Hello there.
So I am a mechanical engineering student doing mechatronics and I have encountered a problem in a question I was attempting. Attached is the question and the transfer function i have derived. My specific problem is that I have never dealt with a 3rd degree characteristic equation and I am struggling to answer the question. All of the Controller design questions that I have dealt with have all had a 2nd degree characteristic equation which I can work with. All I need is guidance on how to deal with this type of transfer function. Any help will be appreciated.
Question transfer function

 

All of the transfer functions that you will ever see, besides all of the one you won't can be decomposed into 1st and 2nd order sections. The 1st order section corresponds to a pole on the negative real axis. This pole has a Q of 0.5. You know this because:

\( Q\;=\;\cfrac{1}{2cos\psi},\text{ where }\psi\text{ is the angle the pole makes with the negative real axis.} \)

If you know the transfer function of the 1st order section, you can factor that out of the 3rd order transfer function, and you will be left with a 2nd order section. These you know how to deal with. If you don't know about how to create a Root-Locus plot, now would be a good time to learn.

16.5: Loci of Roots for Third Order Systems - Engineering LibreTexts
Control Systems/Root Locus - Wikibooks, open books for an open world

There will be either two real roots or a complex conjugate pair. The thing you have to watch out for is the condition where the roots of a conjugate pair migrate to the right half plane. This can't happen in a 2nd order system.

 

If you have access to a simulator, you can simulate the transfer function directly.



That is a pretty nasty peak at the low low frequency of 200 millihertz or about 1 radian per second.