free and forced vibrations

Free oscillators

• Undamped oscillator

• Natural frequency, o

• No external forces• e.g. swing

Forced oscillators

• External forces act on oscillator• Forced/driven oscillator

• Resonance: driving = o

• e.g. loudspeakervibrates in response to oscillating electric signal (driver)

• Barton's pendulums

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

Damping

• Resonant Amplitude limited by damping forces

• At resonance:rate of energy supply = WD against damping forces

• Increasing damping reduces sharpness + strength of resonance

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

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 = 103

Watch quartz crystal = 105

* This is only an approximation. Q-factor does have a complicated precise mathematical definition

Resonance and damping

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

• 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

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 )

Model aircraft being tested for resonance in a

wind tunnel