Table Of ContentFlexible Multibody
Dynamics
Flexible Multibody
Dynamics
Efficient Formulations with Applications
Second Edition
Arun K. Banerjee
Second edition published 2022
by CRC Press
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and by CRC Press
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© 2022 Arun K. Banerjee
First edition published by Wiley in 2016
CRC Press is an imprint of Taylor & Francis Group, LLC
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Library of Congress Cataloging‑in‑Publication Data
Names: Banerjee, Arun K., author.
Title: Flexible multibody dynamics : efficient formulations with applications /
Arun Banerjee.
Description: Second edition. | Boca Raton, FL : CRC Press, [2022] |
Includes index. | Identifiers: LCCN 2021046654 (print) | LCCN 2021046655 (ebook) | ISBN
9781032139197 (hbk) | ISBN 9781032139289 (pbk) | ISBN 9781003231523
(ebk)
Subjects: LCSH: Machinery, Dynamics of. | Multibody systems--Mathematical
models.
Classification: LCC TJ173 .B235 2022 (print) | LCC TJ173 (ebook) | DDC
621.8/11--dc23
LC record available at https://lccn.loc.gov/2021046654
LC ebook record available at https://lccn.loc.gov/2021046655
ISBN: 978-1-032-13919-7 (hbk)
ISBN: 978-1-032-13928-9 (pbk)
ISBN: 978-1-003-23152-3 (ebk)
DOI: 10.1201/9781003231523
Typeset in Times
by Deanta Global Publishing Services, Chennai, India
Alpona, Chhoto, and Buddho,…
and
Professor Thomas Kane, from whose magnificent books I learned
to do dynamics, and whose expectation “to do many interesting
problems together”, is what started me on the path and inspired me
to move on, to develop a body of research that this book contains.
Contents
Preface.................................................................................................................xv
About the Author ..............................................................................................xvii
Introduction Background Material on Dynamics and Vibrations .....................1
I.1 Introduction ..........................................................................1
I.2 Direction Cosine Matrix .......................................................1
I.3 Fundamental Theorem on Differentiation of a Vector
in Two Frames ......................................................................2
I.4 Quaternions ..........................................................................4
I.5 Four Basic Theorems in Kinematics ....................................6
I.6 Generalized Coordinates and Generalized Speeds ..............7
I.7 Partial Velocities and Partial Angular Velocities: Key
Components in Kane’s Method ............................................8
I.8 Definition of Inertia Force and Inertia Torque .....................9
I.9 Vibration of an Elastic Body: Mode Shapes,
Frequencies, Modal Effective Mass, and Model
Reduction: The Eigenvalue Problem ..................................10
I.9.1 Reduction of Degrees of Freedom ........................11
References .....................................................................................14
Chapter 1 Derivation of Equations of Motion ................................................15
1.1 Available Analytical Methods and the Reason for
Choosing Kane’s Method ...................................................15
1.2 Kane’s Method of Deriving Equations of Motion ..............16
1.3 Kane’s Equations of Motion ...............................................18
1.3.1 Simple Example: Equations for a Double
Pendulum ...............................................................19
1.3.2 Complex Example: Equations of Motion for
a Spinning Spacecraft with Three Rotors, a
Tank with Thruster Fuel Slosh, and a Nutation
Damper ..................................................................21
1.3.3 Comparison to Derivation of Equations of
Motion by Lagrange’s Method: Application to
the Same Complex Example Shown in Figure 1.2 ...27
1.3.4 Boltzmann–Hamel Equations ...............................29
1.3.5 Gibbs Equations ....................................................31
1.3.5.1 Reader’s Exercise ...................................32
1.4 Kane’s Method of Direct Derivation of Linearized
Dynamical Equation ...........................................................32
vii
viii Contents
1.5 Prematurely Linearized Equations and a Posteriori
Correction by ad hoc Addition of Geometric Stiffness
Due to Inertia Loads ...........................................................36
1.6 Kane’s Equations with Undetermined Multipliers for
Constrained Motion ............................................................38
1.6.1 Summary of Equations of Motion with
Undetermined Multipliers for Constraints ............40
1.6.2 A Simple Application ............................................40
Appendix A: Guideline for Choosing Efficient Motion
Variables in Kane’s Method ................................................42
A.1 Customary Choice of Generalized Speeds ...........44
A.2 Efficient Choice of Generalized Speeds ................45
Appendix B: Sliding Impact with Friction of a Nose Cap on
a Package of Parachute .......................................................45
Problem Set 1 ................................................................................49
References .....................................................................................49
Chapter 2 Deployment, Station-Keeping, and Retrieval of a Flexible
Tether Connecting a Satellite to the Shuttle ..................................51
2.1 Equations of Motion for Deployment of a Satellite
Tethered to the Space Shuttle .............................................52
2.1.1 Kinematical Equations ..........................................53
2.1.2 Dynamical Equations ............................................54
2.1.3 Simulation Results .................................................59
2.2 Thruster Augmented Retrieval of a Tethered Satellite
to the Orbiting Shuttle ........................................................60
2.2.1 Dynamical Equations ............................................61
2.2.2 Simulation Results .................................................73
2.2.3 Conclusion .............................................................73
2.3 Dynamics and Control of Station-Keeping of the
Shuttle-Tethered Satellite ....................................................73
2.4 Pointing Control with Tethers as Actuators of a Space
Station–Supported Platform ...............................................76
2.4.1 Non-Linear Equations of Motion ..........................77
2.4.2 Linearized Equations ............................................78
2.4.3 Control Law Design ..............................................80
Appendix 2.A Formation Flying with Multiple Tethered
Satellites .............................................................................82
Appendix 2.B Orbit Boosting of Tethered Satellite Systems
by Electrodynamic Forces ..................................................85
Problem Set 2 ................................................................................88
References .....................................................................................91
Contents ix
Chapter 3 Kane’s Method of Linearization Applied to the Dynamics of
a Beam in Large Overall Motion ..................................................93
3.1 Non-Linear Beam Kinematics with Neutral Axis
Stretch, Shear, and Torsion .................................................93
3.2 Non-Linear Partial Velocities and Partial Angular
Velocities, and Their Linearization ..................................100
3.3 Use of Kane’s Method for Direct Derivation of
Linearized Dynamical Equations .....................................102
3.4 Simulation Results for a Space-Based Robotic
Manipulator ......................................................................109
3.5 Erroneous Results Obtained Using Vibration Modes
with Large Rotation in Conventional Analysis .................110
Problem Set 3 ...............................................................................112
References ....................................................................................115
Chapter 4 Dynamics of a Plate in Large Overall Motion .............................117
4.1 Beginning at the End: Simulation Results .........................117
4.2 Application of Kane’s Methodology for Linearization
without Deriving the Non-Linear Equations .....................118
4.3 Simulation Algorithm .......................................................126
4.4 Conclusion ........................................................................128
Appendix 4 Specialized Modal Integrals ....................................128
Problem Set 4 ..............................................................................130
References ...................................................................................132
Chapter 5 Dynamics of an Arbitrary Flexible Body in Large Overall
Motion .........................................................................................133
5.1 Dynamical Equations for General Structures with the
Use of Vibration Modes ....................................................134
5.2 Compensating for Premature Linearization by Adding
Geometric Stiffness due to Inertia Loads ........................137
5.2.1 Rigid Body Kinematical Equations ....................142
5.3 Summary of the Algorithm ..............................................142
5.4 Crucial Test and Validation of the Theory in Application .....143
5.5 Conclusion ........................................................................148
Problem Set 5 ..............................................................................150
References ...................................................................................150
Chapter 6 Flexible Multibody Dynamics: Dense Matrix Formulation ........153
6.1 Flexible Body System in a Tree Topology ........................153
6.2 Kinematics of a Joint in a Flexible Multibody System .....153
