GATE 2026 AE – Question 33
The response $x(t)$ of a freely vibrating single degree of freedom underdamped system is given below. In the equation, $A$ and $\phi$ are constants. The damping ratio of the system is ________ (rounded off to 3 decimal places).
$x(t) = A e^{-5t} \sin(10t + \phi)$
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Show answer and explanation
Correct answer: 0.437 to 0.457
Explanation
Compare with $x = A e^{-\zeta\omega_n t}\sin(\omega_d t + \phi)$: $\zeta\omega_n = 5$ and $\omega_d = \omega_n\sqrt{1 - \zeta^2} = 10$. Then $\omega_n^2 = 5^2 + 10^2 = 125$, $\omega_n = 11.18$, and $\zeta = 5/11.18 = 0.447$.