Question

A centrifugal pump of mass 500 kg is mounted on a flexible support causing the support to deflect by 6 mm. The ratio of damping coefficient to the critical damping coefficient for the support is found to be 0.15. When the pump running at 500 rev/min, an amplitude of vibration of 5 mm is measured. Analyse the value of disturbing force created by the pump during its operation.

Please show answer step by step.

Answer 1

Give, m = 500kg               $\delta$ = 6mm               $\zeta$ = 0.15               N = 500rpm                    A = 5mm

We know for a mass supported by spring, mg = k$\delta$$\Rightarrow$k = mg/$\delta$$\Rightarrow$ k = 817.5 KN/m

We Know, $\omega$ = 2$\pi$N/60 = 52.36 rad/sec

Taking for max response as r is not given

For max response, r = $\sqrt{1-2\zeta^{2} }$ $\Rightarrow$ r = 0.97724

We know, r = $\omega$ /$\omega_{n}$   $\Rightarrow$ $\omega_{n}$ = 52.36/0.97724 $\Rightarrow$ $\omega_{n}$= 53.58 rad/sec

c = 2m $\omega_{n}$ x $\zeta$ = 9912.3 N/m/s

We Know for a mass supported by spring,

A= F/k (k - mw2)2 + (cw)2

$\Rightarrow 5 \times 10^{-3} = \frac{F/817500}{\sqrt{(817500-500\times 52.36 ^{2})^{2}+(9912.3\times 52.36)^{2}}}$

$\Rightarrow 5 \times 10^{-3}\times758612.815 = F/817500$

$\Rightarrow F = 3793.064 \times 817500 = 3100.82MN$