### Q as Energy Stored over Energy Dissipated

Yet another meaning for is as follows [20, p. 326]

where the resonator is freely decaying (unexcited).

Proof. The total stored energy at time is equal to the total energy of the remaining response. After an impulse at time 0, the stored energy in a second-order resonator is

The energy dissipated in the first period is , where

Assuming as before, so that

Assuming further that , we obtain

This is the energy dissipated in one cycle. Dividing this into the total stored energy at time zero, , gives

whence

as claimed. Note that this rule of thumb requires , while the one of the previous section only required .

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