flexural vibration of rotating cracked timoshenko beam

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2006 12: 1271Journal of Vibration and ControlSamer Masoud Al-Said, Malak Naji and Adnan A. Al-Shukry

Flexural Vibration of Rotating Cracked Timoshenko Beam

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Flexural Vibration of Rotating Cracked TimoshenkoBeam

SAMER MASOUD AL-SAIDMALAK NAJIADNAN A. AL-SHUKRYMechanical Engineering Department, Jordan University of Science and Technology, P. O. Box3030, Irbid 22110, Jordan ([email protected])

(Received 13 December 2005� accepted 9 August 2006)

Abstract: In this study, a simple model is proposed that describes the flexural vibration characteristics of arotating cracked Timoshenko beam. The cracked beam is modeled using two uniform segments connectedby a massless torsional spring at the crack location. The equation of motion is derived using Lagrange’smethod in conjunction with the assumed mode method. The proposed model is used to study the effect ofcrack depth, shear deformation, and rotation speed on the dynamic characteristics of the beam, and comparethe results with those obtained from the widely used Euler-Bernoulli beam model. It is shown that for thesame crack depth the proposed model has a higher reduction in frequency compared to that of Euler-Bernoullibeam. Model verification is carried out using three dimensional finite element analysis, which reveals goodagreement with the assumed mode results.

Key words: Cracked beam, crack identification, rotating beam vibration, nondestructive test

NOMENCLATURE

D Crack depth ratio (a/H)E Modulus of elasticityG Shear modulusI Area moment of inertiaJ Beam mass moment of inertia per unit length J � � I = m K2

K Cross-section radius of gyrationL Beam lengthmb Mass per unit length of the beamM Internal beam momentT Beam kinetic energyU Beam potential energy

t� Reference time, ��

�AL4

E I

y Total beam deflection� Beam cross-sectional angle of distortion due to shear� Cross-section shear factor.

Journal of V ibrat ion and Control, 12 (11): 1 271–1 28 7, 2 00 6 DOI: 1 0.1 177 / 10 775 46 30 607 16 94

��2006 SAGE Publications

at Afyon Kocatepe Universitesi on May 16, 2014jvc.sagepub.comDownloaded from


1272 S. M. AL-SAID ET AL.

� Beam cross-sectional angle of rotation due to bending� Dimensionless crack flexibility� Angular speed� Dimensionless time, � t

T �� Dimensionless coordinate, � x

LU1 Shear parameter� beam bending to shear rigidity ratio, � E I

�G AL2

U2 Inertia parameter, � K 2

L2

f i , qi i-th generalized coordinate for total deflection & pure bendingYi , i i-th mode shape for total deflection & pure bending� Dimensionless angular speedU1�D�� Non-dimensional natural frequency of rotating Timoshenko beamU1�D�0 Non-dimensional natural frequency of non-rotating cracked beamU1�0�� Non-dimensional natural frequency of rotating un-cracked beamU1�0�0 Non-dimensional natural frequency of non-rotating un-cracked beam.co Interaction term between crack depth ratio and beam rotating speed�� Total derivative with respect to �

1. INTRODUCTION

There are many factors that can induce a crack in a mechanical structure, such as generalenvironmental attacks, erosion, corrosion, and fatigue, as well as mechanical accidents. Thefact that many factors can contribute to the formation of a crack, along with the potentiallycatastrophic consequences, has made structural integrity testing an extremely active area ofresearch. Despite advances in the theory and technology of nondestructive testing, investi-gating the integrity of a structure is a labor-intensive and time-consuming process that shouldonly be performed when really needed. One approach for reducing inspection-related down-time and cost is to equip a machine with a failure early warning device. Such an onlinedevice monitors crack-related abnormalities in the behavior of a machine. If the device givesstrong indications that a crack is present, an advisory message is issued to the operator totake the machine out of service and have it tested. A good understanding of the dynamics ofcracked structures is essential for the construction of such early warning devices. There ismore than one way in which a crack in a structure may manifest its presence. For example,the presence of a crack may be deduced from local variations in stiffness which affect themechanical behavior of the entire structure, or through changes in the natural frequenciesand mode shapes of a beam-like structure. In addition to detecting the presence of a crack,these indicators may also be used to measure the extent of the damage and even to locate it.

Several researchers have investigated the effect of cracks on the dynamic characteris-tics of structural elements. Mengchang and Renji (1997) suggested an approximate methodfor the analysis of vibrating cracked beams based on a linear-spring model in conjunctionwith the Euler-Bernoulli beam theory. The characteristic equation for vibration in the beamwas derived using modal analysis and principles of fracture mechanics. Parhi and Behera,(1997) investigated the behavior of a cracked, vibrating beam with moving mass, using theRunge-Kutta method to solve the differential equations describing the dynamic deflectionof a cantilever beam. Horibe (1996) used the boundary integral equation method (BIEM)



FLEXURAL VIBRATION OF ROTATING CRACKED TIMOSHENKO BEAM 1273

to analyze large deflections in cracked beams. In his study, the crack was simulated usingan equivalent rotational spring connecting the two segments of a cracked beam. He appliedthe BIEM method to each segment of the beam and used an iterative method to solve theresulting non-linear boundary integral equations. He also investigated the influence of crackdepth and location on the deflection of the center of the beam and the resulting stress distri-bution. Ruotolo et al. (1996) studied the dynamic response of a cracked cantilevered beamto harmonic excitation. The study was performed using a finite element model called theclosing crack model. Undamaged parts of the beam were modeled using an Euler-type finiteelement having two degrees of freedom (DoF), namely transverse displacement and rotation,at each node. Both fully open and fully closed cracks were used to represent the damagedelement.

Chondros et al. (1998) developed a theory for modeling lateral vibration of crackedEuler-Bernoulli continuous beams with single- or double-edge open cracks. They derivedthe governing differential equation of motion and the corresponding boundary condition ofthe cracked beam assuming a one-dimensional continuum. By describing a displacementfield in the neighborhood of the crack, fracture mechanics was used to model the crack ascontinuously flexibile. Chen and Jeng (1991) used a finite element model to analyze thedynamic behavior of pre-twist rotating blades with a single edge crack. They investigatedthe influence of crack location and crack depth on the natural frequencies of the blade.

Gounaris and Papadopoulos (1997) proposed a new method for crack identification in abeam. They studied a model that has a transverse surface crack assumed to be always open.Their method is based on the observation that the eigenmodes of any cracked structure aredifferent from those of an un-cracked one, and suggested that the mode shape differencescould be correlated with the crack depth and location. The correlated differences were theratio of two amplitude measurements in two positions and the distance of the node of thevibrating mode from the left end of the beam. To show the effect of crack depth on thetransverse vibration characteristics, Masoud et al. (1998) derived a model for a pre-stressedfixed-fixed cracked beam using modal analysis. They also studied the interaction betweenthe crack depth and axial load, and the effect of this interaction on the natural frequency ofthe system� the theoretical results obtained were verified experimentally.

A cantilever beam with a transverse edge crack was studied by Chati et al. (1997) usingmodal analysis. The nonlinearity resulting from the crack opening and closing was modeledas a piecewise linear system utilizing the idea of bilinear frequencies. The finite elementmethod was used to obtain the natural frequencies in each linear region, and a perturba-tion method was used to obtain the non-linear normal modes of vibration and the associatedperiod of motion. Chen and Chen (1995) studied the stability of a rotating cracked shaft sub-jected to axial compressive end force. They discussed the influence of existing open crackson the natural whirling speeds of the shaft. Their results revealed that when a cracked shaft issubjected to an increasing end load, two principal instability regions of different types (diver-gence and flutter) appear. Hasan (1995) used a perturbation method to evaluate the first orderperturbation of the eigenfrequencies of a beam on an elastic foundation. The local flexibilityintroduced by the crack in the cracked section was represented by a massless tortional springwhose stiffness depends on the severity of the crack. He found that the magnitude of changein the eigenfrequencies is a function of the severity and the location of the crack. Sunder-meter and Weaver (1995) used the weak non-linear character of a cracked vibrating beam to




determine crack location and depth. Tsai and Wang (1997) studied vibration of a rotor withmultiple cracks. Christides and Barr (1984) derived a differential equation, along with theassociated boundary conditions, for a nominally uniform Euler-Bernoulli beam containingone or more pairs of symmetric cracks.

In order to simplify analysis, most of the work in the literature uses the Euler-Bernoullimodel to describe cracked beams, abolishing the shear deformation effect. Taking this effectinto account is important for accurate design, especially for short beams such as gas turbineblades or when studying high modes of vibration. Methods that take this effect into accountuse complicated techniques such as finite element or continuum mechanics methods.

In the work reported here, a simple mathematical model that accounts for shear defor-mation is proposed for a rotating cracked beam. The assumed mode method combined withLagrange’s technique is used to derive the governing equations of motion for the beam. Tworotating Timoshenko beams connected by a massless torsional spring are used to simulatethe cracked beam. The effect of crack depth, rotating speed, and shear deformation on thebeam’s natural frequencies is investigated. The proposed model is verified for a rotatingcracked beam using 3-D finite element analysis. The commercial finite element softwareCOSMOS/M (version 2.5) is used in the simulation.

2. MATHEMATICAL MODELING

The focus here is on the flexural free vibration of double-sided cracked cantilever Timo-shenko beam with a uniform rectangular cross-section. The beam is assumed to rotate inthe horizontal (x-z) plane at a constant angular speed (�) with a crack located at its midspan(Figure 1). The cracked beam is simulated using two uniform Timoshenko beams connectedby a mass-less torsion spring at the crack location (Figure 2). The beam is assumed tohave negligible structural damping. Rotation of the beam about an axis causes variable axialload:

P �x� t �L�

x

mb�2�d� � 1

2mb�

2L2

�1� x2

L2

�(1)

To write the equation of motion governing the vibration of a Timoshenko beam, the beam isassumed to have a constant cross-sectional area (A), mass per unit length (mb), area momentof inertia (I) and modulus of elasticity (E). The total deflection y(x, t) of the beam at anypoint (x) can be considered in two parts� one caused by pure bending and the other by pureshear. Hence, the slope of the deflection curve (y(x)) may be written as

�y

�x� � �x� t � � �x� t (2)

where� �x� t is the angle of rotation due to bending and �(x, t) is the angle of distortion dueto shear. The linear and angular deflections are assumed to be small. The relation betweenthe bending moment (M) and the bending deflection is given by




Figure 1. Rotating Beam Model� a) Physical Model, b) Mathematical Model.

Figure 2. Free body diagram of beam element.




T � 1

2

L�0

mb

��y �x� t

�t

�2

dx � 1

2

L�0

J

��

�t

�2

dx (10)

The potential energy U of the beam due to elastic bending, shear deformation, variable axialforce and the crack is

U � 1

2

L�0

E I

��

�x

�2

dx � 1

2

L�0

�G A

��y

�x� �

�2

dx �L�

0

P �ds � dx � 1

2

M2

Kcr(11)

where 1K cr

is equivalent to the spring flexibility used to represent the crack (Haisty andSpringer, 1988)

1

Kcr� 9�D2

B H 2 E

�0�5033� 0�9022D � 3�412D2 � 3�181D3 � 5�793D4

D � a

H , a is the crack depth, H is the half beam depth, and B is the beam width. The term�ds � dx represents the change in the horizontal projection of an element of length (ds):

ds � dx ��dx 2 �

��y �x� t

�x

�2

�dx 2 1�2

� dx �� 1

2

��y �x� t

�x

�2

dx (12)

Substituting equations (3) and (12) into equation (11) gives

U � 1

2

L�0

E I

��

�x

�2

dx � 1

2

L�0

�G A

��y

�x� �

�2

dx

� 1

2

L�0

p

��y

�x

�2

� 1

2

�E I 2

Kcr

��

�x

�2

(13)

Using the y and � definitions (equations (7) and (8)), the kinetic and potential energies maybe written as

T � 1

2

L�0

mb

n�i�1

n�j�1

�qi �q j Yi Y j dx � 1

2

L�0

Jn�

i�1

n�j�1

�fi �f ji j dx (14)

U � 1

2

L�0

E In�

i�1

n�j�1

fi f ji

j dx




� 1

2

L�0

�G An�

i�1

n�j�1

�qi Y

i � fii

�q j Y

j � f j j

dx

�L�

0

p�x n�

i�1

n�j�1

qi q j Yi Y

j dx � 1

2

n�i�1

n�j�1

�E I 2

Kcrfi f j

i

j

��x�0�5L

(15)

By substituting equations (14) and (15) into Lagrange’s equation (9), two coupled equationsdescribing beam flexural vibration are obtained. These equations may be written in the non-dimensional form�

U1 [W]nxn [0]nxn

[0]nxn U2 [N]nxn

��

q�

nx1��f�

nx1

��

�[V]nxn �U1�

2[P]nxn � [R]nxn

� [R]nxn [S]nxn � [N]nxn �� [E]nxn

� q�nx1

f�nx1

� 0� (16)

where

Wr j �L�

0

Yr Y j dx� Nr j �L�

0

r j dx� Sr j �L�

0

r

j dx�

Vr j �L�

0

Y r Y

j dx� Rr j �L�

0

Y r j dx� Pr j �

L�0

p�x Y r Y

j dx�

Er j � r

j

��x�0�5

� U1 � E I

�G AL2� U2 � K 2

L2

The non-dimensional flexibility (�) of a symmetric double-sided crack was shown in Ma-soud et al. (1998) as a function of the crack depth ratio D:

� � 6�D2 H

L

��5033� �9022D � 3�412D2 � 3�181D3 � 5�793D4

(17)

Equation (16) is a system of second order linear differential equations with constant co-efficients. The solution of this system of equations is a typical eigenvalue problem.

3.1. The Assumed Function

To obtain a good approximate solution of the above system, the proposed solution of the non-rotating cracked Timoshenko beam is used as the assumed function. Since the effect of thecrack is localized to the immediate neighborhood of the cracked beam, the beam is treated astwo uniform segments connected by a torsion spring at the crack location (Figure 1). The left-




hand segment of the beam is designated by subscript (1), and that on the right by subscript(2). Equations (5) and (6) hold true for the two beam segments far from the crack position.Assuming that there is no angular rotation, and that the motion is harmonic, equations (5)and (6) may be represented in non-dimensional form as:

d4Y

d� 4 � �2 �U1 �U2 d2Y

d� 2 � �2�1� �2U1U2

Y � 0 (18)

d4

d� 4 � �2 �U1 �U2 d2

d� 2 � �2�1� �2U1U2

� 0 (19)

The solution of these equations is found to be

Y �� C1 sin�� C2 cos�� C3 sinh �� C4 cosh �� (20)

�� C 1 cos�� C

2 sin�� C 3 cosh �� C

4 sinh �� (21)

where

��

�

��

1�2

� �U1 �U2 �

��U1 �U2

2 � 4

�2

� 12

12

(22)

In order to find the coefficients of each function Y and as well as the constants � and �,the case of a clamped free beam is considered (Figure 1). The boundary conditions appliedto the clamped end are that the lateral motion and slope due to bending both equal to zero�that is:

Y1 �0 � 0 1 �0 � 0�

For compatibility of displacement, moment, and shear force of both segments at the cracklocation, the following must hold:

Y1

�1

2

�� Y2

�1

2

��

1

�1

2

��

2

�1

2

��

dY1

d�

�1

2

��1

�1

2

�� dY2

d�

�1

2

��2

�1

2

�The angular displacement between the two segments of the beam related to the bendingmoment at the crack section is given by

2�1�2 �1�1�2 � � 2�1�2

Finally, at the free end of the beam, the moment and shear force vanish, giving




2 �1 � 0�

dY2

d�

��1 �2

��1 � 0

Substituting equations (20) and (21) into the above boundary conditions for the cantileverbeam and satisfying the non-rotating beam governing equations of motion, eight linear alge-braic homogenous equations are obtained. These equations may be written in matrix formas:

[A�] Ci� � 0� i � 1� 2� � � � 8

To calculate the natural frequencies of the cracked beam, the determinant of [A�] is set tozero. The corresponding coefficients �Ci are then calculated. The resulting frequencies andcoefficients are substituted in equations (20) and (21) to determine the mode shapes for totaldeflection of the cracked beam (Y �� ) and the beam’s slope due to pure bending (�� ).These functions are then substituted into equation (16) to find the governing equations ofmotion for a rotating cracked Timoshenko beam.

4. RESULTS AND DISCUSSION

The proposed mathematical model is used here to simulate a rotating cracked Timoshenkobeam. The results obtained are for a crack placed at the mid-span of the beam, but similarresults may be obtained regardless of the location of the crack. Two steel beams with rec-tangular cross-sections are used (E � 207G Pa, G � 79G Pa, � � 7830Kg�m3, � � 0�3).The first beam has a low shear deformation effect (U1 = 6�5� 10�4) and its dimensions (inmeters) are: width = 0.01, height = 0.01, length = 0.2. The second beam has a large sheardeformation effect (U1 = 1�637� 10�2) and the dimensions (in meters) width = 0.1, height =0.1, length = 0.4. Only the first three modes (n = 3) are used to compute the beam deflection(y) and bending (�). Therefore, the system of equations governing motion of the crackedbeam (equation 16) has a 6x6 dimensionality. In order to test the fidelity of the model, a threedimensional finite element analysis is performed on the above beam using the commercialpackage COSMOS/M version 2.5. The finite element model and the mesh layout are shownin Figure 3. A twenty-node solid brick element is used in meshing the beam in volumes farfrom the crack. Volumes in the vicinity of the cracked are meshed using a ten-node tetrahe-dral element. Each node has three translational degrees of freedom in the nodal x, y and zcoordinates.

The mathematical model reveals that the frequency of the beam is a nonlinear function ofcrack depth, speed of rotation, and shear deformation. In order to study the effect of rotationand crack depth on the natural frequencies of the system, a two-dimensional Taylor seriesexpansion is performed:

U1�D�� U1�0�0 ��U1�D��

�D

��0D�0

�D � �U1�D��

��

��0D�0

��

� �2 U1�D��

�D��

��0D�0

�D�� 1

2

�2U1�D��

�D2

��0D�0

�D2




Figure 3. Finite element model� a) Three dimensional solid model of the rotating cracked beam� b) Frontview for the cracked beam� c) Meshing lay out.

or

U1�D�� U1�D�0 � U1�0�� U1�0�0 � co

where

U1�D�0 � U1�0�0 ��U1�D��

�D

��0D�0

�D � 1

2

�2U1�D��

�D2

��0D�0

�D2 � ��

U1�0�� U1�0�0 ��U1�D��

��

��0D�0

�� 1

2

�2U1�D��

��2

��0D�0

��2 � ��

and

co � �2U1�D��

�D��

��0D�0

�D��

where (U1�D��) is the non-dimensional natural frequency of the cracked rotating Timo-shenko beam, (U1�D�0) is the non-dimensional natural frequency of the non-rotating crackedTimoshenko beam, (U1�0��) is the non-dimensional natural frequency of the rotating un-cracked Timoshenko beam, and (U1�0�0) is the non-dimensional natural frequency of thenon-rotating un-cracked Timoshenko beam. The interaction effect between crack depth ratioand beam rotating speed on the natural frequency appears as (co):




Figure 5. As Figure (4) but for significant shear deformation effect U1 = 0.01637, (0.1*0.1*0.4).

Figure 6. Effects of speed of rotation and crack depth ratio on the interaction term for the first two modesof a steel beam with significant shear deformation effect U1 = 0.01637, (0.1*0.1*0.4). (a) First mode� (b)

Second model. Crack depth ratio (D): �� 0.1, – � – � – 0.2, – – – – – 0.3, — ��— 0.4, ——— 0.5.




Figure7.Effectsofsheardeformationandspeedofrotationonnaturalfrequenciesofanintactbeam. (a)Firstmode�(b)Secondmode.Speedofrotation�: — — — 2 , – – – – – 4 , ��6 , – �–�– 8 , — ��—

10.incalculatingthefrequencychangeobtainedbysubtractingthefrequencyrelatedtothe crackedrotatingbeamfromthatfortheintactone:U1�D��U1�0��U1�D�0�U1�0�0�c o Asthespeedofrotationincreases,thistermapproachesasignificantconstantvalueforthe firstmode.However,itcontinuestoincreaseforthesecondmode.InFigure7theeffect ofsheardeformationU1onthefrequencydifferencebetweenthe rotating cracked and intact beamsforthefirsttwonaturalfrequenciesisshownfordifferentspeedsofrotation(

�).This figuredemonstratesthatforarotatingEuler-Bernoullibeamthefirstnaturalfrequencyis higherthanthatofaTimoshenkobeamforallvaluesofthespeedofrotation.Thebehavior ofthesecondnaturalfrequencyexhibitsadifferentpattern:Forlowvaluesofshearparameter (atallspeedsofrotation),thenaturalfrequencyoftheEuler-Bernoullibeamislessthanthat oftheTimoshenkobeam.Figures8and9showtheeffectoftheshearparameter(U1)onthe beam’snaturalfrequencyforthenon-rotatingandrotatingcrackedbeamcasesrespectively. Itcanbeseenfromthefiguresthatthenaturalfrequenciesofthebeamarenonlinearfunctions ofboththecrackdepthratioandtheshearparameter,andalsothatthenormalizedfrequency differencebetweenthecrackedTimoshenkobeamandthecrackedEuler-Bernoullibeamis amonotonicallyincreasingfunctionofthecrackdepth.BycomparingFigures8and9it canbeconcludedthattheabovedifferenceishigherforthenon-rotatingbeamthanforthe rotatingcase.

a t A f y o n K o c a t e p e U n i v e r s i t e s i o n M a y 1 6 , 1 2 0 1 4 j v c . s a g e p u b . c o m D o w n l o a d e d f r o m



Figure 8. Effects of shear deformation and crack depth ratio on natural frequencies of a non-rotatingbeam. (a) First mode� (b) Second mode. Crack depth ratio D: �� 0.1, – � – � – 0.2, – – – – 0.3, — � � —0.4, ——— 0.5.



5. CONCLUSION

In this paper, a simple mathematical model is proposed for describing the lateral free vibra-tion of a rotating cracked Timoshenko beam. The accuracy of the model was establishedusing a three dimensional finite element analysis performed using the commercial COS-MOS/M software version 2.5. The proposed model was used to study the variations in thecharacteristics of the beam under the influence of different factors such as speed of rotation,crack depth, and shear deformation. It was found that there is an uncertainty (due to theinteraction co between the speed of rotation and the crack depth ratio) in determining thereduction in a beam’s natural frequencies due to the presence of a crack if this is calculatedby subtracting the frequency of the intact rotating beam from that of the cracked one. It wasalso found that for increasing speed of rotation this term approaches a constant for the firstnatural frequency, while for the second mode, the terms become proportional to the speed ofrotation. The effect of shear deformation, which is important for describing the character-istics of short beams as well as accurate high mode characterization, has also been studied.It was found that for a long beam, a higher vibration mode has a higher sensitivity to sheardeformation. Finally, one important application of the proposed method lies in its abilityto accurately identify a crack in a rotating beam by matching the measured set of naturalfrequencies of the beam with the ones calculated using the model.

REFERENCES

Chati, M., Rand, R., and Mukherjee, S., 1997, “Modal analysis of cracked beam,” Journal of Sound and Vibration207, 249–270.

Chen, L. W. and Chen, H. W., 1995, “Stability analysis of cracked shaft subjected to the end load,” Journal of Soundand Vibration 188, 497–513.

Chen, L. W. and Jeng, C. H., 1991, “Vibrational analysis of cracked pre-twisted blades,” Computers and Structures46, 133–140.

Chondros, T. G., Dimarogonas, A. D., and Tao, J., 1998, “Continuous cracked beam vibration theory,” Journal ofSound and Vibration 215, 35–46.

Christides, S. and Barr, A. D. S., 1984, “One-dimensional theory of cracked Bernoulli-Euler beam,” InternationalJournal of Mechanical Sciences 26, 639–648.

Gounaris, G. D. and Papadopoulos, C. A., 1997, “Analytical and experimental crack identification of beam struc-tures in air or in fluid,” Computers and Structures 65, 633–639.

Haisty, B. S. and Springer, W. T., 1988, “A general beam element for use in damage assessment of complex struc-tures,” Journal of Vibration, Acoustics, Stress, and Reliability in Design 110, 389–394.

Hasan, W. M., 1995, “Crack detection from the vibration of the eigenfrequencies of a beam on elastic foundation,”Engineering Fracture Mechanics 52, 409–421.

Horibe, T., 1996, “Large-deflection analysis of cracked beam by boundary integral equation method,” Japan Societyof Mechanical Engineers Part A, 62, 1996, 1834–1839.

Masoud, S., Jarrah, M. A., and Al-Maamory, M., 1998, “Effect of crack depth on the natural frequency of a pre-stressed fixed beam, Journal of Sound and Vibration 214, 201–212.

Mengcheng, C. and Renji, T., 1997, “Approximate method of response analysis of vibration for cracked beam,”Applied Mathematics and Mechanics 18, 221.

Parhi, D. R. and Behera A. K., 1997, “Dynamic deflection of a cracked beam with moving mass,” Proceedings OfThe I nstit ution Of Mechanical Engine ers, Part C, Journal of Mechanical Engineering Science 21 1, 77–87.

Ruotolo, R., Surace, C., Crespo P., and Storer, D., 1996, “Harmonic analysis of the vibrations of a cantileveredbeam with a closing crack,” Computers and Structures 61, 1057–1074.




Sundermeter, J. N. and Weaver R. L., 1995, “On crack identification and characterization in a beam by nonlinearvibration analysis,” Journal of Sound And Vibration 183, 857–871.

Tsai, T. C. and Wang, Y. Z., 1997, “The vibration of multi-crack rotor,” International Journal of Mechanical Sciences39, 1037–1053.



flexural vibration of rotating cracked timoshenko beam

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