Question

A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to the instrument. Assuming be transmitted to the intrument. Assuming that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N.m) of the required mounting pad is

Answer 1

Given,
$\omega =2\pi f=2\pi \times 60=120\pi ,m=1kg$
$\frac{F_{TR}}{F_0}=\frac{\sqrt{1+(2\xi r{)}^2}}{\sqrt{(1-r^2{)}^2+(2\xi r{)}^2}}$
No damping,
$\therefore \xi =0$
$\frac{F_{TR}}{F_0}=0.05$
$0.05=\frac{1}{r^2-1}$
$r^2-1=20$
$r=\sqrt{21}$
$\frac{\omega }{{\omega }_n}=\sqrt{21};\frac{120\pi }{{\omega }_n}=\sqrt{21}$
${\omega }_n=82.266$ rad/s
$\sqrt{\frac{k}{m}}=82.266$
$\sqrt{\frac{k}{1}}=6767.728$
$k=6767.72$ N/m