Electrical – How to find damping ratio of a 4th order system

damping-factor

For example in this 4th order transfer function how the damping ratio would be calculated?
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in fact I` m encountered with this problem: If I didnt realize the concept of problem plz guide me:

problem: calculate gain magnitude (k) if damper ratio of closed loop system is 0.7 (it means zita=0.7):

.

open loop transfer function:

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the expansion of open loop system fraction:

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and now how much is zita?
quadratic faraction isn t standard form because there s a 's' in numerator, am I right?

how much is k due to the zita has a magnitude of 0.7?
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Best Answer

The damping ratio is a parameter, usually denoted by ζ (zeta),1 that characterizes the frequency response of a second order ordinary differential equation.

The quote above is taken from Wikipedia: Damping ratio. In other words it relates to a 2nd order transfer function and not a 4th order system. Having said that, if it is possible to reduce the denominator to two multiplying equations each of the form: -

\$s^2 +2s\zeta \omega_n + \omega_n^2\$ (where \$\zeta\$ is damping ratio and \$\omega_n\$ is natural resonant frequency)

Then it would have some meaning.

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