Forced oscillation
An oscillator being driven by a periodic external force, settling into a steady state at the drive frequency.
Definition
A forced oscillator is any oscillating system being pushed by an external periodic force — a child on a swing getting nudged in time, an electrical circuit fed a sine-wave voltage, a mass on a spring jostled by a motor. The governing equation adds a driving term F₀·cos(ω_d·t) to the homogeneous oscillator equation, so the motion is a sum of two pieces: a transient that decays away at the system's natural frequency, and a steady state that oscillates forever at the drive frequency.
Which piece dominates depends on how long you watch and how much damping the system has. Over many cycles, damping kills the transient and only the steady state survives. The system's own natural frequency is no longer audible in the motion; the driver dictates everything.
The amplitude of the steady state depends on how close the drive frequency ω_d is to the system's natural frequency ω₀. Far from resonance the response is small; near resonance it can become enormous. This is how radios tune to stations, how MRI machines excite specific nuclear spins, and how a shattered wine glass meets its end.