the frequency-response curve
Drive a system at a slow rhythm and it barely stirs; speed up the rhythm toward its favourite rate and it answers with huge swings; push faster still and it can barely keep up and the swings shrink again. Plot the size of the steady response against the driving frequency and you get a curve with a hump in the middle. That hump-shaped plot is the frequency-response curve — the fingerprint of how a system reacts to every possible driving frequency.
The steady-state amplitude of m x'' + c x' + k x = F_0 cos(omega t) works out to R(omega) = F_0 / sqrt((k - m omega^2)^2 + (c omega)^2). Read it as a function of the driving frequency omega: when omega is small, R starts near F_0/k (the static deflection); as omega approaches the natural frequency, the (k - m omega^2) term nearly vanishes and R shoots up to a peak; for large omega the denominator grows and R falls back toward zero. Alongside the amplitude there is a companion phase-lag curve, the angle phi by which the response trails the force, sweeping from near 0 to near 180 degrees as omega passes through the peak.
This single curve is how engineers see a system's whole personality at a glance: where it is sensitive, where it is deaf, and how sharp its resonance is. A lightly damped system has a tall, narrow spike (it strongly favours one frequency — good for tuning a radio, dangerous for a bridge); heavy damping flattens the curve into a gentle, broad bump (it responds modestly across a wide band). The height and sharpness of the peak are governed entirely by the damping ratio.
For F_0 = 1, m = 1, k = 100, c = 2, the amplitude R(omega) = 1/sqrt((100 - omega^2)^2 + (2 omega)^2) is tiny for small omega, rockets to a tall narrow peak near omega ≈ 10, then sinks toward zero for large omega.
Small damping (c = 2) makes a sharp, tall resonance peak near omega = 10.
The amplitude peak does not sit exactly at the natural frequency omega_0 when damping is present — it shifts a touch lower, to omega = omega_0 sqrt(1 - 2 zeta^2). For light damping the shift is negligible, but it is not zero, and for zeta above 1/sqrt(2) there is no peak at all.