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Critical Speed of the Shaft

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Critical Speed of the Shaft

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  1. Critical speed of shaft

  2. Critical/whirling speed of shaft • The centre of gravity of the pulley or gear mounted on a shaft is at certain distance from the axis of rotation. • Due to this, the shaft is subjected to centrifugal force which willbend the shaft which will further increase the distance of centre of gravity from the axis of rotation. • The bending of shaft not only depends upon the distance between C.G of the shaft and gears, but alsodepends upon the speed at which the shaft rotates. • The speed at which the shaft runs so that the additional deflection of the shaft from the axis of rotation becomes infinite, is known as critical or whirling speed.

  3. 1. A shaft 1.5 m long, supported in flexible bearings at the ends carries two wheels each of 50 kg mass. One wheel is situated at the centre of the shaft and the other at a distance of 375 mm from the centre towards left. The shaft is hollow of external diameter 75 mmand internal diameter 40 mm. The density of the shaft materialis7700 kg/m3 and itsmodulusofelasticity is 200 GN/m2. Find the lowest whirling speed of the shaft, taking into account the mass of the shaft. Given: l = 1.5 m ; m1 = m2 = 50 kg ; d1 = 75 mm = 0.075 m ; d2 = 40 mm = 0.04 m ; ρ = 7700 kg/m3 E = 200 GN/m2 = 200 × 109 N/m2

  4. Moment of inertia of shaft • Mass per unit length

  5. Deflection of beam at C • Deflection of beam at D

  6. Static deflection due to weight of shaft • Natural frequency of shaft • Critical speed of shaft • Nc = 32.4 x 60 = 1944 rpm

  7. Given • d = 5 mm = 0.005 m ; • l = 200 mm = 0.2 m ; • m = 50 kg ; • E = 200 GN/m2 = 200 × 109 N/m2 • Moment of inertia of shaft 2. . A vertical shaft of 5 mm diameter is 200 mm long and is supported in long bearings at its ends. A disc of mass 50 kg is attached to the centre of the shaft. Neglecting any increase in stiffness due to the attachment of the disc to the shaft, find the critical speed of rotation

  8. Static deflection of shaft (fixed shaft-Long bearings) • d = Wa3b3/3EIl3 • = 50 x 9.81 x (0.1)3 x(0.1)3 / 3 x200 × 109x 30.7 x 10-12 • = 3.33 x 10-3 m • Frequency of the shaft • fn • Critical speed of shaft • Nc = 8.64 x 60 = 518.4 rpm ~ 520 rpm

  9. A vertical steel shaft 15 mm diameter is held in long bearings 1 metre apart and carries at its middle a disc of mass 15 kg. The eccentricity of the centre of gravity of the disc from the centre of the rotor is 0.30 mm. The modulus of elasticity for the shaft material is 200 GN/m2 and the permissible stress is 70 MN/m2. Determine : 1. The critical speed of the shaft and 2. The range of speed over which it is unsafe to run the shaft. Neglect the mass of the shaft

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