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phy293_l05.page1 - A ω m = A p γ 2 2 2 ω 2-γ 2 2 γ 2 =...

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PHY293 Oscillations Lecture #5 September 21, 2009 1. Second problem set now assigned (http://www.physics.utoronto.ca/ phy293h1f/waves/phy293 ps2.pdf) Start of material 1. Resonance in Forced Oscillations Looking at A ( ω ) we find a frequency where the amplitude is maximal As γ is reduced the peak gets higher and narrower [see plots shown in class] The peak frequency can be found by differentiation: A ( ω ) = A 0 p ( ω 2 0 - ω 2 ) 2 + ( γω ) 2 stationary at dA ( ω ) = 0 d [( ω 2 0 - ω 2 ) 2 + ( γω ) 2 ] = 0 This is the condition for the minimum or maximum and gives: 2( ω 2 0 - ω 2 )( - 2 ω ) + 2 γ 2 ω = 0 2 ω (2 ω 2 - 2 ω 2 0 + γ 2 ) = 0 Has two real solutions: ω = [0 , p ω 2 0 - γ 2 / 2] The first solution is the local minimum at ω = 0 ( A 0 ) This second solution is a maximum at: ω m = q ω 2 0 - γ 2 / 2 Note that for γ 2 < 2 ω 2 0 there is no resonance This is not quite the condition for critically damped solutions This one is squared while the critical damping of a free-oscillator is a statement about ω 0 and γ without squares) Resonance only happens for low damping At this resonance frequency the amplitude is
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Unformatted text preview: A ( ω m ) = A p ( γ 2 / 2) 2 + ( ω 2-γ 2 / 2) γ 2 = A p ω 2 γ 2-γ 4 / 4 = A ω γ 1 q 1-γ 2 4 ω 2 2. Phase Lag • Up to now we’ve focused on the amplitude of the driven oscillations: A ( ω ) • There is also physics in the phase lag: X ( t ) = A ( ω ) cos( ωt-δ ); tan δ = ωγ ( ω 2-ω 2 ) • The phase is a smooth connection between two regimes [see plots shown in class]: (a) At low driving frequencies δ → ◦ The mass and platform move in unison (rigid regime): X ( t ) = A cos( ωt ) for ω ± ω ◦ In this regime A ( ω ) → A ω 2 ω 2 = A ◦ The response of the system is determined by the stiffness of the spring (b) At high frequency ( ω ² ω ) A ( ω m ) = A p ( γ 2 / 2) 2 + ( ω 2-γ 2 / 2) γ 2-→ A ω 2 ω 2 ◦ And the phase (for ω ² ω : tan δ = ωγ ( ω 2-ω 2 )-→-γ ω...
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