Question

If the maximum static deflection of a shaft is 5 mm, then the estimated critical speed using Rayleigh–Ritz method is ____________ RPM (rounded off to nearest integer).

Hint:

In Rayleigh–Ritz critical speed estimation, only the maximum static deflection is needed. Smaller deflection means higher critical speed.

Answer 1

Correct Answer: 422

For a rotating shaft, Rayleigh’s method gives the first critical speed as:
\[ \omega_n = \sqrt{\frac{g}{\delta_{\max}}} \]
where:
\( g = 9.81\ \text{m/s}^2 \)
\( \delta_{\max} = 5\ \text{mm} = 0.005\ \text{m} \)
Substitute the values:
\[ \omega_n = \sqrt{\frac{9.81}{0.005}} \]
\[ \omega_n = \sqrt{1962} = 44.29\ \text{rad/s} \]
Convert radial speed to RPM using:
\[ N = \frac{\omega_n \times 60}{2\pi} \]
\[ N = \frac{44.29 \times 60}{6.283} \]
\[ N = 423.16\ \text{RPM} \]
Rounded to the nearest integer:
\[ N \approx 423\ \text{RPM} \]

Final Answer: 423\ \text{RPM}