\(\omega_a=4\omega_d\).
\({\cfrac { d^2 T }{ d\, t^2 }}=-\omega_d T_o\)
\(\xi^2=\cfrac{1}{2}(1-\cfrac{\omega_a}{\cfrac{1}{4}T_o})\)
\(\xi\) can be reduced to zero and beyond.
For
\(\omega_a>\cfrac{1}{4}T_o\) for a rectified waveform driving force,
\(\xi\) is complex. What happens when \(\xi\) is complex?
Consider the energy loss in a damped system,
\(F_{loss}=-2p.\cfrac{d\,x}{d\,t}\)
\(E_{loss}=\int{-F_{loss}}dx=\int{2p.\cfrac{d\,x}{d\,t}}dx=\int{2\xi\omega_o\cfrac{d\,x}{d\,t}}\cfrac{d\,x}{d\,t}dt\)
\(\cfrac{d\,E_{loss}}{d\,t}=2\xi\omega_o\left\{\cfrac{d\,x}{d\,t}\right\}^2\)
Since \(\cfrac{d\,x}{d\,t}\) has a time component, a complex \(i\xi\) would mean a \(\pi/2\) phase delay in \(\cfrac{d\,E_{loss}}{d\,t}\).
\(2i\xi\omega_o\left\{\cfrac{d\,x}{d\,t}\right\}^2=2\xi\omega_o\left\{\cfrac{d\,x}{d\,t}\right\}^2e^{i\pi/2}=2\xi\omega_o\left|\cfrac{d\,x}{d\,t}\right|^2e^{i2wt}e^{i\pi/2}=2\xi\omega_o\left|\cfrac{d\,x}{d\,t}\right|^2e^{i(2wt+\pi/2)}\)
