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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?

Respuesta :

Answer:

[tex]\zeta =0.69026[/tex]

Explanation:

We have given no more than 5% overshoot

So maximum overshoot = 5 % =0.05

Maximum overshoot is given by [tex]M_P=e^\frac{-\pi \zeta }{\sqrt{1-\zeta ^2}}[/tex]

Here [tex]\zeta[/tex] is the damping ratio

We know that [tex]cos\Theta =\zeta[/tex]

So [tex]\sqrt{1-\zeta ^2}=sin\Theta[/tex]

So [tex]0.05=e^{-\pi cot\Theta }[/tex]

[tex]ln0.05={-\pi cot\Theta }[/tex]

[tex]0.954={ cot\Theta }[/tex]

[tex]\Theta =46.3485[/tex]

[tex]cos\Theta =0.69026[/tex]

[tex]\zeta =0.69026[/tex]

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