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How to find the value of the parameter $a$ in this transfer function?

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: $$G(s)=\frac{a}{m\omega_{n}^{2}}(\frac{\omega_{n}^{2}}{s^{2}+2\omega_{n}\zeta+\omega_{n}^{2}})$$

I was thinking that I could writeZx: $$2\omega_{n}^{2}\zeta=4$$ and $$\omega_{n}^{2}=a$$ asd And then solve for a. Would this be possible?

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migrated from electronics.stackexchange.com Mar 6 '12 at 16:40

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marked as duplicate by Qmechanic, dmckee Mar 6 '12 at 18:19

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This is off topic for this forum without a direct electronics application. –  tyblu Mar 1 '12 at 4:59
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Canonical 2nd order tf form is close to the factor within the brackets of G(s), so you can do what you propose with two small changes: \$2\omega\zeta s=4\$ (not \$2\omega^2\zeta=4\$). –  tyblu Mar 1 '12 at 5:05
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Err, I made a small error. Oh well. No one will ever see it. ;) –  tyblu Mar 1 '12 at 5:14
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The situation where this question was migrated from electronics and created a duplicate is why cross-posting is discouraged. In the future, please try one site at a time. –  dmckee Mar 6 '12 at 18:22
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1 Answer 1

A general second order equation is given as

                 G(s)= ω^2/(s^2+ 2ζω+ω^2 )

Hence, ω = √a,

   2ζω = 4

Solving yeild a=8.163 rad/sec

So, I guess you were headed in the right direction. :)

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