Damping Coefficient and Resonant Frequency of a 2nd order circuit

[gfd43tg]

Hello,

I am having major confusion for how to find the damping coefficient, α, and the resonance frequency, ωo, for a second order circuit, in general. I know that there are tables for standard circuits like a series RLC and parallel RLC giving those values, but I am certain that I need to know how to solve for them in general.

The professors notes did not even show how to do the series RLC circuit without a ton of handwaving to the point of being unfollowable. The textbook (he wrote it) does not have it in full detail either. It just essentially presents the results. He made it very clear to us that we need to know how to do it for either 2 inductors separated by resistors or 2 capacitors separated by resistors, and tables for those are not in the textbook.

I know that once I clean up the circuit through my KCL equations, I will eventually get a 2nd order DE that looks like this

v'' + av' + bv = c

From here, I am stuck with how I should proceed to find those parameters so I can determine if my circuit is over or underdamped, or critically damped.

Thank you

 

[gfd43tg]

To be clear, I absolutely need to make a table that is the same as this (from textbook)

For the first two circuits in this figure

So I can put them on my cheat sheet. I always wondered why go through so much effort to make these tables, but then explicitly tell us that neither are going to be on the exam, and it will be something else, but I digress.