Question

A vibrating mass of 300 kg mounted on a massless support by a spring of stiffness 40,000 N>m and a damper of unknown damping coefficient is observed to vibrate with a 10-mm amplitude while the support vibration has a maximum amplitude of only 2.5 mm (at resonance). Calculate the damping constant and the amplitude of the force on the base.

Answer 1

Answer:

400 N

Explanation:

$\text { Given: } m=300 \mathrm{~kg}, k=40,000 \mathrm{~N} / \mathrm{m}, \omega_{b}=\omega_{n}(r=1), X=10 \mathrm{~mm}, Y=2.5 \mathrm{~mm}$ .

Find damping constant

$\frac{X}{Y}=\left[\frac{1+(2 \zeta r)^{2}}{\left[\left(1-r^{2}\right)^{2}+(2 \zeta r)^{2}\right.}\right]^{1 / 2} \\ \left.\frac{10}{2.5}=\frac{\left\lceil 1+4\zeta^{2}\right]^{1 / 2}}{4 \zeta^{2}}\right] \\ 16=\frac{1+4 \zeta^{-2}}{4 \zeta^{2}} \\ \zeta^{2}=\frac{1}{60}=\frac{c^{2}}{4 k m} \\ c=\sqrt{\frac{4(40,000)(300)}{60}} \\ c=894.4 \mathrm{~kg} / \mathrm{s}$

Amplitude of force on base:

$F_{T}=k Y r^{2}\left[\frac{1+(2 \zeta r)^{2}}{\left(1-r^{2}\right)^{2}+(2 \zeta r)^{2}}\right]^{1 / 2}$

substituting the values in above formula we get

F_T = 400 N