Question

A 110 kg machine is mounted on an elastic foundation of stiffness 2 x106 When operating at 150 rad, the machine is subject to a harmonic force of magnitude 1500 N. The steady-state amplitude of the machine is measured as 1.9 mm. What is the damping ratio of the foundation?

Answer 1

Natural frequency, $\omega_n=\sqrt{\frac{k}{m}}=\sqrt{\frac{2*10^6}{110}}=134.84\frac{rad}{s}$

$A=\frac{F_0/k}{\sqrt{[1-(\frac{\omega}{\omega_n})^2]^2+[\frac{2\xi\omega}{\omega_n}]^2}}$

$1.9*10^{-3}=\frac{1500/2*10^6}{\sqrt{[1-(\frac{\omega}{\omega_n})^2]^2+[\frac{2\xi\omega}{\omega_n}]^2}}$

${[1-(\frac{\omega}{\omega_n})^2]^2+[\frac{2\xi\omega}{\omega_n}]^2}=\frac{1500}{2*10^6*1.9*10^{-3}}=\frac{15}{38}$

$[2.2248*\xi]^2=0.39473-0.05640$

$2.2248*\xi=\sqrt{0.33834}$

$\xi=0.2614$