I J A I C R, 4(1), 2012, pp. 13-20 *Corresponding author: [email protected], [email protected]Prediction of Cracks Using Fea Analysis and Fuzzy Logic Approach 1 D. R. Parhi, Manoj Kumar Muni and Chinmaya Sahu 1 Dept. of Mechanical Engg., National Institute of Technology-Rourkela, Odisha-769008 Generally damage in a structural element may occur due to normal operations, accidents, deterioration or severe natural events such as earth quake or storms. During operation, all structures are subjected to degenerative effects that may cause initiation of structural defects such as cracks which, as time progresses, lead to the catastrophic failure or breakdown of the structure. There is change in natural frequencies and mode shapes takes place due to different orientation of the cracks. By examining these changes, crack position and its magnitude can be identified. The analysis has been done by using finite element method with the help of ANSYS workbench and CATIA V5. It has been noticed that when the crack depth increases natural frequency decreases and also there is a deviation in the mode shape. Fuzzy-logic with triangular, Gaussian and mixture of triangular and Gaussian membership function has been used for detection of crack by considering frequencies as input parameter. The FUZZY logic controller used here comprises of Three input and two output parameters. The Input parameters are first three natural frequencies and the output parameters are relative crack location and relative crack depth. A series of FUZZY rules are derived from vibration parameters which are finally used for prediction of crack location and its depth. By comparing the fuzzy result with the experimental result it is observed that the fuzzy controller can predict the relative crack location and relative crack depth in a very accurate manner with a very small percentage of error. INTRODUCTION Cracks are a potential source of catastrophic failure in mechanical machines and in civil structures and in aerospace engineering. To avoid the failure caused by cracks, many researchers have performed extensive investigations over the years to develop structural integrity monitoring techniques. Most of the techniques are based on vibration measurement and analysis because, in most cases, vibration based methods can offer an effective and convenient way to detect fatigue cracks in structures. Eftekhari et al. [1] presented an analytical model of cracked cantilever beams made of functionally graded materials (FGMS).They implemented the local flexibility concept to model the open crack and modeled the open edge crack as a mass less elastic rotational spring. Cheng et al. [2] have investigated on vibration characteristics of cracked rotating tapered beam by the P-version finite element method. They used shape functions with shifted Legendre orthogonal polynomials to represent the transverse displacement field. Waghulde et al. [3] have researched about mechanical vibrations induced in flexible structures such as beams and plates through various optimization functions and control theories to optimize transient response dynamic characteristics. They used piezoelectric materials, sensors, actuators to reduce these vibrations. Finite element method is used to model the structure with piezoelectric material and genetic algorithm is applied to find optimal placement of a number of piezoelectric sensors with objective functions of minimize control force, displacement, velocity. Liu Yong-Guang. [4] has presented a new damage index named smeared cracked model damage index (SCMDI) based on the mean curvature of beam type structure in time domain to detect cracks. They introduced the smeared cracked model in damage mechanics and demonstrated SCMDI is more robust and high efficiency in crack detection. SCMDI is used to identify possible locations and corresponding severities of cracks in the beam type structure. Rezaee et al. [5] have followed mechanical energy balance method for free vibration analysis of cracked cantilever beam. They modeled the cracked beam as a fatigue crack by considering the effect of opening and closing of crack during the beam vibration. Sadeghian et al. [6] have used a beam with different types of cross section or structural discontinuities such as crack along its length and provided a special kind of frequency analysis for cracked and stepped beam located on
Prediction of Cracks Using Fea Analysis andFuzzy Logic Approach
1D. R. Parhi, Manoj Kumar Muni and Chinmaya Sahu1Dept. of Mechanical Engg., National Institute of Technology-Rourkela, Odisha-769008
Generally damage in a structural element may occur due to normal operations, accidents, deterioration or severe naturalevents such as earth quake or storms. During operation, all structures are subjected to degenerative effects that may causeinitiation of structural defects such as cracks which, as time progresses, lead to the catastrophic failure or breakdown of thestructure. There is change in natural frequencies and mode shapes takes place due to different orientation of the cracks. Byexamining these changes, crack position and its magnitude can be identified. The analysis has been done by using finiteelement method with the help of ANSYS workbench and CATIA V5. It has been noticed that when the crack depth increasesnatural frequency decreases and also there is a deviation in the mode shape. Fuzzy-logic with triangular, Gaussian andmixture of triangular and Gaussian membership function has been used for detection of crack by considering frequencies asinput parameter. The FUZZY logic controller used here comprises of Three input and two output parameters. The Inputparameters are first three natural frequencies and the output parameters are relative crack location and relative crackdepth. A series of FUZZY rules are derived from vibration parameters which are finally used for prediction of crack locationand its depth. By comparing the fuzzy result with the experimental result it is observed that the fuzzy controller can predictthe relative crack location and relative crack depth in a very accurate manner with a very small percentage of error.
INTRODUCTION
Cracks are a potential source of catastrophic failure inmechanical machines and in civil structures and inaerospace engineering. To avoid the failure caused bycracks, many researchers have performed extensiveinvestigations over the years to develop structuralintegrity monitoring techniques. Most of the techniquesare based on vibration measurement and analysisbecause, in most cases, vibration based methods canoffer an effective and convenient way to detect fatiguecracks in structures.
Eftekhari et al. [1] presented an analytical modelof cracked cantilever beams made of functionallygraded materials (FGMS).They implemented the localflexibility concept to model the open crack andmodeled the open edge crack as a mass less elasticrotational spring. Cheng et al. [2] have investigatedon vibration characteristics of cracked rotating taperedbeam by the P-version finite element method. Theyused shape functions with shifted Legendre orthogonalpolynomials to represent the transverse displacementfield. Waghulde et al. [3] have researched aboutmechanical vibrations induced in flexible structuressuch as beams and plates through various optimization
functions and control theories to optimize transientresponse dynamic characteristics. They usedpiezoelectric materials, sensors, actuators to reducethese vibrations. Finite element method is used tomodel the structure with piezoelectric material andgenetic algorithm is applied to find optimal placementof a number of piezoelectric sensors with objectivefunctions of minimize control force, displacement,velocity. Liu Yong-Guang. [4] has presented a newdamage index named smeared cracked model damageindex (SCMDI) based on the mean curvature of beamtype structure in time domain to detect cracks. Theyintroduced the smeared cracked model in damagemechanics and demonstrated SCMDI is more robustand high efficiency in crack detection. SCMDI is usedto identify possible locations and correspondingseverities of cracks in the beam type structure.
Rezaee et al. [5] have followed mechanical energybalance method for free vibration analysis of crackedcantilever beam. They modeled the cracked beam as afatigue crack by considering the effect of opening andclosing of crack during the beam vibration. Sadeghianet al. [6] have used a beam with different types of crosssection or structural discontinuities such as crack alongits length and provided a special kind of frequencyanalysis for cracked and stepped beam located on
elastic foundations. They considered D’Alembert’sapproach for the solution of wave differential equationsand utilized the technique of wave method which ismainly depended on the study of transmission of waveand reflection of waves colliding to a barrier. Rezaeiet al. [7] have demonstrated the effectiveness of adamage index as the EMD energy damage index forcrack identification of beam. They utilized theempirical mode decomposition for health assessmentof the system based on its vibrational data. Rezaee etal. [8] have proposed a analytical method for vibrationanalysis of a cracked simply supported beam. Theyconsidered a nonlinear model for the fatigue crack andsolved the governing equation of motion of the crackedbeam by using perturbation method. Buezas et al. [9]have provided a novel technique for the crack detectionin structural elements by means of a genetic algorithmoptimization method. The cracked model taken intoaccount the existence of contact between the interfacesof the crack. They addressed bi- and three dimensionalmodels to handle the dynamics of a structural elementwith a transverse breathing crack. The methodology isnot restricted to beam-like structures since it can beapplied to any arbitrary shaped 3D element. Elshafeyet al. [10] have discussed the damage detection inoffshore jacket platforms subjected to random loadsusing a combined method of random decrementsignature and neural networks. They used the randomdecrement technique to extract the free decay of thestructure.
THEORETICAL FIGURE OF BEAM
selected. In the geometry tab, the CATIA V5 model isimported (Right Click -> Import geometry). In themeshing sizing option is added. Then the GenerateMesh option is selected. In the details of sizing thewhole imported model is selected (confirmed byclicking APPLY). The size is changed from default to1mm. In the next step the use of ‘fixed support’ optionis made. After that one face is selected (Confirmed byclicking APPLY tab). Then the ‘solve option’ is selectedwhich shows the modal frequencies. Each modalfrequency or the whole modal frequency was selectedto evaluate the mode shape. This will create all modeshapes.
TABULATION OF FREQUENCIES FOR DIFFERENTMODE SHAPES CONSIDERING CRACK ANDWITHOUT CRACK
Frequency without crack
Mode Frequency without crack
1 5.1571
2 32.161
3 90.321
FREQUENCY WHEN CRACK IS AT 50MM FROMFIXED END
Mode Frequency (in Hz) with crack depth in mm
0.5 1 1.5 2 2.5 3
1 5.1527 5.1414 5.126 5.0972 5.0513 4.9702
2 32.144 32.097 32.033 31.916 31.735 31.423
3 90.282 90.203 90.104 89.907 89.611 89.111
Similarly frequencies for different crack locationand crack depth have been calculated.
Figure 1: Cantilever Beam with Dimensions
DEVELOPMENT OF MODEL IN CATIA V5
Models of cantilever beam have been developed byusing CATIA V5 with different crack size and atdifferent crack location. Crack depth varied from0.5mm to 3.0mm with an interval of 0.5mm. Similarlycrack location was carried from 50mm to 700mm fromfixed end with an interval of 50mm.
PROCEDURE FOR ANALYSIS
Modal analysis is selected from the analysis systemsavailable in the work bench. In the “Outline ofschematic” of engineering data, structural steel is Figure 2: For First Mode
Prediction of Cracks using FEA Analysis and Fuzzy Logic Approach 15
ANALYSIS OF THE FUZZY CONTROLLER
The FUZZY controller developed has got three inputparameters and two output parameters. The linguisticterms used for the inputes are as follows.
• The linguistic term used for the inputs are asfollows
1. First natural frequency=”FNF”
2. Second natural frequency=”SNF”
3. Third natural frequency=”TNF”
• The linguistic term used for the output are asfollows
1. Relative crack location=”RCL”
2. Relative crack depth=”RCD”
The FUZZY controller used in the present text isshown in fig. 5. The membership functions usedare triangular, Gaussian and a mixture of both.The linguistic terms for all the membership functionsused in the FUZZY controller are described in thetable 1.
Table 1Description of FUZZY Linguistic Terms
Membership Linguistic TermsFunctions Name
f1L Low Value of Natural Frequency for FirstMode of Vibration
f1m Medium Value of Natural Frequency for FirstMode of Vibration
f1h High Value of Natural Frequency for FirstMode of Vibration
f2L Low Value of Natural Frequency for SecondMode of Vibration
f2m Medium Value of Natural Frequency forSecond Mode of Vibration
f2h High Value of Natural Frequency for SecondMode of Vibration
f3L Low Value of Natural Frequency forThirdMode of Vibration
f3m Medium Value of Natural Frequency for ThirdMode of Vibration
f3h High Value of Natural Frequency for ThirdMode of Vibration
LcL Low Value of crack location
mcL Medium Value of Crack Location
hcL High Value of Crack Location
Lcd Low Value of Crack Depth
mcd Medium Value of Crack Depth
hcd High Value of Crack Depth
Table 2FUZZY Rules used in FUZZY Controller
Sl. No. Some of rules used in fuzzy controller
1 If FNF is f1L,SNF is f2L,TNF is f3L, then RCL is LcL,RCD is hcd
2 If FNF is f1L,SNF is f2L,TNF is f3m, then RCL is LcL,RCD is hcd
3 If FNF is f1m,SNF is f2m,TNF is f3h, then RCL is LcL,RCD is hcd
4 If FNF is f1m,SNF is f2L,TNF is f3h, then RCL is LcL,RCD is hcd
5 If FNF is f1h,SNF is f2m,TNF is f3h, then RCL is LcL,RCD is hcd
Figure 3: For Second Mode
Figure 4: For Third Mode
Figure 5
contd. table 2
16 D. R. Parhi, Manoj Kumar Muni & Chinmaya Sahu
6 If FNF is f1L,SNF is f2m,TNF is f3h, then RCL is LcL,RCD is hcd
7 If FNF is f1L,SNF is f2h,TNF is f3h, then RCL is LcL,RCD is hcd
8 If FNF is f1m,SNF is f2h,TNF is f3h, then RCL is LcL,RCD is hcd
9 If FNF is f1L,SNF is f2h,TNF is f3m, then RCL is LcL,RCD is hcd
10 If FNF is f1m,SNF is f2h,TNF is f3L, then RCL is LcL,RCD is hcd
11 If FNF is f1h,SNF is f2h,TNF is f3L, then RCL is mcL,RCD is hcd
12 IF FNF is f1m,SNF is f2L,TNF is f3m, then RCL is mcL,RCD is hcd
13 If FNF is f1h,SNF is f2m,TNF is f3h, then RCL is mcL,RCD is hcd
14 If FNF is f1m,SNF is f2m,TNF is f3L, then RCL is LcL,RCD is hcd
15 If FNF is f1h,SNF is f2L,TNF is f3h, then RCL is mcL,RCD is hcd
16 If FNF is f1h,SNF is f2L,TNF is f3m, then RCL is hcL,RCD is hcd
17 If FNF is f1h,SNF is f2L,TNF is f3L, then RCL is hcL,RCD is hcd
18 If FNF is f1h,SNF is f2m,TNF is f3L, then RCL is hcL,RCD is hcd
19 If FNF is f1h,SNF is f2h,TNF is f3h, then RCL is LcL,RCD is Lcd
20 If FNF is f1h,SNF is f2h,TNF is f3m, then RCL is hcl,RCD is mcd
21 If FNF is f1h,SNF is f2m,TNF is f3m, then RCL is mcL,RCD is mcd
FUZZY controller for finding out crack depth and cracklocation
TRIANGULAR MEMBERSHIP FUNCTION
Triangular Membership functions are shown in thefig. 6
Sl. No. Some of rules used in fuzzy controller
Figure 6a: Membership Functions for Natural Frequency forFirst Mode of Vibration
Figure 6b: Membership Functions for Natural Frequency forSecond Mode of Vibration
Figure 6c: Membership Functions for Natural Frequency forThird Mode of Vibration
Figure 6d: Membership Functions for RelativeCrack Location
Figure 6e: Membership Functions for RelativeCrack Depth
Prediction of Cracks using FEA Analysis and Fuzzy Logic Approach 17
The inputs to the FUZZY controller are First,Second and Third natural frequency. The outputsfrom the fuzzy controller are relative crack depthand relative crack location. 21 numbers of thefuzzy rules out of several fuzzy rules are listed in thetable 2.
Resultant values of relative crack depth and relativecrack location when
Figure 7a: Membership Functions for Natural Frequency forFirst Mode of Vibration
Figure 7b: Membership Functions for Natural Frequency forSecond Mode of Vibration
GAUSIAN MEMBERSHIP FUNCTION
Figure 7c: Membership Functions for Natural Frequency forThird Mode of Vibration
Figure 7d: Membership Functions for RelativeCrack Location
Figure 7e: Membership Functions for RelativeCrack Depth
18 D. R. Parhi, Manoj Kumar Muni & Chinmaya Sahu
The linguistic terms and rules used are same as incase of triangular membership function.
Resultant values of relative crack depth and relativecrack location when
FNF = 5.05, SNF = 31.7, TNF = 89.5, then RCL =281 and RCD = 2.25
The linguistic terms and rules used are same as incase of triangular membership function.
Resultant values of relative crack depth and relativecrack location when
FNF = 5.05, SNF = 31.7, TNF = 89.5, then RCL =265 and RCD = 2.25
COMBINATION OF TRIANGULAR AND GAUSIANMEMBERSHIP FUNCTION
Figure 8a: Membership Functions for Natural Frequency forFirst Mode of Vibration
Figure 8b: Membership Functions for Natural Frequency forSecond Mode of Vibration
Figure 8c: Membership Functions for Natural Frequency forThird Mode of Vibration
Figure 8d: Membership Functions for RelativeCrack Location
Figure 8e: Membership Functions for RelativeCrack Depth
Prediction of Cracks using FEA Analysis and Fuzzy Logic Approach 19
RESULT AND DISCUSSION
Discussion is based upon the outputs of ANSYSworkbench. With the help of ANSYS workbench, forcantilever beam with different crack location anddifferent crack depth natural frequencies for differentmode shapes have been calculated. From these outputsgraphs have been plotted by taking crack depth in x-axis and frequency in y-axis and shown in fig. 2., fig.3. and fig. 4. for first mode, second mode and thirdmode of vibration respectively. it can be noticed thatwith increase in crack depth (at a certain crack location)frequency of vibration decreases for first mode, secondmode and third mode of vibration and also it isobserved that there are significant variations in modeshapes at the vicinity of crack location due to presenceof crack. To find out relative crack location and relativecrack depth fuzzy controller with triangular, Gaussianand combination of both membership functions havebeen used. Table 1 shows Description of FuzzyLinguistic terms and Table 2 represents fuzzy rules usedin Fuzzy controller. For a given set of inputs whentriangular membership function is used the output weare getting is approximately equal to the theoreticalone having 9.6% of error. For a given set of inputswhen Gaussian membership function is used the outputwe are getting is approximately equal to the theoreticalone having 12.4% of error. For a given set of inputswhen combination of both triangular and Gaussianmembership function is used the output we are gettingis approximately equal to the theoretical one having6.0% of error.
CONCLUSION
From the results and discussions the followingconclusions has been drawn
• Significant changes in natural frequencies andmode shapes of the vibrating beam areobserved at the vicinity of crack location.
• The fuzzy controller is developed withtriangular, Gaussian and combination of bothmembership functions.
• The input parameters to the fuzzy controllerare first three natural frequencies. The outputsfrom the fuzzy controllers are relative cracklocation and relative crack depth.
• By comparing the fuzzy results with thetheoretical results it is observed that the fuzzycontroller can predict the relative cracklocation and crack depth in a very accuratemanner with having small percentage of error.
• The crack depth and crack location of a beamcan be predicted by the developed fuzzycontroller in nanoseconds thereby saving aconsiderable amount of computational time.
References
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[3] Waghulde, K. B., Kumar Bimlesh., Garse, T. D. and Patil,M. M. (2011), ‘Vibration Analysis and Control ofCantilever Plate by using Finite Element Analysis’,Journal of Engineering Research and Studies, 2, 36-42.
[4] Liu, Yong-Guang (2011), ‘Cracks Detection of BeamType Structures based on Curvature in Time Domainusing Smeared Crack Model’, IEEE Publication.
[5] Rezaee, Mousa. and Hassannejad, Reza (2010), ‘A NewApproach to Free Vibration Analysis of a Beam with aBreathing Crack based on Mechanical Energy BalanceMethod’, Acta Mechanica Solida Sinica, 24, 185-194.
[6] Saddeghian, M. and Ekhteraei Toussi, H. (2010),‘Frequency Analysis for a Timoshenko Beam Locatedon Elastic Foundation’, IJE Transactions, 24, 87-105.
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20 D. R. Parhi, Manoj Kumar Muni & Chinmaya Sahu
[8] Rezaee, Mousa and Hassannejad, Reza (2010), ‘FreeVibration Analysis of Simply Supported Beam withBreathing Crack using Perturbation Method’, ActaMechanica Solida Sinica, 23, 459-470.
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[10] Elshafey Ahmed, A., Haddara Mahmoud, R. andMarzouk, H. (2010), ‘Damage Detection in OffshoreStructures using Neural Networks’, Marine Structures,23, 131-145.
