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