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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.

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 [

Natural frequency-based crack identification methods employ the natural frequency change as the basic feature [

Crack detection methods have been developed based on measured mode shapes directly or indirectly [

As an efficient index of crack, fractal dimension is also introduced to detect crack for beam and plate structures by Hadjileontiadis et al. [

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.

Figure

Cracked cantilever beam: (a) geometry, (b) model of cracked beam.

As shown in Figure

The global stiffness matrix

The free vibration problem of plate is expressed as a generalized eigen-problem:

In the present, the command EIG in software MATLAB is employed to solve (

The concept of FD and its relevant mathematical model were originally introduced by Mandelbrot [

This method can also be applied to calculate the FD of the mode shapes. Since it exhibits high-noise insusceptibility [

This paper proposes a hybrid method to detect crack locations based on wavelet transform and FD for beam structure. The steps of the proposed method are followed.

The cracked beam is modeled using the B-spline wavelet on the interval (BSWI) finite element method. The beam crack is modeled according to linear elastic fracture mechanics theory. The first several mode shapes of the cracked beam is acquired from eigen formulation.

By Db4 wavelet transform, the wavelet coefficients are calculated for the mode shape. Applying the wavelet coefficients and reconstructing the original mode shapes, approximation signal in scaling space and detailed signals in wavelet spaces are gained.

FD estimation method is used to calculate FD values of detailed signal.

The mode shapes, approximation and detailed signals, and FD are plotted in the geometry space of the beam structure. The crack locations can be identified by the peak points of the signals along the beam length.

Numerical simulation is carried out to verify the effectiveness of the proposed method for a cantilevered steel beam with two cracks. The cantilever beam length ^{11} N/m^{2} and material density ^{3}. The crack parameters are

The BSWI4_{3} Euler beam element is used as approximation bases to model the cracked beam, where 4 and subscript 3 denote the order and the level of the BSWI wavelet. In the simulation, we use 20 BSWI4_{3} Euler beam element (184 DOFs). The left of the beam is fixed and its right is free. In this paper, we do not consider damping.

The first three mode shapes of the cracked beam are acquired from eigen formulation. For comparison purpose, mode curvature-based method and the proposed method are adopted to detected crack locations.

The size of sliding window is a crucial parameter, which significantly affects the results of crack identification. And the selection of the size of sliding window is usually done by trial and error. In this paper, the size of sliding window is set as 5 mm.

Mode curvature-based method is applied to the first and third mode shapes, respectively. The detection results are show in Figure

Crack location identification results using mode curvature-based method. (a) The first mode shape. (b) The third mode shape.

The proposed method is used to detect crack locations. The detection results of crack locations are showed in Figures

Crack location identification results using the proposed method for the first mode shape. (a) The first mode shape S and its wavelet decomposition at level 2. (b) FD of the detailed signal

Crack location identification results using the proposed method for the third mode shape. (a) The third mode shape S and its wavelet decomposition at level 2. (b) FD of the detailed signal

The proposed method is directly applied to third mode shape, crack locations are shadowed by two false peaks induced by two inflexions of the third mode shape, as showed in Figure

By comparison, it is concluded that the proposed method can detect crack locations more accurately than mode curvature-based method and wavelet-based method.

Noise immunity is a vital characteristic to crack detection method. To study noise immunity of the proposed method, Gaussian random white noise with SNR = 80 and 100 dB is added to mode shape to simulate noise environment, where SNR is abbreviations of signal-to-noise ratio. A linear mapping as shown in (_{1} and low-frequency approximation signal A and detail signal D_{2}. In 80 and 100 dB noise level, the detection results are showed in Figures

Crack location identification results in 80 dB noise level for the first mode shape. (a) Noise contaminated the first mode shape S_{N} and its wavelet decomposition at level 2. (b) FD of the detailed signal

Crack location identification results in 100 dB noise level for the first mode shape. (a) Noise contaminated the first mode shape S_{N} and its wavelet decomposition at level 2. (b) FD of the detailed signal

Crack location identification results in 80 dB noise level for the third mode shape. (a) Noise contaminated the first mode shape S_{N} and its wavelet decomposition at level 2. (b) FD of the detailed signal

Crack location identification results in 100 dB noise level for the third mode shape. (a) Noise contaminated the first mode shape S_{N} and its wavelet decomposition at level 2. (b) FD of the detailed signal

In this section, an experiment is conducted to validate the proposed method on steel cantilever beam with two cracks. The experimental setup is shown in Figure

Experimental setup.

The geometry of the cantilever beam is shown in Figure ^{11} N/m^{2} and material density ^{3}. The artificial cracks are prepared on the numerical milling machine. The parameters of two cracks are

The beam is fixed in its left by a holder and connected with electrodynamic mode shaker JMJ-5 near the right, which excites the investigated specimen by the random noise signal. The displacements are acquired using scanning laser Doppler vibrometer (LDV) Polytec PSV-400 connected with a vibrometer controller OFV-5000. 41 equidistant measuring points are arranged along the cantilever beam. The frequency bandwidth is defined within the range of 0–2.5 kHz, the resolution is 1 Hz, the sampling frequency is 6.4 kHz, and the sampling time is 40 ms.

The first several mode shapes can be acquired by the vibrometer-dedicated software. The first mode shape and the third mode shape are processed based on the proposed method. The experiment results are showed in the Figures

Crack location identification results in experimental circumstances. (a) The first measured mode shape S and its wavelet decomposition at level 2. (b) FD of the detailed signal

Crack location identification results in experimental circumstances. (a) The third measured mode shape S and its wavelet decomposition at level 2. (b) FD of the detailed signal

In the present study, a hybrid method is presented to detect crack locations using wavelet transform and fractal dimension (FD) for beam structures. Wavelet transform is applied to decompose the mode shapes of beam structures. To improve the sensitivity of location detection, FD estimation method is employed to analyze detailed signal of the mode shape. For comparison purpose, curvature mode shape based method is also used to identify the crack locations. Numerical simulations and experimental investigations are carried out to test effectiveness of the proposed method for a cantilever beam with two cracks. The results indicate that, using wavelet decomposition or curvature mode alone, the locations of small cracks are not detected for beam structures but are identified accurately according to FD singularity of detailed signal. At the same time, the proposed method performs well below 80 dB level noise and can been extended more complex structures. It is worth to point out that one mode shape should be identified using experimental modal analysis, which is only suitable for offline crack detection. However, for the structures under working condition, many mode shapes will be measured simultaneously to generate the operating deflection shape (ODS). Therefore, the further work is to extend the present approach for online detection of cracks in structures.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Basic Research Program of China (“973” Program) no. 2011CB706805, the National Natural Science Foundation of China nos. 51475356, and Zhejiang Province Public Welfare Project no. 2014C31103.