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 ...
View
Full
Document
This note was uploaded on 07/10/2011 for the course PHY 293 taught by Professor Pierresavaria during the Fall '07 term at University of Toronto.
 Fall '07
 PierreSavaria
 mechanics, Force, Inertia, Statistical Mechanics

Click to edit the document details