Elements Of The Theory Of Resonance - Illustrated By The Motion Of A Pendulum

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ELEMENTS OF THE THEORY OF RESONANCE ILLUSTRATED BY THE MOTION OF A PENDULUM By E. W. BROJKN, Sc. D, F. R. S. Josiah ff Gibbs Professor of Mathematics Tale University CAMBRIDGE AT THE UNIVERSITY PRESS 1932 PRINTED IN THE UNITED STATES OF AMERICA CONTENTS Introduction ...... page t d x I. The Equation - K 2 sin x o . . . 1 1 Method of Approximation . . . .18 2 v The Generalized Equation - f o . 20 II. Solutions of III. Application to the Pendulum with an Oscillating Support . . . . . .36 IV. Applications to the Mutual Influence of Two Pendulums ...... 43 V. Solutions of v x m f, 53 VI. The Effect of Friction ..... 56 PREFACE This pamphlet contains an attempt to describe and analyse the phenomena which are peculiar to resonance in an elementary manner. Although the subject is of fundamental importance in many mechanical problems it has received but little attention in the text books. The treatment, which consists of an investigation of the changes which take place in the amplitude and phase of a vibrating element under certain types of forces, appears to be convenient both for the mathematical development and for the interpretation of the results. The issue of The Rice Institute Pamphlet, Vol. xix, No. i, in the present form, is made with the permission of the Trustees of the Institute and the co-operation of the Cambridge University Press, and is due to the hope that it may prove to be useful to students and others who may not have easy access to the publications of the Institute. My thanks are due to Professor C. G. Darwin for his helpful criticisms, especially in the mode of presentation of a subject in which the method of approach and certain of the results appear to have some degreeof novelty. ERNEST W. BROWN April 1932 ELEMENTS OF THE THEORY OF RESONANCE ILLUSTRATED BY THE MOTION OF A PENDULUM 1 INTRODUCTION 1. The original observations which gave rise to the word resonance were audible sounds which are familiar to most of us. A note is struck on a piano or other musical instrument and some body a wall or another musical instrument will take up the same note and will continue to sound it even after the original source is stopped. The megaphone takes up the vibrations of the air produced by the voice and gives them out again, concentrated in a particular direction. A vibrating body, for example, the cylinders in the engine of an automobile, will set up vibrations in other parts of the car, certain vibrations being noticeable at certain speeds and others at other speeds. To all these phenomena the term resonance is applied. But the actual nature of the phenomena is not always the same. The megaphone and loud speaker are designed to take up and emit any vibration within a certain range of frequency. On the other hand, the audible vibrations of a stretched wire are confined to a limited number of sounds as long as we keep the tension unaltered, and these bear definite relations to one another to produce resonance in such a wire, it is necessary to sound a note with a frequency 1 A course of lectures, given at the Rice Institute, April 22, 23, and 24, 1931, by Ernest William Brown, D. Sc., F. R. S., Josiah W. Gibbs Professor of Mathematics in Yale University 1 2 Introduction very near that of one of these modes of vibration. These illustrations bring up the necessity for a definition of what are usually called the natural frequencies or periods of vibration of anysystem. 2. A stretched wire when made to vibrate appears to give out the same note, in general, however it may be struck or when the bow of the violin is drawn across it. An ordinary drinking glass exhibits the same phenomenon it gives out the same note when struck as when the finger is wetted and run around the top edge of the glass. The periods of these sounds are called natural periods of vibration they depend only on the construction of the mechanism and not on the manner in which its vibrations are started. But this last statement is not exact...