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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 outofphase 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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 Fall '07
 PierreSavaria
 mechanics, Force, Inertia, Statistical Mechanics

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