I'm trying to find the amplitude of steady state response of the following differential equation:
$$\ddot{x}+2p\dot x + {\omega_0}^2x=\cos(\omega t)$$
A particular solution is
$$x_p=\Re{\dfrac{e^{i\omega t}}{\omega_0^2 – \omega^2 + i2p\omega}} $$
The amplitude at steady state is then $$A=\dfrac{1}{\sqrt{(\omega_0^2 – \omega^2)^2 + (2p\omega)^2}}$$
The denominator has minimum value when $\omega^2 =\omega_0^2 – 2p^2 $:
$$A=\dfrac{1}{2p\sqrt{\omega_0^2-p^2}}$$
This expression seems to suggest that the amplitude goes to infinity as $p$ approaches $\omega_0$.
But amplitude has to be finite(from other examples of LRC tank circuit etc).
Pretty sure I'm wrong but not able to see where. Any help?
Best Answer
Notice something very important...
When $p=w_0$, then you have (per your previous equation)
$w^2=-w_0^2$, which is to me an impossibility.
In the more correct order, since you have the previous equation, you have constraints on the system that prevents this to happen.
Put into other words, you find a solution for the maximum of $A$ that has no sense existing with $p$ close to $w_0$...