H. AbdiDepartment of Mechanical

and Industrial Engineering,

Northeastern University,

Boston, MA 02115

H. Nayeb-HashemiDepartment of Mechanical

and Industrial Engineering,

Northeastern University,

Boston, MA 02115

A. M. S. HamoudaDepartment of Mechanical

and Industrial Engineering,

Qatar University,

Doha 2713, Qatar

A. Vaziri1Department of Mechanical

and Industrial Engineering,

Northeastern University,

Boston, MA 02115

e-mail: vaziri@coe.neu.edu

Torsional Dynamic Responseof a Shaft With Longitudinaland Circumferential CracksTurbo generator shafts are often subjected to cyclic torsion resulting in formation oflarge longitudinal cracks as well as circumferential cracks. The presence of these crackscould greatly impact the shaft resonance frequencies. In this paper, dynamic response ofa shaft with longitudinal and circumferential cracks is investigated through a comprehen-sive analytical study. The longitudinally cracked section of the shaft was modeled as anuncracked shaft with reduced torsional rigidity. Torsional rigidity correction factor (i.e.,the ratio of torsional rigidity of the cracked shaft to that of the uncracked shaft) wasobtained from finite element analysis and was shown to be only a function of crack depthto the shaft radius. The resonance frequency and frictional energy loss of a shaft with alongitudinal crack were found little affected by the presence of the crack as long as thecrack depth was less than 20% of the shaft radius even if the entire shaft is cracked longi-tudinally. Moreover, we showed that the longitudinal crack location could be more con-veniently identified by monitoring the slope of the torsional response along the shaft. Thecircumferential crack was modeled as a torsional spring with a torsional damping. Thetorsion spring and damping constants were obtained using fracture mechanics. For ashaft with both a longitudinal crack and a circumferential crack, the resonance frequencywas governed by the longitudinal crack when the circumferential crack depth was lessthan 30% of the shaft radius. [DOI: 10.1115/1.4028609]

Keywords: longitudinal crack, circumferential crack, cyclic torsion, shaft resonancefrequency

1 Introduction

Since turbine rotor failures at Southern California Edison’sMohave station in 1970 and 1971, preventing such a failure hasreceived a significant attention. In order to prevent shaft failure,both dynamic response of the shaft due to the disturbance in elec-trical line and subsequent fatigue crack initiation and propagationmust be investigated. Electrical lines disturbance subjects theturbo generator shafts to a cyclic torsion with large amplitudes of-ten causing a plastic deformation. It has been observed that such aphenomena causes formation of large longitudinal cracks alongthe shaft axis which constitutes its weak planes due to alignmentof inclusions along the shaft axis when it is manufactured byextrusion. Circumferential cracks are also initiated in these shafts,Fig. 1. However, their growth rate is significantly lower than thelongitudinal cracks due to the formation of a complex fracturesurface. Dimarogonas and Massouros first studied the effects ofcircumferential cracks on the torsional dynamic response of shafts[1,2]. They modeled cracked shaft as two solid shafts connectedin the cracked region with a linear torsional spring. The local flex-ibility matrix of the shaft due to the presence of the crack wascomputed using energy release rates concept. Papadopoulos andDimarogonas extended the work to show the coupling of longitu-dinal and bending vibrations of a rotating shaft with a transversesurface crack which formed a basis for crack detection [3]. It isworth noting that they considered only transverse thumb nailcracks; however, in this paper, we investigated dynamic responseof a shaft with the presence of both longitudinal and circumferen-tial cracks. Chondros et al. [4–13] studied the bending and tor-sional vibrations of circumferentially cracked shafts. They

derived the equations of motion using Hu–Washizu–Barr varia-tional formulation. Fracture mechanics was used to model thecrack as a continuous flexibility, and Rayleigh quotient was usedto approximate the natural frequencies of the cracked rod. All theabove studies show that the resonance frequencies of a shaft aresignificantly changed with the presence of such cracks in the shaftwhich allows for detection of crack and estimation of its location[5,11,13–18]. In addition, Shih and Chung studied the vibrationsof a flexible connecting rod with the breathing crack in a slider-crank mechanism and observed a similar pattern of changingresonance frequencies [19].

Crack surface interactions play an important role in determiningthe torsional dynamic response of shafts. Breathing crack theoryhas been developed to quantify the effect of frictional energy losson the shaft resonance frequencies [1,13]. Nayeb-Hashemi et al.[20–24], studied mode III crack growth behavior in commercialrotor steel. They observed that in contrast to the circumferentialcrack growth which results in a complex fracture surface resem-bling factory roof, longitudinal cracks form a smooth fracturesurface. Crack surface interactions in the circumferential crackgrowth reduce the effective stress intensity factor at the crack tip.This reduction in the stress intensity factor is almost absent in thelongitudinal crack, due to much smoother crack surfaces. Vaziriand Hashemi [25–27] in a series of investigations studied thedynamic response of shafts with circumferential cracks consider-ing the complexity of the cracked surfaces. They investigated theeffect of crack surface morphology (factory roof, pitch height andwave length) on the effective stress intensity factor and energyloss and showed the effective stress intensity factor depends onthe applied stress intensity factor and fracture surface morpholo-gies. The crack surface interactions diminish at high applied stressintensity factors. They also derived the loss factor associated withinteraction of crack surfaces and crack tip cyclic plasticity. Theyillustrated for small applied stress intensity factors crack surfaceinteractions dominates the energy loss and for high applied stress

1Corresponding author.Contributed by the Technical Committee on Vibration and Sound of ASME for

publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript receivedNovember 5, 2013; final manuscript received September 17, 2014; published onlineOctober 6, 2014. Assoc. Editor: Yukio Ishida.

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intensity factors plasticity dominates the energy loss. Their resultsshow that the natural frequency of the circumferentially crackedshafts decreases with the length of the crack; however, if thecracked surface interaction is severe, the natural frequencyapproaches to that of uncracked shafts.

Although there are substantial investigations on the effects ofcircumferential cracks on the torsional dynamic response of theshafts, it seems that the literature lacks a comprehensive investiga-tion on the effects of longitudinal cracks and their interaction withthe circumferential cracks. This paper addresses the vibrationalcharacteristics of shafts with longitudinal cracks as well as cir-cumferential cracks. The results presented here indicate thatdynamic response of a shaft could be significantly compromisedwith the presence of the longitudinal cracks. In Sec. 2, the theoreti-cal method to model longitudinal and circumferential cracks isdescribed, equations of motion of different segments of the shaft arederived, and corresponding boundary conditions are specified. InSec. 3, dynamic response of a shaft with a longitudinal crack onlyand then a shaft with a circumferential crack only and finally a shaftwith both longitudinal and circumferential cracks are discussedusing the results from a comprehensive analytic parametric study.

2 Theoretical and Numerical Investigations

A shaft with a longitudinal crack is modeled as an uncrackedshaft with the same radius but with reduced torsional rigidity. Ashaft with a circumferential crack is modeled as two solid shaftsconnected by a torsional spring and a torsional damper at thecrack location. In Sec. 2, the equations of motion of a shaft withlongitudinal and circumferential cracks under cyclic torsion arederived and corresponding boundary conditions are specified.

2.1 Torsional Rigidity of a Shaft With a LongitudinalCrack. Torsional rigidity of a shaft with a longitudinal crack withvarious crack depths was obtained by finite element analysis usingABAQUS finite element code. Figure 2(a) shows typical shaft geom-etry and its boundary conditions. The shaft was fixed at one end

and attached to a rigid plate at the other end which prevents thewarping of its cross section. The rigid plate was subjected to tor-sion while its rotation was monitored. Singular crack tip elementswere used to simulate crack tip singularity. Optimum mesh sizewas obtained by mesh refinements. The analysis was performedfor various shaft radius R and crack depth CL. The torsional rigid-ity of the cracked shaft is obtained from

T ¼ GJ�dhdx

(1)

where T is the applied torque, GJ* is the torsional rigidity of thecracked shaft, and dh=dx is the rotation gradient along the shaft.The torsional rigidity of the cracked shaft is related to thetorsional rigidity of the uncracked shaft by defining

GJ� ¼ GJ � f (2)

where GJ is the torsional rigidity of the uncracked shaft (G is theshear modulus and J is the area polar moment of inertia) and f isthe correction factor. Here we assumed crack surface interactionis very small, thus it will not affect the shaft torsional rigidity.The shaft torsional rigidity is thus reduced with the presence ofthe longitudinal crack. Figure 2(b) shows the correction factor f isindependent of the shaft radius and is only a function of longitudi-nal crack depth to the shaft radius, CL/R. Using the curve fittingtechnique, the torsional rigidity correction factor is expressed as

f ¼ fCL

R

� �¼ 3:76

CL

R

� �5

�12:82CL

R

� �4

þ16:16CL

R

� �2

� 8:05CL

R

� �2

þ0:15CL

R

� �þ 1

(3)

In order to analyze the torsional vibrations of a shaft with a longi-tudinal crack, the cracked shaft is modeled as an equivalentuncracked shaft with the same radius and reduced torsional rigid-ity of GJ � f , Fig. 2(b).

2.2 Frictional Energy Loss Between LongitudinallyCracked Surfaces and Evaluation of the Corresponding LossFactor. The effect of energy loss due to the crack surface interac-tions is taken into account by considering shear modulus as acomplex number, Gð1þ igLÞ, where gL is the loss factor associ-ated with the crack surface interactions. The shear displacementalong the crack at each cross section is presented by Broberg[5,28] as

Fig. 2 (a) Finite element model of a shaft with a longitudinal crack subjected to torsion and(b) torsional rigidity correction factor of a shaft with a longitudinal crack computed by finiteelement technique, Eq. (3)

Fig. 1 Circumferential and longitudinal cracks formation in ashaft subjected to cyclic torsion

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u3 ¼2kIII

GffiffiffiffiffiffiffiffiffipCL

p ðC2L � y2Þ0:5 (4)

As shown in Fig. 3, y is the distance from outer surface, CL is thecrack depth, G is the shear modulus of the shaft, and kIII is theapplied mode III stress intensity factor which is given as [28]

kIII ¼ GgðR;CLÞdhdx

(5)

Here, dh=dx is the rotation gradient along the shaft and

gðR;CLÞ ¼CLð2R� CLÞ2

4pðR� CLÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pRCL

2R� CL

r4RðR� CLÞ þ 3C2

L

2CL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðR� CLÞ

p atan2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðR� CLÞ

pCL

!� 3

" #

(6)

Substituting Eq. (5) into Eq. (4) results in

u3 ¼2gðR;CLÞ

dhdxffiffiffiffiffiffiffiffiffi

pCL

p ðC2L � y2Þ0:5 (7)

The interaction of asperities between the mutual crack surfacescreates shear stress along the cracked surfaces. By assuming aconstant shear stress profile between cracked surfaces such ass ¼ asY where sY is the material shear strength and a is a con-stant, the frictional energy loss per unit length of the shaft in acycle can be evaluated from

DW ¼ 4

ðCL

0

ðasYdyÞðu3Þ (8)

The energy loss in a cycle can be found from Eqs. (7) and (8) ispresented as

DW ¼ 2ffiffiffipp

C1:5L asYgðR;CLÞ

dhdx

� �max

(9)

The potential energy per unit length of the shaft is obtained as

W ¼ð

1

2Gðs2ÞdA ¼

ð1

2GGr

dhdx

� �2

dA ¼ GJ�

2

dhdx

� �2

(10)

The loss factor is defined as the ratio of energy loss per maximumpotential energy of the structure in each cycle

gL ¼DW

2pWmax

(11)

Substituting Eqs. (9) and (10) into Eq. (11) yields

gL ¼2C1:5

L asYgðR;CLÞffiffiffipp

GJ�dhdx

� �max

(12)

Using the shaft torsion constitutive equations, the loss factor ispresented as

gL ¼2C1:5

L agðR;CLÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

T

TY

� �sf

J

R

� � (13)

Here, TY is the minimum torque required to initiate yielding in theuncracked shaft. Figure 3 shows the loss factor as a function ofCL=R. Energy loss for crack depths less than 20% of the shaftradius is very small even with a significant friction ða ¼ 0:9Þbetween crack surfaces.

2.3 Circumferential Crack Modeling. In order to evaluatethe torsional dynamic response of a shaft with a circumferentialcrack, the shaft is divided into two segments connected by a tor-sion spring and a torsion damping, Fig. 4(a). The torsional stiff-ness and damping constants of the spring are evaluated byconsidering the energy loss at the crack region due to the cyclicplastic deformation at the crack tip and the frictional energy lossdue to the crack surface interactions using fracture mechanics.The details of this analysis are presented in Refs. [25,26].

The torsion spring is defined as

Kc ¼ Ktð1þ igcÞ (14)

Here, Kt is the torsional stiffness and gc is the circumferentialcrack loss factor. The torsional stiffness Kt is obtained from

Fig. 3 Longitudinal energy loss factor in a shaft subjected tocyclic torsion for various crack surface interactions, Eq. (13)

Fig. 4 (a) Schematic model of a shaft with a circumferentialcrack and its corresponding model with torsional spring anddamping and (b) schematic model of a shaft with both longitu-dinal and circumferential cracks

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Kt ¼ GR3MðcÞ (15)

Here

MðcÞ ¼ pc2:5

4ð1� cÞ 1þ 1

2cþ 3

8c2 þ 5

16c3 þ 32

128c4 þ 0:208c5

� �(16)

and c ¼ RC=R where RCis the distance from the shaft center tocrack tip in the cracked section, Fig. 4(a).

Crack surface interactions of circumferential cracks are rela-tively more pronounced than those of longitudinal cracks. There-fore, it is expected that energy loss factor of the circumferentialcrack to be greater than that of the longitudinal crack.

2.4 Dynamic Response of a Shaft With Both Circumferen-tial and Longitudinal Cracks. The torsional dynamic responseof a cracked turbo generator shaft subjected to a harmonictorsional loading at its free end, TðtÞ ¼ T0eixt, is evaluated byconsidering the equation of motion of each segment shown inFig. 4(b) and their corresponding boundary conditions.

The equation of motion for the uncracked section can bepresented as

@2h@x2¼ q

@2h@t2

(17)

Here, G and q are the shear modulus and density and h is the shaftcross section rotation. The equation of motion for the shaft with alongitudinal crack is presented as

Gð1þ igLÞf ðCL

RÞ @

2h@x2¼ q

@2h@t2

(18)

The steady state cross section rotation hnðx; tÞ corresponding toeach segment when the shaft is subjected to harmonic torsionalloading is in the form of

hnðx:tÞ ¼ �hnðxÞeixt (19)

Here, subscript n refers to the segment number. Two nondimen-sionalized parameters are introduced as

e ¼ x

L(20)

an ¼qL2x2

Gn(21)

Using these nondimensionalized parameters, the equation ofmotion of each uncracked segment of the shaft can be presentedas

d2 �hn

de2þ an

�hn ¼ 0 (22)

While the equation of motion for longitudinal cracked section canbe presented as

ð1þ igLÞfCL

R

� �d2 �hn

de2þ an

�hn ¼ 0 (23)

Solution of Eqs. (22) and (23) is in the form of

�hnðeÞ ¼ ReX2

m¼1

AnmeSnme

!(24)

where Sn1 and Sn2 are the roots of the characteristic equation foreach of the Eqs. (22) and (23) and Anm are constants obtainedfrom the boundary conditions of each segment shown in Fig. 4(b)which are in the following form:

at e1 ¼ 0; �h1ðe1Þ ¼ 0 (25)

at e2 ¼ L1=L; �h1ðe2Þ ¼ �h2ðe2Þ (26)

at e2 ¼ L1=L; T1ðe2; tÞ ¼ T2ðe2; tÞ (27)

at e3 ¼ L2=L; Teðe3; tÞ ¼ Kt½h3ðe3; tÞ � h2ðe3; tÞ� (28)

at e3 ¼ L2=L; T2ðe3; tÞ ¼ T3ðe3; tÞ (29)

at e4 ¼ L3=L; �h3ðe4Þ ¼ �h4ðe4Þ (30)

at e4 ¼ L3=L; T3ðe4; tÞ ¼ T4ðe4; tÞ (31)

at e5 ¼ 1; Tðe5; tÞ ¼ T0eixt (32)

Here, Tðe; tÞ is related to hðe; tÞ using Eq. (1). If circumferentialcrack is located before or after the longitudinal crack, the aboveboundary conditions should be adjusted accordingly.

3 Results and Discussions

Table 1 show the shaft geometry and material properties usedin this study. First, the effects of the presence of longitudinalcrack only or circumferential crack only on the shaft dynamicresponse are studied. The results are then extended when both ofthese cracks are present in the shaft. These cracks could be at adistance from each other or simultaneously be present in a crosssection, such as shown in Fig. 4(b). Experimental investigationshave shown that longitudinal crack forms first in a shaft subjectedto cyclic torsion followed by formation of the circumferentialcrack. These cracks may eventually interact as shown in Fig. 1[19–23].

3.1 Longitudinal Crack Only. The effects of crack lengthand depth and crack surface interactions on the first resonance fre-quency are presented. The results could be used in detecting crackdepth and its location in the shaft.

3.1.1 Crack Length and Depth. Assuming the longitudinalcrack is located in the middle of the shaft and crack surface inter-actions are minimal (i.e., gL ¼ 0), MATLAB-based codes weredeveloped to analyze the dynamic response of the cracked shaft.Figure 5 shows the effect of crack length and depth on the firstresonance frequency. The results indicate that both crack lengthand crack depth could have a significant effect on the firstresonance frequency of the shaft. For a small crack depthðCL=R � 0:2Þ, the longitudinal crack length has apparently negli-gible effect on the shaft first resonance frequency. This is eventrue if the entire shaft is cracked longitudinally. The effect ofcrack depth is more pronounced when CL=R � 0:3. Moreover,with a small crack length ðlL=L � 0:1Þ, longitudinal crack doesnot change the shaft resonance frequency significantly unlessCL=R � 0:6.

3.1.2 Crack Surface Interactions. The effect of crack surfaceinteractions is taken into account by considering energy loss factor

Table 1 Shaft geometry and material properties

Material 4140 steelYoung’s modulus, E 200 GPaDensity, q 7850 Kg/m3

Yield stress, Y 420 MPaPoisson’s ratio, v 0.3Radius, R 0.1 mLength, L 2 m

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term in the shaft torsional rigidity. Figure 6(a) shows the effectsof the crack surface interactions on the peak response frequency.The results indicate that for a significant energy loss between lon-gitudinal crack surfaces, there is no identifiable frequency wherepeak response occurs. This is analogues to the response of a singledegree of freedom system with a high damping ratio. Furthermore,Fig. 6(b) shows for severe crack surface interactions, the peakresponse of the shaft approaches to that of the uncracked shaft.For CL=R � 0:6 and a ¼ 0:9, the peak response frequency of thecracked shaft is the same as that of the uncracked shaft. Moreover,if CL=R � 0:6 and a ¼ 0:9, the cracked shaft could even becomestiffer than the uncracked shaft and its peak response frequencycould be as high as 1.2 times of that of the uncracked shaft.

However, Fig. 6(c) illustrates for small crack surface interactionswhen 0 � a � 0:1, the peak response of the cracked shaft is notsignificantly altered from that of the uncracked shaft. As men-tioned before, for the same crack depth, energy loss factor of a cir-cumferential crack is greater than that of a longitudinal crack. Ouranalysis in Sec. 2.2 showed that the loss factor for a longitudinalcrack cannot be higher than 11 for typical torsional loads whilethe corresponding loss factor value for circumferential crackscould be as high as 50 for the same crack depth size. However,small energy loss in the longitudinal crack region can have signifi-cant effects on the dynamic response of the shaft since its effectsare more pronounced over much larger longitudinal crack alongthe shaft.

Fig. 5 (a) The effects of longitudinal crack depth located in the middle of the shaft on its firstresonance frequency for various crack length and (b) the effects of longitudinal crack lengthon the shaft first resonance frequency for various crack depth

Fig. 6 (a) Effects of energy loss factor on the frequency response of a shaft with a longitudi-nal crack, (b) effects of the energy loss factor of a longitudinal crack on the first resonance fre-quency when 0 £ a £ 0:9, and (c) effects of the energy loss factor of a longitudinal crack on thefirst resonance frequency when 0 £ a £ 0:1

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3.1.3 Mode Shapes and Crack Location Detection. Assumingthe crack suraface interactions are minimal (i.e., gL ¼ 0), the firstmode shape of a shaft with a longitudinal crack located at its mid-section is depicted in Fig. 7(a). The results indicate that modeshapes are not significantly altered by the presence of a longitudi-nal crack especially if the crack depth is small. In contrast to themode shape, the slope of the mode shape could be used as a sensi-tive parameter to find the location of the longitudinal crack in ashaft, Fig. 7(b).

3.2 Circumferential Crack Only. The effects of the circum-ferential crack depth and location and crack surface interactionson the dynamic response of the shaft are investigated.

3.2.1 Crack Depth and Location. Assuming the crack sura-face interactions are minimal (i.e., gC ¼ 0), MATLAB-based codeswere developed to analyze the dynamic response of the crackedshaft. Figures 8(a) and 8(b) show the effect of crack depth andlocation on the shaft resonance frequency. Results indicate thatboth crack depth and location could significantly affect the shaftresonance frequency. As the crack gets closer to the shaft free endits effect on the first resonance frequency becomes less pro-nounced. For Xc=L ¼ 0:9 and Cc=R � 0:75, the circumferentialcrack has a negligible effect on the shaft resonance frequency.The results also indicate that for Cc=R � 0:3., the shaft resonancefrequency is very close to that of the uncracked shaft independentof crack location. Furthermore, for circumferential crack withcrack depth of Cc=R � 0:9, the natural frequency of the shaft isnot affected by the crack location for cracks located at the dis-tance of Xc=L � 0:5 and the shaft resonance frequency is between10% and 15% of that of uncracked shaft. The first resonance fre-quency of the circumferential cracked shafts obtained in this

investigation is in close agreement with those obtained by Dimar-ogonas, and Massouros [1].

3.2.2 Crack Surface Interactions. Figure 9(a) shows theeffects of the crack surface interactions on the frequency responseof a shaft with a circumferential crack. The frequency and ampli-tude where the peak response occurs significantly depend on thecrack surface interactions. It is notable that for severe crack sur-face interactions, the shaft resonance frequency approaches thenatural frequency of the uncracked shaft. As shown in Fig. 9(b),for gC � 6:5 and CC=R � 0:7 the resonance frequency of thecracked shaft is very close to that of the uncracked shaft and it isvery little dependent on the crack depth. For CC=R � 0:3, theeffect of the crack surface interactions is negligible; however, for0:5 � CC=R � 0:7, the effect of crack surface interactions is sig-nificant when gC � 5. In the case that the crack is very deep (i.e.,CC=R � 0:9), even severe crack surface interactions cannot com-pensate the loss of the shaft stiffness. For these cases, the shaftpeak response frequency is only about 40% of uncracked shaftwhen gc ¼ 20.

3.3 Simultaneous Longitudinal and CircumferentialCracks. Assuming crack surface interactions are minimal (i.e.,gL ¼ 0, gc ¼ 0), Dynamic response of a shaft with both longitudi-nal and circumferential cracks is investigated. For simplicity it isassumed that entire shaft is cracked longitudinally (i.e., lL=L ¼ 1)which is plausible given the fact that longitudinal crack growsmuch faster along the shaft than a circumferential crack. Theeffects of longitudinal crack depth, circumferential crack depthand its location on the dynamic response of the cracked shaft arepresented in Fig. 10. The results show the resonance frequency ofa shaft with both longitudinal and circumferential cracks may belittle affected by the presence of the circumferential crack when

Fig. 7 (a) First mode shape of a shaft with a longitudinal crack and (b) first derivate of the firstmode shape with respect to position

Fig. 8 (a) The effects of circumferential crack depth on the shaft first resonance frequencyfor various crack locations and (b) the effects of circumferential crack location on the shaftfirst resonance frequency for various crack depth

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CC=R � 0:3. The shaft resonance frequency apparently is gov-erned by the presence of the longitudinal crack for any circumfer-ential crack location. The effect of the circumferential cracklocation and its depth is more pronounced when CC=R � 0:3. ForCC=R � 0:9, longitudinal crack does not have a significant effecton the shaft resonance frequency compared to that of a shaft withonly a circumferential crack. However, as the circumferentialcrack location moves toward the free end of the shaft, the effect ofthe longitudinal crack becomes more pronounced.

4 Conclusions

Longitudinal and circumferential cracks often form in turbogenerator shafts under cyclic torsion. The presence of such cracks

could significantly impact the shaft dynamic response. A paramet-ric study was carried out to understand the effect of these crackson the shaft resonance frequency. Frequency response of a longi-tudinally cracked shaft was investigated by modeling the crackedsection as a shaft with a reduced torsional rigidity. A correctionfactor was defined as the torsional rigidity of the cracked shaft tothat of the uncracked shaft. The torsional rigidity correction factorof the shaft was found to be just a function of crack depth to theshaft radius. The effect of the crack surface interactions was takeninto the consideration and was found to be very small for crackdepths less than 20% of the shaft radius. The results show that theresonance frequency of the shaft may be little affected by the pres-ence of a longitudinal crack when CL=R � 0:2 even when the

Fig. 9 (a) Effects of energy loss factor on the frequency response of a shaft with a circumfer-ential crack and (b) effects of the energy loss factor of a circumferential crack on the shaft firstresonance frequency for different crack depth

Fig. 10 The effects of circumferential crack depth on the shaft resonance frequency for vari-ous longitudinal crack depths when (a) Xc=L 5 0:3, (b) Xc=L 5 0:6, and (c) Xc=L 5 0:9

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entire shaft is cracked. The effect of the longitudinal crack on theshaft peak response frequency is more pronounced whenCL=R � 0:2. The effect of the crack surface interactions of a lon-gitudinal crack in a shaft on the peak response frequency is alsoinvestigated. For small crack surface interactions, gL � 1:2, theeffect of the crack surface interactions could be ignored in thedynamic analysis of the shaft for any crack depth. The results alsoshow that the peak response frequency approaches or may exceedthat of uncracked shaft for severe crack surface interactions.

The circumferentially cracked section was modeled as a torsionalspring connecting two solid shafts. The torsional spring constant andcrack surface interactions energy loss were obtained using fracturemechanics. The results show that circumferential crack depth andlocation and energy loss factor are important parameters affectingthe dynamic response of the shaft. The effect of circumferentialcrack location on the resonance frequency of the shaft is more pro-nounced for crack length of Cc=R � 0:3. For severe crack surfaceinteractions, the peak response frequency of the shaft approachesthat of the uncracked shaft but in contrast to the longitudinal crack,it never exceeds that. The study of both circumferential and longitu-dinal cracks on the shaft resonance frequency shows that the circum-ferential crack may not have a significant effect on the shaftresonance frequency for Cc=R � 0:3 and the resonance frequencymay be governed just by the longitudinal crack in this case. How-ever, for Cc=R � 0:9, the circumferential crack plays a significantrole and the effect of the longitudinal crack on the shaft resonancefrequency could be ignored.

Acknowledgment

This paper was made possible by the support of NPRP 09-145-2-061 grant from Qatar National Research Fund (QNRF). The state-ments made herein are solely the responsibility of the authors.

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