Question 4.5: It was observed that a machine with a mass M = 600 kg has an...
It was observed that a machine with a mass M = 600 kg has an amplitude of vibration of 0.01 m when the operating speed is 10 Hz. If the machine iscritically damped with an equivalent stiffness coefficient k of 10 × 10³ N/m, determine the amount of unbalance me.
The forcing frequency ω_{f} is
ω_{f} = 2πf = 2π(10) = 20π = 62.832 rad/s
The circular frequency ω of the machine is
ω = \sqrt{\frac{k}{M}} = \sqrt{\frac{10 × 10³}{600}} = 4.082 rad/s
The frequency ratio r is then given by
r = \frac{ω_{f}}{ω} = \frac{62.832}{4.082} = 15.392
Since the machine is critically damped, ξ = 1. The magnification factor β_{r} can be determined by using Eq. 4.75 as
β_{r} = \frac{r²}{\sqrt{(1 − r²)² + (2rξ)²}} = \frac{(15.392)²}{\sqrt{[1 − (15.392)²]² + [2(15.392)(1)]²}} = 0.9958
Thus, the amplitude is
0.01 =(\frac{me}{M}) β_{r} = (\frac{me}{M}) (0.9958)
which implies that
\frac{me}{M}= 0.01004
That is, the amount of unbalance me is
me = 0.01004(M) = 0.01004 × 600 = 6.0253 kg · m