Hi there Froskoy,
First thing is you can analyze this circuit in the frequency domain and come out with a transfer function H(s).
Once you have that, it will have a denominator in this form:
a*s^2+b*s+c
What you can do next is get the s^2 term alone without any coefficient by dividing by 'a', this gives us:
s^2+s*b/a+c/a
which with simple change of variables can be written:
s^2+B*s+C
But this can also be written in this new form:
s^2+2*d*wn*s+wn^2
and to get this far all we had to do was equate B and C as:
B=2*d*wn
C=wn^2
This form is especially useful because 'd' is the damping factor, so all we have to do now is solve for the damping factor 'd' knowing B and C. Since B is made up of factors of wn and 2 besides 'd', if we divide by the square root of C we'll get 2*d, from which all we have to do is divide by 2 and we'll have the damping factor 'd' and can use the greater than or less than 1 criterion to determine what form of response we'll have.
So the damping factor comes out to:
d=B/(2*sqrt(C))
So the idea is to analyze the circuit for it's transfer function, then simplify the denominator and solve for the damping factor. This works for any second order transfer function regardless of the connection of the components.