It is desired to keep the amplitude distortion at more than 4%. Determine the natural frequency of vibration if damping factor is 0.05.

$N_1$ = 500 rpm

$w_{min} = \frac{2 \pi \times 500}{60} = 52.36$ rad/sec

$N_2$ = 1500 rpm

$w_{max} = \frac{2 \pi \times 500}{60} = 157.07$ rad/sec

$\xi = 0.5$

Error > 4%

i.e. $\frac{Z}{Y} = 1.04$

NOTE: when error < 4%

$\frac{Z}{Y} = 0.96$

$\frac{Z}{Y} = \frac{r^2}{\sqrt{ (1-r^2)^2 + (2 \zeta r)^2}}$

$1.04 = \frac{r}{\sqrt{ (1-r^2)^2 + r^2 }} = r^2$

$\sqrt{ (1-r^2)^2 + r^2} = 0.9245 r^4$

$1 – 2 r^2 + r^4 + r^2 = 0.9245 r^4$

$r^4 – 0.9245 r^4 + r^2 + 1 = 0$

Put $r^2 = r$

$-.075 r^2 – r + 1 = 0$

$\therefore$ $r^2 = 1.088$ OR $r^2 = 12.24$

r = 1.043 OR r = 3.49

$w_n = \frac{w_{min}}{r_{max}}$

$= \frac{52.36}{3.49} = 14.96$ rad/sec