Question

A precision instrument package (m = 3 kg) needs to be mounted on a surface vibrating at 70 Hz. It is desired that only 10% of the base surface vibration amplitude be transmitted to the instrument. Assume 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 kN/m) of the required mounting pad is

• A. 34.7
• B. 65.2
• C. 72.1
• D. 52.7

Answer 1

The correct option is D 52.7

$\frac{\sqrt{1+2\xi {\left(\frac{\omega }{{\omega }_n}\right)}^2}}{\sqrt{{(1-{\left(\frac{\omega }{{\omega }_n}\right)}^2)}^2+2\xi {\left(\frac{\omega }{{\omega }_n}\right)}^2}}$

$\omega =2\pi f=2\pi \times 70=439.6\text{rad/sec}$

$\xi =\frac{c}{2\sqrt{mk}}=\frac{0}{2\sqrt{mk}}=0(\because \text{no damping)}$

$m=3\text{kg}$

$TR=0.1$

$0.1=\frac{1}{\pm (1-{\left(\frac{\omega }{{\omega }_n}\right)}^2)}$

$\Rightarrow 1-{\left(\frac{\omega }{{\omega }_n}\right)}^2=-10(\because \text{'-' sign only will give posibble values )}$

${\omega }_n^2=\frac{{\omega }^2}{11}=17568$

$k=m{\omega }_n^2=52.7\text{kN/m}$