I have the following spring-mass ODE solutions: $$1:\;\;\;x(t)=-3\sin(2t)+4\cos(2t)+12t\sin(t)$$ $$2:\;\;\;x(t)=6e^{-t}\cos(3t)-3e^{-t}\sin(2t)+40\sin(7t)$$ How is is possible to figure if each one is damped (and the kind of damping?), if there is resonance, if there is a transient component, and if there is a steady state component?
I'm going to assume that both are damped (not sure though) so for the first one I think I can tell that there is resonance because of the t in the 3rd term, however, I am not sure about the other components. For the second one, I don't see anything that points to resonance.
I'm just not sure how to pick these problems apart if someone could explain it clearly I would be forever grateful to you!
The first function solves $\ddot{x}+4x=24\cos t$, which is undamped forced harmonic oscillation. Of course, any $x$ would be consistent with either damped or undamped motion depending on the forcing.
The second function is $x=fe^{-t}+40\sin 7t$ with $f:=6\cos 3t-3\sin 2t$. You'll want to double-check my calculations, but $$\ddot{x}+\gamma\dot{x}+\omega^2 x=(\ddot{f}+(\gamma-2)\dot{f}+(1-\gamma+\omega^2) f)e^{-t}+40(\omega^2-49)\sin 7t+280\gamma\cos 7t.$$The $e^{-t}$ coefficient is $$-18(\gamma -2)\sin 3t-6(\gamma -2)\cos 2t+6(-8-\gamma+\omega^2)\cos 3t-3(-3-\gamma+\omega^2)\sin 2t.$$Damping implies $\gamma >0$. With $\gamma=2$, we can delete all but one term through a suitable choice of $\omega^2$. However, any choice of the parameters requires the forcing to include more than just frequency-$7$ terms, and a three-frequency forcing is consistent with $\gamma=0$.