Quality Factor and Bandwidth

The bandwidth of the resonance is typically described using the quantity $${Q = \frac{f_0}{\Delta f}}$$, for quality factor. A high-Q has a sharp resonance, and a low-Q has a broad resonance curve. For a vibrator set into motion and left to vibrate freely, its decay time is proportional to the Q of its resonance.

In terms of our mechanical system, given the resistance $${\alpha = \frac{R}{2m}}$$, the quality factor is:

$$Q = \frac{\omega_0}{2\alpha}.$$