Quality factor

The largest response to a periodic driving force occurs when the natural solutions are underdamped and the drive frequency is close to the natural frequency $\omega_0=\sqrt{k/m}$. It is convenient to define a quality factor,

At critical damping, $\frac{k}{m}=\frac{b^2}{4m^2}$, and the quality factor is $Q=\frac{1}{2}$. The system is overdamped for $Q < \frac{1}{2}$ and underdamped for $Q > \frac{1}{2}$. For lightly damped systems, $b^2 << 4mk$, $Q$ has the interpretation that it is $\pi$ times the number of periods of the oscillation that occur in the time it takes for the amplitude of an undriven system to drop by a factor of $1/e$. Mathematically this is $Q\approx\frac{\pi\tau}{T}$. If the drive frequency is very close to the natural frequency for an oscillator with a large $Q$, the amplitude of the oscillations can be very large.

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.