forced vibrations of damped single degree freedom systems?
in a spring mass damper system subjected to a periodic force. the response of the system only depends on the force and the restoring forces caused due to the spring is ignored because the damper is assumed to have taken care of it. now it's here that my confusion originates. in a damped free vibration (under damping) when a initial displacement is given to the mass, the mass vibrates at a damping frequency and the amplitude eventually vanishes. the mass completes a few oscillations before the vibration completely dies out.
but in a forced vibration with damping, when the force is applied the vibrations take place at the frequency of the periodic force, so what happens to the oscillations caused due to the damper(in under damping). in my book says those vibrations due to damping die out quickly, but they atleast need to oscillate atleast 2-3 times before dying out, and how can they do that while the periodic force is still being applied. how can the mass vibrate at two frequencies( damping and periodic force's) at the same time?