Numerical and Experimental Analysis of Cracked Cantilever

Journal of Mechanical Engineering Research and Developments

ISSN: 1024-1752

CODEN: JERDFO

Vol. 43, No.5, pp. 415-427

Published Year 2020

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Numerical and Experimental Analysis of Cracked Cantilever

Beam under Free Vibration

Ali A. Al-Saffar*, Raghad Azeez Neamah, Luay S. Alansari,

Mechanical Engineering Department – Faculty of Engineering – University of Kufa, Iraq

*Corresponding author E-mail: [email protected]

ABSTRACT: The topic of this work is to discover the variations of depth, shapes and locations of the crack on

the natural frequency of cantilever circler shaft. Thirty-two cracked Aluminium shafts have 25 mm diameter and

45 mm length were experimentally tested with 1,2,4,5 mm crack depths and different positions on the beam were

applied. Three different finite element models, using ANSYS – APDL (17.2) were built to simulate the crack in

the cantilever shaft. The results showed that increasing the depth of crack reduces the frequency of shaft for same

mode number. While, at the small crack depth, there is no influence of crack position on the first and second

modes of frequencies.

KEYWORDS: cracked beam, natural frequency, ANSYS software

INTRODUCTION

Beams are extensively used in various engineering application fields, like machinery, aerospace, automobile and

else. [1–4]. Beams such as structures are treated with static and /or dynamic loading. Due to different causes like

wear and temperature fluctuation, cracks could cause failure [5]. If these cracks are not discovered early, an

unexpected failure like catastrophic consequences may be obtained [6,7]. The method of destruction of the crack

in cantilever shaft is crucial not only for save operation conditions but also restoring performance of the system.

All structures have rigid nature can cause some irregularities which may lead to progress of crack. The dynamic

properties of the beam will be changed if it suffers from damages. The stiffness, natural frequency, mode shapes

and else can be affected by development of cracks. Most researchers have simulated the cracks as a linear system

because they have assumed that cracks are always opening during vibration of the beam. Therefore, they used

mode shapes and natural frequencies to determine the crack severity. However, these linear response features are

not valid when detecting fatigue cracks due to the crack breathing effect in reality [8]. During the previous years,

a number of scholars have utilized the various nonlinear characteristics of beams with breathing cracks by

adopting analytical solutions [9–11], numerical procedures [12-14] and experimental techniques [15-17].

In the opening crack model, Sahu and Rohini [18] considered a square area of mild steel specimen to examine the

act of cracks on the free vibration of cantilever shaft. FEM in MATLAB environment was used and from theory

of linear elastic fracture, a local flexibility matrix was added to the total flexibility matrix. Chandra et al [19]

investigated the fracture behaviour of columns and cracked beams as a result of crack propagation using the FEA.

Orhan [20] studied the behaviour of cantilever shaft for free and forced vibration, influence of two edge cracks

was taking into account to observe the crack reaction on the frequency parameters. Prathamesh [21] also observed

the changes in the natural frequencies due to using different boundary conditions and crack depth and location.

High-pass filters and domain decomposition method were utilised [22] to examine the natural frequency and

destruction observation of Euler-Bernoulli cracked beams. The proposed method was accurate and effective and

experimental work was performed to verify it. A number of papers were carried out by researchers [23-26] to

specify the crack type implementation on the free vibration of beam theoretically and experimentally.

Euler-Bernoulli and Timoshenko theories were a doubted to calculate the free and forced vibration for the

functional graded material (FGM) cracked beams by Yang et al [27] and Ke et al [28]. Applying of open edge

cracks for various boundary conditions on the frequency of FGM beam were also investigated. For multiple

cracked beams, Aydin [29] and Wei et al [30] found the frequency equations of bending vibration of FGM cracked

beams in the third order mode. The technique of transfer matrix was used by the authors based on random number

mailto:[email protected]

Numerical and Experimental Analysis of Cracked Cantilever Beam under Free Vibration

416

of cracks. Sherafatnia et al [31] and Rakideh et al [32] organized the frequency equations where they manipulated

the gained results to arrange a neural network that can establish properties of cracks on shaft. The FEA was also

employed to determine the frequency of the FGM beam relying on size and location of cracks [33-35]. In present

work, variation of the location, shape and depth of the crack on the natural frequency and mode shape of the

cantilever beam where studied experimentally. A finite element analysis (FEA) using ANSYS software 17.2 was

adopted to verify the results.

Problem Description and Experimental Work

In this work, the natural frequency of cracked and un-cracked cantilever aluminium shaft was measured

experimentally. Aluminium shafts with diameter (25 mm) and length (45 cm) were applied. The position of crack

was (5, 10, 15, 20, 25, 30, 35 and 40) mm far from the fixed end, and the depth of crack was (1, 2, 4 and 5) mm.

The density and the Young Modulus of the shafts were measured experimentally in the lab and these properties

are reviewed in Table (1).

Table 1. The Properties of the Aluminium shafts.

Property of material Modulus of Elasticity (E) Poisson's Ratio(ν) Density (ρ)

Aluminum 70 GPa. 0.3 2700 m

3

To measure the frequencies of the intact and cracked shafts, the accelerometer model (SN 151779) was utilized

to obtain the response of the shaft. Impact hammer model (086C03) was employed to excite the shaft and an

amplifier model (480E09) was used to amplify the accelerometer signal and send it, the output signal, to an

oscilloscope model (ADS 1202CL+). The experimental rig diagram is shown in Figure. 1.


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Figure 1. Experimental Rig diagram.

In additional to the intact cantilever shaft, 32 cracked cantilever beams with various crack positions and depths

were also performed to obtain the frequencies experimentally and these cases can be summarized in Table (2).

Table 2. Crack Position and Crack Depth of the Cracked Cantilever Shaft Used in this Work.

No. Length of Shaft (cm) Crack Position from Fixed End (cm.) Crack Depth (mm.)

1

45

5 1, 2, 4 and 5

2 10 1, 2, 4 and 5

3 15 1, 2, 4 and 5

4 20 1, 2, 4 and 5

5 25 1, 2, 4 and 5

6 30 1, 2, 4 and 5

7 35 1, 2, 4 and 5

8 40 1, 2, 4 and 5

Theoretical Work

Three finite element models, using ANSYS – APDL (17.2) were built to simulate the crack in cantilever shaft

with an element SOLID187 was used. The SOLID187 is a 3-D high order element with 10-nodes and it has a

quadratic displacement behavior and well suited to model the irregular mesh. The number of elements were

between 25000 and 40000 and the number of nodes were between 15000 and 45000. It is well known that the

crack is a source of stress concentration which could affect the frequency of the cantilever beam. Therefore, in

these three models the variation of the natural frequency due to changing the crack shape was investigated. As

well as, to the variation in the crack shape, the depth and the position of the crack were also observed. These three

models are:

First Model

The crack shape is a rectangular shape as can be seen in Figure. 2. Uniform depth along the crack thickness was

assumed where the crack thickness is 1 mm. There are two angles of stress concentration with an angle equal to

(90o).


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Figure 2. The Crack Shape of the First Model.

Second Model

In this model, the crack shape is a triangular shape, Figure. 3. The crack tip was assumed to be lies at the maximum

depth of crack as a result of machining process by a cutter tool and the crack thickness is (1 mm). The crack tip is

the source of stress concentration with an angle smaller than (90o) when the depth of crack increases the angle of

stress concentration decreases with fixed crack thickness.

Figure 3. The Crack Shape of the Second Model.

Third Model

The crack shape in this model is a combination of rectangular and triangular shapes as can be seen in Figure. 4.

The crack tip was assumed to be lies at the maximum depth of crack and the crack thickness is (1 mm). The crack

tip is the source of stress concentration with angle (2α). It is assumed that the value of angle (α) is (30, 45 and

60o).

Geometry of Crack

α=30o.


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α=45o.

α=60o.

Figure 4. The Crack Geometry of the Third Model and Mesh When α= 30, 45 and 60o.

RESULTS AND DISCUSSION

Figure 5 shows the comparison between the experimental and numerical first mode frequency when the depth of

crack increases for different crack positions. Numerically, when the position of crack is (5cm) near to fixed end,

the first mode frequency decreases with increasing the depth of crack. Also, the results of the second model are

the lower results comparing with the first and third models. In third model, the first mode of frequency increases

when the angle (α) raises. The same behaviour can be seen when the position of cracks is 10 ,15 and 20 cm close

to fixed end. When the position of crack is larger than (20 cm), there is no influence for the depth of crack on the

first mode natural frequency. The crack means reducing in cross section area (i.e. reducing in second moment of

inertia) and this leads to reduce the stiffness of shaft at this crack position. The crack depth has an effect on the

second mode natural frequency which is shown in Figure. 6.

The second mode frequency decreases when the depth of crack becomes greater and the crack position increases

up to (35 cm). Also, Figure. 7 shows the relation between the depth of crack and the third mode natural frequency.

Where, it can be seen that the third mode frequency decreases with increasing the depth of crack. When the crack

depth increases the reducing in stiffness get bigger. Therefore, this fact is noted in Figures. 5 to 7, but with

limitations. One of these limitations is the crack position and mode number. The decreasing rate of second mode

frequency due to varying depth of crack, depends on the crack position. Also, the behaviour of the second Model

differs than other models and depends on the crack position. Generally, the crack causes reducing in stiffness of

shaft and when the crack lies close to fixed end, the reducing in stiffness of shaft will be larger than that close to

free end. Therefore, the frequency reduces when the crack occurs close by fixed end.

As a results, when the crack depth is 1 or 2 mm, the first and second mode frequencies are constant at any position

of crack as shown in Figures.8 and 9. While the first and second modes of frequency decrease when the crack

position becomes closer to the fixed end at crack depth is 4 and 5 mm. Also, the variation rate of first and second


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mode frequencies increase when the depth of crack increases. In Figure. 10, there are irregularities in the results

of third mode natural frequency. This irregularity happens because of increasing the number of modes.

p c =5 cm. p c =10 cm.

p c =15 cm. p c =20 cm.

p c =25 cm. p c =30 cm.

p c =35 cm. p c =40 cm.

Figure 5. First mode frequency against depth of crack.


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p/c : position of crack

p c =5 cm. p c =10 cm.

p c =15 cm. p c =20 cm.

p c =25 cm. p c =30 cm.

p c =35 cm. p c =40 cm.

Figure 6. Second mode frequency against depth of crack.


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p c =5 cm. p c =10 cm.

p c =15 cm. p c =20 cm.

p c =25 cm. p c =30 cm.

p c =35 cm. p c =40 cm.

Figure 7. Third mode frequency against depth of crack.


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d c =1 mm. d c =2 mm.

d c =4 mm. d c =5 mm.

Figure 8. First mode frequency against position of crack.

d/c: depth of crack

d c =1 mm. d c =2 mm.


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d c =4 mm. d c =5 mm.

Figure 9. Second mode frequency against position of crack

d c =1 mm. d c =2 mm.

d c =4 mm. d c =5 mm.

Figure 10. Third mode frequency against position of crack.

CONCLUSIONS

From the results, the following points can be concluded:

1) The frequencies of cracked shaft depend on the crack depth, when the crack depth increases the frequency

decreases for the same mode number.

2) When the mode number increases, the crack depth effects on the natural frequencies increase. In other

word, the variation rate of natural frequencies increases when the mode number increases at the same

crack depth.


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3) The frequencies of cantilever shaft depend on the crack position, the first, second and third mode

frequencies decreases when the crack lies close to the fixed end for the large crack depth.

4) For the small crack depth, there is no effect of crack position on the first and second modes of frequencies.

5) Finally, the third model is suitable for crack simulation in shaft and the effect of crack angle (2α) is

sharply appear in this model.

REFERENCES

[1] J. Liu, Y.M. Shao, and W.D. Zhu, "Free vibration analysis of a cantilever beam with a slant edge crack". Proc.

Inst.Mech. Eng. Part C J. Eng. Mech. Eng. Sci., Vol. 231, Pp. 823–843, 2016.

[2] R. Zarfam, and A.R. Khaloo, "Vibration control of beams on elastic foundation under a moving vehicle and

random lateral excitations". J. Sound Vibr., Vol. 331, Pp. 1217–1232, 2012.

[3] X.F. Li, "Free Vibration of Axially Loaded Shear Beams Carrying Elastically Restrained Lumped-Tip Masses

via Asymptotic Timoshenko Beam Theory". J. Eng. Mech., Vol. 139, Pp. 418–428, 2013.

[4] V.V. Matveev, and O.A. Bovsunovsky, "Vibration-based diagnostics of fatigue damage of beam-like

structures". J. Sound Vibr., Vol. 249, Pp. 23–40, 2002.

[5] L. Chen, J.L. Jiang, Z.C. Fan, X.D. Chen, and T.C. Yang, "A new model for life prediction of fatigue–creep

Interaction". Int. J. Fatigue., Vol. 29, Pp. 615–619, 2007.

[6] D.L. Yang, Y.L. Liu, S.B. Li, J. Tao, C. Liu, and J.H. Yi, "Fatigue crack growth prediction of 7075 aluminum

alloy based on the GMSVR model optimized by the artificial bee colony algorithm". Eng. Comput., Vol. 34,

Pp. 1034–1053, 2017.

[7] T.G. Chondros, A.D. Dimarogonas, and J. Yao, "A continuous cracked beam vibration theory". J. Sound Vibr.,

Vol. 215, Pp. 17–34, 1998.

[8] S.M. Cheng, X.J. Wu, and W. Wallace, "Vibrational response of a beam with a breathing crack". J. Sound

Vibr., Vol. 225, Pp. 201–208, 1999.

[9] V.V. Matveev, O.E. Boginich, and A.P. Yakovlev, "Approximate analytical method for determining the

vibration-diagnostic parameter indicating the presence of a crack in a distributed-parameter elastic system at

super- and subharmonic resonances". Strength Mater. Engl. Tr., Vol. 42, Pp. 528–543, 2010.

[10] A.P. Bovsunovskii, C. Surace, and O.A. Bovsunovskii, "The act of damping and force application point on

the non-linear dynamic behavior of a cracked beam at sub- and super-resonance vibrations". Strength Mater.

Engl. Tr., Vol. 38, Pp. 492–497, 2006.

[11] R. Ruotolo, C. Surace, P. Crespo, and D. Storer, "Harmonic analysis of the vibrations of a cantilevered beam

with a closing crack". Comput. Struct. Vol. 61, Pp. 1057–1074, 1996.

[12] U. Andreaus, P. Casini, and F. Vestroni, "Non-linear dynamics of a cracked cantilever beam under harmonic

Excitation". Int. J. Non-Linear Mech., Vol. 42, Pp. 566–575, 2007.

[13] H. Long , Y. Liu, and K. Liu, "Nonlinear Vibration Analysis of a Beam with a Breathing Crack". Appl. Sci.

Vol. 9, Pp. 3874, 2019. doi:10.3390/app9183874 www.mdpi.com/journal/applsci.


426

[14] U. Andreaus, and P. Baragatti, "Cracked beam identification by numerically analysing the nonlinear

behaviour of the harmonically forced response". J. Sound Vibr., Vol. 330, Pp. 721–742, 2011.

[15] A.P. Bovsunovsky, and C. Surace, "Considerations regarding superharmonic vibrations of a cracked beam

and the variation in damping caused by the presence of the crack". J. Sound Vibr., Vol. 288, Pp. 865–886,

2005.

[16] U. Andreaus, and P. Baragatti, "Experimental damage detection of cracked beams by using nonlinear

characteristics of forced response". Mech. Syst. Signal Proc., Vol. 31, Pp. 382–404, 2012.

[17] P. Gudmundson, "The dynamic behavior of slender structures with cross-sectional cracks". J. Mech. Phys.,

Solids, Vol. 31, Pp. 329–345, 2010.

[18] S. Sahu, B. Rohini, "Free Vibration Analysis of Cantilever Beams with Transvers Open Cracks".

Proceedings of National Conf. on Recent Innovations in Science Engineering & Technology, 13th Sept,

Bengaluru, India, Pp. 23-25, 2015.

[19] K. Chandra, M. Kumar, "Finite element analysis for fracture behaviour of cracked beam-column". Finite

element in analysis and design, Vol. 40, Pp 1773-1789, 2004.

[20] S. Orhan, "Analysis of free and forced vibration of cracked cantilever beam". NDT and E International, 40,

p 443-450, 2007.

[21] M.J. Prathamesh, "Free vibration analysis of cracked beam. Journal of Engineering research and

applications", Vol. 3, No. 6, Pp. 1172-1176, 2013.

[22] M. Qibo, (2016). "Vibration Analysis of Cracked Beams Using Adomian Decomposition Method and Non-

Baseline Damage Detection via High-Pass Filters". International Journal of Acoustics and Vibration, Vol. 21,

No. 2, Pp. 170-177.

[23] V. Khalkar, and S. Ramachandran, "Comparative vibration study of EN 8 and EN 47cracked cantilever

beam". Journal of Vibroengineering, Vol. 19, No.1, Pp. 246-259, 2017.

[24] I. Khan, D. Parhi, "Finite Element Analysis of Double Cracked Beam and its Experimental Validation".

Procedia Engineering, Vol. 51, Pp. 703 – 708, 2013.

[25] D. Agarwallaa, and D. Parhi, "Effect of Crack on Modal Parameters of a Cantilever Beam Subjected to

Vibration". Procedia Engineering, Vol. 51, Pp. 665 – 669, 2013.

[26] R. Behera, A. Pandey, and D. Parhi, "Numerical and Experimental Verification of a Method for Prognosis of

Inclined Edge Crack in Cantilever Beam Based on Synthesis of Mode Shapes". Procedia Technology, Vol.

14, Pp. 67–74, 2014.

[27] J. Yang, Y. Chen, Y. Xiang, and L. Jia, "Free and forced vibration of cracked inhomogeneous beams under

an axial force and a moving load. Journal of Sound and Vibration", Vol. 312, Pp. 166-181.


427

[28] L. Ke, J. Yang, S. Kitipornchia, and Y. Xiang, "Flexural vibration and elastic buckling of a cracked

Timoshenko beam made of functionally graded materials". Mechanics of advanced materials and structures,

Vol. 16, Pp. 488-502, 2012.

[29] K. Aydin, "Free vibration of functional graded beams with arbitrary number of cracks". European journal of

mechanics, A/solid, Vol. 42, Pp. 112-124, 2013.

[30] D. Wei, Y. Liu, and Z. Xiang, "An analytical method for free vibration analysis of functionally graded beams

with edge cracks". Journal of sound and vibration, Vol. 331, Pp. 1685-1700, 2012.

[31] K. Sherafatnia, G. Farrahi, and S. Faghidian, "Analytic approach to free vibration and bucking analysis of

functionally graded beams with edge cracks using four engineering beam theories". International journal of

engineering, Vol. 27, No. 6, Pp. 979-990, 2014.

[32] M. Rakideh, M. Dardel, and M. Pashaei, "Crack detection of Timoshenko using vibration behaviour and

neural network". International journal of Engineering, Vol. 26, Pp. 1433-1444, 2013.

[33] H. Ziou, "Numerical modelling of a Timoshenko FGM beam using the finite element method". International

journal of structure engineering, Vol. 7, No. 3, Pp. 239-261, 2016.

[34] Z. Yu, and F. Chu, "Identification of crack in functionally graded material beams using the p-version of finite

element method". Journal of Sound and Vibration, Vol. 325, No. (1-2), Pp. 69-84, 2009.

[35] A. Banerjee, B. Panigrahi, and G. Pohit, "Crack modelling and detection in Timoshenko FGM beam under

transverse vibration using frequency contour and response surface model with GA". Non-destructive testing

and evaluation, 2015. DOI.10.1080/10589759.2015.1071812.

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Numerical and Experimental Analysis of Cracked Cantilever