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Associate Professor: C. H.LIAO
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
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).
1.
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
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 = F0/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