Question

$\text{In a vibration isolation system, the transmissibility will be equal to unity for all values of damping factor, if}\frac{\omega }{{\omega }_n}\text{is}$

• A. equal to one
• B. $\text{equal to}\sqrt{2}$
• C. $\text{less than}\sqrt{2}$
• D. $\text{greater than}\sqrt{2}$

Answer 1

The correct option is B $\text{equal to}\sqrt{2}$

$T.R=\frac{F_t}{F_0}=\frac{\sqrt{1+{\left(2\xi \frac{\omega }{{\omega }_n}\right)}^2}}{\sqrt{{[1-{\left(\frac{\omega }{{\omega }_n}\right)}^2]}^2}+{\left(2\xi \frac{\omega }{{\omega }_n}\right)}^2}$

$\therefore 1+{\left(2\xi \frac{\omega }{{\omega }_n}\right)}^2={[1-{\left(\frac{\omega }{{\omega }_n}\right)}^2]}^2+{\left(2\xi \frac{\omega }{{\omega }_n}\right)}^2$

$\therefore 1={[1-{\left(\frac{\omega }{{\omega }_n}\right)}^2]}^2$

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

$\therefore {\left(\frac{\omega }{{\omega }_n}\right)}^2=2\text{or}0$

But it cannot be zero

$\therefore {\left(\frac{\omega }{{\omega }_n}\right)}^2=2$

$\therefore \frac{\omega }{{\omega }_n}=\sqrt{2}$