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Unformatted text preview: d V i b t i d Forced Vibrations and esonance I Resonance I Harmonic Forcing hy should we care about this topic Why should we care about this topic Tacoma Narrows Bridge – 1940  42 mph wind drives oscillations Harmonic Forcing k F cos ω t k m ω = 2 d x Newton II: 2 2 cos s m kx F t dt d x k x F t ω = − + = 2 cos m k x F t dt ω + = – note that there are two frequencies: ω , the natural frequency, and ω , the driving frequency – we can expect that the motion will be a superposition of ω and ω 0 – transient motion → steadystate motion ω ω + ω – let’s look at the steadystate motion Harmonic Forcing k 2 x F cos ω t k m ω = 2 cos d x m k x F t dt ω + = – focus on two cases ¡ ω << ω 0 ⇒ oscillates at ~ ω with A ~ F /k = F /m ω 2 ⇒ response controlled by the spring stiffness ¡ ω >> ω x rm dominated by m(d 2 /dt 2 term ⇒ kx term dominated by m(d x/dt ) term ⇒ response governed by the inertia of the system ⇒ small A oscillation out of phase with the driving force...
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 Spring '08
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 Force, tacoma narrows bridge, Harmonic Forcing

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