Detection of Crack Using Genetic Algorithm - Semantic Scholar

Detection of Crack Using Genetic Algorithm Mitesh J. Mungla, Dharmendra S. Sharma and Reena R. Trivedi Abstract--- Genetic algorithm based intelligent search of crack parameters (location and its severity) in cantilever beam has been presented in the paper. Algorithm is prepared to find global minima of fitness function which is function of measured and theoretical natural frequencies. Theoretical model of cracked beam has been derived based on Euler-Bernoulli beam theory. The crack, open and uniform depth, is modeled as rotational spring connecting two segments of an integrated beam. For validation of proposed method, the cracks are generated at various locations on specimens (made of aluminum alloys (6061-T6) using wire cut electro discharge machining process (WEDM). The results are found to be in good agreement with the actual crack parameters. Keywords--- Crack Detection, Euler-Bernoulli Beam, NonDestructive Technique (NDT), Genetic Algorithm (GA), Rotational Spring

input for artificial neural networks (ANNs) to predict crack location and severity in beam like structure and laminates have been presented by Sahin and Shenoi [4], [5]. GA based crack detection was studied in beam-like structure using natural frequency based data input by Vakil-Baghmisheh et al. [7] and wavelet based elements by Xiang et al. [9]. Vakil-Baghmisheh et al. [8] prepared algorithm using PSO for crack detection in cantilever beam. In present paper, natural frequency data of beam-like cantilever structure are coupled with genetic algorithm based search mechanism for identification of location and depth of the crack. The present work is validated through number of experiments. II.

I.

INTRODUCTION

L

AST few decades, many non-conventional nondestructive techniques (NDTs) have been developed for detection of crack location. Especially, vibration based NDTs have opened new research dimension and become one of the potential alternative for identification of crack because of ability to overcome many limitations of conventional NDT. Local crack presence in the component globally affects modal parameters of it. Based on that reference, many researchers attempted to develop mechanism for detection of single/multiple cracks in last two and half decades. Adams et al. [1] and [2] simulated crack as rotational spring and established base for detection of crack for longitudinal vibration of bar. Many researchers explored the field for different components with vivid combinations and try to generalize methodology and applicability to most of all mechanical and civil engineering components, not only for crack detection but also for health monitoring and fault diagnostic purposes. Dimarogonas [3] has widely reviewed literatures related to crack modeling and crack detection strategies in various mechanical/structural components like bar, beam, turbine blades, shaft etc. Similarly, in the last decade or so, many researchers have employed stochastic methods like artificial neural networks (ANN), genetic algorithms (GAs), particle swarm optimization (PSO) etc. for identification of crack parameters. Combination of vibration (natural frequency) based data as Mitesh J. Mungla, Associate Professor, Mechanical Engineering Department, Institute of Technology and Management Universe, Vadodara, Gujarat, India. E-mail:[email protected] Dharmendra S. Sharma, Professor & Head, Mechanical Engineering Department, Faculty of Technology, M S University, Vadodara, Gujarat, India. E-mail:[email protected] Reena R. Trivedi, Associate Professor, Mechanical Engineering Department, Institute of Technology, Nirma University, Ahmedabad-382481, Gujarat, India. E-mail:[email protected]

FORMULATION OF CRACKED BEAM

The governing equation of transverse vibration of the uniform intact beam can be derived using Euler-Bernoulli beam equation as, 4

y( x

where

y( x

)

)

4

4

y(

)

(1)

0

is lateral deflection of beam,

is normalized location, x is distance of beam

L 2

measured from fixed end,

Ar L

4

1/4

,

EI

where is natural frequency of the beam and , E, A r, L and I total length and moment of inertia of the beam respectively. The (lateral deflection) solution of uncracked beam (1) can be expressed as follows. y(

)

A1 c o s h (

)

B1 s in h (

D 1 si n (

)

C 1 co s (

)

(2)

)

The open crack existing on the beam can be modeled as rotational spring having stiffness Kr This rotational spring separates an intact beam into two segments connected through it shown in figure 1. Thus, stiffness K r varies from zero (when thorough crack) to infinite (no crack).

Equations (3) and (4) along with boundary conditions (7)(10) and compatibility conditions (11)-(14) yields eight characteristic simultaneous equations: Ag

Bg

0

(15)

where,

Ag

Figure 1: Clamped-Free Beam with Single Crack The mode shape of left and right segments of the beam can be defined as, y1 (

)

A1 c o sh (

)

B 1 si n h (

D 1 si n ( y2 (

)

A 2 c o sh (

)

)

)

(3)

C 2 co s (

)

)

EI

3 7 .2 2 6 ( a / t )

2

5

3 .9 5 ( a / t )

3

0

0

0

0

0

Ch(

)

Sh(

)

C(

)

S(

Sh (

)

C h(

)

S(

)

C(

Ch(

)

Sh(

)

C(

S(

)

)

Sh (

)

C(

)

)

Sh (

)

C h(

)

S(

)

)

C h(

)

Sh(

)

C(

0

C h( )

S h( )

C(

0

0

0

0

Sh(

C h(

S(

M

2

M

3

K 4

C1

)

S h(

D1

K

)

Bg

A1

B1

A2

Sh(

)

Ch(

),

M

2

C(

),

M

4

)

C h(

B2

)

C2

K

S( )

S(

)

) )

S(

)

)

C(

)

0

M

0

)

0

1

0

C h(

0

M

K 1

K

S(

3

For

1 6 .3 7 5 ( a / t )

6

7 6 .8 1( a / t ) - 1 2 6 .9 ( a / t )

8

0

1

K

S(

)

C(

)

C(

)

D2

)

Ch(

)

cosh(

C(

)

cos(

),

K

Ch( K

)

C(

)

Sh(

)

s in h (

S(

)

s in (

),

Sh (

),

S(

),

), ),

(5)

5 .3 4 6 hf ( a t )

1 .8 6 2 4 ( a / t )

0

0

where

(4)

Where a is crack depth, t is beam thickness, f ( a / t ) is local compliance function, which is computed from strain energy density function, f (a / t )

1

1

M

M

Moreover, the relationship between rotational spring constant and crack depth ratio ( a / t ) from strain energy density function [6] calculated as: Kr

0

0

)

)

B 2 sinh (

D 2 si n (

C1 c o s(

1

9

+ 1 7 2 ( a / t ) - 1 4 3 .9 7 ( a / t ) + 6 6 .5 6 ( a / t )

4

7

(6)

10

The boundary conditions of Clamped-Free beam as, y1 (

0)

0,

(7)

y 2 ''(

1)

0,

(8)

y1 '(

0)

0,

(9)

y 2 '''(

1)

0,

Additionally, continuity conditions at crack

(10) S,

y1 ( S )

y2 (S )

(11)

y1 ''( S )

y 2 ''( S )

(12)

non-trivial

solutions,

frequencies of cracked beam. Thus, III.

Ag

0

yield

f(

, ,K)

0.

natural

GENETIC ALGORITHM

GA is an optimization approach which quickly searches global maxima/minima of complex problem and works based on survival of fittest principle. The flowchart of algorithm for the presented problem is shown in figure 2. In presented work, algorithm initiates with random initial selection of chromosomes (initial population). Each chromosome constituted with string of normalized crack location 0 S 1 and crack severity 0 1 , where a t . Using forward method, the theoretical natural frequencies are obtained from the model (15) for a chromosome. Algorithm is prepared to minimize the fitness function, that is defined as; 5

F i tn es s f u n c tio n

a b s N Ti i 1

NM

i

(16)

y1 '''( S ) y 2 '''( S ) (13) where N T i are the first five theoretical natural However, crack in the beam affects the slope pattern of the beam and hence, additional compatibility condition needs to frequencies and N M i are the first five natural frequencies, be defined at crack location as: measured through experiments. Zero value of fitness function indicates exact match of theoretical and measured frequencies y1 '( S ) y1 ''( S ) y 2 '( S ) (14) which is almost difficult to achieve. In such condition, K convergence value of fitness function needs to adopt. where K K r L E I is non-dimensional stiffness of the rotational spring representing crack. At crack location, S .

IV.

To validate the integrity of the presented study, test-set up, consists of a) vibration isolation platform, b) accelerometers, c) data acquisition system and d) data analysis software, is employed to carry out experiments with clamped-free condition.

Generate chromosomes

Calculates theoretical natural frequencies (NT) for each chromosome

Feed measured Natural frequenci es (NM)

RESULT AND DISCUSSIONS

Generate new population

Four beam (three cracked and one intact) specimens of 270 mm long aluminium alloy material (6061-T6) having cross section (40 mm width and 5 mm thickness) are prepared for the study shown in figure 3. The cracks at various locations (50 mm, 150 mm and 200 mm from free ends) of 1 mm, 2 mm and 3 mm depth on the three specimens are generated using wire cut electro discharge machining process (W-EDM).

Mutation Fitness evaluati on Crossov er

Convergenc e check?

Reproducti on

Display results [Fitness function, Normalized crack location (S), Crack depth ratio (a/t)]

Figure 2: Flow Chart of Algorithm GA inherently search global maxima/minima based on evolution mechanism. After first convergence check, the initial population is then operated by three main operators; reproduction, crossover and mutation to create a next generation of chromosomes. The reproduction operator selects better chromosomes than previous generations from mating pool and the crossover operator procreates different individuals for next generations by combining chromosomes from two individuals of the previous generation. Here, new chromosomes are created by exchanging information among strings of the mating pool. The mutation operator alters a string locally expecting a better chromosome. Even though none of these claims is guaranteed and/or tested while creating a chromosome. It is expected that if bad chromosomes are created the reproduction operator will eliminate them in the next generation and if good chromosomes are created, they will be having better probability for selection. The subsequent population is evaluated and tested till desired convergence reached.

Figure 3: Intact and Cracked Specimens of Aluminum Alloy Measured natural frequencies of cracked aluminum beam obtained using multichannel data acquisition system; they have been presented in Table 1. Algorithm is prepared on MATLAB software. Many runs, with combinations of different alternatives of three operators, have been executed to obtain prompt and accurate convergence. In present case, parents are selected based on highest probability (roulette wheel) from mating pool. Each pair (two chromosomes) procreates two offspring through single point crossover. To avoid local minima and infiltration of unwanted chromosomes (those matches with their parents), adaptive feasible operator randomly generates directions that are adaptive with respect to the last successful or unsuccessful generation. It is found that fitness function converges within 50 generations. Convergence of fitness function of test point 2 is displayed in figure 4. Normalized crack locations and normalized crack severities for test point 2 and 3 have been displayed after 50 generations in figure 5 and 6 respectively. Estimated crack location and crack depth ratio of all test points are shown in Table 2.

Table 1: First Five Measured Natural Frequencies of Test Points Normalised Crack

Measured Natural Frequency (Hz)

Test Point No.

S

a/t

NM1

NM2

NM3

1

0.2593

0.2

57.430

361.605

2

0.4444

0.2

57.605

3

0.8148

0.2

4

0.2593

5 6

NM4

NM5

1008.380

1976.42

3279.88

359.795

1011.920

1976.685

3271.13

57.715

361.415

1009.020

0.4

56.445

361.175

993.750

1971.73 1950.490

3260.785 3276.360

0.4444

0.4

57.205

353.400

1004.825

1950.685

3240.545

0.8148

0.4

52.715

360.320

1007.850

1929.065

3200.115

7

0.2593

0.6

54.725

360.025

956.425

1892.990

3268.465

8

0.4444

0.6

56.030

336.575

1000.940

1887.920

3173.235

9

0.8148

0.6

57.690

356.935

956.665

1822.850

3077.870

Table 2: Actual and Estimated Crack Parameters of Aluminum Alloys

Test Point No.

Norm Crack Loc.

Crack Depth Ratio

Est. Norm Crack Loc

Error (%)

Est. Crack Depth Ratio

Error (%)

1

0.2593

0.20

0.270

4.13

0.211

5.50

2

0.4444

0.20

0.450

1.26

0.215

7.50

3

0.8148

0.20

0.771

-5.38

0.216

8.00

4

0.2593

0.40

0.261

0.66

0.408

2.00

5

0.4444

0.40

0.448

0.81

0.408

2.00

6

0.8148

0.40

0.789

-3.13

0.414

3.55

7

0.2593

0.60

0.260

0.27

0.612

2.00

8

0.4444

0.60

0.446

0.36

0.611

1.83

9

0.8148

0.60

0.791

-2.92

0.610

1.67

Figure 4: Convergence of Fitness Function of Test Point 2

Figure 5: Crack Parameters of Test Point 2 after 50 Generations

Figure 6: Crack Parameters of Test Point 3 after 50 Generations

Material properties, used in the theoretical formulation, are slight different than that of actual properties. To increase the accuracy of prediction of crack parameters, correction to material properties ( ) of beam material need to be provided. Correction factors of material properties can be obtained by comparing measured and theoretical natural frequency of intact beam.

[5]

Formulation of clamped-free cracked beam is only applicable to isotropic and homogeneous materials with only open and single crack.

[8]

[6] [7]

[9]

V.

CONCLUSION AND FUTURE ENHANCEMENT

It has been observed that error in prediction of crack location and its severity using GA is within 5.5 % and 8 % respectively. However, increase the number of natural frequency in fitness function reduces error. But, at the same time, it increases computation time. Following further enhancements of work can be possible. 1.

Multiple crack locations can be also predicted by combination of vibration parameter and GA.

2.

Formulation and identification mechanism can be extended to inclined crack in various other mechanical/structural components like shaft, pipe, rotor blades etc.

3.

The formulation can be further extended for different end fixities (i. e. Free-Free, Fixed-Fixed, Fixed-Free etc.). ACKNOWLEDGMENT

The authors would like to thank anonymous reviewers and also welcome their valuable suggestions and comments to improve quality of paper. REFERENCES [1]

[2]

[3]

[4]

R.D. Adams, D. Walton, J.E. Flitcroft and D. Short Vibration Testing as a Non-Destructive Test Tool for Composite Materials American Society for Testing and Material, Composite Reliability, ASTM STP, Philadelphia, Vol. 580, Pp. 159-175, 1975. R.D. Adams, P. Cawley, C.J. Pye and B.J. Stone, A Vibration Technique for Non-Destructively Assessing the Integrity of Structures Journal of Mechanical Engineering Science, Vol. 20, No. 2, Pp. 93-110, 1978. A.D. Dimarogonas, Vibration of Cracked Structures: A State of the Art Engineering Fracture Mechanics, Vol. 55, No. 5, Pp. 831-857, 1996. M. Sahin, and R.A. Shenoi, Quantification and Localisation of Damage in Beam-Like Structures by Using Artificial Neural Networks with Experimental Validation , Engineering Structures, Vol. 25, Pp. 17851802, 2003.

M. Sahin and R.A. Shenoi, Vibration-Based Damage Identification in Beam-Like Composite Laminates, Proceeding of Institute of Mechanical Engineers, Part-C, Journal of Mechanical Engineering Science, Vol. 217, No. 6, Pp. 661-676, 2003. A.D. Dimarogonas and S.A. Paipetis, Analytical Methods in Rotor Dynamics Elsevier Applied Science London, 1983. M.T. Vakil-Baghmisheh, M. Peimani, M.H. Sadeghi and M.M. Ettefagh, Crack Detection in Beam-Like Structures using Genetic Algorithms Applied Soft Computing, Vol. 8, Pp. 1150-1160, 2008. M.T. Vakil-Baghmisheh, M. Peimani, M.H. Sadeghi, M.M. Ettefagh and A.F. Tabrizi, A Hybrid Particle Swarm-Nelder-Mead Optimization For Crack Detection In Cantilever Beams Applied Soft Computing, Vol. 12, Pp. 2217-2226, 2012. J. Xiang, Y. Zhong, X. Chen and Z. He, Crack Detection in a Shaft by Combination of Wavelet-Based Elements and Genetic Algorithm International Journal of Solids and Structures, Vol. 45, Pp. 4782-4795, 2008.

Mitesh J Mungla 9) born in Junagadh, Gujarat, India on 18.05.1979. He has completed Master of Engineering from L D College of Engineering, Ahmedabad, Gujarat, India in 2003. He is currently pursuing Ph D from Nirma University, Ahmedabad, Gujarat, India. He initiated his career as lecturer at CKPCET, Surat, Gujarat, India during 2003 to 2008. He worked as assistant professor at Institute of Technology at Nirma University, Ahmedabad, Gujarat during 2008 to 2010. From 2010 to 2014, he served as associate professor at Gandhinagar Institute of Technology, Gandhinagar, Gujarat. He is currently working as associate professor at ITM Universe, Vadodara, Gujarat since February 2014. He is life time member of Indian Society of Technical Education (LM69669). He has published five national and international conference papers and two journal papers. Currently, he is working in the field of vibration based nonconventional non-destructive techniques (NC-NDT) for detection of crack. (Email: [email protected])