European Consortium for Mathematics in Industry E. Eich-Soellner and C. FUhrer Numerical Methods in Multibody Dynamics

European Consortium for Mathematics in Industry E. Eich-Soellner and C. FUhrer Numerical Methods in Multibody Dynamics

European Consortium for Mathematics in Industry Edited by Leif Arkeryd, Goteborg Heinz Engl, Linz Antonio Fasano, Firenze Robert M. M. Mattheij, Eindhoven Pekka Neittaanmaki, Jyvaskyla Helmut Neunzert, Kaiserslautern Within Europe a number of academic groups have accepted their responsibility towards European industry and have proposed to found a European Consortium for Mathematics in Industry (ECMI) as an expression of this responsibility. One of the activities of ECMI is the publication of books, which reflect its general philosophy; the texts of the series will help in promoting the use of mathematics in industry and in educating mathematicians for industry. They will consider different fields of applications, present casestudies, i'ntroduce new mathematical concepts in their (elation to practical applications. They shall also represent the variety of the European mathematical traditions, for example practical asymptotics and differential equations in Britian, sophisticated numerical analysis from France, powerful computation in Germany, novel discrete mathematics in Holland, elegant real analysis from Italy. They will demonstrate that all these branches of mathematics are applicable to real problems, and industry and universities in any country can clearly benefit from the skills of the complete range of European applied mathematics.

Numerical Methods in Multibody Dynamics Edda Eich-Soellner Fachbereich Informatik / Mathematik Fachhochschule Mlinchen, Germany Claus FUhrer Department of Computer Science and Numerical Analysis Lund University, Sweden Springer Fachmedien Wiesbaden GmbH 1998

Prof. Dr. rer. nat. Edda Eich-Soellner Fachbereich InformatiklMathematik Fachhochschule MUnchen LothstraBe 34 0-80335 MUnchen Germany e-mail: edda.eich@informatik.fh-muenchen.de WWW: http://www.informatik.fh-muenchen.delprofessoren/eddaeich-soellnerl Docent Dr. rer. nat. Claus FUhrer Department of Computer Science and Numerical Analysis Lund University Box 118 S-22100 Lund Sweden e-mail: c1aus@dna.lth.se WWW: http://www.dna.lth.se/home/claus_fuhrer/ Die Deutsche Bibliothek - CIP-Einheitsaufnahme Eich-SoeJlner, Edda: Numerical methods in multi body dynamics I Edda Eich-Soellner; Claus FUhrer. (European Consortium for Mathematics in Industry) ISBN 978-3-663-09830-0 ISBN 978-3-663-09828-7 (ebook) DOI 10.1007/978-3-663-09828-7 Copyright 1998 by Springer Fachmedien Wiesbaden Originally published by B. G. Teubner Stuttgart in 1998 All rights reserved No part of this book may be reproduced by any means, or transmitted, or translated into a machine language without the written permission of the publisher. Cover design by Peter Pfitz Stuttgart.

Preface Numerical Analysis is an interdisciplinary topic which develops its strength only when viewed in close connection with applications. Nowadays, mechanical engineers having computer simulation as a daily engineering tool have to learn more and more techniques from that field. Mathematicians, on the other hand, are increasingly confronted with the need for developing special purpose methods and codes. This requires a broad interdisciplinary understanding and a sense for model-method interactions. With this monograph we give an introduction to selected topics of Numerical Analysis based on these facts. We dedicate our presentations to an interesting discipline in computational engineering: multibody dynamics. Though the basic ideas and methods apply to other engineering fields too, we emphasize on having one homogeneous class of applications. Both authors worked through many years in teams developing multibody codes. Interdisciplinary work also includes transferring ideas from one field to the other and a big amount of teaching - and that was the idea of this book. This book is intended for students of mathematics, engineering and computer science, as well as for people already concerned with the solution of related topics in university and industry. After a short introduction to multibody systems and the mathematical formulation of the equations of motion, different numerical methods used to solve simulation tasks are presented. The presentation is supported by a simple model of a truck. This truck model will follow the reader from the title page to the appendix in various versions, specially adapted to the topics. The models used in this book are not intended to be real-life models. They are constructed to demonstrate typical effects and properties occurring in practical simulations. The methods presented include linear algebra methods (linearization, stability analysis of the linear system, constrained linear systems, computation of nominal interaction forces), nonlinear methods (Newton and continuation methods for the computation of equilibrium states), simulation methods (solution of discontinuous ordinary differential and differential algebraic equations) and solution methods for

6 inverse problems (parameter identification). Whenever possible, a more general presentation of the methods is followed by a special view, taking the structure of multibody equations into consideration. Each chapter is divided into sections. Some of the material can be skipped during a first reading. An asterisk (*) in the section title is indicating these parts. Nearly all methods and examples are computed using MATLAB programs and nearly all examples are related to the truck model. Those MATLAB programs which describe the truck itself are given in the appendix for supporting the description of the model. Others are given as fragments in the text, where MATLAB is used as a piece of meta language to describe an algorithm. Some of the examples had been used in universitary and post universitary courses. These can be obtained among other information related to this book via the book's homepage1. We want to thank everyone who has helped us to write this book: our teachers, colleagues, friends and families. January 1998 Edda Eich-Soellner and Claus Fuhrer 1 http://www.teubner.de/cgi-bin/teubner-anzeige.sh?buch_no= 12

Contents 1 Multibody Systems 1.1 What is a Multibody System?.... 1.2 Basic Mathematical Tasks in Multibody Dynamics 1.3 Equations of Motion of Multibody Systems... 1.3.1 Unconstrained Planar Multibody Systems 1.3.2 Constrained Planar Multibody Systems. 1.3.3 Nonholonomic Constraints......... 1.3.4 System with Dynamical Force Elements* 1.3.5 General Systems......... 1.4 Relative and Absolute Coordinates.. 1.4.1 Mixed Coordinate Formulation* 1.5 Linearization of Equations of Motion. 1.6 Nominal Interaction Forces.... 11 11 13 14 14 17 20 21 21 23 26 30 32 2 Linear Systems 2.1 State Space Form of Linear Constrained Systems 2.2 Numerical Reduction to State Space Form. 2.3 Constrained Least Squares Problems 2.3.1 Pseudo-Inverses.... 2.3.2 Numerical Computation...... 42 2.3.3 Underdetermined Linear Systems. 45 2.4 The Transition Matrix... 50 2.5 The Frequency Response.......... 51 2.5.1 Numerical Methods for Computing the Frequency Response. 54 2.6 Linear Constant Coefficient DAEs... 56 2.6.1 The Matrix Pencil*................. 59 2.6.2 The Matrix Pencil and the Solution of the DAE* 61 2.6.3 Construction of the Drazin Inverse*.. 2.7 DAEs and the Generalized Eigenvalue Problem 2.7.1 State Space Form and the Drazin ODE* 35 35 37 39 41 64 71 73

8 CONTENTS 3 Nonlinear Equations 3.1 Static Equilibrium Position 3.2 Solvability of Nonlinear Equations 3.3 Fixed Point Iteration........ 3.4 Newton's Method.... 3.5 Numerical Computation of Jacobians. 3.6 Reevaluation of the Jacobian... 3.7 Limitations and Extensions...... 3.8 Continuation Methods in Equilibrium Computation 3.8.1 Globalizing the convergence of Newton's Method 3.8.2 Theoretical Background.......... 3.8.3 Basic Concepts of Continuation Methods 4 Explicit Ordinary Differential Equations 4.1 Linear Multistep Methods.... 4.1.1 Adams Methods.......... 4.1.2 Backward Differentiation Formulas (BDF) 4.1.3 General Form of Multistep Methods 4.1.4 Accuracy of a Multistep Method 4.1.5 Order and Step Size Selection.. 4.1.6 Solving the Corrector Equations 4.2 Explicit Runge-Kutta Methods 4.2.1 The Order of a Runge-Kutta Method 4.2.2 Embedded Methods for Error Estimation 4.2.3 Stability of Runge-Kutta Methods 4.3 Implicit Runge-Kutta Methods.......... 4.3.1 Collocation Methods.... 4.3.2 Corrector Equations in Implicit Runge-Kutta Methods 4.3.3 Accuracy of Implicit Runge-Kutta Methods. 4.3.4 Stability of Implicit Runge-Kutta Methods 4.4 Stiff Systems.... 4.5 Continuous Representation.... 5 Implicit Ordinary Differential Equations 5.1 Implicit ODEs.... 5.1.1 Types of Implicit ODEs.... 5.1.2 Existence and Uniqueness of Solutions 5.1.3 Sensitivity under Perturbations 5.1.4 ODEs with Invariants.... 5.2 Linear Multistep Methods for DAEs.... 5.2.1 Euler's Method and Adams-Moulton for DAEs 5.2.2 Multistep Discretization Schemes for DAEs 5.2.3 Convergence of Multistep Methods.... 77 77 79 80 81 84 86 87 88 88 90 91 95 96 96 99 101.102.110.114 118 120 121 124 124 125 127 130 131 132 136 139 139 139 140 143 146 149 149 155 156

CONTENTS 9 5.2.4 Solving the Corrector Equations in Discretized DAEs. 162 5.3 Stabilization and Projection Methods. 164 5.3.1 Coordinate Projection Method.......... 165 5.3.2 Implicit State Space Form............ 166 5.3.3 Projection Methods for ODEs with Invariants.. 171 5.4 One Step Methods....... 176 5.4.1 Runge-Kutta Methods. 176 5.4.2 Half-Explicit Methods. 179 5.5 Contact Problems as DAEs.. 185 5.5.1 Single Point Contact of Planar Bodies. 185 5.5.2 Problems................. 190 6 ODEs with Discontinuities 193 6.1 Difficulties with Discontinuities........... 194 6.2 Differential Equations with Switching Conditions. 196 6.3 Switching Algorithm............... 198 6.3.1 Test for Occurrence of a Discontinuity... 198 6.3.2 Localization of Switching Points.... 198 6.3.3 Changing the Right Hand Side and Restarting. 202 6.3.4 Adaptation of the Discretization......... 202 6.3.5 A Model Problem for Setting up a Switching Logic.. 204 6.3.6 Aspects of Realization........... 206 6.3.7 Other Methods for Discontinuous ODEs. 208 6.3.8 DAEs with Discontinuities........ 209 6.4 Coulomb Friction: Difficulties with Switching. 6.4.1 Introductory Example.......... 6.4.2 Coulomb Friction: Mathematical Background 6.5 Other Special Classes of Discontinuities....... 6.5.1 Time Events.... 6.5.2 Structure Varying Contact Problems with Friction 6.5.3 Computer-controlled Systems.... 6.5.4 Hysteresis... 6.5.5 Approximations by Piecewise Smooth Functions 6.5.6 Implementation of Switching Algorithms..210.211. 213.224.224.224.228. 231.232.241 7 Parameter Identification Problems 7.1 Problem Formulation.... 7.2 Numerical Solution.... 7.2.1 Elimination of the ODE: Integration 7.2.2 GauB-Newton Methods.... 7.2.3 Evaluation of Functions and Jacobians. 7.2.4 Summary of the Algorithm.. 7.2.5 The Boundary Value Approach... 243.244.247.247.248.250.253.254

10 CONTENTS 7.3 Extensions... 7.3.1 Differential Algebraic Equations 7.3.2 Discontinuous Systems.... A The Truck Model A.l Data ofthe Truck Model. A.2 Model of a Nonlinear Pneumatic Spring.. A.3 Matlab m-files for the Unconstrained Truck Bibliography Index.259.259.264 267.267.268.271 277 287