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Electrical & Mechanical Oscillations OS Jones ELECTRICAL AND MECHANICAL OSCILLATIONS An Introduction BY D. S. JONES ROUTLEDGE AND KEGAN PAUL LONDON First published 1961 by Routledge & Kegan Paul Ltd Broadway House, 68-74 Carter Lane London, E.C.4 Printed in Great Britain by Latimer, Trend & Co Ltd, Plymouth © D. S. Jones 1961 No part of this book may be reproduced in any form without permission from the publisher, except for the quotation of brief passages in criticism Second impression 1965 Contents page vii 1 1 1.2. The equilibrium of a conservative system with one degree offreedom 5 1.3. Stability 9 1.4. The principle of virtual work 13 Exercises 18 Preface 1. Equilibrium 1.1. Introduction 2. Harmonic Oscillations 20 2.1. The motion of a conservative system near equilibrium 20 2.2. Examples 23 2.3. Forced oscillations 30 Exercises 36 3. The Effect of Friction 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 39 Introduction Free oscillations with viscous damping Forced oscillations with viscous damping The production of heat Frequency measurement Forces due to mountings Amplitude measurement Operational solution 39 41 44 48 49 Exercises 55 v 50 51 53 CONTENTS page 57 4. Electrical Circuits 4.1. 4.2. 4.3. 4.4. 4.5. The fundamental c£rcuit elements Kt"rchhoff's laws The electro-mechanical analogy Complex impedances Operat£onal impedances 57 59 64 65 Exercises 71 5. Non-Linear Oscillations 5.1. Introduction 5.2. The phase plane 5.3. Equilibrium 5.4. Slightly non-linear damping 5.5. Relaxation oscillations 5.6. Forced oscillations Exercises 69 74 74 76 77 83 90 95 99 Answers to Exercises 10