Stability of Cracked Rotorsin the Coupled Vibration Mode
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Transcript of Stability of Cracked Rotorsin the Coupled Vibration Mode
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7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode
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G. A . Papadopou los
A. D. Dimarogonas
Department ofMechan ica l E ng ineer ing ,
Wash ing ton Un ivers i ty ,
S t . Lou is , MO63130
StabilityofCracked Rotors in the
Coupled Vibrat ion Mode
A
transverse
surface crack is known to add to a shaft a local flexibility due to the
stress strain singularityin thevicinityof the crack tip. This flexibility can be
represented, in the
general case
by wayof
a
6x6
compliance
matrix
describing
the
localflexibility in a short shaft element w hich includes thecrack.This matrix has
off diagonal terms which cause coupling along
the
directions whichareindicated
by
the off-diagonal terms.Inaddition, w hen the shaft rotates the crack opens and
closes. Then the differential
equations
o f
motion have periodically varying
stiffness
coefficients and the solution ca n be expressed as a sumofharmonicfunctions of
time.
A
method for the
determination
of
the intervals
of
instability
of the first and
ofsecondkind
is
developed.
The
results have been presented
in
stability charts
in
the
frequency vs.depth of thecrackdomain. Thecouplingeffect due to thecrack leads
to veryinteresting resultssuch as
new
frequencies andvibration modes.
1 Introduction
Propagating fatigue cracks can have detrimental effects on
the reliabilityof rotating machinery such as turbomachinery,
helicopter rotors, process machinery, etc. In turbomachinery
for power generation, for example, the energy crisis of the ear
ly 70's led toaconsiderable slow-down in the erection of new
power stations. The bulk of the power equipment in this coun
try was commissioned in the 1960's and with a design life of 30
years, it is now in the third and last decade of its life. It is ex
pected that failures duetolow-cycle fatigue initiated cracks
will beamajor causeof machine failures in the next decade.
The existing methodsof crack detection arebasedon the
monitoring of a vibration having twice the critical frequency.
However, this approach has not been used for m onitoring and
diagnosis because such vibration can be caused by other fac
tors.
The discoveryofcoupling modes can lead to an unam
biguous diagnosis because such modes can be initiated only by
surface c racks.
It is known th at the presence of a transverse crack in a struc
tural member introduces local flexibility, which
can be
described by way of a local flexibility matrix, the dimension of
which depends
on
the number
of
degrees
of
freedom con
sidered. The maximum rank
of
such
a
matrix is 6 x 6 .
Such a ma trix was first introduced for beams of rectangular
cross-section with transverse surface cracks by Dimarogonas
and Paipetis (1983)for5 degreesof freedom neglecting tor
sion. They have also reviewed the work on cracked roto rs and
stationary structures.
More recently, Nelson
and
Nataraj (1986) introduced
a
Finite Element methodology and a time-varying (regularly
closing) crack together with a series expansion of the solution
Contr ibuted
by the
Technical Comm ittee
on
Vibra t ion
and
Sound
for
publication
in the JOURNAL OF VIBRATION, ACOUSTICS, STRESS, AND RELIABILITY
IN DESIGN. Manuscript received June 25, 1987.
to yield a better understanding of the subcritical resonance at
integer fractionsof the rotor's critical speed.
In the area of the vibration of cracked stationary structures,
Dimarogonas (1976) introduced thelocal flexibility of the
cracked sectionof the shaft for vibration analysisofbeams
and Chondros and Dimarogonas (1979) for plates. Reduced
section approximations have been used by Kirmser (1944) and
Petroski (1981, 1984), among others, for theapproximate
evaluation of the change of the fundamental frequency due to
existence of the crack.
Coupling between different vibration modes due to cracks
has been first reported by Dimarogonas and Paipetis (1983)
who developeda 5x5 flexibility matrixfor abeamofrec
tangular cross-section. Nondiagonal termsinsuchamatrix
would indicate static and dynamic coupling. This observation
led to analytical andexperimental verification for the ex
istence
of
substan tial and identifiable cou pling between axial
and bending vibration, (Papadopoulos andD imarogonas,
1987) just as between torsional and bending vibration in a sta
tionary shaft (Papado poulos and D imarogonas, 1987) and in a
fractured bone (Nikiforidis et al. 1986).
2
Flexibility
of a
Cracked Shaft
A shaft of diameter
D ,
length
L,
m aterial properties
E
and
v
hasacrack of depth d=a/Dat location =L\/L (Fig. 1).
It is known that the presenceof a transverse surface crack
on a shaft element introduces a local flexibility (Dimarogonas,
1976).
This flexibility hasbeen calculated andexpressed
(Papadopoulos and Dimaro gonas, 1987) by way of a full 6 x 6
compliance matrix due to general loading 6 degreesof
freedom) (Fig.1).
Here three degreesoffreedom, i.e., bending P
4
, P
5
in the
two main directions ofthe cracked section and extension in the
direction of the forceP
x
, will be used.
356 / Vo l .
110, JULY 1988 Transactions of the AS MECopyright 1988 by ASME
wnloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
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7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode
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F i g .
1
.9
.8
.7
.6
.5
.4
.3
.2
C , I - 7
L
\5
^ c
I 0 '
3
10-2 lO ' 10 10'
D imens ion less Comp l i ance
F ig .2(a)
102
10-3 10-2 10-1 10 101
Dimens ionless Compl iance
F ig .
2(b)
The 3 x 3 com pliance matrix for the crack is,
10 2
[ J
=
C
55
/R C
i4
/R C
5)
C 45 / /V C 4 4 / / ? C 4J
'14
C / J
'1 1
J
105
9
8
7
6
5
4
3
2
C
4 -
/ ^ M 4
C45
103
(1)
Wh ere , F
0
=AE/(\-v
2
), A is the shaft cross-section sur
face, R=D/2, and C,-, dimen sionless com plianc e function s of
the relative crack depth a (Fig. 2).
Matrix [C
cr
] relates the two bending moments about x and y
and the extensional force along z to the respective relative
{
F i g .
3
ro tat ions
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Crack
Fig.
4
w h e r e
[ C ^ ]
=
2F
C
M
/R
C,
5
/R
Qi
C
54
/R C
55
/R C
51
C,
C,, i?
T hus du r ing the ro ta t ion the s t i f fness va r ie s wi th t ime
or
wi th
t h e a n g l e . T h i s v a r i a t i o n m a y bee x p r e s s e d byw ayof a t r u n
ca ted , cos ine se r ie s .
[K\ = [K(wt)\ = {K
0
\
+ [K^cos wt + [K
2
]cos 2oit+
[AT
3
]cos
3cot+[K
4
]cos
Aut
4
M =
l
K
J
C0 S
v
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7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode
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\^^^M [
^ A A H W / K J ^ V X W J ^
0 50 100
Frequency
Fig. 7
In the problem at hand , the dimension of m atrix
[D]
is 18 X18
and the computation of the determinants was a modest prob
lem for the computer, to the extent that an exhaustive search
was performed in the (o>,a)plane.
4 Experimental Evidenc e
In 1983, a 300 MW steam turbine of the Lavrion Station of
the Public Power Corporation, G reece, made by ALSTHOM,
experienced violent vibration on the LP rotor of unknown
cause. The velocity vibration probes, located near the bear
ings, were used by the authors to obtain information for the
rotor vibration. The transducer output was analyzed on a
spectrum analyzer and a typical such spectrum is shown in Fig.
7,corresponding to bearing No. 6 of the machine. There was
immediate evidence for existence of a crack. Both the very
high 2 per rev. component and the 12 and 6 Hz peaks cor
responding to one half and one quarter of the critical speed,
respectively, suggested the existence of a deep crack. The
crack could not be precisely located, nor its magnitude precise
ly estimated. Based on the test results the company decided to
dismantle the turbine and an 120 degrees crack was found on
the Low Pressure rotor. The rotor was repaired by the
manufacturer and run subsequently without problems.
On the spectrum of Fig. 7 some other vibration frequencies
were noticed, which could not be related to any of the higher
bending modes. In particular, a succession of 44, 88, 176 Hz
frequencies suggested a relation to either torsional or
longitudinal natural vibration frequencies. Eventually, it was
found that these frequencies were close to subharmonics of the
355 Hz fundamental longitudinal natural frequency
(calculated). Since the vibration measured was in a lateral
direction, it was concluded that the crack resulted in coupling
of longitudinal and lateral vibration. This could only be ex
plained on the basis of the coupling terms in the cracked shaft
flexibility matrix. Indeed, the stability chart of Figs. 5 and 6
show such behavior. Several subharmonics of the longitudinal
and lateral natural frequencies are noticed as thresholds of in
stability. In fact, this analysis did not take damping into ac
count. Therefore, in certain areas of the (CJ, a) plane the
analysis predicts instability which for linear systems means
that vibration will increase beyond bound. Damping and
nonlinearities limit the vibration amplitudes and therefore in
the areas of instability vibration at the respective frequency is
expected, which is substantiated by the test results in Fig. 7.
5 Conclusion
It was shown that a surface crack on a rotating shaft can
yield a variety of unstable areas of operation due to the cou
pling of lateral and longitudinal vibration. This coupling can
only be related to the existence of cracks and can be used for
crack identification. It can also supplement the utilization of
twice the speed of rotation and half the critical speed signals in
the process of crack identification.
This analysis can be extended to surface and solid structures
with very interesting consequences. Applications can be en
visaged in the diagnosis of nonunion healing in cracked bones,
diagnosis of cracks and cracked welds in pressure vessels, reac
tor containment vessels, reactor piping, and finally in real time
quality control of material production processes.
References
Dimarogonas,
A. D., 1976, Vibration Engineering, West Publishers, St . Paul.
Dimarogonas, A. D., and Paipetis , S. A., 1983, Analytical M ethods in Motor
Dynamics, Elsevier-Applied Science Publishers, London .
Dimarogonas, A. D., and Papado poul os, C. A., 1983, "Vibratio n of Crack
ed Shafts in Bending," Journal of Sound and Vibration, Vol. 91, No. 4, pp.
538-593.
Kirmser, P. G ., 1944, "Th e Effect of Discontinuit ies on the Natural Frequen
cy of Beams," Proc.
ASTM
Vol. 44, pp. 897-904.
Nelson, H. D., and
Nata ra j ,
C , 1986, "The Dynamics of a Rotor System
with a Cracked Shaft ," ASME
JOURNAL OF VIBRATION, ACOUSTICS, STRESS, AND
RELIABILITY IN D E S I G N , Vol. 108, pp. 189-196.
Nikiforidis, G ., Bezerianos, A., and Dima rogona s, A. D., 1986, "Vibra tion
Analysis of Bone Healing under Plast ic Cast ," IEEE/SBMI
Conf.,
F t . Worth ,
Texas.
Papadopoulos , C. A., and Dimarogonas, A. D., 1987, "Coupled
Longitudinal and Bending Vibration of Rotating Sha fts,"
Journal of Sound and
Vibration, Vol. 117, No . 1, pp . 81-93.
P a pa dopou los , C. A., and Dimarogonas, A. D., 1987, "Coupling of Bend
ing and Torsional Vibration of a Cracked Timoshenko Shaft ," Ingenieur Ar-
chiv,
Vol. 57, pp. 257-266.
Papadopoulos , C. A., and Dimarogonas, A. D., 1987, "Coupled
Longitudinal and Bending Vibrations of a Cracked Shaft ," ASME
JOURNAL OF
VIBRATION,
ACOUSTICS, STRESS
AN D
RELIABILITY
IN
D E S I G N ,
Vol. 110, pp. 1-8.
Petroski , H. J ., 1984, "On the Cracked Bell ," Journal of Sound and Vibra
tion, Vol. 96, No. 4, pp. 485-493.
Pe troski , H . J ., 1981, "Simple Static and Dynamic Models for the Cracked
Elastic Beam," Int. Journal of Fracture, Vol. 17, pp. R71-R76 .
J o u r n a l o f V ib r a t i o n , A c o u s t i c s , S t r e s s , a n d R e l i a b il i ty i n D e s i g n JULY 1 988 , Vo l . 11 0 /3 59
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