Free And Forced Vibrations

1. free and forced vibrations

2. Free oscillators
•Undamped oscillator
•Natural frequency, ¦
o
•No external forces
•e.g. swing

3. Forced oscillators
•External forces act on oscillator
•Forced/driven oscillator
•Resonance: driving ¦ = ¦
o
•e.g. loudspeaker
vibrates in response to oscillating
electric signal (driver)
•Barton's pendulums

4. Problems
•Resonance driver applies forces that
continually supply energy to oscillator ®
increasing amplitude
•A increases indefinitely unless energy
transferred away
•Severe case: A limit reached when oscillator
destroys itself
•e.g. wine glass shatters when opera singer
reaches particular note

5. Damping
•Resonant Amplitude limited by damping
forces
•At resonance:
rate of energy supply = WD against damping
forces
•Increasing damping reduces sharpness
+ strength of resonance

6. Types of damping
•Oscillating masses lose energy
\amplitude decreases
•Light damping: amplitude decreases
gradually
•Critical damping: amplitude decreases
can decrease to zero without oscillation.
Minimum time = T/4

7. Q-factors
•Measure of resonance
•Definition: number free oscillations a free
oscillator competes before decaying to zero*
•Light damping = large Q factor
•Heavy damping = small Q factor
–e.g. Car = 1
Guitar string = 10
3
Watch quartz crystal = 10
5
* This is only an approximation. Q-factor does have
a complicated precise mathematical definition

8. Resonance and damping

9. Examples
•Pushing a child on a swing – maximum A
when pushing ¦ = ¦
o
•Tuning a radio – electrical resonance occurs
when ¦
o
of tuning circuit adjusted to match ¦
of incoming signal
•Pipe instruments - column of air forced to
vibrate. If reed ¦ = ¦
o
of column loud sound
produced
•Rotating machinery – e.g. washing machine.
An out of balance drum will result in violent
vibrations at certain speeds

10. •Investigation strength of chemical bonds –
EM radiation = oscillating electrical
disturbance. When incident on a crystal ions
subjected to oscillating electrical force. At
correct frequency ions ions oscillated by
resonance
•Energy is absorbed from radiation, the
frequency of which can be measured with a
spectrometer

11. Unwanted resonance
•Structures/machinery
•Results in destruction
•Damping
•Changing ¦
o
of object by changing its
mass
•Change stiffness of supports (\moving
resonant ¦ away from driving ¦)

12. Model aircraft being tested for resonance in a wind
tunnel