[mechanisms and machine science] vibration engineering and technology of machinery volume 23 ||...

Analysis of Coupled Transverse and AxialVibrations of Euler Bernoulliand Timoshenko Beams with LongitudinalCrack for Its Detection

Jeslin Thalapil and S.K. Maiti

Abstract This paper presents a method to analyze the vibration of monolithicbeams with longitudinal cracks for its detection. Both forward problem of deter-mination of natural frequencies knowing the beam and crack geometry details aswell as inverse problem of detection of crack with the knowledge of changes in thebeam natural frequencies has been examined. Both long (Euler-Bernoulli) and short(Timoshenko) beams have been studied. For modeling a crack located at the freeend of a cantilever, the beam is divided into three segments. For an internal cracklocated away from the free end of the beam, it is split into four segments. In bothcases, two of the segments take care of beam portions above and below the crack.The cracked segments are constrained to have the same transverse displacementsbut different axial movements. The modeling shows good accuracy for both theforward and inverse problems. The formulation predicts the first five fundamentalnatural frequencies in the forward problems with a maximum difference of 5 % withreference to finite element solutions for short beams with edge or inner cracks.Further, in the case of inverse problems, edge crack with sizes varying 5 to 50 % ofthe beam length has been detected with errors less than 3 % in both short as well aslong beams. In the case of inner crack located at the mid-span with sizes varyingfrom 5 to 45 % of the beam length has been detected with errors less than 3 % inlocation and 6 % in size. The results thus show encouraging possibility ofexploitation of the proposed method for crack detection in practice.

Keywords Structural health monitoring � Vibration of delaminated beams �Coupled vibrations � Longitudinal crack detection

J. Thalapil � S.K. Maiti (&)Department of Mechanical Engineering, Indian Institute of Technology, Mumbai, Indiae-mail: [email protected]

J. Thalapile-mail: [email protected]

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_39

433

1 Introduction

Laminated beams are extensively used in aircraft, spacecraft and space structuresbecause of high strength to weight ratio and stiffness to weight ratio. Ramkumaret al. [1] first proposed the model for vibration analysis of composite beams. Themodel was based on four Timoshenko beams. Wang et al. [2] examined the freevibrations of an isotropic beam with a through-width delamination by using fourEuler–Bernoulli beams connected at the delamination boundaries. In their formu-lation they considered the coupling effect of longitudinal and flexural motionsbetween delaminated layers. He assumed that the delaminated layers deformed‘freely’ without touching each other. This is termed as ‘free model theory’, whichwas shown to be physically inadmissible. Mujumdar and Suryanarayan [3] thenproposed a ‘constrained model theory’ where the delaminated layers are assumed tokiss each other, but are allowed to slide over each other. This model was physicallyadmissible and the results showed good accuracy with the experimental results.Tracy and Pardoen [4] presented similar constrained model for a simply supportedcomposite beam. Shu and Fan [5] extended this constrained model for the modelingof bi-material beams. Shen and Grady [6] found the separation of delaminatedsegments in their experiments. Luo and Hanagud [7] proposed an analytical modelbased on the Timoshenko beam theory, which uses piecewise-linear springs tosimulate the ‘opening’ and ‘closing’ behavior between the delaminated surfaces.The model combined both free and constrained model, with the consideration thatfor free model the spring stiffness would be zero and infinity for constrained model.Saravanos and Hopkins [8] developed an analytical solution for predicting naturalfrequencies, mode shapes and modal damping of a delaminated composite beambased on a general laminate theory which involves kinematic assumptions repre-senting the discontinuities in the in-plane and through-the-thickness displacementsacross each delamination crack. Shu et al. [9] presented an analytical solution tostudy a sandwich beam with double delaminations. His study emphasized theinfluence of the contact mode, ‘free’ and ‘constrained’, between the delaminatedlayers and the local deformation near the two fronts of the delamination. Shu andMai [10] investigated the local deformation near the delamination fronts andidentified the ‘rigid connector’ and the ‘soft connector’ conditions. A ‘rigid con-nector’ considers the differential stretching between the delaminated layers, while a‘soft connector’ neglects the differential stretching. Lestari and Hanagud [11]studied a composite beam with multiple delaminations by using the Euler–Bernoullibeam theory with piecewise-linear springs to simulate the ‘opening’ and ‘closing’behavior between the delaminated surfaces. Shu and Della [12] studied the com-posite beams with multiple delaminations by using the ‘free’ and ‘constrained’models. Their study emphasized the influence of the multiple delaminations on thefirst and second frequencies and the corresponding mode shapes of the beams.

Large volumes of investigation have been reported involving crack(s) perpen-dicular to the longitudinal direction of a beam. Both forward and inverse problemshave been addressed. The problem of longitudinal crack(s), particularly their

434 J. Thalapil and S.K. Maiti

detection, has not received that much of attention. The study of longitudinal crackshas been mostly driven by delamination in layered isotropic and composite beams.For the analysis of problems associated with such cracks, both free and constrainedmodels have been proposed. Out of these two, the constrained model appears togive more accurate results. This paper examines the possibility of detecting lon-gitudinal crack in monolithic long and short beams.

2 Mathematical Formulation for Long Beams

Beams with longitudinal crack show a coupling of axial and transverse vibration[1]. Figure 1 shows a typical beam with a longitudinal crack. The governingequation of transverse vibration, considering a Euler-Bernoulli beam, is given by

d2

dx2EI

d2Wdx2

� �� x2qAW ¼ 0; or

d4W

dn4� k4W ¼ 0 ð1Þ

where W is the transverse displacement, ω is the natural frequency of vibration, E isthe modulus of elasticity, I is the second moment of inertia, ρ is the density of the

beam, A is the cross sectional area, k4 ¼ qAL4x2�EI is the non-dimensionless

frequency parameter and n ¼ x=L is the non-dimensionless length.The governing equation of longitudinal vibration of the same beam is given by

d2Udx2

þ qEx2U ¼ 0; or

d2U

dn2þ p2U ¼ 0 ð2Þ

whereU is the axial displacement and p2 ¼ qL2x2=E.The solutions to Eqs. 1 and 2 can be written as follows.

W ¼ C1 cos knð Þ þ C2 cosh knð Þ þ C3 sin knð Þ þ C4 sinh knð ÞU ¼ A1 cos pnð Þ þ A2 sin pnð Þ ð3Þ

a

dc

x,

y,W

L

d

ξ

Fig. 1 Long beam with longitudinal crack

Analysis of Coupled Transverse and Axial Vibrations… 435

The beam with a longitudinal crack at depth dc Fig.1 from top fibre can bedivided into four segments as shown in Fig. 2. The vibration of each segment willbe guided by equations of the type (1) and (2). However, there will be someconstraints on their deformation as per the ‘constrained model theory’ [3].

The solutions to the equations of motion, Eqs. 1 and 2 for each segment can bewritten as follows.

For Segment 1,

W1 ¼ C1 cos k1nð Þ þ C2 cosh k1nð Þ þ C3 sin k1nð Þ þ C4 sinh k1nð ÞU1 ¼ C13 cos pnð Þ þ C14 sin pnð Þ

where k41 ¼qA1L4

EI1x2

ð4Þ

The segments 2 (ABEF) and 3 (ABCD) kiss each other while undergoingtransverse vibrations. Hence, they have the same vertical displacements. But, theyhave different axial displacements.

W2 ¼ W3 ¼ C5 cos k2nð Þ þ C6 cosh k2nð Þ þ C7 sin k2nð Þ þ C8 sinh k2nð ÞU2 ¼ C15 cos pnð Þ þ C16 sin pnð ÞU3 ¼ C17 cos pnð Þ þ C18 sin pnð Þ

where k42 ¼q A2 þ A3ð ÞL4EðI2 þ I3Þ x2

ð5Þ

Segment 4 has the same form as that of segment 1.

Seg. 1

Seg. 2

Seg. 3

d

d3

y, W

x,ξ

ξ=0 ξ=β ξ=α ξ=1

d2

Seg. 4a

Seg.2

Seg.4

Seg.1

Seg.3

A B

CD

EF

G H

I J

Fig. 2 Long beam with longitudinal beam


W4 ¼ C9 cos k1nð Þ þ C10 cosh k1nð Þ þ C11 sin k1nð Þ þ C12 sinh k1nð ÞU4 ¼ C19 cos pnð Þ þ C20 sin pnð Þ ð6Þ

where A1;A2 andA3 are the cross sectional areas of segment 1, 2 and 3 respectivelyand I1; I2 and I3 are the area moments of inertia of the respective segments about anaxis passing through the centers of the cross-sections and normal to the plane of thepaper.

Using the boundary conditions at ξ = 0 and ξ = 1 for a cantilever beam,

W1 ¼ 0;W01 ¼ 0;U1 ¼ 0 forn ¼ 0 ð7Þ

W004 ¼ 0;W

0004 ¼ 0;U

04 ¼ 0 forn ¼ 1 ð8Þ

Further at the left end A of the crack (n ¼ bÞ, there is a continuity of transversedisplacement, transverse slope, shear forces, axial forces.

W1 ¼ W2; W01 ¼ W

02

I1W0001 ¼ I2 þ I3ð ÞW 000

2 ; A1U01 ¼ A2U

02 þ A3U

03

ð9Þ

The jump conditions in axial displacement between A and G as well as A and Iand balance of moments at the same location (n ¼ bÞ can be expressed as follows.

U2 ¼ U1 � d32L

W01;U3 ¼ U1 þ d2

2LW

01

I1W001 ¼ I2 þ I3ð ÞW 00

2 �A2d3L2

U02 þ

A3d2L2

U03

ð10Þ

Similarly at the right end B of the crack (n ¼ aÞ, the continuity conditions are

W2 ¼ W4;W02 ¼ W

04

I1W0004 ¼ I2 þ I3ð ÞW 000

2 ;A1U04 ¼ A2U

02 þ A3U

03

ð11Þ

At the same location, the jump conditions are

U2 ¼ U4 � d32L

W04;U3 ¼ U4 þ d2

2LW

04

I1W004 ¼ I2 þ I3ð ÞW 00

2 �A2d3L2

U02 þ

A3d2L2

U03

ð12Þ

Using Eqs. 7 to 12, the characteristic equation and the characteristic determinantQj j of size 20� 20 is obtained. The characteristic equation is given by Qj j ¼ 0.


2.1 Mathematical Formulation for Short Beams

The governing equation of transverse vibration of a short beam [13] is given by

EIo2wox2

þ k0 oyox

� w

� �AG� qI

o2wot2

¼ 0

qo2yot2

� k0 o2yox2

� owox

� �G ¼ 0

ð13Þ

where y is the transverse deflection and w is the angle of rotation due to bendingmoment.

The solutions to these equations are [14]

Y ¼ C1 cosh bpnð Þ þ C2 sinh bpnð Þ þ C3 cos bqnð Þ þ C4 sin bqnð ÞW ¼ C

01 sinh bpnð Þ þ C

02 cosh bpnð Þ þ C

03 sin bqnð Þ þ C

04 cos bqnð Þ ð14Þ

where Y is amplitude of yðx; tÞ;W is amplitude of w, n is the non-dimensional co-ordinate, and

n ¼ xL; b2 ¼ qAL4x2

EI; r2 ¼ I

AL2; s2 ¼ EI

k0AGL2

p

q

�¼ 1ffiffiffi

2p � r2 þ s2

þ r2 � s2 2þ 4

b2

� �12

" #12 ð15Þ

The governing equation of longitudinal vibration of a similar beam is given by

d2Udx2

þ qEx2U ¼ 0 or

d2U

dn2þ k2U ¼ 0 ð16Þ

where k2 ¼ qL2x2=E and U is the axial displacement.The solution to for axial vibration is given by

U ¼ A1 cos knð Þ þ A2 sin knð Þ ð17Þ

A beam with a crack is divided into 4 segments as shown in Fig. 2. The‘constrained model theory’ [3] is again invoked. That is, segments 2 (ABEF) and 3(ABCD) have the same transverse displacement but different axial displacements.The solutions to all the segments are given below.


Segment 1,

Y1 ¼ C1 cosh b1p1nð Þ þ C2 sinh b1p1nð Þ þ C3 cos b1q1nð Þ þ C4 sin b1q1nð ÞW1 ¼ C

01 sinh b1p1nð Þ þ C

02 cosh b1p1nð Þ þ C

03 sin b1q1nð Þ þ C

04 cos b1q1nð Þ

U1 ¼ C13 cos knð Þ þ C14 sin knð Þð18Þ

where

b21 ¼qA1L4x2

EI1; r21 ¼

I1A1L2

; s21 ¼EI1

k0A1GL2

Segments 2 and 3 have identical transverse displacements but different axialdisplacements. Hence,

Y3 ¼ Y2 ¼ C5 cosh b2p2nð Þ þ C6 sinh b2p2nð Þ þ C7 cos b2q2nð Þ þ C8 sin b2q2nð ÞW3 ¼ W2 ¼ C




08 cos b2q2nð Þ

U2 ¼ C15 cos knð Þ þ C16 sin knð ÞU3 ¼ C17 cos knð Þ þ C18 sin knð Þ

ð19Þ

where

b22 ¼q A2 þ A3ð ÞL4x2

E I2 þ I3ð Þ ; r22 ¼I2 þ I3ð Þ

A2 þ A3ð ÞL2 ; s22 ¼

E I2 þ I3ð Þk0 A2 þ A3ð ÞGL2

Solutions for segment 4 are given by

Y4 ¼ C9 cosh b1p1nð Þ þ C10 sinh b1p1nð Þ þ C11 cos b1q1nð Þ þ C12 sin b1q1nð ÞW4 ¼ C




012 cos b1q1nð Þ

U4 ¼ C19 cos knð Þ þ C20 sin knð Þð20Þ

The relations between Ci and C0i in the above equations is given by

C0i ¼

bLp2 þ s2

pCi; for i ¼ 1; 2; 5; 6; 9; 10

C0i ¼ � b

Lq2 � s2

qCi; for i ¼ 3; 7; 11

C0i ¼

bLq2 � s2

qCi; for i ¼ 4; 8; 12

ð21Þ


Using the boundary conditions at n ¼ 0 and n ¼ 1 for a cantilever beam,

Y1 ¼ 0;W1 ¼ 0;U1 ¼ 0; for n ¼ 0

W04 ¼ 0;

1LY

04 �W4 ¼ 0;U

04 ¼ 0; for n ¼ 1

ð22Þ

The second equation is connected with the shear force. Further at the left side Aof the crack, n ¼ b, there is a continuity of transverse displacement, slope, shearforce, axial force. Therefore,

Y1 ¼ Y2; W1 ¼ W2

A11LdY1db

�W1

� �¼ A2 þ A3ð Þ 1

LdY2db

�W2

� �A1U

01 ¼ A2U

02 þ A3U

03

ð23Þ

At the crack location, the beam segments 2 and 3 have different axial dis-placements. The jump conditions in axial displacements at A and G as well as Aand I and balance of moments can be expressed as follows.

U2 ¼ U1 � d32W1;U3 ¼ U1 þ d2

2W1

I1W01 ¼ I2 þ I3ð ÞW0

2 �A2d3L2

U02 þ

A3d2L2

U03

ð24Þ

Similarly at the right end B of the crackn ¼ a, the continuity conditions aregiven by

Y2 ¼ Y4;W2 ¼ W4

A2 þ A3ð Þ 1LdY2db

�W2

� �¼ A1

1LdY4db

�W4

� �A2U

02 þ A3U

03 ¼ A1U

04

ð25Þ

Similarly, the jump conditions at the right end B of the crack at the same locationare as follows.

U2 ¼ U4 � d32W4;U3 ¼ U4 þ d2

2W4

I1W04 ¼ I2 þ I3ð ÞW0

2 �A2d3L2

U02 þ

A3d2L2

U03

ð26Þ

Using the Eqs. 22 to 26, the characteristic equation of vibration of the crackedbeam is obtained. The characteristic determinant Qj j is of size 20� 20. It can besolved to obtain the natural frequencies of a beam with a longitudinal crack.


2.2 Forward Problem

The characteristic equation Qj j ¼ 0 is solved to obtain the natural frequencies of thecracked beam. Case studies were considered for long beam with the followingdimensions and material properties: Length, L = 1200 mm, Width, B = 12.5 mm,Depth, D = 100 mm, E = 210 GPa, density, ρ = 7860 kg/m3. A typical variation ofln Qj jwithx=x0

, where x20 ¼ EI�

qAL4 is shown in Fig. 3. The plot shows the var-

iation for long beams with longitudinal cracks at various depths from the top fibre.The forward problem has also been done for short beams with the followinggeometric and material data: Length, L = 100 mm, Depth, D = 25 mm, Width,B = 12.5 mm, E = 210 GPa, density, ρ = 7860 kg/m3.

2.3 Inverse Problem

The determination of crack location with the knowledge of natural frequencies hasbeen done for long and short beams with longitudinal axial crack. The characteristicdeterminant Q of size 20� 20 is a function of the crack location parameters and thenatural frequency, i.e. Q ¼ Qðb; a;xÞ. The characteristic determinant is plottedvarying crack parameters β and α, each from 0 to 1. The contour plot technique ofMATLAB is used to plot the value of Q for three or more measured naturalfrequencies and the intersection of three or more such plots gives the location andsize of the crack. A typical contour plot is shown in Fig. 4. It is desirable to use four

Fig. 3 Variation of ln Qj jwith x=x0for cantilever beams with longitudinal crack


or more modal frequencies for a better confirmation and accuracy of the location ofcrack and its size. In applications, the detection method therefore requires mea-surement of at least the first four transverse natural frequencies of the beam with acrack and the corresponding uncracked beam. In order to ensure a common inter-section point of these plots, it is important that the modulus of elasticity ‘E’, whichgoes as input to the characteristic equation of beam with a crack for each mode, iscalculated using the procedure given by Nandwana and Maiti [15]. According tothis procedure two sets of frequency parameters are involved, the theoretical fre-quency parameter for the uncracked beam for a particular modekuct, and themeasured or computed frequency parameter of uncracked beam for the same modekucm.kuct for cantilever or simply supported beam is available in standard texts.kucm, on the other hand, is determined from the measured or computed uncrackednatural frequency (xucm) through k4ucm ¼ xucmð Þ2 qAL4=EI

. These two parame-

ters must be the same, but they are generally found to differ. This calls for a minoradjustment or zero setting. The modulus of elasticity (E) is so adjusted thatk4uct ¼ xucmð Þ2 qAL4=EI

. This gives a value for E, which is given as input to the

characteristic equation of beam with a crack. It may be noted that the value of Echanges from one mode to another.

For case studies, the natural frequencies were computed using ANSYS 13.0software. The beam was modelled using SOLID 183 elements. The zone around thecrack tip is discretized using quarter point singular elements.

Fig. 4 Contour plot of Qj j ¼ 0 with β and α for cantilever beam with longitudinal crack


Tab

le1

Com

parisonof

analytical

andANSY

Sfrequenciesforlong

cantileverbeam

with

long

itudinalcrack

d 2= d

αSize,a =L

Firstnaturalfrequencyω1(H

z)Second

naturalfrequencyω2(H

z)Third

naturalfrequencyω3(H

z)

Analytics

ANSY

S%

Error

Analytics

ANSY

S%

Error

Analytics

ANSY

S%

Error

Uncracked

beam

57.98

57.73

0.43

363.37

350.8

3.58

1017

.593

8.9

8.37

0.1

0.6

0.1

57.98

57.72

0.45

363.36

350.7

3.60

1015

.193

6.7

8.37

0.8

0.3

57.89

57.63

0.46

361.78

349.2

3.58

1006

.092

8.9

8.30

0.9

0.4

57.82

57.55

0.47

358.61

346.2

3.56

1003

.192

7.1

8.20

0.3

0.6

0.1

57.96

57.69

0.47

363.33

350.7

3.59

1008

.392

8.3

8.63

0.7

0.2

57.84

57.54

0.52

362.30

349.6

3.61

980.96

901.7

8.79

0.9

0.4

57.25

56.90

0.61

343.66

331.3

3.73

955.0

882.9

8.17

0.4

0.7

0.2

57.76

57.46

0.53

361.82

349.2

3.61

962.77

885.6

8.70

0.8

0.3

57.39

57.05

0.60

353.45

340.8

3.70

937.46

866.6

8.17

0.9

0.4

56.90

56.53

0.65

335.06

323.2

3.64

924.49

857.7

7.78

0.5

0.6

0.1

57.97

57.73

0.42

363.03

350.7

3.50

1014

.493

8.6

8.08

0.7

0.2

57.97

57.71

0.46

359.8

347.8

3.43

970.94

899.2

7.97

0.8

0.3

57.79

57.53

0.45

339.32

327.5

3.59

818.05

764.9

6.93

GeometricandMaterialData:

L=1,20

0mm,b=12

.5mm,d=10

0mm,b¼

0:5,

E=21

0GPa,ρ=7,86

0kg

/m3


3 Results and Discussions

Table 1 presents the results of the forward problem for a long cantilever beam. Theforward problem has been studied for crack with left end at β = 0.5. The location ofthe right end of the crack α varies from 0.6 to 0.9 and depth ratio d2=d (Fig. 2)varies from 0.1 to 0.5. That is, non-dimensionless crack size a=L varies from 0.1 to0.4.

Figure 5 shows the variation of the first four natural frequencies of a short beamwith longitudinal axial crack with its left tip A at β = 0.5 and the crack sizea=Lvaries from 0.05 to 0.45.

Table 2 presents the results of the forward problem for short beams with lon-gitudinal crack. Results include cracks with left tip at β = 0.5, and the crack sizea=L varying from 0.1 to 0.4 and depth ratio g ¼ d1=d varies from 0.1 to 0.4. Theresults of the inverse problem of detection of longitudinal axial cracks in long andshort beams are presented in Tables 3 and 4 respectively. The natural frequencieswere again obtained using ANSYS 13.0.

In Table 3, results are presented for long beams with cracks at different locationsand sizes. The error in prediction of crack location is 4 % and in size is 2 %.

In Table 4, results are presented for short beams with cracks at different locationsand sizes. The error in prediction of crack location is 3 % and in size is 6 %.

Fig. 5 Variation of natural frequencies for a short beam with longitudinal crack for β = 0.5


Tab

le2

Com

parisonof

analytical

andANSY

Sfrequenciesforshortcantileverbeam

swith

long

itudinalcrack

Crack

locatio

n1stMod

e,ω1(H

z)2n

dMod

e,ω2(H

z)3rdMod

e,ω3(H

z)

ηα

a =L

Analytics

ANSY

S%

Error

Analytics

ANSY

S%

Error

Analytics

ANSY

S%

Error

0.1

0.6

0.1

1991

.619

97.7

0.31

1007

5.5

1015

1.1

0.74

2305

6.3

2327

3.10

0.93

0.7

0.2

1990

.819

96.2

0.27

1007

2.9

1014

4.8

0.71

2301

2.4

2318

2.70

0.73

0.8

0.3

1989

.219

93.7

0.23

1005

6.5

1011

3.9

0.57

2299

0.8

2313

9.50

0.64

0.2

0.6

0.1

1991

.41

1996

.79

0.27

1007

5.51

1015

1.00

0.74

2303

7.44

2317

8.50

0.61

0.7

0.2

1989

.48

1992

.47

0.15

1006

9.02

1013

3.30

0.63

2292

6.04

2291

2.20

0.06

0.8

0.3

1985

.45

1985

.91

0.02

1002

.91

1005

4.50

0.26

2286

6.87

2282

1.80

0.20

0.9

0.4

1980

.00

1978

.06

0.10

9923

.27

9889

.33

0.34

2284

0.69

2277

1.90

0.30

0.3

0.6

0.1

1991

.13

1995

.79

0.23

1007

5.41

1015

0.30

0.74

2301

0.77

2308

2.00

0.31

0.7

0.2

1987

.58

1988

.08

0.03

1006

3.41

1011

9.50

0.55

2280

0.34

2259

3.20

0.92

0.8

0.3

1980

.18

1976

.58

0.18

9987

.40

9986

.46

0.01

2267

6.19

2241

1.20

1.18

0.9

0.4

1970

.21

1962

.68

0.38

9797

.10

9708

.90

0.91

2263

5.51

2235

7.90

1.24

0.4

0.6

0.1

1990

.82

1995

.07

0.21

1007

5.31

1015

0.50

0.74

2298

0.92

2301

9.40

0.17

0.7

0.2

1985

.47

1985

.02

0.02

1005

7.11

1011

2.30

0.55

2265

4.94

2239

3.50

1.17

0.8

0.3

974.35

1968

.81

0.28

9942

.05

9931

.98

0.10

2243

9.90

2206

0.60

1.72

0.9

0.4

1959

.43

1949

.54

0.51

9659

.61

9564

.78

0.99

2238

9.73

2201

0.70

1.72

GeometricandMaterialData:

L=10

0mm,b=12

.5mm,D

=25

mm,b¼

0:5,

E=21

0GPa,ρ=7,86

0kg

/m3


Table 3 Prediction of crack parameters for long cantilever beam with axial crack using contourplots

Actual parameters Measured natural frequencies(Hz)

Predicted crack parameters

Locationβ

Sizea=L

ω1 ω2 ω3 ω4 Locationβ*

%Error

Sizea=L*

%Error

0.5 0.05 57.7 350.8 935.7 1735.6 0.507 0.69 0.054 0.389

0.5 0.1 57.6 350.7 923.1 1716.4 0.503 0.3 0.108 0.838

0.5 0.15 57.5 350.4 901.8 1645.1 0.497 0.26 0.158 0.831

0.5 0.2 57.4 349.0 878.7 1525.5 0.498 0.21 0.209 0.869

0.5 0.25 57.2 345.6 862.2 1401.1 0.498 0.17 0.259 0.878

0.5 0.3 56.9 339.5 856.2 1300.1 0.498 0.15 0.309 0.878

0.5 0.35 56.6 330.5 855.5 1234.3 0.500 0.02 0.360 0.983

0.5 0.4 56.3 319.7 846.5 1207.1 0.531 3.06 0.394 0.555

0.5 0.45 56.0 308.7 824.0 1203.4 0.534 3.36 0.437 1.251

0.1 0.1 57.6 345.4 915.7 1706.4 0.115 1.49 0.116 1.576

0.2 0.1 57.6 346.9 934.9 1726.2 0.204 0.38 0.114 1.427

0.3 0.1 57.6 348.8 936.2 1683.5 0.292 0.78 0.112 1.193

0.4 0.1 57.6 350.3 923.3 1723.7 0.387 1.25 0.11 0.951

0.6 0.1 57.6 350.1 935.6 1677.7 0.595 0.76 0.108 0.837

0.7 0.1 57.7 349.5 937.4 1724.7 0.697 0.30 0.103 0.341

0.8 0.1 57.7 349.6 930.4 1723.1 0.808 0.79 0.097 0.254

Geometric and Material Data: L = 1,200 mm, b = 12.5 mm, D = 100 mm, E = 210 GPa,ρ = 7,860 kg/m3

Table 4 Prediction of axial crack location in short cantilever beams using contour plots

Actual parameters Measured natural frequency (Hz) Predicted crack parameters

Locationβ

Sizea=L


%Error

Sizea=L*

%Error

0.5 0.05 1997 10150.3 23240.2 37615.2 0.487 1.264 0.091 4.06

0.5 0.1 1994.5 10149.2 22994.6 37102.9 0.478 2.16 0.145 4.47

0.5 0.2 1983.63 10108 22312.7 33683.1 0.473 2.724 0.250 4.99

0.5 0.25 1975.34 10036.1 22052.3 31366.4 0.473 2.728 0.301 5.07

0.5 0.3 1965.7 9910.8 21925 29208.5 0.473 2.728 0.351 5.09

0.5 0.4 1944.19 9510.31 21874.6 25937.4 0.473 2.737 0.452 5.23

0.2 0.1 1992.72 10007.4 23286.4 37429.3 0.176 2.391 0.145 4.46

0.3 0.1 1993.15 10082.1 23220.4 37068.8 0.271 2.926 0.145 4.51

0.4 1993.77

(continued)


4 Conclusions

1. Beams with longitudinal crack show a coupling of transverse and axial vibra-tions. A method has been developed for the study of vibration characteristics ofbeams with longitudinal crack.

2. The accuracy of this method in prediction of natural frequencies is illustrated bycase studies involving both long and short beams with crack. The natural fre-quencies show good match with the ANSYS results.

3. The model has been used to predict the parameters of a longitudinal axialinternal crack in both long and short beams. The range of crack size consideredis 5 to 45%. In case of long beams, the maximum error in prediction of cracklocation is 4 % and the error in prediction of crack size is 2 %. The same errorsin the case of short beams are 3 and 6 % respectively.

References

1. Ramkumar RL, Kulkarni SV, Pipes RB (1979) Free vibration frequencies of a delaminatedbeam. In: Reinforced Plastics/composites institute, 34th annual technical conference; 1979.The Society of the Plastics Industry Inc. 22-E, pp 1–5

2. Wang JTS, Liu YY, Gibby JA (1982) Vibration of split beams. J Sound Vib 1982 84(4):491–502

3. Mujumdar PM, Suryanarayan S (1988) Flexural vibrations of beams with Delaminations.Journal of Sound and Vibration 125(3):441–461

4. Tracy JJ, Pardoen GC (1989) Effect of delamination on the natural frequencies of compositelaminates. J Compos Mater 23(12):1200–1215

5. Shu D, Fan H (1996) Free vibration of abimaterial split beam. Compos: Part B 27(1):79–846. Shen MHH, Grady JE (1992) Free vibrations of delaminated beams. AIAA J 30(5):1361–13707. Luo H, Hanagud S (2000) Dynamics of delaminated beams. Int J Solids Struct 37

(10):1501–15198. Saravanos DA, Hopkins DA (1996) Effects of delaminations on the damped characteristics of

composite laminates: analytical and experiments. J Sound Vib 192(5):977–993

Table 4 (continued)

Actual parameters Measured natural frequency (Hz) Predicted crack parameters

Locationβ

Sizea=L


%Error

Sizea=L*

%Error

0.1 10136.1 22948.5 37640.3 0.374 2.562 0.145 4.48

0.5 0.1 1994.59 10149.2 22994.6 37102.9 0.478 2.187 0.144 4.44

0.6 0.1 1995.56 10130.5 23264.1 36544.3 0.575 2.547 0.147 4.69

0.7 0.1 19955 10111.5 23299.6 37376.1 0.672 2.817 0.147 4.74

0.8 0.1 1997.42 10117.5 23182.1 37605.9 0.775 2.543 0.147 4.70

Geometric and material data: L = 100 mm, B = 12.5 mm, D = 25 mm, E = 210 GPa, ρ = 7,860 kg/m3 , υ = 0.3


9. Shu D (1995) Vibration of sandwich beams with double delaminations. Compos Sci Technol54(1):101–109

10. Shu D, Mai YW (1993) Buckling of delaminated composites re-examined. Compos SciTechnol 47(1):35–41

11. Lestari W, Hanagud S (1999) Health monitoring of structures: multiple delamination dynamicsin composite beams. In: 40th AIAA/ASME/ASCE/AHS/ASC Structures, structural dynamicsand materials conference and adaptive structures forum

12. Shu D, Della CN (2004) Vibrations of multiple delaminated beams. Compos Struct 64(3):467–477

13. Timoshenko SP (1928) Vibration problems in engineering. D. Van Nostrand Company INC,London

14. Huang TC (1961) The effect of rotatory inertia and of shear deformation on the frequency andnormal mode equations of uniform beams with simple end conditions. J Appl Mech28:579–584

15. Nandwana BP, Maiti SK (1997) Modelling of vibration of beam in presence of inclined edgeor internal crack for its possible detection based on frequency measurements. Eng Fract Mech58(3):193–205


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