The Spring–Mass–Damper

Free vibration: how it rings down

Pull the mass and let go. With no damping it would oscillate forever at ωₙ = √(k/m). Add damping and the damping ratio ζ = c/(2√(km)) decides the character. Underdamped (ζ < 1): it overshoots and oscillates inside a decaying envelope — a plucked guitar string, a car going over a bump. Critically damped (ζ = 1): it returns to rest in the shortest possible time with no overshoot — exactly what you want from a screen door closer. Overdamped (ζ > 1): it crawls back slowly, like a hinge full of molasses. Drag c across the three regimes and watch the time history change shape.

Forced vibration and resonance

Now drive it with an oscillating force. After the transient dies, the mass settles into a steady oscillation at the driving frequency — but the amplitude depends on how close that frequency is to ωₙ. The magnification factor is 1/√((1−r²)² + (2ζr)²), the curve in the bottom plot. Sweep r toward 1 and the response peaks — at resonance, a tiny force produces an enormous motion, limited only by damping. This is the Tacoma Narrows bridge, a wine glass shattering to a note, and why every rotating machine is balanced to keep its operating speed away from its natural frequencies.

Less damping, sharper peak

The height of the resonance peak is roughly 1/(2ζ) — so halving the damping doubles the worst-case amplitude and makes the peak razor-sharp. With very low ζ you have to tune r almost exactly to ωₙ to catch it, but when you do, the motion is violent. Set damping near zero, switch to forced mode, and inch r toward 1.0.