Dynamics of Multibody Systems

231 Pages·2008·2.187 MB·English

by  Jens Wittenburg (auth.)

JensWittenburg
Dynamicsof MultibodySystems
Jens Wittenburg
Dynamics
of Multibody Systems
SecondEdition
123
ProfessorDr.-Ing.JensWittenburg
UniversityofKarlsruhe(TH)
InstituteofEngineeringMechanics
Kaiserstrasse
Karlsruhe,Germany
Email:[email protected]
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Preface
Preface to the Second Edition
The first edition of this book published thirty years ago by Teubner had the
title Dynamics of Systems of Rigid Bodies [97]. Soon after publication the
term multibody system became the name of this new and rapidly developing
branch of engineering mechanics. For this reason, the second edition pub-
lished by Springer appears under the title Dynamics of Multibody Systems.
Because of the success of the first edition (translations into Russian (1980),
Chinese (1986) and Vietnamese (2000); use as textbook in advanced courses
in Germany and abroad) little material has been added in the new edition.
In Chaps.1–4nothing has changedexceptforthe incorporationofshortsec-
tions on quaternions and on raccording axodes. Chapters 5 and 6 have been
rewritteninanewform.Bothchaptersarestilldevotedtomultibodysystems
composedofrigidbodieswithfrictionlessjoints.Manyyearsofteachinghave
ledtosimplermathematicalformulationsinvariousplaces.Also,theorderof
topics has changed. Multibody systems with spherical joints and with equa-
tionsofmotionallowingpurelyanalyticalinvestigationsarenolongertreated
first but last. The emphasis is placed on a general formalism for multibody
systems with arbitrary joints and with arbitrary system structure. This for-
malism has found important engineering applications in many branches of
industry. The first software tool based on the formalism was a FORTRAN
program written by the author in 1975 for Daimler-Benz AG for simulating
the dynamics of a human dummy in car accidents (passenger inside the car
or pedestrian outside). Wolz [106] created the software tool MESA VERDE
(MEchanism, SAtellite, VEhicle, Robot Dynamics Equations). Its charac-
teristic feature is the generation of kinematics and dynamics equations in
symbolic form.Using the same formalismSalecker[71],Wei[91],Weber [89],
Bu¨hrle [11] and Reif [62] developed equations of motion as well as software
toolsformultibodysystemscomposedofflexiblebodiesandforsystemswith
VIII Preface
electrical and hydraulic components. As a result of collaboration with IPG
Automotive, Karlsruhe MESA VERDE-generated kinematics and dynamics
equations for vehicles became the backbone of IPG’s CarMaker(cid:2) product
range, which has become a powerful tool for vehicle dynamics analysis and
for Hardware-in-the-Loop testing of vehicle electronic control systems. Car-
MakeristhebasisofAVLInMotion(cid:2) whichisusedforHardware-in-theLoop
development and testing of engines and entire powertrains.MESA VERDE-
generatedequationsareusedinthesoftwaretoolFADYNAdevelopedbyIPG
for Daimler-Chrysler.MESA VERDE is also used by Renault, PSA Peugeot
Citroen and Opel.
It is a pleasure to thank Prof. Lothar Gaul for encouraging Springer as
well as the author to publish this second edition. The author is indebted
to Gu¨nther Stelzner and to Christian Simonides for their frequent advice in
using TEX and to Marc Hiller for producing the data of all figures. Finally,
I would like to thank the publisher for their technical advice and for their
patience in waiting for the completion of the manuscript.
Karlsruhe,
June 2007 Jens Wittenburg
Preface to the First Edition
A system of rigid bodies in the sense of this book may be any finite num-
ber of rigid bodies interconnected in some arbitrary fashion by joints with
ideal holonomic, nonholonomic, scleronomic and/or rheonomic constraints.
Typical examples are the solar system, mechanisms in machines and living
mechanisms such as the human body provided its individual members can
be considered as rigid. Investigations into the dynamics of any such system
require the formulation of nonlinear equations of motion, of energy expres-
sions, kinematic relationships and other quantities. It is common practice to
develop these for eachsystem separately and to consider the labor necessary
for deriving, for example, equations of motion from Lagrange’s equation, as
inevitable. It is the main purpose of this book to describe in detail a formal-
ism which substantially simplifies this task. The formalism is generalin that
it provides mathematical expressions and equations which are valid for any
system of rigidbodies. It is flexible in that it leavesthe choice of generalized
coordinatestothe user.Atthe sametime itis soexplicitthatits application
to any particular system requires only little more than a specification of the
system geometry. The book is addressed to advanced graduate students and
to researchworkers.It tries to attract the interestof the theoretician as well
as of the practitioner.
The first four out of six chapters are concerned with basic principles and
with classicalmaterial.InChap.1 the readeris made familiar with symbolic
Preface IX
vectorandtensornotationwhichisusedthroughoutthisbookforitscompact
form.Inordertofacilitatethetransitionfromsymbolicallywrittenequations
to scalar coordinate equations matrices of vector and tensor coordinates are
introduced. Transformationrules for such matrices are discussed, and meth-
ods are developed for translating compound vector-tensor expressions from
symbolic into scalar coordinate form. For the purpose of compact formula-
tions of systems of symbolically written equations matrices are introduced
whose elements are vectors or tensors. Generalized multiplication rules for
such matrices are defined.
InChap.2onrigidbodykinematicsdirectioncosines,Eulerangles,Bryan
angles and Euler parameters are discussed. The notion of angular velocity
is introduced, and kinematic differential equations are developed which re-
late the angular velocity to the rate of change of generalized coordinates. In
Chap.3basicprinciplesofrigidbodydynamicsarediscussed.Thedefinitions
ofbothkineticenergyandangularmomentumleadstotheintroductionofthe
inertiatensor.Formulationsofthelawofangularmomentumforarigidbody
arederivedfromEuler’saxiomandalsofromd’Alembert’sprinciple.Because
of severe limitations on the length of the manuscript only those subjects are
covered which are necessary for the later chapters. Other important topics
such as cyclic variables or quasicoordinates, for example, had to be left out.
In Chap. 4 some classical problems of rigid body mechanics are treated for
which closed-form solutions exist. Chapter 5 which makes up one half of the
bookisdevotedtothepresentationofageneralformalismforthedynamicsof
systems of rigid bodies. Kinematic relationships, nonlinear equations of mo-
tion,energyexpressionsandotherquantitiesaredevelopedwhicharesuitable
for both numericaland nonnumericalinvestigations.The unform description
validfor anysystemofrigidbodies restsprimarilyonthe applicationofcon-
cepts of graphtheory (the firstapplicationto mechanics atthe time of[66]).
Thismathematicaltoolincombinationwithmatrixandsymbolicvectorand
tensornotationleadstoexpressionswhichcaneasilybe interpretedinphysi-
calterms.Theusefulnessoftheformalismisdemonstratedbymeansofsome
illustrative examples of nontrivial nature. Chapter 6 deals with phenomena
which occur when a multibody system is subject to a collision either with
another system or between two of its own bodies. Instantaneous changes of
velocitiesandinternalimpulsesinjointsbetweenbodiescausedbysuchcolli-
sions are determined. The investigationreveals an interesting analogy to the
law of Maxwell and Betti in elastostatics.
The material presented in subsections 1, 2, 4, 6, 8 and 9 of Sect. 5.2 was
developed in close cooperation with Prof. R.E. Roberson (Univ. of Calif. at
San Diego) with whom the author has a continuous exchange of ideas and
results since 1965. Numerous mathematical relationships resulted from long
discussionssothatauthorshipisnotclaimedbyanyoneperson.Itisapleas-
ant opportunity to express my gratitude for this fruitful cooperation. I also
X Preface
thank Dr. L. Lilov (Bulgarian Academy of Sciences) with whom I enjoyed
close cooperation on the subject. He had a leading role in applying methods
ofanalyticalmechanics(subjectofSect.5.2.8)andhecontributedimportant
ideas to Sect. 5.2.5.Finally, I thank the publishers for their kind patience in
waiting for the completion of the manuscript.
Hannover,
February 1977 Jens Wittenburg
Contents
1 Mathematical Notation ................................... 1
2 Rigid Body Kinematics ................................... 9
2.1 Generalized Coordinates of Angular Orientation............ 9
2.1.1 Euler Angles..................................... 9
2.1.2 Bryan Angles .................................... 12
2.1.3 Rotation Tensor ................................. 14
2.1.4 Euler–Rodrigues Parameters....................... 18
2.1.5 Euler–Rodrigues Parameters in Terms of Euler Angles 19
2.1.6 Quaternions ..................................... 20
2.2 Kinematics of Continuous Motion ........................ 23
2.2.1 Angular Velocity. Angular Acceleration ............. 23
2.2.2 Inverse Motion................................... 26
2.2.3 Instantaneous Screw Axis. Raccording Axodes ....... 27
2.3 Kinematic Differential Equations ......................... 32
2.3.1 Direction Cosines................................. 32
2.3.2 Euler Angles..................................... 33
2.3.3 Bryan Angles .................................... 33
2.3.4 Euler–Rodrigues Parameters....................... 34
3 Basic Principles of Rigid Body Dynamics ................. 37
3.1 Kinetic Energy......................................... 37
3.2 Angular Momentum .................................... 39
3.3 Properties of Moments and of Products of Inertia........... 40
3.3.1 Change of Reference Point. Reference Base Unchanged 40
3.3.2 Change of Reference Base. Reference Point Unchanged 41
3.3.3 Principal Axes. Principal Moments of Inertia ........ 42
3.3.4 Invariants. Inequalities ............................ 43
3.4 Angular Momentum Theorem............................ 44
3.5 Principle of Virtual Power ............................... 47
XII Contents
4 Classical Problems of Rigid Body Mechanics ............. 49
4.1 Unsymmetric Torque-Free Rigid Body .................... 49
4.1.1 Polhodes. Permanent Rotations .................... 50
4.1.2 Poinsot’s Geometric Interpretation of the Motion..... 52
4.1.3 Solution of Euler’s Equations of Motion ............. 53
4.1.4 Solution of the Kinematic Differential Equations ..... 55
4.2 Symmetric Torque-Free Rigid Body....................... 58
4.3 Self-Excited Symmetric Rigid Body....................... 60
4.4 Symmetric Heavy Top .................................. 62
4.5 Symmetric Heavy Body in a Cardan Suspension............ 70
4.6 Gyrostat. General Considerations......................... 72
4.7 Torque-Free Gyrostat ................................... 77
4.7.1 Polhodes. Permanent Rotations .................... 78
4.7.2 Solution of the Dynamic Equations of Motion........ 80
5 General Multibody Systems .............................. 89
5.1 Definition of Goals...................................... 89
5.2 Elements of Multibody Systems .......................... 91
5.3 Interconnection Structure of Multibody Systems............ 94
5.3.1 Directed System Graph. Associated Matrices......... 95
5.3.2 Directed Graphs with Tree Structure ............... 101
5.3.3 Regular Tree Graphs.............................. 102
5.4 Principle of Virtual Power for Multibody Systems .......... 105
5.4.1 Systems Without Constraints to Inertial Space....... 105
5.4.2 Generalized Coordinates .......................... 107
5.5 Systems with Tree Structure ............................. 109
5.5.1 Kinematics of Individual Joints .................... 109
5.5.2 Kinematics of Entire Systems ...................... 113
5.5.3 Equations of Motion .............................. 116
5.5.4 Augmented Bodies ............................... 118
5.5.5 Force Elements................................... 121
5.5.6 Constraint Forces and Torques in Joints............. 124
5.5.7 Software Tools ................................... 126
5.6 Systems with Closed Kinematic Chains.................... 129
5.6.1 Removal of Joints. Holonomic Constraints ........... 129
5.6.2 Duplication of Bodies ............................. 131
5.6.3 Controlled Joint Variables ......................... 132
5.6.4 Nonholonomic Constraints......................... 135
5.6.5 Constraint Forces and Torques in Joints............. 135
5.6.6 Illustrative Examples ............................. 136
5.6.6.1 Planar Fourbar........................... 136
5.6.6.2 OrthogonalBricard Mechanism............. 137
5.6.6.3 Stewart Platform ......................... 142
5.6.6.4 Table on Wheels.......................... 147
5.7 Systems with Spherical Joints............................ 150