Critical spoed of shaft

1. 1
Critical speeds of shafts
A.CHANDRA MOHANA RAO
02-02-2015

2. 2
Introduction
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 the
Figure

3. 3
Introduction
Whirling of shafts
Shaft
disc
bearings
Problems in shaft and a rotor
systems:
(i) Unbalance in rotor/disc
(ii) Improper assembly
(iii) Weaker bearings

4. 4
Introduction
Whirling of shafts
Unbalance in rotor / disc
Top view of a rotor
rotor
Geometric
centre
Mass
centre
e
m
For perfect balancing
(i) Mass centre (centre
of gravity) has to co-
inside with the
geometric centre
(ii) m.e = unbalance =0

5. 5
Introduction
Whirling of shafts
Unbalance in rotor / disc
Top view of a disc
e
m

Centrifugal force

6. 6
Whirling of shafts
Static dynamic

7. 7
Whirling of shafts
Top view of the disc
P- Geometric center
G- centre of gravity
O- center of rotation
O P G
d e
Rotating 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 large
deflection at the
center

8. 8
Whirling is defined as the rotation of plane
made by the bent shaft and line of centers
of bearings as shown in Figure
Whirling of shafts

9. 9
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
L
L/2

10. 10
Whirling of shafts
neglecting damping
O P G
d e
P- Geometric center
G- centre of gravity
O- center of rotation
e- eccentricity
d- deflection of shaft
Centrifugal force
e)
(d
mω2

Top view of the disc
Restoring force
(spring force)
K.d


11. 11
Whirling of shafts
neglecting damping
Kd
e)
(d
mω2


Equating both the forces
2
2
mω
K
e
mω
d


2
2
r
1
er
d


Divide numerator and
denominator by K

12. 12
Whirling of shafts
neglecting damping
2
2
r
1
er
d


It is observed from above equation that theoretically,
the deflection of the shaft tends to infinity when r =1,
i.e =n.
The speed of the shaft under this condition is referred
as critical speed of shaft.

13. 13
2
2
r
1
r
e
d


Whirling of shafts
neglecting damping
0 1 2 3 4
0
1
2
3
4
d/e
/n
(r)
Critical speed

14. 14
Whirling of shafts
neglecting damping
If r <1 Below critical speed
d is +ve
which indicates that disc
rotates about O ( centre of
rotation) and O and G
(Centre of gravity) are
opposite each other
2
2
r
1
er
d


G
P
O
Top view of the disc

15. 15
Whirling of shafts
neglecting damping
If r >1 Above critical speed
d is –ve
d  -e, which indicates O,
approaches G and disc
rotates about center of
gravity.
2
2
r
1
er
d


G
P
O
Top view of the disc

16. 16
Important
It is desired to run the shaft at speed much higher
than the natural frequency of the shaft rotor system,
which has reduced whirling of shaft.
Whirling of shafts
neglecting damping

17. 17
Whirling of shafts with damping
Damping is the resistance to motion
Air or
Oil
For the analysis of the systems
with damping an additional
assumption is made, i.e the
external damping force is
proportional to the velocity of the
disc at geometric center.

18. 18
Whirling of shafts with damping
Force diagram
O
P
G
Kd cd
b
d
e
mb
ω2
Three forces acting on the shaft under equilibrium:
(i) centrifugal fore at G acts racially outwards
(ii) restoring force at point P acts radialy inwards and
(iii) damping force at P acts radialy outwards.

19. 19
Whirling of shafts with damping
Top view of the disc at time t
x
y
O
P
G (xg,yg)
y
x
d
e


t
t
e.cosω
x
xg 

t
e.sinω
y
yg 


20. 20
Whirling of shafts with damping
The equation of motion for the system in X - direction is:
0
Kx
x
c
x
m g 

 


0
Kx
x
c
)
e.cosω
ω
x
m( 2



 

 t
t
e.cosω
mω
Kx
x
c
x
m 2


 


The equation of motion for the system Y - direction is:
t
e.sinω
mω
Ky
y
c
y
m 2


 


F

21. 21
The governing equation of motion of the system is:
Solution of governing differential equation
(t)
x
(t)
x
x(t) p
c 

Let, x(t), the steady state solution of equation of motion is:
)
Xcos(ω
x(t) ψ

 t
Above Eqn has to satisfy governing Eqn.
Transient solution Steady state solution
Whirling of shafts with damping
t
e.cosω
mω
Kx
x
c
x
m 2


 



22. 22
Vectorial representation of forces
Reference axis
KX-m2X
O
A
B F
t
Impressed force
KX

Spring force
cX
Damping
force m2X
Inertia force
X

Displacement
vector
Whirling of shafts with damping

23. 23
Whirling of shafts with damping
The steady state response of the system in x, horizontal
direction is :
    e
mω
cω
X
mω
KX 2
2
2
2


 X
From triangle OAB
  e
2
mω
2
cω
2
2
mω
K
2
X 






















   2
2
2
2
cω
mω
K
e
mω
X




24. 24
2
2
2
2
)
(2ξ
)
r
(1
e
r
X
r



2
2
2
2
K
cω
K
mω
1
K
e
mω
X

















Whirling of shafts with damping
Dividing by K

25. 25
Whirling of shafts with damping
The steady state response of the system in x,
horizontal direction is :
   
ψ)
cos(ω
2ξ
r
1
x(t)
2
2
2



 t
r
er2
Similarly, the steady state response of the system in y,
Vertical direction is :
   
ψ)
sin(ω
2ξ
r
1
y(t)
2
2
2



 t
r
er2

26. 26
Whirling of shafts with damping
The deflection of shaft is :
x
y
O
P
G (xg,yg)
y
x
d
e


t
2
2
y
x
d 

   2
2
2
2
2ξ
r
1
er
d
r




27. 27
Whirling of shafts with damping
The deflection of shaft is :
   2
2
2
2
2ξ
r
1
r
e
d
r



0 1 2 3 4
0
1
2
3
4
=0.0
=0.1
=0.2
=0.3
=0.4
=0.5
=0.707
=1
d/e
/n
(r)
Critical speed

28. 28
Whirling of shafts with damping
The phase angle is :







 
2
1
r
1
2ξ
tan
ψ
r
0 1 2 3 4 5
0
20
40
60
80
100
120
140
160
180
=1.0
=0.707
=0.5
=0.2
=0.1
=0
Phase
angle,

/r
(r)

29. 29
Summary
Due unbalance in a shaft-rotor system, rotating shafts tend
to bend out at certain speed and whirl in an undesired
manner
Whirling is defined as the rotation of plane made by the
bent shaft and line of centers of bearings
Theoretically, the deflection of the shaft tends to infinity
when r =1, i.e =n.
The speed of the shaft under this condition is referred as
critical speed of shaft.

30. 30
Summary
It is desired to run the shaft at speed much higher than the
natural frequency of the shaft rotor system
Theory indicates that at higher speeds the shaft tries to
rotate at centre of gravity, and deflection of the shaft is
negligible