Confusion regarding finding closed loop transfer function and damping constant from open-loop transfer function

Question

I have second order transfer function $$ G(s) = \frac{1.247}{s^2+9.76s+23.8}$$ G(s) is in the forward path of a unity feedback system.

How do I find the closed loop transfer function and damping constant from this? Now I know that the equation to find the closed loop transfer function is $$TF = \frac{G}{1+G}$$

If I use that here, I get $$TF = \frac{1.247}{s^2+9.76s+25.047}$$ and It is not in the form of a standard second order system which is $$TF = \frac{\omega_n^2}{s^2+2\delta\omega_ns+\omega_n^2}$$

That is, the numerator in TF is not 25.047. So How do I calculate damping ratio in this situation? Can I still take 25.047 as the square of natural frequency and equate that with the coefficient of s to find damping ratio?

Answer 1

As user215805 stated, the standard form is $$TF=K\frac{\omega^2_n}{s^2+2\delta\omega_ns+\omega^2_n}$$ And you have $$TF=\frac{1.247}{s^2+9.76s+25.047}$$ This means that in your case $$\omega^2_n=25.047$$ Which you already know, but notice that from the standard form you can deduce that $$K\omega^2_n=1.247$$ And simply $$K=\frac{1.247}{\omega^2_n}=\frac{1.247}{25.047}≈0.05$$ So now you have $$TF=0.05\frac{25.047}{s^2+9.76s+25.047}$$ Similarly you can deduce that $$2\delta\omega_n=9.76$$ And therefore $$\delta=\frac{9.76}{2\omega_n}≈0.975$$ And so you have reached the standard form $$TF=0.05\frac{25.047}{s^2+2(0.975)\sqrt{25.047}s+25.047}$$