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class10 - example Arya example 3.2 1.4 Quality Factor The...

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example: Arya example 3.2. 1.4 Quality Factor The Q -value is a dimensionless quantity that represents the degree of damping of an oscilla- tor. It is 2 π times the ratio of the energy stored to the average energy loss per period. Call P the rate at which energy is dissipated, and the time period of oscillation is T 1 = 2 π / ω 1 , then Q = 2 π E P 2 π / ω 1 = E P/ ω 1 (67) since for an underdamped oscillator, 1 / ω 1 is the time of motion for 1 radian. Thus Q is also the ratio of the energy stored in the oscillator to the average energy dissipated per radian. For a lightly damped oscillator, Q is large, as follows: The energy of a lightly damped oscillator is given by equation 65: E ( t ) = E 0 e - 2 γ t (68) and recalling equation 66, we can calculate the energy dissipated in the time Δ t : Δ E = | dE/dt | Δ t = 2 γ E Δ t (69) If we then specify Δ t as the time for 1 radian of oscillation Q = E Δ E = E 2 γ E/ ω 1 = ω 1 2 γ (70) 1.5 Sinusoidal driving forces
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