Free vibration

2.  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 will bend 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 also depends 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. 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. 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
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

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