Driven Oscillator

2 min read#oscillations

Push a damped oscillator with a periodic force and, after the transients fade, it forgets its own rhythm and dances to yours.

Contents

Transient plus steady state
Amplitude depends on tuning
See also

Driven Oscillator

Left alone, a Damped Oscillator eventually stops. Keep pushing it with a periodic external force and it settles instead into a steady rhythm set by the drive, not by itself. With a sinusoidal forcing at angular frequency $\omega$ , the equation gains a right-hand side:

m\ddot x + c\dot x + kx = F_0\cos(\omega t).

This is one of the most consequential equations in physics and engineering — it governs everything from a child on a swing to the tuning circuit in a radio.

Transient plus steady state

The solution splits cleanly into two pieces:

x(t) = \underbrace{x_h(t)}_{\text{transient}} + \underbrace{x_p(t)}_{\text{steady state}}.

The transient $x_h$ is the damped oscillator's own response to being disturbed. It decays as $e^{-\zeta\omega_0 t}$ and is gone after a few time constants — the system forgetting its initial conditions.
The steady state $x_p = A(\omega)\cos(\omega t - \delta)$ oscillates forever at the driving frequency, lagging it by a phase $\delta$ .

Amplitude depends on tuning

The steady-state amplitude

A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + (c\omega/m)^2}}

is small when you drive far above or below the natural frequency $\omega_0$ , and it swells dramatically when $\omega$ approaches $\omega_0$ . That peak is Resonance — important enough to deserve its own page. Push the system harder, into the nonlinear regime, and steady oscillation can detach from the drive entirely, becoming a self-sustained Limit Cycle.

Driven Oscillator

Transient plus steady state

Amplitude depends on tuning

See also