Damped oscillation is oscillatory motion whose amplitude decreases because energy is dissipated (friction, drag, internal losses). The most common linear model assumes viscous damping, where the resistive force is proportional to velocity.

Viscous (linear) damping model

For a mass–spring oscillator:

m x¨ + c x˙ + k x = 0

Useful parameters are

ω0 = sqrt(k/m)
β = c/(2m)
ζ = β/ω0 = c/(2 sqrt(m k))

where ω0 is the undamped natural angular frequency and ζ is the damping ratio.

Damping regimes

Underdamped (ζ < 1): oscillates with an exponentially decaying envelope
Critical damping (ζ = 1): fastest return without oscillation
Overdamped (ζ > 1): non-oscillatory, slower return

Underdamped solution

When ζ < 1:

x(t) = A e^{-β t} cos(ω_d t + φ)
ω_d = sqrt(ω0^2 - β^2) = ω0 sqrt(1 - ζ^2)

For small damping, ω_d ≈ ω0, so the period changes only slightly while the amplitude decays as e^{-β t}.

Energy decay and Q factor

In light damping, mechanical energy approximately follows

E(t) ∝ e^{-2β t}
Q ≈ ω0/(2β) = 1/(2ζ)

A larger Q means weaker damping and a sharper resonance peak.

Driven response (forced oscillation)

With a sinusoidal drive F0 cos(ω t):

m x¨ + c x˙ + k x = F0 cos(ω t)
A(ω) = F0 / sqrt((k - m ω^2)^2 + (c ω)^2)

Damping limits resonance amplitude and introduces a phase lag.

Beyond the linear model

Real losses can be nonlinear (dry friction, quadratic drag, hysteresis). Then decay is not purely exponential and the effective damping can depend on amplitude.

Common pitfalls

Small damping mainly reduces amplitude; it does not necessarily change the period much.
When ζ ≥ 1 the motion no longer oscillates.

Damped Oscillation