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# lecture_7 - Lecture 7 Dissipated energy Forced damped...

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Dissipated energy. Forced damped vibration of a linear mass-damper-spring system con- sists of two parts: the exponetialy decaying part and an oscillatory part. The exponentially decaying part is the homogeneous solution to the forced damped vibration equation of motion, stemming from New- ton’s law. If there was no forcing the system would end up ultimately (when t = ) an equilibrium position. In the presence of forcing, the system ends up oscillating permanently, at the same frequency as the forcing, but with a changed phase of the response. The oscillatory part of the solution is x = Xsin ( ωt - φ ) where X is the amplitude of the response to forcing and φ is the phase of the response. We would like to ﬁnd out how much energy is neces- sary to put into the system to keep it oscillating with that response. We will ﬁnd that by considering the work of the dissipation force over a cycle of the forced oscillation. Note that ˙ x = ωXcos ( ωt - φ ) = ± ωX p 1 - sin 2 ( ωt - φ ) = ± ωX r 1 - x 2 X 2 = ± ω X 2 - x 2 and the dissipative force is F d = c ˙ x If we integrate it over a cycle of oscillations, we get: W d = I F d dx = I c ˙ xdx = I c dx dt dx dt dt = I c ˙ x 2 dt = 2 X 2 Z 2 π ω 0 cos 2 ( ωt -

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lecture_7 - Lecture 7 Dissipated energy Forced damped...

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