Damped Harmonic Oscillator: Under, Critical, Over

Three forces, one equation

A mass m on a spring of stiffness k, with a damper of coefficient c resisting its velocity, obeys one linear second-order differential equation. It is arguably the most useful equation in all of engineering, because so many real systems reduce to it: car suspensions, building sway under wind, analogue meter needles, RLC electrical circuits, even a struck guitar string's fundamental mode.

m·x'' + c·x' + k·x = 0

m x''  = inertia term (mass times acceleration)
c x'   = damping force, proportional to velocity, always opposing motion
k x    = spring restoring force, proportional to displacement

The two numbers that describe every case

Dividing through by m and defining the natural frequency ω₀ = √(k/m) and the damping ratio ζ = c / (2√(mk)) rewrites the equation in a form where the behaviour depends on exactly one number, ζ, regardless of the specific m, c and k:

x'' + 2ζω₀ x' + ω₀² x = 0

characteristic roots:  r = -ζω₀ ± ω₀√(ζ² - 1)

ω₀ sets the speed of the motion — how fast it would oscillate with no damping at all. ζ sets its character: whether the roots are complex (oscillation) or real (no oscillation), and it does so independent of how large or small ω₀ is.

Underdamped: ζ < 1

The roots are a complex-conjugate pair, giving an oscillation at the damped frequency ω_d = ω₀√(1-ζ²), wrapped inside a decaying exponential envelope:

x(t) = A · e^(-ζω₀t) · cos(ω_d t + φ)

The system swings past equilibrium, overshoots, swings back, overshoots less, and so on — each successive peak is smaller than the last by the same fixed ratio (the logarithmic decrement), which is exactly how ζ is measured experimentally from a recorded ringdown. Most real-world springs, tuning forks and lightly-damped structures live here.

Critically damped: ζ = 1

The two roots become one repeated real root, r = -ω₀, and the system returns to equilibrium in the shortest possible time without ever overshooting. This is the boundary case engineers reach for whenever overshoot itself is the problem: analogue galvanometer needles, some door closers, and the shock absorbers on precision instruments are tuned as close to ζ = 1 as the design allows.

Overdamped: ζ > 1

Two distinct real roots give a sum of two decaying exponentials with no oscillation at all — but the slower of the two roots dominates the long-term return, so an overdamped system is actually slower to settle than a critically damped one, despite having "more" damping. This is a common source of confusion: more resistance to motion is not the same as faster settling. A door closer set too strong takes visibly longer to latch than one tuned to critical damping.

Where each regime is chosen on purpose

Car suspensions are deliberately underdamped, typically ζ ≈ 0.2-0.4, because a little softness after a bump feels better and keeps the tyre in contact with the road longer than a stiffer, critically-damped setup would. Seismographs and precision balances are tuned near critical damping so a reading settles quickly without ringing. Overdamped systems show up wherever a slow, controlled, non-oscillating return matters more than speed — a heavily damped door, or a fluid-filled dashpot meant to smooth out a jolt entirely.

Frequently asked questions

Try it live

Everything above runs in your browser — open Damped Oscillator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.