Derive the differential equation of free forced and damped oscillation
Answers
DAMPED OSCILLATIONS
To date our discussion of SHM has assumed that the motion is frictionless, the total energy (kinetic plus potential) remains constant and the motion will continue forever. Of course in real world situations this is not the case, frictional forces are always present such that, without external intervention, oscillating systems will always come to rest. The frictional (damping) force is often proportional (but opposite in direction) to the velocity of the oscillating body such that
resonance eqn1
where b is the damping constant.
This differential equation has solutions
res eqn2
where eqn3 when the damping is small (small b).
Notice that this solution represents oscillatory motion with an exponentially decreasing amplitude
figure 1
See damped oscillation applet courtesy, Davidson College, North Carolina.
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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