Applied Soft Computing 12 (2012) 2217–2226 Contents lists available at SciVerse ScienceDirect Applied Soft Computing j ourna l ho mepage: www.elsevier.com/locate/asoc A hybrid particle swarm–Nelder–Mead optimization method for crack detection in cantilever beams M.T. Vakil Baghmisheh a,∗ , Mansour Peimani a , Morteza Homayoun Sadeghi b , Mir Mohammad Ettefagh b , Aysa Fakheri Tabrizi a a ICT Research Center, Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran b Research Laboratory of Vibration and Modal Analysis, Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran a r t i c l e i n f o Article history: Received 1 September 2009 Received in revised form 22 April 2011 Accepted 18 March 2012 Available online 30 March 2012 Keywords: Crack detection Structural health monitoring Natural frequency Nelder–Mead simplex Particle swarm optimization a b s t r a c t The estimation of a crack location and depth in a cantilever beam is formulated as an optimization problem and the optimal location and depth are found by minimizing the cost function which is based on the difference of the first four measured and calculated natural frequencies. Calculated natural frequencies are obtained using a rotational spring model of the crack, and measured natural frequencies are obtained by using cracked beam frequency response and modal analysis. A hybrid particle swarm–Nelder–Mead (PS–NM) algorithm is used for estimating the crack location and depth. The hybrid PS–NM is made-up of a modified particle swarm optimization algorithm (PSO), aimed to identify the most promising areas, and a Nelder–Mead simplex algorithm (NM) for performing local search within these areas. The PS–NM results are compared with those obtained by the PSO, a hybrid genetic–Nelder–Mead algorithm (GA–NM) and a neural network (NN). The proposed PS–NM method outperforms other methods in terms of speed and accuracy. The average estimation errors for crack location and depth are (0.06%, 0%) for the PS–NM, however, (0.09%, 0%), (0.46%, 0.54%) and (0.39%, 1.66%) for the GA–NM, the PSO and the NN methods, respectively. 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 estimation errors are (9.24%, 8.56%) for the PS–NM, but (9.64%, 9.50%), (10.89%, 10.89%), (11.53%, 11.64%) for the GA–NM, the PSO and the NN methods, respectively. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Crack is a type of damage that often occurs on the elements of structures, and in the case of negligence may cause serious prob- lems; thus a crack should be detected in an early stage before developing. The detection of structural damages such as cracks is still an active area of the research in the field of structural health monitoring [1–3]. For non-destructive test and health-monitoring, various approaches have been studied including both conventional and modern methods. The conventional methods are well-developed, implemented in widely marketed equipments, and accepted by the industry and regulatory agencies as practically applicable nonde- structive evaluation (NDE) methods [4]. The modern NDE methods ∗ Corresponding author. Tel.: +98 411 4794043; fax: +98 411 3300829. E-mail addresses: mvakil@tabrizu.ac.ir, vakil baghmisheh@yahoo.com (M.-T. Vakil-Baghmisheh), mpeimani@gmail.com (M. Peimani), morteza@tabrizu.ac.ir (M.H. Sadeghi), ettefagh@tabrizu.ac.ir (M.M. Ettefagh), aysa.fakheri@gmail.com (A.F. Tabrizi). are still under development, implemented in a limited manner in some equipment and not fully accepted by the industry and reg- ulatory agencies as practically applicable methods. One of these modern methods is the vibration-based inspection methodology [5]. A comprehensive survey of available literature on crack detec- tion technology has been carried out by Richardson [6]. This paper discusses various methods for structural integrity monitoring of nuclear power plants, large civil engineering structures, rotating machinery, etc. A more comprehensive survey was presented later by Doebling et al. [7]. This paper reviews numerous publications available on identifying structural damages via vibration-based tests. It categorizes various methods according to the measured data and the technique of analysis. Salawu presents a review of the literature which has utilized variations of natural frequencies [8]. Sahin and Shenoi have presented a damage detection algorithm using a combination of global and local vibration-based data as input to an artificial NN [9]. Krawczuk has used the wave prop- agation approach combined with a genetic algorithm for damage detection in beam-like structures [10]. In [11,12] the authors have used the genetic algorithms for crack detection in a cantilever beam. 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2012.03.030