Vibrations & Dynamics · Learn
One mass, one spring, one dashpot — the equation behind every wobble in engineering. Two numbers, the natural frequency and the damping ratio, run the entire show.
Newton's second law for a mass on a spring with a damper, pushed by a force $F(t)$:
Inertia + damping + stiffness = force. Divide by $m$ and two natural parameters appear.
$\omega_n$ is how fast it would oscillate with no damping; $\zeta$ is the dimensionless damping that sets the character of the response.
Pull the mass and release ($F=0$). The damping ratio decides what happens:
| Regime | $\zeta$ | Motion |
|---|---|---|
| Underdamped | $<1$ | oscillates inside a decaying envelope at $\omega_d=\omega_n\sqrt{1-\zeta^2}$ |
| Critically damped | $=1$ | fastest return to rest, no overshoot |
| Overdamped | $>1$ | slow, sluggish creep back |
Critical damping is the engineering sweet spot — it's what you want from a door closer or a measurement needle: settle fast, don't overshoot.
Drive it with $F_0\sin(\omega t)$. After the transient dies, the mass oscillates at the driving frequency, but the amplitude depends on how close $\omega$ is to $\omega_n$. With $r=\omega/\omega_n$, the magnification factor is
which peaks near $r=1$: at resonance, a small force produces an enormous motion, limited only by damping (peak height $\approx 1/2\zeta$). This is the Tacoma Narrows bridge, a wine glass shattering to a note, and why every rotating machine is balanced to keep its speed away from $\omega_n$.
Drag $c$ across the three regimes and watch the time history change shape; switch to forced mode and sweep $r$ toward 1 to climb the resonance peak. Open the lab →
EngineeringCandy · Learn · the theory behind the playground