Fundamentals of Multibody Dynamics

Transcription

1

2 Farid Amirouche Fundamentals of Multibody Dynamics Theory and Applications Birkhäuser Boston Basel Berlin

3 Farid M. L. Amirouche The University of Illinois at Chicago Department of Mechanical and Industrial Engineering Chicago, IL U.S.A. Cover design by Alex Gerasev. AMS Subject Classifications: 70Exx, 70E05, 70E15, 70E17, 70E18, 70E20, 70E45, 70E50, 70E55, 70E60, 70E99, 70Fxx, 70F07, 70F10, 70F16, 70F20, 70F25, 70F35, 70G10, 70G40, 70G45, 70G55, 70G60, 70G65, 70H03, 70H05, 70H08, 70H09, 70H20, 70H25, 70H40, 70H45, 70Jxx, 70J10, 70J50, 70K43, 70M20, 93Dxx Library of Congress Cataloging-in-Publication Data Amirouche, Farid M. L. Fundamentals of multibody dynamics : theory and applications / Farid Amirouche. p. cm. Includes bibliographical references and index. ISBN (acid-free paper) 1. Dynamics. 2. Kinematics. I. Title. QA845.A dc ISBN eisbn Printed on acid-free paper. ISBN c 2006 Birkhäuser Boston Based on the author s previous edition, Computational Methods in Multibody Dynamics, Prentice-Hall, Englewood Cliffs, NJ, All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (KeS/EB) SPIN

4 To my son and daughter Larby and Anissa

5 Contents Preface... xiii 1 Particle Dynamics: The Principle of Newton s Second Law Introduction Vectors Scalar Product Vector Product Derivative of a Vector Velocity and Acceleration in Several Coordinates Dynamics of a System of Particles Work and Kinetic Energy Conservation of Energy Conservative Forces Principle of Impulse and Momentum Angular Momentum Principle of Virtual Work Problems Rigid-Body Kinematics Introduction Vectors Differentiation First Derivatives and Partial Derivatives Definition of a Derivative Definition of Partial Derivatives Total Derivative of a Vector Generalized Coordinates Cartesian Coordinates Euler Angles and Direction Cosines Euler and Rodriguez Formula Euler Parameters Rodriguez Parameters Angular Velocity... 68

6 viii Contents 2.7 Further Derivation of the Angular Velocity Angular Velocity and Euler Angles Angular Velocity and Euler Parameters Simple Angular Velocity Angular Velocity and Intermediate Reference Frames Angular Acceleration Velocity and Acceleration of a Fixed Point on a Rigid Body Velocity and Acceleration of a Moving Point on a Rigid Body Small Rotations Problems Kinematics for General Multibody Systems Introduction Configuration Graphs for Treelike Multibody Systems Generalized Coordinates Partitioning Transformation Matrices and Their Derivatives for N-Interconnected Rigid Bodies Transformation Matrices Time Derivatives of Transformation Matrices Angular Velocities and Partial Angular Velocities Angular Accelerations Generalized Speeds Mass Center Velocities of Arbitrary Interconnected Rigid Bodies in Treelike Systems Mass Center Accelerations of Arbitrary Interconnected Rigid Bodies in Treelike Systems Modeling of Joints Free Joint Ball-and-Socket Joint Revolute Joint Problems Modeling of Forces in Multibody Systems Introduction Forces, Moments and Equivalence Force Systems Generalized Active Force Modeling of Springs and Dampers at the Joints Contact Forces Gravitational Forces Generalized Inertia Forces Inertia Properties Second Moment Product of Inertia Properties of the Product of Inertia Inertia Dyadic Parallel Axes Theorem

7 Contents ix 4.9 Problems Equations of Motion of Multibody Systems Introduction Equations of Motion Derivation of Kane s Equations Through the Principle of Virtual Work Principle of Virtual Work Automated Form of the Equations of Motion Matrix Representation of the Equations of Motion Problems Project Hamilton Lagrange and Gibbs Appel Equations Introduction Energy Equations Kinetic Energy Energy of a Rigid Body Work, Potential Energy and Generalized Forces Lagrange s Equations Application of Lagrange Equations to Multibody Systems Relationship Between Kane s and Lagrange Equations Gibbs Appel Equations Hamilton s Equations Problems Handling of Constraints in Multibody Systems Dynamics Introduction Holonomic Constraints Nonholonomic Constraints Constrained Multibody Systems The Augmented Method Coordinate Reduction Pseudo Upper Triangular Decomposition Method The Zero-Eigenvalue Theorem and Embedding Method The Zero-Eigenvalue Theorem Embedding Method Relation Between the PUTD and the Zero-Eigenvalue Theorem Derivation of the Constraint Equation for Closed Loops Prescribed Motion Evaluation of the Constraint Forces The Rolling Coin Problem: Kinematics, Forces and Constraints Analysis and Simulation of Human Locomotion Problems

8 x Contents 8 Numerical Stability of Constrained Multibody Systems Introduction Baumgarte Stability Method Numerical Solution of a Constrained System s Equations Effect of Constraint Differentiation Reduction to State-Space Form Modified PUTD Method: A Gaussian Approach Regularization of the Vanishing Constraints: The Amirouche Ider Stabilization Method Regularization of Linearly Dependent Constraints: The Amirouche Ider Method Linearization and Vibration Analysis of Multibody Systems Introduction Linearization of the Equations of Motion Free Vibration of Continuous Beams: Natural Mode Shapes and Frequencies Transverse Vibration Longitudinal Vibration Torsional Vibration The Eigenvalue Problem Rayleigh Ritz Method Assumed Modes Method Forced System Response and Selection of Mode Shapes Selection of Mode Shapes Numerical Methods for Eigenvalue Problems Jacobi Method Subspace Iteration Method Problems Dynamics of Multibody Systems with Terminal Flexible Links Introduction Method of Motion Overlapping Case 1: Transverse Vibration of the Flexible Body Case 2: Longitudinal Vibration of the Flexible Beam Case 3: Transverse Vibration of a Flexible Link Considering Inertia Force and Gravity Case 4: Torsional Vibration of the Flexible Link Derivation of the Equations of Motion Using the Finite-Element Method Equations of Motion of an Elastic Beam Undergoing Large Rotation: A 2D FE Formulation General Equations of Motion of Multibody Systems with Flexible Terminal Links Six-Dimensional Beam Element Modeling of Terminal Flexible Links in MBS

9 Contents xi 10.7 Analysis of Elastic Beams with Time-Variant Boundary Conditions Problems Dynamic Analysis of Multiple Flexible-Body Systems Introduction Topology and Kinematics of Flexible Treelike Systems Kinetics An Illustrative Example Reduction of the Equations of Motion Through Modal Analysis Effect of Geometric Stiffening Geometric Stiffness Matrix of an Isoparametric Brick Element Dynamic Simulations: Applications Dynamic Simulation of a Space-Based Robotic Manipulator Using Beam Elements Dynamic Simulation of a Three-Flexible-Link Robot Using Three-Dimensional Brick Elements Summary Problems Modeling of Flexibility Effects Using the Boundary-Element Method Introduction Model Description and Notation Partial Velocities Generalized Inertia Forces Generalized Active Forces Generalized Constraint Forces Equations of Motion of a Continuum Body Linearization of the Equations of Motion Weighted Residual Statement Body Forces Formulation of the Nonlinear Stiffness Matrix Nonlinear Stiffness Matrix due to Boundary Integral Nonlinear Stiffness Matrix due to DRM Integral General Equations of Motion Numerical Example Conclusion Appendix A: Multibody Dynamics Flowchart for the Construction of the Equations of Motion with Constraints Appendix B: Centroid Location and Area Moment of Inertia Appendix C: Center of Gravity and Mass Moment of Inertia of Homogeneous Solids

10 xii Contents Appendix D: Symbols Description Appendix E: Units and Conversion References Index

11 Preface Multibody dynamics has grown in the past two decades to become an important tool in the design, prototyping and simulation of complex articulated mechanical systems. Its versatility in analyzing a broad range of applications has made it an attractive feature in our teaching curriculum. Here at our university we offer two courses in multibody dynamics that are given in subsequent semesters. The first course deals mainly with rigid-body dynamics whereas the second course is devoted to constrained systems and the effects of flexibility in multibody dynamics. This book brings together brilliant concepts of dynamics that combine the efforts of many researchers in the field of mechanics. The book s strength lies in its use of matrices in generating the kinematic coefficients associated with the formulation of the governing equations of motion. The book outlines the most effective ways of handling constraints and discusses in great detail how to use both finite-element and boundary-element methods to study the effects of deformation and vibration of structures during the course of certain maneuvers. A large number of examples worked out in their entirety follow every section. This is a learning-by-example approach that has been very instrumental in my teaching and research over the years. This book is the result of many years of research and teaching of multibody dynamics. In essence, it is a revision of my previous book entitled Computational Methods in Multibody Dynamics published in 1991 by Prentice Hall. I have added a new chapter on particle dynamics, which serves as a review of previous dynamical principles that the student needs to be familiar with. An extensive review of the kinematics of a rigid body is presented in Chapter 2 together with a Matlab listing at the end of the chapter that deals with Euler angles and Euler parameters. The remainder of the chapters deal directly with treelike structures with open and closed loops and prescribed motions including flexibility effects. The book bridges the gap between dynamics and engineering applications such as robotics, mechanics, and biosystems and puts into perspective the importance of modeling in the solution of these problems. The first seven chapters form the first part of a course in multibody dynamics where students are only expected to learn how to apply the developed procedures and principles to obtain the equations of motion. The remaining four chapters form the second part of the course where students are exposed to some concepts in vibration

12 xiv Preface and finite-element methods. The first semester is aimed at junior/senior-level students and first-year graduate students, whereas the second is primarily for advanced seniors and graduate students. The book is divided into twelve chapters. Chapter 3 is dedicated to the kinematics of rigid bodies and of interconnected multibody systems, in particular with treelike structure. The kinematic expressions are derived from the assumptions that each body can undergo all possible relative motions, including three rotations and three translations. Each chapter is divided into subsections in which several worked-out problems are presented to highlight the use of matrices in the representation of the kinematics of more than a one-body system. Chapter 4 is devoted to the analysis of forces; this includes springs and dampers and concludes with some inertial properties. The chapter includes some key examples on the contribution of body forces and torques to the formation of generalized active forces. Chapter 5 presents the formulation of the equations of motion making use of previously derived kinematic expressions from previous chapters and Kane s equations, Huston s tensor approach and matrix procedures developed by the author. The end of the chapter provides a special application using the techniques developed in this book. Furthermore, a flowchart used in the Matlab program is given for students who are interested in writing their own code. Student projects of this sort are useful and suggested as part of their homework. In Chapter 6 the energy equations derived by Lagrange, Hamilton and Gibbs equations are presented with examples to show the effectiveness of all the methods. The governing equations of motion of treelike multibody systems are given both in matrix form and in tensor (an index type of notation) form ready for computer implementation. In Chapter 7 holonomic or nonholonomic constraints are studied extensively in the context of constrained multibody systems, where several methods for coordinate reduction through orthogonal complementary arrays are introduced. The pseudoupper triangular method, the zero-eigenvalue theorem, and embedding methods are studied, together with some useful engineering examples. Prescribed motions and constraint force evaluations are also presented. The stability of the constraints as they change from 2D to 3D and the handling of zero constraints form the basis of Chapter 8. Chapter 9 is used as a review and preparation for the analysis of flexible multibody systems. Linearization of the equations of motion and the eigenvalue problem solution to continuous and discrete systems are studied. The next two chapters are used for the formulation of the equations of motion using both the procedures derived in Chapter 5 combined with finite-element methods. This includes treelike systems with terminal flexible links and systems with more than one flexible body, respectively. The final chapter introduces boundary-element methods to multibody dynamics. We developed this concept back in 1996 and it was the subject of a Ph.D. thesis of one of my former students, M. Kerdjouj. It is an excellent idea that needs to be explored further by our researchers and students as it may lead to a more satisfactory way of handling surface flexibility and body deformation in multibody systems (MBS). This book would not have been possible without the help of current and former students during the past fifteen years. I am indebted to Jia Tongyia (Tony) and Kemal

13 Preface xv S. Ider, Mohamed Kerdjoudj, Shareef Nazer, and Tung C.W (Roger), Rick Tong, Xie Mingjun and Tajiri Gordon for their stimulating thoughts through the years of their Ph.D. thesis research. I am especially thankful for the help of my students Francisco Romero, Carlos Lopez, Pedro Gonzalez, David McNeil, Ravikumar Vardrajan, Dr. Mark Gonzalez, Joe Solomon and Nikhil Kulkarni during the course of the multibody dynamics course. I would like to acknowledge Mrs. Irena Zivkovic for her help in the initial text change from Microsoft Word to Latex. It is certainly the dedication of Surya Pratap Rai in helping with the final manuscript organization and editing that made the book possible. I would like to acknowledge the help of my current students Carlos Lopez, Jude Martin, Khurram Mahmudi and Ivan Zivkovic for their attentive help with the figures in Chapters 3 5. Furthermore, I want to extend my sincere thanks to the reviewers who have taken the time out of their busy schedules to help with the review of some of the chapters of the book. My special thanks to Paul Mitiguy, Lazslo Palkovics, Moustafa El-Gindy, Kurt Anderson, Arun Banerjee, Jan Leuridan and Peter Brett as well as others who have contributed directly or indirectly to the content of the book. Lastly, I would like to thank my family for their continuous love and support and the joy they bring into my life. I would like to mention that my father, whose memory lives on, is still an inspiration to my devotion and work. I am also blessed to have my mother s timeless moral support. To all those who crave knowledge in dynamics, I hope this book will stimulate your thoughts and ideas; and for that I wish you success and prosperity. Farid Amirouche Chicago, Illinois 2004