phy293_l07.page4 - (b) There are three frequency regimes...

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Unformatted text preview: (b) There are three frequency regimes for forced oscillations low ω Resonant ω High ω δ→0 δ = π/2 δ→π ω2 A → a0 A = a0 Q √ 1 2 A = a0 ω0 2 1−1/4Q Stiffness dominated spring/damping balanced Inertia dominated (c) Velocity Response • Resonance at ω = ω0 ˙ X (t) = −ωA(ω ) sin(ωt − δ ) ≡ −V (ω ) sin(ωt − δ ) V (ω ) = 2 a0 ω0 2 (ω0 − ω 2 )2 /ω 2 + γ 2 2 • For low ω , V → 0; for high ω V → a0 ω0 /ω • At resonance the reactive elements of the system cancel each other out • The system becomes easy to drive: viscosity/damping and restoring force/spring work together resulting in resonance (d) Power Absorbed by an oscillator • Power averaged over a complete cycle is only absorbed by out-of-phase component of solution < P (ω ) >= Pmax γ2 2 (ω0 − ω 2 )2 /ω 2 + γ 2 3 • Resonant at ω = ω0 , < Pmax >= 1 ma2 ω0 Q 0 2 2 γ /4 • At high Q: < P >= Pmax (ω0 −ω)2 +γ 2 /4 • In either case the full width at half maximum of the power curve is γ and Q = ω0 /γ (e) Summary of Resonance Frequency Peak Value Comments Amplitude 2 ω = ω = ω0 − γ 2 /2 a0 Q Am = q 1 Velocity ω = ω0 Vm = a0 ω0 Q ω ω0 2 A → a0 ω0 /ω 2 δ→π reactive elements cancel Easy to drive 1− 4Q2 Pm Power ω = ω0 3 = 1 ma2 ω0 Q 0 2 FWHM = γ = ω0 /Q Narrow for large Q ...
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