Question

Overshoot for an underdamped system is defined as the maximum displacement of the system at the end of its first half cycle. What is the minimum damping ratio for a system such that it is subject to no more than 5% overshoot?

Answer 1

`\zeta =0.69026`

Step-by-step explanation:

We have given no more than 5% overshoot

So maximum overshoot = 5 % =0.05

Maximum overshoot is given by
`M_P=e^(-\pi \zeta )/(√(1-\zeta ^2))`

Here
`\zeta` is the damping ratio

We know that
`cos\Theta =\zeta`

So
`√(1-\zeta ^2)=sin\Theta`

So
`0.05=e^(-\pi cot\Theta )`

`ln0.05={-\pi cot\Theta }`

`0.954={ cot\Theta }`

`\Theta =46.3485`

`cos\Theta =0.69026`

`\zeta =0.69026`