GATE 2023 AE – Question 51
A single degree-of-freedom spring-mass-damper system has viscous damping ratio of 0.1. The mass is given an initial displacement of 10 cm without imparting any velocity. After exactly two complete cycles of oscillation (i.e., after time 2 dT , where dT is the period of the damped vibration), the amplitude of the displacement is ______ cm. (round off to two decimal place)
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Correct answer: 2.82 to 2.84
Explanation
**Logarithmic decrement.** For a damped free vibration released from rest, the amplitude after each damped period $T_d$ is reduced by the same factor:
$$\frac{x_n}{x_{n+1}}=e^{\delta},\qquad\delta=\frac{2\pi\zeta}{\sqrt{1-\zeta^2}} .$$
**Numbers.** $\zeta=0.1$:
$$\delta=\frac{2\pi(0.1)}{\sqrt{1-0.01}}=\frac{0.6283}{0.99499}=0.6315 .$$
**After two complete cycles** the factor is $e^{-2\delta}$:
$$x=10\ \text{cm}\times e^{-1.263}=10\times0.2829=\mathbf{2.83\ cm}.$$