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1. Mechanical Vibrations
MEC4110
Unit I
Dr. Arshad Hussain Khan
Department of Mechanical Engineering
Zakir Husain College of Engg. & Technology
Aligarh Muslim University, Aligarh

2. Contents
• Overview & Introduction
• Parts of a vibrating system
• Degrees of Freedom
• Discrete and Continuous Systems
• Classification of vibrations
Lecture 1

3. Syllabus

4. Introduction
Any periodic motion of the particles of an elastic body or medium in alternately
opposite directions from the position of equilibrium when that equilibrium has been
disturbed is referred to as vibration. Alternatively “Any motion that repeats itself after
an interval of time is called vibration or oscillation”.
Most human activities involve vibration in one form or other:
• We hear because our eardrums vibrate and see because light waves undergo
vibration.
• Breathing is associated with the vibration of lungs and walking involves
(periodic) oscillatory motion of legs and hands. Human speech requires the
oscillatory motion of larynges (and tongues).
Vibration studies have huge engineering applications in the design of machines,
foundations, structures, engines, turbines, and control systems.
• The structural/machine component subjected to vibration can fail because of
material fatigue resulting from the cyclic variation of the induced stress.
• Vibration may cause rapid wear of machine parts such as bearings and gears
and also creates excessive noise.
• Vibration can loosen fasteners such as nuts in machines. In metal cutting
process vibration can cause chatter, which leads to a poor surface finish.

5. Vibration of a system involves the transfer of its potential energy to kinetic energy and
vice-versa, alternately.
For a damped system, some energy is dissipated in each cycle of vibration and must
be replaced by an external source if steady state of vibration is to be maintained.
Vibrating system in general will have:
• Means for storing potential energy (spring or elasticity)
• Means for storing kinetic energy (mass or inertia),
• Means by which energy is gradually lost (damper).
Elementary Parts of Vibrating Systems
Vibration of a simple pendulum.
Bob of mass m be released after being given an angular displacement at position 1.
The velocity and K.E. is zero at 1. But the P.E. is mgl (1 - cos q) w.r.t. to datum 2.
The gravitational force mg induces a torque mgl sinq about the point O, the bob starts
swinging to the left from position 1.
The bob experiences angular acceleration in the clockwise direction, and by the time it
reaches position 2, all of its P.E. gets converted into K.E.. Hence the bob will not stop in
position 2 but will continue to swing to position 3.
However, as it passes the mean position 2, a counterclockwise torque due to gravity starts
acting on the bob and causes the bob to decelerate.
All the kinetic energy of the bob will be converted to potential energy at 3.
Again due to the gravity torque, the bob continues to attain a counterclockwise velocity.
This process keeps repeating, and the pendulum will have oscillatory motion.
Some energy is dissipated in each cycle of vibration due to damping by the air.

6. Minimum number of independent coordinates required to determine completely the positions of all parts
of a system at any instant of time defines the number of DOF of the system.
Degrees of Freedom
Discrete and Continuous Systems.
Systems with a finite number of degrees of freedom are called discrete or lumped parameter systems, and
those with an infinite number of degrees of freedom are called continuous or distributed systems.
Continuous systems are most of the time approximated as discrete systems. Although treatment of a
system as continuous gives exact results, the analytical methods available for dealing with continuous
systems are limited.
Single DOF
Two DOF
Three DOF

7. • Free and Forced Vibration:
Free Vibration No external force acts on the system. The system is left to vibrate on its own, after giving initial
disturbance.
Forced Vibration System is subjected to an external force (often, a repeating type of force). If the frequency
of the external force coincides with one of the natural frequencies of the system, a condition
known as resonance occurs leading to large oscillations.
• Undamped and Damped Vibration
In undamped vibrations no energy is lost or dissipated in friction or other resistance during vibrations.
In damped vibrations energy is lost or dissipated.
• Linear and Nonlinear Vibration
If all the basic components of a vibratory system the spring, the mass, and the damper behave linearly, it is
referred to as linear vibration.
If, cases where any of the basic components behave nonlinearly, the vibration is called nonlinear
vibration. The differential equations governing the behaviour may be linear or nonlinear as the case may
be. The superposition principle is not valid for nonlinear systems.
• Deterministic and Random Vibration
In case of deterministic vibrations, the magnitude of the excitation/input (force or motion) acting on a vibratory
system is known(deterministic) at any given time.
If the excitation is nondeterministic or random, the vibration is referred to as random and the response is
described in terms of statistical quantities.
Classification of Vibration

8. Thank you and
queries?