Springs / Differential Equations?

Question

Consider an undamped spring which is being periodically forced with amplitude F and frequency w (to avoid confusion w is always omega - in this question). The equation governing the spring is
x(with two dots on top of the x) + k^(2)x = Fsin wt
By considering the form of the solution for different valus of k and w and, in particular, any special cases. Can you explain the phenomena of resonance?
Hint : Don't put values in for k and w. Look for a value of w in terms of k.

Harder - try repeating the question with a damping term.

Answer 1

x'' + k^2 x = F sin ωt

The solution to the homogenous DE x'' + k^2 x = 0 is
x = A sin kt + B cos kt

A particular solution for the non-homogeneous term F sin ωt will be of form
x_p = C sin ωt + D cos ωt
(which is obviously bounded in amplitude)
EXCEPT if sin ωt or cos ωt terms exist in the complementary solution above. This will happen if ω = (±) k.
In the case where ω = k (or -k), the particular solution becomes
x_p = Ct sin ωt + Dt cos ωt
The average amplitude of this is therefore proportional to t. So we will get a sinusoidal solution with increasing amplitude, which will dominate the motion as t becomes large; this is the cause of resonance.