Oscillations
Pham Tan Thi, Ph.D.!
Department of Biomedical Engineering!
Faculty of Applied Science!
Ho Chi Minh University of Technology
What is an Oscillation?
• Any motion that repeats itself
• Described with reference to an
equilibrium position
where the net force is zero, and a
restoring force
which acts to return object to equilibrium
• Characterize by:
- Periodic (T) or frequency (f) or angular frequency (ω)
- Amplitude (A)
Simple Harmonic Oscillation/Motion
F=−kx
F=md2x
dt2
d2x
dt2=−
k
mx=−ω2x
x=Acos(ωt+φ)
v=−Aωsin(ωt+φ)
a=−Aω2cos(ωt+φ)
Total force exerts to object given by Hooke’s law
Newton’s second law of motion:
then
ω2
=
k
m
where
General solution for motion
Velocity
Acceleration
Angular frequency
x
0: glider displaced
to the left from the
equilibrium position.
Fx
0, so ax
0:
compressed spring
pushes glider toward
equilibrium position.
Fx
ax
Fx
x50: The relaxed spring exerts no force on the
glider, so the glider has zero acceleration.
(b)
Ox
y
x
n
mg
y
(a)
x
x
y
x
n
mg
y
x
0: glider displaced
to the right from the
equilibrium position.
Fx
0, so ax
0:
stretched spring
pulls glider toward
equilibrium position.
Fx
ax
Fx
(c)
x
x
y
x
n
mg
y
Phase of the motion
Period and Frequency
f=
1
T
T=
1
f
or
Period (T): is the time to complete one full cycle, or one oscillation [s]!
Frequency (f): is the number of cycles per second [s-1]!
The relation between Frequency and Period:!
x
x
x
x
x
x
x
x
x
ax!2amax
ax
52
amax
ax5amax
x52Ax5Ax 50
v
x50
v
x50
v
x50
ax
v
x
ax
v
x
ax
v
x
ax
v
x
2A2A/2 A/2 A0
ax50
ax50
x
v
x52vmax
v
x5vmax
of SHM.
Angular Frequency
The oscillation frequency is measured in cycles per second, Hertz.!
We may also define an angular frequency ω, in radians per second, to
describe the oscillation!
The position of an object oscillating with simple harmonic motion can then
be written as !
then, the maximum speed of this object is!
ω(in rad/s) = 2π
T=2πf
x(t)=Acosωt
vmax =
2πA
T=2πfA =ωA