Forced vibrations of damped single degree freedom systems?

AI Thread Summary
In a spring mass damper system subjected to a periodic force, the response is influenced by the periodic force while the effects of the spring are considered negligible due to the damper's role. The confusion arises in understanding how the system oscillates at both the damping frequency and the frequency of the applied force. During damped free vibrations, the mass oscillates and the amplitude decreases over time, while in forced vibrations, the system responds primarily at the frequency of the external force. The damping effects do cause oscillations that eventually diminish, but they may still occur alongside the forced vibrations. Clarification on this dual-frequency behavior is essential for understanding the dynamics of damped forced vibrations.
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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?
 
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I'm sorry I don't really understand your question but have a read of this wikipedia page (in case you haven't done so already):

http://en.wikipedia.org/wiki/Damping

Hope it helps!
 
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