INTRODUCTION

Have you ever seen a Thanjavur dancing doll (in Tamil, 'Thanjavur thalayatti bommai'), a well-known Indian craft figure whose head and body, once disturbed, rock gently to and fro until the movement gradually dies away? The same repeated back-and-forth pattern shows up constantly in everyday life: our arms and legs swing forward and backward as we walk, and a cradle rocked by a mother to lull a child to sleep swings the same way. Motions of this kind -- movement that repeats itself, back and forth, about a fixed rest position -- are called oscillatory or vibratory motion. The same pattern even appears at the atomic scale: when a solid is heated, its atoms do not fly apart but vibrate with greater amplitude about their own equilibrium (mean) positions. Because vibration of this kind governs how buildings and mechanical equipment respond to disturbance, the study of oscillatory motion has direct engineering importance, well beyond describing a swinging pendulum.

Motion in nature falls broadly into two classes: motion that repeats itself at equal intervals of time, called periodic motion, and motion that never repeats in this regular way, called non-periodic motion. A large and important sub-class of periodic motion is oscillatory (or vibratory) motion, in which a body moves repeatedly back and forth about a fixed reference or mean position. This unit builds up the mathematics of oscillatory motion step by step: starting from periodic and oscillatory motion in general, it specialises to Simple Harmonic Motion (SHM) -- the simplest and most important oscillatory motion, in which the restoring force/acceleration is directly proportional to the displacement from the mean position and always directed towards it. From this single defining idea the unit develops the equations for displacement, velocity and acceleration in SHM; angular SHM; the linear harmonic oscillator (spring-mass systems, series and parallel spring combinations, the simple pendulum, and liquid oscillating in a U-tube); the energy stored in SHM; and finally the different types of oscillations -- free, damped, maintained, forced -- and the special, technologically important case of resonance.