7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode

1/4

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

7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode

2/4

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

7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode

3/4

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

7/24/2019 Stability of Cracked Rotorsin the Coupled Vibration Mode

4/4

\^^^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

wnloaded From: http://vibrationacoustics asmedigitalcollection asme org/ on 01/28/2016 Terms of Use: http://www asme org/about asme/terms of use