Question 4.8: A vibrometer is used to measure the amplitude of a vibrating...
A vibrometer is used to measure the amplitude of a vibrating machine. It was observed that the machine is vibrating at a frequency of 16 Hz. The natural frequency and damping ratio of the vibrometer are, respectively, 8Hz and 0.7. The reading of the vibrometer indicates that the amplitude of vibration is 1.5 cm. Determine the correct value for the amplitude of vibration of the machine.
The magnification factor β_{r} is
β_{r} = \frac{r²}{\sqrt{(1 − r²)² + (2rξ)²}}
where ξ = 0.7 and
r = \frac{ω_{f}}{ω} = \frac{16}{8} = 2
It follows that
β_{r} = \frac{(2)²}{\sqrt{[1 − (2)²]² + (2 × 2 × 0.7)²}} = 0.97474
Since z = Y_{0}β_{r} \sin(ω_f t − ψ), the amplitude of oscillation of the machine is
Y_{0} = \frac{1.5}{β_{r}} = \frac{1.5}{0.97474} = 1.538872
The percentage error is 2.5915%. The error in measurement can be reduced by using another vibrometer with lower natural frequency.