phy293_l05.page1 - A ( m ) = A p ( 2 / 2) 2 + ( 2- 2 / 2) 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 ( m ) = A p ( 2 / 2) 2 + ( 2- 2 / 2) 2 = A p 2 2- 4 / 4 = A 1 q 1- 2 4 2 2. Phase Lag Up to now weve focused on the amplitude of the driven oscillations: A ( ) There is also physics in the phase lag: X ( t ) = A ( ) cos( t- ); tan = ( 2- 2 ) The phase is a smooth connection between two regimes [see plots shown in class]: (a) At low driving frequencies The mass and platform move in unison (rigid regime): X ( t ) = A cos( t ) for In this regime A ( ) A 2 2 = A The response of the system is determined by the stiffness of the spring (b) At high frequency ( ) A ( m ) = A p ( 2 / 2) 2 + ( 2- 2 / 2) 2- A 2 2 And the phase (for : tan = ( 2- 2 )--...
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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- Toronto.

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