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Chapter 1
Mechanical vibration/ Oscillatory motion
Examples of oscillatory motion:
a.The motion of a pendulum
b.The vibration of a stringed musical instrument
c.The molecules oscillating about their equilibrium positions
d.The electromagnetic waves
What is mechanical vibration?
The periodic motion of mechanical systems.
The reciprocating motion of a body about a certain
position.
A body doing mechanical vibration might be considered
as a particle. This particle is usually called a vibrator.
Equilibrium position: the position where the net force on
the vibrator equals zero.
Harmonic vibration :The displacement of the vibrator
away from the equilibrium position is a cosine or
sine function.
The damping: the friction force, viscous force of the
medium and other resistance acted on the vibrator
Un-damped harmonic vibration
Harmonic vibration
Simple harmonic vibration
Forced vibration
§1-1 Simple harmonic vibration
The force that acts on the vibrator is proportional to the
displacement of the vibrator from the equilibrium position
and is always directed toward the equilibrium position.
F  kx
The force is also called restoring force.
1. Examples of simple harmonic vibration
1) The level spring vibrator
f  kx
k

F
O
x
2) The vertical spring vibrator
The net force on the vibrator:
k
f  mg  f ela
f ela  k  x0  x 
At point O:
mg  kx0  0
f  kx
O'
x0
O
x
x
3) The simple pendulum
f  mg sin 
Because the angle is very small
x
sin     tg 
l
Small angle approximation
mg
f  mg sin   
x   kx
l
f
4) The physical / compound pendulum
The torque about O provided by the
gravity is:
M  mgd sin 
M  mgd  k
k  mgd
Quasi-elastic coefficient
)θ
Small angle approximation
O
d
d sin 
CM
2. Simple harmonic vibration function/ mathematical
Presentation of simple harmonic motion
2
Newton’s second law
In harmonic vibration:
d x
f  ma  m 2
dt
f  kx
2
d x
kx  m 2
dt
2
d x k

x

0
2
dt
m
2
d x k
 x0
2
m
dt
k
Assuming:   2
m
2
d x
2


x

0
2
dt
The dynamical equation of simple harmonic oscillator
2
d x
2
 x  0
2
dt
Generally:
Solve the equation, we have:
x  C1 sint  C2 cos t
x  A cos t   
We often use the cosine function.
or:
x  A sin(t   )
For the physical pendulum:
  0 cos t  
x  A cos t    Simple harmonic vibration function
The simple harmonic vibration function describes the
variation of displacement of the vibrator with time.
The simple harmonic vibration is a periodic motion.
x(t)
Simple harmonic vibration curve
Spring Oscillator
k
t=0
t=T/4
O 
F

v 0
x

vmax
x
t=T/2
t=3T/4
t=T

vmax

v 0
x
x
x
t
x
3. Amplitude, Period / Frequency, phase constant
x  A cos t   
Integral constant A: the maximum value of the position of
the vibration in either the positive or negative x direction.
Amplitude: A
The amplitude is determined by the initial condition of the
harmonic motion.
x  A cos t   
Angular Frequency ω
k
2

m
k

m
The simple harmonic vibration is a periodic motion.
Period T: the time to finish one complete vibration
Frequency ν/f: the number of complete vibration performed
in unit time.
T
2

k

m
2
m
T
 2

k
1
1
f  
T 2
k
m
The period and the frequency depend only on the mass of the
particle and the force constant.
The period and the frequency depend only on the
natural quantity of the system.
For the Spring Oscillator:
1
1
f  
T 2
k
m
For the simple pendulum:
1
1
f  
T 2
g
l
x  A cos t   
Phase angle / phase constant
Along with the amplitude A, the phase angle is determined
by the initial condition of the harmonic motion.
initial condition: the position and the velocity of the particle
at t=0.
x  A cos t   
Phase of the motion
When t=0

 ( t )  t  
 (0)  
Initial phase
The simple harmonic vibration is a periodic motion.
x  t  T   x  t   A cos t   
cos t     cos   t  T    
 cos t  2   
4. Energy considerations in simple harmonic motion
The net force on the oscillator is:
It is conservative force.
f  kx
The mechanical energy
of the oscillator must be
conserved.
E  Ek  E p  C
x  A cos t   
1 2
Ek  mv 
2
1 2
E p  kx 
2
dx
v
  A sin t   
dt
1
2 2
2
mA  sin t   
2
1 2
2
kA cos t   
2
1 2
E  Ek  E p  A  m 2 sin 2 t     k cos2 t    
2
k
2
m  m  k
m
1 2
E  Ek  E p  kA
2
The rules to distinguish simple harmonic motion
1) The restoring force of the particle is:
f  kx
2) The dynamical equation of the particle is:
d 2x
2


x0
2
dt
3) The kinetic equation of the particle is:
x  A cos t   
The characters of simple harmonic motion
1) Simple harmonic vibration is periodic motion
2) The state of the oscillator is determined by the parameters
of amplitude A, angular frequency ωand phase angle φ.
3) ω is determined by the natural quantities of the system.
A andφare determined by the system and the initial condition
of the system.
• If a particle undergoes SHM with
amplitude 0.15m, what is the total distance
it travels in one period?
• How could you double the max speed of a
SHO?