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# Free And Forced Vibrations

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### Free And Forced Vibrations

1. 1. free and forced vibrations
2. 2. Free oscillators <ul><li>Undamped oscillator </li></ul><ul><li>Natural frequency,  o </li></ul><ul><li>No external forces </li></ul><ul><li>e.g. swing </li></ul>
3. 3. Forced oscillators <ul><li>External forces act on oscillator </li></ul><ul><li>Forced/driven oscillator </li></ul><ul><li>Resonance: driving  =  o </li></ul><ul><li>e.g. loudspeaker </li></ul><ul><li>vibrates in response to oscillating electric signal (driver) </li></ul><ul><li>Barton's pendulums </li></ul>
4. 4. Problems <ul><li>Resonance driver applies forces that continually supply energy to oscillator  increasing amplitude </li></ul><ul><li>A increases indefinitely unless energy transferred away </li></ul><ul><li>Severe case: A limit reached when oscillator destroys itself </li></ul><ul><li>e.g. wine glass shatters when opera singer reaches particular note </li></ul>
5. 5. Damping <ul><li>Resonant Amplitude limited by damping forces </li></ul><ul><li>At resonance: </li></ul><ul><li>rate of energy supply = WD against damping forces </li></ul><ul><li>Increasing damping reduces sharpness + strength of resonance </li></ul>
6. 6. Types of damping <ul><li>Oscillating masses lose energy  amplitude decreases </li></ul><ul><li>Light damping: amplitude decreases gradually </li></ul><ul><li>Critical damping: amplitude decreases can decrease to zero without oscillation. Minimum time = T/4 </li></ul>
7. 7. Q-factors <ul><li>Measure of resonance </li></ul><ul><li>Definition: number free oscillations a free oscillator competes before decaying to zero * </li></ul><ul><li>Light damping = large Q factor </li></ul><ul><li>Heavy damping = small Q factor </li></ul><ul><ul><li>e.g. Car = 1 </li></ul></ul><ul><ul><li>Guitar string = 10 3 </li></ul></ul><ul><ul><li>Watch quartz crystal = 10 5 </li></ul></ul><ul><li>* This is only an approximation. Q-factor does have a complicated precise mathematical definition </li></ul>
8. 8. Resonance and damping
9. 9. Examples <ul><li>Pushing a child on a swing – maximum A when pushing  =  o </li></ul><ul><li>Tuning a radio – electrical resonance occurs when  o of tuning circuit adjusted to match  of incoming signal </li></ul><ul><li>Pipe instruments - column of air forced to vibrate. If reed  =  o of column loud sound produced </li></ul><ul><li>Rotating machinery – e.g. washing machine. An out of balance drum will result in violent vibrations at certain speeds </li></ul>
10. 10. <ul><li>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 </li></ul><ul><li>Energy is absorbed from radiation, the frequency of which can be measured with a spectrometer </li></ul>
11. 11. Unwanted resonance <ul><li>Structures/machinery </li></ul><ul><li>Results in destruction </li></ul><ul><li>Damping </li></ul><ul><li>Changing  o of object by changing its mass </li></ul><ul><li>Change stiffness of supports (  moving resonant  away from driving  ) </li></ul>
12. 12. Model aircraft being tested for resonance in a wind tunnel