Critical Speeds of Shafts tutorial 1.ppt

Critical spoed of shaft

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

2. 2
Introduction
Whirling of shafts
Inengineering,wehaveseenmany
applicationsofshaftandarotorsystem.
Powertransmittingshaftsalwayshave
eithergear,pulley,sprocket,rotorora
discattachedtoashaftasshowninthe
Figure

3. 3
Introduction
Whirling of shafts
Shaft
disc
bearings
Problemsinshaftandarotor
systems:
(i)Unbalanceinrotor/disc
(ii)Improperassembly
(iii)Weakerbearings

4. 4
Introduction
Whirling of shafts
Unbalanceinrotor/disc
Topviewofarotor
rotor
Geometric
centre
Mass
centre
e
m
Forperfectbalancing
(i)Masscentre(centre
ofgravity)hastoco-
insidewiththe
geometriccentre
(ii)m.e=unbalance=0

5. 5
Introduction
Whirling of shafts
Unbalanceinrotor/disc
Topviewofadisc
e
m

Centrifugalforce

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
OPG
de
Rotatingshaftstend
tobendoutatcertain
speedandwhirlinan
undesiredmanner,
whichaffectsthe
workingofmachine
andtheshaftmay
alsofailduetolarge
deflectionatthe
center

8. 8
Whirlingisdefinedastherotationofplane
madebythebentshaftandlineofcenters
ofbearingsasshowninFigure
Whirling of shafts

9. 9
Whirling of shafts
neglecting damping
Assumptions
(i)thediscatthemid-spanhasan
unbalance
(ii)theshaftinertiaisnegligible
andtheshaftstiffnessissame
inalldirections
(iii)anyinternaldampingis
neglected
L
L/2

10. 10
Whirling of shafts
neglecting damping
OPG
de
P-Geometric center
G-centre of gravity
O-center of rotation
e-eccentricity
d-deflection of shaft
Centrifugalforcee)(dmω
2

Top view of the disc
Restoringforce
(springforce)
K.d


11. 11
Whirling of shafts
neglecting dampingKde)(dmω
2

Equating both the forces2
2
mωK
emω
d

 2
2
r1
er
d


Divide numerator and
denominator by K

12. 12
Whirling of shafts
neglecting damping2
2
r1
er
d


Itisobservedfromaboveequationthattheoretically,
thedeflectionoftheshafttendstoinfinitywhenr=1,
i.e=
n.
Thespeedoftheshaftunderthisconditionisreferred
ascriticalspeedofshaft.

13. 132
2
r1
r
e
d


Whirling of shafts
neglecting damping0 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
whichindicatesthatdisc
rotatesaboutO(centreof
rotation)andOandG
(Centreofgravity)are
oppositeeachother2
2
r1
er
d


GP
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 Gand disc
rotates about center of
gravity.2
2
r1
er
d


GP
O
Top view of the disc

16. 16
Important
Itisdesiredtoruntheshaftatspeedmuchhigher
thanthenaturalfrequencyoftheshaftrotorsystem,
whichhasreducedwhirlingofshaft.
Whirling of shafts
neglecting damping

17. 17
Whirling of shafts with damping
Dampingistheresistancetomotion
Airor
Oil
Fortheanalysisofthesystems
withdampinganadditional
assumptionismade,i.ethe
externaldampingforceis
proportionaltothevelocityofthe
discatgeometriccenter.

18. 18
Whirling of shafts with damping
Force diagram
O
P
G
Kd
cd
b
d
embω
2
Threeforcesactingontheshaftunderequilibrium:
(i)centrifugalforeatGactsraciallyoutwards
(ii)restoringforceatpointPactsradialyinwardsand
(iii)dampingforceatPactsradialyoutwards.

19. 19
Whirling of shafts with damping
Top view of the disc at time t
x
y
O
P
G (x
g,y
g)
y
x
d
e


tte.cosωxx
g
 te.sinωyy
g


20. 20
Whirling of shafts with damping
The equation of motion for the system in X -direction is:0Kxxcxm
g
 0Kxxc)e.cosωωxm(
2
  t te.cosωmωKxxcxm
2

The equation of motion for the system Y-direction is:te.sinωmωKyycym
2

F

21. 21
The governing equation of motion of the system is:
Solution of governing differential equation(t)x(t)xx(t)
pc

Let, x(t), the steady state solution of equation of motion is: )Xcos(ωx(t) ψ t
Above Eqn has to satisfy governing Eqn.
Transient solutionSteady state solution
Whirling of shafts with dampingte.cosωmωKxxcxm
2


22. 22
Vectorial representation of forces
Reference axis
KX-m
2
X
O
A
B
F
t
Impressed force
KX

Spring force
cX
Damping
force
m
2
X
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 :  emωcωXmωKX
222
2
 X
From triangle OAB e
2
mω
2
cω
2
2
mωK
2
X 



















  
22
2
2
cωmωK
emω
X



24. 24222
2
)(2ξ)r(1
er
X
r
 22
2
2
K
cω
K
mω
1
K
emω
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,
horizontaldirection is : 
ψ)cos(ω
2ξr1
x(t)
22
2


 t
r
er
2
Similarly, the steady state response of the system in y,
Vertical direction is : 
ψ)sin(ω
2ξr1
y(t)
22
2


 t
r
er
2

26. 26
Whirling of shafts with damping
The deflection of shaft is :
x
y
O
P
G (x
g,y
g)
y
x
d
e


t22
yxd   
22
2
2
2ξr1
er
d
r


27. 27
Whirling of shafts with damping
The deflection of shaft is : 
22
2
2
2ξr1
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
r1
2ξ
tanψ
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
Dueunbalanceinashaft-rotorsystem,rotatingshaftstend
tobendoutatcertainspeedandwhirlinanundesired
manner
Whirlingisdefinedastherotationofplanemadebythe
bentshaftandlineofcentersofbearings
Theoretically,thedeflectionoftheshafttendstoinfinity
whenr=1,i.e=
n.
Thespeedoftheshaftunderthisconditionisreferredas
criticalspeedofshaft.

30. 30
Summary
Itisdesiredtoruntheshaftatspeedmuchhigherthanthe
naturalfrequencyoftheshaftrotorsystem
Theoryindicatesthatathigherspeedstheshafttriesto
rotateatcentreofgravity,anddeflectionoftheshaftis
negligible