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Engineering Mechanics III
Department of Mechanical Engineering
1.Theory of Machines By R.S. Khurmi and J.K. Gupta
2.Mechanical Vibration By A.H. Church
What is Vibration?
Vibration can be considered to be the oscillation or repetitive motion of an
object around an equilibrium position.
How Vibration Occurs?
When elastic bodies such as a spring, a beam and a shaft are displaced
from the equilibrium position by the application of external forces, and
then released, they execute a vibratory motion. This is due to the
reason that, when a body is displaced, the internal forces in the form of
elastic or strain energy are present in the body. At release, these forces
bring the body to its original position. When the body reaches the
equilibrium position, the whole of the elastic or strain energy is converted
into kinetic energy due to which the body continues to move in the
opposite direction. The whole of the kinetic energy is again converted into
strain energy due to which the body again returns to the equilibrium
position. In this way, the vibratory motion is repeated indefinitely.
Terms Used in Vibratory Motion
Period of Vibration or Time Period: It is the time interval after which the motion is repeated itself. The period of vibration is usually expressed in seconds.
Cycle: It is the motion completed during one time period.
Frequency: It is the number of cycles described in one second. In S.I. units, the frequency is expressed in hertz (briefly written as Hz) which is equal to one cycle per second.
Types of Vibratory Motion
1. Free or natural vibrations.
2. Forced vibrations.
3. Damped vibrations.
Free or natural vibrations When no external force acts on the body, after giving it an initial displacement, then the body is said to
be under free or natural vibrations. The frequency of the free vibrations is called free or natural
When the body vibrates under the influence of external force, then the body is said to be under forced
vibrations. The external force applied to the body is a periodic disturbing force created by unbalance.
The vibrations have the same frequency as the applied force.
When there is a reduction in amplitude over every cycle of vibration, the motion is said to be damped
vibration. This is due to the fact that a certain amount of energy possessed by the vibrating system is
always dissipated in overcoming frictional resistances to the motion.
Types of Free Vibration
1. Longitudinal vibrations
2. Transverse vibrations
3. Torsional vibrations
When the particles of the shaft or disc moves parallel to the axis of the
shaft, as shown in Fig. then the vibrations are known as longitudinal
In this case, the shaft is elongated and shortened alternately and thus
the tensile and compressive stresses are induced alternately in the
When the particles of the shaft or disc move approximately
perpendicular to the axis of the shaft, as shown in Fig. then the
vibrations are known as transverse vibrations.
In this case, the shaft is straight and bent alternately and bending
stresses are induced in the shaft.
When the particles of the shaft or disc move in a circle about
the axis of the shaft, as shown in Fig. then the vibrations are
known as torsional vibrations.
In this case, the shaft is twisted and untwisted alternately and
the torsional shear stresses are induced in the shaft.
Natural Frequency of Free Longitudinal Vibrations
1. Equilibrium Method
2. Energy Method
3. Rayleigh’s Method
s = Stiffness of the constraint. It is the force required to
produce unit displacement in the direction of vibration. It is
usually expressed in N/m.
m = Mass of the body suspended from the constraint in kg
W = Weight of the body in newtons = m.g
= Static deflection of the spring in metres due to weight W Newtons
x = Displacement given to the body by the external force, in metres
Equilibrium Method(Contd.) In the equilibrium position,
the gravitational pull W = m.g
The spring force, W = s
Since the mass is now displaced from its equilibrium
position by a distance x, as shown in Fig. and is then
released, therefore after time t,
Equilibrium Method(Contd.) Accelerating force = Mass × Acceleration
xd m (2)
Equating equation (1) and (2),
the fundamental equation of simple harmonic motion is
Comparing equations (3) and (4), we have
Time period, s
m t p
and natural frequency,
Taking the value of g as 9.81 m/s2 and in metres,
The value of static deflection may be found out from the
given conditions of the problem. For longitudinal
vibrations, it may be obtained by the relation,
AE fn 4985.0
0)...( EPEK dt
We know that kinetic energy, 2)(
and potential energy, 2
0 (.. sxx
1 ( 22 sx
Rayleigh’s Method In this method, the maximum kinetic energy at the mean position is equal to the maximum potential
energy (or strain energy) at the extreme position. Assuming the motion executed by the vibration to be
simple harmonic, then
Natural Frequency of Free Transverse Vibrations
Natural Frequency of Free Transverse Vibrations(Contd.)