Elastic Multibody Dynamics: A Direct Ritz Approach (Intelligent Systems, Control and Automation: Science and Engineering)

Elastic Multibody Dynamics

International Series on

INTELLIGENT SYSTEMS, CONTROL, AND AUTOMATION: SCIENCE AND ENGINEERING VOLUME 35

Editor Professor S. G. Tzafestas, National Technical University of Athens, Greece

Editorial Advisory Board Professor P. Antsaklis, University of Notre Dame, IN, U.S.A. Professor P. Borne, Ecole Centrale de Lille, France Professor D. G. Caldwell, University of Salford, U.K. Professor C. S. Chen, University of Akron, Ohio, U.S.A. Professor T. Fukuda, Nagoya University, Japan Professor F. Harashima, University of Tokyo, Tokyo, Japan Professor S. Monaco, University La Sapienza, Rome, Italy Professor G. Schmidt, Technical University of Munich, Germany Professor N. K. Sinha, Mc Master University, Hamilton, Ontario, Canada Professor D. Tabak, George Mason University, Fairfax, Virginia, U.S.A. Professor K. Valavanis, University of South Florida, U.S.A.

H. Bremer

Elastic Multibody Dynamics A Direct Ritz Approach

H. Bremer Johannes Kepler University Linz Austria

ISBN-13: 978-1-4020-8679-3

e-ISBN-13: 978-1-4020-8680-9

Library of Congress Control Number: 2008929579 © 2008 Springer Science+Business Media, B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com

TABLE OF CONTENTS

1. INTRODUCTION 1.1 Background 1.2 Contents

1 1 5

2. AXIOMS AND PRINCIPLES 2.1 Axioms 2.2 Principles – the “Differential” Form 2.3 Minimal Representation 2.3.1 Virtual Displacements and Variations 2.3.2 Minimal Coordinates and Minimal Velocities 2.3.3 The Transitivity Equation 2.4 The Central Equation of Dynamics 2.5 Principles – the “Minimal” Form 2.6 Rheonomic and Non-holonomic Constraints 2.7 Conclusions

7 7 7 15 16 18 19 20 22 25 26

3. KINEMATICS 3.1 Translation and Rotation 3.1.1 Rotation Axis and Rotation Angle 3.1.2 Transformation Matrices 3.1.2.1 Rotation Vector Representation 3.1.2.2 Cardan Angle Representation 3.1.2.3 Euler Angle Representation 3.1.3 Comparison 3.2 Velocities 3.2.1 Angular Velocity 3.2.1.1 General Properties 3.2.1.2 Rotation Vector Representation 3.2.1.3 Cardan Angle Representation 3.2.1.4 Euler Angle Representation

29 29 30 33 33 34 36 36 39 40 40 41 42 43

v

Table of Contents

vi

3.3

3.4 3.5 3.6

State Space 3.3.1 Kinematic Differential Equations 3.3.1.1 Rotation Vector Representation 3.3.1.2 Cardan Angle Representation 3.3.1.3 Euler Angle Representation 3.3.2 Summary Rotations Accelerations Topology – the Kinematic Chain Discussion

4. RIGID MULTIBODY SYSTEMS 4.1 Modeling Aspects 4.1.1 On Mass Point Dynamics 4.1.2 The Rigidity Condition 4.2 Multibody Systems 4.2.1 Kinetic Energy 4.2.2 Potentials 4.2.2.1 Gravitation 4.2.2.2 Springs 4.2.3 Rayleigh’s Function 4.2.4 Transitivity Equation 4.2.5 The Projection Equation 4.3 The Triangle of Methods 4.3.1 Analytical Methods 4.3.2 Synthetic Procedure(s) 4.3.3 Analytical vs. Synthetic Method(s) 4.4 Subsystems 4.4.1 Basic Element: The Rigid Body 4.4.1.1 Spatial Motion 4.4.1.2 Plane Motion 4.4.2 Subsystem Assemblage 4.4.2.1 Absolute Velocities 4.4.2.2 Relative Velocities 4.4.2.3 Prismatic Joint/Revolute Joint – Spatial Motion 4.4.3 Synthesis 4.4.3.1 Minimal Representation 4.4.3.2 Recursive Representation 4.5 Constraints 4.5.1 Inner Constraints 4.5.2 Additional Constraints 4.5.2.1 Jacobi Equation 4.5.2.2 Minimal Representation 4.5.2.3 Recursive Representation 4.5.2.4 Constraint Stabilization

44 45 45 46 46 47 49 50 56 59 59 59 61 65 65 66 66 67 68 69 70 71 71 73 75 78 78 78 82 85 85 88 89 91 93 95 100 100 101 102 103 104 107

Table of Contents

4.6

4.7

Segmentation: Elastic Body Representation 4.6.1 Chain and Thread (Plane Motion) 4.6.2 Chain, Thread, and Beam Conclusion

vii

109 109 111 113

5. ELASTIC MULTIBODY SYSTEMS – THE PARTIAL DIFFERENTIAL EQUATIONS 5.1 Elastic Potential 5.1.1 Linear Elasticity 5.1.2 Inner Constraints, Classification of Elastic Bodies 5.1.3 Disk and Plate 5.1.4 Beam 5.2 Kinetic Energy 5.3 Checking Procedures 5.3.1 Hamilton’s Principle and the Analytical Methods 5.3.2 Projection Equation 5.4 Single Elastic Body – Small Motion Amplitudes 5.4.1 Beams 5.4.2 Shells and Plates 5.5 Single Body – Gross Motion 5.5.1 The Elastic Rotor 5.5.2 The Helicopter Blade (1) 5.6 Dynamical Stiffening 5.6.1 The Cauchy Stress Tensor 5.6.2 The Trefftz (or 2nd Piola-Kirchhoff) Stress Tensor 5.6.3 Second-Order Beam Displacement Fields 5.6.4 Dynamical Stiffening Matrix 5.6.5 The Helicopter Blade (2) 5.7 Multibody Systems – Gross Motion 5.7.1 The Kinematic Chain 5.7.2 Minimal Velocities 5.7.3 Motion Equations 5.7.3.1 Dynamical Stiffening 5.7.3.2 Equations of Motion 5.7.4 Boundary Conditions 5.8 Conclusion

115 115 116 117 119 121 123 124 124 131 133 133 153 159 159 162 167 167 167 173 177 183 190 190 193 194 195 196 210 214

6. ELASTIC MULTIBODY SYSTEMS – THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS 6.1 Galerkin Method 6.1.1 Direct Galerkin Method 6.1.2 Extended Galerkin Method 6.2 (Direct) Ritz Method 6.3 Rayleigh Quotient

219 219 219 224 225 229

Table of Contents

viii

6.4

6.5

6.6

6.7 6.8 6.9

Single Elastic Body – Small Motion Amplitudes 6.4.1 Plate 6.4.1.1 Equations of motion 6.4.1.2 Basics 6.4.1.3 Shape Functions: Spatial Separation Approach 6.4.1.4 Expansion in Terms of Beam Functions 6.4.1.5 Convergence and Solution 6.4.2 Torsional Shaft 6.4.2.1 Eigenfunctions 6.4.2.2 Motion Equations 6.4.2.3 Shape Functions 6.4.3 Change-Over Gear Single Elastic Body – Gross Motion 6.5.1 The Elastic Rotor 6.5.1.1 Rheonomic Constraint 6.5.1.2 Choice of Shape Functions – Prolate Rotor (Ω = 0) 6.5.1.3 Choice of Shape Functions – Oblate Rotor (Ω = 0) 6.5.1.4 Configuration Space and State Space (Ω 6= 0) 6.5.1.5 The Laval- (or Jeffcott-) Rotor 6.5.1.6 Rotor with Fixed Point 6.5.1.7 Elastic Rotor Properties Gross Motion – Dynamical Stiffening (Ritz Approach) 6.6.1 Rotating Beam – One-Link Elastic Robot 6.6.1.1 Mass Matrix 6.6.1.2 Restoring Matrix 6.6.1.3 Equations of Motion 6.6.2 Translating Beam – Elastic TT-Robot 6.6.2.1 Mass Matrix 6.6.2.2 Restoring Matrix 6.6.2.3 Equations of Motion 6.6.2.4 Simplified System The Mass Matrix Reconsidered (Ritz Approach) The G-Matrix Reconsidered (Ritz Approach) Conclusions

7. ELASTIC MULTIBODY SYSTEMS – ORDINARY DIFFERENTIAL EQUATIONS 7.1 Summary Procedure 7.1.1 Rigid Multibody Systems 7.1.2 Elastic Multibody Systems 7.2 Mixed Rigid-Elastic Multibody Systems 7.3 Applications 7.3.1 Prismatic Joint – The Telescoping Arm 7.3.1.1 On Mass Distribution: Tip Body Influence

235 235 235 236 238 239 243 246 246 247 249 253 255 255 257 259 264 271 272 277 283 290 291 293 293 295 296 305 305 306 306 311 315 321 327 327 327 329 335 338 338 339

Table of Contents

7.4

7.5 7.6

7.3.1.2 Subsystem Equations 7.3.1.3 The Kinematic Chain 7.3.2 Revolute Joint 7.3.2.1 Subsystem Equations 7.3.2.2 The Kinematic Chain 7.3.3 Spatial Motion 7.3.4 Plane Motion Plane Motion – Recalculation 7.4.1 Minimal Velocities and Projection 7.4.2 Subsystem Matrices 7.4.3 Dynamical Stiffening 7.4.4 The Kinematic Chain Reduced Number of Shape Functions: Controlled Systems Remark on Controlled Systems

ix

340 345 346 346 347 348 350 352 352 354 359 361 369 380

8. A SHORT EXCURSION INTO STABILITY AND CONTROL 8.1 Optimality 8.1.1 Results from Classical Optimization Theory 8.1.2 Riccati- (or LQR-) Control 8.1.3 Control Parameter Optimization 8.2 Stability 8.3 Linear Time-Invariant Systems 8.3.1 Fundamental (or Transition) Matrix 8.3.2 Theorem of Cayley and Hamilton 8.3.3 Stability Theorem for Mechanical Systems 8.4 Stabilization of Mechanical Systems 8.5 Observers 8.5.1 Basic Notation 8.5.2 Complete State Observer for Control 8.5.3 Disturbance Suppression (“High Gain Observer”) 8.5.4 Disturbance Observation 8.6 Decentralized Control 8.7 On Control Input Variables

383 383 387 388 390 391 395 395 399 401 406 411 411 413 416 419 423 426

References

431

List of Symbols

437

Index

445

Chapter 1 INTRODUCTION

1.

Background

This textbook is an introduction to and exploration of a number of core topics in the field of applied mechanics. Mechanics, in both its theoretical and applied contexts, is, like all scientific endeavors, a human construct. It reflects the personalities, thoughts, errors, and successes of its creators. We therefore provide some personal information about each of these individuals when their names arise for the first time in this book. This should enable the reader to piece together a cultural-historical picture of the field’s origins and development. This does not mean that we are writing history. Nevertheless, some remarks putting individuals and ideas in context are necessary in order to make clear what we are speaking about – and what we are not speaking about. At the end of the 19th century, technical universities were established everywhere in Europe in an almost euphoric manner. But the practice of technical mechanics itself, as one of the basics of technical development, was in a desolate state, due largely to the refusal of its practitioners to recognize the influence of kinetics on motion. They were correct to the extend that then current mechanical systems moved with small velocities where kinetics does not play a significant role. But they had failed to keep up with developments in the science underlying their craft and were unable to keep pace with the speeds of such systems as the steam engine. Into this critical situation, Karl Heun introduced a significant, and shortly to become famous, paper (about 120 pages in length) on the kinetic problems in scientific engineering (K. Heun, 1900). Through the influence of the renowned mathematician Felix Klein, Heun attained a professorship in Karlsruhe (Germany) (M.v.Renteln, 1989, 1992). Georg Hamel became his assistant, working for him from 1903 till 1905, during which time period he published his habilitation thesis (G. Hamel, 1904a). It was due to Felix 1

2

1 Introduction

Klein that outstanding scientists like these could eventually bring engineering and natural science back together. Hamel’s contributions enabled access to mechanics on a deep level by citing the work of, and giving full and fair credit to, important researchers that had preceded him. For those readers who are interested in more background and historical development we can therefore recommend Hamel’s writings [e.g. (G. Hamel, 1949)] as the work of a contemporarian of that time, or, predating him, the works of (J.L. de Lagrange, 1813 – 1815). No one less an an authority than Louis Poinsot found its historical part outstanding (I. Szab´o 1979b). Or, for a look backwards from a contemporary viewpoint, one may consult the thoughtful contributions of John Papastavridis, e.g. his famous treatise on Analytical Mechanics (J.G. Papastavridis, 2002). “Mechanics is one part of physics” (A. F¨oppl, 1921). Physics itself, during the nineteenth century, underwent an abstraction process. Following in the footsteps of Ludwig Boltzmann (L. Boltzmann, 1890), physicists turned to the construction of equations or systems of equations that allowed one to calculate real phenomena without any interpretation of the physical reality. According to Gustav Kirchhoff, the equation itself is the explanation. However, in mechanics, such a process is extremely dangerous. For example, constraint forces that do not contribute to motion may never be perceived, which can lead to severe misinterpretations. Even Heinrich Hertz, who rejected the concept of “forces” for theoretical-philosophical purposes, concedes that this is not suitable for practical use (H. Hertz, 1894). We will therefore not renounce interpretability as one of the basic concepts in applied mechanics. But philosophers and professional historians seem to follow Kirchhoff’s guideline by claiming a “theory in a modern sense to be an uninterpreted mathematical model that can be shown to be in partial agreement with physical reality” (H. Pulte, 1998). Continuing with “Lagrange’s m´echanique analytique was a reductionist exercise in two parts, on statics and dynamics (itself held to be reduced to statics via d’Alembert’s Principle) ... as usual with an overambitious exercise, it failed to meet its norms” (I. Grattan-Guiness, 1990). Pulte cites eminent witnesses for corroboration: (C.G.J. Jacobi, 1847) and (L. Poinsot, 1837). However, Jacobi tried to find a mathematical proof for a mechanical principle and Poinsot attempted to deal with scientific falsehoods. According to Pulte, Lagrange’s “analytical methods are in contrast to the foundation of mechanics (Descartes, Leibniz, Newton) . . .” Descartes’ and Leibniz’s errors are well documented, e.g. (I. Szab´o, 1979a,b). Newton and contemporarians were not able to solve practical problems in mechanics; its basics had not yet been laid. [(E.J. Dijksterhuis, 1950) compares the attempt to detect the momentum theorem in Newton’s Principia from 1687 with the tale of “the emperor’s new clothes”.] Clearly, from the engineering viewpoint, we can not share the philosophers’ opinion and will, on the

1.1

Background

3

contrary, use the Lagrangean concept as a fundamental principle. Moreover, we need the momentum theorem in a calculable form, the breakthrough for which came with Leonhard Euler and his textbook mechanica sive motus scientia anayticae exposita (mechanics, or the science of motion, analytically represented) from 1736. Its great success is based on the fact that geometrical methods are consistently replaced with Leibniz’s calculus1 . Euler here uses the “mass point” which his execution is based on. (This concept, although insufficient, is often used even today2 ). The momentum theorem, in the form as it is used today, appeared in 1750 in d´ecouverte d’un nouveau principe de la m´ecanique (discovery of a new mechanical principle, printed 1752). Euler here enlarges his conception to systems of point masses, including liquids (K. Simonyi, 1986), and, for a rigid body, he derives the momentum of momentum theorem. But he obviously feels that something is missing in his concept. Thus, in 1775, he publishes his famous nova methodus motum corporum rigidorum determinandi (new method to determine rigid body motions). Here, thirty nine years after his first textbook, we find an axiomatic basis of mechanics: the momentum theorem and, independently, the momentum of momentum theorem. These, along with Lagrange’s Principle, form a basis for constructing a considerable body of methods in dynamics. These are, till today, as modern as can be. It was in the sixties of the last century (the beginning of the so-called “space age”) when computers entered investigations. Until then, according to Kurt Magnus3 , it had been a question of individual taste, or personal training, which method was selected for the derivation of motion equations. The reason for this statement is, of course, that a calculation by hand allows for a limited number of degrees of freedom only. These times have passed, and today’s applied mechanics is strongly related to the use of modern computers. Here, the choice of method gains in importance. Karl Heun, with his paper from 1900, had focused attention on the analytical methods. These are, in today’s notation, those methods which decompose (“analyze”) energies into its components, i.e., displacements and (generalized) forces. Gustav Hamel enlarged this concept (the Lagrangean equations of the second kind) to non-holonomic systems. On the other hand, Euler’s 1 Also in the contributions of L’Hopital, ˆ

´ Varignon, the Bernoullis, Jacob Hermann, see (I. Szabo, 1979a), (K. Simonyi, 1986), (H. Iro, 2006). 2 “The best known, and in textbooks most popular, kind of description works with the mass point as a basic element and constructs with reasonably simple means something which is then called mechanics. One avoids thus any difficulty with the one weighty but seldom articulated axiom that those principles one has swindled out of particle dynamics are now also applicable to continua. I will therefore totally renounce particle dynamics in the following. What one understands about particle dynamics is nothing but the centerof-gravity theorem”, (G. Hamel, 1909). 3 Pioneer in rocket control, imprisoned in the former USSR from 1947 to 1954. Forced to work as a scientist there, too.

4

1 Introduction

method (the application of momentum theorems along with his cut principle) was also used, which implies the difficulty of eliminating the (generalized) constraint forces. This method is called the “synthetic” one (“synthesize” = put together). It was no wonder that, in this situation, the discussion on “best method” became controversial. The way out here is to establish a methodology which allows a fair comparison of procedures. Such a methodology may for instance be based on the Central Equation of Dynamics which follows from Lagrange’s Principle. The result is obvious: as long as the equations of motion are requested, then that method is best suited in which every step of calculation is carried out in advance, as far as possible, such that one has only to introduce the problem-specific quantities. This is the Projection Equation. We introduce that terminology because it projects the (generalized) forces into the unconstrained space where the motion takes place. This clear and easy concept, nevertheless, took some years for establishment. The first difficulty arose with the fact that, in a general representation, one had to deal with nonintegrable velocities. Since at least translational velocities are integrable when represented in an inertial frame, one firstly concentrated on these. This, however, is far too restrictive. Along with the use of minimal velocities (as a linear combination of the time derivatives of the minimal coordinates) which need not be integrable, the Projection Equation obtained its final form, based on arbitrary reference frames. This leads to a powerful method which we are going to use. Applied to (rigid) multibody systems (MBS), it leads directly to a subsystem representation. The individual subsystem equations are eventually joined together by means of a (transposed) global functional matrix which results in an upper (block-) triangular form. The equations are thus represented in a Gaussform. This gives access to either the (minimal form of the) equations of motion or to a recursive representation (Gauss elimination). Rigid body modeling is also used for elastic multibody systems (EMBS), referred to then as “lumped parameter procedure”, e.g. (L. Meirovitch, 1990), or “finite segmentation method”, e.g. (R.L. Huston, 1981). One motivation for this foregoing may be seen with the development of computers on the one hand and with the upcoming interest in flexible body interaction during the first days of the space age when numerical MBS-codes began to be established. These are meanwhile sophisticated enough to allow for the treatment of almost every kind of rigid MBS. But finite rigid body modeling cannot, in general, always be recommended for EMBS. As will be shown, one would need about one hundred rigid spring-interconnected links just to represent the first three beam bending eigenfrequencies and eigenforms with a certain accuracy. For such cases it is of course much easier to operate with deformation functions. Single elastic body dynamics thereby motivate the use of spatial operators for calculation of the (partial differential) equations of motion and the assigned boundary conditions. Thereby, for almost the majority of technical applications, the deformation am-

1.2

Contents

5

plitudes remain small w.r.t. the reference frame of the undeformed state. But the latter may undergo large motions (denoted as “rigid body motion”). Special care must therefore be taken for the partial linearization w.r.t. the deformations, leading to so-called “dynamical stiffening effects”. On the basis of the Projection Equation we proceed to elastic multibody systems to obtain a straightforward procedure for the (interconnected partial and ordinary differential) equations of motion and for the assigned boundary conditions. The motion equations for elastic variables hereby appear in the form of spatial operators, those of the rigid body coordinates in a Gauss-form, and the boundary conditions as a combination of these. But, at least for elastic multibody systems, an analytical solution is virtually impossible to achieve. We therefore switch over to approximative solutions where the spatial operators are applied to a Ritz series expansion. The application is thereby restricted to uniaxial (beamlike) elastic bodies for subsystem elements. This constricts the range of flexible body considerations but covers a huge field of applications. The (displacement related) Finite Element (FE) procedure can be seen as the Ritz procedure itself (K.J. Bathe, 1982). The shape functions which are needed for a direct Ritz approach can thus of course be calculated from FE codes. Using FE as a preprocessor for EMBS is not a new idea, e.g. (K. Sorge, H. Bremer, F. Pfeiffer 1993). In continuation one needs then the nodal point discretization along with form functions for interpolation. This is, however, assumed to be known and will not be repeated here. On the other hand, EMBS can of course also be seen from the FE viewpoint as in (M. G´eradin, A. Cardona, 2001), for instance. As always, special viewpoints reveal special advantages for different classes of mechanical models. Mechanics, both pure and applied, remains a spirited science.

2.

Contents

The book is organized as follows. In Chapter 2 the axioms and principles we are going to use are shortly reported and the Central Equation is presented. Chapter 3 deals with kinematics and the representation of the kinematic chain for moving reference frames. Chapter 4 considers rigid multibody systems The rigid body is thereby defined such that, by relaxation of the rigidity constraints, one can later directly proceed to elastic bodies. The Central Equation yields a methodology from which the Projection Equation is selected for the derivation of motion equations. The model description is then decomposed into subsystems which leads to a minimal representation and to a recursive representation, respectively, of the equations of motion. Here the typical structure of the subsystem matrices is presented which will run through all the following. A discussion on constraint forces (later to be used for control input) completes this chapter.

6

1 Introduction

The results are applied to elastic subsystems in Chapter 5. For single elastic body systems it is shown that the non-orthogonal rotation axes resulting from directional derivatives in the analytical procedures may cause problems. This justifies once more the use of the Projection Equation. Using the deformation functions and their derivatives for description motivates us to additionally introduce spatial operators. One obtains a straight-on procedure for the interconnected partial and ordinary differential equations and the corresponding boundary conditions. But, obviously, an analytical solution is restricted to simple systems. The eigenfunctions of these, however, may be used as shape functions for approximative solutions. Therefore, in Chapter 6, we pass to the so-called weak formulation and use a direct Ritz approach, applied to simple elastic body systems to begin with. Ritz’s original contribution to the study of plate vibrations is used to demonstrate frequency bounds and eigenform convergence. A torsional shaft as a basic element in gear dynamics shows how the number of shape functions might increase and how the auxiliary function concept may suitably be used. Elastic rotor dynamics demonstrate significant simplification of the motion equations when using shape functions which are orthogonal w.r.t. the mass matrix. The one-link elastic robot as well as the elastic TT-robot are discussed in order to call attention to typical effects in fast-moving elastic systems. With these preparations the procedure is eventually applied in Chapter 7 to elastic multibody systems, where the (controlled) elastic robot serves for a typical example. The method itself is summarized and retraced for plane motion in section 4. Although plane motion is a rather simple case, it can nevertheless give insight for those readers who prefer quick reading without trespassing on the previous chapters in detail. The mathematical prerequisites are (basic) calculus and vector analysis. Linear differential equations and eigenvalue problems are subsumed in a short excursion into stability and control (Chapter 8), starting with the fundamentals of calculus of variations (which, in most European universities, does not belong in basic courses). This chapter, without any claim for completeness, is mainly addressed to graduate students.

Chapter 2 AXIOMS AND PRINCIPLES

1.

Axioms

On condition of nonnegative mass m, steady one-dimensional time t and homogeneous three-dimensional space, we adopt as axioms the momentum theorem [see eq.(2.1)] and the momentum of momentum theorem (L. Euler, 1775)1 .

2.

Principles – the “Differential” Form

The principles, mainly the Lagrangean2 type which the following methodology will be based on, are then to be looked at as applications of the basic axioms along with some kind of modeling considerations. Modeling is the process of abstracting our natural surroundings into the artificial world of human thinking which can hardly be seen as axiomatic. Thus, the principles themselves do not represent axioms once the momentum theorems are chosen. [Clearly, this is only one possible viewpoint. One may also choose principles for axioms and then interpret the momentum theorems to be nonaxiomatic consequences (C.G.J. Jacobi, 1847)3 ]. The chosen axioms appear then as unprovable, sometimes referred to as “natural laws” but better characterized as nature exhibiting rules which, over the years, humans have been able to extract and formulate from observations and daily experience [after (A. F¨oppl, 1921)4 ]. The principles, although nonaxiomatic, are nevertheless unprovable, too, because of its model dependent character. These axiomatic rules and principles should not be inter-mingled [as Jacobi 1 Leonhard

Euler, *1707 in Basel, †1783 in St. Petersburg de Lagrange, *1736 in Turin, †1813 in Paris 3 Carl Gustav Jacob Jacobi, *1804 in Potsdam, †1851 in Berlin 4 August F¨ oppl, *1854 in Groß-Umstadt, †1924 in Ammerland 2 Joseph-Louis

7

8

2 Axioms and Principles

himself did in a masterful piece that balanced this mixture by always asking for proofs of the nonaxiomatic parts, (C.G.J. Jacobi, 1847)]. A fundamental necessity for doing mechanics is to clearly define its basic assumptions, even in an “applied” textbook, in order to establish a reliable basis on which a methodology can be built, and to reveal hidden sources not only of misinterpretations but also of possible mistakes. (And, not least, to respect the literary property of its authors). Let us start with d’Alembert’s Principle5 . It reads [Trait´e de dynamique, 1743, p.58 (K. Heun, 1901)6 ]: “Decompose the motions (momentums) a, b, c which are impressed on each body (particle) into each two others a, α; b, β; c, γ etc. in such a way that, if only the motions a, b, c would have been impressed, the bodies would be able to maintain their motion without hindering each other and, if one would impress on these only the motions α, β, γ, then the system would stay at rest”.

b

s

a α y

y

β

Figure 2.1. On d’Alembert’s Principle (Double Pendulum)

The wording “would stay at rest” states that the momentums α, β, γ equilibrate. Thus, d’Alembert’s Principle gives insight into the system characteristics but the difficulty is to find the corresponding momentums. Here, Lagrange offers a concept. Lagrange’s Principle [“which until recently has been confused with d’Alembert’s Principle” (K. Heun, 1913)] has basically nothing in common with d’Alembert’s Principle except the same basic constraint considerations. Lagrange makes use, in a first interpretation (see also p.13), of the fact that the constraint forces do not perform work due to the missing displacements in constraint force directions (compare Fig. 2.1 for instance). Moreover, for the formulation of the corresponding work expression, he in5 Jean 6 Karl

le Rond d’Alembert, *1717 in Paris, †1783 in Paris Heun, *1859 in Wiesbaden, †1929 in Karlsruhe

2.2

9

Principles – the “Differential” Form

troduces virtual velocities (“vitesses virtuelles”) which are in agreement with the constraints but kept arbitrary in order to conclude on the real motions. In today’s notation, his Principle [established in (J.L de Lagrange, 1764)] reads Z

(dm¨r − dfe )T δr = 0,

(2.1)

(S)

where dm is a particle of the system (S) under consideration, r is its position vector, the dot represents time derivation. The bracket in eq.(2.1) represents the linear momentum theorem. Here, the forces acting on (the mass center of) dm are df = dfe + dfc where the constraint forces dfc vanish and the impressed (work performing) forces dfe remain. It is worth, at this step of evaluation, having a look at Lagrange’s own explanations: 1. “First I have to emphasize that I introduced a new characteristic δ. Here, δZ shall express a differential w.r.t. Z which is not the same as dZ but which is nevertheless built with the same rules” (J.L de Lagrange, 1762). 2. “The virtual velocity is the velocity which will really take place when the equilibrium is disturbed” (J.L de Lagrange, 1815), TI,I,16. 3. “Because d and δ represent totally independent differentials w.r.t. variations, dδx, dδy, dδz cannot differ from δdx, δdy, δdz” (J.L de Lagrange, 1815), TII,III,39. 4. “It is, for the value of δP , not allowed to simply change dδP into δdP etc.” (J.L de Lagrange, 1815), TII,IX,C1,1. The first explanation, comparing d and δ, seems to imply that the virtuals are to be considered infinitesimal. Indeed, if one compares eq.(2.1) with the real work Z

dW =

(dm¨r − dfe )T dr = 0,

(2.2)

(S)

then the δr in eq.(2.1) has to be looked at as infinitesimal. And it is, in this interpretation, no longer a virtual velocity but a virtual displacement. Integration of eq.(2.2) yields the (real, finite) work which will be accomplished under the action of impressed forces Zr Z

W = o (S)

e T

(dm¨r − df ) dr =

t( Z r) Z o (S)

(dm¨r − dfe )T

dr dt = 0. dt

(2.3)

10

2 Axioms and Principles

Differentiating eq.(2.3) w.r.t. time yields the (real) work rate (or power)

˙ = W

Z

(dm¨r − dfe )T r˙ = 0.

(2.4)

(S)

Looking for the (unknown) motion we are next going to replace r˙ by a comparison function:

r˙ → r˙ + δ 0 r˙ :

Z (S)

⇒

Z

Z

(dm¨r − dfe )T δ 0 r˙ +

(dm¨r − dfe )T r˙ = 0

(S)

(dm¨r − dfe )T δ 0 r˙ = 0

(S)

^ Z

(dm¨r − dfe )T r˙ = 0.

(S)

(2.5) In a physical interpretation, the δ 0 r˙ is a virtual velocity which agrees with Lagrange but which is not at all restricted to the infinitesimal case. Of course, the δ 0 r˙ has to be chosen from the same (allowed) space which the real r˙ belongs to. This space is restricted by constraints which are suitably described via an implicit function Φ(r) = 0, Φ ∈ IRm , m ≥ 1.

(2.6)

Thus, both the real and the virtual velocity have to satisfy the same condition: 

∂Φ ∂r





r˙ = 0,

∂Φ ∂r



δ 0 r˙ = 0.

(2.7)

2.2

11

Principles – the “Differential” Form

Obviously, r˙ is perpendicular to the  T normal vector (∂Φ/∂r) of the (ge∂Φ neralized) surface which is defi∂r ned by the constraint equation(s) τ r Φ(r) = 0. In other words: r˙ is an element from the tangent plane τ of Φ = 0 at r(t). Because δ 0 r˙ has to belong to the same space, it is also an Φ=0 element from the tangent plane, or, simply spoken, a tangent vector. The r brilliant background in Lagranr ge’s conception is that the (real) r˙ , resulting from given initial conditions, is well defined as one prescribed Figure 2.2. Constraint tangent vector, but δ 0 r˙ is kept totally arbitrary and does not coincide with r˙ . Let us recall Lagrange’s explanation Nr.1 (p.9): δr is not the same as dr but is calculated with the same rules. There is still a gap in explanation because Lagrange speaks about “virtual velocities” but uses a mathematical notation which is nowadays referred to as “virtual displacement”. The usual way to calculate the latter is 

Φ(r) = 0 ⇒ δΦ =

∂Φ



δr = 01

(2.8)

∂r [see eq.(2.19)] which yields exactly the same procedure as in eq.(2.7) except the change in notation δ 0 r˙ → δr. However, since eq.(2.7) is a homogeneous equation, one may call them “virtuals” anyhow. If one takes it for instance as an acceleration, one is left with Gauss’ Principle8 (C.F. Gauss, 1829) in its “differential” form, Z

(dm¨r − dfe )T δ 00¨r = 0.

(2.9)

(S)

1 The differentiation rules are not unified in literature. We define with the bold symbol a matrix or a column vector. Consequently, since for m = 1, eq.(2.8) represents a scalar product, differentiation of a scalar w.r.t. a vector yields a row vector. Differentiation of a vector (m > 1) w.r.t. a vector then yields a matrix with elements ∂Φi /∂rj (i: row index, j: column index). 8 Carl Friedrich Gauss, *1777 in Braunschweig, †1855 in G¨ ottingen

12

2 Axioms and Principles

“It may possibly be of some interest”, says Jourdain, “to establish yet a third principle of mechanics, which, so to speak, fills up the gap between [Lagrange’s] and [Gauss’] and which is expressed” Z

(dm¨r − dfe )T δ 0 r˙ = 0.

(2.10)

(S)

(P.E.B. Jourdain, 1909) [see also eq.(2.5)] which is called Jourdain’s Principle9 . The notations δ 0 and δ 00 are introduced here to distinguish between the special kind of “variations” which are used to stay in agreement with eq.(2.8), Lagrange δr 6= 0, Jourdain δ 0 r = 0, δ 0 r˙ 6= 0, Gauss δ 00 r = 0, δ 00 r˙ = 0, δ 00¨r 6= 0.

(2.11)

One has then with eq.(2.6), 



∂Φ δr = 0 (a),  ∂r  ˙ ˙ ˙ Φ(r, r˙ ) = 0 ⇒ δ Φ = ∂ Φ δ 0 r˙ = 0 (b),  ∂ r˙  ¨ ¨ r˙ , ¨r) = 0 ⇒ δ Φ ¨ = ∂Φ Φ(r, δ 00¨r = 0 (c). ∂¨r ⇒ δΦ =

Φ(r) = 0

(2.12)

˙ = (∂Φ/∂r)˙r which can be partially diffeHowever, eq.(2.6) yields Φ ˙ r˙ ) = (∂Φ/∂r). Analogously, one obtains rentiated w.r.t. r˙ yielding (∂ Φ/∂ ¨ ˙ ˙ r˙ )¨r by differentiation w.r.t. ¨r, the identity from Φ = (∂ Φ/∂r)˙r + (∂ Φ/∂ ¨ r) = (∂ Φ/∂ ˙ r˙ ) and so forth. Thus, with (∂ Φ/∂¨ 

∂Φ ∂r



=

˙ ∂Φ ∂ r˙

!

=

¨ ∂Φ

!

...,

(2.13)

∂¨r

(“an auxiliary relation which is sometimes attributed to Helmholtz10 ” (G. Hamel, 1949)11 and which henceforth will be called Helmholtz’ auxiliary equation) all the results from eq.(2.12) reduce to one and the same, but with different naming for the “virtuals”. However, due to the special rules in eq.(2.11), 9 Philip

Edward Bertrand Jourdain, *1879 in Ashbourne, †1919 in Crookham Ludwig Ferdinand von Helmholtz, *1821 in Potsdam, †1894 in Charlottenburg 11 Georg Karl Wilhelm Hamel, *1877 in D¨ uren, †1954 in Landshut 10 Hermann

2.2

13

Principles – the “Differential” Form

the interpretation of any assigned physical dimension is meaningless; a kinematic interpretation in that sense does not hold. All the considered virtuals are nothing but tangent vectors. Going back to eq.(2.1), a second interpretation would therefore be “projection into the unconstrained space”, because the assumption of δr being infinitesimal does not enter the calculation at any step, while, for arbitrary magnitude of δr, the interpretation of a work no longer holds. However, the notion “virtual work” is established and will be retained in the sense of a definition. “Virtual” and “infinitesimal” are terms that for centuries, and even in the present, have misled many people. They are the closest thing in dynamics to black magic, and they are the heart of Lagrange’s approach.... If you can construct a good virtual displacement vector, you can do good business with it, says for instance Th. Kane in (P. Radetsky, 1986). However, it is by no means to be chosen but on the contrary is to be kept arbitrary to later conclude on the motion equations! More than a hundred years before Kane, L. Poinsot12 obviously shared this opinion: He replaced the virtual displacements (“virtual velocities”) in eq.(2.1) by the real velocities, thus coming out with eq.[(2.5), right side], and states: “This principle, in this form, does not leave any trace of those ideas of infinitesimal small motions, and of disturbing equilibrium, which seems strange for the problem, and which leaves something obscure in one’s mind” (L. Poinsot, 1837). Clearly, [eq.(2.5), right side] is correct but misses the goal. Let, for simplicity, the impressed forces dfe belong to a potential dV (r); then eq.[(2.5), right side] yields

Z

T

¨r dm˙r +

(S)

Z (S)

d d dV r˙ = dr dt

Z

d 1 T r˙ dm˙r + 2 dt

(S)

Z

dV =

d (T + V ) = 0 dt

(S)

(2.14) [T: kinetic energy, see eq.(4.17)]. Poinsot’s Principle (he himself calls it a Principle) then, for conservative systems, represents the principle of energy conservation which obviously holds. However, because of the fact that the virtual displacements (the arbitrary tangent vectors) now coincide with the actual velocities, one can in general not conclude on the corresponding motion equations from eq.(2.14).

12 Louis

Poinsot, *1777 in Paris, †1859 in Paris

14

2 Axioms and Principles

v r˙ rϕ˙

vP

6

S t Figure 2.3.

r ϕ

Orbit -

Kepler’s Problem

This fact is easily demonstrated by a simple example, for instance Kepler’s13 problem: Let S be the central (inertially fixed) star and P the orbiting planet. The velocity of the planet, tangential to the orbit, consists of the “elongation” r˙ of the position vector and of the “convective part” rϕ˙ where ϕ represents the angle between the horizontal and the position vector (see Fig. 2.3), leading to i 1 1 h T = mv 2 = m r˙ 2 + r2 ϕ˙ 2 , 2 2

while the potential is, with Hooke15 and Newton16 , V = − rc (c: constant). One has then h i d c [T + V ] = r˙ m¨ r + mrϕ˙ 2 + 2 + ϕ˙ mr2 ϕ¨ = 0. dt r 



Everything is correct up to here. The dimension of power is “velocity (r) ˙ times force” and “angular velocity (ϕ) ˙ times moment of force”, respectively. Since r˙ 6= 0, ϕ˙ 6= 0 for the general case, one might think of setting the square brackets individually equal to zero. However, what one obtains then are not the corresponding force and moment balances (motion equations). The reason is, that “Coriolis17 forces” do not perform work and therefore a` priori do not enter energies, but their influence on motion, in any rotating system, may be dramatic (think of weatherhforecast on the rotating earth for instance). If in i c 2 the present example one sets m¨ r + mrϕ˙ + 2 = 0 and takes into account r P that an observer on the earth “feels” m¨ r = fi with the “forces” fi , then the “centrifugal force” fc = −mrϕ˙ 2 has obviously a wrong sign because daily experience tells us that a “centrifugal force” acts outwards, in positive r-direction.

13 Friedrich

Johannes Kepler, *1571 in Weil der Stadt, †1630 in Regensburg Hooke, *1635 in Isle of Wight, †1703 in London 16 Sir Isaac Newton, *1643 in Woolthorpe, †1727 in Kensington 17 Gaspard Gustave de Coriolis, *1792 in Paris, †1843 in Paris 15 Robert

2.3

Minimal Representation

15

Conclusions Virtual displacements are arbitrary tangent vectors of the constraint plane. They are well defined and by no means obscure. There is no need to assume them infinitesimal. As for the moment only “holonomic” constraints (2.6) enter considerations (“holo-nomic”: “totally-regular”), all mentioned Principles are identical, except Poinsot’s, which misses the goal in deriving motion equations. For the same reason, the (frequent) statement “in case of scleronomic (i.e. time-independent) constraints, dr and δr coincide” is wrong. The (frequent) statement “virtual displacements are infinitesimally small and happen infinitely fast” is incomprehensible, hence rejectable. There is, according to (G. Hamel, 1949), no need to establish others than Lagrange’s Principle, and we are going to proceed with eq.(2.1). It is not quite clear when the term “virtual velocities” was replaced with “virtual displacement”. Ernst Mach18 uses it already in (E. Mach, 1873), referring to statics19 . The reason might be that statics does not know velocities. Lagrange himself, with his statement si l’on imagine que tous ces corps se meuvent, durant un instant quelconque dt, par les espaces ds, ds’, ds”, ... avec les vitesses v,v’,v”, ...20 [(J.L de Lagrange, 1764), p.10], implies “infinitesimal displacements”. This, however, would not have been necessary at all.

3.

Minimal Representation

If a system is constrained by certain conditions, then it is often suitable to use coordinates q which implicitly fulfill the constraints. This leads to a problem representation in minimal configuration. Following historical development, we will call these coordinates minimal coordinates: (J.L de Lagrange, 1764) introduced them as “smallest number of undetermined variables”, or in (J.L de Lagrange, 1780) as “variables of minimal number”. Our motivation is, moreover, that we will also need corresponding velocities s˙ which in analogy will be called minimal velocities. Gustav Hamel uses the notation Lagrangean coordinates for minimal coordinates (G. Hamel, 1949), thus also following history, while “generalized coordinates”, introduced by (H. Lamb, 1920)21 , is also 18 Ernst

Mach, *1838 in Chirlitz, †1916 in Haar to (G. Hamel, 1949), the principle of virtual work in statics had been established by Johann Bernoulli in 1717 while its rudimentary roots go back to Aristotle, *384 B.C. in Stagira, †322 B.C. in Chalkis. 20 if one imagines that all these bodies move, within an arbitrary time interval dt, through the spaces ds, ds’, ds”,... with the velocities v,v’,v”,... 21 Horace Lamb, *1849 in Stockport, †1934 in Cambridge 19 According

16

2 Axioms and Principles

widespread. However, mathematicians often use that notion in another sense (e.g. for a mix of cartesian and polar coordinates). We therefore consider the terms minimal coordinate and minimal velocity better suited for our context.

3.1.

Virtual Displacements and Variations

Clearly, the virtual displacements from eq.(2.8) define Lagrange’s Principle (2.1). However, it has simultaneously to fulfill the constraints Z

(dm¨r − dfe )T δr = 0

^

Φ(r) = 0.

(2.15)

(S)

To combine the two parts of eq.(2.15) one may directly insert q: Φ(r(q)) := Φ(q) ≡ 0.

(2.16)

Since q is defined such that it implicitly fulfills the constraints, eq.(2.16) is identically zero, or, in other words: for q, no constraints remain. The question is then how q enters [(2.15), left side]. Here, calculation rules are needed for the δ-process. Starting with the minimal (unconstrained) space, the question is which assumptions can be made on δq. This step is by no means trivial (see subsection 3.3). The easiest procedure is to adopt some basic considerations from variational calculus (this also gives sense to the use of the variation symbol δ for virtual displacements): q = q∗ + δq = q∗ + η, η |tt1o = 0,

(2.17)

where q∗ denotes the true solution (from initial conditions). Hence, one can define the “deviations” in the sense of a Taylor22 series expansion w.r.t. the variational parameter  yielding for r, dr r(q, t) ⇒ δr = d 

"



= =0

∂r ∂q ∂q ∂

#



+ =0

and for Φ, along with eq.(2.18), 22 Brook

Taylor, *1685 in Edmonton, †1731 in London

∂r ∂t ∂t ∂



= =0

∂r ∂q

δq

(2.18)

2.3

17

Minimal Representation

dΦ Φ(r, t) = 0 ⇒ δΦ = d 





=

∂Φ



∂r

=0

∂r

!



δq =

∂q

∂Φ



δr = 0

∂r (2.19)

in agreement with eq.(2.8) and with Lagrange’s explanations Nr.1 and Nr.2 (page 9), where the “disturbed equilibrium” characterizes an arbitrary δr in comparison to the true solution dr. Notice, however, that these results are not self-evident but depend on the chosen calculation rule (2.17). One may easily verify that other assumptions are also possible because q is totally unrestricted. Along with eq.(2.18), Lagrange’s Principle for a minimal representation reads Z (S)

e T

(dm¨r − df )

∂r

!

δq = 0

^

Φ(r(q)) ≡ 0,

(2.20)

∂q

where the minimal coordinates are calculated from the constraints Φ(r) = 0. Remarks. The Taylor expansion is “complete”: due to the linear approach, no higher order terms arise and | δq | (and thus | δr |) is arbitrary in magnitude. This is in agreement with variational calculus where, looking for an (unknown) optimal solution Jopt in terms of a Taylor series w.r.t. the optimum (J = Jopt + (∂J/∂q)opt δq + · · ·), the δq has to be kept arbitrary in magnitude because the (unknown) Jopt may be far away from J. However, this is a totally different question. The only solution required here is the tangent vector δr. In optimization, using eq.(2.17), the problem is shifted to the question of what class of functions η have to enter consideration. To give a simple example: if, in a club, not every club member is known, one can never decide who of them is the tallest one (P. Funk, 1962)23 . However, this question does not arise in the present context at all, because not one special but any (“allowed”) tangent vector enters the calculation. Except for the definition of virtual displacements, the (sometimes so-called) differential principles of dynamics have nothing in common with the usual calculus of variation. In fact, adding non-holonomic constraints to the socalled Hamilton-Principle24 (see Chapter 5) by means of Lagrangean 23 Paul 24 Sir

Georg Funk, *1886 in Vienna, †1969 in Vienna William Rowan Hamilton, *1805 in Dublin, †1865 at Dunsink

18

2 Axioms and Principles

multipliers leads to wrong results when treated with the rules of variational calculus [see (G. Hamel, 1935), (J.G. Papastavridis, 1997)] although Hamilton’s Principle is not at all restricted to holonomic systems as often but erroneously stated.

3.2.

Minimal Coordinates and Minimal Velocities

Looking for a minimal representation one has to assume r = r(s) which will, in general, be a nonlinear vector function of quasi-coordinates s. (Notice that quasi-coordinates themselves do not enter our calculation at any step but only their differentials which are well defined.) The (holonomic) constraints (2.6) are, for the nontrivial case, also nonlinear functions. Its time derivative, however, is linear w.r.t. r˙ . One has, in the first step, for r˙ , 

r = r(s) ⇒ r˙ =

∂r



s˙

(2.21)

∂s which interrelates r˙ and s˙ by the unknown Jacobian(∂r/∂s). This Jacobian is obtained from the time derivative of the constraints and its subsequent partial derivation w.r.t. s˙ , yielding ˙ = Φ = Φ(r) = 0 ⇒ Φ



∂Φ





r˙ = 0 ⇒

∂r

∂Φ



∂ r˙ ∂ s˙

∂r



=0

(2.22)

because (∂Φ/∂r) (which is of course known) cannot depend on velocities. Partial derivation of eq.(2.21) w.r.t. s˙ yields (∂ r˙ /∂ s˙ ) = (∂r/∂s) (the aforementioned Helmholtz auxiliary equation). Thus, one obtains the required Jacobian as a solution of eq.(2.22) which leads to 

r˙ =

∂ r˙ ∂ s˙





s˙ ⇒ s˙ =

∂ s˙



∂ r˙

r˙

(2.23)

with the generalized inverse 

∂ s˙ ∂ r˙

"



=

∂ r˙ ∂ s˙

T 

∂ r˙ ∂ s˙

#−1 

∂ r˙ ∂ s˙

T

(2.24)

which always exists for non-redundant constraints. Several solutions exist for eq.(2.22) and the corresponding s˙ will in general not be integrable. Consequently, as a result of considering the time derivatives, one obtains in the first step a vector of minimal velocities. However, since Φ(r) = 0 represents a purely geometric condition which only depends on

2.3

19

Minimal Representation

˙ of integrable position, the solution manifold {˙s} contains a sub-manifold {q} solutions. Concentrating on these, one has for eq.(2.22), along with Helmholtz’s auxiliary equation, 

∂Φ



∂ r˙

!

∂ q˙

∂r

= 0 ⇒ r˙ =

∂ r˙

!

˙ q.

∂ q˙

(2.25)

Insertion of eq.(2.25) into eq.(2.23) yields 

s˙ =

∂s

∂r



∂r

!

q˙ = H(q)q˙ ∈ IRf

(2.26)

∂q

(f : degree(s) of freedom).

Conclusion Minimal velocities consist of linear combinations of the time derivatives of the minimal coordinates. They need not be integrable (non-holonomic velocity variables). A special solution is H = E (E: identity matrix) leading to holonomic velocity variables. One may interpret the result (2.26) as follows: minimal position is characterized by minimal coordinates q. For the velocity representation one may use their time derivatives, but of course also any unique linear combination of these. The latter is more general and inevitable in case of non-holonomic constraints.

3.3.

The Transitivity Equation

Using δq in the sense of eq.(2.17) answers the essential question: Are the doperation (the total differentiation) and the δ-operation interchangeable? First, looking at r = r(q), one obtains by elementary calculation

dδri − δdri =

X

∂ 2 ri

k,j

∂qj ∂qk

−

∂ 2 ri ∂qk ∂qj

!

dqk δqj +

X ∂ri j

∂qj

(dδqj − δdqj ) (2.27)

where  = const. ⇒ dδq − δdq = d(η) − dη = 0.

(2.28)

Then, eq.(2.27) reduces to the integrability condition which is obviously fulfilled just from modeling considerations: except in science fiction an instantaneous dislocated de- and re-materialization of material points has not yet been observed. Thus,

20

2 Axioms and Principles

dδr − δdr = 0

(2.29)

(see Lagrange’s explanation Nr.3, p.9). However, for non integrable s = s(q) one is left with the “transitivity equation” dδsi − δdsi =

X

∂ 2 si

k,j

∂qj ∂qk

∂ 2 si

−

∂qk ∂qj

!

dqk δqj 6= 0

(2.30)

(see Lagrange’s explanation Nr.4, p.9). It is worth mentioning that Lagrange (in the cited reference) considered rotations: there, dP, dQ, dR refer to ω1 dt, ω2 dt, ω3 dt where (ω1 ω2 ω3 )T = ω ∈ IR3 represents angular velocities. ˙ is a typical representative for a non-holonomic velocity As well known, ω(q, q) variable (see Chapter 3).

Conclusion Recall once more Poinsot’s contribution: for d and δ being the same, the transitivity equation dδ( ) − δd( ) always vanishes, which is obviously false. In order to calculate the transitivity equation one has to choose a rule for the δ’s. This is not at all self-evident. Applying the transitivity equation to the position vector r (of the considered mass element), i.e., dδ(r) − δd(r), for instance, yields zero only if one assumes δq in the sense of variational calculus. Here, (A.I. Lur’e, 1968)25 for instance, was misled with his statement “avec l’´egalit´e e´ vidente” of dδ(r) − δd(r) = 0 because this kind of “self-evidence” already implies the use of the δq’s in the above sense and automatically eliminates other procedures (like the Lie group approach for instance which requires, in the sense of (G. Hamel, 1904a), the interchangeability w.r.t. s instead of q. However, this theory will not be pursued here).

4.

The Central Equation of Dynamics

Looking at Lagrange’s Principle (2.1), one may extract one-time derivation from dm¨rT δr. Along with the product rule one obtains Z

dm¨rT δr − δW =

(S)

d dt

Z (S)

dm˙rT δr −

Z

dm˙rT

d δr − δW = 0 dt

(2.31)

(S)

where δW abbreviates δrT dfe . The last integral term in eq.(2.31) recalls the variation of kinetic energy T [see eq.(2.14), eq.(4.17)]: R

25 Anatolii

Isakovich Lur’e, *1901 in Mogilev-on-Dnjepr, †1980 in St. Petersburg

2.4

21

The Central Equation of Dynamics

δT −

Z

dm˙rT δ

dr = 0. dt

(2.32)

(S)

Subtracting eq.(2.32) from eq.(2.31) yields 

d   dt

 Z

Z

dm˙rT δr − δT − 

dm˙rT



dδr − δdr = δW. dt 

(2.33)

(S)

(S)

Along with eq.(2.29), theR term with [dδr − δdr]/dt may be omitted. If one formulates the momentum (S) dm˙rT in terms of a quadratic form derivative, and if one furthermore inserts δr = δr(s) = (∂r/∂s)δs, one obtains 

d   dt

 Z

(S)

∂

1 ∂r  r˙ dm˙rT δs − δT = δW. ∂ r˙ 2 ∂s 



(2.34)

Applying Helmholtz’s auxiliary equation to (∂r/∂s) yields 

d   dt

 Z

(S)

1 ∂ r˙  r˙ dm˙rT δs − δT = δW. ∂ r˙ 2 ∂ s˙ ∂





(2.35)

The integrand here corresponds to the chain rule applied to the quadratic form. Abandoning the chain rule one may directly differentiate w.r.t s˙ . Then all s-depending terms may be extracted from the integral, having in mind that the minimal velocities s˙ do not depend on mass distribution: 



d ∂    dt ∂ s˙

 Z



1   r˙ dm˙rT  δs − δT = δW. 2

(2.36)

(S)

The integral term herein is the kinetic energy T . One obtains thus for eq.(2.31), d dt



∂T ∂ s˙





δs − δT − δW = 0

(2.37)

in terms of non-holonomic variables s˙ which shall be called the Central Equation in analogy to Heun and Hamel [(G. Hamel, 1904b), (H. Bremer, 1988)].

22

5.

2 Axioms and Principles

Principles – the “Minimal” Form

Since the Central Equation contains a time derivation for the first summation term, one may integrate eq.(2.37) (over a fixed time domain to avoid unknown integration constants). If one sets the variations at the time boundaries equal to zero (which is possible although not compelling), one obtains 

∂T ∂ s˙

t1



δs

=0 to

^

Zt1

(δT + δW )dt = 0.

(2.38)

to

Here, no restrictions are to be made on holonomic or non-holonomic variables. Notice, however, that the first step of evaluation usually requires an integration by parts. For this purpose, δ˙s has to be replaced by d[δs]/dt taking eq.(2.30) into account. This step is carried out in eqs.(5.23/24). For the moment we reduce considerations to holonomic variables q˙ and additionally assume conservative systems (δW = −δV ) yielding Zt1

(T − V )dt = 0.

δ

(2.39)

to

This relation is called Hamilton’s Principle. [According to Felix Klein26 it had already been known to Lagrange (F. Klein, 1926).] Eq.(2.39) is sometimes also referred to as the Principle of Least “Action” (“quantit´e d’action”). With this title it had been published by Maupertuis27 , first in R T spring 1744 postulating a minimum for r˙ dr and two years later as Z

m˙rT dr → min.

(2.40)

r for a single mass m (Maupertuis’ Principle). But is eq.(2.40) really a minimum? Leibniz28 expressed his discomfort already in 1708 in a letter to J. Hermann29 emphasizing that eq.(2.40) yields a minimum or a maximum (I. Szab´o, 1979a)30 . L. Euler also published eq.(2.40) (autumn 1744) but under the restriction of energy conservation, 26 Felix

Christian Klein, *1849 in D¨usseldorf, †1925 in G¨ottingen Louis Moreau de Maupertuis, *1698 in St.-Malo, †1759 in Basel 28 Gottfried Wilhelm von Leibniz, *1646 in Leipzig, †1716 in Hannover 29 Jakob Hermann, *1678 in Basel, †1733 in Basel 30 Istvan Szabo ´ , *1906 in Orosh´aza, †1980 in Berlin 27 Pierre

2.5

23

Principles – the “Minimal” Form Z

Z

m˙rT dr =

r

m˙rT r˙ dt → min.

^

(T + V ) = const. = Eo ,

(2.41)

t

(see eq.(2.14)). This leads to Z

m˙rT r˙ dt =

t

Z

2T dt → min.

^

(T + V ) = const. = Eo

t

⇒

Z

(T + Eo − V )dt → min ⇒ δ

t

Z

(T − V )dt = 0

(2.42)

t

for a single mass. Jacobi adopts this expression for a system of particles in the form Z √ √

Z

2T dt = 2 t

T

T dt2

qX √ Z p = 2 Eo − V [m dqT dq]i ,

t

(2.43)

t

(C.G.J. Jacobi, 1847). Jacobi’s Principle does thus not contain anything new and Leibniz’ doubts (minimum or maximum) are still cogent. However, this question can be answered by a simple example: Consider a spring-mass system (mass m, spring coefficient k) with the energies T = mq˙2 /2, V = k q 2 /2 R and the abbreviation (T − V )dt = J. Eq.(2.39) then yields, along with an integration by parts, Zt1

Zt1

to

to

(mqδ ˙ q˙ − kqδq)dt = −

δJ(q) =

[m¨ q + kq] δqdt = 0

(2.44)

p

with solution q(t) = (q˙o /ω) sin(ωt), ω = k/m, if qo = 0 is assumed. If J(q) really represents a minimum, then J(q) − J(q) > 0 ∀ q = q + η must hold, yielding J(q) = J(q + η) = 1 2 |

Zt1 h

2

mq˙ − kq

2

i

Zt1

dt +

to

to

{z

J(q)

}

1 [mq˙η˙ − kqη] dt + 2 |

Zt1 h

i

mη˙ 2 − kη 2 dt = 0.

to

{z

J(η)

}

(2.45)

24

2 Axioms and Principles

as can easily be seen by an integration by parts: R The middle term vanishes t1

(m¨ q + kq)ηdt + [mqη] ˙ to = 0 because of eq.(2.44) and η(to ) = 0, η(t1 ) = 0. Thus, eq.(2.45) leads to the minimum condition !

J(q) = J(q) + J(η) ⇒ J(q) − J(q) = J(η) > 0.

(2.46)

For to = 0 and t1 arbitrary, one may choose η =  sin(π t/t1 ) yielding for eq.(2.46),

J(η) = !

m 2

Zt1 



η˙ 2 − ω 2 η 2 dt =

o

π



 2

 2

π

mt1 



!2

!

− ω2 > 0

t1

 

⇒ t1
g variables for the chosen procedure and insert the constraints afterward. This is obvious when considering Maggi’s procedure but also holds for Hamel’s equations. The most adequate procedure is here to use Hamel’s variables s˙ T = (˙sTind. s˙ Tdep. ) ∈ IRf (ind.: independent, dep.: dependent) where the dependent variables represent the constraints. Only after carrying out the prescribed calculation steps may one set these equal to zero. This is because the partial derivations (∂T /∂ s˙ i ) are needed which would lead to a loss of information if s˙ Tdep. ∈ IRf −g is set to zero in advance. This fact is easily demonstrated with eq.(4.27) which reads in components " X n

#

"

#

X ∂T (dδsν − δdsν ) ∂T d ∂T − − Qn δsn + = 0. dt ∂ s˙ n ∂sn dt ν ∂ s˙ ν

(4.37)

The value of (dδsν −δdsν ) is found in the transitivity equation (2.30). Along with q˙k =

X

(∂qk /∂sµ )s˙ µ , δqi =

µ

X

(∂qi /∂sn )δsn ,

(4.38)

n

δsn is factored out in eq.(4.37) and yields the (explicit form of the) Hamel equations "

#

X ∂T ∂T d ∂T − − Qn + s˙ µ βνµ,n = 0, n = 1 · · · f dt ∂ s˙ n ∂sn ν,µ ∂ s˙ ν

(4.39)

(G. Hamel, 1904a) where βνµ,n represent the Hamel coefficients βνµ,n

=

X ∂ q˙k ∂ q˙i i,k

∂ s˙ µ ∂ s˙ n

∂ 2 sν ∂qi ∂qk

−

∂ 2 sν ∂qk ∂qi

!

= −βνn,µ .

(4.40)

Evidently, since βνn,n = 0, the generalized momenta (∂T /∂ s˙ k ), k = 1 · · · f, k 6= n influence the nth equation of motion provided that the corresponding βνk,n is nonzero.

76

4 Rigid Multibody Systems

5 By

KA YH H Aq AHH A HH H 
A HH
A HH
 *  γ A vy HHq
 q Bx A  vx  A  Iy sh A c Aq 6

Example: Let s˙ T = (vx vy ωz ) ∈ IR3 and 1 T = s˙ T 2





m 0 0

0 m mc

0 mc Co





vx x ˙s =  vy , q =  y  γ ωz

# s˙ ,

(4.41)

(plane motion of a car, wheel masses neglected, mass center at ce1 , C o : moment of inertia w.r.t. origin, s˙ : velocities in body-fixed components, q: positions in inertial components [see also example 1 on p.83].)

- Ix Figure 4.8.

"

Simplified Car Model

The relationship between s˙ and q˙ is simply ∂s1 ∂q1  ∂s 2 q˙ =   ∂q1 ∂s3 ∂q1



" s˙ =

cos γ − sin γ 0

sin γ cos γ 0

0 0 1

#

∂s1 ∂q2 ∂s2 ∂q2 ∂s3 ∂q2

∂s1 ∂q3 ∂s2  ˙ ∂q3  q ∂s3 ∂q3



(4.42)

and, because [∂s/∂q] is orthonormal in the present case, [∂q/∂s] = [∂s/∂q]T . Hence, all needed differential quotients for calculation of the βνµ,n , ν, µ, n ∈ {1, 2, 3} are known. Because of the skew-symmetry of the β’s, one has only to determine three coefficients for every ν. Moreover, since s˙ 3 is holonomic, β3µ,n = 0 ∀ µ, n. For ν = 1 one obtains β11,2 = 0, β11,3 = − β12,3 = −

∂q1 ∂ 2 s1 ∂s1 ∂q1 ∂q3 ∂q1 ∂ 2 s1

− −

∂s2 ∂q1 ∂q3

∂q2 ∂ 2 s1 ∂s1 ∂q2 ∂q3 ∂q2 ∂ 2 s1

= (− cos γ)(− sin γ) − (sin γ)(cos γ) = 0, = (sin γ)(− sin γ) − (cos γ)(cos γ) = −1

∂s2 ∂q2 ∂q3

(4.43) and, analogously, yields

"

β21,2

=

0, β21,3

=

1, β22,3

= 0. Inserting the nonzero elements into eq.(4.39)

#

  d ∂T ∂T + β23,1 ωz − Q1 δs1 = mv˙ x − mcωz2 − mωz vy − fx δs1 = 0, dt ∂v |{z} ∂vy x −1

"

#

d ∂T ∂T + β13,2 ωz − Q2 δs2 = [mv˙ y + mcω˙ z + mωz vx − fy ] δs2 = 0, dt ∂v |{z} ∂vx y +1

4.3

77

The Triangle of Methods "

#

d ∂T ∂T ∂T + β12,3 vy + β21,3 vx − Q3 δs3 dt ∂ω |{z} ∂v |{z} ∂v z x y −1

+1

= [C o ω˙ z + mcv˙ y + mcvx ωz − Mz ] δs3 = 0.

(4.44)

If the body is not allowed to move with vy (no sliding perpendicular to the path (x, y) of the rear axle in Fig. 4.8), then vy = s˙ 2 = s˙ dep. represents a non-holonomic constraint which is eventually inserted. Along with vy = s˙ 2 = 0 it follows that δs2 = 0 while δs1 and δs3 remain arbitrary. As a result, one obtains the equations of motion



m 0



0 Co



v˙ x ω˙ z

 +



−mc ωz 0

0 mc ωz



vx ωz

 =



fx Mz

(4.45)

.

If one would set vy = 0 in advance, then T would no longer depend on vy , thus (∂T /∂vy ) = 0. One obtains then from eq.(4.44),



m 0

0 Co



v˙ x ω˙ z



 =

fx Mz



(4.46)

. . . which is false for c 6= 0.

4 This difficulty does not arise when using the Projection Equation since here partial derivations of T are no longer needed. 5 Example: For the plane motion of a rigid body the body-fixed coordinate system is used for reference frame. The mass center velocities are vc ∈ IR2 , ω c ∈ IR1 and the forces and torques are, respectively, fe ∈ IR2 , Me ∈ IR1 . The independent s˙ ind. := s˙ = (vx ωz )T is directly used (i.e. the non-holonomic constraint vy = 0 is considered in advance). One has thus

  

vc ωc







vcx :=  vcy  = ωcz

"

1 0 0

0 c 1

#

vx ωz



∂vc ∂ s˙

 

  ⇒    ∂ω c

  = 

"

1 0 0

∂ s˙

0 c 1

# .

(4.47)

The momenta w.r.t. the mass center read



p L

"

 :=

m 0 0

0 m 0

0 0 Cc

# v  " cx m 0  vcy  = 0 m 0

ωcz

0

0 0 Cc

#"

1 0 0

0 c 1

# s˙ .

(4.48)

.

(4.49)

Insertion into eq.(4.34) yields eq.(4.45) directly via



1 0

0 c

0 1

 (" m 0 0

0 mc Cc

#

" ¨s +

0 ωz 0

−ωz 0 0

0 0 0

#"

m 0 0

0 mc Cc

# ) s˙

4 Conclusion The easiest way to determine motion equations is obviously to use a procedure

78

4 Rigid Multibody Systems

in which every step of calculation is implicitly already carried out as far as possible, thus requiring only insertion of problem specific quantities. This yields the longest expression under all available methods. Or the other way round: the shorter the mathematical representation, the more effort in evaluating motion equations. The procedure with minimum effort is thus the last method in Table 4.1 – which is the Projection Equation.

4.

Subsystems

A subsystem formulation is easily obtained from eq.(4.34) by sum splitting and assigning variables y˙ n to every subsystem n. Each subsystem then consists of Nn bodies. The rule of how to combine the subsystems is simply the chain rule of differentiation: N sub  X n=1

 !T T X Nn  ∂v c   ∂ s˙ ∂ y˙ n i=1

∂ y˙ n

∂ω c ∂ y˙ n

!T  " 

e IR p − fe p˙ + ω ˙L + ω e IR L − Me

#  

= 0. i

(4.50) (Notice that p, L are translational and angular momenta with respect to the mass center, each represented in a reference frame R which rotates with ω IR .)

4.1.

Basic Element: The Rigid Body

4.1.1

Spatial Motion

ωr rc ωo o vo

Figure 4.9.

Reference Frame

As eq.(4.50) shows, every individual body i = 1 · · · Nn which belongs to the nth subsystem is characterized by one and the same vector of variables. In order to simplify calculations (and to structurize the problem) it will be useful to first have a look at the dynamics of a single body. For the most general case, a reference coordinate system is chosen which is centered in the “joint” o where the body is attached. The reference frame moves with vo , ω R = ω o .

The body may rotate with ω r relative to the moving frame and translate with r˙ c w.r.t. the joint (index c: mass center, index r: “relative”). The corresponding velocities are collected in a vector of “auxiliary ” (or “intermediate”) variables

4.4

79

Subsystems

T y˙ i = [ vTo ω To r˙ Tc ω Tr ]i . Notice that vo , ω o represent the absolute velocities of the chosen reference frame. One obtains for the n-th subsystem in eq.(4.50) by once more applying the chain rule:

!T   

Nn X

∂ y˙ i

i=1

∂ y˙ n



{z

|

}

T Fi

!T

∂vc ∂ y˙

|

∂ω c ∂ y˙

!T  " i

{z

#  

e o p − fe p˙ + ω e o L − Me L˙ + ω

i

. (4.51)



}

T Fi

We can thus first determine the dynamical equations of each single body and eventually adjust these to the actual subsystem under consideration by T premultiplying with Fi and summarizing. The i-th mass center velocities (which belong to the n-th subsystem) read vo   ω E 0  o  . (4.52)  ˙ rc  0 E i ωr i 

vc ωc

!

e o rc + r˙ c vo + ω ωo + ωr

= i

!

"

E erTc 0 E

= i



#

Thus, when using (vTo ω To r˙ Tc ω Tr )Ti := y˙ i for auxiliary variables, the reT

quired functional matrix Fi in eq.(4.51) is 

T Fi

∂vc = ∂ y˙

!T  " ∂ω c  E erTc =

!T

0

∂ y˙

E 0 0 E

E

i

#T

.

(4.53)

i

Along with the mass center velocities from eq.(4.52) one obtains the momenta p L

!



= i

mE 0 0 Jc

|

{z

 i

vc ωc

!

= Mi Fi y˙ i .

(4.54)

i

}

Mi Insertion into eq.(4.51) yields  !T  ∂vc   ∂ y˙ T

= Fi

"

∂ω c ∂ y˙

!T  " #  e o p − fe ˙ p + ω  e o L − Me  L˙ + ω

˙ ˙ e F y˙ − M F y¨ + M F y˙ + M F y˙ + ΩM

fe Me

i

!#

(4.55) i

80

4 Rigid Multibody Systems 

ei = where Ω T



eo 0 ω eo 0 ω

. Carrying out calculations leads to, along with i

e

F (feT MeT )T = Q for generalized impressed force and corresponding consc traint force Q , e

c

Mi ¨yi + Gi y˙ i − Qi = Qi ,

(4.56)

where mE merTc  me Jo  rc  



T

Mi = F M F i

= 

 mE

merTc Jc

0 



mE 0 merc Jc    , mE 0   Jc

0

(4.57)

i



T ˙ T ˙ e F Gi = F M F + Fi M F + Fi ΩM T

i

eo e oe eo mω mω rTc 2mω 0 c c   o ˙ e oe e o J + J 2me eo ω e o Jc + J˙   mω rc ω rc ω 



=

  mω eo 

e oe mω rTc c e o Jc + J˙ ω

0

eo 2mω

0

0

c e o Jc + J˙ ω

  .  

(4.58)

i

Remarks: Jo is the tensor of moments of inertia with respect to the joint, Jo = Jc + mercerTc . (The Huygens12 -Steiner13 term mercerTc has not to be considered additionally but follows from calculation14 .) T ˙ The first term in eq.(4.58) (first line) yields F M F y˙     0 −mer˙ c 0 0 eo vo 0 0 mω     0 0 me eo ω r ω rcer˙ c 0 0    0 −me o c  = = ˙rc 0 0  eo  0 −me   r˙ c   0 0 mω 0 0 0 ωr 0 0 0 0

0 vo  ωo  0    0   r˙ c  0 ωr 



12 Christiaan

Huygens, *1629 in Den Haag, †1695 in Den Haag Steiner, *1796 in Utzenstorf, †1863 in Bern 14 This result is directly obtained for M(2, 2). Calculating G(2, 2) yields, when multiplied with ω , along o with (R2) on page 443: 13 Jakob

T

T

T

G(2, 2)ω o = [ω e o Jc + merc ω e oerc ]ωo = [ω e o Jc + m(erg e oerc )erc ]ωo = ω e o [Jc + mercerc ]ωo = c ωo + ω T

ω e o Jo ωo because (erg rc ω o ) = −(e rg rc ω o ) = 0. c ω o )(e c ω o )(e

4.4

81

Subsystems T

e M y. ˙ This leads to factor 2 in the third column which may be added to F Ω of G (characterizing the Coriolis-effects).

The second term in eq.(4.58) (first line) takes care of the fact that Jc is not constant because of its representation in the moving reference frac c me R. Time derivation J˙ can, however, be avoided: completing R J˙ = d c dt (ARB B J ABR ) [see eq.(4.21)] with unit matrices yields ˙c RJ

= A˙ RB (ABR ARB ) B Jc ABR + ARB B Jc (ABR ARB ) A˙ BR e r R Jc − R Jc ω er = ω

[see eq.(3.36) for I → R, along with R ω BR = ω r ]. This fact is essential for numerical programming. The intermediate (or auxiliary) variable y˙ ∈ IR12 cannot be interpreted as a vector of minimal velocities for a free body having six degrees of freedom only. Here, kinematics is overdetermined when removing the body from its surroundings unless one defines additional constraints. However, the goal of the present procedure is to use auxiliary variables which make the intermediate calculations as easy as possible to later combine the results simply via eq.(4.51). As known in advance the constraint forces do not contribute when projecting into the minimal space. One has therefore not to take care of these explicitly at any step of the calculation. One obtains, as a result, a dynamical representation which has a clear and simple structure although the dynamical system itself is, of course, highly nonlinear.

Discussion. In order to check the results it is useful to investigate special cases. 1. No relative motion (rc = const, ω r = 0). The corresponding equations of motion (first two lines in eq.(4.57) and eq.(4.58)) read "

mE merTc merc Jo

#

v˙ o ω˙ o

! "

+

e o me e om ω ω rTc e o me e o Jo ω rc ω

#

vo ωo

!

=

Rf

e,o

e,o RM

!

.

(4.59) This case represents the free motion ∈ IR of a rigid body. The reference frame is then body-fixed at the joint. (Notice that G is not skew-symmetric e o Jo .) The applied forces and torques are represented in the reference due to ω frame and refer to the joint (index e: impressed, o: origin (joint)). 6

2. Prescribed rigid body motion: vo = 0, ω o = Ωe3 = const. No relative rotation: ω r = 0. The equations of motion (third line in eq.(4.57) and eq.(4.58)) reduce to e o m˙rc +mω e oe m¨rc +2mω rTc ω o = (R fxe R fye )T =

82

4 Rigid Multibody Systems 

m 0 0 m



x ¨ y¨





0 −2mΩ 2mΩ 0



x˙ y˙





−mΩ2 0 0 −mΩ2





x + + y (4.60) e where m¨ z = R fz is decoupled from the remainder. Eq.(4.60) describes the mass center motion in a rotating frame. It is often used in simplified rotor dynamics (Laval shaft, “mass point”: see Chapter 6, page 273).

3. Prescribed rigid body motion: vo = 0, ω o 6= 0. No relative translation: rc = 0. The equations of motion (last line in eq.(4.57) and eq.(4.58)) are c e o Jc + J˙ )(ω o + ω r ) Jc (ω˙ o + ω˙ r ) + (ω

d c e o [Jc (ω o + ω r )] = R Me,c . [J (ω o + ω r )] + ω (4.61) dt This case corresponds to the rotation ∈ IR3 of a rigid body with a fixpoint (“gyro”) when resolved in a rotating frame according to eq.(4.36), where R ω c = ω o + ω r and R ω IR = ω o . [Eq.(4.61) yields, for instance, eq.(6.165 ˙ γ) a) on page 278 for ω o = Ωe1 , ω r = (0, β, ˙ T .] =

4.1.2 Plane Motion Considering plane motion in the x-y-plane, the mass center velocities read (index i suppressed for simplicity)  





vcx 1 0 −y 1 0 0        0 1 x 0 1 0 v =   cy   0 0 1 0 0 1  ωcz  

vox voy ωoz x˙ y˙ ϕ˙

     ˙  := F y.   

(4.62)

Here, vox , voy , ωoz are the hinge-point velocities, x, y denote the mass center w.r.t. the hinge-point and ϕ represents the relative angle of the body w.r.t. the reference frame which rotates with ωoz . Notice that vox , voy are non-holonomic (see also eq.(4.47) for ϕ = 0, y = 0, x = c = const.). The same as in eq.(4.54), the momenta are 







px vcx m 0 0 m 0 0     ˙  py  =  0 m 0   vcy  =  0 m 0  F y˙ := M F y, c c 0 0 C 0 0 C Lz ωcz (4.63) where C c : moment of inertia w.r.t. the mass center. Because ω r = ϕe ˙ 3 is uniaxial, the time derivative of Jc in G vanishes. The mass matrix according to eq.(4.57) thus reads 





4.4

83

Subsystems      M=   



0 −my m 0 0 m mx 0 m 0   C o −my mx C c   , m 0 0   symmetric m 0  Cc m

(4.64)

and matrix G results in      G=   

0 −m −mx 0 −2m 0 m 0 −my 2m 0 0 mx my 0 2mx 2my 0 0 −m −mx 0 −2m 0 m 0 −my 2m 0 0 0 0 0 0 0 0

      ωoz .   

(4.65)

i

5 Example 1: Let, for the plane motion of a single rigid body, rc = (x 0 0)T , x = c = const. The functional matrix F in eq.(4.62) then reduces to

" F=

1 0 0

0 1 0

0 c 1

x, vox y, voy

#

(4.66)

.

T



fx

fy

Mz

@ @f 

M=

m 0 0

q

F, Figure 4.10.

"

ωoz

rc

@

The subsystem matrices are Q =

t

@ I @

0 m mc

0 mc Co

#

" ,G =

0 m mc

Rigid Body Plane Motion

−m 0 0

−mc 0 0

#

(4.67)

ωoz .

The corresponding vector of “intermediate variables” reads y˙ = (vox voy ωoz )T . If, additionally, the non-holonomic constraint voy = 0 is considered, one is once more left with eq.(4.45):



m 0

0 Co



v˙ ox ω˙ oz



 +

0 mcωoz

−mcωoz 0



vox ωoz



 =

fx Mz

 .

(4.68)

84

4 Rigid Multibody Systems x, vox

   y 

Example 2: If, on the other hand, x 6= const., then the vector of intermediate variables is

  X XXs   x˙  y, voy  yXX X q Xq

ωoz

T y˙ = (vox voy ωoz x) ˙

(4.69)

with corresponding functional matrix

" q

F=

1 0 0

0 1 0

0 x 1

1 0 0

#

(4.70)

.

The subsystem matrices read Figure 4.11.

Relative In-Plane-Translation



m  0 M= 0 m

0 m mx 0

0 mx Co 0





m 0 0   m ,G =  xm 0  0 m

−m 0 0 −m

−xm 0 0 −xm



0 2m  ω , 2xm  oz 0

(4.71)

T

and Q = (fx fy Mz )F. Notice that G is not skew-symmetric due to (the Coriolis-) factor 2 indicating relative translational motion in a rotating frame. Example 3: Relative rotation with rotation axis fixed at the origin needs to consider the constraints

 ϕ˙

x, vox t rc

@ I @

ωoz

@ @

@f

x y



 =c

−y x

q

cos ϕ sin ϕ

"

1 0 0

0 1 0

−y +x 1

T

T

Q=F

#



m  0 ,M =  −my −my

, c = const.

−y +x 1

# .

(4.72)

Along with y˙ = (vox voy ωoz ϕ) ˙ the corresponding matrices read fx fy Mz

ϕ˙



Relative In-Plane-Rotation

"



One obtains thus the functional matrix from eq.(4.62) F=

Figure 4.12.

 =

where

 y, voy



x˙ y˙

0 m mx mx

−my mx Co Co

−my mx  , Co  o C



4.4

85

Subsystems −mωoz 0 myωoz myωoz



0  mωoz G= mxωoz mxωoz

−mxωoz −myωoz 0 0

−mx(2ωoz + ϕ) ˙ −my(2ωoz + ϕ) ˙  . 0 0



(4.73)

4

4.2.

Subsystem Assemblage

The intermediate variables are eventually adapted to the considered subsystem. They are interrelated to the actual describing variables y˙ n by its Jacobians, ∂yi

y˙ i =

∂yn

!

y˙ n = Fi y˙ n ,

(4.74) T

see eq.(4.51). Insertion into eq.(4.56), premultiplication with Fi and summarizing yields the motion equations of nth subsystem "N n X

T Fi Mi Fi

#

¨yn +

"N n X

i=1

i=1 def.

T Fi



# 

Nn

Nn

i=1

i=1

X T e X T c Gi Fi + Mi F˙ i y˙ n − Fi Q i − Fi Qi

= Mn ¨yn + Gn y˙ n − Qen − Qcn = 0.

(4.75)

When combining the single bodies to a subsystem, part of the constraint forces cancel out. Only those will be left which characterize the joining conditions to the neighboring subsystems. Definition: Vector y˙ n shall be called vector of describing (velocity) coordinates. They characterize the most general (and simplest) description of the problem; they are non-holonomic in general. Obviously, the definition of a subsystem is in a wide range arbitrary. The simplest subsystem consists just of one single body. Combining several bodies to one subsystem needs to define one reference frame. This goes along with the choice of y˙ n .

4.2.1 Absolute Velocities Usually, the kinetics result is simpler when choosing absolute velocity components within y˙ n . However, then more effort is, in general, needed for formulation of the impressed forces. The influence of absolute and relative velocities will be discussed with a simple example:

86

4 Rigid Multibody Systems

5 Example: One simple but typical subsystem which consists of two interconnected rigid bodies is a motorgear-arm unit. Considering a DC motor with very small moment of inertia CM along with a harmonic drive gear with very high transmission ratio iG leads to the observation that gear elasticity plays a significant role. Let the motor axis be e3 . One has then simply, for a plane motion, to combine two basic elements according to Fig. 4.10 and Fig. 4.12. In general, absolute velocities simplify the corresponding system matrices. Thus, using vox , voy and ΩA , ΩM (see Fig. 4.13) for variables leads to

x, vox

v γA γM

ωoz = ΩA y, voy

r

ΩM

r

r

y˙ n = (vox voy ΩA ΩM )n .

ΩAp

r

Figure 4.13.

Corresponding subsystem matrices are calculated with (4.66/4.67) for the arm and (4.72/4.73) for the rotor.

Motor-Gear-Arm-Unit

The functional matrices (Jacobians) are obtained from





" vox 1 y˙ A =  voy  = 0 0 ωoz

0 1 0

0 0 1

0 0 0



# y˙ n , y˙ M





vox  voy   = = ωoz   ϕ˙

1 0 0 0

0 1 0 0

0 0 1 −1



0 0  y˙ 0  n 1

(4.76) (Index A: arm, index M : motor, ϕ˙ = ΩM − ΩA ). Assuming vertical gravity direction, vector g reads g(−sinα − cosα)T ∈ IR2 where α˙ = ΩA . One obtains thus the generalized gravity T

force QGr,i = Fi [−mgT 0]Ti with Fi from eq.(4.66). The subsystem matrices read



m  0 Mn =  0 0

0 m mc 0

0 mc o CA 0





0 0   , Gn =  0  CM n

0 m mc 0



QGr,n

−m 0 0 0

−mc 0 0 0



0 0  ω , 0  oz,n 0 n



sin α  cos α  = −mn g  , c cos α  0 n

Qn = QGr,n + QM G,n

(4.77)

where m = mM + mA , mc = mA cA , ωoz = ΩA . The generalized forces QM G represent motor torque and gear elasticity (index M G: motor-gear). The drawback in using absolute

4.4

87

Subsystems

velocities follows from the fact that, in general, impressed forces need more effort in formulation. Here, the gear torque is proportional to (γA − γM ) and has obviously to take the kinematics of the preceding subsystem into account. The velocities of the actual reference frame and of the preceding one are interrelated by the “kinematic chain” eq.(3.79):





vox cos γA  voy   − sin γA  Ω = 0 A 0 ΩM n

|

{z

sin γA cos γA 0 0

}|

y˙ n

Lp sin γA Lp cos γA 1 1





0 vox 0 0   voy   0 + 0   ΩA   1 0 0 n ΩM p

{z

}|

Tnp

{z y˙ p

}|



0   γ˙ A 0  . 0  γ˙ M n iG n | {z } {z } s˙ n Fn

(4.78) Interpretation: Subsystem n is attached to its predecessor p at distance Lp on its x-coordinate. The absolute angular velocity of arm n is then ΩAp + γ˙ An . The absolute motor angular velocity reads ΩAp + iG γ˙ M n : the stator is fixed at arm p and moves with ΩAp . The rotor rotates with ψ˙ n = (∂ψn /∂γM n )γ˙ M n relative to ΩAp where (∂ψn /∂γM n ) = iGn (gear transmission ratio). The superimposed Fn s˙ n then represent the relative velocities contributed by subsystem n. The virtual work performed by motor and gear reads δWM Gn = δψn MM n −kn (γA −γM )n (δγA − δγM )n = δγM n [iG MM + k(γA − γM )]n − δγAn [k(γA − γM )]n (kn : spring coefficient). δ(γA γM )Tn = δsn is directly obtained from eq.(4.78)

Fn s˙ n = −Tnp y˙ p + y˙ n ⇒



δsTn

=

[δyTp

δyTn ]



T −(F+ n Tnp ) + T (Fn )



(4.79)



0 0 1 0 where = is the generalized inverse. Thus, if F+ n and Tnp 0 0 0 1/(iG ) n are not orthogonal, then the generalized motor-gear-force will react on the preceding subsystem for chosen coordinate representation. In the present case one obtains (F+ n Tnp ) =   0 0 1 0 , thus 0 0 1/(iG ) 0 F+ n

n



QM Gn,p

0  0 = 1 0



QM Gn,n

0  0 = 1 0



0   k(γA − γM ) 0  , 1/(iG )  −k(γA − γM ) − iG MM n 0 n



0   −k(γA − γM ) 0   0 k(γA − γM ) + iG MM n 1/(iG ) n

(4.80)

where QM Gn,p signifies that the motor-gear-torque of the nth subsystem reacts on its predecessor p while QM Gn,n represents the corresponding forces on the subsystem itself. This kind of “overlapping effects” can be avoided when using relative velocities.

4

88

4 Rigid Multibody Systems

4.2.2 Relative Velocities The reaction of forces from the i-th subsystem on its predecessor can be suppressed when using relative velocities. This goes along with a dimension amplification. 5 x, vox

v

ΩAr

γA

q

γM

ωF z y, voy

r

r

r

ΩM r

Example: Obviously, subsystem n is “guided” with ΩAp . In general, a system which undergoes a “guiding motion” which does not contribute to the actual velocity degrees of freedom is called a “guided system”. The corresponding “guiding velocities” are characterized by index F (German “F¨uhrung”). Using these for additional variables within y˙ n along with the relative velocities ΩM r , ΩAr (see Fig. 4.14) increases the dimension by one and leads to 1 0 0

0 1 0

0 0 1

0 0 0

0 0 1

1  0 = 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0  1  −1

" FA =



Figure 4.14.

Motor-Gear-Arm-Unit

FM

# ,



for chosen y˙ n = [vox voy ωF z ΩM r ΩAr ]Tn . Applying eq.(4.75) yields the subsystem matrices

   

Mn =

m 0 0 0 0

0 m mc 0 mc

0 mc o CA + CM CM o CA

0 mc o CA 0 o CA



sin α cos α c cos α 0 c cos α



0 0 CM CM 0

   

Qn = −mn g 



    , Gn =   n

0 m mc 0 mc

   + QM G,n , 

−m 0 0 0 0

αn =

−mc 0 0 0 0

n X

0 0 0 0 0

γAi

−mc 0 0 0 0

   ωoz,n ,  n

(4.81)

i=1

n

where m = mM + mA , mc = mA cA . The angular velocity is split into two parts: ωoz = ωF z + ΩAr . Eq.(4.78) then becomes

4.4

Subsystems



 

89

vox cos γA sin γA Lp sin γA  voy   − sin γA cos γA Lp cos γA    0 0 1  ωF z =  ΩM r   0 0 0 0 0 0 ΩAr n

|

{z

y˙ n

}|



vox voy ωF z ΩM r ΩAr

 

}|

{z

} |

0 Lp sin γA 0 Lp cos γA   0 1   0 0 0 0 n

{z

Tnp

y˙ p

0   0    + 0   iG 0 p



0  0  γ˙ M 0 . γ˙ A 0  | {z n} 1 n s˙

{z

Fn

n

}

(4.82) Here, F+ n Tnp = 0, thus

 QM Gn,p = 0,

  

QM Gn,n = 

0 0 0 1/(iG ) 0

0 0 0 0 1

    k(γA − γM ) + iG MM  .  −k(γA − γM )  n

(4.83)

n

4

4.2.3 Prismatic Joint/Revolute Joint – Spatial Motion In conclusion, two typical subsystems are presented: the revolute joint and the prisx, vox matic joint, both considered H H for spatial motion. The moHHv tor axis is e3 (reference fraΩAr me). To start with the prismaq γA tic joint, relative angular velox˙A γM cities are used for variables. The vector of describing vaωF z riables reads y, voy q   vo q ΩM r  ωF     y˙ n =   ΩM r  . (4.84)  ΩAr  x˙ A n Figure 4.15.

Prismatic Joint, Top View

The subsystem equation Mn ¨yn + Gn y˙ n − Qn = Qcn ∈ IR9

(4.85)

is equated with Mi , Gi from eqs.(4.57), (4.58) for i ∈ {A, M } and composed according to eq.(4.75). The JacobiansFi are obtained from

90

4 Rigid Multibody Systems

vo E  0  ωo      r˙ c  =  0 ω r An 0 





vo  ωo       r˙ c  =  ωr M n 





E 0 0 0

 v 0 0 0  o ω F  0 e3 0    ΩM r 0 0 e1    ΩAr 0 0 0 x˙ A 

0 E 0 0

    ,  

 v 0  o  ωF 0    ΩM r 0   ΩAr 0 x˙ A 

0 0 0 E 0 e3 0 0 0 0 e3 −e3

(4.86)

n

    .  

(4.87)

n

Subsystem matrices and generalized forces read   

Mn =  



T

meT1

    Gn =  

(mM + m)ω eo m [ω e oee1 ]T xA 0 e o e2 ]T xA −m [ω T

QM G,n

   =   

0

−m [ω e oee1 ] xA ω e o (JM + JoA ) − [ω e o e3 ]T JM − [ω e o e3 ]T JoA T

−m [ω e o e1 ]





(mM + m)E [me e1 ] xA 0 me2 xA me1 o e3 0  [me e1 ] xA (JM + JoA ) CM e3 CA  T 0 CM e3 CM 0 0  ,  o T o meT2 xA CA e3 0 CA 0 

−m [e e1 ω e o e1 ] xA

0 0 1 0

0 0 0 1

0 0 0 0

0

0

m

(4.88)

n

0 m [ω e o e2 ] xA 2m [ω e o e1 ] CM [ω e o e3 ] CAo [ω e o e3 ] 2m [ee1 ω e o e1 ] xA 0 0 0 0 0 2mωoz 0

−mωoz

0

    ,   n

(4.89)

   +k(γA − γM )/iG + MM     −k(γA − γM )  ,   F  M n

(4.90)

0 0 1



QTGr,n



1 0 0 0 0 0 0 0 1 0 xA 0 xA 0  = −mn gT  0 1 0 0 0 0 1 0 −xA 0 0 0 0 n

(4.91)

where: index A: arm, index M : motor, m = mA , ω o = ω F + ΩAr e3 , MM : motor torque (revolute), FM : motor force (prismatic), JoA = diag{AA , BA +

4.4

91

Subsystems

mx2A , CA + mx2A }, JM = diag{AM , AM , CM }, k: gear spring coefficient, iG : gear transmission ratio. Vector g ∈ IR3 represents gravity in reference frame components. The kinematic relationship between subsystem n and its predecessor p(n) reads, according to eq.(3.79), y˙ n = Tnp y˙ p + Fn s˙ n

(4.92)

where      Tnp y˙ p =    

Anp 0 0 Anp

 "



E erTpn 0 erTpn e3 e1 0 E 0 e3 0 0

0 0 0

 0

0

0 0 0

0 0 0

#



 vo   ω  F    Ω M r  0   ΩAr 0  

0

x˙ A

   ,   p

(4.93)     Fn s˙ n =    

0 0 iG 0

0 0 0 1

0 0 0 0

0

0 1

    γ˙ M      γ˙ A     x˙ A n 

(4.94)

n

xAp e1 , erTpn e3

= xAp e2 . The first two block rows in eq.(4.93) where p rpn = represent the guidance velocities h

rTpn (ω F + ΩAr e3 ) + r˙ pn p von = p vo + e

i p

,

pωF n

= p [ω F + ΩAr e3 ]p .

For interpretation, one may have a look at Fig. 4.15: The guidance velocities which enter the successor are those at location x = xA . These are eventually transformed into the link-fixed successor frame by means of Anp . In order to define y˙ n , Tnp is filled up with zeros in the lower part. The remaining velocities (ΩM r , ΩAr , x˙ A )n are then taken into account by Fn s˙ n . The subsystem equations of a revolute joint are simply obtained for xA = const. by canceling the last rows and columns as indicated in eqs.(4.88) to (4.94).

4.3.

Synthesis

In the last step, the total system is considered. Disposing of the subsystem dynamics, eq.(4.50) takes the form (with Nsub = N for abbreviation)

92

4 Rigid Multibody Systems  N  X ∂ y˙ T n

∂ s˙

n=1

[Mn ¨yn + Gn y˙ n − Qn ] = 0,

(4.95)

or, in matrix form

"

∂ y˙ 1

T 

∂ s˙

∂ y˙ 2

T

∂ s˙ {z

|

FT

 #   ∂ y˙ N T   ···   ∂ s˙ }

M1 ¨y1 + G1 y˙ 1 − Q1 M2 ¨y2 + G2 y˙ 2 − Q2 .. .

    

MN ¨yN + GN y˙ N − QN

(4.96) where s˙ represents the (global) vector of minimal velocities and F is the corresponding (global) functional matrix. It is obtained from the kinematic chain y˙ i = Tip y˙ p + Fi s˙ i

(4.97)

which connects the actual subsystem i with its predecessor p(i). Here, the first subsystem has the immobile inertial system for predecessor: y˙ 1 = F1 s˙ 1 . The second one then yields y˙ 2 = T21 y˙ 1 + F2 s˙ 2 = T21 F1 s˙ 1 + F2 s˙ 2 etc., leading in total to 

y˙ 1  y˙  2  ˙  y3  .  .  .   ..  . y˙ N





F1   T F F2   21 1     T31 F1 T32 F2 F3   .. .. .. = .   . .   . .   .. ..   FN −1 TN 1 F1 · · · · · · TN,N −1 FN −1 FN



s˙ 1   s˙  2  ˙   s3  .  .  .    ..  . s˙ N

       := F˙s    

where Tk,j = Tk,p(k) × Tp(k),p[p(k)] × · · · × Ts(j),j with p(k) as predecessor of k and s(j) as successor of j. Non-marked matrix elements are zero. Thus, choosing s˙ = [˙sT1 s˙ T2 · · · s˙ TN ]T for minimal velocities, F becomes a lower block triangular matrix, leading eq.(4.96) to            

FT1

· · · · · · · · · · · · FT1 TTN 1 .. .. . . T T T Fi · · · · · · Fi TN i .. .. . ··· . .. .. . . FTN

           

M1 ¨y1 + G1 y˙ 1 − Q1 .. . Mi ¨yi + Gi y˙ i − Qi .. . .. . MN ¨yN + GN y˙ N − QN

       = 0.     

(4.98)

4.4

93

Subsystems

4.3.1 Minimal Representation The corresponding equations of motion in minimal representation read M¨s + G˙s − Q = 0 ∈ IRg

where

M = FT blockdiag{M1 · · · MN }F, ˙ G = FT blockdiag{G1 · · · GN }F + FT blockdiag{M1 · · · MN }F, Q = FT [QT1 · · · QTN ]T . (4.99) 5 Example. A simple non-holonomic robot model may consist of a movable basis according to Fig. 4.8 on p.76 with a telescoping arm (plane motion). The non-holonomic constraint is vy = 0, thus

 y˙ 1 =

1 0

0 1



vx γ˙ 1

1y

A K

 = F1 s˙ 1 .

(4.100)

T21

cos γ2  − sin γ2 = 0 0

Aq A A

 

γ2

X XXs A  A  x 2 γ1 A  q * vx  A   q y X X A X Iy q  A   Aq 6

The arm (subsystem 2) is modeled by eqs.(4.69) et seq. once more consisting of one single body only. Thus, eq.(4.74) reduces to y˙ 2 = y˙ 2 = (vox voy ωoz x) ˙ T2 . The kinematic chain (4.97) is built with



 x

q

- Ix



a sin γ2 a cos γ2   1 0

Figure 4.16.

 and F2 s˙ 2 =

0 0

0 0

0 1

1 0

Simple Robot Model

T 

x˙ 2 γ˙ 2



(4.101)

which is easily verified with Fig. 4.16. Symbol a denotes the position of the arm hub in the car-fixed x-direction. From eq.(4.98) one obtains





1 0

      y˙ 1  =  y˙ 2 cos γ2     − sin γ2   0 0

0 1





0 0

0 0

 

         a sin γ2 0 0   a cos γ2   0 0       0 1 1 0

1

0



vx γ˙ 1  := F˙s. x˙ 2  γ˙ 2

(4.102)

94

4 Rigid Multibody Systems

The sub-matrices M1 , G1 , M2 , G2 are given with eqs.(4.68) to (4.71) (once more omitting overlining since the single bodies already represent the corresponding subsystems, F = E). For the minimal representation (4.99), one obtains the system matrices [with ω1 = γ˙ 1 , ω2 = (γ˙ 1 + γ˙ 2 )] s˙ T =

vx

γ˙ 1

x˙ 2

γ˙ 2




,

(4.103) 

m1 + m2

    −m2 x2 sin γ2  M=  m2 cos γ2    −m2 x2 sin γ2

−m2 x2 sin γ2



m2 cos γ2

−m2 x2 sin γ2

C1o + C2o + 2 m2 (a + 2ax2 cos γ2 )

m2 a sin γ2

 C2o +  m2 ax2 cos γ2 

m2 a sin γ2

m2

0

C2o + m2 ax2 cos γ2

0

C2o

  ,    

(4.104) 



 0    m2 x2 ω1 cos γ2 +   G =  m2 aω1 + m1 c1 ω1     m2 ω1 sin γ2  m2 x2 ω1 cos γ2

−m2 aω1 − m1 c1 ω1 −m2 x2 ω2 cos γ2 −2m2 ω2 sin γ2 −m2 x2 ω2 cos γ2 

 

−m2 ax2 γ˙ 2 sin γ2

 2m2 x2 ω2 +  2m2 aω2 cos γ2 −m2 ax2 ω2 sin γ2  , 

−m2 x2 ω2 −m2 aω1 cos γ2

0

−m2 x2 ω2

m2 ax2 ω1 sin γ2

2m2 x2 ω2

0

    

(4.105) 



d 1 . (4.106) R (MR + ML ) 2R (MR − ML ) farm Marm [Here, the car is assumed to be driven by two independent motors with corresponding torques MR , ML acting on the right and on the left rear wheel axle, respectively. The easiest way to determine the corresponding generalized forces is to look at the virtual work: Along with the wheel radius R, the wheel angular velocities β˙ R , β˙ L yield the translational mass center velocities of the right wheel, vR = Rβ˙ R = vx + (d/2) γ˙ 1 , and of the left wheel, vL = Rβ˙ L = vx − (d/2) γ˙ 1 , where d denotes the distance between the wheels. Setting vx := s˙ x (notice that vx is non-holonomic, see example page 76) leads 1 d (MR + ML )δsx + 2R (MR − ML )δγ1 ]. to the virtual work of the drive unit δWD = R QT =

4

4.4

95

Subsystems

4.3.2 Recursive Representation Although the procedure for subsystem synthesis is quite simple, the resulting equations are highly complex, of course. It depends on the aim of investigation if the minimal representation is needed or requested. If one is, in the next step, interested in numerical integration, then the minimal representation might not be suitable. This is mostly the case when high-degree-of-freedom-systems with tree topology are considered. In that case, it will be suitable to make use of the (block-)triangular form of the functional matrix in the sense of a Gaussian elimination algorithm. Obviously, the last line of eq.(4.98), FTN [MN ¨yN + GN y˙ N − QN ] = 0,

(4.107)

can directly be resolved for ¨sN with the aid of eq.(4.97). But ¨sN is then expressed in terms of the unknown ¨yp(N ) . The predecessor equation, however, is premultiplied with FTp as a whole and can, therefore, be arranged in the same structure as eq.(4.107). Resolving for ¨sp now yields an equation which contains the unknown ¨yp[p(N )] . Repeating this procedure till the first, or basic, subsystem, the predecessor of which is the inertial system, yields an equation which no longer has unknowns and which can thus be solved for ¨s1 . The equations for every calculation step read (see the page after next for calculation details) FTi [M∗i ¨yi + G∗i y˙ i − Q∗i ] = 0 eq.(4.97)

n



(4.108) 

o

T ¨si = −M−1 M∗i Tip ¨yp + T˙ ip y˙ p + F˙ i s˙ i + G∗i y˙ i − Q∗i Ri Fi (4.109) with matrices

⇒

def.

MRi = [FTi M∗i Fi ],

def.

Ni =

h

i

T E − M∗i (Fi M−1 Ri Fi )

M∗p = Mp + TTip Ni M∗i Tip (symmetric), ⇒ G∗p = Gp + TTip Ni (G∗i Tip + M∗i T˙ ip ), Q∗p = Qp + TTip Ni (Q∗i − M∗i F˙ i s˙ i − G∗i Fi s˙ i ). The problem is solved in three steps: 1. Determine Tip , T˙ ip , eq.(4.97), i = 1 · · · N upwards (for i = 1 ⇒ p = p(1): System 1 has no predecessor, hence T1p(1) = 0). 2. Determine M∗p , G∗p , Q∗p , eq.(4.111), p = N · · · 1 downwards (for p = N ⇒ i = s(N ): System N has no successor, hence Ts(N )N = 0).

(4.110)

(4.111)

96

4 Rigid Multibody Systems

3. Solve ¨si , eq.(4.109), i = 1 · · · N upwards (for i = 1 ⇒ p = p(1): System 1 has no predecessor, hence T1p(1) = 0). The only inversion needed is that one of the reduced mass matrix MRi in eq.(4.110) which has the dimension of the ith relative degree of freedom. Remarks: At a first glance the computation of matrix Ni in eq.(4.110) looks laborious. However, by nearer inspection Ni has an almost simple structure: decomposition of Fi into its zero and nonzero parts leads, in general, to 

Fi :=

0 C



, Rg[Fi ] = gi , Ci ∈ IRgi ,gi .

(4.112)

i

Decomposition of the actual M∗ in the same manner yields "

M∗i

=

M∗11 M∗12 T

M∗12 M∗22

#

, M∗22,i ∈ IRgi ,gi .

(4.113)

i

The reduced mass matrix and its inverse then read −1 ∗−1 T MRi = CTi M∗22,i Ci ⇒ M−1 Ri = Ci M22,i Ci

−1

.

(4.114)

Insertion into Ni from eq.(4.110) leads to "

Ni =

E −M∗12 M∗−1 22 0 0

#

.

(4.115)

i

On the other hand, M−1 Ri itself is needed for eq.(4.109). It is, however, not −1 really a drawback to first compute M∗−1 22,i and then to combine MRi according to eq.(4.114) because in general Ci is mostly diagonal. In case that several successors exist, then the matrices according to eq.(4.111) need summation, M∗p = Mp +

X

TTip Ni M∗i Tip ,

i∈S

G∗p

= Gp +

X

TTip Ni (G∗i Tip + M∗i T˙ ip ),

(4.116)

i∈S

Q∗p = Qp +

X

TTip Ni (Q∗i − M∗i F˙ i s˙ i − G∗i Fi s˙ i ),

i∈S

where S = {si (p)} denotes the set of successors (H. Gattringer, 2004), see e.g. eq.(3.69).

4.4

97

Subsystems

5 Proof sketch: Let N=3. Focusing on the acceleration terms yields



FT1







y1 + · · · M1 ¨ FT1 TT31 y2 + · · ·  = 0 FT2 TT32   M2 ¨ FT3 M3 ¨ y3 + · · ·

FT1 TT21 FT2

(4.117)

i=3: eq.(4.117) : FT3 [M3 ¨ y3 + · · ·] = 0 ¨ y3 = T32 ¨ y2 + F3¨s3 + · · ·  −1 T ⇒ ¨s3 = − FT3 M3 F3 F3 M3 T32 ¨ y2 + · · ·

(4.118)

i=2:

h

eq.(4.118) : ¨ y3 = E − F3 FT3 M3 F3 eq.(4.117) : FT2

FT2

n



M2 ¨ y2 +

TT32 M3 ¨ y3

M2 + TT32 E − M3 F3 FT3 M3 F3

Replace M2 +

=

i

FT3 M3 T32 ¨ y2 + · · ·

a.



b.

+ ··· = 0 =

h

TT32 N3 M3 T32


−1


−1

i

o

FT3 M3 T32 ¨ y2 ...

M∗2

c. d.

h

where N3 = E − M3 F3 FT3 M3 F3


−1

FT3

(4.119)

i

FT2 [M∗2 ¨ y2 + · · ·] = 0 ¨ y2 = T21 ¨ y1 + F2¨s2 + · · ·  −1 T ∗ ⇒ ¨s2 = − FT2 M∗2 F2 F2 M2 T21 ¨ y1 + · · ·

e.

i=1:

h

eq.(4.119) e. : ¨ y2 = E − F2 FT2 M∗2 F2 FT1



TT21


−1

i

FT2 M∗2 T21 ¨ y1 + · · ·

TT32 M3 ¨ y3




M1 ¨ M2 ¨ + ··· = 0 eq.(4.117) : y1 + y2 + where T31 = T32 T21  eq.(4.119) b. to d. : FT1 M1 ¨ y1 + TT21 M∗2 ¨ y2 + · · · = 0 = FT1

n

h

M1 + TT21 E − M∗2 F2 FT2 M∗2 F2

Replace M1 +

TT21 N2 M∗2 T21

h

=


−1

4

o

FT2 M∗2 T21 ¨ y1 ...

M∗1

where N2 = E − M∗2 F2 FT2 M∗2 F2 FT1 [M∗1 ¨ y1 + · · ·] = 0 ¨ y1 = F1¨s1 + F˙ 1 s˙ 1 ⇒ ¨s1

i

a.



b. c. d.


−1

FT2

i

e.

(4.120)

98

4 Rigid Multibody Systems

5

t

1y

 

t

KA Aq A

Example: Consider a “Multi-link Robot” with direct drive (no gears). The first link (i = 2) rotates relative to the car (i = 1) without relative translation. F1 and F2 are derived from eq.(4.100) and eq.(4.101) for x˙ 2 = 0. Using linkfixed reference frames for the following links (i > 2), the subsystem matrices Mi , Gi are known with eq.(4.66) and eq.(4.67). Joint i is attached at distance Lp to the x-axis of predecessor p, leading to

2x

t



γ2



A A

 t A  γ1 A t q * vx  A   y X XXt A  Iy 2y  A  A q  6

q

1x

" Tip =

cos γi sin γi Lp sin γi − sin γi cos γi Lp cos γi 0 0 1

#

- Ix Figure 4.17.

(4.121)

Multi-link Robot

while Fi s˙ i is simply e3 γ˙ i ( i > 2, plane motion). Thus, for any of the links, one obtains T ∗ M−1 Ri = e3 Mi e3



−1

=

1 M∗i (3, 3)

(4.122)

⇒ For the plane motion, only a scalar inversion is needed.

4 It does, in principle, not matter how many subsystems enter the calculation. For the last example one has g = 2 + nl with nl : number of links. Inserting motor-gear dynamics for the links according to eqs.(4.81)/(4.82) augments the number of degrees of freedom to g = 2(1 + nl ) and needs an almost trivial 2 × 2 matrix inversion. Compared with the “classical” procedure (minimal representation) which needs a g × g inversion demonstrates the fact that the recursive method is indeed best suited for “multiple link systems”. For the plane motion of a chain consider an inertially fixed basic frame (γ1 = −π/2 in Fig. 4.17) under the influence of gravity (acting in x1 -direction). The relative angular velocities γ˙ i are used for minimal velocities. The describing variables are y˙ i = (vox voy ωoz )Ti . Matrix Tip is derived from eq.(4.121), its time derivative is T˙ ip = γ˙ ieeT3 Tip [compare eq.(3.77)]. The subsystem matrices are given with eqs.(4.66)/(4.67). For mi = m, ci = c, Cio = C o ∀ i (identical chain links) the recursion is listed below in the form of a typical FORTRAN subroutine.

4.4

99

Subsystems

5 subroutine f(t,N,x,xdot)

for i = 1 · · · N

Tip , y˙ i = Tip y˙ p + e3 γ˙ i , αi =

i P

γk

k=1

Start: i = 1

Tip = 0

" for p = N · · · 1

M∗p =

" G∗p

=

" Q∗p

=

" Np = Start: p = N

m 0 0

0 m mc

0 m mc

−m 0 0

cos αp − sin αp −c sin αp 1 0 0

0 1 0

0 mc Co

# + TTip Ni M∗i Tip ,

−mc 0 0

# T y˙ p (3) + TTip Ni G∗i M∗i e e3 γ˙ i Tip ,





# mg + TTip Ni [Q∗i − G∗i e3 γ˙ i ] ,

−Mp∗ (1, 3)/Mp∗ (3, 3) −Mp∗ (2, 3)/Mp∗ (3, 3) 0

#

Tip = 0



for i = 1 · · · N

ODE: Start: i = 1

return END

4

vF i = Tip y˙ p : T aF i = Tip ¨ e3 Tip y˙ p yp + γ˙ ie γ¨i = − ∗ 1 eT [M∗i aF i + G∗i (vF i + Fi s˙ i ) − Q∗i ] Mi (3, 3) 3 ¨ yi = aF i + e3 γ¨i Let

xdot(i)=x(i+N) xdot(i+N) = γ¨i vF i = 0, aF i = 0

(4.123)

100

4 Rigid Multibody Systems

5.

Constraints

For a topological chain as well as for a topological tree, too, the problem is now completely characterized either in minimal or in recursive representation. All constraints are hereby considered implicitly.

5.1.

Inner Constraints

The implicit constraint forces can, if needed, simply be determined from the recursion scheme. Let, for brevity, subsystem p consist of a single body. Then, on the free cut body, Fp , Mp act on its origin as well as −Fi , −Mi on the coupling point (coming from the successor i, with negative sign due to counteraction). The latter need transformation into the actual coordinate system by Api = ATip , leading to the total action on body p, Fp Mp

!

Api Fi e rpi Api Fi + Api Mi

−

!

Fp Mp

=

Mp

−

p

i

−Fi

rpi −Mi

Figure 4.18.

       

FT1

i

Aip AiperTip 0 Aip

#T

Qp − TTip Qi .

or, in short:

Fp

! "

i

Counteraction i → p

FT1 TT21 FT1 TT31 · · · FT1 TTN 1 FT2 FT2 TT32 · · · FT2 TTN 2 FT3 · · · FT3 TTN 3 .. .. . . FTN

!

Fi , Mi (4.124)

(4.125)

Next, from all possible generalized cut forces (like additional drive torques etc.) only the constraint forces Q → Qλ are considered which do not accomplish work (Lagrange’s Principle). Thus, projecting these into the minimal configuration by eq.(4.96) along with F from eq.(4.98), the constraint forces cancel out. This leads to 

 λ Q1 − TT21 Qλ2    Qλ − TT Qλ  2 32 3  ..  .  

QλN

     

4.5

101

Constraints      =   

FT1 0 0 .. .

0 FT2 0

0 0 FT3

··· ··· ··· .. .

.. .

0



0 0 0 .. . FTN

 λ Q    1λ   Q2    ..  .   λ





    =   

QN

0 0 .. .

   . 

(4.126)

0

One has thus obtained the local orthogonality condition FTi Qλi = 0 which holds when relative velocities are used for (the components of the) minimal velocities. (Notice that this condition is not true for an arbitrary choice of minimal velocities). One obtains thus, from the free cut body dynamics Mp ¨yp + Gp y˙ p − Qep − Qλp + TTip Qλi = 0,

(4.127)

the equation for the determination of Qλi , by premultiplying eq.(4.127) with FTp : h

i

FTp Mp ¨yp + Gp y˙ p − Qep + TTip Qλi = FTp Qλp = 0.

(4.128)

Considering eq.(4.108) for body p, h

i

FTp M∗p ¨yp + G∗p y˙ p − Q∗p = 0,

(4.129)

one obtains, when matrices (4.111) are inserted, from direct comparison with eq.(4.128), h

FTp Mp ¨yp + Gp y˙ p − Qp n

+ TTip Ni (M∗i (Tip ¨yp + T˙ ip y˙ p + F˙ i s˙ i ) + G∗i y˙ i − Q∗i h

⇒ Qλi = Ni (M∗i (Tip ¨yp + T˙ ip y˙ p + F˙ i s˙ i ) + G∗i y˙ i − Q∗i = M∗i ¨yi + G∗i y˙ i − Q∗i

^

FTi Qλi = 0,

oi

=0

i

(4.130)

[the last expression is obtained by re-inserting ¨si according to eq.(4.109)]. Notice that in eq.(4.130) the ( )∗ -matrices enter the calculation. This means that the constraint force Qλi depends on the dynamics of all its successors. Its actual value can thus be computed as an additional output when using the recursive scheme (4.108) et seq.

5.2.

Additional Constraints

If additional constraints Φ = 0 ∈ IRm arise (such as e.g. closed kinematic loop conditions), then these are explicitly to be taken into account. Vector s˙ in

102

4 Rigid Multibody Systems

eq.(4.99) then no longer represents minimal velocities because its components ˙ The direction of the undergo restrictions; it shall therefore be replaced by y. ˙ y) ˙ T (compare Fig. 2.2); corresponding constraint forces is derived from (∂ Φ/∂ its value enters the calculation via Lagrange parameters λ. Along with g for actual velocity d.o.f., λ has the dimension m and y˙ has the dimension n = g+m. The describing (differential-algebraic) system of equations then reads

˙ ∂Φ

M¨y + Gy˙ − Q − Φ

=

!T

λ = 0 ∈ IRn ,

∂ y˙

(4.131)

0 !

˙ Φ

=

˙ ∂Φ y˙ = 0 ∂ y˙

=

˙ ∂Φ ¨y + d dt ∂ y˙

!

¨ Φ

         (b)          

(a)  

"

˙ ∂Φ ∂ y˙

!#

y˙ = 0

(c)

(4.132)

       and from (b) one obtains the orthogonality relation       !     ˙  ∂ y˙ ∂ Φ ⇒ = 0, (d)   

∂ y˙

∂ s˙

as long as scleronomic constraints are considered (this is the most general case, see section 2.3.3, p.19).

5.2.1 Jacobi Equation Equations (4.131) and (4.132 (c)) may be combined to      

˙ M − ∂Φ ∂ y˙ ! ˙ − ∂Φ 0 ∂ y˙

!T 

 ˙ (Q − Gy)     ¨y  !#  = "  λ ˙  d ∂Φ  y˙

dt

   . 

(4.133)

∂ y˙

This equation shall be called the “Jacobi Equation”. Obviously, Jacobi was the first who formulated it with his words “in order to derive λ one has to differentiate [the constraint equations] twice and to eventually substitute the second time derivatives of the coordinates from the equations of motion” (C.G.J. Jacobi, 1842), p.55. This “substitution” can also directly be done by solving eq.(4.133) with the aid of the corresponding inverse; using the abbreviation

4.5

103

Constraints 

˙ ∂Φ MR =  M−1 ∂ y˙ !

˙ ∂Φ ∂ y˙

!T   ∈ IRm,m

(4.134)

(index R for “reduced”) yields for the inverse of eq.(4.133) (left side) 



 M−1 E −     

˙ ∂Φ ∂ y˙

!T



!

M−1 R

˙ ∂Φ M−1  −M−1 ˙ ∂y

˙ ∂Φ ∂ y˙

!T



 M−1 R     

!

˙ ∂Φ M−1 ∂ y˙

−M−1 R

−M−1 R

(4.135) and thus ¨ ˙ y = M−1 (Q − Gy) − M

−1

˙ ∂Φ

!T 

∂ y˙

"

M−1 R

˙ ∂Φ ∂ y˙

!

"

−1

M

d ˙ + (Q − Gy) dt

˙ ∂Φ ∂ y˙

!# # 

y˙

.

(4.136) The braced term represents the (negative) Lagrangean multiplier which is easily verified with eq.(4.133) and eq.(4.135): "

λ=−

M−1 R

˙ ∂Φ ∂ y˙

!

# −1

M

˙ − (Q − Gy)

"

˙ d ∂Φ ˙ . y) dt ∂ y˙ !

M−1 R

#

(4.137)

5.2.2 Minimal Representation The orthogonality relation (4.132 (d)) may be used to eliminate λ in ˙ s˙ )T , E} eq.(4.133): multiplying eq.(4.133) from the left with blockdiag{(∂ y/∂ ˙ s˙ )˙s yields and inserting y˙ = (∂ y/∂    T ∂ y˙ M   ∂ s˙  !   ˙  − ∂Φ

∂ y˙

   T ∂ y˙ ˙ (Q − Gy)       ∂ s˙  ∂ y˙ ˙ d ∂ y   ¨s + s˙ =  " !#    ∂ s˙ dt ˙ ∂ s ˙   d ∂Φ y˙ 

dt

    .  

(4.138)

∂ y˙

Here, the lower (block-) row, " # " # ˙  d ∂ y˙  ˙ ∂ y˙ ˙ ∂ y˙ ˙ ∂ y˙ ∂Φ d ∂Φ ∂Φ d ∂Φ ¨s − s˙ = − s˙ = 0, − s˙ − dt ∂ y˙ ∂ s˙ dt ∂ y˙ ∂ s˙ ∂ y˙ ∂ s˙ ∂ y˙ dt ∂ s˙ (4.139)

104

4 Rigid Multibody Systems

is identically zero due to the orthogonality condition (4.132(d)), and the first row yields the minimal form "

¨s =

∂ y˙ ∂ s˙

T



M

∂ y˙

#−1 

∂ s˙

∂ y˙

T 



Q−G

∂ s˙

∂ y˙



∂ s˙

d ∂ y˙ s˙ − M s˙ . dt ∂ s˙ (4.140) 

 

In eq.(4.140), constraint forces no longer arise because the motion completely takes place within the tangent plane of constraints. On the other hand, it is often suitable to use an overdetermined number of variables accompanied by the corresponding constraint forces. This is for instance when dealing with structurally variant systems where the constraints only act during finite time, leading to differing numbers of d.o.f. A typical representative here is the contact problem: If contacts are not durably closed but may also reopen under the actual dynamical circumstances.

5.2.3 Recursive Representation When using eq.(4.140) directly, the recursive formulation gets lost. The general structure of the differential-algebraic equations (4.131) to (4.132) leads, of course, to a broad variety of possible numerical algorithms also for kinematic loops in minimal representation. Here, we refer to the special literature [e.g. (M. Hiller, 1997)]. In the present context, the main emphasis is laid on the use of “overdetermined” coordinates along with a constraint stabilization. Remember that the use of relaxed constraints does not reduce the equations of motion since then the corresponding forces belong to the work-performing (impressed) forces. Therefore, y˙ in eq.(4.131) is once more denoted s˙ ∈ IRg where g indicates the actual number of (velocity) d.o.f., while y˙ i (indicated by index) characterizes the ith subsystem. The subsystem describing variables are connected to the minimal velocities by the kinematic chain (4.97) the time derivative of which reads

       

¨y1 ¨y2 ¨y3 .. . ¨yN





       =       



E T21 T31 .. .

E T32

TN 1

···

E ..

. ··· ··· E

      

F1¨s1 F2¨s2 F3¨s3 .. . FN ¨sN

     + ···   

4.5

105

Constraints     ···   

0 T˙ 21 0 ˙ ˙ 0 T32 T21 T32 .. .. . . ˙ ˙ ˙ TN 2 T21 TN 3 T32 · · · TN,N −1 0

       

y˙ 1 y˙ 2 y˙ 3 .. . y˙ N

       

(4.141) where Fi is assumed constant15 . Eq.(4.141) allows us to recursively calculate λ. In the “classical” procedures, eq.(4.131) is used to eliminate ¨y in eq.(4.132 (c)) to come out with λ according to eq.(4.137). However, as eq.(4.137) shows, then the inverse of the total mass matrix is needed. In the present case, however, ¨ y appears as a function of ¨si , i = 1 · · · N , where ¨si is already known from eq.(4.109) in terms of its predecessor ¨yp . Hereby, the forces transform according to eq.(4.111), Q∗p = TTip Ni Q∗i := T∗ip T Q∗i ,

(4.142)

which leads to 

       

Q∗1λ Q∗2λ Q∗3λ .. . Q∗N λ





       =       

E T∗21T E

T∗31T

· · · T∗NT1

T∗32T

· · · T∗NT2

E

· · · T∗NT3 .. .. . . E

                  

!T 

˙ ∂Φ ∂ y˙ 1 !T ˙ ∂Φ ∂ y˙ 2 !T ˙ ∂Φ ∂ y˙ 3 .. . ˙ ∂Φ ∂ y˙ N

!T

          λ.         

(4.143)

˙ y˙ ) · · · (∂ Φ/∂ ˙ y˙ )]T λ In eq.(4.143), Q∗kλ summarizes the action of [(∂ Φ/∂ N 1 upon the kth subsystem. ˙ depends on all y˙ , i = Eq.(4.143) represents the most general case where Φ i 1 · · · N . However, usually the constraints arise in only one of the subsystems (generally the last one, e.g. for the contact problem or for closing a chain). Eq.(4.132 (c)) then becomes 15 For

the last summation term the relation Fi s˙ i = y˙ i − Tip y˙ p (4.97) has been used. Having in mind Tij = Tik Tkj , i < j < k, leads to T˙ ij − T˙ ik Tkj = Tik T˙ kj .

106

4 Rigid Multibody Systems

˙ ∂Φ

¨ = Φ

!

∂ y˙ N

˙ ∂Φ

"

d ¨yN + dt

!#

∂ y˙ N

y˙ N = 0,

(4.144)

where ¨yN reads, according to eq.(4.141),

¨yN

T 

T T y˙ F1¨s1 TTN 1 T˙ 21 TTN 2    .1   ..    . .   .. = .   .. +   ..  , FN ¨sN y˙ N E 0



 







(4.145)

and eq.(4.143) simplifies to        

Q∗1λ Q∗2λ Q∗3λ .. . Q∗N λ





      =      

T∗NT1 T∗NT2 T∗NT3 .. .

       

˙ ∂Φ ∂ y˙ N

!T

λ.

(4.146)

E

Using eq.(4.145) for eq.(4.144) along with ¨si from eq.(4.109) leads to a recursive procedure which needs a dimλ × dimλ matrix inversion instead of g × g: Inserting successively ¨si and ¨yp from the recursion procedure (4.109), one obtains A λ + b = 0,

(4.147)

where A=

"

d b= dt and

˙ ∂Φ ∂ y˙ N

˙ ∂Φ

!"

∂ y˙ N

!#

y˙ N +

˙ ∂Φ ∂ y˙ N

N X

# T T∗N,i Vi T∗N,i

i=1

!

N X

˙ ∂Φ ∂ y˙ N

h

!T

,

(4.148)

i

T∗N,i Vi (Q∗i − G∗i y˙ i ) + NTi+1 T˙ i+1,i y˙ i ,

i=1

T∗N,N := E, T∗N +1,N := 0, T T∗ip = NTi Tip , Vi = Fi M−1 Ri Fi .

(4.149) (4.150)

One has thus found a simple procedure to obtain λ recursively, using the known results from the recursive procedure (4.109). Notice that the only inversion needed is A−1 which has the dimension of λ (e.g., λ ∈ IR1 for the plane contact problem).

4.5

107

Constraints

Remarks: In eqs.(4.148) et seq. a numbering scheme (1, 2, 3 · · · N ) has been applied. For the general case (tree topology), the numberingP scheme according to eq.(3.67) should suitably be used, along with M∗p = Mp + i∈S TTip Ni M∗i Tip etc. where S = {si (p)} denotes the set of successors [see eq.(4.116), p.96]. The same as in eq.(4.98), T∗ij abbreviates T∗ij = T∗i,p(i) × · · · × T∗s(j),j (p: predecessor, s: successor). This term, however, is obtained recursively, without calculating T∗ij as a whole, when running the recursion the first time in order to determine λ. Once λ is known, the recursive calculation is started where the ˙ y˙ )T λ. The additional computer impressed force Q∗N is completed with (∂ Φ/∂ N time, compared to the unconstrained case, requires less than fifty percent (H. Gattringer, 2005).

5.2.4 Constraint Stabilization It is remarkable that the constraint [(4.132)(a)] does not enter the Jacobi equation (4.133) anywhere but only the corresponding acceleration [(4.132)(c)]. The same holds, of course, for eq.(4.147). The fulfillment of the constraint is thus by no means warranted and has to, if need be, forced by additional considerations. To this purpose, the constraints have to be monitored in any case. Once they are violated, one may correct the actual variables numerically using Newton’s iteration scheme applied to the corresponding residuals, [e.g. (P. Betsch, 2004); here, once more, the full range minimal representation is needed]. A quasi-analytical approach is to fulfill the constraints Φ = 0, at least asymptotically, with the use of an asymptotically stable differential equation ¨ + DΦ ˙ + KΦ = 0 ⇒ Φ → 0. Φ

(4.151)

Here, D and K are positive definite matrices [Baumgarte stabilization (J. Baumgarte, 1972), (G.P. Ostermeyer, 1990)]. One obtains thus for (4.132 (c)) −

˙ ∂Φ

!

∂ y˙

"

d ¨y = dt

˙ ∂Φ

!#

˙ + KΦ, y˙ + DΦ

∂ y˙

(4.152)

which leads eq.(4.133) to      

˙ M − ∂Φ ∂ y˙ ! ˙ − ∂Φ 0 ∂ y˙

!T 

     ¨  y = "  λ  d 

dt

˙ (Q − Gy) !

  # . 

˙ ∂Φ ˙ + KΦ y˙ + DΦ ∂ y˙

(4.153) d ˙ y)] ˙ + KΦ in ˙ y˙ with DΦ (∂ Φ/∂ One has therefore just to complete [ dt eq.(4.136) and in eq.(4.149), respectively. It has, however, to be emphasized

108

4 Rigid Multibody Systems

that it is not allowed to set Φ to zero because now Φ represents a variable. That means, from the mechanical point of view, that the constraints are now compliant and the corresponding forces perform work (“principle of relaxation of the constraints”, see also Chapter 5). Obviously, the additional terms which enter eq.(4.152) will influence the system dynamics: Insertion into the equations of motion yields M¨y + Gy˙ − Q = ˙ y) ˙ T (λo + λstab ) where λo corresponds to eq.(4.137) which represents (∂ Φ/∂ ˙ the non-stabilized case and λstab is −M−1 R (DΦ + KΦ). The latter thus not only stabilizes the constraints but simultaneously affects motion. The most critical case is the undamped system. Here, the action of the (generalized) stabilization forces on the system, and the reaction of the system on them, increases each others growth over time since there is no hindering mechanism. Such a behavior was the reason that Baumgarte’s stabilization has often been criticized, e.g. (A. Eiber, W. Schiehlen, 1980). However, when (pervasive) damping is included (which almost always takes place in real mechanical structures), then Baumgarte’s stabilization technique yields reliable results in almost every case. As a conclusion, one may state that Baumgarte’s stabilization technique is a powerful tool for the treatment of closed kinematic chains. It has, however, to be considered carefully. Numerical test calculations are unavoidable.

4.6

109

Segmentation: Elastic Body Representation

6.

Segmentation: Elastic Body Representation

The procedures we have outlined seem to offer a cheap method to handle elastic bodies, too. The Finite Segmentation Method to which we now relate our previous discussion, cuts the elastic body into finite segments which are considered rigid, e.g. (R.L. Huston, 1990). However, the question arises of how many segments will be needed for such a representation. We will discuss this question as a representative example.

6.1.

Chain and Thread (Plane Motion)

t

0

-0.2

-0.4 0 9

-0.6

6 8

5

4

-0.8 7

-1 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 4.19. Stroboscopic View of Chain Oscillations (Bold: Chain Moves to the Right; Thin: Chain Moves to the Left, Dotted: Initial Position)

Fig. 4.19 shows a stroboscopic view of the oscillation of a chain with N = 300 links (recursive numerical integration according to eq.(4.123) along with a 5th-order Runge-Kutta-Fehlberg scheme). As in the preceding example, chain length and chain mass are normalized (Ltot = 1, mtot = 1). The curve numbering n denotes the corresponding observation time tn = n∆t/10 where ∆t = 0.3 s is the integration interval. The initial velocity is zero, the initial

110

4 Rigid Multibody Systems

position (n = 0) is γ1 = 1/N, γ2 · · · γN = 2/N which approaches a circular arc from π to π + 2 with center at (0.5, 0). Repeating integration with N = 500 yields similar behavior. Fig. 4.20 represents a magnification of the (dotted) window in Fig. 4.19 for N = 300. However, comparing with the results for 500 links (Fig. 4.21) indicates slight differences. Therefore, after a long time period, the solutions will diverge significantly16 . One may thus conclude, that, if the chain is used to represent a continuous body, convergence is not yet attained. -0.5

-0.5

-0.55

-0.55

-0.6

-0.6

-0.65

-0.65

-0.7

-0.7

-0.75

-0.75

-0.8 -0.65

-0.6

-0.55

Figure 4.20. Chain, N=300

-0.5

-0.45

-0.8 -0.65

-0.6

-0.55

-0.5

-0.45

Figure 4.21. Chain, N=500

The benefit of the linear [order(N)-] recursion is obvious when compared to the “classical” procedure (calculation of the mass matrix at least ∝ N 2 , inversion ∝ N 3 )17 . However, the question arises whether it will be suitable to augment the link number trying to reach convergence. This question is, from the mechanical viewpoint, ill posed for two reasons: Firstly, one has to emphasize that a chain without any damping in the joints does not exist. Including damping

16 As known from chaos theory, any N -fold pendulum (N ≥ 2) is extremely sensitive to initial conditions in certain energy regions. However, the present investigation is harmless w.r.t. chaos because it takes place at low energy. The end points for the two cases initially differ by 0.076 % and the energy constant by 0.085 %. These are negligible quantities in the present case. 17 Although referring to engineering practice, this argumentation is a bit one-eyed: usually, numerical inversion makes use of a LU-decomposition. In the present case, the mass matrix is already given in an UL-(block-) representation, see eq.(4.99). It is thus not suitable to first determine the mass matrix and eventually to decompose it instead of taking advantage of its special structure.

4.6

111

Segmentation: Elastic Body Representation

will surely lead to convergence for limited N . However, when comparing to an elastic body: how does the damping mechanism look, where to get the corresponding coefficients from? Material damping is an open question (although not very important in the present case as will be seen later). The second question is more important and considers once more the conservative (undamped) case.

6.2.

Chain, Thread, and Beam

The continuous body which is to be represented by a chain without any restoring torques at the joints can only be an ideal thread. Such an ideal thread, however, does not really exist. A “torque-free” thread may be used in a corresponding dynamical environment, for instance in ring spinning machines (textile industry) where the restoring torques play a minor role compared to the centrifugal effects. Under normal circumstances, however, the restoring forces are not negligible, as a simple experiment with common household yarn shows (Fig. 4.22). 1.5 3k

1 0.5 0 -0.5

2k

-1 -1.5

1k

-2 -2.5 0 Figure 4.22. The Yarn Experiment

0.2

0.4

0.6

0.8

Figure 4.23. Calculated Eigenfunctions

1

112

4 Rigid Multibody Systems

The above chain dynamics has therefore rather to be compared to a beam than to a torque-free thread. Such a comparison will, in a certain range, answer the question on the requested number of links. Let, for simplicity, the link masses ∆mi = (m/L)∆L be concentrated at the joints (m: total mass, L: total length, ∆Li = ∆L: identical links). Assuming small motion amplitudes, one obtains for the ith mass center position,

6 x 6

γ3 γ2 x  γ1   x

6

6

w2



 h



w1

Figure 4.24.

w3

-

Chain Kinematics

wi = wi−1 + ∆L

i X

γk

(4.154)

k=1

leading to    

w=

w1 w2 .. .





     = ∆L   

wf

1 0 0 ··· 2 1 0 ··· 3 2 1 ··· .. .. .. . . . . . .

    

γ1 γ2 .. .

    := ∆LTq 

(4.155)

γf

(f : degree of freedom = number of chain links ⇒ ∆L = L/f ). Let furthermore the potential be   1 k V = qT q (4.156) 2 ∆L (k: spring coefficient). Then, the equations of motion read ∆m

∂w ∂q

!T



¨+ w

h i k kL ¨+ q = 0 ⇒ TT T q ∆L m 



1 ∆L

4

q = 0. (4.157)

Solving the eigenvalue Problem for the dimensionless case (kL/m) = 1, "

TT T λ2i +



1 ∆L

4 #

E qi = 0

(4.158)

(qi : ith eigenvector) yields the “Eigenfunctions” weigen,i = ∆LTqi

(4.159)

in form of discrete values as depicted in Fig. 4.23 (solid line) for f = N = 100. For comparison, the eigenfunctions of a clamped-free beam are inserted (dotted

4.7

113

Conclusion

line. For calculation see Table 5.2, p.140). The first eigenfunction does not show perceptible deviations. These start growing with the second eigenfunction, although still in a tolerable range. The calculated chain eigenvalues differ from the exact beam eigenvalues by about 1%, see Table 4.2.

Table 4.2. Numerical Results for a Clamped-free Chain

CHAIN.dat Libraries: Boundary conditions: Number of links: det(M): k(i)L, i=1,5 1.85658694326660 4.64772552974182 7.77652549596182 10.8842263372849 13.9906059162482

7.

eigenvalue Problem: EISPAC Matrix Inversion: NUMREC clamped-free N=100 1.000000000 Deviation (percent) 0.9875E+00 0.9877E+00 0.9960E+00 0.1012E+01 0.1037E+01

Conclusion

The main emphasis is, in this Chapter, laid on a subsystem representation in order to break down the required mathematical effort into several subsequent steps which are easy to handle: The single rigid body is considered as the basic element in rigid body dynamics. Its mass center velocities are calculated with the (intermediate) functional matrix F leading to the single body motion equations with suita˙ bly chosen auxiliary variables   yi : T

F =  ∂vc ∂ y˙

!T

∂ω c ∂ y˙

!T

e c  ⇒ Mi ¨ yi + Gi y˙ i − Qi = Qi , [eq.(4.56)].

The single bodies are gathered in a subsystem with the aid of functional matrix Fi leading to the subsystem dynamics along with the describing variables y˙ n! : ∂yi y˙ n := Fi y˙ n ⇒ Mn ¨yn + Gn y˙ n − Qen = Qcn , [eq.(4.75)]. y˙ i = ∂yn (Clearly, if the subsystem consists of one single body, then (∂yi /∂yn ) = E ⇒ y˙ i = y˙ n , Mi = Mn etc.)

114

4 Rigid Multibody Systems

The subsystems are eventually combined into the total system by means of the total (or global) functional matrix F: h

FT (M1 ¨y1 +G1 y˙ 1 −Qe1 )T · · · (MN ¨yN +GN y˙ N −QeN )T

iT

, [eq.(4.98)].

The global functional matrix is hereby obtained from the kinematic chain y˙ i = Tip y˙ p + Fi s˙ i [eq.(4.97)] with the local functional matrices Fi . Using relative velocities avoids “overlapping effects” between the subsystems. h

iT

Using s˙ = s˙ T1 · · · s˙ TN for global minimal velocity yields F as a lower block-triagonal [eq.(4.98)]. One has thus the access to either –

a minimal representation [eq.(4.99)], or

–

a recursive representation [eq.(4.109) et seq.]

of the motion equations. A recursive representation is recommended for high degree of freedom interconnected systems (“tree topology”, see page 50). Considering additional constraints (section 4.5, p.100), a calculation scheme with an “overdetermined” number of variables along with a relaxation of the constraints is determined. It goes along with a Baumgarte stabilization (which has carefully to be controlled). When simultaneously accurate eigenfrequencies and eigenforms are requested up to a certain order, then the rigid body segmentation cannot really be recommended for elastic body representation. As the chain example shows, one would have to deal with about one hundred rigid body d.o.f. for just one bending direction. It should be mentioned that the recommended order-N-recursion is, in principle, not new. The probably first attempt is due to (A.F. Vereshchagin, 1974) as had been pointed out by Schwertassek [see e.g. (R. Schwertassek, O. Wallrapp, 1999)] when the corresponding contributions by (H. Brandl, R. Johanni, M. Otter, 1986) came out. These papers look at single rigid bodies for subsystems and pursue the idea to eliminate the (generalized) constraint forces step by step, starting with the last body, along with successively applying Euler’s cut principle. Clearly, this procedure leads to quite a big amount of equations. An extension to elastic bodies, as for instance done in (H. Bremer, F. Pfeiffer, 1992), results thus in an extreme effort which is susceptible to subtle errors. Compared with this, the procedure outlined here needs only a few lines. This is due to the Projection Equation along with the use of non-holonomic describing variables which yields a clear structured representation of the basic dynamics.

Chapter 5 ELASTIC MULTIBODY SYSTEMS – THE PARTIAL DIFFERENTIAL EQUATIONS 1.

Elastic Potential

In case of an elastic body the former constraint forces λ in eq.(4.16) convert into impressed forces σ that depend “on the deformation of the violated constraint via some constitutive equation” [((J.G. Papastavridis, 2002)], referred to as the Lagrangean Principle of Relaxation of the constraints [(G. Hamel, 1917)]. Assuming a potential Vel to exist, eq.(4.15) becomes c

e

δW → δW = −δVel = −

Z

σ T δ dVol ,

(5.1)

(Vol : volume) where σ ∈ IR6 is called the “stress vector”. It contains the six independent elements of the symmetric stress tensor σ ∈ IR3,3 . Remark on symmetry. Rearranging eq.(4.16) yields  ∂  ∂x     e T  df  T T = (P ◦ λ) =  0   dVol   

0



=

∂ ∂ ∂ ∂x ∂y ∂z



0

0

∂ ∂y

0

0

∂ ∂z





∂ ∂y ∂ ∂x 0



0 ∂ ∂z ∂ ∂y





∂ λ1  λ  T ∂z    2      λ3  0      λ4      ∂   λ5  λ6 ∂x T

λ 1 λ4 λ 6 σxx σxy σxz ◦  λ4 λ2 λ5  := ∇T  σyx σyy σyz  . λ 6 λ5 λ 3 σzx σzy σzz (5.2)

Because P has the dimension “one per length”, λi obtains the dimension “force per area” (pressure, or stress). Obviously, σij = σji ∀ i, j ∈ {x, y, z}: the stress tensor is symmetric. However, 115

116

5 Elastic Multibody Systems – The Partial Differential Equations

symmetry is not self-evident. The common way to derive the stress z tensor is to firstly calculate the 6 ∂σ dz forces which act upon a volume zz σzz + ∂z 2 dxdydz (if considered cartesian) in terms of “stress times corre∂σzy dy sponding area”, e.g. for the zσzy + ∂y 2 direction r {[σzz + (∂σzz /∂z)dz/2]− y [σzz − (∂σzz /∂z)dz/2]}dxdy+ ∂σzy dy σzy − ∂y 2 {[σzy + (∂σzy /∂y)dy/2]− [σzy − (∂σzy /∂y)dy/2]}dxdz+ {[σzx + (∂σzx /∂x)dx/2]− σzz − ∂σzz dz ∂z 2 [σzx − (∂σzx /∂x)dx/2]}dydz Figure 5.1. Stresses in z-direction (last term not depicted Fig. 5.1.) The corresponding Rvirtual work then reads P [(∂σ ij /∂xj )δui ]dVol +δWB i,j RP 1 [compare eq.(4.16)] . Integration parts yields − i,j [σij δ(∂ui /∂xj )]dVol . R by P Assuming σij = σji leads to − [σ δ ]dV where ij = [(∂ui /∂xj ) + ol i,j ij ij (∂uj /∂xi )]/2. This is identical to eq.(5.1) . σij = σji means, from the mechanical point of view, to presume the element torque-free, see Fig. 5.1. Such a presumption, however, has to be considered axiomatic (known as Boltzmann’s2 axiom). The background of why to obtain directly a symmetric tensor by eq.(5.1) is characterized with the remark “no spin” on p.60: no spin also means that the element has been considered torque-free already from the very beginning. It also answers the question of how to come out with two axioms for the Projection Equation (4.34) starting with one axiom only: the second, namely the symmetry of the stress tensor, has implicitly been considered (see also the remarks on p.71). In the sequel we assume the stress tensor symmetric and proceed with σ ∈ IR6 .

1.1.

Linear Elasticity

When relaxing the constraints, the corresponding impressed force σ “mainly depends on the afore forbidden motion” [(G. Hamel, 1949), p.74]. These had been defined by  = 0 for the rigid body (see p.63). The most natural approach is therefore to assume σ directly proportional to the strain  itself. One may thus set σ = H in eq.(5.1) (where H = HT = const: Hooke matrix, linear elasticity) to come out with 1σ

ij is defined here as dfi /dAj with dfi : force in i-direction; dAj : area, characterized by its normal, thus leading to δui for corresponding motion direction. The sequence i, j is not unified in literature. 2 Ludwig Boltzmann, *1844 in Vienna, †1906 in Dniew

5.1

117

Elastic Potential

1 δVel =  H δ dVol ⇒ Vel = T H dVol , 2  where H reads (1 − ν) ν ν 0 0  ν (1 − ν) ν 0 0   E ν ν (1 − ν) 0 0  H=  1  0 0 0 − ν 0 (1 + ν)(1 − 2ν)  2  0 12 − ν 0 0 0 0 0 0 0 0 Z

 Z

T



(5.3) 

0 0   0   , 0   0  1 2 −ν (5.4) [e.g. (W. Schiehlen, 1987)]. Here, ν denotes Poisson’s3 ratio and E is Young’s4 modulus. This seemingly complicated expression becomes an easy interpretation when considering  in terms of σ. Then, 

 = H−1 σ, H−1

1  −ν  1   −ν =  0 E   0 0

−ν 1 −ν 0 0 0



−ν 0 0 0  −ν 0 0 0   1 0 0 0    0 2(1 + ν) 0 0   0 0 2(1 + ν) 0 0 0 0 2(1 + ν)

(5.5)

shows that a uniaxial stress σxx e1 diminishes the cross sectional area by (2) = (3) = −νσxx (like in a rubber band, for instance) where E overtakes the role of a spring coefficient as in eq.(4.24).

1.2.

Inner Constraints, Classification of Elastic Bodies

Remember that the strain  = P ◦ u in eq.(4.13) is related to the deflection u of an arbitrary point of the considered continuum. On the other hand, one is of course interested in the motion of a predefined “hyperplane” like the middle surface of a plate or the middle axis of a beam. Plate and beam, in this sense, are modeled with suitable assumptions. Such assumptions are, for instance, that the cross sectional area of a beam shall remain undeformed during motion (“Bernoulli-hypothesis”)5 or that the “height” of a plate shall not undergo deformations (“Kirchhoff-hypothesis”)6 . The corresponding coordinates are collected in rp ∈ IR3 . When represented in a local element-fixed coordinate system, the components of rp do not change values during motion: rp is invariant. In the undeformed state, rp may be filled up with the missing coordinates (collected in x) to obtain the triple x, y, z which is needed to characterize any 3 Simon-Denis

Poisson, *1781 in Pithiviers, †1840 in Paris Young, *1173 in Milverton, †1829 in London 5 Jakob I Bernoulli, *1655 in Basel, †1705 in Basel 6 Gustav Robert Kirchhoff, *1824 in K¨ onigsberg, †1887 in Berlin 4 Thomas

118

5 Elastic Multibody Systems – The Partial Differential Equations

arbitrary point inside the body: (rp + x)T = (x y z). Since rp is invariant, vector x hereby defines the independent spatial variables. Example: considering a plate which is extended in the x-y-plane, the z-coordinate is invariant during motion, while x, y (and time t, of course) are independent variables.

Table 5.1. Classification Rigid Body

Beam

x=

x= T

Plate x= T

(0, 0, 0) rp =

T

x= T

(x, 0, 0)

(x, y, 0)

(x, y, z)T

rp =

rp =

rp =

T

(x, y, z)

(0, y, z)

rp + x =

rp + x =

T

Continuum

T

(0, 0, 0)T

(0, 0, z)

rp + x =

rp + x =

(x, y, z)

(x, y, z)

(x, y, z)

(x, y, z)T

invariant:

invariant:

invariant:

invariant:

dV = dxdydz

dA = dydz

dh = dz

“Rigid Body

“Bernoulli-

“Kirchhoff-

Hypothesis”

Hypothesis”

Hypothesis”

m=

m=

m=

RRR

[

T

R

ρ dV ]

T

ZZ [

ρ dA]dx

|

{z

RR

—

m=

Z [

RRR

ρ dh]dxdy

[ρ] dxdydz

|

}

|

dm = ρA dx

c

RRR

[

c

J =

T ρrp rp dV

ee

]

R

ZZ [

|

ee {z

dJc

}

RR

Z [

|

}

Jc =

J =

T ρrp rp dA]dx

{z

dm = ρ dxdydz

}

dm = ρh dxdy

c

J =

{z

T

ρe rpe rp dh]dxdy

{z

dJc

}

RRR

T

[ρe rpe rp ]dxdydz

|

{z

dJc ≡ 0

}

For the plate, z being undeformed enters the calculation as an “inner constraint”. Consequently, no such inner constraints exist for a pure continuum while the rigid body is totally constrained such that the directions of a deformed state at any point of the body coincide with the directions of the body-fixed coordinate system. From this follows that the tensor of moments of inertia is equal to zero

5.1

119

Elastic Potential

for the continuum (vanishing components of rp ) and non-zero for any other solid. These considerations lead to the following statement: An infinitesimal volume consists basically of three infinitesimal spatial directions and needs, for eq.(5.1), corresponding integration. If, however, invariant components rp exist, then part of the spatial integration can be carried out in advance. We will, nevertheless, retain the notation dm and dJc without making a difference on how many infinitesimal spatial directions these depend, in order not to overload notation. Let u(x, t) ∈ IR3 denote the elastic deflection of the neutral hyperplane. The mass center position is then actually rc = x + u. Along with the “bending angle” ϕ(x, t), the deflection u of an arbitrary point reads e ⇒ u = u + ϕr e p. ϕ = ∇u

1.3.

(5.6)

Disk and Plate

Along with x = (x y 0)T , rp = (0 0 z)T (see Table 5.1) and u = (u v w)T one obtains 

0

0

   ϕ=   

0

0

−∂ ∂y

+∂ ∂x

+∂ ∂y −∂ ∂x 0



    ∂w   u u z  ∂x                v  ⇒ u =  v  −  ∂w  .     z       ∂y   

w

w

0

(5.7) Insertion into eq.(4.13) yields the strain vector             =          

xx yy zz xy yz zx





                      =                    

∂ux ∂x ∂uy ∂y ∂uz ∂z ∂ux + ∂uy ∂y ∂x ∂uy + ∂uz ∂z ∂y ∂uz + ∂ux ∂x ∂z



 ∂u − z ∂ 2 w     ∂x ∂x2     ∂v − z ∂ 2 w     ∂y ∂y 2         0   = 2   ∂v   + ∂u − 2z ∂ w   ∂x ∂y ∂x∂y       − ∂w + ∂w = 0     ∂y ∂y    ∂w ∂w

∂x

−

∂x

=0

           .          

(5.8)

120

5 Elastic Multibody Systems – The Partial Differential Equations

Here, the components 3, 5 and 6 are zero, which leads to the reduced representation



red

    =    

∂u − z ∂ 2 w ∂x ∂x2 ∂v − z ∂ 2 w ∂y ∂y 2 ∂v + ∂u − 2z ∂ 2 w ∂x ∂y ∂x∂y

     .    

(5.9)

From eq.(5.5) it follows with eq.(5.8) that σyz = 0, σzx = 0. Assuming rather small values for σzz , the 3rd, 5th and 6th row and column in eq.(5.5) may be canceled which leads to the reduced Hooke-matrix

1 −ν 0 1 ν 0 1 E   ⇒ Hred = . 0 ν 1 0 =  −ν 1 2 E 1 − ν 0 0 2(1 + ν) 0 0 (1 − ν)/2 (5.10) 

H−1 red







From this, one obtains for eq.(5.3),

1 1 dVel , dVel = κT1 [C dK] κ1 + κT2 2 2  1 Eh3 Eh  ν , D = , dK = (1 − ν 2 ) 12(1 − ν 2 ) 0

Z

Vel = C =

"

κT1

=

∂u ∂v 1 ∂x ∂y 2

∂v ∂x

+

∂u ∂y

!#

"

,

κT2

=

[D dK] κ2 , 

ν 0  dxdy, 1 0 0 2(1 − ν)

∂2w ∂2w ∂2w ∂x2 ∂y 2 ∂x∂y

#

(5.11)

where h is the plate height. Here, the frame origin of the undeformed state is centered at h/2 leading to the fact that all integral terms which contain z linearly vanish. Vector κT = (κT1 κT2 ) is called (the vector of) curvature(s). As eq.(5.11) shows, the in-plane motions u(x, y, t), v(x, y, t) are decoupled from the out-of-plane motion w(x, y, t). Considering the latter separately leads to the Kirchhoff-Plate while the former are referred to as disk motion.

5.1

121

Elastic Potential

1.4.

Beam

The deformation of a beam may be considered as a specification of the two-dimensional plate to a uniaxial structure which goes along with the assumption of undeformable cross sectional area (BernoulliHypothesis). The bending e u from angle ϕ = ∇ eq.(5.7) then reads

     ϕ=   

0

+∂ ∂y −∂ ∂x

0

0

0

+∂ ∂y

+∂ ∂x

0

z, w 6 ! ∂w(x, y)

r

-

y, v

Figure 5.2.



 



u            v  =        w

y := ϑ(x)y

∂y

Torsion of a Beam

∂w ∂y − ∂w ∂x ∂u + ∂v ∂y ∂x





ϑ(x)

      ∂w(x)    :=  −   ∂x     ∂v(x)

     .   

∂x (5.12)

This is because, due to plane and undeformed cross sectional areas, ! (∂u/∂y) = 0 and (∂w/∂y) = const. for every x as Fig. 5.2 indicates. Partial derivation (∂ /∂x) is abbreviated ( )0 in the sequel. Along with rp = (0 y z)T (see Table 5.1) one obtains for eq.(5.6), 

u





0

      0   u= v  + v   

w

w0

−v 0 −w0 0 ϑ



0





         −ϑ   y  =   

0

z

Insertion into eq.(4.13) yields the strain vector

u − w0 z − v 0 y v − ϑz w + ϑy

    . (5.13)  

122

5 Elastic Multibody Systems – The Partial Differential Equations 

           =          

xx yy zz xy yz





                      =                    

zx

∂ux ∂x ∂uy ∂y ∂uz ∂z ∂ux + ∂uy ∂y ∂x ∂uy ∂u + z ∂z ∂y ∂uz + ∂ux ∂x ∂z

   u0 − w00 z − v 00 y        0           0   =     −v 0 + v 0 − ϑ0 z = −ϑ0 z         −ϑ + ϑ = 0      w0 + ϑ0 y − w0 = ϑ0 y

         .        

(5.14)

Components 2,3 and 5 are zero, which leads to the reduced representation 

red

u0 − w00 z − v 00 y

   =   

−ϑ0 z ϑ0 y



    1 0   =  0 −z   0 y 

u0



−y −z   ϑ0  0 0   v 00  0 0 

w00

     .   

(5.15)

Here, the vector of curvatures is κ := [u0 ϑ0 v 00 w00 ]T . From eq.(5.5) it follows with eq.(5.14) that σyz = 0. Assuming rather small values for σyy , σzz , the 2nd, 3rd and 5th row and column in eq.(5.5) may be canceled which leads to the reduced Hooke-matrix 

Hred



E 0 0  = 0 G 0  0 0 G

(5.16)

where G = E/[2(1 + ν)] (“Technical Beam Theory”). One obtains thus for eq.(5.3)

Vel =

1 2

Z L

0 −Ey −Ez    0 0  G(y 2 + z 2 )  dA κ dx 2 Eyz   0 Ey   −Ez 0 Eyz Ez 2 (5.17)

  E   Z   0 κT  −Ey    A





5.2

123

Kinetic Energy

where L: beam R R length, A: Rcross sectional area. For a main axes system where ydA = 0, A A zdA = 0, A yzdA = 0 all non-diagonal elements in eq.(5.17) vanish which leads to 1 dVel , dVel = κT [dK] κ, 2 dK = diag{EA, GID , EIz , EIy }dx, Z

Vel =

κT

u0 ϑ0 v 00 w00



=



(5.18)

where EA is called the longitudinal stiffness, GID the torsional stiffness, EI the bending stiffness. Here, Ii , i = y, z, represent the area moments of inertia. In case of a circular cross sectional area, ID corresponds to the polar moment R 2 of inertia Ip = A (y + z 2 )dA. Numerical values for other cross sectional areas (respecting the effects of warping) are tabulated in the special literature.

2.

Kinetic Energy

The kinetic energy of an elastic body is obtained from eq.(4.20) by a limiting process: The deformable body consists of an infinite number of infinitesimal elements with assigned dm and dJc according to Table 5.1. These are connected to each other by the corresponding inner constraints (compare Fig. 5.3 with Fig. 4.9). Thus, the summation in eq.(4.20) passes over to an integration and yields

˙ ϕ

r

rc ωo o vo

Z

T =

dT = B

1 2

Z 

vTc ω Tc

  dmE

0

0 dJc



vc ωc



B

(5.19)

Figure 5.3.

Elastic Body, e.g. Beam

(B: body) when using the element mass center for reference point. Same as in the rigid body case, the mass center velocities are easily expressed in terms of intermediate (or auxiliary) variables y˙ which are composed of the reference frame velocities and the superimposed relative velocities. However, in case of

124

5 Elastic Multibody Systems – The Partial Differential Equations

an elastic body, these have to be replenished with the curvatures in order to later ˙ T κ˙ T )T . take the potential forces into account: y˙ = (vTo ω To r˙ Tc ϕ In the rigid body case, the intermediate variables lead via y˙ i = (∂yi /∂yn )y˙ n := Fi y˙ n to Fi which projects the motion equations of body i into subsystem n in terms of describing variables y˙ n . Regarding an infinitesimal element dmi out of an elastic body i yields y˙ i = Fi y˙ i (index i) when the deformation functions are used for description. This is because these already consider the inner constraints. The Jacobian F (index suppressed) is hereby chosen to reduce the dimension according to y˙ where y˙ ∈ IRn denotes the actually needed variables out of y˙ ∈ IRn . The columns of F ∈ IRn,n , n ≥ n, then T consist of zeros with only one element being 1, leading to F F = E ∈ IRn,n . Starting with the mass center velocities of an element dm, (vTc ω Tc ) ∈ IR6 , one obtains 

vc ωc

!

=

e o rc + r˙ c vo + ω ˙ ωo + ϕ

! "

=

E erTc 0 E

#    

E 0 0 0 E 0

vo ωo r˙ c ˙ ϕ κ˙

    := Fy, ˙  

T˙ y˙ = Fy˙ ⇔ y˙ = F y, F = blockdiag{Frb , Fel }.

(5.20)

The elastic deflection amplitudes are assumed small. Therefore, the relative ˙ (time derivative of the “bending angle”) in angular velocity ω r is denoted ϕ order to characterize all those velocities which belong to the elastic deformations as holonomic. The guidance motion vo , ω o remains non-holonomic in general. Along with dM = blockdiag{dmE, dJc } one obtains the kinetic energy Z

2T = B

(vTc ω Tc )dM(vTc ω Tc )T =

Z B



T

T



y˙ T F F dM F F

y˙ :=

Z

˙ y˙ T dM y.

B

(5.21)

3. Checking Procedures 3.1. Hamilton’s Principle and the Analytical Methods One of the most popular procedures in the field of elastic body vibrations is the use of Hamilton’s Principle to derive the equations of motion, especially because it automatically yields the corresponding boundary conditions. We are therefore going to check this method first. Because y˙ is non-holonomic in part, an elementary integration cannot be carried out directly. Let the corresponding position variables be z. One has then y˙ = H(z)˙z as a linear combination of the time derivatives of z in the same form

5.3

125

Checking Procedures

as for a minimal representation. Notice, however, that the representation here refers to the describing variables which do not characterize a minimal space. Consider for brevity the conservative case. Non-conservative forces can later easily be inserted. One has then ˙ V = V (κ) : T = T (z, y),

Zt1

[δT − δV ] dt = 0.

(5.22)

to

Mathematical calculation needs three subsequent steps: 1. Variation, 2. Integration by parts w.r.t. time, 3. Integration by parts w.r.t. space. Along with 

∂T ∂z





∂T ∂z

!



!

∂z δy := ∂y !

∂T δy, δz = ∂y !     ∂V δκ = ∂V ∂κ δy := ∂V δy ∂κ ∂κ ∂y ∂y one obtains for the first step Zt1 "

∂T

!

δy +

∂y

to

!

∂T

δ y˙ −

∂ y˙

∂V

!

#

δy dt = 0.

(5.23)

∂y

The next step cannot directly be carried out since y˙ is non-holonomic in part, yielding δdy 6= dδy. This will be the reason for the frequent statement that Hamilton’s Principle is restricted to holonomic systems. Such an argumentation, however, does not hold because extending with dδy/dt − dδy/dt enables us to integrate by parts w.r.t. time, yielding Zt1 " to

∂T ∂y

!

d δy − dt

∂T ∂ y˙

!

δy −

∂T

!

∂ y˙

dδy − δdy − dt 

∂V

!

#

δy dt = 0,

∂y (5.24)

or, according to eq.(4.39) and eq.(4.40), " X n

∂T d ∂T − + dt ∂ y˙ n ∂yn

∂V ∂yn

!

+ ···

126

5 Elastic Multibody Systems – The Partial Differential Equations

···

X ∂T ν,µ

∂ y˙ ν

y˙ µ

X ∂zk ∂zi i,k

∂yµ ∂yn

∂ 2 yν ∂zi ∂zk

−

∂ 2 yν ∂zk ∂zi

!  δyn = 0 (5.25)

(Hamel), or "( X k,n

∂T d ∂T − + dt ∂ z˙k ∂zk

∂V ∂zk

!)

∂zk ∂yn

!#

δyn = 0

(5.26)

(Maggi). Remarks. 1. It has to be emphasized that the above relations represent virtual work. Because bending angle ϕ and curvature κ are composed of the spatial derivatives of rc (inner constraints), the variations δyn are not arbitrary. Consequently, their coefficients do not yet represent the equations of motion. 2. If non-holonomic constraints take place it is not allowed to set these to zero in advance but only after carrying out calculations, see Chapter 4.3.3. One could therefore think of leaving non-holonomic constraints aside and concentrating on minimal coordinates. However, as in rigid multibody systems, non-holonomic describing variables will lead to easier procedures. 3. Why should one use, up to here (step 1 and 2), Hamilton’s Principle instead of directly proceeding with Maggi’s or with Hamel’s equations? The answer is not easy to find. Even in the holonomic case, the results of the first two calculation steps are known in advance from the corresponding Eulerequations of variational calculus ( = Lagrange’s equation of the second kind). Nevertheless, the use of Hamilton’s Principle is most popular. The answer to this question is probably: it has grown into a habit. 4. For the present problem, y˙ and z are local quantities which are assigned to dT and dV , respectively [see eq.(5.11), R Req.(5.18), eq.(5.19)]. Therefore, the energies are to be replaced with dT, dV , followed by an integration by parts w.r.t. the spatial coordinates. This leads to the third step of calculation: A sufficient number of corresponding integrations eventually leads to a representation which contains only the deflection variations. If these are arbitrary, their coefficients represent the equations of motion. 5. Such a conclusion, however, may be problematic when using analytical methods (i.e. those methods which are listed above the Central Equation in Table 4.1), as will be demonstrated by the following example.

5.3

127

Checking Procedures

5 Example: The Elastic Rotor An “Elastic Rotor” is represented by a beam which rotates with angular velocity Ω = α˙ about the longitudinal axis (x-axis). Its mass center vector rTc = [x v(x, t) w(x, t)]T is defined z as a vector function w.r.t. the undefor6 med state (see Fig. 5.4). For the calculation of the kinetic energy we choose *y the following reference coordinate sy  stems: The undeformed frame will be  used to calculate the translation. Consi dering the mass center of a beam ele-x ment (beam slice), there is no need to α˙ calculate its translational velocity in the local element fixed frame. The latter, however, is advantageously used for the rotational kinetic energy because there the corresponding tensor of moments of inFigure 5.4. The Elastic Rotor ertia has constant elements. The “mass matrix” dM = blockdiag{dmE, dJc } in eq.(5.21) may be rewritten dM = c blockdiag{(dm/dVol )E, (dJc /dVol )}dVol where dm = ρdVol ⇒ (dm/dV R ol )R = ρ, dJ R R= T T c erperp dm ⇒ (dJ /dVol ) = ρerperp . The integration needed for eq.(5.21) is B → V = L A ol (A: cross sectional area, L: beam length) and leads eq.(5.19) to

T

1 2

Z

=

Z

=

1 2

vTc ω Tc




L

Z 

ρE 0

A


T

vTc



ωc

ρAE 0



0 T rpe rp ρe 0 ρI

 dA



vc ωc

vc ωc

 dx



(5.27)

dx

L

where ρI = ρdiag{Ix , Iy , Iz }: Ii : area moments of inertia, ρ: mass density. ˙ 1 yields the mass center velocity w.r.t. the undeThe “guidance angular velocity” ω o = αe formed reference frame R: R vc = r˙ c + α˙ e e1 rc . Neglecting torsion one obtains the vector of e c = [0 − w0 v 0 ]T . Except α, bending angles ϕ = ∇r ˙ all position and velocity variables are assumed small. If one chooses a Cardan sequence of angles according to eq.(3.21) one obtains, along with 0 0 the sequence [(α) → (ϕ  y = −w ) →(ϕz = v )], the angular velocity w.r.t. the element fixed ˙ + αe frame E: E ω c = Aϕz Aϕy e1 e2 e3 (ϕ ˙ 1 ). The velocities read explicitly





 

0  ˙ , ω c =  vc =  v˙ − αw  w˙ + αv ˙

1−

v0 2 2

−

−v 0 −w0

w0 2 2



v0



1− 0

0

 02

v 2



  α˙    −w˙ 0  (5.28) 0  0 1

v˙

128

5 Elastic Multibody Systems – The Partial Differential Equations

(see also Table 3.2, page 48, for angular velocity). The elastic potential is derived from eq.(5.18). The variation of T and V are thus

Z δT =

vTc dmδvc

+

ω Tc dJc δω c




ZL

EIz v 00 δv 00 + EIy w00 δw00 dx




, δV =

(5.29)

o

leading to

 Zt1 ZL   δ α[ρI ˙ x α] ˙

Zt1 (δT − δV )dt

=

to

to

o

+

δv [ρAα( ˙ w˙ + αv)] ˙ + δ v˙ [ρA(v˙ − αw)] ˙

+

δw [ρAα(− ˙ v˙ + αw)] ˙ + δ w˙ [ρA(w˙ + αv)] ˙

+

δv 0 ρ(Iy − Ix )α˙ 2 v 0 + ρ(Iy − Ix )α˙ w˙ 0 + δ v˙ 0 −ρIz (w0 α˙ − v˙ 0 )

+

δw0 ρ(Iz − Ix )α˙ 2 w0 − ρIz α˙ v˙ 0 + δ w˙ 0 −ρ(Ix − Iy )v 0 α˙ + ρIz w˙ 0

















# 00

−

00

00

δv EIz v − δw EIy w

00



(5.30)

dx dt.

The underlined terms in eq.(5.30) are next integrated by parts w.r.t. time. The resulting variations at the time boundaries are zero [see eq.(2.40)]. After integration w.r.t. time, the time integral itself is therefore not of interest. One obtains

ZL δT − δV

{δα[−ρIx α] ¨

= o

+

δvρA [α( ˙ w˙ + αv) ˙ − v¨ + α˙ w˙ + αw] ¨

+

δwρA [α(− ˙ v˙ + αw) ˙ −w ¨ − α˙ v˙ − αv)] ¨

+

¨ − v¨0 ) δv0 ρ(Iy −Ix )α˙ 2 v 0 + ρ(Iy −Ix )α˙ w˙ 0 + ρIz (w˙ 0 α˙ + w0 α

+

δw0 ρ(Iz −Ix )α˙ 2 w0 − ρIz α˙ v˙ 0 + ρ(Ix −Iy )v˙ 0 α˙ + ρ(Ix −Iy )v 0 α ¨ − ρIz w ¨0

−

δv 00 EIz v 00 − δw00 EIy w00 dx.











(5.31)

We next characterize those terms which depend on the first and on the second spatial derivative of the variations by once and twice underlining,

ZL δT − δV

{δα[−ρIx α] ¨

= o

+

v + 2α˙ w˙ + αw ¨ + α˙ 2 v) δv ρA(−¨

+

δw ρA(−w ¨ − 2α˙ v˙ − αv ¨ + α˙ 2 w)

+

δv 0 −ρIz v¨0 + ρ(Iz +Iy −Ix )α˙ w˙ 0 + ρ(Iy −Ix )α˙ 2 v 0 + ρIz w0 α ¨

+

δw0 −ρIy w ¨ 0 − ρ(Iz +Iy −Ix )α˙ v˙ 0 + ρ(Iz −Ix )α˙ 2 w0 + ρ(Ix − Iy )v 0 α ¨

−

δv 00 EIz v 00 − δw00 EIy w00 dx.













o

 

(5.32)

5.3

129

Checking Procedures

The underlined terms in eq.(5.32) are integrated by parts, and the twice underlined terms are twice integrated by parts, resp., yielding

ZL  δα[

−ρIx α ¨

o

+

]

n

δv

ρA(−¨ v + 2α˙ w˙ + αw ¨ + α˙ 2 v)

¨ +ρIz v¨00 − ρ(Iz +Iy −Ix )α˙ w˙ 00 − ρ(Iy −Ix )α˙ 2 v 00 − ρIz w00 α − EIz v 00 +

n

δw


00

ρA(−w ¨ − 2α˙ v˙ − αv ¨ − α˙ 2 w)

¨ 00 + ρ(Iz +Iy −Ix )α˙ v˙ 00 − ρ(Iz −Ix )α˙ 2 w00 − ρ(Ix −Iy )v 00 α ¨ +ρIy w − EIy w00

 +





dx

−ρIz v¨0 + ρ(Iz +Iy −Ix )α˙ w˙ 0 + ρ(Iy −Ix )α˙ 2 v 0 + ρIz w0 α ¨


0 

−ρIy w ¨ 0 − ρ(Iz +Iy −Ix )α˙ v˙ 0 + ρ(Iz −Ix )α˙ 2 w0 + ρ(Ix −Iy )v 0 α) ¨



δw

+ EIy w00 +



δv 0

+



0

δw





δv

+ EIz v 00 +


00 o

−EIz v 00 −EIz w


0 




00




L

(5.33)

= 0. o

Usually one derives the partial differential equations, along with the corresponding boundary conditions, directly from eq.(5.33). In the present case, however, one would obtain a rather strange result. This becomes clearer when considering a circular cross sectional area with Ix = 2Iy = 2Iz = 2I. In this case, eq.(5.33) would yield the partial differential equations

h h

ρA(¨ v − 2α˙ w− ˙ αw ¨ − α˙ 2 v) − ρI v¨00 − ρI α˙ 2 v 00 +ρIw00 α ¨ + EIv 00


00 i

ρA(w ¨ + 2α˙ v+ ˙ αv ¨ − α˙ 2 w) − ρI w ¨ 00 − ρI α˙ 2 w00 +ρIv 00 α ¨ + EIy w

= 0,


i 00 00

=0

(5.34) along with the boundary conditions

h

ρI v¨0 + ρI α˙ 2 v 0 − ρIw0 α ¨ − EIz v 00


0 iL

=0

_

v |L o = 0,

o

h

ρI w ¨ 0 − ρI α˙ 2 w0 − ρIv 0 α ¨ − EIy w00


0 iL

=0

_

w |L o = 0,

o

EIv 00



L

EIw00



o

L o

=0

_

v 0 |L o = 0,

=0

_

w0 |L o = 0.

(5.35)

130

5 Elastic Multibody Systems – The Partial Differential Equations

The drive equation, Jx α ¨ = Mx , is decoupled from the remainder and can be considered separately. The circulatory (or nonconservative positional) forces are underlined in eq.(5.34). The translational part is, as expected, skew symmetric (once underlined). The rotational part, however, is symmetric (twice underlined). Changing the Cardan sequence [(α) → (ϕz = v 0 ) → (ϕy = −w0 )] yields the same result for the rotational circulatory forces but with negative sign. The reason for this strange result goes along with partial linearization: The elastic deformation amplitudes are assumed small, but the rigid body motion (here: α) ˙ is not. Assuming small elastic deflections, the virtual work according to eq.(5.33) is correct, but concluding eq.(5.34) from eq.(5.33) is not. The reason lies in the choice of the transformation matrix from the undeformed reference to the elementx 6 fixed coordinate system which is needed for the quadratic representation of ρIx α ¨ kinetic energy. Hereby, any transformation which is built by a sequence of elementary transformations yields a nonorthogonal frame for the corresponding rotation axes (see e.g. Fig. 3.7, p.43). The analytical methods (Hamilton, Lagrange, Maggi, Hamel ...), however, make use of directional derivatives of the kinetic energy especially w.r.t. these N axes. Now, considering an orthonormal 0 z frame which is defined for example by w w˙ 0 , −y the (rotation) axes −w˙ 0 , v˙ 0 and the corv˙ 0 responding normal, the longitudinal momentum ρIx α ¨ has a component in the Figure 5.5. Cardan Sequence δv 0 -direction. Taking care of −ρIx αw ¨ 0 δv 0 in eq.(5.32) leads to ρ(Iz − Ix )αw ¨ 00 δv in eq.(5.33) and thus, for the circular cross section, to the correct result EIv 0000 + ρA(¨ v − 2α˙ w˙ − α˙ 2 v − αw) ¨ − ρI(¨ v 00 + α˙ 2 v 00 )−ρI αw ¨ 00 = 0, EIw0000 + ρA(w ¨ + 2α˙ v˙ − α˙ 2 w + αv) ¨ − ρI(w ¨ 00 + α˙ 2 w00 )+ρI αv ¨ 00 = 0

(5.36) It is thus necessary, at least when using successive elementary transformations, to look for suitable intermediate transformations. 4

Conclusions The amplitudes of elastic deformations are considered small. Using analytical methods, however, the energies are to be calculated up to the second order to obtain linear equations. Analytical methods therefore need much effort as vividly demonstrated by eqs. (5.30) – (5.33) . Considering beam systems which undergo rigid body motions, the latter are, in general, not small. Linearization here can therefore only be carried out partially. Besides the effort which has to be made, the direct use of the Lagrangean procedure yields (for the present example) correct results for the

5.3

131

Checking Procedures

virtual work [eqs.(5.29) to (5.32), holonomic coordinates in the present example]. However, depending on the choice of transformations, (at least part of) the virtual angular displacements belong to a non-orthogonal coordinate system and the corresponding moments are thus not independent. Using holonomic variables seemingly simplifies the Lagrangean procedures when compared to eq.(5.25) or eq.(5.26) but reveals them to be practically useless when non-small rigid body motions take place. Choosing, on the other hand, eq.(5.25) or eq.(5.26) with orthogonal components within δy needs intolerable effort. Result: As far as the equations of motion are required, the direct use of analytical methods should, from the practical point of view, not be chosen for elastic multibody systems. Even the frequently cited advantage to automatically obtain the corresponding boundary conditions, under closer inspection, does not hold. One is thus once more left with the Projection Equation.

3.2.

Projection Equation

The equations of motion of the nth subsystem which consists of Nn rigid bodies is derived from eq.(4.51), see Fig. 4.9, p.78. In comparison, an elastic body consists of an infinite number of infinitesimal P small rigid bodies see Fig. 5.3, R p.123. For the limiting case, one obtains, with → , Nn

Z  Bn

∂ y˙ n

!T  !  ∂v T ∂ y˙ c  ∂ y˙ n  ∂ y˙

T

∂ s˙

!T  " ∂ω c 

∂ y˙

Bn

#

 e o dp−dfe −dfc ˙ ω dp+ ˙ ω e o dL−dMe −dMc  dL+

{z

|

[dM¨y + dGy˙ − dQe − dQc ]n

}

(5.37) where Bn denotesPthe nth elastic body. Eq.(5.37) corresponds to eq.(4.75) for n the limiting case N i=1 → Bn where, in the present example, F is constant. The orthogonal frame of the undeformed state is used for a reference coordinate system, as in Fig. 5.3 for instance. The vector of describing coordinates y˙ is derived from eq.(5.20). The calculation follows the same procedure as in eq.(4.53) et seq. and yields dmE dmerTc  dme rc dJo   

T  T dM = F  dmE dmerc   0 dJc 

0

0



dmE 0 0 dmerc dJc 0   dmE



0

0

dJ

0

0

0   F, c

 0  

0

(5.38)

132

5 Elastic Multibody Systems – The Partial Differential Equations eo e oe eo dmω dmω rTc 2dmω 0 0 c c   o c ˙ ˙ eo ω e o dJ + dJ 2dme eo ω e o dJ + dJ 0   dme rc ω rc ω 



  e dG = F   dmω o   0  T

0

e oe dmω rTc

eo 2dmω

0

e o dJ + dJ˙ ω

0

e o dJ + dJ˙ ω

0

0

0

c

c

c

c

  0   F,  0  

0 (5.39)

 

E

 e  rc T F =  E  0

0

0 E 0 E 0

   T  ⇒ dQe = FT F  

e

df dMe

!

   T  +F    

0 0 0 0  T − ∂dVel ∂κ

     .   

(5.40) ˙ T κ˙ T ) For F = E, these matrices are related to y˙ → y˙ = (vTo ω To r˙ Tc ϕ yielding the same element matrices as in eq.(4.56) except that they refer to the infinitesimal element (dm, dJc ) and that they are completed with a zero column (and row) to take the curvatures into account. Due to the special choice of the intermediate variables one obtains almost simple element matrices. On the other hand, the problem is shifted to the determination of the functional matrix (∂ y˙ n /∂ s˙ )T in eq.(5.37). As the example “elastic rotor” on p.127 shows, one is of course interested in a representation in deformation functions. The describing variables, however, also contain their spatial derivatives. As a consequence, the functional matrix (∂ y˙ n /∂ s˙ ) can not be calculated directly. This problem, however, is solved with the use of the corresponding virtual work: N Z X

δyTn [dM¨y + dGy˙ − dQe − dQc ]n = 0.

(5.41)

n=1B n

The problem of how to calculate the functional matrix is then shifted to the determination of δyn in terms of δs. The describing variables may be characterized by a spatial operator D applied to the minimal velocities s˙ . Let for simplicity N =1: y˙ = D ◦ s˙ (◦: “applied to”). One obtains then from eq.(5.41)

(5.42)

5.4 Z

133

Single Elastic Body – Small Motion Amplitudes T

e

δy [dM¨y + dGy˙ − dQ ] =

B

Z

T

c

Z

δy dQ = B

h

iT

δ D◦s

dQc = 0. (5.43)

B

The spatial derivatives within operator D are related to those components of s˙ which represent (the time derivatives of) the deformation functions. Because these are holonomic, d and δ may be interchanged according to eq.(2.30) which allows for an integration by parts w.r.t. space. One obtains thus Z

h

iT

δ D◦s

c

dQ =

B

Z h B

iT

D ◦ δs

c

Z

dQ =

h

i

δsT DT ◦ dQc + δWbound = 0

B

(5.44) with a new operator D which follows from integration. At first glance, this procedure seems cumbersome. By nearer inspection, however, it turns out to be extremely simple: The operator D is the same as the initial one except that all odd derivations change their sign. Simultaneously, one obtains the operators which determine the (dynamical) boundary conditions by successive degeneration of the derivation grade with once more changing sign. This reflects the properties of continuous integrations by parts with its sign changing in the integrand and the boundary terms. The relation (5.44) will be verified in the following section.

4. Single Elastic Body – Small Motion Amplitudes 4.1. Beams The general vibrations of a beam are characterized by u0 x+u ϑ  0  e c + ϑe1 =  −w0 , κ :=  ϑ . rc =  v , ϕ = ∇r  v 00  w v0 w00 











(5.45)

Here, u(x, t) represents longitudinal stretching, v(x, t), w(x, t) are the bending functions in y and z direction, and ϑ(x, t) is the torsion according to eq.(5.12). All velocity vectors in eq.(5.20) are represented in the orthogonal reference frame R which corresponds to the undeformed state. The reference system itself moves with vo , ω o . Accordinghto Table 5.1, ithe tensor of moments of inertia of a beam slice RR reads dJc = ρerperTp dA dx. Its components are constant when calculated in a local element-fixed coordinate system E. Alongh with ρ = ρ(x), the mass i RR T c e density ρ can be factored out yielding E dJ = ρ rperp dA dx = ρIdx

134

5 Elastic Multibody Systems – The Partial Differential Equations

where I: tensor of area moments of inertia [see also eq.(5.27)]. Since calculations are carried out in the reference system R, E dJc has to be transformed into a representation in R. Because all motion amplitudes are assumed small, the e see eq.(3.30). One (first-order) transformation matrix reads AEI = [E − ϕ], T T o c c obtains thus R dJ = R dJ + ercerc dm where R dJ = AER E dJ AER : Ix (Ix − Iy )v 0 (Ix − Iz )w0 c 0  (Ix − Iy )v Iy (Iy − Iz )ϑ  dx. R dJ = ρ 0 (Ix − Iz )w (Iy − Iz )ϑ Iz 



Left index R is suppressed for simplicity in the sequel. For the Euler-Bernoulli beam, the motion equations for u, v, w, ϑ are decoupled from each other and can thus be calculated separately. Therefore, at first we choose bending in z direction as a representative example. If guidance motion vo , ω o does not take place, eq.(5.45) reduces to r˙ c = e rc = −e2 w˙ 0 and ˙ = ∇˙ e3 w, ˙ ϕ κ = κ = w00 . It is therefore not suitable to use y˙ ∈ IR16 for calculation but to reduce the vector of describing variables to y˙ ∈ IR3 with the aid of F,

z 6

w0

q q

q

-x

w(x, t)

Figure 5.6.

Plane Beam Bending

      

vo ωo r˙ c ˙ ϕ κ˙

      =       |

(5.46)

0 0 0 0 e3 0 0 −e2 0 0 0 0 {z

0 0 0 0 0 1

        |



w˙  w˙ 0  w˙ 00 {z

:= y˙

(5.47)

}

}

F (due to κ ∈ IR4 , all the components of F are in IR3 except for the last row which contains scalar elements). This leads to the element matrices according to eq.(5.38) et seq.,

5.4

135

Single Elastic Body – Small Motion Amplitudes

dfze dm 0 0   dM =  0 dJy 0  , dG = 0, dQe =  −dMye . 0 0 0 −EI w00 dx 







(5.48)

If no additional impressed forces arise except the elastic potential forces, eq.(5.41) reads, for n = 1 and dm = ρAdx, dJy = ρIdx, ZL o

     w ¨ 0   ρAdx 0 0       0    0 ρIdx 0   w ¨ − 0 δw δw0 δw00     0 0 0 −EIw00 dx  w ¨ 00 ZL

= o





ρAw ¨   0 00   ¨ 0 dx = 0 δw δw δw  ρI w EIw00

(5.49)

(A: cross sectional area, I = Iy : area moment of inertia, ρ: (mass-) density). Obviously, w is best suited to be the minimal coordinate. In order to factor out δw, eq.(5.49) is integrated by parts: L

Z

0

0

δw ρI w ¨ dx = o L

Z

L δwρI w ¨0 o



−

L

o



= δw0 EIw00

L



− δw o

∂

EIw00

∂x




∂

δw

o

Z

−

L




L

+

(5.50)

EIw00 dx


∂x ∂2

L

δw o

∂

δw0

o

Z

o

ρI w ¨ 0 dx,

∂x

o

δw00 EIw00 dx = δw0 EIw00 

L

Z

∂x2

EIw00 dx.


(5.51)

Ordering w.r.t. integral and boundary terms yields ZL

δwρAw ¨ + δw0 ρI w ¨ 0 + δw00 EIw00 dx





o

ZL



δw

= o

ρAw ¨ 

+ δw 

+ δw0



2 − ∂ (ρI w ¨ 0 ) + ∂ 2 (EIw00 ) ∂x ∂x

(ρI w ¨0)

− ∂ (EIw00 ) ∂x + (EIw00 )



dx

(5.52)

L o

L

= 0. o

136

5 Elastic Multibody Systems – The Partial Differential Equations

From eq.(5.52) one easily concludes with the aforementioned differential operators:  ZL  o



δw ◦

ZL

 

δw

= o





 

+ δw

1

−∂ ∂x

2 +∂2 ∂x

1

−∂ ∂x

0

  



+ δw0

ρAw ¨ ρI w ¨0 EIw00 ρAw ¨ ρI w ¨0 EIw00  ρAw ¨   ¨0 ◦  ρI w EIw00

∂2 ∂x2

∂ ∂x

1



        ◦

  dx      dx   L        

(5.53)

o

L ρAw ¨       ¨ 0   = 0. ◦  ρI w   00 

0

0

1



EIw

o

Notice that the operators are applied to the corresponding terms as indicated by braces. This operation is not commutative. Considering x ∈ ] 0, L [ , w = 0 is a possible but trivial solution, while at the boundaries either the translational (rotational) deflection or the corresponding force (torque) is zero. Thus, setting the three summation terms in eq.(5.53) individually to zero leads to the following equations: "

∂ ∂x

∂x2

−

1

PDE:

#

∂2





ρAw ¨ dQc ¨ 0  := DT ◦ = 0, ◦  ρI w dx 00 EIw

BC 1/2:  L

w

=0

 _

o



(0



1



ρAw ¨  ∂ ¨0  − ) ◦  ρI w  ∂x EIw00

L

:= BoT ◦

dQc = 0, dx

o

BC 3/4: 

w0

L

=0 o

 _

(0





0



ρAw ¨   ρI w ¨0  1) ◦  EIw00

L

:= B1T ◦

dQc =0 dx

o

(5.54) (PDE: partial differential equation, BC: boundary conditions). The procedure according to eq.(5.44) is thus verified. As a result, one has:

5.4

137

Single Elastic Body – Small Motion Amplitudes

˙ T κ˙ T ], here: Choose for y˙ the necessary elements out of [vTo ω To r˙ Tc ϕ (w˙ w˙ 0 w˙00 )T =



∂ ∂x

1

∂2 ∂x2

T

w˙ := D ◦ s˙ .

Calculate dQc = [dM¨y + dGy˙ − dQe ], here: ¨ ρI w ¨ 0 EIw00 )T dx. dQc = (ρAw The rest follows “automatically”. Carrying out calculations yields the partial differential equations of motion (assuming ρ = const., A = const., I = const., E = const.) ρAw ¨ − ρI w ¨ 00 + EIw0000 = 0

(5.55)

along with the corresponding geometrical boundary conditions (BC 1/3) x = 0, L : w = 0, w0 = 0

(5.56)

and the dynamical boundary conditions (BC 2/4) x = 0, L : ρI w ¨ 0 − EIw000 = 0, EIw00 = 0.

(5.57)

As eq.(5.54) shows, either the geometrical boundary conditions or the corresponding dynamical boundary conditions are to be fulfilled: eq.(5.53) represents the virtual work which has to be zero in total. The penultimate line of eq.(5.53) thereby denotes “translational deflection times transverse force”. With δw being the translation, the corresponding force reads ρI w ¨ 0 −EIw000 . Analogously, with δw0 as angular displacement, EIw00 is the corresponding (bending) moment. Because of the (total) linearization, eq.(5.55) is a linear partial differential equation. In the following, ρA and EI are furthermore assumed constant (homogeneous uniform beam). A solution of the partial differential equation is obtained by separation of the (independent) variables: Let w(x, t) =

X

wi (x) · qi (t).

(5.58)

i

Eq.(5.55) then becomes I ρA wi (x) − wi (x)00 q¨i (t) + EIwi (x)0000 qi (t) = 0. A 



(5.59)

Dividing eq.(5.59) by EIqi (t)[wi (x)−(I/A)wi (x)00 ] and rearranging yields

138

5 Elastic Multibody Systems – The Partial Differential Equations

wi (x)0000 ρA q¨i (t) = −ki4 . =− I 00 EI qi (t) [wi (x) − A wi (x) ]

(5.60)

Because [wi (x)0000 ]/[wi (x) − (I/A)wi (x)00 ] depends on space and [ρA¨ qi ]/[EIqi ] depends on time only, equality can only be verified for both summation terms being individually constant. For convenience, this constant is denoted −ki4 (see eq.(5.64) below). Eq.(5.60) can thus be resolved: EI 4 ρA q¨i (t) = −ki4 ⇒ q¨i (t) + k qi (t) = 0, EI qi (t) ρA i I wi (x)0000 = ki4 ⇒ wi (x)0000 + ki4 wi (x)00 − ki4 wi (x) = 0. I 00 A [wi (x) − A wi (x) ] (5.61) One obtains two ordinary linear differential equations, one for the time and one for the spatial coordinate. These can be solved using the common exponential statement: qi (t) = c exp(λi t) ⇒ λ2i +

EI 4 q˙oi ki = 0 ⇒ qi (t) = qoi cos ωi t + sin ωi t, ρA ωi (5.62)

I 2 β − ki4 = 0, ⇒ wi (x) = · · · (5.63) A i where qoi = qi (0), q˙oi = q(0) ˙ are the initial conditions. The time solution is a harmonic vibration while the spatial solution is somewhat more complicated. A simple solution, however, is obtained when setting ρI → 0 (in other words: the rotational inertia of a beam slice is neglected when compared to its translational inertia). In this case one speaks of the Euler–Bernoulli beam, otherwise of the Rayleigh beam. Neglecting the “Rayleigh effect” is evident for slim beams. Considering at first the Euler– Bernoulli beam, eq.(5.63) becomes wi (x) = exp(βi x) ⇒ βi4 + ki4

wi (x) = exp(βi x) ⇒ βi4 − ki4 = (βi2 − ki2 )(βi2 + ki2 ) = 0 ⇒ βi,1/2 = ±ki , βi,3/4 = ±i ki ⇒ wi (x) = c1i cosh ki x + c2i sinh ki x + c3i cos ki x + c4i sin ki x. (5.64) The constants cij are to be calculated from the boundary conditions, eq.(5.56) and eq.(5.57). According to the type of boundary condition to be fulfilled, one has to consider the zeroth up to the third derivative

5.4

139

Single Elastic Body – Small Motion Amplitudes

w(x) cosh kx  w0 (x)/k   sinh kx     w00 (x)/k 2  =  cosh kx sinh kx w000 (x)/k 3 i 



sinh kx cos kx sin kx c1  c2  cosh kx − sin kx cos kx     . sinh kx − cos kx − sin kx   c3  cosh kx sin kx − cos kx i c4 i (5.65)



 



The fourth derivative yields once more the basic function itself, wi0000 /ki4 = wi ,

(5.66)

see eq.(5.61) for ρI → 0. Equating the four boundary conditions in the form [· · ·]ci = 0, ci = (c1i , c1i , c3i , c4i )T , yields a (transcendental) eigenvalue problem: For a nontrivial solution, cji 6= 0, j ∈ {1, 2, 3, 4} for at least one j, eq.(5.65) can only be fulfilled if det[· · ·] =0 holds. 5 Example: For a beam which is clamped at its left bound [w(0) = 0, w0 (0) = 0] and which can freely move without any force or torque at its right end [EIw00 (L) = 0, EIw000 (L) = 0] one has







0 1 0  0    0  =  cosh kL 0 i sinh kL

0 1 sinh kL cosh kL

1 0 − cos kL sin kL

 



c1 0 1   c2  . − sin kL   c3  − cos kL i c4 i

(5.67)

eigenvalue problem:



1 0  det  cosh kL sinh kL

0 1 sinh kL cosh kL

1 0 − cos kL sin kL



0 1  =0 − sin kL  − cos kL i

(5.68)

= (1 + cos ki L · cosh ki L) = 0 ⇒ ki L = · · ·

eigenvector problem: c3 = −c1 , c1 = 1 (chosen) and c2 = −c4 , c4 := γ. Solving γ from the last row of



1 0   cosh kL sinh kL

0 1 sinh kL cosh kL

yields ⇒ γi = 4

1 0 − cos kL sin kL

h

 



0 1 1   −γ  =0 − sin kL   −1  − cos kL i γ i

sinh kL − sin kL cosh kL + cos kL

i

. i

(5.69)

140

5 Elastic Multibody Systems – The Partial Differential Equations

Table 5.2. Classical Bending Solutions Boundaries left/right eigenvalue equation eigenvalues {kn L}, n = 1 · · · ∞

eigenvector c(γ) γ

free/free

c = [+1 − γ + 1 − γ]T

γ = cosh kL − cos kL sinh kL − sin kL {kn L} = {4.73004, 7.85320, 10.9956, 14.1372, 17.2788, · · · ≈ (2n + 1) π 2} clamped/clamped c = [+1 − γ − 1 + γ]T 1 − (cosh kL)(cos kL) = 0

γ = cosh kL − cos kL sinh kL − sin kL {kn L} = {4.73004, 7.85320, 10.9956, 14.1372, 17.2788, · · · ≈ (2n + 1) π 2} T pinned/pinned c = [0 0 0 γ] √ sin kL = 0 γ= 2 1 − (cosh kL)(cos kL) = 0

{kn L} = {n π} c = [+1 − γ − 1 + γ]T

clamped/free

γ = sinh kL − sin kL cosh kL + cos kL {kn L} = {1.87510, 4.69409, 7.85476, 10.9955, 14.1372, · · · ≈ (2n − 1) π 2} clamped/pinned c = [+1 − γ − 1 + γ]T 1 + (cosh kL)(cos kL) = 0

tan kL − tanh kL = 0

γ = cot kL

{kn L} = {3.92660, 7.06858, 10.2102, 13.1518, 16.4934, · · · ≈ (4n + 1) π 4} [pinned/clamped:

cT = (0 − 1/ sinh kL 0 1/ sin kl)]

free/pinned

c = [+1 − γ + 1 − γ]T

tan kL − tanh kL = 0

γ = cot kL

{kn L} = {3.92660, 7.06858, 10.2102, 13.1518, 16.4934, · · · ≈ (4n + 1) π 4} [pinned/free:

cT = (0 − 1/ sinh kL 0 − 1/ sin kl)]

Eigenfunction: wi (x) = q(cosh kx sinh kx cos kx sin kx)i ci EI k2 Eigenfrequency: ωi = ρA i

5.4

141

Single Elastic Body – Small Motion Amplitudes

Table 5.3. Bending Eigenfunctions (Euler-Bernoulli beam)

pinned-clamped

clamped-clamped

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

pinned-pinned

pinned-free (zero mode not depicted)

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

1

-2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

142

5 Elastic Multibody Systems – The Partial Differential Equations

The important cases are “clamped–free” (which characterizes e.g. a beam which is fixed at a revolute joint) and “free–free”. The latter is e.g. used in a prismatic joint where the beam has free ends but can only move with small amplitudes w.r.t. the joint-fixed coordinate system.

Table 5.4. Bending Eigenfunctions (Euler-Bernoulli beam), ctd.

clamped-free

free-free (zero modes not depicted)

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 -2

-2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

The eigenfunctions that belong to the boundary condition pinned-free also contain a rigid body mode (zero eigenvalue, free rotation about the pinned joint) which is not depicted in Table 5.3. It has to be emphasized that due to linearization also the free rotation mode has to be considered small. It can thus be characterized by the function wrb = x (index rb: “rigid body”). However, in multi beam systems, such a mode does not come into play because, in general, the rigid body motions are not small. Due to these, the cases listed in Table 5.3 are of minor interest.

5.4

143

Single Elastic Body – Small Motion Amplitudes

The equations for torsion and stretching are obtained for 

0 vo  0  ωo      e1  r˙ c  =      0  ϕ ˙    0 κ˙ 0 



0 0 0 e1 0 0

0 0 0 0 e1 0



0    0  u˙    0  ϑ˙    0  ,  0  u˙    e2  ϑ˙ 0 0

u˙ ϑ˙ u˙ 0 ϑ˙ 0



1  0     ∂ =   ∂x  0 



0 1 0

     

∂ ∂x

u˙ ϑ˙

!

:= D ◦ s˙

(5.70) yielding dQc = (ρA¨ u ρIx ϑ¨ EAu0 GID ϑ0 )T dx and the equations of motion ∂ dQc  1 0 − ∂x 0 T D ◦ = dx 0 1 0 −∂ ∂x 



ρA¨ u  ¨  ρIx ϑ   EAu0 GID ϑ0 

   = 

ρA¨ u − EAu00 ρIx ϑ¨ − GID ϑ00

!

=0 (5.71)

along with the boundary conditions  

x = 0, L :



0 0 1 0    0 0 0 1 

ρA¨ u ρIx ϑ¨ EAu0 GID ϑ0

    _ u  EAu0  = 0 = 0. = ϑ GID ϑ0 

(5.72) Separation of variables such as in eq.(5.58) yields second-order linear ordinary differential equations; the solutions for time and space are thus trigonometric. Table 5.5. Longitudinal and Torsional Vibrations Boundaries left/right eigenvalue equation⇒ eigenvalues {kn L}

eigenvector c

free/free sin kL = 0 ⇒ kn L = nπ

c=

clamped/free cos kL = 0 ⇒ kn L = (2n − 1) π 2

c= 0

clamped/clamped sin kL = 0 ⇒ kn L = nπ

c= 0

√

2 0

T



√ T 2



√ T 2

Eigenfunction: ui (x) = (cos kx sin kx)i ci , q ϑi (x) = (cos kx sin kx)i ci Ek Eigenfrequency: Longitudinal Vibrations ωi = ρ i Eigenfrequency: Torsional Vibrations ωi =

q

GID k ρIx i

144

5 Elastic Multibody Systems – The Partial Differential Equations

Conclusions/Properties For linear systems, separation of the independent time variable from the spatial variables always holds if the system under consideration is undamped. One obtains an ordinary differential equation for the time variable. If only one spatial variable exists, the corresponding equation is then, of course, also an ordinary one. However, if more than one spatial variables come into play, then the equation for the spatial variables remains a partial differential equation. This is, for instance, the case for a plate as will be discussed later.

Because the separation statement (5.38) holds for every summation term i, the so-called eigenforms wi ∀ i = 1 · · · ∞ can be calculated independently from each other. Considering the Euler–Bernoulli beam, the virtual work reads, RL 00 according to eq.(5.49), o (δwρAw ¨ + δw EIw00 )dx = 0. If one inserts the separation statement (5.58) along with the eigenfunctions from eq.(5.64) but retains qi (t) for variables in the first step, then the variation reads P δw = j wj (x)δqj (t). (Because the wj ’s are fixed they do not undergo variations.) One obtains thus the virtual work, after dividing by ρA, in the form (" # " # ) RL RL 00 00 P P EI δqj wj wi dx q¨i + ρA wj wi dx qi = 0. o o j i The variations δqj are independent and non-zero in general. Therefore, the assigned have to (" coefficients # " be zero themselves: # ) P

RL

i

o

RL 00 00 wj wi dx q¨i + EI ρA o wj wi dx qi

= 0.

According to eq.(5.61), the differential equation q¨i + [(EI)/(ρA)]ki4 qi = 0 holds for every qi , independently from qj , j 6= i. Therefore, from direct comparison, it follows that the eigenforms wi obey the orthogonality condition (adequately normalized)

  L   L   Z Z L for j = i L ki4 for j = i 00 00    wj wi dx = , wj wi dx = .

0 for j 6= i

o

0 for j 6= i

o

(5.73) R Normalizing oL wi2 dx = L has been carried out in Table 5.5 and Table 5.2. This is most easily verified for pure trigonometric solutions: For clamped√ clamped torsion one has 2 sin(iπx/L) ⇒ √ for instance ϑi (x) = RL√ R1 2 sin(iπx/L) · 2 sin(jπx/L)dx = 2L o o sin(iπx/L) · R1 2 sin(jπx/L)d(x/L) = 0 and 2L o sin (iπx/L)d(x/L) = 2L · (1/2) = L.

5.4

145

Single Elastic Body – Small Motion Amplitudes

For all the bending functions of an Euler–Bernoulli beam which have at least one free end, this normalization leads to ±2 at the free end: clamped/free: ±2 . free/free: ±2 (5.74) For the corresponding bending angles one obtains 

wi (0) :

clamped/free: 0 , wi (L) : free/free: 2

wi0 (L) :



  clamped/free:   

2 sin ki L k sin ki L i 1 + sinh ki L

    free/free:

2 sin ki L k sin ki L i 1 − sinh ki L

,

(5.75)

where for “clamped-free” wi0 (0) = 0 and for “free-free” wi0 (0) = − | wi0 (L) |. It is often useful to consider the differential equation of the eigenforms itself, wi0000 − ki4 wi = 0. Integration yields ZL

wi dx = o

(

 1  000 wi (L) − wi000 (0) : 4 ki

γi clamped/free: −wi000 (0)/ki4 = 2 k i free/free: 0 



.

(5.76)

The integral value is R R thus determined by the boundary conditions only. The term oL xwi dx = oL xwi0000 /ki4 dx yields, with an integration by parts, ZL

xwi dx = o

 

 L 000 1  wi (L) + 4 wi00 (0) − wi00 (L) : 4 ki ki

clamped/free: wi00 (0)/ki4 = 2 k1 i  free/free: 0 

2

.

(5.77)

In damped systems, one may use special damping mechanisms which also lead to variable separation. Consider, for instance, an Euler–Bernoulli beam which undergoes damping according to ρAw ¨ + EIw0000 + d1 w˙ + d2 w˙ 0000 = 0. Along with the separation statement, and after dividing by EIwi qi , it takes the form ρA q¨i + d1 q˙i + d2 wi0000 q˙i = − wi0000 = −k 4 . i wi EI qi EI qi EI wi qi

146

5 Elastic Multibody Systems – The Partial Differential Equations

Since wi0000 − k 4 wi = 0, the (spatial) eigenfunctions are thus not influenced, while the time behavior corresponds to a linear damped oscillator: "

#

d2 ωi2 d1 q¨i + + q˙i + ωi2 qi = 0 ρA EI

where ωi2 =

EI 4 k . ρA i

(5.78)

This kind of damping is often used for material damping. It is, however, pragmatic and has (probably) nothing in common with real material properties. Bernoulli’s7 solution and the wave equation: it might be of some interest to investigate “traveling waves” which are, of course, automatically included in the solution according to eq.(5.58) with eq.(5.64). Let d2 /EI = 2ζ/ωi in eq.(5.78). One obtains then q¨i + 2ζωi q˙i + ωi2 qi = 0, or in state space notation d dt

qi q˙i

!



=

0 1 −ωi2 −2ζωi



qi q˙i

!

: x˙ i = Ai xi

which shows the solution qi q˙i

! −ζωi t

=e

cosνi t + ω2

ζωi νi sinνi t

1 νi sinνi t i cosνi t − ζω νi sinνi t

− νii sinνi t

!

qio q˙io

!

:= Φi xio where xio = (qio q˙io )T are the initial conditions and Φ is called the Fundamental Matrix (or Transition Matrix).The eigenfrequency reads p 2 νi = ωi 1 − ζ . Next, letR a force act Ron the system during a vanishing −dt one concludes that f = pδ − time period (shock). From f dt = p = pδ − with p : “impulse”, δ : Dirac-distribution. The state equation then reads x˙ i = Ai xi + bi− δ where bi = (0 wi (ξ)p/m)T , m = ρAL, (L = 1 assumed), ξ: the location on the beam axis where the force acts. The term wi (ξ) here follows from virtual work [w ¨+ · · ·]δw = f− δ (x − ξ)δw: inserP P ting the separation statement w(x, t) = i wi (x)qi (t), δw = j δqj wj , integrating over the beam length and respecting the orthogonality conditions yields q¨i + 2ζωi q˙i + ωi2 qi = (f /m)wi (ξ). The inhomogeneous state equation is solved with theh “variation i of constants” technique: xi = R t −1 − −1 Φi o Φi bi δ (τ − t1 )dτ = Φi Φi bi , t1 ∈ {0, t}. For t1 → 0 one t1

has Φ−1 i (0) = E yielding xi = Φi bi : bi can thus be interpreted as an initial condition which characterizes an impact. Insertion into Φi bi yields p qi (t) = e−ζωi t ai sin(νi t) with the amplitude ai = wi (ξ) mν , and the total i P −ζωi t ai sin(νi t)wi (x). solution is w(x, t) = i e 7 Daniel

Bernoulli, *1700 in Groningen, †1782 in Basel

5.4

147

Single Elastic Body – Small Motion Amplitudes

y

1.0

0.5

0.5

0

0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

-2.0

-2.0

-2.5

-2.5 -3.0

-3.0 0

0.2

0.4

0.6

0.8

1

Figure 5.7. Impact at x=0.15, thin: t=0.1 ms, bold: t=30 ms ( y in mm, x in m)

0

0.2

0.4

0.6

0.8

x

Figure 5.8. Solution after 30 ms, thin: i=5, dotted i=4 Eigenfunctions

The simplest case is that of pinned-pinned boundaries yielding wi (x) = sin(ki x), ki = iπ, 0 ≤ x ≤ 1. Fig. 5.7 shows the results for an impact acting at x = 0.15. Fifty eigenfunctions were used for the problem. After 30 ms the higher eigensolutions are damped out (ζ = 0.02). The result still contains about five eigenforms (Fig. 5.8). hThe parameters i were chosen such p 2 that the basic bending frequency is f1 = EI/ρAπ /(2π) = 15 Hz. From t = 30 ms on the solutions are plotted in Fig. 5.9 in time steps of ∆t = 1 ms. The amplitudes decrease thereby while the wave travels to the right. At about t = 33.3 ms the wave front reflects at the right bearing and eventually travels to the left with increasing amplitude. This behavior can easily be interpreted by rearranging the solution with 1 νi νi cos ki (x − t) − cos ki (x + t) 2 ki ki 

sin(νi t) sin(ki x) =

where,



148

5 Elastic Multibody Systems – The Partial Differential Equations

y

3.0

2.0

1.0

1.0

1.0

0

0

-1.0

-1.0

-2.0

-3.0

-3.0

-3.0 0

0.2

0.4

0.6

0.8

0

1

Figure 5.9. Stroboscopic View, t ∈ {30, 31, 32, 33, 33.3} ms

0.2

0.4

0.6

x

0.8

Figure 5.10. Stroboscopic View, t ∈ {33.4, 34, 35, 36, 37} ms

η with (νi /ki ) = vi , the first summation term (with the minussign) denotes a motion to the right, and, consequently, the other one a motion to the left. This kind of motion can be interpreted as transportation of information while the individual elements dm themselves move at x = const. For the case with pinned-pinned boundaries, the special equation wi00 = −ki2 wi holds for the undamped case, yielding wi0000 = −ki2 wi00 .

6

y

1

-

ξ -

vt

0 0

0.2

Figure 5.11.

0.4

0.6

0.8

x

“Traveling Frame”

0000 This leads to the nth-mode partial differential equation w ¨n = − EI ρA wn = 2 00 + EI ρA ki wn . One obtains thus

∂wn2 ∂x2

−

1 ∂wn2 νn = 0, vn = , 2 kn vn ∂t2

5.4

149

Single Elastic Body – Small Motion Amplitudes p

which is known as the p p p wave equation. Here, vn kn reads EI/ρA = I/A E/ρ, where E/ρ represents the sound speed for the considered material. To give an impression of the wave speed: for this example one obtains vn = n · 30 m/s. y

3.0

2.0

2.0

1.0

1.0

0

0

-1.0

-1.0

-2.0

-2.0

-3.0

-3.0 0

0.2

0.4

0.6

0.8

0.2

0

1

Figure 5.12. Stroboscopic View, t ∈ {350 · · · 366.5} ms

0.4

0.6

0.8

x

Figure 5.13. Stroboscopic View, t ∈ {370 · · · 382} ms

After t = 350 ms, all higher eigenmodes are damped except mainly the first one. The resulting motion is then a simple oscillation. A (total) decomposition of sin(νi t)wi (x) into traveling waves is not possible in other beam bending cases because then wi (x) = a1i cosh(ki x) + a2i sinh(ki x) + a3i cos(ki x) + a4i sin(ki x) (see Table 5.2). A kind of traveling wave can of course also be observed. Fig. 5.14 shows a stroboscopic view on a clamped beam which undergoes an impact at x = 0.75. Obviously, the shock induced wave travels to the left, relative to one main deformation.

y 2.0

1 ms

0

2 ms

-2.0 3 ms

-4.0 -6.0

4 ms

-8.0

0

0.2

Figure 5.14.

0.4

0.6

0.8

x

Shock at x=0.75, clamped beam

150

5 Elastic Multibody Systems – The Partial Differential Equations

The wave equation along with a solution ansatz in the form w = F (x − vt) + G(x + vt) (F, G twice differentiable) is commonly referred to as the d’Alembert representation (from 1747, investigating string vibrations). The breakthrough came with D.Bernoulli 1753 who noticed the simultaneous existence of different tones which lead him to the superposition principle and consequently to the solution in form of a (Fourier) series. Although Euler doubted this result, he himself later anticipated the Fourier8 expansion (1777, printed posthumously 1798) [after (I. Szab´o, 1979a)]. The solution based on eq.(5.58) is the most general one (see also Ritz’s approximation, Chapter 6). Finally, the question has to be answered whether the neglect of the “Rayleigh effect” has been justified. Table (5.6) depicts the vibration forms of the clamped-free Rayleigh beam9 in comparison to the Euler-Bernoulli beam, for a length to thickness ratio L/D = 10.

Table 5.6. Clamped-Free Eigenfunctions of a Rayleigh Beam (thin) in Comparison to an Euler-Bernoulli Beam (bold); L/D=10

Eigenfunctions 1 to 3

Eigenfunctions 4 and 5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 -2

-2 0

8 Jean

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

Baptiste Joseph Fourier, *1768 in Auxerre, †1830 in Paris

0.8

1

9 The eigenfunctions of the Rayleigh beam are calculated with the Ritz approximation using the “clamped-

free” eigenfunctions of the Euler-Bernoulli beam for shape functions, see Chapter 6.2.

5.4

Single Elastic Body – Small Motion Amplitudes

151

The eigenform deviations of the Rayleigh beam (thin) in comparison to the Euler–Bernoulli beam (bold) become visible from the fourth eigenform on. The eigenfrequencies according to Table 5.2 are ωi,norm. = ki2 when normalized to L = 1, (ρA/EI) = 1. The corresponding values for the Rayleigh beam differ from these by {0.07, 0.50, 1.17, 2.11, 3.24}%. Of course, this result can not be generalized to beams with arbitrary boundary conditions but corresponds to a frequently used case. By this, one slowly approaches z 6 the validity border of beam modeling according to Euler and Bernoulli: A basic rule says w0 that Euler’s approach is suffiq −βB cient up to a ratio of L/D ≥ q 10. This holds especially when q -x q one thinks of controlled systems w(x, t) where considering, say, the first two or three eigenfunctions is Figure 5.15. “The Timoshenko Beam” enough to describe the problem. This is due to the fact that the control input usually yields high pervasive damping throughout all variables. High frequency oscillations then scarcely enter dynamics. For bending oscillation with lower frequency, the “Rayleigheffect” does not play a significant role as Table 5.6 indicates10 . The above mentioned rule, however, also says that for L/D ≤ 10, shear forces begin to play a role. These are not considered in Rayleigh’s nor in Euler’s approach. If shear deformation is taken into account, then translational and bending deformations become independent. The hypothesis of undeformed cross sectional area (as in Fig. 5.15), however, can be saved when introducing a correction factor κ. This factor results from a comparison of the work done by the actual shear forces (which deform the element somewhat S-like) with the work done at an undeformed element but with constant shear angle between neighboring elements. The shear rigidity is then κGA. As can be seen in Fig. 5.15, the shear angle may be calculated as the difference between w0 and the bending angle −βB about the y-axis. The elastic bending potential therefore reads 2 dx/2 while the shear potential is dV = κGA(w 0 + β )2 dx/2. dVB = EIy βB S B Considering spatial motion, the total elastic potential is thus

10 Care must be taken in case of rotating systems: as well known, the so-called external damping stabilizes. This means that the corresponding input has to react against the inertial frame. Inner damping, however, such as material damping which is proportional to the relative velocity w.r.t. the rotating frame, destabilizes the rotor, see Chapter 6, section 5.1.

152

5 Elastic Multibody Systems – The Partial Differential Equations

Vel = +

1 2

ZL n

2

2

2

2

GID ϑ0 + EAu0 + EIy β 0 B + EIz γ 0 B

o

h

κGA (v 0 − γB )2 + (w0 + βB )2

io

dx.

(5.79)

Neglecting shear deformation, the bending angles read βB = −w0 and γB = v 0 . Then, the elastic potential once more becomes the Eulerian approach. Remark: The deformation variables are independent of each other if the elastic axis (“shear center axis”) and the mass center axis coincide. Otherwise, the potential forces, which are defined w.r.t. the elastic axis, affect, by the corresponding vector product, the mass center axis which is the reference axis for the inertia forces. An interpretation is the following: the elastic axis is that one which reacts with pure bending when a force is applied perpendicularly to it. The correction factors are found in (G.R. Cowper, 1966) for instance. One obtains for a rectangle (independent of dimensions) κ = [10(1 + ν)][12 + 11ν]−1 , or for the thin-walled square: κ = [20(1 + ν)][48 + 39ν]−1 , or for a hollow circle h

6(1 + ν) 1 +

Ri Ra


2 i2

κ=

h

(7 + 6ν) 1 +

κ 0.875 0.85 0.825 0.8 0.775 0.75 0.725 0.7 0.675 0.65 0.625 0.6 0.575 0.55 0.525 0.5

Ri Ra


2 i2

+ (20 + 12ν)

Ri Ra


2

ν = 0.2 ν = 0.4 ν = 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 5.16. Hollow Circle Correction Factor κ

Ri /Ra

5.4

Single Elastic Body – Small Motion Amplitudes

153

(Ri : inner, Ra : outer radius. The Poisson ratios correspond to: ν = 0.2: glass, or steel; ν = 0.4 − 0.45: lead). As the results show, the Euler-Bernoulli beam is obviously the most important approach. Shear deformation becomes important in lattice mast structures or sandwich beams. Modeling as a Timoshenko beam11 does not require much additional effort (see the Ritz approach, Chapter 6.2.). Considering rather short beams first of all reveals the influence of  *y  shear deformation. If one admits   z  6 w also hollow thin-walled cross   v sectional areas, then beam mo ϕp deling according to Euler and u  R Bernoulli is violated for qui- x  te another reason: The beam then   becomes a shell, for instance a  cylindrical shell. Clearly, such a shell will not behave like a beam, Figure 5.17. Cylindrical Shell in general. The first bending mode of a thin-walled tube (length L=1, radius R=0.1, thickness (Ra − Ri )=h = 0.0011) is comparable to that of a Rayleigh beam. Without shear correction, the corresponding frequency is about 3% lower. Below this value there arise three circumferential modes (“breathing modes”) with frequencies of 40%, 43% and 73% (in relation to the first bending mode). Such modes do not exist in a beam (A. Truckenbrodt, 1983).

4.2.

Shells and Plates

The technique of integration by parts in the form (5.54) leads to a general scheme for arbitrary cases. The most suitable choice for the vector of describing ˙ κ}, ˙ see variables y˙ is to select the necessary components of {vo , ω o , r˙ c , ϕ, eq.(5.20). Also in case of several independent spatial variables, y˙ can be expressed by a spatial operator D applied to the vector s˙ of minimal velocities: y˙ = D ◦ s˙ . Calculation of y˙ needs the mass center translational and angular velocity as well as the curvatures. Using cylinder coordinates y, ϕ for a cylindrical shell according to Fig. 5.17, one obtains for mass center velocity and angular deformation 11 Usually, this kind of modeling is attributed to Stepan Prokopovich Timoshenko (*1878 in Schpotiwka, †1972 in Wuppertal), see (S.P. Timoshenko, 1921,1922), and the term “Rayleigh beam” to Lord Rayleigh, see (J.W.S. Rayleigh, 1877). However, both effects had already been analyzed by Jacques Antoine Charles Bresse (*1822 in Vienne, †1885 in Paris), see (J.A.C. Bresse, 1859), p. 144 et seq. It was (J. Schmidt, 1987) who focused attention upon this fact.

154

5 Elastic Multibody Systems – The Partial Differential Equations "

r˙ Tc =

#

u˙ "

v˙

w˙ , #

1 ∂w − R ∂ϕ

1 ∂v ∂u ϕ = . (5.80) − R ∂ϕ ∂y ∂y Definition of curvatures follows from the corresponding elastic potential which is always represented by a quadratic form. The elastic potential is derived from eq.(5.11), where dx is to be replaced with Rdϕ and κ = κ1 + κ2 with T

"

κT1

1 = R "

κT2

1 = R2

∂u

∂w

!

1 2

∂v

−w

∂ϕ

∂y

∂2w

∂u

!

∂2w

1 R

∂u 1 ∂v + R ∂ϕ ∂y

!#

∂2w

!#

∂u

,

+ (5.81) + ∂ϕ2 ∂ϕ ∂y 2 ∂y∂ϕ ∂y [see e.g. (K. Washizu, 1968), p.189 and p.197]. ˙ κ˙ in terms of a differential operator applied to s˙ T = Rewriting r˙ c , ϕ, [u(y, ˙ ϕ, t) v(y, ˙ ϕ, t) w(y, ˙ ϕ, t)] is unproblematic but laborious. In order to demonstrate the proposed procedure without loosing generality but shortening calculations, we turn to the Kirchhoff plate in cartesian coordinates: For R → ∞ in Fig. 5.17, one has (1/R) → 0 except for (1/R)(∂/∂ϕ) = [∂/∂(Rϕ)] → (∂/∂x). Before carrying out the limiting process, however, it is y *   suitable to shift the global coor     z dinate system (x, y, z) upwards   6 w v   with R. In contrast to the cy   u lindrical shell where the local - x element-fixed frame changes orientations for different angles ϕ, Figure 5.18. Plate the orientation in a plate remains unchanged at every location in the undeformed state. This leads to a decoupling of (u, v) and w and thus to a decoupling of κ1 and κ2 . Considering w(x, y, t) for deformation, one obtains from eq.(5.80) and eq.(5.81) "

r˙ Tc

=

0

κ =

0

" T

w˙ , ϕ =

∂w ∂y

" T

#

∂2w

∂2w

∂2w

∂x2

∂y 2

∂y∂x

−

∂w

#

0 ,

∂x

#

.

(5.82)

5.4

155

Single Elastic Body – Small Motion Amplitudes

For the potential, one is thus once more left with eq.(5.11) [compare also e.g. (C.L. Dym, I.H. Shames, 1973), p.287 and p.291]. Collecting the components from eq.(5.82) in y˙ yields y˙ T =



w˙

∂ w˙ ∂y

∂ 2 w˙ ∂x2

− ∂ w˙ ∂x

∂ 2 w˙ ∂y 2

∂ 2 w˙ ∂x∂y



.

(5.83)

The minimal velocity s˙ is the deformation velocity w˙ itself, leading to the requested differential operator y˙ T =



1

∂ ∂y

∂2 ∂x2

−∂ ∂x

∂2 ∂y 2

∂2 ∂x∂y



w˙ := (D ◦ s˙ )T .

(5.84)

The same as in case of a beam, operator D yields immediately operator D (which determines the partial differential equations) by changing the sign of odd derivatives. The boundary operators Bir are simultaneously obtained by successive degeneration of the derivation grade along with once more changing signs. However, since x and y are independent, the boundary operators have to be considered separately for x and y and need a second index r. This is because the assigned partial derivation may be ∂/∂x, ∂/∂y or ∂ 2 /∂x∂y. Let r ∈ {x, y, xy}. Then, the necessary derivations may be characterized by Dir , where Dor := 1∀ r. The geometrical boundary conditions are then simply Dir s, starting with i = 0. The degeneration steps i for the corresponding Bir , i = 0, 1, · · · are to be carried out till Bir ≡ 0. Notice that, if ∂ 2 /∂x∂y arises, then the degeneration w.r.t. r = x yields −∂/∂y and the degeneration w.r.t. r = y yields −∂/∂x. This leads, for r = xy, to a “mixed” representation which denotes a “corner condition”. In case of the Kirchhoff plate, the existence of such corner conditions had been pointed out by (H. Lamb, 1890). The resulting operators are listed in Table 5.7.

Next, vector dQc where the operators are applied has to be calculated. Along with      

vo ωo r˙ c ˙ ϕ κ˙





    =     |

0 0 0 0 0 0 0 0 e3 0 0 0 0 e 1 e2 0 0 0 0 E {z

F

  w˙ ∂ w˙   ∂y      − ∂ w˙

∂x κ˙

}|

{z

:= y˙

      

(5.85)

}

[compare eqs. (5.47), (5.70)], the element matrices (5.38) et seq. reduce to

156

5 Elastic Multibody Systems – The Partial Differential Equations

Table 5.7. Operators; Index B: Boundary

" T

D =

1

∂ ∂y

− ∂ ∂x

∂2 ∂x2

∂2 ∂y 2

∂2 ∂x∂y

1

− ∂ ∂y

∂ ∂x

∂2 ∂x2

∂2 ∂y 2

∂2 ∂x∂y

−1

− ∂ ∂x

0

− ∂ ∂y

0

− ∂ ∂y

− ∂ ∂x

" T

D =

" xB :

Dox = 1,

T Box

Doy = 1,

T Boy

0

" yB :

# 0

=

1

0

" xB :

D1x

= ∂ , ∂x

T B1x

D1y

= ∂ , ∂y

T B1y

# 0

=

0

0

1

0

0

" yB :

# 0

=

0

0

0

1

0

" xB , yB :

Doxy = 1,

=

DT ◦ dQc

⇒ PDE ⇒ BC’s

T Boxy

Dir s = 0

W

# 0

dm 0  0 dJ dM =   0 0 0 0 dG = 0,  0 0  0 0 dQe = −   0 0 0 0

0

0

0

0

1

= 0

T Bir ◦ dQc = 0,



#

# 0

=

#

i = 0, 1, · · · , r = x, y, xy · · ·

0 0 dJ 0

0 0  , 0  0 

0 0 w   0 0   (∂w/∂y)   0 0   −(∂w/∂x)  κ 0 DdK 



(5.86)

5.4

157

Single Elastic Body – Small Motion Amplitudes

where, from eq.(5.11), 



1 ν 0  dx dy, 0 dK =  ν 1 0 0 2(1 − ν)

(5.87)

D = [Eh3 ][12(1 − ν 2 )]−1 . The distributed mass is dm = ρhdxdy, and the tensor of moments of inertia dJc is obtained from h

dJc =

+2 Z

−h 2

0 −z 0 0 z 0 1 0 0 h3      z 0 0 −z 0 0 0 1 0  dxdy, ρ dzdxdy = ρ 12 0 0 0 0 0 0 0 0 0 









(5.88) see Table 5.1. The local dJc := diag{dJ, dJ, 0} remains unchanged when transformed into a reference coordinate representation [compare eq.(5.46)]. Carrying out calculations yields 



1

0 h2 12 0

0







w ¨       ∂ w/∂y    ρh 0 0 ¨       −∂ w/∂x ¨ h2   0 c dQ =   dxdy. 12     2  2   ∂ w/∂x 1 ν 0   2 2  D  ν 1    0 ∂ w/∂y 2 0 0 2(1 − ν) ∂ w/∂x∂y 

(5.89)

Applying the operators from Table 5.7 to dQc leads to the plate equations as listed in Table 5.8 Neglecting the rotational inertia (ρh3 )/12 yields the equations which have been offered by Gustav Kirchhoff but extended with the corner conditions at (x, y) = (xB , yB ). Contrary to the beam equations, no analytical solution is known for the Kirchhoff plate except for special cases: the circular plate, all around pinned (G. Kirchhoff, 1850), or the rectangular plate, pinned at every edge or at two opposite edges (W. Voigt, 1883)12 .

12 Woldemar

Voigt, *1850 in Leipzig, †1919 in G¨ottingen

158

5 Elastic Multibody Systems – The Partial Differential Equations

Table 5.8. Plate Vibrations – Partial Differential Equations and Boundary Conditions

 PDE

ρhw ¨−

ρh3 12

"

∂2w ¨ + ∂2w ¨ 2 2 ∂x ∂y

#

"

4 4 4 +D ∂ w + ∂ w +2 ∂ w ∂x4 ∂y 4 ∂x2 ∂y 2

"

w=0

W

(2); yB :

w=0

W

(3); xB :

∂w = 0 W ∂x

(4); yB :

∂w = 0 W ∂y

(1); xB :

3 3 −D ∂ w + (2 − ν) ∂ w 3 ∂x ∂x∂y 2

"

3 3 −D ∂ w + (2 − ν) ∂ w ∂y 3 ∂y∂x2

"

2 2 +D ∂ w + ν ∂ w 2 ∂x ∂y 2

"

2 2 +D ∂ w + ν ∂ w ∂y 2 ∂x2

"

xB , yB :

w=0

W

2 2D(1 − ν) ∂ w ∂x∂y

#

+

=0



ρh3 12



∂w ¨ =0 ∂x



ρh3 12



∂w ¨ =0 ∂y

+

#

#

# =0

# =0

#

! =0

Remark: As geometrical interpretation shows, conditions (1) to (4) represent either vanishing deformations or forces, and vanishing bending angles or moments, respectively, along the edges. These conditions already define a unique solution. For instance, in case of clamped edges, the last condition becomes meaningless since it is already considered with the geometrical conditions (1) to (4). The same statement, however, also holds in case of free edges: Once the PDE is solved with the dynamical conditions (1) to (4), one may insert the result into (3) V ! [x = xB : w = w(y) ⇒ (∂ 2 w/∂x2 ) ≡ 0 (∂ 2 w/∂y 2 ) = 0] and (4) [y = yB : w = !

w(x) ⇒ (∂ 2 w/∂y 2 ) ≡ 0 (∂ 2 w/∂x2 ) = 0] which at the corners can only simultaneously be fulfilled for w = const. Then, the last line does not really represent a condition but a (trivial) consequence from conditions (1) to (4) [see also (W. Ritz, 1909), p.288: Using a corner-fixed coordinate system shows, by Taylor series expansion, that deformations will vanish when approaching the corner].

V

5.5

159

Single Body – Gross Motion

5. Single Body – Gross Motion 5.1. The Elastic Rotor The “elastic rotor” according to Fig. 5.4 is now once more calculated using the Projection Equation. Along with two bending directions and the rigid body angular velocity α˙ one obtains

      

vo ωo r˙ c ˙ ϕ κ˙

      =      

0 0 0 0 0 0 e1 0 0 0 0 0 0 e 2 e3 0 0 0 0 0 0 e 2 e3 0 0 0 0 0 0 e3 0 0 0 0 0 0

0 0 0 0 0 1





         

α˙ v˙ w˙ −w˙ 0 v˙ 0 v˙ 00 w˙ 00

       := Fy˙    

(5.90)

(except the last line, all column elements are in IR3 . Due to κ ∈ IR4 [see e.g. eq.(5.45)], the components of the last row are scalars). The tensor of moments of inertia is derived from eq.(5.46). Torsion is not considered. The element matrices are, with eqs.(5.38) et seq.,

e

dQ = −

∂dV

!T

=− 0

0

0

0

∂y

0

EIz v 00

EIy w00


T

dx. (5.91)

dM/dx =           

ρIx + ρA(v 2 + w2 ) −ρAw ρAv ρ(Ix − Iy )v 0 ρ(Ix − Iz )w0 −ρAw ρA 0 0 0 ρAv 0 ρA 0 0 ρ(Ix − Iy )v 0 0 0 ρIy 0 ρ(Ix − Iz )w0 0 0 0 ρIz 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0



0 0  0   0 ,  0  0 0 (5.92)

160

5 Elastic Multibody Systems – The Partial Differential Equations

dG/dx = 

2ρAv α˙ 2ρAwα˙ ρ(Ix − Iy )v˙ 0 ρ(Ix − Iz )w˙ 0

0

  −ρAv α˙    −ρAwα˙    ρ(Ix − Iy )v˙ 0   −ρ(Ix − Iz )αw ˙ 0   ρ(Ix − Iz )w˙ 0   +ρ(I − I )αv 0  x y ˙   0 

0

0

 

0

−2ρAα˙

0

0

0

0 

2ρAα˙

0

0

0

0

0 

0

0

0

−ρIz α˙

0

0

0

0

ρIy α˙

0

0

0

0

0

0

0

0

0

0

0

0

0



  ,   0    0 

0 (5.93)

Neglecting second-order terms, dQc = dM¨y + dGy˙− dQe reads 



ρIx α ¨   ˙ ρA(−¨ α w + v ¨ ) − ρA( α˙ 2 v + 2α˙ w)     ˙ ρA(+¨ αv + w) ¨ − ρA(α˙ 2 w − 2α˙ v)     ¨ − ρIy w ¨ 0 + ρ(Ix − Iy − Iz )v˙ 0 α˙ − ρ(Ix − Iz )α˙ 2 w0 dx. dQc = ρ(Ix − Iy )v 0 α   ρ(Ix − Iz )w0 α ¨ + ρIz v¨0 + ρ(Ix − Iy − Iz )w˙ 0 α˙ + ρ(Ix − Iy )α˙ 2 v 0      EIz v 00 EIy w00

(5.94) From D,        y˙ =     

α˙ v˙ w˙ −w˙ 0 v˙ 0 v˙ 00 w˙ 00

1 0 0 0

0 1 0 0

0

∂ ∂x ∂2 ∂x2 0

             =   0        0   

0 0 1 ∂ − ∂x 0 0 ∂2 ∂x2

        α˙      v˙  = D ◦ s˙ ,   w˙     

(5.95)

one obtains the operator DT for the determination of motion equations, 

1 0 0



 DT =  0 1 0 

0 0 1

0 0 ∂ ∂x

0 ∂ − ∂x 0

0 ∂2 ∂x2 0

0

 

0  ,  2 ∂ ∂x2

(5.96)

5.5

161

Single Body – Gross Motion

as well as the boundary operators 



0 0 0 0 0 0  ∂ T 0 0 0 0 1 − ∂x Bo =   0 0 0 −1 0 0 

0 B1T =  0 0

0 0 0

0 0 0

0 0 0

0 0 0

0  0  , ∂ − ∂x

(5.97)



0 1 0

0 0 . 1

(5.98)

Applying these operators to eq.(5.94) yields the equations of motion Z

[

ρIx dx]¨ α − Mx = 0,

¨ w) − ρIz v¨00 − ρ(Ix − Iy − Iz )α˙ w˙ 00 ρA(¨ v − 2α˙ w˙ − α˙ 2 v − α −ρ(Ix − Iy )α˙ 2 v 00 −ρ(Ix − Iz )¨ αw00 + (EIz v 00 )00 = 0, ¨ v) − ρIy w ¨ 00 + ρ(Ix − Iy − Iz )α˙ v˙ 00 ρA(w ¨ + 2α˙ v˙ − α˙ 2 w + α −ρ(Ix − Iz )α˙ 2 w00 +ρ(Ix − Iy )¨ αv 00 + (EIy w00 )00 = 0, (5.99) (which are replenished with the drive torque Mx ) and the boundary conditions 

ρIz v¨0 + ρ(Ix − Iy − Iz )α˙ w˙ 0

+ ρ(Ix − Iy )α˙ 2 v 0 +ρ(Ix − Iz )¨ αw0 − (EIz v 00 )0

iL o

=0

_

v

L

= 0,

o



ρIy w ¨ 0 − ρ(Ix − Iy − Iz )α˙ v˙ 0

+ ρ(Ix − Iz )α˙ 2 w0 −ρ(Ix − Iy )¨ αv 0 − (EIy w00 )0 EIz v 00 EIy w

L

o 00 L o

=0

_

v0

L

iL o

=0

_

w

L

= 0,

o

= 0,

o

=0

_

w0

L

= 0.

(5.100)

o

Notice that a second-order evaluation [as in eq.(5.28) for instance] is not required anywhere. Furthermore, because all the components of y˙ have been defined w.r.t. the orthogonal reference coordinate system, one obtains automatically the correct signs [compare eq.(5.36)].

162

5.2.

5 Elastic Multibody Systems – The Partial Differential Equations

The Helicopter Blade (1) A simplified model for a helicopter blade is a flexible beam which rotates about a lateral axis. In the sequel, only the in-planemotion v(x, t) is considered. From

z 6

γ˙

* y   



 

rc = (x v 0)T ,

-x

e c = 0 0 v0 ϕ = ∇r


T

,


T

Figure 5.19.

κ = 0 0 v 00 0 (5.101)

The Helicopter Blade

one obtains       

vo ωo r˙ c ˙ ϕ κ˙

      =      

0 0 0 0 e3 0 0 0 0 e2 0 0 0 0 e3 0 0 0 0 e3 0 0 0 0

        

γ˙ v˙   := Fy˙ v˙ 0  v˙ 00 

(5.102)

which leads to the element matrices ρIz + ρA(x2 + v 2 ) ρAx ρIz  ρAx ρA 0 dM =   ρIz 0 ρIz 0 0 0 

0 2ρAv γ˙  −ρAv γ˙ 0 dG =   0 0 0 0 

∂dVel ∂y

0 0 0 0

0 0   dx, 0  0 

0 0   dx, 0  0

(5.103)



(5.104)

!

= 0

0

0

EIz v 00 dx.


(5.105)

5.5

163

Single Body – Gross Motion

Vector dQc reads13 (second-order terms neglected) dQc = dM¨y + dGy˙ + (∂dVel /∂y)T  (ρIz + ρAx2 )¨ γ + ρAx¨ v + ρIz v¨0  ρA(¨ v + x¨ γ − v γ˙ 2 ) =  0  ρIz (¨ v + γ¨ ) EIz v 00

   dx. 

(5.106)

The differential operator D for y˙ according to eq.(5.102) is 



0 γ˙  ! 1   γ˙  v˙     ∂   y˙ =  = D ◦ s˙ (5.107)  v˙ 0  =   ∂x  v˙   2 v˙ 00 0 ∂2 ∂x which yields the operator D for the determination of motion equation DT ◦ dQc = 0 as well as the operators Bi for the dynamical boundary conditions 





1 0 0



0 DT =  ∂ 0 1 − ∂x "

BoT

=

B1T =



1 0

0 0 0 0

0 0

0 1 0 0

0 ∂2  , ∂x2 0 ∂ − ∂x

#

0 1



0 0

(5.108)

,

(5.109)

.

(5.110)

Applying the operators to eq.(5.106) yields Z

[(ρIz + ρAx2 )¨ γ + ρAx¨ v + ρIz v¨0 ]dx − Mz = 0,

T

Check: the mass center velocities vc = e rc e3 γ˙ + ve ˙ 2 , ω c = (γ˙ + v˙ 0 )e3 yield   γ˙     vc erTc e3 e2 0 0  v˙  = F F y˙ as an entity. With F F = F for ab= v˙ 0 ωc e3 0 e3 0 v˙ 00 n h i o d dM F + Ω e dM F where dM = breviation, one easily proves dM = FT dM F, dG = FT dt 13

e = blockdiag{ee3 , ee3 }γ. ˙ diag{dmE, dJc }, Ω

164

5 Elastic Multibody Systems – The Partial Differential Equations

ρA(¨ v + x¨ γ − v γ˙ 2 ) − ρIz v¨00 + (EIz v 00 )00 = 0, 

0

00 0

L

ρIz (¨ v + γ¨ ) − (EIz v )

=0

_

L

=0

_

v0

o

L

L

= 0,

o

o

EIz v 00

v

= 0,

(5.111)

o

where the first line has been completed with the drive torque Mz . An analytical solution, if exists, can only be obtained with much effort. An approximative solution, however, can easily be achieved with Galerkin’s method if blade modeling is reduced to the Euler–Bernoulli beam (ρIz → 0). Doing so, the hub moment of inertia has to be taken into account. Otherwise the determinant of the mass matrix would tend to zero. (The reason is that the rotating beam would be kinematicly overdetermined)14 . Along with ρIz → 0, the boundary conditions in eq.(5.111) are those of a clamped-free Euler–Bernoulli beam. Eq.(5.111) then reads ZL

[CHub +

ZL

2

(ρAx )]¨ γ+ o

ρAx¨ v dx = Mz , o

ρA(¨ v + x¨ γ − v γ˙ 2 ) + (EIz v 00 )00 = 0, v(0) = 0, v 000 (L) = 0, v 0 (0) = 0, v 00 (L) = 0,

(5.112)

(L: beam length). Galerkin’s approximation method15 makes use of a Ritz series approach v(x, t) = v(x)T qR (t) where v(x) has to fulfill all (i.e. the geometrical and the dynamical) boundary conditions. Here, the clamped-free eigenfunctions from Table 5.2 may be chosen for shape functions. Secondly, inserting v(x, t) = v(x)T qR (t) into the equations of motion, these are eventually weighted with the shape functions v(x) and integrated over the beam length:

+

"

Co ρAxvT dx R R ρAxvdx ρAv vT dx

"

0 0 R R T 0000 EIz v v dx − ρAv vT dxγ˙ 2 0

R

#

γ¨ ¨R q

!

#

γ qR

!

=

1 0

!

Mz . (5.113)

14 See

eq.(5.114) below: the determinant of the mass matrix reads Co /mB − P where [2L/(ki2 L2 )]2 → 1/3 (proof e.g. numerically). 15 See Chapter 6.1.1.

P

[2L/(ki2 L2 )]2

5.5

165

Single Body – Gross Motion

(C o = CHub + ρAL3 /3 = CHub + mB L2 /3 where mB = ρAL: beam R R R mass; = oL ). Using beam eigenfunctions, one has ρAv vT dx = mB E, see eq.(5.73). The (spatial) differential equation vj0000 = kj4 vj (5.61) yields R R R EIvi vj0000 dx = EIvi vj kj4 dx ⇒ EIv v0000 T dx = EIz L diag{ki4 }. The R R term ρAx vi dx is derived from eq.(5.77): ρAx vi dx = 2ρA(1/ki )2 . The (ordinary) differential equation (5.113) reads thus, after dividing by mB ,       



Co mB

2L

1 k12 L2 .. .

1 k22 L2

symmetric

 

1 k32 L2

···      

E



γ¨ ¨R q

!



 0 0   +   0 diag{ EI 3 (ki L)4 − γ˙ 2 }

mB L

    

γ qR

!

1  m B   = Mz . 0 

(5.114) The values for ki L are listed in Table 5.2. Let L = 1, mB = 0.15, EI/mB = 48.61 and CHub /mB = 0.08. Applying a constant torque Mz accelerates γ. ˙ Then, after a certain time, instability arises as Fig. 5.20 shows. However, such an instability does not make sense. Eqs.(5.111) only hold for total linearization, i.e. considering not only small deflection amplitudes but also small rigid body motion γ. ˙ This is easily seen for γ˙ = Ω = const. One obtains then from eq.(5.111) the partial differential equation ρA(¨ v −Ω2 v)+(EIz v 00 )00 = 0, (5.115)

4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 5.20.

Tip Trajectory

166

5 Elastic Multibody Systems – The Partial Differential Equations

which indicates that here the centrifugal force ρAΩ2 v acts outwards, in the direction of v, and thus destabilizes the beam. This is, of course, physical nonsense.

Conclusion For arbitrary rigid body motions, the procedure for the determination of the equations of motion (and the corresponding boundary conditions) have to be extended. The reason for the above incorrect motion equation is that the equations are valid only for total linearization. Then, the deformation amplitudes are to be assumed small as well as the rigid body motion such that quadratic terms vanish. In that case, eq.(5.115) reduces to the simple beam equation (5.55) for ρI → 0. In general, however, the rigid body motions are not small. Considering the corresponding partial linearization (w.r.t the deflections only), the above equations are incomplete: the longitudinal action of the centrifugal force which stretches the beam does not enter the calculation. This is a consequence of the linear deformation approach where longitudinal deformations do not arise. However, by nearer inspection, the element center does not move straight in the y-direction but on an arc yielding also a longitudinal deflection. Considering a chain, for instance, where any individual chain link i with length ∆Li rotates about its joint with angle ϕi , leads locally to an elongation in the x-direction ∆Li (cos ϕi − 1) ' −∆Li ϕ2 /2. P At a certain location n, these deflections sum up to − n ∆Li ϕ2 /2. Con0 sidering R x a0 2beam leads, with ϕ = v , via the limiting process ∆Li → dx to − o v /2 dξ. This kind of deflection is of second order. However, the corresponding component of the centrifugal force is of zero order w.r.t. the elastic deformations, thus yielding first-order terms for the virtual work. For the general case one has to expect more such zero-order forces/torques. It is therefore suitable to examine generally second-order displacement fields.

5.6

167

Dynamical Stiffening

6. Dynamical Stiffening The Cauchy Stress Tensor 6.1. The most general way to determine the second-order displacement fields is to reconsider the elastic potential in the form

δV =

ZZZ X Vol

"

x

∂ui ∂xj

!

+

∂uj

u r

i,j

(5.116) where σij are the components of the so-called Cauchy stress tensor, and the corresponding strain tensor components ij read 1 ij = 2

r

σij δij dxdydz xo ro

!#

, ∂xi (5.117)

Figure 5.21.

On Deformation

{i, j} ∈ {1, 2, 3}, see section 5.1, p.115. The deformation u is once more that of an arbitrary point. Assuming small stress (and strain), the integration in eq.(5.116) can be carried out directly as has been done in section 5.1. This goes along with the assumption that the equilibrium conditions (and the equations of motion, respectively) may approximately be considered for the undeformed element. However, under the influence of large forces or in case of large deformations, the problem arises that the mass element continuously changes its shape.

6.2.

The Trefftz (or 2nd Piola-Kirchhoff) Stress Tensor

To enable integration, the description has to be reduced to a known reference state which is denoted o. In the sequel we choose the undeformed state for reference. This kind of procedure is called the Lagrangean approach (see Fig. 5.22): The actual position of the mass element is r = ro + x = ro + (xo + u).

(5.118)

Deformation Gradient. Derivation of eq.(5.118) w.r.t. xo turns an undeformed line element dxo to dx, dx = Fdxo ,

(5.119)

168

5 Elastic Multibody Systems – The Partial Differential Equations

where the functional matrix F=

∂r

!

"

= E+

∂xo

s

deformed dx

s

s

x u xo s

dxo

s undeformed

∂u

!#

(5.120)

∂xo is called the deformation gradient. Volume Change. Eq.(5.119) represents the mapping of the undeformed line element into the deformed state. With this, the deformation kinematics is, in principle, totally defined. Especially, the volume element may be calculated via the parallelepiped formed by its border lines. It reads, before deformation, dVol,o = (ee1 e2 )T e3 dxo dyo dzo . (5.121)

After deformation, the basis vectors turn to ei dxoi → Fei dxoi . Denoting the column vectors of F by gi [“Gittervektoren” (R. Kappus, 1939) = lattice vectors], F = [g1 | g2 | g3 ], yields the deformed volume element dVol , Figure 5.22.

On the Langrangian Approach

dVol = (eg1 g2 )T g3 dxo dyo dzo = det(F) dVol,o .

(5.122)

Furthermore, due to mass conservation, one has ρo dVol,o = ρ dVol ⇒

ρo = det(F). ρ

(5.123)

Area Change. The (cartesian) normal vectors yield dAoi = eej ek dxjo dxko (i, j, k ∈ {1, 2, 3}, cyclic) ⇒ dAi = egj gk dxjo dxko .

(5.124)

(5.125)

Because gTi egj gk = det(F) and gTj egj gk = gTk egj gk ≡ 0, premultiplying eq.(5.125) with FT = [g1 g2 g3 ]T yields

5.6

169

Dynamical Stiffening 

T

F dAi = det(F) ei dxjo dxko = det(F) dAoi ∀ i ⇒

dAo dA



=

ρ T F . ρo (5.126)

Stress Tensor. Along with the chain rule, the Cauchy stress tensor may be rewritten dfs dA Index s denotes stress. Here, 



σ=



=



dfs dfso

dfso dAo



dfso dAo



dAo . dA 

(5.127)



=P

(5.128)

represents the Trefftz16 stress tensor, mostly referred to as “2nd Piola17 Kirchhoff stress tensor”18 . Along with eq.(5.126) and the force projection dfs = Fdfso one obtains σ = FPFT

ρ , ρo

σ = σ T ⇒ P = PT .

(5.129)

Green19 -Lagrange Strain Tensor. Insertion into eq.(5.116) yields, along with Fij = (∂ri /∂xoj ) according to eq.(5.120),

δVel =

Z X Vol

σmn δmn dVol

m,n

 Z X X ρ  =  ρo m,n Vol

16 Erich

i,j

∂rm ∂xoi

!

Pij

∂rn ∂xoj

! 1  δ

2

∂um ∂xn

!

+

∂un  ∂xm 

dVol

Immanuel Trefftz, *1888 in Leipzig, †1937 in Dresden Piola, *1791 in Milano, †1850 in Giussano 18 This term is obviously due to C.Truesdell (The Classical Field Theories, (new) “Handbuch der Physik” III/1, 1960, p. 553: The equations of motion expressed in terms of a reference state). There he refers to contributions of G.Piola (Milano 1833) and G.Kirchhoff (Vienna 1852). However, in the “Enzyklop¨adie der mathematischen Wissenschaften IV” , Vol.4., p.23: one reads “... Piola tried an approach which has still to be proven.” (C.H. M¨uller, E. Timpe: Grundgleichungen d. math. Elastizit¨atstheorie). These authors refer to a later contribution of Piola from 1848. As a matter of fact, the corresponding results have been published by (E. Trefftz, 1931,1933). E.Trefftz is not mentioned by Truesdell. 19 George Green, *1793 in Sneinton, †1841 in Nottingham 17 Gabrio

170

5 Elastic Multibody Systems – The Partial Differential Equations

Z

= Vol,o

 X

X1

Pij

2

m,n



i,j

∂um

δ

∂un

+

∂xn

!"

∂rm

∂xm

|

!

∂xoi

∂rn

!#  

∂xoj



{z

dVol,o .

}

δij

(5.130) The underbraced term represents the strain components δij assigned to Pij which still contain derivations w.r.t. xi , while a representation in xoi is demanP ded. However, having l (∂rk /∂xP ol )dxol = dxk ∀k in mind [see eq.(5.119)], one may construe a quadratic form i,j δij dxoi dxoj from which the requested ij may easily be extracted: X 1

2

i,j,m,n

=

X1 m,n

δ

=

!

+δ

= δ

2

!#

dxoi dxoj

!#

∂un

dxm dxn

dxm dxn +

X m

X

δdui dxi =

i

  1

∂rn ∂xoj

!

X

i

!

∂xoi

∂xm

∂xm

dδui dxi =

∂rm

∂xm

X ∂δun m

!"

∂un

∂xn

1 X 2 n X

∂um

δ

2

+

∂xn

"

"

=

∂um

X ∂δum

∂xn

n

!

dxn dxm

δdxi dxi = δ

X dx2 i

i

dxT dx − dxTo dxo



i

h

:= δ dxTo G dxo

#

2

i

(5.131)

= δduk [see eq.(2.30)] and, from eq.(5.118), where dδu Pk n (∂um /∂xn )dxn = dum = dxm − dxom ⇒ δdum = δdxm has been used. Eq. (5.131) has been replenished with −δdxoi = 0 to obtain an “objective” strain measure which vanishes for pure rigid body motions: u → 0 ⇒ dx → dxo . The strain tensor G which is assigned to the Trefftz stress tensor is commonly referred to as the Green-Lagrange tensor. Using eq.(5.119) one obtains h

i

δ dxTo G dxo = δ



1 (dxT dx − dxTo dxo ) = δ dxTo 2 



1 1 ∂u ⇒ G = (FT F − E) =  2 2 ∂xo

!

+

∂u





!T

1 T (F F − E) dxo 2 

∂u

+

∂xo

!T

∂xo

∂u ∂xo



! 

or, in components, 1 ij = 2

"

∂ui ∂xoj

!

+

∂uj ∂xoi

!

+

3 X

∂uk

k=1

∂xoi

!

∂uk ∂xoj

!#

. (5.132)

5.6

171

Dynamical Stiffening

(In differential geometry, (dxT dx−dxTo dxo )/2 is called “metric”). The elastic potential according to eq.(5.116) thus becomes Z X Vol

Z

X

σmn δmn dVol =

m,n

Vol,o

Pmn δmn dVol,o .

(5.133)

m,n

Remarks: For elastic multibody systems, the Lagrangean approach is the natural one. The same as in rigid body systems, the actual state is considered displaced w.r.t. the initial state. One may also look at the deformations the other way round which involves the question: what did an actual (for instance rectangular) element look like before deformation? This kind of investigation is referred to as the Eulerian approach. It is suitable e.g. in fluid dynamics, using a control volume where the individual particle itself is not of interest. The corresponding strain tensor is called the Almansi-Tensor. According to (F. Ziegler, 1992), it was contributed by Emilio Almansi20 and Gustav Hamel. Inserting eq.(5.132) into eq.(5.133) yields 1 Vel = 2

ZZZ X 3 i,j=1

"

Pij

∂ui ∂xoj

+

∂uj ∂xoi

+

3 X ∂uk ∂uk k=1 ∂xoi ∂xoj

#

dx1o dx2o dx3o .

(5.134) Obviously, if the strains are small, one may neglect quadratic terms in eq.(5.134). For Pij → σij one is then once more left with eqs.(5.116/117). As a matter of fact, this kind of (so-called geometrical) linearization almost holds in “technical” systems. However, in contrast to small strains “which in metallic materials, within the elastic domain, are always less than some thousandths, the deflections and their first derivatives need not be small. Considering beams, plates and shells, the deformations may be large compared to its thickness without leaving the domain of linear elasticity”, (R. Kappus, 1939). There is thus no contradiction in using Hooke’s material law for large deformations in such cases. If zero-order forces occur, the corresponding stresses Pij := dfoi /dAoj := (o) Pij are zero-order terms, too. (Notice that, according to derivation, these forces are defined for the undeformed state). Then, the quadratic terms in 20 Emilio

Almansi, *1869 in Firenze, †1948 in Firenze

172

5 Elastic Multibody Systems – The Partial Differential Equations

eq.(5.134) are to be considered. The corresponding work Wn (index n for “(geometrical) nonlinear”) reads

1 Wn = − 2

ZZZ X 3

" (o) Pij

i,j=1

3 X ∂uk ∂uk k=1

#

∂xoi ∂xoj

dx1o dx2o dx3o

(5.135)

and leads to the second-order displacement fields. (o)

In contrary to the elastic restoring forces, Pij need not be related to deformations (or their derivatives). Therefore, in this context, (S. Falk, 1969) speaks of “area force density” to emphasize that Pij has basically nothing in common with deformations (as often intuitively assumed). For symmetric tensor Pij = Pji it is once more suitable to calculate the corresponding stress vector  which is assigned to the tensor elements ij . Since ij = ji , one obtains from eq.(5.134) Pij ij + Pji ji = 2Pij ij . Factor (1/2) thus cancels out for i 6= j in eq.(5.132) when vector  is defined according to eq.(4.13) [with once more writing dxi instead of dxoi for brevity]:  = (1) + (2) = (xx yy zz xy = yx yz = zy zx = xz )T                =              

∂ux ∂x





        ∂uy       ∂y         ∂uz     ∂z   +    ∂ux + ∂uy      ∂y ∂x         ∂uy ∂u   z +    ∂z ∂y       

∂uz + ∂ux ∂x ∂z

1 2

"

∂ux ∂x

2



+

∂uy ∂x

2



+

∂uz ∂x

2 # 

    !2 !2 !2    ∂uy ∂uz   1  ∂ux  + +  2 ∂y ∂y ∂y  " # 2  2  2    ∂ux + ∂uy ∂uz 1  +  2 ∂z ∂z ∂z  " #    ∂ux ∂ux + ∂uy ∂uy + ∂uz ∂uz   ∂x ∂y ∂x ∂y ∂x ∂y  " #    ∂ux ∂ux + ∂uy ∂uy + ∂uz ∂uz   ∂y ∂z ∂y ∂z ∂y ∂z      ∂uy ∂uy ∂u ∂u ∂u ∂u x

x

∂x ∂z

+

∂x ∂z

+

z

z

∂x ∂z

which is assigned to p = (Pxx Pyy Pzz Pxy = Pyx Pyz = Pzy Pzx = Pxz )T ∈ IR6 .

(5.136)

5.6

173

Dynamical Stiffening

6.3.

Second-Order Beam Displacement Fields

Considering a beam subz mitted to zero-order forces 6 *y and torques at the tip re  (o) veals P by cutting the  beam perpendicularly at location x. The cut-off r fz , M z part is in equilibrium; the 6 r *  equilibrating forces and  fy , M y r torques at its left end (L − rHHH x x) counteract on the right j f ,M H HH x x j H end x of the beam segment (L−x) x and leads to the equilibrium conditions df/dA = P(o) , Figure 5.23. Loaded beam dM/dA = erp P(o) where rp = (o y z)T [see Table 5.1] and dA = dAe1 , dA = dydz. Using (o) (o) (o) P(o) dA = (Pxx Pyx Pzx )T dA := pT dA yields Z

Z

e rp pdA = −

pdA = f,

(5.137)

erp dA = M = e p rf f + M.

(5.138)

Z

Forces and torques are assumed to act at x = L, y = η, z = ζ (see Fig. 5.23) thus leading to rf = ([L − x] η ζ)T for a lever arm w.r.t. the equilibrium point x. The forces are assumed unit forces; the torques are completed with M = const in order to generate unit torques Mio w.r.t. the neutral axis. One obtains thus 

 







η fz − ζ fy Mx Mxo       M = My  = ζ fx + M yo − fz (L − x)  :=  Myo − fz (L − x) . Mzo + fy (L − x) Mz −η fx + M zo + fy (L − x) (5.139) Assuming p = c1 + c2 x + c3 y + c4 z, ci ∈ IR3

(5.140)

174

5 Elastic Multibody Systems – The Partial Differential Equations

yieldsR for eq.(5.137)R c1 A = f Rwhen a central main axes system is used, since R then dA = A and xdA = ydA = zdA = 0. Eq.(5.138) becomes 

Z







0 0     ∼ [c1 + c2 x + c3 y + c4 z] rp dA = ec3  Iz  + ec4  0  = −M (5.141) Iy 0

with y 2 dA = Iz , z 2 dA = Iy and xydA = yzdA = zxdA = 0. Here, Iy , Iz are the area moments of inertia, see eq.(5.18). The first component of eq.(5.141) reads, with Mx from eq.(5.139), ηfz − ζfy = c33 Iz − c42 Iy . Considering a flatRbeam, which, of course, also belongs to the chosen modeling class, (z → 0 ⇒ z 2 dA = Iy → 0 and ζ → 0) yields ηfz = c33 Iz . One may thus conclude that c33 = ηfz /Iz and c42 = ζfy /Iy . Consequently, eq.(5.140) will be fulfilled with R

R

R

R

R





c1

c2

 



fx 1  =  A  fy  fz



c3 

0    0  0



c4 

−Mz  1 0  Iz  ηfz





My  1 Iy  ζfy  . 0

(5.142)

Eq.(5140) thus yields, along with My , Mz from eq.(5.139),  

p= 

(o) Pxx (o) Pyx (o) Pzx

fx − [M + f (L − x)] y + [M − f (L − x)] z  yo z zo y Iz Iy   A    ζf f y y = . z +    A Iy   fz + ηfz y A Iz (5.143) 



The corresponding second-order strains are calculated from eq.(5.136) by e p according to eq.(5.13). For an inextensible once more inserting u = u + ϕr beam (u(x, t) = 0) one obtains 

(2)

(2)



  00   (v y + w00 z)2 + (v 0 − ϑ0 z)2 + (w0 + ϑ0 y)2 /2  (2)     (v 00 y + w00 z)v 0 + ϑ(w0 + ϑ0 y) = .  yx  = 

xx

(2)

zx

(v 00 y + w00 z)w0 − ϑ(v 0 − ϑ0 z)

(5.144) Eqs.(5.143) and (5.144) lead to the corresponding work (5.135)

5.6

175

Dynamical Stiffening

Wn = −

= −

ZZZ Vol ZL  o

pT (2) dxdydz

1 fx 1 fx 1 1 fx 2 2 2 2 2 Iz v 00 + Iy w00 + Ip ϑ0 + fx (v 0 + w0 ) 2A 2A 2A 2

− Myo ϑ0 v 0 − Mzo ϑ0 w0 + fz (L − x)v 0 ϑ0 − v 0 ϑ 

− fy (L − x)w0 ϑ0 − w0 ϑ 



 #

+

0 00

0

00

0

ηfz w v + ζ fy v w + (ηfz + ζfy ) ϑ ϑ dx,

(5.145)

where Ip = (y 2 + z 2 )dA denotes the polar moment of inertia. The first three summation terms are negligible in comparison to the elastic potential21 . Furthermore, since the actual loads shall be unit forces and torques, one concludes from ηfz − ζfy = Mxo along with | fz |=| fy |= 1 that ηfz = Mxo /2, and ζfy = −Mxo /2. Thus, the last term vanishes and the two penultimate terms go along with Mxo leading to R



ZL

Wn =

(fx o

fy

ZL

+



−(v 0 2 + w0 2 )/2   fz )  +(L − x)ϑ0 w0 + ϑw0  dx −(L − x)ϑ0 v 0 − ϑv 0 (v 0 w00 − w0 v 00 )/2  dx. ϑ0 v 0 Mzo )  0 0 ϑw 

(Mxo Myo o



(2)

(5.146)

Eq.(5.146) represents Wn in the form Wn = fT drc + MT dϕ(2) which still depends on the applied load. Reformulating eq.(5.146) to Wn = R T (2) [(df/dx)T rc + (dM/dx)T ϕ(2) ]dx enables us to determine the secondorder displacements where it does not matter what the actual load distribution (df/dx), (dM/dx) looks like. This is achieved with an integration by parts yielding R

R

R

R

example: compare (fx /2A)Iz v 00 2 dx with the bending potential (E/2)Iz v 00 2 dx according to eq.(5.18): Along with E ' 7 · 1010 N/m2 (aluminum) and a cross sectional area A = 1 cm2 the longitudinal force fx ∼ EA would have to take a value ∼ 70 kN to influence the bending potential by just 1%. 21 Numerical

176

5 Elastic Multibody Systems – The Partial Differential Equations Rx



w0 2 )/2 dξ

− +  o  x  R  + [(x − ξ)ϑ0 w0 − ϑ w0 ] dξ  o   Rx 0 0 0

ZL

(df/dx)T

Wn =

(v 0 2

o

− [(x − ξ)ϑ v − ϑ v ] dξ

     dx   

o

|

{z

Rx

 ZL

(dM/dx)T

+ o

}

r(2) c  +o      

(v 0 w00

−

w0 v 00 )/2 dξ

     dx   

Rx

+ ϑ0 v 0 dξ o Rx

+ ϑ0 w0 dξ

(5.147)

o

|

{z

ϕ [ fx− δ (x − L)

(df/dx)T =

where

}

(2)

fy− δ (x − L)

fz− δ (x − L) ],

δ (x − L) Myo− δ (x − L) Mzo− δ (x − L) ]. (dM/dx)T = [ Mxo− Here, − δ represents the Dirac-distribution (cross lined to distinguish from variation). Eq.(5.147) is confirmed by applying the integration rule ZL

Zx

g(x)

ZL

h(ξ)dξ dx =

o

o

 L  Z h(x)  g(ξ)dξ  dx

o

(5.148)

x

which brings eq.(5.147) back to eq.(5.146). This integration rule plays a central role in elastic beam analysis22 . Remark: The second-order “angle vector” ϕ(2) is sometimes calculated via the so-called Kirchhoff analogy: Calculate first the geometric mean of the angular velocities w.r.t. the inertial frame which is built by a sequence of rotations about the z-y-x-axes and about the y-z-x-axes,

22 Proof:

RL

a0 bdx = [ab]L o −

o

 x R =

R o

a b0 dx; a0 = g(x), b =

o

 g(ξ)dξ + c

o L

RL

·

Rx

L

h(x)

o

−

h(ξ)dξ

 g(x)dx dx +

o

RL o

h(ξ)dξ:

o

o

L R

Rx

h(x)

RL

 x R

o o R x

 g(ξ)dξ + c

h(x)dx

o

 g(ξ)dξ dx =

RL o

h(x)

L R

 g(ξ)dξ dx.

x

According to (M. Braun, 1997), eq.(5.148) represents a special case of Fubini’s theorem [Guido Fubini, *1879 in Venezia, †1943 in New York]. Hamel reports that eq.(5.148) was already known to Johann P.G.L. Dirichlet (*1805 in D¨uren, †1859 in G¨ottingen), see (G. Hamel, 1949), p.279.

5.6

177

Dynamical Stiffening

ω = 12 [(ATβ ATγ + ATγ ATβ )e1 | (E + ATγ )e2 | (E + ATβ )e3 ](α˙ β˙ γ) ˙ T where Aϕi , ϕi ∈ {α, β, γ}, are the elementary rotation matrices according to eq.(3.18) et seq. Linearizing yields ATϕi → (E + eei ϕi ), see eq.(3.30). Along with eei ej = ek (i, j, k cyclic) one obtains ω = [(e1 + γe2 − βe3 ) | (e2 − 12 γe1 ) | (e3 + 12 βe1 ](α˙ β˙ γ) ˙ T. (Such a geometric mean corresponds to the use of “semitangential angles” in Finite Element analysis.) Replace α → ϑ, β → (−w0 ), γ → v 0 . Next, change variables t → x ⇒ ω = (dϕ/dt) → (dϕ/dx) (= ˆ Kirchhoff’s analogy) and integrate to come out with    1 0 1 0  ϑ0 1 − v − w x 2 2 R   ϕ = ϕ(1) + ϕ(2) =  v 0 1 0   −w00 dξ, o w0 0 1 v 00 yielding the second-order angular deflection as in eq.(5.147). Although seemingly easy in derivation, such a procedure needs some more R explanation, especially because ω is non-holonomic such that integrating ωdt does not make senseR [see eq.(3.56)] except for the linear part. The meaning of the nonlinear part, (ωdt/dx)dx, does not become clear by this interpretation. We therefore prefer the derivation via the work performed by unit loads which can, just the other way round, be seen as a validation of the so-called Kirchhoff analogy. Remark: Considering the first-order strain (5.15) for u(x) =  0 one obtains    00 00 0 −w z − v y ϑ RL R T (1) RL R T  RL    −ϑ0 z p  dAdx = p  dAdx = MT  −w00 dx o A o A o +ϑ0 y v 00 RL with M from eq.(5.139). Inserting M and noticing o fy (L − x)v 00 dx = RL 0 o fy v dx (integration by parts according to eq.(5.148), terms containing fz analogously) one obtains  by the same  procedure as above   ϑ 0    T L RL ∂M T  R    ∂f  −w0 dx +  v dx, o o ∂x ∂x w v0 yielding the first-order deflections [compare eq.(5.45)].

6.4.

Dynamical Stiffening Matrix

Reconsidering the Projection Equation (5.37) it becomes clear that, if within  !T  ∂v c   ∂ y˙

∂ω c ∂ y˙

!T  " 

e o dp − dfe − dfc dp˙ + ω e o dL − dMe − dMc dL˙ + ω

# 

(5.149)



e o dp − dfe ) and/or the angular mothe linear momentum balance (dp˙ + ω e e o dL − dM ) contains zero-order terms (w.r.t. the mentum balance (dL˙ + ω deformation variables), then, for correct linearization, the functional matrix

178

5 Elastic Multibody Systems – The Partial Differential Equations

˙ T (∂ω c /∂ y) ˙ T ] needs development up to the first order. Since the de[(∂vc /∂ y) formations rc , ϕ undergo derivation, these need a second-order representation: (o) (1) (2) (o) rc = rc + rc + rc , ϕ = ϕ(o) + ϕ(1) + ϕ(2) , where rc = (x 0 0)T and (o) ϕ ≡ 0. Inserting this representation into eq.(5.20) yields the second-order velocities vc ωc

(o)

!

(1)

(2)

=

 "

=

!

(2) e o [rc + rc + rc ] + r˙ (1) vo + ω c + r˙ c ˙ (1) + ϕ ˙ (2) ωo + ϕ

T E [er(o) r(1) r(2) c +e c +e c ] 0 E

E 0 0 0 E 0

vo ωo r˙ (1) c ˙ (1) ϕ κ˙

#     

    +  

r˙ (2) c ˙ (2) ϕ

!

. (5.150)

Because first-order functional matrices are required, one obtains from eq.(5.150) vc ωc

!

"

=

T E [er(o) r(1) c +e c ] 0 E

E 0 0 0 E 0

{z

|

(2)

#

y˙ +

∂rc /∂y ∂ϕ(2) /∂y

!

˙ (5.151) y.

}

F Hence, F remains unchanged and the additional term may be calculated separately. In the sequel, stretching will be neglected. F in eq.(5.20) is chosen F = E which allows one to calculate the dynamical stiffening matrix for an inextensible beam with the full range of possible deflections: y˙ = (vTo , ω To , | 0, v, ˙ w, ˙ | ˙ −w˙ 0 , v˙0 , | ϑ˙ 0 , v˙ 00 , w˙ 00 ). For a problem with fewer deformations, the nonϑ, interesting components are eventually canceled. Formulating the virtual work related to eq.(5.37) yields, when eq.(5.151) is inserted, for the second-order displacements

−δWn :=

Z

δyTi dKni yi

(5.152)

Bi

 Z

= Bi

δyTi



(2) ∂rc

∂y

!T

∂ϕ(2) ∂y

!T   i|

!(o)

e o dp − dfe dp˙ + ω e o dL − dMe dL˙ + ω {z 

− df

(o) T

dM

(o) T

i

T i

}

5.6

179

Dynamical Stiffening

where dKni denotes the dynamical stiffening matrix [index n for (geometrical) nonlinearity]. Here, (partial) linearization w.r.t. the deformations only needs zero-order reactions which are indicated by upper index (o). (2) Calculating δrc , δϕ(2) from eq.(5.147) yields  

δ

  Zx    = δ   o   

(2) rc

ϕ(2)

− 12 (v 0 2 + w0 2 ) 0 ϑ w0 (x − ξ) − ϑw0 −ϑ0 v 0 (x − ξ) + ϑv 0 1 0 00 2 (v w

− w0 v 00 ) 0 ϑ v0 ϑ0 w 0

      dξ    

(5.153)

(index i suppressed). Partitioning y˙ according to T T y˙ = (vTo ω To | y˙ el ),

(5.154)

eq.(5.153) may be expressed in terms of yel , yTel := (x, v, w, | ϑ, −w0 , v 0 , | ϑ0 , v 00 , w00 ),

(5.155)

which yields (2)

δ

rc ϕ(2)

Zx

!

=

[F1 + F2 (x − ξ)] δyel dξ

(5.156)

o

with functional matrices Fi , i = 1, 2, which depend linearly on yel : 

0 −w0 v0

w0 ϑ 0

−v 0 0 ϑ

0

0

v 00 2

0 0 0 0

0 0

0 −ϑ0

w00 2 ϑ0

0 0 0 0 0 0 0 0 0

    F1 =   0 0    0 0



0 0 0

    F2 =   0    0

0

0

0 0 0

0 0 0

0 0 0

0

− w2

v0 w0

0 0

0



     v0  , 2   0 

(5.157)

0

0 0 0

0 0 0

0 0 0

0 −ϑ0 0

0 0 −ϑ0

0 w0 −v 0

0 0 0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0 0



    . 0    0 

0

(5.158)

180

5 Elastic Multibody Systems – The Partial Differential Equations

Remark: Eq.(5.156) may be integrated for the second and third line23 yielding Rx T 0 0 o [F1b + F2b (x − ξ)] δyel dξ + [0, −x, δ(ϑw |o ), x δ(ϑv |o ), 0, 0, 0] where 

F1b

0 0 0 0 0 0 0 0 0

0 0 0

w0 0 0

−v 0 0 0

0

0

v 00 2

w00 2 ϑ0

    =  0 0    0 0

0 0 0 0



F2b

0 0 0

    =  0    0

0

0 0 0 −ϑ0

0

0 0 0

0 0 0

0 0 0

0

− w2

v0 w0

0 0



     v0  , 2   0 

0

(5.157 b)

0

0 0 0

0 0 0

0 −w00 v 00

0 0 0

0 0 0

0 0 0

0 0 ϑ

0 −ϑ 0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

     .    

(5.158 b)

Obviously, Fi → Fib may advantageously be used for eq.(5.156) when x δ(ϑw0 |o ) and x δ(ϑv 0 |o ) vanish. This is the case for a clamped beam (v 0 (0) = 0, w0 (0) = 0, ϑ(0) = 0) attached to a revolute joint, or for a “forklike” joint (ϑ(0) = 0) but not for a free beam (prismatic joint). Along with eq.(5.156), the virtual work eq.(5.152) reads Z 

δWn =

df

(o) T

dM

(o) T





δrc





δϕ(2)



B

 ! ZL (o) T df = 

dx

o

 ! ZL  (o) T df  =  dx o

dM(o) dx

(o)

dM dx

(2)

 !T   (2) δrc   dx

δϕ(2)

 !T  Zx   [F1 + F2 (x − ξ)] δy dξ dx el 

(5.159)

o

which shows that the dynamical stiffening matrix dKn can not yet be formulated explicitly because δy has to be extracted from the inner integral. However, this is easily achieved by once more using the integration rule eq.(5.148) yielding

23

Rx

[−ϑ w0 + ϑ0 w0 (x − ξ)] dξ =

o x

−

R o

Rx o

ϑw00 (x − ξ)dξ − x [ϑw0 ]o .

d [ϑ(x − ξ)] w0 dξ = − dξ

Rx o

ϑw00 (x − ξ)dξ + ϑw0 (x − ξ) |x o=

5.6

181

Dynamical Stiffening  ZL

δWn =

δyTel

h

FT1

FT2

o

RL

 i x    RL  x

df(o) /dξ dM(o) /dξ

!

df(o) /dξ dM(o) /dξ

!



dξ (ξ − x)dξ

    dx.  

(5.160)

F1 , F2 are linear w.r.t. yel and allow for rearranging eq.(5.160) in the form δWn := −

ZL

"

δyTel

o

#

dKn,el · yel dx dx

(5.161)

yielding "

#

dKn,el := dx

ZL  x

dQn dξ



dξ

(5.162)

where (dQn /dξ) =                                  

 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(o)

0

−

(o) dfy df − z dξ dξ

+

dfx dξ

(o)

+

0 (o)

+

dfx dξ

−

(o)

h

dMz dξ

h

dMy dξ

(o)

symmetric

0

0

    0 0 0     0 0 0     0 0 0   i  (o) (o) dfy dMx  1 + (ξ − x) − 2 0 . dξ dξ  i (o) (o)  dfz dMx  1 0 −2 − (ξ − x) dξ dξ       0 0 0     0 0    0

(5.163) Recall: matrix dKn,el is assigned to yel =

x

v

w

ϑ

−w0

v0

ϑ0

v 00

w00


T

.

182

5 Elastic Multibody Systems – The Partial Differential Equations

Remark: For a clamped beam (or for a fork-like joint), eqs.(5.157 b), (5.158 b) may be used to come out with ZL

(dKn,el /dx) =

(dQn /dξ) dξ,

(dQn /dξ) =

x



 0

0

                           

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 (o) df + x

dξ

0 0 0 (o) df − z (ξ

0

0

0

(o) dMz +

(o)

df + x dξ

dξ

dξ (o) dMy − dξ

1 −2

− x)

(o) dMx

dξ 0

0

symmetric

0

0 0

   0     0  (o)  dfy + (ξ − x)   dξ   . 0   (o) dMx 1  −2  dξ      0    0   0

(5.163 b) Matrices (5.163), (5.163 b) are related to yel according to eq.(5.155). In order to adapt them to the total d.o.f. (including rigid body motions) one has to set T

dKn = blockdiag{0, dKn,el }, dKn,el = Fel dKn,el Fel ,

(5.164)

see eqs. (5.20), (5.152), (5.154). Along with eq.(5.152) one obtains the required zero-order terms in eq.(5.163) [and in eq.(5.163 b), respectively] from the linear and the angular momentum balance for yel ≡ 0 yielding dfo  dx    dMo dx 



    T dm T dm dfe e e o dm ω e oe E dm e ω e x x ˙ v o 1 1 dx    dx dx  vo   dx   dx =  − − ,   e dJ dM ωo ω˙ o e o dJ ω 0 0 dx dx dx (5.165) 











5.6

183

Dynamical Stiffening

when the mass center axis and the elastic axis are assumed to coincide with the x-axis.

6.5.

The Helicopter Blade (2)

The same as in Fig. 5.19, we now focus our attention on the effects of dynamical stiffening. Therefore, to generalize the analysis, a tip force fz is additionally implied. In view of later application, the mass distribution is denoted (∂m/∂x) and the corresponding distributed tensor of inertia reads (∂Jc /∂x) which is derived from eq.(5.46). In comparison to eq.(5.101), torsion is additionally considered which leads eq.(5.45) to



z 6

γ, ˙ Mz

 

* y    

fz r -x

Figure 5.24.





Rotating Beam with Tip Force





x ϑ      rc =  v , ϕ =  0 , κ =   0 v0

0 ϑ0  . v 00  0 



(5.166)



˙ T , κ˙ T by The necessary variables y˙ are extracted from vTo , ω To , r˙ Tc , ϕ  

vo

 ωo   r˙ c   ϕ ˙

κ˙

      =      

0 0 0 0 0 0 e3 0 0 0 0 0 0 e2 0 0 0 0 0 0 e1 e 3 0 0 0 0 0 0 e2 e3 0 0 0 0 0 0

        

γ˙ v˙ ϑ˙ v˙ 0 ϑ˙ 0 v˙ 00

      := Fy˙   

(5.167)

(except the last line, all column elements are in IR3 . Due to κ ∈ IR4 the components of the last row are scalars). The element matrices (5.38) to (5.40) are thus

184

5 Elastic Multibody Systems – The Partial Differential Equations

dM = 

∂Jz + ∂m x2  ∂x ∂x  ∂m x   ∂x        

∂m x ∂x ∂m ∂x 0

0 ∂Jz ∂x 0 0

0 0 0

0 0 ∂Jx ∂x 0 0 0



∂Jz ∂x 0

0 0   0 0  

0 ∂Jz ∂x 0 0

0 0   dx,



(5.168)

 0 0    0 0 

0 0

dG = 0 2 ∂m ∂x v   − ∂m 0  ∂x v   ∂J  −( y − ∂Jz )ϑ 0 −( ∂Jx  ∂x ∂x ∂x   0 0   0 0 0 0 



0

0

0 0

0

0

0 0 



∂J ∂Jy x − ∂xy )v 0 −( ∂J ∂x − ∂x )ϑ 0 0 0 0 0 0

  ˙ 0 0  dx γ,  0 0  0 0

0 0 (5.169)



T

e

dQ = F F

T

  !  e  df  =  e  dM   

eT3 erc eT3 eT2 0 0 eT1 0 eT3 0 0 0 0

          

dfe

!

(5.170)

dMe

where F is known from eq.(5.20). T

Check: Considering the actual mass center velocities in the present case, [vc = e rc e3 γ˙ + ve ˙ 2, ˙ 1 + v˙ 0 e3 ] leads directly to ˙ + γe ˙ = ϑe ωc = ϕ ˙ 3 where ϕ



vc ωc



 =

erTc e3 e3

e2 0

0 e1

0 e3

0 0

0 0

 y˙ = F F y˙

with which the system matrices are easily proved, (see footnote on p.163).

In dG (5.169), all columns except the first one may be set to zero because the corresponding gyroscopic force dG y˙ yields second-order terms for these. Since the beam is clamped at its left end, eq.(5.163 b) is used for dynamical stiffening. By canceling the rows and columns that are assigned to w, w0 , w00

5.6

185

Dynamical Stiffening

and completing with a zero row and column for γ˙ according to eq.(5.164), one obtains 

0   0 

   0 ZL   dKn =   0 dx  x    0   

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

(o) dfx

dξ (o) dM − dξy

0 0

(o)

dM − dξy 0

(o)

z 0 − dfdξ (ξ − x)

0

0

0



    (o)  df z − dξ (ξ − x)     dξ.  0     0   

0

(5.171) ˙ 3, The zero-order terms are calculated by eq.(5.165) with vo ≡ 0, ω o = γe (dfe /dx) = fz− δ (x − L)e3 , (dMe /dx) = Mz− δ (x)e3 . Along with eei ej = ek (i, j, k cyclic) one obtains df(o)  dx 

2 − ∂m γ e2 + ∂m ∂x x¨ ∂x xγ˙ e1  





 

=  

dM(o) dx





∂Jc(o) γ¨ e 3 ∂x



+  

fz− δ (x − L)e3 Mz− δ (x)e3

  .

(5.172)

Eq.(5.172) comprises kinetic reactions and impressed forces. However, considering the dynamical stiffening, there is no need to distinguish these for the moment. The components of eq.(5.172) are to be integrated from x to L yielding, (o) (dfz /dx) = fz− δ (x − L) :

RL RL (o) − x R(dfz /dξ)(ξ − x)dξ =R x fz (ξ − x)δ (ξ − L)dξ L L − −

=

x

fz ξδ (ξ − L)dξ − x

(o)

x

fz δ (ξ − L)dξ = fz (L − x). (o)

The torque (dMy /dx) is zero; the remaining term (dfx /dx) = (∂m/∂x)xγ˙ 2 can be integrated only if the mass distribution (∂m/∂x) is known. Assuming (∂m/∂x) = const = ρ A, as in the “helicopter blade” on p.162, one obtains RL x

(o)

(dfx /dξ)dξ =

RL x

ρAγ˙ 2 ξdξ = ρAγ˙ 2 (L2 − x2 )/2.

The dynamical stiffening matrix (5.171) then reads

186

5 Elastic Multibody Systems – The Partial Differential Equations 

0 0 0

0

ρA γ˙ 2 (L2 − x2 ) 0 0 0 2 0 0 0 0 0 −fz (L − x) 0 0

    dKn  =  dx  0   0 

0 0 0



0 0 0

0 0 0

0 0   0 0   0 −fz (L − x) 

.     

0 0 0

(5.173)

Disposing of all components one may now calculate dM ¨y + dG y˙ + dKn y + (∂dVel /∂y)T − dQe := dQc ,

(5.174)

where the potential forces are extracted from dQe . Doing so, eq.(5.170) reads explicitly 





0 1    0 0   e   − dQ 0   fz δ (x − L)  T T  =F F      1 0 dx      0 0 − 0 Mz δ (x) (5.175) (the impressed force fz does not contribute here). The potential forces are, with dVel according to eq.(5.18), −v  !  0  dfe /dx  0 =  e  0 dM /dx   0 0

∂dVel /dx ∂y

x 1 0 0 0 0

0 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

!

=

0 0 0 0 GID ϑ0 EIz v 00




.

(5.176)

Thus, dQc reads      c dQ =    

(ρIz + ρAx2 )¨ γ + ρAx¨ v + ρIz v¨0 − Mz− δ (x) ρA¨ v + ρAx¨ γ − ρAv γ˙ 2 ρIx ϑ¨ − ρ(Iy − Iz )ϑγ˙ 2 − fz (L − x)v 00 ρIz v¨0 + ρIz γ¨ + ρA ˙ 2 (L2 − x2 )v 0 − Mz− δ (x) 2 γ GID ϑ0 EIz v 00 − fz (L − x)ϑ

      dx.   

(5.177)

5.6

187

Dynamical Stiffening

The next calculation steps are as usual: Define the operator D according to y˙ = D ◦ s˙ ,       y˙ =    

γ˙ v˙ ϑ˙ v˙ 0 ϑ˙ 0 v˙ 00

1 0 0 0



          =       0  

0

0 1 0 ∂ ∂x 0



0 0 1 0

        ∂   ∂x  

∂2 ∂x2



γ˙ v˙   := D ◦ s˙ . ˙ ϑ

(5.178)

0

Changing sign for the odd derivatives yields the differential operator D for the equations of motion, DT ◦ dQc = 0, and successive derivation grade degeneration yields, along with once more changing sign, the operators Bi for the determination of the (dynamical) boundary conditions, Bi T ◦ dQc = 0: 

1 0 0

0  ∂  DT =  0 1 0 − ∂x  0 0 1 0 

0 0 0  0 0 0 BoT =  

0 1

0 0 0

0



0 0 0 B1T =  0 0 0 0 0 0

0

0 ∂2 ∂x2 0

0

−∂ ∂x 0 0 0 −∂ ∂x 1 0

0 0 0

0 0 0

0 1 0

   ,    ,   .

(5.179)

Applying these to eq.(5.177) yields the motion equations ZL

2

ZL

(ρIz + ρAx )dx γ¨ + o

ZL

ρAx¨ v dx + o

ρIz v¨0 dx = Mz ,

o

ρA¨ v + ρAx¨ γ − ρIz v¨00 − ρAv γ˙ 2 −

i0 ρA 2 h 2 γ˙ (L − x2 )v 0 2

− [fz (L − x)ϑ]00 + EIz v 0000 = 0, ρIx ϑ¨ − ρ(Iy − Iz )ϑγ˙ 2 − fz (L − x)v 00 − GID ϑ00 = 0 (5.180) 



188

5 Elastic Multibody Systems – The Partial Differential Equations

and the boundary conditions ρIz γ¨ + ρIz v¨0 + [fz (L − x)ϑ]0

x = 0, L : v = 0

_

x = 0, L : ϑ = 0

_

GID ϑ0 = 0,

x = 0, L : v 0 = 0

_

EIz v 00 − fz (L − x)ϑ = 0.

+ ρA ˙ 2 (L2 − x2 )v 0 − EIz v 000 − Mz− δ (x) = 0, 2 γ

(5.181)

Since the beam is clamped at x = 0, the boundary conditions become x=0:

v=0 ϑ=0 v0 = 0

x = L : ρIz (¨ v 0 + γ¨ ) − fz ϑ − EIz v 000 = 0 GID ϑ0 = 0 (5.182) EIz v 00 = 0.

1.5 1 0.5 0 -0.5 -1 -1.5 -1.5 -1 -0.5 Figure 5.25.

0

0.5

1

1.5

Tip Trajectory, Mz =const.

For fz = 0, the equations for bending and torsion are decoupled. Neglecting furthermore ρIz (⇒ Euler beam), one may once more construe a Galerkin approximation by weighting the motion equations with the shape functions and averaging over the beam length. Then, in comparison to eq.(5.112), second line, the underlined expression in eq.(5.180) has additionally to be taken into account yielding −[(ρA)/2]γ˙ 2 × RL 2 2 0T 0 o [v] [(L − x )v ] dx qR .

Integration by parts yields   ZL

  

ρA 2 T T γ˙ (L2 − x2 )v0 v0 dx − v v0 (L2 − x2 ) |L o  qR  {z } | 2  o

(5.183)

=0

where the boundary terms vanish due to v(0) = 0, v(L)v0 (L)T (L2 − L2 ) = 0. Considering the remaining term in eq.(5.113) avoids (of course) instabilities.

5.6

189

Dynamical Stiffening

Discussion/Classical Results Let, for simplicity, γ˙ = Ω = const. Along with ρIy → 0 and ρIz → 0 one obtains from eq.(5.180) and eq.(5.182) the equations of motion of an Euler beam attached to a rotating base with a tip load at its free end:

PDE



ρA¨ v − ρAΩ2 v + EIz v 0000





ρIx ϑ¨ − GID ϑ00



 0 0 − [fz (L − x)ϑ]00 − ρA Ω2 (L2 − x2 )v 0 2   = , + 00 0 − [fz (L − x)v ] (5.184)         000 0 0 −EIz v − fz ϑ v       EIz v 00 =  0 ,  v 0  =  0 . 0 +GID ϑ ϑ x=0 0 0 x=L (5.185) 

BC’s







Case 1: fz = 0. If the tip force becomes zero, then bending and torsion are independent motions. The “dynamical stiffening ” [second summation term in eq.(5.184)] stabilizes bending (due to the centrifugal force component in the longitudinal direction). If this term is disregarded, then the only remaining centrifugal force component in the transverse direction seems to destabilize the beam. This perception has been commonly attributed to contemporary authors, primarily to (T.R. Kane et al., 1987) who initiated a considerable number of papers on rotating Euler beams. Mostly, the authors hereby refer to “dynamical stiffening” as a new theory instead of saying what it is: correct linearization, or, omitting the corresponding terms, nothing but a severe mistake. Clearly, many authors before treated such problems correctly, e.g. (L. Meirovitch, 1970), (F. Weidenhammer, 1970), (P. Lohmeier, 1974). One should also have in mind that already the great Leonhard Euler was well aware of the effects of longitudinal forces, (L. Euler, 1744). Case 2: Ω = 0. Considering a non-rotating beam with a tip load, bending and torsion remain dependent. This case has for instance been investigated by (E. Mettler, 1947) or, for the static case, by (L. Prandtl, 1899) and by (E. Trefftz, 1933). The dynamical case is e.g. considered in (A. Truckenbrodt, 1980) where fz corresponds to a tip weight, fz = −mtip g. There, the chosen parameters diminish the lowest torsional frequency by about 30 %. Actually, this bendingtorsion coupling weakens the beam.

190

7.

5 Elastic Multibody Systems – The Partial Differential Equations

Multibody Systems – Gross Motion

A multibody system is characterized by eq.(5.41) which after completion with the dynamical stiffening matrix reads24 N Z X i=1B

δyTi [dM¨y + dGy˙ + dKn y − dQe ]i = 0,

(5.186)

i

or, in short, XZ

δyTi dQci = 0.

(5.187)

i B i

The same as in rigid systems, y˙ i enters eq.(5.186) with the kinematic chain h

i

y˙ i = Tip y˙ p

e

+ y˙ i,rel

(5.188)

where e denotes the coupling point between predecessor p and its successor i.

7.1.

The Kinematic Chain

For general spatial motion, any element dm (e.g. a beam slice) moves with vc , ω c where c denotes the mass center of the element. Within body p, one element coincides with the coupling point of the successor i (e.g. the tip element of a beam, once more assuming that the interconnection is located on the xaxis). Characterizing the interconnection point with index e (=“end of” body p), the corresponding velocities represent the guidance velocities for the successor i but represented in frame p:

p

vc ωc

!

= p

pe

vo ωo

!

.

(5.189)

i

The general case is represented by eq.(5.20) by25 vc ωc

!

= Fe y˙ e ,

(5.190)

e

from which the actual describing variables y˙ are extracted via ˙ y˙ = Fy,

(5.191)

24 Recall that K y only contributes to the holonomic part y . Therefore, vector y (without dot) may formally n el be used also in case of non-holonomic rigid body coordinates. 25 When compared to eq.(5.151), the missing term from second-order displacements is once more shifted to the “dynamical stiffening”, see section 7.3.1.

5.7

191

Multibody Systems – Gross Motion

thus leading eq.(5.189) to !

vo ωo

p

h

= Fe Fe

i

y˙ pe .

p

i

(5.192)

For representation in frame i, eq.(5.192) becomes

i

vo ωo

!



= i

Aip 0 0 Aip

h

Fe Fe

i p

y˙ pe .

(5.193)

Disposing of the guidance velocities, eq.(5.193) may be completed with the elastic variables to obtain y˙ i : 



 

vo ω    o    ˙yi =   r˙ c  =      ϕ  ˙  κ˙ i

Aip 0 0 Aip

0

  i  vpi Fe Fe  ω  p   pi      y˙ pe +  r˙ c  .      ϕ ˙ 

h

κ˙

(5.194)

i

The describing variables y˙ are calculated with eq.(5.191). Partitioning into independent parts y˙ T = (y˙ Trb y˙ Tel ) yields "

Frb 0 y˙ i = 0 Fel

#

y˙ rb y˙ el

i

!

⇔ i

y˙ rb y˙ el

! i

"

T

Frb 0 = T 0 Fel

#

y˙ i ,

(5.195)

i

see eq.(5.20). The kinematic chain in terms of describing variables y˙ is thus T obtained by premultiplying eq.(5.194) with F yielding 

T

 Frb 



Aip 0 0 Aip

h

Fe Fe

 y˙ i =   

0

i   p  h i   y˙ pe + y˙ i,rel := Tip y˙ p + y˙ i,rel .  e 

(5.196) Plane Rigid Body Motion. In this case, the rigid body motion is determined by 

y˙ rb





vox eT    T1 =  voy  =  e2 0 ωoz



0  0  T e3

vo ωo

!

T

= Frb y˙ rb

(5.197)

192

5 Elastic Multibody Systems – The Partial Differential Equations

yielding T Frb



Aip 0 0 Aip



T

= Aip Frb

(5.198)

(proof by direct calculation) which reduces Tip to 

Aip [Fp ]e

Tip = 

 T

,

where [Fp ]e := [(Frb F F)p ]e .

0

(5.199)

5 Example: In-Plane Motion. Consider the in-plane deformation v(x, t) of a beam which is moved with vo = vox e1 + voy e2 , ω o = ωoz e3 . Eq.(5.195) yields

   y˙ =  



vo ωo r˙ c ˙ ϕ κ˙



     =   

e1 0 0 0 0 0

e2 0 0 0 0 0

0 e3 0 0 0 0

0 0 e2 0 0 0

0 0 0 e3 0 0

0 0 0 0 e3 0



vox voy ωoz v˙ v˙0 v˙00

     

     Frb 0  y˙ (5.200)  := 0 Fel  

[all elements ∈ IR3 except the last row which contains scalars due to κ ∈ IR4 , see eq.(5.45)]. The Jacobian F is derived from eq.(5.20),

 F=

erTc

E 0

E

E 0

0 E

0 0



where rTc = (x v 0). Calculating F F yields

 

e1 0

FF =

e2 0

erTc e3

0 e3

e2 0

e3

0 0



   =  

1 0 0 0 0 0

0 1 0 0 0 0

−v x 0 0 0 1

0 1 0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

    .  

(5.201)

T

By premultiplication with Frb from eq.(5.200) one obtains



eT1  eT2 0



0 0  [F F] = eT3

"

1 0 0

0 1 0

−v x 1

0 1 0

0 0 1

0 0 0

#

(5.202)

:= [F].

The predecessor values are xpe = LP , vpe = vp (Lp ) yielding

 " Tipe =

Aip [Fp ]e 03×6

#

   =   

cos γpi sin γpi 0 − sin γpi cos γpi 0 0 0 1

!"

1 0 −vpe 0 0 0 0 1 Lp 1 0 0 0 0 1 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

#    .   

(5.203)

5.7

193

Multibody Systems – Gross Motion

Interpretation: Calculating [F]y˙ with eq.(5.202) and y˙ from eq.(5.200) yields ωoz x + v˙ vox − ωoz v(x, t) voy + ωoz x + v(x, ˙ t) ωoz + v˙ 0 (x, t)

! .

(5.204) Taking the tip value x = L, eq.(5.204) represents the guidance velocities for a successor attached at x = L which eventually are to be transformed into the successor frame. (Secondorder effects not yet considered, see footnote on p.190).

y, voy

@ I @ @ v(x, t) x, v ox I @ ωoz v @ 

@ I @ @

ωoz

@ @f Figure 5.26.

q

Velocities at Point x

4

7.2.

Minimal Velocities

The same as in the case of a rigid body system, the minimal velocities are extracted from y˙ i,rel ∈ IRni in eq.(5.188). Partitioning into the independent rigid body motion [y˙ rb ]i,rel ∈ IRnrb,i and the elastic motion [y˙ el ]i,rel ∈ IRnel,i , the rigid body part is simply projected into the minimal space with the corresponding Jacobian, [∂ y˙ rb,rel /∂ s˙ ]i s˙ i := [Frb 0]i s˙ i where Frb,i is the local rigid body functional matrix, the same as in eq.(4.97). The elastic relative motion is once more represented with the spatial operator [0 D]i , "

y˙ i,rel =

Frb

0

0

D

#

◦ s˙ i , Frb,i ∈ IR(nrb ,frb )i , Di ∈ IR(nel ,fel )i ,

(5.205)

i

fi = (frb + fel )i : relative d.o.f.26 This kind of partitioning has already been done implicitly for a single elastic body undergoing gross rigid body motion (section 5.5, e.g. the first row in eq.(5.95), or in eq.(5.107) and eq.(5.178), resp.). The reason to emphasize partitioning here lies in the goal of separating the equations of the rigid body motion and the equations of motion for the deformation functions in a multibody system.

26 In

case of non-holonomic constraints, frb reduces to grb , see eq.(2.28).

194

5 Elastic Multibody Systems – The Partial Differential Equations

7.3.

Motion Equations

Starting the kinematic chain (5.188) with i = 1 which has an immobile inertial system for predecessor one obtains, for an elastic multibody system, y˙ 1 = y˙ 1r , y˙ 2 = [T21 y˙ 1r ]e + y˙ 2r · · · ⇒        

y˙ 1 y˙ 2 y˙ 3 .. . y˙ N





      =      

0 T21 T31 .. .

 

0 T32

0 ..

. ··· ··· 0

TN 1 TN 2

      

      

e

y˙ 1r y˙ 2r y˙ 3r .. . y˙ N r





       +       e

y˙ 1r y˙ 2r y˙ 3r .. . y˙ N r

     (5.206)   

where index rel is abbreviated r for simplicity. Compared to the rigid body case [eq.(4.97)) et seq.], eq.(5.206) consists of two parts. This is due to the fact that the guidance motion consists of discrete values at the coupling points (index e) while the superimposed relative motions are spatial functions, both depending on time. From eq.(5.206) one obtains the virtual displacements δyi which are needed for eq.(5.187). Because the first summation term in eq.(5.206) contains discrete values w.r.t. the spatial variables, the corresponding terms are extracted from the integrals, leading eq.(5.187) to T

0 TT21 TT31 δy1r δy   0 TT32  2r     δy3r   0  .   .   .  

δyN r

e

dQc1 · · · TTN 1 RB 1 T  · · · TN 2   RB2 dQc2  dQc3  · · · TTN 3    B3 .. .  .. . ..  R . c dQ 0 e BN N  R

   X N Z  + δyTir dQci = 0.  B  i=1 i 

(5.207)

Since y˙ ir , y˙ kr , are independent ∀ i, k, one obtains for body i, h

δyTir

N X e k=i+1

TTki

"

where δyir =

Z e

i

Bk

Frb

0

0

D

dQck +

#



◦ i

Z Bi

δsrb δsel

δyTir dQci = 0



∈ IRni ,

(5.208)

(5.209)

i

ni = (nrb + nel )i : number of describing variables of the ith element. (·) e

denotes that the corresponding values are to be taken at the coupling point. The virtual displacements follow from eq.(5.205) with independent minimal velocities for the rigid body motion and the elastic deformations.

5.7

195

Multibody Systems – Gross Motion

7.3.1 Dynamical Stiffening R In general, Bi dQci will contain zero-order terms w.r.t. deformations. One would thus expect to necessarily calculate Tip completely up to the first e

order. However, this task results in some complications because the previously needed second-order displacements are represented by integrals. For example, R (2) the “shortening” of a beam is well known to read rx = − L (v 0 2 +w0 2 )/2dx e

(2)

= −

which leads to δrx

e

R

0 0 0 0 L (v δv + w δw )dx [see eq.(5.147)] with two

of the components in δy that we use here, namely δv 0 and δw0 , within the integrand. At a first glance this seems to prevent us from extracting δy for a representation according to eq.(5.208). However, there is a simple way out. As eq.(5.163) shows, zero-order distributed forces and torques enter the equations via a dynamical stiffening matrix. If in eq.(5.208) one sets all elastic deviations to zero, then one is left with the zero-order reactions (indicated by index (o)). The remaining part belongs to the rigid body motion only (index rb, see also eq.(5.213) for partitioning). Next, the first summation term is inserted into the integral for body i by means of the Dirac distribution (once more cross lined to distinguish from variation). Because yrb does not depend on space, δyTrb can be factored out. For a beam-like structure one obtains    # " #  L " ZLi  (o) N h  X  iT Z k dQ(o) dQi (o)   k Tki dxk − δ (xi − Li ) + dxi . δyTi,rb   rb dxk rb dxi rb   k=i+1  o

o

(5.210) One has thus found the zero-order forces in the directions which are indicated T T by δyi,rb . Since these had been obtained by y˙ rb = Frb y˙ rb = Frb (vTo ω To )T [see eq.(5.20)], one eventually arrives at the previously related zero-order reactions on body i: 

(o)



  dfi # " # L " (o) N h   dx iT Z k dQ(o) X dQi T  (o)  − i  k Frb,i  Tki dxk δ (xi −Li )− . = − (o)  rb dxk rb dxi rb dMi  k=i+1 o dxi (5.211)

The Dirac distribution indicates that the successors of body i react at the beam tip xi = Li . (In case of higher-dimensional elastic domains, eq.(5.209) has to be appropriately amended).

196

5 Elastic Multibody Systems – The Partial Differential Equations

The summation formula (5.211) shows how to arrive at these zero-order reactions: in an open tree topology, the last elastic body (i=N) has no successor. Its reaction on the penultimate body is thus easily calculated with eq.(5.211). Once this reaction is determined, one may regress to the next predecessor and so on. This consideration once more leads to a recursive calculation, from the last participating body to the first one. Additional constraints like the contact problem (closed kinematic loop) may eventually be treated the same way as in rigid body systems. Assuming all zero-order terms correctly taken into account we proceed with eq.(5.208).

7.3.2 Equations of Motion Since δsrb is independent of δsel , one obtains for eq.(5.208) along with eq.(5.209), 



δsTrb FTrb



0

0

 

i

δyTel

N X

 e i



TTki

k=i+1



N X

 k=i+1

Z e

Bk

dQck 

Z Bi



TTki

Z e

Bk

dQck 

h

+ Z

+

δsTrb FTrb





0 Bi

0

D ◦ δsel

i i

T  i

dQci = 0, dQci = 0. (5.212)

First line second summation term: δsTrb FTrb is independent of mass distribution and can thus be factored out. Omitting δsrb leads directly to the rigid body motion equations. Second line first summation term: the deformations are locally constant w.r.t. the independent spatial variables at the coupling points (index e). This term thus contributes to the boundary conditions. Second line second summation term: an integration by parts (with the aid of the afore-defined spatial operators) leads directly to the elastic deformation equations via D along with (part of) the corresponding boundary conditions via Bi . Because of the special structure it will thereby be suitable to partition dQcj and Tip according to dQcj =

[dQ]crb,j [dQ]cel,j

!



, Tip = 

[Tip ]rb [Tip ]el 0

 ,

(5.213)

5.7

197

Multibody Systems – Gross Motion

leading to the equations of motions for the rigid body coordinates [FT1 ]rb [FT1 TT21 ]rb · · · [FT1 TTN 1 ]rb  [FT2 ]rb · · · [FT2 TTN 2 ]rb   .. ..  .  . T [FN ]rb 

  R c B [dQ]rb,1   R 1 [dQ]c   B2 rb,2   ..     R . c

BN [dQ]rb,N

e



 0     0 = .   . . 

   . 

0 (5.214)

One has thus found the same structure as in rigid multibody systems, see eq.(4.98). Obviously, the deformation variables enter the calculation and, in contrast to the deformation coordinate equations, the integral has to be retained. For vanishing deformations, the integrals are solved in closed form leading eq.(5.214) to eq.(4.98). The equations of motion for deformation coordinates for beam-like structures are simply D1T ◦ [dQcel,1 ]  c  D2T ◦ [dQel,2 ]  ..   . 

T ◦ [dQc DN el,N ]





    =   

0 0 .. . 0

   . 

(5.215)

198

5 Elastic Multibody Systems – The Partial Differential Equations

5 Example: The Elastic Double Pendulum – Plane Motion. Consider a double pendulum rotating with γi about the body-fixed zi -axes with superimposed elastic deflection vi , i = 1, 2. The inertial frame is denoted by o, gravity acts in the xo -direction. Torsion and out-of-plane bending are not considered thus leading to plane motion. The vector of describing variables is derived from eq.(5.200): y˙ Ti = (vox voy ωoz | v˙ v˙ 0 v˙ 00 )i . The corresponding element matrices are determined by eqs.(5.38) to (5.40) and eq.(5.163 b) with F from eq.(5.200) , yielding

y1

*  - zo , z1 q

q

*  x2

q

-

z2

A AA U x1

? xo Figure 5.27.

Elastic Double Pendulum

    dMi =   

dm 0 −dm v 0 0 0

0 dm dm x dm 0 0

−dm v dm x dJ o dm x dJ 0

0 dm dm x dm 0 0

0 0 dJ 0 dJ 0



0 0 0 0 0 0

    ,  

(5.216)

i

(dJ o = dmx2 + dJ, dJ = dJz ),

    dGi =   



dKn,i

0 0 0 0

0 0 0 0

0 0 0 0

0 dm dm x dm 0 0

0 0 0 0

−dm 0 dm v 0 0 0

0 0 0 0 

    =  (o)  L  0 0 0 0 R dfx  dξ x

0 0 0 0

0

−dm x −dm v 0 −dm v 0 0

0 0 0 0

−2dm 0 2dm v 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

     ωoi ,  

(5.217)

i



 1    0     −v T T  dxi , [Frb F F]i := [Fi ] =  0    0 dξ 0   0

0

i

0 1 x 1 0 0

0 0 1 0 1 0

       i

(5.218)

5.7

199

Multibody Systems – Gross Motion

˙ ⇒ [F]T projects the impressed forces [see eq.(5.202)]. Recall: [F] = [∂(vcx vcy ωcz )T /∂ y) (fxe fye Mze )T that act on the mass element into the description space. Kinematics. Matrix Tipe is obtained from eq.(5.203) for p = 1, i = 2 and reads

  T21e

=

A21 [F1 ]e



 A21

= 

0

1 0 −v1e 0 1 L1 0 0 1

! A21

0 0 0 1 0 0 0 1 0

!    (5.219) 

0

"

[T21 ]rb [T21 ]el

#

=

with 0

A21 =

e

cos γ12 − sin γ12 0

sin γ12 cos γ12 0

0 0 1

!

(5.220)

(compare also eq.(5.213) for partitioning). γ12 = v10 e + γ2 represents the angle between the two reference systems, each of which is characterized by its undeformed state. The kinematics are recursively calculated in the forward direction, from body 1 to body 2:

y˙ T1 = (0 0 γ˙ 1 | v˙ 1 v˙ 10 v˙ 100 ) = y˙ 1,rel ¨ yT1 = (0 0 γ¨1 | v¨1 v¨10 v¨100 ) = ¨ y1,rel ,

i=1:

(5.221)

  " i=2:

y˙ 2 =

A21 [F1 ]e y˙ 1e

#

0

+ y˙ 2,rel

 A21 = 

−v1 γ˙ 1 L1 γ˙ 1 + v˙ 1 γ˙ 1 + v˙ 10

!  e

0

   +    

0 0 γ˙ 2 v˙ 2 v˙ 20 v˙ 200

    ,  

(5.222) ¨ y2 = d dt



A21 [F1 ]e y˙ 1e

 +¨ y2,rel =

0

   A21  

−v1 γ¨1 − v˙ 1 γ˙ 1 + L1 (γ˙ 1 + v˙ 1 )γ˙ 12 L1 γ¨1 + v¨1 + v1 γ˙ 1 γ˙ 12 γ¨1 + v¨10 0

!  e

   +    

0 0 γ¨2 v¨2 v¨20 v¨200

    ,  

(5.223)

˙ 21 ) = A12 (− 1 ω ˙ 21 = A21 (A12 A where A e 21 ) with 1 ω e 21 = γ˙ 12 ee3 simplifies the calculation [ see eq.(3.36)]. Kinetics. The kinematic relations are recursively calculated in the backward direction, i.e. starting with body 2. From eq.(5.216) one obtains, along with eq.(5.223),

200

5 Elastic Multibody Systems – The Partial Differential Equations

[dM ¨ y]2

#" # ! " dm 0 −dm v cos γ12 sin γ12 0 −v1 γ¨1 − v˙ 1 γ˙ 1 +(L1 γ˙ 1 + v˙ 1 )γ˙ 12 0 dm dm x − sin γ12 cos γ12 0 L1 γ¨1 + v¨1 + v1 γ˙ 1 γ˙ 12     −dm v dm x dJ o 0 0 1 γ¨1 + v¨10 + γ¨2  2 e     !   0    + dm2 v¨2     dm2 x2 v¨2 + dJ2 v¨20     = . " # !   0 dm dm x # " cos γ sin γ12 0 −v1 γ¨1 − v˙ 1 γ˙ 1 +(L1 γ˙ 1 + v˙ 1 )γ˙ 12 12     0 0 dJ − sin γ12 cos γ12 0 L1 γ¨1 + v¨1 + v1 γ˙ 1 γ˙ 12   0   0 0 0 0 0 1 γ¨1 + v¨1 + γ¨2  2 e      !   dm2 v¨2   0 +

dJ2 v¨2 0

(5.224) Eq.(5.217) yields, with eq.(5.222),

˙2 [dG y]

 " 0 −dm −dm x # " cos γ 12 sin γ12 dm 0 −dm v − sin γ 12 cos γ12   dm x dm v 0 0 0  2   !  −2dm2 v˙ 2  + 0 (γ˙ 1 + γ˙ 2 + v˙ 10 )e   −2dm2 v2 v˙ 2 ≈ 0   = "  dm 0 −dm v # " cos γ sin γ12 12   0 0 0 − sin γ12 cos γ12   0 0 0 0 0  2   !  0  +

0 0

0 0 1

0 0 1

#

#

−v1 γ˙ 1 L1 γ˙ 1 + v˙ 1 γ˙ 1 + v˙ 10 + γ˙ 2

!

 (γ˙ 1 + γ˙ 2 +

v˙ 10 )e

            .  !  −v1 γ˙ 1  0 L1 γ˙ 1 + v˙ 1 (γ˙ 1 + γ˙ 2 + v˙ 1 )e    γ˙ 1 + v˙ 10 + γ˙ 2 e      e

(5.225) The impressed generalized forces are obtained from the elastic potential dQTpot = 2 −(∂dVel /∂y) where dVel = EIv 00 dx/2, and from gravity (acting in the xo -direction, see Fig. 5.27),

5.7

201

Multibody Systems – Gross Motion

dQeT = dQTpot + (dfxe dfye dMze )[F], dfe = A2o e1 dm2 g with

" A2o =

cos γo2 − sin γo2 0

sin γo2 cos γo2 0

0 0 1

# , where γo2 = (γ1 + v10 e + γ2 ),

yielding dm2 g cos γo2 −dm2 g sin γo2    −dm g (v cos γ + x sin γ )  2 2 o2 2 o2  

 dQe2 =  

 



.   

−dm2 g sin γo2 0 −EI v200 dx2

(5.226)

Zero-order forces that enter the dynamical stiffening matrix are calculated from eq.(5.165) for plane motion (voz = ωox = ωoy = 0) or directly from eqs.(5.224) et seq., (o) dfx = −(eT1 | 0) [dM¨ y + dGy˙ − dQe ](o) , setting the deformation variables to zero. The dynamical stiffening then reads





0 0 0

          [dKn y]2 =   dx2 . 0 L   R2    2 2 ∂m2 0   g cos(γ ¨ ˙ cos γ +ξ( γ ˙ + γ ˙ ) dξv 1 +γ2 )−L1 γ 1 sin γ2 +L1 γ 2 1 2 1 2  x ∂ξ 2

0

(5.227) Partitioning the equations according to eq.(5.213),

dQc2 = [dMy˙ + dGy˙ − dQe ]2 := yields



dQcrb dQcel

 , 2

202

5 Elastic Multibody Systems – The Partial Differential Equations

[dQ]crb,2



dm cos γ12

−dm v

dm sin γ12

 −dm sin γ12 =  −dm[v cos γ 12

dm cos γ12

−dm[v sin γ12 +x sin γ12 ] −x cos γ12 ]



−dm cos γ12

dm sin γ12

 dm cos γ12 +  −dm[v sin γ 12

dm sin γ12 dm[v cos γ12 +x sin γ12 ]

−x cos γ12 ]

−2dm2 v˙ 2 (γ˙ 1 + γ˙ 2 + v˙ 10 e ) dm2 v¨2 dm2 x2 v¨2 + dJ2 v¨20

+



dm x   dJ o



−v1 γ¨1 − v˙ 1 γ˙ 1 +(L1 γ˙ 1 + v˙ 1 )γ˙ 12 L1 γ¨1 + v¨1 + v1 γ˙ 1 γ˙ 12 γ¨1 + v¨10 + γ¨2

−

e

2

−dm x

 −v1 γ˙ 1 L1 γ˙ 1 + v˙ 1 γ˙ 1 + v˙ 10 + γ˙ 2

−dm v   0



! (γ˙ 1 + γ˙ 2 + v˙ 10 )e e

2

cos γo2 − sin γo2 −v2 cos γo2 − x2 sin γo2

!

!

! dm2 g,

(5.228) [dQ]cel,2 −dm sin γ12 dm cos γ12 dm x

 =



0

0

0

0

dJ  0

2

dm cos γ12 dm sin γ12 −dm v

+

0

0

0

0

0

0



  2

−v1 γ¨1 − v˙ 1 γ˙ 1 +(L1 γ˙ 1 + v˙ 1 )γ˙ 12 L1 γ¨1 + v¨1 + v1 γ˙ 1 γ˙ 12 γ¨1 + v¨10 + γ¨2 −v1 γ˙ 1 L1 γ˙ 1 + v˙ 1 γ˙ 1 + v˙ 10 + γ˙ 2

! e

! (γ˙ 1 + γ˙ 2 + v˙ 10 )e e

dm2 v¨2 + dm2 g sin γo2

0 +  dJ2 v¨2 +





L2



  g cos(γ1 +γ2 )−L1 γ¨1 sin γ2 +L1 γ˙ 12 cos γ2 +ξ(γ˙ 1 + γ˙ 2 )2 ∂m2 dξv20 dx2  .  ∂ξ

R 

x2

EI v200 dx2

(5.229) Here, γ12 is the angle between the reference systems of body 1 and body 2, γ12 = (v10 e + γ2 ), and the angle from the inertial frame to the reference frame of body 2 reads γ02 = (γ1 + v10 e + γ2 ). The trigonometric functions in eq.(5.229) are thus sin γ12 = sin γ2 + v10 e cos γ2 , cos γ12 = cos γ2 − v10 e sin γ2 , sin γo2 = sin(γ1 + γ2 ) + v10 e cos(γ1 + γ2 ). Neglecting second-order deformation terms, eq.(5.229) results in

5.7

203

Multibody Systems – Gross Motion [dQ]cel,2

n



[L1 cos γ2 − (L1 v10 e − v1e ) sin γ2 + x2 ] γ¨1 + x2 γ¨2 +



    [L1 sin γ2 + (L1 v10 e − v1e ) cos γ2 − v2 ] γ˙ 12 − v2 γ˙ 22 +      [2v˙ 1e sin γ2 − 2v2 γ˙ 2 ] γ˙ 1 + g [sin(γ1 + γ2 ) + v10 e cos(γ1 + γ2 )] +    o   ∂m 0 2   (¨ v1e cos γ2 + x2 v¨1e ) + v¨2 ∂x2   =  dx2 .   0 0 ∂J2   + (¨ γ1 + γ¨2 + v¨1e + v¨2 )   ∂x2  LR2     0  2 2 ∂m2 g cos(γ1 +γ2 )−L1 γ¨1 sin γ2 +L1 γ˙ 1 cos γ2 +ξ(γ˙ 1 + γ˙ 2 ) dξ v2   ∂ξ  x2    (EI)2 v200

(5.230) Equations of Motion, i=2. The motion equations forthe deformation variable v2 (x2 , t) are obtained with eq.(5.215) in the form D2T ◦ dQcel,2 /dx2 = 0 where the operator reads D2T =

 1 −

∂ ∂2 ∂x2 ∂x22

 .

Applied to eq.(5.230) this operator yields The Equations of Motion for v2 (x2 , t),

n

[L1 cos γ2 − (L1 v10 e − v1e ) sin γ2 + x2 ] γ¨1 + x2 γ¨2

+ [L1 sin γ2 + (L1 v10 e − v1e )1 cos γ2 − v2 ] γ˙ 12 − v2 γ˙ 22 + [2v˙ 1e sin γ2 − 2v2 γ˙ 2 ] γ˙ 1 + g [sin(γ1 + γ2 ) + v10 e cos(γ1 + γ2 )]

h

− ∂ (¨ γ1 + γ¨2 + v¨10 e ∂x2 − ∂ ∂x2

o

∂m2 ∂x2 i 0 ∂J2 + v¨2 ) ∂x2

+ (¨ v1e cos γ2 + x2 v¨100e ) + v¨2

h LR2  h x2

 g cos(γ1 +γ2 )−L1 γ¨1 sin γ2 +L1 γ˙ 12 cos γ2 +ξ(γ˙ 1 + γ˙ 2 )2 ∂m2 dξ v20 ∂ξ

2 + ∂ 2 (EI)2 v200 = 0. ∂x2

i

i

(5.231)

Quick check: setting v1 = γ1 = 0, one obtains

h

x2 γ¨2 − v2 γ˙ 22 + g sin γ2 + v¨2

− ∂ ∂x2

h LR2  x2

i

h

∂m2 − ∂ (¨ γ2 + v¨20 ) ∂J2 ∂x2 ∂x2 ∂x2

 g cos γ2 +ξ γ˙ 22 ∂m2 dξ v20 ∂ξ

i

i

2 + ∂ 2 (EI)2 v200 = 0 ∂x2

h

i

(5.232)

which for g = 0 and (∂J2 /∂x2 ) = ρIz = const yields eq.(5.180), second line, for ϑ = 0.

204

5 Elastic Multibody Systems – The Partial Differential Equations

The motion equations for the rigid body variables are calculated with eq.(5.214),



[FT1 ]rb 0

[FT1 TT21 ]rb [FT2 ]rb



 R B1

 R e

[dQ]crb,1



[dQ]crb,2

 = 0, [F1 ]rb = [F2 ]rb = e3 .

(5.233)

B2

For i = 2, one obtains thus the equations for γ2 (t) by applying eT3 to eq.(5.228), where v2 cos γ12 + x2 sin γ12 = (v2 + x2 v10 e ) cos γ2 + x2 sin γ2 , v2 sin γ12 − x2 cos γ12 = (v2 + x2 v10 e ) sin γ2 − x2 cos γ2 . R Integration B x2 dm2 yields m2 c2 where c2 represents the mass center distance of body B2 2 when considered rigid. Neglecting second-order deformation terms yields The Equations of Motion for γ2 (t), J2o γ¨2

h



+ J2o + m2 c2 L1 cos γ2 − m2 c2 (L1 v10 e − v1e ) + L1

h



+ m2 c2 L1 sin γ2 + m2 c2 (L1 v10 e − v1e ) + L1 +m2 c2 [¨ v1e cos γ2 + 2γ˙ 1 v˙ 1e sin γ2 ] +

J2o v¨10 e

h

+m2 c2 g sin(γ1 + γ2 ) + g m2 c2 v10 e +

+



R B2



R B2

R B2

i

v2 dm2 sin γ2 γ¨1

i

v2 dm2 cos γ2 γ˙ 12

(x2 v¨2 dm2 + v¨20 dJ2 )

i

R B2

v2 dm2 cos(γ1 + γ2 ) = 0.

(5.234)

Quick check: setting vi = 0 yields J2o γ¨2 + [J2o + m2 c2 L1 cos γ2 ] γ¨1 + [m2 c2 L1 sin γ2 ] γ˙ 12 + m2 c2 g sin(γ1 + γ2 ) = 0,

(5.235) which corresponds to the second component of the well-known equations of motion of a rigid double pendulum in terms of relative angles,



J1o + J2o + m2 L21 + 2m2 c2 L1 cos γ2 J2o + m2 c2 L1 cos γ2 J2o + m2 c2 L1 cos γ2 J2o

 −

γ¨1 γ¨2





γ˙ 22 + 2γ˙ 1 γ˙ 2 m2 c2 L1 sin γ2 −γ˙ 12

 +



(5.236)

(m1 c1 + m2 L1 )g sin γ1 + m2 c2 g sin(γ1 + γ2 ) m2 c2 g sin(γ1 + sin γ2 )



  =

0 0

.

[Eq.(5.236) is obtained from eq.(4.99) with FT =



FT1 FT1 TT21 0 FT2



 =

eT3 (T21 e3 )T 0 eT3



 =



0 0 1 L1 sin γ2 L1 cos γ2 1 , 0 0 0 0 0 1

where Tip follows from eq.(4.121) and the matrices derived from eq.(4.67)].

5.7

205

Multibody Systems – Gross Motion

For i=1, one obtains from eq.(5.216) and eq.(5.217), when ¨ y1 and y˙ 1 are inserted, [dM ¨ y]1

 "

dm 0 −dmv 0 dm dmx −dmv dmx dJ o

     =  "  0 dm dm x #   0 0 dJ 0

0

0

0 0 γ¨1

1

0 0 γ¨1

# 1

+

dm¨ v dJ v¨0 0

! +

0 dm¨ v dm x v¨ + dJ v¨0

!

!  1

!

     ,    

(5.237)

1

˙1 [dG y]

"

0 −dm −dm x  dm 0 −dm v  0  dm x dm v

#

  = "  dm 0 −dm v #   0 0 0 0

0

0

1

1

0 0 γ˙ 1

0 0 γ˙ 1

−2dmv˙ 0 −2dmv v˙ ≈ 0

! γ˙ 1 +

! γ˙ 1 +

0 0 0



!

γ˙ 1  1

!

    .    

(5.238)

The impressed forces consist of gravity and elastic potential forces. With [F1 ] from eq.(5.218) one obtains dQe1 1 0 0  0 1 0   −v x 1   





= 

   0 1 0   0 0 1  0 0 0

1

 cos γ1 − sin γ1 0

!

   dm1 g +    

0 0 0 0 0 −EIv 00 dx



dmg cos γ1 −dmg sin γ1   −dmg(v cos γ + x sin γ ) 1 1   . =     −dmg sin γ1   0 00 −EIv dx 1



       1



(5.239)

Partitioning into the rigid body motion and the elastic deformation parts yields

 

[dQ]crb,1 =  

1 −(¨ γ1 v1 + x1 γ˙ 12 + 2v˙ 1 γ˙ 1 + g cos γ1 ) ∂m ∂x1 1 (¨ γ1 x1 − v1 γ˙ 12 + v¨1 + g sin γ1 ) ∂m ∂x1

∂J o

∂J1 1 (¨ v1 x1 + g(v1 cos γ1 + x1 sin γ1 ) ∂m + γ¨1 ∂x11 + v¨10 ∂x ∂x1 1

   dx1 , (5.240) 

206

5 Elastic Multibody Systems – The Partial Differential Equations 



1 (¨ γ1 x1 − v1 γ˙ 12 + v¨1 + g sin γ1 ) ∂m ∂x1

  

 R L1 dfx(o)
 0   dξ v1  dx1 . x1 dξ 

∂J1 [dQ]cel,1 =  (¨ γ1 + v¨10 ) ∂x + 1

(5.241)

[(EI)1 ] v100

The rigid body motion is determined with eq.(5.233), first line: [FT1 ]rb

R B1

[dQ]crb,1 + [FT1 TT21e ]rb

Also, for the zero-order reactions from eq.(5.211), to eq.(5.219),

[T21e ]Trb

[dQ]crb,2 = 0. B2 (o) T [T21 ]rb B [dQ]crb,2 is 2

R

R

needed. According

reads [T21e ]Trb

=

1 0 −v1e

0 1 L1

0 0 1

! AT21 .

The term AT21 [dQ]rb,2 plays thus a major role and will be determined first. It is obtained by premultiplying eq.(5.228) with AT21 from eq.(5.220) yielding AT21 [dQ]crb,2





dm

0

   = 0 dm   −dm(v cos γ12 −dm(v sin γ12 −x cos γ12 )

+x sin γ12 )



−dm(v cos γ12 ! +x sin γ12 )   −v1 γ¨1 − v˙ 1 γ˙ 1 +(L1 γ˙ 1 + v˙ 1 )γ˙ 12 −dm(v sin γ12  L1 γ¨1 + v¨1 + v1 γ˙ 1 γ˙ 12  −x cos γ12 )  γ¨1 + v¨10 + γ¨2  e o dJ 2

dm(v sin γ12



0 −dm  ! −x cos γ12 )    −v1 γ˙ 1 −dm(v cos γ12   + dm 0 γ˙ o2  L1 γ˙ 1 + v˙ 1 +x sin γ12 )   0  −dm(v sin γ12 dm(v cos γ12  γ˙ 1 + v˙ 1 + γ˙ 2 e −x cos γ12 )

+

0

+x sin γ12 )

2

−2dm2 v˙ 2 γ˙ o2 cos γ12 − dm2 v¨2 sin γ12 −2dm2 v˙ 2 γ˙ o2 sin γ12 + dm2 v¨2 cos γ12 dm2 x2 v¨2 + dJ2 v¨20

! +

− cos γ1 sin γ1 (v2 cos γo2 + x2 sin γo2 )

! dm2 g.

(5.242) Zero-Order Reactions. For the plane motion, Frb is derived from eq.(5.200) which trivially

stiffening, only

(o) dfx (o)



(o)

dfx dx enters the calculation, hence

shows that in this case eq.(5.211) represents

dfx1 = −eT1 dx1





TT21

Z B2

[dQ]c2

(o)

dfy dx

dQc1 δ(x1 − L1 ) + dx1

−

(o)

dMz dx

T

. For the dynamical

(o) .

(5.243)

rb

Since eT1 [T21 ]Trb = eT1 AT21 [see eq.(5.219)], the zero-order force acting on dm1 reduces to

5.7

207

Multibody Systems – Gross Motion (o)  T dfx1 = −eT1 A21 dx1



 dQcrb,1 [dQ]crb,2 −δ(x1 − L1 ) + dx1 B2

Z

(o)

(5.244)

.

Thus, setting the deformations to zero and R collecting the first Rcomponents from eq.(5.242) and eq.(5.240) one obtains, once more using B dm2 = m2 and B x2 dm2 = m2 c2 , 2

2

(o)

dfx1 dx1 =

n +

−m2



0

0

m2

+

n h



m2 c2 cos γ2

o

+m2 g cos γ1 =



m2 c2 sin γ2

( L1 γ˙ 1 γ˙ 2

( 0

−δ(x1 − L1 ) +

h

L1 γ¨1

T (¨ γ1 + γ¨2 ) )

L1 γ˙ 1 γ˙ 2 + L1 γ˙ 12

T (γ˙ 1 + γ˙ 2 )2 )

(x1 γ˙ 12 + g cos γ1 ) ∂m1 ∂x1

i

m2 [c2 (¨ γ1 + γ¨2 ) sin γ2 + L1 γ˙ 12 + c2 (γ˙ 1 + γ˙ 2 )2 cos γ2 + g cos γ1 ]

(x1 γ˙ 12 + g cos γ1 )

i

o

−δ(x1 − L1 )

∂m1 . ∂x1

(5.245) Interpretation: Setting all deformations to zero, the resulting equations correspond to a rigid double pendulum. When, for the rigid double pendulum, body 2 is cut free, then the force acting on body 2 reads



1 0

0 1



0 0

[M2 ¨ y2 + G2 y˙ 2 −

Qe2 ]

  =

fx fy

:= 2 f.

The matrices are derived from eq.(4.67), and y˙ 2 reads y˙ 2 = T21 y˙ 1 + y˙ 2,rel = [L1 γ˙ 1 sin γ2 L1 γ˙ 1 cos γ2 (γ˙ 1 + γ˙ 2 )]T , yielding

  2 f = m2

L1

sin γ2 cos γ2



 γ¨1 +L1

− cos γ2 sin γ2



γ˙ 12 +c2





−(γ˙ 1 + γ˙ 2 )2 −g (¨ γ1 + γ¨2 )



cos(γ1 + γ2 ) − sin(γ1 + γ2 )







.

A representation in a reference system 1 is obtained by

 1f =

cos γ2 sin γ2

− sin γ2 cos γ2

   m2 L 1

0 1

 γ¨1 +L1

−1 0

 2f



=

γ˙ 12 −c2



(γ˙ 1 + γ˙ 2 )2 cos γ2 + (¨ γ1 +¨ γ2 ) sin γ2 −g (γ˙ 1 + γ˙ 2 )2 sin γ2 − (¨ γ1 +¨ γ2 ) cos γ2



cos γ1 − sin γ1

This force acts on body 1 with negative sign (counteraction principle). Thereby, 1 fx represents a constraint force in the case of a rigid double pendulum. In an elastic body, however, it performs work due to the superimposed deflection.

.

208

5 Elastic Multibody Systems – The Partial Differential Equations

Equations of Motion, i=1. The motion equations forthe deformation variable v1 (x1 , t) are obtained with eq.(5.215) in the form D1T ◦ dQcel,1 /dx1 = 0 where the operator reads D1T =

 1 −



∂ ∂2 ∂x1 ∂x21

.

(o)

Applied to eq.(5.241), along with dfx from eq.(5.245), yields

The Equations of Motion for v1 (x1 , t): (¨ γ1 x1 − v1 γ˙ 12 + v¨1 + g sin γ1 ) ∂m1 ∂x1  (¨ γ1 + v¨10 ) ∂J1 + ∂x1

− ∂ ∂x1

hR

L1

x1

i

(g cos γ1 + ξ γ˙ 12 ) ∂m1 dξ v10 ∂ξ

+ m2 c2 (¨ γ1 +¨ γ2 ) sin γ2 +m2 L1 γ˙ 12 +m2 c2 (γ˙ 1 + γ˙ 2 )2 cos γ2 +m2 g cos γ1 v10



2 + ∂2 ∂x1



[EI]1 v100





 = 0.

(5.246)

Quick check: m2 = 0 in eq.(5.246) yields eq.(5.232) when index 1 is replaced with index 2 in eq.(5.232). The equations for γ1 are obtained from eq.(5.233) and read, with [F1 ]rb = e3 ,

eT3

Z

[dQ]crb,1 + eT3 [TT21e ]rb

Z

[dQ]crb,2 = 0.

(5.247)

B2

B1

Since eT3 [TT21e ]rb = (−v1e L1 1)AT21 [see eq.(5.219)], eq.(5.247) becomes

Z (0 0 1)

[dQ]crb,1 + (−v1e L1 1) AT21

B1

Z

[dQ]crb,2 = 0

(5.248)

B2

with [dQ]crb,1 according to eq.(5.240) and AT21

R

R B2

[dQ]crb,2 from eq.(5.242).

For calculating eq.(5.248), B xi dmi = ci mi (mass center displacement ci , total beam i mass mi ) is once more used as well as v2 cos γ12 + x2 sin γ12 ' (v2 + x2 v10 e ) cos γ2 + x2 sin γ2 , v2 sin γ12 − x2 cos γ12 ' (v2 + x2 v10 e ) sin γ2 − x2 cos γ2 ,

5.7

209

Multibody Systems – Gross Motion

which yields the Motion Equation for γ1 :

J1o γ¨1 +

R B1

v¨10 dJ1 +

R B1

v¨1 dm1 + g

h

hR B1

v1 dm1 cos γ1 + m1 c1 sin γ1



+ J2o + m2 L21 + 2m2 c2 L1 cos γ2 −2 m2 c2 (L1 v10 e − v1e ) + L1

h



+ J2o + m2 c2 L1 cos γ2 − m2 c2 (L1 v10 e − v1e ) + L1

i 

R B2



R B2

i

v2 dm2 sin γ2 γ¨1

i

v2 dm2 sin γ2 γ¨2

+m2 (L1 + c2 cos γ2 ) v¨1e + (J2o + m2 c2 L1 ) v¨10 e

h



−2 m2 c2 L1 sin γ2 + m2 c2 (L1 v10 e − v1e ) + L1

h



− m2 c2 L1 sin γ2 + m2 c2 (L1 v10 e − v1e ) + L1



R B2

i

v2 dm2 cos γ2 γ˙ 1 γ˙ 2



R B2

i

v2 dm2 cos γ2 γ˙ 22

−2 [m2 c2 (L1 v˙ 10 e − v˙ 1e ) sin γ2 ] γ˙ 1 − 2 [m2 c2 L1 v˙ 10 e sin γ2 ] γ˙ 2 +

R

h

B2

(x2 + L1 cos γ2 )¨ v2 dm2 +

g m2 v1e cos γ2 +

R B2

v¨20 dJ2 − 2L1 (γ˙ 1 + γ˙2 ) sin γ2

i

R B2

R B2

v˙ 2 dm2

h

i

v2 dm2 cos(γ1 + γ2 ) + m2 g L1 sin γ2 + c2 sin(γ1 + γ2 ) = 0.

(5.249) Quick check: neglecting elastic displacements yields [J1o + J2o + m2 L21 + 2m2 c2 L1 cos γ2 ]¨ γ1 + [J2o + m2 c2 L1 cos γ2 ]¨ γ2 −(2γ˙ 1 γ˙ 2 + γ˙ 22 )m2 c2 L1 sin γ2 +(m1 c1 + m2 L1 )g sin γ1 + m2 c2 g sin(γ1 + γ2 ) = 0,

see eq.(5.236), first line (rigid double pendulum). 4

(5.250)

210

5 Elastic Multibody Systems – The Partial Differential Equations

7.4.

Boundary Conditions

Consider once more eq.(5.212), second line, for beam-like structures. After an integration by parts one has, after partitioning with eq.(5.213),  0=

δyTel

+

δsTel,i

N X

 e

Z h



 i



 Z

[Tki ]Tel

[dQk ]crb 

(5.251)

e

k=i+1

Bk

dQcel DT ◦ dx

i

h



dxi + δsTel,i BoT ◦

i

dQcel dx

 iLi h



T

+ δs0 el,i B1T ◦

i o

dQcel dx

 i Li

.

i o

Bi

The first summation term of the second line is zero (motion equations for the deformation variables). The remainder then characterizes the boundary conditions. However, since yel contains spatial derivatives, the virtuals in eq.(5.251) appear disordered. One has thus first to introduce some “sorting matrices” Sj , j = 0, 1, 2, which go along with δyel = D ◦ δsel . For example, for pure beam bending one has 

δyel

=

1 0 0 ∂ ∂x ∂2 ∂x2 0

          

=

1 0 0 0 0 0

     

0 1 0 0 0 0

0 1 − ∂ ∂x 0 0 ∂2 2 ∂x 

       δv := So δsel + S1 δs0el + S2 δs00el  δw     

      δv   δw +     

0 0 0 1 0 0



0 0 −1 0 0 0



      δv 0   δw0 +     

0 0 0 0 1 0

(5.252)

0 0 0 0 0 1

     δv 00  δw00 .  

Remarks: 1) The sequence of components of yel may of course be chosen in any way. Comparison with the assigned D then automatically yields the correct sorting matrices. 2) Curvatures will not contribute to the boundary conditions.

Insertion of these matrices into eq.(5.251) then yields  h

i

δsTel,i SToi  e

+

T δs0el,i ST1i

i e





N X

Z

[Tki ]Tel

k=i+1

 h

N X

h



[dQk ]crb  + δsTel,i BoT ◦

e

dQcel dx

 i Li

 Z

[Tki ]Tel

k=i+1

i o

Bk

e Bk

h

T



[dQk ]crb  + δs0 el,i B1T ◦

dQcel dx

 i Li i o

= 0.

(5.253)

5.7

211

Multibody Systems – Gross Motion

Both lines in eq.(5.253) are independent, yielding the boundary conditions xi = 0, j = o, 1 : (∂ j s/∂xj )el,i = 0



T Bj,1 ◦ [dQ/dx]cel,1

 T  Bj,2 ◦ [dQ/dx]cel,2   ..  . 





0



      =    

0 .. .

   ,  

T Bj,N ◦ [dQ/dx]cel,N

j j xi = Li , j = o, 1 : (∂ s/∂x )el,i =0

       

 R [dQ]crb,1   RB 1  [dQ]crb,2 · · · [STj2 TTN 2 ]el  B2   ..  .. ..  . . .  R c

0 [STj1 TT21 ]el · · · [STj1 TTN 1 ]el 0

0

e

BN

[dQ]rb,N

_

 

(5.254)

0

_

T Bj,1 ◦ [dQ/dx]cel,1

 

0



  T   Bj,2 ◦ [dQ/dx]cel,2   + ..   .  

      =    

0 .. .

   .  

T Bj,N ◦ [dQ/dx]cel,N

0

(5.255) We have thus once more found a simple recursive scheme for determination of the boundary conditions. Vector dQci is determined from i = 1 to i = N upwards, and eq.(5.255) is then resolved from N to 1 downwards. [Notice that [dQ/dx]cel,i includes the dynamical stiffening part from eq.(5.163) along with eq.(5.211)].

212

5 Elastic Multibody Systems – The Partial Differential Equations

5 Example: The Elastic Double Pendulum – Plane Motion The boundary operators read

[Bo ]Ti =



0



− ∂ ∂xi

1

, [B1 ]Ti =



0

0

1



.

(5.256)

Starting with i = 2 (no successor), the boundary conditions for v2 (x2 , t) read



x2 = L2 :



0

1

− ∂ ∂x2

0

0

1

 dQc

= (¨ γ1 + γ¨2 + v¨10 e + v¨20 e ) ∂J2 − (EIv 00 )02e = 0, ∂x2

el,2

dx2

 dQc

el,2

dx2

= (EIv 00 )2e = 0,

v2o = 0, v20 o = 0

x2 = 0 :

(5.257) where index o denotes x2 = 0 and index e represents x2 = L2 and x1 = L1 , respectively. [Quick check: setting γ1 = 0, v1 = 0 yields eq.(5.182)]. For i = 1, body 2 acts as successor. Along with y˙ el = (v˙ v˙ 0 v˙ 00 )T ∈ IRnel (see eq.(5.221), index rel suppressed for brevity), one obtains for eq.(5.252),

δyel =

1 0 0

! δv +

0 1 0

! 0

δv +

0 0 1

! δv 00 := So δsel + S1 δs0el + S2 δs00el .

(5.258)

With [T21 ]Tel from eq.(5.219) one has

STo [T21 ]Tel =

0

=

1

0

1

0




AT21 ,

0

0 0 0

1 0 0

0 1 0

!




0

0 0 0

1 0 0

0 1 0

!




AT21

(5.259) ST1 [T21 ]Tel = =

0

0

1

0

1




AT21 ,

AT21

[as had been mentioned before, the curvature does not contribute:
ST2 [T21 ]Tel = 0 0 0 AT21 ]. From eq.(5.255) one obtains thus the boundary conditions for v1 (x1 , t),

5.7

213

Multibody Systems – Gross Motion  x1 = L1 :

dQ [dQ]crb,2 + (0 1 − ∂ ) el,1 B2 ∂x1 dx1



dQcel,1 dx1



c

(0 1 0)AT21

R



(0 0 1 )AT21

x1 = 0 :

R B2

[dQ]crb,2 + (0

0

1

)

= 0,

= 0,

(5.260)

v1 = 0, v10 = 0.

R

AT21 B [dQ]crb,2 is derived from eq.(5.242) and (d[Q]cel,1 /dx1 ) is derived from eq.(5.241). 2 Carrying out the calculation yields:

x1 = L1 : − ∂ ∂x1



[EI]1 v100e



+ ∂J1 (¨ γ + v¨10 e ) ∂x1 1

h

+ m2 (L1 + c2 cos γ2 ) −

h

+ m2 c2 cos γ2 −

B2



− m2 c2 sin γ2 + −2

h

m2 c2 v˙ 10 e +

B2

v2 dm2 sin γ2 γ¨1

i

R

v2 dm2 sin γ2 γ¨2

+m2 v¨1e + m2 c2 v¨10 e +

h

i

R



R B2

v¨2 dm2 cos γ2

i

R B2

v2 dm2 cos γ2 (γ˙ 1 + γ˙ 2 )2



R B2

i

v˙ 2 dm2 sin γ2 (γ˙ 1 + γ˙ 2 )

+m2 (L1 v10 e − v1e )γ˙ 12 + m2 g(sin γ1 + v10 e cos γ1 ) = 0.

[EI]1 v100e + J2o (¨ γ1 + γ¨2 + v¨10 e ) + m2 c2 cos γ2 v¨1e +

h

h

R

h

h

R

+ m2 c2 L1 cos γ2 − m2 c2 (L1 v10 e − v1e ) + L1 + m2 c2 L1 sin γ2 + m2 c2 (L1 v10 e − v1e ) + L1

h



+ m2 c2 sin(γ1 + γ2 ) + m2 c2 v10 e +

B2

B2



R B2

R B2

(5.261)

(¨ v2 x2 dm2 + v¨20 dJ2 )

i

i

i

i

v2 dm2 sin γ2 γ¨1 v2 dm2 cos γ2 γ˙ 12

i

v2 dm2 cos(γ1 + γ2 ) g

+2m2 c2 sin γ1 v˙ 1e γ˙ 1 = 0.

(5.262)

x1 = 0 :

v1o = 0, v10 o = 0

where v1e := v1 (L1 , t), v1o := v1 (0, t). 4

(5.263)

214

8.

5 Elastic Multibody Systems – The Partial Differential Equations

Conclusion

These simple examples have shown that the use of analytical methods cannot be recommended for the determination of motion equations. This is not only because of the tremendous effort which goes along with its quadratic energy formulations, even in case of linear motions, but also and particularly due to the fact that these methods need directional derivations of the kinetic energy leading to non-orthogonal momentum representation, which may be problematic. Moreover, the use of non-holonomic velocities is even more tedious, and non-holonomic constraints may be inserted only after carrying out calculations for the holonomic case. We are therefore going to use the Projection Equation which had been successfully applied to rigid body systems in the previous Chapter. There, the use of intermediate (or auxiliary) variables enabled us to decrease the required mathematical effort significantly. When compared to rigid multibody systems with its intermediate variables T y˙ i = (vTo ω To r˙ Tc ω Tr )i , these are replenished for elastic body systems with the curvatures required to later take the elastic potential forces into account: T ˙ T κ˙ T )i . y˙ i = (vTo ω To r˙ Tc ϕ Our reference frame moves with vo and ω o while the relative motion of the mass element within that reference frame is characterized by its mass center ˙ The latter is chosen instead of translational velocity r˙ c and its angular velocity ϕ. ˙ as well as r˙ c are time derivatives of holonomic variables ω r to emphasize that ϕ and thus allow for elementary integration that takes into account the deflection variables. This goes along with the assumption of small deformation amplitudes. The rigid body motion, however, remains non-holonomic in general. Since the curvatures do not contribute to the kinetic momentum balances, there is structurally no difference between the motions of a single rigid body e c mi in a moving reference frame, [M y¨ + G y˙ − Q ]i = Qi , and the motion of a mass element dmi , except that the mass element is considered infinitesimal and e c that the mass matrix and the gyroscopic matrix in [dM y¨ + dG y˙ − dQ ]i = dQi are completed with zeros to take the curvatures within y˙ into account. In rigid multibody systems, an adequate number of single bodies is joined together to form a subsystem. This is done with the aid of functional matrices Fi which arise from a chain rule application, y˙ i = (∂yi /∂yn )y˙ n := Fi y˙ n (body i in P n T subsystem n) leading to [M ¨y + G y˙ − Qe ]n = Qcn where Mn = N i=1 Fi Mi Fi etc. (Nn number of bodies in subsystem n). Vector y˙ n is hereby called the vector of describing (velocity) variables for subsystem n. For y˙ i ∈ IRni and y˙ n ∈ IRnn one obtains Fi ∈ IRni ,nn with ni ≤ nn , hence rank [Fi ] = ni . This means, in common words, that the dynamical equation for one single body equated in terms of subsystem variables contains placeholders for the others, and rank[Mn ] = nn is only obtained after summation.

5.8

Conclusion

215

In pure elastic multibody systems, the summation over the number of participating bodies passes to an integration for the individual dmi followed by a summation over the number N of elastic bodies. The vector of describing variables is analogously obtained by y˙ i = (∂yi /∂yi )y˙ i := Fi y˙ i where Fi ∈ IRni ,ni with ni ≤ ni , hence rank [Fi ] = ni . In other words: in pure elastic multibody systems the elastic body itself is considered as a subsystem, and F is chosen to reduce the number of variables to the actual problem under consideration. For mixed rigid/elastic multibody systems there is no problem to eventually introduce placeholders (for instance for a motor-gear-elastic arm-unit, taking the dynamics of the motor and the gear into account) although not pursued here. We will come back to this problem in Chapter 7. In rigid multibody systems, the describing variable y˙ n is obtained from the kinematic chain y˙ n = Tnp y˙ p + y˙ n,rel where y˙ n,rel = (∂ y˙ n,rel /∂ s˙ n )˙sn = Fn s˙ n leads to the minimal velocities s˙ n which the nth subsystem contributes. In elastic multibody systems, the kinematic chain reads y˙ n = [Tnp y˙ p ]e + y˙ n,rel where e denotes the coupling point (“end of” predecessor). The main difference here lies in the fact that [Tnp y˙ p ]e only contains time dependent terms while y˙ n,rel depends on time and space. Because y˙ n,rel contains spatial derivatives (bending angles and curvatures), a functional matrix according to Fn = (∂ y˙ n,rel /∂ s˙ n ) can not be calculated directly. The vector of relative variables y˙ n,rel is instead connected to the vector of minimal velocities s˙ n by a spatial operator: y˙ n,rel = D ◦ s˙ n . The rigid body case which is characterized by Fig. 4.9 on page 78 looks similar to the elastic body in Fig. 5.3 on page 123. Indeed, within the scope of modeling requirements, the finite body as well as the infinitesimal element are taken to be rigid. One basic difference, however, is that the rigid body subsystem connects single bodies mi with the corresponding elements (springs, dampers, joints, etc.) while, in an elastic body, the dmi are interconnected by “inner constraints” which are a` -priori taken into account with the corresponding hypotheses (listed in Table 5.1, page 118). The single finite rigid bodies are recorded with individual r˙ ci (t), ω ri (t) (along with the corresponding constraints) while the single elements of the elastic body are calculated with one and the ˙ same functions r˙ c (x, t), ϕ(x, t) which take the inner constraints into account. The index needed for the rigid body case, y˙ i = Fi y˙ (n=1 dropped for brevity) is thus omitted for the elastic body. ˙ the element matrices dM are simple and eaDue to the special choice of y, sy to obtain, but the problem is shifted to what the functional matrix is to be replaced with. This task, however, is easily solved considering the corresponding virtual work. Integration by parts leads directly to the spatial operators D, Bi (i ∈ {0, 1} in case of a beam) for the determination of the partial differential equations and the associated (dynamical) boundary conditions. These operators follow from the chosen D by changing sign for odd derivatives and

216

5 Elastic Multibody Systems – The Partial Differential Equations

successively degenerating the derivation grade along with once more changing sign. The result is not really surprising because it reflects the procedure of integrating by parts with its consecutive sign changing. However, applying these operators directly saves an immense amount of effort. This can, for instance, be seen in the plate equations on page 157: there, D is chosen according to the sequence y˙ T = (w˙ (∂ w/∂y) ˙ –(∂ w/∂x) ˙ κ˙ T ) where w(x, y, t) is the deformation function. The operator is applied to dQc according to eq.(5.89). Since dQc reads [dM ¨y + dG y˙ − dQe ], one easily interprets the first three components as the linear and the two angular momentums, while the last four are simply DdKκ, once we have disposed of the elastic potential dVel = κT DdK κ/2. Vector dQc can thus be formulated directly without any further calculation, and the plate equations are derived on, say, half a page. The use of dQc for [dM ¨y + dG y˙ − dQe ] is because of space requirements. It should not go unnoticed that this relation is by no means trivial: it does not represent an identity but an equilibrium condition since separating [dM ¨y + dG y˙ − dQe ] from surroundings yields zero only when completed with the corresponding (generalized) constraint forces. However, the constraint forces are not of interest because [dM ¨ y + dG y˙ − dQe ] is eventually re-inserted. The resulting motion equations are correct so far if the rigid body motion is assumed sufficiently slow. In multibody systems, however, this assumption does not hold in general. Then, because linearization is only valid w.r.t. the elastic deformation amplitudes, special care must be taken to obtain correct motion equations. Fortunately, this problem can be treated separately leading to an additional generalized restoring force (dQe → dQe + dKn y, index n for (geometrically) nonlinear, dKn : “dynamical stiffening matrix”). The procedure, itself, i.e. the application of differential operators, remains unaltered. An analytical solution of the obtained differential equations is, for the general case, virtually impossible to achieve. As has been demonstrated, under some additional assumptions an approximative solution may be accessible with the use of Galerkin’s method. The results are eventually applied to elastic multibody systems. The procedure is straightforward: the motion equations for the rigid body coordinates are obtained from eq.(5.214) which appears in the same mathematical structure as in rigid multibody systems. Disposing of the zero-order reactions [eq.(5.211)] for the dynamical stiffening calculation, one may directly calculate the deformation equations by applying the corresponding differential operators [eq.(5.215)]. The foregoing is demonstrated step by step for the plane motion of an elastic pendulum, which clearly demonstrates that the derivation of the (interconnected ordinary and partial) differential equations of motion needs some effort. This effort, however, is almost negligible when compared to a derivation via Hamilton’s principle, for instance.

5.8

217

Conclusion

Nevertheless, the consideration of elastic multibody systems in the form of coupled ordinary and partial differential equations seems to lead to a dead end, which is mainly due to the boundary conditions [eq.(5.254), eq.(5.255)]. These will scarcely be fulfilled by any chosen set of shape functions, making Galerkin’s (direct) method inapplicable. Considering the elastic double pendulum, one must furthermore state that this is the simplest representation of an elastic multibody system that can be found. Turning the pendulum in Fig. 5.27 by 90o around the negative yo -axis where now gravity acts in the negative zo -direction shows that torsion will unavoidably occur due to the forearm weight. This leads already to spatial motion. [Such modeling is used in robotics for SCARA27 -type elastic systems where, in contrast to rigid body models, the gripper motion is no longer decoupled from the arm rotations]. Thus, for multibody systems, one has to expect even more complicated motion equations and, particularly, boundary conditions. In view of (numerical) solutions, we will therefore proceed directly to the (so-called) “weak” ordinary differential equations of motion.

z ,z

6o 1 * y1   p 

A K A Ap

p  

x2

Figure 5.28. Fexible SCARA Basis

27 “Selective

Compliance Assembly Robot Arm”

pH H HH j

x1

Chapter 6 ELASTIC MULTIBODY SYSTEMS – THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS

1. Galerkin Method 1.1. Direct Galerkin Method Boris Galerkin1 published his method in (B.G. Galerkin, 1915). For a special case, the procedure had already been anticipated by I.G. Bubnov in 1913. The method is therefore sometimes referred to as the Bubnov-Galerkin method. According to (M.S. Smirnov, 2003), Galerkin’s method “provides the approximate solution of a variational problem without minimization of an energy functional but using transformed differential equations”; its original application field is mainly characterized by his original paper entitled series solution of some problems in elastic equilibrium of rods and plates. The procedure is, however, not restricted to linear elasticity problems but obviously reveals its main power in the field of approximate solutions of nonlinear systems, thus extended by G.I. Petrov to nonlinear problems in gas dynamics, hydrodynamics and aerodynamics in 1940 (referred to then as the Galerkin-Petrov method). It is sometimes also called the Ritz-Galerkin method, based on a comparison with W. Ritz’s contribution from 1909. The probably easiest access to these methods (and to their distinction) is to once more look at Lagrange’s Principle (2.1) (the principle of virtual work). For a single elastic body (to simplify matters) the motion equation (or generalized force balance) reads DT ◦ dQc = 0: dQc = dM¨y + dGy˙ − dQe ⇒ DT ◦ [dM¨ y + dGy˙ − dQe ] = 0

(6.1)

(potential forces as well as non-potential impressed forces and dynamical stiffening forces, if they occur, are assumed to be taken into account by dQe ). The 1 Boris

Grigoryevitch Galerkin, *1871 in Polotsk, †1945 in Moscow

219

220 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations vector of describing variables y˙ in eq.(6.1) is computed with the aid of the differential operator D applied to the vector of minimal velocities s˙ . [For example, D, D and dQc are given with eq.(5.84), Table 5.7 and eq.(5.89) for a plate, or with eq.(5.178), eq.(5.179) and eq.(5.177) for a (rotating) beam]. Notice that the application of DT in eq.(6.1) reveals the motion equations in terms of s˙ , i.e. the minimal representation. An analytical solution is, in general, not achievable. One may thus use a series expansion in the form s˙ ' Φ(x)T y˙ R (t) where x represents the independent spatial variables (see Table 5.1) and index R denotes “Ritz”. Hereby, only the deformations are influenced; the quasi-coordinates of rigid body motion remain unaltered. The '-sign demonstrates the approximation character at least for finite-dimensional yR . Eq.(6.1) thus becomes h

i

y˙ = D ◦ s˙ ' D ◦ Φ(x)T y˙ R := Ψ(x)T y˙ R , h

i

⇒ DT ◦ dM ΨT ¨yR + dG ΨT y˙ R − dQe = rn 6= 0

(6.2) (6.3)

(rn : residual for yR ∈ IRn ). Clearly, for any chosen set of shape functions the motion equation will not be fulfilled. Nevertheless, for the choice of shape functions there are obviously four basic requirements to be considered: The shape functions have to be mutually independent. They have to fulfill (every) boundary condition. Boundary conditions are always characterized by implicit (functional) expressions; they either represent kinematic relations which have to vanish at the bounds [for a beam: e.g. pinned (no deflection) or clamped (no deflection nor bending angle)] or kinetic (force and torque) balances to be zero. Thus, the shape functions must obey some additional geometrical demands in order to not contradict possible motion (e.g. considering the free end of a beam needs nonzero deflection2 ). The set of used shape functions has to be complete3 . 2 This is because the eigenfunctions will later be represented by a linear combination of chosen shape functions. Such a combination will of course fail if all the shape function are zero at the free end. 3 Consider for instance a symmetric double span bridge of length L with two torque-free joints at the ends and one in the middle. The set of functions wn = sin(2nπx/L) fulfills the boundary conditions [no 00 = 0) at the ends (0, L) as well as in the middle deflections (wn = 0) and torque-free joints (EI wn L/2] but is mathematically incomplete due to missing sin[(2n − 1)πx]/L). Indeed, such a bridge can also behave like two independent half length beams which are clamped at L/2. The corresponding deformations need hyperbolic functions for representation as Table 5.2 shows. Choosing geometrically correct wn = sin(2nπx/L) for shape functions will thus leave out part of the eigenmodes.

6.1

221

Galerkin Method

Evidently, even disposing of adequate admissible shape functions, eq.(6.3) will not be fulfilled anyhow. However, formulating the virtual work for eq.(6.1) and eq.(6.3) yields Z

δsT DT ◦ [dM¨y + dGy˙ − dQe ] = 0

(6.4)

B !

= δyTR

Z

h

B

i

Φ(x) DT ◦ dM ΨT ¨yR + dG ΨT y˙ R − dQe = 0.

(6.5)

Eq.(6.4) yields zero because of the motion equations (6.1) themselves. Eq.(6.5), however, is set to zero and characterizes a postulation, namely to fulfill Lagrange’s Principle under any conditions. Here, yRi is used as a minimal coordinate; δyR may thus be omitted yielding the (direct) Galerkin method Z B

h

i

Φ(x) DT ◦ dM ΨT ¨yR + dG ΨT y˙ R − dQe = 0,

(6.6)

which comprises the above-mentioned “transformed differential equations”. R Since eq.(6.6) represents an orthogonality condition, i.e., B Φ · rn = 0, Galerkin’s method is also called an error orthogonalization method. Comparable to the basic Lagrange formulation (2.15), which yields motions orthogonal to the constraint directions, one automatically obtains here, once the shape functions are chosen, the (approximated) motions which are orthogonal to the residuals. The latter may hereby be interpreted as constraint forces, see eq.(6.3). The procedure is thus, in this interpretation, identical to Lagrange’s observation that real motion takes place perpendicular to the (generalized) constraint forces, i.e. within the tangent plane of constraints. The corresponding unconstrained free motion direction is indicated by the tangent vector s˙ (minimal velocity) which is unknown in the present case. However, once the basic demands for Φ(x) are fulfilled, the infinite Ritz series s˙ = ΦT y˙ R represents the real solution, and the question arises how well a finite Ritz series approximates the solution and how fast is its convergence.

222 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 5 Example: pinned-free beam, first deflection mode. w x L


The bending eigenfunctions of a free-pinned beam are known from Table 5.2, last case:

1.5 1

wi (ξ) = − sinh(ki Lξ)/ sinh(ki L) − sin(ki Lξ)/ sin(ki L)

0.5 0

(6.7)

-0.5

where ξ = x/L. In Fig. 6.1, the bold plot shows the first deformation mode w1 (ξ) while the thin curve represents a second-order polynomial (aξ + bξ 2 ) which approximates the eigenfunctions w1 at the best: Let aT = (a b), f(ξ) = (ξ ξ 2 ). Then, using (aξ + bξ 2 ) = fT a for approximation, the resulting (quadratic) mean error reads

-1 -1.5 -2 0

0.2

Figure 6.1.

0.4

x L

0.8

0.6

Pinned-Free Beam

1 ∆ = 2 2

Z1

(w1 − aT f)2 dξ

(6.8)

o

where a is the vector of free variables to be calculated for minimal error. Differentiation w.r.t. a yields

∂∆

∆2 → min :

=

∂a 2

Z1

2

 1  Z Z1 (w1 − aT f) fT dξ = 0 ⇒  f fT dξ  a = w1 f dξ.

o

T

o

(6.9)

o

2

For a ∈ IR and f = (ξ ξ ) one obtains

 ⇒

1 3

1 4



1 4

1 5





a b



 R1 o

=

R1 o

w1 ξdξ



(6.10)

. 2

w1 ξ dξ

By inserting w1 = w10000 /(k1 L)4 , where (∂w/∂ξ) = w0 is used to simplify notation, the right-hand side can be integrated by parts yielding

 R1 o



R1 o

w10000 ξdξ

w10000 ξ 2 dξ

which reduces to

  =

w1 (0)00 − w1 (1)00 + w1 (1)000 2 [w1 (1)0 − w1 (0)0 ] − 2w1 (1)00 + w1 (1)000

 

(6.11)

6.1

223

Galerkin Method 



0



0

0

 , ( )0 = ∂ ,

(6.12)

∂ξ

2 [w1 (1) − w1 (0) ]

due to the boundary conditions w1 (0) = w100 (0) = 0, w100 (1) = w1000 (1) = 0. One achieves thus

a b

!



1 240  5 = (k1 L)4 1 −4

1 −4



1 3





0 2 [w1 (1)0 − w1 (0)0 ]

 = 6.6934

1

!

− 43

(6.13)

for the nearest approximation to w1 (ξ), On the other hand, the eigenfunction has to be considered unknown, of course. Therefore, if one chooses (without advance knowledge, just “by chance”) wapprox. = κ(ξ − 4ξ 2 /3) for R1 2 shape function and normalizes o wapprox. dξ = 1, one obtains a

!

1

√ = 45

!

(6.14)

. − 43

b

1 With Φ(x)T = Φ(x) = [aT f(x/L)] ∈ IR √ and a from eq.(6.14) for shape function, one should expect a good approximation because 45 differs from 6.6934 by about 0.2% only. The vector of describing variables for plane beam bending reads yT = (1 ∂ /∂x ∂ 2 /∂x2 ) ◦ w = D ◦ w. One obtains then from eq.(6.2),

1







   ∂  y =  ∂x  w '   ∂2 ∂x2

(

1



   ∂   ∂x  Φ(x)  

) yR (t) := Ψ(x)T yR .

(6.15)

∂2 ∂x2

Applied to dM, dG, dQe from eq.(5.48) yields, for dfze = 0, dMye = 0, DT ◦ dM ΨT ¨ yR + dG ΨT y˙ R − dQe



 =

− ∂ ∂x

1

∂2 ∂x2



ρAΦ¨ yR ρIΦ0 y¨R EIΦ00 yR



! dx = r 6= 0, ( )0 =

∂ . ∂x

(6.16)

According to eq.(6.6) the differential equation reads

ZL

Φ [(ρAΦ − ρIΦ00 )¨ yR + EIΦ0000 yR ]dx = 0.

(6.17)

o

Let, for simplicity, (EI)/(ρA) = 1, ρI → 0. Then, inserting the chosen shape function RL R1 along with o Φ2 dx = L o Φ2 dξ = L yields [ρAL] y¨R + 0 = 0

(6.18)

224 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations which does obviously not approximate the pinned-free beam at all. Clearly, because Φ is chosen of second order, Φ0000 vanishes everywhere. The problem, however, arises from the fact that Φ00 (x = L) 6= 0 violates the basic requirements. 4

1.2.

Extended Galerkin Method

The interpretation of eq.(6.6) as a result of the virtual work (6.5) leads to the necessity to also consider the virtual work which is achieved by forces and torques at the boundaries. This yields for eq.(6.4) and eq.(6.5), considering a beam-like structure, dG dQe dM ¨y + y˙ − dx dx dx dx o    dG dQe L dM ¨y + + δsT BoT ◦ y˙ − dx dx dx o   e L dG dQ dM T ¨y + y˙ − + δs0 B1T ◦ =0 dx dx dx o L

Z

!

=

δyTR

"Z

L

o

δsT DT ◦





(6.19)

dG T dQe dM T Ψ ¨yR + Ψ y˙ R − ΦD ◦ dx dx dx dx 

T



dG T dQe L dM T Ψ ¨yR + Ψ y˙ R − dx dx dx o   e L  dG dQ dM ΨT ¨yR + ΨT y˙ R − + Φ0 B1T ◦ = 0. dx dx dx o (6.20) 

+ Φ BoT ◦





In eq.(6.19), all the summation terms are zero because of eq.(6.3) and because of the fulfillment of boundary conditions. If the chosen shape functions do not fulfill all the boundary conditions, then in eq.(6.20) at least part of the second and third line are non-zero. However, because now the (virtual work achieved by the) boundary terms are taken into account by the approximation procedure itself, the demands on the choice of shape functions are restricted to pure geometrical ones. 5 Example: pinned-free beam, first deflection mode. Completing eq.(6.15) et seq. with the boundary terms yields

Z

L

 Φ

o

1

− ∂ ∂x

∂2 ∂x2



ρAΦ¨ yR ρIΦ0 y¨R EIΦ00 yR

! dx + · · ·

6.2

225

(Direct) Ritz Method ( ··· +

Φ

h

( Φ0

+

0

1

− ∂ ∂x

i



 0

0

1

ρAΦ¨ yR ρIΦ0 y¨R EIΦ00 yR

!)L

ρAΦ¨ yR ρIΦ0 y¨R EIΦ00 yR

!)L

o

= 0.

(6.21)

o

√ L = 1, (EI)/(ρA) = 1, ρI → 0. Using again Φ = 45(ξ − 4ξ 2 /3) = √ Let, for brevity, 2 000 0000 45(x − 4x /3) for shape function, Φ and Φ vanish, and one obtains 1

Z

 1

Φ o

( +

Φ

h

( Φ0

+

Z

0

− ∂ ∂x 1

∂2 ∂x2



− ∂ ∂x

i



 0

1 2

=

0



1

h

0

Φ dx y¨R + Φ Φ

00

i1

Φ¨ yR 0 Φ00 yR

!

Φ¨ yR 0 Φ00 yR

!)1

Φ¨ yR 0 Φ00 yR

!)1

dx

o

o

yR = y¨R + 320 yR = 0.

(6.22)

o

o

√ Eq.(6.22) corresponds to a linear oscillator with frequency ω = 320. The exact value is found in Table 5.2 [for (EI)/(ρA) = 1] to be k12 = 3.92662 . The approximated ω is thus 16 % bigger than the exact one. This result is not too bad for only one shape function. It proves the above statement that there must be an approximation for the exact solution because the chosen shape function fulfills the (necessary) minimization requirements according to eq.(6.9). 4

2.

(Direct) Ritz Method

For eq.(6.20), the boundary terms may be shifted to the integrand with an integration by parts yielding δyTR

Z o

L

dG T dQe dM T Ψ ¨yR + Ψ y˙ R − Ψ(x) dx = 0. dx dx dx 



(6.23)

This result is not really surprising because first the boundary terms had been extracted by an integration by parts and these are now re-integrated. The knowledge of Bi is not necessary for the approximative determination of motion. The general case is characterized by eq.(5.186), XZ i B i

δyTi [dM¨y + dGy˙ + dKn y − dQe ]i = 0.

(6.24)

226 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Using eq.(6.2) for every body under consideration yields directly X

δyTRi

i

Z

h

Ψi dM ΨT ¨yR + dG ΨT y˙ R + dKn ΨT yR − dQe

i i

= 0. (6.25)

Bi

We will call eq.(6.25) the “direct Ritz method”. 5 Example: Pinned–free beam. Actually, the “first mode” mentioned above does not account for the free motion explicitly. By nearer inspection, however, the heretofore connected “zero mode” has implicitly been considered. √ This means, that the used shape function Φ = 45(ξ −4ξ/3) is not really “arbitrary” but already contains some necessary basic requirements. Let us therefore have a look at the pinned-free beam as a freely rotating beam with superimposed deflection assuming all motions linearizable small. The rotation is then assigned to Φo = ξ yRo , and Φ1 = ξ 2 yR1 is chosen for relative deformation, thus w ' (ξ ξ 2 )yR := ΦT yR , yR ∈ IR2 , Φ ∈ IR2 . One obtains then for eq.(6.2) instead of eq.(6.15), 1



   ∂  y =  ∂x  w '   ∂2 ∂x2

1



 (



   ∂  T  ∂x  Φ(x)  



ξ

ξ2



yR (t) = 

1 L

2 ξ L 2 L2

 yR := Ψ(x)T yR

)

0

∂2 ∂x2

(6.26)

where (∂/∂x) = (∂/∂ξ)(∂ξ/∂x) = (∂/∂ξ)(1/L) has been used. Applied to eq.(6.25) with matrices (5.48) et seq. and the elastic potential (5.18) yields, for ρI → 0 and after dividing by ρA,

1 ρA

Z1 Ψ

1 dM T Ψ Ldξ = L dξ

o

Z1 " ξ ξ

2

o

1 0 0

#

1 L

0

2 ξ L

2 L2

0 0 0

0 0 0

!



ξ

ξ2



 

1 L

2 ξ L 2 L2

  dξ,

0

(6.27)

1 ρA

Z1 Ψ

1 dK T Ψ Ldξ = L dξ

o

Z1 " ξ ξ

2

o

1 L

0

2 ξ L

2 L2

#

0 0 0

0 0 0

0 0 1

!



ξ

ξ2

 

1 L

2 ξ L 2 L2

0

  EI  ρA dξ,

(6.28) which leads to the differential equation

 1 3  1 4

1  4



0  ¨yR + EI 4  ρAL 1 0 5

0

  yR = 0.

4

(6.29)

6.2

227

(Direct) Ritz Method

Solving the eigenvalue-eigenvector problem yields

 λ2o

0, λ21

=

EI = 320 , X = [xo x1 ] =  ρAL4

1

1

0

4 −3



(6.30)

,

where the λ’s are the eigenvalues and X is the modal matrix consisting of the eigenvectors xi , i = 0, 1. Modal transformation with yR := Xη diagonalizes the system matrices: 1  4

 1 3 XT  1 4

 1 3 =



0 EI T   X¨ η+ X ρAL4 1 0 5



0



0

3

0

 0

(  1 3  

45



0 1

0

0

1

4



64 9



0  η¨ + EI 4  ρAL 1 0 45

  η¨

=



0



+

0 EI  ρAL4 0

  Xη

0

EI   η¨ + ρAL4 1 0 0 45 Eq.(6.31) may be normalized w.r.t. the mass matrix:



0

0

(6.31)

 η = 0.



0

)

η

64 9



(6.32)

 η = 0. 320

The normalization factors are suitably assigned to the modal matrix in order to directly achieve normalization by applying the modal transformation. This leads to



1

1

0

4 −3



 √

3

 0











1 1 √ √  ⇒ xo = 3   , x1 = 45  . √ 4 0 −3 45 0

(6.33)

We obtain thus the above used shape function which was characterized by eq.(6.14) with the eigenvector x1 , and automatically the eigenfunction for the free rotation mode: wo (x) = xTo f = √ 3x/L. Consider once more eq.(6.25), for the problem under consideration:

ZL o

dM T Ψ yR + Ψ dx¨ dx

ZL Ψ

dK T Ψ dxyR = 0. dx

(6.34)

o

The modal transformation diagonalizes the matrices. Using the (w.r.t. the mass matrix) normalized eigenvectors yields

 L   L  Z Z dM dK XT  Ψ ΨT dx X = E, XT  Ψ ΨT dx X = diag{ωi2 } dx

o

dx

o

(6.35)

228 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations where ωi represent the eigenfrequencies. Since X has constant elements, one may also write

ZL 

dM X Ψ dx T





T



Ψ X dx = E,

o

ZL 

XT Ψ



dK dx





ΨT X

dx = diag{ωi2 }

(6.36)

o

where Ψ from eq.(6.2) yields (with once more X = const) Ψ(x)T X = [D ◦ Φ(x)T ]X = D ◦ [Φ(x)T X].

(6.37)

Hence, ΦT X := wTapprox approximates the eigenfunctions wT : If one would have used eigenfunctions for eq.(6.26) et seq., then one would automatically obtain a diagonal mass matrix and a diagonal restoring matrix. This is why the eigenfunctions are those functions which fulfill the dynamical equations independently of each other, see eq.(5.73). For ρI → 0 and having (ξ ξ 2 )T = Φ in mind, eq.(6.27) and eq.(6.28) may be rewritten

 L   L   L   L  Z Z Z Z 1  dM T   1  dK T  EI  T Ψ Φ ΦT dx, Ψ Φ00 Φ00 dx . Ψ dx = Ψ dx =

ρA

dx

ρA

o

dx

o

ρA

o

o

(6.38) The differential equation (6.29) thus reads

 L   L  Z Z EI T T 00 00  Φ Φ dx ¨yR +  Φ Φ dx yR = 0. ρA

o

(6.39)

o T

Using eigenfunctions (wo w1 ) = w for Φ yields the exact solution:

 L   L  Z Z EI  w wT dx ¨yR +  w00 w00 T dx yR = ¨yR + EI 4 diag{0, 3.92662 }yR = 0. ρA

ρAL

o

o

(6.40) The main difference in using shape functions and eigenfunctions is that the latter are independent of each other and thus allow for separate calculation of every mode (up to infinity). Using shape functions, however, leads always to an eigenvalue problem with nondiagonal matrices. One has thus to cut off the matrices at a finite number n to make the eigenvalue problem calculable. On the other hand, also the physical structure has to have a finite number of d.o.f. Otherwise one would come into severe contradictions in basic modeling. To see this, one may once more have a look at the basic definition of the mass element on p.63. The partial differential equations imply an infinite number of modes, but then one comes to the absurdity that a structure has more modes than it has molecules, or that for sufficient high frequency the particles in the structure move faster than the speed of light (P.C. Hughes, 1987). One concludes thus that the partial differential equation is the mathematical model of the mechanical model “elastic body”. No doubt, it is useful but should not be overestimated. Remark: Setting yRi = xi sin ωi t for the ith eigensolution and premultiplying eq.(6.39) with the ith (transposed) eigenvector itself leads to

 L   L  Z Z EI T  Φ00 Φ00 dx xi = 0 ⇒ −xTi  Φ ΦT dx xi ωi2 + xTi ρA

o

o

6.3

229

Rayleigh Quotient    

1 T x 2 i

ωi2 = 







L EI R Φ00 Φ00 T dx x i ρA  o

1 T x 2 i

L R



T

(6.41)

 

Φ Φ dx xi

o

which may be interpreted as maximum potential energy over maximum kinetic energy related to ωi2 . Considering xi → x (and, consequently, ωi → ω) unknown leads to

   

1 T x 2

ω 2 := R(x) = 

 



L EI R Φ00 Φ00 T dx x  ρA 

1 T x 2

 Lo R



Φ ΦT dx x

. 

(6.42)

o

This quotient is known as the Rayleigh quotient (J.W.S. Rayleigh, 1877). Differentiating with respect to x yields with



∂ω 2 ∂x

T

 = xT

ZL

−1

 L Z

EI  Φ ΦT dx x  ρA

o



00 T

Φ00 Φ

 L   Z dx x−ω 2  Φ ΦT dx x = 0

o

o

(6.43) once more the eigenvector problem for eq.(6.39). (∂R(x)/∂x) = 0 thus represents a stationarity condition for ω 2 or at least a necessary condition for minimality. 4

3.

Rayleigh Quotient

When applied to “time-free” systems, the original Ritz method (see sec.4.1) is often referred to as the Rayleigh-Ritz method. “Time-free” means to consider an undamped system for which the Rayleigh quotient can be resolved in the sense of eq.(6.42): xT Kx , (6.44) xT Mx M = MT > 0, K = KT ≥ 0. For finite x ∈ IRf , the variable vector x may be represented by a linear combination of the eigenvectors, x = Xc, where X = [x1 · · · xf ]: modal matrix, c: coefficient vector. Normalizing XT M X = E ⇒ XT K X = diag{ωi2 } then leads to R(x) =

R(x) =

c1 xT1 Kx1 c1 + c2 xT2 Kx2 c2 + · · · + cf xTf Kxf cf c1 xT1 Mx1 c1 + c2 xT2 Mx2 c2 + · · · + cf xTf Mxf cf

= ω12

c21 + c22



ω2 ω1

2

+ · · · + c2f

c21 + c22 + · · · + c2f



 ωf 2 ω1

.

(6.45)

230 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Ordering the eigenfrequencies ω1 ≤ ω2 ≤ · · · ≤ ωf shows that eq.(6.45) possesses a minimum for c2 = 0, · · · , cf = 0 yielding the lowest frequency ω12 (also referred to as Rayleigh’s Principle). Since then all ci are zero except the first one, the minimum corresponds to the first eigenvector x → x1 in eq.(6.44). [As easily seen by the same argumentation, maximum R(x) yields the highest eigenfrequency.] Once the first eigenvector x1 is found one may choose x such that xT Mx1 = 0. From the eigenvector problem xT [−Mω 2 + K]x1 = 0 also follows xT Kx1 = 0 leading eq.(6.45) to

R(x) =

xT K[x1 c1 + Kx2 c2 + · · · + Kxf cf ] xT M[x1 c1 + Mx2 c2 + · · · + Mxf cf ]

= ω22

c22 + c23



ω3 ω2

2

+ · · · + c2f



 ωf 2 ω2

c22 + c23 + · · · + c2f

(6.46)

where the first eigenvalue solution is suppressed due to the orthogonality condition xT Mx1 = 0. This procedure can be continued stepwise for increasing eigenfrequencies and eigenvectors and gives the basis for several methods to solve the eigenvector-eigenvalue problem. 5 Example. For demonstration purposes a finite-dimensional system is considered: plane motion of a chain under the influence of gravity [p.106]. Absolute angles γi are suitably chosen for description, vertical. The potential is then, for small amplitudes, VG = Pfmeasured from the 2 1 mg∆L (f + 1 − i)γ where (1 − cos γi ) ' γi2 /2 has been used. The (horizontal) i 2 i=1 Pi position wi of the mass point m is wi = ∆L k=1 γk (equal-valued ∆Li = ∆L, mi = m). Collecting γ1 · · · γf in q one obtains, in vectorial notation,



f

 f −1

 

..

1  VG = qT Kq, K = mg∆L  2 

    ;  

. ..



. 1

 1   1  . w = ∆LTq, T =   ..  .  . . 1

0 1 .. . .. . 1

··· ..

0

.

···

1 ···

0 .. . .. . 0 1

     ,   

(6.47)

¨ + (∂VG /∂w)T ] = δqT [m∆L2 TT T¨ yielding the motion equations via δwT [mw q + Kq] = 0 ⇒ M¨ q + Kq = 0:

6.3 

231

Rayleigh Quotient f

f −1 f −1

     symmetric

··· ··· .. .

2 ··· 2

1 1 .. . 1 1





f

0 f −1

   1   q¨ +   ν2    symmetric

··· ··· .. .

··· ···

0 0 .. . 0 1

2

    q = 0, ν 2 = g .  ∆L 

(6.48) ¨/ν 2 = (∂ 2 q/∂τ 2 ) to get rid of mechanical We use next the time scaling τ = νt ⇒ q parameters and reduce consideration to f = 3 (triple pendulum) in order to obtain an analytical solution for the eigenvalue problem

" 2

[−Mω + K]x =

3(1 − ω 2 ) −2ω 2 −ω 2

−2ω 2 2(1 − ω 2 ) −ω 2

−ω 2 −ω 2 (1 − ω 2 )

#

(6.49)

x=0:

det[−Mω 2 + K] = ω 6 − 9ω 4 + 18ω 2 + 6 = 0   p √ 3 2k 1 1 2 ⇒ ωk = 3 + 2 3 3 cos arccos √ + αk , αk = π, k = 1, 2, 3. 3 3 3

(6.50) Once the eigenvalues are known, these are inserted into eq.(6.49) to obtain the eigenvectors

"

3(1 − ω 2 ) −2ω 2 −ω 2



k1

⇒



−2ω 2 2(1 − ω 2 ) −ω 2



= k2

2

ω ω2 − 2



−ω 2 −ω 2 (1 − ω 2 )

#





1−3 1−

1 k1 k2

! =

1 2 ω2





0 0 0

!



(6.51)

 

−2 + 3 1 −

1 ω2



where the first component has been freely chosen. The numerical (rounding off) values are

ω22 = 2.294280,

ω12 = 0.415775,

x1 =

1 1.2921 1.6312

! ,

x2 =

1 0.3527 −2.3979

ω32 = 6.289945,

! ,

x3 =

1 −1.6450 0.7669

(6.52)

! .

232 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations With the same choice on the first component of x one obtains the Rayleigh quotient with M and K from eq.(6.48) for f = 3 where k1 , k2 appear as variables. In Fig. 6.2 the Rayleigh quotient is plotted over k1 , k2 ; it reads explicitly

R 3 1 -1

-4

-2

0

Figure 6.2.

2

k1

-4

-2

0

2

4

k2

Rayleigh Quotient, Triple Pendulum

R=

3 + 2k12 + k22 xT Kx = 2 T 3 + 2k1 + k22 + 2k1 k2 + 4k1 − 2k2 x Mx

(6.53)

and shows a maximum of about 6. The minimum is, on closer inspection, of course at k1 = 1.2921, k2 = 1.6312 where R has its minimum value of 0.4158. Once disposing of the first eigenfrequency Rmin = ω12 and the corresponding eigenvector x1 = (1 k1 k2 )T , the next variable to be used for the Rayleigh quotient is chosen orthogonal to Mx1 . Let Mx1 = (a b 1)T (one of the components may once more freely be chosen because of xT Mx1 = 0). One obtains then

 x=

1 k1 −a − bk1

a



! 



 : xT Mx1 = (1 k1 − a − bk1 )   b  = 0 ∀ k1 .

(6.54)

1 Inserting x reduces the problem dimension by 1. The corresponding Rayleigh quotient reads

R(k1 ) =

(3 + a2 ) + 2abk1 + (2 + b2 )k12 . (3 − 2a + a ) + (4 − 2b − 2a + 2ab)k1 + (2 − 2b + b2 )k12 2

(6.55)

6.3

233

Rayleigh Quotient R

The numerical values for the triple pendulum are



a

 

1.8391



     b  =  1.5842  .     1

1

The corresponding Rayleigh quotient in Fig. 6.3 shows a minimum with value 2.2943 at k1 = 0.3527. One obtains the second eigenvector from eq.(6.54): xT2 = [1

6 5.5 5 4.5 4 3.5 3 2.5 2 -4

Figure 6.3.

-2

0

2

k1

2nd Rayleigh Quotient, Triple Pendulum

0.3527 − (1.8391 + 1.5842 × 0.3527)] = [1

0.3527 − 2.3979],

(6.56)

compare eq.(6.52). 4

Conclusion For an approximative solution of the partial differential equations one might think of finite differences, for instance. However, for the present class of problems, such a procedure will, if at all, only succeed with intolerable effort due to the boundary conditions. It is therefore suitable to pass to the so-called “weak formulations” which first of all need a Ritz series expansion: Here, the deformation functions (and their spatial derivatives) are separated w.r.t. space and time, which always holds for undamped single elastic bodies. It is thus also justified to use such a separation statement when combining single bodies to an elastic multi body system. However, contrary to the Bernoulli approach [see e.g. eq.(5.58)] where the time functions and the spatial functions lead to two sets of independent ordinary differential equations, the spatial functions are now chosen in advance [and have to fulfill some basic requirements, of course (see page 220)]. One procedure that is based on such considerations is Galerkin’s approach. Here one is left with an ordinary differential equation for those time functions that are assigned to the chosen shape functions. The usual interpretation of Galerkin’s method is that of orthogonalizing the error: once the shape functions

234 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations are selected the corresponding motion equations are such that the related residuals vanish. However, the (direct) Galerkin procedure needs shape functions which fulfill all the boundary conditions. It turns out to be rather impossible to find such shape functions for the general case (see eqs.(5.261), (5.262) for instance). An interpretation of Galerkin’s (direct) method as a representation of the corresponding virtual work is most helpful. It shows that, if the boundary conditions are not fulfilled, the virtual work which is achieved by the boundary forces and torques has additionally to be considered. The shape functions have then only to satisfy geometrical requirements. Moreover, there is no difficulty in this interpretation to add any kind of forces/torques just by their virtual work. However, from the mechanical point of view, the obtained result looks at least unusual because one obtains non-symmetric integral terms and non-symmetric boundary terms for the mass matrix as well as for the restoring matrix, see eq.(6.17) for instance. It should also be emphasized that the determination of the boundary conditions needs some effort for the general case. Integration by parts shifts the boundary quantities into the integrands and brings symmetry back. But having in mind that the boundary terms have first been calculated via integration by parts, its rearrangement by once more integrating by parts demonstrates a kind of useless “circular” calculation. Galerkin’s direct or extended procedure is therefore not recommended for the present class of problems (although frequently used). Avoiding integration by parts in advance leads directly to (what we call) the direct Ritz method. Because the shape functions are prechosen, explicit knowledge of the dynamical boundary conditions (which is essential in the case of partial differential equations) is not required for motion determination. This is because the spatial variables are no longer independent variables. [If, on the other hand, forces and torques at the boundaries are requested for load calculations, then the boundary conditions come into play once more. However, one must be very careful. As the pinned-free-beam-example shows, the second and third derivatives of the approximated w1 yield for boundary values √ eigenfunction 00 00 000 EIw1 (0) = EIw1 (1) = 2EI 45; EIw (0) = EIw000 (1) = 0 which have not much in common with the true values. Therefore, in order to obtain the (inner) forces and torques at any location, the direct use of Euler’s cut principle is recommended (not further pursued)]. We are thus going to implement the direct Ritz method in the sequel. It is looked at as a use of Lagrange’s Principle (virtual work) which immediately leads to the “weak formulation”. The basic requirement for the shape functions is to fulfill the geometrical demands (and to form a complete set, of course). Its choice is then arbitrary in principle. However, when compared to rigid body systems with its finite number of d.o.f., the difference here lies in the basic modeling assumptions (see also the

6.4

235

Single Elastic Body – Small Motion Amplitudes

comments on p.228). Using a finite series expansion has, of course, to be looked at as an approximation. One can easily imagine that, for a truncated series, the number of chosen shape functions will play a significant role. This goes along with the question of what kind of functions should be chosen. Neither question can be answered generally. The direct Ritz approach will therefore firstly be tested for single elastic body systems with small motion amplitudes. Here, it is often suitable to transform the solutions into mutually independent eigenmodes. The investigation of eigenforms with their nodes and amplitudes gives for instance direct information on controllability and observability which can be looked at as a special application of Hautus’ criteria. Convergence can for instance be tested numerically, comparing the eigensolutions for an increasing number of shape functions. In fast-moving elastic multi body systems, however, eigenvectors do not exist (are not defined), and convergence considerations w.r.t. eigenfunctions are, of course, obsolete. The direct Ritz approach here leads to ordinary differential equations which are highly nonlinear w.r.t. the rigid body coordinates. Nevertheless, the representation is obviously correct if the shape functions are admissible and if their number is high enough. The method still involves an error orthogonalization in its background. How many shape functions will be sufficient for adequate description can once more not be answered generally. The brilliant performance of the concept, however, has often been demonstrated in practical applications. Especially in controlled systems, two or three shape functions for each deformation have mostly been found to be sufficient. This is of course due to control input with pervasive damping throughout all vibration modes.

4. Single Elastic Body – Small Motion Amplitudes 4.1. Plate 4.1.1 Equations of motion Collecting admissible shape functions in Φ(x, y)T = (φ1 , φ2 , · · · , φn ) yields for eq.(6.2) along with D from eq.(5.84) y˙ = D ◦ s˙ =

" 

1

∂y

 

⇒Ψ=

∂

Φ

∂Φ ∂y

−

∂

∂2

∂2

∂2

∂x ∂x2 ∂y 2 ∂x∂y − ∂Φ ∂x

∂2Φ ∂x2

∂2Φ ∂y 2

#T

ΦT

 

y˙ R := ΨT y˙ R



∂2Φ ∂x∂y



.

(6.57)

Applied to the mass matrix and the potential restoring matrix according to eq.(5.86) yields

236 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Z

ΨdMΨT := M =

B





Z Z     Lx Ly



ΦT



 ! ρh 0 0 T  ∂Φ    ∂Φ 3 0 Φ −  0 ρh /12  ∂y  ∂y ∂x 0 0 ρh3 /12  ∂ΦT − ∂x Z

∂Φ





       dxdy  ,    

ΨdKΨT := K =

B



  Z Z     Lx Ly 

∂2Φ

∂2Φ

∂2Φ

∂x2

∂y 2

∂x∂y

!





 1 ν 0    0 D ν 1   0 0 2(1 − ν)  

∂ 2 ΦT ∂x2 ∂ 2 ΦT ∂y 2 ∂ 2 ΦT ∂x∂y





         dxdy  .      

(6.58) Solving the eigenvalue/eigenvector problem h

i

h

i

det −ω 2 M + K = 0 ⇒ −ωn2 M + K xn = 0

(6.59)

leads to the nth (approximated) eigenfunction wen (x, y) = xTn Φ(x, y),

(6.60)

see eq.(6.37).

4.1.2 Basics There exists a variety of admissible shape functions for solving the problem. Here, it is most instructive to follow (W. Ritz, 1909) himself. Neglecting the rotational inertia ρh3 /12 (→ “Kirchhoff plate”), the partial differential equation according to Table 5.8 becomes ρhw ¨ + ∆∆w = 0 where ∆ denotes the Laplace operator, ∆ = ∂ 2 ( )/∂x2 + ∂ 2 ( )/∂y 2 . In case of eigensolutions one has w(x, y, t) = sin ω(t − to ) · we (x, y), hence 12ρ(1 − ν 2 ) 2 ω , (6.61) Eh2 along with the boundary equations from Table 5.8. According to Ritz, all these equations can be combined in the postulate that ∆∆we = λwe , λ =

 ZZ

J=

∂ 2 we  ∂x2

!2

+

∂ 2 we ∂y 2

!2

+ 2ν

∂ 2 we ∂ 2 we ∂x2

∂y 2



+ 2(1 − ν)

∂ 2 we 

dxdy

∂x∂y

(6.62)

6.4

237

Single Elastic Body – Small Motion Amplitudes

yields a minimum for we (eigenfunction, solution of plate equations) under the condition Z Z we2 dxdy = a, (6.63) a a given quantity. Following the rule of calculus of variations one obtains λ=

J(we ) min J = a a

(6.64)

for the basic tone (we → we1 ), while for the kth overtone the additional constraints Z Z

2 wek dxdy = a,

Z Z

wek we1 dxdy = 0, · · ·

Z Z

wek we,k−1 dxdy = 0, (6.65)

enter calculation, once more coming out with eq.(6.64) in the form λk =

J(wek ) min J = . a a

(6.66)

Clearly, eq.(6.62) represents the elastic potential [see eq.(5.11)] and eq.(6.63) is the kinetic energy related to λ ∼ ω 2 . The additional constraints (6.65) represent the successive suppression of previously calculated solutions. Thus, eq.(6.66) corresponds to the kth Rayleigh quotient applied to the infinite d.o.f. system. It should be emphasized that Ritz, at this step, expressly refers to (J.W.S. Rayleigh, 1877). “The new method for solving variational problems which we are going to apply emanates from the following interpolation problem: Let φ1 (x, y), φ2 (x, y) · · · φn (x, y) be a set of functions which we assume steady (along with its 1.,2.,3. and 4. differential quotient) inside the plate. We build the expression wn = x1 φ1 + x2 φ2 + · · · + xn φn and demand to determine the xi for given n such that wn differs as little as possible from one of the demanded functions we which fulfill the plate equations and the boundary conditions · · · 4 . Choosing the potential energy for error measure here leads to postulate Jn = Jn (x1 · · · xn ) → min w.r.t. xi , i = 1 · · · n [for arbitrary k under condition (6.65)]. Because J is positive definite, there exists a lower bound λn for increasing n which enables to approximate λ with desired accuracy... Hence, lim n=∞ λn cannot differ from (Jmin /a). [As a consequence, the approximated λ is always greater than the exact one. This is for instance already experienced with eq.(6.22)]. The postulation Jn = Jn (x1 · · · xn ) → min w.r.t. xi can be fulfilled 4 Ritz’s

symbols are replaced with those which have already been used in the present context.

238 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations by polynomials, Fourier series etc... It has, however, to be emphasized that the expansion into polynomials, Fourier series etc. only allow for a finite number of differentiations of their individual series elements such that (which indeed is the case for the expansion w.r.t. um (x)us (y) considered below) they are possibly not differentiable three or four times, and can thus not be verified by insertion into (3) and (5) [i.e. the differential equation and the dynamical boundary conditions]... Abandoning fulfillment of these, on the other hand, leads to an essentially simplified solution representation. ”(W. Ritz, 1909), p.278 to 286. As a conclusion we may state: the original Ritz method implies an optimization procedure w.r.t. distributed mass systems in the form [Vel ]/[Tel /ω 2 ] → min (where V : potential, T : kinetic energy). Insertion of the Ritz ansatz wn = xTn Φ(x, y) with chosen shape functions Φ yields the eigenvalue problem (6.59) for necessary and sufficient minimization conditions. (Sufficiency is demonstrated with Rayleigh’s quotient according to eq.(6.45) when applied to the distributed system. Eq.(6.65) furthermore leads to recursive frequency calculation). The big advantage in this procedure lies in simple structured equations on the one side that automatically involve treatment of the dynamical boundary conditions on the other side. Remark: Applying eq.(6.25) yields directly the plate equations of motion M¨yR + KyR = 0 with matrices (6.58) which lead to the eigenvalue problem (6.59). This motivates us to call the procedure the direct Ritz method.

4.1.3 Shape Functions: Spatial Separation Approach The shape functions used for the Ritz approach have to fulfill geometrical demands. This can be achieved with a (spatial) separation statement Φ(x, y)k = u(x)i v(y)j , i, j ∈ {1, 2, · · · m}, k ∈ {1, 2, · · · n}, Φ(x, y)l = u(x)m v(y)s , m, s ∈ {1, 2, · · · m}, l ∈ {1, 2, · · · n}, ⇒ n = m2 (6.67) n (n: number of shape functions, Φ(x, y) ∈ IR ). This approach focuses our attention on ui (x) and vj (y) for the x-direction and the y-direction separately. Considering a square plate (Lx = Ly = L), one obtains for eq.(6.63) with ξ = x/L, η = y/L, Z Z

2

wk wl dxdy = L L L

Z1

Z1

ui um dξ o

vj vs dη

(6.68)

o

[which corresponds to the element [k, l] of the mass matrix in eq.(6.58) for ρh3 /12 → 0]. Similarly, eq.(6.62) yields

6.4

239

Single Elastic Body – Small Motion Amplitudes

Jkl =

  2 Z1

1 L



o

dξ

vj vs dη o

Z1

∂ 2 vj ∂ 2 vs

o

∂η 2 ∂η 2

ui um dξ

Z1

!

∂ξ 2 ∂ξ 2

o

Z1

+

∂ 2 ui ∂ 2 um

!

dη

 1  Z 2 Z1 Z1 Z1 2 2v 2u ∂ u ∂ ∂ ∂ v i s m j um dξ vj dη + ui dξ vs dη  + 2ν  o

+

∂ξ 2

2(1 − ν)

∂η 2

o

Z1

∂ui ∂um

o

∂ξ ∂ξ

Z1

∂vj ∂vs

o

∂η ∂η

dξ

∂ξ 2

o

dη

o

∂η 2

 

(6.69)



[which corresponds to the element [k, l] of the restoring matrix in eq.(6.58)]. This seemingly costly result becomes simple when using beam functions.

4.1.4 Expansion in Terms of Beam Functions Considering beam functions one has basically Z1

Z1

∂ 2 ui ∂ 2 uj

o

∂ξ 2

ui uj dξ = δij , o

∂ξ 2

!

dξ = (ki L)4 δij

(6.70)

Kronecker’s5

(δij : symbol; see conclusions/properties of beam functions on p.144). The mass matrix thus turns out to be diagonal, and the first two lines of the (non-dimensional) restoring matrix (6.69) also only deliver diagonal elements. The remaining two lines are obtained from an intermediate calculation in nine steps: we first consider an integration by parts, with (∂u/∂ξ) = u0 for abbreviation, Z1

u0i u0j dξ

= −

o

Z1

1

u00i uj dξ + uj u0i , o

(1)

o

= −

Z1

1

u00j ui dξ + ui u0j . o

(2)

o 4 Insertion of the spatial differential equation um = u0000 m /(km L) yields, for the integrals on the right-hand side of step(1) and step(2),

5 Leopold

Kronecker, *1823 in Liegnitz, †1891 in Berlin

240 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations −

Z1

u00i uj dξ

1 (kj L)4

=

o

−

Z1

1 (ki L)4

ui u00j dξ =

o

Z1 o Z1

000 u000 i uj dξ −

1 1 u00i u000 j o, 4 (kj L)

(3)

000 u000 i uj dξ −

1 1 u00j u000 i o. 4 (ki L)

(4)

o

For a beam with free ends which will be used in the sequel the boundary terms in step(3) and step(4) vanish and one obtains for step(1) and step(2), Z1

u0i u0j dξ

=

o

=

Z1

1 (ki L)4

1

000 0 u000 i uj dξ + uj ui ,

(5)

o

o

1 (kj L)4

Z1

1

000 0 u000 j ui dξ + ui uj .

(6)

o

o

Subtracting step(6) from step(5) yields an equation which can be resolved R 000 000 for ui uj dξ: Z1

000 u000 i uj dξ

(ki L)4 (kj L)4 uj u0i (kj L)4 − (ki L)4 

=

o

1 o

−

1 u0j ui o



.

(7)

, i 6= j,

(8)

Inserting step(7) into step(3) yields, for i 6= j, Z1

u00i uj dξ =

o

(ki L)4 u0 ui (kj L)4 − (ki L)4 j 

1 o

− u0i uj

1 o



and from step(1) one obtains with step(8), Z1 o

u0i u0j dξ =

1 (kj L)4 uj u0i (kj L)4 − (ki L)4 

1 o

− (ki L)4 u0j ui

1 o



. (9)

One has thus found all non-diagonal elements R 0 2 (i 6= j) in terms of the boundary values. The only remaining integral ui dξ can be solved by hand and leads approximately to Z1 o

2

u0i dξ ' (ki L)[6 + (ki L)].

(6.71)

6.4

241

Single Elastic Body – Small Motion Amplitudes u0i ki L

ui 3

1.5

2

1

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2 0

0.2

0.4

0.6

0.8

Figure 6.4. Free-Free Beam Functions

3

1.5

x L

2

1

-2 0

0.2

0.4

Figure 6.5.

0.6

0.8

x L

First Spatial Derivatives

The “exact” values which are obtained from a computer program, [2.211603, 1.766168, 1.545587 · · ·] for [i = 1, 2, 3, · · ·], show a deviation from the approximative values of [2.6%, 0.13%, 0.007%, · · ·]. In the same manner, one may approximate step(9) by Z1

i + j odd : o



1 −



u0i u0j dξ ' 8kj L 

kj ki

1−



kj ki

3   4 

(6.72)

with maximum error less than 2%. The required remainder is known from step(1) (which also holds for i = j, of course). As Fig. 6.4 and Fig. 6.5 show, i 6= j: i + j even, yields zero for step(9) since uj (1)u0i (1) − uj (0)u0i (0) = 0 and u0j (1)ui (1) − u0j (0)ui (0) = 0.

Discussion Special care must be taken when solving the integrals numerically due to the rapidly growing hyperbolic function values. In the present case, without additional normalizing strategies, one needs for instance 80 digits to obtain reliable results (MAPLE).

242 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations When considering a plate with free edges, those beam functions which correspond to the zero eigenvalues also have to be considered to obtain a complete set of shape functions. Starting to count the beam functions with 0 one has √ uo = 1, u1 = 2 3(ξ − 0.5). Then, i and j in the above results belong to {2, 3, · · ·}. [ The integral values that arise are known with the basic properties reported on p.144. One concludes that (a) uo and u1 are indeedR beam eigenfunctions because R 1these are indepen1 dent of ui , i > 2: const o ui dξ = 0 (5.76) and const o ξui dξ = 0 (5.77), i > 2. R (b) The missing terms o1 uk (∂ 2 ui /∂ξ 2 )dξ, k = 1, 2, i > 2 are also derived from eqs.(5.76), (5.77) taking notice of u00f f = k 2 ucc (ff: “free-free”, cc: “clamped-clamped), see Table 5.2 ]. Starting with i = 2 (j = 2) for the first relative deformation, the corresponding eigenvalues in Table 5.2 change to ' [n − (1/2)]π. A numerical check shows that these may already be used for eq.(6.71) and eq.(6.72) yielding a maximum error less than 3%. The shape functions used here are (theoretically) not restricted to a plate with free edges. Other boundary conditions can be taken into account by adding one or two additional shape functions for pinned-pinned [u(0) = u(1) = 0] and for clamped-clamped borders, respectively [u(0) = u(1) = V 0 0 u (0) = u0 (1) = 0] (or any combination). This is, however, not always recommendable as easily seen in the case of an all-around pinned plate. Then, the corresponding beam functions are simply trigonometric as Table 5.2 shows. Using these conditions of course saves much effort. However, this simple consideration leads to the concept of “auxiliary functions” which in many cases is very effective. Consider for instance a one-sided clamped beam carrying a tip mass at its free end. Then it is clear that tip mass inertia plays more and more a role for increasing vibration modes. High frequency modes force the tip to remain (nearly) at rest. Thus, adding a shape function which fulfills u(L) = 0 increases convergence significantly (if the tip mass has a non-negligible rotational inertia, then a second suitable “auxiliary function” which fulfills u0 (L) = 0) should be chosen). Is there, in the “computer age”, any benefit in using beam functions for the solution of a plate instead of cubic splines, for instance? The answer is yes, for several reasons: first it leads to analytical statements concerning the solution, which is always more helpful than a data set to be interpreted. Secondly it leads to rapid convergence. This is due to the underlying orthonormality [which is also helpful when formulating the equations because a lot of (off-diagonal) terms are in advance known to vanish, see eq.(6.69)].

6.4

243

Single Elastic Body – Small Motion Amplitudes

4.1.5 Convergence and Solution T Let x u(x) be a series expansion to approximate a certain function f (x). One has then, the same as in eq.(6.9),  1  Z1 Z T  uu dξ  x = f (ξ)udξ. o

(6.73)

o

Using beam functions for u leads to uuT dξ = E. Furthermore, the mth coor4 dinate um obeys the (spatial) differential equation um = u0000 m /(km L) where 0 ( ) = (∂/∂ξ), ξ = x/L. Integration by parts yields for the mth coordinate R



xm

1 (f 00 u0 − f 0 u00 + f u000 ) = m m m 4 (km L)

1 o

−

Z1



f 000 u0m dξ  .

(6.74)

o

Using free-free beam functions, u00 and u000 vanish at the ends and one obtains 

u0 xm (km L)3 = f 00 m km L

1 o

−

Z1 o



u0 f 000 m dξ  km L

(6.75)

where the right-hand side is lower than a fixed number because | u0m /(km L) |≤ 2 (see Fig. 6.5) and f 00 , f 000 are assumed finite and steady. It follows that xm (km L)3 is bounded, or, because km L ' (m − 12 )π, that xm decreases with (1/m3 ) for increasing m. Analogously, a function f (ξ, η) may be expanded into a series f (ξ, η) = xoo uo (ξ)vo (η)+x1o u1 (ξ)vo (η)+xo1 uo (ξ)v1 (η)+x11 u1 (ξ)v1 (η)+· · · (6.76) where Z1 Z1

xmn =

f (ξ, η)um (ξ)vn (η)dξdη

(6.77)

o o

converges with 1/(m3 n3 ). Coming back to the eigenvector formulation (6.59), the shape functions Φ(x, y) have to be defined by inserting ui (x)vj (y) in a suitable way. One could for instance think of inserting uvT column-wise. Here, it is interesting to note that (W. Ritz, 1909) defines the components of Φ according to preexisting symmetry considerations: For even functions w(x, y) w.r.t. x and y only even um (x), vn (y) come into play. If furthermore w(x, y) is symmetric, then xmn = xnm , otherwise xmn = −xnm , etc. This led Ritz to define the components of Φ as um (x)vn (y) ± un (x)vm (y). As a consequence, the series converges rapidly with one dominant component for every eigenvector xn in eq.(6.60). He obtains thus for the basic tone of a square plate (ν = 0.225, glass)

244 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations λ1 = 12.43, we1 = u1 v1 +0.0394(u1 v3 + u3 v1 ) − 0.0040u3 v3 + · · · , (6.78) 1 0 0.5 0.6 0.3

0

0.3

0.6

0

0 0

Figure 6.6. Basic Tone, Eigenform

0.5 Figure 6.7.

1 Nodal Lines

λ2 = 26.40, we2 = uo v2 − u2 vo −0.0129(uo v4 − u4 vo ) − 0.0045(u2 v4 − v2 u4 ) · · · . (6.79) 1 0 0.5

0

0.6 0.3 0.3

0.6

0

Figure 6.8. 1st Overtone, Eigenform

0 0

0.5 Figure 6.9.

1 Nodal Lines

6.4

245

Single Elastic Body – Small Motion Amplitudes

Walther Ritz uses u(x) ∈ IR7 , v(y) ∈ IR7 , thus 49 shape functions, and calculates 46 eigensolutions of the plate with free borders. Fig. 6.10 shows some results of his original contribution. He solves the corresponding eigenvalue by minimizing eq.(6.66), adding the corresponding constraints eq.(6.65) step by step. Due to his dexterous choice of Φ the diagonal terms in the corresponding eigenvalue problem dominate. This leads him to a consecutive scheme to improve the eigenvalues (and eigenvectors), starting with first rough values (e.g., λ1 = 13.95) up to any accuracy (e.g., λ1 = 12, 43: an improvement of 12.23% in only two steps). At the end of his contribution, he outlines the solution for rectangular plates with arbitrary boundary conditions as well as for any polynomially bounded plate and a circular plate. The latter is used to demonstrate that his second calculation step already yields a basic frequency accuracy in the range of 0.5% with much less effort than Kirchhoff’s method (exact solution, restricted to circular plates) needed sixty years before.

Figure 6.10.

Free Plate Nodal Lines

Seulement Ritz est parvenu dans deux cas, celui du probl`eme de Dirichlet et celui de l’´elasticit´e.” ´ 6 in the foreword to (W. Ritz, 1909) Poincare

6 Jules

´, *1854 in Nancy, †1912 in Paris Henri Poincare

246 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations

4.2.

Torsional Shaft

4.2.1 Eigenfunctions The concept of “auxiliary functions” is demonstrated with a torsional shaft carrying disks at its ends. The corresponding moments of inertia are Jo and J1 , while ρIx L = Js represents the moment of inertia of the shaft when considered rigid. The total assembly is driven with γ(t) such that the boundary condition at the left end is “clamped” when ϑ is considered a relative motion w.r.t. γ.

γ

γ ϑ

r

-x

Js Jo

J1

Figure 6.11. Torsional Shaft, Coordinates

The partial differential equation reads ρIx ϑ¨ − GID ϑ00 = 0P[see eq.(5.71)] which leads to, along with the separation statement ϑ(x, t) = ϑei (x)qi (t),

ϑ00ei + ki2 ϑei = 0, ( )0 =

∂( ) ; ∂x

q¨i + ωi2 qi = 0, ωi2 =

GID 2 k . ρIx i

(6.80)

√ Due to the left boundary condition, the ith eigenfunction is ϑei = 2 sin(ki x). On the right end one has to fulfill the equilibrium condition (torque balance) GID ϑ0ei = −J1 ϑ¨ei . Using eq.(6.80) one obtains J1 ϑei q¨i + GID ϑ0ei qi =

h

i

−J1 ωi2 ϑei + GID ϑ0ei qi

J1 2 k ϑei + ϑ0ei GID qi = 0 : ρIx i √ Js 1 . ϑei = 2 sin(ki x) ⇒ tan ki L = J1 ki L 

=



−

(6.81)

6.4

247

Single Elastic Body – Small Motion Amplitudes

Js 1 , The eigenvalues ki L are obtained by the intersections of tan ki L and J 1 ki L see Fig. 6.12. Once having some rough values to start with, the eigenvalues can be calculated up to any desired accuracy by using Newton’s iteration scheme. Taking (Js /J1 ) = 0.5 yields {kn L} = {0.6533, 3.2923, 6.3616, · · · (n − 1) · π}.

1

ϑei

0.8

1

(6.82)

0.5

0.6

0 0.4

-0.5

0.2

-1

0 0

2

4

6

8

10 12 (ki L)

-1.5

0

Figure 6.12. eigenvalue Determination

0.2

0.4

0.6

0.8 (x/L)

Figure 6.13. Eigenfunctions

The rotational inertia J1 at the right shaft end plays an essential role and forces the shaft for higher frequencies to (nearly) react like a clamped-clamped shaft.

4.2.2 Motion Equations The (describing) variables y˙ for the present case read 

1 γ˙  ˙     ϑ = 0 0 ϑ˙ 0 





0 1   ∂  ∂x

γ˙ ϑ˙

!

:= D ◦ s˙ .

(6.83)

Corresponding matrices dM, dQe are obtained from eq.(5.38) and eq.(5.40) with F,

248 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations       

vo ωo r˙ c ˙ ϕ κ˙

      =      

0 0 e1 0 0 0 0 e1 0 0 0 0

0 0 0 0 0 1

        



γ˙  ϑ˙  := F y˙ ϑ˙ 0

(6.84)

where all components of F are ∈ IR3 except the last row which contains scalars (due to κ ∈ IR4 ). One obtains thus7 dJxo dJx 0 dM =  dJx dJx 0  , 0 0 0 





(6.85)



0 0 0 0 . dK =  0 0 0 0 GID

(6.86)

Let the shape functions be 

Φ=

1 0 0 ϑ



.

(6.87)

One obtains then from eq.(6.2), 



1 0   ΨT = D ◦ ΦT =  0 ϑT  T 0 ϑ0

(6.88)

which leads to M¨yR + KyR = h with matrices  Z

M :=

 Js + Jo + J1  

ΨdM ΨT = 

symmetric

RL

 T

T

ρIx ϑ dx + J1 ϑ (L)

o

RL

ρIx ϑ ϑT dx + J1 ϑ(L) ϑT (L)

  , 

o

(6.89)

R

¨ + Ordering terms from the corresponding virtual work δW = [(δγ + δϑ)(dJx /dx)(¨ γ + ϑ) δϑ0 GID ϑ0 ]dx yields directly eq.(6.85) and eq.(6.86). Notice that dJ o = dJ in the present case.

7 Check:

6.4

249

Single Elastic Body – Small Motion Amplitudes

0

 Z

K :=

ΨdK ΨT = 

0 RL

symmetric

 T

GID ϑ0 ϑ0 dx

,

(6.90)

o

hT =





1 0

Mx .

(6.91)

In eqs.(6.89) to (6.91), the rotational inertia distribution (dJx /dx) = ρIx + Jo− δ (x − 0) + J1− δ (x − L) has been considered as well as ϑ(0) = 0. Mx represents the driving torque.

4.2.3 Shape Functions Parameter Dependent Shape Functions. Using the eigenfunctions according to eq.(6.81) for shape functions yields J1 o 1+ J Js + Js √ 2 k1 L .. √. 2 kn L

       M=    

    K=   



√

2 kn L  

    ,    

E

(6.92)



0

    n o ,  GI 2 D (k L) diag J L  i

0 .. .

h

···

0 ···

0

(6.93)

s

0

hT =

√

2 k1 L

1 Js

0 ···

0

i

Mx

(6.94)

where (J1 /Js ) sin ki L = [1/(ki L)] cos ki L from eq.(6.81) has been used. The mass matrix (6.92) is only decoupled in part because eq.(6.81) refers to the clamped shaft while eqs. (6.92) to (6.94) include the superimposed free rotation. The main drawback in using eq.(6.81) for shape functions is, at least for more general considerations (see e.g. section 4.3), that ki depends on the actual parameters Js , J1 .

250 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Parameter Independent Shape Functions. To simplify matters our attention is focused on how to approximate ϑei from eq.(6.81) to be used in the motion equations with matrices (6.89) to (6.91). Here, the question of convergence is once more answered by inspecting  1  Z Z1  ϑ ϑT dξ  x = f (ξ)ϑ dξ o

(6.95)

o

[compare eq.(6.73)] which expands f (ξ) in terms of ϑ by error minimization. Orthogonal Shape Functions. Choosing orthogonal shape functions, as in eq.(6.73), leads the mth component to  1  Z 0 0 (0) ϑ ϑ xm (km L) =  f 0 m dξ − f (0) m  . (6.96) km L km L o

√ One may for instance use ϑm = 2 sin[(m − 12 )π ξ] (“clamped-free”). Then, xm decreases slowly with (1/m) for increasing m. This is because the underlying differential equation ϑi = −ϑ00i /ki2 (which has been used in eq.(6.96) for integration by parts) is of second-order only. A direct check for convergence is carried out for ϑe2 (see Fig. 6.13) with (k2 L) = 3.2923:  Z1   x=2    o

sin 3.2923ξ · sin π2 ξ dξ sin 3.2923ξ · sin 3π 2 ξ dξ .. .  sin 3.2923ξ · sin (2n − 1) π2 ξ dξ


   .  

(6.97)

The mth component reads sin 3.2923 + (2m − 1) π2 sin 3.2923 − (2m − 1) π2 = − 3.2923 − (2m − 1) π2 3.2923 + (2m − 1) π2 "

xm







#

(6.98)

and leads to the kth approximation ϑ2,k

  k √ X π = 2 xm sin (2m − 1) ξ . 2 m=1

(6.99)

As Fig. 6.14 shows, the eigenfunction seems well represented at first glance. However, a magnification in the neighborhood of ξ = 1 shows significant deviations from the eigenfunction even for high numbers k (which are indicated by index: ϑ2,k ).

6.4

251

Single Elastic Body – Small Motion Amplitudes

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

-0.185

-0.19

ϑ2,50

-0.195

-0.2

ϑ2,100

-0.205

ϑ2,500 -0.21

ϑe2

0.2

0

0.4

Figure 6.14.

0.8

0.6

-0.215 0.995 x/L

ϑ2,50

0.997

0.999 x/L

Figure 6.15. Magnification

Non-Orthogonal Shape Functions: Auxiliary Function. To √ increase convergence, an additional √ function (“auxiliary function”) ϑa = 2 sin πξ is inserted yielding ϑT = 2(sin πξ, sin π2 ξ · · ·), hence  1   Z T   ϑ ϑT dξ  = 1 c

c

(6.100)

E

o

where sin π − (2m − 1) π2 sin π + (2m − 1) π2 = − π − (2m − 1) π2 π + (2m − 1) π2 "

cm







#

,

(6.101)

(| cm |= 8/{[4m(m − 1) − 3]π}). The coefficient vector x now reads 

x=2

 1  Z −1  1 cT   c E  o

sin 3.2923ξ · sin (πξ)
dξ sin 3.2923ξ · sin π2 ξ dξ .. .



sin 3.2923ξ · sin (2n − 1) π2 ξ dξ 

  . 

(6.102)



The inverse is derived from eq.(4.116) and may be specified for the present case to

252 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

1 cT c E

−1

1 = 1 − cT c

"

1 −cT −c E + c cT − cT cE

#

(6.103)

(proof by direct calculation). Using c ∈ IR3 yields with rule (R3) on p.443, 

1 cT c E

−1

=

1 1 − cT c



1 −cT −c E + ec ec



, c ∈ IR3 .

(6.104)

-0.185

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

-0.19 -0.195 -0.2 -0.205

ϑe2 ϑ2,4

-0.21 0

0.2

0.4

0.6

0.8

-0.215 0.995

0.997

0.999 x/L

Figure 6.17. Magnification

Figure 6.16. ϑ2,4

One obtains thus ϑ2,4 =

x/L

i √ h P 2 x1 sin πξ + 4m=2 xm sin(2m − 3) π2 ξ with 

        T 1 1 −c  x=  1 − cT c −c E + ec ec      

sin [3.2923 − π] sin [3.2923 + π] +π   3.2923 − π  − 3.2923     sin 3.2923 + π2 sin 3.2923 − π2  − π π  3.2923 + 3.2923 − 2 2 i i  h h  3π 3π . sin 3.2923 + 2  sin 3.2923 − 2  −  3π 3π 3.2923 + 2 i  3.2923 − 2 i h h   sin 3.2923 + 5π sin 3.2923 − 5π  2 2 − 5π 5π 3.2923 + 2 3.2923 − 2 (6.105) 

6.4

Single Elastic Body – Small Motion Amplitudes

253

Conclusions. 1) Special attention must be drawn ϑi,4 to the shape functions being mutual1.5 ly independent. Linearly dependent functions would lead the system to 1 be kinematicly overdetermined (see 0.5 also footnote 14 on p.164). Independence in the present example is 0 represented by (1 − cT c) 6= 0 in eq.(6.103). -0.5 2) With the use of four shape functi-1 ons (including the auxiliary one) one obtains good results already for the -1.5 first three approximated eigenfunc0 0.2 0.4 0.6 0.8 x/L tions as depicted in Fig. 6.18. The eigenfunctions are hereby distinguisFigure 6.18. Approximations (bold) hed by their number of nodes. Of course, the last (fourth) approximated eigenfunction deviates considerably from the corresponding eigenfunction (thin line). This is a direct consequence of Rayleigh’s quotient (6.66) along with the corresponding constraints (6.65). 3) Additional “auxiliary functions” increase convergence significantly. In the present context, it’s use is even necessary when considering more general applications (see below). While for a single shaft-disk element the eigenfunction according to Fig. 6.14 may be considered accurate enough, Fig. 6.15 shows that the end point is only approached for a very high number of shape functions. Moreover, no matter how many shape functions are inserted, the boundary condition (6.81) will never be fulfilled. This is because of the zero slope of the shape functions used at ξ = 1. Although the dynamical boundary conditions need not be fulfilled for the (direct) Ritz approach, such a behavior may have consequences in e.g. multi shaft-disk systems such as in change-over gears.

4.3.

Change-Over Gear

A change-over gear consists of the drive shaft (above left in Fig. 6.19, black) which transmits the input torque through a toothing wheel pair to the countershaft (below, black). The gear wheels of the countershaft mesh with the gear wheels of the output shaft (above, white) without load except the one which is (by handling the gear lever) fixed to the output shaft for the chosen transmission ratio (not depicted in Fig. 6.19). Let the countershaft consist of n interconnected shaft-disk elements of length Li . The input angle γi for each element is then, along with the Ritz coefficients qϑi which are assigned to ϑi ,

254 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations

Figure 6.19.

Change-Over Gear

γ1 = γ1 , γ2 = γ1 + ϑ1 (L1 )T qϑ1 , ··· !

yR1 =

γ1 qϑ1

!

yR2 =

γ2 qϑ2

!

yR3 =

γ3 qϑ3



= 

= 

=

1 0

0 E

0 0

0 0

··· ···



1 0

ϑ1 (L1 )T 0

0 E

0 0

··· ···



1 0

ϑ1 (L1 )T 0

ϑ2 (L2 )T 0

0 E

··· ···



yR := F1 yR , yR := F2 yR , yR := F3 yR ,

··· yTR := ( γ1

qTϑ1

qTϑ2

qTϑN ).

···

(6.106)

One obtains then the motion equation for the countershaft in a typical Finite Element representation: N h X i=1

i

FT M F ¨yR + i

N h X i=1

i

FT K F yR = Mx /Js e1 i

(6.107)

(N : number of disks) with Mi , Ki from eq.(6.89) and eq.(6.90). Here, the relationships at the end of the predecessor element enter the actual one and lead

6.5

255

Single Elastic Body – Gross Motion

to an accumulation of inaccuracies. It will thus be suitable to use e.g. the first three shape functions according to Fig. 6.18. Because these no longer depend on Ji it does not matter how many disks enter the calculation.

5. Single Elastic Body – Gross Motion 5.1. The Elastic Rotor Consider once more the elastic rotor on page 127. The rotor is now specified to have a nonuniform mass distribution and it is fixed to its surrounding by bearings. A technical example is illustrated by Fig. 6.20 where a rigid disk (RR , DR ) is mounted on a flexible shaft. The bearings (indicated by a spring symbol) are assumed identical and rotationally symmetric.

z6 1   y   6

RR

 

q

 PP

    i P PP PP P

PP @ @ R

P

PP PP  PP  PP   q P   q P x 

D L P R@ I @ PP PP

Figure 6.20. Elastic Rotor: Flexible Shaft (dotted), Rigid Disk (bold), Bearings (spring symbol)

Considering at first a uniform shaft, the corresponding matrices dM, dG, dQ are derived from eq.(5.92) et seq. The vector of minimal velocities reads s˙ = (α, ˙ v, ˙ w) ˙ T . The corresponding Ritz-ansatz 



α(t) ˙ 1 0    ˙ t)  = 0 v(x)T  v(x, 0 0 w(x, ˙ t) 





α(t) ˙ 0   T  0  q˙ v (t)  = Φ y˙ R T w(x) q˙ w (t)

yields for eq.(6.2), along with D from eq.(5.95),

(6.108)

256 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

1 0 0 0

       ΨT = D ◦ ΦT =   0    0  

0

0 1 0 0 ∂ ∂x ∂2 ∂x2 0

0 0 1 −∂ ∂x 0 0 ∂2 ∂x2

  1 0    0 vT     0  0     T  0 0 ◦Φ =   0 v0 T     T   0 v00  

0

0 0 wT −w0 T 0 0 w00 T

0

      .     

(6.109)

Applying Ψ to eq.(5.92) et seq., one obtains ΨdMΨT := dM = ρIx  +ρA(v 2 + w2 )  

−ρAwvT +ρ(Ix − Iz )w0 v0 T

   −ρAvw   +ρ(I − I )v0 w0 x z     ρAwv

ρAvwT −ρ(Ix − Iy )v 0 w0 T   

ρAvvT +ρIz v0 v0 T

0

0

ρAwwT +ρIy w0 w0 T

−ρ(Ix − Iy )w0 v 0

    dx, (6.110)     

ΨdGΨT := dG = 

0

     −ρAαvv ˙    +ρ(Ix − Iy )αv ˙ 0v0   +ρ(Ix − Iz )v0 w˙ 0    −ρAαww ˙    +ρ(Ix − Iz )αw ˙ 0 w0

−ρ(Ix − Iy )w0 v˙ 0

2ρAαvv ˙ T +ρ(Ix − Iz )w˙ 0 v0 T

T 2ρAαww ˙  −ρ(Ix − Iy )v˙ 0 w0 T 



−2ρAαvw ˙ T 0

−ρIy v0 w0 T

2ρAαwv ˙ T +ρIz w0 v0 T

0

       dx,       

(6.111)





−dMx /dx   −(0) +− δ (L)] v  dx, ΨdQ := dQ = −  EIz v 00 v00 + kv[δ −(0) +− EIy w00 w00 + kw[δ δ (L)] w

(6.112)

6.5

257

Single Elastic Body – Gross Motion

where eq.(5.91) has been replenished with the driving torque and the bearing stiffness (linear elasticity, spring coefficient k). Here, the dirac-distribution − δ is once more used to indicate that the bearings act at x = 0 and x = L. Notice that normal symbols like v correspond to v(x, t) while bold faced symbols like v represent the vectors of shape functions, v = v(x). The position vector assigned to eqs.(6.110) et seq. reads yR = (α, qTv , qTw )T where α has a finite value while v = vT qv etc. is assumed small. Neglecting higher-order deformations leads to 

ρIx

  −ρAvw    dM =  +ρ(Ix − Iz )v0 w0    ρAwv

0

0



ρAvvT +ρIz v0 v0 T

0

0

ρAwwT +ρIy w0 w0 T

     dx,    

−ρ(Ix − Iy )w0 v 0

0



  −ρAαvv ˙    +ρ(Ix − Iy )αv ˙ 0v0  0 0 dG =   +ρ(Ix − Iz )v w˙    −ρAαww ˙   +ρ(Ix − Iz )αw ˙ 0 w0

−ρ(Ix − Iy )w0 v˙ 0

0

0

0

−2ρAαvw ˙ T −ρIy v0 w0 T

2ρAαwv ˙ T +ρIz w0 v0 T

0

(6.113)

        dx.      

(6.114)

Consequently, the mass matrix results are nonsymmetric. However, the motion equation for α is decoupled from the remainder. This leads to the following observation.

5.1.1 Rheonomic Constraint Since α is decoupled, one may consider its solution as a rheonomic constraint:

Z

(ρIx α ¨ − dMx /dx)dx = 0 ⇒ α˙ =

Zt o

L

R

R

Mx dt + α˙ o := Ω(t) Jx

(6.115)

where L ρIx dx := L (∂Jx /∂x)dx = Jx . Dynamical stiffening requires the calculation of zero-order forces and torques according to eq.(5.165). The only

258 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations (o)

zero-order term here reads dMx = dMx − ρIx α. ¨ This term, however, is zero due to eq.(6.115). Thus, no “dynamical stiffening” arises in the present context. Next, let ! ! q˙ v α˙ y˙ R = where y˙ C = . (6.116) q˙ w y˙ C Rotational symmetry yields ∂Jy

ρIz = ρIy :=

.

(6.117)

∂x

In the same manner, ρA is replaced by (∂m/∂x) in order to extend the mathematical description to non-uniform mass distribution (see Fig. 6.20, for instance). Inserting α˙ = Ω, v = vT qv , w = wT qw into eqs.(6.113), (6.114) and rearranging leads to  h Z

∂m T ∂x vv



i





0 h

0

L

Z    +Ω   L 

∂Jy 0 0 T ∂x v v

+

∂m T ∂x ww

+

∂Jy 0 0 T ∂x w w

i  dx ¨ yC

T −2 ∂m ∂x vw ∂Jy 0 0T  x − 2 ∂x − ∂J  ∂x v w



0



+

∂m T  +2 ∂x wv ∂J 0 0T x 2 ∂xy − ∂J ∂x w v

 Z   ˙  +Ω   L 

0

T − ∂m ∂x vw ∂Jy 0 0T  x − ∂x − ∂J  ∂x v w



0



+

∂m T  + ∂x wv ∂Jy ∂Jx 0 0T ∂x − ∂x w v

0

T − ∂m ∂x vv   ∂J y ∂J 0 0T x Z  − ∂x − ∂x v v



0



+Ω

2

L

R

  dx y˙ C  

   

T − ∂m ∂x ww ∂J 0 0T x − ∂xy − ∂J ∂x w w

0



h

EIy v00 v00 T dx+k vvT |0 +vvT |L

i

  dx yC  

     dx yC  

0

L

+ 

0

R L

h

EIy w00 w00 T dx+k wwT |0 +wwT |L

  i  yC

6.5

259

Single Elastic Body – Gross Motion
T

0 0

=

[vT

where Ω follows from eq.(6.115). has to be taken at x = ζ ∈ {0, L}.

,

(6.118)

|ζ ] indicates that the coordinate function

5.1.2 Choice of Shape Functions – Prolate Rotor (Ω = 0) Due to the same boundary conditions for v(x, t) and w(x, t), the corresponding shape functions are chosen identical: w(x) = v(x). Considering the non-rotating rotor, (Ω = 0) decouples the motions for the y- and z-direction and leads to 

Z 





∂m T ∂Jy 0 0 T ¨v vv + vv dx q ∂x ∂x 

L



 Z

+ 

h

i

T EIy v00 v00 dx + k vvT |0 +vvT |L  qv = 0.

(6.119)

L n For example, free-free beam √ functions vb ∈ IR from Table 5.2 may be used. “Weighting” these with 1/ ρAL yields the restoring matrix

Ko

 1 Z EIy :=  4 o

=

ρAL

∂ 2 vb ∂ξ 2

!

∂ 2 vb ∂ξ 2

!T



i k h T dξ + vb vb |0 +vb vTb |L  mB

h i diagω 2 + k vb vT |0 +vb vT |L b b bi 1···n mB

(6.120)

along with the mass matrix  1 Z ∂Jy /∂ξ ∂m/∂ξ Mo :=   vb vTb + 2 o

mB

mB L

∂vb ∂ξ

!

∂vb ∂ξ

!T    dξ 

(6.121)

where ξ = x/L, ωb : Euler-beam frequency, mB = ρAL (reference value). Of course, the use of such shape functions only makes sense if the rotor has a small extension in the x-direction (oblate rotor). In case of a prolate rotor one may adjust beam functions by vbj (η), 0 (ξ ) · (η − ξ ), vbj (ξ R ) + vbj R R L L 0 R R vbj (ξ R ) · L + vbj (η − L ), B B

0 < η < ξR, LR , ξR < η < ξR + L B LR + ξ < η < LR + 1. R LB LB

(6.122)

260 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations

p6 pp pp pp

6p p p p p pppp

6 pp

pp

ppp

ppp

ppp

pppp

p p p p p pp pppppppppp



v3 6 pp pp pp pp p

pp

pp

pp

ppp

pppp

p p pp pppp p p p p p p p

ppp

ppp

ppp

ppp

pp

pp

pp

pp

pp

-

-

1 − ξR

LR /LB

ppppp ppp

ppppp

-

- ξR

p ppp

ppp ppp

p pppp

ppp pppp

pppppp

p pp pppp p p p p p p p

p p p p p pp pppp p p p p p

ppp

ppp

ppp

pp

pp

pp

pp

pp

η

pp

pp

p

Figure 6.21. Adjustment to Rotor Geometry

Here, the beam functions vbj are normalized in the x-direction with ξ = x/LB ⇒ 0 ≤ ξ ≤ 1. The left rotor end is fixed at ξ R . LB and LR are the beam length and the rotor length, respectively. Considering a free-free shaft, the first two shape functions are vb1 = 1 and vb2 = 2η − 1 (not normalized). Disposing of a set of independent shape functions, any regular combination of these may instead be used for a shape function. For j ≥ 2 one may thus combine vbj with vb1 and vb2 in the form (vbj − cη) to obtain the end point position η = ηend : [vb0 (ξ R ) − c] !

LR + vb (1) − c = (vb − cη)end ∀j. LB

(6.123)

Especially, (vb − cη)end = vb (ξ = 1) yields c = vb0 (ξ R )[1/(1 + LB /LR )] which projects the end point back into the initial position, see Fig. 6.21 bottom. In case of axis-symmetric systems one obtains a set of symmetric and antimetric shape functions the corresponding motion equations of which are decoupled.

6.5

261

Single Elastic Body – Gross Motion

Last, the longitudinal coordinate η used above, which due to stretching ranges from zero to vbi 1 + (LR /LB ), may be norma6 4 lized to ξ with 0 ≤ ξ ≤ 1 to 1 come out with matrices (6.120), 5 (6.121). This yields ξ = ξ/[1 + 0.5 (LR /LB )], or, since ξ = x/LB , ξ = x/(LR +LB ). Doing so, one 0 obtains for instance the first six shape functions for a symmetric -0.5 rotor system where the rigid rotor 3 takes one third of the total length -1 as depicted in Fig. 6.22 (normalized to vbi,max = 1). Notice that ξ 0 0.2 0.4 0.6 0.8 the shape functions which belong to the zero eigenvalues (dotted, Figure 6.22. Prolate Rotor Functions not numbered) are needed for completeness. The corresponding eigenvalue-eigenvector problem h

i

qv = q sin ωo t : −Mo ωo2 + Ko q = 0

(6.124)

yields the modal matrix To = [q1 · · · qn ] which decouples the motion equations (6.119) by TTo Mo To = E,

diag ω 2 . TTo Ko To = i=1···n oi

(6.125)

Since To has constant elements, one may rewrite TTo Mo To as R [(∂m/∂ξ)/(ρA)(TTo vb )(TTo vb )T + · · ·]dξ = E which demonstrates the fact that TTo vb represents the (approximated) eigenfunctions vo for eq.(6.119) (as had already been demonstrated by eq.(6.37) et seq.). Index o here characterizes the non-rotating Rotor (Ω = 0). The eigenfunctions of the non-rotating case will later be used for shape functions in the rotating case. Once the (non-rotating) shape functions are selected, the requested integrals are easily computed R 1 with the trapezoid rule. One obtains for instance the mass matrix element o [(∂m/∂ξ)/mB ]vi · vj dξ from the scalar product

262 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 1 mB



vi (1) vi (2) . . . vi (k1 ) vi (k1 + 1) . . . vi (k2 ) vi (k2 + 1) . . . vi (n)

×



1 (ρA)1 2

                




(ρA)1 ..

        ×        

. 1 [(ρA)1 2

+ (ρA)2 ] (ρA)2 ..

. 1 [(ρA)2 2

+ (ρA)3 ] (ρA)3 ..

. 1 (ρA)3 2

vj (1)

vj (2)

...

vj (k1 )

vj (k1 + 1)

...

vj (k2 )

vj (k2 + 1)

. . . vj (n)


T

.

(6.126) The shape functions are hereby tabulated with n steps for 0 ≤ ξ ≤ 1 and (ρA)m changes its value at vi = vi (km ), m = 1, 2. The rotor is characterized by the total length L, shaft radius RB , rotor length DR , rotor radius RR and vei the material densities ρB and ρR of 1 the shaft and of the rigid “disk”, 1 respectively. Considering an aluminum disk and a steel shaft, the nume0.5 3 rical values are given in Table 6.1. With these one obtains the eigen0 functions as depicted in Fig. 6.23. Of course, the results do not depend on-0.5 4 ly on the actual system parameters 2 but also on the approximative inte-1 gral solution. The corresponding error mainly results from the fact that ξ 0.2 0.4 0.6 0.8 0 the curvature is kept zero for every calculated trapezoid. In order to obFigure 6.23. Prolate Eigenfunctions tain some error information, one may consider a sufficiently small interval ∆ wherein the curvature f 00 = κ turns out to be approximately constant. The corresponding function f is then R f = κξ(ξ − ∆)/2 which yields the integral value o∆ f (ξ)dξ = −κ∆3 /12, additionally to the trapezoid area with κ = 0. The integration error E summarizes for z = (n − 1) (equidistant) intervals with z∆ = 1 to | E |∼| f 00 | ∆2 /12. Obviously, the error can be kept small for sufficient

6.5

263

Single Elastic Body – Gross Motion

small ∆. The curvature, however, grows quadratically for higher modes: R 2 . The consider for instance the mass matrix element f dξ where f = vbi 2 00 0 00 second derivative f is 2(vbi + vbi vbi ) leading to the maximum curvature κmax ∼ 2ki2 = 2{[(2i + 1)/2](π/L)}2 (see Table 5.2 on page 140). For the sixth mode (i = 4, the first two correspond to zero eigenvalues) one obtains approximately | E |max ' 3.7 · 10−4 if ∆ is chosen (1/3) · 10−2 . Although acceptable, one should keep in mind that the error grows quadratically with i.

Table 6.1. Numerical Values Prolate Rotor / Results (Nonrotating)

Libraries:

eigenvalue Problem: EISPAC Matrix Inversion: NUMREC

Numerical Values ([kg-m-s]-units): RR = 1 ; DR = 1 ; RB = 1 ; ρB = 7850; 200 9 3 L L L ρR ρB = 0.344; k = 20000; EI = 51.20; L = 1; 2 LρB = 0.61654 (reference value); mB = πRB ρR mR mB = ρB



RR RB

2

DR = 56.62 L

2 mR RR Jx mB = mB 2 = 1.0161;

2 Jy mR RR mB = mB 2

Eigenfrequencies (in rad/sec):



1+1 2 6

8.148030588; 519.4296473;



DR RR

2 

Jx = 2m B

21.58053668 571.9133410

The subject rotor is kept in ball bearings (see Fig. 6.20). However, if the rigid disk is considered flat (DR → 0), and if the bearings are assumed very strong (k → ∞), then the shape functions result in simple trigonometric ones (Table 5.2, “pinned-pinned”). The spatial integrals can then be calculated analytically (no integration error). Moreover, the corresponding eigensolutions (for the non-rotating rotor) can also be solved analytically which leads to simple accuracy investigations. One is then left with the oblate rotor which will be discussed in detail. As the results for the rotating case will show, the basic quantities of the non-rotating rotor will already give insight into the total system behavior. This fact also holds for the prolate rotor.

264 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 5.1.3 Choice of Shape Functions – Oblate Rotor (Ω = 0) For sufficiently small DR and k → ∞ the shape functions of the non-rotating rotor and their first derivatives read vTb ∂vb

√

=

!T

√

=

2( sin πξ

sin 2πξ

sin 4πξ · · ·),

sin 3πξ

2π(cosπξ 2 cos 2πξ 3 cos 3πξ 4 cos 4πξ · · ·) (6.127)

∂ξ yielding

√

vTb |0.5 = ∂vb

!T

√

=

∂ξ

2(+1

0

2π(0

−2

−1 0

0

· · · ), · · ·),

+4

(6.128)

0.5

R1

along with the integral values

o

Z1

∂vb

o

∂ξ

Z1

∂ 2 vb

o

∂ξ 2

!

vb vTb dξ = E,

∂vb

!T

diag (nπ)2 , dξ = n=1···

∂ξ !

∂ 2 vb ∂ξ 2

!T

diag (nπ)4 . dξ = n=1···

(6.129)

Mass and inertia moment distribution are, with (∂/∂ξ) = (∂/∂x)(∂x/∂ξ) and x = ξL, ∂Jy δ (ξ − 0.5), δ (ξ − 0.5) = mB + mR− = JB + JR− ∂ξ ∂ξ

∂m

(6.130)

leading eqs. (6.120), (6.121) to 

Mo =

E+ 

  JB diag  nπ 2 L mB n=1···

JR mR +  vb vTb |0.5 + mB mB L2 

Ko =

∂vb ∂ξ

!

∂vb ∂ξ



!T

, 0.5

   n o EIy L diag  nπ 4 diag ω 2 = , (6.131) Euler,n n=1··· L mB n=1···

(JR = JRy , JB = JBy = ρIy L, vb vTb |0.5 = vb (ξ = 0.5) vb (ξ = 0.5)T for abbreviation; index B: beam (shaft), index R: (rigid) rotor).

6.5

265

Single Elastic Body – Gross Motion

Rayleigh Beam vs. Euler Beam. Neglecting mR , JR yields the Rayleigh beam for pinned-pinned boundary h conditions. Because i then Mo , Ko are diago2 nal, the eigenvalue problem det −Mo ωRayleigh − Ko = 0 yields "

JB − 1+ mB



nπ L

2 #

2 2 ωRayleigh,n + ωEuler,n = 0,

(6.132)



JB = mB



RB 2

2

=

D2 16

2 ⇒ ωRayleigh,n = 1 −

Since eq.(6.132) is independent for every n, one has found the exact solution for the present case. The L/D-ratio diminishes the (square of the) nth Euler2 frequency ωEuler,n by 

∆= 1+



4 L nπ D

2 −1

× 100



1



4 L 1 + nπ D h



 2 i2  ωEuler,n .

∆[%] L/D = 10

35 30 25

20 (percent). The deviation shown in Fig. 6.24 demonstrates on15 ce more the validity range of 10 the “L/D-rule” (compare also 5 Table 5.6 on page 150). For the L/D = 50 present case, RB /L = 1/200 ⇒ 0 L/D = 100 (see Table 6.1 n 0 8 6 2 4 and/or Table 6.3) causes negligible errors. In other cases, e.g. for Figure 6.24. Frequency Square Deviation the proposed “L/D ≤ 10”-rule, the number n of dynamically interacting modes should be kept small by introducing damping or control mechanisms.

Symmetric and Antimetric Mode Decomposition. As eq.(6.128) shows, it will be reasonable to rearrange the shape function into symmetric and antimetric ones, √ vTb

=

T v0b

=

2(sin πξ sin 3πξ sin 5πξ · · · | sin 2πξ sin 4πξ sin 6πξ · · ·), √ π 2 (cosπξ 3 cos 3πξ 5 cos 5πξ· · · | 2 cos 2πξ 4 cos 4πξ 6 cos 6πξ · · ·) L

(6.133) yielding

266 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

JR mB L2

∂vb ∂ξ

!

∂vb

!T

=2

∂ξ

0.5



mR mR vb vTb |0.5 = 2 mB mB

JR mB



 0    2  2π    L    0 

+1 −1 +1 · · ·



 −1 +1 −1 · · ·   +1 −1 +1 · · ·   . ..  . .  .     0

0  

0

     +1 −2 +3 · · ·  ,  −2 +4 −6 · · ·   +3 −6 +9 · · ·  

.. .

..

   .     0 

.

(6.134)

In eq.(6.134), the upper part belongs to the “symmetric” modes, the lower one to the “antimetric” modes. Because the remainder in Mo from eq.(6.131) is diagonal (and Ko is diagonal as well), the motion equations for the symmetric modes are decoupled from those for the antimetric modes and can be treated separately.

Antimetric Modes. As Fig. 6.24 shows, the beam rotational inertia may be 2 /4). neglected. The rotor rotational inertia (JR /mB ) is (mR /mB )(RR 2 2 Insertion into eq.(6.134) yields 2(mR /mB )(RR /4)(2π/L) = 2 2(mR /mB )(πRR /L) := κ. The motion equations thus read 



    E + κ   

+1 −2 +3 · · · −2 +4 −6 · · · +3 −6 +9 · · · .. .. . .





2 ωEuler,1

    ¨  y+   

 2 ωEuler,2

.. .

2 ωEuler,3

..

.

  y = 0  

(6.135) where

2 ωEuler,n

= (EIy L/mB

)(2nπ/L)4

(antimetric).

Rotational Inertia Influence. The eigenvalue problem for two shape functions can be solved analytically and yields, with ωEuler,1 = ω1o for abbreviation (where “o” signifies JR = 0), 

2 − (1 + κ)ω 2 ω1o 2κω 2 2 2 2κω 16ω1o − (1 + κ)ω 2



2κω 2 2 −[ω1o − (1 + κ)ω 2 ]



!

0 = . 0 (6.136)

6.5

Solving eq.(6.136) for the first eigenfrequency gives already a rough impression of the influence of JR . One obtains 

−

267

Single Elastic Body – Gross Motion

1 2

ω1 ω1o



2

15 1 + 5κ

=

1 2



r

17 + 20κ 1 + 5κ

1+

ω1
2 ω1o

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0



8 16 2 κ+ κ . 5 9

(6.137) As Fig. 6.25 shows, the impact of the additional moment of inertia on the Euler beam is dramatic. Due to the inertia forces during vibration, the rigid rotor tends to “clamp” the shaft in the middle. With increasing inertia, the frequency decreases rapidly.

0

10

5

Figure 6.25.

15

20

κ

Frequency Square over κ

Numerical Results – Antimetric Modes (Ω = 0). In the next step, the eigenvalue problem (6.135) is solved numerically for n=40 shape functions with the parameters according to Table 6.3. The first three (antimetric) eigenfunctions are depicted in Fig. 6.27. The underlying shape functions are shown in Fig. 6.26. vbi

voi

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2 0

0.2

0.4

0.6

0.8

Figure 6.26. Shape Functions vbi

ξ

-2 0

0.2

0.4

Figure 6.27.

0.6

0.8

Eigenfunctions voi

The first six resulting eigenfrequencies are listed in Table 6.2.

ξ

268 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Table 6.2. Eigenfrequencies – Antimetric Modes (Ω = 0)

Libraries:

eigenvalue Problem: EISPAC Matrix Inversion: NUMREC

Eigenfrequencies (in rad/sec):

55.98187056; 1847.457893; 6589.787260;

573.4721736 3853.273392 10057.14721

Table 6.3. Parameter Values Oblate Rotor RB = 1 ; RR = 1 ; DR = 1 ; ρB = 7850; 200 5 50 L L L ρR k → ∞; EI = 51.20; L = 1; ρB = 1.0; 2 mB = πRB LρB = 0.61654 (reference value); ρR mR mB = ρB



RR RB

2

DR = 32.0 L

2 mR RR Jx mB = mB 2 = 0.64;

2 Jy mR RR mB = mB 2



1+1 2 6



DR RR

2 

Jx ' 1 2 mB

Numerical Results – Symmetric Modes (Ω = 0). The motion equations for the symmetric modes are obtained from eq.(6.134) as 



 mR  E + 2  mB

   

+1 −1 +1 · · · −1 +1 −1 · · · +1 −1 +1 · · · .. .. . .





2 ωEuler,1

    y+  ¨   

.. .

 2 ωEuler,2

2 ωEuler,3 .. .

  y = 0  

(6.138) 2 where ωEuler,n = (EIy L/mB )(nπ/L)4 (symmetric). The (symmetric) eigenfunctions are calculated with n = 10 shape functions as depicted in Fig. 6.28. The first three eigenfunctions are shown in Fig. 6.29. The first six eigenfrequencies are listed in Table 6.4.

Table 6.4. Eigenfrequencies – Symmetric Modes (Ω = 0)

Libraries:

eigenvalue Problem: EISPAC Matrix Inversion: NUMREC

Eigenfrequencies (in rad/sec):

11.11371978; 1830.647043; 6537.118146;

566.2062783 3819.838851 9987.063829

6.5

269

Single Elastic Body – Gross Motion

vbi

voi

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2 0

0.2

0.4

0.6

0.8

ξ

-2 0

Figure 6.28. Shape Functions vbi

0.2

0.4

Figure 6.29.

0.6

0.8

ξ

Eigenfunctions voi

Exact Solution – Approximation Accuracy. The exact solutions are available when considering the intermediate conditions for half the rigid disk (or, identically, considering the half length beam with half the disk mass and half the disk inertia, respectively, at the tip). The procedure is listed in Table 6.5 and yields the solution  √  + 2[sin(k x) − γ sinh(k x)] i i i voi (x) = √  − 2[sin(ki x) − γi sinh(ki x)]  √  + 2[sin(k x) − γ sinh(k x)] i i i voi (x) = √  + 2[sin(ki x) − γi sinh(ki x)]



 0≤x≤ L 2 L ≤ x ≤ L  antimetric, 2 

 0≤x≤ L 2 L ≤ x ≤ L  symmetric. 2 (6.139)

q

The frequencies ωi = EIy L/mB ki2 are obtained from the equations for kL/2 (Table 6.5, last line) numerically. For this purpose, the solutions ω := ωRitz from Table 6.2 and Table 6.4, respectively, are used for starting values 

kL 2

v s u 1u mB L = tωRitz

 start

2

EIy

(6.140)

in a Newton-iteration to fulfill the (kL/2)-equations up to an accuracy of 10−13 . The corresponding “exact” frequencies show that the approximative . . ones differ from these by 0.001% = 0.25% (symmetric) and 0.73% = 1.08%

270 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations (antimetric). Table 6.5 furthermore shows that the eigenfrequencies for high values of mR and JR tend both (symmetric and antimetric) to the pinnedclamped case (see Table 5.2). High mass parameter values mean considerable inertia forces which are also achieved by finite mass parameters but higher frequencies. This explains why the symmetric and antimetric frequencies in Table 6.2 and Table 6.4 approach the same values for higher modes. The eigenforms in Fig. 6.29 and Fig. 6.27 do not show perceivable deviations when compared to the exact ones according to eq.(6.139).

Table 6.5. Eigensolution (Ω = 0) y

y

ppppppppppppppppppppppppppppppppp pppp pp ppppp p p p p ppppp pp pppp ppp p p p p p p p pgppgp p

pp p ppppp ppppp pppp ppppp pppp pppp ppp pp p pppp p pp pp p p pp p pp ppp pg pppp p ppgp ppp pp p x p p p p p ppp pp pp pp ppp ppppppppppppppp pppppppppp

6

6

x

ξ= EIy v 000 + 21 mR ω 2 v = 0

v0 = 0

V

x L

=

1 2

: EIy v 00 − 12 JR ω 2 v 0 = 0

V

v=0

V

v=0

ω 2 = [EIy L/mB ]k4 ⇒ v 000 k3

+

(kL) mR 2 mB

v=0

V

v0 k

=0

v 00 k2

−

(kL)3 JR v 0 2 mB L2 k

=0

√ v = 2(sinh kLξ sin kLξ) · (−γ 1)T ⇒

  

kL 2

cosh cosh (kL)mR 2mB

cos

kL + 2

sinh

kL + 2 (kL)mR kL sin 2mB 2

− cos

kL 2

−γ

× ⇒

kL 2

γ=

1


T





 

 

sinh sinh (kL)3 JR 2mB L2

cos(kL/2) cosh(kL/2)

−

1 2

kL 2




tan

kL 2

− tanh

cosh

kL 2



kL − 2 (kL)3 JR kL cos 2 2mB L2

− sin

kL 2

−γ ⇒

det[· · ·] = 0 ⇒ mB mR

sin

kL − 2

×

=0

kL 2

1

γ=


T

 

=0

sin(kL/2) sinh(kL/2)

det[· · ·] = 0 ⇒ kL 2



=0

mB JR /L2

−2

kL 3 2




coth

kL 2

− cot

kL 2



=0

6.5

271

Single Elastic Body – Gross Motion

5.1.4 Configuration Space and State Space (Ω 6= 0) Using the “non-rotating eigenfunctions” vo , wo = vo for eq.(6.118) simplifies the motion equations significantly. One obtains 

¨yC + (Go − 2E)Ω +(Go − E)Ω

2





0 E −E 0

E 0 0 E



yC +

y˙ C + (Go − E)Ω˙

2 diag{ωoi }





E 0 0 E

0 E −E 0



yC



yC = 0

(6.141)

where Z1

Go := o

∂Jx /∂ξ mB L2

∂vo

!

∂ξ

∂vo

!T

dξ =

∂ξ

TTo

Z1 o

∂Jx /∂ξ mB L2

∂vb

!

∂ξ

∂vb

!T

dξ To

∂ξ

(6.142) 2 from eqs.(6.124), (6.125). with To and ωoi The determination of the equations of motion was based on a reference coordinate frame which moves with Ω. This frame is called the co-rotating

6 A K A

co-rotating (C)



A

@ I @

A arbitrary (A) Ω  A *  @ A  ΩA @A  Ω - inertial (I) @A

@

Figure 6.30. Rotor Coordinate Systems, Top View

frame [motivating index C in eq.(6.118) and eq.(6.141)]. Here, Ω is the absolute rotor velocity. Looking for a representation in a coordinate system that rotates with arbitrary angular velocity ΩA needs transformation 

yC = TyA , T =

E cos α E sin α −E sin α E cos α



, α˙ = Ω = Ω − ΩA ,

(6.143)

see Fig. 6.30. Since T

T



E 0 0 E





T=

E 0 0 E



T

,T



0 E −E 0





T=

0 E −E 0



,

272 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

0 E −E 0





0 E −E 0







0 E ¨ α, −E 0 (6.144) the transformation is easy to achieve: inserting yC = TyA and premultiplying T with T yields T˙ =

Tα˙ = T



¨yA + (Go Ω − 2EΩA ) 2 +[diag{ωoi }

0 E −E 0



¨ = −Tα˙ 2 + T ˙ T α,

˙ A) y˙ A + (Go Ω˙ − EΩ 

+ (Go Ω − EΩA )ΩA ]

E 0 0 E



0 E −E 0



yA



yA = 0.

(6.145)

By this, one may easily calculate multi-rotor systems like spur or planetary gear units with flexible shafts where the single rotors undergo constant multiples of a main rotation ΩA to be selected. For the present rotor problem one may extract eq.(6.141) (co-rotating frame, ΩA = Ω) as well as the rotor equations when viewed from the inertial position (ΩA = 0). For numerical investigations, mostly the state space representation (firstorder ODE) is required. Defining8 

s˙ = y˙ A − ΩA

0 E −E 0



yA

(6.146)

one obtains 



0 +EΩA E 0    y y −EΩ 0 0 E A A A   d     = diag  2 ˙  −Go Ω 0 −(Go Ω − EΩA ) dt s˙ − n=1··· ωon s˙  2 diag ˙ 0 − n=1··· ωon +(Go Ω − EΩA ) +Go Ω (6.147) (proof by direct calculation: insertion of eq.(6.146) yields eq.(6.145)). Before evaluating eq.(6.147) numerically we are going to investigate special cases. 



5.1.5 The Laval- (or Jeffcott-) Rotor “The Swedish engineer Laval9 was the first one to show, by practical experiments in 1889, that a shaft can rotate much faster than people at that time 8 Usually,

˙ see eq.(2.26) and eq.(2.28), respectivethe vector of minimal velocities reads s˙ = H(q)q, ly. The fact that s˙ here consists of two summation terms comes from linearization: q → y : ∆˙s = ˙ ˙ ˙ o ∆y˙ where in the present case {∂[H(y)y]/∂ ˙ ˙ o = H(y)o = E. {∂[H(y)y]/∂y} y} y} o ∆y + {∂[H(y)y]/∂ The ∆-symbols are omitted for brevity. 9 Carl Gustaf Patrik de Laval, *1845 in Orsa, †1913 in Stockholm

6.5

273

Single Elastic Body – Gross Motion

believed. When Laval’s results came to light the reaction was disbelief everywhere” (A. F¨oppl, 1923). August F¨ oppl then, in his textbook, adds a physical explanation where he repeats mainly a contribution of his own from 1895. According to (F.C. Nelson, 2003), that paper had been published in a journal which was probably not well known by contemporary rotor dynamicists. In 1916, the Royal Society of London therefore commissioned Henry H. Jeffcott10 to resolve the conflicts between theoretical investigations and practical results. Three years later, Jeffcott published his results in the Philosophical Magazine. Since then, the technical term Jeffcott-Rotor is used in english speaking areas, while in other countries the Laval-Rotor is more common11 . The Jeffcott-Laval-Rotor is the simplest (and therefore most bay sic) elastic rotor model that can 6 ppppppppppppp be obtained from eq.(6.145): It p p p p p p p p p p p p pv pppppp pppppp ppppp p p p p makes use of only one symmep pppp p p p pppp p ppp pppp tric shape function and simultap p pp ph ph p - x, Ω neously neglects the mass of the elastic shaft. Doing so, the correFigure 6.31. Symmetric Rotor Models sponding eigenform may be looked at as vo,symm = 1 along with “bearing springs” at the (massless) shaft. Of course, if mB is neglected, it does not make sense to relate the eigenfunctions on mB as done in eq.(6.120) et seq. Instead, the whole equation is to be multiplied with (mB /mR ) and mB is eventually set to zero. Since matrix Go results in 0 zero because of vo,symm = 0 and E reduces to 1, one obtains from eq.(6.145) for the co-rotating frame "

#

!

˙ Ω2 0 −2Ω 0 −Ω ωo2 − Ω2 0 ¨ yC + y˙ C + y + y = , C +2Ω 0 0 ωo2 − Ω2 C +Ω˙ 0 −Ω˙ (6.148 a) where additionally a small mass center deviation  on the body-fixed y-axis has been assumed. Setting ΩA = 0, one obtains the equation in the inertial frame 





¨ yI +

ωo2 0 0 ωo2

R



yI =

cos Ωdt R sin Ωdt





! 2

Ω +

R

sin Ωdt R − cos Ωt

!

˙ (6.148 b) Ω.

Clearly, from eq.(6.148 b) it follows that Ω = ωo = const is a critical speed 10 Henry Homan Jeffcott, *1877 in County Donegan, †1937 in Walton-on-Thames. In the sequel we use the term Jeffcott-Laval-Rotor (alphabetical order). 11 It is interesting to note that Laval’s experimental setup, together with a stack of conically shaped disks inserted into the separator bowl of a milk centrifuge in around 1890 [so-called alpha-disks, patented by Clemens von Mauchenheim (called von Bechtolsheim) ] was to revolutionize the dairy industry. This gave rise to renaming the “AB Separator Company” (founded in 1883 by Laval and his partner Oscar Lamm Jr. in Stockholm) into the “Alfa-Laval Company” in 1963.

274 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations (resonance) and eq.(6.148 a) would seem to lead to instability for Ω > ωo since then the restoring matrix becomes negative. However, as a matter of fact, once the critical speed is passed, then the Coriolis forces stabilize the rotor: for ˙ = 0 the eigenvector problem reads Ω 

λ2 + ωo2 − Ω2



+2Ωλ



−2Ωλ λ2 + ωo2 − Ω2

!

1 −



(+)

=

i

0 0

!

(6.149)

where the eigenvector follows directly from the eigenvalue equation det[· · ·] = (λ2 + ωo2 − Ω2 )2 + (2Ωλ)2 = 0: setting +1 = −i2 yields12 +

(λ2 + ωo2 − Ω2 ) (−) i(2Ωλ) = 0

(6.150)

which in eq.(6.149) is fulfilled for both components with the inserted eigenvector. Eq.(6.150) has the solutions 

λ1/2 =

+i(+ωo − Ω) , +i(−ωo − Ω)



λ3/4 =

−i(−ωo − Ω) , −i(+ωo − Ω)

(6.151)

while eq.(6.148 b), for the inertial representation, leads simply to λ1/3 = +iωo , λ2/4 = −iωo .

(6.152)

Im(λi ) ωo

Im(λi ) ωo i=3

2

2

1

1

i=3 i=1

i=4

0

0 -1

i=1

i=4

-1

i=2

-2

-2 i=2

-3

0

0.4

Figure 6.32.

0.8

1.2

1.6

Eigenvalues, Rotating

Ω ωo

-3 0

0.4

Figure 6.33.

0.8

1.2

1.6

Ω ωo

Eigenvalues, Inertial

√ same result is obtained when in eq.(6.148 a) the second component is multiplied with i = −1 and both the components are added to define the new variable z = y1 + iy2 which yields the eigenvalue equation (6.150). This is a common procedure in rotor dynamics. 12 The

6.5

275

Single Elastic Body – Gross Motion

The simple representation in eq.(6.148 b) should not mislead us in the interpretation of two decoupled oscillators. Considering the co-rotating frame, the homogeneous solution ( = 0) is obtained by a linear combination of α(1 − i)T exp(λ1 t) + β(1 i)T exp(λ4 t) with α = (a + ib)/2 and β = (a − ib)/2 as well as γ(1 − i)T exp(λ2 t) + δ(1 i)T exp(λ3 t) with γ = (c + id)/2 and δ = (c − id)/2 yielding 

yC = 

+

cos(ωo − Ω)t − sin(ωo − Ω)t sin(ωo − Ω)t cos(ωo − Ω)t

cos(ωo + Ω)t sin(ωo + Ω)t − sin(ωo + Ω)t cos(ωo + Ω)t





a b

c d





.

(6.153)

The corresponding representation in the inertial frame is obtained by transformation:13 

cos(Ω)t − sin(Ω)t yC sin(Ω)t cos(Ω)t



cos(ωo )t − sin(ωo )t sin(ωo )t cos(ωo )t

yI = =





a b





cos(ωo )t sin(ωo )t + − sin(ωo )t cos(ωo )t





c . d (6.154)

Clearly, for the co-rotating frame, one has four elementary solutions, thus four constants a, b, c, d are needed. Transformation into inertial representations shows, that these are still present. The reason is that, in case of a rotor, one has simultaneously velocity and positional initial conditions which are not independent. Rearranging leads eq.(6.153) to 

yC = 

+

a b



c d





a b



cos(+ωo − Ω)t + 

cos(−ωo − Ω)t +

−b a



−d c



sin(+ωo − Ω)t sin(−ωo − Ω)t,

(6.155)

and eq.(6.154) yields yI =





cos(+ωo )t +

−b a



sin(+ωo )t + · · ·

13 There is not much effort required for this transformation: compared with elementary transformations [see eq.(3.18)], the first matrix in eq.(6.153) may be interpreted as AT β where β = (ωo − Ω)t and the second

as AT where γ = (ωo + Ω)t. In eq.(6.154), these are premultiplied with AT α , α = Ωt. A subsequent (−γ) T T transformation AT α Aβ with the same rotation axis yields A(α+β) , where (α + β) = (Ω + ωo − Ω)t = ωo t. The same holds for the second summation term where (α − γ) = [Ω − (ωo + Ω)]t = −ωo t.

276 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

··· +

c d





cos(−ωo )t +

−d c



sin(−ωo )t.

(6.156)

Eq.(6.156) has a simple interpretation: When viewed at the rotor from the inertial stand point, then the rotor mass center rotates on a circle either in the same sense as Ω does (positive whirl) or against (negative whirl) which depends on the sign of Im(λ) in eq.(6.152). Compared to eq.(6.155), one has the same relation except the fact that a “co-rotating observer” does not notice the rotor speed.

Acceleration. Since, for  = 0, Ω˙ does not arise in eq.(6.148 b), the JeffcottLaval-Rotor is insensitive against acceleration for the inertial representation. One may therefore use eq.(6.148 b) to determine the solution in the inertial frame. Then, if needed, this solution can be transformed into any other rotating frame. Of course, Ω = Ω(t) then depends on time. Damping. Damping plays a major role in rotor dynamics. There are, in general, two sources of damping. The one is due to the Iz bearings. Because these act 6 from “outside”, it is called exterior damping. The bearings are assumed axis-symmetric; the sketch shows an exem≈ dex ple of one of the surrounding u r spring-damper combinations Iy ≈ k which act between the rotor Ω axis (bold point) and the housing (bold circle). The bearing coefficients of the total bearing are k (restoring parameter) and dex (damping paFigure 6.34. Damping Mechanism, Top View rameter). Another damping source emanates from inner mechanisms, like material damping, or arise from movable parts in the system, or from the reaction of a liquid, as for instance mentioned in the footnote on page 273 (Laval’s milk centrifuge), indicated by a dotted circle in Fig. 6.34. Such reactions are proportional to the relative rotor velocity y˙ C when represented in the co-rotating frame. The corresponding damping coefficient is consequently denoted din (“inner damping”). The exterior damping acts against the inertial surrounding and is therefore proportional to the absolute rotor velocity. In the co-rotating frame

6.5

277

Single Elastic Body – Gross Motion

one obtains the generalized damping force Qd , 



−Qd = dex y˙ C +

0 −Ω +Ω 0





yC + din y˙ C ,

(6.157)

which after transformation with eq.(6.143) yields 

(din+dex )



1 0 0 1



y˙ A +[din (Ω−ΩA )−dex ΩA ]

0 1 −1 0



yA := Dy˙ A +NyA . (6.158)

Here, D = DT is called the damping matrix and N = −NT the matrix of nonconservative restoring forces (or circulatory forces). Eq.(6.158) shows that either for dex = 0, ΩA = Ω (co-rotating frame), or for din = 0, ΩA = 0 (inertial frame), the N-matrix vanishes. In these cases, the stability theorems (see Chapter 8) hold. Completing eq.(6.145) with D and N yields for the Jeffcott-Laval-Rotor with constant rotor speed dex = 0, ΩA = Ω (co-rotating frame) 

1 0 0 1





¨yC + 2Ω

0 −1 1 0





y˙ C + din

1 0 0 1



y˙ C + (ωo2 − Ω2 )



1 0 0 1



yC = 0, (6.159)

and for the second case din = 0, ΩA = 0 (inertial frame) 

1 0 0 1





¨yI + dex

1 0 0 1



y˙ I + ωo2



1 0 0 1



yI = 0.

(6.160)

In case of pervasive damping (which in this case is trivially fulfilled with D > 0), (asymptotic) stability is obtained for K > 0 (independent of the remainder), otherwise the system is unstable. This is immediately seen in eq.(6.160) which represents a stable oscillator. From eq.(6.159), however, it follows that the rotor becomes unstable in the overcritical state [Ω > ωo ⇒ K < 0 (see Table 8.3, page 405)]: Stabilization due to Coriolis-forces, as in eq.(6.148 a), is destroyed by inner damping for sufficiently fast rotating rotors. Stabilization can only be brought back by exterior damping which has to be taken into account by bearing design (or active control).

5.1.6 Rotor with Fixed Point The Jeffcott-Laval-Rotor is restricted to the first symmetric mode. However, one should have in mind that also antimetric modes will occur, leading to a second (and, consequently subsequent higher) “critical speed(s)”. The second

278 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations (i.e. first antimetric) mode does, in the Jeffcott-Laval-Model, not cause contradictions only when additionally the rigid rotor is considered a mass point (i.e. neglecting rotational inertia, see Fig. 6.31). However, as Fig. 6.25 on page 267 shows, its influence is not negligible. Symmetric modeling is obviously disadvantageous since gyroscopic torques do affect the rotor stability. A Rotor with Fixed Point with its pure antimetric modes is the counterpart to the JeffcottLaval-Rotor. Analogously to the symmetric mode in the y 6 previous section, consideration of the first antimetric mode needs multiplication with p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p mB L2 /JRy to eventually nep p ppppppppppppp pppppppp p p p p p h p ph p ppppppppppppp p p p x, Ω p p ppppppppppppppppppppppppppppppppppppppppppp glect the beam mass mB . The first antimetric mode is 0 normalized to vo,anti = 1 which leads the Go -matrix Figure 6.35. Antimetric Rotor Models from eq.(6.142), mB L2 Go = JRy along with

Z1 o

∂Jx /∂ξ mB L2

∂vo

!

!T

∂vo

∂ξ

dξ,

(6.161)

∂ξ

(∂Jx /∂ξ) = JBx + JRx− δ (ξ − 0.5)

to

"

2 JBx

mB L Because of

JRy

JRx + JRy

#

=

mB →0

JRx JRy

!

(6.162)



:=

A . B 

(6.163)

(∂m/∂ξ) = mB + mR− δ (ξ − 0.5),

(6.164)

the first summation term in the mass matrix (6.121) vanishes for mB → 0 since vo,anti |0.5 = 0. When premultiplied with (mB L2 /JRx ) the mass matrix obtains the value 1 as requested. The frequency square results ωo2 = (kL2 )/(2JRy ) when vo,anti = ξ − 0.5 ⇒ v 2 |o +v 2 |1 = 1/2 is inserted into eq.(6.120). One obtains thus from eq.(6.145) for the co-rotating frame (ΩA = Ω), 











0 1 0 1 10 y˙ + [ BA −1]Ω˙ y + [( BA −1)Ω2 +ωo2 ] y =0 −1 0 C −1 0 C 01 C (6.165 a) which for the inertial frame (ΩA = 0) reduces to

¨ yC + [ BA −2]Ω



¨yI + [ ]Ω A B

0 1 −1 0



y˙ I + [ ]Ω˙ A B



0 1 −1 0





yI + [ ] ωo2

1 0 0 1



yI = 0. (6.165 b)

6.5

279

Single Elastic Body – Gross Motion 6 α˙ = Ω

β Im(λi ) ωo

i=3

4 3 2 1 0 -1 -2 -3 -4 -5

γ actual state

r

nominal state

i=1 i=4

i=2

0

0.4

0.8

Figure 6.37.

Figure 6.36. Antimetric Rotor Model

1.2

1.6

Ω ωo

Eigenvalues, Inertial

The eigenvector-eigenvalue equation for eq.(6.165 b) (inertial representation, constant rotor speed) now reads  

λ2 + ωo2

A +B Ωλ

A −B Ωλ

λ2

+

ωo2

 

!

1 + (−)

i

0 0

=

!

(6.166)

with the eigenvalue equation +

(λ2 + ωo2 ) (−) i(

A Ωλ) = 0. B

(6.167)

Eq.(6.167) has the solutions

λ1/2

λ3/4

   q A A  2   −i 2B Ω 1 − 1 + 4(ωo / B Ω)  , = q    −i A Ω 1 + 1 + 4(ωo / A Ω)2 2B B    q A A  2   +i 2B Ω 1 + 1 + 4(ωo / B Ω)  , = q    +i A Ω 1 − 1 + 4(ωo / A Ω)2 2B

as depicted in Fig. 6.37 for A/B = 2 (oblate rotor).

B

(6.168)

280 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Technical Approximation and Acceleration. For sufficiently high angular momentum L = AΩ, the square root in eq.(6.168) may be expressed by a first-order power series, [1 + 2ωo2 /(AΩ/B)2 ], leading to 

λ1/2 '

+iωo2 /[(A/B)Ω] , −i(A/B)Ω



λ3/4 '

+i(A/B)Ω . −iωo2 /[(A/B)Ω]

(6.169)

The corresponding frequencies are called the “nutation frequency” ωN ' (A/B)Ω and the “precession frequency” ωP ' ωo2 /[(A/B)Ω]. The approximation in eq.(6.169) is confirmed by the following consideration. Let eq.(6.165 b) be M¨yI + Gy˙ I + (K + N)yI = 0

(6.170)

where M = E, G =

   0 +1  A B

Ω

−1 0

, N=

   0 +1  A ˙ B

Ω

−1 0

, K = ωo2 E.


(6.171) As Fig. 6.37 shows, precession is a rather slow motion for sufficiently high Ω. Here, the acceleration terms do not contribute much. On the other hand, considering fast nutation, the mass forces will equilibrate with the gyroscopic terms. In case of Ω = const. one obtains the technical approximation precession: Gy˙ I + KyI ' 0, nutation: M¨yI + Gy˙ I ' 0

(6.172)

(K. Magnus, 1971). Precession is thus mainly characterized by M¨yI ' 0 and nutation by KyI ' 0, respectively. This consideration is confirmed with the corresponding eigenvalue problems: Setting λk = iωk , k ∈ {P, N }, one obtains precession:  ωo2  det A +i B ΩωP

nutation:   A −ωN −i B Ω  = 0,  ωN = 0 det  A 2 +i B Ω −ωN ωo (6.173) which yield the frequencies according to eq.(6.169). The real solution is obtained by linear combination of the four fundamental solutions: 

yI =

A −i B ΩωP

cos(ωN t) − sin(ωN t) sin(ωN t) cos(ωN t)





a b





+

cos(−ωP t) − sin(−ωP t) sin(−ωP t) cos(−ωP t)





c . d (6.174)

[Transformation into the co-rotating frame yields once more ωN → ωN − Ω, −ωP → (−ωP − Ω) (compare footnote p.275 for calculation)]. Here, precession corresponds to negative and nutation to positive

6.5

281

Single Elastic Body – Gross Motion

whirl. Comparing Fig. 6.37 with Fig. 6.33 shows that these are well distinguished for antimetric modes. Moreover, as known from rotor dynamics, an imbalance will only lead to resonance for Ω = ωeigen where ωeigen belongs to positive whirl. Precession is thus uncritical in this case14 . ˙ is present in the inertial representation as well as in any rotating fraSince Ω me representation, the solution will depend on rotor acceleration. Considering eq.(6.165 b) one may expect a time dependent solution in the form 2

yI = c(ao + a1 t + a2 t + · · ·) exp

Zt

λdt

(6.175)

o

which by factorization leads to Zt

yI := c exp

λdt, c = const.

(6.176)

o

Insertion into eq.(6.165 b) with matrices eq.(6.171) yields h

i

M(λ˙ + λ2 ) + Gλ + (K + N) c = 0.

(6.177)

The technical approximation according to eq.(6.172) reads precession, M¨yI ≈ 0 : det[Gλ + (K + N)] = 0 ˙ ⇒ λ = ±iωP − Ω/Ω ⇒ exp

Z

where ωP = ωo2 /[(A/B)Ω]



λdt = exp −

Z

1 exp ±i Ω 

=





(dΩ/dt)(1/Ω)dt exp ±i

Z



ωP dt



Z

ωP dt

14 This is easily confirmed by solving eq.(6.165 b) with a right-hand side like in eq.(6.148 b) where (Ω − ω ) p

cancels out for the inhomogeneous solution. At first glance, the name “precession” for negative whirl is peculiar. It was introduced when observing a gyro, the rotation axis of which is inclined with angle γ against the vertical. If the mass center lies on the body-fixed rotation axis at distance c from the fixed point, then the gyro axis itself rotates with ωp about the vertical: Let the vertical be x, pointing against gravity, and let the corresponding y-axis lie in the drawing plane. The momentum vector in this frame then reads LT = (AΩ cos γ, AΩ sin γ, 0). The gravity torque is (mgc sin γ)e3 . Considering the momentum theorem (4.36) with ω IR = ωp e1 leads to ωpe e1 L = (mgc sin γ)e3 for Ω = const, γ = const with solution ωp = [mgc/B]/[(A/B)Ω]. Thus, c > 0 (statical instability) leads to ωp > 0 (dynamical stability with preceding motion). This belongs to the “curiosities” of gyrodynamics. Statical stability (such as from bearing stiffness) changes the sign and yields negative “pre”cession.

282 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations nutation, KyI ≈ 0 : det[M(λ˙ + λ2 ) + Gλ + N] = 0 A˙ A A ⇒ λ˙ + λ2 ± i Ωλ ± i Ω = 0 ⇒ λ = ∓i Ω := ∓ωN B B B  Z  Z ⇒ exp

λdt = exp ±i

ωN dt .

(6.178)

The real solution, R

cos(R ωN dt) − sin(R ωN dt) sin( ωN dt) cos( ωN dt)



R R    cos(− R ωP dt) − sin(−R ωP dt) h 1 i c , Ω sin(− ωP dt) cos(− ωP dt) d

yI = +

R





a b



(6.179)

shows that the precession amplitudes will be stabilized for accelerating rotor speed but destabilized for deceleration. [This behavior has been confirmed by numerical investigations (H. Bremer, 1987)]. The main result here is that stabilization/destabilization goes with 1/Ω which will be overplayed by damping which enters the solution exponentially.

Damping. The same as in the case of a symmetric Jeffcott-Laval-Rotor, stability/instability is determined by the restoring matrix a) for din in the absence of dex in a co-rotating frame, eq.(6.165 a), and b) for dex in the absence of din in an inertial frame, eq.(6.165 b). The latter shows once more that external damping leads always to asymptotic stability. Inner damping, however, can only destabilize the system if A − 1 Ω2 + ωo2 < 0, (6.180) B see eq.(6.165 a). Thus, an oblate rotor (A > B) will never be destabilized by inner damping but always leads to asymptotic stability15 . 





free gyro (ωo2 = 0) is unstable due to (pervasive) inner damping if A < B. This was the reason for the instability of the first US-satellite in 1958 (Explorer I). The earth (as a liquid filled gyro) is, on the contrary, asymptotically stable since it represents an oblate ellipsoid (A > B). This was already known to William Thomson [Lord Kelvin of Largs] (*1824 in Belfast, †1907 in Netherhall) in 1870. Thus, the stability of Laval’s milk centrifuge depends strongly on the bowl shape and on the bearing characteristics. 15 A

6.5

Single Elastic Body – Gross Motion

283

5.1.7 Elastic Rotor Properties The Jeffcott-Laval-Rotor is often used for basic investigations of elastic rotors. It offered, in its time, a breakthrough in rotor dynamics, showing that a rotor can be operated with much higher speed than expected. The reason for this misconception was, according to (F.C. Nelson, 2003), due to W. Rankine16 who used an “unfortunate model”17 for his investigations in 1869. The Jeffcott-Laval-Rotor, however, is quite popular in rotor dynamics even today. It is often used to examine various kinds of unsymmetries. The influence of gyroscopic torques is hereby omitted by reducing the rotor to a point mass model. Such an action, however, seems quite self-contradictory. A rotor system moves in a combination of symmetric (i.e. translational) and antimetric (i.e.rotational) modes w.r.t. its bearings. If the rotor system is beyond that geometrically symmetric, then the corresponding motion equations are decoupled (see eq.(6.133) et seq.). The results from section 5.1.5 and 5.1.6 then represent an elastic rotor described by (only) two shape functions. This is, of course, a rather rough model, insufficient for real investigations, but it nevertheless leads to basic physical insights. These basics are, for a rigid rotor system: We call the symmetric modes “pendulum oscillations” (although they are not independent for the two translational motion directions, see p.275). They are independent of rotor acceleration. In an inertial frame representation, the corresponding eigenfrequencies do not change with increasing Ω. This is not the case for the antimetric modes. The frequencies which coincide for Ω = 0 ramify into a positive (+) and a negative (−) whirl branch with increasing rotor speed. They depend on rotor acceleration leading to destabilizing effects when the rotor is decelerated. However, taking damping into account (which in a real system always takes place), these effects are negligible. This validates the “quasi-static” concept (i.e. increasing Ω while ˙ is neglected). Ω Inner damping destabilizes the symmetric modes. The antimetric modes, however, can only be destabilized for a prolate rotor with sufficiently weak bearings. Oblate rotors are asymptotically stabilized by inner damping18 .

16 William

John Macquorn Rankine, *1820 in Edinburgh, †1872 in Glasgow some kind of a rotating massless guide mechanism with a point mass inside which is attached to the origin by a spring. Because the guide prevents relative velocity perpendicular to the spring elongation, Coriolis forces which are needed for stabilization cannot take place in Rankine’s model. 18 This leads to the demand that, whenever possible, a rotor should be shaped oblate (washing machines, Laval milk separators, any kind of centrifuges). 17 I.e.

284 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Exterior damping stabilizes motion, symmetric as well as antimetric modes. Exterior damping (e.g. bearing damping or active control) can prevail over inner damping destabilization. Imbalance resonance can only arise for positive whirl motion (“nutation”). Thus, a rigid oblate rotor will only lead to resonance for its symmetric mode because the nutation obeys the (A/B)Ω-law: in an eigenfrequency over a rotor speed plot, no intersection between ω(+) = (A/B)Ω and ω = Ω occurs for (A/B) > 0. Being aware of these problems and of the stabilizing/destabilizing mechanisms it is reasonable to continue with the undamped model in order to save effort. The basic qualities of the non-rotating rotor mainly define its total behavior. The knowledge of ωoi and (A/B) leads at least approximately to solutions for sufficiently high Ω without any additional calculation. This holds also for an elastic rotor system. Considering the rigid rotor, the eigenvectors are simply (1 i)T [symmetric modes: eq.(6.149) (Jeffcott-Laval), antimetric modes: eq.(6.166) (Rotor with Fixed Point)]. The corresponding motion equations are represented by eq.(6.145) for yA ∈ IR2 where Go vanishes for the symmetric modes. The motion equation for the elastic rotor needs yA ∈ IR2n where n denotes the number of shape functions for each bending direction. Because of the same mathematical structure of the motion equations one concludes, from comparison with the rigid case, with an eigenvector that reads yA = (qT i qT )T , q ∈ IRn . Inserting yA = yA exp(iωt) into eq.(6.145) yields the (reduced) eigenvector problem h

i

diag ω 2 + Go Ω (ΩA − ω) q = 0 ∈ IRn . − (ΩA − ω)2 E + i=1···n oi

(6.181)

For ΩA = Ω, eq.(6.181) corresponds to the co-rotating representation (ω := ωc ) and ΩA = 0 yields the inertial representation (ω := ωI ), respectively. Solving the eigenvalue problem for the co-rotating representation yields ωc which fulfills eq.(6.181), h

i

diag ω 2 + Go Ω (Ω − ωc ) q = 0. − (Ω − ωc )2 E + i=1···n oi

Setting (Ω − ωc ) = −ωI yields the eigenvector equation for inertial representation19 without changing the eigenvector q. Therefore, one obtains the eigenforms veigen,i = qTi vo independently of the used coordinate frame. (Here, vo 19 Compare

for instance eq.(6.153) et seq.

6.5

Single Elastic Body – Gross Motion

285

represent the shape functions which in the present case had been selected as the eigenforms of the non-rotating rotor, Ω = 0 : vo = TTo vb , vb : beam functions). For Ω 6= 0, the eigensolutions reveal positive or negative whirl when looked at from the inertial system. The sign may be detected by solving the eigenvalues yA2 for the co-rotating representation and calculating ωIi = 6 (ωci − Ω), or from the eigenvector problem for the inertial representation: The above ansatz t=0 - y yA = (qT i qT )T exp(iωt) A1 yields, along with its conjugate t>0 complex part, the real solution yTA = 2(qT cos ωt − qT sin ωt). This solution shows negative whirl for positive ω (see Fig. 6.38 for q ∈ IR1 ). The whirl sense Figure 6.38. Re(q) > 0, Im(q) > 0 thus depends on the sign of ω. However, numerical eigenvector codes do not offer such signs but simply calculate the real part of the eigenvalue (which is zero here) and the imaginary part. The corresponding eigenvector then denotes positive and negative whirl: V Re(q) > 0 Im(q) > 0 corresponds to yTA = 2(qT cos ωt −qT sin ωt) and denotes negative whirl. If ω changes its sign we obtain yA = (qTv iqTv )T exp(−iωt) where its conjugate complex part reads (qTv − i qTv )T exp(iωt). Since the computer calculates the eigenvector for +iω, the corresponding V Re(q) > 0 Im(q) < 0 denote positive whirl. The elastic rotor from section 5.1 is calculated with the parameters listed in Table 6.3. Compared with the results from rigid rotor dynamics (see Fig. 6.39), once more the symmetric eigenforms (dotted in Fig. 6.39 and Fig. 6.40) are not influenced by Ω. The main differences between rigid and elastic rotor behavior are: Antimetric frequencies split into two branches, (+) and (−). The lowest antimetric frequency splits into precession (−) and nutation (+). With increasing Ω, the next higher antimetric (+)-frequency overtakes the role of nutation etc. After that, the actual (+)- frequency approaches the next higher antimetric (−)-frequency.

286 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Fig. 6.41 shows the corresponding eigenforms with positive whirl. The bold curves 2 and 4 are the antimetric eigenfunctions which correspond to ω2 and ω4 in Fig. 6.40 when counted from the lowest frequency on, ω1 < ω2 < ω3 < ω4 · · ·. The thin curves are the symmetric eigenfunctions. Analogously, Fig. 6.42 depicts the eigenfunctions with negative whirl. Here, also, the knowledge of ωoi and the (A/B)-ratio yields insight into the rotor behavior, see Fig. 6.43 [(A/B)Ω dotted]. Intersections of the ω = Ωline with antimetric (+)-frequencies indicate resonance. For a rigid rotor with A > B (Fig. 6.39), there is only one resonance (the symmetric mode) to be overdriven to reach operational speed. In the elastic case, the digressive time history of the positive whirl frequencies, once the next higher ones have overtaken the role of nutation, destroys the positive (A/B)Ω-effect for the antimetric modes. However, one would anyway expect resonance at the indicated locations because of the symmetric frequencies which approach the antimetric ones for higher modes (not distinguishable in the plot. The first three symmetric mode frequencies are 1.77 Hz, 90.11 Hz, 291.36 Hz, see Table 6.4). Remarks: Bearing damping (external damping) is necessary to guarantee a smooth rotor behavior. Passive or active controlled damping reacts, of course, on the actual deviation velocity which occurs at the bearing locations. However, using the Ritz ansatz, the basic model for control calculation is always a truncated one. Inserting the calculated control into the real experiment (or, for checking results, into a calculation model which uses some more shape functions), it can (and will) occur that the control excites those modes which are missing in the control calculation model (control spillover20 ). It may also occur that the calculated measurement is (due to an insufficient number of calculated modes) not representative of the real measurement which is used to put the control law into practice (measurement spillover). Both problems can be solved by collocation (measurement at the same location where the control input acts) along with output control instead of individual mode control [(H. Bremer, F. Pfeiffer, 1992), see Chapter 7.6 page 380]. It has also to be mentioned that controllability depends on the actuator location. A control force for instance becomes inactive when acting at an oscillation node: the resulting work is zero due to missing displacement. In case of non axis-symmetric elastic rotors one has to expect node shifting with increasing Ω which influences controllability.

20 This effect was the reason for the instability of the Mariner 10 spacecraft in 1973. However, once the physical background was detected the spacecraft was saved.

6.5

287

Single Elastic Body – Gross Motion

ωi

ωi 80

⊕

80

⊕

⊕

60

60 40

40

A BΩ

20

20



0

0 0

10

20

30

40

0

Ω

10

Figure 6.40.

Figure 6.39. ωi over Ω, Rigid Rotor

2

2

1.5

1.5

1

1

0.5

0.5

20

30

40

Ω

ωi over Ω, Elastic Rotor

0

0 2

-0.5

1

-0.5 -1

-1 Ω = 50 Hz

-1.5

4

-2 0

0.2

0.4

0.6

0.8

Figure 6.41. Eigenforms (+)

x L

3

Ω = 50 Hz

-1.5 -2 0

0.2

0.4

Figure 6.42.

0.6

0.8

Eigenforms (-)

x L

288 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations ωi 450

⊕

400 ω=Ω

350 ⊕

300 250 200 150

⊕

100 50



0 0

50

100

150

200

250

300

Ω

Figure 6.43. ωi (in Hz) versus Ω (in Hz), [A/B] = 2 (Oblate Rotor)

The basic procedure for the elastic rotor is summarized in Table 6.6: 1. Ω = 0: Determine mass matrix and restoring matrix using vb (beam functions, or any other admissible functions), solve the eigenvalue-eigenvectorproblem. 2. Ω = 0: Calculate Go and, if they occur, additional generalized forces Qoy and Qoz , respectively. Example: In case of discrete forces acting at ξ = η one obtains simply (∂fj /∂ξ) = fj−δ(ξ − η) ⇒ Qoj = TTo vb |η [fj /mB ], j = y, z. Example: if these forces are achieved by a magnetic bearing (active control) they read fj = kgap,j ∆j + kcurrent,j Ij , j = y, z
where ∆j = vb |Tη To qj : gap, Ij : current, kgap,j , kcurrent,j : bearing coefficients.

3. Ω 6= 0: Specify ΩA according to the actual aim of investigation. Example: magnetic bearing (active control). Since the bearing is inertially fixed: ΩA = 0 (inertial representation).

6.5

289

Single Elastic Body – Gross Motion

Table 6.6. Elastic Rotor – State Space Equations Ω=0:

 1 Z Mo := 

∂Jy /∂ξ ∂m/∂ξ vb vTb + mB mB L2

o

∂vb

!

∂vb

∂ξ



!T !

dξ  ∈ IRn,n

∂ξ

  1  2   2 T Z   k EIy ∂ vb ∂ vb Ko :=  dξ + vb vTb |0 +vb vTb |L  ∈ IRn,n 4 ρAL

∂ξ 2

mB

∂ξ 2

o

diag ω 2 −Mo ωo2 + Ko q = 0 ∈ IRn ⇒ To = [q1 · · · qn ] , TTo Mo To = E ⇒ TTo Ko To = i=1···n oi







Go :=

TTo

Z1

∂Jx /∂ξ mB L2

∂vb

Qoy :=

TTo

∂fy

vb mB

∂ξ

o

Qoz :=

TTo

Z1 "

!

∂fz

vb mB

∂ξ

o

!

∂vb

!T

∂ξ

∂ξ

1 + mB L

∂vb

o

Z1 "

!

dξ To ∈ IRn,n

!

∂ξ

1 − mB L

∂vb ∂ξ



∂Mz

!# dξ ∈ IRn

∂ξ

!

∂My

!# dξ ∈ IRn

∂ξ

Ω 6= 0 : qv yA =

qw

! ∈ IR2n , s˙ = y˙ A − ΩA



0 −E

E 0



yA ∈ IR2n , x =

yA

! ∈ IR4n

s˙

x˙ = Ax + b



0

  −EΩA   A=   diag  2 − i=1···n ωoi  0

+EΩA

E

0

0

0

0

diag ω − i=1···n oi



2

+(Go Ω − EΩA )

0



  0     E  0       ,b =          −(Go Ω − EΩA )  Qoy   0

Qoz

290 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations

6.

Gross Motion – Dynamical Stiffening (Ritz Approach)

The same as in eq.(5.154), the elastic coordinates are separated from the rigid body coordinates, T T y˙ = (vTo ω To | y˙ el ),

(6.182)

where, for maximum possible deflections (except stretching), one has ˙ −w˙ 0 , v˙ 0 , | ϑ˙ 0 , v˙ 00 , w˙ 00 )T y˙ el = (0, v, ˙ w, ˙ | ϑ, 

0 1 0 0

0

  = 0 0 1 0 −∂  ∂x 

0 0 0 1

0

∂2 ∂x2 0

∂ ∂x 0

0

0

∂ ∂x

0

0

T    2  ∂   2  ∂x 

0

0

= D ◦ s˙ el .



v˙ w˙   ϑ˙ (6.183)

Using the Ritz expansion 

vT v   w = 0 ϑ 0 



0 wT 0



0 qv  0   qw  := Φ(x)T yR,el qϑ ϑT 

(6.184)

yields, for eq.(6.183), T q˙ v 0 v 0 0 0 v0 0 v00 0  0 0 w 0 −w0 0 0 0 w00   q˙ w  q˙ ϑ 0 0 0 ϑ 0 0 ϑ0 0 0



y˙ el

=

 

:= ΨTel y˙ R,el .



(6.185)

Insertion of this result into eq.(5.163) and premultiplying with Ψel yields the dynamical stiffness matrix which is assigned to the elastic variables Ψel dKn,el ΨTel

(6.186)

and which eventually has to be replenished with zeros to take the rigid body coordinates into account: " T

ΨdKn Ψ =

0

0

0 Ψel dKn,el ΨTel

#

.

One obtains for the general case according to eq.(5.163),

(6.187)

6.6

291

Gross Motion – Dynamical Stiffening (Ritz Approach)

Ψel dKn,el ΨTel = 

RL dfx(o)

0 0T

dξv v   x dξ              symmetric  

RL 1 dMx(o) x

2

dξ

00

0

dξ(v w −v w

RL dfx(o) dξ

x

0T

dξw0 w0

00 T

T

   (o) RL dfz(o) dMy 0 0T ) dξv ϑ  (ξ−x)− dξ dξ  x  RL dfz(o)  0 T − dξv ϑ  dξ  x      dx L (o) (o) R dfy dM 0 0 T z − dξw ϑ  (ξ−x)+ dξ dξ  x  RL dfy(o)  0 T + dξw ϑ  dξ  x  0

(6.188 a) which for a left side clamped boundary (or for a “fork-like bearing”) reduces to Ψel dKn,el ΨTel =                  

RL dfx(o) x

dξ

0 0T

dξv v

RL 1 dMx(o) x

2

dξ

00

0T

0

dξ(v w −v w

00 T

)

RL dfz(o) dξ

x

−

x

symmetric

dξ

dξw0 w0

T

−

(x−ξ)dξv ϑ

RL dMy(o) x

RL dfx(o)

 00

dξ

RL dfy(o) dξ

x

−

T

(x−ξ)dξw00 ϑT

RL dMz(o) x

dξv0 ϑ0

T

dξ

dξw0 ϑ0

T

         dx,        

0

(6.188 b) see eq.(5.163 b).

6.1.

Rotating Beam – One-Link Elastic Robot

Consider once more the rotating beam in Chapter 5, section 6.5. When serving as a robot model, the rigid body angle will in general not undergo permanent rotations like in a helicopter blade. It will therefore be suitable to reconsider the resulting equations and look for simplifications. A single link elastic robot according to Fig. 6.44 may be looked at as a basic contribution to elastic robots (A. Truckenbrodt, 1980). The influence of possibly attached mechanisms is taken into account by a heavy payload thus concentrating on the forearm for first investigations. The aim is then to optimize slewing maneuvers under the influence of the tip load.

292 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations

Figure 6.44.

The tip load plays a major role in this example. Actually it leads to a coupling of bending and torsion and diminishes the torsional frequency when compared to a simple Euler beam. (See also the remark on p.189: for an aluminum beam with dimensions 1000 × 50 × 5 mm3 (beam mass 0.71 kg) with a tip mass of 10.0 kg the torsional frequency breakdown is about 30 %).

One-Link Robot

The payload acts like a constant force: fz (x) = fz− δ (x − L), where fz = −m tip g. The differential operator D is derived from eq.(5.178): 

1 0 0 0

     y˙ =     0 

0

0 1 0 ∂ ∂x 0 ∂2 ∂x2

0 0 1 0 ∂ ∂x 0

          



γ˙ v˙   = D ◦ s˙ . ˙ ϑ

(6.189)

Along with the Ritz-ansatz 1 0 s˙ '  0 v(x)T 0 0 





γ(t) ˙ 0  T   q˙ v (t)  0  = Φ(x) y˙ R T ϑ(x) q˙ ϑ (t)

(6.190)

one obtains      T T Ψ = [D ◦ Φ(x) ] =     

1 0 0 vT 0 0 0 v0 T 0 0 0 v00 T

0 0 ϑT 0 T ϑ0 0

         

which is used to calculate the system matrices according to eq.(6.25).

(6.191)

6.6

293

Gross Motion – Dynamical Stiffening (Ritz Approach)

6.1.1 Mass Matrix R T dM The mass matrix reads oL Ψ dM dx Ψ dx with dx from eq.(5.168) yielding    ZL   M=    o 



xvT ∂m + ∂Jz v0 T ∂x ∂x ∂J ∂m T vv + z v0 v0 T ∂x ∂x 0

∂Jz + ∂m x2 ∂x ∂x ∂m + ∂Jz v0 xv ∂x ∂x 0



0 0 ∂Jx ϑ ϑT ∂x

    dx.   

(6.192)

Here, m tip is automatically included with (∂m/∂x).

6.1.2 Restoring Matrix As eq.(5.169) shows, every matrix element in dG is multiplied with an elastic deformation. Thus, in dG y˙ all components except the first one are negligible when y˙ from eq.(5.167) is inserted. The remainder is just a vector which may be rearranged in terms of yR when the Ritz-ansatz is inserted: (dG/dx)y˙ = 0 0 0 0    ∂m ∂m 2 T − ∂x γv(x, 0 ˙ t)    0 − ∂x γ˙ v    ∂J  ∂Jy   ∂J ∂J −( − ∂xz )γϑ(x, ˙ t)γ˙ =  0 0 −( ∂xy − ∂xz )γ˙ 2 ϑT  ∂x     0 0 0 0       0 0 0 0 0 0 0 0 





    γ    q  v .   qϑ  

(6.193) Projecting into the Ritz space by Ψ yields  L  Z dG T   Ψ Ψ dx y˙ R := KG yR ; o

ZL

KG = o



dx

0 0  ∂m 2 T  0 − ∂x γ˙ v v  0 0



0  0  dx.  ∂J ∂J y T 2 z ( ∂x − ∂x )γ˙ ϑ ϑ

(6.194)

Remark: The fact that dGΨT y˙ R yields (part of) the restoring matrix is obviously a question of interpretation. Without rearrangement one would equivalently

294 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations obtain ZL o



0 ∂m γv  ˙ vT qv −  ∂x ∂Jy ( ∂Jz − )γϑ ˙ ϑ T qϑ ∂x ∂x



0 0

0 0   dx

0

0

γ˙ q˙ v q˙ ϑ

! .

This is a consequence of the problem being nonlinear w.r.t. the rigid body coordinates. Eq.(6.194) has a simpler interpretation because it contains (part of) the so-called centrifugal forces ∝ γ˙ 2 . Notice, however, that these “forces” belong to accelerations, sometimes referred to as “reaction forces”. They should not be mixed up with the the impressed forces. Also, the concept of “centrifugal fields” seems a bit misleading when compared to “real” impressed fields like gravitation or any other potential field V . The latter yields, with eq.(5.176), ZL

∂dVel

Ψ

∂y

o

ZL

KV = o

!T



0 0  00 00 T  0 EIz v v 0 0

:= KV yR , 

0  0  dx. T GID ϑ0 ϑ0

(6.195)

The remaining terms emanate from dynamical stiffening. Since w(x, t) is not considered, the second (block-) row and column in eq.(6.188 b) is canceled. (o) Along with dMy = 0 one obtains for eq.(6.187) ZL

ΨdKn ΨT dx := Kn yR ,

o



0

0

0

RL dfx(o) ZL   dξv0 v0 T  0 dξ Kn =  x  RL dfz(o) o 

0

x

dξ (x −

RL dfz(o) x

  

00 T dξ (x − ξ)dξv ϑ   dx.

ξ)dξϑ v00 T

0

 

(6.196) The integrals read as follows: RL dfz(o) x

RL

− dξ (x − ξ)dξ = x fz δ (ξ − L)(x − ξ)dξ L

L

x

x

R R = x fz− δ (ξ − L)dξ− fz ξ− δ (ξ − L)dξ = fz (x − L) where fz = −m tip g.

6.6

295

Gross Motion – Dynamical Stiffening (Ritz Approach) RL dfx(o)

RL h

i

2 − dξ dξ = x ρAξ + m tip ξ δ (ξ − L) γ˙ dξ

x

h

i

= 12 ρA(L2 − x2 ) + mtip L γ˙ 2 , compare eq.(5.173). Since mtip >> ρAL, the first summation term is neglected in the sequel. For the same reason, RL

RL h

o

o

(∂m/∂x)γ˙ 2 v vT dx =

i

ρA + m tip− δ (x − L) γ˙ 2 v vT dx

in eq.(6.194) is replaced with γ˙ 2 m tip v(L)v(L)T =

RL o

γ˙ 2 L1 m tip v(L)v(L)T dx.

One obtains thus the restoring matrix K = KG + KV + Kn , 

0

   ZL   K=  0  o    

0 00 00 T EI  z v v +     vL T γ˙ 2 m tip L v0 v0 T − vLL L

0



0

m tip g(L − x)v00 ϑT T

m tip g(L − x)ϑv00 T

where vL = v(L).



       dx     

GID ϑ0 ϑ0 + ∂Jz − ∂Jy γ˙ 2 ϑ ϑT ∂x ∂x (6.197)

6.1.3 Equations of Motion The equations of motion read M¨yR + KyR = Bu, B = [1 0 qTv

qTw )T ,

0]T ,

(6.198)

where yR = (γ u = u = Mz . The right-hand side is easily confirmed by the virtual work δWdrive = δϕ |x=0 Mz , ϕ |x=0 = γ + v 0 |x=0 . The beam is clamped at its left end, v 0 |x=0 = 0, thus δWdrive = δγ Mz . In case of the one-link robot it is obvious that the centrifugal terms in Kn will not play a significant role, since γ˙ is bounded and appears for short periods of time only. On the other hand, as already mentioned, mtip g influences the motions significantly. Thus, concerning the dynamical stiffening matrix, it will be reasonable to distinguish between kinetic reactions (which might be neglected) and impressed forces (which cannot be neglected).

296 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations

6.2.

Translating Beam – Elastic TT-Robot

The translating beam as depicted in Fig. 6.45 yields a basic model for an elastic TT-robot. Here, TT denotes “translation-translation” and refers to the gripper movement. The gripper is characterized by mGr . It has only one y 6 (relative) degree of freedom in the vertical z-direction. q Arm modeling corresponds  Ao to an Euler beam. As vp long as no contact with η p qA-6 ? q the surface takes place, the x, ξ a gripper motion is decoupled r r from the arm movement. In 6 q q case of contact, however, x mGr m dm the normal force reacts on - 2b  the arm vibrations, and during motion, a torsional torque will be induced by Figure 6.45. Coordinates and Dimensions friction. The arm (index A) is assumed to be flat shaped such that bending only occurs in the y-direction. Elastic deflections are thus v(ξ, t) and ϑ(ξ, t) where ξ is the beam length coordinate. The arm itself moves relative to the prismatic joint (index p) with qA (t). The reference frame which is going to be used is fixed at the prismatic joint and moves in the y-direction with velocity vP . Joint and arm are driven by rack drives via DC motors with harmonic drive gears, the elasticity of which is considered by linear springs, see Fig. 6.46 and Fig. 6.47, respectively. z 6 6 z q

Ωp q @ Rp @ q

- vA = q˙A

RA

- vp = q˙p

y

@ @q

ΩA

-

Figure 6.46. Joint Drive

Figure 6.47.

Arm Drive

-x

6.6

297

Gross Motion – Dynamical Stiffening (Ritz Approach)

Along with the elastic deflections v(ξ, t) = vT qv , ϑ(ξ, t) = ϑT qϑ and the drive velocities ΩA = γ˙ A , Ωp = γ˙ p one has y˙ TR = (γ˙ p γ˙ A q˙p q˙A q˙ Tv q˙ Tϑ ).

(6.199)

For the sake of simplicity the drive unit and the arm may be looked at as subsystems which are eventually combined to form the total system. According to Chapter 4, a single body i is characterized by auxiliary variables y˙ i which are then introduced into the nth subsystem by y˙ i = (∂yi /∂y)y˙ n = Fi y˙ n (body i in subsystem n). In the present case we proceed directly to the Ritz space with y˙ n = y˙ R .

Subsystem “Drive Unit”. Let the motor angular velocities be ψ˙ k , k ∈ {p, A}. The corresponding gear ouput velocity γ˙ k is obtained from ψ˙ k =

∂ψk

!

def.

γ˙ k = ik Ωk , k ∈ {p, A},

(6.200) ∂γk where ik represent the gear transmission ratios. The gear elasticity is represented by the potential 

Vk =

kk 2 k∈{p,A} X



qk − γk Rk

2

kp

k − Rp

0

   0 1 T  = ydr  kp 2   − Rp  

0

p

kA 0 kA −R

A

0 kp Rp2 0

0 kA −R A 0 kA 2 RA

       ydr    

(6.201)

where   γp     γA   = E4×4 0 0  ydr =    qp     qA 



γp γA qp qA qv qϑ

      := Fdr yR .   

(6.202)

In eq.(6.201), kk represent spring coefficients and Rk are characteristic radii, see Fig. 6.46 for instance. The generalized drive torques are obtained from the corresponding virtual work δWdr = (δψp δψA ) · (Mp MA )T = (δγp δγA ) · (ip Mp iA MA )T .

298 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations The tensors of moments of inertia read Jk = diag{Ak , Bk , Ck }, k ∈ {p, A}, where, due to rotational symmetry, Bp = Cp and CA = AA . The drive dynamics equation thus reads 

Ap i2p

   0      0  

0

0

0 0





 kp

    2 BA iA 0 0    0   ydr+ kp ¨  −  Rp 0 mp 0     

0

0 0

0

k − Rp

0

p

kA 0 kA −R

A

0 kp Rp2







0   ip Mp     k A −R  iA MA   A  ydr−  . (6.203)  0  0    

0



kA  2 RA



0



(Eq.(6.203) is not zero because of the constraint forces which will arise when the drive unit is cut free. However, these will vanish when the total system is considered and are thus not of interest.)

Subsystem “Arm”. Bearing. The arm is held in a ball bearing spindle which allows movement in the x-direction. The coupling forces between the bearing and the moving arm may be calculated by y 6 the corresponding spring forces and the related damping. The spring forces are proportional to the relative t t t t t elongations ∆rbearing,i . They r are to be summarized for x t t t t t every “spindle ring” as depicted in Fig. 6.48. IndeFigure 6.48. Bearing pendent of the actual sum which is chosen for the spring elements, one may additionally assume that the beam curvature within the bearing [−a + a], [−b + b] (compare Fig. 6.45) will not be significantly influenced by the bearing forces for sufficiently small b. Then the bearing modeling may be simplified to only one translational and one rotational spring proportional to v(ξ, t) and v 0 (ξ, t) at y = 0, superimposed by a torsional spring for ϑ(ξ, t). As Fig. 6.45 shows, position and angular deflection vectors for any dm read rT =

[ (qA + ξ − qAo ) v(ξ, t)

ϕT = [

ϑ(ξ, t)

0

0

],

v 0 (ξ, t)

]

(6.204)

6.6

299

Gross Motion – Dynamical Stiffening (Ritz Approach)

when viewed from the reference frame (x, y, z). The bearing center is located at x = xbearing = 0 and yields ξbearing = (qAo − qA ) when looked at from the moving beam coordinate system (ξ, η, ζ). The relative translational and rotational elongations are ∆rbearing = rbearing − rbearing,o

with rbearing,o = 0,

∆ϕbearing = ϕbearing − ϕbearing,o with ϕbearing,o = 0,

(6.205)

hence ∂r

∆rbearing =

!

· yA ,

∂yA

∂ϕ

∆ϕbearing =

!

· yA

∂yA

ξbearing

(6.206)

ξbearing

where   qp     qA   = 0 0 E(2+2n)×(2+2n)  yA =    q   v   qϑ 



γp γA qp qA qv qϑ

      := FA yR   

(6.207)

is used for describing variables. Here, 2n represents the number of shape functions for qv ∈ IRn , qϑ ∈ IRn . Premultiplying ∆rbearing with KT = diag{0, kT , 0} and ∆ϕbearing with KR = diag{kϑ , 0, kR } yields the spring forces and torques which are eventually projected into the arm configuration space by the corresponding Jacobians,  

!T

∂r ∂yA

KT

!

∂r

+

∂yA

∂ϕ

!

!T

KR

∂yA

∂ϕ  · yA := Kk yA , (6.208) ∂yA ξbearing

leading to the bearing restoring matrix    

Kk = 

0 0



0 kT v vT + kR v0 v0 T 0

0 kϑ ϑ ϑT

   

(6.209) ξbearing

with ki , i ∈ {T, R, ϑ} for spring coefficients. The shape functions are to be taken at ξbearing = qAo − qA (t) where qA itself is a state variable.

300 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Although derivation of the spring forces and torques seems more complicated than via the corresponding potential, it leads immediately to the hereto related damping. Here, one has to take into account that the beam is moving, thus ξbearing depends on time. Since damping is proportional to the relative velocity, one obtains from eq.(6.206), ∆˙rbearing =

∂r ∂yA

!

· y˙ A + ξbearing

∂

∂r

∂ξ

∂yA

!

· yA ξ˙bearing ,

(6.210)

ξbearing

˙ bearing . Premultiplying eq.(6.210) with DT = and the same holds for ∆ϕ ˙ bearing with DR = diag{dϑ , 0, dR }) diag{0, dT , 0} (and analogously ∆ϕ and projecting into the arm configuration space yields Dd y˙ A + (Kd + Nd )yA

(6.211)

where    

Dd = 

0 dT v vT + dR 0

0

0



v0 v0 T

   

0 dϑ ϑ ϑT

(6.212) ξbearing

and, since ξ˙bearing = −q˙A ,    

(Kd + Nd ) = −q˙A 

0 0



0 dT v v0 T + dR v0 v00 T 0

0 T dϑ ϑ ϑ0

   

(6.213) ξbearing

with di , i ∈ {T, R, ϑ} for damper coefficients. Notice that (6.213) is not symmetric. Decomposition into its symmetric part (Kd ) and its skew symmetric part (Nd ) yields conservative and non-conservative restoring forces21 . 21 These effects are well known in vehicle dynamics. Let a vehicle model consist of (massless) wheels which are attached to the (rigid) car body by a spring and a damper. When moving over a bridge with horizontal velocity q˙x , then the damper velocity is q˙y − [(∂w/∂t) − (∂w/∂x)x] ˙ x=qx where qy is the vertical car position and w(x, t) corresponds to the bridge vibration. The reaction on the bridge is e.g. obtained by the virtual work δw([(∂w/∂t) + (∂w/∂x)x] ˙ x=qx − q˙y ). In terms of Ritz coefficients qw for minimal Tq 0 T xq ˙ [ww + ww ˙ w − wq˙y ]x=qx . In the present case one may interpret coordinates one obtains δqT w w the bearing traveling over the beam with negative speed. The bearing motion q˙p itself does not come into play because v(ξ, t) is already moved with qp .

6.6

301

Gross Motion – Dynamical Stiffening (Ritz Approach)

Beam Dynamics. The matrices dM etc. which correspond to the general motion of a beam are obtained from eq.(5.38) et seq. with F = E. For restricted motion, these are reduced with the actual F, see eq.(5.20). The corresponding describing variables y˙ are then calculated in terms of minimal velocities using the spatial operator: y˙ = D ◦ s˙ , see eq.(5.42). At last, the Ritz ansatz s˙ ' Φ(x)T y˙ A according to eq.(6.2) is inserted: 



vo   ωo       r˙ c  =      ϕ ˙    κ˙ 



|

e2 0 0 0 0 0

e1 0 0 0 0 0

0 0 e2 0 0 0

0 0 0 e1 0 0

0 0 0 e3 0 0

0 0 0 0 e2 0

{z

F

1  0  0    0  0 

0 0 1 0 0 0 1 0 0 0 0 0 0 ∂ ∂x  e3   0 0 0 0  } 0 0 ∂2 2 ∂x {z |



0 0    0  1  1  0    0 0 ∂   0 | ∂x   0

0 1 0 0

0 0 vT 0 {z

0 0 0 ϑT

Φ(x)T



 q˙p     q˙A   .  q˙ v

q˙ ϑ

}

}

D

(6.214)

In F, all components are from IR3 except the last line which contains scalars. This is because of κT = (u0 ϑ0 v 00 w00 ) ∈ IR4 . Furthermore, r˙ Tc = (u˙ v˙ w) ˙ ∈ IR3 denotes the relative mass center velocity of the element dm and T ˙ =(ϑ˙ − w˙ 0 ϕ v˙ 0 ) ∈ IR3 is its relative angular velocity, both w.r.t. the chosen reference frame which moves with vo ∈ IR3 and ω o ∈ IR3 . With the notation [D ◦ ΦT ] = ΨT from eq.(6.2) one obtains 

e2 e1 0 0 0 0 0 0

    FΨT =      0

0 0 e2 vT e3 v0 T

0 0 0 e1 ϑT T

     ,    

(6.215)

0 e3 v00 T e2 ϑ0 0 0 0 0 from which the mass matrix is calculated. (Notice that the last two rows do not contribute because they correspond to curvatures.) As a result, one has  ZL

MA = o

m

0



 0 m  T dM T ΨF FΨ dξ =  R  dξ  v dm 0 

0

0

R

vT dm 0

R v vT dm R 0 0T

0



0

      

0 + v v dJz R 0 ϑ ϑT dJx

A

(6.216)

302 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations with dM from eq.(5.38). Since ω o is zero, GA = 022 . Dynamical Stiffening. 

Since ω o = 0, eq.(5.165) reduces to 





0 q¨A ∂m  df    0 q ¨ =− +  p    dξ ∂ξ 0 fN− δ (ξ − L) − (∂m/∂ξ)g (o)

(6.217)

which represents the zero-order forces acting on the beam. fN comprises the normal forces at ξ = L (contact force and gripper weight). The beam is characterized by (ξ, η, ζ), variable x is therefore replaced by ξ. The direct influence of impressed forces on the system is obtained by projecting F from eq.(5.40) with eq.(6.125) into the actual configuration space, 

0  1 =  0 0 

T

T



ΨF F

e

df dMe



1 0 v 0

0 0 0 0 0 0 0 ϑ

0 0    0 0    0 0 v   0 0  

dfxe dfye dfze dMxe dMye dMze

     ,   

(6.218)

which shows the fact that dfze does not contribute but only enters the equations via the dynamical stiffening matrix. Since the beam has free ends, eq.(6.188 a) is to be used which, after canceling the second (block-) row and column yields for eq.(6.187), ΨdKn ΨT := dKn =  0

        0 

RL dfx(o)

RL dfz(o)

ξ

ξ

0 0T dζ dζv v

symmetric



0 (o)

RL 0 ϑ0 T − dfz dζv0 ϑT (ζ −ξ)dζv dζ dζ ξ

0

     dx.    

(6.219) 22 Check:

with and momentum of momentum, respectively, one obtains the mass Pp and L for momentum [δri dp˙ i + δϕi dL˙ i ], i ∈ {x, y, z}. Here, δrx = δqA , dp˙ x = dm¨ qA ; matrix from i Tq ¨ v, d p ˙ = dm(¨ q + v ); δr = 0; δry = δqp + δqT y p z v v T¨ T 0 0T ¨ ˙ ˙ δϕx = δqT ϑ ϑ, dLx = dJx ϑ qϑ ; δϕy = 0; δϕz = δqv v , dLz = dJz v qv , yielding T¨ T T T ¨v )] + δqA [dm¨ ¨v ) + dJz v0 v0 T q ¨v ] + δqT qp + v q qA ] + δqv [(dm(v¨ qp + vv q δqp [(dm(¨ ϑ [dJx ϑϑ qϑ ].

6.6

303

Gross Motion – Dynamical Stiffening (Ritz Approach)

(Since the beam coordinate has changed to ξ we use ζ for integration variable.) The integrals show the following solutions: RL dfx(o)

RL

qA dζ = −ρA¨ qA (L − ξ), dζ dζ = − ξ ρA¨

ξ

RL dfz(o)

dζ (ζ −ξ)dζ

ξ

=

RL

(L−ξ) RL R fN ζ − δ (ζ − L)dζ − ξ fN− δ (ζ − L)dζ − ρAg(ζ − ξ)d(ζ − ξ)

ξ

ξ

ξ−ξ

= fN (L − ξ) − 12 ρAg(L − ξ)2 , (o)

RL RL RL z − dfdζ δ (ζ − L)dζ + ρAgdζ = −fN + ρAg(L − ξ). dζ = − fN− ξ ξ ξ

Because the dynamical stiffening matrix undergoes integration over the beam length, one obtains, for the part that depends on the impressed forces, ZL

F∗im :=

T

fN [(L−ξ)v0 ϑ0 −v0 ϑT ]dξ −

o

ZL o

1 T ρAg(L−ξ)[ (L−ξ)v0 ϑ0 −v0 ϑT ]dξ. 2 (6.220)

Eq.(6.220) can be reformulated as F∗im =

ZL o

fN v0

ZL i i d h 1 d h ρAgv0 (L − ξ)ϑT dξ − (L − ξ)2 ϑT dξ (6.221) dξ 2 dξ o

which allows for an integration by parts yielding ∗

ZL 

Fim = o

h i 1 1 ρAg(L − ξ)2 − fN (L − ξ) v00 ϑT dξ+ mA g − fN L v0 ϑT . o 2 2 (6.222) 





The first summation term corresponds to the result one would obtain from eq.(6.188 b) (clamped-free beam). Due to the free ends of the arm, the boundary terms in eq.(6.222) do not vanish. The dynamical stiffening thus reads

304 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

0

   Kn =    0



0 RL

ρA¨ qA (ξ − L)v0 v0 T dξ 0

o

0



0

      +     0

0



0 0 [F∗im ]T

   [Fim ]    ∗

0

(6.223) with F∗im according to eq.(6.220) et seq. for abbreviation. The dynamical stiffening matrix in eq.(6.223) is separated into its dynamical part and its impressed force part. The same as in eq.(6.194), the dynamical part is not unique in representation due to nonlinearity. It reads, when multiplied with yA , 



0 0

 0    0  q 0 p   0    q       A  L R =       T 0 0 q RL      ρA¨ qA (ξ − L)v v dξqv  v 0 v0 T dξ 0  0 0   ρA¨ q (ξ − L)v A o   q o ϑ 0 0 0 0 0 (6.224) and may be rearranged to

0 0 0 0 



0  0 

0 0

   RL   0 ρA(ξ − L)v0 v0 T dξqv  o

0

0



0 0  q¨p 0 0     q¨  A  ¨   qv 0 0   ¨ϑ q 0 0

   . 

(6.225)

In the interpretation of eq.(6.225), the stiffening part now belongs to the accelerations and is thus to be assigned to the mass matrix instead of to the restoring matrix. Because the mass matrix later needs inversion, this kind of reorganization is necessary. Interpretation of eq.(6.225) as (part of the) local mass matrix leads to nonsymmetry which seems rather unusual. However, this fact only results from (parR tial) linearization. Indeed, retaining the second-order term in qA − oξ v 0 2 dζ/2 R yields q¨A − oξ v 0 v¨0 dζ + · · ·. One obtains then for the momentum change RL R R R R R qA dξ − oL ρA oξ v 0 v¨0 dζdξ = oL ρAdζ q¨A − oL v 0 v¨0 ξL ρAdζdξ which o ρA¨ yields, when the Ritz ansatz is inserted, mA q¨A + qTv oL ρA(ξ − L)v0 v0 T dξ¨ qv . The second summation term clearly belongs to the second component in eq.(6.225). Insertion of these relations into eq.(6.225) yields R

6.6 

305

Gross Motion – Dynamical Stiffening (Ritz Approach)

0

0

0

0



  RL q¨p 0 T ρA(ξ − L)v0 v0 T dξ 0  q 0   v   q¨A  o    q  L ¨v   R T   0 ρA(ξ − L)v0 v0 dξq 0 0 q v ¨ϑ  o

0

0

0

   yA .  := Madd ¨ 

0

(6.226) and brings symmetry back. Of course, the inserted term does not have any influence on the system solution as long as the linearization assumptions hold. Nevertheless, mass matrix symmetry is important for stability theorems (Chapter 8).

6.2.1 Mass Matrix The total system is now composed using eqs.(6.202) and (6.207), ydr = Fdr yR , yA = FA yR , T

(6.227)

T

yielding the mass matrix M = [Fdr Mdr Fdr + FA (MA + Madd )FA ] with Mdr from eq.(6.203) and MA from eq.(6.216) along with eq.(6.226): M=                   



Ap i2p 0

BA i2A

0

0

mp +mA

0

0

0

0

0

0

0

R

vdm 0

symmetric mA R

(ξ −L)v0 v0 T dm qv 0

R T vv dm R 0 0T

+ v v dJz 0

R

ϑϑT dJx

         .        

(6.228)

6.2.2

Restoring Matrix T

By the same procedure one obtains the restoring matrix K = [Fdr Kdr Fdr + T FA (Kk +Kn )FA ] with Kdr from eq.(6.203) and Kk from eq.(6.209) along with the second summation term in eq.(6.223), completed with elastic potential forces:

306 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations K= 

kp  0   kp −  Rp                 

0



kA kp Rp2

0 kA −R

0

A

kA 2 RA R

0

0

0

0

0

0

0

h

EIz v00 v00 T dξ+

kT vvT +kR v0 v0 T

i ξbearing

[F∗im ]T

0

    symmetric                 R T 0 0 GID ϑ ϑ dξ  

+kϑ ϑϑT |ξbearing

(6.229) where ξbearing = (qAo − qA ). Damping according to eq.(6.211) has been neglected, having in mind that the robot system will be controlled actively. Remark: Because Fdr , FA consist of unit and zero matrices, the mass matrix and the restoring matrix can simply and easily be written down with an “overlapping” for qp and qA .

6.2.3 Equations of Motion The equations of motion read M¨yR + KyR = Bu + Qdisturbance ,

(6.230)

where yR = T

B =

 

γp γA qp qA qTv ip 0

0 iA

0 0

0 0

qTϑ 0 0

T

0 0

,



, u=

Mp MA

!

(6.231)

.

Here, Qdisturbance will for example arise when the gripper contacts the surface.

6.2.4 Simplified System In order to obtain a quick overview, the drive gears are assumed stiff. Additionally, the contact force is neglected as well as the torsional degree of freedom. This leads to the constraint 



Rp γp − qp   Φ =  RA γA − qA  = 0 ⇒ ϑ

6.6

307

Gross Motion – Dynamical Stiffening (Ritz Approach)

1 Rp 0 1 0 0 0

 !

∂Φ

∂yR

∂yR



!

∂q

Rp

= 0 0

  0 0 0    −1 0 0    0 0 E   

0 −1 RA 0 0 0

0 1 RA 0 1 0 0

0 0 0 0 E 0

       = 0.    

(6.232) Here, (∂Φ/∂yR )T λ represents the corresponding constraint forces which are eliminated when premultiplied with (∂yR /∂q)T due to the orthogonality condition (6.232) (see also Chapter 4, sect.5.2). The new variables q are obtained from y˙ R = (∂yR /∂q)q˙ ⇒ q˙ = (∂yR /∂q)+ y˙ R with the generalized inverse (∂yR /∂q)+ ,

0

Rp2 1 + Rp2

0

RA 2 1 + RA

0

0

0

0



Rp   1 + Rp2

 q˙ =    

0

 0     0   

0

2 RA 2 0 1 + RA

0

0

E

γ˙ p γ˙ A q˙p q˙A q˙ v q˙ ϑ

     q˙p    q˙A  . =   q˙ v 

(6.233) The result is holonomic (i.e. eq.(6.233) can be integrated w.r.t. time)23 . Insertion of yR = (∂yR /∂q)q and its derivatives into eq.(6.230) yields, after premultiplication with (∂yR /∂q)T : Mred :=





ip  Ap Rp    

!T

∂q

M

∂yR

!

=

∂q

2

+mp +mA 0

R

∂yR

ρAv dξ

R

0 2

iA + mA qTv BA R A R ρA(ξ −L)v0 v0 T dξqv 

R

ρAvT dξ



  , ρA(ξ −L)v0 v0 T dξ   R T

ρAv v dξ

(6.234)

23 When deriving new variables via the explicit constraint equation, it is not self-evident to obtain holonomic variables even in case of holonomic constraints, see Chap.2 Calculating eq.(6.233) one obtains [Ri γ˙ i + Ri2 q˙i ]/[1 + Ri2 ], i ∈ {p, A}. The constraints yield Ri γ˙ i = q˙i , thus [Ri γ˙ i + Ri2 q˙i ]/[1 + Ri2 ] = [q˙i + Ri2 q˙i ]/[1 + Ri2 ] = q˙i .

308 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

Kred :=

∂yR

!T

K

∂yR

∂q

!

∂q

0 0 0 0 



0 0

   ,   

 = R   EIz v00 v00 T dξ+ i 0 0 h T 0 0T

kT vv + kR v v

ξ=(qAo −qA )

(6.235)  

Bred :=

∂yR ∂q

!T

  B=  

ip Rp 0 0





0 

iA RA 0

   ,  

(6.236)

dJz neglected (Euler beam).

Control Requirements. The main point of interest is the tip motion in “world coordinates”, i.e. the inertial x-y plane. Here, a semicircular path (radius R = 0.85) is considered for a typical maneuver, qpd = −R · cos(πt/T ), qAd = L + R · sin(πt/T ), index d for “desired”. The beam length is L = 2 (all measures taken in the kg-m-s-system). T = 1.5 is chosen for maneuver tivi me. The motor is located at distance 3 2 2 qAo = 0.9, the maneuver starts with 1.5 qA = 0. The beam measures 2.0 x 1 0.005 x 0.05 and yields a mass of 1 0.5 mA = 2 (aluminum); bending ri0 gidity is EIz = 36.5. The bearing spring coefficients are kT = kR = -0.5 2 · 104 . Motor moments of inertia -1 range typically in JM ' 2·10−5 for -1.5 4 J = A , B with a transmission p M A -2 0 0.2 0.4 0.6 0.8 ξ/L ratio of i ' 100 and a transmission radius R ' 0.1. We consider first a rigid beam which is characterized with the shape functions i=1,2 (“riFigure 6.49. Shape Functions 1 to 4 gid body modes”). T Without normalization they read v = [1 (ξ − L2 )] with ξ for beam length coordinate. The hereto assigned Ritz coefficients are denoted by y, ϕ. Carrying out integration yields

6.6

Gross Motion – Dynamical Stiffening (Ritz Approach)





 Ap         

ip Rp

2



+ (mp + mA )

0 0 0

0

0

        



0 

0 

ip R p Mp

mA

0

   i L A + mA 0 −mA 2 ϕ  BA R  A    0 mA 0       2 −m L ϕ 0 m L

mA

0  0 

0

2



0



309

A

 

A

2

0 0

kT



qAo −qA − L2



iA R A MA 









2

kT qAo −qA − L2 



12

0 0

kT

q¨p q¨A  + y¨  ϕ¨

kT qAo −qA − L2 +kR

          

qp qA  = y  ϕ 

T

0

0

.

(6.237)

This result is easily proved with the mass center coordinates for a rigid bar, T ˙ 3 . Taking the I rAc = [(−qAo + qA + c cos ϕ) (qp + y) 0] and I ω A = ϕe joint into account yields I vpc = q˙p e2 , and the motor angular velocities are qAo  characterized by I ω dr,p = C ϕ [−(ip /Rp )e1 ] q˙p and d 6 I ω dr,A = [(iA /RA )e2 ] q˙A . The Projection Equation y (4.34), p.71, yields the qA mass matrix having in mind that the moment of p p p p p p p p p p p p inertia of a rigid homogeneous bar w.r.t. its mass Figure 6.50. Rigid Bar Coordinates center reads mL2 /12. [L is the beam length and the mass center C is located at c = L/2]. Although ϕ is assumed small, rAc has to be considered up to the second order to obtain first-order Jacobians. The restoring matrix is obtained from the potential V = kT [y − (−qAo + qA + c cos ϕ) tan ϕ]2 /2 + kR ϕ2 /2 ≈ kT [y − (−qAo + qA + c)ϕ]2 /2 + kR ϕ2 /2. Notice that in eq.(6.237) qp and qA represent gross motions which are superimposed by small deviations y, ϕ. It is common practice in control to neglect the deviations in the first step in order to derive an open loop control which

310 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations achieves the gross motion. Then, if need be, a closed loop control is superimposed in a second step to get rid of possible disturbances. The open loop control is easily extracted from eq.(6.237). Neglecting y¨ and ϕ¨ one may resolve the first two rows for Mp and MA yielding  

Mp MA



 Ap

=



ip Rp





R + (mp +mA ) i p p





0



iA + m RA BA R A iA A 

0





  



q¨pd . q¨Ad

(6.238)

y Rigid Beam Desired Path

Elastic Beam Desired Path

+0.5

−0.5

2

2.2

2.4

2.6

2.8

Figure 6.51. Open Loop Control, Rigid Bar Tip Motion

2

2.2

2.4

2.6

2.8 x

Figure 6.52. Open Loop Control, Elastic Beam Tip Motion

The control torque (6.238) is used for the rigid bar as well as for the elastic beam with four shape functions according to Fig. 6.49. As the results show, an open loop control might be accepted for the rigid system. Although not really convincing it is common practice in robotics up to now to dispense with closed loop control. However, the elastic system does not reach the aim even allusively. Here, the closed loop control becomes indispensable.

6.7

311

The Mass Matrix Reconsidered (Ritz Approach)

7.

The Mass Matrix Reconsidered (Ritz Approach)

As indicated by eq.(6.225), the zero-order acceleration reactions are now rearranged to obtain the corresponding (part of the) mass matrix. This is easily done: recall eq.(5.160):  ZL

δWn =

δyTel

h

FT1

FT2

o

RL

 i x    RL  x

df (o) /dξ dM(o) /dξ

!

df (o) /dξ dM(o) /dξ

!



dξ (ξ − x)dξ

    dx  

(6.239)

where F1 , F2 are given with eqs.(5.157) and (5.158) for the general case and with eqs.(5.157 b) and (5.158 b) for the one-side clamped beam, respectively. The virtual displacement δyel = ΨTel δyR,el is derived from eq.(6.185). The zero-order acceleration reactions are extracted from eq.(5.165): 

d dx

f (o) M(o)

! acc.



E dm eeT1 dm dx x   v˙ o   dx  = − .   ω ˙o dJ 0 dx

(6.240)

Insertion of δyel yields the requested additional term for the mass matrix which arises from gross accelerations acting on the elastic deflections, 

Madd =

0 MTNL MNL 0



,

(6.241)

where E dm eeT1 dm dξ ξ   RL  dξ 



ZL

MNL = o

h

Ψel FT1

     x dJ 0 i dξ T  F2    dm eT dm ξ  L E dξ e 1 dξ  R     x

0

dJ dξ





 dξ 

       dx.     

   (ξ − x)dξ 

(6.242) Remarks: 1. Insertion of δyTel = δyTR,el Ψel into eq.(6.239) indicates that the calculated terms belong to the elastic coordinates yR,el which are resembled in the lower part of eq.(6.241). The upper part has been symmetrisized as in eq.(6.226). Although this part is of no influence because it

312 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations only reveals second-order negligible terms, knowledge of the mass matrix being symmetric is essential for application of the stability theorems. Also, numerical procedures often make use of symmetries advantageously (e.g. Cholesky decomposition). 2. Of course, the Ritz ansatz has to be inserted into F1 , F2 yielding for the general case Ψel FT1 =



T

−v0 v0 qv

T T 1 (v0 w00 −v00 w0 )qw 2 T T 1 (w00 v0 −w0 v00 )qv 2

v0 ϑTqϑ

0

  −w0 w0 Tq −w0 ϑTq w ϑ 

0

T

T

−ϑw0 qw ϑv0 qv

0

0

T

v0 ϑ0 qϑ



0



T

w0 ϑ0 qϑ  , 0 0T 0 0T ϑ v qv ϑ w qw 0

(6.243 a) Ψel FT2

   

=

T

−v0 ϑ0 qϑ

0

0

0

T

0

0

0

0

T

−ϑ0 v0 qv

0

0

0

0

0

0

w0 ϑ0 qϑ

0

ϑ 0 w0 q w

T

  , 

(6.244 a) which for the left side clamped boundary condition (or for the “fork-like joint”) reduces to Ψel FT1 =



T

−v0 v0 qv

  −w0 w0 Tq w  0

v0 ϑ0 qϑ

0

T T 1 (v0 w00 −v00 w0 )qw 2 T T 1 (w00 v0 −w0 v00 )qv 2

0

0

ϑ0 v0 qv

0

0

0 0

T

0 T



0



T w0 ϑ0 qϑ  , T

ϑ0 w0 qw

(6.243 b) Ψel FT2 =

   

0

0

v00 ϑTqϑ

0

0

0

0

−w00 ϑTqϑ

0

0

0

0

0

−ϑw00 qw

0

0

0

T

T

ϑv00 qv

  . 

(6.244 b) 3. Eq.(6.242) can be calculated explicitly if the mass distribution (dm/dx) [and, consequently, the moment of inertia distribution (dJ/dx)] is known. For a homogeneous beam with constant cross sectional area the requested integrals in eq.(6.242) are

RL

dξ = (L − x),

x

ξdξ = 12 (L2 − x2 ) = 12 (L + x)(L − x),

x

RL

(ξ − x)dξ =

x

RL

1 (L 2

− x)2 ,

RL

ξ(ξ − x)dξ =

L3 3

−

L2 x 2

+

x3 , 6

x

yielding for eq.(6.242), along with

L3 3

−

L2 x 2

+

x3 6

= 23 (L + x2 ) 12 (L − x)2 ,

6.7

The Mass Matrix Reconsidered (Ritz Approach) " # ZL T ρAE ρAe e1 12 (L + x) T MNL =

Ψel F1

0

ρI

313 (L − x)dx

o

1 + 2

ZL

" Ψel FT2

T

ρAE

ρAe e1 23 (L + x2 )

0

ρI

# (L − x)2 dx.

(6.245)

o

Considering a beam with additionally attached rigid bodies, this representation should be retained. In such a case one may simply replace ρA → ρA +

P

mη−δ(x − xη ) as well as ρI → ρI +

η

P − Jη δ(x − xη ) η

in order to conclude the attached bodies influence. 4. On the other hand, neglecting ρI (Euler beam) simplifies the equations significantly (allowing, with the same argument as above, consideration of additionally fixed point masses). One obtains then

ZL MNL =

"

T

Ψel FT1

ρAE

ρAe e1 12 (L + x)

0

0

ZL

"

# (L − x)dx

o

1 + 2

Ψel FT2

T

ρAE

ρAe e1 23 (L + x2 )

0

0

# (L − x)2 dx

(6.246)

o

which shows the fact that the last three columns in eq.(6.243 a) [and eq.(6.243 b), respectively] do not contribute. The (part of the) additional mass matrix then reads explicitly (free boundaries)





+v0 ϑTqϑ (L−x)

0 0T

 −v v qv (L−x)  0 T   − 12 v0 ϑ0 qϑ (L−x)2     0 0 0 ZL   0 T −w ϑ qϑ (L−x)  0 0T  MNL = ρA−w w qw (L−x) 0 dx 0 0 0 2 1 0 0T   + 2 w ϑ qϑ (L−x)   o  000 0T 0T   +ϑw qw (L−x) −ϑv qv (L−x)   0 T

T

+ 21 ϑ0 w0 qw (L−x)2 − 12 ϑ0 v0 qv (L−x)2

  0   000 ZL    + ρA 0 0 0 0   o 000   0



− 12 v0 ϑTqϑ (L2 −x2 ) T

3

2

3

+v0 ϑ0 qϑ ( L3 − L2 x + x6 ) 0 T

− 21 ϑv0 qv (L2 −x2 ) T

3

2

3

+ϑ0 v0 qv ( L3 − L2 x + x6 )

0

     − 21 w0 ϑTqϑ (L2 −x2 )  dx 0 0T L3 L2 x x3 +w ϑ qϑ ( 3 − 2 + 6 )    0T 2 2 1  − 2 ϑw qw (L −x )  T

3

2

3

+ϑ0 w0 qw ( L3 − L2 x + x6 )

(6.247)

314 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations (which has been split into two parts because of space requirements). Eq.(6.247) consists of two types of functions which are characterized by element (2,2) and element (2,6). Let for simplicity ϑTqϑ = ϑ. Element (2,2) may be reformulated as RL RL d 1 [−w0 ϑ(L−x) + w0 ϑ0 12 (L−x)2 ]dx = o w0 dx [ϑ 2 (L−x)2 ]dx o L2 [w0 ϑ]x=0 . Analogously, element (2,6) yields 2 3 2 3 3 2 L L d [−w0 ϑ 12 (L2 − x2 ) + w0 ϑ0 ( L3 − L2 x + x6 )]dx = o w0 dx [ϑ ( L3 − L2 x o 3 2 3 3 L − o w00 ϑ[( L3 − L2 x + x6 )]dx − L3 [w0 ϑ]x=0 .

=−

RL o

w00 ϑ 12 (L−x)2 dx −

R

R

3

+ x6 )]dx =

R

Thus, with an integration by parts, eq.(6.247) yields

 L

Z MNL =

T

−v0 v0 qv

  

1 00 T v ϑ qϑ (L−x) 2

0

ρA −w0 w0 Tqw − 1 w00 ϑTqϑ (L−x)

o

0

 L

Z +

0 0 0

  

ρA 0 0 0 o



0 0 0

0

  

+ρAL2  0 0

0

2

T − 21 ϑw00 qw (L−x)

0 0 0

− 12 w0 ϑT qϑ T 1 ϑw0 2

−v00 ϑTqϑ

−ϑv

qw

T

−w ϑ qϑ

0 00 T

T

−ϑw00 qw

qv

qϑ

0 T

− 12 ϑv0 qv



  0 0 0  (L−x)dx  0 0 0



0 00

1 0 T vϑ 2

0

T 1 ϑv00 qv (L−x) 2

0 0 0

 3  L L2 x x3 + )dx ( − 2 6  3

0

− L3 v0 ϑT qϑ

0

0

0

− L3 ϑv0 qv



0

  

− L3 w0 ϑT qϑ  . T

T

− L3 ϑw0 qw

x=o

(6.248) 5. The (part of the) additional mass matrix which is obtained for the one-side clamped beam is obtained from eq.(6.248) by setting the boundary terms to zero. This leads to considerable simplifications. [Of course, the same result is obtained from eq.(6.246) with eqs.(6.243 b), (6.244 b)].

Notice that MNL has the dimension (nv +nw +nϑ )×6: It denotes the influence of (v˙ To ω˙ To ) ∈ IR6 on qv ∈ IRnv , qw ∈ IRnw , qϑ ∈ IRnϑ and represents thus the maximum number of d.o.f. For an actual case, the non-considered elastic deflections are to be canceled as well as the non-acting rigid body accelerations. As has been emphasized with the calculation of eq.(6.215), the curvatures do not contribute to the calculation of the zero-order part of the mass matrix, R T M = Ψ[F dM F]ΨT . To include the curvatures in y˙ has been necessary for the derivation of the partial differential equations, Chapter 5, which additionally lead to the necessity to fill up dM, dG, F with corresponding zeros. Removing these once more yields, for eq.(5.38),

6.8

315

The G-Matrix Reconsidered (Ritz Approach)

dmE dmerTc  dme rc dJo  

T dM = F  

dmerTc

 dmE

dJc

0



dmE 0 dmerc dJc   F dmE 0   0

(6.249)

dJc

where rc = (x vT qv wT qv )T . The intermediate (or auxiliary) variables according to eq.(5.20) then also ˙ T )T . As eq.(6.215) shows, the term F Ψ is reduce to y˙ = (vTo ω To r˙ Tc ϕ determined very simply. The easiest access is to calculate y˙ = y˙ for full mobility (F = E),                }



vox voy   e e e 0 0 0 0 0 0  1 2 3    v oz   vo   ω ox  0 0 0  ω o   0 0 0 e1 e2 e3     ωoy   r˙ c  =   T T  0 0 0 0 0 0 e2 v e3 w 0 ωoz    ˙ ϕ  q˙ v  T T T 0 0  −e2 w e1 ϑ 0 0 0 0 0 0 e3 v q˙ w  {z | q˙ ϑ F ΨT for F=E (6.250) (all elements ∈ IR3 , longitudinal stretching neglected). The reduced representation F ΨT , F 6= E is then simply obtained by eliminating the columns that correspond to the non-considered motion. 

8.

The G-Matrix Reconsidered (Ritz Approach)

Although not necessary it will be at least consequent to also rearrange the velocity dependent reaction forces in the same manner. The advantage one obtains is that for sufficiently slow-moving systems the arising GNL may a` priori be neglected as well as MNL . Concerning the dynamical stiffening one is then left with only the impressed zero-order forces and torques. The procedure is equivalent to eq.(6.239) et seq.: the zero-order (velocity dependent) acceleration reactions are extracted from eq.(5.165): 

d dx

f (o) M(o)

! veloc.

 = − 

e oe e o dm ω eT1 dm ω dx dx x

0

e o dJ ω dx

   





vo . ωo

(6.251)

Insertion of δyel yields the required additional term for the G-matrix which arises from gross velocities acting on the elastic deflections,

316 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 

Gadd =

0 0 GNL 0



(6.252)

where 



ZL

GNL =

h

Ψel FT1

o

e o dm ω e oe ω eT1 dm dξ dξ ξ

 RL       x e o dJ ω 0 i dξ T  F2    e o dm ω e oe eT1 dm  L ω dξ dξ ξ  R     x e o dJ ω 0





  dξ 

       dx.     

   (ξ − x)dξ 

dξ

(6.253) The only difference is that there is no necessity to fill eq.(6.252) up with symmetric (or antimetric) elements. Remarks: 1. Eq.(6.253) can be calculated explicitly if the mass distribution (dm/dx) [and, consequently, the moment of inertia distribution (dJ/dx)] is known. For a homogeneous beam with constant cross sectional area one obtains

ZL GNL =

"

T

Ψel FT1

ρAω eo

ρAω e oee1 12 (L + x)

0

ω e o ρI

ZL

"

o

1 + 2

Ψel FT2

# (L − x)dx

T

ρAω eo

ρAω e oee1 23 (L + x2 )

0

ω e o ρI

o

# (L − x)2 dx.

(6.254)

2. Neglecting ρI (Euler beam) yields

ZL GNL =

"

T

Ψel FT1

ρAω eo

ρAω e oee1 12 (L + x)

0

0

ZL

"

# (L − x)dx

o

1 + 2

Ψel FT2

T

ρAω eo

ρAω e oee1 23 (L + x2 )

0

0

# (L − x)2 dx.

(6.255)

o

The (part of the) additional G-matrix then reads explicitly, after integration by parts

6.8

317

The G-Matrix Reconsidered (Ritz Approach) RL

GNL = ρA× o

 ω − 2oy

00

T

2



+ ω2ox v00 ϑTqϑ (L−x)2

0 0T

v ϑ qϑ (L−x) −ωoy v v qv (L−x)   T   +ωoz v0 v0 qv (L−x)  000    ωox 00 T 2  ω   − oz w00 ϑTqϑ (L−x)2 ωoz w0 w0 Tqw (L−x) 2 w ϑ qϑ (L−x) 0 0 0 dx 2   T −ωoy w0 w0 qw (L−x)    000   ωoz 00 T  2  − 2 ϑw qw (L−x) ωox ϑv00 Tq (L−x)2 ωox ϑw00 Tq (L−x)2  v w 2 2 ωoy 00 T 2 −

2

ϑv

qv (L−x)

RL

+ ρA× o



2

 0 000   000  0  000  0

ωoy 2

0 0T

2

v v qv (L −x ) 2

2

ωoy 2

3

    dx    

3

ωoz 2

T

2



T

w0 w0 qw (L2 −x2 ) T

3

+ ω2oz v0 v0 qv (L2 −x2 )

−ωox w00 ϑTqϑ ( L3 − L2 x + x6 ) +

2

ωox v00 ϑTqϑ ( L3 − L2 x + x6 )

2

2

3

T

w0 w0 qw (L2 −x2 ) 2

T

−ωox ϑw00 qw ( L3 − L2 x + x6 )

2

ωox ϑv00 qv ( L3 − L2 x + x6 )

+ρAL2 ×



−

ωoy 2

w0 ϑTqϑ

   − ω2oz w0 ϑTqϑ    − ω2oz ϑw0 Tqw −

ωoy 2

T

ϑv0 qv



ωox 0 T v ϑ qϑ 2

0

0

ωox 0 T w ϑ qϑ 2

ωox ϑv 2

0T

qv

ωox ϑw 2

0

0

ωox L 0 T v ϑ qϑ 3

0

L 0 T − ωox w ϑ qϑ 3

0

0T

qw 0 − ωox L ϑw0 Tqw 3

T ωox L ϑv0 qv 3

      x=0

(6.256) 3. Considering the one-side clamped Euler beam, the boundary values at x = 0 vanish.

Since the curvatures do Rnot contribute to the calculation of the zero-order T part of the G-matrix, G = Ψ[F dG F]ΨT , the corresponding zero elements in eq.(5.39) are also eliminated. This leads to eo e oe eo dmω dmω rTc 2dmω 0 c c   o c ˙ e e eo ω e o dJ + dJ 2dmrc ω eo ω e o dJ + dJ˙   dmrc ω 



dG = 

  dmω eo 

0

e oe dmω rTc c

e o dJc + dJ˙ ω

eo 2dmω

0

0

e o dJc + dJ˙ ω

which, of course, goes along with eq.(6.250).

c

   

(6.257)

318 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations 5 Example: Fast-Moving Euler Beam – Plane Motion. The considered velocity coordinates are vox , voy , ωoz = γ˙ and v(x, ˙ t), see Fig. 6.53. Thus, the columns nr. 3,4,5,8,9 which correspond to the non-considered motions voz , ωox , ωoy , qw , qϑ are eliminated in eq.(6.250) yielding F ΨT = e1  0  0 0

y

f ωo

0 e3 0 0

e2 0 0 0

 vo

matrices R theT ΨF dM F Ψ and T

R

ΨF dG F Ψ are G = computed with. dM is derived from eq.(6.249)) and dG from eq.(6.257)). For the Euler beam, dJc = ρI is neglected. The tensor of moments of inertia w.r.t. the reference coordinate center then reduces T to dJo = e rce rc dm, and dJo e3 2 reads x dm. Hence,

x

γ

Plane Beam Motion

ZL M= o

ZL G= o



1  0 ρA  −vT q 0



0  1 ρA  x v

0 1 x v

−1 0 vT q 0

−vT q x x2 xv −x −vT q 0 −v vT q



0 vT   dx, xvT T vv

(6.259)

−2vT 0   ωoz dx, 2qT v vT 0

(6.260)



  T

ΨF F

T



dfe dMe



(6.258) where M =

Figure 6.53.

0 0  e2 vT  T e3 v0



1

0  := dQe =  −vT qv 0

0 1 x v

0 0 0 0

0 0 0 0

0 0 0 0



 0  0    1  0  v

dfxe dfye dfze dMxe dMye dMze

     (6.261)  

where v(x, t) = vT qv has been inserted for rTc = (x v(x, t) 0). (F follows from eq.(5.40) by eliminating the last block-row since curvatures are no longer being considered). Check: The system matrices are easily obtained using the Projection Equation directly: The (mass center) velocity of any dm is vc = vo + r˙ c + ω e o rc := F y˙ R where F = " # 1 0 −v(x, t) 0 0 1 x vT . Since the momentum of momentum is neglected, the zero0 0 0 0

6.8

319

The G-Matrix Reconsidered (Ritz Approach) order mass matrix and G-matrix are obtained from

R

M=

[FT dmF] and G =

FT [dmF˙ + ω e o dmF].

R

The additional (part of the) mass matrix and (of the) G-matrix are obtained from eq.(6.248) and eq.(6.256), respectively, as follows: 1. Because voz , ωox and ωoy do not arise: cancel columns number 3,4,5. 2. The sequence of Ritz-coefficients is qv , qw , qϑ . Since qw and qϑ are not considered: cancel rows number 2 and 3. Thus,

ZL MNL =





0 0T

−v v qv (L − x)

ρA

0

dx

0

o

Z ⇒ Madd =

0 0 0

0 0 0

qTv v0 v0 (x−L) 0 0

v0 v0 qv (x−L)

0

0

0

  

ρA  o

T

0 0 0

 L

T

    dx, 

(6.262) GNL =

RL

ρA ×

o

" ω − 2oy

00

T

2

v ϑ qϑ (L−x)

ZL

ωox 00 T v ϑ qϑ (L−x)2 2 T +ωoz v0 v0 qv (L−x)

 ρA

2

3

# dx =

T

+ ω2oz v0 v0 qv (L2 −x2 )

0 0T

0

2

ωox v00 ϑTqϑ ( L3 − L2 x + x6 )

ωoz 2

ωoz v v qv (L−x)

ϑ=0



0 0T

2

2

v v qv (L −x )

dx

o



0 Z  0  = ρA  0

⇒ Gadd

o



0 0 0

0 0 0

0 0

v0 v0 qv (L−x)

1 0 0T v v qv (L2 −x2 ) 2

0

L

 0

T

   ωoz dx. 

(6.263) The dynamical stiffening matrix (6.187) is obtained with eq.(6.188 b) when the (block-) rows and columns for qw and qϑ are canceled:

 ZL Kn := o

dKn T Ψ Ψ dx = dx

ZL      o 

0 0 0

0 0 0

0 0 0

0

0

0

0 0 0

RL dfx(o) x

dξ

dξv0 v0



T

    dx.  

(6.264)

320 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations Considering gravitational forces in the inertial x-direction (γ = 0 in Fig. 6.53) , one has

"

cos γ − sin γ 0

sin γ cos γ 0

0 0 1

  #  ρAg  cos γ  0 dx = ρAg  − sin γ dx 0

(6.265)

0

for impressed force acting on the beam element. (Remember that the zero-order forces and torques have been defined for the undeformed reference coordinate system, see p.167). One obtains thus



ZL Kn =

ρAg 



o

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 T (L−x)v0 v0 cos γ

   dx

(6.266)

for eq.(6.260). Furthermore, the forces according to eq.(6.265) act directly on the beam via eq.(6.261) yielding

Qe =

ZL o





cos γ − sin γ   dx. ρAg  −(x sin γ + vT qv cos γ)  −v sin γ

(6.267)

Interpretation: Calculating Madd ¨ yR + Gadd y˙ R yields for the last component RL T 0 0T 2 ρA{v v q (L−x)(− v˙ ox + voy ωoz ) + 21 (L2 −x2 )ωoz }v0 v0 qv }dx o RLRL v T 2 = − o x ρA(v˙ ox − voy ωoz − ωoz ξ)dξ v0 v0 dx qv . (o)

2 Comparing to eq.(6.264) reveals (dfx /dξ)kinetic = −ρA(v˙ ox − voy ωoz − ωoz ξ). Thus, the linear acceleration reaction force enters the equations via Madd while the “convective” and the “centrifugal” reaction forces are correctly taken into account by Gadd .

Equations of Motion. The only remaining forces are those from elastic potential which can easily be formulated as KV yR . Notice that (vox , voy , γ) ˙ are assumed fast motions. The equations of motion (up to the first order w.r.t. the elastic deflections) thus read [M + Madd ]¨ yR + [G + Gadd ]y˙ R + [KV + Kn ]yR = Qe

y˙ TR =

where



ZL [M + Madd ] = o

vox

1  0 ρA   −vT qv 0 0T v v qv (x−L)

voy

0 1 x v

ωoz

−vT q x x2 xv

q˙ Tv




T

(6.268)

,

qTv v0 v0 (x−L) vT xvT vvT

(6.269)

   dx, 

(6.270)

6.9

321

Conclusions ZL

[G+Gadd ] = o

0  1 ρA  x v



−1 0 vT qv T v0 v0 qv (L−x)

−x −2vT −vT q 0   ωoz dx, 0 0 2 2 T 1 0 0T v v qv (L −x ) − v v qv 0 2



(6.271) ZL [KV + Kn ] =

 diag

T

0,

0,

0,

EIz v00 v00 T +ρAg(L − x)v0 v0 cos γ

 dx,

(6.272)

o

Qe =

ZL o





cos γ − sin γ   ρAg  dx. −(x sin γ + vT qv cos γ)  −v sin γ

(6.273)

In eq.(6.270), the element (1,4) has been retained to show symmetry. This is though the corresponding forces are of second-order and do not affect motion. On the contrary, in eq.(6.271) the element (3,4) from eq.(6.261) has been canceled because it leads to negligible terms. 4

9.

Conclusions

From the general methodology (Chapter 4), the Projection Equation (4.34) has been chosen as the best suited procedure. Concerning elastic bodies, the operator D has been introduced which interrelates the describing (velocity) variables y˙ and the minimal velocity variables s˙ by y˙ = D ◦ s˙ . The describing variables y˙ are hereby extracted from the intermediate (auxiliary) variables ˙ ˙ T κ˙ T )T with the aid of the functional matrix F : y˙ = Fy. y˙ = (vTo ω To r˙ Tc ϕ T T Here, (vo ω o ) ∈ IR6 is the rigid body motion of the reference frame origin, ˙ T ) ∈ IR6 the hereto relative translational and rotational velocity of the (˙rTc ϕ mass element and κ˙ ∈ IR4 contains the (time derivative of the) curvatures, all expressed for the most general case (maximum number of d.o.f.). The dot (for time derivation) denotes that the “elastic variables” are holonomic while the rigid body variables are, in general, non-holonomic. Considering a single elastic body, there is no effort to define an operator D for the determination of the partial differential equations just by changing signs for the odd derivatives in D. Also, with successive degeneration of the derivation grade in D along with once more changing sign, the (dynamical) boundary conditions are determined. Combining single bodies into an elastic multibody system needs already more effort, mainly due to the “velocity transmission” from one body to the next. Nevertheless, the foregoing is straightforward

322 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations and leads obviously to the simplest procedure for determination of the partial differential equations along with its boundary conditions when compared to analytical methods. However, it turns out to be virtually impossible to find an analytical solution for the problem. Also, an approximative solution with Galerkin’s direct method fails due to the requirement of the shape functions to fulfill all the boundary conditions. We thus proceeded to the direct Ritz-method: Insertion of s˙ ' Φ(x)T y˙ R into y˙ leads to y˙ = D ◦ [Φ(x)T y˙ R ] = [D ◦ Φ(x)T ] y˙ R := ΨT y˙ R . The operator D (as well as the ones for the dynamical boundary conditions) are no longer needed. The procedure itself is easy to apply. The resulting equations, although clearly structured, are highly nonlinear when fast rigid body motions come into play. The question arises whether the approximation converges to the exact solution. Starting with the absence of rigid body motions, the consideration of a plate with free edges gives basic information. It is shown that: The calculated eigenfrequencies approach the exact ones from above with an existing lower bound. Thus, the series does converge. On condition that the series for the basic eigenform we (x, y) is calculated with the use of beam functions ui (x) and vj (y) [i.e., we (x, y) ' Pn Pm 3 3 i=1 j=1 xij ui (x)vj (y)], the influence of xnm decreases with 1/(m n ) and shows, how the series converges. Considering higher modes, more and more constraints come successively into play that correspond to the Rayleigh quotient where the previously calculated eigensolutions are suppressed. Thus, concerning higher modes, the series for these is more and more truncated. As a consequence, the higher the approximated mode the worse is its convergence. Ritz’s original contribution gives valuable hints on how to operate with elastic bodies. Besides the above mentioned ones it shows, by closer inspection, that it is not necessary for every used shape function to fulfill the (geometrical) boundary condition itself because only the total series has to fulfill them. This may be achieved with only few functions and leads to locally defined shape functions (for instance to the use of the aforementioned cubic splines, or, not least, to the Finite-Element-Method. According to (C.C. Zienkiewicz, 1975), the latter was established around 1950, starting intuitively, thus obviously without knowledge of Ritz’s paper). On the other hand, the plate problem shows how to take advantage of beam functions (which belong to globally defined shape functions) and its properties. Here, especially, orthogonality relations come into consideration.

6.9

Conclusions

323

These get lost when locally defined shape functions are used. But since the latter are represented by polynomials (cubic splines, Hermite polynomials, Legendre polynomials, · · ·), integration and differentiation here results simply in changing the coefficients for the corresponding power series that is best suited for numerical treatment. (One might think to use this advantage also in globally defined shape functions. One possibility would be to achieve the required independency of higher shape functions by multiplication of a basic polynomial (to start with) with the independent spatial variable x. This is, however, dangerous because then the resulting mass matrix determinant tends to zero, see Chapter 7). The question whether the already derived plate equations for free edges can be saved for other, say for instance clamped, edge conditions leads to the concept of auxiliary functions: Because Ritz’s method requires fulfillment of the geometrical boundary conditions only, it is possible to superimpose one (or few) additional shape functions which alone fulfill(s) the new arising boundary requirements. However, care must be taken with regard to the independence of the used shape functions. A typical application is the torsional shaft carrying disks at its ends. It may be for instance used as a basic element in gear dynamics. This model is used to demonstrate: how the number of requested shape functions might increase, and how useful the concept of auxiliary functions is. Next, the elastic rotor is reconsidered. Since the rotor drive equation is decoupled from the (two-directional) rotor bending equations, its result represents a rheonomic constraint [α−Ω(t)] ˙ = 0. The rotor equations are eventually transformed into a reference frame which rotates with arbitrary angular velocity ΩA . This enables us to consider multi-rotor systems as well as the single rotor for different representations. The corresponding state equations result extremely simply when the eigenfunctions of the non-rotating rotor are used for description. Considering a prolate rotor, adjusted beam functions are employed and the corresponding system matrices are equated with the trapezoid rule for integration. As pointed out, the numerical error can be kept small. However, the exact frequencies and eigenforms are unknown and convergence can only be averaged by comparing the results for an increasing number of shape functions. On the other hand, the eigenfunctions and frequencies of an oblate rotor are known and can be used to prove convergence. The spatial integration for the determination of the system matrices is solved exactly, using once more beam eigenfunctions and its orthogonality relations. Reliable results are obtained, for

324 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations the present case, with ten shape functions for the symmetric modes and forty shape functions for the antimetric modes. The antimetric modes thus represent the more critical case. It gives once more an impression of how the number of requested shape voi function might increase. But, as 1 in the case of a torsional shaft, a superimposed auxiliary function which takes an imaginary clamping at the rotor location into account improves convergence significantly. This is already indicated with only three shape functions and one auxiliary function (Fig. 6.54, thin: exact solution). 0 Stability considerations for the elastic rotor show essential dif0 0.1 0.2 0.3 0.4 x/L ferences when compared to the Figure 6.54. Antimetric Eigenfunctions 1,2 rigid case. The same as in rigid rotors, the intersections Ω = ω(+) (eigenfrequencies with positive whirl) indicate resonance. But the [A/B]Ω-curve is no longer an asymptote for the nutation frequency. Due to the nutation transfer from one frequency to the next, A/B > 1 cannot prevail from resonance as it does in the rigid case. As has been shown with the Jeffcott-Laval-Rotor, gyroscopic stabilization is (in the undamped case) achieved with Coriolis-forces. These are represented by a skew symmetric matrix assigned to the minimal velocities, see eq.(6.148 a). A basic requirement is therefore, that its determinant does not vanish. This can only be achieved with an equal number of degrees of freedom. Therefore, for an elastic rotor, the same number of eigenfunctions should be used for each of the two bending directions. Up to here, the rigid body motion does not lead to dynamical stiffening effects. The situation changes when the rotating beam is examined. Contrary to the “Helicopter Blade” with its permanent rotation, we use here the rotating beam as a (rough) basic model for an elastic robot where forearm, gripper etc. are represented by a heavy tip mass and concentrate on slewing for typical manueuvres. A special look is once more taken at the (nonlinear) G-matrix: due to the nonlinearity of the problem, the representation is not unique. Rearrangement here leads to a restoring matrix proportional to the square of the angular velocity. In a physical interpretation, this term represents part of the “centrifugal” effects. The missing part emanates from the dynamical stiffening

6.9

Conclusions

325

matrix which is calculated with the zero-order “centrifugal force” (and with the tip weight, of course). Clearly, from the viewpoint of homogeneity of representation, the total “centrifugal” effect should be gathered in one term, preferably the corresponding symmetric restoring matrix. The “centrifugal effects” are gyroscopic. Rearrangement into terms of restoring forces should therefore consequently be called “gyroscopic restoring forces”. But since in linear theory G and K were already introduced for “gyroscopic” and (conservative) “restoring” matrices without differentiating between them we will also adopt this convention for linear equations. As long as the partial differential equations had been considered, there was in the first instance no need to distinguish between kinetic reactions and impressed forces and torques for the “dynamical stiffening”. However, the situation changes for the next application example: the Elastic TT-Robot. Here, the robot arm is driven with fast longitudinal acceleration q¨A . Since qA itself represents a state variable, its acceleration needs assignment to the mass matrix which needs inversion to obtain the state equations. This fact makes reconsideration of the dynamical stiffening matrix unavoidable and leads to an additional part Madd for the mass matrix. Consequently, the nonlinear velocity dependent accelerations are assigned to the (nonlinear) G-matrix with Gadd . One is then left with pure impressed forces and torques for the determination of the dynamical stiffening matrix Kn . The TT-Robot represents a basic model for a telescoping arm. One possibility thereby is to model the ball bearing spindle with a certain number of translational springs as the inspection of the bearing intuitively suggests. A simplification is obtained when the translational springs are replaced by one central spring with a superimposed torsional spring. Such a simplification will at least hold if the geometrical gear extension is not too large. Large extension along with high stiffness may be solved the following way: when seen from the bearing fixed reference system, left and right beam ends are ξL = −(qA + b − qAo ) and (L − ξL ), respectively, see Fig. 6.45. Here, L is the total arm length. The R R R required integrals then split into oL → oξL (t) + oL−ξL (t) where left and right (clamped-free) beam segments enter the calculation separately. For the partial differential equations there was a basic need to include the ˙ This led to the replenishment of curvatures for intermediate variables y˙ = Fy. dM etc. with corresponding zeros. In a logical continuation, we inserted the Ritz ansatz into the vector of describing variables, y˙ = [D ◦ Φ(x)T ] y˙ R := ΨT y˙ R emphasizing that the curvatures do not contribute when calculating M etc. That means, however, that the curvature components in Ψ need not be calculated explicitly. Thus, for the direct Ritz procedure, there is no need to consider curvatures within ΨT and FΨT , respectively.

326 6 Elastic Multibody Systems – The Subsystem Ordinary Differential Equations The zeros in dM etc. are therefore once more eliminated. Within the Ritz procedure, all the effects which curvatures can generate are completely taken into account with Madd , Gadd and Kn , except potential forces. The latter, however, are directly computed from the elastic potential without any effort. The resulting F[D ◦ Φ(x)T ] = FΨT looks very simple (see eq.(6.215) for instance). The result is even easier to obtain when, as has been done with the dynamical stiffening matrix, at first the general case F = E is considered from which the actual FΨT is directly obtained by canceling the rows that refer to the nonconsidered motions. The proposed procedure for the treatment of ordinary differential equations is straightforward. Therefore, after all, one may pose the question: what have the partial differential equations been good for? The answer is, of course, manifold. At first, one should not forget that they represent the minimal form of motion equations. Spillover effects as mentioned on page 286 will not arise in a minimal description. Moreover, as the shaft-disk example showed, the number of required shape functions may increase significantly when using the Ritz approach. One should also have in mind that the applied beam functions for shape functions have been obtained from the corresponding PDE’s, of course. Thus, wherever possible, one should take advantage of the partial differential equations. One application additionally to be mentioned is, for instance, piezo layer design in laminated beams or plates (C. Zehetner, 2005). Considering elastic multibody systems, however, there is obviously no way to directly obtain a solution from its coupled ordinary and partial differential equations along with the corresponding boundary conditions.

Chapter 7 ELASTIC MULTIBODY SYSTEMS – ORDINARY DIFFERENTIAL EQUATIONS

1. Summary Procedure 1.1. Rigid Multibody Systems Within the scope of modeling, the element dm of an elastic body (like dm = ρAdx for a beam element for instance) is considered rigid. This suggests that we first look at a rigid multibody system. After a comparison of methods (Chapter 4), The Projection Equation "   # N  X ∂vci T ∂ω ci T i=1

∂ s˙

∂ s˙

e IR p − fe p˙ + ω e IR L − Me L˙ + ω

!

= 0, s˙ = H(q)q˙

(7.1 a)

i

[eq.(4.34)] has been selected as the most suitable procedure. Our main motivation for this choice is not only the fact that analytical methods need an enormous effort and may lead to misinterpretations, but also that non-holonomic minimal velocities s˙ (which represent a linear combination of the time derivatives of the minimal, or generalized, coordinates q) may be used directly in eq.(7.1 a). This gives also direct access to non-holonomic constraints. All vectors are represented in a suitable reference frame R. Consequently, the functional matrix (∂ω ci /∂ s˙ ) is equated with the absolute angular velocity of the body under consideration, while the “convective” accelerations contain only the (absolute) angular velocity ω IR of the chosen reference frame R. In most applications it is useful to compose a mechanical system of Nsub subsystems. Let the nth subsystem thereby consist of Nn bodies. One obtains then, by sum splitting, N sub  X n=1

 !T T X Nn  ∂v c   ∂ s˙ ∂ y˙ n i=1

∂ y˙ n

∂ω c

!T  "

∂ y˙ n 327



e IR p − fe p˙ + ω e IR L − Me L˙ + ω

#  

=0 i

(7.2 a)

328

7 Elastic Multibody Systems – Ordinary Differential Equations

where the second sum, from 1 to Nn , defines the nth subsystem which goes along with the (one and the same) vector of describing (velocity) variables y˙ n [eq.(4.50)]. Specifying the (not necessarily body-fixed) reference frame origin with O [(vTIR ω TIR ) → (vTo ω To )] yields the mass center velocities and the momenta vc ωc

!

"

= i

T where y˙ i =

p L

E erTc 0 E

!





= i

vo T

E 0 0 E

#

y˙ i := Fi y˙ i i

ωoT

r˙ Tc



vc ωc

mE 0 0 Jc

i



ω Tr

i

,

(7.3 a)

!

:= Mi Fi y˙ i . i

y˙ i is composed with the frame origin velocities (“guidance velocities”) and the superimposed relative velocities of the ith mass center [see eq.(4.52)]. Applying once more the chain rule of differentiation then yields the dynamics of the ith body itself for the general case. It is characterized by the braced term in the following equation: Nn X

∂ y˙ i

i=1

∂ y˙ n |

!T    

{z

}

T Fi

∂vc ∂ y˙

!T

|

∂ω c ∂ y˙

!T  " i

{z

e o p − fe p˙ + ω e o L − Me L˙ + ω

#   i

. (7.4 a)



}

T Fi

Insertion of eq.(7.3 a) leads eq.(7.4 a) to Nn X

T

h

i

Fi Mi ¨yi + Gi y˙ i − Qi ,





T

with Mi = F M F

etc.

(7.5 a)

i

i=1

where M is a simple (block-) diagonal matrix, see eq.(7.3 a). Next, the describing variables for the nth subsystem are chosen [eq.(4.74)]: y˙ i =

∂yi ∂yn

!

y˙ n = Fi y˙ n .

(7.6 a)

Eq.(7.5 a) then becomes Mn ¨yn + Gn y˙ n − Qn ,

with Mn =

Nn h X i=1

T

i

F MF

i

etc.

(7.7 a)

7.1

329

Summary Procedure

[eq.(4.75)]. The final step is to eventually combine the subsystems into a single entity via eq.(7.2 a): N sub  X

∂ y˙ n

T

∂ s˙

n=1

[M¨y + Gy˙ − Q]n = 0.

(7.8 a)

The subsystem matrices read explicitly 

Mn =

Nn X

T Fi

i=1

mE merTc  e Jo  mrc    mE

merTc Jc

0



mE 0  merc Jc 

 Fi , Q = n  0 

mE 0 Jc

Nn X

T T Fi Fi

i=1

Nn X i=1

fe Me



, i

i

eo eo e oe 2mω 0 mω mω rTc c c   o c ˙ e oe e o J + J 2me eo ω e o J + J˙   mω rc ω rc ω 

Gn =





T Fi  

eo  mω

0

e oe mω rTc c e o Jc + J˙ ω

eo 2mω 0

 Fi . 

0  c c ˙ e oJ + J ω i

(7.9 a)

[eq.(4.57), eq.(4.59)]. (The tensor of moments of inertia Jc refers to the mass center, Jo = Jc + mercerTc refers to the frame origin. The impressed forces and torques fe , Me apply to the mass center. All terms are represented in the reference frame.) The further treatment of eq.(7.8 a) is then to formulate the equations of motion either in minimal form according to eq.(4.99) or, preferably for multibody systems, in a recursive representation [see eq.(4.108) et seq.]

1.2.

Elastic Multibody Systems

Considering pure elastic multibody systems, the Projection Equation is obtained by splitting the overall sum in eq.(7.1 a) into a sum over the elastic bodies which is followed by an integration, N sub X

Z "

n=1 B

∂vcn ∂ s˙

T

∂ω cn

T #

∂ s˙

e IR dp − dfe dp˙ + ω ˙ e IR dL − dMe dL + ω

n

!

˙ = 0, s˙ = H(q)q. n

(7.1 b) Here, the elastic body acts as a subsystem itself. It is, once more, suitable to assign describing velocity variables y˙ n to each elastic body via N sub X

  !T Z   T ∂ y˙ ∂v c   ∂ s˙ ∂ y˙

n=1 B

n

!T  " # e IR dp − dfe  ˙ d p + ω  = 0. e IR dL − dMe  dL˙ + ω ∂ y˙

∂ω c

n

(7.2 b)

330

7 Elastic Multibody Systems – Ordinary Differential Equations

The reference frame is specified as the coordinate system of the undeformed subsystem moving with (vTo ω To ). One obtains then the mass center velocities of dm and its momenta vc ωc

!

"

=

T where y˙ =

dp dL

!



=

E erTc 0 E

E 0 0 E

#



ωoT

r˙ Tc

vo T



dmE 0 0 dJc

y˙ := F y˙ ˙T ϕ vc ωc



,

(7.3 b)

!

˙ := dM Fy.

y˙ is composed with the frame origin velocities (“guidance velocities”) and the ˙ instead of superimposed relative velocities of the mass center of dm. Here, ϕ the general expression ω r is used to emphasize that the relative velocities due to elastic deformation are integrable. Since for an elastic body the deflection of every dm is characterized by its deformation function rc (x, y, z, t) [instead of different rci for different mi which belong to one subsystem n], index i is dropped. The actual subsystem is indicated by index n. Applying the chain rule along with eq.(7.3 b) yields for the nth subsystem (general case) Z 

∂ y˙ n

T

∂ s˙

Bn



∂ y˙  ∂ y˙

!T    

| {z }

F

T

∂vc ∂ y˙

!T

∂ω c ∂ y˙

{z

|

F

!T  "

#  e o dp − df dp˙ + ω . e ˙ e o dL − dM  dL + ω e

n

}

T

(7.4 b)

Insertion of eq.(7.3 b) leads eq.(7.4 b) to Z  Bn

∂ y˙ n ∂ s˙

T

T Fn

h



i

dMy¨ + dGy˙ − dQ

n

,

T



with dM = F dMn F

etc. n

(7.5 b) Next, the describing variables y˙ n are chosen for the nth subsystem, specializing the actual problem: y˙ =

∂y ∂yn

!

y˙ n = Fn y˙ n .

(7.6 b)

7.1

331

Summary Procedure

Eq.(7.5 b) then becomes Z  Bn

∂ y˙ n

T

[dM¨y + dGy˙ − dQ]n ,

∂ s˙

h

T

i

with dMn = F dM F

etc.

n

(7.7 b) [eq.(5.37)]. Combining the subsystems to an entity via eq.(7.2 b) yields N sub X

Z 

n=1 B

∂ y˙ n ∂ s˙

T

[dM¨y + dGy˙ − dQ]n .

(7.8 b)

n

The in-between calculation steps have been identical to the rigid body case except that the momenta now refer to the mass element dm instead of the individual mass mi . The matrices dM etc. for eq.(7.8 b) are thus directly obtained from eq.(7.9 a): dmE dmerTc  dme rc dJo 

dmE 0 dmerc dJc  

dmerTc dJc

dmE 0  0 dJc

 T

dM = F  

 dmE

0

 T

 F, dQ = F F 

T

dfe dMe

!

,

eo e oe eo dmω dmω rTc 2dmω 0 c c  dme o c ˙ e e e e e r ω ω dJ + d J 2dm r ω ω dJ + dJ˙    c o o c o o 



T dG = F  

eo  dmω

0

e oe dmω rTc c e o dJc + dJ˙ ω

eo 2dmω 0

 F, 

0  c c e o dJ + dJ˙ ω

(7.9 b) [eq.(5.38) et seq.] for every n (index dropped). The adjustment to the actual elastic body by means of F here corresponds to eliminate those rows and columns that refer to non-contributing deflection variables. The rigid body variables remain unreduced in view of the recursive representation (Gauss-form). The tensor of moments of inertia dJc refers to the mass center while dJo = dJc + dmercerTc refers to the frame origin. The impressed forces and torques dfe , dMe apply to the (mass center of) infinitesimal element dm. All terms are represented in the reference frame (of the undeformed state). The deformation functions are defined as spatial functions w.r.t. this reference frame. Recall: In case of beam-like structures the tensor of moments of inertia reads ρ I (ρ: mass density, I: tensor of area moments of inertia) when represented in an element-fixed (local) frame. One obtains thus, with ∂( )/∂x = ( )0 ,

332

7 Elastic Multibody Systems – Ordinary Differential Equations 

rc







x ϑ(x, t) O(1) e ⇒ =  v(x, t)  , ϕ =  −w0 (x, t)  , ARE = [E − ϕ] w(x, t) v 0 (x, t)

O(1)

Ix (Ix − Iy )v 0 (Ix − Iz )w0 0 Iy (Iy − Iz )ϑ  dx = ρ  (Ix − Iy )v (Ix − Iz )w0 (Iy − Iz )ϑ Iz (7.10) 

dJc = ARE ρI ATRE



[eq.(5.46)]. Comparing eq.(7.8 a) with eq.(7.8 b) (and having in mind that the integral lim P n represents the limiting process Nn →∞ N i=1 ) would yieldRthe same result if the Jacobian (∂ y˙ n /∂ s˙ ) could be extracted from the integral Bn . However, this is not possible for two reasons: y˙ n as well as s˙ depend on the (independent) spatial variables the integration domain is defined with. The minimal velocity s˙ contains the (time derivatives of the) deformation functions, and the describing (velocity) variable y˙ n contains spatial derivatives of s˙ . There is thus no way to formulate the Jacobian (∂ y˙ n /∂ s˙ ) explicitly. The second point leads to the necessity to augment y˙ with the curvatures and to formulate y˙ in terms of spatial operators applied to the minimal velocity, y˙ = D ◦ s˙ ∀ n. The virtual work [corresponding to eq.(7.8 b)] then leads, with an integration by parts, to an almost simple procedure to obtain the partial differential equations along with the associated boundary conditions (Chapter 5). For the treatment of elastic multibody systems, however, there is obviously no way to proceed to an (exact, analytical) solution. We will therefore use the direct Ritz approach for approximative solution: y˙ is separated into the rigid body part y˙ rb and the deformation part y˙ el . The latter is composed with the actually needed components for elastic translation and rotation. These are expressed with the spatial operator: y˙ el = D ◦ s˙ el (here, compared to Chapter 5, the curvatures are omitted). Next, the series expansion ˙ One obtains thus sel ' Φ(x)T q is inserted yielding y˙ el = [D ◦ Φ(x)T ]q. 

y˙ =

y˙ rb y˙ el





=

E 0 0 D ◦ Φ(x)T



y˙ rb q˙



:= Ψ(x)T y˙ R .

(7.11)

In Chapter 6, the operator was defined w.r.t. y˙ as a whole to simplify notation. In the present case, it is reduced to pure elastic components in order to separate the rigid body coordinates from the elastic deflections. The same was done in Chapter 5 considering the elastic double pendulum. Contrary to eq.(5.205),

7.1

333

Summary Procedure

however, here the rigid body part is not reduced to a minimal representation via y˙ rb = Frb s˙ rb in order to provide the recursive representation. From eq.(7.11) one obtains the Jacobian 

∂ y˙ ∂ s˙

!

∂ y˙



=

∂ y˙ R

∂ y˙ R



T



= Ψ(x)

∂ s˙

∂ y˙ R



(7.12)

∂ s˙

for eq.(7.8 b) where (∂ y˙ R /∂ s˙ )T can be extracted from the integral. This is because the minimal velocities which are now used are composed of the rigid minimal velocities and the (time derivative of the) Ritz coefficients q. These do, of course, not depend on spatial variables. Insertion of ¨y = Ψ(x)T ¨yR and y˙ = Ψ(x)T y˙ R yields, for eq.(7.8 b), N sub  X

∂ y˙ Rn

T

[M¨yR + Gy˙ R − QR ]n = 0

∂ s˙

n=1

(7.13)

[compare eq.(7.8 a)] where the subsystem matrices read dmE dmerTc  dme rc dJo 

dmE 0 dmerc dJc  

dmerTc dJc

dmE 0  0 dJc

 Z

M=

[FΨT ]T  

 dmE

0

  [FΨT ], 

eo e oe eo dmω dmω rTc 2dmω 0 c c  dme o c ˙ e e e e e rc ω o ω o dJ + dJ 2dmrc ω o ω o dJ + dJ˙    

Z

G=



[FΨT ]T  

eo  dmω

0

e oe dmω rTc c e o dJc + dJ˙ ω

Z

Q=

T T

[FΨ ] F

eo 2dmω 0

T

dfe dMe

 [FΨT ], 

0  c c e o dJ + dJ˙ ω

!

(7.14)

with F according to eq.(7.3 b). Along with the kinematic chain y˙ Rn = Tnp y˙ Rp + Fn s˙ n , eq.(7.13) takes the form

(7.15)

334

7 Elastic Multibody Systems – Ordinary Differential Equations

FT1 · · · · · · · · · · · · FT1 TTN 1 M1 ¨yR1 + G1 y˙ R1 − QeR1    .. .. ..    . . .     e T T T  Fi · · · · · · Fi TN i   Mi ¨yRi + Gi y˙ Ri − QRi       = 0, .. .. ..    .    . .    .. ..    ..    . . . MN ¨yRN + GN y˙ RN − QeRN FTN (7.16) (Nsub abbreviated N ) see eq.(4.98) et seq. 





Interpretation T ˙ T ). The vector of auxiliary variables reads, with eq.(7.3 b), y˙ = (vo T ω o T r˙ Tc ϕ From eq.(7.6 b) weh havei y˙ = Fy˙ (index dropped), and from eq.(7.11) y˙ = ΨT y˙ R , hence y˙ = FΨT y˙ R . If one inserts the Ritz approximation into rc , ϕ,

then these become functions of q : rc (q), ϕ(q). Along with y˙ TR = (vTo ω To q˙ T ) we get  vo E   ˙y =  ω o  =   r˙ c   0 ˙ ϕ 



0





vo h i   ωo  T y˙ R . = FΨ    ∂rc /∂q q˙ ∂ϕ/∂q

(7.17)

T

Clearly, since F has been introduced such that F F = E [see eq.(5.20)], one T obtains eq.(7.11) from eq.(7.17) by premultiplication with F . The difference in notation is that for eq.(7.11) the non-contributing elastic variables are a` -priori suppressed with D and the Ritz series is inserted then, while in eq.(7.17) the series refers to rc ∈ IR3 , ϕ ∈ IR3 where the system variables are reduced later, with F. In order to derive the partial differential equations for sel it was necessary to formulate the angular deflections and the curvatures in terms of sel which had been achieved with the operator D (Chapter 5). Once the partial differential equations are available, it is but consequential to insert the series approach sel ' Φ(x)T q(t) at this step (Chapter 6). For the weak formulation of elastic multibody systems, however, we prefer the notation according to eq.(7.17) for two reasons: Eq.(7.17) yields [FΨT ] as a whole which may be applied directly to eq.(7.14) [instead of first calculating eq.(7.9 b) and then inserting the series approach]. The system matrices are obtained with a first-order displacement field according to eq.(7.10).

7.2

335

Mixed Rigid-Elastic Multibody Systems

However, as eq.(7.16) shows, the subsystem dynamics may contain zeroorder terms w.r.t. the elastic deflections. Therefore, for correct linearization, Tip in eq.(7.15) needs a first-order representation. The kinematic chain for y˙ R according to eq.(7.15) is obtained in the following way: the velocities at the coupling point (index e) of the predecessor subsystem (index p) are composed of its guidance velocities vop , ω op and the superimposed relative velocities from elastic deflection, (∂rc /∂q) |e q˙ p , (∂ϕ/∂q) |e q˙ p . For the successor i, the coupling point velocities are transformed into the actual frame with Aip which then represent the guidance motion for i. Replenished with q˙ i and superimposed with the relative velocities which the ith frame contributes yields

 #   " Aip 0 E erTc |e (∂rc /∂q) |e          vo 0 Aip 0 E (∂ϕ/∂q) |e vo vrel  p   ω o  + ω rel  . ω o  =      q˙ q˙ i q˙ i   p

0

{z

|

}

Tip

(7.18) Here, for correct linearization, (∂rc /∂q) and (∂ϕ/∂q) need a first-order development and, therefore, rc and ϕ need a second-order representation. The corresponding terms for a beam are given in Chapter 5:

Rx



(v 0 2

w0 2 )/2 dξ



Rx



+  x −o   ϑ +o    x  O(2)  0 R O(2)  0 0 0   rc =  v+ [(x − ξ)ϑ w −ϑ w ] dξ , ϕ =  −w o    x    0 R 0 0 0 w − [(x − ξ)ϑ v −ϑ v ] dξ

(v 0 w00

v

o

−

w0 v 00 )/2 dξ

Rx

+ ϑ0 v 0 dξ o

Rx

+ ϑ0 w0 dξ

    ,   

o

(7.19) see eq.(5.147).

2.

Mixed Rigid-Elastic Multibody Systems

A subsystem that consists of Nn rigid bodies and of Nel elastic bodies is simply obtained by projecting the participating partial systems into one and the same describing space which is characterized by y˙ m :

336 ∂ y˙ n

7 Elastic Multibody Systems – Ordinary Differential Equations !T

[M¨y + Gy˙ − Q]rb,n +

∂ y˙ m

Nel X

∂ y˙ Rk

k=1

∂ y˙ m

!T

[M¨yR + Gy˙ R − QR ]el,k = 0. (7.20)

Here, index rb characterizes the subsystem matrices from eq.(7.8 a) which already contain the Nn rigid bodies and index el stands for the elastic body from eq.(7.13). Along with y˙ n = (∂yn /∂ym )y˙ m and y˙ Rk = (∂yRk /∂ym )y˙ m one obtains the subsystem matrices 

Mm

∂ y˙ n = ∂ y˙ m 

∂ y˙ n ∂ y˙ m

Gm =  

∂ y˙ n ∂ y˙ m

Qm = 

!T

∂ y˙ n ∂ y˙ m

Mrb,n

!T

Grb,n

∂ y˙ n ∂ y˙ m

!T

Qrb,n +

N el P k=1

!

+

N el P k=1

!

+

N el P k=1

∂ y˙ Rk ∂ y˙ m

∂ y˙ Rk ∂ y˙ m ∂ y˙ Rk ∂ y˙ m

!

!T

Mel,k

!T

Gel,k

∂ y˙ Rk  , ∂ y˙ m ! ∂ y˙ Rk  ,

∂ y˙ m



!T

Qel,k  , (7.21)

yielding for the total system N sub  X m=1

∂ y˙ m

T

[M¨y + Gy˙ − Q]m = 0.

∂ s˙

(7.22)

Transformation to y˙ m means to generate “place holders” for the neighboring systems.

Remarks The subsystem matrices (7.14) were generated starting with the most general case which was then step by step reduced to the actual problem under consideration (Chapters 4 to 6), yielding Z

M=

[FΨT ]T dM [FΨT ], etc,

(7.23)

T

where dM = F dM F [see eqs.(7.3 b), (7.5 b)]. dM refers to the mass center velocities. The mass center velocities are eventually projected into the configuration space, indicated by y˙ R (Ritz), hence T

F FΨ =



(∂vc /∂ y˙ R ) (∂ω c /∂ y˙ R )



:= FR ⇒ M =

Z

FTR dM FR , etc.

(7.24)

7.2

Mixed Rigid-Elastic Multibody Systems

337

For very simple cases, FR is directly calculated (e.g. for plane motion, section 4). The Projection Equation yields, along with a Ritz approach, the same equation structure for the resulting (ordinary) differential equations as in the rigid body case. This holds for minimal representation as well as for the recursive calculation which combines the subsystems into a single system. From the physical background, however, there are two main differences to pay attention to: The deflection amplitudes are, for technical applications, assumed small. This does, of course, in general not hold for the rigid body coordinates. A linearization can therefore only be considered in part, leading to the so-called “dynamical stiffening effects”. The corresponding (generalized) forces are taken into account within QeR = QeR,DynStiff + QeR,rest in eq.(7.16) where QeR,DynStiff = −Kn yR . The “dynamical stiffening matrix” Kn is derived from eqs.(6.188 a), p.291. The dynamical stiffening effects refer to zero-order forces and torques w.r.t. the elastic deviations and may consist of dynamical reactions as well as of impressed forces/torques. Rearranging the dynamical reaction forces yields Kn yR = Madd ¨yR + Gadd y˙ R + Kn,rest yR where Kn,rest yR refers to the impressed forces, see Chapter 6, sections 6.7 and 6.8. In fast-moving systems, this rearrangement is necessary since the total subsystem mass matrix needs inversion. In slowly-moving systems, Madd ¨ yR + Gadd y˙ R may be neglected and one is left with the impressed forces only. In assembling the subsystems we also need to consider for Tip to take the interconnections correctly into account. The Ritz approach always goes along with a series truncation which primarily affects the accuracy of the achievable results. Clearly, what we are investigating is always an (adequate) abstraction of the real physical world. This fact, however, should not lead to negligence in calculation. The validity of an early stage series truncation might be ensured in the presence of damping. Damping, however, has to be considered carefully: Passive damping is usually helpful, except the so-called inner damping in rotors. Active damping may be inserted with control, but a control input may also lead to instability via the so-called “spillover”-effects.

338

7 Elastic Multibody Systems – Ordinary Differential Equations

3.

Applications

3.1.

Prismatic Joint – The Telescoping Arm

'$

&% arm

x,vox ,ξ

rdm

y,voy

I @ @ xAo

@

η



γA

@ γM

@ @

@ I @

rotor & gear

@ @

ro

Figure 7.1.

stator

Telescoping Elastic Arm

ωF

The same as in Chapter 4, section 4.2.3, the arm is driven with r˙ o = ∆xe ˙ 1 where ∆x = xA − xAo . However, in contrary to the rigid body case where the arm moves in the x-direction only, the elastic arm now undergoes bending in two directions and torsion. Since rc denotes the mass element center w.r.t. the joint-fixed reference frame, we introduce a new character r dm for the elastic deflection w.r.t. the moving frame (undeformed beam), yielding rc = r dm + ∆xe1 , ∆x = xA − xAo , rTdm = [ξ, v(ξ, t), w(ξ, t)], (7.25) see also Fig. 7.3

The rotation axis of motor and gear output are uni-axial with e3 while the guidance angular velocity which the stator is rotated with is, in general, a spatial motion. Thus, using ω o = ω F + γ˙ A e3 for the arm rotation (reference frame) as well as ω M = ω F + γ˙ M e3 for gear output (see Fig. 7.1) separates the spatial part and the uni-axial part of rotations and simultaneously avoids overlapping effects between the individual subsystems (Chapter 4, section 4.2).

7.3

339

Applications

3.1.1 On Mass Distribution: Tip Body Influence If one looks at the arm from Fig. 7.1 as a predecessor of a subsequent subsystem, then the stator of the successor motor represents a tip body for the predecessor. Let the tip body mass center be located at distance s. The corresponding position vector then reads e r tip = rc (L) + ϕ(L)e 1s (7.26) where L denotes the beam length. (The tip body mass center need not be situated on the extended beam axis as depicted in the sketch although representing the most frequent case).

s

 x,vox

@ I @ y,voy @

ω

@ @ @

Figure 7.2.

q

Extended Shape Function (schematic)

Since r tip has constant values w.r.t. the spatial coordinate x, the tip body is easily taken into account by replacing dm and dJ with δ (x−L−s), dm = ρA + m tip− − dJ = dJ arm + J tip δ (x−L−s).

(7.27)

The x-coordinate is thereby elongated to x ∈ {0, L+s} having in mind that ρA and dJarm are zero for x ∈ {L, L+s}. The same considerations hold for an additional body attached to the origin. The influence of additional bodies at the beam ends is thereby characterized by the corresponding beam boundary conditions. Of course, the tip body also influences the dynamical stiffening. That part of it which belongs to the inertia reactions is discussed in Chapter 6, sections 7 and 8. As eqs. (6.242) and (6.251) show, there arise four basic integrals for an additional tip body. These are listed in Table 7.1 for a uniform homogeneous beam (ρA = const).

340

7 Elastic Multibody Systems – Ordinary Differential Equations

Table 7.1. On the Tip Body Influence: Four Characteristic Integrals L+s L+s Z Z 

 T ρA + m tip−δ(ξ−L−s) dξv0 v0 dx =

o

x

ZL

T

T

ρA(L−x) + m tip v0 v0 dx + m tip v0e v0e s





(7.28)

o L+s L+s

ZL h

Z

Z

o

x

−(ξ−L−s) dξv0 v0 dx = ρAξ + m tip ξδ





T

1 T T ρA(L2 −x2 ) + m tip (L+s) v0 v0 dx + m tip v0e v0e (L+s)s 2

i

(7.29)

o L+s L+s Z Z 

−(ξ−L−s) dξv00 ϑT dx = ρA(ξ −x) + m tip (ξ−x)δ

ZL h

o

x

1 1 ρA(L−x)2 + m tip (L+s−x) v00 ϑT dx + m tip v00e ϑe T s2 2 2

i

(7.30)

o L+s L+s Z Z 

−(ξ−L−s) dξv00 ϑT dx = ρAξ(ξ−x) + m tip ξ(ξ−x)δ

ZL 

o

1 ρA 2



3

x

L2 x x3 L − + 3 2 6



+ m tip (L+s)(L+s−x) v00 ϑT dx +



1 m v00e ϑe T (L+s)s2 2 tip

o

(7.31)

3.1.2 Subsystem Equations Considering extremely fast-moving systems, the additional mass matrix and the additional gyroscopic matrix from Chapter 6 are to be replenished with the tip body part. However, in most cases, such as in elastic robots for instance where the rigid body accelerations are either slow or act for a short time only, the inertia reaction forces within the dynamical stiffening are of minor influence and one is left with the corresponding impressed forces only.

7.3

341

Applications

The describing coordinates for the subsystem “Telescoping Arm” are chosen as x

y˙ R = (vTo , ω TF , ωM r , ΩAr , q˙ T , x˙ A )T

(7.32) with the relative angular velocities ΩAr = γ˙ A (arm) and ωM r = γ˙ M (motor-gear output). The sequence of coordinates in eq.(7.32) is, of course, arbitrary. Adjusting y˙ to y˙ R for the problem under consideration yields, with eq.(7.3 b) and T Fel = E (spatial motion),

 

0 0

rdm rc

@ I @ @ @ @ @ 

∆xe1

Figure 7.3.

E 0 0 0 0 E 0 e3 0 0 0 0

vo   ωo   = ˙y arm =   r˙ c     ˙ ϕ 0 

y

0

0 0 ∂r ∂q ∂ϕ ∂q

Definition of Coordinates

0 0 e1 0





       

vo ωF ωM r ΩAr q˙ x˙ A

    h i  y˙ R .  := FΨT  arm  

(7.33) From eq.(7.25) follows r˙ c = (∂rdm /∂q)q˙ + x˙ A e1 . Index dm is suppressed in eq.(7.33) for simplicity. The motor-gear-unit is obtained from eq.(7.9 a) with

  E vo 0  ωo   = y˙ mot =   0  r˙ c   ωr 0 



0 E 0

0 0 0

0 e3 0

0 0    0 0    0 0  

0 iG e3 −e3 0 0



 

vo ωF ωM r ΩAr q˙ x˙ A

    h i  y˙ R  := F  mot  

(7.34) where Nn = 1 and rc = 0 [compare eq.(4.86) for ΩM r = iG ωM r where iG : gear transmission ratio]. Along with dmM := mM− δ (x−0), dJM := JM− δ (x−0), one obtains

342

7 Elastic Multibody Systems – Ordinary Differential Equations   (dm  +dmM )E

           dM =             

 T rc dm

0

dJo +dJM

dCM iG e3

dJo e3

dCM i2G

0

e

T rc e3 dm

e

dC o

symmetric

∂r dm ∂q

e1 dm

    ∂r  erc ∂q dm  e rc e1 dm  ∂ϕ  +dJc ∂q    0 0   , T ∂r  e3 (e rc dm ∂q rc e1 dm  eT3 e c ∂ϕ  +dJ )  ∂q   T    ∂r dm ∂r  T  ∂q ∂q  ∂r  T   ∂q e1 dm  ∂ϕ ∂ϕ  c + dJ  ∂q ∂q   dm

(7.35)

 (dm  ω e oerTc dm 0 ω e oerTc e3 dm  +dm )ω M eo     c ω e o (dJo +dJM )  erc ω e o dm ω e o dCM iG e3 (ω e o dJo + dJ˙ )e3 c  ˙ +dJ     0 eT3 ω e o dJM iG 0 0  dG = ···   T c T o T o  e3 erc ω e o dm e3 (ω e o dJ +dJ˙ ) 0 e3 ω e o dJ e3    T  T   T ∂r ω ∂r ω  T e e r dm e oerTc dme3 o c  ∂r ω ∂q 0  ∂q T  ∂q e o dm  T c c ∂ϕ ∂ϕ  (ω e o dJc + dJ˙ ) + (ω e o dJc + dJ˙ )e3 +  ∂q ∂q   eT1 ω e o dm eT1 ω e oerTc dm 0 eT1 ω e oerTc e3 dm

7.3

343

Applications

 ∂r dm e o ∂q 2ω

e o e1 dm 2ω

      2e rc ω e o e1 dm       0     T 2e3 e rc ω e o e1 dm        T  ∂r  2 ω e o e1 dm  ∂q    

∂r dm 2e rc ω e o ∂q c e o dJc + dJ˙ ) ∂ϕ +(ω ∂q 0 ···

∂r dm 2eT3 e rc ω e o ∂q c e o dJc + dJ˙ ) ∂ϕ +eT3 (ω ∂q 2 +



∂ϕ ∂q

T



∂r ∂q

T

∂r dm ω e o ∂q c (ω e o dJc + dJ˙ ) ∂ϕ ∂q

∂r dm e o ∂q 2eT1 ω

0

(7.36) where ω o = ω F + ΩAr e3 . The same as in eq.(6.204) et seq., the ball bearing spindle is replaced with a center spring which holds the beam in its position. This position is characterized by 

rbearing







0 ϑ , ϕbearing =  −w0  = v  w ξbearing v0 ξbearing

(7.37)

where ξbearing = −∆x = xAo −xA . Along with the rotational spring coefficients KR = diag{kϑ , kR , kR } and the translational spring coefficients KT = kT E one obtains the bearing restoring matrix Kbearing = "

blockdiag{0, 0, 0, 0,

∂r ∂q

T



KT

∂r ∂q





+

∂ϕ ∂q

T



KR

∂ϕ ∂q

#

, 0} ξbearing

(7.38) while the gear output restoring matrix (spring coefficient kG ) reads 

Kgear = blockdiag{0, 0, kG

1 −1 −1 1



, 0, 0}.

(7.39)

Let the gravity force act in negative e2 -direction. One has then the zero-order force which acts on the mass element

344

7 Elastic Multibody Systems – Ordinary Differential Equations

df(o) = −AiI e2 (dm + dmM )g

(7.40)

where AiI transforms the inertial gravity into the actual frame i. Leaving the inertia reaction forces aside, the dynamical stiffening matrix is obtained from eq.(6.188 a) with eq.(7.40) respecting the tip mass influence according to eqs.(7.28) to (7.31). The generalized gravity force dQgrav follows from eq.(7.40) with eq.(7.33) and F from eq.(7.3 b): 

h

dQgrav = − ΨF

T

i arm

F

T

df(o) 0

!

     = −    

(dm + dmM )E e rc dm 0 eT3 erc dm 

T

∂r dm ∂q dmeT1

       AiI e2 g.    

(7.41) The subsystem matrices (7.35) and (7.36) take the inertia forces and torques, resp., into account. The impressed forces are composed with the bearing and the gear springs, eq.(7.37) and eq.(7.39). They are furthermore completed with the dynamical stiffening Kn yR , see eq.(7.40) et seq. Along with the gravity forces (7.41) and the input torque from eq.(7.34) which is to be replenished with farm for the translational beam motion one obtains [M¨yR + Gy˙ R − Qe − Qinput ]i Z

Mi =

Z

dMi , Gi = Bi

Qei

dGi , Bi

= −(Kbearing + Kgear + Kn )i yRi +

Z Bi

dQgrav,i ,

Qinput,i = (0, 0, iG MM , 0, 0, farm )Ti ,

(7.42)

index i added. The subsystem equations are composed with a first-order Ritz-expansion 0  T = v 0 



∂r ∂q



0 0 wT

0 0   ∂ϕ   0 , = 0 ∂q v0 T 0 



0 −w0 T 0

ϑT qv  0  , q =  qw  . qϑ 0 (7.43) 





7.3

345

Applications

It is obviously not recommendable to fully expand the equations by hand. This is a typical task for the computer.

3.1.3 The Kinematic Chain Once the subsystems are at hand, the next task is to derive the kinematic chain. To this end we consider first the mass center velocity of an element dm which reads, with eq.(7.3 b), 

vc ωc







PP PP Pq   s   yi 

x,vox ,ξ

rdm

y,voy



I @ @ xAo η



γA

@ @ γM

@ @

@ I @ @



  ∂r  vo ∂q     ωo    ∂ϕ 0 E q˙ ∂q (7.45) when the Ritz-ansatz is inserted.

rTc E e

actual frame i

= 

γ Mi

iP xiP P



vo T   e E r E 0 c   ωo   r˙ c  0 E 0 E ˙ ϕ (7.44) and which becomes 

γAi

ωF

@

ro predecessor (index p suppressed)

Figure 7.4.

Two Contiguous Subsystems

If one replaces rc with rpi which connects the origins of frame p (predecessor) and frame i (actual frame), one obtains the guidance velocities of frame i but resolved in frame p. Adjustment to the vector of describing variables yields 



vo ω   F  ∂r e   rTpi 0 erTpi e3 ∂q  1  ω E e vo  Mr      =   Ω  ∂ϕ 0   p ωF i  Ar  0 E 0 e3 ∂q ˙  pe  q x˙ A p 







(7.46)

where, from eq.(7.26) and eq.(7.25), rpi reads e rpi = rdm (L) + ϕ(L)e 1 s + ∆xe1 ,

(7.47)

346

7 Elastic Multibody Systems – Ordinary Differential Equations

see Fig. 7.4. (pe: “predecessor, coupling (or “end”) point). Next, the guidance velocity is to be transformed into the actual frame. Filling the left-hand side vector with the missing components for describing variable and augmenting with the relative velocities from frame i, one obtains  





 

 vo  ω    F       ωM r    =    ΩAr        q˙    x˙ A i       +    

Aip 0 0 Aip



E  

e rTpi

0 E 0

0 0 0 0 0 0 0 0



0 0 0 0

 0 0   0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 E 0

0 0 0 0 0 1

0



e3



 

∂r e 1    ∂q     vo   ∂ϕ 0  ω    F  ∂q  e  ω   Mr        ΩAr  0      q˙   0    x˙ A p  0 

e rTpi e3

0 0 0 0 0

p

   γ˙  M     γ˙ A     q˙    x˙ A

(7.48) i

: y˙ Ri = Tip y˙ Rp + Fi s˙ i [compare eq.(7.18)].

3.2.

 tip body

 arm

x,ξ,vox η R

I @ y,voy @

rdm

I @ @ @

@

@

@

@ @

3.2.1 Subsystem Equations In case of a revolute joint one has to consider the motor-gear-unit at the coordinate origin (“hub”) and a tip body which represents the stator of the subsequent motor. Eq.(7.25) still holds where now ∆x becomes R, see Fig. 7.5. Thus, 

@ hub

Figure 7.5.

Revolute Joint

Revolute Joint



R+ξ   rc =  v(ξ, t)  w(ξ, t)

(7.49)

takes the hub with radius R into account.

7.3

347

Applications

Because xA no longer represents a degree of freedom, the last row and column in dM and dG in eqs.(7.35) and (7.36) vanish for the revolute joint as well as q γ Ai γM i the last component in Q and q xi Qinput . The rest remains unaliP P PP tered because v, w, ϑ (and its PP PP actual frame i derivatives) are zero for x ∈  s {0, R}. The hub only en ters the rigid body part of  yi  equations. The bearing matrix Kbearing does, of course, not x,vox arise. r pi

3.2.2

The Kinematic Chain The kinematic chain y˙ Ri = Tip y˙ Rp + Fi s˙ i . is obtained from eq.(7.48) where    Fi s˙ i =   

0 0 1 0 0

0 0 0 1 0

0 0 0 0 E

I @ y,voy @ γ˙ A

@ @ @ @

γ˙ M

q

q

ΩM r

rotor &gear

 

ωF



 γ˙ M    γ˙ A    q˙

stator predecessor (index p suppressed)

i

(7.50)

Figure 7.6.

Kinematical Chain

and Tip y˙ Rp =    Aip   0         



0 Aip 



E  

e rTpi

0

0 E 0

e rTpi e3

e3 

0 0 0 0 0 0 0 0  0 0 0 0 0

 0 0





∂r  ∂q   ∂ϕ  ∂q e

            

vo  ωF     ωM r     ΩAr  q˙ p 



(7.51)

p

Recall that (∂r/∂q) and (∂ϕ/∂q) are to be expanded to the first order, using the second-order displacements according to eq.(7.19).

348

7 Elastic Multibody Systems – Ordinary Differential Equations

3.3.

Spatial Motion

x4

y5

t j

y4

After disposing of the subsystems, there is no problem in combining them into an entire model. The sketch shows schematically a mobile robot with three motor-gear-elastic armsubsystems. (The first two frames are used for the chassis and the pylon). The describing variables for each arm are

t
 x3 j t
x5
vo Y H
y3 HH H
t ω F = ω o − ΩAr e3  r '$ '$ ΩAr r &%

Figure 7.7.

translation of frame origin guidance rotation of frame origin relative arm angular velocity r ΩM r = iG ωM r relative motor angular velocity &% ωM r relative gear output velocity iG gear transmission ratio q˙ T = (q˙ Tv q˙ Tw q˙ Tϑ ) Ritz variables

Mobile Elastic Robot

(7.52)

and the minimal velocities are composed of the relative velocities,

s˙ T = (vx γ˙ 1 | γ˙ 2 | ωM r3 ΩAr3 q˙ T3 | ωM r4 ΩAr4 q˙ T4 | ωM r5 ΩAr5 q˙ T5 ), (7.53) where vx , γ1 represent the chassis motion [see eq.(4.100)], γ2 characterizes the pylon rotation [see eq.(4.101) for x˙ 2 = 0]. ωM ri and ΩAri are the gear output and the arm angular velocity, qi represents bending and torsion, i = 3 · · · 5. There is obviously no problem in adding a pylon gear, if need be, as well as any other kind of model refinement. The numerical simulation is easily carried out with the recursive scheme (4.107) et seq. It has successfully been applied for a two-arm elastic robot (stationary, blocked wheels) as shown in Fig. 7.8. There, the elastic deflections are measured by strain gauges (which explains the amount of cables); furthermore, the joint angles and angular velocities are disposable for control. Fig. 7.9 represents a typical cooperation situation between two robots. The left is the elastic one carrying a (compliant) pencil which has to follow perpendicularly to a plane which is moved by the right (rigid, commercial) robot (“Master-Slave-Problem”).

7.3

349

Applications

Figure 7.8.

Figure 7.9.

Elastic Robot

Master-Slave Cooperation

The elastic robot has to follow the master’s motion autonomously with prescribed normal force history (constant in this example, Fig. 7.10, left). The master trajectory is here a triangle with about 60 cm side length which is run through in 6 sec for one orbit. The maximum position deviation is ca. 5 mm (Fig. 7.10,

350

7 Elastic Multibody Systems – Ordinary Differential Equations

right), mainly caused by the non-considered surface friction, (R. Mitterhuber, 2005).

Figure 7.10. Left: Normal Force (in N) over time (in s); Right: In-Plane Position (in mm)

3.4.

Plane Motion

Plane motion is considered in the x-y-plane. The elastic deformation is v(x, t) = v(x)T q(t), and the reference frame moves with vox , voy , ωoz . Using y˙ TR = (vox voy ωoz q˙ T )

(7.54)

for description yields



e1 e2 0 0 vo  ωo   0   0 0 e3 y˙ =   r˙ c  =   0 0 0 vT e2 ˙ ϕ 0 0 0 v 0 T e3 





vox    voy   := [FΨT ]y˙ .  R ωoz   q˙ 

(7.55)

Along with rTc = (x vT q 0) and ω o = ωoz e3 , one obtains, from eq.(7.14), 

dm

Z  0  M=  T  −v qdm

0

0 dm xdm vdm

−vT qdm xdm 2 x dm + dC xvdm + v0 dC

0 vT dm xvT dm + v0 T dC vvT dm + v0 v0 T dC

   , 

7.3

351

Applications 

G=

Z    

0 −dm −xdm −2vT dm T +dm 0 −v qdm 0 T T +xdm +v qdm 0 +2q v vT dm T +vdm 0 −v v qdm 0 

Z   Q=  

eT1 eT2 0 0

0 0 eT3 0

0 0 0 veT2

0 0 0 v0 eT3



    ωoz , 

E 0    e dfe   rc E     E 0  dMe 0 E 

(7.56)

with [FΨT ] from eq.(7.55) and F from eq.(7.3 b). Considering gravity for impressed force, one has dfe = −(sin α cos α 0)T dmg yielding sin α   cos α   (x cos α − vT q sin α) dm g, v cos α 

Qgrav = −

Z



where α represents the absolute angle when measured from the horizontal. The generalized potential forces are obtained from the elastic potential, 

Qel.pot = −

∂ V˙ el ∂ y˙ R

!T

Z   =−  

0 0 0 00 EIv v00 T q

    dx, 

hence Q = Qgrav + Qel.pot .

(7.57)

From the above results one obtains the zero-order forces where the dynamical stiffening matrix is equated with Z

h

dfx(o) = −eT1 M¨yR + Gy˙ R − Qgrav − Qel.pot = −

Z

i(o)

2 (v˙ ox − voy ωoz − xωoz + g sin α)dm.

(7.58)

Index (o) here signifies that all deformations are set to zero. The results will be specified for a revolute joint-motor-gear-arm-unit in the next section. We will thereby use the opportunity to recalculate eqs.(7.56) to (7.58) in order to give some basic insight for those readers who prefer a quick reading, without going into the previous chapters in detail.

352

4.

7 Elastic Multibody Systems – Ordinary Differential Equations

Plane Motion – Recalculation

Our main emphasis had been on structurizing the problem for spatial motion. Spatial motion is the general case, unavoidable when bending and torsion simultaneously occur (see e.g. Fig. 5.28. p.217). Plane motion is thus a very special case but may be looked at as a simple example to reveal the physical backgrounds. The main questions here are: 4.1 When considering the momentum theorems: how to contend with constraint forces? (Minimal (non-holonomic) velocities, Projection). 4.2 What do the subsystem matrices look like? Where do the Coriolis effects (the factor 2) come from? (Structurizing). 4.3 Gross motion with superimposed small deflections – what are the consequences of linearizing in parts? (Dynamical Stiffening ). 4.4 How is a mechanical model combined when using subsystems? (The kinematic chain, Gauss-form, minimal representation, recursive representation).

4.1.

Minimal Velocities and Projection

Considering plane motion, the momentum theorems applied to a free-cut mass element read dfxe dfxc dpx dpx d e   dpy  + ω      eo dfy dfyc  = 0. (7.59) dpy − − dt e dLz dLz dMz dMzc 

















Here, index e refers to impressed (work accomplishing) forces and torques, and c represents the (not work accomplishing) constraint forces and torques. (Notice that also the Coriolis forces do not contribute to energy but nevertheless affect motion). Furthermore, ω o is the absolute angular velocity of the reference e o is the 3×3 skew-symmetric spin tensor assigned to coordinate frame, and ω ω o (providing the “vector product”). dpx , dpy are the linear momenta and dLz is the moment of momentum, respectively. They read 













dpx dm 0 0 vcx vcx  dpy  =  0 dm 0   vcy  := dM  vcy  0 0 dC dLz ωcz ωcz

(7.60)

where subindex c denotes the mass center (of the mass element). dm is its mass and dC its mass moment of inertia (w.r.t. the rotation axis z).

7.4

353

Plane Motion – Recalculation

The mass center velocities are next represented in terms of the velocities of the e o rc + r˙ c , reference frame and the superimposed relative velocities: vc = vo + ω ω c = ω o + ϕe ˙ 3 (index o: reference frame origin):  

vcx 1 0 −ry 1 0 0    vcy  =  0 1 +rx 0 1 0     0 0 1 0 0 1  ωcz i 







Here, rTc = (rx ry 0) and ϕT = (0 0 ϕ) characterize the mass center deviation of an element dm w.r.t. the (moving) reference frame. They are approximated with a Ritz series, rc = rc (q), ϕ = ϕ(q), ωoz where q(t) represents the Ritz coefficients, yielding ˙ r˙x = (∂rx /q)q, ˙ r˙y = (∂ry /q)q, ˙ ϕ˙ = (∂ϕ / q)q. Insertion into eq.(7.61) yields 

vcx





1 0 −ry +rx 0 0 1

 vcy  =  0 1

ωcz

i

vox voy ωoz r˙x r˙y ϕ˙

     .   

(7.61)

y, voy 6

ϕ=v 0

q q

q

ry =v(x,t)

Figure 7.11.

-

x, vox

Plane Beam Bending

v (∂rx /∂q)  ox  voy  (∂ry /∂q)    ωoz  := FR y˙ R . (∂ϕ/∂q) q˙ 





(7.62)

(Notice that the rigid body variables vox , voy are, in general, non-holonomic). Vector y˙ R is used for minimal velocities. The Projection Equation for the plane motion of a single beam then reads T dfxe ∂vcx /∂ y˙ R dpx dpx d  ∂vcy /∂ y˙ R    dpy  + ω e o  dpy  −  dfye  = 0. dt ∂ωcz /∂ y˙ R dLz dLz dMze (7.63)

 Z B

 













Eq.(7.63) projects the motion into the unconstrained direction indicated by y˙ R . The constraint forces are thus suppressed [compare eq.(7.59)].

354

4.2.

7 Elastic Multibody Systems – Ordinary Differential Equations

Subsystem Matrices

The beam according to Fig. 7.11 is used as a subsystem. The derivation of the subsystem matrices needs first-order velocities w.r.t. elastic deflections. One obtains, along with the series expansion v(x, t) = v(x)T q(t), 







x 0 rx ∂rx /∂q  vT q  ^   vT   ry  O(1)  ∂ry /∂q =  =   . ∂ϕ/∂q ϕ v0 T v0 T q 







(7.64)

Eq.(7.64) leads the functional matrix FR in eq.(7.62) to 



1 0 −vT q 0 ∂vcx /∂ y˙ R  x vT  FR =  ∂vcy /∂ y˙ R  =  0 1 . T 0 ∂ωcz /∂ y˙ R 0 0 1 v 



(7.65)

Along with dM from eq.(7.60) one obtains the momenta  vox dm 0 −vT qdm 0 dpx  voy      dpy  = dM FR y˙ =  0 dm xdm vT dm   R  ωoz  . T 0 dLz 0 0 dC v dC q˙ (7.66) 









Insertion into the Projection Equation (7.63) yields Z " |

#



Z

eo FTR dMFR ¨yR + FTR ω {z

dM

}

|





Mass Matrix The functional matrix (7.65) along with dM FR from eq.(7.66) yields 1  0 dM =   −vT q 0 



Z dfx  d  dMFR + dMFR y˙ R − FTR  dfy . dt dMz {z } {z } | dG dQ (7.67) 

  0 0  T qdm dm 0 −v 0 1 0   xdm vT dm   0 dm   x 1 0 T dC 0 0 dC v v v0

7.4

355

Plane Motion – Recalculation    

= 

dm 0 T −v qdm 0

0 dm xdm vdm

−vT qdm xdm 2 x dm + dC xvdm + v0 dC



0  vT dm  . xvT dm + v0 T dC  vvT dm + v0 v0 T dC (7.68)

Gyroscopic Matrix The reference frame rotates with ω o = ωoz e3 , yielding for dGy˙ R from eq.(7.67) dGy˙ R =  1 0  0 1   −vT q x 0 v 1  0 +   −vT q 0 

0 1 x v

  v  0  ox T qdm dm 0 −v 0 0 −1 0  voy  0   T   1 0 0  ωoz  0 dm xdm  v dm   ωoz  1 T 0 0 0 0 0 0 dC v dC q˙ v0 



 v 0  ˙ 0  ox  0 0 −vT qdm  0  voy  00 0 0 ωoz  . 1 00 0 0 q˙ v0 







(7.69)

Carrying out the calculation for d(dM FR )/dt y˙ R [second summation term ˙ ωoz which can be rearranged to in dG from eq.(7.67)] leads to [−vT qdm] ˙ thus [−vT dm] is assigned to the last component of y˙ R and [−vT dmωoz ] q, ωoz is factored out. Doing so one obtains for eq.(7.69), 1  0 =   −vT q 0

0 1 x v

 0  vox 0 −dm −xdm −vT dm   0   dm 0 −vT qdm  ωoz  voy  0 ωoz  1 0 0 0 0 0 q˙ v

1  0 +   −vT q 0

0 1 x v

 0  vox 0 0 0 −vT dm  voy  0  0 0 0   ωoz  0 ωoz  , 1 0 0 0 0 q˙ v0



dGy˙ R















(7.70)

hence 1  0 dG =   −vT q 0 

 0 0  0 −dm −xdm −2vT dm  1 0   ωoz dm 0 −vT qdm 0 x 1  0 0 0 0 v v0 

356

7 Elastic Multibody Systems – Ordinary Differential Equations    

= 

0 −dm −xdm −2vT dm +dm 0 −vT qdm 0 T T +xdm +v qdm 0 +2q v vT dm T +vdm 0 −v v qdm 0

    ωoz 6= −dGT . 

(7.71)

Applied Forces Assume a beam acts against gravity. Let furthermore the rigid body angle α be measured from the horizontal such that, for the non-rotated beam, the x-axis coincides with the horizontal. One has then first to transform −dm ge2 into the reference frame. Then, because the gravity force applies to the mass center of dm, this result is to be projected into the description space by means of the functional matrix (7.65), yielding 1  0 =   −vT q 0 

dQgrav

0 1 x v

  0  cos α sin α 0 0 0   − sin α cos α 0  −dm g  1 0 0 1 0 v0 

sin α   cos α  = − (x cos α − vT q sin α) dm g, v cos α 



(7.72)

where α represents the absolute angle against the horizontal. Furthermore, from the elastic potential dVel = 12 qT EIv00 v00 T q dx, one obtains 

dQel.pot = −

∂dV˙ el ∂ y˙ R

!T

  

= −

0 0 0 00 EIv v00 T q

    dx. 

(7.73)

Spatial Integration If the mass distribution dm/dx (and the moment of inertia distribution dC/dx) are known, spatial integration can be carried out. To this end, consider a beam that is specified to carry additional masses at the origin (“hub”) and at its end (“tip mass”). This enables us to later use it as an element in a motor-gear-armunit with a revolute joint. In order to take the hub and the tip body into account, “clamped-free” beam functions are used which are extended to the reference frame origin (x = 0) on the one side and to the tip body mass center (x = R + L + s) on the other side.

7.4

357

Plane Motion – Recalculation  tip body

s

 arm

η R I @ y,voy @

rdm

@ I @

@ v(ξ,t) I x,ξ,vox @ @q

x,vox v(x)

@

I @ y,voy @

@

@

@

@

@

R

@ @

@ @

α

q

@ ωoz

hub

Figure 7.13.

Figure 7.12. Revolute Joint

Shape Function v(x)

Here, L represents the beam length. The beam itself is considered homogeneous (density ρ = const.) with constant cross sectional area A. Along with the shape functions (which are defined w.r.t. the moving reference frame), x≤R v(x)i , , vi0 := v 0 (x)i , for R < x ≤ R + L   0 R+L<x≤R+L+s v(L)i + v 0 (L)i x v (L)i (7.74) the impressed forces read vi :=

 0



 0

−mg sin α





            −mg cos α           R   Qgrav =  g ρAvT dξ sin αq , Q =  el.pot       +gm tip v T sin αq − gmc cos α     e       R    −g ρAvdξ cos α 

−gm tip v ecos α

and one obtains for eq.(7.62),

0

0

0

                

− EIv00 v00 T dξq (7.75) R

358

7 Elastic Multibody Systems – Ordinary Differential Equations

−qT ρAvdξ −qT m tip v e R

  m

0

      m    M=     symmetric      

mc

Co



0      R  ρAvT dξ  T  +m tip v e   ,  R R T T 0 ρAxv dξ + ρIv dξ     +m tip (xvT ) e + Ctip v0 Te    R R  T T 0 0 ρAv v dξ + ρIv v dξ 

+m tip (v vT ) e + Ctip (v0 v0 T ) e (7.76) R RL where the remaining integral refers to the beam, (.)dξ = o (.)dξ. The elongations are x = R + ξ, x e = (R + L + s) and the deflections at the end point read v e = [v(L) + v0 (L)s], v0 e = v0 (L), see eq.(7.74) and Fig. 7.13, respectively. m is the total mass, C o is the total moment of inertia of the undeformed system w.r.t. the frame origin, and index c indicates the mass center of the total (undeformed) system. Recall: eq.(7.76) is assigned to the sequence ¨ ¨T ). yTR = (v˙ ox v˙ oy ω˙ oz q Accordingly, one obtains for the gyroscopic matrix (7.71), 

−2 ρAvT dξ −2m tip v Te R

0

−m

−mc

0

−m tip (v vT ) e q

     R  −qT ρAvdξ  +m 0 0  −qT m tip v e   G=  R R  + ρAvT dξq +2qT ρAv vT dξ   +mc 0  +m tip v Te q +2qT m tip (v vT ) e    R R   + ρAvdξ − ρAv vT dξq

+m tip v e

0

            ωoz .          

(7.77)

Eqs. (7.75) to (7.77) show that it might be helpful for the integration to retain the general structures in order to determine, for comparison, the influence of the tip mass. Once the spatial integration is done, one may neglect ρI in eq.(7.76) (Euler-Beam) as well as the (block-) element (3,4) in eq.(7.77) which creates

7.4

359

Plane Motion – Recalculation

second-order terms. The remaining integral values are analytically derived from eqs.(5.73) to (5.77), p.144 et seq. From eqs.(7.75) to (7.77) one has n

M¨yR + Gy˙ R − Qgrav − Qel.pot

o

(7.78)

for subsystem description which is valid as long as the reference frame moves slowly and the tip mass is sufficiently small.

4.3.

Dynamical Stiffening

The dynamical stiffening matrix from eq.(6.188 a) reduces to



dKn :=

0 0 0 dKn



0

=

0

 



0 L+s R ξ

(o)

dfx dζv0 v0 T  dξ. dζ 

(7.79)

(The integration variable is changed from ξ to ζ since ξ is already used for beam coordinate, see Fig. 7.12; the total length is L+ s, see Fig. 7.13). The zero-order (o) force dfx is obtained from eq.(7.78), Z

h

dfx(o) = −eT1 M¨yR + Gy˙ R − Qgrav − Qel.pot = −

Z

i(o)

2 (v˙ ox − voy ωoz − xωoz + g sin α)(dm/dζ)dζ (7.80)

δ (ξ − L − s), − δ : Dirac-function. Index (o) where (dm/dξ) = ρA + mtip− indicates that all deformations are set equal to zero. The shape functions have been defined for x ∈ {0, R + L + s}, see Fig. 7.13. Thereby, the hub does not contribute (v = 0, v 0 = 0 ∀ x ∈ {0, R}). From eq.(7.80) one obtains, with x = R + ξ, L+s R ξ

(o)

dfx dζ = −v˙ + v ω + ω 2 R − g sin α
ox oy oz oz dζ

2

h

+

h

i

+ ω2oz (L + ξ) (L−ξ) ρA i

2 R − g sin α +ω 2 (L + s) m . −v˙ ox + voy ωoz + ωoz tip oz




(7.81) This result is used for the dynamical stiffening matrix (7.79) which still needs an integration over the whole length {0, L + s}: L+s Z L+s Z

Kn = o

ξ

(o)

dfx T dζv0 v0 dξ. dζ

(7.82)

360

7 Elastic Multibody Systems – Ordinary Differential Equations

The integrals are derived from eq.(7.28) and eq.(7.29). Using the abbreviations

[∂m∗ /∂q]1 = [∂m∗ /∂q]2 =

h RL  o

h RL  o



i

ρA (L − ξ) +m tip v0 v0 T dξ+m tip vL0 v0LT s , ρA

 i (L2 − ξ 2 ) +m tip v0 v0 T dξ+ m tip vL0 v0LT s 2(L + s)

(7.83)

leads eq.(7.82) to

Kn = 2 − g sin α] + [∂m∗ /∂q] [(L + s)ω 2 ] = [∂m∗ /∂q]1 [−v˙ ox + ωoz voy + Rωoz oz 2 2 [∂m∗ /∂q]1 [−v˙ ox + ωoz voy − g sin α] + [∂m∗ /∂q]3 ωoz

where [∂m∗ /∂q]3 = [∂m∗ /∂q]1 R + [∂m∗ /∂q]2 [(L + s).

(7.84)

The dynamical stiffening results then in



K n yR =           

0 0 0 Kn



yR = 

0 0 0 n

o

n

2 [∂m∗ /∂q]1 q − v˙ ox +ωoz voy − g sin α + [∂m∗ /∂q]3 q ωoz

     .    o

(7.85) Because vox is a state variable itself, and because later the mass matrix needs inversion, [∂m∗ /q]1 qv˙ ox has to be assigned to the mass matrix rather than to the restoring matrix. Doing so, it will also be consequent to assign the remaining velocity terms to the gyroscopic matrix, yielding

7.4

361

Plane Motion – Recalculation

0 0 0 " # ∗ ∂m − q ∂q 1



Madd

   =  

0 0 0



Gadd

   =  

0



0 0

0

0

0

symmetric 0

   ,  

(7.86)

0 0 " 0 # ∂m∗ q ∂q 1

0 0 " 0 # ∂m∗ q ∂q 3

0 0 0 0

     ωoz .  

(7.87)

See also Chapter 6, sections 7 and 8. The mass matrix is symmetrized with terms which only contribute for negligible second-order deviations. For the “dynamical stiffening”, one is then left with the impressed forces only1 ,     Kn :=   

0 0 0

0 0 0

0 0 0

0

0

0

0 0 " 0 # ∗ − ∂m ∂q 1

     g sin α.  

(7.88)

Assembling the elastic potential forces as R −Qel.pot = blockdiag{0, 0, 0, EIv00 v00 T dx}yR := Kel.pot yR yields the subsystem dynamics n

o

[M + Madd ]¨yR + [G + Gadd ]y˙ R + [Kn + Kel.pot ]yR − Qgrav .

4.4.

(7.89)

The Kinematic Chain

In order to use eq.(7.89) as a subsystem, the reference frame velocities are to be specified. Assuming the predecessor velocities (index p) to be known,

1 Separate treatment of the dynamical stiffening terms according to kinetic reactions and impressed forces is not only necessary but also suitable. This is because in slowly-moving systems one may a` -priori neglect the kinetic part (additional mass and gyroscopic matrix).

362

7 Elastic Multibody Systems – Ordinary Differential Equations

one obtains the actual velocities (index i) via the kinematic chain according to eq.(7.18), 1 0 −ry (∂rx /∂q) vox  voy   0 1 rx (∂ry /∂q)      +     0 0 1 (∂ϕ/∂q) ωoz q˙ p 0 0 0 0 pe



  

vox  voy   Aip     ωoz  =  q˙ i 



| {z }

y˙ Ri



|

}|

{z

Tip



{z

y˙ Rp



}

|

0 0 1 0

0   0   γ˙ . 0  q˙ i E | {z } {z } s˙ i Fi (7.90) 

Eq.(7.90) represents the guidance motion coming from the predecessor p at the coupling point e (transformed into the actual frame with Aip ) and the superimposed relative velocities Fi s˙ i which the actual subsystem contributes. The transformation matrix 



cos γpi sin γpi 0 Aip =  − sin γpi cos γpi 0  0 0 1

(7.91)

is calculated with the relative angle γpi between frame p and frame i: γpi = γi + (v0 Te q)p . Here, γi is the relative (“rigid body”-) angle of frame i and (v0 Te q)p represents the bending angle of the predecessor at the coupling point. For a correct linearization, matrix Tip has to be developed to the first order. Thus, applying the second-order displacements from eq.(7.19), the columns in eq.(7.90) are equated with  



xe



 



−qT

 rx ∂rx /∂q   O(1)   T  ^ ry  O(1)  ∂ry /∂q =  = v e q     ∂ϕ/∂q ϕ pe

v0 Te q

pe

p

RL o

v0 v0 T dξ

−

v Te v0 Te

qT (v0 v0 T )

 e

s     p

(7.92) where x e = (R + L + s), v e = (vL + v0L s), v0 e = v0 L . Starting with i = 1 which has the immobile inertial system as predecessor, we have y˙ R1 = F1 s˙ 1 . Then, for i = 2, one obtains y˙ R2 = T21 y˙ R1 + F2 s˙ 2 etc., leading in total to

7.4

363

Plane Motion – Recalculation         |



y˙ R1 y˙ R2 y˙ R3 .. .



      =      

y˙ RN {z

:= y˙ R

}

F1 T21 F1 T31 F1 .. .



F2 T32 F2 F3 ..

. · · · FN

T N 1 F1 · · ·

      

{z

|

s˙ 1 s˙ 2 s˙ 3 .. .

}|

:= F

s˙ N {z

:= s˙

       

(7.93)

}

for a system consisting of N subsystems. (Non-marked elements are zero.) Using (˙sT1 · · · s˙ TN )T for minimal velocities then leads to FT1 FT1 TT21 FT1 TT31 · · · FT1 TTN 1  FT2 FT2 TT32 · · · FT2 TTN 2    FT3 · · · FT3 TTN 3  ..  .. .  . FTN 

[M + Madd ]1 ¨yR1 + · · · [M + Madd ]2 ¨yR2 + · · · [M + Madd ]3 ¨yR3 + · · · .. .

       

     = 0.   

[M + Madd ]N ¨yRN + · · · (7.94)

Eq.(7.94) gives access to a minimal representation [by insertion of ¨yR = ˙ s etc. from eq.(7.93)] or to a recursive solution scheme in the sense of a F¨s + F˙ Gaussian elimination procedure, see Chapter 4, p.92 et seq. 5 Example: The Elastic Double Pendulum – Plane Motion. The motion equations of an elastic double pendulum (s = 0, R = 0) read



[FT1 ] [FT1 TT21 ] 0 [FT2 ]

   ˙ [M + Madd ]¨yR + [G + Gadd ]yR + [Kn + Kel.pot ]yR − Qgrav 1 = 0. [M + Madd ]¨ yR + [G + Gadd ]y˙ R + [Kn + Kel.pot ]yR − Qgrav

2

(7.95)

The kinematic chain (7.90) is calculated with

rx ry ϕ

!





L ^ =  vTL q  T v0L q

O(1)

1e

1

 "

∂rx /∂q ∂ry /∂q ∂ϕ/∂q

# O(1)

 −q

=  

1e

T

RL o

0 0T



v v dx 

vTL T v0L

 , γ12 = γ2 + v0L1T q1 .  1

(7.96)

Using the results from the previous section, the elastic double pendulum is trivially calculated (see also eq.(6.268) et seq. for a subsystem equation).

364

7 Elastic Multibody Systems – Ordinary Differential Equations

Fig. 7.15 shows the double pendulum motion in comparison to a rigid system with initial conditions γ1o = π/2, γ2o = 0 (relative angles) and a slight impact at the outer end in the x-direction to induce bending vibrations. The motion is depicted at times tn = n∆t, ∆t = 0.1, n = 0, 1, . . . , 14. Parameter values are those from Table 7.2, p.372.

0

0

0.4

0.4

0.8

0.8

x 0

0.4

0.8

y

Figure 7.14. Rigid Double Pendulum

x 0 Figure 7.15.

0.4

y

0.8

Elastic Double Pendulum

Considering the zero-order terms, one has



Qograv,i



mg cos α i X  −mg sin α  = , αi = γk , i = 1, 2  −gmc sin α k=1 0 i

(7.97)

where, as in eq.(5.216) et seq. (elastic double pendulum PDE’s), α is calculated w.r.t. the vertical axis. For the kinetic part one obtains



m

0 m  o Mi =  symmetric

0 mc Co

0 0 −m −mc 0  0 0  o , Gi =  skew-symmetric 0 0  0 i







0 0 ω . 0  oz,i 0 i

(7.98)

As eq.(7.95) shows, the zero-order terms from subsystem nr.2 act on subsystem nr.1. Along o with γ12 → γ12 = γ2 one has



y˙ oR2







L1 sin γ2 γ˙ 1 L1 (sin γ2 γ¨1 + cos γ2 γ˙ 1 γ˙ 2 )  L1 (cos γ2 γ¨1 − sin γ2 γ˙ 1 γ˙ 2 )   L1 cos γ2 γ˙ 1  ¨o = ,y =  (¨ γ1 + γ¨1 ) (γ˙ 1 + γ˙ 1 )  R2  0 0

which leads to

(7.99)

7.4

365

Plane Motion – Recalculation Mo ¨ yoR + Go y˙ oR − Qograv



 2

=

m2 [L1 (sin γ2 γ¨1 − L1 cos γ2 γ˙ 12 − c2 (γ˙ 1 + γ˙ 1 )2 − g cos(γ1 + γ2 )]  m2 [L1 (cos γ2 γ¨1 + L1 sin γ2 γ˙ 12 − c2 (¨γ1 + γ¨1 ) + g sin(γ1 + γ2 )]   C o (¨γ + γ¨ ) + m c [L (cos γ γ¨ + L sin γ γ˙ 2 + g sin(γ + γ )]  . 1 2 2 2 1 2 1 1 2 1 1 2 2 0





(7.100) This vector is premultiplied with FT1 TT21 yielding −v T q

"

e

−

RL o

T

v0 v0 dxq

L v

e

1 v0

0

# ×

e

0

1

−m2 [c2 (sin γ2 (¨ γ1 + γ¨2 ) + L1 γ˙ 12 + c2 (γ˙ 1 + γ˙ 1 )2 cos γ2 + g cos γ1 ]  m2 [c2 (cos γ2 (¨γ1 + γ¨2 ) + L1 γ¨1 − c2 (γ˙ 1 + γ˙ 1 )2 sin γ2 + g sin γ1 ]   C o (¨γ + γ¨ ) + m c [L (cos γ γ¨ + L sin γ γ˙ 2 + g sin(γ + γ )]  . 1 2 2 2 1 2 1 1 2 1 1 2 2 0





The most interesting part here is the one that arises from (∂rx /∂q)T1 = −

ZL1

R

(7.101) T

v01 v0 1 dx q1 :

T

m2 [c2 sin γ2 (¨ γ1 + γ¨2 ) + L1 γ˙ 12 + c2 (γ˙ 1 + γ˙ 1 )2 cos γ2 + g cos γ1 ]v01 v0 1 dx q1 .

(7.102)

o

Premultiplying with δqT1 (virtual work), re-inserting the Ritz ansatz (δqT1 v01 T v0 1 q1 = v10 ) and integration by parts yields

=

δv10 ,

ZL1 −

δv1

n


 ∂ dx m2 c2 sin γ2 (¨ γ1 + γ¨2 ) + L1 γ˙ 12 + c2 (γ˙ 1 + γ˙ 1 )2 cos γ2 + g cos γ1 v10 ∂x1 o

o



+ δv1 m2



c2 sin γ2 (¨ γ1 + γ¨2 ) + L1 γ˙ 12 + c2 (γ˙ 1 + γ˙ 1 )2 cos γ2 + g cos γ1 v10

see also eq.(5.211) and eq.(5.245). 4




 L1 o

,

(7.103)

366

7 Elastic Multibody Systems – Ordinary Differential Equations

Subsystem Specification For an application-oriented approach we now expand the subsystem to a motorgear-arm-unit (compare subsection 3.2) undergoing plane motion. The specification to a motor-gear-arm-subsystem is done with a transformation FTrel M Frel where Frel is obtained from

vox 1 0  voy   0 1     ωoz  =  0 0 0 0 q˙ 





0 0 1 0

0 0 0 0

0 0 1 0

0 0 0 E

        



vox voy ωF ωM r ΩAr q˙

     := Frel y˙ R   

(7.104)

with new y˙ R . Here,the index rel indicates the use of relative angular velocities w.r.t. the guidance velocity ωF . This choice of variables is suitable to avoid overlapping effects between the subsystems (see p.87 et seq.). Carrying out the transformation for Mtot = FTrel [M + Madd ] Frel and Gtot = FTrel [G + Gadd ] Frel one obtains, after replenishment with the motor equations, Mtot =

  m                                  

R

0

−qT ρAvdξ −qT m tip v

"

R

0

e

−qT ρAvdξ −qT m tip v

−qT

e

∗

∂m ∂q

R m

mc

0

mc

#T  1

T

ρAv dξ +m tip v T e

R

o

C + CM

CM iG

C

o

T

ρAxv dξ +m tip (xvT ) e +Ctip v0 T e

CM i2G

0

0

R

symmetric

C

o

ρAxvT dξ +m tip (xvT ) e +Ctip v0 T e

R

ρAv vT dξ +m tip (v vT ) T

+Ctip (v0 v0 )

e

                 ,                 

e

(7.105)

7.4

367

Plane Motion – Recalculation

Gtot /ωoz =

R



−m

0

     m       mc      0      mc     R  ρAvdξ  +m tip v

−mc T

R

T

−q ρAvdξ −qT m tip v

0

+ ρAvT dξq +m tip v T q

0

0

0

−2 ρAvT dξ −2m tip v T  e

−mc

0



−q ρAvdξ −qT m tip v

0

0

0

0

0

0

0

0

0

0

0

0

0

− ρAv vT dξq −m tip (v vT ) q e +[∂m∗ /∂q]3 q

0

e

             .             

R

e

R

e

R

+ ρAvT dξq +m tip v T q e

R

h

e

∗

∂m ∂q

i

− ρAv vT dξq −m tip (v vT ) q e +[∂m∗ /∂q]3 q

q 1

R

(7.106) Here, m is the total mass and c represents its mass center distance from the frame origin when considered undeformed. The underlined terms are those which arise from dynamical stiffening with [∂m∗ /∂q]1 , [∂m∗ /∂q]3 according to eq.(7.84). The subsystem matrices are assigned to y˙ R as defined with eq.(7.104). The gear potential is assumed Vgear = kG (γA −γM )2 /2. The restoring matrix then reads Ktot = FTrel Kn Frel +

           

0

∂ ∂ [∂Vgear /∂q]T + [∂Vel.pot /∂q]T = ∂q ∂q

0 0 symmetric

0 0 0

0 0 0 kG

0 0 0 −kG kG



0 0 0 0 0

RL o

T

EIv00 v00 dξ h i ∗ g sin α − ∂m ∂q 1

          

(7.107) Pi where, for the ith subsystem, αi = k=1 γAk (absolute angle). The corresponding gravity force from eq.(7.75) reads Qgrav,tot → FTrel Qgrav ,

368

7 Elastic Multibody Systems – Ordinary Differential Equations           =         

Qgrav,tot



−mg sin α

    R  T g ρAv dξ sin αq  − gmc cos α  T  +gm tip v e sin αq  , 0   R  T g ρAv dξ sin αq  − gmc cos α  T +gm tip v e sin αq   R  −g ρAvdξ cos α 

−mg cos α

(7.108)

−gm tip v ecos α

while the input vector is 0

Qinput =

0

0

iG

0

0


T

MM .

(7.109)

The kinematic chain y˙ Ri = Tip y˙ Rp + Fi s˙ i is calculated with 1 0 −ry 0 −ry (∂rx /∂q)  Aip  0 1 +rx 0 +rx (∂ry /∂q)       0 0 1 0 1 (∂ϕ/∂q)   

 



      

Tip =  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

     Fi =    

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 E

       

,

(7.110)

pe

     γ˙ A      , s˙ i =  γ˙ M  ,   q˙  i

(7.111)

where the elements of Tip are computed with eq.(7.91) and eq.(7.92). One has thus n

Mtot ¨yR + Gtot y˙ R + Ktot yR − Qgrav,tot − Qinput

for subsystem representation.

o i

(7.112)

7.5

5.

369

Reduced Number of Shape Functions: Controlled Systems

Reduced Number of Shape Functions: Controlled Systems

It is obvious, from the theoretical point of view, that the required number of shape functions can not generally be determined, at least for an undamped system with arbitrary motion. In controlled systems (where the control induces pervasive damping, see Chapter 8) one may be sure that, under certain conditions, the use of only a few shape functions is sufficient. Practical experience says that for an elastic robot, for instance, two shape functions for each deformation (two bending directions, one torsion) will be sufficient. Control may be split into an open-loop part (which achieves the gross motion) and a closed-loop which is indispensable in view of disturbances. Both are put in execution with the corresponding (generalized) motor forces. From the counteraction principle, according to eq.(4.125) we know how constraint forces Qλi enter the motion equations: FT1 FT1 TT21 FT1 TT31 · · · FT1 TTN 1  FT2 FT2 TT32 · · · FT2 TTN 2    FT3 · · · FT3 TTN 3  ..  .. .  . FTN 



  y1 + · · ·] − Qλ1 + TT21 Qλ2  [M1 ¨ λ λ T  [M ¨   2 y2 + · · ·] − Q2 + T32 Q3   =0 ..   .   

[MN ¨yN + · · ·] − QλN

(7.113) (index R suppressed, Mtot,i → Mi for simplicity). Insertion of the desired path into eq.(7.113), y˙ i (q, s˙ ) → y˙ id (t) etc., corresponds to a rheonomic constraint. The desired motions s˙ id , qid are known in advance, of course, and so are ¨yid , y˙ id which follow from the kinematic chain. The motor torques QM i are eventually obtained by premultiplying the constraint forces with FTi , which cancels all those constraints that do not contribute to motion. One obtains thus the recursion scheme 1 = 1 · · · N : y˙ id p = N · · · 1 : Qλp

h

= Tip y˙ p

i

h

d

+ Fid s˙ id ,

= Mp ¨yp + Gp y˙ p − Qp

i

h

d

+ TTip

i d

Qλi ,

(7.114)

QM p = FTpd Qλp , once more starting with p = N (last system) where Tip = 0. Example: For the chain example on page 98 one has Qλp = (fxλ fyλ Mzλ )Tp where the first two components comprise “ordinary” constraint forces (forcing the system to behave like a chain) while the last one corresponds to the rheonomic constraint. λ := Q The local functional matrix is Fp = e3 yielding Mzp M p. Inserting the open loop control from eq.(7.114) into eq.(7.113) yields

370

7 Elastic Multibody Systems – Ordinary Differential Equations

FT1 FT1 TT21 FT1 TT31 · · · FT1 TTN 1  FT2 FT2 TT32 · · · FT2 TTN 2    FT3 · · · FT3 TTN 3  ..  .. .  . FTN 

  y1 + · · ·]−[M1 ¨y1 + · · ·]d  [M1 ¨  [M y ¨ y2 + · · ·]d 2  2 + · · ·]−[M2 ¨  ..  .  



[MN ¨yN + · · ·]−[MN ¨yN + · · ·]d

  =0  

(7.115) ˙ ˙ which shows the particular solution y(t) = yd etc. Obviously, the system will not follow the desired path when a disturbance takes place. For instance, if one applies an impact at the beginning of observation (i.e. a velocity initial condition), then the trajectory will diverge (characterized by the homogeneous solution part) – unless it is forced to join the desired path with an additional input, i.e. a closed loop. One has thus to augment eq.(7.114) with the corresponding closed loop input, for instance QM p → QM p + D(˙sd − s˙ )p +K(qd −q)p with suitable D and K, respectively. Or concentrate on QM p = Mp ¨ydp + D(˙sd − s˙ )p + K(qd − q)p + f where f comprises the disturbances and the (nonlinear) coupling terms which might be compensated for by means of an observer, etc. These considerations open the whole field of (optimal, or suitable) control, see Chapter 8. If one extracts the Qλi from eq.(7.113), then one obtains FT1 FT1 TT21 FT1 TT31 · · · FT1 TTN 1  FT2 FT2 TT32 · · · FT2 TTN 2    FT3 · · · FT3 TTN 3  ..  .. .  . FTN 

       

FT1

0 FT2

0 ··· 0 ··· FT3 · · · .. .

0 0 0 .. . FTN



 y1 + · · ·]  [M1 ¨   [M y 2 ¨2 + · · ·]   ..  .  

[MN ¨yN + · · ·]

   =  



 λ   QM 1  Q1   Qλ   Q    2   M2   .  =  . .   .   ..   .    

QλN

(7.116)

QM N

˙ s etc. once more yields the minimal representation Insertion of ¨y = F¨s + F˙ (see eq.(4.99), p.93) M¨s + G˙s − Q = QM , QTM = (QTM 1 · · · QTM N )

(7.117)

where Q comprises the generalized impressed forces except the motor torques. Eq.(7.117) may suitably be used for low-dimensional systems. It will next be applied to a simple example in order to test the control influence on the requested number of shape functions.

7.5

371

Reduced Number of Shape Functions: Controlled Systems

5 Example: Elastic Robot

x2

An articulated elastic robot is considered to demonstrate a control influence on an elastic multi body system. The vertical angular velocity ωv is hereby kept zero (plane motion). The robot then consists of two subsystems according to Fig. 7.6 on page 347. This corresponds to four rigid body coordinates, i.e., γA (arm angle) and γM (gear output angle) for each subsystem which are superimposed with elastic deflections. Parameter values are listed in Table 7.2. Here, the bending rigidity is chosen about ten times less than it was for the laboratory robot in Fig. 7.8: EI = 95N m2 represents a rather weak beam, particularly when rigid bodies are attached which decrease the bending frequencies significantly.

   ωv



γ2

u

q 0 v1e

  y2

y1

q



P i PP  m Pu 1 x1   BMB     B γ1  u q B 

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ Figure 7.16.

Articulated Elastic Robot

Basic Considerations In the sequel we will use q˙ for minimal velocities (holonomic system). Furthermore, the motor torques are directly used for control input [Qmi := ui = iGi MM i , iGi : gear transmission ratio, MM i : motor torque]. If one assembles Gq˙ − Q := g to simplify notation, then eq.(7.117) reads ˙ = B(uol + ucl ), qT = (qTM qTA qTe ) M(q)¨ q + g(q, q)

(7.118)

(index M : motor, A: arm, e: elastic). The input uol represents the open loop part and ucl is the closed loop control. The open loop branch achieves the “gross motion”. Because the input refers to the motor coordinates only one has T

(7.119)

B = [E 0 0 ] .

Let us assume that the robot follows a reference path (index R) with small deviations yM , yA , qe where qe are the Ritz-coefficients:









qR yM q = qo + y =  qR  +  yA . 0 qe

(7.120)

372

7 Elastic Multibody Systems – Ordinary Differential Equations

Table 7.2. Parameter Values “Elastic Robot” Shoulder

Elbow

mM 1

Gripper

z

mM 2

mtip2

mtip1

Motor

Gear

Tip Mass

“Rigid Beam”

Beam

i = 1, 2 : kGi = 104

mM i = 1 −4

CM i = 9 · 10

iG = 105

mtip,i = 0.5

ρi = 2.6 · 103 (aluminum)

kG → ∞

10

Ctip,i ' 0

Ei = 7 · 10

height: hi = 0.0095

hi = 0.0600

width: bi = 0.019

bi = 0.0030

Length: Li = 0.5

Li = 0.5

Libraries:

MATLAB/SIMULINK

Partitioning M and g yields for a Taylor expansion w.r.t. the reference path



MM M  MTM A MTM e

MM A MAA MTAe













¨R q g0M uol MM e ¨R  +  g0A  −  0  + MAe   q Mee 0 0 g0e

{z −ho

| 

}

(7.121)

ucl Mo ¨ y + Po y˙ + Qo y =  0  0

where

 Po =

∂g ∂ q˙



 , Qo = o

∂[M¨ q + g] ∂q

 .

(7.122)

o

Since the deviations from qo which arise from arm and from gear elasticity can only be controlled with the joint motors, one has to presume controllability for eq.(7.118) with eq.(7.119). This means that uol is pervasive throughout all coordinates such that ho cannot vanish. It has thus to be treated as an error vector, the time history of which is, however, completely known from reference path planning: Let the deviations y in eq.(7.120) and eq.(7.121), respectively, vanish. One has then the virtual displacements δqT = δqTR [E E 0] which leads eq.(7.121) for ucl = 0 to

7.5

373

Reduced Number of Shape Functions: Controlled Systems     T δq [E E 0]   

[MM M + MM A ]











 g0M uol   T       ¨R + MM A + MAA  q g0A 0 − =0    T  0 g0e T MM e + MAe

(7.123)

from which the open loop control uol is determined: ¨R + (g0M + g0A ). ⇒ uol = MM M + MM A + MTM A + MAA q





(7.124)

This control refers to a reference model with reduced d.o.f. (stiff gears and arms). It is calculated for a desired path according to eq.(8.95). (Using the recursive representation leads to the same results when, in a first step to obtain uol , the gear and the elastic degrees of freedom are locked). If one drives the system (7.121) with uol from eq.(7.124), then one obtains the special structure for ho ,







¨R + g0A MTM A + MAA q

−




"

 
 def.  ¨R + g0A  = −ho =  + MTM A + MAA q  T 
T +

¨R + g0e MM e + MAe q

−fR +fR +feR

#

(7.125)

,

and the equations of motion (7.121) become

" Mo ¨ y + Po y˙ + Qo y +

−fR +fR +feR

#





ucl =  0 . 0

(7.126)

The aim of any controlled robot path is to generate a prescribed gripper position and orientation zTG = (rTG ϕTG ) ∈ IR6 [G: gripper, rG : gripper position, ϕG : gripper orientation (e.g. Cardan angles)]. A (small) deviation ∆zG from the desired values is characterized by a first-order Taylor expansion. (Notice that the gripper position does not depend on the motor coordinates when defined as in Fig. 7.6). This deviation shall be compensated with a (first-order) correction ∆zGR for the reference path which is calculated with the reduced reference model

 ∂v G  ∂ q˙ A ∆zG =   ∂ω G

∂vG ∂vG    ∂ q˙ e  ∂ q˙ R   yA , ∆zGR =   ∆qR ,   qe ∂ω G ∂ω G  ∂ q˙ e ∂ q˙ R R



∂ q˙ A

|

{z



}

FG



|

{z

FGR

(7.127)

}

hence !

∆zGR + ∆zG = 0 ⇒ ∆qR = −F+ GR FG



yA qe

 ⇒ ∆qR = [−FAc − Fec ]



yA qe

 := −Fc

yA qe





(7.128)

(( )+ : generalized inverse, c: “correction”, Fc partitioned according to the arm and to the elastic coordinates).

374

7 Elastic Multibody Systems – Ordinary Differential Equations

In eq.(7.128), the deviations y = (yTA qTe )T are, in principal, to be determined from the differential equations of motion (7.126) for every point in time. Such a procedure is expensive. We therefore look for a quasi-static equilibrium position (index Q: quasi-static) which involves the assumption that, for path correction, the deviation velocities and accelerations do not play an essential role: yQ 6= 0, y˙ Q ' 0, ¨ yQ ' 0. One obtains then from eq.(7.126), with Qo suitably partitioned,

"

−fR +fR +feR

#

" +

 ⇒

QM M QAM QeM

QM A QAA QeA





yAQ qeQ

=−

QM e QAe Qee

QAA QeA

# 0   0   yAQ  =  0  0

qeQ

−1 

QAe Qee

fR feR



(7.129)

,

and eq.(7.128) approximately changes to

 ∆qR = [−FAc − Fec ]

yAQ qeQ



(7.130)

.

There is a specific feature in eq.(7.129): the motor deviation yM Q is set to zero. That means that the arm angle deviations and the elastic deflections are measured w.r.t. the actual value qM . To control the motor angles themselves in a suitable way is a task of the closed loop control. Up to here, the correction term from eq.(7.130) refers to the corrected path. Insertion of (−fTR fTR fTeR )T = −Qo yQ from eq.(7.129) into the motion equations (7.126) yields

(7.131)

Mo ¨ y + Po y˙ + Qo (y − yQ ) = Bucl

where (y−yQ ) → 0 appears as objective function. But eq.(7.131) refers to the new reference path qR + ∆qR which would need to recalculate the corresponding open loop control. Eq.(7.120), for the new reference, reads









qR + ∆qR yM q =  qR + ∆qR  +  yA . 0 qe

(7.132)

However, rewriting eq.(7.132) as









qR yM + ∆qR q =  qR  +  yA + ∆qR  0 qe

(7.133)

leads to the interpretation to retain the “old” reference qR and to assign the corrections to the closed-loop branch. The corresponding quasi-static equilibrium then becomes the target value



0

yd =  yAQ qeQ







" ∆qR −FAc  +  ∆qR  = E − FAc 0 0

−Fec −Fec E

#

yAQ qeQ



(7.134)

which is (exponentially) achieved with a PD motor feedback ucl = −P (yM − yM d ) − D(y˙ M − y˙ M d ) (with scalar P, D for simplicity). Mostly, the motor angles are measured with encoders. The corresponding angular velocities may then for instance be obtained with the use of a “high ˆ˙ M , Chapter 8. gain observer”, q˙ M → q

7.5

375

Reduced Number of Shape Functions: Controlled Systems

There are two more specialties in this context to be mentioned. First, one usually defines the control aim to cancel the deviations y. However, in the present example the elastic deflections (which have been characterized by the Ritz coefficients) are also collected within y. It does not make sense to require the deflections to vanish. This is, under the action of the open loop control (which also includes gravity) virtually impossible with a joint motor. It is therefore suitable to also control the elastic deviations w.r.t. its quasi-static equilibrium which is determined with eq.(7.129). The elastic deformations are measured with strain gauges which deliver curvature signals. To extract the Ritz coefficients from these is expensive. The apparently most efficient procedure is to calculate the curvatures κ from κ = (∂κ/∂qe )qe and to use κ for control. Along with the Ritz series expansion, (∂κ/∂qe ) depends on space only. Since measurements usually are taken at discrete locations, (∂κ/∂qe ) results in a constant for every measurement point. Concerning elastic control, the state feedback then becomes an output feedback. Second, one basic problem in joint control is friction. In case that friction passes into stiction, controllability gets lost. It can be overcome with a high motor angle feedback gain which then, however, will result in intolerable oscillations once stiction is transcended. One possibility to solve this problem is once more to use an observer. Another one is to feed the curvatures forward instead of to feed them back. (Obviously, the overall stability has then carefully to be considered.) The control scheme is resumed in Table 7.3 and Fig. 7.17. Table 7.3. “Elastic Robot” Control Scheme

˙ = B(uol + ucl ) M(q)¨ q + g(q, q) eq.(7.118)

plant

open-loop control

closed-loop control

where

and





¨R + (g0M + g0A ) uol = MM M + MM A + MTM A + MAA q eq.(7.124) ucl = −P (yM − yM d ) − D(y˙ M − y˙ M d ) − Kκ (κ − κd )


T

yM = (qR − qM ), yM d = −Fc yTAQ qTeQ eq.(7.129), eq.(7.130) κ=



∂κ ∂qe



qe , κ d =





∂κ q eQ ∂qe eq.(7.130)

376

7 Elastic Multibody Systems – Ordinary Differential Equations

zGR

r

RRK

r

r

r

qo q˙ o ¨o q

e r e r e 6 OLC 6 6

CM

r

e- ROBOT 6 e 6

JC

r

r

ERK

zG

-

q q˙ κ

? e ? e

EC  e

6

−κd

Open Loop Control

Closed Loop Control

Figure 7.17. “Elastic Robot” Control Block Diagram Input: zGR Gripper Reference Trajectory RRK: Rigid Robot Kinematics, CM : Correction Model, OLC: Open-Loop Control, ROBOT: Robot Plant, JC: Joint Control, EC: Elasticity Control, ERK: Elastic Robot Kinematics Output: zG Gripper Trajectory

7.5

377

Reduced Number of Shape Functions: Controlled Systems

Computation Shape Functions 1

Two spline functions (third-order polynomials) which fulfill the boundary conditions along with two free conditions (normalizing conditions) are used for shape functions: s1 = 27 4

  2 x L

   2

x s2 = 3 L

−

−2

 3  x L

0.8

s2

0.6 , 0.4

 3  x L

s1

. 0.2

They are depicted in Fig. 7.18. The robot control concept is based on several basic assumptions which are checked by simulation:

0 0

0.2

Figure 7.18.

0.4

0.6

0.8

x L

Shape Functions

1. All deviations from the reference path qR remain small, eq.(7.120). The reference is calculated according to eq.(8.95) with T = 0.8 for the maneuver in Fig. 7.19. The motion is depicted at times tn = n∆t, ∆t = 0.1, n = 0, 1, · · · 14. The deviations are shown in Fig. 7.20 (dotted: desired value). 2. Velocities and accelerations are assumed negligible for the path correction [“quasi-static equilibrium”, eq.(7.129)]. This leads to the motor target values yM d = −[FAc | Fec ](yTAQ qTeQ )T [see eq.(7.134), gripper orientation not considered]. With two shape functions (two Ritzcoordinates) for each subsystem one obtains

"

1

0

0

0

1

−1 L

[FAc | Fec ] =

1 L 1 L

0

0

0

1 L

# .

(7.135)

The resulting yM d = (yM 1d yM 2d )T are depicted in Fig. 7.21; articulately they show the additional acceleration for t < T /2 to overcome the rigid body inertia delay and the countersteering for t > T /2 to avoid the overshoot which then would arise. 3. The elastic deflection can not be controlled to zero. [The (quasi-static) equilibrium qeQ does not vanish and consequently enters eq.(7.135)]. The corresponding curvatures (which are assumed to be measured at the bearing points) are fed forward according to Fig. 7.22. The figure also shows that the equilibrium curvatures at the end of the maneuver are nonzero (and negative).

378

7 Elastic Multibody Systems – Ordinary Differential Equations

Maneuver

0

0.8

reference angle

0.7 0.6 0.4

0.5 0.4 0.3 0.2

0.8

0.1 y 0

0.4

0.8

x

0 0

0.4

Figure 7.20.

Figure 7.19. Maneuver

0.8

1.2

1.6

t

Motor Angles (Shoulder)

Control Inputs

0.05

0.1

0.04

0.05

0

0.03 yM 1d

-0.05

0.02

-0.1

yM 2d

0.01

κ2

κ1

-0.15

0

-0.2

-0.01

-0.25

-0.02

-0.3

0 Figure 7.21. loop)

0.4

0.8

1.2

1.6

Motor Target Values (closed

t

0

0.4

Figure 7.22. Values)

0.8

1.2

1.6

t

Curvatures (dotted:Target

7.5

379

Reduced Number of Shape Functions: Controlled Systems

4. The closed-loop control is achieved with a PD motor angle feedback which is superimposed with a curvature feed-forward loop: P = 600 D = 30 . Kκ = −100

ucl = −P (yM − yM d ) − D(y˙ M − y˙ M d ) − Kκ (κ − κd ),

(7.136)

5. The results have been obtained with two shape functions for each subsystem. The simulation validates the assumptions the control scheme is based on. The question whether two shape functions are sufficient for the present problem is answered by augmenting the number of shape functions. For a quick check one may construe a third shape function just with s = s1 x and a fourth one with s = s1 x(1 − x), see Fig. 7.23. This corresponds to a duplication of the roots of s1 at x and (1 − x) – and goes along with a strict warning: theoretically one could produce as many independent shape functions as wanted just by consecutive root duplication. However, then, as a well-known fact, the problem becomes ill-conditioned. This is easily demonstrated with vRi = ξ i , i = 1, 2, . . .. The corresponding mass matrix 1 has the elements M (i, j) = ρAL o vi vj dξ = (ρAL)/(i + j + 1). With growing i and j, M tends to singularity. Fig. 7.24 shows the decrease of det M: for n = 4 shape functions one has already det M = (ρAL) · 0.7101 · 10−12 . Thus, for a quick check, s alone may be used. The obtained result does not show noticeable differences in simulation, even when the control is calculated with two shape functions while three shape functions are used for simulation. This is not self-evident (“spillover” problem). Hence, two shape functions are adequate for the present problem. Improvement

1

detM

0.8

s1

10−5

s2

0.6 s1

10−15

s1

10−25

0.4 0.2 0 0

0.2

0.4

0.6

0.8

Figure 7.23. Root Duplication

x L

0

1

Figure 7.24.

2

3

4

5

6

n

Determinant Breakdown

6. The results have also been proved experimentally for a comparable case, see (B. Gebler, 1987), (U. Kleemann, 1989) which once more validates the outlined procedure. The big progress since then is twofold:

380

7 Elastic Multibody Systems – Ordinary Differential Equations The methodical conditioning of (mechanical) procedures (which is the topic of this monograph. At that time, quite an amount of effort was spent in the derivation of the motion equations just for one special case) The tremendous advance in computer science. The control concept from Fig. 7.17 was initially separated into two parts: The open loop was considered for (time consuming) off-line computation. The results were then tabulated for different trajectories which entered the real-time closed-loop control, e.g. (H. Bremer, 1991). Such a separation is no longer necessary; in addition to the proposed recursive formulation we approach more and more closed real-time applications, see e.g. (H. Gattringer, 2006).

4

6.

Remark on Controlled Systems

As the above example shows, only very few shape functions are needed in controlled systems. This is, of course, due to control induced damping. However, a finite number of shape functions indicates a series truncation which for control may cause problems. Closed-loop control refers to the deviations w.r.t. the path that is generated by open-loop control. For the corresponding state equation one may think of applying a Riccati control with eq.(8.20). Clearly, this control guarantees asymptotic stability for the underlying mathematical model. This model is built with a finite number of Ritz coefficients. If one adds, for simulation, some more coefficients but retains the former control model, then stability is by no means warranted. The same situation arises in experiments where the control mechanism is calculated with a finite model while “nature” does not care about series truncation. In such a state feedback, u = −R−1 BT PR x := −Kx [see eq.(8.20)], the dimension of K depends on the number of Ritz coefficients. Considering the corresponding measurements which put the control into practice, it is almost tedious to extract the Ritz coefficients from them. The direct feedback of the measurements themselves reduces the dimension of the feedback matrix. Sometimes, at least when the control forces (or torques) act directly on a shaft for instance, one can draw a direct conclusion on stability. This is the case for the symmetric elastic rotor from Table 6.6, p.289. For an inertial representation (ΩA = 0) one has the motion equations



¨ y +Ω





0 +Go ˙  y+ −Go 0

diag ω 2

i=1···n

0

oi



0 y = diag ω 2

i=1···n

oi



vc 0 0 vc



fy fz



(7.137) where vc := To vb |η /mB is used for abbreviation, η indicates where the forces act and c stands for “control”. The forces shall be achieved with a magnetic

7.6

381

Remark on Controlled Systems

bearing, fi = kgap vTc qi + kcurrent Ii , i ∈ {y, z}, and kcurrent Ii := ui is used for control input. One obtains then for eq.(7.137), 



0 +Go ¨y + Ω y˙ + −Go 0

"

K 0 0 K

#



y=

vc 0 0 vc



uy uz



(7.138)

diag ω 2 − kgap vc vT . where K = i=1···n c oi When the measurement device is integrated within the bearing, then vm = vc (so-called “collocation”, index m: measurement). Along with 

vTc 0 0 0   0 vTc 0 0 C=  0 0 vTc 0 0 0 0 vTc



0  0  +k1 −k2 +k3 −k4  , , K = +k2 +k1 +k4 +k3 0   vc (7.139) ˆ = (A−BKC), an iterative minimization of eq.(8.29) can be carried out where A with some suitable starting values for K, enters the Lyapunov equation (8.28) for PL = const, controllability presumed. The four values in K are sufficient, since Ky represents the four possible (linear) mechanical forces, i.e. restoring forces (k1 ), damping forces (k3 ), gyroscopic forces (k4 ) and circulatory forces (k2 ), see also (U. Kleemann, 1981). Inserting the obtained results into eq.(7.138) yields the homogeneous equation "

0  0 , B=  vc 0 



+k3 vc vTc Go Ω−k4 vc vTc ¨ y+ T −Go Ω+k4 vc vc +k3 vc vTc



#

"

K+k1 vc vTc −k2 vc vTc ˙ y+ +k2 vc vTc K+k1 vc vTc



#

y=0

(7.140) which is asymptotically stable due to fulfillment of the Lyapunov equation. Premultiplying eq.(7.140) with y˙ T and having in mind that yT = (qTv qTw ) and qTv vc = v(η, t) := vc , qTw vc = w(η, t) := wc yields d dt

n n X 1 X 2 2 2 2 2 2 (q˙vi + q˙w ) + (ωoi qvi + ωoi qwi ) + (k1 − kgap )(vc2 + wc2 ) i 2 i=1 i=1

( "

#)

+k3 (v˙ c2 + w˙ c2 ) + k2 (w˙ c vc − v˙ c wc ) = 0,

(7.141)

382

7 Elastic Multibody Systems – Ordinary Differential Equations

hence n n X 1 X 2 2 2 2 2 2 U := (q˙vi + q˙w ) + (ωoi qvi + ωoi qwi ) + (k1 − kgap )(vc2 + wc2 ) , i 2 i=1 i=1

"

#

d U = −k3 (v˙ c2 + w˙ c2 ) − k2 (w˙ c vc − v˙ c wc ) (7.142) dt where U serves for a Lyapunov function, see eq.(8.31). Clearly, dU/dt < 0 holds for the chosen calculation model. This is quite trivial. But: The system under consideration is asymptotically stable, independent of series truncation, as long as k2 = 0. It remains asymptotically stable for k2 6= 0 if (w˙ c vc − v˙ c wc ) for more and more summation terms in the Ritz series does not change sign. It follows that, with an abdication of optimization, a simple PD-controller (where k2 = 0, k4 = 0) is uncritical as long as collocation is feasible. In conclusion, we emphasize that the advantage of needing only a few shape functions for the control problem has to be paid for with careful investigation of the spillover problem.

Chapter 8 A SHORT EXCURSION INTO STABILITY AND CONTROL

1.

Optimality

Fundamentals of Classical Optimization (Calculus of Variations ) Basic Problem Zx1

J=

f (y, y 0 )dx → min, xo , x1 fixed, ( )0 = d( )/dx.

(8.1)

xo

Assumption: yopt exists. 

Approach: y = yopt +  · η = yopt +

dy d



, η(xo ) = 0, η(x1 ) = 0 =0

⇒ Definition: δ( ) = (d( )/d)=0 · . (Remarks: δy is arbitrary in magnitude (complete Taylor expansion). The problem is turned algebraic but shifted to the investigation of what classes of (arbitrary) functions are to be considered (compare remarks on p. 17 et seq.). We assume steady and continuously differentiable functions in the sequel.) Necessary Condition x R1

"

#

∂f δy + ∂f δy 0 dx = δJ = 0 xo ∂y " # ∂y " # x1 " # x R1 ∂f ∂f ∂f ∂f ∂f d d =0⇒ − dx 0 δydx + − dx 0 = 0 δy xo ∂y ∂y ∂y ∂y ∂y xo |

{z

}

[(∂f /∂y)( · η)]xx1o = 0 (8.2) 383

384

8 A Short Excursion into Stability and Control

(fundamental lemma of calculus of variation1 ⇒ Euler’s differential equation of variation). Sufficient Condition

(J − Jopt ) =

 Zx1  xo



∂J f (y, y 0 ) −   ∂x 

∂J



+

∂y

opt

! opt

  y 0  dx > 0. 

(8.3)

1) Determination of the differential quotients: variation of the upper limit yields

J = J(x1 (), ) =

x1R() xo

"

1 = f (x1 ) ∂x ∂ +

x R1 xo

f (y, y 0 )dx, δJ = dJ d h

i

∂J ∂x1 + ∂J ·  = ∂x ∂ =0 1 ∂ h

i

·

=0

#

∂ 0 ∂ f (y, y )dx

· . =0

(8.4) Setting (∂x1 /∂)=0 ·  = δx1 (variation at the right end with  6= 0 !) yields after an integration by parts: x1Z (=0)"

δJ = f (x1 )δx1 + xo

#

∂f d ∂f δydx + 0 δy=0 − 0 dx ∂y ∂y ∂y

∂f

(8.5)

[notice the upper integral bound x1 ( = 0). Interpretation: a Taylor expansion, e.g. f (x, y) = fo + (∂f /∂x)o ∆x + (∂f /∂y)o ∆y + · · ·, needs the same reference point o for every summation term]. Using the right end point for reference d 0 0 y(x1 (), ) ·  = y=0 δx1 + δy=0 ⇒ δy=0 = δy() − y=0 δx1 d =0 (8.6) 

δy() =



1 “Let η be steady and continuously differentiable, vanishing at x , x . Then, for M being a steady function in o 1 x1 [xo , x1 ], it follows that M ηdx = 0 ⇒ M ≡ 0 in [xo , x1 ] because: assume M > 0 for ξ ∈ [xo , x1 ]. x

R

o

Since M is a steady function we can define an interval [ξ1 , ξ2 ] 3 ξ such that M > 0 within [ξ1 , ξ2 ]. Now, choose η ≡ 0 outside [ξ1 , ξ2 ] and η = (x − ξ1 )2 (x − ξ2 )2 inside R x[ξ11 , ξ2 ]. This function has steady derivatives in [xo , x1 ] and vanishes at xo , x1 . Nevertheless, it yields M ηdx > 0 which is in

Rx

xo

1 contradiction to M ηdx = 0. It is thus impossible that M 6≡ 0 for any ξ ∈ [xo , x1 ].” [(O. Bolza, xo 1909), pp. 25,26]. Bolza was one of Weierstraß’ students. Weierstraß’ lectures are unpublished.

8.1

385

Optimality

yields "

!

#

"

∂f

∂f

!

x1Z (=0)"

#

#

d ∂f y δx1 + δy() + δydx. − δJ = f − 0 0 dx ∂y 0 ∂y ∂y x1 ∂y x1 xo (8.7) 2) When inserting the solutions yopt from the Euler differential equation (8.2) (necessary condition), then the integral term vanishes. By comparison of coefficients we obtain ∂f

0

"

f−

dJopt =

y

∂y 0

|

#

"

0

dx + }

def

∂f

|

(∂J/∂x)opt = −Hopt

!#

∂y 0

opt

{z

∂f

dy.

(8.8)

opt

{z

}

(∂J/∂y)opt

(Remark: inserting the Euler solution now defines the set of curves. Eq.(8.8) has thus nothing more in common with variations but defines the increase of Jopt by (infinitesimal) upper limit offset. δ is therefore replaced with d.) H is the “Hamilton function”. Since optimality is requested for all values of x, the Hamilton function (∂J/∂x)opt = −Hopt has necessarily to be zero. Insertion of Hopt = 0 into eq.(8.3) leads to Hopt = 0, (J − Jopt ) =

x R1 xo

 f (y, y 0 ) −

∂f ∂y 0



!

R1 def x

y 0  dx = opt

Edx > 0 ⇒ E ≥ 0

xo

(8.9) (E: Weierstraß’ excess function2 ). (Remark: The determination of an extremal via E ≥ 0 is not easy at all because “every” possible comparison function has to be discussed.) Constraints x R1

J=

x" o

⇒

x R1 xo

f (y, y 0 )dx → min #

V

g(y, y 0 ) = 0

∂f δy + ∂f δy 0 dx = 0 V ∂y ∂y 0

"

#

∂g δy + ∂g δy 0 = 0. ∂y ∂y 0

(8.10)

(The variations δy, δy 0 are no longer arbitrary!). Lagrange’s trick: multiply the constraint variation with a new variable λ and integrate it, summarize and 2 Karl

Theodor Weierstraß, *1815 in Ostenfelde, †1897 in Berlin

386

8 A Short Excursion into Stability and Control

integrate by parts: x R1

"

#

"

#

∂f δy + ∂f δy 0 dx + xR1 λ ∂g δy + ∂g δy 0 dx xo ∂y xo ∂y 0 ∂y ∂y 0 " # x R1 ∂ ! (f + λg) − d ∂ (f + λg) δydx = 0. = xo ∂y | {z } dx ∂y 0

(8.11)

L

In common words: determine λ such that the square bracket in the second line def

becomes zero, making δy arbitrary. (f + λg) = L is called the Lagrange function. Remark: in case the constraint is defined as an integral over the domain (xo , x1 ) (“isoperimetric problem”), then Lagrange’s trick only works if λ does not depend on x. However, this is indeed the case for such problems, as can easily be shown by inserting a new variable as done in the next paragraph. Free Upper Limit Rx

def

f (y, y 0 )dx = q ⇒ q 0 = f (y, y 0 )

xo

⇒ J = q(x1 )

V

q(x1 ) +

x R1

λo (q 0 − f )dx → min.

(8.12)

xo

(Trick: the free upper limit x is replaced with a fixed limit x1 where (any!) x1 is accompanied by a corresponding constraint). The variation of q yields δq(x1 ) +

x R1 xo

(−λ0o )δqdx + λo δq(x1 ) = 0 ⇒ λ0o = 0, (1 + λo )δq(x1 ) = 0

⇒ λo = −1. (8.13) The Dynamical Problem

Let x →ht, y → x, f (y, y 0 ) → f (x, u), constraint x˙ − a(x, u) = 0 i T ˙ ⇒ L = λ (x − a(x, u)) + λo (q˙ − f (x, u)) . Variable q is of no interest since its variation is already accounted for in eq.(8.13). Along with λo = −1 one obtains for the remaining variables xT = (xT uT λT ), L(x) = λT (x˙ − a) + f = λT x˙ − (λT a − f ) |

{z

H

}





(8.14)

where the H follows from eq.(8.8),  −H = L −



∂L ∂ x˙

 

x˙ = λT x˙ − (λT a − f ) − ( λT





x˙  u˙  = −(λT a − f ) 0 0) λ˙

8.1

387

Optimality

d ∂L ⇒ Euler-Eq. (8.2) for f → L (necessary): ∂L − dt ∂x ∂ x˙    d (λT 0 0) = 0T − dt = (0 0 x˙ T ) − ∂H ∂H ∂H ∂x ∂u ∂λ



Weierstraß’ Function (8.9) for f → L (sufficient): E = L −

(8.15)

∂L ∂ x˙



x˙ ≥ 0, opt

[x˙ = a(x) inserted] !

⇒ −H + aT (λ − λopt ) ≥ 0. (8.16)



Optimal Values according to (8.8) :

∂J ∂x





= L− opt



 

∂L x˙ ∂ x˙

opt

= −Hopt = −H(xopt , uopt , λopt ) = 0. (8.17) The results are summarized in Table 8.1.

1.1.

Results from Classical Optimization Theory

Comment: In Table 8.1 we presume autonomous systems (the time t does not appear explicitly). This is, for mechanical systems, the most general case (compare Chapter 3, section 3.3). As the canonical equations show (left column) we furthermore presume steadiness. The first canonical equation is thereby not of interest (it re-delivers the state equations). The right column yields for λ = λopt but arbitrary x and u: E = −H > 0 ⇒ H(x, u, λopt ) < 0. The optimal value (left column) is H(xopt , uopt , λopt ) = Hopt = 0. Hence, Hopt represents a maximum w.r.t. the “section plane” λ = λopt . The third canonical equation can therefore more precisely be specified as (∂H/∂u) = 0 ⇒ Hopt = maxH w.r.t. u. Obviously, this condition is still only necessary, and, up to here, valid for steady and continuously differentiable u. The control range U in (3∗ ) has therefore to be considered unrestricted. However, Pontryagin3 and coworkers have been lucky to prove its validity for restricted U as well (“Pontryagin’s Maximum Principle”, around 1950),

3 Lev

Semenovich Pontryagin, *1908 in Moscow, †1988 in Moscow

388

8 A Short Excursion into Stability and Control

Table 8.1. Results from Classical Optimization Theory x˙ = a(x, u). Determine u such that J =

R

f (x, u)dt → min

H = λT a(x, u) − f

E = −H(x, u, λ) + aT (λ − λopt )

necessary

sufficient

H(xopt , uopt , λopt ) = 0 “Canonical Equations” (Euler):

 +

 −

∂H ∂λ



∂H ∂x



 ±

∂H ∂u

⇒ Hopt =

= x˙ T

(1)

T = λ˙

(2)

“Excess Function” (Weierstraß): E>0

 =0

(3)

(3∗ )

max H u∈U

⇐

e.g. (P.C. M¨uller, K. Popp 1972). This is an essential extension, because a timeoptimal control, for instance, with unresticted domain U would of course lead to u → ∞. In engineering, it is (successful) tradition to use the canonical equations (e.g. LQR-control) as well as the maximum principle (e.g. time-optimal control) for control calculation, based on necessary conditions. This demonstrates once more the necessity to prove the results by simulation (which does, of course, not include the proof of mathematical models).

1.2.

Riccati- (or LQR-) Control

Let f dt with f = (xT Qx + uT Ru)/2 be minimized for u from the linear plant x˙ = Ax + Bu. Here, Q = QT ≥ 0, R = RT > 0 are suitably chosen weighting matrices with (A, Q) being observable [see eq.(8.72) and eq.(8.75), respectively]. The Hamilton-function R

t

R J = 12 (xT Qx + uT Ru)dt → min to ⇒ H = λT (Ax + Bu) − (xT Qx + uT Ru)/2

(8.18)

8.1

389

Optimality

yields the optimal (open-loop) control T



= λ˙ = −AT λ + Qx − ∂H ∂x  T (8.19) = 0 = BT λ − Ru ⇒ u = R−1 BT λ ± ∂H ∂u as the solution of a differential equation for λ. In order to obtain a maximum ! for H w.r.t. u, (∂ 2 H/∂u∂uT ) = −R, R has to be positive-definite. The optimal closed-loop control is calculated with the ansatz λ = −PR x. Insertion of λ = −PR x into eq.(8.19) yields, along with x˙ = Ax + Bu, u = −R−1 BT PR x, P˙ R + AT PR + PR A − PR BR−1 BT PR + Q = 0,

(8.20)

and the optimal criterion value is4 1 Jopt = x(to )T PR x(to ). 2

(8.21)

The optimal (state feedback) control follows from the solution PR = PTR of the Riccati5 matrix differential equation (8.20). For time-invariant systems, the ansatz PR = const. is sufficient, leading to the corresponding algebraic Riccati equation. In linear mechanical systems, the state equation reads d dt



q s˙





=

0 H+ (q) −M−1 (K + N) −M−1 (D + G)



q s˙





+

0



u. M−1 B (8.22) Let, for simplicity, H(q) = E (holonomic system). We have then, with u from eq.(8.20), the equations of motion M¨ q + (D + G)q˙ + (K + N)q = −Qcontr. q − Pcontr. q˙

(8.23)

d T ˙ Insert x˙ = Ax + Bu and λ ˙ = −AT λ + Qx to come out with at dt [x λ] = [x˙ T λ] + [xT λ]. d T T AT λ + uT BT λ] + [xT (−AT λ + Qx)] = uT BT λ + xT QT x = uT Ru + xT Qx because [x λ] = [x dt t ˙ of BT λ = Ru. If we consider the criterion 21 x(t1 )T Qe x(t1 )+ 1 [(xT Qx+uT Ru)+λT (x−Ax−Bu)]dt t 4 Look

R

o

with an additional weighting for the end state, then the variation for x with subsequent integration by parts Rt T ˙ + λT A − xT Q)δxdt = 0 ⇒ λ(t1 ) = −Q x(t1 ). The above yields xT Qe δx |t1 +λT δx |tt1o − 1 (λ e to

formulated criterion with final state weighting yields 1 T x PR x |to . 2

1 T x Qe x |t1 2

+ 12

R t1

d (xT λ)dt to dt

or, with λ = −PR x, Jopt = Next, let Qe →0 to obtain eq.(8.21). 5 Jacopo Francesco Riccati, *1676 in Venetia, †1754 in Treviso

= − 12 xT λ |to

390

8 A Short Excursion into Stability and Control

where Qcontr. , Pcontr. are suitable abbreviations. These are general matrices; splitting them into its symmetric and antimetric parts, Qcontr. = (K + N)contr. and Pcontr. = (D + G)contr. , one obtains M¨ q + (D + G)tot q˙ + (K + N)tot q = 0, Dtot = +DTtot = D + Dcontr. , Gtot = −GTtot = G + Gcontr. ,

(8.24)

T

Ktot = +Ktot = K + Kcontr. , Ntot = −NTtot = N + Ncontr. .

1.3.

Control Parameter Optimization

The question arises whether the control R T structure u = −[Qcontr. | Pcontr. ]x can a` -priori be considered. In this case, (x Qx)dt/2 → min is for instance used ˆ Here, once more Q = QT ≥ 0 is w.r.t. x˙ = (A − B[Qcontr. | Pcontr. ])x := Ax. ˆ Q) being observable. The Hamilton function suitably chosen with (A, t

R J = 12 (xT Qx)dt → min to ˆ − (xT Qx)/2 ⇒ H = λT Ax

(8.25)

yields 

−

∂H

T

ˆ T λ + Qx, = λ˙ = −A

(8.26)

∂x but (∂H/∂u) does not exist. Let λ = −PL x.

(8.27)

Eq.(8.26) then becomes the Lyapunov6 matrix differential equation ˆ + Q = 0, PL = PT , ˆ T PL + PL A P˙ L + A L

(8.28)

where P˙ L = 0 is sufficient for linear time-invariant systems (yielding the algebraic Lyapunov matrix equation). The optimal criterion value is7 1 Jopt = x(to )T PL x(to ). 2 6 Aleksandr 7 with

Michailovich Lyapunov, *1857 in Jaroslaw, †1918 in Odessa the same argumentation as for the Riccati control

(8.29)

8.2

391

Stability

Thus, starting with suitable values for [Qcontr. | Pcontr. ], eq.(8.28) is solved iteratively to minimize eq.(8.29). The motion equations have the same structure as eq.(8.24). Due to the optimization process, the (asymptotic) stability of eq.(8.24) is self-evident.

2.

Stability

The most general consideration is due to Lyapunov: Consider the first-order differential equation system (“state equation”) [M¨s + G˙s − Qe ] = 0, s˙ = H(q)q˙ d ⇒ dt

q s˙

H+ s˙

!

=

!

:= x˙ = a(x, t)

M−1 [Qe − G˙s]

(8.30)

(H+ : generalized inverse) with equilibrium state x = xequi = 0. (Remark: if xequi 6= 0: transform eq.(8.30) with x := x − xequi ). Then, if a (positive definite) function U (x, t) exists such that 

 0

(8.31)

the equilibrium (in short, the system) is called asymptotically stable for (dU/dt) < 0, and marginally stable for (dU/dt) = 0, and unstable for (dU/dt) > 0. That means, in common words, that after any disturbance the system under consideration will come to rest (asymptotic stability) or it will move with limited k x k ∀ t (marginal stability), or k x k will grow infinitely (instability). For scleronomic systems, U (x) is sufficient [⇒ (∂U/∂t) = 0]. Plausibility: Consider the linear (one d.o.f.) system y¨ + 2Dy˙ + ω 2 y = 0, or, in state space representation with xT = (y y), ˙ 

x˙ =

0 1 −ω 2 −2D



x.

(8.32)

A possible choice for U is the sum of kinetic and potential energy, U = T + V = (y˙ 2 + ω 2 y 2 )/2 which reads in terms of x, 1 U = xT 2



ω2 0 0 1



x.

(8.33)

392

8 A Short Excursion into Stability and Control

One obtains then, for eq.(8.31), ∂U

Ax = xT



∂x

ω2 0 0 1



0 1 2 −ω −2D



x = −Dy˙ 2 .

(8.34)

The solution of the linear damped oscillator is known with y(t) = √ exp(−Dt)[yo cos νt + (y˙ o /ν) sin νt], ν = ω 2 − D2 . Its (asymptotic) stability or instability clearly depends on the sign of D; the system remains marginally stable for vanishing D. The crucial point for the general (nonlinear) case is to find an adequate Lyapunov function U (x). However, considering a Taylor series in the form x˙ = a(x) = Ax + h(x),

(8.35)

then the asymptotic stability/instability of x˙ = Ax

(8.36)

yields asymptotic stability/instability of the original nonlinear system x˙ = a(x) as long as lim kxk→0

k h(x) k =0 kxk

(8.37)

holds, which in mechanical systems is always true (P.C. M¨uller 1977). (The above statements also hold for rheonomic systems.) Since in controlled systems one is mostly interested in asymptotic stability, the above considerations deliver a very powerful tool. If, on the other hand, the linearized system is only marginally stable, then h(x) in eq.(8.35) plays an essential role. 5 Example: Consider a two d.o.f. oscillator according to Fig. 8.1 with a stable equilibrium position at q = 0, β = 0. For m1 = m2 = 1 where m1 represents a homogeneous bar and m2 a point mass (moment of inertia neglected) as well as kβ = 3gL, kq = (9/2)(g/L), one obtains the equations of motion



L2 + (L + q)2 0 3 0 1



β¨ q¨

 +

3 L + q)g sin β + 3gLβ 2(L + q)q˙β˙ − ( 2 9g −(L + q)β˙ 2 − (1 − cos β)g + 2L q

!

 =

0 0

 .

(8.38)

8.2

393

Stability

Here, linearization would yield two decoupled (i.e. independent) linear oscillators. The resulting motion, however, shows a strong interrelation between the two variables. The in-plane motion results from the fact that kq , kβ are chosen such that the frequencies of the linearized systems are ωq /ωβ = 2. If the q-oscillator would be driven by a strong motor which cancels the feedback from β, then one would have parametric resonance (instability. Such a case happened for instance in 1976 when testing an active controlled Maglev (=magnetic levitated) vehicle. m2 then represents the vehicle and m1 the elevated guideway). In the present case, the energy input for q and β interchange periodically as Fig. 8.2 demonstrates.

Iz

 z  

m2

 r    kq 

R=L+q

   r m1

kβ

   h P P

-

Ix

PP q x P

Figure 8.1.

q, β

Iz

1

5.5

0.5

5

0

4.5

-0.5

4

-1

3.5

Two d.o.f. Oscillator

3

-1.5 0 10 20 30 40 50 60 70 t

Figure 8.2. Trajectories q(t) and β(t) (bold) over Time t [q(0)=1.5, β(0)=0]

4

β

6

-3

-2

-1

0

1

2

Ix

Figure 8.3. Motion of m2 in the Inertial Plane [L=4.5]

394

8 A Short Excursion into Stability and Control

A second example is the free rotation of a gyro. Here it is well known that a rotation around the middle axis of moment of inertia is “unstable”. 5 Example: Consider a rigid body rotating rapidly around the z-axis with ωz = Ω and undergoing small angular velocity disturbances ωx , ωy . The corresponding equations of motions are, in the absence of impressed torques, L˙ + ω e R L = 0. In terms of body-fixed coordinates one obtains, in components, Aω˙ x − (B − C)Ωωy = 0, B ω˙ y − (C − A)Ωωx = 0, ˙ − (A − B)ωx ωy = 0. CΩ

(8.39)

Since ωx , ωy are small values, its product may be neglected yielding Ω ' const. Differentiating the first equation w.r.t. time and inserting ω˙ y from the second equation yields ω ¨ x + [(B − C)(A − C)/AB]Ω2 ωx = 0

(8.40)

which represents an unstable oscillator for A < C < B or B < C < A. Indeed, for A : B : C = 2 : 4 : 3, Fig. 8.4 indicates instability for the beginning of motion. However, regarding the trajectories over a longer time period, one can not really speak about instability. One has thus to be very cautious with the term “instability”. The critical case, namely h(x) playing a significant role for undamped mechanical systems, refers to special effects but can never cause infinite growth due to limited energy.

ωi

ωi Ω/10

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

5 0 ωx -5 ωy -10 0

0.2 0.4 0.6 0.8

1

Figure 8.4. Angular Velocities ωi (t) over a Short Time Interval, Ω(0) = 10.

4

Ω

t

0

2

4

6

8

t

Figure 8.5. Angular Velocities ωi (t) over a Longer Time Interval, i ∈ {x, y}

8.3

395

Linear Time-Invariant Systems

3.

Linear Time-Invariant Systems

In case of holonomic systems [with H(q) = E, compare eqs. (8.22), (8.23)], the equations of motion contain full information. In this case, one obtains some basic assertions already from the motion equations. This motivates us to concentrate on these in the sequel. In case that non-holonomic velocities are used for mathematical description (inevitable in non-holonomic systems but admissible in holonomic systems) one has to apply the corresponding state equation results. The linear homogeneous differential equations read [M¨ q + (D + G)q˙ + (K + N)q] = 0, q ∈ IRf , d dt

q q˙

!



=

0 E −1 −1 −M (K + N) −M (D + G)



q q˙

(8.41)

!

: x˙ = Ax,

(8.42) where the individual matrices are split into its symmetric and antimetric parts, M = +MT > 0 : mass matrix, D = +DT : damping matrix, G = −GT : gyroscopic matrix, K = +KT : conservative restoring matrix, N = −NT : nonconservative restoring matrix.

(8.43)

Notice that the total gyroscopic terms from nonlinear equations now contain skew-symmetric parts (as from Coriolis effects, matrix G) and symmetric parts (as from centrifugal effects, matrix K) [see also the remarks on p.325]. Furthermore, these are replenished with terms coming from impressed forces and torques as from potentials (matrix K) or from dampers (matrix D). The appearance of a skew-symmetric N-matrix often depends on the choice of reference frame (see e.g. inner or outer damping in rotors, p.277). Thus, except N, the matrices according to eq.(8.43) have always a clear interpretation.

3.1.

Fundamental (or Transition) Matrix

The solution for eq.(8.41) and eq.(8.42), respectively, is obtained with the eigenvector-eigenvalue approach q = qeλ t ∈ IRf ⇒ [Mλ2 + (D + G)λ + (K + N)]q = 0, x = xeλ t ∈ IRn=2f ⇒ [λE − A]x = 0, ⇒ det[Mλ2 + (D + G)λ + (K + N)] = 0 λi , i = 1 · · · 2f [eigenvalue], T T T ⇒ xi = (qi λi qi ) [eigenvector(s)].

_

det[λE − A] = 0 (8.44)

396

8 A Short Excursion into Stability and Control

There exist n = 2f different eigenvectors if λi 6= λj ∀ i, j, or λi is a v-times multiple eigenvalue and d = n − Rank(λi E − A) = v. In this case, the solution is expressed with the modal matrix X and a vector of constants c ∈ IRn , diag eλi t . x(t) = XeΛ t c where X = [x1 , x2 · · · xn ], eΛ t = i=1···n n

o

(8.45)

A transformation with the modal matrix X decouples the state equation, diag {λi } x = Xξ ⇒ ξ˙ = X−1 AXξ = Λξ, Λ = i=1···n

(8.46)

where Λ = X−1 AX follows from the eigenvector problem (8.44), xi λi = Axi , when all the solutions are considered: [x1 · · · xn ]diag{λi } = A[x1 · · · xn ]. If λi is a v-times multiple eigenvalue and d = n − Rank(λi E − A) < v, then only d < v eigenvectors exist and the missing part for the ith solution is built with v − d independent partial solutions according to h

i

x(i+1) = x(i+1) + x(i) t eλi t , h

2

i

x(i+2) = x(i+2) + x(i+1) t + x(i) t2! eλi t , ··· h i t(v−d) x(i+v−d) = x(i+v−d) + x(i+v−d−1) t + · · · + x(i) (v−d)! eλi t .

(8.47)

Insertion of eq.(8.47) into the state equation x˙ = Ax and setting the terms which are assigned to tk , k = 0, · · · (v − d) individually to zero leads to the vector chain (λi E − A)x(i+1) = −x(i) , .. . (λi E − A)x(i+v−d) = −x(i+v−d−1)

(8.48)

for the determination of v − d so-called “principal vectors”. Let, for simplicity, i = 1. Along with xi = xi exp{λi }, eq.(8.48) may be rewritten in the form

8.3

397

Linear Time-Invariant Systems 

λ1

1 λ1

  

[x1 x2 x3 · · ·] 

0 1 λ1



..

   = A[x1 x2 x3 · · ·] : XJ = AX 

(8.49)

.

where J is a Jordan8 matrix containing v − d − 1 times the value 1 in the upper secondary diagonal of Λ where the v − d multiple eigenvalue arises. A modal transformation X1 AX = J does thus not decouple the system equations in this case. 5 Example: Consider a gyro with fixed point in the absence of restoring forces [ωo2 = 0 in eq.(6.165 b)]. The state matrix reads



0  0 A=  0

0 0 0

0

0

1 0 0 A + Ω B



 0 1    − A Ω  ⇒ λ1/2 = 0, x1 =  B 0







1 0   ,x = 0  2  0

0 1  . 0  0

(8.50)

Here, rank(λ1/2 E − A) = 2, v = 2, d = 4 − 2 = 2: two eigenvectors exist for the double eigenvalue λ1/2 . The physical interpretation is that a torque-free spinning gyro will not topple after a disturbance but will react with an oscillation (“nutation”). For the non-spinning gyro (Ω = 0), the state equation is decoupled for α and β and its ˙ T . Thus, setting Ω = 0 and canceling the second and fourth column derivatives in x = (α β α˙ β) and row in eq.(8.50) we are left with the state equation for α, α˙ yielding

 A=

0 0

1 0

 ⇒ λ1/2 = 0, rank(λ1/2 E − A) = 1,

 (λ1/2 E − A)x1 =

 (λ1/2 E − A)x2 =

0 0

−1 0



0 0

−1 0



x11 x12



x21 x22



λ1/2 t

⇒ x(t) = [x1 c1 + (x1 t + x2 )c2 ]e

 =

 =−

 =

1 0

0 0



1 0



 ⇒ x1 =

 ⇒ x2 =







α α˙



 =

αo 0



,



0 1

 c2 .

(8.51)



 +

α˙ o t α˙ o



4 8 Marie

0 1

t+

Clearly, c1 = αo , c2 = α˙ o (initial conditions), thus x(t) =





1 0

c1 +

1 0

Ennemond Camille Jordan, *1838 in Lyon, †1922 in Paris

.

(8.52)

398

8 A Short Excursion into Stability and Control

The solution for the general case is x(t) = XeJ t c.

(8.53)

Verification for f = 2, λ1 = λ2 = λ, d = 1:

"

eλ t 0

x(t) = x1 eλ t c1 + [x2 + x1 t] eλ t c2 = [x1 x2 ]

teλ t eλ t

#



c1 c2

series expansion: 2 3 eλ t = 1 + λ t + λ2 t + λ3 t + · · · , 2! 3! 3 3 2 t · eλ t = t + λ t2 + λ2 t + · · · = t + 2λ t + 3λ2 t + · · · 2! 2! 3!

" ⇒

eλ t teλ t 0 eλ t

#

 =









λ2 2λ 0 λ2



| {z }

|

{z

}

1 0 λ 1 + t+ 0 1 0 λ

| {z } E

2

J

t2 + 2!

J



λ3 3λ2 0 λ3



|

{z

}

3

t3 + · · · = eJ t . 3!

J

From eq.(8.53) it follows for the initial conditions xo that x(t = 0) = xo = Xc ⇒ c = X−1 xo ⇒ x(t) = XeJ t X−1 xo .

(8.54)

A series expansion yields 2 2 XeJ t X−1 = X E + Jt + J t2! + . . . X−1





2 = XX−1 + XJX−1 t + (XJ X−1 ) (X JX −1 ) t2! · · · = eAt (8.55)

|

{z

E

}

where XJX−1 = A follows from eq.(8.49). Notice that in case of different eigenvalues or n − rank(λi E − A) = di = vi the Jordan matrix reduces to the diagonal matrix of eigenvalues, J → Λ, see eq.(8.45). The solution of x˙ = Ax is thus x(t) = [XeJt X−1 ]xo = [eAt ]xo := Φxo

(8.56)

where Φ is called the Fundamental Matrix. Its properties follow directly from its series expansion: Φ(t = 0) ˙ Φ Φ (t1 + t2 ) Φ−1 (t)

= = = =

E, ΦA = AΦ, Φ (t1 ) Φ (t2 ) , Φ(−t).

(8.57)

8.3

399

Linear Time-Invariant Systems

With the knowledge of Φ also the solution of an inhomogeneous state equation x˙ = Ax + b

(8.58)

is obtained using the technique of variation of constants: let the particular soluti˙ on be xpart = Φc ⇒ x˙ part = Φc+Φ˙ c = AΦc+Φ˙c [see eq.(8.57)]. Insertion of this solution into eq.(8.58) yields c˙ = Φ−1 b. One obtains thus for the solution of eq.(8.58)9 , Zt

x(t) = xhom + xpart = Φxo + Φ

Φ−1 bdτ.

(8.59)

o

[No additional integration constant arises in this representation since the initial condition is defined by x(t = 0) = xo . Remark: for the particular solution it is mostly easier to use a direct approach, based on the actual form of b(t).]

3.2.

Theorem of Cayley and Hamilton

The theorem of Cayley10 and Hamilton reads: any A ∈ IRn,n with constant coefficients fulfills its own characteristic polynomial P : det(λE − A) = P (λ) = an + an−1 λ + an−2 λ2 + · · · + λn = 0, P (A) = an E + an−1 A + an−2 A2 + · · · + An = 0. (8.60) Proof: Pre- and post-multiplication with X−1 and X, respectively, and inserting unit matrices of the form XX−1 yields X−1 [an E an E an E

+an−1 A +an−1 X−1 AX +an−1 J

+an−2 A2 +an−2 X−1 A(XX−1 )AX 2 +an−2 J

+ · · · + An ]X = +··· = n ··· + J = 0

(8.61)

where X−1 AX = J follows from eq.(8.49). In eq.(8.61), last line, the sum over the diagonal elements (i, i) represents the characteristic polynomial P (λ) |λ=λi = 0. J to the power of k reads, m k k | . The non-diagonal elements in for the non-diagonal elements, J (i, i + m) = 1 ∂ λ m! ∂λm λ=λi

9 Two

particular solutions x1 and x2 which fulfill eq.(8.58) yield by subtraction (x˙ 1 − x˙ 2 ) = A(x1 − x2 ) which represents the homogeneous solution xhom = (x1 − x2 ). From this follows x1 = xhom + x2 for any x1 , x2 , thus x = xhom + xpart . The adaption to the problem under consideration is achieved by the initial condition xo which has to fulfill eq.(8.58). 10 Arthur Cayley, *1821 in Yorkshire, †1895 in Cambridge

400

8 A Short Excursion into Stability and Control

n 1 ∂ m [a + a = 0, m = 1 · · · (v − d), n−1 λ + · · · + λ ] m! ∂λm n λ=λi which is the characteristic result for functions with multiple poles.

eq.(8.61) summarize to

n

o

Conclusion: An = −a1 An−1 · · · − an E, n+1 n = −a1 A A  · · · − an A  = −a1 −a1 An−1 − · · · − an E − a2 An−1 − · · · − an A, An+2 = · · ·

(8.62)

⇒ any polynomial in A with grade > (n − 1) can be substituted with a polynomial with grade (n − 1). Application: Frobenius11 transformation. Resolve eq.(8.60), second line, for An = An−1 A and premultiply with (arbitrary) tT ∈ IRn to obtain      

tT E tT A .. . tT An−1

|

{z

T





   A =      }

0 0 .. .

1 0



···  ···  

0 1

−an · · · · · · −a1

|

{z

   }|

AF

tT E tT A .. . tT An−1 {z

T

     

(8.63)

}

⇒ TA = AF T. The last row of the Frobenius matrix AF contains the negative characteristic coefficients ai in descending order. The inhomogeneous state equation (8.58) with b → bu (single input u) then transforms to x˙ = Ax + bu, xF = Tx ⇒ x˙ F = TAT−1 xF + Tbu = AF xF + en u (8.64) when for the transformed input, h

iT

Tb = b A b · · · An−1 b

!

t = en ⇒ t

(8.65)

is used to calculate t. T T One may also resolve eq.(8.60), second line, for An = An−1 AT and premultiply with (arbitrary) tT ∈ IRn to obtain

11 Ferdinand

Georg Frobenius, *1849 in Charlottenburg, †1917 in Berlin

8.3

401

Linear Time-Invariant Systems

tT E tT AT .. .

     

tT An−1 {z

|

T T

T



T

0 0 .. .



   T A =     

1 0

tT E t T AT .. .



···  ···  

−an · · · · · · −a1 {z

|

}

0 1

  

tT An−1

}|

AF

     

T

{z

(8.66)

}

T

T

T

⇒ T AT = AF T ⇒ AT = TATF . Using TxB = x for transformation yields −1

x˙ B = T

−1

ATxB + T

−1

bu = ATF xB + T

(8.67)

bu

where now the transformed (single) output y = cT x = cT TxB may be normalized to eTn xB to obtain t. The corresponding forms are called “normal forms”. They are listed in Table 8.2 (SISO: single input - single output). Matrices Wcontr. and Wobs. are called the controllability and the observability matrix (of the second kind).

Table 8.2. Normal Forms in Linear Control Theory (SISO) Control Normal Form

Observer Normal Form

x˙ = Ax + bu,

y = cT x

Tx = xF

TxB = x

x˙ F = AF xF + Tbu, y = cT T−1 xF

x˙ B = ATF xB + T

!

b A b A2 b · · · A(n−1) b

|

3.3.

{z

WTcontr.

bu, y = cT TxB

!

cT T = eTn

Tb = en



−1

T }

t = en

h

T

T

i

tT c ATc A2 c · · · A(n−1) c = eTn

|

{z

Wobs.

}

Stability Theorem for Mechanical Systems

Following eq.(8.31), the stability analysis of (linear time-invariant) mechanical systems can be specified using U =k x k= xT Rx, R = RT > 0, d k x k= x˙ T Rx + xT Rx˙ = −xT Sx, S = ST > 0 U˙ = dt

(8.68)

402

8 A Short Excursion into Stability and Control

to prove asymptotic stability. Insertion of x˙ = Ax yields h

i

xT AT R + R A + S x = 0

∀ x ⇒ AT R+R A + S = 0 , S = ST > 0. (8.69)

[Lyapunov equation, see also eq.(8.28)]. The demand on S to be positive definite is restrictive. Here, the dynamical interactions are not yet taken into account. However, when formulating the state vector x by means of modal coordinates, h

i

xT Sx = ξT XT S X ξ > 0 ⇒ XT S X > 0,

(8.70)

then the postulate for XT S X to be positive is replaced by the postulate that SX does not contain a zero column. With the interpretation of y = Sx being a measurement vector, one obtains x˙ = A x ,

y = Sx = S X ξ

(eigenvector problem) which corresponds to tion: If

Hautus’12

(8.71) controllability condi-

(λi E − A) xi = 0 ⇒ S xi 6= 0 ∀ i, i = 1 · · · n,

(8.72)

then the system is called (totally) (A, S)-observable. If XT S X is regular, then also 

T



X SX



eJt X−1 = XT S eAt

(8.73)

is regular. If we look at the Fundamental Matrix eAt as an infinite series it can be replaced with an (n − 1)-grade polynomial [see eq.(8.62)]: XT S (β0 E + β1 A + · · ·



+βn−1 An−1 = 

XT

(β0 E β1 E · · ·

   βn−1 E)    

S SA SA2 .. . SAn−1

     .   

(8.74)

Since XT is regular and since there exists a unique solution for the βi ’s, regularity is warranted if 12 Malo

L.J. Hautus, Prof.em., Eindhoven

8.3

403

Linear Time-Invariant Systems h

i

T

rank ST AT ST . . . An−1 ST = n

(8.75)

which represents Kalman’s13 observability condition (compare also Table 8.2 for single output). The problem of controllability is easily solved when focusing on λi 6= λj for the most interesting case14 . Considering the inhomogeneous system x˙ = A x + b, b = Bu, ξ˙ = Λξ + X−1 Bu,

(8.76)

the system will totally be influenced by the input u only if X−1 B does not contain a zero row. Having in mind that the inverse modal matrix can be formulated with the so-called left eigenvectors15 , one obtains the Hautus citerion yTi (λi E − A) = 0 ⇒ yTi B 6= 0 ∀ i = 1 · · · n,

(8.77)

for (A, B)-controllability and the corresponding Kalman criterion is obtained by comparison of eq.(8.72) and eq.(8.75) with eq.(8.77): replace ST with B and AT with A to come out with h

i

rank B A B · · · An−1 B = n.

(8.78)

(compare also Table 8.2 for single input). With these results, eq.(8.69) can be replaced with AT R + R A = −S , S = ST ≥ 0 , (A, S) observable ⇒ asymptotic stability, (A, S) not observable ⇒ marginal stability.

(8.79)

Specializing R yields 

R=

13 Rudolf

14 Although

K 0 0 M



⇒ AT R + RA = −S = −



0 −N N 2D



. (8.80)

E. Kalman, Prof.em., Z¨urich the results are not restricted to this. Let for instance x(t1 ) = 0 which, for eq.(8.59), is achieved



 R t 



T

−1

1 Φ−1 B Φ−1 B dτ xo . Precondition for the existence of a unique by u = − Φ−1 B o control u which brings xo back into the zero state within a limited time interval [0, t1 ] is the regularity of

R t1 

Φ−1 B



Φ−1 B

T

dτ . Formulating ˙ = −Φ−1 A Φ−1 B as a Taylor series for t ∈ [0, t1 ] and having in mind that d[Φ−1 ]/dt = −Φ−2 Φ leads also to the Rank-condition (8.78). 15 From the eigenvalue problem (8.44) we obtain XΛ = AX and ΛX−1 = X−1 A. The kth solution from T the first expression yields xk λk = Axk and the ith solution of the second expression yields yT i λk = yi A T T where the ith row of X−1 is denoted yT . Mutually multiplying with y and x yields (λ − λ )y x = 0 i k k k i i i for i 6= k which may be normalized to 1 for i = k. The yT i are called “left eigenvectors” because the corresponding eigenvector equation reads yT (λE − A) = 0. It is solved the usual way by transposing, (λE − AT )y = 0, which represents the eigenvector problem for AT . the so-called controllability matrix of the first kind, Wcontr. =





o

404

8 A Short Excursion into Stability and Control

Conclusion: For N ≡ 0, D = DT > 0, the required controllability condition . 0 0 .. 0 − KT M−1 D T .. T T  =n = rank rank S . A S .. −1 T 0 D . 0 − (D + G) M D (8.81) > is fulfilled for K < 0. Thus, the restoring matrix K decides on asymptotic stability/instability, independent from G (stability theorem of Thomson-TaitChetaev)16 . If the damping matrix is only semidefinite, D ≥ 0, then the state equation may be split into two parts, 



x˙ =





0 E −M−1 K −M−1 G







x+

0 −M−1 D



y˙ := A0 x + B0 u , (8.82)

which leads to the same stability results if the (A0 , B0 ) −controllability condition [pervasive damping , (P.C. M¨uller, 1971)] is fulfilled (stability theorem of Thomson-Tait-Chetaev-M¨ uller17 . Remark: the controllability condition is of course trivially fulfilled for D > 0). The controllability condition for non-gyroscopic systems (G ≡ 0) reduces to the controllability of (M−1 K), (M−1 D) as can be seen when applying eq.(8.78). If the system is undamped (D ≡ 0), then a system with K < 0 can be stabilized by gyroscopic effects if M = MT > 0 , K = KT < 0 , |det G| 6= 0 and sufficiently large, (8.83) which is proved by calculating the eigenvalues (stability theorem of ThomsonTait). |det G| 6= 0 means that an even number of coordinates is needed to make the gyroscopic stabilization work. However, stability gets lost when (pervasive) damping arises. In non-gyroscopic undamped systems (G ≡ 0, D ≡ 0) the restoring matrix K decides over (marginal) stability/instability (stability theorem of Dirichlet). The results are listed in Table 8.3.

16 Nikolay 17 Peter

Guryevich Chetaev, *1902 in Karaduli, †1959 in Moscow Christian M¨ uller, Prof.em., Wuppertal

8.3

405

Linear Time-Invariant Systems

Table 8.3. Stability of Linear Time-Invariant Mechanical Systems M¨ q + (D + G)q˙ + Ky = 0, M = MT > 0 D = DT ≥ 0

W

K>0

asymptotic stability

(Ao , Bo )-controllable

K0

marginal stability

K < 0, D ≡ 0, G ≡ 0

k det G k>> 1

marginal stability

K>0

marginal stability

K 1 does not significantly improve the estimation. 4

We thus proceed with Z = 0 which, for more than one disturbance function w ∈ IRm , m > 1, is generalized to Z ≡ 0 ∈ IRm . This leads eq.(8.105) to x˙ z˙

!

=

a(x, t) + Fz 0

!

:= x˙ = a(x, t),

x ∈ IRn ,

z ∈ IRm (8.109)

where the observer theory is applied. The measurement equation that is formally assigned to eq.(8.109) reads y = [C 0] x := Cx.

(8.110)

ˆx = S1 ξ + S2 y,

(8.111)

ξ˙ = Dξ + Ly,

(8.112)

Along with the approach

the augmented state in eq.(8.109) is estimated with eq.(8.111) where S1 ξ contains (at least) the non-measured quantities and where the estimation xˆ is corrected by the measurements y themselves. ξ is calculated with a linear differential equation (8.112) and also adapted with y. Furthermore, ξ belongs to a subspace from x which is characterized by map T. After a sufficiently long time, ξ has to coincide with Tx and ˆx has to pass over into x yielding two conditions: lim t→∞

(ξ − Tx) = 0,

lim t→∞

(x − ˆx) = 0.

(8.113) (8.114)

Condition (8.113) is enforced with d (ξ − Tx) = D(ξ − Tx), dt

D asymptotically stable.

(8.115)

8.5

413

Observers

Calculation of eq.(8.115) along with the approach (8.112) and the measurement (8.110) yields the condition Dξ + LCx − Ta(x, t)

= Dξ − DTx ⇒ (LC + DT)x − Ta(x, t) = 0. (8.116)

For eq.(8.114), the limiting case t → ∞ is considered where x = ˆx and ξ = Tx must hold. Along with eq.(8.111) one obtains x = xˆ

= S1 ξ + S2 Cx ⇒ [S1 T + S2 C − E]x = 0

(8.117)

as a second condition, independent of a(x, t). The general structure with S1 , S2 , T gives access to minimal observers where only the non-measured state variables enter the estimation procedure (e.g. (H. Bremer, 1988); not further pursued). Conclusion: an asymptotic stable matrix D guarantees the convergence of the estimation with the approach according to eq.(8.111) and eq.(8.112). As in Lyapunov’s stability theorems (8.35) et seq., the asymptotic stability of a nonlinear plant can be forced with a linear “controller”.

5.2.

Complete State Observer for Control

Considering a linear time-invariant plant one obtains a (complete) state observer (no unknown disturbances ⇒ no state amplification): eqs.(8.35),(8.109): x˙ = Ax (state equation, A assumed constant, h(x) → 0 due to control, see below), eq.(8.110): y = Cx (measurement equation), eq.(8.111): S1 = E, S2 = 0, (chosen) ⇒ ˆx = ξ, eq.(8.117): ⇒ T = E, eq.(8.112): ˆx˙ = Dˆx + LCx, eq.(8.116): D = A − LC ⇒ ˆx˙ = (A − LC)ˆx + LCx.

414

8 A Short Excursion into Stability and Control

Here, Cˆx represents the estimated measureC ment ˆy. Interpretation: x y b A The (physical) plant −K B x˙ = Ax is rebuilt with ˆ x R b -b b r the observer. Compa−C - ? ˆy 6 6 ring the latter with the real plant, its correct A phase is enforced by L the difference of the real and estimated measurement which is fed Figure 8.8. Complete State Observer back with a suitable gain matrix L. With Bu as control input and u = −Kx for control law (state feedback) with (part of) x unknown, the latter is obtained from the observer (dotted in Fig. 8.8). Since the control stabilizes the system asymptotically, the use of the linearized state equation is justified. The linear plant along with the observer equation r

R

d x A −BK x = ˆ ˆx x LC (A − BK − LC) dt (which follows directly from Fig. 8.8) can be transformed with 







x ∆







=

E 0 −E E



x ˆx



(8.118)



(8.119)

(where ∆ = (ˆx − x) represents the estimation error) yielding d dt



x ∆





=

A − BK −BK 0 (A − LC)



x ∆



.

(8.120)

Eq.(8.120) shows the fact that the eigenvalues of the controlled plant (A − BK) are decoupled from those of the observer (A − LC) (separation statement) and allows us to calculate control and observer separately. The main difference hereby is that for the control problem, K is to be calculated while for the observer problem the unknown is L. So to speak, within ˙ = (A − LC)∆, the pair LC appears in the wrong sequence. The solution to ∆ this problem is to look at the adjoint equation which is obtained by transpositi˙ = (AT − CT LT )∆ whereby LT overtakes the role of a (fictive) control on: ∆ input matrix. The solution of the corresponding Riccati equation (8.20) (with R = E and PR = const. for simplicity) then yields the required L: APR + PR AT − PR CT CPR + Q = 0 ⇒ LT = CPR .

(8.121)

8.5

415

Observers

Such a transposition was also the basis for the Observer Normal Form in Table 8.2. For the single output problem one has (A − lcT ) to have in advance chosen eigenvalues by means of l. The solution is obtained by direct comparison with eq.(8.90): l = Pd (A)t where WTobs. t = en . (8.122) Here, Pd (A) represents the characteristic polynomial for A calculated with the characteristic coefficients di of the desired poles. Remark: The fundamentals of observer theory, commonly referred to as Luenberger22 observers, were laid by (D.G. Luenberger, 1964,1966). 5 Example: Observer Gain Calculation. Consider

 A=

0 0

1 0



, C = cT = [1

(8.123)

0].

A) Riccati-Equation for determination of L = l ∈ IR2 along with Q = diag{0, Q2 }. The adjoint equation uses AT for “system matrix”. Precondition is the (AT , Q)-observability which reads, with eq.(8.75), A → AT , ST → QT = Q :

 rank[ Q | AQ ] = rank

0 0

Q2 0

0 Q2

0 0



(8.124)

= 2.

From eq.(8.121) one obtains

APR + PR AT − PR c cT PR + Q = 0 ⇒ PR =

 √

2Q Q

Q √ Q 2Q

 ,

(8.125)

and the gain matrix reads

p

lT = (l1

l2 ) = cT PR = [

2Q

Q].

(8.126)

∆

(8.127)

The estimation error is obtained from eq.(8.120) in the form ˙ = (A − lcT )∆ = ∆



−l1 −l2

1 0



yielding the eigenvalues det[λE − (A − lcT )] = λ2 + l1 λ + l2 = 0



⇒ λ1/2

22 David

l1 =− 1± 2

G. Luenberger, Prof.em, Stanford

r

4l2 1− 2 l1

r

 =−

Q (1 ± i). 2

(8.128) (8.129)

416

8 A Short Excursion into Stability and Control

For Q = 2 · 104 one obtains for instance λ1/2 = −100 · (1 ± i) and the gains are (l1 (2 · 102 2 · 104 ).

l2 ) =

B) Pole Placement. According to eq.(8.122) one obtains Wobs. = [c | AT c] =



1 0

0 1





0 0

1 0

 

⇒t=

0 1



(8.130)

,

hence

 l = Pd (A)t = d2



1 0

0 1



 + d1

0 1



 =



d1 d2

(8.131)

with the characteristic coefficients di from the desired poles λ1 , λ2 , (λ − λ1 )(λ − λ2 ) = λ2 − (λ1 + λ2 )λ + λ1 λ2 = λ2 + d1 λ + d2 ,

(8.132)

thus

 l=

d1 d2



 =

−(λ1 + λ2 ) λ1 λ2



(8.133)

.

p

[If the desired poles are those from eq.(8.129), then −(λ1 + λ2 ) = 2 (Q/2)(1 + 1) brings eq.(8.133) back to eq.(8.126)].

Q/2 and λ1 λ2 =

C) Stabilization. Without explicit knowledge of the resulting eigenvalues, (A − lcT ) can be asymptotically stabilized using the Hurwitz criterion on page 405: the characteristic equation (8.128) yields   l1 0 H= (8.134) 1 l2 ⇒ l1 > 0, l2 > 0, (detH = l1 l2 > 0) is sufficient for asymptotic stability. 4

5.3.

Disturbance Suppression (“High Gain Observer”)

The nonlinear equations of motion of a mechanical system, M¨s + G˙s − Q = f,

(8.135)

can be decomposed in the form diag{Moi }¨s + ∆M¨s + G˙s − Q = f

(8.136)

(P.C. M¨uller, 1995) where Moi = const may be chosen in any way. Here, the ith component reads Moi s¨i +

g X j=1

∆Mij s¨j + (G˙s − Q)i = f i

(8.137)

8.5

417

Observers

where now in 

s¨i =



g X

1  − ∆Mij s¨j − (G˙s − Q)i + f i  := f i , Moi j=1

(8.138)

f i contains all interaction terms as well as all disturbances. Using s˙ = q˙ for simplicity, one obtains d dt

q q˙

!



=

0 1 0 0

q q˙



!

0 1

+

!

f

(8.139)

(index i suppressed to simplify notation). If one applies the state observer according to the previous example (which does not take care of disturbances), then eq.(8.118) yields d dt

x ˆx

!



=

A lcT

0 (A − lcT )



x ˆ x

!

+

f 0

!

(8.140)

where f = (0 1)T f. A transformation with eq.(8.119) becomes ˙ = (A − lcT )∆ − f, ∆

(8.141)

˙ 1 = −l1 ∆1 + ∆2 , ∆ ˙ 2 = −l2 ∆1 − f . ∆

(8.142)

or, in components,

Now, let l2 = pl2o . One obtains then for the second component ˙2 f ∆ = −l2o ∆1 − p p

(8.143)

with ∆1 being asymptotically stable. This shows the fact that f remains without noticeable influence for sufficiently high p. The main advantage is that it does not matter what f explicitly looks like. Good experience has been achieved for the hardware realization of joint control in a biped walking machine, (H. Gattringer, 2006).

418

8 A Short Excursion into Stability and Control

5 Example: Single Joint Consider a joint model as depicted in Fig. 8.9. The equations of motion are known with eq.(4.81) et seq. when vox , voy , ωF z is set equal to zero and the motor torque in eq.(4.83) is replaced with the friction torque Mfrict . The friction torque reads Mfrict = −µFN Rγ˙ M / | γ˙ M | where µ represents the friction coefficient, R is a characteristic radius and FN denotes a normal force. Furthermore, the equation shall undergo a congruence transformation with

γA γM

q  R-

 Figure 8.9.



CM i2G 0

 0

o CA

Joint Model



γ¨M γ¨A

 =

iG 0

0 1



γ˙ M γ˙ A



to come out with

 +





ΩM ΩA

!

k∆γ

+

−iG Mfrict

−k∆γ + mA cA g cos γA

!

0

 

0 =  0

(8.144) where ∆γ = (γM − γA ) represents the gear twisting.

µ µo

M Mmax

0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0.6 0.4 0.2 0 0

0.4

0.8

1.2

1.6

Figure 8.10. Friction Coefficient

vrel

0

1

2

3

Figure 8.11. Friction Torque (M = Mfrict )

4

t

8.5

419

Observers

According to Stribeck23 , there exists a small region of dry friction in the neighborhood of vrel = 0 where vrel characterizes the relative velocity between the contacting bodies followed by mixed friction with decreasing characteristic, followed once more with increasing viscous friction (not depicted in Fig. 8.10. The region of dry friction is often neglected.) The chosen maneuver is characterized by the initial condition (qo , q˙ o ) ≡ 0 (see Fig. 8.9) which induces a friction torque according to Fig. 8.11. The “High-Gain-Observer” ˆ x˙ = T Aˆ x − lc (ˆ x − x) [2nd component of eq.(8.140)] reads, for the motor angles,

4 3 2 γˆ˙ M

γ˙ M

1 0

d dt



γˆM γˆ˙

 

0 1 = 0 0

M



γˆM γˆ˙

 -1

M

-2

  −

l1 l2

(ˆ γM −γM )

γM , γˆM

-3

(8.145) with γM for input variable (measurement in hardware application and corresponding result from eq.(8.144) in simulation, respectively).

-4 0

Figure 8.12.

1

3

2

t

4

Motor Variables and Estimation

The deviation between γ˙ M as depicted in Fig. 8.12 (solid line) and the estimate γˆ˙ M (dotted) is negligible and even not noticeable for γM , γˆM . The high-gain value l2 is, of course, limited due to unavoidable measurement noise amplification. The values used here conform with those from (H. Gattringer, 2006) which have been proved in practice. 4

5.4.

Disturbance Observation

Modeling the disturbances, the corresponding augmented state equation (8.109) will enter our calculation. Assuming a complete (augmented) state observer, the procedure is the same as in Fig. 8.8 where now 

x=

x z





,A=

A F 0 0





, C = [C 0], L =

Lx Lz



(8.146)

(A = const.) enters our calculation, once more assuming h(x) → 0 under the action of control. The observer equation then reads xˆ˙ = (A − L C)xˆ + L C x (+Bu) 23 Richard

Stribeck, *1861 Stuttgart, †1950 in Babelsberg

(8.147)

420

8 A Short Excursion into Stability and Control

(B = [BT 0]T , adjusting dimensions). Splitting eq.(8.147) into its components according to x yields, along with the real (physical) plant for x, x˙ = Ax + f (+Bu), ˆx˙ = Aˆx + Fˆz −Lx C(ˆx − x) (+Bu), ˆz˙ = −Lz C(ˆx − x), f b? 6 b

R

r

x

A

−Kx

B b

b -b 6 6

R

r

ˆx

(8.148)

leading to the block diagram (8.13). The control input is thereby C split into two parts, the state control and the y disturbance control. A transformation according to eq.(8.119) with −C - b? ˆy (x ∆ ˆz)T =

A 

 

E 0 0 x r b  −E E 0  ˆ Lx x 6 ˆf ˆz 0 0 E R Lz (8.149) F r shows that −f enters −Kz B ˆz the equation for ∆ as an inhomogeneous inFigure 8.13. Disturbance Observer put: Let the control input consist of two parts: Bu = −BKx ˆx − BKz ˆz. Eq.(8.148) then yields 











x x f A −BKx −BKz d        ˆ ˆ x x 0 L C A − L C − BK F − BK = +       x x x z dt Lz C −Lz C 0 ˆz ˆz 0 (8.150) which after transformation with eq.(8.149) becomes 













x x f A − BKx −BKx −BKz d        0 A − Lx C F  ∆ =  ∆  +  −f . dt 0 −Lz C 0 ˆz ˆz 0 (8.151) 

Here, the state control ux = −Kx xˆ stabilizes the system asymptotically and the question arises whether uz = −Kz ˆz eliminates the disturbance f. This problem is solved for the following presumptions:

8.5

421

Observers

According to eq.(8.104) et seq., f is composed with basic disturbances w which enter the state equation via the input matrix W. Thereby, w itself is a combination of basic functions z: w = Uz. Let U be the unit matrix, hence f = Wz = Fz. Let f enter the state equations with the same input as the control u does: f = Bf, ˆf = Fˆz, F = B. [This is a special but frequent case in mechanical systems. For the general case see e.g. (P.C. M¨uller, 1995), (S. Engell, 1995)]. T

In mechanical systems, B has the special structure B = [0 B M−1 ]T where M = MT [see eq.(8.22)]. We consider the case that only position variables are measured, leading to CB = 0. One obtains then from eq.(8.151), second and third components ˙ = (A − Lx C)∆ + B(ˆz − f), ∆

(8.152 a)

ˆz˙ = −Lz C∆.

(8.152 b)

Let, as in eq.(8.143), Lz = pLzo . Eq.(8.152 b) then becomes ˆz˙ = −Lzo C∆ ⇒ C∆ → 0 p

(8.153)

for sufficiently high p. From eq.(8.152 a) one obtains ˙ = CA∆ → 0, C∆ ¨ C∆ = CA2 ∆ → 0 ···

(8.154)

(having CB = 0 in mind) which leads to Wobs. ∆ → 0 ⇒ ∆ → 0

(8.155)

since Ak with k > n − 1 can only produce linearly dependent terms, see Cayley-Hamilton’s theorem, page 399. The observability matrix Wobs. is nonzero as a basic prerequisite, thus ∆ → 0. Since Bf 6= 0, it follows from eq.(8.152 a) that ˆz → f for sufficiently high gain matrix Lz ,

(8.156)

and the first component from eq.(8.150) (state equation) yields x˙ = Ax − BKx ˆx − BKz ˆz + Bf ⇒ Kz = E.

(8.157)

422

8 A Short Excursion into Stability and Control

5 Example: Observer Gain Calculation. Consider

 A=

0 0

1 0



, C = cT = [1



0 1





bf 0



0 1 0

#

0], F = B = b =

(8.158)

.

The estimation error is obtained from eq.(8.151), d dt





A − l x cT −lz cT

∆ zˆ



 =

A − l x cT −lz cT

b 0



∆ zˆ

 −

(8.159)

where b 0

"



T

= (A − l c ) =

−lx1 −lx2 −lz

1 0 0

(8.160)

has to be asymptotically stable. A precondition is the observability of the augmented state representation (8.146),

" A=

0 0 0

1 0 0

0 1 0

# , C = [1

0

h

T

0] : Wobs. = C

T

T

C A

T

2T

C A

i

= E ∈ IR3,3 .

(8.161) A) Pole Placement. Eq.(8.161) yields Wobs. t = e3 ⇒ t = e3 , thus

l=

lx1 lx2 lz

!

" = Pd (A)t = d3 E + d2

0 1 0 0 0 1 0 0 0

! + d1

0 0 1 0 0 0 0 0 0

!#

0 0 1

! =

d1 d2 d3

!

(8.162) with the characteristic coefficients di of the desired poles λ1 , λ2 , λ3 ,

(λ − λ1 )(λ − λ2 )(λ − λ3 ) =

λ3 − (λ1 + λ2 + λ3 )λ2 + (λ1 λ2 + λ2 λ3 + λ3 λ1 )λ − λ1 λ2 λ3

=

λ3 + d1 λ2 + d2 λ + d3 = 0.

(8.163)

B) Stabilization. Without explicit knowledge of the resulting eigenvalues, (A − l cT ) can be asymptotically stabilized using the Hurwitz-Stodola-criterion: the characteristic equation for eq.(8.160)) reads λ3 + lx1 λ2 + lx2 λ + lz = λ3 + a1 λ2 + a2 λ + a3 = 0 and leads to

" H=

lx1 1 0

lz lx2 lx1

0 0 lz

# ⇒

a1 = lx1 , H2 = lx1 lx2 − lz , a3 = lz ⇒ lx1 > 0, lx2 > 0, lx1 lx2 > lz .

(8.164)

8.6

423

Decentralized Control

Example: Joint Control. Applying the disturbance observer (8.147), d dt

  ˆ x ˆz



AF = 00

  ˆ x ˆz



ˆ f, f 10



Lx + (x − ˆ x), Lz

5

to the joint from Fig. 8.9 yields 0



 "

γˆM 0 1 0 d γˆ˙  = 0 0 1 M dt 0 0 0 zˆ −

lx1 lx2 lz

#  γˆM γˆ˙ M 

-5

zˆ

!

-10 (ˆ γM −γM ).

(8.165)

-15 0

1

2

3

4

t

Without disturbance compensation, one may use zˆ to determine the coupling terms and the Figure 8.14. Disturbance Estimation generalized disturbances within the system. Fig.(8.14) represents the estimated [Mfrict iG − k∆γ]/(CM i2G ) (dotted) in comparison to the real value (from numerical simulation, solid line). In order to obtain a good result one needs quite high gains as eq.(8.153) et seq. indicate. 4

6.

Decentralized Control

As long as the system is asymptotically stabilized, the use of the linearized equations is justified. However, exact knowledge of the system parameters is, in general, a basic requirement. An exact linearization by state feedback compensates for nonlinearities exactly but is sensitive to parameter deviations in general and will obviously fail in case of, e.g. friction forces due to the differentiation process. To the contrary, an observer does not compensate for nonlinearities exactly but exponentially with a dynamical process. It does not cause problems even when friction arises. But, in order to eliminate the frictional forces (only), the observer also needs knowledge of the system parameters. If the control aspect comes into consideration, then one is additionally interested to enforce a desired behavior. Decomposing the mechanical motion equation (8.135) according to eq.(8.138) and augmenting with the control input leads to q¨i =

1 Bi fi + ui Moi Moi

(8.166)

424

8 A Short Excursion into Stability and Control

where f as a whole is taken to be a disturbance to be estimated (P.C. M¨uller, 1995). If B is assumed B = diag{B i }, the control reads ui =

1 ˆ (Moi vi − f i ), vi = q¨di + dcontr. (q˙di − q˙i ) + kcontr. (qdi − qi ) (8.167) Bi

[compare eq.(8.99)] and leads eq.(8.166) to 1 ˆ fi − fi . Moi 

q¨i = vi +



(8.168)

Notice that this kind of control is independent of the knowledge of system parameters. If parameter deviations take place, these enter the (unknown) disturbance, which is estimated by the observer while the remaining system dynamics is not influenced. The proposed method has successfully been applied to robot dynamics, the experimental results of which are reported in (R.Hu, P.C.M¨uller, 1997). Clearly, feasibility depends on the controllability of the plant with the proposed input. In the above consideration where B = diag{B i } is assumed, controllability is trivially fulfilled. So it is for the single joint with friction where the disturbance enters the equations with the same input as the control torque does. Of course, also eq.(8.100) is controllable with u. But, seemingly, a frictional torque (which enters the motor equation, i.e. the second component) can not directly be compensated for by Gu which appears in the third component. However, with a decomposition according to eq.(8.136), one obtains for eq.(8.100) (under precondition K = diag{ki }, T = diag{Ti }, G = diag{Gi }) P

Moi q¨i +hqi + P ∆Mij q¨j +hpi + Mp,ij p¨j +hτ i +Ti Q˙ M i

= +ki (pi − qi ), = −ki (pi − qi ) +QM i , = −QM i +Gi ui . (8.169)

Adding these equations leads once more to eq.(8.166) et seq. with new f i . Applied to robotic systems, for instance, the result shows that no matter if the gears are compliant or not, or if the torques can directly be used as input (with a suitable preamplifier, yielding small time constants Ti ) or if the voltage enters the equations for control, a decentralized joint torque according to eq.(8.167) is applicable. The (dynamical) coupling terms between the single coordinates remain unknown and are estimated by the observer. The choice of A according to eq.(8.123) has thus not been trivial at all. One obtains by this a control for each joint independently, independent from parameter variations, i.e. a robust decentralized control.

8.6

425

Decentralized Control

5 Example: Joint Control. Consider a singe joint according to Fig. 8.9. The gear output angle is stabilized for an end position of γAd = π/2 using eq.(8.167) (slewing maneuver, Fig. 8.15). QM is directly used for control input (no motor dynamics considered). One obtains then from eq.(8.169) the “disturbance” (friction torque and coupling terms)

 f=



o CA CM i2G o − 1 k∆γ + M − mgc cos γA − CA γ¨M o CA CM iG frict



(8.170)

o (where Moi = CA ). This term remains unknown for hardware realization while, for simulation purposes, it is computed with eq.(8.144).

8 7

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

6 5 4 3 γA , γˆA

2 1 0

γ˙ A , γˆ˙ A

-1 0

1

2

3

4

Figure 8.15. Slewing Maneuver for γA (dotted: γM )

t

0

1

2

3

4

t

Figure 8.16. Estimated and “Real” Variables (from Simulation)

It has often been stated that implementation of the disturbance feedback diminishes the gains considerably. The reason is the counteraction of f in eq.(8.152 a) and eq.(8.157) [e.g. (S. Engell, 1995), p.148]. Indeed, for the present example the gains are reduced to ten percent of the previous l-values to obtain comparable accuracies (Fig. 8.16).

4

426

8 A Short Excursion into Stability and Control

7.

On Control Input Variables

Obviously, a mechanical system is controlled by forces and torques. These are generated by actuators which often depend on additional state variables (like voltage and current in electric motors). In case of nonnegligible feedback between the mechanical state variables and the additional ones, the dynamics of the latter has to be taken into account. This is achieved with a generalization of the Central Equation (2.39). Replace, for the general case, the momentum of the mass element by dm˙r → dp = dp(˙r) without further specification. Lagrange’s Principle (2.1) then reads Z

Z

e T

[dm¨r − df ] δr = 0 →

(S)

[dp˙ − dfe ]T δr = 0

(8.171)

d δr − δW = 0, dt

(8.172)

(S)

which may be reformulated d dt

Z

Z

dpT δr −

(S)

dpT

(S)

or, with dδr = δdr [see eq.(2.30)], d dt

Z

dpT δr −

(S)

Z

dpT δ r˙ − δW = 0.

(8.173)

(S)

Next, the momentum vector is replaced with r˙ 6

T∗ =

Rr˙

dpT =

p dr˙

o

∂ ∂ r˙

dr˙ T =

Rp

R

-

p

Energy and Co-Energy

dpT δ r˙ =

(S)

R (S)

dpT δr =

R ∂dT ∗ R δdT ∗ = δT ∗ , δ r˙ = (S) ∂ r˙ (S)   R ∂dT ∗ ∂ r˙ R ∂dT ∗ (S)

∂ r˙ ∂ s˙

(r˙ )

∂ ∂ r˙

dT ∗

where dT ∗ denotes the kinetic co-energy of the mass element. Since dp is a function of r˙ only, we also have dT ∗ = dT ∗ (˙r), hence

o

Figure 8.17.

dpT d˙r =

(8.174)

r˙ dp

dp

Z

δs =

(S)

∂ s˙

δs =





∂T ∗ δs. ∂ s˙

(8.175)

8.7

427

On Control Input Variables

Here, Helmholtz’ auxiliary equation (2.13) has been used, (∂r/∂s) = (∂ r˙ /∂ s˙ ). Notice that s˙ does not depend on mass distribution and is therefore extracted from the integral. Thus, redefinition of the Central Equation of Dynamics in the form d ∂T ∗ − δT ∗ − δW = 0 dt ∂ s˙ 



(8.176)

(which might be called the Central Equation of Mechatronics) gives access to any kind of “non-mechanical” generalized forces. However, instead of restarting calculations, one should take advantage of the following: ∗ The mechanical part has already been calculated (notice that Tmech = Tmech for non-relativistic mechanics).

The additional state obeys, in general, very simple dynamics. (E.g. nonholonomicity is unknown in heat transfer, electrical networks, fluidic networks.) One can thus, by means of the virtual work, augment the motion equation with any of the procedures listed in Table 4.1 which follow from the central equation for T → T ∗ . If the mechanical part is already known from M¨s + G˙s − Qe , one ˙ obtains for instance, with the augmented vector of minimal velocities s, 

d M0 ¨ G0 ˙ Qe s+ s− + 0 0 0 0 0 dt 













∂Ladd ∂Ladd T − −Q∗add ∂ s˙ ∂s  



T

= 0.

(8.177) ∗ − V Notice that the Lagrangean Ladd = Tadd add refers to the additional dynamics only and contains almost simple expressions. Q∗add comprises the corresponding non-conservative generalized “forces”. Considering an electrical network for representative example, it consists of inductors and capacitors as well as of nonconservative elements like voltage sources and resistors. The energy is calculated with the time integral of current I and voltage U . Along with Q for charge and Φ for magnetic flux one has Z

W =

Z

U Idt =

Z

U dQ =

∗

IdΦ ⇒ W =

Z

Z

QdU =

ΦdI. (8.178)

Using current and voltage for nonmechanical variables leads to ∗ Wmag = oI ΦdI (inductor), Wel = [if linear : Φ = LI, U = Q/C]

R

RQ o

U dQ (capacitor)

(8.179)

428

8 A Short Excursion into Stability and Control ∗ ∗ ⇒ T ∗ = Tmech + Wmag , δW = −δVmech − δWel + δWnc

(8.180)

where δWnc represents the nonconservative portion (corresponding “forces” are, e.g. damping forces for the mechanical part, voltage for the electrical part.) For the use of eq.(8.177), all mechanical parts are already calculated such ∗ that the Lagrangean reduces to Ladd = Wmag − Wel . 5 Example: Vertical motion of a MAGLEV (Magnetic Levitation Vehicle). For the linear case we  ∗ have Φ = LI ⇒ Wmag = LI 2 /2 (I: current, L: inductance) and Q∗ = ucont − RI (u: voltage, R: resistance) ⇒ Ladd = LI 2 /2.

mg ?

actual position

y6 p

m

vehicle

fm 6 f

?m

guideway

6

desired position

R

b p

b b

6 y p

ucont

L

magnet

so

b

Figure 8.18. Simplified MAGLEV Model T L depends on the gap s = so −y. One obtains thus, along with sT = (y Q)T , s˙ = (y˙ I)T :



m¨ y + mg 0

"

 +

d dt

0 [LI]

 −

!  # ∂ LI 2 /2 ∂s 0 ∂s ∂y − =0 ucont − RI 0

(8.181) where [∂s/∂y] = [∂(soR− y)/∂y] = −1. Interpretation: the supplied magnetic energy y [L(y)I 2 ]/2 is stored in - o fm dy. The magnetic force fm counteracting on the vehicle is Ry fm = (∂[ o fm dy]/∂y) = (∂[L(y)I 2 /2]/∂y). Linearising w.r.t. the desired position so , y = yo + ∆y, I = Io + ∆I, yo = 0

^

(∂[LI 2 /2]/∂s)o + mg = 0

(8.182)

yields the control plant

d dt

∆y ∆y˙ ∆I

!



0  ky = m 0

1 0 kI Lo



0  − kmI  −R Lo

∆y ∆y˙ ∆I

!





0 0  ∆ureg  + 1 Lo

(8.183)

where ky = [(∂ 2 L/∂s2 ) I 2 /2]o , kI = [(∂L/∂s) I]o . Here, the magnet voltage acts as control input.

8.7

429

On Control Input Variables

The second component of eq.(8.183) comprises the actual force ∆F = m∆¨ y yielding

xnew :=

∆y ∆y˙ ∆F

!

" =

1 0 ky

0 1 0

0 0 −kI

#

∆y ∆y˙ ∆I

!

(8.184)

:= Txold

which transforms eq.(8.183) to x˙ new = TAT−1 xnew + Tb∆ureg d = dt

∆y ∆y˙ ∆F

!



0  0 = ky R Lo

1 0 k2 ky − LI o



0 1  m  R −L

∆y ∆y˙ ∆F

!

o





0 +  0  ∆ureg . − kI Lo

(8.185)

For control purposes it is mostly easier to measure y¨ (by means of accelerometers) than currents. Resolving eq.(8.185) for ∆F yields



Lo R



∆F˙ + ∆F + h(∆y, ∆y) ˙ =

 where h(∆y, ∆y) ˙ = −ky ∆y +



kI R



∆ureg

kI2 Lo − ky R R

 ∆y. ˙

(8.186)

4 Eq.(8.186) yields, by generalization, the special structure of the motor forces and torques, respectively, T Q˙ M + QM + h(x) = Gu,

(8.187)

where T is the time constant, x represents the system state vector and G is the motor gain which refers to the input u. It should be mentioned that an analogy between electrical (thermodynamical, hydraulical, pneumatical etc.) networks and plane motion mechanics is often helpful, see e.g. (C.H. Crandall et al., 1968), (V. Hadwich, 1998), (B. Heimann et al., 1998).

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LIST OF SYMBOLS

The symbols we use in this book are related to the corresponding ISO-Norm. As far as possible, they represent common abbreviations (with English (and Latin, respectively) origin such as v: velocity, a: acceleration, T: transformation, or German origin such as A: Abbildung, Z: (Gaussian) Zwang, F : F¨uhrung). Vectors and matrices are denoted by bold-faced letters (capital: matrices, small: vectors, in general). A special case is given for the zero element(s): Introducing a zero matrix (0) would consequently need a zero vector (o) which can easily be confused with the letter bold o. Using the slash for a zero matrix (0/ ) and for a zero vector representation (o/) may lead to confusion with the boldfaced Greek “phi”. We therefore define the non-bold zero symbol 0 to generally represent scalar, vector or matrix values where the corresponding dimension follows from the context. 0 a b c d e f

g h i j k l, or L

zero element (∈ IRm,n , m, n ∈ ZZ) (absolute) acceleration (∈ IR3 ) (nonlinear) state function (∈ IRf +g ) inhomogeneous state input, or disturbance (∈ IRf +g ) coefficient vector (∈ IRn ), transposed measurement matrix (SISO) (outer) differential symbol, distance unit vector (∈ IR3 ) force (∈ IR3 ), number of positional d.o.f. (∈ IR1 ), function (∈ IR1 ) disturbance (∈ IRf +g ) gravity vector (∈ IR3 ), | g | = 9.81 for Middle Europe (see p.67), number of velocity d.o.f. (∈ IR1 ) height √ imaginary unit ( −1), counting index counting index counting index, spring coefficient, feedback coefficient length 437

438 m n m

p

q q r s˙ s

t u

u v w we x x x ˆ x, ˆx y ˆ y y˙ i y˙ n y z

List of Symbols number of nonholonomic constraints counting index, “geometrical nonlinearity” index, state dimension (n = f + g) mass, counting index, number of holonomic constraints momentum (∈ IR3 ), generalized momentum (Hamilton, ∈ IRg ), Rodrigues parameter (∈ IR3 ), predecessor index (∈ IR1 ) vector of minimal coordinates (∈ IRf ), Euler parameter (∈ IR4 ) f eigenvector (∈ IR  )  position vector ∈ IR3 vector of minimal velocities (∈ IRg ) and s˙ i (∈ IRgi ), resp. quasi-coordinate (∈ IRg , assigned to s˙ ), tip body mass center (∈ IR3 ), successor index (∈ IR1 ) time rotation axis (∈ IR3 ), neutral axis displacement: (u1 , u2 , u3 )T or (u, v, w)T , vector of beam shape functions (∈ IRn ), (electrical) voltage (∈ IR1 ) displacement of an arbitrary point: (u1 , u2 , u3 )T or (u, v, w)T (absolute) velocity (∈ IR3 ), vector of beam shape functions ∈ IRn vector of beam shape functions (∈ IRn ), disturbance vector (∈ IRf +g ) eigenfunction (plate) spatial variable vector of independent spatial variables (right) eigenvector, augmented state vector (observer) state estimations (observer) left eigenvector estimated measurement (observer) vector of “intermediate”, or auxiliary, velocity variables describing velocity variables (subsystem n) spatial variable (non-minimal) position variables, y˙ = H(z)˙z, vector of basic disturbance functions (observer), spatial variable (∈ IR1 )

List of Symbols A

A B B B Bi C C

D D Dir D D E F F F

Gi Gn G G H

I ID J J

transformation matrix (∈ IR3,3 ), area (∈ IR1 ), area (= normal) vector (∈ IR3 ), system matrix (∈ IRn,n , n = f + g) or Jx : moment of inertia, x-axis control input matrix (state space) control input matrix (configuration space) or Jy : moment of inertia, y-axis operator, yields boundary conditions measurement matrix or Jz : moment of inertia, z-axis, longitudinal stiffness (plate or disk), mass center, (electrical) capacity damping matrix bending stiffness (plate) spatial operator (∂ i /∂xir ) operator, defines variables operator, yields partial differential equations unit matrix, Young’s modulus (∈ IR1 ) (intermediate) functional matrix (single body) (intermediate) functional matrix (subsystem) (general) functional matrix, deformation gradient, disturbance input matrix (F = WU) gyroscopic matrix (single body) gyroscopic matrix (subsystem) gyroscopic matrix (∈ IRg,g ), shear modulus (∈ IR1 ) Green-Lagrange strain tensor ˙ H ∈ IRg,f ), minimal velocity coefficient matrix (˙s = H(q)q, 6,6 Hooke’s matrix (∈ IR ), Hamilton function (∈ IR1 ) tensor of area moments of inertia (∈ IR3,3 ), (electrical) current (∈ IR1 ) torsional area moment of inertia tensor of mass moments of inertia (∈ IR3,3 ), Jacobian (e.g. ∈ IR6N,g ) Jordan matrix

439

440 K dKn L

Mi Mn MRi M N N P Pij P P Q Qn Q

R

R(x) Si T

T∗ Tip T∗ip U

V

List of Symbols restoring matrix (∈ IRg,g ) state feedback matrix (∈ IRn,n , n = f + g) dynamical stiffening matrix momentum of momentum (∈ IR3 ), observer gain matrix, inductance (∈ IR1 ) mass matrix (single body) mass matrix (subsystem) reduced mass matrix (see p. 95) moment (∈ IR3 ), mass matrix (∈ IRg,g ) number of bodies in MBS nonconservative restoring matrix (∈ IRg,g ), recursion matrix (page 95) linearization: velocity dependent matrix (∈ IRg,g ) power (∈ IR1 ) Trefftz (or 2nd Piola-Kirchoff) stresses Trefftz (or 2nd Piola-Kirchoff) stress tensor strain operator (∈ IR6,3 ) generalized force, single body generalized force (subsystem) linearization: position dependent matrix (∈ IRg,g ), generalized force (∈ IRg ), weighting matrix (∈ IRn,n , n = f + g) rotation matrix (∈ IR3,3 ), weighting matrix (∈ IRn,n , n = f + g), radius (∈ IR1 ), (electrical) resistance (∈ IR1 ) Rayleigh quotient estimation matrices (observer, i = 1, 2) transformation matrix (general, e.g. ∈ IR6,6 ), kinetic energy (∈ IR1 ) maneuver time (∈ IR1 ) kinetic co-energy (∈ IR1 ) kinematical chain: y˙ i = Tip y˙ p + Fi s˙ i = NTi Tip (see p. 105) basic disturbance function coefficient matrix (Uz = w), Lyapunov function (∈ IR1 ), (electrical) voltage potential, volume (also Vol )

List of Symbols Vi X W Z

α α, β, γ γ δ − δ 

 ij ij ϑ κ

λ ν π ρ σ σij σ τ ϕ ψ, θ, ϕ ψ˙

441

= FTi M∗i Fi (see p. 106) modal matrix disturbance input matrix, work (∈ IR1 ) disturbance matrix (˙z = Z z), Gaussian constraint (“Zwang”, ∈ IR1 ) (absolute) angular acceleration (∈ IR3 ), degree of nonlinearity (∈ IR1 ) Cardan angles universal gravity constant variation symbol Dirac-distribution Cauchy strain vector (∈ IR6 ), variational parameter (∈ IR1 ), rotor unbalance (∈ IR1 ) Green-Lagrange strain vector (∈ IR6 ) Cauchy strains Green-Lagrange strains vector of beam shape functions (torsion) (∈ IRn ), torsion (∈ IR1 ) (vector of) curvatures, shear correction factor (∈ IR1 ), rotor to shaft inertia relation (∈ IR1 ) vector of Lagrangean multipliers, eigenvalue (∈ IR1 ) Poisson number, frequency quasi-coordinates (∈ IR3 , assigned to ω) mass density, spring elongation stress vector (∈ IR6 ), Cauchy stresses Cauchy stress tensor tangential plane vector of bending angles (∈ IR3 ), rotation angle (∈ IR1 ) Euler angles rotation vector (∈ IR3 ) column matrix of Cardan and Euler angles, resp. (∈ IR3 ), motor angular velocity (∈ IR1 )

List of Symbols

442 ω

angular velocity (∈ IR3 ), frequency (∈ IR1 )

∆

difference, Laplace operator Nabla operator = Θ(f1 , f2 ) defines rotation vector quantities (∈ IR3,3 ) diagonal matrix of eigenvalues = 0 (holonomic) constraint (∈ IRm ), fundamental (or transition) matrix (∈ IRn,n , n = f + g), (matrix of) shape functions magnetic flux (∈ IR1 ) := D ◦ Φ(x): (matrix of) shape function derivatives = 0 (nonholonomic) constraint (∈ IRm ) angular velocity

∇ Θ Λ Φ

Ψ ˙ Ψ Ω Indices:

upper (˙) time derivation f ( ) spin matrix (∈ IR3,3 )

upper right ( )0 spatial derivation e “impressed” c “constrained”, or “w.r.t. mass center” T transposed

middle right ◦ “applied to’

lower right free index, p e.g. “point”, e.g. “predecessor” xy relativity, e.g. between x and y

lower left: Basis, e.g. R reference (general) B body-fixed frame I, o inertial frame

List of Symbols

443

Vectors are generally defined as column vectors. Row vectors are thus obtained by transposition (upper right index T ). Their component representation is characterized by the lower left index (indicating the chosen base). A single lower right index remains free for actual considerations, while an index pair in general indicates relativity. (For instance: vector R rAB connects points A and B (and the origins of frame A and frame B, resp.) in a component representation of frame R). For transformation matrices ∈ IR3,3 , the lower right index pair is to be read from right to left in the sense of “transforming from – to”. (For instance: AAB characterizes the mapping of a vector determined in frame B into its representation in frame A). One obtains thus simple connection rules such that, for transformations, the indices appear in pairs: AAB AB C AC D D r transforms vector r from component representation in frame D via C and B into components of frame A. Using the spin matrix representation e ab for vector product a × b yields the following useful relationships.

(R3)

e =b eT a e ab = −ba e +b ee e−b ee e abc ca + ece ab = 0 = (e ab a−e afb)c e = abT − aT b E e ab

(R4)

e afb = abT − baT

(R1) (R2)

The “dot” for time derivation follows the definition f˙ =

df lim =∆t→0 [f (t + ∆t) − f (t)]/∆t dt

for any function f . This holds also in case that f is a vector component.

INDEX

calculus of variations, 237, 383 canonical equation, 387 Cardan angles, 35, 38 Cauchy, 60, 167, 169 Cayley, 399, 421 central equation, 4, 20–22, 26, 59, 69, 71, 126, 426, 427 change-over gear, 253 Chetaev, 404 closed-loop control, 389 comparison function, 10, 385 constraint force, 2, 4, 8, 9, 64, 71, 73, 80, 81, 85, 100–102, 104, 114, 115, 207, 216, 221, 298, 307, 369 constraints, 9, 10, 15–19, 24–26, 32, 34, 59, 60, 68, 72, 75, 77, 81, 83, 84, 100–102, 104, 105, 107, 108, 114–118, 123, 124, 126, 214, 215, 221, 237, 245, 253, 257, 306, 307, 322, 323, 369, 385, 386 continuity condition, 63 control, 151, 235, 277, 284, 286, 288, 308– 310, 337, 348, 369–371, 373– 375, 377–382, 387–390, 401, 403, 407, 409, 410, 413, 414, 417, 419–421, 423–425, 428, 429 controllability, 235, 286, 372, 375, 381, 401–404, 424 convergence, 110, 111, 221, 235, 242, 243, 250, 251, 253, 322–324, 413 Coriolis, 14, 72, 81, 84, 274, 277, 283, 324, 395, 409

acceleration, 49, 97, 107, 276, 280, 281, 283, 311, 315, 320, 325, 377 Ackermann, 406 Almansi, 171 analytical methods, 2, 3, 71, 126, 130, 131, 214, 322, 327 Appell, 71, 72 Aristotle, 15 auxiliary equation, 12, 18, 19, 21, 70, 72, 427 auxiliary function, 242, 246, 251, 253, 323, 324 auxiliary variables, 79, 81, 113, 214, 297, 315, 321, 334 axioms, 7, 71, 116 backlash, 410 Baumgarte, 107, 108, 114 Bernoulli, 3, 15, 117, 121, 134, 138, 144–146, 150, 151, 153, 164, 233 Boltzmann, 116 Bolza, 384 Born, 60 boundary conditions, 5, 124, 129, 131, 133, 136–139, 142, 143, 145, 151, 155, 158, 161, 163, 164, 166, 187, 188, 196, 210–212, 215, 217, 220, 223, 224, 233, 234, 237, 238, 242, 245, 246, 253, 259, 265, 312, 321–323, 326, 332, 339, 377 Bryan, 35 Bubnov, 219

445

446 d’Alembert, 2, 8, 150 decentralized control, 423, 424 ˆ pital, 3, 47 de l’Ho Descartes, 2 describing (velocity) coordinates, 85 describing variables, 85, 89, 98, 104, 113, 114, 124–126, 132, 134, 153, 190, 191, 194, 198, 220, 223, 247, 301, 321, 328, 330, 345, 348 Dirac, 176, 206, 207, 249, 256, 295 disk, 120, 253, 255, 262, 263, 269, 326 double pendulum, 8, 204, 207, 209, 364 dynamical stiffening, 167, 183–185, 201, 206, 211, 216, 219, 257, 294, 303, 315, 324, 339, 344, 352, 360, 361, 367 dynamical stiffening matrix, 177–180, 185, 190, 195, 201, 295, 302–304, 319, 325, 326, 344, 359 eigenfunction, 111–113, 140, 142–144, 146, 147, 150, 151, 164, 165, 220, 222, 223, 227, 228, 234, 235, 237, 242, 246, 249, 250, 253, 261, 262, 267, 268, 271, 273, 286, 323, 324 eigenvalue, 31, 32, 37, 112, 113, 139, 140, 142, 143, 227, 228, 230, 231, 236, 238, 242, 245, 247, 261, 263, 265–268, 274, 279, 280, 284, 285, 288, 395–398, 403, 404, 406, 414–416, 422 eigenvector, 31, 112, 140, 143, 227–233, 235, 243, 245, 274, 279, 284, 285, 288, 395–397, 403 eigenvector problem, 31, 37, 139, 227, 229, 230, 236, 261, 274, 284, 285, 396, 402, 403 elastic double pendulum, 198, 212, 217, 332, 363, 364 elastic potential, 115, 128, 135, 151, 152, 154, 167, 171, 175, 200, 205, 214, 216, 226, 237, 305, 320, 326, 356 elastic robot, 6, 291, 324, 348, 349, 369, 371 elastic rotor, 127, 159, 255, 273, 283–285, 288, 323, 324, 380 elastic TT-robot, 296, 325

Index Euler, 3, 7, 22, 24, 26, 36, 38, 39, 41, 43, 72, 114, 126, 134, 138, 144, 145, 150–153, 164, 171, 188, 189, 234, 259, 264–268, 292, 296, 308, 313, 316–318, 358, 384, 385, 387, 388 exact linearization, 410, 423 excess function, 385 F¨ oppl, 2, 7, 273 feedback, 67, 374, 375, 380, 393, 426 finite elements, 5, 322 finite segmentation, 109 FORTRAN, 98 Fourier, 150, 238 Frobenius, 400, 406, 409, 411 fundamental lemma, 384 Funk, 17 Galerkin, 68, 164, 188, 216, 217, 219, 221, 233, 234, 322 Gauss, 11, 12, 24, 26, 68, 72, 95, 352 generalized inverse, 18, 87, 307, 373, 391 Gibbs, 71, 72 gravitation potential, 66 Green, 169, 170 guidance motion, 55, 124, 134, 194, 335 guidance velocities, 55, 91, 190, 191, 193, 335, 345, 346, 366 Hamel, 1–3, 12, 21, 69, 72, 75, 126, 130, 176 Hamilton, 17, 18, 22, 26, 72, 124–126, 130, 216, 385, 388, 390, 399, 421 Hautus, 235, 402, 403 helicopter blade, 162, 183, 291 Helmholtz, 12, 18, 19, 21, 70, 72, 427 Hermann, 3, 12, 22, 44 Hertz, 2, 25 Heun, 3, 8, 21 holonomic, 18, 19, 22, 25, 26, 72, 76, 124– 126, 131, 133, 190, 214, 307, 321, 371, 389, 395 Hooke, 14, 116, 120, 122, 171 Hurwitz, 405, 416, 422 Huygens, 80 impact, 146, 147, 149, 267, 364, 370

Index impressed forces, 9, 13, 52, 87, 115, 135, 185, 199, 205, 219, 294, 295, 302, 303, 325, 329, 331, 337, 340, 344, 357, 361, 395 intermediate variables, 81, 84, 85, 124, 132, 214, 315, 321, 325 Jacobi, 2, 7, 23, 24, 26, 72, 102, 107 Jacobian, 18, 71, 85, 86, 89, 124, 192, 193, 299, 309, 332, 333 Jeffcott, 272, 278, 283, 284, 324 Jordan, 397, 398 Jourdain, 12, 25 Kalman, 403 Kane, 13, 73 Kappus, 168, 171 Kepler, 14, 29 kinematic chain, 92, 93, 104, 114, 190, 191, 194, 215, 333, 335, 345, 347, 352, 362, 363, 368, 369 kinetic energy, 13, 20, 21, 29, 65, 69, 70, 123, 124, 127, 130, 214, 229, 237, 238 Kirchhoff, 2, 117, 120, 154, 155, 157, 169, 176, 177, 236, 245 Klein, 1, 22 Kronecker, 239 Lagrange, 2, 3, 7–9, 11–13, 15–17, 20, 22, 44, 63, 64, 72, 73, 100, 102, 103, 126, 130, 131, 169–171, 219, 221, 234, 385, 386, 426– 428 Lamb, 15, 155 Laval, 82, 272, 273, 276–278, 282–284, 324 Legendre, 72, 323 Leibniz, 2, 3, 22, 23 Lie, 20, 52, 281 linear mechanical systems, 389 Luenberger, 404 Lur’e, 20 Lyapunov, 381, 382, 390, 392, 402, 413 Mach, 15 Maggi, 72, 75, 126, 130 magnetic bearing, 288, 381 magnetic levitation vehicle, 428 Magnus, 3, 36, 280

447 mass element, 20, 63–66, 167, 199, 214, 228, 321, 331, 338, 343, 426 mass point, 3, 59, 60, 230, 278 Maupertuis, 22, 26, 72 minimal coordinates, 4, 15–17, 19, 126, 135, 300 minimal representation, 17, 18, 93–95, 98, 103, 107, 114, 125, 220, 333, 337, 352, 370 minimal velocities, 4, 15, 16, 18, 19, 21, 69, 70, 81, 92, 98, 101, 102, 104, 114, 132, 153, 155, 193, 194, 215, 220, 221, 255, 272, 301, 321, 324, 327, 332, 333, 348, 371, 427 mobile robot, 348 modal matrix, 31, 227, 229, 261, 396, 403 modeling, 4, 7, 19, 25, 29, 38, 174, 215, 217, 278, 296, 298, 327, 419 moments of inertia, 65, 80, 123, 127, 133, 134, 157, 159, 174, 246, 298, 308, 329, 331 momentum theorem, 2, 3, 7, 9, 71, 281 moving reference frame, 39, 49, 81, 214, 357 M¨ uller, 404 multiplier, 18, 103 Newton, 2, 14, 107, 247, 269 non-holonomic, 3, 17, 19, 20, 22, 25, 26, 69, 72, 75, 77, 82, 83, 85, 93, 94, 114, 124–126, 177, 190, 214, 321, 327, 395 nutation, 36, 280, 282, 284–286, 324, 397 observability, 235, 401, 403, 411, 415, 421, 422 observer, 14, 370, 375, 401, 410, 412–415, 417, 419, 422–424 one-link elastic robot, 291 optimal control, 388, 389 optimality, 385 optimization, 17, 238, 382, 383, 391 order-n-recursion, 114 ordinary differential equation, 144, 233 output control, 286 partial differential equation, 61, 136, 137, 144, 165, 228, 236, 246 particle, 59

448 pervasive damping, 108, 151, 235, 277, 282, 369, 404 Petrov, 219 Piola, 167, 169 plane motion, 63, 76, 77, 82, 83, 86, 93, 98, 109, 120, 192, 198, 201, 206, 212, 216, 230, 318, 337, 352, 353, 363, 371, 393, 429 plate, 117, 118, 120, 121, 144, 154, 155, 157, 216, 220, 237, 238, 242, 243, 245, 322, 323 ´, 245 Poincare Poinsot, 2, 13, 15, 20, 72 point mechanics, 60 Poisson, 117 Pontryagin, 387 potential, 13, 14, 60, 66–69, 112, 115, 124, 151, 152, 155, 175, 186, 219, 230, 235, 238, 294, 297, 300, 309, 326 potential energy, 229, 237, 391 power, 10, 219, 280, 323, 399 precession, 36, 280–282, 285 predecessor, 50, 53, 55, 58, 87, 91, 92, 95, 96, 98, 105, 107, 190, 192, 194, 196, 215, 254, 335, 339, 345, 362 principle, 2–4, 8, 9, 11–13, 15–18, 20, 22– 26, 63, 64, 68, 72, 98, 100, 114, 115, 124–126, 150, 168, 207, 216, 219, 221, 230, 234, 369, 388, 426 prismatic joint, 142, 180, 296, 338 Projection Equation, 4, 5, 70, 71, 77, 78, 114, 116, 131, 159, 177, 214, 309, 318, 321, 327, 329, 337, 353, 354 quasi-coordinates, 18, 220 Rayleigh, 68, 138, 150, 151, 153, 229, 230, 232, 233, 237, 238, 253, 265, 322 recursive representation, 4, 95, 100, 104, 114, 329, 333, 352, 373 reduced mass matrix, 96 reference frame, 5, 33, 44, 45, 55, 57, 58, 65, 66, 70, 71, 77–79, 81, 82, 85, 87, 91, 123, 127, 133, 202, 214,

Index 296, 299, 301, 321, 323, 327– 331, 338, 355, 356, 395 Resal, 36, 38 residual, 220 revolute joint, 57, 91, 142, 180, 346, 347 rheonomic, 25, 257, 323, 369, 392 Riccati, 380, 388–390, 414, 415 Riemann, 47 rigidity condition, 59, 61 Ritz, 150, 153, 164, 219, 221, 226, 229, 233–238, 245, 253, 255, 269, 286, 290, 292, 293, 297, 300, 301, 304, 308, 312, 319, 322, 323, 325, 326, 332, 333, 336, 337, 344, 345, 348, 365, 371, 375, 377, 380, 382 Rodrigues, 41 rotating beam, 164, 183, 226, 291, 324 rotation angle, 34, 37–39 rotation axis, 33, 34, 37, 39, 275, 281, 338 Schwarz, 44 scleronomic, 15, 25, 72, 102, 391 second-order displacements, 166, 167, 172, 175, 178, 195, 337 shape functions, 150, 164, 188, 217, 220, 221, 224, 228, 233–236, 238, 242, 243, 245, 248–251, 253, 255, 257, 259–264, 266–268, 283–286, 299, 308, 310, 322– 324, 326, 357, 359, 369, 377, 379, 380, 382 shell, 153, 154 SISO, 401 sorting matrices, 210 spatial operators, 64, 132, 153, 193, 196, 215, 301, 332 spillover, 286, 326, 379, 382 spring potential, 67 stability, 67, 277, 278, 281, 282, 305, 312, 324, 375, 380, 391, 392, 401– 405, 407, 413, 416 state equation, 146, 380, 389, 395–397, 399, 400, 404, 411, 413, 414, 419, 421 state feedback, 375, 380, 389, 410, 414, 423 Steiner, 80 Stodola, 405, 422 strain, 63, 116, 117, 119, 121, 167, 170, 171, 177, 348, 375

Index strain tensor, 167, 169–171 stress, 115, 117, 167, 169, 172 stress tensor, 115, 116, 167, 169, 170 Stribeck, 419 structurally variant systems, 104 subsystem, 78, 79, 83–95, 98, 100, 104, 105, 113, 114, 124, 131, 214, 215, 297, 327–331, 335–339, 341, 344, 345, 348, 354, 363, 364, 366, 367, 371, 377, 379 successor, 53, 91, 92, 95, 96, 100, 101, 107, 190, 193, 195, 196, 212, 335, 339 synthetic procedure, 73 Szabo, 22 Tait, 35, 404 Taylor, 16, 17, 67, 158, 372, 373, 383, 384, 392, 403 telescoping arm, 57, 93, 325 Timoshenko, 151, 153 tip body, 339, 340, 346, 356 topological chain, 100 topological tree, 53, 100 topology matrix, 50–53 torsional shaft, 246, 323, 324 transitivity equation, 20, 69, 72, 75

449 Trefftz, 167, 169, 170 variation, 9, 16, 17, 20, 22, 24, 69, 125, 126, 128, 144, 176, 195, 384– 386, 389, 399, 424 variational calculus, 16–18, 20, 24, 68, 126 Varignon, 3 virtual acceleration, 11 virtual displacements, 9, 11, 13, 15–17, 24, 25, 194, 311, 372 virtual velocities, 9, 11, 13 virtual work, 15, 87, 94, 116, 126, 130–132, 137, 144, 146, 166, 178, 180, 215, 219, 221, 224, 234, 248, 295, 297, 300, 332, 365, 427 Voigt, 157 walking machine, 417 wave equation, 146, 149, 150 Weierstrass, 384, 385, 388 whirl, 276, 281, 283–286, 324 work, 8–10, 13, 14, 64–66, 72, 100, 104, 108, 151, 172, 174, 177, 207, 286 Young, 117

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