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7/26/2019 Whirling of Shafts
http://slidepdf.com/reader/full/whirling-of-shafts 1/31
Introductio
n
Whirling of shafts
In engineering, we have seen many
applications of shaft and a rotor system.
Power transmitting shafts always have
either gear, pulley, sprocket, rotor or a
disc attached to a shaft as shown in theFigure
7/26/2019 Whirling of Shafts
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Introductio
n
Whirling of shafts
Shaft
disc
bearings
Problems in shaft and a rotor
systems:
(i !nbalance in rotor"disc
(ii Improper assembly
(iii #eaker bearings
7/26/2019 Whirling of Shafts
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Introductio
n
Whirling of shafts
Unbalance in rotor / disc
$op view of a rotor
rotor
%eometric
centre
&ass
centree
m
For perfect balancing
(i &ass centre (centre
of gravity has to co'
inside with the
geometric centre
(ii m.e unbalance )
7/26/2019 Whirling of Shafts
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Introductio
n
Whirling of shafts
Unbalance in rotor / disc
$op view of a disc
e
m
ω
*entrifugal force
7/26/2019 Whirling of Shafts
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Whirling of shafts
Static dynamic
7/26/2019 Whirling of Shafts
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Whirling of shafts
$op view of the disc
P' %eometric center
%' centre of gravity
+' center of rotation
+ P %
d e
otating shafts tend
to bend out at certain
speed and whirl in an
undesired manner,which affects the
working of machine
and the shaft may
also fail due to largedeflection at the
center
7/26/2019 Whirling of Shafts
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Whirling is defined as the rotation of planemade by the bent shaft and line of centers
of bearings as shown in Figure
Whirling of shafts
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Whirling of shafts
neglecting damping
Assumptions
(i the disc at the mid'span has an
unbalance
(ii the shaft inertia is negligible
and the shaft stiffness is same
in all directions
(iii any internal damping is
neglected
-
-"
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Whirling of shafts
neglecting damping
+ P %
d e
P' %eometric center
%' centre of gravity
+' center of rotation
e' eccentricity
d' deflection of shaft
*entrifugal force
$op view of the disc
estoring force(spring force) /.d ω
e(dm0+
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Whirling of shafts
neglecting damping
12uating both the forces
3ivide numerator and
denominator by /
/de(dm0=+
m0/
em0d
−
=
r 4
er d
−
=
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Whirling of shafts
neglecting damping
It is observed from above e2uation that theoretically,
the deflection of the shaft tends to infinity when r 4,
i.e ωωn.
$he speed of the shaft under this condition is referred
as critical speed of shaft .
r 4
er d
−
=
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Whirling of shafts
neglecting damping
*ritical speed
r 4
r
e
d
−
=
) 4 5 6
)
4
5
6
d " e
ω"ωn (r
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Whirling of shafts
neglecting damping
If r 74 8elow critical speed
d is 9ve
which indicates that disc
rotates about + ( centre of
rotation and + and % (*entre of gravity are
opposite each other
%P
+
$op view of the disc
r 4
er d
−
=
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Whirling of shafts
neglecting damping
If r 4 ;bove critical speed
d is <ve
d → 'e, which indicates +,
approaches % and discrotates about center of
gravity.
%P
+
$op view of the disc
r 4
er d
−
=
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Important
It is desired to run the shaft at speed much higher
than the natural fre2uency of the shaft rotor system,
which has reduced whirling of shaft.
Whirling of shafts
neglecting damping
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Whirling of shafts with damping
Damping is the resistance to motion
;ir or
+il
For the analysis of the systems
with damping an additional
assumption is made, i.e the
e=ternal damping force isproportional to the velocity of the
disc at geometric center.
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Whirling of shafts with damping
Force diagram
+
P
%
/d cωd
b
d
e
$hree forces acting on the shaft under e2uilibrium:
(i centrifugal fore at % acts racially outwards
(ii restoring force at point P acts radialy inwards and
(iii damping force at P acts radialy outwards.
mb0
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Whirling of shafts with damping
$op view of the disc at time t
=
y
+
P
% (=g,yg
y
x
d
e
φ
ψ ωt
t e.cos0==g +=t e.sin0yyg +=
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Whirling of shafts with damping
$he e2uation of motion for the system in > ' direction is:
$he e2uation of motion for the system ? ' direction is:
F
)/==c=m g =++
)/==ce.cos00=m(
=++− t
t e.cos0m0/==c=m =++
t e.sin0m0/yycym =++
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$he governing e2uation of motion of the system is:
Solution of governing differential e2uation
-et, =(t, the steady state solution of e2uation of motion is:
;bove 12n has to satisfy governing 12n.
$ransient solution Steady state solution
Whirling of shafts with damping
(t=(t==(t pc +=
>cos(0=(t @−= t
t e.cos0m0/==c=m =++
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Aectorial representation of forces
eference a=is
KX-m 2 X
O
A
B F
t
Impressed force
KX
Spring force
c X
Damping
force m 2 X
Inertia force
X
Displacementvector
Whirling of shafts with damping
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Whirling of shafts with damping
$he steady state response of the system in x, horiBontal direction is :
From triangle +;8
( ) ( ) em0c0>m0/> =+− >
( ) em0c0
m0/> =+−
( ) ( )
c0m0/
em0>
+−=
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Whirling of shafts with damping
3ividing by /
(Cr (4
er
> r +−=
/c0
/m04
/
em0
>
+
−
=
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Whirling of shafts with damping
$he steady state response of the system in x,horiBontal direction is :
Similarly, the steady state response of the system in y,
Aertical direction is :
( ) ( )
@cos(0
Cr 4
=(t
−
+−
= t
r
er
( ) ( )@sin(0
Cr 4y(t
−+−
= t
r
er
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Whirling of shafts with damping
$he deflection of shaft is :
=
y
+
P
% (=g,yg
y
x
d
e
φ
ψ ωt
y=d +=
( ) ( )
Cr 4
er d
r +−
=
7/26/2019 Whirling of Shafts
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Whirling of shafts with damping
$he deflection of shaft is :
*ritical speed
( ) ( )
Cr 4
r
e
d
r +−
=
) 4 5 6
)
4
5
6ξ).)
ξ).4
ξ).
ξ).5
ξ).6
ξ).D
ξ).E)E
ξ4
d " e
ω"ωn (r
7/26/2019 Whirling of Shafts
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Whirling of shafts with damping
$he phase angle is :
−= −
4
r 4
Ctan@
r
) 4 5 6 D)
)
6)
)
G)
4))
4)
46)
4)
4G)
ξ4.)
ξ).E)Eξ).D
ξ).ξ).4ξ)
P h a s e
a n g l e , ψ
ω"ωr (r
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Summary
3ue unbalance in a shaft'rotor system, rotating shafts tend
to bend out at certain speed and whirl in an undesired
manner
#hirling is defined as the rotation of plane made by thebent shaft and line of centers of bearings
$heoretically, the deflection of the shaft tends to infinity
when r 4, i.e ωωn.$he speed of the shaft under this condition is referred as
critical speed of shaft.
7/26/2019 Whirling of Shafts
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Summary
It is desired to run the shaft at speed much higher than the
natural fre2uency of the shaft rotor system
$heory indicates that at higher speeds the shaft tries to
rotate at centre of gravity, and deflection of the shaft is
negligible
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Theory questions
#hat do you understand by critical speed of shafts.
3erive the necessary relations and thus, e=plain what ishappening in the system carrying a shaft having an
unbalanced disc at its centre is operated above and
below critical speed.
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Theory questions
$he speed of the
shaft under the
condition when
r 4, i.e ωωn isreferred as
critical speed of
shaft .
3erive the relation
*ritical speed
r 4
r
e
d
−
=) 4 5 6
)
4
5
6
d " e
ω"ωn (r