Resonance problem involving Laplace transformations

[tjr39]

Homework Statement

The sine wave $$sin(t)$$ will only drive the harmonic oscillator $$y'' + \omega ^2 y$$ into resonance when $$\omega = 1$$. For what values of $$\omega$$ will the half- and full-wave rectified sine waves drive the harmonic oscillator into resonance.

Homework Equations

The Attempt at a Solution

Starting with the half-wave rectified sine wave;

Taking the Laplace transform of both sides and rearranging for Y(s);

$$Y(s)= \frac{1+e^{-s\pi}}{(s^2+1)(1-e^{-s2\pi})(s^2+ \omega ^2)} + \frac{sy(0)+y'(0)}{s^2+\omega^2}$$


From here I think I need to find the poles of Y(s) but I am unsure what to do with the $$(1-e^{-s2\pi})$$ in the denominator of the first term. Similar problem when looking at the full-wave rectified sine curve.

 

[foxjwill]

Ok, first of all, this is the math part of the forum, so if you want help solving an ODE, you'll have to actually tell us what it is -- we won't necessarily know what you mean simply by giving a physical description.