The Q value of a resonance is defined to be Q = (R)/ where (R) is the resonance

Question

The Q value of a resonance is defined to be Q = (R)/

where (R) is the resonance frequency and is related to the width of resonance peak. The larger the Q value the larger and sharper the resonance peak. Specifically, the width is defined as the difference in between the two points that are 1/ 2 smaller than the maximum resonance amplitude.

(a) Find the maximum amplitude.

(b) Find the two points where the resonance curve has 1/ 2 of the maximum amplitude.

(c) Assuming that the damping is light, <<0, find to first order in . Calculate Q. You should find Q 0/2 .

Explanation / Answer

•Resonance frequency R º the frequency at which the amplitude D() is a maximum. Þ (dD/d) = 0     

Solving for = R gives:        R = [02 - 22]½

•Clearly the resonance frequency R < 0

where 0 = (k/m)½ is the natural” frequency of oscillator! How much less obviously depends on the size of the damping constant !

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