87) in (3.83) and applying the standard Bubnov-Galerkin procedure, gives, after dropping the time arguments, ¨1 + ¯ ˙ 1 + λ2 W1 + F (W1 + W2 ) + F (W1 − W2 ) = 0 W ζW ¨2 + ¯ ˙ 2 + 16λ2 W2 + F (W1 + W2 ) − F (W1 − W2 ) = 0 W ζW where ζ¯ = and

3c π 4 EI , λ2 = 3 3m + Alρ l (3m + Alρ) √ √ 3 3 F (z) = f z 3m + Alρ 2

(3.88)

(3.89)

(3.90)

are constant parameters and a re-scaled restoring force function, respectively.

3.7 Normal and Local Mode Coordinates

77

Equations (3.88) are decoupled in the linear terms related to the elastic beam centre line, whereas the modal coupling is due to the spring nonlinearities included in F (z). Local mode coordinates. Alternatively, the model can be discretized by introducing the ‘local mode’ coordinates determined by the spring locations ui (t) = w(t, yi ); i = 1, 2

(3.91)

Taking into account (3.86) and (3.87), and substituting in (3.91) reveals simple links between the normal and local coordinates as √ √ 3 3 (W1 + W2 ), u2 = (W1 − W2 ) (3.92) u1 = 2 2 or, inversely,

√

√ 3 3 (u1 + u2 ), W2 = (u1 − u2 ) W1 = (3.93) 3 3 Substituting (3.93) in (3.87), gives the ‘local mode expansion’ for the beam’s centre line πy πy + u2 (t)ψ2 (3.94) w(t, y) = u1 (t)ψ1 l l where the local mode shape functions are

√

3 11 ψ1 (x) sin x = (3.95) ψ2 (x) sin 2x 3 1 −1

Both normal and local mode shape functions are shown in Figs. 3.11 and 3.12, respectively. Transformation (3.95) can be generalized for a greater number of modes. Note that functions (3.95) satisfy the following ortogonality condition π π ψi (x) ψj (x) dx = δij (3.96) 3 0 where δij is the Kronecker symbol. However, the diﬀerential equations of motion for u1 (t) and u2 (t) are obtained directly by substituting (3.93) in (3.88) and making obvious algebraic manipulations that gives √ √ u ¨1 + ζ¯u˙ 1 + (λ2 /2)(17u1 − 15u2 ) + 3F (2u1 / 3) = 0 (3.97) √ √ 2 u ¨2 + ζ¯u˙ 2 − (λ /2)(15u1 − 17u2 ) + 3F (2u2 / 3) = 0 This kind of discretization seems to be similar to that given by the ﬁnite element approaches.

78

3 Nonsmooth Processes as Asymptotic Limits

1.0 sin x 0.5 sin 2 x 0.0 0.5 1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Fig. 3.11 Normal mode shape functions; here and below dashed lines correspond to second mode.

1.0 2 x

1 x 0.5 0.0 0.5 1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Fig. 3.12 Local mode shape functions.

Further, equations (3.97) are represented as a set of ﬁrst order equations u˙ 1 = v1 u˙ 2 = v2 v˙ 1 = −ω 2 u1 + ε[ω 2 u2 − ζv1 − p(u1 )] v˙ 2 = −ω 2 u2 + ε[ω 2 u1 − ζv2 − p(u2 )] where ω=

17 15 2k + λ2 , k = F (0), ε = 2 2

2 ¯ζ λ , ζ= ω ε

(3.98)

3.7 Normal and Local Mode Coordinates

and

79

√ √ √ 2 3 3 2 3 F ui − kui = βu3i p(ui ) = ε 3 3

are new constant parameters and the nonlinear component of the spring characteristic; it is assumed that the damping coeﬃcient and the nonlinear component are small enough to provide the order of magnitude ζ = O(1) and p(ui ) = O(1). For calculation purposes, the spring characteristic is taken in the form F (u) = u + (4/3)u3 which brings the nonlinearity parameter β to the form β=

64ω 2 135λ2

(3.99)

Equations (3.97), and analogously (3.98), possess advantages for transient analysis because the corresponding linearized system has the same natural frequencies and the nonlinear components are decoupled. As a result, the one-frequency perturbation tool becomes applicable. The corresponding amplitude-phase variables are introduced as follows ui = αi (t) cos[ωt + δi (t)] vi = −ωαi (t) sin[ωt + δi (t)]

(3.100)

(i = 1, 2) Substituting (3.100) in (3.98) and applying the averaging procedure with respect to the fast phase z = ωt, gives ε α˙ 1 = − [ζα1 + ωα2 sin(δ1 − δ2 )] 2 ε α˙ 2 = − [ζα2 − ωα1 sin(δ1 − δ2 )] 2 εω α2 3εβ 2 α cos(δ1 − δ2 ) + δ˙ 1 = − 2 α1 8ω 1 εω α1 3εβ 2 α cos(δ1 − δ2 ) + δ˙ 2 = − 2 α2 8ω 2

(3.101)

The result of the work of [157] as well as further analysis show that the localization may occur as the system vibrates in the out-of-phase mode, u1 (t) ≡ −u2 (t). In order to investigate the dynamics near the out-of-phase vibration mode, let us introduce three new variables s, ρ and θ, α1 = −s(t) + ρ(t) α2 = s(t) + ρ(t) δ2 = δ1 + Δ(t)

(3.102)

80

3 Nonsmooth Processes as Asymptotic Limits

where s and ρ characterize the amplitudes of the out-of-phase and in-phase modes, respectively, and Δ is a phase shift between the local modes so that the variables ρ and Δ describe small deviations from the out-of-phase mode. Substituting (3.102) in (3.101), linearizing the result with respect to ρ and Δ and then eliminating the phase variable Δ, gives

1 2 3 2 2 2 (3.103) ρ ¨ + εζ ρ˙ + ε ω + ζ − βs ρ = 0 4 4 whereas the equation obtained for s gives the solution

1 s = s0 exp − εζt 2

(3.104)

In the case |ζ| 1, equation (3.103) describes an oscillator with a slow varying frequency. Making the frequency ‘frozen’ enables one of the determining roots of the corresponding ‘characteristic equation’ 1 3 2 2 k1,2 = ε − ζ ± i ω − βs (3.105) 2 4 If the viscosity is negative, ζ < 0, then equations (3.103) through (3.105) qualitatively describe the transition to the local mode as the system energy increases. In particular, expression (3.105) shows that when the amplitude of the out-phase mode, which is associated with s, becomes large enough then the amplitude of the in-phase mode, ρ, looses its oscillatory character and grows monotonically. As a result, one of the local mode increases its amplitude, whereas another one decays; see expressions (3.102). This is an onset of the dynamic transition to a localized mode. The corresponding critical time follows from explicit solution of (3.104) and (3.105) t∗ =

4ω 2 1 ln ε|ζ| 3βs20

(3.106)

In order to provide numerical evidence for the dynamic transition from normal to local mode vibrations, let us introduce an indicator of the energy partition calculated as ⎧ ⎨ −1 if E1 = 0 and E2 = 0 E1 − E2 (3.107) = 0 if E1 = E2 P = ⎩ E1 + E2 1 if E1 = 0 and E2 = 0 where Ei = (vi2 + ω 2 u2i )/2 is the total energy of i-th oscillator under the condition ε = 0.

3.8 Local Mode Interaction Dynamics

81

Quantity (3.107) is varying within the interval −1 ≤ P ≤ 1. The ends of the interval obviously correspond to the local modes, whereas its center P = 0 corresponds to the normal modes. The time history of the energy partition (3.107) is illustrated by Fig. 3.13. The following parameters were taken for numerical simulations: λ = 0.05, ¯ ζ = −0.002, k = 1.0, ω = 2k + (17/2)λ2 = 1.4217, β = 32/(9ε) = 383.29, ¯ = −0.2156. The initial normal mode ε = 15(λ/ω)2 /2, and therefore ζ = ζ/ε amplitudes at zero velocities are W1 (0) = 0.0001 and W2 (0) = −0.003. The critical time estimate based on expression (3.106) t∗ = 3474.29 is in quite a good match with Fig. 3.13.

0.0 0.2 P

0.4 0.6 0.8 1.0

t 0

1000

2000

3000

4000

5000

t

Fig. 3.13 ‘Sudden’ transition from normal to local mode vibration as the system energy has reached its critical value.

3.8

Local Mode Interaction Dynamics

Let us introduce new variables, K, θ and Δ, as follows

1 π α1 = K(t) cos θ(t) + 2 4

π 1 α2 = K(t) sin θ(t) + 2 4 Δ = δ2 − δ1 + π

(3.108) (3.109)

Further, considering the local mode total energies Ei under no interaction condition, and taking into account (3.100), (3.108) and (3.110), gives Ei =

1 2 (v + ω 2 u2i ); i = 1, 2 2 i

(3.110)

82

3 Nonsmooth Processes as Asymptotic Limits

1 2 2 1 ω (α1 + α22 ) = ω 2 K 2 2 1 1 ΔE = E1 − E2 = ω 2 (α21 − α22 ) = − ω 2 K sin θ 2 2 E = E1 + E2 =

(3.111) (3.112)

The variable K therefore is proportional to the total energy of the degenerated system, whereas the phase angle θ characterizes the energy partition between the local modes (3.107) as follows P =

ΔE = − sin θ E

(3.113)

The third variable (3.109) describes the phase shift in the high-frequency vibrations between the local modes so that Δ = 0 corresponds to the outphase motions of the masses attached to the beam; note the diﬀerence with (3.102). Diﬀerentiating (3.109), (3.111) and (3.112), and enforcing equations (3.101), gives dκ ζ =− κ dt1 ω dθ = sin Δ dt1 dΔ = − cos Δ tan θ + κ sin θ dt1

(3.114)

where t1 = εωt is a new temporal argument, and κ=

3β K 8ω 2

(3.115)

In the conservative case, ζ = 0, the ﬁrst equation in (3.114) gives the energy integral κ = const, whereas another two equations admit the integral 1 G = − cos Δ cos θ + κ cos 2θ = const 4

(3.116)

This particular case matches the results obtained for a linearly coupled set of Duﬃng’s oscillators in [101], and later reproduced in [158] however by means of diﬀerent complex variable approaches. In particular, it was shown in [158] that the last two equations in (3.114) are equivalent to a strongly nonlinear conservative oscillator d2 θ tan θ =0 (3.117) + G2 dt21 cos2 θ as κ → 0.

3.8 Local Mode Interaction Dynamics

83

Oscillator (3.117) appears to be exactly solvable with general solution θ = arcsin[sin θ0 sin(|G|t1 / cos θ0 )]

(3.118)

where θ0 is the amplitude and another arbitrary constant can be introduced as a temporal shift. As already mentioned in this chapter, oscillator (3.117) was considered in [78] and [122] as a phenomenological model for quite diﬀerent kinds of problems. Since no direct physical meaning of such a unique ‘restoring force characteristic’ was found, the fact of exact solvability not attracted much attention for quite a long time. In the case κ = 0, but still ζ = 0, some perturbation occurs on the righthand side of (3.117); the corresponding perturbation tool based on the actionangle variables was introduced in [158]. Let us show now that equations (3.114) can describe the transition to local modes under the assumption of small negative viscosity |ζ/ω| 1 and ζ < 0

(3.119)

Under condition (3.119), the factor κ in the third equation of (3.114) can be viewed as a slowly growing quasi constant. In this case, making κ ‘frozen’ and linearizing the last two equations in (3.114) near the equilibrium (θ, Δ) = (0, 0), gives d2 θ + (1 − κ)θ = 0 (3.120) dt21 Note that small θ and Δ bring the original system close to the out-of-phase vibration mode as follows from (3.113) and (3.109). When the growing energy parameter κ passes the critical point κ = 1, the type of equilibrium is changing from a focus to a saddle point and thus the variable θ becomes exponentially growing. Practically, however, the exponential growth will be suppressed by the nonlinearity. As a result two limit phase trajectories (separatrix loops) occur around two new stable equilibrium points. These two points represent two new stable modes of the original system - local modes. The phase plane diagrams for sub- and super-critical energy levels are shown in Figs. 3.14 and 3.15, respectively. Note that the equilibrium subjected to such qualitative change corresponds to the out-of-phase vibration mode of the model, whereas another two equilibrium points, (θ, Δ) = (0, ±π), correspond to the same in-phase mode and remain stable. Therefore, out-of-phase vibrations appear to be less favorable to energy equipartition as the energy reaches its critical level. Note that links between localization and speciﬁcs of phase trajectories was discussed also in [101] based on the system of two coupled Duﬃng oscillators. In particular, the limit phase trajectories were interpreted as nonlinear beats of inﬁnitely long period, keeping the energy near one of the two oscillators.

84

3 Nonsmooth Processes as Asymptotic Limits

4 I 2

O

0

2 I 4 1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ Fig. 3.14 Phase plane structure at undercritical system energy, κ = 0.5.

4 I 2

L

0

L

2 I 4 1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ Fig. 3.15 Phase plane structure at postcritical system energy, κ = 1.5.

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

85

As follows from expression (3.113), the growth of θ increases the energy unbalance between the local modes, and that is onset of the mode localization. The corresponding critical time, at which the localization begins, is obtained from (3.115) as follows 8ω 2 1 3β ∗ ∗ ln K exp (ε |ζ| t ) = 1 =⇒ t = 0 8ω 2 ε |ζ| 3βK0

(3.121)

As follows from (3.102) and (3.111), K0 = 2(s20 +ρ20 ) therefore, under the condition ρ20 1, expressions (3.106) and (3.121) give the same result. Note that the developed analytical approach, describing the local mode interaction in terms of the energy and phase variables, appears to be independent of the individual features of the illustrating model and this can be used in other similar cases. Compared to publications [101] and [158] introducing the same set of descriptive variables, K, θ and Δ, current approach has distinctive features as follows: 1) Instead of general mass-spring models, the elastic beam supported by nonlinear springs is considered in this work. This provides clear geometrical interpretations for both the normal and local modes through the corresponding shape functions of the beam centre line. 2) Instead of using a quite complicated system reduction in terms of complex coordinates, it is shown that the same result can be achieved by means of the traditional set of amplitude-phase variables and one-frequency averaging procedure. 3) The non-conservative case is considered in order to describe qualitative changes in the dynamics as the total energy of the system adiabatically increases or decreases. Based on such a generalization, new quantitative and qualitative results are obtained. In particular, explicit expressions have been obtained for the critical time at which onset of the localization occurs. The phenomenon is explained in terms of the related phase-plane diagram subjected to a qualitative change (center-saddle transition) as the total energy of the system reaches its critical level.

3.9

Auto-localized Modes in Nonlinear Coupled Oscillators

Below, the term ‘auto-localized’ means that the system itself may come into the nonlinear local mode regime and stay there regardless initial energy distribution among its particles. As follows from the Poincare’s recurrence theorem, such phenomena are rather impossible within the class of conservative systems [9]. However, interactions between the system particles can be designed in speciﬁc ways in order to achieve desired phenomena. It is assumed

86

3 Nonsmooth Processes as Asymptotic Limits

that such a design can be implemented practically by using speciﬁc electric circuits and possibly mechanical actuators. On macro-levels, the autolocalization may help to optimize vibration suppression. Some results from the previous publication [155] are reproduced below after some notation modiﬁcations in order to make the description coherent with the current text. Let us consider an array of N harmonic oscillators such that each of the oscillators interacts with only the nearest neighbors. The corresponding diﬀerential equations of motion are of the form x ¨j + Ω 2 xj = β(xj−1 − 2xj + xj+1 )+ + α[(Ej − Ej−1 )Ej−1 − (Ej+1 − Ej )Ej+1 ]x˙ j

(3.122)

1 2 (x˙ + Ω 2 x2j ); j = 1, ..., N (3.123) 2 j where Ej = Ej (t) is the total energy of the j-th oscillator under the boundary conditions of ﬁxed ends E0 (t) ≡ EN +1 (t) ≡ 0, and Ω, β, and α are constant parameters of the model. On the right-hand side of equation (3.122), two groups of terms describe coupling between the oscillators. If α = 0 then the only linear coupling remains. In this case, under special initial conditions, N diﬀerent coherent periodic motions i.e. linear normal modes, can take place. It is well known that any other motion is combined of the linear normal mode motions, whereas the energy is conserved on each of the modes the way it was initially distributed between the modes. In other words, no energy localization is possible on individual particles if α = 0. Another group of terms, including the common factor α, has the opposite to linear elastic interaction eﬀect. These nonlinear terms are to simulate possible ‘competition’ between the oscillators, in other words, one-way energy ﬂow to the neighbor whose energy is lager. Such kind of interaction dominates when the total system energy is large enough to essentially involve high degrees of the coordinates and velocities. For future analysis let us introduce the complex conjugate variables {Aj (t) , A¯j (t)} into equations (3.122) according to relationships (3.37) and (3.38). In term of the complex amplitudes, the total energy of individual oscillator (3.123), excluding the energy of coupling, takes the form Ej =

Ej =

1 2 ¯ 1 2 Ω Aj Aj = Ω 2 |Aj | 2 2

(3.124)

When β = α = 0 the system is decomposed into the N uncoupled oscillators, and one has a constant solution in the new variables. In general case, substituting (3.37) in (3.122), taking into account (3.38), and applying averaging with respect to the phase z = Ωt, gives the following set of equations (the complex conjugate set is omitted below)

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

87

iβ A˙ j = − (Aj−1 − 2Aj + Aj+1 )+ 2Ω αΩ 4 2 2 2 2 2 2 + |Aj | − |Aj−1 | |Aj−1 | − |Aj+1 | − |Aj | |Aj+1 | Aj (3.125) 8 Let us consider ﬁrst, the simplest model of two coupled oscillators (N = 2). In this case, system (3.125) is reduced to iβ αΩ 4 A˙ 1 = − (A2 − 2A1 ) + |A1 |2 − |A2 |2 |A2 |2 A1 2Ω 8 4 iβ αΩ A˙ 2 = − (A1 − 2A2 ) + |A2 |2 − |A1 |2 |A1 |2 A2 2Ω 8

(3.126)

Despite of the presence velocities x˙ j in the original equations (3.122), system (3.126) still has the integral 2

2

K = |A1 | + |A2 | = 2(E1 + E2 )/Ω 2 = const. As a result, the dimension of system’ phase space is reduced by introducing the angular variables ϕ1 (t), ϕ2 (t) and ψ(t), √ √ A2 = K sin ψ exp(iϕ2 ) (3.127) A1 = K cos ψ exp(iϕ1 ), where the angle ψ determines the energy distribution between the oscillators as follows E2 |A2 | = (3.128) tan ψ = |A1 | E1 0 ≤ ψ < π/2 Substituting (3.127) into (3.126) and considering separately real and imaginary parts, gives β β − tan ψ cos (ϕ2 − ϕ1 ) Ω 2Ω β β − cot ψ cos (ϕ2 − ϕ1 ) ϕ˙ 2 = Ω 2Ω 1 β sin (ϕ2 − ϕ1 ) − αK 2 Ω 4 sin 4ψ ψ˙ = − 2Ω 32

ϕ˙ 1 =

(3.129)

Introducing the phase shift Δ = ϕ2 −ϕ1 and new temporal variable p = Ωt/β, gives dΔ = − cot 2ψ cos Δ dp dψ 1 = − (sin Δ + λ sin 4ψ) dp 2

(3.130)

88

3 Nonsmooth Processes as Asymptotic Limits Π

2

Π

4

0 Π

2

0

Π

2

Fig. 3.16 Low energy transition to the nonsmooth limit cycle; numerical solution obtained for the following system parameter and initial conditions: λ = 0.2; Δ(0) = 0.0, Ψ (0) = π/4 + 0.1.

where λ is a dimensionless parameter linked with the total energy of both oscillators as follows λ=

Ωα αK 2 Ω 5 = (E1 + E2 )2 16β 4β

(3.131)

System (3.130) is periodic with respect to both phase coordinates Δ and ψ, as a result, its phase plane has periodic cell-wise structure. Let us consider just one cell, π π! π (3.132) R0 = − < Δ < , 0 < Ψ < 2 2 2 including the equilibrium (critical) point (Δ, ψ) = (0, π/4)

(3.133)

As follows from (3.127) and (3.128), at point (3.133), both oscillators vibrate in-phase with the same energy, E1 = E2 . Linearized (near (3.133)) system (3.130) has the following couple of roots of characteristic equation (3.134) r1,2 = λ ± i 1 − λ2

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

89

Π

2

Π

4

0 0

25 p

50

0

25 p

50

Π

2

0

Π

2

Fig. 3.17 Low energy transition to the “impact” limit cycle of phase variables at λ = 0.2.

Expression (3.134) determines the ‘low energy’ interval 0 < λ < 1 with a qualitatively similar system behavior. Equilibrium point (3.133) is unstable for positive λ while no other equilibrium points exist within the rectangular (3.132). As a result, the system trajectory is eventually attracted to the boundary of rectangular R0 (3.132) as shown in Figs. 3.16 and 3.18. This is a periodic limit cycle whose period is found in a closed form, π/2 P =2 0

dψ −2 1 − λ sin 4ψ

0

π/2

2π dψ = √ 1 + λ sin 4ψ 1 − λ2

(3.135)

where the horizontal pieces of the boundary ∂R0 have zero contribution as those passed momentarily by the system (3.130). This is conﬁrmed also by the diagrams in Figs. 3.17 and 3.19 showing step-wise jumps of the variable Δ(p) in steady state limits.

90

3 Nonsmooth Processes as Asymptotic Limits Π

2

Π

4

0 Π

2

0

Π

2

Fig. 3.18 Transition to the nonsmooth limit cycle under the energy level approaching its critical value; the numerical solution obtained for the following system parameter and initial conditions: λ = 0.8; Δ(0) = 0.0, Ψ (0) = π/4 + 0.1

Expression (3.135) shows that P → ∞ as λ → 1. The inﬁnity long period means that there is only one-way energy ﬂow in the system, in other words, the energy is eventually be localized on one of the oscillators. The corresponding total critical energy value is determined by substituting λ = 1 in (3.131). This gives β (3.136) = E∗ E1 + E2 = 2 Ωα If E1 + E2 < E ∗ then periodic energy exchange with the period T = βP/Ω takes place, but no localization is possible. Therefore, in order to be localized on one of the oscillators, the total system energy must be large enough. Interestingly enough, the transition to localized mode of this model happens through non-smooth limit cycle along which the dynamics of phase variables, Ψ and Δ, resembles the behavior of coordinate and velocity of impact oscillator2; see Figs. 3.17 and 3.19.

2

As already mentioned, the possibility of ‘vibro-impact dynamics’ of phase variables was noticed later in [101] when considering another model of nonlinear beats.

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

91

Π

2

Π

4

0 0

15 p

30

0

15 p

30

Π

2

0

Π

2

Fig. 3.19 Transition to the “impact” limit cycle of phase variables at λ = 0.8.

Chapter 4

Nonsmooth Temporal Transformations (NSTT)

Abstract. In this chapter, diﬀerent versions of nonsmooth argument substitutions, speciﬁcally - nonsmooth time, are introduced with proofs of the related identities. Basic rules for algebraic, diﬀerential and integral manipulations are described. In particular, ﬁnal subsections show how to implement nonsmooth argument substitutions in the diﬀerential equations. These impose two principal features on the dynamical systems by generating speciﬁc algebraic structures and switching formulations to boundary-value problems. Notice that the transformation itself imply no constraints on dynamical systems and easily applies to both smooth and nonsmooth systems. Any further steps, however, should account for physical properties of the related systems. Indeed, linear coordinate transformations can signiﬁcantly simplify the linear dynamic problems. Weakly nonlinear coordinate transformations often play the major role in the quasi-linear theory. Further, dynamical systems with discontinuities can be simpliﬁed by means of appropriate non-smooth transformations of variables.

4.1

Non-smooth Time Transformations

Major features induced by nonsmooth temporal substitutions can be brieﬂy listed as follows: • Introducing non-smooth temporal variables, in particular triangular sine wave, brings the coordinates into the algebra of hyperbolic numbers1 ; • Under appropriate conditions, diﬀerentiation or integration of the coordinates keeps the result within the same algebra and therefore eases the corresponding manipulations with the dynamic systems; • Explicit time argument can be used together with the nonsmooth time in order to describe amplitude and/or frequency modulated processes. 1

As mentioned in Introduction, - complex numbers X + Y e, where e2 = 1; see details in the next subsections.

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 93–129, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

94

4 Nonsmooth Temporal Transformations (NSTT)

Notice that the transformation itself is a preliminary stage of analysis ﬁnalized by speciﬁc boundary value problems on standard intervals. Then, appropriate methods must be applied to the boundary value problems according to the related physical content.

4.1.1

Positive Time

To begin with a simple illustration of the non-smooth positive time, consider the series

1 1 t t (4.1) ln 2 cosh = + exp (−t) − exp (−2t) + exp (−3t) − ... 2 2 2 3 which is convergent for t ≥ 0 but obviously becomes divergent if t < 0; the convergence is conditional at t = 0. Nevertheless, replacing t −→ |t|, makes this series convergent for any t as follows

t |t| ln 2 cosh ≡ ln 2 cosh 2 2 1 1 |t| + exp (− |t|) − exp (−2 |t|) + exp (−3 |t|) − ... (4.2) = 2 2 3 An ‘side eﬀect’ of such manipulation is that ﬁnite sums of the series are losing diﬀerentiability at t = 0, whereas the original function is smooth everywhere. However, it will be shown later that the series can be rearranged in such a manner that any truncated series becomes diﬀerentiable at t = 0 as many times as needed. Note that the transformation t −→ |t| is non-invertible. As a result, the manipulation illustrated by example (4.2) may not work for other cases. In the above example though, the substitution t −→ |t| holds for both positive and negative time t due to evenness of the original function. In order to extend the above idea on the general case, let us represent the time argument in the form . (4.3) t = |t| |t| . 2

Now, taking into account the relationship (|t| ) = 1, gives sequentially t2n = |t|2n

and t2n+1 = (|t|)2n+1 |t|. , n = 1, 2, ...

(4.4)

Example 1. Combining identities (4.4) and the power series expansion of the exponential function, gives |t|3 |t|2 . exp (t) ≡ exp (|t| |t| ) = 1 + + ... + |t| + + ... |t|. 2! 3!

4.1 Non-smooth Time Transformations

95

or exp (t) = cosh (|t|) + sinh (|t|) |t|.

(4.5)

Expression (4.5) represents the exponential function as a sum of the even and odd components. Let us take a general function, x (t), and assume that .

x (t) = x (|t| |t| ) = X (|t|) + Y (|t|) |t|

.

(4.6)

This gives the following set of equations for the components, X (|t|) and Y (|t|), x (|t|) = X (|t|) + Y (|t|) x (− |t|) = X (|t|) − Y (|t|)

.

for |t| = 1 . for |t| = −1

and, ﬁnally, 1 [x (|t|) + x (− |t|)] 2 1 Y (|t|) = [x (|t|) − x (− |t|)] 2

(4.7)

X (|t|) =

Therefore, identity (4.6) holds under condition (4.7). It is seen that the righthand side of expression (4.6) represents an element of the algebra with the . basis {1, |t| }. The corresponding ‘table of products’ is generated by the re. 2 lationship (|t| ) = 1. This leads to other useful algebraic relationships, for instance, . X + Y |t| = 0 ⇐⇒ {X = 0, Y = 0} and .

.

f (X + Y |t| ) = Rf (X, Y ) + If (X, Y ) |t|

(4.8)

where 1 [f (X + Y ) + f (X − Y )] 2 1 If (X, Y ) = [f (X + Y ) − f (X − Y )] 2

Rf (X, Y ) =

(4.9)

Therefore, introducing the nonsmooth positive time |t|, imposes speciﬁc complexiﬁcation on the system coordinate x (t) −→ {X (|t|) , Y (|t|)}

(4.10)

As shown in diﬀerent sections through this chapter, such complexiﬁcations always accompany non-smooth argument substitutions.

96

4.1.2

4 Nonsmooth Temporal Transformations (NSTT)

‘Single-Tooth’ Substitution

A a simple generalization of the positive time, |t|, let us introduce the temporal shift, say a, as follows s = s (t) = |t − a|

(4.11)

Obviously, s˙ = −1 for t − a < 0 and s˙ = 1 for t − a > 0 , and therefore s˙ 2 = 1 for, at least, almost all t. Proposition 1. A general function x (t) can be represented in the form x (t) = X (s) + Y (s) s˙

(4.12)

where s = s (t) is given by (4.11). Proof. By analogy to (4.3), t = a + ss˙

(4.13)

Substituting (4.13) in (4.12), gives x (a − s) = X (s) − Y (s)

for s˙ = −1

x (a + s) = X (s) + Y (s)

for s˙ = 1

and thus, 1 [x (a + s) + x (a − s)] 2 1 Y (s) = [x (a + s) − x (a − s)] 2

X (s) =

(4.14)

Therefore, identity (4.12) holds for any x (t) under condition (4.14).

4.1.3

‘Broken Time’ Substitution

Another generalization of the positive time |t|, is given by the piece-wise linear function v1 t for t ≤ 0 (4.15) τ= v2 t for t ≥ 0 where v1 = v2 . The inverse relationship can be represented in the form t = A (τ ) + B (τ ) τ˙

where A=

1 1 + v1 v2

τ

and B = −

(4.16) 1 τ v1 v2

4.1 Non-smooth Time Transformations

97

and τ˙ 2 = −v1 v2 + (v1 + v2 ) τ˙

(4.17)

Further, applying some function x to both sides of equality (4.16), gives x (t) = X (τ ) + Y (τ ) τ˙

(4.18)

where both components X and Y are determined analogously to (4.6) through (4.7) in the form

1 τ τ X (τ ) = v2 x − v1 x v2 − v1 v1 v2

τ τ 1 x −x (4.19) Y (τ ) = − v2 − v1 v1 v2 Now, assuming that another function f is applied to both sides of (4.18), gives (4.20) f (x) = Rf (X, Y ) + If (X, Y ) τ˙ where 1 [v2 f (X + Y v1 ) − v1 f (X + Y v2 )] v2 − v1 1 If = − [f (X + Y v1 ) − f (X + Y v2 )] v2 − v1

Rf =

(4.21)

Comparing the right-hand sides of expressions (4.16), (4.18) and (4.20), shows that the algebraic structure generated by the nonsmooth time substitutions is preserved after diﬀerent functional manipulations with the corresponding elements of the algebra.

4.1.4

Sawtooth Sine Transformation

Let us consider the periodic version of nonsmooth time (4.11) which is based on the sawtooth sine - triangular wave function t for − 1 ≤ t ≤ 1 ∀t (4.22) τ (t) = , τ (t) = τ (4 + t) −t + 2 for 1 ≤ t ≤ 3 Recall that function (4.22) describes position of a bead oscillating between two perfectly stiﬀ parallel barriers with no energy loss. The distance between the barriers is two spatial units, and the velocity vector has the unit length and is normal to the barrier planes. In other words, function (4.22) describes the motion of the standard impact oscillator; see Chapter 1 for illustrations. This function can be also expressed in the closed form by means of the trigonometric functions as

98

4 Nonsmooth Temporal Transformations (NSTT)

τ (t) = (2/π)arcsin[sin(πt/2)]

(4.23)

The period is normalized to four in order to provide the unit slope, τ˙ 2 = 1

(4.24)

Proposition 2. [133] Any periodic process, whose period is normalized to T = 4, is expressed through the sawtooth sine τ (t) and the rectangular cosine τ˙ (t) in the form x (t) = X (τ ) + Y (τ ) τ˙ (4.25) where the X- and Y -components are given by 1 [x (τ ) + x (2 − τ )] 2 1 Y (τ ) = [x (τ ) − x (2 − τ )] 2

X (τ ) =

(4.26)

In other words, any periodic process is uniquely expressed through the state variables, such as the coordinate τ and velocity τ˙ , of the standard impact oscillator. Proof [144]. It can be veriﬁed by inspection that t = 1 + (τ − 1) τ˙

if − 1 < t < 3

(4.27)

Note that the function τ˙ (t) has a step-wise discontinuity at t = 1, which is suppressed however by the continuous factor, τ (t) − 1, of zero value at t = 1. Based on this remark, property (4.24) will be considered as true everywhere on the interval −1 < t < 3, since τ˙ is either explicitly or implicitly accompanied by the factor τ (t) − 1 whenever it appears in algebraic manipulations. For instance, identity 2

t2 = 1 + (τ − 1) + 2 (τ − 1) τ˙ holds everywhere on the interval −1 < t < 3. Further, applying the method of mathematical induction, gives tn = An (τ ) + Bn (τ )τ˙

(4.28)

where n is any positive integer, An and Bn are polynomials of the degree n or n − 1. Expression (4.28) shows that any analytic function x(t), that admits the power series expansion on the interval −1 < t < 3, can be represented in the form x(t) = x [1 + (τ − 1) τ˙ ] = X (τ ) + Y (τ ) τ˙ (4.29) where X (τ ) and Y (τ ) are power series of τ .

4.1 Non-smooth Time Transformations

99

Now, let us assume that expression (4.29) holds even though the function x(t) is not analytic. In this case, one must show that the X- and Y components can be determined with no power series expansions. Indeed, since either τ˙ = 1 or τ˙ = −1 on the entire interval −1 < t < 3, except may be the point t = 1, then expression (4.29) gives two equations, x (τ ) = X (τ ) + Y (τ ) and x (2 − τ ) = X (τ ) − Y (τ ). Solving these equations for X and Y and substituting the solution in (4.29), gives the identity, x(t) =

1 1 [x (τ ) + x (2 − τ )] + [x (τ ) − x (2 − τ )] τ˙ 2 2

(4.30)

which obviously holds on the interval −1 < t < 3, except may be t = 1. If the function x(t) is continuous at t = 1 then identity (4.30) is also true at t = 1 because the step-wise discontinuity of the rectangular cosine τ˙ (t) at t = 1 is suppressed by the factor 2Y (τ ) = x (τ ) − x (2 − τ ) → x (1 − 0) − x (1 + 0) → 0

(4.31)

as t → 1 ± 0; see deﬁnition (4.22). Now if the function x(t) is periodic of the period T = 4 then identity (4.30) holds almost everywhere on the interval −∞ < t < ∞, because the right-hand side of (4.30) depends on the time argument t through the pair of periodic functions τ (t) and τ˙ (t) of the same period T = 4. Finally, if the function x(t) is continuous also at t = −1, and thus t = 3 due to the periodicity, then identity (4.30) is true for every t. So let us consider the point t = −1 at which the function τ˙ (t) has the step-wise discontinuity. Taking into account that τ (−1 ± 0) = −1 + 0 and the periodicity condition x(t) = x(t − 4), gives 2Y (τ ) = x[τ (−1 ± 0)] − x[2 − τ (−1 ± 0)] (4.32) = x (−1 + 0) − x (3 − 0) = x (−1 + 0) − x (−1 − 0) → 0 as t → −1 ± 0 due to continuity of the function x(t) at t = −1. Therefore, the step-wise discontinuity of the function τ˙ (t) at the point t = −1 is suppressed as well due to (4.32). It is proved that if x(t) is continuous then identity (4.30) or (4.25) holds everywhere on the interval −∞ < t < ∞. The periodic version of conditions (4.31) and (4.32) is Y |τ =±1 = 0 (4.33) Remark 1. If the function x(t) is step-wise discontinuous at the instances Λ={t : τ (t) = ±1} then limits (4.31) and (4.32) are non-zero. In this case, the discontinuities of the function τ˙ (t) are not suppressed but describe the real behavior of the original function x(t).

100

4 Nonsmooth Temporal Transformations (NSTT)

Proposition 3. Elements (4.25) belong to the algebra of hyperbolic numbers due to (4.24). As a result, f (X + Y τ˙ ) = Rf + If τ˙

(4.34)

1 [f (X + Y ) + f (X − Y )] 2 1 If = [f (X + Y ) − f (X − Y )] 2

Rf =

provided that f (X ± Y ) are deﬁned. Relationship (4.34) can be easily veriﬁed by setting sequentially τ˙ = 1 and τ˙ = −1. The right-hand side of expression (4.25) therefore admits interpretation as a speciﬁc (hyperbolic) ‘complex number’ with ‘real’ X and ‘imaginary’ Y components. The corresponding ‘imaginary unity’ τ˙ creates the cycling group so that τ˙ 2 = 1, τ˙ 3 = τ˙ , τ˙ 4 = 1,... . This remark signiﬁcantly simpliﬁes all mathematical manipulations with representation (4.25). For example, 2

(X + Y τ˙ ) = X 2 + Y 2 + 2XY τ˙ 3 (X + Y τ˙ ) = X 3 + 3XY 2 + Y 3 + 3Y X 2 τ˙ The right-hand sides of the above expressions appear to have the same hyperbolic structure of the element X + Y τ˙ . Another example resembles Euler formula in the complex analysis, however hyperbolic functions are involved as follows exp (X + Y τ˙ ) = exp (X) [cosh (Y ) + sinh (Y ) τ˙ ]

(4.35)

Further, by taking into account that τ˙ 2 = 1, one obtains X 1 Y = 2 − 2 τ˙ 2 X + Y τ˙ X −Y X −Y2

if X = ±Y

This example explains why division is not always possible, and this is essential diﬀerence with the conventional complex analysis. Very often, the triangular wave function is supposed to depend on some phase variable, say ϕ(t). In all such cases, the ‘imaginary element’ is given by the derivative with respect to the phase variable τ (ϕ). In order to avoid confusion, let us introduce a new notation for the rectangular cosine of the normalized period and amplitude as follows e(ϕ) = τ (ϕ). Now the basic relation (4.36) e2 = 1 holds regardless argument of the sawtooth sine. Example 2. Introduce the sawtooth time into the functions sin t and cos t.

4.1 Non-smooth Time Transformations

101

Solution. In order to deal with the period normalized to four, let us introduce new argument ϕ = 2t/π as

π π 2t ϕ sin t = sin = sin 2 π 2 Now, applying representation (4.25) to the new argument ϕ , gives π π ! π 1 sin τ + sin (2 − τ ) = sin τ 2 2 2 2 ! 1 π π Y (τ ) = sin τ − sin (2 − τ ) = 0 2 2 2

X (τ ) =

and therefore

π sin t ≡ sin τ 2

Analogously,

π cos t = cos τ 2

2t π

2t π

2t e π

Example 3. Combining the results from previous example with the Euler formula for complex exponential functions, gives

kπ kπ τ i + cos τ e for k = 1, 3, 5, ... exp (ikt) = sin 2 2

kπ kπ exp (ikt) = cos τ + sin τ ie for k = 0, 2, 4, ... 2 2 where τ and e still depend on the same argument 2t/π. The right-hand sides of these equalities can be viewed as elements of a more complicated algebra, z = α + βe + γi + δei, with the basis elements {1, e, i, ei}. The corresponding table of products is given by × 1 e i ei

4.1.5

1 1 e i ei

e e 1 ei i

i i ei −1 −e

ei ei i −e −1

Links between NSTT and Matrix Algebras

Hyperbolic numbers represent a very simple example of so-called Cliﬀord geometric algebras which is isomorphic to a matrix algebra. For instance, it is known [105] that complex numbers associate with skew-symmetric 2 × 2 matrixes with equal diagonal entries, whereas the hyperbolic numbers correspond to the symmetric matrixes as follows

102

4 Nonsmooth Temporal Transformations (NSTT)

a + ib ←→ X + eY ←→

a b −b a XY Y X

The above correspondences are isomorphisms because both summations and multiplications with the numbers associate with the corresponding matrix operations, for instance (X1 + eY1 )(X2 + eY2 ) = (X1 X2 + Y1 Y2 ) + (X1 Y2 + Y1 X2 )e

X1 Y1 X2 Y2 X1 X2 + Y1 Y2 X1 Y2 + Y1 X2 = Y1 X1 Y2 X2 X1 Y2 + Y1 X2 X1 X2 + Y1 Y2

(4.37) (4.38)

As was already mentioned, the algebra of hyperbolic numbers and Cliﬀord algebras were developed in an abstract way as manipulations with speciﬁc numbers with no relation to any nonsmooth transformations. However, the uncovered links may appear to be useful for conducting automatic symbolic manipulations with computers. Indeed, most of the corresponding packages have built-in tools to handling the matrix operations whereas deﬁning the operations with hyperbolic numbers may require some programming work.

4.1.6

Diﬀerentiation and Integration Rules

From the point of view of applications to the diﬀerential equations, it is essential that, under some conditions, diﬀerential operations also preserve the hyperbolic structure of elements [133]. For example, x˙ = Y + X τ˙

+ Y τ¨

(4.39)

where primes mean derivatives with respect to τ . The term underlined in (4.39) should be ignored whenever the function x(t) is continuous and therefore condition (4.33) is satisﬁed. Indeed, the derivative τ¨ represents the periodic system of pulses, acting at those points Λ where the factor Y is equal to zero due to (4.33), τ¨ = 2

∞

[δ(t + 1 − 4k) − δ(t − 1 − 4k)]

k=−∞

In a similar manner, one can sequentially consider high-order derivatives. For example, the second derivative is given by x ¨ = X + Y τ˙

(4.40)

X |τ =±1 = 0

(4.41)

under the boundary condition

4.1 Non-smooth Time Transformations

103

Further, an arbitrary odd derivative is given by x(2k−1) (t) = Y (2k−1) (τ ) + X (2k−1) (τ ) e

(4.42)

provided that Y (τ ) |τ =±1 = 0 X (τ ) |τ =±1 = 0 Y

(2k−2)

(τ ) |τ =±1

... =0

(4.43)

Also an arbitrary even derivative is x(2k) (t) = X (2k) (τ ) + Y (2k) (τ ) e

(4.44)

if Y (τ ) |τ =±1 = 0 X (τ ) |τ =±1 = 0 ... X

(2k−1)

(4.45)

(τ ) |τ =±1 = 0

Finally, integration also gives the hyperbolic ‘number’ (X + Y τ˙ )dt = Q + P τ˙ whose components are Q(τ ) =

(4.46)

τ

Y dτ + C

and

τ

P (τ ) =

Xdτ −1

0

where C is an arbitrary constant, and the following condition must be satisﬁed 1 X(τ )dτ = 0

(4.47)

−1

Relationship (4.46) can be easily veriﬁed by diﬀerentiation with respect to t. The role of condition (4.47) is to provide zero mean value of the integrand in (4.46).

4.1.7

NSTT Averaging

Lemma 1. Let x(t) be a general periodic function of the period T = 4a so that presentation (4.25) holds

104

4 Nonsmooth Temporal Transformations (NSTT)

x(t) = X(τ (φ)) + Y (τ (φ))e(φ)

(4.48)

where φ = t/a, e(φ) = τ (φ) and 1 [x (aτ ) + x (2a − aτ )] 2 1 Y (τ ) = [x (aτ ) − x (2a − aτ )] 2

X (τ ) =

(4.49)

Then the mean value of x(t) over its period is 1 T

T

0

1 x(t)dt = 2

1

X(τ )dτ

(4.50)

−1

In other words, the ‘imaginary’ component Y e of the ‘hyperbolic number’ (4.48) gives zero contribution into the mean value. Proof. With no loss of generality, let us assume that a = 1 and thus T = 4. Then, during one period −1 < t < 3, τ (t) = e(t) = 1 and dt = dτ for − 1 < t < 1 τ (t) = e(t) = −1 and dt = −dτ for 1 < t < 3 Taking into account (4.48), gives 1 T =

1 4

0

T

1 x(t)dt = 4

1

−1

[X(τ ) + Y (τ )]dτ −

1 4

1

x(t)dt + −1

3

x(t)dt 1

1

−1

[X(τ ) − Y (τ )]dτ =

1 2

1

X(τ )dτ −1

Example 4. Let x(t) be a periodic function of the general period T . Taking into account (4.48) and the above remark, gives x2 = (X + Y e)2 = X 2 + Y 2 + 2XY e Then, based on the above Lemma, < x2 >≡

1 T

T

x2 dt = 0

1 2

1

(X 2 + Y 2 )dτ

(4.51)

−1

Example 5. Consider the case x(t) = A sin t + B cos t, where T = 2π and a = π/2. In this case, relationships (4.56) give X(τ ) = A sin(πτ /2) and Y (τ ) = A cos(πτ /2). Therefore, (4.51) gives < x2 >= (A2 + B 2 )/2.

4.1 Non-smooth Time Transformations

4.1.8

105

Generalizations on Asymmetrical Sawtooth Wave

It has been shown in [138] and [149] that an arbitrary periodic function x (t) whose period is normalized to four can still be represented in the form (4.25), even though the triangular wave τ - saw-tooth sine - has asymmetrical slopes such that t/ (1 − γ) for − 1 + γ ≤ t ≤ 1 − γ τ (t, γ) = (4.52) (−t + 2) / (1 + γ) for 1 − γ ≤ t ≤ 3 + γ ∂τ (t, γ) = e (t, γ) ∂t

τ˙ (t, γ) =

(4.53)

where γ ∈ (−1, 1) is a parameter characterizing the inclination of the saw as shown in Fig. 4.1. Τ 1 Γ

1

2

3

4

5

6

7

8

5

6

7

8

t

1 e 1 Γ1

1

1

1 Γ

1

2

3

4

t

Fig. 4.1 Asymmetric saw-tooth sine and rectangular cosine (γ = 0.5).

In this case, the inverse transformation of time on the period has the form

1 1 1 − γ2 +e +[2−(1+γ) τ ] −e (4.54) t= [(1−γ) τ ] 2 1+γ 1−γ (−1 + γ ≤ t ≤ 3 + γ) By using this expression, one can show that x (t) = X [τ (t, γ)] + Y [τ (t, γ)] e (t, γ)

(4.55)

106

4 Nonsmooth Temporal Transformations (NSTT)

1 1 1 X= x [(1 − γ) τ ] + x [2 − (1 + γ) τ ] 2α 1 + γ 1−γ 1 {x [(1 − γ) τ ] − x [2 − (1 + γ) τ ]} Y = 2α

(4.56)

e2 = α + βe

(4.57) where α = 1/ 1 − γ 2 and β = 2γα. As compared to (4.36), relationship (4.57) somewhat complicates algebraic and diﬀerential operations so that, respectively, f (X + Y e) = Rf (X, Y ) + If (X, Y ) e

(4.58)

1 1 1 f (Z+ ) + f (Z− ) Rf (X, Y ) = 2α 1 + γ 1−γ 1 [f (Z+ ) − f (Z− )] If (X, Y ) = 2α Z± = X ±

Y 1∓γ

and x˙ = αY + (X + βY ) e + Y

∂e ∂t

(4.59)

where f (Z+ ) and f (Z− ) are deﬁned, primes indicate diﬀerentiation with respect to τ , and ∞ ∂e = 2α [δ (t + 1 − γ − 4k) − δ (t − 1 + γ − 4k)] ∂t

(4.60)

k=−∞

If the function x (t) is continuous at time instances Λ = {t : τ (t, γ) = ±1} then the singular term ∂e/∂t is suppressed by the condition Y |t∈Λ = Y |τ =±1 = 0 analogously to the symmetric case. Otherwise, the periodic singular term remains in expression (4.59). It is convenient to calculate high-order derivatives by means of the matrixoperator D acting on the vector-column (X, Y )T sequentially as follows

0α d D= 1 β dτ

X αY D = (4.61) Y X + βY

X αX + αβY D2 = βX + (α + β 2 )Y Y

4.1 Non-smooth Time Transformations

107

Therefore, applying D and D2 , gives x˙ = αY + (X + βY )e x ¨ = αX + αβY + [βX + (α + β 2 )Y ]e under the conditions Y |τ =±1 = 0

(X + βY )|τ =±1 = 0 Note that rules (4.61) are still valid in the symmetric case γ = 0, when the diﬀerentiation matrix operator takes the form

01 d D= 1 0 dτ Finally, the result of integration (4.46) has the components Q=

Y (τ ) −

β X (τ ) dτ α

and P =

1 α

τ X (ξ) dξ −1

under the zero-mean value condition (4.47).

4.1.9

Multiple Frequency Case

Practical applications of NSTT to the class of multiple frequency motions face problems similar by its nature to those caused small denominators in quasilinear approaches. However, formal generalizations of the basic identities are quite simple as illustrated below. Let Pϕ and Nϕ be operators acting on some periodic function x(ϕ) of the period T = 4 as follows 1 {x[τ (ϕ)] + x[2 − τ (ϕ)]} ≡ X[τ (ϕ)] 2 1 Nϕx (ϕ) = {x[τ (ϕ)] − x[2 − τ (ϕ)]} ≡ Y [τ (ϕ)] 2 Pϕx (ϕ) =

(4.62)

where functions X (τ ) and Y (τ ) are deﬁned according to (4.26). In such notations, representation (4.25) takes the form x(ϕ) = (Pϕ + eNϕ)x(ϕ)

(4.63)

where e (ϕ) = τ (ϕ). Let us consider now a function of multiple arguments x = x (ϕ1 , ..., ϕn ) of the period T = 4 with respect to each of its n arguments. This function describes a multiple frequency quasi-periodic process if ϕ1 = ω1 t,..., ϕn = ωn t

108

4 Nonsmooth Temporal Transformations (NSTT)

with a set of positive incommensurable numbers ω1 ,..., ωn . In this case, transformation (4.63) is independently applicable to each of the n arguments as follows n (Pϕj + ej Nϕj )x (ϕ1 , ..., ϕn ) (4.64) x (ϕ1 , ..., ϕn ) = j=1

where new notations τj = τ (ϕj ) and ej = τ (ϕj ) are introduced. Note that all the operators included in the product are commutating as those applied to diﬀerent arguments of the function. Moreover, the number of co-factors of the product can diﬀer from one to n. Let us consider the case of two arguments in details. Introducing the notations e0 ≡ 1 and e3 = e1 e2 , and taking the product as shown in (4.64), gives x (ϕ1 , ϕ2 ) = X (τ1 , τ2 ) e0 + Y (τ1 , τ2 ) e1 + Z (τ1 , τ2 ) e2 + W (τ1 , τ2 ) e3 (4.65) where X = Pϕ1 Pϕ2 x (ϕ1 , ϕ2 ) Y = Nϕ1 Pϕ2 x (ϕ1 , ϕ2 ) Z = Pϕ1 Nϕ2 x (ϕ1 , ϕ2 ) W = Nϕ1 Nϕ2 x (ϕ1 , ϕ2 ) The basis of elements (4.65) obeys the table of products × e0 e1 e2 e3

e0 e0 e1 e2 e3

e1 e1 e0 e3 e2

e2 e2 e3 e0 e1

e3 e3 e2 e1 e0

(4.66)

Further, applying some function f to the element (4.65), gives f (Xe0 + Y e1 + Ze2 + W e3 ) = Rf e0 + If1 e1 + If2 e2 + If3 e3

(4.67)

where the coeﬃcients at the right-hand side are deﬁned by the system of linear equations Rf + If1 + If2 + If3 = f (X + Y + Z + W ) Rf + If1 − If2 − If3 = f (X + Y − Z − W ) Rf − If1 + If2 − If3 = f (X − Y + Z − W ) Rf −

If1

−

If2

+

If3

= f (X − Y − Z + W )

(4.68)

4.2 Idempotent Basis Generated by the Triangular Sine-Wave

109

Equations (4.68) are obtained by substituting diﬀerent combinations of e1 = ±1 and e3 = ±1 in (4.67). Some application of the multiple frequency case will be illustrated at the end of Chapter 5 and also in Chapter 14.

4.2

Idempotent Basis Generated by the Triangular Sine-Wave

4.2.1

Deﬁnitions and Algebraic Rules

In addition to the standard basis {1, e}, the hyperbolic plane has another natural basis {e+ , e− }associated with the two isotropic lines separating the hyperbolic quadrants as described in Chapter 1. The transition from one basis to another is given by (see Fig. 4.2) 1 (1 + e) 2 1 e− = (1 − e) 2

e+ =

(4.69)

or 1 = e+ + e− e = e+ − e−

(4.70)

Therefore, x = X + Y e = X(e+ + e− ) + Y (e+ − e− ) = (X + Y )e+ + (X − Y )e− where x = x(t) is any periodic function whose period is normalized to T = 4, and therefore e = e(t). Taking into account (4.26), gives x = X+ (τ )e+ + X− (τ )e−

(4.71)

where τ = τ (t), e+ = e+ (t), e− = e− (t), and X+ = X + Y = x(τ ) X− = X − Y = x(2 − τ )

(4.72)

The elements e+ and e− are mutually annihilating, and they are called idempotents because e+ e− = 0 e2− = e− e2+

= e+

(4.73)

110

4 Nonsmooth Temporal Transformations (NSTT)

Fig. 4.2 The standard basis and idempotent basis (on the left and right, respectively.)

As follows from (4.70), ee+ = e+ and ee− = −e− . Obviously, the matrix representation of (4.71) is

X+ 0 X+ e+ + X− e− ←→ 0 X− Properties (4.73) make the idempotent basis very convenient to use in diﬀerent calculations. For instance, for any real number α, α α (X+ e+ + X− e− )α = X+ e+ + X− e−

(4.74)

One can extend relationship (4.74) on a general function f as follows f (X+ e+ + X− e− ) = f (X+ )e+ + f (X− )e−

(4.75)

provided that f (X+ ) and f (X− ) are deﬁned. As seen from (4.75), any function is acting as linear if applied to a hyperbolic element represented in the idempotent basis. As a result, the major advantage of using the idempotent basis is that the diﬀerential equations of motion in terms of the components X+ (τ ) and X− (τ ) appear to be decoupled regardless nonlinearity. However, the corresponding boundary value problems still remain coupled because the idempotent basis makes

4.2 Idempotent Basis Generated by the Triangular Sine-Wave

111

unfortunately boundary conditions coupled. The nature of this fact is explained in the next subsection. The time variable t can be represented in the idempotent basis during one period of a periodic process, say T = 4. This can be shown by substituting (4.70) in (4.27), where τ˙ (t) = e(t), and then conducting some manipulations as follows t = 1 + (τ − 1) e = (e+ + e− ) + (τ − 1) (e+ − e− ) (−1 < t < 3) = τ e+ + (2 − τ )e−

(4.76)

Further, using the basis’ properties (4.73), gives, for instance, t2 = τ 2 e+ + (2 − τ )2 e− tα = τ α e+ + (2 − τ )α e− (α > 0)

(4.77)

Note that expressions (4.77) provide the direct proof of representation (4.71), at least, in the class of analytical periodic functions x(t).

4.2.2

Time Derivatives in the Idempotent Basis

Taking into account deﬁnitions (4.69) and (4.70), gives ﬁrst time derivatives of the basis elements 1 (4.78) e˙ + = −e˙ − = e˙ 2 where ∞ e˙ + = [δ(t + 1 − 4k) − δ(t − 1 − 4k)] k=−∞

Therefore, diﬀerentiating (4.71) gives x˙ = X+ ee+ + X− ee− + X+ e˙ + + X− e˙ − 1 = X+ e+ − X− e− + (X+ − X− )e˙ 2

Assuming the continuity condition

gives

(X+ − X− )|τ =±1 = 0

(4.79)

e+ − X− e− x˙ = X+

(4.80)

Analogously, assuming the continuity condition for x(t), ˙ + X− )|τ =±1 = 0 (X+

gives then

e+ + X− e− x¨ = X+

(4.81)

112

4 Nonsmooth Temporal Transformations (NSTT)

Example 6. Suppose the equation x˙ = f (x), where x(t) ∈ Rn , has a family of periodic solutions of the period T = 4a. Introducing the sawtooth time τ = τ (t/a) and using the idempotent basis, gives e+ − X− e− ) (X+

1 = f (X+ e+ + X− e− ) = f (X+ )e+ + f (X− )e− a

or X+ = af (X+ ) X− = −af (X− )

(4.82)

under condition (4.79). As mentioned above, the sawtooth time substitution implemented in the idempotent basis leads to the decoupled set of equations such as (4.82), whereas boundary condition (4.79) is coupled. Note that it is practically suﬃcient to solve only ﬁrst equation of system (4.82). Then solution of the second equation is obtained by making replacement τ → −τ in the ﬁrst solution.

4.3

Idempotent Basis Generated by Asymmetric Triangular Wave

4.3.1

Deﬁnition and Algebraic Properties

Let us introduce basic algebraic manipulations with the idempotent basis generated by the asymmetric triangular (sawtooth) wave (4.52). The standard basis is given by the set {1, e} and shown on the left in Fig. 4.3, where e = e(t, γ) is deﬁned by (4.53) periodic function with step-wise discontinuities at the points Λγ = {t : τ (t, γ) = ±1}. In this case, we introduce the idempotent basis as follows 1 [1 − γ + (1 − γ 2 )e] 2 1 e− = [1 + γ − (1 − γ 2 )e] 2 e+ =

(4.83)

or, inversely, 1 = e+ + e− 1 1 e= e+ − e− 1−γ 1+γ

(4.84)

where e+ = e+ (t, γ) and e− = e− (t, γ), and the parameter of asymmetry γ is included in order to normalize the range of change for the basis elements as 0 ≤ e+ ≤ 1 and 0 ≤ e− ≤ 1; see the diagrams on the left in Fig. 4.3.

4.3 Idempotent Basis Generated by Asymmetric Triangular Wave

113

Fig. 4.3 The standard basis and idempotent basis (on the left and right, respectively) generated by the asymmetric triangular wave with γ = 0.3.

Deﬁnition (4.83) brings (4.54) to the form t = (1 − γ) τ e+ + [2 − (1 + γ) τ ] e−

(4.85)

If γ = 0 then deﬁnition (4.83) becomes equivalent to (4.69). However, the elements e+ and e− are mutually annihilating, and create idempotents regardless the magnitude of γ so that table of products (4.73) still holds for any γ ∈ (−1, 1). Also, as follows from (4.73) and (4.84), 1 e+ 1−γ 1 e− ee− = − 1+γ ee+ =

Now, substituting (4.84) into identity (4.55), gives

1 1 x = X + Y e = X(e+ + e− ) + Y e+ − e− 1−γ 1+γ

1 1 = X+ Y e+ + X − Y e− 1−γ 1+γ

(4.86)

(4.87)

where x = x(t) is any periodic function whose period is normalized to T = 4.

114

4 Nonsmooth Temporal Transformations (NSTT)

Substituting (4.56) in (4.87) and conducting algebraic manipulations, gives x = X+ (τ, γ)e+ + X− (τ, γ)e−

(4.88)

where τ = τ (t, γ) is deﬁned above by (4.52), and X+ (τ, γ) = x((1 − γ)τ )

and

X− (τ, γ) = x(2 − (1 + γ)τ )

(4.89)

Obviously, basic algebraic properties (4.74) and (4.75) remain valid.

4.3.2

Diﬀerentiation Rules

Taking into account deﬁnitions (4.83) and (4.84), gives ﬁrst time derivatives of the basis elements ∂e+ ∂e− 1 ∂e =− = (1 − γ 2 ) ∂t ∂t 2 ∂t

(4.90)

where ∂e/∂t is a sequence of Dirac δ-functions taking eﬀect whenever t ∈ Λγ. Taking into account that ∂τ /∂t = e and diﬀerentiating (4.88), gives x˙ =

∂e+ ∂e− ∂X− ∂X+ ee+ + ee− + X+ + X− ∂τ ∂τ ∂t ∂t

(4.91)

Substituting (4.86) and (4.90) in (4.91), brings (4.91) to the form x˙ =

1 ∂X+ 1 ∂X− 1 ∂e e+ − e− + (1 − γ 2 )(X+ − X− ) 1 − γ ∂τ 1 + γ ∂τ 2 ∂t

(4.92)

Assuming the continuity of x(t), leads to boundary conditions (X+ − X− )|τ =±1 = 0

(4.93)

These conditions eliminate the singular term ∂e/∂t from expression (4.92), which, as a result, takes the form x˙ =

1 ∂X+ 1 ∂X− e+ − e− 1 − γ ∂τ 1 + γ ∂τ

(4.94)

As seen from (4.94), the algebraic structure of representation (4.88) is preserved after the diﬀerentiation. In other words, under continuity condition (4.93), the result of diﬀerentiation just leads to the following replacements in (4.88) 1 ∂X+ 1 ∂X− X+ → and X− → − (4.95) 1 − γ ∂τ 1 + γ ∂τ Property (4.95) enables one of writing down high derivatives iteratively by making substitutions (4.95) in (4.92). For instance, second derivative is given by

4.3 Idempotent Basis Generated by Asymmetric Triangular Wave

x ¨=

1 1 ∂ 2 X+ ∂ 2 X− e+ + e− 2 2 2 (1 − γ) ∂τ (1 + γ) ∂τ 2

1 ∂X+ 1 ∂X− ∂e 1 + + (1 − γ 2 ) 2 1 − γ ∂τ 1 + γ ∂τ ∂t

The singular term ∂e/∂t in (4.96) is eliminated by condition

1 ∂X+ 1 ∂X− + |τ =±1 = 0 1 − γ ∂τ 1 + γ ∂τ

115

(4.96)

(4.97)

As a result, the second derivative x¨ takes the form of expansion with respect to the basis {e+ , e− }. Such a process can be continued as soon as derivatives remain continuous in Λγ . Otherwise, time derivatives of the function e(t, γ) acquire non-formal meaning and must be present in the entire expression.

4.3.3

Oscillators in the Idempotent Basis

In order to illustrate the use of idempotent basis for vibration problems, consider oscillator x ¨ + f (x) = pe+ (t/a, γ) + qe− (t/a, γ)

(4.98)

where p and q are constant amplitudes of the external loading represented in the idempotent basis with the period T = 4a. No speciﬁc conditions are imposed on the characteristic f (x), however, it is assumed that oscillator (4.98) possesses periodic solution with the period of external loading, T . Such a solution admits the form (4.88), where τ = τ (t/a, γ). Due to the presence of time scaling factor, expressions for time derivatives derived in the previous subsection must be modiﬁed by the replacement ∂τ → a∂τ . Now, substituting (4.88) in (4.98), taking into account (4.93) through (4.97) and the basic algebraic property of idempotents, f (X+ e+ + X− e− ) = f (X+ )e+ + f (X− )e− , gives equations 1 ∂ 2 X+ + f (X+ ) = p (1 − γ)2 a2 ∂τ 2 ∂ 2 X− 1 + f (X− ) = q (1 + γ)2 a2 ∂τ 2

(4.99) (4.100)

under the boundary conditions (4.93) and (4.97). The obvious advantage of using the idempotent basis is that equations (4.99) and (4.100) are in a better match with the form of original equation (4.98), appear to be decoupled and therefore can be solved in a similar way. Although the entire boundary value problem is still coupled through the boundary conditions, the problem caused by coupling is eased, however,

116

4 Nonsmooth Temporal Transformations (NSTT)

since algorithms for solving equations are often more complicated than those applied to the boundary conditions.

4.3.4

Integration in the Idempotent Basis

Let x(t) be a periodic function of the period normalized to T = 4 with zero mean value on the period T . Then, integrating the right-hand side of (4.88), gives (4.101) [X+ (τ, γ)e+ + X− (τ, γ)e− ]dt = P+ (τ, γ)e+ + P− (τ, γ)e− where τ P+ (τ, γ) = (1 − γ)

X+ (z, γ)dz + C

−1

τ

P− (τ, γ) = −(1 + γ)

X− (z, γ)dz + C

(4.102)

−1

Note that both expressions in (4.102) have the same arbitrary constant of integration C. As a result, substituting (4.102) in (4.101) and taking into account (4.84), gives a single constant of integration as follows: Ce+ + Ce− = C(e+ + e− ) = C. Proof of (4.102) is obtained by taking time derivative of both sides of equality (4.101). Under the condition of continuity of the right-hand side of (4.101), the diﬀerentiation leads to the boundary value problem 1 ∂P+ (τ, γ) = X+ (τ, γ) 1−γ ∂τ 1 ∂P− (τ, γ) − = X− (τ, γ) 1+γ ∂τ (P+ − P− )|τ =±1 = 0

(4.103)

This boundary value problem has solution (4.102) under the following condition though 1 (1 − γ) −1

1 X+ (τ, γ)dτ + (1 + γ)

X− (τ, γ)dτ = 0

(4.104)

−1

The meaning of this condition is clariﬁed by substituting (4.89) in (4.104) and taking into account the standard properties of deﬁnite integrals as follows

4.4 Discussions, Remarks and Justiﬁcations

1

117

1 x[(1 − γ)τ ]d[(1 − γ)τ ] +

−1 (1−γ)

(1−γ)

x(z)dz −

=

x[2 − (1 + γ)τ ]d[(1 + γ)τ ] −1

−(1−γ)

3+γ

x(z)dz =

T x(z)dz =

−1+γ

2+(1+γ)

x(t)dt = 0 0

In other words, condition (4.104) requires zero mean value for x(t) on the period. Otherwise, the result of integration would appear to be out of the class of periodic functions by making the algebraic structure of right-hand side of (4.101) invalid.

4.4

Discussions, Remarks and Justiﬁcations

Let us show that the saw-tooth temporal argument τ (t) associates with the very general group properties of the conservative oscillator d2 x + f (x) = 0 dt2

(4.105)

Suppose that another oscillating time parameter, say g, is introduced as x(t) = X[g (t)], where g (t) is a periodic function, which is not necessarily sawtooth. This kind of substitution brings the diﬀerential equation of motion to the form

2 2 dg d X d2 g dX + f (X) = 0 (4.106) + dt dg 2 dt2 dg A side eﬀect of such substitution is that, generally speaking, the system has lost its original Newtonian form. In order to keep the diﬀerential equation of motion in its original Newtonian form, the following conditions must be imposed on equation (4.106)

dg dt

2 =1

and

d2 g dX =0 dt2 dg

(4.107)

Conditions (4.107) are satisﬁed by g =α+t

or g = β − t

(4.108)

where α and β are arbitrary constants. Relationships (4.108) simply represent the group of time transformations which is admitted by the original system (4.105). Let us recall however that the function g (t) was assumed to be periodic. Therefore, according to (4.108), g (t) must be a piece-wise linear periodic function. The simplest one-frequency case is given by the triangular wave, g (t) ≡ τ (t). In this case the ﬁrst equality of (4.107) is equivalent to the basic algebraic relationship (4.24) whereas the

118

4 Nonsmooth Temporal Transformations (NSTT)

second equality holds for all t under the condition (4.41). As a result, the new equation (4.106) takes the same form as (4.105). Therefore, the triangular wave time substitution τ (t) possesses the unique property among all periodic time substitutions, namely, it preserves the form of diﬀerential equations of conservative oscillators.

4.4.1

Remarks on Nonsmooth Solutions in the Classical Dynamics

Problems of classical dynamics are usually formulated in terms of second order diﬀerential equations of motion with respect to the system coordinates. Therefore, the corresponding solutions must be at least twice continuously diﬀerentiable functions of time. The existence and uniqueness theorem imposes also special conditions on the system characteristics and external forcing functions. As a result, using discontinuities or distributions for modeling dynamical systems takes the corresponding diﬀerential equations out of frames of the classic theory of diﬀerential equations. This requires additional examination of such formulations in order to insure that solutions actually exist. One of the NSTT roles is to bring models back into the area of classic theory. However, some preliminary analyses of correctness of original models is still required especially in non-linear cases. In this and next subsections, some details and examples are introduced on nonlinear formulations with discontinuities and δ-functions. For illustration purposes, let us consider a one-dimensional forced oscillator x ¨ + f (x, x) ˙ = q (t)

(4.109)

where the functions q (t), f (x, x) ˙ and ∂f (x, x) ˙ /∂ x˙ must be continuous according to the Cauchy theorem. Let the system be subjected to an impulsive external loading applied at t = t0 . In this case, the external forcing function can be expressed by the Dirac δ-function2 x ¨ + f (x, x) ˙ = Iδ (t − t0 ) (4.110) where I is a constant linear momentum per unit mass. The diﬀerential equation of motion (4.110) cannot be treated within the classic theory of diﬀerential equations. It is also impossible to ﬁnd a solution in the class of twice diﬀerentiable functions. However, a physically meaningful interpretation of equation (4.110) is suggested by the theory of distributions. For example, in terms of function-theoretic approaches, distributions are not classical functions but linear functionals acting on manifolds of appropriate diﬀerentiable functions {ϕ (t)}. Further, equalities are thought to be integral 2

The Dirac δ-function belongs to the class of so-called generalized functions introduced by Sergei Sobolev in 1935 and re-introduced by Laurent Schwartz in the late 1940s, who developed a theory of distributions.

4.4 Discussions, Remarks and Justiﬁcations

119

identities rather then point-wise relationships. Therefore, a ‘real meaning’ of equation (4.110) is given by the identity ∞

∞ δ (t − t0 ) ϕ (t) dt

[¨ x + f (x, x)] ˙ ϕ (t) dt = I −∞

(4.111)

−∞

which is supposed to be true for any ‘testing function’ ϕ (t). Taking into account the deﬁnition of δ-function and integrating by parts, gives ∞ [−x˙ ϕ˙ (t) + f (x, x) ˙ ϕ (t)] dt = Iϕ (t0 ) (4.112) −∞

On one hand, this expression allows to weaken the smoothness condition on the class of solutions x (t). On the other hand, the integral identity (4.111) appears to have quite clear physical meaning since its both sides represent works done on arbitrary virtual displacements ϕ (t). Note, that the solution x (t) itself is not considered as a functional unless the function f is linear. For example, let us consider the case f (x, x) ˙ = 2λx˙ +ω 2 x, where λ and ω are constant parameters. In this case, by proceeding with integration by parts in equation (4.112), one obtains ∞

ϕ ¨ (t) − 2λϕ˙ (t) + ω 2 ϕ (t) x (t) dt = Iϕ (t0 )

−∞

Therefore, x (t) generates a linear functional of the form

"∞

(...) x (t) dt and

−∞

thus can be qualiﬁed as a generalized solution3 . In nonlinear cases, the concept of generalized solution does not work, but equation (4.112) still holds and enables one to use the concept of ‘weak solution’ [46]. This concept is involved to justify manipulations with the nonsmooth functions at intermediate steps of the approach.

4.4.2

Caratheodory Equation

Since δ-function is deﬁned as a linear functional then nonlinear operations with δ-functions are generally ‘illegal.’ However, δ-functions still can participate in nonlinear diﬀerential equations in a linear way - as summands. 3

Sometimes the very presence of δ-functions in any nonlinear equation is rejected by the reason that nonlinearity is incompatible with the notion of linear functionals. In diﬀerential equations of motion, however, singular forces and the corresponding accelerations usually participate as summands, whereas velocities and coordinates include no δ-type singularities and thus can be subjected to nonlinear operations.

120

4 Nonsmooth Temporal Transformations (NSTT)

Such cases were examined in details by Caratheodory [45]. Let us consider the diﬀerential equation x˙ = f (t, x), where the right-hand side satisﬁes the Caratheodory conditions. Namely, in the domain D of the (t, x) space: the function f (t, x) is deﬁned and continuous with respect x for almost all t; the function f (t, x) is measurable in t for each x; and |f (t, x)| ≤ m (t), where function m (t) is summable. The above conditions are less restrictive than those required by the classical existence theorem namely the function f (t, x) is allowed to be step-wise discontinuous in t. Such an extension becomes possible if the right-hand side of the equivalent integral equation is calculated by Lebesgue, t f (ξ, x (ξ)) dξ x (t) − x (t0 ) = t0

Now, let us the right-hand side of the equation includes the δ-impulse as a summand x˙ = f (t, x) + νδ (t − a) (4.113) where a and v are constant parameters. In this case, the right-hand side does not satisfy any more the Caratheodory conditions. However, changing the variable x (t) −→ y (t) :

x (t) = y (t) + vH (t − a)

(4.114)

where H is the Heaviside’s unit-step function, brings the diﬀerential equation back into the form y˙ = f [t, y + vH (t − a)] ≡ F (t, y)

(4.115)

where the Dirac’s function has been eliminated due to the equality H˙ (t − a) = δ (t − a); as a result the right-hand side, F (t, y), satisﬁes the Caratheodory conditions. Note that, due to the presence of discontinuous function H (t − a), the right hand side of the transformed equation, F (t, y), is generally piecewise continuous in t, even though the function f (t, x) may be continuous. Relationships (4.113) through (4.115) admit direct extensions on the vector space so that many mechanical and physical models can be represented in the form (4.113). Example 7. Parametrically excited Duﬃng’s oscillator subjected to external impact at t = a is described by equation u ¨ + ω02 (1 + ε cos 2t) u − βu3 = 2pδ (t − a)

(4.116)

Equation (4.116) is transformed to (4.113) by introducing the vector-functions as follows

4.4 Discussions, Remarks and Justiﬁcations

121

u x1 0 = , ν= u˙ x2 2p

x2 f (t, x) = βx31 − ω02 (1 + ε cos 2t) x1 x=

Example 8. Center lines of linearly elastic beams resting on elastic foundations can be described by the dimensionless diﬀerential equation d4 W + γ (ξ) W = qδ (ξ − a) dξ 4

(4.117)

where W = W (ξ) is the center line coordinate, γ (ξ) is a variable stiﬀness of the foundation, q is a transverse force localized at ξ = a. Equation (4.117) is brought to the form (4.113) by considering t as a spatial coordinate ξ and introducing the matrixes ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ W 0 100 0 ⎢ dW/dξ ⎥ ⎢0 ⎥ ⎢0⎥ 0 1 0 ⎥ ⎥ x, ⎥ x=⎢ f (ξ, x) = ⎢ ν=⎢ ⎣ d2 W/dξ 2 ⎦ , ⎣0 ⎣0⎦ 0 0 1⎦ d3 W/dξ 3 −γ (ξ) 0 0 0 q In order to completely formulate the problem, the corresponding boundary conditions must be added. Let us show now that introducing the nonsmooth argument (4.11) eliminates the delta-pulse in equation (4.113) in such a way that no step-wise discontinuity occurs in the new equations. Indeed, substituting (4.12) and (4.13) into equation (4.113) and taking into account expression s¨ = 2δ (t − a), gives Y + Rf (s, X, Y ) + [X + If (s, X, Y )] s˙ + (2Y − v) δ (t − a) = 0

(4.118)

where expressions 1 [f (a + s, X + Y ) + f (a − s, X − Y )] 2 1 If = [f (a + s, X + Y ) − f (a − s, X − Y )] 2

Rf =

are obtained analogously to (4.34). Eliminating the delta-pulse in (4.118) and equating separately both components of the remaining hyperbolic element to zero, gives the boundary-value problem X + If (s, X, Y ) = 0 Y + Rf (s, X, Y ) = 0 v Y |s=0 = 2

(4.119)

122

4 Nonsmooth Temporal Transformations (NSTT)

Although equations (4.119) have a more complicated form as compared to (4.113), the step-wise discontinuous function H (t − a) is not involved any more, also it may become important that the new argument s is half-limited: 0 ≤ s < ∞.

4.4.3

Other Versions of Periodic Time Substitutions

In this subsection, it will be shown that the symmetric sawtooth sine generates the most simple algebraic structure among other possible versions of non-invertible time substitutions. Note that periodic temporal arguments have been considered in the literature for a long time. The main idea of these approaches is to make a new temporal argument limited and therefore expand class of usable algorithms. For example, the harmonic transformation of time in combination with the power series method was used by Ince [70] for investigation of periodic motions. In particular, by introducing the new variables τ = sin t and x (t) = X[τ (t)] (4.120) in the Mathieu equation, x ¨ + (a + b cos 2t) x = 0 one obtains the equation 1 − τ 2 Xτ2 − τ Xτ + a + b − 2bτ 2 X = 0 which admits periodic solutions in terms of the power series with respect to the new temporal argument |τ | ≤ 1 [195]. Transformation (4.120) with the power series methods were employed for non-linear vibrating systems as well [198], [110], [163]. Non-harmonic time transformations dealing with Jacobian functions can be also found in the literature [21]. As it is known, however, such transformations of time are restricted by special cases and cannot be applied to any periodic motion. From the mathematical point of view, it is caused by the fact that an inverse transformation does not exist on the whole period. In this section, diﬀerent versions of periodic time are introduced in such a way that the corresponding transformations are valid for any periodic motion. This is reached by a special complexiﬁcation of the coordinates. Let us start with a generalization of substitution (4.120). For the sake of compactness, notations τ = τ (ϕ) ≡ sin ϕ

and

e = e (ϕ) ≡ cos ϕ

(4.121)

will be used below, where e (ϕ) = τ (ϕ). Proposition 4. Any suﬃciently smooth periodic function x (ϕ) of the period T = 2π, can be represented in the form

4.4 Discussions, Remarks and Justiﬁcations

123

x (ϕ) = X[τ (ϕ)] + Y [τ (ϕ)]e (ϕ)

(4.122)

where X and Y are of the power series form with respect to τ . Proof. By collecting separately terms with odd and even wave numbers in the corresponding Fourier series, one obtains ∞

x (ϕ) =

A0 + [A2n cos 2nϕ + A2n−1 cos (2n − 1) ϕ] 2 n=1 +

∞

[B2n sin 2nϕ + B2n−1 sin (2n − 1) ϕ]

(4.123)

n=1

Then, introducing notations (4.121) into the tabulated expressions [52] (Formulas No. 1.332), gives n n cos 2nϕ = a2i τ 2i , cos[(2n − 1) ϕ] = a2i−1 τ 2i−2 e i=0

sin 2nϕ =

n

i=1

b2i−2 τ 2i−1

e,

sin[(2n − 1) ϕ] =

i=1

n

b2i−1 τ 2i−1(4.124)

i=1

where the coeﬃcients are listed in [52]. Substituting (4.124) in (4.123) and reordering the terms, completes the proof. As it is seen from identities (4.124), the second component in representation (4.122) is due to the odd cosine-terms and even sine-terms of the Fourier expansion. Note that combination (4.122) possesses algebraic properties similar to those generated by the sawtooth time substitution. Namely, diﬀerentiation, integration or any suﬃciently smooth function of representation (4.122) gives an element of the same two-component structure as (4.122). This is due to the fact that none of the listed above operations destroys periodicity of the function, and hence, identity (4.122) can be applied to the result of the operations as well. Practically, results of the operations are obtained by taking into account the trigonometric identities τ (ϕ) = e (ϕ) , e (ϕ) = −τ (ϕ)

(4.125)

e2 = 1 − τ 2

(4.126)

and Note also that the components of representation (4.122) apparently are linearly independent and thus the whole combination is zero if and only if its both components are zero.

124

4 Nonsmooth Temporal Transformations (NSTT)

In order to illustrate manipulations with (4.122), let us introduce the temporal argument τ = sin ωt into the Duﬃng oscillator with no linear stiﬀness term [186] (4.127) x¨ + ζ x˙ + x3 = F sin ωt Considering periodic solutions and taking into account (4.125) and (4.126) on every step of the transformation, gives 1 − τ 2 X − τ X ω 2 + ζ 1 − τ 2 Y − τ Y ω (4.128) +3 1 − τ 2 XY 2 + X 3 = F τ 1 − τ 2 Y − 3τ Y − Y ω 2 + ζX ω +3X 2 Y + 1 − τ 2 Y 3 = 0

(4.129)

The unknown functions X and Y must satisfy conditions of analytical continuation on the boundaries of the interval −1 ≤ τ ≤ 1. These conditions are obtained by substituting τ = ±1 in equations (4.128) and (4.129) as follows (4.130) −τ X ω 2 − τ ζY ω + X 3 − F τ |τ =±1 = 0

− (3τ Y + Y ) ω 2 + ζX ω + 3X 2 Y |τ =±1 = 0

(4.131)

The above system does not admit a family of solutions on which Y (τ ) ≡ 0 due to the damping, therefore transformation (4.120) is not valid in this case. Interestingly enough, the form of representation (4.122) remains the same in the case of triangular wave, although the basic algebraic operation (4.126) is diﬀerent. For the comparison reason, let us consider equation (4.127) with the forcing function F τ (t/a), where τ is assumed to be the triangular wave, i.e. sawtooth sine of the period 4a. Let us represent periodic solutions in the form (4.122), where the new temporal argument is the sawtooth sine (4.22) of the phase variable ϕ = t/a. Then, substituting (4.122) into equation (4.127) and considering the result as a two-component element of the algebra, one obtains the boundary value problem X a−2 + ζY a−1 + XY 2 + X 3 = F τ

(4.132)

Y a−2 + ζX a−1 + 3Y X 2 + Y 3 = 0

(4.133)

Y |τ =±1 = 0,

X |τ =±1 = 0

(4.134)

where the boundary conditions (4.134) stay for elimination of the periodic series of Dirac functions from the ﬁrst and second derivatives of the coordinate.

4.4 Discussions, Remarks and Justiﬁcations

4.4.4

125

General Case of Non-invertible Time and Its Physical Meaning

Let us consider now a general class of functions {τ (ϕ) , e (ϕ)} produced by the conservative oscillator x ¨ + Π (x) = 0, where Π (x) is the potential energy of the oscillator. In order to make the amplitude normalized to unity, let us normalize the potential energy and the phase variable as P (x) = Π (x) /Π (1) 2Π (1)t, respectively. As a result, the diﬀerential equation of and ϕ = motion and the energy integral take the form, respectively,

1 d2 x + P (x) = 0 dϕ2 2

and

dx dϕ

2 = 1 − P (x)

(4.135)

Let x = τ (ϕ) be the system coordinate determined implicitly from the energy integral τ(ϕ) ds =ϕ (4.136) 1 − P (s) 0

Then second expression from (4.135) gives e2 = 1 − P (τ ) where e (ϕ) = τ (ϕ)

and

1 e (ϕ) = − P (τ ) 2

(4.137)

(4.138)

Now let us formulate (without proof) Proposition 5. Any periodic function x (ϕ) whose period is normalized to 1 T =4 0

ds 1 − P (s)

can be represented in the form (4.122), where the functions τ (ϕ) and e (ϕ) are given by (4.136) and (4.138). For example, one can transform equation (4.127) based on the potential energy function P (x) = x2n . Then the boundary value problems (4.128) through (4.131) and (4.132) through (4.134) can be derived as particular cases n = 1 and n −→ ∞, respectively.

4.4.5

NSTT and Cnoidal Waves

There are other methodological sources of approximate solutions in terms of power series with respect to diﬀerent type of periodic functions. In case of oscillators with even nonlinearities, it was suggested to approximate solutions by power series of the elliptic Jacobi cn-function [36]

126

4 Nonsmooth Temporal Transformations (NSTT)

x(t) = A +

N

Bn cn2n (ωt + αn , k 2 )

(4.139)

n=1

where A and Bn are constants, ω is the frequency parameter, and k is the modulus. The reason for using series (4.139) is that the Jacobi function eﬀectively captures strongly unharmonic temporal shapes of periodic motions near separatrix loops. However, mathematical properties of Jacobi functions may require certain eﬀorts while manipulating with solutions especially when the solutions are involved into further perturbation procedures. Note the particular case of expression (4.139) N = 1 resembles the well known case of cnoidal wave [184]. x(t) = −

E(m) + dn2 [2K(m)t] K(m)

(4.140)

where K(m) and E(m) are compete elliptic integrals of ﬁrst kind and second kind, respectively, and m = k 2 is the parameter (Jacobi), the period is normalized to unity. If the parameter m is small enough, function (4.140) takes almost harmonic shapes, however, when m is getting larger then x(t) describes so-called cnoidal waves as shown in Fig. 4.4. From the physical point of view, function (4.140) with speciﬁc scaling factors exactly describes temporal behavior of the interaction force between particles of the Toda lattice f = a[exp(−br) − 1],

ab > 0

(4.141)

where r is the distance between adjacent particles, a and b arbitrary parameters. Periodic function (4.140) admits exact Fourier series expansions x(t) =

∞ l cos(2πlt) π2 [K(m)]2 sinh[πlK(1 − m)/K(m)]

(4.142)

l=1

Interestingly enough, function (4.140) can be also represented as a sum of localized waves (solitons), however, shifted one with respect other on the same interval x(t) =−

∞ π2 π + cosh−2 [π(t−l)K(m)/K(1−m)] 2K(m)K(1 − m) [2K(1 − m)]2 l=−∞

(4.143) Practical reasons for using either of series (4.142) or (4.143) is that terms of both series consist of elementary functions4 of time, while executing 4

Actually, exponential functions with real and imaginary exponents.

4.4 Discussions, Remarks and Justiﬁcations

127

Fig. 4.4 Transition from quasi-harmonic to cnoidal wave.

diﬀerent principles of approximation. Namely, each term of the Fourier series is carrying global information about the periodic process, whereas each term of series (4.143) provides just local description in some interval near the time point determined by the number of term, l. Although both series describe the process exactly from the theoretical standpoint, the only limited number of terms is possible to keep in calculations. As a result, the above mentioned diﬀerence in principles of approximation may become essential as discussed further in this subsection. Now, adapting transformation (4.71) to the case T = 1 and then applying the result to periodic function (4.143), gives xn,p (t) = −

π2 π(e+ + e− ) + 2K(m)K(1 − m) [2K(1 − m)]2

×{

n

cosh−2 [λ(m)(τ − 4l)]e+ +

l=−n

λ(m) =

πK(m) 4K(1 − m)

p+1 l=−p

(4.144) cosh−2 [λ(m)(τ − 2 + 4l)]e− }

128

4 Nonsmooth Temporal Transformations (NSTT)

Fig. 4.5 NSTT of the periodic cnoidal wave at diﬀerent Jacobi parameters m: 3-three terms truncation (n = 0, p = 0), 5-ﬁve terms truncation (n = 1, p = 0), and ∞ - exact expression.

where the inﬁnite limits of summation have been replaced by the integers n > 0 and p > 0; the oscillating triangular wave time, τ = τ (4t), and the idempotent basis e+ = [1 + e(4t)]/2 and e− = [1 − e(4t)]/2 are introduced; substitution 1 = e+ + e− has been made in order to emphasize that the entire expression (4.144) has the form of hyperbolic number represented in the idempotent basis.

4.4 Discussions, Remarks and Justiﬁcations

129

Obviously, x∞,∞ (t) = x(t), so that, in this case, (4.144) becomes exact equivalent to (4.143). However, convergence properties of series (4.144) are signiﬁcantly improved as compared to (4.143). Indeed, as time t is growing in (4.143), one must switch from one term to another to keep suﬃcient precision of approximation so that on the inﬁnite time interval, −∞ < t < ∞, the entire series (4.143) is needed. In contrast, the temporal argument of series (4.144) is always bounded by the standard interval −1 ≤ τ ≤ 1, whereas the periodicity of wave is captured by the basis functions rather than multiple terms of the series. As a result, terms of series (4.144) are exponentially decaying as the summation index increases. As follows from the diagrams in Fig. 4.5, for the range of Jacobi parameter m > 0.2 just ﬁve terms in (4.144) are enough to capture both quasi-harmonic and cnoidal wave shapes with a very good precision. For greater parameters m, when the wave becomes essentially cnoidal, only three terms provide quite a perfect match with the exact shape. As seen from the upper fragment in Fig. 4.5, the three term approximation gives essential discontinuous error for very small Jacobi parameter m = 0.08. However, in this almost harmonic range, it is more eﬀective to use few or even one term Fourier series (4.142) rather than series (4.143). Another essential advantage of series (4.144) is that, in the cnoidal range, diﬀerent algebraic manipulations with truncated series are essentially eased due to the idempotent properties.

Chapter 5

Sawtooth Power Series

Abstract. In this chapter, we introduce polynomials and power series expansions with respect to the triangular sine-wave. These can be used for approximations of periodic signals and unknown periodic solutions of dynamical systems. Such approximations may appear to be eﬀective in those cases when trigonometric series converge slowly due to step-wise discontinuities or spikes. Another reason for using polynomial expansions is that they are usually more convenient for algebraic manipulations. If the process under consideration is smooth then suﬃcient class of smoothness of approximations is achieved by imposing speciﬁc constraints on the coeﬃcients. Other equations for the coeﬃcients may appear either as a result of optimization procedures, that minimize the error of approximation, or as an outcome of iterative procedures dictated by the diﬀerential equations of motion. It is also shown in this chapter that using operators Lie associated with dynamical systems essentially facilitates construction of the periodic power series.

5.1

Manipulations with the Series

5.1.1

Smoothing Procedures

Consider a smooth periodic function x (t) of the period T = 4 represented in the form x (t) = X (τ )+Y (τ ) τ˙ , where τ = τ (t). Obviously, both components, X and Y , admit power series expansions with respect to the argument τ with no loss of periodicity in the original time t. However, keeping the ﬁnite number of terms in such series may violate the smoothness conditions at τ = ±1. Indeed, functions and their truncated series may behave diﬀerently near the boundaries τ = ±1. Therefore, let us introduce formal algorithms that can be applied to the truncated series in order to improve their smoothness properties. First, consider the expansion X (τ ) =

2N X (i) (0) i=0

i!

τ i + O τ 2N +1

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 131–144, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(5.1)

132

5 Sawtooth Power Series

Obviously, the power terms τ i and, generally speaking, ﬁnite sums in (5.1) are non-diﬀerentiable at the set of of points Λ = {t : τ (t) = ±1}. In contrast, the binomials τ i+2 τi − ; i = 1, 2, ... (5.2) φi (τ ) = i i+2 are twice diﬀerentiable with respect to t on Λ that can be veriﬁed by taking derivatives with respect to t. For instance, ﬁrst two generalized derivatives are dφi (τ ) = τ i−1 (1 − τ 2 )τ˙ (t) dt d2 φi (τ ) = (i − 1) τ i−2 − (i + 1) τ i dt2 where the term τ i−1 (1 − τ 2 )¨ τ has been removed from the second derivative due to the presence factor 1 − τ 2 that takes zero value whenever τ¨ = 0. It is seen therefore that functions (5.2) are twice continuously diﬀerentiable with respect to t even though each of the functions is combined of two nondiﬀerentiable terms. This is achieved by the speciﬁc choice for the coeﬃcients and signs in (5.2) such that the power terms have same jumps of slopes but with opposite signs. Now, considering (5.2) as equations with respect to diﬀerent powers of τ , gives, respectively odd and even powers, as τ 2N +3 τ = φ1 (τ ) + ... + φ2N +1 (τ ) + 2N + 3

τ 2N +3 τ 3 = 3 φ3 (τ ) + ... + φ2N +1 (τ ) + 2N + 3

2N +3 τ τ 2N +1 = (2N + 1) φ2N +1 (τ ) + 2N + 3

(5.3)

and τ = 2 φ2 (τ ) + ... + φ2N +2 (τ ) + 4 τ = 4 φ4 (τ ) + ... + φ2N +2 (τ ) +

τ 2N +3 2N = 2N φ2N (τ ) + τ 2N + 2 2

τ 2N +2 2N + 2 τ 2N +2 2N + 2

(5.4)

where N is an arbitrary positive integer. Substituting (5.3) and (5.4) in (5.1) and setting N → ∞, gives X (τ ) = X (0) +

∞ i (2k−1) X (0) i=1 k=1

X (2k) (0) φ2i−1 (τ ) + φ2i (τ ) (2k − 2)! (2k − 1)!

(5.5)

5.1 Manipulations with the Series

133

In contrast to (5.1), particular sums of series (5.5) are twice continuously diﬀerentiable functions of t. Since the Y -component usually appears with the step-wise discontinuous factor τ˙ (t), the related series must be re-organized in a somewhat diﬀerent way by taking into account the corresponding necessary condition of continuity: Y (±1) = 0. In this case, appropriate polynomials are designed as ψi (τ ) = τ i − τ i+2 ; i = 0, 1, 2, ...

(5.6)

These polynomials provide continuity for the term ψi [τ (t)]τ˙ (t) as well as its ﬁrst derivative d[ψi (τ )τ˙ ] = ψi (τ ) τ˙ 2 = iτ i−1 − (i + 2) τ i+1 dt where the term ψi (τ ) τ¨ has been eliminated due to the boundary conditions ψi (±1) = 0. However, second derivative appears to be a step-wise discontinuous function at every time t ∈ Λ, d2 [ψi (τ )τ˙ ] = i (i − 1) τ i−2 − (i + 2) (i + 1) τ i τ˙ 2 dt Therefore, particular sums of the re-built series Y (τ ) τ˙ =

∞ i (2k−1) Y (0) i=0 k=0

(2k − 1)!

ψ2i−1 (τ ) +

Y (2k) (0) ψ2i (τ ) τ˙ (2k)!

(5.7)

are less smooth than those obtained for the X-component (5.5). Combining (5.5) and (5.7) and considering an arbitrary period, T = 4a, gives x (t) = X (0) +

∞ i=1

KX (i)

τi τ i+2 − i i+2

+e

∞

KY (i) τ i − τ i+2

(5.8)

i=0

where τ =τ (t/a) and e =τ (t/a), and the coeﬃcients are calculated as follows KX (2i − s) =

i X (2k−s) (0) (2k − 1 − s)! k=1

KY (2i − s) =

i Y (2k−s) (0) k=0

(s = 0, 1)

(2k − s)!

(5.9)

134

5 Sawtooth Power Series

Example 9 ln 3 π 2 + ln (2 + cos t) = 2 12 +

τ2 τ4 − 2 4

ln 3 1 − τ2 + 2

+

ln 3 π 2 π4 − + 2 12 192

+

π2 π4 − 12 48

ln 3 π 2 − 2 12

τ4 τ6 − 4 6

+ ...

2 τ − τ4

4 τ − τ 6 + ... e

where τ = τ (2t/π) and e = τ (2t/π). Series (5.8) may represent periodic solutions of dynamical systems. Formal solutions are obtained by taking into account the diﬀerential equations of motion when calculating the derivatives in coeﬃcients (5.9). It can be done by means of the operator Lie as discussed in the next section. Finally, the above smoothness procedures can be repeated as many times as needed until necessary smoothness of the particular sums is achieved. For instance, the expressions (5.2) and (5.6) reveal next steps of the smoothing as φi (τ ) =

ϕi (τ ) =

τi τ i+2 − ∈ C2 (τ i ) |τ =1 (τ i+2 ) |τ =1 φi (τ ) (3) φi

(τ ) |τ =1

−

φi+2 (τ ) (3) φi+2 (τ ) |τ =1

∈ C4

(5.10)

and, ψi (τ ) τ˙ = τ i − τ i+2 τ˙ ∈ C 1 χi (τ ) τ˙ =

ψi (τ ) (2)

ψi (τ ) |τ =1

−

ψi+2 (τ ) (2)

ψi+2 (τ ) |τ =1

τ˙ ∈ C 3

(5.11)

respectively, where the symbol C indicates the class of smoothness in the interval −∞ < t < ∞. Second sets of equations in (5.10) and (5.11) have to be inverted for φi (τ ) and ψi (τ ) in a similar to (5.3) and (5.4) way. Then, by using the corresponding expressions, one can introduce functions ϕi (τ ) and χi (τ ) into series (5.5) and (5.7), respectively.

5.2 Sawtooth Series for Normal Modes

5.2

135

Sawtooth Series for Normal Modes

5.2.1

Periodic Version of Lie Series

Lie series with respect to the physical time parameter are considered below. Note that the corresponding procedure essentially diﬀers of those used for asymptotic integration of the diﬀerential equations of motion, where Lie series associate with a small parameter of perturbation [62] , [204]. Let us consider the diﬀerential equation of motion with respect to the position vector-function x (t) ∈ Rn x ¨ + f (x, x, ˙ t) = 0

(5.12)

under the initial conditions x0 = x|t=0

and x˙ 0 = x| ˙ t=0 = v0

where vector-function f is assumed to be diﬀerentiable with respect to every argument as many times as needed. The standard Cauchy form of dynamical system (5.12) with respect to the coordinate and velocity vectors is x˙ = v v˙ = −f (x, v, t) t˙ = 1

(5.13)

Then, the dynamics of system (5.13) can be locally described by the Lie series [70]

1 (5.14) x = exp[(t − t0 )G]x0 ≡ 1 + (t − t0 )G + (t − t0 )2 G2 + · · · x0 2! G = v0 ·

∂ ∂ ∂ − f (x0 , v0 , t0 ) · + ∂x0 ∂v0 ∂t0

(5.15)

where G is Lie operator associated with system (5.13), and {x0 , v0 , t0 } is some initial point in the system’ phase space. The dot between two quantities indicates dot products. For example, v0 ·

∂ ∂ ∂ ≡ v01 + · · · + v0n ∂x0 ∂x01 ∂x0n

Series (5.14) is simply Taylor series whose coeﬃcients are calculated by enforcing equations (5.13). Unfortunately, this general idea is still of little use for oscillatory processes probably due to locality of expansion (5.14). In other words, even entire expansion (5.14) does not explicitly reveal such global characteristics of oscillations as their amplitude and period. Moreover, the

136

5 Sawtooth Power Series

corresponding truncated series produce increasingly growing errors as the time t runs away from the selected initial point t0 . In order to overcome these disadvantages, it is suggested to adapt the Lie series solution for the class of periodic motions as follows. Theorem 1. [145] Assume that system (5.13) admits a periodic solution x(t) of the period T = 4a so that x(t + 4a) = x(t) for any t, and some point {x0 , v0 , t0 } belongs to this solution. Then such a solution can be expressed in the form x = exp(aG){cosh[a(τ − 1)G] + e sinh[a(τ − 1)G]}x0

(5.16)

where τ and e are saw-tooth sine and rectangular cosine, whose periods are normalized to four and amplitudes are normalized to unity as τ (ϕ) = (2/π) arcsin sin(πϕ/2)

(5.17)

e(ϕ) = sgn[ cos(πϕ/2)]

(5.18)

and, respectively, and ϕ = (t−t0 )/a is a re-scaled time. If, in addition, the solution is odd with respect to one half of the period, x(t+2a) = −x(t), then expression (5.16) simpliﬁes to x = [sinh(aτ G) + e cosh(aτ G)]x0

1 1 ≡ aτ G + (aτ G)3 + · · · x0 + e 1 + (aτ G)2 + · · · x0 3! 2!

(5.19)

Proof of expression (5.16) is obtained by substituting the identity [144] ϕ = 1 + [τ (ϕ) − 1]e(ϕ),

(−1 < ϕ < 3)

(5.20)

in (5.14) and taking into account that e2 = 1

(5.21)

at almost every time instance1 . In order to prove the particular case (5.19), one should keep in mind that exp(2aG)x0 = x(t0 + 2a) = −x0 , as it follows from (5.14), and the oddness condition assumed. Note that τ and e are indeed quite simple piece-wise linear functions; the above analytical expressions (5.17) and (5.18) just deﬁne them in the unitform which enables one to avoid conditioning of computation in the original temporal scale, t0 ≤ t < ∞. This possibility becomes essential when the dynamics includes some evolutionary component. 1

The set of isolated points {ϕ : τ (ϕ) = ±1}appears to have no eﬀect on the results [144].

5.2 Sawtooth Series for Normal Modes

137

Physical meaning of relationship (5.20) is that, during the whole period, the time variable ϕ is expressed through the coordinate τ and velocity e of a classic particle freely oscillating between the two absolutely stiﬀ barriers with no energy loss. Due to (5.21), this relationship possesses the algebraic structure of ‘hyperbolic complex numbers’ as revealed by (5.16). Let us outline possible applications of expressions (5.16) and (5.19). For the sake of simplicity, consider the particular case (5.19). Of course, formal expression (5.19) does not guarantee the existence of periodic solutions. In case some periodic solution does exist, one should be able to ﬁnd the corresponding vectors x0 and v0 from appropriate conditions. In autonomous case, the scalar parameter, a, is also unknown and must be determined. The related conditions are formulated as a requirement of smoothness of expression (5.19), which is generally non-smooth or even discontinuous due to the presence of non-smooth and discontinuous functions τ and e, respectively. The ‘smoothing’ relations are obtained by eliminating the step-wise discontinuities of the coordinate and velocity vectors imposing the constraints cosh(aG)x0 = 0 cosh(aG)v0 = 0

(5.22)

In autonomous case, algebraic equations (5.22) represent a nonlinear eigenvalue problem, where a is an eigen-value, and {x0 , v0 } is a combined (state) eigen-vector. By narrowing the class of periodic motions to those on which the system passes its trajectory twice in the conﬁgurations space during the same period, one obtains a subclass of normal mode motions. For more physically meaningful deﬁnitions and discussions, see reference [190]. Let us formulate the corresponding problem based on the periodic Lie series solutions. Consider the vibrating system x ¨ + f (x) = 0,

x ∈ Rn

(5.23)

where f (−x) = −f (x), and the initial conditions are x|t=0 = x0 = 0 and x| ˙ t=0 = v0 . The normal mode solutions of system (5.23) are obtained as a particular case of (5.19) and (5.22) x = sinh(aτ G)x0 |x0 =0

(5.24)

cosh(aG)v0 |x0 =0

(5.25)

where the initial vector x0 = 0 is substituted into the expressions only after all degrees of the diﬀerential operator

138

5 Sawtooth Power Series

G = v0 ·

∂ ∂ − f (x0 ) · ∂x0 ∂v0

have been applied. Relationship (5.24) can be interpreted as a parametric equation of normal mode trajectories of the system with the parameter interval −1 ≤ τ ≤ 1. Let us illustrate relationships (5.24) and (5.25) based on the linear system so that the result could be compared with the well known conventional solution. Example 10. Suppose that f (x) = Kx, where K is positively deﬁned symmetric n × n-matrix with eigen-system {v0 , ω 2 } so that Kv0 = ω 2 v0 . In this case, by applying the operator G twice, one obtains that v0 is also an eigen-vector of the operator G2 , namely, G2 v0 = −ω 2 v0 . Then, keeping in mind the power series form of expressions (5.24) and (5.25) as those in (5.19) and sequentially applying the operator G2 , gives x = (v0 /ω) sin(aωτ ) and cos(aω) = 0, respectively. Notably, the last equation shows that there exist an inﬁnite number of roots {a} related to the same eigen-frequency ω! However, it is easily to ﬁnd that all the roots produce the same solution in terms of the original time t. The minimal quarter of the period is a = π/(2ω), therefore x = (v0 /ω) sin(πτ /2), and τ = (2/π) arcsin sin ωt. Nonlinear cases and the related problems dealing with truncated expansions of (5.25) will be further discussed in a full-length paper.

5.3

Lie Series of Transformed Systems

5.3.1

Second-Order Non-autonomous Systems

In the previous section, the sawtooth temporal argument was introduced into Lie series solutions with respect to time t. Alternatively, the sawtooth argument can be introduced ﬁrst into the diﬀerential equations of motion before the Lie series procedure is applied. Let the dynamical system be described by the set of second-order equations (5.12), where the vector-function f is periodic with respect to t with the period T = 4a. Periodic motions of the period T are considered. Note that, in the autonomous case, the period is a priory unknown and thus must be determined. Following the rules introduced in Chapter 4 (see also Chapters 8 and 12 for further details) and making the substitutions t → τ (t/a) and x (t) = X(τ ) + Y (τ )e in (5.12), gives the boundary value problem X + a2 Rf (X, Y, X , Y , τ ) = 0 Y + a2 If (X, Y, X , Y , τ ) = 0

(5.26)

5.3 Lie Series of Transformed Systems

139

Y |τ =±1 = 0, X |τ =±1 = 0

(5.27)

where 1 Rf = {f [X + Y, (Y + X ) /a, aτ ] ± f [X − Y, (Y − X ) /a, 2a − aτ ]} If 2 (5.28) Analogously to (5.15), the operator Lie of system (5.26) at some initial point is represented as ∂ ∂ ∂ + Y0 · − a2 Rf (X0 , Y0 , X0 , Y0 , τ0 ) · ∂X0 ∂Y0 ∂X0 ∂ ∂ −a2 If (X0 , Y0 , X0 , Y0 , τ0 ) · + ∂Y0 ∂τ0

G = X0 ·

The idea of Lie series enables one of representing solution of the system (5.26) in the power series form with respect to the sawtooth argument τ under the initial conditions at P0 (X0 , Y0 , X0 , Y0 , τ0 ); it is assumed that τ0 = 0. Then, the corresponding solution of the original equation is represented in the form x (t) = X(X0 , X0 , Y0 , Y0 , τ ) + Y (X0 , X0 , Y0 , Y0 , τ )e

(5.29)

where τ = τ (t/a) and e = τ (t/a). Now, the arbitrary quantities X0 , X0 , Y0 , and Y0 can be determined from the smoothness conditions (5.27) Y (X0 , X0 , Y0 , Y0 , τ )|τ =±1 = 0,

∂X(X0 , X0 , Y0 , Y0 , τ ) |τ =±1 = 0 ∂τ

(5.30)

However, instead of solving equations (5.30), the smoothing procedure described in the previous section can be applied. In this case, conditions (5.30) are satisﬁed automatically, but the initial quantities X0 , X0 , Y0 , and Y0 remain undetermined. This enables one of improving the solution’ smoothness by imposing stronger smoothness conditions as follows ∂ 2 Y (X0 , X0 , Y0 , Y0 , τ ) |τ =±1 = 0, ∂τ 2

∂ 3 X(X0 , X0 , Y0 , Y0 , τ ) |τ =±1 = 0 ∂τ 3 (5.31) Equations (5.30) or (5.31) for X0 , X0 , Y0 and Y0 are generally quite complicated. In diﬀerent particular cases however, the problem can be simpliﬁed by taking into account possible symmetries of the system. So let us consider equation x ¨ + f (x, t) = 0

(5.32)

where the vector-function f (x, t) is odd with respect to the positional vector x and even with respect to the quarter of the period t = a. In this case, boundary-value problem (5.26) and (5.27) reduces to

140

5 Sawtooth Power Series

X + a2 f (X, aτ ) = 0 X |τ =1 = 0

(5.33)

Y ≡0 The operator Lie and the corresponding solution are, respectively, G = X and

∂ ∂ ∂ − a2 f (X, aτ ) + ∂X ∂X ∂τ

∞ i τ 2i+1 G2k−1 X|τ =0 τ 2i−1 − ≡ X(X0 , τ ) x (t) = (2k − 2)! 2i − 1 2i + 1 i=1 k=1

where τ = τ (t/a), the initial position is X0 = 0, whereas the constant vector X0 is determined from equation ∂ 3 X(X0 , τ ) |τ =1 = 0 ∂τ 3 Example 11. Consider now a two degree of freedom oscillating system

d2 x1 ζ (2x1 − x2 ) + x31 − P sin Ωt 0 + = 0 ζ (2x2 − x1 ) + x32 dt2 x2 In this case, the parameter a = π/(2Ω) is the quarter of the period of the external forcing function. The operator Lie of the system is ∂ ∂ πτ ∂ + X2 − a2 ζ (2X1 − X2 ) + X13 − P sin ∂X1 ∂X2 2 ∂X1 ∂ ∂ −a2 [ζ (2X2 − X1 ) + X23 ] + ∂X2 ∂τ

G = X1

Let the initial position be X0 = 0. Then, introducing the notation X0 = T [A1 , A2 ] and calculating the coeﬃcients of the ﬁrst three terms of series (5.8), gives

3

τ5 τ3 τ 1 X1 = τ − − A1 + A1 + (a2 P π − 4 a2 ζ A1 + 2a2 ζ A2 ) 3 3 5 4

5 7 τ τ 1 + − A1 + (a2 P π − 4 a2 ζ A1 + 2a2 ζ A2 ) 5 7 4 1 2 [a P π 3 + 8a4 P π ζ − 8A1 5 a4 ζ 2 − 6 a2 A1 2 + 32 a4 ζ 2 A2 ] − 192 and

3 τ τ5 τ3 1 X2 = τ − − A2 + A2 + (a2 ζ A1 − 2 a2 ζ A2 ) 3 3 5 2

5.3 Lie Series of Transformed Systems

141

τ5 1 τ7 − A2 + (a2 ζ A1 − 2 a2 ζ A2 ) 5 7 2 1 4 + [a P π ζ − 8 a4 ζ 2 A1 + 2A2 5 a4 ζ 2 − 6 a2 A2 2 ] 48

+

In this case, equations (5.31) give two scalar equations for A1 and A2 , ∂ 3 X1 (A1 , A2 , τ ) |τ =1 = 0, ∂τ 3

∂ 3 X2 (A1 , A2 , τ ) |τ =1 = 0 ∂τ 3

(5.34)

where the ﬁrst and the second equations guarantee smoothness of the ac¨2 (t), respectively. Intersections of curves celeration components x ¨1 (t) and x (5.34) on the Cartesian plane A1 A2 allow to locate the roots, as it is shown√in Figs. 5.1 and 5.2. For instance, setting ζ = 1.0, P = 0.2 and Ω = 1.01, gives (A1 , A2 ) = (0.7711, 0.6677). This solution corresponds to a weakly nonlinear perturbation of the in-phase √ linear mode. Similarly, in the neighborhood of the second frequency Ω = 3.01, one ﬁnds the solution (A1 , A2 ) = (0.5971, −0.8181), which is close to the out-of-phase linear mode; see Fig. 5.1. Interestingly enough, Fig. 5.2 shows that there are also two roots, (2.3315, 2.2864) and (−2.2065, −2.2548), corresponding to in-phase vibrations even though the frequency of the external forcing function is close to that of the out-phase linear mode. This eﬀect may take place due to large amplitudes such that the in-phase nonlinear frequency becomes close to the out-of-phase linear frequency. Since the initial conditions, corresponding to the periodic regimes are known, one can integrate the diﬀerential equations of motion numerically in order to check the analytical solutions. In the latter example, one would obtain that both results are perfectly matching. It must be noted, however, that some of the periodic solutions may appear to be unstable. As a result, even a very small imperfection in the initial conditions will lead to a signiﬁcant divergence of the results. Moreover, the ‘tails’ of polynomial expansions may give roots which do not correspond to any solution. Nevertheless, the above approaches seems to make sense as compared to those based on the direct Fourier expansions. Indeed, the number of algebraic equations in (5.22), (5.30) or (5.31) is independent on the number of terms in the series.

5.3.2

NSTT of Lagrangian and Hamiltonian Equations

In diﬀerent theoretical areas, describing systems in terms of the analytical dynamics brings some advantages because, until certain stage, it is suﬃcient to deal with a single scalar function such as Lagrangian or Hamiltonian rather then manipulate with a set of diﬀerential equations. Suppose the sawtooth temporal argument is introduced into Lagrangian or Hamiltonian. Let us show that the corresponding diﬀerential equations of

142

5 Sawtooth Power Series

Fig. 5.1 The curves of smoothness for ¨1 (t) and x ¨2 (t), respec√ the accelerations x tively thin and solid lines, when Ω = 1.01.

Fig. 5.2 The curves of smoothness for ¨1 (t) and x ¨2 (t), respec√ the accelerations x tively thin and solid lines, when Ω = 3.01.

5.3 Lie Series of Transformed Systems

143

motion are still derivable from the transformed functions in the standard way of the analytical dynamics. Consider ﬁrst Lagrangian that depends on time t periodically with the period T = 4a, L = L(x, x, ˙ t) (5.35) On the manifold of smooth periodic motions of the same period T , the vectorfunction x(t) is represented as x (t) = X (τ ) + Y (τ ) e, where τ = τ (t/a) and e = e(t/a). As a result, Lagrangian (5.35) takes the form L(x, x, ˙ t) = RL (X, Y, X , Y , τ ) + IL (X, Y, X , Y , τ ) e

(5.36)

where both components on the right-hand side are determined analogously to expressions (5.28). It can be veriﬁed by inspection that the diﬀerential equations of periodic motions can be represented now in any of the two equivalent forms d dτ d dτ

∂RL ∂RL =0 − ∂X ∂X ∂RL ∂RL =0 − ∂Y ∂Y

or d ∂IL ∂IL =0 − dτ ∂Y ∂Y d ∂IL ∂IL =0 − dτ ∂X ∂X Now let us consider the Hamiltonian H = H(p, q, t)

(5.37)

which may be periodic with respect to time with the period T = 4a. On the manifold of periodic motions of the period T , the coordinates and the linear momenta are represented as q = X (τ ) + Y (τ ) e

and p = U (τ ) + V (τ ) e

respectively, where τ = τ (t/a) and e = e (t/a). Let us transform the Hamiltonian (5.37) as H −→ aH(p, q, t) = RaH (U, V, X, Y, τ ) + IaH (U, V, X, Y, τ ) e where

RaH IaH

=

a [H (U + V, X + Y, aτ ) ± H (U − V, X − Y, 2a − aτ )] 2

144

5 Sawtooth Power Series

The corresponding diﬀerential equations of periodic motions are ∂RaH ∂RaH , U = − ∂U ∂X ∂R ∂R aH aH , V =− Y = ∂V ∂Y

X =

or ∂IaH ∂IaH , V =− ∂V ∂X ∂IaH ∂IaH , U =− Y = ∂U ∂Y

X =

Besides, in autonomous cases, the operator Lie associates with the Poisson bracket, for instance, GX = {X, RaH } Therefore, introducing the sawtooth temporal argument for periodic motions preserves both Lagrangian and Hamiltonian structures of the diﬀerential equations of motion.

5.3.3

Remark on Multiple Argument Cases

In multiple frequency cases, the smoothing procedures can be applied sequentially to each of the arguments. For instance, in the case of two arguments, τ1 = τ (t/a1 ) and τ2 = τ (t/a2 ) , the X-component would take the form

i j N2 N1 τ1 τ1i+2 τ2j+2 τ2 X (τ1 , τ2 ) = X (0, 0) + − − (5.38) KX (i, j) i i+2 j j+2 i=1 i=1 where KX (i, j) are constant coeﬃcients. Analytical manipulations with such 2D polynomials as (5.38) are quite complicated because algebraic and diﬀerential operations aﬀect the structure of binomials. However, let us bring attention to a possibility of using polynomial expansions (5.38), and the related complete versions, for diﬀerent kinds of approximations by determining coeﬃcients of expansions through appropriate numerical optimization procedures. Besides of multiple phase cases in dynamics, similar expansions may be useful for multidimensional spatial problems dealing, for instance, with elastic cell-wise periodic structures; see also Chapter 14 for possible formulations. In such cases, the triangular sine waves depend on the spatial coordinates, say x and y as τ1 = τ (x/a1 ) and τ2 = τ (y/a2 ). In static problems, the parameters a1 and a2 are usually given by design, whereas the coeﬃcients can be obtained through variational principles by minimizing the corresponding functionals.

Chapter 6

NSTT for Linear and Piecewise-Linear Systems

Abstract. Remind that the tool of nonsmooth argument substitutions was introduced ﬁrst to describe strongly nonlinear vibrations whose temporal mode shapes are asymptotically close to non-smooth ones. Such cases are known to be most diﬃcult for analyses because diﬀerent quasi-harmonic methods are already ineﬀective whereas nonsmooth mapping tools are still inapplicable. It is quite clear however that the non-smooth arguments can be introduced regardless the strength of nonlinearity or the form of dynamical systems in general. For instance, it is shown in this chapter that the non-smooth substitutions can essentially simplify analyses of diﬀerent linear models with non-smooth or discontinuous inputs. It is also shown that, in piecewise-linear cases, the nonsmooth temporal transformation provides an automatic matching the motions from diﬀerent subspaces of constant stiﬀness and justiﬁes quasi-linear asymptotic solutions for the speciﬁc nonsmooth case of piece-wise linear characteristics.

6.1

Free Harmonic Oscillator: Temporal Quantization of Solutions

Introducing the sawtooth temporal argument into the diﬀerential equations of motion may bring some speciﬁc features into the corresponding solutions. For illustrating purposes, let us consider the harmonic oscillator x ¨ + ω02 x = 0

(6.1)

First, let us obtain exact general solution of the oscillator (6.1) in terms of the sawtooth temporal argument by using the substitution x = X (τ ) + Y (τ ) e V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 145–178, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(6.2)

146

6 NSTT for Linear and Piecewise-Linear Systems

where τ = τ (t/a) and e = e (t/a) are the standard triangular and rectangular wave functions, respectively. Substituting (6.2) in (6.1), gives the boundary value problem a−2 X (τ ) + ω02 X (τ ) = 0 a−2 Y (τ ) + ω02 Y (τ ) = 0

(6.3) (6.4)

X (±1) = 0, Y (±1) = 0

(6.5)

By considering the parameter a as an eigen value of the problem, one obtains the set of eigen values and the corresponding solutions as, respectively, aj =

and Xj = sin

jπ 2ω0

jπτ jπτ + ϕj , Yj = cos − ϕj 2 2

(6.6)

(6.7)

where ϕj = (π/4) [1 + (−1)j ] , τ = τ (t/aj ), and j is any positive real integer. Therefore, introducing the sawtooth oscillating time produced the discrete family of solutions for harmonic oscillator (6.1). The nature of such kind of quantization is due to the speciﬁc temporal symmetry of periodic motions. In other words, the quantization is associated with a multiple choice for the period Tj = 4aj = jT

(6.8)

where T = 2π/ω0 is the natural period of oscillator (6.1). In terms of the original temporal variable t, the number j plays no role for the temporal mode shape, given by

jπ 2ω0 t x (t) = A sin τ (6.9) + ϕj 2 jπ

2ω0 t 2ω0 t jπ τ +B cos − ϕj e 2 jπ jπ where A and B are arbitrary constants, and x(t) is the same harmonic wave regardless the number j. In this section, the free linear oscillator was considered for illustrating purposes. Of course, there is no other pragmatic reason for introducing the sawtooth time into equation (6.1). The situation drastically changes however in non-autonomous cases of non-smooth or discontinuous inputs. It is shown below that, in such cases, the sawtooth time variable can help to facilitate determining particular solutions. The eﬀect of ‘temporal quantization’ represented by expression (6.9), which seems to be just identical transformation in the autonomous case, acquires helpful meaning at the presence of external

6.2 Non-autonomous Case

147

excitations. For instance, according to (6.9) the so-called combination resonances will appear to be an inherent property of oscillators.

6.2

Non-autonomous Case

6.2.1

Standard Basis

Consider the linear harmonic oscillator under the external forcing described by the linear combination of triangular and rectangular wave functions

t t 2 x¨ + ω0 x = F τ + Ge (6.10) a a where F and G are constant amplitudes, and a is a quarter of the period. Substituting (6.2) in (6.10), leads to the boundary value problem a−2 X (τ ) + ω02 X (τ ) = F τ a−2 Y (τ ) + ω02 Y (τ ) = G

(6.11) (6.12)

under the boundary conditions (6.5). In contrast to autonomous case (6.1), the parameter a is known. However, the equations (6.11) and (6.12) are non-homogeneous, and thus a non-zero solution exists for any a and can be found in few elementary steps. As a result, the particular periodic solution of the original equation (6.10) takes the form t F sin [aω0 τ (t/a)] xp (t) = X (τ ) + Y (τ ) e = 2 τ − ω0 a aω0 cos aω0 cos [aω0 τ (t/a)] t G e (6.13) + 2 1− ω0 cos aω0 a The corresponding general solution is x (t) = A cos (ω0 t − ϕ) + xp (t), where A and ϕ are arbitrary amplitude and phase parameters. Note that solution (6.13) immediately shows all possible resonance combinations aω0 = (2k + 1) π/2 or ω0 = 2k + 1 (6.14) Ω where k = 1, 2, 3... , and Ω = 2π/T = π/(2a) is the principal circular frequency of the external forcing. It is interesting to compare the solution (6.13) with those obtained by the conventional methods such as Fourier series. So, taking into account expansion,

∞ t (2k + 1) πt 8 (−1)k τ (6.15) sin = 2 2 a π 2a (2k + 1) k=0

148

6 NSTT for Linear and Piecewise-Linear Systems

gives the particular solution of equation (6.10) in the form xp (t) =

∞ k=0

ω02

−

1 (2k+1)π 2a

2 ×

(6.16)

k

k

8F (−1)

4G (−1) (2k + 1) πt (2k + 1) πt × + cos 2 sin 2 2a π (2k + 1) 2a π (2k + 1)

Solution (6.16) indicates the same resonance conditions, (6.14). However, inﬁnite trigonometric series are less convenient for calculations, especially when dealing with derivatives of the solutions; indeed, diﬀerentiation slows down convergence of series (6.16).

6.2.2

Idempotent Basis

Consider the linear oscillator including viscous damping under the rectangular wave external loading

t x ¨ + 2ζω0 x˙ + ω02 x = pe (6.17) a The purpose is to obtain periodic (particular) solution with the period of external loading T = 4a. Recall that the idempotent basis is introduced by means of the linear transformation (see Chapters 1 and 4) {1, e} −→ {e+ , e− } :

e± =

1 (1 ± e) 2

(6.18)

or, inversely, 1 = e+ + e− and e = e+ − e− , where e2± = e± and e+ e− = 0. Now, the periodic solution and external loading are represented in the form x(t) = U (τ )e+ + V (τ )e−

(6.19)

pe = p(e+ − e− ) where e± = e± (t/a), and U (τ ) and V (τ ) are unknown functions of the triangular wave τ = τ (t/a). Substituting (6.19) in (6.17), and sequentially eliminating derivatives of the rectangular wave e(t/a) as described in Chapter 4, gives equations U + 2ζωaU + (ωa)2 U = pa2 V − 2ζωaV + (ωa)2 V = −pa2

(6.20)

and boundary conditions (U − V )|τ =±1 = 0 (U + V )|τ =±1 = 0

(6.21)

6.3 Systems under Periodic Pulsed Excitation

149

All the coeﬃcients and right-hand sides of both equations in (6.20) are constant, and the equations are decoupled. As a result, solution of boundary value problem (6.20) and (6.21) is easily obtained in the form U (τ ) =

2p exp(−ατ ) p − (6.22) 2 2 ω βω (cos 2β + cosh 2α) ×[cos β cosh α(β cos βτ + α sin βτ ) + sin β sinh α(α cos βτ − β sin βτ )]

2p exp(ατ ) p (6.23) + ω2 βω 2 (cos 2β + cosh 2α) ×[cos β cosh α(β cos βτ − α sin βτ ) + sin β sinh α(α cos βτ + β sin βτ )] where α = ωaζ and β = ωa 1 − ζ 2 . Substituting (6.22) and (6.23) in (6.19), gives closed form particular solution of original equation (6.17). Transition to the original temporal variable is given by the functions τ (ϕ) = (2/π) arcsin[sin(πt/2)] and e(ϕ) = sgn[cos(πt/2)]. Since the system under consideration is linear, the general solution of equation (6.17) can be obtained by adding general equation of the corresponding equation with zero right-hand side. Finally, note that neither trigonometric expansions nor any integral transforms were involved into the solution procedure. V (τ ) = −

6.3

Systems under Periodic Pulsed Excitation

Instantaneous impulses acting on a mechanical system can be modeled either by imposing speciﬁc matching conditions on the system state vector at pulse times or by introducing Dirac’s functions into the diﬀerential equations of motion. The ﬁrst approach deals with the diﬀerential equations of a free system separately between the impulses, therefore a sequence of systems under the matching conditions are considered. The second method gives a single set of equations over the whole time interval without any conditions of matching. In this case however the analysis can be carried out correctly in terms of distributions, which unfortunately requires additional mathematical justiﬁcations in non-linear cases. Both of the above approaches are actually employed for diﬀerent quantitative and qualitative analyses. The analytical tool, which is described below, on the one hand, eliminates the singular terms from the equations and, on the other hand, brings solutions to the unit-form of a single analytic expression for the whole time interval.

6.3.1

Regular Periodic Impulses

Introducing the sawtooth temporal argument may signiﬁcantly simplify solutions whenever loading functions are combined of the triangular wave and

150

6 NSTT for Linear and Piecewise-Linear Systems

its derivatives. For instance, let us seek a particular solution of the ﬁrst order diﬀerential equation v˙ + λv = μ

∞

[δ (t + 1 − 4k) − δ (t − 1 − 4k)]

(6.24)

k=−∞

where λ and μ are constant parameters. For positive λ, equation (6.24) describes the velocity of a particle moving in a viscous media under the periodic impulsive force. The corresponding physical model is shown in Fig. 6.1, where the freely moving massive tank experiences perfectly elastic reﬂections from the stiﬀ obstacles. By scaling the variables, one can bring the diﬀerential equation of motion of the particle to the form (6.24), where v (t) = x˙ (t).

Fig. 6.1 If the particle’ mass is very small compared to the total mass of the tank then the inertia force applied to the particle inside the tank has the periodic pulse-wise character.

First, note that the right-hand side of equation (6.24) can be expressed through the generalized derivative of the rectangular wave function as follows v˙ + λv =

μ e˙ (t) 2

(6.25)

Now let us represent the particular solution in the form v (t) = X (τ (t)) + Y (τ (t)) e (t)

(6.26)

Substituting (6.26) in (6.25), gives μ e˙ (t) = 0 Y + λX + (X + λY ) e (t) + Y − 2

(6.27)

Apparently, the elements {1, e} and e˙ in combination (6.27) are linearly independent as functions of diﬀerent classes of smoothness. Therefore,

6.3 Systems under Periodic Pulsed Excitation

Y + λX = 0,

X + λY = 0,

151

Y |τ =±1 =

μ 2

(6.28)

In contrast to equation (6.24) or (6.25), boundary value problem (6.28) includes no discontinuities whereas the new independent variable belongs to the standard interval, −1 ≤ τ ≤ 1. Solving the boundary value problem (6.28) and taking into account substitution (6.26), gives periodic solution of equation (6.24) in the form v = X +Ye =

μ (− sinh λτ + e cosh λτ ) 2 cosh λ

or

μ exp [−λτ (t) e (t)] e (t) (6.29) 2 cosh λ Fig. 6.2 illustrates solution (6.29) for μ = 0.2 and diﬀerent magnitudes of λ. v=

0.2 Λ 0.4 0.1

v

0.0

Λ 1.9

0.1 0.2

0

2

4

6

8

t

Fig. 6.2 The family of discontinuos periodic solutions.

Note that the discontinuous solution v (t) is described by the unit-form expression (6.29) through the two elementary functions τ (t) and e (t).

6.3.2

Harmonic Oscillator under the Periodic Impulsive Loading

Let us consider the harmonic oscillator subjected to periodic pulses x ¨ + ω02 x = 2p

∞

[δ (ωt + 1 − 4k) − δ (ωt − 1 − 4k)]

k=−∞

where p, ω0 and ω are constant parameters.

(6.30)

152

6 NSTT for Linear and Piecewise-Linear Systems

The right-hand side of equation (6.30) can be expressed through ﬁrst derivative of the rectangular wave as follows x ¨ + ω02 x = p

de (ωt) d (ωt)

(6.31)

Let us seek a periodic solution of the period T = 4/ω in the form x (t) = X (τ (ωt)) + Y (τ (ωt)) e (ωt)

(6.32)

Substituting (6.32) in (6.31) under the necessary condition of continuity for x (t), gives de (ωt) ω 2 X + ω02 X + ω 2 Y + ω02 Y e + ω 2 X − p =0 d (ωt)

(6.33)

Analogously to the previous subsection, equation (6.33) gives the boundary value problem X +

ω 2 0

ω

X = 0,

X |τ =±1

p = 2, ω

Y +

ω 2 0

ω

Y =0

(6.34)

Y |τ =±1 = 0

Solving boundary value problem (6.34) and taking into account (6.32), gives the periodic solution of the original equation (6.30) in the form x = X (τ (ωt)) =

p sin [(ω0 /ω) τ (ωt)] ωω0 cos (ω0 /ω)

(6.35)

where Y ≡ 0. Solution (6.35) is continuous, but nonsmooth at those times t where τ (ωt) = ±1. All possible resonances are given by ω=

2 ω0 ; π k

k = 1, 3, 5, ...

(6.36)

where the factor 2/π is due to diﬀerent normalization of the periods for trigonometric and sawtooth sines. Now let us consider the case of viscous damping described by the diﬀerential equation of motion x ¨ + 2ζ x˙ + ω02 x = p where ζ is the damping factor.

de (ωt) d (ωt)

(6.37)

6.3 Systems under Periodic Pulsed Excitation

153

In this case, the boundary value problem becomes coupled ω 2 ζ 0 X + 2 Y + X=0 ω ω 2 ζ ω0 Y =0 Y + 2 X + ω ω

(6.38)

p , Y |τ =±1 = 0 ω2 As a result, the periodic solution has both X and Y components X |τ =±1 =

p 2 2 cos β cosh α + sin2 β sinh2 α ×[cosh α cos β cosh ατ sin βτ − sinh α sin β sinh ατ cos βτ

x = X +Ye=

βω 2

(6.39)

+ (sinh α cos βτ cosh ατ sin β − sinh ατ sin βτ cosh α cos β) e] where τ = τ (ωt), e = e (ωt); α = ζ/ω and β = ω02 − ζ 2 ω.

0.2 0.1 x 0.0 0.1 0.2 0

2

4 t

6

8

Fig. 6.3 Response of the damped harmonic oscillator under the periodic impulsive excitation for p = 0.1, ζ = 0.5, ω0 = 4 and ω = 0.2 (low-frequency pulses.)

Figs. 6.3 through 6.5 illustrate qualitatively diﬀerent responses of the system when varying the input frequency. In diﬀerent proportions, the responses combine properties of the harmonic damped motion and the non-smooth motion due to the impulsive loading. For instance, when ω >> ω0 and ω >> ζ, the system is near the limit of a free particle under the periodic impulsive force. In this case, the boundary value problem is reduced to X = 0,

Y = 0;

X |τ =±1 =

p , ω2

Y |τ =±1 = 0

(6.40)

154

6 NSTT for Linear and Piecewise-Linear Systems

0.06 0.04 0.02 x 0.00 0.02 0.04 0.06 0

2

4 t

6

8

Fig. 6.4 System response on ‘resonance’ pulses ω = (2/π)ω0 = 2.5465.

0.003 0.002 0.001 x 0.000 0.001 0.002 0.003 0

2

4 t

6

8

Fig. 6.5 Response on high-frequency pulses; ω = 6.

This gives the triangular temporal shape of the motion, x = pτ (ωt) /ω 2 , which is approached by the time history record on Fig. 6.5. Finally, let us consider N -degrees-of-freedom system My ¨ + Ky = p

de (ωt) d (ωt)

(6.41)

where y (t) is N -dimensional vector-function, p is a constant vector, M and K are constant N × N mass and stiﬀness matrixes respectively.

6.3 Systems under Periodic Pulsed Excitation

155

Let {e1 , ..., eN } and ω1 ,...,ωN be the normal mode basis vectors and the corresponding natural frequencies, respectively, such that Kej = ωj2 M ej ,

eTk M ej = δkj

for any k = 1, ..., N and j = 1, ..., N . Introducing the principal coordinates xj (t), y=

N

xj (t) ej

(6.42)

j=1

gives a decoupled set of impulsively forced harmonic oscillators of the form (6.31), de (ωt) (6.43) x¨j + ωj2 xj = pj d (ωt) where pj = eTj p. Therefore, making use of solution (6.35) for each of the oscillators (6.43) and taking into account (6.42), gives y=

N (eTj p)ej sin [(ωj /ω) τ (ωt)] ωωj cos (ωj /ω) j=1

(6.44)

The corresponding resonances are determined by the condition ω=

2 ωj π k

where k = 1, 3, 5, ... and j = 1, ..., N .

6.3.3

Periodic Impulses with a Temporal ‘Dipole’ Shift

Let us consider the impulsive excitation with a dipole shift of pulse times. In this case, the right-hand side of equation (6.25) can be expressed by second derivative of the saw-tooth function with some incline described the parameter γ as shown in Fig. 6.6 v˙ + λv = p =

∂ 2 τ (ωt, γ) 2

∂ (ωt) ∞ 2p

1 − γ2

=p

k=−∞

∂e (ωt, γ) ∂ (ωt)

(6.45)

[δ (ωt + 1 − γ − 4k) − δ (ωt − 1 + γ − 4k)]

156

6 NSTT for Linear and Piecewise-Linear Systems

Fig. 6.6 Basic NSTT asymmetric wave functions.

Based on the NSTT identities introduced in Chapter 4, periodic solutions of equation (6.45) still can be represented in the form v = X (τ ) + Y (τ ) e

(6.46)

where τ = τ (ωt, γ) and e = e (ωt, γ); see Fig. 6.6 for graphic illustrations. Substituting (6.46) in equation (6.45), gives ωαY + λX + [ω (X + βY ) + λY ] e + (ωY − p)

∂e(ωt, γ) =0 ∂(ωt)

(6.47)

where α = 1/ 1 − γ 2 , β = 2γα, and the identity e2 = α + βe has been taken into account. Equation (6.47) is equivalent to the boundary-value problem ω (X + βY ) = −λY ωαY = −λX

(6.48)

ωY |τ =±1 = p The corresponding solution is

λ cosh ωλ τ λ sinh ωλ τ λ p cosh γ τ + sinh γ exp γ Y = ω ω ω ω cosh ωλ sinh ωλ ωα X=− Y (6.49) λ where the X-component is deﬁned by diﬀerentiation due to the second equation in (6.48).

6.4 Parametric Excitation

6.4

157

Parametric Excitation

In this section, two diﬀerent cases of parametric excitation are considered based on relatively simple linear models. Piecewise-constant and impulsive excitations are described by means of the functions e(ωt, γ) and ∂e(ωt, γ)/∂ (ωt), respectively. There are at least two reasons for using NSTT as a preliminary analytical step. First, NSTT automatically gives conditions for matching solutions at discontinuity points. Second, due to the automatic matching through the NSTT functions, the corresponding solutions appear to be in the closed form that is important feature when further manipulations with the solutions are required by problem formulations.

6.4.1

Piecewise-Constant Excitation

Let us consider the linear oscillator under periodic piecewise-constant excitation (6.50) x¨ + ω02 [1 + εe(ωt, γ)]x = 0 where ω0 , ω, γ and ε are constant parameters. We will seek periodic solutions with the period of excitation T = 4/ω in the form x = X (τ ) + Y (τ ) e (6.51) where τ = τ (ωt, γ) and e = e(ωt, γ). As follows from the form of equation (6.50), the acceleration x ¨ may have step-wise discontinuities due to the presence of the function e(ωt, γ), whereas the coordinate x (t) and the velocity x˙ (t) must be continuous. So neither velocity x˙ (t) nor acceleration x ¨ (t) can include Dirac δ-functions. Taking ﬁrst derivative of (6.51), gives ∂e(ωt, γ) ω (6.52) x˙ (t) = αY + (X + βY )e + Y ∂ (ωt) where the last term, that consists of the periodic sequence of δ-functions, must be excluded by imposing the boundary condition for Y -component Y |τ =±1 = 0

(6.53)

Under condition (6.53), the second derivative takes the form

x ¨ (t) = ω 2 [α(X + βY )] + ω 2 [βX + (α + β 2 )Y ]e ∂e (ωt, γ) +ω 2 (X + βY ) ∂ (ωt)

(6.54)

In this case, the singular term, which is underlined in (6.54), is eliminated by condition

158

6 NSTT for Linear and Piecewise-Linear Systems

(X + βY ) |τ =±1 = 0

(6.55)

Substituting (6.51) and (6.54) in the diﬀerential equation of motion (6.50) and taking into account the algebraic properties, brings the left-hand side of the equation to the algebraic form {· · ·} + {· · ·}e. Then, setting separately each of the two algebraic components to zero, gives the set of diﬀerential equations for X (τ ) and Y (τ ) in the following matrix form

α αβ β α + β2

X Y

+ r2

1 αε ε 1 + βε

X Y

=0

(6.56)

where r = ω0 /ω. Further, any particular solution of linear diﬀerential equations with constant coeﬃcients (6.56) can be represented in the exponential form X 1 =B exp (λτ ) (6.57) Y μ where B, μ and λ are constant parameters. Substituting (6.57) in (6.56), leads to a characteristic equation which two pairs of roots determined by the relationships 2 λ2 = − (1 − γ) ε − (1 − γ) r2 ≡ ±k 2 (6.58) 2 λ2 = (1 + γ) ε − (1 + γ) r2 ≡ ±l2 where signs of the notations ±k 2 and ±l2 depend on the parameters ε and γ. Let us consider the case of negative signs, when the following condition holds − (1 − γ) < ε < (1 + γ) (6.59) Due to condition (6.59), the stiﬀness coeﬃcient in equation (6.50) is always positive, whereas (6.58) gives λ = ±ki and λ = ±li. As a result, the general solution of equations (6.56) takes the form X = B1 sin kτ + B2 cos kτ + B3 sin lτ + B4 cos lτ Y = μ1 (B1 sin kτ + B2 cos kτ ) + μ2 (B3 sin lτ + B4 cos lτ )

(6.60)

where B1 ,...,B4 are arbitrary constants, and μ1 = −

1 αk 2 − r2 α βk 2 − εr2

and

μ2 = −

1 αl2 − r2 α βl2 − εr2

Substituting (6.60) in boundary conditions (6.53) and (6.55), gives the homogeneous set of four linear algebraic equations with respect to the arbitrary

6.4 Parametric Excitation

159

constants. Setting the corresponding determinant to zero, gives condition for non-zero solutions in the form [μ1 (1 + βμ2 ) l cos k sin l − μ2 (1 + βμ1 ) k cos l sin k] × [μ1 (1 + βμ2 ) l cos l sin k − μ2 (1 + βμ1 ) k cos k sin l] = 0

(6.61)

10

8

10Ε

6

4

2

0 0

2

4

6

8

10

r Fig. 6.7 Instability zones for piecewise constant parametric excitation when γ = 0.7

One the parameter plane, ε − r, equation (6.61) describes the family of curves separating stability and instability zones as shown in Fig. 6.7, where the instability zones are shadowed.

6.4.2

Parametric Impulsive Excitation

Let us consider the case of parametric impulsive excitation whose temporal shape is given by ﬁrst derivative of the basic function, e (ωt, γ),

∂e (ωt, γ) 2 x ¨ + ω0 1 + ε x=0 (6.62) ∂ (ωt) This case was considered in [138] based on the saw-tooth transformation of time. In particular, it was shown that the periodic solutions of the period T = 4/ω exists under the condition

160

6 NSTT for Linear and Piecewise-Linear Systems

2 2r2 1 − γ 2 sin2 2r p = cos 4r − cos 4γr 2

(6.63)

where r = ω0 /ω and p = εr2 .

1.0 0.8 0.6 p 0.4 0.2 0.0

0

2

4

6

8

10

r

Fig. 6.8 ‘Collapse’ of the instability zones at γ = 1/5: each ﬁfth zone is missing; here and below, only the upper half-plane is shown due to the symmetry.

1.0 0.8 0.6 p 0.4 0.2 0.0

0

2

4

6

8

10

r

Fig. 6.9 γ = 1/2: each second zone is missing.

The dependence of p on r for ﬁxed γ has the branched zone-like structure which is typical for diﬀerent cases of parametrically excited oscillators. Interestingly enough, diﬀerent subsequences of zones may disappear as the parameter γ varies. For instance, if γ = 1/5 then each ﬁfth zone is missing and, if γ = 1/2 then each second zone is missing; see Figs. 6.8 and 6.9, respectively. Such an eﬀect was discussed in [138].

6.4 Parametric Excitation

6.4.3

161

General Case of Periodic Parametric Excitation

Below, the problem formulation only is discussed for the case of periodic parametric loading with both regular and singular components. It is assumed that there are two discontinuities and singularities on each period located at the same points. The diﬀerential equation of motion is represented in the vector form

∂e x¨ + Q (τ ) + P (τ ) e + p x=0 (6.64) ∂ϕ where x(t) ∈ Rn is the coordinates vector-column,τ = τ (ϕ, γ), e = e (ϕ, γ), ϕ = ωt is the phase variable, p is a constant n × n matrix, and Q (τ (ϕ, γ)) and P (τ (ϕ, γ)) are periodic matrixes of the period T = 4 with respect to the phase ϕ. In equation (6.64), the ﬁrst two terms of the coeﬃcient can represent any periodic function q (ϕ) with step-wise discontinuities on Λ = {t : τ (ϕ, γ) = ±1}. In case the original function q (ϕ) is continuous, one has P = 0 on Λ. Let us represent periodic solutions of the period T = 4 in the form (6.51). Substituting (6.51) in equations (6.64), taking into account the equality e2 = α + βe, the necessary condition of continuity of the vector function x (t), (6.53), and using (6.52) and (6.54) gives equations ω 2 (αX + αβY ) + QX + αP Y = 0 ω 2 α + β 2 Y + βX + P X + QY + βP Y = 0 and the boundary condition 2 ω (X + βY ) + pX |τ =±1 = 0

(6.65)

(6.66)

where, in the case of ﬁxed sign of impulses, the matrix p should be provided with the factor sgn(τ ). Together with (6.53), relations (6.65) and (6.66) represent a boundaryvalue problem for determining the vector functions X and Y and the corresponding conditions for existence of periodic solutions. Note that substitution (6.51) in equation (6.64) generates the speciﬁc term e∂e/∂ϕ. Let us show that, within the theory of distributions, this terms can be interpreted as follows 1 ∂e ∂e = β (6.67) e ∂ϕ 2 ∂ϕ First, note that, at this point, the relationship (6.67) is a result of formal diﬀerentiation of both sides of the relation e2 = α + βe with respect to the phase ϕ. To justify (6.67), let us assume that ω = 1 so that ϕ ≡ t and consider expression (6.53) locally, near the point t = 1 − γ, which is a typical point of the entire set of discontinuity points Λ = {t : τ (t) = ±1}.

162

6 NSTT for Linear and Piecewise-Linear Systems

Generally speaking, the ‘product’ f (t)δ(t) requires the function f (t) to be at least continuous at t = 0. However, it is possible to provide the left-hand side of (6.67) with a certain meaning due to the fact that both terms of the product are generated by the same sequence of smooth functions. In order to illustrate the above remark and prove equality (6.67), let us consider a family of smooth functions {δε (t)} such that ε δε (t) dt = 1

(6.68)

−ε

for all positive ε, and δε (t) = 0 outside the interval −ε < t < ε. Therefore, in terms of weak limits, δε (t) → δ (t) as ε → 0. Now, sequences of smooth functions approximating e and ∂e/∂t in the neighborhood of point t = 1 − γ can be chosen as, respectively, eε =

β 1 − θε (t − 1 + γ) 1−γ γ

where θε (t) =

"t −∞

and

β ∂eε = − δε (t − 1 + γ) ∂t γ

(6.69)

δε (ξ) dξ and −1 + γ < t < 3 + γ.

Based on the above deﬁnitions for eε and ∂eε /∂t, one has eε → e and ∂eε /∂t → ∂e/∂t as ε → 0 in the interval −1 + γ < t < 3 + γ. Substituting (6.69) in equality (6.67) instead of e and ∂e/∂ϕ, reduces the problem to the proof of identity θε δ ε =

1 δε 2

(6.70)

as ε → 0. For simplicity reason, let us move the origin to the point t = 1 − γ and show that the left-hand side of (6.70) gives δ (t) /2 as ε → 0 in the sense of weak limit. First, the area bounded by θε δε is ε ε 1 dθε 1 dt = θε2 |ε−ε = θε δε dt = θε dt 2 2 −ε −ε Then, let φ (t) belongs to the class of continuous testing functions, which is usually considered in the theory of distributions. By deﬁnition, in some ε-neighborhood of the point t = 0, one has | φ (t) − φ (0) |< 2η, where η is as small as needed whenever ε is suﬃciently small. Therefore, ε ε 1 | θε (t) δε (t) φ (t) dt − φ (0) |≤ θε (t) δε (t) | φ (t) − φ (0) | dt ≤ η 2 −ε −ε

6.5 Input-Output Systems

In other words,

163

ε

−ε

θε (t) δε (t) φ (t) dt →

1 φ (0) 2

as ε → 0. This completes the proof.

6.5

Input-Output Systems

The input-output form of dynamical systems may be convenient for diﬀerent reasons, for instance, when dealing with control problems. In many linear cases, input-output systems are represented in the form of a single high order equation an

dn y dy dm u du + ... + a1 + a0 y = bm m + ... + b1 + b0 u n dt dt dt dt

(6.71)

where u = u(t) and y = y(t) are input and output, respectively, and an , ... , a1 , a0 , bm , ... ,b1 , b0 are constant coeﬃcients. For illustration purposes, a two-degrees-of-freedom model as shown in Fig. 6.10 is considered, although the general case (6.71) can be handled in the same way.

Fig. 6.10 Two mass-spring model.

Eliminating x2 (t) from the system, gives a single higher-order equation with respect to the another coordinate, x1 (t), in the form k1 k2 d4 x1 d3 x1 m1 d2 x1 c1 dx1 + + c1 3 + (k1 + k2 + k2 ) 2 + k2 x1 4 dt dt m2 dt m2 dt m2 d2 F1 k2 = + F1 (6.72) 2 dt m2 m1

System (6.72) is a particular case of (6.71), where n = 4 and m = 2. Let us consider the step-wise discontinuous periodic function F1 (t) = u(t) = e(ωt) and represent equation (6.72) in the form

164

6 NSTT for Linear and Piecewise-Linear Systems

a4

d4 y dy + ... + a1 + a0 y = b2 ω 2 e + b1 ωe + b0 e 4 dt dt

(6.73)

where ≡ d/d(ωt), and all the coeﬃcients and variables are identiﬁed by comparing (6.72) to (6.73). The right-hand side of equation (6.73) contains discontinuous and singular functions, therefore equation (6.73) must be treated in terms of distributions. Nevertheless, let us show that, on the manifold of periodic solutions, equation (6.73) is equivalent to some classic boundary-value problem. Let us represent the output in the form y(t) = X(τ ) + Y (τ )e

(6.74)

where τ = τ (ωt) and e = e(ωt). When diﬀerentiating expression (6.74) step-by-step one should eliminate the singular term e in the ﬁrst two derivatives by sequentially setting boundary conditions as follows dy = (Y + X e)ω, dt d2 y = (X + Y e)ω 2 , dt2

Y |τ =±1 = 0

(6.75)

X |τ =±1 = 0

However, it is dictated by the form of the input in (6.73), that the singular terms e and e must be preserved on the next two steps given by d3 y = (Y + X e + Y e )ω 3 dt3 d4 y = (X (4) + Y (4) e + X e + Y e )ω 4 dt4

(6.76)

The fourth-order derivative in (6.76) takes into account the equality ee = 0, which easily follows from (6.53) in the symmetric case β = 0. Substituting (6.75) and (6.76) in (6.73), and considering {1, e, e, e } as a linearly independent basis, gives equations a4 ω 4 X IV + a3 ω 3 Y + a2 ω 2 X + a1 ωY + a0 X = 0 a4 ω 4 Y IV + a3 ω 3 X + a2 ω 2 Y + a1 ωX + a0 Y = b0

(6.77)

under the boundary conditions at τ = ±1: Y = 0, b2 ω 2 Y = , a4

X = 0 ω 3 X

1 = a4

a3 b1 − b2 a4

(6.78)

In contrast to equation (6.73), the boundary value problem (6.77) and (6.78) does not include discontinuous terms any more.

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

165

Although the number of equations in (6.77) is doubled as compared to (6.73), such a complication is rather formal due to the symmetry of the equations. Indeed, introducing the new variables, U = X +Y and V = X −Y , decouples system (6.77) in such a way that the corresponding roots of the characteristic equations diﬀer just by signs. (Besides, this fact reveals the possibility of using the idempotent basis for decoupling the resultant set of equations as discussed in Chapter 4 and will be discussed later in this chapter.) In addition, the type of the symmetry suggests that X(τ ) and Y (τ ) are odd and even functions, respectively. This enables one of reducing the general form of solution to a family of solutions with four arbitrary constants

2 α α βj βj j j τ sin τ + Bj sinh τ cos τ (6.79) Aj cosh X= ω ω ω ω j=1

2 α α βj βj b0 j j τ sin τ + Bj cosh τ cos τ + Y = Aj sinh ω ω ω ω a 0 j=1

where αj ± βj i are complex conjugate roots of the characteristic equation a4 p4 + ... + a1 p + a0 = 0

(6.80)

The assumption that both of the roots are complex reﬂects the physical meaning of the example, however other cases would lead to even less complicated expressions. Finally, substituting (6.79) in (6.78) gives a linear algebraic set of four independent equations with respect to four constants: A1 , A2 , B1 and B2 . Although the corresponding analytical solution is easy to obtain by using the R standard Mathematica commands, the result is somewhat complicated for reproduction. Practically, it may be reasonable to determine the constants by setting the system parameters to their numerical values moreover that only numerical solution are often possible for characteristic equations.

6.6

Piecewise-Linear Oscillators with Asymmetric Characteristics

Piecewise-linear oscillators are often considered as ﬁnite degrees-of-freedom models of cracked elastic structures [32],[2],[192], but may occur also due to speciﬁc design solutions. In many cases, the corresponded periodic solutions can be combined of diﬀerent pieces of linear solutions valid for two diﬀerent subspaces of the conﬁguration space [33], [75], [192]. In this section, it will be shown that the nonsmooth transformation of time results in a closed form analytical solution matching both pieces of the solution automatically by means of elementary functions.

166

6.6.1

6 NSTT for Linear and Piecewise-Linear Systems

Amplitude-Phase Equations

Let us consider a piece-wise linear oscillator of the form m¨ q + k[1 − εH(q)]q = 0

(6.81)

where H(q) is Heaviside unit-step function, m and k are mass and stiﬀness parameters, respectively, and |ε| 1. Therefore, k− = k and k+ = k(1 − ε) are elastic stiﬀness of the oscillator for q < 0 and q > 0, respectively. The exact general solution of oscillator (6.81) can be obtained by satisfying the continuity conditions for q and q˙ at the matching point q = 0, where the characteristic has a break. Such approaches are often facing quite challenging algebraic problems, however, as the number of degrees of freedom increases or external forces are involved. This is mainly due to the fact that times of crossing the point q = 0 are a priory unknown. In this section, it will be shown that, applying a combination of asymptotic expansions with respect to ε and nonsmooth temporal transformations, gives a unit-form solution for oscillator (6.81) with a possibility of generalization on the normal mode motions of multiple degrees-of-freedom systems. In particular, the nonsmooth temporal transformation: 1) provides an automatic matching the motions from diﬀerent subspaces of constant stiﬀness, and 2) justiﬁes quasi-linear asymptotic solutions for the speciﬁc nonsmooth case of piece-wise linear characteristics. Let us clarify the above two remarks. Introducing the notation ω 2 = k/m, brings equation (6.81) to the standard form of a weakly non-linear oscillator q¨ + ω 2 q = εω 2 H(q)q

(6.82)

The non-linear perturbation on the right-hand side of oscillator (6.82) is a continuous but non-smooth function of the coordinate q. Since the major algorithms of quasi-linear theory assume smoothness of non-linear perturbations, then such algorithms are not applicable in this case unless appropriate modiﬁcations and extensions have been made. Even though deriving ﬁrst-order asymptotic solutions usually require no diﬀerentiation of characteristics, dealing with two pieces of the solution may complicate any further stages. Let us show that combining quasi-linear methods of asymptotic integration, such as Krylov-Bogolyubov averaging, with nonsmooth temporal transformations results in a closed form analytical solution for piece-wise linear oscillator (6.81). Note that oscillator (6.81) plays an illustrative role for the approach developed below. Then a more complicated case will be considered. At this stage, let us introduce the amplitude-phase coordinates {A(t), ϕ(t)} on the phase plane of oscillator (6.81) through relationships

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

q = A cos ϕ q˙ = −ωA sin ϕ

167

(6.83)

The following compatibility condition is imposed on transformation (6.83) A˙ cos ϕ − A sin ϕϕ˙ = −ωA sin ϕ

(6.84)

Substituting (6.83) in (6.82) and taking into account (6.84), gives 1 A˙ = − εωAH(A cos ϕ) sin 2ϕ 2 ϕ˙ = ω − εωH(A cos ϕ) cos2 ϕ

(6.85)

The right-hand sides of equations (6.85) are 2π-periodic with respect to the phase variable, ϕ. Therefore, nonsmooth transformation of the phase variable applies through the couple of functions τ = τ (2ϕ/π)

and

e = e(2ϕ/π)

(6.86)

Assuming that A ≥ 0 and taking into account the obvious identities, sin ϕ = sin(πτ /2) cos ϕ = cos(πτ /2)e H(A cos ϕ) = (1 + e)/2 e2 = 1

(6.87)

brings (6.85) to the form 1 A˙ = − εω(1 + e)A sin πτ 4 1 πτ ϕ˙ = ω − εω(1 + e) cos2 2 2

(6.88) (6.89)

Note that the right-hand sides of (6.88) and (6.89) are nonsmooth but continuous with respect to the phase ϕ since the step-wise discontinuities of the rectangular cosine e(2ϕ/π) are suppressed by the factors sin πτ and cos2 (πτ /2), respectively.

6.6.2

Amplitude Solution

Let us show that equation (6.88) has an exact 2π-periodic solution with respect to the phase variable, ϕ. According to the idea of NSTT, any periodic solution can be represented in the form A = X(τ ) + Y (τ )e (6.90) where τ and e are deﬁned by (6.86).

168

6 NSTT for Linear and Piecewise-Linear Systems

Substituting (6.90) in (6.88) and taking into account (6.89), gives boundaryvalue problem (X − Y ) = 0 επ (X + Y ) sin πτ =− X +Y 4 1 − ε cos2

πτ 2

Y |τ =±1 = 0

(6.91) (6.92)

where ≡ d/dτ . Solution of the boundary value problem, (6.91) and (6.92), is obtained by elementary integration. Then representation (6.90) gives A(ϕ) = α[1 + ζ(τ )] − α[1 − ζ(τ )]e πτ −1/2 ) ζ(τ ) = (1 − ε cos2 2

(6.93)

where τ = τ (2ϕ/π), e = e(2ϕ/π), and α is an arbitrary positive constant. Note that solution (6.93) exactly captures the amplitude in both subspaces q < 0 and q > 0. However, the temporal mode shape and the period essentially depend on the phase variable ϕ described by equation (6.89). Generally speaking, the phase equation (6.89) admits exact integration, but the result would appear to have implicit form. Alternatively, it is shown below that solution for the phase variable can be approximated by asymptotic series in the explicit form 1 ϕ = φ − ε[πτ + (1 + e) sin πτ ] 8 1 2 − ε {4(2 − cos πτ )(πτ + sin πτ ) 128 −[4πτ (1 + cos πτ ) − 8 sin πτ + sin 2πτ ]e} + O(ε3 )

(6.94)

where τ = τ (2φ/π), e = e(2φ/π), and 3 1 φ = ω[1 − ε − ε2 + O(ε3 )]t 4 32

(6.95)

Note that the functions τ and e in (6.93) and (6.94) depend on the diﬀerent arguments.

6.6.3

Phase Solution

In this subsection, a second-order asymptotic procedure for phase equations with non-smooth periodic perturbations is introduced. If applied to equation (6.89), the developed algorithm gives solution (6.94).

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

169

Let us consider some phase equation of the general form ϕ˙ = ω[1 + εf (ϕ)]

(6.96)

where f (ϕ) is a 2π-periodic, nonsmooth or even step-wise discontinuous function, and ε is a small parameter, |ε| 1. Using the basic NSTT identity for f (ϕ), brings equation (6.96) to the form ϕ˙ = ω + εω{G[τ (2ϕ/π)] + M [τ (2ϕ/π)]e}

(6.97)

where the functions G(τ ) and M (τ ) are expressed through f (ϕ). Note that the class of smoothness of the periodic perturbation in equation (6.97) depends on the behavior of functions G(τ ) and M (τ ) and their derivatives at the boundaries τ = ±1. If, for instance, M (±1) = 0 then the perturbation is step-wise discontinuous whenever τ (2ϕ/π) = ±1. Let us introduce the asymptotic procedure for equation (6.97). Note that, in case ε = 0, the right-hand side of equation (6.97) is constant. So, following the idea of asymptotic integration, let us ﬁnd phase transformation ϕ = φ + εF1 (φ) + ε2 F2 (φ) + ...

(6.98)

where functions Fi (φ) are such that the new phase variable also has a constant temporal rate even though ε = 0. In other words, transformation (6.98) should bring equation (6.97) to the form (6.99) φ˙ = ω(1 + εγ1 + ε2 γ2 + ...) where γi are constant coeﬃcients to be determined together with Fi (φ) during the asymptotic procedure. Note that the procedure, which is described below, has several speciﬁc features due to the presence of nonsmooth periodic functions. In particular, high-order approximations require a non-conventional interpretation for power series expansions; see the next subsection for the related remarks. Other modiﬁcations occur already in the leading order approximation. Substituting (6.98) into equation (6.97), then enforcing equation (6.99) and collecting the terms of order ε, gives F1 (φ) = G(τ ) + eM (τ ) − γ1

(6.100)

where the triangular and rectangular waves depend now on the new phase variable φ as τ = τ (2φ/π) and e = e(2φ/π), respectively. According to the conventional averaging procedure, the constant γ1 is selected to achieve zero mean on the right-hand side of equation (6.100) and thus provide periodicity of solution, F1 (φ). In the algorithm below, the periodicity is due to the form of representation for periodic solutions, whereas the operator of averaging occurs automatically from the corresponding conditions of smoothness that is boundary conditions for the solution components.

170

6 NSTT for Linear and Piecewise-Linear Systems

So we seek solution of equation (6.100) in the form F1 (φ) = U1 (τ ) + eV1 (τ )

(6.101)

Substituting (6.101) in (6.100) and following the NSTT procedure, gives the boundary-value problem π M (τ ) 2 π V1 (τ ) = [G(τ ) − γ1 ] 2 V1 (±1) = 0 U1 (τ ) =

(6.102)

Note that there are two conditions on the function V1 (τ ) described by the ﬁrst-order diﬀerential equation in (6.102). However, there is a choice for γ1 , which is to satisfy one of the two conditions. As a result, solution of boundaryvalue problem (6.102) is obtained by integration in the form π τ U1 (τ ) = M (z)dz 2 0 π τ [G(z) − γ1 ]dz (6.103) V1 (τ ) = 2 −1 1 1 G(τ )dτ γ1 = 2 −1 Further, collecting the terms of order ε2 , gives F2 (φ) = G2 (τ ) + eM2 (τ ) + P2 (τ )e − γ2

(6.104)

where 2 [U1 (τ )G (τ ) + V1 (τ )M (τ )] − M (τ )γ1 π 2 G2 (τ ) = U1 (τ )M (τ ) − G(τ )γ1 + γ12 π 2 P2 (τ ) = U1 (τ )M (τ ) π e ≡ de(2φ/π)/d(2φ/π)

M2 (τ ) =

(6.105)

In contrast to ﬁrst-order equation (6.100), equation (6.104) includes the singular term P2 (τ )e produced by the power series expansion of the perturbation in equation (6.97). If the perturbation is smooth then P2 (±1) = 0 and such singular term disappear; see the example below for illustration. Nevertheless, the second-order approximation remains valid even in discontinuous case, when P2 (±1) = 0.

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

171

So let us represent solution of equation (6.104) in the form F2 (φ) = U2 (τ ) + eV2 (τ )

(6.106)

Then, substituting (6.106) in (6.104), gives boundary-value problem π M2 (τ ) 2 π V2 (τ ) = [G2 (τ ) − γ2 ] 2 π V2 (±1) = P2 (±1) 2 U2 (τ ) =

(6.107)

In contrast to (6.102), boundary-value problem (6.107) has, generally speaking, non-homogeneous boundary conditions for V2 . These conditions compensate the singular term e from diﬀerential equation (6.104). As a result equations (6.107) are free of any singularities and admit solution analogously to ﬁrst-order equations (6.102), π τ M2 (z)dz U2 (τ ) = 2 0 π τ π [G2 (z) − γ2 ]dz + P2 (−1) (6.108) V2 (τ ) = 2 −1 2 1 1 1 G2 (τ )dτ + [P2 (−1) − P2 (1)] γ2 = 2 −1 2 Now, we return to the illustrating model. In particular case (6.89), one has 1 πτ G(τ ) ≡ M (τ ) ≡ − cos2 2 2

(6.109)

and G(±1) = M (±1) = 0 G (±1) = M (±1) = 0 G (±1) = M (±1) = −π 2 /4

(6.110)

where ≡ d/dτ . First two lines of conditions (6.110) provide continuity for the right hand side of (6.97) and its ﬁrst derivative at those ϕ where τ (2ϕ/π) = ±1. As follows from (6.105), for this class of smoothness one has P2 (±1) = 0 and thus no singular terms occur in the ﬁrst two steps of asymptotic procedure. Finally, taking into account (6.109) and (6.110) and conducting integration in (6.103) and (6.108), brings solution (6.98) to the form (6.94) and (6.95).

172

6.6.4

6 NSTT for Linear and Piecewise-Linear Systems

Remarks on Generalized Taylor Expansions

Nonsmoothness of the triangular sine is similar to that function |t| has at zero. So let us consider its formal power series |t + ε| = |t| + |t| ε +

1 2 |t| ε + ... 2!

(6.111)

where ε > 0 and −∞ < t < ∞, and prime indicates Schwartz derivative. It is clear that equality (6.111) has no regular point-wise meaning. For instance, equality (6.111) is obviously not true on the interval −ε < t < 0. In addition, the right-hand side of (6.111) is uncertain at t = 0, whereas the left-hand side gives ε. Nevertheless, let us show that equality (6.111) admits a generalized interpretation and holds in terms of distributions. Let ψ(t) be a test function in terms of the distribution theory, more precisely, ψ(t) is inﬁnitely diﬀerentiable with compact support that is identically zero outside of some bounded interval. Integrating by parts and then shifting the variable of integration, gives ∞

1 |t| + |t| ε + |t| ε2 + ... ψ(t)dt 2! −∞

∞ 1 = |t| ψ(t) − ψ (t)ε + ψ (t)ε2 − ... dt (6.112) 2! −∞ ∞ ∞ |t|ψ(t − ε)dt = |t + ε|ψ(t)dt = −∞

−∞

Therefore, equality (6.111) holds in the integral sense of distributions.

2

1 q

0

-1

-2 0

10

20 t

30

40

Fig. 6.11 Second-order asymptotic and numerical solutions shown by solid and dashed lines, respectively

6.7 Multiple Degrees-of-Freedom Case

173

Fig. 6.11 compares analytical solution (6.83), (6.93) and (6.94) shown by the solid line and numerical solution shown by the dashed line. As expected, the amplitude show the perfect match, whereas some phase shift develops after several cycles.

6.7

Multiple Degrees-of-Freedom Case

Let us consider a multiple degrees-of-freedom piecewise-linear system of the form Mx ¨ + Kx = εH(Sx)Bx (6.113) where x(t) ∈ Rn is a vector-function of the system coordinates, M is a mass matrix, H denotes the Heaviside unit-step function, S is a normal vector to the plane splitting the conﬁguration space into two parts with diﬀerent elastic properties, so that the stiﬀness matrix is K when Sx < 0 and K − εB when Sx > 0. It is assumed that the stiﬀness jump is small, |ε| 1. The number of possible iterations of the classic perturbation tools usually depends on a class of smoothness of the perturbation. The perturbation term on the right-hand side of (6.113) is continuous but nonsmooth. Therefore, only ﬁrst-order asymptotic solution can be obtained within the classic theory of diﬀerential equations. Moreover, the piecewise character of the perturbation complicates the form of the solution due to the necessity of matching the diﬀerent pieces of the solution.

Fig. 6.12 Two degrees-of-freedom piecewise-linear system as a model of a rod with a small crack.

However, we show that the idea of nonsmooth time transformation gives a unit-form solution by automatically matching the pieces of solution in two diﬀerent conﬁguration subspaces with diﬀerent stiﬀness properties. Let seek a 2π-periodic normal mode solution of (6.113) with respect to the phase ϕ in the form x(ϕ) = Aj cos ϕ + εx(1) (ϕ) + O(ε2 ) ) ϕ = ωj 1 + εγ (1) + O(ε2 )t

(6.114)

174

6 NSTT for Linear and Piecewise-Linear Systems

where ωj and Aj are arbitrary eigen frequency and eigen vector (normal mode) of the linearized system: (−ωj2 M + K)Aj = 0;

j = 1,...,n

(6.115)

Substituting (6.114) in (6.113), taking into account (6.87) and (6.115) in the ﬁrst order of ε, gives

1 d2 x(1) 1 πτ (1) (1) ωj2 M BA BA (6.116) + Kx = + ( + γ KA )e cos j j j 2 dϕ 2 2 2 where τ = τ (2ϕ/π), e = e(2ϕ/π), and the relationship (1 + εγ (1) )−1 = 1 − εγ (1) + O(ε2 ) has been used. Since the function x(1) (ϕ) is sought to be 2π-periodic with respect to ϕ, we represent it in the form x(1) = X(τ ) + Y (τ )e

(6.117)

This gives the boundary-value problem

2 2ωj 1 πτ M X + KX = BAj cos , X |τ =±1 = 0 π 2 2

2 2ωj 1 πτ BAj + γ (1) KAj cos M Y + KY = π 2 2 Y |τ =±1 = 0

(6.118) (6.119)

Representing the corresponding solution in terms of the normal mode coordinates n n X= Ai Xi (τ ), Y = Ai Yi (τ ) (6.120) i=1

i=1

and taking into account M -orthogonality of the set of eigen vectors, gives

2 2ωj πτ , Xi |τ =±1 = 0 Xi + ωi2 Xi = βij cos (6.121) π 2

2 2ωj πτ , Yi |τ =±1 = 0 (6.122) Yi + ωi2 Yi = (βij + γ (1) κij ) cos π 2

where βij =

1 Ai BAj , 2 Ai M Ai

κij =

Ai KAj Ai M Ai

(6.123)

are dimensionless coeﬃcients. Note that despite of the similar representation for solution (6.114), there is a noticeable diﬀerence between the classic Poincare-Lindshtedt method and current procedure due to (6.117). Namely, according to the

6.7 Multiple Degrees-of-Freedom Case

175

Poincare-Lindshtedt method, the frequency correction term γ (1) is to kill the so called secular terms in the asymptotic expansions. In our case, the secular terms appear to be periodic due to the ‘built in’ periodicity of the new temporal argument. However, periodicity of solutions is provided by the existence of solutions of the boundary-value problems, such as (6.121) and (6.122). Due to the linearity, the existence of solutions allows the direct veriﬁcation. So if i = j then both problems (6.121) and (6.122) are solved in the standard way with no presence of γ (1) because κij = 0. The corresponding solution is given by

ωj βij πωi τ πωi πτ − cos csc Xi = 2 cos (6.124) ωi − ωj2 2 ωi 2ωj 2ωj Yi =

ωi2

βij πτ cos 2 − ωj 2

(6.125)

In the particular case i = j, problem (6.121) still has a solution, but problem (6.122) generally speaking does not. Fortunately, in this case, we have κjj = 0 and thus the problem is set to have the trivial solution by condition γ (1) = −

βjj κjj

(6.126)

Therefore, πβjj Xj = 4ωj2

2 πτ πτ + cos τ sin 2 π 2

Yj = 0

(6.127) (6.128)

So expressions (6.117), (6.120), and (6.124) through (6.128) completely determine the ﬁrst order approximation x(1) (ϕ). Let us consider the example of mass-spring model m1 x ¨1 + (k1 + k2 )x1 − k2 x2 = εk1 H(x1 )x1 ¨2 − k2 x1 + (k2 + k3 )x2 = 0 m2 x

(6.129)

Equations (6.111) can be represented in the form (6.113), where

m1 0 k + k2 −k2 k 0 M = , K= 1 , B= 1 −k2 k2 + k3 0 m2 0 0 x1 , S= 10 x= x2 In this case, the ﬁrst-order asymptotic solution for the in-phase (j = 1) and out-of-phase (j = 2) takes the form, respectively,

176

6 NSTT for Linear and Piecewise-Linear Systems

επ 2 πτ πτ πτ + cos + τ sin + O(ε2 ) 2 16 π 2 2 πτ εk1 πτ πτ e cos − + cos (6.130) x2 = e cos 2 8k2 2 2

−1/2 k2 k2 πτ k2 π / sin + O(ε2 ) − 1+2 cos 1+2 1+2 k1 k1 2 k1 2

x1 = e cos

ϕ= and

k1 m

ε 1 − + O(ε2 )t 4

εk1 πτ k2 πτ πτ x1 = −e cos + + cos − 1+2 e cos 2 8k2 2 2 k1 πτ k2 k2 π / 1+2 / 1+2 × cos / sin + O(ε2 ) (6.131) 2 k1 2 k1

εk1 π 2 πτ πτ πτ + cos + τ sin x2 = e cos + O(ε2 ) 2 16(k1 + 2k2 ) π 2 2 * k1 + 2k2 εk1 + O(ε2 )t 1− ϕ= m 4(k1 + 2k2 )

where it is assumed that m1 = m2 = m. Solutions (6.130) and (6.100) show that a bi-linearity may have quite diﬀerent eﬀect on diﬀerent modes. In particular, solution (6.130) reveals the possibility of internal resonances, when

πω2 k2 ω2 sin = 1+2 (6.132) = 0, 2ω1 ω1 k1 If, for instance, the system is close to the frequency ratio ω2 /ω1 = 2 then the in phase mode may be aﬀected signiﬁcantly by a crack even under very small magnitudes of the parameter ε. In contrary, solution (6.100) has the denominator sin[(π/2)ω1 /ω2 ], which is never close to zero because 0 < ω1 /ω2 < 1. Therefore, in current asymptotic approximation, the inﬂuence of crack on the out-of phase mode is always of order ε provided that k2 /k1 = O(1). The inﬂuence of the bilinear stiﬀness on inphase mode trajectories in the closed to internal resonance case is seen from Fig. 6.13, where both analytical and numerical solutions are shown for comparison reasons. The frequency ratio ω2 /ω1 = 2.0025 is achieved by conditioning the spring stiﬀness parameters as follows k2 = (3/2)k1 + 0.005.

6.8 The Amplitude-Phase Problem in the Idempotent Basis

177

1.0

0.5 Ε 0.01 x2

Ε0

0.0

0.5

1.0 1.0

0.5

0.0

0.5

1.0

x1 Fig. 6.13 The inﬂuence of a small “crack” ε = 0.01 on the in-phase mode trajectory near the frequency ratio ω2 /ω1 = 2; the dashed line shows the numerical solution, and the thin solid line corresponds to the perfect (linear) case ε = 0.

6.8

The Amplitude-Phase Problem in the Idempotent Basis

Recall that the idempotent basis is given by e+ = (1+e)/2 and e− = (1−e)/2 so that e2+ = e+ , e2− = e− and e+ e− = 0. Equations (6.88) and (6.89) therefore take the form 1 A˙ = − εωe+ A sin πτ 2 πτ ϕ˙ = ω − εωe+ cos2 2

(6.133) (6.134)

Let us represent the amplitude as a function of ϕ in the form A(ϕ) = X+ (τ )e+ + X− (τ )e−

(6.135)

where e+ = e+ (2ϕ/π), e− = e− (2ϕ/π) and τ = τ (2ϕ/π). Substituting (6.135) in (6.133) and taking into account (6.134), gives 1 πτ 2 (X+ e+ − X− ) = − εωe+ (X+ e+ + X− e− ) sin πτ e− )(ω − εωe+ cos2 π 2 2

178

6 NSTT for Linear and Piecewise-Linear Systems

or (1 − ε cos2

πτ π )X+ = − εX+ sin πτ 2 4 =0 X−

(6.136)

under the boundary condition (X+ − X− )|τ =±1 = 0

(6.137)

The boundary-value problem (6.136) and (6.137) admits exact solution so that (6.135) gives ﬁnally A(ϕ) = α[(1 − ε cos2

πτ −1/2 ) e+ + e− ] 2

where α is an arbitrary positive constant.

(6.138)

Chapter 7

Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

Abstract. In this chapter, two variable expansions are introduced, where the fast temporal scale is represented by the triangular wave. In contrast to the conventional two variable procedure, the diﬀerential equations for slow dynamics emerge from the boundary conditions eliminating discontinuities rather than resonance terms. For illustrating purposes, an impulsively loaded single degree-of-freedom model with qubic damping and no elastic force is considered. Further, the method is applied to the Duﬃng’s oscillator under the periodic impulsive excitation whose principal frequency is close to the linear resonance frequency. Note that the illustrating models are weakly nonlinear, nevertheless the triangular wave temporal argument adequately captures speciﬁcs of the impulsive loading and, as a result, provides closed form asymptotic solutions. Following the conventional quasi-harmonic approaches, in such cases, would face generalized Fourier expansions for the external forcing function with no certain leading term.

7.1

Nonsmooth Two Variables Method

Solutions of Cauchy problems under arbitrary initial conditions are, generally speaking, non-periodic. However, in case of amplitude or frequency modulated oscillating motions, the idea of averaging can be involved in order to obtain the corresponding solutions. A proper formalization can be developed by introducing slow temporal variables in addition to the fast oscillating time τ . Consider, for instance, ﬁrst order impulsively loaded system v˙ + εv 3 = p

de (t) dt

(7.1)

under the initial condition v |t=0 = v 0

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 179–193, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(7.2)

180

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

where ε is a small enough positive parameter, and p is a constant parameter characterizing the strength of pulses. The Cauchy problem (7.1) and (7.2) describes one-dimensional motions of a material point under the periodic series of impulses and non-linear damping forces. Note that any standard formulation of the idea of averaging is diﬃcult to apply directly to equation (7.1) since its right hand side is neither smooth nor even continues function of time. Analyzing the both sides of equation (7.1), shows that solutions should have step-wise discontinuities at the points Λ = {t : τ (t) = ±1}. Let us represent such solutions in the form (7.3) v = X τ, t0 + Y τ, t0 e where τ = τ (t), e = e (t) and t0 = εt. Therefore, the solution is assumed to depend upon the two time scales, such as the fast oscillating time τ and the slow time t0 . Substituting now (7.3) into (7.1), gives

∂X ∂Y 3 2 +ε + X + 3XY ∂τ ∂t0

∂X ∂Y 2 3 + +ε + 3Y X + Y e + (Y − p)e˙ = 0 ∂τ ∂t0 This ﬁnally leads to the set of partial diﬀerential equations

∂Y ∂X 2 3 = −ε + 3Y X + Y ∂τ ∂t0

∂X ∂Y 3 2 = −ε + X + 3XY ∂τ ∂t0

(7.4)

under boundary conditions Y |τ =±1 = p

(7.5)

Following the standard procedure of the two-variable expansions, we represent solution of the boundary value problem (7.4) and (7.5) is in the form of asymptotic series X=

∞

εi Xi τ, t0 ,

i=0

Y =

∞

εi Yi τ, t0

(7.6)

i=0

Substituting (7.6) into (7.4) and (7.5) and matching coeﬃcients of the same powers of ε, gives a sequence of simpliﬁed boundary value problems. In particular, zero order problem is ∂X0 = 0, ∂τ

∂Y0 =0 ∂τ

7.1 Nonsmooth Two Variables Method

181

Y0 |τ =±1 = p Both equations and the boundary condition are satisﬁed by solution X0 = A0 t0 , Y0 = p

(7.7)

where A0 is an arbitrary function of the slow time. Taking into account (7.7), gives the following ﬁrst order problem ∂X1 = − 3pA20 + p3 ∂τ

dA0 ∂Y1 3 2 =− + A0 + 3A0 p ∂τ dt0 Y1 | τ =±1 = 0 Integrating equations (7.8), gives general solution X1 = − 3pA20 + p3 τ + A1 t0

dA0 3 2 Y1 = − + A0 + 3A0 p τ + C1 t0 dt0 The boundary condition for Y1 in (7.8) dictates C1 t0 ≡ 0 and

(7.8)

(7.9)

dA0 + A30 + 3A0 p2 = 0 (7.10) dt0 Therefore Y1 ≡ 0, whereas function A1 t0 in the expression for X1 remains unknown. Equation (7.10) admits exact solution in the following explicit form

2 0 −1/2 1 A0 = − 2 + const · exp 6p t 3p

Finally, the leading order asymptotic solution of the Caushy problem is written as

2 −1/2 0 −2 1 1 exp 6p + v − p εt + pe (t) + O (ε) v(t) = ± − 2 + 3p 3p2 (7.11) where plus or minus sign are taken if v 0 − p > 0 or v 0 − p < 0, respectively. Substituting asymptotic solution (7.11) in equation (7.1), leads to the error εR(εt)e (t), where R(εt) is a function of slow time scale. Although the error is of the same order of magnitude as the perturbation, it relates to the singular component of the equation with a scaling factor of order one. In other words, for such kind of systems, regular and singular error components

182

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

0.8 0.6 v 0.4 0.2 0.0 0

10

20 t

30

40

Fig. 7.1 Velocity of the nonlinearly damped material point under the pulsed periodic excitation.

should be considered separately. Note that the ﬁrst term in expression (7.11) is a transient response of the system that vanishes as time increases; see Fig. 7.1 for illustration. In conclusion, it is seen from solution (7.11) that the amplitude of impulses may have a strong eﬀect on the time scale of transient response of the system.

7.2

Resonances in the Duﬃng’s Oscillator under Impulsive Loading

Let us consider the Duﬃng’s oscillator under the periodic impulsive excitation

de (ωt) 2 3 x ¨ + ω0 x = −ε x − p (7.12) d (ωt) Introducing the detuning parameter σ into the diﬀerential equation (7.12) as ω02 = gives

nπ 2 ω + εσ 2

nπ 2 de (ωt) 3 ω x = −ε σx + x − p x¨ + 2 d (ωt)

(7.13)

(7.14)

where n is an odd positive integer. Recall that, in the near resonance condition (7.13), the factor π/2 occurs due to the speciﬁc normalization of the triangular sine wave period, T = 4.

7.2 Resonances in the Duﬃng’s Oscillator under Impulsive Loading

Let us represent solutions of equation (7.14) in the form x(t) = X τ (ωt) , t0 + Y τ (ωt) , t0 e (ωt)

183

(7.15)

where t0 = εt is the slow time used in the previous section. Based on the continuity condition for x (t),

one obtains

Y |τ =±1 = 0

(7.16)

∂Y ∂X ∂Y ∂X x˙ = ω +ε 0 + ω +ε 0 e ∂τ ∂t ∂τ ∂t

(7.17)

Then, substituting (7.15) and (7.17) in (7.14), eventually gives the boundary value problem (7.16), ∂X |τ =±1 = εp (7.18) ω2 ∂τ and nπ 2 2 ∂2X ∂2Y 2∂ X ω X + ε + ω 2 2 + 2εω ∂τ ∂τ ∂t0 ∂t02 2 3 2 = −ε σX + X + 3Y X nπ 2 2 ∂2Y ∂2X 2∂ Y ω Y + ε + (7.19) ω 2 2 + 2εω ∂τ ∂τ ∂t0 ∂t02 2 = −ε σY + Y 3 + 3X 2 Y Note that, due to substitution (7.15), the boundary value problem, (7.16), (7.18) and (7.19), includes no singular functions. Therefore, further averaging procedures can be correctly applied. Following the formalism of two variable expansions, we seek asymptotic solutions in the form of power series X = X0 τ, t0 + εX1 τ, t0 + ε2 X2 τ, t0 + ... Y = Y0 τ, t0 + εY1 τ, t0 + ε2 Y2 τ, t0 + ... (7.20) Substituting (7.20) in (7.16), (7.18) and (7.19), and matching coeﬃcients of the same degrees of ε, gives a series of boundary value problems, where the leading order problem is ∂ 2 X0 nπ 2 + X0 = 0, ∂τ 2 2 ∂ 2 Y0 nπ 2 + Y0 = 0, ∂τ 2 2

∂X0 |τ =±1 = 0 ∂τ Y0 |τ =±1 = 0

(7.21)

These equations and the boundary conditions are satisﬁes at any t0 by solution

184

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

nπ nπ τ, Y0 = D0 t0 cos τ (7.22) X0 = A0 t0 sin 2 2 where A0 t0 and D0 t0 are arbitrary functions of the slow time scale to be determined on the next step of iteration. So, collecting all terms of order ε, gives the boundary value problem of the next asymptotic order as 2

nπ 2 ∂ 2 Y0 2 ∂ X1 3 2 + X1 = − 2ω + σX0 + X0 + 3Y0 X0 ω ∂τ 2 2 ∂τ ∂t0 ∂X1 ω2 |τ =±1 = p (7.23) ∂τ ω2

∂ 2 Y1 nπ 2 ∂ 2 X0 3 2 + Y + σY + Y + 3X Y = − 2ω 1 0 0 0 0 ∂τ 2 2 ∂τ ∂t0 Y1 |τ =±1 = 0

The right-hand sides of equations (7.23) of the slow time implicitly depend t0 through still unknown functions A0 t0 and D0 t0 , and their derivatives A0 t0 and D0 t0 . Solution of the diﬀerential equations (7.23) is found to be nπτ nπτ + B1 t0 cos X1 = A1 t0 sin 2 2

1 A0 nπτ 3 2 3 2 + σ + A0 + D0 − D0 τ cos ω nπω 4 4 2

A0 3 1 nπτ 7 − σ + A20 + D02 − D0 sin nπω nπω 8 8 2 A0 3nπτ − 2 2 2 A20 − 3D02 sin 8n π ω 2

(7.24)

and nπτ nπτ Y1 = C1 t0 sin + D1 t0 cos 2

2 1 D0 nπτ 3 2 3 2 − σ + A0 + D0 + A0 τ sin ω nπω 4 4 2

D0 1 7 nπτ 3 − σ + A20 + D02 + A0 cos nπω nπω 8 8 2 2 D0 3nπτ − 2 2 2 3A0 − D02 cos 8n π ω 2 Now the boundary condition for X1 gives B1 t0 ≡ 0

(7.25)

7.3 Strongly Nonlinear Oscillator under Periodic Pulses

and

nπ A0 3 2 2p 2 A0 + D0 + sin D0 = σ+ nπω 4 nπω 2

185

(7.26)

Then, the boundary condition for Y1 gives C1 t0 ≡ 0 and

A0 = −

D0 3 2 A0 + D02 σ+ nπω 4

(7.27)

In current approximation, another two arbitrary functions of the slow time scale can be chosen as A1 t0 ≡ 0 and D1 t0 ≡ 0, at least, because the same type of terms are included already into the generating solution (7.22). Substituting derivatives from (7.26) and (7.27) in (8.24) and (7.25) and taking into account (7.22), gives ε nπτ nπτ 1 − 2 2 2 A20 − 3D02 cos2 X = A0 sin 2 2n π ω 2 2εp nπτ nπτ nπ − 2 2 2 sin nπτ cos − sin + O(ε2 ) (7.28) n π ω 2 2 2

Y = D0 cos

ε nπτ nπτ 1 + 2 2 2 3A20 − D02 sin2 + O(ε2 ) 2 2n π ω 2

(7.29)

Substituting (7.28) and (8.87) in (7.15), ﬁnally gives the closed form approximate solution x(t), despite of the discontinuous impulsive loading. Figs. 7.2, 7.3, and 7.4 illustrate the results of calculations under the following parameters and initial conditions: n = 1, ω = 1.0, σ = 0.1, p = 1.0, ε = 0.1, A0 (0) = 0.1, D0 (0) = 0.0. The coordinate-velocity diagram shows two velocity jumps per one cycle due to the periodic impulsive excitation.

7.3

Strongly Nonlinear Oscillator under Periodic Pulses

Let us consider the case of strongly nonlinear exactly solvable oscillator described in Chapter 3 by adding periodic impulsive loading on the right-hand side as follows F de (t/a) a d (t/a) ∞ = 2F (−1)k δ[t − (2k − 1)a]

x ¨ + tan x + tan3 x =

k=−∞

(7.30)

186

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

1.5 1.0 B0

0.5 0.0 0.5 A0

1.0 1.5 2.0 0

5

10

15 t

20

25

0

Fig. 7.2 The amplitudes A0 (t0 ) and D0 (t0 ) in the slow time scale, t0 = εt, shown by the gray and black lines, respectively.

2 1 x

0 1 2 0

50

100

150

200

250

t Fig. 7.3 The time history of the oscillator near the principal resonance.

Here 2F > 0 and 4a = T > 0 characterize the amplitude and period of the impulsive loading. Making substitutions x = X(τ ) and τ = τ (t/a) in equation (7.30), gives

d2 X dX de (t/a) a−2 2 + tan X + tan3 X = a−2 aF − dτ dτ d (t/a)

7.3 Strongly Nonlinear Oscillator under Periodic Pulses

187

3 2 1 v 0 1 2 3 2

1

0 x

1

2

Fig. 7.4 The coordinate-velocity plane trajectory on the interval 0 < t < 40.

or

d2 X + tan X + tan3 X = 0 dτ 2 under the following continuity condition for x(t), a−2

dX |τ =±1 = aF dτ

(7.31)

(7.32)

Boundary value problem (7.31) and (7.32) describe the class of steady state periodic motions of the period T . With reference to Chapter 3, equation (7.31) has exact solution of the form X(τ ) = arcsin[sin A sin(aτ / cos A)]

(7.33)

188

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

where A is an arbitrary constant, which is suﬃcient to satisfy both conditions in (7.32) due to the symmetry of solution (7.33). Substituting (7.33) in (7.32) and conducting analytical manipulations with elementary functions, gives α2 F T = 4α kπ ± arccos , (k = 1, 2, 3, ...) (7.34) (1 − α2 )(1 − α2 F 2 ) α2 F T = 4α arccos , (1 − α2 )(1 − α2 F 2 )

(k = 0)

(7.35)

where T = 4a is the period of forced vibration, and α = cos A.

35 30 25 T

20 15

k3

10

k2

5

k1

0

k0 1.2

1.1

1.3 A

1.4

1.5

Fig. 7.5 First few branches of the period versus amplitude diagram under the ﬁxed pulse amplitudes, F = 1.8.

It can be shown that solutions (7.34) and (7.35) exist in the interval 1 π arccos √ = Amin < A < Amax = 2 2 1+F

(7.36)

Fig. 7.5 illustrates the sequence of branches of solutions (7.34) and (7.35) at diﬀerent numbers k, and the parameter F = 1.8. The diagram gives such combinations of the period and amplitude at which oscillator (7.30) can have periodic motions with the period of external impulses, T = 4a. The upper and

7.4 Impact Oscillators under Impulsive Loading

189

lower branches of each loop correspond to plus and minus signs in expression (7.34), respectively. Solution (7.35) corresponds to the number k = 0 and has the only upper branch. For the selected magnitude of F , the minimal amplitude is found to be Amin = 1.0637, which corresponds to the left edges of the amplitude-period loops in Fig. 7.5. As a result, further slight increase of the amplitude is accompanied by bifurcation of the solutions as shown in Fig.7.6. The diagram includes only ﬁrst three couples of new solutions (k = 1, 2, 3) from the inﬁnite set of solutions. Further evolution of temporal shapes of solutions, related to the amplitude increase, is illustrated by Figs. 7.7 and 7.8. As follows from these diagrams, the inﬂuence of external pulses on the temporal shapes is decreasing as the amplitude grows. This is the result of dominating the restoring force over the external pulses. When the amplitude becomes close to its maximum Amax = π/2, the oscillator itself generates high-frequency impacts; see Chapter 3 for details.

7.4

Impact Oscillators under Impulsive Loading

Consider nonlinear oscillator under periodic pulse excitation of the following general form de (ωt) (7.37) x ¨ + 2ζ x˙ + f (x, ωt) = p d (ωt) where ζ, ω and p are constant parameters, and the function f (x, ωt) is periodic with respect to the argument ωt with the period equal to four. It is assumed that the coordinate x is subjected to either the constraint condition 0 ≤ x (t) (7.38) or − 1 ≤ x (t) ≤ 1

(7.39)

According to the idea of non-smooth transformation of positional coordinates [199], [204] (see also the end Chapter 1 of this book), the constraints (7.38) or (7.39) are eliminated by unfolding the coordinate x (t) −→ l (t) : x = S (l)

(7.40)

where S (l) ≡ |l| in case (7.38) or S (l) ≡ τ (l) in case (7.39), so that the new coordinate l belongs to the entire inﬁnite interval, l (t) ∈ (−∞, ∞), in contrast to (7.38) or (7.39). Taking into account the identity [S (l)]2 = 1, gives the ﬁnal result of transformation (7.40) in the form dl de (ωt) d2 l + 2ζ + S (l) f [S (l) , ωt] = pS (l) 2 dt dt d (ωt)

(7.41)

where, as mentioned above, the new coordinate l (t) is free of any constraints.

190

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

x 1.0

k1

0.5 0.5

0.5 1

2

3

4

t

5

0.5

1.0 k2

0.5

2

3

4

5

6

2

4

6

8

10 12

t

x 1.0 0.5

2

4

6

8

10 12

t 0.5

1.0

t

1.0

x 1.0

k3

0.5 0.5

1

1.0

x 1.0

0.5

x 1.0

x 1.0 0.5

5

10

15

1.0

t 0.5

5

10

15

t

1.0

Fig. 7.6 Periodic solutions corresponding to the ﬁrst three closed loops of the diagram shown in Fig. 7.5 at A = 1.0647; the left and right columns relate to lower and upper branches of the loops, respectively.

Note that the right-hand side of equation (7.41) makes sense within the distribution theory under the condition that times of interaction with constraints (7.38) or (7.39) never coincide with the external pulse times. Consider now the class of periodic motions of the period of external loading, T = 4/ω. As follows from Chapter 6 and the previous sections of this chapter, within such class of motions, the external pulses are eliminated by introducing the new time argument as follows t −→ τ : τ = τ (ωt) ,

l = X (τ ) + Y (τ ) e

(7.42)

7.4 Impact Oscillators under Impulsive Loading

x

x

1.0

k1

0.5 0.5

0.5 1.0 1.5 2.0 2.5 3.0

t 0.5

1

2

3

4

5

6

7

t

1.0

x

k2

1.0 0.5

x 1.0 0.5

2

0.5

4

6

8

t 0.5

1.0

2

4

6

8

10 12

t

1.0

x

x k3

1.0 0.5

1.0

1.0 0.5

1.0

0.5

191

1.0 0.5

2

4

6

8 10 12

t 0.5

5

10

15

t

1.0 Fig. 7.7 Same as Fig. 7.6 for A = 1.15.

So, substituting (7.42) into equation (7.41), gives ω 2 X (τ ) + 2ωζY (τ ) + R (X, Y, τ ) + ω 2 Y (τ ) + 2ωζX (τ ) + I (X, Y, τ ) e de (ωt) = pRS (X, Y ) + pIS (X, Y ) e − ω 2 X (τ ) d (ωt)

(7.43)

where 1 R = {S (X + Y ) f [S (X + Y ) , τ ] ± S (X − Y ) f [S (X − Y ) , 2 − τ ]} I 2

192

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

x

x k1

1.0 0.5 0.5

0.5 1.0

1.0

1.0 0.5

t

1.5

0.5 1.0

x

3

t

4

1

2

3

4

5

1.0 0.5

t

2

0.5 1.0

x

4

6

8

t

x k3

1.0 0.5 0.5 1.0

2

x k2

1.0 0.5 0.5 1.0

1

2

4

6

8

1.0 0.5

t 0.5 1.0

2

4

6

8

10

t

Fig. 7.8 Same as Fig. 7.6 for A = 1.3.

and

RS IS

=

1 {S (X + Y ) ± S (X − Y )} 2

under the necessary condition of continuity for l (t), Y (±1) = 0

(7.44)

Equation (7.43) is equivalent to the following boundary value problem ω 2 X (τ ) + 2ωζY (τ ) + R (X, Y, τ ) = 0 ω 2 Y (τ ) + 2ωζX (τ ) + I (X, Y, τ ) = 0

(7.45)

[X (τ ) − ω −2 pS (X)]|τ =±1 = 0

(7.46)

7.4 Impact Oscillators under Impulsive Loading

193

where boundary condition (7.46) has been simpliﬁed by enforcing condition (7.44). After solution of the boundary value problem (7.44) through (7.46) has been obtained, the original coordinate is given by composition of transformations (7.40) and (7.42) as x (t) = S (X (τ ) + Y (τ ) e)

(7.47)

where τ = τ (ωt) and e = e (ωt). The advantage of such a boundary value problem formulation is that the eﬀect of both internal and external pulses are captured by the composition of two transformations (7.47). As a result, the ﬁnal system is free of any singular terms. Some particular cases and examples were considered earlier in [140].

Chapter 8

Strongly Nonlinear Vibrations

Abstract. This chapter presents analytical successive approximations algorithms for diﬀerent oscillators with strongly nonlinear characteristics. In general terms, such algorithms approximate temporal mode shapes of vibrations by polynomials and other simple functions of the triangular sine wave. In order to develope the algorithms, the triangular wave is introduced into dynamical systems as a new temporal argument. The corresponding manipulations with dynamical systems are described in the ﬁrst three sections. Then the description focuses on the algorithm implementations for diﬀerent essentially unharmonic cases including oscillators whose characteristics may approach nonsmooth or even discontinuous limits.

8.1

Periodic Solutions for First Order Dynamical Systems

Let us consider a dynamical system described by ﬁrst-order diﬀerential equation with respect to the vector-function x(t) ∈ Rn , x˙ = f (x)

(8.1)

where f (x) is a continuous vector-function, and the over dot indicates time derivative. We consider the class of periodic motions of the period T = 4a. Note that, in the autonomous case, the period is a priori unknown. Periodic solutions usually require speciﬁc, a priory unknown, initial conditions. Practically, however, such kind of the speciﬁc initial conditions are determined in a backward way, after some periodic family of solutions is obtained under the assumption of periodicity. In our case, the assumption of periodicity is imposed automatically by the form of representation for periodic solutions x = X(τ ) + Y (τ )e V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 195–239, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(8.2)

196

8 Strongly Nonlinear Vibrations

where τ = τ (t/a) and e = e(t/a) are the standard triangular sine and rectangular cosine, respectively, and X(τ ) and Y (τ ) are unknown components of the solution. Substituting (8.2) in (8.1), gives (Y − aRf ) + (X − aIf )e + Y e = 0 where 1 [f (X + Y ) + f (X − Y )] 2 1 If = If (X, Y ) = [f (X + Y ) − f (X − Y )] 2

Rf = Rf (X, Y ) =

Eliminating the periodic singular term e = de(t/a)/d(t/a) by means of the boundary condition for Y (τ ), gives the non-linear boundary value problem on the standard interval, −1 ≤ τ ≤ 1, Y = aRf (X, Y ) X = aIf (X, Y ) Y |τ =±1 = 0

(8.3)

Note that, the entire interval −1 ≤ τ ≤ 1 is completely covered by a half of the period, −a ≤ t ≤ a, however, representation (8.2) unfolds the corresponding fragment on the entire time interval −∞ < t < ∞.

8.2

Second Order Dynamical Systems

Consider now the diﬀerential equation of motion in the standard Newtonian form x ¨ + f (x, x, ˙ t) = 0 (8.4) where x(t) ∈ Rn is a positional vector-function, and the vector-function f is assumed to be suﬃciently smooth and periodic with respect to the explicit time t with the period T = 4a; the autonomous case, when the period is a priory unknown, is under consideration as well. Substituting (8.2) into (8.4), using the diﬀerential and algebraic properties of substitution (8.2), and imposing the boundary (smoothness) conditions, gives (X + a2 Rf ) + (Y + a2 If )e = 0 where

X − Y 1 X + Y , aτ + f X − Y, − , 2a − aτ Rf = f X + Y, 2 a a (8.5)

8.2 Second Order Dynamical Systems

197

X − Y 1 X + Y , aτ − f X − Y, − , 2a − aτ f X + Y, 2 a a (8.6) This leads to the boundary value problem If =

X + a2 Rf (X, Y, X , Y , τ ) = 0, X |τ =±1 = 0

(8.7)

Y + a2 If (X, Y, X , Y , τ ) = 0, Y |τ =±1 = 0

(8.8)

Let us discuss the form of equations (8.7) and (8.8). Firstly, despite of the obvious formal complication, equations (8.7) and (8.8) possess certain symmetries dictated by substitution (8.2). For instance, introducing the new unknown variables, U (τ ) = X(τ ) + Y (τ ) V (τ ) = X(τ ) − Y (τ ) brings the boundary value problem, (8.7) and (8.8), to the form, in which the diﬀerential equations are decoupled at cost of coupling the boundary conditions though, U + a2 f (U, U /a, 2a − aτ ) = 0 V + a2 f (V, − V /a, 2a − aτ ) = 0

(8.9)

U + V |τ =±1 = 0 U − V |τ =±1 = 0 In case when analytical methods are applied, the diﬀerential equations (8.9) for U (τ ) and V (τ ) can be usually solved in a similar way. Nevertheless, the previous boundary value problem, (8.7) and (8.8), may appear to have some advantages for analyses. For instance, the problem may admit families of solutions with either Y (τ ) ≡ 0 or X(τ ) ≡ 0. Secondly, due to substitution (8.2), the major qualitative property of solutions, such as periodicity, is a priory captured by the new argument, τ . As a result, the following simpliﬁed system can be employed as a generating model for analytical algorithms of successive approximations X = 0 Y

(8.10)

=0

Indeed, substituting obvious solutions of equations (8.10) in (8.2), gives a family of nonsmooth periodic motions with respect to the original time parameter, t, x(t) = X(0) + X (0)τ (t/a) + (Y (0) + Y (0)τ (t/a))e(t/a)

(8.11)

198

8 Strongly Nonlinear Vibrations

Thirdly, from the physical standpoint, linear equations (8.10) describe a strongly nonlinear (nonsmooth) generating model. In particular, if Y (0) = 0 and Y (0) = 0, then vector-function (8.11) describes vibrations of basic vibroimpact models. The analytical algorithms developed in few sections below are based on the idea of approximation of smooth vibrating systems by the basic vibroimpact models. In other words, the triangular sine wave is assumed to be a dominant component of temporal mode shapes of vibrations. Such an idea indeed follows the analogy with the quasi harmonic approaches. In particular, the harmonic balance method approximates vibrating systems by eﬀective harmonic oscillators regardless types or magnitudes of the system nonlinearities. This is justiﬁed by the fact that Fourier coeﬃcients usually decay in a fast enough rate so that, for instance, second term can be considered as a small correction to the ﬁrst term. The corresponding “small parameter” is therefore hidden in the iterative procedure itself rather than explicitly present in the diﬀerential equations of motion. Finally, equations (8.10) make sense due to the temporal substitution t −→ τ (t/a). In terms of the original variables, the corresponding equation, x ¨ = 0, contains too little information about the original system (8.4) and captures no global properties of the dynamics.

8.3

Periodic Solutions of Conservative Systems

8.3.1

The Vibroimpact Approximation

Let us consider the case of n-degrees-of-freedom conservative system x¨ + f (x) = 0

(8.12)

where f (x) is an odd analytical vector-function of the positional vectorcolumn x(t) ∈ Rn . A one-parameter family of periodic solutions will be built such that X(−τ ) ≡ −X(τ ). Since equation (8.12) admits the group of time translations then another arbitrary parameter can be always added to the time variable. Taking into account the symmetry of system (8.12), enables one of considering the particular case of substitution (8.2) x(t) = X(τ (t/a)), Y ≡ 0

(8.13)

Based on the conditions assumed, the boundary value problem (8.7) and (8.8) is reduced to the following one X + a2 f (X) = 0 X |τ =1 = 0

(8.14)

8.3 Periodic Solutions of Conservative Systems

199

We seek solutions of the boundary value problem (8.14) in the form of series of successive approximations X = X 0 (τ ) + X 1 (τ ) + X 2 (τ ) + ...

(8.15)

a2 = h0 (1 + γ1 + γ2 + ...)

(8.16)

In order to organize the corresponding iterative procedure, it is assumed that O( X i ) O( X i+1 ) O(γi+1 ) O(γi+2 )

(8.17)

(i = 0, 1, 2, ...) where the norm of vector-functions is deﬁned by X =max X Rn . τ

Based on assumptions (8.17), series (8.15) and (8.16) generate the sequences of equations and boundary conditions as, respectively, X 0 = 0

(8.18)

X 1 = −h0 f (X 0 ) X

2

= −h0 [γ1 f (X ) + 0

fx (X 0 )X 1 ]

(8.19) (8.20)

··· and (X 0 + X 1 )|τ =1 = 0

(8.21)

X 2 |τ =1 = 0

(8.22)

··· Note that condition (8.21) includes ﬁrst two approximations as the only way to proceed with a non-zero generating solution. In particular, the generating solution is found from equation (8.18) in the form X 0 = A0 τ

(8.23)

where A0 ∈ Rn is an arbitrary constant vector, and the oddness condition has been enforced in order to set to zero another constant vector. In line with the discussion at the end of the previous section, solution (8.23) describes a multi-dimensional vibro-impact oscillator between two absolutely stiﬀ and perfectly elastic barriers such that A0 is the normal vector to both barriers. Direction of the vector A0 will be deﬁned on the next step of successive approximations, whereas its length will appear to be coupled with the parameter h0 by some relationship due to boundary condition (8.21).

200

8 Strongly Nonlinear Vibrations

So substituting (8.23) in (8.19) and integrating, gives solution τ X = A τ − h0 1

(τ − ξ)f (A0 ξ)dξ

1

(8.24)

0

where A1 is another arbitrary constant vector. Note that the ﬁrst term in expression (8.24) is similar to generating solution (8.23) and thus contributes nothing new into the entire solution within the ﬁrst two steps of the procedure. Therefore, we take A1 = 0 and then substitute the combination X 0 + X 1 in the boundary condition (8.21). This gives a nonlinear eigen value problem with respect to the vector A0 in the form 1 1 0 f (A0 τ )dτ = A (8.25) h0 0

Equation (8.25) represents a set of n scalar equations relating the components of vector A0 and the parameter h0 . The combination A0 and 1/h0 will be interpreted as an eigen vector and eigen value of the nonlinear eigen vector problem (8.25). Taking scalar product of both sides of equation (8.25) with A0T , gives h0 = A0T

A0T A0 "1 f (A0 τ )dτ

(8.26)

0

where the upper index T stays for transpose operation. In order to clarify the meaning of expressions (8.25) and (8.26), let us consider the linear case f (x) ≡ Kx, where K is an n × n stiﬀness matrix. The corresponding relationships will diﬀer from those of the exact linear theory by speciﬁc constant factors because the temporal mode shape of vibrations is not exact but approximated by the triangular sine wave. Nevertheless, in nonlinear cases, expression (8.26) can provide estimates for amplitudefrequency response characteristics. Further, integrating equation (8.20), gives τ X = A τ − h0 2

2

(τ − ξ)[γ1 f (A0 ξ) + fx (A0 ξ)X 1 (ξ)]dξ

(8.27)

0

where A2 is an arbitrary constant vector, and fx ( A0 ξ ) is the n × n − matrix of ﬁrst partial derivatives (Jacobian).

8.3 Periodic Solutions of Conservative Systems

201

Then boundary condition (8.22) gives 1 2

A = h0

[γ1 f (A0 τ ) + fx (A0 τ )X 1 (τ )]dτ

(8.28)

0

where the coeﬃcient γ1 is yet unknown. In order to determine the coeﬃcient γ1 , an additional condition for the vector A2 can be imposed, for instance, as follows A0T A2 = 0

(8.29)

This condition means that the vector A2 must be orthogonal to the corresponding vector of the generating solution A0 in order to keep the amplitude ﬁxed. Substituting (8.28) in (8.29), gives A0T γ1 = −

"1

fx (A0 τ )X 1 dτ

0

A0T

"1

(8.30) f (A0 τ )dτ

0

This completes the second step of successive approximations. All the further steps can be passed in the same way. In general terms, convergence properties of the above procedure are due to the following integral operator ⎧ 1 ⎫ τ ⎨ ⎬ F [X] ≡ a2 τ f (X(ξ))dξ + ξf (X(ξ))dξ (8.31) ⎩ ⎭ τ

0

where A0T a2 = h 0

"1

f (A0 τ )dτ

0

A0T

"1

(8.32) f (X)dτ

0

Based on deﬁnition (8.31), the original boundary value problem (8.14) admits representation in the form X = F [X]. Therefore, the convergence condition is [X 0 ]δX

FX <1 (8.33)

δX

where δX is an arbitrary vector-function from a small enough neighborhood of X 0 .

202

8 Strongly Nonlinear Vibrations

In the case of linearized system, condition (8.33) leads to the set of inequalities ωi /ωj < 1 for all i = j, where ωj is the eigen frequency of the linear normal mode, which is chosen to be a generating solution.

8.3.2

One Degree-of-Freedom General Conservative Oscillator

In the one-degree-of-freedom case with odd characteristic, the boundary condition at τ = 1 is reduced to a single equation, which is sequentially satisﬁed by the factor h0 and terms γ1 , γ2 ,... of series (8.16). As a result, the process of successive approximations therefore eases by setting A0 = A and Ai = 0 for i = 1, 2,... . Let us introduce notations hi = h0 γi and represent series (8.15) and (8.16) in the form X = X0 (τ ) + X1 (τ ) + X2 (τ ) + ... a2 = h0 + h1 + h2 + ...

(8.34)

where Xi (τ ) are scalar functions of the triangular sine wave τ = τ (t/a). Due to the reduction of one-dimensional case, all terms of the expansions are iteratively determined by the explicit relationships. First two steps of the iterative procedure are coupled by the smoothing boundary condition (8.21) that provides the leading order smooth estimate for the temporal mode shape by coupling the parameters, h0 = h0 (A), as follows X0 = Aτ

(8.35)

τ X1 = −h0

(τ − ξ)f (Aξ)dξ 0

1 h0 = A/

f (Aξ)dξ

(8.36)

0

All the next steps of the procedure are passed then in a similar way based on relationships Xi = −

i

τ

hi−1 = −

i−1

(τ − ξ)Ri−j dξ

hj−1

j=1

0

αi−j hj−1

j=1

(i = 2, 3, ...)

(8.37)

8.3 Periodic Solutions of Conservative Systems

203

where the coeﬃcients and integrands are generated by means of the formal auxiliary parameter, ε, as follows 1 αi =

1 Ri dξ/

0

R0 dξ

(8.38)

0

1 di f (X 0 + εX 1 + ε2 X 2 + · · ·) Ri = |ε=0 i! dεi (i = 0, 1, 2, . . .) The way of using the parameter ε is in compliance with assumptions (8.17). Respectively, such parameter splits the restoring force according to (8.38) and then disappears from expressions. The convergence of such iterative series of successive approximations is illustrated by the following example. Example 12. Let us consider the oscillator x ¨ + xm = 0 where m is an odd positive integer. This oscillator was already discussed in Chapter 3 under the notation m = 2n − 1. Now, applying two iterations according to the above scheme, (8.36) and (8.37), gives solution

2m+3 τ m+2 m τ τ m+2 X =A τ− + − − R3 − R4 − · · · m + 2 2(m + 2) 2m + 3 m + 2 (8.39)

2 m + 1 m (m + 1) a2 = m−1 1 + 1+ + r3 + r4 + · · · (8.40) A 2(m + 2) (m + 2)(2m + 3) where expressions m+2

m |τ | i−1 2 (m + 2)2 m 0 < ri (m) < i 2 (m + 2)

0 < Ri (m, τ ) <

(8.41)

provide estimates for high order terms of the successive approximations. In particular, expressions (8.41) indicate that series (8.39) and (8.40) may converge quite slowly. However, the asymptotic of large exponents m essentially improves precision of the truncated series even though ﬁrst few terms of the series are included. The temporal mode shapes of diﬀerent iterations are shown in Fig. 8.1, whereas Fig. 8.2 illustrates the period as a function of the number m under the ﬁxed energy in the ﬁrst iteration only. For comparison reason, the exact result and ﬁrst order approximation according to the harmonic balance method are also shown in the diagram. In particular, Fig. 8.1 shows that high-order iterations are localized near the time points corresponding to amplitude positions of the oscillator. For instance, the ﬁrst

204

8 Strongly Nonlinear Vibrations

Fig. 8.1 The ﬁrst three terms of iteration (thin lines) and their sum (solid line) for the temporal mode shape of the oscillator x ¨ + x5 = 0.

6

Period, T

5

harmonic balance

4

exact solution 3 sawtooth 0

10

20 30 Degree of nonlinearity, m

40

50

Fig. 8.2 The period of the oscillator with power characteristic at diﬀerent exponents m obtained by three diﬀerent methods under the total energy E = 2.

iteration X1 just compensates the discontinuities of slope of the generating solution X0 with a minor eﬀect on the rest of the triangular sine wave. Further, Fig. 8.2 conﬁrms that expansion (8.40) gives a better estimate for the

8.3 Periodic Solutions of Conservative Systems

205

period than the harmonic balance as the exponent m increases. Note that the entire series (8.39) and (8.40) are not asymptotic with respect to m or 1/m in the sense of Poincare, however the series perfectly capture the asymptotic of impact oscillator as m → ∞.

8.3.3

A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes

As another example of conservative oscillator, we consider a single mass vibrating system illustrated by Fig. 8.3 and described by the Lagrangian mw˙ 2 − kl2 L= 2

w2 1+ 2 −1 l

2

Here, m is mass, k is the linear stiﬀness of each spring, l is the length of each spring at the equilibrium position at which the springs are horizontal, and w is the particle vertical coordinate. In terms of the dimensionless coordinate x = w/l and phase ϕ = (2k/m)1/2 t, the corresponding diﬀerential equation of motion takes the form x d2 x +x− √ =0 dϕ2 1 + x2

(8.42)

Then, applying substitution (8.13) as x = X(τ ) and τ = τ (ϕ/a), leads to the boundary problem

X ≡ −hf (X) X = −h X − √ 1 + X2 (8.43) X |τ =1 = 0 where h = a2 .

wt l

l

Fig. 8.3 The system which becomes weakly non-linear at large amplitudes.

206

8 Strongly Nonlinear Vibrations

First two steps of the successive approximation procedure give X0 = Aτ −1 √ 1 + A2 − 1 1 − h0 = 2 A2 and

h0 1 3 2 X1 (τ ) = − 2 (Aτ ) + 2Aτ − Aτ 1 + (Aτ ) − arcsinh(Aτ ) (8.44) 2A 3

9A − A3 h0 6A + A3 1 + √ γ1 = 3 − 1+ √ arcsinh(A) A 12 6 1 + A2 1 + A2

Interestingly enough, this model is essentially nonlinear at small amplitudes, but it becomes linear as the amplitudes are inﬁnitely large. Indeed, taking the corresponding limits, shows that 1 X1 3 → − τ 5 , h0 A2 → 8, γ1 → A 5 10

as A → 0

(8.45)

and

X1 1 1 → − τ 3 , h0 → 2, γ1 → as A → ∞ (8.46) A 3 6 The asymptotic (8.45) obviously corresponds the nonlinear oscillator, whereas the limit case (8.46) associates with the harmonic oscillator. Nevertheless, solution (8.44) is valid for both large and small amplitudes. Both limit cases follow from equation (8.42). In the case of small amplitudes, |x| 1, by using the estimate (1 + x2 )−1/2 ∼ 1 − x2 /2 in equation (8.42), we obtain d2 x/dϕ2 + x3 /2 = 0

(8.47)

In the case of large amplitudes, it follows even from Fig. 8.3 that the distance between the spring ﬁxed ends becomes negligible if as compared to l. As a result, the mechanical model becomes eﬀectively close to a mass-spring oscillator of mass m with a single spring of stiﬀness 2k. In terms of the diﬀerential equations of motion, we also obtain the corresponding limit from exact equation (8.42) by√assuming that, during ‘most of the time of vibration cycle,’ |x| 1 and thus 1 + x2 ∼ |x|. As a result, equation (8.42) is replaced by (8.48) d2 x/dϕ2 + x − sgn(x) = 0 where the term sgn(x) has to be neglected due to the same condition |x| 1 that gives the standard linear oscillator. Alternatively, the discontinuous term in equation (8.48) can be saved and then considered as a perturbation of the harmonic oscillator. Note that equation (8.48) admits another form as follows

8.3 Periodic Solutions of Conservative Systems

d2 x/dϕ2 + sgn(x)(|x| − 1) = 0

207

(8.49)

The restoring force characteristic of oscillator (8.49) represents a particular case of the characteristic, p(x) = sgn(x)f (|x|), which is considered later in this chapter.

8.3.4

Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero

Let us consider the case of symmetric exponentially growing restoring force characteristic with a step-wise discontinuity at zero such that exp(x) for x > 0 f (x) = (8.50) − exp(−x) for x < 0 Although force (8.50) has no certain value at the point x = 0, this still can play the role of equilibrium position. From the physical standpoint, this is equilibrium of a small bead at the bottom of V -shaped potential well. The local dynamics in a small neighborhood of such type of equilibria is considered later in this chapter; see the text to Fig. 8.13. It follows from (8.50) that f (−x) = −f (x). Therefore, periodic motions of the corresponding oscillator can be described by the function x = X(τ ), where X(−τ ) = −X(τ ) and τ = τ (t/a). In terms of these NSTT variables, the oscillator of a unit mass is described by the boundary value problem for τ ∈ (0, 1] X + h exp(X) = 0, X |τ =1 = 0 X − h exp(−X) = 0, X |τ =−1 = 0 for τ ∈ [−1, 0)

(8.51)

where h = a2 . This problem is exactly solvable, and solution that satisﬁes the continuity of state condition at τ = 0 has the form X± (τ ) ≡ Aτ ± 2 ln (1 + exp(−A))

h exp(±Aτ − A) ∓2 ln 1 + 2h0

(8.52)

where X+ and X− are taken for positive and negative subintervals of τ , respectively, and 2A2 (8.53) h = 2h0 = exp(A)[1 + exp(−A)]2 Note that both the diﬀerential equation of oscillator and its solution admit unit form representations as, respectively, x ¨ + sgn(x) exp(|x|) = 0

(8.54)

208

and

8 Strongly Nonlinear Vibrations

X = sgn(τ ) A|τ | + 2 ln

1 + exp(−A) 1 + (h/(2h0 )) exp(A|τ | − A)

(8.55)

The parameter h is obtained from equation

or 1−

h h0

X |τ =1 = 0

(8.56)

−1 h =0 1+ 2h0

(8.57)

This exactly solvable case can play the role of a majorant for evaluation of convergence properties of successive approximations. For that reason, we introduce a formal ‘small’ parameter, ε = 1, and represent solution of equation (8.57) in the form ε −1 h = 2h0 = εh0 1 − (8.58) 2 Taking into account (8.58), brings (8.55) to the form

1 − (ε/2) (1 − exp(−A)) X = sgn(τ ) A|τ | + 2 ln (8.59) 1 − (ε/2) (1 − exp(A|τ | − A)) It can be shown by direct calculations that the power series expansions of (8.58) and (8.59) with respect to ε lead to the same series as those obtained by means of the iterative procedure introduced in this section for a general one-degree-of-freedom oscillator. Moreover, the structure of expression (8.58) suggests that considering the modiﬁed series, h=

εh0 1 − ελ1 − ε2 λ2 − ...

(8.60)

leads to the exact value h already on the second step of the procedure. This fact can be employed for other cases in order to improve eﬃciency of the successive approximation series (8.16). For instance, according to the idea of Pad`e transform [19], the following equality must hold in every order of ε εh0 = εh0 (1 + εγ1 + ε2 γ2 − ...) 1 − ελ1 − ε2 λ2 − ...

(8.61)

This is equivalent to (1 + εγ1 + ε2 γ2 − ...)(1 − ελ1 − ε2 λ2 − ...) = 1

(8.62)

Taking the product of series on the left-hand side of (8.62) and considering diﬀerent orders of ε, generates a sequence of equations for the coeﬃcients

8.3 Periodic Solutions of Conservative Systems

209

λ1 , λ2 , ... . Then, substituting the corresponding solutions in (8.60), gives a particular case of Pad`e transform of (8.16) in the form h=

εh0 1 − εγ1 − ε2 (γ2 − γ12 ) − ...

(8.63)

In many cases, expansion (8.63) appears to be more eﬀective than (8.16). Note that it is also possible to organize the successive approximations procedure by using expansion (8.60) instead of (8.16).

8.3.5

A Generalized Case of Odd Characteristics

This subsection deals with some generalization of the standard one-degreeof-freedom conservative oscillator x¨ + f (x) = 0

(8.64)

where f (x) is a smooth odd characteristic, f (−x) = −f (x)

(8.65)

It was shown in this chapter that periodic solutions of the oscillator (8.64) admit the form x = X(τ ), where X(−τ ) = −X(τ )

(8.66)

and, in addition, X(τ )τ ≥ 0 for −1 ≤ τ ≤ 1. We consider now the following class of oscillators x ¨ + sgn(x)f (|x|) = 0

(8.67)

In the case of odd characteristic, f (x), oscillator (8.67) is equivalent to the original one (8.64). The extension is due to the fact that the oscillator (8.67) always has an odd characteristic regardless whether or not the function f (x) itself is odd. In general case, however, the characteristic sgn(x)f (|x|) may appear to be nonsmooth at the equilibrium point x = 0. As a result, direct implementation of iterative procedures with high-order derivatives of oscillator’ characteristics becomes quite limited. Nevertheless, as illustrated below, the group properties of equation (8.67) can help to eﬀectively build solution of equation (8.67) based on solution of equation x ¨ + f (x) = 0 for x > 0 by ignoring the point x = 0. Obviously, if P (x) is the potential energy of oscillator (8.64), then P (|x|) is the potential energy corresponding to oscillator (8.67). The following example explains why equation (8.67) covers a broader class of oscillators than (8.64). Example 13. x ¨+sgn(x)|x|3/2 = 0 is an oscillator, but x¨ + x3/2 = 0 is not; see also Chapter 3 for the related discussion.

210

8 Strongly Nonlinear Vibrations

Based on the transition from equation (8.64) to equation (8.67) and the general symmetry properties (8.65) and (8.66), we introduce the following representation for periodic solutions of equation (8.67) x = sgn(τ )X(|τ |)

(8.68)

Such an extension enables one to obtain ‘closed form’ analytical solutions for a large number of oscillators by a simple adaptation of already known solutions, X(τ ), for diﬀerent cases of smooth characteristics. Example 14. Applying transformation (8.68) to solution (8.39), which was derived for the power form characteristic xα with an odd positive exponent α = m, gives

2α+3 α |τ | |τ |α+2 |τ |α+2 + − X = Asgn(τ ) |τ | − (8.69) α+2 2(α + 2) 2α + 3 α+2 where τ = τ (t/a) and the expansion (8.40) for a2 requires only the replacement m → α. Expansion (8.69) represents an approximate solution of the equation (8.70) x ¨ + sgn(x)|x|α = 0 where the notation α substitutes m in order to emphasize that the new exponent can take any positive real value, such as even, odd, rational or irrational. Figs. 8.4 and 8.5 illustrate solution (8.69) compared to numerical solution for two diﬀerent exponents α and the same parameter A = 1. As both ﬁgures show, the analytical and numerical solutions are in a better match under the large exponent α due to the inﬂuence of vibroimpact asymptotic, α → ∞.

0.6

3 2

0.4 0.2 x

0.0 0.2 0.4 0.6 0

1

2

3

4

5

6

7

t

Fig. 8.4 Analytical and numerical solutions of the modiﬁed oscillator shown by continuous and dashed lines respectively.

8.4 Periodic Motions Close to Separatrix Loop

211

1.0 2010 0.5

x

0.0

0.5

1.0 0

5

10

15

20

25

30

35

t

Fig. 8.5 Analytical and numerical solutions of the generalized oscillator under the large exponent α.

The common feature of the algorithms and examples of this section is that generating solutions for successive approximations are represented by triangular sine waves of proper amplitudes and periods. Such generating solutions belong to the ‘real’ component of the hyperbolic ‘number,’ x = X + Y e. In contrast, the next section introduces algorithms of successive approximations based on the ‘imaginary’ component. It will be seen that these two approaches have diﬀerent physical contents.

8.4

Periodic Motions Close to Separatrix Loop

In this section, the classic mathematical pendulum is considered as an example, although the developed algorithm may be applicable to other cases of one degree-of-freedom systems with multiple equilibrium positions. So we illustrate the algorithm based on the diﬀerential equation of motion x ¨ + sin x = 0

(8.71)

It is assumed that the pendulum oscillates inside the separatrix loop around the stable equilibrium (x, x) ˙ = (0, 0) in between two physically identical unstable saddle-points (x, x) ˙ = (±π, 0). The separatrix loop also represents trajectory of the system with the total energy π Es ≡

sin xdx = 2 0

(8.72)

212

8 Strongly Nonlinear Vibrations

The pendulum remains close to the separatrix loop in the area of periodic motions if the total energy E is within the range 0<1−

E << 1 Es

(8.73)

Let us show that, under condition (8.73), a successive approximation solution can be derived from the particular case of boundary value problem (8.7) and (8.8). Such particular case is given by setting X ≡ 0 so that x(t) = Y (τ (t/a))e(t/a)

(8.74)

Substituting (8.74) in equation (8.71) and following the NSTT procedure, yields Y = −a2 sin Y (8.75) and Y |τ =1 = 0, Y (−τ ) ≡ Y (τ )

(8.76)

We seek solution of the boundary value problem, (8.75) and (8.76), in the form of series of successive approximations Y = π + εY1 (τ ) + ε3 Y3 (τ ) + ε5 Y5 (τ ) + ... a2 = p2 / 1 − ε2 λ2 − ε4 λ4 − ...

(8.77) (8.78)

where ε = 1 is an auxiliary “parameter” that helps to organize the iterative process. According to the formal expansion (8.77), the generating solution is represented by the rectangular cosine of the amplitude π, x(t) = πe(t/a), which is a step-wise discontinuous function. Such temporal mode shapes occur near the separatrix loop in the natural time scale of the pendulum because the system spends most of the time during one period near the unstable equilibrium positions x = π and its physically identical, x = −π. Therefore, expansion (8.77) is designed to be a high-energy expansion near the unstable equilibrium, rather than around the stable equilibrium position, x = 0. Substituting expansions (8.77) and (8.78) into the equation (8.75) and collecting terms with the same power of ε, leads to the sequence of equations d2 Y1 − p2 Y1 = 0 dτ 2

1 3 d2 Y3 2 2 − p Y3 = p λ2 Y1 − Y1 dτ 2 6

(8.79) (8.80)

2 d2 Y5 λ2 3 1 5 1 2 2 2 λ Y Y Y Y − p Y = p + λ − + + λ Y − Y (8.81) 5 4 1 2 3 3 2 dτ 2 6 1 120 1 2 1 ...

8.4 Periodic Motions Close to Separatrix Loop

213

Further, the family even solutions of equation (8.79) can be represented in the form cosh pτ Y1 = −A (8.82) cosh p where A is an arbitrary constant accompanied by the factor − cosh−1 p, which is convenient for further calculations due to the relationship Y1 (1) = −A. In particular, this provides the same order of magnitude for the arbitrary constant A as the parameter p and period T = 4a both go to inﬁnity. Further procedure is formally similar to the standard Poincare-Lindstedt algorithm for nonlinear conservative oscillators with positive linear stiﬀness. For example, substituting (8.82) into the right part of the equation (8.80), gives a ‘resonance term’ on the right-hand side of the equation, which is proportional to cosh pτ . This generates ‘hyperbolic secular terms’ of the form τ cosh pτ and τ cosh pτ in the particular solution of equation (8.80). Occurrence of such terms can be prevented, however, analogously to the PoincareLindstedt method by setting λ2 =

A2 8 cosh2 p

(8.83)

As a result, the particular solution of equation (8.80) takes the form Y3 =

A3 cosh 3pτ 192 cosh3 p

(8.84)

At the next stage, equation (8.81) gives solution

A5 1 Y5 = cosh 5pτ cosh 3pτ − 5 4096 cosh5 p under the condition λ4 = −

3A4 512 cosh4 p

(8.85)

(8.86)

Substituting (8.82) through (8.83) in (8.77) and (8.78), and setting ε = 1, gives approximate solution A3 cosh 3pτ A cosh pτ + cosh p 192 cosh3 p

A5 1 cosh 5pτ + cosh 3pτ − 5 4096 cosh5 p Y =π−

and

h=p 1− 2

where τ = τ (t/a) and a =

√

h.

3A4 A2 + 2 8 cosh p 512 cosh4 p

(8.87)

−1 (8.88)

214

8 Strongly Nonlinear Vibrations

The truncated series of successive approximations (8.87) and (8.88) depend upon two parameters, A and p, coupled by the boundary (continuity) condition (8.76) as follows

A3 cosh 3p A5 1 cosh 5p (8.89) A=π+ + cosh 3p − 5 192 cosh3 p 4096 cosh5 p Equation (8.89) should be interpreted as implicit function A = A(p) near the point A = π. Therefore, expansions (8.87) and (8.88) represent a oneparameter family of periodic solutions with the parameter p.

2 1

x 0 1 2 0

5

10

15

t Fig. 8.6 Analytical and numerical solutions of the period T=8.5 (p=2) shown by solid and thin lines, respectively.

Note that the equation (8.71) admits the group of time translations. As a result, another arbitrary parameter, say t0 , is introduced by substitution t → t + t0 . Figures 8.6 and 8.7 show that the analytical and numerical solutions are matching better for larger periods as the system trajectory becomes closer to the separatrix loop.

8.5

Self-excited Oscillator

This section illustrates the case when both X and Y components of solutions participate in the iterative process. In particular, we consider periodic self-sustained vibrations described the diﬀerential equation of motion x ¨ + g(x)x˙ + f (x) = 0

(8.90)

8.5 Self-excited Oscillator

215

3 2 1

x 0 1 2 3 0

10

20

30

40

t Fig. 8.7 Analytical and numerical solutions of the period T=24.5 (p=6) shown by the solid and thin lines, respectively.

where f (x) and g(x) are analytic functions, such that (Lienard’ conditions) a) G(x) =

"x

g(u)du is an odd function such that G(0) = G(±μ) = 0 for some

0

μ > 0, b) G(x) → ∞ if x → ∞, and G(x) is a monotonously increasing function for x > μ, c) f (x) is an odd function such that f (x) > 0 for x > 0. The above conditions guarantee that system (8.90) has a single stable limit cycle. In this case, the boundary-value problem (8.7) and (8.8) takes the form X = −a2 Rf − a (Rg Y + Ig X ) ≡ −εFX , X |τ =±1 = 0

(8.91)

Y = −a2 If − a (Ig Y + Rg X ) ≡ −εFY , Y |τ =±1 = 0 where the period of limit cycle T = 4a is unknown, expressions Rf and If as well as Rg and Ig are obtained by applying (8.5) and (8.6) to each of the functions f (x) and g(x), and notations εFX and εFY are introduced with the formal factor ε = 1 for further convenience. We seek solution of the boundary value problem (8.91) in the form of series of successive approximations X = X0 (τ ) + εX1 (τ ) + ε2 X2 (τ ) + · · ·

(8.92)

Y = Y0 (τ ) + εY1 (τ ) + ε Y2 (τ ) + · · · 2

a = q0 + εq1 + ε2 q2 + · · ·

(8.93)

216

8 Strongly Nonlinear Vibrations

Further solution procedure can be simpliﬁed by taking into account the symmetry properties X(−τ ) ≡ −X(τ ) and Y (−τ ) ≡ Y (τ ) due to the above conditions (a) through (c). Substituting (8.92) and (8.93) into (8.91) and matching the coeﬃcients of the respective powers of ε, gives the following sequence of boundary value problems X0 = 0 Y0 = 0, Y0 |τ =1 = 0

(8.94)

X1 = −FX,0 , (X0 + X1 ) |τ =1 = 0 Y1 = −FY,0 , Y1 |τ =1 = 0

(8.95)

Xi+1 = −FX,i , Xi+1 |τ =1 = 0 Yi+1 = −FY,i , Yi+1 |τ =1 = 0 (i = 1, 2, ...)

(8.96)

where

1 di FX 1 di FY | , F = |ε=0 (8.97) ε=0 Y,i i! dεi i! dεi Note that zero-order and ﬁrst-order approximations are coupled through the boundary condition for X-component in (8.95), whereas no boundary condition is imposed on X0 in (8.94). This speciﬁc represents a formalization of the physical assumption regarding the dominating component in the temporal mode shape of vibration, which is assumed to be close to the sawtooth sine rather than rectangular cosine. As a result, the generating system (8.94) gives solution (8.98) X0 = Aτ , Y0 ≡ 0 FX,i =

where A is an arbitrary constant. Substituting (8.98) in the right-hand side of equations (8.95) and integrating, yields τ X1 = −q02

τ (τ − ξ)f (Aξ)dξ,

Y1 = −Aq0

0

(τ − ξ)g(Aξ)dξ

(8.99)

0

Then, substituting (8.99) in the boundary conditions in (8.95), gives the following two equations for parameters q0 and A 1 q02

1 (1 − ξ)g(Aξ)dξ = 0

f (Aξ)dξ = A, 0

0

(8.100)

8.5 Self-excited Oscillator

217

Relationships (8.98) through (8.100) complete ﬁrst basic steps of the iterative procedure. Further iterations are organized then in a similar way as follows τ Xi+1 = −

τ (τ − ξ)FX,i (ξ)dξ,

Yi+1 = −

0

ζ dζ

1

FY,i (ξ)dξ

(8.101)

0

1 FX,i (ξ)dξ = 0; i = 1, 2, ....

(8.102)

0

Note that the boundary conditions for Yi+1 are satisﬁed automatically due to the lower limit of the outer integral in (8.101), whereas the boundary conditions for Xi+1 generates equations (8.102) for determining the coeﬃcients of series (8.93). Practically, high-order approximations can be obtained by using computer systems of symbolic manipulations. Example 15. Consider the self-excited oscillator with the power form stiﬀness of the degree m = 3, x ¨ + (bx2 − 1)x˙ + x3 = 0 In this case, g(x) = bx2 − 1 and f (x) = x3 . Conducting elementary integrations in (8.100), gives the algebraic system 1 2 3 q A = A, 4 0

1 q0 A 6 − bA2 = 0 12

with non-trivial solution q0 =

2b , A= 3

6 b

As a result, integrating (8.99), yields 6 τ5 , Y1 = τ 2 − τ 4 X1 = − b 5 All further steps of the procedure are conducted according to the same scheme (8.101) and (8.102). For instance, ﬁrst two steps of the procedure give approximate solution τ5 1 6 {τ − + [105τ 9 + 900τ 7 b − 21τ 5 (70b + 9) + 350τ 3 b]} x= b 5 3150 1 [20 − 43τ 2 + 20τ 4 + 216τ 6 ]}e +(1 − τ 2 ){τ 2 − 420 and the period

b 3 T = 4a = 8 1+ 6 20

218

8 Strongly Nonlinear Vibrations

Two more steps of the procedure correct the above expression for the period as follows

b 3 2960b + 2121 7367360b2 + 4554992b + 8659035 + T = 4a = 8 1+ + 6 20 50400 605404800 Figs.8.8 and 8.9 show limit cycle trajectories described by the analytical solutions in one and two iterations, respectively. For comparison, the numerical solution for transition to the limit cycle is also presented. Then, Fig. 8.10 illustrates dependence of the quarter of period parameter, a = T /4, versus the quantity b−1/2 , which can be viewed as an estimate for the amplitude of limit cycle.

v 3

b 1.0 m3

2

1

x

0 1

analytical, 1 iteration numerical

2 3 2

1

0

1

2

Fig. 8.8 Trajectories of numerical solution and the one iteration analytical limitcycle solution.

8.6

Strongly Nonlinear Oscillator with Viscous Damping

This section describes the successive approximation procedure combined with the asymptotic of small energy dissipation that leads to a slow amplitude decay. The scheme of the algorithm is closed to that was introduced earlier in [137].

8.6 Strongly Nonlinear Oscillator with Viscous Damping

219

v 3

b 1.0 m3

2

1

x

0 analytical, 2 iterations 1

numerical

2 3 1.5

1.0

0.5

0.0

0.5

1.0

1.5

Fig. 8.9 Trajectories of numerical solution and approximate (two iterations) analytical limit-cycle solution. 5

4

3

a analytical, four iterations

2

numerical estimates 1

0

0

2

4

6

8

12

b

Fig. 8.10 Illustration of convergence of the iterative procedure on the parameter plane.

220

8 Strongly Nonlinear Vibrations

So consider a strongly nonlinear oscillator under the viscous damping x ¨ + 2μx˙ + f (x) = 0

(8.103)

where f (x) is an odd function such that xf (x) ≥ 0, and 0 < μ << 1. The idea of two variable expansions will be employed below in combination with the sawtooth time substitution. Let us assume that τ (ϕ) is a ‘fast’ time scale, whose phase ϕ depends on the ‘slow’ time scale t0 = μt according to the following diﬀerential equation ϕ˙ = ω(t0 )

(8.104)

where the right-hand side is a priory unknown. Let us represent unknown solution of equation (8.103) in the form x = x(ϕ, t0 ) = X(τ (ϕ), t0 ) + Y (τ (ϕ), t0 )e(ϕ)

(8.105)

Substituting (8.105) in equation (8.103), and imposing ‘smoothness conditions,’ ∂X |τ =±1 = 0 (8.106) Y |τ =±1 = 0, ∂τ gives two partial diﬀerential equations ∂Y ∂2X − μ2 LX = −Rf − μH 2 ∂τ ∂τ ∂X ∂2Y ω 2 2 = −If − μH − μ2 LY ∂τ ∂τ

ω2

(8.107)

where, as follows from (8.5) and (8.6), 1 [f (X + Y ) + f (X − Y )] 2 1 If = If (X, Y ) = [f (X + Y ) − f (X − Y )] 2

Rf = Rf (X, Y ) =

and two linear diﬀerential operators are introduced

∂ dω H ≡ 2ω 1 + 0 + 0 ∂t dt ∂2 ∂ L ≡ 02 + 2 0 ∂t ∂t

(8.108)

Note that the fast and slow temporal scales are associated with diﬀerent physical processes developed in the system. The slow energy dissipation process is represented by the explicit small parameter μ, but there is no explicit parameter associated with perturbations of the triangular sine wave, which is supposed to be a generating solution of the iterative process. However, as

8.6 Strongly Nonlinear Oscillator with Viscous Damping

221

discussed earlier in this section, introducing the sawtooth temporal argument, implies that the entire right-hand side of system (8.107) is small. Otherwise, the triangular sine wave cannot be considered as a dominating component of the temporal mode shape of oscillations. Recall that, in a similar way, selecting the harmonic wave as a dominating solution in quasi harmonic approaches implies that nonlinearities are small regardless system’ parameters. Therefore, iterative procedure for boundary value problem (8.106) and (8.107) should incorporate two diﬀerent procedures, as those described above in this section, and a proper asymptotic procedure related to the dissipation process. Once again, the quasi-harmonic methods face similar situation when dealing with weakly nonlinear systems under small damping conditions. For instance, if being applied to such cases, the method of multiple scales accounts for both unharmonicity and dissipation, after appropriate assumption regarding the relation between non-linearity and damping parameters has been made. Very often though, such parameters are assumed to be of the same order of magnitude. As to the boundary value problem (8.106) and (8.107), similar assumption can be introduced by providing the terms Rf and If , and the parameter μ with the same formal ‘small factor’ ε = 1. Then, the multiple scales or two variables expansions can be organized by using the auxiliary parameter ε [137]. In the case of linear oscillator, such an algorithm recovers the exact solution of the linear diﬀerential equation of motion however in the speciﬁc form ε(1 − εμ2 ) ω exp(−t0 ) 0 sin τ (ωt) (8.109) x = X(τ, t ) = C ω ε(1 − εμ2 ) and ω2 =

ε(1 − εμ2 ) 4 arcsin2 ε/2

where C is an arbitrary constant, and another arbitrary constant can be introduced through the arbitrary time shift. Note that solution (8.109) includes no Y -component because, at every stage of the iterative process, it appears to be possible to satisfy condition H(∂X/∂τ ) = 0

(8.110)

Therefore, the second equation of system (8.107) is satisﬁed by setting Y ≡ 0. Practically, condition (8.110) generates the common factor exp(−t0 ) for all successive approximations. In general nonlinear case, however, it is rather impossible to satisfy condition (8.110) at every stage of the process, but it still works for leading order approximate solutions. Example 16. Consider the weakly damped oscillator of the m degree power form restoring force characteristic

222

8 Strongly Nonlinear Vibrations

x¨ + 2μx˙ + xm = 0

(8.111)

At this stage, the exponent m is an odd positive number. (It will be shown later that a broader class of power characteristics can also be considered.) Note that, for this kind of oscillators, whether or not the damping is small depends on the level of amplitude and the exponent m. This is due to the fact that, under small enough amplitudes, the elastic force becomes negligible regardless the magnitude of damping. By assuming that the inﬂuence of damping is negligible during one cycle of vibration, one can use expressions (8.40) for estimations of magnitudes of damping and elastic forces. As a result, the condition of ‘small damping’ derives in the form μ2

1 (m + 1)Am−1 4

One step of the procedure gives approximate solution [137]

−4μt τ m+2 x = Cexp τ− m+3 m+2 where τ = τ (ϕ) and the phase variable is approximated by

m−1 t ϕ = ϕ∞ 1 − exp −2μ m+3 ϕ∞ =

(8.112)

(8.113)

(8.114)

1 m + 3 C (m−1)/2 2μ m − 1 2(m + 1)

Interestingly enough, the above approximate solution predicts that the oscillator makes only a ﬁnite number of waves as m > 1 and t → ∞. Figs. 8.11 and 8.12 illustrate damped responses of the oscillator with two diﬀerent degrees of nonlinearity, m = 3 and m = 7, respectively. As follows from the diagrams, the approximate analytical solution and numerical one are matching relatively well, especially at higher exponent, m = 7. In particular, this justiﬁes the idea of using the sawtooth wave in strongly nonlinear cases, when the oscillator becomes close to the standard vibroimpact model.

8.6.1

Remark on NSTT Combined with Two Variables Expansion

In general, the iterative process of sawtooth expansions and the averaging procedure can be separated. Moreover, the stage of sawtooth time substitution does not impose any speciﬁc method of analyses. So let us apply the two variables method directly to the nonlinear boundary value problem (8.106) and (8.107) by means of the asymptotic series X = X0 (τ, t0 ) + μX1 (τ, t0 ) + μ2 X2 (τ, t0 ) + · · · Y = Y0 (τ, t0 ) + μY1 (τ, t0 ) + μ2 Y2 (τ, t0 ) + · · ·

(8.115)

8.6 Strongly Nonlinear Oscillator with Viscous Damping

223

Fig. 8.11 Approximate analytical and numerical solutions of the damped oscillator with cubic power form characteristic.

Fig. 8.12 Approximate analytical and numerical solutions of the damped oscillator with the seven-th degree power form characteristic.

224

8 Strongly Nonlinear Vibrations

and ω = ω0 (t0 ) + μω1 (t0 ) + μ2 ω2 (t0 ) + · · · H = H0 + μH1 + μ2 H2 + ... (8.116) where Hi = 2ωi 1 + ∂/∂t0 + dωi /dt0 . Substituting (8.115) and (8.116) into (8.106) and (8.107) and matching coeﬃcients of like powers of μ, gives, in particular, ∂ 2 X0 ∂X0 (0) |τ =±1 = 0 = −Rf (X0 , Y0 ), ∂τ 2 ∂τ 2 ∂ Y0 (0) ω02 = −If (X0 , Y0 ), Y0 |τ =±1 = 0 ∂τ 2

ω02

(8.117)

and ω02

∂ 2 X1 ∂ 2 X0 ∂Y0 (1) = −R (X , Y , X , Y ) − 2ω ω − H0 0 0 1 1 0 1 f ∂τ 2 ∂τ 2 ∂τ

∂X1 |τ =±1 = 0 ∂τ ∂ 2 Y1 ∂ 2 Y0 ∂X0 (1) ω02 (8.118) = −If (X0 , Y0 , X1 , Y1 ) − 2ω0 ω1 − H0 2 2 ∂τ ∂τ ∂τ Y1 |τ =±1 = 0 (0)

(1)

(0)

(1)

where Rf , Rf , ... and If ,If , ... are determined by the expansions (0)

(1)

(2)

R = Rf + μRf + μ2 Rf + ... (0)

(1)

(2)

I = If + μIf + μ2 If + ... By taking into account the assumptions on f (x) in equation (8.103), one can represent solution of problem (8.117) in the following general form X0 = X0 (τ, A, ω0 ),

Y0 ≡ 0

(8.119)

where A = A(t0 ) is an arbitrary function of the slow time scale, which is coupled with the frequency ω0 through the boundary condition ∂X0 (τ, A, ω0 ) |τ =1 = 0 ∂τ

(8.120)

In general, this relationship determines the implicit function ω0 = ω0 (t0 ). Now substituting solution (8.119) in (8.118), gives equations ω02

∂ 2 X1 ∂ 2 X0 + f (X )X = −2ω ω 0 1 0 1 ∂τ 2 ∂τ 2

(8.121)

8.6 Strongly Nonlinear Oscillator with Viscous Damping

225

and

∂ 2 Y1 ∂X0 (8.122) + f (X0 )Y1 = −H0 2 ∂τ ∂τ Let us consider equation (8.122). The best choice would be achieved by setting the right-hand side to zero and therefore making possible the solution Y1 ≡ 0 which is consistent with zero-order solution (8.119) and provides a better smoothness property of the corresponding solution at this stage; see condition (8.110). Note that the right-hand side cannot be always made zero for any τ unless the zero-order solution admits separation of the variables t0 and τ. However, it is still possible to ‘minimize’ the right-hand side of equation (8.122) by making it orthogonal to solution of the corresponding homogeneous equation, ∂X0 /∂τ , in other words, ω02

1 2

1 −1

∂X0 ∂X0 H0 dτ ≡ ∂τ ∂τ

.

∂X0 ∂X0 H0 ∂τ ∂τ

/ =0

(8.123)

Taking into account the expression H0 = 2ω0 1 + ∂/∂t0 + dω0 /dt0 and condition (8.123), gives 0

2 1 ∂X0 (8.124) = C exp(−2t0 ) ω0 ∂τ where C is an arbitrary constant. It can be shown that the ‘minimization condition’ (8.123) occurs also in a rigorous mathematical way based on the boundary conditions for Y1 ; at least, this can be easily veriﬁed in the linear case f (x) ≡ x.

8.6.2

Oscillator with Two Nonsmooth Limits

Consider the following generalization of equation (8.111) x¨ + 2μx˙ + sgn(x)|x|α = 0

(8.125)

where α is a non-negative real number; see the comments to equation (8.70). In this case, zero-order solution (8.119) can be obtained in the form (8.69), x(t) = A(t0 )sgn(τ (ϕ)) (8.126)

α |τ (ϕ)|α+2 |τ (ϕ)|α+2 |τ (ϕ)|2α+3 + − × |τ (ϕ)| − α+2 2(α + 2) 2α + 3 α+2 ϕ(t) ˙ = ω0 (t0 ),

t0 = μt

226

8 Strongly Nonlinear Vibrations

where the functions A(t0 ) and ω0 (t0 ) are coupled by relation (8.40) as follows A(α−1)/2 1 ω0 = = √ a α+1

1+

−1/2 α (α + 1)2 1+ 2(α + 2) (α + 2)(2α + 3)

Equations (8.127) and (8.124) admit exact solution

4μt α−1 A = C exp − t , ϕ = ϕ∞ 1 − exp −2μ 3+α α+3

(8.127)

(8.128)

where C is a new arbitrary constant and ϕ∞

C (α−1)/2 (2 + α) 2(3 + 2α) 1 α+3 = 2μ α − 1 (α + 1) (7α3 + 31α2 + 47α + 24)

(8.129)

It follows from expressions (8.128) and (8.129) that the linear system α = 1 plays the role of a boundary between the two strongly nonlinear areas N0 = {α : 0 ≤ α << 1} and N∞ = {α : 1 << α < ∞}

(8.130)

In other words, we show that α = 1 separates two qualitatively diﬀerent regions of the dynamics determined by the inﬂuence of diﬀerent nonsmooth limits of the potential well; see Fig. 8.13 for illustration. In particular, if α > 1 then the phase variable ϕ has the ﬁnite limit ϕ∞ as t → ∞. In contrast, if α < 1 then the phase with its temporal rate are exponentially growing, as the amplitude decays and the system approaches the bottom of the potential well. The physical meaning of this eﬀect is most clear from the limit case α → 0, which is discussed below. Figs. 8.14, 8.15 and 8.16, 8.17 illustrate solution (8.126) through (8.129) for large and small exponents α, respectively. The diagrams suggest quite a good match with numerical solution in both branches of the exponent (8.130). The numerical solutions shown by dashed lines were obtained by the stanR dard solver NDSolve built in Mathematica . Fig. 8.14 also shows that some divergence between the curves occurs when the amplitude is decreased to the level about A = 0.6. Below this level, the condition of small damping (8.112) is not guaranteed any more. In contrast, the curves are in a better match for smaller amplitudes if α < 1, see Fig. 8.16. In this case, the amplitude decay just strengthens condition (8.112). The phase plane diagrams shown in Figs. 8.15 and 8.17 have qualitatively diﬀerent shapes as dictated by the inﬂuence of diﬀerent nonsmooth limits of the potential well, see Fig. 8.13. Let us show now that solution (8.126) captures both nonsmooth limits α → 0 and α → ∞. For a physically meaningful transition to the limits, let us express the ˙ t=0 , arbitrary parameter C through the initial velocity v0 = x|

8.6 Strongly Nonlinear Oscillator with Viscous Damping

227

1/(α+1) v02 (α + 1) 7α3 + 31α2 + 47α + 24 C= 2(α + 2)2 (2α + 3)

and consider the two diﬀerent cases. 1) As α → ∞, the solution (8.126) through (8.129) gives x = τ (ϕ) v0 [1 − exp (−2μt)] ϕ= 2μ

(8.131)

Solution (8.131) exactly describes the system motion in the square potential well. 2) When α → 0, expressions (8.126) through (8.129) are reduced to

|τ (ϕ)| 4 2 x = v0 exp − μt τ (ϕ) 1 − (8.132) 3 2

2 3 μt − 1 exp ϕ= 2μv0 3 where the identity sgn[τ (ϕ)]|τ (ϕ)| ≡ τ (ϕ) has been taken into account. If, in addition μ = 0, then solution (8.132) also exactly describes the system dynamics with another nonsmooth limit of the potential energy, |x|, as shown in Fig. 8.13. However, if μ = 0 then substituting solution (8.132) into the diﬀerential equation of motion gives an error O(μ2 ). In terms of ﬁrstorder asymptotic solutions, the error of order μ occurs on the time period of order 1/μ. Therefore, solution (8.132) exactly captures the carrying shape of the vibration, but gives only asymptotic estimate for the exponential decay.

Fig. 8.13 Potential energy representation for the two limit oscillators.

228

8 Strongly Nonlinear Vibrations

Fig. 8.14 Temporal mode shape of the vibration for α ∈ N∞ , C = 1.5 and μ = 0.04; here and below, the dashed line represents numerical solutions.

4

2 v 0 2 4 1.0

0.5

0.0 x

0.5

1.0

Fig. 8.15 Phase plane diagram for α ∈ N∞ .

Note that the error of solution (8.126) is due to the error of the iterative procedure for elastic vibrations and the error of asymptotic for energy dissipation. As shown above, the error of successive approximations vanishes as either α → ∞ or α → 0, but the error of asymptotic vanishes only as α → ∞. Finally, let us discuss the qualitative diﬀerence of the dynamics in the parameter intervals N0 and N∞ . As follows from equation (8.128), for α ∈ N0 , the phase of vibration and the corresponding frequency are exponentially increasing in the slow time scale μt . The physical meaning of this phenomenon

8.6 Strongly Nonlinear Oscillator with Viscous Damping

1.0

229

Α 1 14 analytical, numerical

0.5

x 0.0 0.5 1.0 0

20

40

60

80

100

120

140

t Fig. 8.16 Temporal mode shape of the vibration for α ∈ N0 , C = 2.5 and μ = 0.04.

Fig. 8.17 Phase plane diagram for α ∈ N0 .

becomes most clear in the limit case α = 0. In this case, according to solution (8.132), the amplitude and frequency are, respectively

2 4 1 1 v02 exp − μt and ϕ˙ = μt = exp (8.133) A(μt) = 2 3 v0 3 2A(μt) Expressions (8.133) describe increasingly rapid vibrations -‘dither’- near the corner of the potential energy |x| as the amplitude approaches zero. This result is conﬁrmed by the much earlier analysis of the corresponding conservative case [78]. In contrast, when α ∈ N∞ , the oscillator makes a limited

230

8 Strongly Nonlinear Vibrations

number of cycles such that the phase ϕ remains bounded for any time t. Again, the most clear interpretation is obtained in the limit case α → ∞, when, as follows from (8.131), the phase variable ϕ(t) represents the total distance passed by the particle by time t, and ϕ˙ = v is the absolute value of the velocity. Since the barriers are perfectly elastic the particle reﬂects with no energy loss, the velocity v (t) remains continuous function of time described by the linear diﬀerential equation v˙ + 2μv = 0 or ϕ ¨ + 2μϕ˙ = 0. ˙ = v0 , one obtains exactly Under the initial conditions ϕ(0) = 0 and ϕ(0) solution (8.131). In conclusion, the explicit analytical solution for a class of strongly nonlinear oscillators with viscous damping is introduced. Two diﬀerent nonsmooth functions involved into the solution are associated with two diﬀerent nonsmooth limits of the oscillator. As a result, the solution is drastically simpliﬁed to give the best match with numerical tests if approaching any of the two limits.

8.7

Bouncing Ball

In this section, we consider a small ball of mass m falling under the gravity force onto a horizontal plane. In addition to the gravity, the ball is subjected to the linear damping with the coeﬃcient c. Impacts with the plane are inelastic with the restitution coeﬃcient κ. The vertical coordinate z(t) therefore is described by the following equations of motion m¨ z = −mg − cz˙ (z = 0) z˙+ = −κz˙− (z = 0)

(8.134)

where z˙− and z˙+ are velocities right before and immediately after the impact, respectively. Let h be a natural spatial scale of the system. This can be, for instance, the maximal height that has been reached by the ball during the very ﬁrst cycle. Introducing the coordinate transformation z = h|x| and re-scaling the time as t = h/gp, brings equations (8.134) to the form [204] d2 x dx + sgnx = 0 + 2μ dp2 dp

dx dx dx − = (1 − κ) dp + dp − dp − *

where 1 c μ= 2m

h g

(8.135)

8.7 Bouncing Ball

231

In the particular case κ = 1, solution (8.132) becomes applicable to equation (8.135). Then, returning back to the original notations of equation (8.134), gives

1 2 2c t |τ (ϕ)| − τ (ϕ) (8.136) z(t) = C exp − 3m 2 c g 3m exp t −1 ϕ(t) = C c 3m where C is a new arbitrary constant. If C = z˙02 /g then solution (8.136) satisﬁes the speciﬁc initial conditions z(0) = 0 and z(0) ˙ = z˙0 . One more arbitrary constant can be introduced by shifting the time, t− > t + t0 , that would allow to consider non-zero initial height of the ball. In order to compare solution (8.136) with numerical solution, let us represent equation (8.134) in the form z˙ = u c u˙ = −g − u m

(8.137)

where u+ = −κu− whenever z = 0. Further, introduce new unknown state variables {s, v} according to [73] z = ssgn(s) u = sgn(s)[1 − ksgn(sv)]v 1−κ k= 1+κ

(8.138)

Applying (8.138) to (8.137), gives1 s˙ = [1 − ksgn(sv)]v g c [sgn(s) + ksgn(v)] v˙ = − v − m 1 − k2

(8.139)

As compared to (8.137), system (8.138) automatically accounts for the velocity jump condition at z = 0. In other words, transformation (8.138) makes the strong nonlinearity (due to non-elastic impacts) explicitly present in (8.139). From the standpoint of numerical procedures, such a transformation enables one of using built-in solvers of diﬀerent packages with no impact conditioning at z = 0. Note that, in the particular case κ = 1 (k = 0), system (8.139) becomes equivalent to (8.135). In this particular case, the direct numerical integration R of equations (8.139), using the NDSolve procedure built in Mathematica 1

Note that diﬀerentiation of sgn-functions will produce Dirac’s delta-functions with eﬀectively zero factors however.

232

8 Strongly Nonlinear Vibrations

package, gives solution in a perfect match with analytical solution (8.136); see Fig.8.18. Let us assume now that the energy loss happens only due to impact interactions of the ball with the plane z = 0. The question is whether or not solution (8.136) can still be adapted by interpreting the parameter c as some “eﬀective damping coeﬃcient,” such that the energy loss between two impacts is equal to that happens in one impact.

0.05 0.04 0.03

analytical, numerical

z 0.02 0.01 0.00 0

1

2

3

4

5

6

t Fig. 8.18 NSTT analytical and direct numerical solutions for the height of bouncing ball under the linear dissipation condition; m = 1.0, c = 0.5, z(0) = 0 and z(0) ˙ = 1.0.

Assuming that the damping is small enough and using the classical parabolic approximation for the ball height during one cycle, z(t) = −gt2 /2+ v0 t, yields 2v0 /g 1 2cef f v03 cef f = m(1 − κ2 )v02 z˙ 2 dt = 3g 2 0 or 3mg cef f = (1 − κ2 ) (8.140) 4v0 Expression (8.140) shows that, in this case, eﬀective linear damping cannot be introduced on the entire time interval since the “initial velocity” v0 de˙ in (8.140), provides a good creases from cycle to cycle. Choosing v0 = z(0) enough match between analytical (8.136) and numerical solutions during ﬁrst few cycles of the motion, as follows from Fig. 8.19. Then, the divergence between the curves accelerates. The result can be improved by using some

8.7 Bouncing Ball

233

0.05 0.04 0.03

analytical

z

numerical 0.02 0.01 0.00 0

1

2

3

4

5

6

t Fig. 8.19 NSTT analytical solution with eﬀective damping corresponding to the coeﬃcient of restitution κ = 0.97, and direct numerical solutions for the height of bouncing ball; the parameters are: m = 1.0, c = 0.0, z(0) = 0 and z(0) ˙ = 1.0.

0.05 0.04 0.03

numerical

z

analytical

0.02 0.01 0.00 0

1

2

3

4

5

6

t Fig. 8.20 NSTT analytical solution with eﬀective damping adjusted by “variable initial velocity,” and direct numerical solutions for the height of bouncing ball; the parameters are: m = 1.0, c = 0.0, κ = 0.97, z(0) = 0 and z(0) ˙ = 1.0.

estimate for the “initial velocity” decay based on solution (8.136), where the eﬀective damping is still constant, however. This one-step iteration gives c ef f t (8.141) c˜ef f = cef f exp 3m Fig. 8.20 shows a better match between numerical and analytical solutions achieved due to modiﬁcation (8.141).

234

8.8

8 Strongly Nonlinear Vibrations

The Kicked Rotor Model

The so-called kicked rotor model is introduced in physics as a relatively simple essentially nonlinear model for chaotic behavior of systems, where one variable may be either bounded or unbounded in phase space [95], [20], [71]. The kicked rotor is described, in some units, by the Hamiltonian H=

∞ 1 2 I + K cos θ δ (t − n) 2 n=−∞

(8.142)

where I is the angular momentum, θ is the conjugate angle, and K is the stochasticity parameter that determines qualitative features of the dynamics. The sequence of pulses in (8.142) can be expressed through the sawtooth sine τ = τ (2t − 1) in the form, ∞

1 δ (t − n) = − τ (2t − 1) τ (2t − 1) 2 n=−∞

(8.143)

where primes denotes diﬀerentiation with respect to the entire argument of a function, 2t − 1. Note that the only role of the ﬁrst multiplier, τ (2t − 1), on the right-hand side is to provide pulses with the same sign. Therefore, (8.142) takes the form H=

1 2 1 I − Kτ (2t − 1) τ (2t − 1) cos θ 2 2

(8.144)

The corresponding diﬀerential equation of motion is 1 ¨ θ = − Kτ τ sin θ 2

(8.145)

We seek a family of solutions with the period T = 2 by introducing the sawtooth time argument τ , θ = θ(τ ) (8.146) Substituting (8.146) in (8.145), gives

dθ 1 d2 θ Kτ sin θ + = − τ dτ 2 8 dτ Eliminating the singular term τ , leads to the boundary condition dθ 1 |τ =±1 = − Kτ sin θ|τ =±1 dτ 8

(8.147)

8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

235

and the diﬀerential equation d2 θ =0 dτ 2 The boundary-value problem (8.147) and (8.148) admits solution θ = Aτ (2t − 1) + B

(8.148)

(8.149)

where A and B appear to be coupled by the set of equations

or

1 A = − K sin(A ± B) 8

(8.150)

1 A = − K sin A cos B 8

(8.151)

cos A sin B = 0

(8.152)

and In particular, equation (8.151) show that the number of periodic solutions of the period T = 2 is growing as the parameter K increases.

8.9

Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

Consider the case of asymmetric restoring force characteristics described by two pieces of smooth monotone increasing functions f (x) and g(x) of which one or both may be nonlinear (Fig. 8.21) f (x), −∞ < x < x1 (8.153) F (x) = g(x), x1 < x < ∞ It is assumed that the entire characteristic F (x) is at least continuous in the interval −∞ < x < ∞, in other words, the following matching condition holds f (x1 ) = g(x1 ) (8.154) and f (0) = 0. Expression (8.153) admits the unit form F (x) = f (x) + [g(x) − f (x)]H(x − x1 )

(8.155)

where H indicates the unit-step Heaviside function. Let us outline analytical procedure for periodic solutions of equation x ¨ + F (x) = 0

(8.156)

236

8 Strongly Nonlinear Vibrations

Fig. 8.21 Asymmetric piece-wise nonlinear restoring force characteristic.

Introducing the triangular wave temporal argument τ = τ (t/a) with unknown scaling parameter a and making the substitution x = X(τ ) in equation (8.156), gives boundary value problem X + hF (X) = 0

(8.157)

X |τ =±1 = 0

(8.158)

where, as usually, h = a2 = (T /4)2 . Following the physical reasoning discussed in Introduction and the second section of this chapter, we seek solution in the form of successive approximation series with no explicit small parameter X = X0 (τ ) + X1 (τ ) + X2 (τ ) + ...

(8.159)

h = h0 + h1 + ... where the generating solution X0 (τ ) obeys the diﬀerential equation X0 (τ ) = 0 and therefore describes the dynamics of a two-parameter family of free impact oscillators with arbitrary amplitudes A0 and the origin shift as follows X0 (τ, τ1 , A0 ) = x1 + A0 (τ − τ1 )

(8.160)

The idea behind approximation (8.160) is to choose its parameters in order to make the motion x(t) = X0 (τ (t/a)) in ‘some sense’2 close to that of oscillator (8.156). The form of (8.160) implies that the system is at the matching point whenever τ = τ1 . Next term of series (8.159) is found then from the equation X1 = −h0 F (X0 ) in the integral form 2

Note that the harmonic balance analogy of generating solution (8.160) would be x(t) = A sin ωt + B.

8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

237

τ X1 (τ, τ1 , A0 , A1 ) = A1 (τ − τ1 ) − h0

(τ − ξ)F (X0 (ξ, τ1 , A0 ))dξ

(8.161)

τ1

where the combination of arbitrary constants is similar to that in generating solution (8.160). Since X1 (τ1 , τ1 , A0 , A1 ) = 0 then τ = τ1 corresponds to the matching point x = x1 in ﬁrst two terms of the successive approximations, (8.160) and (8.161), τ X = x1 + (A0 + A1 )(τ − τ1 ) − h0

(τ − ξ)F (X0 (ξ, τ1 , A0 ))dξ

(8.162)

τ1

Therefore, after the ﬁrst correction to generating solution (8.160), we have three arbitrary constants A0 , A1 and τ1 , and one still unknown parameter h0 related to the period of free vibration. Now, substituting (8.162) in boundary conditions (8.158), gives 1 A0 + A1 − h0

F (X0 (τ, τ1 , A0 ))dτ = 0 τ1 τ1

A0 + A1 + h0

F (X0 (τ, τ1 , A0 ))dτ = 0

(8.163)

−1

where the variable of integration ξ has been formally replaced by τ . Equations (8.163) are equivalent to 1 F (X0 (τ, τ1 , A0 ))dτ = 0

(8.164)

F (X0 (τ, τ1 , A0 ))dτ = h−1 0 (A0 + A1 )

(8.165)

−1

1 τ1

If no more iterations are planned then we set A1 = 0, because the corresponding term contributes nothing qualitatively new into the approximate solution. In this case, equation (8.164) is used to express τ1 through another constant provides A0 and the matching point coordinate x1 . Then, equation (8.165) √ the link between the parameters of amplitude A0 and period T = 4 h0 . The next step of iteration employs the parameter A1 , however. The form of diﬀerential equation for the next step of procedure is analogous to (8.20), where γ1 h0 = h1 . Therefore, on the next step, the general solution is given by

238

8 Strongly Nonlinear Vibrations

τ X2 = −h1

(τ − η)F (X0 (η, τ1 , A0 ))dη τ1 τ

−h0

(8.166)

(τ − η)F (X0 (η, τ1 , A0 ))X1 (η, τ1 , A0 , A1 )dη + A2 (τ − τ1 )

τ1

where zero lower limit of integration provides X2 (τ1 ) = 0, and the prime indicates derivative with respect to the entire argument, F (X0 ) ≡ dF (X0 )/dX0 . Note that the characteristic F (x) is, generally speaking, nonsmooth at the matching point x1 so that the derivative F (x) may not exist at x = x1 . Although integration of step-wise discontinuous functions is still possible, the current procedure is designed to avoid calculating derivatives of the characteristic at the point x1 . For that reason, the lower limit of integration in (8.166) is associated with the non-smoothness point by expression (8.160). In addition, as follows from (8.161), the uncertainty F (x1 ) in the integrand is suppressed by zero factor X1 (τ1 ) = 0. Obviously, on the next step of iteration, this factor will accompany the second derivative F (x1 ), whereas the ﬁrst derivative acquires the factor X2 (τ1 ). Now, applying boundary conditions (8.158) to (8.166), yields 1 −h1

F (X0 (τ, τ1 , A0 ))dτ τ1

1 − h0

F (X0 (τ, τ1 , A0 ))X1 (τ, τ1 , A0 , A1 )dτ + A2 = 0

(8.167)

τ1

τ1 h1

F (X0 (τ, τ1 , A0 ))dτ

−1

τ1 + h0

F (X0 (τ, τ1 , A0 ))X1 (τ, τ1 , A0 , A1 )dτ + A2 = 0

(8.168)

−1

Substracting the both sides of equation (8.167) from (8.168) and taking into account equation (8.164), gives 1 −1

F (X0 (τ, τ1 , A0 ))X1 (τ, τ1 , A0 , A1 )dτ = 0

(8.169)

8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

239

Then, taking into account (8.165), brings equation (8.167) to the form 1

−1 F (X0 (τ, τ1 , A0 ))X1 (τ )dτ = −h1 h−2 0 (A0 + A1 ) + h0 A2

(8.170)

τ1

If no more iterations needed, then we set A2 = 0 in expression (8.166) and equation (8.170). Further, substituting (8.161) in (8.169), gives equation for A1 , whereas (8.170) gives equation for h1 . This completes two steps of successive approximations, although the algebraic problem still persists, and its complexity depends on the functions f (x) and g(x).

Chapter 9

Strongly Nonlinear Waves

Abstract. This short chapter deals with modulated waves described by strongly nonlinear Klein-Gordon equation. Mechanical equivalent for this model can be represented by the inﬁnite linear string on elastic foundation. In this case, the principal (carrying) wave can be approximated by polynomials or other elementary functions of the triangular sine wave based on the algorithms of the previous chapter. Then the diﬀerential equations for slow varying modulating components are derived by using the Whitham assumptions and the speciﬁc boundary conditions associated with nonsmooth argument substitutions.

9.1

Wave Processes in One-Dimensional Systems

A one-dimensional wave can be described by the function u = u(θ, x, t)

(9.1)

where θ = θ(x, t) is a phase variable, x and t are the coordinate and time, respectively. In the stationary case, the phase variable is usually the only argument of the wave shape function u = u(θ), where θ = ωt − kx with temporal and spatial wave numbers, ω and k, respectively, so that ∂θ =ω ∂t

and

∂θ = −k ∂x

(9.2)

Function (9.1) includes also the variables x and t explicitly in order to describe wave modulation eﬀects. Therefore, we assume that the dependence upon explicit x and t is slow. Such an assumption can be formalized by means of the formal auxiliary parameter ε [194], [116] u = u(θ, x0 , t0 ) where x0 = εx, t0 = εt, and relationships (9.2) take now the form V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 241–244, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(9.3)

242

9 Strongly Nonlinear Waves

∂θ = ω(x0 , t0 ) and ∂t

∂θ = −k(x0 , t0 ) ∂x

(9.4)

Let the wave shape function u be periodic with respect to θ with the period T = 4. In this case, the sawtooth phase argument can be introduced as follows τ = τ (θ), (9.5) u = U (τ, x0 , t0 ) + V (τ, x0 , t0 )e where e = e(θ) = τ (θ) is the rectangular cosine of the phase variable.

9.2

Klein-Gordon Equation

For illustration purposes, we consider the Klein-Gordon equation ∂2u ∂2u − 2 + f (u) = 0 ∂t2 ∂x

(9.6)

where f (u) is an odd function. Substituting (9.5) in (9.6) and taking into account (9.4), gives 2 ∂2U ∂V ω − k2 − ε2 LU = −Rf (U, V ) − εH ∂τ 2 ∂τ 2 ∂ 2V ∂U ω − k2 − ε2 LV = −If (U, V ) − εH 2 ∂τ ∂τ ∂U |τ =±1 = 0 ∂τ V |τ =±1 = 0 where

(9.7)

(9.8)

∂ ∂ ∂k ∂ω H ≡ 2 ω 0 +k 0 + 0 + 0 ∂t ∂x ∂t ∂x ∂2 ∂2 L ≡ 02 − 02 ∂t ∂x

and 1 [f (U + V ) + f (U − V )] 2 1 If = [f (U + V ) − f (U − V )] 2

Rf =

Analogously to the case of strongly nonlinear damped oscillator considered in the previous chapter, we represent solution in the form of series

9.2 Klein-Gordon Equation

243

U (τ, x0 , t0 ) = U0 (τ, x0 , t0 ) + εU1 (τ, x0 , t0 ) + ... V (τ, x0 , t0 ) = V0 (τ, x0 , t0 ) + εV1 (τ, x0 , t0 ) + ...

(9.9)

Substituting (9.9) into (9.7) and (9.9) and equating coeﬃcients of like powers of ε, gives in ﬁrst two steps 2 ∂ 2 U0 ω − k2 = −f (U0 ) ∂τ 2 V0 ≡ 0 ∂U0 |τ =1 = 0 ∂τ where U0 (−τ ) = −U0 (τ ) due to oddness of the function f (U0 ), and 2 ∂ 2 U1 ω − k2 = −f (U0 )U1 ∂τ 2 2 ∂ 2 V1 ∂U0 ω − k2 = −f (U0 )V1 − H ∂τ 2 ∂τ ∂U1 |τ =±1 = 0 ∂τ V1 |τ =±1 = 0

(9.10) (9.11)

(9.12)

(9.13)

The generating solution is obtained from (9.10) in the following general form U0 = U0 (τ, A, ω 2 − k 2 )

(9.14)

where A, ω, and k are functions of the slow variables x0 and t0 . The functions A(x0 , t0 ), ω(x0 , t0 ), and k(x0 , t0 ) are coupled by the boundary condition (9.11) ∂U0 (τ, A, ω 2 − k 2 ) |τ =1 = 0 (9.15) ∂τ and by the condition of solvability of the boundary value problem (9.12) for V1 , . / 1 ∂U0 ∂U0 ∂U0 ∂U0 H dτ = 2 H =0 ∂τ ∂τ ∂τ ∂τ −1

or

0

0

2 1 2 1 ∂U0 ∂U0 ∂ ∂ ω + 0 k =0 ∂t0 ∂τ ∂x ∂τ

(9.16)

Example 17. Consider the equation ∂2u ∂2u − 2 + sgn(u)|u|α = 0 ∂t2 ∂x

(9.17)

244

9 Strongly Nonlinear Waves

where α is a real positive number. In this case, the generating solution is obtained analogously to the case of oscillator with two non-smooth limits, considered in Chapter 8, by replacing ω02 with ω 2 − k 2 . In particular, Aα−1 ω −k = α+1 2

2

1+

−1 α (α + 1)2 1+ 2(α + 2) (α + 2)(2α + 3)

(9.18)

whereas equation (9.16) gives ∂(ωA2 ) ∂(kA2 ) + =0 ∂t ∂x

(9.19)

Equations (9.18) and (9.19) have been written in terms of the original time and coordinate by setting the auxiliary parameter to unity, ε = 1. However, according to the basic assumption, A = A (t, x), ω = ω (t, x) and k = k (t, x) still should be treated as slow functions as compared to τ (θ).

Chapter 10

Impact Modes and Parameter Variations

Abstract. In this chapter, a new parameter variation and averaging tools are introduced for impact modes. It is also shown that a speciﬁc combination of two impact modes gives another impact mode1 . The corresponding manipulations with impact modes become possible due to the availability of closed form exact solutions obtained by means of the triangular sine temporal substitution for impulsively loaded and vibroimpact systems. In particular, the idea of Van-der-Pol and averaging tool are adapted for the case of impact oscillator. For illustrating purposes, a model of coupled harmonic and impact oscillators is considered. Further, mass-spring systems with multiple impacting particles are considered in order to illustrate impact localization phenomena on high-energy levels.

10.1

An Introductory Example

Vibration modes with impacts have been under study for several years [191], [197], [15], [156]. In practical terms, such studies deal with the dynamics of elastic structures whose amplitudes are limited by stiﬀ constraints. These may be designed intentionally or occur due to a deterioration of joints. As a result, such kinds of dynamics are often accompanied by a rattling noise or dither during operating regimes of vehicles or machine tools. From the theoretical standpoint, the interest to such problems is driven by the question what happens to linear normal modes as the energy of elastic vibrations becomes suﬃcient for reaching the constraints. Interestingly enough, some of the analytical approaches developed in the area recently found applications in molecular dynamics [49]. However, due to strong nonlinearities of the impact dynamics, most of the results relates to periodic particular solutions according to the idea of nonlinear normal modes [190]. Let us recall that the importance of linear normal modes is emphasized by the linear superposition principle as 1

Notice that the number of impact modes depends on the number of constraints and therefore can signiﬁcantly exceed the number of degrees of freedom.

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 245–264, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

246

10 Impact Modes and Parameter Variations

well as the parameter variation and averaging methods for weakly nonlinear cases. Let us consider a one degree of freedom free harmonic oscillator between two absolutely rigid barriers. A mechanical model of such an oscillator can be represented as a mass-spring model with two-sided amplitude limiters as shown in Fig. 10.1 (a).

Fig. 10.1 The oscillator with bilateral rigid barriers (on the left) is replaced by the oscillator under the periodic series of external impulses (on the right).

The interaction with the barriers at x = ±Δ is assumed to be perfectly elastic, and the system is represented in the form x ¨ + ω02 x = 0,

|x| ≤ Δ

(10.1)

Since the normal mode regimes are periodic by their deﬁnition then the reaction of constraints can be treated as a periodic series of external impulses acting on the masses of the system.

10.1 An Introductory Example

247

Applying this remark to the one degree of freedom system as it is shown in Fig. 10.1 (b), the related diﬀerential equation of motion is written in the linear form x ¨ + ω02 x = 2p

∞

[δ (ωt + 1 − 4k + α) − δ (ωt − 1 − 4k + α)] (10.2)

k=−∞

= pτ (ωt + α) where δ (ξ) is the Dirac function, τ (ξ) is the triangular sine wave, 2p, ω, and α will be interpreted as arbitrary parameters. For further convenience, the right-hand side of equation (10.2) is expressed through second-order generalized derivative of the triangular sine wave with respect to the entire argument, ωt + α. The parameter ω will be called a frequency parameter, although it diﬀers by the factor π/2 from the regular trigonometric frequency, Ω = (π/2) ω. Both parameters, ω and Ω, may be used below. In contrast to system (10.1), the auxiliary system (10.2) is linear but not completely equivalent to the original one; see the analyses below. Representing unknown steady-state periodic solution in the form, x = X (τ ) , τ = τ (ωt + α)

(10.3)

gives the boundary value problem with no singular terms, ω 2 X (τ ) + ω02 X (τ ) = 0

X (τ ) |τ =±1 = pω

(10.4) −2

and the related solution is represented in the saw-tooth time form [139] x=

p sin [(ω0 /ω) τ (ωt + α)] ωω0 cos (ω0 /ω)

(10.5)

This solution can be veriﬁed by direct substitution of expression (10.5) into the equation of motion (10.2). A connection between solution (10.5) and vibration of the original system with stiﬀ constraints is established by imposing the conditions: • The impulses on the right-hand side of equation (10.2) act when the mass strikes the limiters, x = ±Δ if τ = ±1 ⇐⇒ τ = 0

(10.6)

• The system cannot penetrate through the limiters, therefore, |x| ≤ Δ for all τ ∈ [−1, 1]

(10.7)

248

10 Impact Modes and Parameter Variations

Substituting solution (10.5) into condition (10.6), determines the parameter p, (10.8) p = Δωω0 cot (ω0 /ω) Substituting now (10.8) in (10.5), gives solution in the ﬁnal form x(t) = Δ

sin [(ω0 /ω) τ (ωt + α)] sin (ω0 /ω)

(10.9)

Obviously, solution (10.9) satisﬁes condition (10.6) automatically. The related parameter p (10.8) will further be treated as an “eigen value” of the nonlinear (impact) problem. Other parameters, ω and α, are determined by the initial conditions. Let us assume that x (0) = 0 and thus α = 0. As a result, the total energy of the oscillator per unit mass is expressed through the initial velocity as 2 E = [x˙ (0)] /2. Then, taking into account (10.9) and making some analytical manipulations, gives

1 ω 2 Δ2 ω0 = ± arccos 1 − 0 + kπ (10.10) ω 2 E where k = 0,1, ... . The right-hand side of expression (10.10) gives a sequence of real numbers if the total energy is above its critical value, E ≥ E∗ = ω02 Δ2 /2, so that the oscillator can reach the constraints. However, not all of the real numbers ω lead to real motions of the original system. Indeed, since the auxiliary system (10.2) has no limiters then condition (10.6) does not guarantee that the oscillator will remain inside the region |x| ≤ Δ during the period of vibration. Therefore, condition (10.7) must be veriﬁed as well. Such a veriﬁcation implemented for solution (10.9) shows that condition (10.7) is satisﬁed only for the smallest root in set (10.10). Fig. 10.2 illustrates the temporal mode shapes corresponding to the ﬁrst two roots ω0 /ω. It is seen that the second solution (on the right), violates condition (10.7) while satisﬁes condition (10.6). Note that the above approach can be applied to the case of unilateral limiters. Let us remove, for instance, the left limiter and consider the oscillator (10.1) under the condition x ≤ Δ. In this case, the boundary conditions in (10.4) should be modiﬁed as X (τ ) |τ =±1 = ±pω −2

(10.11)

Such a periodic change of sign eﬀectively switches the directions of positive δ-pulses on the right-hand side of equation (10.2). As a result, the solution takes the form, cos [(ω0 /ω) τ (ωt + α)] x(t) = Δ (10.12) cos (ω0 /ω)

10.2 Parameter Variation and Averaging

249

a

2 1 x

b

2 1 x

0

0

1

1

2

2 0

5

10

15

20

0

5

t

10

15

20

t

Fig. 10.2 Real (a) and “phantom” (b) solutions corresponding to the ﬁrst (smallest) and second roots respectively, (ω0 /ω)1 = 1.1502 and (ω0 /ω)2 = −1.1502 + π. The total energy level is E = 1.2E∗ .

where the period and the related basic frequency are T = 2/ω and Ω = πω, respectively.

10.2

Parameter Variation and Averaging

In order to illustrate the idea of parameter variations for solution (10.9), let us include the viscous damping into the model and represent the diﬀerential equations of motion between the constraints in the form x˙ = v v˙ = −2ζω0 v − ω02 x

(10.13)

where ζ is the damping ratio parameter. In this case, the triangular wave frequency ω in solution (10.9) is not constant any more, although the amplitude of the vibration remains constant as long as the oscillator is in the impact regime. The corresponding parameter variation is implemented as a change of the state variables {x(t), v(t)} → {γ(t), φ(t)}, dictated by solution (10.9) sin(γτ (φ)) sin γ cos(γτ (φ)) v = ω0 Δ e(φ) sin γ

x=Δ

(10.14)

250

10 Impact Modes and Parameter Variations

where τ = τ (φ) and e = τ (φ) depend upon the fast phase φ = φ(t), and γ = γ(t) determines a relatively slow evolution of the temporal mode shape of the vibration. Substituting (10.14) in (10.13), gives still exact equations γ˙ = 2ζω0 cos2 γτ tan γ ω0 [1 + eζ(sin 2γτ − 2τ cos2 γτ tan γ)] φ˙ = γ

(10.15)

Below, the ﬁrst-order averaging procedure is applied. Notice that the righthand side of equations (10.15) is periodic with respect to the phase φ. However, as proved in Chapter4, the averaging can be conducted with respect to the variable τ over its interval −1 ≤ τ ≤ 1. As a result, one obtains

1 sin 2γ tan γ γ˙ = ζω0 1 + 2γ 8 = 2ζω0 γ + ζω0 γ 5 + O(γ 6 ) 45 ω 0 (10.16) φ˙ = γ Keeping the leading-order term only on the right-hand side of the ﬁrst equation in (10.16), gives solution γ = γ0 exp(2ζω0 t) 1 φ= [1 − exp(−2ζω0 t)] 2ζγ0

(10.17)

A simple asymptotic analysis of expressions (10.14) and the remark after expression (10.10) give the parameter interval, 0 < γ < π/2, within which the impact dynamics takes place. The vibration mode shapes close to the triangular wave near the left edge of the interval, but, as the energy dissipates and the parameter γ approaches π/2, vibrations become close to harmonic. The total energy is expressed through the parameter γ in the form E(t) =

1 ω02 Δ2 2 sin2 γ(t)

(10.18)

The duration of the impact stage of the dynamics is estimated via solution (10.17),

π 1 ln γ(tmax ) = π/2 =⇒ tmax = (10.19) 2ζω0 2γ0 where γ0 = γ(0). As follows from Fig. 10.3, the above averaging procedure leads to practically no error of the time history record within the entire interval of validity of the approach. However, the analytical solution based on the reduced model

10.2 Parameter Variation and Averaging

251

1.0

0.5

x

0.0

0.5

1.0 0

10

a

20

30

40

t

1.0

numerical, averaging 0.5

x

0.0

analytical

0.5

1.0 80

b

82

84

86

88

90

t

Fig. 10.3 The time history of the impact oscillator according to exact equations, those after the averaging, and the analytical closed form solution; the parameters are γ0 = 0.2, ζ = 0.01, ω0 = 1.0, and Δ = 1.0.

gives some deviation from the exact curve at the end of the impact stage of the dynamics. Notice that there is no impact interactions with the constraints for t > tmax , where the model becomes harmonic oscillator whose amplitude exponentially decays due to the energy dissipation. At this stage, the

252

10 Impact Modes and Parameter Variations

transformation (10.14) is not valid any more nor there is any need in transformations. However, a question occurs about such solutions that would be capable of describing both impact and non-impact stages within the same unit-form expressions.

10.3

A Two-Degrees-of-Freedom Model

Let us obtain ﬁrst the impact mode solutions for the model shown in Fig. 10.4 under no damping condition, c = 0. For the sake of simplicity, let us also assume that k1 = k2 = k. On the normal mode (impact) motions, the system can be eﬀectively replaced by x ¨1 + ω02 (2x1 − x2 ) = pe (ωt) x ¨2 + ω02 (2x2 − x1 ) = 0

(10.20)

where ω02 = k/m, and the parameters ω and p must provide the following condition (10.21) |x1 | ≤ Δ The impact mode solution is represented in the form xn (t) = Xn (τ )

(10.22)

where τ = τ (ωt) and n = 1, 2. Substituting (10.22) in (10.20) and eliminating the singular term e (ωt), gives the linear boundary value problem ω 2 X1 + ω02 (2X1 − X2 ) = 0 ω 2 X2 + ω02 (2X2 − X1 ) = 0

(10.23)

X1 |τ =±1 = pω −2 X2 |τ =±1 = 0 The corresponding solution has the form p sin(ω0 τ /ω) + X1 = 2ωω0 cos(ω0 /ω) p sin(ω0 τ /ω) X2 = − 2ωω0 cos(ω0 /ω)

√ 3 sin( 3ω0 τ /ω) √ 3 cos( 3ω0 /ω) √ √ 3 sin( 3ω0 τ /ω) √ 3 cos( 3ω0 /ω)

(10.24)

√

(10.25)

where the ‘eigen value’ p is determined by substituting X1 into X1 |τ =±1 = ±Δ

(10.26)

10.4 Averaging in the 2DOF System

253

This gives √ √ p = 2ωω0 Δ[tan(ω0 /ω) + ( 3/3) tan( 3ω0 /ω)]−1

(10.27)

In order to insure that solution (10.25) and (10.27) describe real motions, condition (10.7) must be veriﬁed. As follows from detailed considerations below, condition (10.7) is satisﬁed at least in the frequency range (π/2) ω > ω2 , where ω2 is the frequency of the out-of-phase mode of the related no impact system.

10.4

Averaging in the 2DOF System

Let us consider now the model shown in Fig. 10.4 by assuming that a relatively slow energy dissipation is possible due to the viscous damping. In order to introduce the corresponding parameter variation technique, let us represent the diﬀerential equations of motion in the following general form x˙ 1 = v1 v˙ 1 = −f1 (x1 , x2 , v1 , v2 ) + pe (φ) x˙ 2 = v2

(10.28)

v˙ 2 = −f2 (x1 , x2 , v1 , v2 ) where pe (φ) = pτ (φ) and f1 (x1 , x2 , v1 , v2 ) = 2ζΩ1 v1 + Ω12 x1 + β(x1 − x2 ) f2 (x1 , x2 , v1 , v2 ) = Ω22 x2 − β(x1 − x2 )

(10.29)

are impact and linear force components per unit mass, β = k/m is the parameter of coupling, ζ = c/(2Ω1 m) is the damping ratio, and Ωi = ki /m (i = 1, 2).

Fig. 10.4 The two-degrees-of-freedom model with viscous damping in the impact subcomponent.

254

10 Impact Modes and Parameter Variations

The idea of parameter variations is implemented below as a change of the state variables, {x1 (t), v1 (t), x2 (t), v2 (t)} → {γ(t), φ(t), A(t), B(t)}, according to expressions sin(γτ (φ)) sin γ cos(γτ (φ)) e(φ) v1 = Ω1 Δ sin γ

πτ (φ) πτ (φ) x2 = A sin + B cos e(φ) 2 2

πτ (φ) πτ (φ) + Ω1 A cos e(φ) v2 = −Ω1 B sin 2 2

x1 = Δ

(10.30)

It is assumed in (10.30) that the principal frequency of the vibration, dφ(t)/dt, is dictated by the impact subcomponent rather then by a natural frequency of the corresponding linearized system. However, the scaling factor Ω1 is still used in order to indicate the dominant oscillator of the dynamic process under consideration. Notice that x1 and v1 are transformed analogously to (10.14), whereas the transformation of x2 are v2 is based on the standard general solution of the harmonic oscillator represented however in the non-smooth temporal transformation form. Substituting (10.30) in (10.28), gives e cos γτ (f1 sin γ − Ω12 Δ sin γτ ) tan γ Ω1 Δ 1 [f1 sin γ(sin γτ − τ cos γτ tan γ) φ˙ = γΩ1 Δ +Ω12 Δ cos γτ (cos γτ + τ sin γτ tan γ)] 1 1 A˙ = − Ω1 B(1 − cos πτ ) + πB φ˙ 2 2 1 1 πτ +( Ω1 A sin πτ − )e f2 cos 2 Ω1 2 1 1 B˙ = Ω1 A(1 + cos πτ ) − πAφ˙ 2 2 1 1 πτ + f2 sin − e Ω1 B sin πτ Ω1 2 2 γ˙ =

(10.31)

where the functions f1 and f2 are expressed through the new variables by substitution (10.30) in (10.29), and the impact term pe (φ) has been eliminated by setting (compare with (10.8)) ˙ p = p(t) = Δφ(t)Ω 1 cot γ(t)

(10.32)

Further reduction of system (10.31) includes two major steps, such as averaging with respect to the fast phase φ and applying the power series expansion

10.4 Averaging in the 2DOF System

255

with respect to the parameter γ. Since the periodic functions in equations (10.31) are expressed through the triangular sine wave τ (φ) then the averaging can be implemented by considering τ as an argument of the averaging as described in Chapter 4. Then, keeping the leading-order terms of the power series expansions with respect to γ, gives γ˙ = 2ζΩ1 γ Ω1 φ˙ = γ

β + Ω12 + Ω22 π A˙ = −B − φ˙ 2Ω1 2

2 2 β + Ω1 + Ω2 4βΔ π B˙ = A − φ˙ − 2 2Ω1 2 π Ω1

(10.33)

Approximate equations (10.33) describe only one-way interaction between the oscillators so that the ﬁrst two equations can be easily solved analytically; see the above one degree-of-freedom case. Then, substituting the result into the next two equations, gives a linear set of equations with variable coeﬃcients for A(t) and B(t), which can also be considered analytically. Let us skip such kind of analysis but illustrate the ﬁnal result in Figs. 10.5 and 10.6 for

0.00020

reduced model numerical

0.00015

Ρ

averaging 0.00010

0.00005

0.00000 0

20

40

60

80

t Fig. 10.5 The total energy of the second oscillator at relatively weak coupling, β = 0.01

256

10 Impact Modes and Parameter Variations

0.007

numerical

0.006

averaging

0.005 0.004

Ρ 0.003

reduced model

0.002 0.001 0.000 0

20

40

60

80

t Fig. 10.6 The total energy of the second oscillator at stronger coupling, β = 0.05

two diﬀerent strengths of the coupling β. The initial conditions and other parameters are selected as follows: γ(0) = 0.2, φ(0) = 0.0, A(0) = −0.01, B(0) = 0, ζ = 0.01, Ω1 = Ω2 = 1.0, Δ = 1.0. The diagrams show the energy versus time of the second oscillator based on numerical solutions for three diﬀerent equations, such as exact equations (10.31), the equations obtained by averaging (not described here), and the reduced set (10.33). The solutions are in quite a good match most of the time interval, however, the solution of truncated set (10.33) show some error near the end of the interval. This happens because the parameter γ is slowly approaching its limit magnitude π/2, at which the ﬁrst oscillator stops interacting with the constraints and the entire system becomes linear. Remind that equations (10.33) were obtained by truncating the polynomial expansions in the neighborhood of γ = 0; as a result, the accuracy of the equations is low near the point γ = π/2. However, the precision can be signiﬁcantly improved by keeping few more terms of the power series with respect to γ.

10.5

Impact Modes in Multiple Degrees of Freedom Systems

Let us consider the N -degrees-of-freedom conservative system described by T the coordinate vector x = (x1 , ..., xN ) ∈ RN . The corresponding mass-spring model is shown in Fig. 10.7. It is assumed that displacements of the ath mass

10.5 Impact Modes in Multiple Degrees of Freedom Systems

257

Fig. 10.7 The mass-spring model of a discrete elastic system with displacement limiters.

is limited by perfectly stiﬀ elastic constraints, such that |xa | ≤ Δa or, in the matrix form, T Ia x ≤ Δa (10.34)

T where Ia = 0, ..., 1, ..., 0 . a

Inside the domain (10.34), the diﬀerential equations of motion are assumed to be linear, Mx ¨ + Kx = 0 (10.35) where M and K are constant mass and stiﬀness N ×N -matrixes, respectively. The form of matrix equation (10.35) however is general enough to describe diﬀerent models, not necessarily mass-spring chains, see the ﬁrst example below. In order to obtain impact mode solutions, the system (10.34) and (10.35) is replaced by the following impulsively forced linear system under no constraints condition (10.36) Mx ¨ + Kx = pIa τ (ωt + α) where p is a priory unknown ‘eigen-value,’ and ω and α are arbitrary constant parameters. A family of periodic solutions of the period T = 4/ω can be found as a linear superposition of solutions (10.5) for each of the N linear modes of system (10.35) with appropriate replacement of the parameters, N (eTj Ia )ej sin[(ωj /ω)τ (ωt + α)] x(t) = p ωωj cos(ωj /ω) j=1

(10.37)

where ej and ωj are the jth normal mode and the natural frequency of linear system (10.35); it is assumed that ωi < ωj when i < j, and the linear normal modes are normalized as

258

10 Impact Modes and Parameter Variations

eTj M ei = δji

(10.38)

where δji is the Kronecker symbol. The impulses act at those time instances when the ath mass interacts with the constraints, in other terms, ITa x = ±Δa when τ = ±1

(10.39)

Substituting (10.37) into (10.39), one obtains the related “eigen-value,” ⎡ p = Δa ω ⎣

N j=1

⎤−1 (eTj Ia )2

tan(ωj /ω) ⎦ ωj

(10.40)

where eTj Ia is the ath component of the jth linear mode vector. Substitution (10.40) into (10.37) gives a two parameter family of the periodic solutions for the impact modes. The parameter α is an arbitrary phase shift, whereas the frequency parameter ω is subjected to some restrictions due to condition (10.34). As shown in [156], condition (10.34) is satisﬁed when the principal frequency of vibration, Ω = (π/2) ω, exceeds the highest frequency of the linear spectrum, Ω > ωN . The corresponding impact mode represents an extension of the highest linear normal mode, in which any two neighboring masses vibrate out-of-phase. Such an impact mode becomes spatially localized as Ω → ∞. This result was obtained also by qualitative methods in [197]. Condition (10.34) may be satisﬁed also when the frequency Ω is located in a small enough right neighborhood of any frequency ωj and, in addition, ωi /ωj is not an odd number for all i = j. The idea of the proof is to ﬁnd such cases when ITa x is a monotonic function of τ on the interval −1 ≤ τ ≤ 1, and hence condition (10.39) at the boundaries guarantees that inequality (10.34) holds inside the entire interval. Generally, the impact modes appear to have a quite complicated spectral structure. Therefore, a detailed investigation may be required in order to formulate necessary and suﬃcient conditions of impact mode existence. However, a suﬃcient condition of non-existence can be formulated by using the physical meaning of the parameter p. Namely, if an impact mode exists then the inequality p (ω) > 0 holds. Indeed, the parameter p (10.40) cannot be negative for any real impact vibrating regime as a reaction of constraint, because it cannot be directed to a barrier. Thus impact modes can not exist when p (ω) < 0.

10.5.1

A Double-Pendulum with Amplitude Limiters

The top mass m1 of a free double-pendulum, as shown in Fig. 10.8, oscillates between the two absolutely stiﬀ constraints providing small angular amplitudes of the top pendulum,

10.5 Impact Modes in Multiple Degrees of Freedom Systems

259

|ϕ1 | ≤ Δ1 1

Fig. 10.8 Double pendulum with bilateral constraints.

Assuming that the angle ϕ2 is also small enough, brings the diﬀerential equations of motion between the barriers to the linear form μ2 ϕ ¨1 + ϕ ¨ 2 + μ2 ϕ1 = 0 ϕ ¨1 + ϕ ¨ 2 + ϕ2 = 0 where μ2 = 1 + m1 /m2 , over dots indicate diﬀerentiation with respect to the new temporal argument, t¯ = g/lt, and l is the length of each rod. In this case, the linear mode vectors and natural frequencies are

μ 1 1 , ω1 = e1 = μ+1 2μ (μ + 1) μ

μ 1 1 e2 = , ω2 = −μ μ − 1 2μ (μ − 1)

260

10 Impact Modes and Parameter Variations

where the modal vectors satisfy the orthogonality condition, eTj M ei = δji , with respect to the mass matrix

2 μ 1 M= 1 1 As a result, taking into account that Ia = I1 , Δa = Δ1 , and N = 2, brings the formal solution (10.37) and (10.40) to the form

ϕ1 −1 = Δ1 [tan (ω1 /ω) + (ω2 /ω1 ) tan (ω2 /ω)] × ϕ2

sin [(ω2 /ω) τ (ωt + α)] 1 sin [(ω1 /ω) τ (ωt + α)] ω2 1 + μ cos (ω1 /ω) ω1 −μ cos (ω2 /ω) where the relationships μ − 1 = μω2−2 and μ + 1 = μω1−2 have been used in manipulations. Note that the frequency of ﬁrst impact mode must be close enough to the ﬁrst linear frequency ω1 so that it is still away from the left neighborhood of the next frequency, ω2 . In current case, ω2 is the highest frequency, therefore its right “neighborhood” has no upper boundary. As a result, the highest impact mode becomes spatially localized as its frequency parameter grows. The localization admits explicit estimation by the asymptotic expansion

−1 ω2 ω2 ϕ2 ω1 ω2 ω2 ω1 − + |τ =1 = μ tan tan tan tan ϕ1 ω ω1 ω ω ω1 ω

ω12 − ω22 2ω12 ω22 ω −2 + O ω −4 =μ 2 1+ (10.41) ω1 + ω22 3 (ω12 + ω22 ) 1 = −1 − ω −2 − O ω −4 3 It follows from (10.41) that (ϕ2 /ϕ1 ) |τ =1 → −1 as ω → ∞, so that the amplitude of the bottom mass becomes negligibly small whereas the upper mass has the amplitude determined by the angular limiters.

10.5.2

A Mass-Spring Chain under Constraint Conditions

Let us consider a mass-spring chain of N identical particles under the constraint condition x ¨n +

k (−xn−1 + 2xn − xn+1 ) = 0; m x0 = xN +1 = 0;

|xa | ≤ Δa ;

n = 1, ..., N

(10.42)

1

(10.43)

10.5 Impact Modes in Multiple Degrees of Freedom Systems

261

where k and m are the stiﬀness of each spring and the mass of each particle, respectively. In this case, the corresponding linear modes and their frequencies are described exactly by expressions

T N πj 2 πj , ..., sin (10.44) ej = sin N +1 N +1 N +1 πj ωj = ωN +1 sin 2 (N + 1) where notation2 ωN +1 = 2 k/m has been introduced, and the basis vectors are normalized so that eTj ei = δji . The ath component of the jth normal mode vector is therefore aπj 2 T ej Ia = sin (10.45) N +1 N +1 Let us show that the impact mode periodic solution (10.37) and (10.40) becomes localized as ω → ∞. First, replacing the trigonometric functions by their asymptotic estimates sin(ωj τ /ω) ∼ ωj τ /ω, cos(ωj /ω) ∼ 1, and tan(ωj /ω) ∼ ωj /ω, gives 2N xb ∼ Δa

T T j=1 (ej Ia )(ej Ib ) τ 2N T 2 j=1 (ej Ia )

(ωt + α)

as

ω→∞

(10.46)

In this particular example, the sums can easily be evaluated. So, taking into account expression (10.45) and the standard trigonometric sums [52], gives N

bπj 2 aπj sin = δab sin N + 1 j=1 N +1 N +1

(10.47)

1 2 ω (−δa,b−1 + 2δa,b − δa,b+1 ) 2 N +1

(10.48)

N

(eTj Ia )(eTj Ib ) =

j=1 N j=1

(eTj Ia )(eTj Ib )ωj2 =

Substituting then (10.47) in (10.46), gives xb ∼ Δa δab τ (ωt + α)

as

ω→∞

(10.49)

Expression (10.49) shows that the ath particle of the chain vibrates according to the saw-tooth temporal mode shape with the inﬁnitely large frequency, 2

Note that this is just a suitable notation since the (N + 1)th frequency does not physically exist.

262

10 Impact Modes and Parameter Variations

whereas all the other particles are at rest. Therefore, the impact mode becomes spatially localized as ω → ∞.

10.6

Systems with Multiple Impacting Particles

Let us consider the case of two particles, say the ath and bth, under the constraint conditions. These conditions are |xa | ≤ Δa and |xb | ≤ Δb or, in the vector notations, T I x ≤ Δa and IT x ≤ Δb (10.50) a b In this case, the impulsive excitation on the right-hand side of the auxiliary equation must act on both ath and bth particles, so that the equation takes the form Mx ¨ + Kx = (pa Ia + pb Ib )τ (ωt + α) (10.51) where pa and pb are parameters to be determined. The related solution includes terms related to pa and pb , and can be represented in the form x(t) =

N pa (eTj Ia )ej + pb (eTj Ib )ej sin(ωj τ (ωt + α) /ω) ωωj cos(ωj /ω) j=1

(10.52)

Following the idea of normal modes, let us assume that the impact mode periodic motion is accompanied by synchronous impacts of both particles with the constraints according to conditions ITa x = ±Δa and ITb x = ±Δb when τ = ±1

(10.53)

Substituting (10.52) in (10.53), gives linear algebraic equations with respect to pa and pb in the form kaa pa + kab pb = Δa kba pa + kbb pb = Δb where kab =

N (eTj Ia )(eTj Ib ) j=1

ωωj

tan

(10.54)

ωj ω

(10.55)

Expressions (10.52) and (10.54) give a formal impact mode solution, which indeed describes an impact mode when the determinant of system (10.54) is non-zero and condition (10.50) holds. Solution (10.52) can be viewed as a strongly non-linear superposition of the two basic impact modes with a single impacting mass.

10.6 Systems with Multiple Impacting Particles

263

Let us consider the higher frequency domain, (π/2) ω >> ωN , for the above example of mass-spring system. In this case, tan(ωj /ω) > 0 for all j = 1, ..., N , and therefore, the coeﬃcients kab (10.55) create a Gram matrix with non-zero determinant [23]. The asymptotic estimation below conﬁrms this conclusion. Namely, assuming that (π/2) ω >> ωN , provides that all the ratios ωj /ω are small enough to justify the following asymptotic estimate

ωj τ 1 ωj 3 sin[(ωj /ω)τ ] τ3 = + (10.56) τ− + O ω −5 cos(ωj /ω) ω 2 ω 3 Then substituting expression (10.56) in (10.52) and (10.55) for an arbitrary cth particle, gives

1 ωN +1 2 τ3 xc = (Pa δac + Pb δbc ) τ − τ− × (10.57) 4 ω 3 −4 × [Pa (δa,c−1 − 2δac + δa,c+1 ) + Pb (δb,c−1 − 2δbc + δb,c+1 )] + O ω where xc = xT Ic and τ = τ (ωt + α), and the parameters {Pa , Pb } = ω −2 {pa , pb } are determined by the linear algebraic system Kaa Pa + Kab Pb = Δa Kba Pa + Kbb Pb = Δb

(10.58)

where Kab = ω 2 kab = δab −

1 ωN +1 2 (δa,b−1 − 2δab + δa,b+1 ) + O ω −4 (10.59) 6 ω

As follows from (10.59), equations (10.58) have the solution Pa = Δa and Pb = Δb as ω → ∞. In this limit, the vibration energy becomes localized on the two particles vibrating between the barriers with the saw-tooth temporal shape, (10.60) xc ∼ (Δa δac + Δb δbc ) τ (ωt + α) as ω → ∞ According to (10.60), the particles with numbers c = a and c = b are at rest. However, they oscillate with amplitudes of diﬀerent orders of ω −2 when the parameter ω takes a large but ﬁnite value. Expansion (10.57) shows that the temporal mode shapes of particles c = a and c = b are non-smooth and getting close to the triangular sine wave as the frequency ω increases. The temporal mode shapes of the nearest particles, c = a ± 1 and c = b ± 1, have amplitudes of order ω −2 and appear to be twice continuously diﬀerentiable with respect to time, t. In particular, calculating directly ﬁrst two derivatives, gives

264

10 Impact Modes and Parameter Variations

τ3 d τ− = ω 1 − τ 2 τ ∈ C 1 (R) dt 3

2 d τ3 2 2 2 −2τ (τ τ = −ω 2 2τ ∈ C (R) ) + 1 − τ τ − = ω dt2 3 where prime denotes diﬀerentiation with respect to the whole argument of the saw-tooth sine, τ ≡ dτ /d (ωt + α), and the term underlined is zero3 because 1 − τ 2 = 0 on the set of points {t : τ (ωt + α) = ±1}, where τ = 0.

3

Namely, it gives zero contribution into the related integrals of the theory of distributions.

Chapter 11

Principal Trajectories of Forced Vibrations

Abstract. As shown earlier by Zhuravlev (1992) that harmonically loaded linear conservative systems possess an alternative physically reasonable basis, which is generally diﬀerent from that associated with conventional principal coordinates. Brieﬂy, such a basis determines directions of harmonic loads along which the system response is equivalent to a single oscillator. The corresponding deﬁnition (principal directions of forced vibrations) is loosing sense in nonlinear case, when the linear tool of eigen vectors becomes inapplicable. However, it will be shown in this chapter that nonlinear formulation is still possible in terms of eigen vector-functions of time given by NSTT boundary value problems. Physical meaning of the corresponding nonlinear deﬁnitions for both discrete and continual models is discussed.

11.1

Introductory Remarks

The theory of linear normal modes deﬁnes a natural basis in the conﬁguration space of linear conservative systems. The corresponding directions are associated with a set of independent harmonic oscillators. The number of such oscillators is inﬁnite, if the original system is continuous. In the later case, the modal analysis provides reduction of a continuous system to the related discrete set of harmonic oscillators. As it is known, the normal modes are deﬁned for a class of unforced systems, therefore only initial conditions select those oscillators that will be excited during the dynamical process. Practically, a normal mode regime must be supported by some external loading due to inevitable energy dissipation. However, the theory does not identify directly such external forces. Let ψj (y) be, for instance, the jth mode shape of a beam. Generally speaking, the external loading of the same proﬁle, ψj (y), will excite not only the jth mode but also some others, unless the mass per unit length of the beam is constant. From the mathematical viewpoint, this is due to the mass density, say ρ(y), participating as a weighting factor in the orthogonality condition V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 265–273, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

266

11 Principal Trajectories of Forced Vibrations

< ψi (y)ρ(y)ψj (y) >= 0,

i = j

(11.1)

The question therefore is what kind of external force must be applied to a mechanical system in order to generate a normal mode type of motion when all the system particles coherently vibrate with the same frequency? Following reference [203], let us consider ﬁrst the linear case assuming that the linear n-degree-of-freedom forced system oscillates as a single harmonic oscillator in such a manner that the coordinates vector x (t) and the force vector p (t) are collinear to the same constant vector q with a constant length ratio μ as follows (11.2) x = q sin ωt, p = μq sin ωt In the case of forced vibration, the frequency ω is rather predetermined by the external loading and therefore should not play the role of eigen-value. It was shown in [203] that the coeﬃcient of proportionality μ can play such a role instead. In a regular case, the coeﬃcient μ has exactly n eigen-values, whereas the vector q determines the corresponding ‘principal directions’ according to the deﬁnition of reference [203]. Note that the principal directions are always orthogonal regardless the mass matrix of the system. Such an approach therefore determines a new natural basis for external forces from the standpoint of system considered. This, of-course, should not be viewed as a substitute for the theory of normal modes, however, some non-autonomous problems can be naturally solved by making use of the above complementary basis. In nonlinear cases, deﬁnition (11.2) is unapplicable and the above notion of principal directions loses its sense. However, it was shown in [136] that the basic idea still can be generalized by considering trajectories instead of directions. Also a mixed spatio-temporal consideration must be applied since spatial and temporal coordinates are not separable in nonlinear cases and the related vibration and forcing are generally neither harmonic in time not similar in space. There are some practically important formulations of the problem for the case of nonlinear forced vibration, which could be qualiﬁed as inverse or semi-inverse approaches. The related methods select practically reasonable external forces that generates simple enough dynamics. For example, Harvey [57] considered ‘natural forcing functions’ proportional to the non-linear restoring force of the forced Duﬃng oscillator. The notion of ‘exact steady state’ was deﬁned by Rosenberg [168] for a strongly nonlinear single degree of freedom system as a vibration with the cosine-wave temporal shape of the period of external force. The corresponding forcing function is determined under some initial conditions. Kinney and Rosenberg [80] considered systems with many degrees of freedom.

11.2 Principal Directions of Linear Forced Systems

11.2

267

Principal Directions of Linear Forced Systems

Let us illustrate ﬁrst the basic idea of reference [203] by considering the linear n-degree-of-freedom forced system Mx ¨ + Kx = p (ωt) , x (t) ∈ Rn

(11.3)

where M and K are constant mass and stiﬀness n× n -matrixes, respectively; p (ωt) is a periodic vector-force of the period T = 2π with respect to ωt, and the upper dot means diﬀerentiation with respect to time, t. Substituting (11.2) in (11.3), gives the eigen-value problem with respect to the parameter μ and vector q in the form − ω 2 M q + Kq = μq

(11.4)

Let q = vs and μ = μs be the sth eigen-vector and eigen-value respectively, s = 1, ..., n. The eigen-vectors vs are orthogonal and can be normalized by condition (11.5) viT vj = δij where δij is the Kronecker symbol. Therefore, the set of vectors vs determine a natural basis for the case of forced vibrations. Let, for instance, the external force be p = Q sin ωt, where Q ∈ Rn is an arbitrary constant vector. In this case, the corresponding steady-state (particular) solution is written as vsT Q vs sin ωt (11.6) x= μs s Now, let es and ωs be conventional linear normal modes and natural frequencies of the system. (The related eigen-value problem is obtained from (11.4) by setting μ = 0.) As follows from the linear theory, the normal mode vectors are orthogonal with respect to the mass matrix M so that the normalization condition can be represented in the form eTi M ej = δij Using the normal mode basis for the above steady-state, gives eTs Q x= es sin ωt ωs2 − ω 2 s

(11.7)

(11.8)

Since the uniqueness theorem holds, expansions (11.6) and (11.8) must represent the same solution, and therefore, eTs Q vsT Q vs = es (11.9) μs ωs2 − ω 2 s s

268

11 Principal Trajectories of Forced Vibrations

Let the external force amplitude vector Q be directed along one of the principal directions. Then, expansion (11.6) will include only one term, whereas expansion (11.8) still includes all n terms. Now, let us consider the case, when the mass matrix is equal to the identity matrix, M = E. In this particular case, expression (11.4) takes the standard form of the eigen-value problem for normal modes with respect to the eigenvalue parameter ω 2 + μ, (11.10) − ω 2 + μ Eq + Kq = 0 As follows from (11.10), the eigen-values of free and forced vibration are coupled by expression ω 2 + μs = ωs2 , s = 1, ..., n

(11.11)

It is seen that each eigen-value of forced vibration, μs = ωs2 − ω 2 , is a monotonically decreasing functions of the external frequency ω with only one zero at ω = ωs .

11.3

Deﬁnition for Principal Trajectories of Nonlinear Discrete Systems

Let us consider the nonlinear case Mx ¨ + Kx + εf (x) = p (ωt) ,

x (t) ∈ Rn

(11.12)

where f (x) is an analytic nonlinear vector-function such that f (−x) = −f (x), ε is a small positive parameter, and the forcing function and matrixes are deﬁned in equation (11.3). If ε = 0, then the concept of principal directions of forced vibrations is not applicable any more, however it is still possible to consider principal trajectories instead based on the following Deﬁnition 1. Trajectories of periodic motions of the period T = 2π/ω on which mechanical system (11.12) behaves as a Newtonian particle in Rn , namely the external force and acceleration vectors are coupled by the Newton second law, m¨ x (t) = p (ωt) (11.13) will be called principal trajectories of forced vibrations. In equation (11.13), m is a priory unknown eﬀective mass parameter. The eﬀective mass m and the force p (ωt) must be chosen in order to make equations (11.12) and (11.13) compatible. Note that, in the linear case, the above deﬁnition still gives principal directions of forced vibrations (11.2) after representing the mass parameter as follows

11.4 Asymptotic Expansions for Principal Trajectories

269

μ (11.14) ω2 Indeed, substituting expression x (t) = q sin ωt in equation (11.13) and taking into account expression (11.14), gives deﬁnition (11.2) in the form p =μx. In contrast to linear case (11.2), however, deﬁnition (11.13) allows non-harmonic temporal shapes. Current deﬁnition itself does not imply that the system is weakly nonlinear. However, if the parameter ε is small then explicit solutions can be obtained in terms of conventional asymptotic expansions as described in the next section. As mentioned, the notion of principal trajectories seems to relate to the idea of ‘natural forcing functions’ introduced in [57] for the Duﬃng oscillator. Let us consider now a multidimensional case from that point of view. Applying deﬁnition (11.13) to the general nonlinear system m=−

Mx ¨ + F (x) = p (ωt)

(11.15)

and eliminating the acceleration, gives the external forcing vector-function as a linear transformation of the restoring force in the form, −1

1 F (x) p (ωt) = E − M m

(11.16)

where the matrix of the transformation includes the eﬀective mass parameter m. Relationship (11.16) can be viewed as a vector version of the concept of natural forcing functions. On the other hand, using the deﬁnition for principal trajectories and excluding the external forcing vector p (ωt) from the equation of motion, gives an auxiliary free system described by the diﬀerential equation of motion (M − mE) x ¨ + F (x) = 0 The idea of transforming the forced problem to a free vibration problem by imposing the form of excitation was used also in [31] with illustrations on two degrees of freedom systems based on an essentially diﬀerent methodology though.

11.4

Asymptotic Expansions for Principal Trajectories

In order to make equations (11.12) and (11.13) compatible, let us eliminate the forcing vector-function p (ωt) and thus consider equation Mx ¨ + Kx + εf (x) = m¨ x (t)

(11.17)

270

11 Principal Trajectories of Forced Vibrations

A family of periodic solutions, that give principal directions of linearized system as ε → 0, will be considered. Let us represent such solutions (principal trajectories) in the following parametric form x = X (τ )

(11.18)

where is τ = τ ((2ω/π)t) is the triangular sine wave of the period of external loading, T = 2π/ω. Substituting (11.18) into (11.17), gives

LX+εf (X) =

L≡

2ω π 2ω π

2

mX

2 M

(11.19)

d2 +K dτ 2

under the boundary condition X (τ ) |τ =±1 = 0

(11.20)

As mentioned above, the temporal and spatial variables generally are not separable any more in nonlinear cases, therefore it is impossible to obtain an exact nonlinear version of the eigenvector problem (11.4). As a result, both temporal and spatial mode shapes must be corrected on each step of the related asymptotic process as described below. Remind that the diﬀerential operator L in equation (11.19) includes the frequency parameter ω ﬁxed, whereas the mass m is an eigen value to be determined. Let ma and ea (τ ) be the eigen value and eigen vector of the linearized problem, ε = 0, respectively,

2 2ω ea Lea = ma π ea | τ =±1 = 0

(11.21)

where the index a = {s, j} consists of spatial and temporal mode shape numbers, s = 1, ..., n and j = 1, ..., respectively. The scalar product of any two vector-functions x = x (τ ) and y = y (τ ) will be deﬁned as follows 1 1 T x, y = x ydτ 2 −1 Let us represent solution of the weakly nonlinear eigen value problem (11.19) and (11.20) in the following form of asymptotic expansions

11.5 Deﬁnition for Principal Modes of Continuous Systems

271

X(τ ) = Aea (τ ) + εX(1) (τ ) + O ε2 m = ma + εη1 + O ε2

(11.22)

Then substituting (11.22) in (11.19) and (11.20), and matching the coeﬃcients of the ﬁrst order of ε, gives equation

(1)

LX

−

2ω π

2 (1)

ma X

= −f (Aea ) +

2ω π

2

η1 Aea

(11.23)

and boundary condition X(1) |τ =±1 = 0

(11.24)

Following the idea of perturbations for eigen-value problems [83], let us represent solution of equation (11.23) (1) ab eb (τ ) (11.25) X(1) = b=a (1)

where b = {r, i} is a double index, ab are yet unknown constant coeﬃcients, and boundary condition (11.24) is automatically satisﬁed. Let us assume the following normalization condition for the eigen vectorfunctions 0, b = a (11.26) ea (τ ), eb (τ ) = 1, b=a Substituting (11.25) in (11.23) and taking into account (11.26), determines (1) the coeﬃcients ab and η1 . As a result, expansions (11.22) give ﬁrst-order asymptotic solution π 2 e , f (Ae ) e b a b + O ε2 2ω mb − ma b=a π 2 e , f (Ae ) a a + O ε2 m = ma − ε 2ω A X = Aea + ε

(11.27)

As follows from the form of solution (11.27), all the coeﬃcients are uniquely determined under the condition that ma = mb for a = b. The possibility of degeneration, namely ma = mb for a = b, depends on the inner properties of the system and the frequency parameter ω. The related examples were considered earlier [136], [141].

11.5

Deﬁnition for Principal Modes of Continuous Systems

Let us consider a one-dimensional elastic system whose vibration is described by some function u = u(t, y). For certainty reason, let us consider a

272

11 Principal Trajectories of Forced Vibrations

non-linear string of the length l under external distributed loading described by the partial diﬀerential equation and boundary conditions Lu+εf [u] = p(ωt, y),

0

u(t, 0) = u(t, l) = 0 L ≡ ρ(y)

2

(11.28) (11.29)

2

∂ ∂ −T 2 2 ∂t ∂y

(11.30)

where L is the diﬀerential self-adjoint operator of linear string, ρ(y) is a mass per unit length parameter, T is a constant tensile force, f [u] is a nonlinear operator acting in the corresponding function space of conﬁgurations, ε is a small parameter, and p(ωt, y) is the external forcing function, which is assumed to be 2π-periodic with respect to ωt. Now keeping in mind expressions (11.28) through (11.30), let us introduce Deﬁnition 2. Periodic forced vibrations of a continuous system, in which the system motion is equivalent to a particle in the function space of conﬁgurations described by the second Newton law, σ

∂ 2 u(t, y) = p(ωt, y) ∂t2

(11.31)

will be called a principal mode of forced vibration. In one-dimensional cases, σ is a priory unknown eﬀective mass per unit length. Substituting (11.31) in (11.28), gives the following partial diﬀerential equation for principal modes of forced vibrations Lu+εf [u] = σ

∂2u ∂t2

(11.32)

Introducing the triangular wave time substitution as τ = τ ((2ω/π)t) and u(t, y) = U (τ, y), gives

LU +εf (U ) =

L≡

2ω π 2ω π

2 σ 2

∂2U ∂τ 2

ρ(y)

∂2 ∂2 − T ∂τ 2 ∂y 2

(11.33)

The boundary conditions are formulated for both temporal and spatial variables as U (τ, 0) = U (τ, l) = 0 (11.34) and, ∂U (τ, l) |τ =±1 = 0 ∂τ respectively.

(11.35)

11.5 Deﬁnition for Principal Modes of Continuous Systems

273

In this case, the scalar product of two functions U = U (τ, y) and V = V (τ, y) from the conﬁguration space can be deﬁned as U ,V =

1 2l

1

l

U V dτ dy −1

(11.36)

0

Further, a weakly nonlinear asymptotic procedure can be developed analogously to the above discrete case.

Chapter 12

NSTT and Shooting Method for Periodic Motions

Abstract. In this chapter, two-dimensional shooting diagrams are introduced for visualization of manifolds of periodic solutions and their bifurcations. A general class of nonlinear oscillators under smooth, nonsmooth, and impulsive loadings is considered. The corresponding boundary value problems are formulated by introducing the triangular wave temporal argument. The Duﬃng oscillator with no linear stiﬀness (Ueda circuit) is considered for illustration. It is shown that the temporal mode shape of the loading is responsible for qualitative features of the dynamics, such as transitions from regular and random motions. The important role of unstable periodic orbits is discussed.

12.1

Introductory Remarks

Periodic solutions and their bifurcation diagrams often reveal important qualitative features of the dynamics even though the system motion is not expected to be periodic. In particular, the number of periodic orbits, their distribution and properties reveal the structure of chaotic orbits; see, for instance, works [12], [53], [123], and references therein. Direct numerical tools for detection and construction of periodic orbits based on the mapping approach can be found in the book [125] and paper [54]. Diﬀerent formulations in terms of boundary value problems for ordinary diﬀerential equations are described in [10]. Theoretical and applied results regarding periodic motions, bifurcations and chaos are reported in [13] and [196]. In this chapter, a special two-dimensional visualization of the shooting method is introduced in order to incorporate the general two component NSTT as a preliminary analytical stage [144]. Note that the same approach using the one-component NSTT was suggested earlier in [154] and impleR mented in Mathematica interface [18]. In particular, subharmonic orbits of the forced pendulum and bifurcation diagrams were obtained by examining the shooting curves and their zeros.

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 275–294, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

276

12 NSTT and Shooting Method for Periodic Motions

Let us consider a multi-dimensional oscillator described be the diﬀerential equation F (¨ x, x, ˙ x, t) = 0 (12.1) where x (t) ∈ Rn , and the vector-function F ∈ Rn is periodic with respect to time t with the period T = 4a. In this work, diﬀerent kind of temporal discontinuity in the diﬀerential equations of motion will be considered. In order to satisfy the related mathematical requirements, the left-hand side of equation (12.1) must be interpreted in terms distributions [178] ∞ F (¨ x, x, ˙ x, t)ϕ (t) dt = 0

(12.2)

−∞

where ϕ (t) is any suﬃciently smooth testing function. However, at this point, let us assume that the function F (¨ x, x, ˙ x, t) is regular with no singular terms involved. Let us consider periodic solutions of the period T by means of the coordinate complexiﬁcation (NSTT) x → {X, Y } :

x = X (τ ) + Y (τ ) τ

(12.3)

whereτ = τ (t/a) the triangular sine wave of the period T = 4a, and τ = dτ (t/a) /d (t/a) is its ﬁrst generalized derivative, which is a step-wise discontinuous function at the time instances Λ = {t : τ (t/a) = ±1}

(12.4)

As discussed in this book, the above discontinuities can be suppressed by the condition Y (±1) = 0, which is the necessary condition of continuity of the original coordinates x(t). Then, assuming for a while no inﬁnite discontinuities in F , substituting (12.3) in (12.1), using the algebraic properties of representation (12.3) as well as other NSTT rules, gives the boundary value problem

X + Y X + Y X + Y , , aτ = 0 (12.5) , F a2 a a

X − Y X − Y X − Y F , − , , 2a − aτ = 0 (12.6) a2 a a Y |τ =±1 = 0,

X |τ =±1 = 0

(12.7)

where the prime used with X and Y means diﬀerentiation with respect to τ . Both equations (12.5) and (12.6) are easily derived by the corresponding algebraic manipulations, whereas boundary conditions (12.7) represent the result of elimination of the singular term τ (t/a) when substituting (12.3) in

12.2 Problem Formulation

277

(12.1). In some cases, such a singular term can be employed though in order to eliminate singularities from original equations; see below. Despite of a relatively complicated form of equations (12.5) and (12.6), the new formulation brings some advantages due to the fact that the new temporal variable τ is bounded and automatically accounts for periodicity of solutions regardless their temporal shapes. This property appears to be important in those cases when the solutions do not represent a ﬁnal stage of investigation but must be used for further analyses. The dimension increase is often compensated by an eﬀective decrease of the temporal interval of the problem, since the range −1 τ 1 is covered by the original time domain −a t a, which is twice shorter than the whole period T = 4a. Moreover, there are many cases when the number of equations can be reduced to that of the original system due to the symmetry of equations. If, for instance, the vector-function F (¨ x, x, ˙ x, t) is even with respect to the velocity x˙ or includes no velocity at all, and the explicit dependence on time t produces zero ‘imaginary component’ then boundary value problem (12.5) through (12.7) admits a family of solutions on which

X X X , , aτ = 0 (12.8) Y ≡ 0, F , a2 a a X |τ =±1 = 0

(12.9)

The particular case (12.8) and (12.9) was investigated numerically by the shooting method in [154] and [177] based on a single- and multiple-degrees-of freedom systems, respectively. It should be noted that no special requirements are imposed on numerical methods or packages for solving the above boundary-value problems. However, the shooting algorithm in the MatheR matica interface provides a physically meaningful way of visualization of periodic solutions due to the speciﬁc combination analytical and numerical commands.

12.2

Problem Formulation

Let assume now that the system loading may include a periodic series of Dirac δ-pulses acting at times Λ(12.4). As is known [45], Dirac δ-functions can participate in nonlinear diﬀerential equations only as summands because nonlinear manipulations with δ-functions are physically meaningless, except special concepts [106]. Therefore, the original equation (12.1) must be concretized as x¨ + f (x, x, ˙ t) = q (t) (12.10) where q (t) = Q (τ (t/a)) + P (τ (t/a)) τ (t/a) + p (τ (t/a)) τ (t/a)

(12.11)

278

12 NSTT and Shooting Method for Periodic Motions

and τ (t/a) = d2 τ (t/a) /d (t/a)2

∞

t t + 1 − 4k − δ − 1 − 4k =2 δ a a

(12.12)

k=−∞

In equation (12.10), the function f (x, x, ˙ t) may still include parametric terms of the period T = 4a with possible step-wise discontinuities on Λ. The acceleration x ¨ also participates as a summand, since it must have the same kind of singularities as the external forcing function, q (t). According to the distribution theory [165], p (τ (t/a)) must be at least continuous on Λ, otherwise the ‘product’ p (τ ) τ cannot be treated as a distribution. Note, that behavior of the function p(τ (t/a)) between the times Λ is arbitrary, since only values p (−1) and p (1) contribute into the expression p (τ (t/a)) τ (t/a) (12.13)

∞ t t + 1 − 4k − p (1) δ − 1 − 4k =2 p (−1) δ a a k=−∞

The numbers p (−1) and p (1) control the ‘amplitudes’ and directions of the δ−pulses. For example, all the pulses can be positively co-directed by setting p (τ ) = − sign τ . Remark 2. Expressions (12.3) and (12.11) represent particular cases of the truncated series q (t) =

N

Pk (τ (t/a)) dk τ (t/a) /d (t/a)k

(12.14)

k=0

where Pk (τ (t/a)) must be at least k − 2 times continuously diﬀerentiable in the neighborhood of points t = ±a. Although, physical interpretation of the higher order terms in (12.14) is not straightforward, such terms still can occur after reducing the number of equations from the entire system. In case (12.10) and (12.11), one has N = 2 and therefore the velocity vector x˙ must be step-wise discontinuous. Further, if N = 3 then the velocity x˙ includes singular terms and the function f (x, x, ˙ t) in equation (12.10) must be linear ˙ t) must be linear also with respect to x. ˙ If N = 4 then the function f (x, x, with respect to the position vector x provided that any parametric terms are suﬃciently smooth functions of time. Therefore, only linear systems can be considered if N ≥ 4. Since the basis elements {1, τ , τ } represent functions of diﬀerent classes of smoothness, then substituting (12.3) and (12.11) in (12.10), gives separately a−2 X + Rf (X, Y, X , Y , τ ) = Q (τ )

(12.15)

12.3 Sample Problems and Discussion

and

279

a−2 Y + If (X, Y, X , Y , τ ) = P (τ )

(12.16)

a−2 X |τ =±1 = p (±1)

(12.17)

where

X − Y 1 X + Y Rf , aτ ± f X − Y, − , 2a − aτ = f X + Y, If 2 a a (12.18) Note that the singular term a−1 Y τ is eliminated from the velocity vector x˙ (t) by imposing another boundary condition Y |τ =±1 = 0

(12.19)

The boundary value problem (12.15) through (12.19) includes no singular or discontinuous functions, therefore standard numerical codes and packages can be applied with no speciﬁc constraints on their choice. In general, equations (12.15) and (12.16) are coupled. Although the equations can be decoupled by introducing the new unknown functions, X(τ ) + Y (τ ) and X(τ ) − Y (τ ), the boundary conditions will become coupled. There are two special cases, however, when the entire problem can be reduced. If, for instance, f (x, x, ˙ t) = f (x, −x, ˙ 2a − t), and P (τ ) ≡ 0 then the problem admits a family of solutions such that Y ≡ 0, a−2 X + f (X, X /a, aτ ) = Q (τ )

(12.20)

under the boundary condition (12.17). In case f (x, x, ˙ t) = −f (−x, x, ˙ 2a − t), Q (τ ) ≡ 0 and p (τ ) ≡ 0, then one can consider another family of solutions on which X ≡ 0, a−2 Y + f (Y, Y /a, aτ ) = P (τ )

(12.21)

under the boundary condition (12.19). This chapter nevertheless focuses on the general two-component problem (12.15) through (12.19).

12.3

Sample Problems and Discussion

12.3.1

Smooth Loading

The Duﬃng-Ueda oscillator [186] under the periodic loading of diﬀerent temporal shapes will be considered below. Let us start with the standard case of sine-wave voltage x¨ + ζ x˙ + x3 = B sin ωt where ζ, B, and ω are constant parameters.

(12.22)

280

12 NSTT and Shooting Method for Periodic Motions

In this case, the diﬀerential equations (12.15) and (12.16) take the form a−2 X + ζa−1 Y + X 3 + 3XY 2 = B sin a−2 Y + ζa−1 X + Y 3 + 3X 2 Y = 0

πτ 2

(12.23) (12.24)

where a = π/(2ω) is a quarter of the loading period, and the boundary conditions are Y |τ =±1 = 0, X |τ =±1 = 0 (12.25) The shooting method can be applied now as follows. First, the diﬀerential equations (12.23) and (12.24) are solved under the initial conditions X(−1) = g, X (−1) = 0

Y (−1) = 0, Y (−1) = h

(12.26) (12.27)

where g and h are numbers to be determined in order to satisfy boundary conditions (12.25). Let us represent solution of the initial-value problem (12.23), (12.24), (12.26) and (12.27) in the following general form X = X(τ ; g, h), Y = Y (τ ; g, h)

(12.28)

By the idea of shooting method, the initial value problem (12.23), (12.24), (12.26) and (12.27) must be iteratively solved multiple times at diﬀerent g and h until suﬃcient precision has been reached for boundary conditions (12.25) at right end τ = 1, ∂X(τ ; g, h) |τ =1 ≡ G(g, h) = 0 ∂τ Y (τ ; g, h)|τ =1 ≡ H(g, h) = 0

(12.29)

When dealing with the particular cases (12.20) or (12.21), such a procedure is not diﬃcult since one has only one equation with a singe unknown, G(g) = 0 or H(h) = 0. Multidimensional cases, such as (12.29), appear to be more diﬃcult and time consuming. From this point of view, the important feature of Mathematica is that it is possible to program the functions G(g, h) and H(g, h) ‘explicitly’ in such a way that the arguments g and h are included into the numerical solver of diﬀerential equations. This can be done as follows. First, the numerical solution is deﬁned as a function of the arguments g and h according to the command sol[g , h ]:=NDSolve[{eqX, eqY, X[-1]==g, X [-1] == 0, Y[-1] == 0, Y [-1] == h}, {X,Y}, {τ ,-1,1}]; where eqX and eqY are equations (12.23) and (12.24), respectively.

12.3 Sample Problems and Discussion

281

Then, the functions G(g, h) and H(g, h) are deﬁned as follows G[g , h ]:=X [1]/.sol[g, h][[1]]; H[g , h ]:=Y[1]/.sol[g, h][[1]]; As a result, the functions G(g, h) and H(g, h) can be considered as usual functions of two arguments. In particular, intersections of two manifolds (12.29) can be located and determined by using the commands ContourPlot and FindRoot, respectively. Each of the determined roots of equations (12.29) represents a periodic solution of the original equation. If the loading amplitude B is a control parameter, then the evolution of diagrams G(g, h; B) = 0 and H(g, h; B) = 0 represents the corresponding structural changes in the set of periodic solutions. Fig. 12.1 gives an example of such a diagram. The parameters were chosen in order to provide conditions for the ‘randomly transitional’ process in terms of work [186]. The diagram clearly shows ﬁve intersections between the two diﬀerent families of curves. The corresponding solutions of the input period T = 4a = 2π are shown in Figs. 12.2 and 12.3. Direct numerical solutions the corresponding estimates for Floquet multipliers show that ﬁrst four periodic solutions, (a) through (d), are unstable, and only one solution (e) is stable. Solution (e) was detected by direct analog and digital computer simulations reported in [186], whereas solutions (a) through (d) were unlisted. Instead, a ‘non-reproducible trajectory’ as a realization of the ‘randomly transitional’ process was represented in the xv-plane. Such a trajectory can be treated as a chaotic drift around the ﬁrst three unstable motions (a), (b) and (c). However, high order periodic solutions may also aﬀect the dynamics of chaotic drift [12]. Fig. 12.4 shows what actually happens when trying to numerically reproduce an unstable periodic orbit, say (a). Neither the shooting algorithm nor computer codes allow to perfectly introduce the initial conditions, therefore it is unlikely that the oscillator will remain on the unstable orbit. After few cycles, the system leaves the orbit (a) for the ‘randomly transitional’ drift around the all three unstable orbits (a), (b) and (c) with ‘no certain choice’ between them. The long-term time history and the corresponding spectrogram of this motion, represented in Fig. 12.4, conﬁrm its random character during quite a long period of time. Although preliminary qualitative information about stability or instability of periodic solutions can be obtained by direct numerical tests, one can quantify stability properties based on the well known Floquet theory in terms of the characteristic multipliers [111], [56]. In order to remind the principals, let us consider periodic solution x(t) of the equation (12.10),where f (x, x, ˙ t) = f (x, x, ˙ t + T ), q (t) = q (t + T ), and the period is T = 4a.

282

12 NSTT and Shooting Method for Periodic Motions

20

10

d c h

0

e

b a

10

20 4

2

0 g

2

4

Fig. 12.1 The curves G(g, h) = 0 (continuous) and H(g, h) = 0 (dashed) and their intersections for the Ueda oscillator under the sine-wave input and the following parameters: ζ = 0.1, B = 12, and ω = 1 (0.1592 Hz).

A variation of the solution x(t), say u(t), is described by the linear diﬀerential equation with periodic coeﬃcients u ¨ + q1 (t)u˙ + q2 (t)u = 0 where q1 (t) = ∂f (x, x, ˙ t)/∂ x˙ is assumed to be independent on x, ˙ and q2 (t) = ∂f (x, x, ˙ t)/∂x. Then substitution ⎛ ⎞ t 1 u = y(t) exp ⎝− q1 (z)dz ⎠ 2 0

12.3 Sample Problems and Discussion

283

x

x

6

6

a

4

b

4

2

2 1

2

2

3

4

5

6

t

1

2

4

4

6

6 x

2

3

4

5

6

t

x

6

6

c

4

d

4

2

2 1

2

2

3

4

5

6

t 2

4

4

6

6

1

2

3

4

5

6

t

x 6

e

4 2 2

1

2

3

4

5

6

t

4 6 Fig. 12.2 The temporal mode shapes of periodic solutions.

gives y¨ + p(t)y = 0 where

(12.30)

1 1 p(t) = q2 (t) − [q1 (t)]2 − q˙1 (t) 4 2 As known from the Floquet theory, stability of the solution x(t) is determined by the Floquet multipliers μ1,2 = A ± A2 − 1 (12.31)

284

12 NSTT and Shooting Method for Periodic Motions

v 6 4 2 0 2 4 6

v a

b

5 x

0

x

5 3 2 1 0

1

2

3 2 1 0

3

v

1

2

3

v c

5 0

d

5 x

5

0

x

5 3 2 1 0

1

2

3 2 1 0

3

1

2

3

v 6 4 2 0 2 4 6

e x

3 2 1

0

1

2

3

Fig. 12.3 The projections of periodic trajectories on xv-planes.

where A = [y1 (T ) + y˙ 2 (T )]/2, and y1 (t) and y2 (t) are two fundamental solutions of equation (12.30) given by the initial conditions y1 (0) = 1, y˙ 1 (0) = 0 y2 (0) = 0, y˙ 2 (0) = 1 Based on the number A, the solution x(t) is unstable if A2 > 1, and it is stable if A2 < 1. In case A2 = 1, there exist a periodic solution of equation (12.30).

12.3 Sample Problems and Discussion

285

x

5

0

−5

50

100

150 Time

200

250

300

2.5

Frequency

2 1.5 1 0.5 0

0

10

20

30

40

50

Time

Fig. 12.4 The time history record and its spectrogram (in Hz) for Ueda oscillator after the direct numerical integration. The parameters are: ζ = 0.1 and B = 12.0.

Now, let x(t) be a periodic solution of equation (12.22). The corresponding variational equation is (12.32) u ¨ + ζ u˙ + 3x2 u = 0 where u = u(t) is a small variation of the solution x = x(t). After the standard substitution u(t) = y(t) exp(−ζt/2), equation (12.32) takes the form

ζ2 2 y¨ + 3x − y=0 4 Taking into account the form of solution (12.3), gives the variational equation with periodic coeﬃcient y¨ + [U (τ (t/a)) + V (τ (t/a))τ (t/a)]y = 0

(12.33)

where U (τ ) = 3X 2 (τ ) + 3Y 2 (τ ) − ζ 2 /4 and V (τ ) = 6X(τ )Y (τ ). Note that the periodic coeﬃcient in equation (12.32) is continuous with respect to time t since V (±1) = 0 due to the boundary conditions (12.25). By using the numerical solutions of equation (12.33), one obtains the number A for every solution Aa = 10.5155, Ab = −2.63747, Ac = −2.63749 Ad = 1.70201, and Ae = 0.143507

(12.34)

286

12 NSTT and Shooting Method for Periodic Motions

where the index correspond to the type of periodic solution of the original equation; see Figs. 12.1, 12.2, and 12.3. These numbers conﬁrm that only solution (e) is stable.

12.3.2

Step-Wise Discontinuous Input

Let us consider now the case of discontinuous periodic input of the rectangular cosine temporal shape x ¨ + ζ x˙ + x3 = B

dτ (t/a) d (t/a)

(12.35)

where a is a quarter of the input period. In this case, the right-hand side of equations (12.23) and (12.24) are modiﬁed so that the equations take the form a−2 X + ζa−1 Y + X 3 + 3XY 2 = 0 a−2 Y + ζa−1 X + Y 3 + 3X 2 Y = B

(12.36) (12.37)

under the homogeneous boundary conditions (12.25). Fig. 12.5 shows the shooting diagram under the ﬁxed parameters ω = π/(2a) = 1 (0.1592 Hz), B = 7.4 and ζ = 0.05. In this case, there are seven intersections between the two families of curves and therefore seven periodic solutions of the period T = 4a as shown in Figs. 12.6 and 12.7. Note that, under the same parameters, the system response on the rectangular cosine input shows new features compared to those under the sine-wave input [144]. For example, after few cycles along the orbit (a), the system starts its drift around the ﬁrst three solutions, (a), (b) and (c). At this stage, the dynamics resembles that under the sine-wave input. Further, however, after several random ‘jumps’ between the orbits (a), (b) and (c), the system becomes eventually attracted by the stable orbit (e). The direct numerical solution, represented in Fig. 12.8, clearly shows all three stages of the time and spectral histories of the dynamics. In order to clarify stability, the Floquet theory can be applied analogously to the case of the sine-wave input.

12.3.3

Impulsive Loading

Let us consider the same oscillator loaded by the periodic series of pulses x ¨ + ζ x˙ + x3 = pτ

∞

t t + 1 − 4k − δ − 1 − 4k = 2p δ a a k=−∞

where p = p (±1) = B = const.

(12.38)

12.3 Sample Problems and Discussion

287

40 f

20 e

h

0 a b

c d 20

g

40 4

2

0 g

2

4

Fig. 12.5 The curves G(g, h) = 0 (continuous) and H(g, h) = 0 (dashed) for the Ueda oscillator under the step-wise input and the following parameters: ζ = 0.05, B = 7.4, and ω = 1.

In this case, both equations (12.36) and (12.37) should have zero righthand side, however, the non-homogeneous version of the boundary condition (12.17) must be imposed in order to eliminate the pulses. The second expression in (12.26) and the ﬁrst one in (12.29) must be modiﬁed as X (−1) =a2 B=[π/(2ω)]2 B and G (g, h)=a2 B=[π/(2ω)]2 B, respectively. Therefore, the singular terms are eliminated from the system due to the sawtooth time, and the shooting procedure can be applied in the same fashion as that under the smooth input. The shooting diagram and the corresponding periodic solutions are shown in Figures 12.9, and 12.10 and 12.11, respectively. The projections of the phase trajectories show discontinuities of the velocity on the xv-plane caused by the external pulses. The last four projections,

288

12 NSTT and Shooting Method for Periodic Motions

x

x

6 4 2

6 4 2

a

1

2 4 6

2

3

4

5

6

t

2

3

4

5

6

t

x 6 4 2

c

1

2

3

4

5

6

d

t

1

2 4 6

x

2

3

4

5

6

t

x

6 4 2

6 4 2

e

1

2 4 6

1

2 4 6

x 6 4 2 2 4 6

b

2

3

4

5

6

t 2 4 6

f

1

2

3

4

5

6

t

x 6 4 2 2 4 6

g

1

2

3

4

5

6

t

Fig. 12.6 The temporal mode shapes of periodic solutions og Ueda circuit under the step-wise input.

(h) through (k), can be qualiﬁed as ‘quasi free’ vibrations sustained by the pulses. In the shooting diagram represented in Fig. 12.9, the related

12.3 Sample Problems and Discussion

289

v

v

4 2 0 2 4

a x

3 2 1

0

1

2

6 4 2 0 2 4 6

b x

3 2 1

3

v

0

1

2

3

v 10

6 4 2 0 2 4 6

5

c x

d

0

x

5 3 2 1

0

1

2

10 4

3

2

v 5

e

0

x

5 4

2

2

4

v

10

10

0

0

2

20 10 0 10 20

f

6 4 2

4

0

2

x

4

6

v 20 10 0 10 20

g

6 4 2

0

2

x

4

6

Fig. 12.7 The projections of periodic trajectories on xv-planes.

intersections are diﬃcult to determine due to a very small angle between the intersecting curves.

290

12 NSTT and Shooting Method for Periodic Motions

x

5

0

−5

50

100

150 Time

200

250

300

2.5

Frequency

2 1.5 1 0.5 0

0

10

20

30

40

50

Time

Fig. 12.8 The time history record and its spectrogram (in Hz) for evolution of the solution (a) under the step-wise voltage of the amplitude B = 7.4.

12.4

Other Applications

12.4.1

Periodic Solutions of the Period - n

The above sections deal with periodic solutions with the input period T = 4a. In order to capture ‘subharmonic’ solutions of the period nT , the components of representation (12.3) must be taken in the form 1 [x (naτ ) + x (2na − naτ )] 2 1 Y (τ ) = [x (naτ ) − x (2na − naτ )] 2

X (τ ) =

(12.39)

where τ = τ (t/(na)). For instance, applying (12.39) to the sine wave sin ωt, gives nπτ nπτ 1 sin + sin nπ − 2 2 2 1 nπτ nπτ + sin − sin nπ − τ 2 2 2 7 8 nπτ 1 1 + (−1)n+1 + 1 − (−1)n+1 τ = sin 2 2

sin ωt =

where τ = dτ (t/(na))/d(t/(na)) and a = π/(2ω).

(12.40)

12.4 Other Applications

291

30

k

j 20

10 g i h

0

h

d

a

c f

b

e

10

20

30 6

4

2

0

2

4

6

g

Fig. 12.9 The curves G(g, h) = 0 (continuous) and H(g, h) = 0 (dashed) for Ueda circuit under the impulsive input and parameters: ζ = 0.05, B = 1.4, and ω = 1.

According to representation (12.40), equations (12.23) and (12.24) must be modiﬁed as follows (na)−2 X + ζ(na)−1 Y + X 3 + 3XY 2 nπτ B 1 + (−1)n+1 sin = 2 2 (na)−2 Y + ζ(na)−1 X + Y 3 + 3X 2 Y B nπτ 1 − (−1)n+1 sin = 2 2

(12.41)

292

12 NSTT and Shooting Method for Periodic Motions x

x a

1.5 1.0 0.5

b

2

2

1 2 3 4 5 6

t

1

1

2

3

4

5

6

t

d

e

1

2

3

4

5

6

t

1

1

2

3

4

5

6

x 2 1 1

1 2

2

3

4

5

6

h

1

1 2 3

x

6

t

2

3

4

5

6

t

x

3 2 1 t

1

1 2

x g

5

f

t

2

2

4

2 1

1

1

3

x

2

1

2

2 x

2

1

1

2

x

c

1

1

0.5 1.0 1.5

2

3

4

5

6

i

3 2 1 t 1 2 3

1

2

3

4

5

6

t

x

6 4 2 2 4 6

x

6 4 2

j

1

2

3

4

5

6

t 2 4 6

k

1

2

3

4

5

6

t

Fig. 12.10 The temporal mode shapes of periodic solutions og Ueda circuit under the impulsive input.

The right-hand side of these equations shows that direct replacement a → na in (12.23) and (12.24) would not work. If n = 1 then equations (12.41) take the form (12.23) and (12.24), but if n > 1 equations (12.41) can give new solutions in addition to those described by equations (12.23) and (12.24). The corresponding calculations however become time consuming and give complicated diagrams as the number n increases.

12.4 Other Applications

293

v

v

v

4

2 1 0 1 2

a x

4

b

2 0

x

2 2 1 0

v

1

4

2

d x

0

x

2

2

4 2 1 0

1

2 1 0

2

v 6 4 2 0 2 4

2 1 0

1

2

v e

2

0 4

x

v 4

4 2

0 2

4

1.5 1.0 0.50.00.51.01.5

c

2

1

2

4 2 0 2 4 6

f x

2 1 0

v

1

2

v

10 g

h

5 x

0

x

5 2 1 0

1

10

2

v 20 10 0 10 20

i

5 0

x

5 321 0 1 2 3

321 0 1 2 3

v j x

6 4 2 0 2 4 6

20 10 0 10 20

k x

6 4 2 0 2 4 6

Fig. 12.11 The projections of periodic trajectories on xv-planes.

12.4.2

Two-Degrees-of-Freedom Systems

Using the above two-dimensional geometrization of shooting diagrams enables one of considering special cases of two-degrees-of-freedom systems based on equations (12.20) or (12.21). For example, equation (12.20) can be treated as an equation with respect to the two-component vector-function X = {X1 (τ ), X2 (τ )}. Such an interpretation leads to two scalar equations a−2 X1 + f1 (X1 , X2 , X1 /a, X2 /a, aτ ) = Q1 (τ ) a−2 X2 + f2 (X1 , X2 , X1 /a, X2 /a, aτ ) = Q2 (τ )

(12.42)

294

12 NSTT and Shooting Method for Periodic Motions

In this case, the shooting procedure should be based on the initial conditions at τ = −1, X1 (−1) = g, X1 (−1) = 0 X2 (−1) = h, X2 (−1) = 0

(12.43)

where the numbers g and h are determined to satisfy the boundary conditions on the right end of the interval −1 ≤ τ ≤ 1, ∂X1 (τ ; g, h) |τ =1 ≡ G(g, h) = 0 ∂τ ∂X2 (τ ; g, h) |τ =1 ≡ H(g, h) = 0 ∂τ

(12.44)

In this case, every solution g and h of system (12.44) gives the initial position on the conﬁguration plane X1 X2 at which the system starts with zero velocity its periodic motion of the period T = 4a.

12.4.3

The Autonomous Case

The nonlinear normal modes represent an important class of periodic motions. The related references and description of analytical methods can be found in [100] and [190]. Analogously to the linear theory, the basic nonlinear normal mode solutions are given by the class of autonomous conservative systems. In this case, equations (12.42) take the form a−2 X1 + f1 (X1 , X2 ) = 0 a−2 X2 + f2 (X1 , X2 ) = 0

(12.45)

The form of equations (12.45) is easier than (12.42), but the parameter a becomes unknown. It is possible to avoid determining the parameter a by considering it as a control parameter for tracking the evolution of shooting diagrams. Alternatively, the parameter a can be considered as a shooting parameter by imposing one constraint on the parameters g or h. Let us consider, for instance, the system trajectories in the conﬁguration plane X1 X2 . Introducing the amplitude A = g 2 + h2 in (12.43), gives X1 (−1) = A cos ϕ X2 (−1) = A sin ϕ where the angle ϕ (0 ≤ ϕ < 2π) together with the parameter a can play the role of a new unknowns to be determined by shooting whereas the amplitude A is considered as a control parameter.

Chapter 13

Essentially Non-periodic Processes

Abstract. This chapter describes a possible physical basis for NSTT in case of essentially non-periodic processes. The physical time is structurised to match the one-dimensional dynamics of rigid-body chain of identical particles. Namely, the continuos ‘global’ time is associated with the propagation of linear momentum, whereas a sequence of non-smooth ‘local’ times describe behavious of individual physical particles. Such an idea helps to incorporate temporal symmetries of the dynamics into diﬀerential equations of motion in many other cases of regular or irregular sequences of internal impacts or external pulses. Since the local times are bounded, a much wider set of analytical tools becomes possible, wereas matching conditions are generated automatically by the corresponding time substitution.

13.1

Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles

The periodic version of NSTT employs basis functions generated by the most simple impact oscillator. This is based the fact that the triangular sine and rectangular cosine waves capture general temporal symmetries of periodic processes regardless speciﬁcs of individual vibrating systems. Below, a nonperiodic pair of nonsmooth functions is considered, such as the ramp function, s (t; d) =

1 (d + |t| − |t − d|) 2

(13.1)

and its ﬁrst order generalized derivative, s˙ (t; d), with respect to the temporal argument, t; see Figs.13.1 and 13.2, respectively. Such kind of functions play an important role in signal analyses [74]. Possible physical interpretation of these functions is represented in Fig. 13.3. Namely, the function s (t, d) can be treated as a coordinate of a particle, say a very small perfectly stiﬀ bead, initially located at the origin x = 0. At the time instance t = 0, this bead is struck by the identical bead with the velocity v = 1. After the linear momentum exchange, the reference bead starts moving until it stopped by the third bead x = d; in our case d = 1. V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 295–303, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

296

13 Essentially Non-periodic Processes

s 1.0 0.8 0.6 0.4 0.2 1.0

0.5

0.5

1.0

1.5

2.0

t

Fig. 13.1 The unit slope ramp function at d = 1.0

s 1.0 0.8 0.6 0.4 0.2 1.0

0.5

0.5

1.0

1.5

2.0

t

Fig. 13.2 First derivative of the ramp function

Fig. 13.3 Physical meaning of the ramp function: s(t; 1) describes position of the bead struck by another bead from the left and moving until it strikes the next bead of the same mass.

Now, let us consider an inﬁnite chain of the identical perfectly stiﬀ beads located regularly on a straight line at the points xi (i = 0, 1, ...). No energy loss is assumed so that any currently moving bead has the same velocity. As a result, the linear momentum is translated with the constant speed v = 1, whereas the beads are interacting at the time instances ti = xi . Making the temporal shift t → t − ti in function (13.1), gives si (t) = s (t − ti , di ) = where di = ti+1 − ti .

1 (di + |t − ti | − |t − ti+1 |) 2

(13.2)

13.1 Nonsmooth Time Decomposition and Pulse Propagation

297

Due to the unit velocity, function (13.2) can play the role of ‘local’ time for the bead moving within the interval xi < x < xi+1 during the ‘global’ time interval ti < t < ti+1 . The term “local” means that the temporal variable si starts at zero when the “global” time, t, has reached the point t = ti . In other words, the global temporal variable is associated with the linear momentum, whereas all the local temporal variables are “attached” to the physical bodies. For any sequence of time instances, Λ={t0 , t1 , ...}, the global time, t ∈ [t0 , ∞), can be expressed through the sequence of local times, {si }, in the form ∞ (ti + si ) s˙ i (13.3) t= i=0

where the derivatives s˙ i satisfy the relationship s˙ i s˙ j = s˙ i δij

(13.4)

Practically, (13.3) is always a ﬁnite sum because temporal intervals of physical processes always have ﬁnite upper bounds. Equality (13.3) can be easily veriﬁed within the arbitrary interval ti < t < ti+1 , by using deﬁnitions (13.1) and (13.2), but the set of boundary points Λ still require some attention. So formally diﬀerentiating both sides of equality (13.3) with respect to t and taking into account (13.4), gives 1= =

∞ i=0 ∞

s˙ i +

∞

(ti + si ) [δ (t − ti ) − δ (t − ti+1 )]

(13.5)

i=0

s˙ i + t0 δ (t − t0 )

i=0

where the relationship (ti+1 + si+1 (ti+1 ) − ti − si (ti+1 )) = 0 has been used for calculations. Therefore, for almost all t > t0 , expression (13.5) gives 1=

∞

s˙ i

(13.6)

i=0

Equality (13.6) holds based on the deﬁnition for s˙ i as illustrated by Fig. 13.2, therefore expansion (13.3) also holds for almost all t > t0 at least, and its time derivative includes no singular functions. Note that the right-hand side of expansion (13.3) can be viewed as an element of algebra with the basis {si } and multiplication rule (13.4). This signiﬁcantly eases diﬀerent manipulations with the temporal variable (13.3), for instance

298

13 Essentially Non-periodic Processes

n

t =

∞

n

(ti + si ) s˙ i , n = 1, 2, ...

(13.7)

i=0

or, generally, x (t) =

∞

x (ti + si ) s˙ i =

i=0

∞

Xi (si ) s˙ i

(13.8)

i=0

Since the right-hand sides of (13.7) and (13.8) have the same structure as the argument t itself, then considering further a general function g, gives g (x) =

∞

g (Xi ) s˙ i

(13.9)

i=0

Now, diﬀerentiating (13.8) with respect to time t, and taking into account that si (ti ) = 0 and si−1 (ti ) = di−1 , gives x˙ (t) = =

∞ i=0 ∞

Xi (si ) s˙ i + Xi (si ) s˙ i +

i=0

∞ i=0 ∞

Xi (si ) [δ (t − ti ) − δ (t − ti+1 )]

(13.10)

[Xi (0) − Xi−1 (di−1 )] δ (t − ti )

i=0

where X−1 (d−1 ) = 0. Therefore, all the δ- pulses are eliminated from (13.10) under condition, which can be qualiﬁed as a necessary condition of continuity for x (t), Xi (0) − Xi−1 (di−1 ) = 0

(13.11)

Under condition (13.11), the derivative x˙ (t) has the same algebraic structure as the function x (t) itself. As a result, transformation (13.3) can be applied to a general class of dynamical systems. Moreover, in case of impulsively loaded systems, the sequences of δ-pulses in (13.10) can be utilized for eliminating the corresponding singularities from dynamical systems.

13.2

Impulsively Loaded Dynamical Systems

Let us consider a dynamical system subjected to an arbitrary sequence of δ-impulses, applied to the system at time instances Λ = {t0 , t1 , ...}, x˙ = f (x, t) +

∞

pi δ (t − ti ) ,

x (t) ∈ Rn

(13.12)

i=0

x ≡ 0,

t < t0

(13.13)

where f (x, t) is a suﬃciently smooth vector-function, pi are vectors characterizing magnitudes and directions of the impulses.

13.2 Impulsively Loaded Dynamical Systems

299

In particular case, when t0 = 0, and pi = 0 (i = 1, ... ), system (13.12) and (13.13) becomes equivalent to the following initial value problem x˙ = f (x, t)

(13.14)

x (0) = p0

(13.15)

Below, solution of the initial value problem (13.12) and (13.13) is introduced in the speciﬁc form based on the operator Lie associated with dynamical system (13.15) ∂ ∂ + ∂x ∂t ∂ ∂ ∂ = f1 (x, t) + .... + fn (x, t) + ∂x1 ∂xn ∂t

A = f (x, t)

(13.16)

It is known, for instance, that the exponent of operator (13.16) produces temporal shifts as follows ezA f (x (t) , t) = f (x (t + z) , t + z) (13.17)

∂f (x, t) ∂ (x, t) = f (x, t) + f (x, t) + z + O(z 2 ) ∂x ∂t Proposition 6. Solution of the initial value problem (13.12) and (13.13) can be represented in the form x (t) =

∞

[ai−1 + pi + F (ai−1 + pi , ti , si (t))] s˙ i (t)

(13.18)

i=0

where ai = x (ti+1 ) is the sequence of constant vectors determined by the mapping a−1 = 0 ai = ai−1 + pi + F (ai−1 + pi , ti , di ) ;

(13.19) i = 1, 2, ...

and the function F is deﬁned by

z

ezA f (x, t) dz

F (x, t, z) =

(13.20)

0

where A is the operator Lie (13.16). Proof. Substituting vector analogs of expressions (13.3), (13.8) and (13.10) into the diﬀerential equation of motion (13.12) and taking into account (13.9), gives ∞ {[Xi (si ) − f (Xi (si ) , ti + si )]s˙ i + (13.21) i=0

300

13 Essentially Non-periodic Processes

[Xi (0) − Xi−1 (di−1 ) − pi ]δ (t − ti )} = 0

(13.22)

The left-hand side of expression (13.22) includes both regular and singular terms. Moreover, the basis elements s˙ i are linearly independent, and all the δ-pulses are acting at diﬀerent time instances.. Therefore, equation (13.22) gives (13.23) Xi (si ) = f (Xi (si ) , ti + si ) Xi (0) = Xi−1 (di−1 ) + pi = ai−1 + pi

(13.24)

where a−1 = 0 and ai = Xi (di ) (i = 0, 1, 2, ...). Equation (13.23) can be represented in the integral form si Xi (si ) = Xi (0) +

f (Xi (z) , ti + z) dz

(13.25)

0

Since the variable of integration is limited by the interval 0 ≤ z ≤ si , the integrand in (13.25) can be approximated by the easy to integrate Maclaurin’s series with respect to z. Moreover, such a series can be represented in the convenient form of Lie series based on the fact that Xi (z) are coordinates of the dynamical system with the operator Lie (13.16). As a result, all the coeﬃcients of power series are expressed through the “initial conditions” at z = 0 (t = ti ) by enforcing the form of the dynamical system. This eliminates all high-order time derivatives from the coeﬃcients of the power series. means of the right-hand side of the dynamical system; no high order derivatives of the coordinates are included any more into the coeﬃcients of the series. So, taking into account the notation Xi (si ) = x (ti + si ), and expressions (13.17) and (13.20), brings (13.25) to the form si ezA f (x (ti ) , ti ) dz

Xi (si ) = Xi (0) + 0

= Xi (0) + F (x (ti ) , ti , si ) = Xi (0) + F (Xi (0) , ti , si )

(13.26)

Substituting now Xi (0) from (13.24) in (13.26), gives Xi (si ) = ai−1 + pi + F (ai−1 + pi , ti , si )

(13.27)

Finally, substituting (13.27) in expansion (13.8), gives (13.18). Then, substituting si = di in (13.27) gives (13.19) and thus completes the proof. Solution (13.18) and (13.19) should be viewed as a semi-analytic solution, since some numerical tool is required for calculating the discrete mapping (13.19). The central role here belongs to the function s (t; d) (13.1), which is automatically matching all the neighboring pieces of the solution.

13.2 Impulsively Loaded Dynamical Systems

301

Note that the distances di between times Λ are not necessary small, however, the precision of the solution can be improved by increasing the number of terms of the Lie series ezA f (x, t) with respect to z, rather than reducing the distances di .

13.2.1

Harmonic Oscillator under Sequential Impulses

In order to estimate precision of the above procedure, let us consider the particular case in which function (13.20) can be calculated exactly in the closed form due to the presence of exact analytical solution in between the pulses Λ. The diﬀerential equation of motion on the entire time range is x ¨ + 2ζω x˙ + ω 2 x =

∞

pi δ (t − ti )

(13.28)

i=0

In this case, the function f (x, t) in equation (13.12) becomes

x2 f (x) = −2ζωx2 − ω 2 x1

(13.29)

Using the identity ezA f (x (t) , t) = f (x (t + z) , t + z) and the exact analytical solution of the corresponding free oscillator, gives both components of the vector-function (13.20) in the form ) ) ζe−z ζ ω 2 2 −z ζ ω cos(z 1 − ζ ω) + sin(z 1 − ζ ω) − 1 x1 F1 (x; z) = e 1 − ζ2 ) e−z ζ ω + sin(z 1 − ζ 2 ω) x2 ω 1 − ζ2 ) ωe−z ζ ω F2 (x; z) = − sin(z 1 − ζ 2 ω) x1 (13.30) 2 1−ζ ) ) ζ e−z ζ ω 2 2 −z ζ ω + e cos(z 1 − ζ ω) − sin(z 1 − ζ ω) − 1 x2 1 − ζ2 In this particular case, properties of mapping (13.19) depend on the following determinant 1 + ∂F1 /∂x1 ∂F1 /∂x2 = e−2di ζω J = (13.31) ∂F2 /∂x1 1 + ∂F2 /∂x2 Let us introduce the relative error δ = |J − Jappr | /J

(13.32)

302

13 Essentially Non-periodic Processes

where Jappr is an approximate determinant based on the Lie series expansion (13.17). Figs. 13.4 and 13.5 show diagrams for the relative error δ versus the distance d between any two neighboring impulse times when the highest order terms kept in Lie series (13.17) are O(z 2 ) and O(z 3 ), respectively.

0.030 0.025 0.020

Δ 0.015 0.010 0.005 0.000 0.0

0.2

0.4

0.6

0.8

d Fig. 13.4 Relative error of the determinant based on the truncated Lie series including terms of order O(z 2 ).

Fig. 13.5 Relative error of the determinant based on the truncated Lie series including terms of order O(z 3 ).

As follows from the diagrams, precision of the discrete mapping essentially depends on both the distance between pulse times and the number of terms kept in the Lie series. As a result, the error due to a large distance can be reduced by increasing the number of terms in the Lie series.

13.2 Impulsively Loaded Dynamical Systems

13.2.2

303

Random Suppression of Chaos

A speciﬁc case of the Duﬃng oscillator with no linear stiﬀness under sine modulated random impulses was considered in [142]. The corresponding differential equation of motion is represented in the form x ¨ + ζ x˙ + x3 = B sin t

∞

δ (t − ti )

(13.33)

i=0

where ζ is a constant linear damping coeﬃcient, and B is the amplitude of modulation. Distances between any two sequential impulse times are given by di = ti+1 − ti =

π (1 + βηi ) 12

where ηi are random real numbers homogeneously distributed on the interval [−1, 1], and β is a small positive number, 0 < β 1. Introducing the state vector x = (x, x) ˙ T ≡ (x1 , x2 )T , brings system (13.33) to the standard form (13.12), where

x2 0 f (x) = = , p i B sin ti −ζx2 − x31 Note that oscillator (13.33) represents of-course a modiﬁed version of the well known oscillator, x¨ + ζ x˙ + x3 = B sin t, considered ﬁrst by Ueda [186] as a model of nonlinear inductor in electrical circuits - the Ueda circuit. In particular, the result of work [186], as well as many further investigations of similar models, reveal the existence of stochastic attractors often illustrated by the Poincare diagrams [113]. Similar diagrams obtained under non-regular snapshots can be qualiﬁed as ‘stroboscopic’ diagrams. The results of the computer simulations described in [142] show that some irregularity of the pulse times can be used for the purposes of a more clear observation of the system orbits in the stroboscopic diagrams. When repeatedly executing the numerical code, under the same input conditions, such a small disorder in the input results some times in a less noisy and more organized stroboscopic diagrams. However, such phenomenon itself was found to be a random event whose ‘appearance’ depends on the level of pulse randomization as well as the number of iterations.

Chapter 14

Spatially-Oscillating Structures

Abstract. This chapter illustrates applications of nonsmooth argument substitutions to modeling spatially oscillating structures such as one-dimensional elastic rods with periodic discrete inclusions, and two- or three-dimensional acoustic media with periodic nonsmooth boundary sources of waves. Whenever the corresponding global spatial domains are inﬁnite or cyclical, the related analytical manipulations are similar to those conducted with dynamical systems. The idea of averaging is implemented through the two-variable expansions, where the fast scale is represented by the triangular periodic wave. Such an approach results in closed form analytical solutions despite of the presence discrete inclusions or external discontinuous loads.

14.1

Periodic Nonsmooth Structures

Static and dynamic problems of elasticity dealing with non-smooth periodic structures are often considered in the literature due to their practical importance; see recent reviews [76] for introduction and references. Such problems can also be considered by means of the non-smooth argument transformations. In this case though, the transformation must be applied to the spatial independent variable related to coordinate along which the structure under consideration is periodic. Such an approach was introduced in [135] for a string on discrete elastic foundation, although the complete description of the tool was given later for the corresponding nonlinear case [151], [171], [152], ∞ y y ∂2u ∂2u ρ 2 − T 2 + 2f (u) δ − 1 − 2k = q , y, t (14.1) ∂t ∂y ε ε k=−∞

−∞ < y < ∞ where ρ is the mass density per unit length, T is tension, q(y/ε, y, t) is the body force or external loading, which is assumed to be periodic in the ‘fast scale’ y/ε (0 < ε 1) with the period normalized to four, and similar assumption is made with respect to the transverse displacement of the string u = u(y/ε, y, t); see Fig. 14.1 for illustration. V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 305–337, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

306

14 Spatially-Oscillating Structures

Note that equation (14.1) has no point-wise interpretation due to the presence of Dirac δ-function. Both sides of the equation therefore must be viewed as distributions producing the same output if applied to the same testing function. Correctness of such type of modeling was intensively discussed in the literature; see, for instance, [46]. Omitting details, the series of δ-functions in equation (14.1) has certain meaning if the function f (u(y/ε, y, t)) is at least continuous at y = ε(1 + 2k) for every k. Such a continuity condition easily follows from the very form of equation (14.1). Indeed, if the displacement u were step-wise discontinuous at y = ε(1 + 2k) then the derivative ∂ 2 u/∂y 2 would produce uncompensated derivatives of δ-function. So the displacement u is at least continuous function of the coordinate y that is in match with the physical meaning of model. Moreover, it will be shown below that appropriate nonsmooth substitution for the spatial argument y/ε eliminates the singularities from equation (14.1). As a result, new equations can be considered within the classical theory. In equation (14.1), the co-directed set of localized forces generated by the elastic springs can be expressed through second derivative of the triangular wave τ = τ (y/ε) in the form 2f (u)

∞ k=−∞

δ

y ε

− 1 − 2k = −f (u)sgn(τ )τ

(14.2)

Both the transverse displacement and the external loading are expressed in terms of the non-smooth variables as follows u = U (τ, y, t) + V (τ, y, t)τ q = Q(τ, y, t) + P (τ, y, t)τ

(14.3)

Finally, substituting (14.2) and (14.3) in equation (14.1) and using the diﬀerential and algebraic rules of the non-smooth argument transformation, gives the diﬀerential equations and boundary conditions as, respectively,

2 ∂2V ∂2U Q(τ, y, t) ∂2U 2 ρ ∂ U +ε = −2ε − − (14.4) ∂τ 2 ∂y∂τ T ∂t2 ∂y 2 T

2 ∂2U ∂2V P (τ, y, t) ∂2V 2 ρ ∂ V + ε = −2ε − − (14.5) 2 2 2 ∂τ ∂y∂τ T ∂t ∂y T and V |τ =±1 = 0 ∂U ε2 |τ =±1 = ∓ f (U )|τ =±1 ∂τ T

(14.6) (14.7)

where the form of boundary condition (14.7) was eased by taking into account condition (14.6).

14.1 Periodic Nonsmooth Structures

307

Further analysis is based on representing solutions in the form of asymptotic series U (τ, y, t) = V (τ, y, t) =

∞ k=0 ∞

εk Uk (τ, y, t) εk Vk (τ, y, t)

(14.8)

k=0

Problem formulations based on equations (14.4) through (14.7) possess certain advantage as compared to equation (14.1). For instance, the inﬂuence of inﬁnite discontinuities in equation (14.1) is captured by the form of substitution (14.3) so that no discontinuities are present in the new equations. Also, the structural discreteness of the nonlinear elastic foundation (14.2) is associated with the new spatial variable τ restricted in the range −1 ≤ τ ≤ 1. As a result, conventional perturbation tools become applicable. Moreover, polynomial expansions with respect to the coordinate τ will not be aﬀecting the regularity of asymptotic expansions (14.8) in terms of the original fast scale y/ε. Note that partial diﬀerential equations (14.4) and (14.5) are coupled whereas boundary conditions (14.6) and (14.7) are decoupled with respect to the unknowns U and V . However, introducing new unknown functions, say X = U + V and Y = U − V , decouples equations (14.4) and (14.5) as follows

2 ∂2X ∂2X Q+P ∂2X 2 ρ ∂ X + ε = −2ε − − =0 (14.9) ∂τ 2 ∂y∂τ T ∂t2 ∂y 2 T

2 ∂2Y ∂2Y Q−P ∂2Y 2 ρ ∂ Y + ε = 2ε − − =0 (14.10) ∂τ 2 ∂y∂τ T ∂t2 ∂y 2 T Equations (14.9) and (14.10) are of the same structure, except opposite signs of the partial derivatives. The buckling of a circular ring loaded by a discrete regular set of concentrated compressive forces was considered in [189]; see Fig.14.1. Taking into account identity (14.2), the diﬀerential equation of equilibrium of such ring can be represented in the form y(t) ˙ = f (y(t)) − λ

∞

δ

k=−∞

≡ f (y(t)) + 0 ≤ t ≤ 2π

t − 1 − 2k a

(14.11)

λ sgn(τ )τ 2

where t = s/R is the arc length of the undeformed ring axis per radius, y(t) is a six-component vector-function characterizing elastic states of the ring,

308

14 Spatially-Oscillating Structures

Fig. 14.1 Linear string on the discrete regular set of nonlinearly-elastic springs.

τ = τ (t/a) is the triangular sine wave of the period T = 4a and prime means its Schwartz derivative, a = ε/2 is a small parameter as compared to unity, λ is a dimensionless parameter which is proportional to the localized forces P applied to the ring, and conditions of periodicity are imposed on the elements of vector-function y(t). Let us represent the unknown vector-function in the form y = X(t, τ ) + Y (t, τ )τ Substituting then (14.12) in equation (14.11), gives

1 ∂X 1 ∂Y ∂X ∂Y + − R(X, Y ) + + − I(X, Y ) τ a ∂τ ∂t a ∂τ ∂t

1 λ Y − sgn(τ ) τ = 0 + a 2 or

ε ∂Y ∂X = + R(X, Y ) − ∂τ 2 ∂t

∂X ε ∂Y = + I(X, Y ) − ∂τ 2 ∂t 1 Y |τ =±1 = ± λε 4 where 1 [f (X + Y ) + f (X − Y )] 2 1 I(X, Y ) = [f (X + Y ) − f (X − Y )] 2

R(X, Y ) =

(14.12)

(14.13)

(14.14) (14.15)

14.1 Periodic Nonsmooth Structures

309

Fig. 14.2 The circular ring under discrete regular set of compressive radial forces.

The mechanical model, which is shown in Fig. 14.3, was considered in [150] based on the generalized (asymmetric) version of the triangular wave. In contrast to the model shown in Fig. 14.1, the sprigs are linearly elastic and shifted in dipole-wise manner so that the diﬀerential equation of motion with respect to the string deﬂection u = u(t, y) has the form ρ

∂ 2 u ku ∂ 2 τ (y/a, γ) ∂ 2u 2 (1 − γ − T − )sgn(τ (y/a, γ)) =0 (14.16) ∂t2 ∂y 2 a ∂(y/a)2 −∞ < y < ∞

where the triangular wave with diﬀerent positive and negative slopes is given by z/ (1 − γ) for − 1 + γ ≤ z ≤ 1 − γ τ = τ (z, γ) = (14.17) (−z + 2) / (1 + γ) for 1 − γ ≤ z ≤ 3 + γ −1 < γ < 1 Schwartz derivatives of function (14.17) satisfy the following relationships

∂τ (z, γ) ∂z

2 =α+β

∂τ (z, γ) ∂z

∂τ (z, γ) ∂ 2 τ (z, γ) 1 ∂ 2 τ (z, γ) β = ∂z ∂z 2 2 ∂z 2

(14.18)

(14.19)

310

14 Spatially-Oscillating Structures

∞ ∂ 2 τ (z, γ) = 2α [δ (z + 1 − γ − 4k) − δ (z − 1 + γ − 4k)] ∂z 2 k=−∞ where α = 1/ 1 − γ 2 and β = 2γα. Let us represent the string deﬂection in the form

u = U (τ, y, t) + V (τ, y, t)τ

(14.20)

(14.21)

where τ = τ (y/a, γ) and τ = ∂τ (y/a, γ)/∂(y/a). The components of representation (14.21), U and V , depend on the coordinate y both explicitly and through the triangular wave function τ in such a way that the complete partial derivative ∂u/∂y is equivalent to applying the diﬀerential matrix operator

∂/∂y (α/a)∂/∂τ D= (14.22) (1/a)∂/∂τ (β/a)∂/∂τ +∂/∂y to the vector-column of the components U and V ∂u U ⇐⇒ D V ∂y

(14.23)

under the condition V |τ =±1 = 0

(14.24)

The regular part of the second derivative can be calculated by means of the relationship ∂2u 2 U ⇐⇒ D (14.25) V ∂y 2 However, second derivative of the triangular wave function must be preserved in order to eliminate the same kind of singularity from equation (14.16). So, substituting (14.21) in (14.16) and collecting separately terms related to diﬀerent elements of the basis {1, τ }, gives ρ

×

α ∂2 ∂2 + a2 ∂τ 2 ∂y 2

U+

∂2U −T ∂t2 αβ ∂ 2 2α ∂ 2 + a2 ∂τ 2 a ∂τ ∂y

V =0

(14.26)

∂2V ρ 2 −T ∂t

β ∂2 α + β2 ∂2 ∂2 2 ∂2 2β ∂ 2 + + + V =0 × U + a2 ∂τ 2 a ∂τ ∂y a2 ∂τ 2 a ∂τ ∂y ∂y 2 (14.27)

14.1 Periodic Nonsmooth Structures

under the additional to (14.24) condition

∂V T ∂U k(1 − γ 2 ) +β U |τ =±1 − 2 |τ =±1 = ± a ∂τ ∂τ a

311

(14.28)

Note that some terms have been eliminated from condition (14.28) by taking into account condition (14.24). Further analysis of the boundary value problem (14.24) and (14.26) through (14.28) can be implemented by using the asymptotic approach [151] considering a as a small parameter.

Fig. 14.3 Linear string on the discrete periodic set of linearly-elastic dipole-wise shifted springs of the stiﬀness k.

Similar version of the transformation was employed for statics of layered composites in [179]. The idea is to represent the elastic constants in the form λ∗ = λ(1 + lτ ) μ∗ = μ(1 + mτ ) τ = ∂τ (x/a, γ) /∂(x/a)

(14.29)

where x is the spatial coordinate, which is perpendicular to the layer boundaries, and λ, μ, l, and m are such constants that λ∗ and μ∗ remain positive on the period of elastic structure, T = 4a. Respectively, the vector of elastic displacements is represented in the form {u, v, w} = {U (1) , V (1) , W (1) } + {U (2) , V (2) , W (2) }τ

(14.30)

where U (i) = U (i) (τ, y, z), V (i) = V (i) (τ, y, z), and W (i) = W (i) (τ, y, z). Then, both representations, (14.29) and (14.30), are substituted into the static equations of three-dimensional elastic bodies with variable elastic

312

14 Spatially-Oscillating Structures

constants, and the NSTT algebraic and diﬀerential manipulations applied in order to formulate boundary value problems, however with constant elasticity parameters, for the components of representation (14.30).

14.2

Averaging for One-Dimensional Periodic Structures

Let us consider a one-dimensional 4ε-periodic structure whose static elastic states are described by the vector-function z(y) ∈ Rn that depends upon the longitudinal coordinate y attached to the undeformed structure. The number of vector’ components n can always be increased so that the diﬀerential equation of equilibrium takes the form of ﬁrst-order diﬀerential equation, for instance, as follows z (y) = f (z(y), ϕ, y) + p(y)τ (ϕ)

(14.31)

Here the spatial scale ϕ = y/ε associates with the structural periodicity, the vector-function f (z, ϕ, y) ∈ Rn is continuous with respect to z and y, but it is allowed to be step-wise discontinuous with respect to ϕ at the points {ϕ : τ (ϕ) = ±1}, and p(y) ∈ Rn is a continuous vector-function describing the amplitude modulation of the localized loading. Example 18. In terms of the matrixes,

u(y) 0 z= , p(y) = v(y) q(y)/(2ε)

v(y)/{EF [1 + ατ (y/ε)]} f = 0

(14.32)

equation (14.31) describes an elastic rod whose cross sectional area is a piecewise constant periodic function of the longitudinal coordinate as it is shown in Fig. 14.4 Indeed, substituting (14.32) in (14.31) and eliminating then v(y), gives the second-order diﬀerential equation of equilibrium for the rod du q(y) y d y τ . (14.33) EF 1 + ατ = dy ε dy 2ε ε

Fig. 14.4 An elastic rod with a periodic non-smoothly varying cross sectional area and concentrated loading.

14.3 Two Variable Expansions

313

The averaging technique is described below based on the general form of equation (14.31).

14.3

Two Variable Expansions

The idea of two variable expansions will be used by considering the saw-tooth oscillating coordinate τ = τ (y/ε) and the original coordinate y 0 ≡ y as fast and slow spatial scales, respectively, provided that the following assumption holds ε << 1 (14.34) Let us represent solutions of equation (14.31) in the form z = X(τ, y 0 ) + Y (τ, y 0 )τ

(14.35)

Substituting (14.35) in equation (14.31), gives

∂X ∂Y ∂X ∂Y +ε + ε − R − I + τ + [Y − εp(y 0 )]τ = 0 f f ∂τ ∂y 0 ∂τ ∂y 0 (14.36) where 1 Rf X, Y, τ, y 0 = [f (X + Y, τ, y 0 ) ± f (X − Y, 2 − τ, y 0 )] (14.37) If X, Y, τ, y 0 2 Expression (14.36) is equivalent to the following boundary value problem with no discontinuities

∂Y ∂X +ε − I =0 (14.38) f ∂τ ∂y 0

∂X ∂Y +ε − R =0 f ∂τ ∂y 0 Y |τ =±1 = εp(y 0 )

(14.39)

Let us represent solutions of the boundary value problem, (14.38) and (14.39), in the form of asymptotic series with respect to ε, X= Y =

∞ i=0 ∞

εi X i (τ, y 0 ) εi Y i (τ, y 0 )

i=0

where the functions X i and Y i are to be sequentially determined.

(14.40)

314

14 Spatially-Oscillating Structures

Substituting (14.40) in (14.37), generates power series expansions Rf = Rf0 + εRf1 + ε2 Rf2 + ...

(14.41)

If = If0 + εIf1 + ε2 If2 + ... where the following notations are used Rfi =

1 ∂ i Rf |ε=0 , i! ∂εi

Ifi =

1 ∂ i If |ε=0 i! ∂εi

Substituting (14.40) in (14.38) and (14.39), and matching the coeﬃcients of the same powers of ε, gives the corresponding sequence of equations and boundary conditions. In particular, zero-order problem takes the form ∂Y 0 ∂X 0 = 0, =0 ∂τ ∂τ

(14.42)

Y 0 |τ =±1 = 0

(14.43)

and As follows from (14.42), the generating solution is independent on the fast oscillating scale τ . Therefore, taking into account (14.43), gives solution X 0 = A0 (y 0 ),

Y0 ≡0

(14.44)

where A0 is an arbitrary vector-function of the slow co-ordinate that will be determined on the next step of the asymptotic procedure. So, collecting the terms of order ε, gives the diﬀerential equations and boundary conditions in the form, respectively, ∂Y 0 ∂X 1 = If0 − = If (A0 , 0, τ, y 0 ) ∂τ ∂y 0 ∂Y 1 ∂X 0 dA0 0 0 = Rf0 − = R (A , 0, τ, y ) − f ∂τ ∂y 0 dy 0

(14.45) (14.46)

and Y 1 |τ =±1 = p(y 0 )

(14.47)

Integrating equations (14.45) and (14.46), gives ﬁrst-order terms of the asymptotic solution τ 1

If0 dτ + A1 (y 0 )

X = 0

τ Rf0 dτ −

Y1 = −1

dA0 (τ + 1) + p(y 0 ) dy 0

(14.48)

14.4 Second Order Equations

315

where A1 is a new arbitrary vector-function of the slow spatial scale y 0 , and the limits of integration for Y 1 are chosen in such a manner that boundary condition (14.47) is satisﬁed automatically at the point τ = −1 whereas another point, τ = 1, gives equation dA0 1 = dy 0 2

1 Rf0 dτ ≡< Rf0 >

(14.49)

−1

Note that the ‘slow scale’ equation (14.49) was obtained by satisfying the boundary condition in contrast to the conventional scheme of two variable expansions in which such kind of equations are obtained by eliminating the so-called ‘resonance terms.’ Enforcing now equation (14.49), brings the component Y 1 to the ﬁnal form τ (Rf0 − < Rf0 >)dτ + p(y 0 )

1

Y =

(14.50)

−1

At this stage, expressions (14.48) through (14.50) determine the ﬁrst-order terms of the asymptotic solution, however, the slow-scale vector-function A1 (y 0 ) still remains unknown. The corresponding ordinary diﬀerential equation is obtained on the next stage from the boundary condition for Y 2 and can be represented in the form 0 1 ∂Rf0 dA1 = (14.51) A1 + F 1 (A0 , y 0 ) dy 0 ∂A0 where ∂Rf0 /∂A0 is the Jacobian matrix, and the vector-function F 1 is known. Note, that equation (14.51) is linear. Moreover, on the next steps, equations for the vector-functions A2 , A3 ,... will be of the same linear structure, including the same Jacobian matrix.

14.4

Second Order Equations

Let us consider now the second order diﬀerential equation with respect to the vector-function z(y) ∈ Rn , however, in the linear form z (y) + [q(ϕ, y) + p(y)τ (ϕ)]z = g(ϕ, y) + r(y)τ (ϕ)

(14.52)

where q and p are n× n-matrixes, g and r are n-dimensional vector-functions, and ϕ = y/ε is the fast spatial scale. Based on the assumptions of the previous section, the functions q and g and solutions of equation (14.52) can be represented in the form, respectively,

316

14 Spatially-Oscillating Structures

q(ϕ, y) = Q(τ (ϕ), y 0 ) + P (τ (ϕ), y 0 )τ (ϕ) g(ϕ, y) = G(τ (ϕ), y 0 ) + F (τ (ϕ), y 0 )τ (ϕ) and

(14.53)

z(y) = X(τ (ϕ), y 0 ) + Y (τ (ϕ), y 0 )τ (ϕ)

where y 0 ≡ y represents the slow spatial scale. This leads to the boundary value problem

2 ∂ X ∂2X ∂2Y 2 = −2ε −ε + QX + P Y − G ∂τ 2 ∂τ ∂y 0 ∂y 02

∂2Y ∂2Y ∂2X 2 = −2ε −ε + P X + QY − F ∂τ 2 ∂τ ∂y 0 ∂y 02

(14.54) (14.55)

and ∂X |τ =±1 = ε2 [r(y) − p(y)X]|τ =±1 ∂τ Y |τ =±1 = 0

(14.56)

Further, representing the solution of the boundary value problem (14.54), (14.55) and (14.56) in the form of asymptotic series (14.40), gives a sequence of boundary value problems, in which ﬁrst two steps appear to have quite trivial solutions, such as X 0 = B 0 (y 0 ), Y 0 ≡ 0 and X 1 ≡ 0, Y 1 ≡ 0 where B 0 is an arbitrary function of the slow argument y 0 . As a result, ﬁrst two non-trivial steps of the averaging procedure give z(y) = B 0 (y 0 ) + ε2 [X 2 (τ (ϕ), y 0 ) + Y 2 (τ (ϕ), y 0 )τ (ϕ)] + O(ε3 )

(14.57)

where, in second-order of ε, the solution components are τ (τ − ξ)[G(ξ, y 0 )− < G(τ, y 0 ) >

2

X =

(14.58)

−1

−(Q(ξ, y 0 )− < Q(τ, y 0 ) >)B 0 ]dξ + (r − pB 0 )τ + B 2 (y 0 ) τ 2 (14.59) Y = [(τ − ξ)(F (ξ, y 0 ) − P (ξ, y 0 )B 0 ) −1

− < (1 − τ )(F (τ, y 0 ) − P (τ, y 0 )B 0 ) >]dξ

14.4 Second Order Equations

317

Here, notation < • > means averaging with respect to τ as deﬁned in (14.49), the vector-function B 0 = B 0 (y 0 ) satisﬁes equation d2 B 0 + < Q(τ, y 0 ) > B 0 =< G(τ, y 0 ) > dy 02

(14.60)

The new arbitrary function of the slow coordinate, B 2 (y 0 ), has to be deﬁned on the next step of the procedure. Note that the δ-loads generated by the derivative τ (ϕ) are switching their directions twice per one period of the triangular wave. In many practical cases though the direction of impulses may remain constant. The corresponding reformulation of the problem can be implemented by introducing the factor -sgn(τ ) into the diﬀerential equation as follows z (y) + [q(ϕ, y) − p(y)sgn(τ )τ (ϕ)]z = g(ϕ, y) − r(y)sgn(τ )τ (ϕ) (14.61) Now, in equation (14.61), the term sgn(τ )τ (ϕ) generates δ-loads of the same direction, whereas the boundary condition (14.56) for the X-component takes the form ∂X |τ =±1 = ∓ε2 [r(y) − p(y)X]|τ =±1 (14.62) ∂τ The form of expressions (14.58) and (14.60) is modiﬁed as, respectively, τ (τ − ξ)[G(ξ, y 0 )− < G(τ, y 0 ) >

2

X =

(14.63)

−1

−(Q(ξ, y 0 )− < Q(τ, y 0 ) >)B 0 ]dξ −

τ2 (r − pB 0 ) + B 2 (y 0 ) 2

and, d2 B 0 + (< Q(τ, y 0 ) > +p)B 0 =< G(τ, y 0 ) > +r dy 02

(14.64)

and the component Y 2 is still described by (14.59). Example 19. Let us consider an inﬁnite beam resting on a discrete foundation represented by the periodic set of linearly-elastic springs of stiﬀness c. The corresponding diﬀerential equation of equilibrium is D

∞ x d4 w cw x − 1 − 2k = q + δ load dx4 a a L k=−∞

−∞ < x < ∞ Let us introduce the following dimensionless values y=

y w x , ϕ= , W = L ε a

(14.65)

318

14 Spatially-Oscillating Structures

γ=

cL4 L4 , ψ(y) = qload (y) aD aD

where ε = a/L << 1. As a result the above equation (14.65) for the beam’ center line takes the form d4 W 1 − γsgn[τ (ϕ)]τ (ϕ) W = ψ(y) 4 dy 2

(14.66)

This equation becomes equivalent to (14.61), after the substitutions

0 W (y) , g= ≡G z= ψ(y) W (y)

0 0 0 −1 p= , q= ≡Q γ/2 0 00 P ≡ 0, F ≡ 0, r ≡ 0 After the corresponding calculations in (14.63) and (14.59), one ﬁnally obtains

0 2 1 2 y 0 2 z = B (y) + ε pτ B (y) + B (y) + O ε3 (14.67) 2 ε where the vector-function B 0 = [B10 , B20 ]T is determined from equation (14.64). In terms of the vector B 0 components, equation (14.64) reads

0

d2 B10 0 −1 B1 0 + = (14.68) B20 γ/2 0 ψ(y) dy 2 B20 The ﬁrst equation in (14.68) gives B20 = d2 B10 /dy 2 , whereas the second equation after elimination of B20 takes the form d4 B10 γ + B10 = ψ(y) 4 dy 2

(14.69)

Equation (14.69) is indeed obtained by averaging equation (14.66) with respect to the fast spatial scale ϕ. In other words, equation (14.69) describes an elastic beam resting on eﬀective continuous elastic foundation. In order to illustrate the asymptotic solution, let us consider the beam under the harmonic transverse loading, qload (x/L) ≡ q0 sin (πx/L), where q0 =const. Taking into account only the leading order ‘slow’ and ‘fast’ components, gives the bending moment in terms of the original variables M (x) = D

d2 w dx2

= −M0

a 2 x πx τ2 1−γ sin 2πL a L

(14.70)

14.5 Acoustic Waves from Non-smooth Periodic Boundary Sources

319

where M0 = 2q0 π 2 L2 /(2π 4 + γ). The bending moment diagram is given in Fig. 14.5.

1.0

MxMo

0.5 0.0 0.5 1.0 0

1

2

3

4

5

6

x Fig. 14.5 Bending moment of the beam on the discrete elastic foundation; numerical values of the parameters are as follows: L = π, a = 0.2, and γ = 1948.0

Finally, note that the homogenization procedure described in [24] gives the averaged equation in the slow spatial scale and a so-called ‘cell problem’ in the fast scale. In the above approach, the analog of cell problem associates with the fast oscillating spatial scale given by the triangular wave function τ (y/ε). As a result, solution for the cell problem automatically unfolds on the entire structure so that the fast and slow components of elastic states are eventually expressed though the same coordinate.

14.5

Acoustic Waves from Non-smooth Periodic Boundary Sources

This section deals with two-dimensional acoustic waves propagating from a discontinuous periodic source located at the boundary of half-inﬁnite space. It is shown that introducing the triangular wave function as a speciﬁc spatial coordinate naturally eliminates discontinuities from the boundary condition associated with the active boundary. For illustrating purposes, let us consider the case of two-dimensional stationary waves propagating in the half-inﬁnite media from a piecewise-linear periodic boundary source as shown in Fig. 14.6.

320

14 Spatially-Oscillating Structures

Fig. 14.6 The model

Let us describe acoustic waves by the linear wave equation in the standard form 1 ∂2P ∂2P ∂ 2P ∂2P = + + (14.71) c2f ∂t2 ∂x2 ∂y 2 ∂z 2 where P is a pressure deviation from the static equilibrium pressure; x, y, z and t are spatial coordinates and time, respectively, and cf is the speed of sound in the media. Further, the plane problem is considered when P = P (t, y, z), and therefore, ∂ 2 P/∂x2 = 0. Such an assumption can be justiﬁed by suﬃciently long piezoelectric rods whose characteristics are constant along the x-coordinate. Suppose that the pressure generated by the rods near the boundary is P0 = A sin ωt, where A and ω are constant amplitude and frequency, respectively. Let the boundary condition at z = 0 to have the form P0 (t) for (4n − 1) a ≤ y ≤ (4n + 1) a (14.72) P (t, y, 0) = 0 for (4n + 1) a ≤ y ≤ (4n + 3) a n = 0, ±1, ±2, ... Note that, based on what is actually known near the ﬂuid-source interface, the boundary condition can also be formulated for pressure derivatives. From the mathematical standpoint, this do not aﬀect much the solution procedure though. Let us seek the steady-state solution, which is periodic with respect to t and y and remains bounded as z → ∞. Since the boundary condition is periodic along y-coordinate with period T = 4a, then, according to the idea of non-smooth argument transformation, the triangular wave periodic coordinate is introduced as y → τ (y/a). As a result, the boundary condition (14.72) and yet unknown solution are represented as, respectively, y 1 1 (14.73) P (t, y, z)|z=0 = P0 (t) + P0 (t) τ 2 2 a and

P (t, y, z) = P1 (t, τ (y/a), z) + P2 (t, τ (y/a), z)τ (y/a)

(14.74)

where the components P1 and P2 are considered as new unknown functions.

14.5 Acoustic Waves from Non-smooth Periodic Boundary Sources

321

Taking into account the expression [τ (y/a)]2 = 1, gives ﬁrst generalized derivative of the original unknown function in the form y 1 ∂P2 1 ∂P1 y 1 ∂P = + τ + P2 τ ∂y a ∂τ a ∂τ a a a

(14.75)

Since the function P (t, y, z) has to be continuous with respect to y in the unbounded open region z > 0, then the periodic singular term τ in (14.75) must be eliminated by imposing condition P2 |τ =±1 = 0

(14.76)

Analogously, second derivative takes the form ∂2P 1 ∂ 2 P1 1 ∂ 2 P2 y = + τ ∂y 2 a2 ∂τ 2 a2 ∂τ 2 a

(14.77)

under the condition

∂P1 |τ =±1 = 0 (14.78) ∂τ Note that both derivatives, (14.75) and (14.77), as well as the original function (14.74) appear to have the same algebraic structure of hyperbolic numbers. Obviously, diﬀerentiation with respect to t and z preserve such a structure as well. As a result, substituting the second derivatives into diﬀerential equation (14.71) and collecting separately terms related to each of the basis elements {1, τ }, gives two partial diﬀerential equations for the components of representation (14.74) 1 ∂ 2 Pi ∂ 2 Pi 1 ∂ 2 Pi = + c2f ∂t2 a2 ∂τ 2 ∂z 2

(14.79)

(i = 1, 2) Substituting then (14.74) in (14.73), gives the corresponding set of boundary conditions 1 1 Pi (t, τ, z) |z=0 = P0 (t) = A sin ωt (14.80) 2 2 Now equations (14.79) and boundary conditions (14.76), (14.78) and (14.80) constitute two independent boundary value problems for the components P1 and P2 . However, the result achieved is that no discontinuous functions are present any more in the boundary conditions. Solving the above boundary value problems by the standard method of separation of variables, gives ﬁnally1 1

Derived by S. Pavlyshyn.

322

14 Spatially-Oscillating Structures

P (t, y, z) =

z 1 A sin ω t − + 2 cf

A

m (−1)k−1 1 y sin ω (t − Kk z) cos k − πτ + (14.81) (k − 1/2) π 2 a k=1

∞ k−1 y (−1) 1 y sin ωt exp (−χk z) cos k − πτ τ (k − 1/2) π 2 a a

k=m+1

where *

2 1 π 2 − k− , 2 a *

2 2 ω 1 π 2 − , χk = k− 2 a cf

Kk =

ω cf

2

k = 1, ..., m

k = m + 1, ...

and m is the maximum number at which the expression under the ﬁrst square root is still positive.

2 P

6 0 2

4

1

z 0

2 1 y 2 3

0

Fig. 14.7 Acoustic wave surface for the set of parameters : cf = 10.0, a = 1, ω = 172, t = 3, and A = 2.

A three-dimensional illustration of solution (14.81) is given by Figs. 14.7 and 14.8 for two diﬀerent magnitudes of the frequency ω. Besides, it is seen

14.6 Spatio-temporal Periodicity

323

Fig. 14.8 Contour plot of the wave ﬁeld shown in Fig. 14.7.

that shorter waves are carrying the information about the discreetness of the wave source for a longer distance from the source. In conclusion, solution (14.81) could be of-course obtained in terms of the standard trigonometric expansions by applying the method of separation of variables directly to the original problem, (14.71) and (14.72). However, the derivation of solution (14.81) implies no integration of discontinuous functions, since all the discontinuities have been captured in advance by transformation (14.74). It is also worth to note that, the terms of series (14.81) are calculated on the standard interval, −1 ≤ τ ≤ 1, which is covered by one half of the total period, whereas the standard Fourier expansions must be built over the entire period. This is due to the fact that representation (14.74) automatically unfolds the half-period domain on the inﬁnite spatial interval.

14.6

Spatio-temporal Periodicity

As a possible generalization of the approach, let us consider, for instance, the boundary condition in the form P (t, y, z)|z=0 = f (t, y)

(14.82)

324

14 Spatially-Oscillating Structures

Fig. 14.9 Acoustic wave surface for the set of parameters: cf = 10.0, a = 1, ω = 86, t = 3, and A = 2.

Fig. 14.10 Contour plot of the wave ﬁeld shown in Fig. 14.9.

14.6 Spatio-temporal Periodicity

325

where the function f is periodic with temporal period Tt = 2π/ω and spatial period Ty = 4a. Introducing the triangular wave spatial argument, τy = τ (y/a), gives f (t, y) = F1 (t, τ (y/a)) + F2 (t, τ (y/a))τ (y/a)

(14.83)

where 1 [f (t, aτy ) + f (t, 2a − aτy )] 2 1 F2 (t, τy ) = [f (t, aτy ) − f (t, 2a − aτy )] 2

F1 (t, τy ) =

(14.84)

In a similar way, introducing the triangular wave temporal argument, τt = τ (2ωt/π), into both of the components, F1 and F2 , gives eventually expression of the form f (t, y) = f0 (τt , τy )e0 + f1 (τt , τy )e1 + f2 (τt , τy )e2 + f3 (τt , τy )e3

(14.85)

where components fi (τt , τy ) are uniquely determined by rule (14.84) applied to each of the two arguments, and the following basis is introduced e0 = 1 e1 = τ (2ωt/π) e2 = τ (y/a) e3 = e1 e2

(14.86)

Basis (14.86) obeys the table of products × e0 e1 e2 e3

e0 1 e1 e2 e3

e1 e1 1 e3 e2

e2 e2 e3 1 e1

e3 e3 e2 e1 1

(14.87)

Now, the acoustic pressure is represented in the similar to (14.85) form P (t, y, z) = P0 (τt , τy , z)e0 + P1 (τt , τy , z)e1 + P2 (τt , τy , z)e2 + P3 (τt , τy , z)e3 (14.88) Regarding the problem described in the previous section, the components of representation (14.88) can be obtained as an exercise by introducing the argument τt directly into solution (14.81). However, formulations based on representation (14.88) become technically reasonable whenever the boundary pressure is adequately described by the functions τt and τy or their diﬀerent combinations, for instance, polynomials. In such cases, polynomial approximations with respect to the bounded arguments may appear to be more eﬀective as compared to Fourier expansions.

326

14 Spatially-Oscillating Structures

Let, for instance, P0 (t) describes the periodic sequence of rectangular spikes of the amplitude A,

2ω 1 1 t ≡ A(1 + e1 ) (14.89) P0 (t) = A 1 + τ 2 π 2 Then, boundary condition (14.73) takes the form 1 A(1 + e1 )(1 + e2 ) 4 1 ≡ A(e0 + e1 + e2 + e3 ) 4

P (t, y, z)|z=0 =

(14.90)

where the basis elements {e0 , e1 , e2 , e3 } are given by (14.86), and the table of products (14.87) is taken into account. Now, substituting representation (14.88) in (14.90), gives the boundary conditions for its components at z = 0 as follows Pi (τt , τy , 0) =

1 A; i = 0,...,3 4

Finally, the three-dimensional case can be considered by adding periodicity of the source along the x-direction at the boundary z = 0 and introducing the corresponding triangular wave argument, say τx . The corresponding rules for algebraic manipulations would be analogous to those generated by the arguments τt and τy . However, necessary details are illustrated below on another model.

14.7

Membrane on a Two-Dimensional Periodic Foundation

Consider an inﬁnite membrane resting on a linearly elastic foundation of the stiﬀness K(x, y) under the transverse load q(x, y). Assuming that both the stiﬀness K and load q are measured per unit membrane tension T , the partial diﬀerential equation of equilibrium is represented in the form the Δu − K(x, y)u = q(x, y) ∂2 ∂2 Δ= + 2 2 ∂x ∂y

(14.91)

where u = u(x, y) is the membrane transverse deﬂection. The foundation is assumed to be step-wise discontinuous and periodic along each of the coordinates as described by the function K(x, y) =

x y k 1 + τ 1 + τ 4 a b

(14.92)

14.7 Membrane on a Two-Dimensional Periodic Foundation

327

Fig. 14.11 Fragment of the map of periodic elastic foundation; a = 1.0 and b = 2.0.

With reference to Fig. 14.11, function (14.92) is deﬁned on the inﬁnite plane, such that 0 (x, y) ∈ any “dark ﬁeld” K(x, y) = (14.93) k (x, y) ∈ any “light ﬁeld” In the same way, Fig. 14.12 provides maps for the elements of basis e0 = 1 e1 = τ (x/a) e2 = τ (y/b) e3 = e1 e2

(14.94)

The above table of products (14.87) still valid for basis (14.94). As a result, function (14.92) takes eventually the form K(x, y) =

k (e0 + e1 + e2 + e3 ) 4

(14.95)

Now let us represent the membrane deﬂection in the form u(x, y) = X(τx , τy , x, y)e0 + Y (τx , τy , x, y)e1 +Z(τx , τy , x, y)e2 + W (τx , τy , x, y)e3

(14.96)

328

14 Spatially-Oscillating Structures

Fig. 14.12 The standard basis map: each of the elements is equal to unity within light domains and zero within dark domains.

where τx = τ (x/a) and τy = τ (y/b) are triangular waves whose lengths are determined by the periods of foundation along x− and y− direction, respectively; scales of the explicitly present variables, x and y, are associated with the scales of loading q(x, y), which is assumed to be slow as compared to the spatial rate of foundation. Note that both linear and non-linear algebraic manipulations with combinations of type (14.96) are dictated by the table of products (14.87). For example, taking into account (14.95) and (14.96), gives Ku =

k (X + Y + Z + W )(e0 + e1 + e2 + e3 ) 4

(14.97)

14.7 Membrane on a Two-Dimensional Periodic Foundation

329

High-order derivatives of (14.96) are simpliﬁed by using the table of products (14.87) and introducing speciﬁc diﬀerential operators as follows. First, using the chain rule, gives 1 dτx = τ (x/a) = dx a 1 dτy = τ (y/a) = dy b

1 e1 a 1 e2 b

(14.98) (14.99)

Then, taking into account (14.87), (14.94), (14.98) and (14.99), gives ﬁrst derivatives of (14.96) in the form

1 ∂Y 1 ∂X ∂u ∂X ∂Y = + + e0 + e1 ∂x a ∂τx ∂x a ∂τx ∂x

1 ∂W ∂Z 1 ∂Z ∂W + + + (14.100) e2 + e3 a ∂τx ∂x a ∂τx ∂x de1 (x/a) 1 +(Y + W e2 ) d(x/a) a ∂u = ∂y

1 ∂Z 1 ∂W ∂X ∂Y + + e0 + e1 b ∂τy ∂y b ∂τy ∂y

1 ∂X 1 ∂Y ∂Z ∂W + + + e2 + e3 b ∂τy ∂y b ∂τy ∂y de2 (y/b) 1 +(Z + W e1 ) d(y/b) b

(14.101)

Last addends in (14.100) and (14.101) include derivatives of the step-wise discontinuous functions e1 (x/a) and e2 (y/b). Such derivatives are expressed through Dirac δ-functions and therefore must be excluded from the expressions (14.100) and (14.101) due to continuity of the original function u(x, y). The δ-functions are eliminated under the boundary conditions Y |τx =±1 = 0 W |τx =±1 = 0

(14.102)

and Z|τy =±1 = 0 W |τy =±1 = 0

(14.103)

The rest of terms in (14.100) and (14.101) represent linear combinations of the basis {e0 , e1 , e2 , e3 }. In order to formalize the diﬀerentiation procedure, let us associate expansion (14.96) with the vector-column

330

14 Spatially-Oscillating Structures

⎡

⎤ X ⎢Y ⎥ ⎥ u =⎢ ⎣Z ⎦ W

(14.104)

In a similar way, let us introduce the vector-columns ux and uy associated with derivatives (14.100) and (14.101) under conditions (14.102) and (14.103), respectively,2 ux = Dx u uy where

⎡

∂/∂x ⎢ a−1 ∂/∂τx Dx = ⎢ ⎣0 0 and

⎡

∂/∂y ⎢0 Dy = ⎢ ⎣ b−1 ∂/∂τy 0

(14.105)

= Dy u

a−1 ∂/∂τx ∂/∂x 0 0

0 0 ∂/∂x a−1 ∂/∂τx

⎤ 0 ⎥ 0 ⎥ a−1 ∂/∂τx ⎦ ∂/∂x

0 ∂/∂y 0 b−1 ∂/∂τy

b−1 ∂/∂τy 0 ∂/∂y 0

⎤ 0 b−1 ∂/∂τy ⎥ ⎥ ⎦ 0 ∂/∂y

(14.106)

(14.107)

These diﬀerential matrix operators automatically generate high-order derivatives of combination (14.96) provided that necessary smoothness (boundary) conditions hold. For instance, the components of expansion for Δu are given by the elements of vector-column (D2x + D2y )u under conditions

1 ∂X ∂Y + |τx =±1 = 0 a ∂τx ∂x

∂W 1 ∂Z + |τx =±1 = 0 a ∂τx ∂x

(14.108)

1 ∂X ∂Z + |τy =±1 = 0 b ∂τy ∂y

1 ∂Y ∂W + |τy =±1 = 0 b ∂τy ∂y

(14.109)

Consider now the particular case a = b = ε 1. Following the diﬀerentiation and algebraic manipulation rules as introduced above, and substituting (14.96) in (14.91), gives 2

Note that ux is not ∂u/∂x.

14.7 Membrane on a Two-Dimensional Periodic Foundation

Δτ X + 2ε

Δτ Y + 2ε

∂2Y ∂ 2Z + ∂τx ∂x ∂τy ∂y ∂2X ∂ 2W + ∂τx ∂x ∂τy ∂y

331

+ ε2 (ΔX − F ) = ε2 q(x, y) + ε2 (ΔY − F ) = 0

2 ∂ W ∂2X + Δτ Z + 2ε + ε2 (ΔZ − F ) = 0 ∂τx ∂x ∂τy ∂y

2 ∂ Z ∂2Y + Δτ W + 2ε + ε2 (ΔW − F ) = 0 ∂τx ∂x ∂τy ∂y

(14.110)

where Δτ = ∂ 2 /∂τx2 + ∂ 2 /∂τy2 , Δ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 , and symbol F denotes the following group of terms related to the elastic foundation F ≡

1 k(X + Y + Z + W ) 4

(14.111)

Boundary conditions (14.108) and (14.109) can be simpliﬁed due to (14.102) and (14.103). As a result, the complete set of boundary conditions takes the form ∂X |τ =±1 = 0, ∂τx x Y |τx =±1 = 0,

∂X |τ =±1 = 0 ∂τy y ∂Y |τ =±1 = 0 ∂τy y

∂Z |τ =±1 = 0, Z|τy =±1 = 0 ∂τx x W |τx =±1 = 0, W |τy =±1 = 0

(14.112)

Note that equations (14.110) have constant coeﬃcients whereas the step-wise discontinuities of foundation have been absorbed by the triangular wave arguments τx and τy . The corresponding boundary conditions (14.112) generate a so-called “cell problem” [76] within the rectangular domain {−1 ≤ τx ≤ 1, −1 ≤ τy ≤ 1}. Therefore, the arguments τx and τy locally describe the fast varying component of membrane shape. In contrast, the explicitly present coordinates x and y describe the slow component in the inﬁnite plane {−∞ < x < ∞, −∞ < y < ∞}. Finally, the increase in number of equations (14.110) of-course complicates solution procedures from the technical standpoint. However, the obvious symmetry of equations helps to ease the corresponding derivations. For instance, equations (14.110) can be decoupled by introducing new unknown functions Ui = Ui (τx , τy , x, y) (i = 1, .., 4) as follows U1 = X + Y + Z + W U2 = X + Y − Z − W U3 = X − Y + Z − W U4 = X − Y − Z + W

(14.113)

332

14 Spatially-Oscillating Structures

Linear transformation (14.113) however makes boundary conditions (14.112) coupled. New boundary conditions are given by the inverse substitution in (14.112) 1 (U1 + U2 + U3 + U4 ) 4 1 Y = (U1 + U2 − U3 − U4 ) 4 1 Z = (U1 − U2 + U3 − U4 ) 4 1 W = (U1 − U2 − U3 + U4 ) 4 X=

(14.114)

Note that transformation (14.113) can be eﬀectively incorporated at the very beginning of transformations by using the idempotent basis as described in the next section.

14.8

The Idempotent Basis for Two-Dimensional Structures

The two-dimensional idempotent basis is introduced as follows 1 (e0 + e1 )(e0 + e2 ) = 4 1 − i2 = e+ 1 e2 = (e0 + e1 )(e0 − e2 ) = 4 1 − + i3 = e1 e2 = (e0 − e1 )(e0 + e2 ) = 4 1 − − i4 = e1 e2 = (e0 − e1 )(e0 − e2 ) = 4

+ i1 = e+ 1 e2 =

1 (e0 + e1 + e2 + e3 ) 4 1 (e0 + e1 − e2 − e3 ) 4 1 (e0 − e1 + e2 − e3 ) (14.115) 4 1 (e0 − e1 − e2 + e3 ) 4

where the standard basis ei is deﬁned by (14.94) and the table of products (14.87), and the following notations for one-dimensional idempotent basis are used 1 (e0 ± ek ) 2 − e+ k ek = 0; (k = 1, 2) e± k =

(14.116)

The main reason for using basis (14.115) is that its table of products has the normalized diagonal form ik in = δkn (14.117) where δkn is the Kronecker symbol. The geometrical meaning of property (14.117) follows from the maps in Fig. 14.13.

14.8 The Idempotent Basis for Two-Dimensional Structures

333

Fig. 14.13 The map of idempotent basis: each of the elements is equal to unity within light domains and zero within dark domains.

In this basis, representation (14.96) takes the form u(x, y) =

4

Uk (τx , τy , x, y)ik

(14.118)

k=1

As a result, 4 4 f( Uk ik ) = f (Uk )ik k=1

(14.119)

k=1

where f is practically any function, linear or nonlinear. First-order partial derivatives of representation (14.119) are obtained as follows

334

=

14 Spatially-Oscillating Structures

∂u(x, y) ∂x 4 1 ∂Uk (τx , τy , x, y) k=1

a

∂τx

(14.120)

∂Uk (τx , τy , x, y) ∂ik ik + Uk (τx , τy , x, y) e1 ik + ∂x ∂x

Further, taking into account (14.115) and (14.116), gives − + + e1 i1 = (e+ 1 − e1 )e1 e2 = i1

− + − e1 i2 = (e+ 1 − e1 )e1 e2 = i2 − − + e1 i3 = (e+ 1 − e1 )e1 e2 = −i3

− − − e1 i4 = (e+ 1 − e1 )e1 e2 = −i4

and ∂i1 ∂x ∂i2 ∂x ∂i3 ∂x ∂i4 ∂x

∂e+ 1 + e ∂x 2 ∂e+ = 1 e− ∂x 2 ∂e− = 1 e+ ∂x 2 ∂e− = 1 e− ∂x 2 =

1 ∂e1 + e 2 ∂x 2 1 ∂e1 − e = 2 ∂x 2 1 ∂e1 + e =− 2 ∂x 2 1 ∂e1 − e =− 2 ∂x 2

=

As a result, derivative (14.120) takes the form

1 ∂U1 1 ∂U2 ∂U1 ∂U2 ∂u = + + i1 + i2 ∂x a ∂τx ∂x a ∂τx ∂x

1 ∂U3 ∂U3 1 ∂U4 ∂U4 + − + + i3 + − i4 a ∂τx ∂x a ∂τx ∂x ∂e1 + 1 ∂e1 − 1 e + (U2 − U4 ) e + (U1 − U3 ) 2 ∂x 2 2 ∂x 2 Analogously, one obtains

1 ∂U1 ∂u ∂U1 1 ∂U2 ∂U2 = + + i1 + − i2 ∂y b ∂τy ∂y b ∂τy ∂y

1 ∂U3 ∂U3 1 ∂U4 ∂U4 + + + i3 + − i4 b ∂τy ∂y b ∂τy ∂y ∂e2 + 1 ∂e2 − 1 e + (U3 − U4 ) e + (U1 − U2 ) 2 ∂y 1 2 ∂y 1

(14.121)

(14.122)

14.8 The Idempotent Basis for Two-Dimensional Structures

335

Let us introduce vector, associated with expansion (14.118), and the corresponding diﬀerential matrix operators as, respectively, ⎡ ⎤ U1 ⎢ U2 ⎥ ⎥ (14.123) u =⎢ ⎣ U3 ⎦ U4 and ⎡1

∂ a ∂τx

+

⎢0 Dx = ⎢ ⎣0

∂ ∂x

0 1 ∂ a ∂τx

+

∂ ∂x

0 0

0 ⎡1

∂ b ∂τy

⎢0 ⎢ Dy = ⎢ ⎣0 0

+

∂ ∂y

0 − 1b ∂τ∂y + 0 0

0 0 − a1 ∂τ∂x + 0

∂ ∂y

∂ ∂x

0 0 1 ∂ b ∂τy

0

+

∂ ∂y

0 0 0 − a1 ∂τ∂x + 0 0 0 − 1b ∂τ∂y +

⎤ ⎥ ⎥ ⎦

(14.124)

∂ ∂x

⎤ ⎥ ⎥ ⎥ ⎦

(14.125)

∂ ∂y

Substituting (14.118) into the original equation (14.91), using the diﬀerentiation rules for idempotent basis, and assuming that a = b = ε, gives

2 ∂ U1 ∂ 2 U1 Δτ U1 + 2ε + + ε2 ΔU1 = ε2 [q(x, y) + kU1 ] ∂τx ∂x ∂τy ∂y

2 ∂ U2 ∂ 2 U2 − Δτ U2 + 2ε + ε2 ΔU2 = ε2 q(x, y) ∂τx ∂x ∂τy ∂y

2 ∂ 2 U3 ∂ U3 − Δτ U3 − 2ε (14.126) + ε2 ΔU3 = ε2 q(x, y) ∂τx ∂x ∂τy ∂y

2 ∂ U4 ∂ 2 U4 + Δτ U4 − 2ε + ε2 ΔU4 = ε2 q(x, y) ∂τx ∂x ∂τy ∂y where the notations Δτ and Δ have the same meaning as those in equations (14.110). Equations (14.126) are decoupled, at cost of coupling the boundary conditions though ∂(U1 − U3 ) |τx =±1 ∂τx (U1 − U3 )|τx =±1 ∂(U2 − U4 ) |τx =±1 ∂τx (U2 − U4 )|τx =±1

∂(U1 − U2 ) |τy =±1 = 0 ∂τy = 0, (U1 − U2 )|τy =±1 = 0 ∂(U3 − U4 ) = 0, |τy =±1 = 0 ∂τy = 0, (U3 − U4 )|τy =±1 = 0 = 0,

(14.127)

336

14 Spatially-Oscillating Structures

Note that both boundary value problems (14.110) through (14.112) and (14.126) through (14.127) implement the transition from two to four spatial arguments: {x, y} → {τx , τy , x, y}. The arguments τx and τy naturally relate to cell problems and incorporate the corresponding elastic components within the class of closed form solutions.

Fig. 14.14 The map of idempotent basis generated by the asymmetric triangular waves with parameters: a = b = 1.0, γ1 = 0.2 and γ2 = 0.6; each of the elements is equal to unity within light domains and zero within dark domains.

Finally, let us introduce two-dimensional idempotent basis generated by the triangular asymmetric wave; see Fig. 14.14. First, following deﬁnitions of Chapter 4, let us introduce one-dimensional idempotents associated with x and y coordinates

14.8 The Idempotent Basis for Two-Dimensional Structures

1 [1 − γi + (1 − γi2 )ei ] 2 1 [1 + γi − (1 − γi2 )ei ] e− i = 2 (i = 1, 2)

337

e+ i =

(14.128)

where e1 = ∂τ (x/a, γ1 )/∂(x/a) and e2 = ∂τ (y/b, γ2 )/∂(y/b). + Now the two-dimensional idempotent basis is given by i1 = e+ 1 e2 , i2 = + − − + − − e1 e2 , i3 = e1 e2 and i4 = e1 e2 , although further expansions shown in (14.115) are not valid any more.

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Appendix

351

APPENDIX 1: MathematicaR (Version 6) notebook for sawtooth power-series solutions for the oscillator t,t x xm 0. This module builds sawtooth power series solutions of the oscillator. The module is easy to modify on polynomial characteristics of the oscillator. n the number of iterations m the exponent STSRn, m : Modulef, h, RHS, Ε, suc, x, X, H, fx : xm ; n

h Εi Hi ; i0 n

RHS Ε h f Εi Xi Ξ ; i0

suc CancelCoefficientNormalSeriesRHS, Ε, 0, n, TableΕi , i, 1, n ; X0 Ξ : A Ξ; DoXk Τ Integratesuck Ξ Τ, Ξ, 0, Τ, Assumptions Τ 0 && Rem 1, k, 1, n; H0 H0 . Solve Τ X0 Τ X1 Τ 0 . Τ 1, H0 1; DoHk Hk . Solve Τ Xk 1 Τ 0 . Τ 1, Hk 1, k, 1, n; TableXk Τ, k, 0, n, TableHk , k, 0, n 1

Successive approximations (execution): n 3; AnalyticalSolution STSRn, Α; DoXi1 AnalyticalSolution1i, i, 1, n 1; DoHi1 AnalyticalSolution2i, i, 1, n; DoPrint"X", i, "", Xi

Simplify, i, 0, n DoPrint"H", i, "", Hi

Simplify, i, 0, n 1 X0A Τ X1

X2

A Τ2Α 2Α A2 Α Α Τ A Τ1Α AΑ 3 2 Α 2 Α Τ A ΤΑ 2 2 Α2 3 2 Α

A12 Α Α Τ2Α A2 Α 1 Α2 4 3 Α AΑ Α 8 10 Α 3 Α2 Τ A ΤΑ 2 Α 3 Α2 Α3 Τ2 A Τ2 Α X3 2 2 Α3 3 2 Α 4 3 Α

352

Appendix

H0A1Α 1 Α H1

H2

A1Α Α 1 Α 2 2 Α A1Α Α 1 Α3 2 2 Α2 3 2 Α

Successive approximation truncated series: n

n1

x Xi ; h Hi ; v i0

i0

1 h

Τ x;

Appendix

353

APPENDIX 2: MathematicaR (Version 6) notebook for sawtooth power-series expansion of periodic functions. In[1]:=

f ft periodic function of the period T m "length" of the series; see the example below smoothness True or False; use smoothness True to smooth the series TauSeriesf, T, m, smoothness : Modulea,s1, s2, NLX, NLY,RE, IM, der, TauSpectrum, a T 4; s1 t a Τ; s2 t 2a a Τ; 1 1 RE f . s1 f . s2; IM f . s1 f . s2; 2 2 derF : ApplyD, F, Τ ; NLX NestListder, RE, 2m 1 . Τ 0; NLY NestListder, IM, 2m 1 . Τ 0; i

DoKX2i 1 k1 i

KY2i 1 k0 i

KY2i k0

Factorial2k 2

, KX2i k1

NLY2k 1 Factorial2k

,

, i, 1, m ; 2m

i1 2m

2m

Y KYiΤi Τi 2 , X i0

k1

NLYkΤk1

2m

k1

In[2]:=

Factorial2k 1

NLY2k Factorial2k 1

Ifsmoothness, X RE . Τ 0 KXi

Y

NLX2k 1

i

NLX2k

Factorialk 1

Τi i

Τi 2 i 2

,

NLXkΤk1 Factorialk 1

,

Example of usage ft : Sin

Πt 2

3

T 4; In[4]:=

Expansions with no smoothing procedure fnosmootht, m : X Y e . TauSeriesft, T, m, False

. Τ Τ4t T, e e4t T fnosmootht, 1 fnosmootht, 2

,

354

Appendix

fnosmootht, 3 fnosmootht, 4 fnosmootht, 5 Out[5]=

0 1

Out[6]=

8 1 8 1 8 1 8

64 1 64

Π5 Τt5

Π5 Τt5

Π5 Τt5

13 Π7 Τt7 15 360 13 Π7 Τt7 15 360

41 Π9 Τt9 1 548 288

Expansions with smoothing procedure fsmootht, m : X Y e . TauSeriesft, T, m, True

. Τ Τ4t T, e e4t T fsmootht, fsmootht, fsmootht, fsmootht,

1 2 3 4

0 3

Out[12]=

8 3 Out[13]=

8 3 Out[14]=

8

In[15]:=

1

Π3 Τt3

Out[9]=

64

Π3 Τt3

Out[8]=

Out[11]=

1

Π3 Τt3

Out[7]=

In[10]:=

Π3 Τt3

Π3

Π3

Π3

3 Π3 8

Τt3 3 Τt3 3 Τt3 3

5 Π5 64

Τt5 5 Τt5 5 Τt5 5

3 Π3 8 3 Π3 8

91 Π7

Τt7

15 360

7

5 Π5

Τt5

64

5

5 Π5

Τt5

64

5

Τt7 7 Τt7 7

Τt9 9

Defining the basis functions 2 Π ΤΞ : ArcSin Sin Ξ

; Π 2 Π eΞ : Sign Cos Ξ

; 2

Appendix In[17]:=

355

Convergence with no smoothing

PlotEvaluateft, fnosmootht, 2, fnosmootht, 3, fnosmootht, 4, fnosmootht, 5, t, 0, 2 T, PlotStyle Thickness.005, Color Black, Thickness.002, Dashing0.01, 0.01, Color Black, Thickness.002,Color Black,Thickness.002,Color Black, Thickness.003, Color Black, AxesLabel "t", "x", PlotRange All x 4

2

Out[17]=

2

4

6

t

8

2

4 In[18]:=

Convergence with smoothing

PlotEvaluateft, fsmootht, 2, fsmootht, 3, fsmootht, 4, t, 0, 2 T, PlotStyle Thickness.005, Color Black, Thickness.002, Dashing0.01, 0.01, Color Black, Thickness.002, Color Black, Thickness.003, Color Black, AxesLabel "t", "x", PlotRange All x 1.5 1.0 0.5 Out[18]=

2 0.5 1.0 1.5

4

6

8

t

Appendix

357

APPENDIX 3: MathematicaR (Version 4) notebook. NSTT & Shooting method for periodic solutions of the oscillator d 2 x dt2

Ζ

dx dt

t

Ε x3 B e a .

px, v, t : Ζ v Ε x3 ; Π a ; 2Ω Substitution xt XΤt a YΤt aet a applied to the equation of motion: eqX

1 a2

Τ,Τ XΤ

1 2

pXΤ YΤ, eqY

1 a2

Τ,Τ YΤ

1 2

pXΤ YΤ,

pXΤ YΤ, 1 a

a

a

Τ XΤ Τ YΤ, a Τ

Τ XΤ Τ YΤ, 2 a a Τ

pXΤ YΤ, 1

1

1 a

0

Simplify;

Τ XΤ Τ YΤ, a Τ

Τ XΤ Τ YΤ, 2 a a Τ

B

Simplify;

Parameters: Ω 1.0; Ζ 0.05; Ε 1.0; B 7.4; Clearg, h; dg 3; dh 20; solg, h : NDSolveeqX, eqY, X1 g, X '1 0, Y1 0, Y '1 h, X, Y, Τ, 1, 1, MaxSteps Infinity; Xng, h : X '1 . solg, h1; Yng, h : Y1 . solg, h1; plx ContourPlotXng, h, g, dg, dg, h, dh, dh, Contours 0, ContourShading False, FrameLabel "g", "h", PlotPoints 100, RotateLabel False, DisplayFunction Identity; ply ContourPlotYng, h,g, dg, dg,h, dh, dh, Contours 0, ContourShading False, FrameLabel "g", "h", PlotPoints 100, RotateLabel False, ContourStyle Dashing0.01, 0.01, DisplayFunction Identity; pxy Showplx, ply, DisplayFunction $DisplayFunction;

358

Appendix

15

10

5

0 h -5

-10

-15

-20 -3

-2

-1

0 g

Magnified portion of the diagram:

1

2

3

Showpxy, PlotRange 0.3, 0, 5, 4;

-4

-4.2

-4.4 h -4.6

-4.8

-0.25

-0.2

-0.15 g

-0.1

-0.05

0

Appendix

359

Find one of the roots: g, h g, h . FindRootXng, h 0, Yng, h 0, g, 0.3, 0, h, 5, 4 0.113351, 4.51957

Check precision: sln NDSolveeqX, eqY, X1 g, X '1 0, Y1 0, Y '1 h, X, Y, Τ, 1, 1; Y1, X '1 . sln1 2.09795 109 , 1.47496 109 Introduce the sawtooth sine and the rectangular cosine: 2 Π Τt : ArcSin Sin t

; Π 2 Π et : Sign Cos t

; 2 Graphic output compared to solution of the related Cauchy problem: xt : XΤt a YΤt a et a . sln1; 1

Y 'Τt a X 'Τt a et a . sln1; a xxtt PlotEvaluatext, t, 0, 4 a, PlotRange All,

vt :

AxesLabel "t", "x", PlotStyle Thickness.009, TextStyle FontSize 14, DisplayFunction Identity; xxvv ParametricPlotEvaluatext, vt, t, 0, 4a, PlotRange All, AxesLabel "x", "v", Frame True, PlotStyle Thickness.009, TextStyle FontSize 14, DisplayFunction Identity; Cleary; Tmax 4a; dirsol NDSolve t,t yt Ζ t yt Ε yt3 B et a, y0 x0, y '0 v0, y, t, 0, Tmax, MaxSteps Infinity ; y y . dirsol1; yyvv ParametricPlotEvaluateyt, y 't, t, 0, Tmax, PlotRange All, AxesLabel "x", "v", PlotStyle Thickness.001, TextStyle FontSize 14, PlotPoints 500, Frame True, DisplayFunction Identity; ShowGraphicsArrayxxtt, xxvv, yyvv;

360

Appendix

x 3 2 1 t -1 -2 -3

1

2

3

4

5

6

v 4 2 0 -2 -4

x

-3 -2 -1 0

1

2

3

v 4 2 0 -2 -4

x

-3 -2 -1 0

1

2

3

Appendix

361

APPENDIX 4: MathematicaR notebook. Conducts NSTT of the T-periodic differential equation of the form t,t x f x, t x, t 0. In[1]:=

see examples below for the meaning of inputs

TauTransformequation, function, argument, period, basis : Modulesub, eqn, RE, IM, sub function XΤ YΤe,

argument function Y 'Τ X 'Τe a,

argument,argument function X ''Τ Y ''Τe a2 ; Q Partequation, 1 . sub; 1 eqn Q . t a Τ Q . t 2a a Τ 2

1 2

Q . t a Τ Q . t 2a a Τ e . a period 4;

EQNX Simplify EQNY Simplify subid XΤ

1 2 1 2 1 2

eqn . e 1 eqn . e 1 ; eqn . e 1 eqn . e 1 ; X Τ X Τ, YΤ

1

1 2 1

X Τ X Τ,

Τ X Τ Τ X Τ, Y 'Τ Τ X Τ Τ X Τ, 2 2 1 1 X ''Τ Τ,Τ X Τ X Τ, Y ''Τ Τ,Τ X Τ X Τ; 2 2 IDP SimplifyEQNX EQNY . subid; IDM SimplifyEQNX EQNY . subid; Ifbasis 1, EQNX 0, EQNY 0, Y1 0, Y1 0, X '1 0, X '1 0, IDP 0, IDM 0,X 1 X 1 0, X 'Τ

X 1 X 1 0, X '1 X '1 0, X '1 X '1 0

EXAMPLES In[2]:=

In[3]:=

Out[3]=

If basis 1, then the output is created in the standard basis 1,e, otherwise the output is in the idempotent basis e ,e 1 e 2,1e 2 TauTransformx ''t xt3 0, xt, t, 4a, 1

XΤ3 3 XΤ YΤ2

X Τ a2

0, 3 XΤ2 YΤ YΤ3

Y1 0, Y1 0, X 1 0, X 1 0

Y Τ a2

0,

362 In[4]:=

Out[4]=

Appendix TauTransformx ''t xt3 0, xt, t, 4a, 0 X Τ

X Τ3

a2

0, X Τ3

X Τ a2

0, X 1 X 1 0,

X 1 X 1 0, X 1 X 1 0, X 1 X 1 0 In[5]:=

Out[5]=

TauTransformx ''t xt3 P Sint 0, xt, t, 2Π, 0

P Sin

ΠΤ 2

X Τ3

4 X Τ Π2

0,P Sin

ΠΤ 2

X Τ3

4 X Τ Π2

0,

X 1 X 1 0, X 1 X 1 0, X 1 X 1

0, X 1 X 1 0 In[6]:=

Out[6]=

TauTransformx ''t xt3 P Sint 0, xt, t, 2Π, 1

P Sin YΤ3

In[7]:=

In[8]:=

Out[8]=

ΠΤ

XΤ3 3 XΤ YΤ2

2 4 Y Τ Π2

4 X Τ Π2

0, Y1 0, Y1 0, X 1 0, X 1 0

If present, the function ee4t T must be shown with no argument; derivatives de dt are not allowed by this code, but necessary generalizations are easy to implement TauTransformx ''t 1 Α e xt Β xt3 0, xt, t, 4a, 1

XΤ Β XΤ3 Α YΤ 3 Β XΤ YΤ2 Α XΤ YΤ 3 Β XΤ2 YΤ Β YΤ3 Y1 0, X 1 0, X 1 0

In[9]:=

Out[9]=

0, 3 XΤ2 YΤ

X Τ a2 Y Τ a2

0,

0, Y1 0,

TauTransformx ''t 1 Α e xt Β xt3 0, xt, t, 4a, 0

1 Α X Τ Β X Τ3

X Τ a2

X Τ a2

0, 1 Α X Τ Β X Τ3

0, X 1 X 1 0, X 1 X 1 0, X 1

X 1 0, X 1 X 1 0

PROJECT: Using the idempotent basis, find T-periodic solution of the linear oscillator under external and parametric rectangular wave periodic excitation of the period T=4,

Appendix

363 1

t,t x 1 2 etΩ2 x pet. In[10]:=

Formulates boundary value problem in the idempotent basis:

T 4; bvp TauTransformx ''t Ω2 1

Out[11]=

p

3 2

Ω2 X Τ X Τ 0, p

1

e xt p e 0, xt, t, T, 0

2 1

Ω2 X Τ X Τ 0,

2

X 1 X 1 0, X 1 X 1 0, X 1 X 1 0, X 1 X 1 0 In[12]:=

Solves the boundary value problem analytically: sol DSolvebvp, X Τ, X Τ, Τ

Simplify 2p

3 Cos

Ω 2

Out[12]=

3

Sin

2 Ω Cos

3

2 Τ Ω Sin

Ω 2

X Τ

, 3 Ω2

X Τ 2 p 3

3 Cos

Ω

3 Cos

2

Ω

Sin

3

Sin

2

2 3

3 Cos

Cos

In[13]:=

3

2 Ω 4 Cos

2 3 2

Ω

Ω Sin

Ω

Ω 4

3

2 Ω Sin

3 Cos

ΤΩ

Ω 2

Sin

2 3 Ω2

3 Cos

Ω

Sin

2

2 Ω Sin

3

2 Ω Cos

3 2 3 2

Ω

Ω

2

Extracting two components of the solution: X Τ X Τ . sol1; X Τ X Τ . sol1;

In[15]:=

In[17]:=

Back to the original xt variables: 1 1 x X Τ 1 e X Τ 1 e . Τ Τt, e et; 2 2 1 1 1 v Τ X Τ 1 e Τ X Τ 1 e .Τ Τt,e et; T 4 2 2 Basis functions:

364

Appendix Τt :

2 Π

ArcSin Sin

et : Sign Cos

Π 2

In[19]:=

Π 2

t

;

t

;

Parameters and graphic output: Ω 20.0; p 1.0; PlotEvaluatex, t, 0, 8, PlotRange All, AxesLabel "t", "x ", PlotStyle Thickness.005

x 0.010

0.005

Out[21]=

2

4

6

8

t

0.005 0.010

In[22]:=

Validating the code by durect numerical solution: x0 x . t 0; v0 v . t 0; 1 dirsol NDSolvey ''t Ω2 1

e t ytp et 0,y0 x0, 2 y '0 v0, y, t, 0, 50, MaxSteps Infinity ; y y . dirsol1;

In[26]:=

PlotEvaluateyt, t, 0, 8, PlotRange All, AxesLabel "x", "v", PlotStyle Thickness.003 v 0.010

0.005

Out[26]=

2 0.005 0.010

4

6

8

x

Lecture Notes in Applied and Computational Mechanics Edited by F. Pfeiffer and P. Wriggers Further volumes of this series found on our homepage: springer.com Vol. 52: Pilipchuk, V.N. Nonlinear Dynamics: Between Linear and Impact Limits 364 p. 2010 [978-3-642-12798-4]

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Nonlinear Dynamics Between Linear and Impact Limits

Valery N. Pilipchuk

123

Valery N. Pilipchuk Professor (Research) Mechanical Engineering Wayne State University 5050 Anthony Wayne Dr., 2140 Detroit, Michigan 48202 USA E-mail: [email protected]

ISBN: 978-3-642-12798-4

e-ISBN: 978-3-642-12799-1

DOI 10.1007/ 978-3-642-12799-1 Lecture Notes in Applied and Computational Mechanics

ISSN 1613-7736 e-ISSN 1860-0816

Library of Congress Control Number: 2010925555 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 9876543210 springer.com

Preface

The main objective of this book is to introduce a uniﬁed physical basis for analyses of vibrations with essentially unharmonic, non-smooth or may be discontinuous time shapes. It is known that possible transitions to nonsmooth limits can make investigations especially diﬃcult. This is due to the fact that the dynamic methods were originally developed within the paradigm of smooth motions based on the classical theory of diﬀerential equations. From the physical standpoint, these represent low-energy approaches to modeling dynamical systems. Although the impact dynamics has also quite a long pre-history, any kind of non-smooth behavior is often viewed as an exemption rather than a rule. Similarly, the classical theory of diﬀerential equations usually avoids non-diﬀerentiable and discontinuous functions. To-date, however, many theoretical and applied areas cover high-energy phenomena accompanied by strongly non-linear spatio-temporal behaviors making the classical smooth methods ineﬃcient in many cases. For instance, such phenomena occur when dealing with dynamical systems under constraint conditions, friction-induced vibrations, structural damages due to cracks, liquid sloshing impacts, and numerous problems of nonlinear physics. Similarly to the wellknown analogy between mechanical and electrical harmonic oscillators, some electronic instruments include so-called Schmitt trigger circuits generating nonsmooth signals whose temporal shapes resemble mechanical vibro-impact processes. In many such cases, it is still possible to adapt diﬀerent smooth methods of the dynamic analyses through strongly non-linear algebraic manipulations with state vectors or by splitting the phase space into multiple domains based on the system speciﬁcs. As a result, the related formulations are often reduced to discrete mappings in a wide range of the dynamics from periodic to stochastic. Possible alternatives to such approaches can be built on generating models developing essentially nonlinear/unharmonic behaviors as their inherent properties. Such models must be general and simple enough in order to play the role of physical basis. As shown in this book, new generating systems can be found by intentionally imposing the ‘worst case scenario’ on conventional methods in anticipation that failure of one asymptotic may

VI

Preface

point to its complementary counterpart. However, the related mathematical formalizations are seldom straightforward and require new principles. For instance, the tool developed here employs nonsmooth (impact) systems as a basis to describing not only impact but also smooth or even linear dynamics. This is built on the idea of non-smooth time substitutions/transformations (NSTT) proposed originally for strongly nonlinear but still smooth models. On the author’s view, the methodological role of NSTT is to reveal explicit links between impact dynamics and hyperbolic algebras analogously to the link between harmonic vibrations and conventional complex analyses. In particular, this book gives the ﬁrst systematic description for NSTT and related analytical and numerical algorithms. The text focuses on methodologies and discussions of their physical and mathematical basics. Detailed applications are mostly excluded from this book, however, necessary references on journal publications are provided. The material of this book was prepared during several years of work at Dnipropetrovsk National University and Technological University (Ukraine), Wayne State University, and Research and Development of General Motors Corporation (Michigan, USA). The author greatly appreciates discussions on diﬀerent subjects related to this book during diﬀerent periods of time with Professors I.V. Andrianov, R.A. Ibrahim (Wayne State University), L.I. Manevich (Russian Academy of Science), A.A.Martynyuk (National Academy of Science of Ukraine), Yu.V. Mikhlin (Kharkov Polytechnic Institute), A.F. Vakakis (University of Illinois), A.A. Zevin (National Academy of Science of Ukraine), and V.F. Zhuravlev (Russian Academy of Science). The author also would like to thank Dr. Kelly Cooper (ONR) for her support and attitude to fundamental research that was very helpful at the ﬁnal stage of work on this book.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Brief Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Asymptotic Meaning of the Approach . . . . . . . . . . . . . . . . . . . . 1.2.1 Two Simple Limits of Lyapunov Oscillator . . . . . . . . . . 1.2.2 Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis . . . . . . . . . . . . . . . . . . . 1.3 Quick ‘Tutorial’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Remarks on the Basic Functions . . . . . . . . . . . . . . . . . . . 1.3.2 Viscous Dynamics under the Sawtooth Forcing . . . . . . 1.3.3 The Rectangular Cosine Input . . . . . . . . . . . . . . . . . . . . 1.3.4 Oscillatory Pipe Flow Model . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Periodic Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Strongly Nonlinear Oscillator . . . . . . . . . . . . . . . . . . . . . 1.4 Geometrical Views on Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Geometrical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Nonlinear Equations and Nonlinear Phenomena . . . . . 1.4.3 Rigid-Body Motions and Linear Systems . . . . . . . . . . . 1.4.4 Remarks on the Multi-dimensional Case . . . . . . . . . . . . 1.4.5 Elementary Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Example of Simpliﬁcation in Nonsmooth Limit . . . . . . 1.4.7 Non-smooth Time Arguments . . . . . . . . . . . . . . . . . . . . . 1.4.8 Further Examples and Discussion . . . . . . . . . . . . . . . . . . 1.4.9 Diﬀerential Equations of Motion and Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Non-smooth Coordinate Transformations . . . . . . . . . . . . . . . . . 1.5.1 Caratheodory Substitution . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Transformation of Positional Variables . . . . . . . . . . . . . 1.5.3 Transformation of State Variables . . . . . . . . . . . . . . . . .

1 1 4 4 6 9 9 9 11 12 15 15 17 17 19 21 23 24 25 26 28 30 33 33 33 35

VIII

2

3

4

Contents

Smooth Oscillating Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linear and Weakly Non-linear Approaches . . . . . . . . . . . . . . . . 2.2 A Brief Overview of Smooth Methods . . . . . . . . . . . . . . . . . . . . 2.2.1 Periodic Motions of Quasi Linear Systems . . . . . . . . . . 2.2.2 The Idea of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Averaging Algorithm for Essentially Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Averaging in Complex Variables . . . . . . . . . . . . . . . . . . . 2.2.5 Lie Group Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 38 39

Nonsmooth Processes as Asymptotic Limits . . . . . . . . . . . . . 3.1 Lyapunov’ Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlinear Oscillators Solvable in Elementary Functions . . . . . 3.2.1 Hardening Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Localized Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Softening Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Nonsmoothness Hiden in Smooth Processes . . . . . . . . . . . . . . . 3.3.1 Nonlinear Beats Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Asymptotic of Equipartition . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Asymptotic of Dominants . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Necessary Condition of Energy Trapping . . . . . . . . . . . 3.4.4 Suﬃcient Condition of Energy Trapping . . . . . . . . . . . . 3.5 Transition from Normal to Local Modes . . . . . . . . . . . . . . . . . . 3.6 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Normal and Local Mode Coordinates . . . . . . . . . . . . . . . . . . . . 3.8 Local Mode Interaction Dynamics . . . . . . . . . . . . . . . . . . . . . . . 3.9 Auto-localized Modes in Nonlinear Coupled Oscillators . . . . .

51 51 54 56 59 60 61 62

Nonsmooth Temporal Transformations (NSTT) . . . . . . . . . . 4.1 Non-smooth Time Transformations . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Positive Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 ‘Single-Tooth’ Substitution . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 ‘Broken Time’ Substitution . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Sawtooth Sine Transformation . . . . . . . . . . . . . . . . . . . . 4.1.5 Links between NSTT and Matrix Algebras . . . . . . . . . 4.1.6 Diﬀerentiation and Integration Rules . . . . . . . . . . . . . . . 4.1.7 NSTT Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.8 Generalizations on Asymmetrical Sawtooth Wave . . . . 4.1.9 Multiple Frequency Case . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Idempotent Basis Generated by the Triangular Sine-Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Deﬁnitions and Algebraic Rules . . . . . . . . . . . . . . . . . . . 4.2.2 Time Derivatives in the Idempotent Basis . . . . . . . . . .

93 93 94 96 96 97 101 102 103 105 107

41 43 44

64 69 71 73 74 74 74 76 81 85

109 109 111

Contents

4.3 Idempotent Basis Generated by Asymmetric Triangular Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Deﬁnition and Algebraic Properties . . . . . . . . . . . . . . . . 4.3.2 Diﬀerentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Oscillators in the Idempotent Basis . . . . . . . . . . . . . . . . 4.3.4 Integration in the Idempotent Basis . . . . . . . . . . . . . . . 4.4 Discussions, Remarks and Justiﬁcations . . . . . . . . . . . . . . . . . . 4.4.1 Remarks on Nonsmooth Solutions in the Classical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Caratheodory Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Other Versions of Periodic Time Substitutions . . . . . . 4.4.4 General Case of Non-invertible Time and Its Physical Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 NSTT and Cnoidal Waves . . . . . . . . . . . . . . . . . . . . . . . .

IX

112 112 114 115 116 117 118 119 122 125 125

5

Sawtooth Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Manipulations with the Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Smoothing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sawtooth Series for Normal Modes . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Periodic Version of Lie Series . . . . . . . . . . . . . . . . . . . . . 5.3 Lie Series of Transformed Systems . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Second-Order Non-autonomous Systems . . . . . . . . . . . . 5.3.2 NSTT of Lagrangian and Hamiltonian Equations . . . . 5.3.3 Remark on Multiple Argument Cases . . . . . . . . . . . . . .

131 131 131 135 135 138 138 141 144

6

NSTT for Linear and Piecewise-Linear Systems . . . . . . . . . 6.1 Free Harmonic Oscillator: Temporal Quantization of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Non-autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Standard Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Idempotent Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Systems under Periodic Pulsed Excitation . . . . . . . . . . . . . . . . 6.3.1 Regular Periodic Impulses . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Harmonic Oscillator under the Periodic Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Periodic Impulses with a Temporal ‘Dipole’ Shift . . . . 6.4 Parametric Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Piecewise-Constant Excitation . . . . . . . . . . . . . . . . . . . . 6.4.2 Parametric Impulsive Excitation . . . . . . . . . . . . . . . . . . 6.4.3 General Case of Periodic Parametric Excitation . . . . . 6.5 Input-Output Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Amplitude-Phase Equations . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Amplitude Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 147 147 148 149 149 151 155 157 157 159 161 163 165 166 167

X

Contents

6.6.3 Phase Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Remarks on Generalized Taylor Expansions . . . . . . . . . 6.7 Multiple Degrees-of-Freedom Case . . . . . . . . . . . . . . . . . . . . . . . 6.8 The Amplitude-Phase Problem in the Idempotent Basis . . . . 7

8

9

Periodic and Transient Nonlinear Dynamics under Discontinuous Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Nonsmooth Two Variables Method . . . . . . . . . . . . . . . . . . . . . . 7.2 Resonances in the Duﬃng’s Oscillator under Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Strongly Nonlinear Oscillator under Periodic Pulses . . . . . . . . 7.4 Impact Oscillators under Impulsive Loading . . . . . . . . . . . . . . Strongly Nonlinear Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Periodic Solutions for First Order Dynamical Systems . . . . . . 8.2 Second Order Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 8.3 Periodic Solutions of Conservative Systems . . . . . . . . . . . . . . . 8.3.1 The Vibroimpact Approximation . . . . . . . . . . . . . . . . . . 8.3.2 One Degree-of-Freedom General Conservative Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero . . . . . . . . . . . . . . . . . . . 8.3.5 A Generalized Case of Odd Characteristics . . . . . . . . . 8.4 Periodic Motions Close to Separatrix Loop . . . . . . . . . . . . . . . 8.5 Self-excited Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Strongly Nonlinear Oscillator with Viscous Damping . . . . . . . 8.6.1 Remark on NSTT Combined with Two Variables Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Oscillator with Two Nonsmooth Limits . . . . . . . . . . . . 8.7 Bouncing Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 The Kicked Rotor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 172 173 177 179 179 182 185 189 195 195 196 198 198 202 205 207 209 211 214 218 222 225 230 234 235

Strongly Nonlinear Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.1 Wave Processes in One-Dimensional Systems . . . . . . . . . . . . . . 241 9.2 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

10 Impact Modes and Parameter Variations . . . . . . . . . . . . . . . . 10.1 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Parameter Variation and Averaging . . . . . . . . . . . . . . . . . . . . . . 10.3 A Two-Degrees-of-Freedom Model . . . . . . . . . . . . . . . . . . . . . . . 10.4 Averaging in the 2DOF System . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Impact Modes in Multiple Degrees of Freedom Systems . . . .

245 245 249 252 253 256

Contents

XI

10.5.1 A Double-Pendulum with Amplitude Limiters . . . . . . . 258 10.5.2 A Mass-Spring Chain under Constraint Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 10.6 Systems with Multiple Impacting Particles . . . . . . . . . . . . . . . . 262 11 Principal Trajectories of Forced Vibrations . . . . . . . . . . . . . . 11.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Principal Directions of Linear Forced Systems . . . . . . . . . . . . . 11.3 Deﬁnition for Principal Trajectories of Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Asymptotic Expansions for Principal Trajectories . . . . . . . . . . 11.5 Deﬁnition for Principal Modes of Continuous Systems . . . . . .

265 265 267

12 NSTT and Shooting Method for Periodic Motions . . . . . . . 12.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Sample Problems and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Smooth Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Step-Wise Discontinuous Input . . . . . . . . . . . . . . . . . . . . 12.3.3 Impulsive Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Periodic Solutions of the Period - n . . . . . . . . . . . . . . . . 12.4.2 Two-Degrees-of-Freedom Systems . . . . . . . . . . . . . . . . . 12.4.3 The Autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 275 277 279 279 286 286 290 290 293 294

13 Essentially Non-periodic Processes . . . . . . . . . . . . . . . . . . . . . . 13.1 Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Impulsively Loaded Dynamical Systems . . . . . . . . . . . . . . . . . . 13.2.1 Harmonic Oscillator under Sequential Impulses . . . . . . 13.2.2 Random Suppression of Chaos . . . . . . . . . . . . . . . . . . . .

295

14 Spatially-Oscillating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Periodic Nonsmooth Structures . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Averaging for One-Dimensional Periodic Structures . . . . . . . . 14.3 Two Variable Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Acoustic Waves from Non-smooth Periodic Boundary Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Spatio-temporal Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Membrane on a Two-Dimensional Periodic Foundation . . . . . 14.8 The Idempotent Basis for Two-Dimensional Structures . . . . .

268 269 271

295 298 301 303 305 305 312 313 315 319 323 326 332

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Chapter 1

Introduction

Abstract. This chapter contains physical and mathematical preliminaries with diﬀerent introductory remarks. Although some of the statements are informal and rather intuitive, they nevertheless provide hints on selecting the generating models and corresponding analytical techniques. The idea is that simplicity of a mathematical formalism is caused by hidden links between the corresponding generating models and subgroups of rigid-body motions. Such motions may be qualiﬁed indeed as elementary macro-dynamic phenomena developed in the physical space. For instance, since rigid-body rotations are associated with sine waves and therefore (smooth) harmonic analyses then translations and mirror-wise reﬂections must reveal adequate tools for strongly unharmonic and nonsmooth approaches. This viewpoint is illustrated by physical examples, problem formulations and solutions.

1.1

Brief Literature Overview

Analytical methods of conventional nonlinear dynamics are based on the classical theory of diﬀerential equations dealing with smooth coordinate transformations, asymptotic integrations and averaging [96], [27], [87], [124], [204], [160], [79], [50], [114], [86], [8], [50], [161], [77], [89], [115], [120], [117], [116], [204], [97], [62]. The corresponding solutions often include quasi harmonic expansions as a generic feature that explicitly points to the physical basis of these methods namely - the harmonic oscillator. Generally speaking, some of the techniques are also applicable to dynamical systems close to integrable but not necessarily linear. However, nonlinear generating solutions are seldom available in closed form [9]. As a result, strongly nonlinear methods usually target speciﬁc situations and are rather diﬃcult to use in other cases. Generating models for strongly nonlinear analytical tools with a wide range of applicability must obviously 1) capture the most common features of oscillating processes regardless their nonlinear speciﬁcs, 2) possess simple enough solutions in order to provide eﬃciency of perturbation schemes, and 3) V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 1–36, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

2

1 Introduction

describe essentially nonlinear phenomena out of the scope of the weakly nonlinear methods. So the key notion of the present work suggests possible recipes for selecting such models among so-called non-smooth systems while keeping the class of smooth motions still within the range of applicability. Note that diﬀerent non-smooth cases have been also considered for several decades by practical and theoretical reasons [81], [39], [17], [42], [127], [72], [196], [82], [204], [172], [45], [65], [204], [169], [29], [14], [93], [30], [180], [51], [206], [94], [44], [173], [25], [162], [55], [185], [85], [130], [129], [128], [1], [68], [69], [37]. On the physical point of view, this kind of modeling essentially employs the idea of perfect spatio-temporal localization of strong nonlinearities or impulsive loadings. For instance, sudden jumps of restoring force characteristics are represented by absolutely stiﬀ constraints under the assumption that the dynamics in between the constraints is smooth and simple enough to describe. As a result, the system dynamics is discretized in terms of mappings and matchings diﬀerent pieces of solutions. The present work however is rather close to another group of methods dealing with the diﬀerential equations of motion on entire time intervals despite of discontinuity and/or nonsmoothness points. Such methods are developed to satisfy the matching conditions automatically by means of speciﬁc coordinate transformations on preliminary stages of study. To some extent, these can be qualiﬁed as adaptations of the diﬀerential equations of motion for further studies by another methods. Among such kind of transformations, the Caratheodory substitution [45] can be mentioned ﬁrst. This linear substitution, which includes the unit-step Heaviside function, eliminates Dirac deltafunctions participating as summands in diﬀerential equations. Much later, non-smooth nonlinear coordinate transformations were suggested in [199], [204] for the class of impact systems. This strongly nonlinear transformation eﬀectively eliminates stiﬀ barriers by unfolding the system conﬁguration space in a mirror-wise manner with respect to the barrier planes. A similar idea was implemented later regarding the system phase space [72] in order to resolve certain problems related to non-elastic impacts. Some technical details and discussions on these methods are included below in Section 5 for comparison reason. In contrast, the present approach employs time histories of impact systems as new time arguments. Originally such a nonsmooth temporal transformation (NSTT), or a sawtooth oscillating time1 , was introduced for strongly nonlinear but smooth periodic motions with certain temporal symmetries [132]. Then, it was shown that such an approach still works for general cases by generating speciﬁc algebraic structures in terms of the coordinates [134], [144]. The occurrence of such algebraic structures seems to be essential 1

Here and further, the term ‘sawtooth’ is used for periodic piecewise-linear functions regardless symmetries of their teeth; in most cases, it is the triangular sine wave.

1.1 Brief Literature Overview

3

feature of the approach since it justiﬁes and simpliﬁes analytical manipulations with non-invertible temporal substitutions such as NSTT. Further, the technique was applied to diﬀerent problems of theoretical and applied mechanics [15], [101], [104], [103], [47], [189], [92], [91], [175], [170], [177], [171], [16], [179], [109], [181], [67]. Basic ideas and some of the tools were adapted earlier for the nonlinear normal mode analyses and included in the monograph [190]. However, much of the material presented in this book is either new or signiﬁcantly updated. The entire text is organized as follows. As mentioned in the Preamble, this Chapter 1 contains physical and mathematical preliminaries and diﬀerent introductory remarks. Chapter 2 gives a brief overview of selected analytical methods for smooth oscillating processes. The description focuses on the ideas and technical details that are used further in combinations with non-smooth approaches. In particular, the method of asymptotic integration of the diﬀerential equations of motion based on the Hausdorf equation for Lie operators is reproduced. Chapter 3 includes diﬀerent examples of smooth vibrating systems that, under some conditions, show close to non-smooth or even non-smooth time histories. Such cases are usually most diﬃcult for conventional analyses. Much of the content focuses on nonlinear beat and localization phenomena. Recent interest to this area is driven by the idea of nonlinear energy absorption. Chapter 4 provides a new description of NSTT with proofs of the basic identities and rules for algebraic, diﬀerential and integral manipulations. In particular, ﬁnal subsections show how to introduce nonsmooth arguments into the diﬀerential equations. Such manipulation imposes two main features on the dynamical systems namely generates speciﬁc algebraic structures for unknown functions and switches formulations to boundary-value problems. Notice that the transformation itself imposes no constraints on dynamical systems and easily applies to both smooth and nonsmooth systems. Any further steps, however, should account for physical properties of the related systems. Chapter 5 preliminary illustrates the NSTT’ advantages by introducing power-series expansions for general periodic processes. This becomes possible because the new temporal argument - the triangular sine wave - is itself periodic in the original time. Therefore, such expansions can be viewed as some alternative to Fourier series for processing periodic signals especially with step-wise discontinuities. Then a periodic version of the Lie series is introduced. As a result, formal analytical solutions for normal mode motions of dynamical systems are obtained. Chapters 6, 7 and 8 describe NSTT based analytical tools for linear, weakly nonlinear and strongly nonlinear vibrating systems, respectively. In particular, applying NSTT to linear and weakly nonlinear systems may be very eﬀective in those cases when nonsmooth loadings are present and thus harmonic or quasi harmonic approaches require multiple term expansions for capturing essential features of the dynamics.

4

1 Introduction

Chapters 9 through 11 deal with the concept of nonlinear normal modes. In particular, it is shown that NSTT leads to adequate formulations of the normal mode problem for impact systems. Also the idea of nonlinear normal modes for the case of forced vibration is formulated in terms of NSTT. Chapter 12 presents a semi-analytical approach combining NSTT with the shooting method that essentially extends the area of applications. Chapter 13 describes a possible physical basis for NSTT in case of essentially non-periodic processes. Finally, Chapter 14 illustrates diﬀerent applications to spatially oscillating structures such as one-dimensional elastic rods with periodic discrete inclusions and two-dimensional media with a periodic nonsmooth boundary source of waves.

1.2

Asymptotic Meaning of the Approach

1.2.1

Two Simple Limits of Lyapunov Oscillator

Let us introduce some preliminaries and remarks based on a one-degree-offreedom model as shown at the top of Fig. 1.1.2 This relatively simple model however depicts the gradual evolution from linear to extremely nonlinear dynamics as the exponent n runs from unity to inﬁnity. Notably, all the temporal mode shapes of the oscillator are described by special functions except two boundaries of the interval 1 ≤ n < ∞. Both boundaries represent simple asymptotic limits within the class of elementary functions. Consider ﬁrst the limit of harmonic oscillator, n = 1, that generates the sine and cosine waves as illustrated on the left column of the diagram in Fig. 1.1. From the very general standpoint, a widely known convenience of using this couple of functions can be explained by their certain link to the group of rigid-body motions namely the subgroup of rotations. The standard complex plane representation and the Euler formula can be mentioned here as related tools. Further, taking the linear combination of harmonic waves with diﬀerent frequencies and keeping in mind the idea of parameter variations invokes the area of harmonic and quasi harmonic analyses for both signal processing and dynamical systems. Such tools therefore represent complicated dynamic processes as a combination of the very simple rigid-body rotations with certain angular speeds. Let us consider now another limit n = ∞, when the restoring force vanishes inside the interval −1 < x < 1 but becomes inﬁnitely growing as the system reaches the potential barriers at x = ±1. The physical meaning of this limit is introduced at the top of the right column of the diagram. Despite of the strong nonlinearity caused by impacts, the limit oscillator is also 2

Note that oscillators with power-form characteristics were considered for quite a long time since possibly Lyapunov who obtained such oscillators while investigating degenerated cases in dynamic stability problems; see Chapter 3 for details.

1.2 Asymptotic Meaning of the Approach

5

..

x x2 n1 0 n

Harmonic oscillator

Impact oscillator

Sine and cosine waves Rotation group

Triangular and rectangular waves Translation and reflection groups

1

1

..

x x2 n1 0 n1

1

sin t cos t

Π

2Π

Τt

t et

1

2

4

Conventional complex numbers

Hyperbolic complex numbers

z x yi i2 1

x X Τ Y Τ e e2 1; e Τ t

Fourier series

Power series

k0 Ak cos kt Bk sin kt Quasi harmonic methods

k0

1 k

X k 0 Τ k

1 k

t

Y k 0 Τ k e

Strongly nonlinear methods

Fig. 1.1 Two asymptotic limits described by elementary functions associated with subgroups of rigid-body motions.

described by quite simple elementary functions such as triangular sine and rectangular cosine, say τ (t) and τ˙ (t). These two are associated with another subgroup of the rigid-body motions namely translation and reﬂection. Therefore, analogously to the above case n = 1, the upper limit n = ∞ can play the same fundamental role by generating an hierarchy of alternative to quasi harmonic tools. On ﬁrst look, such alternative tools can still be developed within the paradigm of Fourier expansions by taking appropriate multiple frequency combinations of τ (t) and τ˙ (t). Although such approaches could work for signal processing, it is unclear how to deal with a large number of singularities if substituting the related expansions into diﬀerential equations of motion. Indeed nonsmoothness of such Fourier basis seems to contradict

6

1 Introduction

Fig. 1.2 Complex and hyperbolic planes are shown on the left and right, respectively; in contrast to the circle, each of the hyperbola branches is covered exactly once as the hyperbolic angle is varying in the inﬁnite interval.

the very language of dynamical systems involving time derivatives.3 In addition, the Fourier expansions are closely linked to the linear superposition principle which is inadequate to nonlinear cases. Finally, the major role of Fig. 1.1 is to convince the reader that any presence of functions τ (t) and τ˙ (t) in further developed analytical algorithms is not a simple match of diﬀerent pieces of solutions4 . On the contrary, it has its real physical basis and invokes speciﬁc mathematical tools.

1.2.2

Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis

So instead let us introduce another formalism dealing with same basic frequency but adjusting the triangular sine shape to the desired one through 3

4

Generally speaking, it is possible to start with an expansion for high-order derivatives and then come backward to coordinates by integrations. However, algebraic complexity of such approaches may overshadow any advantages as compared to the regular Fourier expansions. It is quite clear that harmonic waves can also be interpreted as those combined of the same pieces of curves, but such a viewpoint would eliminate much of the vibration theory and many physical eﬀects.

1.2 Asymptotic Meaning of the Approach

7

polynomials and other simple functions of the argument τ . Such formalism is based on the following statement [133]: Any periodic process x (t) of the period normalized to T = 4 can be expressed through the dynamic state of the impact oscillator, {τ (t), τ˙ (t)}, in the form of ‘hyperbolic complex number’ x = X (τ ) + Y (τ ) τ˙

(1.1)

On the right-hand side, the functions X and Y are easily expressed through the original function x(t), if this function is known; see Chapter 4 for further details. In case x(t) is an unknown periodic motion of some dynamical system, equations for X and Y components are obtained by substituting (1.1) into the corresponding diﬀerential equation of motion. Then either analytical or numerical procedures can be applied. For instance, one may seek solutions in the form of power series with respect to the ‘oscillating time’ τ . Therefore expression (1.1) can be qualiﬁed as non-smooth time transformation, t → τ , on the manifold of periodic motions. Note that the structure of hyperbolic numbers has been known for quite a long time mostly as a formal extension of the regular complex numbers with no relation to either vibrating systems or nonsmooth functions; see the references below. In the mathematical literature, such an extension (which is often regarded to as a simple particular case of Cliﬀord’s algebras5) is introduced as follows. Whereas the algebraic equation p2 = 1 has the real number solutions p = ±1, the existence of a unipotent u is assumed such that u = +1 and u = −1 but u2 = 1. Then by considering the elements {1, u} as a standard basis, any hyperbolic number w ∈ H can be written in the form w = x + uy, where x and y are real numbers [176]. The hyperbolic √ conjugate of w is deﬁned by w− = x−uy so that |w|H = ww− is the norm of w. Fig. 1.2 illustrates the diﬀerence between complex and hyperbolic planes. More details and some, rather abstract, applications of this algebraic theory can be found in references [90], [176], [7], [187]. As to applied areas, the same kind of algebraic structures occurred in hydrodynamics in connection with characteristics of partial diﬀerential equations [90]. In our case, the unipotent is not a number but the discontinuous function of certain physical nature i.e. the rectangular cosine wave e(t). Indeed since t is running then there is no unique choice for the magnitude of e, whereas always6 e2 = 1. Therefore, identity (1.1) generates the hyperbolic structure from the very general properties of periodic processes. Note that both terms on the right-hand side of (1.1) are essential as those responsible for components with diﬀerent temporal symmetries. For instance, 5 6

William Kingdon Cliﬀord (1845-1879), English mathematician who, in particular, developed the idea that space may not be independent of time. More precisely, for almost any t.

8

1 Introduction

Fig. 1.3 Geometrical interpretation of the particular case with a sine-wave temporal symmetry: observing the coordinate x does not allow to conclude which of the two temporal variables, τ or t, is actually ‘running.’

Fig. 1.3 illustrates geometrical meaning of the particular case of temporal symmetry with no Y -component, where x(t) ≡ x[aτ (t/a)] = X(τ ). It is important to note that, under some conditions on X and Y , combination (1.1) can be of any class of smoothness even though the couple {τ, τ˙ } has singularities at such time instances t where τ = ±1. Finally, let us mention that the hyperbolic plane has another natural basis associated with the two isotropic lines separating the hyperbolic quadrants as shown in Fig. 1.2. The transition from one basis to another is given by e± = (1 ± e)/2 or, inversely, 1 = e+ + e− and e = e+ − e− . Therefore, x = X + Y e = X(e+ + e− ) + Y (e+ − e− ) = (X + Y )e+ + (X − Y )e− ≡ X+ (τ )e+ + X− (τ )e− where x = x(t) any periodic function as deﬁned in (1.1). whose period is normalized to T = 4. On one hand, the advantage is that the elements e+ and e− are mutually annihilating (idempotents) so that e+ e− = 0, e2− = e− and e2+ = e+ . It is clear also that ee+ = e+ and ee− = −e− . Due to the annihilation property, the idempotent basis signiﬁcantly eases diﬀerent algebraic manipulations, for instance, 2 2 e+ + X− e− (X+ e+ + X− e− )2 = X+ On the other hand, this basis usually couples the corresponding smoothness (boundary) conditions; see Chapter 4 for further details and examples.

1.3 Quick ‘Tutorial’

1.3

9

Quick ‘Tutorial’

1.3.1

Remarks on the Basic Functions

First note that the basic functions, τ (t) and τ˙ (t) are expressed through the standard elementary functions in the closed form as τ (t) = and τ˙ (t) = cos

πt 2 arcsin sin π 2

−1 πt πt = e(t) cos 2 2

(1.2)

(1.3)

Obviously τ (t) is a triangular sine wave whose amplitude is unity and the period is T = 4, whereas e(t) is a rectangular cosine wave with step-wise discontinuities as shown in Fig. 1.1. Therefore, expression (1.3) holds ‘almost everywhere.’ For calculating purposes, it can be represented in the form e(t) = sgn [cos(πt/2)]. Practically, for given t, one can calculate the functions τ (t) and e(t) based on their piecewise-linear graphs even with no tables nor calculators involved. Necessary mathematical details regarding the discontinuities of e(t) are discussed later; see also [144]. At this stage however, solutions can be compared with exact solutions obtained in a diﬀerent way or numerical ones in order to validate the technique. Moreover, during the derivations, there is no need in keeping in mind expressions (1.2) and (1.3). It is suﬃcient to take into account linear independence of the elements 1, e and e, ˙ as those from diﬀerent classes of smoothness, and the following properties τ˙ = e e˙ = 2

∞

[δ (t + 1 − 4k) − δ (t − 1 − 4k)]

(1.4)

k=−∞

e2 = 1 where, strictly speaking, the equalities should be interpreted in terms of the distribution theory due to the presence of singularities that occur whenever τ = ±1. Let us consider few examples illustrating the technique and giving a hint regarding those cases when the NSTT is reasonable to apply.

1.3.2

Viscous Dynamics under the Sawtooth Forcing

Consider ﬁrst the very simple one-dimensional case. A light particle in a viscous ﬂuid is subjected to the sawtooth force so that the diﬀerential equation of motion is reduced to

10

1 Introduction

x(t) ˙ = qτ (t)

(1.5)

where q represents the force amplitude per unit coeﬃcient of viscosity. The general solution of equation (1.5) is given by

t

x(t) = x(0) + q

τ (ϕ)dϕ

(1.6)

0

where x(0) is an arbitrary initial position, and a part of the problem, which is calculating the integral, still persists. Now let us represent the particular (periodic) solution in the form x = X(τ ) + Y (τ )e

(1.7)

where τ = τ (t) and e = e(t). Substituting (1.7) in (1.5) and taking into account properties (1.4), gives Y (τ ) − qτ + X (τ )e + Y (τ )e˙ = 0

(1.8)

Based on the linear independence of 1, e and e, ˙ equation (1.8) is equivalent to the boundary value problem Y (τ ) = qτ ,

X (τ ) = 0,

Y (±1) = 0

(1.9)

where the boundary condition provides zero factor for all δ−functions of the derivative e. ˙ In contrast to (1.5), the integration of equation (1.9) is strightforward since the variable of integration is τ . So substituting obvious solution of problem (1.9) into (1.7), gives general solution of equation (1.5) q x = C + (τ 2 − 1)e 2

(1.10)

where C is an arbitrary constant. Then taking into account that τ (0) = 0 and e(0) = 1, gives x(0) = C − q/2 and therefore solution (1.10) takes the form x = x(0) +

q q + (τ 2 − 1)e 2 2

(1.11)

As compared to the direct approach (1.6), the NSTT allowed for conducting the integration of the diﬀerential equation of motion with no dealing with the piece-wise structure of the integrand. Moreover, comparing solutions (1.6) and (1.11), gives the result of direct integration (1.6) in the form

t

τ (ϕ)dϕ = 0

1 {1 + [τ 2 (t) − 1]e(t)} 2

(1.12)

1.3 Quick ‘Tutorial’

11

As a simple test, expression (1.12) can be veriﬁed by taking ﬁrst derivative of the both sides. Note that there are two boundary conditions for Y in (1.9). Since the Y − component of the solution appeared to be even with respect to τ , then both of the conditions are satisﬁed even though one arbitrary constant only is available for Y . If the forcing function in (1.5) were of even degree with respect to τ , for instance, qτ 2 , then the corresponding boundary-value problem would have no solution. This fact is explained by the absence of periodic solutions under the loading qτ 2 , which has a non-zero mean value, whereas representation (1.7) imposes periodicity on the function x(t). However, the NSTT is applicable in case of any odd degree polynomial of τ on the right-hand side of equation (1.5), x(t) ˙ =

n

qk τ 2k−1 (t)

(1.13)

k=1

where qk are constant coeﬃcients. Analogously to the above particular case, taking into account the modiﬁed equation, n qk τ 2k−1 Y (τ ) = k=1

gives periodic solution n n qk qk 2k x = x(0) + + (τ − 1) e 2k 2k k=1

(1.14)

k=1

This solution is veriﬁed by the direct substitution of (1.14) in (1.13).

1.3.3

The Rectangular Cosine Input

Consider the case of step-wise discontinuous periodic loading x(t) ˙ = pe(t)

(1.15)

Substituting (1.7) in (1.15), gives Y + (X − p)e + Y e˙ = 0 The boundary value problem therefore takes the form Y (τ ) = 0,

X (τ ) = p,

Y (±1) = 0

(1.16)

In this case, Y ≡ 0 and the solution of equation (1.15) is x = x(0) + pτ (t)

(1.17)

12

1 Introduction

In this simple case, the above solution could be written though based on the deﬁnition of e(t) (1.3). As a generalization of the right-hand side of equation (1.15), let us consider equation m pi τ i (t) e(t) (1.18) x(t) ˙ = i=0

where pi are constant coeﬃcients. In this case, the periodic solution x = x(0) +

m pi i+1 τ i + 1 i=0

(1.19)

does exist for both odd and even exponents of the polynomial in (1.18). Finally, combining the right-hand sides of equations (1.13) and (1.18) and considering the corresponding inﬁnite series, gives ∞ ∞ 2k−1 i qk τ (t) + pi τ (t) e(t) ≡ Q(τ ) + P (τ )e (1.20) f (t) = k=1

i=0

It will be shown later expansion (1.20) represents a very general class of zero mean periodic functions with the period normalized to T = 4. The corresponding periodic solution of the equation x(t) ˙ = f (t) is obtained by combining solutions (1.14) and (1.19).

1.3.4

Oscillatory Pipe Flow Model

As a possible application, consider a simpliﬁed model of pipe ﬂow driven by a regularly repeating rectangular pressure wave [183]. The pipe ﬂow is assumed to have lumped inertance L and quadratic-law resistance with the coeﬃcient K. The ﬂow Q(t) is described by ﬁrst-order non-linear diﬀerential equation LQ˙ = P0 + P1 e(t/a) − KQ2

(1.21)

where P0 and P1 are constants characterizing the pressure drop, and 4a is the period of the rectangular pressure wave. Introducing parameters k = K/L, p0 = P0 /L and p1 = P1 /L, brings equation (1.21) to the form Q˙ + kQ2 = p0 + p1 e(t/a)

(1.22)

Temporal behavior of the ﬂow essentially depends on the model parameters and initial conditions. Let us assume however that the ﬂow becomes eventually periodic with the period of rectangular pressure wave. The objective is to ﬁnd the average steady state ﬂow. The corresponding periodic solution can be represented in the form

1.3 Quick ‘Tutorial’

13

Q(t) = X(τ ) + Y (τ )e

(1.23)

where τ = τ (t/a) and e = e(t/a) are triangular and rectangular waves, respectively, with the period T = 4a. Substituting (1.23) in (1.22), gives a−1 (Y + X e + Y e ) + k(X 2 + Y 2 + 2XY e) = p0 + p1 e

(1.24)

where e = de(t/a)/d(t/a), and therefore Y + ak(X 2 + Y 2 ) = ap0 X + 2akXY = ap1

(1.25)

Y (±1) = 0 Introducing the new unknowns U = X + Y and V = X − Y , brings the boundary value problem (1.25) to the form U + akU 2 = aF V − akV 2 = −aG

(1.26)

U (±1) − V (±1) = 0 where F = p0 + p1 and G = p0 − p1 are constant. Both equations in (1.26) are separable and thus admit general solutions of the form √ 2 exp(−2a kF τ ) F √ U (τ, C1 ) = 1− k C1 + exp(−2a kF τ ) √ 2 exp(−2a kGτ ) G √ (1.27) 1− V (τ, C2 ) = − k C2 + exp(−2a kGτ ) where C1 and C2 are arbitrary constants of integration to be determined from the boundary conditions U (1, C1 ) − V (1, C2 ) = 0 U (−1, C1 ) − V (−1, C2 ) = 0

(1.28)

Equations (1.28) appear to be nonlinear algebraic set of equations that, generally speaking, admits multiple solutions of which some solutions may be real, whereas others are complex. Each real solution for the constants C1 and C2 gives a periodic solution of diﬀerential equation (1.22). However, simple numerical tests show that some of the periodic solutions may appear to be unstable. (Detailed parametric study and stability analysis go beyond of this introductory illustration.) After a set of the arbitrary constants has been

14

1 Introduction

determined, the pipe ﬂow function can be represented in either standard or idempotent basis as follows 1 1 [U + V ] + [U − V ]e 2 2 = U (τ, C1 )e+ + V (τ, C2 )e−

Q(t) = X + Y e =

(1.29)

where τ = τ (t/a), and e± = [1 ± e(t/a)]/2 are elements of the idempotent basis; algebraic properties and further details of transition from one basis to another are discussed in Chapter 4. Also, it will be shown in Chapter 4 that, temporal mean value of a periodic function can be found by averaging its X(τ )-component with respect to τ . Applying this statement to solution (1.29), gives the average steady state ﬂow G F − k k 0 −1 √ √ 1 [1 + C1 exp(2a kF )][1 + C2 exp(−2a kG)] √ √ + ln 4ak [1 + C1 exp(−2a kF ][1 + C2 exp(2a kG)]

1 < Q >≡ T

T

1 Q(t)dt = 2

10

T8

9

T4

1

1 X(τ )dτ = 2

(1.30)

Q 8 7 6 0.0

0.5

1.0 tT

1.5

2.0

Fig. 1.4 Proﬁles of steady state pipe ﬂows obtained for two diﬀerent periods of the rectangular pressure wave.

Fig. 1.4 shows what happens to the steady state ﬂow proﬁle as the period of pressure wave becomes twice longer. The model parameters are k = 0.03, p0 = 2.0, and p1 = 1.5. In cases T = 4 and T = 8, the arbitrary constants are C1 = 8.2248, C2 = −0.3125 and C1 = 11.7200, C2 = −0.2651, respectively. Note that the average ﬂow for the period T = 4 is < Q >= 8.12292, whereas the longer period T = 8 gives somewhat smaller average, < Q >= 8.02996.

1.3 Quick ‘Tutorial’

1.3.5

15

Periodic Impulsive Loading

Let us consider the one-dimensional motion of a material point under the periodic impulsive loading inside the linearly viscous ﬂuid. The corresponding diﬀerential equation of motion can be represented in the form v˙ + λv = 2p

∞

[δ (t + 1 − 4k) − δ (t − 1 − 4k)] = pe˙

(1.31)

k=−∞

where λ and p are a coeﬃcient of viscosity and a half of the pulse amplitude per unit mass, respectively. Equation (1.31) has a simple form involving the distribution. Such kind of problem is usually solved by applying either the generalized Fourier series or Laplace transform, or by considering the equation between the pulses by matching diﬀerent pieces of the solution at the pulse times. Alternatively, the solution can be obtained in few quick steps by using identity (1.1). Indeed, substituting (1.1) in (1.31) and taking into account (1.4), gives (Y + λX) + (X + λY ) e + (Y − p)e˙ = 0 or

Y + λX = 0,

X + λY = 0,

Y |τ =±1 = p

(1.32)

Solving boundary value problem (1.32), gives the closed form particular solution p (sinh λτ − e cosh λτ ) (1.33) v=− cosh λ The entire general solution is obtained by adding the term with an arbitrary constant, C exp(−λt), due to linearity of equation (1.31).

1.3.6

Strongly Nonlinear Oscillator

Now let us consider the nonlinear oscillator shown in Fig. 1.1, x ¨ + x2n−1 = 0

(1.34)

where n is an arbitrary positive integer. Note that this example gives the asymptotic basis for introducing the sawtooth temporal argument τ justiﬁed by the limit n → ∞, in which x(t) takes the triangular sine shape [132]. The root of the mathematical problem here is that the limit is nonsmooth, whereas large but ﬁnite numbers n generate close to the sawtooth but smooth oscillations. Let us show that changing the temporal variable t → τ facilitates a natural transition to the limit n → ∞. Taking into account the temporal symmetry of the oscillation, reduces the substitution to the particular case X(−τ ) ≡ −X(τ ) and Y ≡ 0 so that

16

1 Introduction

x = X(τ ),

τ = τ (t/a)

(1.35)

where a = T /4 is an unknown quarter of the period. Substituting (1.35) in (1.34) and taking into account (1.4), gives 1 (X + X e ) + X 2n−1 = 0 a2

(1.36)

Due to the oddness of X(τ ), equation (1.36) is equivalent to a one-point boundary value problem X = −a2 X 2n−1 ,

X |τ =1 = 0

(1.37)

Now the idea is to take advantage of the fact that the new temporal argument is periodic and bounded, −1 ≤ τ ≤ 1. So, in order to design an analogous of the quasi harmonic approaches, successive iterations can be applied. This tool requires no small parameter to be present explicitly. As a result, one generally must rely on convergence of the procedure. Although the convergence is usually slow7 , the physical basis of the convergence can always be determined whenever the problem has indeed some physical content. Note that the harmonic balance method also uses no explicit small parameter by assuming however that temporal mode shapes are close to harmonic. In other words, high frequency terms just correct but not exceed the principal harmonic. In the present case, solution is approximated by the triangular sine, which is corrected by higher powers of the same triangular sine. On the physical point of view, the model under consideration has to be close to the impact oscillator rather then the harmonic one. In terms of the new time variable τ , such an assumption simply means that the right-hand side of the diﬀerential equation of motion (1.37) is small enough to justify the following generating system X0 = 0

(1.38)

Indeed, this equation describes a family of impact oscillators with the triangular sine wave time histories X0 = Aτ (t/a)

(1.39)

where A is an arbitrary constant and another constant is zero due to the symmetry X(−τ ) ≡ −X(τ ). There are diﬀerent ways to formalizing the entire procedure. For instance, next term of the series can be obtained by substituting the generating solution (1.39) into the right-hand side of equation (1.37) and then integrating twice so that τ 2n+1 (1.40) X1 = Aτ − a2 A2n−1 2n(2n + 1) 7

This is rather a side eﬀect of the generality of successive approximation techniques.

1.4 Geometrical Views on Nonlinearity

17

Note that the linear term Aτ occurred again as a result of integration in (1.37). Keeping the same arbitrary constant A automatically includes generating solution (1.39) into ﬁrst-order approximation. This represents a certain mathematical constraint, which actually enables one of determining the parameter a by satisfying the boundary condition in (1.37). This gives a2 = and therefore

2n A2n−2

τ 2n+1 X1 = A τ − 2n + 1

(1.41)

(1.42)

High-order algorithms and the corresponding error estimates are described in Chapter 8. In particular, expression (1.41) is sequentially improved as follows

2n 2n − 1 2 a = 2n−2 1 + A 4n + 2 Note that analogous successive approximation algorithms can be applied to equation (1.37) with a quite general right-hand side. In the above case, however, the physical meaning of the parameter, n, shown in Fig. 1.1, directly relates to the idea of algorithm. As a result, solution (1.42) performs better as the exponent n increases. Note that, a physically meaningful transition to the asymptotic limit n → ∞ can be implemented by replacing the parameter A in (1.42) with the initial velocity v0 = A/a which in contrast to A is independent of n. Finally, despite of the manipulations with nonsmooth and discontinuous functions, solution (1.42) is twice continuously diﬀerentiable with respect to the argument t; this can be veriﬁed by taking ﬁrst two formal derivatives of (1.42). Generally speaking, the boundary value problem, such as (1.37), may appear to be complicated for any analytical method. In such cases, a combination of NSTT with the shooting method can be eﬀectively used as a semi-analytical approach [153], [144].

1.4

Geometrical Views on Nonlinearity

1.4.1

Geometrical Example

As shown in the previous sections, the asymptotic of linearity has a strongly nonlinear but simple enough counterpart so that both may complement each other as possible approaches to vibration problems. In other words, two couples of basic functions, {sin t, cos t} and {τ (t), e(t)}, naturally associate with opposite boundaries of parameter intervals of certain physical systems. Note that the oscillator (1.34) is not a unique example showing such a property.

18

1 Introduction

Fig. 1.5 Rigid-body rotation that generates both sine and triangular waves within the same class of elementary functions.

Another example is given by a strongly nonlinear oscillator which is exactly solvable in terms of elementary functions [78], [122]. Below, a geometrical interpretation for this case is introduced. Let us consider a geometrical model generating the sine and triangular sawtooth waves as two asymptotic limits of the same family of periodic functions. With reference to Fig. 1.5, the distance between two ﬁxed points C and O is equal to unity. A disc of the radius CP = α is rotating around its geometrical center C with some angular speed ω so that the angle between CO and CP is ϕ = ωt. The edge point P of the rotating disc has its image Q on the co-centered circle of the unit radius. The image is obtained under the condition that, during the rotation, P Q remains parallel to CO. Then position of the image Q is given by either its Cartesian coordinate Y or by the arc length y as follows Y = α sin ϕ (1.43) y = arcsin(α sin ϕ)

(1.44)

Expression (1.44) is obtained by equating projections of CP and CQ on OY and taking into account that the angle QCO is equal to the corresponding arc length y. Obviously, (1.44) becomes equivalent to (1.43) and thus describes the harmonic sine wave as α → 0.

1.4 Geometrical Views on Nonlinearity

19

Suppose now that α → 1 − 0. In this case, (1.44) takes the triangular sine wave shape

2ϕ π y→ τ (1.45) 2 π Interestingly enough, both types of observation of the above rigid-body model are associated with diﬀerent elastic oscillators. Namely, on one hand, as a function of time, expression (1.43) satisﬁes equation Y¨ + ω 2 Y = 0

(1.46)

On the other hand, function y(t) (1.44) satisﬁes the diﬀerential equation of motion of conservative oscillator tan y =0 cos2 y

(1.47)

α2 + ω −2 = 1

(1.48)

y¨ + under condition,

In particular, condition (1.48) implies that the frequency ω depends upon the disc radius monotonically in such a way that 1 ≤ ω < ∞ as 0 ≤ α < 1. Therefore, nonsmooth limit (1.45) is reached under the inﬁnitely large frequency ω, when the total energy of oscillator (1.47) also becomes inﬁnitely large, tan2 y y˙ 2 + →∞ (1.49) H= 2 2 The reasons for considering oscillator (1.47) as a generating model in nonlinear dynamics will be described later. At this stage, let us bring attention to the fact that both equations (1.46) and (1.47) actually describe the same rigid-body rotation but in diﬀerent coordinate systems.

1.4.2

Nonlinear Equations and Nonlinear Phenomena

The purpose of next few subsections (1.4.2 through 1.4.6) is to ﬁnd some basis for the idea of using elementary nonsmooth systems as generating systems in nonlinear dynamics. The discussion below may provide a hint for selecting nonlinear generating systems within the class of nonsmooth systems even though some statements of the discussion may have no rigorous proof. As a starting point, it could be useful to clarify why the harmonic oscillator represents a good generating model. Indeed, deﬁning the nonlinearity regardless mathematical formalizations appears to be a challenging task. Namely it is diﬃcult to ﬁnd appropriate physical principals for such a deﬁnition that would qualify nonlinearity as a natural phenomenon rather than the speciﬁc form of mathematical expressions. Brieﬂy speaking, nonlinear phenomena may occur at high energy levels. The following scenario is described in the book [126]: “...once the power or violence of a system is increased, it leaves

20

1 Introduction

the familiar linear region and enters the more complex world of nonlinear eﬀects: rivers become turbulent, ampliﬁers overload and distort, chemicals explode, machines go into uncontrollable oscillations, plates buckle, metals fracture, and structures collapse.” The author goes even further by saying that “within such an approach, mind may no longer appear as an alien stuﬀ in a mechanical universe; rather the operation of mind will have resonances to the transformations of matter, and indeed, the two will be found to emerge from a deeper ground.” However, mathematical approaches can still be helpful based on the following logical identity Nonlinear systems = All systems - Linear systems

(1.50)

Although this relationship seems to bring no physical contents, nevertheless it shows how physical understanding of nonlinearity can possibly be achieved. In other words, by clarifying the physical basis that provides the well known simplicity of the linear systems and harmonic analyses, one could ﬁnd what actually makes systems nonlinear. The standard mathematical deﬁnition for linear systems based on the superposition principle. Generally, a linear system can be deﬁned in terms of operators. Let L be an operator acting on state or may be coordinate vectors {q}. Then the operator L is linear, if for any two states, say q1 and q2 , and any two constants, C1 and C2 , the following relationship holds L(C1 q1 + C2 q2 ) = C1 Lq1 + C2 Lq2

(1.51)

Now, according to (1.50), a system is nonlinear if it is not linear. This mathematical deﬁnition, however, may appear to be confusing from physical viewpoints. For instance, based on deﬁnition (1.51), system 2

ρ ¨(t) − ρ(t) ϕ(t) ˙ = ρ(t) [sin 2 ϕ(t) − 2] ρ(t)¨ ϕ(t) + 2ρ(t) ˙ ϕ(t) ˙ = ρ(t) cos 2 ϕ(t)

(1.52)

does not satisfy its conditions and therefore is nonlinear, whereas x ¨1 (t) + 2x1 (t) − x2 (t) = 0 x ¨2 (t) + 2x2 (t) − x1 (t) = 0 is a linear system whose operator reads

2 2 d /dt + 2 −1 L= −1 d2 /dt2 + 2

(1.53)

(1.54)

so that Lq = 0, where q = [x1 (t) , x2 (t)]T . The linear superposition principle is therefore applicable to equations (1.53) whereas it does not work for equations in the nonlinear form (1.52).

1.4 Geometrical Views on Nonlinearity

21

However, both sets of equations (1.52) and (1.53) describe the same mechanical system in polar and Cartesian coordinates, respectively. On the plane of conﬁgurations, the corresponding coordinate transformation is given by x1 = ρ cos ϕ,

x2 = ρ sin ϕ

In this example, it is clear that the system is linear from the physical point of view; indeed nonlinearity of equations (1.52) is due to speciﬁc choice for coordinates. Therefore a ‘physical deﬁnition’ for linear systems must specify the type of reference coordinate system, for instance, as follows: A mechanical system is linear if its diﬀerential equations of motions in Cartesian coordinates are linear 8 . This deﬁnition involves already some physical principles since the Cartesian coordinates are uniquely associated with general properties of the physical space. Note that the above deﬁnition still involves the mathematical notion of coordinate systems. As known from observations however, some dynamic phenomena are perceived as nonlinear with no explicit system coordinates. So next two subsections make an attempt to clarify why such a perception of nonlinearity is actually possible.

1.4.3

Rigid-Body Motions and Linear Systems

As mentioned at the beginning of this introduction, sine and cosine waves are associated with the subgroup of rigid-body rotations. On the other hand, same temporal shapes describe vibrations of deformable linearly elastic bodies such as harmonic oscillators. Therefore, one-dimensional dynamics generated by linearly elastic restoring forces can be represented as free rigid-body rotations however in the two-dimensional space. In other words, linearly elastic forces are eﬀectively eliminated by expanding the dimension of space. Therefore, linear dynamics can be viewed as kinematics of freely rotating discs, where the number of discs is the number of modes. Such a viewpoint can be formalized as follows. Let z be a complex vector frozen into a rigid body (disc), whose position is observed in the empty space. Note that the notion of ‘position’ becomes quite vague if there is only one body. Therefore it is assumed that the observer represents another physical body, which is a single point. Then a straight line between this point and the center of the disc gives a reference line for determining the vector z direction. Since the disc is assumed to be rigid then z z¯ = z z¯ 8

(1.55)

This deﬁnition was suggested by V.Ph.Zhuravlev during a private discusion at International Conference “Nonlinear Phenomena,” Moscow, 1989.

22

1 Introduction

where the over bar means complex conjugate, and the prime denotes any new position of the body. Expression (1.55) recalls mathematical objects generated by the Galilean group of rotations. The corresponding operator G is introduced through relationship (1.56) z = Gz where G must depend on some parameter, say ϕ, characterizing the transition z → z. Substituting (1.56) in (1.55), gives 2

¯ = |G| = 1 =⇒ G = exp(iϕ), GG

i2 = −1

(1.57)

Since ϕ is the only parameter of the only one ‘process’ then ϕ must be qualiﬁed as time with possibly some scaling factor. Introducing an arbitrary scaling factor, say ω, gives then G = exp (iωt) = cos ωt + i sin ωt

(1.58)

Now a simple calculation shows that

d d ¯=0 − iω G = 0 and + iω G dt dt ¯ satisfy the diﬀerential equation of harTherefore both operators, G and G, monic oscillator associated with the operator

d d d2 − iω + iω = 2 + ω 2 (1.59) dt dt dt By representing the scaling factor in the form ω = k/m, one can interpret then m and k as, respectively, mass and stiﬀness of a mass-spring model associated with the ‘product’ of clockwise and counter-clockwise rotations of the disc. Let us consider now n rotating discs with the angular frequencies {ω1 , ..., ωn } by generalizing product (1.59) as follows

d − iω1 dt

n 2 d d d d 2 + iω1 ... − iωn + iωn = + ω j dt dt dt dt2 j=1

(1.60) Operator (1.60) represents an arbitrary n-degrees-of-freedom linear elastic system. Replacing d/dt → λ, gives the corresponding characteristic equation in the form n 2 λ + ωj2 = 0 (1.61) j=1

1.4 Geometrical Views on Nonlinearity

23

Finally, let us consider the asymptotic limit n → ∞. In order to calculate the limit, the frequency dependence on its index j must be speciﬁed. If, for instance, ωj = ja with some constant a then equation (1.61) gives sinh

πλ =0 a

(1.62)

as n → ∞. Equation (1.62) associates with a linearly elastic string of the length l = π with two ﬁxed ends, ∂ 2 u (t, x) ∂ 2 u (t, x) − a2 = 0; 2 ∂t ∂x2

u (t, 0) = u (t, π)

(1.63)

Therefore, it is shown that basic linearly elastic dynamic models can be logically obtained from the kinematics of freely rotating discs or even from a more fundamental concept given by expression (1.55). To some extent, this may explain why nonlinearity is perceptible as a physical phenomenon. Indeed the linear dynamics is simply matching with the usual perception (experience) of the space. All other motions give therefore an impression of unusual, in other words, nonlinear phenomena.

1.4.4

Remarks on the Multi-dimensional Case

The multi-dimensional example below gives another (purely geometrical) viewpoint on the link between rigid-body motions and the notion of linearity. So, by deﬁnition, the mapping f : Rn −→ Rn is an isometry if relation

f (v) − f (w) = v − w

(1.64)

holds for all v, w ∈ Rn . Deﬁnition (1.64) requires distances between any two images and their preimages be same; this is a generalization of relationship (1.55). Let us consider such isometries with a ﬁxed point O in Rn , that is symbolically f (O) = O. Let us show that an isometry that ﬁxes the origin is a linear mapping f (av + bw) = af (v) + bf (w)

(1.65)

where a and b are arbitrary scalars. First, note that the relations

f (v) = v = −v = f (−v)

and

f (v) − f (−v) = v− (−v) = 2 v

hold due to (1.64), and thus f (v) and f (−v) are antipodal: f (−v) = −f (v).

24

1 Introduction

Second, taking into account the latter result and deﬁnition (1.64), transforms the polarization identity as follows 1 2 2 2

v + w − v − w

2 1 =

f (v) − f (−w) 2 − f (v) 2 − f (−w) 2 2 1 2 2 2 =

f (v) + f (w) − f (v) − f (w)

2 = f (v) · f (w)

v·w =

Therefore, the inner product is preserved. Now, if {ui }is orthogonal basis in Rn , then {f (ui )} is another orthogonal basis. Taking the dot product of the both sides of identity (1.65) with the arbitrary basis vector f (ui ), gives f (av + bw) · f (ui ) = af (v) · f (ui ) + bf (w) · f (ui ) or (av + bw) · ui ≡ av · ui + bw · ui The above relation proves identity (1.65) in terms of the coordinates associated with the basis {f (ui )}. All linear isometries of Rn are denoted by the symbol O (n). By choosing a basis for Rn , one can represent every element of O (n) as a matrix. It can be shown that O (n) consists of all n × n matrixes such that A−1 = AT : 1 = det I = det AA−1 = det AAT = det A det AT = (det A)2

(1.66)

Relationship (1.66) is therefore a multidimensional matrix analogy of the relationship (1.57).

1.4.5

Elementary Nonlinearities

Let us consider now the temporal coordinate of Galilean spatio-temporal continuum. In particular, the idea of ‘perfectly rigid time’ is expressed by relationship

2 dt =1 (1.67) dt2 = dt2 ⇐⇒ dt By considering (1.67) as a diﬀerential equation with respect to t = s (t), one obtains two possible solutions t = ± (t − a) where a is an arbitrary temporal shift.

(1.68)

1.4 Geometrical Views on Nonlinearity

25

Now, combining diﬀerent branches of (1.68) for t < a and t > a, gives t = s (t) = |t − a|

(1.69)

Function (1.69) also satisﬁes equation (1.67), for all t except may be single point t = a, where the classic derivative of s (t) has no certain value. Namely, equality holds for almost all t except may be t = a. Nevertheless, solution (1.69) admits an obvious physical interpretation since it describes a free material point moving in the space splitted into two subspaces by a perfectly stiﬀ plane, were the normal to the plane velocity is scaled by (1.70) s˙ 2 = 1 Since the perception of empty space rejects any built in stiﬀ planes then a sudden V -turn of the particle will represent unusual, in other words, strongly non-linear event. This is obviously the most elementary nonlinear event whose simplicity nevertheless will be employed further in less trivial cases.

1.4.6

Example of Simpliﬁcation in Nonsmooth Limit

The idea that transition to most severe nonlinearity may actually simplify a system response ﬁnds its support in very diﬀerent examples. Consider, for instance, the diﬀerential equation of motion

2x (1.71) x ¨ = exp − 2 The nonlinear force on the right-hand side gives the ‘rigid body’ limit as the parameter approaches zero. Let us assume that x˙ = −1 and x → ∞ as t → −∞. In this case, equation (1.71) has exact solution

t−a (1.72) x = 2 ln cosh 2 where a is an arbitrary parameter such that x(a) ˙ = 0; see Fig. 1.6, where a = 1. Now let us consider the asymptotic limit → 0. Taking into account evenness of cosh-function, brings solution (1.72) to the form

|t − a| s 2 2 (1.73) x = ln cosh ln cosh = 2 2 This simple manipulation represents a useful preliminary step for asymptotic estimates because the new temporal argument, s, remains always positive as the original time runs in the interval −∞ < t < ∞.

26

1 Introduction x 2.0 Ε0.1 Ε0.5

1.5

Ε1.0 1.0

0.5

1

1

2

3

t

Fig. 1.6 Time histories of the particle when the barrier approaches rigid body limit, → 0.

Further algebraic manipulations bring solution (1.73) to the form

2s 2 x = s + ln (1.74) 1 + exp − 2 2 Taking into account that 0 < exp −2s/2 < 1, gives the ‘rigid-body’ limit x → s = |t − a|

(1.75)

as → 0. The above example shows that (1.69) can be viewed as a natural time associated with the temporal symmetry of the V-turn. Note however that the new temporal argument is nonsmooth and non-invertible with respect to the original time. Therefore, using (1.69) for temporal substitutions in the diﬀerential equations of motion has certain speciﬁcs as discussed below.

1.4.7

Non-smooth Time Arguments

As mentioned above, piece-wise linear function (1.69), s(t), can play the role of natural temporal argument associated with the elementary nonlinear events such as dynamic V-turns caused by potential barriers. Namely the temporal symmetry of V-turns is captured by the function s(t) regardless possible shapes of the potential barriers. Therefore, using the function s(t) as a temporal argument should simplify analyses since the global information about V-turns is a priory built into the new argument. The corresponding time substitution however is not straightforward since its inverse version does not exist in the form t = t(s) but requires the following generalization t = t(s, s). ˙ Below, this case is discussed ﬁrst. Then the periodic version is

1.4 Geometrical Views on Nonlinearity

27

introduced that reveals structural similarity of expressions as far as periodic motions are in fact regular sequences of v-turns. Non-periodic case. Let us start with the inverse version of (1.69), which includes both the image s and its time derivative s˙ as follows t = t(s, s) ˙ = a + ss˙

(1.76)

Now relation (1.70) reveals that combination (1.76) represents speciﬁc algebraic structure with the basis {1, s}. ˙ In other words, time belongs to the algebra of hyperbolic ‘complex numbers’ with the table of products generated by (1.70). For instance, taking (1.76) squared, gives another hyperbolic ‘number’ (1.77) t2 = a2 + s2 + 2ass˙ Moreover, it is easy to prove for any function x(t) that x (t) = X (s) + Y (s) s˙

(1.78)

where 1 [x (a + s) + x (a − s)] 2 1 Y (s) = [x (a + s) − x (a − s)] 2

X (s) =

For instance, setting a = 0 in the case x(t) = exp t, gives exp (ss) ˙ = cosh s + s˙ sinh s

(1.79)

In contrast to the conventional complex algebra, division is not always possible, so the following relationship is meaningless for |a| = s or t = 0: 1 (a − ss) ˙ a 1 s = = = 2 − 2 s˙ t a + ss˙ (a + ss) ˙ (a − ss) ˙ a − s2 a − s2 In particular case of even function x (t), with respect to t = a, one has Y (s) ≡ 0; see (1.73) for example. Periodic case. The periodic version of nonsmooth time transformations as that preliminary introduced in Section 1 involves triangular and rectangular waves described by the periodic piecewise-linear function πt 2 t for − 1 ≤ t ≤ 1 ∀t , τ (t) = τ (4 + t) τ (t) = arcsin sin = −t + 2 for 1 ≤ t ≤ 3 π 2 and its Schwartz derivative e(t) = τ˙ (t); see Fig. 1.1. Remind that from the physical point of view, the functions τ (t) and e (t) describe dynamic states of a particle oscillating between two perfectly rigid

28

1 Introduction

Fig. 1.7 Mechanical model generating the saw-tooth sine and rectangular cosine, τ (t) and τ˙ (t)

planes with no energy loss as shown in Fig. 1.7, where spatio-temporal coordinates are normalized so that dt2 = dτ 2 ⇐⇒ e2 = τ˙ 2 = 1

(1.80)

Note that relationship (1.80) is a periodic version of (1.70) whereas, for any periodic function of the period T = 4, representation (1.1) is a periodic version of (1.78). In other words, any periodic motion uniquely associates with the standard impact vibration, for instance as follows x (t) = A sin

πt πτ πτ πt + B cos = A sin + B cos e 2 2 2 2

(1.81)

where τ = τ (t), e = τ˙ (t), and A and B are arbitrary constants. Exercise 1. By taking sequentially derivatives, show that diﬀerentiation keeps the result within the set of hyperbolic numbers provided that the corresponding process is smooth enough. In particular, π 2n−1 πτ πτ πτ πτ −B sin + B cos e /dt2n−1 = + A cos e d2n−1 A sin 2 2 2 2 2 πτ πτ πτ πτ π 2n A sin + B cos e /dt2n = + B cos e d2n A sin 2 2 2 2 2

1.4.8

Further Examples and Discussion

The Fourier analysis is based on the standard trigonometric pair of functions {sin ωt, cos ωt} or {exp (iωt) , exp (−iωt)} so that periodic processes are described by linear combinations of these functions with appropriate frequencies {ωk }. For instance, the dynamic states of impact oscillators are represented as follows

8 1 1 3πt 5πt πt τ (t) = 2 sin − 2 sin + 2 sin − ... π 2 3 2 5 2

4 3πt 1 5πt πt 1 e (t) = − cos + cos − ... (1.82) cos π 2 3 2 5 2

1.4 Geometrical Views on Nonlinearity

29

The well known convenience of Fourier series for handling partial and ordinary diﬀerential equations is due to the fact that exp(iωt) is an eigen function of the time derivative, in other words, d exp(iωt)/dt = iω exp(iωt). This important advantage is unfortunately missing when using the set {τ (ωk t), e(ωk t)} as a nonsmooth basis for Fourier expansions. The alternative way is based on power series expansions. Indeed, in terms of the oscillating time τ , polynomial approximations can be applied with no periodicity loss. For example, πτ πτ 1 πτ 3 1 πτ 5 πt = sin = − + − ... 2 2 2 3! 2 5! 2

πτ 1 πτ 2 πt 1 πτ 4 = cos e= 1− cos + − ... e 2 2 2! 2 4! 2 sin

(1.83)

While being polynomials, truncated series of (1.83) preserve periodicity at cost of smoothness loss though; see Fig. 1.8 for explanation. Fortunately, nonsmoothness times Λ = {t : τ (t) = ±1} are same for every term of the series and this enables one of smoothing the series by re-ordering their terms as follows

3 πτ π π π3 τ τ5 πt τ3 = sin = + − sin τ− + 2 2 2 3 2 16 3 5

3 5 5 7 π π π τ τ + + − + + ... (1.84) 2 16 768 5 7

πτ π2 2 πt 2 = cos e = [1 − τ + 1 − τ − τ4 cos 2 2 8

π2 π4 4 + 1− + τ − τ 6 + ...]e 8 384 Fig. 1.9 illustrates both convergence and smoothness of the transformed series as compared with the diagrams in Fig.1.8. Moreover, it will be shown that the power series of τ can be re-ordered in such a manner that their particular sums become as smooth as necessary. Still the convenience of such kind of series is that they can accurately approximate non-smooth or close to them processes just by ﬁrst few terms. Finally, there are many physical processes easily described by very simple combinations of the triangular sine and rectangular cosine waves, which otherwise would require quite long Fourier sums; see Fig. 1.10, for examples.

30

1 Introduction

Fig. 1.8 Truncated Maclaurin NSTT expansions with respect to τ for sin(πt/2) and cos(πt/2) including one, two, and three ﬁrst terms of the expansions as indicated on the curves.

1.4.9

Diﬀerential Equations of Motion and Distributions

Any nonsmooth substitution in the diﬀerential equations of motion must be carefully examined since diﬀerentiation is involved. In many physically meaningful cases though suﬃcient justiﬁcations can be achieved by understanding equalities as integral identities. For illustration purposes, let us consider conservative oscillator (1.85) x ¨ + Px (x) = 0 where P (x) is the potential energy.

1.4 Geometrical Views on Nonlinearity

31

Fig. 1.9 First term of the modiﬁed NSTT series with respect to τ for sin(πt/2) and cos(πt/2); 0 - original functions, 1 - ﬁrst term of the series.

Classical pointwise interpretations of diﬀerential equations require no discontinuities to occur on the left-hand side of (1.85). Therefore, ‘real motions’ must be described by at least twice continuously diﬀerentiable functions of time, x (t) ∈ C 2 (R). Practically, however, physical systems cannot be observed at every time instance. In other words, point-wise deﬁnition of equality (1.85) appears to be somewhat restrictive from the physical standpoint. As an extension of classical approaches, the left-hand side of (1.85) can be considered as some force producing zero work on arbitrary path variations δx during the observation interval, t1 < t < t2 , so that

32

1 Introduction

t2

(¨ x + Px (x)) δxdt = 0

(1.86)

1 2 x˙ − P (x) dt = −δS = 0 2

(1.87)

t1

or

t2

−δ t1

where S is the action by Hamilton.

Fig. 1.10 Sample temporal shapes of periodic processes easily described by combinations of the triangular sine and rectangular cosine

1.5 Non-smooth Coordinate Transformations

33

Expression (1.87) represents the Hamiltonian principle, where the integral can be calculated for less smooth functions x (t) ∈ C (R). Therefore, it is possible to essentially extend the class of solutions by ‘keeping in mind’ interpretation (1.86) when considering equation (1.85). Note that equation (1.85) may be strongly nonlinear, however the highest derivative must participate in a linear way as a summand, otherwise transition from (1.86) to (1.87) may become impossible.

1.5

Non-smooth Coordinate Transformations

1.5.1

Caratheodory Substitution

In general, including Dirac’s δ-functions in nonlinear diﬀerential equations complicates mathematical justiﬁcations of the modeling. It is important however how δ-functions participate in equations [46]. For instance, let the differential equation include δ-impulse as a summand x˙ = kx3 + qδ (t − t1 )

(1.88)

where k, q and t1 are constant parameters. In this case, the δ-input generates step-wise discontinuity of the response x(t) at t = t1 , however the nonlinear operation in (1.88) still can be conducted. Moreover, the δ-pulse is easily excluded from equation (1.88) by substitution (1.89) x (t) = y (t) + qH (t − t1 ) where y (t) is a new unknown function and H (t − t1 ) is the unit-step Heaviside function. Indeed, substituting (1.89) in (1.88) and taking into account that H˙ (t − t1 ) = δ (t − t1 ), H 2 (t − t1 ) = H (t − t1 ), and H 3 (t − t1 ) = H (t − t1 ), gives y˙ = k[y 3 + (q 3 + 3yq 2 + 3y 2 q)H (t − t1 )]

(1.90)

where y (t) is now continuous on the interval of consideration including the point t1 .

1.5.2

Transformation of Positional Variables

Now let us discuss the method of nonsmooth coordinate transformation (NSCT) for mechanical systems with perfectly stiﬀ constraints [200]. According to this approach, the new coordinates are introduced in order to automatically satisfy the constraint conditions. Let us reproduce the idea based on one-degree-of-freedom Langrangian system L=

1 2 x˙ − P (x) 2

(1.91)

34

1 Introduction

whose motion is limited by interval −1≤x≤1

(1.92)

It is assumed that the particle collides with the obstacles x = ±1 with no energy loss. Note that relationships (1.91) and (1.92) give no unique diﬀerential equation of motion on the entire time domain since every collision indicates transition from one system to another. Indeed, even though the form of Lagrangian (1.91) remains the same, one must deal with diﬀerent solutions before and after the collision, say x− (t) and x+ (t). Matching such solutions is usually not straightforward since the collision time is a priory unknown. The major reason for applying NSCT is that it gives a single diﬀerential equation of motion for the entire time interval with no constraint conditions. As a result, the mentioned above matching procedure simply becomes unnecessary. In addition, after the transformation, a new system appears to be well suited for averaging since no impact forces are involved any more. In the above case (1.91) and (1.92), the NSCT is introduced as follows9 x = τ (s)

(1.93)

where s(t) is a new positional coordinate. Substituting (1.93) into (1.91), gives L=

1 1 [τ (s)s] ˙ 2 − P (τ (s)) = s˙ 2 − P (τ (s)) 2 2

(1.94)

where the prime indicates diﬀerentiation with respect to s, and the relationship [τ (s)]2 = e2 (s) = 1 has been taken into account. Now, condition (1.92) is satisﬁed automatically since −1 ≤ τ (s) ≤ 1, whereas the Hamiltonian principle gives the diﬀerential equation of motion with no constrains dP e(s) = 0 (1.95) s¨ + dτ On geometrical point of view, transformation (1.93) unfolds the conﬁguration space by switching the coordinate direction on opposite whenever the particle collides with an obstacle. As a result, the new conﬁguration space acquires a cell-wise non-local structure as illustrated in [191] and [148]. This usually makes the diﬀerential equation essentially nonlinear even when the system in between the constraints is linear. Let, for instance, the potential energy function represents the harmonic oscillator, P (x) = ω 2 x2 /2. Then the coordinate transformation (1.93) brings system x¨ + ω 2 x = 0, 9

−1≤x≤1

(1.96)

Note that both notations and normalization for the period for the triangular wave function diﬀer from the original work [200].

1.5 Non-smooth Coordinate Transformations

35

to the form s¨ + ω 2 τ (s)e(s) = 0,

−∞<s<∞

(1.97)

Note that, as a side eﬀect of the elimination of constraints, the diﬀerential equation lost its linearity. In contrast to (1.97) however, the linear diﬀerential equation in (1.96) does not describe the entire system, which is strongly non linear from the very beginning due to impact interactions with constraints. Finally, results of comparison between NSCT and NSTT can be summarized as follows [143]: Coordinate transformation

Time transformation

Unfolds the space with no eﬀect on Folds the time, and generates the the time variable hyperbolic algebraic structures in space Targets rigid barriers (constraints) Barriers do no matter Dynamic regime independent Assumes certain temporal symmetries of motion Applies to a function (image) Transforms an argument (preimage) Essentially changes the ODE struc- Preserves most structural properties ture, for instance, linear on strongly of ODEs nonlinear As follows from the table, NSCT is quite opposite to NSTT from the very diﬀerent viewpoints.

1.5.3

Transformation of State Variables

Under some conditions, the NSCT can still be adapted to the case of nonelastic interactions with constraints in a purely geometrical way [204]. However, generalized approaches to such cases should involve both coordinates and velocities [73], [72]. For illustrating purposes, let us consider the case of harmonic oscillator under the constraint condition x˙ = Ax,

x1 > 0

where x = [x1 (t), x2 (t)]T is the state vector such that x2 = x˙ 1 , and

0 1 A= −ω 2 0

(1.98)

(1.99)

It is assumed that every collision with the constraint x1 = 0 results in a momentary energy loss characterized by the restitution coeﬃcient κ. The idea is to unfold the phase space in such way that the energy loss would automatically occur whenever the system crosses a preimage of the line x1 = 0. The corresponding transformation is

36

1 Introduction

x = Sy where y = [s(t), v(t)]T is a new state vector, and

10 S= sgn(s) 0 1 − ksgn(sv)

(1.100)

(1.101)

where k = (1 − κ)/(1 + κ). Transformation (1.100) is of-course strongly nonlinear due to non-smooth dependence (1.101). Substituting (1.100) in (1.98), gives equation −1

y˙ = (S

AS)y

(1.102)

In the component-wise form, expressions (1.100) and (1.102) are written as, respectively, x1 = x1 (s, v) ≡ ssgn(s) x2 = x2 (s, v) ≡ sgn(s)[1 − ksgn(sv)]v

(1.103)

and s˙ = [1 − ksgn(sv)]v v˙ = −ω 2 s[1 + ksgn(sv)]/(1 − k 2 )

(1.104)

Now both unknown components of the state vector are continuous, whereas speciﬁcs of non-elastic collisions are captured by transformation (1.103). Finally, consider the general case of one-degree-of-freedom nonlinear oscillator x˙ 1 = x2 x˙ 2 = −f (x1 , x2 , t)

(1.105)

with non-elastic constraint x1 = 0 of the restitution coeﬃcient κ. Applying transformation (1.103) to system (1.105), gives s˙ = [1 − ksgn(sv)]v v˙ = −f (x1 (s, v), x2 (s, v), t)sgn(s)[1 + ksgn(sv)]/(1 − k 2 )

(1.106)

Although the above illustrations is one-dimensional, similar coordinate transformations can be introduced also for multiple degree-of-freedom systems in which one of the coordinates is normal to the constraint. The corresponding analytical manipulations can be conducted in terms of Routh descriptive functions such that the normal to the constraint coordinate is Lagrangian whereas other generalized coordinates and associated momenta are Hamiltonian.

Chapter 2

Smooth Oscillating Processes

Abstract. This chapter gives a brief overview of selected analytical methods for smooth oscillating processes. Most of such methods are indeed quasilinear. In other words, the corresponding technical implementations employ harmonic oscillators as generating models. The description focuses only on the ideas and technical details that are further combined with non-smooth methods. As most eﬀective way, procedures of asymptotic integration of the diﬀerential equations of motion bring original systems to such simple form that further solution becomes straightforward. In particular, the method of asymptotic integration of the diﬀerential equations of motion based on the Hausdorﬀ equation for operators Lie is reproduced.

2.1

Linear and Weakly Non-linear Approaches

By both practical and theoretical reasons, the quantitative methods of dynamics were developed ﬁrst for smooth processes. As a rule, smooth oscillations can be directly observed under no special conditions. For instance, projection of any ﬁxed point of a body rotating with constant angular speed, makes a perfect impression about harmonic oscillations. Interestingly, in 1693, Leibniz derived the diﬀerential equation for sine geometrically by considering a circle. Much later, original analytical ideas of nonlinear vibrations emerged from the celestial mechanics considering perturbations of circular orbits of rigid-body motions rather than any mass-spring oscillators. Robert Hooke (1635-1703) was probably ﬁrst who suggested the basic elastic mass-spring model, whereas Galileo and Huygens were investigating the pendulum. Later, d’Alambert, Daniel Bernoulli and Euler considered a one-dimensional continual model of a string. It was found that the vibrating string represents the inﬁnity of harmonic oscillators corresponding to diﬀerent mode shapes of the string. It is well known that a serious discussion arised about whether or not V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 37–49, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

38

2 Smooth Oscillating Processes

the sum of smooth functions, such as sines, can represent a non-smooth shape of the string. These discussions were ﬁnalized by the Fourier theorem. Let us reproduce the result for a periodic function of time f (t) of the period T in the complex form f (t) =

∞

ck exp(iωk t)

(2.1)

k=−∞

1 ck = T

T /2

−T /2

f (t) exp(−iωk t)dt, ωk =

2π k T

This relation generates a one-to-one mapping between function and its Fourier coeﬃcients f (t) ←→ {...c−2 , c−1 , c1 , c2 , ...}

(2.2)

Note that mathematical expressions (2.1) do not necessarily imply that the periodic process f (t) must be produced by linear systems even though the right-hand side of (2.1) combines free vibrations of linear oscillators, in other words - ‘rigid-body rotations’ as discussed in Chapter 1. Therefore, the Fourier analysis and associated analytical tools provides a ‘linear language’ for nonlinear systems regardless speciﬁcs of algorithm implementations. Indeed, most quantitative methods for weakly nonlinear periodic motions, actually estimate Fourier coeﬃcients of the corresponding solutions. As a result, on one hand, such tools possess a high level of generality. On the other hand, even ‘elementary’ strongly nonlinear phenomena (as qualiﬁed in Chapter 1) may become quite diﬃcult to describe in terms of the ‘linear language.’ Nevertheless, the quantitative theory of nonlinear vibration has been advanced by new asymptotic techniques developed originally for solving nonlinear diﬀerential equations. Most traditional methods are essentially based on perturbation or averaging methods [50]. Similar results can be obtained within the theory of Poincare’ normal forms [118], which retains resonance terms, whereas all non-resonance terms are eliminated by means of a coordinate transformation. Such a normal form is qualiﬁed as the simplest possible form of the equations of motion.

2.2

A Brief Overview of Smooth Methods

2.2.1

Periodic Motions of Quasi Linear Systems

Consider a weakly unharmonic oscillator of the form x ¨ + ω02 x = εf (x, x) ˙

(2.3)

2.2 A Brief Overview of Smooth Methods

39

where ε is a small parameter, 0 < ε 1, and f (x, x) ˙ is smooth enough function. Periodic solutions of equation (2.3) can be found by splitting the nonlinear system into the sequence of linear oscillators by means of the power series x = x0 + εx1 + ε2 x2 + ...

(2.4)

The perturbation on the right-hand side of equation (2.3) changes the principal frequency of the oscillator so that ω 2 = ω02 (1 + εγ1 + ε2 γ2 + ...)

(2.5)

The new frequency is introduced explicitly into the diﬀerential equation of motion by re-scaling the temporal argument ϕ = ωt

(2.6)

As a result, series (2.4) appears to be composed of trigonometric functions of multiple phases ϕ, 2ϕ, 3ϕ,... . A similar idea was implemented by Lyapunov for systems of ﬁrst-order equations, for instance x˙ 1 = a11 x1 + a12 x2 + f1 (x1 , x2 )

(2.7)

x˙ 2 = a21 x1 + a22 x2 + f2 (x1 , x2 ) where f1 and f2 are nonlinear functions. It is assumed that system (2.7) admits ﬁrst analytical integral and the corresponding linearized system has only periodic solutions. Then periodic solutions of (2.7) admit power series expansions with respect to the amplitude parameter. There exist at least two extensions of Lyapunov’s theory, such as local and global approaches to nonlinear normal modes, see for instance [100], [190], [119].

2.2.2

The Idea of Averaging

Let us illustrate diﬀerent implementations of the idea of averaging by reproducing some technical details. The following description focuses on such tools that remains applicable to non-hamiltonian systems. Following Van-der-Pol’s approach, let us transform system (2.3) by chang˙ → {a, ϕ}: ing the variables {x, x} x = a cos ϕ,

x˙ = −aω0 sin ϕ

(2.8)

40

2 Smooth Oscillating Processes

As a result, one obtains ε f (a cos ϕ, −aω0 sin ϕ) sin ϕ ω0 ε f (a cos ϕ, −aω0 sin ϕ) cos ϕ ϕ˙ = ω0 − ω0 a

a˙ = −

(2.9)

Despite of a formal complexity, system (2.9) has essential advantage due to diﬀerent time scales of the new variables. This enables one of eliminating the fast phase ϕ on the right-hand side of the system by applying the averaging 2π 1 · · · dϕ < · · · >ϕ ≡ 2π 0 as follows ε < f (a cos ϕ, −aω0 sin ϕ) sin ϕ >ϕ ω0 ε < f (a cos ϕ, −aω0 sin ϕ) cos ϕ >ϕ ϕ˙ = ω0 − ω0 a a˙ = −

(2.10)

Solutions of system (2.10) are considered then as approximate solutions of the original system (2.9). This method was essentially generalized in thirties [28] by incorporating the Lindstedt-Poincare and Van-der-Pol’s ideas as follows. Let us consider the general system with one fast phase x˙ = εX(x, y) y˙ = ω(x) + εY (x, y)

(2.11)

where y and x are scalar and vector variables respectively. In contrast to (2.9), the frequency in (2.11) depends on the slow vectorfunction x. Sometimes, such kind of systems is called essentially nonlinear since the condition ε = 0 does not make the frequency state independent. However, if ε = 0 then system (2.11) has no fast phase on the right-hand side. The problem is to ﬁnd close to identical transformation x = q + εu(q, ψ) + O(ε2 ) y = ψ + εv(q, ψ) + O(ε2 )

(2.12)

which eliminates the fast phase entirely from the system by bringing equation (2.11) to the form q˙ = εA(q) + O(ε2 ) ψ˙ = ω0 (q) + εω1 (q) + O(ε2 )

(2.13)

2.2 A Brief Overview of Smooth Methods

41

This problem is solved by substituting expansions (2.12) into equations (2.11) and enforcing then equations (2.13).

2.2.3

Averaging Algorithm for Essentially Nonlinear Systems

In order to illustrate the corresponding procedure, let us specify system (2.11) as follows l˙ = μR(μl, s, ϕ, θ) s˙ = μ2 S(μl, s, ϕ, θ) θ˙ = μQ(l, s) + μ2 Θ(μl, s, ϕ, θ)

(2.14)

ϕ˙ = Ω(μl, s) + μ2 G(μl, s, ϕ, θ) where all the coordinates and functions are scalars, and μ 1. Such kind of equations may occur when considering ‘essentially nonlinear’ systems under diﬀerent resonance conditions. This is the reason for using another notation for small parameter. √ In resonance cases, original small parameters are often modiﬁed as μ = ε to capture speciﬁcs of the dynamics near resonance surfaces [9], [205]. The basic approximation is obtained from system (2.14) by applying the averaging procedure directly to the terms of order μ on the right hand side. This gives, l˙ = μ < R(0, s, ϕ, θ) >ϕ θ˙ = μQ(l, s)

(2.15)

where s should be considered as a constant. System (2.15) is easily integrated and the result is known to give an error of order μ on time intervals of order 1/μ. In many cases however, ﬁrst approximation gives incomplete characterizations of systems. In order to illustrate the basic stages of second approximation, consider the ﬁrst equation only. It is suﬃcient for illustration of the procedure, which is sequentially applied in the same way to other equations. Let us represent the ﬁrst equation of system (2.14) in the form l˙ = μ < R(0, s, ϕ, θ) >ϕ +μ[R(0, s, ϕ, θ)− < R(0, s, ϕ, θ) >ϕ ] (2.16) +μ2 lRμl (0, s, ϕ, θ) + O(μ3 )

Following the idea of averaging, one eliminates the second term on the righthand side by means of the coordinate transformation l = q + μf (q, s, ϕ, θ)

(2.17)

42

2 Smooth Oscillating Processes

Then, substituting (2.17) into (2.16), gives

∂f ∂f ∂f ∂f ˙ q˙ + μ q˙ + s˙ + ϕ˙ + θ ∂q ∂s ∂ϕ ∂θ = μ < R(0, s, ϕ, θ) > ϕ + μ[R(0, s, ϕ, θ)− < R(0, s, ϕ, θ) > ϕ] (2.18) +μ2 qRμl (0, s, ϕ, θ) + O(μ3 ) Now the fast phase ϕ is eliminated from the equation in ﬁrst order of μ by taking into account (2.14) and imposing condition ∂f Ω(0, s) = R(0, s, ϕ, θ)− < R(0, s, ϕ, θ) >ϕ ∂ϕ

(2.19)

Further, f (q, s, ϕ, θ) is independent of q because the terms of order μ on the right-hand side of equation (2.18) are independent of q. As a result, equation (2.18) takes the form q˙ = μ < R(0, s, ϕ, θ) > ϕ + μ2 qRμl (0, s, ϕ, θ)

∂f ∂f 2 −μ qΩ (0, s) + Q(q, s) + O(μ3 ) ∂ϕ μl ∂θ

(2.20)

Since the fast phase ϕ is eliminated from the terms of order μ then the averaging procedure is applied to the terms of order μ2 analogously to the ﬁrst stage of the method. As follows from (2.19), < ∂f /∂ϕ >ϕ =< ∂f /∂θ >ϕ = 0, therefore after the averaging, equation (2.20) takes the form q˙ = μ < R(0, s, ϕ, θ) >ϕ +μ2 < qRμl (0, s, ϕ, θ) >ϕ +O(μ3 )

or q˙ = μ < R(μq, s, ϕ, θ) >ϕ +O(μ3 )

(2.21)

The second approximation therefore is obtained by applying the operator of averaging to original equation (2.14). Note however that the meaning of the coordinate q is now diﬀerent. Namely, the original coordinate l is expressed through the new coordinate q by relationship (2.17), which due to (2.19) takes the form μ l=q+ Ω(μq, s)

ϕ (R(μq, s, ϕ, θ)− < R(μq, s, ϕ, θ) >ϕ )dϕ + O(μ2 ) (2.22) 0

In this expression, the variable μq was put back into the expression R instead of zero. Although such a manipulation has no eﬀect on the order of approximation, the new form is in a better match with the form of equation (2.21). Expressions (2.21) and (2.22) summarize the averaging procedure in second order of μ.

2.2 A Brief Overview of Smooth Methods

43

Note that the case of multiple fast phases turns out to be more complicated in many respects due to the well known problem of small denominators.

2.2.4

Averaging in Complex Variables

In the physical literature, vibration problems are usually considered in terms of complex variables [89]. The idea of using the complex variables may be suggested by the standard manipulations of the variation of constants for oscillator (2.3) as follows. If ε = 0 then general solution of equation (2.3) is represented in the complex form 1 x = [A exp(iω0 t) + A¯ exp(−iω0 t)] (2.23) 2 where A and A¯ are arbitrary complex conjugate constants. The velocity is x˙ =

iω0 [A exp(iω0 t) − A¯ exp(−iω0 t)] 2

(2.24)

If ε = 0 then the constants are assumed to be time dependent whereas expressions (2.23) and (2.24) are considered as a change of the state variables ¯ {x, x} ˙ → {A, A}

(2.25)

under the compatibility condition dA dA¯ exp(iω0 t) + exp(−iω0 t) = 0 dt dt

(2.26)

By solving equations (2.23) and (2.24) with respect to A one obtains A=

1 exp(−iω0 t)(x˙ + iω0 x) iω0

(2.27)

Similar kind of complex amplitudes is used in both physics [89] and nonlinear mechanics [102]. Now equation (2.3) gives dA ε = exp(−iω0 t)f dt iω0

(2.28)

where f = f (x, x) ˙ is expressed trough (2.23) and (2.24). Equation (2.28) is still exactly equivalent to (2.3). If the parameter ε is small then the amplitude A is slow, and one can apply the averaging

44

2 Smooth Oscillating Processes

ε dA = dt 2πi

2π/ω 0

exp(−iω0 t)f dt

(2.29)

0

On theoretical point of view, complex amplitudes may bring some convenience compared to the traditional Van-der-Pol variables. Firstly, until the certain stage of manipulations, it is usually possible to keep only one equation since another one is its complex conjugate. Secondly, such a symmetry of the equations helps sometimes to reveal interesting features of the dynamics. Note that the above manipulations remain valid in degenerated cases of multiple degrees of freedom systems. For instance, equation (2.3) can be interpreted as a vector equation with the scalar factor ω02 .

2.2.5

Lie Group Approaches

The one-parameter Lie1 group approaches are motivated by the idea of matching the tool and the object of study as explained in works [201] and [204]. Brieﬂy, it is suggested to seek transformation (2.12) among solutions of dynamical systems rather than the class of the arbitrary nonlinear transformations. Original materials and overviews of the mathematical structure of Lie groups, Lie algebras and Lie transforms with applications to nonlinear differential equations can be found in [38], [61], [22], [95], [35]. An essential ingredient of this version is the Hausdorﬀ formula, which relates the Lie group operators of the original and new systems, and the operator of coordinate transformation. According to [202] and [204], most of the averaging techniques just reproduce this formula, each time implicitly, during the transformation process. But, there is no need of doing this, since it is reasonable to start the transformation using Hausdorﬀ’s relationship. The corresponding algorithms therefore enable one of optimizing the number of manipulations for high-order approximations of asymptotic integration. The theory of Lie groups deals with a set of transformations. In other words, some dynamical system y˙ = f (y, ε) is transformed into its simplest form z˙ = g (z, ε) by means of a coordinate transformation y → z produced by solution z = z(y, ε) of the third dynamical system dz = T (z, ε) , dε

z |ε=0 = y

where the choice for vector-function T (z, ε) depends upon desired properties of the transformed system. 1

Marius Sophus Lie ( 1842-1899 ), Norwegian mathematician; diﬀerent mathematical objects are named after him, for instance, groups, operators, algebras, and series.

2.2 A Brief Overview of Smooth Methods

45

As mentioned, one of the advantages of the group formulation is that it speciﬁes a general class of near identical transformations. Speciﬁcally, one should select the expression z = z(y, ε) among solutions of a dynamical system, but not among all classes of the near identical transformations. Another basic advantage is that all manipulations of the scheme can be done in linear terms of the monomial Lie group operators. Moreover, the result of transformation in general terms of operators is well-known and is given by the Hausdorﬀ formula. The description below presents all the stages starting with the traditional Newtonian form of the diﬀerential equations of motion as implemented in [147]. The original system will be reduced to its normal form by Poincare. In terms of the principal coordinates qk , a nonlinear dynamical system of n-degrees of freedom may be described by a set of n + 1 autonomous diﬀerential equations written in the standard form q¨k + ωk2 qk = εFk (q1 , ..., qn+1 , q˙1 , ..., q˙n+1 );

k = 1, ..., n + 1

(2.30)

where an overdot denotes diﬀerentiation with respect to time t, ε is a small parameter, and an external excitation has been replaced by the coordinate qn+1 . The functions Fk include all nonlinear terms and possibly parametric excitation terms, and ωk are the principal mode frequencies. It is assumed that the functions Fk admit Taylor expansions near zero. The Poincare normal form theory deals with sets of ﬁrst-order diﬀerential equations written in terms of normal form coordinates. In this case it is convenient to transform the n + 1 second-order diﬀerential equations (2.30) into n + 1 ﬁrst-order diﬀerential equations plus their conjugate set. This can be done by introducing the complex coordinates yk = q˙k + iωk qk qk =

1 (yk − y¯k ), 2iωk

q˙k =

(2.31) 1 (yk + y¯k ) 2

(2.32)

Introducing the transformation (2.31) into the equations of motion (2.30), gives d (q˙k + iωk qk ) − iωk (q˙k + iωk qk ) = dt dyk − iωk yk = εFk (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 ) = dt

q¨k + ωk2 qk =

or y˙ k = iωk yk + εFk (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 )

(2.33)

and the corresponding complex conjugate (cc) set of equations, where the functions Fk (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 ) are obtained by substituting (2.32) in the right hand side of equation (2.30). These terms can be represented in the polynomial form

46

2 Smooth Oscillating Processes

Fk =

m

l

n+1 l1 n+1 Fkσ y1m1 · · · yn+1 y¯1 · · · y¯n+1

(2.34)

|σ|=2,3,...

where the Taylor coeﬃcients are Fkσ =

∂ |σ| Fk 1 | m ln+1 y=0 n+1 σ! ∂y1m1 · · · ∂yn+1 ∂ y¯1l1 · · · ∂ y¯n+1

and multiple-index notations have been introduced as follows σ = {m1 , ..., mn+1 , l1 , ..., ln+1 } |σ| = m1 + · · · + mn+1 + l1 + · · · + ln+1 σ! = m1 ! · · · mn+1 !l1 · · · ln+1 ! Equations (2.33) correspond to the standard form, which is ready for analysis in terms of Lie group operators. To apply the theory of the Lie groups we rewrite equations (2.33) in the form (2.35) y˙ = Ay, A = A0 + εA1 where y = (y1 , ..., yn+1 ; y¯1 , ..., y¯n+1 )T , and A0 =

n+1 k=1

iωk yk

∂ + cc ∂yk

and

A1 =

n+1

Fk

k=1

∂ + cc ∂yk

(2.36)

are operators of linear and nonlinear components of the system, respectively. In order to bring the equations of motion to their simplest (Poincare) form, we introduce the coordinate transformation y → z in the Lie series form y = e−εU z = z−εU z+

ε2 2 U z − ... 2!

(2.37)

T

where z = (z1 , ..., zn+1 ; z¯1 , ..., z¯n+1 ) , and the operator of transformation U is represented in the power series form with respect to the small parameter ε U = U0 + εU1 + · · ·

(2.38)

The coeﬃcients of this series are Uj =

n+1 k=1

Tj,k

∂ + cc ∂zk

(2.39)

where Tj,k = Tj,k (z1 , ..., zn+1 ; z¯1 , ..., z¯n+1 ) are unknown functions to be determined.

(2.40)

2.2 A Brief Overview of Smooth Methods

47

One of the advantages of this process is that the inverse coordinate transformation to the form (2.37) can be easily written as eεU y = z

(2.41)

where one should simply replaces z with y in the operator of transformation, U. If ε = 0, transformation (2.37) becomes identical, y = exp (0) z = z. In this case, equation (2.35) has already the simplest linear form and there is no need to transform the system. For ε = 0 transformation (2.37) converts the system (2.35) into the following one: z˙ = Bz

(2.42)

where the new operator B is given by the Hausdorﬀ formula [22]: B = A + ε [A, U ] +

ε2 [[A, U ] , U ] + ... 2!

(2.43)

where [A, U ] = AU − U A is the commutator of operators A and U . An optimized iterative algorithm for high-order solutions of equation (2.43) was suggested in [202] and [204]. In order to illustrate just the leading order terms of asymptotic expansions, let us follow the direct procedure though. Substituting the power series expansions for A and U given by relations (2.35) and (2.38) into (2.43) gives B = A0 + ε (A1 + [A0 , U0 ])

1 2 +ε [A0 , U1 ] + [A1 , U0 ] + [[A0 , U0 ] , U0 ] + ... 2!

(2.44)

A simple calculation gives B=

n+1

{iωk zk + ε [Fk + (A0 − iωk ) T0,k ]}

k=1

∂ + O ε2 + cc ∂zk

(2.45)

where the terms of order ε2 have been ignored, Fk = Fk |y→z and A0 = A0 |y→z . The above relationships show that a transformation of the system y˙ = Ay → z˙ = Bz can be considered in terms of operators A → B. In order to bring the system into its normal form, one must eliminate as many nonlinear terms as possible from the transformed system such that the system dynamic characteristics are preserved. It follows from (2.45) that all nonlinear terms of order ε could be eliminated under the condition

48

2 Smooth Oscillating Processes

Fk + (A0 − iωk ) T0,k = 0 Representing the unknown functions in the polynomial form mn+1 l1 ln+1 σ m1 T0,k z1 · · · zn+1 z¯1 · · · z¯n+1 T0,k =

(2.46)

|σ|=2,3,...

and taking into account (2.34), gives m1 mn+1 l1 ln+1 σ Fkσ + iΔσk T0,k z1 · · · zn+1 z¯1 · · · z¯n+1 Fk + (A0 − iωk ) T0,k = |σ|=2,3,...

where Δσk = (m1 − l1 − δ1k ) ω1 + ... + (mn+1 − ln+1 − δn+1,k ) ωn+1 m

(2.47) l

n+1 l1 n+1 To reach zero-th coeﬃcient of the monomial z1m1 · · · zn+1 z¯1 · · · z¯n+1 , one must put Fσ σ = i kσ T0,k Δk

under the condition that Δσk = 0. If Δσk = 0 for some k and σ then the corresponding nonlinear term cannot be eliminated from the transformed equation since it is qualiﬁed as a resonance term. Finally, the result of transformation is summarized as follows. The original set: mn+1 l1 ln+1 Fkσ y1m1 · · · yn+1 y¯1 · · · y¯n+1 (2.48) y˙ k = iωk yk + ε |σ|=2,3,...

The transformation of coordinates:

yk = zk − ε

|σ|=2,3,... Δσ k =0

i

Fkσ m1 mn+1 l1 ln+1 z1 · · · zn+1 z¯1 · · · z¯n+1 + O ε2 σ Δk

(2.49)

The transformed set: z˙k = iωk zk + ε

mn+1 l1 ln+1 Fkσ z1m1 · · · zn+1 z¯1 · · · z¯n+1 + O ε2

(2.50)

|σ|=2,3,... Δσ k =0

Equations (2.50) represent the normal form of the system, where the summation is much simpler than that in the original set (2.48). Namely, the summation in (2.50) contains only those terms that give rise to resonance while the ﬁrst term on the right hand side stands for the fast component of the

2.2 A Brief Overview of Smooth Methods

49

motion. The fast component of the motion can be extracted by introducing the complex amplitudes ak (t) as follows zk = ak (t) exp(iωk t)

(2.51)

Substituting (2.51) into (2.50) and taking into account the resonance condition, Δσk = 0, gives mn+1 l1 ln+1 1 (2.52) Fkσ am ¯1 ...¯ an+1 + O ε2 a˙ k = ε 1 ...an+1 a |σ|=2,3,... Δσ k =0

System (2.52) describes the dynamics in terms of slowly varying amplitudes and, as a result, reveals global properties of the dynamics in a much easier way then the original system. After solution of system (2.52) is obtained, the coordinate transformations (2.51) and (2.49) can interpret the result in terms of the original coordinates.

Chapter 3

Nonsmooth Processes as Asymptotic Limits

Abstract. In this chapter, we consider diﬀerent physical models generating both smooth and nonsmooth temporal mode shapes as appropriate conditions occur. The objective is to bring attention to the fact that nonsmooth processes may naturally occur as high-energy asymptotics in diﬀerent oscillatory models with no intentionally introduced stiﬀ constraints or external impacts. In other words, nonsmooth temporal mode shapes may be as natural as sine waves generated by oscillators under low-energy conditions. Essentially nonlinear phenomena, such as nonlinear beats and energy localization are also considered. In particular, it is shown that energy exchange between two oscillators may possess hidden nonsmooth behaviors.

3.1

Lyapunov’ Oscillator

Let us consider a family of oscillators described by the diﬀerential equation x ¨ + x2n−1 = 0

(3.1)

where n is a positive integer. In the particular case n = 1 one has the simplest harmonic oscillator. However, when n > 1 the system becomes essentially nonlinear and cannot be linearized within the class of vibrating systems. Moreover, as the parameter n increases, the temporal mode shape of oscillator (3.1) while remaining smooth is gradually approaching the triangular wave non-smooth limit. In general, such transition represents a challenging problem from both physical and mathematical viewpoints. Therefore, it is important to understand some basic cases, such as oscillator (3.1) and those considered in the next section. These special cases admit exact solutions, so that it is possible to actually see how smooth motions are approaching the non-smooth limit. There are also methodological reasons for considering equation (3.1) as a simple example of oscillators including both linear and strongly nonlinear cases. It is known that, for an arbitrary positive integer n, general solution of equation (3.1) V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 51–91, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

52

3 Nonsmooth Processes as Asymptotic Limits

can be expressed in terms of special Lyapunov’s function [96], [77], [56] such as snθ and csθ as deﬁned by expressions1

θ=

snθ

1 − nz 2

1−2n 2n

dz,

cs2n θ + n sn2 θ = 1

0

These functions possess the properties cs0 = 1,

sn0 = 0,

dsnθ = cs2n−1 θ, dθ

dcsθ = −snθ dθ

The period is given by √ T =4 n

1 0

1 √ dx π Γ 2n √ =2 n Γ n+1 1 − x2n 2n

Now, the general solution of equation (3.1) can be written as x = Acs An−1 t + α

(3.2)

where A and α are arbitrary constants. Note that the scaling factors A and An−1 are easily predictable based on the form of equation (3.1). Indeed, equation (3.1) admits the group of transformations x = A¯ x(t¯), where t¯ = An−1 t. For n = 1 the functions snθ and csθ give the standard pair of trigonometric functions sin θ and cos θ, respectively. Interestingly enough, the strongly nonlinear limit n → ∞ also gives a quite simple pair of periodic functions. Despite some mathematical challenges, this case admits interpretation by means of the total energy x2n 1 x˙ 2 + = (3.3) 2 2n 2 where the number 1/2 on the right-hand side corresponds to the initial conditions x (0) = 0 and x˙ (0) = 1. Taking into account that the coordinate of the oscillator reaches its amplitude value at zero kinetic energy, gives the estimate −n1/(2n) ≤ x (t) ≤ n1/(2n) for any time t. Since n1/(2n) −→ 1 as n −→ ∞ then the limit motion is restricted by the interval −1 ≤ x (t) ≤ 1. Inside of this interval, the second term on the left-hand side of expression (3.3) vanishes and hence, x˙ = ±1 or x = ±t + α± , where α± are constants. By manipulating with the signs and constants one can construct the sawtooth sine τ (t) - triangular wave - since there is no other way to providing the periodicity condition. 1

Another version of special functions for equation (3.1) was considered in [167].

3.1 Lyapunov’ Oscillator

53

So the family of oscillators (3.1) includes the two quite simple cases associated with the boundaries of the interval 1 ≤ n < ∞. Respectively, one has the two couples of periodic functions {x, x} ˙ = {sint, cost},

if n = 1

(3.4)

if n → ∞

(3.5)

and {x, x} ˙ → {τ (t), τ˙ (t)},

where τ˙ (t) is a generalized derivative of the sawtooth sine and will be named as a rectangular cosine. Earlier, the power-form characteristics with integer exponents were employed for phenomenological modeling amplitude limiters of vibrating elastic structures [191] and illustrations of impact asymptotics [132], [137]. It should be noted that such phenomenological approaches to impact modeling are designed to capture the integral eﬀect of interaction with physical constraints bypassing local details of the dynamics near constraints. Such details ﬁrst of all depend upon both the vibrating body and amplitude limiter physical properties. In many cases, Hertz model of interaction may be adequate to describe the local dynamics near constraint surfaces [64]. Note that direct replacement of the characteristic x2n−1 by the Hertzian restoring force kx3/2 in (3.1) gives no oscillator. The equation, x¨ + kx3/2 = 0

(3.6)

which is a particular case of that used in [64], must be obviously accompanied by the condition 0 ≤ x, where x = 0 corresponds to the state at which the moving body and constraint barely touch each other with still zero interaction force. The following modiﬁcation brings system (3.6) into the class of oscillators with odd characteristics x ¨ + ksgn(x)|x|3/2 = 0

(3.7)

However, oscillator (3.7) essentially diﬀers from oscillator (3.1) since equation (3.7) describes no gap (clearance) between the left and right constraint surfaces. In other words models (3.1) and (3.7) still represent physically diﬀerent situations. The gap 2Δ with its center at the origin x = 0 can be introduced in equation (3.7) as follows x ¨ + k[H(x − Δ)|x − Δ|3/2 − H(−x − Δ)|x + Δ|3/2 ] = 0

(3.8)

where H means Heaviside unit-step function. This is a generalization of model (3.7), which is now obtained from (3.8) by setting Δ = 0. Equation (3.8) can be viewed as a physical impact oscillator that accounts for elastic properties of its components. As compared to

54

3 Nonsmooth Processes as Asymptotic Limits

phenomenological model (3.1), equation (3.8) was obtained on certain physical basis given by the Hertz contact theory. Finally, oscillators with power-form characteristics, including their generalizations, can be found in physical literature[26], [121], [58], [60], [98] and diﬀerent areas of applied mathematics and mechanics [112], [11], [3], [4], [43], [107], [5], [34], [6], [63], [193], [48], [108], [164]. In [146], the power-form restoring forces were introduced to simulate the eﬀect of liquid container’ walls on liquid sloshing impacts; see also review article [66].

3.2

Nonlinear Oscillators Solvable in Elementary Functions

A class of strongly nonlinear oscillators admitting surprisingly simple exact general solutions at any level of the total energy is described below. Although the fact of exact solvability of these oscillators has been known for quite a long time [78], it did not attract much attention possibly due to the speciﬁc form of the oscillator characteristics with uncertain physical interpretations. It is clear however that, in a phenomenological way, such characteristics capture suﬃciently general physical situations with hardening and softening behavior of the elastic forces. For instance, these oscillators were recently used as a phenomenological basis for describing diﬀerent practically important physical and mechanical systems [122], [40], [41]. The hardening characteristic is close to linear for relatively small amplitudes but becomes inﬁnity growing as the amplitude approaches certain limits. As a result, the corresponding temporal mode of vibration changes its shape from smooth quasi harmonic to nonsmooth triangular wave. In contrast, the softening characteristic behaves in a non-monotonic way such that the vibration shape is approaching the rectangular wave as larger amplitudes are considered. Earlier, amplitude-phase equations were obtained for a coupled array of the hardening oscillators [157]. It will be shown below that such oscillators admit explicit introduction of the action-angle variables within the class of elementary functions. As a result, conventional averaging procedures become applicable to a wide range of nonlinear motions including transitions from high- to low-energy dynamics. In particular, analytical solutions are obtained under small damping conditions. These solutions show a good match with the corresponding numerical solutions at any energy level even within the ﬁrstorder asymptotic approximation. Hardening and softening cases of these oscillators are, respectively, H=

tan2 x tan x p2 + ⇒x ¨+ =0 2 2 cos2 x

(3.9)

and

tanh2 x tanh x p2 + ⇒x ¨+ =0 2 2 cosh2 x where p = x˙ is the linear momentum of the Hamiltonian H. H=

(3.10)

3.2 Nonlinear Oscillators Solvable in Elementary Functions

55

Further objectives are to investigate the high-energy asymptotics with transitions to nonsmooth temporal mode shapes and to show that both of the above oscillators can play the role of generating systems for regular perturbation procedures within the class of elementary functions. tan x cos2 x 100

50

Π

2

Π

Π

4

4

Π

2

x

50

100 Fig. 3.1 Hardening characteristic.

tanh x cosh2 x 0.5

Π

2

Π

Π

4

4

Π

2

x

0.5 Fig. 3.2 Softening characteristic.

Notice that oscillators (3.9) and (3.10) complement each other as those with stiﬀ and soft characteristics represented in Figs. 3.1 and 3.2, respectively. These oscillators can be represented also in the form x ¨ + tan x + tan3 x = 0

(3.11)

56

3 Nonsmooth Processes as Asymptotic Limits

x ¨ + tanh x − tanh3 x = 0

(3.12)

Further analyses of equations (3.11) and (3.12) can be quite easily conducted by means of substitutions q = tan x and q = tanh x, respectively. Interestingly enough, oscillators (3.11) and (3.12) without the qubic terms, namely x¨ + tan x = 0 and x ¨ + tanh x = 0, were considered by Timoshenko and Yang [182]. But, despite of the simpliﬁed form, the corresponding solutions were found to be special functions.

3.2.1

Hardening Case

Consider ﬁrst stiﬀ oscillator (3.9), whose solution is

t x = arcsin sin A sin cos A

(3.13)

where A is an arbitrary constant, and another constant is introduced as an arbitrary time shift t− > t + const., since the equations admits the group of temporal shifts. Therefore, (3.13) represents a general periodic solution of the period T = 2π cos A, and the total energy is expressed through the amplitude A as E=

1 tan2 A 2

(3.14)

In zero energy limit, when the amplitude A is close to zero, the oscillator linearizes whereas solution (3.13) gives the corresponding sine-wave temporal shape. On the other hand, the energy becomes inﬁnitely large as the parameter A approaches the upper limit π/2. In this case, the period vanishes while the oscillation takes the triangular wave shape, as it is seen from expression (3.13). Fig. 3.3 illustrates the evolution of the vibration shape in the normalized coordinates. Below, the action-angle variables are introduced in terms of elementary functions. This enables one of considering non-periodic motions by using exact solution (3.13) as a starting point of the averaging procedure. For a single degree-of-freedom conservative oscillator, the action coordinate I is known to be the area bounded by the system’ trajectory on the phase plane divided by 2π whereas the angle ϕ coordinate is simply phase angle [8], [124]. In the case of stiﬀ oscillator (3.9), one obtains 1 1 I= pdx = −1 (3.15) 2π cos A and, ϕ= respectively.

t cos A

(3.16)

3.2 Nonlinear Oscillators Solvable in Elementary Functions

57

1.0 Α0.001 0.5 Α0.999 Ns

0.0

0.5

1.0 0

1

2

3

4

5

6

Fig. 3.3 Normalized temporal mode shapes of the stiﬀ oscillator, N s(ϕ, α) = arcsin(α sin ϕ)/ arcsin α.

The original coordinate and the velocity are expressed by the action-angle variables as follows [158] √ √ (1 + I) 2I + I 2 cos ϕ 2I + I 2 (3.17) sin ϕ , p = x = arcsin 1+I 1 + (2I + I 2 ) cos2 ϕ In order to observe the convenience of action-angle coordinates, let us chose the Hamiltonian description of the oscillator. Taking into account expressions (3.14) and (3.15), and eliminating the amplitude A, gives the total energy and thus the Hamiltonian in the form 1 H = I + I2 2

(3.18)

The corresponding diﬀerential equations of motion are derived as follows ∂H =1+I ∂I ∂H I˙ = − =0 ∂ϕ

ϕ˙ =

(3.19)

As it is seen, the diﬀerential equation of the oscillator (3.9) takes the linear form with respect to the action-angle coordinates, and thus possess the exact general solution (3.20) I = I0 , ϕ = (1 + I0 ) t + ϕ0 where I0 > 0 and ϕ0 are arbitrary constants. By substituting (3.20) in (3.17), one can express the solution via the original coordinates. The meaning of the initial action is clear from the energy relationship

58

3 Nonsmooth Processes as Asymptotic Limits

1 1 E = I0 + I02 = tan2 A 2 2

(3.21)

Note that the linearity of the Hamiltonian equations is due to the speciﬁc strongly non-linear form of the coordinate transformation (3.17). In other words, the system nonlinearity has been ‘absorbed’ in a purely geometric way by the nonlinear coordinate transformation. As mentioned at the beginning, simplicity of the transformed system and that of the corresponding solution can be essentially employed for the purpose of perturbation analysis. Let us consider, however, the diﬀerential equation of motion in the Newtonian form x ¨+

tan x = εf (x, x) ˙ cos2 x

(3.22)

were ε is a small parameter. This system is weakly non-hamiltonian. However, it is still possible to consider expressions (3.17) as a change of the coordinates {x, p} −→ {I, ϕ} by imposing the compatibility condition dx/dt = p, where x and p must be taken from (3.17). This gives εf (x, p) sin ϕ (1 + I) (2I + I 2 ) [1 + (2I + I 2 ) cos2 ϕ] √ εf (x, p) 2I + I 2 cos ϕ I˙ = 1 + (2I + I 2 ) cos2 ϕ

ϕ˙ = 1 + I −

(3.23)

where the function f (x, p) must be expressed through the action-angle coordinates by means of (3.17). For instance, in the case of linear damping, f (x, p) ≡ −p, one obtains ε cos ϕ sin ϕ 1 + (2I + I 2 ) cos2 ϕ ε (1 + I) 2I + I 2 cos2 ϕ ˙ I=− 1 + (2I + I 2 ) cos2 ϕ

ϕ˙ = 1 + I +

(3.24)

At this stage, let us implement just one step of the procedure and evaluate its eﬀectiveness. Applying the operator of averaging with respect to the phase variable gives the corresponding ﬁrst-order averaged system in the linear form ϕ˙ = 1 + I,

I˙ = −εI

(3.25)

Substituting the general solution of system (3.25) in (3.17), ﬁnally gives

2I0 exp (−εt) + I02 exp (−2εt) 1 − exp (−εt) sin t + I0 + ϕ0 x = arcsin 1 + I0 exp (−εt) ε (3.26)

3.2 Nonlinear Oscillators Solvable in Elementary Functions

59

where I0 and ϕ0 are arbitrary constants. The corresponding time history records and phase plane diagrams for diﬀerent damping coeﬃcients are shown in Fig. 3.4. Even the ﬁrst order approximation appears to be perfectly matching with numerical solution for all range of amplitudes. The analytical and numerical curves can be distinguished only at relatively large magnitudes of the damping parameter ε. Also, the diagrams show that the temporal mode shape is gradually changing from the triangular to harmonic as time increases and thus the amplitude decays. 1.5 1.0 0.5 x 0.0 0.5 1.0 1.5

0

10

20

30

40

50

1.5 1.0 0.5 x 0.0 0.5 1.0 1.5

0

10

t

v

30

40

50

1.0

1.5

t

10

10

5

5

0

v 0

5

5

10 1.5 1.0 0.5 0.0 x

20

0.5

1.0

1.5

1.5 1.0 0.5 0.0 x

0.5

Fig. 3.4 The dynamics of the linearly damped stiﬀ oscillator under the initial conditions I0 = 10 and ϕ0 = 0, and two diﬀerent damping parameters: ε = 0.2 (on the left,) and ε = 0.8 (on the right.) Analytical and numerical solutions show a slight mismatch only on the top right diagram.

3.2.2

Localized Damping

Let us consider the case of nonlinear damping x ¨+

tan x + 2εx˙ tan2 x = 0 cos2 x

(3.27)

In this case, the perturbation is given by f (x, p) ≡ −2p tan2 x. Such a damping is rapidly growing near the boundaries of the interval −π/2 ≤ x ≤ π/2, but it becomes negligible when the amplitude is small, |x| << 1.

60

3 Nonsmooth Processes as Asymptotic Limits

In the action-angle coordinates, ﬁrst order averaging gives ϕ˙ = 1 + I, and thus ϕ=t+

I˙ = −εI 2

1 ln (1 + εI0 t) + ϕ0 , ε

I=

I0 1 + εI0 t

Using the coordinate transformation (3.17), gives solution

I0 (2 + I0 + 2εI0 t) ln(1 + εI0 t) sin t + + ϕ0 x = arcsin 1 + I0 + εI0 t ε

(3.28)

where I0 and ϕ0 are arbitrary constants. Note that the amplitude decay of solutions (3.26) and (3.28) is qualitatively diﬀerent. For instance, the amplitude of vibration (3.28) originally decays in a fast rate and then becomes very slow. In contrast, the amplitude of vibration (3.26) ﬁrst decays slowly then the decay rate abruptly increases and then slows down again.

3.2.3

Softening Case

Let us consider now softening oscillator (3.10), whose exact solution is

t x = arc sinh sinh A sin (3.29) cosh A As Fig. 3.5 shows, the high-energy vibration shape approaches the rectangular wave and thus essentially diﬀers of that observed in the stiﬀ case. Based on solution (3.29), the action-angle coordinates are introduced by means of expressions √ √ (1 − I) 2I − I 2 cos ϕ 2I − I 2 sin ϕ , p = (3.30) x = arc sinh 1−I 1 − (2I − I 2 ) cos2 ϕ where

1 (3.31) cosh A Nevertheless, all the analytical manipulations are analogous to those in the stiﬀ case. For instance, taking into account (3.31), gives the total energy as a function of the action coordinate I =1−

E=

1 1 tanh2 A = I − I 2 2 2

(3.32)

3.3 Nonsmoothness Hiden in Smooth Processes

61

1.0 Α106 0.5 Α0.001 Nh

0.0

0.5

1.0 0

1

2

3

4

5

6

Fig. 3.5 Normalized temporal mode N h(ϕ, α) =arcsinh(α sin ϕ)/arcsinh α.

shapes

of

the

soft

oscillator,

In the presence of viscous damping, x ¨+

tanh x = −εx˙ cosh2 x

(3.33)

one obtains, compare to (3.25), ϕ˙ = 1 − I,

I˙ = −εI

(3.34)

and general solution of the original equation takes the form

2I0 exp (−εt) − I02 exp (−2εt) 1 − exp (−εt) sin t−I0 +ϕ0 x = arc sinh 1 − I0 exp (−εt) ε (3.35) The corresponding time history records and phase plane diagrams are shown in Fig. 3.6 for diﬀerent damping coeﬃcients. The ﬁrst order approximation appears to perfectly match the corresponding numerical solution for all range of amplitudes, unless the initial action I0 approaches the magnitude 1. As follows from expressions (3.32), this magnitude corresponds to the maximum value of the total energy of the oscillator. Note that the energy of the hardening oscillator has no maximum.

3.3

Nonsmoothness Hiden in Smooth Processes

In this section, we consider nonlinear beats phenomena as another source of nonsmooth behavior that brings certain physical meaning to oscillator (3.22). Note that nonlinear beats became of growing interest just few decades ago

62

3 Nonsmooth Processes as Asymptotic Limits

0.8 0.6 0.4 0.2 x 0.0 0.2 0.4 0.6

0.6 0.4 x 0.2 0.0 0.2 0

10

20

30

40

50

0

10

20

t 1.0

40

50

1.0

0.5 v

30 t

0.5 v

0.0

0.0

0.5 0.60.40.2 0.0 0.2 0.4 0.6 0.8 x

0.5 0.2

0.0

0.2 x

0.4

0.6

Fig. 3.6 The dynamics of the linearly damped softening oscillator under the initial conditions I0 = 0.5, ϕ0 = 0, and two diﬀerent damping parameters: ε = 0.2 (on the left,) and ε = 0.8 (on the right.) Analytical and numerical curves practically coincide.

from diﬀerent viewpoints of physics and nonlinear dynamics [84], [59], [99], [188], [88]. Interestingly enough, phase variables of interacting oscillators with close natural frequencies may show non-smoothness of temporal behavior during the beating [59], for instance similar to that of a vibro-impact process [101], [104].

3.3.1

Nonlinear Beats Model

Consider two linearly coupled Duﬃng oscillators x ¨1 + Ω 2 x1 = −β(x1 − x2 ) − αx31 ≡ f1 x ¨2 + Ω 2 x2 = −β(x2 − x1 ) − αx32 ≡ f2

(3.36)

where α and β are nonlinearity and linear coupling parameters, respectively. Let us introduce complex coordinates Aj (t) and A¯j (t) as follows 1 [Aj exp(iΩt) + A¯j exp(−iΩt)] 2 iΩ x˙ j = [Aj exp(iΩt) − A¯j exp(−iΩt)] 2 xj =

where j = 1, 2.

(3.37)

3.3 Nonsmoothness Hiden in Smooth Processes

63

Convenience of using the complex amplitudes for linear and nonlinear mode analyses have been known for a long time [131], [174]. In physical literature though, complex amplitudes are introduced more often as vectors rotating on complex phase planes of oscillators, but the resultant equations are usually similar to those obtained below. Expressions (3.37) implement indeed a complex version of the parameter variation method based on the solution of the corresponding linear system. The related compatibility condition is imposed in the form dAj dA¯j exp(iΩt) + exp(−iΩt) = 0 dt dt

(3.38)

By substituting (3.37) in (3.36) and taking into account (3.38) gives dAj 1 = exp(−iΩt)fj dt iΩ

(3.39)

Assuming that the coupling and nonlinearity parameters are suﬃciently small and averaging the right-hand side with respect to Ωt, gives

βi 3α 2 ¯ A˙ 1 = A1 A1 A1 − A2 + 2Ω 4β

βi 3α 2 ¯ ˙ A A2 A2 = −A1 + A2 + 2Ω 4β 2

(3.40)

Further, following work [104], a slowly rotating subcomponent on the complex plane is eliminated by means of substitution

Aj = ψj (t) exp

iβ t 2Ω

Substituting (3.41) into (3.40), gives

˙ψ = − βi ψ − 3 α ψ 2 ψ ¯ 1 2 2Ω 4β 1 1

˙ψ = − βi ψ − 3 α ψ 2 ψ ¯ 2 1 2Ω 4β 2 2

(3.41)

(3.42)

This system has two integrals as follows ¯ +ψ ψ ¯ K = ψ1 ψ 1 2 2

(3.43)

64

3 Nonsmooth Processes as Asymptotic Limits

3α 4 4 ¯ +ψ ψ ¯ |ψ 1 | + |ψ 2 | G = ψ1 ψ 2 2 1− 8β Integral (3.43) admits substitution

√ 1 π ψ 1 = K cos θ(t) + exp[i δ1 (t)] 2 4

√ 1 π ψ 2 = K sin θ(t) + exp[i δ 2 (t)] 2 4

(3.44)

(3.45)

Substituting (3.45) in (3.42) and (3.44), gives G = −K cos Δ cos θ −

3 K2 α (3 − cos 2 θ) = const. 32 β

(3.46)

β 3K α Δ˙ = − cos Δ tan θ + sin θ (3.47) Ω 8Ω β sin Δ (3.48) θ˙ = Ω where Δ = δ 2 − δ 1 + π. It will be shown below that the phase variable θ, which determines the process of energy ﬂow between the oscillators, is described by the oscillator (3.22). Equations (3.47)-(3.48) are similar to those obtained in [101], [104], where it was noticed that temporal shapes of the phase variables θ and Δ may resemble the behavior of the state variables of impact oscillator. This observation seems to be important since it is hard to expect any “impact oscillators” in weakly nonlinear systems of type (3.36). Below, the corresponding ‘conservative oscillator’ admitting the impact limit will be explicitly obtained and analyzed by using the methodology described in the previous section. However, the approach below deals with a general class of nonlinear restoring force characteristics admitting powerseries expansions. It will be shown also that equations (3.47)-(3.48) can be derived by introducing the standard set of amplitude-phase variables and applying then the traditional one fast phase averaging technique.

3.4

Nonlinear Beat Dynamics: The Standard Averaging Approach

Let us consider two identical linearly coupled oscillators u ¨1 + b(u1 − u2 ) + p(u1 ) = 0 u ¨2 + b(u2 − u1 ) + p(u2 ) = 0

(3.49)

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

65

where b is the coupling stiﬀness per unit mass, and p(u) is the restoring force characteristic, which is assumed to be an analytic function that admits a power series expansion. Assuming that the system has equilibrium at zero and introducing the notation Ω2 = b + k ε = b/Ω 2 = b/(b + k)

(3.50)

f (u) = [(b + k)/b][p(u) − ku] where f (u) is a nonlinear component of the characteristic, and k=p’(0). Note that the power series expansion for f (u) starts from at least second degree of u. Therefore, the order of magnitude of the function f (u) can be manipulated by making appropriate assumptions as to the magnitude of the total energy of the system. Taking into account the above notations and introducing the velocities v1 (t) and v2 (t) brings the original system to the form u˙ 1 = v1 u˙ 2 = v2 v˙ 1 = −Ω 2 u1 + ε[Ω 2 u2 − f (u1 )]

(3.51)

v˙ 2 = −Ω u2 + ε[Ω u1 − f (u2 )] 2

2

As ε → 0, the system degenerates into two identical harmonic oscillators whose total energies are conserved of-course since neither damping nor external loading are present. At non-zero ε, the oscillators become non-linear and interact with each other in such a way that one of the oscillators is loaded by the force proportional to the displacement of another oscillator. Since system (3.51) is still perfectly symmetric and conservative, it is reasonable to assume a relatively slow periodic energy exchange between the oscillators. In order to describe this process in physically meaningful terms, let us introduce new set of variables as follows {u1 , v1 , u2 , v2 }− > {K, ϕ, δ1 , δ2 }: √ u1 = K cos ϕ cos(Ωt + δ1 ) √ v1 = − KΩ cos ϕ sin(Ωt + δ1 ) √ (3.52) u2 = K sin ϕ cos(Ωt + δ2 ) √ v2 = − KΩ sin ϕ sin(Ωt + δ2 ) In case ε = 0 and constant {K, ϕ, δ1 , δ2 }, expressions (3.52) gives an exact general solution of system (3.51). Therefore, relationships (3.52) simply implement the idea of parameter variations; the corresponding diﬀerential equations will be given below.

66

3 Nonsmooth Processes as Asymptotic Limits

Now, in order to track the oscillator energies during the vibration process, let us use quantities 1 2 (v + Ω 2 u21 ) = 2 1 1 E2 = (v22 + Ω 2 u22 ) = 2

E1 =

1 2 Ω K cos2 ϕ 2 1 2 Ω K sin2 ϕ 2

(3.53)

and

1 2 Ω K (3.54) 2 Besides, expressions (3.53) and (3.54) clarify the physical meaning of the variables K and ϕ participating in transformation (3.52), where other two variables, δ1 and δ2 , are phases of the vibrating oscillators. In particular, K is proportional to the total energy of the decoupled and linearized oscillators, whereas the phase ϕ characterizes the energy split between the oscillators. In case ε = 0, the energy parameter K will have small temporal ﬂuctuations due to coupling and nonlinear terms in (3.51). Nevertheless expressions (3.53) and (3.54) still will be used as the energy related quantities for characterization of the energy exchange process between the oscillators. In order to conduct the transition to the new variables, let us substitute (3.52) in (3.51) and then solve the set of equations with respect to the derivatives as follows √ 2ε K ˙ K = −εKΩ sin 2ϕ sin(2Ωt + δ1 + δ2 ) + Ω √ ×{f [ K cos ϕ cos(Ωt + δ1 )] cos ϕ sin(Ωt + δ1 ) √ +f [ K sin ϕ cos(Ωt + δ2 )] sin ϕ sin(Ωt + δ2 )} ε ε ϕ˙ = Ω[sin(δ1 − δ2 ) − cos 2ϕ sin(2Ωt + δ1 + δ2 )] − √ 2 KΩ √ ×{f [ K cos ϕ cos(Ωt + δ1 )] sin ϕ sin(Ωt + δ1 ) √ (3.55) −f [ K sin ϕ cos(Ωt + δ2 )] cos ϕ sin(Ωt + δ2 )} δ˙ 1 = −εΩ cos(Ωt + δ1 ) cos(Ωt + δ2 ) tan ϕ √ ε cos(Ωt + δ1 ) sec ϕf [ K cos ϕ cos(Ωt + δ1 )] +√ KΩ δ˙ 2 = −εΩ cos(Ωt + δ2 ) cos(Ωt + δ1 ) cot ϕ √ ε cos(Ωt + δ2 ) csc ϕf [ K sin ϕ cos(Ωt + δ2 )] +√ KΩ E0 = E1 + E2 =

System (3.55) is still an exact equivalent to system (3.49) and represents a standard dynamic system with a single fast phase, ψ = Ωt. As a next natural stage, let us apply the direct averaging to the right-hand side of (3.55) with respect to the fast phase ψ:

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

K˙ = 0 ε ϕ˙ = Ω sin(δ1 − δ2 ) 2 ε ε δ˙ 1 = − Ω cos(δ1 − δ2 ) tan ϕ + F1 (K, ϕ) 2 Ω ˙δ 2 = − ε Ω cos(δ1 − δ2 ) cot ϕ + ε F2 (K, ϕ) 2 Ω

67

(3.56)

where the residue theorem has been applied so that 2π √ 1 1 F1 (K, ϕ) = √ f ( K cos ϕ cos ψ) cos ψdψ K cos ϕ 2π 0 √ 1 1 1 1 1 = √ Res{f [ K cos ϕ (z + )] (z + 2 )} 2 z 2 z K cos ϕ 2π √ 1 1 F2 (K, ϕ) = √ f ( K sin ϕ cos ψ) cos ψdψ K sin ϕ 2π 0 √ 1 1 1 1 1 Res{f [ K sin ϕ (z + )] (z + 2 )} = √ 2 z 2 z K sin ϕ First equation in (3.56) shows that the energy parameter K introduced in (3.54) remains averagely constant regardless the magnitude of coupling and nonlinearity parameter ε. This gives a justiﬁcation for using quantities (3.53) and (3.54) for describing the energy exchange between the oscillators: indeed, neither the coupling nor nonlinear stiﬀness in (3.51) can accumulate the energy during one vibration cycle. Further complete description of the dynamics can be conducted in terms of the two phase shift parameters, Δ(t) and θ(t), introduced as follows δ2 = δ1 + Δ − π 1 1 ϕ= θ+ π 2 4

(3.57)

Substituting (3.57) in (3.56) and introducing the slow time parameter t1 = εΩt, gives dθ = sin Δ dt1 dΔ = − cos Δ tan θ + F (θ) dt1

(3.58)

where

1 1 1 1 1 [F2 (K, θ + π) − F1 (K, θ + π)] Ω2 2 4 2 4 It can be shown that system (3.58) has the integral F (θ) =

G = − cos Δ cos θ + h(θ) = const.

(3.59)

68

3 Nonsmooth Processes as Asymptotic Limits

where h(θ) = −

F (θ) cos θdθ

(3.60)

Now let us show that system (3.58) can be reduced to a single strongly nonlinear oscillator with respect to the coordinate θ. Taking time derivative of both sides of the ﬁrst equation in (3.58) and eliminating from the result dΔ/dt1 and cos Δ by means of the second equation in (3.58) and the integral of motion, (3.59), respectively, gives tan θ d2 θ = R(θ) + dp2 cos2 θ

(3.61)

where p = |G|t1 = ε|G|Ωt is a new slow temporal argument, θ = θ(p) and F (θ) tan θ R(θ) = G−2 h(θ)[2G − h(θ)] 2 − [G − h(θ)] (3.62) cos θ cos θ Equation (3.61) represents a principal equation describing the energy exchange in the coupled set of oscillators (3.49) through the phase shift (3.57). Note that the right-hand side in (3.61) is due to only nonlinearity associated with the nonlinear stiﬀness f (u); see relationships (3.50) and (3.51). In case R(θ) = 0, equation (3.61) has exact analytical solution

p (3.63) θ(p) = arcsin sin θ0 sin cos θ0 where θ0 is the amplitude of θ, whereas another constant can be introduced as an arbitrary temporal shift, which is admitted by equation (3.61). Generally speaking, it is still possible to ﬁnd implicit solutions of equation (3.61) in terms of quadratures for nonzero R(θ). In most cases, however, the corresponding expressions appear to be technically complicated for analyses. Therefore secondary asymptotic approaches to oscillator (3.61) may be reasonable for understanding its behaviors. For illustrating purposes, let us consider the example when the nonlinear stiﬀness in (3.50)-(3.51) consists of cubic and ﬁfth-degree terms f (x) = α3 x3 + α5 x5

(3.64)

In this case, integration in (3.60) gives h=

K(6α3 + 5Kα5 ) cos2 θ 32Ω 2

(3.65)

whereas equation (3.61) takes the form tan θ d2 θ = μ sin 2θ + 2 dp cos2 θ

(3.66)

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

69

where

K 2 (6α3 + 5Kα5 )2 (3.67) 2048G2 Ω 4 Substituting (3.57) in (3.53), gives the corresponding energy values versus phase θ: μ=

1 KΩ 2 (1 − sin θ) 4 1 E2 = KΩ 2 (1 + sin θ) 4 E1 =

3.4.1

(3.68)

Asymptotic of Equipartition

For suﬃciently small amplitudes of θ, equation (3.66) can be reduced to the following Duﬃng equation d2 θ 4 + (1 − 2μ)θ + (1 + μ)θ3 = 0 dp2 3

(3.69)

As follows from (3.68) on physical point of view, the equilibrium point θ = 0 corresponds to equal energy distribution between the oscillators (3.49). So when the linear stiﬀness is positive 1 − 2μ > 0, equation (3.69) describes periodic energy exchange between the oscillators (3.49) provided that the initial energy distribution is close to equal and the cubic approximation for the characteristic is justiﬁed. The period of the energy exchange process can be easily estimated based on the corresponding solution of equation (3.69). However, the linear stiﬀness becomes negative when μ>

1 2

(3.70)

Condition (3.70) says that the equal energy distribution may become unstable if the parameter μ is suﬃciently large. As a result, system (3.69) can stay in one of the two new stable equilibrium positions so that a larger portion of the total energy is localized on one of the two identical oscillators (3.49). Note that the equilibrium points of system (3.69) correspond to nonlinear normal mode regimes. The phase-plane diagrams of oscillator (3.66) are shown in Figs. 3.7, 3.8, and 3.9 for diﬀerent magnitudes of μ. It is seen how two more equilibria occur when μ exceeds the critical value (3.70). Note that condition (3.70) is obviously necessary but not suﬃcient to guarantee that the localization will actually occur. Necessary and suﬃcient conditions will be discussed below. In order to determine the initial state of oscillator (3.66) and its parameter μ, let us consider transformation (3.52) at t = 0. With no loss of generality,

70

3 Nonsmooth Processes as Asymptotic Limits 3

2

dΘdp

1

0

1 2 3

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ

Fig. 3.7 Periodic energy exchange case, μ = 0.2. 3

2

dΘdp

1

0

1 2 3

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ

Fig. 3.8 Energy trapping bifurcation, μ = 0.5

one can select δ1 (0) = 0. Then, taking into account expressions (3.57), (3.53) and (3.54), gives

√ π θ(0) u1 (0) = K cos + 4 2 v1 (0) = 0

√ π θ(0) u2 (0) = − K cos Δ(0) sin + (3.71) 4 2

√ θ(0) π + v2 (0) = KΩ sin Δ(0) sin 4 2

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

71

3

2

dΘdp

1

0

1 2 3

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ

Fig. 3.9 Super-critical energy trapping diagrams, μ = 0.9

and

θ(0) = Δ(0) = K= G=

E1 (0) − E2 (0) − arcsin E1 (0) + E2 (0)

v2 (0) − arctan Ωu2 (0) 2 [E1 (0) + E2 (0)] Ω2 K(6α3 + 5Kα5 ) − cos Δ(0) cos θ(0) + cos2 θ(0) 32Ω 2

(3.72)

where 1 2 v1 (0) + Ω 2 u21 (0) 2 1 2 v2 (0) + Ω 2 u22 (0) E2 (0) = 2 E1 (0) =

Since the initial velocity of the ﬁrst oscillator is ﬁxed v1 (0) = 0 then the system initial state is determined by the three quantities K, θ(0), and Δ(0); see (3.71). Alternatively, one can specify u1 (0), u2 (0) and v2 (0) and then ﬁnd K, θ(0), and Δ(0) from (3.72).

3.4.2

Asymptotic of Dominants

As the amplitude of θ is getting closer to π/2 then the phase θ oscillations acquire nonsmooth temporal shapes. Expression (3.63), for instance, shows that, at amplitudes near π/2, the energy exchange phase will be close to the triangular wave shape with a relatively small wave-length. In this case,

72

3 Nonsmooth Processes as Asymptotic Limits

Duﬃng equation (3.69) seems to be not an adequate model. So assuming that μ is suﬃciently small, let us introduce action-angle variables as described earlier in reference [158] √ 2I + I 2 sin φ θ = arcsin 1+I √ (1 + I) 2I + I 2 cos φ θ = (3.73) 1 + (2I + I 2 ) cos2 φ Substituting (3.73) in (3.66) under the compatibility condition θ = dθ/dp, gives still exact equivalent of oscillator (3.66) dI I(2 + I) =μ sin 2φ dp (1 + I)2 dφ 2μ = 1+I − sin2 φ dp (1 + I)3

(3.74)

Note that the coordinate transformation (3.73) still would be valid for general case (3.61) although with diﬀerent to (3.74) result. In contrast to (3.66), system (3.74) is weakly nonlinear with a very simple solution at μ = 0. In [158], the direct averaging was applied to the right-hand side of (3.74) in order to obtain the ﬁrst-order solution. The idea of averaging can be also implemented as asymptotic integration of system (3.74) by means of the coordinate transformation J(2 + J) cos 2ψ + O(μ2 ) 2(1 + J)3 (J 2 + 2J − 2) φ = ψ−μ sin 2ψ + O(μ2 ) 4(1 + J)4 I = J −μ

(3.75)

Transformation (3.75) is obtained from the condition eliminating the fast phase ϕ from the terms of order μ on the right-hand side in such a way that the new system takes the form dJ = O(μ2 ) dp μ dψ = 1+J − + O(μ2 ) dp (1 + J)3 System (3.76) is easily integrated as follows

μ ψ = 1+J − p (1 + J)3 where J = const.

(3.76)

(3.77)

3.4 Nonlinear Beat Dynamics: The Standard Averaging Approach

73

Then the reversed chain of transformations back to (3.73) is applied. Solution (3.73) through (3.77) appears to have a good match with the corresponding numerical solution when μ << 1/2; see (3.70) for interpretation. The solution can work well even under condition (3.70), but the corresponding initial conditions must keep the oscillator out of the triple equilibria region. The oscillators’s motion within such region is better approximated by Duﬃng’s equation (3.69). On physical point of view, the applicability loss for solution (3.73) through (3.77) is due to the energy localization phenomenon, which is not captured by the above solution despite of its strong nonlinearity. Indeed, the term, which is responsible for occurring the triple equilibria is ignored in the leading-order approximation.

3.4.3

Necessary Condition of Energy Trapping

On physical point of view, necessary condition of localization is the presence of triple equilibrium positions of oscillator (3.66) within the basic interval −π/2 < θ < π/2, which is provided by condition (3.70). However, to guarantee the energy localization, the initial conditions must keep the oscillator within one of the two branches of the separatrix loop. Let us bring both of the above conditions to the explicit form. Necessary condition. For simplicity reason, let us consider the case of cubic nonlinearity and introduce two dimensionless parameters ν= and

3E0 α3 8Ω 4

ζ=

ΔE0 E0

(3.78)

2 (3.79)

where E0 = E1 (0) + E2 (0) is the total initial energy deﬁned by (3.54), and ΔE0 = E1 (0) − E2 (0), so that √ ν estimates the weight of nonlinearity in the system dynamics, whereas ζ is the initial energy disbalance per total initial energy E0 . Calculating the constants K and G from (3.72) and making algebraic manipulations, eventually brings the necessary condition of localization (3.70) to the form (2 − ζ)ν ζν < cos Δ(0) < √ (3.80) −√ 1−ζ 1−ζ Let us recall that condition (3.80) provides the presence of two stable equilibrium position of oscillator (3.66) near the origin (θ, dθ/dp) = (0, 0) which itself becomes unstable. However (3.80) does no guarantee that oscillator (3.66) is in the neighborhood of a stable equilibrium condition.

74

3 Nonsmooth Processes as Asymptotic Limits

3.4.4

Suﬃcient Condition of Energy Trapping

Now let us deﬁne that the energy localization takes place whenever the initial conditions keep oscillator (3.66) within one of the two separatrix loops surrounding the stable equilibrium points. A manifold of such initial conditions is obtained from ﬁrst integral of oscillator (3.66) as follows

dθ dp

2 + tan2 θ + μ cos 2θ < μ

(3.81)

Taking into account equation (3.58), dθ/dp = |G|−1 sin Δ, and calculating the left - hand side of (3.81) at p = 0, gives (3.82) ζ[cos Δ(0) − ν 1 − ζ]2 + sin2 Δ(0) < ζν 2 Note that both conditions (3.80) and (3.82) include the same set of parameters, such as the initial phase shift from the out-of-phase mode, Δ(0), the parameter characterizing the initial energy distinction between the oscillators, ζ, and the parameter characterizing the total energy and thus strength of nonlinearity, ν.

3.5

Transition from Normal to Local Modes

The transient mode localization phenomenon is considered in a mechanical model combined of a simply supported beam and transverse nonlinear springs with hardening characteristics. Two diﬀerent approaches to the model reduction, such as normal and local mode representations for the beam’s center line, are discussed. It is concluded that the local mode discretization brings advantages for the transient localization analysis. Based on the speciﬁc coordinate transformations and the idea of averaging, explicit equations describing the energy exchange between the local modes and the corresponding localization conditions are obtained. It was shown that when the energy is slowly pumped into the system then, at some point, the energy equipartition around the system suddenly breaks and one of the local modes becomes the dominant energy receiver. The phenomenon is interpreted in terms of the related phase-plane diagram which shows qualitative changes near the image of out-phase mode as the total energy of the system has reached its critical level. A simple closed form expression is obtained for the corresponding critical time estimate. The text below is an update of recent publication by the author [159].

3.6

System Description

The model under investigation represents a simply supported elastic beam of length l with two masses attached to the beam and connected to the base

3.6 System Description

75

Fig. 3.10 The mechanical model admitting both normal and local mode motions; all the springs have hardening restoring force characteristics.

by nonlinear springs; see Fig. 3.10. The corresponding diﬀerential equation of motion and boundary conditions are, respectively, ρA

∂4w ∂2w + EI 4 = f1 (t)δ(y − y1 ) + f2 (t)δ(y − y2 ) 2 ∂t ∂y

and w(t, y)|y=0,l = 0,

∂ 2 w(t, y) |y=0,l = 0 ∂y 2

(3.83)

(3.84)

where fi (t) = −f [w(t, yi )] − c

∂w(t, yi ) ∂ 2 w(t, yi ) ; i = 1, 2 −m ∂t ∂t2

(3.85)

are transverse forces applied to the beam from masses attached at the two points y = y1,2 . It will be assumed that the model is perfectly symmetric with respect to y = l/2 so that the springs are attached at points

76

3 Nonsmooth Processes as Asymptotic Limits

y1 = l/3 and y2 = 2l/3

(3.86)

Below we consider the case of the hardening restoring force characteristics of the springs and show that, under appropriate conditions, a slow energy in-ﬂow leads to the localization of vibration modes. As a result, the system energy is spontaneously shifted to either the left or right side of the beam - the symmetry break. The adiabatically ‘slow’ energy increase means that the energy source has a minor or no direct eﬀect on the mode shapes. For simulation purposes, such an energy in-ﬂow is provided by the assumption that the viscous damping coeﬃcient c is suﬃciently small and negative; the physical basis for such an assumption was discussed in [158], [157]. This remark, which is substantiated below by the corresponding numerical values of the parameters, is important to follow, otherwise the phenomenon, which is the focus of this paper, may not be developed. In contrast, the dissipation (c > 0) can lead to a spontaneous dynamic transition from local to normal modes, when the total energy reaches its sub-critical level. Note that the presence of Dirac δ-functions in equation (3.83) requires a generalized interpretation of the diﬀerential equation of motion in terms of distributions [166]. The corresponding compliance is provided by further model reduction based on the Bubnov-Galerkin approach, which actually switches from the point-wise to integral interpretation of equations.

3.7

Normal and Local Mode Coordinates

Normal mode coordinates. Let us evaluate two possible ways to discretizing the model (3.83). In this paper, the reduced-order case of two degrees-offreedom is considered, when the conventional normal mode representation for the boundary value problem (3.83) - (3.84) is w(t, y) = W1 (t) sin

πy 2πy + W2 (t) sin l l

(3.87)

Substituting (3.87) in (3.83) and applying the standard Bubnov-Galerkin procedure, gives, after dropping the time arguments, ¨1 + ¯ ˙ 1 + λ2 W1 + F (W1 + W2 ) + F (W1 − W2 ) = 0 W ζW ¨2 + ¯ ˙ 2 + 16λ2 W2 + F (W1 + W2 ) − F (W1 − W2 ) = 0 W ζW where ζ¯ = and

3c π 4 EI , λ2 = 3 3m + Alρ l (3m + Alρ) √ √ 3 3 F (z) = f z 3m + Alρ 2

(3.88)

(3.89)

(3.90)

are constant parameters and a re-scaled restoring force function, respectively.

3.7 Normal and Local Mode Coordinates

77

Equations (3.88) are decoupled in the linear terms related to the elastic beam centre line, whereas the modal coupling is due to the spring nonlinearities included in F (z). Local mode coordinates. Alternatively, the model can be discretized by introducing the ‘local mode’ coordinates determined by the spring locations ui (t) = w(t, yi ); i = 1, 2

(3.91)

Taking into account (3.86) and (3.87), and substituting in (3.91) reveals simple links between the normal and local coordinates as √ √ 3 3 (W1 + W2 ), u2 = (W1 − W2 ) (3.92) u1 = 2 2 or, inversely,

√

√ 3 3 (u1 + u2 ), W2 = (u1 − u2 ) W1 = (3.93) 3 3 Substituting (3.93) in (3.87), gives the ‘local mode expansion’ for the beam’s centre line πy πy + u2 (t)ψ2 (3.94) w(t, y) = u1 (t)ψ1 l l where the local mode shape functions are

√

3 11 ψ1 (x) sin x = (3.95) ψ2 (x) sin 2x 3 1 −1

Both normal and local mode shape functions are shown in Figs. 3.11 and 3.12, respectively. Transformation (3.95) can be generalized for a greater number of modes. Note that functions (3.95) satisfy the following ortogonality condition π π ψi (x) ψj (x) dx = δij (3.96) 3 0 where δij is the Kronecker symbol. However, the diﬀerential equations of motion for u1 (t) and u2 (t) are obtained directly by substituting (3.93) in (3.88) and making obvious algebraic manipulations that gives √ √ u ¨1 + ζ¯u˙ 1 + (λ2 /2)(17u1 − 15u2 ) + 3F (2u1 / 3) = 0 (3.97) √ √ 2 u ¨2 + ζ¯u˙ 2 − (λ /2)(15u1 − 17u2 ) + 3F (2u2 / 3) = 0 This kind of discretization seems to be similar to that given by the ﬁnite element approaches.

78

3 Nonsmooth Processes as Asymptotic Limits

1.0 sin x 0.5 sin 2 x 0.0 0.5 1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Fig. 3.11 Normal mode shape functions; here and below dashed lines correspond to second mode.

1.0 2 x

1 x 0.5 0.0 0.5 1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Fig. 3.12 Local mode shape functions.

Further, equations (3.97) are represented as a set of ﬁrst order equations u˙ 1 = v1 u˙ 2 = v2 v˙ 1 = −ω 2 u1 + ε[ω 2 u2 − ζv1 − p(u1 )] v˙ 2 = −ω 2 u2 + ε[ω 2 u1 − ζv2 − p(u2 )] where ω=

17 15 2k + λ2 , k = F (0), ε = 2 2

2 ¯ζ λ , ζ= ω ε

(3.98)

3.7 Normal and Local Mode Coordinates

and

79

√ √ √ 2 3 3 2 3 F ui − kui = βu3i p(ui ) = ε 3 3

are new constant parameters and the nonlinear component of the spring characteristic; it is assumed that the damping coeﬃcient and the nonlinear component are small enough to provide the order of magnitude ζ = O(1) and p(ui ) = O(1). For calculation purposes, the spring characteristic is taken in the form F (u) = u + (4/3)u3 which brings the nonlinearity parameter β to the form β=

64ω 2 135λ2

(3.99)

Equations (3.97), and analogously (3.98), possess advantages for transient analysis because the corresponding linearized system has the same natural frequencies and the nonlinear components are decoupled. As a result, the one-frequency perturbation tool becomes applicable. The corresponding amplitude-phase variables are introduced as follows ui = αi (t) cos[ωt + δi (t)] vi = −ωαi (t) sin[ωt + δi (t)]

(3.100)

(i = 1, 2) Substituting (3.100) in (3.98) and applying the averaging procedure with respect to the fast phase z = ωt, gives ε α˙ 1 = − [ζα1 + ωα2 sin(δ1 − δ2 )] 2 ε α˙ 2 = − [ζα2 − ωα1 sin(δ1 − δ2 )] 2 εω α2 3εβ 2 α cos(δ1 − δ2 ) + δ˙ 1 = − 2 α1 8ω 1 εω α1 3εβ 2 α cos(δ1 − δ2 ) + δ˙ 2 = − 2 α2 8ω 2

(3.101)

The result of the work of [157] as well as further analysis show that the localization may occur as the system vibrates in the out-of-phase mode, u1 (t) ≡ −u2 (t). In order to investigate the dynamics near the out-of-phase vibration mode, let us introduce three new variables s, ρ and θ, α1 = −s(t) + ρ(t) α2 = s(t) + ρ(t) δ2 = δ1 + Δ(t)

(3.102)

80

3 Nonsmooth Processes as Asymptotic Limits

where s and ρ characterize the amplitudes of the out-of-phase and in-phase modes, respectively, and Δ is a phase shift between the local modes so that the variables ρ and Δ describe small deviations from the out-of-phase mode. Substituting (3.102) in (3.101), linearizing the result with respect to ρ and Δ and then eliminating the phase variable Δ, gives

1 2 3 2 2 2 (3.103) ρ ¨ + εζ ρ˙ + ε ω + ζ − βs ρ = 0 4 4 whereas the equation obtained for s gives the solution

1 s = s0 exp − εζt 2

(3.104)

In the case |ζ| 1, equation (3.103) describes an oscillator with a slow varying frequency. Making the frequency ‘frozen’ enables one of the determining roots of the corresponding ‘characteristic equation’ 1 3 2 2 k1,2 = ε − ζ ± i ω − βs (3.105) 2 4 If the viscosity is negative, ζ < 0, then equations (3.103) through (3.105) qualitatively describe the transition to the local mode as the system energy increases. In particular, expression (3.105) shows that when the amplitude of the out-phase mode, which is associated with s, becomes large enough then the amplitude of the in-phase mode, ρ, looses its oscillatory character and grows monotonically. As a result, one of the local mode increases its amplitude, whereas another one decays; see expressions (3.102). This is an onset of the dynamic transition to a localized mode. The corresponding critical time follows from explicit solution of (3.104) and (3.105) t∗ =

4ω 2 1 ln ε|ζ| 3βs20

(3.106)

In order to provide numerical evidence for the dynamic transition from normal to local mode vibrations, let us introduce an indicator of the energy partition calculated as ⎧ ⎨ −1 if E1 = 0 and E2 = 0 E1 − E2 (3.107) = 0 if E1 = E2 P = ⎩ E1 + E2 1 if E1 = 0 and E2 = 0 where Ei = (vi2 + ω 2 u2i )/2 is the total energy of i-th oscillator under the condition ε = 0.

3.8 Local Mode Interaction Dynamics

81

Quantity (3.107) is varying within the interval −1 ≤ P ≤ 1. The ends of the interval obviously correspond to the local modes, whereas its center P = 0 corresponds to the normal modes. The time history of the energy partition (3.107) is illustrated by Fig. 3.13. The following parameters were taken for numerical simulations: λ = 0.05, ¯ ζ = −0.002, k = 1.0, ω = 2k + (17/2)λ2 = 1.4217, β = 32/(9ε) = 383.29, ¯ = −0.2156. The initial normal mode ε = 15(λ/ω)2 /2, and therefore ζ = ζ/ε amplitudes at zero velocities are W1 (0) = 0.0001 and W2 (0) = −0.003. The critical time estimate based on expression (3.106) t∗ = 3474.29 is in quite a good match with Fig. 3.13.

0.0 0.2 P

0.4 0.6 0.8 1.0

t 0

1000

2000

3000

4000

5000

t

Fig. 3.13 ‘Sudden’ transition from normal to local mode vibration as the system energy has reached its critical value.

3.8

Local Mode Interaction Dynamics

Let us introduce new variables, K, θ and Δ, as follows

1 π α1 = K(t) cos θ(t) + 2 4

π 1 α2 = K(t) sin θ(t) + 2 4 Δ = δ2 − δ1 + π

(3.108) (3.109)

Further, considering the local mode total energies Ei under no interaction condition, and taking into account (3.100), (3.108) and (3.110), gives Ei =

1 2 (v + ω 2 u2i ); i = 1, 2 2 i

(3.110)

82

3 Nonsmooth Processes as Asymptotic Limits

1 2 2 1 ω (α1 + α22 ) = ω 2 K 2 2 1 1 ΔE = E1 − E2 = ω 2 (α21 − α22 ) = − ω 2 K sin θ 2 2 E = E1 + E2 =

(3.111) (3.112)

The variable K therefore is proportional to the total energy of the degenerated system, whereas the phase angle θ characterizes the energy partition between the local modes (3.107) as follows P =

ΔE = − sin θ E

(3.113)

The third variable (3.109) describes the phase shift in the high-frequency vibrations between the local modes so that Δ = 0 corresponds to the outphase motions of the masses attached to the beam; note the diﬀerence with (3.102). Diﬀerentiating (3.109), (3.111) and (3.112), and enforcing equations (3.101), gives dκ ζ =− κ dt1 ω dθ = sin Δ dt1 dΔ = − cos Δ tan θ + κ sin θ dt1

(3.114)

where t1 = εωt is a new temporal argument, and κ=

3β K 8ω 2

(3.115)

In the conservative case, ζ = 0, the ﬁrst equation in (3.114) gives the energy integral κ = const, whereas another two equations admit the integral 1 G = − cos Δ cos θ + κ cos 2θ = const 4

(3.116)

This particular case matches the results obtained for a linearly coupled set of Duﬃng’s oscillators in [101], and later reproduced in [158] however by means of diﬀerent complex variable approaches. In particular, it was shown in [158] that the last two equations in (3.114) are equivalent to a strongly nonlinear conservative oscillator d2 θ tan θ =0 (3.117) + G2 dt21 cos2 θ as κ → 0.

3.8 Local Mode Interaction Dynamics

83

Oscillator (3.117) appears to be exactly solvable with general solution θ = arcsin[sin θ0 sin(|G|t1 / cos θ0 )]

(3.118)

where θ0 is the amplitude and another arbitrary constant can be introduced as a temporal shift. As already mentioned in this chapter, oscillator (3.117) was considered in [78] and [122] as a phenomenological model for quite diﬀerent kinds of problems. Since no direct physical meaning of such a unique ‘restoring force characteristic’ was found, the fact of exact solvability not attracted much attention for quite a long time. In the case κ = 0, but still ζ = 0, some perturbation occurs on the righthand side of (3.117); the corresponding perturbation tool based on the actionangle variables was introduced in [158]. Let us show now that equations (3.114) can describe the transition to local modes under the assumption of small negative viscosity |ζ/ω| 1 and ζ < 0

(3.119)

Under condition (3.119), the factor κ in the third equation of (3.114) can be viewed as a slowly growing quasi constant. In this case, making κ ‘frozen’ and linearizing the last two equations in (3.114) near the equilibrium (θ, Δ) = (0, 0), gives d2 θ + (1 − κ)θ = 0 (3.120) dt21 Note that small θ and Δ bring the original system close to the out-of-phase vibration mode as follows from (3.113) and (3.109). When the growing energy parameter κ passes the critical point κ = 1, the type of equilibrium is changing from a focus to a saddle point and thus the variable θ becomes exponentially growing. Practically, however, the exponential growth will be suppressed by the nonlinearity. As a result two limit phase trajectories (separatrix loops) occur around two new stable equilibrium points. These two points represent two new stable modes of the original system - local modes. The phase plane diagrams for sub- and super-critical energy levels are shown in Figs. 3.14 and 3.15, respectively. Note that the equilibrium subjected to such qualitative change corresponds to the out-of-phase vibration mode of the model, whereas another two equilibrium points, (θ, Δ) = (0, ±π), correspond to the same in-phase mode and remain stable. Therefore, out-of-phase vibrations appear to be less favorable to energy equipartition as the energy reaches its critical level. Note that links between localization and speciﬁcs of phase trajectories was discussed also in [101] based on the system of two coupled Duﬃng oscillators. In particular, the limit phase trajectories were interpreted as nonlinear beats of inﬁnitely long period, keeping the energy near one of the two oscillators.

84

3 Nonsmooth Processes as Asymptotic Limits

4 I 2

O

0

2 I 4 1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ Fig. 3.14 Phase plane structure at undercritical system energy, κ = 0.5.

4 I 2

L

0

L

2 I 4 1.5

1.0

0.5

0.0

0.5

1.0

1.5

Θ Fig. 3.15 Phase plane structure at postcritical system energy, κ = 1.5.

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

85

As follows from expression (3.113), the growth of θ increases the energy unbalance between the local modes, and that is onset of the mode localization. The corresponding critical time, at which the localization begins, is obtained from (3.115) as follows 8ω 2 1 3β ∗ ∗ ln K exp (ε |ζ| t ) = 1 =⇒ t = 0 8ω 2 ε |ζ| 3βK0

(3.121)

As follows from (3.102) and (3.111), K0 = 2(s20 +ρ20 ) therefore, under the condition ρ20 1, expressions (3.106) and (3.121) give the same result. Note that the developed analytical approach, describing the local mode interaction in terms of the energy and phase variables, appears to be independent of the individual features of the illustrating model and this can be used in other similar cases. Compared to publications [101] and [158] introducing the same set of descriptive variables, K, θ and Δ, current approach has distinctive features as follows: 1) Instead of general mass-spring models, the elastic beam supported by nonlinear springs is considered in this work. This provides clear geometrical interpretations for both the normal and local modes through the corresponding shape functions of the beam centre line. 2) Instead of using a quite complicated system reduction in terms of complex coordinates, it is shown that the same result can be achieved by means of the traditional set of amplitude-phase variables and one-frequency averaging procedure. 3) The non-conservative case is considered in order to describe qualitative changes in the dynamics as the total energy of the system adiabatically increases or decreases. Based on such a generalization, new quantitative and qualitative results are obtained. In particular, explicit expressions have been obtained for the critical time at which onset of the localization occurs. The phenomenon is explained in terms of the related phase-plane diagram subjected to a qualitative change (center-saddle transition) as the total energy of the system reaches its critical level.

3.9

Auto-localized Modes in Nonlinear Coupled Oscillators

Below, the term ‘auto-localized’ means that the system itself may come into the nonlinear local mode regime and stay there regardless initial energy distribution among its particles. As follows from the Poincare’s recurrence theorem, such phenomena are rather impossible within the class of conservative systems [9]. However, interactions between the system particles can be designed in speciﬁc ways in order to achieve desired phenomena. It is assumed

86

3 Nonsmooth Processes as Asymptotic Limits

that such a design can be implemented practically by using speciﬁc electric circuits and possibly mechanical actuators. On macro-levels, the autolocalization may help to optimize vibration suppression. Some results from the previous publication [155] are reproduced below after some notation modiﬁcations in order to make the description coherent with the current text. Let us consider an array of N harmonic oscillators such that each of the oscillators interacts with only the nearest neighbors. The corresponding diﬀerential equations of motion are of the form x ¨j + Ω 2 xj = β(xj−1 − 2xj + xj+1 )+ + α[(Ej − Ej−1 )Ej−1 − (Ej+1 − Ej )Ej+1 ]x˙ j

(3.122)

1 2 (x˙ + Ω 2 x2j ); j = 1, ..., N (3.123) 2 j where Ej = Ej (t) is the total energy of the j-th oscillator under the boundary conditions of ﬁxed ends E0 (t) ≡ EN +1 (t) ≡ 0, and Ω, β, and α are constant parameters of the model. On the right-hand side of equation (3.122), two groups of terms describe coupling between the oscillators. If α = 0 then the only linear coupling remains. In this case, under special initial conditions, N diﬀerent coherent periodic motions i.e. linear normal modes, can take place. It is well known that any other motion is combined of the linear normal mode motions, whereas the energy is conserved on each of the modes the way it was initially distributed between the modes. In other words, no energy localization is possible on individual particles if α = 0. Another group of terms, including the common factor α, has the opposite to linear elastic interaction eﬀect. These nonlinear terms are to simulate possible ‘competition’ between the oscillators, in other words, one-way energy ﬂow to the neighbor whose energy is lager. Such kind of interaction dominates when the total system energy is large enough to essentially involve high degrees of the coordinates and velocities. For future analysis let us introduce the complex conjugate variables {Aj (t) , A¯j (t)} into equations (3.122) according to relationships (3.37) and (3.38). In term of the complex amplitudes, the total energy of individual oscillator (3.123), excluding the energy of coupling, takes the form Ej =

Ej =

1 2 ¯ 1 2 Ω Aj Aj = Ω 2 |Aj | 2 2

(3.124)

When β = α = 0 the system is decomposed into the N uncoupled oscillators, and one has a constant solution in the new variables. In general case, substituting (3.37) in (3.122), taking into account (3.38), and applying averaging with respect to the phase z = Ωt, gives the following set of equations (the complex conjugate set is omitted below)

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

87

iβ A˙ j = − (Aj−1 − 2Aj + Aj+1 )+ 2Ω αΩ 4 2 2 2 2 2 2 + |Aj | − |Aj−1 | |Aj−1 | − |Aj+1 | − |Aj | |Aj+1 | Aj (3.125) 8 Let us consider ﬁrst, the simplest model of two coupled oscillators (N = 2). In this case, system (3.125) is reduced to iβ αΩ 4 A˙ 1 = − (A2 − 2A1 ) + |A1 |2 − |A2 |2 |A2 |2 A1 2Ω 8 4 iβ αΩ A˙ 2 = − (A1 − 2A2 ) + |A2 |2 − |A1 |2 |A1 |2 A2 2Ω 8

(3.126)

Despite of the presence velocities x˙ j in the original equations (3.122), system (3.126) still has the integral 2

2

K = |A1 | + |A2 | = 2(E1 + E2 )/Ω 2 = const. As a result, the dimension of system’ phase space is reduced by introducing the angular variables ϕ1 (t), ϕ2 (t) and ψ(t), √ √ A2 = K sin ψ exp(iϕ2 ) (3.127) A1 = K cos ψ exp(iϕ1 ), where the angle ψ determines the energy distribution between the oscillators as follows E2 |A2 | = (3.128) tan ψ = |A1 | E1 0 ≤ ψ < π/2 Substituting (3.127) into (3.126) and considering separately real and imaginary parts, gives β β − tan ψ cos (ϕ2 − ϕ1 ) Ω 2Ω β β − cot ψ cos (ϕ2 − ϕ1 ) ϕ˙ 2 = Ω 2Ω 1 β sin (ϕ2 − ϕ1 ) − αK 2 Ω 4 sin 4ψ ψ˙ = − 2Ω 32

ϕ˙ 1 =

(3.129)

Introducing the phase shift Δ = ϕ2 −ϕ1 and new temporal variable p = Ωt/β, gives dΔ = − cot 2ψ cos Δ dp dψ 1 = − (sin Δ + λ sin 4ψ) dp 2

(3.130)

88

3 Nonsmooth Processes as Asymptotic Limits Π

2

Π

4

0 Π

2

0

Π

2

Fig. 3.16 Low energy transition to the nonsmooth limit cycle; numerical solution obtained for the following system parameter and initial conditions: λ = 0.2; Δ(0) = 0.0, Ψ (0) = π/4 + 0.1.

where λ is a dimensionless parameter linked with the total energy of both oscillators as follows λ=

Ωα αK 2 Ω 5 = (E1 + E2 )2 16β 4β

(3.131)

System (3.130) is periodic with respect to both phase coordinates Δ and ψ, as a result, its phase plane has periodic cell-wise structure. Let us consider just one cell, π π! π (3.132) R0 = − < Δ < , 0 < Ψ < 2 2 2 including the equilibrium (critical) point (Δ, ψ) = (0, π/4)

(3.133)

As follows from (3.127) and (3.128), at point (3.133), both oscillators vibrate in-phase with the same energy, E1 = E2 . Linearized (near (3.133)) system (3.130) has the following couple of roots of characteristic equation (3.134) r1,2 = λ ± i 1 − λ2

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

89

Π

2

Π

4

0 0

25 p

50

0

25 p

50

Π

2

0

Π

2

Fig. 3.17 Low energy transition to the “impact” limit cycle of phase variables at λ = 0.2.

Expression (3.134) determines the ‘low energy’ interval 0 < λ < 1 with a qualitatively similar system behavior. Equilibrium point (3.133) is unstable for positive λ while no other equilibrium points exist within the rectangular (3.132). As a result, the system trajectory is eventually attracted to the boundary of rectangular R0 (3.132) as shown in Figs. 3.16 and 3.18. This is a periodic limit cycle whose period is found in a closed form, π/2 P =2 0

dψ −2 1 − λ sin 4ψ

0

π/2

2π dψ = √ 1 + λ sin 4ψ 1 − λ2

(3.135)

where the horizontal pieces of the boundary ∂R0 have zero contribution as those passed momentarily by the system (3.130). This is conﬁrmed also by the diagrams in Figs. 3.17 and 3.19 showing step-wise jumps of the variable Δ(p) in steady state limits.

90

3 Nonsmooth Processes as Asymptotic Limits Π

2

Π

4

0 Π

2

0

Π

2

Fig. 3.18 Transition to the nonsmooth limit cycle under the energy level approaching its critical value; the numerical solution obtained for the following system parameter and initial conditions: λ = 0.8; Δ(0) = 0.0, Ψ (0) = π/4 + 0.1

Expression (3.135) shows that P → ∞ as λ → 1. The inﬁnity long period means that there is only one-way energy ﬂow in the system, in other words, the energy is eventually be localized on one of the oscillators. The corresponding total critical energy value is determined by substituting λ = 1 in (3.131). This gives β (3.136) = E∗ E1 + E2 = 2 Ωα If E1 + E2 < E ∗ then periodic energy exchange with the period T = βP/Ω takes place, but no localization is possible. Therefore, in order to be localized on one of the oscillators, the total system energy must be large enough. Interestingly enough, the transition to localized mode of this model happens through non-smooth limit cycle along which the dynamics of phase variables, Ψ and Δ, resembles the behavior of coordinate and velocity of impact oscillator2; see Figs. 3.17 and 3.19.

2

As already mentioned, the possibility of ‘vibro-impact dynamics’ of phase variables was noticed later in [101] when considering another model of nonlinear beats.

3.9 Auto-localized Modes in Nonlinear Coupled Oscillators

91

Π

2

Π

4

0 0

15 p

30

0

15 p

30

Π

2

0

Π

2

Fig. 3.19 Transition to the “impact” limit cycle of phase variables at λ = 0.8.

Chapter 4

Nonsmooth Temporal Transformations (NSTT)

Abstract. In this chapter, diﬀerent versions of nonsmooth argument substitutions, speciﬁcally - nonsmooth time, are introduced with proofs of the related identities. Basic rules for algebraic, diﬀerential and integral manipulations are described. In particular, ﬁnal subsections show how to implement nonsmooth argument substitutions in the diﬀerential equations. These impose two principal features on the dynamical systems by generating speciﬁc algebraic structures and switching formulations to boundary-value problems. Notice that the transformation itself imply no constraints on dynamical systems and easily applies to both smooth and nonsmooth systems. Any further steps, however, should account for physical properties of the related systems. Indeed, linear coordinate transformations can signiﬁcantly simplify the linear dynamic problems. Weakly nonlinear coordinate transformations often play the major role in the quasi-linear theory. Further, dynamical systems with discontinuities can be simpliﬁed by means of appropriate non-smooth transformations of variables.

4.1

Non-smooth Time Transformations

Major features induced by nonsmooth temporal substitutions can be brieﬂy listed as follows: • Introducing non-smooth temporal variables, in particular triangular sine wave, brings the coordinates into the algebra of hyperbolic numbers1 ; • Under appropriate conditions, diﬀerentiation or integration of the coordinates keeps the result within the same algebra and therefore eases the corresponding manipulations with the dynamic systems; • Explicit time argument can be used together with the nonsmooth time in order to describe amplitude and/or frequency modulated processes. 1

As mentioned in Introduction, - complex numbers X + Y e, where e2 = 1; see details in the next subsections.

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 93–129, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

94

4 Nonsmooth Temporal Transformations (NSTT)

Notice that the transformation itself is a preliminary stage of analysis ﬁnalized by speciﬁc boundary value problems on standard intervals. Then, appropriate methods must be applied to the boundary value problems according to the related physical content.

4.1.1

Positive Time

To begin with a simple illustration of the non-smooth positive time, consider the series

1 1 t t (4.1) ln 2 cosh = + exp (−t) − exp (−2t) + exp (−3t) − ... 2 2 2 3 which is convergent for t ≥ 0 but obviously becomes divergent if t < 0; the convergence is conditional at t = 0. Nevertheless, replacing t −→ |t|, makes this series convergent for any t as follows

t |t| ln 2 cosh ≡ ln 2 cosh 2 2 1 1 |t| + exp (− |t|) − exp (−2 |t|) + exp (−3 |t|) − ... (4.2) = 2 2 3 An ‘side eﬀect’ of such manipulation is that ﬁnite sums of the series are losing diﬀerentiability at t = 0, whereas the original function is smooth everywhere. However, it will be shown later that the series can be rearranged in such a manner that any truncated series becomes diﬀerentiable at t = 0 as many times as needed. Note that the transformation t −→ |t| is non-invertible. As a result, the manipulation illustrated by example (4.2) may not work for other cases. In the above example though, the substitution t −→ |t| holds for both positive and negative time t due to evenness of the original function. In order to extend the above idea on the general case, let us represent the time argument in the form . (4.3) t = |t| |t| . 2

Now, taking into account the relationship (|t| ) = 1, gives sequentially t2n = |t|2n

and t2n+1 = (|t|)2n+1 |t|. , n = 1, 2, ...

(4.4)

Example 1. Combining identities (4.4) and the power series expansion of the exponential function, gives |t|3 |t|2 . exp (t) ≡ exp (|t| |t| ) = 1 + + ... + |t| + + ... |t|. 2! 3!

4.1 Non-smooth Time Transformations

95

or exp (t) = cosh (|t|) + sinh (|t|) |t|.

(4.5)

Expression (4.5) represents the exponential function as a sum of the even and odd components. Let us take a general function, x (t), and assume that .

x (t) = x (|t| |t| ) = X (|t|) + Y (|t|) |t|

.

(4.6)

This gives the following set of equations for the components, X (|t|) and Y (|t|), x (|t|) = X (|t|) + Y (|t|) x (− |t|) = X (|t|) − Y (|t|)

.

for |t| = 1 . for |t| = −1

and, ﬁnally, 1 [x (|t|) + x (− |t|)] 2 1 Y (|t|) = [x (|t|) − x (− |t|)] 2

(4.7)

X (|t|) =

Therefore, identity (4.6) holds under condition (4.7). It is seen that the righthand side of expression (4.6) represents an element of the algebra with the . basis {1, |t| }. The corresponding ‘table of products’ is generated by the re. 2 lationship (|t| ) = 1. This leads to other useful algebraic relationships, for instance, . X + Y |t| = 0 ⇐⇒ {X = 0, Y = 0} and .

.

f (X + Y |t| ) = Rf (X, Y ) + If (X, Y ) |t|

(4.8)

where 1 [f (X + Y ) + f (X − Y )] 2 1 If (X, Y ) = [f (X + Y ) − f (X − Y )] 2

Rf (X, Y ) =

(4.9)

Therefore, introducing the nonsmooth positive time |t|, imposes speciﬁc complexiﬁcation on the system coordinate x (t) −→ {X (|t|) , Y (|t|)}

(4.10)

As shown in diﬀerent sections through this chapter, such complexiﬁcations always accompany non-smooth argument substitutions.

96

4.1.2

4 Nonsmooth Temporal Transformations (NSTT)

‘Single-Tooth’ Substitution

A a simple generalization of the positive time, |t|, let us introduce the temporal shift, say a, as follows s = s (t) = |t − a|

(4.11)

Obviously, s˙ = −1 for t − a < 0 and s˙ = 1 for t − a > 0 , and therefore s˙ 2 = 1 for, at least, almost all t. Proposition 1. A general function x (t) can be represented in the form x (t) = X (s) + Y (s) s˙

(4.12)

where s = s (t) is given by (4.11). Proof. By analogy to (4.3), t = a + ss˙

(4.13)

Substituting (4.13) in (4.12), gives x (a − s) = X (s) − Y (s)

for s˙ = −1

x (a + s) = X (s) + Y (s)

for s˙ = 1

and thus, 1 [x (a + s) + x (a − s)] 2 1 Y (s) = [x (a + s) − x (a − s)] 2

X (s) =

(4.14)

Therefore, identity (4.12) holds for any x (t) under condition (4.14).

4.1.3

‘Broken Time’ Substitution

Another generalization of the positive time |t|, is given by the piece-wise linear function v1 t for t ≤ 0 (4.15) τ= v2 t for t ≥ 0 where v1 = v2 . The inverse relationship can be represented in the form t = A (τ ) + B (τ ) τ˙

where A=

1 1 + v1 v2

τ

and B = −

(4.16) 1 τ v1 v2

4.1 Non-smooth Time Transformations

97

and τ˙ 2 = −v1 v2 + (v1 + v2 ) τ˙

(4.17)

Further, applying some function x to both sides of equality (4.16), gives x (t) = X (τ ) + Y (τ ) τ˙

(4.18)

where both components X and Y are determined analogously to (4.6) through (4.7) in the form

1 τ τ X (τ ) = v2 x − v1 x v2 − v1 v1 v2

τ τ 1 x −x (4.19) Y (τ ) = − v2 − v1 v1 v2 Now, assuming that another function f is applied to both sides of (4.18), gives (4.20) f (x) = Rf (X, Y ) + If (X, Y ) τ˙ where 1 [v2 f (X + Y v1 ) − v1 f (X + Y v2 )] v2 − v1 1 If = − [f (X + Y v1 ) − f (X + Y v2 )] v2 − v1

Rf =

(4.21)

Comparing the right-hand sides of expressions (4.16), (4.18) and (4.20), shows that the algebraic structure generated by the nonsmooth time substitutions is preserved after diﬀerent functional manipulations with the corresponding elements of the algebra.

4.1.4

Sawtooth Sine Transformation

Let us consider the periodic version of nonsmooth time (4.11) which is based on the sawtooth sine - triangular wave function t for − 1 ≤ t ≤ 1 ∀t (4.22) τ (t) = , τ (t) = τ (4 + t) −t + 2 for 1 ≤ t ≤ 3 Recall that function (4.22) describes position of a bead oscillating between two perfectly stiﬀ parallel barriers with no energy loss. The distance between the barriers is two spatial units, and the velocity vector has the unit length and is normal to the barrier planes. In other words, function (4.22) describes the motion of the standard impact oscillator; see Chapter 1 for illustrations. This function can be also expressed in the closed form by means of the trigonometric functions as

98

4 Nonsmooth Temporal Transformations (NSTT)

τ (t) = (2/π)arcsin[sin(πt/2)]

(4.23)

The period is normalized to four in order to provide the unit slope, τ˙ 2 = 1

(4.24)

Proposition 2. [133] Any periodic process, whose period is normalized to T = 4, is expressed through the sawtooth sine τ (t) and the rectangular cosine τ˙ (t) in the form x (t) = X (τ ) + Y (τ ) τ˙ (4.25) where the X- and Y -components are given by 1 [x (τ ) + x (2 − τ )] 2 1 Y (τ ) = [x (τ ) − x (2 − τ )] 2

X (τ ) =

(4.26)

In other words, any periodic process is uniquely expressed through the state variables, such as the coordinate τ and velocity τ˙ , of the standard impact oscillator. Proof [144]. It can be veriﬁed by inspection that t = 1 + (τ − 1) τ˙

if − 1 < t < 3

(4.27)

Note that the function τ˙ (t) has a step-wise discontinuity at t = 1, which is suppressed however by the continuous factor, τ (t) − 1, of zero value at t = 1. Based on this remark, property (4.24) will be considered as true everywhere on the interval −1 < t < 3, since τ˙ is either explicitly or implicitly accompanied by the factor τ (t) − 1 whenever it appears in algebraic manipulations. For instance, identity 2

t2 = 1 + (τ − 1) + 2 (τ − 1) τ˙ holds everywhere on the interval −1 < t < 3. Further, applying the method of mathematical induction, gives tn = An (τ ) + Bn (τ )τ˙

(4.28)

where n is any positive integer, An and Bn are polynomials of the degree n or n − 1. Expression (4.28) shows that any analytic function x(t), that admits the power series expansion on the interval −1 < t < 3, can be represented in the form x(t) = x [1 + (τ − 1) τ˙ ] = X (τ ) + Y (τ ) τ˙ (4.29) where X (τ ) and Y (τ ) are power series of τ .

4.1 Non-smooth Time Transformations

99

Now, let us assume that expression (4.29) holds even though the function x(t) is not analytic. In this case, one must show that the X- and Y components can be determined with no power series expansions. Indeed, since either τ˙ = 1 or τ˙ = −1 on the entire interval −1 < t < 3, except may be the point t = 1, then expression (4.29) gives two equations, x (τ ) = X (τ ) + Y (τ ) and x (2 − τ ) = X (τ ) − Y (τ ). Solving these equations for X and Y and substituting the solution in (4.29), gives the identity, x(t) =

1 1 [x (τ ) + x (2 − τ )] + [x (τ ) − x (2 − τ )] τ˙ 2 2

(4.30)

which obviously holds on the interval −1 < t < 3, except may be t = 1. If the function x(t) is continuous at t = 1 then identity (4.30) is also true at t = 1 because the step-wise discontinuity of the rectangular cosine τ˙ (t) at t = 1 is suppressed by the factor 2Y (τ ) = x (τ ) − x (2 − τ ) → x (1 − 0) − x (1 + 0) → 0

(4.31)

as t → 1 ± 0; see deﬁnition (4.22). Now if the function x(t) is periodic of the period T = 4 then identity (4.30) holds almost everywhere on the interval −∞ < t < ∞, because the right-hand side of (4.30) depends on the time argument t through the pair of periodic functions τ (t) and τ˙ (t) of the same period T = 4. Finally, if the function x(t) is continuous also at t = −1, and thus t = 3 due to the periodicity, then identity (4.30) is true for every t. So let us consider the point t = −1 at which the function τ˙ (t) has the step-wise discontinuity. Taking into account that τ (−1 ± 0) = −1 + 0 and the periodicity condition x(t) = x(t − 4), gives 2Y (τ ) = x[τ (−1 ± 0)] − x[2 − τ (−1 ± 0)] (4.32) = x (−1 + 0) − x (3 − 0) = x (−1 + 0) − x (−1 − 0) → 0 as t → −1 ± 0 due to continuity of the function x(t) at t = −1. Therefore, the step-wise discontinuity of the function τ˙ (t) at the point t = −1 is suppressed as well due to (4.32). It is proved that if x(t) is continuous then identity (4.30) or (4.25) holds everywhere on the interval −∞ < t < ∞. The periodic version of conditions (4.31) and (4.32) is Y |τ =±1 = 0 (4.33) Remark 1. If the function x(t) is step-wise discontinuous at the instances Λ={t : τ (t) = ±1} then limits (4.31) and (4.32) are non-zero. In this case, the discontinuities of the function τ˙ (t) are not suppressed but describe the real behavior of the original function x(t).

100

4 Nonsmooth Temporal Transformations (NSTT)

Proposition 3. Elements (4.25) belong to the algebra of hyperbolic numbers due to (4.24). As a result, f (X + Y τ˙ ) = Rf + If τ˙

(4.34)

1 [f (X + Y ) + f (X − Y )] 2 1 If = [f (X + Y ) − f (X − Y )] 2

Rf =

provided that f (X ± Y ) are deﬁned. Relationship (4.34) can be easily veriﬁed by setting sequentially τ˙ = 1 and τ˙ = −1. The right-hand side of expression (4.25) therefore admits interpretation as a speciﬁc (hyperbolic) ‘complex number’ with ‘real’ X and ‘imaginary’ Y components. The corresponding ‘imaginary unity’ τ˙ creates the cycling group so that τ˙ 2 = 1, τ˙ 3 = τ˙ , τ˙ 4 = 1,... . This remark signiﬁcantly simpliﬁes all mathematical manipulations with representation (4.25). For example, 2

(X + Y τ˙ ) = X 2 + Y 2 + 2XY τ˙ 3 (X + Y τ˙ ) = X 3 + 3XY 2 + Y 3 + 3Y X 2 τ˙ The right-hand sides of the above expressions appear to have the same hyperbolic structure of the element X + Y τ˙ . Another example resembles Euler formula in the complex analysis, however hyperbolic functions are involved as follows exp (X + Y τ˙ ) = exp (X) [cosh (Y ) + sinh (Y ) τ˙ ]

(4.35)

Further, by taking into account that τ˙ 2 = 1, one obtains X 1 Y = 2 − 2 τ˙ 2 X + Y τ˙ X −Y X −Y2

if X = ±Y

This example explains why division is not always possible, and this is essential diﬀerence with the conventional complex analysis. Very often, the triangular wave function is supposed to depend on some phase variable, say ϕ(t). In all such cases, the ‘imaginary element’ is given by the derivative with respect to the phase variable τ (ϕ). In order to avoid confusion, let us introduce a new notation for the rectangular cosine of the normalized period and amplitude as follows e(ϕ) = τ (ϕ). Now the basic relation (4.36) e2 = 1 holds regardless argument of the sawtooth sine. Example 2. Introduce the sawtooth time into the functions sin t and cos t.

4.1 Non-smooth Time Transformations

101

Solution. In order to deal with the period normalized to four, let us introduce new argument ϕ = 2t/π as

π π 2t ϕ sin t = sin = sin 2 π 2 Now, applying representation (4.25) to the new argument ϕ , gives π π ! π 1 sin τ + sin (2 − τ ) = sin τ 2 2 2 2 ! 1 π π Y (τ ) = sin τ − sin (2 − τ ) = 0 2 2 2

X (τ ) =

and therefore

π sin t ≡ sin τ 2

Analogously,

π cos t = cos τ 2

2t π

2t π

2t e π

Example 3. Combining the results from previous example with the Euler formula for complex exponential functions, gives

kπ kπ τ i + cos τ e for k = 1, 3, 5, ... exp (ikt) = sin 2 2

kπ kπ exp (ikt) = cos τ + sin τ ie for k = 0, 2, 4, ... 2 2 where τ and e still depend on the same argument 2t/π. The right-hand sides of these equalities can be viewed as elements of a more complicated algebra, z = α + βe + γi + δei, with the basis elements {1, e, i, ei}. The corresponding table of products is given by × 1 e i ei

4.1.5

1 1 e i ei

e e 1 ei i

i i ei −1 −e

ei ei i −e −1

Links between NSTT and Matrix Algebras

Hyperbolic numbers represent a very simple example of so-called Cliﬀord geometric algebras which is isomorphic to a matrix algebra. For instance, it is known [105] that complex numbers associate with skew-symmetric 2 × 2 matrixes with equal diagonal entries, whereas the hyperbolic numbers correspond to the symmetric matrixes as follows

102

4 Nonsmooth Temporal Transformations (NSTT)

a + ib ←→ X + eY ←→

a b −b a XY Y X

The above correspondences are isomorphisms because both summations and multiplications with the numbers associate with the corresponding matrix operations, for instance (X1 + eY1 )(X2 + eY2 ) = (X1 X2 + Y1 Y2 ) + (X1 Y2 + Y1 X2 )e

X1 Y1 X2 Y2 X1 X2 + Y1 Y2 X1 Y2 + Y1 X2 = Y1 X1 Y2 X2 X1 Y2 + Y1 X2 X1 X2 + Y1 Y2

(4.37) (4.38)

As was already mentioned, the algebra of hyperbolic numbers and Cliﬀord algebras were developed in an abstract way as manipulations with speciﬁc numbers with no relation to any nonsmooth transformations. However, the uncovered links may appear to be useful for conducting automatic symbolic manipulations with computers. Indeed, most of the corresponding packages have built-in tools to handling the matrix operations whereas deﬁning the operations with hyperbolic numbers may require some programming work.

4.1.6

Diﬀerentiation and Integration Rules

From the point of view of applications to the diﬀerential equations, it is essential that, under some conditions, diﬀerential operations also preserve the hyperbolic structure of elements [133]. For example, x˙ = Y + X τ˙

+ Y τ¨

(4.39)

where primes mean derivatives with respect to τ . The term underlined in (4.39) should be ignored whenever the function x(t) is continuous and therefore condition (4.33) is satisﬁed. Indeed, the derivative τ¨ represents the periodic system of pulses, acting at those points Λ where the factor Y is equal to zero due to (4.33), τ¨ = 2

∞

[δ(t + 1 − 4k) − δ(t − 1 − 4k)]

k=−∞

In a similar manner, one can sequentially consider high-order derivatives. For example, the second derivative is given by x ¨ = X + Y τ˙

(4.40)

X |τ =±1 = 0

(4.41)

under the boundary condition

4.1 Non-smooth Time Transformations

103

Further, an arbitrary odd derivative is given by x(2k−1) (t) = Y (2k−1) (τ ) + X (2k−1) (τ ) e

(4.42)

provided that Y (τ ) |τ =±1 = 0 X (τ ) |τ =±1 = 0 Y

(2k−2)

(τ ) |τ =±1

... =0

(4.43)

Also an arbitrary even derivative is x(2k) (t) = X (2k) (τ ) + Y (2k) (τ ) e

(4.44)

if Y (τ ) |τ =±1 = 0 X (τ ) |τ =±1 = 0 ... X

(2k−1)

(4.45)

(τ ) |τ =±1 = 0

Finally, integration also gives the hyperbolic ‘number’ (X + Y τ˙ )dt = Q + P τ˙ whose components are Q(τ ) =

(4.46)

τ

Y dτ + C

and

τ

P (τ ) =

Xdτ −1

0

where C is an arbitrary constant, and the following condition must be satisﬁed 1 X(τ )dτ = 0

(4.47)

−1

Relationship (4.46) can be easily veriﬁed by diﬀerentiation with respect to t. The role of condition (4.47) is to provide zero mean value of the integrand in (4.46).

4.1.7

NSTT Averaging

Lemma 1. Let x(t) be a general periodic function of the period T = 4a so that presentation (4.25) holds

104

4 Nonsmooth Temporal Transformations (NSTT)

x(t) = X(τ (φ)) + Y (τ (φ))e(φ)

(4.48)

where φ = t/a, e(φ) = τ (φ) and 1 [x (aτ ) + x (2a − aτ )] 2 1 Y (τ ) = [x (aτ ) − x (2a − aτ )] 2

X (τ ) =

(4.49)

Then the mean value of x(t) over its period is 1 T

T

0

1 x(t)dt = 2

1

X(τ )dτ

(4.50)

−1

In other words, the ‘imaginary’ component Y e of the ‘hyperbolic number’ (4.48) gives zero contribution into the mean value. Proof. With no loss of generality, let us assume that a = 1 and thus T = 4. Then, during one period −1 < t < 3, τ (t) = e(t) = 1 and dt = dτ for − 1 < t < 1 τ (t) = e(t) = −1 and dt = −dτ for 1 < t < 3 Taking into account (4.48), gives 1 T =

1 4

0

T

1 x(t)dt = 4

1

−1

[X(τ ) + Y (τ )]dτ −

1 4

1

x(t)dt + −1

3

x(t)dt 1

1

−1

[X(τ ) − Y (τ )]dτ =

1 2

1

X(τ )dτ −1

Example 4. Let x(t) be a periodic function of the general period T . Taking into account (4.48) and the above remark, gives x2 = (X + Y e)2 = X 2 + Y 2 + 2XY e Then, based on the above Lemma, < x2 >≡

1 T

T

x2 dt = 0

1 2

1

(X 2 + Y 2 )dτ

(4.51)

−1

Example 5. Consider the case x(t) = A sin t + B cos t, where T = 2π and a = π/2. In this case, relationships (4.56) give X(τ ) = A sin(πτ /2) and Y (τ ) = A cos(πτ /2). Therefore, (4.51) gives < x2 >= (A2 + B 2 )/2.

4.1 Non-smooth Time Transformations

4.1.8

105

Generalizations on Asymmetrical Sawtooth Wave

It has been shown in [138] and [149] that an arbitrary periodic function x (t) whose period is normalized to four can still be represented in the form (4.25), even though the triangular wave τ - saw-tooth sine - has asymmetrical slopes such that t/ (1 − γ) for − 1 + γ ≤ t ≤ 1 − γ τ (t, γ) = (4.52) (−t + 2) / (1 + γ) for 1 − γ ≤ t ≤ 3 + γ ∂τ (t, γ) = e (t, γ) ∂t

τ˙ (t, γ) =

(4.53)

where γ ∈ (−1, 1) is a parameter characterizing the inclination of the saw as shown in Fig. 4.1. Τ 1 Γ

1

2

3

4

5

6

7

8

5

6

7

8

t

1 e 1 Γ1

1

1

1 Γ

1

2

3

4

t

Fig. 4.1 Asymmetric saw-tooth sine and rectangular cosine (γ = 0.5).

In this case, the inverse transformation of time on the period has the form

1 1 1 − γ2 +e +[2−(1+γ) τ ] −e (4.54) t= [(1−γ) τ ] 2 1+γ 1−γ (−1 + γ ≤ t ≤ 3 + γ) By using this expression, one can show that x (t) = X [τ (t, γ)] + Y [τ (t, γ)] e (t, γ)

(4.55)

106

4 Nonsmooth Temporal Transformations (NSTT)

1 1 1 X= x [(1 − γ) τ ] + x [2 − (1 + γ) τ ] 2α 1 + γ 1−γ 1 {x [(1 − γ) τ ] − x [2 − (1 + γ) τ ]} Y = 2α

(4.56)

e2 = α + βe

(4.57) where α = 1/ 1 − γ 2 and β = 2γα. As compared to (4.36), relationship (4.57) somewhat complicates algebraic and diﬀerential operations so that, respectively, f (X + Y e) = Rf (X, Y ) + If (X, Y ) e

(4.58)

1 1 1 f (Z+ ) + f (Z− ) Rf (X, Y ) = 2α 1 + γ 1−γ 1 [f (Z+ ) − f (Z− )] If (X, Y ) = 2α Z± = X ±

Y 1∓γ

and x˙ = αY + (X + βY ) e + Y

∂e ∂t

(4.59)

where f (Z+ ) and f (Z− ) are deﬁned, primes indicate diﬀerentiation with respect to τ , and ∞ ∂e = 2α [δ (t + 1 − γ − 4k) − δ (t − 1 + γ − 4k)] ∂t

(4.60)

k=−∞

If the function x (t) is continuous at time instances Λ = {t : τ (t, γ) = ±1} then the singular term ∂e/∂t is suppressed by the condition Y |t∈Λ = Y |τ =±1 = 0 analogously to the symmetric case. Otherwise, the periodic singular term remains in expression (4.59). It is convenient to calculate high-order derivatives by means of the matrixoperator D acting on the vector-column (X, Y )T sequentially as follows

0α d D= 1 β dτ

X αY D = (4.61) Y X + βY

X αX + αβY D2 = βX + (α + β 2 )Y Y

4.1 Non-smooth Time Transformations

107

Therefore, applying D and D2 , gives x˙ = αY + (X + βY )e x ¨ = αX + αβY + [βX + (α + β 2 )Y ]e under the conditions Y |τ =±1 = 0

(X + βY )|τ =±1 = 0 Note that rules (4.61) are still valid in the symmetric case γ = 0, when the diﬀerentiation matrix operator takes the form

01 d D= 1 0 dτ Finally, the result of integration (4.46) has the components Q=

Y (τ ) −

β X (τ ) dτ α

and P =

1 α

τ X (ξ) dξ −1

under the zero-mean value condition (4.47).

4.1.9

Multiple Frequency Case

Practical applications of NSTT to the class of multiple frequency motions face problems similar by its nature to those caused small denominators in quasilinear approaches. However, formal generalizations of the basic identities are quite simple as illustrated below. Let Pϕ and Nϕ be operators acting on some periodic function x(ϕ) of the period T = 4 as follows 1 {x[τ (ϕ)] + x[2 − τ (ϕ)]} ≡ X[τ (ϕ)] 2 1 Nϕx (ϕ) = {x[τ (ϕ)] − x[2 − τ (ϕ)]} ≡ Y [τ (ϕ)] 2 Pϕx (ϕ) =

(4.62)

where functions X (τ ) and Y (τ ) are deﬁned according to (4.26). In such notations, representation (4.25) takes the form x(ϕ) = (Pϕ + eNϕ)x(ϕ)

(4.63)

where e (ϕ) = τ (ϕ). Let us consider now a function of multiple arguments x = x (ϕ1 , ..., ϕn ) of the period T = 4 with respect to each of its n arguments. This function describes a multiple frequency quasi-periodic process if ϕ1 = ω1 t,..., ϕn = ωn t

108

4 Nonsmooth Temporal Transformations (NSTT)

with a set of positive incommensurable numbers ω1 ,..., ωn . In this case, transformation (4.63) is independently applicable to each of the n arguments as follows n (Pϕj + ej Nϕj )x (ϕ1 , ..., ϕn ) (4.64) x (ϕ1 , ..., ϕn ) = j=1

where new notations τj = τ (ϕj ) and ej = τ (ϕj ) are introduced. Note that all the operators included in the product are commutating as those applied to diﬀerent arguments of the function. Moreover, the number of co-factors of the product can diﬀer from one to n. Let us consider the case of two arguments in details. Introducing the notations e0 ≡ 1 and e3 = e1 e2 , and taking the product as shown in (4.64), gives x (ϕ1 , ϕ2 ) = X (τ1 , τ2 ) e0 + Y (τ1 , τ2 ) e1 + Z (τ1 , τ2 ) e2 + W (τ1 , τ2 ) e3 (4.65) where X = Pϕ1 Pϕ2 x (ϕ1 , ϕ2 ) Y = Nϕ1 Pϕ2 x (ϕ1 , ϕ2 ) Z = Pϕ1 Nϕ2 x (ϕ1 , ϕ2 ) W = Nϕ1 Nϕ2 x (ϕ1 , ϕ2 ) The basis of elements (4.65) obeys the table of products × e0 e1 e2 e3

e0 e0 e1 e2 e3

e1 e1 e0 e3 e2

e2 e2 e3 e0 e1

e3 e3 e2 e1 e0

(4.66)

Further, applying some function f to the element (4.65), gives f (Xe0 + Y e1 + Ze2 + W e3 ) = Rf e0 + If1 e1 + If2 e2 + If3 e3

(4.67)

where the coeﬃcients at the right-hand side are deﬁned by the system of linear equations Rf + If1 + If2 + If3 = f (X + Y + Z + W ) Rf + If1 − If2 − If3 = f (X + Y − Z − W ) Rf − If1 + If2 − If3 = f (X − Y + Z − W ) Rf −

If1

−

If2

+

If3

= f (X − Y − Z + W )

(4.68)

4.2 Idempotent Basis Generated by the Triangular Sine-Wave

109

Equations (4.68) are obtained by substituting diﬀerent combinations of e1 = ±1 and e3 = ±1 in (4.67). Some application of the multiple frequency case will be illustrated at the end of Chapter 5 and also in Chapter 14.

4.2

Idempotent Basis Generated by the Triangular Sine-Wave

4.2.1

Deﬁnitions and Algebraic Rules

In addition to the standard basis {1, e}, the hyperbolic plane has another natural basis {e+ , e− }associated with the two isotropic lines separating the hyperbolic quadrants as described in Chapter 1. The transition from one basis to another is given by (see Fig. 4.2) 1 (1 + e) 2 1 e− = (1 − e) 2

e+ =

(4.69)

or 1 = e+ + e− e = e+ − e−

(4.70)

Therefore, x = X + Y e = X(e+ + e− ) + Y (e+ − e− ) = (X + Y )e+ + (X − Y )e− where x = x(t) is any periodic function whose period is normalized to T = 4, and therefore e = e(t). Taking into account (4.26), gives x = X+ (τ )e+ + X− (τ )e−

(4.71)

where τ = τ (t), e+ = e+ (t), e− = e− (t), and X+ = X + Y = x(τ ) X− = X − Y = x(2 − τ )

(4.72)

The elements e+ and e− are mutually annihilating, and they are called idempotents because e+ e− = 0 e2− = e− e2+

= e+

(4.73)

110

4 Nonsmooth Temporal Transformations (NSTT)

Fig. 4.2 The standard basis and idempotent basis (on the left and right, respectively.)

As follows from (4.70), ee+ = e+ and ee− = −e− . Obviously, the matrix representation of (4.71) is

X+ 0 X+ e+ + X− e− ←→ 0 X− Properties (4.73) make the idempotent basis very convenient to use in diﬀerent calculations. For instance, for any real number α, α α (X+ e+ + X− e− )α = X+ e+ + X− e−

(4.74)

One can extend relationship (4.74) on a general function f as follows f (X+ e+ + X− e− ) = f (X+ )e+ + f (X− )e−

(4.75)

provided that f (X+ ) and f (X− ) are deﬁned. As seen from (4.75), any function is acting as linear if applied to a hyperbolic element represented in the idempotent basis. As a result, the major advantage of using the idempotent basis is that the diﬀerential equations of motion in terms of the components X+ (τ ) and X− (τ ) appear to be decoupled regardless nonlinearity. However, the corresponding boundary value problems still remain coupled because the idempotent basis makes

4.2 Idempotent Basis Generated by the Triangular Sine-Wave

111

unfortunately boundary conditions coupled. The nature of this fact is explained in the next subsection. The time variable t can be represented in the idempotent basis during one period of a periodic process, say T = 4. This can be shown by substituting (4.70) in (4.27), where τ˙ (t) = e(t), and then conducting some manipulations as follows t = 1 + (τ − 1) e = (e+ + e− ) + (τ − 1) (e+ − e− ) (−1 < t < 3) = τ e+ + (2 − τ )e−

(4.76)

Further, using the basis’ properties (4.73), gives, for instance, t2 = τ 2 e+ + (2 − τ )2 e− tα = τ α e+ + (2 − τ )α e− (α > 0)

(4.77)

Note that expressions (4.77) provide the direct proof of representation (4.71), at least, in the class of analytical periodic functions x(t).

4.2.2

Time Derivatives in the Idempotent Basis

Taking into account deﬁnitions (4.69) and (4.70), gives ﬁrst time derivatives of the basis elements 1 (4.78) e˙ + = −e˙ − = e˙ 2 where ∞ e˙ + = [δ(t + 1 − 4k) − δ(t − 1 − 4k)] k=−∞

Therefore, diﬀerentiating (4.71) gives x˙ = X+ ee+ + X− ee− + X+ e˙ + + X− e˙ − 1 = X+ e+ − X− e− + (X+ − X− )e˙ 2

Assuming the continuity condition

gives

(X+ − X− )|τ =±1 = 0

(4.79)

e+ − X− e− x˙ = X+

(4.80)

Analogously, assuming the continuity condition for x(t), ˙ + X− )|τ =±1 = 0 (X+

gives then

e+ + X− e− x¨ = X+

(4.81)

112

4 Nonsmooth Temporal Transformations (NSTT)

Example 6. Suppose the equation x˙ = f (x), where x(t) ∈ Rn , has a family of periodic solutions of the period T = 4a. Introducing the sawtooth time τ = τ (t/a) and using the idempotent basis, gives e+ − X− e− ) (X+

1 = f (X+ e+ + X− e− ) = f (X+ )e+ + f (X− )e− a

or X+ = af (X+ ) X− = −af (X− )

(4.82)

under condition (4.79). As mentioned above, the sawtooth time substitution implemented in the idempotent basis leads to the decoupled set of equations such as (4.82), whereas boundary condition (4.79) is coupled. Note that it is practically suﬃcient to solve only ﬁrst equation of system (4.82). Then solution of the second equation is obtained by making replacement τ → −τ in the ﬁrst solution.

4.3

Idempotent Basis Generated by Asymmetric Triangular Wave

4.3.1

Deﬁnition and Algebraic Properties

Let us introduce basic algebraic manipulations with the idempotent basis generated by the asymmetric triangular (sawtooth) wave (4.52). The standard basis is given by the set {1, e} and shown on the left in Fig. 4.3, where e = e(t, γ) is deﬁned by (4.53) periodic function with step-wise discontinuities at the points Λγ = {t : τ (t, γ) = ±1}. In this case, we introduce the idempotent basis as follows 1 [1 − γ + (1 − γ 2 )e] 2 1 e− = [1 + γ − (1 − γ 2 )e] 2 e+ =

(4.83)

or, inversely, 1 = e+ + e− 1 1 e= e+ − e− 1−γ 1+γ

(4.84)

where e+ = e+ (t, γ) and e− = e− (t, γ), and the parameter of asymmetry γ is included in order to normalize the range of change for the basis elements as 0 ≤ e+ ≤ 1 and 0 ≤ e− ≤ 1; see the diagrams on the left in Fig. 4.3.

4.3 Idempotent Basis Generated by Asymmetric Triangular Wave

113

Fig. 4.3 The standard basis and idempotent basis (on the left and right, respectively) generated by the asymmetric triangular wave with γ = 0.3.

Deﬁnition (4.83) brings (4.54) to the form t = (1 − γ) τ e+ + [2 − (1 + γ) τ ] e−

(4.85)

If γ = 0 then deﬁnition (4.83) becomes equivalent to (4.69). However, the elements e+ and e− are mutually annihilating, and create idempotents regardless the magnitude of γ so that table of products (4.73) still holds for any γ ∈ (−1, 1). Also, as follows from (4.73) and (4.84), 1 e+ 1−γ 1 e− ee− = − 1+γ ee+ =

Now, substituting (4.84) into identity (4.55), gives

1 1 x = X + Y e = X(e+ + e− ) + Y e+ − e− 1−γ 1+γ

1 1 = X+ Y e+ + X − Y e− 1−γ 1+γ

(4.86)

(4.87)

where x = x(t) is any periodic function whose period is normalized to T = 4.

114

4 Nonsmooth Temporal Transformations (NSTT)

Substituting (4.56) in (4.87) and conducting algebraic manipulations, gives x = X+ (τ, γ)e+ + X− (τ, γ)e−

(4.88)

where τ = τ (t, γ) is deﬁned above by (4.52), and X+ (τ, γ) = x((1 − γ)τ )

and

X− (τ, γ) = x(2 − (1 + γ)τ )

(4.89)

Obviously, basic algebraic properties (4.74) and (4.75) remain valid.

4.3.2

Diﬀerentiation Rules

Taking into account deﬁnitions (4.83) and (4.84), gives ﬁrst time derivatives of the basis elements ∂e+ ∂e− 1 ∂e =− = (1 − γ 2 ) ∂t ∂t 2 ∂t

(4.90)

where ∂e/∂t is a sequence of Dirac δ-functions taking eﬀect whenever t ∈ Λγ. Taking into account that ∂τ /∂t = e and diﬀerentiating (4.88), gives x˙ =

∂e+ ∂e− ∂X− ∂X+ ee+ + ee− + X+ + X− ∂τ ∂τ ∂t ∂t

(4.91)

Substituting (4.86) and (4.90) in (4.91), brings (4.91) to the form x˙ =

1 ∂X+ 1 ∂X− 1 ∂e e+ − e− + (1 − γ 2 )(X+ − X− ) 1 − γ ∂τ 1 + γ ∂τ 2 ∂t

(4.92)

Assuming the continuity of x(t), leads to boundary conditions (X+ − X− )|τ =±1 = 0

(4.93)

These conditions eliminate the singular term ∂e/∂t from expression (4.92), which, as a result, takes the form x˙ =

1 ∂X+ 1 ∂X− e+ − e− 1 − γ ∂τ 1 + γ ∂τ

(4.94)

As seen from (4.94), the algebraic structure of representation (4.88) is preserved after the diﬀerentiation. In other words, under continuity condition (4.93), the result of diﬀerentiation just leads to the following replacements in (4.88) 1 ∂X+ 1 ∂X− X+ → and X− → − (4.95) 1 − γ ∂τ 1 + γ ∂τ Property (4.95) enables one of writing down high derivatives iteratively by making substitutions (4.95) in (4.92). For instance, second derivative is given by

4.3 Idempotent Basis Generated by Asymmetric Triangular Wave

x ¨=

1 1 ∂ 2 X+ ∂ 2 X− e+ + e− 2 2 2 (1 − γ) ∂τ (1 + γ) ∂τ 2

1 ∂X+ 1 ∂X− ∂e 1 + + (1 − γ 2 ) 2 1 − γ ∂τ 1 + γ ∂τ ∂t

The singular term ∂e/∂t in (4.96) is eliminated by condition

1 ∂X+ 1 ∂X− + |τ =±1 = 0 1 − γ ∂τ 1 + γ ∂τ

115

(4.96)

(4.97)

As a result, the second derivative x¨ takes the form of expansion with respect to the basis {e+ , e− }. Such a process can be continued as soon as derivatives remain continuous in Λγ . Otherwise, time derivatives of the function e(t, γ) acquire non-formal meaning and must be present in the entire expression.

4.3.3

Oscillators in the Idempotent Basis

In order to illustrate the use of idempotent basis for vibration problems, consider oscillator x ¨ + f (x) = pe+ (t/a, γ) + qe− (t/a, γ)

(4.98)

where p and q are constant amplitudes of the external loading represented in the idempotent basis with the period T = 4a. No speciﬁc conditions are imposed on the characteristic f (x), however, it is assumed that oscillator (4.98) possesses periodic solution with the period of external loading, T . Such a solution admits the form (4.88), where τ = τ (t/a, γ). Due to the presence of time scaling factor, expressions for time derivatives derived in the previous subsection must be modiﬁed by the replacement ∂τ → a∂τ . Now, substituting (4.88) in (4.98), taking into account (4.93) through (4.97) and the basic algebraic property of idempotents, f (X+ e+ + X− e− ) = f (X+ )e+ + f (X− )e− , gives equations 1 ∂ 2 X+ + f (X+ ) = p (1 − γ)2 a2 ∂τ 2 ∂ 2 X− 1 + f (X− ) = q (1 + γ)2 a2 ∂τ 2

(4.99) (4.100)

under the boundary conditions (4.93) and (4.97). The obvious advantage of using the idempotent basis is that equations (4.99) and (4.100) are in a better match with the form of original equation (4.98), appear to be decoupled and therefore can be solved in a similar way. Although the entire boundary value problem is still coupled through the boundary conditions, the problem caused by coupling is eased, however,

116

4 Nonsmooth Temporal Transformations (NSTT)

since algorithms for solving equations are often more complicated than those applied to the boundary conditions.

4.3.4

Integration in the Idempotent Basis

Let x(t) be a periodic function of the period normalized to T = 4 with zero mean value on the period T . Then, integrating the right-hand side of (4.88), gives (4.101) [X+ (τ, γ)e+ + X− (τ, γ)e− ]dt = P+ (τ, γ)e+ + P− (τ, γ)e− where τ P+ (τ, γ) = (1 − γ)

X+ (z, γ)dz + C

−1

τ

P− (τ, γ) = −(1 + γ)

X− (z, γ)dz + C

(4.102)

−1

Note that both expressions in (4.102) have the same arbitrary constant of integration C. As a result, substituting (4.102) in (4.101) and taking into account (4.84), gives a single constant of integration as follows: Ce+ + Ce− = C(e+ + e− ) = C. Proof of (4.102) is obtained by taking time derivative of both sides of equality (4.101). Under the condition of continuity of the right-hand side of (4.101), the diﬀerentiation leads to the boundary value problem 1 ∂P+ (τ, γ) = X+ (τ, γ) 1−γ ∂τ 1 ∂P− (τ, γ) − = X− (τ, γ) 1+γ ∂τ (P+ − P− )|τ =±1 = 0

(4.103)

This boundary value problem has solution (4.102) under the following condition though 1 (1 − γ) −1

1 X+ (τ, γ)dτ + (1 + γ)

X− (τ, γ)dτ = 0

(4.104)

−1

The meaning of this condition is clariﬁed by substituting (4.89) in (4.104) and taking into account the standard properties of deﬁnite integrals as follows

4.4 Discussions, Remarks and Justiﬁcations

1

117

1 x[(1 − γ)τ ]d[(1 − γ)τ ] +

−1 (1−γ)

(1−γ)

x(z)dz −

=

x[2 − (1 + γ)τ ]d[(1 + γ)τ ] −1

−(1−γ)

3+γ

x(z)dz =

T x(z)dz =

−1+γ

2+(1+γ)

x(t)dt = 0 0

In other words, condition (4.104) requires zero mean value for x(t) on the period. Otherwise, the result of integration would appear to be out of the class of periodic functions by making the algebraic structure of right-hand side of (4.101) invalid.

4.4

Discussions, Remarks and Justiﬁcations

Let us show that the saw-tooth temporal argument τ (t) associates with the very general group properties of the conservative oscillator d2 x + f (x) = 0 dt2

(4.105)

Suppose that another oscillating time parameter, say g, is introduced as x(t) = X[g (t)], where g (t) is a periodic function, which is not necessarily sawtooth. This kind of substitution brings the diﬀerential equation of motion to the form

2 2 dg d X d2 g dX + f (X) = 0 (4.106) + dt dg 2 dt2 dg A side eﬀect of such substitution is that, generally speaking, the system has lost its original Newtonian form. In order to keep the diﬀerential equation of motion in its original Newtonian form, the following conditions must be imposed on equation (4.106)

dg dt

2 =1

and

d2 g dX =0 dt2 dg

(4.107)

Conditions (4.107) are satisﬁed by g =α+t

or g = β − t

(4.108)

where α and β are arbitrary constants. Relationships (4.108) simply represent the group of time transformations which is admitted by the original system (4.105). Let us recall however that the function g (t) was assumed to be periodic. Therefore, according to (4.108), g (t) must be a piece-wise linear periodic function. The simplest one-frequency case is given by the triangular wave, g (t) ≡ τ (t). In this case the ﬁrst equality of (4.107) is equivalent to the basic algebraic relationship (4.24) whereas the

118

4 Nonsmooth Temporal Transformations (NSTT)

second equality holds for all t under the condition (4.41). As a result, the new equation (4.106) takes the same form as (4.105). Therefore, the triangular wave time substitution τ (t) possesses the unique property among all periodic time substitutions, namely, it preserves the form of diﬀerential equations of conservative oscillators.

4.4.1

Remarks on Nonsmooth Solutions in the Classical Dynamics

Problems of classical dynamics are usually formulated in terms of second order diﬀerential equations of motion with respect to the system coordinates. Therefore, the corresponding solutions must be at least twice continuously diﬀerentiable functions of time. The existence and uniqueness theorem imposes also special conditions on the system characteristics and external forcing functions. As a result, using discontinuities or distributions for modeling dynamical systems takes the corresponding diﬀerential equations out of frames of the classic theory of diﬀerential equations. This requires additional examination of such formulations in order to insure that solutions actually exist. One of the NSTT roles is to bring models back into the area of classic theory. However, some preliminary analyses of correctness of original models is still required especially in non-linear cases. In this and next subsections, some details and examples are introduced on nonlinear formulations with discontinuities and δ-functions. For illustration purposes, let us consider a one-dimensional forced oscillator x ¨ + f (x, x) ˙ = q (t)

(4.109)

where the functions q (t), f (x, x) ˙ and ∂f (x, x) ˙ /∂ x˙ must be continuous according to the Cauchy theorem. Let the system be subjected to an impulsive external loading applied at t = t0 . In this case, the external forcing function can be expressed by the Dirac δ-function2 x ¨ + f (x, x) ˙ = Iδ (t − t0 ) (4.110) where I is a constant linear momentum per unit mass. The diﬀerential equation of motion (4.110) cannot be treated within the classic theory of diﬀerential equations. It is also impossible to ﬁnd a solution in the class of twice diﬀerentiable functions. However, a physically meaningful interpretation of equation (4.110) is suggested by the theory of distributions. For example, in terms of function-theoretic approaches, distributions are not classical functions but linear functionals acting on manifolds of appropriate diﬀerentiable functions {ϕ (t)}. Further, equalities are thought to be integral 2

The Dirac δ-function belongs to the class of so-called generalized functions introduced by Sergei Sobolev in 1935 and re-introduced by Laurent Schwartz in the late 1940s, who developed a theory of distributions.

4.4 Discussions, Remarks and Justiﬁcations

119

identities rather then point-wise relationships. Therefore, a ‘real meaning’ of equation (4.110) is given by the identity ∞

∞ δ (t − t0 ) ϕ (t) dt

[¨ x + f (x, x)] ˙ ϕ (t) dt = I −∞

(4.111)

−∞

which is supposed to be true for any ‘testing function’ ϕ (t). Taking into account the deﬁnition of δ-function and integrating by parts, gives ∞ [−x˙ ϕ˙ (t) + f (x, x) ˙ ϕ (t)] dt = Iϕ (t0 ) (4.112) −∞

On one hand, this expression allows to weaken the smoothness condition on the class of solutions x (t). On the other hand, the integral identity (4.111) appears to have quite clear physical meaning since its both sides represent works done on arbitrary virtual displacements ϕ (t). Note, that the solution x (t) itself is not considered as a functional unless the function f is linear. For example, let us consider the case f (x, x) ˙ = 2λx˙ +ω 2 x, where λ and ω are constant parameters. In this case, by proceeding with integration by parts in equation (4.112), one obtains ∞

ϕ ¨ (t) − 2λϕ˙ (t) + ω 2 ϕ (t) x (t) dt = Iϕ (t0 )

−∞

Therefore, x (t) generates a linear functional of the form

"∞

(...) x (t) dt and

−∞

thus can be qualiﬁed as a generalized solution3 . In nonlinear cases, the concept of generalized solution does not work, but equation (4.112) still holds and enables one to use the concept of ‘weak solution’ [46]. This concept is involved to justify manipulations with the nonsmooth functions at intermediate steps of the approach.

4.4.2

Caratheodory Equation

Since δ-function is deﬁned as a linear functional then nonlinear operations with δ-functions are generally ‘illegal.’ However, δ-functions still can participate in nonlinear diﬀerential equations in a linear way - as summands. 3

Sometimes the very presence of δ-functions in any nonlinear equation is rejected by the reason that nonlinearity is incompatible with the notion of linear functionals. In diﬀerential equations of motion, however, singular forces and the corresponding accelerations usually participate as summands, whereas velocities and coordinates include no δ-type singularities and thus can be subjected to nonlinear operations.

120

4 Nonsmooth Temporal Transformations (NSTT)

Such cases were examined in details by Caratheodory [45]. Let us consider the diﬀerential equation x˙ = f (t, x), where the right-hand side satisﬁes the Caratheodory conditions. Namely, in the domain D of the (t, x) space: the function f (t, x) is deﬁned and continuous with respect x for almost all t; the function f (t, x) is measurable in t for each x; and |f (t, x)| ≤ m (t), where function m (t) is summable. The above conditions are less restrictive than those required by the classical existence theorem namely the function f (t, x) is allowed to be step-wise discontinuous in t. Such an extension becomes possible if the right-hand side of the equivalent integral equation is calculated by Lebesgue, t f (ξ, x (ξ)) dξ x (t) − x (t0 ) = t0

Now, let us the right-hand side of the equation includes the δ-impulse as a summand x˙ = f (t, x) + νδ (t − a) (4.113) where a and v are constant parameters. In this case, the right-hand side does not satisfy any more the Caratheodory conditions. However, changing the variable x (t) −→ y (t) :

x (t) = y (t) + vH (t − a)

(4.114)

where H is the Heaviside’s unit-step function, brings the diﬀerential equation back into the form y˙ = f [t, y + vH (t − a)] ≡ F (t, y)

(4.115)

where the Dirac’s function has been eliminated due to the equality H˙ (t − a) = δ (t − a); as a result the right-hand side, F (t, y), satisﬁes the Caratheodory conditions. Note that, due to the presence of discontinuous function H (t − a), the right hand side of the transformed equation, F (t, y), is generally piecewise continuous in t, even though the function f (t, x) may be continuous. Relationships (4.113) through (4.115) admit direct extensions on the vector space so that many mechanical and physical models can be represented in the form (4.113). Example 7. Parametrically excited Duﬃng’s oscillator subjected to external impact at t = a is described by equation u ¨ + ω02 (1 + ε cos 2t) u − βu3 = 2pδ (t − a)

(4.116)

Equation (4.116) is transformed to (4.113) by introducing the vector-functions as follows

4.4 Discussions, Remarks and Justiﬁcations

121

u x1 0 = , ν= u˙ x2 2p

x2 f (t, x) = βx31 − ω02 (1 + ε cos 2t) x1 x=

Example 8. Center lines of linearly elastic beams resting on elastic foundations can be described by the dimensionless diﬀerential equation d4 W + γ (ξ) W = qδ (ξ − a) dξ 4

(4.117)

where W = W (ξ) is the center line coordinate, γ (ξ) is a variable stiﬀness of the foundation, q is a transverse force localized at ξ = a. Equation (4.117) is brought to the form (4.113) by considering t as a spatial coordinate ξ and introducing the matrixes ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ W 0 100 0 ⎢ dW/dξ ⎥ ⎢0 ⎥ ⎢0⎥ 0 1 0 ⎥ ⎥ x, ⎥ x=⎢ f (ξ, x) = ⎢ ν=⎢ ⎣ d2 W/dξ 2 ⎦ , ⎣0 ⎣0⎦ 0 0 1⎦ d3 W/dξ 3 −γ (ξ) 0 0 0 q In order to completely formulate the problem, the corresponding boundary conditions must be added. Let us show now that introducing the nonsmooth argument (4.11) eliminates the delta-pulse in equation (4.113) in such a way that no step-wise discontinuity occurs in the new equations. Indeed, substituting (4.12) and (4.13) into equation (4.113) and taking into account expression s¨ = 2δ (t − a), gives Y + Rf (s, X, Y ) + [X + If (s, X, Y )] s˙ + (2Y − v) δ (t − a) = 0

(4.118)

where expressions 1 [f (a + s, X + Y ) + f (a − s, X − Y )] 2 1 If = [f (a + s, X + Y ) − f (a − s, X − Y )] 2

Rf =

are obtained analogously to (4.34). Eliminating the delta-pulse in (4.118) and equating separately both components of the remaining hyperbolic element to zero, gives the boundary-value problem X + If (s, X, Y ) = 0 Y + Rf (s, X, Y ) = 0 v Y |s=0 = 2

(4.119)

122

4 Nonsmooth Temporal Transformations (NSTT)

Although equations (4.119) have a more complicated form as compared to (4.113), the step-wise discontinuous function H (t − a) is not involved any more, also it may become important that the new argument s is half-limited: 0 ≤ s < ∞.

4.4.3

Other Versions of Periodic Time Substitutions

In this subsection, it will be shown that the symmetric sawtooth sine generates the most simple algebraic structure among other possible versions of non-invertible time substitutions. Note that periodic temporal arguments have been considered in the literature for a long time. The main idea of these approaches is to make a new temporal argument limited and therefore expand class of usable algorithms. For example, the harmonic transformation of time in combination with the power series method was used by Ince [70] for investigation of periodic motions. In particular, by introducing the new variables τ = sin t and x (t) = X[τ (t)] (4.120) in the Mathieu equation, x ¨ + (a + b cos 2t) x = 0 one obtains the equation 1 − τ 2 Xτ2 − τ Xτ + a + b − 2bτ 2 X = 0 which admits periodic solutions in terms of the power series with respect to the new temporal argument |τ | ≤ 1 [195]. Transformation (4.120) with the power series methods were employed for non-linear vibrating systems as well [198], [110], [163]. Non-harmonic time transformations dealing with Jacobian functions can be also found in the literature [21]. As it is known, however, such transformations of time are restricted by special cases and cannot be applied to any periodic motion. From the mathematical point of view, it is caused by the fact that an inverse transformation does not exist on the whole period. In this section, diﬀerent versions of periodic time are introduced in such a way that the corresponding transformations are valid for any periodic motion. This is reached by a special complexiﬁcation of the coordinates. Let us start with a generalization of substitution (4.120). For the sake of compactness, notations τ = τ (ϕ) ≡ sin ϕ

and

e = e (ϕ) ≡ cos ϕ

(4.121)

will be used below, where e (ϕ) = τ (ϕ). Proposition 4. Any suﬃciently smooth periodic function x (ϕ) of the period T = 2π, can be represented in the form

4.4 Discussions, Remarks and Justiﬁcations

123

x (ϕ) = X[τ (ϕ)] + Y [τ (ϕ)]e (ϕ)

(4.122)

where X and Y are of the power series form with respect to τ . Proof. By collecting separately terms with odd and even wave numbers in the corresponding Fourier series, one obtains ∞

x (ϕ) =

A0 + [A2n cos 2nϕ + A2n−1 cos (2n − 1) ϕ] 2 n=1 +

∞

[B2n sin 2nϕ + B2n−1 sin (2n − 1) ϕ]

(4.123)

n=1

Then, introducing notations (4.121) into the tabulated expressions [52] (Formulas No. 1.332), gives n n cos 2nϕ = a2i τ 2i , cos[(2n − 1) ϕ] = a2i−1 τ 2i−2 e i=0

sin 2nϕ =

n

i=1

b2i−2 τ 2i−1

e,

sin[(2n − 1) ϕ] =

i=1

n

b2i−1 τ 2i−1(4.124)

i=1

where the coeﬃcients are listed in [52]. Substituting (4.124) in (4.123) and reordering the terms, completes the proof. As it is seen from identities (4.124), the second component in representation (4.122) is due to the odd cosine-terms and even sine-terms of the Fourier expansion. Note that combination (4.122) possesses algebraic properties similar to those generated by the sawtooth time substitution. Namely, diﬀerentiation, integration or any suﬃciently smooth function of representation (4.122) gives an element of the same two-component structure as (4.122). This is due to the fact that none of the listed above operations destroys periodicity of the function, and hence, identity (4.122) can be applied to the result of the operations as well. Practically, results of the operations are obtained by taking into account the trigonometric identities τ (ϕ) = e (ϕ) , e (ϕ) = −τ (ϕ)

(4.125)

e2 = 1 − τ 2

(4.126)

and Note also that the components of representation (4.122) apparently are linearly independent and thus the whole combination is zero if and only if its both components are zero.

124

4 Nonsmooth Temporal Transformations (NSTT)

In order to illustrate manipulations with (4.122), let us introduce the temporal argument τ = sin ωt into the Duﬃng oscillator with no linear stiﬀness term [186] (4.127) x¨ + ζ x˙ + x3 = F sin ωt Considering periodic solutions and taking into account (4.125) and (4.126) on every step of the transformation, gives 1 − τ 2 X − τ X ω 2 + ζ 1 − τ 2 Y − τ Y ω (4.128) +3 1 − τ 2 XY 2 + X 3 = F τ 1 − τ 2 Y − 3τ Y − Y ω 2 + ζX ω +3X 2 Y + 1 − τ 2 Y 3 = 0

(4.129)

The unknown functions X and Y must satisfy conditions of analytical continuation on the boundaries of the interval −1 ≤ τ ≤ 1. These conditions are obtained by substituting τ = ±1 in equations (4.128) and (4.129) as follows (4.130) −τ X ω 2 − τ ζY ω + X 3 − F τ |τ =±1 = 0

− (3τ Y + Y ) ω 2 + ζX ω + 3X 2 Y |τ =±1 = 0

(4.131)

The above system does not admit a family of solutions on which Y (τ ) ≡ 0 due to the damping, therefore transformation (4.120) is not valid in this case. Interestingly enough, the form of representation (4.122) remains the same in the case of triangular wave, although the basic algebraic operation (4.126) is diﬀerent. For the comparison reason, let us consider equation (4.127) with the forcing function F τ (t/a), where τ is assumed to be the triangular wave, i.e. sawtooth sine of the period 4a. Let us represent periodic solutions in the form (4.122), where the new temporal argument is the sawtooth sine (4.22) of the phase variable ϕ = t/a. Then, substituting (4.122) into equation (4.127) and considering the result as a two-component element of the algebra, one obtains the boundary value problem X a−2 + ζY a−1 + XY 2 + X 3 = F τ

(4.132)

Y a−2 + ζX a−1 + 3Y X 2 + Y 3 = 0

(4.133)

Y |τ =±1 = 0,

X |τ =±1 = 0

(4.134)

where the boundary conditions (4.134) stay for elimination of the periodic series of Dirac functions from the ﬁrst and second derivatives of the coordinate.

4.4 Discussions, Remarks and Justiﬁcations

4.4.4

125

General Case of Non-invertible Time and Its Physical Meaning

Let us consider now a general class of functions {τ (ϕ) , e (ϕ)} produced by the conservative oscillator x ¨ + Π (x) = 0, where Π (x) is the potential energy of the oscillator. In order to make the amplitude normalized to unity, let us normalize the potential energy and the phase variable as P (x) = Π (x) /Π (1) 2Π (1)t, respectively. As a result, the diﬀerential equation of and ϕ = motion and the energy integral take the form, respectively,

1 d2 x + P (x) = 0 dϕ2 2

and

dx dϕ

2 = 1 − P (x)

(4.135)

Let x = τ (ϕ) be the system coordinate determined implicitly from the energy integral τ(ϕ) ds =ϕ (4.136) 1 − P (s) 0

Then second expression from (4.135) gives e2 = 1 − P (τ ) where e (ϕ) = τ (ϕ)

and

1 e (ϕ) = − P (τ ) 2

(4.137)

(4.138)

Now let us formulate (without proof) Proposition 5. Any periodic function x (ϕ) whose period is normalized to 1 T =4 0

ds 1 − P (s)

can be represented in the form (4.122), where the functions τ (ϕ) and e (ϕ) are given by (4.136) and (4.138). For example, one can transform equation (4.127) based on the potential energy function P (x) = x2n . Then the boundary value problems (4.128) through (4.131) and (4.132) through (4.134) can be derived as particular cases n = 1 and n −→ ∞, respectively.

4.4.5

NSTT and Cnoidal Waves

There are other methodological sources of approximate solutions in terms of power series with respect to diﬀerent type of periodic functions. In case of oscillators with even nonlinearities, it was suggested to approximate solutions by power series of the elliptic Jacobi cn-function [36]

126

4 Nonsmooth Temporal Transformations (NSTT)

x(t) = A +

N

Bn cn2n (ωt + αn , k 2 )

(4.139)

n=1

where A and Bn are constants, ω is the frequency parameter, and k is the modulus. The reason for using series (4.139) is that the Jacobi function eﬀectively captures strongly unharmonic temporal shapes of periodic motions near separatrix loops. However, mathematical properties of Jacobi functions may require certain eﬀorts while manipulating with solutions especially when the solutions are involved into further perturbation procedures. Note the particular case of expression (4.139) N = 1 resembles the well known case of cnoidal wave [184]. x(t) = −

E(m) + dn2 [2K(m)t] K(m)

(4.140)

where K(m) and E(m) are compete elliptic integrals of ﬁrst kind and second kind, respectively, and m = k 2 is the parameter (Jacobi), the period is normalized to unity. If the parameter m is small enough, function (4.140) takes almost harmonic shapes, however, when m is getting larger then x(t) describes so-called cnoidal waves as shown in Fig. 4.4. From the physical point of view, function (4.140) with speciﬁc scaling factors exactly describes temporal behavior of the interaction force between particles of the Toda lattice f = a[exp(−br) − 1],

ab > 0

(4.141)

where r is the distance between adjacent particles, a and b arbitrary parameters. Periodic function (4.140) admits exact Fourier series expansions x(t) =

∞ l cos(2πlt) π2 [K(m)]2 sinh[πlK(1 − m)/K(m)]

(4.142)

l=1

Interestingly enough, function (4.140) can be also represented as a sum of localized waves (solitons), however, shifted one with respect other on the same interval x(t) =−

∞ π2 π + cosh−2 [π(t−l)K(m)/K(1−m)] 2K(m)K(1 − m) [2K(1 − m)]2 l=−∞

(4.143) Practical reasons for using either of series (4.142) or (4.143) is that terms of both series consist of elementary functions4 of time, while executing 4

Actually, exponential functions with real and imaginary exponents.

4.4 Discussions, Remarks and Justiﬁcations

127

Fig. 4.4 Transition from quasi-harmonic to cnoidal wave.

diﬀerent principles of approximation. Namely, each term of the Fourier series is carrying global information about the periodic process, whereas each term of series (4.143) provides just local description in some interval near the time point determined by the number of term, l. Although both series describe the process exactly from the theoretical standpoint, the only limited number of terms is possible to keep in calculations. As a result, the above mentioned diﬀerence in principles of approximation may become essential as discussed further in this subsection. Now, adapting transformation (4.71) to the case T = 1 and then applying the result to periodic function (4.143), gives xn,p (t) = −

π2 π(e+ + e− ) + 2K(m)K(1 − m) [2K(1 − m)]2

×{

n

cosh−2 [λ(m)(τ − 4l)]e+ +

l=−n

λ(m) =

πK(m) 4K(1 − m)

p+1 l=−p

(4.144) cosh−2 [λ(m)(τ − 2 + 4l)]e− }

128

4 Nonsmooth Temporal Transformations (NSTT)

Fig. 4.5 NSTT of the periodic cnoidal wave at diﬀerent Jacobi parameters m: 3-three terms truncation (n = 0, p = 0), 5-ﬁve terms truncation (n = 1, p = 0), and ∞ - exact expression.

where the inﬁnite limits of summation have been replaced by the integers n > 0 and p > 0; the oscillating triangular wave time, τ = τ (4t), and the idempotent basis e+ = [1 + e(4t)]/2 and e− = [1 − e(4t)]/2 are introduced; substitution 1 = e+ + e− has been made in order to emphasize that the entire expression (4.144) has the form of hyperbolic number represented in the idempotent basis.

4.4 Discussions, Remarks and Justiﬁcations

129

Obviously, x∞,∞ (t) = x(t), so that, in this case, (4.144) becomes exact equivalent to (4.143). However, convergence properties of series (4.144) are signiﬁcantly improved as compared to (4.143). Indeed, as time t is growing in (4.143), one must switch from one term to another to keep suﬃcient precision of approximation so that on the inﬁnite time interval, −∞ < t < ∞, the entire series (4.143) is needed. In contrast, the temporal argument of series (4.144) is always bounded by the standard interval −1 ≤ τ ≤ 1, whereas the periodicity of wave is captured by the basis functions rather than multiple terms of the series. As a result, terms of series (4.144) are exponentially decaying as the summation index increases. As follows from the diagrams in Fig. 4.5, for the range of Jacobi parameter m > 0.2 just ﬁve terms in (4.144) are enough to capture both quasi-harmonic and cnoidal wave shapes with a very good precision. For greater parameters m, when the wave becomes essentially cnoidal, only three terms provide quite a perfect match with the exact shape. As seen from the upper fragment in Fig. 4.5, the three term approximation gives essential discontinuous error for very small Jacobi parameter m = 0.08. However, in this almost harmonic range, it is more eﬀective to use few or even one term Fourier series (4.142) rather than series (4.143). Another essential advantage of series (4.144) is that, in the cnoidal range, diﬀerent algebraic manipulations with truncated series are essentially eased due to the idempotent properties.

Chapter 5

Sawtooth Power Series

Abstract. In this chapter, we introduce polynomials and power series expansions with respect to the triangular sine-wave. These can be used for approximations of periodic signals and unknown periodic solutions of dynamical systems. Such approximations may appear to be eﬀective in those cases when trigonometric series converge slowly due to step-wise discontinuities or spikes. Another reason for using polynomial expansions is that they are usually more convenient for algebraic manipulations. If the process under consideration is smooth then suﬃcient class of smoothness of approximations is achieved by imposing speciﬁc constraints on the coeﬃcients. Other equations for the coeﬃcients may appear either as a result of optimization procedures, that minimize the error of approximation, or as an outcome of iterative procedures dictated by the diﬀerential equations of motion. It is also shown in this chapter that using operators Lie associated with dynamical systems essentially facilitates construction of the periodic power series.

5.1

Manipulations with the Series

5.1.1

Smoothing Procedures

Consider a smooth periodic function x (t) of the period T = 4 represented in the form x (t) = X (τ )+Y (τ ) τ˙ , where τ = τ (t). Obviously, both components, X and Y , admit power series expansions with respect to the argument τ with no loss of periodicity in the original time t. However, keeping the ﬁnite number of terms in such series may violate the smoothness conditions at τ = ±1. Indeed, functions and their truncated series may behave diﬀerently near the boundaries τ = ±1. Therefore, let us introduce formal algorithms that can be applied to the truncated series in order to improve their smoothness properties. First, consider the expansion X (τ ) =

2N X (i) (0) i=0

i!

τ i + O τ 2N +1

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 131–144, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(5.1)

132

5 Sawtooth Power Series

Obviously, the power terms τ i and, generally speaking, ﬁnite sums in (5.1) are non-diﬀerentiable at the set of of points Λ = {t : τ (t) = ±1}. In contrast, the binomials τ i+2 τi − ; i = 1, 2, ... (5.2) φi (τ ) = i i+2 are twice diﬀerentiable with respect to t on Λ that can be veriﬁed by taking derivatives with respect to t. For instance, ﬁrst two generalized derivatives are dφi (τ ) = τ i−1 (1 − τ 2 )τ˙ (t) dt d2 φi (τ ) = (i − 1) τ i−2 − (i + 1) τ i dt2 where the term τ i−1 (1 − τ 2 )¨ τ has been removed from the second derivative due to the presence factor 1 − τ 2 that takes zero value whenever τ¨ = 0. It is seen therefore that functions (5.2) are twice continuously diﬀerentiable with respect to t even though each of the functions is combined of two nondiﬀerentiable terms. This is achieved by the speciﬁc choice for the coeﬃcients and signs in (5.2) such that the power terms have same jumps of slopes but with opposite signs. Now, considering (5.2) as equations with respect to diﬀerent powers of τ , gives, respectively odd and even powers, as τ 2N +3 τ = φ1 (τ ) + ... + φ2N +1 (τ ) + 2N + 3

τ 2N +3 τ 3 = 3 φ3 (τ ) + ... + φ2N +1 (τ ) + 2N + 3

2N +3 τ τ 2N +1 = (2N + 1) φ2N +1 (τ ) + 2N + 3

(5.3)

and τ = 2 φ2 (τ ) + ... + φ2N +2 (τ ) + 4 τ = 4 φ4 (τ ) + ... + φ2N +2 (τ ) +

τ 2N +3 2N = 2N φ2N (τ ) + τ 2N + 2 2

τ 2N +2 2N + 2 τ 2N +2 2N + 2

(5.4)

where N is an arbitrary positive integer. Substituting (5.3) and (5.4) in (5.1) and setting N → ∞, gives X (τ ) = X (0) +

∞ i (2k−1) X (0) i=1 k=1

X (2k) (0) φ2i−1 (τ ) + φ2i (τ ) (2k − 2)! (2k − 1)!

(5.5)

5.1 Manipulations with the Series

133

In contrast to (5.1), particular sums of series (5.5) are twice continuously diﬀerentiable functions of t. Since the Y -component usually appears with the step-wise discontinuous factor τ˙ (t), the related series must be re-organized in a somewhat diﬀerent way by taking into account the corresponding necessary condition of continuity: Y (±1) = 0. In this case, appropriate polynomials are designed as ψi (τ ) = τ i − τ i+2 ; i = 0, 1, 2, ...

(5.6)

These polynomials provide continuity for the term ψi [τ (t)]τ˙ (t) as well as its ﬁrst derivative d[ψi (τ )τ˙ ] = ψi (τ ) τ˙ 2 = iτ i−1 − (i + 2) τ i+1 dt where the term ψi (τ ) τ¨ has been eliminated due to the boundary conditions ψi (±1) = 0. However, second derivative appears to be a step-wise discontinuous function at every time t ∈ Λ, d2 [ψi (τ )τ˙ ] = i (i − 1) τ i−2 − (i + 2) (i + 1) τ i τ˙ 2 dt Therefore, particular sums of the re-built series Y (τ ) τ˙ =

∞ i (2k−1) Y (0) i=0 k=0

(2k − 1)!

ψ2i−1 (τ ) +

Y (2k) (0) ψ2i (τ ) τ˙ (2k)!

(5.7)

are less smooth than those obtained for the X-component (5.5). Combining (5.5) and (5.7) and considering an arbitrary period, T = 4a, gives x (t) = X (0) +

∞ i=1

KX (i)

τi τ i+2 − i i+2

+e

∞

KY (i) τ i − τ i+2

(5.8)

i=0

where τ =τ (t/a) and e =τ (t/a), and the coeﬃcients are calculated as follows KX (2i − s) =

i X (2k−s) (0) (2k − 1 − s)! k=1

KY (2i − s) =

i Y (2k−s) (0) k=0

(s = 0, 1)

(2k − s)!

(5.9)

134

5 Sawtooth Power Series

Example 9 ln 3 π 2 + ln (2 + cos t) = 2 12 +

τ2 τ4 − 2 4

ln 3 1 − τ2 + 2

+

ln 3 π 2 π4 − + 2 12 192

+

π2 π4 − 12 48

ln 3 π 2 − 2 12

τ4 τ6 − 4 6

+ ...

2 τ − τ4

4 τ − τ 6 + ... e

where τ = τ (2t/π) and e = τ (2t/π). Series (5.8) may represent periodic solutions of dynamical systems. Formal solutions are obtained by taking into account the diﬀerential equations of motion when calculating the derivatives in coeﬃcients (5.9). It can be done by means of the operator Lie as discussed in the next section. Finally, the above smoothness procedures can be repeated as many times as needed until necessary smoothness of the particular sums is achieved. For instance, the expressions (5.2) and (5.6) reveal next steps of the smoothing as φi (τ ) =

ϕi (τ ) =

τi τ i+2 − ∈ C2 (τ i ) |τ =1 (τ i+2 ) |τ =1 φi (τ ) (3) φi

(τ ) |τ =1

−

φi+2 (τ ) (3) φi+2 (τ ) |τ =1

∈ C4

(5.10)

and, ψi (τ ) τ˙ = τ i − τ i+2 τ˙ ∈ C 1 χi (τ ) τ˙ =

ψi (τ ) (2)

ψi (τ ) |τ =1

−

ψi+2 (τ ) (2)

ψi+2 (τ ) |τ =1

τ˙ ∈ C 3

(5.11)

respectively, where the symbol C indicates the class of smoothness in the interval −∞ < t < ∞. Second sets of equations in (5.10) and (5.11) have to be inverted for φi (τ ) and ψi (τ ) in a similar to (5.3) and (5.4) way. Then, by using the corresponding expressions, one can introduce functions ϕi (τ ) and χi (τ ) into series (5.5) and (5.7), respectively.

5.2 Sawtooth Series for Normal Modes

5.2

135

Sawtooth Series for Normal Modes

5.2.1

Periodic Version of Lie Series

Lie series with respect to the physical time parameter are considered below. Note that the corresponding procedure essentially diﬀers of those used for asymptotic integration of the diﬀerential equations of motion, where Lie series associate with a small parameter of perturbation [62] , [204]. Let us consider the diﬀerential equation of motion with respect to the position vector-function x (t) ∈ Rn x ¨ + f (x, x, ˙ t) = 0

(5.12)

under the initial conditions x0 = x|t=0

and x˙ 0 = x| ˙ t=0 = v0

where vector-function f is assumed to be diﬀerentiable with respect to every argument as many times as needed. The standard Cauchy form of dynamical system (5.12) with respect to the coordinate and velocity vectors is x˙ = v v˙ = −f (x, v, t) t˙ = 1

(5.13)

Then, the dynamics of system (5.13) can be locally described by the Lie series [70]

1 (5.14) x = exp[(t − t0 )G]x0 ≡ 1 + (t − t0 )G + (t − t0 )2 G2 + · · · x0 2! G = v0 ·

∂ ∂ ∂ − f (x0 , v0 , t0 ) · + ∂x0 ∂v0 ∂t0

(5.15)

where G is Lie operator associated with system (5.13), and {x0 , v0 , t0 } is some initial point in the system’ phase space. The dot between two quantities indicates dot products. For example, v0 ·

∂ ∂ ∂ ≡ v01 + · · · + v0n ∂x0 ∂x01 ∂x0n

Series (5.14) is simply Taylor series whose coeﬃcients are calculated by enforcing equations (5.13). Unfortunately, this general idea is still of little use for oscillatory processes probably due to locality of expansion (5.14). In other words, even entire expansion (5.14) does not explicitly reveal such global characteristics of oscillations as their amplitude and period. Moreover, the

136

5 Sawtooth Power Series

corresponding truncated series produce increasingly growing errors as the time t runs away from the selected initial point t0 . In order to overcome these disadvantages, it is suggested to adapt the Lie series solution for the class of periodic motions as follows. Theorem 1. [145] Assume that system (5.13) admits a periodic solution x(t) of the period T = 4a so that x(t + 4a) = x(t) for any t, and some point {x0 , v0 , t0 } belongs to this solution. Then such a solution can be expressed in the form x = exp(aG){cosh[a(τ − 1)G] + e sinh[a(τ − 1)G]}x0

(5.16)

where τ and e are saw-tooth sine and rectangular cosine, whose periods are normalized to four and amplitudes are normalized to unity as τ (ϕ) = (2/π) arcsin sin(πϕ/2)

(5.17)

e(ϕ) = sgn[ cos(πϕ/2)]

(5.18)

and, respectively, and ϕ = (t−t0 )/a is a re-scaled time. If, in addition, the solution is odd with respect to one half of the period, x(t+2a) = −x(t), then expression (5.16) simpliﬁes to x = [sinh(aτ G) + e cosh(aτ G)]x0

1 1 ≡ aτ G + (aτ G)3 + · · · x0 + e 1 + (aτ G)2 + · · · x0 3! 2!

(5.19)

Proof of expression (5.16) is obtained by substituting the identity [144] ϕ = 1 + [τ (ϕ) − 1]e(ϕ),

(−1 < ϕ < 3)

(5.20)

in (5.14) and taking into account that e2 = 1

(5.21)

at almost every time instance1 . In order to prove the particular case (5.19), one should keep in mind that exp(2aG)x0 = x(t0 + 2a) = −x0 , as it follows from (5.14), and the oddness condition assumed. Note that τ and e are indeed quite simple piece-wise linear functions; the above analytical expressions (5.17) and (5.18) just deﬁne them in the unitform which enables one to avoid conditioning of computation in the original temporal scale, t0 ≤ t < ∞. This possibility becomes essential when the dynamics includes some evolutionary component. 1

The set of isolated points {ϕ : τ (ϕ) = ±1}appears to have no eﬀect on the results [144].

5.2 Sawtooth Series for Normal Modes

137

Physical meaning of relationship (5.20) is that, during the whole period, the time variable ϕ is expressed through the coordinate τ and velocity e of a classic particle freely oscillating between the two absolutely stiﬀ barriers with no energy loss. Due to (5.21), this relationship possesses the algebraic structure of ‘hyperbolic complex numbers’ as revealed by (5.16). Let us outline possible applications of expressions (5.16) and (5.19). For the sake of simplicity, consider the particular case (5.19). Of course, formal expression (5.19) does not guarantee the existence of periodic solutions. In case some periodic solution does exist, one should be able to ﬁnd the corresponding vectors x0 and v0 from appropriate conditions. In autonomous case, the scalar parameter, a, is also unknown and must be determined. The related conditions are formulated as a requirement of smoothness of expression (5.19), which is generally non-smooth or even discontinuous due to the presence of non-smooth and discontinuous functions τ and e, respectively. The ‘smoothing’ relations are obtained by eliminating the step-wise discontinuities of the coordinate and velocity vectors imposing the constraints cosh(aG)x0 = 0 cosh(aG)v0 = 0

(5.22)

In autonomous case, algebraic equations (5.22) represent a nonlinear eigenvalue problem, where a is an eigen-value, and {x0 , v0 } is a combined (state) eigen-vector. By narrowing the class of periodic motions to those on which the system passes its trajectory twice in the conﬁgurations space during the same period, one obtains a subclass of normal mode motions. For more physically meaningful deﬁnitions and discussions, see reference [190]. Let us formulate the corresponding problem based on the periodic Lie series solutions. Consider the vibrating system x ¨ + f (x) = 0,

x ∈ Rn

(5.23)

where f (−x) = −f (x), and the initial conditions are x|t=0 = x0 = 0 and x| ˙ t=0 = v0 . The normal mode solutions of system (5.23) are obtained as a particular case of (5.19) and (5.22) x = sinh(aτ G)x0 |x0 =0

(5.24)

cosh(aG)v0 |x0 =0

(5.25)

where the initial vector x0 = 0 is substituted into the expressions only after all degrees of the diﬀerential operator

138

5 Sawtooth Power Series

G = v0 ·

∂ ∂ − f (x0 ) · ∂x0 ∂v0

have been applied. Relationship (5.24) can be interpreted as a parametric equation of normal mode trajectories of the system with the parameter interval −1 ≤ τ ≤ 1. Let us illustrate relationships (5.24) and (5.25) based on the linear system so that the result could be compared with the well known conventional solution. Example 10. Suppose that f (x) = Kx, where K is positively deﬁned symmetric n × n-matrix with eigen-system {v0 , ω 2 } so that Kv0 = ω 2 v0 . In this case, by applying the operator G twice, one obtains that v0 is also an eigen-vector of the operator G2 , namely, G2 v0 = −ω 2 v0 . Then, keeping in mind the power series form of expressions (5.24) and (5.25) as those in (5.19) and sequentially applying the operator G2 , gives x = (v0 /ω) sin(aωτ ) and cos(aω) = 0, respectively. Notably, the last equation shows that there exist an inﬁnite number of roots {a} related to the same eigen-frequency ω! However, it is easily to ﬁnd that all the roots produce the same solution in terms of the original time t. The minimal quarter of the period is a = π/(2ω), therefore x = (v0 /ω) sin(πτ /2), and τ = (2/π) arcsin sin ωt. Nonlinear cases and the related problems dealing with truncated expansions of (5.25) will be further discussed in a full-length paper.

5.3

Lie Series of Transformed Systems

5.3.1

Second-Order Non-autonomous Systems

In the previous section, the sawtooth temporal argument was introduced into Lie series solutions with respect to time t. Alternatively, the sawtooth argument can be introduced ﬁrst into the diﬀerential equations of motion before the Lie series procedure is applied. Let the dynamical system be described by the set of second-order equations (5.12), where the vector-function f is periodic with respect to t with the period T = 4a. Periodic motions of the period T are considered. Note that, in the autonomous case, the period is a priory unknown and thus must be determined. Following the rules introduced in Chapter 4 (see also Chapters 8 and 12 for further details) and making the substitutions t → τ (t/a) and x (t) = X(τ ) + Y (τ )e in (5.12), gives the boundary value problem X + a2 Rf (X, Y, X , Y , τ ) = 0 Y + a2 If (X, Y, X , Y , τ ) = 0

(5.26)

5.3 Lie Series of Transformed Systems

139

Y |τ =±1 = 0, X |τ =±1 = 0

(5.27)

where 1 Rf = {f [X + Y, (Y + X ) /a, aτ ] ± f [X − Y, (Y − X ) /a, 2a − aτ ]} If 2 (5.28) Analogously to (5.15), the operator Lie of system (5.26) at some initial point is represented as ∂ ∂ ∂ + Y0 · − a2 Rf (X0 , Y0 , X0 , Y0 , τ0 ) · ∂X0 ∂Y0 ∂X0 ∂ ∂ −a2 If (X0 , Y0 , X0 , Y0 , τ0 ) · + ∂Y0 ∂τ0

G = X0 ·

The idea of Lie series enables one of representing solution of the system (5.26) in the power series form with respect to the sawtooth argument τ under the initial conditions at P0 (X0 , Y0 , X0 , Y0 , τ0 ); it is assumed that τ0 = 0. Then, the corresponding solution of the original equation is represented in the form x (t) = X(X0 , X0 , Y0 , Y0 , τ ) + Y (X0 , X0 , Y0 , Y0 , τ )e

(5.29)

where τ = τ (t/a) and e = τ (t/a). Now, the arbitrary quantities X0 , X0 , Y0 , and Y0 can be determined from the smoothness conditions (5.27) Y (X0 , X0 , Y0 , Y0 , τ )|τ =±1 = 0,

∂X(X0 , X0 , Y0 , Y0 , τ ) |τ =±1 = 0 ∂τ

(5.30)

However, instead of solving equations (5.30), the smoothing procedure described in the previous section can be applied. In this case, conditions (5.30) are satisﬁed automatically, but the initial quantities X0 , X0 , Y0 , and Y0 remain undetermined. This enables one of improving the solution’ smoothness by imposing stronger smoothness conditions as follows ∂ 2 Y (X0 , X0 , Y0 , Y0 , τ ) |τ =±1 = 0, ∂τ 2

∂ 3 X(X0 , X0 , Y0 , Y0 , τ ) |τ =±1 = 0 ∂τ 3 (5.31) Equations (5.30) or (5.31) for X0 , X0 , Y0 and Y0 are generally quite complicated. In diﬀerent particular cases however, the problem can be simpliﬁed by taking into account possible symmetries of the system. So let us consider equation x ¨ + f (x, t) = 0

(5.32)

where the vector-function f (x, t) is odd with respect to the positional vector x and even with respect to the quarter of the period t = a. In this case, boundary-value problem (5.26) and (5.27) reduces to

140

5 Sawtooth Power Series

X + a2 f (X, aτ ) = 0 X |τ =1 = 0

(5.33)

Y ≡0 The operator Lie and the corresponding solution are, respectively, G = X and

∂ ∂ ∂ − a2 f (X, aτ ) + ∂X ∂X ∂τ

∞ i τ 2i+1 G2k−1 X|τ =0 τ 2i−1 − ≡ X(X0 , τ ) x (t) = (2k − 2)! 2i − 1 2i + 1 i=1 k=1

where τ = τ (t/a), the initial position is X0 = 0, whereas the constant vector X0 is determined from equation ∂ 3 X(X0 , τ ) |τ =1 = 0 ∂τ 3 Example 11. Consider now a two degree of freedom oscillating system

d2 x1 ζ (2x1 − x2 ) + x31 − P sin Ωt 0 + = 0 ζ (2x2 − x1 ) + x32 dt2 x2 In this case, the parameter a = π/(2Ω) is the quarter of the period of the external forcing function. The operator Lie of the system is ∂ ∂ πτ ∂ + X2 − a2 ζ (2X1 − X2 ) + X13 − P sin ∂X1 ∂X2 2 ∂X1 ∂ ∂ −a2 [ζ (2X2 − X1 ) + X23 ] + ∂X2 ∂τ

G = X1

Let the initial position be X0 = 0. Then, introducing the notation X0 = T [A1 , A2 ] and calculating the coeﬃcients of the ﬁrst three terms of series (5.8), gives

3

τ5 τ3 τ 1 X1 = τ − − A1 + A1 + (a2 P π − 4 a2 ζ A1 + 2a2 ζ A2 ) 3 3 5 4

5 7 τ τ 1 + − A1 + (a2 P π − 4 a2 ζ A1 + 2a2 ζ A2 ) 5 7 4 1 2 [a P π 3 + 8a4 P π ζ − 8A1 5 a4 ζ 2 − 6 a2 A1 2 + 32 a4 ζ 2 A2 ] − 192 and

3 τ τ5 τ3 1 X2 = τ − − A2 + A2 + (a2 ζ A1 − 2 a2 ζ A2 ) 3 3 5 2

5.3 Lie Series of Transformed Systems

141

τ5 1 τ7 − A2 + (a2 ζ A1 − 2 a2 ζ A2 ) 5 7 2 1 4 + [a P π ζ − 8 a4 ζ 2 A1 + 2A2 5 a4 ζ 2 − 6 a2 A2 2 ] 48

+

In this case, equations (5.31) give two scalar equations for A1 and A2 , ∂ 3 X1 (A1 , A2 , τ ) |τ =1 = 0, ∂τ 3

∂ 3 X2 (A1 , A2 , τ ) |τ =1 = 0 ∂τ 3

(5.34)

where the ﬁrst and the second equations guarantee smoothness of the ac¨2 (t), respectively. Intersections of curves celeration components x ¨1 (t) and x (5.34) on the Cartesian plane A1 A2 allow to locate the roots, as it is shown√in Figs. 5.1 and 5.2. For instance, setting ζ = 1.0, P = 0.2 and Ω = 1.01, gives (A1 , A2 ) = (0.7711, 0.6677). This solution corresponds to a weakly nonlinear perturbation of the in-phase √ linear mode. Similarly, in the neighborhood of the second frequency Ω = 3.01, one ﬁnds the solution (A1 , A2 ) = (0.5971, −0.8181), which is close to the out-of-phase linear mode; see Fig. 5.1. Interestingly enough, Fig. 5.2 shows that there are also two roots, (2.3315, 2.2864) and (−2.2065, −2.2548), corresponding to in-phase vibrations even though the frequency of the external forcing function is close to that of the out-phase linear mode. This eﬀect may take place due to large amplitudes such that the in-phase nonlinear frequency becomes close to the out-of-phase linear frequency. Since the initial conditions, corresponding to the periodic regimes are known, one can integrate the diﬀerential equations of motion numerically in order to check the analytical solutions. In the latter example, one would obtain that both results are perfectly matching. It must be noted, however, that some of the periodic solutions may appear to be unstable. As a result, even a very small imperfection in the initial conditions will lead to a signiﬁcant divergence of the results. Moreover, the ‘tails’ of polynomial expansions may give roots which do not correspond to any solution. Nevertheless, the above approaches seems to make sense as compared to those based on the direct Fourier expansions. Indeed, the number of algebraic equations in (5.22), (5.30) or (5.31) is independent on the number of terms in the series.

5.3.2

NSTT of Lagrangian and Hamiltonian Equations

In diﬀerent theoretical areas, describing systems in terms of the analytical dynamics brings some advantages because, until certain stage, it is suﬃcient to deal with a single scalar function such as Lagrangian or Hamiltonian rather then manipulate with a set of diﬀerential equations. Suppose the sawtooth temporal argument is introduced into Lagrangian or Hamiltonian. Let us show that the corresponding diﬀerential equations of

142

5 Sawtooth Power Series

Fig. 5.1 The curves of smoothness for ¨1 (t) and x ¨2 (t), respec√ the accelerations x tively thin and solid lines, when Ω = 1.01.

Fig. 5.2 The curves of smoothness for ¨1 (t) and x ¨2 (t), respec√ the accelerations x tively thin and solid lines, when Ω = 3.01.

5.3 Lie Series of Transformed Systems

143

motion are still derivable from the transformed functions in the standard way of the analytical dynamics. Consider ﬁrst Lagrangian that depends on time t periodically with the period T = 4a, L = L(x, x, ˙ t) (5.35) On the manifold of smooth periodic motions of the same period T , the vectorfunction x(t) is represented as x (t) = X (τ ) + Y (τ ) e, where τ = τ (t/a) and e = e(t/a). As a result, Lagrangian (5.35) takes the form L(x, x, ˙ t) = RL (X, Y, X , Y , τ ) + IL (X, Y, X , Y , τ ) e

(5.36)

where both components on the right-hand side are determined analogously to expressions (5.28). It can be veriﬁed by inspection that the diﬀerential equations of periodic motions can be represented now in any of the two equivalent forms d dτ d dτ

∂RL ∂RL =0 − ∂X ∂X ∂RL ∂RL =0 − ∂Y ∂Y

or d ∂IL ∂IL =0 − dτ ∂Y ∂Y d ∂IL ∂IL =0 − dτ ∂X ∂X Now let us consider the Hamiltonian H = H(p, q, t)

(5.37)

which may be periodic with respect to time with the period T = 4a. On the manifold of periodic motions of the period T , the coordinates and the linear momenta are represented as q = X (τ ) + Y (τ ) e

and p = U (τ ) + V (τ ) e

respectively, where τ = τ (t/a) and e = e (t/a). Let us transform the Hamiltonian (5.37) as H −→ aH(p, q, t) = RaH (U, V, X, Y, τ ) + IaH (U, V, X, Y, τ ) e where

RaH IaH

=

a [H (U + V, X + Y, aτ ) ± H (U − V, X − Y, 2a − aτ )] 2

144

5 Sawtooth Power Series

The corresponding diﬀerential equations of periodic motions are ∂RaH ∂RaH , U = − ∂U ∂X ∂R ∂R aH aH , V =− Y = ∂V ∂Y

X =

or ∂IaH ∂IaH , V =− ∂V ∂X ∂IaH ∂IaH , U =− Y = ∂U ∂Y

X =

Besides, in autonomous cases, the operator Lie associates with the Poisson bracket, for instance, GX = {X, RaH } Therefore, introducing the sawtooth temporal argument for periodic motions preserves both Lagrangian and Hamiltonian structures of the diﬀerential equations of motion.

5.3.3

Remark on Multiple Argument Cases

In multiple frequency cases, the smoothing procedures can be applied sequentially to each of the arguments. For instance, in the case of two arguments, τ1 = τ (t/a1 ) and τ2 = τ (t/a2 ) , the X-component would take the form

i j N2 N1 τ1 τ1i+2 τ2j+2 τ2 X (τ1 , τ2 ) = X (0, 0) + − − (5.38) KX (i, j) i i+2 j j+2 i=1 i=1 where KX (i, j) are constant coeﬃcients. Analytical manipulations with such 2D polynomials as (5.38) are quite complicated because algebraic and diﬀerential operations aﬀect the structure of binomials. However, let us bring attention to a possibility of using polynomial expansions (5.38), and the related complete versions, for diﬀerent kinds of approximations by determining coeﬃcients of expansions through appropriate numerical optimization procedures. Besides of multiple phase cases in dynamics, similar expansions may be useful for multidimensional spatial problems dealing, for instance, with elastic cell-wise periodic structures; see also Chapter 14 for possible formulations. In such cases, the triangular sine waves depend on the spatial coordinates, say x and y as τ1 = τ (x/a1 ) and τ2 = τ (y/a2 ). In static problems, the parameters a1 and a2 are usually given by design, whereas the coeﬃcients can be obtained through variational principles by minimizing the corresponding functionals.

Chapter 6

NSTT for Linear and Piecewise-Linear Systems

Abstract. Remind that the tool of nonsmooth argument substitutions was introduced ﬁrst to describe strongly nonlinear vibrations whose temporal mode shapes are asymptotically close to non-smooth ones. Such cases are known to be most diﬃcult for analyses because diﬀerent quasi-harmonic methods are already ineﬀective whereas nonsmooth mapping tools are still inapplicable. It is quite clear however that the non-smooth arguments can be introduced regardless the strength of nonlinearity or the form of dynamical systems in general. For instance, it is shown in this chapter that the non-smooth substitutions can essentially simplify analyses of diﬀerent linear models with non-smooth or discontinuous inputs. It is also shown that, in piecewise-linear cases, the nonsmooth temporal transformation provides an automatic matching the motions from diﬀerent subspaces of constant stiﬀness and justiﬁes quasi-linear asymptotic solutions for the speciﬁc nonsmooth case of piece-wise linear characteristics.

6.1

Free Harmonic Oscillator: Temporal Quantization of Solutions

Introducing the sawtooth temporal argument into the diﬀerential equations of motion may bring some speciﬁc features into the corresponding solutions. For illustrating purposes, let us consider the harmonic oscillator x ¨ + ω02 x = 0

(6.1)

First, let us obtain exact general solution of the oscillator (6.1) in terms of the sawtooth temporal argument by using the substitution x = X (τ ) + Y (τ ) e V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 145–178, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(6.2)

146

6 NSTT for Linear and Piecewise-Linear Systems

where τ = τ (t/a) and e = e (t/a) are the standard triangular and rectangular wave functions, respectively. Substituting (6.2) in (6.1), gives the boundary value problem a−2 X (τ ) + ω02 X (τ ) = 0 a−2 Y (τ ) + ω02 Y (τ ) = 0

(6.3) (6.4)

X (±1) = 0, Y (±1) = 0

(6.5)

By considering the parameter a as an eigen value of the problem, one obtains the set of eigen values and the corresponding solutions as, respectively, aj =

and Xj = sin

jπ 2ω0

jπτ jπτ + ϕj , Yj = cos − ϕj 2 2

(6.6)

(6.7)

where ϕj = (π/4) [1 + (−1)j ] , τ = τ (t/aj ), and j is any positive real integer. Therefore, introducing the sawtooth oscillating time produced the discrete family of solutions for harmonic oscillator (6.1). The nature of such kind of quantization is due to the speciﬁc temporal symmetry of periodic motions. In other words, the quantization is associated with a multiple choice for the period Tj = 4aj = jT

(6.8)

where T = 2π/ω0 is the natural period of oscillator (6.1). In terms of the original temporal variable t, the number j plays no role for the temporal mode shape, given by

jπ 2ω0 t x (t) = A sin τ (6.9) + ϕj 2 jπ

2ω0 t 2ω0 t jπ τ +B cos − ϕj e 2 jπ jπ where A and B are arbitrary constants, and x(t) is the same harmonic wave regardless the number j. In this section, the free linear oscillator was considered for illustrating purposes. Of course, there is no other pragmatic reason for introducing the sawtooth time into equation (6.1). The situation drastically changes however in non-autonomous cases of non-smooth or discontinuous inputs. It is shown below that, in such cases, the sawtooth time variable can help to facilitate determining particular solutions. The eﬀect of ‘temporal quantization’ represented by expression (6.9), which seems to be just identical transformation in the autonomous case, acquires helpful meaning at the presence of external

6.2 Non-autonomous Case

147

excitations. For instance, according to (6.9) the so-called combination resonances will appear to be an inherent property of oscillators.

6.2

Non-autonomous Case

6.2.1

Standard Basis

Consider the linear harmonic oscillator under the external forcing described by the linear combination of triangular and rectangular wave functions

t t 2 x¨ + ω0 x = F τ + Ge (6.10) a a where F and G are constant amplitudes, and a is a quarter of the period. Substituting (6.2) in (6.10), leads to the boundary value problem a−2 X (τ ) + ω02 X (τ ) = F τ a−2 Y (τ ) + ω02 Y (τ ) = G

(6.11) (6.12)

under the boundary conditions (6.5). In contrast to autonomous case (6.1), the parameter a is known. However, the equations (6.11) and (6.12) are non-homogeneous, and thus a non-zero solution exists for any a and can be found in few elementary steps. As a result, the particular periodic solution of the original equation (6.10) takes the form t F sin [aω0 τ (t/a)] xp (t) = X (τ ) + Y (τ ) e = 2 τ − ω0 a aω0 cos aω0 cos [aω0 τ (t/a)] t G e (6.13) + 2 1− ω0 cos aω0 a The corresponding general solution is x (t) = A cos (ω0 t − ϕ) + xp (t), where A and ϕ are arbitrary amplitude and phase parameters. Note that solution (6.13) immediately shows all possible resonance combinations aω0 = (2k + 1) π/2 or ω0 = 2k + 1 (6.14) Ω where k = 1, 2, 3... , and Ω = 2π/T = π/(2a) is the principal circular frequency of the external forcing. It is interesting to compare the solution (6.13) with those obtained by the conventional methods such as Fourier series. So, taking into account expansion,

∞ t (2k + 1) πt 8 (−1)k τ (6.15) sin = 2 2 a π 2a (2k + 1) k=0

148

6 NSTT for Linear and Piecewise-Linear Systems

gives the particular solution of equation (6.10) in the form xp (t) =

∞ k=0

ω02

−

1 (2k+1)π 2a

2 ×

(6.16)

k

k

8F (−1)

4G (−1) (2k + 1) πt (2k + 1) πt × + cos 2 sin 2 2a π (2k + 1) 2a π (2k + 1)

Solution (6.16) indicates the same resonance conditions, (6.14). However, inﬁnite trigonometric series are less convenient for calculations, especially when dealing with derivatives of the solutions; indeed, diﬀerentiation slows down convergence of series (6.16).

6.2.2

Idempotent Basis

Consider the linear oscillator including viscous damping under the rectangular wave external loading

t x ¨ + 2ζω0 x˙ + ω02 x = pe (6.17) a The purpose is to obtain periodic (particular) solution with the period of external loading T = 4a. Recall that the idempotent basis is introduced by means of the linear transformation (see Chapters 1 and 4) {1, e} −→ {e+ , e− } :

e± =

1 (1 ± e) 2

(6.18)

or, inversely, 1 = e+ + e− and e = e+ − e− , where e2± = e± and e+ e− = 0. Now, the periodic solution and external loading are represented in the form x(t) = U (τ )e+ + V (τ )e−

(6.19)

pe = p(e+ − e− ) where e± = e± (t/a), and U (τ ) and V (τ ) are unknown functions of the triangular wave τ = τ (t/a). Substituting (6.19) in (6.17), and sequentially eliminating derivatives of the rectangular wave e(t/a) as described in Chapter 4, gives equations U + 2ζωaU + (ωa)2 U = pa2 V − 2ζωaV + (ωa)2 V = −pa2

(6.20)

and boundary conditions (U − V )|τ =±1 = 0 (U + V )|τ =±1 = 0

(6.21)

6.3 Systems under Periodic Pulsed Excitation

149

All the coeﬃcients and right-hand sides of both equations in (6.20) are constant, and the equations are decoupled. As a result, solution of boundary value problem (6.20) and (6.21) is easily obtained in the form U (τ ) =

2p exp(−ατ ) p − (6.22) 2 2 ω βω (cos 2β + cosh 2α) ×[cos β cosh α(β cos βτ + α sin βτ ) + sin β sinh α(α cos βτ − β sin βτ )]

2p exp(ατ ) p (6.23) + ω2 βω 2 (cos 2β + cosh 2α) ×[cos β cosh α(β cos βτ − α sin βτ ) + sin β sinh α(α cos βτ + β sin βτ )] where α = ωaζ and β = ωa 1 − ζ 2 . Substituting (6.22) and (6.23) in (6.19), gives closed form particular solution of original equation (6.17). Transition to the original temporal variable is given by the functions τ (ϕ) = (2/π) arcsin[sin(πt/2)] and e(ϕ) = sgn[cos(πt/2)]. Since the system under consideration is linear, the general solution of equation (6.17) can be obtained by adding general equation of the corresponding equation with zero right-hand side. Finally, note that neither trigonometric expansions nor any integral transforms were involved into the solution procedure. V (τ ) = −

6.3

Systems under Periodic Pulsed Excitation

Instantaneous impulses acting on a mechanical system can be modeled either by imposing speciﬁc matching conditions on the system state vector at pulse times or by introducing Dirac’s functions into the diﬀerential equations of motion. The ﬁrst approach deals with the diﬀerential equations of a free system separately between the impulses, therefore a sequence of systems under the matching conditions are considered. The second method gives a single set of equations over the whole time interval without any conditions of matching. In this case however the analysis can be carried out correctly in terms of distributions, which unfortunately requires additional mathematical justiﬁcations in non-linear cases. Both of the above approaches are actually employed for diﬀerent quantitative and qualitative analyses. The analytical tool, which is described below, on the one hand, eliminates the singular terms from the equations and, on the other hand, brings solutions to the unit-form of a single analytic expression for the whole time interval.

6.3.1

Regular Periodic Impulses

Introducing the sawtooth temporal argument may signiﬁcantly simplify solutions whenever loading functions are combined of the triangular wave and

150

6 NSTT for Linear and Piecewise-Linear Systems

its derivatives. For instance, let us seek a particular solution of the ﬁrst order diﬀerential equation v˙ + λv = μ

∞

[δ (t + 1 − 4k) − δ (t − 1 − 4k)]

(6.24)

k=−∞

where λ and μ are constant parameters. For positive λ, equation (6.24) describes the velocity of a particle moving in a viscous media under the periodic impulsive force. The corresponding physical model is shown in Fig. 6.1, where the freely moving massive tank experiences perfectly elastic reﬂections from the stiﬀ obstacles. By scaling the variables, one can bring the diﬀerential equation of motion of the particle to the form (6.24), where v (t) = x˙ (t).

Fig. 6.1 If the particle’ mass is very small compared to the total mass of the tank then the inertia force applied to the particle inside the tank has the periodic pulse-wise character.

First, note that the right-hand side of equation (6.24) can be expressed through the generalized derivative of the rectangular wave function as follows v˙ + λv =

μ e˙ (t) 2

(6.25)

Now let us represent the particular solution in the form v (t) = X (τ (t)) + Y (τ (t)) e (t)

(6.26)

Substituting (6.26) in (6.25), gives μ e˙ (t) = 0 Y + λX + (X + λY ) e (t) + Y − 2

(6.27)

Apparently, the elements {1, e} and e˙ in combination (6.27) are linearly independent as functions of diﬀerent classes of smoothness. Therefore,

6.3 Systems under Periodic Pulsed Excitation

Y + λX = 0,

X + λY = 0,

151

Y |τ =±1 =

μ 2

(6.28)

In contrast to equation (6.24) or (6.25), boundary value problem (6.28) includes no discontinuities whereas the new independent variable belongs to the standard interval, −1 ≤ τ ≤ 1. Solving the boundary value problem (6.28) and taking into account substitution (6.26), gives periodic solution of equation (6.24) in the form v = X +Ye =

μ (− sinh λτ + e cosh λτ ) 2 cosh λ

or

μ exp [−λτ (t) e (t)] e (t) (6.29) 2 cosh λ Fig. 6.2 illustrates solution (6.29) for μ = 0.2 and diﬀerent magnitudes of λ. v=

0.2 Λ 0.4 0.1

v

0.0

Λ 1.9

0.1 0.2

0

2

4

6

8

t

Fig. 6.2 The family of discontinuos periodic solutions.

Note that the discontinuous solution v (t) is described by the unit-form expression (6.29) through the two elementary functions τ (t) and e (t).

6.3.2

Harmonic Oscillator under the Periodic Impulsive Loading

Let us consider the harmonic oscillator subjected to periodic pulses x ¨ + ω02 x = 2p

∞

[δ (ωt + 1 − 4k) − δ (ωt − 1 − 4k)]

k=−∞

where p, ω0 and ω are constant parameters.

(6.30)

152

6 NSTT for Linear and Piecewise-Linear Systems

The right-hand side of equation (6.30) can be expressed through ﬁrst derivative of the rectangular wave as follows x ¨ + ω02 x = p

de (ωt) d (ωt)

(6.31)

Let us seek a periodic solution of the period T = 4/ω in the form x (t) = X (τ (ωt)) + Y (τ (ωt)) e (ωt)

(6.32)

Substituting (6.32) in (6.31) under the necessary condition of continuity for x (t), gives de (ωt) ω 2 X + ω02 X + ω 2 Y + ω02 Y e + ω 2 X − p =0 d (ωt)

(6.33)

Analogously to the previous subsection, equation (6.33) gives the boundary value problem X +

ω 2 0

ω

X = 0,

X |τ =±1

p = 2, ω

Y +

ω 2 0

ω

Y =0

(6.34)

Y |τ =±1 = 0

Solving boundary value problem (6.34) and taking into account (6.32), gives the periodic solution of the original equation (6.30) in the form x = X (τ (ωt)) =

p sin [(ω0 /ω) τ (ωt)] ωω0 cos (ω0 /ω)

(6.35)

where Y ≡ 0. Solution (6.35) is continuous, but nonsmooth at those times t where τ (ωt) = ±1. All possible resonances are given by ω=

2 ω0 ; π k

k = 1, 3, 5, ...

(6.36)

where the factor 2/π is due to diﬀerent normalization of the periods for trigonometric and sawtooth sines. Now let us consider the case of viscous damping described by the diﬀerential equation of motion x ¨ + 2ζ x˙ + ω02 x = p where ζ is the damping factor.

de (ωt) d (ωt)

(6.37)

6.3 Systems under Periodic Pulsed Excitation

153

In this case, the boundary value problem becomes coupled ω 2 ζ 0 X + 2 Y + X=0 ω ω 2 ζ ω0 Y =0 Y + 2 X + ω ω

(6.38)

p , Y |τ =±1 = 0 ω2 As a result, the periodic solution has both X and Y components X |τ =±1 =

p 2 2 cos β cosh α + sin2 β sinh2 α ×[cosh α cos β cosh ατ sin βτ − sinh α sin β sinh ατ cos βτ

x = X +Ye=

βω 2

(6.39)

+ (sinh α cos βτ cosh ατ sin β − sinh ατ sin βτ cosh α cos β) e] where τ = τ (ωt), e = e (ωt); α = ζ/ω and β = ω02 − ζ 2 ω.

0.2 0.1 x 0.0 0.1 0.2 0

2

4 t

6

8

Fig. 6.3 Response of the damped harmonic oscillator under the periodic impulsive excitation for p = 0.1, ζ = 0.5, ω0 = 4 and ω = 0.2 (low-frequency pulses.)

Figs. 6.3 through 6.5 illustrate qualitatively diﬀerent responses of the system when varying the input frequency. In diﬀerent proportions, the responses combine properties of the harmonic damped motion and the non-smooth motion due to the impulsive loading. For instance, when ω >> ω0 and ω >> ζ, the system is near the limit of a free particle under the periodic impulsive force. In this case, the boundary value problem is reduced to X = 0,

Y = 0;

X |τ =±1 =

p , ω2

Y |τ =±1 = 0

(6.40)

154

6 NSTT for Linear and Piecewise-Linear Systems

0.06 0.04 0.02 x 0.00 0.02 0.04 0.06 0

2

4 t

6

8

Fig. 6.4 System response on ‘resonance’ pulses ω = (2/π)ω0 = 2.5465.

0.003 0.002 0.001 x 0.000 0.001 0.002 0.003 0

2

4 t

6

8

Fig. 6.5 Response on high-frequency pulses; ω = 6.

This gives the triangular temporal shape of the motion, x = pτ (ωt) /ω 2 , which is approached by the time history record on Fig. 6.5. Finally, let us consider N -degrees-of-freedom system My ¨ + Ky = p

de (ωt) d (ωt)

(6.41)

where y (t) is N -dimensional vector-function, p is a constant vector, M and K are constant N × N mass and stiﬀness matrixes respectively.

6.3 Systems under Periodic Pulsed Excitation

155

Let {e1 , ..., eN } and ω1 ,...,ωN be the normal mode basis vectors and the corresponding natural frequencies, respectively, such that Kej = ωj2 M ej ,

eTk M ej = δkj

for any k = 1, ..., N and j = 1, ..., N . Introducing the principal coordinates xj (t), y=

N

xj (t) ej

(6.42)

j=1

gives a decoupled set of impulsively forced harmonic oscillators of the form (6.31), de (ωt) (6.43) x¨j + ωj2 xj = pj d (ωt) where pj = eTj p. Therefore, making use of solution (6.35) for each of the oscillators (6.43) and taking into account (6.42), gives y=

N (eTj p)ej sin [(ωj /ω) τ (ωt)] ωωj cos (ωj /ω) j=1

(6.44)

The corresponding resonances are determined by the condition ω=

2 ωj π k

where k = 1, 3, 5, ... and j = 1, ..., N .

6.3.3

Periodic Impulses with a Temporal ‘Dipole’ Shift

Let us consider the impulsive excitation with a dipole shift of pulse times. In this case, the right-hand side of equation (6.25) can be expressed by second derivative of the saw-tooth function with some incline described the parameter γ as shown in Fig. 6.6 v˙ + λv = p =

∂ 2 τ (ωt, γ) 2

∂ (ωt) ∞ 2p

1 − γ2

=p

k=−∞

∂e (ωt, γ) ∂ (ωt)

(6.45)

[δ (ωt + 1 − γ − 4k) − δ (ωt − 1 + γ − 4k)]

156

6 NSTT for Linear and Piecewise-Linear Systems

Fig. 6.6 Basic NSTT asymmetric wave functions.

Based on the NSTT identities introduced in Chapter 4, periodic solutions of equation (6.45) still can be represented in the form v = X (τ ) + Y (τ ) e

(6.46)

where τ = τ (ωt, γ) and e = e (ωt, γ); see Fig. 6.6 for graphic illustrations. Substituting (6.46) in equation (6.45), gives ωαY + λX + [ω (X + βY ) + λY ] e + (ωY − p)

∂e(ωt, γ) =0 ∂(ωt)

(6.47)

where α = 1/ 1 − γ 2 , β = 2γα, and the identity e2 = α + βe has been taken into account. Equation (6.47) is equivalent to the boundary-value problem ω (X + βY ) = −λY ωαY = −λX

(6.48)

ωY |τ =±1 = p The corresponding solution is

λ cosh ωλ τ λ sinh ωλ τ λ p cosh γ τ + sinh γ exp γ Y = ω ω ω ω cosh ωλ sinh ωλ ωα X=− Y (6.49) λ where the X-component is deﬁned by diﬀerentiation due to the second equation in (6.48).

6.4 Parametric Excitation

6.4

157

Parametric Excitation

In this section, two diﬀerent cases of parametric excitation are considered based on relatively simple linear models. Piecewise-constant and impulsive excitations are described by means of the functions e(ωt, γ) and ∂e(ωt, γ)/∂ (ωt), respectively. There are at least two reasons for using NSTT as a preliminary analytical step. First, NSTT automatically gives conditions for matching solutions at discontinuity points. Second, due to the automatic matching through the NSTT functions, the corresponding solutions appear to be in the closed form that is important feature when further manipulations with the solutions are required by problem formulations.

6.4.1

Piecewise-Constant Excitation

Let us consider the linear oscillator under periodic piecewise-constant excitation (6.50) x¨ + ω02 [1 + εe(ωt, γ)]x = 0 where ω0 , ω, γ and ε are constant parameters. We will seek periodic solutions with the period of excitation T = 4/ω in the form x = X (τ ) + Y (τ ) e (6.51) where τ = τ (ωt, γ) and e = e(ωt, γ). As follows from the form of equation (6.50), the acceleration x ¨ may have step-wise discontinuities due to the presence of the function e(ωt, γ), whereas the coordinate x (t) and the velocity x˙ (t) must be continuous. So neither velocity x˙ (t) nor acceleration x ¨ (t) can include Dirac δ-functions. Taking ﬁrst derivative of (6.51), gives ∂e(ωt, γ) ω (6.52) x˙ (t) = αY + (X + βY )e + Y ∂ (ωt) where the last term, that consists of the periodic sequence of δ-functions, must be excluded by imposing the boundary condition for Y -component Y |τ =±1 = 0

(6.53)

Under condition (6.53), the second derivative takes the form

x ¨ (t) = ω 2 [α(X + βY )] + ω 2 [βX + (α + β 2 )Y ]e ∂e (ωt, γ) +ω 2 (X + βY ) ∂ (ωt)

(6.54)

In this case, the singular term, which is underlined in (6.54), is eliminated by condition

158

6 NSTT for Linear and Piecewise-Linear Systems

(X + βY ) |τ =±1 = 0

(6.55)

Substituting (6.51) and (6.54) in the diﬀerential equation of motion (6.50) and taking into account the algebraic properties, brings the left-hand side of the equation to the algebraic form {· · ·} + {· · ·}e. Then, setting separately each of the two algebraic components to zero, gives the set of diﬀerential equations for X (τ ) and Y (τ ) in the following matrix form

α αβ β α + β2

X Y

+ r2

1 αε ε 1 + βε

X Y

=0

(6.56)

where r = ω0 /ω. Further, any particular solution of linear diﬀerential equations with constant coeﬃcients (6.56) can be represented in the exponential form X 1 =B exp (λτ ) (6.57) Y μ where B, μ and λ are constant parameters. Substituting (6.57) in (6.56), leads to a characteristic equation which two pairs of roots determined by the relationships 2 λ2 = − (1 − γ) ε − (1 − γ) r2 ≡ ±k 2 (6.58) 2 λ2 = (1 + γ) ε − (1 + γ) r2 ≡ ±l2 where signs of the notations ±k 2 and ±l2 depend on the parameters ε and γ. Let us consider the case of negative signs, when the following condition holds − (1 − γ) < ε < (1 + γ) (6.59) Due to condition (6.59), the stiﬀness coeﬃcient in equation (6.50) is always positive, whereas (6.58) gives λ = ±ki and λ = ±li. As a result, the general solution of equations (6.56) takes the form X = B1 sin kτ + B2 cos kτ + B3 sin lτ + B4 cos lτ Y = μ1 (B1 sin kτ + B2 cos kτ ) + μ2 (B3 sin lτ + B4 cos lτ )

(6.60)

where B1 ,...,B4 are arbitrary constants, and μ1 = −

1 αk 2 − r2 α βk 2 − εr2

and

μ2 = −

1 αl2 − r2 α βl2 − εr2

Substituting (6.60) in boundary conditions (6.53) and (6.55), gives the homogeneous set of four linear algebraic equations with respect to the arbitrary

6.4 Parametric Excitation

159

constants. Setting the corresponding determinant to zero, gives condition for non-zero solutions in the form [μ1 (1 + βμ2 ) l cos k sin l − μ2 (1 + βμ1 ) k cos l sin k] × [μ1 (1 + βμ2 ) l cos l sin k − μ2 (1 + βμ1 ) k cos k sin l] = 0

(6.61)

10

8

10Ε

6

4

2

0 0

2

4

6

8

10

r Fig. 6.7 Instability zones for piecewise constant parametric excitation when γ = 0.7

One the parameter plane, ε − r, equation (6.61) describes the family of curves separating stability and instability zones as shown in Fig. 6.7, where the instability zones are shadowed.

6.4.2

Parametric Impulsive Excitation

Let us consider the case of parametric impulsive excitation whose temporal shape is given by ﬁrst derivative of the basic function, e (ωt, γ),

∂e (ωt, γ) 2 x ¨ + ω0 1 + ε x=0 (6.62) ∂ (ωt) This case was considered in [138] based on the saw-tooth transformation of time. In particular, it was shown that the periodic solutions of the period T = 4/ω exists under the condition

160

6 NSTT for Linear and Piecewise-Linear Systems

2 2r2 1 − γ 2 sin2 2r p = cos 4r − cos 4γr 2

(6.63)

where r = ω0 /ω and p = εr2 .

1.0 0.8 0.6 p 0.4 0.2 0.0

0

2

4

6

8

10

r

Fig. 6.8 ‘Collapse’ of the instability zones at γ = 1/5: each ﬁfth zone is missing; here and below, only the upper half-plane is shown due to the symmetry.

1.0 0.8 0.6 p 0.4 0.2 0.0

0

2

4

6

8

10

r

Fig. 6.9 γ = 1/2: each second zone is missing.

The dependence of p on r for ﬁxed γ has the branched zone-like structure which is typical for diﬀerent cases of parametrically excited oscillators. Interestingly enough, diﬀerent subsequences of zones may disappear as the parameter γ varies. For instance, if γ = 1/5 then each ﬁfth zone is missing and, if γ = 1/2 then each second zone is missing; see Figs. 6.8 and 6.9, respectively. Such an eﬀect was discussed in [138].

6.4 Parametric Excitation

6.4.3

161

General Case of Periodic Parametric Excitation

Below, the problem formulation only is discussed for the case of periodic parametric loading with both regular and singular components. It is assumed that there are two discontinuities and singularities on each period located at the same points. The diﬀerential equation of motion is represented in the vector form

∂e x¨ + Q (τ ) + P (τ ) e + p x=0 (6.64) ∂ϕ where x(t) ∈ Rn is the coordinates vector-column,τ = τ (ϕ, γ), e = e (ϕ, γ), ϕ = ωt is the phase variable, p is a constant n × n matrix, and Q (τ (ϕ, γ)) and P (τ (ϕ, γ)) are periodic matrixes of the period T = 4 with respect to the phase ϕ. In equation (6.64), the ﬁrst two terms of the coeﬃcient can represent any periodic function q (ϕ) with step-wise discontinuities on Λ = {t : τ (ϕ, γ) = ±1}. In case the original function q (ϕ) is continuous, one has P = 0 on Λ. Let us represent periodic solutions of the period T = 4 in the form (6.51). Substituting (6.51) in equations (6.64), taking into account the equality e2 = α + βe, the necessary condition of continuity of the vector function x (t), (6.53), and using (6.52) and (6.54) gives equations ω 2 (αX + αβY ) + QX + αP Y = 0 ω 2 α + β 2 Y + βX + P X + QY + βP Y = 0 and the boundary condition 2 ω (X + βY ) + pX |τ =±1 = 0

(6.65)

(6.66)

where, in the case of ﬁxed sign of impulses, the matrix p should be provided with the factor sgn(τ ). Together with (6.53), relations (6.65) and (6.66) represent a boundaryvalue problem for determining the vector functions X and Y and the corresponding conditions for existence of periodic solutions. Note that substitution (6.51) in equation (6.64) generates the speciﬁc term e∂e/∂ϕ. Let us show that, within the theory of distributions, this terms can be interpreted as follows 1 ∂e ∂e = β (6.67) e ∂ϕ 2 ∂ϕ First, note that, at this point, the relationship (6.67) is a result of formal diﬀerentiation of both sides of the relation e2 = α + βe with respect to the phase ϕ. To justify (6.67), let us assume that ω = 1 so that ϕ ≡ t and consider expression (6.53) locally, near the point t = 1 − γ, which is a typical point of the entire set of discontinuity points Λ = {t : τ (t) = ±1}.

162

6 NSTT for Linear and Piecewise-Linear Systems

Generally speaking, the ‘product’ f (t)δ(t) requires the function f (t) to be at least continuous at t = 0. However, it is possible to provide the left-hand side of (6.67) with a certain meaning due to the fact that both terms of the product are generated by the same sequence of smooth functions. In order to illustrate the above remark and prove equality (6.67), let us consider a family of smooth functions {δε (t)} such that ε δε (t) dt = 1

(6.68)

−ε

for all positive ε, and δε (t) = 0 outside the interval −ε < t < ε. Therefore, in terms of weak limits, δε (t) → δ (t) as ε → 0. Now, sequences of smooth functions approximating e and ∂e/∂t in the neighborhood of point t = 1 − γ can be chosen as, respectively, eε =

β 1 − θε (t − 1 + γ) 1−γ γ

where θε (t) =

"t −∞

and

β ∂eε = − δε (t − 1 + γ) ∂t γ

(6.69)

δε (ξ) dξ and −1 + γ < t < 3 + γ.

Based on the above deﬁnitions for eε and ∂eε /∂t, one has eε → e and ∂eε /∂t → ∂e/∂t as ε → 0 in the interval −1 + γ < t < 3 + γ. Substituting (6.69) in equality (6.67) instead of e and ∂e/∂ϕ, reduces the problem to the proof of identity θε δ ε =

1 δε 2

(6.70)

as ε → 0. For simplicity reason, let us move the origin to the point t = 1 − γ and show that the left-hand side of (6.70) gives δ (t) /2 as ε → 0 in the sense of weak limit. First, the area bounded by θε δε is ε ε 1 dθε 1 dt = θε2 |ε−ε = θε δε dt = θε dt 2 2 −ε −ε Then, let φ (t) belongs to the class of continuous testing functions, which is usually considered in the theory of distributions. By deﬁnition, in some ε-neighborhood of the point t = 0, one has | φ (t) − φ (0) |< 2η, where η is as small as needed whenever ε is suﬃciently small. Therefore, ε ε 1 | θε (t) δε (t) φ (t) dt − φ (0) |≤ θε (t) δε (t) | φ (t) − φ (0) | dt ≤ η 2 −ε −ε

6.5 Input-Output Systems

In other words,

163

ε

−ε

θε (t) δε (t) φ (t) dt →

1 φ (0) 2

as ε → 0. This completes the proof.

6.5

Input-Output Systems

The input-output form of dynamical systems may be convenient for diﬀerent reasons, for instance, when dealing with control problems. In many linear cases, input-output systems are represented in the form of a single high order equation an

dn y dy dm u du + ... + a1 + a0 y = bm m + ... + b1 + b0 u n dt dt dt dt

(6.71)

where u = u(t) and y = y(t) are input and output, respectively, and an , ... , a1 , a0 , bm , ... ,b1 , b0 are constant coeﬃcients. For illustration purposes, a two-degrees-of-freedom model as shown in Fig. 6.10 is considered, although the general case (6.71) can be handled in the same way.

Fig. 6.10 Two mass-spring model.

Eliminating x2 (t) from the system, gives a single higher-order equation with respect to the another coordinate, x1 (t), in the form k1 k2 d4 x1 d3 x1 m1 d2 x1 c1 dx1 + + c1 3 + (k1 + k2 + k2 ) 2 + k2 x1 4 dt dt m2 dt m2 dt m2 d2 F1 k2 = + F1 (6.72) 2 dt m2 m1

System (6.72) is a particular case of (6.71), where n = 4 and m = 2. Let us consider the step-wise discontinuous periodic function F1 (t) = u(t) = e(ωt) and represent equation (6.72) in the form

164

6 NSTT for Linear and Piecewise-Linear Systems

a4

d4 y dy + ... + a1 + a0 y = b2 ω 2 e + b1 ωe + b0 e 4 dt dt

(6.73)

where ≡ d/d(ωt), and all the coeﬃcients and variables are identiﬁed by comparing (6.72) to (6.73). The right-hand side of equation (6.73) contains discontinuous and singular functions, therefore equation (6.73) must be treated in terms of distributions. Nevertheless, let us show that, on the manifold of periodic solutions, equation (6.73) is equivalent to some classic boundary-value problem. Let us represent the output in the form y(t) = X(τ ) + Y (τ )e

(6.74)

where τ = τ (ωt) and e = e(ωt). When diﬀerentiating expression (6.74) step-by-step one should eliminate the singular term e in the ﬁrst two derivatives by sequentially setting boundary conditions as follows dy = (Y + X e)ω, dt d2 y = (X + Y e)ω 2 , dt2

Y |τ =±1 = 0

(6.75)

X |τ =±1 = 0

However, it is dictated by the form of the input in (6.73), that the singular terms e and e must be preserved on the next two steps given by d3 y = (Y + X e + Y e )ω 3 dt3 d4 y = (X (4) + Y (4) e + X e + Y e )ω 4 dt4

(6.76)

The fourth-order derivative in (6.76) takes into account the equality ee = 0, which easily follows from (6.53) in the symmetric case β = 0. Substituting (6.75) and (6.76) in (6.73), and considering {1, e, e, e } as a linearly independent basis, gives equations a4 ω 4 X IV + a3 ω 3 Y + a2 ω 2 X + a1 ωY + a0 X = 0 a4 ω 4 Y IV + a3 ω 3 X + a2 ω 2 Y + a1 ωX + a0 Y = b0

(6.77)

under the boundary conditions at τ = ±1: Y = 0, b2 ω 2 Y = , a4

X = 0 ω 3 X

1 = a4

a3 b1 − b2 a4

(6.78)

In contrast to equation (6.73), the boundary value problem (6.77) and (6.78) does not include discontinuous terms any more.

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

165

Although the number of equations in (6.77) is doubled as compared to (6.73), such a complication is rather formal due to the symmetry of the equations. Indeed, introducing the new variables, U = X +Y and V = X −Y , decouples system (6.77) in such a way that the corresponding roots of the characteristic equations diﬀer just by signs. (Besides, this fact reveals the possibility of using the idempotent basis for decoupling the resultant set of equations as discussed in Chapter 4 and will be discussed later in this chapter.) In addition, the type of the symmetry suggests that X(τ ) and Y (τ ) are odd and even functions, respectively. This enables one of reducing the general form of solution to a family of solutions with four arbitrary constants

2 α α βj βj j j τ sin τ + Bj sinh τ cos τ (6.79) Aj cosh X= ω ω ω ω j=1

2 α α βj βj b0 j j τ sin τ + Bj cosh τ cos τ + Y = Aj sinh ω ω ω ω a 0 j=1

where αj ± βj i are complex conjugate roots of the characteristic equation a4 p4 + ... + a1 p + a0 = 0

(6.80)

The assumption that both of the roots are complex reﬂects the physical meaning of the example, however other cases would lead to even less complicated expressions. Finally, substituting (6.79) in (6.78) gives a linear algebraic set of four independent equations with respect to four constants: A1 , A2 , B1 and B2 . Although the corresponding analytical solution is easy to obtain by using the R standard Mathematica commands, the result is somewhat complicated for reproduction. Practically, it may be reasonable to determine the constants by setting the system parameters to their numerical values moreover that only numerical solution are often possible for characteristic equations.

6.6

Piecewise-Linear Oscillators with Asymmetric Characteristics

Piecewise-linear oscillators are often considered as ﬁnite degrees-of-freedom models of cracked elastic structures [32],[2],[192], but may occur also due to speciﬁc design solutions. In many cases, the corresponded periodic solutions can be combined of diﬀerent pieces of linear solutions valid for two diﬀerent subspaces of the conﬁguration space [33], [75], [192]. In this section, it will be shown that the nonsmooth transformation of time results in a closed form analytical solution matching both pieces of the solution automatically by means of elementary functions.

166

6.6.1

6 NSTT for Linear and Piecewise-Linear Systems

Amplitude-Phase Equations

Let us consider a piece-wise linear oscillator of the form m¨ q + k[1 − εH(q)]q = 0

(6.81)

where H(q) is Heaviside unit-step function, m and k are mass and stiﬀness parameters, respectively, and |ε| 1. Therefore, k− = k and k+ = k(1 − ε) are elastic stiﬀness of the oscillator for q < 0 and q > 0, respectively. The exact general solution of oscillator (6.81) can be obtained by satisfying the continuity conditions for q and q˙ at the matching point q = 0, where the characteristic has a break. Such approaches are often facing quite challenging algebraic problems, however, as the number of degrees of freedom increases or external forces are involved. This is mainly due to the fact that times of crossing the point q = 0 are a priory unknown. In this section, it will be shown that, applying a combination of asymptotic expansions with respect to ε and nonsmooth temporal transformations, gives a unit-form solution for oscillator (6.81) with a possibility of generalization on the normal mode motions of multiple degrees-of-freedom systems. In particular, the nonsmooth temporal transformation: 1) provides an automatic matching the motions from diﬀerent subspaces of constant stiﬀness, and 2) justiﬁes quasi-linear asymptotic solutions for the speciﬁc nonsmooth case of piece-wise linear characteristics. Let us clarify the above two remarks. Introducing the notation ω 2 = k/m, brings equation (6.81) to the standard form of a weakly non-linear oscillator q¨ + ω 2 q = εω 2 H(q)q

(6.82)

The non-linear perturbation on the right-hand side of oscillator (6.82) is a continuous but non-smooth function of the coordinate q. Since the major algorithms of quasi-linear theory assume smoothness of non-linear perturbations, then such algorithms are not applicable in this case unless appropriate modiﬁcations and extensions have been made. Even though deriving ﬁrst-order asymptotic solutions usually require no diﬀerentiation of characteristics, dealing with two pieces of the solution may complicate any further stages. Let us show that combining quasi-linear methods of asymptotic integration, such as Krylov-Bogolyubov averaging, with nonsmooth temporal transformations results in a closed form analytical solution for piece-wise linear oscillator (6.81). Note that oscillator (6.81) plays an illustrative role for the approach developed below. Then a more complicated case will be considered. At this stage, let us introduce the amplitude-phase coordinates {A(t), ϕ(t)} on the phase plane of oscillator (6.81) through relationships

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

q = A cos ϕ q˙ = −ωA sin ϕ

167

(6.83)

The following compatibility condition is imposed on transformation (6.83) A˙ cos ϕ − A sin ϕϕ˙ = −ωA sin ϕ

(6.84)

Substituting (6.83) in (6.82) and taking into account (6.84), gives 1 A˙ = − εωAH(A cos ϕ) sin 2ϕ 2 ϕ˙ = ω − εωH(A cos ϕ) cos2 ϕ

(6.85)

The right-hand sides of equations (6.85) are 2π-periodic with respect to the phase variable, ϕ. Therefore, nonsmooth transformation of the phase variable applies through the couple of functions τ = τ (2ϕ/π)

and

e = e(2ϕ/π)

(6.86)

Assuming that A ≥ 0 and taking into account the obvious identities, sin ϕ = sin(πτ /2) cos ϕ = cos(πτ /2)e H(A cos ϕ) = (1 + e)/2 e2 = 1

(6.87)

brings (6.85) to the form 1 A˙ = − εω(1 + e)A sin πτ 4 1 πτ ϕ˙ = ω − εω(1 + e) cos2 2 2

(6.88) (6.89)

Note that the right-hand sides of (6.88) and (6.89) are nonsmooth but continuous with respect to the phase ϕ since the step-wise discontinuities of the rectangular cosine e(2ϕ/π) are suppressed by the factors sin πτ and cos2 (πτ /2), respectively.

6.6.2

Amplitude Solution

Let us show that equation (6.88) has an exact 2π-periodic solution with respect to the phase variable, ϕ. According to the idea of NSTT, any periodic solution can be represented in the form A = X(τ ) + Y (τ )e (6.90) where τ and e are deﬁned by (6.86).

168

6 NSTT for Linear and Piecewise-Linear Systems

Substituting (6.90) in (6.88) and taking into account (6.89), gives boundaryvalue problem (X − Y ) = 0 επ (X + Y ) sin πτ =− X +Y 4 1 − ε cos2

πτ 2

Y |τ =±1 = 0

(6.91) (6.92)

where ≡ d/dτ . Solution of the boundary value problem, (6.91) and (6.92), is obtained by elementary integration. Then representation (6.90) gives A(ϕ) = α[1 + ζ(τ )] − α[1 − ζ(τ )]e πτ −1/2 ) ζ(τ ) = (1 − ε cos2 2

(6.93)

where τ = τ (2ϕ/π), e = e(2ϕ/π), and α is an arbitrary positive constant. Note that solution (6.93) exactly captures the amplitude in both subspaces q < 0 and q > 0. However, the temporal mode shape and the period essentially depend on the phase variable ϕ described by equation (6.89). Generally speaking, the phase equation (6.89) admits exact integration, but the result would appear to have implicit form. Alternatively, it is shown below that solution for the phase variable can be approximated by asymptotic series in the explicit form 1 ϕ = φ − ε[πτ + (1 + e) sin πτ ] 8 1 2 − ε {4(2 − cos πτ )(πτ + sin πτ ) 128 −[4πτ (1 + cos πτ ) − 8 sin πτ + sin 2πτ ]e} + O(ε3 )

(6.94)

where τ = τ (2φ/π), e = e(2φ/π), and 3 1 φ = ω[1 − ε − ε2 + O(ε3 )]t 4 32

(6.95)

Note that the functions τ and e in (6.93) and (6.94) depend on the diﬀerent arguments.

6.6.3

Phase Solution

In this subsection, a second-order asymptotic procedure for phase equations with non-smooth periodic perturbations is introduced. If applied to equation (6.89), the developed algorithm gives solution (6.94).

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

169

Let us consider some phase equation of the general form ϕ˙ = ω[1 + εf (ϕ)]

(6.96)

where f (ϕ) is a 2π-periodic, nonsmooth or even step-wise discontinuous function, and ε is a small parameter, |ε| 1. Using the basic NSTT identity for f (ϕ), brings equation (6.96) to the form ϕ˙ = ω + εω{G[τ (2ϕ/π)] + M [τ (2ϕ/π)]e}

(6.97)

where the functions G(τ ) and M (τ ) are expressed through f (ϕ). Note that the class of smoothness of the periodic perturbation in equation (6.97) depends on the behavior of functions G(τ ) and M (τ ) and their derivatives at the boundaries τ = ±1. If, for instance, M (±1) = 0 then the perturbation is step-wise discontinuous whenever τ (2ϕ/π) = ±1. Let us introduce the asymptotic procedure for equation (6.97). Note that, in case ε = 0, the right-hand side of equation (6.97) is constant. So, following the idea of asymptotic integration, let us ﬁnd phase transformation ϕ = φ + εF1 (φ) + ε2 F2 (φ) + ...

(6.98)

where functions Fi (φ) are such that the new phase variable also has a constant temporal rate even though ε = 0. In other words, transformation (6.98) should bring equation (6.97) to the form (6.99) φ˙ = ω(1 + εγ1 + ε2 γ2 + ...) where γi are constant coeﬃcients to be determined together with Fi (φ) during the asymptotic procedure. Note that the procedure, which is described below, has several speciﬁc features due to the presence of nonsmooth periodic functions. In particular, high-order approximations require a non-conventional interpretation for power series expansions; see the next subsection for the related remarks. Other modiﬁcations occur already in the leading order approximation. Substituting (6.98) into equation (6.97), then enforcing equation (6.99) and collecting the terms of order ε, gives F1 (φ) = G(τ ) + eM (τ ) − γ1

(6.100)

where the triangular and rectangular waves depend now on the new phase variable φ as τ = τ (2φ/π) and e = e(2φ/π), respectively. According to the conventional averaging procedure, the constant γ1 is selected to achieve zero mean on the right-hand side of equation (6.100) and thus provide periodicity of solution, F1 (φ). In the algorithm below, the periodicity is due to the form of representation for periodic solutions, whereas the operator of averaging occurs automatically from the corresponding conditions of smoothness that is boundary conditions for the solution components.

170

6 NSTT for Linear and Piecewise-Linear Systems

So we seek solution of equation (6.100) in the form F1 (φ) = U1 (τ ) + eV1 (τ )

(6.101)

Substituting (6.101) in (6.100) and following the NSTT procedure, gives the boundary-value problem π M (τ ) 2 π V1 (τ ) = [G(τ ) − γ1 ] 2 V1 (±1) = 0 U1 (τ ) =

(6.102)

Note that there are two conditions on the function V1 (τ ) described by the ﬁrst-order diﬀerential equation in (6.102). However, there is a choice for γ1 , which is to satisfy one of the two conditions. As a result, solution of boundaryvalue problem (6.102) is obtained by integration in the form π τ U1 (τ ) = M (z)dz 2 0 π τ [G(z) − γ1 ]dz (6.103) V1 (τ ) = 2 −1 1 1 G(τ )dτ γ1 = 2 −1 Further, collecting the terms of order ε2 , gives F2 (φ) = G2 (τ ) + eM2 (τ ) + P2 (τ )e − γ2

(6.104)

where 2 [U1 (τ )G (τ ) + V1 (τ )M (τ )] − M (τ )γ1 π 2 G2 (τ ) = U1 (τ )M (τ ) − G(τ )γ1 + γ12 π 2 P2 (τ ) = U1 (τ )M (τ ) π e ≡ de(2φ/π)/d(2φ/π)

M2 (τ ) =

(6.105)

In contrast to ﬁrst-order equation (6.100), equation (6.104) includes the singular term P2 (τ )e produced by the power series expansion of the perturbation in equation (6.97). If the perturbation is smooth then P2 (±1) = 0 and such singular term disappear; see the example below for illustration. Nevertheless, the second-order approximation remains valid even in discontinuous case, when P2 (±1) = 0.

6.6 Piecewise-Linear Oscillators with Asymmetric Characteristics

171

So let us represent solution of equation (6.104) in the form F2 (φ) = U2 (τ ) + eV2 (τ )

(6.106)

Then, substituting (6.106) in (6.104), gives boundary-value problem π M2 (τ ) 2 π V2 (τ ) = [G2 (τ ) − γ2 ] 2 π V2 (±1) = P2 (±1) 2 U2 (τ ) =

(6.107)

In contrast to (6.102), boundary-value problem (6.107) has, generally speaking, non-homogeneous boundary conditions for V2 . These conditions compensate the singular term e from diﬀerential equation (6.104). As a result equations (6.107) are free of any singularities and admit solution analogously to ﬁrst-order equations (6.102), π τ M2 (z)dz U2 (τ ) = 2 0 π τ π [G2 (z) − γ2 ]dz + P2 (−1) (6.108) V2 (τ ) = 2 −1 2 1 1 1 G2 (τ )dτ + [P2 (−1) − P2 (1)] γ2 = 2 −1 2 Now, we return to the illustrating model. In particular case (6.89), one has 1 πτ G(τ ) ≡ M (τ ) ≡ − cos2 2 2

(6.109)

and G(±1) = M (±1) = 0 G (±1) = M (±1) = 0 G (±1) = M (±1) = −π 2 /4

(6.110)

where ≡ d/dτ . First two lines of conditions (6.110) provide continuity for the right hand side of (6.97) and its ﬁrst derivative at those ϕ where τ (2ϕ/π) = ±1. As follows from (6.105), for this class of smoothness one has P2 (±1) = 0 and thus no singular terms occur in the ﬁrst two steps of asymptotic procedure. Finally, taking into account (6.109) and (6.110) and conducting integration in (6.103) and (6.108), brings solution (6.98) to the form (6.94) and (6.95).

172

6.6.4

6 NSTT for Linear and Piecewise-Linear Systems

Remarks on Generalized Taylor Expansions

Nonsmoothness of the triangular sine is similar to that function |t| has at zero. So let us consider its formal power series |t + ε| = |t| + |t| ε +

1 2 |t| ε + ... 2!

(6.111)

where ε > 0 and −∞ < t < ∞, and prime indicates Schwartz derivative. It is clear that equality (6.111) has no regular point-wise meaning. For instance, equality (6.111) is obviously not true on the interval −ε < t < 0. In addition, the right-hand side of (6.111) is uncertain at t = 0, whereas the left-hand side gives ε. Nevertheless, let us show that equality (6.111) admits a generalized interpretation and holds in terms of distributions. Let ψ(t) be a test function in terms of the distribution theory, more precisely, ψ(t) is inﬁnitely diﬀerentiable with compact support that is identically zero outside of some bounded interval. Integrating by parts and then shifting the variable of integration, gives ∞

1 |t| + |t| ε + |t| ε2 + ... ψ(t)dt 2! −∞

∞ 1 = |t| ψ(t) − ψ (t)ε + ψ (t)ε2 − ... dt (6.112) 2! −∞ ∞ ∞ |t|ψ(t − ε)dt = |t + ε|ψ(t)dt = −∞

−∞

Therefore, equality (6.111) holds in the integral sense of distributions.

2

1 q

0

-1

-2 0

10

20 t

30

40

Fig. 6.11 Second-order asymptotic and numerical solutions shown by solid and dashed lines, respectively

6.7 Multiple Degrees-of-Freedom Case

173

Fig. 6.11 compares analytical solution (6.83), (6.93) and (6.94) shown by the solid line and numerical solution shown by the dashed line. As expected, the amplitude show the perfect match, whereas some phase shift develops after several cycles.

6.7

Multiple Degrees-of-Freedom Case

Let us consider a multiple degrees-of-freedom piecewise-linear system of the form Mx ¨ + Kx = εH(Sx)Bx (6.113) where x(t) ∈ Rn is a vector-function of the system coordinates, M is a mass matrix, H denotes the Heaviside unit-step function, S is a normal vector to the plane splitting the conﬁguration space into two parts with diﬀerent elastic properties, so that the stiﬀness matrix is K when Sx < 0 and K − εB when Sx > 0. It is assumed that the stiﬀness jump is small, |ε| 1. The number of possible iterations of the classic perturbation tools usually depends on a class of smoothness of the perturbation. The perturbation term on the right-hand side of (6.113) is continuous but nonsmooth. Therefore, only ﬁrst-order asymptotic solution can be obtained within the classic theory of diﬀerential equations. Moreover, the piecewise character of the perturbation complicates the form of the solution due to the necessity of matching the diﬀerent pieces of the solution.

Fig. 6.12 Two degrees-of-freedom piecewise-linear system as a model of a rod with a small crack.

However, we show that the idea of nonsmooth time transformation gives a unit-form solution by automatically matching the pieces of solution in two diﬀerent conﬁguration subspaces with diﬀerent stiﬀness properties. Let seek a 2π-periodic normal mode solution of (6.113) with respect to the phase ϕ in the form x(ϕ) = Aj cos ϕ + εx(1) (ϕ) + O(ε2 ) ) ϕ = ωj 1 + εγ (1) + O(ε2 )t

(6.114)

174

6 NSTT for Linear and Piecewise-Linear Systems

where ωj and Aj are arbitrary eigen frequency and eigen vector (normal mode) of the linearized system: (−ωj2 M + K)Aj = 0;

j = 1,...,n

(6.115)

Substituting (6.114) in (6.113), taking into account (6.87) and (6.115) in the ﬁrst order of ε, gives

1 d2 x(1) 1 πτ (1) (1) ωj2 M BA BA (6.116) + Kx = + ( + γ KA )e cos j j j 2 dϕ 2 2 2 where τ = τ (2ϕ/π), e = e(2ϕ/π), and the relationship (1 + εγ (1) )−1 = 1 − εγ (1) + O(ε2 ) has been used. Since the function x(1) (ϕ) is sought to be 2π-periodic with respect to ϕ, we represent it in the form x(1) = X(τ ) + Y (τ )e

(6.117)

This gives the boundary-value problem

2 2ωj 1 πτ M X + KX = BAj cos , X |τ =±1 = 0 π 2 2

2 2ωj 1 πτ BAj + γ (1) KAj cos M Y + KY = π 2 2 Y |τ =±1 = 0

(6.118) (6.119)

Representing the corresponding solution in terms of the normal mode coordinates n n X= Ai Xi (τ ), Y = Ai Yi (τ ) (6.120) i=1

i=1

and taking into account M -orthogonality of the set of eigen vectors, gives

2 2ωj πτ , Xi |τ =±1 = 0 Xi + ωi2 Xi = βij cos (6.121) π 2

2 2ωj πτ , Yi |τ =±1 = 0 (6.122) Yi + ωi2 Yi = (βij + γ (1) κij ) cos π 2

where βij =

1 Ai BAj , 2 Ai M Ai

κij =

Ai KAj Ai M Ai

(6.123)

are dimensionless coeﬃcients. Note that despite of the similar representation for solution (6.114), there is a noticeable diﬀerence between the classic Poincare-Lindshtedt method and current procedure due to (6.117). Namely, according to the

6.7 Multiple Degrees-of-Freedom Case

175

Poincare-Lindshtedt method, the frequency correction term γ (1) is to kill the so called secular terms in the asymptotic expansions. In our case, the secular terms appear to be periodic due to the ‘built in’ periodicity of the new temporal argument. However, periodicity of solutions is provided by the existence of solutions of the boundary-value problems, such as (6.121) and (6.122). Due to the linearity, the existence of solutions allows the direct veriﬁcation. So if i = j then both problems (6.121) and (6.122) are solved in the standard way with no presence of γ (1) because κij = 0. The corresponding solution is given by

ωj βij πωi τ πωi πτ − cos csc Xi = 2 cos (6.124) ωi − ωj2 2 ωi 2ωj 2ωj Yi =

ωi2

βij πτ cos 2 − ωj 2

(6.125)

In the particular case i = j, problem (6.121) still has a solution, but problem (6.122) generally speaking does not. Fortunately, in this case, we have κjj = 0 and thus the problem is set to have the trivial solution by condition γ (1) = −

βjj κjj

(6.126)

Therefore, πβjj Xj = 4ωj2

2 πτ πτ + cos τ sin 2 π 2

Yj = 0

(6.127) (6.128)

So expressions (6.117), (6.120), and (6.124) through (6.128) completely determine the ﬁrst order approximation x(1) (ϕ). Let us consider the example of mass-spring model m1 x ¨1 + (k1 + k2 )x1 − k2 x2 = εk1 H(x1 )x1 ¨2 − k2 x1 + (k2 + k3 )x2 = 0 m2 x

(6.129)

Equations (6.111) can be represented in the form (6.113), where

m1 0 k + k2 −k2 k 0 M = , K= 1 , B= 1 −k2 k2 + k3 0 m2 0 0 x1 , S= 10 x= x2 In this case, the ﬁrst-order asymptotic solution for the in-phase (j = 1) and out-of-phase (j = 2) takes the form, respectively,

176

6 NSTT for Linear and Piecewise-Linear Systems

επ 2 πτ πτ πτ + cos + τ sin + O(ε2 ) 2 16 π 2 2 πτ εk1 πτ πτ e cos − + cos (6.130) x2 = e cos 2 8k2 2 2

−1/2 k2 k2 πτ k2 π / sin + O(ε2 ) − 1+2 cos 1+2 1+2 k1 k1 2 k1 2

x1 = e cos

ϕ= and

k1 m

ε 1 − + O(ε2 )t 4

εk1 πτ k2 πτ πτ x1 = −e cos + + cos − 1+2 e cos 2 8k2 2 2 k1 πτ k2 k2 π / 1+2 / 1+2 × cos / sin + O(ε2 ) (6.131) 2 k1 2 k1

εk1 π 2 πτ πτ πτ + cos + τ sin x2 = e cos + O(ε2 ) 2 16(k1 + 2k2 ) π 2 2 * k1 + 2k2 εk1 + O(ε2 )t 1− ϕ= m 4(k1 + 2k2 )

where it is assumed that m1 = m2 = m. Solutions (6.130) and (6.100) show that a bi-linearity may have quite diﬀerent eﬀect on diﬀerent modes. In particular, solution (6.130) reveals the possibility of internal resonances, when

πω2 k2 ω2 sin = 1+2 (6.132) = 0, 2ω1 ω1 k1 If, for instance, the system is close to the frequency ratio ω2 /ω1 = 2 then the in phase mode may be aﬀected signiﬁcantly by a crack even under very small magnitudes of the parameter ε. In contrary, solution (6.100) has the denominator sin[(π/2)ω1 /ω2 ], which is never close to zero because 0 < ω1 /ω2 < 1. Therefore, in current asymptotic approximation, the inﬂuence of crack on the out-of phase mode is always of order ε provided that k2 /k1 = O(1). The inﬂuence of the bilinear stiﬀness on inphase mode trajectories in the closed to internal resonance case is seen from Fig. 6.13, where both analytical and numerical solutions are shown for comparison reasons. The frequency ratio ω2 /ω1 = 2.0025 is achieved by conditioning the spring stiﬀness parameters as follows k2 = (3/2)k1 + 0.005.

6.8 The Amplitude-Phase Problem in the Idempotent Basis

177

1.0

0.5 Ε 0.01 x2

Ε0

0.0

0.5

1.0 1.0

0.5

0.0

0.5

1.0

x1 Fig. 6.13 The inﬂuence of a small “crack” ε = 0.01 on the in-phase mode trajectory near the frequency ratio ω2 /ω1 = 2; the dashed line shows the numerical solution, and the thin solid line corresponds to the perfect (linear) case ε = 0.

6.8

The Amplitude-Phase Problem in the Idempotent Basis

Recall that the idempotent basis is given by e+ = (1+e)/2 and e− = (1−e)/2 so that e2+ = e+ , e2− = e− and e+ e− = 0. Equations (6.88) and (6.89) therefore take the form 1 A˙ = − εωe+ A sin πτ 2 πτ ϕ˙ = ω − εωe+ cos2 2

(6.133) (6.134)

Let us represent the amplitude as a function of ϕ in the form A(ϕ) = X+ (τ )e+ + X− (τ )e−

(6.135)

where e+ = e+ (2ϕ/π), e− = e− (2ϕ/π) and τ = τ (2ϕ/π). Substituting (6.135) in (6.133) and taking into account (6.134), gives 1 πτ 2 (X+ e+ − X− ) = − εωe+ (X+ e+ + X− e− ) sin πτ e− )(ω − εωe+ cos2 π 2 2

178

6 NSTT for Linear and Piecewise-Linear Systems

or (1 − ε cos2

πτ π )X+ = − εX+ sin πτ 2 4 =0 X−

(6.136)

under the boundary condition (X+ − X− )|τ =±1 = 0

(6.137)

The boundary-value problem (6.136) and (6.137) admits exact solution so that (6.135) gives ﬁnally A(ϕ) = α[(1 − ε cos2

πτ −1/2 ) e+ + e− ] 2

where α is an arbitrary positive constant.

(6.138)

Chapter 7

Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

Abstract. In this chapter, two variable expansions are introduced, where the fast temporal scale is represented by the triangular wave. In contrast to the conventional two variable procedure, the diﬀerential equations for slow dynamics emerge from the boundary conditions eliminating discontinuities rather than resonance terms. For illustrating purposes, an impulsively loaded single degree-of-freedom model with qubic damping and no elastic force is considered. Further, the method is applied to the Duﬃng’s oscillator under the periodic impulsive excitation whose principal frequency is close to the linear resonance frequency. Note that the illustrating models are weakly nonlinear, nevertheless the triangular wave temporal argument adequately captures speciﬁcs of the impulsive loading and, as a result, provides closed form asymptotic solutions. Following the conventional quasi-harmonic approaches, in such cases, would face generalized Fourier expansions for the external forcing function with no certain leading term.

7.1

Nonsmooth Two Variables Method

Solutions of Cauchy problems under arbitrary initial conditions are, generally speaking, non-periodic. However, in case of amplitude or frequency modulated oscillating motions, the idea of averaging can be involved in order to obtain the corresponding solutions. A proper formalization can be developed by introducing slow temporal variables in addition to the fast oscillating time τ . Consider, for instance, ﬁrst order impulsively loaded system v˙ + εv 3 = p

de (t) dt

(7.1)

under the initial condition v |t=0 = v 0

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 179–193, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(7.2)

180

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

where ε is a small enough positive parameter, and p is a constant parameter characterizing the strength of pulses. The Cauchy problem (7.1) and (7.2) describes one-dimensional motions of a material point under the periodic series of impulses and non-linear damping forces. Note that any standard formulation of the idea of averaging is diﬃcult to apply directly to equation (7.1) since its right hand side is neither smooth nor even continues function of time. Analyzing the both sides of equation (7.1), shows that solutions should have step-wise discontinuities at the points Λ = {t : τ (t) = ±1}. Let us represent such solutions in the form (7.3) v = X τ, t0 + Y τ, t0 e where τ = τ (t), e = e (t) and t0 = εt. Therefore, the solution is assumed to depend upon the two time scales, such as the fast oscillating time τ and the slow time t0 . Substituting now (7.3) into (7.1), gives

∂X ∂Y 3 2 +ε + X + 3XY ∂τ ∂t0

∂X ∂Y 2 3 + +ε + 3Y X + Y e + (Y − p)e˙ = 0 ∂τ ∂t0 This ﬁnally leads to the set of partial diﬀerential equations

∂Y ∂X 2 3 = −ε + 3Y X + Y ∂τ ∂t0

∂X ∂Y 3 2 = −ε + X + 3XY ∂τ ∂t0

(7.4)

under boundary conditions Y |τ =±1 = p

(7.5)

Following the standard procedure of the two-variable expansions, we represent solution of the boundary value problem (7.4) and (7.5) is in the form of asymptotic series X=

∞

εi Xi τ, t0 ,

i=0

Y =

∞

εi Yi τ, t0

(7.6)

i=0

Substituting (7.6) into (7.4) and (7.5) and matching coeﬃcients of the same powers of ε, gives a sequence of simpliﬁed boundary value problems. In particular, zero order problem is ∂X0 = 0, ∂τ

∂Y0 =0 ∂τ

7.1 Nonsmooth Two Variables Method

181

Y0 |τ =±1 = p Both equations and the boundary condition are satisﬁed by solution X0 = A0 t0 , Y0 = p

(7.7)

where A0 is an arbitrary function of the slow time. Taking into account (7.7), gives the following ﬁrst order problem ∂X1 = − 3pA20 + p3 ∂τ

dA0 ∂Y1 3 2 =− + A0 + 3A0 p ∂τ dt0 Y1 | τ =±1 = 0 Integrating equations (7.8), gives general solution X1 = − 3pA20 + p3 τ + A1 t0

dA0 3 2 Y1 = − + A0 + 3A0 p τ + C1 t0 dt0 The boundary condition for Y1 in (7.8) dictates C1 t0 ≡ 0 and

(7.8)

(7.9)

dA0 + A30 + 3A0 p2 = 0 (7.10) dt0 Therefore Y1 ≡ 0, whereas function A1 t0 in the expression for X1 remains unknown. Equation (7.10) admits exact solution in the following explicit form

2 0 −1/2 1 A0 = − 2 + const · exp 6p t 3p

Finally, the leading order asymptotic solution of the Caushy problem is written as

2 −1/2 0 −2 1 1 exp 6p + v − p εt + pe (t) + O (ε) v(t) = ± − 2 + 3p 3p2 (7.11) where plus or minus sign are taken if v 0 − p > 0 or v 0 − p < 0, respectively. Substituting asymptotic solution (7.11) in equation (7.1), leads to the error εR(εt)e (t), where R(εt) is a function of slow time scale. Although the error is of the same order of magnitude as the perturbation, it relates to the singular component of the equation with a scaling factor of order one. In other words, for such kind of systems, regular and singular error components

182

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

0.8 0.6 v 0.4 0.2 0.0 0

10

20 t

30

40

Fig. 7.1 Velocity of the nonlinearly damped material point under the pulsed periodic excitation.

should be considered separately. Note that the ﬁrst term in expression (7.11) is a transient response of the system that vanishes as time increases; see Fig. 7.1 for illustration. In conclusion, it is seen from solution (7.11) that the amplitude of impulses may have a strong eﬀect on the time scale of transient response of the system.

7.2

Resonances in the Duﬃng’s Oscillator under Impulsive Loading

Let us consider the Duﬃng’s oscillator under the periodic impulsive excitation

de (ωt) 2 3 x ¨ + ω0 x = −ε x − p (7.12) d (ωt) Introducing the detuning parameter σ into the diﬀerential equation (7.12) as ω02 = gives

nπ 2 ω + εσ 2

nπ 2 de (ωt) 3 ω x = −ε σx + x − p x¨ + 2 d (ωt)

(7.13)

(7.14)

where n is an odd positive integer. Recall that, in the near resonance condition (7.13), the factor π/2 occurs due to the speciﬁc normalization of the triangular sine wave period, T = 4.

7.2 Resonances in the Duﬃng’s Oscillator under Impulsive Loading

Let us represent solutions of equation (7.14) in the form x(t) = X τ (ωt) , t0 + Y τ (ωt) , t0 e (ωt)

183

(7.15)

where t0 = εt is the slow time used in the previous section. Based on the continuity condition for x (t),

one obtains

Y |τ =±1 = 0

(7.16)

∂Y ∂X ∂Y ∂X x˙ = ω +ε 0 + ω +ε 0 e ∂τ ∂t ∂τ ∂t

(7.17)

Then, substituting (7.15) and (7.17) in (7.14), eventually gives the boundary value problem (7.16), ∂X |τ =±1 = εp (7.18) ω2 ∂τ and nπ 2 2 ∂2X ∂2Y 2∂ X ω X + ε + ω 2 2 + 2εω ∂τ ∂τ ∂t0 ∂t02 2 3 2 = −ε σX + X + 3Y X nπ 2 2 ∂2Y ∂2X 2∂ Y ω Y + ε + (7.19) ω 2 2 + 2εω ∂τ ∂τ ∂t0 ∂t02 2 = −ε σY + Y 3 + 3X 2 Y Note that, due to substitution (7.15), the boundary value problem, (7.16), (7.18) and (7.19), includes no singular functions. Therefore, further averaging procedures can be correctly applied. Following the formalism of two variable expansions, we seek asymptotic solutions in the form of power series X = X0 τ, t0 + εX1 τ, t0 + ε2 X2 τ, t0 + ... Y = Y0 τ, t0 + εY1 τ, t0 + ε2 Y2 τ, t0 + ... (7.20) Substituting (7.20) in (7.16), (7.18) and (7.19), and matching coeﬃcients of the same degrees of ε, gives a series of boundary value problems, where the leading order problem is ∂ 2 X0 nπ 2 + X0 = 0, ∂τ 2 2 ∂ 2 Y0 nπ 2 + Y0 = 0, ∂τ 2 2

∂X0 |τ =±1 = 0 ∂τ Y0 |τ =±1 = 0

(7.21)

These equations and the boundary conditions are satisﬁes at any t0 by solution

184

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

nπ nπ τ, Y0 = D0 t0 cos τ (7.22) X0 = A0 t0 sin 2 2 where A0 t0 and D0 t0 are arbitrary functions of the slow time scale to be determined on the next step of iteration. So, collecting all terms of order ε, gives the boundary value problem of the next asymptotic order as 2

nπ 2 ∂ 2 Y0 2 ∂ X1 3 2 + X1 = − 2ω + σX0 + X0 + 3Y0 X0 ω ∂τ 2 2 ∂τ ∂t0 ∂X1 ω2 |τ =±1 = p (7.23) ∂τ ω2

∂ 2 Y1 nπ 2 ∂ 2 X0 3 2 + Y + σY + Y + 3X Y = − 2ω 1 0 0 0 0 ∂τ 2 2 ∂τ ∂t0 Y1 |τ =±1 = 0

The right-hand sides of equations (7.23) of the slow time implicitly depend t0 through still unknown functions A0 t0 and D0 t0 , and their derivatives A0 t0 and D0 t0 . Solution of the diﬀerential equations (7.23) is found to be nπτ nπτ + B1 t0 cos X1 = A1 t0 sin 2 2

1 A0 nπτ 3 2 3 2 + σ + A0 + D0 − D0 τ cos ω nπω 4 4 2

A0 3 1 nπτ 7 − σ + A20 + D02 − D0 sin nπω nπω 8 8 2 A0 3nπτ − 2 2 2 A20 − 3D02 sin 8n π ω 2

(7.24)

and nπτ nπτ Y1 = C1 t0 sin + D1 t0 cos 2

2 1 D0 nπτ 3 2 3 2 − σ + A0 + D0 + A0 τ sin ω nπω 4 4 2

D0 1 7 nπτ 3 − σ + A20 + D02 + A0 cos nπω nπω 8 8 2 2 D0 3nπτ − 2 2 2 3A0 − D02 cos 8n π ω 2 Now the boundary condition for X1 gives B1 t0 ≡ 0

(7.25)

7.3 Strongly Nonlinear Oscillator under Periodic Pulses

and

nπ A0 3 2 2p 2 A0 + D0 + sin D0 = σ+ nπω 4 nπω 2

185

(7.26)

Then, the boundary condition for Y1 gives C1 t0 ≡ 0 and

A0 = −

D0 3 2 A0 + D02 σ+ nπω 4

(7.27)

In current approximation, another two arbitrary functions of the slow time scale can be chosen as A1 t0 ≡ 0 and D1 t0 ≡ 0, at least, because the same type of terms are included already into the generating solution (7.22). Substituting derivatives from (7.26) and (7.27) in (8.24) and (7.25) and taking into account (7.22), gives ε nπτ nπτ 1 − 2 2 2 A20 − 3D02 cos2 X = A0 sin 2 2n π ω 2 2εp nπτ nπτ nπ − 2 2 2 sin nπτ cos − sin + O(ε2 ) (7.28) n π ω 2 2 2

Y = D0 cos

ε nπτ nπτ 1 + 2 2 2 3A20 − D02 sin2 + O(ε2 ) 2 2n π ω 2

(7.29)

Substituting (7.28) and (8.87) in (7.15), ﬁnally gives the closed form approximate solution x(t), despite of the discontinuous impulsive loading. Figs. 7.2, 7.3, and 7.4 illustrate the results of calculations under the following parameters and initial conditions: n = 1, ω = 1.0, σ = 0.1, p = 1.0, ε = 0.1, A0 (0) = 0.1, D0 (0) = 0.0. The coordinate-velocity diagram shows two velocity jumps per one cycle due to the periodic impulsive excitation.

7.3

Strongly Nonlinear Oscillator under Periodic Pulses

Let us consider the case of strongly nonlinear exactly solvable oscillator described in Chapter 3 by adding periodic impulsive loading on the right-hand side as follows F de (t/a) a d (t/a) ∞ = 2F (−1)k δ[t − (2k − 1)a]

x ¨ + tan x + tan3 x =

k=−∞

(7.30)

186

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

1.5 1.0 B0

0.5 0.0 0.5 A0

1.0 1.5 2.0 0

5

10

15 t

20

25

0

Fig. 7.2 The amplitudes A0 (t0 ) and D0 (t0 ) in the slow time scale, t0 = εt, shown by the gray and black lines, respectively.

2 1 x

0 1 2 0

50

100

150

200

250

t Fig. 7.3 The time history of the oscillator near the principal resonance.

Here 2F > 0 and 4a = T > 0 characterize the amplitude and period of the impulsive loading. Making substitutions x = X(τ ) and τ = τ (t/a) in equation (7.30), gives

d2 X dX de (t/a) a−2 2 + tan X + tan3 X = a−2 aF − dτ dτ d (t/a)

7.3 Strongly Nonlinear Oscillator under Periodic Pulses

187

3 2 1 v 0 1 2 3 2

1

0 x

1

2

Fig. 7.4 The coordinate-velocity plane trajectory on the interval 0 < t < 40.

or

d2 X + tan X + tan3 X = 0 dτ 2 under the following continuity condition for x(t), a−2

dX |τ =±1 = aF dτ

(7.31)

(7.32)

Boundary value problem (7.31) and (7.32) describe the class of steady state periodic motions of the period T . With reference to Chapter 3, equation (7.31) has exact solution of the form X(τ ) = arcsin[sin A sin(aτ / cos A)]

(7.33)

188

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

where A is an arbitrary constant, which is suﬃcient to satisfy both conditions in (7.32) due to the symmetry of solution (7.33). Substituting (7.33) in (7.32) and conducting analytical manipulations with elementary functions, gives α2 F T = 4α kπ ± arccos , (k = 1, 2, 3, ...) (7.34) (1 − α2 )(1 − α2 F 2 ) α2 F T = 4α arccos , (1 − α2 )(1 − α2 F 2 )

(k = 0)

(7.35)

where T = 4a is the period of forced vibration, and α = cos A.

35 30 25 T

20 15

k3

10

k2

5

k1

0

k0 1.2

1.1

1.3 A

1.4

1.5

Fig. 7.5 First few branches of the period versus amplitude diagram under the ﬁxed pulse amplitudes, F = 1.8.

It can be shown that solutions (7.34) and (7.35) exist in the interval 1 π arccos √ = Amin < A < Amax = 2 2 1+F

(7.36)

Fig. 7.5 illustrates the sequence of branches of solutions (7.34) and (7.35) at diﬀerent numbers k, and the parameter F = 1.8. The diagram gives such combinations of the period and amplitude at which oscillator (7.30) can have periodic motions with the period of external impulses, T = 4a. The upper and

7.4 Impact Oscillators under Impulsive Loading

189

lower branches of each loop correspond to plus and minus signs in expression (7.34), respectively. Solution (7.35) corresponds to the number k = 0 and has the only upper branch. For the selected magnitude of F , the minimal amplitude is found to be Amin = 1.0637, which corresponds to the left edges of the amplitude-period loops in Fig. 7.5. As a result, further slight increase of the amplitude is accompanied by bifurcation of the solutions as shown in Fig.7.6. The diagram includes only ﬁrst three couples of new solutions (k = 1, 2, 3) from the inﬁnite set of solutions. Further evolution of temporal shapes of solutions, related to the amplitude increase, is illustrated by Figs. 7.7 and 7.8. As follows from these diagrams, the inﬂuence of external pulses on the temporal shapes is decreasing as the amplitude grows. This is the result of dominating the restoring force over the external pulses. When the amplitude becomes close to its maximum Amax = π/2, the oscillator itself generates high-frequency impacts; see Chapter 3 for details.

7.4

Impact Oscillators under Impulsive Loading

Consider nonlinear oscillator under periodic pulse excitation of the following general form de (ωt) (7.37) x ¨ + 2ζ x˙ + f (x, ωt) = p d (ωt) where ζ, ω and p are constant parameters, and the function f (x, ωt) is periodic with respect to the argument ωt with the period equal to four. It is assumed that the coordinate x is subjected to either the constraint condition 0 ≤ x (t) (7.38) or − 1 ≤ x (t) ≤ 1

(7.39)

According to the idea of non-smooth transformation of positional coordinates [199], [204] (see also the end Chapter 1 of this book), the constraints (7.38) or (7.39) are eliminated by unfolding the coordinate x (t) −→ l (t) : x = S (l)

(7.40)

where S (l) ≡ |l| in case (7.38) or S (l) ≡ τ (l) in case (7.39), so that the new coordinate l belongs to the entire inﬁnite interval, l (t) ∈ (−∞, ∞), in contrast to (7.38) or (7.39). Taking into account the identity [S (l)]2 = 1, gives the ﬁnal result of transformation (7.40) in the form dl de (ωt) d2 l + 2ζ + S (l) f [S (l) , ωt] = pS (l) 2 dt dt d (ωt)

(7.41)

where, as mentioned above, the new coordinate l (t) is free of any constraints.

190

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

x 1.0

k1

0.5 0.5

0.5 1

2

3

4

t

5

0.5

1.0 k2

0.5

2

3

4

5

6

2

4

6

8

10 12

t

x 1.0 0.5

2

4

6

8

10 12

t 0.5

1.0

t

1.0

x 1.0

k3

0.5 0.5

1

1.0

x 1.0

0.5

x 1.0

x 1.0 0.5

5

10

15

1.0

t 0.5

5

10

15

t

1.0

Fig. 7.6 Periodic solutions corresponding to the ﬁrst three closed loops of the diagram shown in Fig. 7.5 at A = 1.0647; the left and right columns relate to lower and upper branches of the loops, respectively.

Note that the right-hand side of equation (7.41) makes sense within the distribution theory under the condition that times of interaction with constraints (7.38) or (7.39) never coincide with the external pulse times. Consider now the class of periodic motions of the period of external loading, T = 4/ω. As follows from Chapter 6 and the previous sections of this chapter, within such class of motions, the external pulses are eliminated by introducing the new time argument as follows t −→ τ : τ = τ (ωt) ,

l = X (τ ) + Y (τ ) e

(7.42)

7.4 Impact Oscillators under Impulsive Loading

x

x

1.0

k1

0.5 0.5

0.5 1.0 1.5 2.0 2.5 3.0

t 0.5

1

2

3

4

5

6

7

t

1.0

x

k2

1.0 0.5

x 1.0 0.5

2

0.5

4

6

8

t 0.5

1.0

2

4

6

8

10 12

t

1.0

x

x k3

1.0 0.5

1.0

1.0 0.5

1.0

0.5

191

1.0 0.5

2

4

6

8 10 12

t 0.5

5

10

15

t

1.0 Fig. 7.7 Same as Fig. 7.6 for A = 1.15.

So, substituting (7.42) into equation (7.41), gives ω 2 X (τ ) + 2ωζY (τ ) + R (X, Y, τ ) + ω 2 Y (τ ) + 2ωζX (τ ) + I (X, Y, τ ) e de (ωt) = pRS (X, Y ) + pIS (X, Y ) e − ω 2 X (τ ) d (ωt)

(7.43)

where 1 R = {S (X + Y ) f [S (X + Y ) , τ ] ± S (X − Y ) f [S (X − Y ) , 2 − τ ]} I 2

192

7 Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

x

x k1

1.0 0.5 0.5

0.5 1.0

1.0

1.0 0.5

t

1.5

0.5 1.0

x

3

t

4

1

2

3

4

5

1.0 0.5

t

2

0.5 1.0

x

4

6

8

t

x k3

1.0 0.5 0.5 1.0

2

x k2

1.0 0.5 0.5 1.0

1

2

4

6

8

1.0 0.5

t 0.5 1.0

2

4

6

8

10

t

Fig. 7.8 Same as Fig. 7.6 for A = 1.3.

and

RS IS

=

1 {S (X + Y ) ± S (X − Y )} 2

under the necessary condition of continuity for l (t), Y (±1) = 0

(7.44)

Equation (7.43) is equivalent to the following boundary value problem ω 2 X (τ ) + 2ωζY (τ ) + R (X, Y, τ ) = 0 ω 2 Y (τ ) + 2ωζX (τ ) + I (X, Y, τ ) = 0

(7.45)

[X (τ ) − ω −2 pS (X)]|τ =±1 = 0

(7.46)

7.4 Impact Oscillators under Impulsive Loading

193

where boundary condition (7.46) has been simpliﬁed by enforcing condition (7.44). After solution of the boundary value problem (7.44) through (7.46) has been obtained, the original coordinate is given by composition of transformations (7.40) and (7.42) as x (t) = S (X (τ ) + Y (τ ) e)

(7.47)

where τ = τ (ωt) and e = e (ωt). The advantage of such a boundary value problem formulation is that the eﬀect of both internal and external pulses are captured by the composition of two transformations (7.47). As a result, the ﬁnal system is free of any singular terms. Some particular cases and examples were considered earlier in [140].

Chapter 8

Strongly Nonlinear Vibrations

Abstract. This chapter presents analytical successive approximations algorithms for diﬀerent oscillators with strongly nonlinear characteristics. In general terms, such algorithms approximate temporal mode shapes of vibrations by polynomials and other simple functions of the triangular sine wave. In order to develope the algorithms, the triangular wave is introduced into dynamical systems as a new temporal argument. The corresponding manipulations with dynamical systems are described in the ﬁrst three sections. Then the description focuses on the algorithm implementations for diﬀerent essentially unharmonic cases including oscillators whose characteristics may approach nonsmooth or even discontinuous limits.

8.1

Periodic Solutions for First Order Dynamical Systems

Let us consider a dynamical system described by ﬁrst-order diﬀerential equation with respect to the vector-function x(t) ∈ Rn , x˙ = f (x)

(8.1)

where f (x) is a continuous vector-function, and the over dot indicates time derivative. We consider the class of periodic motions of the period T = 4a. Note that, in the autonomous case, the period is a priori unknown. Periodic solutions usually require speciﬁc, a priory unknown, initial conditions. Practically, however, such kind of the speciﬁc initial conditions are determined in a backward way, after some periodic family of solutions is obtained under the assumption of periodicity. In our case, the assumption of periodicity is imposed automatically by the form of representation for periodic solutions x = X(τ ) + Y (τ )e V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 195–239, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(8.2)

196

8 Strongly Nonlinear Vibrations

where τ = τ (t/a) and e = e(t/a) are the standard triangular sine and rectangular cosine, respectively, and X(τ ) and Y (τ ) are unknown components of the solution. Substituting (8.2) in (8.1), gives (Y − aRf ) + (X − aIf )e + Y e = 0 where 1 [f (X + Y ) + f (X − Y )] 2 1 If = If (X, Y ) = [f (X + Y ) − f (X − Y )] 2

Rf = Rf (X, Y ) =

Eliminating the periodic singular term e = de(t/a)/d(t/a) by means of the boundary condition for Y (τ ), gives the non-linear boundary value problem on the standard interval, −1 ≤ τ ≤ 1, Y = aRf (X, Y ) X = aIf (X, Y ) Y |τ =±1 = 0

(8.3)

Note that, the entire interval −1 ≤ τ ≤ 1 is completely covered by a half of the period, −a ≤ t ≤ a, however, representation (8.2) unfolds the corresponding fragment on the entire time interval −∞ < t < ∞.

8.2

Second Order Dynamical Systems

Consider now the diﬀerential equation of motion in the standard Newtonian form x ¨ + f (x, x, ˙ t) = 0 (8.4) where x(t) ∈ Rn is a positional vector-function, and the vector-function f is assumed to be suﬃciently smooth and periodic with respect to the explicit time t with the period T = 4a; the autonomous case, when the period is a priory unknown, is under consideration as well. Substituting (8.2) into (8.4), using the diﬀerential and algebraic properties of substitution (8.2), and imposing the boundary (smoothness) conditions, gives (X + a2 Rf ) + (Y + a2 If )e = 0 where

X − Y 1 X + Y , aτ + f X − Y, − , 2a − aτ Rf = f X + Y, 2 a a (8.5)

8.2 Second Order Dynamical Systems

197

X − Y 1 X + Y , aτ − f X − Y, − , 2a − aτ f X + Y, 2 a a (8.6) This leads to the boundary value problem If =

X + a2 Rf (X, Y, X , Y , τ ) = 0, X |τ =±1 = 0

(8.7)

Y + a2 If (X, Y, X , Y , τ ) = 0, Y |τ =±1 = 0

(8.8)

Let us discuss the form of equations (8.7) and (8.8). Firstly, despite of the obvious formal complication, equations (8.7) and (8.8) possess certain symmetries dictated by substitution (8.2). For instance, introducing the new unknown variables, U (τ ) = X(τ ) + Y (τ ) V (τ ) = X(τ ) − Y (τ ) brings the boundary value problem, (8.7) and (8.8), to the form, in which the diﬀerential equations are decoupled at cost of coupling the boundary conditions though, U + a2 f (U, U /a, 2a − aτ ) = 0 V + a2 f (V, − V /a, 2a − aτ ) = 0

(8.9)

U + V |τ =±1 = 0 U − V |τ =±1 = 0 In case when analytical methods are applied, the diﬀerential equations (8.9) for U (τ ) and V (τ ) can be usually solved in a similar way. Nevertheless, the previous boundary value problem, (8.7) and (8.8), may appear to have some advantages for analyses. For instance, the problem may admit families of solutions with either Y (τ ) ≡ 0 or X(τ ) ≡ 0. Secondly, due to substitution (8.2), the major qualitative property of solutions, such as periodicity, is a priory captured by the new argument, τ . As a result, the following simpliﬁed system can be employed as a generating model for analytical algorithms of successive approximations X = 0 Y

(8.10)

=0

Indeed, substituting obvious solutions of equations (8.10) in (8.2), gives a family of nonsmooth periodic motions with respect to the original time parameter, t, x(t) = X(0) + X (0)τ (t/a) + (Y (0) + Y (0)τ (t/a))e(t/a)

(8.11)

198

8 Strongly Nonlinear Vibrations

Thirdly, from the physical standpoint, linear equations (8.10) describe a strongly nonlinear (nonsmooth) generating model. In particular, if Y (0) = 0 and Y (0) = 0, then vector-function (8.11) describes vibrations of basic vibroimpact models. The analytical algorithms developed in few sections below are based on the idea of approximation of smooth vibrating systems by the basic vibroimpact models. In other words, the triangular sine wave is assumed to be a dominant component of temporal mode shapes of vibrations. Such an idea indeed follows the analogy with the quasi harmonic approaches. In particular, the harmonic balance method approximates vibrating systems by eﬀective harmonic oscillators regardless types or magnitudes of the system nonlinearities. This is justiﬁed by the fact that Fourier coeﬃcients usually decay in a fast enough rate so that, for instance, second term can be considered as a small correction to the ﬁrst term. The corresponding “small parameter” is therefore hidden in the iterative procedure itself rather than explicitly present in the diﬀerential equations of motion. Finally, equations (8.10) make sense due to the temporal substitution t −→ τ (t/a). In terms of the original variables, the corresponding equation, x ¨ = 0, contains too little information about the original system (8.4) and captures no global properties of the dynamics.

8.3

Periodic Solutions of Conservative Systems

8.3.1

The Vibroimpact Approximation

Let us consider the case of n-degrees-of-freedom conservative system x¨ + f (x) = 0

(8.12)

where f (x) is an odd analytical vector-function of the positional vectorcolumn x(t) ∈ Rn . A one-parameter family of periodic solutions will be built such that X(−τ ) ≡ −X(τ ). Since equation (8.12) admits the group of time translations then another arbitrary parameter can be always added to the time variable. Taking into account the symmetry of system (8.12), enables one of considering the particular case of substitution (8.2) x(t) = X(τ (t/a)), Y ≡ 0

(8.13)

Based on the conditions assumed, the boundary value problem (8.7) and (8.8) is reduced to the following one X + a2 f (X) = 0 X |τ =1 = 0

(8.14)

8.3 Periodic Solutions of Conservative Systems

199

We seek solutions of the boundary value problem (8.14) in the form of series of successive approximations X = X 0 (τ ) + X 1 (τ ) + X 2 (τ ) + ...

(8.15)

a2 = h0 (1 + γ1 + γ2 + ...)

(8.16)

In order to organize the corresponding iterative procedure, it is assumed that O( X i ) O( X i+1 ) O(γi+1 ) O(γi+2 )

(8.17)

(i = 0, 1, 2, ...) where the norm of vector-functions is deﬁned by X =max X Rn . τ

Based on assumptions (8.17), series (8.15) and (8.16) generate the sequences of equations and boundary conditions as, respectively, X 0 = 0

(8.18)

X 1 = −h0 f (X 0 ) X

2

= −h0 [γ1 f (X ) + 0

fx (X 0 )X 1 ]

(8.19) (8.20)

··· and (X 0 + X 1 )|τ =1 = 0

(8.21)

X 2 |τ =1 = 0

(8.22)

··· Note that condition (8.21) includes ﬁrst two approximations as the only way to proceed with a non-zero generating solution. In particular, the generating solution is found from equation (8.18) in the form X 0 = A0 τ

(8.23)

where A0 ∈ Rn is an arbitrary constant vector, and the oddness condition has been enforced in order to set to zero another constant vector. In line with the discussion at the end of the previous section, solution (8.23) describes a multi-dimensional vibro-impact oscillator between two absolutely stiﬀ and perfectly elastic barriers such that A0 is the normal vector to both barriers. Direction of the vector A0 will be deﬁned on the next step of successive approximations, whereas its length will appear to be coupled with the parameter h0 by some relationship due to boundary condition (8.21).

200

8 Strongly Nonlinear Vibrations

So substituting (8.23) in (8.19) and integrating, gives solution τ X = A τ − h0 1

(τ − ξ)f (A0 ξ)dξ

1

(8.24)

0

where A1 is another arbitrary constant vector. Note that the ﬁrst term in expression (8.24) is similar to generating solution (8.23) and thus contributes nothing new into the entire solution within the ﬁrst two steps of the procedure. Therefore, we take A1 = 0 and then substitute the combination X 0 + X 1 in the boundary condition (8.21). This gives a nonlinear eigen value problem with respect to the vector A0 in the form 1 1 0 f (A0 τ )dτ = A (8.25) h0 0

Equation (8.25) represents a set of n scalar equations relating the components of vector A0 and the parameter h0 . The combination A0 and 1/h0 will be interpreted as an eigen vector and eigen value of the nonlinear eigen vector problem (8.25). Taking scalar product of both sides of equation (8.25) with A0T , gives h0 = A0T

A0T A0 "1 f (A0 τ )dτ

(8.26)

0

where the upper index T stays for transpose operation. In order to clarify the meaning of expressions (8.25) and (8.26), let us consider the linear case f (x) ≡ Kx, where K is an n × n stiﬀness matrix. The corresponding relationships will diﬀer from those of the exact linear theory by speciﬁc constant factors because the temporal mode shape of vibrations is not exact but approximated by the triangular sine wave. Nevertheless, in nonlinear cases, expression (8.26) can provide estimates for amplitudefrequency response characteristics. Further, integrating equation (8.20), gives τ X = A τ − h0 2

2

(τ − ξ)[γ1 f (A0 ξ) + fx (A0 ξ)X 1 (ξ)]dξ

(8.27)

0

where A2 is an arbitrary constant vector, and fx ( A0 ξ ) is the n × n − matrix of ﬁrst partial derivatives (Jacobian).

8.3 Periodic Solutions of Conservative Systems

201

Then boundary condition (8.22) gives 1 2

A = h0

[γ1 f (A0 τ ) + fx (A0 τ )X 1 (τ )]dτ

(8.28)

0

where the coeﬃcient γ1 is yet unknown. In order to determine the coeﬃcient γ1 , an additional condition for the vector A2 can be imposed, for instance, as follows A0T A2 = 0

(8.29)

This condition means that the vector A2 must be orthogonal to the corresponding vector of the generating solution A0 in order to keep the amplitude ﬁxed. Substituting (8.28) in (8.29), gives A0T γ1 = −

"1

fx (A0 τ )X 1 dτ

0

A0T

"1

(8.30) f (A0 τ )dτ

0

This completes the second step of successive approximations. All the further steps can be passed in the same way. In general terms, convergence properties of the above procedure are due to the following integral operator ⎧ 1 ⎫ τ ⎨ ⎬ F [X] ≡ a2 τ f (X(ξ))dξ + ξf (X(ξ))dξ (8.31) ⎩ ⎭ τ

0

where A0T a2 = h 0

"1

f (A0 τ )dτ

0

A0T

"1

(8.32) f (X)dτ

0

Based on deﬁnition (8.31), the original boundary value problem (8.14) admits representation in the form X = F [X]. Therefore, the convergence condition is [X 0 ]δX

FX <1 (8.33)

δX

where δX is an arbitrary vector-function from a small enough neighborhood of X 0 .

202

8 Strongly Nonlinear Vibrations

In the case of linearized system, condition (8.33) leads to the set of inequalities ωi /ωj < 1 for all i = j, where ωj is the eigen frequency of the linear normal mode, which is chosen to be a generating solution.

8.3.2

One Degree-of-Freedom General Conservative Oscillator

In the one-degree-of-freedom case with odd characteristic, the boundary condition at τ = 1 is reduced to a single equation, which is sequentially satisﬁed by the factor h0 and terms γ1 , γ2 ,... of series (8.16). As a result, the process of successive approximations therefore eases by setting A0 = A and Ai = 0 for i = 1, 2,... . Let us introduce notations hi = h0 γi and represent series (8.15) and (8.16) in the form X = X0 (τ ) + X1 (τ ) + X2 (τ ) + ... a2 = h0 + h1 + h2 + ...

(8.34)

where Xi (τ ) are scalar functions of the triangular sine wave τ = τ (t/a). Due to the reduction of one-dimensional case, all terms of the expansions are iteratively determined by the explicit relationships. First two steps of the iterative procedure are coupled by the smoothing boundary condition (8.21) that provides the leading order smooth estimate for the temporal mode shape by coupling the parameters, h0 = h0 (A), as follows X0 = Aτ

(8.35)

τ X1 = −h0

(τ − ξ)f (Aξ)dξ 0

1 h0 = A/

f (Aξ)dξ

(8.36)

0

All the next steps of the procedure are passed then in a similar way based on relationships Xi = −

i

τ

hi−1 = −

i−1

(τ − ξ)Ri−j dξ

hj−1

j=1

0

αi−j hj−1

j=1

(i = 2, 3, ...)

(8.37)

8.3 Periodic Solutions of Conservative Systems

203

where the coeﬃcients and integrands are generated by means of the formal auxiliary parameter, ε, as follows 1 αi =

1 Ri dξ/

0

R0 dξ

(8.38)

0

1 di f (X 0 + εX 1 + ε2 X 2 + · · ·) Ri = |ε=0 i! dεi (i = 0, 1, 2, . . .) The way of using the parameter ε is in compliance with assumptions (8.17). Respectively, such parameter splits the restoring force according to (8.38) and then disappears from expressions. The convergence of such iterative series of successive approximations is illustrated by the following example. Example 12. Let us consider the oscillator x ¨ + xm = 0 where m is an odd positive integer. This oscillator was already discussed in Chapter 3 under the notation m = 2n − 1. Now, applying two iterations according to the above scheme, (8.36) and (8.37), gives solution

2m+3 τ m+2 m τ τ m+2 X =A τ− + − − R3 − R4 − · · · m + 2 2(m + 2) 2m + 3 m + 2 (8.39)

2 m + 1 m (m + 1) a2 = m−1 1 + 1+ + r3 + r4 + · · · (8.40) A 2(m + 2) (m + 2)(2m + 3) where expressions m+2

m |τ | i−1 2 (m + 2)2 m 0 < ri (m) < i 2 (m + 2)

0 < Ri (m, τ ) <

(8.41)

provide estimates for high order terms of the successive approximations. In particular, expressions (8.41) indicate that series (8.39) and (8.40) may converge quite slowly. However, the asymptotic of large exponents m essentially improves precision of the truncated series even though ﬁrst few terms of the series are included. The temporal mode shapes of diﬀerent iterations are shown in Fig. 8.1, whereas Fig. 8.2 illustrates the period as a function of the number m under the ﬁxed energy in the ﬁrst iteration only. For comparison reason, the exact result and ﬁrst order approximation according to the harmonic balance method are also shown in the diagram. In particular, Fig. 8.1 shows that high-order iterations are localized near the time points corresponding to amplitude positions of the oscillator. For instance, the ﬁrst

204

8 Strongly Nonlinear Vibrations

Fig. 8.1 The ﬁrst three terms of iteration (thin lines) and their sum (solid line) for the temporal mode shape of the oscillator x ¨ + x5 = 0.

6

Period, T

5

harmonic balance

4

exact solution 3 sawtooth 0

10

20 30 Degree of nonlinearity, m

40

50

Fig. 8.2 The period of the oscillator with power characteristic at diﬀerent exponents m obtained by three diﬀerent methods under the total energy E = 2.

iteration X1 just compensates the discontinuities of slope of the generating solution X0 with a minor eﬀect on the rest of the triangular sine wave. Further, Fig. 8.2 conﬁrms that expansion (8.40) gives a better estimate for the

8.3 Periodic Solutions of Conservative Systems

205

period than the harmonic balance as the exponent m increases. Note that the entire series (8.39) and (8.40) are not asymptotic with respect to m or 1/m in the sense of Poincare, however the series perfectly capture the asymptotic of impact oscillator as m → ∞.

8.3.3

A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes

As another example of conservative oscillator, we consider a single mass vibrating system illustrated by Fig. 8.3 and described by the Lagrangian mw˙ 2 − kl2 L= 2

w2 1+ 2 −1 l

2

Here, m is mass, k is the linear stiﬀness of each spring, l is the length of each spring at the equilibrium position at which the springs are horizontal, and w is the particle vertical coordinate. In terms of the dimensionless coordinate x = w/l and phase ϕ = (2k/m)1/2 t, the corresponding diﬀerential equation of motion takes the form x d2 x +x− √ =0 dϕ2 1 + x2

(8.42)

Then, applying substitution (8.13) as x = X(τ ) and τ = τ (ϕ/a), leads to the boundary problem

X ≡ −hf (X) X = −h X − √ 1 + X2 (8.43) X |τ =1 = 0 where h = a2 .

wt l

l

Fig. 8.3 The system which becomes weakly non-linear at large amplitudes.

206

8 Strongly Nonlinear Vibrations

First two steps of the successive approximation procedure give X0 = Aτ −1 √ 1 + A2 − 1 1 − h0 = 2 A2 and

h0 1 3 2 X1 (τ ) = − 2 (Aτ ) + 2Aτ − Aτ 1 + (Aτ ) − arcsinh(Aτ ) (8.44) 2A 3

9A − A3 h0 6A + A3 1 + √ γ1 = 3 − 1+ √ arcsinh(A) A 12 6 1 + A2 1 + A2

Interestingly enough, this model is essentially nonlinear at small amplitudes, but it becomes linear as the amplitudes are inﬁnitely large. Indeed, taking the corresponding limits, shows that 1 X1 3 → − τ 5 , h0 A2 → 8, γ1 → A 5 10

as A → 0

(8.45)

and

X1 1 1 → − τ 3 , h0 → 2, γ1 → as A → ∞ (8.46) A 3 6 The asymptotic (8.45) obviously corresponds the nonlinear oscillator, whereas the limit case (8.46) associates with the harmonic oscillator. Nevertheless, solution (8.44) is valid for both large and small amplitudes. Both limit cases follow from equation (8.42). In the case of small amplitudes, |x| 1, by using the estimate (1 + x2 )−1/2 ∼ 1 − x2 /2 in equation (8.42), we obtain d2 x/dϕ2 + x3 /2 = 0

(8.47)

In the case of large amplitudes, it follows even from Fig. 8.3 that the distance between the spring ﬁxed ends becomes negligible if as compared to l. As a result, the mechanical model becomes eﬀectively close to a mass-spring oscillator of mass m with a single spring of stiﬀness 2k. In terms of the diﬀerential equations of motion, we also obtain the corresponding limit from exact equation (8.42) by√assuming that, during ‘most of the time of vibration cycle,’ |x| 1 and thus 1 + x2 ∼ |x|. As a result, equation (8.42) is replaced by (8.48) d2 x/dϕ2 + x − sgn(x) = 0 where the term sgn(x) has to be neglected due to the same condition |x| 1 that gives the standard linear oscillator. Alternatively, the discontinuous term in equation (8.48) can be saved and then considered as a perturbation of the harmonic oscillator. Note that equation (8.48) admits another form as follows

8.3 Periodic Solutions of Conservative Systems

d2 x/dϕ2 + sgn(x)(|x| − 1) = 0

207

(8.49)

The restoring force characteristic of oscillator (8.49) represents a particular case of the characteristic, p(x) = sgn(x)f (|x|), which is considered later in this chapter.

8.3.4

Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero

Let us consider the case of symmetric exponentially growing restoring force characteristic with a step-wise discontinuity at zero such that exp(x) for x > 0 f (x) = (8.50) − exp(−x) for x < 0 Although force (8.50) has no certain value at the point x = 0, this still can play the role of equilibrium position. From the physical standpoint, this is equilibrium of a small bead at the bottom of V -shaped potential well. The local dynamics in a small neighborhood of such type of equilibria is considered later in this chapter; see the text to Fig. 8.13. It follows from (8.50) that f (−x) = −f (x). Therefore, periodic motions of the corresponding oscillator can be described by the function x = X(τ ), where X(−τ ) = −X(τ ) and τ = τ (t/a). In terms of these NSTT variables, the oscillator of a unit mass is described by the boundary value problem for τ ∈ (0, 1] X + h exp(X) = 0, X |τ =1 = 0 X − h exp(−X) = 0, X |τ =−1 = 0 for τ ∈ [−1, 0)

(8.51)

where h = a2 . This problem is exactly solvable, and solution that satisﬁes the continuity of state condition at τ = 0 has the form X± (τ ) ≡ Aτ ± 2 ln (1 + exp(−A))

h exp(±Aτ − A) ∓2 ln 1 + 2h0

(8.52)

where X+ and X− are taken for positive and negative subintervals of τ , respectively, and 2A2 (8.53) h = 2h0 = exp(A)[1 + exp(−A)]2 Note that both the diﬀerential equation of oscillator and its solution admit unit form representations as, respectively, x ¨ + sgn(x) exp(|x|) = 0

(8.54)

208

and

8 Strongly Nonlinear Vibrations

X = sgn(τ ) A|τ | + 2 ln

1 + exp(−A) 1 + (h/(2h0 )) exp(A|τ | − A)

(8.55)

The parameter h is obtained from equation

or 1−

h h0

X |τ =1 = 0

(8.56)

−1 h =0 1+ 2h0

(8.57)

This exactly solvable case can play the role of a majorant for evaluation of convergence properties of successive approximations. For that reason, we introduce a formal ‘small’ parameter, ε = 1, and represent solution of equation (8.57) in the form ε −1 h = 2h0 = εh0 1 − (8.58) 2 Taking into account (8.58), brings (8.55) to the form

1 − (ε/2) (1 − exp(−A)) X = sgn(τ ) A|τ | + 2 ln (8.59) 1 − (ε/2) (1 − exp(A|τ | − A)) It can be shown by direct calculations that the power series expansions of (8.58) and (8.59) with respect to ε lead to the same series as those obtained by means of the iterative procedure introduced in this section for a general one-degree-of-freedom oscillator. Moreover, the structure of expression (8.58) suggests that considering the modiﬁed series, h=

εh0 1 − ελ1 − ε2 λ2 − ...

(8.60)

leads to the exact value h already on the second step of the procedure. This fact can be employed for other cases in order to improve eﬃciency of the successive approximation series (8.16). For instance, according to the idea of Pad`e transform [19], the following equality must hold in every order of ε εh0 = εh0 (1 + εγ1 + ε2 γ2 − ...) 1 − ελ1 − ε2 λ2 − ...

(8.61)

This is equivalent to (1 + εγ1 + ε2 γ2 − ...)(1 − ελ1 − ε2 λ2 − ...) = 1

(8.62)

Taking the product of series on the left-hand side of (8.62) and considering diﬀerent orders of ε, generates a sequence of equations for the coeﬃcients

8.3 Periodic Solutions of Conservative Systems

209

λ1 , λ2 , ... . Then, substituting the corresponding solutions in (8.60), gives a particular case of Pad`e transform of (8.16) in the form h=

εh0 1 − εγ1 − ε2 (γ2 − γ12 ) − ...

(8.63)

In many cases, expansion (8.63) appears to be more eﬀective than (8.16). Note that it is also possible to organize the successive approximations procedure by using expansion (8.60) instead of (8.16).

8.3.5

A Generalized Case of Odd Characteristics

This subsection deals with some generalization of the standard one-degreeof-freedom conservative oscillator x¨ + f (x) = 0

(8.64)

where f (x) is a smooth odd characteristic, f (−x) = −f (x)

(8.65)

It was shown in this chapter that periodic solutions of the oscillator (8.64) admit the form x = X(τ ), where X(−τ ) = −X(τ )

(8.66)

and, in addition, X(τ )τ ≥ 0 for −1 ≤ τ ≤ 1. We consider now the following class of oscillators x ¨ + sgn(x)f (|x|) = 0

(8.67)

In the case of odd characteristic, f (x), oscillator (8.67) is equivalent to the original one (8.64). The extension is due to the fact that the oscillator (8.67) always has an odd characteristic regardless whether or not the function f (x) itself is odd. In general case, however, the characteristic sgn(x)f (|x|) may appear to be nonsmooth at the equilibrium point x = 0. As a result, direct implementation of iterative procedures with high-order derivatives of oscillator’ characteristics becomes quite limited. Nevertheless, as illustrated below, the group properties of equation (8.67) can help to eﬀectively build solution of equation (8.67) based on solution of equation x ¨ + f (x) = 0 for x > 0 by ignoring the point x = 0. Obviously, if P (x) is the potential energy of oscillator (8.64), then P (|x|) is the potential energy corresponding to oscillator (8.67). The following example explains why equation (8.67) covers a broader class of oscillators than (8.64). Example 13. x ¨+sgn(x)|x|3/2 = 0 is an oscillator, but x¨ + x3/2 = 0 is not; see also Chapter 3 for the related discussion.

210

8 Strongly Nonlinear Vibrations

Based on the transition from equation (8.64) to equation (8.67) and the general symmetry properties (8.65) and (8.66), we introduce the following representation for periodic solutions of equation (8.67) x = sgn(τ )X(|τ |)

(8.68)

Such an extension enables one to obtain ‘closed form’ analytical solutions for a large number of oscillators by a simple adaptation of already known solutions, X(τ ), for diﬀerent cases of smooth characteristics. Example 14. Applying transformation (8.68) to solution (8.39), which was derived for the power form characteristic xα with an odd positive exponent α = m, gives

2α+3 α |τ | |τ |α+2 |τ |α+2 + − X = Asgn(τ ) |τ | − (8.69) α+2 2(α + 2) 2α + 3 α+2 where τ = τ (t/a) and the expansion (8.40) for a2 requires only the replacement m → α. Expansion (8.69) represents an approximate solution of the equation (8.70) x ¨ + sgn(x)|x|α = 0 where the notation α substitutes m in order to emphasize that the new exponent can take any positive real value, such as even, odd, rational or irrational. Figs. 8.4 and 8.5 illustrate solution (8.69) compared to numerical solution for two diﬀerent exponents α and the same parameter A = 1. As both ﬁgures show, the analytical and numerical solutions are in a better match under the large exponent α due to the inﬂuence of vibroimpact asymptotic, α → ∞.

0.6

3 2

0.4 0.2 x

0.0 0.2 0.4 0.6 0

1

2

3

4

5

6

7

t

Fig. 8.4 Analytical and numerical solutions of the modiﬁed oscillator shown by continuous and dashed lines respectively.

8.4 Periodic Motions Close to Separatrix Loop

211

1.0 2010 0.5

x

0.0

0.5

1.0 0

5

10

15

20

25

30

35

t

Fig. 8.5 Analytical and numerical solutions of the generalized oscillator under the large exponent α.

The common feature of the algorithms and examples of this section is that generating solutions for successive approximations are represented by triangular sine waves of proper amplitudes and periods. Such generating solutions belong to the ‘real’ component of the hyperbolic ‘number,’ x = X + Y e. In contrast, the next section introduces algorithms of successive approximations based on the ‘imaginary’ component. It will be seen that these two approaches have diﬀerent physical contents.

8.4

Periodic Motions Close to Separatrix Loop

In this section, the classic mathematical pendulum is considered as an example, although the developed algorithm may be applicable to other cases of one degree-of-freedom systems with multiple equilibrium positions. So we illustrate the algorithm based on the diﬀerential equation of motion x ¨ + sin x = 0

(8.71)

It is assumed that the pendulum oscillates inside the separatrix loop around the stable equilibrium (x, x) ˙ = (0, 0) in between two physically identical unstable saddle-points (x, x) ˙ = (±π, 0). The separatrix loop also represents trajectory of the system with the total energy π Es ≡

sin xdx = 2 0

(8.72)

212

8 Strongly Nonlinear Vibrations

The pendulum remains close to the separatrix loop in the area of periodic motions if the total energy E is within the range 0<1−

E << 1 Es

(8.73)

Let us show that, under condition (8.73), a successive approximation solution can be derived from the particular case of boundary value problem (8.7) and (8.8). Such particular case is given by setting X ≡ 0 so that x(t) = Y (τ (t/a))e(t/a)

(8.74)

Substituting (8.74) in equation (8.71) and following the NSTT procedure, yields Y = −a2 sin Y (8.75) and Y |τ =1 = 0, Y (−τ ) ≡ Y (τ )

(8.76)

We seek solution of the boundary value problem, (8.75) and (8.76), in the form of series of successive approximations Y = π + εY1 (τ ) + ε3 Y3 (τ ) + ε5 Y5 (τ ) + ... a2 = p2 / 1 − ε2 λ2 − ε4 λ4 − ...

(8.77) (8.78)

where ε = 1 is an auxiliary “parameter” that helps to organize the iterative process. According to the formal expansion (8.77), the generating solution is represented by the rectangular cosine of the amplitude π, x(t) = πe(t/a), which is a step-wise discontinuous function. Such temporal mode shapes occur near the separatrix loop in the natural time scale of the pendulum because the system spends most of the time during one period near the unstable equilibrium positions x = π and its physically identical, x = −π. Therefore, expansion (8.77) is designed to be a high-energy expansion near the unstable equilibrium, rather than around the stable equilibrium position, x = 0. Substituting expansions (8.77) and (8.78) into the equation (8.75) and collecting terms with the same power of ε, leads to the sequence of equations d2 Y1 − p2 Y1 = 0 dτ 2

1 3 d2 Y3 2 2 − p Y3 = p λ2 Y1 − Y1 dτ 2 6

(8.79) (8.80)

2 d2 Y5 λ2 3 1 5 1 2 2 2 λ Y Y Y Y − p Y = p + λ − + + λ Y − Y (8.81) 5 4 1 2 3 3 2 dτ 2 6 1 120 1 2 1 ...

8.4 Periodic Motions Close to Separatrix Loop

213

Further, the family even solutions of equation (8.79) can be represented in the form cosh pτ Y1 = −A (8.82) cosh p where A is an arbitrary constant accompanied by the factor − cosh−1 p, which is convenient for further calculations due to the relationship Y1 (1) = −A. In particular, this provides the same order of magnitude for the arbitrary constant A as the parameter p and period T = 4a both go to inﬁnity. Further procedure is formally similar to the standard Poincare-Lindstedt algorithm for nonlinear conservative oscillators with positive linear stiﬀness. For example, substituting (8.82) into the right part of the equation (8.80), gives a ‘resonance term’ on the right-hand side of the equation, which is proportional to cosh pτ . This generates ‘hyperbolic secular terms’ of the form τ cosh pτ and τ cosh pτ in the particular solution of equation (8.80). Occurrence of such terms can be prevented, however, analogously to the PoincareLindstedt method by setting λ2 =

A2 8 cosh2 p

(8.83)

As a result, the particular solution of equation (8.80) takes the form Y3 =

A3 cosh 3pτ 192 cosh3 p

(8.84)

At the next stage, equation (8.81) gives solution

A5 1 Y5 = cosh 5pτ cosh 3pτ − 5 4096 cosh5 p under the condition λ4 = −

3A4 512 cosh4 p

(8.85)

(8.86)

Substituting (8.82) through (8.83) in (8.77) and (8.78), and setting ε = 1, gives approximate solution A3 cosh 3pτ A cosh pτ + cosh p 192 cosh3 p

A5 1 cosh 5pτ + cosh 3pτ − 5 4096 cosh5 p Y =π−

and

h=p 1− 2

where τ = τ (t/a) and a =

√

h.

3A4 A2 + 2 8 cosh p 512 cosh4 p

(8.87)

−1 (8.88)

214

8 Strongly Nonlinear Vibrations

The truncated series of successive approximations (8.87) and (8.88) depend upon two parameters, A and p, coupled by the boundary (continuity) condition (8.76) as follows

A3 cosh 3p A5 1 cosh 5p (8.89) A=π+ + cosh 3p − 5 192 cosh3 p 4096 cosh5 p Equation (8.89) should be interpreted as implicit function A = A(p) near the point A = π. Therefore, expansions (8.87) and (8.88) represent a oneparameter family of periodic solutions with the parameter p.

2 1

x 0 1 2 0

5

10

15

t Fig. 8.6 Analytical and numerical solutions of the period T=8.5 (p=2) shown by solid and thin lines, respectively.

Note that the equation (8.71) admits the group of time translations. As a result, another arbitrary parameter, say t0 , is introduced by substitution t → t + t0 . Figures 8.6 and 8.7 show that the analytical and numerical solutions are matching better for larger periods as the system trajectory becomes closer to the separatrix loop.

8.5

Self-excited Oscillator

This section illustrates the case when both X and Y components of solutions participate in the iterative process. In particular, we consider periodic self-sustained vibrations described the diﬀerential equation of motion x ¨ + g(x)x˙ + f (x) = 0

(8.90)

8.5 Self-excited Oscillator

215

3 2 1

x 0 1 2 3 0

10

20

30

40

t Fig. 8.7 Analytical and numerical solutions of the period T=24.5 (p=6) shown by the solid and thin lines, respectively.

where f (x) and g(x) are analytic functions, such that (Lienard’ conditions) a) G(x) =

"x

g(u)du is an odd function such that G(0) = G(±μ) = 0 for some

0

μ > 0, b) G(x) → ∞ if x → ∞, and G(x) is a monotonously increasing function for x > μ, c) f (x) is an odd function such that f (x) > 0 for x > 0. The above conditions guarantee that system (8.90) has a single stable limit cycle. In this case, the boundary-value problem (8.7) and (8.8) takes the form X = −a2 Rf − a (Rg Y + Ig X ) ≡ −εFX , X |τ =±1 = 0

(8.91)

Y = −a2 If − a (Ig Y + Rg X ) ≡ −εFY , Y |τ =±1 = 0 where the period of limit cycle T = 4a is unknown, expressions Rf and If as well as Rg and Ig are obtained by applying (8.5) and (8.6) to each of the functions f (x) and g(x), and notations εFX and εFY are introduced with the formal factor ε = 1 for further convenience. We seek solution of the boundary value problem (8.91) in the form of series of successive approximations X = X0 (τ ) + εX1 (τ ) + ε2 X2 (τ ) + · · ·

(8.92)

Y = Y0 (τ ) + εY1 (τ ) + ε Y2 (τ ) + · · · 2

a = q0 + εq1 + ε2 q2 + · · ·

(8.93)

216

8 Strongly Nonlinear Vibrations

Further solution procedure can be simpliﬁed by taking into account the symmetry properties X(−τ ) ≡ −X(τ ) and Y (−τ ) ≡ Y (τ ) due to the above conditions (a) through (c). Substituting (8.92) and (8.93) into (8.91) and matching the coeﬃcients of the respective powers of ε, gives the following sequence of boundary value problems X0 = 0 Y0 = 0, Y0 |τ =1 = 0

(8.94)

X1 = −FX,0 , (X0 + X1 ) |τ =1 = 0 Y1 = −FY,0 , Y1 |τ =1 = 0

(8.95)

Xi+1 = −FX,i , Xi+1 |τ =1 = 0 Yi+1 = −FY,i , Yi+1 |τ =1 = 0 (i = 1, 2, ...)

(8.96)

where

1 di FX 1 di FY | , F = |ε=0 (8.97) ε=0 Y,i i! dεi i! dεi Note that zero-order and ﬁrst-order approximations are coupled through the boundary condition for X-component in (8.95), whereas no boundary condition is imposed on X0 in (8.94). This speciﬁc represents a formalization of the physical assumption regarding the dominating component in the temporal mode shape of vibration, which is assumed to be close to the sawtooth sine rather than rectangular cosine. As a result, the generating system (8.94) gives solution (8.98) X0 = Aτ , Y0 ≡ 0 FX,i =

where A is an arbitrary constant. Substituting (8.98) in the right-hand side of equations (8.95) and integrating, yields τ X1 = −q02

τ (τ − ξ)f (Aξ)dξ,

Y1 = −Aq0

0

(τ − ξ)g(Aξ)dξ

(8.99)

0

Then, substituting (8.99) in the boundary conditions in (8.95), gives the following two equations for parameters q0 and A 1 q02

1 (1 − ξ)g(Aξ)dξ = 0

f (Aξ)dξ = A, 0

0

(8.100)

8.5 Self-excited Oscillator

217

Relationships (8.98) through (8.100) complete ﬁrst basic steps of the iterative procedure. Further iterations are organized then in a similar way as follows τ Xi+1 = −

τ (τ − ξ)FX,i (ξ)dξ,

Yi+1 = −

0

ζ dζ

1

FY,i (ξ)dξ

(8.101)

0

1 FX,i (ξ)dξ = 0; i = 1, 2, ....

(8.102)

0

Note that the boundary conditions for Yi+1 are satisﬁed automatically due to the lower limit of the outer integral in (8.101), whereas the boundary conditions for Xi+1 generates equations (8.102) for determining the coeﬃcients of series (8.93). Practically, high-order approximations can be obtained by using computer systems of symbolic manipulations. Example 15. Consider the self-excited oscillator with the power form stiﬀness of the degree m = 3, x ¨ + (bx2 − 1)x˙ + x3 = 0 In this case, g(x) = bx2 − 1 and f (x) = x3 . Conducting elementary integrations in (8.100), gives the algebraic system 1 2 3 q A = A, 4 0

1 q0 A 6 − bA2 = 0 12

with non-trivial solution q0 =

2b , A= 3

6 b

As a result, integrating (8.99), yields 6 τ5 , Y1 = τ 2 − τ 4 X1 = − b 5 All further steps of the procedure are conducted according to the same scheme (8.101) and (8.102). For instance, ﬁrst two steps of the procedure give approximate solution τ5 1 6 {τ − + [105τ 9 + 900τ 7 b − 21τ 5 (70b + 9) + 350τ 3 b]} x= b 5 3150 1 [20 − 43τ 2 + 20τ 4 + 216τ 6 ]}e +(1 − τ 2 ){τ 2 − 420 and the period

b 3 T = 4a = 8 1+ 6 20

218

8 Strongly Nonlinear Vibrations

Two more steps of the procedure correct the above expression for the period as follows

b 3 2960b + 2121 7367360b2 + 4554992b + 8659035 + T = 4a = 8 1+ + 6 20 50400 605404800 Figs.8.8 and 8.9 show limit cycle trajectories described by the analytical solutions in one and two iterations, respectively. For comparison, the numerical solution for transition to the limit cycle is also presented. Then, Fig. 8.10 illustrates dependence of the quarter of period parameter, a = T /4, versus the quantity b−1/2 , which can be viewed as an estimate for the amplitude of limit cycle.

v 3

b 1.0 m3

2

1

x

0 1

analytical, 1 iteration numerical

2 3 2

1

0

1

2

Fig. 8.8 Trajectories of numerical solution and the one iteration analytical limitcycle solution.

8.6

Strongly Nonlinear Oscillator with Viscous Damping

This section describes the successive approximation procedure combined with the asymptotic of small energy dissipation that leads to a slow amplitude decay. The scheme of the algorithm is closed to that was introduced earlier in [137].

8.6 Strongly Nonlinear Oscillator with Viscous Damping

219

v 3

b 1.0 m3

2

1

x

0 analytical, 2 iterations 1

numerical

2 3 1.5

1.0

0.5

0.0

0.5

1.0

1.5

Fig. 8.9 Trajectories of numerical solution and approximate (two iterations) analytical limit-cycle solution. 5

4

3

a analytical, four iterations

2

numerical estimates 1

0

0

2

4

6

8

12

b

Fig. 8.10 Illustration of convergence of the iterative procedure on the parameter plane.

220

8 Strongly Nonlinear Vibrations

So consider a strongly nonlinear oscillator under the viscous damping x ¨ + 2μx˙ + f (x) = 0

(8.103)

where f (x) is an odd function such that xf (x) ≥ 0, and 0 < μ << 1. The idea of two variable expansions will be employed below in combination with the sawtooth time substitution. Let us assume that τ (ϕ) is a ‘fast’ time scale, whose phase ϕ depends on the ‘slow’ time scale t0 = μt according to the following diﬀerential equation ϕ˙ = ω(t0 )

(8.104)

where the right-hand side is a priory unknown. Let us represent unknown solution of equation (8.103) in the form x = x(ϕ, t0 ) = X(τ (ϕ), t0 ) + Y (τ (ϕ), t0 )e(ϕ)

(8.105)

Substituting (8.105) in equation (8.103), and imposing ‘smoothness conditions,’ ∂X |τ =±1 = 0 (8.106) Y |τ =±1 = 0, ∂τ gives two partial diﬀerential equations ∂Y ∂2X − μ2 LX = −Rf − μH 2 ∂τ ∂τ ∂X ∂2Y ω 2 2 = −If − μH − μ2 LY ∂τ ∂τ

ω2

(8.107)

where, as follows from (8.5) and (8.6), 1 [f (X + Y ) + f (X − Y )] 2 1 If = If (X, Y ) = [f (X + Y ) − f (X − Y )] 2

Rf = Rf (X, Y ) =

and two linear diﬀerential operators are introduced

∂ dω H ≡ 2ω 1 + 0 + 0 ∂t dt ∂2 ∂ L ≡ 02 + 2 0 ∂t ∂t

(8.108)

Note that the fast and slow temporal scales are associated with diﬀerent physical processes developed in the system. The slow energy dissipation process is represented by the explicit small parameter μ, but there is no explicit parameter associated with perturbations of the triangular sine wave, which is supposed to be a generating solution of the iterative process. However, as

8.6 Strongly Nonlinear Oscillator with Viscous Damping

221

discussed earlier in this section, introducing the sawtooth temporal argument, implies that the entire right-hand side of system (8.107) is small. Otherwise, the triangular sine wave cannot be considered as a dominating component of the temporal mode shape of oscillations. Recall that, in a similar way, selecting the harmonic wave as a dominating solution in quasi harmonic approaches implies that nonlinearities are small regardless system’ parameters. Therefore, iterative procedure for boundary value problem (8.106) and (8.107) should incorporate two diﬀerent procedures, as those described above in this section, and a proper asymptotic procedure related to the dissipation process. Once again, the quasi-harmonic methods face similar situation when dealing with weakly nonlinear systems under small damping conditions. For instance, if being applied to such cases, the method of multiple scales accounts for both unharmonicity and dissipation, after appropriate assumption regarding the relation between non-linearity and damping parameters has been made. Very often though, such parameters are assumed to be of the same order of magnitude. As to the boundary value problem (8.106) and (8.107), similar assumption can be introduced by providing the terms Rf and If , and the parameter μ with the same formal ‘small factor’ ε = 1. Then, the multiple scales or two variables expansions can be organized by using the auxiliary parameter ε [137]. In the case of linear oscillator, such an algorithm recovers the exact solution of the linear diﬀerential equation of motion however in the speciﬁc form ε(1 − εμ2 ) ω exp(−t0 ) 0 sin τ (ωt) (8.109) x = X(τ, t ) = C ω ε(1 − εμ2 ) and ω2 =

ε(1 − εμ2 ) 4 arcsin2 ε/2

where C is an arbitrary constant, and another arbitrary constant can be introduced through the arbitrary time shift. Note that solution (8.109) includes no Y -component because, at every stage of the iterative process, it appears to be possible to satisfy condition H(∂X/∂τ ) = 0

(8.110)

Therefore, the second equation of system (8.107) is satisﬁed by setting Y ≡ 0. Practically, condition (8.110) generates the common factor exp(−t0 ) for all successive approximations. In general nonlinear case, however, it is rather impossible to satisfy condition (8.110) at every stage of the process, but it still works for leading order approximate solutions. Example 16. Consider the weakly damped oscillator of the m degree power form restoring force characteristic

222

8 Strongly Nonlinear Vibrations

x¨ + 2μx˙ + xm = 0

(8.111)

At this stage, the exponent m is an odd positive number. (It will be shown later that a broader class of power characteristics can also be considered.) Note that, for this kind of oscillators, whether or not the damping is small depends on the level of amplitude and the exponent m. This is due to the fact that, under small enough amplitudes, the elastic force becomes negligible regardless the magnitude of damping. By assuming that the inﬂuence of damping is negligible during one cycle of vibration, one can use expressions (8.40) for estimations of magnitudes of damping and elastic forces. As a result, the condition of ‘small damping’ derives in the form μ2

1 (m + 1)Am−1 4

One step of the procedure gives approximate solution [137]

−4μt τ m+2 x = Cexp τ− m+3 m+2 where τ = τ (ϕ) and the phase variable is approximated by

m−1 t ϕ = ϕ∞ 1 − exp −2μ m+3 ϕ∞ =

(8.112)

(8.113)

(8.114)

1 m + 3 C (m−1)/2 2μ m − 1 2(m + 1)

Interestingly enough, the above approximate solution predicts that the oscillator makes only a ﬁnite number of waves as m > 1 and t → ∞. Figs. 8.11 and 8.12 illustrate damped responses of the oscillator with two diﬀerent degrees of nonlinearity, m = 3 and m = 7, respectively. As follows from the diagrams, the approximate analytical solution and numerical one are matching relatively well, especially at higher exponent, m = 7. In particular, this justiﬁes the idea of using the sawtooth wave in strongly nonlinear cases, when the oscillator becomes close to the standard vibroimpact model.

8.6.1

Remark on NSTT Combined with Two Variables Expansion

In general, the iterative process of sawtooth expansions and the averaging procedure can be separated. Moreover, the stage of sawtooth time substitution does not impose any speciﬁc method of analyses. So let us apply the two variables method directly to the nonlinear boundary value problem (8.106) and (8.107) by means of the asymptotic series X = X0 (τ, t0 ) + μX1 (τ, t0 ) + μ2 X2 (τ, t0 ) + · · · Y = Y0 (τ, t0 ) + μY1 (τ, t0 ) + μ2 Y2 (τ, t0 ) + · · ·

(8.115)

8.6 Strongly Nonlinear Oscillator with Viscous Damping

223

Fig. 8.11 Approximate analytical and numerical solutions of the damped oscillator with cubic power form characteristic.

Fig. 8.12 Approximate analytical and numerical solutions of the damped oscillator with the seven-th degree power form characteristic.

224

8 Strongly Nonlinear Vibrations

and ω = ω0 (t0 ) + μω1 (t0 ) + μ2 ω2 (t0 ) + · · · H = H0 + μH1 + μ2 H2 + ... (8.116) where Hi = 2ωi 1 + ∂/∂t0 + dωi /dt0 . Substituting (8.115) and (8.116) into (8.106) and (8.107) and matching coeﬃcients of like powers of μ, gives, in particular, ∂ 2 X0 ∂X0 (0) |τ =±1 = 0 = −Rf (X0 , Y0 ), ∂τ 2 ∂τ 2 ∂ Y0 (0) ω02 = −If (X0 , Y0 ), Y0 |τ =±1 = 0 ∂τ 2

ω02

(8.117)

and ω02

∂ 2 X1 ∂ 2 X0 ∂Y0 (1) = −R (X , Y , X , Y ) − 2ω ω − H0 0 0 1 1 0 1 f ∂τ 2 ∂τ 2 ∂τ

∂X1 |τ =±1 = 0 ∂τ ∂ 2 Y1 ∂ 2 Y0 ∂X0 (1) ω02 (8.118) = −If (X0 , Y0 , X1 , Y1 ) − 2ω0 ω1 − H0 2 2 ∂τ ∂τ ∂τ Y1 |τ =±1 = 0 (0)

(1)

(0)

(1)

where Rf , Rf , ... and If ,If , ... are determined by the expansions (0)

(1)

(2)

R = Rf + μRf + μ2 Rf + ... (0)

(1)

(2)

I = If + μIf + μ2 If + ... By taking into account the assumptions on f (x) in equation (8.103), one can represent solution of problem (8.117) in the following general form X0 = X0 (τ, A, ω0 ),

Y0 ≡ 0

(8.119)

where A = A(t0 ) is an arbitrary function of the slow time scale, which is coupled with the frequency ω0 through the boundary condition ∂X0 (τ, A, ω0 ) |τ =1 = 0 ∂τ

(8.120)

In general, this relationship determines the implicit function ω0 = ω0 (t0 ). Now substituting solution (8.119) in (8.118), gives equations ω02

∂ 2 X1 ∂ 2 X0 + f (X )X = −2ω ω 0 1 0 1 ∂τ 2 ∂τ 2

(8.121)

8.6 Strongly Nonlinear Oscillator with Viscous Damping

225

and

∂ 2 Y1 ∂X0 (8.122) + f (X0 )Y1 = −H0 2 ∂τ ∂τ Let us consider equation (8.122). The best choice would be achieved by setting the right-hand side to zero and therefore making possible the solution Y1 ≡ 0 which is consistent with zero-order solution (8.119) and provides a better smoothness property of the corresponding solution at this stage; see condition (8.110). Note that the right-hand side cannot be always made zero for any τ unless the zero-order solution admits separation of the variables t0 and τ. However, it is still possible to ‘minimize’ the right-hand side of equation (8.122) by making it orthogonal to solution of the corresponding homogeneous equation, ∂X0 /∂τ , in other words, ω02

1 2

1 −1

∂X0 ∂X0 H0 dτ ≡ ∂τ ∂τ

.

∂X0 ∂X0 H0 ∂τ ∂τ

/ =0

(8.123)

Taking into account the expression H0 = 2ω0 1 + ∂/∂t0 + dω0 /dt0 and condition (8.123), gives 0

2 1 ∂X0 (8.124) = C exp(−2t0 ) ω0 ∂τ where C is an arbitrary constant. It can be shown that the ‘minimization condition’ (8.123) occurs also in a rigorous mathematical way based on the boundary conditions for Y1 ; at least, this can be easily veriﬁed in the linear case f (x) ≡ x.

8.6.2

Oscillator with Two Nonsmooth Limits

Consider the following generalization of equation (8.111) x¨ + 2μx˙ + sgn(x)|x|α = 0

(8.125)

where α is a non-negative real number; see the comments to equation (8.70). In this case, zero-order solution (8.119) can be obtained in the form (8.69), x(t) = A(t0 )sgn(τ (ϕ)) (8.126)

α |τ (ϕ)|α+2 |τ (ϕ)|α+2 |τ (ϕ)|2α+3 + − × |τ (ϕ)| − α+2 2(α + 2) 2α + 3 α+2 ϕ(t) ˙ = ω0 (t0 ),

t0 = μt

226

8 Strongly Nonlinear Vibrations

where the functions A(t0 ) and ω0 (t0 ) are coupled by relation (8.40) as follows A(α−1)/2 1 ω0 = = √ a α+1

1+

−1/2 α (α + 1)2 1+ 2(α + 2) (α + 2)(2α + 3)

Equations (8.127) and (8.124) admit exact solution

4μt α−1 A = C exp − t , ϕ = ϕ∞ 1 − exp −2μ 3+α α+3

(8.127)

(8.128)

where C is a new arbitrary constant and ϕ∞

C (α−1)/2 (2 + α) 2(3 + 2α) 1 α+3 = 2μ α − 1 (α + 1) (7α3 + 31α2 + 47α + 24)

(8.129)

It follows from expressions (8.128) and (8.129) that the linear system α = 1 plays the role of a boundary between the two strongly nonlinear areas N0 = {α : 0 ≤ α << 1} and N∞ = {α : 1 << α < ∞}

(8.130)

In other words, we show that α = 1 separates two qualitatively diﬀerent regions of the dynamics determined by the inﬂuence of diﬀerent nonsmooth limits of the potential well; see Fig. 8.13 for illustration. In particular, if α > 1 then the phase variable ϕ has the ﬁnite limit ϕ∞ as t → ∞. In contrast, if α < 1 then the phase with its temporal rate are exponentially growing, as the amplitude decays and the system approaches the bottom of the potential well. The physical meaning of this eﬀect is most clear from the limit case α → 0, which is discussed below. Figs. 8.14, 8.15 and 8.16, 8.17 illustrate solution (8.126) through (8.129) for large and small exponents α, respectively. The diagrams suggest quite a good match with numerical solution in both branches of the exponent (8.130). The numerical solutions shown by dashed lines were obtained by the stanR dard solver NDSolve built in Mathematica . Fig. 8.14 also shows that some divergence between the curves occurs when the amplitude is decreased to the level about A = 0.6. Below this level, the condition of small damping (8.112) is not guaranteed any more. In contrast, the curves are in a better match for smaller amplitudes if α < 1, see Fig. 8.16. In this case, the amplitude decay just strengthens condition (8.112). The phase plane diagrams shown in Figs. 8.15 and 8.17 have qualitatively diﬀerent shapes as dictated by the inﬂuence of diﬀerent nonsmooth limits of the potential well, see Fig. 8.13. Let us show now that solution (8.126) captures both nonsmooth limits α → 0 and α → ∞. For a physically meaningful transition to the limits, let us express the ˙ t=0 , arbitrary parameter C through the initial velocity v0 = x|

8.6 Strongly Nonlinear Oscillator with Viscous Damping

227

1/(α+1) v02 (α + 1) 7α3 + 31α2 + 47α + 24 C= 2(α + 2)2 (2α + 3)

and consider the two diﬀerent cases. 1) As α → ∞, the solution (8.126) through (8.129) gives x = τ (ϕ) v0 [1 − exp (−2μt)] ϕ= 2μ

(8.131)

Solution (8.131) exactly describes the system motion in the square potential well. 2) When α → 0, expressions (8.126) through (8.129) are reduced to

|τ (ϕ)| 4 2 x = v0 exp − μt τ (ϕ) 1 − (8.132) 3 2

2 3 μt − 1 exp ϕ= 2μv0 3 where the identity sgn[τ (ϕ)]|τ (ϕ)| ≡ τ (ϕ) has been taken into account. If, in addition μ = 0, then solution (8.132) also exactly describes the system dynamics with another nonsmooth limit of the potential energy, |x|, as shown in Fig. 8.13. However, if μ = 0 then substituting solution (8.132) into the diﬀerential equation of motion gives an error O(μ2 ). In terms of ﬁrstorder asymptotic solutions, the error of order μ occurs on the time period of order 1/μ. Therefore, solution (8.132) exactly captures the carrying shape of the vibration, but gives only asymptotic estimate for the exponential decay.

Fig. 8.13 Potential energy representation for the two limit oscillators.

228

8 Strongly Nonlinear Vibrations

Fig. 8.14 Temporal mode shape of the vibration for α ∈ N∞ , C = 1.5 and μ = 0.04; here and below, the dashed line represents numerical solutions.

4

2 v 0 2 4 1.0

0.5

0.0 x

0.5

1.0

Fig. 8.15 Phase plane diagram for α ∈ N∞ .

Note that the error of solution (8.126) is due to the error of the iterative procedure for elastic vibrations and the error of asymptotic for energy dissipation. As shown above, the error of successive approximations vanishes as either α → ∞ or α → 0, but the error of asymptotic vanishes only as α → ∞. Finally, let us discuss the qualitative diﬀerence of the dynamics in the parameter intervals N0 and N∞ . As follows from equation (8.128), for α ∈ N0 , the phase of vibration and the corresponding frequency are exponentially increasing in the slow time scale μt . The physical meaning of this phenomenon

8.6 Strongly Nonlinear Oscillator with Viscous Damping

1.0

229

Α 1 14 analytical, numerical

0.5

x 0.0 0.5 1.0 0

20

40

60

80

100

120

140

t Fig. 8.16 Temporal mode shape of the vibration for α ∈ N0 , C = 2.5 and μ = 0.04.

Fig. 8.17 Phase plane diagram for α ∈ N0 .

becomes most clear in the limit case α = 0. In this case, according to solution (8.132), the amplitude and frequency are, respectively

2 4 1 1 v02 exp − μt and ϕ˙ = μt = exp (8.133) A(μt) = 2 3 v0 3 2A(μt) Expressions (8.133) describe increasingly rapid vibrations -‘dither’- near the corner of the potential energy |x| as the amplitude approaches zero. This result is conﬁrmed by the much earlier analysis of the corresponding conservative case [78]. In contrast, when α ∈ N∞ , the oscillator makes a limited

230

8 Strongly Nonlinear Vibrations

number of cycles such that the phase ϕ remains bounded for any time t. Again, the most clear interpretation is obtained in the limit case α → ∞, when, as follows from (8.131), the phase variable ϕ(t) represents the total distance passed by the particle by time t, and ϕ˙ = v is the absolute value of the velocity. Since the barriers are perfectly elastic the particle reﬂects with no energy loss, the velocity v (t) remains continuous function of time described by the linear diﬀerential equation v˙ + 2μv = 0 or ϕ ¨ + 2μϕ˙ = 0. ˙ = v0 , one obtains exactly Under the initial conditions ϕ(0) = 0 and ϕ(0) solution (8.131). In conclusion, the explicit analytical solution for a class of strongly nonlinear oscillators with viscous damping is introduced. Two diﬀerent nonsmooth functions involved into the solution are associated with two diﬀerent nonsmooth limits of the oscillator. As a result, the solution is drastically simpliﬁed to give the best match with numerical tests if approaching any of the two limits.

8.7

Bouncing Ball

In this section, we consider a small ball of mass m falling under the gravity force onto a horizontal plane. In addition to the gravity, the ball is subjected to the linear damping with the coeﬃcient c. Impacts with the plane are inelastic with the restitution coeﬃcient κ. The vertical coordinate z(t) therefore is described by the following equations of motion m¨ z = −mg − cz˙ (z = 0) z˙+ = −κz˙− (z = 0)

(8.134)

where z˙− and z˙+ are velocities right before and immediately after the impact, respectively. Let h be a natural spatial scale of the system. This can be, for instance, the maximal height that has been reached by the ball during the very ﬁrst cycle. Introducing the coordinate transformation z = h|x| and re-scaling the time as t = h/gp, brings equations (8.134) to the form [204] d2 x dx + sgnx = 0 + 2μ dp2 dp

dx dx dx − = (1 − κ) dp + dp − dp − *

where 1 c μ= 2m

h g

(8.135)

8.7 Bouncing Ball

231

In the particular case κ = 1, solution (8.132) becomes applicable to equation (8.135). Then, returning back to the original notations of equation (8.134), gives

1 2 2c t |τ (ϕ)| − τ (ϕ) (8.136) z(t) = C exp − 3m 2 c g 3m exp t −1 ϕ(t) = C c 3m where C is a new arbitrary constant. If C = z˙02 /g then solution (8.136) satisﬁes the speciﬁc initial conditions z(0) = 0 and z(0) ˙ = z˙0 . One more arbitrary constant can be introduced by shifting the time, t− > t + t0 , that would allow to consider non-zero initial height of the ball. In order to compare solution (8.136) with numerical solution, let us represent equation (8.134) in the form z˙ = u c u˙ = −g − u m

(8.137)

where u+ = −κu− whenever z = 0. Further, introduce new unknown state variables {s, v} according to [73] z = ssgn(s) u = sgn(s)[1 − ksgn(sv)]v 1−κ k= 1+κ

(8.138)

Applying (8.138) to (8.137), gives1 s˙ = [1 − ksgn(sv)]v g c [sgn(s) + ksgn(v)] v˙ = − v − m 1 − k2

(8.139)

As compared to (8.137), system (8.138) automatically accounts for the velocity jump condition at z = 0. In other words, transformation (8.138) makes the strong nonlinearity (due to non-elastic impacts) explicitly present in (8.139). From the standpoint of numerical procedures, such a transformation enables one of using built-in solvers of diﬀerent packages with no impact conditioning at z = 0. Note that, in the particular case κ = 1 (k = 0), system (8.139) becomes equivalent to (8.135). In this particular case, the direct numerical integration R of equations (8.139), using the NDSolve procedure built in Mathematica 1

Note that diﬀerentiation of sgn-functions will produce Dirac’s delta-functions with eﬀectively zero factors however.

232

8 Strongly Nonlinear Vibrations

package, gives solution in a perfect match with analytical solution (8.136); see Fig.8.18. Let us assume now that the energy loss happens only due to impact interactions of the ball with the plane z = 0. The question is whether or not solution (8.136) can still be adapted by interpreting the parameter c as some “eﬀective damping coeﬃcient,” such that the energy loss between two impacts is equal to that happens in one impact.

0.05 0.04 0.03

analytical, numerical

z 0.02 0.01 0.00 0

1

2

3

4

5

6

t Fig. 8.18 NSTT analytical and direct numerical solutions for the height of bouncing ball under the linear dissipation condition; m = 1.0, c = 0.5, z(0) = 0 and z(0) ˙ = 1.0.

Assuming that the damping is small enough and using the classical parabolic approximation for the ball height during one cycle, z(t) = −gt2 /2+ v0 t, yields 2v0 /g 1 2cef f v03 cef f = m(1 − κ2 )v02 z˙ 2 dt = 3g 2 0 or 3mg cef f = (1 − κ2 ) (8.140) 4v0 Expression (8.140) shows that, in this case, eﬀective linear damping cannot be introduced on the entire time interval since the “initial velocity” v0 de˙ in (8.140), provides a good creases from cycle to cycle. Choosing v0 = z(0) enough match between analytical (8.136) and numerical solutions during ﬁrst few cycles of the motion, as follows from Fig. 8.19. Then, the divergence between the curves accelerates. The result can be improved by using some

8.7 Bouncing Ball

233

0.05 0.04 0.03

analytical

z

numerical 0.02 0.01 0.00 0

1

2

3

4

5

6

t Fig. 8.19 NSTT analytical solution with eﬀective damping corresponding to the coeﬃcient of restitution κ = 0.97, and direct numerical solutions for the height of bouncing ball; the parameters are: m = 1.0, c = 0.0, z(0) = 0 and z(0) ˙ = 1.0.

0.05 0.04 0.03

numerical

z

analytical

0.02 0.01 0.00 0

1

2

3

4

5

6

t Fig. 8.20 NSTT analytical solution with eﬀective damping adjusted by “variable initial velocity,” and direct numerical solutions for the height of bouncing ball; the parameters are: m = 1.0, c = 0.0, κ = 0.97, z(0) = 0 and z(0) ˙ = 1.0.

estimate for the “initial velocity” decay based on solution (8.136), where the eﬀective damping is still constant, however. This one-step iteration gives c ef f t (8.141) c˜ef f = cef f exp 3m Fig. 8.20 shows a better match between numerical and analytical solutions achieved due to modiﬁcation (8.141).

234

8.8

8 Strongly Nonlinear Vibrations

The Kicked Rotor Model

The so-called kicked rotor model is introduced in physics as a relatively simple essentially nonlinear model for chaotic behavior of systems, where one variable may be either bounded or unbounded in phase space [95], [20], [71]. The kicked rotor is described, in some units, by the Hamiltonian H=

∞ 1 2 I + K cos θ δ (t − n) 2 n=−∞

(8.142)

where I is the angular momentum, θ is the conjugate angle, and K is the stochasticity parameter that determines qualitative features of the dynamics. The sequence of pulses in (8.142) can be expressed through the sawtooth sine τ = τ (2t − 1) in the form, ∞

1 δ (t − n) = − τ (2t − 1) τ (2t − 1) 2 n=−∞

(8.143)

where primes denotes diﬀerentiation with respect to the entire argument of a function, 2t − 1. Note that the only role of the ﬁrst multiplier, τ (2t − 1), on the right-hand side is to provide pulses with the same sign. Therefore, (8.142) takes the form H=

1 2 1 I − Kτ (2t − 1) τ (2t − 1) cos θ 2 2

(8.144)

The corresponding diﬀerential equation of motion is 1 ¨ θ = − Kτ τ sin θ 2

(8.145)

We seek a family of solutions with the period T = 2 by introducing the sawtooth time argument τ , θ = θ(τ ) (8.146) Substituting (8.146) in (8.145), gives

dθ 1 d2 θ Kτ sin θ + = − τ dτ 2 8 dτ Eliminating the singular term τ , leads to the boundary condition dθ 1 |τ =±1 = − Kτ sin θ|τ =±1 dτ 8

(8.147)

8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

235

and the diﬀerential equation d2 θ =0 dτ 2 The boundary-value problem (8.147) and (8.148) admits solution θ = Aτ (2t − 1) + B

(8.148)

(8.149)

where A and B appear to be coupled by the set of equations

or

1 A = − K sin(A ± B) 8

(8.150)

1 A = − K sin A cos B 8

(8.151)

cos A sin B = 0

(8.152)

and In particular, equation (8.151) show that the number of periodic solutions of the period T = 2 is growing as the parameter K increases.

8.9

Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

Consider the case of asymmetric restoring force characteristics described by two pieces of smooth monotone increasing functions f (x) and g(x) of which one or both may be nonlinear (Fig. 8.21) f (x), −∞ < x < x1 (8.153) F (x) = g(x), x1 < x < ∞ It is assumed that the entire characteristic F (x) is at least continuous in the interval −∞ < x < ∞, in other words, the following matching condition holds f (x1 ) = g(x1 ) (8.154) and f (0) = 0. Expression (8.153) admits the unit form F (x) = f (x) + [g(x) − f (x)]H(x − x1 )

(8.155)

where H indicates the unit-step Heaviside function. Let us outline analytical procedure for periodic solutions of equation x ¨ + F (x) = 0

(8.156)

236

8 Strongly Nonlinear Vibrations

Fig. 8.21 Asymmetric piece-wise nonlinear restoring force characteristic.

Introducing the triangular wave temporal argument τ = τ (t/a) with unknown scaling parameter a and making the substitution x = X(τ ) in equation (8.156), gives boundary value problem X + hF (X) = 0

(8.157)

X |τ =±1 = 0

(8.158)

where, as usually, h = a2 = (T /4)2 . Following the physical reasoning discussed in Introduction and the second section of this chapter, we seek solution in the form of successive approximation series with no explicit small parameter X = X0 (τ ) + X1 (τ ) + X2 (τ ) + ...

(8.159)

h = h0 + h1 + ... where the generating solution X0 (τ ) obeys the diﬀerential equation X0 (τ ) = 0 and therefore describes the dynamics of a two-parameter family of free impact oscillators with arbitrary amplitudes A0 and the origin shift as follows X0 (τ, τ1 , A0 ) = x1 + A0 (τ − τ1 )

(8.160)

The idea behind approximation (8.160) is to choose its parameters in order to make the motion x(t) = X0 (τ (t/a)) in ‘some sense’2 close to that of oscillator (8.156). The form of (8.160) implies that the system is at the matching point whenever τ = τ1 . Next term of series (8.159) is found then from the equation X1 = −h0 F (X0 ) in the integral form 2

Note that the harmonic balance analogy of generating solution (8.160) would be x(t) = A sin ωt + B.

8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

237

τ X1 (τ, τ1 , A0 , A1 ) = A1 (τ − τ1 ) − h0

(τ − ξ)F (X0 (ξ, τ1 , A0 ))dξ

(8.161)

τ1

where the combination of arbitrary constants is similar to that in generating solution (8.160). Since X1 (τ1 , τ1 , A0 , A1 ) = 0 then τ = τ1 corresponds to the matching point x = x1 in ﬁrst two terms of the successive approximations, (8.160) and (8.161), τ X = x1 + (A0 + A1 )(τ − τ1 ) − h0

(τ − ξ)F (X0 (ξ, τ1 , A0 ))dξ

(8.162)

τ1

Therefore, after the ﬁrst correction to generating solution (8.160), we have three arbitrary constants A0 , A1 and τ1 , and one still unknown parameter h0 related to the period of free vibration. Now, substituting (8.162) in boundary conditions (8.158), gives 1 A0 + A1 − h0

F (X0 (τ, τ1 , A0 ))dτ = 0 τ1 τ1

A0 + A1 + h0

F (X0 (τ, τ1 , A0 ))dτ = 0

(8.163)

−1

where the variable of integration ξ has been formally replaced by τ . Equations (8.163) are equivalent to 1 F (X0 (τ, τ1 , A0 ))dτ = 0

(8.164)

F (X0 (τ, τ1 , A0 ))dτ = h−1 0 (A0 + A1 )

(8.165)

−1

1 τ1

If no more iterations are planned then we set A1 = 0, because the corresponding term contributes nothing qualitatively new into the approximate solution. In this case, equation (8.164) is used to express τ1 through another constant provides A0 and the matching point coordinate x1 . Then, equation (8.165) √ the link between the parameters of amplitude A0 and period T = 4 h0 . The next step of iteration employs the parameter A1 , however. The form of diﬀerential equation for the next step of procedure is analogous to (8.20), where γ1 h0 = h1 . Therefore, on the next step, the general solution is given by

238

8 Strongly Nonlinear Vibrations

τ X2 = −h1

(τ − η)F (X0 (η, τ1 , A0 ))dη τ1 τ

−h0

(8.166)

(τ − η)F (X0 (η, τ1 , A0 ))X1 (η, τ1 , A0 , A1 )dη + A2 (τ − τ1 )

τ1

where zero lower limit of integration provides X2 (τ1 ) = 0, and the prime indicates derivative with respect to the entire argument, F (X0 ) ≡ dF (X0 )/dX0 . Note that the characteristic F (x) is, generally speaking, nonsmooth at the matching point x1 so that the derivative F (x) may not exist at x = x1 . Although integration of step-wise discontinuous functions is still possible, the current procedure is designed to avoid calculating derivatives of the characteristic at the point x1 . For that reason, the lower limit of integration in (8.166) is associated with the non-smoothness point by expression (8.160). In addition, as follows from (8.161), the uncertainty F (x1 ) in the integrand is suppressed by zero factor X1 (τ1 ) = 0. Obviously, on the next step of iteration, this factor will accompany the second derivative F (x1 ), whereas the ﬁrst derivative acquires the factor X2 (τ1 ). Now, applying boundary conditions (8.158) to (8.166), yields 1 −h1

F (X0 (τ, τ1 , A0 ))dτ τ1

1 − h0

F (X0 (τ, τ1 , A0 ))X1 (τ, τ1 , A0 , A1 )dτ + A2 = 0

(8.167)

τ1

τ1 h1

F (X0 (τ, τ1 , A0 ))dτ

−1

τ1 + h0

F (X0 (τ, τ1 , A0 ))X1 (τ, τ1 , A0 , A1 )dτ + A2 = 0

(8.168)

−1

Substracting the both sides of equation (8.167) from (8.168) and taking into account equation (8.164), gives 1 −1

F (X0 (τ, τ1 , A0 ))X1 (τ, τ1 , A0 , A1 )dτ = 0

(8.169)

8.9 Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics

239

Then, taking into account (8.165), brings equation (8.167) to the form 1

−1 F (X0 (τ, τ1 , A0 ))X1 (τ )dτ = −h1 h−2 0 (A0 + A1 ) + h0 A2

(8.170)

τ1

If no more iterations needed, then we set A2 = 0 in expression (8.166) and equation (8.170). Further, substituting (8.161) in (8.169), gives equation for A1 , whereas (8.170) gives equation for h1 . This completes two steps of successive approximations, although the algebraic problem still persists, and its complexity depends on the functions f (x) and g(x).

Chapter 9

Strongly Nonlinear Waves

Abstract. This short chapter deals with modulated waves described by strongly nonlinear Klein-Gordon equation. Mechanical equivalent for this model can be represented by the inﬁnite linear string on elastic foundation. In this case, the principal (carrying) wave can be approximated by polynomials or other elementary functions of the triangular sine wave based on the algorithms of the previous chapter. Then the diﬀerential equations for slow varying modulating components are derived by using the Whitham assumptions and the speciﬁc boundary conditions associated with nonsmooth argument substitutions.

9.1

Wave Processes in One-Dimensional Systems

A one-dimensional wave can be described by the function u = u(θ, x, t)

(9.1)

where θ = θ(x, t) is a phase variable, x and t are the coordinate and time, respectively. In the stationary case, the phase variable is usually the only argument of the wave shape function u = u(θ), where θ = ωt − kx with temporal and spatial wave numbers, ω and k, respectively, so that ∂θ =ω ∂t

and

∂θ = −k ∂x

(9.2)

Function (9.1) includes also the variables x and t explicitly in order to describe wave modulation eﬀects. Therefore, we assume that the dependence upon explicit x and t is slow. Such an assumption can be formalized by means of the formal auxiliary parameter ε [194], [116] u = u(θ, x0 , t0 ) where x0 = εx, t0 = εt, and relationships (9.2) take now the form V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 241–244, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(9.3)

242

9 Strongly Nonlinear Waves

∂θ = ω(x0 , t0 ) and ∂t

∂θ = −k(x0 , t0 ) ∂x

(9.4)

Let the wave shape function u be periodic with respect to θ with the period T = 4. In this case, the sawtooth phase argument can be introduced as follows τ = τ (θ), (9.5) u = U (τ, x0 , t0 ) + V (τ, x0 , t0 )e where e = e(θ) = τ (θ) is the rectangular cosine of the phase variable.

9.2

Klein-Gordon Equation

For illustration purposes, we consider the Klein-Gordon equation ∂2u ∂2u − 2 + f (u) = 0 ∂t2 ∂x

(9.6)

where f (u) is an odd function. Substituting (9.5) in (9.6) and taking into account (9.4), gives 2 ∂2U ∂V ω − k2 − ε2 LU = −Rf (U, V ) − εH ∂τ 2 ∂τ 2 ∂ 2V ∂U ω − k2 − ε2 LV = −If (U, V ) − εH 2 ∂τ ∂τ ∂U |τ =±1 = 0 ∂τ V |τ =±1 = 0 where

(9.7)

(9.8)

∂ ∂ ∂k ∂ω H ≡ 2 ω 0 +k 0 + 0 + 0 ∂t ∂x ∂t ∂x ∂2 ∂2 L ≡ 02 − 02 ∂t ∂x

and 1 [f (U + V ) + f (U − V )] 2 1 If = [f (U + V ) − f (U − V )] 2

Rf =

Analogously to the case of strongly nonlinear damped oscillator considered in the previous chapter, we represent solution in the form of series

9.2 Klein-Gordon Equation

243

U (τ, x0 , t0 ) = U0 (τ, x0 , t0 ) + εU1 (τ, x0 , t0 ) + ... V (τ, x0 , t0 ) = V0 (τ, x0 , t0 ) + εV1 (τ, x0 , t0 ) + ...

(9.9)

Substituting (9.9) into (9.7) and (9.9) and equating coeﬃcients of like powers of ε, gives in ﬁrst two steps 2 ∂ 2 U0 ω − k2 = −f (U0 ) ∂τ 2 V0 ≡ 0 ∂U0 |τ =1 = 0 ∂τ where U0 (−τ ) = −U0 (τ ) due to oddness of the function f (U0 ), and 2 ∂ 2 U1 ω − k2 = −f (U0 )U1 ∂τ 2 2 ∂ 2 V1 ∂U0 ω − k2 = −f (U0 )V1 − H ∂τ 2 ∂τ ∂U1 |τ =±1 = 0 ∂τ V1 |τ =±1 = 0

(9.10) (9.11)

(9.12)

(9.13)

The generating solution is obtained from (9.10) in the following general form U0 = U0 (τ, A, ω 2 − k 2 )

(9.14)

where A, ω, and k are functions of the slow variables x0 and t0 . The functions A(x0 , t0 ), ω(x0 , t0 ), and k(x0 , t0 ) are coupled by the boundary condition (9.11) ∂U0 (τ, A, ω 2 − k 2 ) |τ =1 = 0 (9.15) ∂τ and by the condition of solvability of the boundary value problem (9.12) for V1 , . / 1 ∂U0 ∂U0 ∂U0 ∂U0 H dτ = 2 H =0 ∂τ ∂τ ∂τ ∂τ −1

or

0

0

2 1 2 1 ∂U0 ∂U0 ∂ ∂ ω + 0 k =0 ∂t0 ∂τ ∂x ∂τ

(9.16)

Example 17. Consider the equation ∂2u ∂2u − 2 + sgn(u)|u|α = 0 ∂t2 ∂x

(9.17)

244

9 Strongly Nonlinear Waves

where α is a real positive number. In this case, the generating solution is obtained analogously to the case of oscillator with two non-smooth limits, considered in Chapter 8, by replacing ω02 with ω 2 − k 2 . In particular, Aα−1 ω −k = α+1 2

2

1+

−1 α (α + 1)2 1+ 2(α + 2) (α + 2)(2α + 3)

(9.18)

whereas equation (9.16) gives ∂(ωA2 ) ∂(kA2 ) + =0 ∂t ∂x

(9.19)

Equations (9.18) and (9.19) have been written in terms of the original time and coordinate by setting the auxiliary parameter to unity, ε = 1. However, according to the basic assumption, A = A (t, x), ω = ω (t, x) and k = k (t, x) still should be treated as slow functions as compared to τ (θ).

Chapter 10

Impact Modes and Parameter Variations

Abstract. In this chapter, a new parameter variation and averaging tools are introduced for impact modes. It is also shown that a speciﬁc combination of two impact modes gives another impact mode1 . The corresponding manipulations with impact modes become possible due to the availability of closed form exact solutions obtained by means of the triangular sine temporal substitution for impulsively loaded and vibroimpact systems. In particular, the idea of Van-der-Pol and averaging tool are adapted for the case of impact oscillator. For illustrating purposes, a model of coupled harmonic and impact oscillators is considered. Further, mass-spring systems with multiple impacting particles are considered in order to illustrate impact localization phenomena on high-energy levels.

10.1

An Introductory Example

Vibration modes with impacts have been under study for several years [191], [197], [15], [156]. In practical terms, such studies deal with the dynamics of elastic structures whose amplitudes are limited by stiﬀ constraints. These may be designed intentionally or occur due to a deterioration of joints. As a result, such kinds of dynamics are often accompanied by a rattling noise or dither during operating regimes of vehicles or machine tools. From the theoretical standpoint, the interest to such problems is driven by the question what happens to linear normal modes as the energy of elastic vibrations becomes suﬃcient for reaching the constraints. Interestingly enough, some of the analytical approaches developed in the area recently found applications in molecular dynamics [49]. However, due to strong nonlinearities of the impact dynamics, most of the results relates to periodic particular solutions according to the idea of nonlinear normal modes [190]. Let us recall that the importance of linear normal modes is emphasized by the linear superposition principle as 1

Notice that the number of impact modes depends on the number of constraints and therefore can signiﬁcantly exceed the number of degrees of freedom.

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 245–264, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

246

10 Impact Modes and Parameter Variations

well as the parameter variation and averaging methods for weakly nonlinear cases. Let us consider a one degree of freedom free harmonic oscillator between two absolutely rigid barriers. A mechanical model of such an oscillator can be represented as a mass-spring model with two-sided amplitude limiters as shown in Fig. 10.1 (a).

Fig. 10.1 The oscillator with bilateral rigid barriers (on the left) is replaced by the oscillator under the periodic series of external impulses (on the right).

The interaction with the barriers at x = ±Δ is assumed to be perfectly elastic, and the system is represented in the form x ¨ + ω02 x = 0,

|x| ≤ Δ

(10.1)

Since the normal mode regimes are periodic by their deﬁnition then the reaction of constraints can be treated as a periodic series of external impulses acting on the masses of the system.

10.1 An Introductory Example

247

Applying this remark to the one degree of freedom system as it is shown in Fig. 10.1 (b), the related diﬀerential equation of motion is written in the linear form x ¨ + ω02 x = 2p

∞

[δ (ωt + 1 − 4k + α) − δ (ωt − 1 − 4k + α)] (10.2)

k=−∞

= pτ (ωt + α) where δ (ξ) is the Dirac function, τ (ξ) is the triangular sine wave, 2p, ω, and α will be interpreted as arbitrary parameters. For further convenience, the right-hand side of equation (10.2) is expressed through second-order generalized derivative of the triangular sine wave with respect to the entire argument, ωt + α. The parameter ω will be called a frequency parameter, although it diﬀers by the factor π/2 from the regular trigonometric frequency, Ω = (π/2) ω. Both parameters, ω and Ω, may be used below. In contrast to system (10.1), the auxiliary system (10.2) is linear but not completely equivalent to the original one; see the analyses below. Representing unknown steady-state periodic solution in the form, x = X (τ ) , τ = τ (ωt + α)

(10.3)

gives the boundary value problem with no singular terms, ω 2 X (τ ) + ω02 X (τ ) = 0

X (τ ) |τ =±1 = pω

(10.4) −2

and the related solution is represented in the saw-tooth time form [139] x=

p sin [(ω0 /ω) τ (ωt + α)] ωω0 cos (ω0 /ω)

(10.5)

This solution can be veriﬁed by direct substitution of expression (10.5) into the equation of motion (10.2). A connection between solution (10.5) and vibration of the original system with stiﬀ constraints is established by imposing the conditions: • The impulses on the right-hand side of equation (10.2) act when the mass strikes the limiters, x = ±Δ if τ = ±1 ⇐⇒ τ = 0

(10.6)

• The system cannot penetrate through the limiters, therefore, |x| ≤ Δ for all τ ∈ [−1, 1]

(10.7)

248

10 Impact Modes and Parameter Variations

Substituting solution (10.5) into condition (10.6), determines the parameter p, (10.8) p = Δωω0 cot (ω0 /ω) Substituting now (10.8) in (10.5), gives solution in the ﬁnal form x(t) = Δ

sin [(ω0 /ω) τ (ωt + α)] sin (ω0 /ω)

(10.9)

Obviously, solution (10.9) satisﬁes condition (10.6) automatically. The related parameter p (10.8) will further be treated as an “eigen value” of the nonlinear (impact) problem. Other parameters, ω and α, are determined by the initial conditions. Let us assume that x (0) = 0 and thus α = 0. As a result, the total energy of the oscillator per unit mass is expressed through the initial velocity as 2 E = [x˙ (0)] /2. Then, taking into account (10.9) and making some analytical manipulations, gives

1 ω 2 Δ2 ω0 = ± arccos 1 − 0 + kπ (10.10) ω 2 E where k = 0,1, ... . The right-hand side of expression (10.10) gives a sequence of real numbers if the total energy is above its critical value, E ≥ E∗ = ω02 Δ2 /2, so that the oscillator can reach the constraints. However, not all of the real numbers ω lead to real motions of the original system. Indeed, since the auxiliary system (10.2) has no limiters then condition (10.6) does not guarantee that the oscillator will remain inside the region |x| ≤ Δ during the period of vibration. Therefore, condition (10.7) must be veriﬁed as well. Such a veriﬁcation implemented for solution (10.9) shows that condition (10.7) is satisﬁed only for the smallest root in set (10.10). Fig. 10.2 illustrates the temporal mode shapes corresponding to the ﬁrst two roots ω0 /ω. It is seen that the second solution (on the right), violates condition (10.7) while satisﬁes condition (10.6). Note that the above approach can be applied to the case of unilateral limiters. Let us remove, for instance, the left limiter and consider the oscillator (10.1) under the condition x ≤ Δ. In this case, the boundary conditions in (10.4) should be modiﬁed as X (τ ) |τ =±1 = ±pω −2

(10.11)

Such a periodic change of sign eﬀectively switches the directions of positive δ-pulses on the right-hand side of equation (10.2). As a result, the solution takes the form, cos [(ω0 /ω) τ (ωt + α)] x(t) = Δ (10.12) cos (ω0 /ω)

10.2 Parameter Variation and Averaging

249

a

2 1 x

b

2 1 x

0

0

1

1

2

2 0

5

10

15

20

0

5

t

10

15

20

t

Fig. 10.2 Real (a) and “phantom” (b) solutions corresponding to the ﬁrst (smallest) and second roots respectively, (ω0 /ω)1 = 1.1502 and (ω0 /ω)2 = −1.1502 + π. The total energy level is E = 1.2E∗ .

where the period and the related basic frequency are T = 2/ω and Ω = πω, respectively.

10.2

Parameter Variation and Averaging

In order to illustrate the idea of parameter variations for solution (10.9), let us include the viscous damping into the model and represent the diﬀerential equations of motion between the constraints in the form x˙ = v v˙ = −2ζω0 v − ω02 x

(10.13)

where ζ is the damping ratio parameter. In this case, the triangular wave frequency ω in solution (10.9) is not constant any more, although the amplitude of the vibration remains constant as long as the oscillator is in the impact regime. The corresponding parameter variation is implemented as a change of the state variables {x(t), v(t)} → {γ(t), φ(t)}, dictated by solution (10.9) sin(γτ (φ)) sin γ cos(γτ (φ)) v = ω0 Δ e(φ) sin γ

x=Δ

(10.14)

250

10 Impact Modes and Parameter Variations

where τ = τ (φ) and e = τ (φ) depend upon the fast phase φ = φ(t), and γ = γ(t) determines a relatively slow evolution of the temporal mode shape of the vibration. Substituting (10.14) in (10.13), gives still exact equations γ˙ = 2ζω0 cos2 γτ tan γ ω0 [1 + eζ(sin 2γτ − 2τ cos2 γτ tan γ)] φ˙ = γ

(10.15)

Below, the ﬁrst-order averaging procedure is applied. Notice that the righthand side of equations (10.15) is periodic with respect to the phase φ. However, as proved in Chapter4, the averaging can be conducted with respect to the variable τ over its interval −1 ≤ τ ≤ 1. As a result, one obtains

1 sin 2γ tan γ γ˙ = ζω0 1 + 2γ 8 = 2ζω0 γ + ζω0 γ 5 + O(γ 6 ) 45 ω 0 (10.16) φ˙ = γ Keeping the leading-order term only on the right-hand side of the ﬁrst equation in (10.16), gives solution γ = γ0 exp(2ζω0 t) 1 φ= [1 − exp(−2ζω0 t)] 2ζγ0

(10.17)

A simple asymptotic analysis of expressions (10.14) and the remark after expression (10.10) give the parameter interval, 0 < γ < π/2, within which the impact dynamics takes place. The vibration mode shapes close to the triangular wave near the left edge of the interval, but, as the energy dissipates and the parameter γ approaches π/2, vibrations become close to harmonic. The total energy is expressed through the parameter γ in the form E(t) =

1 ω02 Δ2 2 sin2 γ(t)

(10.18)

The duration of the impact stage of the dynamics is estimated via solution (10.17),

π 1 ln γ(tmax ) = π/2 =⇒ tmax = (10.19) 2ζω0 2γ0 where γ0 = γ(0). As follows from Fig. 10.3, the above averaging procedure leads to practically no error of the time history record within the entire interval of validity of the approach. However, the analytical solution based on the reduced model

10.2 Parameter Variation and Averaging

251

1.0

0.5

x

0.0

0.5

1.0 0

10

a

20

30

40

t

1.0

numerical, averaging 0.5

x

0.0

analytical

0.5

1.0 80

b

82

84

86

88

90

t

Fig. 10.3 The time history of the impact oscillator according to exact equations, those after the averaging, and the analytical closed form solution; the parameters are γ0 = 0.2, ζ = 0.01, ω0 = 1.0, and Δ = 1.0.

gives some deviation from the exact curve at the end of the impact stage of the dynamics. Notice that there is no impact interactions with the constraints for t > tmax , where the model becomes harmonic oscillator whose amplitude exponentially decays due to the energy dissipation. At this stage, the

252

10 Impact Modes and Parameter Variations

transformation (10.14) is not valid any more nor there is any need in transformations. However, a question occurs about such solutions that would be capable of describing both impact and non-impact stages within the same unit-form expressions.

10.3

A Two-Degrees-of-Freedom Model

Let us obtain ﬁrst the impact mode solutions for the model shown in Fig. 10.4 under no damping condition, c = 0. For the sake of simplicity, let us also assume that k1 = k2 = k. On the normal mode (impact) motions, the system can be eﬀectively replaced by x ¨1 + ω02 (2x1 − x2 ) = pe (ωt) x ¨2 + ω02 (2x2 − x1 ) = 0

(10.20)

where ω02 = k/m, and the parameters ω and p must provide the following condition (10.21) |x1 | ≤ Δ The impact mode solution is represented in the form xn (t) = Xn (τ )

(10.22)

where τ = τ (ωt) and n = 1, 2. Substituting (10.22) in (10.20) and eliminating the singular term e (ωt), gives the linear boundary value problem ω 2 X1 + ω02 (2X1 − X2 ) = 0 ω 2 X2 + ω02 (2X2 − X1 ) = 0

(10.23)

X1 |τ =±1 = pω −2 X2 |τ =±1 = 0 The corresponding solution has the form p sin(ω0 τ /ω) + X1 = 2ωω0 cos(ω0 /ω) p sin(ω0 τ /ω) X2 = − 2ωω0 cos(ω0 /ω)

√ 3 sin( 3ω0 τ /ω) √ 3 cos( 3ω0 /ω) √ √ 3 sin( 3ω0 τ /ω) √ 3 cos( 3ω0 /ω)

(10.24)

√

(10.25)

where the ‘eigen value’ p is determined by substituting X1 into X1 |τ =±1 = ±Δ

(10.26)

10.4 Averaging in the 2DOF System

253

This gives √ √ p = 2ωω0 Δ[tan(ω0 /ω) + ( 3/3) tan( 3ω0 /ω)]−1

(10.27)

In order to insure that solution (10.25) and (10.27) describe real motions, condition (10.7) must be veriﬁed. As follows from detailed considerations below, condition (10.7) is satisﬁed at least in the frequency range (π/2) ω > ω2 , where ω2 is the frequency of the out-of-phase mode of the related no impact system.

10.4

Averaging in the 2DOF System

Let us consider now the model shown in Fig. 10.4 by assuming that a relatively slow energy dissipation is possible due to the viscous damping. In order to introduce the corresponding parameter variation technique, let us represent the diﬀerential equations of motion in the following general form x˙ 1 = v1 v˙ 1 = −f1 (x1 , x2 , v1 , v2 ) + pe (φ) x˙ 2 = v2

(10.28)

v˙ 2 = −f2 (x1 , x2 , v1 , v2 ) where pe (φ) = pτ (φ) and f1 (x1 , x2 , v1 , v2 ) = 2ζΩ1 v1 + Ω12 x1 + β(x1 − x2 ) f2 (x1 , x2 , v1 , v2 ) = Ω22 x2 − β(x1 − x2 )

(10.29)

are impact and linear force components per unit mass, β = k/m is the parameter of coupling, ζ = c/(2Ω1 m) is the damping ratio, and Ωi = ki /m (i = 1, 2).

Fig. 10.4 The two-degrees-of-freedom model with viscous damping in the impact subcomponent.

254

10 Impact Modes and Parameter Variations

The idea of parameter variations is implemented below as a change of the state variables, {x1 (t), v1 (t), x2 (t), v2 (t)} → {γ(t), φ(t), A(t), B(t)}, according to expressions sin(γτ (φ)) sin γ cos(γτ (φ)) e(φ) v1 = Ω1 Δ sin γ

πτ (φ) πτ (φ) x2 = A sin + B cos e(φ) 2 2

πτ (φ) πτ (φ) + Ω1 A cos e(φ) v2 = −Ω1 B sin 2 2

x1 = Δ

(10.30)

It is assumed in (10.30) that the principal frequency of the vibration, dφ(t)/dt, is dictated by the impact subcomponent rather then by a natural frequency of the corresponding linearized system. However, the scaling factor Ω1 is still used in order to indicate the dominant oscillator of the dynamic process under consideration. Notice that x1 and v1 are transformed analogously to (10.14), whereas the transformation of x2 are v2 is based on the standard general solution of the harmonic oscillator represented however in the non-smooth temporal transformation form. Substituting (10.30) in (10.28), gives e cos γτ (f1 sin γ − Ω12 Δ sin γτ ) tan γ Ω1 Δ 1 [f1 sin γ(sin γτ − τ cos γτ tan γ) φ˙ = γΩ1 Δ +Ω12 Δ cos γτ (cos γτ + τ sin γτ tan γ)] 1 1 A˙ = − Ω1 B(1 − cos πτ ) + πB φ˙ 2 2 1 1 πτ +( Ω1 A sin πτ − )e f2 cos 2 Ω1 2 1 1 B˙ = Ω1 A(1 + cos πτ ) − πAφ˙ 2 2 1 1 πτ + f2 sin − e Ω1 B sin πτ Ω1 2 2 γ˙ =

(10.31)

where the functions f1 and f2 are expressed through the new variables by substitution (10.30) in (10.29), and the impact term pe (φ) has been eliminated by setting (compare with (10.8)) ˙ p = p(t) = Δφ(t)Ω 1 cot γ(t)

(10.32)

Further reduction of system (10.31) includes two major steps, such as averaging with respect to the fast phase φ and applying the power series expansion

10.4 Averaging in the 2DOF System

255

with respect to the parameter γ. Since the periodic functions in equations (10.31) are expressed through the triangular sine wave τ (φ) then the averaging can be implemented by considering τ as an argument of the averaging as described in Chapter 4. Then, keeping the leading-order terms of the power series expansions with respect to γ, gives γ˙ = 2ζΩ1 γ Ω1 φ˙ = γ

β + Ω12 + Ω22 π A˙ = −B − φ˙ 2Ω1 2

2 2 β + Ω1 + Ω2 4βΔ π B˙ = A − φ˙ − 2 2Ω1 2 π Ω1

(10.33)

Approximate equations (10.33) describe only one-way interaction between the oscillators so that the ﬁrst two equations can be easily solved analytically; see the above one degree-of-freedom case. Then, substituting the result into the next two equations, gives a linear set of equations with variable coeﬃcients for A(t) and B(t), which can also be considered analytically. Let us skip such kind of analysis but illustrate the ﬁnal result in Figs. 10.5 and 10.6 for

0.00020

reduced model numerical

0.00015

Ρ

averaging 0.00010

0.00005

0.00000 0

20

40

60

80

t Fig. 10.5 The total energy of the second oscillator at relatively weak coupling, β = 0.01

256

10 Impact Modes and Parameter Variations

0.007

numerical

0.006

averaging

0.005 0.004

Ρ 0.003

reduced model

0.002 0.001 0.000 0

20

40

60

80

t Fig. 10.6 The total energy of the second oscillator at stronger coupling, β = 0.05

two diﬀerent strengths of the coupling β. The initial conditions and other parameters are selected as follows: γ(0) = 0.2, φ(0) = 0.0, A(0) = −0.01, B(0) = 0, ζ = 0.01, Ω1 = Ω2 = 1.0, Δ = 1.0. The diagrams show the energy versus time of the second oscillator based on numerical solutions for three diﬀerent equations, such as exact equations (10.31), the equations obtained by averaging (not described here), and the reduced set (10.33). The solutions are in quite a good match most of the time interval, however, the solution of truncated set (10.33) show some error near the end of the interval. This happens because the parameter γ is slowly approaching its limit magnitude π/2, at which the ﬁrst oscillator stops interacting with the constraints and the entire system becomes linear. Remind that equations (10.33) were obtained by truncating the polynomial expansions in the neighborhood of γ = 0; as a result, the accuracy of the equations is low near the point γ = π/2. However, the precision can be signiﬁcantly improved by keeping few more terms of the power series with respect to γ.

10.5

Impact Modes in Multiple Degrees of Freedom Systems

Let us consider the N -degrees-of-freedom conservative system described by T the coordinate vector x = (x1 , ..., xN ) ∈ RN . The corresponding mass-spring model is shown in Fig. 10.7. It is assumed that displacements of the ath mass

10.5 Impact Modes in Multiple Degrees of Freedom Systems

257

Fig. 10.7 The mass-spring model of a discrete elastic system with displacement limiters.

is limited by perfectly stiﬀ elastic constraints, such that |xa | ≤ Δa or, in the matrix form, T Ia x ≤ Δa (10.34)

T where Ia = 0, ..., 1, ..., 0 . a

Inside the domain (10.34), the diﬀerential equations of motion are assumed to be linear, Mx ¨ + Kx = 0 (10.35) where M and K are constant mass and stiﬀness N ×N -matrixes, respectively. The form of matrix equation (10.35) however is general enough to describe diﬀerent models, not necessarily mass-spring chains, see the ﬁrst example below. In order to obtain impact mode solutions, the system (10.34) and (10.35) is replaced by the following impulsively forced linear system under no constraints condition (10.36) Mx ¨ + Kx = pIa τ (ωt + α) where p is a priory unknown ‘eigen-value,’ and ω and α are arbitrary constant parameters. A family of periodic solutions of the period T = 4/ω can be found as a linear superposition of solutions (10.5) for each of the N linear modes of system (10.35) with appropriate replacement of the parameters, N (eTj Ia )ej sin[(ωj /ω)τ (ωt + α)] x(t) = p ωωj cos(ωj /ω) j=1

(10.37)

where ej and ωj are the jth normal mode and the natural frequency of linear system (10.35); it is assumed that ωi < ωj when i < j, and the linear normal modes are normalized as

258

10 Impact Modes and Parameter Variations

eTj M ei = δji

(10.38)

where δji is the Kronecker symbol. The impulses act at those time instances when the ath mass interacts with the constraints, in other terms, ITa x = ±Δa when τ = ±1

(10.39)

Substituting (10.37) into (10.39), one obtains the related “eigen-value,” ⎡ p = Δa ω ⎣

N j=1

⎤−1 (eTj Ia )2

tan(ωj /ω) ⎦ ωj

(10.40)

where eTj Ia is the ath component of the jth linear mode vector. Substitution (10.40) into (10.37) gives a two parameter family of the periodic solutions for the impact modes. The parameter α is an arbitrary phase shift, whereas the frequency parameter ω is subjected to some restrictions due to condition (10.34). As shown in [156], condition (10.34) is satisﬁed when the principal frequency of vibration, Ω = (π/2) ω, exceeds the highest frequency of the linear spectrum, Ω > ωN . The corresponding impact mode represents an extension of the highest linear normal mode, in which any two neighboring masses vibrate out-of-phase. Such an impact mode becomes spatially localized as Ω → ∞. This result was obtained also by qualitative methods in [197]. Condition (10.34) may be satisﬁed also when the frequency Ω is located in a small enough right neighborhood of any frequency ωj and, in addition, ωi /ωj is not an odd number for all i = j. The idea of the proof is to ﬁnd such cases when ITa x is a monotonic function of τ on the interval −1 ≤ τ ≤ 1, and hence condition (10.39) at the boundaries guarantees that inequality (10.34) holds inside the entire interval. Generally, the impact modes appear to have a quite complicated spectral structure. Therefore, a detailed investigation may be required in order to formulate necessary and suﬃcient conditions of impact mode existence. However, a suﬃcient condition of non-existence can be formulated by using the physical meaning of the parameter p. Namely, if an impact mode exists then the inequality p (ω) > 0 holds. Indeed, the parameter p (10.40) cannot be negative for any real impact vibrating regime as a reaction of constraint, because it cannot be directed to a barrier. Thus impact modes can not exist when p (ω) < 0.

10.5.1

A Double-Pendulum with Amplitude Limiters

The top mass m1 of a free double-pendulum, as shown in Fig. 10.8, oscillates between the two absolutely stiﬀ constraints providing small angular amplitudes of the top pendulum,

10.5 Impact Modes in Multiple Degrees of Freedom Systems

259

|ϕ1 | ≤ Δ1 1

Fig. 10.8 Double pendulum with bilateral constraints.

Assuming that the angle ϕ2 is also small enough, brings the diﬀerential equations of motion between the barriers to the linear form μ2 ϕ ¨1 + ϕ ¨ 2 + μ2 ϕ1 = 0 ϕ ¨1 + ϕ ¨ 2 + ϕ2 = 0 where μ2 = 1 + m1 /m2 , over dots indicate diﬀerentiation with respect to the new temporal argument, t¯ = g/lt, and l is the length of each rod. In this case, the linear mode vectors and natural frequencies are

μ 1 1 , ω1 = e1 = μ+1 2μ (μ + 1) μ

μ 1 1 e2 = , ω2 = −μ μ − 1 2μ (μ − 1)

260

10 Impact Modes and Parameter Variations

where the modal vectors satisfy the orthogonality condition, eTj M ei = δji , with respect to the mass matrix

2 μ 1 M= 1 1 As a result, taking into account that Ia = I1 , Δa = Δ1 , and N = 2, brings the formal solution (10.37) and (10.40) to the form

ϕ1 −1 = Δ1 [tan (ω1 /ω) + (ω2 /ω1 ) tan (ω2 /ω)] × ϕ2

sin [(ω2 /ω) τ (ωt + α)] 1 sin [(ω1 /ω) τ (ωt + α)] ω2 1 + μ cos (ω1 /ω) ω1 −μ cos (ω2 /ω) where the relationships μ − 1 = μω2−2 and μ + 1 = μω1−2 have been used in manipulations. Note that the frequency of ﬁrst impact mode must be close enough to the ﬁrst linear frequency ω1 so that it is still away from the left neighborhood of the next frequency, ω2 . In current case, ω2 is the highest frequency, therefore its right “neighborhood” has no upper boundary. As a result, the highest impact mode becomes spatially localized as its frequency parameter grows. The localization admits explicit estimation by the asymptotic expansion

−1 ω2 ω2 ϕ2 ω1 ω2 ω2 ω1 − + |τ =1 = μ tan tan tan tan ϕ1 ω ω1 ω ω ω1 ω

ω12 − ω22 2ω12 ω22 ω −2 + O ω −4 =μ 2 1+ (10.41) ω1 + ω22 3 (ω12 + ω22 ) 1 = −1 − ω −2 − O ω −4 3 It follows from (10.41) that (ϕ2 /ϕ1 ) |τ =1 → −1 as ω → ∞, so that the amplitude of the bottom mass becomes negligibly small whereas the upper mass has the amplitude determined by the angular limiters.

10.5.2

A Mass-Spring Chain under Constraint Conditions

Let us consider a mass-spring chain of N identical particles under the constraint condition x ¨n +

k (−xn−1 + 2xn − xn+1 ) = 0; m x0 = xN +1 = 0;

|xa | ≤ Δa ;

n = 1, ..., N

(10.42)

1

(10.43)

10.5 Impact Modes in Multiple Degrees of Freedom Systems

261

where k and m are the stiﬀness of each spring and the mass of each particle, respectively. In this case, the corresponding linear modes and their frequencies are described exactly by expressions

T N πj 2 πj , ..., sin (10.44) ej = sin N +1 N +1 N +1 πj ωj = ωN +1 sin 2 (N + 1) where notation2 ωN +1 = 2 k/m has been introduced, and the basis vectors are normalized so that eTj ei = δji . The ath component of the jth normal mode vector is therefore aπj 2 T ej Ia = sin (10.45) N +1 N +1 Let us show that the impact mode periodic solution (10.37) and (10.40) becomes localized as ω → ∞. First, replacing the trigonometric functions by their asymptotic estimates sin(ωj τ /ω) ∼ ωj τ /ω, cos(ωj /ω) ∼ 1, and tan(ωj /ω) ∼ ωj /ω, gives 2N xb ∼ Δa

T T j=1 (ej Ia )(ej Ib ) τ 2N T 2 j=1 (ej Ia )

(ωt + α)

as

ω→∞

(10.46)

In this particular example, the sums can easily be evaluated. So, taking into account expression (10.45) and the standard trigonometric sums [52], gives N

bπj 2 aπj sin = δab sin N + 1 j=1 N +1 N +1

(10.47)

1 2 ω (−δa,b−1 + 2δa,b − δa,b+1 ) 2 N +1

(10.48)

N

(eTj Ia )(eTj Ib ) =

j=1 N j=1

(eTj Ia )(eTj Ib )ωj2 =

Substituting then (10.47) in (10.46), gives xb ∼ Δa δab τ (ωt + α)

as

ω→∞

(10.49)

Expression (10.49) shows that the ath particle of the chain vibrates according to the saw-tooth temporal mode shape with the inﬁnitely large frequency, 2

Note that this is just a suitable notation since the (N + 1)th frequency does not physically exist.

262

10 Impact Modes and Parameter Variations

whereas all the other particles are at rest. Therefore, the impact mode becomes spatially localized as ω → ∞.

10.6

Systems with Multiple Impacting Particles

Let us consider the case of two particles, say the ath and bth, under the constraint conditions. These conditions are |xa | ≤ Δa and |xb | ≤ Δb or, in the vector notations, T I x ≤ Δa and IT x ≤ Δb (10.50) a b In this case, the impulsive excitation on the right-hand side of the auxiliary equation must act on both ath and bth particles, so that the equation takes the form Mx ¨ + Kx = (pa Ia + pb Ib )τ (ωt + α) (10.51) where pa and pb are parameters to be determined. The related solution includes terms related to pa and pb , and can be represented in the form x(t) =

N pa (eTj Ia )ej + pb (eTj Ib )ej sin(ωj τ (ωt + α) /ω) ωωj cos(ωj /ω) j=1

(10.52)

Following the idea of normal modes, let us assume that the impact mode periodic motion is accompanied by synchronous impacts of both particles with the constraints according to conditions ITa x = ±Δa and ITb x = ±Δb when τ = ±1

(10.53)

Substituting (10.52) in (10.53), gives linear algebraic equations with respect to pa and pb in the form kaa pa + kab pb = Δa kba pa + kbb pb = Δb where kab =

N (eTj Ia )(eTj Ib ) j=1

ωωj

tan

(10.54)

ωj ω

(10.55)

Expressions (10.52) and (10.54) give a formal impact mode solution, which indeed describes an impact mode when the determinant of system (10.54) is non-zero and condition (10.50) holds. Solution (10.52) can be viewed as a strongly non-linear superposition of the two basic impact modes with a single impacting mass.

10.6 Systems with Multiple Impacting Particles

263

Let us consider the higher frequency domain, (π/2) ω >> ωN , for the above example of mass-spring system. In this case, tan(ωj /ω) > 0 for all j = 1, ..., N , and therefore, the coeﬃcients kab (10.55) create a Gram matrix with non-zero determinant [23]. The asymptotic estimation below conﬁrms this conclusion. Namely, assuming that (π/2) ω >> ωN , provides that all the ratios ωj /ω are small enough to justify the following asymptotic estimate

ωj τ 1 ωj 3 sin[(ωj /ω)τ ] τ3 = + (10.56) τ− + O ω −5 cos(ωj /ω) ω 2 ω 3 Then substituting expression (10.56) in (10.52) and (10.55) for an arbitrary cth particle, gives

1 ωN +1 2 τ3 xc = (Pa δac + Pb δbc ) τ − τ− × (10.57) 4 ω 3 −4 × [Pa (δa,c−1 − 2δac + δa,c+1 ) + Pb (δb,c−1 − 2δbc + δb,c+1 )] + O ω where xc = xT Ic and τ = τ (ωt + α), and the parameters {Pa , Pb } = ω −2 {pa , pb } are determined by the linear algebraic system Kaa Pa + Kab Pb = Δa Kba Pa + Kbb Pb = Δb

(10.58)

where Kab = ω 2 kab = δab −

1 ωN +1 2 (δa,b−1 − 2δab + δa,b+1 ) + O ω −4 (10.59) 6 ω

As follows from (10.59), equations (10.58) have the solution Pa = Δa and Pb = Δb as ω → ∞. In this limit, the vibration energy becomes localized on the two particles vibrating between the barriers with the saw-tooth temporal shape, (10.60) xc ∼ (Δa δac + Δb δbc ) τ (ωt + α) as ω → ∞ According to (10.60), the particles with numbers c = a and c = b are at rest. However, they oscillate with amplitudes of diﬀerent orders of ω −2 when the parameter ω takes a large but ﬁnite value. Expansion (10.57) shows that the temporal mode shapes of particles c = a and c = b are non-smooth and getting close to the triangular sine wave as the frequency ω increases. The temporal mode shapes of the nearest particles, c = a ± 1 and c = b ± 1, have amplitudes of order ω −2 and appear to be twice continuously diﬀerentiable with respect to time, t. In particular, calculating directly ﬁrst two derivatives, gives

264

10 Impact Modes and Parameter Variations

τ3 d τ− = ω 1 − τ 2 τ ∈ C 1 (R) dt 3

2 d τ3 2 2 2 −2τ (τ τ = −ω 2 2τ ∈ C (R) ) + 1 − τ τ − = ω dt2 3 where prime denotes diﬀerentiation with respect to the whole argument of the saw-tooth sine, τ ≡ dτ /d (ωt + α), and the term underlined is zero3 because 1 − τ 2 = 0 on the set of points {t : τ (ωt + α) = ±1}, where τ = 0.

3

Namely, it gives zero contribution into the related integrals of the theory of distributions.

Chapter 11

Principal Trajectories of Forced Vibrations

Abstract. As shown earlier by Zhuravlev (1992) that harmonically loaded linear conservative systems possess an alternative physically reasonable basis, which is generally diﬀerent from that associated with conventional principal coordinates. Brieﬂy, such a basis determines directions of harmonic loads along which the system response is equivalent to a single oscillator. The corresponding deﬁnition (principal directions of forced vibrations) is loosing sense in nonlinear case, when the linear tool of eigen vectors becomes inapplicable. However, it will be shown in this chapter that nonlinear formulation is still possible in terms of eigen vector-functions of time given by NSTT boundary value problems. Physical meaning of the corresponding nonlinear deﬁnitions for both discrete and continual models is discussed.

11.1

Introductory Remarks

The theory of linear normal modes deﬁnes a natural basis in the conﬁguration space of linear conservative systems. The corresponding directions are associated with a set of independent harmonic oscillators. The number of such oscillators is inﬁnite, if the original system is continuous. In the later case, the modal analysis provides reduction of a continuous system to the related discrete set of harmonic oscillators. As it is known, the normal modes are deﬁned for a class of unforced systems, therefore only initial conditions select those oscillators that will be excited during the dynamical process. Practically, a normal mode regime must be supported by some external loading due to inevitable energy dissipation. However, the theory does not identify directly such external forces. Let ψj (y) be, for instance, the jth mode shape of a beam. Generally speaking, the external loading of the same proﬁle, ψj (y), will excite not only the jth mode but also some others, unless the mass per unit length of the beam is constant. From the mathematical viewpoint, this is due to the mass density, say ρ(y), participating as a weighting factor in the orthogonality condition V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 265–273, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

266

11 Principal Trajectories of Forced Vibrations

< ψi (y)ρ(y)ψj (y) >= 0,

i = j

(11.1)

The question therefore is what kind of external force must be applied to a mechanical system in order to generate a normal mode type of motion when all the system particles coherently vibrate with the same frequency? Following reference [203], let us consider ﬁrst the linear case assuming that the linear n-degree-of-freedom forced system oscillates as a single harmonic oscillator in such a manner that the coordinates vector x (t) and the force vector p (t) are collinear to the same constant vector q with a constant length ratio μ as follows (11.2) x = q sin ωt, p = μq sin ωt In the case of forced vibration, the frequency ω is rather predetermined by the external loading and therefore should not play the role of eigen-value. It was shown in [203] that the coeﬃcient of proportionality μ can play such a role instead. In a regular case, the coeﬃcient μ has exactly n eigen-values, whereas the vector q determines the corresponding ‘principal directions’ according to the deﬁnition of reference [203]. Note that the principal directions are always orthogonal regardless the mass matrix of the system. Such an approach therefore determines a new natural basis for external forces from the standpoint of system considered. This, of-course, should not be viewed as a substitute for the theory of normal modes, however, some non-autonomous problems can be naturally solved by making use of the above complementary basis. In nonlinear cases, deﬁnition (11.2) is unapplicable and the above notion of principal directions loses its sense. However, it was shown in [136] that the basic idea still can be generalized by considering trajectories instead of directions. Also a mixed spatio-temporal consideration must be applied since spatial and temporal coordinates are not separable in nonlinear cases and the related vibration and forcing are generally neither harmonic in time not similar in space. There are some practically important formulations of the problem for the case of nonlinear forced vibration, which could be qualiﬁed as inverse or semi-inverse approaches. The related methods select practically reasonable external forces that generates simple enough dynamics. For example, Harvey [57] considered ‘natural forcing functions’ proportional to the non-linear restoring force of the forced Duﬃng oscillator. The notion of ‘exact steady state’ was deﬁned by Rosenberg [168] for a strongly nonlinear single degree of freedom system as a vibration with the cosine-wave temporal shape of the period of external force. The corresponding forcing function is determined under some initial conditions. Kinney and Rosenberg [80] considered systems with many degrees of freedom.

11.2 Principal Directions of Linear Forced Systems

11.2

267

Principal Directions of Linear Forced Systems

Let us illustrate ﬁrst the basic idea of reference [203] by considering the linear n-degree-of-freedom forced system Mx ¨ + Kx = p (ωt) , x (t) ∈ Rn

(11.3)

where M and K are constant mass and stiﬀness n× n -matrixes, respectively; p (ωt) is a periodic vector-force of the period T = 2π with respect to ωt, and the upper dot means diﬀerentiation with respect to time, t. Substituting (11.2) in (11.3), gives the eigen-value problem with respect to the parameter μ and vector q in the form − ω 2 M q + Kq = μq

(11.4)

Let q = vs and μ = μs be the sth eigen-vector and eigen-value respectively, s = 1, ..., n. The eigen-vectors vs are orthogonal and can be normalized by condition (11.5) viT vj = δij where δij is the Kronecker symbol. Therefore, the set of vectors vs determine a natural basis for the case of forced vibrations. Let, for instance, the external force be p = Q sin ωt, where Q ∈ Rn is an arbitrary constant vector. In this case, the corresponding steady-state (particular) solution is written as vsT Q vs sin ωt (11.6) x= μs s Now, let es and ωs be conventional linear normal modes and natural frequencies of the system. (The related eigen-value problem is obtained from (11.4) by setting μ = 0.) As follows from the linear theory, the normal mode vectors are orthogonal with respect to the mass matrix M so that the normalization condition can be represented in the form eTi M ej = δij Using the normal mode basis for the above steady-state, gives eTs Q x= es sin ωt ωs2 − ω 2 s

(11.7)

(11.8)

Since the uniqueness theorem holds, expansions (11.6) and (11.8) must represent the same solution, and therefore, eTs Q vsT Q vs = es (11.9) μs ωs2 − ω 2 s s

268

11 Principal Trajectories of Forced Vibrations

Let the external force amplitude vector Q be directed along one of the principal directions. Then, expansion (11.6) will include only one term, whereas expansion (11.8) still includes all n terms. Now, let us consider the case, when the mass matrix is equal to the identity matrix, M = E. In this particular case, expression (11.4) takes the standard form of the eigen-value problem for normal modes with respect to the eigenvalue parameter ω 2 + μ, (11.10) − ω 2 + μ Eq + Kq = 0 As follows from (11.10), the eigen-values of free and forced vibration are coupled by expression ω 2 + μs = ωs2 , s = 1, ..., n

(11.11)

It is seen that each eigen-value of forced vibration, μs = ωs2 − ω 2 , is a monotonically decreasing functions of the external frequency ω with only one zero at ω = ωs .

11.3

Deﬁnition for Principal Trajectories of Nonlinear Discrete Systems

Let us consider the nonlinear case Mx ¨ + Kx + εf (x) = p (ωt) ,

x (t) ∈ Rn

(11.12)

where f (x) is an analytic nonlinear vector-function such that f (−x) = −f (x), ε is a small positive parameter, and the forcing function and matrixes are deﬁned in equation (11.3). If ε = 0, then the concept of principal directions of forced vibrations is not applicable any more, however it is still possible to consider principal trajectories instead based on the following Deﬁnition 1. Trajectories of periodic motions of the period T = 2π/ω on which mechanical system (11.12) behaves as a Newtonian particle in Rn , namely the external force and acceleration vectors are coupled by the Newton second law, m¨ x (t) = p (ωt) (11.13) will be called principal trajectories of forced vibrations. In equation (11.13), m is a priory unknown eﬀective mass parameter. The eﬀective mass m and the force p (ωt) must be chosen in order to make equations (11.12) and (11.13) compatible. Note that, in the linear case, the above deﬁnition still gives principal directions of forced vibrations (11.2) after representing the mass parameter as follows

11.4 Asymptotic Expansions for Principal Trajectories

269

μ (11.14) ω2 Indeed, substituting expression x (t) = q sin ωt in equation (11.13) and taking into account expression (11.14), gives deﬁnition (11.2) in the form p =μx. In contrast to linear case (11.2), however, deﬁnition (11.13) allows non-harmonic temporal shapes. Current deﬁnition itself does not imply that the system is weakly nonlinear. However, if the parameter ε is small then explicit solutions can be obtained in terms of conventional asymptotic expansions as described in the next section. As mentioned, the notion of principal trajectories seems to relate to the idea of ‘natural forcing functions’ introduced in [57] for the Duﬃng oscillator. Let us consider now a multidimensional case from that point of view. Applying deﬁnition (11.13) to the general nonlinear system m=−

Mx ¨ + F (x) = p (ωt)

(11.15)

and eliminating the acceleration, gives the external forcing vector-function as a linear transformation of the restoring force in the form, −1

1 F (x) p (ωt) = E − M m

(11.16)

where the matrix of the transformation includes the eﬀective mass parameter m. Relationship (11.16) can be viewed as a vector version of the concept of natural forcing functions. On the other hand, using the deﬁnition for principal trajectories and excluding the external forcing vector p (ωt) from the equation of motion, gives an auxiliary free system described by the diﬀerential equation of motion (M − mE) x ¨ + F (x) = 0 The idea of transforming the forced problem to a free vibration problem by imposing the form of excitation was used also in [31] with illustrations on two degrees of freedom systems based on an essentially diﬀerent methodology though.

11.4

Asymptotic Expansions for Principal Trajectories

In order to make equations (11.12) and (11.13) compatible, let us eliminate the forcing vector-function p (ωt) and thus consider equation Mx ¨ + Kx + εf (x) = m¨ x (t)

(11.17)

270

11 Principal Trajectories of Forced Vibrations

A family of periodic solutions, that give principal directions of linearized system as ε → 0, will be considered. Let us represent such solutions (principal trajectories) in the following parametric form x = X (τ )

(11.18)

where is τ = τ ((2ω/π)t) is the triangular sine wave of the period of external loading, T = 2π/ω. Substituting (11.18) into (11.17), gives

LX+εf (X) =

L≡

2ω π 2ω π

2

mX

2 M

(11.19)

d2 +K dτ 2

under the boundary condition X (τ ) |τ =±1 = 0

(11.20)

As mentioned above, the temporal and spatial variables generally are not separable any more in nonlinear cases, therefore it is impossible to obtain an exact nonlinear version of the eigenvector problem (11.4). As a result, both temporal and spatial mode shapes must be corrected on each step of the related asymptotic process as described below. Remind that the diﬀerential operator L in equation (11.19) includes the frequency parameter ω ﬁxed, whereas the mass m is an eigen value to be determined. Let ma and ea (τ ) be the eigen value and eigen vector of the linearized problem, ε = 0, respectively,

2 2ω ea Lea = ma π ea | τ =±1 = 0

(11.21)

where the index a = {s, j} consists of spatial and temporal mode shape numbers, s = 1, ..., n and j = 1, ..., respectively. The scalar product of any two vector-functions x = x (τ ) and y = y (τ ) will be deﬁned as follows 1 1 T x, y = x ydτ 2 −1 Let us represent solution of the weakly nonlinear eigen value problem (11.19) and (11.20) in the following form of asymptotic expansions

11.5 Deﬁnition for Principal Modes of Continuous Systems

271

X(τ ) = Aea (τ ) + εX(1) (τ ) + O ε2 m = ma + εη1 + O ε2

(11.22)

Then substituting (11.22) in (11.19) and (11.20), and matching the coeﬃcients of the ﬁrst order of ε, gives equation

(1)

LX

−

2ω π

2 (1)

ma X

= −f (Aea ) +

2ω π

2

η1 Aea

(11.23)

and boundary condition X(1) |τ =±1 = 0

(11.24)

Following the idea of perturbations for eigen-value problems [83], let us represent solution of equation (11.23) (1) ab eb (τ ) (11.25) X(1) = b=a (1)

where b = {r, i} is a double index, ab are yet unknown constant coeﬃcients, and boundary condition (11.24) is automatically satisﬁed. Let us assume the following normalization condition for the eigen vectorfunctions 0, b = a (11.26) ea (τ ), eb (τ ) = 1, b=a Substituting (11.25) in (11.23) and taking into account (11.26), determines (1) the coeﬃcients ab and η1 . As a result, expansions (11.22) give ﬁrst-order asymptotic solution π 2 e , f (Ae ) e b a b + O ε2 2ω mb − ma b=a π 2 e , f (Ae ) a a + O ε2 m = ma − ε 2ω A X = Aea + ε

(11.27)

As follows from the form of solution (11.27), all the coeﬃcients are uniquely determined under the condition that ma = mb for a = b. The possibility of degeneration, namely ma = mb for a = b, depends on the inner properties of the system and the frequency parameter ω. The related examples were considered earlier [136], [141].

11.5

Deﬁnition for Principal Modes of Continuous Systems

Let us consider a one-dimensional elastic system whose vibration is described by some function u = u(t, y). For certainty reason, let us consider a

272

11 Principal Trajectories of Forced Vibrations

non-linear string of the length l under external distributed loading described by the partial diﬀerential equation and boundary conditions Lu+εf [u] = p(ωt, y),

0

u(t, 0) = u(t, l) = 0 L ≡ ρ(y)

2

(11.28) (11.29)

2

∂ ∂ −T 2 2 ∂t ∂y

(11.30)

where L is the diﬀerential self-adjoint operator of linear string, ρ(y) is a mass per unit length parameter, T is a constant tensile force, f [u] is a nonlinear operator acting in the corresponding function space of conﬁgurations, ε is a small parameter, and p(ωt, y) is the external forcing function, which is assumed to be 2π-periodic with respect to ωt. Now keeping in mind expressions (11.28) through (11.30), let us introduce Deﬁnition 2. Periodic forced vibrations of a continuous system, in which the system motion is equivalent to a particle in the function space of conﬁgurations described by the second Newton law, σ

∂ 2 u(t, y) = p(ωt, y) ∂t2

(11.31)

will be called a principal mode of forced vibration. In one-dimensional cases, σ is a priory unknown eﬀective mass per unit length. Substituting (11.31) in (11.28), gives the following partial diﬀerential equation for principal modes of forced vibrations Lu+εf [u] = σ

∂2u ∂t2

(11.32)

Introducing the triangular wave time substitution as τ = τ ((2ω/π)t) and u(t, y) = U (τ, y), gives

LU +εf (U ) =

L≡

2ω π 2ω π

2 σ 2

∂2U ∂τ 2

ρ(y)

∂2 ∂2 − T ∂τ 2 ∂y 2

(11.33)

The boundary conditions are formulated for both temporal and spatial variables as U (τ, 0) = U (τ, l) = 0 (11.34) and, ∂U (τ, l) |τ =±1 = 0 ∂τ respectively.

(11.35)

11.5 Deﬁnition for Principal Modes of Continuous Systems

273

In this case, the scalar product of two functions U = U (τ, y) and V = V (τ, y) from the conﬁguration space can be deﬁned as U ,V =

1 2l

1

l

U V dτ dy −1

(11.36)

0

Further, a weakly nonlinear asymptotic procedure can be developed analogously to the above discrete case.

Chapter 12

NSTT and Shooting Method for Periodic Motions

Abstract. In this chapter, two-dimensional shooting diagrams are introduced for visualization of manifolds of periodic solutions and their bifurcations. A general class of nonlinear oscillators under smooth, nonsmooth, and impulsive loadings is considered. The corresponding boundary value problems are formulated by introducing the triangular wave temporal argument. The Duﬃng oscillator with no linear stiﬀness (Ueda circuit) is considered for illustration. It is shown that the temporal mode shape of the loading is responsible for qualitative features of the dynamics, such as transitions from regular and random motions. The important role of unstable periodic orbits is discussed.

12.1

Introductory Remarks

Periodic solutions and their bifurcation diagrams often reveal important qualitative features of the dynamics even though the system motion is not expected to be periodic. In particular, the number of periodic orbits, their distribution and properties reveal the structure of chaotic orbits; see, for instance, works [12], [53], [123], and references therein. Direct numerical tools for detection and construction of periodic orbits based on the mapping approach can be found in the book [125] and paper [54]. Diﬀerent formulations in terms of boundary value problems for ordinary diﬀerential equations are described in [10]. Theoretical and applied results regarding periodic motions, bifurcations and chaos are reported in [13] and [196]. In this chapter, a special two-dimensional visualization of the shooting method is introduced in order to incorporate the general two component NSTT as a preliminary analytical stage [144]. Note that the same approach using the one-component NSTT was suggested earlier in [154] and impleR mented in Mathematica interface [18]. In particular, subharmonic orbits of the forced pendulum and bifurcation diagrams were obtained by examining the shooting curves and their zeros.

V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 275–294, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

276

12 NSTT and Shooting Method for Periodic Motions

Let us consider a multi-dimensional oscillator described be the diﬀerential equation F (¨ x, x, ˙ x, t) = 0 (12.1) where x (t) ∈ Rn , and the vector-function F ∈ Rn is periodic with respect to time t with the period T = 4a. In this work, diﬀerent kind of temporal discontinuity in the diﬀerential equations of motion will be considered. In order to satisfy the related mathematical requirements, the left-hand side of equation (12.1) must be interpreted in terms distributions [178] ∞ F (¨ x, x, ˙ x, t)ϕ (t) dt = 0

(12.2)

−∞

where ϕ (t) is any suﬃciently smooth testing function. However, at this point, let us assume that the function F (¨ x, x, ˙ x, t) is regular with no singular terms involved. Let us consider periodic solutions of the period T by means of the coordinate complexiﬁcation (NSTT) x → {X, Y } :

x = X (τ ) + Y (τ ) τ

(12.3)

whereτ = τ (t/a) the triangular sine wave of the period T = 4a, and τ = dτ (t/a) /d (t/a) is its ﬁrst generalized derivative, which is a step-wise discontinuous function at the time instances Λ = {t : τ (t/a) = ±1}

(12.4)

As discussed in this book, the above discontinuities can be suppressed by the condition Y (±1) = 0, which is the necessary condition of continuity of the original coordinates x(t). Then, assuming for a while no inﬁnite discontinuities in F , substituting (12.3) in (12.1), using the algebraic properties of representation (12.3) as well as other NSTT rules, gives the boundary value problem

X + Y X + Y X + Y , , aτ = 0 (12.5) , F a2 a a

X − Y X − Y X − Y F , − , , 2a − aτ = 0 (12.6) a2 a a Y |τ =±1 = 0,

X |τ =±1 = 0

(12.7)

where the prime used with X and Y means diﬀerentiation with respect to τ . Both equations (12.5) and (12.6) are easily derived by the corresponding algebraic manipulations, whereas boundary conditions (12.7) represent the result of elimination of the singular term τ (t/a) when substituting (12.3) in

12.2 Problem Formulation

277

(12.1). In some cases, such a singular term can be employed though in order to eliminate singularities from original equations; see below. Despite of a relatively complicated form of equations (12.5) and (12.6), the new formulation brings some advantages due to the fact that the new temporal variable τ is bounded and automatically accounts for periodicity of solutions regardless their temporal shapes. This property appears to be important in those cases when the solutions do not represent a ﬁnal stage of investigation but must be used for further analyses. The dimension increase is often compensated by an eﬀective decrease of the temporal interval of the problem, since the range −1 τ 1 is covered by the original time domain −a t a, which is twice shorter than the whole period T = 4a. Moreover, there are many cases when the number of equations can be reduced to that of the original system due to the symmetry of equations. If, for instance, the vector-function F (¨ x, x, ˙ x, t) is even with respect to the velocity x˙ or includes no velocity at all, and the explicit dependence on time t produces zero ‘imaginary component’ then boundary value problem (12.5) through (12.7) admits a family of solutions on which

X X X , , aτ = 0 (12.8) Y ≡ 0, F , a2 a a X |τ =±1 = 0

(12.9)

The particular case (12.8) and (12.9) was investigated numerically by the shooting method in [154] and [177] based on a single- and multiple-degrees-of freedom systems, respectively. It should be noted that no special requirements are imposed on numerical methods or packages for solving the above boundary-value problems. However, the shooting algorithm in the MatheR matica interface provides a physically meaningful way of visualization of periodic solutions due to the speciﬁc combination analytical and numerical commands.

12.2

Problem Formulation

Let assume now that the system loading may include a periodic series of Dirac δ-pulses acting at times Λ(12.4). As is known [45], Dirac δ-functions can participate in nonlinear diﬀerential equations only as summands because nonlinear manipulations with δ-functions are physically meaningless, except special concepts [106]. Therefore, the original equation (12.1) must be concretized as x¨ + f (x, x, ˙ t) = q (t) (12.10) where q (t) = Q (τ (t/a)) + P (τ (t/a)) τ (t/a) + p (τ (t/a)) τ (t/a)

(12.11)

278

12 NSTT and Shooting Method for Periodic Motions

and τ (t/a) = d2 τ (t/a) /d (t/a)2

∞

t t + 1 − 4k − δ − 1 − 4k =2 δ a a

(12.12)

k=−∞

In equation (12.10), the function f (x, x, ˙ t) may still include parametric terms of the period T = 4a with possible step-wise discontinuities on Λ. The acceleration x ¨ also participates as a summand, since it must have the same kind of singularities as the external forcing function, q (t). According to the distribution theory [165], p (τ (t/a)) must be at least continuous on Λ, otherwise the ‘product’ p (τ ) τ cannot be treated as a distribution. Note, that behavior of the function p(τ (t/a)) between the times Λ is arbitrary, since only values p (−1) and p (1) contribute into the expression p (τ (t/a)) τ (t/a) (12.13)

∞ t t + 1 − 4k − p (1) δ − 1 − 4k =2 p (−1) δ a a k=−∞

The numbers p (−1) and p (1) control the ‘amplitudes’ and directions of the δ−pulses. For example, all the pulses can be positively co-directed by setting p (τ ) = − sign τ . Remark 2. Expressions (12.3) and (12.11) represent particular cases of the truncated series q (t) =

N

Pk (τ (t/a)) dk τ (t/a) /d (t/a)k

(12.14)

k=0

where Pk (τ (t/a)) must be at least k − 2 times continuously diﬀerentiable in the neighborhood of points t = ±a. Although, physical interpretation of the higher order terms in (12.14) is not straightforward, such terms still can occur after reducing the number of equations from the entire system. In case (12.10) and (12.11), one has N = 2 and therefore the velocity vector x˙ must be step-wise discontinuous. Further, if N = 3 then the velocity x˙ includes singular terms and the function f (x, x, ˙ t) in equation (12.10) must be linear ˙ t) must be linear also with respect to x. ˙ If N = 4 then the function f (x, x, with respect to the position vector x provided that any parametric terms are suﬃciently smooth functions of time. Therefore, only linear systems can be considered if N ≥ 4. Since the basis elements {1, τ , τ } represent functions of diﬀerent classes of smoothness, then substituting (12.3) and (12.11) in (12.10), gives separately a−2 X + Rf (X, Y, X , Y , τ ) = Q (τ )

(12.15)

12.3 Sample Problems and Discussion

and

279

a−2 Y + If (X, Y, X , Y , τ ) = P (τ )

(12.16)

a−2 X |τ =±1 = p (±1)

(12.17)

where

X − Y 1 X + Y Rf , aτ ± f X − Y, − , 2a − aτ = f X + Y, If 2 a a (12.18) Note that the singular term a−1 Y τ is eliminated from the velocity vector x˙ (t) by imposing another boundary condition Y |τ =±1 = 0

(12.19)

The boundary value problem (12.15) through (12.19) includes no singular or discontinuous functions, therefore standard numerical codes and packages can be applied with no speciﬁc constraints on their choice. In general, equations (12.15) and (12.16) are coupled. Although the equations can be decoupled by introducing the new unknown functions, X(τ ) + Y (τ ) and X(τ ) − Y (τ ), the boundary conditions will become coupled. There are two special cases, however, when the entire problem can be reduced. If, for instance, f (x, x, ˙ t) = f (x, −x, ˙ 2a − t), and P (τ ) ≡ 0 then the problem admits a family of solutions such that Y ≡ 0, a−2 X + f (X, X /a, aτ ) = Q (τ )

(12.20)

under the boundary condition (12.17). In case f (x, x, ˙ t) = −f (−x, x, ˙ 2a − t), Q (τ ) ≡ 0 and p (τ ) ≡ 0, then one can consider another family of solutions on which X ≡ 0, a−2 Y + f (Y, Y /a, aτ ) = P (τ )

(12.21)

under the boundary condition (12.19). This chapter nevertheless focuses on the general two-component problem (12.15) through (12.19).

12.3

Sample Problems and Discussion

12.3.1

Smooth Loading

The Duﬃng-Ueda oscillator [186] under the periodic loading of diﬀerent temporal shapes will be considered below. Let us start with the standard case of sine-wave voltage x¨ + ζ x˙ + x3 = B sin ωt where ζ, B, and ω are constant parameters.

(12.22)

280

12 NSTT and Shooting Method for Periodic Motions

In this case, the diﬀerential equations (12.15) and (12.16) take the form a−2 X + ζa−1 Y + X 3 + 3XY 2 = B sin a−2 Y + ζa−1 X + Y 3 + 3X 2 Y = 0

πτ 2

(12.23) (12.24)

where a = π/(2ω) is a quarter of the loading period, and the boundary conditions are Y |τ =±1 = 0, X |τ =±1 = 0 (12.25) The shooting method can be applied now as follows. First, the diﬀerential equations (12.23) and (12.24) are solved under the initial conditions X(−1) = g, X (−1) = 0

Y (−1) = 0, Y (−1) = h

(12.26) (12.27)

where g and h are numbers to be determined in order to satisfy boundary conditions (12.25). Let us represent solution of the initial-value problem (12.23), (12.24), (12.26) and (12.27) in the following general form X = X(τ ; g, h), Y = Y (τ ; g, h)

(12.28)

By the idea of shooting method, the initial value problem (12.23), (12.24), (12.26) and (12.27) must be iteratively solved multiple times at diﬀerent g and h until suﬃcient precision has been reached for boundary conditions (12.25) at right end τ = 1, ∂X(τ ; g, h) |τ =1 ≡ G(g, h) = 0 ∂τ Y (τ ; g, h)|τ =1 ≡ H(g, h) = 0

(12.29)

When dealing with the particular cases (12.20) or (12.21), such a procedure is not diﬃcult since one has only one equation with a singe unknown, G(g) = 0 or H(h) = 0. Multidimensional cases, such as (12.29), appear to be more diﬃcult and time consuming. From this point of view, the important feature of Mathematica is that it is possible to program the functions G(g, h) and H(g, h) ‘explicitly’ in such a way that the arguments g and h are included into the numerical solver of diﬀerential equations. This can be done as follows. First, the numerical solution is deﬁned as a function of the arguments g and h according to the command sol[g , h ]:=NDSolve[{eqX, eqY, X[-1]==g, X [-1] == 0, Y[-1] == 0, Y [-1] == h}, {X,Y}, {τ ,-1,1}]; where eqX and eqY are equations (12.23) and (12.24), respectively.

12.3 Sample Problems and Discussion

281

Then, the functions G(g, h) and H(g, h) are deﬁned as follows G[g , h ]:=X [1]/.sol[g, h][[1]]; H[g , h ]:=Y[1]/.sol[g, h][[1]]; As a result, the functions G(g, h) and H(g, h) can be considered as usual functions of two arguments. In particular, intersections of two manifolds (12.29) can be located and determined by using the commands ContourPlot and FindRoot, respectively. Each of the determined roots of equations (12.29) represents a periodic solution of the original equation. If the loading amplitude B is a control parameter, then the evolution of diagrams G(g, h; B) = 0 and H(g, h; B) = 0 represents the corresponding structural changes in the set of periodic solutions. Fig. 12.1 gives an example of such a diagram. The parameters were chosen in order to provide conditions for the ‘randomly transitional’ process in terms of work [186]. The diagram clearly shows ﬁve intersections between the two diﬀerent families of curves. The corresponding solutions of the input period T = 4a = 2π are shown in Figs. 12.2 and 12.3. Direct numerical solutions the corresponding estimates for Floquet multipliers show that ﬁrst four periodic solutions, (a) through (d), are unstable, and only one solution (e) is stable. Solution (e) was detected by direct analog and digital computer simulations reported in [186], whereas solutions (a) through (d) were unlisted. Instead, a ‘non-reproducible trajectory’ as a realization of the ‘randomly transitional’ process was represented in the xv-plane. Such a trajectory can be treated as a chaotic drift around the ﬁrst three unstable motions (a), (b) and (c). However, high order periodic solutions may also aﬀect the dynamics of chaotic drift [12]. Fig. 12.4 shows what actually happens when trying to numerically reproduce an unstable periodic orbit, say (a). Neither the shooting algorithm nor computer codes allow to perfectly introduce the initial conditions, therefore it is unlikely that the oscillator will remain on the unstable orbit. After few cycles, the system leaves the orbit (a) for the ‘randomly transitional’ drift around the all three unstable orbits (a), (b) and (c) with ‘no certain choice’ between them. The long-term time history and the corresponding spectrogram of this motion, represented in Fig. 12.4, conﬁrm its random character during quite a long period of time. Although preliminary qualitative information about stability or instability of periodic solutions can be obtained by direct numerical tests, one can quantify stability properties based on the well known Floquet theory in terms of the characteristic multipliers [111], [56]. In order to remind the principals, let us consider periodic solution x(t) of the equation (12.10),where f (x, x, ˙ t) = f (x, x, ˙ t + T ), q (t) = q (t + T ), and the period is T = 4a.

282

12 NSTT and Shooting Method for Periodic Motions

20

10

d c h

0

e

b a

10

20 4

2

0 g

2

4

Fig. 12.1 The curves G(g, h) = 0 (continuous) and H(g, h) = 0 (dashed) and their intersections for the Ueda oscillator under the sine-wave input and the following parameters: ζ = 0.1, B = 12, and ω = 1 (0.1592 Hz).

A variation of the solution x(t), say u(t), is described by the linear diﬀerential equation with periodic coeﬃcients u ¨ + q1 (t)u˙ + q2 (t)u = 0 where q1 (t) = ∂f (x, x, ˙ t)/∂ x˙ is assumed to be independent on x, ˙ and q2 (t) = ∂f (x, x, ˙ t)/∂x. Then substitution ⎛ ⎞ t 1 u = y(t) exp ⎝− q1 (z)dz ⎠ 2 0

12.3 Sample Problems and Discussion

283

x

x

6

6

a

4

b

4

2

2 1

2

2

3

4

5

6

t

1

2

4

4

6

6 x

2

3

4

5

6

t

x

6

6

c

4

d

4

2

2 1

2

2

3

4

5

6

t 2

4

4

6

6

1

2

3

4

5

6

t

x 6

e

4 2 2

1

2

3

4

5

6

t

4 6 Fig. 12.2 The temporal mode shapes of periodic solutions.

gives y¨ + p(t)y = 0 where

(12.30)

1 1 p(t) = q2 (t) − [q1 (t)]2 − q˙1 (t) 4 2 As known from the Floquet theory, stability of the solution x(t) is determined by the Floquet multipliers μ1,2 = A ± A2 − 1 (12.31)

284

12 NSTT and Shooting Method for Periodic Motions

v 6 4 2 0 2 4 6

v a

b

5 x

0

x

5 3 2 1 0

1

2

3 2 1 0

3

v

1

2

3

v c

5 0

d

5 x

5

0

x

5 3 2 1 0

1

2

3 2 1 0

3

1

2

3

v 6 4 2 0 2 4 6

e x

3 2 1

0

1

2

3

Fig. 12.3 The projections of periodic trajectories on xv-planes.

where A = [y1 (T ) + y˙ 2 (T )]/2, and y1 (t) and y2 (t) are two fundamental solutions of equation (12.30) given by the initial conditions y1 (0) = 1, y˙ 1 (0) = 0 y2 (0) = 0, y˙ 2 (0) = 1 Based on the number A, the solution x(t) is unstable if A2 > 1, and it is stable if A2 < 1. In case A2 = 1, there exist a periodic solution of equation (12.30).

12.3 Sample Problems and Discussion

285

x

5

0

−5

50

100

150 Time

200

250

300

2.5

Frequency

2 1.5 1 0.5 0

0

10

20

30

40

50

Time

Fig. 12.4 The time history record and its spectrogram (in Hz) for Ueda oscillator after the direct numerical integration. The parameters are: ζ = 0.1 and B = 12.0.

Now, let x(t) be a periodic solution of equation (12.22). The corresponding variational equation is (12.32) u ¨ + ζ u˙ + 3x2 u = 0 where u = u(t) is a small variation of the solution x = x(t). After the standard substitution u(t) = y(t) exp(−ζt/2), equation (12.32) takes the form

ζ2 2 y¨ + 3x − y=0 4 Taking into account the form of solution (12.3), gives the variational equation with periodic coeﬃcient y¨ + [U (τ (t/a)) + V (τ (t/a))τ (t/a)]y = 0

(12.33)

where U (τ ) = 3X 2 (τ ) + 3Y 2 (τ ) − ζ 2 /4 and V (τ ) = 6X(τ )Y (τ ). Note that the periodic coeﬃcient in equation (12.32) is continuous with respect to time t since V (±1) = 0 due to the boundary conditions (12.25). By using the numerical solutions of equation (12.33), one obtains the number A for every solution Aa = 10.5155, Ab = −2.63747, Ac = −2.63749 Ad = 1.70201, and Ae = 0.143507

(12.34)

286

12 NSTT and Shooting Method for Periodic Motions

where the index correspond to the type of periodic solution of the original equation; see Figs. 12.1, 12.2, and 12.3. These numbers conﬁrm that only solution (e) is stable.

12.3.2

Step-Wise Discontinuous Input

Let us consider now the case of discontinuous periodic input of the rectangular cosine temporal shape x ¨ + ζ x˙ + x3 = B

dτ (t/a) d (t/a)

(12.35)

where a is a quarter of the input period. In this case, the right-hand side of equations (12.23) and (12.24) are modiﬁed so that the equations take the form a−2 X + ζa−1 Y + X 3 + 3XY 2 = 0 a−2 Y + ζa−1 X + Y 3 + 3X 2 Y = B

(12.36) (12.37)

under the homogeneous boundary conditions (12.25). Fig. 12.5 shows the shooting diagram under the ﬁxed parameters ω = π/(2a) = 1 (0.1592 Hz), B = 7.4 and ζ = 0.05. In this case, there are seven intersections between the two families of curves and therefore seven periodic solutions of the period T = 4a as shown in Figs. 12.6 and 12.7. Note that, under the same parameters, the system response on the rectangular cosine input shows new features compared to those under the sine-wave input [144]. For example, after few cycles along the orbit (a), the system starts its drift around the ﬁrst three solutions, (a), (b) and (c). At this stage, the dynamics resembles that under the sine-wave input. Further, however, after several random ‘jumps’ between the orbits (a), (b) and (c), the system becomes eventually attracted by the stable orbit (e). The direct numerical solution, represented in Fig. 12.8, clearly shows all three stages of the time and spectral histories of the dynamics. In order to clarify stability, the Floquet theory can be applied analogously to the case of the sine-wave input.

12.3.3

Impulsive Loading

Let us consider the same oscillator loaded by the periodic series of pulses x ¨ + ζ x˙ + x3 = pτ

∞

t t + 1 − 4k − δ − 1 − 4k = 2p δ a a k=−∞

where p = p (±1) = B = const.

(12.38)

12.3 Sample Problems and Discussion

287

40 f

20 e

h

0 a b

c d 20

g

40 4

2

0 g

2

4

Fig. 12.5 The curves G(g, h) = 0 (continuous) and H(g, h) = 0 (dashed) for the Ueda oscillator under the step-wise input and the following parameters: ζ = 0.05, B = 7.4, and ω = 1.

In this case, both equations (12.36) and (12.37) should have zero righthand side, however, the non-homogeneous version of the boundary condition (12.17) must be imposed in order to eliminate the pulses. The second expression in (12.26) and the ﬁrst one in (12.29) must be modiﬁed as X (−1) =a2 B=[π/(2ω)]2 B and G (g, h)=a2 B=[π/(2ω)]2 B, respectively. Therefore, the singular terms are eliminated from the system due to the sawtooth time, and the shooting procedure can be applied in the same fashion as that under the smooth input. The shooting diagram and the corresponding periodic solutions are shown in Figures 12.9, and 12.10 and 12.11, respectively. The projections of the phase trajectories show discontinuities of the velocity on the xv-plane caused by the external pulses. The last four projections,

288

12 NSTT and Shooting Method for Periodic Motions

x

x

6 4 2

6 4 2

a

1

2 4 6

2

3

4

5

6

t

2

3

4

5

6

t

x 6 4 2

c

1

2

3

4

5

6

d

t

1

2 4 6

x

2

3

4

5

6

t

x

6 4 2

6 4 2

e

1

2 4 6

1

2 4 6

x 6 4 2 2 4 6

b

2

3

4

5

6

t 2 4 6

f

1

2

3

4

5

6

t

x 6 4 2 2 4 6

g

1

2

3

4

5

6

t

Fig. 12.6 The temporal mode shapes of periodic solutions og Ueda circuit under the step-wise input.

(h) through (k), can be qualiﬁed as ‘quasi free’ vibrations sustained by the pulses. In the shooting diagram represented in Fig. 12.9, the related

12.3 Sample Problems and Discussion

289

v

v

4 2 0 2 4

a x

3 2 1

0

1

2

6 4 2 0 2 4 6

b x

3 2 1

3

v

0

1

2

3

v 10

6 4 2 0 2 4 6

5

c x

d

0

x

5 3 2 1

0

1

2

10 4

3

2

v 5

e

0

x

5 4

2

2

4

v

10

10

0

0

2

20 10 0 10 20

f

6 4 2

4

0

2

x

4

6

v 20 10 0 10 20

g

6 4 2

0

2

x

4

6

Fig. 12.7 The projections of periodic trajectories on xv-planes.

intersections are diﬃcult to determine due to a very small angle between the intersecting curves.

290

12 NSTT and Shooting Method for Periodic Motions

x

5

0

−5

50

100

150 Time

200

250

300

2.5

Frequency

2 1.5 1 0.5 0

0

10

20

30

40

50

Time

Fig. 12.8 The time history record and its spectrogram (in Hz) for evolution of the solution (a) under the step-wise voltage of the amplitude B = 7.4.

12.4

Other Applications

12.4.1

Periodic Solutions of the Period - n

The above sections deal with periodic solutions with the input period T = 4a. In order to capture ‘subharmonic’ solutions of the period nT , the components of representation (12.3) must be taken in the form 1 [x (naτ ) + x (2na − naτ )] 2 1 Y (τ ) = [x (naτ ) − x (2na − naτ )] 2

X (τ ) =

(12.39)

where τ = τ (t/(na)). For instance, applying (12.39) to the sine wave sin ωt, gives nπτ nπτ 1 sin + sin nπ − 2 2 2 1 nπτ nπτ + sin − sin nπ − τ 2 2 2 7 8 nπτ 1 1 + (−1)n+1 + 1 − (−1)n+1 τ = sin 2 2

sin ωt =

where τ = dτ (t/(na))/d(t/(na)) and a = π/(2ω).

(12.40)

12.4 Other Applications

291

30

k

j 20

10 g i h

0

h

d

a

c f

b

e

10

20

30 6

4

2

0

2

4

6

g

Fig. 12.9 The curves G(g, h) = 0 (continuous) and H(g, h) = 0 (dashed) for Ueda circuit under the impulsive input and parameters: ζ = 0.05, B = 1.4, and ω = 1.

According to representation (12.40), equations (12.23) and (12.24) must be modiﬁed as follows (na)−2 X + ζ(na)−1 Y + X 3 + 3XY 2 nπτ B 1 + (−1)n+1 sin = 2 2 (na)−2 Y + ζ(na)−1 X + Y 3 + 3X 2 Y B nπτ 1 − (−1)n+1 sin = 2 2

(12.41)

292

12 NSTT and Shooting Method for Periodic Motions x

x a

1.5 1.0 0.5

b

2

2

1 2 3 4 5 6

t

1

1

2

3

4

5

6

t

d

e

1

2

3

4

5

6

t

1

1

2

3

4

5

6

x 2 1 1

1 2

2

3

4

5

6

h

1

1 2 3

x

6

t

2

3

4

5

6

t

x

3 2 1 t

1

1 2

x g

5

f

t

2

2

4

2 1

1

1

3

x

2

1

2

2 x

2

1

1

2

x

c

1

1

0.5 1.0 1.5

2

3

4

5

6

i

3 2 1 t 1 2 3

1

2

3

4

5

6

t

x

6 4 2 2 4 6

x

6 4 2

j

1

2

3

4

5

6

t 2 4 6

k

1

2

3

4

5

6

t

Fig. 12.10 The temporal mode shapes of periodic solutions og Ueda circuit under the impulsive input.

The right-hand side of these equations shows that direct replacement a → na in (12.23) and (12.24) would not work. If n = 1 then equations (12.41) take the form (12.23) and (12.24), but if n > 1 equations (12.41) can give new solutions in addition to those described by equations (12.23) and (12.24). The corresponding calculations however become time consuming and give complicated diagrams as the number n increases.

12.4 Other Applications

293

v

v

v

4

2 1 0 1 2

a x

4

b

2 0

x

2 2 1 0

v

1

4

2

d x

0

x

2

2

4 2 1 0

1

2 1 0

2

v 6 4 2 0 2 4

2 1 0

1

2

v e

2

0 4

x

v 4

4 2

0 2

4

1.5 1.0 0.50.00.51.01.5

c

2

1

2

4 2 0 2 4 6

f x

2 1 0

v

1

2

v

10 g

h

5 x

0

x

5 2 1 0

1

10

2

v 20 10 0 10 20

i

5 0

x

5 321 0 1 2 3

321 0 1 2 3

v j x

6 4 2 0 2 4 6

20 10 0 10 20

k x

6 4 2 0 2 4 6

Fig. 12.11 The projections of periodic trajectories on xv-planes.

12.4.2

Two-Degrees-of-Freedom Systems

Using the above two-dimensional geometrization of shooting diagrams enables one of considering special cases of two-degrees-of-freedom systems based on equations (12.20) or (12.21). For example, equation (12.20) can be treated as an equation with respect to the two-component vector-function X = {X1 (τ ), X2 (τ )}. Such an interpretation leads to two scalar equations a−2 X1 + f1 (X1 , X2 , X1 /a, X2 /a, aτ ) = Q1 (τ ) a−2 X2 + f2 (X1 , X2 , X1 /a, X2 /a, aτ ) = Q2 (τ )

(12.42)

294

12 NSTT and Shooting Method for Periodic Motions

In this case, the shooting procedure should be based on the initial conditions at τ = −1, X1 (−1) = g, X1 (−1) = 0 X2 (−1) = h, X2 (−1) = 0

(12.43)

where the numbers g and h are determined to satisfy the boundary conditions on the right end of the interval −1 ≤ τ ≤ 1, ∂X1 (τ ; g, h) |τ =1 ≡ G(g, h) = 0 ∂τ ∂X2 (τ ; g, h) |τ =1 ≡ H(g, h) = 0 ∂τ

(12.44)

In this case, every solution g and h of system (12.44) gives the initial position on the conﬁguration plane X1 X2 at which the system starts with zero velocity its periodic motion of the period T = 4a.

12.4.3

The Autonomous Case

The nonlinear normal modes represent an important class of periodic motions. The related references and description of analytical methods can be found in [100] and [190]. Analogously to the linear theory, the basic nonlinear normal mode solutions are given by the class of autonomous conservative systems. In this case, equations (12.42) take the form a−2 X1 + f1 (X1 , X2 ) = 0 a−2 X2 + f2 (X1 , X2 ) = 0

(12.45)

The form of equations (12.45) is easier than (12.42), but the parameter a becomes unknown. It is possible to avoid determining the parameter a by considering it as a control parameter for tracking the evolution of shooting diagrams. Alternatively, the parameter a can be considered as a shooting parameter by imposing one constraint on the parameters g or h. Let us consider, for instance, the system trajectories in the conﬁguration plane X1 X2 . Introducing the amplitude A = g 2 + h2 in (12.43), gives X1 (−1) = A cos ϕ X2 (−1) = A sin ϕ where the angle ϕ (0 ≤ ϕ < 2π) together with the parameter a can play the role of a new unknowns to be determined by shooting whereas the amplitude A is considered as a control parameter.

Chapter 13

Essentially Non-periodic Processes

Abstract. This chapter describes a possible physical basis for NSTT in case of essentially non-periodic processes. The physical time is structurised to match the one-dimensional dynamics of rigid-body chain of identical particles. Namely, the continuos ‘global’ time is associated with the propagation of linear momentum, whereas a sequence of non-smooth ‘local’ times describe behavious of individual physical particles. Such an idea helps to incorporate temporal symmetries of the dynamics into diﬀerential equations of motion in many other cases of regular or irregular sequences of internal impacts or external pulses. Since the local times are bounded, a much wider set of analytical tools becomes possible, wereas matching conditions are generated automatically by the corresponding time substitution.

13.1

Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles

The periodic version of NSTT employs basis functions generated by the most simple impact oscillator. This is based the fact that the triangular sine and rectangular cosine waves capture general temporal symmetries of periodic processes regardless speciﬁcs of individual vibrating systems. Below, a nonperiodic pair of nonsmooth functions is considered, such as the ramp function, s (t; d) =

1 (d + |t| − |t − d|) 2

(13.1)

and its ﬁrst order generalized derivative, s˙ (t; d), with respect to the temporal argument, t; see Figs.13.1 and 13.2, respectively. Such kind of functions play an important role in signal analyses [74]. Possible physical interpretation of these functions is represented in Fig. 13.3. Namely, the function s (t, d) can be treated as a coordinate of a particle, say a very small perfectly stiﬀ bead, initially located at the origin x = 0. At the time instance t = 0, this bead is struck by the identical bead with the velocity v = 1. After the linear momentum exchange, the reference bead starts moving until it stopped by the third bead x = d; in our case d = 1. V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 295–303, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

296

13 Essentially Non-periodic Processes

s 1.0 0.8 0.6 0.4 0.2 1.0

0.5

0.5

1.0

1.5

2.0

t

Fig. 13.1 The unit slope ramp function at d = 1.0

s 1.0 0.8 0.6 0.4 0.2 1.0

0.5

0.5

1.0

1.5

2.0

t

Fig. 13.2 First derivative of the ramp function

Fig. 13.3 Physical meaning of the ramp function: s(t; 1) describes position of the bead struck by another bead from the left and moving until it strikes the next bead of the same mass.

Now, let us consider an inﬁnite chain of the identical perfectly stiﬀ beads located regularly on a straight line at the points xi (i = 0, 1, ...). No energy loss is assumed so that any currently moving bead has the same velocity. As a result, the linear momentum is translated with the constant speed v = 1, whereas the beads are interacting at the time instances ti = xi . Making the temporal shift t → t − ti in function (13.1), gives si (t) = s (t − ti , di ) = where di = ti+1 − ti .

1 (di + |t − ti | − |t − ti+1 |) 2

(13.2)

13.1 Nonsmooth Time Decomposition and Pulse Propagation

297

Due to the unit velocity, function (13.2) can play the role of ‘local’ time for the bead moving within the interval xi < x < xi+1 during the ‘global’ time interval ti < t < ti+1 . The term “local” means that the temporal variable si starts at zero when the “global” time, t, has reached the point t = ti . In other words, the global temporal variable is associated with the linear momentum, whereas all the local temporal variables are “attached” to the physical bodies. For any sequence of time instances, Λ={t0 , t1 , ...}, the global time, t ∈ [t0 , ∞), can be expressed through the sequence of local times, {si }, in the form ∞ (ti + si ) s˙ i (13.3) t= i=0

where the derivatives s˙ i satisfy the relationship s˙ i s˙ j = s˙ i δij

(13.4)

Practically, (13.3) is always a ﬁnite sum because temporal intervals of physical processes always have ﬁnite upper bounds. Equality (13.3) can be easily veriﬁed within the arbitrary interval ti < t < ti+1 , by using deﬁnitions (13.1) and (13.2), but the set of boundary points Λ still require some attention. So formally diﬀerentiating both sides of equality (13.3) with respect to t and taking into account (13.4), gives 1= =

∞ i=0 ∞

s˙ i +

∞

(ti + si ) [δ (t − ti ) − δ (t − ti+1 )]

(13.5)

i=0

s˙ i + t0 δ (t − t0 )

i=0

where the relationship (ti+1 + si+1 (ti+1 ) − ti − si (ti+1 )) = 0 has been used for calculations. Therefore, for almost all t > t0 , expression (13.5) gives 1=

∞

s˙ i

(13.6)

i=0

Equality (13.6) holds based on the deﬁnition for s˙ i as illustrated by Fig. 13.2, therefore expansion (13.3) also holds for almost all t > t0 at least, and its time derivative includes no singular functions. Note that the right-hand side of expansion (13.3) can be viewed as an element of algebra with the basis {si } and multiplication rule (13.4). This signiﬁcantly eases diﬀerent manipulations with the temporal variable (13.3), for instance

298

13 Essentially Non-periodic Processes

n

t =

∞

n

(ti + si ) s˙ i , n = 1, 2, ...

(13.7)

i=0

or, generally, x (t) =

∞

x (ti + si ) s˙ i =

i=0

∞

Xi (si ) s˙ i

(13.8)

i=0

Since the right-hand sides of (13.7) and (13.8) have the same structure as the argument t itself, then considering further a general function g, gives g (x) =

∞

g (Xi ) s˙ i

(13.9)

i=0

Now, diﬀerentiating (13.8) with respect to time t, and taking into account that si (ti ) = 0 and si−1 (ti ) = di−1 , gives x˙ (t) = =

∞ i=0 ∞

Xi (si ) s˙ i + Xi (si ) s˙ i +

i=0

∞ i=0 ∞

Xi (si ) [δ (t − ti ) − δ (t − ti+1 )]

(13.10)

[Xi (0) − Xi−1 (di−1 )] δ (t − ti )

i=0

where X−1 (d−1 ) = 0. Therefore, all the δ- pulses are eliminated from (13.10) under condition, which can be qualiﬁed as a necessary condition of continuity for x (t), Xi (0) − Xi−1 (di−1 ) = 0

(13.11)

Under condition (13.11), the derivative x˙ (t) has the same algebraic structure as the function x (t) itself. As a result, transformation (13.3) can be applied to a general class of dynamical systems. Moreover, in case of impulsively loaded systems, the sequences of δ-pulses in (13.10) can be utilized for eliminating the corresponding singularities from dynamical systems.

13.2

Impulsively Loaded Dynamical Systems

Let us consider a dynamical system subjected to an arbitrary sequence of δ-impulses, applied to the system at time instances Λ = {t0 , t1 , ...}, x˙ = f (x, t) +

∞

pi δ (t − ti ) ,

x (t) ∈ Rn

(13.12)

i=0

x ≡ 0,

t < t0

(13.13)

where f (x, t) is a suﬃciently smooth vector-function, pi are vectors characterizing magnitudes and directions of the impulses.

13.2 Impulsively Loaded Dynamical Systems

299

In particular case, when t0 = 0, and pi = 0 (i = 1, ... ), system (13.12) and (13.13) becomes equivalent to the following initial value problem x˙ = f (x, t)

(13.14)

x (0) = p0

(13.15)

Below, solution of the initial value problem (13.12) and (13.13) is introduced in the speciﬁc form based on the operator Lie associated with dynamical system (13.15) ∂ ∂ + ∂x ∂t ∂ ∂ ∂ = f1 (x, t) + .... + fn (x, t) + ∂x1 ∂xn ∂t

A = f (x, t)

(13.16)

It is known, for instance, that the exponent of operator (13.16) produces temporal shifts as follows ezA f (x (t) , t) = f (x (t + z) , t + z) (13.17)

∂f (x, t) ∂ (x, t) = f (x, t) + f (x, t) + z + O(z 2 ) ∂x ∂t Proposition 6. Solution of the initial value problem (13.12) and (13.13) can be represented in the form x (t) =

∞

[ai−1 + pi + F (ai−1 + pi , ti , si (t))] s˙ i (t)

(13.18)

i=0

where ai = x (ti+1 ) is the sequence of constant vectors determined by the mapping a−1 = 0 ai = ai−1 + pi + F (ai−1 + pi , ti , di ) ;

(13.19) i = 1, 2, ...

and the function F is deﬁned by

z

ezA f (x, t) dz

F (x, t, z) =

(13.20)

0

where A is the operator Lie (13.16). Proof. Substituting vector analogs of expressions (13.3), (13.8) and (13.10) into the diﬀerential equation of motion (13.12) and taking into account (13.9), gives ∞ {[Xi (si ) − f (Xi (si ) , ti + si )]s˙ i + (13.21) i=0

300

13 Essentially Non-periodic Processes

[Xi (0) − Xi−1 (di−1 ) − pi ]δ (t − ti )} = 0

(13.22)

The left-hand side of expression (13.22) includes both regular and singular terms. Moreover, the basis elements s˙ i are linearly independent, and all the δ-pulses are acting at diﬀerent time instances.. Therefore, equation (13.22) gives (13.23) Xi (si ) = f (Xi (si ) , ti + si ) Xi (0) = Xi−1 (di−1 ) + pi = ai−1 + pi

(13.24)

where a−1 = 0 and ai = Xi (di ) (i = 0, 1, 2, ...). Equation (13.23) can be represented in the integral form si Xi (si ) = Xi (0) +

f (Xi (z) , ti + z) dz

(13.25)

0

Since the variable of integration is limited by the interval 0 ≤ z ≤ si , the integrand in (13.25) can be approximated by the easy to integrate Maclaurin’s series with respect to z. Moreover, such a series can be represented in the convenient form of Lie series based on the fact that Xi (z) are coordinates of the dynamical system with the operator Lie (13.16). As a result, all the coeﬃcients of power series are expressed through the “initial conditions” at z = 0 (t = ti ) by enforcing the form of the dynamical system. This eliminates all high-order time derivatives from the coeﬃcients of the power series. means of the right-hand side of the dynamical system; no high order derivatives of the coordinates are included any more into the coeﬃcients of the series. So, taking into account the notation Xi (si ) = x (ti + si ), and expressions (13.17) and (13.20), brings (13.25) to the form si ezA f (x (ti ) , ti ) dz

Xi (si ) = Xi (0) + 0

= Xi (0) + F (x (ti ) , ti , si ) = Xi (0) + F (Xi (0) , ti , si )

(13.26)

Substituting now Xi (0) from (13.24) in (13.26), gives Xi (si ) = ai−1 + pi + F (ai−1 + pi , ti , si )

(13.27)

Finally, substituting (13.27) in expansion (13.8), gives (13.18). Then, substituting si = di in (13.27) gives (13.19) and thus completes the proof. Solution (13.18) and (13.19) should be viewed as a semi-analytic solution, since some numerical tool is required for calculating the discrete mapping (13.19). The central role here belongs to the function s (t; d) (13.1), which is automatically matching all the neighboring pieces of the solution.

13.2 Impulsively Loaded Dynamical Systems

301

Note that the distances di between times Λ are not necessary small, however, the precision of the solution can be improved by increasing the number of terms of the Lie series ezA f (x, t) with respect to z, rather than reducing the distances di .

13.2.1

Harmonic Oscillator under Sequential Impulses

In order to estimate precision of the above procedure, let us consider the particular case in which function (13.20) can be calculated exactly in the closed form due to the presence of exact analytical solution in between the pulses Λ. The diﬀerential equation of motion on the entire time range is x ¨ + 2ζω x˙ + ω 2 x =

∞

pi δ (t − ti )

(13.28)

i=0

In this case, the function f (x, t) in equation (13.12) becomes

x2 f (x) = −2ζωx2 − ω 2 x1

(13.29)

Using the identity ezA f (x (t) , t) = f (x (t + z) , t + z) and the exact analytical solution of the corresponding free oscillator, gives both components of the vector-function (13.20) in the form ) ) ζe−z ζ ω 2 2 −z ζ ω cos(z 1 − ζ ω) + sin(z 1 − ζ ω) − 1 x1 F1 (x; z) = e 1 − ζ2 ) e−z ζ ω + sin(z 1 − ζ 2 ω) x2 ω 1 − ζ2 ) ωe−z ζ ω F2 (x; z) = − sin(z 1 − ζ 2 ω) x1 (13.30) 2 1−ζ ) ) ζ e−z ζ ω 2 2 −z ζ ω + e cos(z 1 − ζ ω) − sin(z 1 − ζ ω) − 1 x2 1 − ζ2 In this particular case, properties of mapping (13.19) depend on the following determinant 1 + ∂F1 /∂x1 ∂F1 /∂x2 = e−2di ζω J = (13.31) ∂F2 /∂x1 1 + ∂F2 /∂x2 Let us introduce the relative error δ = |J − Jappr | /J

(13.32)

302

13 Essentially Non-periodic Processes

where Jappr is an approximate determinant based on the Lie series expansion (13.17). Figs. 13.4 and 13.5 show diagrams for the relative error δ versus the distance d between any two neighboring impulse times when the highest order terms kept in Lie series (13.17) are O(z 2 ) and O(z 3 ), respectively.

0.030 0.025 0.020

Δ 0.015 0.010 0.005 0.000 0.0

0.2

0.4

0.6

0.8

d Fig. 13.4 Relative error of the determinant based on the truncated Lie series including terms of order O(z 2 ).

Fig. 13.5 Relative error of the determinant based on the truncated Lie series including terms of order O(z 3 ).

As follows from the diagrams, precision of the discrete mapping essentially depends on both the distance between pulse times and the number of terms kept in the Lie series. As a result, the error due to a large distance can be reduced by increasing the number of terms in the Lie series.

13.2 Impulsively Loaded Dynamical Systems

13.2.2

303

Random Suppression of Chaos

A speciﬁc case of the Duﬃng oscillator with no linear stiﬀness under sine modulated random impulses was considered in [142]. The corresponding differential equation of motion is represented in the form x ¨ + ζ x˙ + x3 = B sin t

∞

δ (t − ti )

(13.33)

i=0

where ζ is a constant linear damping coeﬃcient, and B is the amplitude of modulation. Distances between any two sequential impulse times are given by di = ti+1 − ti =

π (1 + βηi ) 12

where ηi are random real numbers homogeneously distributed on the interval [−1, 1], and β is a small positive number, 0 < β 1. Introducing the state vector x = (x, x) ˙ T ≡ (x1 , x2 )T , brings system (13.33) to the standard form (13.12), where

x2 0 f (x) = = , p i B sin ti −ζx2 − x31 Note that oscillator (13.33) represents of-course a modiﬁed version of the well known oscillator, x¨ + ζ x˙ + x3 = B sin t, considered ﬁrst by Ueda [186] as a model of nonlinear inductor in electrical circuits - the Ueda circuit. In particular, the result of work [186], as well as many further investigations of similar models, reveal the existence of stochastic attractors often illustrated by the Poincare diagrams [113]. Similar diagrams obtained under non-regular snapshots can be qualiﬁed as ‘stroboscopic’ diagrams. The results of the computer simulations described in [142] show that some irregularity of the pulse times can be used for the purposes of a more clear observation of the system orbits in the stroboscopic diagrams. When repeatedly executing the numerical code, under the same input conditions, such a small disorder in the input results some times in a less noisy and more organized stroboscopic diagrams. However, such phenomenon itself was found to be a random event whose ‘appearance’ depends on the level of pulse randomization as well as the number of iterations.

Chapter 14

Spatially-Oscillating Structures

Abstract. This chapter illustrates applications of nonsmooth argument substitutions to modeling spatially oscillating structures such as one-dimensional elastic rods with periodic discrete inclusions, and two- or three-dimensional acoustic media with periodic nonsmooth boundary sources of waves. Whenever the corresponding global spatial domains are inﬁnite or cyclical, the related analytical manipulations are similar to those conducted with dynamical systems. The idea of averaging is implemented through the two-variable expansions, where the fast scale is represented by the triangular periodic wave. Such an approach results in closed form analytical solutions despite of the presence discrete inclusions or external discontinuous loads.

14.1

Periodic Nonsmooth Structures

Static and dynamic problems of elasticity dealing with non-smooth periodic structures are often considered in the literature due to their practical importance; see recent reviews [76] for introduction and references. Such problems can also be considered by means of the non-smooth argument transformations. In this case though, the transformation must be applied to the spatial independent variable related to coordinate along which the structure under consideration is periodic. Such an approach was introduced in [135] for a string on discrete elastic foundation, although the complete description of the tool was given later for the corresponding nonlinear case [151], [171], [152], ∞ y y ∂2u ∂2u ρ 2 − T 2 + 2f (u) δ − 1 − 2k = q , y, t (14.1) ∂t ∂y ε ε k=−∞

−∞ < y < ∞ where ρ is the mass density per unit length, T is tension, q(y/ε, y, t) is the body force or external loading, which is assumed to be periodic in the ‘fast scale’ y/ε (0 < ε 1) with the period normalized to four, and similar assumption is made with respect to the transverse displacement of the string u = u(y/ε, y, t); see Fig. 14.1 for illustration. V.N. Pilipchuk: Nonlinear Dynamics, LNACM 52, pp. 305–337, 2010. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

306

14 Spatially-Oscillating Structures

Note that equation (14.1) has no point-wise interpretation due to the presence of Dirac δ-function. Both sides of the equation therefore must be viewed as distributions producing the same output if applied to the same testing function. Correctness of such type of modeling was intensively discussed in the literature; see, for instance, [46]. Omitting details, the series of δ-functions in equation (14.1) has certain meaning if the function f (u(y/ε, y, t)) is at least continuous at y = ε(1 + 2k) for every k. Such a continuity condition easily follows from the very form of equation (14.1). Indeed, if the displacement u were step-wise discontinuous at y = ε(1 + 2k) then the derivative ∂ 2 u/∂y 2 would produce uncompensated derivatives of δ-function. So the displacement u is at least continuous function of the coordinate y that is in match with the physical meaning of model. Moreover, it will be shown below that appropriate nonsmooth substitution for the spatial argument y/ε eliminates the singularities from equation (14.1). As a result, new equations can be considered within the classical theory. In equation (14.1), the co-directed set of localized forces generated by the elastic springs can be expressed through second derivative of the triangular wave τ = τ (y/ε) in the form 2f (u)

∞ k=−∞

δ

y ε

− 1 − 2k = −f (u)sgn(τ )τ

(14.2)

Both the transverse displacement and the external loading are expressed in terms of the non-smooth variables as follows u = U (τ, y, t) + V (τ, y, t)τ q = Q(τ, y, t) + P (τ, y, t)τ

(14.3)

Finally, substituting (14.2) and (14.3) in equation (14.1) and using the diﬀerential and algebraic rules of the non-smooth argument transformation, gives the diﬀerential equations and boundary conditions as, respectively,

2 ∂2V ∂2U Q(τ, y, t) ∂2U 2 ρ ∂ U +ε = −2ε − − (14.4) ∂τ 2 ∂y∂τ T ∂t2 ∂y 2 T

2 ∂2U ∂2V P (τ, y, t) ∂2V 2 ρ ∂ V + ε = −2ε − − (14.5) 2 2 2 ∂τ ∂y∂τ T ∂t ∂y T and V |τ =±1 = 0 ∂U ε2 |τ =±1 = ∓ f (U )|τ =±1 ∂τ T

(14.6) (14.7)

where the form of boundary condition (14.7) was eased by taking into account condition (14.6).

14.1 Periodic Nonsmooth Structures

307

Further analysis is based on representing solutions in the form of asymptotic series U (τ, y, t) = V (τ, y, t) =

∞ k=0 ∞

εk Uk (τ, y, t) εk Vk (τ, y, t)

(14.8)

k=0

Problem formulations based on equations (14.4) through (14.7) possess certain advantage as compared to equation (14.1). For instance, the inﬂuence of inﬁnite discontinuities in equation (14.1) is captured by the form of substitution (14.3) so that no discontinuities are present in the new equations. Also, the structural discreteness of the nonlinear elastic foundation (14.2) is associated with the new spatial variable τ restricted in the range −1 ≤ τ ≤ 1. As a result, conventional perturbation tools become applicable. Moreover, polynomial expansions with respect to the coordinate τ will not be aﬀecting the regularity of asymptotic expansions (14.8) in terms of the original fast scale y/ε. Note that partial diﬀerential equations (14.4) and (14.5) are coupled whereas boundary conditions (14.6) and (14.7) are decoupled with respect to the unknowns U and V . However, introducing new unknown functions, say X = U + V and Y = U − V , decouples equations (14.4) and (14.5) as follows

2 ∂2X ∂2X Q+P ∂2X 2 ρ ∂ X + ε = −2ε − − =0 (14.9) ∂τ 2 ∂y∂τ T ∂t2 ∂y 2 T

2 ∂2Y ∂2Y Q−P ∂2Y 2 ρ ∂ Y + ε = 2ε − − =0 (14.10) ∂τ 2 ∂y∂τ T ∂t2 ∂y 2 T Equations (14.9) and (14.10) are of the same structure, except opposite signs of the partial derivatives. The buckling of a circular ring loaded by a discrete regular set of concentrated compressive forces was considered in [189]; see Fig.14.1. Taking into account identity (14.2), the diﬀerential equation of equilibrium of such ring can be represented in the form y(t) ˙ = f (y(t)) − λ

∞

δ

k=−∞

≡ f (y(t)) + 0 ≤ t ≤ 2π

t − 1 − 2k a

(14.11)

λ sgn(τ )τ 2

where t = s/R is the arc length of the undeformed ring axis per radius, y(t) is a six-component vector-function characterizing elastic states of the ring,

308

14 Spatially-Oscillating Structures

Fig. 14.1 Linear string on the discrete regular set of nonlinearly-elastic springs.

τ = τ (t/a) is the triangular sine wave of the period T = 4a and prime means its Schwartz derivative, a = ε/2 is a small parameter as compared to unity, λ is a dimensionless parameter which is proportional to the localized forces P applied to the ring, and conditions of periodicity are imposed on the elements of vector-function y(t). Let us represent the unknown vector-function in the form y = X(t, τ ) + Y (t, τ )τ Substituting then (14.12) in equation (14.11), gives

1 ∂X 1 ∂Y ∂X ∂Y + − R(X, Y ) + + − I(X, Y ) τ a ∂τ ∂t a ∂τ ∂t

1 λ Y − sgn(τ ) τ = 0 + a 2 or

ε ∂Y ∂X = + R(X, Y ) − ∂τ 2 ∂t

∂X ε ∂Y = + I(X, Y ) − ∂τ 2 ∂t 1 Y |τ =±1 = ± λε 4 where 1 [f (X + Y ) + f (X − Y )] 2 1 I(X, Y ) = [f (X + Y ) − f (X − Y )] 2

R(X, Y ) =

(14.12)

(14.13)

(14.14) (14.15)

14.1 Periodic Nonsmooth Structures

309

Fig. 14.2 The circular ring under discrete regular set of compressive radial forces.

The mechanical model, which is shown in Fig. 14.3, was considered in [150] based on the generalized (asymmetric) version of the triangular wave. In contrast to the model shown in Fig. 14.1, the sprigs are linearly elastic and shifted in dipole-wise manner so that the diﬀerential equation of motion with respect to the string deﬂection u = u(t, y) has the form ρ

∂ 2 u ku ∂ 2 τ (y/a, γ) ∂ 2u 2 (1 − γ − T − )sgn(τ (y/a, γ)) =0 (14.16) ∂t2 ∂y 2 a ∂(y/a)2 −∞ < y < ∞

where the triangular wave with diﬀerent positive and negative slopes is given by z/ (1 − γ) for − 1 + γ ≤ z ≤ 1 − γ τ = τ (z, γ) = (14.17) (−z + 2) / (1 + γ) for 1 − γ ≤ z ≤ 3 + γ −1 < γ < 1 Schwartz derivatives of function (14.17) satisfy the following relationships

∂τ (z, γ) ∂z

2 =α+β

∂τ (z, γ) ∂z

∂τ (z, γ) ∂ 2 τ (z, γ) 1 ∂ 2 τ (z, γ) β = ∂z ∂z 2 2 ∂z 2

(14.18)

(14.19)

310

14 Spatially-Oscillating Structures

∞ ∂ 2 τ (z, γ) = 2α [δ (z + 1 − γ − 4k) − δ (z − 1 + γ − 4k)] ∂z 2 k=−∞ where α = 1/ 1 − γ 2 and β = 2γα. Let us represent the string deﬂection in the form

u = U (τ, y, t) + V (τ, y, t)τ

(14.20)

(14.21)

where τ = τ (y/a, γ) and τ = ∂τ (y/a, γ)/∂(y/a). The components of representation (14.21), U and V , depend on the coordinate y both explicitly and through the triangular wave function τ in such a way that the complete partial derivative ∂u/∂y is equivalent to applying the diﬀerential matrix operator

∂/∂y (α/a)∂/∂τ D= (14.22) (1/a)∂/∂τ (β/a)∂/∂τ +∂/∂y to the vector-column of the components U and V ∂u U ⇐⇒ D V ∂y

(14.23)

under the condition V |τ =±1 = 0

(14.24)

The regular part of the second derivative can be calculated by means of the relationship ∂2u 2 U ⇐⇒ D (14.25) V ∂y 2 However, second derivative of the triangular wave function must be preserved in order to eliminate the same kind of singularity from equation (14.16). So, substituting (14.21) in (14.16) and collecting separately terms related to diﬀerent elements of the basis {1, τ }, gives ρ

×

α ∂2 ∂2 + a2 ∂τ 2 ∂y 2

U+

∂2U −T ∂t2 αβ ∂ 2 2α ∂ 2 + a2 ∂τ 2 a ∂τ ∂y

V =0

(14.26)

∂2V ρ 2 −T ∂t

β ∂2 α + β2 ∂2 ∂2 2 ∂2 2β ∂ 2 + + + V =0 × U + a2 ∂τ 2 a ∂τ ∂y a2 ∂τ 2 a ∂τ ∂y ∂y 2 (14.27)

14.1 Periodic Nonsmooth Structures

under the additional to (14.24) condition

∂V T ∂U k(1 − γ 2 ) +β U |τ =±1 − 2 |τ =±1 = ± a ∂τ ∂τ a

311

(14.28)

Note that some terms have been eliminated from condition (14.28) by taking into account condition (14.24). Further analysis of the boundary value problem (14.24) and (14.26) through (14.28) can be implemented by using the asymptotic approach [151] considering a as a small parameter.

Fig. 14.3 Linear string on the discrete periodic set of linearly-elastic dipole-wise shifted springs of the stiﬀness k.

Similar version of the transformation was employed for statics of layered composites in [179]. The idea is to represent the elastic constants in the form λ∗ = λ(1 + lτ ) μ∗ = μ(1 + mτ ) τ = ∂τ (x/a, γ) /∂(x/a)

(14.29)

where x is the spatial coordinate, which is perpendicular to the layer boundaries, and λ, μ, l, and m are such constants that λ∗ and μ∗ remain positive on the period of elastic structure, T = 4a. Respectively, the vector of elastic displacements is represented in the form {u, v, w} = {U (1) , V (1) , W (1) } + {U (2) , V (2) , W (2) }τ

(14.30)

where U (i) = U (i) (τ, y, z), V (i) = V (i) (τ, y, z), and W (i) = W (i) (τ, y, z). Then, both representations, (14.29) and (14.30), are substituted into the static equations of three-dimensional elastic bodies with variable elastic

312

14 Spatially-Oscillating Structures

constants, and the NSTT algebraic and diﬀerential manipulations applied in order to formulate boundary value problems, however with constant elasticity parameters, for the components of representation (14.30).

14.2

Averaging for One-Dimensional Periodic Structures

Let us consider a one-dimensional 4ε-periodic structure whose static elastic states are described by the vector-function z(y) ∈ Rn that depends upon the longitudinal coordinate y attached to the undeformed structure. The number of vector’ components n can always be increased so that the diﬀerential equation of equilibrium takes the form of ﬁrst-order diﬀerential equation, for instance, as follows z (y) = f (z(y), ϕ, y) + p(y)τ (ϕ)

(14.31)

Here the spatial scale ϕ = y/ε associates with the structural periodicity, the vector-function f (z, ϕ, y) ∈ Rn is continuous with respect to z and y, but it is allowed to be step-wise discontinuous with respect to ϕ at the points {ϕ : τ (ϕ) = ±1}, and p(y) ∈ Rn is a continuous vector-function describing the amplitude modulation of the localized loading. Example 18. In terms of the matrixes,

u(y) 0 z= , p(y) = v(y) q(y)/(2ε)

v(y)/{EF [1 + ατ (y/ε)]} f = 0

(14.32)

equation (14.31) describes an elastic rod whose cross sectional area is a piecewise constant periodic function of the longitudinal coordinate as it is shown in Fig. 14.4 Indeed, substituting (14.32) in (14.31) and eliminating then v(y), gives the second-order diﬀerential equation of equilibrium for the rod du q(y) y d y τ . (14.33) EF 1 + ατ = dy ε dy 2ε ε

Fig. 14.4 An elastic rod with a periodic non-smoothly varying cross sectional area and concentrated loading.

14.3 Two Variable Expansions

313

The averaging technique is described below based on the general form of equation (14.31).

14.3

Two Variable Expansions

The idea of two variable expansions will be used by considering the saw-tooth oscillating coordinate τ = τ (y/ε) and the original coordinate y 0 ≡ y as fast and slow spatial scales, respectively, provided that the following assumption holds ε << 1 (14.34) Let us represent solutions of equation (14.31) in the form z = X(τ, y 0 ) + Y (τ, y 0 )τ

(14.35)

Substituting (14.35) in equation (14.31), gives

∂X ∂Y ∂X ∂Y +ε + ε − R − I + τ + [Y − εp(y 0 )]τ = 0 f f ∂τ ∂y 0 ∂τ ∂y 0 (14.36) where 1 Rf X, Y, τ, y 0 = [f (X + Y, τ, y 0 ) ± f (X − Y, 2 − τ, y 0 )] (14.37) If X, Y, τ, y 0 2 Expression (14.36) is equivalent to the following boundary value problem with no discontinuities

∂Y ∂X +ε − I =0 (14.38) f ∂τ ∂y 0

∂X ∂Y +ε − R =0 f ∂τ ∂y 0 Y |τ =±1 = εp(y 0 )

(14.39)

Let us represent solutions of the boundary value problem, (14.38) and (14.39), in the form of asymptotic series with respect to ε, X= Y =

∞ i=0 ∞

εi X i (τ, y 0 ) εi Y i (τ, y 0 )

i=0

where the functions X i and Y i are to be sequentially determined.

(14.40)

314

14 Spatially-Oscillating Structures

Substituting (14.40) in (14.37), generates power series expansions Rf = Rf0 + εRf1 + ε2 Rf2 + ...

(14.41)

If = If0 + εIf1 + ε2 If2 + ... where the following notations are used Rfi =

1 ∂ i Rf |ε=0 , i! ∂εi

Ifi =

1 ∂ i If |ε=0 i! ∂εi

Substituting (14.40) in (14.38) and (14.39), and matching the coeﬃcients of the same powers of ε, gives the corresponding sequence of equations and boundary conditions. In particular, zero-order problem takes the form ∂Y 0 ∂X 0 = 0, =0 ∂τ ∂τ

(14.42)

Y 0 |τ =±1 = 0

(14.43)

and As follows from (14.42), the generating solution is independent on the fast oscillating scale τ . Therefore, taking into account (14.43), gives solution X 0 = A0 (y 0 ),

Y0 ≡0

(14.44)

where A0 is an arbitrary vector-function of the slow co-ordinate that will be determined on the next step of the asymptotic procedure. So, collecting the terms of order ε, gives the diﬀerential equations and boundary conditions in the form, respectively, ∂Y 0 ∂X 1 = If0 − = If (A0 , 0, τ, y 0 ) ∂τ ∂y 0 ∂Y 1 ∂X 0 dA0 0 0 = Rf0 − = R (A , 0, τ, y ) − f ∂τ ∂y 0 dy 0

(14.45) (14.46)

and Y 1 |τ =±1 = p(y 0 )

(14.47)

Integrating equations (14.45) and (14.46), gives ﬁrst-order terms of the asymptotic solution τ 1

If0 dτ + A1 (y 0 )

X = 0

τ Rf0 dτ −

Y1 = −1

dA0 (τ + 1) + p(y 0 ) dy 0

(14.48)

14.4 Second Order Equations

315

where A1 is a new arbitrary vector-function of the slow spatial scale y 0 , and the limits of integration for Y 1 are chosen in such a manner that boundary condition (14.47) is satisﬁed automatically at the point τ = −1 whereas another point, τ = 1, gives equation dA0 1 = dy 0 2

1 Rf0 dτ ≡< Rf0 >

(14.49)

−1

Note that the ‘slow scale’ equation (14.49) was obtained by satisfying the boundary condition in contrast to the conventional scheme of two variable expansions in which such kind of equations are obtained by eliminating the so-called ‘resonance terms.’ Enforcing now equation (14.49), brings the component Y 1 to the ﬁnal form τ (Rf0 − < Rf0 >)dτ + p(y 0 )

1

Y =

(14.50)

−1

At this stage, expressions (14.48) through (14.50) determine the ﬁrst-order terms of the asymptotic solution, however, the slow-scale vector-function A1 (y 0 ) still remains unknown. The corresponding ordinary diﬀerential equation is obtained on the next stage from the boundary condition for Y 2 and can be represented in the form 0 1 ∂Rf0 dA1 = (14.51) A1 + F 1 (A0 , y 0 ) dy 0 ∂A0 where ∂Rf0 /∂A0 is the Jacobian matrix, and the vector-function F 1 is known. Note, that equation (14.51) is linear. Moreover, on the next steps, equations for the vector-functions A2 , A3 ,... will be of the same linear structure, including the same Jacobian matrix.

14.4

Second Order Equations

Let us consider now the second order diﬀerential equation with respect to the vector-function z(y) ∈ Rn , however, in the linear form z (y) + [q(ϕ, y) + p(y)τ (ϕ)]z = g(ϕ, y) + r(y)τ (ϕ)

(14.52)

where q and p are n× n-matrixes, g and r are n-dimensional vector-functions, and ϕ = y/ε is the fast spatial scale. Based on the assumptions of the previous section, the functions q and g and solutions of equation (14.52) can be represented in the form, respectively,

316

14 Spatially-Oscillating Structures

q(ϕ, y) = Q(τ (ϕ), y 0 ) + P (τ (ϕ), y 0 )τ (ϕ) g(ϕ, y) = G(τ (ϕ), y 0 ) + F (τ (ϕ), y 0 )τ (ϕ) and

(14.53)

z(y) = X(τ (ϕ), y 0 ) + Y (τ (ϕ), y 0 )τ (ϕ)

where y 0 ≡ y represents the slow spatial scale. This leads to the boundary value problem

2 ∂ X ∂2X ∂2Y 2 = −2ε −ε + QX + P Y − G ∂τ 2 ∂τ ∂y 0 ∂y 02

∂2Y ∂2Y ∂2X 2 = −2ε −ε + P X + QY − F ∂τ 2 ∂τ ∂y 0 ∂y 02

(14.54) (14.55)

and ∂X |τ =±1 = ε2 [r(y) − p(y)X]|τ =±1 ∂τ Y |τ =±1 = 0

(14.56)

Further, representing the solution of the boundary value problem (14.54), (14.55) and (14.56) in the form of asymptotic series (14.40), gives a sequence of boundary value problems, in which ﬁrst two steps appear to have quite trivial solutions, such as X 0 = B 0 (y 0 ), Y 0 ≡ 0 and X 1 ≡ 0, Y 1 ≡ 0 where B 0 is an arbitrary function of the slow argument y 0 . As a result, ﬁrst two non-trivial steps of the averaging procedure give z(y) = B 0 (y 0 ) + ε2 [X 2 (τ (ϕ), y 0 ) + Y 2 (τ (ϕ), y 0 )τ (ϕ)] + O(ε3 )

(14.57)

where, in second-order of ε, the solution components are τ (τ − ξ)[G(ξ, y 0 )− < G(τ, y 0 ) >

2

X =

(14.58)

−1

−(Q(ξ, y 0 )− < Q(τ, y 0 ) >)B 0 ]dξ + (r − pB 0 )τ + B 2 (y 0 ) τ 2 (14.59) Y = [(τ − ξ)(F (ξ, y 0 ) − P (ξ, y 0 )B 0 ) −1

− < (1 − τ )(F (τ, y 0 ) − P (τ, y 0 )B 0 ) >]dξ

14.4 Second Order Equations

317

Here, notation < • > means averaging with respect to τ as deﬁned in (14.49), the vector-function B 0 = B 0 (y 0 ) satisﬁes equation d2 B 0 + < Q(τ, y 0 ) > B 0 =< G(τ, y 0 ) > dy 02

(14.60)

The new arbitrary function of the slow coordinate, B 2 (y 0 ), has to be deﬁned on the next step of the procedure. Note that the δ-loads generated by the derivative τ (ϕ) are switching their directions twice per one period of the triangular wave. In many practical cases though the direction of impulses may remain constant. The corresponding reformulation of the problem can be implemented by introducing the factor -sgn(τ ) into the diﬀerential equation as follows z (y) + [q(ϕ, y) − p(y)sgn(τ )τ (ϕ)]z = g(ϕ, y) − r(y)sgn(τ )τ (ϕ) (14.61) Now, in equation (14.61), the term sgn(τ )τ (ϕ) generates δ-loads of the same direction, whereas the boundary condition (14.56) for the X-component takes the form ∂X |τ =±1 = ∓ε2 [r(y) − p(y)X]|τ =±1 (14.62) ∂τ The form of expressions (14.58) and (14.60) is modiﬁed as, respectively, τ (τ − ξ)[G(ξ, y 0 )− < G(τ, y 0 ) >

2

X =

(14.63)

−1

−(Q(ξ, y 0 )− < Q(τ, y 0 ) >)B 0 ]dξ −

τ2 (r − pB 0 ) + B 2 (y 0 ) 2

and, d2 B 0 + (< Q(τ, y 0 ) > +p)B 0 =< G(τ, y 0 ) > +r dy 02

(14.64)

and the component Y 2 is still described by (14.59). Example 19. Let us consider an inﬁnite beam resting on a discrete foundation represented by the periodic set of linearly-elastic springs of stiﬀness c. The corresponding diﬀerential equation of equilibrium is D

∞ x d4 w cw x − 1 − 2k = q + δ load dx4 a a L k=−∞

−∞ < x < ∞ Let us introduce the following dimensionless values y=

y w x , ϕ= , W = L ε a

(14.65)

318

14 Spatially-Oscillating Structures

γ=

cL4 L4 , ψ(y) = qload (y) aD aD

where ε = a/L << 1. As a result the above equation (14.65) for the beam’ center line takes the form d4 W 1 − γsgn[τ (ϕ)]τ (ϕ) W = ψ(y) 4 dy 2

(14.66)

This equation becomes equivalent to (14.61), after the substitutions

0 W (y) , g= ≡G z= ψ(y) W (y)

0 0 0 −1 p= , q= ≡Q γ/2 0 00 P ≡ 0, F ≡ 0, r ≡ 0 After the corresponding calculations in (14.63) and (14.59), one ﬁnally obtains

0 2 1 2 y 0 2 z = B (y) + ε pτ B (y) + B (y) + O ε3 (14.67) 2 ε where the vector-function B 0 = [B10 , B20 ]T is determined from equation (14.64). In terms of the vector B 0 components, equation (14.64) reads

0

d2 B10 0 −1 B1 0 + = (14.68) B20 γ/2 0 ψ(y) dy 2 B20 The ﬁrst equation in (14.68) gives B20 = d2 B10 /dy 2 , whereas the second equation after elimination of B20 takes the form d4 B10 γ + B10 = ψ(y) 4 dy 2

(14.69)

Equation (14.69) is indeed obtained by averaging equation (14.66) with respect to the fast spatial scale ϕ. In other words, equation (14.69) describes an elastic beam resting on eﬀective continuous elastic foundation. In order to illustrate the asymptotic solution, let us consider the beam under the harmonic transverse loading, qload (x/L) ≡ q0 sin (πx/L), where q0 =const. Taking into account only the leading order ‘slow’ and ‘fast’ components, gives the bending moment in terms of the original variables M (x) = D

d2 w dx2

= −M0

a 2 x πx τ2 1−γ sin 2πL a L

(14.70)

14.5 Acoustic Waves from Non-smooth Periodic Boundary Sources

319

where M0 = 2q0 π 2 L2 /(2π 4 + γ). The bending moment diagram is given in Fig. 14.5.

1.0

MxMo

0.5 0.0 0.5 1.0 0

1

2

3

4

5

6

x Fig. 14.5 Bending moment of the beam on the discrete elastic foundation; numerical values of the parameters are as follows: L = π, a = 0.2, and γ = 1948.0

Finally, note that the homogenization procedure described in [24] gives the averaged equation in the slow spatial scale and a so-called ‘cell problem’ in the fast scale. In the above approach, the analog of cell problem associates with the fast oscillating spatial scale given by the triangular wave function τ (y/ε). As a result, solution for the cell problem automatically unfolds on the entire structure so that the fast and slow components of elastic states are eventually expressed though the same coordinate.

14.5

Acoustic Waves from Non-smooth Periodic Boundary Sources

This section deals with two-dimensional acoustic waves propagating from a discontinuous periodic source located at the boundary of half-inﬁnite space. It is shown that introducing the triangular wave function as a speciﬁc spatial coordinate naturally eliminates discontinuities from the boundary condition associated with the active boundary. For illustrating purposes, let us consider the case of two-dimensional stationary waves propagating in the half-inﬁnite media from a piecewise-linear periodic boundary source as shown in Fig. 14.6.

320

14 Spatially-Oscillating Structures

Fig. 14.6 The model

Let us describe acoustic waves by the linear wave equation in the standard form 1 ∂2P ∂2P ∂ 2P ∂2P = + + (14.71) c2f ∂t2 ∂x2 ∂y 2 ∂z 2 where P is a pressure deviation from the static equilibrium pressure; x, y, z and t are spatial coordinates and time, respectively, and cf is the speed of sound in the media. Further, the plane problem is considered when P = P (t, y, z), and therefore, ∂ 2 P/∂x2 = 0. Such an assumption can be justiﬁed by suﬃciently long piezoelectric rods whose characteristics are constant along the x-coordinate. Suppose that the pressure generated by the rods near the boundary is P0 = A sin ωt, where A and ω are constant amplitude and frequency, respectively. Let the boundary condition at z = 0 to have the form P0 (t) for (4n − 1) a ≤ y ≤ (4n + 1) a (14.72) P (t, y, 0) = 0 for (4n + 1) a ≤ y ≤ (4n + 3) a n = 0, ±1, ±2, ... Note that, based on what is actually known near the ﬂuid-source interface, the boundary condition can also be formulated for pressure derivatives. From the mathematical standpoint, this do not aﬀect much the solution procedure though. Let us seek the steady-state solution, which is periodic with respect to t and y and remains bounded as z → ∞. Since the boundary condition is periodic along y-coordinate with period T = 4a, then, according to the idea of non-smooth argument transformation, the triangular wave periodic coordinate is introduced as y → τ (y/a). As a result, the boundary condition (14.72) and yet unknown solution are represented as, respectively, y 1 1 (14.73) P (t, y, z)|z=0 = P0 (t) + P0 (t) τ 2 2 a and

P (t, y, z) = P1 (t, τ (y/a), z) + P2 (t, τ (y/a), z)τ (y/a)

(14.74)

where the components P1 and P2 are considered as new unknown functions.

14.5 Acoustic Waves from Non-smooth Periodic Boundary Sources

321

Taking into account the expression [τ (y/a)]2 = 1, gives ﬁrst generalized derivative of the original unknown function in the form y 1 ∂P2 1 ∂P1 y 1 ∂P = + τ + P2 τ ∂y a ∂τ a ∂τ a a a

(14.75)

Since the function P (t, y, z) has to be continuous with respect to y in the unbounded open region z > 0, then the periodic singular term τ in (14.75) must be eliminated by imposing condition P2 |τ =±1 = 0

(14.76)

Analogously, second derivative takes the form ∂2P 1 ∂ 2 P1 1 ∂ 2 P2 y = + τ ∂y 2 a2 ∂τ 2 a2 ∂τ 2 a

(14.77)

under the condition

∂P1 |τ =±1 = 0 (14.78) ∂τ Note that both derivatives, (14.75) and (14.77), as well as the original function (14.74) appear to have the same algebraic structure of hyperbolic numbers. Obviously, diﬀerentiation with respect to t and z preserve such a structure as well. As a result, substituting the second derivatives into diﬀerential equation (14.71) and collecting separately terms related to each of the basis elements {1, τ }, gives two partial diﬀerential equations for the components of representation (14.74) 1 ∂ 2 Pi ∂ 2 Pi 1 ∂ 2 Pi = + c2f ∂t2 a2 ∂τ 2 ∂z 2

(14.79)

(i = 1, 2) Substituting then (14.74) in (14.73), gives the corresponding set of boundary conditions 1 1 Pi (t, τ, z) |z=0 = P0 (t) = A sin ωt (14.80) 2 2 Now equations (14.79) and boundary conditions (14.76), (14.78) and (14.80) constitute two independent boundary value problems for the components P1 and P2 . However, the result achieved is that no discontinuous functions are present any more in the boundary conditions. Solving the above boundary value problems by the standard method of separation of variables, gives ﬁnally1 1

Derived by S. Pavlyshyn.

322

14 Spatially-Oscillating Structures

P (t, y, z) =

z 1 A sin ω t − + 2 cf

A

m (−1)k−1 1 y sin ω (t − Kk z) cos k − πτ + (14.81) (k − 1/2) π 2 a k=1

∞ k−1 y (−1) 1 y sin ωt exp (−χk z) cos k − πτ τ (k − 1/2) π 2 a a

k=m+1

where *

2 1 π 2 − k− , 2 a *

2 2 ω 1 π 2 − , χk = k− 2 a cf

Kk =

ω cf

2

k = 1, ..., m

k = m + 1, ...

and m is the maximum number at which the expression under the ﬁrst square root is still positive.

2 P

6 0 2

4

1

z 0

2 1 y 2 3

0

Fig. 14.7 Acoustic wave surface for the set of parameters : cf = 10.0, a = 1, ω = 172, t = 3, and A = 2.

A three-dimensional illustration of solution (14.81) is given by Figs. 14.7 and 14.8 for two diﬀerent magnitudes of the frequency ω. Besides, it is seen

14.6 Spatio-temporal Periodicity

323

Fig. 14.8 Contour plot of the wave ﬁeld shown in Fig. 14.7.

that shorter waves are carrying the information about the discreetness of the wave source for a longer distance from the source. In conclusion, solution (14.81) could be of-course obtained in terms of the standard trigonometric expansions by applying the method of separation of variables directly to the original problem, (14.71) and (14.72). However, the derivation of solution (14.81) implies no integration of discontinuous functions, since all the discontinuities have been captured in advance by transformation (14.74). It is also worth to note that, the terms of series (14.81) are calculated on the standard interval, −1 ≤ τ ≤ 1, which is covered by one half of the total period, whereas the standard Fourier expansions must be built over the entire period. This is due to the fact that representation (14.74) automatically unfolds the half-period domain on the inﬁnite spatial interval.

14.6

Spatio-temporal Periodicity

As a possible generalization of the approach, let us consider, for instance, the boundary condition in the form P (t, y, z)|z=0 = f (t, y)

(14.82)

324

14 Spatially-Oscillating Structures

Fig. 14.9 Acoustic wave surface for the set of parameters: cf = 10.0, a = 1, ω = 86, t = 3, and A = 2.

Fig. 14.10 Contour plot of the wave ﬁeld shown in Fig. 14.9.

14.6 Spatio-temporal Periodicity

325

where the function f is periodic with temporal period Tt = 2π/ω and spatial period Ty = 4a. Introducing the triangular wave spatial argument, τy = τ (y/a), gives f (t, y) = F1 (t, τ (y/a)) + F2 (t, τ (y/a))τ (y/a)

(14.83)

where 1 [f (t, aτy ) + f (t, 2a − aτy )] 2 1 F2 (t, τy ) = [f (t, aτy ) − f (t, 2a − aτy )] 2

F1 (t, τy ) =

(14.84)

In a similar way, introducing the triangular wave temporal argument, τt = τ (2ωt/π), into both of the components, F1 and F2 , gives eventually expression of the form f (t, y) = f0 (τt , τy )e0 + f1 (τt , τy )e1 + f2 (τt , τy )e2 + f3 (τt , τy )e3

(14.85)

where components fi (τt , τy ) are uniquely determined by rule (14.84) applied to each of the two arguments, and the following basis is introduced e0 = 1 e1 = τ (2ωt/π) e2 = τ (y/a) e3 = e1 e2

(14.86)

Basis (14.86) obeys the table of products × e0 e1 e2 e3

e0 1 e1 e2 e3

e1 e1 1 e3 e2

e2 e2 e3 1 e1

e3 e3 e2 e1 1

(14.87)

Now, the acoustic pressure is represented in the similar to (14.85) form P (t, y, z) = P0 (τt , τy , z)e0 + P1 (τt , τy , z)e1 + P2 (τt , τy , z)e2 + P3 (τt , τy , z)e3 (14.88) Regarding the problem described in the previous section, the components of representation (14.88) can be obtained as an exercise by introducing the argument τt directly into solution (14.81). However, formulations based on representation (14.88) become technically reasonable whenever the boundary pressure is adequately described by the functions τt and τy or their diﬀerent combinations, for instance, polynomials. In such cases, polynomial approximations with respect to the bounded arguments may appear to be more eﬀective as compared to Fourier expansions.

326

14 Spatially-Oscillating Structures

Let, for instance, P0 (t) describes the periodic sequence of rectangular spikes of the amplitude A,

2ω 1 1 t ≡ A(1 + e1 ) (14.89) P0 (t) = A 1 + τ 2 π 2 Then, boundary condition (14.73) takes the form 1 A(1 + e1 )(1 + e2 ) 4 1 ≡ A(e0 + e1 + e2 + e3 ) 4

P (t, y, z)|z=0 =

(14.90)

where the basis elements {e0 , e1 , e2 , e3 } are given by (14.86), and the table of products (14.87) is taken into account. Now, substituting representation (14.88) in (14.90), gives the boundary conditions for its components at z = 0 as follows Pi (τt , τy , 0) =

1 A; i = 0,...,3 4

Finally, the three-dimensional case can be considered by adding periodicity of the source along the x-direction at the boundary z = 0 and introducing the corresponding triangular wave argument, say τx . The corresponding rules for algebraic manipulations would be analogous to those generated by the arguments τt and τy . However, necessary details are illustrated below on another model.

14.7

Membrane on a Two-Dimensional Periodic Foundation

Consider an inﬁnite membrane resting on a linearly elastic foundation of the stiﬀness K(x, y) under the transverse load q(x, y). Assuming that both the stiﬀness K and load q are measured per unit membrane tension T , the partial diﬀerential equation of equilibrium is represented in the form the Δu − K(x, y)u = q(x, y) ∂2 ∂2 Δ= + 2 2 ∂x ∂y

(14.91)

where u = u(x, y) is the membrane transverse deﬂection. The foundation is assumed to be step-wise discontinuous and periodic along each of the coordinates as described by the function K(x, y) =

x y k 1 + τ 1 + τ 4 a b

(14.92)

14.7 Membrane on a Two-Dimensional Periodic Foundation

327

Fig. 14.11 Fragment of the map of periodic elastic foundation; a = 1.0 and b = 2.0.

With reference to Fig. 14.11, function (14.92) is deﬁned on the inﬁnite plane, such that 0 (x, y) ∈ any “dark ﬁeld” K(x, y) = (14.93) k (x, y) ∈ any “light ﬁeld” In the same way, Fig. 14.12 provides maps for the elements of basis e0 = 1 e1 = τ (x/a) e2 = τ (y/b) e3 = e1 e2

(14.94)

The above table of products (14.87) still valid for basis (14.94). As a result, function (14.92) takes eventually the form K(x, y) =

k (e0 + e1 + e2 + e3 ) 4

(14.95)

Now let us represent the membrane deﬂection in the form u(x, y) = X(τx , τy , x, y)e0 + Y (τx , τy , x, y)e1 +Z(τx , τy , x, y)e2 + W (τx , τy , x, y)e3

(14.96)

328

14 Spatially-Oscillating Structures

Fig. 14.12 The standard basis map: each of the elements is equal to unity within light domains and zero within dark domains.

where τx = τ (x/a) and τy = τ (y/b) are triangular waves whose lengths are determined by the periods of foundation along x− and y− direction, respectively; scales of the explicitly present variables, x and y, are associated with the scales of loading q(x, y), which is assumed to be slow as compared to the spatial rate of foundation. Note that both linear and non-linear algebraic manipulations with combinations of type (14.96) are dictated by the table of products (14.87). For example, taking into account (14.95) and (14.96), gives Ku =

k (X + Y + Z + W )(e0 + e1 + e2 + e3 ) 4

(14.97)

14.7 Membrane on a Two-Dimensional Periodic Foundation

329

High-order derivatives of (14.96) are simpliﬁed by using the table of products (14.87) and introducing speciﬁc diﬀerential operators as follows. First, using the chain rule, gives 1 dτx = τ (x/a) = dx a 1 dτy = τ (y/a) = dy b

1 e1 a 1 e2 b

(14.98) (14.99)

Then, taking into account (14.87), (14.94), (14.98) and (14.99), gives ﬁrst derivatives of (14.96) in the form

1 ∂Y 1 ∂X ∂u ∂X ∂Y = + + e0 + e1 ∂x a ∂τx ∂x a ∂τx ∂x

1 ∂W ∂Z 1 ∂Z ∂W + + + (14.100) e2 + e3 a ∂τx ∂x a ∂τx ∂x de1 (x/a) 1 +(Y + W e2 ) d(x/a) a ∂u = ∂y

1 ∂Z 1 ∂W ∂X ∂Y + + e0 + e1 b ∂τy ∂y b ∂τy ∂y

1 ∂X 1 ∂Y ∂Z ∂W + + + e2 + e3 b ∂τy ∂y b ∂τy ∂y de2 (y/b) 1 +(Z + W e1 ) d(y/b) b

(14.101)

Last addends in (14.100) and (14.101) include derivatives of the step-wise discontinuous functions e1 (x/a) and e2 (y/b). Such derivatives are expressed through Dirac δ-functions and therefore must be excluded from the expressions (14.100) and (14.101) due to continuity of the original function u(x, y). The δ-functions are eliminated under the boundary conditions Y |τx =±1 = 0 W |τx =±1 = 0

(14.102)

and Z|τy =±1 = 0 W |τy =±1 = 0

(14.103)

The rest of terms in (14.100) and (14.101) represent linear combinations of the basis {e0 , e1 , e2 , e3 }. In order to formalize the diﬀerentiation procedure, let us associate expansion (14.96) with the vector-column

330

14 Spatially-Oscillating Structures

⎡

⎤ X ⎢Y ⎥ ⎥ u =⎢ ⎣Z ⎦ W

(14.104)

In a similar way, let us introduce the vector-columns ux and uy associated with derivatives (14.100) and (14.101) under conditions (14.102) and (14.103), respectively,2 ux = Dx u uy where

⎡

∂/∂x ⎢ a−1 ∂/∂τx Dx = ⎢ ⎣0 0 and

⎡

∂/∂y ⎢0 Dy = ⎢ ⎣ b−1 ∂/∂τy 0

(14.105)

= Dy u

a−1 ∂/∂τx ∂/∂x 0 0

0 0 ∂/∂x a−1 ∂/∂τx

⎤ 0 ⎥ 0 ⎥ a−1 ∂/∂τx ⎦ ∂/∂x

0 ∂/∂y 0 b−1 ∂/∂τy

b−1 ∂/∂τy 0 ∂/∂y 0

⎤ 0 b−1 ∂/∂τy ⎥ ⎥ ⎦ 0 ∂/∂y

(14.106)

(14.107)

These diﬀerential matrix operators automatically generate high-order derivatives of combination (14.96) provided that necessary smoothness (boundary) conditions hold. For instance, the components of expansion for Δu are given by the elements of vector-column (D2x + D2y )u under conditions

1 ∂X ∂Y + |τx =±1 = 0 a ∂τx ∂x

∂W 1 ∂Z + |τx =±1 = 0 a ∂τx ∂x

(14.108)

1 ∂X ∂Z + |τy =±1 = 0 b ∂τy ∂y

1 ∂Y ∂W + |τy =±1 = 0 b ∂τy ∂y

(14.109)

Consider now the particular case a = b = ε 1. Following the diﬀerentiation and algebraic manipulation rules as introduced above, and substituting (14.96) in (14.91), gives 2

Note that ux is not ∂u/∂x.

14.7 Membrane on a Two-Dimensional Periodic Foundation

Δτ X + 2ε

Δτ Y + 2ε

∂2Y ∂ 2Z + ∂τx ∂x ∂τy ∂y ∂2X ∂ 2W + ∂τx ∂x ∂τy ∂y

331

+ ε2 (ΔX − F ) = ε2 q(x, y) + ε2 (ΔY − F ) = 0

2 ∂ W ∂2X + Δτ Z + 2ε + ε2 (ΔZ − F ) = 0 ∂τx ∂x ∂τy ∂y

2 ∂ Z ∂2Y + Δτ W + 2ε + ε2 (ΔW − F ) = 0 ∂τx ∂x ∂τy ∂y

(14.110)

where Δτ = ∂ 2 /∂τx2 + ∂ 2 /∂τy2 , Δ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 , and symbol F denotes the following group of terms related to the elastic foundation F ≡

1 k(X + Y + Z + W ) 4

(14.111)

Boundary conditions (14.108) and (14.109) can be simpliﬁed due to (14.102) and (14.103). As a result, the complete set of boundary conditions takes the form ∂X |τ =±1 = 0, ∂τx x Y |τx =±1 = 0,

∂X |τ =±1 = 0 ∂τy y ∂Y |τ =±1 = 0 ∂τy y

∂Z |τ =±1 = 0, Z|τy =±1 = 0 ∂τx x W |τx =±1 = 0, W |τy =±1 = 0

(14.112)

Note that equations (14.110) have constant coeﬃcients whereas the step-wise discontinuities of foundation have been absorbed by the triangular wave arguments τx and τy . The corresponding boundary conditions (14.112) generate a so-called “cell problem” [76] within the rectangular domain {−1 ≤ τx ≤ 1, −1 ≤ τy ≤ 1}. Therefore, the arguments τx and τy locally describe the fast varying component of membrane shape. In contrast, the explicitly present coordinates x and y describe the slow component in the inﬁnite plane {−∞ < x < ∞, −∞ < y < ∞}. Finally, the increase in number of equations (14.110) of-course complicates solution procedures from the technical standpoint. However, the obvious symmetry of equations helps to ease the corresponding derivations. For instance, equations (14.110) can be decoupled by introducing new unknown functions Ui = Ui (τx , τy , x, y) (i = 1, .., 4) as follows U1 = X + Y + Z + W U2 = X + Y − Z − W U3 = X − Y + Z − W U4 = X − Y − Z + W

(14.113)

332

14 Spatially-Oscillating Structures

Linear transformation (14.113) however makes boundary conditions (14.112) coupled. New boundary conditions are given by the inverse substitution in (14.112) 1 (U1 + U2 + U3 + U4 ) 4 1 Y = (U1 + U2 − U3 − U4 ) 4 1 Z = (U1 − U2 + U3 − U4 ) 4 1 W = (U1 − U2 − U3 + U4 ) 4 X=

(14.114)

Note that transformation (14.113) can be eﬀectively incorporated at the very beginning of transformations by using the idempotent basis as described in the next section.

14.8

The Idempotent Basis for Two-Dimensional Structures

The two-dimensional idempotent basis is introduced as follows 1 (e0 + e1 )(e0 + e2 ) = 4 1 − i2 = e+ 1 e2 = (e0 + e1 )(e0 − e2 ) = 4 1 − + i3 = e1 e2 = (e0 − e1 )(e0 + e2 ) = 4 1 − − i4 = e1 e2 = (e0 − e1 )(e0 − e2 ) = 4

+ i1 = e+ 1 e2 =

1 (e0 + e1 + e2 + e3 ) 4 1 (e0 + e1 − e2 − e3 ) 4 1 (e0 − e1 + e2 − e3 ) (14.115) 4 1 (e0 − e1 − e2 + e3 ) 4

where the standard basis ei is deﬁned by (14.94) and the table of products (14.87), and the following notations for one-dimensional idempotent basis are used 1 (e0 ± ek ) 2 − e+ k ek = 0; (k = 1, 2) e± k =

(14.116)

The main reason for using basis (14.115) is that its table of products has the normalized diagonal form ik in = δkn (14.117) where δkn is the Kronecker symbol. The geometrical meaning of property (14.117) follows from the maps in Fig. 14.13.

14.8 The Idempotent Basis for Two-Dimensional Structures

333

Fig. 14.13 The map of idempotent basis: each of the elements is equal to unity within light domains and zero within dark domains.

In this basis, representation (14.96) takes the form u(x, y) =

4

Uk (τx , τy , x, y)ik

(14.118)

k=1

As a result, 4 4 f( Uk ik ) = f (Uk )ik k=1

(14.119)

k=1

where f is practically any function, linear or nonlinear. First-order partial derivatives of representation (14.119) are obtained as follows

334

=

14 Spatially-Oscillating Structures

∂u(x, y) ∂x 4 1 ∂Uk (τx , τy , x, y) k=1

a

∂τx

(14.120)

∂Uk (τx , τy , x, y) ∂ik ik + Uk (τx , τy , x, y) e1 ik + ∂x ∂x

Further, taking into account (14.115) and (14.116), gives − + + e1 i1 = (e+ 1 − e1 )e1 e2 = i1

− + − e1 i2 = (e+ 1 − e1 )e1 e2 = i2 − − + e1 i3 = (e+ 1 − e1 )e1 e2 = −i3

− − − e1 i4 = (e+ 1 − e1 )e1 e2 = −i4

and ∂i1 ∂x ∂i2 ∂x ∂i3 ∂x ∂i4 ∂x

∂e+ 1 + e ∂x 2 ∂e+ = 1 e− ∂x 2 ∂e− = 1 e+ ∂x 2 ∂e− = 1 e− ∂x 2 =

1 ∂e1 + e 2 ∂x 2 1 ∂e1 − e = 2 ∂x 2 1 ∂e1 + e =− 2 ∂x 2 1 ∂e1 − e =− 2 ∂x 2

=

As a result, derivative (14.120) takes the form

1 ∂U1 1 ∂U2 ∂U1 ∂U2 ∂u = + + i1 + i2 ∂x a ∂τx ∂x a ∂τx ∂x

1 ∂U3 ∂U3 1 ∂U4 ∂U4 + − + + i3 + − i4 a ∂τx ∂x a ∂τx ∂x ∂e1 + 1 ∂e1 − 1 e + (U2 − U4 ) e + (U1 − U3 ) 2 ∂x 2 2 ∂x 2 Analogously, one obtains

1 ∂U1 ∂u ∂U1 1 ∂U2 ∂U2 = + + i1 + − i2 ∂y b ∂τy ∂y b ∂τy ∂y

1 ∂U3 ∂U3 1 ∂U4 ∂U4 + + + i3 + − i4 b ∂τy ∂y b ∂τy ∂y ∂e2 + 1 ∂e2 − 1 e + (U3 − U4 ) e + (U1 − U2 ) 2 ∂y 1 2 ∂y 1

(14.121)

(14.122)

14.8 The Idempotent Basis for Two-Dimensional Structures

335

Let us introduce vector, associated with expansion (14.118), and the corresponding diﬀerential matrix operators as, respectively, ⎡ ⎤ U1 ⎢ U2 ⎥ ⎥ (14.123) u =⎢ ⎣ U3 ⎦ U4 and ⎡1

∂ a ∂τx

+

⎢0 Dx = ⎢ ⎣0

∂ ∂x

0 1 ∂ a ∂τx

+

∂ ∂x

0 0

0 ⎡1

∂ b ∂τy

⎢0 ⎢ Dy = ⎢ ⎣0 0

+

∂ ∂y

0 − 1b ∂τ∂y + 0 0

0 0 − a1 ∂τ∂x + 0

∂ ∂y

∂ ∂x

0 0 1 ∂ b ∂τy

0

+

∂ ∂y

0 0 0 − a1 ∂τ∂x + 0 0 0 − 1b ∂τ∂y +

⎤ ⎥ ⎥ ⎦

(14.124)

∂ ∂x

⎤ ⎥ ⎥ ⎥ ⎦

(14.125)

∂ ∂y

Substituting (14.118) into the original equation (14.91), using the diﬀerentiation rules for idempotent basis, and assuming that a = b = ε, gives

2 ∂ U1 ∂ 2 U1 Δτ U1 + 2ε + + ε2 ΔU1 = ε2 [q(x, y) + kU1 ] ∂τx ∂x ∂τy ∂y

2 ∂ U2 ∂ 2 U2 − Δτ U2 + 2ε + ε2 ΔU2 = ε2 q(x, y) ∂τx ∂x ∂τy ∂y

2 ∂ 2 U3 ∂ U3 − Δτ U3 − 2ε (14.126) + ε2 ΔU3 = ε2 q(x, y) ∂τx ∂x ∂τy ∂y

2 ∂ U4 ∂ 2 U4 + Δτ U4 − 2ε + ε2 ΔU4 = ε2 q(x, y) ∂τx ∂x ∂τy ∂y where the notations Δτ and Δ have the same meaning as those in equations (14.110). Equations (14.126) are decoupled, at cost of coupling the boundary conditions though ∂(U1 − U3 ) |τx =±1 ∂τx (U1 − U3 )|τx =±1 ∂(U2 − U4 ) |τx =±1 ∂τx (U2 − U4 )|τx =±1

∂(U1 − U2 ) |τy =±1 = 0 ∂τy = 0, (U1 − U2 )|τy =±1 = 0 ∂(U3 − U4 ) = 0, |τy =±1 = 0 ∂τy = 0, (U3 − U4 )|τy =±1 = 0 = 0,

(14.127)

336

14 Spatially-Oscillating Structures

Note that both boundary value problems (14.110) through (14.112) and (14.126) through (14.127) implement the transition from two to four spatial arguments: {x, y} → {τx , τy , x, y}. The arguments τx and τy naturally relate to cell problems and incorporate the corresponding elastic components within the class of closed form solutions.

Fig. 14.14 The map of idempotent basis generated by the asymmetric triangular waves with parameters: a = b = 1.0, γ1 = 0.2 and γ2 = 0.6; each of the elements is equal to unity within light domains and zero within dark domains.

Finally, let us introduce two-dimensional idempotent basis generated by the triangular asymmetric wave; see Fig. 14.14. First, following deﬁnitions of Chapter 4, let us introduce one-dimensional idempotents associated with x and y coordinates

14.8 The Idempotent Basis for Two-Dimensional Structures

1 [1 − γi + (1 − γi2 )ei ] 2 1 [1 + γi − (1 − γi2 )ei ] e− i = 2 (i = 1, 2)

337

e+ i =

(14.128)

where e1 = ∂τ (x/a, γ1 )/∂(x/a) and e2 = ∂τ (y/b, γ2 )/∂(y/b). + Now the two-dimensional idempotent basis is given by i1 = e+ 1 e2 , i2 = + − − + − − e1 e2 , i3 = e1 e2 and i4 = e1 e2 , although further expansions shown in (14.115) are not valid any more.

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Appendix

351

APPENDIX 1: MathematicaR (Version 6) notebook for sawtooth power-series solutions for the oscillator t,t x xm 0. This module builds sawtooth power series solutions of the oscillator. The module is easy to modify on polynomial characteristics of the oscillator. n the number of iterations m the exponent STSRn, m : Modulef, h, RHS, Ε, suc, x, X, H, fx : xm ; n

h Εi Hi ; i0 n

RHS Ε h f Εi Xi Ξ ; i0

suc CancelCoefficientNormalSeriesRHS, Ε, 0, n, TableΕi , i, 1, n ; X0 Ξ : A Ξ; DoXk Τ Integratesuck Ξ Τ, Ξ, 0, Τ, Assumptions Τ 0 && Rem 1, k, 1, n; H0 H0 . Solve Τ X0 Τ X1 Τ 0 . Τ 1, H0 1; DoHk Hk . Solve Τ Xk 1 Τ 0 . Τ 1, Hk 1, k, 1, n; TableXk Τ, k, 0, n, TableHk , k, 0, n 1

Successive approximations (execution): n 3; AnalyticalSolution STSRn, Α; DoXi1 AnalyticalSolution1i, i, 1, n 1; DoHi1 AnalyticalSolution2i, i, 1, n; DoPrint"X", i, "", Xi

Simplify, i, 0, n DoPrint"H", i, "", Hi

Simplify, i, 0, n 1 X0A Τ X1

X2

A Τ2Α 2Α A2 Α Α Τ A Τ1Α AΑ 3 2 Α 2 Α Τ A ΤΑ 2 2 Α2 3 2 Α

A12 Α Α Τ2Α A2 Α 1 Α2 4 3 Α AΑ Α 8 10 Α 3 Α2 Τ A ΤΑ 2 Α 3 Α2 Α3 Τ2 A Τ2 Α X3 2 2 Α3 3 2 Α 4 3 Α

352

Appendix

H0A1Α 1 Α H1

H2

A1Α Α 1 Α 2 2 Α A1Α Α 1 Α3 2 2 Α2 3 2 Α

Successive approximation truncated series: n

n1

x Xi ; h Hi ; v i0

i0

1 h

Τ x;

Appendix

353

APPENDIX 2: MathematicaR (Version 6) notebook for sawtooth power-series expansion of periodic functions. In[1]:=

f ft periodic function of the period T m "length" of the series; see the example below smoothness True or False; use smoothness True to smooth the series TauSeriesf, T, m, smoothness : Modulea,s1, s2, NLX, NLY,RE, IM, der, TauSpectrum, a T 4; s1 t a Τ; s2 t 2a a Τ; 1 1 RE f . s1 f . s2; IM f . s1 f . s2; 2 2 derF : ApplyD, F, Τ ; NLX NestListder, RE, 2m 1 . Τ 0; NLY NestListder, IM, 2m 1 . Τ 0; i

DoKX2i 1 k1 i

KY2i 1 k0 i

KY2i k0

Factorial2k 2

, KX2i k1

NLY2k 1 Factorial2k

,

, i, 1, m ; 2m

i1 2m

2m

Y KYiΤi Τi 2 , X i0

k1

NLYkΤk1

2m

k1

In[2]:=

Factorial2k 1

NLY2k Factorial2k 1

Ifsmoothness, X RE . Τ 0 KXi

Y

NLX2k 1

i

NLX2k

Factorialk 1

Τi i

Τi 2 i 2

,

NLXkΤk1 Factorialk 1

,

Example of usage ft : Sin

Πt 2

3

T 4; In[4]:=

Expansions with no smoothing procedure fnosmootht, m : X Y e . TauSeriesft, T, m, False

. Τ Τ4t T, e e4t T fnosmootht, 1 fnosmootht, 2

,

354

Appendix

fnosmootht, 3 fnosmootht, 4 fnosmootht, 5 Out[5]=

0 1

Out[6]=

8 1 8 1 8 1 8

64 1 64

Π5 Τt5

Π5 Τt5

Π5 Τt5

13 Π7 Τt7 15 360 13 Π7 Τt7 15 360

41 Π9 Τt9 1 548 288

Expansions with smoothing procedure fsmootht, m : X Y e . TauSeriesft, T, m, True

. Τ Τ4t T, e e4t T fsmootht, fsmootht, fsmootht, fsmootht,

1 2 3 4

0 3

Out[12]=

8 3 Out[13]=

8 3 Out[14]=

8

In[15]:=

1

Π3 Τt3

Out[9]=

64

Π3 Τt3

Out[8]=

Out[11]=

1

Π3 Τt3

Out[7]=

In[10]:=

Π3 Τt3

Π3

Π3

Π3

3 Π3 8

Τt3 3 Τt3 3 Τt3 3

5 Π5 64

Τt5 5 Τt5 5 Τt5 5

3 Π3 8 3 Π3 8

91 Π7

Τt7

15 360

7

5 Π5

Τt5

64

5

5 Π5

Τt5

64

5

Τt7 7 Τt7 7

Τt9 9

Defining the basis functions 2 Π ΤΞ : ArcSin Sin Ξ

; Π 2 Π eΞ : Sign Cos Ξ

; 2

Appendix In[17]:=

355

Convergence with no smoothing

PlotEvaluateft, fnosmootht, 2, fnosmootht, 3, fnosmootht, 4, fnosmootht, 5, t, 0, 2 T, PlotStyle Thickness.005, Color Black, Thickness.002, Dashing0.01, 0.01, Color Black, Thickness.002,Color Black,Thickness.002,Color Black, Thickness.003, Color Black, AxesLabel "t", "x", PlotRange All x 4

2

Out[17]=

2

4

6

t

8

2

4 In[18]:=

Convergence with smoothing

PlotEvaluateft, fsmootht, 2, fsmootht, 3, fsmootht, 4, t, 0, 2 T, PlotStyle Thickness.005, Color Black, Thickness.002, Dashing0.01, 0.01, Color Black, Thickness.002, Color Black, Thickness.003, Color Black, AxesLabel "t", "x", PlotRange All x 1.5 1.0 0.5 Out[18]=

2 0.5 1.0 1.5

4

6

8

t

Appendix

357

APPENDIX 3: MathematicaR (Version 4) notebook. NSTT & Shooting method for periodic solutions of the oscillator d 2 x dt2

Ζ

dx dt

t

Ε x3 B e a .

px, v, t : Ζ v Ε x3 ; Π a ; 2Ω Substitution xt XΤt a YΤt aet a applied to the equation of motion: eqX

1 a2

Τ,Τ XΤ

1 2

pXΤ YΤ, eqY

1 a2

Τ,Τ YΤ

1 2

pXΤ YΤ,

pXΤ YΤ, 1 a

a

a

Τ XΤ Τ YΤ, a Τ

Τ XΤ Τ YΤ, 2 a a Τ

pXΤ YΤ, 1

1

1 a

0

Simplify;

Τ XΤ Τ YΤ, a Τ

Τ XΤ Τ YΤ, 2 a a Τ

B

Simplify;

Parameters: Ω 1.0; Ζ 0.05; Ε 1.0; B 7.4; Clearg, h; dg 3; dh 20; solg, h : NDSolveeqX, eqY, X1 g, X '1 0, Y1 0, Y '1 h, X, Y, Τ, 1, 1, MaxSteps Infinity; Xng, h : X '1 . solg, h1; Yng, h : Y1 . solg, h1; plx ContourPlotXng, h, g, dg, dg, h, dh, dh, Contours 0, ContourShading False, FrameLabel "g", "h", PlotPoints 100, RotateLabel False, DisplayFunction Identity; ply ContourPlotYng, h,g, dg, dg,h, dh, dh, Contours 0, ContourShading False, FrameLabel "g", "h", PlotPoints 100, RotateLabel False, ContourStyle Dashing0.01, 0.01, DisplayFunction Identity; pxy Showplx, ply, DisplayFunction $DisplayFunction;

358

Appendix

15

10

5

0 h -5

-10

-15

-20 -3

-2

-1

0 g

Magnified portion of the diagram:

1

2

3

Showpxy, PlotRange 0.3, 0, 5, 4;

-4

-4.2

-4.4 h -4.6

-4.8

-0.25

-0.2

-0.15 g

-0.1

-0.05

0

Appendix

359

Find one of the roots: g, h g, h . FindRootXng, h 0, Yng, h 0, g, 0.3, 0, h, 5, 4 0.113351, 4.51957

Check precision: sln NDSolveeqX, eqY, X1 g, X '1 0, Y1 0, Y '1 h, X, Y, Τ, 1, 1; Y1, X '1 . sln1 2.09795 109 , 1.47496 109 Introduce the sawtooth sine and the rectangular cosine: 2 Π Τt : ArcSin Sin t

; Π 2 Π et : Sign Cos t

; 2 Graphic output compared to solution of the related Cauchy problem: xt : XΤt a YΤt a et a . sln1; 1

Y 'Τt a X 'Τt a et a . sln1; a xxtt PlotEvaluatext, t, 0, 4 a, PlotRange All,

vt :

AxesLabel "t", "x", PlotStyle Thickness.009, TextStyle FontSize 14, DisplayFunction Identity; xxvv ParametricPlotEvaluatext, vt, t, 0, 4a, PlotRange All, AxesLabel "x", "v", Frame True, PlotStyle Thickness.009, TextStyle FontSize 14, DisplayFunction Identity; Cleary; Tmax 4a; dirsol NDSolve t,t yt Ζ t yt Ε yt3 B et a, y0 x0, y '0 v0, y, t, 0, Tmax, MaxSteps Infinity ; y y . dirsol1; yyvv ParametricPlotEvaluateyt, y 't, t, 0, Tmax, PlotRange All, AxesLabel "x", "v", PlotStyle Thickness.001, TextStyle FontSize 14, PlotPoints 500, Frame True, DisplayFunction Identity; ShowGraphicsArrayxxtt, xxvv, yyvv;

360

Appendix

x 3 2 1 t -1 -2 -3

1

2

3

4

5

6

v 4 2 0 -2 -4

x

-3 -2 -1 0

1

2

3

v 4 2 0 -2 -4

x

-3 -2 -1 0

1

2

3

Appendix

361

APPENDIX 4: MathematicaR notebook. Conducts NSTT of the T-periodic differential equation of the form t,t x f x, t x, t 0. In[1]:=

see examples below for the meaning of inputs

TauTransformequation, function, argument, period, basis : Modulesub, eqn, RE, IM, sub function XΤ YΤe,

argument function Y 'Τ X 'Τe a,

argument,argument function X ''Τ Y ''Τe a2 ; Q Partequation, 1 . sub; 1 eqn Q . t a Τ Q . t 2a a Τ 2

1 2

Q . t a Τ Q . t 2a a Τ e . a period 4;

EQNX Simplify EQNY Simplify subid XΤ

1 2 1 2 1 2

eqn . e 1 eqn . e 1 ; eqn . e 1 eqn . e 1 ; X Τ X Τ, YΤ

1

1 2 1

X Τ X Τ,

Τ X Τ Τ X Τ, Y 'Τ Τ X Τ Τ X Τ, 2 2 1 1 X ''Τ Τ,Τ X Τ X Τ, Y ''Τ Τ,Τ X Τ X Τ; 2 2 IDP SimplifyEQNX EQNY . subid; IDM SimplifyEQNX EQNY . subid; Ifbasis 1, EQNX 0, EQNY 0, Y1 0, Y1 0, X '1 0, X '1 0, IDP 0, IDM 0,X 1 X 1 0, X 'Τ

X 1 X 1 0, X '1 X '1 0, X '1 X '1 0

EXAMPLES In[2]:=

In[3]:=

Out[3]=

If basis 1, then the output is created in the standard basis 1,e, otherwise the output is in the idempotent basis e ,e 1 e 2,1e 2 TauTransformx ''t xt3 0, xt, t, 4a, 1

XΤ3 3 XΤ YΤ2

X Τ a2

0, 3 XΤ2 YΤ YΤ3

Y1 0, Y1 0, X 1 0, X 1 0

Y Τ a2

0,

362 In[4]:=

Out[4]=

Appendix TauTransformx ''t xt3 0, xt, t, 4a, 0 X Τ

X Τ3

a2

0, X Τ3

X Τ a2

0, X 1 X 1 0,

X 1 X 1 0, X 1 X 1 0, X 1 X 1 0 In[5]:=

Out[5]=

TauTransformx ''t xt3 P Sint 0, xt, t, 2Π, 0

P Sin

ΠΤ 2

X Τ3

4 X Τ Π2

0,P Sin

ΠΤ 2

X Τ3

4 X Τ Π2

0,

X 1 X 1 0, X 1 X 1 0, X 1 X 1

0, X 1 X 1 0 In[6]:=

Out[6]=

TauTransformx ''t xt3 P Sint 0, xt, t, 2Π, 1

P Sin YΤ3

In[7]:=

In[8]:=

Out[8]=

ΠΤ

XΤ3 3 XΤ YΤ2

2 4 Y Τ Π2

4 X Τ Π2

0, Y1 0, Y1 0, X 1 0, X 1 0

If present, the function ee4t T must be shown with no argument; derivatives de dt are not allowed by this code, but necessary generalizations are easy to implement TauTransformx ''t 1 Α e xt Β xt3 0, xt, t, 4a, 1

XΤ Β XΤ3 Α YΤ 3 Β XΤ YΤ2 Α XΤ YΤ 3 Β XΤ2 YΤ Β YΤ3 Y1 0, X 1 0, X 1 0

In[9]:=

Out[9]=

0, 3 XΤ2 YΤ

X Τ a2 Y Τ a2

0,

0, Y1 0,

TauTransformx ''t 1 Α e xt Β xt3 0, xt, t, 4a, 0

1 Α X Τ Β X Τ3

X Τ a2

X Τ a2

0, 1 Α X Τ Β X Τ3

0, X 1 X 1 0, X 1 X 1 0, X 1

X 1 0, X 1 X 1 0

PROJECT: Using the idempotent basis, find T-periodic solution of the linear oscillator under external and parametric rectangular wave periodic excitation of the period T=4,

Appendix

363 1

t,t x 1 2 etΩ2 x pet. In[10]:=

Formulates boundary value problem in the idempotent basis:

T 4; bvp TauTransformx ''t Ω2 1

Out[11]=

p

3 2

Ω2 X Τ X Τ 0, p

1

e xt p e 0, xt, t, T, 0

2 1

Ω2 X Τ X Τ 0,

2

X 1 X 1 0, X 1 X 1 0, X 1 X 1 0, X 1 X 1 0 In[12]:=

Solves the boundary value problem analytically: sol DSolvebvp, X Τ, X Τ, Τ

Simplify 2p

3 Cos

Ω 2

Out[12]=

3

Sin

2 Ω Cos

3

2 Τ Ω Sin

Ω 2

X Τ

, 3 Ω2

X Τ 2 p 3

3 Cos

Ω

3 Cos

2

Ω

Sin

3

Sin

2

2 3

3 Cos

Cos

In[13]:=

3

2 Ω 4 Cos

2 3 2

Ω

Ω Sin

Ω

Ω 4

3

2 Ω Sin

3 Cos

ΤΩ

Ω 2

Sin

2 3 Ω2

3 Cos

Ω

Sin

2

2 Ω Sin

3

2 Ω Cos

3 2 3 2

Ω

Ω

2

Extracting two components of the solution: X Τ X Τ . sol1; X Τ X Τ . sol1;

In[15]:=

In[17]:=

Back to the original xt variables: 1 1 x X Τ 1 e X Τ 1 e . Τ Τt, e et; 2 2 1 1 1 v Τ X Τ 1 e Τ X Τ 1 e .Τ Τt,e et; T 4 2 2 Basis functions:

364

Appendix Τt :

2 Π

ArcSin Sin

et : Sign Cos

Π 2

In[19]:=

Π 2

t

;

t

;

Parameters and graphic output: Ω 20.0; p 1.0; PlotEvaluatex, t, 0, 8, PlotRange All, AxesLabel "t", "x ", PlotStyle Thickness.005

x 0.010

0.005

Out[21]=

2

4

6

8

t

0.005 0.010

In[22]:=

Validating the code by durect numerical solution: x0 x . t 0; v0 v . t 0; 1 dirsol NDSolvey ''t Ω2 1

e t ytp et 0,y0 x0, 2 y '0 v0, y, t, 0, 50, MaxSteps Infinity ; y y . dirsol1;

In[26]:=

PlotEvaluateyt, t, 0, 8, PlotRange All, AxesLabel "x", "v", PlotStyle Thickness.003 v 0.010

0.005

Out[26]=

2 0.005 0.010

4

6

8

x

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