FREE VIBRATION
Module 2
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VIBRATION
• A body is said to vibrate if it has a to-and-fro motion.
• vibrations are due to elastic forces.
• Whenever a body is displaced from its equilibrium position,
work is done on the elastic constraints of the forces on the
body and is stored as stain energy.
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DEFINITIONS
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TYPES OF VIBRATIONS
Longitudinal Vibrations
If the shaft is elongated and shortened so that the same moves up and down
resulting in tensile and compressive stresses in the shaft, the vibrations are
said to be longitudinal.
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Transverse Vibrations
• When the shaft is bent alternately and tensile and compressive stresses due
to bending result, the vibrations are said to be transverse. The particles of the
body move approximately perpendicular to its axis.
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Torsional Vibrations
When the shaft is twisted and untwisted alternately and torsional shear
stresses are induced, the vibrations are known as torsional vibrations. The
particles of the body move in a circle about the axis of the shaft.
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BASIC FEATURES OF VIBRATING SYSTEMS
For mathematical analysis of a vibratory system, it is necessary to have an
idealized model of the same which appropriately represents the system.
Basic Elements
inertial and restoring and damping elements
Inertial elements
These are represented by masses for rectilinear motion and moment of inertia for angular
motion.
Restoring Elements
Massless linear or torsional springs represent the restoring elements for rectilinear and
torsional motions respectively.
Damping Elements
Massless dampers of rigid elements may be considered for energy dissipation in a
system.
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Inertia element
Restoring element
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Damping Element
DEGREES OF FREEDOM
The number of independent coordinates required to describe a vibratory system
is known as its degree of freedom.
Single-degree-of-freedom systems.
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DEGREES OF FREEDOM
Two-degree-of-freedom systems.
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DEGREES OF FREEDOM
Infinite degrees of freedom.
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FORCED VIBRATION
Module 2
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HARMONICALLY EXCITED VIBRATION
UNDAMPED SYSTEM
The mass is subjected to an oscillating force
Forces acting on the mass at any instant
𝑘
𝑘
---------------1
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Complete solution of this Equation consists of two
parts
• Complementary function (CF)
• Particular integral (PI)
CF is the solution of the equation
𝑘
The particular solution;
---------------2
𝑥! =
The Exciting force F is harmonic the particular solution 𝑥! 𝑡 also harmonic and has the
same frequency 𝜔
Thus we assume the solution is in the form
𝑥! = 𝑋 sin(𝜔𝑡)
---------------3
where X is a constant that denotes the maximum amplitude of 𝑥!
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𝑥! = 𝑥 = 𝑋 sin(𝜔𝑡)
𝑥!̇ = 𝑋 𝜔 cos(𝜔𝑡)
𝑥!̈ = −𝑋 𝜔" sin(𝜔𝑡)
Substitute the values of 𝑥! , 𝑥!̈ In equation 1, and solve for X, we get
Substitute the value of X in equation 2 we get, Particular integral, PI
𝐹#
𝑥! =
sin(𝜔𝑡)
"
𝑘 − 𝑚𝜔
Multiplying the numerator and denominator by 1/k
𝐹#3
𝑘
𝑥! =
sin(𝜔𝑡)
𝑘3 − 𝑚3 𝜔 "
𝑘
𝑘
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Where, 𝜔$" = %⁄&
Multiplying the numerator and denominator by 1/k
𝐹"$
𝑘 sin(𝜔𝑡)
𝑥! =
𝜔 $
1−
𝜔#
The complete solution is
𝐹#3
𝑘 sin(𝜔𝑡)
𝑥' + 𝑥! = 𝑋𝑠𝑖𝑛 𝜔$ 𝑡 + 𝜙 +
𝜔 "
1−
𝜔$
Thus the resultant motion is the sum of two harmonics. The constants X and
𝜙 of the first harmonic are obtained from the initial conditions.
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HARMONICALLY EXCITED VIBRATION
DAMPED SYSTEM
• The mass is subjected to an oscillating force
• Forces acting on the mass at any instant
kx
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kx
kx
----------1
kx
Complete solution of this Equation consists of two parts
• Complementary function (CF)
• Particular integral (PI)
CF is the solution of the equation
𝑚𝑥̈ + 𝑐 𝑥̇ + 𝑘𝑥 = 0
𝑥! =
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The Exciting force F is harmonic the particular solution 𝑥" 𝑡
also harmonic, we assume it in the form of
𝑥" 𝑡 = 𝑋𝑠𝑖𝑛 𝜔𝑡 − 𝜙
𝑥!̇ = 𝑋𝑐𝑜𝑠 𝜔𝑡 − 𝜙 . 𝜔
𝑥!̈ = −𝑋𝜔0 𝑠𝑖𝑛 𝜔𝑡 − 𝜙
Substitute the values of 𝑥! , 𝑥!̈ In equation 1, and solve for X, we get
𝑚 −𝑋𝜔0 𝑠𝑖𝑛 𝜔𝑡 − 𝜙
+ 𝑐 𝑋𝑐𝑜𝑠 𝜔𝑡 − 𝜙 . 𝜔 + 𝑘𝑋𝑠𝑖𝑛 𝜔𝑡 − 𝜙 = 𝐹1 sin(𝜔𝑡)
𝑘 − 𝑚𝜔0 𝑋𝑠𝑖𝑛 𝜔𝑡 − 𝜙 + 𝑐𝜔𝑋𝑐𝑜𝑠 𝜔𝑡 − 𝜙 = 𝐹1 sin(𝜔𝑡)
We know that
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𝑋 𝑘 − 𝑚𝜔! . [sin 𝜔𝑡 cos 𝜙 − cos 𝜔𝑡 sin 𝜙 +
𝑐𝜔[cos 𝜔𝑡 cos 𝜙 + sin 𝜔𝑡 sin 𝜙 = 𝐹" sin 𝜔𝑡
Equating the coefficients of cos 𝜔𝑡 and sin 𝜔𝑡 on both
sides of the resulting equation, we obtain
----------2
---------3
From 2
From 3
𝐹!
---------4
𝑋=
𝑘 − 𝑚𝜔 " cos 𝜙 + 𝑐𝜔 sin 𝜙
𝑐𝜔𝑋 cos 𝜙
"
= 𝑋 𝑘 − 𝑚𝜔
sin 𝜙
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Substitute these values in equation 4 and simplify
We get
𝑋=
𝐹!
𝑘 − 𝑚𝜔 " " + 𝑐𝜔 "
Multiplying the numerator and denominator by 1/k
𝑋=
𝐹#5
𝑘
𝜔
1− 𝜔
$
" "
𝑐𝜔 "
+
𝑘
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#$
%
=
# #; $
#;
.
%
= 𝜁2 𝑘𝑚
$
%
= 𝜁2
$
<⁄
=
∵ 𝑐% = 2 𝑘𝑚
𝐹#+
𝑘
𝑋=
𝜔
1−
𝜔$
% %
𝜔 %
+ 𝜁2
𝜔$
This is the amplitude for damped forced vibration with
constant harmonic excitation.
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#2
3$ = 𝛿%& , is the static deflection of the spring under a force 𝐹'
The frequency of the steady- state forced vibration is the same as that of the
impressed vibration
𝜙 𝑖𝑠 𝑡ℎ𝑒 𝑝ℎ𝑎𝑠𝑒 𝑙𝑎𝑔 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑡𝑜 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣𝑒𝑐𝑡𝑜𝑟
𝑋=
From Equation 5
𝜙 = 𝑡𝑎𝑛&'
𝛿()
𝜔 "
1− 𝜔
$
"
𝜔 "
+ 𝜁2 𝜔
$
𝑐𝜔
𝑘
𝑚𝜔 %
1−
𝑘
= 𝑡𝑎𝑛&'
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𝜔
𝜁2
𝜔$
𝜔 %
1−
𝜔$
Particular solution
𝑥" 𝑡 = 𝑋 𝑠𝑖𝑛 𝜔𝑡 − 𝜙
𝐹19
𝑘
𝑥! 𝑡 =
𝜔 0
1−
𝜔3
0
𝑠𝑖𝑛 𝜔𝑡 − 𝜙
𝜔 0
+ 𝜁2
𝜔3
∵ 𝑥 = 𝑥# +𝑥'
= 𝑋𝑒
()$A *
𝐹#
𝑠𝑖𝑛 𝜔* 𝑡 + 𝜙+ +
1−
𝜔
𝜔$
⁄
𝑘
" "
+ 𝜁2
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𝜔
𝜔$
"
𝑠𝑖𝑛 𝜔𝑡 − 𝜙
𝑥%
𝑥&
𝑥 = 𝑥4 +𝑥!
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VIBRATION ISOLATION AND TRANSMISSIBILITY
• Machines having unbalanced force produces vibration.
• This vibration is transmitted to the foundation.
• To reduce this vibration we use spring/Dampers or vibration
isolating material.
“Transmissibility is defined as the ratio of the force
transmitted (to the foundation) to the force applied. It is a
measure of the effectiveness of the vibration isolating
material”
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Transmitted force is the vector sum of the spring force (𝑘𝑋()* )and
damping force (cω𝑋()* ).
𝐹' =
(𝑘𝑋)" +(cω𝑋)"
𝐹' = 𝑋 (𝑘)" +(cω)"
𝐹#
𝐹' =
(𝑘)" +(cω)"
𝑘 − 𝑚𝜔 " " + 𝑐𝜔 "
𝐹+ =
𝐹#
𝑐 %
1 + ( ω)
𝑘
𝑚 % %
𝑐
1− 𝜔
+ ( 𝜔 )%
𝑘
𝑘
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𝐹#
𝐹+ =
𝜔 %
1 + (2𝜁
)
𝜔$
𝜔
1−
𝜔$
Transmissibility,
𝐹+
𝜀=
=
𝐹#
% %
𝜔 %
+ (2𝜁
)
𝜔$
𝜔 %
1 + (2𝜁
)
𝜔$
𝜔
1−
𝜔$
% %
𝜔 %
+ (2𝜁
)
𝜔$
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At resonance
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WHIRLING OF SHAFTS
When a rotor is mounted on a shaft, its centre of mass does not usually coincide
with the centre line of the shaft.
Therefore, when the shaft rotates, it is subjected to a centrifugal force which
makes the shaft bend in the direction of eccentricity of the centre of mass.
This further increases the eccentricity, and hence the centrifugal force.
In this way, the effect is cumulative and ultimately the shaft may even fail. The
bending of the shaft depends upon the eccentricity of the centre of mass of the
rotor as also upon the speed at which the shaft rotates.
“Critical or whirling or whipping speed is the speed at which the shaft tends to
vibrate violently in the transverse direction.”
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Thus when ω = 𝜔! , the deflection y is infinitely large
(resonance occurs) and the speed ω is the critical speed, i.e.
If the speed of the shaft is increased rapidly beyond the critical speed, ω > 𝜔!
"! $
or ( ) < 1 or y is negative.
#
This means that the shaft deflects in the opposite direction.
As the speed continues to increase, y approaches the value e or the centre of
mass of the rotor approaches the centre line of rotation. This principle is used in
running high-speed turbines by speeding up the rotor rapidly or beyond the
critical speed.
When y approaches the value of e, the rotor runs steadily.
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