Associate Professor: C. H.LIAO

Contents:

3.1 Introduction 993.2 Simple Harmonic Oscillator 1003.3 Harmonic Oscillations in Two Dimensions

1043.4 Phase Diagrams 1063.5 Damped Oscillations 1083.6 Sinusoidal Driving Forces 1173.7 Physical Systems 1233.8 Principle of Superposition-Fourier Series 1263.9 The Response of Linear Oscillators to

Impulsive Forcing 129

3.1 Introduction

1. If the particle is displaced from the origin (in either direction), a certain force tends to restore the particle to its original position. An example is an atom in a long molecular chain.

2. The restoring force is, in general, some complicated function of the displacement and perhaps of the particle's velocity or even of some higher time derivative of the position coordinate.

3. We consider here only cases in which the restoring force F is a function only of the displacement: F = F(x).

3.2 Simple Harmonic Oscillator

the kinetic energy

The incremental amount of work dW

ωo represents the angular frequency of the motion, which is related to the frequency νo

by

Find the angular velocity and period of oscillation of a solid sphere of mass mand radius R about a point on its surface. See Figure 3-l.

Sol.: The equation of motion for θ is

the restoring force is

where

We may consider the quantities x(t) and to be the coordinates ofa point in a two-dimensional space, called phase space.

3.5 Damped Oscillations

The motion represented by the simple harmonic oscillator is termed a free oscillation;

The general solution is:

The envelope of the displacement versus time curve is given by

Sol.:

Sol.:

The simplest case of driven oscillation is that in which an external driving force varying harmonically with time is applied to the oscillator

where A = F0/m and where w is the angular frequency of the driving force

3.7 Physical SystemsWe stated in the introduction to this chapter

that linear oscillations apply to more systems than just the small oscillations of the mass-spring and the simple pendulum.

We can apply our mechanical system analog to acoustic systems. In this case, the air molecules vibrate.

Sol.:

Sol.:

The quantity in parentheses on the left-hand side is a linear operator, which we may represent by L.

In the usual physical case in which F(t) is periodic with period τ = 2π /ω

F(t) has a period τ

*Fourier's theorem for any arbitrary periodic function can be expressed as:

Sol.:

Thanks for your attention.

Problem discussion.Problem:3-1, 3-6, 3-10, 3-14, 3-19, 3-24, 3-26, 3-29, 3-

32, 3-37, 3-43