Quality factor

The largest response to the drive force occurs when the natural solutions are underdamped and the drive frequency is the same as the natural frequency $\omega_0=\sqrt{k/m}$. This condition is called a resonance. For lightly damped systems, the drive frequency has to be very close to the natural frequency and the amplitude of the oscillations can be very large. The quality factor $Q=\frac{\sqrt{mk}}{b}$ describes how strong the resonance is. $Q$ is the number of periods of the oscillation occur in the time it takes for the amplitude of the oscillations to drop by a factor of $1/e$. For $Q<\frac{1}{2}$, the solutions are overdamped and the resonance frequency is zero. For $Q > \frac{1}{2}$, the system is underdamped and the resonance gets stronger for higher $Q$. The quality factor can also be expressed as $Q=\frac{\pi\tau}{T}$, where $T=2\pi/\omega_0$ is the period of the undamped oscillations and $\tau$ describes the exponential decay.

The amplitude of the response $|A|/F_0 = 1/\rho$ that is observed after the transient response has decayed to zero is plotted below as a function of frequency. For small damping $Q >> 1$, there is a sharp resonance at $\omega_0$ and $Q = \omega_0/\Delta\omega$ where $\Delta\omega$ is the full width at half maximum. For low frequencies $(\omega <\omega_0)$, the response $x$ is in phase with the driving force. At high frequencies $(\omega >\omega_0)$, the response is out of phase with the driving force.