GATE 2022 AE – Question 62
A damper with damping coefficient, c, is attached to a mass of 5 kg and spring of stiffness 5 kN/m as shown in figure. The system undergoes under-damped oscillations. If the ratio of the 3rd amplitude to the 4th amplitude of oscillations is 1.5, the value of c is ____ Ns/m (rounded off to the nearest integer).
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Correct answer: 20
Explanation
**Logarithmic decrement.** For an under-damped system, successive amplitudes decrease by a constant ratio:
$$\delta=\ln\frac{x_n}{x_{n+1}}=\ln1.5=0.4055 .$$
This is true for any pair of consecutive amplitudes, including the 3rd and the 4th.
**Damping ratio.** From $\delta=\dfrac{2\pi\zeta}{\sqrt{1-\zeta^2}}$:
$$\zeta=\frac{\delta}{\sqrt{4\pi^2+\delta^2}}=\frac{0.4055}{\sqrt{39.478+0.1644}}=\frac{0.4055}{6.2962}=0.0644 .$$
**Damping coefficient.** The critical damping is $c_c=2\sqrt{km}=2\sqrt{5000\times5}=316.2$ Ns/m:
$$c=\zeta\,c_c=0.0644\times316.2=20.4\approx\mathbf{20\ Ns/m}.$$