This presentation gives an introduction to mechanical vibration or Theory of Vibration for BE courses. Presentation is prepared as per the syllabus of VTU.For any suggestions and criticisms please mail to: hareeshang@gmail.com or visit:ww.hareeshang.wikifoundry.com. Thanks for watching this presentation. Hareesha N G

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Hareesha N G, Asst. Prof, DSCE, BLore-78

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3. BASIC CONCEPTS OF VIBRATION
• All bodies having mass and elasticity are capable of producing vibration.
• The mass is inherent of the body and elasticity causes relative motion
among its parts.
• When body particles are displaced by the application of external force, the
internal forces in the form of elastic energy are present in the body.
• These forces try to bring the body to its original position.
• At equilibrium position, the whole of the elastic energy is converted into
kinetic energy and body continues to move in the opposite direction
because of it.
• The whole of the kinetic energy is again converted into elastic or strain
energy due to which the body again returns to the equilibrium position.
• In this way, vibratory motion is repeated indefinitely and exchange of
energy takes place.
• Thus, any motion which repeats itself after an interval of time is called
vibration or oscillation.
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4. BASIC CONCEPTS OF VIBRATION (Cntd..)
• The swinging of simple pendulum as shown in figure 1 is an
example of vibration or oscillation as the motion of ball is to and fro
from its mean position repeatedly.
• The main reasons of vibration are as follows :
– Unbalanced centrifugal force in the system. This is caused because of non-
uniform material distribution in a rotating machine element.
– Elastic nature of the system.
– External excitation applied on the system.
– Winds may cause vibrations of certain systems such as electricity lines,
telephone lines, etc.
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5. IMPORTANCE OF VIBRATION STUDY IN
ENGINEERING
• The structures designed to support the high speed engines and turbines are
subjected to vibration.
• Due to faulty design and poor manufacture, there is unbalance in the
engines which causes excessive and unpleasant stresses in the rotating
system because of vibration.
• The vibration causes rapid wear of machine parts such as bearings and
gears.
• Unwanted vibrations may cause loosening of parts from the machine.
• Because of improper design or material distribution, the wheels of
locomotive can leave the track due to excessive vibration which results in
accident or heavy loss.
• Many buildings, structures and bridges fall because of vibration.
• If the frequency of excitation coincides with one of the natural frequencies
of the system, a condition of resonance is reached, and dangerously large
oscillations may occur which may result in the mechanical failure of the
system.
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6. IMPORTANCE OF VIBRATION STUDY IN
ENGINEERING (Contd…)
• Vibration can be used for useful purposes such as
vibration testing equipments, vibratory conveyors,
hoppers, sieves and compactors.
• Vibration is found very fruitful in mechanical workshops
such as in improving the efficiency of machining, casting,
forging and welding techniques, musical instruments and
earthquakes for geological research.
• It is useful for the propagation of sound.
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7. IMPORTANCE OF VIBRATION STUDY IN
ENGINEERING (Contd…)
Thus undesirable vibrations should be eliminated or reduced
upto certain extent by the following methods :
– Removing external excitation, if possible
– Using shock absorbers.
– Dynamic absorbers.
– Resting the system on proper vibration isolators.
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8. DEFINITIONS
• Periodic motion: A motion which repeats itself after equal intervals of time.
• Time period: Time taken to complete one cycle.
• Frequency: Number of cycles per unit time.
• Amplitude: The maximum displacement of a vibrating body from its equilibrium
position.
• Natural frequency: When no external force acting on the system after giving it
an initial displacement, the body vibrates. These vibrations are called free
vibrations and their frequency as natural frequency. It is expressed in rad/sec or
Hertz.
• Fundamental Mode of Vibration: The fundamental mode of vibration of a
system is the mode having the lowest natural frequency.
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10. SHM Demonstration
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11. DEFINITIONS (Contd..)
Simple Harmonic Motion:
• The motion of a body to and fro about a fixed point is called simple
harmonic motion.
• The motion is periodic .
• The motion of a simple pendulum is simple harmonic in nature.
• A body having simple harmonic motion is represented by the
equation
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x Asin( ) A sin( t )
 
 
2
2 2
2
dx
A cos( t )
dt
and
d x
A sin( t ) x
dt
 
  

   

13. DEFINITIONS (Contd..)
Damping:
– It is the resistance to the motion of a vibrating body.
– The vibrations associated with this resistance are known as damped
vibrations.
Phase difference:
– Suppose there are two vectors x1 and x2 having frequencies ω rad/sec, each
The vibrating motions can be expressed as
x1=A1.sin (ωt)
x2=A2.sin (ωt+φ)
– In the above equation the term φ is known an the phase difference.
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15. DEFINITIONS (Contd..)
Resonance:
– When the frequency of external excitation is equal to the natural
frequency of a vibrating body, the amplitude of vibration
becomes excessively large.
– This concept is known as resonance.
Mechanical systems:
– The systems consisting of mass, stiffness and damping are
known as mechanical systems.
Continuous and Discrete Systems:
– Most of the mechanical systems include elastic members which
have infinite number of degree of freedom.
– Such systems are called continuous systems.
– Continuous systems are also known as distributed systems.
– Cantilever, simply supported beam etc. are the examples of such
systems.
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16. DEGREE OF FREEDOM:
• The minimum number of independent coordinates required to specify the
motion of a system at any instant is known as degrees of freedom of the
system.
• In general, it is equal to the number of independent displacements that are
possible.
• This number varies from zero to infinity.
• The one, two and three degrees of freedom systems are shown in figure 2.
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In single degree of freedom there is only one
independent coordinate x1 to specify the
configuration as shown in figure (a). Similarly,
there are two (x1, x2). and three coordinates (x1,
x2 and x3) for two and three degrees of freedom
systems as shown in figure (b) and (c)

17. PARTS OF A VIBRATING SYSTEM
• A vibratory system basically consists of three elements, namely the mass,
the spring and damper.
• In a vibrating body, there is exchange of energy from one form to another.
• Energy is stored by mass in the form of kinetic energy (1/2 mv2), in the
spring in the form of potential energy (1/2 kx ) and dissipated in the
damper in the form of heat energy which opposes the motion of the
system.
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2 3
1
1. Inertia (stores kinetic energy) –Mass (m)
2. Elasticity (stores potential energy) – Spring
(k)
3. Energy Dissipation- Damper (C)

18. PARTS OF A VIBRATING SYSTEM
• Energy enters the system with the application of external force known as
excitation.
• The excitation disturbs the mass from its mean position and the mass goes
up and down from the mean position.
• The kinetic energy is converted into potential energy and potential energy
into kinetic energy. This sequence goes on repeating and the system
continues to vibrate.
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19. PARTS OF A VIBRATING SYSTEM (contd..)
• At the same time damping force (cv) acts on the mass and opposes
its motion. Thus some energy is dissipated in each cycle of
vibration due to damping.
• The free vibrations die out and the system remains at its static
equilibrium position.
• A basic vibratory system is shown in figure .
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20. F = cv
Velocity
Displacement Acceleration
x
v a
k
c
m
m
F = kx F = ma
Mechanical Parameters and Components

21. TYPES OF VIBRATION
Some of the important types of vibration are as follows
1) Free and Forced Vibration
• After disturbing the system, the external excitation is removed. then the
system vibrates on its own. This type of vibration is known as free
vibration.
– Simple pendulum is one of the examples.
• The vibration which is under the influence of external force is called
forced vibration.
– Machine tools, electric bells etc. are the suitable examples.
2) Linear and Non-linear Vibration
• In a system., if mass, spring and damper behave in a linear manner, the
vibrations caused are known as linear in nature.
– Linear vibrations are governed by linear differential equations.
– They follow the law of superposition.
• On the other hand, if any of the basic components of a vibratory system
behaves non-linearly, the vibration is called non-linear.
– Linear vibration becomes, non-linear for very large amplitude of vibration.
– It does not follow the law of superposition.
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22. TYPES OF VIBRATION (Contd..)
3) Damped and Un-damped Vibration
• If the vibratory system has a damper, the motion of the system will
be opposed by it and the energy of the system will be dissipated in
friction.
• This type of vibration is called damped vibration.
• On the contrary, the system having no damper is known as un-
damped vibration.
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n
2

d
2

Overdamped ( 1)
 
Underdamped ( 1)
 
Underdamped ( 0 )
 
Critically
damped ( 1)
 

23. TYPES OF VIBRATION (Contd..)
4) Deterministic and Random Vibration
• If in the vibratory system, the amount of external excitation is
known in magnitude, it causes deterministic vibration.
• Contrary to it, the non-deterministic vibrations are known as
random vibrations.
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24. TYPES OF VIBRATION (Contd..)
5) Longitudinal, Transverse and Torsional Vibrations
• Figure represents a body of mass m carried on one end of a weightless spindle,
the other end being fixed. If the mass m moves up and down parallel to the
spindle axis, it is said to execute longitudinal vibrations as shown in figure (a).
• When the particles of the body or shaft move approximately perpendicular to the
axis of the shaft, as shown in figure (b), the vibration so caused are known as
transverse.
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If the spindle gets alternately
twisted and untwisted on account
of vibratory motion of the
suspended disc, it is called to be
undergoing torsional vibrations
as shown in figure (c).

25. TYPES OF VIBRATION (Contd..)
6) Transient Vibration
• In ideal systems the free vibrations continue indefinitely as there is
no damping.
• The amplitude of vibration decays continuously because of
damping (in a real system) and vanishes ultimately.
• Such vibration in a real system is called transient vibration.
• Systems with finite number of degrees of freedom are called
discrete or lumped systems.
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26. SIMPLE HARMONIC MOTION:
• The motion of a body to and fro about a fixed point is called simple
harmonic motion.
• The motion is periodic .
• The motion of a simple pendulum is simple harmonic in nature.
• A body having simple harmonic motion is represented by the
equation
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SIMPLE HARMONIC MOTION:

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x Asin( ) A sin( t )
 
 
2
2 2
2
dx
A cos( t )
dt
and
d x
A sin( t ) x
dt
 
  

   
SIMPLE HARMONIC MOTION:

29. PHENOMENON OF BEATS
• When two harmonic motions pass through some point in a medium
simultaneously, the resultant displacement at that point is the vector
sum of the displacement due to two component motions.
• This superposition of motion is called interference.
• The phenomenon of beat occurs as a result of interference between
two waves of slightly different frequencies moving along the same
straight line in the same direction.
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http://historicaltuning.co
m/BeatsVideo.mp4

30. PHENOMENON OF BEATS (contd…)
• Consider that at particular time, the two wave motions are in the
same phase.
• At this stage the resultant amplitude of vibration will be maximum.
• On the other hand, when the two motions are not in phase with each
other, they produce minimum amplitude of vibration.
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Again after some time the two
motions are in phase and produce
maximum amplitude and then
minimum amplitude.
This process goes on repeating and
the resultant amplitude
continuously keeps on changing
from maximum to minimum. This
phenomenon is known as beat.
https://www.youtube.com/
watch?v=5hxQDAmdNWE

31. MATLAB Program for solving Harmonic analysis
Problems
clc
clear all
syms t
w=20*pi;
n=1:6;
x1(t)=20*t;
x2(t)=-20*t+2;
x(t)=x1(t)+x2(t);
a0=20*(int(x1(t),t,0,0.05)+int(x2(t),t,0.05,0.1))
an=20*(int((x1(t)*cos(w*n*t)),t,0,0.05)+(int((x2(t)*cos(w*n*t)),t,
0.05,0.10)))
bn=20*(int((x1(t)*sin(w*n*t)),t,0,0.05)+(int((x2(t)*sin(w*n*t)),t,
0.05,0.10)))
% a0/2 =1
% an =[ -4/pi^2, 0, -4/(9*pi^2), 0, -4/(25*pi^2), 0]
%bn =[ 0, 0, 0, 0, 0, 0]
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34. %%
% Problem number-2
clc
clear all
syms t
w=10*pi;
n=1:6;
x1(t)=-20*t+2; % 0<t<0.2
a0=20*(int(x1(t),t,0,0.2))
an=10*(int((x1(t)*cos(w*n*t)),t,0,0.2))
bn=10*(int((x1(t)*sin(w*n*t)),t,0,0.2))
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35. %%
% Problem number-3
clc
clear all
syms t
w=100*pi;
n=1:6;
x1(t)=abs(sin(100*pi*t));
a0=50*(int(x1(t),t,0,0.02))
an=100*(int((x1(t)*cos(w*n*t)),t,0,0.02))
bn=100*(int((x1(t)*sin(w*n*t)),t,0,0.02))
% This is the solution
% a0 = 2/pi
% an = [ 0, -4/(3*pi), 0, -4/(15*pi), 0, -4/(35*pi)]
% an= -4/[(n^2-1)*pi] for even vlaues of n an=0 for odd
values of n.
% bn = [ 0, 0, 0, 0, 0, 0]
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