Consider a single degree of freedom system comprising a mass M, supported on a spring and a dashpot as shown in the figure.

If the amplitude of the free vibration response reduces from 8 mm to 1.5 mm in 3 cycles, the damping ratio of the system is______ (round off to three decimal places).

Answer :

Correct answer is : 0.088

$\frac{x_0}{x_3}=\frac{8}{1.5}=\frac{80}{15}$

$\frac{x_0}{x_1}\cdot \frac{x_1}{x_2}\cdot \frac{x_2}{x_3}=\frac{80}{15}$

$e^{\delta }\cdot e^{\delta }\cdot e^{\delta }=\frac{80}{15}\Rightarrow e^{3\delta }=\frac{80}{15}$

$3\delta =\ln \left(\frac{80}{15}\right)\Rightarrow \delta =0.55799$

$\frac{2\pi \xi }{\sqrt{1-{\xi }^2}}=0.5579\Rightarrow \xi =0.088$