Engineering Acoustics

2Vibration of Simple and Continuous Systems

2.1 Introduction

The vibrations in machines and structures result in oscillatory motion that propagates in air and/or water and that is known as sound. The simplest type of oscillation in vibration and sound phenomena is known as simple harmonic motion, which can be shown to be sinusoidal in time. Simple harmonic motion is of academic interest because it is easy to treat and manipulate mathematically; but it is also of practical interest. Most musical instruments make tones that are approximately periodic and simple harmonic in nature. Some machines (such as electric motors, fans, gears, etc.) vibrate and make sounds that have pure tone components. Musical instruments and machines normally produce several pure tones simultaneously. The simplest vibration to analyze is that of a mass–spring–damper system. This elementary system is a useful model for the study of many simple vibration problems. In this chapter we will discuss some simple theory that is useful in the control of noise and vibration. For more extensive discussions on sound and vibration fundamentals, the reader is referred to more detailed treatments available in several books [1–7]. We start off by discussing simple harmonic motion. This is because very often oscillatory motion, whether it be the vibration of a body or the propagation of a sound wave, is like this idealized case. Next, we introduce the ideas of period, frequency, phase, displacement, velocity, and acceleration. ...