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, f_{D} , is proportional to the velocity, v, with a damping constant, b, then

f_{D}=-bv

The equation of motion for this system is:

md^{2}x/dt^{2 } + 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' =√(ω_{0}^{2} - d^{2 }) 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+ω_{0}t)*e^{-}^{ω}_{0}^{t}

Case III. If (b/2m) > √((k/m), it is called over damped case.

The equation becomes:

x = [(e^{ω}^{t }_{+ }e^{-}^{ω}^{t})+(d/ω)*(e^{ω}^{t }-e^{-}^{ω}^{t})]/2

where ω =√(d^{2}-ω_{0}^{2} )