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This page intentionally left blank ENGINEERING DYNAMICS Engineering Dynamics is a new treatment of kinematics and classical and analytical dynamics based on Ginsberg’s popular Advanced Engineering Dynamics Second Edition. Like its predecessor, this book conveys physical and analytical understanding of the basic principles of dynamics but is more comprehensive and better addresses real-world complexities. Every section has been rewritten, and many topics have been added or enhanced. Several derivations are new, and others have been reworked to make them more accessible, general, and elegant. Many new examples are provided, and those that were retained have been reworked. They all use a careful pedagogical structure that mirrors the text presentation. Instructors will appreciate the significant enhancement of the number and variety of homework exercises. All of the text illustrations have been redrawn to enhance their clarity. Jerry Ginsberg began his academic career at Purdue University in 1969, and he was a Fulbright–Hays Advanced Research Fellow in 1975. He moved to Georgia Tech in 1980, where he became the first holder of the Woodruff Chair in Mechanical Systems in 1988. Professor Ginsberg has worked in a broad range of areas in mechanical vibrations and acoustics for which he developed and applied specialized mathematical and computational solutions that provide greater insight compared to standard numerical techniques. Professor Ginsberg is the author of more than 150 technical and archival papers and the highly regarded textbooks: Advanced Engineering Dynamics, Mechanical and Structural Vibrations, Statics, and Dynamics (the last two with Joseph Genin), as well as two chapters in Nonlinear Acoustics. He is a Fellow of the Acoustical Society of America and of the American Society of Mechanical Engineers, and he has served as an associate editor of the Journal of the Acoustical Society and of the ASME Journal of Vibration and Acoustics. He received the Georgia Tech Distinguished Professor Award in 1994, the Archie Higdon Distinguished Educator Award from ASEE in 1998, the Trent–Crede Medal from ASA in 2005, and the Per Bruel Gold Medal from ASME in 2007. He has delivered a number of distinguished lectures, including the 2001 ASME Rayleigh Lecture and the 2003 Special Lecture for the Noise Control and Acoustics Division of ASME, as well as keynote speeches at several meetings, including the Second International Congress on Dynamics, Vibrations, and Control in Beijing in 2006. Engineering Dynamics JERRY GINSBERG Georgia Institute of Technology CAMBRIDGE UNIVERSITY PRESS Cambridge New York Melbourne Madrid Cape Town Singapore São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org 9780521883030 © Jerry Ginsberg 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-47872-7 eBook (EBL) ISBN-13 978-0-521-88303-0 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface. xiii Acknowledgments. xv 1 Basic Considerations. 1 1.1 Vector Operations 1.1.1 Algebra and Computations 1.1.2 Vector Calculus—Velocity and Acceleration 1.2 Newtonian Mechanics 1.2.1 Newton’s Laws 1.2.2 Systems of Units 1.2.3 Energy and Momentum 1.3 Biographical Perspective 2 Particle Kinematics. 30 2.1 Path Variables 2.1.1 Tangent and Normal Components 2.1.2 Parametric Description of Curves 2.1.3 Binormal Direction and Torsion of a Curve 2.2 Rectangular Cartesian Coordinates 2.3 Curvilinear Coordinates 2.3.1 Cylindrical and Polar Coordinates 2.3.2 Spherical Coordinates 2.3.3 Arbitrary Curvilinear Coordinates 2.4 Mixed Kinematical Descriptions 3 Relative Motion. 91 3.1 Coordinate Transformations 3.1.1 Rotation Transformations 3.1.2 Rotation Sequences 3.2 Displacement 3.3 Time Derivatives 3.4 Angular Velocity and Acceleration 3.4.1 Analytical Description 3.4.2 Procedure v vi Contents 3.5 Velocity and Acceleration Analysis Using a Moving Reference Frame 3.6 Observations from a Moving Reference Frame 4 Kinematics of Constrained Rigid Bodies. 173 4.1 General Equations 4.2 Eulerian Angles 4.3 Interconnections and Linkages 4.4 Rolling 5 Inertial Effects for a Rigid Body. 228 5.1 Linear and Angular Momentum 5.1.1 System of Particles 5.1.2 Rigid Body—Basic Equations 5.1.3 Kinetic Energy 5.2 Inertia Properties 5.2.1 Moments and Products of Inertia 5.2.2 Transformations 5.2.3 Inertia Ellipsoid 5.2.4 Principal Axes 5.3 Rate of Change of Angular Momentum 6 Newton–Euler Equations of Motion. 296 6.1 Fundamental Equations 6.1.1 Basic Considerations 6.1.2 Procedural Steps 6.2 Planar Motion 6.3 Newton–Euler Equations for a System 6.4 Momentum and Energy Principles 6.4.1 Impulse–Momentum Principles 6.4.2 Work–Energy Principles 6.4.3 Collisions of Rigid Bodies 7 Introduction to Analytical Mechanics. 391 7.1 Background 7.1.1 Principle of Virtual Work 7.1.2 Principle of Dynamic Virtual Work 7.2 Generalized Coordinates and Kinematical Constraints 7.2.1 Selection of Generalized Coordinates 7.2.2 Constraint Equations 7.2.3 Configuration Space 7.3 Evaluation of Virtual Displacements 7.3.1 Analytical Method 7.3.2 Kinematical Method last_pages_______________________________ of a unit vector, 74, 93–94 of a vector, 3, 43, 45, 95–96 Composite shape, for inertia properties, 253 Computational techniques, augmented method, 511, 519 constraint stabilization, 520–521 embedded method, 516–519 for holonomic systems, 466–470 integrated multiplier, 510 orthogonal complement, 513–515, 519 Condition number, 515 Configuration constraint, see constraint equations configuration Configuration space, 408–411, 414, 416, 427, 448, 554, 557, 560 Conservation of energy, 348, 658 of generalized momentum, 583, 667 for a spinning top, of momentum, 21, 341, 644 of the Hamiltonian function, 578–580 Conservative force, 345–346, 348 virtual work of, 444–445 Constrained generalized coordinates, 398, 400, 409, 432, 437, 456, 544 equations of motion for, 492–494, 508 initial conditions for, 521 Constraint equations, 183, 398, 400–405 acatastatic, 412 catastatic, 412 configuration Configuration space, 408–411, 414, 416, 427, 448, 554, 557, 560 Conservation of energy, 348, 658 of generalized momentum, 583, 667 for a spinning top, of momentum, 21, 341, 644 of the Hamiltonian function, 578–580 Conservative force, 345–346, 348 virtual work of, 444–445 Constrained generalized coordinates, 398, 400, 409, 432, 437, 456, 544 equations of motion for, 492–494, 508 initial conditions for, 521 Constraint equations, 183, 398, 400–405 acatastatic, 412 catastatic, 412 configuration Configuration space, 408–411, 414, 416, 427, 448, 554, 557, 560 Conservation of energy, 348, 658 of generalized momentum, 583, 667 for a spinning top, of momentum, 21, 341, 644 of the Hamiltonian function, 578–580 Conservative force, 345–346, 348 virtual work of, 444–445 Constraint condition ball-and-socket joint, 184, 189, 234 collar, 186–187 for planar motion, 128, 183–184, 186, 364 pin, 185–187 rolling, 200–206 slider, see collar see also constraint equation Constraint force, 349, 350, 379, 391, 431–437, 449 contribution to generalized forces, 550, 493, 495, 536, 599 see also reaction Constraint matrix, see Jacobian constraint matrix Constraint stabilization method, 520–521 Coordinate system, 3, 6, 10, 13, 95 global, 128, 129, 135, 136, 275 Coordinates affine, 28 Cartesian, see Cartesian coordinates change due to rotation, 96 curvilinear, see curvilinear coordinates cylindrical, see cylindrical coordinates extrinsic, 30, 45 generalized, see generalized coordinates hyperbolic-elliptic, 68–69 ignorable, 455, 482–485, 657 intrinsic, see path variables quasi-, see quasi-coordinates right-handed, 6, 95, 274 spherical, see spherical coordinates Coriolis acceleration, 54, 61, 71, 135, 571 in motion relative to the Earth, 150–151 Coriolis, G., 24, 54 Curve parametric representation of, 38–39 properties of, 32–33, 43–44 Curvilinear coordinates, 64–67, 69–71 see also coordinates Cycloidal path, 89, 201–202, 661 Cylindrical coordinates, 51–54, 134 and Lagrange’s equations, 450 d’Alembert’s inertial force, 14, 391–392, 394, 620 Degrees of freedom, 183, 188, 397, 400, 404–405, 518, 552, 562, 585, 591 Derivative of a unit vector, 32–33, 43–44, 52–53, 60, 65–67, 123, 126, 127 of a vector, 9–10, 123 of angular momentum, 20, 232, 275–280 of angular velocity, see angular acceleration relative to a moving reference frame, 11, 122, 206 Differential equations of motion, Gibbs-Appell, 600 Hamilton’s, 575 Lagrange’s, 449, 508 see also computational techniques, Differential-algebraic equation, 509 Dimensional homogeneity, 15, 16 Direction angles, 92–93 Direction cosines, 92–95 Index, 721 for principal axes, 264, 268–270 of an equivalent axis of rotation, 107–110 Displacement definition, 17, 113 Eulerian, 115 infinitesimal, 17, 18, 119–121 kinematically admissible, 410 Lagrangian, 115 of a rigid body, 173– Ключевые слова: xp yp, yball governor, ax, rate, 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