Question

A mass of 50 kg is suspended from a spring of stiffness 10 kN/m. It is set oscillating and it is observed that two successive oscillations have amplitudes of 10 mm and 1 mm. Determine the damping ratio.

• A. 0.315
• B. 0.328
• C. 0.344
• D. 0.353

Answer 1

The correct option is C 0.344

For successive amplitudes $m=1$
amplitude reduction factor
$=ln\left(\frac{x_1}{x_2}\right)=ln\left(\frac{10}{1}\right)=ln10=2.3$
amplitude reduction factor $=\frac{2\pi \delta m}{\sqrt{1-{\delta }^2}}$
$\Rightarrow \frac{2\pi \delta m}{\sqrt{1-{\delta }^2}}=2.3$
squaring both sides
$\frac{39.478{\delta }^2}{1-{\delta }^2}=5.29\Rightarrow {\delta }^2=0.118\Rightarrow \delta =0.344$