Table Of ContentPlanar Multibody Dynamics
Planar Multibody Dynamics
Formulation, Programming with MATLAB®,
and Applications
Second Edition
Parviz E. Nikravesh
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the
accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular
use of the MATLAB® software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2019 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-138-09612-7 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the
validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copy-
right holders of all material reproduced in this publication and apologize to copyright holders if permission to publish
in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know
so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or
utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including pho-
tocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission
from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA
01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users.
For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been
arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Library of Congress Cataloging‑in‑Publication Data
Names: Nikravesh, Parviz E., 1946- author.
Title: Planar multibody dynamics : formulation, programming with MATLAB®, and
applications / Parviz Nikravesh.
Description: Second edition. | Boca Raton : Taylor & Francis, CRC Press,
2018. | Includes bibliographical references.
Identifiers: LCCN 2018017483| ISBN 9781138096127 (hardback : alk. paper) |
ISBN 9781315105437 (e-book)
Subjects: LCSH: Machinery, Kinematics of. | Machinery, Dynamics of.
Classification: LCC TJ175 .N527 2018 | DDC 621.8/11—dc23
LC record available at https://lccn.loc.gov/2018017483
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
This book is dedicated to the memory of
Dr. Moris Nikravesh
“The Medicine Man”
Contents
Preface ...........................................................................................................................................xiii
Acknowledgments .....................................................................................................................xvii
Author ...........................................................................................................................................xix
1. Introduction .............................................................................................................................1
1.1 Multibody Mechanical Systems ..................................................................................1
1.2 Types of Analyses .........................................................................................................2
1.3 Methods of Formulation ..............................................................................................2
1.4 Computer Programs .....................................................................................................9
2. Preliminaries .........................................................................................................................13
2.1 Reference Axes ............................................................................................................13
2.2 Scalars and Vectors .....................................................................................................13
2.2.1 Arrays ..............................................................................................................20
2.3 Matrices ........................................................................................................................21
2.3.1 Matrix Operations..........................................................................................23
2.3.2 Linear Algebraic Equations ..........................................................................26
2.4 Vector, Array, and Matrix Differentiation ...............................................................27
2.4.1 Time Derivatives ............................................................................................27
2.4.2 Partial Derivatives .........................................................................................28
2.5 Equations and Expressions ........................................................................................31
2.5.1 Compact and Expanded Forms ...................................................................31
2.6 Problems .......................................................................................................................32
3. Fundamentals of Planar Kinematics ................................................................................37
3.1 A Particle ......................................................................................................................37
3.1.1 Kinematics of a Particle.................................................................................37
3.2 Kinematics of a Rigid Body .......................................................................................38
3.2.1 Coordinates of a Body ...................................................................................39
3.2.2 Velocity of a Body ..........................................................................................43
3.2.3 Acceleration of a Body ...................................................................................44
3.3 Definitions ....................................................................................................................46
3.3.1 Array of Coordinates .....................................................................................46
3.3.2 Degrees of Freedom .......................................................................................48
3.3.3 Constraint Equations .....................................................................................49
3.3.4 Kinematic Joints .............................................................................................51
3.4 Problems .......................................................................................................................53
4. Fundamentals of Planar Dynamics ..................................................................................57
4.1 Newton’s Laws of Motion ..........................................................................................57
4.2 Particle Dynamics .......................................................................................................58
4.2.1 Dynamics of a System of Particles...............................................................58
4.3 Rigid body Dynamics .................................................................................................61
4.3.1 Moment of a Force and Torque ....................................................................62
vii
viii Contents
4.3.2 Centroidal Equations of Motion ..................................................................63
4.3.3 Noncentroidal Equations of Motion ............................................................67
4.4 Multibody Dynamics ..................................................................................................69
4.4.1 Applied Forces ................................................................................................69
4.4.2 Reaction Forces ...............................................................................................75
4.5 Friction Force ...............................................................................................................76
4.5.1 Wheel and Tire ...............................................................................................80
4.5.2 Motor and Driver ...........................................................................................82
4.6 Work and Energy ........................................................................................................83
4.7 Problems .......................................................................................................................84
References ...............................................................................................................................88
5. Vector Kinematics .................................................................................................................89
5.1 Use of Vectors ..............................................................................................................89
5.1.1 Unit Vectors ....................................................................................................90
5.1.2 Types of Vectors .............................................................................................92
5.2 Open-Chain Systems ..................................................................................................94
5.3 Closed-Chain Systems ................................................................................................97
5.3.1 Slider-Crank Mechanism ..............................................................................97
5.3.2 Four-Bar Mechanism ...................................................................................101
5.3.3 Six-Bar Quick-Return Mechanism .............................................................105
5.3.4 Six-Bar Dwell Mechanism ..........................................................................107
5.3.5 Complete Kinematic Analysis ....................................................................108
5.4 Problems .....................................................................................................................108
6. Free-Body Diagram ............................................................................................................113
6.1 FBD Examples ............................................................................................................113
6.1.1 Two-Body System (Unconstrained) ...........................................................113
6.1.2 Two-Body System (Constrained) ...............................................................116
6.1.3 Sliding Pendulum ........................................................................................118
6.1.4 Slider-Crank Mechanism ............................................................................120
6.1.5 Four-Bar Mechanism ...................................................................................122
6.2 Equations of Motion .................................................................................................126
6.3 Force Analysis ...........................................................................................................126
6.3.1 Slider-Crank Mechanism ............................................................................127
6.3.2 F our-Bar Mechanism ...................................................................................128
6.3.3 G eneralization of Force Analysis...............................................................130
6.4 Problems .....................................................................................................................130
7. Body-Coordinate Formulation .........................................................................................135
7.1 General Procedure ....................................................................................................135
7.2 Kinematic Joints ........................................................................................................137
7.2.1 Revolute (Pin) Joint ......................................................................................139
7.2.2 Translational (Sliding) Joint ........................................................................141
7.2.3 Revolute–Revolute Joint ..............................................................................142
7.2.4 Revolute–Translational Joint .......................................................................144
7.2.5 Rigid Joint .....................................................................................................145
7.2.6 Simple Constraints .......................................................................................146
Contents ix
7.2.7 Circular Disc .................................................................................................146
7.2.8 Driver Constraints .......................................................................................147
7.2.9 System Jacobian ............................................................................................148
7.3 Unconstrained Equations of Motion ......................................................................149
7.4 Constrained Equations of Motion ..........................................................................152
7.4.1 R eaction Forces and Lagrange Multipliers...............................................153
7.5 Total Energy ...............................................................................................................160
7.6 Problems .....................................................................................................................161
8. Body-Coordinate Simulation Program ..........................................................................169
8.1 Application Examples ...............................................................................................170
8.1.1 Double A-Arm Suspension .........................................................................170
8.1.2 MacPherson Suspension .............................................................................173
8.1.3 Cart.................................................................................................................177
8.1.4 Conveyor Belt and Friction .........................................................................180
8.1.5 Rod Impacting Ground ...............................................................................182
8.2 Problems .....................................................................................................................183
9. Joint-Coordinate Formulation ..........................................................................................185
9.1 Definitions ..................................................................................................................185
9.1.1 Joint Coordinate and Joint Reference Point ..............................................186
9.1.2 Recursive Kinematics ..................................................................................188
9.2 Open-Chain Systems ................................................................................................190
9.2.1 Absolute Angle .............................................................................................195
9.2.2 Equations of Motion ....................................................................................197
9.3 Closed-Chain Systems ..............................................................................................203
9.3.1 Cut-Joint Constraints ...................................................................................203
9.3.2 Equations of Motion ....................................................................................206
9.3.3 Jacobian Matrix ............................................................................................213
9.3.4 Initial Conditions .........................................................................................217
9.3.5 Reaction Forces .............................................................................................218
9.3.6 Driver Constraint .........................................................................................220
9.4 A MATLAB Program ................................................................................................221
9.5 Problems .....................................................................................................................222
10. Point-Coordinate Formulation .........................................................................................229
10.1 Classical Method .......................................................................................................229
10.2 Primary and Stationary Points ................................................................................231
10.3 Constraints .................................................................................................................233
10.3.1 Length Constraint ........................................................................................235
10.3.2 Angle Constraints ........................................................................................235
10.3.3 S imple Constraints .......................................................................................236
10.4 Secondary Points .......................................................................................................238
10.5 Equations of Motion .................................................................................................241
10.6 Force and Torque Distribution ................................................................................242
10.7 Mass Distribution......................................................................................................246
10.8 Mass Condensation ...................................................................................................248
10.8.1 Two Primary Points .....................................................................................248
