Short Notes: Vibration Isolation | Short Notes for Mechanical Engineering PDF Download

 Page 1


Vibration Isolation
Importance of vibration isolation
• When an unbalanced machine is installed on the foundation, it produces 
vibration in the foundation.
• So in order to prevent these vibrations or to minimize the transmission of 
forces to the foundation, vibration isolation is important.
Methods of isolating the vibration
• High speed engines / machines mounted on foundation and supports cause 
vibrations of excessive amplitude because of the unbalanced forces.
• It can be minimized by providing "spring - damper" etc.
• The materials used for vibration isolation are rubber, felt cork, etc. these are 
placed between the foundations and vibrating body.
Force Transmissibility
• Consider the machine mounted in springs as shown in fig. if the spring used 
are helical and of steel, damping may be neglected.
• However, if leaf springs or rubber and cork padding are used, damping case, 
spring force can be considered to be of two parts-one proportional to 
displacement and the other proportional to velocity.
• So our model of fig. can be used to represent realistic conditions.
Page 2


Vibration Isolation
Importance of vibration isolation
• When an unbalanced machine is installed on the foundation, it produces 
vibration in the foundation.
• So in order to prevent these vibrations or to minimize the transmission of 
forces to the foundation, vibration isolation is important.
Methods of isolating the vibration
• High speed engines / machines mounted on foundation and supports cause 
vibrations of excessive amplitude because of the unbalanced forces.
• It can be minimized by providing "spring - damper" etc.
• The materials used for vibration isolation are rubber, felt cork, etc. these are 
placed between the foundations and vibrating body.
Force Transmissibility
• Consider the machine mounted in springs as shown in fig. if the spring used 
are helical and of steel, damping may be neglected.
• However, if leaf springs or rubber and cork padding are used, damping case, 
spring force can be considered to be of two parts-one proportional to 
displacement and the other proportional to velocity.
• So our model of fig. can be used to represent realistic conditions.
• The force transmitted to the foundations is by spring (KX) and by dashpot 
(ccjX) because these are the only connections.
• Spring force and damping force vectors are at right angles. So the amplitude 
of force transmitted.
F tr= ^/(K X ): + (c^X ):
• The ratio of the amplitude of the force transmitted to that of the exciting force 
is called transmissibility of force and is
Which is non-dimensional quantity.
T ra n s m is s ib ility o f M otio n
• Let us find the motion of the mass m in fig. when the support is given a 
motion
y = yO sin ut -* (1)
Page 3


Vibration Isolation
Importance of vibration isolation
• When an unbalanced machine is installed on the foundation, it produces 
vibration in the foundation.
• So in order to prevent these vibrations or to minimize the transmission of 
forces to the foundation, vibration isolation is important.
Methods of isolating the vibration
• High speed engines / machines mounted on foundation and supports cause 
vibrations of excessive amplitude because of the unbalanced forces.
• It can be minimized by providing "spring - damper" etc.
• The materials used for vibration isolation are rubber, felt cork, etc. these are 
placed between the foundations and vibrating body.
Force Transmissibility
• Consider the machine mounted in springs as shown in fig. if the spring used 
are helical and of steel, damping may be neglected.
• However, if leaf springs or rubber and cork padding are used, damping case, 
spring force can be considered to be of two parts-one proportional to 
displacement and the other proportional to velocity.
• So our model of fig. can be used to represent realistic conditions.
• The force transmitted to the foundations is by spring (KX) and by dashpot 
(ccjX) because these are the only connections.
• Spring force and damping force vectors are at right angles. So the amplitude 
of force transmitted.
F tr= ^/(K X ): + (c^X ):
• The ratio of the amplitude of the force transmitted to that of the exciting force 
is called transmissibility of force and is
Which is non-dimensional quantity.
T ra n s m is s ib ility o f M otio n
• Let us find the motion of the mass m in fig. when the support is given a 
motion
y = yO sin ut -* (1)
sinw t
• It is clear that because of the presence of damping the mass m will have a 
displacement lagging behind that of the support.
Let its motion be
x = X sin (ut - < p ) -* (2)
The equation of motion of mass m is
mx+c(x-y)+K(x-y) = 0 -»(3)
substituting for x, x and x from equation (2) and for y and y from equation (1), 
equation (3) yields.
CU A
(») Relative positions of force vectors (b) Triangle of forces
sin(u.-t — o) — Cu.X Sin u;t-o + —
-K X sin((w t— o) — Cu;y0 sin . d - -
9
+ Kv0 sin ut=0
• Each one of the terms in equation represents a force. It may therefore be 
represented by a rotating vector.
Plotting the vector gives the diagram of fig. 
From the triangle of forces
X =
m „- — c u
— * (4)
0 + 3 = tan 1— —— ^ — * • (5)
K = m u.*
3= tan'1
C A
K
- ( 6)
Page 4


Vibration Isolation
Importance of vibration isolation
• When an unbalanced machine is installed on the foundation, it produces 
vibration in the foundation.
• So in order to prevent these vibrations or to minimize the transmission of 
forces to the foundation, vibration isolation is important.
Methods of isolating the vibration
• High speed engines / machines mounted on foundation and supports cause 
vibrations of excessive amplitude because of the unbalanced forces.
• It can be minimized by providing "spring - damper" etc.
• The materials used for vibration isolation are rubber, felt cork, etc. these are 
placed between the foundations and vibrating body.
Force Transmissibility
• Consider the machine mounted in springs as shown in fig. if the spring used 
are helical and of steel, damping may be neglected.
• However, if leaf springs or rubber and cork padding are used, damping case, 
spring force can be considered to be of two parts-one proportional to 
displacement and the other proportional to velocity.
• So our model of fig. can be used to represent realistic conditions.
• The force transmitted to the foundations is by spring (KX) and by dashpot 
(ccjX) because these are the only connections.
• Spring force and damping force vectors are at right angles. So the amplitude 
of force transmitted.
F tr= ^/(K X ): + (c^X ):
• The ratio of the amplitude of the force transmitted to that of the exciting force 
is called transmissibility of force and is
Which is non-dimensional quantity.
T ra n s m is s ib ility o f M otio n
• Let us find the motion of the mass m in fig. when the support is given a 
motion
y = yO sin ut -* (1)
sinw t
• It is clear that because of the presence of damping the mass m will have a 
displacement lagging behind that of the support.
Let its motion be
x = X sin (ut - < p ) -* (2)
The equation of motion of mass m is
mx+c(x-y)+K(x-y) = 0 -»(3)
substituting for x, x and x from equation (2) and for y and y from equation (1), 
equation (3) yields.
CU A
(») Relative positions of force vectors (b) Triangle of forces
sin(u.-t — o) — Cu.X Sin u;t-o + —
-K X sin((w t— o) — Cu;y0 sin . d - -
9
+ Kv0 sin ut=0
• Each one of the terms in equation represents a force. It may therefore be 
represented by a rotating vector.
Plotting the vector gives the diagram of fig. 
From the triangle of forces
X =
m „- — c u
— * (4)
0 + 3 = tan 1— —— ^ — * • (5)
K = m u.*
3= tan'1
C A
K
- ( 6)
By transforming the above equations to the non-dimensional form, the following 
are obtained
tr —
- ( 7 )
1 —
j^tan 2c —
- ( 8 )
- ( 9 )
Equation, giving the expression for transmissibility o motion is the same as 
equation (7) giving the expression for force transmissibility. 
Transmissibility
• Transmissibility is defined as the ratio of the force transmitted to the force 
applied. It is a measure of the effectiveness of the vibration isolating material.
Transmissibility
F: y j l + V Z r ) 1
Fo y j( l- r:);+(2#r)*
Transmissibilrly curve using damping ratio
Page 5


Vibration Isolation
Importance of vibration isolation
• When an unbalanced machine is installed on the foundation, it produces 
vibration in the foundation.
• So in order to prevent these vibrations or to minimize the transmission of 
forces to the foundation, vibration isolation is important.
Methods of isolating the vibration
• High speed engines / machines mounted on foundation and supports cause 
vibrations of excessive amplitude because of the unbalanced forces.
• It can be minimized by providing "spring - damper" etc.
• The materials used for vibration isolation are rubber, felt cork, etc. these are 
placed between the foundations and vibrating body.
Force Transmissibility
• Consider the machine mounted in springs as shown in fig. if the spring used 
are helical and of steel, damping may be neglected.
• However, if leaf springs or rubber and cork padding are used, damping case, 
spring force can be considered to be of two parts-one proportional to 
displacement and the other proportional to velocity.
• So our model of fig. can be used to represent realistic conditions.
• The force transmitted to the foundations is by spring (KX) and by dashpot 
(ccjX) because these are the only connections.
• Spring force and damping force vectors are at right angles. So the amplitude 
of force transmitted.
F tr= ^/(K X ): + (c^X ):
• The ratio of the amplitude of the force transmitted to that of the exciting force 
is called transmissibility of force and is
Which is non-dimensional quantity.
T ra n s m is s ib ility o f M otio n
• Let us find the motion of the mass m in fig. when the support is given a 
motion
y = yO sin ut -* (1)
sinw t
• It is clear that because of the presence of damping the mass m will have a 
displacement lagging behind that of the support.
Let its motion be
x = X sin (ut - < p ) -* (2)
The equation of motion of mass m is
mx+c(x-y)+K(x-y) = 0 -»(3)
substituting for x, x and x from equation (2) and for y and y from equation (1), 
equation (3) yields.
CU A
(») Relative positions of force vectors (b) Triangle of forces
sin(u.-t — o) — Cu.X Sin u;t-o + —
-K X sin((w t— o) — Cu;y0 sin . d - -
9
+ Kv0 sin ut=0
• Each one of the terms in equation represents a force. It may therefore be 
represented by a rotating vector.
Plotting the vector gives the diagram of fig. 
From the triangle of forces
X =
m „- — c u
— * (4)
0 + 3 = tan 1— —— ^ — * • (5)
K = m u.*
3= tan'1
C A
K
- ( 6)
By transforming the above equations to the non-dimensional form, the following 
are obtained
tr —
- ( 7 )
1 —
j^tan 2c —
- ( 8 )
- ( 9 )
Equation, giving the expression for transmissibility o motion is the same as 
equation (7) giving the expression for force transmissibility. 
Transmissibility
• Transmissibility is defined as the ratio of the force transmitted to the force 
applied. It is a measure of the effectiveness of the vibration isolating material.
Transmissibility
F: y j l + V Z r ) 1
Fo y j( l- r:);+(2#r)*
Transmissibilrly curve using damping ratio
At resonance
r =
When no damper is used £ = 0
t
V l+ (2 * r ):
1
1
1 -
, ft J
Transmissibility curve with l 0.1 to 1
• Frequency response and phase relationship of single degree of freedom 
system with base excitation transmissibility and phase relatively of a system. 
° At
c o = V2 co r (s = 1 ),
the response is same as the magnitude of the excitation amplitude for 
the values of damping.
° For
c o > 4 2 a (s< 1):
the maximum experiences lower amplitude of vibration than the base 
excitation.
° For
c o < ¦j2aH(s>l)1Ft >F0
damping is not useful, actually, it increases the amplitude for a given 
system.
° At resonance o j = u)n , the phase angle is not 90°.
A Single Degree of Freedom System Excited by Support Motion
¥ H(* )=
1+ (2 grT 
( l - r ) : + (2 ^ r):
iz r
tan 6 =
1- r* + (2 ^ r) 
H(c j) is maximum when
r - 7 7 V ' / 1 + ^ ~ 1