GATE 2022 ME (ME2) – Question 31
For a dynamical system governed by $\ddot x+2\zeta\omega_n\dot x+\omega_n^2x=0$, the damping ratio $\zeta$ is equal to $\frac{1}{2\pi}\log_e2$. The displacement x of this system is measured during a hammer test. A displacement peak in the positive displacement direction is measured to be 4 mm. Neglecting higher powers (>1) of the damping ratio, the displacement at the next peak in the positive direction will be ______ mm (in integer).
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Correct answer: 2
Explanation
**Step 1: logarithmic decrement.** For light damping, with the damping ratio small (neglecting powers of $\zeta$ above 1),
$$\delta=\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}\approx2\pi\zeta .$$
**Step 2: substitute** $\zeta=\dfrac1{2\pi}\ln2$:
$$\delta\approx2\pi\cdot\frac{\ln2}{2\pi}=\ln2 .$$
**Step 3: ratio of successive peaks.** $\delta=\ln(x_n/x_{n+1})$, so $x_n/x_{n+1}=2$.
**Step 4: next peak.**
$$x_{n+1}=\frac{4}{2}=\mathbf{2\ mm}.$$