All shafts act as beams (simply supported or cantilevers) when stationary. All beams deflect with or without external loads. Thus rotating shafts will also deflect during rotation. The deflection causes unbalancing of the mass of the shaft & mass of the pulleys mounted. The unbalanced mass of the rotating object causes vibration. These vibrations increase with the increase of shaft speed of rotation. The rotational speed becomes numerically equal to the natural frequency of vibration. That speed is the critical speed.
Factors on which critical speed depends
(i) Type of support (simply supported or cantilever type)
(ii) Length of the shaft
(iii) Diameter of the shaft
(iv) Stiffness of the shaft
(v) Mass of shaft
(vi) Other masses mounted on the shaft
(vii) Magnitude of deflection
(viii) Magnitude of unbalance of the masses with respect to the axis of rotation
(ix) Type and the amount of damping available in the system
It is absolutely necessary to calculate the critical speed of a rotating shaft to avoid excessive noise and vibration.
Calculation of the critical speed (Nc)
calculating the critical speed requires natural frequency. There are two methods to calculate the natural frequency of vibration i.e. Rayleigh- Ritz method (slightly overestimates) and Dun Kerley’s method (slightly underestimates). For any orientation of the shaft, use Rayleigh–Ritz equation.
Natural frequency, f = critical speed (Nc)
Natural frequency f = Nc = (30/ π) (g / δst)0.5
g = acceleration due to gravity (9.81 m/s2)
δst = Shaft vertical static deflection when placed horizontally
Nc is in RPM
NOTE: As rotational speed increase, δst also increase and hence natural frequency also change. However some value of rotational speed becomes equal in magnitude to the natural frequency of vibration. This rotational speed is the Critical Speed. At this speed, shaft may come out of the bearing. Avoid critical speed It at all costs.
From actual practice and experience, there is a suggestion. That the maximum operating speed should not exceed 75% of the critical speed.
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