Shock and Vibration

Volume 2015, Article ID 832763, 10 pages

http://dx.doi.org/10.1155/2015/832763

## Identification of Crack Location in Beam Structures Using Wavelet Transform and Fractal Dimension

^{1}The State Key Laboratory for Manufacturing Systems Engineering, Xi’an 710049, China^{2}School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China^{3}School of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou 325035, China

Received 13 December 2014; Accepted 20 January 2015

Academic Editor: Yanxue Wang

Copyright © 2015 Yong-Ying Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Identification of structural crack location has become an intensely investigated subject due to its practical importance. In this paper, a hybrid method is presented to detect crack locations using wavelet transform and fractal dimension (FD) for beam structures. Wavelet transform is employed to decompose the mode shape of the cracked beam. In many cases, small crack location cannot be identified from approximation signal and detailed signals. And FD estimation method is applied to calculate FD parameters of detailed signals. The crack locations will be detected accurately by FD singularity of the detailed signals. The effectiveness of the proposed method is validated by numerical simulations and experimental investigations for a cantilever beam. The results indicate that the proposed method is feasible and can been extended to more complex structures.

#### 1. Introduction

Crack identification has gained increasing attentions from the scientific and engineering domains since the unpredicted structural failure may cause catastrophic, economic, and life loss. In the past several decades, many crack detection methods and techniques have been developed. Among them, nondestructive detecting technique is reliable and effective in maintaining safety and integrity of structures [1, 2]. However, most nondestructive crack identification methods, such as ultrasonic methods and X-ray methods, are costly and time-consuming for large structures. In the last several decades, a lot of research efforts have been made to develop an effective approach to detect cracks in structures. The vibration-based crack detection methods are developed to overcome these difficulties [3, 4]. The fundamental idea of vibration-based crack identification is to detect the crack-induced changes in the physical properties, which will lead to detectable changes in mode properties [5–7].

Natural frequency-based crack identification methods employ the natural frequency change as the basic feature [8–11]. The natural frequency-based method is attractive because the natural frequencies can be conveniently measured from just a few accessible points. By aid of the frequency, many crack identification methods were developed [12–16]. Although frequency is usually regarded as an easy-obtained characteristic with satisfying accuracy even for complex structures, its properties of lumped parameter greatly restrict wide application of frequency-based crack detection methods. Another limitation is that the crack identification problem is often ill-posed even without noise pollution, which leads to nonuniqueness of the solutions of crack location and severity. First, it is obvious that crack with same severity in symmetric locations of a symmetric structure will cause identical frequency changes. Furthermore, crack with different severity in different locations can also produce identical changes in a few measured natural frequencies.

Crack detection methods have been developed based on measured mode shapes directly or indirectly [17–19]. These methods are roughly divided into two categories. The traditional methods based on mode shape try to construct a relationship between crack location and mode shape change by a finite element method or experimental test. However, they need mode shape data from intact structures. Because of the development of modern signal processing techniques, modern signal processing methods can be employed to extract singularity information of the mode shape in the cracked structures [16, 20–22]. Singularity detection using mode shape is attractive mainly because it is possible to detect crack location without the a priori knowledge in the cracked zones. However, small crack has a minor effect on the mode shape. So singularity information of the mode shape is not identified directly without further analysis. Wavelet transform is a useful tool to detect local singularity characteristic of many different kinds of signals through decomposing the signals [23–27]. These methods treat mode shape data as a signal in spatial domain, and they use spatial wavelet transform technique to detect the irregularity of the signal caused by crack. To detect the singularities using wavelet transform, the mode shapes of cracked structure should be acquired first. Then the wavelet coefficients are calculated from the mode shapes, and the wavelet coefficients are plotted in the full region for each level of wavelets. Finally the distributions of the wavelet coefficients at each level are examined, and the peaks or sudden changes in the distributions of the wavelet coefficients are applied to locate the crack positions. The limitation of wavelet transform methods is very common; that is, there are serious boundary effects. In many cases, the irregularity of the signal caused by crack is not detected using wavelet transform alone.

As an efficient index of crack, fractal dimension is also introduced to detect crack for beam and plate structures by Hadjileontiadis et al. [28, 29]. Due to its simplicity of the fractal dimension (FD) evaluation and its robustness against noise, a FD-based technique has been developed to identify cracks of structures [28–31]. Crack location is determined by a peak on the FD curve through the local irregularity of the fundamental mode shape introduced by the crack. If the higher mode shapes were considered, this method might give misleading information.

In the present work, a hybrid method is presented to detect crack locations based on wavelet transform and fractal dimension (FD) for beam structures, which can enhance the high sensitivity to the singularities induced by structural crack. Using Db4 wavelet decomposition, approximation signal and wavelet detailed signals of the mode shape are obtained. Then FD estimation method is used to calculate FD parameter of detailed signal. The crack location can be determined by the abrupt change of the FD along the beam length. Numerical simulation and experimental investigation are performed to testify the present method. In the numerical simulation, the mode shapes of the beam with crack are obtained based on wavelet-based Euler beam elements using B-spline wavelet on the interval (BSWI), whereas in the experimental investigation, the mode shapes are measured using Polytec Vibrometer PSV-400.

#### 2. BSWI Finite Element Model of Cracked Beam

Figure 1(a) shows a model of cracked cantilever beam with dimensions of length and uniform cross section (where is the width and is the depth of beam), two open cracks of depth locates at away from the clamped end, where the subscript expresses the serial number of cracks. Since the linear rotational spring model can describe open crack effectively, present work is based on this model (as show in Figure 1(b)). The stiffness of rotational springs can be written as where is the modulus of elasticity of beams, is the depth of beams, and is a dimensionless local compliance function expressed as follows [32]: To express briefly, we denote two dimensionless parameters to describe cracks: relative crack size and normalized location .