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Damped Harmonic Motion
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             Almost everyone has an intuitive understanding of the playground swing, and so it is a good first example. If the person in the swing is neither “pumping” nor being pushed, and if frictional losses are small, one has a simple harmonic oscillator, at least for small amplitudes. If the rider drags his or her feet then there is damping.
 
 
The Damped Harmonic Oscillator:
 
If the damping force, fD , is proportional to the velocity, v, with a damping constant, b, then
 
                                                 fD=-bv
 
The equation of motion for this system is:
 
                                                md2x/dt2  + bdx/dt  + kx = 0            
 
 
There are three cases depending on the degree of damping discussed below.
 
 
 
 
Case I.   If b is small enough such that (b/2m) < √((k/m), then this is called under damped case.
 
The equation becomes:
 
x= A e-(b/2m)t cos (ω' t + φ) 
 
Where, ω' =√((k/m) - (b/2m)2 )
         Or, ω' =√(ω02 - d2 ) with d = b/2m, called damping factor.
 
 
 
Case II.   If (b/2m) = √((k/m), then it is critically damped case.
 
The equation becomes:
 
x =A(1+ω0t)*e-ω0t
 
 
 
Case III.   If (b/2m) > √((k/m), it is called over damped case.
 
The equation becomes:
 
x = [(eωt + e-ωt)+(d/ω)*(eωt -e-ωt)]/2
where ω =√(d2-ω02 )

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