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

Answer:

$\zeta =0.69026$

Explanation:

We have given no more than 5% overshoot

So maximum overshoot = 5 % =0.05

Maximum overshoot is given by $M_P=e^\frac{-\pi \zeta }{\sqrt{1-\zeta ^2}}$

Here $\zeta$ is the damping ratio

We know that $cos\Theta =\zeta$

So $\sqrt{1-\zeta ^2}=sin\Theta$

So $0.05=e^{-\pi cot\Theta }$

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

$0.954={ cot\Theta }$

$\Theta =46.3485$

$cos\Theta =0.69026$

$\zeta =0.69026$