driven oscillation
Driven oscillation is what happens when you keep pushing an oscillator at a steady rhythm from the outside. Picture pushing a child on a swing again and again, a washing machine shuddering as its motor spins, or a loudspeaker cone shoved back and forth by the electrical signal feeding it. It answers: how does an oscillator respond when an external periodic force keeps supplying energy?
Precisely, an external periodic force such as F(t) = F_0 cos(omega_d t) is applied at a driving frequency omega_d. At first there is a messy transient, but after that dies out the system settles into a steady-state oscillation at the driving frequency omega_d, not at its own natural frequency. The steady amplitude depends on how close omega_d is to the system's natural frequency omega_0, and also on how much damping there is.
The response grows dramatically when the driving frequency approaches the natural frequency; that near-match is resonance, the reason a well-timed push builds a swing higher and higher. One point beginners often miss: in the steady state the oscillator moves at the driver's frequency, however far that is from its own preference. What changes with the mismatch is the amplitude of the response and the phase lag between the push and the motion.
Push a swing (natural period about 2 s) once every 2 s, in time with its motion, and it builds up hugely. Push at some other rhythm, say every 0.7 s, and the swing just jitters with a small, awkward response.
In steady state a driven oscillator moves at the driving frequency; its amplitude peaks near the natural frequency.
In steady state the oscillator moves at the driving frequency, not its own natural frequency; the mismatch changes the amplitude and the phase lag, not the frequency of the response.