Associate Professor: C. H.L IAO

Contents:  3.1 Introduction 99  3.2 Simple Harmonic Oscillator 100  3.3 Harmonic Oscillations in Two Dimensions 104  3.4 Phase Diagrams 106  3.5 Damped Oscillations 108  3.6 Sinusoidal Driving Forces 117  3.7 Physical Systems 123  3.8 Principle of Superposition-Fourier Series 126  3.9 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

ω o represents the angular frequency of the motion, which is related to the frequency ν o by

Ex. 3-1 Find the angular velocity and period of oscillation of a solid sphere of mass m and radius R about a point on its surface. See Figure 3-l. Sol.: The equation of motion for θ is

3.3 Harmonic Oscillations in 2-D (2- Dimensions) the restoring force is where

Conclusions:

3.4 Phase Diagrams We may consider the quantities x(t) and to be the coordinates of a 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:

Underdamped Motion The envelope of the displacement versus time curve is given by

Ex. 3-2 Sol.:

Critically Damped Motion

Overdamped Motion

Ex. 2-3 Sol.:

3.6 Sinusoidal Driving Forces 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 = F 0 /m and where w is the angular frequency of the driving force

Resonance Phenomena

3.7 Physical Systems We 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.

The hanging mass-spring system V.S. equivalent electrical circuit.

Ex.3-4 Sol.:

Ex. 3-5 Sol.:

3.8 Principle of Superposition-Fourier Series The quantity in parentheses on the left-hand side is a linear operator, which we may represent by L.

Periodic function 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:

Ex. 3-6 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

3-1

3-6

3-10

3-14

3-19

3-24

3-26

3-29

3-32

3-37

3-43