Identification of Crack in a Cantilever Beam using Improved PSO Algorithm

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1 IJIRST International Journal for Innovative Research in Science & Technology Volume 1 Issue 11 April 2015 ISSN (online): Identification of Crack in a Cantilever Beam using Improved PSO Algorithm Cletus Sam. C A. Asharaf Ali M. Tech Student Assistant Professor Department of Mechanical Engineering Department of Mechanical Engineering SRM University, Kattankulathur, India SRM University, Kattankulathur, India Abstract The estimation of a crack location and depth in a cantilever beam structure is formulated as an optimization problem and the optimal location and depth are found by minimizing the objective function. The identification of crack is done by adopting an improved PSO Algorithm. This new method combines Particle Swarm Optimization and Swallow Swarm Optimization that helps in minimizing the objective function with comparatively less time compared to standard PSO Algorithm. The fitness value of a PSO algorithm is calculated based on mean squared error (MSE) simulated response and actual response. The performance evaluation is done in frequency domain analysis. Finally, the results obtained using improved PSO algorithm is compared with the original PSO Algorithm to determine the level of accuracy. Keywords: Cracks Identification, Fracture Mechanics, Frequency Domain, Optimization, Particle Swarm Optimization, Swallow Swarm Optimization, Hybrid Particle Swallow Swarm Optimization I. INTRODUCTION The vibration based damage identification methods are promoted in their use over nondestructive evaluation methods (viz. visual, acoustic or ultrasonic, magnetic field methods) mainly because of simplicity. Dimarogonas [1] presented a review on crack damage identification in beams and turbine rotors along with the cracked structural vibration theories. Doebling et al. [2] summarized the vibration based damage identification methods to detect, locate and characterize a crack damage in structural and mechanical systems by examining changes in measured vibration response. Several research have been done on published on modeling and detection of cracks in beam like structures using numerous Finite Element Analysis (FEA) models as single crack identification [3 6] and multiple crack identification [7], but still there is a lag in the development of a rigorous cracked beam vibration theory as well as an efficient method [1,7] to detect the multiple cracks with good accuracy and reduced computational effort. Crack or local defect in a structural member introduces local flexibility that affects the dynamic response of the structure. Monitoring the dynamics response of the structure is an effective aid to identify the location and quantification of the crack in structures. Dimarogonas [1] stated the advances and future work direction on the vibration of the cracked structures in a review. Doebling [2] presented a literature review on detection, location and characterization of the structural damage through techniques that examine the changes in the measured structural vibration response. Krawczuk [3] developed a finite element model for cantilever beam with a crack in middle based on elasto-plastic fracture mechanics and concluded that the inertia matrix does not affect the natural frequencies in numerical point of view. Viola [7] developed the finite element model for cracked Timoshenko beam with open crack and derived the stiffness and consistent mass matrices. Qian [3] derived the element stiffness matrix of a beam with a crack from an integration of stress intensity factors. The crack in terms of location and depth in cantilever beam was identified by auto-regressive-moving-average method in time domain analysis. Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture. When material damage like micro-cracks and voids grow in size and become localized, the averaging procedure can no longer be applied and discontinuities must be taken into account. This localization results in a macroscopic crack, which may grow very fast, resulting in global failure. All rights reserved by 454

2 Fig. 1: Tensile Test with Axial Elongation and Fracture Detection of cracks is done by experimental techniques, ranging from simple and cheap to sophisticated and expensive. Experimental Fracture Mechanics (EFM) is about the use and development of hardware and procedures, not only for crack detection, but, moreover, for the accurate determination of its geometry and loading conditions. The crack growth criterion can also be based on the stress state at the crack tip. This stress field can be determined analytically. It is characterized by the stress intensity factor. The resulting crack growth criterion is referred to as local, because attention is focused at a small material volume at the crack tip. Assumption of linear elastic material behavior leads to infinite stresses at the crack tip. In reality this is obviously not possible: plastic deformation will occur in the crack tip region. Using yield criteria (Von Mises,Tresca) the crack tip plastic zone can be determined. When this zone is small enough (Small Scale Yielding, SSY), LEFM concepts can be used. II. SWALLOW SWARM OPTIMIZATION (SSO) SSO has common features with PSO but also several significant differences. Three types of particles are considered: leader, explorer and aimless particles. Leader particles are categorized into two types: Local Leaders (LL) that conduct the related internal subcolonies and show a local optimum point, and Head Leader (HL) that is responsible for the leadership of the entire colony and indicates the global optimum point. Explorer particles, that represent the largest part of the population, take care of the exploration of design space. In each iteration(k), particles play different roles according to their type. Each swallow arriving at an extreme point emits a special sound to guide the group toward there. If that place is the best in the design space, that particle becomes the Head Leader, HL (k).if the particle reaches a good position (yet not the best) compared with its neighboring particles, it is chosen as a local leader, LL (k). Fig. 2: Types of Particles and Movements explorer particles Otherwise, the particle is an explorer one and has to change its position in the search space. The new position of explorer particles, Xi (k+1), is Xi (k) considering VHLi (k+1) (change velocity vector of particle toward Head Leader), and VLLi (k+1) (change velocity vector of particle toward Local Leader). The change velocity of explorer particles toward head leader and corresponding local leader are dynamically adjusted according to the current velocity vector of the particle toward leaders All rights reserved by 455

3 (VHLi (k) and VLLi (k), respectively), best self-experienced position (X (k) ), and leader s position (HL (k) and LL (k), respectively). This is shown schematically in Figure 1.2and modeled mathematically as follows: - (6) - (7) - (8) - (9) Where HL, HL, LL and LL are acceleration control coefficients adaptively definedwhile rand() is a random number uniformly distributed in (0,1). Aimless particles o (i)also carry out exploration but have nothing to do with head leader and local leaders. They simply move back and forth with respect to their previous positions by displacing by a random fraction of the allowable step defined by the upper and lower bound of design variables. That is: [ { } ] - (10) III. HYBRID PARTICLE SWALLOW SWARM OPTIMIZATION (HPSSO) HPSSO includes two important features of SSO added to the basic PSO formulation: considering a specific number of subcolonies and a certain number of particles for specific task. Similar to SSO, there are leaders (global leader and local leaders), explorers, and aimless particles. The size of population N is specified along with the number of subcolonies N subcolony and aimless particles N aimless ; the number of explorer particles (Ne.p) is determined as a consequence. HPSSO starts with a set of particles randomly positioned in the design space and with random velocities. The position and velocity of each particle are progressively updated to search the optimum. In each iteration, particles are sorted based on the value of the cost function (usually, pseudo-cost function including penalty terms or fitness). The best particle is set as the head leader and N subcolony subsequent particles are set as local leaders going from top to bottom. N aimless particles are then selected from the worst ones going from bottom to top. The remaining particles are set as explorers. The search of each explorer particle is performed by adding the updated velocity vector to the current position of the particle. Compared with PSO, the velocity vector includes an extra term to account for the contribution of local leaders: -(11) -(12) Where is the local leader of the subcolony including the ith particle, r 3 is a uniform random number in the (0,1) interval and c 3 is the learning factor controlling the influence of the proximity cognition. In each iteration, the position of a particle included in a subcolony can be changed so as to move away from the current local leader and join the leader of another group. The distance between each explorer particle and local leaders is used to determine the related subcolony so that each explorer particle can be placed near the closest local leader. That is:, i1,2,.,n ep, j1,2,.,n sub-colony -(13) -(14) Where ng is the number of design variables and disti,jis the distance between the ith explorer particle and jth local leader. Three possible options are considered for aimless particles: (i) they perform just a random search in the same way as it is done in SSO; (ii) they perform a local search in the neighborhood of local leaders; (iii) they perform a dynamic search in the neighborhood of the global leader. If option (ii) is chosen, the number of aimless particles coincides with the number of subcolonies and hence an aimless particle should be defined for each subcolony. In this case, the distance between the worst particles and local leaders is the criterion to assign each aimless particle to its corresponding subcolony. This strategy is most effective in structural optimization problems, while making aimless particles to search in the neighborhood of the global leader is good for mathematical optimization problems. Aimless particles perform their search in the neighborhood of the local leader of their subcolony or the head leader of the population according to the following rules: All rights reserved by 456

4 Fig. 3: Schematic Representation of the Transition from the Current Iteration to Subsequent Iteration { [ ] - (15) whererand(-1,1) is a uniform random number between -1 and 1; mins and maxs, respectively, are the lower k and upper bound of design variables; is a parameter defined to generate the effective search range about global optimum or local leaders. That is: - (16) Where max and min, respectively, are the values of in the first and last iterations of the algorithm, set in the present study as 0.01 and 0.001; iter is the number of the current iteration; itermaxis the total number of optimization iterations. Figure 1.3illustrates the transition between two consecutive generations. The population is updated by: (i) copying first the head leader and local leaders from one generation to the next (in some way, this can be interpreted as an elitist strategy); (ii) performing search with explorer particles to move population toward the best regions of design space (exploration phase or global search); (iii) performing a dynamic local search with aimless particles in the neighborhood of the head leader or local leaders. The flowchart of the HPSSO algorithm is presented in Figure 1.4. IV. CRACKED STRUCTURAL MEMBER FINITE ELEMENT MODEL The structural member in the present study is considered as Euler-Bernoulli beam element having length L e with an open crack of depth a at a distance L c from the left end. A cracked beam finite element model based on elasto-plastic fracture mechanics and finite element method developed by Krawczuk et al. (2000) and Viola et al. (2001, 2002) is considered in the present study to model the crack damage. The element has two nodes with two degrees of freedom (transverse displacement, and rotations ) at each node as shown in figure below. The cracked beam is divided into two parts with a rotational mass less spring of length zero having a torsional rigidity (G) that is dependent on the crack geometry and beam cross-section. It is given (Viola et al., 2001) as follows where E is Young s modulus, H is height, B is breadth of rectangular beam cross-section and β B / H.The function ( ) depends on the dimensional crack ratio and it can be represented as: ( ) 2 ( ), (0 < 0.5 ) - (18) And ( ) ( 1.32 / (1- )2 ) , ( 0.5 < 1 ). - (19) The cracked element is taken as a one-dimensional continuum with an elastic hinge at the fracturing section, having a rotational constant D chosen to simulate the crack effect. The principal coordinate system (O xy ) centered at the left end of the beam with x axis along the beam and (O 1x1y1 ) and (O 2x2y2 ) the other two coordinate systems as shown in figure above. Displacements and rotations are given as follows: - (17) All rights reserved by 457

5 { And { - (20) The four boundary conditions, at the cracked section where the elastic hinge is located are as follows: And [ ] ( ) -(21) Where E is the Young s modulus and is the area moment of inertia of the beam element with rectangular crosssection. Fig. 4: Flowchart of the HPSSO Algorithm All rights reserved by 458

6 Fig. 5: Beam Element with a Crack Modeled As Zero Length Mass-Less Spring The element stiffness matrix [Ke] of the cracked beam element is not explicitly shown in Krawczuk et al.[4] and Viola et al. [7]. Hence, this has been derived here as follows: [ ] [ ] -(22) Where, -,,, And is the flexibility coefficient; when K0 the form of the stiffness matrix is the same as that of the non-cracked beam element. The mass matrix in the present study is considered as same as that of non-cracked beam element because the mass matrix derived from the cracked beam element do not affect much the natural frequencies and mode shapes in the numerical point of view Krawczuk et al.,[4]. The element mass matrix [Me] of the beam element is as follows: [ ] [ ] where, is mass density of the material and is area of the beam element with rectangular cross-section.the corresponding element stiffness matrix [K e ] and mass matrix [M e ] of each structural member respectively, are assembled in the global stiffness matrix [K g ] and mass matrix [M g ] of the structure to simulate the structural vibration response. -(22) All rights reserved by 459

7 V. CRACK IDENTIFICATION After assembling the global stiffness and mass matrices, the problem definition has to be carried out. Initially, a cantilever beam with single crack is considered for detecting the crack damage in terms of crack location and crack depth. The beam structure with dimensions 300 mm length (L), 20 mm breadth (B) and 20 mm height (H) is taken into consideration. The material of the cantilever beam is steel with N/m2 Young s modulus (E) and mass density ( ) of 7850 kg/m3. An open crack of specified depth (a) exists at a distance L1 from the fixed left end of the cantilever beam. The entire beam is divided into five elements with 50 mm, 60 mm, 60 mm, 60 mm and 70 mm respectively from left end to right end. Each element has two nodes with two degrees of freedom (transverse displacement, w and rotation) at each node. The crack may exist at any location in the entirecontinuous span of the cantilever beam. The dimensionless parameters L1/L and a/h respectively for the crack location and depth are assumed to vary in between 0 and 1 from fixed end to free end and from zero depth to full depth respectively. Natural frequencies and mode shapes of the considered cantilever beam are simulated from eigen value equation using assembled global stiffness matrix [K] and mass matrix [M]. Graph 1 refers to the minimization of the problem using PSO Algorithm and Graph 2 refers to the same using Improved PSO Algorithm. On comparison of the two graphs it is evident that the Improved PSO helps in converging by 15 iterations where the PSO Algorithm does the same in 20 iterations. VI. RESULTS AND DISCUSSIONS Fig. 6: PSO Algorithm Crack Detection Fig. 7: Improved PSO Algorithm Crack Detection The results in Table 1 are discussed in detail here. The Absolute Mean Percentage Error(AMPE) of the 12 test damage cases to the input patterns was 24.36% for the crack location (L1) and 76.17% for the crack depth (a) of the single crack identification using conventional PSO Algorithm method. Whereas on using the Improved PSO Algorithm, it is 9.027% for the crack location(l1) and % for the crack depth (a).there is some significant improvement in finding the crack location and crack depth in Improved PSO than the conventional PSO Algorithm. Table -1: Comparative Results of Improved PSO and Conventional PSO Algorithm All rights reserved by 460

8 VII. CONCLUSION Crack identification in terms of crack location and crack depth in a cantilever beam was performed in the frequency domain using the Improved PSO Algorithm. The crack was modeled as a rotational mass-less spring with zero length using developed fracture mechanics and finite element theory. The crack identification results of the Improved PSO method is compared with the existing PSO method in terms of accuracy and computational effort. The results showed a very good significant improvement in the crack identification by the Improved PSO Optimization method in comparison with the other methods. Therefore, the proposed Improved PSO Algorithm method appears to be a successful generalized crack identification technique with very good accuracy combined with less computational effort. Moreover, the implementation is not particularly difficult when compared to other hybrid methods. The proposed method can be tested in future with more complex problems with multiple crack identification. REFERENCES [1] A.D. Dimarogonas, Vibration of cracked structures: a state of the art review, Eng. Fract. Mech. 55 (1996) [2] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage identification methods, Shock Vib. Dig. 30 (1998) [3] G.-L. Qian, S.N.Gu, J.-S. Jiang, The dynamic behaviour and crack detection of a beam with a crack, J. Sound Vib. 138 (1990) [4] M. Krawczuk, A. Zak, W. Ostachowicz, Elastic beam finite element with a transverse elasto-plastic crack, Finite Elem. Anal. Des. 34 (2000) [5] E. Viola, L. Federici, L. Nobile, Detection of crack location using cracked beam element method for structural analysis, Theoretical Appl. Fract. Mech. 36 (2001) [6] E. Viola, L. Nobile, L. Federici, Formulation of cracked beam element for structural analysis, J. Eng. Mech. ASCE 128 (2002) [7] J. Lee, Identification of multiple cracks in a beam using natural frequencies, J. Sound Vib. 320 (2009) [8] R.Machavaram, Identification of crack in a structural member using improved radial basis function neural networks, Vol. 6 No. 2, (2013). All rights reserved by 461

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