I am given a transfer function of a second-order system as: $$G(s)=\frac{a}{s^{2}+4s+a}$$
and I need to find the value of the parameter $a$ that will make the damping coefficient $\zeta=.7$. I am not sure how to do this but I might have found something that might have helped so I am going to take a stab at it. I found a transfer function in the book of a second order spring-mass-damper system with an external applied force in the book as: $$G(s)=\frac{a}{m\omega_{n}^{2}}(\frac{\omega_{n}^{2}}{s^{2}+2\zeta \omega_{n}+\omega_{n}^{2}})$$
I was thinking that I could just write $\omega_{n}^{2}=a$ and $2\omega_{n}\zeta=4$. Then I could just solve for $a$. Is this possible?
Answer
You have the answer.
Consider $2*\zeta*\omega_n = 4$. $\zeta = 0.7$. $\omega_n^2 = a$ What value of $\omega_n$ (or $a$ in your case) satisfies this?
Roughly 8.18 is your answer.
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