1. Topic 4 – Oscillations and Waves
4.2 Damped and Forced Harmonic Motion
2. Damped
● In SHM there is only the one restoring force
acting in the line of the displacement.
● In damped harmonic motion (DHM) an
additional damping force acts in the opposite
direction to the velocity of the object to
dissipate energy and stop the vibrations.
3. Damping Forces
● The damping force acts so as to cause the amplitude of the
vibrations to decay naturally dissipating energy.
● The general equation of this decay is A=A0
e-ˠt
● Here ˠ is a damping factor.
● The system can be under-damped
● This means the system can make more than one full oscillation
before it comes to a stop.
● The system can be over-damped
● The system comes to a stop before it completes one oscillation
● The system can be critically-damped
● The system completes exactly one oscillation before stopping.
5. Damping Forces (beyond Syllabus)
● The general equations governing the motion of
a damped harmonic oscillation are:
x=x0 e
−γ t
cos(ωt+ ϕ)
v=−x0(γ e
−γ t
cos(ωt+ ϕ)+ ωe
−γ t
sin(ωt+ ϕ))
a=−x0 (−γ2
e−γ t
cos(ωt+ ϕ)+ ω2
e− γt
cos(ωt+ ϕ))
6. Natural Frequency
● The frequency with which a system oscillates if
it is started and allowed to move freely is called
its natural frequency.
● Simple harmonic motion occurs at the natural
frequency.
● Often, extra energy is imparted into the system
each oscillation by another external periodic
force.
● This is like a child pushing a swing to keep it going.
● Such a system is said to be a forced harmonic
oscillator.
7. Forced Harmonic Motion
● The equation for forced harmonic motion (with
some damping) would be:
● Here the first part of the equation is the normal
SHM equation with natural frequency ω0
and
amplitude x0
● The second part of the equation is due to the
forcing (driving) force of magnitude F and
driving frequency ω
x=x0e
− γt
cos(ω0t)+
F
mω
2
cos(ωt)
8. Forced Harmonic Motion and Resonance
● As the driving frequency of the system
approaches the natural frequency of the
system, the amplitude of the system increases
dramatically.
● The force adds energy to each swing making
the amplitude continue to increase and
increase.
● When the two frequencies are identical, then
the system is said to be at resonance.
9. Resonance
● The state in which the frequency of the
externally applied periodic force equals the
natural frequency of the system is called
resonance.
● This causes oscillations with large amplitudes.
● Damping causes the maximum amplitude to be
limited.
10. Resonance
-5 0
1 9 5 0
3 9 5 0
5 9 5 0
7 9 5 0
9 9 5 0
1 1 9 5 0
1 3 9 5 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
MaximumAmplitude
D r iv in g F r e q u e n c y
V e r y L ig h t d a m p in g
L ig h t d a m p in g
M e d iu m D a m p in g
H e a v y D a m p in g
11. Dangerous Resonance
● Resonance can be
disastrous
● If a bridge happens to
have a natural frequency
that is in the range of the
frequencies that can be
generated by the wind
then the bridge can
oscillate.
● The bridge can then
vibrate it can collapse!
● This is resonance at its
worst!!!
12. Useful Resonance
● Resonance can be useful.
● A radio is tuned by causing a quartz crystal to
resonate at a particular frequency.
● Wind instruments rely on the resonance of a
vibrating air column to make an audible sound.
– Because of the sharp spike on the frequency response
curve, other frequencies are cancelled out and not heard.
