4.2 damped harmonic motion

About simple damped harmonic Motion

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=A
0
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.

4. Damping Forces

5. Damping Forces (beyond Syllabus)
●The general equations governing the motion of
a damped harmonic oscillation are:
x=x
0
e
−γt
cos(ωt+ϕ)
v=−x
0(γe
−γt
cos(ωt+ϕ)+ωe
−γt
sin(ωt+ϕ))
a=−x
0(−γ
2
e
−γt
cos(ωt+ϕ)+ω
2
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 ω
0
and
amplitude x
0
●The second part of the equation is due to the
forcing (driving) force of magnitude F and
driving frequency ω
x=x
0
e
−γt
cos(ω
0
t)+
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
Maximum Amplitude
D r i v i n g F r e q u e n c y
V e r y L i g h t d a m p i n g
L i g h t d a m p i n g
M e d i u m D a m p i n g
H e a v y D a m p i n 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.