Crack detection in beam-like structures using genetic algorithms
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.
Section snippets
The cracked-beam model
In the first step of examining a new fault detection method, it should be used on a model of a structure with certain faults. In the cracked-beam model, a crack on a beam is modeled by placing a torsional spring in the position of the crack. The spring stiffness Kθ can be calculated from the following formula:where ν is the Poisson coefficient, h is the beam height, a is the crack depth, I is the second moment of area, E is the Young's module, and j(a/h) is the computed by
The genetic algorithm method
A genetic algorithm is a probabilistic search algorithm based on a model of natural evolution. The algorithm has clearly demonstrated its capability to create good approximate solutions in complex optimization problems [19]. The genetic algorithm method is very attractive in comparison with classical methods since it does not require a search within the whole solution space.
To estimate the location and depth of a crack in a structure using natural frequency information, we use genetic
Numerical results
Three crack sizes at 7 different locations along the span of the beam are used to generate the test points, so in total 21 test points (frequency data) with different crack conditions are used to examine the proposed approach. For each test point the algorithm is run from five different initial random points and the answer with the best cost value is selected as the predicted answer. The input natural frequencies of the test points of these crack properties are shown in Table 2. Table 4, Table 5
Conclusions
A genetic algorithm approach has been presented for detecting cracks in beam-like structures. The search process proposed in this paper utilizes binary and continuous genetic algorithms to find the crack location and depth whose natural frequencies have maximum similarity with the input natural frequencies. Also a new cost function based on natural frequencies was presented.
Some guidelines for selecting GA parameters are provided. In comparison with traditional GAs, we use a GA with small
Acknowledgements
This research work has been partially supported by Iran Telecom Research center, under grant no. T/500/4369. The authors would like to thank anonymous reviewers for their valuable suggestions and comments which helped us to improve the quality of the paper.
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