forced vibration
Free vibration is what a system does after one nudge — it rings and fades. Forced vibration is what happens when you keep driving it, pushing a swing on every pass, or feeding alternating current into a circuit. The system no longer relaxes to rest; it settles into the rhythm of whatever is driving it.
The model is m x'' + c x' + k x = F_0 cos(omega t): the same damped oscillator now with a sustained periodic driving force at frequency omega. By the homogeneous-plus-particular structure, the solution has two parts. The transient is the free damped motion (the homogeneous solution), which dies out as e^(-gamma t). The steady-state is the particular solution, a sinusoid at the driving frequency omega — not the system's own natural frequency — with an amplitude and a phase lag set by how close omega is to the natural frequency. After the transient fades, only the steady-state remains.
Two facts make forced vibration central to engineering. First, the long-term response oscillates at the driving frequency, not the natural one — the system forgets its own preferred tune and follows the driver. Second, the steady-state amplitude peaks sharply when omega approaches the natural frequency, the phenomenon of resonance, which can be exploited (radio tuning, musical instruments) or feared (bridges, machinery). Computing the amplitude-versus-frequency response curve is the everyday task of vibration analysis.
For x'' + 4x = 3 cos t, the natural frequency is 2 and the drive is at 1. A trial x_p = A cos t gives -A + 4A = 3, so A = 1; the steady-state is x_p = cos t, oscillating at the DRIVING frequency 1, not at 2.
After transients die, the system oscillates at the driving frequency, not its own natural one.
Transient and steady-state both exist from t = 0; the transient is not 'before' the steady-state in time — they are superposed, and the transient simply becomes negligible once e^(-gamma t) has decayed.