I'm trying to grade a differential equations quiz about forced oscillations. Because I remember very little from my differential equations course, I'm going to appeal to MSE. The question is as follows:
For a forced mass-spring system with equation $mx'' + bx' + kx = F\cos{\omega t}$, investigate the possibility of practical resonance of this system. Find the amplitude $C(\omega)$ of steady periodic forced oscillations with frequency $\omega$. Sketch the graph $C(\omega)$ and find the practical resonance frequency (if any). $m=1, b=6,k=45, F=50$.
Here is my interpretation of what is happening:
First, we have to realize that the amplitude $C(\omega)$ is given by $$C(\omega)=\frac{F}{\sqrt{(k-m\omega^2)^2+(b\omega)^2}}$$
Then we see if $C(\omega)$ is maximized anywhere in the positive reals by setting $C'(\omega)=0$ and solving. After doing so, we get solutions $\omega=0,\pm\sqrt{27}$. Since it only makes sense to analyze positive frequencies, we have that $\omega=\sqrt{27}$. This is called our practical resonance frequency $(\omega^*)$.
Here's where I start to get a bit fuzzy (Let me know if the following is wrong):
We say that the system is $\underline{\text{resonating}}$ if $\omega^*\approx\omega_0$, where $\omega_0 = \sqrt{\frac{k}{m}}$. Since $\omega_0 = \sqrt{45}$ in this problem, and $\sqrt{45}$ is not that close to $\omega^*=\sqrt{27}$, we conclude that the system is not resonating.
Does this idea sound right? Any help would be appreciated.
If you plot the response you get the graph below. Alpha agrees the maximum is at $\omega=\sqrt{27}$. The fact that the maximum response is higher than the response at zero frequency indicates that it is underdamped and resonating. I wouldn't look at how the damped and undamped frequencies relate, rather whether the response peaks. This is a rather weak resonance as shown by the low ratio of maximum response to response at zero frequency (about $1.3$) and by the width of the peak.