forced oscillations and resonance
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FORCED OSCILLATIONS AND RESONANCE
fig3Suppose now that instead of allowing our system to oscillate in isolation we apply a "driving force". For example, in the case of the (vertical) mass on a spring the driving force might be applied by having an external force (F) move the support of the spring up and down. In this case the equation of motion of the mass is given by,
Equation 4
One common situation occurs when the driving force itself oscillates, in which case we may write
equation 5
where omega d is the (angular) frequency of the driving force.
This equation has solutions of the form
eqn7
fig2where the amplitude of these oscillations, B, depends on the parameters of the motion, eqn8
eqn9
The amplitude, B, has a maximum value when eqn10. This is called the resonance condition. Note that at resonance, B, can become extremely large if b is small. (In the diagram at right omega0 is the natural frequency of the oscillations, omega, in the above analysis). In designing physical systems it is very important to identify the system's natural frequencies of vibration and provide sufficient damping in case of resonance. This clearly did not happen in the design of the Tacoma Narrows Bridge (Tacoma Narrows Newsreel) in 1940.
There are many physical and engineering systems where resonance is very important e.g. shock absorbers, earthquakes, loudspeakers, NMR, microwave ovens etc. etc. A very important subject which, unfortunately, we do not have time to discuss in any more detail. sad face
A good example of "coupled" oscillations, where forcing the oscillators causes their oscillations to synchronize may be found here.
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Forced Oscillation ;-
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It is an oscillation imposed upon a body or system by and with the frequency of some external vibrator of sensibly different frequency — opposed to free oscillation.
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Force Resonance ;-
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Resonance is the increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic force whose frequency is equal or very close to the natural undamped frequency of the system.
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