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# Damped Oscillations

Module by: Paul Padley. E-mail the author

Summary: Add friction to a simple harmonic oscillator and see what happens.

## Damped Oscillations

Consider a simple harmonic oscillator that has friction, then the equations of motion must be changed with the addition of a friction term. So we write m 2 x t 2 = k x b x t m 2 x t 2 = k x b x t where b x t b x t is the friction term. Rearranging we obtain: m 2 x t 2 + b x t + k x = 0 m 2 x t 2 + b x t + k x = 0 or 2 x t 2 + γ x t + ω 0 2 x = 0 2 x t 2 + γ x t + ω 0 2 x = 0 Where γ = b m γ = b m and ω 0 2 = k m ω 0 2 = k m Assume a solution of form x = A e i ( p t + α ) x = A e i ( p t + α ) substitute into equation and get ( p 2 + i p γ + ω 0 2 ) A e i ( p t + α ) = 0 ( p 2 + i p γ + ω 0 2 ) A e i ( p t + α ) = 0 so p 2 + i p γ + ω 0 2 = 0 p 2 + i p γ + ω 0 2 = 0 p p must have real and imaginary parts, so rewrite: p = ω + i s p = ω + i s p 2 = ω 2 + 2 i ω s s 2 p 2 = ω 2 + 2 i ω s s 2 So the equation p 2 + i p γ + ω 0 2 = 0 p 2 + i p γ + ω 0 2 = 0 becomes upon substitution: ω 2 2 i ω s + s 2 + i ω γ s γ + ω 0 2 = 0 ω 2 2 i ω s + s 2 + i ω γ s γ + ω 0 2 = 0 This equation implies that the real and imaginary parts are each zero.Separate the real and imaginary partsImaginary parts give: 2 ω s + ω γ = 0 s = γ 2 2 ω s + ω γ = 0 s = γ 2 From Real parts get ω 2 + s 2 s γ + ω 0 2 = 0 ω 2 + s 2 s γ + ω 0 2 = 0 or ω 2 + γ 2 4 γ 2 γ + ω 0 2 = 0 ω 2 + γ 2 4 γ 2 γ + ω 0 2 = 0 ω 2 γ 2 4 + ω 0 2 = 0 ω 2 γ 2 4 + ω 0 2 = 0 Which rearranges to ω 2 = ω 0 2 γ 2 4 ω 2 = ω 0 2 γ 2 4 Thus the solution becomes x = A e γ t / 2 e i ( ω t + α ) x = A e γ t / 2 e i ( ω t + α ) where ω = ω 0 2 γ 2 4 ω = ω 0 2 γ 2 4 Note that this has assumed a frictional damping force. For a more complicated damping force, the result would be different.

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