phy293_l07.page3 - 4 Review of Forced Damped SHO 2 2(a...

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4. Review of Forced Damped SHO (a) Given : ¨ x + γ ˙ x + ω 2 0 = a 0 ω 2 0 cos( ωt ) And two initial conditions (like x (0) = x i and ˙ x (0) = v i We find a solution x ( t ) = x c ( t ) + X ( t ) Where the complementary solutions are I γ < 2 ω 0 x c ( t ) = C 1 e - γt/ 2 cos( ω 0 t + C 2 ) Under-damped ω 0 = p ω 2 0 - γ 2 / 4 II γ = 2 ω 0 x c ( t ) = ( C 1 + C 2 t ) e - γt/ 2 Critically damped III γ > 2 ω 0 x c ( t ) = C 1 e α 1 t + C 2 e - α 2 t Over-damped α 1 , 2 = - γ/ 2 ± p γ 2 / 4 - ω 2 0 In all three cases C 1 , 2 are determined by x i and v i Independent of the drive frequency ω or its amplitude a 0 these contributions die away like e - γt/ 2 (or faster). The steady state solution has an amplitude and phase independent of time
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