Nonlinear Approaches in Engineering Applications
Liming Dai • Reza N. Jazar Editors
Nonlinear Approaches in Engineering Applications
123
Editors Liming Dai Industrial Systems Engineering University of Regina Regina, SK, Canada
[email protected] Reza N. Jazar School of Aerospace, Mechanical, and Manufacturing Engineering RMIT University Melbourne, VIC, Australia
[email protected] ISBN 978-1-4614-1468-1 e-ISBN 978-1-4614-1469-8 DOI 10.1007/978-1-4614-1469-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011942901 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to our wives Xinming and Mojgan
Being smart means: do not keep your smartness at the level that is needed for your job
Preface
The initiation of this book started with the discussions in the ASME 2010 and 2009 Congress in the track of Dynamics Systems and Control, Optimal Approaches in Nonlinear Dynamics and Acoustics, which were organized by the editors. Since the 1980s, many new nonlinear approaches and techniques have been developed and the research on nonlinear science and dynamics have brought new insights in our dealing with nonlinear and complex systems in nature, engineering, and society. Although more and more such approaches and techniques have been brought into research and engineering practices, linearization and simplification are still the dominating approaches existing in physics and engineering. Strictly and precisely speaking, our nature is nonlinear. All the phenomena in nature and all the responses of physics and engineering systems are nonlinear. Linearization means simplification, and it damages the original natural or characteristics of the systems and may lead to inaccuracy, misunderstanding, or even incorrect conclusions in comprehending the physics and engineering systems we are trying to manage. Hooke’s law including the generalized Hooke’s law, for example, is linear and it composes the foundation of linear elasticity and dominates numerous solutions of physical systems and mechanical designs. However, no material is perfectly linear. Although most plain carbon steels are considered as materials obey Hooke’s law in engineering practice, for instance, it is a common knowledge to the material scientists these materials are nonlinear and the constitutive relations of the materials are close to a nonlinear polynomial of third or fifth order. Any single material used in the real world can actually be a nonlinear and complex system, not only due to its material or structure nonlinearity but also due to the inhomogeneousness and anisotropy of the materials, let alone the associated complex external loading and large deformation which may occur in many engineering cases. With the explosively growing body of knowledge and discoveries in nonlinear science, obviously, we are entering an era that nonlinearity and complexity must be taken into consideration in coping with physical and engineering systems in real world. Another challenge facing the scientists and engineers in our time is the generation of the solutions and characterization of the nonlinear systems modeled from the physical systems in reality. It would be greatly beneficial in accurately ix
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evaluating the behavior of nonlinear systems and revealing the actual nature of the systems, with utilization of the existing mathematical tools and analytical means, if the analytical solutions of nonlinear systems could be pursued. Due to the nonlinearity and complexity of the nonlinear systems, unfortunately, it is very difficult or impossible to derive the analytical or closed-loop solutions for the systems. In solving or simulating the nonlinear systems, one may have to rely on approximate or numerical methods, which may only provide approximate results for the systems while errors are unavoidable during the processes of generating the approximate results. Approximation and inaccuracy are the inescapable shadows following the current research and engineering practices involving nonlinearity or nonlinear systems. In the role of the editors as well as the chapter contributors of this book, we have tried to present a collection of chapters showing the theoretically and practically sound nonlinear approaches and their engineering applications in various areas, hoping that this book may provide useful tools and comprehensible examples of solving, modeling, and simulating the nonlinear systems existing in the real world. The carefully selected chapters contained in the present book reflect recent advances in nonlinear approaches and their engineering applications. The book intends to feature in particular the fundamental concepts and approaches of nonlinear science and their applications in engineering and physics fields. It is anticipated that this book may help to promote the development of nonlinear science and nonlinear dynamics in engineering, as well as to stimulate research and applications of nonlinear science and nonlinear dynamics in physics and engineering practices. It is also expected that the book will further enhance the comprehension of nonlinear science and stimulate interactions among scientists and engineers who are interested in nonlinear science and who find that nonlinearity and complexity of systems play an important role in their respective fields. With the theme of the book, nonlinear approaches and engineering applications, the book covers interdisciplinary studies on theories and methods of nonlinear science and their applications in complex systems such as those in nonlinear dynamics, nanotechnology, fluid dynamics, aerospace structure engineering, mechatronics engineering, control engineering, ocean engineering, offshore structure engineering, mechanical engineering, human body dynamics, and material science. Examples include innovative methodology of diagnosing nonlinear characteristics; approach of modeling squeeze-film phenomena in MEM S ; nonlinear modeling of microbeams; new and explicit equations of motion for general nonlinear constrained mechanical systems; study on nonlinear dynamic behaviors of a Fermi oscillator with two periodic excitations; development of active surface control (ASC) architecture for deployable mesh reflectors to be used for satellite communications; nonlinear finite elements approach for modeling nano- and macroscale beam-like materials and structures; approach of modeling a submerged rigid body supported by slack moorings; study on linear and nonlinear viscoelastic materials in terms of constitutive equations, stress relaxation, and strain rate dependency; investigation on the nonlinear hysteresis effect in electromechanical brakes; energy conservative design and nonlinear control of a hopping robot; study on nonlinear and complex
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flow around a cylinder; vibration of truncated conical shells; image-based pose estimation of quadrotor unmanned aerial vehicle (UAV); and nonlinear modeling of head-neck complex system.
Level of the Book This book aims at engineers, scientists, researchers, engineering and physics students of undergraduate and graduate levels, together with the interested individuals in engineering, physics, and mathematics. This chapter-book focuses on application of the nonlinear approaches representing a wide spectrum of disciplines of engineering and science. Throughout the book, great emphases are placed on engineering applications, physical meaning of the nonlinear systems, and methodologies of the approaches in analyzing and solving for the systems. Topics that have been selected are of high interest in engineering and physics. An attempt has been made to expose the engineers and researchers to a broad range of practical topics and approaches. The topics contained in the present book are of specific interest to engineers who are seeking expertise in nonlinear analysis, mathematical modeling of complex systems, optimization of nonlinear systems, nonclassical engineering problems, and future of engineering. The primary audience of this book is the researchers, graduate students and engineers in mechanical engineering, engineering mechanics, civil engineering, aerospace engineering, ocean engineering, mathematics, and science disciplines. In particular, the book can be used as a textbook for the graduate students as well as senior undergraduate students to enhance their knowledge by taking a graduate or advanced undergraduate course in the areas of nonlinear science, dynamics and vibration of continuous system, structure dynamics, and engineering applications of nonlinear science. It can also be utilized as a guide to the readers’ fulfillment in practices. The covered topics are also of interest to engineers who are seeking to expand their expertise in these areas.
Organization of the Book The main structure of the book consists of 15 chapters. Each of the chapters covers an independent topic along the line of nonlinear approach and engineering applications of nonlinear science. The main concepts in nonlinear science and engineering applications are explained fully with necessary derivatives in details. The book and each of the chapters are intended to be organized as essentially self-contained. All necessary concepts, proofs, mathematical background, solutions, methodologies, and references are supplied except for some fundamental knowledge well-known in the general fields of engineering and physics. The readers may
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therefore gain the main concepts of each chapter with as less as possible the need to refer to the concepts of the other chapters. Readers may hence start to read one or more chapters of the book for their own interests.
Method of Presentation The scope of each chapter is clearly outlined and the governing equations are derived with an adequate explanation of the procedures. The covered topics are logically and completely presented without unnecessary overemphasis. The topics are presented in a book form rather than in the style of a handbook. Tables, charts, equations, and references are used in abundance. Proofs and derivations are emphasized in such a way that they can be straightforwardly followed by the readers with fundamental knowledge of engineering science and university physics. The physical model and final results provided in the chapters are accompanied with necessary illustrations and interpretations. Specific information that is required in carrying out the detailed theoretical concepts and modelling processes has been stressed.
Prerequisites The present book is primarily intended for researchers, engineers, and graduate students, so the assumption is that the readers are familiar with the fundamentals of dynamics, calculus, and differential equations associated with dynamics in engineering and physics, as well as a basic knowledge of linear algebra and numerical methods. The presented topics are given in a way to establish as conceptual framework that enables the readers to pursue further advances in the field. Although the governing equations and modelling methodologies will be derived with adequate explanations of the procedures, it is assumed that the readers have a working knowledge of dynamics, university mathematics, and physics together with theory of linear elasticity.
Acknowledgments This book is made available under the close and effective collaborations of all the enthusiastic chapter contributors who have the expertise and experience in various disciplines of nonlinear science and engineering applications. They deserve sincere gratitude for the motivation of creating such book, encouragement in completing the book, scientific and professional attitude in constructing each of the chapters of the book, and the continuous efforts toward improving the quality of the book. Without the collaboration and consistent efforts of the chapter contributors, the completion
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of this book would have been impossible. What we have at the end is a book that we have every reason to be proud of. It has been gratifying to work with the staff of Spinger-Verlag through the development of this book. The assistances provided by the staff members have been valuable and efficient. We thank Spinger-Verlag for their production of an elegant book. Regina, SK, Canada Melbourne, VIC, Australia
Liming Dai Reza N. Jazar
Contents
1
Characterizing Nonlinear Dynamic Systems . . . . . . . .. . . . . . . . . . . . . . . . . . . . Liming Dai and Lu Han
1
2
Nonlinear Modeling of Squeeze-Film Phenomena . .. . . . . . . . . . . . . . . . . . . . Reza N. Jazar
41
3
Nonlinear Mathematical Modeling of Microbeam MEMS . . . . . . . . . . . . Reza N. Jazar
69
4
Complex Motions in a Fermi Oscillator .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105 Yu Guo and Albert C.J. Luo
5
Nonlinear Visco-Elastic Materials . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135 Franz Konstantin Fuss
6
Nonlinear Dynamic Modeling of Nano and Macroscale Systems . . . . . 171 Michael J. Leamy
7
Equilibrium of a Submerged Body with Slack Mooring . . . . . . . . . . . . . . . 211 Brian C. Fabien
8
Nonlinear Deployable Mesh Reflectors.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237 Hang Shi and Bingen Yang
9
Nonlinearity in an Electromechanical Braking System . . . . . . . . . . . . . . . . 265 Reza Hoseinnezhad and Alireza Bab-Hadiashar
10 Nonlinear Dynamics of Incompressible Flow. . . . . . . .. . . . . . . . . . . . . . . . . . . . 283 Jiazhong Zhang 11 Explicit Equation of Motion of Constrained Systems . . . . . . . . . . . . . . . . . . 315 Firdaus E. Udwadia and Thanapat Wanichanon 12 Nonlinear Dynamic of a Rotating Truncated Conical Shell . . . . . . . . . . . 349 Changping Chen
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13 Nonlinear Real-Time Pose Estimation of Quadrotor UAV . . . . . . . . . . . . . 393 Chayatat Ratanasawanya, Mehran Mehrandezh, and Raman Paranjape 14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 421 Klaus Nji and Mehran Mehrandezh 15 Nonlinearities in Human Body Dynamics . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 475 M. Fard, Y. Ohtaki, T. Ishihara, and H. Inooka Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 525
List of Figures
Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 1.12 Fig. 1.13 Fig. 1.14 Fig. 1.15 Fig. 1.16 Fig. 1.17 Fig. 1.18 Fig. 1.19 Fig. 1.20 Fig. 1.21 Fig. 1.22
Procedures of Periodicity Ratio determination . .. . . . . . . . . . . . . . . . . . . . Procedures of statistical hypothesis testing . . . . . .. . . . . . . . . . . . . . . . . . . . Bar chart for the distribution of different categories of motion under different given overlapping boundaries .. . . . . . . . . . . . . . Poincare map at k D 0:6 and B D 10:6 . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Phase diagram at k D 0:6 and B D 10:6 . . . . . . . .. . . . . . . . . . . . . . . . . . . . Poincare map at k D 0:15 and B D 7:2 . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Phase diagram at k D 0:15 and B D 7:2 . . . . . . . .. . . . . . . . . . . . . . . . . . . . Poincare map at k D 0 and B D 7 . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Quasiperiodic phase diagram at k D 0 and B D 7 .. . . . . . . . . . . . . . . . . Poincare map at k D 0:81 and B D 0:02 .. . . . . . .. . . . . . . . . . . . . . . . . . . . Phase diagram at k D 0:81 and B D 0:02 . . . . . . .. . . . . . . . . . . . . . . . . . . . Poincare map of a nonperiodic case at k D 0:37 and B D 8:9.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Phase diagram at k D 0:37 and B D 8:9 . . . . . . . .. . . . . . . . . . . . . . . . . . . . Poincare map of a nonperiodic motion at k D 0:09 and B D 11:9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Poincare map of a nonperiodic motion at k D 0:09 and B D 11:9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Poincare map at k D 0:68 and B D 5:3 . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Phase diagram at k D 0:68 and B D 5:3 . . . . . . . .. . . . . . . . . . . . . . . . . . . . Critical region diagram of binomial distribution . . . . . . . . . . . . . . . . . . . . Periodic–quasiperiodic–chaotic diagram for Duffing system under given overlapping boundary of 107 . . . . . . . . . . . . . . . . . . Periodic–quasiperiodic–chaotic diagram for Duffing system under given overlapping boundary 103 . . . . . . . . . . . . . . . . . . . . . Periodic–quasiperiodic–chaotic diagram for Duffing system under given overlapping boundary of 108 . . . . . . . . . . . . . . . . . . Periodicities of the responses of Duffing system under the given overlapping boundary of 107 . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10 11 17 21 22 22 23 23 23 24 24 25 25 27 27 28 28 30 32 33 34 34 xvii
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List of Figures
Fig. 1.23 Three-dimensional periodic–quasiperiodic–chaotic diagram for Duffing system . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 1.24 Phase diagram at k D 0:17, B D 10, and PRD 1 . . . . . . . . . . . . . . . . . . . Fig. 1.25 Phase diagram at k D 0:16, B D 10, and PRD 0:3127 .. . . . . . . . . . . . Fig. 1.26 Phase diagram at k D 0:15, B D 10, and PRD 0:0403 .. . . . . . . . . . . . Fig. 1.27 Phase diagram at k D 0:14, B D 10, and PRD 0:0 .. . . . . . . . . . . . . . . . Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16
Fig. 3.1 Fig. 3.2 Fig. 3.3
A microcantilever resonator and resistance of the pressurized gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A microcantilever and a clamed-clamped model of microresonator to keep the capacitor’s plates parallel .. . . . . . . . . . . . . . Effect of variation of polarization voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of polarization voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of polarization voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of polarization voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of excitation voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of excitation voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of excitation voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of excitation voltage on frequency response of the microbeam . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of squeeze-film damping on frequency response of the microbeam . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of squeeze-film damping on frequency response of the microbeam . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of squeeze-film damping on frequency response of the microbeam . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of variation of squeeze-film damping on frequency response of the microbeam . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients . . . . Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients . . . . A clamped– clamped model of a MEMS and its voltage connections .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A clamped– clamped model of a MEMS to keep the capacitor’s plates parallel .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Various examples of connecting a microbeam to the ground . . . . . . .
35 36 36 36 37
43 46 56 57 57 58 58 59 59 60 60 61 61 62 62 63
73 73 74
List of Figures
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Fig. 3.4 Fig. 3.5 Fig. 3.6
74 83
Alternative examples of connecting a microbeam to the ground . . . A microcantilever-based MEMS . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Effect of the variation of excitation and polarization voltages on peak amplitude at resonance . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.7 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.8 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.9 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.10 Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients . . . . Fig. 3.11 Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients . . . . Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4
Fig. 4.5
Fig. 4.6
Fig. 4.7
Fig. 4.8
Fig. 4.9
Mechanical model of Fermi oscillator . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Absolute domains and boundaries without stick: (a) Bottom oscillator, (b) top oscillator, and (c) particle .. . . . . . . . . . . Absolute domains and boundaries with stick: (a) bottom oscillator, (b) top oscillator, and (c) particle .. . . .. . . . . . . . . . . . . . . . . . . . Domains and boundaries definition relative to the bottom oscillator: (a) .z; zP/-plane for bottom oscillator, (b) .z; zP/-plane for bottom oscillator, (c) -plane for particle, and (d) .z; zP/-plane for particle . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Domains and boundaries definition relative to the top oscillator: (a) -plane for top oscillator, (b) .z; zP/-plane for top oscillator, (c) .z; zP/-plane for particle, and (d) .z; zP/-plane for particle .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Switching sets and generic mappings for nonstick motion in absolute coordinates: (a) bottom oscillator, (b) top oscillator, and (c) particle . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Switching sets and generic mappings for stick motion in absolute coordinates: (a) bottom oscillator, (b) top oscillator, and (c) particle .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Bifurcation scenario of varying restitution coefficient: (a) displacement of particle, (b) velocity of particle, and (c) switching phase. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1) .. . . . . . . . . . . . . . . . . . . Analytical prediction of varying the restitution coefficient of impact: (a) switching displacement of particle, (b) switching velocity of particle; (c) switching phase. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1) . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
95 95 96 96 99 99 106 107 109
110
111
117
118
123
124
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List of Figures
Fig. 4.10 Zoomed Analytical prediction of varying the restitution coefficient of impact: (a) switching displacement of particle, (b) switching velocity of particle; (c) switching phase. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1) . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.11 Zoomed Analytical prediction of varying the restitution coefficient of impact: (a) real part of eigenvalues, (b) imaginary part of eigenvalues, and (c) magnitude of eigenvalues. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1) . . . . . . . . . . . . . . . . . . . . Fig. 4.12 Analytical prediction of varying the restitution coefficient of impact: (a) real part of eigenvalues, (b) imaginary part of eigenvalues, and (c) magnitude of eigenvalues. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1) . . . . . . . . . . . . . . . . . . . . Fig. 4.13 Periodic motion: (a) displacement time history, (b) velocity time history, and (c) trajectory of particle with moving boundaries. (Q.1/ D Q.2/ D 12:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, e .1/ D e .2/ D 0:1, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1 ). .1/ .3/ Initial conditions: x0 D x0 D 0:5575740, .1/ .3/ .2/ xP 0 D xP 0 D 1:5959726, x0 D 0:0613454, .2/ xP 0 D 1:6725197 for t0 D 0:0227104 . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.14 Periodic motion: (a) acceleration along displacement, (b) acceleration time history and (c) jerk time history. (Q.1/ D Q.2/ D 12:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, e .1/ D e .2/ D 0:1, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1). .1/ .3/ Initial conditions: x0 D x0 D 0:5575740, .1/ .3/ .2/ xP 0 D xP 0 D 1:5959726, x0 D 0:0613454, .2/ xP 0 D 1:6725197 for t0 D 0:0227104 . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
125
126
128
129
130
List of Figures
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Fig. 4.15 Chaotic motion: (a) displacement time history, (b)velocity time history, (c) Poincare map .3/ .3/ of xk ; yk . (Q.1/ D Q.2/ D 20:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1 ). The initial conditions .1/ .1/ are x0 D 0:941968193, xP 0 D 1:86109613, .2/ .3/ .2/ x0 D x0 D 1:45709118, xP 0 D 1:82722043, and .3/ xP 0 D 10:7134817 for t0 D 0:289711605 .. . . . . ... . . . . . . . . . . . . . . . . . . 131 .4/
.4/
Fig. 4.16 Chaotic motion: (a) Poincare map of xk ; yk , .3/ (b) Poincare map of xk mod .th ; 2h/ , .3/ and (c) Poincare map of yk mod .th ; 2h/ . (Q.1/ D Q.2/ D 20:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1). The .1/ initial conditions are x0 D 0:941968193, .1/ .2/ .3/ xP 0 D 1:86109613, x0 D x0 D 1:45709118, .2/ .3/ xP 0 D 1:82722043, and xP 0 D 10:7134817 for t0 D 0:289711605 132 .1/ Fig. 4.17 Parameter map ( D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1) . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133 Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5
Linear visco-elastic models; (a) Zener model (standard linear solid of Voight form), (b) standard linear solid of Maxwell form, (c) Wiechert model (generalised Maxwell model, note that the first damper 1 is rigid) .. . . . . . . . . . . . . Stress relaxations of power and logarithmic models, as well as of Maxwell standard linear solid (MSLS) on a graph with linear axes.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stress relaxations of power and logarithmic models, as well as of Maxwell standard linear solid (MSLS) on a single-log graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stress relaxations of power and logarithmic models, as well as of Maxwell standard linear solid (MSLS) on a double-log graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ramp strain followed by constant strain. 4t D time period of the ramp segment, "0 D maximal strain applied .. . . . . . . . .
137
141
141
142 144
xxii
Fig. 5.6
Fig. 5.7
Fig. 5.8
Fig. 5.9
Fig. 5.10 Fig. 5.11
Fig. 5.12
Fig. 5.13
Fig. 5.14
Fig. 5.15
List of Figures
Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a single-log graph; bold curves: relaxation after applying a Heaviside strain function, dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a single-log graph; dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a double-log graph; bold curves: relaxation after applying a Heaviside strain function, dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a double-log graph; dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Stress relaxations of power model and Wiechert model (Prony series 1, (5.36)–(5.39)) after Heaviside strain function . . . . . Stress relaxations of power model and Wiechert model (Prony series 2, (5.42)–(5.45)) after applying a ramp and constant strain according to Fig. 5.5 and (5.31); D stress increasing with ramp strain . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Power model (R D 10; D 0:3) at different strain rates (1:667s1 ; 2:5s1 ; 5s1 ): stress against strain (loading and unloading segment; three stress values at 0:4 strain are marked) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Power model (R D 10; D 0:3) at different strain rates (1:667s 1; 2:5s 1 ; 5s 1 ); modulus against strain (loading only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Power model (R D 10; D 0:3) at different strain rates (1:667s1 ; 2:5s1 ; 5s1 ); three log modulus values at 0.4 strain against ln "P0 =", linear fit with intercept at ln 10 and gradient of 0:3 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Calculation of parameters from Fig. 5.13; elasticity constant R against stress. Solid curve: running average filter; sudden increase of noise between strain 0:35 and 0:4 is due to the numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri) .. . . . . . . . . . . . . . .
145
146
147
148 149
150
156
156
157
157
List of Figures
Fig. 5.16 Calculation of parameters from Fig. 5.13; viscosity constant against strain. Solid curve: running average filter; sudden increase of noise between strain 0:35 and 0:4 is due to the numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri) .. . . . . . . . . . . . . . . Fig. 5.17 Calculation of parameters from Fig. 5.13; Difference of correlation coefficients of log fit (rL ) and power fit (rP ), positive difference indicates better fit of power function; solid curve: running average filter; sudden increase of noise between strain 0:35 and 0:4 is due to the numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri).. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.18 Determination of R and ; TP 930 (rapid prototyping material), stress relaxation after initial ramp strain at a strain rate of 0:26s1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.19 Determination of R and ; DM 9750 (rapid prototyping material) modulus data obtained at three different strain rates (noisy modulus data were intentionally not filtered) . . . . . . . . . . Fig. 5.20 R data of DM 9750 (rapid prototyping material); horizontal line: R constant from Fig. 5.19; dots: intercept from sliding window filter with window width 9; bold grey curve: R after applying running average filter of window width 33 .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.21 data of DM 9750 (rapid prototyping material); horizontal line: constants from Fig. 5.19; dots: gradient from sliding window filter with window width 9; bold grey curve: after applying running average filter of window width 33 .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.22 Properties of rapid prototyping materials; R against hardness Shore D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.23 Properties of rapid prototyping materials; against hardness Shore D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.24 Properties of rapid prototyping materials; against R .. . . . . . . . . . . . . Fig. 5.25 Properties of rapid prototyping materials; R= against hardness Shore D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.26 Properties of a cricket ball with five cork layers; (a) R and against deflection .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.27 Properties of a cricket ball with five cork layers; stiffness against deflection, the black solid lines show the experimental data, and the gray circles indicate the stiffness recalculated from R, , deflection and deflection rate with (5.77) . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
xxiii
158
159
160
161
161
162 163 163 164 164 165
165
xxiv
List of Figures
Fig. 5.28 Properties of a cricket ball with five cork layers; force against deflection, the black solid lines show the experimental data, the black dashed lines (“1”) indicate the force calculated from (5.79), and the grey dashed lines (“2”) represent the numerical integration of the stiffness recalculated from (5.77) . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 166 Fig. 6.1
Geometry of the intrinsic beam in the undeformed and deformed configurations .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.2 Small differential element used to derive a relative position vector in terms of the intrinsic metrics net curvature and strain fl. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.3 Comparisons of strain histories predicted by an ABAQUS beam (circular markers) and the intrinsic beam (lines) finite element models for the case of simultaneous twisting and bending loads . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.4 Comparisons of bending curvature histories predicted by an ABAQUS beam (circular markers) and the intrinsic beam (lines) finite element models for the case of simultaneous twisting and bending loads . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.5 Comparisons of torsion histories predicted by an ABAQUS beam (a) and the intrinsic beam (b) finite element models for the case of simultaneous twisting and bending loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.6 Geometry of the graphene sheet; representative volume element; final nanotube configuration .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.7 Snapshots of the dynamic response of a (10; 10) armchair nanotube to a follower end load spaced at intervals of 5 103 time units (a) for the loading phase starting at time 0:0 and ending at 8:0 104 and (b) the unloading phase starting at time 8:0 104 and ending at time 1:5 104 . Note that the force is rendered only twice in (a) for illustration purposes, but is present at each snapshot .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.8 Configurations employed in developing the helical spring model. Material points along a straight reference configuration ref are mapped to an initial configuration 0 via initial curvature and strain K0 , 0 . Similar mappings hold for a deformed configuration f . The latter two configurations are related by mappings which invoke net curvature and strain, KO , O . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.9 FFT of KO 2 at node 30 – example fixed–fixed spring discretized using 48 elements (97 nodes) . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.10 Modal convergence with number of intrinsic finite elements used for first example spring – longitudinal modes .. . . . . .
172
174
183
184
185 186
191
193 198 198
List of Figures
Fig. 6.11 Modal convergence with number of intrinsic finite elements used for first example spring – transverse modes . . . . . . . . . Fig. 6.12 FFT of KO 2 at node 30 – example fixed-free spring discretized using 48 elements (97 nodes) . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.13 Initial spring configurations resulting from application of moderate-magnitude impulsive longitudinal loading. Ensuing motion is on the same order of magnitude as the configurations depicted . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.14 FFT of KO 2 at node 30 following application of moderate loading – example fixed-free spring discretized using 48 elements (97 nodes) . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.15 Time histories of curvature KO 2 at node 30 following application of harmonic loading – example fixed-free spring discretized using 48 elements (97 nodes). Two loading cases are depicted: 52:43 Hz, which is a potential secondary resonance, and 60:0 Hz where no resonances are expected . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.16 Time histories of curvature KO 2 at node 30 following application of harmonic loading – example fixed-free spring discretized using 48 elements (97 nodes). Three loading cases are depicted: 52:43 Hz, 60:0 Hz, and 63:1 Hz . . . . . . . . Fig. 6.17 Initial spring configurations resulting from application of large-magnitude impulsive transverse loading .. . . . . . . . . . . . . . . . . . . Fig. 6.18 FFT of KO 2 at node 30 following application of large-amplitude loading.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
xxv
199 200
201
201
202
202 203 204
Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7 Fig. 7.8 Fig. 7.9 Fig. 7.10 Fig. 7.11 Fig. 7.12 Fig. 7.13 Fig. 7.14
Slack mooring system . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Coordinate transformation .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Mooring line.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Model node .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Mooring line network .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Rigid body force/torques . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Unit vectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Redundant line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Redundant line equilibrium position . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Case 1: No stream velocity . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Case 2: Stream velocity vN D Œ0; Z 1:4 ; 0T . . . . . . . .. . . . . . . . . . . . . . . . . . . .
212 214 214 215 215 216 218 219 219 220 232 232 233 234
Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4
Configuration of a deployable mesh reflector.. . .. . . . . . . . . . . . . . . . . . . . 3D truss model of mesh reflectors .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A truss element in undeformed and deformed configurations . . . . . . Nodal displacements and force balance in local coordinate . . . . . . . .
238 238 244 245
xxvi
List of Figures
Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 Fig. 8.10 Fig. 8.11 Fig. 8.12 Fig. 8.13 Fig. 8.14 Fig. 8.15 Fig. 8.16 Fig. 8.17 Fig. 8.18
An example of deployable mesh reflector . . . . . . .. . . . . . . . . . . . . . . . . . . . Optimal vertical external loads .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Optimal original length of members . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Initial shape of the mesh reflector .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Actual deformed shape of the mesh reflector . . . .. . . . . . . . . . . . . . . . . . . . Element tensions at the deformed configuration.. . . . . . . . . . . . . . . . . . . . 1st mode of the mesh reflector: side-wing mode (x direction).. . . . . 2nd mode of the mesh reflector: side-wing mode (y direction).. . . . 3rd mode of the mesh reflector: up-and-down mode . . . . . . . . . . . . . . . . 4th mode of the mesh reflector: breathing mode . . . . . . . . . . . . . . . . . . . . Impact of thermal effects on natural frequencies .. . . . . . . . . . . . . . . . . . . Impact of external load on natural frequencies . .. . . . . . . . . . . . . . . . . . . . R & D progress of ASC architecture .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3D plots of the designate shape for the 835-node model .. . . . . . . . . . .
254 255 255 256 256 257 258 258 258 259 260 260 261 261
Fig. 9.1 Fig. 9.2
General structure of a brake-by-wire system . . . .. . . . . . . . . . . . . . . . . . . . Block diagram of the e-caliper control system in a typical EMB design .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The diagram of the electromechanical brake caliper.1 stator field winding, 2 brake pads, 3 ball-screw, 4 planetary gear-set, 5 thrust bearing, 6 clamp force sensor location, 7 resolver location, 8 permanent rotor magnet location, 9 load distribution plate, 10 nut, 11 caliper bridge .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Arrangement of the strain gauges in each load cell around the load sleeve and Strain gauge electrical circuitry . . . . . . . . A hysteretic behaviour is observed when the clamp force measurements are plotted versus actuator displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A picture of the ball-screw and the load sleeve with the key on it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Modelling the hysteretic slippage as a parallel combination of saturating elasto-slip elements .. . . . . . . . . . . . . . . . . . . . Stiffness curves of the spring elements in the model of Fig. 9.7 .. . . Modelling the hysteretic slippage as a parallel combination of saturating elasto-slip elements . .. . . . . . . . . . . . . . . . . . . . An example of characteristic curve parameter updating upon the termination of a local hysteresis cycle and start of the next cycle .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A flowchart for the proposed automatic calibration technique using the hysteresis model (9.5) . . . . . .. . . . . . . . . . . . . . . . . . . . A picture of the experimental setup . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Position of the actuator during the experiment . .. . . . . . . . . . . . . . . . . . . . Variation of characteristic curve .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
266
Fig. 9.3
Fig. 9.4 Fig. 9.5
Fig. 9.6 Fig. 9.7 Fig. 9.8 Fig. 9.9 Fig. 9.10
Fig. 9.11 Fig. 9.12 Fig. 9.13 Fig. 9.14
267
268 269
270 270 272 272 273
276 277 278 279 279
List of Figures
xxvii
Fig. 9.15 Clamp force measurements by the external and internal sensors during the experiment: The internal sensor measurement has been corrected by calculating and removing its hysteresis part .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 280 Fig. 9.16 Error of the force estimates with respect to their true values . . . . . . . 280 Fig. 10.1 Fig. 10.2 Fig. 10.3 Fig. 10.4 Fig. 10.5
Fig. 10.6 Fig. 10.7
Fig. 10.8 Fig. 10.9 Fig. 10.10 Fig. 10.11 Fig. 10.12
Fig. 10.13 Fig. 10.14 Fig. 10.15 Fig. 10.16 Fig. 10.17
Fig. 10.18 Fig. 10.19
Schematic diagram of flow around circular cylinder . . . . . . . . . . . . . . . . Schematic diagram of characteristic . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Wake patterns at low Re number . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The relation between Re and St (Henderson 1997) .. . . . . . . . . . . . . . . . . Saddle point (left) and center (right) in two-dimensional incompressible flows (Bisgaard 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The cusp bifurcation, a loop is appearing or disappearing in the flow . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Homoclinic connection of a saddle point (left, structurally stable) and heteroclinic connection (right, structurally unstable) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Streamline topology bifurcation scenarios at Re D 100 (Bisgaard 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Vortex shedding in a period at Re D 100, and the time has been nondimensionalized . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Vortex shedding in a period at Re D 100, and the time has been nondimensionalized . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Two dimensional steady flow separation .. . . . . . . .. . . . . . . . . . . . . . . . . . . . Movements of unstable manifold in one period (T D 1). (a) Flow with slip boundary conditions. (b) Flow with nonslip boundary conditions .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . Pressure contour of the mean flow . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Pressure contour of POD modes . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Bifurcation diagram of the system with respect to the Reynolds number.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Eigenvalues of the Jacobian matrix for different Reynolds number.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Time history of first mode for different mode number. (a) Re D 20, (b) Re D 100, (c) Re D 200, (d) Re D 200 (enlarge) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A schematic of the curve of neutral stability . . . .. . . . . . . . . . . . . . . . . . . . Curves of neutral stability for laminar-boundary profiles with different shape factors, after Schlichting and Ulrich (Schlichting and Gersten 2000) .. . . . .. . . . . . . . . . . . . . . . . . . .
284 286 291 291
293 294
294 295 296 297 298
300 303 304 305 305
306 310
311
Fig. 11.1 A wheel rolling down an inclined plane under gravity .. . . . . . . . . . . . . 333 Fig. 11.2 A two degree-of-freedom multi-body system . . .. . . . . . . . . . . . . . . . . . . . 335
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Fig. 11.3 Decomposition of the multi-body system shown in Fig. 11.2 using more than two coordinates.. . . . . .. . . . . . . . . . . . . . . . . . . . 337 Fig. 12.1 An element on the middle surface of shell in a relative coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.2 Geometry of truncated conical shell . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.3 Comparison of nonlinear frequency of simply supported cylindrical shell (˝1 D 0, R=L D 1:0, R= h D 25, ' D 0, n D 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.4 Effects of angle velocity on nonlinear frequency of simply supported truncated conical shell (R=L D 1:0, R= h D 25, ' D 30, n D 3) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.5 Effects of boundary conditions on nonlinear frequency of truncated conical shell (˝1 D 1, R=L D 0:5, R= h D 20, ' D 30, n D 3) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.6 Effect of radius-to-length ratio on nonlinear frequency of clamped truncated conical shell (˝1 D 1, R= h D 20, ' D 30, n D 3) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.7 Effect of semi-vertex angle on nonlinear frequency of simply supported truncated conical shell (˝1 D 1, R=L D 0:5, R= h D 20, n D 3).. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.8 Physical relationship of the truncated conical shallow shell . . . . . . . . Fig. 12.9 (a) Time history curve. (b) Phase plane curve. (c) Poincare map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.10 (a) Time history curve. (b) Phase plane curve. (c) Poincare map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.11 (a) Time history curve. (b) Phase plane curve. (c) Poincare map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.12 Amplitude frequency response curves of a truncated conical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.13 Main dynamic area of instability of a truncated conical shell . . . . . . Fig. 12.14 Internal response between the (1,1) modal and (1,3) modal on the transverse direction of the rotary shell .. . . . . . . . . . . . . . . Fig. 12.15 Internal response between the (1,1) modal and (1,3) modal on the circle direction of the rotary shell .. . . . . . . . . . . . . . . . . . . . Fig. 12.16 Internal response between the (1,1) modal and (1,3) modal on the longitudinal direction of the rotary shell .. . . . . . . . . . . . . Fig. 12.17 The amplitude and frequency response curves of the rotary shell corresponding to the (2, 2) modal vibration (˝1 =˝r D 0:4) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.18 The internal resonance characters between the (1, 1) modal and (2, 2) modal of the system as well as the effects of rotary speed to the amplitude and frequency response curves of the rotary shell corresponding to (2, 2) modal vibration (˝1 =˝r D 0:4) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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Fig. 12.19 The amplitude and frequency response curves when the shell is oscillating with (1, 1) modal (f22 D 0, a12 D 0:3) . . . . . . . . . Fig. 12.20 The internal response between the modal (1,1) and modal (2, 2) of the shell (f22 D 8, a12 D 0:5) . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.21 The effect of the rotary speed ˝1 on the main dynamic unstable area of the system (H D 0:1, 2 D 0:6) . . . . . . . . . . . . . . . . . . . Fig. 12.22 The effects of the thickness of the cylindrical shell H on the main dynamic unstable area of the system (˝1 D 0:1, 2 D 0:4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.1 World (W ) and Quadrotor-fixed (Q) frames axes and sign convention. Rotation around the axes of frame Q is expressed using Euler angles. The two frames are related through rotation matrix (W RQ / and translation vector (W tQ / .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.2 Model of the quadrotor (top view) showing direction of propeller rotation and reactive torques (i ) . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.3 Model of the quadrotor (side view) for deriving roll and pitch equations of motion .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.4 (a) Quadrotor with IR reflectors attached to the front, left, and right ends (b) trackable object seen in software and its virtual center defined in the middle . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.5 A camera mounted at the front of the quadrotor .. . . . . . . . . . . . . . . . . . . . Fig. 13.6 Rotation and translation matrices calculated by classicPOSIT relating object frame (O/ to camera frame (C ) .. . . . . Fig. 13.7 Target object with dimensions and object frame defined .. . . . . . . . . . . Fig. 13.8 Target object feature points extraction – Image processing flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.9 Image processing results (a) acquired image (b) segmented LED location (c) LED detected (d) segmented window location (e) corners detected in image (f) detected corners of window before sort (g) five feature points detected after sort and undistortion .. . . . . . . . . . . . . Fig. 13.10 Hash tables for undistortion of image coordinates (a) Horizontal image coordinate, u (b) Vertical image coordinate, v .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.11 6DOF pose estimation from detected feature point locations .. . . . . . Fig. 13.12 Real-time pose estimation test setup showing all four coordinate frames and OptiTrack cameras .. . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.13 Comparison of X-translation estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.14 Comparison of Y -translation estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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Fig. 13.15 Comparison of Z-translation estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.16 Comparison of roll angle estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.17 Comparison of pitch angle estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.18 Comparison of yaw angle estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.19 Error of classicPOSIT-based translational estimates compared to OptiTrack-based values . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.20 Error of classicPOSIT-based rotational estimates compared to OptiTrack-based values . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.1 Fig. 14.2 Fig. 14.3 Fig. 14.4 Fig. 14.5 Fig. 14.6 Fig. 14.7 Fig. 14.8 Fig. 14.9 Fig. 14.10 Fig. 14.11 Fig. 14.12 Fig. 14.13 Fig. 14.14 Fig. 14.15 Fig. 14.16
Fig. 14.17 Fig. 14.18
Configuration of the entire experimental setup . .. . . . . . . . . . . . . . . . . . . . Physical structure of the system .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ideal modeling structure . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Coordinate frames.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The equilibrium manifold . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Equilibrium input voltage to actuator . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Locus of transfer function zeros are is varied from 0ı to 90ı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Overview of system identification.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Physical configuration of balancing mechanism.. . . . . . . . . . . . . . . . . . . . Configuration of the entire experimental setup . .. . . . . . . . . . . . . . . . . . . . System used for identification .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Comparison between real and simulation data for D 20ı using raw data, Fit = 1743I 85:03 . . . .. . . . . . . . . . . . . . . . . . . . Comparison between real and simulation data for D 20ı using estimated data, Fit = 88.0846;92.2398 . . . . . . . . . . . . . . Comparison between real and simulation data for D 30ı using raw data, Fit = 1386I 82:9.. . . . .. . . . . . . . . . . . . . . . . . . . Comparison between real and simulation data for D 30ı using estimated data, Fit D 87.8;98.6 . .. . . . . . . . . . . . . . . . . . . . Comparison between real and simulation data for D 40ı using raw data, Fit D 97:9I 82:0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Comparison between real and simulation data for D 40ı using estimated data, Fit D 83.3;99.1 . .. . . . . . . . . . . . . . . . . . . . Comparison between real and simulation data for D 50ı using raw data, Fit D 561:6I 80:5 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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Fig. 14.19 Comparison between real and simulation data for D 50ı using estimated data, Fit D 82.2;97.8 . .. . . . . . . . . . . . . . . . . . . . Fig. 14.20 Comparison between real and simulation data for D 60ı using raw data, Fit D 344:4I 76:7 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.21 Comparison between real and simulation data for D 60ı using estimated data, Fit D 83.3;98.0 . .. . . . . . . . . . . . . . . . . . . . Fig. 14.22 Pole-Zero locus using estimated parameters .. . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.23 Simulation response of FL vs. LQR control in response to constant disturbance.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.24 Simulation result of FL vs. LQR control in response to a sinusoidal disturbance . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.25 Simulation result of FL vs. LQR in response to two pulses.. . . . . . . . Fig. 14.26 Simulation result of FL vs. LQR control in response to nonzero initial conditions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.27 Simulation result of FL vs. LQR control in response to nonzero initial conditions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.28 Simulation result of FL vs. LQR control in response to nonzero initial conditions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.29 Experimental result of FL control in response to two pulses . . . . . . . Fig. 15.1 The schematic drawing of the passive and active mechanisms responsible for stabilizing the HNC. The passive part (single-inverted pendulum model with inertia and viscoelastic elements) is considered as a physical model in this study . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 15.2 Experiment design for measuring the dynamics of Head–Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 15.3 (a) A double-degree-of-freedom model of the HNC with its schematic draw. (b) The angular displacements, 1 and 2 , of the model . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 15.4 The mean values of the coherency functions for four subjects. Each graph belongs to one subject . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 15.5 Comparison of the experimental (Exp.) and simulated (Sim.) results of the magnitudes of the transfer functions for four subjects. Note that the # sign indicates the subject number . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 15.6 Comparison of the experimental (Exp.) and simulated (Sim.) results of the phases of the transfer functions for four subjects. Note that the # sign indicates the subject number . . . Fig. 15.7 Comparison of the experimental (Exp.) and simulated (Sim.) outputs (lower panel) for one subject in the time domain when the input (the upper panel) is a Guassian random vibration (validation of the model) .. . . . .. . . . . . . . . . . . . . . . . . . . Fig. 15.8 HNC passive and active elements . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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Fig. 15.9 Fig. 15.10 Fig. 15.11 Fig. 15.12 Fig. 15.13 Fig. 15.14 Fig. 15.15
Fig. 15.16
Fig. 15.17
Fig. 15.18
Fig. 15.19
Fig. 15.20
List of Figures
The schematic drawing of the experiment design . . . . . . . . . . . . . . . . . . . A random phase multisine excitation signal . . . . .. . . . . . . . . . . . . . . . . . . . The power spectrum of the multisine input signal .. . . . . . . . . . . . . . . . . . The block diagram of the HNC in the AV and AN conditions .. . . . . The magnitudes of the HNC transfer function in three conditions of P, AV, and AN . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The phases of the HNC transfer function in three conditions of P, AV, and AN . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Comparison of the experimental (Exp.) and simulated (Sim.) results of the magnitudes of the HNC transfer functions, corresponding to the nine subjects, in the P condition.. . Comparison of the experimental (Exp.) and simulated (Sim.) results of the phases of the HNC transfer functions, corresponding to the nine subjects, in the P condition.. . Comparison of the experimental (Exp.) and simulated (Sim.) results of the magnitudes of the HNC transfer functions, corresponding to the nine subjects, in the AV condition Comparison of the experimental (Exp.) and simulated (Sim.) results of the phases of the HNC transfer functions, corresponding to the nine subjects, in the AV condition Comparison of the experimental (Exp.) and simulated (Sim.) results of the magnitudes of the HNC transfer functions, corresponding to the nine subjects, in the AN condition Comparison of the experimental (Exp.) and simulated (Sim.) results of the phases of the HNC transfer functions, corresponding to the nine subjects, in the AN condition
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Chapter 1
Characterizing Nonlinear Dynamic Systems Periodicity Ratio and Statistic Hypothesis Liming Dai and Lu Han
Abstract This chapter is on analyzing nonlinear dynamic systems with employment of Periodicity Ratio and a statistic hypothesis. By introducing the statistical hypothesis, the efficiency and accuracy of diagnosing the nonlinear characteristics of dynamic systems are improved. A new approach of accurately determining the Periodicity Ratio is developed. Overlapping points in a Poincare map are verified on a statistically sound basis. The characteristics of nonlinear systems are investigated by using the present approach. The numerical results generated by the approach are compared with those of the conventional approaches. The distinguished advantages of the approach are presented.
1.1 Introduction Setting up proper criteria for identifying dynamical systems’ characteristics is crucial in nonlinear dynamics. The techniques providing high efficiency and accuracy in diagnosing and quantifying different characteristics such as chaos, periodicity, quasiperiodicity, and other nonlinear characteristics are always demanded in studying nonlinear dynamic systems. Based on the current literature, Lyapunov exponent approach is probably the most popular approach (Wolf et al. 1985; Parks 1992; Nayfeh and Balachandran 2004) due to its efficiency and simplicity. Lyapunov exponents measures the sensitivity of a system to initial conditions and therefore classifies the system’s responses as either convergent or divergent. However, Lyapunov exponent can only describe whether a system is convergent or divergent. This brings limitations in applying Lyapunov exponents in analyzing a nonlinear system for its characteristics. By the recent literature in the field (Dai and Wang 2008), convergent responses of a system may not necessarily be
L. Dai () • L. Han Industrial Systems Engineering, University of Regina, Regina, SK, Canada S4S 0A2 e-mail:
[email protected];
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 1, © Springer Science+Business Media, LLC 2012
1
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periodic, and divergent responses may not necessarily be chaos, let alone the other characteristics such as quasiperiodic or nonperiodic motions. The approach with employment of Periodicity Ratio for diagnosing nonlinear characteristics such as chaotic, periodic, quasiperiodic, and nonperiodic behavior of a nonlinear dynamic system was first introduced by Dai and Singh (1997) and (1995) and used for analyzing the behavior of versatile nonlinear dynamic systems (Dai and Singh 1998; Dai 2008; Lu et al. 2000). Compared to other classic methods, this newly developed approach can be used to diagnose not only the periodic or chaotic motion of the systems but also the motions between pure periodic and chaotic motion such as quasiperiodic motion and nonperiodic motion (Dai 2008; Dai and Wang 2008). In fact, a Periodicity Ratio determined can be used to quantitatively describe how close a corresponding motion of a dynamic system is to a pure periodic one. The higher the periodicity of a dynamic system is, the higher value of the Periodicity Ratio will be, until it reaches its maximum possible value that is a unity, representing a perfect periodic case. In addition, with the Periodicity Ratio approach, the quasiperiodic and chaotic motion of the system can be easily distinguished from the regular motions of the system. This provides a great advantage in analyzing the complex behavior of a nonlinear dynamic system. However, the accuracy of the Periodicity Ratio determined by the approach of Dai and Singh depends on accurate determination of the periodically overlapping points in Poincare maps. Moreover, the process of determining for the overlapping points, necessary for quantifying the Periodicity Ratio values, is a numerical approach which relies on the dynamic system itself and the computational system together with the numerical method used. With the conventional approach introduced by Dai and Singh, therefore, the overlapping points numerically determined always have deviations. This may reduce the accuracy of the Periodicity Ratio determination and even reduce the liability of some of the results to be used for diagnosing the nonlinear behaviors. To implement the Periodicity Ratio to diagnose the nonlinear behavior of a dynamic system with higher efficiency and accuracy, a practically sound improvement is needed to reduce the deviations which are caused in calculating for the overlapping points to be used for determining the Periodicity Ratios. In numerically evaluating the Periodicity Ratios, in practice, the following are found significantly affecting the accuracy and reliability of the Periodicity Ratios determined. 1. A considerably large number of points in a Poincare map for a nonlinear dynamic system are to be examined and compared in the process of determining the Periodicity Ratios. 2. The accuracy and reliability of the Periodicity Ratios numerically determined relies heavily on the accuracy and reliability of the numerical methods used. 3. The accuracy and reliability of the Periodicity Ratios numerically determined may also rely on the accuracy and reliability of the physical devices with which the numerical calculations are performed.
1 Characterizing Nonlinear Dynamic Systems
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4. Complexity of the dynamic system affects the numerical results, especially the stability of the results. 5. Errors of numerical calculations used in finding the numerical solutions of nonlinear dynamic systems, therefore the overlapping points in a Poincare map, are unavoidable and accumulating during the numerical calculations. 6. Nonlinear responses especially chaos are sensitive to initial conditions and system parameters, and they are greatly unpredictable if the responses of the system are considered over a large time period. 7. The overlapping points determined numerically for Poincare maps are affected by the duration of stable state of the responses of a nonlinear system, which is actually used for determining the Periodicity Ratios. The initial instable state of the system is usually ignored in determining for the overlapping points and therefore the Periodicity Ratios. 8. Uncertainties and stochastic effects exist in the numerical calculation processes; these may cause unexpected and random numerical results. With these considerations, it is natural to employ a reliable statistic approach to respond to the concerns listed above; hence, increasing the efficiency and accuracy of the numerical procedures, reducing the deviations in verifying the overlapping points, and therefore increasing the accuracy and reliability of the Periodicity Ratios are to be determined. A statistical hypothesis test presented is a methodology of making decisions based on the samples collected from a controlled experiment or an uncontrolled observational study to examine the properness and accuracy of the overlapping boundary to be used for determining the Periodicity Ratios. An index known as p-value needs to be calculated and to be used in the test. This p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. The null hypothesis can be rejected when the p-value is less than or equal to a significance level set beforehand. When the null hypothesis is rejected, the result is said to be statistically significant. In this chapter, a new approach employing a statistical hypothesis method based on a binomial distribution will be developed for determining the overlapping points in a Poincare map and establishing an overlapping boundary for a group of overlapping points considered. With this approach, more accurate Periodicity Ratios are expected in comparing with that of the conventional approaches.
1.2 Periodicity Ratio (PR) Determination with Statistical Significance 1.2.1 Basic Definition of PR Index The approach of Periodicity Ratio can be used to efficiently identify the behaviors of motions for nonlinear dynamic systems. The Periodicity Ratio or PR, which is
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used as an index with values between 0 and 1 inclusive, is determined through an examination of the overlapping points in a Poincare map with respect to the total number of points generated by the Poincare sections in creating a Poincare map. As convention, it is believed the motion of a dynamic system can be classified into four different categories of motion (Dai and Singh 1997; Dai 2008): periodic motion, nonperiodic motion, chaotic motion, and quasiperiodic motion. It is widely acknowledged, for a steady state of a periodic motion of a dynamic system, the corresponding Poincare map consists of a finite number of visible points (Dai and Singh 1997; Nayfeh and Balachandran 2004). In a perfect periodic case, all the points in the Poincare map are overlapping points. On the other hand, the points in the Poincare map of a chaotic case are distributed in an unpredictable manner, which results in no or very few overlapping points in the corresponding Poincare map, out of a significantly large number of points generated for plotting the Poincare map. A quasiperiodic case may also contain a negligibly small number of overlapping points; however, the system in this case shows certain regularity and predictability, in contrasting to that of a chaotic case. Based on the findings of Dai and Singh (1997) and (1995), the PR value of nonperiodic motion, neither chaos nor perfect periodic cases, should be less than 1 but greater than 0. The PR index therefore quantitatively describes the periodicity of the responses of a dynamic system. The more periodic a dynamic system is, the closer the corresponding PR value is to 1. When PR value approaches zero, the corresponding system has no periodicity at all and therefore represents either chaotic or quasiperiodic responses to the system. As per Dai (2008) and Dai and Singh (1997); (1995) for a complete periodic system, there should be a finite number of j points appearing in the corresponding Poincare map and all the other points overlap with these visible points. The determination of whether or not a point in the Poincare map is an overlapping point is based on the following equations: Xki D x.t0 C kT / x.t0 C iT / XP ki D x.t P 0 C kT / x.t P 0 C iT /
(1.1)
where x is the displacement of the system, t0 a given time, k an integer in the range of 1 k j , and T the period of the periodic motion. If n is designated as the overall number of points generated for the Poincare map, i in the above equation is then an integer satisfying 1 k np. The point .xi ; xP i / is regarded as an overlapping point of the kth point .xi ; xP i / only if Xkj D 0 and XP kj D 0. The total number of points which overlap the kth point in the corresponding Poincare map can be calculated by the following equation: ( .k/ D
n X i Dk
) Q.Xki / Q.XP ki / P
n X i Dk
! ŒQ.Xki / Q.XP ki / 1 ;
(1.2)
1 Characterizing Nonlinear Dynamic Systems
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where Q.y/ and P .z/ are step functions expressible in the form Q.y/ D and
P .z/ D
1; 0;
if y D 0 if y ¤ 0
(1.3)
1; 0;
if z D 0 : if z ¤ 0
(1.4)
Designating NOP as the total number of overlapping points, the equation developed for determining NOP can be expressed by NOP D .1/ C
n X
.k/ P
kD2
k1 Y
! fXkl C Xkl g :
(1.5)
lD1
If the response of a dynamic system is completely periodic, all the points for the Poincare map must be overlapping points and the corresponding NOP in this case can be expressed as n X NOP D .k/: (1.6) kD1
In those cases overlapping points may not necessarily be periodic points; further determination for periodically overlapping points needs to be considered. Consider the kth group overlapping points among the j groups. The time span between the i th point in the kth group and the i C 1th point in the same group can be defined as Tk;i D tk;i C1 tk;i :
(1.7)
And the time span between the i C 2th point and the i C 1th point is then Tk;i C1 D tk;i C2 tk;i C1 :
(1.8)
Consider the kth group among the j groups overlapping points. Assuming there are q overlapping points in the kth group, with the above equations defined, the number of periodic points with identical time spans as that of the i th and i C 1th points in the kth group can be determined by employing the following formula: .i / D
( q2 X hDi
) Q.Tk;i Tk;hC1 /
q2 X hDi
! Q.Tk;i Tk;hC1 / C 1:
(1.9)
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The overall number of periodic points in the kth group can thus be calculated by .k/ D .1/ C
q1 X
.i / P
i D2
iY D1
! fTk;i Tk;h g :
(1.10)
hD1
Then the total number of periodically overlapping points (denoted as NPP) among the entire points in the Poincare map is expressible as NPP D .1/ C
j X
.k/:
(1.11)
kD1
And the Periodicity Ratio can be defined as the following: D lim
n!1
NPP n
(1.12)
where NPP is the number of periodically overlapping points or periodic points in a Poincare map and n is designated as the overall number of points generated for the Poincare map. The characteristics of dynamic systems’ motion can thus be determined by their corresponding PR values. Specifically, four categories of characteristics can be classified with implementation of the PR values: 1. The systems whose PR values are 1. By the definition of Periodicity Ratio, this implies that all the points in its corresponding Poincare map are periodic points and they are also overlapping points. Accordingly, the motion of this case can only be periodic. 2. The systems whose PR values are 0 and their responses show some regularity and there are not as sensitive to initial conditions as that of chaotic cases. These systems are therefore identified as quasiperiodic systems. From all the systems with 0 PR values, the quasiperiodic systems can be distinguished by the least square fitting method together with Lyapunov exponents (Han et al. (2009)). 3. The systems whose PR values are 0 and they are not quasiperiodic systems are the systems showing great nonperiodicity and they are sensitive to initial conditions. These systems are classified as chaotic systems. 4. The systems whose PR values are less than 1 yet greater than 0. These systems are classified as nonperiodic or irregular systems. For the systems falling in this category, the larger is its PR value, the closer will be its behavior to periodic motion. The behaviors of these systems are in between perfect periodic and perfect chaos. The greatest advantage of the Periodicity Ratio method is that the PR value can be used as a single value index in diagnosing the periodicity and, therefore, can be used to determine the periodic character of a dynamic system. The closer the PR value is to 1, the more periodic would be the system. Chaos is a case that the motion of
1 Characterizing Nonlinear Dynamic Systems
7
the dynamic system is completely nonperiodic. Therefore, its Periodicity Ratio is 0. Moreover, the Periodicity Ratio reveals the fact that there are an infinite number of patterns of motion in between chaos and periodic responses for a nonlinear dynamic system. The PR value therefore quantifies how close a motion of a dynamic system is to a purely periodic system. Based on the definition of the Periodicity Ratio, it can be seen that the PR value actually depends upon whether a point generated by a Poincare section is overlapping with the others, or whether the distance between a point in a Poincare map and a visible point in the same map is 0 or not. By definition, the Euclidean distance between point p1 and p2 is the length of the line segment connecting them. In Cartesian coordinates, if p1 D .m1 ; m2 ; : : : ; mn / and p2 D .n1 ; n2; : : : ; nn / are two points in Euclidean n-space, then the distance between these two points are p d.p1 ; p2 / D d.p2 ; p1 / D .m1 n1 /2 C .m1 n1 /2 C .mn nn /2 v u n uX D t .mi ni /2 : (1.13) i D1
Therefore, the distance between a point considered and a visible point in a Poincare map can be considered as standing for the Euclidean distance defined in the same dimension as the given nonlinear dynamical system. Theoretically, the distance between two overlapped points should be perfectly 0. However, in the actual numerical computation, the numerical solution of a dynamic system can only be approximate and errors due to numerical calculation are unavoidable. Plus, all the effects of numerical calculation, as described previously, affect the distances among the overlapping points. A perfect state of overlapping points with zero distance therefore rarely exists in practice. Several factors may affect the appearance of the computational deviations for the points generated for Poincare maps such as the implementation of the numerical method, the essential differences of computers, and the computation complexity caused by nonlinear differential systems. Based on the numerical simulations performed in this research and those reported in the literature (Ueda 1980a,b; Nayfeh and Mook 1989; Nayfeh and Balachandran 2004; Dai 2008), following are conclusive for periodic cases in nonlinear dynamic systems: 1. Periodic responses such as harmonic oscillations and periodic motions of multiple periods commonly exist in nonlinear dynamic systems. 2. Under certain conditions (initial conditions and system parameters), responses of a nonlinear dynamic system can be highly stable and perfectly periodic. 3. The points generated by Poincare sections do appear periodically in the periodic cases of a nonlinear system, and they form finite numbers of visible points overlapping numerous points in the corresponding Poincare map. However, deviations between a visible point and its overlapped points do exist though they are very small. The overlapping points are actually a group of points in the vicinity of a visible point in a Poincare map.
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To identify the overlapping points numerically determined, practically, one may have to set a tiny area at the vicinity of a visible point (usually the first point generated by Poincare section after the responses of the corresponding nonlinear dynamic system become stable). Any point periodically appears and falls in this area can be considered as an overlapping point of the corresponding visible point. As per the pioneer works of Dai and Singh, conventionally, this tiny area is selected on the basis of comparing the distances of the points near a visible point. A small circle of constant radius is then determined to include most of the points nearby in the circle and to be used for evaluating all the points with respect to all the visible points in a Poincare map. This implies that the distance between a visible point and another point needs to be examined. Denote d0 as the distance and designates it as overlapping boundary. The distance d0 can then be used as a criterion. If a point examined has a distance from the visible point smaller than d0 , it is considered as overlapping point. For numerical calculations, a function C.d / can be defined as ( 1; if d d0 C.d / D (1.14) 0; if d > d0 to examine the distances among the points in a Poincare map. However, as can be seen from the following sections, determination of the overlapping boundary d0 can be challenging in practice. Conventionally, d0 is selected intuitively based on simple comparisons of the numerical results. The selection is rather random and may cause missing of some overlapping points or miscount some of the overlapping points. With the conventional approach, a question needs to be answered is always: How small should this circle be? Or, what should be the proper radius of the circle to include all the representative overlapping points therefore to optimize the determination of the corresponding Periodicity Ratio? A statistical approach in evaluating the overlapping points seems a natural choice to answer the question. This approach should take into considerations of the approximation and variations of numerical results and the deviations caused by numerical calculations and the dynamic system itself. To improve the accuracy and reliability of the overlapping points to be counted, therefore the Periodicity Ratio to be determined, a development of a novel approach for calculating the overlapping points on a practically and statistically sound basis is therefore necessary. It should be noted that the statistical approach should be a determination for a “nominate point” based on the points periodically appear and with a short enough distances from each other, rather than the “first visible point” in a Poincare map. And the statistical approach should also be a determination for all the overlapping points which are within the circle of radius d0 centered at the nominate point.
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1.2.2 PR Index Determination with Statistical Hypothesis Testing 1.2.2.1 General Procedures of PR Determination The approach with Periodicity Ratio (PR) can be employed to detect characteristics of motion of nonlinear dynamic systems that can be classified as quasiperiodicity, chaos, periodicity, and nonperiodicity. The characteristics are specified by their corresponding PR values which are determined by the total number of periodic points NPP as shown in (1.11). Conventionally, in numerically determining the PR values, the following procedures are necessary: 1. Numerically solving the dynamic system with employment of a numerical method, such as Runge–Kutta method and PT method (Dai and Singh 2003) 2. With Poincare sections, determine the total number of points NP for plotting a Poincare map 3. From all the points found in the procedure above, find out all the points overlapping with the others 4. Among all the overlapping points, select all the points that periodically appear, NPP, the number of periodic points 5. Determine the PR value, corresponding to the solution of the system with specific initial conditions and system parameters Procedure 3 requires a process of determining for all the overlapping points with the criterion d0 the overlapping boundary. With the conventional approach, as mentioned previously, d0 is determined rather intuitively that may only ensure that most of the points with close enough distances are included in a small circle of constant radius. In determining the periodic points as stated in procedure 4, for a nonautonomous nonlinear differential system subjected to an external source of excitation, a periodic solution of the system may have a period which is either a multiple or integer submultiples of the period 2=!, where ! is the frequency of the external source of excitation (Guckenheimer and Holmes 1983; Nayfeh and Balachandran 2004). With the concept of Poincare map, Poincare sections can be used to create the points for plotting a Poincare map. The successive points on the Poincare section can be denoted as fXt0 ; Xt0 CT ; : : : ; Xt0 CN T g. If the system finally leads to a periodic solution, then, after a given period of time, all the points of the Poincare map˚ will converge to a finite number of individual points which must have the form Xm ; XmC1 ; : : : ; XmCq . These individual points in the Poincare map are actually the visible points shown in the map. In fact, in this case of periodic response of the system, each of the visible points in the Poincare map is actually overlapping with many points periodically appearing. In other words, at the steady state of a periodic case, the number of visible points in the Poincare map should be a constant and will not change with increase of time
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Fig. 1.1 Procedures of Periodicity Ratio determination
Numerical method selection for accuracy and efficiency
Select numerical parameters such as time step and convergence accuracy
Stabilized solution after truncation of the first xx cycles of calculations
Selection of an overlapping boundary
Save as nonoverlapping points
No
Overlapping point?
Yes
Save as nonoverlapping and nonperiodic points
No Periodic point?
Yes PR value determined
in numerically simulating the dynamic system and generating the points for the Poincare map. The increase of numerical simulation time in this case may only increase the total number of points n used for generating the Poincare map, but not the visible points. Thus, any overlapping point Xp in a Poincare map would be a periodic point ˚ with XmCl among Xm ; XmC1 ; : : : ; XmCq .0 l m/ if and only if the following condition is satisfied: P
q X lD0
(
Y
!
q1
P
hD0
Q.XpCh XmClCh /
l Y
!) ! Q.Xpk XmClCk /
D 1:
kD1
(1.15) Detailed procedures of Periodicity Ratio determination and conditions applied during the processes of determination are shown in the flow chart of Fig. 1.1. As can be seen from the procedures above, for determining the PR values, proper selection of overlapping boundary d0 is critical to the accuracy of the PR values determined. However, in the conventional approach, selection of the overlapping boundary is usually a procedure of trail and error. Subjective effects
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Fig. 1.2 Procedures of statistical hypothesis testing
Set null hypothesis (H0) and alternative hypothesis (H1)
Set the size of samples and significance level
Construct a test statistic
Determine a critical value to set a cutoff point for the test statistic
Calculate the p-value
p-value smaller than the given significance level Yes
Do not reject null hypothesis (H0)
No
Reject null hypothesis (H0)
are involved in the process of the selection. Different person may use different overlapping boundaries. This may affect the accuracy and reliability of the PR values determined.
1.2.2.2 Fundamentals of Statistical Hypothesis Testing Methodology A reliable statistical approach may help to improve the rationality of the overlapping boundary determination and perform the determination efficiently and accurately on a scientifically sound basis, so that the PR values can be determined with higher accuracy. With this consideration, a statistical hypothesis testing technique is developed. Before describing the application of the statistical hypothesis testing methodology, some fundamentals related to the methodology are necessary. For the sake of clarity and convenience, the fundamentals will be described in association with the follow chart shown in Fig. 1.2 which illustrates the general steps used in the statistical hypothesis testing.
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Following Fisher’s concept (Fisher 1966; Lehmann and Romano 2005), null hypothesis defined in this chapter is a statistical hypothesis that is tested for possible rejection under the assumption that it is true (usually those observations are the result of chance). The concept of hypothesis on the other hand is contrary to the null hypothesis; the observation in this case is the result of a real effect. The hypothesis and null hypothesis need to be set before the test starts. After the hypothesis and null hypothesis are set, the number of samples required for the test need to be determined. This is usually a fraction of the total population of the samples. A significant level also needs to be set. The significance level, designated as ˛, is the probability of making a Type I error. A Type I error is a decision in favor of the alternative hypothesis when, in fact, the null hypothesis is true. A Type II error is a decision to fail to reject the null hypothesis when, in fact, the null hypothesis is false. A critical value is also needed to set the cutoff point for the test statistic. If the value of the test statistic computed from the sample data is beyond the critical value, the decision may be made to reject the null hypothesis in favor of the alternative hypothesis. The p-value is a probability that a test statistic at least as significant as the one observed will be obtained, assuming that the null hypothesis is true. Whether to reject the null hypothesis is based on a comparison of the p-value with the significance level set in advance. If the p-value is less than the significance level, reject H0 . Otherwise, fail to reject H0 .
1.2.2.3 Testing a Given Overlapping Boundary The above general steps will be followed to determine the overlapping boundary. To investigate the various behavior of a given nonlinear dynamic system, the Periodicity Ratio defined previously will be applied to the system with the system parameters of desired range. The different states of motion of the system corresponding to different parameters are classified as periodicity, nonperiodicity, quasiperiodicity, and chaos according to the corresponding PR values. All the obtained cases (states of motion) determined with the implementation of the Periodicity Ratios reflecting a specified overlapping boundary are taken from the whole population and to be used in the statistical hypothesis test conduction. These motions will be regarded as type-determined motions since they are already classified into the four categories of motion. With the utilization of the overlapping boundary selected, the Periodicity Ratios can be tentatively calculated. Correspondingly, as per the PR determination procedures described, the states of the motions can be determined with the Periodicity Ratios obtained. In other words, a periodic–quasiperiodic–chaotic diagram corresponding to the Periodicity Ratio determined can be plotted. Each point in the periodic–quasiperiodic–chaotic diagram represents a state of motion for the dynamic system considered.
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However, the states of motion and the Periodicity Ratios determined need to be verified or reevaluated for their correctness. This reevaluation can be performed by checking the actual motion of the states determined, with Poincare maps and phase diagrams. For example, if a state determined is periodic but the Poincare map or phase diagram of the motion does not show the periodicity, the overlapping boundary and the corresponding Periodicity Ratio are not proper. And the category of the state is then proven to be incorrect. If the Poincare map or phase diagram does show that the motion of the state is indeed periodic, the category of motion determined is correct, and the selection of the overlapping boundary and the corresponding PR value are proper. Obviously, the selection of overlapping boundary must be statistically sound. Those states determined in the incorrect catalogue by the PR index are regarded as misjudged states. The statistical hypothesis will be implemented to determine the catalogue of the whole population of system states obtained by the PR index under given overlapping boundary. The number of states determined may reach a certain percentage (denoted as p percentage here) out of the population. The null hypothesis is therefore stated as the following. H0 : The percentage of misjudged states within all the states obtained per the PR values, based on the given overlapping boundary d0 , cannot exceed p percentage. Then a decision must be made as to whether to accept or reject the given value d0 , as the selected overlapping boundary. Since it is difficult to investigate all the system’s actual behaviors with large population of different parameters, the samples need to be selected from the total samples available on a statistically sound basis. A sample test is therefore necessary, and a statistical decision should be made to either reject or accept the Hypotheses H0 . In the statistical hypothesis test, a critical value k needs to be designed according to the sample and the test statistic as well as the required test significance level. If the number of misjudged states in the samples selected exceeds the critical value, the original hypothesis therefore the tentative overlapping boundary should be rejected. The actual test procedures are as follows. Considering n state samples from the whole population of type-determined motions obtained per the PR values which are determined on the basis of a given boundary d0 . Event A: “The number of incorrectly determined states of motion within the sample states obtained per the PR values with implementation of a given boundary d0 is greater than k.” Practically, event A is equivalent to a case in which the number of incorrectly determined states within the samples determined is k C 1 or k C 2 or k C 3, . . . , or even n. Denote Pp .A/ as the probability of the occurrence of event A when the percentage of the incorrect determined states of motion of the entire population is p. Therefore, Pp .A/ represents the probability of H0 being rejected. Denote event Aj as “the number of incorrectly determined states of motion within the samples obtained by the PR values, based on a given overlapping boundary d0 is j .”
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Denote Pp .AkCi / as the probability of the occurrence of event AkCi when the percentage of the incorrectly determined states of motion of the entire population is p. As defined in event A, one may conclude n [
AD
Aj
(1.16)
j DkC1
and 0 Pp .A/ D Pp @
n [
1 Aj A D
j DkC1
0
n X
Pp .Aj / Pp @
j DkC1
n [
1 Aj A
(1.17)
j DkC1
Since AkC1 , AkC2 , AkC3 ,. . . , An are all independent events, thus 0 Pp @
n [
1 Aj A D 0
(1.18)
j DkC1
and
n X
Pp .A/ D
Pp .Aj /:
(1.19)
j DkC1
As the statistical hypothesis involves correct and incorrect diagnoses with PR values, to determine for the probability Pp .A/, a binomial distribution can be implemented. Binomial distribution is a discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments which are specific as determining if the states determined fall in the correct category of motion. By the reference (Lehmann and Romano 2005) for the binomial distribution, the probability of the occurrence of event Aj can be calculated by n Pp .Aj / D p j .1 p/nj j
(1.20)
and it can be proved (Lehmann and Romano 2005) that Pp .Aj / D
n X n p j .1 p/nj : j
(1.21)
j DkC1
To avoid high chance of incorrectly rejecting the null hypotheses H0 , a proper value needs to be assigned to the test significance level ˛. Usually, this value is chosen either 0:05 or 0:1 (Lehmann and Romano 2005). The critical value of k can be determined once the binomial distribution is determined provided that the probability Pp .A/ D ˛ is known. Also, Pp .A/ relies on the known value of n.
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If there are k or more misjudged samples among all the samples obtained from the PR values with given overlapping boundary d0 , then d0 is not an appropriate overlapping boundary as the null hypothesis should be rejected.
1.3 Determination of Proper Overlapping Boundary for the Duffing Dynamic System The Duffing dynamic system is a popular system in the area of nonlinear dynamics. This system is governed by Duffing equation in the following form (Ueda 1980a,b): xR C k xP C x 3 D B cos t;
(1.22)
where B and k are system parameters relating to energy dissipation and amplitude of external excitations, respectively. Duffing system has very rich nonlinear dynamic behaviors within its sizeable number of changing parameters. The conventional tools, such as phase diagrams, wave curves, Lyapunov exponents and Poincare maps, are very useful in analyzing the behaviors. However, each of the tools may reveal merely one state of behavior for specific initial conditions and given system parameters. It can be extremely tedious or almost impossible to analyze the global characteristics of the system through plotting out all the diagrams or curves, as there may be a numerous or almost infinite number of data to deal with if a large range of system parameters or initial conditions need to be considered. Taking the advantage of a single value index, Periodicity Ratio is employed to generate a much detailed periodic–quasiperiodic–chaotic region diagram for Duffing dynamic system, without plotting any figures (Dai and Singh 1997, 1998; Dai 2008). However, the periodic–quasiperiodic–chaotic region diagram initiated by Dai and Singh was based on Periodicity Ratios determined by an overlapping boundary which was selected rather straight and intuitive. More accurate and probably more detailed periodic–quasiperiodic–chaotic region diagram can be expected if a statistically sound overlapping boundary can be determined for the Duffing system. With its typical nonlinear behaviors, Duffing system is perfect for demonstrating the application of statistical hypothesis testing technique described above. To achieve an entire picture of the application, an investigation on a large population of the system responses under different parameters of the Duffing system shown in (1.22) is considered. This Duffing system was used by Ueda (1980a,b) who reported the first time the periodic-chaotic diagram that was popular in the field of nonlinear dynamics. Dai and Singh (1997, 1998) also employed this very system and reported their periodic–quasiperiodic–chaotic diagram that is much accurate and detailed in comparison with that of Ueda. Quasiperiodic responses of the system were included in their diagram.
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For the sake of clear comparison, about the same ranges of system parameters used by Ueda and Dai and Singh are used in this section. Specifically, the parameter B is allowed to vary from 0 to 15 with an increment of 0:1 and k varies from 0 to 1 by an increment of 0:01. As such, more than 15,000 states under different parametric values are to be examined, which is the whole population to follow the test to be conducted. Denote s0 and v0 as the initial displacement and velocity of the dynamic system, respectively. For each state, corresponding to the specified system parameters and initial conditions, the first 50 cycles are omitted to eliminate the perturbation caused by the initial conditions. And 451 cycles are computed to determine the PR values of each state of the system.
1.3.1 Effects of Overlapping Boundaries on Diagnosis of Nonlinear Characteristics With implementation of Periodicity Ratios, accuracy and reliability of the characteristic diagnosis of a nonlinear system depend on the accuracy and reliability of the PR values determined. By the definition of Periodicity Ratio, the PR values are heavily relying on the accuracy of the overlapping points used, and the overlapping points are determined on the proper selection of the overlapping boundary. Therefore, selection of a proper overlapping boundary is crucial to the accuracy of the characterization of nonlinear systems with the Periodicity Ratio. The effects of the overlapping boundary on the accuracy of PR values are to be investigated in this section. To demonstrate the effects of overlapping boundary on the PR values produced with the overlapping boundary, and therefore the system’s characteristics of motion classified by the PR values, different overlapping boundaries are tested for evaluating the effects of overlapping boundaries. For the sake of accuracy of the Periodicity Ratios to be determined, in general, the overlapping boundary to be selected must be small enough and properly determined. In the research of this chapter, for the performance of the statistic hypothesis test, eight different values of 101 , 102 , 103 , 104 , 105 , 106 , 107 , 108 are selected as the overlapping boundaries di . The selection of the overlapping boundaries in this range is rather wide and such selection is for the sake of demonstrating the application of the statistic hypothesis approach in determining for an optimal overlapping boundary. As desired, less number of overlapping boundaries can be used in practice. For each of the eight overlapping boundaries selected, PR values are determined as per the PR value calculation procedures described previously. With the PR values determined, all the characteristics of the motion are diagnosed for the system. Therefore, eight different diagnosing results are obtained. Figure 1.3 shows the results with respect to the number of states of motion and the specific overlapping
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Fig. 1.3 Bar chart for the distribution of different categories of motion under different given overlapping boundaries
boundary used. It should be noticed that the value of horizontal axis of the figure is decreasing in a natural logarithmic fashion. The numbers shown on the horizontal axis are the absolute values of the exponents. The upward diagonal area in the bars of the figure represents the portion of the quasiperiodic cases, outlined diamond area the portion of chaotic cases, dashed vertical area the portion of nonperiodic cases, and the dotted area the portion of periodic cases, among all the motion states of the system. As can be seen from the bar chart in the figure, the selection of overlapping boundary indeed and significantly affects the accuracy of the Periodicity Ratios, which in turn affect the diagnosing results for the dynamic system. From the bar chart in Fig. 1.3, the following can be concluded: 1. Under any given overlapping boundary, the number of periodic states occupies the majority of the bar. This implies that periodic responses dominating the responses of the system in the ranges of parameters and initial conditions considered. 2. The portion of periodic responses diagnosed is reduced as the value of the overlapping boundary decreases. The percentages corresponding to the other three types of states increase as the overlapping boundary decreases. 3. When the overlapping boundary is smaller than 103, the portion occupied by chaos remains almost constant. 4. When the overlapping boundary is smaller or greater than 103, the percentage of nonperiodic portion increases monotonically. However, the variation of nonperiodic cases is relatively small, except when the overlapping boundary becomes extremely small. 5. Chaotic portion shifts up when the value of overlapping boundary decreases while quasiperiodic portion increases. However, shifting rate is gradually stabilized as the boundary becomes smaller than 105 . 6. Chaos and quasiperiodicity of the system cannot be detected if the overlapping boundary used is too large.
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It should be noticed that there are cases that the maximum distances between some points in the Poincare map are smaller than the larger overlapping boundary but larger than the small boundary. This implies that some points previously classified as periodic overlapped points may no longer be considered as periodic points when the overlapping boundary becomes smaller. The phenomena revealed in the graph can be analyzed in details as the following: • When the overlapping boundary selected is too large, some of the chaotic points can be mistakenly included in one or all of the other three categories of motions. One may imagine, when the overlapping boundary is large, more points would be included in a larger circle with the larger overlapping boundary as radius. Therefore, more points would fall into the circle and counted as the overlapping points. This may significantly increase the possibility of counting these points as periodic points. As a result, it may mistakenly generate some cases of periodic or the other two types of motions other than chaos. • On the other hand, if the overlapping boundary is too small, less overlapping points as periodic points would be counted. In this case, some of the periodic points may be considered as nonperiodic just because they are slightly out of the small circle quarantined by the small overlapping boundary, though they appear periodically. • One may also conclude from the results shown in the graph that most of the periodic cases evaluated are very stable. During the numerical simulation, they precisely appear periodically, even when the overlapping boundary becomes very small. However, some of the periodic points are not as stable. They may slightly vary with time, though they also appear periodically. • When the overlapping boundary is small enough, the chaotic portion of the system becomes stable and not sensitive to the variations of the overlapping boundary. In chaotic cases, as can be concluded, the dominative portion of the points generated from Periodicity Ratio calculation is distributed in a random manner, and periodic points can hardly be detected. This conclusion matches with the characteristics of chaotic responses of a nonlinear system. As can be seen from the discussion above, the overlapping boundary should be properly determined as they are critical to the accuracy of the Periodicity Ratio. Larger or smaller overlapping boundary may result in less accurate Periodicity Ratio and, in return, lead to inaccurate or even incorrect results in diagnosing the characteristics of a nonlinear dynamic system.
1.3.2 Random Sampling per Different Overlapping Boundaries As discussed in the above section, the selection of overlapping boundary plays an important role in accurately determining for the PR values. Now the problem left is how an appropriate overlapping boundary can be selected on a scientifically sound basis. The application of the statistical hypothesis testing technique in selecting an
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optimal overlapping boundary is to be demonstrated though the implementation of the technique in the Duffing system. With the utilization of a given overlapping boundary, when the parameter B of the Duffing system varies from 0 to 15 with an increment of 0:1 and k varies from 0 to 1 by an increment of 0:01, the corresponding PR values are determined for the system. The system’s behaviors are then classified into the four catalogues of motion to produce the desired population of samples for the test. As eight different overlapping boundaries are considered associating with the eight groups of PR values determined, eight groups of sample population can be made available for the test. Among each of the sample populations, a proper number of samples need to be selected for each of the four categories of motion. The motions of such samples selected are then compared to the actual behaviors of the system. Since there are four different catalogues of motion for the system, namely periodic motion, nonperiodic motion, quasiperiodic motion, and chaotic motion, one may have to consider that how many samples need to be selected from each category of motion, to best fulfill the requirement of the statistic hypothesis test. Moreover, the samples selected must be highly representative to the samples of a specific population considered. Based on a group of PR values corresponding to a specific overlapping boundary, the number of samples in each catalogue of motion can be determined from a sample population that has been classified into the four categories of motion. To ensure the representativity of the samples to be used for the test, the number of samples of each categories of motion is determined based on the proportion of each of the four categories of motion to the entire sample population considered. With the present approach, the samples to be used for the test are taken from a specific sample population corresponding to an overlapping boundary. For the sake of simplicity and clarity, the number of samples to be used for the test is 1% of the sample population specified. The samples are divided into four groups to reflect the four categories of motion, and the proportion of the samples of a group to the total samples selected is the same as the proportion of the categorized motion to the entire population. For a given overlapping boundary di , assume n samples need to be selected from the entire population. Designate Sli as the number of samples to be used for the test corresponding to the lth category of motion. Based on the description above, Sli can be expressed by the following equations: S1i D
nPi ; TN
S2i D
nQi ; TN
S3i D
nCi ; TN
S4i D
nNPi ; TN
(1.23)
where l 2 f1; 2; 3; 4g, i 2 f1; 2; 3; 4; 5; 6; 7; 8g, and Pi denotes the number of periodic points under a given boundary di , Qi denotes the number of quasiperiodic points under a given boundary di , Ci denotes the number of chaotic points under a given boundary di , NPi denotes the number of nonperiodic points under a given boundary di , and TN is the number of the entire population.
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1.3.3 Sampling Method Once the number of samples required is determined, a proper method of selecting the samples from the sample population specified with a given overlapping boundary needs to be utilized. This section focuses on how the samples under each category of motion can be collected with a given overlapping boundary. Since the range of B is from 0 to 15, where the increment is 0:1, and k ranges from 0 to 1 by an increment of 0:01, the two parameters B and k can be considered to follow a uniform distribution of Œ0 W 0:1 W 15 and Œ0 W 0:01 W 1, respectively. When the two values of B and k are specfied, the system is uniquely determined. Obtaining samples of the system means to determine the samples of the two parameters. Since the distributions of the two parameters can be independent, one may firstly select a value for B from its uniform distribution Œ0 W 0:1 W 15 and then a value for k from Œ0 W 0:01 W 1. For representativity of the samples to be selected, the samples need to be selected in a random manner. In other words, a proper pair of B and k values needs to be selected randomly. Considering that a range of integer from 1 to 151 can be associated with the integers in Œ0 W 0:1 W 15 and 1 to 101 can be associated with the integers in Œ0 W 0:01 W 1, a random number can be generated among these groups of integers. In fact, the random numbers can be generated by any means of random number generators. Since the statistic hypothesis approach presented is relying on the numerical methods for obtaining solutions for the nonlinear dynamic system, the random numbers used in this chapter are generated through a package available in Matlab. Assuming a random integer I1 is initially generated from Œ1 W 151, a specific B value denoted as B0 can be determined by B0 D .I1 1/ =10. Similarly, a specific k can be determined as k0 D .I2 1/ =100, where I2 is the random integer generated from Œ1 W 100. Once the pair .B0 ; k0 / is determined, the corresponding solution of motion is readily available. The overlapping points can then be calculated with a given overlapping boundary di . With the overlapping boundary determined, a Periodicity Ratio can be calculated corresponding to the Duffing system specified by .B0 ; k0 /. With this Periodicity Ratio, the category of the motion specified can be diagnosed. One may then count one sample for this category. Consequently, with the same procedure, find a solution of the system with a new pair .B0 ; k0 / randomly selected via the same procedures. Utilizing the same overlapping boundary, a new Periodicity Ratio is calculated for this solution, and the category of motion for this case can be diagnosed. One sample for this category of motion is then counted. This process should be repeated until each and every required number of samples, which are determined through the procedures described in Sect. 1.3.2, are determined for all four categories of motion. The only restriction on randomization is that the number of samples of any specific category of motion must satisfy the demanding number.
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2.6502
2.6502
2.6502
2.6502
2.6502 3.1098
3.1098
3.1098
3.1098
3.1098
3.1098
Fig. 1.4 Poincare map at k D 0:6 and B D 10:6
1.3.4 Finalizing Overlapping Boundary Under Visual Comparison and Statistical Quantity Computation The most reliable and evidential tools available for visualizing the responses of nonlinear dynamic systems are phase diagrams and Poincare maps. With the other means, it would be difficult to identify the categories of motion for a dynamic system with confidence. For example, Lyapunov exponent approach may be used to comprehensively determine convergence and divergence of a motion but it cannot further classify whether a convergent motion is periodic or quasiperiodic. For each sample finally determined through the statistical hypothesis tests, its corresponding motion can be graphically described by a Poincare map and phase diagram. These two types of diagrams are convenient for being used to identify its actual category of motion. In comparing the actual category of motion with that have been determined by the PR values tested, one may easily judge whether the category of motion determined by the PR value is correct or not. By comparing the total number of samples correctly and incorrectly diagnosed by the tested PR values for each of the eight groups of samples corresponding to the eight overlapping boundaries, an optimal overlapping boundary can be expected. Figure 1.4 shows a Poincare map for a case corresponding to the Duffing system with k D 0:6 and B D 10:6. As per all the eight PR values corresponding to the eight different overlapping boundaries, the samples tested for this case all lead to periodic motion, or the PR values are straightly a unity. Indeed, as can be seen from this Poincare map and its phase diagram shown in Fig. 1.5, the motion is a perfect periodic motion. Another case is shown in Fig. 1.6 for the system with k D 0.15 and B D 7 : 2. From the Poincare map in Fig. 1.6 and its phase diagram in Fig. 1.7, this is a chaotic case. The PR values obtained for this case are 0:5600, 0:0200, 0, 0, 0, 0; 0, 0, corresponding to the eight overlapping boundaries in the order of 101 , 102 , 103 , 104 , 105 , 106, 107, 108. Obviously, this chaotic case cannot be diagnosed
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Fig. 1.5 Phase diagram at k D 0:6 and B D 10:6
5 4 3 2 1 0 -1 -2 1.8
2
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Fig. 1.6 Poincare map at k D 0:15 and B D 7:2
if the overlapping boundary selected is too large .di > 102/. This is not hard to understand. As the overlapping boundary used is too large, more points in the Poincare map can be counted as “periodic points” although they are not repeating periodically with identical displacements and velocities. A small amount of such “periodic points” will lead the miscounting. As a result, the chaotic case would be counted as a nonchaotic case. Quasiperiodic motions are the responses of the system, which are not perfectly periodic but show some regularity and predictability. A quasiperiodic case can be identified by Periodicity Ratio together with the least square method or Lyapunov exponent. As an example, a typical quasiperiodic case tested is shown in Figs. 1.8 and 1.9. When the overlapping boundaries selected are extremely large .di > 101 /, the case is recognized as nonperiodic. (In the case of di D 101, the PR value calculated is 0:3950). However, when the overlapping boundaries are smaller than 101 , all the results diagnosed are quasiperiodic.
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6 4 2 0 -2 -4 -6 -4
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Fig. 1.7 Phase diagram at k D 0:15 and B D 7:2
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Fig. 1.8 Poincare map at k D 0 and B D 7
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5
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1
Fig. 1.9 Quasiperiodic phase diagram at k D 0 and B D 7
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9.9
x 10-3
9.89 9.88 9.87 9.86 9.85 9.84 9.83 9.82 -0.046
-0.044
-0.042
-0.04
-0.038
-0.036
-0.034
-0.032
Fig. 1.10 Poincare map at k D 0:81 and B D 0:02
0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.05
-0.045
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
Fig. 1.11 Phase diagram at k D 0:81 and B D 0:02
Nevertheless, in many cases, proper diagnosis of quasiperiodic states requires the overlapping boundary to be properly small. Such case is shown in Figs. 1.10 and 1.11. As can be seen from Fig. 1.10, the points shown in the Poincare map are localized in a small area, but they are not appearing periodically as a periodic case. The displacement and velocity may vary slightly each time it appears. If the overlapping boundary selected is too large, di > 103 , the case can be counted as perfect periodic. When the overlapping boundary is small enough, as d i < 105 , this case can then be diagnosed as a quasiperiodic case as it should be. Nonperiodic cases do not include chaos or quasiperiodic motions. They certainly show some periodicity but not completely periodic. The Periodicity Ratios of nonperiodic motions are in between 1 and 0, but never be 1 or 0. For many nonperiodic cases, they can be interchanged with periodic cases, especially in the
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1 0.5 0 -0.5 -1 -1.5 2.7
2.75
2.8
2.85
2.9
2.95
Fig. 1.12 Poincare map of a nonperiodic case at k D 0:37 and B D 8:9
6 4 2 0 -2 -4 -6 -4
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Fig. 1.13 Phase diagram at k D 0:37 and B D 8:9
areas of transition, if the overlapping boundary is not properly selected. A case of nonperiodic motion is shown in Figs. 1.12 and 1.13. The PR values for this case are found as 1:0000, 0:9600, 0:7600, 0:6200, 0:4867, 0:3733, 0:2467, 0:1267 corresponding to the eight overlapping boundaries selected. Based on the eight different overlapping boundaries selected, eight different groups of PR values are determined. Each group of PR values is used to diagnose the types of motion for the Duffing system. With the PR values determined, the types of motion diagnosed may not necessarily be all correct in comparing with the actual motions of the system. Among all the samples selected per the statistical hypothesis tests, there are always certain number of states diagnosed by the PR values correctly matching with the actual sates of motion and some others are not correct. In Table 1.1, all the tested results of motion states are listed corresponding to the PR values determined with the overlapping boundaries selected. In the first column of each of the eight
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L. Dai and L. Han Table 1.1 Comparison of samples obtained via the PR values corresponding to the eight overlapping boundaries selected Type Sample Incorrectly Type Sample Incorrectly (PR) number diagnosed (PR) number diagnosed 101 Q C P NP Total
0 1 140 13 154
0 0 5 11 16
102 Q C P NP Total
1 4 138 11 154
0 0 3 9 12
103 Q C P NP Total
2 8 136 9 155
0 0 2 5 7
104 Q C P NP Total
3 10 132 10 155
0 0 2 6 8
105 Q C P NP Total
4 10 130 11 155
0 0 1 5 6
106 Q C P NP Total
5 10 127 12 154
0 1 0 7 8
1 0 1 1 3
108 Q C P NP Total
6 10 105 33 154
2 0 0 16 18
107 Q C P NP Total
6 10 123 16 155
groups of test results, Q stands for the quasiperiodic states of motion, C the chaotic states, P the periodic states, and NP designates the motions of nonperiodic states. In the table, 101 , 102, 103 , 104, 105 , 106 , 107, 108 are the sizes of the overlapping boundaries utilized for the eight groups of tests. In the second column of each of the eight groups of test results, the number of samples is tabulated for the motions of each of the four categories. The third column shows the number of samples whose diagnosed results as per the PR values employed are different from the actual state of motion; they are therefore the number of samples that are incorrectly diagnosed corresponding to the four categories of motion. As can be seen from Table 1:1, 154 or 155 samples are taken for the tests from all the states of motion corresponding to each of the eight overlapping boundaries. It should be noticed that 1% of samples is taken from each types of motion states. As the number of samples must be an integer, the total samples used for the testes may vary from 154 to 155 for each group of tests corresponding to a specified overlapping boundary.
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-1.2115 -1.212 -1.2125 -1.213 -1.2135 -1.214 -1.2145 0.837
0.8375
0.838
0.8385
Fig. 1.14 Poincare map of a nonperiodic motion at k D 0:09 and B D 11:9 6 4 2 0 -2 -4 -6 -4
-3
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-1
0
1
2
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Fig. 1.15 Poincare map of a nonperiodic motion at k D 0:09 and B D 11:9
As can be seen from Table 1:1, the total number of incorrectly determined samples varies from case to case corresponding to the eight overlapping boundaries. This implies that the number of incorrectly determined samples can be minimized if a proper overlapping boundary can be identified. It may also be seen from Table 1:1 that most of the sample tests are falling in the first columns and significantly small number of samples are in the second columns for each of the eight overlapping boundaries used for the tests. This is due to the fact that the overlapping boundaries used are fairly small and not too far from the optimal overlapping boundary to be determined. The actual motion corresponding to an incorrectly determined sample implies that the overlapping boundary used is too small or too large than the optimal overlapping boundary. Based on the data obtained in the tests with diagnosed results tabulated in Table 1:1, under some relatively larger overlapping boundaries, say d0 D 101 , 102 , 103 , or 104 , the periodic states incorrectly diagnosed are mainly nonperiodic points in reality. Figures 1.14 and 1.15 show a typical example of this
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2.11
2.12
2.13
2.14
2.15
2.16
2.17
Fig. 1.16 Poincare map at k D 0:68 and B D 5:3
3 2 1 0 -1 -2 -3 -3
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-1
0
1
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3
Fig. 1.17 Phase diagram at k D 0:68 and B D 5:3
case. As can be seen from Fig. 1.14, Poincare map of this case, the points are not actually overlapped. This implies that the motion is not precisely and periodically repeating itself. Corresponding phase diagram of this case is shown in Fig. 1.15. Although the phase diagram seems a case very close to a diagram of periodic case by naked eyes, there are slight variations observed at the localized areas as the curves in the figure are not perfectly overlapping at all the places. Therefore, this case is a case of nonperiodic motion but not periodic. However, this nonperiodic case is fairly close to periodic. As can be seen from Fig. 1.14, all the points in the Poincare map are localized in a small area of 0:0030:0015 and the points actually appear periodically as per the tests. This is why the phase diagram seems a perfect periodic case by naked eyes. Several samples diagnosed as nonperiodic are actually quasiperiodic cases, if the overlapping boundaries are not properly chosen. This is shown in Figs. 1.16 and 1.17. Although the phase diagram indicates some similarities to a nonperiodic case, there are no periodic points found in the test.
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In performing the test, there are some other points worth to be emphasized. • It is evident in the tests that the incorrectly diagnosed chaotic and quasiperiodic cases are relatively small, regardless of the selection of the overlapping boundaries. This is true as there are very few periodic points for both chaotic and quasiperiodic cases. • Most periodic samples are pretty stable. Their states of motion will not change with change of the overlapping boundaries. • Rapid increase of nonperiodic samples is usually an indicator that the overlapping boundary used is probably too small. • There exists an optimal overlapping boundary for a system considered. However, no extremely perfect overlapping boundary can be found to exclude all the incorrectly diagnosed cases, regardless of how many samples are used and how many overlapping boundaries are to be selected for the tests. This is due to the limitations of the numerical calculations, the complexity of the dynamic system, the external excitation on the dynamic system, duration of the stable motion considered, statistical techniques used, and the other factors involved. To evaluate and judge which overlapping boundary is the optimal one among the eight overlapping boundaries selected, as indicated in Sect. 1.2.2.3, the binomial distribution can be employed. In implementing the binomial distribution, the following need to be confirmed. 1. There are n samples need to be tested 2. Every test for a sample may produce merely one specific result (correct or incorrect diagnosis) 3. The accuracy p for each test is a constant 4. All the tests are independent of each other In the tests demonstrated above, all these must be confirmed such that the random samples may follow the binomial distribution. Also, to apply the equations of statistical calculations described in Sect. 1.2.2.3 and the binomial distribution, the percentage of incorrectly diagnosed samples and the level of significance must be decided beforehand. For the sake of simplification, the percentage of incorrectly diagnosed samples is set 1%, i.e., p D 0:01; this implies 99% of confidence level. With this confidence level, the statistical tests for the overlapping boundary determination should satisfy for most of the dynamic systems. As a convention, the level of significance used for the Duffing system is 0:05. In other words, the significance level should be greater than or equal to 0:05. Under the conditions and the data provided, a diagram illustrating the critical region of a binomial distribution with p D 0:01 is constructed as shown in Fig. 1.18. The red diamonds in the figure represent the binomial distribution when n D 155 and the blue crosses denote the binomial distribution when n D 154. In the figure, vertical axis illustrates the probability of the null hypothesis H0 being rejected, and the horizontal axis is the k value indicating the critical value representing the cutoff point for the test statistic. For the practice of the statistical hypothesis test in Duffing system, the probability of hypothesis H0 being rejected
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Fig. 1.18 Critical region diagram of binomial distribution
is Pp .A/ that can be calculated by (1.21). The index k is used as a single value criterion which determines whether H0 should be rejected or not. The combination of the probability and k value determines the properness of the selection of the overlapping boundary. This is the advantage of the binomial distribution diagram. When p D 0:01, the figure provides a clear view regarding the critical region for a test significance level of 0:05. As the significant level required is greater than or equal to 0:05, when k 5, the critical region is reached. Accordingly, the corresponding hypothesis made in this case should therefore be rejected. At k D 4, the property Pp .A/ D 0:0507 is found as per the discussion in Sect. 1.2.2.3 for the number of samples selected. This implies that k should not be greater than 4 to avoid the hypothesis to be rejected. In other words, if five or more incorrectly determined states of motion are detected within the limit number of samples, the PR values calculated with the boundary di selected is not acceptable. From Table 1:1, only the test associated with the overlapping boundary of 107 satisfies the required k value; therefore, this overlapping boundary is the optimal among all the overlapping boundaries tested, as this overlapping boundary provides 99% accurate satisfaction under a level of significance of 0:05 that are set forth previously. It should be noticed in statistical hypothesis test practice, however, there may be more than one overlapping boundaries that satisfy the significance level and accuracy requirements. In this case, the best overlapping boundary should be the optimal one to be accepted. From the discussions of the previous two sections associated with the test results shown in Table 1:1 and Fig. 1.18, one may recognize that the Periodicity Ratio assures a fairly high accuracy when periodic motions are considered. From the sample data used, at least 99% of the samples assume periodic motions by the PR values are indeed reflecting periodic motions in reality. In fact, with the 0:05 level of significance, this conclusion stands even for relatively large boundaries (five incorrectly determined cases out of 140 samples assumed periodic with the PR values under an overlapping boundary of 101).
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This is mainly due the high stability of the periodic motions. Furthermore, the samples of periodic motions include only periodic points in Poincare maps; those points overlapped but not periodically overlapped are rejected from the periodic catalogue. The tests under a given boundary are not only to determine whether the distance between different points is less than the overlapping boundary but also to determine whether the character of those overlapping points is periodically overlapped. Periodic overlapping is a unique characteristic of periodic points. As can be seen from the discussion above, in practice, it is not necessary to compare all the samples with the Poincare maps and phase diagrams, if the following points are taken into consideration: • The samples of pure periodic motions are highly stable. A great majority of them can actually be counted as correctly diagnosed cases, without performing the test. • The samples of chaotic and quasiperiodic motions are also stable and can be easily identified, so long as the overlapping boundaries used are not extremely large. • The incorrectly diagnosed samples are likely located at the vicinities of transitional areas. Therefore, in practice, one may only need to focus on these areas and test the few samples selected from these areas. With these points taken in mind, it will significantly reduce the processing time for the statistical hypothesis test while fulfill the requirements for highly accurate Periodicity Ratios. From the discussions in the sections above, it can be concluded that the statistic approach presented significantly improves the accuracy of the Periodicity Ratio approach in diagnosing the characteristics of nonlinear dynamic systems, on a solid scientific basis. Evidently, higher accuracy of the PR values may help to reveal more accurate and detailed information of the dynamic systems, especially in the areas of transition from one type of motion to the other. This will be further discussed in the following section with examples. Obviously, to establish this improvement in accuracy requires further efforts, i.e., the statistical hypothesis test technique. Accuracy requirements vary from case to case and depend on the requirements of the problems to be solved. This is especially true in engineering applications.
1.4 Effects of Overlapping Boundaries on the Overall Nonlinear Behaviors of Duffing System As discussed in the previous sections, the overlapping boundary of high accuracy may result in highly accurate Periodicity Ratios which in turn provide high quality diagnosis of nonlinear behaviors for the dynamic systems. So far, only the individual state of motion is considered with the accuracy of overlapping boundaries. In this section, the effects of the overlapping boundaries on the overall behaviors of the Duffing system with respect to a wide range of system parameters are investigated.
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Fig. 1.19 Periodic–quasiperiodic–chaotic diagram for Duffing system under given overlapping boundary of 107
Probably the most widely cited graphical analysis on the overall behaviors of Duffing system is Ueda (1979, 1980a,b, 1992) work. Ueda is the pioneer who initiated the diagram which illustrated the periodic and chaotic behavior in a 2-D diagram. Such a diagram greatly helps to identify the nonlinear behaviors of a dynamic system from the perspective of initial conditions and a range of system parameters, though Ueda’s diagram was rather rough and less accurate. Also, the quasiperiodic behaviors of the Duffing system were missed from Ueda’s work. However, as the Periodicity Ratios rely on the overlapping boundary selected, the accuracy of such analysis and therefore the accuracy of the periodic–quasiperiodic– chaotic diagram plotted may be affected by the selection of the overlapping boundary. A periodic–quasiperiodic–chaotic diagram shown in Fig. 1.19 is plotted for the Duffing system with an overlapping boundary of 107. This diagram is much accurate and detailed in comparing with that of Ueda. The initial conditions used for plotting the diagram is s0 D 2 and v0 D 0, which represent the initial displacement and initial velocity, respectively. Each state (a point) in the diagram corresponds to a set of specified system parameters and initial conditions. The first 50 cycles are omitted to eliminate the perturbation caused by the initial conditions, and 451 cycles are computed to determine the PR value of the very state of the system. Same as the whole population investigated in the statistical hypothesis test before, parameter B varies from 0 to 15 with an increment of 0:1 and k from 0 to 1 by an increment of 0:01. The green diamonds in the diagram indicate chaotic cases while blue dots
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Fig. 1.20 Periodic–quasiperiodic–chaotic diagram for Duffing system under given overlapping boundary 103
represent periodic cases, and red crosses are the quasiperiodic cases. The white blank areas are occupied by those irregular or nonperiodic states whose PR values are between 0 and 1. A similar diagram with overlapping boundary of 103 is plotted in Fig. 1.20. Obviously, this diagram is less accurate in comparing with Fig. 1.19, as some quasiperiodic states are missing and some nonperiodic cases are counted as periodic. This is due to the larger overlapping boundary selected. This large overlapping boundary leads to the improper diagnosis. For the purpose of comparison, a periodic–quasiperiodic–chaotic diagram with overlapping boundary of 108 is plotted and shown in Fig. 1.21. As can be seen from the figure, many periodic cases are counted as nonperiodic and some of the quasiperiodic cases are considered as chaos. The improper diagnosis is due to the extremely small overlapping boundary used. As can be seen from the periodic–quasiperiodic–chaotic diagrams, the periodic cases of the system are indeed stable as most of the periodic areas in the diagrams are identical. As one may conclude, the Periodicity Ratio is suitable for quantitatively describing the periodicity of a dynamic system. In fact, various periodic behaviors of a system can be conveniently analyzed by Periodicity Ratios. The periodicities of the Duffing system are shown in Fig. 1.22. With this diagram, one may easily identify the periodicities of each state of the system and visualize the distributions of the periodicities. In Fig. 1.22, each point in the grey areas of the diagram represents a steady-state periodic motion for the nonlinear system. Different grey levels are used in the figure to distinguish the different periodicities, and the different grey levels correspondingly represent different stable states of periodic responses of the system.
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Fig. 1.21 Periodic–quasiperiodic–chaotic diagram for Duffing system under given overlapping boundary of 108
Fig. 1.22 Periodicities of the responses of Duffing system under the given overlapping boundary of 107
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Fig. 1.23 Three-dimensional periodic–quasiperiodic–chaotic diagram for Duffing system
The periodic regions are classified into more than 20 catalogues corresponding to more than 20 grey levels, as shown in the figure, based on their number of true periodicities and distinguishability. The periodic regions not distinguishable are ignored. The specific legends shown as a vertical bar can be found in the figure, where numerical numbers indicate the number of periodicity. One of some interesting phenomena found from this diagram is that the number of periodicities in most of the regions consistently and consecutively increases from 1 to 14 and the increment is accurately 1. It can also be found from the figure that the responses of the system in these regions are perfectly periodic, no irregular responses. Each grey level in the region represents a type of periodic response, and the boundaries of the colors are clearly laid out without transitional areas. The boundaries are critical to the periodicities of the system. The number of periodicity changes by one right after crossing a boundary and maintains unchanged inside the boundaries enclosing a single grey level. Based on the tests performed, these interesting phenomena can be clearly identified only if the overlapping boundary is properly selected. With the overlapping boundary and the Periodicity Ratios of high accuracy, transitional areas between two types of motion can be clearly distinguished. This is important for analyzing the transitional behaviors of these areas (Ueda 1992; Zhang and Yang 2007; Holmes and Rand (1976); Feng et al. (2006)). For the sake of clarity with respect to Periodicity Ratio, a 3-D periodic–quasiperiodic–chaotic diagram for the Duffing system is plotted as shown in Fig. 1.23. In the diagram, the color legends represent the PR values from 0 to 1. With this diagram, the transitional areas can be easily identified and the variations of the states of motion are actually quantified with the PR values, as can be visualized. From the diagram, for example, one may easily identify a transitional area from periodic to chaotic motions. Figures 1.24–1.27 show the phase diagrams taken from this area.
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Fig. 1.24 Phase diagram at k D 0:17, B D 10, and PRD 1
8 6 4 2 0 -2 -4 -6 -8 -4
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Fig. 1.25 Phase diagram at k D 0:16, B D 10, and PRD 0:3127
8 6 4 2 0 -2 -4 -6 -8 -4
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Fig. 1.26 Phase diagram at k D 0:15, B D 10, and PRD 0:0403
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8 6 4 2 0 -2 -4 -6 -8 -4
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Fig. 1.27 Phase diagram at k D 0:14, B D 10, and PRD 0:0
With the 3-D diagram, one may identify quantitatively with the Periodicity Ratios how the motion is gradually changed from a perfect periodic motion to a perfect chaotic motion. Again, accurate identification on the transition depends on the accurate determination of the overlapping boundary and corresponding Periodicity Ratios.
1.5 Key Symbols A B Ci B0 ; k0 H0 TN Pi P ./ Pp .A/ Q ./ Qi T X XP NPi PR NOP NPP
Designated event in the hypothesis test Amplitude of external excitation for Duffing system The number of chaotic points under a given boundary The selected parameters for the samples Null hypothesis The number of the entire population in the statistic test The number of periodic points under a given boundary Step functions The probability of hypothesis H( ) being rejected Step functions The number of quasiperiodic points under a given boundary Cycle, period Coordinate of point in Poincare map Coordinate of points in Poincare map The number of nonperiodic points under a given boundary Periodicity Ratio Number of overlapping points in the Poincare map Number of periodically overlapping points in the Poincare map
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d0 d .p1 ; p2 / k
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p p1 ; p2 s0 t v0 x xP y z
Overlapping boundary Distance between two points in Euclidean n-space Energy dissipation coefficient of Duffing system, integer, critical value indicating the cutoff point for the test statistic Sample number for each given overlapping boundary, integer, total number of points in a population The percentage of misjudged samples of a population Two points in Euclidean n-space Initial displacement of Duffing system Time Initial velocity of Duffing system Displacement Velocity Variable Variable
Greek ˛ !
Significance level Periodicity ratio Overall number of periodic points The frequency of excitation for a dynamical system Number of periodic points Number of overlapping points
n
References Dai L, Singh MC (1995) Periodicity ratio in diagnosing chaotic vibrations. In: Proceedings of the 15th Canadian congress of applied mechanics 1, pp 390–391. Dai L, Singh MC (1997) Diagnosis of periodic and chaotic responses in vibratory systems. J Acoust Soc Am 102(6):3361–3371. Dai L (2008) Nonlinear dynamics of piecewise constant systems and implementation of piecewise constant arguments. World Scientific, Singapore. Dai L, Singh MC (1998) Periodic, quasiperiodic, and chaotic behavior of a driven froude pendulum. Non-lin Mech 33(6):947–965. Dai L, Singh MC (2003) A new approach with piecewise-constant arguments toapproximate and numerical solutions of oscillatory problems. J Sound Vibrat 263:535–548. Dai L, Wang G (2008) Implementation of periodicity ratio in analyzing nonlinear dynamic systems: a comparison with lyapunov exponent. J Comput and Nonlin Dyn 3(1):338–349. Feng Z, Chen G, Hsu S (2006) Qualitative study of the damped Duffing equation and applications. Discrete and Continuous Dynamical Systems 6(5):1097–1112. Fisher RA (1966) The design of experiments, 8th edn. Hafner, Edinburgh. George EPB, Hunter WG, Hunter JS (1978) Statistics for experimenters: An introduction to design, data analysis, and model building. Wiley, New York.
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Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag New York, LLC. Han L, Dai L, Zhang HY (2009) Periodicity and nonlinearity of nonlinear dynamical systems under periodic excitation. ASME 2009 international mechanical engineering congress & exposition. Lake Buena Vista, USA. Holmes P, Rand D (1976) The bifurcations of Duffin’s equation: an application of catastrophe theory. J. Sound Vib 44:237–253. Lehmann EL, Romano JP (2005) Testing statistical hypotheses. 3rd edn. Springer, New York. Lu L, Lu Z, Shi Z (2000) The periodicity of the chaotic motion of a tension-slack oscillator in Hausdorff metric spaces. Mech Res Comm 27(4):503–510. Nayfeh AH, Mook DT (1989) Non-linear Oscillation, Wiley, New York. Nayfeh AH, Balachandran B (2004) Applied nonlinear dynamics: analytical, computational, and experimental methods non-linear oscillation. Wiley, New York. Parks PC (1992) Lyapunov’s stability theory – 100 years on. IMA J Math Contr Inf 9:275–303. Ueda Y (1980) Explosion of strange attractors exhibited by Duffing’s equation. In: Helleman RHG (ed) Nonlinear dynamics, 422–434. Ueda Y (1979) Randomly transitional phenomena in the system governed by Duffing’s equation. J Stat Phys 20:181–196. Ueda Y (1992) The road to chaos. Aerial Press, New York. Ueda Y (1980) Steady motions exhibited by Duffing’s equations: A picture book of regular and chaotic motions. In: Holmes PJ (ed) New approaches to nonlinear problems in dynamics, 311–322. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D: Nonlin Phenom 16(3):285–317. Zhang M, Yang J, (2007) Bifurcations and chaos in duffing’s equation. Acta Math Applicat Sinica (English series) 23(4):665–684.
Chapter 2
Nonlinear Modeling of Squeeze-Film Phenomena in Microbeam MEMS Reza N. Jazar
Abstract Oscillating microplates attached to microbeams is the main part of many microresonators and micro-electro-mechanical systems (MEMS). The sque-ezefilm phenomena appears when the microplate is vibrating in a viscose medium. The phenomena can potentially change the design point and performance of the micro-system, although its effects on MEMS dynamic are considered secondary compared to main mechanical and electrical forces. In this investigation, we model the squeeze-film phenomena and present two nonlinear mathematical functions to define and model the restoring and damping behaviors of squeeze-film phenomena. Accepting an analytical approach, we present the mathematical modeling of microresonator dynamic and develop effective equations to be utilized to study the electrically actuated microresonators. Then employing the averaging perturbation method, we determine the frequency response of the microbeam and examine the effects of parameters on the resonator’s dynamics. The nonlinear model for MEMS includes the initial deflection due to polarization voltage, mid-plane stretching, and axial loads, as well as the nonlinear displacement coupling of the actuating electric force. The main purpose of this chapter is to present an applied model to simulate the squeeze-film phenomena, and introduce their design parameters.
2.1 Introduction and Background This investigation presents two mathematical functions to describe stiffness and damping characteristics of squeeze-film phenomena. Micromechanical devices usually obey a complex set of equations of motion to model the tight coupling
R.N. Jazar () School of Aerospace, Mechanical, and Manufacturing Engineering, RMIT University, Melbourne, Victoria, Australia e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 2, © Springer Science+Business Media, LLC 2012
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of multiple energy domains of the system. Direct solution of these equations based on fully meshed structures is computationally intensive, making it difficult to use in system-level simulation. In order to perform rapid design prediction and optimization, accurate, and easy-to-use reduced-order MEMS device models must be built (Chen et al. 2004). In order to overcome the high computational cost problem and provide a fast and efficient system level simulation, the MEMS design community has converged to construct the low-order models. A common approach is based on parameterized lumped element representation for MEMS devices (Mukherjee et al. 1999; Fedder 1994). Cantilevers have shown to be reliable structures for resonators, in both macro and micro scales. However, shrinking increases the strength-to-weight ratio and allows devices to be constructed with aspect ratios that would not be possible in the macro domain. Also, surface tension and viscous forces become increasingly significant at smaller dimensions (Madou 2002). These viscous forces present a challenge in observing the oscillation of the microbeams even at pressures of a few hundred m torr (Harmany 2003). Furthermore, when the beam deflection is large, the linearized model deviates from the device nonlinear model significantly. To overcome the difficulties of modeling the interactions of so many phenomena that are involved in MEMS, a reduced-order nonlinear model is necessary. MEMS devices have mainly been designing by trial and error. As a result, MEMS design process requires several iterations before the required performance of a given device are satisfied. Experiments must be utilized to validate the theoretical prediction and clarify the order of contribution of the involved physical phenomena. Experimental results depend on the accuracy of the experimental devices and on the skills and knowledge of the experimenter. On the other hand, the FEM software are limited because they are time consuming, cumbersome, expensive, and they use numerous variables to represent the state of the system, where most of those variables are not important to the designer (Younis et al. 2003; Nayfeh and Younis 2004a,b). Conversely, the reduced-order models, known as micromodels, need to be expanded and improved as a basis for prediction and optimization tool of the proposed behavior. Reduced-order models are developed to capture the most significant characteristics of an MEMS behavior in a few variables (Younis et al. 2003; Mahmoudian et al. 2004). Typical MEMS devices employ a parallel capacitor of which one electrode is fixed and the other is attached to a microcantilever or microbeam. The unfixed electrode is a massive (compare to the mass of the microbeam) microplate capable to vibrate by using the flexibility of the microbeam. The microplate serves as a mechanical resonator. It is actuated electrically and its motion can be controlled and detected by capacitive changes. The fundamental resonant frequency of the vibrating microelement is sensitive to the strain and shifts with external loads. The shift can be converted to an electric signal in the capacitor, and can be utilized to measure the cause of external load (Younis et al. 2003). These microbeam systems have quite a wide range of application as sensors and actuators. They can be utilized to create highly sensitive force and mass sensors. These structures are also utilized in
2 Nonlinear Modeling of Squeeze-Film Phenomena
Micro
43
cantil
ever ss f ma
Proo
as ed g
suriz
Pres
Fig. 2.1 A microcantilever resonator and resistance of the pressurized gas
atomic force microscopy (AFM) and scanning probe microscopy (SPM), which can probe extremely small features with a high resolution. These microbeams can also be integrated into MEMS flow, pressure, and biochemical sensors (Gupta et al. 2003). Squeeze-film refers to thin-layer gas between the microplates of the capacitor as is shown in Fig. 2.1. The analysis of this phenomenon usually employs the Reynolds equation for isothermal incompressible fluid film. Isothermal equation makes it independent of the thermal effects. Finite element methods also make another basis to study the squeeze-film phenomena. However, finite element method approach, especially commercial software has problem to tackle with the nonlinear effects of many physical parameters (Younis et al. 2003; Younis 2004). There exist both, damping and stiffness factors in the squeeze-film phenomena. The factors arise from the effects of the surrounding gas medium. The damping factors are related to the way in which energy is transfer to the gas medium through thermal effects. Therefore, the quality factor of the cantilevers will be affected by squeezefilm damping effects. The stiffness factor, however, appears because the air under the microplate is acting similar to a visco-elastic member that would exhibit resistance to compression. In special circumstances, this would cause an increase in the effective spring constant of the system, which would outweigh the added damping effects and actually shift the resonant frequency up at increasing pressures (Yang et al. 1997a; Yang and Senturia 1997b; Yang 1998; Younis 2001; Harmany 2003). Many studies have been reported in the late 20th and the begining of 21st century on squeeze-film damping, most notably by Starr (1990); Andrews et al. (1993); Andrews and Harris (1995); Yang and Senturia (1996); Yang et al. (1997a); Yang and Senturia (1997b); Veijola et al. (1998); Veijola and Mattila (2001); AbdelRahman et al. (2002); Younis et al. (2003). Analytical studies are based almost exclusively on a linearized form of the Reynolds equation, solutions to which have been derived for rectangular and circular geometries under the simplified boundary condition that the acoustic pressure vanishes at the structure’s perimeter (Langlois 1962; Griffin et al. 1966; Blech 1983). Modifications to the equation have been made to account for more realistic boundary conditions (Darling et al. 1998) as well as slip flow at the solid–fluid interface (Sun et al. 2002) and the inclusion of
44
R.N. Jazar
damping holes in the proof mass (Bao et al. 2003). Nonetheless, analytical solutions are inherently limited to simple geometries and numerical approaches are required to model more complex structures (Houlihan and Kraft 2005). In Chen and White (2000) and Chen and Kang (2000), the system first linearizes around an equilibrium point, and then extracts a subspace for reduced-order modeling. The states of the initial nonlinear system are projected onto the reduced state space. Nayfeh and Younis (2004a,b) and Abdel-Rahman et al. (2004) have provided a very well and extensive review and analysis on the attempts for expressing the effects of squeeze-film in MEMS dynamics simulating the squeeze-film effect on a microbeam with an externally distributed pressure. From a mathematical modeling viewpoint, most of the previous modeling have come up with equivalent mass, spring, viscosity, or damping affected by squeezefilm. Among them, Zhang et al. (2004) is a notable, where the authors have tried to model a clamped–clamped microbeam under two external loads; the electric and distributed squeezed film pressure. Assuming that the pressure is governed by the isothermal incompressible Reynolds equation they solved the problem using a Fourier series approach. It means they assumed a linear electric force. They modeled the effects of squeeze-film, by defining an effective mass and effective damping parameters called squeeze inertia and squeeze damping numbers expressed by two complicated series. Then the squeeze numbers were used in an equivalent linear mass-spring-dashpot resonator. Their analysis and approach has the shortcoming of assuming a linear electric force. The squeeze-film has no apparent effect within the order of displacements that the electric force may be assumed linear. There have been several works which have dealt with the steady-state oscillation of such parametric and externally forced microbeams. The modeling and application of microbeam-based MEMS are described in some references such as Younis et al. (2003); Younis and Nayfeh (2003); Younis (2001, 2004); Malatkar (2003); Najar et al. (2005), using different viewpoints. However, the effects of slowly varying quantities in microbeams-based MEMS have not been fully investigated; specifically, when the unperturbed system contains a homoclinic manifold, which separates regions of qualitatively different behaviors. Nayfeh and co-workers have studied electrically actuated clamped–clamped microbeams in capacitive microswitches subject to a full nonlinear model (Abdel-Rahman et al. 2004, 2002; Zhao et al. 2004; Vogl and Nayfeh 2005; Nayfeh et al. 2005; Younis and Nayfeh 2005a,b). They focused on the damping mechanisms in microbeams (Nayfeh and Younis 2004a,b), and showed that squeeze-film and thermoelastic damping are the main sources of damping in microbeams vibration. Thermo-elastic effect is usually modeled by an equivalent viscous damping, while the squeeze-film phenomenon is analyzed by a coupled elesto-fluid problem. The analysis of electric actuated microresonators often proceeds by abridging the system to a linearly damped Mathieu equation, with or without a synchronized forcing term. Then the resonance conditions of the equation are approximated and
2 Nonlinear Modeling of Squeeze-Film Phenomena
45
carried out using the Mathieu stability diagram. Although such an analysis might be valid on short timescales within a small domain of control parameter space in the vicinity of primary resonance, the long-term behavior of the system can show significant differences. Even if the critical frequency ratios are assumed to be known a priori, these assumptions can fail to account for qualitative changes in the dynamical behavior of the system. We present two mathematical models to simulate stiffness and damping effects of squeeze-film. Having a mathematical description simplifies the analysis and provides the power of prediction. A reliable mathematical model provides the ability to develop prototypes and design the best performance. Squeeze-film phenomena in microresonators provide mechanisms by which the system can undergo unexpected and undesired qualitative changes in their dynamical behavior and lose their functionality. However, inclusion of these phenomena can improve the design and functionality of microresonators.
2.2 Mathematical Modeling of Microresonators 2.2.1 Reduced-Model of Microresonators Electrostatic actuation is achieved by applying a voltage difference between opposite electrodes of the variable capacitor. The induced electrostatic force deforms the capacitor until they are balanced by the restoring mechanical forces. The electric load is composed of a direct current (DC ) polarization voltage, vp , and an alternative current (AC ) actuating voltage. Polarization voltage vp , makes the system more complicated with interesting behaviors, and probably more effective and controllable microresonators. Electrostatic force of polarization voltage introduces collapse load or pull-in effect, where the mechanical restoring force of the microbeam can no longer resist the opposing electric forces. So a continuous increase in deflection occurs that leads to a short circuit in the electric field (Younis 2004). The polarization voltage controls the position and stability of equilibria. More specifically, the position and stability behavior of the origin is important, since it is to be act as the center of oscillation. It becomes unstable for large values of the polarization voltage vp , but because of nonlinearity and excitation, as soon as the resonance causes the amplitude to increase, period dependency of the amplitude causes the resonance to detune and decrease its tendency to produce large amplitude. In other words, nonlinearity helps to limit the amplitude at resonance. A fact that will not be seen in the linearized model and will be shown in the following sections. Analysis shows that electric force affects the restoring force with a softening effect. Therefore, the electric force tends to shift the natural frequencies to lower values. One-dimensional electrostatic force, fe , per unit length of the beam is
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R.N. Jazar
a -w x
m d
+ vp -
v=vi sin(ω t)
b
-w x
m d
+ vp -
v=vi sin(ω t) Fig. 2.2 A microcantilever and a clamed-clamped model of microresonator to keep the capacitor’s plates parallel
fe D
2 " 0 a v vp 2 .d w/2
v D vi sin !t;
(2.1) (2.2)
where "0 D 8:854187817620 1012 A2 s4 = kg m3 is permittivity in vacuum, A is the area of the microplate, and w D w.x; t/ is the lateral displacement of the microbeam. The complete microresonator is composed of a beam resonator, a ground plane underneath, and one (or more) capacitive transducer electrode/s. A DC -bias voltage, vp , is applied to the resonator while an AC excitation voltage is applied to its underlying ground plane/s. A single capacitor clamped–clamped and a cantilever model of the resonator are illustrated in Fig. 2.2. 2 The inertia force per unit length of a vibrating microbeam is fi D @@tw2 , where is the mass per unit length. Bending, axial force, and mid-plane stretching are the restoring forces in microbeams flexing. The mechanical restoring forces of microbeams are usually hardening and therefore, they tend to shift the natural frequencies to higher values. The restoring force due to rigidity of the microbeam 4 is fr D EI @@xw4 and the restoring force due to internal tension of the microbeam is 2
ft D P @@xw2 . However, the internal tension force, P , is because of theRincreasing the length due to deflection mid-plane stretching PL= .Ea0 / D ı D .ds dx/ R 1 L @w 2 dx (Esmailzadeh et al. 1997; Nayfeh and Mook 1979). 2 0 @x
2 Nonlinear Modeling of Squeeze-Film Phenomena
47
The viscous damping fv D c @w is the most common sources of energy @t dissipation. However, viscous damping is usually a substitution for internal, thermal, structural, intrinsic and squeeze-film dampings. In order to analyze squeeze-film damping, we assume no such equivalent viscous damping is present.
2.2.2 Modeling the Squeeze Film Phenomena Generally speaking, the efficiency of actuation and the sensitivity of motion detection of the microresonators improves by decreasing the distance between the capacitor electrodes, and increasing the effective area of the electrodes (White 2002; Lyshevski 2001; Zhang et al. 2004). Under these conditions, the squeezefilm phenomena appear if the MEMS device is not in a vacuumed capsule. Although a vast amount of microresonators are fabricated to operate in a partially vacuumed capsule, there are still a lot of applications that the resonator must work in direct contact with air or other fluids (White 2002). Squeeze-film phenomena are the result of the massive movement of the fluid underneath the plate, which is resisted by the viscosity of the fluid. A non-uniform pressure distribution gives rise underneath the movable plate, which acts as a spring and a damper. The equivalent spring and damping rates are dependent on the frequency and amplitude of oscillation. It is estimated that the damping force is more important at low frequencies, whereas the spring force is more significant at high frequencies (Nayfeh and Younis 2004a,b). To analyze the squeeze-film effects, the non-uniform pressure distribution of the gas film may be added as an external force to the equation of lateral motion of the microbeam. The pressure, which is dependent on the distance between electroplates, must then be found using fluid dynamics. Hence, the phenomenon is a coupled elasto-fluid problem. Some researchers have used the incompressible isothermal Reynolds equation to solve the coupled elasto-fluid problem approximately (Younis and Nayfeh 2003; Nayfeh and Younis 2004a,b; Shi et al. 1996; Zhang et al. 2004; Hung and Senturia 1999). Using this approach, the squeeze-film load is evaluated by R b=2 the resultant of the pressure distribution fp D b=2 p dy, and pressure is calculated using Reynolds equation r w3 prp D @.pw/ where, D 12= .1 C 6K/ is the @t effective viscosity (Burgdorfer 1959; Yang 1998), K is the Knudsen number, and b is the width of the microelectrode plate (Younis 2004; Lyshevski 2001; Rewienski 2003). A better analysis for squeeze-film effects might be coupling the oscillating system with heat convections to make an elasto-thermo-fluid problem, and utilizing compressible and non-isothermal fluid dynamics. However, such an analysis would be time consuming, cumbersome, and case dependent and in general does not provide general rules of design for such systems. Therefore, micromresonator devices obey a complex set of equations that must account for the tight coupling of multiple energy domains of the system. Direct solution of these equations based on fully meshed structures is computationally
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intensive, making it difficult to use in system-level simulation. In order to perform rapid design prediction and optimization, accurate and easy-to-use reduced-order MEMS device models must be built (Chen et al. 2004). In order to overcome the high computational cost problem and thus to provide fast and efficient system level simulation, the MEMS design community has converged upon the technique of solving MEMS device dynamical behaviors by constructing low-order device models to match with the direct analysis. A common approach for doing this is based on parameterized lumped element representation for MEMS devices (Mukherjee et al. 1999). When the beam bends, the pressure distribution of the ambient air under the microplate increases. This pressure increase produces a backward pressure force that damps the beam motion. The dynamic behavior of this coupled electromechanical fluidic system can be modeled with the beam equation and the Reynold’s squeeze-film damping equation (Hung and Senturia 1999; Gupta and Senturia 1997; Pan et al. 1998; Shi et al. 1996; Starr 1990; Yang and Senturia 1996; Younis 2004) as @2 w @w Ea0 @4 w 2 Cc C EI 4 D @t @t @x L Z C
1 2 b=2
Z
L 0
p dy C
b=2
r w3 prp D
!
2
@2 w @x 2 2 "0 a v vp
@w @x
12 @ .pw0 / ; 1 C 6K @t
dx
2 .d w/2
(2.3) (2.4)
where w.x; t/ is the height of the beam above the substrate, and p.x; y; t/ is the pressure distribution under the beam. (2.3) is distributed along the x-axis. Nominal value of the parameters involved are: beam length L D 610 m, width b D 40 m, thickness h D 2:2 m, initial gap d D 2:3 m, material Young’s modulus E D 149G Pa, density =.hb/ D 2330 kg= m3 , air viscosity D 1:82 105 kg= ms, moment of inertia I D bh3 =12, Kundsen’s number K D =w, and the ambient air Pa D 1:013 105 Pa (Chen et al. 2004). The mean-free path of the gas D 0:064 m. The size and mass of the microplate is a D 150 150 m2 and m D 1:15 1010 kg. The effective viscosity was under investigation for more than a century (Yang 1998), while determined by Burgdorfer (1959). Equations (2.3) and (2.4) show that simulating the dynamic behavior of the device involves squeeze-film, mechanical and electrostatic components. The system is nonlinear due to the nonlinear nature of the squeeze-film force, nonlinear rigidity of mid-plane stretching, and the nonlinear electrostatic force. The Reynolds equation is a combination of the Navier–Stokes equation, the continuity equation, and the state equation Yang (1998). The derivation of the isothermal Reynolds equation starts with integrating the Navier–Stokes equation in y-axes, then obtaining mass flow equations which are functions of the pressure gradient in x and y directions. Then substituting the results into the continuity equation and obtain a
2 Nonlinear Modeling of Squeeze-Film Phenomena
49
primitive form of the Reynolds equation, which is a function of density and, is temperature dependence. Under the assumption of isothermal conditions, we can further eliminate the density term by using the equation of state and thus obtain the isothermal Reynolds equation. The Reynolds equation derived under these assumptions: 1. The fluid is Newtonian. 2. The fluid obeys the ideal gas law. 3. The inertia and body forces are negligible compared to the viscous and pressure forces. 4. The variation of pressure across the fluid film is negligibly small. 5. The flow is laminar. 6. The width of the gap separating the two plates, where the gas is trapped inside, is very small compared to the lateral extent of the plates. 7. The fluid can be treated as continuum and does not slip at the boundaries. 8. The system is isothermal. Since there are no analytical solutions to (2.3) and (2.4), some researchers necessarily solved the equations numerically under more simplifications. The Reynolds equation can be linearized around the ambient pressure and gap size (Nayfeh and Younis 2004a,b). Now, ignoring the nonlinearities in equation reduces the problem to a couple of linear partial differential equations, which can still be solved numerically only. Based on these assumption, it is estimated that the squeeze-film damping force is more important at low frequencies, whereas the squeeze-film spring force is more significant at high frequencies (Nayfeh and Younis 2004a,b). In case of a rectangular parallel plate oscillating in normal direction with a given small harmonic motion, and the pressure boundary conditions on the edges of the plates are equal to ambient pressure, the analytical solutions of pressure force components of the linearized Reynolds equation are found by Blech (1983). However, in microresonator dynamic analysis, the displacement oscillation of the microbeam is what the analysis must find and therefore, the Blech analysis can only be used as a rough estimation.
2.2.2.1 Damping Model We separate the two characteristics of the squeeze-film phenomena and model the damping and stiffness effects individually. Motion of the microplate and flow of the gas underneath is similar to the function of a decoupler plate in hydraulic engine mounts (Golnaraghi and Jazar 2001; Jazar 2002). More specifically, the squeeze-film damping is qualitatively similar to the function of decoupler plate in hydraulic engine mounts, which is an amplitude-dependent damping (Christopherson and Jazar 2005). It means squeeze-film damping effect is a positive
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R.N. Jazar
phenomenon to isolate the vibration of microplate from the substrate. Following Golnaraghi and Jazar (2001), and utilizing the aforementioned similarity, we model the squeeze-film damping force, fsd , by a cubic function, fsd D cs w2
@w @t
(2.5)
where the coefficient cs is assumed constant and must be evaluated experimentally. The equation expresses a stiffness model because of the displacement term @w=@t, and it is nonlinear because the coefficient cs w2 of @w=@t is displacement dependent. The equation expresses a damping model of the velocity term because @w=@t, and it is nonlinear because the coefficient cs w2 of @w=@t is displacement dependent. The displacement dependency is of second degree to indicate that the damping coefficient approaches high values when the electrodes get close to each other. However, the amplitude of vibration is practically less than the gap size and hence, when the electrodes get close the speed approaches zero and makes squeezefilm damping zero. Considering these physical constraints and dynamic behavior of the damping, a general damping function can be fsd D
cs0
w 2i C2 @w 2i C1 d
i 2N
@t
(2.6)
where the function (2.5) is the simpest model.
2.2.2.2 Stiffness Model In the simplest case, we present the following fifth degree function to simulate the spring force, fss , of the squeeze-film phenomenon, simply because at low amplitudes, w 0, the fluid layer is not strongly squeezed and there is no considerable resistance. On the other hand, at high amplitudes, w d , there is not much fluid to react as a spring. In addition, speed is proportionally related to the squeezeness of trapped fluid.
@w fss D ks .d w/ w @t 2
2 (2.7)
The coefficient ks assumed constant and must be evaluated experimentally. The coefficients cs and ks are dependent on geometry as well as dynamic properties of the fluid, but assumed to be independent of kinematics of the microbeam such as displacement and velocity. The equation expresses a stiffnes model because of the displacement term w, and it is nonlinear because the coefficient ks .d w/2 .@w=@t/2 is displacement and velocity dependent. The displacement dependency is of second degree to indicate that the stiffness coefficient approaches low values when the electrodes get close
2 Nonlinear Modeling of Squeeze-Film Phenomena
51
to each other. It is velocity dependent to indicate that the stiffness effect of the film is higher at higher speeds and is zero at no speed. Considering these physical constraints and dynamic behavior of the stiffness, a general stiffness function can be 2i C2 @w i 2N (2.8) fss D ks0 .d w/2i C2 w2i C1 @t where the function (2.7) is the simpest model.
2.3 Equation of Motion Including mathematical modeling of squeeze-film effects, the following equation is to be utilized for dynamic analysis of the microresonator behavior using w D w.x; t/ for lateral displacement of a microbeam. 2 @2 w @4 w @w 2 2 @w 2 C EI 4 C cs w C ks .d w/ w @t @x @t @t 2 Z L 2 ! 2 "0 a v vp Ea0 1 @w @w D dx C : L 2 0 @x @x 2 2 .d w/2
(2.9)
We define the following variables to make the equation of motion dimensionless. The parameter n is a constant depending on mode shape of the microbeam. D !1 t !1 D zD yD Y D rx D rD a1 D
n2 L2
(2.10)
s EI
x L w d w0 d ! !x ! !1 "0 aL4 2d 3 EI
(2.11) (2.12) (2.13) (2.14) (2.15) (2.16)
(2.17)
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R.N. Jazar
a0 d 2 n I
(2.18)
cs d 2 L2 a4 D p EI
(2.19)
a3 D
a5 D
ks d 4 :
(2.20)
Using these parameters, the equation of motion transforms to the following dimensionless equation. 2 @2 y @4 y @y 2 2 @y C a C C a y .1 y/ y 4 5 @ 2 @z4 @ @ Z 2 v vp @2 y 1 1 @y 2 D a3 2 dz C a1 : @z 2 0 @z .1 y/2
(2.21)
We apply a separation solution, y D Y ./ ' .z/, and assume a first harmonic function as the mode shape of the deflected microbeam. By accepting a first harmonic shape function, the temporal function Y ./ would then represent the maximum deflection of the microbeam, which is the middle point for symmetric boundary conditions, and the tip point for microcantilever. Therefore, the differential equation describing the evolution of the temporal function Y ./ would be YR C Y C Y 3 C a4 Y 2 YP C a5 .1 Y / Y YP 2 p 1 .˛ C ˇ/ C 2 D 2˛ˇ sin .r/ ˇ cos .2r/ ; .1 Y /2
(2.22)
where ˛ D a1 v2p
p 2 2˛ˇ D 2a1 vp vi a1 ˇ D v2i 2 D n2 a3 Z 1 1 @' 2 2 dz: n D 2 0 @z
(2.23) (2.24) (2.25) (2.26) (2.27)
The first harmonic mode shape to satisfy the required boundary conditions for a microcantilever is z (2.28) ' .z/ D 1 cos 2
2 Nonlinear Modeling of Squeeze-Film Phenomena
53
and therefore, nD
: 4
(2.29)
(2.22) is an effective and applied reduced model of the microresonator including the squeeze-film phenomena as well as geometric nonlinearities and exact electrostatic actuating force.
2.4 Initial Deflection The cubic stiffness term in (2.22) is a result of mid-plane stretch. Let us assume mechanical restoring force is denoted by fm D Y C Y 3 , 0 Y 1. Applying a polarization voltage affects the equilibrium positions of the system and bends the microbeam. Consequently, the rest position of the microbeam would not be at Y D 0. Assume the polarized MEMS has the rest point at Y D Y0 , 0 Y0 1, instead of Y D 0. This will translate the origin of measuring fm and introduces a new mechanical restoring force as fm D 1 3Y02 Y 3Y0 Y 2 C Y 3 , 0 Y 1. Therefore, a second degree restoring force must be added to the equations of motion to include the initial bending of the microbeam. Note that including second degree restoring force makes the system asymmetric, which generates its own problems when we try to solve the system using perturbations and other approximate methods. The initial displacement Y0 is a function of polarization voltage and can be determined by searching for equilibria of the MEMS. The equilibria would be found by ignoring time variations in (2.22)
1 C Y 2 .1 Y /2 Y D ˛:
Polarization voltage and initial bending of the microbeam introduce a new problem. When the polarization voltage of the inactive MEMS is not zero, the clearance between the two plates of the capacitor would not be d , and therefore, the limit of Y would be less than 1. It might be better to redefine the equations in order to have the bent rest point always as a zero equilibrium position (Mahmoudian et al. 2004). In this case, the gap size, d , is a function of polarization voltage.
2.5 Mathematical Analysis Steady-state response of the system can be utilized to determine the sensitivity of the steady-state response of the system to parameter variation, as well as frequency shifts and amplitude changes. As is seen from (2.22), nonlinearity is due to electrostatic actuation, mid-plane stretching, squeeze-film damping and squeeze-film stiffness. To clarify the effects of squeeze-film phenomena, we assume
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R.N. Jazar
the system is linear everywhere except for squeeze-film effects and we keep the squeeze-film stiffness and damping as presented in (2.5) and (2.7). This system is governed by the following forced nonlinear Mathieu equation. YR C a4 Y 2 YP C a5 .1 Y /2 Y YP 2 p C 1 2 ˛ C ˇ C 2 2˛ˇ sin .r/ ˇ cos .2r/ p D ˛ C 2 2˛ˇ sin .r/ C 2ˇ sin2 .r/ :
(2.30)
Following the averaging method, in order to find the amplitude of steady state oscillation of the microbeam around resonance and taking care of initial displacement due to polarization, we assume a solution in the following form Y D A0 C A ./ sin .r C YP D A ./ r cos .r C
.t//
(2.31)
.t// :
(2.32)
These solutions are applied provided that AP ./ sin .r C
.t// C A ./ P .t/ cos .r C
.t// D 0:
(2.33)
Substituting (2.31) and (2.32) in (2.30) leads to an equation, which must be utilized to find the secular term A0 D
˛Cˇ 1 2˛ 2ˇ
(2.34)
That equation along with (2.33) must be used to derive a couple of first-order differential equations for AP ./ and P .t/, say AP ./ D f1 .a4 ; a5 ; ˛; ˇ; r; A; '/
(2.35)
A ./ P ./ D f2 .a4 ; a5 ; ˛; ˇ; r; A; '/;
(2.36)
where ' ./ D r C
./:
(2.37)
Assuming AP ./ and P ./ are slow variables and their average remains constant during one cycle, we substitute their equations with their integral over one period of oscillation. Z 2 A2 2 P A ./ D A ./ d' D A0 C rAa4 C ˇA sin 2 r 4 0 p (2.38) 2 2˛ˇ .1 C 2A0 / sin r
2 Nonlinear Modeling of Squeeze-Film Phenomena
Z
2
./ D 0
P .t/ d' D rA
55
1C
3A20
A2 A0 C 2 2
A
CA 1 r 2 2˛ 2ˇ ˇ cos2 p : 2 2˛ˇ .1 C 2A0 / cos rA
3r
2
4
a5
(2.39)
At steady-state conditions, A ./ and ./ must not vary in time and therefore, the left-hand sides of (2.38) and (2.39) are zero. Eliminating ./ provides a relationship between the parameters of the system to have a periodic steady-state response with frequency r. Z1 r 4 C Z2 r 2 Z3 .Z4 C Z5 /
p Z6 C Z7 D 0:
(2.40)
The parameters Zi , i D 1; 2; : : : ; 16, are dynamic parameters of the system and are independent of the excitation frequency. Z1 D a52 ˇ2 A8 A4 C 12A20 16A0 4 A2 C4 1 C 9A40 24A30 C 22A20 8A0 (2.41) C16a5 ˇ2 A6 A2 6A20 C 9A 2 C 64ˇ2 A4 Z2 D 16a5 ˇ2 A6 .2ˇ C 2˛ 1/ 6A20 C 8A0 A2 2 C4ˇ 2 A8 a42 C 8a5 C 32ˇ 2 A6 a42 A20 C 2a5 3A20 C 4A0 C 1 (2.42) Z3 D 64ˇ 2 A4 .1 4˛/ 256ˇ 3 A2 ˛ .2A0 C 1/2 p Z4 D 128ˇ 2 A2 A0 2˛ˇ p Z5 D 64ˇ 2 A2 2˛ˇ Z6 D a5 ˇA4 r 2 2 C 6A20 C A2 8A0 C 32˛ˇA0 .1 C A0 / C8˛ˇ 1 2A2 C 8ˇA2 1 r 2 ˇ Z7 D 64ˇ 2 A4 8˛ˇ C 3ˇ 2 4ˇ C 4˛ 2 :
(2.43) (2.44) (2.45)
(2.46) (2.47)
Note that nonzero excitation is necessary to have a nontrivial response. The effect of variation of each parameter on the dynamic response of the MEMS will be examined in the next section. Since the frequency response equation cannot be solved for any parameters, a numerical analysis is needed to evaluate the overall effect of parameters. In what follows, the analytic description (2.40) is utilized to describe the behavior of steady-state dynamic of the system around resonance.
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2.6 Dynamic Analysis The steady-state response of microresonators reflect important behavior of resonator-based MEMS. The resonance frequency determines the amount of force the structure can exert while the frequency response establishes the pick amplitude of oscillation. These characteristics indirectly affect design parameters such as sensitivity, quality factor, switching frequency, and external noise (Sudipto and Aluru 2004). Amplitude of oscillation of the microresonator at steadystate conditions is determined by (2.40) indicating that its dynamics governs by polarization voltage parameter ˛, alternative excitation voltage parameters ˇ, the excitation frequency ratio r, as well as the squeeze-film damping and stiffness parameters a4 , and a5 . In order to demonstrate the dependency of steady-state behavior of the MEMS, we graphically illustrate the frequency response for various parameters. The nominal values of a sample microcantilever and adapted from Yang et al. (2002); Khaled et al. (2003), although because of the dimensionless forms of the equations, any other microresonator can be analyzed as well. Figures 2.3–2.6 depicts the effect of variation of polarization voltage for a set of parameters. The amplitude of steady-state oscillation increases by increasing the polarization voltage. Almost no squeeze-film effect is shown in Fig. 2.3, where a4 is set to a small value and a5 is set to zero. Since we eliminated every kind of damping except the squeeze-film damping, a small squeeze-film damping is needed to simulate dissipation mechanisms presented in the system. Low squeeze-film damping is similar to viscous damping and shows no nonlinear effects qualitatively. Squeeze-film stiffness affects the restoring force of the system in a nonlinear fashion as shown in Fig. 2.4. This effect is important for resonator mass sensors which have been analyzed and designed by assuming constant stiffness. Increasing the damping diminishes the response of the system and dominates the nonlinear
Fig. 2.3 Effect of variation of polarization voltage on frequency response of the microbeam
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Fig. 2.4 Effect of variation of polarization voltage on frequency response of the microbeam
Fig. 2.5 Effect of variation of polarization voltage on frequency response of the microbeam
effects of little stiffness. This effect is shown in Fig. 2.5. Figure 2.6 shows the frequency response of the microbeam with a little damping and a high squeeze-film stiffness effects. It shows that squeeze-film stiffness shifts the resonance to higher frequencies and can introduce a resonance amplitude dependency as well as jump and instability occurrence. The peak value of the frequency response at resonance is a monotonically increasing function of the polarization voltage. Therefore, there must be a maximum acceptable ˛, due to constraint Y < 1. It is shown in Figs. 2.7–2.10 that the response of the system to variation of the excitation voltage has the same pattern as changing the polarization voltage.
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Fig. 2.6 Effect of variation of polarization voltage on frequency response of the microbeam
Fig. 2.7 Effect of variation of excitation voltage on frequency response of the microbeam
More specifically, the peak value increases and the resonant frequency shifts to higher frequencies when the amplitude of the excitation voltage increases. Squeezefilm damping and stiffness also have the same effects as described for polarization voltage variation. There must also be a higher limit for the excitation voltage to have oscillation within the gap size limit. Variation of the squeeze-film damping is illustrated in Figs. 2.11and 2.12. Increasing damping diminishes the amplitude of the oscillation as expected. It is known that decreasing the damping increases the quality factor of the system and is not considered a positive phenomenon. Therefore, damping has two main roles in MEMS dynamics. It should exist to control the system to stay within the physical
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Fig. 2.8 Effect of variation of excitation voltage on frequency response of the microbeam
Fig. 2.9 Effect of variation of excitation voltage on frequency response of the microbeam
limit Y < 1; and on the other hand, it must be as low as possible to provide high quality factor (Younis 2004). In Fig. 2.11, the squeeze-film stiffness is ignored, while there is some stiffness in Fig. 2.12. The effect of backbone tilting is obvious when squeeze-film stiffness is present. Variation of the squeeze-film stiffness, which is equivalent to oscillating in a viscouser media, is shown in Figs. 2.13 and 2.14. Comparing Figs. 2.13 and 2.14 indicates that introducing damping reduces the pick amplitude but keeps the qualitative behavior of the system. Variation of the stiffness has almost no or little effect on the pick amplitude; however, bending the backbone curve is a significant effect. Hence, there must be a maximum allowable stiffness to avoid jump.
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Fig. 2.10 Effect of variation of excitation voltage on frequency response of the microbeam
Fig. 2.11 Effect of variation of squeeze-film damping on frequency response of the microbeam
Frequency response analysis has shown that the squeeze-film stiffness has the most impact on resonance frequency and affects as a hardening spring. However, polarization and excitation voltages also have a softening effect and shift the resonance frequency to lower values. On the other hand, squeeze-film damping has high effects on decreasing the peak amplitudes with least affect on resonance frequency shift. The presented model cannot be validated without looking at the equivalent linear damping and stiffness of the model. In other words, the most important fact showing the proposed dynamic can model the squeeze-film phenomena effectively is the
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Fig. 2.12 Effect of variation of squeeze-film damping on frequency response of the microbeam
Fig. 2.13 Effect of variation of squeeze-film damping on frequency response of the microbeam
equivalent linear stiffness and damping, predicted for the first time by Blech (1983). Figures 2.15–2.16 depict the equivalent damping and stiffness, where keq , and ceq are traditional linear stiffness and damping per unit length of the microbeam. The equivalent stiffness keq , and damping ceq , are evaluated numerically by comparing the frequency response of the modeled microresonator with a microresonator having linear damping and stiffness. The equivalent damping is illustrated in Fig. 2.15 and the equivalent stiffness is depicted in Fig. 2.16. The results are quite comparable and in agreement with what presented by Blech (1983). The squeeze-film stiffness
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Fig. 2.14 Effect of variation of squeeze-film damping on frequency response of the microbeam
Fig. 2.15 Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients
ks and damping cs coefficients, hidden in a5 and a4 , are proportional to their linear equivalents keq and ceq . The equivalent squeeze-film damping is higher around resonance and decreases beyond that, justifying a decreasing effect on resonant amplitude. However, the equivalent squeeze-film stiffness is a monotonically increasing function of both frequency and squeeze-film stiffness coefficient. Therefore, stiffness has more effect at high frequencies, while damping affects the system at low and resonance frequencies.
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Fig. 2.16 Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients
2.7 Conclusion and Future Work This work introduces a mathematical model to simulate the effects of squeeze-film phenomena on dynamics and performance of microbeam-based electromechanical resonators. Squeeze-film phenomena, also referred to as “squeeze-film damping,” strongly influence the performance of microelectro-mechanical systems, and therefore is an important parameter to be considered in design and control of such systems. To improve the efficiency of actuation and sensitivity, the gap between the capacitor plates is usually kept minimized creating more squeeze-film resistance to the motion of the microplate in the viscous fluid. The squeeze-film damping is a deep-rooted multidisciplinary subject in dynamics of thermo-fluidsolid interactions. In this research, the squeeze-film parameters are modeled by two functions in terms of equivalent damping and the overall stiffness of the system. The changes in the stiffness and damping forces, which are, respectively, modeled by a fifth and a third degree nonlinear functions, are assumed to be frequency and amplitude dependent. The dynamics model for microresonators focusing on modeling of the impacts due to squeeze-film in performance of such systems is analyzed utilizing a dimensionless equation of motion and employing averaging perturbation method. The frequency-amplitude relationship of the system is presented by an implicit equation. A sensitivity analysis has been performed to illustrate the effect of the phenomena in steady-state response, frequency shift, and pick amplitude of the oscillation. It is shown that the damping has little effect on resonant shift but diminishes the pick amplitude of oscillation significantly. The nonlinear characteristic of the stiffness model affects the steady-state dynamic behavior of the system dramatically. Due to hardening behavior, the backbone of the frequency response bends to higher frequencies and introduces jump in high values. Validation of the model has
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been done solely by evaluating the equivalent frequency-dependent stiffness and damping. It was shown that the behavior of such equivalent coefficients qualitatively match with reported results. The mathematical modeling is consistent to enhance the exploration as well as design and control of MEMS more effectively. To show the direction of next steps in improvement of the presented theory, it should be noted that the development of the dynamics model and the theoretical sensitivity analysis needs to be along with conduction of proper experiments. Note that the presented models do not necessarily require micro-scale systems to be applied. Hence, the model can be confirmed and coefficients may be evaluated utilizing a meso scale resonator, although the squeeze-film phenomena are significant in micro scale. Dynamics of microresonators must also be examined under the effect of squeeze-film phenomenon in time space to analyze and validate transient responses. The introduced coefficients, which are assumed to be independent of the frequency and amplitude of oscillation, must be determined for individual cases to determine its dependency to the viscosity of media, as well as geometry of the microbeam. In the presented analysis, the nonlinearities of the flexural rigidity and electrostatic forces are ignored to clarify the influence of the squeeze-film phenomena. In another attempt, these nonlinearities must be included to be compared with the effects of phenomena.
2.8 Key Symbols ai a a0 A A0 b c cs d E fe fi fm fr fsd fss ft fv fTd
Dimensionless parameters of equation of motion Effective area of electrode plate Cross-sectional area of the beam Amplitude Initial deflection Width of the microbeam Viscous damping rate Thermal damping force coefficient Gap size Young modulus Electric force Inertia force Mechanical restoring force Bending restoring force Squeeze-film damping force Squeeze-film stiffness force Tension force Viscous damping force Thermal damping force
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fTs fv h I K ks l lT L m n P r s t vp V w w0 x x0 xd y Y z Z
Thermal stiffness force Viscous damping force Thickness of microbeam and microplate Cross-sectional second moment Knudsen number Squeeze-film stiffness force coefficient Linear dimension Thermal diffusion length Length of microbeam or microcantilever Mass Mode shape parameter Longitudinal internal tension force Dimensionless excitation frequency Curved beam length Time Polarization voltage Volume Lateral displacement Maximum lateral displacement Longitudinal coordinate Initial longitudinal stretch Dynamic longitudinal stretch Lateral coordinate, dimensionless lateral displacement Dimensionless amplitude of oscillations at maximum Longitudinal dimensionless coordinate Short notation parameter
Greek ˛ Polarization voltage parameter ˇ Alternative voltage parameter Viscosity Coefficient of cubic mechanical stiffness force Thickness of microplate Conductivity ' Mode shape function Phase angle "0 Permittivity in vacuum Dimensionless time Symbol overdot
d ./ =dt
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References Abdel-Rahman EA, Nayfeh AH, Younis MI (2004) Finite amplitude motions of beam resonators and their stability. J Comput Theoret Nanosci 1(4):385–391 Abdel-Rahman EM, Younis MI, Nayfeh AH (2002) Characterization of the mechanicsl behavior of an electrically actuated microbeam. J Micromech Microeng 12:759–766 Andrews M, Harris I, Turner G (1993) A comparison of squeeze-film theory with measurements on a microstructure. Sensors Actuators A 36:79–87 Andrews MK, Harris PD (1995) Damping and gas viscosity measurements using a microstructure. Sensors Actuators A 49:103–108 Bao M, Yang H, Sun Y, French PJ (2003) Modified Reynolds equation and analytical analysis of squeeze-film air damping of perforated structures. J Micromech Microeng 13:795–800 Blech JJ (1983) On isothermal squeeze-films. J Lubric Technol 105:615–620 Burgdorfer A (1959) The influence of the molecular mean free path on the performance of hydrodynamic gas lubricated bearing. J Basic Eng 81:94–99 Chen J, Kang SM (2000) An algorithm for automatic model reduction of nonlinear MEMS devices. Proceedings of IEEE International Symposium Circuits and Systems, 28–31 May 2000, pp 445–448 Chen J, Kang S, Zou J, Liu C, Schutt-Ain´e JE (2004) Reduced-order modeling of weakly nonlinear MEMS devices with Taylor-series expansion and Arnoldi approach. J Microelectromech Syst 13(3):441–451 Chen Y, White J (2000) A quadratic method for nonlinear model order reduction. Proceedings of the International Symposium on Modeling and Simulation of Microsystem Conference, March 2000 Christopherson J, Jazar GN (2005) Optimization of classical hydraulic engine mounts based on RMS method. J Shock Vibr 12(12):119–147 Darling RB, Hivick C, Xu J (1998) Compact analytical modeling of squeeze-film damping with arbitrary venting conditions using a green’s function approach. Sensors Actuators A 70:32–41 Esmailzadeh E, Mehri B, Reza NJ (1997) Existence of periodic solution for equation of motion of simple beams with harmonically variable length. J Vibr Acoustics 119:485–488 Fedder GK (1994) Simulation Micromech Syst Ph.D. dissertation, University of California at Berkeley Golnaraghi MF, Jazar RN (2001) Development and analysis of a simplified nonlinear model of a hydraulic engine mount. J Vibr Control 7(4):495–526 Griffin WS, Richardson HH, Yamanami S (1966) A study of fluid squeeze-film damping. ASME J Basic Eng D 88, 451–456. Gupta A, Denton JP, McNally H, Bashar R (2003) NovelFabrication method for surface micromachined thin single-crystal silicon cantilever beams. J Microelectromech Syst 12(2):185 Gupta RK, Senturia SD (1997) Pull-in time dynamics as a measure of absolute pressure. Proceedings of the tenth annual international workshop on micro electro mechanical systems, New York, NY, pp 290–294 Harmany Z (2003) Effects of vacuum pressure on the response characteristics on MEMS cantilever structures. NSF EE REU PENN STATE Ann Res J I:54–64 Houlihan R, Kraft M (2005) Modeling squeeze-film effects in a MEMS accelerometer with a levitated proof mass. J Micromech Microeng 15:893–902 Hung ES, Senturia SD (1999) Generating efficient dynamical models for microelectromechanical systems from a few finite element simulation runs. J Microelectromech Syst 8:280–289 Jazar RN, Golnaraghi MF (2002) Nonlinear modeling, experimental verification, and theoretical analysis of a hydraulic engine mount. J Vibr Control 8(1):87–116 Khaled ARA, Vafai K, Yang M, Zhang X, Ozkan CS (2003) Analysis, control and augmentation of microcantilever deflections in bio-sensing systems. Sensors Actuators B 7092:1–13 Langlois WE (1962) Isothermal squeeze-films. Q Appl Math 20:131–150
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Lyshevski SE (2001) Nano- and microelectromechanical systems, fundamentals of nano- and microengineering. CRC Press, Boca Raton, Florida Madou MJ (2002) Fundamentals of microfabrication: the science of miniaturization, 2nd edn. CRC Press, Boca Raton, FL Mahmoudian N, Aagaah MR, Jazar RN, Mahinfalah M (2004) Dynamics of a micro electro mechanical system (MEMS). International conference on MEMS, NANO, and smart systems, Banff, Alberta - Canada, 25–27 August 2004 Malatkar P (2003) Nonlinear vibrations of cantilever beams and plates. Ph.D. Thesis in Mechanical Engineering, Virginia Polytechnic Institute and State University Mukherjee T, Fedder GK, Blanton RD (1999) Hierarchical design and test of integrated microsystems. IEEE Design Test 16:18–27 Najar F, Choura S, El-Borgi S, Abdel-Rahman EM, Nayfeh AH (2005) Modeling and design of variable-geometry electrostatic microactuators. J Micromech Microeng 15(3):419–429 Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York Nayfeh AH, Younis MI (2004a) Modeling and simulations of thermoelastic damping in microplates. J Micromech Microeng 14:1711–1717 Nayfeh AH and Younis MI (2004b) A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping. J Micromech Microeng 14:170–181 Nayfeh AH, Younis MI, Abdel-Rahman EA (2005) Reduced-order modeling of MEMS. Third MIT conference on computational fluid and solid mechanics, Cambridge, MA, 14–17 June 2005 Pan F, Kubby J, Peeters E, Tan A, Mukherjee S (1998) Squeeze film damping effect on the dynamic response of a MEMS torsion mirror. J Micromech Microeng 8(3i):200–208 Rewienski MJ (2003) A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems. Ph.D. thesis, Department of Electrical Engineering, Massachusetts Institute of Technology Shi F, Ramesh P, Mukherjee S (1996) Dynamic analysis of micro-electro-mechanical systems. Int J Numer Meth Eng 39(24):4119–4139 Starr JB (1990) Squeeze-film damping in solid-state accelerometers. Proceedings of technical digest IEEE solid-state sensors and actuators workshop, Hilton Head Island, SC, pp 44–47 Sudipto K, Aluru NR (2004) Full-Lagrangian schemes for dynamic analysis of electrostatic MEMS. J Microelectromech Syst 13(5):737–758 Sun Y, Chan WK, Liu N (2002) A slip model with molecular dynamics. J Micromech Microeng 12:316–322 Veijola T, Mattila T (2001) Compact squeezed-film damping model for perforated surface. Proceedings of IEEE 11th international conference on solid-state sensors, actuators and microsystems, pp 1506–1509 Veijola T, Kuisma H, Lahdenper¨a J (1998) The influence of gas-surface interaction on gas-film damping in a silicon accelerometer. Sensors Actuators A 66:83–92 Vogl GW, Nayfeh AH (2005) A reduced-order model for electrically actuated clamped circular plates. J Micromech Microeng 15:684–690 White A (2002) Review of some current research in microelectromechanical systems (MEMS) with defence applications, DSTO Aeronautical and Maritime Research Laboratory, Fishermans Bend Vic, Australia, pp 10 Yang YJ (1998) Squeeze-film damping for MEMS structures. MS Thesis, Electrical Engineering, Massachusetts Institute of Technology Yang YJ, Gretillat M-A, Senturia SD (1997) Effect of Air damping on the dynamics of nonuniform deformations of microstructures, international conference on solid-state. Sensors and Actuators. Chicago, 16–19 June 1997, pp 1094–1096 Yang JL, Ono T, Esashi M (2002) Energy dissipation in submicrometer thick single-crystal 116 cantilevers. J Microelectromech Syst 11(6):775–783 Yang Y-J, Senturia SD (1996) Numerical simulation of compressible squeezed-film damping. Proceedings of solid-state sensor and actuator workshop pp 76–79
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Yang YJ, Senturia SD (1997) Effect of air damping on the dynamics of nonuniform deformations of microstructures. Proceedings of IEEE international conference on solid-state sensors and actuators, New York, NY, pp 1093–1096 Younis MI (2001) Investigation of the mechanical behavior of micro-beam-based MEMS devices. MS. thesis in Mechanical Engineering, Virginia Polytechnic Institute and State University, December 2001 Younis MI (2004) Modeling and simulation of micrielectromecanical system in multi-physics fields. Ph.D. thesis, Mechanicsl Engineering, Virginia Polytechnic Institute and State University Younis MI, Abdel-Rahman EM, Nayfeh A (2003) A reduced-order model for electrically actuated microbeam-based MEMS. J Microelectromech Syst 12(5):672–680 Younis MI, Nayfeh AH (2003) A study of the nonlinear response of a resonant microbeam to electric actuation. J Nonlinear Dyn 31:91–117. Younis MI, Nayfeh AH (2005a) Modeling squeeze-film damping of electrostatically actuated microplates undergoing large deflections. ASME 20th biennial conference on mechanical vibration and noise, 5th international conference on multibody systems, nonlinear dynamics and control, DETC2005-84421, Long Beach, CA, 24–28 September 2005 Younis MI, Nayfeh AH (2005b) Dynamic analysis of MEMS resonators under primary-resonance excitation. ASME 20th biennial conference on mechanical vibration and noise, DETC200584146, Long Beach, CA, 24–28 September 2005 Zhang C, Xu G, Jiang Q (2004) Characterization of the squeeze-film damping effect on the quality factor of a microbeam resonator. J Micromech Microeng 14:1302–1306 Zhao Z, Dankowicz H, Reddy CK, Nayfeh AH (2004) Modelling and simulation methodology for impact microactuators. J Micromech Microeng 14:775–784
Chapter 3
Nonlinear Mathematical Modeling of Microbeam MEMS Reza N. Jazar
Abstract Microbeams and microcantilevers are the main part of many MEMS. There are several body and contact forces affecting a vibrating microbeam. Among them, there are some forces appearing to be more significant in micro and nanosize scales. Accepting an analytical approach, we present the mathematical modeling of microresonators dynamics and develop effective equations to be utilized to study the electrically actuated microresonators. The presented nonlinear model includes the initial deflection due to polarization voltage, mid-plane stretching, and axial loads as well as the nonlinear displacement coupling of electric force. It also includes the thermal and squeeze-film phenomena. The equations are nondimensionalized and the design parameters are developed. In order to have a set of equations, depending on depth of accuracy and difficulty, we present equations of motion for linearized and different level of nonlinearity. The simulation method makes it easy for investigators to pick the appropriate equation depending on their design and application. It is shown that the equation of motion for microresonators is highly nonlinear, parametric, and externally excited. The most important phenomena affecting the motion of microbeam-based and microcantilever-based microresonators are reviewed in this chapter and the corresponding forces are introduced. The mechanical and electrical forces are the primary forces that cause the microresonators work. There are also two specific phenomena: squeeze-film and thermal damping, that their effects on MEMS dynamics are considered secondary compared to mechanical and electrical forces. Some tertiary phenomena such as van der Waals, Casimir, and fringing field effects are also introduced.
R.N. Jazar () School of Aerospace, Mechanical, and Manufacturing Engineering, RMIT University, Melbourne, Vic, Australia e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 3, © Springer Science+Business Media, LLC 2012
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There are a few reported investigations on secondary phenomena, and their effects are defined. However, based on some reported theoretical and experimental results, we qualitatively analyze them and present two nonlinear functions to define the restoring and damping behavior of squeeze-film. In addition, we use two Lorentzian functions to describe the restoring and damping forces caused by thermal phenomena.
3.1 Introduction Micro-electro-mechanical systems (MEMSs) are devices in micro scale to transfer an electrical signal to a mechanical movement or vice versa. Although every microelectro-mechanical system may be called MEMS, we use “MEMS” in this chapter only to refer to microsystems that are based on microbeam and microcantilevers. Microbeam-based MEMS are generally strain sensitive transducers that provide the basis for temperature, strain, acceleration, vibrations, and other sensors. They may also be used as microactuators. Several techniques, such as optical, acoustic, magnetic, piezoresistive, and capacitive methods, have been developed to excite or detect the MEMS vibrations and change their steady-state amplitudes. Among them, the electric excitation and detection, based on capacitive and resonance frequency, are shown their reliabilities and will remain dominant in near future. Although the application of MEMS has been growing up since 1980, MEMS devices and systems have mainly been designing by trial and error. A prototype of proposed MEMS is tested and modified until the desired performance is achieved. However, for optimal design, a reliable mathematical model must be developed and examined. Then, using software developed prototypes, the best performance and design can be achieved theoretically. Therefore, the final fabrication and testing would be faster and more effective. Manufacturing in MEMS technology is based on silicon, and manufacturing processes are based on photolithography, growth, deposition, doping, and surface micromachining. Indeed, advanced simulation and modeling tools for MEMS design are needed because most MEMS devices are presently modeled using weak analytical tools, resulting in a relatively inaccurate prediction of performance behavior. As a result, MEMS design process requires several iterations before the performance requirements of a given device are finally satisfied. Moreover, the stability and parametric domain of applicability of such MEMS are not clearly understood. This chapter focuses on analytical approaches to develop a reliable mathematical model of microbeam-based and microcantilever-based MEMS for better designs. As will be shown shortly, the mathematical models of MEMS are highly nonlinear and parametric. Having a correct mathematical model provides the power of prediction of their behavior. Younis et al. (2003) have divided the investigators on the MEMS into two groups, experimenters and modelers. There is also a middle group trying
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to utilize available computer software, usually finite element related, and model the microsystems. Although improvement and progress in all groups are important, this report is to improve the modeling of MEMS. In academia, experiments must be used as a secondary tool to validate the theoretical prediction, and clarify the order of contribution of the involved physical phenomena. Experiment can make the modeling procedure more exact and more effective, provided to be joined with modeling. However, experimental results depend on the accuracy of the experimental devices, and even more prominently depend on the skills and knowledge of the experimenter. On the other hand, the finite element method (FEM) software are limited because they are time-consuming, cumbersome, expensive, and they use numerous variables to represent the state of the system, where most of those variables are not important to the designer (Younis et al. 2003; Younis 2004). Conversely, the reduced-order models, known as micromodels, need to be expanded and improved in order to be used as the basis for prediction and optimization tool of the proposed behavior. Reduced-order models are developed to capture the most significant characteristics of MEMS behavior in a few variables (Nayfeh and Younis 2004). Typical MEMS devices employ a parallel capacitor, in which one electrode is fixed and the other is allowed to move using some flexibility. The movable electrode, fabricated in the form of microbeam, microplate, or microcantilever, serves as a mechanical resonator. It is actuated electrically and its motion can be detected by capacitive changes. The fundamental resonant frequency of the vibrating microelement is sensitive to the axial strain. External loads such as pressure, temperature, force, and acceleration generate axial strains and shift the natural frequencies. This shift can be converted to an electric signal in the capacitance to measure the interested physical quantity (Younis and Nayfeh 2003). Most of the mathematical models utilized by experimenters can be improved by considering the coupling between the electric forces and the structural displacement. Simple models are valid only for very small displacements. As a result, their prediction and expected behavior diverge from experimental results significantly, when the actuating voltage and/or displacement increase. The fundamental characteristics of the electric actuated MEMS are due to nonlinearities in electrostatic excitation. This report starts with detailed introduction of physical phenomena which affect the MEMS dynamics. We split the physical phenomena contributed in MEMS dynamics into three groups in order of their effects and importance. The effects are expressed explicitly by using existed or proposed analytic expressions. The general equation of motion will be developed in for MEMS, by including primary and secondary important phenomena. The next section is the main part of this investigation. It provides a basis for further analysis and simulation of MEMS dynamics. In Sect. 3.3, the general equation is simplified to provide a set of applied different equations by linearizing the system or eliminating the secondary phenomena.
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3.2 Important Phenomena 3.2.1 Electric Load The electric load and the mechanical restoring force govern the behavior of MEMS. Electromagnetic actuation is a prevalent method of deriving MEMS. Electromagnetic actuation is achieved by applying a voltage difference between opposite electrodes of the variable capacitor. The induced electromagnetic force deforms the capacitor until they are balanced by the restoring mechanical forces. The electric load is composed of a direct current (DC) polarization voltage, vp , and an alternative current (AC) actuating voltage, v D vi cos !t. The DC voltage applies an electrostatic force on the microbeam and usually changes the equilibrium position/s. The polarization voltage has an upper limit beyond which the mechanical restoring force can no longer resist its opposing force. This limit is called collapse load, pull-in load, critical load, or break load. The collapse load corresponds to a saddle-node (Jazar et al. 2009), or transcritical bifurcation (Mahmoudian et al. 2004) depending on the system. The electric force affects the restoring force with a softening effect. Therefore, the electric force tends to shift the natural frequencies to the lower values. The electrostatic forces are found by computing the spatial derivative of electrostatic energy. Then, one-dimensional electrostatic force, fe , between two electrodes would be fe D
2 "0 a v vp 2 .d w/2
v D vi sin !t;
;
(3.1) (3.2)
where "0 D 8:854187817620 1012 A2 s4 = kg m3 is the permittivity in vacuum, a is the effective area of the microplate, and w D w.x; t/ is the lateral displacement of the microbeam. The complete microresonator is composed of a beam resonator, a ground plane underneath in contact with the beam, and one (or more) capacitive transducer electrode/s. To bias and excite the device, a DC-bias voltage, vp , is applied to the resonator while an AC excitation voltage is applied to its underlying ground plane/s. A double capacitors clamped–clamped model of the resonator is illustrated in Fig. 3.1. When the AC excitation frequency is close to the fundamental resonance frequency of the resonator, the resonator begins to oscillate, creating a time-varying capacitance C "0 a C D (3.3) d w between the resonator and the electrode/s (Nguyen 1995). It is usually assumed that the beam length is much greater than the electrode lengths. Therefore, the variation
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vo
Fig. 3.1 A clamped– clamped model of a MEMS and its voltage connections
+ vp -
Fig. 3.2 A clamped– clamped model of a MEMS to keep the capacitor’s plates parallel
+ vp x -
d
vi
v=vi sin(ω t) of the beam deflection across the length of the electrodes is ignorable. It is a reasonable assumption since the actual fabrication of MEMS can be improved to be uniform as shown in Fig. 3.2.
3.2.2 Restoring Force A MEMS restoring force is composed of three components: bending, axial force, and mid-plane stretching. The mechanical restoring force of microbeams is usually hardening and, therefore, it tends to shift the natural frequencies to higher frequencies. Decreasing the size of MEMS makes them lighter and much stiffer. Consequently, their natural frequencies shift up significantly. Because of the fabrication ease, rectangular cross-sections often turns out to be the most relevant; although in NEMS, using carbon nanotube (with circular crosssection) as a beam element is becoming more common. The mechanical element of MEMS is a microbeam, with its behavior depending on shape (dimension, I , . . . ), environment (temperature, humidity, load, . . . ), and mechanical properties (Young modulus E, Poisson ratio , . . . ). Examples of microbeams are illustrated in Fig. 3.3. Although the only difference among the mechanical elements shown in Fig. 3.3 is their boundary conditions, it is traditional to call the beams under clampedfree boundary conditions, microcantilever, and all the others, microbeam. Besides the current designs of microbeams, it is possible to fabricate the microbeams as illustrated in Fig. 3.4, using more flexible boundary conditions. Using flexible
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Fig. 3.3 Various examples of connecting a microbeam to the ground
x microcantilever y
x
clamped-clamped microbeam y
x
clamped-simple microbeam y
x
simple-simple microbeam
Fig. 3.4 Alternative examples of connecting a microbeam to the ground
y x
y x
y
y
x
x
ground connection reduces the resonant frequency significantly. Hence, it is possible to set the resonant frequency of the microbeam by designing a proper flexible connection. This fact can improve the MEMS maintenance. Apart from static deformations, mechanical changes can also be well-observed in dynamic behavior. The resonance frequency increases by increasing the stress, a well-known effect from microscopic string instruments. Due to scaling law, it is reasonable to ignore the rotary inertia, and shear deformation effects. Hence, accepting an Euler–Bernoulli model provides a very well approximation for the dynamic behavior of MEMS.
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The inertia force per unit length of a vibrating microbeam is fi D
@2 w @t 2
(3.4)
where is the mass per unit length. The restoring force per unit length is dependent on two terms. First, the restoring force due to rigidity of the microbeam fr D EI
@4 w @x 4
(3.5)
and second, the restoring force due to internal tension of the microbeam, ft D P
@2 w : @x 2
(3.6)
However, the internal tension force, P , is composed by the initial stretch x0 , possible dynamic variable length xd , and increasing the length due to deflection, ı, which is called mid-plane stretching. A proper method for showing the internal tension is PL D x0 C ı C xd cos .!x t/ ; EA0
(3.7)
where Z ıD
Z
L
.ds dx/ D 0
1 2
Z 0
L
@w @x
0s @ 1C
@w @x
2
1 1A dx
2 dx:
(3.8)
It is reasonable to assume x0 D xd D 0, since the common source of vibrations in MEMS is electric actuation, and they fabricated with zero initial stretch. The base excitation and initial stretch have not been considered in literature thoroughly so far. However, it is quite possible to build prestressed microbeams to postpone the collapse load or compensate un-modeled phenomena. Note that any change in temperature and residual stresses can make longitudinal stretch and justifies considering x0 . In addition, we might need to excite a microbeam by applying a dynamic elongation x .t/ D xd cos .!x t/ : (3.9) We may assume the initial elongation is proportional to the temperature of the beam. Temperature of the beam is proportional to Lorentzian function of excitation frequency as explained (Jazar 2006).
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3.2.3 Damping Damping strongly affects the dynamics, control, performance, and design of MEMS. Influence of damping on the dynamic of MEMS depends on their design and operating conditions. There are several energy dissipation mechanisms in MEMS devices. Acoustic radiation or thermoelastic damping, internal, structural, intrinsic losses, and viscous damping are the most common sources of energy dissipations. Among them, the viscous damping force fv D c
@w @t
(3.10)
is the simplest important model and unavoidable one, where c is the damping coefficient per unit length of the microbeam. Mostly, damping and dissipation mechanisms are mixed and coupled (Nayfeh and Younis 2004). However, the viscous damping may be assumed as a substitute or complementary for different damping mechanisms. Generally speaking, the efficiency of actuation and the sensitivity of motion detection of MEMS improve by decreasing the distance between the capacitor electrodes and increasing the area of the electrodes. Under these conditions, the squeeze-film damping appears if the MEMS device is not in a vacuumed capsule. Squeeze-film damping is a result of the massive movement of the fluid underneath the plate, which is resisted by the viscosity of the fluid. A nonuniform pressure distribution gives rise underneath the movable plate, which may act as a spring and/or a damper. The equivalent spring and damping rates are dependent on the frequency of vibration. It is estimated that the damping force is more important at low frequencies, whereas the spring force is more significant at high frequencies (Nayfeh and Younis 2004). To analyze the squeeze-film damping, the nonuniform pressure distribution of the gas film may be added as an external force to the equation of lateral motion of the microbeam. The pressure, which is dependent on the distance between electroplates, then must be found using fluid dynamics. Hence, the phenomenon is a coupled elasto-fluid problem. Some researchers have used the incompressible isothermal Reynolds equation to solve the coupled elastofluid problem approximately (Casimir 1948; Ding et al. 2000; Jazar and Golnaraghi 2002). In this approach, the squeeze-film load is evaluated by: Z fs D
b=2
p dy
(3.11)
b=2
and pressure is calculated using the following equation @ .pw/ r .1 C 6K/ w3 prp D 12 ; @t
(3.12)
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where K is the Knudsen number. An extension to this analysis might be coupling the system with heat convections to make an elasto-thermo-fluid problem, and use nonisothermal and compressible fluid dynamics. The motion of the moving electrode in a viscous fluid is models similar to the motion of the decoupler plate in hydraulic engine mounts (Jazar and Golnaraghi 2002). Following Golnaraghi and Jazar (2001) and Jazar et al. (2006), the squeezefilm damping force, fsd , by a cubic function
@w fsd D ks .d w/ w @t
2 ;
(3.13)
where the coefficient cs must be evaluated experimentally (Jazar and Mahinfalah 2006). Stiffness behavior of the squeeze-film has a negative rate, since it decreases by increasing frequency. However, no negative polynomial, x, x 2 , x 3 , : : :, can describe this behavior properly. The reason is that at low amplitude, x 0, the fluid layer is not much squeezed and there is not much resistance; on the other hand, at high amplitude, x d , there is not much fluid to react as a spring. The following function is presented to simulate the spring force, fss , of the squeeze-film fss D ks .d w/ w;
(3.14)
where the coefficient ks must be evaluated experimentally (Jazar et al. 2006). The coefficients cs and ks are dependent on geometry as well as dynamic properties of the fluid. They are assumed to be independent of kinematics of the microbeam such as displacement and velocity (Lifshitz and Roukes 2000). It is desired to design and construct systems with very little loss of energy and very high quality factor Q. Quality factor is defined as inverse of the fraction of energy lost per radian of vibration. The quality factor of MEMS decreases monotonically with reducing size. Sound waves, such as wave traveling through an elastic material, called thermoelastic damping are another energy dissipation mechanism in MEMS. Thermoelastic damping, also known as thermomechanical noise is a consequence of the microbeam being in thermal equilibrium with its environment. Energy dissipation in the microbeam causes the stored elastomechanical energy to leak away and be converted into heat. Thermoelastic damping depends on the thermodynamic properties of the material which are functions of temperature. Thermoelastic damping is proportional to frequency; hence, when the principal natural frequency increases while the size of devices decreases, the thermoelastic damping becomes significant (Lifshitz and Roukes 2000). Material linear thermal expansion coefficient ˛T D
1 @L L @T
(3.15)
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R.N. Jazar
is the simplest macroscopic parameter that couples length change (strain) with temperature change. The coupling of the strain field to a temperature field provides an energy dissipation mechanism. We call this damping mechanism temperature relaxation, which warms every vibrating thermoelastic solid up, and allows a free system to relax back to rest. Generally, the material of the beam will be soften by heat, resulting in a softening stiffness rate. Relaxation of the thermoelastic solid is achieved through the irreversible flow of heat driven by local temperature gradients produced by strain field. Thermoelastic damping introduces an upper limit to the quality factor. Torsional modes of beams involve no local volume change, and therefore, thermoelastic damping is a function of transversal vibrations. Due to Zener approximate theory, the thermoelastic damping is significant when the frequency of vibration, !, satisfy the condition !Z D 1, where Z is the Zener relaxation time, b is the width of the beam, and T is the thermal diffusivity of the beam material. Zener relaxation time, Z , is related to the quality factor due to QZ1 D
!Z E˛T2 T0 ; Cp 1 C .!Z /2
(3.16)
where Cp is the heat capacity per unit volume of the beam material, T0 is the uniform temperature of the beam, and E is Young modulus (Lifshitz and Roukes 2000). A better approximation is provided by Lifshitz and Roukes (2000) as follows: 6E˛T2 T0 D Cp r !i Db ; 2T
QZ1
1 1 sinh C sin 2 3 cosh C cos
;
(3.17) (3.18)
where, !i is the i thp natural frequency of the microbeam and is dimensionless and proportional to ! . This expression has a maximum at 2:225, equal to Q1 = E˛T2 T0 =Cp D 0:494. Note that the maximum is independent of the dimension of the microbeam. It is a function of temperature, E, ˛T , and Cp . The natural or eigen frequencies for beams in terms of the beam dimensions are s !i D ai
2
EI ; L4
(3.19)
where I is the area geometric moment of the cross-section (cb 2 =12 for rectangular), is the mass per unit length of the beam, and ai is a number depending on the boundary of the beam (Tadayon et al. 2006). Therefore, 2 D p conditions p 2 2 2 2 ai b = 4 3L lT , where lT D =E is thermal diffusion length. The values of lT are measured experimentally for the material of the beam and tabulated.
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The analytic equation (3.13) can be approximated by the following simpler equation with less than 1% error (Lifshitz and Roukes 2000) 2 E˛T2 T0 L ; Cp 2 =2 x L .x/ D ; 1 C x2 QZ1 D
(3.20) (3.21)
where L is called Lorentzian function. p For a linear mass-spring-dashpot oscillator, the quality factor Q is Q D km=c. Therefore, a thermal damping force fTd D cT
!=!1
@w 1 C .!=!1 / @t 2
(3.22)
can be defined to simulate the damping force corresponding to thermal warming up. Thermal damping force introduces a new character, since it is a function of excitation frequency with a maximum at fundamental resonance frequency (Jazar et al. 2005; Jazar et al. 2009). In addition, the effect of stiffness softening of the microbeam is also a frequency dependent characteristic. So, a similar softening function is presented to define this behavior (Jazar et al. 2009). More specifically, a negative Lorentzian function of excitation frequency fTs D kT
!=!1 1 C .!=!1 /2
w
(3.23)
determines the drop in linear stiffness force, fr D EI @4 w=@x 4 (Rastgaar et al. 2005; Jazar 2006). The breaking frequency of the thermal stiffness softening is at the fundamental resonance frequency (Meirovitch 1996). Air damping is proportional to the surface/volume ratio of the moving object, and square of velocity. Therefore, it might be more important in micro and nanoscales. The drag force fD on a microplate immersed in a fluid with speed v D @w=@t is fD D
2 1 @w ; CD a 2 @t
(3.24)
where a is the density of the fluid, a is the wetted area, and CD is the drag coefficient depending on the Reynolds number. Generally speaking, air damping can be modeled or substituted very well with traditional viscous damping since the maximum velocity of microbeams is not too much. This is also confirmed by new analysis based on one-dimensional and three-dimensional Stokes flow models. Although by vacuuming the housing of MEMS we can eliminate the fluid drag, there are many applications that MEMS must work in a liquid environment.
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3.2.4 van der Waals Force The van der Waals force resulting from the interaction between instantaneous dipole moments for atoms can be modeled in a simple form by: fW D
ACW 2 6 .d w/3
;
(3.25)
where CW is a constant character of interactions between the two atoms. Due to very short range of application, the van der Waals force is ignorable in MEMS while it is significant in NEMS. However, van der Waals force contributes the stiction and adhesion of contacting surfaces. The contribution of van der Waals force in vibrating MEMS can be modeled by a correction factor for the collapse load or correction factor for effective gap size. We do not consider the van der Waals force in this investigation.
3.2.5 Casimir Effect In addition to the electrostatic interaction of the capacitive coupling electrodes, they are also coupled via the Casimir effect. This effect, which was predicted by Casimir (1948), is a quantum effect between the ground-state energy of the electromagnetic field and boundary conditions that lead to an observable force between macroscopic bodies. Casimir force, similar to the van der Waals force, is a short range force that can play a major role in modern NEMS when the distance between neighboring electrodes is in sub-micron length scale. The Casimir force eases the collapse phenomenon, resulting the movable electrode sticks to the stationary electrode. It shrinks the basin of attraction of the trivial equilibrium and limits the range of stable operation of the system. The Casimir force is closely related to van der Waals attraction between dielectric bodies. However, in principal terms, they are quite different (Laumoreaux 1997). Casimir effect is among those phenomena that cause malfunctioning in MEMS due to stiction. Stiction is the collapse of movable elements into nearby surfaces, resulting in their permanent adhesion. In theory, the vacuum takes the form of tiny particles that are constantly forming and disappearing. Normally, the vacuum is filled with particles of almost any wavelength, but for two uncharged metallic plates and very close together, longer wavelength would exclude. The extra waves outside the plates would then generate a force that tended to push them together. The generalized zero point energy per unit area between two parallel plates with infinite conductivity and a distance d apart is given by: hc0 VCas D ; (3.26) 1440 .d w/3
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where h is the Planck’s constant, and c0 is the speed of light. This potential results in the Casimir force fCas D
hc0 Ab 480 .d w/
4
D
Rb .d w/4
R D 1:3 1027 N m2 :
;
(3.27) (3.28)
The Casimir force is valid for d < 100 nm. There is a correction due to finite conductivity (Laumoreaux 1999), temperature (Laumoreaux 1997), and surface inhomogeneity and roughness (Bordag et al. 1995) of the plates to calculate the effective Casimir force fCe D c T r fCas .d / ; (3.29) where c , T , and r are the correction coefficients for conductivity, temperature, and inhomogeneity, respectively. It is shown that c , T , and r are proportional to different exponents up to 1=d 4 , d 4 , and 1=d 4 , respectively (Ding et al. 2000). Generally speaking, the Casimir force is much lower than the other involved forces and we may ignore in MEMS analysis. The contribution of Casimir force in MEMS dynamic can also be modeled by a correction factor for the collapse load or correction factor for the gap size.
3.2.6 Fringing Field Effect In many electromagnetic actuated MEMS, the gap between the electrodes is not negligible relative to the lateral dimensions of the microplate. A uniform magnetic field cannot drop abruptly to zero at the edges of the effective area of the microplate. Therefore, in actual situation, there is always a fringing field existing. In general, the effect of fringing field is considered in modifying the capacitance of the capacitor. The capacitance including the fringing field is a function of d=L, and =d , where is the thickness of the microplate. Many different formulae for computing fringing field appear in the literature, which can be reviewed in (Leus and Elata 2004). However, the first-order approximation of the effect of the fringing field is expressed by a moderate electric force using af as the fringing field coefficient. 2 af v vp ff D .d w/
(3.30)
3.2.7 Sensing Principles Once the microbeam shows a static or dynamic deflection, there are several methods of sensing this deflection. One of the sensing methods is the strain gauge principle.
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The electrical resistance of a piece of metal depends on its size and shape. For a rod with cross-sectional area a, length L, and conductivity , the resistance R is given by: 1 RD : (3.31)
L An external load changes the resistance due to both a change in length and in crosssection area, by: dR D .1 C 2v/ " R The ratio of the relative change of the resistance per unit strain is indicated by the gauge factor G. Typical values of G are between 1 and 2. This number is good enough to make strain gauge load cells with a precision of 1:100000. This method of deflection detection is not very successful for nonmetallic and semi-conductive materials. The second method of sensing displacement, which is the main method in MEMS, is using electrostatics. Displacement of MEMS changes the geometry of the corresponding capacitance. This change can be detected either by a variation in voltage or by the change in capacitance. Other sensing methods are based on temperature, and heat conduction effects, since the heat transfer through the medium filling the gap between two close microplates influences the pressure of the medium. Optical read-out is also a very successful method of deflection sensing especially in stationary MEMS.
3.2.8 Scaling Law There is a huge transition going from macro to micro due to various scaling laws. The most important is Galileo’s square-cube law which says: if l is a measuring dimension, then volume is proportional to l 3 while surface is proportional to l 2 . Therefore, scaling down from l D 1 to l D 1=10, means surface forces will play 1000 times more important volume forces. In addition, mechanical strength of an object is dependent on the cross-section of the object and is proportional to the square of the linear dimension. The weight of a structure increases with the cube of its dimensions, but the area of the load-bearing sections increases with the square of the dimensions. Consequently, an elementary structure which is scaled upwards may eventually fail under its own weight. Thus, the best material to design a beam spring to tolerate a given deflection without yielding would be one which has the highest ratio of (yield/modulus). Due to the scaling law, gravity force does not play important role in MEMS, except when sensors are especially aimed at measuring these effects. Diffusion and heat transfer increases by a factor of 1= l 2 . Hence, thermal processes will
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83
be 100 times faster when dimension reduces 10 times. In addition, capillary condensation, van der Waals forces, and electrostatic forces are also increased significantly by reducing the size.
3.3 Mathematical Model Figure 3.5 illustrates a microcantilever and a clamped–clamped microbeam MEMS configurations. In what follows, we develop the mathematical model for only clamped–clamped, clamped–free, and simple–simple microbeams boundary conditions. However, the procedure is quite general and can be extended and applied to other configurations. The equation describing lateral vibrations of beams is @w @2 @2 w C 2 2 Cc @t @t @x
X @w @2 w @ P C fe ; EI 2 C f D @x @x @x
(3.32)
where (3.32) simplifies to the following equation when the beam’s geometry and internal tension are uniform.
@2 w @w @2 w @4 w X C c C f D P C fe : C EI @t 2 @t @x 4 @x 2
(3.33)
P The term f is the summation of the thermal, squeeze-film, and internal tension restoring forces applied on a unit length of the beam. Different forces applied on MEMS are summarized in Table 3:1.
a
y
x d
+ vp -
m
v = vi sin (ωt)
b
y
x
m d
+ vp -
v = vi sin (ωt)
Fig. 3.5 A microcantilever-based MEMS
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R.N. Jazar Table 3.1 Effective external forces acting on MEMS @2 w .x; t / fi D Inertia @t 2 @4 w .x; t / fr D EI Bending restoring 4 @x 2 "0 A v vp Electrostatic fe D 2 .d w .x; t //2 @w .x; t / fv D c Viscous damping @t EA0 @2 w .x; t / .x0 C ı C xd cos .!t // ft D Internal tension L @x 2 !=!1 @w .x; t / Thermal damping fTd D cT @t 1 C .!=!1 /2 !=!1 fTs D kT w .x; t / Thermal stiffness /2 1 C .!=! 1 2 @w Squeeze-film damping fsd D ks .d w/ w @t fss D ks .d w/ w Squeeze-film stiffness
Based on Table 3:1, and including the most important forces, the general equation of motion for MEMS would be
2 @w @w @2 w @4 w 2 @w C c C c w .d w/ w C EI C k s s @t 2 @t @x 4 @t @t @w !=!1 w kT 1 C .!=!1 / @t 1 C .!=!1 /2 ! Z 1 L @w 2 EA0 @2 w x0 C xd cos .!t/ C dx D L 2 0 @x @x 2 2 " 0 A v vp C 2 .d w/2 CcT
!=!1
2
(3.34)
We define the following variables to make the equation of motion dimensionless. D !1 t !1 D
n2 L2
x L w yD d zD
(3.35)
s EI
(3.36) (3.37) (3.38)
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85
w0 d ! rx D !x ! rD !1 Y D
(3.39) (3.40) (3.41)
"0 aL4 2d 3 EI
(3.42)
cL2 a2 D p EI
(3.43)
A0 L2 n I
(3.44)
cs d 2 L2 a4 D p EI
(3.45)
a1 D
a3 D
a5 D
ks d 4
(3.46)
cT L2 a6 D p EI
(3.47)
kT L4 EI Z 1 1 @' 2 n2 D dz 2 0 @z a7 D
(3.48) (3.49)
The parameter n is a constant depending on mode shape of the microbeam. Using these parameters, the equation of motion transforms to the following dimensionless equation. @2 y @4 y @y @y C a2 C 4 C a4 y 2 C a5 .1 y/ y 2 @ @ @z @ Ca6
r @y r y a7 2 1 C r @ 1 C r2
1 D a3 z0 C zd cos .rx / C 2 2 v vp Ca1 .1 y/2
Z 0
1
@y @z
2
! dz
@y @
2
@2 y @y 2 (3.50)
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R.N. Jazar
x0 La3 xd zd D La3 z0 D
(3.51) (3.52)
The parameters ai , i D 1; 2; 3; : : : ; 7 are defined in order of importance, although the importance of different phenomena depends on design and application. Hence, when we need to simplify (not linearize) the equation of motion by ignoring the least importance phenomenon, we may drop the term corresponding to the highest ai first. We apply a separation solution y D Y ./ ' .z/ ;
(3.53)
where the spatial function ' .z/ is a mode shape function that must satisfy the boundary conditions. In the following sections, we assume a first harmonic function as the mode shape for a deflected microbeam. This assumption is reasonable due to high resonance frequency of microbeams. We also ignore the dynamic excitation due to longitudinal elongation and apply zd D 0. By accepting a first harmonic shape function, the temporal function Y ./ would then represent the maximum deflection of the microbeam, which is the middle point for symmetric boundary conditions, and the tip point for microcantilever. In case of asymmetric boundary conditions, the function Y ./ must be defined depending on the coordinate system and the appropriate first mode shape. Therefore, the differential equation describing the evolution of the temporal function for a microbeam would be YR C h C a6
r 1 C r2
YP C 1 C b3 a7
r 1 C r2
Y
C Y 3 C a4 Y 2 YP C a5 .1 Y / Y p 1 2˛ˇ sin .r/ ˇ cos .2r/ ; D .˛ C ˇ/ C 2 .1 Y /2
(3.54)
where
p
h D a2
(3.55)
˛D
(3.56)
a1 v2p
2 2˛ˇ D 2a1 vp vi a1 ˇ D v2i 2
(3.57) (3.58)
D n2 a3
(3.59)
b3 D a3 z0
(3.60)
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3.3.1 Clamped–Clamped Microbeam Considering a clamped–clamped microbeam requires the following boundary conditions y .1; / D 0
(3.61)
y .0; / D 0
(3.62)
@ y .1; / D 0 @z
(3.63)
@ y .0; / D 0: @z
(3.64)
The following mode shape satisfies the required boundary conditions ' .z/ D and therefore, n2 D
1 2
Z
1 .1 cos .2z// 2 1
0
@' @z
2 dz D
2 : 4
(3.65)
(3.66)
3.3.2 Simple–Simple Microbeam In case of a simple–simple microbeam, the boundary conditions would be y .1; / D 0
(3.67)
y .0; / D 0
(3.68)
@ y .1; / D 0 @z2
(3.69)
@2 y .0; / D 0: @z2
(3.70)
2
The first harmonic mode shape satisfying the required boundary conditions is ' .z/ D sin .z/ and therefore, n2 D
1 2
Z 0
1
@' @z
2 dz D
(3.71)
2 : 4
(3.72)
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3.3.3 Microcantilever A microcantilever is a microbeam with the following boundary conditions. y .0; / D 0
(3.73)
@ y .0; / D 0 @z
(3.74)
@2 y .1; / D 0 @z2
(3.75)
@3 y .1; / D 0 @z3 z ' .z/ D 1 cos 2 and therefore, n2 D
1 2
Z 0
1
@' @z
2 dz D
(3.76) (3.77) 2 : 16
(3.78)
3.3.4 Initial Deflection In the previous sections, it was shown that the stiffness related to the mechanical element is composed by a linear and a cubic term. The cubic term is a result of mid-plane stretch of the microbeam. Let us assume this mechanical restoring force is denoted by: fm D Y C Y 3 0 Y 1: (3.79) As long as the inactive MEMS has Y D 0 as the rest point, the symmetric restoring force fm is applicable. However, applying a polarization voltage will affect the equilibrium positions of the system and bend the microbeam. Consequently, the rest position of the MEMS would not be at Y D 0. Assume the polarized MEMS has the rest point Y D Y0 , 0 Y0 1, instead of Y D 0. This will translate the origin of measuring fm and introduces a new mechanical restoring force as: fm D 1 C 3 Y02 Y 3 Y0 Y 2 C Y 3
0 Y 1:
(3.80)
Therefore, a second-degree restoring force must be added to the equations of motion to include the initial bending of the microbeam. Note that including seconddegree restoring force makes the system asymmetric, which generates its own problems when we try to solve the system using perturbations and other approximate methods.
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The initial displacement Y0 is a function of the polarization voltage and can be determined by searching for equilibria of the MEMS in each individual application (Jazar et al. 2009). For the cases that are studied in this report, the equilibria would be found by solving the following equation Y .1 Y /2 C Y 3 .1 Y /2 C a5 Y .1 Y / .1 Y /2 D ˛:
(3.81)
Polarization voltage and initial bending of the microbeams introduce another problem. When the polarization voltage of the inactive MEMS is not zero, the clearance between two plates of the capacitor would not be d , and therefore, the limit of Y would be less than 1. Sometimes it might be better to redefine the equations in order to have the bent rest point always as a zero equilibrium position. In this case, the gap size d is not constant, but a function of polarization voltage. This kind of modeling and associated equations of motion have been examined by Mahmoudian et al. (2004).
3.4 Simplified Models In this section, the equation of motion for different situations of the MEMS are derived and compared. It is possible (and actually has partially done by few researchers) that the nonlinear electromagnetic force be expanded in a Taylor series expansion (Younis et al. 2003). Then, depending on the order of acceptable nonlinearities and interested phenomena, different equations can be developed.
3.4.1 Linear and Nonpolarized Model Consider the case in which vp D 0. This is a practical model to eliminate those problems caused by shifting the rest point to have an asymmetric vibrating system. Eliminating the polarization voltage simplifies the governing equation to YR C h C a6 D
r 1 C r2
1 .1 Y /2
YP C 1 a7
r 1 C r2
Y C Y 3 C a4 Y 2 YP C a5 .1 Y / Y
.ˇ ˇ cos .2r// :
(3.82)
Series expansion of the nonlinear electromagnetic excitation indicates that 1 .1 Y /
2
D 1 C 2Y C 3Y 2 C 4Y 3 C 5Y 4 C 6Y 5 C O Y 6 :
(3.83)
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R.N. Jazar
Assuming Y 1 and ignoring the thermal and squeeze-film phenomena, convert the equation of motion to a forced Mathieu-type differential equation: YR C hYP C .1 2ˇ C 2ˇ cos .2r// Y D 2ˇ sin2 .r/ :
(3.84)
Note that thermal effects are linear in Y with frequency dependent coefficients. Therefore, it is possible to consider the thermal effects even in linear model and model the system with the following equation: YR C h C a6
r 1 C r2
YP C 1 a7
r 2ˇ C 2ˇ cos .2r/ Y 1 C r2
D 2ˇ sin2 .r/ :
(3.85)
The difference between model (3.84) and (3.85) is the existence of frequency response, because (3.84) does not show a frequency response while the model (3.85) does.
3.4.2 Nonlinear and Nonpolarized Model As the amplitude of excitation increases, the linear approximation of the model is no longer appropriate. Hence, we have to adapt a nonlinear approximation. Accepting second-degree nonlinear terms, and ignoring the thermal and squeeze-film effects, produces the following equation which is an externally force nonlinear parametric system. YR C hYP C .1 2ˇ C 2ˇ cos .2r// Y 3 .ˇ ˇ cos .2r// Y 2 D 2ˇ sin2 .r/ : (3.86) However, except for squeeze-film damping, the other dynamic forces are either first or second-degree of the amplitude. Therefore, the following equation would be a better approximation up to second-degree. YR C h C a6
r 1 C r2
YP C 1 a7
r C a 2ˇ C 2ˇ cos .2r/ Y 5 1 C r2
.a5 C 3ˇ 3ˇ cos .2r// Y 2 D 2ˇ sin2 .r/ :
(3.87)
Accepting third-degree approximation, with and without the effects of thermal and squeeze-film, generates the following models respectively. YR C hYP C .1 2ˇ C 2ˇ cos .2r// Y 3 .ˇ ˇ cos .2r// Y 2 . 4ˇ C 4ˇ cos .2r// Y 3 D 2ˇ sin2 .r/ ;
(3.88)
3 Nonlinear Mathematical Modeling of Microbeam MEMS
R Y C h C a6
r 1 C r2
P Y C 1 a7
91
r C a5 2ˇ C 2ˇ cos .2r/ Y 1 C r2
C a4 Y 2 YP .a5 C 3ˇ 3ˇ cos .2r// Y 2 C . 4ˇ C 4ˇ cos .2r// Y 3 D 2ˇ sin2 .r/ :
(3.89)
Finally, (3.82) illustrates the full nonlinear model including the thermal and squeeze-film effects. However, ignoring these effects must be analyzed based on the following equation which is a forced Duffing equation. YR C hYP C Y Y 3 D
1 .1 Y /2
.ˇ ˇ cos .2r// :
(3.90)
3.4.3 Linear and Polarized Model Dynamics behavior of the MEMS is more interesting when a polarization voltage is present. Assuming Y 1, and noting that polarization voltage must be less than the collapse voltage, and the thermal and squeeze-film effects are ignorable, the governing equation of the MEMS is p YR C hYP C 1 2˛ 2ˇ C 2ˇ cos .2r/ 4 2˛ˇ sin .2r/ Y p (3.91) D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ : We may include the thermal effects in linear model to derive the following equation for polarized model of MEMS dynamic. This model is an appropriate equation to investigate the effects of thermal damping and temperature relaxation in linear systems. r r R P Y C h C a6 2˛ 2ˇ C 2ˇ cos .2r/ Y C 1 a7 1 C r2 1 C r2 p 4 2˛ˇ sin .2r/ Y p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ :
(3.92)
3.4.4 Nonlinear and Polarized Model The most complete analysis of the MEMS is performed when the nonlinearity is included. If the second-degree nonlinearity is accepted, the MEMS equation of motion, without thermal and squeeze-film effects, becomes
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p YR C hYP C 1 2˛ 2ˇ C 2ˇ cos .2r/ 4 2˛ˇ sin .r/ Y p C3 ˛ C ˇ ˇ cos .2r/ C 2 2˛ˇ sin .r/ Y 2 p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ ;
(3.93)
where after including the thermal effect would be YR C h C a6
YP C 1 a7
r C a 5 Y 1 C r2 p 2 .˛ C ˇ ˇ cos .2r// C 4 2˛ˇ sin .2r/ Y a p 5 C3 C ˛ C ˇ ˇ cos .2r/ C 2 2˛ˇ sin .r/ Y 2 3 p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ : r 1 C r2
(3.94)
By increasing the effect of nonlinearity, i.e., by including the effect of Y 3 , and ignoring thermal and squeeze-film effects, the equation of motion would be in the form of p YR C hYP C 1 2˛ 2ˇ C 2ˇ cos .2r/ 4 2˛ˇ sin .2r/ Y p 3 ˛ C ˇ ˇ cos .2r/ C 2 2˛ˇ sin .r/ Y 2 p C 4˛ 4ˇ C 8ˇ cos .2r/ 8 2˛ˇ sin .r/ Y 3 p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ ; (3.95) where after including third-order effects of thermal and squeeze-film becomes YR C h C a6
r 1 C r2
YP
p r C a5 2˛ 2ˇ C 2ˇ cos .2r/ 4 2˛ˇ sin .r/ Y C 1 a7 1 C r2 a p 5 C ˛ C ˇ ˇ cos .2r/ C 2 2˛ˇ sin .r/ Y 2 C a4 Y 2 YP 3 3 p C 4˛ 4ˇ C 8ˇ cos .2r/ 8 2˛ˇ sin .r/ Y 3 p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ : (3.96)
In general, when there is no restriction on the term 1= .1 Y /2 , the equation of motion would be (3.54).
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3.5 Mathematical Analysis In order to show how the mathematical models presented for thermal and squeezefilm phenomena controls the behavior of a microresonator, we analyze their effects individually. Let us ignore every nonlinearity except those belonging to these phenomena to show their effects more clear.
3.5.1 Thermal Effects Considering thermal effects, ignoring the squeeze-film phenomena, and adapting a linear approximation for the electrostatic force, the equation describing lateral vibrations of the microbeam can be summarized to the following equation when the beam’s geometry is uniform. YR C h C a6
r 1 C r2
YP C 1 a7
r 2˛ 2ˇ C 2ˇ cos .2r/ 1 C r2 p 4 2˛ˇ sin .r/ Y
p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ :
(3.97)
In order to find the amplitude of oscillation, we assume a solution in the following form Y D A0 C A ./ sin .r C YP D A ./ r cos .r C
.t// ;
.t// :
(3.98) (3.99)
Applying the averaging method provides .˛ C ˇ/ 1 C r 2 A0 D .1 2˛ 2ˇ/ .1 C r 2 / a7 r
(3.100)
and the following frequency response. p Z1 r 12 C Z2 r 10 C Z3 r 9 C Z4 r 8 C Z5 r 7 C Z6 Z7 Z8 r 6 p CZ9 r 5 C Z10 Z11 Z8 r 4 C Z12 r 3 p p C Z13 Z11 Z8 r 2 C Z14 r C Z15 Z16 Z8 D 0; (3.101)
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where Zi are dynamic parameters of the system not related to the excitation frequency. Z1 D 4ˇ 2 A4
Z2 D 4ˇ A 4ˇ C 2 C 4˛ C h 2
4
2
Z3 D 8ˇ 2 A4 .a7 C ha6 / Z4 D 4ˇ 2 A4 a6 1 C 42˛ C 4h2 C 4˛ 2 C ˇ .3ˇ C 8˛ C 12/ 16˛ˇ 3 A2 .4A0 .A0 C 1/ C 1/ Z5 D 8ˇ 2 A4 .2a7 .˛ C 1/ C 3ha6 / Z6 D 4ˇ 2 A4 8˛ .˛ C 1/ C 2 3h2 2 C 2a62 C a72 64˛ˇ 3 A2 .4A0 .A0 C 1/ C 1/ Z7 D 16ˇ 2 A2 .A0 C 1/ Z8 D ˛ˇ 2 1 C r 2 2˛ .A0 C 1/ A2 1 C r 2 .ˇ C 2˛/ ˛ˇ2 1 C r 2 A2 r a7 C r 3 1 Z9 D 24ˇ 2 A4 .2a7 .ˇ C ˛/ C ha6 / Z10 D 4ˇ 2 A4 ˇ 3 .18ˇ C 8 .6˛ 1// C 4h2 1 C a62 C 2a72 C32˛ˇ 2 A4 .2˛ 1/ 96˛ˇ 3 A2 .4A0 .A0 C 1/ C 1/ Z11 D 48ˇ 2 A2 .2A0 C 1/
(3.102) (3.103) (3.104)
(3.105) (3.106)
(3.107) (3.108)
(3.109) (3.110)
(3.111) (3.112)
(3.113) Z12 D 8ˇ A .4a7 .ˇ C ˛/ C ha6 2a7 / Z13 D 4ˇ 2 A4 ˇ 2 .12ˇ C 4 .8˛ 3// C 4˛ .4˛ 3/ C h2 C 2 C a72 2
4
64˛ˇ 3 A2 .4A0 .A0 C 1/ C 1/ Z14 D 8ˇ 2 A4 a7 .2 .ˇ C ˛/ 1/
(3.114) (3.115)
Z15 D 4ˇ A .ˇ .4ˇ C 4 .2˛ 1// C 4˛ .˛ 1/ C 1/ 2
Z16 D
4
16˛ˇ3 A2 .4A0 .A0 C 1/ C 1/
(3.116)
1 Z11 3
(3.117)
Z17 D 2ˇ C 4˛ C h2 Z18 D 2 a7 C a6 h2
(3.118)
Z19 D .2˛ C ˇ/ .2˛ C ˇ C 2/ C a62 C 2h2 2
(3.120)
Z20 D 2 .a6 h C a7 .2˛ C ˇ//
(3.121)
(3.119)
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Fig. 3.6 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance
Fig. 3.7 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance
Because of the hard limits on oscillation, the peak amplitude should be less than d . The design criteria for the electric actuated microcantilever resonator would be a relationship between dynamic parameters that produce a peak value of y equal to one. Keeping A D 1, a design surface in the parameter space .˛; ˇ; h/ can be defined for a fixed value of a6 and a7 . Figures 3.6–3.9 illustrate the behavior of peak value of amplitude Ap , by varying the microresonator parameters. The figure is based on a nominal microresonator with properties indicated in Table 3:2, (Leus and Elata 2004; Yang et al. 2002; Kaajakari et al. 2004; Khaled et al. 2003). It can be
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Fig. 3.8 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance
Fig. 3.9 Effect of the variation of excitation and polarization voltages on peak amplitude at resonance
seen that Ap is a nonlinear monotonically increasing function of both polarization and excitation voltages, while it is a decreasing function of damping. More detail on thermal effects on microresonators can be seen in (Jazar and Mahinfalah 2006; Tadayon et al. 2006; Jazar et al. 2006).
3 Nonlinear Mathematical Modeling of Microbeam MEMS Table 3.2 Nominal values of the dynamic parameters of the MEMS
97
m c d ˛ A ˇ
1 1011 kg 1 1018 N s= m 2:0 m 5:53125 105 v2p 200 50 m 2:765625 105 v2i
3.5.2 Squeeze-film Effects Ignoring the thermal phenomena while considering squeeze-film effects, and adapting a linear approximation for the electrostatic force, the equation describing lateral vibrations of the uniform microbeam can be summarized to YR C a4 Y 2 YP C a5 .1 Y /2 Y YP 2 p C 1 2 ˛ C ˇ ˇ cos .2r/ C 2 2˛ˇ sin .r/ Y p D ˛ C 2ˇ sin2 .r/ C 2 2˛ˇ sin .2r/ :
(3.122)
Following the same approach and accepting (3.98) and (3.99), an approximate solution for steady-state amplitude around resonance would be found as: A0 D
˛Cˇ 1 2˛ 2ˇ
(3.123)
and the frequency response as: Z20 r 4 C Z31 r 3 Z23 .Z33 C Z34 /
p
Z35 C Z36 D 0;
(3.124)
where Z21 D .2˛ C ˇ 1/2 C a72 C h2
(3.125)
Z22 D 2a7 .2˛ C ˇ 1/
(3.126)
Z23 D .2˛ C ˇ 1/ C 1
(3.127)
Z24 D Z17 C 2ˇ
(3.128)
Z25 D Z20 C 2ˇ .1 C 2˛/
(3.129)
2
Z26 D Z20 C 2a7 ˇ
(3.130)
Z27 D Z21 C 2ˇ .4˛ C 2ˇ 1/
(3.131)
Z28 D Z27 C 2a7 ˇ
(3.132)
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Z29 D Z23 C 2ˇ .2˛ C ˇ 1/ Z30 D a52 ˇ 2 A8 A4 C 4 3A20 4A0 1 C4a52 ˇ2 A8 1 C 9A40 24A30 C 22A20 8A0 C16a5 ˇ 2 A6 A2 6A20 C 9A0 2 C 64ˇ 2 A4 Z31 D 16a5 ˇ 2 A6 .2˛ C 2ˇ 1/ 8A0 6A20 A2 2 C4ˇ2 A8 a42 C 8a5 C32ˇ 2 A6 a42 A20 C 2a5 3A20 C 4A0 C 1 Z32 D 64ˇ 2 A4 .1 4˛/ 256ˇ 3 A2 ˛ .2A0 C 1/ p Z33 D 128ˇ 2A2 A0 2˛ˇ p Z34 D 64ˇ 2 A2 2˛ˇ Z35 D a5 ˇA4 r 2 8A0 6A20 A2 2 C 32˛ˇA0 .1 C A0 / C8˛ˇ 1 2A2 C 8ˇA2 1 r 2 ˇ Z36 D 64ˇ 2 A4 8ˇ˛ C 3ˇ2 4ˇ C 4˛ 2
(3.133)
(3.134)
(3.135) (3.136) (3.137) (3.138)
(3.139) (3.140)
Frequency response analysis shows that the squeeze-film stiffness has the most impact on resonance frequency and affects as a hardening spring (Younis et al. 2003; Younis 2004; Jazar et al. 2006). However, polarization and excitation voltages also have a softening effect and shift the resonance to lower frequencies. On the other hand, squeeze-film damping has high effects on decreasing the peak amplitudes with least effect on resonance frequency shift. The equivalent linear damping and stiffness for squeeze-film model are two factors that can be utilized for validation and comparison. In other words, the parameters showing the proposed dynamic can model the squeeze-film phenomena effectively are equivalent to the linear stiffness and damping, predicted (Blech 1983). Figures 3.10 and 3.11 depict the equivalent damping and stiffness, where keq and ceq are traditional linear stiffness and damping per unit length of the microbeam. The equivalent stiffness keq , and damping ceq , are evaluated numerically by comparing the frequency response of the modeled microresonator with a microresonator having linear damping and stiffness. The results are quite comparable and in agreement with what presented by Blech (1983) and Younis (2004). The squeeze-film stiffness ks and damping cs coefficients, hidden in a5 and a4 , are proportional to their linear equivalents of keq and ceq . The equivalent squeeze-film damping is higher around resonance and decreases beyond that, justifying a decreasing effect on resonant amplitude. However, the equivalent squeeze-film stiffness is a monotonically increasing function of both frequency and squeeze-film stiffness coefficients. Therefore, stiffness has more effect at high frequencies, while damping affects the system at low and resonance frequencies.
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Fig. 3.10 Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients
Fig. 3.11 Equivalent linear damping and stiffness as a function of frequency and squeeze-film damping and stiffness coefficients
3.6 Summary We have reviewed the effective phenomena in MEMS dynamics. Effective phenomena can be divided into three groups in order of their importance: primary, secondary, and tertiary. The effective phenomena have been translated to effective force per unit length of the microbeam. The primary forces are inertia, bending restoring force, and electrostatic force. The electromagnetic force introduces the most nonlinear term into the equation of motion of the microbeam, since it is
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proportional to the inverse square of displacement. In addition, the electromagnetic force introduces a harmonically time varying coefficient that makes the equations parametric coupling due to and AC actuating current. Therefore, the MEMS vibrating motion is a nonlinear and parametric problem inherently. The secondary forces are mid-plane stretching and internal tension restoring force, viscous damping, thermal stiffness and damping forces, squeeze-film stiffness and damping forces. The stretching introduces a cubic term which makes the restoring force to be a third-degree of displacement. The stretching combined with the bending restoring force is equivalent to a hardening symmetric spring. However, when the system is using a polarization voltage to activate the system statically, the symmetric behavior of the hardening spring eliminates, since the rest position of the microbeam would be a bent situation. This symmetry breaking introduces a seconddegree term into the restoring force, which makes the solution of the overall motion of the system, problematic. The polarization voltage is brought in the MEMS system to amplify sensitivity of the system. The vibrating electrode is a microplate attached to a microbeam or microcantilever in order to have a movable suspension. The vibrating microplate underneath fluid introduces several effects; among them there is a mechanical effect that affects the vibration and dynamic behavior of MEMS. The damping effect of the phenomenon is called squeeze-film damping in the literature. Squeeze-film phenomena also introduces a spring-type behavior. The most important contribution of this investigation is modeling and simulation of these two characteristics of the phenomena. We have shown that the damping behavior of the squeeze-film can be simulated by a cubic term into the equation of motion. Squeeze-film damping is highly dependent on the displacement, make it more damped at higher amplitudes. The spring-like behavior of this phenomenon is simulated with a second-degree equation, with a maximum restoring force at the half gap amplitude. The novelty of simulation of the mechanical effects of squeeze-film is modeling the restoring and damping using polynomial functions independently. The thermal properties of the microbeam is another secondary phenomenon that contributes into damping system due to warming and heat energy dissipation, and into restoring system due to heat softening the material of the microbeam. We have called the heat softening temperature relaxation, to remind the character of stiffness change. Both effects of thermal behavior are frequency dependent. An analytic research has shown that the thermal damping can be described by a suitable Lorantzian function. So, we have simulated the damping and stiffness effects using two independent Lorantzian functions. The thermal damping force is positive while the thermal restoring force is negative, since the thermal damping dissipates energy and the thermal stiffness decreases the mechanical stiffness. The tertiary group of phenomena are usually insignificant in NEMS due to appearance in the nanoscale gap size. These are: Casimir and van der Waals forces, which are proportional to very small coefficients time inverse cubic and inverse quadratic functions of gap size. There is also another geometry dependent phenomenon called fringing field effect. The contribution of this event in the equation of motion lessens by increasing the size of microplates.
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By simulating the primary and secondary forces, the most comprehensive model of MEMS has been developed, and it is shown that its dynamic behavior is governed by a highly nonlinear ordinary differential equation (ODE) including parametrically and externally excitations. Having the general equation makes it easy to extract the required equation for different application. It also provides the potential of investigating the different phenomena individually and compare their contributions. For this propose, a set of reduced equation have been derived according to nonlinearity inserted by the phenomena. It has also been shown that the system is described by a parametrically and externally excited equation. Therefore, the system must be analyzed to develop its parametric space showing stable and unstable regions divided by transient curves. It is an open area of research in MEMS dynamics. The system also gives an idea about possible frequency response, and hence stability analysis of the corresponding frequency response. It is another open window for future research.
3.7 Key Symbols ai A A0 b c c0 cT cs C CD Cp d E f fCas fCe fD fe ff fi fr fsd fss ft
Dimensionless parameters of equation of motion Effective area of electrode plate Cross-sectional area Width of the microbeam Viscous damping rate Speed of light Squeeze-film damping force coefficient Thermal damping force coefficient Capacitance Drag coefficient Heat capacity per unit volume Gap size Young modulus External force Casimir force Effective Casimir force Drag force Electric force Fringing field force Inertia force Bending restoring force Squeeze-film damping force Squeeze-film stiffness force Tension force
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fTd fTs fv fw G h I K ks kT l lT L L m n P Q r R R s t T T0 v v vp VCas w x x0 xd y Y z
Thermal damping force Thermal stiffness force Viscous damping force van der Waals force Resistance gauge factor Dimensionless damping rate, Planck constant Cross-sectional second moment Knudsen number Squeeze-film stiffness force coefficient Thermal stiffness force coefficient Linear dimension Thermal diffusion length Length of microbeam or microcantilever Lorantzian function Mass Mode shape parameter Longitudinal internal tension force Quality factor Dimensionless excitation frequency Electrical resistance Casimir force constant Curved beam length Time Temperature Uniform temperature Speed Alternative excitation voltage Polarization voltage Casimir potential energy Lateral displacement Longitudinal coordinate Initial longitudinal stretch Dynamic longitudinal stretch Lateral coordinate, dimensionless lateral displacement Dimensionless amplitude of oscillations at maximum Longitudinal dimensionless coordinate
Greek ˛ ˛i ˛T
Polarization voltage parameter Frequency number of microbeam Thermal expansion coefficient
3 Nonlinear Mathematical Modeling of Microbeam MEMS
ˇ T ı c r T
' a ! !i !x z "0
Alternative voltage parameter Thermal diffusivity Mid-plane stretch Coefficient of cubic mechanical stiffness force Thickness of microplate Conductivity correction coefficient Inhomogenity correction coefficient Thermal correction coefficient Conductivity Mode shape function Poisson ratio Mass per unit length Fluid volumetric density Frequency of alternative voltage i th resonance frequency Frequency of the longitudinal dynamic stretching Zener relaxation time Permittivity in vacuum Dimensionless time
Symbol overdot
d ./ =dt
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References Blech JJ (1983), On isothermal squeeze-films. J Lubric Technol 105:615–620. Bordag M, Kilmchitskaya GL, Mostepanenko VM (1995) Corrections to the Casimir force between plates with stochastic surfaces. Phys Lett 200:95–102. Casimir HBG (1948) On the attraction between two perfectly conducting plates. Proc K Ned Akad Wet 51:793–795. Ding JN, Meng YG, Wen SZ (2000) Mechanical stability and sticking in model microelectromechaincal systems (MEMS) under Casimir force. Int J Nonlinear Sci Numer Simul 1:373–378. Golnaraghi MF, Jazar RN (2001) Development and analysis of a simplified nonlinear model of a hydraulic engine mount. J Vibrat Contr 7(4):495–526. Jazar RN (2006) Mathematical modeling and simulation of thermoelastic effects in flexural microcantilever resonators dynamics. J Vibrat Contr 12(2):139–163. Jazar RN, Golnaraghi MF (2002) Nonlinear modeling, experimental verification, and theoretical analysis of a hydraulic engine mount. J Vibrat Contr 8(1):87–116. Jazar RN, Mahinfalah M, Khazaei A, Alimi MH (2006) Squeeze-film phenomena in microresonators dynamics: a mathematical modeling point of view. ASME international mechanical engineering congress and exposition, Chicago, IL, November 5–10. Jazar RN, Mahinfalah M (2006) Squeeze-film damping in microresonators dynamics. Nonlinear science and complexity conference, Beijing, China, August 07–12.
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Jazar RN, Mahinfalah M, Mahmoudian N, Aagaah MR (2009) Effects of nonlinearities on the steady state dynamic behavior of electric actuated microcantilever-based resonators. J Vibrat Contr 15(9):1283–1306. Jazar RN, Mahinfalah M, Rastgaar Aagaah M, Mahmoudian N, Khazaei A, Alimi MH (2005) Mathematical modeling of thermal effects in steady state dynamics of microresonators using Lorentzian function: part 1 – thermal damping. ASME international mechanical engineering congress and exposition, Orlando, FL, November 5–11. Kaajakari V, Mattila T, Oja A, Sepp¨a H (2004) Nonlinear limits for single-crystal silicon microresonators. IEEE J Microelectromech Syst 13(5):715–724. Khaled ARA, Vafai K, Yang M, Zhang X, Ozkan CS (2003) Analysis, control and augmentation of microcantilever deflections in bio-sensing systems. Sens Actuat B 94:103–115. Laumoreaux SK (1999) Calculation of the Casimir force between imperfectly conducting plates. Phys Rev A 59:3149–3153. Laumoreaux SK (1997) Determination of the Casimir force in the 0.6 to 6 m range. Phys Rev Lett 78:5–8. Leus V, Elata D (2004) Fringing field effect in electrostatic actuators. Technical report ETR-20042, TECHNION, Israel Institute of Technology, Faculty of Mechanical Engineering. Lifshitz R, Roukes ML (2000) Thermoelastic damping in micro- and nanomechanical systems. Phys Rev 61(8):5600–5609. Mahmoudian N, Rastgaar Aagaah M, Jazar RN, Mahinfalah M (2004) Dynamics of a micro electro mechanical system (MEMS), The 2004 international conference on MEMS, NANO, and smart systems, Banff, Alberta, Canada, August 25–27. Meirovitch L (1996) Principles and technologies of vibrations, Prentice Hall, New Jersey. Nayfeh AH, Younis MI (2004) A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping. J Micromech Microeng 14:170–181. Nguyen CTC (1995) Micromechanical resonators for oscillators and filters. Proc IEEE international ultrasonic symposium, Seattle, WA, Nov. 7–10 pp 489–499. Rastgaar AM, Mahmoudian N, Jazar RN, Mahinfalah M, Khazaei A, Alimi MH (2005), Mathematical modeling of thermal effects in steady state dynamics of microresonators using Lorentzian function: part 2 - temperature relaxation. ASME international mechanical engineering congress and exposition, Orlando, FL, November 5–11. Tadayon MA, Sayyaadi H, Jazar RN (2006) Nonlinear modeling and simulation of thermal effects in microcantilever resonators dynamic. Int MEMS Conf 2006 (iMEMS2006), Singapore, 9–12 May. Yang J, Ono T, Esashi M (2002) Energy dissipation in submicrometer thick single-cristal silicon cantilevers. J Microelectromech Syst 11(6):775–783. Younis MI (2004) Modeling and simulation of micrielectromecanical system in multi-physics fields. Ph.D. thesis, Mechanical Engineering, Virginia Polytechnic Institute and State University. Younis MI, Abdel-Rahman EM, Nayfeh A (2003) A reduced-order model for electrically actuated microbeam-based MEMS. J Microelectromech Syst 12(5):672–680. Younis MI, Nayfeh AH (2003) A study of the nonlinear response of a resonant microbeam to electric actuation. J Nonlin Dyn 31:91–117. Zhang C, Xu G, Jiang Q (2004) Characterization of the squeeze-film damping effect on the quality factor of a microbeam resonator. J Micromech Microeng 14:1302–1306.
Chapter 4
Complex Motions in a Fermi Oscillator Yu Guo and Albert C.J. Luo
Abstract The nonlinear dynamic behaviors of a Fermi oscillator with two periodic excitations are discussed using the theory of discontinuous dynamical systems. The mechanism of motion switchability in such a system is addressed, and the periodic and chaotic motions for such an oscillator are studied via the mapping structures. A parameter map for different motions is presented.
4.1 Physical Problems Consider the Fermi accelerator consists of a particle moving vertically between two periodically excited oscillators. The mass in each oscillator m.˛/ , .˛ 2 f1; 2g/ is connected with a spring of constant k .˛/ and a damper of coefficient c .˛/ to the fixed wall. Both oscillators are driven with periodic excitation force F .˛/ .t/, as shown in Fig. 4.1. The mass of particle is m.3/ and the restitution coefficients of impact for the bottom and top oscillators are e .1/ and e .2/ , respectively. The gap between the equilibrium positions of the two oscillators is h. If the particle does not travel together with any of the oscillators, the corresponding motion is called the nonstick motion. The equations of motion for nonstick motion are given by the Newton’s law, i.e., ) xR .3/ D g; 2 (4.1) .˛/ xR .˛/ C 2 .˛/ xP .˛/ C ! .˛/ x .˛/ D Q cos .˛/ t; m.˛/ where .˛/ D c .˛/ =2m.˛/ ; ! .˛/ D xR
.i /
is the acceleration, xP
.i /
is the velocity, and x
q k .˛/ =m.˛/
.i /
(4.2)
is the displacement (i D 1; 2; 3).
Y. Guo () • A.C.J. Luo Southern Illinois University Edwardsville, Edwardsville, Illinois, USA e-mail:
[email protected];
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 4, © Springer Science+Business Media, LLC 2012
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Fig. 4.1 Mechanical model of Fermi oscillator
k
c (2)
(2)
F (2) (t) = Q (2) cosW (2)t
x(2)
m(2) e(2)
m(3)
h
e(1) m(1) k(1)
x(3) x(4)
F (1) (t) = Q (1) cosW (1)t
c (1)
If the particle stays on one of the two oscillators and travels together, this motion is called a stick motion. The equations of motion for stick motion are 9 2 xR .˛/ C 2 .˛/ xP .˛/ C ! .˛/ x .˛/ D A.˛/ cos .˛/ t; = ; 2 .˛/ N N xR .0/ C 2d .0/ xP .0/ C ! .0/ x .0/ D A0 cos .˛/ t; ;
(4.3)
where xR .0/ , xP .0/ , and x .0/ are the acceleration, velocity, and displacement for both the ball and oscillator, respectively. N Q.˛/ N m.3/ C m.˛/ q N N N =m.3/ C m.˛/ N D c .˛/ =2.m.3/ C m.˛/ /; ! .0/ D k .˛/
A.˛/ D d .0/
Q.˛/ m.˛/
.˛/ N
and A0 D
˛N D 2 if ˛ D 1 and 1 if ˛ D 2:
(4.4)
The impact relation among the particle and the oscillators is expressed as .3/
.˛/
.3/ .˛/ D x I xC D xC D x .3/
1 .3/ .˛/ .3/ .˛/ Œm.3/ xP C m.˛/ xP m.˛/ e .˛/ .xP xP /; m.3/ m.˛/ 1 .3/ .˛/ .3/ .˛/ D .3/ .˛/ Œm.3/ xP C m.˛/ xP C m.3/ e .˛/ .xP xP /: m m
xP C D .˛/
xP C
(4.5)
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4.2 Discontinuous Analysis As in Luo and Guo (2010a), due to the discontinuity of the system, the domains and boundaries in absolute coordinate system are introduced as sketched in Fig. 4.2. The origin of the absolute coordinate is set at the equilibrium position of the bottom .1/ .2/ oscillator. The absolute domains 1 and 1 for the bottom and top oscillators .3/ and domain 1 for the particle without stick are defined as 9 .1/ 1 D f.x .1/ ; xP .1/ /jx .1/ 2 .1; x .3/ /g; > > = .2/
1 D f.x .2/ ; xP .2/ /jx .2/ 2 .x .3/ ; C1/g;
.3/ 1
a
D f.x ; xP /jx .3/
.3/
.3/
2 .x ; x /g: .1/
b
x (1)
∂
(1) 1
.2/
x (2)
(3) 1( +∞ )
∂
x (1)
x (2) ∂
∂
(1) 1( −∞ )
c ∂
x (3)
∂
(3) 1( −∞ )
x (2)
x(2) (3) 1( −∞ )
(3) 1( +∞)
(3) 1
x(1)
(2) 1( −∞ )
(2) 1
x (1)
x (1)
(4.6)
> > ;
x (3)
x(2)
Fig. 4.2 Absolute domains and boundaries without stick: (a) Bottom oscillator, (b) top oscillator, and (c) particle
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The absolute boundaries are defined as o 9 n .i / i @1.C1/ D .x .i / ; xP .i / /j'1.C1/ x .i / x .iN/ D 0; xP .i / ¤ xP .iN/ ; = o n .j / j @1.1/ D .x .j / ; xP .j / /j'1.1/ x .j / x .jN/ D 0; xP .j / ¤ xP .jN/ ; ;
(4.7)
where i D 1; 2 and {N D 3; 2 with (j D 1; 3 and jN D 3; 1). The domains are represented by hatched areas, and the boundaries are depicted by dashed and solid .2/ .3/ curves in Fig. 4.2. The boundaries of @1.C1/ and @1.C1/ are curves at x .2/ D .1/
.3/
x .3/ and the boundaries @1.1/ and @1.1/ are curves at x .1/ D x .2/ . For stick .i /
.i /
motion, the absolute domains 0 and 0 (i D 1; 2; 3) for the two oscillators and particle are n o 9 .1/ .3/ 0 D .x .1/ ; xP .1/ /jx .1/ 2 .xcr ; x .2/ /; xP .1/ D xP .3/ ; > > > > n o > > > .2/ .3/ 0 D .x .2/ ; xP .2/ /jx .2/ 2 .x .1/ ; xcr /; xP .2/ D xP .3/ ; > > > > ˇ > > .1/ ˇ > .3/ 2.1;x /;x .3/ Dx .1/ > .3/ x P P cr .3/ .3/ ˇ > ; 0 D .x ; xP / ˇ .3/ .2/ = or x 2.xcr ;C1/;xP .3/ DxP .2/ o n (4.8) ˇ .1/ .3/ 1 D .x .1/ ; xP .1/ /ˇx .1/ 2 .1; xcr /; xP .3/ ¤ xP .1/ ; > > > > o > n > ˇ > .2/ .3/ 1 D .x .2/ ; xP .2/ /ˇx .2/ 2 .xcr ; C1/; xP .3/ ¤ xP .2/ ; > > > > > ˇ > .1/ .2/ > ˇ .3/ .3/ .1/ > .3/ .3/ .3/ ˇx 2.xcr ;x /;xP ¤xP > 1 D .x ; xP / ˇ .3/ .1/ .2/ .3/ .2/ ; ; or x 2.x ;xcr /;xP ¤xP .i / .3/ .˛/ where xcr is for appearance and vanishing of stick motion with xcr D xcr and .3/ .˛/ xcr D xcr , and ˛ D 1; 2 are for stick on the bottom and top, respectively. The .i / .i / domains 1 and 0 are presented by hatched and shaded regions in Fig. 4.3. The corresponding absolute boundaries are given by dashed curves, and the stick boundaries are defined as .1/ .1/ .3/ .3/ 9 @10 D f.x .1/ ; xP .1/ /j'10 x .1/ xcr D 0; xP .1/ D xP cr g; > > > .2/ .1/ .3/ .3/ = @10 D f.x .2/ ; xP .2/ /j'10 x .2/ xcr D 0; xP .2/ D xP cr g; (4.9) ˇ .3/ > .1/ .1/ ˇ .3/ .3/ > .3/ > .3/ .3/ ˇ'10 x xcr D0;xP DxP cr ; : @10 D .x ; xP / ˇ .3/ .3/ .2/ .2/ or '10 x xcr D0;xP .3/ DxP cr
The vectors for state variables and vector fields are .i /
.i /
.i /
.i /
.i /
.i /
Xœ D .xœ ; xP œ /T ; fœ D .xP œ ; Fœ /T for .i D 1; 2; 3 and œ D 0; 1/;
(4.10)
where i D 1; 2; 3 represents the bottom, top oscillators, and the particle, respectively; and œ D 0; 1 stands for the stick or nonstick domains. Then equation of absolute motion is P .i / D f.i / .X.i / ; t/ X œ œ œ
for i D 1; 2; 3 and œ D 0; 1:
(4.11)
4 Complex Motions in a Fermi Oscillator
a
x(1)
109
b (1) 10
(1) 1
(3) 1( )
x(2)
(2) 10
(3) 1( )
(1) 0
(2) 0
x
(2) 1
(1)
x (2) x(1)
xcr(1)
xcr(2)
x (2)
c
x(3)
(3) 0
xcr(1)
(3) 10
(3) 10
(3) 1
(3) 0
x(3)
xcr(2)
Fig. 4.3 Absolute domains and boundaries with stick: (a) bottom oscillator, (b) top oscillator, and (c) particle
For nonstick motion, .i /
.i /
.3/
.3/
F1 .x1 ; t/ D 2 .i / xP .i / .! .i / /2 x .i / C A.i / cos .i / t; .i D 1; 2/ F1 .x1 ; t/ D g:
) (4.12)
For stick motion, .˛/ N
.˛/ N
.i /
.i /
N .˛/ N 2 .˛/ N N F1 .x1 ; t/ D 2 .˛/ xP N .! .˛/ / x N C A.˛/ cos .˛/ t
F0 .x0 ; t/ D 2d .0/ xP .0/ .! .0/ /2 x .0/ C A.˛/ cos .˛/ t
) .i D ˛; 3/: (4.13)
For simplicity, the relative displacement, velocity, and acceleration between the particle and the bottom or top oscillators are defined as z.i / D x .i / x .N{ / , zP.i / D xP .i / xP .N{ / and zR.i / D xR .i / xR .N{ / , where i D ˛; 3 and iN D 3; ˛ represent the particle
110
Y. Guo and A.C.J. Luo
a (1) 1(
b
z (1)
z (1)
(3) 1( )
)
(1) 10 (1) 10
(1) 1
(1) 0
,
(1) 1
(1) 0
z (1)
z(1) (1) 01
c (3) 1( )
(1) 01
d
z (3)
z (3)
(3) 1( )
(3) 01
(3) 10
,
(3) 01
(3) 0
(3) 1
z(3)
(3) 0
(3) 1
z (3)
(3) 10
Fig. 4.4 Domains and boundaries definition relative to the bottom oscillator: (a) .z; zP/-plane for bottom oscillator, (b) .z; Pz/-plane for bottom oscillator, (c) -plane for particle, and (d) .z; Pz/-plane for particle
and one of the two oscillators, accordingly. The relative domains and boundaries for the particle and oscillators are then defined as sketched in Figs. 4.4 and 4.5 for the motion relative to bottom or top oscillators. The stick domain and boundaries in the relative phase space becomes a point in Figs. 4.4a, c and 4.5a, c. Therefore, the stick domains and boundaries in the relative velocity and acceleration (i.e. zP .i / , zR.i / ) plane is presented in Figs. 4.4b, d and 4.5b, d. The filled regions represent the stick motion domains, while the shaded regions indicate the nonstick motion domains. .i / .i / The domains 0 and 1 for the relative motions for the particle and the two oscillators are
4 Complex Motions in a Fermi Oscillator
a
111
b
z (2)
z (2)
(2) 1( )
(3) 1( )
(2) 10
(2) 10
Ω1(2)
(2) 0
,
Ω1(2)
z(2)
Ω(2) 0
(2) 01
c (3) 1( )
z (2)
(2) 01
d
z (3)
z (3)
(3) 1( ) (3) 01 (3) 10
Ω1(3)
,
(3) 0
z
Ω(3) 0
(3)
Ω1(3)
(3) 01
z (3)
(3) 10
Fig. 4.5 Domains and boundaries definition relative to the top oscillator: (a) -plane for top oscillator, (b) .z; Pz/-plane for top oscillator, (c) .z; Pz/-plane for particle, and (d) .z; zP/-plane for particle
9 .i / 0 D f.z.i / ; zP .i / /jPz.i / D 0; z.i / D 0g; > > > > .1/ = 1 D f.z.1/ ; zP.1/ /jz.1/ 2 .1; 0/g; > .2/ .2/ .2/ P /jz.2/ 2 .0; C1/g; > 1 D f.z ; z > ˇ .3/ .1/ .3/ o > n > > .3/ .3/ .3/ ˇz 2.x x ;0/; 1 D .z ; zP / ˇor z.3/ 2.0;x .2/ x .3/ / : ; .i /
.i /
.i /
.i /
(4.14)
The boundaries @1.C1/ , @1.1/ , @10 and @01 for the particle associated with the bottom or top oscillators are 9 .i / .i / @1.1/ D f.z.i / ; zP .i / /j'1.1/ z.i / D 0; zP .i / ¤ 0g; > > = .j / .j / .j / .j / .j / .j / (4.15) @1.C1/ D f.z ; zP /j'1.C1/ z D 0; zP ¤ 0g; > > .l/ .l/ .l/ .l/ .l/ ; .l/ .l/ @10 D @01 D f.z ; zP /j'10 zPcr D 0; zcr D 0g;
112
Y. Guo and A.C.J. Luo .3/
.3/
where i D 1; 3, j D 2; 3, l D 1; 2; 3. @1.1/ and @1.C1/ are the impact chatter boundaries for the particle relative to the bottom or top oscillators, respectively. .3/ .3/ .1/ @10 and @01 are the stick motion boundaries for the particle. @1.1/ and .2/
@1.C1/ are the impact-chatter boundaries for the bottom or top oscillators, .1/
.1/
respectively. @10 and @01 are the stick motion boundaries for the bottom .2/ .2/ oscillator. @10 and @01 are the stick motion boundaries for the top oscillator. The relative vectors for state variables and relative vector fields are .i /
.i /
.i /
.i /
.i /
.i /
zœ D .zœ ; zPœ /T ; gœ D .Pzœ ; gœ /T ;
(4.16)
where i D 1; 2 are the bottom and top oscillators, respectively; i D 3 are for the particle. œ D 0; 1 gives the corresponding stick and nonstick domains. For i D 1; 2; 3 and œ D 0; 1, the equations of relative motion are .˛/ N
.˛/ N
.˛/ N
zP œ D gœ .xœ ; t/; .˛/ .˛/ .˛/ .3/ zP œ D gœ .zœ ; xœ ; t/; .3/ zP œ .i /
.i /
where xP œ D fœ
D
.3/ .3/ .˛/ gœ .zœ ; xœ ; t/;
9 > > = (4.17)
> > ;
.i / xœ ; t .
1. For nonstick motion, the relative forces per unit mass are .˛/ N
.˛/ N
.˛/ N
.˛/ N
.˛/ N
.˛/
.˛/
.˛/
.˛/
.˛/
N N 2 N N g1 .z1 ; x1 ; t/ D 2 .˛/ xP 1 .! .˛/ / x1 C A.˛/ cos .˛/ t;
g1 .z1 ; x1 ; t/ D 2 .˛/ xP 1 .! .˛/ /2 x1 C A.˛/ cos .˛/ t C g; .3/ .3/ .˛/ g1 .z1 ; x1 ; t/
D g C
.˛/ 2 .˛/ xP 1
C
.˛/ .! .˛/ /2 x1
A
.˛/
.˛/
cos t:
9 > > = > > ;
(4.18)
2. For stick motion, the relative velocities and relative forces per unit mass are .3/
.˛/
zP0 D zP0 D 0 .˛/ N
.˛/ N
.˛/ N
.˛/ N
.˛/ N
N N 2 N N g1 .z1 ; x1 ; t/ D 2 .˛/ xP 1 .! .˛/ / x1 C A.˛/ cos .˛/ t; .˛/ .˛/ .˛/ g0 .z1 ; x1 ; t/
D
.3/ .3/ .˛/ g0 .z1 ; x1 ; t/
D 0:
9 > > = > > ;
(4.19)
4.3 Mechanism of Motion Switchability To discuss the motion switchability mechanism of the Fermi oscillator, the normal vector of the boundary relative to the bottom or top oscillator is n@˛ˇ D r'˛ˇ D
@'˛ˇ @'˛ˇ ; @z @Pz
T ;
(4.20)
4 Complex Motions in a Fermi Oscillator
113
where r D .@=@z; @=@Pz; /T . n@.3/ and n@.3/ are the normal vectors of the 10 01 stick boundaries, n@.3/ and n@.3/ are the normal vectors of impact chatter 1.1/
1.C1/
boundaries. Thus, n@.3/ D n@.3/ D .0; 1/T ; 10
and n@.3/
01
1.1/
D n@.3/
D .1; 0/T :
(4.21)
1.C1/
The zero-order and first-order G-functions for the stick boundaries relative to the bottom or top oscillators are introduced from Luo (2008, 2009) G
.0;0/
G
.1;0/
.˛/ @01 .˛/
@10
.˛/
.3/
.z0 ; x0 ; tm˙ / D nT
.˛/ @01
.˛/
.3/
.z1 ; x1 ; tm˙ / D nT
.˛/
@10
.˛/
.3/
.˛/
.3/
:g.˛/ .z0 ; x0 ; tm˙ /; :g.˛/ .z1 ; x1 ; tm˙ /;
9 > > > > > > > > =
.˛/ .3/ :Dg.˛/ .z0 ; x0 ; tm˙ /; > > > > > > > .1;1/ .˛/ .3/ T .˛/ .˛/ .3/ ; G .˛/ .z1 ; x1 ; tm˙ / D n .˛/ :Dg .z1 ; x1 ; tm˙ /; >
G
.0;1/
.˛/
@01
.˛/
.3/
.z0 ; x0 ; tm˙ / D nT
@10
@10
.0;0/
.3/ .˛/ G .3/ .z0 ; x0 ; tm˙ / @
D nT
.3/ @01
01
G
.1;0/
.3/
.3/
@10
.˛/
.z1 ; x1 ; tm˙ / D nT
.3/
@10
.3/
.˛/
.3/
.˛/
:g.3/ .z0 ; x0 ; tm˙ /; :g.3/ .z1 ; x1 ; tm˙ /;
9 > > > > > > > > =
.3/ .˛/ :Dg.3/ .z0 ; x0 ; tm˙ /; > > > > > > > .1;1/ .3/ .˛/ T .3/ .3/ .˛/ ; G .3/ .z1 ; x1 ; tm˙ / D n .3/ :Dg .z1 ; x1 ; tm˙ /: >
G
(4.22)
.˛/
@01
.0;1/
.3/
.3/ @01
.˛/
.z0 ; x0 ; tm˙ / D nT
(4.23)
.3/ @01
@10
@10
Notice that tm is the switching time of the motion on the corresponding boundary and tm˙ D tm ˙ 0 which represents the motion on each side of the boundary in different domains. The G-functions for the impact chatter boundaries are G
.0;1/
G
.1;1/
G
.0;1/
G
.1;1/
.3/ @1.C1/ .3/
@1.C1/ .3/ @1.1/ .3/
@1.1/
.3/
.2/
.3/ @1.C1/
.3/
.2/
.z1 ; x1 ; tm˙ / D nT
.3/
@1.C1/
.3/
.1/
.3/ @1.1/
.3/
.1/
.3/
@1.1/
.2/
:g.3/ .z1 ; x1 ; tm˙ /;
> > > > > > > .3/ .1/ .3/ :Dg .z1 ; x1 ; tm˙ /: > ; .3/
.z1 ; x1 ; tm˙ / D nT .z1 ; x1 ; tm˙ / D nT
9 > > > > > > .3/ .3/ .2/ > :Dg .z1 ; x1 ; tm˙ /; > = .3/
.z1 ; x1 ; tm˙ / D nT
.1/
:g.3/ .z1 ; x1 ; tm˙ /;
.i /
.i /
(4.24)
From the passable flow condition from domain 1 to 0 in Luo (2008); Luo and Guo (2010a), the analytical conditions for stick motion on bottom or top oscillators are 9 .0;1/ .˛/ .3/ .1/˛ G .˛/ .z1 ; x1 ; tm / < 0; > = @10
˛
.1/ G
.0;0/ .˛/
@01
.z0 ; x0 ; tmC / < 0: > ; .˛/
.3/
114
Y. Guo and A.C.J. Luo
.1/˛ G .1/˛ G
.0;1/ .3/ @10
.0;0/ .3/ @01
9 .3/ .˛/ .z1 ; x1 ; tm / > 0; > = (4.25)
.z0 ; x0 ; tmC / > 0: > ; .3/
.˛/
Therefore, .˛/
.˛/
.3/
.˛/
.˛/
.3/
.3/
.3/
.˛/
.3/
.3/
.˛/
)
.1/˛ g1 .z1 ; x1 ; tm / < 0; .1/˛ g0 .z0 ; x0 ; tmC / < 0:
)
.1/˛ g1 .z1 ; x1 ; tm / > 0;
(4.26)
.1/˛ g0 .z0 ; x0 ; tmC / > 0:
The foregoing conditions produce the existence conditions of stick motion on bottom or top oscillators, i.e., xR .1/ .tm˙ / > xR .3/ .tm˙ / D g; for the bottom xR .2/ .tm˙ / < xR .3/ .tm˙ / D g; for the top.
(4.27)
The physical meaning of the above conditions is that the acceleration of the bottom oscillator xR .1/ .tm˙ / should be larger than the particle’s acceleration of xR .3/ .tm˙ / D g in order for the particle to stick on the bottom oscillator. However, the acceleration of the top oscillator xR .2/ .tm˙ / should be less than the particle acceleration in order for the particle to stick on the top oscillator. The criteria for .˛/ vanishing of the stick motion from the bottom or top oscillator @01 at are from Luo (2008, 2009), 9 .0;0/ .˛/ .3/ .0;1/ .˛/ .3/ G .˛/ .z0 ; x0 ; tm / D 0; .1/˛ G .˛/ .z0 ; x0 ; tm / > 0I > = @01
@01
.1;0/
.˛/ .3/ G .˛/ .z1 ; x1 ; tmC / @
D 0; .1/ G ˛
01
G
.0;0/
.3/
.˛/
.1;1/ .˛/
@01
˛ .3/ .z0 ; x0 ; tm / D 0; .1/ G
@01
.z1 ; x1 ; tmC / > 0: > ;
.3/ .˛/ G .3/ .z0 ; x0 ; tmC / @ 01
D 0; .1/ G ˛
.3/
9 .3/ .˛/ > .z ; x ; t / < 0I = m .3/ 0 0
.0;1/
@01
.1;0/
.˛/
.1;1/ .3/ @01
.3/ .˛/ .z0 ; x0 ; tmC / < 0: > ;
(4.28)
.˛/
From the foregoing equations, the relative force relations for @01 are .˛/
.˛/
.3/
.˛/
.˛/
.3/
.˛/
.˛/
.3/
.˛/
.˛/
.3/
.3/
.3/
.˛/
.3/
.3/
.˛/
.3/
.3/
.˛/
.3/
.3/
.˛/
g0 .z0 ; x0 ; tm / D 0; .1/˛ dtd g0 .z0 ; x0 ; tm / > 0I g1 .z1 ; x1 ; tmC / D 0; .1/˛ dtd g1 .z1 ; x1 ; tmC / < 0: g0 .z0 ; x0 ; tm / D 0; .1/˛ dtd g0 .z0 ; x0 ; tm / < 0I g1 .z1 ; x1 ; tmC / D 0; .1/˛ dtd g1 .z1 ; x1 ; tmC / > 0:
)
) (4.29)
4 Complex Motions in a Fermi Oscillator
115
With the relative acceleration and jerk, one gets 9 .1/ .3/ «m˙ < x «m˙ D 0 for the bottom, = xR .1/ .tm˙ / D xR .3/ .tm˙ / D g; x .2/
.3/
«m˙ > x «m˙ D 0 for the top. xR .2/ .tm˙ / D xR .3/ .tm˙ / D g; x
;
(4.30)
Using the G-functions of the flow to each boundary, the conditions of grazing motions are from Luo (2008, 2009), i.e., .1/˛ G
.1;0/
.˛/
.˛/
@1.1/ ˛ .1;1/
and .1/ G .1/˛ G
9 > > > > > > .˛/ .3/ .˛/ > .Z1 ; X1 ; tm˙ / > 0 for @1.1/ I > =
.˛/
@1.1/
.1;0/
.˛/
.1/˛ G
.˛/
.˛/
@1.C1/
.1;0/
.3/
.3/ @1.1/
and .1/˛ G .1/˛ G
> > > > > > .˛/ .3/ .˛/ > .Z1 ; X1 ; tm˙ / > 0 for @1.C1/ : > ; .3/
.Z ; X1 ; tm˙ / D 0;
1 @1.C1/ ˛ .1;1/
and .1/ G
.3/
.Z1 ; X1 ; tm˙ / D 0;
.1;1/ .3/
.3/
.3/ @1.C1/
and .1/˛ G
9 > > > > > > .˛/ .˛/ .˛/ > .Z1 ; X1 ; tm˙ / < 0 for @1.1/ ; > = .˛/
.Z1 ; X1 ; tm˙ / D 0;
@1.1/
.1;0/
(4.31)
.˛/
.Z1 ; X1 ; tm˙ / D 0;
.1;1/ .3/
@1.C1/
.3/
.˛/
.Z1 ; X1 tm˙ / < 0
> > > > > > .˛/ > for @1.C1/ : > ;
(4.32)
So the grazing motion conditions on the bottom and top for the nonstick motion boundaries are ) .1/ .3/ xP .3/ D xP .1/ and xR .1/ < xR .3/ D g for @1.1/ ; @1.1/ I (4.33) .2/ .3/ xP .3/ D xP .2/ and xR .2/ > xR .3/ D g for @1.C1/ ; @1.C1/ : The grazing conditions for stick motion boundaries are from Luo (2008, 2009), i.e., G
.1;0/
G
.0;0/
.˛/
.˛/ @10
G G
.1;0/ .3/ @10
.0;0/ .˛/
@01
.1;1/
.i / @10
.˛/
.˛/
@01
.3/
.Z1 ; X1 ; tm˙ / D 0; and .1/˛ G .3/
.Z0 ; X0 ; tm˙ / D 0; and .1/˛ G
9 .˛/ .3/ .˛/ .Z1 ; X1 ; tm˙ / > 0 for @10 ; > =
.0;1/ .˛/
@01
.3/
.˛/
.Z1 ; X1 ; tm˙ / D 0; and .1/˛ G .3/
.˛/
.Z0 ; X0 ; tm˙ / D 0; and .1/˛ G
.1;1/ .i / @10
.0;1/ .˛/
@01
.˛/ .3/ .˛/ .Z0 ; X0 ; tm˙ / < 0 for @01 : > ;
9 .3/ .˛/ .˛/ .Z1 ; X1 ; tm˙ / < 0 for @10 ; > =
.3/ .˛/ .˛/ .Z0 ; X0 ; tm˙ / > 0 for @01 : > ;
(4.34)
116
Y. Guo and A.C.J. Luo
The corresponding accelerations and jerks should satisfy the following relations. .1/
.3/
.1/
.3/
.2/
.3/
.2/
.3/
«.1/ < x «.3/ D 0 for @10 ; @10 ; xR .1/ D xR .3/ D g and x xR .1/ D xR .3/ D g and x «.1/ > x «.3/ D 0 for @01 ; @01 ; «.2/ > x «.3/ D 0 for @10 ; @10 ; xR .2/ D xR .3/ D g and x xR .2/ D xR .3/ D g and x «.2/ < x «.3/ D 0 for @01 ; @10 ;
) )
9 > > for bottomI > > = for top:
> > > > ;
(4.35)
4.4 Periodic Motions and Stability From boundaries, the switching sets for the Fermi oscillator without stick are P 1.1/
P 1.C1/
D
P.1/
N P.2/
1.1/
9 > > o> > .1/ > = ¤ xP k >
N P.3/
1.1/
1.1/
ˇ n ˇ .3/ .1/ .1/ .2/ .2/ .3/ .3/ .1/ .3/ D .xk ; xP k ; xk ; xP k ; xk ; xP k ; tk /ˇxk D xkC ; xP k (4.36) N P.2/ N P.3/ P.1/ > D 1.C1/ > 1.C1/ 1.C1/ > ˇ n o> > ˇ .3/ .1/ .1/ .2/ .2/ .3/ .3/ .2/ .3/ .2/ > D .xk ; xP k ; xk ; xP k ; xk ; xP k ; tk /ˇxk D xkC ; xP k ¤ xP k ;
where the switching sets
P.i / 1.1/
and
P.i / 1.C1/
.i /
are defined on boundary @1.1/
.i /
and @1.C1/ , respectively. The corresponding definitions for the top and bottom oscillators plus the particle are given as n ˇ o 9 P.i / ˇ .˛/ .i / .i / .˛/ N .˛/ .˛/ N .i / xk ; xP k ; tk ˇxk D xk ; xP k ¤ xP k @1.1/ ; ˛D1; 3; = 1.1/ D ˇ o n (4.37) P.i / ˇ .˛/ .i / .i / .˛/ N .˛/ .˛/ N .i / xk ; xP k ; tk ˇxk D xk ; xP k ¤ xP k @1.C1/ ; ˛D2; 3:; 1.C1/ D Thus, the generic mappings for motions without stick motion are P1 W †1.1/ ! †1.C1/ ; P2 W †1.C1/ ! †1.1/ ; P3 W †1.1/ ! †1.1/ ; P4 W †1.C1/ ! †1.C1/ :
(4.38)
From the above definitions, the switching subsets and the sub-mappings without stick motion are sketched in Fig. 4.6a, b for the bottom and top oscillators, respectively. In Fig. 4.6c, the sub-mappings without stick motion for the particle are presented. P1 D ..1/ P1 ;.2/ P1 ;.3/ P1 /; P2 D ..1/ P2 ;.2/ P2 ;.3/ P2 /; P3 D ..1/ P3 ;.2/ P3 ;.3/ P1 /; P4 D ..1/ P4 ;.2/ P4 ;.3/ P4 /; .i /
.i /
.i /
.i /
.i /
.i /
.i /
.i /
.i /
(4.39)
P1 W †1.1/ ! †1.C1/ ;.i / P2 W †1.C1/ ! †1.1/ ;
.i /
P3 W †1.1/ ! †1.C1/ ;.i / P4 W †1.1/ ! †1.C1/ ;
(4.40)
4 Complex Motions in a Fermi Oscillator
a
Σ
(1) 1(
b
x(1)
)
117
Σ
(3) 1( )
x(2)
Σ1((2)
Ω1(1) (1)
(1)
Σ1((2)
) (2)
)
P1
P2
P4
Σ1((3)
x(1) (1)
(1)
(2)
)
P3
(2)
P4 (2)
P1
Σ1((1) c
P3
x(2)
P2
Ω1(2) )
x(3)
Σ1((3)
Σ1((3)
)
)
Ω
(3) 1
(3)
P1 (3)
(3)
P4
x(3)
P3 (3)
P2
Fig. 4.6 Switching sets and generic mappings for nonstick motion in absolute coordinates: (a) bottom oscillator, (b) top oscillator, and (c) particle
Similarly, the switching subsets and the sub-mappings with stick motion are presented in Fig. 4.7a, b for the bottom and top oscillators, respectively. The submappings with stick motion for the particle are sketched in Fig. 4.7c. From the boundaries in (4.7) and (4.9), the switching sets of the Fermi oscillator with stick motion are defined as 9 .1/ N .2/ N .3/ > ˛ †10 D ˛ †10 ˛ †10 ˛ †10 > ˇ n o > > > ˇ .3/ .˛/ .3/ .˛/ .˛/ N .˛/ N .˛/ .˛/ .3/ .3/ > D .xk ; xP k ; xk ; xP k ; xk ; xP k ; tk /ˇxk Dxk ; xP k DxP k ; > > > 9> N .2/ N .3/ > .1/ > = > †1.1/ D†1.1/ †1.1/ †1.1/ > > ˇ n o > (4.41) > ˇ .3/ .1/ .3/ .1/ .1/ .2/ .2/ .3/ .3/ .1/ > D .xk ; xP k ; xk ; xP k ; xk ; xP k ; tk /ˇxk Dxk ; xP k ¤ xP k ;=> > > > > N .2/ N .3/ .1/ > > > †1.C1/ D†1.C1/ †1.C1/ †1.C1/ > > > > > ˇ o > n > > > ˇ .3/ .2/ .3/ .1/ .1/ .2/ .2/ .3/ .3/ .2/ > ; D .xk ; xP k ; xk ; xP k ; xk ; xP k ; tk /ˇxk Dxk ; xP k ¤ xP k ; ;
118
Y. Guo and A.C.J. Luo
a
x(1)
Σ1((1)
)
(1)
(1)
Σ1((3)
(1) Σ10
Ω1(1)
b x(2) )
Σ1((3)
Ω (1) 0
P2
)
Ω
Ω (2) 0
P5 (2)
x (1) (1)
P1 (2)
P4
P3 (2)
P1
P3
x (2)
P5 (2)
c
)
(2) 1
(2) (1)
P4
(1)
Σ1((2)
(2) Σ10
P2
x(3) Ω1(3)
Ω (3) 0 (3)
P1 (3)
(3)
P5 (3)
Ω (3) 0
P4
x(3) (3)
P3
(3)
P6
P2
Fig. 4.7 Switching sets and generic mappings for stick motion in absolute coordinates: (a) bottom oscillator, (b) top oscillator, and (c) particle
P where ˛ D 1; 2 with ˛N D 2; 1. The switching set ˛ 10 is defined on the boundary @10 . Thus, the generic mappings for the stick motion are defined as .1/
.2/
P1 W †10 ! †1.C1/ ; or †1.1/ ! †10 ; .1/
.2/
P2 W †1.C1/ ! †10 ; or †10 ! †1.1/ ; .1/
.2/
P3 W †1.1/ ! †10 ; †10 ! †1.C1/ ; .1/
.1/
.2/
.2/
P5 W †10 ! †10 ; and P6 W †10 ! †10 :
(4.42)
where the global mappings P1 and P2 will map from one switching set to another. The local mappings of (P3 , P4 , P5 , and P6 ) map from one switching set to itself, as
4 Complex Motions in a Fermi Oscillator
119
in Figs. 4.6 and 4.7. From the above definitions, the governing equations for generic mapping Pj , (j D 1; 2; 3; 4) can be expressed by f.j / Yk ; YkC1 D 0 for Pj :
(4.43)
with T .j / .j / .j / .j / .j / .j / ; f.j / ; D f1 ; f2 ; f3 ; f4 ; f5 ; f6 T .1/ .1/ .2/ .2/ .3/ .3/ Yk D xk ; xP k ; xk ; xP k ; xk ; xP k ; tk T .1/ .1/ .2/ .2/ .3/ .3/ YkC1 D xkC1 xP kC1 ; xkC1 ; xP kC1 ; xkC1 ; xP kC1 ; tkC1
(4.44)
9 .3/ .1/ .3/ .2/ xk D xk and xkC1 D xkC1 for P1 ; > > > > > .3/ .2/ .3/ .1/ x D x and x Dx for P2 ; = k .3/ xk .3/ xk
D D
k .1/ xk .2/ xk
kC1 .3/ and xkC1 .3/ and xkC1
D D
kC1 .1/ xkC1 .2/ xkC1
.˛/
The governing equations for the stick mapping P5 .j /
.j /
(4.45)
for P3 ; > > > > > for P4 ; ; .˛/
and P6
can be expressed as
.˛/
f.j /.Zk ; ZkC1 / D 0 for Pj .j D 5; 6/
(4.46)
and T .j / .˛/ N .˛/ N .3/ .3/ Zk D xk ; xP k ; xk ; xP k ; tk T .j / .˛/ N .˛/ N .3/ .3/ ZkC1 D xkC1 ; xP kC1 ; xkC1 ; xP kC1 ; tkC1 f.j / D .0/ .˛/ f5 .3/
.0/
.˛/ N f1 ; .˛/
.0/
.˛/ N f2 ;
.0/
.3/f1 ;
.0/
.0/
.3/f2 ;
T
.˛/f5
.˛/
D g1 .0; XkC1 ; tkC1 /: .˛/
.3/
.˛/
.3/
.˛/
.3/
.˛/
xk D xk and xP k D xP k ; xkC1 D xkC1 and xP kC1 D xP kC1 ; .˛/
)
.˛/
.1/˛ xR k < .1/˛C1 g; and .1/˛ xR kC1 > .1/˛C1 g; with j D 5; 6 for .˛; ˛/ N D .1; 2/ and .2; 1/:
(4.47)
The notation for mapping action is introduced as Pjk jk1 :::j1 D Pjk ı Pjk1 ı ı Pj1 ;
(4.48)
120
Y. Guo and A.C.J. Luo
where jk 2 f1; 2; 3; 4; 5; 6g is a positive integer. For a motion with m-time repeated mapping structure of Pj1 j2 jk , the total mapping structure can be expressed as .m/ Pjk jk1 :::j1 D Pjk ı Pjk1 ı ı Pj1 ı ı Pjk ı Pjk1 ı ı Pj1 DP.jk jk1 :::j1 /m: „ ƒ‚ … m
(4.49)
Consider a motion with a generalized map,
P D P4ml 0k3l 2k2l 3nl 1k1l ı ı P4m1 0k31 2k21 3n1 1k11 D P .4ml 0k3l 2k2l 3nl 1k1l /:::.4m1 0k31 2k21 3n1 1k11 / ; „ ƒ‚ … „ ƒ‚ … lterms
lterms
where jjs 2 f0; 1g and ms ; ns 2 N; (s D 1; 2; : : : ; l ). Define vectors
(4.50)
Xk .Xk1 ; Xk2 ; Xk3 ; Xk4 ; Xk5 ; Xk6 /T ; .1/
.1/
.2/
.2/
.3/
.3/
Xk1 2 fxk ; xP k ; xk ; xP k ; xk ; xP k ; tg;
(4.51)
and Yk .Yk1 ; Yk2 ; Yk3 ; Yk4 ; Yk5 ; Yk6 /T ; .1/
.1/
.2/
.2/
.3/
.3/
Yk1 2 fyk ; yPk ; yk ; yPk ; yk ; yPk ; tg
(4.52)
The motion pertaining to the mapping structure in (4.50) can be determined by YkCPl
sD1 .ms Ck3s Ck2s Cns Ck1s /
DP Xk D P ml k3l k2l nl k1l X .4 0 2 3 1 / : : : .4m1 0k31 2k21 3n1 1k11 / k ƒ‚ … „ lterms
(4.53)
From the algebraic equations for generic mappings in (4.44)–(4.47), one can obtain a set of nonlinear algebraic equations for such a mapping structure, i.e., 9 > f.1/ .Xk ; YkC1 / D 0; : : : ; f.3/ .XkCk1l ; YkCk1l C1 / D 0; : : : ; > > = .2/ f .XkCk1l Cnl ; YkCk1l Cnl C1 / D 0; : : : ; (4.54) .4/ f .XkCPl .ms Ck3s Ck2s Cns Ck1s /1 ; YkCPl .ms Ck3s Ck2s Cns Ck1s / / D 0 > > sD1 sD1 > ; YkC D XkC : where D 1; : : : ;
Xl sD1
.ms C k3s C k2s C ns C k1s / 1
(4.55)
The periodic motion pertaining to such a mapping requires YkCPl
sD1 .ns Ck3s Cms Ck2s Ck1s /
D Xk
(4.56)
4 Complex Motions in a Fermi Oscillator
or x
.i /
kC .i /
xP
kC
t
121
Pl
9 .i / D xk ; =
Pl
.i / D xP k ;
sD1 .ms Ck3s Ck2s Cns Ck1s /
.i /
kC
sD1 .ms Ck3s Ck2s Cns Ck1s /
for i = 1, 2, 3
.i /
Pl
sD1 .ms Ck3s Ck2s Cns Ck1s /
D tk C 2N :
(4.57)
Solving (4.54)–(4.57) generates the switching sets of periodic motion relative to the mapping structure in (4.50). Once the switching points for a specific periodic motion is obtained, its local stability and bifurcation analysis can be completed through the corresponding Jacobian matrix. For instance, the Jacobian matrix of the mapping structure in (4.53) is computed, i.e., DP D DP
.4ml 0k3l 2k2l 3nl 1k1l /:::.4m1 0k31 2k21 3n1 1k11 /
„
ƒ‚
…
lterms
D
l Y
.ms /
DP4
.k3s /
DP0
.k2s /
DP2
.ms /
DP3
.k1s /
DP1
;
(4.58)
sD1
where
@Y C1 DP D @X for D k; k C 1; : : : ; k C
Xl sD1
66
@Y. C1/i D @Xj
:
(4.59)
66
.ms C k3s C k2s C ns C k1s / 1
(4.60)
and all the Jacobian matrix components can be computed through (4.54). The variational equation for a set of switching points o n (4.61) Xk ; YkC1 ; : : : ; XkC†l .m Ck Ck Cn Ck / sD1
s
3s
2s
s
1s
is YkC†l
sD1 .ms Ck3s Ck2s Cns Ck1s /
DP .Xk /Xk
(4.62)
Xk
(4.63)
If YkC†l
sD1 .ms Ck3s Ck2s Cns Ck1s /
the Eigenvalues are computed by ˇ ˇ ˇDP .X / Iˇ D 0 k
(4.64)
If all jœi j < 1 for .i D 1; 2; : : : ; 6/, the periodic motion is stable. If one of jœi j < 1 for ˇ ˇ(i 2 f1; 2; : : : ; 6g), the periodic motion is unstable. If one of jœi j D 1 and ˇœj ˇ < 1 for (i; j 2 f1; 2; : : : ; 6g and j ¤ i ), the period-doubling bifurcation of ˇ ˇ periodic motion occurs. If one of jœi j D 1 and ˇœj ˇ < 1 for (i; j 2 f1; 2; : : : ; 6g and j ¤ i ), the saddle-node bifurcation of the periodic motion occurs. If jœ1;2;3;4 j < 1
122
Y. Guo and A.C.J. Luo
with the complex eigenvalues of jœ5;6 j D 1, the Neimark bifurcation of the periodic motion occurs. However, the Eigenvalue analysis cannot be used to predict sticking and grazing motions. Both of them should be determined through the normal vector fields, and the stick motion is determined by (4.25) and the grazing bifurcation is determined by (4.32) or (4.34).
4.5 Illustrations Setting e .1/ D e .2/ D e, the bifurcation scenario of varying e for the Fermi oscillator is presented in Fig. 4.8. The system parameters are Q.1/ D Q .2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1. The switching displacement, velocity, and phase of the particle versus the restitution coefficient e are shown in Fig. 4.8a–c, respectively. The acronyms “PD,” “GB,” and “NB” indicate the period-doubling bifurcation, grazing bifurcation, and Neimark bifurcation, respectively. The shaded areas are for regions of periodic motion. For e 2 .0:0; 0:58/, the impact charter with stick motion exists. In other words, the particle is undergoing the periodic motion, where stick motion with top or bottom oscillator occurs after impact chattering. At e D 0:158, 0:219 and 0:58, grazing bifurcations occur where the current periodic motion disappears, and another different periodic motion starts. Using the same parameters, the analytical prediction of periodic motions with varying the restitution coefficient is obtained as in Fig. 4.9. The displacement, velocity, and switching phase of the particle versus the coefficient of restitution are shown in Fig. 4.9a–c, respectively. The solid and dotted curves represent the stable and unstable solutions, respectively. The gray area indicates stable periodic motions, and the yellow region represents complex periodic motions with coexisting solutions, which is zoomed in Figs. 4.10 and 4.11. The acronyms “PD,” “NB,” “USN ,” and “GB” represent the period doubling bifurcation, Neimark bifurcation, unstable saddle node bifurcation, and grazing bifurcation, respectively. For e 2 .0:0; 0:58/, the periodic motion of impact chatter with stick exists. And there are three grazing bifurcations existing at e D 0:158, 0:219, and 0:58. For e 2 .0:58; 0:693/, the stable periodic motions with mapping structures of P213m (m D 4; 5; : : : ; n ) exist, where increases as decreases. Each branch of these periodic motions becomes unstable with a Neimark bifurcation and then disappears with an unstable saddle node bifurcation on the right end, while on the left end motion becomes unstable with a period doubling bifurcation as presented in Figs. 4.10 and 4.11. The blue area in Figs. 4.10 and 4.11 shows an example of such solutions. Also in this region there is a branch of periodic motion with mapping structure of P 2133 2 , . / which becomes unstable after the period doubling bifurcation at e D 0:628, and the unstable solution disappears after the grazing bifurcation at e D 0:115. The stable P 2133 2 motion disappear at the saddle node bifurcation at e D 0:771, which is . / corresponding to the period doubling bifurcation of the stable P2133 motion when the motion becomes unstable. Similarly, as e increases, the stable P2133 motion becomes
4 Complex Motions in a Fermi Oscillator
a
GB GB
1.2
GB
NB PD
Displacment of the Particle x(3)
x(3) k+3
P153i 2 0.6
P153i264 j
P2133 P213m
Stick Motion
i, j =1, 2,...,n
P(2133 )2
m = 4,5,...,n (3) xk+2
0.0
(3) xk+4 (3) xk+1
-0.6 0.0
b
Velocity of the Particle y(3)
0.2
0.4 0.6 Coefficient of Restitution e
GB GB
7.0
GB
0.0
0.8
xk(3)
1.0
NB PD (3) yk+3 (3) yk+ 2
3.5
(3) yk+1
P153i 264 j
i, j =1,2,...,n
-3.5
Stick Motion
P2133
P153i 2 y
(3) k
P213m
P(2133 )2
m = 4,5,...,n
-7.0
(3) yk+4
-10.5 0.0
c
0.2
0.4 0.6 Coefficient of Restitution e GB
GB GB
1.0
tk Stick Motion
P153i 2
4
i , j =1, 2,...,n
tk+4 P2133
P213m
P153i264 j 2
0.8
NB PD
6
Switching Phase Mod (Ω, t2π)
Fig. 4.8 Bifurcation scenario of varying restitution coefficient: (a) displacement of particle, (b) velocity of particle, and (c) switching phase. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1)
123
m = 4,5,...,n
tk+3
P(2133 )2 tk+2
0 0.0
tk+1 0.2
0.4 0.6 Coefficient of Restitution e
0.8
1.0
124 Fig. 4.9 Analytical prediction of varying the restitution coefficient of impact: (a) switching displacement of particle, (b) switching velocity of particle; (c) switching phase. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1)
Y. Guo and A.C.J. Luo
4 Complex Motions in a Fermi Oscillator Fig. 4.10 Zoomed Analytical prediction of varying the restitution coefficient of impact: (a) switching displacement of particle, (b) switching velocity of particle; (c) switching phase. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1)
125
126 Fig. 4.11 Zoomed Analytical prediction of varying the restitution coefficient of impact: (a) real part of eigenvalues, (b) imaginary part of eigenvalues, and (c) magnitude of eigenvalues. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1)
Y. Guo and A.C.J. Luo
4 Complex Motions in a Fermi Oscillator
127
unstable after the Neimark bifurcation at e D 0:939. Finally, the unstable solution disappears after the unstable saddle node bifurcation at e D 0:954. The real parts, imaginary parts, and magnitudes of the eigenvalues are illustrated in Fig. 4.12a–c, respectively, and the zoomed views are in Fig. 4.11a–c. Using the same parameters, a periodic motion of P36 2646 15 is illustrated with .1/ .3/ e D 0:1 in Figs. 4.13 and 4.14. The initial conditions are given as x0 D x0 D .1/ .3/ .2/ .2/ 0:5575740, xP 0 D xP 0 D 1:5959726, x0 D 0:0613450, xP 0 D 1:6725197 for t0 D 0:0227104. The time histories of displacement and velocity are presented in Fig. 4.13a, b, respectively. The black solid curves give the motion of the bottom and top oscillators. The red solid curve depicts the motion of the particle. The yellow shaded area indicates the region of stick motion, and the black circles represent the switching points of the motion. The particle with the bottom oscillator (P3 ) impacts six times, and the stick motion is formed with the bottom oscillator (P36 ). The particle will freely flight. The particle with the top oscillator impacts six times (P46 ). After that, the stick motion with the top oscillator (P6 ) is formed. This forms a complete periodic motion. Discontinuity of the velocities can be observed from Fig. 4.13b. The velocities of the bottom and top oscillators are very close to each other, and they do not change much after impact because the mass of the particle is much smaller than the two oscillators. The particle trajectory in phase plane with moving boundaries is presented in Fig. 4.13c. The black solid curves are the moving boundaries, and the red solid curve represents the motion of the particle. The discontinuity due to impacts is also observed for both of the moving boundaries and the motion of the particle. For the onset and vanishing conditions of stick motion, the acceleration distribution along the particle displacement, the time histories of acceleration and jerk are presented in Fig. 4.14a–c, respectively. After impacting six times with the bottom oscillator, the velocities of particle and bottom oscillator become equal, and the acceleration of the bottom oscillator is greater than the acceleration of particle (g), thus the onset condition of stick motion with the bottom oscillator (P5 ) is satisfied. Thus, the particle starts to move together with the bottom oscillator. This motion will continue until the forces per unit mass (or acceleration) equal to g again. At the same time, the two jerks are less than zero, which satisfies the vanishing condition of stick motion on bottom. Thus, the motion relative to P5 switches into the motion relative to P1 . The particle impacts six times with the top oscillator until the velocity of the particle equal to that of the top oscillator. At the same time, the acceleration of the top oscillator is less than the one of the particle (g), and the onset condition of the stick motion on the top (P6 ) satisfies. Thus, the particle starts moving together with the top oscillator until their acceleration equals to g again. The jerks are less than zero, which satisfies the vanishing condition of the stick motion on top. Therefore, the particle separates with the top oscillator and switches into the free flight motion in domain 1 until the particle impacts with the bottom oscillator. The chaotic motion is illustrated in Figs. 4.15 and 4.16 under the same parameters .1/ .1/ with e D 0:95. The initial conditions are x0 D 0:941968193, xP 0 D 1:86109613,
128 Fig. 4.12 Analytical prediction of varying the restitution coefficient of impact: (a) real part of eigenvalues, (b) imaginary part of eigenvalues, and (c) magnitude of eigenvalues. (Q.1/ D Q.2/ D 12:0, .2/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1)
Y. Guo and A.C.J. Luo
4 Complex Motions in a Fermi Oscillator
129
Fig. 4.13 Periodic motion: (a) displacement time history, (b) velocity time history, and (c) trajectory of particle with moving boundaries. (Q.1/ D Q.2/ D 12:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, e .1/ D e .2/ D 0:1, h D 0:5, k .1/ D k .2/ D 80:0, .1/ .3/ .1/ .3/ c .1/ D c .2/ D 0:1 ). Initial conditions: x0 D x0 D 0:5575740, xP 0 D xP 0 D 1:5959726, .2/ .2/ x0 D 0:0613454, xP 0 D 1:6725197 for t0 D 0:0227104
130
Y. Guo and A.C.J. Luo
Fig. 4.14 Periodic motion: (a) acceleration along displacement, (b) acceleration time history and (c) jerk time history. (Q.1/ D Q.2/ D 12:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1; 0, m.3/ D 0:01, e .1/ D e .2/ D 0:1, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1). Initial conditions: .1/ .3/ .1/ .3/ .2/ .2/ x0 D x0 D 0:5575740, xP0 D xP 0 D 1:5959726, x0 D 0:0613454, xP0 D 1:6725197 for t0 D 0:0227104
4 Complex Motions in a Fermi Oscillator Fig. 4.15 Chaotic motion: (a) displacement time history, (b) velocity time history, (c) map of Poincare .3/
.3/
xk ; yk
a
1.2
x(2) x(3)
.
(Q.1/ D Q.2/ D 20:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1 ). The initial conditions are .1/ x0 D 0:941968193, .1/ xP 0 D 1:86109613, .2/ .3/ x0 D x0 D 1:45709118, .2/ xP 0 D 1:82722043, and .3/ xP 0 D 10:7134817 for t0 D 0:289711605
131
0.6
0.0
x(1)
-0.6 0
b
1
2
3
4
5
3
4
5
Time t
10 y(3) 5 y(
0
)
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-10 0
c
1
2 Time t
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-6
-12 -0.6
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Fig. 4.16 Chaotic motion: (a) Poincare map of .4/
a
7.0
.4/
xk ; yk , (b) Poincare map .3/ of xk mod .th ; 2h/ , and (c) Poincare map of .3/ yk mod .th ; 2h/ . (Q.1/ D Q.2/ D 20:0, .1/ D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1). The initial conditions are .1/ x0 D 0:941968193, .1/ xP 0 D 1:86109613, .2/ .3/ x0 D x0 D 1:45709118, .2/ xP 0 D 1:82722043, and .3/ xP 0 D 10:7134817 for t0 D 0:289711605
x(1)
3.5
x(2)
0.0
-3.5
-7.0 -0.6
0.0
0.6
Displacement of the Oscillators x(
b
1.2 )
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0.6
0.0
-0.6 0
c
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2 3 4 5 Switching Phase Mod(Ωt,2π)
6
2 3 4 5 Switching Phase Mod(Ωt,2π)
6
12
6
0
-6
-12 0
1
4 Complex Motions in a Fermi Oscillator Fig. 4.17 Parameter map (.1/ D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1)
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30
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P41322 P(41322) 2
P41332 Complex Motion
P2136
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P(32421)2
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P2135
P2132
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5 0 0.0
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.2/
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x0 D 1:45709118, xP 0 D 1:82722043, x0 D 1:45709118, and xP 0 D 10:7134817 for t0 D 0:289711605. The time histories of displacements and velocities are presented in Fig. 4.15a, b, respectively. The thin solid curves depict the motions of the bottom and top oscillators, and the thick solid curve represents the motion of particle. The switching sections for the particle, bottom and top oscillators in phase plane are also shown in Figs. 4.15c and 4.16a, respectively. The switching sections for particle’s displacement and velocity versus switching phase are presented in Fig. 4.16b, c, respectively. The invariant set of such a chaotic motion is presented. A parameter map of excitation amplitude Q versus the coefficient of restitution e is presented in Fig. 4.17, where Q runs from zero to thirty and e runs from zero to one. Other parameters are given as .1/ D .2/ D 10:0, m.1/ D m.2/ D 1:0, m.3/ D 0:01, h D 0:5, k .1/ D k .2/ D 80:0, c .1/ D c .2/ D 0:1. The gray areas give the complex periodic motions with impact chattering and stick. The white region means where the motion does not interact with any of the boundaries. The chaotic motion region is represented by the gray colored regions. All the other areas represent periodic motions with different mapping structures. From the previous illustrations, the analytical conditions for stick and grazing motions to the boundaries for the Fermi oscillator with two periodic excitations were illustrated. The mechanism of stick and nonstick motion of the particle with the two driving oscillators was demonstrated. Bifurcation scenarios showed the complex motions existing in such Fermi oscillator. Finally, a detailed parameter map of different types of motions is illustrated. Other recent results relative to the Fermi oscillator can be referred to Luo and Guo (2009, 2010b).
References Luo ACJ (2008) A theory for flow switchability in discontinuous dynamical systems. Nonlin Anal Hybrid Syst 2(4):1030–1061 Luo ACJ (2009) Discontinuous dynamical systems on time-varying domains. HEP-Springer, Dordrecht
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Luo ACJ, Guo Y (2009) Motion switching and chaos of a particle in a generalized Fermiacceleration oscillator. Mathematical Problems in Engineering, Article ID 298906, pp 1–40 Luo ACJ, Guo Y (2010a) Switching mechanism and complex motions in an ex-tended Fermiacceleration oscillator. J Comput Nonlin Dyn 5(4):1–14 Luo ACJ, Guo Y (2010b) Switchability and bifurcation of motions in a double-excited Fermiacceleration oscillator. Proceedings of the 2010 ASME international me-chanical engineering congress and exposition, IMECE2010-39165
Chapter 5
Nonlinear Visco-Elastic Materials Stress Relaxation and Strain Rate Dependency Franz Konstantin Fuss
Introduction Linear visco-elastic models in the simplest form require three parameters, two springs and one damper, as two-parameter models, i.e. Maxwell and Kelvin–Voight models, either relax stress or creep, but are not capable of doing both. Threeelement linear models can be expanded by adding further linear elements (Wiechert model). Nonlinearity can be attributed to linear models, e.g., by power Hertzian springs and exponential functions applied to springs and dampers. Still, the basic structure of such models is linear, as long as they consist of springs and dampers. In visco-elastic materials such as polymers, the Young’s modulus is commonly used for the materials’ stiffness characterisation, often without referring to the strain rate applied. Dependency of the Young’s modulus on the strain rate is an inherent property of visco-elasticity. By definition, the Young’s modulus is determined at small strain as is the Poisson’s ratio. There is a vast amount of literature available on nonlinear visco-elastic models – e.g., Findley et al. (1948); Lakes (2009); Tschoegl (1989); Wineman (2009). Nonlinear visco-elastic models with a defined number of parameters are for example: Fung (1972, 1981) quasi-linear visco-elastic model (QLV ) with 5 parameters; 8-parameter QLV by Toms et al. (2002); Leaderman (1943) 3-parameter logarithmic model; Phillips (1905) 2-parameter logarithmic model; 3-parameter power models by Findley and Khosla (1956); Nutting (1921); 2-parameter power models by Oza et al. (2003); Hingorani et al. (2004); Duenwald et al. (2009); and (g) 5-parameter hyperbolic sine model by Findley et al. (1948).
(a) (b) (c) (d) (e) (f)
F.K. Fuss () School of Aerospace, Mechanical, and Manufacturing Engineering, RMIT University, Melbourne, Victoria, Australia e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 5, © Springer Science+Business Media, LLC 2012
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The simplest models are represented by two parameters, the “elasticity parameter,” R, and the “viscosity parameter,” , in the form of power or logarithmic functions, determined by stress relaxation experiments. This chapter addresses three specific problems and provides solutions thereof: (a) Which type of model philosophy is preferable for visco-elastic material characterisation: generalised linear or nonlinear models? (b) Is the stress relaxation a suitable method for material characterisation? (c) Is the Young’s modulus appropriate for characterising visco-elastic materials? Furthermore, in order to solve these problems, this chapter provides the mathematical basis for linear and nonlinear visco-elastic materials in terms of constitutive equations, stress relaxation and strain rate dependency.
5.1 Constitutive Equations and Stress Relaxation Function of Visco-Elastic Models This section introduces the constitutive equations of linear models, Zener model (standard linear solid SLS of Voight form), SLS of Maxwell form, and generalised Maxwell SLS or Wiechert model (Wiechert 1889, 1893), derived from first principles, and the constitutive equations of nonlinear models, power and logarithmic law models, from stress relaxation.
5.1.1 Zener Model (SLS of Voight Form) The Zener model consists of a Kelvin–Voight model (spring and dashpot in parallel) and a series spring (Fig. 5.1a) sharing the same stress. From Fig. 5.1a " D " 1 C "2
(5.1)
D "1 R1
(5.2)
D "2 R2 C "P2 ;
(5.3)
where R denotes the spring constants and the coefficient of viscosity. Taking the Laplace transform of (5.1)–(5.3), eliminating "1 and "2 by substitution, and solving for O yields the constitutive equation of the Zener model O D "O
R1 R2 C sR1 ; R1 C R2 C s
where the caret (^ ) denotes the transformed parameter.
(5.4)
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Fig. 5.1 Linear visco-elastic models; (a) Zener model (standard linear solid of Voight form), (b) standard linear solid of Maxwell form, (c) Wiechert model (generalised Maxwell model, note that the first damper 1 is rigid)
The equation for stress relaxation results from applying a constant strain "0 to the model through a Heaviside function H.t/ " D "0 H.t/
(5.5)
the Laplace transform of which is "O D
"0 : s
(5.6)
By substituting (5.6) into (5.4), we obtain O D "0
R1 R2 C sR1 s .R1 C R2 C s/
(5.7)
the inverse Laplace transform of which yields the function of stress relaxation t
R1 CR2
R2 C R1 e D R1 "0 R1 C R2
;
(5.8)
where the stress is normalised to the constant strain "0 . Equation (5.8) represents the exponential decay of stress with time, typical for linear models.
5.1.2 SLS of Maxwell Form The Maxwell SLS consists of a Maxwell model (spring and dashpot in series) and a parallel spring (Fig. 5.1b) sharing the same strain.
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From Fig. 5.1b D 1 C 2 1 D R1 " "P D
2 P 2 C : R2
(5.9) (5.10) (5.11)
Taking the Laplace transform of (5.9)–(5.11), eliminating 1 and 2 by substitution, and solving for O yields the constitutive equation of the Maxwell SLS O D "O
sR1 C sR2 C R1 R2 : s C R2
(5.12)
By substituting (5.6) into (5.12), we obtain O D "0
sR1 C sR2 C R1 R2 s .s C R2 /
(5.13)
the inverse Laplace transform of which yields the function of stress relaxation R t 2 D R1 C R2 e : "0
(5.14)
Like in the Zener model, the stress decays exponentially with time.
5.1.3 Wiechert Model The Wiechert model is an expansion of the Maxwell SLS, consisting of a finite number of Maxwell models in parallel with an additional parallel spring (Fig. 5.1c), all of them sharing the same strain. From Fig. 5.1c n X D i ; (5.15) i D1
where n denotes the number of parallel units 1 D R1 " "P D
2 P 3 3 P n n P 2 C D C D D C : R2 2 R3 3 Rn n
(5.16) (5.17)
Considering the parallel spring as a Maxwell model with infinite viscosity (1 D 1, rigid dashpot; Fig. 5.1d), (5.16) and (5.17) merge to "P D
1 P 2 2 P n n P 1 C D C D D C : R1 1 R2 2 Rn n
(5.18)
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The constitutive equation of the Wiechert model is obtained after Laplace transform of (5.18) O D "O
n X si Ri : si C Ri i D1
(5.19)
The stress relaxation function results from applying a Heaviside strain function and taking inverse Laplace transform n
R X t i D Ri e i ; "0 i D1
(5.20)
which is an exponential Prony series expansion, commonly used in finite element software for fitting a Wiechert model to the stress relaxation function of nonlinear visco-elastic materials. Alternatively, we can modify (5.20) to D "0
"0
C 1
n X
Ri e
t
Ri i
;
(5.21)
i D2
where .="0 /1 is the stress, normalised to the constant strain, at infinite times, i.e. R1 .
5.1.4 Nonlinear Power Law Model The power law model is characterised by a power decay of stress with time t: D R t : "0
(5.22)
Stress is normalised to the constant strain "0 applied by the Heaviside function H.t/ of (5.5). Taking Laplace transform of (5.22) yields O D "0 R
. C 1/ ; s C1
(5.23)
where denotes the Gamma function. By substituting (5.6) into (5.23), we obtain the constitutive equation of the power law of nonlinear visco-elasticity: O D s "O R .1 / :
(5.24)
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(5.24) reveals the intrinsic properties of the power law model (Fuss 2008): (a) 0 < 1, as the Gamma function in (5.24) approaches infinity when approaches 1 (b) the stress is the th (fractional) derivative of the strain ", times the constant R.1 /.
5.1.5 Nonlinear Log Law Model The logarithmic law model is characterised by a logarithmic decay of stress with time t: D R ln t; (5.25) "0 where “ln” denotes the natural logarithm. Stress is normalised to the constant strain "0 applied by the Heaviside function H.t/ of (5.5). Taking Laplace transform of (5.25) yields R ln s ; (5.26) O D "0 "0 s s s where denotes the Euler–Mascheroni constant (0:577215665 : : :). By substituting (5.6) into (5.26), we obtain the constitutive equation of the logarithmic law of nonlinear visco-elasticity: O D "O R C "O . C ln s/ :
(5.27)
5.2 Stress Relaxation: Rule of Thumb for Model Selection For characterising a nonlinear material or structure, an appropriate model has to be selected and the parameters (elasticity constant R, viscosity constant ) have to be quantified. The appropriate model is theoretically selected by exclusion from the function of the stress relaxation. In a graph with linear axes, the three functions, exponential, power and logarithmic, are hardly distinguishable (Fig. 5.2). In a single-log graph, the log function of (5.25) appears linear (Fig. 5.3), the power function is curved upwards, and the exponential function becomes s-shaped. In a double-log graph, the power function of (5.22) appears linear (Fig. 5.4), the log function is curved downwards, and the exponential function is still s-shaped. If single-log or doublelog graphs do not linearise the stress relaxation data, but rather show an s-shaped curve asymptoting to maximal and minimal values, then the function is exponential and a linear model would be selected, provided that perfectly linear materials exist in reality.
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Fig. 5.2 Stress relaxations of power and logarithmic models, as well as of Maxwell standard linear solid (MSLS) on a graph with linear axes
Fig. 5.3 Stress relaxations of power and logarithmic models, as well as of Maxwell standard linear solid (MSLS) on a single-log graph
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Fig. 5.4 Stress relaxations of power and logarithmic models, as well as of Maxwell standard linear solid (MSLS) on a double-log graph
Writing (5.22) of the power model in logarithmic form ln
D ln R ln t "0
(5.28)
shows that a linear fit to ln.="0 / as a function of ln t is defined by the gradient and the intercept ln R. Equation (5.28) of the power model is similar to (5.25) of the log model. In the log model, a linear fit to ="0 as a function of ln t is defined by the gradient and the intercept R. This analysis allows for the quantification of and R.
5.3 Problems to be Solved 5.3.1 Generalised Linear or Nonlinear Models The linear alternative to nonlinear models is the Wiechert model (as an exponential Prony series expansion). When ignoring nonlinearity, there is no need of distinguishing between power and log law models. Furthermore, exponential Prony series
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expansion is the choice for Finite Element Modeling (FEM ), implemented in the software solutions. It will be explored subsequently, whether the number of linear elements is suitable for material characterization, and whether the Wiechert model is applicable to stress relaxation tests.
5.3.2 Stress Relaxation Tests The rule of thumb outlined above is a simplified process which does not apply in practice that easily. The rule of thumb refers to these stress relaxation data only, which are obtained under experimental conditions, which, in turn, are identical to the model conditions. This is actually not the case, as “a step change in strain . . . is not possible to be performed experimentally” Kohandel et al. (2008). Stress relaxation is modelled by applying a Heaviside (unit step) function to the strain applied to the material. A unit step strain cannot be reproduced by material testing machines, as their strain rate and acceleration of the cross head is limited. Instead, in material testing, a ramp function is initially applied, bringing the test specimen to the desired strain, which is subsequently kept constant. This fact can make logarithmic law materials appear like power law materials, as will be shown below. Changing the experimental procedure means abandoning stress relaxation experiments in favour of something else. Creep experiments are in essence the same as stress relaxation experiments, the difference being that an instantaneous stress has to be applied (which is only possible in theory) and that the strain has to be continuously changed subsequently by the testing machine, in order to keep the stress constant. Cyclic experiments have an initial transient phase followed by a steady state and moreover introduce a further variable: the cycle frequency. It will be explored subsequently, whether simple tensile and compression experiments might provide the solution. These tests are required for determining the Young’s modulus.
5.3.3 Young’s Modulus of Nonlinear Visco-Elastic Materials The Young’s modulus is the ratio of stress to strain at small strains. It corresponds to the stiffness of a Hookean spring, the ratio of force to deflection. The Young’s modulus, however, is strain rate dependent in visco-elastic materials. Thus, the specific strain rate, at which the Young’s modulus was determined, has to be quoted. It will be explored subsequently, whether elasticity and viscosity parameters, R and , suffice for material characterisation, and whether the elasticity parameter could replace the Young’s modulus.
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5.4 Actual Test Conditions vs. Heaviside Strain As mentioned above, by definition, stress relaxation hinges on a unit step Heaviside strain function. In practice, material testing machines cannot apply a Heaviside strain function with a unit impulse (Dirac delta) strain rate function. The required crosshead speed and its derivatives would be too high. Hence, the loading part is replaced by a ramp strain with the strain rate "P0 , which in turn is followed by constant strain "0 , which is the maximal strain applied to the test specimen (Fig. 5.5): f1 .t/ D "P 0 t f2 .t/ D "0
0t t
"0 "P0
"0 "P0
"0 D 4t "P0
(5.29) (5.30)
the Laplace transform of which is L .f1 ; f2 / D "P0
1 e s4t ; s 2 4t
(5.31)
where 4t is the time delay (time period of the ramp segment). Replacing "O of the constitutive (5.4), (5.12), (5.19), (5.24), and (5.27) by (5.31) provides the transformed function of the different models after applying the function shown in Fig. 5.5. The inverse Laplace transform of these transformed equations is calculated numerically (e.g. with Scientist 3:0 by Micromath, Saint Louis, Missouri) and yields the ratio of to "0 , comparable to (5.8), (5.14), (5.20), (5.22), and (5.25). Figures 5.6–5.9 show the difference between the theoretical stress relaxation (based on a Heaviside function) and the actual stress relaxation when applying the function of (5.31). The initial ramp strain causes the stress to overshoot the theoretical stress relaxation curve. In materials with log law properties, this fact
Fig. 5.5 Ramp strain followed by constant strain. 4t D time period of the ramp segment, "0 D maximal strain applied
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Fig. 5.6 Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a single-log graph; bold curves: relaxation after applying a Heaviside strain function, dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31)
(upward curved initial segment) can pretend power law properties, especially when the exposure time of the stress relaxation is not sufficiently long (Fig. 5.7). The functions resulting from (5.31) exist only in the transformed state without an analytical solution to the inverse problem. Still, the rule of thumb can be applied by keeping the exposure to the initial ramp function short (high crosshead speeds), and by removing data of small times. This method requires recording the stress relaxation over large times, and delivers the (constant) viscosity parameter only for that specific strain, which was selected for stress relaxation.
5.5 Prony Series Expansion of a Log or Power Law Stress Relaxation As mentioned above, nonlinear visco-elastic stress relaxations are converted to a QLV model, by fitting the Wiechert model’s stress relaxation to the experimental data. For this purpose, (5.20) is re-written for better understanding: R R R t 1 t 2 t n D R1 e 1 C R2 e 2 C C Rn e n : "0
(5.32)
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Fig. 5.7 Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a single-log graph; dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31)
Note that 1 D 1 (rigid dashpot; Fig. 5.1c), which renders the argument of the first exponential function in (5.32) zero. Subsequently, a four-spring (n D 4) Wiechert model of the form R R R R t 1 t 2 t 3 t 4 D R1 e 1 C R2 e 2 C R3 e 3 C R4 e 4 "0
(5.33)
is fit to the stress relaxation of a power law material with R D 10, and D 0:3. Both the theoretical and the experimental stress relaxation are considered in order to understand the effect of experimental conditions. The theoretical stress relaxation follows (5.22), whereas the experimental one is modelled by replacing "O of the constitutive (5.24) with (5.31). Equation (5.33) has to be solved for four (n) unknowns (R1 ; : : : ; R4 ) by preselecting four (n) time points and three (n 1) ratios of R to : t1 D 0:1 R2 D 0:1 2
t2 D 1
t3 D 10
R3 D 2:2 3
t4 D 100 R4 D 10: 4
(5.34) (5.35)
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Fig. 5.8 Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a double-log graph; bold curves: relaxation after applying a Heaviside strain function, dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31)
The preselection process requires four time points which should be equally spaced, considering the logarithmic time scale, as well as three R= ratios with different viscosities. For the theoretical power law stress relaxation, substituting these data into (5.33) yields 10 0:10:3 D R1 C R2 e 0:10:1 C R3 e 2:20:1 C R4 e 100:1 10 1
0:3
D R1 C R2 e
0:11
C R3 e
2:21
C R4 e
101
10 100:3 D R1 C R2 e 0:110 C R3 e 2:210 C R4 e 1010 10 100
0:3
D R1 C R2 e
0:1100
C R3 e
2:2100
C R4 e
10100
(5.36) (5.37) (5.38) (5.39)
and solving for R1 ; : : : ; R4 provides R1 D 2:512
R2 D 6:797
R3 D 12:078
R4 D 2:772
(5.40)
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Fig. 5.9 Stress relaxations of power and logarithmic models, as well as of Maxwell (MSLS) on a double-log graph; dashed curves D family of stress relaxations after applying a ramp and constant strain according to Fig. 5.5 and (5.31)
2 ; : : : ; 4 is obtained from the R= ratios: 2 D 67:965
3 D 5:491
4 D 0:277
1 D 1
(5.41)
For the experimental power law stress relaxation according to the strain function of Fig. 5.5, substituting the time and R= data into (5.33) yields 28:46 D R1 C R2 e 0:10:1 C R3 e 2:20:1 C R4 e 100:1 10:157 D R1 C R2 e
0:11
C R3 e
2:21
C R4 e
101
10 100:3 D R1 C R2 e 0:110 C R3 e 2:210 C R4 e 1010 10 100
0:3
D R1 C R2 e
0:1100
C R3 e
2:2100
C R4 e
10100
(5.42) (5.43) (5.44) (5.45)
solving for R1 ; : : : ; R4 delivers R1 D 2:512
R2 D 6:797
R3 D 13:489
R4 D 22:818
(5.46)
1 D 1
(5.47)
2 ; : : : ; 4 is obtained from the R= ratios: 2 D 67:965
3 D 6:131
4 D 2:282
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Fig. 5.10 Stress relaxations of power model and Wiechert model (Prony series 1, (5.36)–(5.39)) after Heaviside strain function
Inserting R1 ; : : : ; R4 and 2 ; : : : ; 4 in (5.33) yields the stress relaxation function of the Prony series expansion of the power law stress relaxation (R D 10; D 0:3). All functions, the theoretical and experimental stress relaxations of the power model and the corresponding Wiechert models (Prony series) are shown in Figs. 5.10 and 5.11. The four-spring Wiechert model is an approximation of the power model’s stress relaxation within the time points specified. The larger the number of springs, the more accurate is the approximation. Nevertheless, the stress, normalised to the constant strain applied, ranges only between the sum of all spring constants and the one of R1 . Thus, inaccuracy of data at small times affects the modulus of the material at high strain rates, where the modulus equals the sum of all spring constants, as will be shown below. Furthermore, highly accurate Prony series expansion requires a high number n of parallel springs, which returns n spring constants and n 1 coefficients of viscosity, i.e. 2n 1 parameters, which is hardly suitable for comparing the viscous and elastic properties of materials. Prony series expansion is certainly suitable for FEM software, where the actual characterisation and comparison of materials is not required. Increasing the Wiechert model’s elements to large numbers is actually a finite element approach in itself. Still, care has to be taken as to the stress overshoot, shown in Figs. 5.6–5.9.
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Fig. 5.11 Stress relaxations of power model and Wiechert model (Prony series 2, (5.42)–(5.45)) after applying a ramp and constant strain according to Fig. 5.5 and (5.31); D stress increasing with ramp strain
5.6 Strain Rate Dependency Strain rate dependency is an inherent characteristic of visco-elastic materials: the faster the strain rate "P 0 , the stiffer the material. Constant strain rate tests are among standard testing protocols of material testing machines (ramp up, ramp down) and the usefulness of constant strain rate tests for characterisation of nonlinear visco-elastic materials is explored subsequently. The strain rate dependency will be analysed by the following steps: (a) a constant strain rate "P 0 is applied to the constitutive equations by replacing "O with a ramp function "P0 t, i.e. "P0 (5.48) s2 (b) After taking the inverse Laplace transform of the modified constitutive equation, D f .t/; (c) The time t is replaced by t D "="P0 , which results in D f ."/; (d) Taking the strain derivative of stress yields the modulus E: E D f ."/; "O .s/ D
This procedure is justified, if, and only if, R and are constants and do not change with strain.
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5.6.1 Zener Model Substituting (5.48) into (5.4) yields O D "P 0
R1 R2 C sR1 s 2 .R1 C R2 C s/
(5.49)
the inverse Laplace transform of which yields R CR2 1 t 1 R12 1 e B R1 R2 t C C: t D "P0 B 2 @ R1 C R2 C A .R1 C R2 / 0
(5.50)
Replacing t by the ratio "="P0 : " D
" R1 CR2 "R1 R2 "P0 R12 "P 0 R12 "P 0 C e : R1 C R2 .R1 C R2 /2 .R1 C R2 /2
(5.51)
After differentiating with strain and simplifying
" R1 CR2
R1 R2 C R12 e "P0 P " D E" D R1 C R2
:
(5.52)
At zero or infinite strain rate
" R1 CR2
R1 R2 C R12 e "P0 lim "P0 !0 R1 C R2 !0
" R1 CR2
R1 R2 C R12 e "P0 lim "P0 !1 R1 C R2
D
R1 R2 R1 C R2
D R1
(5.53)
(5.54)
!1
the dashpot does not transmit any load or is rigid, respectively, and the Zener model reduces to two series springs or the series spring R1 , respectively. The Zener model exemplifies a typical feature, common to all linear models, namely that the modulus E has a minimum and maximum value, unequal 0 and 1, respectively.
5.6.2 Maxwell SLS Substituting (5.48) into (5.12) yields O D "P0
sR1 C sR2 C R1 R2 s 2 .R2 C s/
(5.55)
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the inverse Laplace transform of which yields R t 2 : t D "P 0 R1 t C 1 e
(5.56)
Replacing t by the ratio "="P0 : " D "R1 C "P 0 "P 0 e
" R2 "P0
:
(5.57)
After differentiating with strain and simplifying P " D E" D R1 C R2 e
" R2 "P0
:
(5.58)
At zero or infinite strain rate " R2 lim R1 C R2 e "P0 D R1
(5.59)
"P0 !0 !0
" R2 " P lim R1 C R2 e 0 D R1 C R2 ;
"P0 !1 !1
(5.60)
the dashpot does not transmit any load or is rigid, respectively, and the Maxwell SLS reduces to the parallel spring R1 or the two parallel springs, respectively. Similar to the Zener model, the Maxwell SLS shows the same behaviour of 0 < Emin < Emax < 1.
5.6.3 Wiechert Model Expanding (5.58) to n parallel springs E" D R1 C
n X
Rj e
" Rj "P 0 j
:
(5.61)
j D2
At zero or infinite strain rate 0 lim @R1 C
"P0 !0 !0
0
lim @R1 C
"P0 !1 !1
n X
Rj
" Rj e "P0 j
j D2 n X j D2
Rj e
" Rj "P0 j
1 A D R1 1 AD
n X j D1
(5.62)
Rj :
(5.63)
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5.6.4 Nonlinear Power Law Model Substituting (5.48) into (5.24) yields O D s 2 "P0 R .1 / :
(5.64)
After rearranging and applying the recursion formula of the Gamma function O D "P0 R
.1 / 1 .1 / "P 0 R .2 / D "P 0 R D s 2 1 s 2 1 s 2
(5.65)
the inverse Laplace transform of which yields R 1 t : 1
(5.66)
R "1 R D "P0 "1 : 1 1 "P 1
(5.67)
t D "P0 Replacing t by the ratio "="P0 : " D "P0
After differentiating with strain and simplifying
P " D E" D R"P0 " :
(5.68)
Equation (5.68) has the same structure as the stress relaxation of (5.22): the independent variable to the power of multiplied by a constant. Thus, the relationship between modulus and strain rate follows a power law too. At zero or infinite strain rates, the modulus is zero or infinite, respectively: 0 E 1. Equation (5.68) can be re-written as ln E" D ln R ln " C ln "P0 ;
(5.69)
which has the same structure as (5.73) as shown below.
5.6.5 Nonlinear Log Law Model Substituting (5.48) into (5.27) yields O D "P0
R ln s " P 0 s2 s s s
(5.70)
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F.K. Fuss
the inverse Laplace transform of which yields t D "P0 Rt "P0 .t C t ln t/ :
(5.71)
Replacing t by the ratio "="P0 and simplifying: " D "R C " " ln " C " ln "P0 :
(5.72)
After differentiating with strain and simplifying P " D E" D R ln " C ln "P0 :
(5.73)
Equation (5.73) has the same structure as the stress relaxation of (5.25): a log function of the independent variable multiplied by . Thus, the relationship between modulus and strain rate follows a log law too. At zero or infinite strain rates, the modulus is infinite, 1 or C1, respectively. Consequently, at very small strain rates, the modulus becomes theoretically negative according to (5.73). The modulus E is zero if " "P D R : (5.74) e If the ratio R to is 2 or 10, and the strain is 0:01, then E D 0 at a strain rate of 1:35 m"=s or 1:63 m"= h, respectively. It has to be considered that the log model is a model only and thus certain conditions are not practicable in reality.
5.6.6 Derivation of R and from Nonlinear Models Equations (5.69) and (5.73) provide the viscosity parameter , as well as the (strain rate independent) elasticity parameter R over a large range of strains (the resolution of which depends only on the data sampling frequency), and delivers a faster and more accurate method of extracting the viscosity and elasticity parameters from simple tension or compression experiments. Rewriting (5.69) and (5.73) as "P0 C ln RP " "P 0 E" D L ln C RL "
ln E" D P ln
(5.75) (5.76)
shows the principle of deriving R and , where the subscripts P and L refer to power and log models, respectively. When plotting ln E (power law) or E (log law) against the independent variable, ln .P"0 ="/, the intercept and gradient of the linear regression are R and , respectively.
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Apparently, this procedure poses a problem: the independent variable equals the natural logarithm of the ratio of "P0 to ". Yet, at which strain " should the ratio be taken? This does not seem to be an obstacle first, as the Young’s modulus E (usually obtained at small strains) is a constant regardless of the amount of strain. Yet, if E is constant, "P 0 is constant and " is variable, then R and become a function of ", which contravenes, as mentioned above, the model conditions, which are constant R and . This apparent dilemma explains the nature of nonlinear visco-elastic materials: the Young’s modulus E cannot be constant in visco-elastic materials. It is commonly accepted that E changes with the strain rate "P 0 . Yet, E is often uncritically expressed as a constant without giving proper reference to the strain rate "P0 at which E was obtained. Equally, E changes with strain ", especially at small strains, where the change is maximal (as will be shown below in Fig. 5.14). E changing with " and "P 0 keeps R and constant. Consequently, in nonlinear visco-elastic materials the (inconstant) Young’s modulus E should be replaced by R and (RP and P , or RL and L ).
5.7 Method of Deriving Visco-Elastic Parameters from Compression or Tensile Experiments 5.7.1 Principle of Deriving R and from Nonlinear Models Figure 5.12 shows the stress–strain curves of a power model (R D 10; D 0:3). Instead of applying (5.67), the stress was calculated with a numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri), and the modulus E (Fig. 5.13) was determined by numerical differentiation. As detailed above, the Young’s modulus E changes with strain ". R and (Figs. 5.14–5.16) were recalculated from (5.75). In spite of two consecutive numerical processes, the calculated R and are in good agreement with the original parameters. The decision which model, log or power, to apply is determined conveniently by applying both, and calculating the correlation coefficient r of the linear fit function of (5.75), rP , and (5.76), rL . If rP > rL , i.e. rP rL > 1 (Fig. 5.17), then a power model fits better than a logarithmic one.
5.7.2 Application to Visco-Elastic Materials This case report shows the practical application of analytic nonlinear viscoelasticity to polymeric composite materials of different hardness. These types of materials, produced by Objet Geometries (Billerica, Massachusetts), are used for rapid prototyping on Connex 3D printing systems (Objet Geometries, Billerica, Massachusetts). There are two base materials (Tango Plus TP 930, Vero White
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F.K. Fuss
Fig. 5.12 Power model (R D 10; D 0:3) at different strain rates (1:667s1 ; 2:5s1 ; 5s1 ): stress against strain (loading and unloading segment; three stress values at 0:4 strain are marked)
Fig. 5.13 Power model (R D 10; D 0:3) at different strain rates (1:667s 1 ; 2:5s 1 ; 5s 1 ); modulus against strain (loading only)
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Fig. 5.14 Power model (R D 10; D 0:3) at different strain rates (1:667s1 ; 2:5s1 ; 5s1 ); three log modulus values at 0.4 strain against ln "P0 =", linear fit with intercept at ln 10 and gradient of 0:3
Fig. 5.15 Calculation of parameters from Fig. 5.13; elasticity constant R against stress. Solid curve: running average filter; sudden increase of noise between strain 0:35 and 0:4 is due to the numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri)
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F.K. Fuss
Fig. 5.16 Calculation of parameters from Fig. 5.13; viscosity constant against strain. Solid curve: running average filter; sudden increase of noise between strain 0:35 and 0:4 is due to the numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri)
V W 830; Table 5.1) with hard and soft modulus. Different mixtures of the two base materials result in composites of different hardness. The materials are combined by layers of alternate hardness, where the layer thickness ratio of hard to soft material determines the specific hardness. Within one layer, the two materials are arranged in a checkerboard pattern. Consequently, the two materials are arranged in series in all three directions. If two logarithmic law materials are arranged in series, then the relationship of R and is nonlinear. If differences between the two extreme R and are large, then the nonlinear relationship approaches a power curve. Large R-differences are intended for rapid prototyping materials in order to cover a wide range of hardness. In logarithmic law materials, R and are linked together as the degree of viscosity is not determined by alone (in contrast to power law materials). When increasing R, must increase as well in order to keep the level of viscosity, e.g. in terms of relative energy loss. In logarithmic law materials, the degree of viscosity is rather related to the ratio of R to . Thus, logarithmic law materials with strikingly different R are expected to have -differences in a comparable magnitude. A power law relationship between R and is hence expected. The first question is which model to select. Applying the conventional way of stress relaxation experiments, the data curve upwards in a single logarithmic coordinate system (Fig. 5.18), which indicates the power law (cf. Fig. 5.3). Yet, as already known from Sect. 5.4, an initial overshoot is expected, when applying
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Fig. 5.17 Calculation of parameters from Fig. 5.13; Difference of correlation coefficients of log fit (rL ) and power fit (rP ), positive difference indicates better fit of power function; solid curve: running average filter; sudden increase of noise between strain 0:35 and 0:4 is due to the numerical inverse Laplace solver (Scientist 3:0 by Micromath, Saint Louis, Missouri) Table 5.1 Rapid prototyping materials; data marked with an asterisk are taken from Objet Geometries (2011); Shore A was converted to Shore D according to the conversion table from Thermal Tech Equipment (2011), by fitting an exponential equation to the data (Shore D D 2:386e 0:0314ShoreA ) Material Hardness Hardness L RL =L code Shore A Shore D RL V W 830 DM 9795 DM 9785 DM 9770 DM 9760 DM 9750 DM 9740 TP 930
.100/ 95 85 70 60 50 40 27
83 47:2 34:5 21:5 15:7 11:5 8:4 5:6
4:83 108 3:55 107 1:17 107 4:72 106 2:76 106 1:65 106 1:11 106 7:83 105
2:14 108 1:28 107 3:77 106 1:30 106 6:98 105 3:81 105 2:17 105 1:30 105
2:254 2:773 3:106 3:624 3:9531 4:324 5:098 6:011
a ramp strain followed by a constant strain, and the data curve asymptotically approaches the theoretical fit function obtained when applying a Heaviside strain function. Due to this fact, the decision between log and power model is conveniently based on fitting a log and power function (Fig. 5.18) to the data segment at large times (e.g. from 300 s to 600 s in Fig. 5.18). As seen in Fig. 5.18, at smaller times,
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F.K. Fuss
Fig. 5.18 Determination of R and ; TP 930 (rapid prototyping material), stress relaxation after initial ramp strain at a strain rate of 0:26s1
the data points are located below the power fit but above the log fit. This clearly indicates a logarithmic law material as the fit curve to any data segment can never intersect any other part of the stress relaxation curve (if obtained from consecutive ramp and constant strain) due to asymptotic behaviour. The log fit in Fig. 5.18 is still not suitable as the slope and intercept are expected to change slightly at very large times. Alternatively, the modulus E, obtained from the numerical strain derivative of the stress at smaller strain " and at least three different strain rates, is plotted against "P 0 =" (Fig. 5.19). A log and power fit provides a clue for model selection by mere inspection, and the coefficient of correlation .r/ or determination .r 2 / supports decision making, even if the data are noisy (Fig. 5.19). R and are directly obtained from the appropriate fit function, and this procedure is repeated for each material (Table 5.1). Alternatively to fitting the data as shown in Fig. 5.19, the data can be processed by calculating the intercept and gradient from a sliding window filter and subsequent smoothing of R and by a running average filter (Figs. 5.20 and 5.21). The sliding window filter calculates R and only for a specific strain range, defined by the width of the window. The test specimens (100 400 400 mm) of the materials identified in Table 5.1 were subjected to compression tests at three different strain rates (0:266s1 ; 0:0266s1; 0:00266s1). The modulus E was determined numerically, and R and obtained from (5.76), according to the procedure shown in Fig. 5.19.
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Fig. 5.19 Determination of R and ; DM 9750 (rapid prototyping material) modulus data obtained at three different strain rates (noisy modulus data were intentionally not filtered)
Fig. 5.20 R data of DM 9750 (rapid prototyping material); horizontal line: R constant from Fig. 5.19; dots: intercept from sliding window filter with window width 9; bold grey curve: R after applying running average filter of window width 33
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F.K. Fuss
Fig. 5.21 data of DM 9750 (rapid prototyping material); horizontal line: constants from Fig. 5.19; dots: gradient from sliding window filter with window width 9; bold grey curve: after applying running average filter of window width 33
The function of R and against hardness is almost exponential (Figs. 5.22 and 5.23), whereas R= against hardness and against R are power functions (Figs. 5.24 and 5.25), as anticipated above.
5.7.3 Application to Visco-Elastic Structures Structures with varying cross-section or composition in the direction of the compressive or tensile force can be analysed with the same methods as visco-elastic materials. However, structures are characterised by stiffness k changing with deflection x. Strictly speaking, this does not match the model conditions, which are based on constant R and . Nevertheless, characterisation of the stiffness with R and (Fig. 5.26) for comparative reasons is useful, considering that further application of R and is limited: (a) R and can be used for recalculating the stiffness k at different deflection rates xP 0 (Fig. 5.27) xP 0 C ln RP x xP 0 C RL kx D L ln x
ln kx D P ln
(5.77) (5.78)
5 Nonlinear Visco-Elastic Materials
Fig. 5.22 Properties of rapid prototyping materials; R against hardness Shore D
Fig. 5.23 Properties of rapid prototyping materials; against hardness Shore D
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164
Fig. 5.24 Properties of rapid prototyping materials; against R
Fig. 5.25 Properties of rapid prototyping materials; R= against hardness Shore D
F.K. Fuss
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Fig. 5.26 Properties of a cricket ball with five cork layers; (a) R and against deflection
Fig. 5.27 Properties of a cricket ball with five cork layers; stiffness against deflection, the black solid lines show the experimental data, and the gray circles indicate the stiffness recalculated from R, , deflection and deflection rate with (5.77)
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Fig. 5.28 Properties of a cricket ball with five cork layers; force against deflection, the black solid lines show the experimental data, the black dashed lines (“1”) indicate the force calculated from (5.79), and the grey dashed lines (“2”) represent the numerical integration of the stiffness recalculated from (5.77)
(b) R and cannot be used for calculating the force F (Fig. 5.28) from
Fx D xP 0 x 1 L and
RP 1 P
xP 0 Fx D x RL C L 1 C ln x
(5.79)
(5.80)
as (5.79) and (5.80) deliver a different result due to conflict with the basic model conditions; instead, the force has to be determined from the stiffness k, (5.77) or (5.78) by numerical integration. Figures 5.26–5.28 show R and as well as the stiffness k and the force F of a cricket ball against deflection. The ball (Kookaburra Turf, 5 cork layers) was compressed at three different crosshead speeds perpendicular to the plane of the seam. The viscosity parameter is almost constant over most of the deflection range, whereas R increases with deflection due to the ball’s spherical shape. The average of rP rL was positive (C0:0055), which indicates a power model.
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5.8 Summary In this chapter, three questions were addressed: (a) Quasi-linear or nonlinear models, what are the advantages and disadvantages? (b) If nonlinear, which method of deriving the material properties (elastic and viscous constants) is appropriate – stress relaxation, or an alternative method? (c) Is the Young’s modulus, apart from the fact that it represents elasticity only and not viscosity, appropriate for characterising nonlinear visco-elastic materials?
5.8.1 Quasi-Linear or Nonlinear? The stress relaxation of the Wiechert model (whose structure is an exponential Prony series expansion commonly used in FEM software) has decisive disadvantages when fitted to the stress relaxation of a nonlinear visco-elastic model: a) missing relaxation data during the ramp strain (Fig. 5.5) of the experimental conditions and thus at small times; b) inaccurate fit at the beginning of the constant strain due to overshooting stress; and c) for highly accurate fit, a large number of parallel Maxwell models is required, which is certainly suitable for FEM but not for characterisation of nonlinear visco-elastic models. The fact that relaxation data are missing at small times and that relaxation data are inaccurate at intermediate times affects the modulus at large strain rates. This is due to the fact that R1 of the Wiechert model is (1) the asymptotic stress, normalised to the constant strain, of the stress relaxation at large times, as well as (2) the asymptotic modulus at strain rates approaching zero. As the stress relaxation values of Heaviside strain and ramp plus constant strain conditions are equal at large times, the modulus at small strain rates remains unaffected. However, the modulus at large strain rates is affected by the overshoot due to ramp plus constant strain conditions. This problem can be easily solved by using the method proposed in Sect. 5.7, calculating the stress relaxation from R and obtained from this method, and fitting a Wiechert model of a large number of individual Maxwell models to the stress relaxation data over a large time range. This solution is also suitable for FEM and should be implemented in the software. By no means must the time axis be corrected, in a sense that the time (4t in Fig. 5.5) at peak stress is set to zero. This procedure seriously underestimates the material stiffness.
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5.8.2 Stress Relaxation or Alternative Nonlinear Methods? The actual test conditions for stress relaxation are not based on a Heaviside strain, but rather on an initial ramp strain followed by a constant strain. This causes the stress to overshoot initially and subsequently to asymptotically approach the stress levels expected from Heaviside strain conditions. There are two solutions to this problem: (a) Record the stress relaxation for large times and fit a power or log function only to the very last segment of the time axis; or (b) Instead of keeping the standard test conditions of ramp plus constant strain, apply a general ramp strain, and determine the nonlinear visco-elastic material properties, R and , from compression or tensile tests at different strain rates. The second solution determines R and from the intercept and gradient, respectively, of modulus E data (log model) or ln E (power model) against the natural logarithm of the ratio of "P0 to ". This solution offers the following advantages: (a) More data over a range of strains is available for curve fitting (filtering of E data might be necessary at smaller strain rates); (b) R and data determined serve for calculating the stress relaxation and subsequently fitting a Wiechert model to the stress relaxation data for FEM purposes (as mentioned above); and (c) It is also less time consuming than stress relaxation tests, especially when considering that large times are required to improve the accuracy of these tests. The alternative method, deriving R and from compression or tensile tests at different deflection rates, is only of limited use for structures which exhibit a change of R and with deflection, as the model conditions are based on constant R and . Nevertheless, this procedure is useful for comparative characterisation purposes and allows recalculating the modulus E at different deflection rates. However, the force F cannot be recalculated from (5.79) and (5.80), the equivalent of (5.67) and (5.72), if R and are not constant. Instead, F has to be calculated from the stiffness k by integrating with deflection x.
5.8.3 Young’s Modulus or Alternative Parameters? In visco-elastic materials, the Young’s modulus E is not only strain rate dependent but also strain dependent. Thus, both the strain rate and the strain at which E was determined has to be referenced for material characterisation. E does not provide any information on viscosity. Hence, E is unsuitable for characterisation of viscoelastic materials. Thus, it is suggested in this chapter that the Young’s modulus E is replaced by the elastic and viscous constants R and . For characterising a material with R and , three issues have to be considered:
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(a) The alternative method suggested above is not applicable for deriving R and equivalent to a tangent modulus, as this term implies R and changing with strain; (b) R and are expected to be temperature dependent in many materials such as polymers; (c) The modulus E as a function of " can only be recalculated from (5.68) and (5.73) within the range of strain rates originally used for determining R and ; extrapolation outside this range is justified for maximally half the range on either side of the original strain rate window.
5.9 Key Symbols E e F n H.t/ k L ln R R1;2;:::;n RP RL rP rL s t 4t x xP " "0 "O "P "P0
Young’s modulus Exponential function Force Number of parallel Maxwell bodies in series Heaviside function of time Stiffness Laplace transform operator Natural logarithm Elasticity / stiffness parameter Strain rate independent modulus Spring constant of spring 1; 2; : : : ; n Strain rate independent modulus of power law model Strain rate independent modulus of logarithmic law model Coefficient of correlation of linear fit function (power law) According to (5.75) Coefficient of correlation of linear fit function (power law) According to (5.76) Complex variable of transformed functions Time Time delay of constant strain of stress relaxation Deflection Deflection rate Greek Gamma function Euler–Mascheroni constant (0:577215665 : : :) Strain Constant strain of stress relaxation Transformed strain Strain rate Constant strain rate of ramp strain Viscosity parameter / coefficient of viscosity
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Coefficient of viscosity of damper 1; 2; : : : ; n Viscosity parameter of power law model Viscosity parameter of logarithmic law model Stress Transformed stress Stress rate
References Duenwald SE, Vanderby R, Lakes RS (2009) Constitutive equations for ligament and other soft tissue: evaluation by experiment. Acta Mech 205:23–33 Findley WN, Adams CH, Worley WJ (1948) The effect of temperature on the creep of two laminated plastics as interpreted by the hyperbolic-sine law and activation energy theory. Proc ASTM 48:1217–1228 Findley WN, Khosla G (1956) An equation for tension creep of three unfilled thermoplastics. SPE J 12:20–25 Fung YC (1981) Biomechanics: mechanical properties of living tissues. Springer, New York Fung YC (1972) Stress-strain-history relations of soft tissues in simple elongation. In: Fung YC, Perrone N, Anliker M (eds) Biomechanics: its foundations and objectives. Prentice-Hall, Englewood Cliffs, NJ Fuss FK (2008) Cricket balls: construction, nonlinear visco-elastic properties, quality control and implications for the game. Sports Technol 1:41–55 Hingorani R, Provenzano P, Lakes RS, Escarcega A, Vanderby R (2004) Nonlinear viscoelasticity in rabbit medial collateral ligament. Ann Biomed Eng 32:306–312 Kohandel M, Sivaloganathan S, Tenti G (2008) Estimation of the quasi-linear viscoelastic parameters using a genetic algorithm. Math Comp Modelling 47:266–270 Lakes R (2009) Viscoelastic materials. Cambridge University Press, Cambridge Leaderman H (1943) Elastic and creep properties of filamentous materials. Textile Foundation, Washington DC Nutting PG (1921) A study of elastic viscous deformation. Proc ASTM 21:1162–1171 Objet Geometries (2011) Digital materials. http://www.objet.com/Pages /Digital Materials . Accessed 25 May 2011 Oza A, Lakes RS, Vanderby R (2003) Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal. Rheol Acta 42:557–568 Phillips P (1905) The slow stretch in India rubber, glass and metal wires when subjected to a constant pull. Phil Mag 9:513–531 Thermal Tech Equipment (2011) Shore Durometer conversion chart. http://www.ttequip.com/ KnowledgeLibrary/TechPageShoreDurometerCon-versionChart.htm. Accessed 25 May 2011 Toms SR, Dakin GJ, Lemons JE, Eberhardt AW (2002) Quasi-linear viscoelastic behavior of the human periodontal ligament. J Biomech 35:1411–1415 Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior: an introduction. Springer, Berlin Wiechert E (1893) Gesetze der elastischen Nachwirkung f¨ur constante Temperatur. Ann Phys 286:335–348, 546–570 Wiechert E (1889) Ueber elastische Nachwirkung, Dissertation, K¨onigsberg University, Germany Wineman A (2009) Nonlinear viscoelastic solids – a review. Math Mech Solids 14:300–366
Chapter 6
Nonlinear Dynamic Modeling of Nano and Macroscale Systems Using Finite Elements and an Intrinsic Beam Formulation Michael J. Leamy
Abstract This chapter presents a nonlinear finite element approach for modeling nano and macroscale beam-like materials and structures. The work is based on a series of three articles published by the author and his co-workers from 2007 to 2010 (Leamy 2007; Leamy and Lee 2009; Leamy 2010). The chapter begins by describing a finite element procedure for discretizing a set of nonlinear, intrinsic beam equations. These equations are notable for their use of curvature and strain, vice rotational and displacement, field variables. Included in the finite element development is a discussion of zero energy modes and their remediation. Attention then turns to application of the approach in studying dynamics of systems at the nanoscale (e.g., carbon nanotubes) through to the macroscale (e.g., helical springs).
6.1 Intrinsic Beam Formulation Borri and Mantegazza (1985), presented the first geometrically exact intrinsic formulation for the dynamics of an initially curved and twisted beam. Their formulation departed from conventional approaches by using curvature and strain metrics to describe the beam’s configuration away from an initially curved and twisted reference configuration. Hodges and his co-workers further developed the approach and retained the intrinsic metrics through to solution of the equations(Shang et al. 1999; Patil et al. 1999, 2000; Hodges et al. 2002, 1996; Cesnik and Shin 2001; Hodges 2003). Their equations were later used as the basis for a finite element solution procedure introduced by Leamy and co-workers (Leamy 2007; Leamy and Lee 2009), described later and used throughout this chapter.
M.J. Leamy () School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 6, © Springer Science+Business Media, LLC 2012
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M.J. Leamy undeformed
Fig. 6.1 Geometry of the intrinsic beam in the undeformed and deformed configurations
b3
deformed
b2
B3
b1 x1
B2 B1
s R
r z2
z3 z1
Figure 6.1 depicts the geometry of an initially curved and twisted beam in which position along the center-line of the beam is given by r and for which a set of orthogonal unit basis vectors bi are used to locate points away from the center-line. The center-line tangent unit vector is denoted by b1 while b2 , b3 denote unit vectors fixed in the beam’s cross-section. Distance along the center-line is denoted by x1 while off-center-line points have at least one nonzero x2 , x3 . Spatial changes of this triad are given by b0i D kbi where k denotes the curvature vector with components k1 measuring the initial twist and k2 , k3 measuring curvature components along the b2 , b3 directions. A new center-line R in the deformed configuration is measured by arc distance s. Unit vectors in the deformed configuration are given by Bi . With the presence of nonzero cross-sectional shear, B1 is not necessarily tangent to R. Instead, B2 and B3 are considered to be unit vectors in the direction of convected b2 and b3 while B1 is defined by B1 D B2 B3 . For small deformations, the x1 spatial changes in the Bi basis can be expressed using a curvature vector as B0i D K Bi . In addition to bending and twisting, the center-line is allowed to stretch, as indicated by strain component ”11 , and the cross-section is allowed to shear in both transverse directions, as indicated by ”12 and ”13 . The strain components are stored in a strain vector denoted by ” D Œ”11 ; 2”12 ; 2”13 T . From the described kinematics, a set of intrinsic equations of motion can be derived which govern the beam’s response to external loads (Hodges 1990; Hodges 2003), F0 C K F C f D PP C XP;
(6.1)
P C H C V P; M C K M C .e1 C ”/ F C m D H
(6.2)
0
where F denotes the internal force resultant, M the internal moment resultant, P the linear momentum per unit length (associated with velocity V), and H the angular momentum per unit length (associated with angular velocity ). The net distributed forces per unit length and the net distributed moments per unit length are captured by f and m, respectively. The unit vector e1 is given by Π1 0 0 T . Note that these equations are nonlinear, with the nonlinearity appearing at a low order.
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A second set of nonlinear kinematic constraints relates x1 -spatial changes in the velocity V and angular velocity to time derivatives of the net curvature (i.e., K k) and the strain ”, 0 C K D P
(6.3)
0
V C K V C .e1 C ”/ D ”P
(6.4)
The general momenta and velocities are related through the mass per-unit-length ; cross-sectional mass moments and product of inertia i2 , i3 , i23 ; and centroidal offsets from the center-line xN 2 , xN 3 , 9 2 9 8 38 ˆ 0 0 0 xN 3 xN 2 ˆ V1 > P1 > > > ˆ ˆ > 6 > ˆ ˆ > > 7ˆ ˆ > > ˆ ˆ P 0 0 x N 0 0 V 2 3 2 > > 6 ˆ 7 ˆ > 6 > ˆ ˆ = = 7< < 0 xN 2 P3 0 0 7 V2 6 0 D6 (6.5) 7 6 0 ˆ xN 3 xN 2 i2 C i3 H1 > 0 0 7ˆ 1 > > > ˆ ˆ > > 6 ˆ 7 ˆ ˆ > > ˆ ˆ 4 xN 3 H2 > 0 0 0 i2 i23 5 ˆ 2 > > > ˆ ˆ > > ˆ ˆ ; ; : : H3 xN 2 0 0 0 i23 i3 3 where subscripts refer to the Bi basis vectors. Specification of (generally anisotropic) constitutive relationships linking internal stress resultants (F; M) and deformation metrics (; ”) completes the formulation. A convenient approach, especially when linking to atomistic systems as described later, is to introduce a strain energy function u per unit length such that FD
@u @”
and
MD
@u : @
(6.6)
Equations (6.2)–(6.6) constitute a completed formulation for determining the beam’s configuration as a function of time. Explicit information about the centerline’s position and the cross-section’s rotation, although not necessary, can be included by first introducing a set of rotation parameters (or the like) (Hodges 2006). This can facilitate the inclusion of non-follower loading – note that substitution of constants for f and m in (6.2) yields follower loads. A set of Rodrigues parameters of the form 8 9 < 1 = D 2 (6.7) : ; 3 can be introduced which capture at each location x1 and time t the rotation of the Bi basis relative to the bi basis. A direction cosine matrix relating the Bi basis to the bi basis, Cij .x1 ; t/ Bi bi , can be expressed in-turn as: CD
1 14 T I Q C 12 T 1 C 14 T
:
(6.8)
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Fig. 6.2 Small differential element used to derive a relative position vector in terms of the intrinsic metrics net curvature and strain ”
R*(x1; a2, a3) z2
R(x1) R*(x1+ dx1; b2, b3)
z1
The velocity of the center-line and the derivatives of the Rodrigues parameters are then governed by: uP D CT V v !u Q (6.9) and
1 1 P D I C Q C T . C!/; 2 4
(6.10)
where v denotes the initial velocity and ! Q denotes the skew-symmetric form of the initial angular velocity. For use in atomistic modeling (e.g., carbon nanotubes), the kinematics need to be worked out further to arrive at an intrinsic representation of the relative position vector of two points in the beam spanning a small center-line difference dx1 . This allows, for example, relating the strain energy u appearing in (6.6) to general potential functions requiring explicit knowledge of distances and angles – e.g., interatomic or force potentials. Figure 6.2 illustrates the three position vectors required to develop the intrinsic relative position of two points spanning a small center-line difference dx1 . Included is the position vector in the deformed configuration R .x1 / of any point on the cross-section originally occupying material point .x1 ; x2 .x1 / D a2 ; x3 .x1 / D a3 / and the position vector R .x1 C x2 / of any second point originally occupying material point .x1 C dx1 ; x2 .x1 C dx1 / D b2 ; x3 .x1 C dx1 / D b3 /. It is convenient to express R .x1 / using the center-line position R .x1 / in the deformed configuration, R .x1 I a2 ; a3 / D R.x1 / C a2 B2 .x1 / C a3 B3 .x1 /
(6.11)
such that R .x1 C dx1 / can be expressed as: R .x1 C dx1 I b2 ; b3 / D R.x1 C dx1 / C b2 B2 .x1 C dx1 / C b3 B3 .x1 C dx1 /: (6.12) To calculate a relative position vector using R .x1 / and R .x1 C dx1 /, R .x1 C dx1 / must be expressed in terms of the basis vectors Bi at x1 . This can be accomplished using a Taylor expansion applied to both Bi .x1 C dx1 / and
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R .x1 C dx1 /. Note that it is critical to expand both the position vector and the basis vectors, as opposed to just the position vector when deriving the local canonical form of a space curve. For an illustration, consider a bending deformation in which the change in length of an arc segment away from the center-line could not be accounted for if B2 .x1 C dx1 /, B3 .x1 C dx1 / remained oriented with B2 .x1 /, B3 .x1 /. The basis vectors can be expanded to O dx13 as, 1 Bi .x1 C dx1 / D Bi .x1 / C B0i .x1 /dx1 C B00i .x1 /dx12 C O dx13 2 1 D Bi .x1 / C .K.x1 / Bi .x1 //dx1 C .K.x1 / .K.x1 / Bi .x1 // 2 3 0 CK .x1 / Bi .x1 / dx1 C O dx1 (6.13) and the center-line position can be expanded in a similar manner,
where
1 R.x1 C dx1 / D R.x1 / C R0 .x1 /dx1 C R00 .x1 /dx12 C O.dx13 /; 2
(6.14)
R0 .x1 / D .1 C ”11 /B1 .x1 / C 2”12 B2 .x1 / C 2”13 B3 .x1 /:
(6.15)
Using both expansions the final expression for R .x1 C dx1 / is given as, R .x1 C dx1 I b2 ; b3 / D Œ.1 C ”11 C b3 K2 b2 K3 /dx1 B1 .x1 / CŒb2 C .2”12 b3 K1 /dx1 B2 .x1 / C Œb3 C .2”13 C b2 K1 /dx1 B3 .x1 / CO.dx12 /:
(6.16)
From (6.11) and (6.16), the relative position of two points in the deformed configuration can now be stated as, ra;b D Œ.1 C ”11 C b3 K2 b2 K3 /dx1 B1 .x1 / CŒ.b2 a2 / C .2”12 b3 K1 /dx1 B2 .x1 / CŒ.b3 a3 / C .2”13 C b2 K1 /dx1 B3 .x1 / C O.dx21 /:
(6.17)
Equation (6.17) is notable in that a relative position vector has been expressed purely in terms of the intrinsic state variables curvature and strain (i.e., without the need for displacement information). For materials obeying a general constitutive law, which could include dependence on strain and strain-rate, (6.17) provides the necessary starting point for developing moment–curvature, moment–strain, force– strain, and force–curvature relationships. For example, the normal strain of any line segment connecting two similar points in the cross-section (b2 D a2 and b3 D a3 ) is given by:
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”11 D
D
k r.x1 ;x2 ;x3 /;.x1 Cdx1 ;x2 ;x3 / k2 dx21 2dx21 1 .1 C ”11 C x3 K2 x2 K3 /2 C .2”12 x3 K1 /2 C .2”13 C x2 K1 /2 1 : 2 (6.18)
When translated to a stress and integrated over a cross-section, (6.18) yields the force and moment resultants as a function of curvature and strain. (6.17) has additional utility in problems modeled using an interatomic or pseudo potential, such as nanotubes, polymers, or proteins. There, (6.17) allows the necessary deformed bond lengths and angles to be computed based solely on the intrinsic metrics.
6.2 Finite Element Formulation The governing equations are now specialized to the case of a beam whose mass center is located on the center-line (i.e., xN 2 xN 3 D 0), and whose cross-section has appropriate symmetry (i23 0) – the latter being satisfied by an x1 -axis of symmetry, or an x1 -x2 or x1 -x3 plane of symmetry, or other special cases. This case is considered captures many systems of interest and allows for an explicit formulation where a diagonal mass matrix is requisite. With the above simplifications the governing equations reduce to, P D F0 C K F C f V; V 2 3 3 2 P1 i 1 1 i1 4 i2 P 2 5 D M0 C K M C .e1 C ”/ F C m 4 i2 2 5 P3 i3 i3 3 P D 0 C K K
(6.19) (6.20) (6.21)
0
”P D V C K V C .e1 C ”/ ;
(6.22)
where i1 i2 C i3 has been introduced. The governing equations as stated earlier, with an appropriate selection of boundary conditions, denote the strong form. The weak form is more convenient from a finite element standpoint and can be developed from the strong using virtual velocity and angular velocity measures ıV and ı. Unlike the actual velocities, the virtual velocities are chosen to satisfy homogenous boundary conditions at the domain ends. Taking the inner product of each equation with the appropriate virtual quantity yields the following weak form, Z
P ıV dx1 D V
Z
.F0 C K F C f V/ ıV dx1
(6.23)
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3 2 3 P 1 T ı1 i1 4 i2 P 2 5 4 ı2 5 dx1 P3 i3 ı3 2
Z
Z D
0
2
31T 2 3 i1 1 ı1 @M0 CK MC.e1 C”/ F C m 4 i2 2 5A 4 ı2 5 dx1 (6.24) i3 3 ı3 Z Z P ı dx1 D .0 C K / ı dx1 K (6.25) Z
Z ”P ıV dx1 D
.V0 C K V C .e1 C ”/ / ıV dx1 :
(6.26)
Spatial derivatives of the noninterpolated internal forces can be shifted to the virtual velocities through an integration by parts operation. Applying the homogenous boundary conditions to the virtual velocities allows the first two governing equations to be rewritten as, Z Z Z P ıV dx1 D F ıV0 dx1 C .K F C f V/ ıV dx1 (6.27) V Z
3 2 3 2 03 P 1 T ı1 Z i1 ı1 4 i2 P 2 5 4 ı2 5 dx1 D M 4 ı0 5 dx1 2 P3 i3 ı3 ı0 2
3
Z C
2
0
31T 2 3 i 1 1 ı1 @K M C .e1 C ”/ F C m 4 i2 2 5A 4 ı2 5 dx1 : (6.28) i 3 3 ı3
Next the center-line distance x1 , the deformation state measures (V, , K, ”), and the virtual velocities (ıV, ı) are interpolated for an element, x1 D Œ0; l, in the usual manner using shape functions N ./ and nodal values, x1 D NI ./x1I
V D NI ./VI
D NI ./I
K D NI ./KI
” D NI ./” I
ıV D NJ ./ıV J
ı D NJ ./ıJ ;
(6.29)
where represents a natural coordinate assuming values from 1 to 1 and where nodal quantities are indicated by a superscript I or J ranging from 1 to n, the number of element nodes. Repeated indices denote summation in the usual sense. Introducing the interpolated quantities into the governing equations, evaluating inner products, and recognizing that the expressions must hold for all allowable virtual velocity fields yields semidiscrete equations, aJI VPkI D FQkJ c JIK Ii VjK eijk
(6.30)
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0
1
C P Ik D MQ kJ c JIK Ii B ik @ ij K A eijk j „ƒ‚… „ƒ‚…
aJI
no sum on k
(6.31)
no sum on j
aJI KP kI D b JI Ik C c JIK KiI K j eijk
(6.32)
aJI ”P kI D b JI VkI C c JIK KiI VjK eijk C c JIK ”iI C ıi1 K j eijk
(6.33)
where Z1 a D JI
Z1 NJ NI Jd
1
c
D
NJ NI; 1
Z1 JIK
b D JI
NJ NI NK Jd 1
ˇ ˇ ˇ dx1 ˇ ˇ ˇ J Dˇ d ˇ
@ Jd @x1
(6.34)
are quantities which can be calculated initially and reused at each later time interval), eijk denotes the permutation index operator and ıi1 denotes the Kronecker delta, and equivalent force and moment terms requiring integration are represented by the quantities FQkJ and MQ kJ given explicitly by, FQkJ
Z1 D
.Fk NJ0 C NI NJ KiI Fj eijk C fk NJ /J d
(6.35)
1
MQ kJ
Z1 D
.Mk NJ0 C NI NJ KiI Mj eijk C NI NJ .”iI C ıi1 /Fj eijk C mk NJ /Jd
1
(6.36) In all expressions, the free indices J and k indicate 3n nonlinear first-order equations. In the temporal integration of the semidiscrete equations, [(6.35), (6.36)] are generally computed using Gauss integration. In certain simplifying cases, the constitutive modeling (6.6) and external forcing allow [(6.35), (6.36)] to be integrated in closed-form at a significant savings in computation.
6.3 Three-Noded Element 6.3.1 Shape Functions and Constants The formal finite element formulation detailed in Sect. 6.2 can been implemented in the form of a three-noded element integrated temporally using a predictor– corrector implementation of the generalized-˛ method (Chung and Hulbert 1993).
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The Lagrange polynomials used for a three-noded element’s shape functions are given as follows, N1 D
1 . 1/ 2
N2 D 1 2
N3 D
1 . C 1/; 2
(6.37)
where each shape function evaluates to one at its home location and zero at other nodal locations. A number of quantities can now be evaluated. The expressions aJI , b JI , and c JIK are functions of undeformed arc length x1 and therefore need only be computed once and retained for the entire length of the simulation. These values are easily tabulated from (6.34).
6.3.2 Treatment of Zero-Energy Modes The formulation presented admits zero-energy modes through nodal shapes which contribute to zero equivalent forces FQkJ and moments MQ kJ . Specifically, the first term appearing in the right-hand side of (6.35, 6.36) may evaluate to zero for specific nodal shapes. For example, a typical quadratic form of energy appearing in the constitutive model (6.6) will result in forces and moments proportional to their corresponding strain and curvature (i.e., Fk D Ck ”k and Mk D Dk Kk , no sum on k implied). Using this information and the shape functions (6.37), three equations will result determining uniquely the three nodal strain values comprising a zero-energy mode: 2 03 2 3 Z1 N1 0 (6.38) Ck N1 ”k1 C N2 ”k2 C N3 ”k3 4 N20 5 J d D 4 0 5: 0 N30 1 A similar set of equations holds for the equivalent moments and the nodal curvatures. For the Lagrange polynomials chosen, the zero-energy nodal shapes satisfying (6.38) and the moment equivalent are given by ”k1 D ”k3 D ”k2 =2 and Kk1 D Kk3 D Kk2 =2. Thus, the nodal shape giving rise to zero energy modes is T h D 2 1 2 . The zero-energy deficiency can be remediated by introducing restoring forces proportional to a shape H reciprocal to h. Defining H requires the determination of two other primary shapes orthogonal (by choice) to H. For example, it is desirable that the restoring force (and hence H) be orthogonal to the rigid body shape r D T T 0 0 0 and the constant strain/moment shape b D 1 1 1 . Note that being orthogonal to the rigid body shape is satisfied identically. This, conveniently, allows us to choose another shape with which to be orthogonal – we choose the linearly T increasing strain/moment shape l D 1 2 3 . With these choices, the normalized T reciprocal shape is given by H D 1=6 1=3 1=6 such that H h D 1 and H r D H b D H l D 0. Zero-energy control is then accomplished by monitoring the nodal strains/curvatures for each element and applying proportional restoring forces in the shape of H.
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6.3.3 Temporal Integration A predictor–corrector instantiation of the generalized-˛ method (Chung and Hulbert 1993; Gobat and Grosenbaugh 2001) is used to integrate in time the semidiscrete equations while lumping all mass-like matrices. It is worth noting that this lumping is performed for the kinematic equations [(6.30)–(6.33)] as well. To prepare the equations for integration, [(6.30)–(6.33)] are cast into the standard form: P D Fext .X; t/ Fint .X; t/; (6.39) ŒMX where X is an array containing nodal velocities, angular velocities, curvatures, and strains; ŒM is a generalized mass matrix diagonalized using the “special lumping technique” (Hodges 1987); and Fext and Fint store the internal and external force/moment terms on the right-hand-side of [(6.30)–(6.33)], including any concentrated forces/moments applied on the nodes. To this standard form is introduced the mass and stiffness ˛-parameters .˛m ; ˛k / and the state at time ti (and associated superscript i ), P i C˛m ŒMX P i D1 DFext .Xi ; ti /.1˛k /Fint .X; ti /˛k Fint .Xi 1 ; ti 1 /: .1j˛m /ŒMX (6.40) The generalized-˛ approach uses a Newmark-like update equation for the solution array, i 1 Pi ; P ”X Xi D Xi 1 C ti X (6.41) where ” is an algorithm control parameter and 4ti represents the current time step. Q i is In this work, the integration in time proceeds with two steps. First, a predictor X computed via, P i 1 D ..1 ˛m /ŒM/1 ˛m ŒMX P i 2 C Fext .Xi 1 ; ti 1 / X .1 ˛k /Fint .Xi 1 ; ti 1 / ˛k Fint .Xi 2 ; ti 2 /
(6.42)
Q i D Xi 1 C ti .1 ”/X P i 1 : X
(6.43)
Note that the inverse appearing in [(6.42)–(6.43)] is trivial because the mass matrix has been diagonalized. Second, a corrector step is taken to arrive at the updated array Xi , QP i D .1 ˛ /ŒM/1 .˛ ŒMX P i 1 /j C Fext .X Q i ; ti / X m m Q i ; ti / ˛k Fint .Xi 1; ti 1 / .1 ˛k /Fint .X
(6.44)
i Q i C ti ” X PQ : Xi D X
(6.45) Pi
From (6.42-6.45) it can be seen that the velocity X also follows a predictorcorrector route to calculation that is out of phase with the solution array Xi - i.e.,
6 Nonlinear Dynamic Modeling of Nano and Macroscale Systems Table l 75 C1 108 Ey 106
6.1 Parameters for the validation study i1 i2 1 105 1:67 106 8:33 105 C2 C3 D1 3:32 107 3:32 107 6:67 108 v b H 0:25 10 10
i3 8:33 105 D2 8:33 108 Fe 5 105
181
zc 108 D3 8:33 108 Te 12 105
P i 1 occurs. The nodal during the predictor calculation for Xi , the correction to X displacements (6.9) and Rodrigues parameters (6.10) can be updated using a similar explicit predictor-corrector scheme, when needed.
6.3.4 Validation Results This section presents results generated using the intrinsic beam finite element approach versus those obtained from the commercial finite element package ABAQUS (Hibbit et al. 2011). Considered is a homogeneous, initially straight, isotropic rectangular cantilevered beam of cross-sectional dimensions b and h, with a suddenly applied tip load (of magnitude Fe ) whose components in the b2 and b3 directions are equal – note that the forcing is not a follower load and hence requires calculation of the direction cosines matrix at the final node. In addition, the end experiences a suddenly applied twisting moment Te . The parameter space for the beam and loading are given in Table 6.1. The integration parameters used in all intrinsic beam simulations presented are ” D 0:5, ˛m D 0:0, and ˛k D 0:5. The zero-energy control parameter is denoted by zc . Comparisons are made to an ABAQUS model composed of 32-noded beam elements (termed B31 elements in ABAQUS). To match the number of degrees of freedom, 15 intrinsic beam elements are simulated. The intrinsic beam requires calculation of cross-sectional mass moments and product of inertia i1 ; i2 ; i3 which follow from the area moments/products of inertia multiplied through by the mass per unit length . Bending stiffness (D2 and D3 ) and torsional stiffness (D1 ) also use the area moments/products of inertia, but multiplied through by Young’s modulus Ey and shear modulus G, respectively. The axial stiffness (C1 ) is given as Ey bh. The correction factors for the two shear stiffness (C2 and C3 ) are systematically determined for an arbitrary cross-section composed of an arbitrary material (Yu and Hodges 2004). For uniform rectangular beams, these correction factors are given by, 6 k2 D C 5 6 k3 D C 5
1C 1C
"
2
4
"
2
4
1 18 X tan h.m / 1 5 5 m1 m5
#
# 1 1 18 X tan h.n 1 / ; 5 5 n1 n5
(6.46)
(6.47)
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where denotes the ratio of b to h. The calculations require evaluation of an infinite sum, however the series converges such that the final computed values for the beam considered are k2 D k3 D 1:20557. The shear stiffness follow as C2 D Gbh=k2 and C3 D Gbh=k3 . Simulation results from the ABAQUS and intrinsic beam model appear in Fig. 6.3. This figure presents direct comparison of the three components of the strain vector at seven evenly spaced locations along the center-line and spanning the length of the beam. While these strain quantities are solution variables in the intrinsic formulation, and hence computed directly, they are derived quantities in the ABAQUS simulation, which instead uses nodal displacements and rotations as solution variables. Nevertheless, all three strain components show good agreement with some minor discrepancies isolated to a few points in time. Note that the results show agreement in time-histories characterized by broad-band frequency content. The good agreement continues for the bending curvatures K2 and K3 , as shown in Fig. 6.4. These are also derived quantities in the ABAQUS simulation. The final comparison concerns the twist K1 at evenly spaced points on the center-line. These results are presented in Fig. 6.5. Due to some mismatch in both amplitudes and frequencies, a single figure with both ABAQUS and intrinsic continua results presented together does not highlight the differences. Instead, they are presented separately. The general behavior in both is similar. The end node exhibits a relatively flat-line response, while the other nodes form ‘peaks’ which rise above it. Each peak is composed of step-like responses of the nodal twists, where the nodes starting from the fixed end show the largest sustained amplitude, and nodes after show decreasing time of amplitude sustainment such that the node at x1 D 62:5 appears as a spike in the middle of each peak. As the simulation progresses, both set of results show a marked increase in the frequency content. Some clear differences are also present. The ABAQUS results predict a greater amplitude (approximately 15%) in the global response initially, which abates toward the end of the simulation. The fundamental frequency predicted by ABAQUS has a very small difference from that predicted by the intrinsic beam model. These differences may be explained by the presence of a warping moment in the ABAQUS model vice the absence in the intrinsic beam model. For closed cross-sections, as in the present case, ABAQUS assumes no warping prevention and neglects axial strains due to warping. Although there is no warping prevention, ABAQUS formulates a nonvanishing warping moment applied as an additional torque around the centroid (ABAQUS Theory Manual, 1998). This additional torque is likely responsible for the differences in K1 observed, particularly the small amplitude differences.
6.4 Bulk Dynamic Response of Carbon Nanotubes Next is described a multiscale approach for obtaining the constitutive relationship required to use the intrinsic beam finite element simulator to analyze the bulk dynamic response of carbon nanotubes, a promising fiber-like element valued for its
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Fig. 6.3 Comparisons of strain histories predicted by an ABAQUS beam (circular markers) and the intrinsic beam (lines) finite element models for the case of simultaneous twisting and bending loads
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Fig. 6.4 Comparisons of bending curvature histories predicted by an ABAQUS beam (circular markers) and the intrinsic beam (lines) finite element models for the case of simultaneous twisting and bending loads
unique mechanical, chemical, and electrical properties (Peter 1999). A connection is made between interatomic potential energy governing the motion of individual atoms, to the strain energy per unit length required in (6.6). The interatomic potential requires information about bond lengths and angles, while the intrinsic formulation knows only curvatures and strains. However, (6.17) can be used to determine these quantities locally for a representative volume element, thereby enabling the multiscale approach. The resulting simulator is shown to accurately capture bulk (or zero temperature) dynamic response with a significant savings in computational effort as compared to more traditional approaches, such as molecular dynamics.
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Fig. 6.5 Comparisons of torsion histories predicted by an ABAQUS beam (a) and the intrinsic beam (b) finite element models for the case of simultaneous twisting and bending loads
6.4.1 Atomistic-Based Constitutive Modeling A carbon nanotube can be described as a graphene sheet rolled about a particular direction C described by the so-called chiral angle , as shown in Fig. 6.6. Graphene is a particular crystalline lattice form of carbon in which each carbon atom is bonded to three neighboring carbon atoms, forming a hexagonal arrangement. In Fig. 6.2, straight line segments depict the hybridized sp 2 bonds between the carbon atoms, while the carbon atoms themselves (not shown) exist at the intersections of the line segments. Accordingly, each hexagon holds six carbon atoms.
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Fig. 6.6 Geometry of the graphene sheet; representative volume element; final nanotube configuration
In general, a crystalline lattice in its reference (unloaded) configuration is comprised of a number of interpenetrating Bravais lattices whose points are given by: X D Mi ai C pk i D 1; 2; 3; M i 2 Z;
(6.48)
where X is the lattice intersection point, ai is the linearly independent (although not necessarily orthogonal) lattice vector or Bravais base vector, and pk is the shift vector for the inner atoms. For N C 1 atoms in the basis, the index k runs from 0 to N . For graphene, two Bravais lattices are present and therefore k equals one. Note that as graphene is a planar crystal, only the base vectors a1 and a2 need be ˚ These base considered, where each has an undeformed length lo equal to 2:46 A. vectors can be used to define the chiral vector C W .n; m/. The length of the chiral vector is given by: p C D jCk D l0 n2 C nm C m2 (6.49)
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which when divided by 2 yields the nanotube radius r. Due to periodicity of the lattice, each choice of C defines a unit cell, which is defined to be the smallest rectangle defined by C, and a translate of C, such that all four corners of the unit cell coincide with an atomic lattice point. The translation vector is given by T whose length is well documented, see for example (Peter 1999), and is given by, p p 3C =dH T D jTk D l0 n02 C n0 m0 C m02 D p 3C =.3dH/
n m ¤ 3zdH ; n m D 3zdH
(6.50)
where dH denotes the highest common divisor of n and m and z denotes any integer. The height h and width w of the unit cell will be used interchangeably with T and C . Note that many stable carbon nanotube configurations are known to exist which result in a variety of admissible radii (r’s) and chiralities ( ’s). Two configurations in particular are the armchair tubes [30ı chiral angle ; C W .n; n/] and the zig-zag tubes [0ı chiral angle ; C W .n; 0/]. The stored potential energy of an atomistic system can be modeled using an appropriate atomistic potential function. These potentials typically see application in Molecular Dynamics (MD) simulations, but have also recently been applied to reduced-order or continuum-like models (Zhang et al. 2004; Belytschko et al. 2002; Arroyo and Belytschko 2003). For a given set of interacting atoms, the atomistic potential function computes the atomistic energy based on bond lengths (i.e., twobody potentials), and in many cases, bond lengths and bond angles (i.e., three-body potentials). An alternative means to compute atomic energy is through use of the so-called Tight-binding models, which are simplified quantum-mechanical models. Tight-binding models are not pursued herein. Examples of commonly used potentials for carbon systems include the Morse potential, a two-body potential, and the Modified Morse and Brenner potentials (Brenner 1990), three-body potentials. Although commonly accepted for carbon nanotubes, the Brenner potential does not exhibit a clear separation between the bond-generated energy and the angle-generated energy, which is necessary for calculating the energy of the representative volume element (see Fig. 6.6 and later discussion). A recent Modified Morse potential (Belytchsko et al., 2002) does have this feature, and is therefore the potential chosen for this work. The potential is detailed in Sect. 6.4.2.
6.4.2 Modified Morse Potential The atomistic potential chosen for this study is a Modified Morse Potential (Belytchsko et al., 2002), although it is noted that the formulation is not dependent on any specific atomistic potential. As discussed in (Belytchsko et al., 2002), if the classical Morse Potential is to be used for modeling CNT’s, a three-body term accounting for angular position must be included in order to stabilize a tubular position. As such, the modified potential then takes the form,
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E D Estretch C Eangle n o 2 Estretch D De 1 e ˇ.rr0 / 1 Eangle D
1 k . 0 /2 1 C ksextic . 0 /4 ; 2
(6.51) (6.52) (6.53)
where Estretch is the bond energy due to bond stretch, Eangle the bond energy due to bond angle-bending, r the length of the bond, and is the current angle of the adjacent bond. The bond lengths and bond angles needed in (6.53) are found using (6.17). For example, the bond length rab between atoms a and b is given by krab k and the bond angle abc between atoms a, b, and c is given by: cos abc D
rab rab : krab kkrab k
(6.54)
Note that the locations of the atoms are dependent on the chiral angle chosen, or equivalently, by the pair .n; m/. The parameters used in all simulations presented herein correspond to sp2 bonds and are given by: r0 D 1:39 1010 m
De D 6:03105 1019 Nm
ˇ D 2:625 1010 m1
0 D 2:094rad
k D 0:9 1018 Nm=rad2
ksextic D 0:754 rad4 :
(6.55)
Performance of this potential for strains below 10% has been shown to compare very well (Belytchsko et al., 2002) to the more commonly accepted Brenner potential (Brenner 1990) – the advantage of adopting the Modified Morse Potential is that the stretching and angular contributions are distinct, which is important when forming a representative volume element, as discussed later.
6.4.3 Representative Volume Element The connection between the deformation state variables (V; ; ; ”) and the stored atomic energy can be made using representative volume elements (rve’s) at predefined locations x1 . The rve is instrumental in allowing atomic motion to be sampled (and averaged), thereby reducing the order of the model, without the need to calculate the total atomic energy present among all the atoms. Strain energy density per unit length u is connected to rve atomistic energy E rve as follows, uD
E rve ; l rve
(6.56)
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where l rve is a characteristic length of the rve. Note that for each x1 in which the internal forces F and moments M are to be calculated, several rve’s should be evaluated (and energy averaged) corresponding to several locations on the nanotube surface. In this way, using bending as an example, stretching of atomic bonds at one location on the perimeter, and compression of atomic bonds at an opposing location, is appropriately captured. Following the development of Arroyo and Belytschko (2003), a four-atom rve is chosen as shown in Fig. 6.2. In contrast to Arroyo and Belytschko (2003), the present approach requires a strain energy density per unit length, which is developed as follows. The chosen rve consists of three bond lengths and three bond angles covering completely the three bond length and angle varieties in each graphene hexagon. However, for the graphene hexagons, every bond length is shared by two hexagons, while each bond angle is unique to each hexagon. As such, the three rve bonds represent the three net bond lengths contained in a single graphene hexagon, while the three rve angles represent only half of the net bond angles in the same graphene hexagon. This dictates that an energy per unit area be defined as: e rve
rve rve Estretch C 2Eangle E rve D AHex AHex
(6.57)
rve where Estretch is calculated from the Modified Morse Potential summing the stretch rve energy from the three rve bond lengths, Eangle calculated summing the anglebending energy using the three rve bond angles, and AHex represents the area of the ˚ 2 ). The final desired expression for energy per unit graphene hexagon (5:019743 A length u can now be formed from the unit cell dimensions width w and height h,
uD
rve rve Estretch C 2Eangle e rve h w w D e rve w D h AHex
(6.58)
For completeness of the discussion, the rve characteristic length is identified from (6.56) and (6.58) as l rve D AHex =w. To calculate the internal forces and moments, derivatives of u with respect to the deformations ” and must be formed. These derivatives can be calculated as follows, ! rve rve @Eangle @Estretch @u w @rij @ijk FD ; (6.59) D C2 @” AHex @rij @” @ijk @” ! rve rve @Eangle @Estretch @u @rij @ijk w MD ; (6.60) D C2 @ AHex @rij @ @ijk @ where i; j; k are indices representing the four rve atoms such that the nonzero bond lengths include only fr12 ; r13 ; r14 g and the nonzero bond angles ijk include only f123 ; 124 ; 134 g. These quantities can be calculated from the deformation state
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M.J. Leamy Table 6.2 Parameter space for the (10; 10) armchair nanotube used in the large-deformation bulk dynamic response simulation l r i 1 ˚ ˚ ˚ ˚ A A amu= A amu A 900 i2 ˚ amu= A
6:875 i3 ˚ amu A
195:35 dV i hp ˚ amu e V= A
9:233 103 d h p i ˚ amu e V A
4:617 103
4:617 103
0
0
dK p e V=amu
d” p
0
0
e V=amu
using the procedure described in Sect. 6.4.2, while the complexity associated with finding closed-form expressions for their derivatives with respect to ” and require finite difference approximations in the numerical implementation. It is worth noting that at this stage of the development the link between bulk-scale response and atomistics has been accomplished without the need for assuming a reliance on bulk-scale material descriptors, such as Young’s Modulus, as witnessed by their absence in the stress resultants .F; M/ given by [(6.59), (6.60)]. On a final note, it is often necessary to allow inner displacements to occur within the rve during large deformations, or else the simulation is too stiff. Interested readers can find details in Leamy (2007).
6.4.4 Bulk Dynamic Response Validation studies confirming accurate equilibrium and dynamic response predictions using the multiscale carbon nanotube simulator can be found in Leamy (2007). Here, simulation results are presented highlighting large-deformation response of a (10; 10) armchair nanotube to end-loading similar in spirit to that applied by an atomic force microscope in a mechanical testing procedure. The parameter space for this nanotube is given in Table 6.2. The end load simulated increases linearly to ˚ at time 1:0 104 units before decreasing a maximum value of 37:5 103e V= A linearly to zero starting at time 8:0 104 units and ending at time 1:0 105 units. A rigid constraint fixes the nanotube at its opposite end. Results are presented as a visual rendering of the nanotube at evenly spaced intervals in time. Due to the use of curvatures and strains in the state description, the location of the nanotube’s configuration in space requires additional processing over a standard Cartesian description. Specifically, points are first generated sequentially starting from the nanotube end (its center-line location in space and the orientation of its basis vectors Bi being specified) using the Taylor expansion for the centerline R .x1 C dx1 /, for points on the nanotube surface R .x1 C dx1 /, and for the
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Fig. 6.7 Snapshots of the dynamic response of a (10; 10) armchair nanotube to a follower end load spaced at intervals of 5 103 time units (a) for the loading phase starting at time 0:0 and ending at 8:0 104 and (b) the unloading phase starting at time 8:0 104 and ending at time 1:5 104 . Note that the force is rendered only twice in (a) for illustration purposes, but is present at each snapshot
updated basis vectors Bi .x1 C dx1 /. The required curvatures and strains at any point between the element nodes are found from the nodal values and the interpolation functions. This procedure is carried out from one end to the other in increments dx1 and establishes a user-defined number of circumferential points around the crosssection and along the nanotube axis. Quadrilaterals spanning dx1 are then defined R using the points and are rendered with the aid of OpenGL library calls. As depicted in snapshots from the nanotube visualization shown in Fig. 6.7, the forced (10; 10) nanotube exhibits a complex dynamic response during the simulated assembly procedure. Initially, the nanotube “winds up” and deformation is localized to its forced end. As time proceeds, a more characteristic tip-loaded bending deformation develops in which the center-line of the nanotube displays its greatest curvature at the supported end, and near-absent curvature at the loading end. After
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time 3:0 104 time units [snapshot 6 in subfigure (a)] the follower load contains a component of force in the negative vertical direction, causing the nanotube’s configuration to flatten and resemble a horseshoe. During this time tip oscillations are present. Following the release of the load starting at time 8:0 104 units, the nanotube exhibits large oscillations with prominent second and third bending mode content.
6.5 Nonlinear Dynamic Modeling of Helical Springs The intrinsic beam finite element simulator is next employed to efficiently study dynamically loaded helical springs. Convergence studies demonstrate that a sparse number of elements accurately capture spring dynamic response, with more elements required to resolve higher frequency content, as expected. Presented results also document rich, amplitude-dependent frequency response. In particular, moderate loading amplitudes lead to the presence of secondary resonances (Nayfeh and Mook 1979) (not captured by linearized models), while large loading amplitudes lead to complex dynamics and transverse buckling. Helical springs are ubiquitous in machine elements where they are employed to store and release energy, absorb shock, or maintain a force between contacting surfaces. Their application in systems undergoing dynamic operation, such as engine valve springs and vehicle suspension systems, are of particular interest in this section due to the possibility of resonant behavior (or “spring surge”). The existence of primary, and potentially secondary, resonances must be accounted for in helical spring design in order to avoid undesirable large displacements and fatigue failure. Three configurations are employed in describing the kinematics of the helical coil spring – see Fig. 6.8. The first, termed the reference configuration and denoted by ref represents the spring in a straightened configuration with zero strains and zero curvatures. The second configuration, termed the initial configuration and denoted by 0 , represents the spring in its unstressed configuration characterized by an initial state of strain and curvature, K0 , ” 0 . The third configuration, termed the deformed configuration and denoted by f , depicts the spring in a state in which internal forces and moments arise. Relative to ref , it is characterized by a state of strain and curvature denoted by Kf , ” f . Alternatively, it can be characterized by additional curvature and strain (KO ; ”) O away from 0 , where KO D Kf K0 and ”O D ” f ” 0 . A set of unit basis vectors employed in each configuration allows for a specific component representation of strain and curvature to be used in each configuration (see Fig. 6.1): Cartesian unit vectors ŒI1 ; I2 ; I3 in ref , basis vectors ŒB01 ; B02 ; B03 in 0 , and basis vectors ŒBf1 ; Bf2 ; Bf3 in f . The basis vectors in 0 and f are considered embedded in the cross-section of the helical spring such that B02 and Bf2 align with a material line along I2 in ref (see Fig. 6.2); similarly, B03 and Bf3 align with a material line along I3 in ref . The remaining unit vectors, B01 and Bf1 , are defined as completing an othonormal set.
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Fig. 6.8 Configurations employed in developing the helical spring model. Material points along a straight reference configuration ref are mapped to an initial configuration 0 via initial curvature and strain K0 , ” 0 . Similar mappings hold for a deformed configuration f . The latter two configurations are related by mappings which invoke net curvature and strain, KO , ”O
The strains and curvatures in 0 and f are defined implicitly using expressions for the spatial rate-of-change of the centerline position and the triad of basis vectors, R00 D .1 C ”011 /B01 C 2”012 B02 C 2”013 B03 ;
(6.61)
R0f D .1 C ”f11 /Bf1 C 2”f12 Bf2 C 2”f13 Bf3 ;
(6.62)
B00
(6.63)
D K 0 B0 ;
B0f D Kf Bf ;
(6.64)
where R0 .x1 / denotes position of the centerline in ˝0 , Rf .x1 / denotes position of the centerline in ˝f , and the strain vectors are related to the strain components via T ” f D ”f11 2”f12 2”f13 T ” 0 D ”011 2”012 2”013 :
(6.65) (6.66)
6.5.1 Constitutive Modeling The required constitutive relationships for the helical springs considered herein are derived next using strain measures derived from (6.16)–(6.17) and interpreted from the three configurations presented earlier.
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6.5.1.1 Initial Configuration 0 Metric tensor components can be employed to track the deformation of line segments in the body of the helical spring, both in terms of stretch and angular change. Frequently, these components are associated with scalar products of convected basis vectors, but this is not the usage herein because the basis vectors ŒB1 ; B2 ; B3 do not strictly convect. Instead, the components are found directly from oriented line segments. The metric tensor components in the initial configuration, G0ij , result from consideration of an oriented line segment dxref in ˝ref originating at a point in the cross-section .x1 , x2 , x3 / and terminating at .x1 C dx1 , x2 C dx2 , x3 C dx3 /. From (6.16), this line segment is oriented in ˝0 as follows, dx0 D R0 .x1 C dx1 ; x2 C dx2 ; x3 C dx3 / R0 .x1 ; x2 ; x3 / D Œ.1 C ”011 C x3 K02 x2 K03 /dx1 B01 .x1 / C Œdx2 C .2”012 x3 K01 / dx1 B02 .x1 /
C Œdx3 C .2”013 C x2 K01 / dx1 B03 .x1 / C O dx 2 :
(6.67)
From this expression, convected I1 , I2 , and I3 originating from any location in the cross-section (x2 , x3 ) are identified as, G01 D .1 C ”011 C x3 K02 x2 K03 /B01 .x1 / C .2”012 x3 K01 / B02 .x1 / C .2”013 C x2 K01 / B03 .x1 / ;
(6.68)
G02 D B02 .x1 /;
(6.69)
G03 D B03 .x1 /;
(6.70)
where these expressions are obtained from the multipliers of dx1 , dx2 , dx3 in (6.67). The metric tensor components then arise from the definitions G0ij G0i G0j . These components can be used to define strain tensor, such as the Green an appropriate Lagrange strain tensor: e0ij G0ij ıij =2 where ıij denotes the Kronecker delta and arises in the strain due to the fact that I 1 , I2 , and I3 are an orthonormal set, 1 .1 C ”011 C x3 K02 x2 K03 /2 2 1 1 1 C .2”012 x3 K01 /2 C .2”013 C x2 K01 /2 2 2 2 1 D .2”012 x3 K01 / 2 1 D .2”013 C x2 K01 / 2 D e033 D e023 D 0:
e 01 D
e 02 e 03 e022
(6.71) (6.72) (6.73) (6.74)
Note that these strain components are consistent with those found in other works and neglect any contribution due to warping of the cross-section.
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6.5.1.2 Final Configuration f The strain components for the final configuration follows closely that of the initial configuration ˝0 . In fact, the convected I1 , I2 , and I3 vectors can be recovered from (6.68) to (6.70) by replacing ”0ij , K0i , and B0i with ”fij , Kfi , and Bfi : Gf1 D .1 C ”f11 C x3 Kf2 x2 Kf3 /Bf1 .x1 / C 2”f12 x3 Kf1 Bf2 .x1 / C 2”f13 C x2 Kf1 Bf3 .x1 /
(6.75)
Gf2 D Bf2 .x1 /
(6.76)
Gf3 D Bf3 .x1 /
(6.77)
The strain components in f are defined such that they measure changes in stretch and orientation of line segments initially along G0i and then convected to Gfi . This requires accounting for the strain present in 0 such that the Green–Lagrange strain tensor takes the form efij Gfij G0ij =2. Note that this results in 0 having zero stress when using Hooke’s Law with efij , as desired. The internal stress resultants follow. If the net curvature and strain metrics KO D Kf K0 and ”O D ” f ” 0 are introduced, and the stress resultants are linearized for small KO and ”O (with arbitrarily large K0 and ” 0 ), the desired constitutive relationships are given by, Z
r4 O 2G x3 ef12 C x2 ef13 dA D G K1 2 A Z
r4 O D Ex3 ef11 dA D E K2 C 2K02 ”O 12 C 2”012 KO 2 4 A
r4 E 2K01 ”O 12 C 2”012 KO 1 4 Z
r4 O D Ex2 e011 dA D E K3 C 2K03 ”O 13 C 2”013 KO 3 4 A
4 r E 2K01 ”O 13 C 2”013 KO 1 4
Mf1 D Mf2
Mf3
(6.78)
(6.79)
(6.80)
Z Ff1 D
Eef11 dA A
D
r2 E .4”O 11 C 4 .”011 ”O 11 C 4”012 ”O 12 C 4”013 ”O 13 // 4
r2 2 C Er 2K01 KO 1 C K02 KO 2 C K03 KO 3 4
(6.81)
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Z Ff2 D Z
2Gef12 dA D
2 r 2 G ”O 12 k
(6.82)
2Gef13 dA D
2 r 2 G ”O 13 ; k
(6.83)
A
Ff3 D A
where r denotes the cross-sectional radius, G D E=.2 C 2 / denotes the shear modulus and where transverse shear resultants include shear correction factors k. Note that these expressions indicate linear coupling of net strains and curvatures in the axial force Ff1 and bending moments Mf2 and Mf3 . For circular cross-sections, a recent estimate Yu and Hodges (2004) for the shear correction factor is given by: kD
8 2 C 7 C 14 6 .1 C /2
:
(6.84)
6.5.2 Results This section presents simulation results for the forced response of two example springs. First, the springs are subjected to impact-like forcing and simulated afterward to recover free response. For small amplitude forcing, Fourier transforms of the resultant free response yield the natural frequencies. For larger amplitude forcing, Fourier transforms reveal additional frequencies in the response which can be identified with nonlinear effects such as super-harmonic, subharmonic, and combination frequencies. For a fixed-free supported spring, small harmonic excitation at these other frequencies is explored to determine which yield large response, and hence can be identified with secondary resonances. The discussion of natural frequencies and small motions is followed-up by discussion of an example loading state inducing large dynamic motions in the fixed-free spring, leading to complex dynamics and transverse buckling. 6.5.2.1 Fixed–Fixed Example Spring: Natural Frequencies The parameter space for the first spring considered is given in Table 6.3. Simulations are performed using fixed–fixed boundary conditions. Parameters in Table 6.3 not previously defined include the spring outer radius R, number of turns n, height h, and helical angle ˛. These parameters lead to calculation of the initial curvature K02 , twist K01 , and length l from the expressions, cos2 ˛ R cos ˛ sin ˛ K01 D R p l D 2 Rn 1 C tan2 ˛:
K 02 D
(6.85) (6.86) (6.87)
6 Nonlinear Dynamic Modeling of Nano and Macroscale Systems Table 6.3 Parameters for the fixed–fixed example spring
Table 6.4 Calculated quantities for the fixed–fixed example spring
K 01 1 m 29:485 G N= m2 7:923 1010
r Œ mm 0:5 ˛ Œdeg 8:5744
R Πmm 5:0 E ΠN= m2 2:06 1011
K 021 m 195:55 K
l Πmm 241:46 i2 Πkg m 3:878 1010
1:176
197
n 7:6 0:3
h Πmm 36:0
Πkg= m3 7900:0
Πkg= m 6:2 103 i3 Πkg m 3:878 1010
These and other computed quantities appear in Table 6.4. Note that due to symmetry of the helical spring cross-section, the product of inertia i23 and centroidal offsets xN 2 and xN 3 are identically zero. The natural frequencies are determined for the spring using an impact-like set of small-magnitude forces which set the spring into motion, and then are removed in order to allow for free response. Two sets of forces are used. The first yields primarily longitudinal motion of the spring and consists of nodes along the center coil being forced in the positive B02 direction. The second yields primarily transverse motion and consists of upper coils being forced in the negative B02 direction, and lower coils being forced in the positive B02 direction. In both sets of forces, the magnitude of the forcing is kept small such that small displacements ensue and near-linear behavior results. Natural frequencies are calculated from the computed free response using fast Fourier transforms (FFTs) of the nodal variables (nodal curvatures, strains, velocities, etc.), which identify the frequency content. In practice, only an FFT of the second component of curvature at a node a small distance from the center of the spring is sufficient to capture all frequencies. An example FFT result is shown in Fig. 6.9 for a transversely loaded fixed–fixed spring. This spring has been discretized using 48 three-noded elements (97 total nodes). The FFT is taken at node 30. The FFT peaks in this figure yield the desired natural frequencies. A large number of simulations were carried out for the first example spring, exciting it in either longitudinal or transverse motion, calculating the curvature FFTs, and then cataloging the frequencies. The results of these simulations are summarized in Figs. 6.10 and 6.11. Figure 6.10 plots the calculated longitudinal frequencies as a function of four discretization cases: 12, 17, 24, and 48 elements. Figure 6.11 is similar, but plots the transverse frequencies. It is evident from these figures that (1) for the first eight modes, convergence is reached in almost all cases with the use of 24 elements and (2) modal convergence is approached from
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Fig. 6.9 FFT of KO 2 at node 30 – example fixed–fixed spring discretized using 48 elements (97 nodes) Fig. 6.10 Modal convergence with number of intrinsic finite elements used for first example spring – longitudinal modes
Longitudinal Modal Convergence 1000 900
877.4
877.4
524.9
524.9
836.2
800 700
Hz
600
524.9
524.9
500 400 300
426.3
426.3
426.3
426.3
375.4
390.6
392.2
17
24
48
323.5
200 100 0 12
Number of Elements Mode 1
Mode 3
Mode 4
Mode 6
“below” – i.e., the frequencies increase with increasing number of elements until convergence is reached. The latter point is quite unlike traditional displacementbased finite ele-ment approaches, which are known to converge from “above.” It can also be noted that some of the modes display convergence with very few elements, including the 3rd and 4th modes (both longitudinal) and the 8th mode (transverse). This rapid convergence of select modes would appear to be a result of interpolating curvature as it has not been noted in other more-traditional models.
6 Nonlinear Dynamic Modeling of Nano and Macroscale Systems Fig. 6.11 Modal convergence with number of intrinsic finite elements used for first example spring – transverse modes
199
Transverse Modal Convergence 1200 1000
Hz
800 600
1033
1036
1036
909.4
910.9
912.5
836.2
860.6
863.6
375.4
395.2
396.7
17
24
48
935.4
906.4
630.2
400 343.3
200 0 12
Number of Elements Mode 2
Table 6.5 Parameters for the fixed–fixed example spring
r Œ mm 6:0 ˛ Œdeg 7:44
Mode 5
R Πmm 65:0 E N= m2 2:09 1011
Mode 7
n 6:0 0:28
Mode 8
h Πmm 320:0
kg= m3 7800:0
The natural frequencies for this example have also been cataloged in other works. For example, using the psuedospectral method with 50 collocation points, Lee (2007) computed the first eight natural frequencies to be Œ393:5, 396:1, 462:9, 525:7, 863:8, 877:0, 913:8 Hz. Note that these frequencies agree very well with those presented herein.
6.5.2.2 Fixed-Free Example Spring: Primary and Secondary Resonances A second example of a fixed-free spring is explored next. This spring has also been studied previously (Lee and Thompson 2001; Lee 2007). The interest herein is on predicting natural frequencies, as in the last example, and uncovering nonlinear response under moderate forcing. The spring parameter set is defined in Table 6.5, with computed properties appearing in Table 6.6. The identification of natural frequencies for this spring is performed with a procedure similar to the one described for the example fixed–fixed spring. Figure 6.12 presents an FFT of the second curvature component at node 30 for the longitudinally loaded spring. The natural frequencies identified from this plot are Œ9:497, 24:16, 42:85, 63:10, 71:17, 72:34 Hz. The transverse loading uncovers additional frequencies at Œ9:485, 21:36, 42:09 Hz, for a total of nine natural
200 Table 6.6 Calculated quantities for the fixed–fixed example spring
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K 01 1 m 1:975 G N= m2 8:164 1010
K 021 m 15:127 K 1:175
l Πmm 2:471 i2 Πkg m 7:939 106
Πkg= m 0:882 i3 Πkg m 7:939 106
Fig. 6.12 FFT of KO 2 at node 30 – example fixed-free spring discretized using 48 elements (97 nodes)
frequencies. The first eight frequencies computed using 50 collocation points and the psuedospectral finite element method (Lee and Thompson 2001) are given as Œ9:4718, 9:4997, 21:358, 24:170, 42:100, 42:857, 63:107, 71:205 Hz. Comparing the two methods, good agreement can again be documented. Next, the fixed-free spring is subjected to moderate amplitude forcing in order to study nonlinear response character. The longitudinal loading case chosen results in spring motion depicted in Fig. 6.13. Note that the amplitude of motion is that which can be reasonably experienced in typical applications. An FFT of the ensuing free response is provided in Fig. 6.14. Evident in this FFT are two effects that can be associated with nonlinear response. The first is shifting of the natural frequencies as a result of increasing response amplitude. In particular, the natural frequency at 9:497 Hz, previously identified using small-amplitude loading, has shifted upward to 9:631 Hz; similarly, 42:85 Hz has shifted downward to 42:76 Hz. Second, and more importantly, additional frequencies are present in the response. For the frequency range considered, two significant additional frequencies have appeared at 11:76 Hz and 52:43 Hz. In nonlinear systems these are typically associated with super-harmonic, subharmonic, or combination frequencies. Furthermore, forcing at these frequencies can often lead to relatively large response termed secondary resonances.
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Fig. 6.13 Initial spring configurations resulting from application of moderate-magnitude impulsive longitudinal loading. Ensuing motion is on the same order of magnitude as the configurations depicted
Fig. 6.14 FFT of KO 2 at node 30 following application of moderate loading – example fixed-free spring discretized using 48 elements (97 nodes)
To determine if these additional frequencies will give rise to large response, simulations were performed with harmonic forcing at each. The magnitude of the forcing was decreased by a factor of 50 from the magnitude of the moderate impulsive loading, and was sustained throughout the length of the simulations. This is a relatively severe test for secondary resonances because the motion amplitude under this forcing, away from a resonance, is very small. At this level
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Fig. 6.15 Time histories of curvature KO 2 at node 30 following application of harmonic loading – example fixed-free spring discretized using 48 elements (97 nodes). Two loading cases are depicted: 52:43 Hz, which is a potential secondary resonance, and 60:0 Hz where no resonances are expected
Fig. 6.16 Time histories of curvature KO 2 at node 30 following application of harmonic loading – example fixed-free spring discretized using 48 elements (97 nodes). Three loading cases are depicted: 52:43 Hz, 60:0 Hz, and 63:1 Hz
of forcing, harmonic loading at 11:76 Hz frequency fails to result in resonance-like motions. However, loading at 52:43 Hz does, as is evident in Fig. 6.15. This figure compares the curvature time history for 52:43 Hz loading versus 60:0 Hz loading,
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Fig. 6.17 Initial spring configurations resulting from application of large-magnitude impulsive transverse loading
which is close to 52:43 Hz but is where no resonances are expected. Comparing the responses, it can be observed that the motion from the 52:43 Hz excitation has hall-marks of resonance, including an initial exponential-like growth leading to large response. This is missing from the 60:0 Hz response, which results in significantly less amplitude motion than the 52:43 Hz loading. Finally, it is of interest to compare the motion at the suspected 52:43 Hz secondary resonance with motion at a primary resonance. Figure 6.16 provides a comparison of curvature time-histories at 52:43 Hz, 60:0 Hz, and 63:1 Hz (the location of a natural frequency). It is evident in this figure that the suspected secondary resonance, while yielding large motions, yields motion amplitudes on the order of 30% associated with primary resonance. It is typical in nonlinear systems for the secondary resonances to result in smaller amplitude motion than primary resonances, on the order of that observed here. This does not, however,
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Fig. 6.18 FFT of KO 2 at node 30 following application of large-amplitude loading
minimize their importance as the motions may be large enough to be problematic in applications, leading to undesirable coil motions such as self-contact and premature fatigue failure.
6.5.2.3 Fixed-Free Example Spring: Large Dynamic Response Next, the spring considered in previous section (now with fixed-free boundary conditions) is subjected to large amplitude forcing in order to highlight complex dynamic motions which can be captured using the presented approach. The model uses 48 elements (97 sequentially numbered nodes) and is subjected to nodal forcing similar to that used to excite transverse modes: nodes 2–12 are each excited by 2:0 Newton loads in the positive B02 direction, while nodes 86–96 are excited by 2:0 Newton loads in the negative B02 direction. Note that node 1 is located at the fixed end (x1 D 0), and node 97 is located at the free end. The loading time-history is such that the load magnitudes are ramped-up from zero to their final values in 0:5 ms, and then ramped-down to zero in a further 0:5 ms, at which point the spring undergoes free response. The initial configurations that result are given in Fig. 6.17 at 0:5 ms increments until a time of 5:5 ms. Observed in these configurations is large-amplitude response with complex dynamics, to include transverse buckling (see final configuration in figure) at a point in time past when the load has been ramped down. The complex dynamics observed in Fig. 6.17 is accompanied by broad frequency content (see Fig. 6.18). Note that the simulation has been carried-out to a final time of 0:5 s in order to compute the FFT in Fig. 6.18, indicating good stability of the combined discretization technique and integration approach.
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6.6 Concluding Remarks This chapter has presented a nonlinear finite element approach for dynamic modeling of nano and macroscale materials and structures based on an intrinsic beam formulation. The approach differs from more traditional approaches in that curvatures and strains have been chosen for the field variables. When constitutive models are developed from atomistic potentials, the resulting multiscale simulator quickly and accurately captures bulk dynamic response of nanoscale materials. When coupled to more traditional constitutive models, the resulting simulator accurately captures the dynamic response of precurved and twisted macroscale structures using a sparse number of elements. These properties may make the approach attractive for studying materials and structures across many scales, particularly those materials and structures exhibiting highly curved and twisted dynamic configurations.
6.7 Key Symbols a2 ; a3 AHex ˛ .˛m ; ˛k / H .t/ aJI ai b bi b JI Bi B0i Bfi c JIK C C C1 C2 ; C3 D1 D2 ; D3 ıij ı ıJ
Material points at location x1 along x2 , x3 Area of the graphene hexagon, (CNT) Helical angle, (Spring) Mass and stiffness ˛-parameters Material points at location x1 C dx1 along x2 , x3 Finite element inertia matrix coefficients Bravais base vectors Beam width Basis vectors in initial configuration (i D 1; 2; 3) Finite element stiffness matrix elements Basis vectors in the deformed configuration (i D 1; 2; 3) Basis vector in the initial configuration (i D 1; 2; 3), (Spring) Basis vector in the deformed configuration (i D 1; 2; 3), Finite element higher-order stiffness matrix elements Length of the chiral vector, C D kCk, (CNT) Direction cosine matrix; or chiral vector, (CNT) Axial stiffness Shear stiffnesses Torsional stiffness Bending stiffnesses Kronecker delta operator Virtual angular velocity J th nodal virtual angular velocity
206
ıV ıV J 4ti eijk e rve e0ij efij e1 E Estretch Eangle E rve Ev f FQ J k
Ffi F Fe Fext Fint G G0ij Gfij G0ij Gfij ” ” ”0 ”f ”O ”I h h H i1 i1 ; i2 ; i3 Ii J k2 ; k3 k
M.J. Leamy
Virtual velocity J th nodal virtual velocity Time step at time ti Permutation index operator Energy per unit length in rve, (CNT) Strain component in ˝0 , (Spring) Strain component in ˝f , (Spring) T Unit vector 1 0 0 Atomic energy, (CNT) Bond energy due to bond stretch, (CNT) Bond energy due to bond angle-bending, (CNT) rve atomistic energy, (CNT) Young’s modulus Net distributed force per unit length Finite element equivalent force Force component in ˝f , (Spring) Internal force resultant Tip load force magnitude External force/moment finite element terms Internal force/moment finite element terms Shear modulus Metric tensor component in ˝0 , (Spring) Metric tensor component in ˝f , (Spring) Basis vector in ˝0 arising from convected Ij , (Spring) Basis vector in ˝f arising from convected Ij , (Spring) Newmark algorithm control parameter Strain vector Initial state strain, (Spring) Deformed state strain, (Spring) Net strain, (Spring) I th nodal strain Beam height; unit cell height, (CNT); spring height, (Spring) T Nodal shape for zero energy modes h D 2 1 2 Angular momentum per unit length; or shape reciprocal to h Polar moment of inertia per unit length, i1 i2 C i3 Cross-sectional mass moments and product of inertia Cartesian basis vector, (Spring) Jacobian determinant Shear correction factors Shear correction factor, (Spring) Net curvature K k
6 Nonlinear Dynamic Modeling of Nano and Macroscale Systems
K K0 Kf O K KI l l0 l rve m m Mfi MQ J k
M ŒM n N N1 ./ !Q I ref 0 f pk
P r r ra;b rab R R .x1 / R0 .x1 / Rf .x1 / R .x1 I a2 ; a3 /
Curvature vector in deformed configuration Initial state curvature, (Spring) Deformed state curvature, (Spring) Net curvature, (Spring) I th nodal curvature Beam length; spring length, (Spring) Length of Bravais base vectors, (CNT) Characteristic length of the rve, (CNT) Chiral index referencing a2 , (CNT) Net distributed moments per unit length Moment component in ˝f , (Spring) Finite element equivalent moment Internal moment resultant Generalized mass matrix The mass per-unit-length Chiral index referencing a1 (CNT) Number of atoms minus one in basis (CNT); number of turns I th shape function Skew-symmetric form of the initial angular velocity Angular velocity I th nodal angular velocity Reference configuration, (Spring) Initial configuration, (Spring) Deformed configuration, (Spring) Shift vectors for the inner atoms, (CNT) Poisson ratio Linear momentum per unit length Chiral angle, (CNT) Radius of nanotube, (CNT) Position vector in initial configuration Relative position in the deformed configuration Bond length between atoms a and b, (CNT) Spring outer radius, (Spring) Center-line position in the deformed configuration Position of the centerline in ˝0 , (Spring) Position of the centerline in ˝f , (Spring) Position vector in the deformed configuration on the cross-section originally occupying material point .x1 ; x2 .x1 / D a2 ; x3 .x1 / D a3 / Ratio of b to h for computation of shear correction factors; density, (Spring)
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Time Time at step i Magnitude of translation vector, T D kTk, (CNT) Translation vector, (CNT) Tip load twisting moment magnitude Rodrigues parameters Bond angle, (CNT) Bond angle between atoms a, b, and c, (CNT) Strain energy function per unit length Velocity Ith nodal velocity Unit cell width, (CNT) Center-line location in initial configuration I th nodal location Cross-section location in initial configuration Centroidal offsets from the center-line State array containing nodal velocities, angular velocities, curvatures, strains State predictor Natural coordinate
References Arroyo M, Belytschko T .2003/ A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes. Mech Mater 35:193–215 Belytschko T, Xiao SP, Schatz GC, Ruogg RS .2002/Atomistic simulations of nanotube fracture. Phys Rev B 65:235430 Borri M, Mantegazza P .1985/ Some contributions on structural and dynamic modeling of helicopter rotor blades. I’Aerotecnica Missili e Spazio 64(9):143–154 Brenner DW .1990/ Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 42(15):9458–9471 Cesnik CES, Shin S .2001/ On the modeling of integrally actuated helicopter blades. Int J Solids Struct 38:1765–1789 Chung J, Hulbert GM .1993/ A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized- method. J Appl Mech 60(2):371–375 Gobat JI, Grosenbaugh MA .2001/ Application of the generalized-˛ method to the time integration of the cable dynamics equations. Comput Methods Appl Mech Eng 190:4817–4829 Hibbitt, Karlsson, Sorensen, (2011), ABAQUS Theory Manual v6.10, Dassault Syst`emes Inc. Hodges DH (1990) A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int J Solids Struct 26(11):1253–1273 Hodges DH .2003/ Geomertrically-exact, intrinsic theory for dynamics of curved and twisted, anisotropic beams. AIAA J 41(6):1131–1137 Hodges DH .2006/ Nonlinear composite beam theory.AIAA, Reston, VA Hodges DH, Patil MJ, Chae S .2002/ Effect of thrust on bending-torsion flutter of wings. J Aircraft 39(2):371–376
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Hodges DH, Shang X, Cesnik CES .1996/ Finite element solution of nonlinear intrinsic equations for curved composite beams. J Am Helicopter Soc 41(4):313–321 Hughes TJR .1987/ The finite element method – linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs, NJ, USA Leamy MJ .2007/ Bulk dynamic response modeling of carbon nanotubes using an intrinsic finite element formulation incorporating interatomic potentials. Int J Solids Struct 44(3–4):874–894 Leamy MJ .2010/ Intrinsic finite element modeling of nonlinear dynamic response in helical springs. In: Proceedings of the ASME 2010 international mechanical engineering congress and exposition, Paper No. IMECE2010-37434, Vancouver, British Columbia, Canada, 12–18 November 2010 Leamy MJ, Lee C-Y .2009/ Dynamic response of intrinsic continua for use in biological and molecular modeling: explicit finite element formulation. Int J Numer Methods Eng 80(9): 1171–1195 Lee J, Thompson D .2001/ Dynamic stiffness formulation, free vibration and wave motion of helical springs. J Sound Vibration 239(2):297–320 Lee J .2007/ Free vibration analysis of cylindrical helical springs by the pseudospectral method. J Sound Vibration 302(1–2):185–196 Nayfeh AH, Mook DT .1979/ Nonlinear oscillations. Wiley, New York Patil MJ, Hodges DH, Cesnik CES .1999/ Nonlinear aeroelasticity and flight dynamics of highaltitude long-endurance aircraft. In: Proceeding of the 40th structures, structural dynamics and material conference, AIAA Paper 99-1470, Saint Louis, Missouri, 12–15 April 1999, pp 2224–2232 Patil MJ, Hodges DH, Cesnik CES .2000/ Nonlinear aeroelastic analysis of complete aircraft in subsonic flow. J Aircraft 37(5):753–760 Peter JFH .1999/ Carbon nanotubes and related structures. Cambridge University Press, Cambridge, UK Shang X, Hodges DH, Peters DA .1999/ Aeroelastic stability of composite hingeless rotors in hover with finite-state unsteady aerodynamics. J Am Helicopter Soc 44(3):206–221 Yu WB, Hodges DH .2004/ Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. J Appl Mech Trans ASME 71(1):15–23 Zhang P, Jiang H, Huang Y, Geubelle PH, Hwang KC .2004/ An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. J Mech Phys Solids 52:977–998
Chapter 7
Equilibrium of a Submerged Body with Slack Mooring Brian C. Fabien
Abstract This chapter presents an approach to the modeling of a submerged rigid body that is supported by slack moorings. Our main goal is to determine the static equilibrium configuration of these systems. The submerged body and mooring lines are subject to various loads including: gravity, buoyancy, viscous drag due to a constant stream flow, applied forces, and applied torques. The elastic mooring lines in the system can only support tensile loads, and are arranged in a network so that some mooring lines may be completely slack. Here we use a Lagrangian mechanics framework to determine the equations of equilibrium of the system. It is shown that the equations of equilibrium can be solved via solving a sequence of nonlinear programming (NLP) problems. The resultant NLP problems are solved using a sequential quadratic programming technique that minimizes the exact L1 penalty function. The chapter presents some examples to illustrate the behavior of slack moored systems that are submerged in a steady stream.
7.1 Introduction There is considerable interest in the development of ocean renewable energy (Polayge 2009), in particular, hydrokinetic energy systems that extract power from flowing streams. These streams are due to river flows or changes in tidal currents. An important factor in determining the economic viability of these systems is the cost of the mooring system. At shallow water depths, less than 60 m, monopile foundations can be used to construct hydrokinetic energy systems that are economically competitive with other types of energy generation systems. For deep water installations, i.e., water depths greater than 60 m, compliant moorings are necessary to ensure that these hydrokinetic energy systems remain cost competitive B.C. Fabien () Department of Mechanical Engineering, University of Washington, Seattle, Washington, USA e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 7, © Springer Science+Business Media, LLC 2012
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with other forms of energy generation. To facilitate the development of deep water hydrokinetic energy systems, engineers require computational tools that can access the behavior of compliant mooring systems. The main goal of the study presented here is to determine a stable static equilibrium configuration of slack moored submerged bodies. These bodies represent kydrokinetic turbines that are moored in deep water. Some contributions of this chapter include: (1) the use of a Lagrangian mechanics framework to model the system; (2) the treatment of the mooring line slackness, and sea floor impenetrability as nonholonomic (inequality) constraints; and (3) considering a network of possibly redundant mooring lines (i.e., some mooring lines may be completely slack). Nikravesh and Srinisasan (1985) also uses a Lagrangian mechanics approach to find the static equilibrium of mechanical systems. However, the models developed in (Nikravesh and Srinisasan 1985) assume that the system is conservative, and all constraints are holonomic. In Schulz and Pellegrino (2000) conservative systems with inequality constraints are considered. In this chapter we consider systems that contain nonconservative forces, and are subject to inequality constraints. Section 7.2 gives a general description of the type of system that will be modeled. Using a Lagrangian mechanics framework the system model is developed in Sect. 7.3. The equations of equilibrium are presented in Sect. 7.4. This section also shows that equations of equilibrium can be solved via a sequence of optimization problems. Section 7.5 presents a robust and efficient method for solving the optimization problems developed in Sect. 7.4. Section 7.6 presents some examples to illustrates the equilibrium behavior of slack moored systems. The chapter concludes in Sect. 7.7 with a discussion of the results obtained, and the directions of future study.
7.2 System Description A schematic of the types of systems that will be modeled is shown in Fig. 7.1. This diagram shows a submerged body that is anchored to the sea floor via a network of mooring lines. The system is in a tidal stream that has a steady velocity profile. Here, it is assumed that the body is submerged far from the surface, and hence we neglect the influence of surface waves.
Fig. 7.1 Slack mooring system
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The body is rigid, with a center of mass and a center of buoyancy that may not be coincident. Moreover, the body can have different drag force coefficients about its principal axes. The submerged body is supported by NL mooring lines. These elastic mooring lines can only support tensile loads, and are subject to gravity and buoyancy forces, as well as viscous drag forces due to the stream velocity. We require that the mooring lines and the rigid body satisfy the sea floor impenetrability constraint. Specifically, all points on the mooring lines and the rigid body must remain on or above the sea floor. A unique feature of the model developed here is that we allow a network of mooring lines where some mooring lines may become completely slack in certain equilibrium configurations. This feature will allow the system designer to evaluate the efficacy of using a network of mooring lines to accommodate varying stream velocity directions. In such a system some mooring lines may be slack for one prevailing stream direction, but become taut when the stream direction changes. Thus, the system can be designed to passively control the equilibrium configuration of the system under varying stream flow conditions. It should be noted that (Raman-Nair and Baddour 2002) discusses a problem similar to the one being considered here. However, there are some noteworthy differences that can be highlighted. The modeling approach in Raman-Nair and Baddour (2002) is based on Kane’s equations, and although multiple mooring lines are modeled, a network of mooring lines (i.e., mooring lines attached to each other) is not considered. In addition, Raman-Nair and Baddour (2002) only considers an approximation of the sea floor impenetrability constraint.
7.3 System Model The system model is developed using a Lagrangian mechanics approach. Section 7.3.1 establishes the model coordinate systems, and defines the displacement variables used to determine the equilibrium configuration. Section 7.3.2 develops the kinematic constraint equations associated with the network of mooring lines. Section 7.3.3 develops expressions for the potential energy of the system, and the virtual work done by the nonconservative forces.
7.3.1 Kinematics 7.3.1.1 Rigid Body Kinematics All system displacements are measured relative to the fixed rectangular coordinate system X -Y -Z, with origin on the sea floor, and with the Z-axis directed vertically upward. Here, X -Y -Z represents an inertial reference frame. Such a system is illustrated in Fig. 7.2.
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Fig. 7.2 Coordinate system
Fig. 7.3 Coordinate transformation
The kinematics of the body will be described using the body-fixed rectangular coordinate system, x-y-z. The origin of the x-y-z system is located at the center of mass of the body, and is aligned with the principal axes of inertia. The x-y-z system can both translate and rotate with respect to the inertial coordinate system X -Y -Z. The position of the center of mass of the body is then defined by the vector cN D ŒcX ; cY ; cZ T . Here, cX , cY , and cZ represent the coordinates of the center of mass of the body with respect to the inertial coordinate system X -Y -Z. The orientation of the body with respect to the inertial coordinate system is defined using the 1-2-3 Euler angles (Fabien 2009). In particular, let ˛, ˇ, and represent three successive, finite rotations, about the x-axis, y-axis, and z-axis of intermediate frames. Then, the coordinate transformation matrix from the x-y-z body-fixed system to the X -Y -Z system is given by: 2 32 32 3 c s 0 cˇ 0 sˇ 1 0 0 (7.1) T .˛; ˇ; / D 4 s c 0 5 4 0 1 0 5 4 0 c˛ s˛ 5 ; 0 0 1 sˇ 0 cˇ 0 s˛ c˛ where s˛ D sin ˛, c˛ D cos ˛, etc. Hence, the six displacement variables cX ; cY ; cZ ; ˛; ˇ, and , determine the position and orientation of the rigid body. Now, suppose pN 0 is a point in the x-y-z body-fixed system with coordinates pN 0 D 0 Œpx ; py0 ; pz0 T . Then, the coordinates of pN 0 with respect to the inertial coordinate system is given by: pN D cN C T .˛; ˇ; /pN 0 ; (7.2) where pN D ŒpX ; pY ; pZ T . This vector equation is illustrated graphically in Fig. 7.3.
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Fig. 7.4 Mooring line
Fig. 7.5 Model node
7.3.2 Mooring Line Network 7.3.2.1 Mooring Line Kinematics The i th mooring line in the system is divided into Ni segments using Ni C 1 nodes pNi;0 , pNi;1 , : : :, pNNi (see Fig. 7.4). Each of these nodes represents a point in the X Y -Z system; namely, pNi;j D ŒpX;i;j ; pY;i;j ; pZ;i;j T , j D 0; 1; : : : ; Ni . Thus, there are 3.Ni C 1/ displacement variables associated with the i th mooring line. In order to construct a network of mooring lines our system model uses three additional types of nodes. These node types are described with the aid of Fig. 7.5. The node types defined here are: • Body-fixed nodes. These nodes are attached to the body, and are fixed points in the x-y-z system, e.g., bN10 and bN20 . The coordinates of the node bN10 in the x-y-z body-fixed system are 0 0 0 T bN10 D Œbx1 ; by1 ; bz1 : (7.3) Using (7.1) we see that the coordinates of the node bN10 in the inertial frame, X Y -Z, are (7.4) bN1 D cN C T .˛; ˇ; /bN10 : Similarly, the coordinates of the node bN20 in the inertial frame are bN2 D cN C T .˛; ˇ; /bN20 :
(7.5)
• Free nodes. The i th free node has coordinates rNi D ŒrX;i ; rY;i ; rZ;i T
(7.6)
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Fig. 7.6 Mooring line network
in the inertial frame. The NF free nodes are massless, and are used to establish interconnections between different mooring lines. • Sea floor nodes. The i th sea floor node has coordinates sNi D ŒsX;i ; sY;i ; sZ;i T
(7.7)
in the inertial frame. These nodes are fixed in the inertial frame, and are used to define locations where the mooring lines attach to the sea floor. Notice that body-fixed and sea floor nodes do not introduce any new displacement variables to the system. However, each free node adds three displacement variables to the system, i.e., rX;i ; rY;i , and rZ;i . Using these nodes it is easy to define the network of mooring lines that support the rigid body in the system. An example of a mooring line network is shown in Fig. 7.6. There, mooring line 1 connects the body-fixed node bN20 to the free node rN1 ; mooring line 2 connects the body-fixed node bN10 to the free node rN1 ; and mooring line 3 connects the free node rN1 to the sea floor node sN1 . Each of these mooring line definitions introduces two holonomic displacement constraints to the system model. One set of constraints establish the connection at the first node on the line, and the second set of constraints establish the connection at the last node on the line. In particular, as the first node on mooring line 1 is coincident with the body-fixed node bN20 we have the displacement constraints 1;0 D cN C T .˛; ˇ; /bN20 pN1;0 D 0:
(7.8)
The terms cN C T .˛; ˇ; /bN20 define the coordinates of the body-fixed node bN20 in the inertial frame. Mooring line 1 is divided into N1 segments, so the last node on the line is pN1;N1 . As node pN1;N1 is coincident with the free node rN1 we have the displacement constraints 1;N1 D pN1;N1 rN1 D 0:
(7.9)
Similarly, the displacement constraints associated with line 2 are 2;0 D cN C T .˛; ˇ; /bN10 pN2;0 D 0 2;N2 D pN2;N2 rN1 D 0:
(7.10) (7.11)
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The displacement constraints associated with line 3 are 3;0 D rN1 pN3;0 D 0
(7.12)
3;N3 D pN3;N3 sN1 D 0:
(7.13)
In general, for the i th mooring line, i;0 2 R3 defines the displacement constraints at the first node on the line, and i;Ni 2 R3 defines the displacement constraints at the last node on the line.
7.3.3 Kinetics This section describes the various forces and torques that act on the rigid body and mooring lines. A goal of this study is to determine a stable equilibrium configuration of the system. To accomplish this task it will be advantageous to write the work done by the conservative forces and torques in terms of a potential energy function. This, in fact, is the approach that will be used throughout this section. Here, we also note that the viscous drag forces are treated as nonconservative forces. This is because it is not always possible to write these forces in terms of a potential for arbitrary stream velocities. 7.3.3.1 Rigid Body Kinetics The rigid body is subjected to inertia forces, viscous drag forces, and applied forces and moments (see Fig. 7.7). Each of these loads will be considered in turn. Gravity and Buoyancy Here, it is assumed that the center of mass (c) N and the center of buoyancy (cNb0 ) need not be collocated. The virtual work done by the gravity force is mgıcZ , where m is the mass of the body, g the acceleration due to gravity, and ıcZ is the virtual displacement of the center of mass in the Z direction. The virtual work done by the buoyancy force is f Vb gıcbZ , where f is the fluid density, Vb the volume of the body, and ıcbZ is the virtual displacement of the center of buoyancy in the Z direction. Thus, the potential energy contribution due to gravity and buoyancy forces on the rigid body are Vgb D mgc Z f Vb gcbZ ; (7.14) where cbZ is the third component of the vector cNb D cN C T .˛; ˇ; /cNb0 .
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Fig. 7.7 Rigid body force/torques
Viscous Drag Forces Let dN 0 2 R3 denote the viscous drag force acting on the body, in the body-fixed frame. Using Morrison’s formula (Raman-Nair and Baddour 2002) these forces are given by: 2 3 Ax CDx jv0x jv0x 1 dN 0 D f 4 Ay CDy jv0y jv0y 5 ; (7.15) 2 0 0 Az CDz jvz jvz where Ax ; Ay , and Az are the projected surface area of the body in the body-fixed x, y, and z directions, respectively. The drag coefficients in the x, y, and z directions are CDx ; CDy , and CDz , respectively. The components of the stream velocity vector, in the vicinity of the body, are v0x ; v0y , and v0z . (These velocity components are given with respect to the body-fixed frame.) Thus, the drag force on the body in the inertial frame is written as (7.16) dN D T .˛; ˇ; /dN 0 : Also note that if vN 2 R3 is the stream velocity in the inertial frame then the velocity in the body-fixed frame is computed using vN 0 D T .˛; ˇ; /T vN .
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Fig. 7.8 Segment
Fig. 7.9 Extension
The virtual work done by this viscous drag force on the body is ıWd D dX ıcX C dY ıcY C dZ ıcZ ;
(7.17)
where dX ; dY , and dZ are the elements of the vector dN , and ıcX ; ıcY , and ıcZ are the virtual displacements of the center of mass c. N Applied Forces and Torques These applied forces represent the efforts due to actuators on the body. The work done by these constant forces and torques can be written in terms of the potential energy function Vf D fX cX fY cY fZ cZ ˛ ˛ ˇ ˇ ;
(7.18)
where fX ; fY , and fZ are the applied forces with respect to the inertial frame, and ˛ ; ˇ , and are the applied torques. Note that Œ˛ ; ˇ ; T D T .˛; ˇ; /T ŒX ; Y ; Z T , where X ; Y , and Z are the body torques, with respect to the inertial frame. 7.3.3.2 Mooring Line Kinetics Mooring Line Tension The work done by the tension in each segment of the mooring line can be written in terms of a potential energy function. Let Li;j denote the undeformed length of the j th segment of mooring line i (see Fig. 7.8). Let li;j denote the deformed length of the j th segment of mooring line i (see Fig. 7.9). Then, the potential energy stored in this segment is written as: Vi;j D
1 2 ki;j li;j ; 2
(7.19)
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Fig. 7.10 Unit vectors
where the stiffness of the segment is ki;j D Ai;j Ei =Li;j and the extension of the segment is li;j D li;j Li;j . In these expressions, Ai;j is the cross sectional area of the mooring line segment and Ei is the modulus of elasticity p of the mooring line. The deformed length of the line segment is given by li;j D kpNi;j pNi;j 1 k2 . Note that the mooring line can only support tensile loads. To account for this ‘slackness condition’ we will impose the inequality constraints li;j 0, for i D 1; 2; : : : ; NL , and j D 1; 2; : : : ; Ni . The mooring line model used here is appropriate for chain type lines. To model cable type mooring lines we will need to include bending and torsional stiffness elements at each node (Huang 1994). Gravity and Buoyancy The potential energy due to the gravity and buoyancy forces acting on the j th segment of mooring line i is written as: Vgbi;j D wi;j uZ;i;j ;
(7.20)
where wi;j D mi;j g C f Ai;j Li;j , mi;j is the mass of the segment, and uZ;i;j is the third component of the vector uN i;j D .pNi;j C pNi;j 1 /=2. This approach assumes the net gravity and buoyancy forces act at the center of the undeformed mooring line segment. This assumption will be valid if the extension of the line segment is small. Viscous Drag Forces Let vN i;j be the stream velocity in the vicinity of segment j on mooring line i . Depending on the orientation of the segment relative to the stream velocity, the fluid flows over different areas of the segment. In particular, part of the fluid can flow normal to the segment and part of the fluid can flow parallel to the segment. The fluid velocity can be written in terms of unit vectors that are parallel and normal to the line segment. Specifically, vN i;j D ai;j lOi;j C bi;j nO i;j ;
(7.21)
where lOi;j is the unit vector in the direction from pNi;j 1 to pN i;j , and nO i;j is a unit vector orthogonal to lOi;j (see Fig. 7.10). The coefficient ai;j is the speed of the fluid velocity parallel to the segment and is found by the equation ai;j D vN Ti;j lOi;j , where lOi;j D .pNi;j pNi;j 1 /=kpNi;j pNi;j 1 k.
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2 2 The coefficient bi;j can then be found using the fact that kNvi;j k2 D ai;j C bi;j . In O addition, the normal unit vector can be selected as nO i;j D .Nvi;j ai;j li;j /=bi;j . The viscous drag force on the line segment is then
1 1 dNi;j D f Al;i;j CDl;i;j jai;j jai;j lOi;j C f An;i;j CDn;i;j jbi;j jbi;j nO i;j ; 2 2
(7.22)
where Al;i;j and An;i;j are the line segment parallel and normal cross sectional areas, respectively. The drag coefficients in the parallel and normal directions are CDl;i;j and CDn;i;j , respectively. The virtual work done by the drag force dNi;j D ŒdX;i;j ; dY;i;j ; dZ;i;j T is ıWd;i;j D
1 1 dX;i;j .ıpX;i;j C ıpX;i;j 1 / C dY;i;j .ıpY;i;j C ıpY;i;j 1 / 2 2 1 (7.23) C dZ;i;j .ıpZ;i;j C ıpZ;i;j 1 /: 2
Here, ıpX;i;j ; ıpY;i;j , and ıpZ;i;j are the virtual displacements associated with node pNi;j , and ıpX;i;j 1 ; ıpY;i;j 1 , and ıpZ;i;j 1 are the virtual displacements associated with node pNi;j 1 . Hence, we distribute the drag force uniformly between nodes pNi;j 1 and pNi;j .
7.4 Equilibrium Equations 7.4.1 Generalized Displacements Using the kinematic and kinetic expressions developed earlier we define x 2 Rnx to be the vector of generalized displacements. Specifically, x D ŒcX ; cY ; cZ ; ˛; ˇ; ; rX;1 ; rY;1 ; rZ;1 ; : : : ; rX;NF ; rY;NF ; rZ;NF ; pX;1;0 ; pY;1;0 ; pZ;1;0 ; : : : ; pX;NL ;NNL ; pY;NL ;NNL ; pZ;NL ;NNL ; l1;1 ; l1;2 ; : : : ; l1;N1 ; : : : ; lNL ;NNL T The elements of x are; • The rigid body translations: cX ; cY ; cZ . • The rigid body rotations: ˛; ˇ; .
(7.24)
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• The free node displacements: rX;i ; rY;i ; rZ;i ; i D 1; 2; : : : ; NF . • The mooring line node displacements: pX;i;j ; pY;i;j ; pZ;i;j ; i D 1; 2; : : : ; NL ; j D 0; 1; 2; : : : ; Ni . • The mooring line segment lengths: li;j ; i D 1; 2; : : : ; NL ; j D 1; 2; : : : ; Ni . P L PNL Hence, x has dimension nx D 6 C 3NF C 3 N i D1 .Ni C 1/ C i D1 Ni .
7.4.2 Equality Constraints As noted in the model development, the displacement variables in x are not all independent. The equality constraints that relate the displacement variables can be itemized as follows. • The mooring line terminal constraints. There are six interconnection constraints associated with each mooring line. Specifically, the three equations defined by i;0 D 0, and the three equations defined by i;Ni D 0, i D 1; 2; : : : ; NL . (See Sect. 7.3.2). • The mooring line segment length constraints. The length of each line segment is defined by the equation li;j D
q kpNi;j pNi;j 1 k2
(7.25)
for i D 1; 2; : : : ; NL , j D 1; 2; : : : ; Ni . (See Sect. 7.3.3.2). These equality constraints can be put in a vector h.x/ 2 Rnh which has the form 2 6 6 6 6 6 6 6 6 6 6 6 6 h.x/ D 6 6 6 6 6 6 6 6 6 6 6 6 4
1;0 1;N1 2;0 2;N2 :: : NL ;0 p NL ;NNL l1;1 kpN1;1 pN1;0 k2 :: : p l1;N1 kpN1;N1 pN1;N1 1 k2 :: : q lNL ;NNL kpNNL ;NNL pNNL ;NNL 1 k2
Hence, h.x/ has dimension nh D 6NL C
PNL
i D1 .
3 7 7 7 7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 7 7 7 5
(7.26)
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7.4.3 Inequality Constraints The generalized displacements x must also satisfy some inequality constraints. In particular, • The line segment length constraints: li;j 0, i D 1; 2; : : : ; NL , j D 1; 2; : : : ; Ni (the line segments can not support compressive loads). • The sea floor impenetrability constraints: cZ 0 (the center of mass of the body must be on or above the sea floor). rZ;i 0, i D 1; 2; : : : ; NF (the free nodes must be on or above the sea floor). pZ;i;j 0, i D 1; 2; : : : ; NL , j D 0; 1; 2; : : : ; Ni (the mooring line nodes must be on or above the sea floor). These inequality constraints can be put in a vector g.x/ 2 Rng such that 2
l1;1 :: :
3
6 6 6 6 lN ;N L NL 6 6 cZ 6 6 6 rZ;1 g.x/ D 6 :: 6 6 : 6 6 rZ;NF 6 6 pZ;1;0 6 6 :: 4 : pZ;NL ;NNL In which case ng D 1 C NF C
PNL
i D1 .2Ni
7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 5
(7.27)
C 1/.
7.4.4 Equilibrium Conditions The total potential energy of the system is V .x/ D Vgb .x/ C Vf .x/ C
NL X Ni X
Vi;j .x/ C Vgb;i;j .x/:
(7.28)
i D1 j D1
This equation simply combines the potential energies derived in the previous section. (See (7.14), (7.18), (7.19), and (7.20.) The virtual work done by the conservative forces is thus (7.29) ıWc .x/ D ec .x/T ıx
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where ec .x/ D @V .x/=@x are the conservative generalized efforts (i.e., forces and torques). The virtual work done by the nonconservative forces is ıWnc .x/ D ıWd .x/ C
NL X Ni X
ıWd;i;j .x/ D enc .x/T ıx;
(7.30)
i D1 j D1
where enc .x/ are the nonconservative generalized efforts. This equation simply combines the virtual work expressions (7.17) and (7.23). Let x 2 Rnx be an equilibrium configuration for the system. In an equilibrium configuration the total virtual work (ıW .x/ D ıWc .x/CıWnc .x/) must vanish, and the constraints must be satisfied. Therefore, ıW .x / D ıWc .x/ C ıWnc .x/ D ec .x /T ıx C enc .x /T ıx D 0
(7.31)
and the constraints hi .x / D 0, i D 1; 2; : : : ; nh , and gi .x / 0, i D 1; 2; : : : ; ng . Moreover, this must hold for all admissible virtual displacements ıx. The virtual displacements are admissible if the first variation of h.x/ D 0 and g.x/ 0 are satisfied at x D x . That is, the admissible virtual displacements ıx satisfy ıh.x / D h.x / C rh.x /T ıx D 0
(7.32)
ıg.x / D g.x / C rg.x /T ıx 0:
(7.33)
Here rh.x/ D
[email protected]/=@x/ 2 Rnx nh is the Jacobian of the equality constraints, and rg.x/ D
[email protected]/=@x/ 2 Rnx ng is the Jacobian of the inequality constraints. By invoking the Theorem of Lagrange Multipliers (Lemma 5:3 in (Hestenes 1966)) we find the following necessary conditions for an equilibrium. If x is an equilibrium configuration then there are vectors 2 Rnh , 2 Rng , i 0, i D 1; 2; : : : ; ng , such that ec .x / enc .x / C rh.x / C rg.x / D 0
hi .x / D 0; i D 1; 2; : : : ; nh gi .x / 0;
i gi .x / D 0;
i D 1; 2; : : : ; ng :
(7.34) (7.35) (7.36)
This is a system of nonlinear equations that includes inequality constraints. The elements of the vector are the Lagrange multipliers associated with the equality constraints. The elements of the vector are the Lagrange multipliers associated with the inequality constraints. The multipliers must be nonnegative, and the last condition in (7.36) indicates that if i > 0 then gi .x / D 0, i D 1; 2; : : : ; ng .
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7.4.5 An Equivalent NLP Problem In the absence of nonconservative efforts the equilibrium conditions (7.36) can be found from the following nonlinear programming (NLP) problem min fV .x/ j hj .x/ D 0; j 2 E; gi .x/ 0; i 2 Ig:
x2Rnx
(7.37)
Here, E D f1; 2; : : : ; nh g and I D f1; 2; : : : ; ng g. The problem (7.37) seeks to find a x 2 Rnx that minimizes the potential energy function V .x/, subject to the equality constraints h.x/ D 0 and the inequality constraints g.x/ 0. In this case, (with enc .x/ D 0), the (7.36) represents the first-order necessary conditions for an extremum of problem (7.37). In fact, if x is a minimum for the nonlinear programming problem (7.37) then x is a stable equilibrium configuration of the conservative system (enc .x/ D 0). On the other hand, if x is a maximum, it also satisfies (7.36) but in this case it is an unstable equilibrium configuration. We will use a modified version of the NLP problem (7.37) to find solutions to the equilibrium conditions (7.36) when enc .x/ ¤ 0. To develop this strategy, let x .k/ denote the k-th approximate solution to the equilibrium conditions (7.36). Then, x .kC1/ is determined by solving the NLP problem min ff .x/ j hj .x/ D 0; j 2 E; gi .x/ 0; i 2 Ig;
x2Rnx
(7.38)
where f .x/ D V .x/ enc .x .k/ /T x. This problem finds a stable equilibrium of a system with the modified potential energy function f .x/ D V .x/enc .x .k/ /T x, and the same constraints as those in (7.36). Thus, in (7.38) the nonconservative efforts are treated as constant, and the elements of enc are determined using the previous solution. Starting with some initial estimate x .0/ we solve a sequence of problems (7.38), for k D 1; 2; : : :, until kx .kC1/ x .k/ k is sufficiently small. If x is a stable equilibrium for (7.36), and x .0/ is sufficiently close to x then it can be shown that limk!1 x .k/ D x . A robust and efficient method for solving the NLP problem (7.38) is given in Algorithm 1, the solution is developed in the next section.
7.5 Solution of the NLP Problem This section presents a sequential quadratic programming technique (SQP) for the solution of the NLP problem (7.38). The algorithm is based on the minimization of the L1 exact penalty function. Here, the search directions used to minimize the penalty function are determined by solving strictly convex quadratic programming (QP) problems, that are always feasible.
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Algorithm 1 Inputs: x .0/ , .0/ , .0/ , B .0/ D I , O D .0/ , O > 0, 0 < ˇ < 1=2, 0 < < 1, and a convergence tolerance t > 0. Output: x , , and kT .x ; ; /k t . for kD 0; 1; do if T x .k/ ; .k/ ; .k/ t , break. Solve QPsc x .k/ ; B .k/ O ˇ to find d .k/ ; w.k/ ; .kC1/ and .kC1/ ˚ ˇ ; O ; .k/ .k/ .k/ ˇ ˇ ; gi x ; j 2 E; i 2 I Set D max o; hj x if w.k/ .k/ > 0 then ˇ ˇ P ˇ .kC1/ ˇ P .kC1/ Set .kC1/ D j 2E ˇj ˇ C j 2I j else ˇ ˇ P ˇ .kC1/ ˇ P .kC1/ Set .kC1/ D max ; O C ˇ C j 2I j j 2E ˇj end if Set O D .kC1/ if x .k/ C d .k/ ; .kC1/ x .k/ 4 x .k/ ; .kC1/ ; d .k/ then x .kC1/ D x .k/ C d .k/ else if c x .k/ t and QPsc B .k/ ; d .k/ ; x .k/ is feasible then .k/ .k/ .k/ .k/ Solve QPsc B ; d ; x to find dN else dN .k/ D 0 end if if dN .k/ > d .k/ , set dN .k/ D 0 ˚ Compute the first number ˛ .k/ in the sequence 1; ˇ; ˇ 2 ; where 2 x .k/ C ˛ .k/ d .k/ C ˛ .k/ dN .k/ ; .kC1/ x .k/ ˛ .k/ 4 x .k/ ; .kC1/ ; d .k/ 2 Set x .kC1/ D x .k/ C ˛ .k/ d .k/ C ˛ .k/ dN .k/ end if end if Update B .kC1/ end for x D x .k/ , D .k/ , D .k/
It is well known that, for some problems, the use of exact penalty functions prevents the SQP method from achieving a superlinear rate of convergence. In particular, these algorithms are unable to accept full steps sizes in the vicinity of the solution. This phenomena is known as the “Maratos effect.” Here, we use second-order corrections to improve the global and local convergence behavior of the algorithm.
7.5.1 Definitions and Basic Assumptions Let B.x/ D fj j gj .x/ D 0g denote the set of active inequality constraints at some point x. We assume that NLP has the following properties;
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1. The functions f .x/, hj .x/, j 2 E, and gj .x/, j 2 I are at least twice continuously differentiable. 2. At the local minimum, x , the vectors rhj .x /, j 2 E and rgj .x /, j 2 B.x / are linearly independent, where rhj .x/ D @hj .x/=@x and rgj .x/ D @gj .x/=@x. Define the scalar Lagrangian function as L.x; ; / D f .x/ C T h.x/ C T g.x/, where, 2 Rnh and 2 Rng are Lagrange multipliers associated with the equality and inequality constraints, respectively. If the point x 2 Rnx is a local minimum of NLP then there exists vectors 2 Rnh and 2 Rng such that rL.x ; ; / D rf .x / C rh.x / C rg.x / D 0
(7.39)
hj .x / D 0; j 2 E
(7.40)
j gj .x /
D 0;
j
0; gj .x / 0; j 2 I;
(7.41)
where rL.x; ; / D @L.x; ; /=@x and rf .x/ D @f .x/=@x. The system of equations (7.41) are called the first-order necessary conditions, or the Karush-KuhnTucker (KKT) conditions. Moreover, a point x that satisfies (7.41) is called a KKT point. The Hessian of the Lagrangian is defined as H.x; ; / D @L2 .x; ; /=@x 2 2 nx nx R . For the class on NLP problems considered in this chapter we require that the local minimum, x , satisfy the second-order sufficient conditions (Bertsekas 1982). That is, at x the conditions in (7.41) hold, and j > 0; j 2 B.x /I d T H.x ; ; /d > 0
(7.42)
for all d ¤ 0, where DhTj .x /d D 0; j 2 E, Dgj .x /T d D 0; j 2 B.x /.
7.5.2 The L1 Exact Penalty Function A well known strategy for solving the problem NLP is to replace (7.38) with the unconstrained minimization problem min .x; / D f .x/ C maxf0; jhj .x/j; gi .x/g; j 2 E; i 2 I
x2Rnx
(7.43)
where > 0 is a suitably chosen scalar penalty function weight. The function .x; / is the L1 exact penalty function. Under P the conditions P stated above, (i.e., (7.41) and (7.42)), it can be shown that if > . j 2E jj j C j 2I j / then x is a local minimum of the function .x; /. (See Proposition 4:1 in Bertsekas (1982), or Theorem 14:5:1 in Conn et al. (2000).) Hence, the problem (7.43) is of great utility in finding solutions to NLP.
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We note that .x; / is nondifferentiable; however, given a vector d 2 Rnx and a scalar ˛ > 0, sufficiently small, it can be shown that .x C ˛d; / .x; / ˛ .x; ; d / C O.˛2 /;
(7.44)
where .x; ; d / D rf .x/T d C Œmaxf0; jhj .x/ C rhj .x/T d j gi .x/ C rgi .x/T d g maxf0; jhj .x/j; gi .x/g
(7.45)
j 2 E; i 2 I. Here, O.˛ 2 / implies that as ˛ ! 0 there is a constant 1 > M > 0 such that O.˛2 / ˛ 2 M . Thus, for a given point x, and penalty function weight > 0; if there is a d ¤ 0 such that .x; ; d / < 0, then there is an ˛ > 0 that will yield .x C ˛d; / < .x; /, in which case d is called a descent direction for the penalty function. Using such descent directions we can carry out the unconstrained minimization (7.43). In Fabien (2008) it is shown that a descent direction for the penalty function .x; / can be determined by solving the strictly convex quadratic programming problem QPsc.x; B; O ; / O W
min
d 2Rnx ;w2R
1 T 1 O d Bd C rf .x/T d C O w2 C w 2 2
(7.46)
subject to jhj .x/ C rhj .x/T d j w; j 2 E
(7.47)
gj .x/ C rgj .x/ d w; j 2 I
(7.48)
T
w 0;
(7.49)
where B 2 Rnx nx is a symmetric positive definite approximation to the Hessian of the Lagrangian, and O > 0 and O > 0 are suitably chosen scalars. A very important property of QPsc is that we can always find a feasible solution, even in cases where the linear constraints hj .x/ C Dhj .x/T d D 0; j 2 E and gj .x/ C Dgj .x/T d 0; j 2 I are inconsistent.
7.5.2.1 A Descent Direction Let .x/ D max.0; jhj .x/j; gi .x//, j 2 E, and i 2 I. Let d ¤ 0 be the solution to QPsc, with j , j 2 E, and i , i 2 I being the corresponding Lagrange multipliers. Then, d is a descent direction for the exact penalty function .x; /, if is selected according to the rule:
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( D
P P j j C j 2I j if .w .x// > 0 Pj 2E j P max.; O . j 2E jj j C j 2I j C // otherwise:
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(7.50)
This result is derived in Fabien (2008).
7.5.2.2 Second-order Correction One approach to avoiding the Maratos effect is to perform the approximate minimization of .x; / along an arc that attempts to maintain constraint feasibility (Bertsekas 1982; Panier and Tits 1991). Specifically, we find the vector dN 2 Rnx that solves the QP problem QPsoc.x; B; d / W min
dN 2Rnx
1 Œd C dN T BŒd C dN C Df .x/T Œd C dN 2
(7.51)
subject to Dhj .x/T dN C hj .x C d / D 0; j 2 E
(7.52)
Dgj .x/T dN C gj .x C d / 0; j 2 I:
(7.53)
Note that in QPsoc the constraints are evaluated at x C d , where d is the solution to QPsc. The increment dN is often called a second-order correction step, and can be viewed as the Newton step required to restore feasibility of the constraints starting from the point x C d . If the constraints are linearly independent at x then it can be shown that kdN k D O.kd k2 / (Bertsekas 1982; Panier and Tits 1991). With this second-order correction available, we determine the next iterate in the SQP algorithm by minimizing the penalty function along the arc x C ˛d C ˛2 dN . A particular implementation of this arc search is given below.
7.5.3 A Globally Convergent Algorithm The NLP solution technique described in Algorithm 1 is basically a descent search method, where the L1 exact penalty function is minimized using a backtracking line search. The iteration counter for the algorithm is k, and x .k/ is used to denote x at the k-th iteration. The inputs to Algorithm 1 are; x .0/ an estimate to the local minimum; .0/ , .0/ estimates of the Lagrange multipliers associated with the equality and inequality constraints, respectively; B .0/ a symmetric positive definite matrix; O D .0/ an estimate of the penalty function weight; > 0 a small constant used in adjusting the penalty function weight; O the weight used in QPsc; ˇ and constants used in the line search; and a convergence tolerance t .
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The output from the algorithm is a KKT point x , and the corresponding Lagrange multipliers and . In step 2 the algorithm determines if the first-order necessary conditions, (7.41), are satisfied to within the desired tolerance. This is done computing the norm of the vector 3 2 rL.x .k/ ; .k/ ; .k/ / 5; T .x .k/ ; .k/ ; .k/ / D 4 (7.54) h.x .k/ / .k/
g.x / where rL.x .k/ ; .k/ ; .k/ / D rf .x .k/ / C rh.x .k/ /.k/ C rg.x .k/ / .k/ , and D .k/ .k/ .k/ diag. 1 , 2 , : : :, ng /. Note that in addition to requiring that kT k t , we .k/ also need j 0, j 2 I for .x .k/ ; .k/ ; .k/ / to satisfy the first-order necessary conditions (7.41). If x .k/ is not a KKT point, the algorithm computes a search direction by solving QPsc(x .k/ ,B .k/ , , O ). O Steps 4 through 10 update the penalty function weight, .kC1/ , .k/ such that (1) d is a descent direction for .x .k/ ; .kC1/ /, and (2) the sequence f.kC1/g is monotone nondecreasing. Steps 11 13 indicate that the new iterate is taken as x .kC1/ D x .k/ C d .k/ if there is sufficient decrease in the penalty function. Otherwise, an attempt is made to compute a second-order correction. The second-order correction is taken as dN .k/ D 0 if one of the following conditions is satisfied 1. kc.x .k/ /k1 > t , where
c.x
.k/
h.x .k/ / /D gBC .x .k/ /
(7.55)
where B C D B C .x .k/ / D fj j gj .x .k/ / > 0g, and gBC .x .k/ / is a subvector of g.x .k/ / whose elements are determined by the index set B C . Note that if this condition holds there is no need to solve QPsoc. 2. QPsoc(x .k/ ; B .k/ ; d .k/ ) has inconsistent constraints. 3. kdN .k/ k > kd .k/ k. Once d .k/ and dN .k/ have been determined the algorithm performs an Armijo (Bertsekas 1982) type backtracking line (arc) search in step 20 to find the next iterate. The algorithm updates the positive definite matrix B .k/ using the well known BFGS formula with Powell’s modification Powell (1978), i.e., B .kC1/ D B .k/
B .k/ pp T B .k/ rr T C p T B .k/ p pT r
(7.56)
r D q C .1 /B .k/ p
(7.57)
p D x .kC1/ x .k/
(7.58)
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q D DL.x .kC1/ ; .kC1/ ; .kC1/ / DL.x .k/ ; .kC1/ ; .kC1/ / 8 1; if p T q 0:2p T B .k/ p; ˆ < D T .k/ B p ˆ : T0:8p ; otherwise: .k/ p B p pT q
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(7.59) (7.60)
This update formula guarantees that B .kC1/ is symmetric positive definite if B .k/ is symmetric positive definite. We note that the problems QPsc and QPsoc are both strictly convex. Here, we use the dual feasible, active set algorithm developed by Goldfarb and Idnani (1983) to solve these quadratic programming problems. Using the assumptions stated above is possible to establish that the algorithm has one of the following outcomes; A. The penalty parameter is increased an infinite number of times, i.e., .k/ ! 1 as k ! 1. B. The penalty parameter is increased a finite number of times and limk!1 .x .k/ ; .kC1/ / ! 1. C. The penalty parameter is increased a finite number of times and the sequence f.x .k/ , .k/ , .k/ /g converges to a point .x ; ; / that satisfies the first-order necessary conditions (7.41). Outcome (A) indicates that the sequence f.x .k/ ; .k/ ; .k/ /g approaches a point where the Lagrange multipliers are undefined, or the sequence diverges. Outcome (B) can occur if f .x .k/ / decreases faster than the penalty term, .x .k/ /. In practice this outcome can usually be avoided by selecting .0/ sufficiently large. Outcome (C ) is a desirable result as it indicates that x .k/ converges to a KKT point of the NLP.
7.6 Examples This section presents two examples to illustrate the modeling and solution methodology described above. The first example considers a network of mooring lines where one line is redundant. The second example determines the equilibrium position of a submerged buoy in a stream. In these examples we set the convergence tolerance for the NLP problem to be t D 104 .
7.6.1 Redundant Mooring Line Model A schematic of the system to be considered is shown in Fig. 7.11. This model consists of: (a) two sea floor nodes sN1 D Œ1; 0; 0T and sN2 D Œ1; 0; 0T ; (b) a free node rN1 with initial position Œ0; 0; 0:2T ; (c) a body-fixed node bN10 D Œ0; 0; 0;
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Fig. 7.11 Redundant line
Fig. 7.12 Redundant line equilibrium position
(d) mooring line 1 connect bN10 and rN1 ; (e) mooring line 2 connect rN1 and sN1 ; (f) mooring line 3 connect rN1 and sN2 ; (g) an applied force fZ D 50 N at the center of mass of the body. Here, we select the line stiffness as AE=L D 109 m-Pa for all the mooring lines. For each mooring line the submerged weight per unit length is 0:1 N/m. In this model the undeformed lengths of the lines are L1 D 1 m, L2 D 1 m, and L3 D 5 m. Each line is divided into ten segments, i.e., N1 D N2 D N3 D 10. This model is designed to illustrate the behavior of the system when there are redundant mooring lines, and the elastic deflection of the taut lines are negligible. Clearly, from the model description, mooring line 3 will not support any load, and thus will be completely slack. This can be observed in Fig. 7.12 which shows the equilibrium solution obtained for this problem. In this result mooring lines 1 and 2 are taut, and are directed vertically. Mooring line 3 is slack, with 1 m of the line hanging vertically downward, and the remaining 4 m on the sea floor. It should be noted that the arrangement of the mooring line on the sea floor is arbitrary so long as the length constraints (l3;j 0), and impenetrability constraints (pZ;3;j 0) are satisfied.
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Fig. 7.13 Case 1: No stream velocity
7.6.2 Submerged Buoy in a Stream In this model a submerged buoy is supported by three mooring lines, that connect bN10 D Œ0; 0; 0T the sea floor nodes, sN1 D Œ10; 0; 0T , sN2 D Œ5; 8:66; 0T , sN3 D Œ5; 8:66; 0T , respectively. Line 1 connects bN10 to sN1 , line 2 connects bN10 to sN2 , and line 3 connects bN10 to sN3 . The mooring lines have undeformed lengths L1 D L2 D L3 D 17:77 m. Each line is divided into 20 segments, and has stiffness AE=L D 109 m Pa. The submerged weight per unit length is 9:81 N/m. The mooring lines have diameter 0:035 m, and drag coefficients CDn D CDl D 1. The buoy has mass 0:001 kg, effective cross sectional area Ax D Ay D Az D 1 m2 , and drag coefficients CDx D CDy D CDz D 0:1. A vertical force fZ D 522:97 N is applied to the center of mass of the buoy. Here, we consider two equilibrium problems. In case (1) the stream velocity is zero, and in case (2) the stream velocity is defined as vN D Œ0; Z 1:4 ; 0T m/s. Thus, in case (2) the prevailing stream velocity is along the Y -axis and increases with elevation. The equilibrium solutions obtained for these problems are shown in Figs. 7.13 and 7.14. In case 1, the equilibrium position for the center of mass of the buoy is bN D Œ0; 0; 13:465T m. The numerical algorithm required 215 iterations to converge to this solution starting from an initial estimate bN D Œ0; 0; 25T , and taken as straight line segments connecting to the sea floor nodes. In this case all the nodes of the mooring lines are above the sea floor (except the last nodes on each line). The results obtained here match those obtained in (Raman 2002) for a similar problem.
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Fig. 7.14 Case 2: Stream velocity vN D Œ0; Z 1:4 ; 0T
In case 2, the equilibrium position for the center of mass of the buoy is bN D Œ0:625, 3:083, 12:222T m. From Fig. 7.14 it can be seen that the mooring lines undergo some significant deformation relative to case 1. In fact, a significant portion off line 2 is on the sea floor, and a somewhat smaller portion of line 1 is also on the sea floor. Line 3 makes no contact with the sea floor (except at the last node). This solution required 769 iterations of the numerical algorithm. Here, we use the same initial estimate as in case 1. There is no analytical solution for this problem. However, solving the problem a second time with twice the mesh density yields an insignificant change in the equilibrium position.
7.7 Summary This chapter presented an approach to the modeling of a submerged body that is supported by a network of elastic mooring lines. The mooring lines can only support tensile loads and may become slack in certain equilibrium configurations. This slackness condition introduces inequality constraints into the model definition. Additional inequality constraints arise from the requirement that the mooring lines and the rigid body cannot penetrate the sea floor. The mooring line model used here is a good approximation of chain type mooring lines that do not support bending or torsion loads. For cable type mooring lines, that provide some bending and torsional stiffness, the model must be updated accordingly. Future work will consider the dynamic behavior of this system. This would include the effects of time varying currents and the influence of waves. Additionally,
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we plan to investigate the feasibility of using active controls to regulate the position and orientation of the submerged body. This would be accomplished by using actuators to adjust the lengths of some or all of the mooring lines.
7.8 Key Symbols Al;i;j ; An;i;j Ax ; Ay ; Az bN 0 ; bNi i
cN CDl;i;j ; CDn;i;j CDx ; CDy ; CDz dNi;j dN ; dN 0 f; h; g Li;j ; li;j L m mi;j Ni NF NL pNi;j rNi sNi T .˛; ˇ; / V Vgb Vf Vi;j Vgbi;j Vb v0x ; v0y ; v0z vN i;j
Mooring line segment effective area Rigid body effective area The position of body fixed node i The position of the rigid body center of mass Mooring line segment drag coefficients Rigid body drag coefficients Drag force on the j -th segment of the i -th mooring line Rigid body drag force NLP cost function, equality constraints and inequality constraints The undeformed and deformed length of The j -th segment of the i -th mooring Lagrangian The mass of the rigid body The mass of the j -th segment of the i -th mooring The number of segments in mooring line i The number of free nodes The number of mooring lines The position of the j -th node on the i -th mooring The position of the i -th free node The position of sea floor node i Coordinate transformation matrix The total potential energy (P.E.) of the system P.E. of the rigid body due to gravity and buoyancy P.E. of the rigid body due to applied forces and torques P.E. of the j -th segment of the i -th mooring line due to stiffness P.E. of the j -th segment of the i -th mooring line due to gravity and buoyancy Volume of the rigid body Stream velocity at the rigid body center of mass Stream velocity at the j -th segment of the i -th mooring
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Greek ˛; ˇ; ıW ıWc ; ıWnc ıWd ; ıWd;i;j ; f i;0 ; i;Ni
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Rigid body Euler angles Total virtual work Virtual work of conservative and nonconservative forces Virtual work due to drag forces L1 penalty function Lagrange multipliers for the NLP Fluid density The displacement constraint at end nodes of line i
References Bertsekas DP (1982) Constrained optimization and lagrange multiplier methods. Academic Press, New York Conn AR, Gould NIM, Toint PL (2000) Trust-region methods, MPS-SIAM Series on Optimization, SIAM Fabien BC (2008) Parameter optimization using the L1 exact penalty function and strictly convex quadratic programming problems. Appl Math Comput 198:833–848 Fabien BC (2009) Analytical systems dynamics: modeling and simulation. Springer-Verlag, Berlin, Heidelberg, New York Goldfarb D, Idnani A (1983) A numerically stable dual method for solving strictly convex quadratic programs. Math Prog 27:1–33 Hestenes M (1966) Calculus of variations and optimal control theory. Wiley, New York Huang S (1994) Dynamic analysis of three-dimensional marine cables. Ocean Eng 21:587–605 Nikravesh P, Srinisasan M (1985) Generalized co-ordinate partitioning in static equilibrium analysis of large-scale mechanical systems. Int J Numer Methods Eng 21:451–464 Panier E, Tits A (1991) Avoiding the Maratos effect by means of nonmonotone line search I. General constrained problems. SIAM J Numer Anal 28:1183–1195 Polayge BL (2009) Hydrodynamic effects of kinetic power extraction by in-stream tidal turbines. Ph.D. Dissertation, University of Washington Powell M (1978) A fast algorithm for nonlinearly constrained optimization calculations. In: Watson GA (ed) Numerical analysis, Dundee 1977, Lecture notes in mathematics, No. 630, Springer, Berlin, Heidelberg, New York, pp. 144–157 Raman-Nair W, Baddour RE (2002) Three-dimensional coupled dynamics of a buoy and multiple mooring lines: formulation and algorithm. Quart J Mech Appl Math 55:179–207 Schulz M, Pellegrino S (2000) Equilibrium paths of mechanical systems with unilateral constraints I: theory. Proceedings: Mathematical, Physical and Engineering Sciences, vol 456, pp. 2223–2242
Chapter 8
Nonlinear Deployable Mesh Reflectors Design, Modeling and Analysis Hang Shi and Bingen Yang
Abstract A reflector is a structural device that receives and reflects electromagnetic signals. A reflector normally has a dish shape as working surface and is supported by another structure (commonly a truss) behind. Unlike reflectors on the ground, when a reflector is installed onto a satellite or a space shuttle and used in space, many crucial requirements must be considered, one of which requires that the reflector has to be deployable. Because the size of a space reflector is usually much larger than the spacecraft that carries it, the reflector must be first folded into a small volume on the ground that can be stored inside the spacecraft, and then be deployed into the space after the spacecraft has been launched onto the designated orbit. After the deployment is completed, the reflector will produce and automatically maintain a working surface (aperture) with tolerant surface errors. Due to this particular feature, such structural devices are called deployable reflectors. As one of many types of deployable reflectors, deployable mesh reflectors have broad space applications, and have brought continuously interest in academia and industry in the past. Deployable mesh reflectors have been used in several renowned projects, such as ETS VIII for satellite communication, MBSAT for global broadcasting, “NEXRAD in Space (NIS)” mission for remote sensing and climate forecasting and GEO-mobile satellites by Boeing for mobile communications (Thomson 2002; Natori et al. 1993; Meguro et al. 1999; Im et al. 2003). Deployable mesh reflectors are also envisioned for many other applications such as high data rate deep space communications, Earth and planetary radars, and RF astronomy observations. Figure 8.1 illustrates a structure design for deployable mesh reflectors that has been considered by NASA engineers and studied in our research. The reflector is
H. Shi () • B. Yang Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, USA e-mail:
[email protected];
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 8, © Springer Science+Business Media, LLC 2012
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Fig. 8.1 Configuration of a deployable mesh reflector
Fig. 8.2 3D truss model of mesh reflectors
supported by the flat truss on the boundary and the working surface is constructed by the mesh and the front net. The nodes of the front net and rear net are connected by tension ties, where the actuators are installed. Those actuators properly adjust the length of the tension ties, so as to generate and maintain the desired working surface during the deployment and the in-space mission. According to the structural configuration in Fig. 8.1, the mesh reflector can be modeled as a nonlinear truss structure (shown in Fig. 8.2), whose elements can only sustain axial tension stress. The structure is fixed on the boundary and the working surface is formed by the truss elements. The tension ties are connected to the nodes, which provide the vertical external loads because of the symmetry between the front net and the rear net in the configuration under the concern. There are two crucial factors in performance assessment of deployable mesh reflectors: the aperture size of the reflector (mostly in term of the diameter) and the root-mean-square (RMS) value of the surface error. According to the antenna theory,
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larger-sized reflectors are capable of transmitting greater amount of data with higher resolution, and the smaller surface RMS error implies broader frequency bandwidth of the transmitted signals. The characteristic ratio, which is defined as the ratio of the reflector diameter to surface RMS error, is one of the key parameters to evaluate the performance of the mesh surface. Obviously, to increase the characteristic ratio, we can either increase the diameter of the reflector or decrease the surface RMS error; both of which, however, will enlarge the number of mesh cells, and increase the complexity and difficulty in design and manufacturing of this kind of reflectors. Therefore, development of large-sized deployable reflectors with small surface RMS errors, although in urgent demand due to the stringent requirements on surface performance to serve signal with high accuracy, has been a challenge for years. Previous investigations (Hedgepeth 1982a,b) have shown the performance limitation due to thermoelastic strain and manufacturing errors of materials in passive structure. It has been suggested that active surface (shape) control becomes necessary to improve the surface performance of deployable space reflectors for space missions and other applications. In this chapter, we present the results from our research project on developing the active surface control (ASC) architecture by using nonlinear modeling and analysis techniques. The remaining of the chapter is arranged as follows: Sect. 8.1 specifically states the objectives of the research. Section 8.2 presents the theoretical formulation of problem modeling and analysis. Then the numerical results and discussions will be provided on a sampled deployable mesh reflector in Sect. 8.3. Finally, Sect. 8.4 addresses some remarks on the progress of development of ASC architecture and the future research direction, and then concludes the chapter.
8.1 Problem Statements To develop the ASC architecture, the following three problems will be addressed in the chapter: 1. For a given working profile of the mesh reflector, it is first to determine a proper initial (untensioned and undeformed) profile of the mesh reflector and the vertical external tension loads that are applied at the truss nodes during the deployment of the reflector. The criterion is that the shape of the surface reflector after deployment matches the desired shape with a minimum error. Also considered in the design/analysis process is a requirement that deployed stress or axial tension load within the mesh support structure fall within a specified range. 2. Applying the design results of the first problem such as the optimal original lengths of the members and vertical external forces at the nodes of structure, it is crucial to determine the actual deformation of the mesh reflector and validate that the actual working shape after deployment is close to the desired shape with a minimum error. This task requires the development of a nonlinear static model of the truss structure and a solving algorithm for this nonlinear problem.
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3. As long as the design and the static analysis are achieved, the elastic-thermal dynamic behavior of the mesh reflector needs to be investigated and the properties such as natural frequencies and mode shapes should be discussed. Therefore, a nonlinear dynamic model that describes the elastic-thermal properties of mesh reflectors will be developed. The proposed model shall be useful for future development of feedback control algorithms for surface error compensation for space deployable mesh reflectors.
8.2 Modeling and Analysis 8.2.1 Determination of Initial Profile By given a desired working profile or a deformed shape of the mesh reflector, the initial undeformed profile of a truss structure is determined by examining the force balance of the deformed truss with the desired surface, which will yield the tension forces of the truss members and required external vertical loads. Assume that the truss structure has n nodes and m members, and that only vertical external loads are applied to those nodes, as shown in Fig. 8.2. Consider node i of a truss structure. A typical truss member k that is connected at the node which has nodes i and j as its two ends. These nodes can be expressed by position vectors rEi D xi;d eE1 C yi;d eE2 C zi;d eE3
(8.1)
rEj D xj;d eE1 C yj;d eE2 C zj;d eE3 ;
(8.2)
where eE1 ; eE2 ; eE3 are the base vectors of a global coordinate system xyz, and the coordinates xi ; yi ; zi and xj;d ; yj;d ; zj;d are from a desired surface of the reflector. Then it is obvious that the deformed element is defined by the vector below lEk D rEj rEi D lx;k eE1 C ly;k eE2 C lz;k eE3
(8.3)
lx;k D xj;d xi;d ; ly;k D yj;d yi;d ; lz;k D zj;d zi;d
(8.4)
in which and the unit vector in the longitudinal direction of the deformed member is 1 ˇEk lEk D ˇx;k eE1 C ˇy;k eE2 C ˇz;k eE3 lk
(8.5)
ˇx;k D lx;k = lk ; ˇy;k D ly;k = lk ; ˇz;k D lz;k = lk
(8.6)
with
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while ˇx;k ; ˇy;k ; ˇz;k are the direction cosines, and lk is deformed length of the element, which is calculated by lk D
q 2 2 2 lx;k C ly;k C lz;k :
(8.7)
Vector Ek can be expressed as Ek D .eE1 eE2 eE3 /f gk ;
with
T f gk D ˇx;k ˇy;k ˇz;k :
(8.8)
Let an external load Pi be applied at node i in the negative z-direction. Force balance at node i , which involves the axial forces Tk for all those members that are connected to the node and Pi , yields the following equilibrium equation X f gk Tk D fBZ gPi ;
(8.9)
k
where fBz g D Œ0 0 1T :
(8.10)
Force balance at all the nodes of the structure gives the equilibrium equation ŒM fT g D ŒBfP g;
(8.11)
where fT g D ŒT1 : : : Tm T ; fP g D ŒP1 : : : Pn T ŒB D diag.fBz gfBz g : : : fBz g/ 2 R3nn
(8.12)
and matrix ŒM 2 R3nm is composed of the coefficients of f gk of all the elements. Here, fT g is a vector of member tension forces, and fP gk a vector of external vertical loads. The truss structure can be either statically indeterminate (m > 3n) or statically determinate (m D 3n). By the theory of linear algebraic equations, the solution of (8.9) can be written as ( fT g D
ŒV fˇg C fT gp
for m > 3n
fT gp
for m D 3n
;
(8.13)
where fT gp is a particular solution of (8.9), and ŒV D Œfvg1 : : : fvgm3n ; fˇg D Œˇ1 : : : ˇm3n T
(8.14)
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with fvgi being all null vectors of ŒM , and ˇi being arbitrary constants. By pseudo inverse, one particular solution is given by fT gP D ŒmT .ŒM ŒM T /1 ŒBfP g:
(8.15)
The particular solution given by (8.15) always exists because has a full rank. It follows from (8.13) and (8.15) that the member tension forces can be expressed as fT g D ŒC fX g;
(8.16)
where ( ŒC D
fX g D
ŒV ŒM T .ŒM ŒM T /1 ŒB T 1
ŒM .ŒM ŒM / ŒB ( T fˇgT fP gT for m > 3n T
fP g
for m > 3n for m D 3n
for m D 3n
:
(8.17)
Vectors fˇg and fP g, as shall be seen, can be obtained by an optimization scheme. As mentioned in the Problem Statement, selection of an initial profile for the truss structure must assure that the member tension forces fall in a specific range for proper operation of the reflector. This means that the tension forces determined by the approach given in (8.17) should satisfy the condition fT gmin < fT g < fT gmax;
(8.18)
fT gmax D h f1m g; fT gmin D 1 f1m g
(8.19)
where with l and h being the specified lower and upper bound, and flm g being an m-vector whose elements are all ones. Also, all the external loads must be downward for proper mounting of the truss structure (Fig. 8.2), which implies fP g < 0:
(8.20)
In design of a mesh reflector, it is desirable to have a minimized (narrowest) tension band h l . In this work, a quadratic programming formulation is introduced to implant the above-mentioned conditions. It is developed based on the equilibrium equation (8.17) of the entire structure. By (8.16), the constraint conditions (8.18), (8.19) and (8.20) together can be cast in the following inequality ŒAfX g < fbg with
3 3 2 ŒC fT gmax ŒA D 4 ŒC 5 ; fbg D 4 fT gmin 5 f0g ŒE 2
(8.21)
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( ŒE D
243
ŒŒ0 ŒI
for m > 3n
ŒI
for m D 3n
:
(8.22)
Define fT g D ŒLfT g; where
2 m1 6 ŒL D 4
m
:: :
m1
1 m
(8.23) 3 7 5:
(8.24)
m1 m
Under the condition (8.18), it can be shown that 1 .max min / 1 jfT g jj1 2 jjfT g jj2 ; 2
(8.25)
where 1 and 2 are positive constants, and max D max Tk
(8.26)
min D min Tk
(8.27)
1km
1km
Because max min h l
(8.28) 2 minimizing the tension band max min can be achieved by min 12 fT g 2 . This leads to the following quadratic programming problem 1 T fX g ŒH fX g min fX g 2 subject to ŒAfX g < fbg;
(8.29)
ŒH D .ŒLŒC /T .ŒLŒC /:
(8.30)
where by (8.16) and (8.24),
Obviously, (8.29) is the standard quadratic programming problems, which have been well studied. In these optimization problems, ŒH are always semi-positive definite and the constraints are convex. As such, global optimization can be achieved via many methods (Nocedal and Wright 2006). Once the member tension forces fT g are obtained by the proposed optimization procedure, the original (undeformed) length of each truss member can be determined by Lk D
EAk lk Tk C EAk
(8.31)
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Fig. 8.3 A truss element in undeformed and deformed configurations
where Lk is the undeformed length of member k, and lk is the length of the deformed member as given in (8.7). Here, without the loss of generality, a linear strain-stress relation for the truss members has been used and is the Young’s modulus of the kth member. As a summary, the above solution procedure takes the following three steps: • • • •
Step 1. Obtain fX g via solving the quadratic programming problem (8.29); Step 2. With the fX g, compute the external loads by (8.17); and Step 3. Determine the member tension forces fT g by (8.16); Step 4. Calculate the original length of every member Lk by (8.31).
8.2.2 Nonlinear Static Analysis 8.2.2.1 Formulation of Nonlinear Model As it is discussed in the Introduction, a mesh reflector can be abstracted as threedimensional truss structure, of which each element is modeled as bar and is only able to afford tension of elongation. The deploying process of the reflector is described by the deformation of the truss structure. Following the same assumptions and notations in previous subsection, the deformation process of this element is shown in Fig. 8.3. Here, is defined as the vector of node REi under undeformed configuration. Hence, the node positions of deformed element can be described as fYd gk D fXi;d Yi;d Zi;d Xj;k Yj;d Zj;d gT with k presenting the index number of the element.
(8.32)
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Fig. 8.4 Nodal displacements and force balance in local coordinate
In local coordinate of the element, the relationship between nodal displacement and forces is shown in Fig. 8.4. From the geometry, the elongation of the element is Lk D lk Lk
(8.33)
in which Lk is the original length of element. According to elasticity theory, the relation between stress and strain is k D Ek " k
(8.34)
with k D
Nk Ak
"k D
lk lk
(8.35)
while Nk is the axial force of the element and Ak is the area of cross-section. In general, Ek could be nonlinear and a function of strain as Ek D Ek ."k /:
(8.36)
Substituting (8.33), (8.35), and (8.36) into (8.34), we have Nk D Ek Ak
lk Lk : Lk
(8.37)
From the force balance of the element, the nodal force vector fqk g in local coordinates is
1 qi; k fqk g D Nk : D (8.38) 1 qj; k Hence, "p # .xj;d xi;d /2 C .yj;d yi;d /2 C .zj;d zi;d /2 1 Ek Ak 1 : fqk g D 1 Lk
(8.39) By applying standard coordinate transformation, the nodal forces of the element in the global coordinate xyz are obtained as fQk g D ŒTˇ;k T fqk g;
(8.40)
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where
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ˇx;k ˇy;k ˇz;k 0 0 0 ŒTˇ;k D 0 0 0 ˇx;k ˇy;k ˇz;k
(8.41)
fQk g D fqx;i qy;i qz;i qx;j qy;j qz;j gT :
(8.42)
It can be noticed that the right side of (8.39) has˚two independent variables: original
length and node positions in deformed shape ydef;k , and can be re-written as a nonlinear function of them, fQk g D ffk .Lk ; fyd gk /g ŒTˇ;k T fqk g:
(8.43)
Finally, force balance at all the nodes of the structure leads to the equilibrium equation in global coordinates, fQg D ff .fLorg g; fydef g/g
(8.44)
fQg D fqx;1 qy;1 qz;1 : : : qx;i qy;i qz;i : : : qx;n qy;n qz;n gT
(8.45)
in which
fLorg g D fL1 : : : Li : : : Lm gT
(8.46)
fydef g D fx1;d y1;d z1;d : : : xi;d yi;d zi;d : : : xn;d yn;d zn;d g : T
(8.47)
The nonlinear model of (8.44) has two advantages. It has considered the geometry and elasticity nonlinearity at the same time, which results in the generality of the model. Furthermore, the equation of system describes the relationship between original lengths of elements, deformed shape profile and external forces, without requiring any information of initial shape profile, where the structure is static under zero external loads. This may provide significant convenience when the shape of the reflector is geometrically complex and unknown. Let˚flini g be defined as the element length vector in initial profile of the structure, when Lorg D flini g, meaning each element is fully extended without any inner tension, the formulation of the equilibrium equation in this special case could be different from above and results in the following model of structure in (8.48), which is discussed in Yang et al. (2008). ŒKNL fDg C fUNL g D fQg
(8.48)
8.2.2.2 Algorithm of Solving the Nonlinear Model After the model of the reflector is derived, the solution of (8.44) needs to be obtained. In the case that the original length vector and deformed shape profile of the structure are ˚known,
the external force vector is trivial to calculate, by plugging in the value of Lorg and fydef g into the right-hand side of (8.44). However, when some known external loads are applied on the structure, how to obtain original
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lengths of elements or the deformed shape profile can be troublesome and therefore comes this algorithm of nonlinear solver. Depending on the objectives of the calculation, either original length vector or the deformed shape profile, it has two slightly different formulations on the problem.
˚ Case 1: fydef g is unknown while Lorg and fQg are given. Define the function to be the 2-norm of the (8.44), F D
1 jjff .fLorg g; fydef g/g fQgjj2 : 2
(8.49)
Then to solve (8.44) is to solve the following unconstraint optimization problem min F .fydef g/:
(8.50)
In this chapter, the framework of trust region method is adapted. The updating policy of trust region radius is suggested by Nocedal and Wright (2006) and the only unspecified aspects left in the framework are the cost function formulation and the solution computed for each step of iteration according to (Yang et al 2008). By using quadratic model, the sub-problem of the (8.51) in each step of iteration is to minimize the following cost function, 1 Fj .fydef gj / C fgy gj fpy g C fpy gT Œr 2 Fj .fydef gj /fpy g; 2
(8.51)
dFj .fydef gj / D ff .fydef g/gTj fQgT ŒJdef;y j dfydef gj
(8.52)
˚ where gy j is 1 3n a row vector as the gradient function and the term of 2 r Fj fydef gj is the Hessian matrix in iteration. The gradient function at each step can be obtained by fgy gj D
in which Jdef;y j is the Jacobian matrix of the structure at iteration defined by (8.53) ŒJdef;y j D
dff .fydef g/gj dfydef gj
(8.53)
and can be linearly assembled from the local Jacobian matrices that are derived from (8.54) at each element. dffk .fydef g/gj dfydef gj d.ŒTˇ;k Tj / 1 T 1 d.Ek;j Ak;j / D "k;j Ek;j Ak;j "k;j C ŒTˇ;k j 1 1 dfydef gj dfydef gj d"k;j 1 Ek;j Ak;j C ŒTˇ;k Tj : (8.54) 1 dfydef gj
ŒJdef;y;k j D
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To avoid calculating the second derivative of the objective function, a symmetric matrix is introduced in Levenberg–Marquardt method (Shultz et al. 1985) as the approximation of Hessian, (8.55) ŒBy j D ŒJdef;y Tj ŒJdef;y j Œr 2 Fj .fydef gj /: ˚ Since Fj fydef gj is constant when py varies, finally the cost function is 1 my .fpy g/ D fgy gj fpy g C fpy gT ŒBy j fpy g: 2
(8.56)
˚
Case 2: Lorg is unknown while fydef g and fQg are given.
˚
Comparing to (8.50), the optimization problem is changed to treat Lorg as optimizing variable, min F .fLorg g/:
(8.57)
Hence, by following the similar derivation of (8.51)–(8.56), the cost function can be obtained as 1 mL .fpL g/ D fgL gj fpL g C fpL gT ŒBL j fpL g; (8.58) 2 where dFk .fLorg gj / fgL gj D D ff .fLorg g/gTj fQgT ŒJdef;L j (8.59) dfLorg gj fBL gj D ŒJdef;L Tj ŒJdef;L j
(8.60)
dff .fLorg g/gj dfLorg gj
(8.61)
ŒJdef;L j D
and the Jacobian matrix is assembled from the local Jacobian matrices calculated from (8.62) ŒJdef;L;k j D
D
dffk .fLorg g/gj dfLorg gj d ŒTˇ;k Tj 1
Ek;j Ak;j "k;j C 1 dfLorg gj d"k;j T 1 CŒTˇ;k j : Ek;j Ak;j 1 dfLorg gj
ŒTˇ;k Tj
1 d.Ek;j / Ak;j "k;j 1 dfLorg gj (8.62)
Although the cost functions are different for different cases, the same approach is used to obtain the exact step solution. The suboptimization problem always has a global minimum. However, since ŒBj may not always be positive definite, the solution which minimizes (8.56) and (8.58) may not be unique. Based on the previous research More (1978), an exact global solution of the trust region problem
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for each step can be concluded into the following four cases and for simplicity of the expression, the subscripts of previous vectors and matrices will be dropped in the (8.63)–(8.66). Set 4 to be the trust region radius in each step, and to be undetermined scalars. The exact solution fp g for the optimization (8.56) and (8.58) is • When ŒB is positive definite and k ŒB1 fggT k 4, then fp g D ŒB1 fggT
(8.63)
• When ŒB is positive definite and k ŒB1 fggT k > 4, then fp g D .ŒB C ŒI /1 fggT
2 Œ0; C1/
(8.64)
• When ŒB is indefinite and fggfq1 g ¤ 0 where fq1 g is the eigenvector corresponding to the smallest eigenvalue 1 , or when ŒB is indefinite, fggfq1 g D 0 and k .ŒB C 1 ŒI /1 fggT k 4, fp g D .ŒB C ŒI /1 fggT
2 Œ1 ; C1/
(8.65)
• When ŒB is indefinite, and k .ŒB C 1 ŒI /1 fggT k < 4 fp g D .ŒB C 1 ŒI /1 fggT C
fq1 g jfq1 gj
(8.66)
while and satisfy the equation kfp gk D 4. Under the assumption that F is twice continuously differentiable and bounded, and ŒB is also bounded, the previous study (More 1978; Sorensen and More 1983) has implied that by calculating exact solutions of iterations, the algorithm of nonlinear solver will always converge to the first-order stationary point, where kfggk ˚ D 0. It should be noticed that the mapping from fQg to fydef g or from fQg to Lorg defined by (8.44) is not always a one-to-one mapping. In (8.56) and (8.58) when ŒB is positive definite, the subproblem always has only one solution and the (8.50) and (8.57) will have the unique global minimizer. However, when ŒB is semi-positive definite which sharps the problem to be nonconvex, the subproblem may have more than one solution and the algorithm only leads to local minima. This latter case is especially common while the norm of fQg is zero or small enough.
8.2.3 Nonlinear Dynamic Model To serve the purpose of developing active surface control algorithm, it is necessary to investigate the dynamic characteristics of deployable mesh reflectors. Therefore, a control-orientated dynamic model is first formulated below based on the principles
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of structural dynamics (Meirovitch 1997; Nayfeh and Pai 2004; Pai 2007; Yang 2005). Since the mesh reflector in consideration is viewed as a 3-D truss structure, based on previous assumption and formulation, the strain of the element can be obtained by substituting (8.33) into (8.35), p .xi;d xj;d /2 C .yi;d yj;d /2 C .zi;d zj;d /2 Lk "k .fyd gk / D : (8.67) Lk Since the temperature changes is one of the major factors which affect the surface accuracy of mesh reflectors, besides investigating the nonlinear dynamic vibrations of structure, the dynamic thermal distortion also needs to be considered. For a single truss element k, the thermal strains "therm;k under temperature changes is defined as "therm;k D ˛k Tk ;
(8.68)
where ˛k is the thermal coefficient of the material and 4Tk is the temperature variation of the member. Then the total strain of kth element of the structure is the summary of thermal strain "therm;k and elastic strain "elast;k by assuming no expansions due to other factors such as piezoelectric effects, etc.: "t;k D
lk Lk D "therm;k C "elast;k Lk
(8.69)
in which lk is the deformed length of the element and Lk is the original (undeformed) length. Because of the in-space working environments of the reflector, the gravity is the ignored and the potential energy all comes from the elastic energy of the material. The potential energy of the single element due to elastic deformation under thermal effects is then Z LZ " Vk D Ek ."elast;k /"elast;k A."t;k ; s/d"elast;k ds 0
Z D
0
L
Z
0 "t;k "therm;k
Ek ."/"A." C "therm:k;s /d"ds
(8.70)
0
with Ek ."elast;k / to be Young’s modulus and A ."t;k ; s/ as the cross-section area at axial location s. Hence, we have the variation of the elastic energy as Z ıVk D .Ek ."t;k "therm;k //."t;k "therm;k /
L
A."t;k ; s/ds 0
@"t;k ıfyd gk (8.71) @fyd gk
in which fyd gk is defined in (8.32) as the deformed nodal coordinates of the kth element. Define fyc gk is the deformed coordinates of center of mass shown in (8.72) fyc gk D fxc;k yc;k zc;k gT :
(8.72)
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The kinematic energy is T D T t C Tr ;
(8.73)
where Tt is the kinetic energy due to translation, and Tr due to rotation about the longitudinal axis of the element. Because the moment of inertia of the element about the longitudinal axis is small, Tr 0. The center of mass of the element is RL mk .s/sds sc;k D 0 D ˛c;k L; (8.74) Mk where Mk is the total mass of the element. Then the velocity of the center of mass is obtained as xP c;k D ˛c;k xP j;d C .l ˛c;k /xP i;d yPc;k D ˛c;k yPj;d C .l ˛c;k /yPi;d zPc;k D ˛c;k zPj;d C .l ˛c;k /Pzi;d
(8.75)
which in a matrix format is
with
fyPc gk D Œ˛k fyPd gk
(8.76)
3 1 ˛c;k 0 0 ˛c;k 0 0 Œ˛k D 4 0 1 ˛c;k 0 0 ˛c;k 0 5 : 0 0 1 ˛c;k 0 0 ˛c;k
(8.77)
2
Therefore, the kinematic energy of the element is Tk D Tt;k D
1 2 2 Mk .xP c;k C yPc;k C zP2c;k / 2
and the variation of the kinematic energy of kth element is @fyc gk T T fyRc gk : ıTk D ıfyd gk Mk @fyd gk
(8.78)
(8.79)
Under the assumption that no damping effects within structure, the virtual work of external nodal forces at the kth element is ıWnc;k D ıfyd gTk fFk g;
(8.80)
where fFk g is the external force vector. For the entire structure, we have ıV D
m X 1
ıVk D ıfydef gT
m X @"t;k T @fydef g 1
Z
.Ek ."t;k "therm;k // ."t;k "therm;k /
L
A."t;k ; s/ds 0
(8.81)
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ıT D
m X
ıTk D ıfydef gT
m X
1
Mk
1
ıWnc D
m X
@fyc gk @fydef g
T fyRc gk
ıWnc;k D ıfydef gT fQg;
(8.82)
(8.83)
1
where m is the number of elements of the truss, fydef g is the global coordinate vector of all nodes (after deformation), and fQg is the vector of the external forces applied at the nodes. By the extended Hamilton principle Z T2 .ıWnc ıV C ıT /dt D 0 (8.84) T1
1 n oT @fyc gk M f y R fQg k @fydef g c gk C Z T2 B C B m n 1 oT T P C B @" ıfydef g t;k B .Ek ."t;k "therm;k //."t;k "therm;k / C dt D 0: @fydef g T1 @ A R1 L 0 A."t;k ; s/ds 0
m P
(8.85) It follows that the nonlinear equations of motion of the deployable mesh reflector are in the matrix form
m X @fyc gk T Mk fyRc gk @fydef g 1
C
m X 1
Mk
@"t;k @fydef g
T
Z
L
.Ek ."t;k "therm;k //."t;k "therm;k /
A."t;k ; s/ds D fQg:
0
(8.86) From (8.75) & (8.76), it can be concluded that f¨yc gk is a linear function of fydef g, fyRc gk D Œk fyRdef g;
(8.87)
@fyd gk ; @fydef g
(8.88)
where Œk D Œ˛k
and it is a constant matrix. Thus, (8.86) can be rewritten as
Z L m X @"t;k T .Ek ."t;k "therm;k //."t;k "therm;k / A."t;k ; s/ds D fQg ŒM fyRdef gC @fydef g 0 1 (8.89) with ! m X ŒM D (8.90) Mk ŒTk Œk : 1
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8.2.4 Linearization To develop the feedback active shape controller on deployable mesh reflectors, it is very common to linearize the nonlinear model (8.89) at the nonlinear static equilibriums. The static equilibrium of the model f¯ydef g is the solution of (8.91) by setting f¨ydef g D 0 in (8.89),
Z L m X @"t;k T .Ek ."t;k "therm;k //."t;k "therm;k / A."t;k ; s/ds D fQg: (8.91) @fydef g 0 1 The nonlinear static equilibrium f¯ydef g cannot be solved analytically in general, but the nonlinear solving algorithm proposed in the Sect. 8.2.2.2 is able to numerically solve the (8.91) accurately and sufficiently. Since fydef g D fyNdef g C fydef g;
(8.92)
where f4ydef g is a small perturbation, according to the perturbation techniques, (8.89) can be linearized as ŒM fyRdef g C ŒKtherm fydef g D fQg;
(8.93)
where ŒKtherm D
X @"t;k T @ .Ek ."t;k "therm;k //."t;k "therm;k / @fydef g @fydef g ˇ Z L ˇ A."t;k ; s/ds ˇˇ : (8.94) 0
fydef g
Observing (8.89) and (8.93), it can be seen that the coefficients of the second-order derivative are a constant matrix, which reveals the fact that the thermal distortion does not affect the mass matrix. Therefore, the eigenvalue problem of the linearized model described by (8.93) is to solve 2 !i ŒM C ŒKtherm fgi D 0; (8.95) where !i is the i th natural frequency of the deployable mesh reflector, and fgi is the corresponding mode shape. The natural frequencies (eigenvalues) of the structure are the roots of the characteristic equation def .ŒKtherm !i2 ŒM / D 0:
(8.96)
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8.3 Numerical Results 8.3.1 Setup of the Mesh Reflector Example The proposed design algorithm, static model, and dynamic analysis are implemented on a sampled deployable mesh reflector shown in Fig. 8.5, which is modeled as a truss structure with 37 nodes and 90 members. The reflector is deployed into a spherical surface of diameter D D 30 m and height H D 11:18 m, which is deployed by properly determined external loads that are applied in vertical (negative z) direction. The truss elements of the structure have the nonlinear strain-stress relation with ( E0 " > 0 E."/ D : (8.97) 0 "0 For the numerical simulation, the longitudinal rigidity is E0 A D 1:1121 105 N.
8.3.2 Optimal Design Results Considering the desired working configuration in Fig. 8.5, and the inner tension requirement in (8.98), h D 9lbs;
l D 2lbs
(8.98)
z
in which h and l are the upper bound and lower bound of the member tension, respectively, the first step is to formulate the quadratic programming problem in (8.29) and solve the unknown vector fX g. Then in the second step, vertical external load vector fP gopt is obtained by (8.17). Since the boundary of the reflector is fixed, the external loads applied on the boundary nodes do not have any contribution to the deformation of the structure. Therefore, only the results of the external loads on the nodes which are outside the boundary of the reflector are plotted in Fig. 8.6.
0 -1 -2 -3 10 5 0
x
-5 -10
0 -4 -2 -8 -6 y
Fig. 8.5 An example of deployable mesh reflector
2
4
6
8
8 Nonlinear Deployable Mesh Reflectors Fig. 8.6 Optimal vertical external loads
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External load (lbs)
-1.5 -1.6 -1.7 -1.8 -1.9 -2 0
2
4
6
8
10
12
14
16
18
20
Node number
Fig. 8.7 Optimal original length of members Member length (m)
4.5
4
3.5
3 0
10
20
30
40
50
60
70
80
90
Member number
Finally, from (8.16) and (8.31), the undeformed length of each element is calculated and also plotted in the Fig. 8.7. Since the optimization in the first step is feasible and globally converges, the vertical external load vector fP gopt and undeformed length vector fLgopt are the optimal design results of the initial profile, which provide the minimum tension band of the structure.
8.3.3 Static Deformation and Verification Based on the optimal original length vector fLgopt previously obtained, the nonlinear static model of the mesh reflector in (8.44) can be formulated. In this case, there are two unknowns in the nonlinear equation: external loads fQg and actual deformed shape fydef g. By assigning different external loads, two different shapes can be obtained: the initial shape where no external loads are applied on the nodes of the reflector, or fQg D 0
(8.99)
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Fig. 8.8 Initial shape of the mesh reflector
o o o
z
10
o
0
o
5
o
-2
0
-4 -10
-5
-5 0
x
Fig. 8.9 Actual deformed shape of the mesh reflector
y
-10
5
10
z
0 -2 -4 10 5 0
y
-5 -10 -10
-5
0
5
10
x
and the actual deformed shape where the optimal vertical external load from the design results are applied on the structure, or fQg D ŒBfP gopt :
(8.100)
Therefore, using the nonlinear solving algorithm presented in Sect. 8.2.2 to solve the nonlinear static equation of (8.44) under two conditions (8.99) and (8.100), both the initial shape and the actual deformed shape of the reflector are calculated and plotted in Figs. 8.8 and 8.9, respectively. It should be mentioned that the initial shape and deformed shape are plotted separately since two graphs will cover each other due to small deformation of the reflector. It is obvious that both initial shape and deformed shape are six-fold symmetric to the origin of X Y plane. It also needs to be pointed out that in Fig. 8.8, the lines with circles indicate that the original lengths of these elements are larger than the lengths of lines themselves, and therefore those members will not afford any inner tension due to the nonlinear elasticity in (8.97) and are dangling in the initial shape. Once the actual deformed shape is ready, it is very important to verify the tension requirement defined in (8.101). l < Ti < h :
(8.101)
8 Nonlinear Deployable Mesh Reflectors Fig. 8.10 Element tensions at the deformed configuration
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Member Tension (lbs)
2.5 2.4 2.3 2.2 2.1 2 1.9
0
10
20
30
40
50
60
70
80
Member number
By checking every member tension shown in Fig. 8.10, the maximum inner tension is 2:3823 lbf and the minimum one is 2 lbf. The tension band of the entire mesh reflector is as narrow as 0:3823 lbf, which means the member tensions are not only satisfying the design requirement in (8.101) but also quite uniform. Furthermore, to verify the surface error between actual deformed shape and desired shape, the absolute surface error and relative surface error are defined and calculated in (8.102) and (8.103), "absolute D jjfydef gact fydef gdes jj D 8:2530 1015
(8.102)
jjfydef gact fydef gdes jj D 2:3410 1016 : jjfydef gdes jj
(8.103)
"relative D
8.3.4 Dynamic Analysis Considering the deployable mesh reflector setup in Sect. 8.3.1, it is further assumed that the Poisson’s effect of elements is ignored and each member has uniform temperature distribution. At first, the dynamic analysis is carried out without any temperature variation in the reflector. Then the nonlinear dynamic model in (8.89) can be derived and it is linearized at the deformed configuration under optimal external loads obtained from Sect. 8.3.3. Therefore, the eigenvalue problem in (8.95) is formulated and the natural frequencies of linearized model at the equilibrium configuration are calculated. The range of !i is from 13:6346 rad= s to 195:6108 rad= s, with the first four being !1 D !2 D 13:6346 rad=s
(8.104)
!3 D 14:7368 rad=s;
(8.105)
!4 D 16:0460 rad=s:
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Fig. 8.11 1st mode of the mesh reflector: side-wing mode (x direction)
Fig. 8.12 2nd mode of the mesh reflector: side-wing mode (y direction)
Fig. 8.13 3rd mode of the mesh reflector: up-and-down mode
Due to the axis symmetry of the reflector, repeated natural frequencies appear in pairs (for instance, the first two). The first four mode shapes of the reflector are plotted in Figs. 8.11–8.14 respectively, where the solid lines portray mode shapes and dotted lines represent the equilibrium configuration of the truss.
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Fig. 8.14 4th mode of the mesh reflector: breathing mode
By closely observing the first four modes, each of them has very specific behavior pattern. In the first mode, the lowest node maintains its coordinates while other nodes slightly slide to the positive x direction, which makes the mode similar to a side-wing movement. The second mode is also a side-wing mode, except that its nodes move to the negative y direction. Those two modes are the least energyconsuming modes because all the nodes except the lowest one only have major perturbations in one dimension. In the third mode, the nodes move up and down. Although the movement remains mainly in one dimension, it consumes more energy because of the larger perturbation magnitude which is caused by the movement of the lowest node. The nodes in the fourth mode perturb in three dimensions and the reflector expands like a breathing lung, which requires the most energy comparing all first four modes. Since the reflector is usually installed onto the spacecraft which orbits Earth in space, the amount of sunlight which shoots on the structure determines the surface temperature of the reflector. For one orbiting cycle, the range of the temperature variations could be from around 250 K to 350 K depending on the altitudes of the different orbits, while the period time of each cycle varies from one hour to 24 h. Then the speed of temperature variation is very slow and the temperature of the reflector changes gradually. Therefore, it is practicable and reasonable to treat the structural dynamic distortion under thermal variation as a quasi-static process. To investigate the impact of temperature variation on the dynamics of the mesh reflector, it is assumed that in certain period of time the reflector experiences the overall range of 100 K temperature variation. Since the temperature of structure changes gradually, without changing the external loads, the reflector statically distorted under the thermal effects and the nonlinear static equilibrium of the structure can be calculated every 5 K temperature variation from (8.91). Then multiple eigenvalue problems of (8.95) are formulated by linearizing the nonlinear dynamic model in (8.89) at the nonlinear equilibriums. Solving all the eigenvalue problems will provide many sets of natural frequencies of the reflector at different temperatures. Finally, we can plot all the distinct natural frequencies together versus temperature variation, as shown below. In Fig. 8.15, it indicates that the natural frequencies of the system will increase when the temperature of the structure increases and this increasing curve may not be very smooth.
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Fig. 8.15 Impact of thermal effects on natural frequencies
ω1 ω3 ω4 ω6 ω7 ω9 ω10
Natural Frequency (rad/s)
20 19 18 17 16 15 14 13 -100 -90
-80
-70
-60
-50
-40
-30
-20
-10
0
Nature Frequency of 1st Mode (rad/s)
Temperature Variation (K)
ΔP ΔP ΔP ΔP ΔP
13.82 13.8 13.78
= = = = =
10N 20N 30N 40N 50N
13.76 13.74 13.72 13.7 13.68 13.66 13.64 -50
-45
-40
-35
-30
-25
-20
-15
-10
Temperature Variation (K)
Fig. 8.16 Impact of external load on natural frequencies
Besides thermal effects, impact of the external loads on the natural frequencies is also interesting to observe. If we increase the magnitude of each vertical external load by 10 N each time and repeatedly obtain the natural frequencies at the nonlinear static equilibrium at every 10 K temperature variation, then Fig. 8.16 can be generated by only plot the first nature frequency at each nonlinear equilibrium. From the plot, it can be concluded that increasing the magnitude of external loads will result higher natural frequencies and the external loads may have larger impact on the dynamic properties than the temperature variation of the reflector does.
8.4 Final Remarks Figure 8.17 shows the R&D roadmap of Active Surface Control architecture. The research work presented in this chapter covers the optimal design, nonlinear solver, coupled thermoelastic nonlinear model, and dynamic analysis based on the
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Optimal design of initial profile
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Nonlinear solver (static analysis)
Linearized models
Implementation and validation
NL dynamic model
Thermo-elastic model
Active surface control
Fig. 8.17 R & D progress of ASC architecture
Fig. 8.18 3D plots of the designate shape for the 835-node model
linearization. More details and examples can be found in the authors’ paper (Yang et al. 2009; Shi et al. 2010; Shi 2011a,b). According to Fig. 8.17, the ongoing efforts of the research go into two directions: the fully development of ASC architecture and the large model validations. Based on the proposed methods, static and dynamic behaviors of the deployable mesh reflectors are accurately modeled and analyzed, which revealed important properties of the reflector structure itself. The methods and results will be used to design the active surface control algorithms via state feedback. Meanwhile, the presented methods shall be implemented on an 835-node model for a 30 m reflector (shown in Fig. 8.18) even before the control algorithm is ready to use. This numerical implementation will involve much more complex structure modeling and analysis, and will be very close to the actual engineering applications, which can prove the efficiency and strength of the methods. Once the active feedback control algorithm is developed, it will be assembled into the simulation and demonstrate the improvement on high surface performance of deployable mesh reflectors. Acknowledgements This work was a result of the authors’ previous projects partially sponsored by NASA’s Jet Propulsion Laboratory.
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8.5 Key Symbols Ak Ek F j k ŒKtherm lk Lk fLarg g m Mk ŒMmass n fp g fP g fQg fT g fyc gk fyd gk fydef g fyNdef g
Cross-section area of the member Young’s modulus of the member Cost function of optimization problem Index of the optimization iteration Index of the structure elements Stiffness matrix of the linearized dynamics Deformed length of the member Undeformed length of the member Original length vector of the structure Number of the structure members Total mass of the member Mass matrix of the linearized dynamics Number of the structure nodes Exact solution of the optimization in each iteration Vector of external vertical load Nodal force vector in global coordinates Vector of member tension forces Deformed coordinates of mass center of the element Deformed shape profile of the member Deformed shape profile of the structure Static equilibrium of deformed shape
Greek ˛k ˇx;k ˇy;k ˇz;k l h ıV ıWnc ıT 4 4Tk "k "elast;k "therm;k fgi !i
Thermal coefficient of the member X Component of direction cosine of the member Y Component of direction cosine of the member Z Component of direction cosine of the member Lower design bound of member tension Upper design bound of member tension Variation of the potential energy Variation of the non-conservative virtual work Variation of the kinematic energy Trust region radius in each optimization iteration Temperature variation at the element Strain of the member Elastic strain of the member Thermal strain of the member Eigenvector of linearized dynamic model Natural frequency of linearized dynamic model
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References Hedgepeth JM (1982) Accuracy potentials for large space antenna reflectors with passive structure. J Spacecraft Rockets 19(3):211–217 Hedgepeth JM (1982) Influence of fabrication tolerances on the surface accuracy of large antenna structures. AIAA J 20(5):680–686 Im E, Smith EA, Durden SL, Huang J, Rahmat-Samii Y, Lou M (2003) Instrument Concept of NEXRAD in space (NIS) – a geostationary radar for hurricane studies. Proceedings of internnational geoscience and remote sensing symposium, Toulouse, France Meguro A, Tsujihata A, Hamamoto N (1999) The 13 m aperture space antenna reflectors for engineering test satellite VIII, Antennas and Propagation Society International Symposium, IEEE, vol 3, pp 1520–1523 Meirovitch L (1997) Principles and techniques of vibrations. Prentice Hall, Englewood Cliffs, NJ, USA More JJ (1978) The Levenberg-Marquardt algorithm: implementation and theory. In: Watson G (ed) Numerical analysis. Springer-Verlag, Berlin, Heidelberg, New York, pp 105–116 Natori MC, Takano T, Inoue T, Noda T (1993) Design and development of a deployable mesh antenna for MUSES-B spacecraft. 34th AIAA /ASME /ASCE /AHS /ASC Structures, Structural Dynamics and Materials Conference, La Jolla, CA, 19–22 April 1993 Nayfeh AH, Pai PF (2004) Linear and nonlinear structural mechanics. Wiley, New York Nocedal J, Wright SJ (2006) Numerical optimization. Springer, Berlin, Heidelberg, New York Pai PF (2007) Highly flexible structures: modeling, computation, and experimentation. AIAA, New York Shultz GA, Schnabel RB, Byrd RH (1985) A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM J Numer Anal 22(1):47–67 Sorensen DC, More JJ (1983) Computing a trust region step. SIAM J Scientific Statist Comput 4(3):553–572 Shi H, Yang B, Thomson M, Fang H (2010) Nonlinear modeling and optimal initial profile solution for deployable mesh reflectors. 51th AIAA/ASME/ASCE /AHS/ASC structures, structural dynamics and materials conference, Orlando, FL Shi H, Yang B, Thomson M, Fang H (2011a) A nonlinear dynamic model for deployable mesh reflectors. 52th AIAA/ASME/ASCE /AHS/ASC structures, structural dynamics and materials conference, Denver, CO Shi H, Yang B, Thomson M, Fang H (2011b) Coupled elastic-thermal dynamics of deployable mesh reflectors. 52th AIAA/ASME/ASCE /AHS /ASC structures, structural dynamics and materials conference, Denver, CO Thomson M (2002) AstroMeshTM deployable reflectors for Ku and Ka band commercial satellites. 20th AIAA international communication satellite systems conference and ex-hibit, Montreal, Quebec Yang B (2005) Stress, strain, and structural dynamics. Elsevier, Amsterdam Yang B, Shi H, Thomson M, Fang H (2008) Nonlinear modeling and surface mounting optimization for extremely large deployable mesh antenna reflector. Proceedings of ASME 2008 international mechanical congress and explosion, Boston, MA Yang B, Shi H, Thomson M, Fang H (2009) Optimal mounting of deployable mesh reflectors via static modeling and quadratic programming. 50th AIAA/ASME/ASCE /AHS/ASC structures, structural dynamics and materials conference, Palm Spring, CA
Chapter 9
Nonlinearity in an Electromechanical Braking System Development of a Smart Caliper Reza Hoseinnezhad and Alireza Bab-Hadiashar
Abstract In electromechanical brakes, central controllers require accurate information about the clamp force between brake pad and disc as a function of pad displacement. This function is usually denoted as characteristic curve of the caliper. In a typical electromechanical braking system, clamp force measurements vary with actuator displacements in a hysteretic manner. Due to ageing, temperature and other environmental variations, the hysteretic characteristic curve of calliper varies with time. Therefore, automatic caliper calibration in real time is vital for high-performance braking action and vehicle safety. Due to memory and processing power limitations, the calibration technique should be memory efficient and of low computational complexity. This chapter investigates the hysteresis as a nonlinear effect in the electromechanical brakes, and describes a technique to parametrically model this effect. This technique is a simple and memory-efficient real-time calibration method in which a Maxwell-slip model is fitted to the data samples around each hysteresis cycle. Experimental results from the data recorded in various temperatures show that this technique results in clamp force measurements with less than 0:7% error over the range of clamp force variations. It is also shown that by using these measurements, the characteristic curve can be accurately calibrated in real time.
9.1 Introduction Electromechanical brakes (EMBs) replace traditional mechanical and hydraulic linkages with electric actuators and computer control systems. Many vehicle manufacturers are engaged in research or collaborative development programs focussed in
R. Hoseinnezhad () • A. Bab-Hadiashar School of Aerospace, Mechanical, and Manufacturing Engineering, RMIT University, Melbourne, Victoria, Australia e-mail:
[email protected];
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 9, © Springer Science+Business Media, LLC 2012
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Fig. 9.1 General structure of a brake-by-wire system
this area. The requirements for reliability and safety in automotive braking systems are just as stringent as the ones they are replacing. On the other hand, EMB components should be cost-wise competitive with conventional technologies. Therefore, there are always limited scope for hardware redundancy, which means the EMBs should be inherently safe and reliable (Hoseinnezhad and Bab-Hadiashar 2006b; Jonner et al. 1996; Park and Heo 2004; Rooks et al. 2003; Schenk et al. 1995). To achieve accurate and stable control of the vehicles equipped by EMBs, we need to attain accurate models for various nonlinear static and dynamic processes involved in such braking systems. Using our earlier works (Hoseinnezhad 2006; Hoseinnezhad and Bab-Hadiashar 2005, 2006a; Hoseinnezhad et al. 2008; Saric et al. 2008), we present one of the most important nonlinearities existing in electric calipers. Before focusing on the nonlinear phenomena, we present a quick review of general structure and components of common EMB system designs. A general diagram of a brake-by-wire system is shown in Fig. 9.1 (Hoseinnezhad and Bab-Hadiashar 2009). A typical system includes four principal components: a central brake controller (also called central control unit or CCU), a sensing/measurement apparatus for driver’s brake demand, brake units in four corners of the vehicle, and a communication network. More specifically, any EMB system includes electromechanical brake calipers (e-calipers) with embedded brake torque controllers at each vehicle corner, wheel speed and vehicle motion
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Fig. 9.2 Block diagram of the e-caliper control system in a typical EMB design
sensors, a central controller unit, and a human–machine interface, such as an instrumented brake pedal, all communicating via a fault-tolerant communications network (Hoseinnezhad 2006; Hoseinnezhad and Bab-Hadiashar 2005, 2006a; Park and Heo 2004; Saric et al. 2006). Figure 9.2 shows a block diagram of the e-caliper control system which includes the connections between the central control unit (CCU) and one of the e-calipers in the EMB system. The CCU is the central vehicle dynamic control unit and generates the brake commands required to perform high-level braking tasks such as anti-skid braking (ABS), vehicle stability control (VSC), or traction control (TC). These commands are sent to the four e-calipers via a communication network. These brake commands are in the form of the desired clamp force to be generated by each e-caliper. Such commands are usually generated in CCU by processing the clamp force and displacement measurements in the calipers and the wheel speed measurements. A local controller in each caliper regulates the electric current that drives the brake actuator (Hoseinnezhad et al. 2008; Saric et al. 2008). A schematic diagram of an e-caliper developed by PBR Australia (Hoseinnezhad et al. 2008; Saric et al. 2008) is shown in Fig. 9.3. In this design, the rotational displacement of the brake actuator is converted to transitional displacement of a ball-screw through a planetary gear-set. This causes the load sleeve to push the brake pad toward the brake disc and generate the clamp force.
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Fig. 9.3 The diagram of the electromechanical brake caliper.1 stator field winding, 2 brake pads, 3 ball-screw, 4 planetary gear-set, 5 thrust bearing, 6 clamp force sensor location, 7 resolver location, 8 permanent rotor magnet location, 9 load distribution plate, 10 nut, 11 caliper bridge
Measurement of the position and speed of the actuator and the resulting clamp force in the caliper are safety-critical tasks in an EMB system, because those measurements are the key variables used by the CCU to generate the brake commands. The position and speed of the actuators are measured by resolvers (as shown in Fig. 9.2). The techniques required for obtaining accurate and robust estimates of position and speed as well as automatic calibration of the resolver have already been developed (Hoseinnezhad 2006; Hoseinnezhad and Bab-Hadiashar 2005, 2006a). Both techniques are efficient in terms of their accuracy, memory usage and computational complexity, and can be implemented in real time. Thus, we assume that reliable and accurate measurements for the position and speed of the actuators in Figs. 9.2 and 9.3 would be available. The CCU and the caliper local controllers require accurate knowledge of the characteristic curve of the calipers, i.e. the profile of the clamp force versus pad displacement. In addition, accurate characteristic curve of e-calipers can be utilised to calculate clamp force estimates from the displacements measured by resolvers. Fusion of the direct force measurements given by the sensors with their alternative estimates from the actuator position can result in more reliable clamp force measurements, which enhances the performance of the brake control and system safety (Saric et al. 2006; Schwarz et al. 1998, 1999) In the above design, rotational displacement of the caliper actuator is transformed to pad movement via the planetary gear-set and ball-screw and based on the kinematics of this transformation, actuator and pad displacements are almost proportional. Therefore, without noteworthy loss of accuracy, we study the profile of clamp force measurement against the actuator position in place of the pad movement.
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9.2 The Hysteresis Effect Characteristic curve varies with ageing and environmental conditions (e.g., temperature and humidity) and should be accurately calibrated in real time. Such a calibration can only be performed by utilising recent samples of the measured forces and their corresponding displacements. Therefore, the accuracy of e-caliper calibration significantly depends on the accuracy of clamp force measurements. Since the stress in the load sleeve is almost uniformly distributed over the cross section, for the purpose of measurement of the very large loads experienced in a brake caliper, the load sleeve can be considered as an axially loaded spring element. Thus, load cells are used to measure the clamp force. Figure 9.4 shows the arrangement of the strain gauges in a typical load cell on the load sleeve and their electrical circuitry. In a load cell, adjacent strain gauges are connected in opposite bridge arms to remove bending strains resulted from off-axis or transverse components of forces. Moreover, the strain gauges are oriented transversely to desensitise the bridge output to temperature changes Window (1992). As shown in Hoseinnezhad et al. (2008) when the load cell measurements are plotted against the actuator displacement, the result involves hysteresis around the true characteristic curve of the caliper similar to the one shown in Fig. 9.5 (Hoseinnezhad et al. 2008). This hysteresis is caused by the presliding component of the friction that exists between the key (placed to prevent the load sleeve from rotating with the ball-screw) and its keyway inside the housing of the load sleeve – see Fig. 9.6 (Hoseinnezhad et al. 2008). To obtain accurate clamp force measurements for caliper calibration and control, the hysteretic friction component of the force measurements provided by the load cells should be detected. The friction modelling and estimation procedure applied by the system should be simple and efficient in terms of its required memory and computational power. This is because of the limits of the processing power of central control unit (CCU in Fig. 9.2) and the available memory in the system. In addition, besides the identification of the hysteresis part and removing it from the measured clamp force (to obtain a reliable measurement) and calibration of the characteristic curve, there are many other complicated processing jobs to be performed by the CCU using its available memory and computational power. Some examples are vehicle state estimation, ABS, VSC and TC.
Fig. 9.4 Arrangement of the strain gauges in each load cell around the load sleeve and Strain gauge electrical circuitry
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Fig. 9.5 A hysteretic behaviour is observed when the clamp force measurements are plotted versus actuator displacement Fig. 9.6 A picture of the ball-screw and the load sleeve with the key on it
In the next sections, a memory and computational efficient technique is presented for identification of the hysteresis part of the measurements, extracting reliable estimates of clamp forces, and real-time calibration of characteristic curve using the estimated clamp forces. In Sect. 9.3 different hysteresis models are reviewed and an appropriate model (with the desired accuracy and computational complexity) is selected to be applied for extracting the true clamp force from the hysteretic measurements. A memory and computationally efficient technique for automatic tuning of the characteristic curve of the caliper is explained in Sect. 9.4. Experimental results are presented in Sects. 9.5 and 9.6 concludes the chapter.
9.3 Maxwell-Slip Hysteresis Modelling As it is observed in the example shown in Fig. 9.5 (Hoseinnezhad et al. 2008), the clamp force measured by the strain gauges (hereafter, the set of the six load cells are called internal clamp force sensor or internal sensor for short) is comprised of two
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parts: The real clamp force that causes the axial load on the load sleeve and changes the resistance of the strain gauges, and the sliding friction force between the key on the load sleeve and the keyway in its housing. To obtain an accurate measurement of the clamp force, the friction part should be estimated and removed. Since this friction force causes the hysteresis phenomenon, a hysteresis model can be used to estimate the friction force. Hsu and Ngo (1997) have introduced a Hammerstein configuration, which includes a Hammerstein-based dynamic model for hysteresis. This model includes a nonlinear static block followed by a linear dynamic block, and is applied to model the rate-dependent and temperature-dependent hysteresis phenomenon. Li and Tan (2004) have applied a neural network to estimate the influence of hysteresis for adaptive control of a nonlinear system which involves hysteresis. The above two approaches are too complicated to be implemented in real-time brakeby-wire systems. Oh et al. (2005) have analysed the Dahl, LuGre, and Maxwell-slip friction models as Duhem hysteresis models, classifying each model as either a generalised or a semilinear Duhem model. Here, we follow their unified treatment of Duhembased friction models to investigate the friction-induced hysteresis in e-calipers. Through some experiments (explained in Sect. 9.5), it was shown in Hoseinnezhad et al. (2008) that by using the Maxwell-slip model to capture the hysteresis part of the load cell measurements, clamp force estimates with sufficient accuracy can be obtained. The minimum accuracy of clamp force measurement – required by high level braking functions in CCU – is around 99% accuracy over the range of clamp force variations 0–40 kN. In Maxwell-slip model, the hysteretic slippage is modelled as M zeromass elasto-slip elements connected in parallel as shown in Figs. 9.7 and 9.8 (Hoseinnezhad et al. 2008). Each element of this model is characterised by its stiffness Ki , position xi .t/, spring deflection ıi .t/ D x.t/ xi .t/ and maximum spring deflection i (before the element i starts to slip). The input displacement x.t/ is common to all elements. The total hysteretic friction force ff is given by the summation of all operators’ spring forces: ff .t/ D
M X
Ki i ı i .t/;
(9.1)
i D1
where ı i .t/ is given by: ( ı i .t/ D
ıi .t / i
if jıi .t/j < i (stick) sgn .ıi .t// if jıi .t/j i (slip)
and the dynamics of the element position xi .t/ is as follows: if jıi .t/j < i (stick) xi .t/ xi .t C 1/ D x.t/ sgn .ıi .t// i if jıi .t/j i (slip)
(9.2)
(9.3)
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Fig. 9.7 Modelling the hysteretic slippage as a parallel combination of saturating elasto-slip elements
Fig. 9.8 Stiffness curves of the spring elements in the model of Fig. 9.7
Figure 9.9 shows the friction force ff given by the above model plotted versus the displacement x. WePobserve that the PMmodel results in a hysteretic friction force varying within ΠM K ; C i D1 i i i D1 Ki i , and the more the number of elements M are, the smoother the curve is. The centre of the hysteresis curve along the x axis depends on the location of the actuator at the beginning of the hysteresis cycle (Hoseinnezhad et al. 2008). To model the hysteresis in the e-caliper application, it is assumed that the M points are evenly distributed over the maximum sticking displacement max : i D
i max I i D 1; : : : ; M: M
(9.4)
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Fig. 9.9 Modelling the hysteretic slippage as a parallel combination of saturating elasto-slip elements
The maximum sticking displacement can be calculated from some previously recorded force-displacement measurements: It is the displacement at which the maximum deflection from the characteristic curve is observed. This deflection is caused by the pre-sliding friction and for larger displacements, the measured clamp force follows a path almost parallel with the characteristic curve.
9.4 Characteristic Curve Calibration: Algorithm In our previous works (Saric et al. 2006), we have shown that the characteristic curve of the e-caliper can be modelled by a third-order polynomial. The measurements given by the internal clamp force sensor are also modelled as the sum of two parts: The clamping component and the friction component. The clamping component is a third-order polynomial function of the displacement (given by the characteristic curve model) and the friction component is given by Maxwell-slip model as described in Sect. 9.3. fOc .t/ D Ax.t/3 C Bx.t/2 C C x.t/ C D C
M X i D1
Ki i ı i .t/:
(9.5)
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The parameters f1 ; : : : ; M g are assumed to be known a priori by using (9.4). Characteristic curve parameters A, B, C and D, and the linear coefficients of the hysteresis model fK1 ; : : : ; KM g are determined by fitting an ensemble of data to the above model simply by using least-squares technique. An appropriate ensemble of data samples f.x.i /; fc .i //g should contain points around a full hysteresis cycle on the force-displacement plot. As the data samples are sequentially received by the CCU through the communication network, the CCU should be able to detect the starting point of a hysteresis cycle so as to start recording data samples till the end of the cycle (before the next starting point). Before a new hysteresis cycle begins, the position coordinate x is decreasing with time (data points are moving on the lower half of the current hysteresis cycle as shown in Fig. 9.9) and as soon as a new cycle starts, the x coordinate begins to increase with time. Therefore, it was suggested in Hoseinnezhad et al. (2008) that a starting point is detected as below: x.i / > x.i n0 / ) x.i / is a starting point. (9.6) x.i n0 / < x.i 2n0 / The parameter n0 prevents the incorrect detection of a starting point due to the fluctuations of displacement signals (caused by noise). However, there is a tradeoff, as a large n0 would result in late detection of the starting point of hysteresis cycles. An appropriate value for n0 depends on the application specific factors such as sampling rate, signal-to-noise ratio of the position measurements, and nominal and maximum actuator speeds, and can be determined by trial and error through experiments. Because of memory limitations, recording of all data samples in a hysteresis cycle is not feasible. Assume that only L samples of f.x.i /; fc .i //g pairs out of the data samples in each hysteresis cycle can be recorded for determining the parameters of the model (9.5). An iterative method is needed for optimal selection and recording of the data samples as they are consecutively received by the central controller via the communication network. The recorded data samples should be distributed around the hysteresis cycle as evenly as possible. For this purpose, the mutual distances between the recorded samples in the force-displacement plane need to be maximised. The following iterative method to perform this maximisation was used in Hoseinnezhad et al. (2008) while choosing the data samples for model fitting. When a starting point is detected, the next first L data samples f.x.i /; fc .i //g are recorded and denoted by fxtemp .1/; : : : ; xtemp .L/g
(9.7)
fftemp .1/; : : : ; ftemp .L/g:
(9.8)
and
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Upon receiving the next data sample by the CCU, that sample is also recorded and denoted by xtemp .L C 1/; ftemp .L C 1/. A normalised geometric distance between two consecutive data samples is defined as: s ftemp .k C 1/ ftemp .k/ 2 xtemp .k C 1/ xtemp .k/ 2 C ; (9.9) dk D Xmax Xmin Fmax Fmin where 1 k L and Fmax and Fmin are the upper and lower bounds of clamp force variations, and Xmax and Xmin are similar quantities for displacement and in practice, they can be determined off-line. Let dj be the smallest distance among the L mutual distances, which can be easily recorded and updated iteratively as new samples arrive. If j D L, then the new sample (the .L C 1/-th sample) is too close to its previous sample and is not recorded. If j < L, then the normalised geometric distance between the j -th and .j C1/-th samples is the smallest distance. Therefore, the last L j data samples are left-shifted in the memory and the new sample is stored as the L-th location: First left-shift: ftemp .j C 1/ ! ftemp .j / xtemp .j C 1/ ! xtemp .j / Second left-shift: ftemp .j C 2/ ! ftemp .j C 1/ xtemp .j C 2/ ! xtemp .j C 1/
:
:: : L-th left-shift: ftemp .L C 1/ ! ftemp .L/ xtemp .L C 1/ ! xtemp .L/ The above scheme is repeated until the cycle finishes and the starting point of a new cycle is detected. As it is shown in Fig. 9.5, a full cycle may include hysteretic variations around a small part of the whole characteristic curve of the caliper. Therefore, the use of least squares for fitting the model (9.5) to the data recorded from a single cycle will only locally enhance the characteristic curve. To resolve this issue, considering memory limitations, it was suggested in Hoseinnezhad et al. (2008) to select N C 1 points (N is assumed to predetermined based on the available memory space) with their x coordinates evenly distributed over the whole range of variations of displacements ŒXmin ; Xmax , and those are called principal fitting points or PF points for short. The displacement coordinates of PF points are given by: fXmin ; Xmin C ı; Xmin C 2ı; : : : ; Xmax g with ı D .Xmax Xmin /=N . The force coordinates of PF points are calculated using a third-order polynomial model with the last updated values of A; B; C and D as its parameters. When a local hysteresis cycle finishes and the next one starts, a new set of parameters, say A0 ; B 0 ; C 0 and D 0 are estimated. For each of the PF points whose displacement coordinate is between the minimum and
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Fig. 9.10 An example of characteristic curve parameter updating upon the termination of a local hysteresis cycle and start of the next cycle
maximum range of the local hysteresis cycle, the force coordinate is replaced with the measure given by the new parameters A0 ; B 0 ; C 0 and D 0 . Then a third order polynomial is fitted to the PF points and the parameters A; B; C and D are updated (see Fig. 9.10). A detailed flowchart of the complete algorithm for real-time calibration of caliper characteristic curve is shown in Fig. 9.11. There is an Initialisation block in which an initial set of model (and other required) parameters of the proposed technique are inputted, the first hysteresis cycle is detected and the locations of hysteresis elements fx0 .1/; : : : ; x0 .M /g are initialised. Then in the Iterative Parameter Updating block, the next hysteresis cycles are detected and the characteristic curve parameters are updated iteratively, as next data samples become available to the CCU.
9.5 Experimental Results The performance of the proposed real-time calibration technique was examined through a series of experiments conducted using the e-caliper of the EMB system developed at PBR Australia (Hoseinnezhad and Bab-Hadiashar 2005; Hoseinnezhad et al. 2008; Saric et al. 2006). The e-caliper was placed in an environmental chamber, which provided a controlled temperature and humidity. Figure 9.12 shows a picture of the experimental setup. The PC is running Vector CANape under Windows XP. It also controls the e-Caliper and records the position, temperature and clamp force measurements provided by caliper sensors. CANape also controls the 42V power supply (a Delta Elektronika SM70-45D power supply for the e-caliper actuator and brake-by-wire circuitry) via a standard National Instruments DAQ break out box connected to a PCI-MIO16E4 PCI card installed in the PC. An external force sensor has been also used to measure and record the true clamp force between the brake pad and brake disc.
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Fig. 9.11 A flowchart for the proposed automatic calibration technique using the hysteresis model (9.5)
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Fig. 9.12 A picture of the experimental setup
In each experiment, the measurements provided by the internal and external force sensors and the position sensor (resolver) were recorded in a specific temperature. In a total number of fourteen experiments, the temperature varied between 28ı C and 54ı C (with 2ı C increments). In each experiment, the e-caliper was commanded by CANape to follow a number of consecutive sinusoidal displacements with increasing amplitudes, as shown in Fig. 9.13. The force-displacement plot shown in Fig. 9.5 includes the data recordings at 40ı C. To show the variations of characteristic curve with temperature, in Fig. 9.14, the true clamp forces are plotted against actuator displacements in three different temperatures. By using the proposed technique to estimate the friction force and correct the hysteretic part of the clamp force measurements provided by the internal sensors, the true clamp force is estimated from the internal measurements, and used to calibrate the characteristic curve. Figure 9.15 shows the true clamp forces (given by external measurements) and the estimates from the internal measurements, plotted versus time. It was observed that the force estimates obtained from the internal measurements by the proposed technique closely follow the external clamp force measurements. To quantify the accuracy of clamp force estimates, the error of the force estimates is calculated with respect to their true values and plotted versus time as shown in Fig. 9.16. It was observed in Hoseinnezhad et al. (2008) that the clamp force measurement error does not exceed 0:27 kN. According to the technical specifications of the EMB design
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Fig. 9.13 Position of the actuator during the experiment
Fig. 9.14 Variation of characteristic curve
developed at PBR Australia (Hoseinnezhad 2006; Hoseinnezhad and Bab-Hadiashar 2005; Hoseinnezhad et al. 2008; Saric et al. 2006), the high level braking modules require clamp force measurements with a maximum error of 1% over the range 040 kN. This error limit was devised by EMB design experts through multiple tests of the EMB prototype in various road conditions and braking scenarios. A maximum error of 0:27 kN achieved in those experiments (Hoseinnezhad et al. 2008) was equivalent to 0:7% error over the range 0–40 kN and therefore, the presented method is suitable for clamp force measurement and caliper calibration in an EMB system.
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Fig. 9.15 Clamp force measurements by the external and internal sensors during the experiment: The internal sensor measurement has been corrected by calculating and removing its hysteresis part
Fig. 9.16 Error of the force estimates with respect to their true values
9.6 Conclusions This chapter introduced a hysteretic nonlinearity existing in the relationship between force and position measurements. A real-time calibration technique for EMB calipers, based on parametrically modelling the hysteresis and adaptively estimating the model parameters, was presented. The proposed method is computationally inexpensive and memory efficient and can be easily implemented into an electromechanical braking system. In this method, upon the starting of each hysteresis
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cycle, a clamp force model is fitted to the data samples recorded from the previous hysteresis cycle. The clamp force model includes a Maxwell-slip model for the hysteresis caused by friction. As a result, a set of model parameters estimates for the characteristic curve are obtained. Then this model is applied to update the characteristic curve over the whole range of force-displacement variations. In a series of experiments, a brake-by-wire caliper was controlled to follow a sinusoidal displacement pattern in different temperatures and the displacement data and force sensor readings were recorded, along with the true clamp force measured by an external force sensor. The proposed technique was applied to extract the true clamp force from the hysteretic internal force sensor readings and to update the characteristic curve in real-time. The results showed a clamp force measurement error of less than 0:7% over the range of 0–40 kN, and using these measurements, the characteristic curve was automatically calibrated in real time with desirable accuracy.
9.7 Key Symbols A; B; C and D ABS CCU EMB Fmin and Fmax Ki L M n0 xi .t/ Xmin and Xmax fx.i /; fc .i /g fxtemp .i /; ftemp .i /gL i D1 Greek ıi .t/ ıNi .t/ i max
Characteristic curve parameters Antilock (or Antiskid) Braking System Central Control Unit Electromechanical Brake Upper and lower bounds of clamp force The stiffness of the i -th elasto-slip elements The number of data samples to estimate the hysteresis model parameters The number of elasto-slip elements Half the number of past data smaples used for deteciton of a starting point The position of the i -th elasto-slip elements Upper and lower bounds of displacement An ensemble of data points around a full hysteresis cycle on the force-displacement plot Data samples used for batch training to estimate the hysteresis model parameters
The spring deflection of the i -th elasto-slip elements The normalised spring deflection of the i -th elasto-slip elements The maximum spring deflection of the i -th elasto-slip elements The maximum sticking displacement
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References Hoseinnezhad R (2006) Position sensing in by-wire brake callipers using resolvers. IEEE Trans Vehicular Tech 55(3):924–932 Hoseinnezhad R, Bab-Hadiashar A (2005) Missing data compensation for safety-critical components in a drive-by-wire system. IEEE Trans Vehicular Tech 54(4):1304–1311 Hoseinnezhad R, Bab-Hadiashar A (2006) Calibration of resolver sensors in electro-mechanical braking systems: a modified recursive weighted least squares approach. IEEE Trans Indust Elect 54(2):1052–1060 Hoseinnezhad R, Bab-Hadiashar A (2006) Fusion of redundant information in brake-by-wire systems, using a fuzzy voter. J. Adv Inform Fusion (JAIF) 1(1):35–45 Hoseinnezhad R, Bab-Hadiashar A, Rocco T (2008) Real-time clamp force measurement in electro-mechanical brake calipers. IEEE Trans Vehicular Technol 57(2):770–777 Hoseinnezhad R, Bab-Hadiashar A (2009) Recent patents on measurement and estimation in brakeby-wire technology. Recent Patents Elect Eng 2(1):54–64 Hsu JT, Ngo KDT (1997) A hammerstein-based dynamic model for hysteresis phenomenon. IEEE Trans Power Electron 12(3):406–413 Jonner W, Winner H, Dreilich L, Schunck E (1996) Electrohydraulic brake system – the first approach to brake-by-wire technology. SAE Technical Paper 960991 Li C, Tan Y (2004) Adaptive control based on neural estimation for systems with unknown hysteresis. Proceedings of IEEE conference on control applications, Taipei, Taiwan, pp 1509–1514 Oh J, Padthe AK, Bernstein DS, Rizos DD, Fassois SD (2005) Duhem models for hysteresis in sliding and presliding friction. Proceedings of the 44th IEEE conference on decision and control, and the European control conference 2005, Seville, Spain, pp 8132–8137 Park K, Heo SJ (2004) A study on the brake-by-wire system using hardware-in-the-loop simulation. Int J Vehicle Des 36(1):38–49 Rooks O, Armbruster M, Buchli S, Sulzmann A, Spiegelberg G, Kiencke U (2003) Redundancy management for drive-by-wire computer systems. Lecture notes in computer sciences 2788 – The 22nd international conference on computer safety, reliability and security, Edinburgh, Scotland, pp 249–262 Saric S, Bab-Hadiashar A, Hoseinnezhad R (2006) A sensor fusion approach to estimate clamp force in brake-by-wire systems. Proceedings of IEEE 63rd Vehicular Technical Conference (VTC’2006), Melbourne, Australia Saric S, Bab-Hadiashar A, Hoseinnezhad R (2008) Clamp-force estimation for a brake-by-wire system: a sensor-fusion approach. IEEE Trans Vehicular Technol 57(2):778–786 Schenk DE, Wells RL, Miller JE (1995) Intelligent braking for current and future vehicles. SAE Technical Paper 950762 Schwarz R, Isermann R, Bohm J, Nell J, Rieth P (1998) Modelling and control of an electromechanical disk brake. SAE Technical Paper 980600 Schwarz R, Isermann R, Bohm J, Nell J, Rieth P (1999) Clamping force estimation for a brakebywire actuator. SAE Technical Paper 990482 Window AL (1992) Strain gauge technology, 2nd edn. Springer, New York
Chapter 10
Nonlinear Dynamics of Incompressible Flow Complexity in Two-Dimensional Flows Around Circular Cylinder Jiazhong Zhang
Abstract Fluid flow around a bluff body is common in engineering applications, and the examples include flows past offshore structures, heat exchangers, power transmission lines etc. In particular, the flow around circular cylinders is encountered frequently in engineering and has been investigated theoretically, experimentally, and numerically over the past decades. When the Reynolds number Re exceeds a critical value, the boundary layer will separate from the body and form an unsteady flow pattern. At a certain Re number, the flow pattern will become unstable, and a symmetry-breaking will occur. The ensuing lift, drag and flow-induced vibration will be changed dramatically, at a certain degree. In engineering, the unsteady flow pattern around the body have great influence on the aerodynamics performance, for example, the enhancement of lift of airfoil or blades in turbo-machinery, reduction of the drag, suppression of flow-induced vibration, reinforcement of heat transfer in heat-exchangers, etc. In a sense, the flow around circular cylinders is a classical problem, since there exists a very rich variety of nonlinear phenomena in such infinite-dimensional or continuous dynamic system, despite the simple geometries involved. The vortex is the fundamental point in the aerodynamics performance, and hence flow control has attracted a great deal of interest in the past. In this chapter, nonlinear dynamics, including bifurcation and stability, are applied to the analysis of the complex nonlinear phenomena in the flows around a circular cylinder. In particular, the pattern formation, flow separation, stability and bifurcation of the flow, and the drag reduction by perturbation etc. are studied numerically in detail, and some fundamental understandings of the complex phenomena are gained. Additionally, the numerical approaches implemented in the analysis are also presented.
J. Zhang () School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Peoples Republic of China e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 10, © Springer Science+Business Media, LLC 2012
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10.1 Governing Equations Figure 10.1 shows the sketch of streamline of flow around a circular cylinder. Considering xi D .x; y/T 2 t R2 at time t 2 .0; T / as the spatial domain, the governing equation for two-dimensional unsteady incompressible flow can be derived as @ui C uj ui;j im;m D 0 (10.1) @t ui;i D 0:
(10.2)
where and ui are the density and velocity of the fluid, respectively, and ij the stress tensor, which satisfies ij D pI ij C 2"ij ;
(10.3)
1 .ui;j C uj;i /; 2
(10.4)
with the rate-of-strain tensor "ij "ij D
where p and are the pressure and the dynamic viscosity, respectively, and Iij the identity tensor. Let L be the characteristic scale, uL be the characteristic velocity, and dimensionless variables are defined as x ; L p p D 2 ; u L
x D
y ; L u u D ; uL
y D
uL t ; L v v D : uL t D
(10.5)
For the sake of simplicity, the “” is dropped here, and the governing equations become
Fig. 10.1 Schematic diagram of flow around circular cylinder
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@ui D0 @xi @ui @p 1 @2 ui @ui D C C uj @t @xj @xi Re @xi @xj Re D
uL L :
(10.6)
The boundary conditions in this problem are listed as the following, • The boundary on which the velocity needs to be specified is denoted by .t /g t D @t ui D .ui /g
on .t /g t D @t :
(10.7)
• The boundary on which the tractions are prescribed is denoted by .t /h t . These are the conditions assumed to be imposed on the remaining part of the boundaries of the domain, nj ij D .i /h
on .t /h t D @t :
(10.8)
As for the initial condition, the initial velocity field should be specified with divergence-free condition in the domain t , ui .xi ; 0/ D u0i .xi /
(10.9)
10.2 Numerical Methods 10.2.1 CBS Scheme Solving original Navier–Stokes equations using Standard Finite Element Method, the numerical solution will oscillate and become unstable due to the effect of nonlinear convective terms, as Reynolds number is higher. In order to overcome such difficulty, new coordinates xi0 D xi0 .xi ; t/ along the characteristics are introduced (Zienkiewicz and Codina 1995; Zienkiewicz et al. 1995). After the coordinate transformation, the convective terms will be removed, and the resulting equations are the simple diffusion equations, which can be efficiently approached by the Standard Finite Element Method. On the characteristics, it satisfies dxi0 D dxi ci dt D 0, where ci is convective velocity, namely, ci D ui , that is, @x 0 @xi0 D ci ; i D 1: @t @xi
(10.10)
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Fig. 10.2 Schematic diagram of characteristic
ϕ (xk)
tn+1
t
Characteristics xk
ϕ (xk – ck)
tn
δ k = ckΔt
According to the coordinates transformation governed by (10.10), (10.6) can be rewritten in the following form. @ui D0 @xi0
(10.11)
1 @2 u i @p @ui : D 0 C @t @xi Re @xj0 @xj0
(10.12)
Since the continuity equation remains the same as the original one after the transformation, only momentum equation is considered in this Section. The discretization of (10.12) along the characteristic gives (see Fig. 10.2) " # ˇ ˇ ˇ ˇ 1 nC1 ˇ nˇ ui ˇ ui ˇ t xk .xk ık / nC1 ˇ n ˇ ˇ ˇ @p 1 @2 ui @p 1 @2 u i ˇ ˇ D C C .1 / C ; ˇ @xi Re @xj @xj @xi Re @xj @xj ˇ.xk ık / xk (10.13) where is the relax factor, 2 .0; 1/, xk ık , xk is the spatial coordinates along the characteristic at the nth and n C 1th time step n ˇ or level, respectively. ˇ 2 ˇ n nˇ Let ık D c 4t, D 0 and u and @p C 1 @ ui can be ˇ k
i .xk ık /
@xi
Re @xj @xj
.xk ık /
expanded by Taylor series at xk . Then (10.13) becomes @ @p n 1 @2 uni uinC1 uni D t cj .ui /n C @xj @xi Re @xj @xj @ t @p n 1 @2 uni @ C .cj ui /n C C CO.t/2 ; ck 2 @xk @xj @xi Re @xj @xj
(10.14)
where the second term on r.h.s is the stabilized term introduced by the transformation.
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10.2.2 Finite Element Techniques After deriving Navier–Stokes equation using CBS scheme in Sect. 10.2.1, there are three steps for solving the resulting equations by introducing an operator split procedure in temporal discretization before spatial discretization. First, ignore the pressure term in (10.14) to get an intermediate velocity, and then solve the continuity equation with intermediate velocity to get the pressure distribution. Finally, correct the velocity in momentum equation with obtained pressure distribution. Let Uin be the unknown velocity at the nth time step. Define the following relationships, 1 unC D .1 1 /uni C 1 uinC1 i
(10.15)
UinC1 D .1 1 /Uin C 1 UinC1 :
(10.16)
The CBS procedures are summed up as below, • Step 1. obtain intermediate velocities. @ 1 @2 uni Ui Uin D t uj .Ui /n C @xj Re @xj @xj t 2 @ @ 1 @2 uni C : uk .uj Ui /n C 2 @xk @xj Re @xj @xj
(10.17)
• Step 2. solve the continuity equation. Because Ui satisfies relationship (10.16), yields UinC1 D .1 1 /Uin C 1 UinC1 n @ @p n t 2 @p n D .1 1 /Ui C 1 Ui t C : ck @xi 2 @xk @xi
(10.18)
Then, dropping the last term in (10.18), the continuity equation becomes @2 p n @Ui 1 @Uin 1 : 1 D C .1 1 / @xi @xi t @xi @xi
(10.19)
• Step 3. correct the velocities with obtained pressure. UinC1 Ui D t
@p n : @xi
(10.20)
Equations (10.17), (10.19), and (10.20) are obtained from the time discretization for Navier–Stokes equations with CBS scheme, and then Standard Finite Element Method is applied to these equations. The domain of flow field is meshed by
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unstructured triangular element. Then linear shape functions in the equal order are used for velocity and pressure, i.e. Ui D N Ui ;
p D N p;
Ui D NUi ;
p D Np;
(10.21)
where N D fN1 N2 N3 g is the linear shape function vector in an element. UN i , p, N 4UN i , 4pN velocities, pressure and their increments in the corresponding element, respectively. Substitute (10.21) into (10.17), (10.19), and (10.20), and approach (10.17), (10.19), and (10.20) by the Standard Finite Element Method. Dropping the superscript at nth time step, the matrices of the three steps are obtained as the following, • Step 1. obtain intermediate velocities. t 2 ŒKu UN i ; Ui Ui D tM 1ŒCu UN i C K i C f i C 2
(10.22)
where Z M D
N T Nd; t
Z
Cu D
NT t
@ .Ncj /d; @xj
Z
Ku D
N T ck
@ @xk
@ Ncj d; @xj
t
Z
f i D
N T ij nj dt :
(10.23)
t
• Step 2. solve the continuity equation.
t1 H p D GU i C 1 GU i fp ; where Z GD
@N T Nd; @xi
t
Z
H D t
@N T @N d; @xi @xi
(10.24)
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Table 10.1 Comparisons between numerical and experimental results at Re D 10;40;100;185 Re Source Lv s Cd 10 Ding (2004) 0:252 29:6 2:85 Takami and Keller (1969) 0:25 29:7 3:18 Tuann and Olson (1978) 0:252 30:0 3:07 Present 0:2663 28:959 2:8420 40
Re 100
185
Ding (2004) Takami and Keller (1969) Tuann and Olson (1978) Present Source Ding (2004) Liu et al. (1998) Present
2:1 2:32 2:20 2:1355 Si
54:8 53:6 53:5 53:356 Clr:m:s ˙0:339 ˙0:28 ˙0:311
1:675 1:536 1:713 1:5423 CN d
Guilmineau and Queutey (2002) Lu and Dalton (1996) Present
0:19 0:195 0:195
1:28 1:31 1:4034
0:443 0:4886
1:325 ˙ 0:008 1:35 ˙ 0:012 1:300 ˙ 0:014
Lv is the length of vortex, s the separated angle, Si Strouhal number
Z fp D t
@p nTi dt : N T U i C 1 U i t @xi
(10.25)
t
• Step 3. correct the velocities with obtained pressure. nC1
U i
D U i C M 1 tŒGp :
(10.26)
10.2.3 Verifications Let the freestream velocity be the characteristic velocity, and diameter of cylinder be the characteristic length. The domain of flow field in this problem is a rectangular region of 26 12, in which the center of a circular cylinder is located at the point .0; 0/, and its dimensionless diameter is 1. The unstructured triangular mesh of computational domain is generated using Easymesh software (Niceno 2004). The computation is carried out at Re D 10; 40; 100; 185 for the verification. The length of vortex, separated angle, lift coefficient, drag coefficient and the Strouhal number (Si ) are listed in Table 10.1, and they are in good agreement with the existing numerical and experimental results (Ding 2004; Takami and Keller 1969; Tuann and Olson 1978; Liu et al. 1998; Lu and Dalton 1996; Guilmineau and Queutey 2002).
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10.3 Flow Patterns 10.3.1 Wake Pattern Formation at Low Re Number Cylinder wake, especially in the incompressible flow, has been always one of the classic problems in fluid mechanics. In the past decades, a lot of experiments, theoretical analyses and numerical simulations have been conducted by scholars. A detailed description of cylinder wake may refer to Williamson (1996). Because of the complexity of the problem, the fundamental nature of wake pattern formation and its evolution has not yet been fully understood. Currently, the three-dimensional vortex shedding has become the point of the research (Henderson 1997; Barkley et al. 2000; Henderson and Barkley 1996). In numerical simulation, DNS (Direct Numerical Simulation) (Scardovelli and Zaleski 1999) is the powerful tool to investigate the three-dimensional vortex shedding, and Floquet analysis (Barkley and Henderson 1996) is also introduced to analyze the secondary instability of the flow in theoretical analysis. In this section, Re number is introduced and defined as Re D U1 D=v, where U1 is the incoming flow velocity, D the diameter of the cylinder. The wake patterns develop and evolve as the Re number increases, and is described in detail as follow. For Re < 6: Two-dimensional steady flow creeps around the cylinder and no vortex structures are observed in the wake. For 6 < Re < 47: A pair of symmetrically attached vortex structures appear at the rear of the cylinder, the size of the vortex structures increases as the Re number increases. For 47 < Re < 190: Hopf bifurcation occurs in the flow field and periodical vortex shedding develops in the wake. The vortex shedding frequency increases as the Re number increases. For 190 < Re < 260: Three-dimensional vortex shedding appears in the cylinder wake. Mode A with spanwise wavelength D 3:96D develops at about Re 190 and Mode B with spanwise wavelength D 1:0D develops at about Re 260.
10.3.2 The Discontinuities in the Relation Between Re Number and St Number In order to investigate the relation between the vortex shedding frequency and Re number, the S t number is introduced and defined as S t D fD=U1 , where f is the vortex shedding frequency. As shown in Figs. 10.3 and 10.4, no vortex shedding
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Fig. 10.3 Wake patterns at low Re number
Fig. 10.4 The relation between Re and St (Henderson 1997)
0.25 2D Experimental fit
St ≡ fd/u∞
0.20
Re’2 = 260 0.15 Re2 = 190 Re1 = 47 Steady 0.10 10
100
Re
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occurs at low Re number (Re < p 47) and thus S t D 0. The FKE-relation (Fey et al. 1998) S t .Re/ D b1 C b2 = Re is valid as the Re number is in the interval [47,180]. The first discontinuity occurs at Re D 190 because of the appearance of the three-dimensional vortex structure Mode A. The second discontinuity occurs at Re D 260 due to the transition from Mode A to Mode B.
10.3.3 Streamline Topology Bifurcation in the Wake Streamline topology method is powerful to investigate the mechanism of the vortex shedding. The process of vortex shedding from a cylinder has been interpreted in terms of instantaneous streamlines by Perry et al. (1982). Bakker (1991) also used the topology method to analyze the bifurcations of streamline patterns. Some brief introduction of streamline topology method is given as follow.
10.3.3.1 Stream Function and Hamiltonian System For the flow field, the relation of displacement and velocity of a fluid particle is written as ˙ D V(t; X): X (10.27) At time t D t0 , the streamline can be obtained by integration of the following equation ˙ D V(t0 ; X): X (10.28) By introducing the stream function xP D
, the velocity is then given as, @ ; @y
yP D
@ @x
(10.29)
From the view point of dynamical system, the above equation is an autonomous Hamiltonian system.
10.3.3.2 Stagnation Points and Flow Structures For the dynamical system defined by (10.27), the stagnation point can be determined by the following equation v.x0 / D 0:
(10.30)
If the coordinate transformation x D x x0 is introduced, then the stagnation point will always locate at x0 D .0; 0/ in the new coordinate, and the Taylor expansion of the velocity v in the neighborhood of the stagnation point x0 reads x˙ D Jx C O.jxj2 /;
(10.31)
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Fig. 10.5 Saddle point (left) and center (right) in two-dimensional incompressible flows (Bisgaard 2005)
where J is the Jacobian matrix of v evaluated at x0 D .0; 0/, it can be written as " JD
@u @x @ @x
@u @y @ @y
#
" D
@2 @x@y 2 @@x 2
@2 @y 2 @2 @x@y
# :
(10.32)
For the linearized system, (10.31) can be reduced as x˙ D Jx:
(10.33)
The characteristic equation of Jacobian matrix is 2 Tr.J/ C Det.J/ D 0
(10.34)
For incompressible fluid, we have T r .J/ D 0. Therefore the eigenvalues of Jacobian matrix can be written as p 1;2 D ˙2 Det.J/: (10.35) There exists three possibilities, the eigenvalues are, respectively, real, complex conjugate, and zero, and the stagnation points are correspondingly saddle point, center, and degenerate stagnation point. For det .J/ < 0, 2 D 1 ¤ 0, the stagnation point is saddle point. For det .J/ > 0, 1;2 D ˙i !, the stagnation point is center. For det .J/ D 0, 1;2 D 0, the stagnation point is degenerate stagnation point. The sketch of streamlines around saddle point and center is shown in Fig. 10.5. For two-dimensional streamline topology, there are two kinds of structurally unstable flows, one is a flow containing degenerate stagnation points, which means the cusp bifurcation (shown in Fig. 10.6) occurs at the stagnation point. Another structurally unstable flow is a flow containing so-called heteroclinic connection (shown in Fig. 10.7). On the other hand, flows containing homoclinic connection and general stagnation point (saddle point and center) are structurally stable.
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Fig. 10.6 The cusp bifurcation, a loop is appearing or disappearing in the flow
Fig. 10.7 Homoclinic connection of a saddle point (left, structurally stable) and heteroclinic connection (right, structurally unstable)
More complicated mathematical skills such as canonical transformation and unfolding of the degenerate stagnation point are needed for the further investigation of the mechanism of vortex shedding, more details of theory analysis of the topology method may refer to Brons et al. (2007); Bisgaard (2005). The schematic of streamline topology bifurcation scenario is given in Fig. 10.8. Some numerical simulation results by CBS method are shown in Figs. 10.9 and 10.10 to compare with the bifurcation scenario.
10.4 Dynamics in Flow Separation 10.4.1 Flow Separation and Criteria 10.4.1.1 Flow Separation in Physical Meaning Flow separation is the origin of separated vortical flow. In physical, flow separation has many definitions but none of them is exact and clear. In this section, we use the definition given in reference (Wu et al. 2006) and defined the separation as the process that fluid elements adjacent to the wall no longer move along the wall but turn into the interior of fluid (Wu et al. 2006). This definition allows us to study flow separation at an infinitesimal neighborhood of separation point. Therefore, in the study of flow separation, the first thing need to do is locating the separation point, which is called separation criteria.
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Fig. 10.8 Streamline topology bifurcation scenarios at Re D 100 (Bisgaard 2005)
10.4.1.2 Separation Criteria Flow separation has different criteria based on different definitions. Earlier criteria for separation are based on the Eulerian description of flow, i.e. Prandtl’s criteria, MRS criteria (Sear and Tellionis 1975), and later, they have been proved being meaningless for unsteady flow. Recently, Lagrangian description of flow has been used, and it is found that flow particles separate from boundary and form a material spike. Haller found that this material spike is relevant to the presence of an unstable manifold initiating from a fixed point on the boundary which is separation point. Based on this definition of flow separation, he obtained the separation criteria for general time-dependent flow (Haller 2004).
10.4.2 Flow Separation from Viewpoint of Dynamic System Flow field can be considered as a dynamic system in the form xP D u D f .x; y; t/ yP D D g.x; y; t/
(10.36)
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Fig. 10.9 Vortex shedding in a period at Re D 100, and the time has been nondimensionalized
with the no-slip boundary conditions u.x; 0/ D 0; .x; 0/ D 0;
@ D 0; @y
(10.37)
where y D 0 is boundary wall, and y > 0 is the flow region. By adding an initial condition x .0/ D x0 , y .0/ D y0 , (10.36) is the pathline of flow particle
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Fig. 10.10 Vortex shedding in a period at Re D 100, and the time has been nondimensionalized
initiating from .x0 ; y0 /. Therefore, (10.36) is Lagrangian description of flow. Then the theories from nonlinear dynamics can be used to study the stagnation points and their related manifolds. In order to study the equilibrium of system (10.36), Jacobi matrix is needed to identify the equilibrium. Unlike the equilibrium in interior of fluid, however, every point on no-slip boundary wall is equilibrium of system, thus boundary wall is a nonhyperbolic critical manifold. In topology theory (Bakker 1991), this nonhyperbolic singular property of wall is removed by introducing an equivalent system dx 1 dx D D y 1 f .x; y; t/ D u .x; y; t/ dt y dt 1 dy dy D D y 1 g.x; y; t/ D .x; y; t/; dt y dt
(10.38)
where y D 0 is boundary wall, and flow region is y > 0. In (10.38), the boundary conditions are changed into .x; 0/ D 0;
(10.39)
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Fig. 10.11 Two dimensional steady flow separation
which is the same as slip boundary conditions. Therefore, flow on the boundary can move along the wall, and system governed by (10.38) is equivalent to the flow with slip boundary conditions. By comparing with two systems, it is found that two systems have different timescale near the boundary which leads to different dynamical properties of flow separation. (a) Steady flow separation In steady 2-D flow, there are only two independent variables .x; y/. Using the definition of invariant manifold, there exit invariant manifolds that are coincided with the trajectories of particles and the related streamline on .x; y/ plan. From Fig. 10.11, it can be seen that steady flow separation is characterized as intersection of separation streamline and boundary wall. At the separation point, wall shear is vanished and admitted negative gradient along the wall, according to Prandtl’s criteria for steady separation. From Lagrangian viewpoint, a fluid element initiated from separation point will be squeezed along the wall and stretched in the direction of separation streamline, and a material spike is formed. In dynamic system governed by (10.38), the Jacobi matrix at separation point has two real eigenvalues with different sign, which means that the separation point is a saddle point, and wall and separation streamline are the stable and unstable manifold, respectively. The separation streamline here is rather a pathline than streamline. (b) Unsteady flow When the flow is unsteady, invariant manifold only exists in 3D spaces .x; y; t/. 1D manifold on .x; y/ plan is the intersection of 2D invariant manifold and the plan at instant time. This 1D manifold separates the plan into different parts that no particles can be transported between them, and it will move on the .x; y/ plan with time. The movements and intersection of stable and unstable manifolds are crucial to understand the transport and mixing of two-dimensional flow. In unsteady flow, separation streamline is no longer coincided with pathline. Although separation streamline can still describe unsteady flow separation through evolvement of streamline topology, it cannot show the transport properties in flow separation. Periodic flow is a special unsteady flow that invariant manifolds exist on the Poincare map, related to the period of flow.
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In average theorem, perturbed periodic system possesses a unique hyperbolic periodic orbit with the same stability as the hyperbolic fixed point of averaged system. The periodic flow can be considered as a steady flow under periodic perturbation, and thus possesses a hyperbolic periodic orbit near a regular saddle point of steady flow. Because boundary is an invariant manifold under any perturbation, this hyperbolic orbit lies on the boundary. For periodic flow with slip boundary condition, the separation point is moving along the hyperbolic periodic orbit on boundary. However, in flow with no-slip boundary condition, separation point of steady flow is nonhyperbolic point. When back to system (10.36), the effects of no-slip boundary slows the movements of unstable manifold and hyperbolic orbit down by the factor y. Thus on the boundary hyperbolic orbit becomes a stagnation point, while its unstable manifold will still moves periodically. Therefore, if there is a separation point on the no-slip boundary, it should be fixed. This type of separation is called fixed separation. In order to describe the difference between slip boundary and no-slip boundary, we use the velocity field derived by Ghosh et al. (1998), which is also used as an example in Haller (2004). The velocity field has the form 2 u.x; y; t/ D y 1 C 3y C x2 y 2 C xf .t/ 3 1 (10.40) .x; y; t/ D y xy yf .t/ ; 2 where the wall is at y D 0, and f .t/ D sin .2 t/. The corresponding velocity field with slip boundary condition is 2 u.x; y; t/ D 1 C 3y C x 2 y 2 C xf .t/ 3 1 .x; y; t/ D xy yf .t/: 2
(10.41)
The movements of unstable manifold of flows with slip boundary conditions and no-slip boundary conditions in one period are shown in Fig. 10.12. In flow with slip boundary conditions, there is a hyperbolic periodic orbit on the boundary, while the flow with nonslip boundary conditions has a fixed separation point. In Haller’s fixed separation criteria (Haller 2004), the separation profile is an unstable manifold emerging from the fixed separation point. Then followed by the mathematical derivation (details refer to Haller (2004)), he obtained the necessary and sufficiency conditions of fixed separation for general unsteady flow, which are ˇZ t ˇ ˇ uy .; 0; / ˇˇ ˇ lim sup ˇ d ˇ < 1 (10.42) t !1 t0 .; 0; / Z t Z t uy .; 0; s/ 1 .uxy .; 0; / yy .; 0; // 2 xy .; 0; / ds d D 1; . / t0 t0 .; 0; s/ (10.43)
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Fig. 10.12 Movements of unstable manifold in one period (T D 1). (a) Flow with slip boundary conditions. (b) Flow with nonslip boundary conditions
where .; 0/ is the fixed separation point, y D 0 the no-slip boundary, the dense of fluid. For incompressible flow, the criteria can be simplified as ˇZ t ˇ ˇ ˇ ˇ (10.44) lim sup ˇ uy .; 0; /d ˇˇ < 1 t !1
Z
t0
t
uxy .; 0; /d D 1:
(10.45)
t0
Flow separation criteria based on unstable manifolds are Lagrangian description of flow phenomena (Haller and Poje 1998; Haller 2000; Surana and Haller 2008). Combining with Lagrangian coherent structures in fluid, it is possible to reveal the nature of nonlinear phenomena and flow controlling by using local perturbations on boundary.
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10.5 Stability Analysis of Flow Around a Circular Cylinder 10.5.1 Existing Bifurcation Analysis Flow around circular cylinders presents a paradigm of vortices related to rich nonlinear phenomena. For Re D 5 7, the flow separates and two symmetrical steady vortices appear behind the cylinder. When Re D 45 49, the steady flow loses its stability and a periodic flow occurs. In this regime, vortices are shed periodically from the cylinder, i.e. von Karman Vortex Street. Further, the flow becomes turbulent at the higher Reynolds number. An extensive study has been conducted on this subject for decades. In early 1980s, Tobak and Peake (1982) summarized the early work on the topology of three dimension separated flow. In 1988, Ongoren and Rockwell (1988a); Ongoren and Rockwel (1988b) classified (1) symmetric vortex pair mode (per T) as “the symmetric S mode” and (2) asymmetric modes: 2S (per T), 2S (per 2T), SCP (per 2T) and 2P (per 2T) as “the asymmetric AI, AII, AIII and AIV modes”, respectively, when flow around oscillating cylinders was studied. In 1991, Bakker (1991) used qualitative theory of differential equations to give the possible changes in flow topology near the critical points and applied this method to the study of flow past a circular cylinder. In 1994, Noack and Eckelmann (1994a,b) studied the global stability of the cylinder wake using spectral method, and found that the change to periodic flow is the results from a supercritical Hopf bifurcation. In 2003, from viewpoint of streamline pattern, Hartnack (1999) and Brons and Hartnack (1999) used normal form transformation to simplify the dynamical system of the streamlines, and encapsulate the features of the dynamical system. In this section, the stability analysis of flow around a circular cylinder is performed. At first, a low-dimensional dynamic system is derived using POD modes. Then the nonlinear behavior of the system is studied in details. In particular, the bifurcation and stability analysis of the dynamic system is presented.
10.5.2 Low-Dimensional Dynamic System Using POD Modes For the fluid dynamics, it is an infinite-dimensional system. The nonlinear dynamics can be easily applied to the system with finite dimension. So, the fluid dynamic system should be reduced into a system with finite dimension, before the nonlinear dynamics is used to study the stability and bifurcation of the system. Proper Orthogonal Decomposition, also known as Karhunen–Loeve expansion, is a method to obtain an optimal low-dimensional empirical basis related to the dominant flow features, deriving from an ensemble of high-dimensional experimental or simulation data. Moreover, by projecting the solution of the dynamical system onto the complete space spanned by POD modes, a low-dimensional model can be obtained for further stability analysis.
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The velocities can be written as V D V0 .X/ C
m X
ak .t/'k .X/;
(10.46)
kD1
where v0 .x/ is the mean flow field, 'k .x/ the kth POD mode or quasi-mode, ak .t/ the Fourier coefficient. Project the solution of incompressible Navier–Stokes equations onto the space spanned by POD modes f 'i j i D 1; : : : ; 1g, and yields m m X 1 X d lij aj C qijk aj ak ai D fi aj ; Re; : : : D dt Re j D0
i D 1; : : : ; m;
j;kD0
(10.47) where lij D .'i ; 'j /
(10.48)
qijk D .'i ; 5:.'j ; 'k // Z .'i ; 'j / D 'i :'j d
(10.49) (10.50)
a0 1
(10.51)
!
Thus, a low-dimensional model for fluid dynamic system is derived.
10.5.3 Bifurcation and Stability of Flow Around a Circular Cylinder Bifurcation theory is used to study the changes in qualitative behavior or topological structure of the dynamical system. A bifurcation occurs when a small smooth change for a parameter (parameters) leads to a sudden change of the topological structure or the number of the solution of the system. In the fluid dynamical system, the Reynolds number is usually chosen as bifurcation parameter to study the flow evolution. In this section, the stability of the flow around a circular cylinder is presented, and the equilibrium and periodic solution of the system are given.
10.5.3.1 Stability of Quasi-mode of the System Consider the system of nonlinear ordinary differential equations xP D f .x; /
x 2 R n ; 2 R;
(10.52)
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Fig. 10.13 Pressure contour of the mean flow
where is the bifurcation parameter. The equilibrium points of (10.52) satisfies f .x; / D 0:
(10.53)
The eigenvalues of the Jacobian matrix A D fx .x; /
(10.54)
can be used for stability analysis of the equilibrium. If all the eigenvalues of A have negative real part, then the equilibrium point is stable. If the Jacobian matrix has a complex pair of eigenvalues D r C i i with r > 0 and i ¤ 0 with all other eigenvalues having negative real part, limit cycle occurs. When a complex pair of eigenvalues crosses the imaginary axis, the behavior of the system evolves from a stable equilibrium point to periodic motion. This kind of bifurcation is called a Hopf bifurcation. For the flow around a circular cylinder, the equilibrium of the system (10.47), namely, the steady quasi-mode, can be obtained by the following expression, fi .aj ; Re; : : :/ D
m m X 1 X lij aj C qijk aj ak D 0: Re j 0
(10.55)
j;k0
The stability of the equilibrium or the quasi-mode is studied using the eigenvalues of Jacobian matrix, X @fi 1 lij C 2 D qijk ak : @aj Re m
Dij D
(10.56)
k0
The flow around the circular cylinder at Re D 200 is simulated using CBS scheme (Zienkiewicz and Codina 1995; Zienkiewicz et al. 1995). Two hundred snapshots of the velocities of the flow are chosen as the ensemble, and the POD modes are extracted using the method of snapshots (Holmes et al. 1998). It can be seen from Fig. 10.13, the mean flow has symmetric structure. Furthermore, the pressure contours of the first four modes are presented in Fig. 10.14.
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Fig. 10.14 Pressure contour of POD modes
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Fig. 10.15 Bifurcation diagram of the system with respect to the Reynolds number
Fig. 10.16 Eigenvalues of the Jacobian matrix for different Reynolds number
The stability of the equilibriums of the system is analyzed. The equilibriums of the system and the related Jacobian matrix are obtained from (10.55) and (10.56). Figure 10.15 presents the bifurcation diagram of the system with respect to the Reynolds number from 5 to 300. The eigenvalues of the equilibrium of the system are shown in Fig. 10.16. It can be seen that when Reynolds number is less than 49, all the real parts of the eigenvalues are less than zero, that is, the equilibrium or the quasi-modes of the system is stable. When Re D 49:719, the real part of the eigenvalues cross the imaginary axis. From viewpoint of nonlinear dynamics, the equilibrium of system loses its stability and the system becomes periodic motion.
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a
b
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0.0025
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a1
a1
2-modes 4-modes 6-modes
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2-modes 4-modes 6-modes
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0.0020
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-1 -1.5 0
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c
d
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a1
a1
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-100
-1500 0
594
Time history
Time history
200
400
600
Time history
800
1000
-150 500
520
540
560
580
Time history
Fig. 10.17 Time history of first mode for different mode number. (a) Re D 20, (b) Re D 100, (c) Re D 200, (d) Re D 200 (enlarge)
A Hopf bifurcation occurs at Re D 49:719 since there exists one pair of pure imaginary eigenvalues with all other eigenvalues having nonzero real part. It is the boundary of the topological change of the system. The mode interaction also plays an important role in the stability of the nonlinear dynamical system. From Fig. 10.15, it can be seen that the amplitude of the first mode is rather small when the Reynolds number is less than 155. However, it increases significantly as the Reynolds number is greater than 170. In light of the analysis above, the system for Re D 20 has a stable equilibrium, and the ones for Re D 100 and Re D 200 undergo a periodic motion. However, the increase of the amplitude cannot still be understood. In order to deal with such problem, the cases with three typical Reynolds number, Re D 20, Re D 100 and Re D 200 are simulated. Different numbers of the modes are selected to show the effect of mode interaction on stability of the system, shown in Fig. 10.17. For Re D 20, the systems with different mode numbers undergo the same behavior finally (see Fig. 10.17a). It indicates that the first two modes are “active” in the system while the higher order modes are “inactive.” The behavior of the system can be approached by the first two modes. For Re D 100, the behaviors of the systems with different mode numbers become somewhat different. It implies that higher modes interact
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with the first two modes. However, the first two modes are still dominant in the system, shown in Fig. 10.17b. For Re D 200, the behaviors of the systems with different mode numbers are much different with each other. The higher modes become “active” and strong mode interaction occurs. It can be seen that the behavior of the system becomes divergent when only the first two modes are used, which means the system cannot be approached only by the first two modes. When higher modes are introduced, the behaviors of the system undergo a periodic motion, which agrees with the analysis mentioned above.
10.6 Drag Reduction and Its Control 10.6.1 Drag and Its Nature Flow past the circular cylinder involves many flow features such as flow separation, vortex shedding, Karman vortex street and is of considerable theoretical and practical importance in the fluid dynamics. For this flow problem, drag reduction has attracted a great deal of attention from the consideration of energy saving and structural vibrations, and many flow control methods are available now. In general, the control methods of drag on a bluff body could be divided into two categories: the passive and active control methods (Choi et al. 2008). For the passive control method, drag reduction is achieved by means of various passive devices without power input, such as surface modifications with roughness, dimple, longitudinal groove, splitter plate, and small secondary control cylinder. For the active control method, actuators with power input are used, and flow is controlled by various forcing devices, such as rotary, streamwise, and transverse oscillations of a bluff body; inflow oscillation; electromagnetic forcing; steady blowing/suction; timeperiodic blowing/suction; distributed forcing; and the synthetic jets. Compared with the passive control, the active control methods are influenced slightly by the inflow direction. Moreover, the recently developed active closed-loop control method (He et al. 2000; Li et al. 2003; Bergmann et al. 2005) has made the bluff-body flow control more precisely. It has been confirmed, by numerous experiments and numerical simulations, that both of the passive and active flow control methods are effective in drag reduction. However, their mechanisms are quite different. For passive control methods, the 3D geometric modifications such as helical strake and spanwise wavy surface will introduce a 3D forcing to the flow field of the 2D bluff bodies, and the vortex shedding process and the Karman vortex street can be attenuated. For active control methods, the low energy fluid in the boundary layer is removed or excited by additional excitation, and the separation is delayed and the drag is significantly reduced. Besides the influence on the flow separation and vortex shedding, the passive and active control devices also have great influence on the stability of the boundary
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layer and the free shear layer of the bluff body. Since the laminar and turbulent flow states are essentially different, the transition will lead to an abrupt change of the drag force. Compared with the laminar boundary layer, the mixture in the turbulent boundary layer is more violent, and thus the velocity profile has a higher gradient near the wall, which leads to a higher friction drag. On the other hand, the turbulent mixing motion brings more energy from the outer or main flow into the boundary layer to weaken the adverse pressure gradient. The separation can be significantly delayed, and the form drag of bluff bodies can be decreased. This section makes a brief discussion of the influence of the control methods on the laminar-turbulence transition. The classical stability analysis theory is given first, and then the influences of several control methods are discussed, respectively.
10.6.2 Perturbation and Linear Stability of Flow For the 2D incompressible viscous flow, the governing equations, namely the Navier–Stokes (NS) equations, can be expressed by @ @u C D0 @x @y 2 @u @2 u @u @u 1 @p @ u C 2 C fx Cu C D C @t @x @y @x @x 2 @y 2 @ @ @ 1 @ @ C 2 C fy ; Cu C D C 2 @t @x @y @x @y
(10.57) (10.58) (10.59)
where u and v are the velocity components, the density, p the pressure, the kinematic viscosity and fx and fy the body force components. Assume a given steady basic flow has the form of Uy; V D 0; P .x; y/
(10.60)
u0 .x; y; t/; 0 .x; y; t/; P 0 .x; y; t/;
(10.61)
and the 2D perturbations are
The resulting flow according to (10.60) and (10.61) is u D U C u0 .x; y; t/; D 0 .x; y; t/; P D P 0 .x; y; t/;
(10.62)
Both (10.60) and (10.62) are solutions of the NS equations, and the initial discrepancy between the basic flow and the resulting flow is very small. If the disturbance dies away or amplified in time, the basic flow is then called stable or unstable, respectively.
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Substitute (10.62) into (10.57)–(10.59), ignoring the terms with quadratic perturbation velocities and considering that the basic flow should also satisfy the NS equations, we have @u0 @ 0 C D0 @x @y
2 0 @u0 1 @p 0 @u @u0 dU @2 u0 CU C 0 D C C @t @x dy @x @x 2 @y 2 @ 0 @2 0 @ 0 1 @p 0 @2 0 C : CU D C @t @x @y @x 2 @y 2 Introducing
(10.63) (10.64) (10.65)
a stream function using the relationship u0 D
@ @ ; 0 D @y @x
(10.66)
and further eliminating the pressure p 0 , (10.63) to (10.65) can be reduced to one equation with respect to , which can be expressed by @ @t
@2 @ @2 @ d2 U @2 @2 C U C C @x 2 @y 2 @x @x 2 @y 2 @x dy 2 4 @ @4 @ : D C 2 C @x 4 @x 2 @y 2 @y 4
(10.67)
Stream function of the 2D disturbance can be expanded by Fourier modes, and each mode has the form of .x; y; t/ D '.y/ei.axˇt / ;
(10.68)
where ' .y/ is the amplitude function, ˛ is real and ˇ is complex number, and ˇ D ˇr C iˇi
(10.69)
with (10.69), (10.68) can be rewritten as .x; y; t/ D '.y/eˇi t ei.˛xˇr t / :
(10.70)
It can be seen from (10.70) that the perturbation is represented by the sinusoidal and cosinusoidal waves in the basic flow. The amplitude is ' .y/ eˇi t , the wavelength is D 2 =˛ and the frequency is ˇr . The imaginary number ˇi is the amplification factor, and determines whether the wave grows or die away. If ˇi > 0, the wave is amplified and the basic laminar flow is unstable, if ˇi < 0 the basic flow is stable, and if ˇi D 0 the basic flow is in a state of neutral stability.
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Fig. 10.18 A schematic of the curve of neutral stability
Substituting (10.70) into (10.67), a fourth-order ordinary differential equation can be obtained for the amplitude function ' .y/: i ˇ U .' 00 ˛2 '/ U 00 ' D .' 00 00 2˛ 2 ' 00 C ˛4 '/: ˛ ˛
(10.71)
The nondimensional form of (10.71) can be given by i .' 0000 2˛N 2 ' 00 C ˛N 4 '/; .UN c/.' 00 ˛N 2 '/ UN 00 ' D ˛Re N
(10.72)
where ˇ U1 L U ;c D ; Re D ; ˛N D ˛L UN D U1 ˛U1
(10.73)
and U1 and L are reference velocity and length, respectively. Combined (10.71) with the boundary conditions y D0W
' D 0;
' 0 D 0;
y D1W
' D 0;
' 0 D 0;
(10.74)
the stability of a specified basic flow U .y/ can be analyzed. For a given basic flow, the neutral stability state is very important, and the aim of stability analysis is to determine the curve of neutral stability (see Fig. 10.18). In Fig. 10.18, the point on the curve, where the Reynolds number Reind is smallest, is of special interest, because it gives the indifference Reynolds number below which all modes are damped, while some are amplified above it. With Reind , the position where the laminar boundary layer losses its stability and the onset of transition could be theoretically determined.
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Fig. 10.19 Curves of neutral stability for laminar-boundary profiles with different shape factors, after Schlichting and Ulrich (Schlichting and Gersten 2000)
10.6.3 Influence of the Control Methods on the Drag From the above stability analysis, it can be found that the stability of a flow field in practice is strongly dependent on the velocity profile U.y/ and the perturbations. When U.y/ is changed, the curve of neutral stability in Fig. 10.18 will be affected because the governing equation (i.e., (10.72)) is changed. Schlichting and Ulrich (Takami and Keller 1969) have analyzed the stability of a family of velocity profiles governed by U.y/ ƒ D 2 23 C 4 C .1 /3 ; Ue 6
(10.75)
where Ue is the velocity at the outer edge of the boundary layer, D y=ı and the shape factor of the velocity profile. This velocity profile corresponds to the laminar boundary layer at a flat plate at zero incidence when pressure changes at the outer flow. At the pressure minimum ƒ D 0, and ƒ > 0 implies pressure decrease and ƒ > 0 pressure increase. Figure 10.19 shows the curves of neutral stability under different ƒ. As shown in the figure, the unstable region is increased significantly with the decrease of the shape factor of the velocity profile (or pressure increase). It also indicates that the laminar boundary layers in the pressure drop region are considerably more stable than those in the pressure increase region. Therefore, as the passive flow control methods using the spanwise wavy surface, splitter flat and
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secondary small cylinder and the active methods such as structural oscillation and steady or unsteady suction/blowing will affect the flow field, and their influence on the flow stability are very great. As shown in Fig. 10.18, the stability of a velocity profile is also determined by the perturbation wavelength. If there is no initial perturbation or the wavelength is not beyond the unstable region, the laminar-turbulence transition will not happen even if the Reynolds number is very large. On the contrary, if extra disturbance is imposed on the basic flow in practice, the transition Reynolds number could be decreased. Passive control methods using the wall roughness, dimple and helical wire, and the active methods by inflow oscillation, loudspeaker and synthetic jets, will bring perturbations into the basic flow, and thus are favorable for the laminar-turbulence transition and form drag reduction of the bluff bodies.
10.7 Key Symbols A; B A0 ci D f Iij J L N p pN t Ui UN i U1 ui uL x; y xk
Mode Initial deflection Convective velocity Diameter of the cylinder Vortex shedding frequency Identity tensor Jacobian matrix Characteristic scale Linear shape function vector in an element Pressure Pressure in an element Time Velocity at the nth time step Velocity in an element Incoming flow velocity Velocity of fluid Characteristic velocity Coordinate Spatial coordinates
Greek ˛ ˇ ˇi
Real number Complex number Amplification factor Relax factor Wavelength Dynamic viscosity, bifurcation parameter
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"ij Rate-of-strain tensor Kinematic viscosity Density of fluid fx , fy Body force component ij Stress tensor ' Amplitude function Stream function ƒ Shape factor of velocity profile t Domain
References Barkley D, Tuckerman LS, Golubitsky M (2000) Bifurcation theory for three-dimensional flow in the wake of a circular cylinder. Phys Rev E 61(5):5247–5252 Barkley D, Henderson RD (1996) Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J Fluid Mech 322:215–241 Bakker PG (1991) Bifurcations in flow patterns. Kluwer Academic Publishers, Dordecht Bergmann M, Cordier L, Brancher J, (2005) Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model. Phys Fluids 17:097101 Bisgaard AV (2005) Structures and bifurcation in fluid flows: with application to vortex breakdown and wakes. Technical University of Denmark, Denmark Brøns M, Hartnack J (1999) Streamline topologies near simple degenerate critical points in twodimensional flow away from boundaries. Phys Fluids 11:314 Brøns M, Jakobsen B, Niss K, Bisgaard AV, Voigt LK (2007) Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers. J Fluid Mech 584:23–43 Choi H, Jeon WP, Kim JS (2008) Control of flow over a bluff body. Ann Rev Fluid Mech 40:113–139 Ding H, Shu C, Yeo K, Xu D (2004) Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method. Comput Methods Appl Mech Eng 193:727–744 Fey U, K¨onig M, Eckelmann H (1998) A new Strouhal-Reynolds number relationship for the circular cylinder in the range 47 < Re < 2 105. Phys Fluids 10(7):1547–1549 Ghosh S, Leonard A, Wiggins S (1998) Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field. J Fluid Mech 372:19–163 Guilmineau E, Queutey P (2002) A numerical simulation of vortex shedding from an oscillating circular cylinder. J Fluids Struct 16(6):773–794 Haller G (2000) Finding finite-time invariant manifolds in two dimensional velocity fields. Chaos 10(1):9–108 Haller G (2004) Exact theory of unsteady separation for two-dimensional flows. J Fluid Mech 512:257–311 Haller G, Poje AC (1998) Finite transport in aperiodic flow. Phys D 119:2–380 Hartnack J (1999) Streamline topologies near a fixed wall using normal forms. Acta Mech 136(1):55–75 He JW, Glowinski R, Metcalfe R, Nordlander A, Periaux J (2000) Active control and drag optimization for flow past a circular cylinder. 1: oscillatory cylinder rotation. J Comput Phys 163:83–117 Henderson RD (1997) Nonlinear dynamics and pattern formation in turbulent wake transition. J Fluid Mech 352:65–112
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Henderson RD, Barkley D (1996) Secondary instability in the wake of a circular cylinder. Phys Fluids 8(6):1683–1685 Holmes P, Lumley J, Berkooz G (1998) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge (London/New York) Li Z, Navon I, Hussaini M, Le Dimet F (2003) Optimal control of cylinder wakes via suction and blowing. Comput Fluids 32:149–171 Liu C, Zheng X, Sung C (1998) Preconditioned multigrid methods for unsteady incompressible flows. J Comput Phys 139(1):35–57 Lu X, Dalton C (1996) Calculation of the timing of vortex formation from an oscillating cylinder. J Fluids Struct 10(5):527–541 Niceno B (2004) Easymesh-A two dimensional quality mesh generator. site:http://wwwdinma.univ.trieste.it/nirftc/research/easymesh/ Noack BR, Eckelmann H (1994a) A global stability analysis of the steady and periodic cylinder wake. J Fluid Mech 270:297-330 Noack BR, Eckelmann H (1994b) A low-dimensional Galerkin method for the three di-mensional flow around a circular cylinder. Phys Fluids 6(1):124–143 Ongoren A, Rockwell D (1988a) Flow structure from an oscillating cylinder. Part 1: Me-chanism of phase shift and recovery in the near wake. J Fluid Mech 191:197–223 Ongoren A, Rockwel DL (1988b) Flow structure from an oscillating cylinder. Part 2: Mode competition in the near wake. J Fluid Mech 191:225–245 Perry AE, Chong MS, Lim TT (1982) The vortex-shedding process behind two-dimensional bluff bodies. J Fluid Mech 116:77–90 Schlichting H, Gersten K (2000) Boundary layer theory. Springer, Berlin Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Ann Rev Fluid Mech 31(1):567–603 Sear WR, Tellionis DP (1975) Boundary-layer separation in unsteady flow. SIAM J Appl Math 28:15–235 Surana A, Haller G (2008) Ghost manifolds in slow-fast systems, with applications to unsteady fluid flow separation. Phys D, 237:1507–1529 Takami H, Keller H (1969) Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder. Phys Fluids, II 5:1–6 Tobak M, Peake D (1982) Topology of three-dimensional separated flows. Ann Rev Fluid Mech 14(1):61–85 Tuann S, Olson M (1978) Numerical studies of the flow around a circular cylinder by a finite element method. Comput Fluids 6:219–240 Williamson CHK (1996) Vortex dynamics in the cylinder wake. Ann Rev Fluid Mech 28:477–539 Wu JZ, Ma HY, Zhou MD (2006) Vortex and vorticity dynamics. Springer, New York Zienkiewicz O, Codina R, (1995) A general algorithm for compressible and incompressible flowPart I. The split, characteristic-based scheme. Int J Numer Methods Fluids 20:869–885 Zienkiewicz O, Morgan K, Sai B, Codina R, Vasquez M (1995) A general algorithm for compressible and incompressible flow-Part II. Tests on the explicit form. Int J Numer Methods Fluids 20:887–913
Chapter 11
Explicit Equation of Motion of Constrained Systems Applications to Multi-body Dynamics Firdaus E. Udwadia and Thanapat Wanichanon
Abstract This chapter develops a new, simple, general, and explicit form of the equations of motion for general nonlinear constrained mechanical systems that can have holonomic and/or nonholonomic constraints that may or may not be ideal, and that may contain either positive semi-definite or positive definite mass matrices. This is done through the replacement of the actual unconstrained mechanical system, which may have a positive semi-definite mass matrix, with an unconstrained auxiliary system that is then subjected to the same holonomic and/or nonholonomic constraints as those applied to the actual unconstrained mechanical system. The unconstrained auxiliary system is subjected to the same “given” force as the actual mechanical system, and its mass matrix is appropriately augmented to make it positive definite so that the so-called fundamental equation can then be directly and simply applied to obtain the closed-form acceleration of the actual constrained mechanical system. Furthermore, it is shown that by appropriately augmenting the “given” force that acts on the actual unconstrained mechanical system, the auxiliary system directly provides the constraint force that needs to be imposed on the actual unconstrained mechanical system so that it satisfies the given holonomic and/or nonholonomic constraints. Thus, irrespective of whether the mass matrix of the actual unconstrained mechanical system is positive definite or positive semi-definite, a simple, unified fundamental equation results that give a closed-form representation of both the acceleration of the constrained mechanical system and the constraint force. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, constrained mechanical systems, such as those encountered in multi-body dynamics. Several examples are provided.
F.E. Udwadia () • T. Wanichanon Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, USA e-mail:
[email protected];
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 11, © Springer Science+Business Media, LLC 2012
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11.1 Introduction The description of the motion of constrained nonlinear mechanical systems is an important problem in analytical dynamics that has been worked on by numerous researchers. Appell (1899, 1911); Chataev (1989); Dirac (1964); Gauss (1829); Gibbs (1879); Goldstein (1981); Hamel (1949); Lagrange (1787); Sudarshan and Mukunda (1974) provide a brief sampling of some of the researchers who have made substantial contributions. The importance of developing the equations of motion for constrained systems stems from the fact that the modeling of complex multibody dynamical systems relies very heavily on our ability to incorporate constraints in a seamless and easy manner. Typically, such multi-body systems are described through the use of numerous constraints, and often it may be difficult to know a priori–especially with nonholonomic constraints–even whether the constraints are all independent of one another. While considerable progress has been made in recent years in obtaining the equations of motion for nonlinear constrained systems, several questions remain unanswered at the present time. A significant problem in deriving the equation of motion for constrained mechanical systems arises when the mass matrix of the unconstrained mechanical system is singular. Since the mass matrix then does not have an inverse, standard methods for obtaining the constrained equations of motion, which usually rely on the invertability of the mass matrix, cannot be used. For example, the so-called fundamental equation developed by Udwadia and Kalaba (1992) cannot be directly applied. Observing this, Udwadia and Phohomsiri (2006) derived an explicit equation of motion for such systems with singular mass matrices. However, the structure of their explicit equation differs significantly from their so-called fundamental equation (Udwadia and Kalaba 1992). Recently, by using the concept of an unconstrained auxiliary system, Udwadia and Schutte (2010) developed a simpler explicit equation of motion that has the same form as the socalled fundamental equation, and is valid for systems whose mass matrices may or may not be singular. They do this by augmenting the singular mass matrix of the unconstrained mechanical system by appropriately making it positive definite. However, the structural form of this augmented mass matrix requires considerable computational effort especially when complex multi-body systems are involved. In this chapter, we present a new alternative equation of motion for systems with positive definite and/or positive semi-definite mass matrices that is in many respects superior to that proposed in Udwadia and Schutte (2010). We consider an unconstrained auxiliary system that has a positive definite mass matrix instead of the actual unconstrained mechanical system whose mass matrix may be positive semi-definite. This unconstrained auxiliary system is subjected to the same “given” force as that acting on the actual mechanical system, and when subjected to the same constraints as the actual unconstrained mechanical system, provides in closed form, at each instant of time, the acceleration of the actual constrained mechanical system. Similarly, by suitably augmenting the “given” force that is acting on the actual unconstrained mechanical system, we obtain from the
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auxiliary system the proper constraint force acting on the actual unconstrained mechanical system in closed form. In short, the auxiliary system obtained herein gives the equation of constrained motion of the actual mechanical system in closed form whether or not the mass matrix is singular in a much simpler, more straightforward, and more computationally efficient, manner than in Udwadia and Schutte (2010). The proofs of our results are also much simpler, and they lead to deeper insights into the nature of constrained motion of mechanical systems. We briefly point out the importance of being able to formulate correctly the constrained equations of motion for mechanical systems whose mass matrices are positive semi-definite. When a minimum number of coordinates is employed to describe the (unconstrained) motion of mechanical systems, the corresponding set of Lagrange equations usually yields mass matrices that are nonsingular (Pars 1979). One might thus consider that systems with singular mass matrices are not common in classical dynamics. However, in modeling complex multi-body mechanical systems, it is often helpful to describe such systems with more than the minimum number of required generalized coordinates. And in such situations, the coordinates are then not independent of one another, often yielding systems with positive semi-definite mass matrices. Thus, in general, singular mass matrices can and do arise when one wants more flexibility in modeling complex mechanical systems. The reason that more than the minimum number of generalized coordinates are usually not used in the modeling of complex multi-body systems, though this could often make the modeler’s task much simpler, is that they result in singular mass matrices, and to date systems with such matrices have been difficult to handle within the Lagrangian framework. Several examples are provided in this chapter showing how singular mass matrices can appear in the modeling of constrained mechanical systems. This is the reason it is useful to obtain in closed form the general, explicit equations of motion for constrained mechanical systems whose mass matrices may or may not be singular. Since such systems normally arise when modeling largescale, complex mechanical systems in which the modeler seeks to substantially facilitate his/her work by using more than the minimum number of coordinates to describe the system, it is also important to keep an eye on the computational efficiency of the equations so obtained.
11.2 System Description of General Constrained Mechanical Systems It is useful to conceptualize the description of a constrained mechanical system, S , in a three-step procedure. We do this in the following way: First, we describe the so-called unconstrained mechanical system in which the coordinates are all independent of each other. We do that by considering an unconstrained mechanical system whose motion at any time t can be described,
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using Lagrange’s equation, by M.q; t/qR D Q.q; q; P t/
(11.1)
q.t D 0/ D q0 ; q.t P D 0/ D qP 0 ;
(11.2)
with the initial conditions
where q is the generalized coordinate n-vector; M is an n by n matrix that can be either positive semi-definite .M 0/ or positive definite .M > 0/ at each instant of time; and Q is an n-vector, called the “given” force, which is a known function of q, q, P and t. We shall often refer to the system described by (11.1) as the unconstrained mechanical system S . Second, we impose a set of constraints on this unconstrained description of the system. We suppose that the unconstrained mechanical system is now subjected to the m constraints given by P t/ D 0; i D 1; 2; : : : ; m; 'i .q; q;
(11.3)
where r m equations in the equation set (11.3) are functionally independent. The set of constraints described by (11.3) include all the usual varieties of holonomic and/or nonholonomic constraints, and then some. We shall assume that the initial conditions (11.2) satisfy these m constraints. Therefore, the components of the nvectors q0 and qP0 cannot all be independently assigned. We further assume that the set of constraints (11.3) are smooth enough so that we can differentiate them with respect to time t to obtain the relation A.q; q; P t/qR D b.q; q; P t/;
(11.4)
where A is an m by n matrix whose rank is r, and b is an m-vector. We note that each row of A arises by appropriately differentiating one of the m constraint equations given in (11.3). Using the information in the previous two steps, in the last step we bring together the description of motion of the constrained mechanical system as M.q; t/qR D Q.q; q; P t/ C Q c .q; q; P t/;
(11.5)
where Qc is the constraint force n-vector that arises to ensure that the constraints (11.4) are satisfied at each instant in time. Thus, (11.5) describes the motion of the actual constrained mechanical system, S . In what follows we shall suppress the arguments of the various quantities unless required for clarity. Equation (11.3) provides the kinematical conditions related to the constraints. We now look at the dynamical conditions. The work done by the forces of constraints under virtual displacements at any instant of time t can be expressed as (Udwadia and Kalaba 2001) vT .t/Qc .q; q; P t/ D vT .t/C.q; q; P t/;
(11.6)
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where C .q; q; P t/ is an n-vector describing the nature of the nonideal constraints which is determined by physical observation and/or experimentation, and the virtual displacement vector, v.t/, is any nonzero n-vector that satisfies (Udwadia and Kalaba 1996) A.q; q; P t/v D 0: (11.7) When the mass matrix M in (11.1) is positive definite, the explicit equation of motion of the constrained mechanical system S is given by the so-called fundamental equation (Udwadia and Kalaba 2002) qR D a C M 1=2 B C .b Aa/ C M 1=2 .I B C B/M 1=2 C;
(11.8)
where a D M 1 Q, B D AM 1=2 , and the superscript “C” denotes the Moore– Penrose (MP) inverse of a matrix (Graybill 1983; Penrose 1955; Udwadia and Kalaba 1997). We note that (11.8) is valid (1) whether or not the equality constraints (11.3) are holonomic and/or nonholonomic, (2) whether or not they are nonlinear functions of their arguments, (3) whether or not they are functionally dependent, (4) and whether or not the constraint force is nonideal. We note that the constrained mechanical system S is completely described through the knowledge of the matrices M and A, and the column vectors Q, b, and C . The latter four are functions of q, q, P and t, while the elements of the matrix M are, in general, functions of q and t. In what follows, we shall also denote the acceleration of the constrained system given in (11.8) by qR S .D q/. R However, when the unconstrained mechanical system given by (11.1) is such that the matrix M is singular .M 0/, the above equation cannot always be applied since the matrix M 1=2 may not exist. In that case, (11.8) needs to be replaced by equation (Udwadia and Phohomsiri 2006) qR D
.I AC A/M A
C
QCC b
WD MN C
QCC b
(11.9)
under the proviso that the rank of the matrix MO T D M j AT is n. This rank condition is a necessary and sufficient condition for the constrained mechanical system to have a unique acceleration – a consequence of physical observation of the motion of systems in classical mechanics. However, the form of (11.9) when M is positive definite is noticeably different from the form of the so-called fundamental equation (11.8). A unified equation of motion that is applicable to both these situations is presented in Udwadia and Schutte (2010). They considered an auxiliary system that has a positive definite mass matrix, which is subjected to the same constraint conditions as the actual mechanical system that has a singular mass matrix. This positive definite mass matrix of the auxiliary system is expressed as M C ˛ 2 AC A, where ˛ is any nonzero real number and again the superscript “C” denotes the Moore–Penrose (MP) inverse of the matrix (Udwadia and Schutte 2010). However, the use of the Moore–Penrose (MP) inverse of the matrix A is expensive to compute, especially when the row and column dimensions of A are large.
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In this chapter, we uncover a new general equation of motion for constrained mechanical systems by instead using the augmented mass matrix, MAT G D M C ˛2 AT GA, which is simpler, more general, and directly uses the so-called fundamental equation (11.8). The function ˛ .t/ is an arbitrary, nowhere-zero, sufficiently smooth (C 2 ) real function of time, and G .q; t/ WD N T .q; t/ N .q; t/ is any arbitrary m by m positive definite matrix whose elements are sufficiently smooth functions (C 2 ) of the arguments. Thus, greater generality, simpler results, and greater computational efficiency are herein achieved. Furthermore, the proofs of the various results are much simpler than in Udwadia and Schutte (2010).
11.3 Explicit Equations of Motion for General Constrained Mechanical Systems From physical observation, the acceleration of a system in classical dynamics under a given set of forces and under a given set of initial conditions is known to be uniquely determinable. As shown in Udwadia and Phohomsiri (2006), a necessary and condition for this to occur is that the rank of the matrix MO T D sufficient T is n. We shall therefore assume throughout this chapter that for the M jA constrained systems we consider herein, the matrices M and A are such that this condition is always satisfied. Thus, we assume that the actual constrained mechanical system under consideration is appropriately mathematically modeled and the resulting acceleration of the system can be uniquely found.
11.3.1 Positive Definiteness of the Augmented Mass Matrices Lemma 11.1. : Let M 0, let ˛ .t/ be an arbitrary, nowhere-zero, sufficiently smooth (C 2 ) real function of time, and let G .q; t/ D N T .q; t/ N .q; t/ be any m by m positive definite matrix (Chen 1999) whose elements are sufficiently smooth functions (C 2 ) of the arguments. The n by n augmented mass matrix MAT G WD M C ˛ 2 AT GA is positive definite at each instant of time if and only if the n by n C m matrix MO T D M j AT has rank n at each instant of time. Proof. (a) Consider any fixed instant of time. Assume that MO has rank n; we shall prove that the augmented mass matrix MAT G D M C ˛2 AT GA is positive definite at that instant. We first observe that the matrix MAT G is symmetric since M is symmetric as is AT GA. Since the column space of the matrix A is identical to the column space of ˛A, n D rankMO D rank
M A
D rank
M ˛A
D rank M j ˛AT :
(11.10)
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We shall denote by Col .X / the column space of the matrix X . Since Col .M / D Col M 1=2 , and Col AT D Col AT N T because N is nonsingular, we get (11.11) n D rank M j ˛AT D rank M 1=2 j ˛AT N T WD rank.MQ /; where we have denoted MQ WD M 1=2 j ˛AT N T . Next, we consider the augmented mass matrix MAT G . It can be expressed as MAT G D M C ˛2 AT GA D M C ˛ 2 AT N T NA M 1=2 D M 1=2 j ˛AT N T ˛NA D MQ MQ T 0:
(11.12)
Thus, the n by n matrix MAT G must at least be positive semi-definite. But from (11.11), rank MQ D n, hence MAT G is positive definite. Proof. (b) Consider any fixed instant of time. Assume that MAT G D M C ˛ 2 AT GA is positive definite; we shall prove that MO has full rank n at that instant. Q QT From (11.12) and the assumption that M AT G > 0, we have MAT G D M M > 0, T Q Q Q so that M M has rank n, and hence rank M D n. Since elementary row operations do not change the rank of a matrix, we find that
M 1=2 ˛NA 1=2 T T D rank M j A N :
n D rank.MQ T / D rank
D rank
M 1=2 NA
(11.13)
And also since Col .M / D Col M 1=2 , and Col AT D Col AT N T , we have
T rank M 1=2 j AT N T D rank M j AT D rank M j AT D rank.MO /:
Hence, rank MO D n, and the proof is therefore complete.
(11.14) t u
11.3.2 Explicit Equation for Constrained Acceleration Having the auxiliary mass matrix MAT G , which has been proved to be always a positive definite matrix, we are now ready to begin implementing the explicit equations of motion for the constrained acceleration of the system S that may have a positive semi-definite mass matrix, M 0. We begin by proving a useful result that will be used many times from here on.
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Lemma 11.2. Let AC denote the Moore–Penrose (MP) inverse of the m by n matrix A, then .I AC A/AT D 0:
(11.15)
Proof. .I AC A/AT D AT AC AAT D AT .AC A/T AT D AT AT .AT /C AT D 0:
(11.16)
In the second equality above, we have used the fourth Moore–Penrose (MP) condition (see Graybill (1983); Penrose (1955); Udwadia and Kalaba (1997)) and in the last equality, we have used the first MP condition. This yields the stated result. t u Recall that our actual mechanical system has a mass matrix M that may be positive semi-definite, and since M 1=2 does not exist we encounter difficulty in finding the acceleration of the constrained mechanical system when using the fundamental equation (see (11.8)). However, we note from Lemma 11.1 that the matrix MAT G is always positive definite when MO has rank n. Moreover, this rank condition is a check that our mathematical model appropriately describes a given physical system, since in all physical systems in classical mechanics the acceleration must be uniquely determinable. Were we then to use this matrix MAT G (instead of M ) as the mass matrix of an “appropriate” unconstrained auxiliary system, subjected to the same constraints as the actual unconstrained mechanical system, we would encounter no difficulty in using the fundamental equation to obtain the acceleration of this constrained auxiliary system see (11.8), since the mass matrix of this auxiliary system is positive definite! Our aim then is to define this unconstrained auxiliary system in the “appropriate” manner so that the resultant constrained acceleration it yields upon application of the fundamental equation always coincides with the acceleration of our actual constrained mechanical system. We now proceed to show that this indeed can be done, and we demonstrate how to accomplish this. Consider any unconstrained mechanical system S , (a) whose equation of motion is described by (11.1) where the n by n mass matrix M may be positive semi-definite or positive definite, and whose initial conditions are given in relations (11.2), (b) which is subjected to the m constraints given by (11.4) (or equivalently by (11.3)) that are satisfied by the initial conditions q0 and qP 0 as described by (11.2), and (c) which is subjected to the nonideal constraint that is prescribed by the n-vector C .q; q; P t/ as in (11.6). Recall that we shall always subsume that the actual mechanical system S has the property that MO T D ŒM j AT has rank n at each instant of time.
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Consider further an unconstrained auxiliary system SAT G that has (1) an augmented mass matrix given by MAT G D M.q; t/ C ˛ 2 .t/AT .q; q; P t/G.q; t/A.q; q; P t/ > 0;
(11.17)
where ˛ .t/ is any sufficiently smooth function (C 2 would be sufficient) of time that is nowhere zero, and G .q; t/ D N T .q; t/ N .q; t/ is any m by m positive definite matrix with its elements sufficiently smooth functions (C 2 would be sufficient) of its arguments, and (2) an augmented ‘given’ force defined by QAT G;z .q; q; P t/ D Q.q; q; P t/ C AT .q; q; P t/G.q; t/z.q; q; P t/;
(11.18)
where z .q; q; P t/ is any arbitrary, sufficiently smooth m-vector, (3) so that the equation of motion of this unconstrained auxiliary system is given by MAT G .q; t/qR D Q.q; q; P t/ C AT .q; q; P t/G.q; t/z.q; q; P t/ P t/: WD QAT G;z .q; q;
(11.19)
Similar to the conceptualization stated in Sect. 11.2, the system described by (11.19) is referred to as the unconstrained auxiliary system SAT G . (4) We shall subject this unconstrained auxiliary system SAT G to (a) the same initial conditions, and (b) the same constraints, which the unconstrained mechanical system S is subjected to, as described in items (b) and (c) above. We note from (11.19) that the unconstrained auxiliary system has at each instant of time an augmented mass matrix MAT G obtained by using the mass matrix, M 0, of the unconstrained mechanical system S and augmenting it by ˛2 AT GA; also, the “given” force, QAT G;z .q; q; P t/, which the unconstrained auxiliary system is subjected to, at each instant of time, is obtained by using the “given” force Q .q; q; P t/ that the unconstrained mechanical system S is subjected to, and augmenting it by AT Gz, where the vector z is arbitrary (note that z could be taken to be the zero vector). Thus, the unconstrained auxiliary system SAT G differs from the unconstrained mechanical system S in that at each instant of time (a) it has an augmented mass matrix MAT G , and (b) it is subjected to an augmented “given” force QAT G;z . The two unconstrained systems (S and SAT G ) when subjected to the same set of constraints (both ideal and/or nonideal at each instant of time) and the same set of initial conditions yield, correspondingly, what we shall call the constrained mechanical system and the constrained auxiliary system. Having both the unconstrained mechanical system S and the unconstrained auxiliary system SAT G we shall now show the following result: Result 1: The acceleration of the constrained mechanical system S obtained by considering the unconstrained mechanical system and its constraints as described
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by (a)–(b), is identical with, and directly obtained from, the explicit acceleration of the constrained auxiliary system SAT G obtained by considering the unconstrained auxiliary system and its constraints and initial conditions as described by (1)–(4). Proof. As shown in Udwadia and Phohomsiri (2006), the acceleration, qRS , of the constrained mechanical system S is described by (see (11.9))
.I AC A/M qRs D A
C
QCC b
C QCC N WD M ; b
(11.20)
while the acceleration, qRSAT G , of the constrained auxiliary system SAT G is given by (see also (11.9))
qRSAT G
.I AC A/MAT G D A
C
QAT G;z C C b
T C Q C A Gz C C N WD MA : b (11.21)
Let us consider first the term I AC A M of (11.20). Post-multiplication of both sides of (11.15) by ˛2 GA yields ˛2 .I AC A/AT GA D 0;
(11.22)
so that .I AC A/M D .I AC A/M C ˛2 .I AC A/AT GA D .I AC A/.M C ˛2 AT GA/ D .I AC A/MAT G
(11.23)
Using (11.23) in equation (11.20) thus yields qR s D
.I AC A/MAT G A
C
QCC b
WD MN AC
QCC b
:
(11.24)
We note that the acceleration of the constrained system is still the same even though the mass matrix M 0 is replaced with the augmented mass matrix MAT G > 0 (see equations (11.20) and (11.24)). Since the augmented mass matrix MAT G is 1=2 always positive definite, the matrix MAT G now does exist. The drawback in using the fundamental equation (11.8) is now resolved. Thus, in order to obtain only the acceleration of a constrained system, we can just simply replace the mass matrix M 0 with the augmented mass matrix MAT G > 0 and use the fundamental equation (11.8). Again, pre-multiplying and post-multiplying both sides of (11.15) by MAT G and Gz respectively, we have MAT G .I AC A/AT Gz D 0:
(11.25)
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C Noting that for any matrix X , X C D X T X X T (Penrose 1955), from (11.24), we have T C QCC C QCC C T N N N D MA M A MAT G .I A A/ j A qR s D MA b b D ŒMN AT MN A C ŒMAT G .I AC A/.Q C C / C AT b D ŒMN AT MN A C ŒMAT G .I AC A/.Q C C / C MAT G .I AC A/AT Gz C AT b Q C AT Gz C C D ŒMN AT MN A C ŒMAT G .I AC A/ j AT b Q C AT Gz C C (11.26) D qR sAT G : D MN AC b The third equality above follows from (11.25) and the last from (11.21). This proves the claim. t u Since we know that at each instant of time the acceleration of the constrained mechanical system S is the same as that of the constrained auxiliary system SAT G (see (11.26)), and also that the augmented mass matrix of the system SAT G is positive definite, we can directly apply the so-called fundamental equation (11.8) to the unconstrained auxiliary system described by (11.19) to get qR SAT G and therefore qR explicitly as (Udwadia and Kalaba 2002) 1=2
qRs D aAT G;z C MAT G BACT G .b AaAT G;z / 1=2
1=2
C MAT G .I BACT G BAT G /MAT G C;
(11.27)
where MAT G D M C ˛2 AT GA > 0; QAT G;z D Q C A Gz; T
(11.28) (11.29)
1 1 T aAT G;z D MA1 T G QAT G;z D MAT G Q C MAT G A Gz;
(11.30)
and 1=2
BAT G D AMAT G :
(11.31)
Remark 11.1. We know that when MO has rank n, the acceleration, qR S , of the constrained mechanical system S is unique and is explicitly given by (11.20). And since we have shown that qR S D qR SAT G at each instant of time, the acceleration of the constrained auxiliary system must be independent of the arbitrary (nowhere-zero) scalar function ˛ .t/, the arbitrary m-vector z.t/, and the arbitrary (positive definite) matrix G .q; t/, provided each of these three entities is a sufficiently smooth .C 2 / function of their arguments.
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Remark 11.2. Since ˛, z, and G are arbitrary as just stated, we can further particularize (11.27) by setting ˛ 1, z 0, and G Im in describing our unconstrained auxiliary system. Thus, this unconstrained auxiliary system now has only an augmented mass matrix M C AT A, and it is subjected to the same “given” force as the unconstrained mechanical system S . This unconstrained auxiliary system, when subjected to the same constraints (kinematical and dynamical) and initial conditions as those placed on S , yields the acceleration of the constrained mechanical system S , given by 1=2
1=2
1=2
qR s D aAT C MAT BACT .b AaAT / C MAT .I BACT BAT /MAT C; where
(11.32)
MAT D M C AT A > 0;
(11.33)
aAT D MA1 T Q;
(11.34)
and 1=2
BAT D AMAT
(11.35)
11.3.3 Explicit Equation for Constraint Force So far, we have developed an unconstrained auxiliary system SAT G which always has a positive definite mass matrix, and we have used it in the so-called fundamental equation (11.8) to directly yield the acceleration of the constrained mechanical system S . We now further explore whether the constraint force Qc acting on the unconstrained mechanical system S (that is brought into play by the presence of the constraints (b) and (c) described earlier in Sect. 11.3.2) can be directly adduced from the equation of motion of the constrained auxiliary system SAT G . To show this, we begin by putting forward a useful result. Lemma 11.3.
T T MAT G BACT G AMA1 TG A D A ; 1=2
(11.36)
where MAT G is defined in (11.17) and BAT G is defined in (11.31). Proof. T MAT G BACT G AMA1 TG A 1=2
1=2
1=2
D MAT G BACT G AMAT G MAT G AT D MAT G BACT G BATT G BATT G 1=2
1=2
T D MAT G .BACT G BAT G /T BATT G D MAT G BATT G .BATG /C BATT G 1=2 1=2
D MAT G BATT G D AT :
1=2
(11.37)
In the third equality above, we have used the fourth MP condition and in the fifth we have used the first MP condition.
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From (11.5), we know that once we obtain the constrained acceleration qR D qR S from (11.27) of the mechanical system S , we can determine the constraint force Qc acting on the unconstrained mechanical system S (described by (11.1)) at each instant of time from the relation Q c D M qR Q D M qR S Q:
(11.38)
Alternatively, consider the equation of motion of the constrained auxiliary system SAT G , which can be obtained by pre-multiplying both sides of the (11.27) by MAT G . We have MAT G qRs D Q C AT Gz C MAG BACT G .b AaAT G;z / 1=2
1=2
C MAT G .I BACT G BAT G /MAT G C 1=2
WD QAT G;z C QAc T G;z ;
(11.39)
where 1=2
QAc T G;z D MAT G BACT G .b AaAT G;z /CMAT G .I BACT G BAT G /MAT G C: 1=2
1=2
(11.40)
We notice from (11.39) that under the same set of constraints (both ideal and nonideal) as those acting on the unconstrained mechanical system S , the constraint force acting on the unconstrained auxiliary system SAT G (described by (11.19)) is QAc T G;z ; the explicit expression for this force is given by (11.40). We now explore the connection between Q c and QAc T G;z , our aim being to obtain Q c explicitly from QAc T G;z . We now claim that this can indeed be done by appropriately choosing the m-vector z, which has so far been left arbitrary. To show this, we begin by considering only the third member on the right-hand side of the first equality of (11.39). Expanding it, we have MAT G BACT G .b AaAT G;z / 1=2
T D MAT G BACT G .b AMA1 T G ŒQ C A Gz/ 1=2
T D MAT G BACT G .b AaAT G AMA1 T G A Gz/ 1=2
T D MAT G BACT G .b AaAT G / MAT G BACT G AMA1 T G A Gz 1=2
1=2
D MAT G BACT G .b AaAT G / AT Gz: 1=2
(11.41)
Notice that we have denoted aAT G WD MA1 T G Q in the second equality, and used relation (11.36) in the last equality of the above equation. Using (11.41) in the third member on the right-hand side of (11.39) yields MAT G qRs D Q C MAT G BACT G .b AaAT G / 1=2
1=2
C MAT G .I BACT G BAT G /MAT G C WD Q C QAc T G ; 1=2
(11.42)
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where aAT G D MA1 T G Q;
(11.43)
and 1=2
QAc T G D MAT G BACT G .b AaAT G / C MAT G .I BACT G BAT G /MAT G C: 1=2
1=2
(11.44)
Equation (11.42) shows that the acceleration qR S of the constrained mechanical system S is given by c qRs D MA1 T G ŒQ C QAT G ;
(11.45)
and that it is indeed independent of the arbitrary m-vector z .t/ as remarked in the previous sub-section. Furthermore, equating the right-hand sides of (11.39) with (11.42) (both of which equal MAT G qRS ), we get QAT G;z C QAc T G;z D Q C AT Gz C QAc T G;z D Q C QAc T G ;
(11.46)
so that from the last equality we have QAc T G D QAc T G;z C AT Gz:
(11.47)
We now prove the following result: Result 2: When the unconstrained mechanical system, S , and the unconstrained auxiliary system, SAT G , have the same initial conditions and when they are each subjected to the same (ideal and nonideal) constraints, with the choice of the m-vector, z.t/ D ˛2 .t/b.q; q; P t/;
(11.48)
where ˛ .t/ is a nowhere-zero, sufficiently smooth function of time, the constraint force acting on the unconstrained auxiliary system SAT G is the same as the constraint force acting on the unconstrained mechanical system S at each instant of time. In short, Q c D QAc T G;˛2 b :
(11.49)
Proof. From (11.42), we know that at each instant of time QAc T G D MAT G qR s Q D .M C ˛2 AT GA/qRs Q D M qR s Q C ˛ 2 AT GAqR D Qc C ˛ 2 AT Gb:
(11.50)
In the last equality above, we have used equations (11.38) and (11.4). Substituting (11.50) in (11.47), we get Q c D QAc T G;z C AT Gz ˛2 AT Gb;
(11.51)
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which is the general result that relates the constraint force Q c acting on the unconstrained mechanical system S to the constraint force QAc T G;z acting on the unconstrained auxiliary system SAT G , at each instant of time. Note that in (11.51) ˛ .t/ is any arbitrary nowhere-zero scalar function, the m-vector z .t/ is any arbitrary sufficiently smooth function, and G .q; t/ is any positive definite matrix whose elements are continuous functions of its arguments. Finally, using (11.51), when z D ˛2 b, the result follows. t u Therefore, the force of constraint Q c acting on the unconstrained mechanical system S can also directly be obtained from the force of constraint QAc T G;z acting on the unconstrained auxiliary system SAT G by the appropriate selection of the mvector z .t/ D ˛ 2 .t/ b .q; q; P t/. We have now shown that if one would like to derive the constrained equations of motion of a general mechanical system S that has either a positive semidefinite or positive definite mass matrix, which is subjected to the kinematical constraints AqR D b and the nonideal dynamical constraints described by the nvector C .q; q; P t/ (under the proviso that matrix MO has rank n), one could obtain the (explicit) constrained equation of motion of the mechanical system S by following the three-step conceptualization of constrained motion as follows in terms of a new unconstrained auxiliary system: 1. Description of the unconstrained auxiliary system: (a) Replace the mass matrix M 0 of the actual unconstrained mechanical system S as given in (11.1) with the augmented mass matrix MAT G as given in (11.17); (b) Choose z D ˛ 2 b and replace the “given” force Q of the actual unconstrained mechanical system S with the augmented “given” force QAc T G;˛2 b as defined in (11.18); (c) Use the augmented mass matrix described in (a) and the augmented “given” force described in (b) to obtain (11.19), which describes the unconstrained auxiliary system SAT G ; 2. Description of the constraints: Subject this unconstrained auxiliary system to the same set of constraints (both ideal and nonideal) and initial conditions as the actual unconstrained mechanical system S ; 3. Description of the constrained auxiliary system: Apply the fundamental equation (Udwadia and Kalaba 1992, 2002) (see (11.39)) to the unconstrained auxiliary system described in (1) above, which is subjected to the constraints described in (2). The resulting equation of motion of this constrained auxiliary system has the following two important features: (i) the explicit acceleration of the constrained auxiliary system SAT G , obtained by using (11.27), is the same, at each instant of time, as the explicit acceleration of the constrained mechanical system S , obtained by using (11.9), and
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(ii) at each instant of time, the constraint force Qc (see (11.38)) acting on the unconstrained mechanical system S , because of the presence of the constraints imposed on it, is the same as the constraint force QAc T G;˛2 b (see (11.40)) acting on the unconstrained auxiliary system SAT G , which is described by (11.19). We are thus led to the somewhat surprising conclusion: the dynamics of the actual constrained mechanical system S are completely mimicked by the dynamics of the above-mentioned constrained auxiliary system SAT G . Finally, we point out that if one were interested only in obtaining the acceleration at each instant of time of the constrained mechanical system S , one can use any arbitrary m-vector z.t/ in (11.29) to obtain the augmented “given” force QAc T G;z and then use (11.27); for simplicity, Occam’s razor would then suggest that we might prefer to take z .t/ 0. As pointed out in item (1), part (b), above, if one were, in addition, also interested in finding the correct constraint force acting on the unconstrained mechanical system S , one would need to choose z .t/ D ˛ 2 .t/ b .q; q; P t/ and use equations (11.30) and (11.40). Clearly then, when the constraints are such that b .q; q; P t/ 0, the choice of z .t/ 0 in the augmented “given” force (11.18) in the description of the unconstrained auxiliary system (11.19) is automatically selected. In that case, the use of (11.39) yields, at each instant of time, the correct acceleration of the constrained mechanical system S as well as the correct force of constraint acting on the unconstrained mechanical system S . The approach of the above three-step conceptualization of constrained motion by utilizing the auxiliary system can be summarized as in Tables 11.1–11.3. This table schematically shows how one generates the auxiliary system SAT G from the actual given mechanical system S . Step 1 deals with the description of the unconstrained system S and the corresponding unconstrained auxiliary system SAT G . Instead of using the mass matrix M of an actual mechanical system S that may or may not be singular, we use the mass matrix MAT G for the auxiliary system SAT G , which is positive definite under the proviso that MO has full rank. In addition, we also augment the given force Q of the actual mechanical system S with the term AT Gz in defining the unconstrained auxiliary system. Then the unconstrained acceleration of the auxiliary system can be written as aAT G;z D MA1 T G QAT G;z while the unconstrained acceleration is undefined, as shown, in the case where the mass matrix is singular for the unconstrained mechanical system. In Step 2 – the description of the constraints – while describing of the constraints we apply the same set of (ideal and nonideal) constraints to the auxiliary system SAT G as applied to the actual mechanical system S . In Step 3 – the description of the constrained system – we can obtain the explicit equation of the constrained acceleration of the actual mechanical system by using the unconstrained auxiliary system and the constraints defined in the previous two steps and applying the so-called fundamental equation (Udwadia and Kalaba 1992, 2002) (see (11.27)). The fundamental equation also explicitly gives the constraint force on the actual mechanical system when using the m-vector z D ˛ 2 b in our definition of the unconstrained auxiliary system (see (11.30) and (11.40)).
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Table 11.1 Step 1: Description of unconstrained system System descriptions: mass matrix Actual mechanical system, S: M.q; t / 0; M is an n n matrix Auxiliary system, SAT G here MAT G D M.q; t / C ˛ 2 .t / AT .q; q; P t / G .q; t / A .q; q; P t/ > 0 ˛ .t / ¤ 0 is an arbitrary function of time G .q; t / > 0 is an arbitrary m m matrix System descriptions: given force Actual mechanical system, S: Q .q; q; P t/ Auxiliary system, SAT G here P t / C AT .q; q; QAc T G;z D Q .q; q; P t / G .q; t / z .t / m-vector z .t / is an arbitrary m-vector System descriptions: equation of motion Actual mechanical system, S: M qR D Q Auxiliary system, SAT G : MAT G qRSAT G D QAT G;z System descriptions: unconstrained acceleration Actual mechanical system, S: a D M 1 Q or Undefined Auxiliary system, SAT G : aAT G;z D MA1 T G QAT G;z
Table 11.2 Step 2: Description of constraints System descriptions: description of kinematic constraints Actual mechanical system, S: A .q; q; P t / qR D b .q; q; P t/ A is an m n matrix Auxiliary system, SAT G here A .q; q; P t / qR D b .q; q; P t/ System descriptions: description of non-ideal constraints Actual mechanical system, S: C .q; q; P t/ Auxiliary system, SAT G here C .q; q; P t/
We also note that even though both the actual mechanical system and the auxiliary system approach yield the same dynamical results, as mentioned earlier, in order to obtain a unified explicit equation of constrained motion, we prefer using the auxiliary system approach.
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Table 11.3 Step 3: Description of constrained system System descriptions: equation of motion Actual mechanical system, S: M qR D Q C Qc Auxiliary system, SAT G : MAT G qRSAT G D QAT G:z C QAc T G:z System descriptions: constrained acceleration Actual mechanical system, S: C Q C QC I AC A M qR D b A Auxiliary system, SAT G
1=2 qR D qRSAT G D aAT G:z C MAT G BACT G b AaAT G;z
1=2 1=2 CMAT G I BACT G BAT G MAT G C 1=2
BACT G D AMAT G
˛ .t / ¤ 0 is an arbitrary function of time G .q; t / > 0 is an arbitrary m m matrix z .t / is an arbitrary m-vector System descriptions: constraint force on unconstrained system Actual mechanical system, S: Qc D M qR Q Auxiliary system, SAT G : 1=2 Qc D QAc T G;˛2 b D MAT G BACT G b AaAT G;˛ 2 b
1=2 1=2 CMAT G I BACT G BAT G MAT G C 1=2
BACT G D AMAT G
˛ .t / ¤ 0 is an arbitrary function of time G .q; t / > 0 is an arbitrary m m matrix z .t / D ˛ .t /2 b .q; q; P t/
11.4 Illustrative Examples In this section, five examples are provided to illustrate the results developed and to show their usefulness. The first three examples are purposely chosen to be very simple so that the central ideas in the chapter are better understood. The fourth example is substantive because in it we show how our results can be directly used to obtain in a simple and straightforward manner the nonlinear equations describing the rotational dynamics of a rigid body in terms of quaternions. This result is of considerable use and importance in nonlinear multi-body dynamics. The last example takes the reader even further showing that there can be an unlimited number of mechanical systems which, when subjected to the same constraints, the same externally applied “given” forces, and the same initial conditions, exhibit identical dynamical behavior, making them completely indistinguishable from one another.
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Fig. 11.1 A wheel rolling down an inclined plane under gravity
Example 11.1. Consider a wheel of mass m and radius R rolling on an inclined surface without slipping, as shown in Fig. 11.1, with the gravitational acceleration g downward. The angle of the inclined surface is ˇ, where 0 < ˇ < =2. The system clearly has just one degree of freedom, which can be described by . If y is the vertical displacement of the center of the wheel as it rolls down the inclined plane, the wheel’s potential energy can be simply expressed as V D mgy:
(11.52)
Were we to take and y as the independent generalized coordinates and use the Lagrangian 1 2 1 L D T V D m RP C Ic P 2 C mgy; (11.53) 2 2 where Ic is the moment of inertia around the center of the wheel, to obtain Lagrange’s equations of motion for the unconstrained system (since we are assuming that the coordinates are independent), we would get the relations of the unconstrained system in the form of (11.1) as mR2 C Ic 0 0 R : (11.54) D mg 0 0 yR Note that the mass matrix M describing the unconstrained motion is now singular. This singularity is a consequence of the fact that in reality the system has only one degree of freedom and we are using more than the minimum number of generalized coordinates to describe the system, and pretending that these coordinates are independent. The advantage of doing this is that the unconstrained equations of motion, (11.54), can be trivially written down. However, the two coordinates y and are in reality not independent of one another, since the wheel rolls down without slipping. They are related through the equation of constraint y D R sin ˇ
(11.55)
that must be added to the formulation in order to model the dynamics of the physical situation properly. Differentiating (11.55) twice, we obtain the constraint equation in the form of (11.4), AqR D b, where A D R sin ˇ j 1 (11.56) b D 0:
(11.57)
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Since in this problem the system is ideal, C D 0. The next step is usually done through the use of the fundamental equation (11.8), but in this problem the mass matrix is now positive semi-definite, so the unconstrained acceleration a is undefined and the fundamental equation (11.8) cannot be applied. Thus, we have to use instead the relation (11.9) that can deal with the system when the mass matrix is positive semi-definite. However, in order to use the relation (11.9), we have to make sure that we model the system correctly. We can do that by checking the condition of the uniqueness of the equation of motion – the matrix M j AT has full rank (Udwadia and Phohomsiri 2006) – and so it is. Thus using the relation (11.9), the constrained mechanical system becomes 1 R mgR sin ˇ D : (11.58) yR mR2 C Ic mgR 2 sin2 ˇ We again note that the structure of (11.9) differs widely from the so-called fundamental equation (11.8); (11.9) does not readily lend itself to physical interpretation as does (11.8) (Udwadia and Kalaba 1992, 2002) but the drawback of (11.8) is that it cannot be directly applied when the mass matrix of the system is singular. However, if we use the unconstrained auxiliary system which replaces the mechanical system with its positive semi-definite mass matrix by one that is positive definite, then the general form to obtain the equations of motion of the constrained system, the fundamental equation (Udwadia and Kalaba 1992, 2002) can be handily applied. By choosing ˛ .t/ D 3; z .t/ D ˛2 b D 0 and G .q; t/ D u .t/ > 0 where u .t/ is an arbitrary positive function, we obtain the unconstrained auxiliary system (11.19), where the augmented mass matrix becomes MAT G D M C ˛2 AT GA mR2 C Ic C 9u.t/R2 sin2 ˇ 9u.t/R sin ˇ D >0 9u.t/R sin ˇ 9u.t/
(11.59)
and the augmented “given” force is 0 : D Q C ˛ A Gb D mg
QAT G;˛2 b
2
T
(11.60)
Then using the augmented mass matrix from (11.59) and the augmented “given” force from (11.60) in the fundamental equation (11.39), the equation of motion of the constrained auxiliary system yields the same equation of motion, (11.58), as that of the constrained mechanical system. Furthermore since b D 0, (11.39) also gives the correct constraint force required in the application to the unconstrained mechanical system. This constraint force is given by (11.40) as " # Qc mgR sin ˇ D : (11.61) Qyc mg
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335 x2
Fig. 11.2 A two degree-of-freedom multi-body system
x1
d k2
k1 m1
m2
The advantage of using the auxiliary system is that the so-called fundamental equation, the general form of obtaining the equations for constrained motion, can directly be applied and yields the correct equations of motion for the constrained mechanical system. Example 11.2. Consider a system of two masses, m1 and m2 , connected with springs, k1 and k2 , as shown in Fig. 11.2. Defining xN 1 D x1 l10 and xN 2 D x2 l20 , where l10 and l20 are the unstretched lengths of the springs k1 and k2 , respectively, and using the Lagrangian to obtain Lagrange’s equations of motion for the system, we get the dynamics of the unconstrained system as the form of (11.1) as m1 0 xRN 1 xRN M R 1 WD xN 2 xRN 2 0 m2 .k1 C k2 /xN 1 C k2 xN 2 k2 .l10 C d / D WD Q; (11.62) k2 .xN 1 xN 2 C l10 C d / where d is the length of the mass m1 . Assuming that m1 and m2 are both positive, we then obtain the acceleration # " .k1 Ck2 /xN 1 Ck2 xk N 2 .l10 Cd / xRN 1 m1 D : (11.63) k2 .xN 1 xN 2 Cl10 Cd / xRN 2 m2 We now assume that m1 D 0. The mass matrix on the left-hand side of (11.62) now becomes singular. However, the physical system that has now just one spring connecting mass m2 has a unique acceleration. In order to obtain the equation of motion, our attention is therefore drawn to the need for an additional constraint, so that the condition for determining the acceleration uniquely – namely, that the matrix M j AT has full rank – is satisfied. Considering the location of the mass m1 D 0, we obtain the following constraint relation: k1 xN 1 D k2 .xN 2 xN 1 d /:
(11.64)
Differentiating (11.64) twice with respect to time t and then putting it in the form of (11.4), we get the equation of the constraint
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AqR D b; where A D Œk1 C k2 k2 and b D 0:
(11.65)
We assume that the constraint is ideal so that C D 0. To obtain the equations of motion of this constrained system, which now has a singular mass matrix, we replace the mass matrix of the unconstrained system M 0 with the augmented mass matrix MAT G > 0. Since the system is subjected to only one constraint, an arbitrary positive definite matrix G is now an arbitrary positive function g .q; t/, which can be combined with the nowhere zero function ˛ .t/. Thus, we choose ˛ 2 g D gN where gN .q; t/ is an arbitrary positive function. Furthermore, since b D 0 we choose z .t/ D ˛ 2 b D 0. The augmented mass matrix MAT G of the auxiliary system in relation (11.17) can be then expressed as MAT G D M C ˛2 AT GA .k1 C k2 /2 g.q; N t/ .k1 C k2 /k2 g.q; N t/ > 0: D N t/ m2 C k22 g.q; N t .k1 C k2 /k2 g.q;
(11.66)
The augmented “given” force for the unconstrained auxiliary system is QAT G;z D Q, since z .t/ is chosen to be zero. Using this new augmented mass matrix MAT G instead of M in the fundamental (11.27), we obtain the acceleration of the constrained motion 2 3 " # k1 k 2 xR 1 .xN 2 l10 d / 4 .k1 Ck22 /2 5 D ; (11.67) k1 k2 m2 xR 2 .k1 Ck2 /
which is the correct result. We note that when m1 D 0, one can imagine that the system is now composed of only the mass m2 , with the springs k1 and k2 connected to it in series. As anticipated, the acceleration xR 2 of the mass m2 is given by the second row of (11.67), which is the correct equation, since .xN 2 l10 d / is the total extension of both the springs k1 and k2 . Again, this example illustrates how the actual constrained mechanical system is mimicked by the constrained auxiliary system. In the next example, we will present another approach to deriving the equations of motion of the constrained system in Example 11.2. It will give us an idea about the advantage of using the auxiliary system that can “transform” any system S with a positive semi-definite mass matrix to a system SAT G with a positive definite mass matrix. The derivation of the unconstrained system is simpler and at the same time the so-called fundamental equation can be readily utilized. Example 11.3. Consider the same system of two masses, m1 and m2 , connected with springs, k1 and k2 , that is shown in Fig. 11.3. But for this example, we shall model this system by decomposing it into two separate sub-systems – that is, we consider it as a multi-body system – as shown in Fig. 11.3. The two subsystems are then connected together by the “connection constraint,” q1 D x1 C d where d is the length of the mass m1 . We use the coordinates x1 , q1 and q2 to describe the configuration of the two sub-systems, and first treat these coordinates
11 Explicit Equation of Motion of Constrained Systems Fig. 11.3 Decomposition of the multi-body system shown in Fig. 11.2 using more than two coordinates
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Sub-system 1 x1
d
k1 m1 Sub-system 2 m2
q1 q2
k2
as being independent in order to get the unconstrained equations of motion. We then “connect” the two sub-systems by imposing the constraint q1 D x1 C d to obtain the equations of motion of the composite system shown in Fig. 11.2. Again defining xN 1 D x1 l10 and qN 2 D q2 l20 , where l10 and l20 are the unstretched lengths of the springs k1 and k2 , respectively, and using Lagrange’s equation, the equations of motion for the unconstrained system can be written as the form of (11.1) as 32 3 2 3 2 2 3 xRN 1 m1 0 0 k1 xN 1 xRN 1 (11.68) M 4 qRN 1 5 WD 4 0 m2 m2 5 4 qRN 1 5 D 4 0 5 WD Q: 0 m 2 m2 qRN 2 qRN 2 k2 qN2 Note that these equations for the unconstrained system are almost trivial to obtain. To model the system shown in Fig. 11.2 using these two separate sub-systems, we connect the two sub-systems by using the constraint q1 D x1 C d D xN 1 C l10 C d . Differentiating this constraint twice with respect to time we get the equation of the constraint in the form of (11.4), AqR D b, where A D 1 1 0 b D 0: (11.69) Again by choosing more than the minimum number of coordinates to describe the configuration of the system, and treating them as being independent, we obtain a mass matrix, M , that is singular (positive semi-definite). The description of the constrained mechanical system S thus is provided by the description of the unconstrained system given by (11.68), which has a singular mass matrix, and the constraint AqR D b, where A and b are given by relations (11.69), that is imposed on it. We assume that the constraint is ideal so that C D 0. To obtain the equations of motion for the constrained mechanical system S , we replace the mass matrix of the unconstrained mechanical system M that is positive semi-definite with the augmented mass matrix MAT G that is positive definite of an unconstrained auxiliary system. This augmented mass matrix (see (11.17)) is obtained by again choosing ˛ 2 g D g, N where gN is an arbitrary positive function, as
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2
M AT G
3 m1 C gN gN 0 D M C ˛ 2 AT GA D 4 gN m2 C gN m2 5 > 0: 0 m2 m2
(11.70)
Since b D 0 we choose z .t/ D ˛2 b D 0, and thus QAT G;z D Q. The augmented mass matrix (11.70) is positive definite, so we use the fundamental equation (11.27) to obtain the acceleration of the constrained auxiliary system as 2 k1 xN 1 Ck2 qN2 3 xNR 1 m1 6 k1 xN 1 Ck2 qN2 4 qRN1 5 D 6 m1 4
qRN2 k1 xNm1 Ck2 qN2
3
2
1
k2 qN2 m2
7 7; 5
(11.71)
which is the correct result. This verifies that the dynamic of the constrained mechanical system S is completely mimicked by the dynamic of the constrained auxiliary system SAT G . Although both systems have different mass matrices, the resulting accelerations of the constrained motion of both systems are exactly the same. In the next example, we show how the results obtained in this chapter can be directly applied to obtaining the quaternion equations of rotational motion for rigid bodies in a simple and direct manner. When considering the rotational dynamics of rigid bodies, the use of quaternions removes singularity problems that inevitably arise when using Euler angles. However, the quaternion 4-vector describing a physical rotation is constrained to have unit norm, and hence the equations of motion in terms of quaternions can be considered constrained equations of motion. Example 11.4. Consider a rigid body that has an absolute angular velocity, ! 2 R3 , with respect to an inertial coordinate frame. The components of this angular velocity with respect to its body-fixed coordinate frame whose origin is located at the body’s center of mass are denoted by !1 , !2 , and !3 . Let us assume, without loss of generality, that the body-fixed coordinate axes attached to the rigid body are aligned along its principal axes of inertia, where the principal moments of inertia are given by Ji > 0, i D 1; 2; 3. The rotational kinetic energy of the rigid body is then simply 1 P (11.72) T D ! T J ! D 2PuT E T JE uP D 2uT EP T J Eu; 2 where T (11.73) ! D !1 ; !2 ; !3 J D diag J1 ; J2 ; J3 (11.74) and
T u D u 0 ; u 1 ; u2 ; u3
(11.75)
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is the unit quaternion 4-vector that describes the rotation such that ! D 2E uP D P where 2Eu, 2 3 u1 u0 u3 u2 E D 4 u2 u3 u0 u1 5 : (11.76) u3 u2 u1 u0 We note that the components of u are not independent and are constrained since the quaternion u must have unit norm to represent a physical rotation. Under the assumption, however, that these components are independent, one obtains the unconstrained equations of motion of the system using Lagrange’s equations as M uR WD 4E T JE uR D 8EP T JE uP C u WD Q;
(11.77)
where the 4-vector u in (11.77) represents the generalized impressed quaternion torque. The connection between the generalized torque 4-vector u and the phys T ically applied torque 3-vector B D 1 ; 2 ; 3 , whose components i ; i D 1; 2; 3 are about the body-fixed axes of the rotating body, is known to be given by the relation u D 2E T B :
(11.78)
We note now that the 4 4 matrix M D 4E JE in relation (11.77) of this unconstrained system is singular, since its rank is 3. The unit norm constraint on the quaternion u requires that T
uT u D u20 C u21 C u22 C u23 D 1;
(11.79) which yields A D uT and b D PuT uP . The 4 5 matrix MO T D M j AT has rank 4 since 16E T J 2 E 0 M (11.80) ŒM u D 0 1 uT
is a symmteric matrix whose eigenvalues are 0, 1, 16J12 , 16J22 , and 16J32 . Hence, by Lemma 11.1, the matrix MAT G D M C ˛2 g .t/ u uT given in (11.17) is positive definite, where the arbitrary function g .t/ > 0, 8t and we can choose ˛ to be any positive constant. Using (11.27), we obtain using some algebra the generalized acceleration of the system given by 1 1 1 uR D E T J 1 !J Q ! N .!/u C E T J 1 B ; 2 4 2
(11.81)
where !Q is the usual skew-symmetric matrix obtained from the 3-vector ! and N .!/ is the norm of !. In addition, by virtue of the presence of the unit norm constraint, if we would like to obtain the constraint torque (Qc ) applied to the unconstrained mechanical system (11.77) directly by using the unconstrained auxiliary system, which is obtained by replacing the mass matrix M in (11.77) with MAT G , by Result 2 and (11.18), we would also have to augment the “given” torque Q in (11.77) as
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QAT G;˛2 b D Q ˛ 2 g.t/uN uP ;
(11.82)
where N .Pu/ D uP T uP is the norm of uP . Then using (11.30) and (11.40) the constraint torque for the rotational motion of a rigid body is explicitly obtained by Q c D QAc T G;˛2 b D 2.! T J !/u;
(11.83)
which is the correct constraint torque that the unconstrained system (11.77) is subjected to when the constraint (11.79) is imposed on it. Thus far we have been careful to choose examples that contain a positive semidefinite mass matrix .M 0/. In the next example, we consider a system under the condition that the mass matrix M is positive definite .M > 0/. In this situation, the usual approach for obtaining the constrained equations of motion, the fundamental equation (11.8) is applicable and there is no need to think of the auxiliary system to obtain the equations of motion of the constrained system. However, this example will show that two uncoupled nonlinear systems which can be modeled separately can be altered to a coupled nonlinear auxiliary system that can be much more complicated. Both coupled and uncoupled systems will yield exactly the same constrained dynamics; however, the auxiliary system can hardly be discerned as one that would yield the same dynamics. Example 11.5. Consider two particles of masses m1 and m2 that have no impressed forces acting on them and move in the x y plane. The coordinates of mass m1 in an inertial frame of reference are .x1 ; y1 / and those of mass m2 are .x2 ; y2 /. The vector of coordinates describing the configuration of the system is therefore given T by q D x1 ; y1 ; x2 ; y2 . The two masses are independently constrained to (1) trace elliptical trajectories (with a common focus) with eccentricities "1 and "2 , q xi2 C yi2 D "i xi C pi ; i D 1; 2; (11.84) where the constant pi D "i l ¤ 0, and (2) move along the ellipses so that the sectorial areas which they trace per unit of time are constants, xi yPi yi xP i D ci ; i D 1; 2; (11.85) where ci is a constant. To obtain the relation of the constraints in the matrix form as in (11.4), we differentiate equations (11.84) and (11.85) appropriately with respect to time and obtain the constraint AqR D b;
(11.86)
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where the matrices 2 x1 r1 "1 6 0 AD6 4 y1 0
3 2 2 23 y1 0 0 c1 =r1 6 c 2 =r 2 7 0 x2 r2 "2 y2 7 7 and b D 6 2 2 7 (11.87) 4 0 5 x1 0 05 x2 0 y2 0 q Note that we have used ri D xi2 C yi2 , i D 1; 2 in (11.87). Assuming that the mass m1 moves along the inner ellipse, and the mass m2 moves along the outer ellipse, the mass matrix M D Diagf m1 ; m1 ; m2 ; m2 g
(11.88)
is a constant diagonal matrix. Since there are no “given” forces, the 4-vector Q D Di agf 0; 0; 0; 0 g. The constraints are assumed to be ideal so that C D 0. The dynamical system S comprises the unconstrained mechanical system described by the equation M qR D Q D 0, which is subjected to the constraint equations (11.86) and (11.87). The mass matrix is invertible, and we can use (11.8), to directly obtain the acceleration of the constrained mechanical system S , which is 1 x1 c12 qR D p1 r1 r12
1 y1 c12 p1 r1 r12
1 x2 c22 p2 r2 r22
1 y2 c22 p2 r2 r22
T :
(11.89)
We shall now show that the same acceleration (11.89) of the constrained motion results from a multitude of other systems that are subjected to the same constraints (described by equations (11.86) and (11.87)) as the dynamical system S . As an illustration, we consider the unconstrained auxiliary system (11.19) by replacing the mass matrix M of the unconstrained mechanical system S with the augmented mass matrix MAT G as given in (11.17). We choose, for simplicity, ˛ .t/ D 2 for all time t, z D ˛2 b and the positive definite matrix 3 2 4 1 0 0 61 2 0 17 7 (11.90) G.t/ D 6 40 0 2 05 0 1 0 1 for all t. We then have the augmented mass matrix of the auxiliary system given by 2
MAT G
11 6 12 D M C ˛2 AT GA D 6 4 13 14
12 22 23 24
13 23 33 34
3 14 24 7 7; 34 5 44
(11.91)
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where 11 D m1 C 16.x1 r1 "1 /2 C 8y12 ; 13 D 4.x1 r1 "1 /.x2 r2 "2 /; 22 D m1 C 8x12 C 16y12 ;
12 D 8y1 .x1 2r1 "1 /
14 D 4y2 .x1 r1 "1 /
23 D 4y1 .x2 r2 "2 /;
24 D 4y1 y2
33 D m2 C 4.x2 r2 "2 / C 4.x2 r2 "2 y2 / 2
2
34 D 4y2 .x2 r2 "2 / C 4.x2 C y2 /.x2 r2 "2 y2 / 44 D m2 C 4.x2 C y2 /2 C 4y22 ;
(11.92)
and the augmented “given” force described by QAT G;˛2 b D Q C ˛ 2 AT Gb 2 3 .x1 r1 "1 /.4c12 r22 C c22 r12 / 7 4 6 .4c12 r22 C c22 r12 /y1 7: D 2 26 2 2 2 2 2 2 4 r1 r2 .x2 r2 "2 /.c1 r2 C 2c2 r1 / c2 r1 y2 5 c22 r12 .x2 C y2 / C .c12 r22 C c22 r12 /y 2
(11.93)
When this unconstrained auxiliary system is subjected to the same constraints as the system S , namely, the constraints given by equations (11.86) and (11.87), using equations (11.30) and (11.40) with C D 0 we obtain the explicit constraint force QAc T G:˛2 b acting on the unconstrained auxiliary system, which is, as expected, the same as the constraint force Q c acting on the unconstrained mechanical system S , and which is given by Q c D QAc T G;˛2 b D MAT G BACT G .b AaAT G;˛2 b / .m1 y1 c12 / m2 x2 c22 .m1 x1 c12 / D 2 2 p2 r2 r22 .p1 r1 r1 / .p1 r1 r1 / 1=2
m2 y2 c22 p2 r2 r22
T :
(11.94)
Using equations (11.91), (11.93), and (11.94), the constrained equations of motion of the system can be obtained from (11.39) as R AT G D QAT G;˛2 b C QAc T G;˛2 b : MAT G qs
(11.95)
Pre-multiplying both sides of (11.95) by MA1 T G , the acceleration of the constrained auxiliary system SAT G is obtained, which as expected, is the same as that given by (11.89) for the constrained mechanical system S: Clearly, other (positive) functions ˛ .t/ and other positive definite matrices G .t/ would give other unconstrained auxiliary systems which, when subjected to same constraints as the system S , would yield motions that would be indistinguishable from those of the constrained system S , assuming that both systems start with the
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same initial conditions. We see that these results follow Results in Sect. 11.3, and illustrate that: (1) The explicit acceleration of the constrained auxiliary system SAT G given in (11.27) is the same as the explicit acceleration of the constrained mechanical system S , and, (2) the constraint force Qc acting on the unconstrained mechanical system S , because of the presence of the constraints imposed on it, is the same as the constraint force QAc T G:˛2 b (see (11.40)) acting on the unconstrained auxiliary system SAT G when the unconstrained auxiliary system is subjected to the same constraints as the unconstrained mechanical system S . It should be noted that while the unconstrained mechanical system S that we started with is simple (it has a diagonal mass matrix M , with Q D 0), the unconstrained auxiliary system appears much more complex and has both inertial coupling, as seen from (11.91), as well as force coupling, as seen from (11.93). However, the constrained motion of both of these unconstrained systems, when subjected to the (same) constraints given by relations (11.86) and (11.87), are identical for any given set of initial conditions!
11.5 Conclusion The main contributions of this chapter are the following: 1. In Lagrangian mechanics, describing mechanical systems with more than the minimum number of required coordinates is helpful in forming the equations of motion of complex mechanical systems since this often requires less labor in the modeling process. The reason that we do not usually use more coordinates than the minimum number is because in doing so we often encounter singular mass matrices and then standard methods for handling such constrained mechanical systems become inapplicable. For example, methods that rely on the invertability of the mass matrix (such as the use of the fundamental equation given in (11.8)) cannot be used. 2. Under the proviso that MO T D M j AT has rank n, in this chapter a unified explicit equation of motion for a general constrained mechanical system has been developed irrespective of whether the mass matrix is positive definite or positive semi-definite (singular). This is accomplished by replacing the actual unconstrained mechanical system S with an unconstrained auxiliary system SAT G , which is obtained by adding ˛ 2 AT Gz to the mass matrix M of the unconstrained mechanical system S , and adding AT Gz to the “given” force Q acting on the unconstrained mechanical system S . The mass matrix, MAT G D M C ˛ 2 AT GA, of this unconstrained auxiliary system SAT G is always positive definite irrespective of whether the mass matrix M is positive semi-definite .M 0/ or positive definite .M > 0/. Thus, by applying the fundamental
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4.
5.
6.
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equation to this unconstrained auxiliary system, which is subjected to the same constraints (and initial conditions) as those imposed on the unconstrained mechanical system S , one directly obtains the acceleration of the constrained mechanical system S . The restriction that MO T D M j AT has full rank n, is not as significant a restriction in analytical dynamics as might appear at first sight, because it is a necessary and sufficient condition that the acceleration of the constrained system be uniquely determinable – a condition that is always satisfied in classical mechanics. In fact it provides a useful check on the modeling being done, especially when dealing with complex multi-body systems. We show that the acceleration of the constrained mechanical system S so obtained through the use of the auxiliary system SAT G is independent of the arbitrarily prescribed: (a) nowhere-zero function ˛ .t/; (b) the m-vector z .t/; and (c) the positive definite matrix G .q; t/, provided these are sufficiently smooth (C 2 ) functions of their arguments. In the special case, when ˛ .t/ D 1, z .t/ D 0, and G D Im , the unconstrained auxiliary system simplifies and is the same as the unconstrained mechanical system S except that the mass matrix of the auxiliary system is obtained by adding AT A to that of the unconstrained mechanical system S . Under identical constraints and initial conditions, the accelerations of the constrained auxiliary system and the constrained mechanical system are identical, and the latter can then be obtained from the former. The constraint force Qc acting on the unconstrained mechanical system S (by virtue of the presence of the constraints) can be obtained directly from the constraint force QAc T G:z acting on the unconstrained auxiliary system from the relation Q c D QAc T G:z C AT Gz ˛2 AT Gb. Furthermore, by choosing z D ˛ 2 b when describing the unconstrained auxiliary system, we obtain the simpler result Q c D QAc T G:˛2 b . Thus, when b .q; q; P t/ 0, the choice of z .t/ 0 for the constrained auxiliary system (11.39) and application of (11.40) give both the acceleration of the constrained mechanical system S as well as the constraint force directly. The results (and their derivations) that have been developed in this chapter differ from those in Udwadia and Schutte (2010) in two important respects. (a) They are simpler, because we do not use the generalized Moore–Penrose (MP) inverse of the matrix A in the determination of the unconstrained auxiliary system. Instead, we simply use its transpose. (b) They are more general because we can incorporate the arbitrary function ˛ .t/ and the arbitrary positive definite matrix G .q; t/ in the creation of our unconstrained auxiliary systems. Besides the simplicity and the aesthetic value that result from these differences, there are substantial practical benefits that accrue. Most importantly, these new results provide a major improvement in terms of computational costs since the computation of the transpose of a matrix is near-costless compared to its generalized inverse; this difference in cost becomes increasingly important as the size of the computational model increases. Another advantage is that the
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flexibility in choosing ˛ .t/ and G .q; t/ can become important from a numerical conditioning point of view, especially when dealing with large, complex multibody systems. 8. The results in this chapter point to deeper aspects of analytical mechanics and show that: (a) given any constrained mechanical system S described by the matrices M .q; t/ 0 and A .q; q; P t/, and the column vectors Q .q; q; P t/, b .q; q; P t/, and C .q; q; P t/, (b) there exists a kind of gage invariance whereby there are infinitely many unconstrained systems with positive definite mass matrices given by MAT G D M.q; t/ C ˛2 .t/AT .q; q; P t/G.q; t/A.q; q; P t/ and “impressed” forces given by QAT G;z .q; q; P t/ D Q.q; q; P t/ C AT .q; q; P t/G.q; t/z.q; q; P t/ which, (c) when subjected to the same constraints (both holonomic and nonholonomic, ideal and nonideal) as those on the given mechanical systems S , and when started with the same initial conditions as the given mechanical system S , (d) will be indistinguishable in their motions from those of the given constrained mechanical system S . The arbitrariness of the (nowhere-zero) function ˛ .t/ and that of the matrix G .q; t/ > 0 ensures this gage invariance.
11.6 Key Symbols A m by n matrix of constraint equations a Unconstrained acceleration of actual system aAT G;z Unconstrained acceleration of auxiliary system B Multiplication of the matrices A and M b m-vector of constraint equations C n-vector of the nonideal constraints c Twice the constant sectorial area d Length of the mass m1 E Quaternion coefficient matrix G Arbitrary m by m positive definite matrix g Gravitational acceleration Ic Central moment of inertia of the wheel J Principal moment of inertia ki Stiffness coefficient of the spring
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l li 0 M m N .!/ p D "l Q Qc q; qs qS AT G R r ri S SATG u x xi y z Greek ˛ ˇ ' ! u B "
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Distance between the directrix and focus of an ellipse Unstretched length of the spring ki an n by n mass matrix Mass of the system Norm of ! Constant n-vector given force n-vector constraint force n-vector generalized coordinate of the actual system n-vector generalized coordinate of the auxiliary system Radius of the wheel Rank of matrix A Radial distance of the particle from focus of the ellipse Actual mechanical system Auxiliary system Unit quaternion 4-vector n-vector virtual displacement Horizontal axis of the system Length of the mass mi from the reference wall Vertical axis of the system Arbitrary, sufficiently smooth m-vector
Arbitrary, nowhere-zero, sufficiently smooth real function of time Angle of the inclined surface constraint Angle of the wheel rolling on an inclined surface angular velocity 4-vector generalized impressed quaternion torque 3-vector physically applied torque eccentricity of the ellipse
Symbol C Moore–Penrose (MP) inverse of a matrix
References Appell P (1899) Sur une forme generale des equations de la dynamique. C R Acad Sci III 129:459–460 Appell P (1911) Example de mouvement d’un point assujeti a une liason exprimee par une relation non lineaire entre les composantes de la vitesse. Rend Circ Mat Palermo 32:48–50 Chataev NG (1989) Theoretical mechanics. Mir Publications, Moscow Chen CT (1999) Linear system theory and design. Oxford University Press, New York, pp 73–75
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Dirac PAM (1964) Lectures in quantum mechanics. Yeshiva University Press, New York Gauss C (1829) Uber ein neues allgemeines grundgesetz der mechanik. J Reine Angew Math 4:232–235 Gibbs W (1879) On the fundamental formulae of dynamics. Am J Math 2:49–64 Goldstein H (1981) Classical mechanics. Addison-Wesley, Reading, MA Graybill F (1983) Matrices with applications in statistics. Wadsworth and Brooks, Belmont, CA, USA Hamel G (1949) Theoretische mechanik: eine einheitliche einfuhrung in die gesamte mechanik. Springer-Verlag, Berlin, New York Lagrange JL (1787) Mechanique analytique. Mme Ve Courcier, Paris Pars LA (1979) A treatise on analytical dynamics. Oxbow Press, Woodridge, CT Penrose R (1955) A generalized inverse of a matrices. Proc Cambridge Philos Soc 51:406–413 Sudarshan ECG, Mukunda N (1974) Classical dynamics: a modern perspective. Wiley, New York Udwadia FE, Kalaba RE (1992) A new perspective on constrained motion. Proceedings of the royal society of London A, vol 439, pp 407–410 Udwadia FE, Kalaba RE (1996) Analytical dynamics: a new approach. Cambridge University Press, Cambridge, pp 82–103 Udwadia FE, Kalaba RE (1997) An alternative proof for greville’s formula. J Opt Theory Appl 94(1):23–28 Udwadia FE, Kalaba RE (2001) Explicit equations of motion for mechanical systems with nonideal constraints. ASME J Appl Mech 68:462–467 Udwadia FE, Kalaba RE (2002) On the foundations of analytical dynamics. Int J Nonlin Mech 37:1079–1090 Udwadia FE, Phohomsiri P (2006) Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proceedings of the royal society of London A, vol 462, pp 2097–2117 Udwadia FE, Schutte AD (2010) Equations of motion for general constrained systems in lagrangian mechanics. Acta Mech 213:111–129
Chapter 12
Nonlinear Dynamic of a Rotating Truncated Conical Shell Changping Chen
Abstract Truncated conical shell is an important structure that has been widely applied in many engineering fields. When the structure is in the rotation motion, the vibration and stability problems of the truncated conical shell appear. In this chapter, dynamic study of rotating truncated conical shell will be engaged. Based on the Hamilton’s principle and the Timoshenko–Mindlin hypotheses on moderately thick plates together with the Galerkin method and the harmonic balance method, nonlinear vibration of the truncated conical moderately thick shell is analyzed. With employment of the Melnikov method, the Homoclinic bifurcation and subharmonic bifurcation of the dynamic system is studied. The parametric oscillation of the truncated conical shallow shell under various rotational speeds is investigated under the conditions that the truncated conical shallow shell is rotating around a single axle and excited by a transverse periodic load. The detailed intercoupling characteristics of the high-order and low-order modals of the truncated conical shallow shell system are disclosed by utilizing the Harmonic Balance Method. The internal response properties of the system are investigated by using the Multiple Scale Method. The nonlinear dynamic stabilities of the system are also analyzed by using the Incrementation Harmonic Balance Method.
Nomenclatures hW LW RW
Thickness of the truncated conical shell Generatrix length of the truncated conical shell Mean radius of the shell at the truncated end
C. Chen () Department of Civil Engineering and Architecture, Xiamen University of Technology, Xiamen, Fujian, People’s Republic of China e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 12, © Springer Science+Business Media, LLC 2012
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˛; ˇ W u; v; w W I W N W ai , ei .i D 1; 2; 3/ W V0 W ¨W !1 , !2 , !3 W "x ; " ; "x ; "xz and " z W x ; and x W u x ; u ; uz W 'W W ˝1 W xW !; ; N kN W QW ˝W ˝N W "W ; k W f W t; W H W EW
C. Chen
Rotations of a normal to the middle surface in the x and directions Displacement components at the middle surface of the truncated conical Inertial reference system Relative coordinate system The coordinate components of the base vectors in systems I and N The velocity vector of the original point of system N The rotational velocity of which system N rotates relatively around inertial system I The angular velocities of the truncated conical shell rotating around the X -, Y - and Z-axis of system N Strains on the middle surface The varying values of the curvature Displacements of any point on the shell Semi-vertex angle of the shell Mass density of the shell Angular velocity of the shell around axle x Axle which crosses the centre and parallel to the symmetry axis of the shell Constant coefficients of the equation Transverse uniform load applied to the shell Periodic variety frequency of the transverse load Frequency of varying rotational speed. Perturbation parameter Coefficients related to !; ; N kN and ˝1 Amplitude of transverse uniform load Time variable Hamilton action The Young’s modulus
12.1 Introduction As a common structure, the truncated conical shell has been widely applied in many fields such as space flight, rocketry, aviation, and submarine technology. The static, dynamic, and stability analysis of such shells in large overall motion is very important for their applications, and has been of considerable research interest in recent years.
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The earliest research on conical shells can be found in Leissa’s work in 1973 (Leissa 1973); the vibrational properties of isotropic conical shells were analyzed in this work. Using Raleigh–Ritz method, Chandraseharan and Ramamurti (1981, 1982) researched free vibration problems of axisymmetric and antisymmetric lamination conical shells. Dumir and Khatri (1986) analyzed the nonlinear dynamic response of an axisymmetric isotropic conical shell by using Galerkin method. Xu (1991) and Xu et al. (1995a) researched the nonlinear free oscillations and post bucking of truncated lamination moderate thickness conical shells and symmetrically laminated moderately thick spherical caps under the same condition while considering the rotary inertia and transverse shearing deformation. More dynamic research papers on truncated conical shells can be found in the literature (Sofiyev and Aksogan 2002; Kayran and Vinson 1990; Olshanskii and Sevriukov 1976; Aksogan and Sofiyev 2002). However, the literature concerned with nonlinear dynamic analysis of shells in noninertial reference systems is scant in comparison with that involving the inertial reference system. The dynamic response of plates in the noninertial reference system was first studied by Kane et al. (1983) by using the Kane equation in 1989. He extended the Kane equation to the motion of deformable bodies and founded a new important approach to researching the dynamic problem of a deformable body. Later, Boutaghou (1992) and Bolin (Chang and Shabana 1990) investigated dynamic problems of beams and plates in large overall motion using Hamilton’s principle and the finite element method. In this chapter, the nonlinear dynamic analysis of rotating truncated conical shell is introduced on the basis of the authors’ previous works (Fu and Chen 2001; Chen and Dai 2006). The first section of this chapter uses the Galerkin method and the harmonic balance method to study the nonlinear vibration of the truncated conical shell of moderately thickness, and discusses the effect of the transverse shear deformation and rotatory inertia on its nonlinear vibration. The second section employs the Melnikov method to discuss the Homoclinic orbit bifurcation and subharmonic bifurcation of the dynamic system of the truncated shallow thin conical shell rotating around an axle. The system’s chaotic motion and parametric oscillations under various rotational speeds are also discussed. Finally, the last section studies the general internal dynamic properties of rotary truncated conical shells of thin thickness while considering the intercoupling of the high-order and low-order modals, with implementation of the Harmonic Balance method and Multiple Scale Method.
12.2 Nonlinear Vibration of Elastic Truncate Shells of Moderate Thickness 12.2.1 Governing Equation Development Consider an elastic shell’s motion in an inertial reference system I . O 0 is the origin point of relative coordinate system N , which fixed in the restraint boundary of the
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Fig. 12.1 An element on the middle surface of shell in a relative coordinate system
shells. The shell is referred to the Cartesian coordinate, where ai , ei .i D 1; 2; 3/ are the coordinate components of the base vectors in systems I and N , respectively. Consider any element on the middle surface of shells, which the orthogonal curves coordinate system .˛; ˇ; / with the base vector ci .i D 1; 2; 3/ flowing on the middle surface of shells shown in Fig. 12.1. Suppose r0 D r0 .˛; ˇ; / under the coordinate system N as the position vector of an arbitrary point P on the middle surface at any time, we consider another point PN , which is in the normal direction on the point P , and the distance between P and PN is r. The position vectors of the point PN before and after deformations are r, r, respectively. Further, suppose uN i , uQ i to be the displacement components of the point PN under the Cartesian coordinate system and the orthogonal curvilinear coordinate system, respectively. From Fig. 12.1, the geometric relationship in the N system can be obtained as follows r D r C uN i ei D r C uQ i ei D r0 C c3 C uQ i ci ;
(12.1)
where r D xi ei , r0 D xi0 ei . Given R0 as the position vector between origin point O of system I and origin point O 0 of system N , R is the position vector after deformation of point PN in the system I , then there is R D R0 C r. Then computing the velocity vector V of the point PM after deformation in the inertial reference system I , we obtain VD
dR D V0 C ¨ r CQui ci dt
(12.2)
in which V0 is the velocity vector of the original point of system N , ¨ is the rotational velocity of which system N rotates relatively around inertial system I , “” denotes partial differentiation with respect to time t. As for shells, the coordinate conversion must be made because curved coordinate system is applied in calculating. Consider a truncated conical moderately thick shell in large overall motion with thickness h, semivertex angle ', generatrix length L, and mean radius R at the truncated end shown in Fig. 12.2. In order to simplify the problem, let us take
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
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Fig. 12.2 Geometry of truncated conical shell
the center of truncated conical shell’s top circle as the original point of system N and system N ’s X -axis locates on the truncated conical shell’s symmetric axis, !1 , !2 , !3 are the angular velocities of the truncated conical shell rotating around the X -, Y -, and Z-axes of system N , respectively. Besides, the relation s D R C x sin
(12.3)
can be obtained from Fig. 12.2. The shell is referred to the coordinate system x, , z, where x is the generatrix direction measured from the truncated boundary, is the circumferential direction, and z is the transverse direction measured from the middle surface. Denoting strains on the middle surface to be "x , " , "x , "xz , and " z , the varying values of the curvature are x , , and x , according to the literature (Xu et al. 1995b), the strain– displacement relationship is recorded below: 1 "x D u;x C w2;x ; 2 1 1 " D .v; w cos C u sin / C 2 w2; ; s 2s 1 "x D .u; C sv;x v sin C w;x w; /; s "xz D ˛ C w;x ; 1 .sˇ C v cos C w; /; s x D ˛;x ;
" z D
1 .ˇ; C ˛ sin /; s 1 D .sˇ;x C ˛; ˇ sin /; s
D x
(12.4)
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in which u, v, and w are displacement components at the middle surface of the truncated conical; ˛, ˇ are rotations of a normal to the middle surface in the x and directions, respectively, and a comma denotes partial differentiation with respect to the corresponding coordinates. Based on the Timoshenko–Mindlin hypothesis about moderately thick plates, the displacements ux , u , uz of any point on the shell may be represented by ux D u.x; y; t/ C z˛.x; y; t/; u D v.x; y; t/ C zˇ.x; y; t/; uz D w.x; y; t/:
(12.5)
Based on the classic theory of elastic plate and shell, as for the truncated conical shell, the relations of the membrane forces Nx , N , Nx moments Mx , M , Mx and transverse shear forces Qx , Qy and the middle surface strains may be written as .Nx ; N / D
Eh ."x C " ; " C "x /; 1 2
Nx D N x D
Eh "x ; 2.1 C /
.Mx ; M / D D.x C ; C x /; Mx D M x D .Qx ; Q / D
Eh3 x ; 24.1 C /
Eh ."xz ; " z /; 24.1 C /
(12.6)
where E is the Young’s modulus, is Poisson’s ratio, and D D Eh3 =12.1 2 / is the flexural rigidity of the shell. R Using the Hamilton principle (Arnod 1980) t0 .ıT ıV ıW /dt D 0, in which ıT , ıV , ıW are the variations of a system’s kinetic energy, elastic potential energy and external work, respectively, the following nonlinear equation sets of motion are established for truncated conical moderately thick shells: shnij ŒRuk nkj C ! 2 .xj0 C nkj uk / !j !k .xk0 C nlk ul / C 2!l ejkl nrk uP r C "l ejkl .xk0 C nrk ur / C Fi D 0 a0j
.i; j; k; l D 1; 2; 3/; s h3 nij Œ'Rk nkj C ! 2 . j C nkj 'k / !j !s . s C nrs ur / 12 C2!s ejps nrp 'Pr C "s ejps . p C nrp ur / C Mi D 0 .i; k; r D 1; 2; 3I j; s; p D 1; 2; 3/;
(12.7)
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where "i , !i , and a0i are angle acceleration, angle velocity and components of acceleration in the system N , respectively, and "i D di =dt, i , nij are conversion coefficients of coordinate system, Fi and Mi are items concerned with internal force and moment, respectively, and their expressions are
."2 ; "2 ; "3 / D . sin ; cos cos ; cos sin /; 2 3 cos sin cos sin sin 5; nij D 4 0 sin cos sin cos cos cos sin
(12.8)
F1 D .sNx /;x N sin C .Nx /; ; F2 D N; C Nx sin C .sNx /;x Q cos ; F3 D .sQx /;x C Q; C N cos ' C .sNx w;x C Nx w; /;x 1 C.Nx# w;x / C .N w; /; ; s M1 D Mx sin ' C sMx;x M sin ' C sMx; sQx M2 D M; C 2Mx sin C sMx;x sQ :
(12.9)
Substituting (12.8) and (12.9) into (12.7), neglecting the terms concerned with the large overall motion system, the resulting equations can be the same as those in the literature (Xu et al. 1995b), which are obtained by using the principle of virtual work. In order to simplify the problems, only the case where truncated conical shell rotates uniformly around the axis X will be considered. In addition, the dimensionless parameters are defined as follows: v x ; "D ; 2 R L s s E S D D 2 C sin ; D t ; 2 L h .1 2 / r h2 .1 2 / h R 1 D ; 2 D ; i D !i ; R L E u ; L
W D
w ; R
H D
h D 1 2 ; L
QD
h.1 2 / q: ER
U D
V D
Ei D "i
h2 .1 2 / ; E (12.10)
Substituting (12.10) into (12.7), the dimensionless equations of nonlinear flexural vibration for the truncated conical moderately thick shell under the condition of rotating uniformly around the axis X are obtained in the following:
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UR C 4 2 1 sin ' VP C 21 . cos2 ' U sin2 ' 2 W sin ' cos ' S / .1 /H 2 sin 1 2 2 U; C 2 W; S.U; C 2 W; W; / S 2 1 1 2 2 V; C U sin 2 W cos C 22 W;2 .2 2 V; s 2s 1 2 H 2 .1 / CU; sin ' 2 W; cos ' C 22 W; W; 2 W;2 sin ' s 2s 2S 1 2 2 V; C .U; 2 2 V; sin C 22 W; W; C 22 W; W; / D 0 2 1 1 H 2 2 P P R
1 sin U 1 cos W 1 V 2 2 V; C U; sin : V 2 2 S 2 22 1 H.1 / 2 W; cos C W; W; C S.U; C 2 W; W; / s 4 S cos 6 2 V; sin C 2 2 S V; C U; C 22 W; W; C 22 W; W; S sin . 2 W; C sˇ C U cos / C .U; C 2 2 V sin C 22 W; W; / D 0 S
2 WR C 4 ˝1 cos VP 1 .S C U sin C 2 W cos / cos H 2 U;
2 2 W; sin 2 C 2 W; 2 C 2 2 V; C U sin 2 W cos C 2 W;2 2 S 2S S CW; H 2 W; U; C 22 W; W; C 2 2 V; C U; sin S sin 22 22 sin 2 2 2 V; 2 W; cos C W; W; W; S 2S 2 S2 22 2 H 2 .1 / CU sin 2 W cos C W W; 2 2 V;
2S ; S 1 H 2 .1 / 2 C .U; 2 2 V sin C 2 W; W; / W; 2 2 V;
S S 1 sin ' C .U; 2 2 V; sin ' C 22 .W; W; C W; W; // 2 .U; S S 2 H .1 / 1 2 2 V sin C 22 W; W; / W; C ˛ sin S 2
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
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H 2 .1 / 1 H 2 W; 1 C W; C ˛; 2 2 V; C U sin 2 2 S2 S H 2 W; 1 22 2 22 2 2 2 V; C U; C W;
W 2 W cos C 2S ; 2 S2 S 22 2 C .U W CU; sin 2 W; cos C ; W; ; C 2 W; W; / S 1 H 2 .1 / W; 2 2 V; C C U; 2 2 V; sin C 22 W; W; S S cos H 2 .1 / 1 CW; W; 2 W; C ˇ; C U; 2S 2 S S H 2 cos 1 2 2 2 V; C U sin 2 W cos C 2 W;2 S 2 S 2S 2 C U; C 2 W; 2 Q D 0 2 2 2 .1 / sin P Ts Ri ˛R C 2˝1 sin ˇ ˝1 sin .cos C ˛ sin / Ts H S 1 ˛; .ˇ; C ˛ sin / C S˛; C .ˇ; C ˛; sin / S .1 / 1 sin ˇ; C .˛; ˇ; sin / .ˇ; C ˛ sin / C S 2S S C6.1 /. 2 W; C ˛/ D 0 1 2 2 1 R Ts Ri ˇ C 2 1 sin ˛P 1 ˇ Ts H ˛; C .ˇ; S S .1 / sin 1 C˛; sin / C ˇ; C .˛; ˇ sin / S S sin 1 .1 / .˛; ˇ sin / ˇ; C ˛; ˇ; sin 2 S S2 2 cos W; C ˇ C U D0 (12.11) C6.1 / S S in which Ts and Ri are the tracing constants, the influences of transverse shear deformation and rotary inertia are considered when Ts D Ri D 1 and neglected when Ts D Ri D 0.
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For example, the boundary conditions for the truncated conical shell with two ends (at D 0; 1), both clamped and both simply supported may be written in the following dimensionless forms: W D ˛ D ˇ D U D 0; V D 0 for clamped ends, M D 0 for simply supported ends;
(12.12)
where M D ˛; C
.ˇ; C ˛ sin '/: S
(12.13)
12.2.2 Solution Methodology A multi-mode analysis is carried out in this work. As usual, a solution of (12.11) is sought in the separable form ŒU; V; W D
1 X 1 X
ŒUmn ./; Vmn ./; Wmn ./ Xm . /Yn ./;
mD1 nD0
Œ˛; ˇ D
1 1 X X
Œ˛mn ./; ˇmn ./ Xm . /Yn ./;
(12.14)
mD1 nD0
where Xm , Yn are given by Xm . / D Am .cosh ˇm cos ˇm / C Bm sinh ˇm C sin ˇm ; Yn ./ D cos n;
(12.15)
in which the constant coefficients Am , Bm , and ˇm can be determined by (12.12) and Am D
sinh ˇm sin ˇm ; cosh ˇm cos ˇm
Bm D 1;
1 cosh ˇm cos ˇm D 0
(12.16)
for the clamped ends, and Am D Bm D 0; ˇm D m ;
(12.17)
for the simply supported ends. According to the orthogonality of functions Xi , Yi , there are
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
Z
1
( Xi Xj d D Hij D
0
Z
2
Yi Yj d D 0
1 .2A2i 2
0; Bi2 C 1/ C
Ai ˇj
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.Bi C 1/;
i ¤ j; i D j;
1 ıij ; 2
(12.18)
where ıij is the Kronecker delta function. Substituting (12.14) into (12.11), multiplying the former three of the resulting equations by Xp . /Yq ./, the fourth by Xp0 . /Yq ./, the Xp . /Yq0 ./, integrating them from 0 to 1 with respect to and from 0 to 2 with respect to , the following five sets of equations about Upq ./, Vpq ./, Wpq ./, ˛pq ./, and ˇpq ./ are obtained: m mn UR pq C C1 1 VPpq C .C2 21 ımp C C3p /Ump C C4pq Vmn C .C5 21 ımp m mnkl CC6p /Wmq C C7pq Wmn Wkl D 0;
(12.19)
m mn VRpq C C8 1 UP pq C C9 1 WP pq C .C10p 21 ımp /Vmq C C11pq Umn mn mn mnkl CC12pq Wmn C C13pq ˇmn C C14pq Wmn Wkl D 0
(12.20)
mn mn mn WR pq C C15 1 VPpq C .C16pq C17 21 ımp ınq /Umn C C18pq Vmn C .C19pq mn mn mnkl C20 21 ımp ınq /Wmn C C21pq ˛mn C C22pq ˇmn C C23pq Umn Wkl mnkl mnkl mnkl CC24pq Vmn Wkl C .C25pq C C26pq /Wmn Wkl mnklrs CC27pq Wmn Wkl Wrs C Qpq D 0
(12.21)
mn mn P mn mn ˛R mn C 1 C29pq C C31pq /˛mn ˇmn C . 21 C30pq C28pq mn mn CC32pq ˇmn C C33pq Wmn D 0
(12.22)
mn R mn mn mn ˇmn C 1 C35pq C34pq ˛P mn C . 21 C36pq C C37pq /ˇmn mn mn mn CC38pq ˛mn C C39pq Umn C C40pq Wmn D 0:
(12.23)
This is a set of complicated nonlinear ordinary differential equations. C1 C40 are the integrating constant coefficients that are not given here. The present value of angular velocity 1 causes the damped items and makes the five sets of equations couple with each other. Here, the harmonic balance method will be used to obtain the relationship of the nonlinear frequency and the amplitude. Thus, the time functions Upq , Vpq , Wpq , ˛pq , and ˇpq are expanded into Fourier series in as Upq D
1 X j D1
.j / .j / .apq cos j C bpq sin j/;
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Vpq D
1 X
.j / .j / .cpq cos j C dpq sin j/;
j D1
Wpq D
1 X
.j / .epq cos j C fpq.j / sin j/;
j D1
˛pq D
1 X
.j / / .gpq cos j C h.j pq sin j/;
j D1
ˇpq D
1 X
.j / .j / .kpq cos j C spq sin j/;
(12.24)
j D1
in which N D ; N N 0/ D . ;
p
E=H 2 .1 2 /.!; N !N 0 /;
(12.25)
N N 0 are dimensionless nonlinear and linear vibration frequencies and !, where , N !N 0 are dimensional nonlinear and linear frequencies, respectively. Substituting (12.24) into (12.19)–(12.23) multiplying both sides of them by cos i and sin i, integrating them from 0 to 2 , a set of simultaneous nonlinear algebraic equations is obtained. According to this set of equations, the relationship between nonlinear vibration frequency and amplitude for a truncated conical moderately thick shell may be determined.
12.2.3 Numerical Results and Discussions In Figs. 12.3–12.6, n is the number of waves along the circumference, ˝N 0 is the dimensionless fundamental linear frequency, wmax = h is the dimensionless transverse amplitude, and wmax = h D R max.W /= h. The elastic constants used in the calculation are v D 0:3, E D 20:6 104 N=mm2 , D 7:85 109 s2 =mm4 and the results obtained by neglecting the effects of transverse shear deformation and rotatory inertia are represented by soled curves (Ts D Ri D 0/ and those considering these effects by dotted curves (Ts D Ri D 1/. Since the results in large overall motion for the nonlinear vibration of truncated conical moderately thick shells are hardly found in the literature, the present results for a truncated conical shell in an inertial reference system with the semi-vertex angle being zero (circular cylindrical shell) are compared in Fig. 12.3 with the available data (Jian and Zhan 1991). The effects of transverse shear deformation and rotatory inertia are not considered for the purpose of comparison. It can be seen that the corresponding response curves are nearly in coincidence.
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell 1.20 Present Literature[6] 1.10 Ω Ω0
Fig. 12.3 Comparison of nonlinear frequency of simply supported cylindrical shell (˝1 D 0, R=L D 1:0, R= h D 25, ' D 0, n D 3)
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1.00
0.90 0.0
0.5
1.0
1.5
2.0
2.5
3.0
wmax / h
Fig. 12.4 Effects of angle velocity on nonlinear frequency of simply supported truncated conical shell (R=L D 1:0, R= h D 25, ' D 30, n D 3)
Fig. 12.5 Effects of boundary conditions on nonlinear frequency of truncated conical shell (˝1 D 1, R=L D 0:5, R= h D 20, ' D 30, n D 3)
Figure 12.4 shows the effects of angle velocity on the amplitude-frequency response curves for a simply supported truncated conical shell. It is obvious that the effects of angular velocity ˝1 in large overall motion reduce the system’s dynamic rigidity, decrease the nonlinear frequency, and soften the response curve trends. The effects of boundary conditions on nonlinear transverse vibration frequency of truncated conical shell in large overall motion are shown in Fig. 12.5, in which the curves exhibit the hardening type of nonlinearity for the clamped shell but softening
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Fig. 12.6 Effect of radius-to-length ratio on nonlinear frequency of clamped truncated conical shell (˝1 D 1, R= h D 20, ' D 30, n D 3)
Fig. 12.7 Effect of semi-vertex angle on nonlinear frequency of simply supported truncated conical shell (˝1 D 1, R=L D 0:5, R= h D 20, n D 3)
type for the simply supported shell. However, because of the existence of damping item produced by the shell’s noninertial motion, there are small fluctuations at the beginning of the curves for both cases. As expected, the effects of transverse shear deformation and rotatory inertia are more significant for the clamped ends than for the simply supported ends. Figure 12.6 shows the amplitude-frequency response curves of a clamped truncated conical shell with different values of the radius-to-length ratio R=L, in large overall motion. It can be seen that, with increasing the ratio, the softening of response curves is more obvious, while effects of transverse shear deformation and rotatory inertia become weaker. The amplitude-frequency response curves for a simply supported truncated conical shell with different semi-vertex anglers ' in large overall motion are shown in Fig. 12.7. The linear frequency of truncated conical shell decreases clearly with increasing '. Figure 12.8 shows the effects of different radius-to-thickness ratios R= h on amplitude-frequency response curves for a simply supported truncated conical shell in large overall motion. Under the condition of constant value R=L, the effects of transverse shear deformation on response curve decrease with the increasing of radius-to-thickness ratio R= h. When R= h tends to a higher value, the truncated conical moderately thick shell is near to the case of thin shell, in which its transverse shear deformation can be omitted.
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Fig. 12.8 Physical relationship of the truncated conical shallow shell
12.3 Nonlinear Vibration and Homoclinic Orbit Bifurcation of a Shallow Rotating Truncated Conical Shell of Thin Thickness 12.3.1 Equation of Motion A truncated shallow conical shell rotating around an axle is considered here and its geometric relationship is shown in Fig. 12.8. The thickness of the shell is h, generatrix length is L, mean radius of the shell at the truncated end is R, semi-vertex angle is ', and the mass density is . The conical shell uniformly rotates around the axle x with an angular velocity ˝1 . According to the previous section, just by considering the transverse motion of the system, the governing equation of the shell can be simplified and written as: N 3 C Q D 0; wR C .! 2 ˝ N 12 /w C kw
(12.26)
where Q is the transverse uniform load applied to the shell, w is the transverse displacement of the shell, and !; ; N kN are the constant coefficients. The expression can be retrieved from the previous section. Transforming the style of (12.26), suppose t D !, Q D "f ! 2 cos ˝, where ˝ is the periodic variety frequency of the transverse load and " is the perturbation parameter. Equation (12.26) can be expressed as: (12.27) wR C .1 /w C kw3 D "f cos ˝; N 2 ; D ˝ in which is the rotating parameter and k D k=! N 12 =! 2 , where > 0; k > 0.
12.3.2 Homoclinic Orbit Bifurcation Suppose x1 D w; x2 D w, P the (12.27) can be translated into the following form: xP 1 D x2 ; xP 2 D . 1/x1 kx13 C "f cos ˝t:
(12.28)
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When " D 0, (12.28) can be simplified to the following forms: xP 1 D x2 ; xP 2 D . 1/x1 kx13 :
(12.29)
Equation (12.29) is the typical Hamilton system, its Hamilton action is: H D
1 1 2 1 x . 1/x12 C kx14 : 2 2 2 4
(12.30)
Taking a singular analysis to the Hamilton system (12.29), suppose xP 1 D 0; xP 2 D 0, then the following equations can be derived from (12.29): x2 D 0; . 1/x1 kx13 D 0;
(12.31)
because > 0; k > 0. According to the above equations, the Hamilton’s singular distribution may have several cases: 1. When > 0,p the system will have three singular points which are point (0, 0) andppoints (˙ . 1/=k; 0/, where point (0, 0) is the saddle point and points (˙ . 1/=k; 0/ are the center points. 2. When < 1, the system will only have one singular point, (0, 0) which is a center point. The path of the phase-plane diagram around the singularity is composed of a set of concentric closed-loop curves. 3. When D 0, the system will only have one singular point, (0, 0) and it is higher than the singular point. According to the singular point analysis above, it can be stated that a Homoclinic orbit exists through saddle point P .0; 0/ in Case 1. In light of (12.30) the Hamilton action H D 0. Therefore, from (12.29) and (12.30), the expressions for the undisturbed Homoclinic orbit can be expressed as: r p 2. 1/ x1 D sech. 1t/; k r p p 2 x2 D . 1/ (12.32) sech. 1t/th. 1t/; k and the corresponding Melnikov function can be given as: Z M.t0 / D
1 1
r D
x2 .t/f cos ˝.t C t0 /dt ˝
2 ˝ ch 2.1/ sin ˝t0 ; Œt0 2 .0; T / : ˝ k ch 1 C1
(12.33)
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365
From the above expression, it can be seen that: 1. When all the values of max M.t0 / and min M.t0 / are positive or negative, the single zero point will not exist in expression (12.33). Therefore, the stable manifold W s and instable manifold W u crossing through the saddle point P will never intersect. 2. When the product of max M.t0 / and min M.t0 / is negative, the simple zero points will form in expression (12.33). Consequently, the cross-sections of stable manifold W s and instable manifold W u crossing saddle point P will intersect at nondegenerate Homoclinic points. The system (12.28) will experience Homoclinic orbit bifurcation and the Smale horse-shoe will appear in the phase plane of the system. Then the system will enter Smale chaotic motion. 3. When either max M.t0 / or min M.t0 / is zero and appears as a secondary zero point in M.t0 / for some t0 the stable manifold W s and instable manifold W u will bring about secondary Homoclinic tangency.
12.3.3 Subharmonic Bifurcation of Resonance Periodic Orbits According to the above analysis, two sets of parametric periodic orbits will appear inside the Homoclinic orbit due to expressions (12.29) and (12.30). The orbits can be expressed as: s # r "r 2. 1/ 1 1 x1p D K t ˙ sn ; .1 C K 2 /k 1 C K2 k # "r "r # r K. 1/ 2 1 1 t dn cn t ; (12.34) x2p D 1 C K2 k 1 C K2 1CK where sn, cn, and dn are the Jacobi elliptic functions, K.p/ is the first complete elliptical integral and p satisfies the following equation: Hp D
p 2 . 1/2 ; .1 C p 2 /k
(12.35)
where Hp is the Hamilton action inside the Homoclinic orbit of the system. The period of the periodic orbit is given by: s 1 C p2 : (12.36) T .p/ D 4K.p/ 1 Therefore, the condition for resonance is: s T .p/ D 4K.p/
m2 1 C p2 D ; 1 n˝
(12.37)
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where m fraction. As p and n are integers which cannot be reduced to a common p 1, the system will primarily resonate; when ˝ D p2 1, the ˝ D system will enter 1=2 subharmonic resonance; when ˝ D 1=2 1, ultraharmonic resonance will occur. The subharmonic Melnikov function corresponding to expression (12.34) can be shown as: Z
nT .p/
M m=n .t0 / D
x2p .t/f cos ˝.t C t0 /dt
0
Z
nT .p/
D 0
( D
m 2 K
K. 1/ 1 C K2
q
r
2s 3 2s 3 2 4 1 5 4 1 5 cn t dn t k 1 C K2 1 C K2 n ¤ 1 and m must be an even number,
0; 2.1/ k.1CK 2 /
cosh
m K 0 2K
cos ˝t0 ; n D 1 and m must be an odd number, (12.38)
where K 0 D K 0 .p/ and K 0 is the secondary complete elliptic integral. Therefore, according to expression (12.38), the following conclusions can be obtained: 1. When n ¤ 1 and m is an even number, M m=n .t0 / will always be zero for any t0 . Accordingly, higher-order even subharmonic bifurcation will occur in the perturbing system. 2. When n D 1 and m is an odd number, there are three cases as follows: a. When any value of max M m=n .t0 / or min M m=n .t0 / is positive or negative, there will not be a simple zero point in M m=n .t0 / for any t0 . Therefore, the m=n harmonic will not take place in the perturbed system (12.28). b. When a simple zero point occurs in the expression for M m=n .t0 /, the perturbed system (12.28) will make a transition into a chaotic state by infinite odd subharmonic bifurcation. c. When either max M m=n .t0 / or min M m=n .t0 / is zero, secondary zero points will exist in the expression for M m=n .t0 /. Hereby, the subharmonic saddle point bifurcation will occur in the perturbed system (12.28).
12.3.4 Numerical Simulation of Chaotic Motion Chaotic motion has specific number characteristics. Therefore, it is important to examine the character of chaotic motion by means of numerical simulation. This can uncover criteria to identify whether or not a steady motion will lead to chaos. The following figures depict the numerical results of the system (12.28) by using a newly developed method called the P–T method (Dai and Singh 2003), the characteristics of the phase plane figure, time history curves and Poincare map are used to judge whether or not chaotic motion will take place.
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell Fig. 12.9 (a) Time history curve. (b) Phase plane curve. (c) Poincare map
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a
Time history curve
b
Phase plane curve
c
Poincare map
1. When the parameters D 13:0, ˝ D 6:0, ultra-harmonic resonance will occur in System (12.28). The other parameters take on the following values v D 0:3, E D 20:6 104 N=mm2 , D 7:85 109 s2 =mm2 , R=L D 0:5, R= h D 25, " D 0:01, f D 100:0, x10 D 0:1, x20 D 0:4. The results are shown in Fig. 12.9. 2. Figure 12.10 shows the principle resonance of the system (12.28) when the parameters are D 8:0, ˝ D 7:0, D 0:3 E D 20:6 104 N=mm2 , D 7:85 109 s2 =mm2 , R=L D 0:5, R= h D 20, " D 0:01, f D 115:0, x20 D 0:5. 3. The results of 1/2 subharmonic resonance are shown in Fig. 12.11 with the following parameters D 7:0, ˝ D 12, D 0:3, E D 20:6 104 N=mm2 , D 7:85 109 s2 =mm2 , R=L D 0:5, R= h D 30, " D 0:01, f D 85:0, x10 D 0:2, x20 D 0:5.
368 Fig. 12.10 (a) Time history curve. (b) Phase plane curve. (c) Poincare map
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a
Time history curve
b
Phase plane curve
c
-6
Poincare map
12.3.5 Nonlinear Parametric Oscillation of the Shell Consider the truncated conical shallow shell shown in Fig. 12.8. Suppose the rotary speed of the shell is no longer a constant but varies with time according to the following form: N ˝1 ./ D cos ˝; (12.39)
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell Fig. 12.11 (a) Time history curve. (b) Phase plane curve. (c) Poincare map
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a
Time history curve
b
Phase plane curve
c
Poincare map
in which is the amplitude of the rotary speed and ˝N is its frequency. Neglect the transverse uniform load applied to the shell and assume that and k are small variables, the governing equation of the shell can be written as: N C "kw3 D 0; wR C ! 2 w C ˛"2 cos2 ˝w
(12.40)
where ˛ is an integral constant of the system. Due to the variation of the rotary speed of the shell, the parametric phenomenon will occur even if the harmonic load on the shell were existent. In the present research, the Multiple Scale Method is employed in (12.40). Two timescales are taken as:
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T0 D ; T1 D ":
(12.41)
The transverse displacement of the shell can be expanded into a power series: w D w0 C "w1 C
(12.42)
Substitute (12.42) into (12.40) and then equate the coefficients with the same power of variable ". The perturbation equations are hereby obtained: "0 W "1 W where D0 D
D02 w0 C ! 2 w0 D 0;
(12.43)
N 0 w0 kw30 ; (12.44) D02 w1 C ! 2 w1 D 2D1 D2 w0 ˛2 cos2 ˝T
@ ; D1 @T0
D
@ . @T1
The solution to (12.43) can be expressed as:
W0 D A.T1 / exp.i!T0 / C cc;
(12.45)
where cc denotes conjugate items in (12.45). Substitute (12.45) into (12.44), the following equation can be obtained: 1 D02 w1 C ! 2 w1 D 2i!A0 .T1 / exp.i!T0 / ˛2 A0 .T1 / exp i.2˝N C !/T0 4 1 2 N0 1 ˛ A .T1 / exp i.2˝N !/T0 ˛2 A.T1 / exp.i!T0 / 4 2 3 2 N (12.46) kŒA exp.3i!T0 / C 3A A exp.i!T0 / C cc: Set the pumping frequency ˝N to approach the natural frequency ! of the shell: ˝N D ! C ":
(12.47)
Here, is a perturbation parameter. Substitute the above expression into (12.46), to erase the permanent items, and the following equation can be achieved: 1 1 2i!A0 .T1 / C ˛2 AN exp.iT1 / C ˛2 A C 3kA2 AN D 0: 4 2
(12.48)
To solve (12.48), expand the variable A.T1 / into the complex forms: A.T1 / D a.T1 / exp iˇ.T1 /:
(12.49)
Substitute (12.49) into (12.48), separate the imaginary part and real part and then equate them, the following equations can be obtained: 1 1 2a!ˇ 0 C ˛2 a cos C ˛2 a C 3ka3 D 0; 4 2
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell Fig. 12.12 Amplitude frequency response curves of a truncated conical shell
371
a
=0
=8 = 15
1 2!˛ 0 C ˛2 a sin D 0; 4
(12.50)
where is the difference phase of the system and its expression as follows: D 2ˇ C T1 :
(12.51)
In the case of stable vibration, the amplitude and phase difference of the system should not vary with time, that is a0 D 0 D 0:
(12.52)
Therefore, (12.50) takes the following forms: a sin D 0; 1 1 a! C ˛2 a cos C ˛2 a C 3ka3 D 0: 4 2
(12.53)
Due to the first (12.53), there are two possible solutions that exist: 1. a D 0, which means the transverse amplitude of the shell is zero. At this time, no response takes place and the solution is trivial. 2. sin D 0; which means D n ; n D 0; 1. According to the second equation of (12.53), the relationship between the amplitude and frequency response of the system can be determined as follows: 1 3ka2 C ˛2 .2 C cos n / ! D 0; 2
.n D 0; 1/:
(12.54)
The amplitude and frequency response curves obtained from (12.54) are shown in Fig. 12.12. The frequency response curves slope toward the right side with increasing frequency. The amplitude and frequency response curves also vary with different amplitudes of angular velocity. The curves move toward the right with an increase of .
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Fig. 12.13 Main dynamic area of instability of a truncated conical shell
R / L = 0.5 R / L = 0.2
During the period that the trivial solution corresponding to a D 0 bifurcates to a nontrivial solution, the effect of varying the parameter on the stability of the system can be determined by the derivative of about a. After obtaining the derivative, the points of intersection between the axes and the tangent of the a curves can be obtained. The points of intersection are also the points, where a D 0. Therefore, set a D 0 in (12.54), the relationship between parameter and parameter can be obtained as follows: 1 2 ˛ .2 ˙ 1/ ! D 0: 2
(12.55)
Figure 12.13 shows the stability curves consisting of and the area contained by the two curves is the area of instability. The main area of instability increases when the ratio of the diameter and length of the shell is increased.
12.4 General Nonlinear Vibration Analysis of a Rotary Truncated Conical Shell of Thin Thickness 12.4.1 Equation of Motion A rotary truncated conical shell is the same to the one in previous section shown in Fig. 12.8, the shell thickness is h, the generatrix length is L, mean radius of the shell at the truncated end is R, the semi-vertex angle is ', and the mass density is . The conical shell uniformly rotates around the axle x with an angular velocity 1 . Based on the governing equations derived from the previous two sections, when the shell uniformly rotates around the x axis, the rotary speeds around the y and z axes and the rotary accelerations around the three axes can all be set to zero. The governing equations thereby can be simplified to the following forms: mn mn mn mn uR pq CC2pq ˝1 vP mn C .˝12 C10pq C C39pq /umn C .˝12 C11pq mn mnkl CC41pq /wmn C C42pq wmn wkl D 0
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
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mn mn mn vR pq CC44pq ˝1 uP mn C C47pq ˝1 wP mn C .˝12 C50pq mn mn CC80pq /vmn C C79pq umn D 0 mn mn mn mn mn ˝1 vP mn C .˝12 C116pq C C149pq /wmn C .˝12 C115pq C C147pq /umn wR pq CC108pq mnkl mnkl mnklst CC150pq umn wkl C C152pq wmn wkl C C153pq wmn wkl wst C Qpq D 0
(12.56)
where C2 to C153 are the integrating constant coefficients. The coefficients can be derived with the system parameters and the geometric and material properties. Qpq is the transverse uniform load applied to the shell.
12.4.2 Internal Resonance Between Low- and High-Order Modals of the System The critical speed of the system is defined as follows: p p p
cr D Min. C14911 =C11611 ; C3911 =C1011 ; C8011 =C50611 /:
(12.57)
In order to maintain stability under static conditions, the rotary speed 1 of the shell is assumed to be less than cr . The case where !13 D 3!11 is considered first. Assume the external excitation is a harmonic force and it can be expressed in the following form: Qpq D fpq cos :
(12.58)
When tends to !13 , 1/3 subharmonic resonance and internal resonance will take place in the rotary system. To solve the (12.56), the Harmonic Balance Method is employed and sets ˛D
1
: 3
Substituting this into (12.56), the following equations can be obtained: 1 2 1 mn mn mn ˝1 Pvmn C .˝12 C10pq C C39pq /umn ˝ uR pq C C2pq 9 3 mn mn mnkl C.˝12 C11pq C C41pq /wmn C C42pq wmn wkl D 0;
1 mn 1 mn 1 2 ˝1 Pumn C C47pq ˝1 w P mn
vR pq C C44pq 9 3 3 mn mn mn C.˝12 C50pq C C80pq /vmn C C79pq umn D 0;
(12.59)
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1 mn 1 2 mn mn ˝1 Pvmn C .˝12 C116pq C C149pq /wmn
wR pq C C108pq 9 3 mn mn mnkl mnkl C.˝12 C115pq C C147pq /umn C C150pq umn wkl C C152pq wmn wkl mnklst CC153pq wmn wkl wst C fpq cos 3˛ D 0:
(12.60)
Suppose the (12.60) have the following forms as series solutions: upq D
1 X
.j / .j / .apq cos j˛ C bpq sin j˛/;
j D1
vpq D
1 X
.j / .j / .cpq cos j˛ C dpq sin j˛/;
j D1
wpq D
1 X
.j / .epq cos j˛ C fpq.j / sin j˛/:
(12.61)
j D1
Substituting the above expressions into (12.60) and performing Galerkin integration, six sets of complex algebraic formulation sets can be obtained, which are coupled to each other. The amplitude corresponding to each mode of the system can be defined as: q .j / .j / / D .apq /2 C .bpq /2 ; A.j pq q .j / .j / .j / D .cpq /2 C .dpq /2 ; Bpq q .j / .j / .j / Cpq D .epq /2 C .fpq /2 : (12.62) Due to the assumption, the internal resonance will occur between the (1,1) modal and (1,3) modal. Their relationships with the external exciting frequency and .j / .j / .j / Apq ; Bpq ; Cpq .j D 1; 3; p D 1; q D 1; 3/ can be decided by implementing a numerical calculation. For the sake of accuracy and numerical convergence, a newly developed numerical method (Dai and Singh 2003) is employed in the numerical calculations to obtain the response of the conical shell. Figure 12.14 shows the internal resonance phenomena between the (1, 1) modal and (1, 3) modal under the external exciting force with varying frequency in the transverse direction of the rotary shell. It can be seen from the figure that the amplitudes on the transverse direction are excited to rather large values because of the internal resonance between the (1, 1) modal and (1, 3) modal. The values of both .1/ .3/ C11 and C13 increase with an increasing frequency ratio =3!11 . However, as the .1/ frequency ratio ˝=3!11 reaches certain range, the value of C11 begins to decrease .3/ and C13 increases at a faster rate. After reaching a multiple of 1.5 of the frequency
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
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Fig. 12.14 Internal response between the (1,1) modal and (1,3) modal on the transverse direction of the rotary shell
Fig. 12.15 Internal response between the (1,1) modal and (1,3) modal on the circle direction of the rotary shell
ratio =3!11 , the system presents instability and the solutions of the system become complex. The internal resonance characteristics between the (1,1) modal and (1,3) modal in the circular direction of the rotary shell are shown in Fig. 12.15. From the figure, we can see that the internal resonance phenomenon between the (1, 1) modal and (1, 3) modal of the shell also occurred in the circular direction because of the .1/ .3/ effects of rotating the shell. The amplitudes B11 and B13 in the circumferential direction are excited to a rather large value due to the internal resonance of the system. Figure 12.16 shows the internal resonance characteristics between the (1,1) modal and (1,3) modal in the longitudinal direction of the rotary shell, which we can see that the internal resonance phenomenon between the (1, 1) modal and (1, 3) modal of the shell also occurred in the longitudinal direction because of the rotating
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Fig. 12.16 Internal response between the (1,1) modal and (1,3) modal on the longitudinal direction of the rotary shell
the shell. Figures 12.15 and 12.16 also showed that the effects of displacements in the circumferential and longitudinal directions are not negligible in the dynamic analysis of the system.
12.4.3 Internal Resonance Analysis of a Rotary Shallow Truncated Conical Shell To deeply expose the dynamic characteristics between the low and high modals of the rotary truncated conical shell, a special case where the semi-vertex angle ' of the shell tends to 90ı is considered. In this case, the governing equations of the system can be simplified to the following forms: mn mn mnkl uR pq C C2pq ˝1 vP mn C C39pq umn C C42pq wmn wkl D 0; mn mn mn mn ˝1 uP mn C .˝12 C50pq C C80pq /vmn C C79pq umn D 0; vR pq C C44pq mn mn mnkl C C149pq /wmn C C150pq umn wkl wR pq C .˝12 C116pq mnklst CC153pq wmn wkl wst C Qpq D 0:
(12.63)
Due to the analytical result in the aforementioned section, the vibration in the transverse direction is essential among the three directions especially when the semi-vertex angle ' closes to 90ı . Therefore, the inertial and gyroscopic items from (12.63) in the circular and longitudinal directions can be ignored. Then the (12.63) can be combined into a set of equations about the transverse direction of the system as follows:
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
377
mm 2 mnklst w R pq C .!pq / wmn C "Dpq wmn wkl wst C Qpq D 0
.m; n; k; l; s; t; p; q D 1; 2; 3; : : :/; in which mn !pq D
q
(12.64)
mn mn ˝12 C116pq C C149pq
(12.65)
are the dimensionless natural frequencies of the rotary truncated conical shallow shell. " is a small constant whose magnitude is consistent with the amplitude of mnklst are nonlinear coefficients of the the system in the transverse direction. "Dpq governing equations and mnij
iNjN
mnklst mnklst klst Dpq D C153pq C42pq C150 =C39ij : iNjN
(12.66)
Therefore, the equations in (12.64) are considered order differential equations with weak cubic nonlinear items. The multiple scale method can be implemented to solve the equations in (12.64). Set the timescale as: T0 D ; T1 D "; : : :
(12.67)
and expand wpq as a power series about ": .1/ 2 wpq D w.0/ pq C "wpq C O." /:
(12.68)
Define the differential operator: D0 D @=@T0 ; D1 D @=@T1 :
(12.69)
The external excitations are also assumed to be harmonic forces and their expression is as follows: Qpq D "fpq cos ; (12.70) in which is the frequency of the harmonic external force. By submitting expressions (12.68) into (12.64) and equating the coefficients of the same power about ", the following perturbation equations can be obtained: 2 .0/ "0 W D02 w.0/ pq C !pN qN wpq D 0;
(12.71) .0/
.0/
2 .1/ .0/ mnklst .0/ "1 W D02 w.1/ wmn wkl wst pq C !pN qN wpq D 2D0 D1 wpq Dpq
fpq cos T0 :
(12.72)
The solution to (12.71) can be expressed as: w.0/ pq D ApNqN .T1 / exp.i!pq T0 / C cc;
(12.73)
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in which cc denotes conjugate items of expression (12.73). Place expression (12.73) into (12.72) and the following equation can be obtained: 2 .1/ 0 mnklst D02 w.1/ ŒAmN nN .T1 / pq C !pNqN wpq D 2i!pq ApN qN .T1 / exp.i!pq T0 / Dpq
exp.i!mn T0 / C ANmN nN .T1 / exp.i!mn T0 / ŒAkN lN.T1 / exp.i!kl T0 / CANkN lN.T1 / exp.i!kl T0 / ŒAsNtN.T1 / exp.i!st T0 / C ANsNtN.T1 / exp.i!st T0 / 1 fpq exp.i T0 / C cc; .m; n; k; l; s; t; p; q D 1; 2; 3; : : :/: 2
(12.74)
It can be seen from (12.74) that, in the general case, multiple modals of the shell are associated with each other when the internal resonance takes place. For the sake of simplification, in the present research, the interactions of internal resonances between the (1, 1), (1, 2), (2, 1), (2, 2) modals of the system are considered. Assume !12 D !21 , according to the theory of vibration the internal resonance of the shell may take place under the condition of !11 C !22 D 2!12 . Therefore: !11 C !22 D 2!12 C "1 :
(12.75)
Here, 1 is a variable tuning parameter. In addition, suppose the external excitation frequency tends to !22 , that is:
D !22 C "2 :
(12.76)
Here, 2 is another variable tuning parameter. By placing (12.75) and (12.76) into (12.74) and removing the permanent items the following equations can be obtained: 221212 2 N 111212 2i !11 A011 12D11 A12 A22 exp.i1 T1 / 3A11 .8D11 A12 AN12 112222 111111 C2D11 A22 AN22 C D11 A11 AN11 D 0; 221112 121212 A11 AN12 A22 exp.i1 T1 / 6A12 .4D12 A12 AN12 2i!12 A012 12D12 121111 122222 C2D12 A11 AN11 C 2D12 A22 AN22 / D 0; 111212 N 221111 A11 A212 exp.i1 T1 / 3A22 .2D22 A11 AN11 2i !22 A022 12D22
1 111212 222222 C8D22 A12 AN12 C D22 A22 AN22 / f22 exp.i2 T1 / D 0: 2 iˇ
(12.77)
Place Apq D apq epN qN ; .p; q D 1; 2/ into (12.77), then separate the real and imaginary parts and equate them: 0 221212 2 111212 2 2!11 a11 ˇ11 12D11 a12 a22 cos 1 24D11 a11 a12 122222 2 111111 3 6D11 a11 a22 3D11 a11 D 0; 0 122222 2 C 12D11 a12 a22 sin 1 D 0; 2!11 a11
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0 221112 121212 3 2!12 a12 ˇ12 12D12 a11 a12 a22 cos 1 24D12 a11 a12 121111 2 122222 2 12D12 a12 a11 12D12 a22 D 0; 0 221112 C 12D12 a11 a12 a22 sin 1 D 0; 2!12 a12 0 111212 2 221111 2 12D22 a11 a12 cos 1 6D22 a22 a11 2!22 a22 ˇ22
1 221212 2 222222 3 24D22 a22 a12 3D22 a22 f22 cos 2 D 0; 2 1 0 111212 2 C 12D22 a11 a12 sin 1 C f22 sin 2 D 0; 2!22 a22 2
(12.78)
where 1 and 2 are the differences in phase of the system and their expressions are as follows: 1 D 2ˇ12 ˇ22 ˇ11 1 T1 ; 2 D ˇ22 C 2 T:
(12.79)
Considering that the equations in (12.78) are the governing equations which decide the amplitude and phase of the shell system, according to the theories of vibration, the following conditions must come into existence under the condition of steadystate motion of the shell: 0 0 0 D a12 D a22 D 0; a11
10 D 20 D 0:
(12.80)
Substituting (12.80) into (12.78), the governing equations decide the steady-state solutions of the system and can be written as follows: 2 a12 a22 sin 1 D 0;
a11 a12 a22 sin 1 D 0; 1 111212 2 12D22 a11 a12 sin 1 C f22 sin 2 D 0; 2 111212 2 221111 2 2!22 a22 2 12D22 a11 a12 cos 1 6D22 a22 a11
1 221212 2 222222 3 24D22 a22 a12 3D22 a22 f22 cos 2 D 0; 2 121111 111111 3 221112 2 !12 D11 /a11 C 12!11 D12 a11 a22 cos 1 3.4!11 D12 122222 112222 2 121212 111212 C6.2!11 D12 !12 D11 /a11 a22 C 24.!11 D12 !12 D11 / 2 221212 2 a11 a12 12!12 D12 a12 a22 cos 1 2!11 !12 a11 .1 C 2 / D 0:
(12.81)
The solutions to the equations in (12.81) can be discussed for three cases as shown below.
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Fig. 12.17 The amplitude and frequency response curves of the rotary shell corresponding to the (2, 2) modal vibration (˝1 =˝r D 0:4) f22 = 3
a 22 f22 = 1 f22 = 0
σ2
12.4.3.1 The Vibrational Amplitude of Modal (1, 2) a12 is Set to Zero The solution can be obtained as sin 2 D 0 from the third equation of (12.81), scilicet, 2 D n , .n D 0; 1/. Then according to the forth and fifth equations of (12.81), the following equations can be written: 1 221111 2 222222 3 2!22 a22 2 bD22 a11 a22 3D22 a22 f22 cos n D 0; 2 112222 2 111111 2 a22 C 3D11 a11 C 2!22 .1 C 2 / D 0: a11 Œ6D11
(12.82)
Supposing a11 D 0, the amplitude and frequency response equation of the system under the condition of a (2,2) modal resonance can be expressed as: 222222 3 2!22 a22 2 3D22 a22 D
1 f22 cos n 2
.n D 0; 1/:
(12.83)
At this time, the zeroth order approximate solutions of the system can be written as follows: .0/
.0/
w11 D w12 D 0; .0/
w22 D a2 exp.i in / C cc
.n D 0; 1/:
(12.84)
The amplitude and frequency response curves for the (2, 2) modal resonance of (12.83) under a different external excitation is shown in Fig. 12.17. It can be seen form Fig. 12.17 that the shape of the amplitude and frequency response curves of the shell slant slightly toward the right and show hardened characteristics. As a consequence, the natural frequency corresponding to the (2, 2) modal is higher than the natural frequency corresponding to the (1, 1) modal. Therefore, the amplitude
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381
Fig. 12.18 The internal resonance characters between the (1, 1) modal and (2, 2) modal of the system as well as the effects of rotary speed to the amplitude and frequency response curves of the rotary shell corresponding to (2, 2) modal vibration (˝1 =˝r D 0:4)
and frequency response interval for the (2, 2) modal is also more narrow than the interval for the (1, 1) modal vibration. From the figure we can also find that the effect of the rotary speed on the amplitude and frequency response curves is not very obvious. Assume a11 ¤ 0, then from equations (12.82), the following equations can be obtained: 1 221111 2 222222 3 a11 a22 3D22 a22 f22 cos n D 0; 2!22 a22 2 bD22 2 112222 2 111111 2 a22 C 3D11 a11 C 2!22 .1 C 2 / D 0; (12.85) 6D11
the expressions for the phase angle are: ˇ11 D .1 C 2 /T1 C '0 ; ˇ22 D 2 T1 n
.n D 0; 1/;
(12.86)
where '0 is constant and can be determined from the initial conditions of the system. Therefore, the zeroth order approximate solutions of the system can be obtained as: .0/
w11 D a11 exp ifŒ!11 ".1 C 2 / C '0 g C cc; .0/
w22 D a22 exp i. n / C cc
.n D 0; 1/:
(12.87)
The amplitude and frequency response curves between a11 , a22 and 2 can be decided from the equations in (12.85) by using an iterative method as shown in Fig. 12.18. It can be seen from Fig. 12.18 that although the external excitation frequency just tends to !22 , the vibration of the (1, 1) modal is excited by the effect of the (2, 2) modal resonance, where this phenomena is called the internal resonance of the system. At this time, the vibration amplitude a11 of the (1, 1) modal
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Fig. 12.19 The amplitude and frequency response curves when the shell is oscillating with (1, 1) modal (f22 D 0, a12 D 0:3)
is higher than the vibrational amplitude a22 of the (2, 2) modal. The values of a11 and a22 increase with 2 . On the other hand, a11 decreases and a22 increases with an increase in rotary speed of the system, which means that the rotary speed of the shell affects the energy exchange when the internal resonance occurs between the (1, 1) and (2, 2) modal. 12.4.3.2 The Vibrational Amplitude of Modal (2, 2) a12 is Set to Zero Substituting a22 D 0 into (12.81), the following equations can be derived: 111212 2 a11 a12 ˙ f22 D 0; 12D22 121111 111111 2 3.4!11 D12 !12 D11 /a11 2!11 !12 .1 C 2 / 121212 111212 2 !12 D11 /a12 D 0: C24.!11 D12
(12.88)
111212 coefficient is equal to zero, the value of f22 must In this case, because the D22 also be set to zero. Otherwise, the value of a22 is not zero. Therefore, once the value of a12 is certain, the response curves between a12 and 2 can be determined from the second equation of (12.88), which can be found from Fig. 12.19. It can be seen from the Fig. 12.19 that a11 increases with an increase in 2 ; increasing the angular velocity 1 of the shell decreases the natural frequency !11 of the system and the starting point moves to left. At the same time, increasing the angular velocity 1 will decrease the vibrational amplitude of the (1, 1) modal. Therefore, an across point will occur on the plane of Fig. 12.19. At that time, because 1 2 D n .n D 0; 1/, and according to the first expression of (12.78) and (12.79), the corresponding expression about the phase can be obtained as follows: 111212 2 111111 2 2!11 ˇP11 24D11 a12 2D11 a11 D 0;
2ˇ12 ˇ11 D .1 C 2 /T1 C n :
(12.89)
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
383
The zeroth order approximate solution of w11 and w12 can also be expressed in the following forms: .0/ w11 D a11 exp iŒ.!11 C "ˇP11 / C '0 C cc; 1 n .0/ P '0 C cc; (12.90) w12 D a12 exp i !12 C ".1 C 2 C ˇ11 / C 2 2
in which '0 can be determined from initial conditions. 12.4.3.3 The Case of sin 1 D 0 At this time, 1 D n .n D 0; 1/. According to the third expression of (12.81), the value of 2 can also be obtained as: 2 D m .m D 0; 1/:
(12.91)
Then the following two equations can be obtained from the last two equations of (12.81): 111212 2 221111 2 a11 a12 cos n 6D22 a22 a11 2!22 a22 2 12D22
1 221212 2 222222 3 24D22 a22 a12 3D22 a22 f22 cos m D 0; 2 121111 111111 3 221112 2 !12 D11 /a11 C 12!11 D12 a11 a22 cos n 3.4!11 D12 122222 112222 2 121212 111212 !12 D11 /a11 a22 C 24.!11 D12 !12 D11 / C6.2!11 D12 2 221212 2 12!12 D12 a12 a22 cos n 2!11 !12 a11 .1 C 2 / D 0: a11 a12
(12.92)
There are three unknown variables a11 ; a12 ; a22 in (12.92). One of them must be given an initial value to obtain the response relationship between 2 and the other two variables. By setting a12 to a nonzero constant, the response curves between 2 and a11 and a22 can be determined as shown in Fig. 12.20. At this time, the amplitudes of three modals of the shell are all nonzero but the internal response also occurs among them. On the other hand, an increase in angler velocity of the shell leads to a reduction in the amplitude of a11 and an increase in the amplitude of variable a22 . Due to 1 D n .n D 0; 1/ and 2 D m .m D 0; 1/, the corresponding expression regarding the phase can be obtained as follows: ˇ22 D "2 C m ; 2ˇ12 ˇ11 D ".1 C 2 / C n :
(12.93)
The zeroth order approximate solution of w11 , w12 and w22 also can be expressed as follows:
384
C. Chen
Fig. 12.20 The internal response between the modal (1,1) and modal (2, 2) of the shell (f22 D 8, a12 D 0:5)
w11 D a11 exp if!11 C "Œ!11 ".1 C 2 / C 2"ˇP12 / n C 2'0 C cc; .0/
w12 D a12 exp iŒ.!12 C "ˇP12 / C '0 C cc; .0/ .0/
w22 D a22 exp i. C m C cc:
(12.94)
12.4.4 Nonlinear Stability Analysis of the Rotary Truncated Conical Shell In the foregoing analysis, the internal oscillatory properties of the rotary thin truncated conical shell are discussed. In this section, the nonlinear dynamic stabilities of the rotary truncated conical shell are analyzed with a small semi-vertex angle ' of the shell under an axial uniform excitational load Npq . This sets the axial excitational load to the following form: Npq D N 0 C N cos ;
(12.95)
in which N 0 is the static part of the axial uniform excitational load, N is the dynamic amplitude of the axial uniform excitional load, is the excitational frequency of the axial uniform excitational load. The transverse vibration is mainly considered and because we research the parameter vibration under an axial uniform excitational load, the transverse load Qpq is also neglected in this case. Therefore, due to (12.56), the governing equations of the rotary truncated conical shell under the axial uniform excitational load can be written as follows: mn mn mnkl C D3pq .N 0 C N cos / wmn C Dpq wmn wkl D1pNqN wR pq C ŒD2pq mnklst CD5pq wmn wkl wst D 0 .m; n; k; l; s; t; p; q D 1; 2; 3; : : :/;
(12.96)
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
385
in which D1pNqN D 1
C47 C108pNqN 21 ; C80pN qN C C50 21
mn mn mn kl mn D2pq D C147pq C C116pq
21 C147pq C41kl =C39kNlN; mn mnkl D Ikl C150pq =C39kNlN; D3pq ijkl
mnkl st mnkl mn mnkl D4pq D C147pq C42st =C39Ns tN C150pq C41ij =C39iNjN C C152pq ; mnij
mnklst mnklst klst D C153pq C42pq C150ij =C39iNjN ; D5pq
(12.97)
where I is the unit matrix. Transforming (12.96) with N D , the following equations can be obtained: mn mn mnkl D1pNqN wR pq C ŒD2pq C D3pq .N 0 C N cos / w N mn C Dpq wmn wkl mnklst CD5pq wmn wkl wst D 0 .m; n; k; l; s; t; p; q D 1; 2; 3; : : :/:
(12.98)
In the above equations, neglecting the first, third, and fourth items, and setting N D 0, N 0 D Ncr , the dimensionless static buckling load Ncr of the shell can be decided. Furthermore, according to the first and the second items of (12.98) and setting N D 0, N 0 D 0, the dimensionless natural vibration frequency 0 of the shell can also be obtained. The Incrementation Harmonic Balance Method (Lau et al. 1982) is employed here to solve (12.98). Assuming the static part of the axial uniform exciting load is equal to zero and setting N D NN C N ; N C ;
D wpq D wN pq C wpq ;
(12.99)
N w N pq ) is one of the critical balance boundary points corresponding in which (NN , , to (12.98), (N , , wpq / are the small perturbation variables corresponding to the variables (N ; ; wpq /. Substituting (12.99) into (12.98), and just remains the linear items about small increments (N , , wpq /, the incremental equations for the system can be expressed in the following forms: mn mn mnkl N 2 D1pNqN w
R pq C .D2pq C NN D3pq cos /w N N mn wkl mn C 2D4pq w mnklst mn N 1pN qN wR pq N D3pq C3D5pq wN mn wN kl wst D R 2
D wN mn cos N
.m; n; k; l; s; t; p; q D 1; 2; 3; : : :/;
(12.100)
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C. Chen
where the expression for R is mn mn mnkl C NN D3pq cos N /w N mn C 2D4pq wN mn wN kl R D Œ˝N 2 D1pNqN wRN pq C .D2pq mnklst C3D5pq wN mn wN kl wN st
.m; n; k; l; s; t; p; q D 1; 2; 3; : : :/:
(12.101)
When the system arrives a new critical balance boundary points after the small perturbation (N , , wpq /, the value of correction R should tend to zero. In order to obtain solutions to (12.100) corresponding to period T D 2 and 2T D 4 , w N pq ./ N and wpq ./ N are extended into Fourier series as follows: 1 X
ŒwN pq ./; N wpq .N / D
.k/ .k/ Œapq ; apq sin
kD1;3;5;:::
C
1 X
k N 2
.k/ .k/ Œbpq ; bpq cos
kD1;3;5;::: 1 X
ŒwN pq ./; N wpq .N / D
.k/ .k/ Œapq ; apq sin
kD2;4;6;:::
C
1 X
k N ; 2
k N 2
.k/ .k/ Œbpq ; bpq cos
kD0;2;4;6;:::
k N : 2
(12.102)
Substituting the above expressions into (12.100) and equating the coefficients with the same items about sin k2N and cos k2N , the following linear algebraic equation set can be expressed as: ŒG f˛g D frg C N fpg C fqg: .k/
(12.103) .k/
In above equation, ŒG is the coefficient matrix, f˛g D fapq ; bpq gT , frg is the column vector about R, fpg and fqg are the column vectors corresponding to the third item and the second item on the right side in (12.100). According to (12.103), the dynamic unstable boundary, consisting of .D N = Ncr / and the ratio of frequency =2 0 , can be decided under an axial uniform excitational load on the shell. During the calculations, the material parameters are 11 11 defined as D 0:3,q D 7:8 108 N s2 =mm4 . Ncr D D211 =D311 , E D 21
11 104 N=mm2 , 0 D D211 =D111 . Figure 12.21 shows the effect of the rotary speed 1 on the main dynamic unstable area of the system. It can be seen from the figure that the position of the main dynamic unstable area moves upward and the area size reduces with the rotation of the shell, which means the right rotation of the shell can enhance the stability of the shell under the excitation of an axial uniform force. The effect of the thickness of the shell H on the main dynamic unstable area of the system is illustrated in Fig. 12.22. It can be seen from Fig. 12.22 that the stability of the
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell Fig. 12.21 The effect of the rotary speed ˝1 on the main dynamic unstable area of the system (H D 0:1, 2 D 0:6)
387
2.8 α=Ω/Ω0 2.6 2.4 2.2 2.0 1.8 Ω = 0.0 Ω = 1.0
1.6 1.4 0.0
Fig. 12.22 The effects of the thickness of the cylindrical shell H on the main dynamic unstable area of the system (˝1 D 0:1, 2 D 0:4)
0.2
λ 0.4
0.6
0.8
1.0
2.6 α=Ω/Ω0
2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8
H=0.25 H=0.15
1.7 1.6 0.0
0.2
λ 0.4
0.6
0.8
1.0
shell increases with an increase in thickness of the shell. Therefore, it is possible to improve the stability of the structure by increasing the thickness of the shell.
Appendix Z mn C2pq D 4 2 sin '
Z Xm Xp
D cos '
(12.104)
Z
Z mn C10pq
Yn Yq
Xm X p
Yn Yq Z
Z mn D 2 cos ' C11pq
(12.105)
Xm Xp
Yn Yq
(12.106)
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C. Chen
Z mn C39pq
Xm0 Xp
D .1 /H sin ' 2
Z sin2 '
Xm Xp
Z
Z
Z Yn Y q C
Yn Yq C sin '
Z
Xm00 Xp
Xm0 Xp
1 S
Z Yn Yq
Z Yn Y q
Z 1 (12.107) Xm Xp 2 Yn00 Yq S Z Z 1 mn C41pq D .1 /H 2 sin ' 2 sin ' cos ' Xm Xp 2 Yn Yq S Z Z 1 (12.108) 2 cos ' Xm0 Xp Yn Yq S Z Z 1 1 mnkl 2 2 0 0 C42pq D .1 /H sin ' 2 Xm Xk Xp Yn Y l Y q 2 S Z Z Z Z 1 1 2 0 00 2 C 2 Xm Xk Xp Yn Yl Yq sin ' 2 Xm Xk Xp 3 Yn0 Yl0 Yq 2 S Z Z Z 1 1 C 22 Xm Xk0 Xp 2 Yn Yl Yq H 2 .1 / 22 Xm0 Xk Xp S 2 Z Z Z 1 1 (12.109) 2 Yn Yl00 Yq C Xm Xk0 Xp 2 Yn0 Yl0 Yq S S Z Z 1 mn C44pq D sin ' Xm Xp Yn Yq (12.110) 2 Z Z Z Z 1 1 H2 mn D cos ' Xm Xp Yn Yq C Yn0 Yq (12.111) sin 2' Xm Xp C47pq 2 S Z Z Z Z 1 mn 2 2 C50pq D Xm Xp Yn Yq H cos ' Xm Xp 2 Yn Yq (12.112) S Z Z Z Z 1 1 H2 mn D Yn0 Yq sin ' Xm Xp 2 Yn0 Yq sin C Xm0 Xp C79pq 2 2 S S Z Z Z 2 Z 1 1 .1 /H Yn0 Yq C sin ' Xm Xp 2 Yn0 Yq (12.113) Xm0 Xp 4 2 S S Z Z 1 1 mn D H 2 Xm Xp 2 Yn00 Yq sin .1 /H 2 C80pq S 2 Z Z Z Z 1 2 00 0 .cos ' Xm Xp Yn Yq C sin ' Xm Xp Yn Y q S Z Z H 4 cos2 ' 1 .1 / Xm00 Xp 2 Yn Yq .1 / sin ' 24 S Z Z Z Z 1 1 0 00 Xm Xp 3 Yn Yq C2 Xm Xp 4 Yn Yq (12.114) S S 1 .1 /H 2 2
Z
1 S2
1 S
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
Z
Z
mn C108pq mn C115pq mn C116pq
mn C147pq
H2 D 4 cos ' Xm Xp Yn Yq sin 2' 6 Z Z 1 D sin 2' Xm Xp Yn Yq 2 2 Z Z D sin2 ' Xm Xp Yn Yq
389
Z
2
Xm0 Xp
1 S
Z Yn Yq (12.115) (12.116)
Z Z Z Z 1 cos ' Xm00 Xp Yn Yq C sin ' cos2 ' Xm0 Xp Yn Yq (12.117) S Z Z Z Z 2 1 1 H cos ' D Yn Y q sin ' Xm Xp 2 Yn Yq C Xm0 Xp 2 S S
H2 12
(12.118) Z mn C149pq D H 2 cos2 '
Xm Xp Z
C.3 / sin ' Z C2
00 Xm Xp
Z
000 Xm Xp
Yn Yq 1 S
Z
Yn00 Yq sin '
H4 12
Z
0000 Xp Xm
Z
Yn Yq C 2 sin2 ' Z
0 Xm Xp
1 S3
Z
Z Yn Yq 00 Xm Xp
1 S2
Z Yn Yq
Yn Y q
Z Z Z 1 1 3 (12.119) Y C 2.1 / sin ' X X Y Y Y n q m p n q S4 S4 Z Z Z Z 1 1 0 Xm D H 2 sin ' Xk0 Xp Yn Y l Y q Yn Yl Yq C Xm Xk0 Xp 2 S S Z Z Z Z 1 0 Yn Yl Yq Xm H 2 Xk00 Xp Yn Yl Yq C sin ' Xm Xk00 Xp S Z Z Z Z 1 00 0 0 Xm Xk Xp Yn Yl Yq C sin ' Xm Xk0 Xp H 2 Yn Yl Yq S Z Z Z Z 1 1 sin2 ' Xm Xk0 Xp 2 Yn Yl Yq / H 2 .1/ Xm Xk0 Xp 2 Yn0 Yl0 Yq S S Z Z Z Z 1 1 1 0 Xm H 2 .1 / Xk Xp 2 Yn0 Yl0 Yq sin ' Xm Xk Xp 3 Yn0 Yl0 Yq 2 S S Z Z Z Z 1 1 0 H 2 sin ' Xm Xk Xp 3 Yn Yl00 Yq C Xm Yn Yl00 Yq Xk Xp 2 S S Z Z Z Z 1 1 0 0 Xk Xp 2 Yn0 Yl0 Yq H 2 sin ' Xm Xk Xp 3 Yn Yl0 Yq C Xm S S Z Z 1 1 (12.120) H 2 .1 / Xm Xk0 Xp 2 Yn00 Yl Yq 2 S Z
C
mnkl C150pq
Z
1 S2
Xm Xp
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Z 1 Yn Y l Y q S2 Z Z 1 C 2 H 2 cos ' Xm Xk00 Xp Yn Y l Y q S Z Z Z Z 1 1 0 0 Xm C 2 H 2 cos ' Xk0 Xp Xk Xp 2 Yn Yl Yq Yn Yl Yq sin ' Xm S S Z Z Z Z 1 1 C 2 cos ' Xm Xk Xp 3 Yn Yl00 Yq C Xm Xk Xp 3 Yn0 Yl0 Yq S S Z Z Z Z 1 2 1 1 0 H 2 cos ' Xm Xk Xp 3 Yn0 Yl0 Yq C Xm Yn Yl Yq Xk0 Xp 2 S S Z
mnkl C152pq D 2 H 2 sin ' cos '
Xm Xk0 Xp
(12.121)
Z Z 1 1 2 2 Yn Yl Yt Yq D 2 H sin ' Xm0 Xk0 Xs0 Xp 2 S Z Z 1 Yn0 Yl0 Yt Yq C Xm Xk Xs0 Xp 3 S Z Z Z Z 1 1 2 2 0 0 00 00 0 0 Xm Xk Xs Xp Yn Yl Yt Yq C Xm Xk Xs Xp 2 Yn Yl Yt Yq 2H 2 S Z Z Z Z 1 Xm0 Xk0 Xs00 Xp Yn Yl Yt Yq C Xm0 Xk Xs0 Xp 2 22 H 2 Yn Yl0 Yt0 Yq S Z Z 1 1 sin ' Yn Yl0 Yt0 Yq Xm0 Xk Xs Xp 3 2 S Z Z 1 2 2 0 0 Yn Yl0 Yt0 Yq Xm Xk Xs Xp 2 2 H .1 / S Z Z 1 1 Yn Yl0 Yt0 Yq Xm00 Xk Xs Xp 2 C 2 S Z Z Z Z 1 1 Yn Yl0 Yt0 Yq 2 sin ' Xm0 Xk Xs Xp 3 Yn Yl0 Yt0 Yq C2 Xm0 Xk0 Xs Xp 2 S S Z Z Z Z 1 1 1 Yn0 Yl0 Yt0 Yq C Xm0 Xk0 Xs Xp 2 Yn Yl Yt00 Yq Xm Xk Xs Xp 4 22 H 2 2 S S Z Z Z Z 1 1 C2 Xm Xk Xs Xp 4 Yn0 Yl00 Yt0 Yq C2 Xm0 Xk0 Xs Xp 2 Yn Yl0 Yt0 Yq S S Z Z 1 1 Xm0 Xk Xs0 Xp 2 Yn0 Yl0 Yt Yq 22 H 2 .1 / 2 S Z Z 1 0 0 00 Yn Yl Yt Yq C Xm Xk Xs Xp 2 (12.122) S
mnklst C153pq
12 Nonlinear Dynamic of a Rotating Truncated Conical Shell
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References Aksogan O, Sofiyev AH (2002) The dynamic stability of a laminated truncated conical shell with variable elasticity moduli and densities subject to a time-dependent external pressure. J Strain Anal Eng Design 37(3):201–210 Arnod VI (1980) Mathematical method of classical mechanics. Springer, Berlin Boutaghou ZE (1992) Dynamic of flexible beams and plates in large overall motion. J Appl Mech 59:991–1004 Chandrasekharan K, Ramamurti V (1981) Axisymmetric free vibrations of laminated conical shell. Proceedings of the international symposium on mechanical behaviour and structures, Media Ottawa, pp 18–21 Chandrasekharan K, Ramamurti V (1982) Asymmetric free vibrations of laminated conical shell. J Mech Design 104:453–462 Chang B, Shabana A (1990) Finite element formulation for the large displacement analysis of plates. J Appl Mech 57:707–717 Chen CP, Dai LM (2006) Nonlinear internal resonance and parameter vibration of a truncated conical thin shell. IMECE2006–13775, ASME Congress, Chicago Dai L, Singh MC (2003) A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems. J Sound Vibr 263(3):535–548 Dumir PC, Khatri KN (1986) Axisymmetric static and dynamic buckling of orthotropic truncated shallow conicels caps. Comput Struct 22:335–342 Fu YM, Chen CP (2001) Nonlinear vibration of elastic truncated conical moderately thick shells in large overall motion. Int J Nonlinear Mech 36(5):763–772 Jian XS, Zhan XZ (1991) Multimode analysis of nonlinear vibration for circular cylindrical shells. J Appl Mech 14(1):5–10 Kane TR, Likins PW, Levinson DA (1983) Spacecraft dynamics. McGrand-Hill, New York Kayran A, Vinson JR (1990) Free vibration analysis of laminated composite truncated circular conical shells. AIAA J 28(7):1259–1269 Lau SL, Cheung YK, Wu SYA (1982) Variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. ASME J Appl Mech 49:849–853 Leissa AW (1973) Vibration of Shells. NASA SP-288 Olshanskii VP, Sevriukov VI (1976) Dynamics of a conical shell under axisymmetric nonstationary loading. Dinamikai Prochnost’ Mashin 23:72–75 Sofiyev AM, Aksogan O (2002) The dynamic stability of a nonhomogeneous orthotropic elastic truncated conical shell under a time dependent external pressure. Struct Eng Mech 13(3):329– 343 Xu CS (1991) Buckling and postbuckling of symmetrically laminated moderately thick spherical caps. Int J Solids Struct 28(9):1171–1184 Xu CS, Xian ZQ, Chia CY (1995a) Nonlinear theory and vibration analysis of laminated truncated thick conical shells. Int J Nonlinear Mech 16(11):139–154 Xu CS, Xian ZQ, Chia CY (1995b) Nonlinear theory and vibration analysis of laminated truncated, thick, conical shells. Int J Nonlinear Mech 16(11):1–15
Chapter 13
Nonlinear Real-Time Pose Estimation of Quadrotor UAV Chayatat Ratanasawanya, Mehran Mehrandezh, and Raman Paranjape
Abstract Object pose (rotation and translation) estimation problem arises in several domains of application. This chapter presents simplified nonlinear equations of motion for a quadrotor derived based on Newtonian mechanics before discussing and comparing the performance of two real-time image-based quadrotor pose estimation methods: one based on a commercial motion capture and tracking system utilizing six infrared (IR) cameras mounted around the test area aliased OptiTrack; the other based on classicPOSIT, an iterative pose estimation algorithm using a single image of a target object taken by an onboard camera. The geometry of the target and intrinsic parameters of the camera are known a priori; the image coordinates of five noncoplanar feature points on the target are extracted through a real-time image processing algorithm for pose estimation. Test results prove that onboard camera pose estimation is an attractive solution for autonomous real-time control of a quadrotor unmanned aerial vehicle (UAV).
13.1 Introduction Unmanned Aerial Vehicles (UAVs) recently draw a great deal of attention within the public and private sectors as a useful tool for environmental surveillance (Allen and Walsh 2008), mitigation, prevention, and timely response to emergency situations, and law enforcement (Murphy and Cycon 1999).
C. Ratanasawanya () • R. Paranjape Ph.D., P.Eng. Electronic Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, 3737 Wascana Parkway, Regina, SK, Canada S4S0A2 e-mail:
[email protected] M. Mehrandezh Ph.D., P.Eng. Industrial Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, 3737 Wascana Parkway, Regina, SK, Canada S4S0A2 e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 13, © Springer Science+Business Media, LLC 2012
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UAVs may include a wide range of aircraft, which can generally be grouped into two categories: fixed-wing and rotary-wing (rotorcraft). Rotorcraft; such as quadrotor helicopters, are preferable over fixed-wing aircraft for surveillance applications because of their agility and ability to hover, take-off and land vertically in small, relatively rugged areas. A Quadrotor has four fixed-pitch propellers configured in two pairs of counter-rotating blades. Its motion in physical space is achieved by varying the speed of the four rotors, rotating its body, and in turn results in translational motion. In this chapter, two real-time image-based quadrotor rotation and translation, collectively called pose, estimation methods are analyzed and compared. Simplified nonlinear equations of motion are also presented. The pose estimated from images can be applied to control the motion of the quadrotor, governed by the dynamics equations. The problem of object pose estimation arises in several domains of application such as localization, visual servoing, and object tracking (Chesi and Hashimoto 2004; Gramegna et al. 2004; Hamel and Mahony 2007; Wei and Lee 2010; Zhang and Wu 2011). Compared to typical inertial, sonar, atmospheric, and GPS-based sensors; vision appears as an appealing alternative for deployment in small UAVs with limited payloads due to its compact size and abundant information in captured images. Vision provides noncontact, nondestructive means for surveillance and inspection. A vision system deployed for localization and pose estimation of a quadrotor could be fixed in the workspace, for example, a ground-based camera provides images for helicopter pose estimation by tracking the positions and areas of five color pads attached underneath a quadrotor helicopter (Altug et al. 2002). A camera could also be onboard the vehicle determining its pose via the use of an object placed in the workspace, such as, a planar object consists of four disjointed circles (Guenard et al. 2008; Bourquardez et al. 2007). For both vision system configurations, the use of a single or multiple cameras are possible. Single camera systems, commonly referred to as monocular vision, have been applied to UAVs, for example, monocular vision mounted on a quadrotor provides its pose estimates by tracking a set of three dark colored targets mounted on a white wall (Mkrtchyan et al. 2009). A Moir´e pattern bearing target is tracked by an onboard monocular vision for pose estimation and operation of UAVs; such as hovering, stationary, and moving platform landings, in close proximity to other craft and landing platforms (Tournier 2006). Multiple camera systems have been implemented for position estimation; for instance, using a pair of onboard cameras as a visual odometer (Amidi 1996). Also, a dual camera system, one ground-based pan and tilt camera and the other one onboard the quadrotor, was used for pose estimation (Altug et al. 2003). Two real-time quadrotor pose estimation methods are presented and compared in this chapter. The first method is based on a commercial-grade motion capture and tracking system. It consists of six Infrared (IR) cameras fixed in the workspace.
13 Nonlinear Real-Time Pose Estimation of Quadrotor UAV
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The system reconstructs three-dimensional (3D) positions from images of IR reflectors attached to the quadrotor taken by the cameras. The second method is based upon an iterative pose estimation algorithm that uses real-time images of a target object taken by monocular vision system onboard the quadrotor. The performance of the second method is compared against that of the first. This chapter is organized as follows. Section 13.2 presents the nonlinear dynamics equations governing the motion of the quadrotor. Section 13.3 explains the operation of the motion tracking system and how pose estimates can be obtained from it. Section 13.4 outlines the pose estimation using a single onboard camera. Experimental setup and results can be found in Sects. 13.5 and 13.6, respectively. Section 13.7 concludes the chapter with a brief discussion.
13.2 Quadrotor Dynamics Dynamics models to be presented were derived for the quadrotor helicopter enclosed in a carbon fiber cage1 (Quanser Inc. 2011) as shown in Fig. 13.1. It is capable of three degrees-of-freedom (DOF) translational and 3DOF rotational motions. Defining a quadrotor-fixed frame, denoted by Q, using the right-hand rule with k-axis pointing toward the back of the quadrotor, the translational position of the
Fig. 13.1 World (W ) and Quadrotor-fixed (Q) frames axes and sign convention. Rotation around the axes of frame Q is expressed using Euler angles. The two frames are related through rotation matrix (W RQ / and translation vector (W tQ /
1
Qball-X4 by Quanser Inc.
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quadrotor is expressed with respect to the world (inertial) frame, W , by a translation vector, W tQ , where: W
tQ D ŒX Y ZT 2 W:
(13.1)
B
Note that the standard notation tA is used to denote a translation vector relating frame A to frame B. The quadrotor orientation is expressed using a rotation matrix, W RQ , with Euler angles: roll (), pitch (), and yaw ( ): 2
C C W RQ D 4 S C S
3 C S S S C S S C C C S C C C S S S S C S C S 5 ; S C C C
(13.2)
where C and S denote cos./ and sin./, respectively. The full nonlinear dynamics equations of a quadrotor can be derived by using both Lagrangian mechanics (Castillo et al. 2004), and Newtonian mechanics (Hamel et al. 2002). The mathematical models presented in the following subsections were derived using Newtonian mechanics with a few simplifying assumptions (Quanser Inc. 2004): • The Quadrotor is perfectly symmetric at its center • Roll and pitch angles are decoupled • Coriolis (Grewal et al. 2007) and Gyroscopic (Ley et al. 2009), effects are negligible • The Quadrotor operates in quasi-stationary state, a state near hover; therefore, roll and pitch angles are small
13.2.1 Actuator Dynamics There are four sets of motors and propellers configured in two counter-rotating pairs. Figure 13.2 shows the direction of rotation of each motor/propeller, mi . The thrust generated by each propeller, Fi , is modeled by a first-order system: Fi D Kp
! Vpi I i D 1; : : : ; 4; sC!
(13.3)
where, Kp is a thrust coefficient, ! the actuator bandwidth, s the Laplace variable, and Vpi is the pulse width modulation (PWM) input to motor mi . As the propellers on the quadrotor are fixed-pitch type; i.e., they cannot change their blade angle, the motion of the quadrotor is induced by varying the speed of the propellers, hence varying the thrust each motor generates. Vpi is varied in order to change the speed of the propellers. The voltage to each motor is related to the voltages required for roll, V , for pitch, V , for yaw, V , and for vertical, VY , motions as follows:
13 Nonlinear Real-Time Pose Estimation of Quadrotor UAV Fig. 13.2 Model of the quadrotor (top view) showing direction of propeller rotation and reactive torques (i )
τ2
τ3
397
m2
FRONT
m3
m4
m1
τ1
F1
Fig. 13.3 Model of the quadrotor (side view) for deriving roll and pitch equations of motion
τ4
m1
lp
F2 m2
Vp1 D VY C V C V ; Vp2 D VY V C V ; Vp3 D VY C V V ; Vp4 D VY V V :
(13.4)
A state variable to represent the actuator dynamics, , is defined as: i D
! Vpi I i D 1; : : : ; 4: sC!
(13.5)
13.2.2 Roll and Pitch Motions With the assumption that roll () and pitch () angles are decoupled and that the quadrotor is perfectly symmetrical at its center, the roll and pitch motions are modeled by using Fig. 13.3 as follows. Let the distance between the center of gravity (c.g.) of the quadrotor and its motor/propeller assembly be lp . The rotation around the c.g. is caused by the difference in the moments generated by thrusts F1 and F2 from motors m1 and m2 , respectively. Therefore,
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J R D F lp ;
where F D F2 F1 :
(13.6)
J R D F lp ;
where F D F4 F3 :
(13.7)
Similarly,
J and J are the moment of inertia for pitch and roll motions, respectively. To accommodate an integrator in the feedback controller design based on the linear quadratic regulator (LQR) technique, two state variables s1 and s2 are further defined: Z Z s1 D dt and s2 D dt: (13.8) Combining (13.3), (13.5), (13.6), and (13.8), the state-space model for pitch dynamics is: 2 P3 2 0 1 0 K l 6 R 7 6 0 0 Jp p 6 7D6 6 4 Pi 5 4 0 0 ! sP1 1 0 0
32 3 2 3 0 0 7 P7 6 7 076 0 7 6 7 C 6 7 Vpi I 0 5 4 i 5 4 ! 5 s1 0 0
i D 1; 2:
(13.9)
i D 3; 4:
(13.10)
Similarly, the model for roll dynamics is 2
3 2 0 P 6 R7 6 6 7 60 6 7D6 6 P 7 6 4 i 5 40 sP2 1
32
3 2 3 0 76 P 7 6 7 K p lp 6 7 6 7 0 J 0 7 6 7 6 0 7 7 7 6 7 C 6 7 Vpi I 0 ! 0 5 4 i 5 4 ! 5 s2 0 0 0 0 1
0
0
13.2.3 Yaw Motion Yaw motion is caused by the difference between reactive torques, torques opposing direct torques generated in the same direction as the rotation of propellers (see Fig. 13.2). They are summed at the center of the quadrotor frame; nonzero sum results in yaw motion. For simplification, reactive torques, i , are assumed to be generated directly by the propellers and is related to the PWM input, Vpi , by: i D K Vpi ;
(13.11)
where, K is the torque coefficient. The yaw motion is modeled by the following equation: J R D 1 C 2 3 4 : (13.12)
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where, J is the moment of inertia for yaw motion. Combining (13.11) and (13.12) gives: J R D K Vp ; where Vp D Vp1 C Vp2 Vp3 Vp4 :
(13.13)
The yaw dynamics can be written in state-space form as follows:
P R
0 1 D 0 0
"
P
C
0
#
K J
Vp :
(13.14)
13.2.4 Vertical Motion The vertical motion of the quadrotor (along the Y axis) results from the throttle, summation of all thrusts generate by the four propellers. Assuming each propeller generates equal thrust, the height dynamics can be written as: Mq YR D 4F cos./ cos./ Mq g;
(13.15)
where Mq is the mass of the quadrotor and g is the earth gravitational acceleration coefficient. With the assumption that the quadrotor operates in quasi-stationary state, (13.15) is linearized by assuming D D 0 and results in the following state-space equation: 2 P Y 6 YR 6 4 Pi
3
2
0 7 6 0 7D6 5 6 40 sP3 1
32 3 2 3 2 3 0 0 0 Y 4Kp 7 6 7 6 7 076 YP 7 Mq 7C6 0 7 Vpi C6 g 7 I 76 4 5 4 4 5 0 5 ! 0 ! 0 5 i s3 0 0 0 0 0 1 0
0
Z s3 D
Y dt:
(13.16)
13.2.5 Lateral Motions Lateral motions of the quadrotor along the latitude (X ) and longitude (Z) axes results from changing the roll and pitch angles hence redirecting components of the throttle. Assuming that the yaw angle is zero, the equations of motion along the X and Z axes can be written as: Mq XR D 4F sin./
(13.17)
Mq ZR D 4F sin./:
(13.18)
and
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Linearize (13.17) and (13.18) by assuming quasi-stationary operation gives the following state-space models: 2 P3 2 0 1 X 6 XR 7 6 0 0 6 7D6 4 Pi 5 6 40 0 sP4 1 0
32 3 2 3 0 0 X 4Kp 76 P 7 6 7 0 76X 7 6 0 7 Mq C Vpi I 7 ! 0 5 4 i 5 4 ! 5 s4 0 0 0 0
Z s4 D
X dt
(13.19)
Z dt:
(13.20)
and 2 P3 2 0 1 0 Z 4K 6 ZR 7 6 0 0 Mqp 6 7D6 6 4 Pi 5 4 0 0 ! sP5 1 0 0
32 3 2 3 0 0 Z 7 6 6 P 07 Z 76 7 6 0 7 7 Vpi I C 7 0 5 4 i 5 4 ! 5 s5 0 0
Z s5 D
13.3 Fixed-Camera Configuration Pose Estimation The 6DOF position and orientation of the quadrotor in the workspace are estimated using a motion capture and tracking system aliased OptiTrack2 . OptiTrack system (NaturalPoint Inc. 2011) consists of six IR emitting cameras surrounding a workspace in which the motion of an object is to be tracked. IR light from the cameras illuminates reflectors (markers) attached to the object. The six IRsensitive cameras take images of the markers and recreate their 3D positions through software (NaturalPoint Inc. 2011) relative to the system’s coordinate frame, defined during camera calibration process. The coordinate system of OptiTrack is assumed to be the world frame, W . OptiTrack claims that its position reading has 2–4 mm accuracy after calibration. OptiTrack operates in two modes: point-cloud and trackable. • Point-cloud mode. The system provides 3D coordinates of each of the markers in its view with respect to the system’s coordinates separately. As each marker is treated as a point in the workspace, its 6DOF information is unobtainable in this mode. • Trackable mode. A group of at least three reflectors attached to a physical object creates a trackable object in software. The system provides 6DOF pose reading of the object at its virtual center with respect to the system’s coordinates. In order to estimate the pose of the quadrotor, three IR reflectors are attached to the front, left, and right ends of the quadrotor cross bars as shown in Fig. 13.4a.
2
Motion capture and tracking system by NaturalPoint Inc.
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Fig. 13.4 (a) Quadrotor with IR reflectors attached to the front, left, and right ends (b) trackable object seen in software and its virtual center defined in the middle
OptiTrack, operating in trackable mode, sees a triangular trackable object defined by the three markers at the corners as shown in Fig. 13.4b with its virtual center defined in the middle. The virtual center is assumed to coincide with the actual c.g. of the quadrotor located at the intersection of the cross bars; therefore, the 6DOF information of the trackable object center in the calibrated workspace estimated by Optitrack is identical to the pose of the quadrotor.
13.4 Onboard Camera Configuration Pose Estimation The 6DOF pose of the quadrotor in the workspace can be estimated alternatively by using real-time images taken by monocular vision system or a single camera onboard the quadrotor. The onboard wireless camera is mounted underneath a crossbar at the front of the helicopter (see Fig. 13.5).
13.4.1 Pose Estimation Algorithm Pose estimation, sometimes referred to as a Perspective-n-Point (PnP) problem. The goal of the problem is to solve for the position and orientation, collectively called pose, of a camera given its intrinsic parameters such as focal length and a set of n correspondences between 3D points and their two-dimensional (2D) projections on image (Moreno-Noguer et al. 2007). Pose estimation algorithms are categorized into two groups: iterative and noniterative methods. • Iterative pose estimation algorithms, such as (David et al. 2004; Horaud et al. 1997), can handle arbitrary values of n and achieve superior accuracy
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Fig. 13.5 A camera mounted at the front of the quadrotor
provided that they converge to a solution. Excellent precision of the approach comes at the price of calculation time and that an initial estimate is required to start the iterative process. Among iterative algorithms, Lu et al.’s approach (Lu et al. 2000) has been claimed the most robust method to outliers, accurate, and computationally efficient. • Noniterative pose estimation algorithms, for example (Dhome et al. 1989; Gao et al. 2003; Moreno-Noguer et al. 2007; Triggs 1999), are faster than iterative ones; although, less accurate. They do not require initial pose estimate to start. Some algorithms are specialized for small fixed values of n (Dhome et al. 1989; Gao et al. 2003); some can deal with any value of n (Moreno-Noguer et al. 2007; Triggs 1999). Moreno-Noguer et. al.’s approach (Moreno-Noguer et al. 2007) shows comparable accuracy to Lu et al.’s method, yet much faster. The pose estimation algorithm chosen for this work is an iterative algorithm called “POSIT.” Four types of POSIT algorithm have been developed and are available on the website3 : classicPOSIT (DeMenthon and Davis 1995), modernPOSIT (David et al. 2004), POSIT for coplanar points (Oberkampf et al. 2002), and softPOSIT (David et al. 2004). The requirements and attributes of the four POSIT algorithms are summarized in Table 13.1. classicPOSIT is implemented because of its simplicity. The algorithm estimates the pose of an object using a single image (DeMenthon and Davis 1995). It is composed of two stages: first, Pose from Orthography and Scaling (POS) determines an initial pose of the object by assuming that the image coordinates of feature points (points of interest) on the object have been obtained by a scaled orthographic projection (SOP); second, POS with Iteration (POSIT) refines the initial pose found 3
http://www.cfar.umd.edu/daniel/Site 2/Code.html
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Table 13.1 Summary of four types of POSIT algorithm Requirements and attributes Minimum number of feature points Feature points are coplanar Model of object must be known Image and object feature points correspondence must be known The origin of object frame must be a feature points Language(s) the code is available in
classicPOSIT 4
modernPOSIT 4
POSIT for coplanar points 4
softPOSIT 4
No Yes
No Yes
Yes Yes
No No
Yes
Yes
Yes
No
Yes
No
Yes
No
C and MATLAB
C and MATLAB
C
MATLAB
Fig. 13.6 Rotation and translation matrices calculated by classicPOSIT relating object frame (O/ to camera frame (C )
C
x
Camera
C
y C
z
x C
C
RO, tO
Center of projection C
y
O z
in the previous stage through iterations and converges on a more accurate pose. SOP is an approximation to perspective projection, which is a common model assumed for cameras. In this approximation, the depths of different points on an object expressed in the camera coordinate frame, C , are not very different from one another, and can all be set to the depth of a reference point on the object. The algorithm further assumes that at least four noncoplanar feature points can be detected in the image and can be matched to the same points on the object. It also assumes that relative geometry between the points and the camera intrinsic parameters are known. The algorithm gives a rotation matrix (C RO / and a translation vector (C tO / of the object coordinate frame, O, which is attached to one of the feature points on the object, with respect to the camera coordinate frame attached to the center of projection of the camera. The result of classicPOSIT can be represented graphically as shown in Fig. 13.6.
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Fig. 13.7 Target object with dimensions and object frame defined
13.4.2 Target Object and Image Features The target object used has two characteristics: first, it is not self-occlusive, i.e., all feature points are available to the camera from any angle, so that the quadrotor pose may be estimated as long as the object is in the field of view of the camera. Second, its feature points can be extracted by an image processing algorithm relatively easily to minimize calculation time. The target object consists of a sheet of black paper and a white light emitting diode (LED) extending from the plane of the paper. The dimensions of the object are shown in Fig. 13.7. Four corners of the paper and the LED form five noncoplanar feature points as per the requirement of the POSIT algorithm. It has the two characteristics stated above: the black together with the LED creates a target object which is not self-occluded and the feature points can be detected easily. The object frame, O, is defined with its origin attached to the LED; x-axis points to the right, y-axis points up, and z-axis points away from the wall.
13.4.3 Real-Time Image Processing The image processing algorithm summarized by the flowchart shown in Fig. 13.8 was developed in Simulink (The MathWorks Inc. 2011). The tasks of the algorithm are: to detect feature points, to rearrange the image coordinates of the detected points, to undistort the coordinates of those points, and to filter large jumps in detected coordinates between two consecutive sample times. The resultant undistorted image coordinates of feature points are passed to classicPOSIT to estimate the pose of the quadrotor.
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Image Acquisition Previous locations
Intensity image at 30 fps
Detect LED
Arrange feature points
Hash tables
Did any of the points move more than threshold?
Detect 4 paper corners
Use previous locations for invalid points
Use current locations for all points
Features location undistortion
Has any of the points been invalid for more than 5 frames? No
Display image and detected feature points
Buffering filter
Yes
No Current locations
1 sample delay
Pass previous locations through Current locations
yes
Reset all to current locations
Previous locations or
Current locations
Feature point locations
Output to pose estimation
Fig. 13.8 Target object feature points extraction – Image processing flowchart
13.4.3.1 Feature Points Detection and Arrangement Intensity images are acquired from the wireless camera at the rate of 30 frames per second (Fig. 13.9a). This original image is used for two processes: one is LED detection; the other is four window corners detection. For LED detection, because the white LED is very bright, it appears as an area with high intensity. Segmentation is done by comparing each pixel in the original image to a threshold, thLED of 0.8: Imagein thLED :
(13.21)
The centroid of the white area in the resultant binary image, shown in Fig. 13.9b, is then determined to represent the LED in image coordinates. This is the first feature point and is color coded in purple (Fig. 13.9c). Detection of the four corners is done in two steps: first, the area representing the window is segmented through thresholding similar to (13.21): Imagein thwindow ;
(13.22)
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Fig. 13.9 Image processing results (a) acquired image (b) segmented LED location (c) LED detected (d) segmented window location (e) corners detected in image (f) detected corners of window before sort (g) five feature points detected after sort and undistortion
where, thwindow of 0.2 is the intensity threshold value for segmentation of window. The result is a binary image shown in Fig. 13.9d. Second, the original image is passed through a corner detection method with the following settings: • • • • • • •
Method: Local intensity comparison (Rosen & Drummond) Intensity comparison threshold: 0.1 Maximum angle to be considered a corner (in degrees): 135.0 Output: Corner location Maximum number of corners: 50 Maximum metric value that indicates a corner: 0.01 Neighborhood size (suppress region around detected corners): [7 7]
Note that Local intensity comparison method was chosen for corner detection to achieve the fastest computation. Up to 50 corners may be detected as the result of the corner detection block; this is illustrated as the white dots in Fig. 13.9e. Unwanted corners are screened out with the help of the window area detected in the first step and left with only four actual corners of the window (Fig. 13.9f). Notice that the four corners are color coded to show the user the ordering of the feature points. The correct order of feature points shall be as shown in Table 13.2; therefore, the feature points detected in Fig. 13.9f are in the wrong order and they must be rearranged.
13 Nonlinear Real-Time Pose Estimation of Quadrotor UAV Table 13.2 Correct feature points order and color code
Feature point # 1 2 3 4 5
Description LED Upper left window corner Lower left window corner Upper right window corner Lower right window corner
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Color code Purple Red Green Blue Yellow
Fig. 13.10 Hash tables for undistortion of image coordinates (a) Horizontal image coordinate, u (b) Vertical image coordinate, v
The detected five feature points are sorted according to their image coordinates to be in the order shown in Table 13.2. The correct order of feature points is crucial for pose estimation as required by classicPOSIT. If the feature points are arranged in wrong order, the pose estimates will also be incorrect. 13.4.3.2 Undistortion of Feature Point Coordinates and Filtering Image undistortion is a process that compensates acquired images for lens distortion. Straight edges appear as curves on image due to imperfection during lens manufacturing. The sorted image coordinates of feature points from the previous step are distorted coordinates as they were determined from the acquired image that has not been compensated for lens distortion. If these coordinates are used directly for pose estimation, the resultant pose estimates are incorrect. Instead of creating a new intensity undistorted image by interpolating the intensity of each pixel from the original distorted image. This is computationally intense and may slow down real-time process significantly. Instead, two hash tables were created to quickly determine undistorted image coordinates directly from the distorted image coordinates found from the previous steps. Two hash tables: one for horizontal image coordinate, u, the other for vertical image coordinate, v, were created by using lens distortion coefficients determined during offline camera calibration (Bouguet 2004). A small section of the tables is shown in Fig. 13.10. Given a distorted image coordinates, for example, .u; v/ D .39; 47/, the first table (Fig. 13.10a) is accessed for its content stored at row 47, column 39, which is the undistorted horizontal coordinate, u. Then, the second table (Fig. 13.10b) is accessed for its content stored at the same location, which is the undistorted vertical coordinate, v. Therefore, undistorted image coordinates is .u; v/ D .4; 16/.
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Fig. 13.11 6DOF pose estimation from detected feature point locations
The undistorted coordinates of the five feature points are displayed on the acquired image to the user (Fig. 13.9g). Notice that the points do not correspond to their actual positions on the image. This is viewed as correct due to undistortion. Feature points filtering is done by using a simple buffering filter to filter out large change in image coordinates of each feature point from its value in the previous sample time. If any of the feature points moved more than 10 pixels between two consecutive samples, the current coordinates of that point is considered as noise and the coordinates of the previous sample time is used. Moreover, if the same point has been invalid for more than five consecutive frames then the point is considered lost and the most recent coordinates is used.
13.4.4 Pose Estimation from Image Features Three components are required for pose estimation using classicPOSIT (DeMenthon and Davis 1995) (see Fig. 13.11): First, the image coordinates of the feature points must be arranged in the following format: 2 3 u 1 v1 6 :: :: 7 (13.23) 4 : : 5; un vn where n is number of correspondences between 3D points and their 2D image projections; in other words, n is number of feature points. In the case of this work, n D 5. This is obtained from image processing described in Sect. 13.4.3. Second, the relative geometry between feature points expressed in the target object frame must be arranged as an n 3 matrix. These coordinates must be listed in the same order as the image coordinates. In the case of the target object with the dimensions shown in Fig. 13.7, this matrix can be written as follows: 2
0 6 0:085725 6 6 6 0:085725 6 4 0:793750 0:793750
0 0:260350 0:295275 0:260350 0:295275
3 0 0:200025 7 7 7 0:200025 7 7 0:200025 5 0:200025
m:
(13.24)
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Third, the focal length, f , and the principal point, i.e., the point where the optical axis intersects the image plane, cc, must be provided. During offline camera calibration (Bouguet 2004), these parameters were determined for an image of size 320 240 pixels to be: f D 325:7336 pixels;
cc D .192; 180/:
(13.25)
13.4.4.1 Coordinate Frames Transformation and Inverse Kinematics Let us recall that OptiTrack reads the pose of the quadrotor in the inertial frame; while classicPOSIT gives the pose of the target object relative to the camera mounted on the quadrotor. One must transform the results of classicPOSIT to reflect the pose of the quadrotor in the world frame in order to compare them to OptiTrackbased measurements. The relationship between the object frame and the camera frame is determined by classicPOSIT in the form of a rotation matrix and a translation vector, C RO and C tO , respectively. These two pieces of information may be written as a homogeneous transformation matrix from object frame to camera frame, C T O , as follows: C
TO D
C
RO 0
33
C
tO 31 : 1
(13.26)
Because the camera is attached to the quadrotor (Fig. 13.5), the transformation matrix between the camera and the quadrotor-fixed frames, Q T C , is a known constant. One writes: 2
1 6 0 Q TC D 6 40 0
0 1 0 0
3 0 0 0 0:038100 7 7: 1 0:323850 5 0 1
(13.27)
The target object frame has been defined to have the same orientation as the inertial frame but attached to the LED (see Fig. 13.12), which does not move; therefore, the transformation matrix of the object frame relative to the world frame, W T O , is a known constant and can be written as: 2
1 6 0 W TO D 6 40 0
0 1 0 0
3 0 0:567020 0 0:809456 7 7: 1 1:596288 5 0 1
(13.28)
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Fig. 13.12 Real-time pose estimation test setup showing all four coordinate frames and OptiTrack cameras
The pose of the quadrotor in the world frame is calculated by: W 1 Q 1 RQ 33 W tQ 31 W TQ D W TO C TO TC D : 0 1
(13.29)
Let W RQ be in the form 3 R11 R12 R13 W RQ D 4 R21 R22 R23 5 : R31 R32 R33 2
(13.30)
The attitude angles are calculated from: 1;2
where
D atan2
R31 ; R11 cos .1;2 / C R21 sin .1;2 /
R21 1 D atan2 R11
R21 and 2 D atan2 R11
(13.31)
:
(13.32)
R23 cos .1;2 / C R13 sin .1;2 / : D atan2 R22 cos .1;2 / R12 sin .1;2 /
(13.33)
Additionally,
1;2
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Note that atan2 is four-quadrant inverse tangent. One of the two of each attitude angle will be smaller than ˙=2 radians, which is the reasonable value. The quadrotor should neither yaw more than 90 degrees for the projection camera that was used otherwise the target object would be lost from the camera view nor it should roll or pitch more than 90 degrees otherwise it would crash. The position X , Y , and Z are obtained from: T W tQ D X Y Z : (13.34) It takes less than 30 ms from when a new image is available to estimate the pose of the quadrotor. New images are available at 30 frames per second that equates to the sampling period of 33 ms. As acquiring new images takes longer than processing them, pose estimation from images taken by an onboard camera can be done in real-time.
13.5 Experimental Setup The two pose estimation methods were tested with the setup shown in Fig. 13.12. The locations of the quadrotor-fixed frame, Q, the camera frame, C , the target object frame, O, and the world (inertial) frame, W , are shown. Note that the object frame has the same orientation as the world frame with Z-axis of the inertial frame points away from the wall to which the target object is attached. The quadrotor was randomly placed (not flown) in 18 locations around the test area of size 4:2 m 3:6 m in an indoor laboratory. At each location, the quadrotor pose was different. The test area was surrounded by six OptiTrack cameras mounted on the walls for real-time measurement of the quadrotor pose relative to its coordinate frame, which was also defined as the inertial frame. The 6DOF pose of the quadrotor were estimated simultaneously 150 times by classicPOSIT from 150 images acquired and processed in real-time. Pose estimated by Optitrack and classicPOSIT were compared after the experiment. As OptiTrack system claims its position readings is accurate up to 2–4 mm, its pose values were used as the reference against which the classicPOSIT pose estimates were compared.
13.6 Results The results of 6DOF pose estimates using classicPOSIT compared to the estimates using OptiTrack of all 18 tests are shown in Figs. 13.13–13.18. Each point on these plots is the mean value of 150 samples. The standard deviations (SD) of OptiTrackbased and classicPOSIT-based pose estimates are shown in Tables 13.3 and 13.4,
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80 60
X-translation (cm)
40 20 0 -20 -40 -60 -80 -100 -120
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Test #
Fig. 13.13 Comparison of X-translation estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test Mean Y-translation of 150 estimates from each test 80
70
Y-translation (cm)
60
50
40
30 OptiTrack-based classicPOSIT-based 20
0
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6
8
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Test #
Fig. 13.14 Comparison of Y -translation estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test
respectively. The trend of the pose estimated from the two methods follow each other quite well for all 6DOF. Among the three translational DOF, the values of X estimated from onboard cameras are always greater than their OptiTrack counterparts (see Fig. 13.13);
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Mean Z-translation of 150 estimates from each test 140 OptiTrack-based classicPOSIT-based
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Z-translation (cm)
80 60 40 20 0 -20 -40
0
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Test #
Fig. 13.15 Comparison of Z-translation estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test Mean roll angle (φ ) of 150 estimates from each test
10 8 6 4
φ (deg)
2 0 -2 -4 -6 -8 -10
OptiTrack-based classicPOSIT-based 0
2
4
6
8
10
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14
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Test #
Fig. 13.16 Comparison of roll angle estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test
unlike the values of estimated Y and Z that in some tests are almost the same as OptiTrack measurements. For example, the values of Y estimate from the two methods (Fig. 13.14) are almost identical in tests 3, 10, and 14. The same behavior is also observed for Z estimates (Fig. 13.15) in tests 2, 6, and 13.
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θ (deg)
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0
-5
-10
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0
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Fig. 13.17 Comparison of pitch angle estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test Mean yaw angle (ψ) of 150 estimates from each test
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OptiTrack-based classicPOSIT-based 20
ψ (deg)
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0
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-20
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Fig. 13.18 Comparison of yaw angle estimated by OptiTrack and by classicPOSIT. Each data point is the mean value calculated from 150 samples in each test
The error between the two estimation methods, shown by the distance between the two data point in each test, is the greatest for X estimates. This is also supported by the error plot for all translational estimates shown in Fig. 13.19. The smallest error for X estimates is 8 cm; while the largest error is 36 cm. The
13 Nonlinear Real-Time Pose Estimation of Quadrotor UAV Table 13.3 Standard deviation (SD) of calculated from 150 samples in each test Test X (cm) Y (cm) Z (cm) 1 0.0010 0.0016 0.0007 2 0.0014 0.0016 0.0012 3 0.0013 0.0013 0.0009 4 0.0015 0.0015 0.0011 5 0.0016 0.0013 0.0009 6 0.0014 0.0013 0.0012 7 0.0011 0.0013 0.0010 8 0.0015 0.0016 0.0007 9 0.0030 0.0156 0.0014 10 0.0046 0.0023 0.0011 11 0.0067 0.0014 0.0013 12 0.0014 0.0033 0.0012 13 0.0016 0.0011 0.0008 14 0.0013 0.0014 0.0009 15 0.0014 0.0013 0.0008 16 0.0013 0.0014 0.0011 17 0.0015 0.0017 0.0011 18 0.0016 0.0013 0.0012
415
OptiTrack-based 6DOF estimates .ı / 0.0020 0.0018 0.0014 0.0018 0.0018 0.0011 0.0014 0.0018 0.0323 0.0035 0.0015 0.0040 0.0010 0.0014 0.0011 0.0021 0.0028 0.0014
.ı / 0.0024 0.0024 0.0019 0.0021 0.0023 0.0018 0.0021 0.0026 0.0263 0.0041 0.0018 0.0057 0.0018 0.0018 0.0018 0.0035 0.0027 0.0020
.ı / 0.0014 0.0017 0.0011 0.0012 0.0011 0.0021 0.0012 0.0010 0.0049 0.0027 0.0069 0.0013 0.0012 0.0009 0.0010 0.0015 0.0009 0.0011
Table 13.4 Standard deviation (SD) of classicPOSIT-based 6DOF estimates calculated from 150 samples in each test Test X (cm) Y (cm) Z (cm) .ı / .ı / .ı / 1 1:0060 1:0235 0.3511 0.1612 0.2929 0.2562 2 15:8091 3:2475 2.1178 0.4117 0.6308 2.3130 3 3:0482 12:6819 3.8319 0.3264 4.1615 0.8439 4 2:0852 2:3903 1.0314 0.3735 0.6895 0.5683 5 5:8156 6:0252 2.0780 0.2521 1.3932 1.3341 6 1:5017 0:2603 0.1703 0.1268 0.1101 0.5728 7 5:2479 0:9041 0.4290 0.1193 0.1080 1.2824 8 4:1441 1:4190 1.9335 0.2411 0.3816 0.9123 9 1:2467 1:6733 0.8854 0.1656 0.4045 0.2592 10 9:0607 5:2706 2.2709 0.3348 1.0534 1.7751 11 5:5611 5:2413 1.6710 0.3425 1.0230 1.0973 12 4:3397 2:1456 0.5539 0.3584 0.5073 0.9673 13 3:5994 0:7600 1.4632 0.3513 0.2661 0.9592 14 2:1597 1:3318 0.4632 0.1705 0.4542 0.6324 15 8:0223 0:7696 3.1985 0.3527 0.1648 1.9195 16 0:7739 1:5774 0.6383 0.1606 0.3886 0.2138 17 0:9564 0:9210 0.8268 0.2017 0.3142 0.3118 18 7:2199 4:5780 1.2445 0.1617 0.8514 1.3624
values of Y translation estimated from onboard camera are the closest to OptiTrack measurements as seen from Fig. 13.19 that the error of all 18 tests remain within 5 cm. The Z estimation error lies in the middle with the greatest error of 14 cm in test 19 and the lowest error of 0.5 cm in test 2.
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For the three rotational DOF, their performances are comparable in that the estimates from classicPOSIT follow the estimates from OptiTrack and that approximately 85% of the errors remain within 4 degrees (see Fig. 13.20). Only the roll estimate from test 13 exceeded the 4-degree error level; while the rest belong to yaw estimates.
13 Nonlinear Real-Time Pose Estimation of Quadrotor UAV Table 13.5 Mean and standard deviation (SD) of error of classicPOSIT-based pose estimates compared to OptiTrack-based values from all 2,700 samples of 18 tests
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Mean 16.5993 3.3956 5.7379 1.2874 1.5493 4.1395
SD 10.9070 3.8116 4.3130 1.1580 1.4160 1.8739
From the SD shown in Table 13.3, one can see that OptiTrack provides very reliable 6DOF pose estimates. Position measurements vary in the order of 0.02 mm among 150 samples in a test; while orientation measurements vary in the order of 0.003 degree. The SD of classicPOSIT-based pose estimates shown in Table 13.4 are much higher compared to those of OptiTrack, especially for X estimates. This means that the estimation values fluctuate even though the helicopter was static during the experiments. This is due partly to image noise and partly to the feature point extraction inaccuracy. OptiTrack is a well-calibrated commercial system and its measurements are not vulnerable to noise. The system uses IR light instead of visible light and it has wired connection to a computer retrieving the measurements. Pose estimated from images taken by the onboard wireless camera, on the other hand, is error prone. The accuracy of the pose estimated by this method is dependent on the accuracy of feature point extraction. If the locations of feature points are extracted with error even by a few pixels, the resultant pose estimates may be off by a considerable value. The wireless image transmission from the camera could also contribute to the pose estimation error because transmission noise is imminent. Although pose estimation using images from onboard camera is neither as accurate nor as tolerable to noise as OptiTrack-based estimates, it does prove to be an attracting alternative. The errors of all 2,700 estimates from 18 tests for all 6DOF are summarized in Table 13.5. The mean errors for Y and Z estimates are lower than 6 cm; while that for X estimates is 16.6 cm. The mean errors for roll and pitch are less than 2 degrees while that for yaw is less than 4.5 degrees. These errors can still be considered quite low and remain within an acceptable bound. The use of onboard camera as the source for pose estimation does have an advantage that OptiTrack system cannot compete; that is, it allows for an autonomous quadrotor operation outdoors.
13.7 Conclusion Simplified nonlinear equations of motions derived using Newtonian mechanics for a quadrotor enclosed in a carbon fiber cage have been presented in this chapter. The equations are linearized by assuming that the quadrotor operates in quasi-stationary state, a state near hover. The linearized equations are rewritten in state-space format;
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some of them have incorporated a state variable to accommodate an integrator for the design of a controller based on LQR technique. Two real-time quadrotor pose estimation methods have been presented. The first method makes use of OptiTrack system consisting of six commercial IR cameras fixed in the workspace and reconstructs the position and orientation of the quadrotor through images of three reflectors attached to it. The second method uses real-time images of a target object taken by a single camera onboard the quadrotor as the source from which quadrotor pose is estimated. Image coordinates of five feature points located on the target are extracted, rearranged, undistorted, and filtered then passed to an iterative pose estimation algorithm called classicPOSIT. Through coordinates transformation and inverse kinematics, the pose of the quadrotor is determined. The results of the two methods are compared. Experimental results show that pose estimation using images from an onboard camera is quite appealing although less accurate and more susceptible to noise than OptiTrack-based pose estimates. The errors of onboard-camera-based pose estimates using the other method as a reference are relatively low and remain within acceptable bounds. Pose estimates from the two methods discussed in this chapter could be used for autonomous closed-loop control of the quadrotor. Position-based feedback control laws would be designed through the dynamic models presented.
References Allen J, Walsh B (2008) Enahnced oil spill surveillance, detection and monitoring through the applied technology of unmanned air systems. In: International oil spill conference, Washington, D.C., 2008. pp 113–120 Altug E, Ostrowski JP, Mahony R (2002) Control of a quadrotor helicopter using visual feedback. In: Proceedings. ICRA ‘02. IEEE international conference on robotics and automation, 2002 vol 71 pp 72–77 Altug E, Ostrowski JP, Taylor CJ (2003) Quadrotor control using dual camera visual feedback. In: Robotics and automation, Proceedings. ICRA ‘03. IEEE international conference on robotics and automation, 14–19 Sept. 2003 2003. vol 4293 pp 4294–4299 Amidi O (1996) An autonomous vision-guided helicopter. Carnegie Mellon University, Pittsburgh, PA Bouguet JY (2004) Camera calibration toolbox for matlab. http://www.vision.caltech.edu/ bouguetj/calib doc/. Bourquardez O, Mahony R, Guenard N, Chaumette F, Hamel T, Eck L (2007) Kinematic visual servo control of a quadrotor aerial vehicle. IRISA, Rennes, France. http://hal.inria.fr/inria00164387/ Castillo P, Dzul A, Lozano R (2004) Real-time stabilization and tracking of a four-rotor mini rotorcraft. Contr Syst Technol IEEE Trans 12(4):510–516 Chesi G, Hashimoto K (2004) A simple technique for improving camera displacement estimation in eye-in-hand visual servoing. Patt Anal Mach Intell, IEEE Trans 26(9):1239–1242 David P, DeMenthon DF, Duraiswami R, Samet H (2004) SoftPOSIT: Simultaneous pose and correspondence determination. Int J Comput Vis 59(3):259–284 DeMenthon DF, Davis LS (1995) Model-based object pose in 25 lines of code. Int J Comput Vis 15(1):123–141
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Dhome M, Richetin M, Lapreste JT, Rives G (1989) Determination of the attitude of 3D objects from a single perspective view. Patt Anal Mach Intell, IEEE Trans 11(12):1265–1278 Gao XS, Hou XR, Tang J, Cheng HF (2003) Complete solution classification for the perspectivethree-point problem. Patt Anal Mach Intell, IEEE Trans 25(8):930–943 Gramegna T, Venturino L, Cicirelli G, Attolico G, Distante A (2004) Optimization of the POSIT algorithm for indoor autonomous navigation. Robot Autonom Syst 48(2–3):145–162 Grewal MS, Weill LR, Andrews AP (2007) Global Positioning Systems, Inertial Navigation, and Integration (2nd Edition). John Wiley & Sons: 326 Guenard N, Hamel T, Mahony R (2008) A practical visual servo control for an unmanned aerial vehicle. Robot, IEEE Trans 24(2):331–340 Hamel T, Mahony R (2007) Image based visual servo control for a class of aerial robotic systems. Automatica 43(11):1975–1983. doi:DOI: 10.1016/j.automatica.2007.03.030 Hamel T, Mahony R, Lozano R, Ostrowski JP (2002) Dynamic modelling and configuration stabilization for an X4-flyer. In: World Congress, Volume# 15 j Part# 1, Barcelona, Spain. Horaud R, Dornaika F, Lamiroy B (1997) Object pose: the link between weak perspective, paraperspective, and full perspective. Int J Comput Vis 22(2):173–189. doi:10.1023/a:1007940112931 Ley W, Wittmann K, Hallmann W (2009) Handbook of Space Technology. John Wiley & Sons: 347–350 Lu CP, Hager GD, Mjolsness E (2000) Fast and globally convergent pose estimation from video images. Patt Anal Mach Intell, IEEE Trans 22(6):610–622 Mkrtchyan AA, Schultz RR, Semke WH (2009) Vision-based autopilot implementation using a quadrotor helicopter. Paper presented at the AIAA Infotech@Aerospace Conference and AIAA Unmanned. . . Unlimited Conference, Seattle, Washington, 6–9 Apr. Moreno-Noguer F, Lepetit V, Fua P (2007) Accurate Non-Iterative O(n) Solution to the PnP Problem. In: Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, 14–21 Oct. 2007 pp 1–8 Murphy DW, Cycon J (1999) Applications for mini VTOL UAV for law enforcement. vol 3577(1). SPIE NaturalPoint Inc. (2011) OptiTrack: optical motion capture and tracking. http://www.naturalpoint. com/optitrack/. Accessed May 22 NaturalPoint Inc. (2011) Tracking Tools: Real-time optical 3D tracking. http://www.naturalpoint. com/optitrack/products/tracking-tools/. Accessed May 22 Oberkampf D, DeMenthon DF, Davis LS (2002) Iterative pose estimation using coplanar points. In: IEEE pp 626–627 Quanser Inc. Quanser Qball-X4 user manual. Document no. 829. Rev. 12 Quanser Inc. (2011) Quanser – advanced research: Qball-X4. http://www.quanser.com/english/ html/UVS Lab/fs Qball X4.htm. Accessed May 22 The MathWorks Inc. (2011) Simulink – simulation and model-based design. http://www. mathworks.com/products/simulink/. Accessed May 22 Tournier GP (2006) Six degree of freedom estimation using monocular vision and moir´e patterns. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology. Accessed May 26 Triggs B (1999) Camera pose and calibration from 4 or 5 known 3D points. In: The proceedings of the seventh IEEE international conference on computer vision, 1999 vol 271 pp 278–284 Wei L, Lee EJ (2010) Multi-pose face recognition using head pose estimation and PCA approach. JDCTA: Int J Digit Content Technol Applicat 4(1):112–122 Zhang Y, Wu L (2011) Face Pose Estimation by Chaotic Artificial Bee Colony. JDCTA: Int J Digit Content Technol Applicat 5(2):55–63
Chapter 14
Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Klaus Nji and Mehran Mehrandezh
Abstract The problem of nonlinear identification and energy conservative control of balance in a monopod is addressed in this chapter. A monopod is emulated using a concave balancing mechanism, referred to as the body, mounted on an inverted pendulum, referred to as the leg, via a hip joint. The body curvature, represented by , can be altered and is elected as the design parameter of interest as it is observed that at an optimal body span angle, opt , certain interesting phenomena transpire: The linearized system is transformed from nonminimum to minimum phase (MP), the conditions for feedback linearization of the nonlinear model satisfied, and minimal mechanical power required for stability of the simulated model is observed. A nonlinear gray-box system identification routine is developed and implemented within MATLAB, to estimate certain immeasurable parameters that arise within the original system dynamics. To estimate the optimal angle opt , another immeasurable parameter, the identification routine is further employed for various values of . A locus of the transfer function zeros is then used to interpolate the value of at which the system achieves MP behavior. At this configuration, the magnitude of the transfer function zeros become much larger in comparison with that of the largest pole and, therefore, has a negligible contribution to the phase characteristics of the system. After the experimental identification of opt , a Linear Quadratic Regulator with integral action is design and implemented to achieve the control objectives, with the design parameters set as close as physically realizable to the optimal setting. A nonlinear controller based on Feedback Linearization is also designed
K. Nji Software Engineering, EMS Global Tracking - a Honeywell Company, 400 Maple Grove, Ottawa, ON, K2V 1B8 e-mail:
[email protected] M. Mehrandezh () P.Eng., Associate Professor, Faculty of Engineering, University of ReginaRegina, SK, S4S 0A2 e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 14, © Springer Science+Business Media, LLC 2012
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and implemented to take advantage of one benefit of this proposed design. The performance of the nonlinear controller is also compared against that of the LQ Regulator. Analysis and documentation of the power savings from the proposed design, in relation to existing prototypes, is a subject of further works. Furthermore, the performance of the developed controllers can be further improved via careful tuning to meet more stringent control objectives.
14.1 Introduction Stein (2003) in his classic Hendrick W. Bode lecture during the 1989 IEEE Conference on Decision and Control, once said “Respect the Unstable” for the task of controlling an unstable system is not only a challenging but can be a dangerous undertaking.
14.1.1 Literature Survey on Hopping Robots Dynamically stable legged robots remain the platform of choice when compared to wheeled and tracked locomotion in terms of versatility, mobility, and speed in a continuous and smooth terrain. Legged robots especially capable of stable hopping also can be used as an efficient option for transportation in low-gravity environments. Most of the hopping robots that have been realized in laboratory, especially during the past two decades, have been dynamically stable in the sense that stability is only achievable via a continuous hopping motion. Static stability on the other hand, while a useful property if successfully incorporated with hopping, is a challenge to achieve in such small legged dynamic structures. Low, self-sufficient power requirements of autonomous dynamic machines and the physics of keeping a hopper’s center of mass within its small support polygon render the task of achieving static stability a challenging one. Nonetheless, the dynamics and balancing of legged systems have been extensively studied over the past two decades, dating back to Matsouka (1979), with Raibert making a major contribution to the field. Together with his colleagues, the latter successfully built one, two, and four-legged robots that performed stable hopping, running, centering gaits, and summersaults (Raibert 1997; Playter 1994). The basic structure of Raibert’s monopod (i.e., one-legged robot), mainly constructed to understand the dynamics of legged locomotion, comprised of a body mass mounted atop a single leg via a hip joint. The hopper was capable of achieving stable hopping using a simple, yet appropriate, tri-partitioned control scheme. In this scheme, forward velocity was controlled using a foot placement algorithm, hopping height by regulating the system energy stored in a linear spring, and balance by using the body to apply torques onto the hip.
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The body in Raibert’s planar monopod was constructed with its center of mass above the hip joint. The physics of this setting is synonymous with the well known inverted pendulum (Spong and Praly 1996a) and, depending on active link, can be compared in structure to the Acrobot (Spong and Praly 1996b), pendubot (Spong and Block 1995), or even the inverted wedge (Hsu 1992). The Acrobot and Pendubots, both two link planar robots, are simply variants of the double inverted pendulum, with the fundamental difference between both underactuated systems residing in the location of the lone actuator as illustrated in Fig. 14.1. On the Acrobot (Spong 1995), the actuator is located at the base of the active link (the elbow). In contrast, the actuator is situated at the base of the first link (the shoulder) on the Pendubot (Spong and Block 1995). The above underactuated mechanical systems have four equilibrium states, although it is more common to find control objectives targeted toward the unstable equilibrium where both links are in the upright position. These systems share several similar and undesirable properties – they are nonminimum phase (NMP) and inherently unstable systems. It is well known that NMP systems, or systems with transfer function zeros on the right side of the s-plane, are difficult to control because of the high control cost to meet almost any control objective (Qui and Davison 1993). The reason for the high control cost associated with NMP systems is the presence of an inverse-response and time-delay within the system dynamics. Dynamic electromechanical systems are especially prone to the detrimental effects arising from NMP zeros given that high control costs typically translate to larger and more powerful motors, larger batteries or larger power sources. The net effect of these larger components is a sluggish and power hungry system, which entirely defeats the purpose of cheap and modern control. When one further considers the Rover deployed to Mars by the Jet Propulsion Laboratory at NASA (NASA Jet Propulsion Laboratory 2004–2005), it immediately becomes evident how an energy conservative strategy is paramount if dynamic machines are to make their place in history.
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Raibert already foresaw the setbacks of power hungry systems after using hydraulic actuators in his first monopod. He subsequently made attempts to reduce the power demands of the earlier monopods by first replacing the hydraulic actuators with electrical ones (Raibert 1997). Numerous attempts have been made to further reducing the overall power demands of the hopping cycle. Based on the assumption of loose coupling between the dynamics of height regulation, balance, and forward velocity as initially proposed by Raibert, the approach has been to make individual amendments on the power demands from each phase of the hopping cycle. For instance, to reduce the power consumption required for balance during stance, efforts have ranged from morphological modifications to the synthesis of more efficient control laws. For instance, Ringrose (Ringrose 1997) built a self-stabilizing running robot with a large circular foot, while Zeglin replaced the linear spring that acted as energy storage in Raibert’s model with a bow string in his Bow Legged robot (Zeglin and Brown 1998). In the latter, the effect was a noticeable reduction in the leg mass, to about 1% of body mass, which allowed the hopper to attain higher hopping heights without compromising power savings. Zeglin also introduced passive stability by placing the body’s center of mass below the hip joint. Papantoniou (1991), Rad and Buehler (1993), and Dummer and Berkemeier (2000) used lighter electric instead of heavy hydraulic actuators. Ahmadi and Buehler added hip and leg compliance, reducing the energy loss during impact at touchdown (Ahmadi and Beuhler 1997). McGeer built systems without sensors, actuators, or computers (McGeer 1995), which were also very energy efficient, although they had inherent control and intelligent limitations. From the perspective of more efficient control, especially in light of the recent advances in nonlinear control, analytical studies on simplified system dynamics were performed by Koditschek and Buhler (1991) and Vakakis et al. (1991). These studies have subsequently aided in the development of higher order, energy efficient, and robust controllers for each phase in the hopping cycle, which can be assumed to be loosely coupled. Examples of these works can be found in Helferty et al. (1989); Mehrandezh et al. (1995); Schwind and Koditschek (1994); Becker et al. (1995), although this list is far from exhaustive.
14.1.2 Contributions to the Literature Contributions of this work are twofold: A power conserving, inherently stable, balancing mechanism of a monopod is proposed in order to achieve the objective of controlling posture during stance. The proposed structure consists of a concave downward-swinging symmetrical body mounted on a single inverted pendulum serving as the leg of a monopod. The balancing structure, also referred to as the body in this chapter, is mounted on the inverted pendulum, also referred to as the leg, at a hip joint. In contrast to some existing balancing prototypes, the center of mass of the structure lies below the hip joint. Furthermore, body movement occurs
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in a plane below the hip joint. We will show that a specific selection of physical parameters results to a unique body angle, one that is later referred to as the optimal angle (opt ). At this optimal configuration, the linear approximation of the proposed idealized model is transformed from a nonminimum to a minimum phase system, and the original nonlinear system dynamics is feedback linearizable. The concept of lowering the body center of mass below the hip, thereby inducing a more stable system, was also used by Gokan et al. in their rope hopping robot (Yoshinda et al. 1994). Such structures with the body center of mass below the hip joint and bodily movements occurring predominantly in the plane below the hip can be classified as “Downward-Swinging” prototypes. Raibert’s planar hopper, the hip and leg compliant hopper built by Ahmadi and Buehler, as well as the energy efficient hopper built by Papantoniou, the Acrobot, Pendubots, and double inverted pendulum for instance, where the body center of mass lies above the hip joint and bodily movement occurs predominantly in the plane above the hip joint, can be classified as a “Upward-Swinging” prototypes. The fundamental difference between the two classes centers around inherent stability of the active link. “Upward-Swinging” prototypes have an inherently unstable active link whereas “Downward-Swinging” Prototype have an inherently stable active link. The second contribution of this work is the synthesis of several control laws that can be further tuned and adapted solely for the purpose of controlling balance in a monopod in the presence of modeling and parametric uncertainty. We are not reinventing the wheel but using proven formulae to synthesize and test different types of controllers: A Linear Quadratic Regulator (LQR) and Feedback Linearizing controller (EFL) (Slotine 1991c). Integral action is incorporated into the controller solutions in an effort to reduce steady state errors. Experimental results with both controllers within the loop are also presented. The reason behind the synthesis of two different control laws is to compare the performance of the classic LQR against nonlinear techniques, such as EFL. Given that EFL control is very sensitive to modeling uncertainty, no overwhelming performance improvements are expected of this controller even though successful simulation runs are obtained. For robustness to parametric and modeling uncertainties, other forms of robust control are being considered. This work gets its motivation from the recent research interest in nonholonomic systems and developments in nonlinear control methodologies. Our inspiration comes from recent projects such as the Mars Rover, the Sony QRIO robot (Sony Corporation 2005), and other advances in Legged and Walking robots. The minimally actuated hopping rover for planetary exploration built by Fiorini et al. (1999) was also a good point of reference.
14.1.3 Outline The remainder of this chapter is organized as follows: Section 14.2 revisits a detailed description of the mechanical structure. In this chapter, the equations governing the dynamics of the system are developed and linearized about the unstable or
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desired equilibrium. Stability analysis of the linearized model is also performed and an investigation of the power requirements is studied with respect to a certain design parameter. Section 14.3 deals exclusively with system identification. A nonlinear gray box technique is used to estimated certain immeasurable parameters that appear in the governing dynamics. Exhaustive experimental results obtained from the identification of the optimal design parameter are also presented. System identification is done at numerous system configurations to arrive at the optimal design solution. Section 14.4 deals with the formulation and synthesis of the Feedback Linearizing Controller with Integral Action. Simulation and a sample experimental result are presented.
14.2 Problem Definition and Mathematical Model 14.2.1 Introduction In this section, an elaborate derivation of the mathematical model for the proposed balancing mechanism of a monopod (i.e., a one-legged robot) is undertaken. First, a detailed physical description of the system is given, followed by a formal definition of the control problem. As in most model-based control exercises, a mechanistic model to predict the dominant system dynamics is formulated with two different types of control inputs. Based on the nature of the control problem, a stability analysis of the linearized system is performed about a desired equilibrium point. A brief analysis of the mechanical power requirement as a function of a certain design parameter concludes the section.
14.2.2 Problem Formulation and Physical Description of System The goal of this project is to design an energy conserving balancing mechanism and control system to maintain balance in a monopod during the stance phase of a hopping cycle. The physical system mainly comprises a concave downward-swinging symmetrical body mounted on an inverted pendulum serving as the noncompliant leg of the monopod. Figure 14.2 presents the real system used in this work. The balancing mechanism, henceforth referred to as the body, is mounted on the pendulum, henceforth referred to as the leg, via a joint, henceforth referred to as the hip. The basic structure is similar to a 2-link planar manipulator but for the fact that motion of the active link occurs predominantly in the southern hemisphere. The southern hemisphere, in this context, refers to the region below the hip joint.
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Fig. 14.2 Physical structure of the system
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The body is constructed using two aluminum bars, connected at an angle, , at the hip joint. At each end of the aluminum bars is affixed a small mass such that this section of the body can be modeled as concentrated point masses. The links are connected to a platform that allows the selection of ten discrete possible values for , which is designated as the design parameter for reasons that will become apparent later. A DC motor attached to the base of the leg provides the torque necessary to induce body motion in response to a disturbance. The motor is mounted at the base of the leg to minimize adverse torques that would arise from its weight if mounted higher up the leg axis. A chain and sprocket transmission transfers the torque from the motor shaft to the body. Incremental optical encoders are used to measure the angular displacements of both links from a defined equilibrium. The leg, 80 cm long and constructed from a 2 400 piece of wood, is free to rotate around a pivot, hence, an inverted pendulum by design. This scenario, which mimics a one-legged hopping robot during the stance phase of a hopping cycle, allows one to greatly simplify the hopper’s dynamics during stance. The fixed nature of the pivot simulates static friction between the foot of the robot and ground. This friction force is typically assumed to prevent slippage when the monopod is titled during stance.
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With the body mounted on the leg via the hip joint, the goal, therefore, is to balance the system around its unstable equilibrium state defined as the point where the leg is vertically up and the body is in its normal stable position. This equilibrium position, defined as a reference in the phase plane is the only point along the trajectory where encoder readings are set to zero.
14.2.3 Mathematical Model with Motor Torque as Input The proposed balancing mechanism consists of a concave downward-swinging body modeled as two concentrated masses, m2 , each located at an equidistance, l2 , from the hip joint (joint 1). The center of gravity of the body is assumed to be located at a distance, lc2 , below the hip joint, as illustrated in Fig. 14.3. An armature controlled Permanent Magnet Direct Current (PMDC) motor of mass, mm , is affixed to the leg at a distance, lm , from the foot (joint 0). Motor toque is transmitted to the body via a chain and sprocket assembly. The leg is assumed to have a uniformly distributed mass, ml , and is of total length l1 . The center of mass of the leg is located a distance, lc1 , from the foot. The moment of inertia of the leg and body about their respective centers of gravity is represented
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Table 14.1 Description of the system physical parameters
Symbol m1 m2 mm l1 l2 IL1 IL2 Im fr1 fr2 fc1 fc2
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Description Mass of leg Mass of body Mass of motor Location of leg COG with respect to foot Location of body COG with respect to hip joint Moment of inertia of leg about its COG Moment of inertia body about its COG Moment of inertia of motor shaft about its axis of rotation Viscous friction coefficient at joint 0 Viscous friction coefficient at joint 1 Coulomb friction coefficient at joint 0 Coulomb friction coefficient at joint 1
by IL1 and IL2 . The mass moment of inertia of the motor’s rotor about its axis of rotation along the shaft is denoted by Im . Viscous friction coefficients for both joints are labeled fr1 and fr2 . Coulomb friction coefficients for both joints are labeled fc1 and fc2 . Half of the angle between the two arms of the body is represented by . We will later show via simulation, that at a specific angle, referred to as opt , the linearized model of the system is transformed from nonminimum to MP, the conditions required for feedback linearization of the nonlinear model are satisfied and minimal mechanical power required for stability during stance is observed. Table 14.1 summarizes the physical parameters of the system. In order to describe the position and orientation of a body in space, a coordinate system or frame is typically attached to the object. The position and orientation of this frame is then described with respect to some reference or base frame. In this work, two coordinate frames are used to describe the position and orientation of both links. The origin of a base coordinate Frame 0 is affixed to the foot and the origin of Frame 1 at the hip joint. Using the Denavit-Hartenberg convention (Sciavicco and Siciliano 1990, 42) axis z0 is chosen along the rotational axis of the joint 0 and axis z1 along the rotational axis of joint 1. Axes y0 and y1 are chosen to represent the vertical axes of both coordinate frames. Axes x0 and x1 are then chosen to complete the right-hand rule. An illustration of the coordinate frame assigned to the model is given later (Fig. 14.4). With respect to the above coordinate frames, the angular displacement of the leg from the vertical (y0 ) is represented by q1 with counterclockwise rotation denoted as positive. The relative angular displacement from the leg to the body’s center of mass is represented by q2 . Defining a state vector q D Œq1 q2 T , the equations of motion governing any robotic manipulator can be written in the form: P qP C Fr qP C Fc sgn.q/ P C G.q/ D ; M.q/qR C C.q; q/
(14.1)
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Fig. 14.4 Coordinate frames y1
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xo
Joint 0
where M.q/ represents the inertia matrix, C.q; q/ P a matrix of Coriolis and centrifugal forces, Fr a diagonal matrix of joint frictional constants, and G.q/ is the gravitational matrix. The governing dynamic equations of the system are obtained using the Langrange formulations given as: d @L @L D i dt @qP i @qi
i D 1; 2
(14.2)
with the Langrange defined as: L DT U
(14.3)
and T and U , respectively, represent the total kinetic and potential energy of the system. qi represent the generalized coordinates (q1 and q2 in our system) and i represents the generalized forces associated with the generalized coordinate qi . However, (Sciavicco and Siciliano 1990, 140) have provided a simpler method of computing the system matrices after defining certain position vectors, which can then be used to obtain the required system Jacobians. Using their method, the inertia matrix can be obtained from the expression .Li /T
B.q/ D mi JP
.Li /
JP
.Li /T
C JO
.Li /
ILi JO
.mi /T
C mmi JP
.mi /
JP
.mi /T
C JO
.mi /T
Imi JO
;
(14.4)
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
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the Coriolis terms from n 1P 1 X @bi k @bj k cij D bij C qP k ; 2 2 @qj @qi
(14.5)
kD1
and the gravitational matrix from X @U D mi JPLii .q/ C mmi JPmii .q/ ; @qi j D1 n
gi D
(14.6)
where mi and mmi respectively represents the mass of each link and associated motor, pi the positional vector of the center of gravity of each link with respect to the origin of the base frame, pmi , the positional vector of the motor center of mass with respect to the base frame, and g0 , the gravity vector. The expression bij represents the elements of the inertia matrix. JO and JP represent a special set of system matrices computed solely from the position vectors of the links Pi . First, we represent the position vector of the origin of the base or reference frame by P0 . P1 and PL , respectively represent the position vectors of frame 1, and the leg center of mass, with respect to the base frame. The position vector of the body center of mass and that of the motor’s center of mass, with respect to the base frame, is represented by Pb and Pm , respectively. PNb represents the position vector of the body center of mass with respect to frame 1. The above vectors are explicitly given below: T P0 D 0 0 0 T P1 D l1 sin q1 l1 cos q1 0 T PL D lc1 sin q1 lc1 cos q1 0 T PNb D lc2 sin.q1 C q2 / lc2 cos.q1 C q2 / 0 Pb D ŒP1 C PN b T Pm D lm sin q1 lm cos q1 0 T Zm D 0 0 1 : For the sake of simplicity, the subscripts s1 , c1 , s2 , c2 , s12 , c12 , will be used from this point forward to represent sin.q1 /, cos.q2 /, sin.q2 /, cos.q2 /, sin.q1 C q2 /, and cos.q1 Cq2 /, respectively. Using the above vectors and the chosen coordinate frame, the following Jacobian matrices can be computed: 3 2 2 3 lc1 c1 0 l1 c1 C lc2 c12 lc2 c12 .l/ .b/ JP D 4lc1 s1 05 (14.7) JP D 4 l1 s1 C lc2 s12 lc2 s12 5 0 0 0 0
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2
.l/ JO
.m/
JP
3 0 0 D 40 05 1 0 2 3 lm c1 0 D 4lm s1 05 0 0
2
.b/ JO
.m/
JO
0 4 D 0 1 2 0 D 40 1
3 0 05 1 3 0 05
(14.8)
(14.9)
gr
. Using the state vector q D Œq1 q2 T and representing the input torque provided by the DC motor by , the equations of motion governing the system in the form of (14.1) is given by: m2 2m1 c2 m3 m1 c2 qR1 m1 s2 x4 m1 s2 .x3 C x4 / qP1 C m1 s2 x3 m3 m1 c2 m4 qR2 0 qP2 m5 s1 C m6 s12 0 qP1 fc1 sgn.qP 1 / 0 fr1 0 C C D ; C m6 s12 0 fr2 qP2 0 fc2 sgn.qP1 / l (14.10) which can be also be expanded to .m2 2m1 c2 /qR1 C .m3 m1 c2 /qR2 C .m1 s2 qP2 C fr1 /qP 1 Cm1 s2 .qP 1 C qP 2 /qP2 C fc1 sgn.qP 1 / m5 s1 C m6 s12 D 0
(14.11)
.m3 m1 c2 /qR1 C m4 qR2 m1 s2 qP 12 C fr2 qP2 C fc2 sgn.qP2 / C m6 s12 D l ; (14.12) where m1 D ml2 l1 lc2 2 2 m2 D ml1 lc1 C Il1 C ml2 l12 C ml2 lc2 C Il2 C mm lm2 C Im 2 m3 D ml2 lc2 C Il2 C Im gr 2 m4 D ml2 lc2 C Il2 C Im gr2
m5 D ml1 glc1 C ml2 gl1 C mm glm m6 D ml2 glc2
(14.13)
and a classic friction model, described by (14.15), has been used to represent the joint friction (Papadopoulos and Chasparis 2002) as follows: Tf D fci sgn.qPi / C fri qP i
(14.14)
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Table 14.2 PMDC motor parameters
for i D 1 and 2, where
Parameter i La Ra km u ka m m
8 ˆ ˆ 0 qP D 0
(14.15)
qP < 0
with fci representing the Coulomb friction and fri representing the viscous friction as described earlier in this section. For the sake of control purposes, it is desirable to express the system input in terms of the supply voltage to the servo amplifier driving the DC motor. In the proceeding section, the dynamics of the motor is presented and subsequently integrated into the above system equations such that the generalized input to the system is the input voltage, u to a servo amplifier.
14.2.4 Mathematical Model with Computed Voltage Signal as Input The dynamics of a PMDC motor is governed by two coupled equations (Kuo 1995): La
dia .t/ C Ra ia .t/ C kb Pm .t/ D ka u.t/ dt m .t/ D km ia .t/;
(14.16) (14.17)
where the variables are defined in Table 14.2. Considering the motor driving the load though a gearing mechanism with ratio gr , gearing efficiency , and ignoring the inertia of the gears, the torque balance for the entire actuator assembly is l .t/ m .t/ D Im Rm .t/ C Bm Pm .t/ C Tf sgn.Pm / C ; gr
(14.18)
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where m is the torque at the motor input shaft, l the torque experienced by load, Bm the motor internal friction constant, and Tf is the motor internal friction torque. Assuming that the armature inductance, La , is relatively smaller than other terms, its effect can be neglected. Further assuming m D gr q2 and combining (14.16), (14.17), and (14.18) results in Tl .t/ D
gr kamp km u.t/gr2 Ra
kb km CBm qP2 .t/gr Tf sgn.qP 2 / gr2 Im qR 2 .t/: Ra (14.19)
Combining the second order differential equations in (14.11) with (14.19) results in .m2 2m1 c2 /qR1 C .m3 m1 c2 /qR2 C .m1 s2 qP2 C fr1 /qP 1 Cm1 s2 .qP 1 C qP 2 /qP2 C fc1 sgn.qP 1 / m5 s1 C m6 s12 D 0
(14.20)
.m11 m12 c2 /qR1 Cm13 qR2 m12 s2 qP 12 Cm14 qP2 Cm15 sgn.qP2 /Cm16 s12 D u; (14.21) where gr kamp km Ra kb km m8 D gr2 C Bm Ra
m7 D
m9 D gr2 Im m10 D Tf gr m3 m11 D m7 m1 m12 D m7 m4 C m 9 m13 D m7 m14 D
fr2 C m8 m7
fc2 C gr Tf m7 m6 D m7
m15 D m16
(14.22)
and u is the required computed control voltage. Equations (14.20) and (14.21) will be linearized and used for stability analysis and controller synthesis.
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14.2.5 System Equilibrium The first step in analyzing any nonlinear system is to linearize it about some nominal operating point. Linearization allows the control engineer to gather local behavior of the system about the nominal equilibrium. Before linearizing the system, however, the nominal operation point which is typically a system equilibrium point must be appropriately defined. An equilibrium point1 is a state in which a system would remain if it were unperturbed by external disturbances (Bernstein 1997). Thus, at the equilibrium point, system velocities are zero (i.e.,) qP1 D qP 2 D qR1 D qR 2 D 0
(14.23)
f1 .q1 ; q2 / D 0
(14.24)
f2 .q1 ; q2 / D 0;
(14.25)
such that
where f1 and f2 represent the equations in (14.20) and (14.21), respectively. The values of the equilibrium points can be achieved by solving (14.24) and (14.25) simultaneously. Performing the mathematics and substituting (14.23) into (14.24) and (14.25) results in the equilibrium equations: m6 sin.q10 C q20 / m5 sin.q10 / D 0
(14.26)
m16 sin.q1 C q20 / C m15 D ueq ;
(14.27)
where q10 and q20 define the equilibrium points of q1 and q2 , respectively. Equation (14.26) defines a continuous equilibrium manifold around which the system can be regulated while (14.27) defines the motor input voltage that is needed to maintain the equilibrium point. Thus, for a given value of q10 , there exists a corresponding value of q20 such that an equilibrium manifold equation is satisfied, i.e., x20 D x10 C sin1
m5 sin.x10 / : m6
(14.28)
A plot of the equilibrium points given by (14.28) is illustrated in Fig. 14.5 with clockwise angular displacement represented by a negative sign. A plot of the input voltage required to maintain each equilibrium position is given by Fig. 14.6. Note that these plots, based on immeasurable system parameters computed using first
1
In the phase plane, an equilibrium point is a singular point.
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q20 (degs)
20 0 −20 −40 −60 −80
−80
−60
−40
−20
0
20
40
60
80
20
40
60
80
q10 (degs)
Fig. 14.5 The equilibrium manifold 1.5
1
ueq (V)
0.5
0
−0.5
−1
−1.5
−80
−60
−40
−20
0
q10 (degs)
Fig. 14.6 Equilibrium input voltage to actuator
principles or from simple experiments, will be repeated after these parameters have been estimated in Section 14.3. Amongst the equilibrium set is the singular or equilibrium point q10 D 0 q20 D 0 ueq Ñ 0;
(14.29)
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which is the only point of interest. The point .q10 D 0; q20 D 0; ueq Ñ 0/ is selected as the nominal operating point2 about which the original nonlinear dynamics, given by (14.20) and (14.21), will be linearized and system stability analysis performed. Worthy of note is a nonzero input voltage required to maintain the system about the nominated nominal equilibrium. This voltage, though less than the motor deadband of 0.3V, is a function of both the Coulomb friction manifested by the leg joint (joint 1) as well as the internal friction torque of the motor. In other words, if the motor has to overcome a larger overall “dry” friction torque within a close neighborhood of the nominal equilibrium, the value of the equilibrium voltage will be larger as well. The equilibrium manifold described by (14.26) and illustrated in Fig. 14.5 describes a one to one mapping between the output variable of interest q1 and the actuated degree of freedom (DOF) q2 . Therefore, a control system, during the process of driving the output q1 to q10 D 0 as defined by the control objectives, will inadvertently drive the state variable q2 to q20 D 0. The system can, therefore, be considered as a single-input-single output (SISO) system allowing one to apply the pole-zero concept in SISO linear theory to analyze system stability.
14.2.6 Open Loop Poles and Zeros of Linearized System One of the objectives of this exercise is to design a balancing mechanism that can be used to maintain an upright posture in a monopod with conservative energy demands. This translates to stabilization of the plant about the singular points defined in (14.29) for the rotational angles of each link, as described in the previous section. Before engaging in controller synthesis, a stability analysis of the open loop plant around the desired equilibrium will be performed to allow for a better understanding and definition of the control objective. Furthermore, this analysis would later provide an insight into the selection of a design parameter – a parameter which influences the overall mechanical power consumption of the system. A linear model of the system is first obtained by linearizing the original dynamics given in (14.20) about the unstable equilibrium position with the states and control input set to zero, (i.e., q1 D q2 D qP1 D qP2 D u D 0). Defining the state matrix x D Œq1 q2 qP 1 qP2 3 the linearized dynamics about x D Œ0 0 0 0 is given as:
2 3
xP D Ao x C Bo u
(14.30)
y D Co x C Do u;
(14.31)
In accordance with the control objectives. Throughout this chapter, the variables qi and xi , i D 1; 2 will be used interchangeably.
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where 2 6 6 6 6 Ao D 6 6 6 4
3
0
0
1
0
0 m16 .m3 m1 / C m13 .m5 m6 /
0 m16 .m3 m1 / m13 m6
0 m1 3fr1
1 m14 .m3 m1 /
.m5 C m6 /.m12 m11 / C m16 .2m1 m2 / m6 .m11 m12 / C m16 .2m1 m2 / fr1 .m11 m12 / m14 .m2 2m1 /
7 7 7 7 7 7 7 5
(14.32) 2
3
0 7 6 0 7 6 6 m3 C m1 7 Co D 1 0 0 0 Bo D 6 Do D 0 7 7 6 4 m 2m 5 2 1 D .m12 m11 /.m3 m1 / C m13 .m2 2m1 /
(14.33)
(14.34)
and the constants mi , i D 1 W 16 are given in (14.13) and (14.22). The open-loop transfer function relating the output of interest, q1 , to the control input u is given by: P .s/ D
Bo .3/s 2 C Bo .4/Ao .3; 2/ Ao .4; 2/Bo .3/ q1 .s/ D 4 ; u.s/ s .Ao .4; 2/ C Ao .3; 1//s 2 C det.Ao /
(14.35)
where the coefficients of the laplace operator in the transfer function are elements of the system triple .Ao ; Bo ; Co / given in (14.32) and (14.33). It can be easily verified that the zeros of the above transfer function are given by the expression: r m6 po D ˙ ; m1 ¤ m3 : (14.36) m 1 m3 A brief discussion on the effects of the location of open loop zeros on control effort follows. The idea is to allow an appreciation of the energy conservative nature of the novel balancing mechanism and choice in design parameters. Before venturing further, it should be noted that the linearized system has one unstable open loop pole, implying an unstable open loop system as expected. The purpose of the control system will be to drive the unstable pole to the stable region of the s-plane. Controllers, on the other hand, cannot influence the location of transfer function zeros. Hence, any attempt to minimize the detrimental effect of unstable transfer function zeros simply by manipulating the physical design will be beneficial to the system. We will see why in the next section.
14.2.7 Control Effort Dependency on Span Angle One of the main contributions of this work is the design of a balancing mechanism for a monopod that consumes less mechanical power as compared to existing
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prototypes. To aid in the selection of a suitable set of design parameters, a qualitative analysis of the mechanical power requirements is first performed simply by observing the location of the open loop transfer function zeros. Before doing so, it is worth reviewing the concepts of internal stability of Linear Time Invariant (LTI) Systems. Even though the original system is nonlinear, it is assumed that system operation occurs predominantly within a close neighborhood to the origin such that the dominant behavior of the system can be represented by the corresponding LTI set of equations in (14.30) and (14.31). A LTI system whose transfer function has strict left half plane (LHP) zeros possesses a stable internal dynamics. Such systems are referred to as a MP system. Definition 14.1. The internal dynamics of a system is the part of the dynamics that is not observable from the external input–output relationship (Slotine 1991d). A NMP system, on the other hand, has transfer function zeros on the right half plane (RHP) or an associated time delay within the system dynamics. Such systems are also characterized by unstable internal dynamics, which represents undesirable phenomena “outside” the input–output relationship, such as violent vibrations of mechanical members. NMP systems are difficult to control because their step response is typically characterized by undershoot. NMP systems also suffer from the Waterbed Effect, meaning that small sensitivity to noise can be achieved over one frequency range only at the expense of large sensitivity in another frequency range (Freudenberg and Looze 1985). Moreover, the application of certain control schemes such as Input–Output Linearization (IOL), a variant of feedback linearization (FL), is possible only if the internal dynamics of the system is stable. Transfer function zeros, however, are intrinsic properties of linear systems. Slotine (1991c) extended the concept of zeros to nonlinear systems in the form of the zero-dynamics. Definition 14.2. Zero dynamics is defined as the internal dynamics of the system when the system output is kept at zero by the input (Slotine 1991a). For linear systems, the roots of the zero dynamics are exactly the zeros of the system transfer function. Thus linear systems with strict LHP zeros poses stable internal dynamics, permitting control effort to be constrained within conservative bounds. For nonlinear systems, on the other hand, the relationship is not very clear, even though it can be shown that local asymptotic stability of the zero dynamics is enough to guarantee local asymptotic stability of the internal dynamics (Slotine 1991b). Stability of the zero dynamics of a nonlinear system can be determined using the Lyaponuv’s Linearization Stability theorem, which states: Theorem 14.3. • If the linearized system is strictly stable (i.e., if all eigenvalues of the system matrix, Ao , are strictly in the left half of the s-plane) then the equilibrium point is asymptotically stable for the actual nonlinear system.
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• If the linearized system is unstable (i.e., at least one of the eigenvalues of the system matrix, Ao , is strictly in the right half of the s-plane) then the equilibrium point is unstable for the actual nonlinear system. • If the linearized system is strictly marginally stable (i.e., if all eigenvalues of the system matrix, Ao , are strictly in the left half of the s-plane but atleast one of them lies on the j! axis) then one cannot conclude anything from the linear approximation, as the equilibrium point may be stable, asymptotically stable, or unstable for the nonlinear system. The zero-dynamics corresponding to the linearized model is simply the linearization of the zero dynamics of the nonlinear model. From the Lyaponuv’s Linearization theorem, one can therefore conclude that local stability of the zero dynamics for the nonlinear model can also be extracted from the location of the transfer function zeros. Thus, if a linear system is NMP, the nonlinear system can also be considered NMP. The same applies for MP linear systems.
14.2.8 Definition of as Design Parameter Referring to (14.36), it can be seen that the system can be made minimum or NMP simply by altering the location of the transfer function zeros. The location of these transfer function zeros is governed by the denominator of (14.36) given that .m6 > 0/, which in turn is a function of the angle between the two arms of the body represented by . Consider, for a moment, the case in which m1 D m3 such that the transfer function zeros are of very large magnitude or nonexistence. Making the simplifying assumption 2 IL2 D m2 .l22 lc2 /
(14.37)
for the moment of inertia of the body and solving from from the expression m1 D m3 using the definitions in (14.10), results in D opt D cos
1
m2 l22 C Im gr m 2 l1 l2
:
(14.38)
This specific value of is referred to as opt for reasons that will become apparent toward the end of this section. First, we will consider the changes in the location of the open-loop transfer function zeros as a function of . Within the range 0 < opt the transfer function (14.35) has one undesirable RHP zero for m3 < m1 . The zeros lie very close to the origin when D 0 and traverse the s-plane, along the real axis toward ˙1 as approaches opt . As such, the linear system is always NMP for < opt .
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Table 14.3 Transfer function zeros for changes in 0ı 10ı 20ı 20ı zeros ˙15:7581 ˙17:1810 ˙26:9791 ˙20:6659i
40ı ˙10:8636i
50ı ˙7:3409i
441
60ı ˙5:2506i
Numerous reports have been presented on the detrimental effect of NMP zeros based on the Bode Integral (Bode 1945). Francis and Zames (1984) qualitatively showed that requiring the magnitude of the sensitivity function, jS.j!/j, to be arbitrarily small over some frequency interval forces it to be large elsewhere. jS.j!/j directly quantifies such feedback properties as output disturbance rejection and robustness to small parameter variation, while the magnitude of complementary sensitivity function jT .j!/j D j1 S.j!/j quantifies the response of the feedback system to sensor noise. Freudenberg and Looze illustrated that large values of jS.j!/j reduce the close loop gain margin (Middleton et al. 2000) further showed that the presence of RHP zeros forces the system to exhibit undershoot to a step response. In summary, NMP zeros are not desirable, especially when low power consumption is a design concern. Within the range opt < 90ı the transfer function given by (14.35) has two complex conjugate zeros along the j! axis for m3 > m1 . Within this range, the system is no longer NMP and could be considered as marginally minimum phase (MMP). Starting at very large magnitudes when D opt , the zeros move toward the origin along the imaginary axis as approaches 90ı . As such, the linear system is always MMP for 90ı > opt . Middleton (1998) studied the effect of zeros in the j! axis and concluded that a system with j! zeros of sufficiently small magnitude relative to the poles will always exhibit overshoot to a step response. They further demonstrated that the presence of an upper limit on the permissible overshoot of the output signal places an effective lower bound on the achievable settling time of the closed loop system when the plant has zeros on or near the imaginary axis. This therefore implies that systems with j! axis zeros of sufficiently large magnitude suffer from neither the aforementioned shortcoming nor the detrimental effect of NMP zeros. Being able to directly influence the phase characteristics of the system by simply varying one variable is the main reason behind the choice of as the design parameter. Table 14.3 lists the exact zero location as is varied from 0ı to 60ı . Figure 14.7 further illustrates the trajectory of the transfer function zeros as is varied from 0ı to 60ı . Starting from D 0, the transfer function zeros traverse the real axis to a very large value when D opt . From the figure it can be deduced that, opt appears to lie between 20ı and 30ı . For values of > opt , these transfer function zeros traverse to the imaginary axis and move closer to the origin as is increased further. In the next section of this chapter an experimental value of opt will be obtained via system identification. Later, the effect of changes in on the control effort required to maintain stability will be analyzed.
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K. Nji and M. Mehrandezh Pole−Zero Map 25
0 10
20 φ opt ≤ φ ≤ 60
20
o
30
15
40 50
10
Imaginary Axis
60 0≤φ≤φ
5
opt
0 −5 −10 −15 −20 −25 −40
−30
−20
−10
0
10
20
30
Real Axis
Fig. 14.7 Locus of transfer function zeros are is varied from 0ı to 90ı
14.3 System Identification A mathematical representation of a physical phenomenon is the very basis of modelbased control design. The method of obtaining this mathematical representation or model depends on the process complexity and the amount of a priori information available on the process dynamics. System identification is a statistical method of obtaining the mathematical model of a process entirely from data or estimating certain meaningful physical parameters that cannot be accurately measured, yet exist in a mathematical model that has been computed from first principles.
14.3.1 Introduction The mathematical model of a physical process or phenomenon is very useful in many disciplines. Mathematical models allow one to gain a better understanding of process dynamics via simulation. If one considers the case of an expensive plant operating in a dangerous regime; manned aircraft, chemical processes, nuclear reactors, and space robotics readily come to mind, mathematical models provide a low cost option that allows the control engineer to gain an insight on controller
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performance once within the loop, thus saving development cost. Models can also be used in optimization problems and to predict events, such as weather patterns, stock prices, interest rates, and much more. All real processes such as discontinuity, hysteresis, static friction, jump phenomena, and many more are nonlinear and their dynamics cannot be exactly described using mathematical relationships. Even when the dominant behavior of the process can be well captured using well known mathematical formulae, there often are certain physical parameters that are not measurable and can only be estimated using field data. For instance, the dynamics of a robotic manipulator can be accurately modeled using kinematic relationships between the links (Craig 1989; Sciavicco and Siciliano 1990). Nonetheless, rigid body parameters, such as the moment of inertia and the center of gravity of each link, are difficult to measure and are typically estimated from input–output data in the process of system identification.
14.3.2 Overview of System Identification System Identification is the method by which a process model is obtained using input–output data and immeasurable parameters within a precomputed mathematical model are estimated. The models used in the identification procedure are classified as White Box, Gray Box, or Black Box, depending on how much prior knowledge is available on the process (Nelles 2001a, 15). Irrespective of the nature of the model being used, the goal of the identification procedure is to obtain a mathematical representation that is capable of capturing the dominant dynamics of the real process as the overall quality of the resulting mathematical representation places an upper bound on the performance of the final problem solution. For instance, the performance of a model-based controller will depend on how well the model captures the essential dynamics of the process it represents. Figure 14.8 briefly illustrates the concept of system identification. Here, the model output yO is compared against that of the real process, y, corrupted with the noise vector n. The objective of the identification procedure is to obtain a model, f , that minimizes the difference between the process and model output, e, according to some performance criterion. Before engaging in the identification of the plant parameters used in this chapter, it is worthwhile reviewing five major steps that make up system identification. 1. Selection of an Input Channel: In the case of multiple input systems, this step involves the selection of inputs that will be used to excite behaviors of interest. In the case of single input systems, such as the one used in this chapter, the choice is obvious. 2. Selecting an excitation signal: The excitation signal is of utmost importance, as it presents the only opportunity to excite the dynamics of interest in the real process.
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Process
+ − Model
Fig. 14.8 Overview of system identification
3. Selecting a model architecture: This is one of the most difficult steps, given that there can exist numerous models that can adequately describe any given process. The final decision is often based on the problem type or intended use of the model. For instance, a decent mathematical model is sufficient for control purposes. 4. Selecting a set of model parameters and estimation of the parameter values: Once a model architecture has been selected from the previous step, the choice of model parameters is a relatively straight forward task. Depending on how the parameters appear in the model structure, they can be estimated using linear or nonlinear optimization techniques. For model architectures derived from empirical formulae (white box), the parameters typically represent meaningful physical quantities. In most robotic applications, these parameters typically include the moment of inertia of links and friction coefficients between joints. Linear optimization techniques are often used to exploit the linearity property of a manipulator model with respect to a suitable set of dynamic parameters (Sciavicco and Siciliano 1990, 143). In the case of underactuated manipulator models, a case in point is the model used in this chapter, obtaining a set of dynamic parameters is not a straight forward proposition; hence, a nonlinear optimization technique is employed, as will be explained later in this section. 5. Validating the model: After the parameters have been determined from the previous step, model integrity is validated using fresh data. This will allow the engineer to predict performance of the overall problem solution based on a fit between the real and simulated process output. The above paragraphs briefly describe system identification. For a more comprehensive coverage of the subject, readers can refer to either of the following sources (Nelles 2001a; Ljung 1999). As mentioned earlier, selection of a model architecture and excitation signal the most crucial steps in system identification. This becomes evident when one considers the following scenario: In Black Box identification, where no prior
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information is available on the process and model parameters have no physical interpretation, the burden of extracting any important process behavior lies entirely on the choice of the excitation signal. Black box models are used when a process cannot be accurately described using empirical–mathematical relationships. In Gray box identification, however, some prior knowledge about the system is available. As previously mentioned, there still are certain parameters in the model that cannot be measured and will have to be estimated using input–output data. Estimation of these parameters is often guided by some prior knowledge, such as an allowable operational range or the type of parametric value. Although prior knowledge facilitates the selection of a model architecture, it does not make the selection of an excitation signal an easier task. Careful thought has to be put into the selection of an excitation signal such that its frequency content excites only the process dynamics of interest and not high frequency unmodeled dynamics commonly found in mechanical systems. Before dwelling further on this, the experimental setup for this project will now be introduced.
14.3.3 Experimental Setup The objective in this chapter is to develop a balancing mechanism and control system that can maintain an upright posture using minimal mechanical power on a 2 DOF, one legged robot. The balancing mechanism or body is inherently stable and constructed such that it lowers the overall center of gravity of the host system. The body is made up of two links with masses concentrated at each end. The links are attached to the hip joint and are separated by angle . The hip joint is constructed such that can be varied, allowing one to study how this design parameter influences the stability properties and mechanical power requirements of the system. An image of the body is given in Fig. 14.9. The balancing mechanism is mounted on a 200 400 piece of wood 80 cm long. This serves as a noncompliant leg of the hopping robot during stance. At the base of the leg is mounted a PMDC motor powered by an Advanced Motion servo amplifier (Advanced Motion Controls 2004–2005). The specifications of the motor and servo amplifier are listed later in this chapter. A chain and sprocket assembly is used to transmit power from the motor to the balancing mechanism. Control signals generated in MATLAB/Simulink (The Mathworks 2004–2005) are communicated to the hardware using Mathworks Real Time Windows Target via a National Instruments Data Acquisition card, NIDAQ 6036E (National Instruments 2004–2005). Rotational angles of both links are measured using incremental optical encoders with a resolution of 2,500 and 1,000 PPR. The entire mechanical system is mounted on a stand as shown in Fig. 14.10.
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joint
link
Fig. 14.9 Physical configuration of balancing mechanism
leg joint 2 body
PC running MATLAB
BNC connector
Servo amp + Power Supply
transmission joint 1 encoders
Fig. 14.10 Configuration of the entire experimental setup
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
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14.3.4 Nonlinear Gray Box Parametric Identification System Identification is the method by which a process model is obtained using input–output data and immeasurable parameters within a precomputed mathematical model are estimated. One of the most difficult tasks in system identification is the selection of a model structure. This task is facilitated if the physical phenomenon can be well described using first principles. An identification procedure based on a model computed using first principles, that is a nonlinear function of the unknown parameters is referred to as Nonlinear Gray Box Identification (Nelles 2001a, 15). The starting point in Nonlinear Gray-box identification is the determination of a continuous-time nonlinear state space model structure of the form: xP D f .t; x.t/; ; u.t//
(14.39)
y D h.t; x.t/; ; u.t/; w.t//;
(14.40)
where f and g are nonlinear functions, x.t/ the state vector, y.t/ the output vector, the unknown parameter vector to be estimated, u.t/ the excitation signal, w.t/ the measurement disturbance signal, assumed to be white for simplicity, and t denotes the time vector. As mentioned earlier, the objective of identification is to find the parameter vector, , that minimizes a cost function according to a certain optimization criterion. A commonly used optimization criterion is minimization of a cost function based on the squared error between the real y.t/ and estimated process output y.t; O /. This cost function is expressed mathematically as
VN D
N 1 X 2 e .t; / N i D1
(14.41)
for a given input–output data, where e which represents the difference between the process output, y, and predicted output, y, O for our purposes will be explicitly given by: e.t; / D y.t/ y.t; O /:
(14.42)
The parameter vector is estimated by applying a prediction error method, which performs a numerical optimization of criterion 14.41 using an iterative search algorithm. Such algorithms require an initial estimate of the parameter vector o , and the quality of the final solution will depend on the choice of initial estimate. First, we select a model structure from which the parameter vector to be estimated can be extrapolated. Although the goal of this work is to stabilize the system about its upright unstable equilibrium, system identification is performed around the stable downward equilibrium for safety and stability reasons. It is assumed that the estimated parameters equally apply around both equilibriums.
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Fig. 14.11 System used for identification
Link 1
l
Link 2
ml2
The model structure chosen is a mathematical model of the robot around its stable downward equilibrium, as shown in Fig. 14.11. In state space form, the dynamics around the stable equilibrium is given as: f .q ; q ; qP ; qP / C g3 .q2 /u qR 1 D 3 1 2 1 2 (14.43) f4 .q1 ; q2 ; qP 1 ; qP 2 / C g4 .q2 /u qR2 where 0
1 .m3 m12 s2 qP12 m3 m14 qP2 m1 c2 m12 s2 qP12 m1 c2 m14 qP2 C m13 qP1 fr1 B C B C B 2m qP m s qP m m s qP 2 m3 .fc2 C m10 /sgn.qP2 / C m m s C m m s C 13 1 1 2 2 13 1 2 2 13 5 1 13 6 12 C B m B C 7 @ m c .f C m /sgn.qP / A 1 2 c2 10 2 m3 m16 s12 m1 c2 m16 s12 C m13 .fc1 /sgn.qP1 // m7 f3 D .m11 m3 C m11 m1 c2 C m12 c2 m3 C m12 c2 /2 m1 m13 m2 2m13 m1 c2
(14.44) 0
1 fc1 sgn.qP1 / ..m1 s1 qP2 C f r1/qP1 C m1 s2 .qP1 C qP2 /qP2 m5 s1 m6 s12 B C m3 C m1 c2 B C B .m2 C 2m1 c2 /.m3 m12 s2 qP 2 m3 m14 qP2 m1 c2 m12 s2 qP 2 m1 c2 m14 qP2 C 1 1 B C B C B C B 2m qP m s qP C m qP f m m s qP 2 m3 .fc2 C m10 /sgn.qP2 / C m m s C B C 13 1 1 2 2 13 1 r1 13 1 2 2 13 5 1 m7 B C @ A m1 c2 .fc2 C m10/sgn.qP2 / m3 m16 s12 m1 c2 m16 s12 C m13 m6 s12 C m13 fc1 sgn.qP1 /// m7 f4 D .m11 m3 C m11 m1 c2 C m12 c2 m3 C m12 c2 /2 m1 m13 m2 2m13 m1 c2
(14.45)
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
g3 D
449
m3 C m1 c2 .m11 m3 C m11 m1 c2 C m12 c2 m3 C m12 c2 /2 m1 m13 m2 2m13 m1 c2 (14.46)
g4 D
m2 2m1 c2 .m11 m3 C m11 m1 c2 C m12 c2 m3 C m12 c2 /2 m1 m13 m2 2m13 m1 c2 (14.47)
and the constants, mi ; i 2 Œ1 W 16 are given again as follows: m1 D ml2 l1 lc2 2 2 m2 D ml1 lc1 C Il1 C ml2 l12 C ml2 lc2 C Il2 C mm lm2 C Im 2 m3 D ml2 lc2 C Il2 C Im gr 2 m4 D ml2 lc2 C Il2 C Im gr2
m5 D ml1 glc1 C ml2 gl1 C mm glm m6 D ml2 glc2 gr kamp km Ra kb km m8 D gr2 C Bm Ra m7 D
m9 D gr2 Im m10 D Tf gr m3 m11 D m7 m1 m12 D m7 m4 C m 9 m13 D m7 m14 D
fr2 C m8 m7
fc2 C gr Tf m7 m6 D m7
m15 D m16
(14.48)
Immeasurable quantities in the above system constants which make up the parameter vector are: T D Lc1 Lc2 IL1 IL2 fr1 fr2 fc1 fc2 (14.49) where all variables were defined in Sect. 14.2.3.
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Estimation of the parameter is done in two steps. In the first step, two initial estimates, 01 and 02 , are obtained using simple experiments and empirical formulae. In the second step, each initial estimate is passed to an optimization routine which searches for a parameter vector that minimizes the sum of squares of the error between the real and simulated output state variables at every time instant. The initial estimate that returns a lowest residual is selected. A chirp signal is used to excite the system for generation of input–output data. Advantages of a chirp signal are persistent excitation and continuity in terms of the amplitude levels and frequency content (Wang p 1992). A chirp signal is also chosen as it delivers a waveform with a crest factor of 2 not too far above the theoretical recommended value of 1 (Pintelon and Schoukens 2001). A crest factor defines the amount of input power delivered by the signal and is defined as the ratio of peak signal amplitude to the effective root mean square (rms) value, where effective means that only the signal power in the frequency range of interest is used in the rms calculation. For instance, signals with an impulsive behavior have a large crest factor and will, for any given peak value, inject much less power into the system than a signal with a lower crest factor. A frequency sweep from 0.5 Hz to 3 Hz is selected, as this is sufficient to excite dominant dynamics of our electromechanical system. Frequency ranges above 3 Hz tend to excite high frequency unmodeled dynamics, such as slackness in the drive train. Selection of a sampling rate is coupled to the systems’ dominant time constant, which also defines the bandwidth. Sampling at rates substantially greater than system bandwidth may lead to data redundancy, numerical issues, and modeling of high frequency artifacts likely due to noise. Sampling at rates lower than system dynamics leads to difficulties in determining correct system parameters as well as the problems introduced by aliasing. A common rule of thumb is to sample signals at 10 times the bandwidth of the system (or bandwidth of interest) for the model (Ljung 1999). Assuming uncertainty exists in the system bandwidth, and provided that a very fast data acquisition environment is available, it is useful to sample as fast as possible and then use digital prefiltering and decimation to reduce the sampling rate to the desired value. Based on the above guidelines, a sampling rate, Ts , of 0.05s is chosen for the identification procedure. The number of data points, N is chosen to minimize the model error based on the relationship p n model error p ; N
(14.50)
where represents the noise variance and n represents the number of parameters to be estimated (Nelles 2001b). Encoder readings representing angular displacements of both links from the stable equilibrium are stored in the MATLAB workspace. The differential equations that describe the plant dynamics given in (14.43)–(14.47) are implemented using Simulink function blocks. The Simulink mathematical model is excited
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Table 14.4 Physical parameters
Symbol m1 m2 l1 l2 g E ka Tc TPK mm Ra La Km Ke Tf Jm gr Ts
Units Kg Kg m m Kg=m=s2 V V/V Nm Nm Kg ˝ mH N-m/A V/rad/s Nm Kg m2 – s
451
Value 1.6330 1.2300 0.4 0.37 9.81 24 2.4 3.5 33 1.017 2.64 2.63 0.046 0.046 8.5e-3 1.6e-5 65.5:1 0.05
with the same chirp signal used on the real plant and a Runge-Kutta numerical integration algorithm solves for the resulting angular displacements qO1 and qO2 . A constrained nonlinear least squares optimization algorithm, lsqnonlin, available in the MATLAB Optimization toolbox (The Mathworks 2004–2005), minimizes the squared error, e 2 , between the real and the simulated output states where e D .qO 1 q1 /2 C .qO 2 q2 /2 , resulting to an optimal solution in . The parameters are validated on fresh data and a measure of the model quality using the estimated parameters is obtained using the expression 0
qP N
1
i D1 .qx .t/ qOx C B fit D 100 @1 qP A; N 2 .q .t/ . q N // x x i D1
.t//2
(14.51)
where x is 1 or 2, depending on the angular position vector, and N is the length of the data set. Measurable system parameters are given in Table 14.4, while Figs. 14.12 and 14.13 below, illustrate the effectiveness of the nonlinear system identification routine with set to 20ı . The experiment is carried out with system parameters given in Table 14.4. Figure 14.12 represents real and simulated data collected before the optimization routine. It can be observed that there is no correlation between the real and the simulated data for link 1, which is also confirmed by a negative fit of 1743.
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1 0 −1 −2
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30
q2 (rads)
4 2 0 −2
0
time(s)
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Fig. 14.12 Comparison between real and simulation data for D 20ı using raw data, Fit = 1743I 85:03
5 0 −5 0
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0.4 0.2 0 Real Simulation
−0.2 −0.4
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0 4
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2 0 −2
0
time(s)
Fig. 14.13 Comparison between real and simulation data for D 20ı using estimated data, Fit = 88.0846;92.2398
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
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14.4 Experimental Identification of opt The objective in this chapter is to develop a balancing mechanism and control system that can maintain an upright posture, using minimal mechanical power, in a 2 DOF, one legged robot. The balancing mechanism comprises of two links separated by an angle with concentrated masses at each end. In Sect. 14.2.7 the effect of a change in on the system power requirements was analyzed. It was observed that to achieve minimal mechanical power simply by altering the physical structure, will have to be set such that the linearized system has transfer function zeros of relatively large magnitude, or no transfer function zeros at all. The value of that corresponds to the lack of transfer function zeros is referred to as opt . A mathematical expression for opt , derived in (14.38), is defined by the equation 2 1 m2 l2 C Im gr D opt D cos : (14.52) m2 l1 l2 The derivation of (14.52) is based on a simplified moment of inertia of the body about its center of gravity, i.e., 2 IL2 D m2 .l22 lc2 /:
(14.53)
The above expression is further based on the simplifying assumption that the center of gravity of the body lies in line and midway between both link masses. These assumptions inadvertently lead to an inaccurate estimation of opt , which could prevent the realization of mechanical power savings. Moreover, IL2 is one parameter that cannot be accurately measured or computed and will have to be estimated via system identification. To obtain a more accurate value for opt , system identification is, therefore, performed for several values of . The resulting parameters are then employed with the linearized model, and the locus of the transfer function zeros is used to obtain a better estimate of opt . Figure 14.13, on the other hand, represents real and simulated data collected after the optimization routine. It can be seen that the optimization routine successfully converges to a good solution as simulated data correlates to the real data. Fit has also dramatically improved for link 1 and link 2 to 88% and 92%, respectively. Specifically, system identification was performed for six settings of : D 20ı ; 30ı , 40ı , 50ı , 60ı . Table 14.5 below shows the parameters estimated from a nonlinear gray box identification routine, while Tables 14.6 and 14.7, respectively, display the fit before and after system identification. Figures 14.14–14.21 illustrate results of system identification for each individual selection of . The plotted results and resulting goodness measures further illustrate the effectiveness of our identification procedure. Using estimated parameters given in Table 14.5, the transfer function zeros of the linearized system are plotted for various values of . As indicated in Fig. 14.22, the system is NMP for D 20ı and 30ı , as there is a transfer function zero on the positive real axis on the s-plane, at each of these settings. For D 40ı ; 50ı ; and 60ı the system is marginally MP as transfer function zeros lie along the imaginary axis.
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Table 14.7 Fit values after system identification
.ı / Link 1 Link 2
Supplied Input (V)
Table 14.6 Fit values before system identification
40 0.0554 0.0430 0.0095 0.0221 0.1438 0.5000 0.0081 0.7555 0.2668
20 1743.6 85.0
20 88.1 92.2
50 0.0631 0.0400 0.0035 0.0252 0.1004 0.4999 0.0219 0.6343 0.2537
30 1386.8 82.96
40 97.9 82.0
30 87.8 98.6
40 83.3 99.1
60 0.0747 0.0253 0.0199 0.0241 0.1000 0.5000 0.0040 0.2156 0.1744
50 561.6 80.5
60 344.4 76.7
50 82.2 97.8
60 83.3 98.0
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1 0 Real Simulation
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Fig. 14.14 Comparison between real and simulation data for D 30ı using raw data, Fit = 1386I 82:9
Therefore, there exists a value of D opt such that transfer function zeros are of very large magnitude or nonexistent. This value is between 30ı and 40ı . In other words, 30ı opt 40ı . For the sake of comparison, the mathematical
Supplied Input (V)
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
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Fig. 14.15 Comparison between real and simulation data for D 30ı using estimated data, Fit D 87.8;98.6
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1 0 −1
Real Simulation
−2 0
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4 2 0 −2
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Fig. 14.16 Comparison between real and simulation data for D 40ı using raw data, Fit D 97:9I 82:0
value of opt computed using (14.38) is 21:5ı . Several sources can be cited for this discrepancy. The predominant contributor, however, lies within the simplification of system dynamics and assumptions that had to be made prior to formulating the nonlinear system equations. Nonetheless, this results further substantiates the theory
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Fig. 14.17 Comparison between real and simulation data for D 40ı using estimated data, Fit D 83.3;99.1
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Fig. 14.18 Comparison between real and simulation data for D 50ı using raw data, Fit D 561:6I 80:5
behind an optimum that will render the linearized system MP (i.e., no transfer function zeros). The physical design only allows discrete selection of angles in multiples of 10ı . As such, a opt is set to 40ı for the rest of the project.
Supplied Input (V)
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
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Fig. 14.19 Comparison between real and simulation data for D 50ı using estimated data, Fit D 82.2;97.8
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Fig. 14.20 Comparison between real and simulation data for D 60ı using raw data, Fit D 344:4I 76:7
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Fig. 14.21 Comparison between real and simulation data for D 60ı using estimated data, Fit D 83.3;98.0
Pole−Zero Map 20
15
30 40 50
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Imaginary Axis
5
0 ≤ φ ≤ φ opt 0
−5
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o
−10
−15 −200
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Fig. 14.22 Pole-Zero locus using estimated parameters
50
100
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14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot Table 14.8 Parameter Link 1 Link 2
Model fit changes for a 40% parametric uncertainty lc1 lc2 IL1 IL2 fr1 fr2 4:87 38:35 0:13 26:06 1:51 2:42 0:01 2:33 0:00 0:01 0:01 0:89
f10 0:08 0:00
f20 7:12 3:02
459
53:19 17:07
14.4.1 Sensitivity Analysis With set to 40ı , a sensitivity analysis is performed for each individual parameter. The procedure involves finding a parameter that produces the greatest deviation from the real model when multiplied by an uncertainty factor. A C40% uncertainty is applied to all the parameters estimated in the previous section and changes in the data fit are recorded as illustrated in Table 14.8. Each column in the table represents a change in model fit when the corresponding parameter is multiplied by a +40% uncertainty. For instance, column 10 represents the change in model fit when the parameter is multiplied by a 40% uncertainty. It can be observed that uncertainty in this parameter introduces the largest modeling uncertainty. The same pattern is observed for negative parametric uncertainty.
14.4.2 Conclusion In this chapter, a customized nonlinear gray-box system identification procedure was successfully used to estimate certain parameters in the dynamic model of a onelegged robotic prototype. The estimated parameters are either impossible or difficult to measure and compute mathematically, forcing their estimation using input–output data. An optimal design angle, referred to as opt , which was defined in Sect. 14.2.7, qualifies as a parameter whose mathematical computation is flawed from over simplification of nonlinear phenomena inherent in the real system. Consequently, this important, yet immeasurable, system parameter was estimated using a nonlinear identification procedure, and was found to lie within the neighborhood of 30ı and 40ı . Given that the physical design only allows discrete selection of angles in multiples of 10ı , is set to 40ı for the remainder of the project. Furthermore, a sensitivity analysis was also performed on the estimated model parameters. It was observed that uncertainty in lc2 and largely account for modeling discrepancy.
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14.5 Feedback Linearizing Control 14.5.1 Introduction The mathematical representation of a process plays a dominant role in modelbased control design. Linear Quadratic Regulator(LQR) is a practical example of a control system based on the mathematical model of the plant. Feedforward controller (Ogunnaike and Ray 1994, 571), Smith Predictor (Ogunnaike and Ray 1994, 605), Inverse-Response Compensator (Ogunnaike and Ray 1994, 613), and the Model Predictive Controller (MPC) (Ogunnaike and Ray 1994, 991), just to name a few, are several examples of practical model-based controllers. Amongst the fundamental problems of system analysis, model-based control solves the “Process Control” problem summarized as follows: Given a desired output behavior and the process model, derive a controller that would enforce the specified output behavior. Performance of the derived controller in meeting the control objectives is, therefore, constrained by how accurately the process model describes the behavior of the real process. Virtually, all real world systems are nonlinear and certain physical phenomena cannot exactly be described using mathematical relationships. A typical strategy in such a case is to linearize the original system dynamics around a desired operating point. Such approaches, which typically result in linear controllers such as LQR, have been successful in industry. This is because most physical processes take on approximately linear behavior as they approach steady state (Ogunnaike and Ray 1994, 625). Linear controllers perform reasonably well for small excursions from steady state. For instance, for large excursions from steady state or when modeling uncertainty exists, performance of a linear control system deteriorates. In this case, more effective alternatives must be considered. Nonlinear control systems, on the other hand, explicitly recognize uncertainty or nonlinearities in the process model. As a result, these control strategies allow larger operation ranges and often offer larger tolerances to plant or modeling uncertainties. So, when the nature of the process demands tight performance specifications, large excursions from steady state, and robustness to modeling uncertainty, nonlinear control is the method of choice. The noticeable increase in academic and industrial research in the area of nonlinear control has been spurred in part by the advantages nonlinear controllers offer over their linear counterparts, although recent advances and availability of low-cost and high-speed microprocessors have also played a major role.
14 Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot
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14.5.2 Overview In this section control of balance in the one-legged robot is attempted using Feedback Linearization (FL), a nonlinear control strategy. Feedback Linearization has been successfully used to address some practical control problems in systems such as high-performance aircraft (Sweriduk et al. 2004), industrial drives (Ha 1990), and spark ignition engines (Guzella and Schmid 1995). Basically, it involves transforming the original nonlinear dynamics of a system into an equivalent linear model via exact state transformation and feedback. In contrast to Jacobian linearization used in LQR control design, for instance, where the effects of nonlinearities in the plant model are ignored, Feedback linearization attempts to deal with nonlinearities via state transformation and feedback. As a result, one naturally should expect better performance in terms of larger operation range. However, because of the fact that exact cancelation of the system nonlinearities requires an accurate model, Feedback Linearization controllers are potentially sensitive to parametric and modeling uncertainty. The remainder of this section presents synthesis of a feedback linearizing controller to maintain an upright posture on a one-legged monopod. Control objectives and the mathematical formulation of the plant are referenced from the previous section in this chapter.
14.5.3 Controller Design The first step in designing a control system for any given physical plant is to derive a meaningful model of the plant (i.e., one that captures the dominant dynamics around the operation range of interest). The basic idea behind FL is to perform a coordinate transformation of the original system dynamics via feedback such that, with respect to the new coordinates, the model is linear. Within this coordinate system, linear control techniques can then be applied. FL applies only to fully actuated systems. However, input-state linearization is a variant that shares similar properties with FL and can be applied to underactuated systems. Consider the original system dynamics qR1 f3 .q1 ; q2 ; qP 1 ; qP 2 / C g3 .q2 /u ; (14.54) D qR2 f4 .q1 ; q2 ; qP 1 ; qP 2 / C g4 .q2 /u where 0
1 .m3 m12 s2 qP12 m3 m14 qP2 C m1 c2 m12 s2 qP12 C m1 c2 m14 qP2 C m13 qP1 fr1 B C B C B C2m qP m s qP C m m s qP 2 m3 .fc2 C m10 /sgn.qP2 / m m s C m m s C 13 1 1 2 2 13 1 2 2 13 5 1 13 6 12 C B m7 B C @ m1 c2 .fc2 C m10 /sgn.qP2 / A C m3 m16 s12 C m1 c2 m16 s12 C m13 .fc1 /sgn.qP1 // m7 f3 D .m11 m3 m11 m1 c2 C m12 c2 m3 m12 c2 /2 m1 m13 m2 C 2m13 m1 c2
(14.55)
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1 fc1 sgn.qP1 / ..m1 s1 qP 2 C f r1/qP1 m1 s2 .qP1 C qP 2 /qP2 C m5 s1 m6 s12 B C m m c 3 1 2 B C B .m2 2m1 c2 /.m3 m12 s2 qP 2 m3 m14 qP2 C m1 c2 m12 s2 qP 2 C m1 c2 m14 qP 2 C 1 1 B C B C B C m .f C m /sgn. q P / 3 c2 10 2 B C2m qP m s qP C m qP f C m m s qP 2 C m m s 13 1 1 2 2 13 1 r1 13 1 2 2 13 5 1 B C m7 B C @ A m1 c2 .fc2 C m10/sgn.qP2 / m3 m16 s12 C C m1 c2 m16 s12 C m13 m6 s12 C m13 fc1 sgn.qP1 /// m7 f4 D .m11 m3 m11 m1 c2 m12 c2 m3 m12 c2 /2 m1 m13 m2 C 2m13 m1 c2
(14.56) g3 D
m3 m1 c2 .m11 m3 m11 m1 c2 m12 c2 m3 m12 c2 /2 m1 m13 m2 C 2m13 m1 c2 (14.57)
g4 D
m2 C 2m1 c2 .m11 m3 m11 m1 c2 m12 c2 m3 m12 c2 /2 m1 m13 m2 C 2m13 m1 c2 (14.58)
and the constants, mi ; i 2 Œ1 W 16 were defined in Sect. 14.2 and u is the computed control input. We want to find a state transformation z D T .x/ and input transformation u D .x/ C .x/v such that the original system dynamics is transformed into an equivalent linear time-invariant dynamics of the form zP D Az C Bv with
2
0 60 AD6 40 0
1 0 0 0
0 1 0 0
3 0 07 7 15
(14.59) 2 3 0 607 7 BD6 405 :
0
(14.60)
1
For the original dynamics (14.54) to be Feedback Linearizable, there must exist a region < such that the following conditions hold: 1. The matrix
M D g adf .g/ ad2f .g/ ad3f .g/
(14.61)
.g/ is full rank where the term adkf .g/ denotes the iterative Lie Bracket Œf; adk1 f @g @f where the Lie Bracket of f and g is defined as Œf; g = @x f @x g. For simplicity of notation, we will represent this control matrix as M D w1 w2 w3 w4 . The above condition is mathematically equivalent to: det.M / ¤ 0:
(14.62)
2. The set of vectors fg; adf .g/; ad2f .g/g is involutive in D @xi j D1 After performing the math, we obtain the following coordinate transformation 2
x1 x3 Tc1 C m6 .sin.x1 / C x2 cos.x1 // C x3 fr1 m5 sin.x1 / m2 C 2m3
3
7 6 7 6 7 2 3 2 3 6 7 6 z1 T1 7 6 7 6 7 6 7 6 7 6z2 7 6T2 7 6 7 6 7D6 7D6 7 4z3 5 4T3 5 6 m6 .cos.x1 / x2 sin.x1 // m5 cos.x1 / cos.x1 / 7 6 x C m x C 3 6 4 7 6 m C 2m m C 2m z4 T4 2 3 2 3 7 6 7 6 4 fr1 .Tc1 C m6 .sin.x1 / C x2 cos.x1 // C x3 fr1 m5 sin.x1 // 5 .m2 C 2m3 /2
(14.74)
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and feedback linearizing control input uO given by (Khalil 2002, 510) uO D
< dT4 ; f > v D .x/v C %.x/; < dT4 ; g > < dT4 ; g >
(14.75)
where 2m3 m2 m6 c cos.x1 / .m6 . sin.x1 / x2 cos.x1 // C m5 sin.x1 //x32 m6 sin.x1 /x3 x4 %.x/ D C m2 C 2m3 .x/ D
fr1 x3 .m6 .cos.x1 / x2 sin.x1 // m5 cos.x1 // C ::: .m2 C 2m3 /2 m6 x3 x4 sin.x1 / fr1 x4 m6 cos.x1 / C C m2 C 2m3 .m2 C 2m3 /2 fr12 m6 .cos.x1 / x2 sin.x1 // m5 cos.x1 / C m2 C 2m3 .m2 C 2m3 /2 Tc1 C m6 .sin.x1 /Cx2 cos.x1 // C x3 fr1 m5 sin.x1 / 2m3 m2 m2 C 2m3 m6 cos.x1 / Resulting from the input and state transformation is the following set of linear equations zP 1 D z2 zP2 D z3 zP3 D z4 zP4 D v:
(14.76)
To force integral action into the control law for the elimination of steady-state error, one can augment the state transformation Z with an integral of x1 such that the augmented system in canonical form will be given by: zP D Az C Bv; where
2R
z1
3
6 z 7 6 1 7 6 7 z D 6 z2 7 6 7 4 z3 5 z4
2 0 60 AD6 40 0
3 0 1 0 0 0 0 1 07 7 0 0 0 15 0 0 0 0
(14.77) 2 3 0 607 6 7 6 7 B D 607 : 6 7 405 1
(14.78)
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For stability, v is chosen as v D KEFL z
(14.79)
such that the matrix ABKEFL is Hurwitz (Khalil 2002, 238). The overall nonlinear control for the original system (14.10), is obtained using (14.79), (14.75), and (14.64).
14.5.4 Simulation In order to obtain a fair comparison, the control gains for both a previously sythesized LQR and FLs are based on a common set of closed loop poles. At this stage of experimentation, the focus is simply to verify the ability of this control solution to stabilize the robot. Integral action is omitted at this stage from the Feedback Linearizing strategy. Using the state and control matrices 3 2 300 0 0 0 6 0 200 0 07 7 QD6 4 0 0 1 05 0 0 0 1
R D 60
(14.80)
for LQR without integral action results in the following controller gains: KLQR D 100:7650 13:9073 24:8712 0:0324 :
(14.81)
Which represent the smallest LQR controller gains that will stabilize the simulated system. Continuous-time closed loop poles corresponding to the above controller gains are 41:2144; 4:0215; 3:1210; and 1:0368. Discrete closed loop poles are not used to synthesize the EFL gains for the sampling effect of the transformation given in (14.76) is not known. Using the Ackermann method (Powell et al. 1990, 245), in conjunction with the above closed loop poles, the nonlinear control gains KEFL without integral action are given as: KEFL D 536:2861 835:4767 357:0555 49:3936 :
(14.82)
Efficacy of the controller in rejecting four types of disturbances will be considered. The first type of disturbance is a constant voltage applied at the servo amplifier input. The second type of disturbance considered is a periodic voltage applied to the input of the servo amplifier. A disturbance comprising of two opposing pulses, 1 s in width, spaced 20 s apart is also considered. Each pulse has an absolute magnitude of 1 V. These disturbances are applied with all system states initially set to zero. Finally, the system states are subjected to nonzero initial values and
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u (V)
1 0 5
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−1
0 −2 10 0
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Fig. 14.23 Simulation response of FL vs. LQR control in response to constant disturbance
controller behavior is monitored. To add some degree of reality, nonlinearities such as deadband, white noise in the control input, and a ZOH element are all included in the Simulink model. Angular velocities estimates are obtained by differentiating positional vectors and applying the result through a first-order low pass filter with a cutoff frequency of 2Hz.
14.5.4.1 Constant Voltage Disturbance Constant disturbance comprises of a constant voltage signal applied persistently to the servo amplifier driving the robot’s DC motor. This disturbance, C, enters the system through the same channel as the control input, i.e., Va D u C C;
(14.83)
where Va is the actual voltage applied to the servo amplifier, u the computed control voltage, and C is the disturbance. A value of C = 0.1V was used in the simulation. It can be observed from Fig. 14.23 that this voltage disturbance forces the FL controller to balance the robot about an equilibrium other than the desired. One can also observe that the closed loop behavior of the system under FL and LQR control is similar. Another observation is that the FL controller is a little more sluggish than the LQR control.
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u (V)
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−100 0
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Fig. 14.24 Simulation result of FL vs. LQR control in response to a sinusoidal disturbance
14.5.4.2 Periodic Voltage Disturbance Periodic disturbance comprises of 1 Hz sinusoidal voltage signal with an amplitude of 2V. As before, this disturbance is persistently applied through the same channel as the control input, i.e., Va D u C 2 sin.2 t/; (14.84) where Va is the actual voltage applied to the servo amplifier and u is the computed control voltage. The simulation result presented in Fig. 14.24, reveals a balanced system, even though the disturbance is hardly attenuated. For such small disturbances, one can hardly distinguish the response of the system under LQR and FL control.
14.5.4.3 Pulsed Voltage Disturbance A disturbance comprising of two opposing voltage pulses, 1 s wide, 20 s apart, each with an absolute magnitude of 1V, is also considered. In contrast to the two previous cases, this disturbance is momentarily applied with the system in open loop before the nonlinear FL is applied. Figure 14.25 reveals an overall satisfactory response of the nonlinear FL controller. One also can observe the similarities in the response between LQR and the FL controller.
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Fig. 14.25 Simulation result of FL vs. LQR in response to two pulses
14.5.4.4 Initial Displacement from Equilibrium State variables are initially set as follows: q1 D 2:5ı , q2 D 10ı , qP 1 D 4ı =s, and qP2 D 5ı =s. Arbitrary initial velocities are chosen such that the first link is moving further away from the equilibrium position (counterclockwise), while the second link is moving clockwise in relation to the first link. As depicted in Fig. 14.26, the FL controller is able to stabilize the system for this set of initial conditions. Again, response of the nonlinear FL controller is very similar to that of the LQR contoller. In an effort to determine which control system offers a greater region of attraction, simulation is carried out with state variables q1 and q2 systematically increased and both controllers run in unison. Response of the FL and LQR controllers always follow the pattern already observed: LQR control offers a quicker response to system disturbances than FL. In terms of the region of attraction, LQR offers a significant advantage, as illustrated in Fig. 14.27. In this case, the initial state disturbances are set to q1 D 3ı , q2 D 10ı , qP1 D 4ı =s, and qP 2 D 5ı =s. While LQR is still able to stabilize the system, the FL control fails. A sensitivity analysis was also performed in Sect. 14.4.1 on the estimated model parameters. It was observed that uncertainty in the moment of inertia of link 2 and the efficiency factor, lc2 and respectively, largely account for modeling discrepancy. To test for controller robustness, a C 40% increase in lc2 and is applied to the model subjected to initial conditions q1 D 2:5ı , q2 D 10ı , qP 1 D 4ı =s, and qP2 D 5ı =s. Both controllers are still able to stabilize the system. Interestingly
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Fig. 14.26 Simulation result of FL vs. LQR control in response to nonzero initial conditions
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Fig. 14.27 Simulation result of FL vs. LQR control in response to nonzero initial conditions
enough, when a C40% uncertainty is introduced in IL1 , the moment of inertia of the leg, the FL controller fails to stabilize the system for the aforementioned set of initial conditions, while LQR is still able to stabilize the system (Fig. 14.28).
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Fig. 14.28 Simulation result of FL vs. LQR control in response to nonzero initial conditions
14.5.5 Experimental Results Figure 14.29 below depicts the response of the FL control in the real system subjected to 2 s interval unit pulses with opposing polarity, spaced 30 s apart. It can be observed that the nonlinear control law is able to stabilize the real system, even though the actual body angle D 40ı is not equal to 30ı opt 40ı . Given that numerous simulation results do not reveal any significant advantage of the FL over a LQR design, limited experimental data was collected with this control in action.
14.5.6 Conclusion In this chapter, a nonlinear Feedback Linearizing (FL) controller was synthesized to maintain balance in a 2 DOF one-legged hopping robot while offering a large region of attraction. Even though the synthesized nonlinear controller was able to stabilize the robot for small disturbances, its performance did leave a lot to be desired, especially when compared against that of LQR control. Specifically, LQR offered a larger region of attraction, had a quicker response to system disturbances, and was less sensitive to parametric and modeling uncertainty when compared against the FL controller. The synthesized nonlinear controller, therefore, did not meet the objectives for which it was original designed.
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Fig. 14.29 Experimental result of FL control in response to two pulses
FL is a powerful control strategy, nonetheless, when an accurate mathematical model of a system is available and the system dynamics is feedback linearizable (Slotine 1991c, 207). Otherwise, as it has been observed, the control does not guarantee the performance inherent in other nonlinear control strategies. When using the robotic system parameters computed using mathematical relationships such that the moment of inertia of link 2 does not violate one of the conditions required for feedback linearizaton, as mentioned earlier in this chapter, the nonlinear FL controller outperforms the LQR. Specifically, for mathematically computed system parameters, the FL controller is able to stabilize the system subjected to initial displacements of q1 as large as 25ı , while the region of attraction for the LQR is restricted to 5ı . The shortcomings portrayed by the FL control can be, therefore, mapped to assumptions that were made for the feedback linearization conditions to be satisfied. In particular, the discrepancy between estimated and mathematically computed system parameters, which violates a condition required for feedback linearization, is a major impediment. To achieve the control objectives of large operation range and robustness in the face of parametric or modeling uncertainty, FL is, therefore, not a suitable candidate for this application. Other nonlinear control strategies must be sought.
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References Advanced Motion Controls. www.a-m-c.com/ Ahmadi M, Beuhler M (1997) Stable control of a simulated one-legged hopping robot with hip and leg compliance. IEEE Trans Robotics Automation 13(1):96–104 Becker G, Pannu S, Kazerooni H (1995) Stability of a one legged robot using -synthesis. IEEE international conference on robotics and automation, Nagoya , Japan Bernstein DS (1997) A Student’s Guide to Classical Control, IEEE Control Systems Magazine., Vol. 17, pp. 96–100 Bode HW (1945) Feedback amplifier. Van Nostrand, New York Craig JJ (1989) Introduction to robotics. Addison Wesley, New York Dummer R, Berkemeier M (2000) Low-energy control of a one-legged robot with 2 degrees of freedom. Proceedings of the IEEE international conference on robotics and automation, San Francisco, CA Fiorini P, Heverly M, Hayati S, Gensler J (1999) A hopping robot for planetary exploration. International proceedings of IEEE aerospace conference, Snowmass, CO, March 1999 Francis BA, Zames G (1984) On optimal sensitivity theory for siso feedback systems. IEEE Trans Auto Control AC-29:9–16 Freudenberg JS, Looze DP (1985) Right half plane poles and zeros and design tradeoffs in feedback systems. IEEE Trans Automatic Control AC-30(6):555–565 Guzella L, Schmid AM (1995) Feedback linearization of spark ignition engines with continuously variable transmission. IEEE transactions on control systems technology, vol 3, March 1995 Ha IJ, Kim DI, Ko MS (1990) Control of an induction motor via feedback linearization with inputoutput decoupling. Int J Control 51(4):863–886 Helferty JJ, Collins JB, Kam M (1989) Adaptive control of a legged robot using an artificial neural network. Proceedings of the American control conferennce, pp 165–168, Fairborn, OH , USA Hsu P (1992) Dynamics and control design project offers taste of real world. IEEE Control Syst Mag 12:31–38 Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River, NJ Koditschek DE, Bhler M (1991) Analysis of a simplified hopping robot. Int J Robotics Res 10(6):587–605 Kuo BC (1995) Automatic control systems. Prentice Hall, Englewood Cliffs, NJ Ljung L (1999) System identification – theory for the user, 2nd edn. PTR Prentice Hall, Upper Saddle River, NJ Matsouka K (1979) A Model of Repetitive Hopping Movements in Man. Proceedings of the Fifth World Congress on Theory of Machines and Mechanisms. American Society of Mechanical Engineers, New York, p.1168–1171. McGeer T (1995) Passive dynamic walking. Int J Robotics Res 9(2):62–82 Mehrandezh M, Surgenor B, Dean S (1995) Jumping height control of an electrically actuated hopping robot: modeling and simulation. IEEE Proceedings of the 34th conference on decision and control, New Orlenas, LA Middleton RH, Goodwin GC, Woodyatt AR, Shim J (1998) Fundamental limitations due to j! axis zeros in siso systems. Technical Report EE9723, February 1998 Middleton R, Freudenberg J, Stefanopoulou A (2000) A survey of inherent design limitations. Proceedings of the American control conference, June 2000 NASA Jet Propulsion Laboratory. Mars exploration rover mission. marsrovers.jpl.nasa.gov. National Instruments NI DAQCard-6036E, 16-Bit, 200kS/s E Series Multifunction DAQ for PCMCIA, http://sine.ni.com/nips/cds/view/p/lang/en/nid/11914 Nelles O (2001a) Nonlinear system identification. Springer-Verlag, New York Nelles O (2001b) Nonlinear system identification, Sect. 7.2.2. Springer-Verlag, Berlin, Heidelberg, New York Ogunnaike BA, Ray WH (1994) Process dynamics. modeling and control, 2nd edn. Oxford University Press, New York
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Papantoniou KV (1991) Control architecture for an electrical actively balanced multi-leg robot based on experiments with a planar one leg machine. IFAC robot control, Vienna, Austria Papadopoulos E, Chasparis G (2002) Analysis and model-based control of servomechanisms with friction. Proc Int Conf Intel Robotics Syst 3(30):2109–2114 Pintelon R, Schoukens J (2001) System identification, a frequency domain approach, 2nd edn. Wiley, New York Playter RR (1994) Passive dynamics in the control of gymnastic maneouvers. PhD thesis, M.I.T., Cambridge, MA Powell JD, Franklin GF, Workman ML (1990) Digital control of dynamic systems, 2nd edn. Addison-Wesley, Rreading, MA Qui L, Davison E (1993) Performance limitations of non-minimum phase systems in the servomechanism problem. Automatica 29(2):337–349 Rad PGH, Buehler M (1993) Design, modeling and control of a hopping robot. IEEE/RSJ Conference on intelligent robots and systems, Yokohama, Japan, July 1993 Raibert MH (1997) Legged robots that balance. MIT Press, Cambridge Ringrose R (1997) Self stabilizing running. PhD thesis, M.I.T., Cambridge, MA Schwind WJ, Koditschek DE (1994) Control of forward velocity for a simplified planar hopping robot. Technical Report, EECS, Department, College of Engineering, Ann Arborm, MI, 12 February 1994 Sciavicco L, Siciliano B (1990) Modelling and control of robotic manipulator, 2nd edn. SpringerVerlag, Berlin, Heidelberg, New York, p 143 Slotine J-JE (1991a) Applied nonlinear control. Prentice Hall, Englewood Cliffs, NJ, pp 226, 253 Slotine J-JE (1991b) Applied nonlinear control. Prentice Hall, Englewood Cliffs, NJ, pp 227–228 Slotine J-JE (1991c) Applied nonlinear control. Prentice Hall, Englewood Cliffs, NJ Slotine J-JE (1991d) Applied nonlinear control. Prentice Hall, Englewood Cliffs, NJ, p 258 Sony Corporation. Sony Dream Robot, (2005) http://en.wikipedia.org/wiki/QRIO Spong MW (1995) The swing-up control problem of the acrobot. IEEE Syst Mag 15:49–55 Spong MW, Block DJ (1995) The pendubot: a mechatronic system for control research and education. Proceedings of the 34th IEEE conference on decision and control, New Orleans, December 1995, pp 555–556 Spong MW, Praly L (1996) Control of underactuated mechanical systems using switching and saturation. Proceedings of the block island workshop on control using logic based switching, Block Island, Rhode Island , USA Spong MW, Praly L (1996) Energy based control of a class of underactuated mechanical systems. IFAC World Congress, July 1996, San Francisco, CA, USA Stein G (2003) Respect the unstable. IEEE Control Syst Mag 0272-1708 23(4):12–25 Sweriduk GD, Menon PK, Vaddi SS (2004) Nonlinear discrete time design methods for missile flight control systems. AIAA Guidance, Navigation and Control Systems, Austin, TX, 11–14 August 2004 The Mathworks. Matlab and simulink for technical computing. www.mathworks.com. Vakakis AF, Burdick JW, Caughey T (1991) An interesting strange attractor in the dynamics of a hopping robot. Int J Robotics Res 10(6):606–618 Wang W (1992) Modeling scheme for longitudinal vehicle control. Proceedings of the 31th IEEE Conference of Decision and Control, vol 1. Tucson, AZ, USA, pp 549–554 Yoshinda H, Gokan M, Yamafuji K (1994) Postural stability and motion control of a rope hopping robot. JSME Int J 37(4) Zeglin G, Brown H (1998) Control of a bow leg hopping robot. Proceedings of the IEEE international conference on robotics and automation, Leuven, Belgium
Chapter 15
Nonlinearities in Human Body Dynamics M. Fard, Y. Ohtaki, T. Ishihara, and H. Inooka
Abstract The nonlinearities in human body dynamics is discussed by focusing on the intrinsic (passive) and reflexive (active) dynamics of head–neck complex (HNC). An experiment is first designed to measure the passive responses of HNC to horizontal fore-and-after vibration. The intrinsic frequency response function between the HNC angular velocity and the trunk horizontal acceleration is obtained in the frequency range of 0:5 10 Hz. It is shown that nonlinear distortions of the human intrinsic responses to the vibration can be characterized by using frequency domain identification method if the excitation is a non-predictive random. The physical dynamic parameters of human HNC are then obtained and discussed. A second experiment is designed to measure the reflexive dynamics of the head in response to the vibration. The reflexive dynamics of the HNC is also modeled and characterized by using frequency domain identification method. The results suggest that the role of human sensory systems, visual, vestibular, and proprioceptive sensors, can be modeled by adding linear control elements to the passive HNC model. It is shown that the passive and active dynamics of the HNC can be characterized by utilizing the experiment and analysis for frequency domain. This allows not only reducing the nonlinear distortion of the measured head dynamics but also characterizing the level of nonlinearities. M.Fard () School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Australia e-mail:
[email protected] Y. Ohtaki Kanagawa Institute of Technology, Japan e-mail:
[email protected] T. Ishihara Fukushima University, Japan e-mail:
[email protected] H. Inooka Tohoku University, Japan e-mail:
[email protected] L. Dai and R.N. Jazar (eds.), Nonlinear Approaches in Engineering Applications, DOI 10.1007/978-1-4614-1469-8 15, © Springer Science+Business Media, LLC 2012
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15.1 Introduction The attempts to summarize the knowledge about the human body dynamics in response to the vibration, merely by introducing a complex nonlinear model, or by defining a single curve representing all responses to all frequency, do not reflect a modern understanding of the effects of vibration on the body. The dynamic response of the different parts of human body to the vibration has been investigated in the literature (Paddan and Griffin 1988ab; Griffin 1996). More attention has been paid on the head or head–neck complex (HNC). The interest in the head can be justified that the head is the vital part of the human body that is associated with the vibration discomfort or that its movement affects the human sensory systems such as visual, vestibular, and proprioceptive sensors. Head–Neck Complex (HNC) may response to the vibration by the use of two strategies called passive (intrinsic responses) and active (reflexive responses). The passive strategy, in which the HNC responses to the vibration in an open-loop manner, is related to the mechanics of the head and the neck, e.g., the viscoelastic properties of the neck. The active strategy is related to the human sensory systems, e.g., vestibular, visual, and proprioceptive sensory systems. The neck muscles can be activated by the receiving information from the central nervous system (CNS), which consists of the brain and the spinal cord, on the velocity, about position, and acceleration of the HNC. The information is carried by motor neurons from the CNS to the neck muscles. In this case, it is said that the HNC responses to the vibration in a closed-loop manner. The active and passive response of the HNC to the vibration has great influence on the perception of the vibration and motion sickness (Keshner 2000; Guitton et al. 1986). Despite the significant roles of the active and passive elements of the HNC, to stabilize it in response to the vibration, little quantitative knowledge exist concerning the roles of these elements in stabilizing the HNC. However, many researches have been done on the dynamics of the HNC in the car collision analyses in which only passive responses of the HNC are taken into consideration. The investigations show that the head has a considerable translational motion (protraction/ retraction) in addition to the rotational (flexion/extension) motion in the high-speed car collision. However, the significance of the translational motion decreases, in comparison with the rotational motion, as the collision speed decreases. The translational motion of the HNC may become inconsiderable when the trunk of the seated human volunteer is exposed to the vibration with a value much less than that occurs in the collision. Conversely, in this case, the rotational (pitch motion) motion of the HNC may become more significant. This may be a reason that the leaning on the seatback, as mentioned earlier, is reported to increase the pitch motion of the head while the seated human body is exposed to the horizontal vibration. The two major mechanisms, passive and active, responsible for stabilizing the HNC are illustrated in Fig. 15.1. This schematic drawing indicates that the mechanical properties of the HNC, e.g., the spring and damping coefficients, and the human sensory systems are generally responsible for stabilization of the HNC
15 Nonlinearities in Human Body Dynamics Fig. 15.1 The schematic drawing of the passive and active mechanisms responsible for stabilizing the HNC. The passive part (single-inverted pendulum model with inertia and viscoelastic elements) is considered as a physical model in this study
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θ k,c I n fo.
lh
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o
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..
u =y in response to the vibration (Guitton et al. 1986; Keshner and Peterson 1995; Keshner 2000). We presume that the structure of this model (Fig. 15.1) is somewhat similar to the single-inverted-pendulum models presented for the human upright standing posture (Ishida and Imai 1987; Peterka 2000). The HNC utilizes not only the human sensory systems (active elements) but also the inertia and viscoelastic properties of the neck (passive elements) play one of the major roles on stabilizing the HNC in response to the vibration (Guitton et al. 1986; Keshner and Peterson 1995; Keshner 2000). Therefore, to quantify the role of the human sensory systems on stabilizing the HNC, it is required to first identify the passive elements of the HNC. Nevertheless, investigators, who are working on quantifying the vestibulocollic reflex (VCR) and cervicocollic reflex (CCR) of the HNC have reported the lack of knowledge about the passive parameters, e.g., viscoelastic parameters of the neck. As mentioned earlier, there are major requirements to study the influences of the trunk vibration to the HNC, as well as to present an explicit model for the HNC passive responses that has potential for future applications. The contribution of the VCR and CCR in stabilizing the HNC is negligible when the human subject keeps the neck muscle relaxed and exposed to a nonpredictive random vibration (Guitton et al. 1986). Hence, under this condition, it is likely possible to measure the passive dynamics of the HNC in the seated human body. In this chapter, a method is presented to characterize the passive (intrinsic) dynamics of the HNC in response to the trunk horizontal vibration. First, we design an experimental method to measure the passive responses of the HNC to the trunk horizontal vibration. Second, a model is introduced to estimate the physical parameters of the HNC, and to predict the response of the system to the trunk horizontal vibration. Third, the frequency domain identification method is used, as a well-suited identification method for the HNC, to obtain the mechanical parameters of the HNC, including the viscoelastic properties of the neck.
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15.2 Method 15.2.1 Experiments The experiments were conducted using an electro-hydraulic vibrator. Its platform (3m3m) had a stroke of 30 cm, which had the capability of producing the signals from DC to 30 Hz. A rigid seat was mounted on the platform. The flat (horizontal) surface of seat had a thin layer of cushion. The wooden seatback was adjustable in height to cover the posterior part of the trunk for each individual subject. It was inclined at an angle of 14ı to the vertical direction. The height of seat surface from the platform was 410 mm (Fig. 15.2). Four male subjects (young and healthy) took part in this study. Their physical characteristics are summarized in Table 15.1. Informed consent was obtained from each subject prior to the participation. None of subjects had any history of neck pain, diseases of cervical spine, neck injury, or musculoskeletal disorders. No subject was on any form of medication at the time of experiment. Each subject was placed on the seat and the trunk of subject was then fixed to the seatback. This has been done using six straps, which were connected to a strong cloth covering anterior part of the trunk. Therefore, it was assumed that the horizontal vibration of the trunk was equal to that of the platform. During the experiment, the subjects were blindfolded and instructed to be relaxed and not to resist or apply any voluntary response by the neck muscles.
Angular rate sensor
HNC pitch motion
Body harness
Accelerometer
Vibration Direction
Fig. 15.2 Experiment design for measuring the dynamics of Head–Neck
Vibrator's table
15 Nonlinearities in Human Body Dynamics Table 15.1 Physical characteristics of subjects
Subject No Age(yr) Height(m) Weight(kg)
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#1 24 1.68 56
#2 24 1.80 60
#3 23 1.70 55
#4 26 1.73 67
Mean (SD) 24.3 (1.3) 1.73 (0.05) 59.5 (5.45)
Note that the # sign indicates the subject number
A surface-mounted accelerometer (Analog Devices ADXL05) attached to the platform to measure the horizontal acceleration of the input to the HNC. An angular rate sensor (Murata ENC-03J) was applied to measure the HNC angular velocity. This sensor was a lightweight type, pasted on the forehead of the subject to measure mainly in mid-sagittal plane, and to provide high-resolution measures of HNC angular velocity. The input was a zero-mean Gaussian random vibration with 1:60 ms2 rms and a nominally flat spectrum. Each test run took 50 s. Each subject underwent this vibration four times individually, but for one of them, one more set of data was collected to use for validating the results. The measured signals, the platform acceleration, and the angular velocity, were sampled at 100 Hz with an A/D converter and both band-pass filtered at 0.5 Hz and 10 Hz with identical second-order Butterworth filters.
15.2.2 Advanced Signal Processing The Fourier spectra of the measured signals can be derived from the time-domain data by the use of fast Fourier transform method (FFT). As, the measured timedomain signals have contaminating noise, the Fourier spectra of the signals may also be corrupted by noise. The measured signals in the frequency domain can be written by: Xm .!k / D X.!k / C M.!k /;
(15.1)
Ym .!k / D Y .!k / C N.!k /;
(15.2)
where the Xm .!k / and Ym .!k / are the measured Fourier spectra of the input and output signal at selected angular frequency !k (k D 1; :::; F ), respectively. The terms X.!k / and Y .!k / are the true values of the input and output spectra, and the M.!k / and N.!k / are the noise on the spectra of the input and output signal, respectively. It is proven in that the noise on the FFT coefficients tends to be uncorrelated and normally distributed with zero mean value when the number of time samples increases. The frequency response function (nonparametric form of the transfer function) of the system H.!k /, at angular frequencies !k , can be derived, from experimental data, by using the cross-spectral density function method (Bendat and Piersol 1980; Bendat 1997; Bendat and Piersol 2000).
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H.!k / D
Gxm ym .!k / Gxm xm .!k /
(15.3)
where Gxm ym .!k / is the cross spectrum of the measured input and the output at angular frequencies !k , and Gxm xm .!k / is the auto spectrum of the measured input. As the input and output noises are not correlated, the influences of the noise on the Gxm ym .!k / is removed during the averaging process of the cross spectrum. Therefore, the Gxm ym .!k / is nearly equal to the true value of the cross spectrum, Gxy .!k /. The (15.3) can be written as H.!k / D
Gxy .k/ ; Gxx .!k / C GMM .!k /
(15.4)
where the Gxx .!k / is the true value of auto spectrum of the input at angular frequencies !k , and the GMM .!k / indicates the auto spectrum of the noise of the input signal. It is seen that the noise has almost no effect on the numerator of the transfer function (15.4), therefore, the random error in the estimated transfer function by the use of cross spectrum is negligible. However, the term of the GMM .!k /, which is generated by the measurement noise, will cause a bias error (systematic error) on the estimated transfer function. As the denumerator of the transfer function (15.4) is a real value (not complex), the bias may have influence on only the magnitude of the transfer function, and it has no effect on the phase of the transfer function. Alternatively, we can approximate the (15.4) to (15.5) by using Taylor series, providing the signal-to-noise ratio (S/N) is greater than one (i.e., GMM .!k /
