tjr39
Sep2-09, 10:02 PM
1. The problem statement, all variables and given/known data
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.
2. Relevant equations
3. 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.
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.
2. Relevant equations
3. 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.