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# lec_12 - October 4 2011 12-1 12 Forced Oscillations We...

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October 4, 2011 12-1 12. Forced Oscillations We consider a mass-spring system in which there is an external oscillating force applied. One model for this is that the support of the top of the spring is oscillating with a certain frequency. The equation of motion becomes m ¨ u + γ ˙ u + ku = F 0 cos ( ωt ) . (1) Let us ﬁnd the general solution using the complex func- tion method. First assume that γ = 0; i.e., there is no friction. The given equation is the real part of the complex equation m ¨ u + ku = F 0 e i ω t . Let ω 0 = r k m be the natural frequency of the unforced equation. If ω 6 = ω 0 , then the general complex solution is u ( t ) = c 1 cos( ω 0 t ) + c 2 sin( ωt ) + Re ( F 0 m ( ) 2 + k e i ω t ) = R cos( ω 0 t - δ ) + Re ( F 0 m ( ) 2 + k e i ω t ) = R cos( ω 0 t - δ ) + F 0 m ( k m - ω 2 ) cos( ωt )

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October 4, 2011 12-2 = R cos( ω 0 t - δ ) + F 0 m ( ω 2 0 - ω 2 ) cos( ωt ) . Resonance:
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lec_12 - October 4 2011 12-1 12 Forced Oscillations We...

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