1903 Undamped Forced Systems

We now look at the pure resonant case for a second-order LTI (Linear Time Invariance) DE. We will use the language of spring-mass systems in order to interpret the results in physical terms, but in fact the mathematics is the same for any second-order LTI DE for which the coefficient of the first derivative is equal to zero.

The problem is thus to find a particular solution the DE $$x''+\omega_0^2x=\cos \omega t\tag{1}$$ Complex replacement: $$z''+\omega_0^2z=e^{i\omega t}\\ x=Re(z)$$ Characteristic polynonial: $$p(s)=s^2+\omega_0^2\\ p'(s)=2s\\ p(i\omega)=\omega_0^2-\omega^2\\ p'(i\omega)=2i\omega$$ Exponential Response formula: $$z_p=\begin{cases} \frac{e^{i\omega t}}{p(i\omega)}&=\frac{e^{i\omega t}}{\omega_0^2-\omega^2}&\text{if }\omega\neq\omega_0\\ \frac{te^{i\omega t}}{p'(i\omega)}&=\frac{te^{i\omega t}}{2i\omega}&\text{if } \omega=\omega_0 \end{cases}\\ x_p=\begin{cases} \frac{\cos \omega t}{\omega_0^2-\omega^2}&\text{if }\omega\neq\omega_0\\ \frac{t\sin \omega_0 t}{2\omega_0}&\text{if } \omega=\omega_0 \end{cases}$$

Resonance and amplitude response of the undamped harmonic oscillator

In $x_p$ the amplitude $=A=A(\omega)=\vert\frac{1}{\omega_0^2-\omega^2}\vert$ is a function of $\omega$.
The plot below shows $A$ as a function of $\omega$. Note, it is similar to the damped amplitude response except the peak is infinitely high. As $\omega$ gets closer to $\omega_0$ the amplitude increases. Tplot is output amplitude vs. input frequency.

When $\omega=\omega_0$ we have $x_p=\frac{t\sin \omega t}{2\omega_0}$. This is called pure resonance (like a swing). The frequency $\omega_0$ is called the resonant or natural frequency of the system.
In the plot below notice that the response is oscillatory but not periodic. The amplitude keeps growing in time (caused by the factor of $t$ in $x_p$).
The plot is output vs. time (for a fixed input frequency).

Reconciling the Resonant and Non-resonant Solutions

Let's find another particular solution to $(1)$. $$x_c=\frac{\cos \omega_0t}{\omega_0^2-\omega^2}$$ is a complementary solution because it is the solution to the associated homogeneous equation.

$x_p-x_c$ still is a particular solution to $(1)$, so a new particular solution is $$x_p=\frac{\cos \omega t}{\omega_0^2-\omega^2}-\frac{\cos \omega_0t}{\omega_0^2-\omega^2}$$

When $\omega$ goes to $\omega_0$, $$\begin{aligned} &\lim_{\omega \to \omega_0} \frac{\cos \omega t}{\omega_0^2-\omega^2}-\frac{\cos \omega_0t}{\omega_0^2-\omega^2}\\ &=\lim_{\omega \to \omega_0} \frac{-\omega\sin \omega t}{-2\omega}\tag*{L'Hôpital's rule}\\ &=\frac{t\sin \omega_0 t}{2\omega_0} \end{aligned}$$ Notes, the variable is $\omega$ not $t$.