. you are here->home->Physical Sciences->Virtual Laboratory: Oscillations->Damped Harmonic Motion . . Damped Harmonic Motion . . 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 )Cite this Simulator:iitk.vlab.co.in,. (2012). Damped Harmonic Motion. Retrieved 24 January 2018, from iitk.vlab.co.in/?sub=27&brch=236&sim=1210&cnt=1 ..... ..... .....