Unit – 1 Vibration
Unit – 1
Fundamental Aspects of Vibrations &
Undamped Free Vibrations
Vibration -
Definition:
Regular oscillatory or periodic motion which repeats itself after a definite interval.
Vibration refers to mechanical oscillations about an equilibrium point.
Oscillation -
Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a
point of equilibrium) or between two or more different states.
Examples include a swinging pendulum and AC power.
Main Causes –
The main causes of setting up vibratory motion in a body are –
(I) Unbalanced centrifugal forces,
(II) Elastic nature of system.
(III) External excitations.
(IV) Winds
(V) Dry friction between mating surfaces.
Advantages –
Advantages of vibration are –
Vibrations are employed to simulate earthquakes for geological investigations.
Vibration may be used for drilling of geo – technical wells.
Vibrations are also employed in agriculture for harvesting by forced vibrations of fruit bearing
trees.
Disadvantages –
Disadvantages of vibration are –
Unwanted vibrations produce unpleasant stresses in machine parts. 01
Unit – 1 Vibration
Many structures, buildings and bridges fall because of vibration.
The vibration causes rapid wear of machine parts such as bearings and gears.
Excessive vibration is also dangerous for human beings.
Types of Vibration –
Vibrations
(According to the Actuating Force)
(A)Free vibration (B) Forced vibration
Damped Undamped Damped Undamped
Vibrations
(According to the Stresses in the Supporting Medium)
(1)Longitudinal Vibrations (2) Transverse Vibration (3) Torsional
Vibration
According to the Actuating Force –
(A) Free Vibration – When after giving the initial displacement, external force is removed, and then
body vibrates on its own.
(B) Forced Vibration – Vibrations which are caused and maintained by a periodic disturbing force.
According to the Stresses in the Supporting Medium – 02
Unit – 1 Vibration
(1)Longitudinal Vibrations – Consider a body of mass „m‟ as shown in fig. carried on one end of a
weightless spindle, the other end being fixed. If the mass moves up and down resulting in direct tension
and compression of spindle, it is said to execute Longitudinal Vibrations.
In this case the different particles constituting the body move in parallel planes.
(2)Transverse Vibration – If the body instead of moving vertically up and down vibrates in the
exaggerates manner as shown in fig., due to bending of the supporting medium, it is said to execute
transverse vibrations.
In this case the particles of shaft or disc move approximately perpendicular to the axis of the shaft.
(3) Torsional Vibration – When shaft gets twisted and untwisted alternatively due to vibratory motion of
suspended body, it is said to be undergoing Torsional Vibrations.
In this case shaft is subjected to Torsional shear stresses and shaft particles moves in a circle about the
axis of shaft as shown in fig.
03
Unit – 1 Vibration
Elementary Parts of Vibrating System –
The elementary parts of Vibrating system are:
(1) Mass (2) Spring (3) Damper
Description of parts:
The mass element is considered as a rigid body which executes vibration and can gain or lose kinetic
energy which is proportional to change in velocity of body.
The Spring element has elasticity and assumed to be of negligible mass.
The Damping element neither has mass nor elasticity.
Vibrating System Elements:
Equation:
The stiffness, of the spring is defined by the relation -
Where,
F is the force applied on the body
x is the displacement produced by the force. 04
Unit – 1 Vibration
Similarly,
Damping is defined by the relation –
Where,
Is the velocity = dx/dt
The equation of motion for such a vibrating system can be written as –
Where,
Is Inertia force
Is damping force
Is spring force.
NOTE –
“Inertia Force – This force essential for a Vibration.”
05
Unit – 1 Vibration
Degree of Freedom –
Definition:
The minimum number of independent coordinates required to determine completely the position of all
parts of a system at any instant of time defines the “Degree of Freedom”.
Classification:
Vibratory systems are often classified on the basis of the degree of freedom they have:
(1) Single Degree of Freedom System
(2) Multi degree of Freedom System
(1) Single Degree-of-Freedom (SDOF) systems –
The simplest vibratory system can be described by a single mass connected to a spring (and possibly a
dashpot). The mass is allowed to travel only along the spring elongation direction. Such systems are
called Single Degree-of-Freedom (SDOF) systems.
Examples Are: A mass supported by a spring and move in one direction, a pendulum oscillating in one
plane, a crank and slider mechanism.
(2) Multi degree of Freedom System -
The systems having more than one degree of freedom are known as “several or multi degree of freedom
system”.
Ajay Choubey 09893924108 06
Unit – 1 Vibration
Examples are : A two – mass, two – spring system and A spring – supported rigid mass, (These two
system has also two degrees of freedom)
Some Important Terms –
(A) Periodic Motion –
A motion which repeats itself after equal intervals of time is known as “periodic motion”.
Example: A bouncing ball, A rocking chair, The earth in its orbit around the sun, a water wave
etc.
(B) Simple Harmonic Motion –
To and fro motion of a body about a fixed point is called “simple harmonic motion”.
Example: Mass on a spring, Mass on a simple pendulum etc.
(C) Time Period –
Time taken to complete one cycle is known as “time period”.
(D) Frequency –
The number of complete cycles of motion in a unit time is known as “Frequency”. Hertz (Hz).
(E) Amplitude –
The maximum displacement of a vibrating body from the mean position is known as
“Amplitude”.
Ajay Choubey 09893924108 07
Unit – 1 Vibration
Fourier series For Harmonic Analysis –
In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a
(possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials).
The study of Fourier series is a branch of Fourier analysis.
Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat
equation in a metal plate.
●The heat equation is an important partial differential equation which describes the distribution of heat (or variation in
temperature) in a given region over time. For a function u(x, y, z, t) of three spatial variables (x, y, z) and the time variable t, the
heat equation is
Also written
Or sometimes
Where α is a positive constant and or denotes the Laplace operator. In the physical problem of temperature
variation, u(x, y, z, t) is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider
the case α = 1.
08
Unit – 1 Vibration
Fourier series:
Periodic signal (P.S.) as a drawing – For this (P.S.) it have particular pattern. So, this is a time period „T‟
for particular signal. (Means exactly same pattern repeating.) After „T‟, pattern will be repeating. So, this
is a periodic signal.
The French mathematician, J. Fourier showed that a periodic motion/signal can be represented by a series
of Sines and cosines that are harmonic.
Let us see, terms x (t) = Periodic signal. And we have known that, the P.S. showed through sines and
cosines. Therefore –
x (t) = a0/2 + a1 cosω1t + a2 cos ω2t + ………. (And then infinite no. likes a3….an) + b1 sinω1t + b2 sin ω2t
+ ………… (We can have infinite such term)
And here ω1 is nothing but 2π/T. (ω1=2π/T) were T is the time period of this P.S.
So, ω1 is the fundamental frequency of this signal. And in general ωn = n. ω1. Where n is the integer
number like 1, 2, and 3….up to ∞.
So, this is called Fourier series. And in general, particular signal take most not periodic such length
assumed periodic and after that it repeat.
Integers:
Integers are like whole numbers, but they also include negative numbers ... but still no fractions
allowed!
09
So, integers can be negative {-1, -2,-3, -4, -5, … }, positive {1, 2, 3, 4, 5, … }, or zero {0}
We can put that all together like this:
Integers = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ...}
Vibration is good for us and bad for us. If we talk about automobile application this type of discomfort
because of undulation of lot of jerk rod.
