FREE VIBRATION
Terms used in vibratory motion
• Period of vibration or time period
• Cycle
• Frequency.
Types of Vibratory Motion
• 1. Free or natural vibrations.
• 2. Forced vibrations.
• 3. Damped vibrations.
Types of Free Vibrations
ill-effects of vibration :
• Accuracy of parts machined in vibrating machine
excessive vibration at resonant condition may
lead to complete failure of the part.
• Vibration lasting for small interval of time may
cause severe damage to the structure- Earth
quake & explosion
• Harmless continuous vibration over a long
period may result in fatigue failure.
• Such failures are observed in all types of
components small & large from valve springs
& crank shaft of an automobile to elevator of
an aero plane or even large draw bridges.
• Excessive vibration of passenger vehicle
• Vibration of cock-pit of an aero plane
• Excessive vibration of hand held machines
may damage human tissues
• Severe vibration jolting of tractors or earth
moving equipment may result in spinal
injuries.
USEFUL VIBRATIONS.
• Vibration conveyors • Impactors• non-destructive testing –crack deduction• Vibratory sieves
Medical Application • imaging of internal organs• tooth cleaning • heart beat
• Music instruments• Time keeping instruments
SOURCES OF VIBRATION Shaft running at near critical speeds Misalignment and bent shaft Damaged rolling element bearings
- balls ,rollers etc Fluid flow
- turbulence and cavitations Damaged and worn out gears Faulty belt drives Oil film whirl and whip in journal bearings Impact Electrically induced vibration
Natural Frequency of Free Longitudinal Vibrations
• 1. Equilibrium Method
• 2. Energy method
• 3. Rayleigh’s method
1. Equilibrium Method
Natural Frequency of Free Transverse Vibrations
Natural Frequency of Free Transverse Vibrations For a Shaft Subjected to a Number of Point Loads
Dunkerley’s methodThe natural frequency of transverse vibration for a shaft carrying
a number of point loads and uniformly distributed load is
obtained from Dunkerley’s empirical formula. According to
this
Therefore, according to Dunkerley’s empirical
formula, the natural frequency of the whole
system,
Problem : #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.
• Solution:
Critical or Whirling Speed of a Shaft
Problem #2:
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,
a 200 GN/m2. Assume the shaft to be freely
supported.
• Solution:
Problem #3:
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/m3 and its modulus of elasticity is
200 GN/m2 . Find the lowest whirling speed of the
shaft, taking into account the mass of the shaft.
• Solution:
therefore mass of the shaft per metre length,
• We know that the static deflection due to a load W
Static deflection due to a mass of 50 kg at C,
• Similarly, static deflection due to a mass of 50 kg at D
Static deflection due to uniformly distributed load or mass of the shaft,
• frequency of transverse vibration,
Since the whirling speed of shaft (Nc) in r.p.s. is equal to the frequency of transverse vibration in Hz, therefore
Frequency of Free Damped Vibrations (Viscous Damping)
1. When the roots are real (overdamping)
2. When the roots are complex conjugate (underdamping)
3. When the roots are equal (critical damping)
Thus the motion is again aperiodic. The critical damping
coefficient (Cc) may be obtained by substituting (Cc) for c
in the condition for critical damping, i.e.
Damping Factor or Damping Ratio
Logarithmic Decrement
Problem #4
The measurements on a mechanical vibrating system
show that it has amass of 8 kg and that the springs can
be combined to give an equivalent spring of stiffness 5.4
N/mm. If the vibrating system have a dashpot attached
which exerts a force of 40 N when the mass has a
velocity of 1 m/s, find : 1. critical damping coefficient, 2.
damping factor, 3. logarithmic decrement, and 4. ratio
of two consecutive amplitudes.
Solution:
Problem #5
A machine of mass 75 kg is mounted on springs and is fitted
with a dashpot to damp out vibrations. There are three
springs each of stiffness 10 N/mm and it is found that the
amplitude of vibration diminishes from 38.4 mm to 6.4 mm
in two complete oscillations. Assuming that the damping
force varies as the velocity, determine : 1. the resistance of
the dashpot at unit velocity ; 2. the ratio of the frequency
of the damped vibration to the frequency of the undamped
vibration ; and 3. the periodic time of the damped
vibration.
Solution
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