Advanced Engineering. Dynamics. H. R. Harrison. T. Nettleton. Formerly Department of Mechanical Engineering & Aeronautics City University London

Transcription

1 Advanced Engineering Dynamics H. R. Harrison Formerly Department of Mechanical Engineering & Aeronautics City University London T. Nettleton Formerly Department of Mechanical Engineering & Aeronautics City University London A member of the Hodder Headline Group LONDON 0 SYDNEY 0 AUCKLAND Copublished in North, Central and South America by John Wiley & Sons Inc., New York 0 Toronto

2 First Published in Great Britain in 1997 by Arnold, a member of the Hodder Headline Group, 338 Euston Road, London NWI 3BH Copublished in North, Central and South America by John Wiley & Sons, Inc., 605 Third Avenue, NewYork, NY H R Harrison & T Nettleton All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronically or mechanically, including photocopying, recording or any information storage or retrieval system, without either prior permission in writing from the publisher or a licence permitting restricted copying. In the United Kingdom such licences are issued by the Copyright Licensing Agency: 90 Tottenham Court Road, London W 1 P 9HE. Whilst the advice and information in this book is believed to be true and accurate at the date of going to press, neither the author[s] nor the publisher can accept any legal responsibility or liability for any errors or omissions that may be made. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN ISBN (Wiley) Typeset in 10/12pt Times by J&L Composition Ltd, Filey, North Yorkshire Printed and bound in Great Britain by J W Arrowsmith Ltd, Bristol

3 Preface The subject referred to as dynamics is usually taken to mean the study of the kinematics and kinetics of particles, rigid bodies and deformable solids. When applied to fluids it is referred to as fluid dynamics or hydrodynamics or aerodynamics and is not covered in this book. The object of this book is to form a bridge between elementary dynamics and advanced specialist applications in engineering. Our aim is to incorporate the terminology and notation used in various disciplines such as road vehicle stability, aircraft stability and robotics. Any one of these topics is worthy of a complete textbook but we shall concentrate on the fundamental principles so that engineering dynamics can be seen as a whole. Chapter 1 is a reappraisal of Newtonian principles to ensure that definitions and symbols are all carefully defined. Chapters 2 and 3 expand into so-called analytical dynamics typified by the methods of Lagrange and by Hamilton s principle. Chapter 4 deals with rigid body dynamics to include gyroscopic phenomena and the stability of spinning bodies. Chapter 5 discusses four types of vehicle: satellites, rockets, aircraft and cars. Each of these highlights different aspects of dynamics. Chapter 6 covers the fundamentals of the dynamics of one-dimensional continuous media. We restrict our discussion to wave propagation in homogeneous, isentropic, linearly elastic solids as this is adequate to show the differences in technique when compared with rigid body dynamics. The methods are best suited to the study of impact and other transient phenomena. The chapter ends with a treatment of strain wave propagation in helical springs. Much of this material has hitherto not been published. Chapter 7 extends the study into three dimensions and discusses the types of wave that can exist within the medium and on its surface. Reflection and refraction are also covered. Exact solutions only exist for a limited number of cases. The majority of engineering problems are best solved by the use of finite element and finite difference methods; these are outside the terms of reference of this book. Chapter 8 forges a link between conventional dynamics and the highly specialized and distinctive approach used in robotics. The Denavit-Hartenberg system is studied as an extension to the kinematics already encountered. Chapter 9 is a brief excursion into the special theory of relativity mainly to define the boundaries of Newtonian dynamics and also to reappraise the fundamental definitions. A practical application of the theory is found in the use of the Doppler effect in light propagation. This forms the basis of velocity measuring equipment which is in regular use.

4 xii Preface There are three appendices. The first is a summary of tensor and matrix algebra. The second concerns analytical dynamics and is included to embrace some methods which are less well known than the classical Lagrangian dynamics and Hamilton s principle. One such approach is that known as the Gibbs-Appell method. The third demonstrates the use of curvilinear co-ordinates with particular reference to vector analysis and second-order tensors. As we have already mentioned, almost every topic covered could well be expanded into a complete text. Many such texts exist and a few of them are listed in the Bibliography which, in tum, leads to a more comprehensive list of references. The important subject of vibration is not dealt with specifically but methods by which the equations of motion can be set up are demonstrated. The fimdamentals of vibration and control are covered in our earlier book The Principles of Engineering Mechanics, 2nd edn, published by Edward Arnold in The author and publisher would like to thank Briiel and Kjaer for information on the Laser Velocity Transducer and SP Tyes UK Limited for data on tyre cornering forces. It is with much personal sadness that I have to inform the reader that my co-author, friend and colleague, Trevor Nettleton, became seriously ill during the early stages of the preparation of this book. He died prematurely of a brain tumour some nine months later. Clearly his involvement in this book is far less than it would have been; I have tried to minimize this loss. Ron Harrison January 1997

5 Contents Preface 1 Newtonian Mechanics Fundamentals Space and time Mass Force Work and power Kinematics of a point Kinetics of a particle Impulse Kinetic energy Potential energy Coriolis s theorem Newton s laws for a group of particles Conservation of momentum Energy for a group of particles The principle of virtual work D Alembert s principle 2 Lagrange s Equations Generalized co-ordinates Proof of Lagrange s equations The dissipation function Kinetic energy Conservation laws Hamilton s equations Rotating frame of reference and velocity-dependent potentials Moving co-ordinates Non-holonomic systems Lagrange s equations for impulsive forces xi

6 viii Contents 3 Hamilton s Principle Derivation of Hamilton s principle 3.3 Application of Hamilton s principle Lagrange s equations derived from Hamilton s principle Illustrative example 4 Rigid Body Motion in Three Dimensions Rotation Angular velocity Kinetics of a rigid body Moment of inertia Euler s equation for rigid body motion Kinetic energy of a rigid body Torque-free motion of a rigid body Stability of torque-free motion Euler s angles The symmetrical body Forced precession Epilogue 5 Dynamics of Vehicles Gravitational potential The two-body problem The central force problem Satellite motion Effects of oblateness Rocket in free space Non-spherical satellite Spinning satellite De-spinning of satellites Stability of aircraft Stability of a road vehicle 6 Impact and One-Dimensional Wave Propagation The one-dimensional wave Longitudinal waves in an elastic prismatic bar Reflection and transmission at a boundary Momentum and energy in a pulse Impact of two bars Constant force applied to a long bar The effect of local deformation on pulse shape Prediction of pulse shape during impact of two bars Impact of a rigid mass on an elastic bar Dispersive waves I

7 Contents ix 6.12 Waves in a uniform beam 6.13 Waves in periodic structures 6.14 Waves in a helical spring 7 Waves in a Three-Dimensional Elastic Solid Strain Stress Elastic constants Equations of motion Wave equation for an elastic solid Plane strain Reflection at a plane surface Surface waves (Rayleigh waves) Conclusion 8 Robot Arm Dynamics Typical arrangements 8.3 Kinematics of robot arms 8.4 Kinetics of a robot arm 9 Relativity Problems The foundations of the special theory of relativity Time dilation and proper time Simultaneity The Doppler effect Velocity The twin paradox Conservation of momentum Relativistic force Impact of two particles The relativistic Lagrangian Conclusion Appendix 1 - Vectors, Tensors and Matrices Appendix 2 - Analytical Dynamics Appendix 3 - Curvilinear Co-ordinate Systems Bibliography 297 Index 299