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Unformatted text preview: Resonance dx m 2 = −kx dt 2 dx m 2 = −kx dt x = A cos ωt + B sin ωt 2 dt x = A cos ωt + B sin ωt m dx
2 2 = − kx ω=
2 k T= m 2π ω Fd = −bv dx = −b dt dx dx m 2 = − kx − b dt dt 2 x = Ce cos ωt Transient solution −α t x = A sin ωt + B cos ωt
[ [ L] sin ωt + [ L] cos ωt = 0 ω = ω −α
2 0 2 0 2 k where ω = m b and α = 2m 2 2 ω0 = α is critical damping Forced Oscillations Fex cos ωt dx m 2 = −kx + Fex cos ωt dt x = A cos ωt
2 ( ω − ω 2 0 2 ) A − a cos ωt = 0 0 Fex where a 0 = m a0 A= 2 2 ω0 − ω Damped forced oscillations dx dx Fex cos ωt − kx − b = m 2 dt dt x = A cos ωt + B sin ωt
2 [ L] cos ωt + [ L] sin ωt = 0 A= (ω (ω a0 ( ω − ω
2 0 2 0 2 −ω 22 ) ) )
2 2 +β ω B= −a0 βω
2 0 −ω 22 +β ω
2 2 Fex where a0 = m b and β = m ...
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This note was uploaded on 11/20/2009 for the course PH ASF taught by Professor Goodstein during the Spring '07 term at Caltech.
 Spring '07
 Goodstein

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