Damped Oscillator

2 min read#oscillations

Add friction to a harmonic oscillator and its energy bleeds away — gently ringing down, just barely settling, or sluggishly creeping home.

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Damped Oscillator

Real oscillators lose energy. Add a velocity-proportional drag — air resistance, internal friction, electrical resistance — to the Simple Harmonic Oscillator and you get the damped oscillator:

m\ddot x + c\dot x + kx = 0.

The new middle term c\dot x always opposes the motion, draining energy on every pass. How the system relaxes depends entirely on how the damping c compares to the stiffness and mass, summarized by the dimensionless damping ratio \zeta = c/(2\sqrt{mk}).

Three regimes

Underdamped (\zeta < 1) — the system still oscillates, but inside a shrinking envelope e^{-\zeta\omega_0 t}. It rings down over many cycles. This is a plucked string or a struck bell.

Critically damped (\zeta = 1) — the fastest possible return to rest without overshooting. Car suspensions and door closers are tuned near here.

Overdamped (\zeta > 1) — so much drag that the system creeps back to equilibrium slowly, never crossing it. Think of a spoon settling in honey.

The plot below shows the underdamped case: an oscillation e^{-0.3t}\cos(3t) caged between its two exponential envelopes.

Underdamped decay
The oscillation (middle curve) is bounded by the decaying envelope ±e^(−0.3t). Each swing is a little smaller than the last.

Watch it ring down

Damping turns the conservative pendulum's perpetual loop into an inward spiral in the Phase Portrait — every orbit shrinks toward the stable Fixed Point at the bottom. Give the simulation below a healthy dose of friction and watch the swings die away:

A pendulum with damping 0.4. Energy leaks out each swing until it hangs still at the bottom.

See also

Related topics

  • ResonanceDrive an oscillator near its natural frequency and the response explodes — the principle behind tuning radios, shattering glass, and toppling bridges.
  • Limit CycleAn isolated closed orbit that a nonlinear system generates and maintains entirely on its own — the mathematics of self-sustained rhythm.
  • Simple Harmonic OscillatorA mass pulled back by a force proportional to its displacement — the linear ideal whose sinusoidal motion underlies every other oscillator.
  • Driven OscillatorPush a damped oscillator with a periodic force and, after the transients fade, it forgets its own rhythm and dances to yours.
  • Phase PortraitThe qualitative picture of all trajectories in state space, revealing a system's behavior at a glance.
  • State SpaceThe space of all possible states of a system, in which each point is a complete snapshot and motion traces a trajectory.

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