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The Driven Oscillator

Module by: Paul Padley. E-mail the author

Summary: We derive the equations for a driven oscillator.

Lets consider a case with a driven or forced oscillator. We now have m x ¨ + k x = F 0 e i ω t m x ¨ + k x = F 0 e i ω t or x ¨ + ω 0 2 x = F 0 m e i ω t . x ¨ + ω 0 2 x = F 0 m e i ω t . Try x = A e i ( ω t + α ) x = A e i ( ω t + α ) then x ˙ = i ω A e i ( ω t + α ) x ˙ = i ω A e i ( ω t + α ) and x ¨ = ω 2 A e i ( ω t + α ) . x ¨ = ω 2 A e i ( ω t + α ) . So now get ω 2 A e i ( ω t + α ) + ω 0 2 A e i ( ω t + α ) = F 0 m e i ω t ω 2 A e i ( ω t + α ) + ω 0 2 A e i ( ω t + α ) = F 0 m e i ω t ( ω 0 2 ω 2 ) A e i ( ω t + α ) = F 0 m e i ω t ( ω 0 2 ω 2 ) A e i ( ω t + α ) = F 0 m e i ω t ( ω 0 2 ω 2 ) A = F 0 m e i α ( ω 0 2 ω 2 ) A = F 0 m e i α ( ω 0 2 ω 2 ) A = F 0 m cos α i F 0 m sin α ( ω 0 2 ω 2 ) A = F 0 m cos α i F 0 m sin α Next you spearate the Real and Imaginary parts. The Imaginary part gives 0 = F 0 m sin α 0 = F 0 m sin α so α = 0 , π α = 0 , π implies cos α = ± 1 cos α = ± 1 The Real part gives ( ω 0 2 ω 2 ) A = F 0 m cos α ( ω 0 2 ω 2 ) A = F 0 m cos α ( ω 0 2 ω 2 ) A = ± F 0 m ( ω 0 2 ω 2 ) A = ± F 0 m A = ± F 0 m ( ω 0 2 ω 2 ) A = ± F 0 m ( ω 0 2 ω 2 ) So a solution is x = F 0 m ( ω 0 2 ω 2 ) e i ( ω t + α ) x = F 0 m ( ω 0 2 ω 2 ) e i ( ω t + α ) where α = 0 , π α = 0 , π

This is an extremely important result, this is the phenomenum of resonance. When you drive an oscillator at its resonant frequency then the amplitude of the oscillation will become huge. In the equation above, it becomes infinite, but in practice there will be some damping that prevents that. You have known this since your childhood, this is how you swing on a swing. If you live in a snowy climate, you know (or at least should know) that a trick to get your car out of a snow bank is is to rock it back and forth - if you get the frequency right you will make the car oscillate with a large amplitude and dislodge it. The electrical analogue is used to tune a radio.

A driven damped oscillator will be given as a homework problem.

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