the quality factor Q
/ Q /
How good is an oscillator at oscillating? A church bell rings for many seconds after one strike; a thud of clay dies instantly. A finely tuned radio picks one station cleanly; a poor one smears them together. The quality factor, written Q, is a single number that captures this 'goodness': how many oscillations a system completes before its energy drains away, and equivalently how sharply it favours its resonant frequency.
Q measures the ratio of energy stored to energy lost per cycle. For the spring-mass-damper system it works out to Q = sqrt(m k) / c = omega_0 m / c, and it ties directly to the damping ratio by Q = 1/(2 zeta). A high Q means light damping: the system rings for a long time (roughly Q oscillations before its amplitude noticeably fades) and its frequency-response peak is tall and narrow. A low Q means heavy damping: the ringing dies in a swing or two and the response curve is a broad, gentle bump.
Engineers reach for Q constantly because it sums up a system's selectivity in one figure. A radio tuner or a laser cavity wants a very high Q (thousands or millions) to be exquisitely frequency-selective; a car suspension wants a low Q (near 1) so it does not keep bouncing. Q also estimates resonance danger: the peak amplification at resonance is roughly Q times the static response, so a structure with Q = 50 can amplify a resonant push fifty-fold — a warning engineers take seriously.
For m = 1, k = 400, c = 2: omega_0 = 20, so Q = sqrt(1·400)/2 = 10. The system rings about 10 cycles before fading, and its resonance peak amplifies roughly tenfold.
Q = 10 means about ten rings and a tenfold resonant amplification.
Q, the damping ratio, and the logarithmic decrement are three views of the same thing (Q = 1/(2 zeta), and delta ≈ pi/Q for light damping). They are not independent quantities — pick whichever is most natural for your problem.