Consider the undamped single-degree-of-freedom system shown below that is excited by harmonic forcing, where ω is the frequency of excitation:
Shown below is an animation showing the amplitude/frequency plot, |X(ω)|, for the particular solution response and the corresponding time history for xP(t) as we sweep through the frequency of excitation ω:
From this animation, you can see the following characteristics of the particular solution discussed in lecture:
- For low frequencies of excitation (ω << ωn), the response is in-phase with the forcing, and the response amplitude approaches that equilibrium displacement due to a constant forcing f0 as ω goes to zero.
- For high frequencies of excitation (ω >> ωn), the response is 180° out-of-phase with the forcing, and the response amplitude tends to zero as ω becomes large.
- For near the “resonant” frequency (ω = ωn), the response amplitude becomes large (in theory, becomes infinite) and the response changes from “in-phase” to “out-of-phase” with respect to the forcing.