UNIT – III
TRANSVERSE VIBRATION
PROBLEMS ON TRANSVERSE VIBRATIONS:
1. A cantilever shaft 50 mm diameter and 300 mm long has a disc of mass 100 kg at its free end. The
Young's modulus for the shaft material is 200 GN/m2. Determine the frequency of longitudinal
and transverse vibrations of the shaft.
2. A shaft of length 0.75 m, supported freely at the ends, is carrying a body of mass 90 kg at 0.25 m
from one end. Find the natural frequency of transverse vibration. Assume E = 200 GN/m 2 and
shaft diameter = 50 mm.
3. A flywheel is mounted on a vertical shaft as shown in Fig. 23.8. The both ends of the shaft are
fixed and its diameter is 50 mm. The flywheel has a mass of 500 kg. Find the natural frequencies
of longitudinal and transverse vibrations. Take E = 200 GN/m2.
Natural Frequency of Free Transverse Vibrations For a Shaft Subjected to a Number of Point Loads
DUNKERLEY’S METHOD:
PROBLEM ON DUNKERLEY’S EQUATION:
1. A shaft 50 mm diameter and 3 metres long is simply supported at the ends and carries three loads
of 1000 N, 1500 N and 750 N at 1 m, 2 m and 2.5 m from the left support. The Young's modulus
for shaft material is 200 GN/m2. Find the frequency of transverse vibration.
CRITICAL OR WHIRLING SPEED OF A SHAFT
PROBLEMS ON CRITICAL SPEED OF A SHAFT:
1. Calculate the whirling speed of a shaft 20 mm diameter and 0.6 m long carrying a mass of 1 kg at
its mid-point. The density of the shaft material is 40 Mg/m3, and Young’s modulus is 200 GN/m2.
Assume the shaft to be freely supported.
2. 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 mm and internal diameter 40
mm. The density of the shaft material is 7700 kg/m 3 and its modulus of elasticity is 200 GN/m 2.
Find the lowest whirling speed of the shaft, taking into account the mass of the shaft.
3. 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 and the maximum
bending stress when the shaft is rotating at 75% of the critical speed. The centre of the disc is 0.25
mm from the geometric axis of the shaft. E = 200 GN/m2.
4. 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/m 2 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. [For a shaft with fixed
3
Wl Wl
end carrying a concentrated load (W) at the centre assume δ= and ¿ , where δ and M
192 EI 8
are maximum deflection and bending moment respectively].
TORSIONAL VIBRATION
PROBLEMS:
1. A shaft of 100 mm diameter and 1 metre long has one of its end fixed and the other end carries a
disc of mass 500 kg at a radius of gyration of 450 mm. The modulus of rigidity for the shaft
material is 80 GN/m2. Determine the frequency of torsional vibrations.
2. A flywheel is mounted on a vertical shaft as shown in Fig. The both ends of a shaft are fixed and
its diameter is 50 mm. The flywheel has a mass of 500 kg and its radius of gyration is 0.5 m. Find
the natural frequency of torsional vibrations, if the modulus of rigidity for the shaft material is
80 GN/m2.
FREE TORSIONAL VIBRATIONS OF A SINGLE ROTOR SYSTEM
FREE TORSIONAL VIBRATIONS OF A TWO ROTOR SYSTEM
FREE TORSIONAL VIBRATIONS OF A THREE ROTOR
SYSTEM
TORSIONALLY EQUIVALENT SHAFT
PROBLEMS:
FREE TORSIONAL VIBRATIONS OF A GEARED SYSTEM
NATURAL FREQUENCY OF TORSIONAL VIBRATION OF A GEARED SYSTEM:
PROBLEMS:
