phy293_l05.page2

# phy293_l05.page2 - This gives 180(tan approaching 0 from...

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This gives δ 180 ( tan δ approaching 0 from below) The mass lags 180 out of phase, ie. it moves in the opposite direction to the driving force This system is inertia dominated The amplitude still falls off at high ω like ω - 2 but it is not due to the damping [see log-log plot shown in class] 3. Velocity Response There is also a resonance in velocity – mass moves faster than forced-end of spring This actually occurs at the natural frequency of the system ˙ X ( t ) = - ωA ( ω ) sin( ωt - δ ) The velocity amplitude is: V ( ω ) = a 0 ωω 2 0 p ( ω 2 0 - ω 2 ) 2 + ( γω ) 2 = a 0 ω 2 0 p ( ω 2 0 - ω 2 ) 2 2 + γ 2 This is maximised when the denominator is minimised: d ( ( ω 2 0 - ω 2 ) 2 ω 2 + γ 2 ) = 0 But the γ 2 term just vanishes from the derivative Taking derivative of a ratio: d dx ( A B ) = A 0 B - B 0 A B 2 gives: 2( ω 2 0 - ω 2 )2 ω ( ω 2 ) - (2 ω )( ω 2 0 - ω 2 ) 2 = 0 Gathering terms together this gives: ( ω 2 0 - ω 2 )[2 ω ](2 ω 2 - ω 2 0 - ω 2 ) = 0 There are three different families of roots here: ω = [0 , ± ω 0 , ± 0 ] Of these only ω = ω 0 is positive and real
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