Crack detection in beam-like structures using genetic algorithms

Abstract

A fault diagnosis method based on genetic algorithms (GAs) and a model of damaged (cracked) structure is proposed. For modeling the cracked-beam structure an analytical model of a cracked cantilever beam is utilized and natural frequencies are obtained through numerical methods. Our method utilizes genetic algorithms to monitor the possible changes in the natural frequencies of the structure. The identification of the crack location and depth in the cantilever beam is formulated as an optimization problem, and binary and continuous genetic algorithms (BGA, CGA) are used to find the optimal location and depth by minimizing the cost function which is based on the difference of measured and calculated natural frequencies. Also we present a new cost function based on natural frequencies. The average values of location and depth prediction errors are 1.02% and 1.98%, respectively, using the BGA. These values become 0.73% and 1.11% for the CGA. To validate the proposed method and investigate the modeling and measurement errors some experimental results are also included. The average values of experimental location and depth prediction errors are 10.57% and 11.19%, respectively, for the BGA. These values become 10.21% and 10.39% for the CGA.

Introduction

Beams are common structures used to carry and transfer high loads in machines and civil structures, and cracks are the main cause of structural failure. Sudden failure during high load operation could be disastrous, thus early crack detection is important. Non-destructive inspection techniques are generally used to investigate the critical changes in the structural parameters, so that an unexpected failure can be predicted. Damage can be detected, quantified and localized by on-line damage assessment techniques, through measuring vibration parameters, which indicate the global conditions of the structures. In general a crack causes a reduction in the stiffness and an increase in the damping of the structure. These changes of physical properties cause a reduction in the natural frequencies and a deviation of the mode shape. Therefore, it is possible to predict the location and the depth of a crack by measuring changes in the vibration parameters. Changes in the natural frequencies are used more often than deviation of mode shapes, since frequencies can be measured more easily than mode shapes, and they are less seriously affected by experimental errors [1].

Damage detection methods of structures based on changes in their vibration properties have been widely employed during the last two decades. Existing methods include those based on examination of changes in natural frequencies, mode shapes or mode shape curvatures. Doebling et al. [2] published a state-of-the-art review on vibration-based damage identification methods. Messina et al. [3] used the sensitivity and a statistical-based method to structural damage detection. Kosmatka and Ricles [4] presented the modal vibration characterization method using the vibratory residual forces and weighted sensitivity analysis. Ratcliffe [5] performed the frequency and curvature-based experiments. Vestroni and Capecchi [6] presented the method for concentrated damage detection based on natural frequency measurement. Gawronski and Sawicki [7] used the method based on modal and sensor norms. Hu et al. [8] presented a method using quadratic programming. Law et al. [9] presented a method for large-scale structures using super-elements with the concept of damage detection orientation modeling. Sahin and Shenoi [10] have presented a damage detection algorithm using a combination of global and local vibration-based data as input to artificial neural networks (ANNs) for location and severity prediction of the damage.

Among various vibration-based damage detection methods described, those based on updating structural model parameters can be reduced to the solution of constrained optimization problems. Comparisons of the updated model parameters with the original correlated model parameters provide an indication of damage and can be used to quantify the location and the extent of the damage. However, for optimization problems in which the objective function has many local maxima and minima, or when the variables are combinations of many discrete and continuous variables, it is difficult to use conventional optimization algorithms such as the conjugate gradient method to obtain the global optimum.

In the last two decades, genetic algorithms (GAs) [11], [12] have been recognized as promising intelligent search techniques for difficult optimization problems. GAs are stochastic search techniques based on the mechanism of natural selection and natural evolution. The genetic algorithms come in two types: binary parameter and real parameter. The mechanism of the binary and continuous genetic algorithms (BGA, CGA) is introduced in [12]. Converting variable values to binary numbers and worrying about the number of bits needed to represent a variable are avoided in the CGA. The CGAs also are more compatible with other optimization algorithms, thus making them easier to combine or hybridize [12]. Haupt and Haupt are not the only ones to reach this conclusion. The conducted experiments by Michalewicz [13] indicate that the floating-point representation is faster, has more consistent results from run to run, and provides a higher precision (especially with large domains where binary coding would require prohibitively long representations). The floating-point representation is robust, accurate, and efficient because it is conceptually closest to the real design space, and moreover, the string length reduces to the number of design variables [14]. It has been reported that the real-coded GAs outperforms binary-coded GAs in many design problems [15].

Krawczuk [16] has used the wave propagation approach combined with a GA for damage detection in beam-like structures. Mares and Surace [17] employed a GA to identify damage in elastic structures. In their study, a modified version of residual force vectors in terms of the stiffness matrix of the damaged structure is chosen as an objective function to be minimized while stiffness reduction factors of all the elements are chosen to be variables. It implicitly means that the number of variables is equal to that of the elements in the finite element (FE) model and therefore the proposed damage detection procedure is time-consuming. Sahoo and Maity [18] train a NN considering the frequency and strain as the input parameters and the location and amount of the damage as the output parameters. They use a GA to select the NN parameters. The number of total runs, i.e. the number of GA generations, population size and the NN training iterations are reported to be around 5, 100, 40 and 2000, respectively, so the number of mean square error (MSE) evaluations is 5 × 100 × 40 × 2000 = 4 × 10⁷ for a clamped free beam. However their results are quite encouraging, but their NN training phase is very time-consuming. In [19] Vakil-Baghmisheh and co-workers use a MLP for estimating the crack location.

In this paper a GA-based procedure for estimating crack location and depth in an aluminum beam is described and some guidelines for selecting the GA parameters are presented. The damage effect is modeled as a torsional spring [20], and to develop the equations of the motion of the cracked beam the Hamilton's principle is used. Then the eigenvalue problem is solved to obtain the natural frequencies of the beam. Another way for evaluating the natural frequencies of a cracked beam is based on the use of the finite elements method [20], but it is less accurate than a continuous model which is used in this paper.

The identification of crack location and depth in a cantilever beam is formulated as an optimization problem, and the binary and continuous genetic algorithms (BGA, CGA) are used to find the optimal location and depth by minimizing the cost function which is based on the difference of measured and calculated frequencies. In comparison with traditional GAs a GA with small population size combined with large mutation rate is used, so the great exploration of the search space is achieved with a small number of cost function evaluations. We iterate the GA from five different initial points (variables) and choose the best answer and in this way the rate of success in finding the global minimum instead of a local one is increased. In order to compare CGA to BGA we have implemented a set of test points. Both GAs have equal population size and maximum number of iterations. The variables in BGA are represented by a total of 21 bits, while in CGA there is no need to convert the variable values. The obtained results demonstrate higher accuracy of the CGA over the BGA. Some practical experiments carried out in the laboratory validate the integrity of the suggested method.

The organization of the paper is as follows: Section 2 illustrates the effect of cracks on natural frequencies. A GA fault diagnosis method is explained in Section 3. In Section 4, the simulation and experimental results are presented. Section 5 presents conclusions.