Shock and Vibration

Volume 2017 (2017), Article ID 7283450, 13 pages

https://doi.org/10.1155/2017/7283450

## A Signal Decomposition Method for Ultrasonic Guided Wave Generated from Debonding Combining Smoothed Pseudo Wigner-Ville Distribution and Vold–Kalman Filter Order Tracking

School of Automotive and Traffic Engineering, Nanjing Forestry University, Nanjing, Jiangsu 210037, China

Correspondence should be addressed to Junhua Wu

Received 15 May 2017; Accepted 3 August 2017; Published 14 September 2017

Academic Editor: Michele Palermo

Copyright © 2017 Junhua Wu 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

Carbon fibre composites have a promising application future of the vehicle, due to its excellent physical properties. Debonding is a major defect of the material. Analyses of wave packets are critical for identification of the defect on ultrasonic nondestructive evaluation and testing. In order to isolate different components of ultrasonic guided waves (GWs), a signal decomposition algorithm combining Smoothed Pseudo Wigner-Ville distribution and Vold–Kalman filter order tracking is presented. In the algorithm, the time-frequency distribution of GW is first obtained by using Smoothed Pseudo Wigner-Ville distribution. The frequencies of different modes are computed based on summation of the time-frequency coefficients in the frequency direction. On the basis of these frequencies, isolation of different modes is done by Vold–Kalman filter order tracking. The results of the simulation signal and the experimental signal reveal that the presented algorithm succeeds in decomposing the multicomponent signal into monocomponents. Even though components overlap in corresponding Fourier spectrum, they can be isolated by using the presented algorithm. So the frequency resolution of the presented method is promising. Based on this, we can do research about defect identification, calculation of the defect size, and locating the position of the defect.

#### 1. Introduction

Carbon fibre composite is widely used in modern industry, such as aerospace domain and military products, because of its high strength and light weight. At present, such a material has been generalized to automotive industry, obviously reducing the weight of automobile. Debonding defect is a major defect of the carbon fibre composites. A great number of investigations of the nondestructive evaluation and testing (NDE/NDT) have done research for this type of defect [1–5].

Currently, ultrasonic guided wave (GW) testing has emerged as a popular NDE/NDT technique. The method can estimate the location, severity, and type of defects. Successful applications of defect identification of carbon fibre composites have been done [3, 6, 7]. However, dispersion effects and noise make ultrasonic testing waves as multicomponent signals, which results in that it is difficult to do NDE/NDT with raw testing waves. Therefore, isolating different components of GW and obtaining the corresponding time-frequency distributions (TFD) are vital for the inspection of the defect.

A number of scholars have done investigations about signal processing methods of GWs. Kercel et al. [8] used Bayesian parameter estimates to isolate multiple modes in GW signals collected from laser ultrasonic testing on a manufacturing assembly line. Cai et al. [9] provided a time-distance domain transform (TDDT) method to interpret the dispersion of Lamb waves, which can result in high spatial resolution images of damage areas. Rizzo and di Scalea utilized Discrete Wavelet Transform (DWT) to extract wavelet domain features for enhanced defect characterization in multiwire strand structures [10]. Gangadharan et al. presented a time reversal technique using GWs to detect damage in an aluminum plate, and good results were achieved [11]. The wavelet analysis is widely used [12–17] in domains; many successful applications of wavelet transform (WT) for GW signals have been done. Li et al. [14] proposed a combined method employing empirical mode decomposition (EMD) and wavelet analysis to attain good time resolution of the response signals. Paget et al. [15] proposed a new damage-detection technique based on WT with a new basis. Yu et al. [16] used the techniques of statistical averaging to reduce global noise and discrete wavelet denoising using a Daubechies wavelet to remove local high-frequency disturbances. Y. Y. Kim and E.-H. Kim [17] evaluated the effectiveness of WT analysis for studying the wave dispersion.

EMD, which can isolate adaptively different components, was proposed by Huang in 1998 [18]. At present, many investigations of theory and application have been done [19–24]. Li et al. [14], Osegueda et al. [20], and Salvino et al. [22] used EMD to process GW signals in plate structures. However, the frequency resolution of EMD is a limitation. Reference [25] reveals that when the ratio between a relatively low frequency and a relatively high frequency is greater than 0.75, two components of a signal cannot be separated.

In 1993, Vold and Leuridan [26] proposed Vold–Kalman filter order tracking (VKF_OT) for the estimation of a single order component. In 1997, they [27] derived a scheme to simultaneously estimate multiple components. Instantaneous frequency of the isolated component is a necessary prior knowledge for VKF_OT. Therefore, we introduce Smoothed Pseudo Wigner-Ville distribution (SPWVD), which can remove the cross-term in frequency direction and time direction of the time-frequency panel, to get instantaneous frequencies of isolated components. We present a signal decomposition method for ultrasonic GWs combining VKF_OT and SPWVD in this paper.

The rest of this paper is organized as follows. Section 2 presents the theories of Smoothed pseudo Wigner-Ville distribution and Vold–Kalman filter order tracking. The principle of algorithm is illustrated in Section 3. Section 4 provides an illustration of the presented method. The details of the experiment are described in Section 5. Section 6 shows the application of the presented algorithm to the experimental signals. Finally, Section 7 concludes.

#### 2. Smoothed Pseudo Wigner-Ville Distribution and Vold–Kalman Filter Order Tracking

##### 2.1. Smoothed Pseudo Wigner-Ville Distribution

Wigner-Ville distribution has a fine time-frequency resolution and can reach the low boundary of Heisenberg uncertainty principle. It is defined as [28] However, for multicomponent signals, it suffers from inevitable interference of cross-terms. SPWVD can remove it in frequency direction and time direction of the time-frequency panel. And the formula of SPWVD is as follows [28]:where and are smoothing window functions in time direction and frequency direction, respectively. is an analyzed signal, and and are time variable and frequency variables, respectively. The time resolution and frequency resolution of SPWVD are promising. Moreover, no interference is in the representation.

##### 2.2. Vold–Kalman Filter Order Tracking

Isolation of different modes is important for defect identification by ultrasonic guided waves. On this basic, we can locate the defect and evaluate the defect size. Therefore, VKF_OT is employed to separate wave packages.

In this paper, the angular-displacement VKF_OT techniques are adapted. The method is used to obtain the tracked components by minimizing the energy of errors for both the structural and data equations by mean of one of the least squares approaches [29].

The th order component can be defined aswhere is the complex envelope and is the complex conjugate of to make a real waveform. It is noted that is a carrier wave and defined aswhere is the speed of the reference axle and is the elapsed angular displacement. The discrete form of (4) can then be written as

###### 2.2.1. The Structural Equation

As the tracked component can be written as (3), where the envelope needs to be computed. Generally, fulfills [29]where is a higher-degree term in . The corresponding discrete forms can be expressed,where is the difference operator, the index is the differentiation order, and physically is a combination of other spectral components and additional measurement noise.

###### 2.2.2. The Data Equation

A measured signal can be taken as a combination of several components, , and measurement noise,where the integral number is the order of spectral components to be tracked and is unwanted spectral components and measurement errors. Each component of interest modulates with a carrier wave .

###### 2.2.3. Calculation of the Tracked Component

Let = 2 and let data length be ; then the calculation matrix form can be expressed as [29]To simultaneously track multiple orders and spectral components such as resonance, it can be extended to all order components of interest as well. Letand then (9) becomeswhere elements in the matrix are column vectors with a length , which is the th order component; are error vectors with a dimension ; and is a matrix with a dimension .

The terms with negative indexes in (8) assure to be a real waveform. is the measured signal with a length of , an error vector with dimension , and consists of carrier signals, asThus, (8) can be rewritten asAs the angular-velocity VKF_OT scheme, we introduce a weighting factor and combine (9) and (13), and thenEquation (14) can be symbolized asThe evaluation of tracked order components is exactly to find a vector fulfillingthat is, . The vector can be written asThe matrix is written aswhere and . Moreover, is written aswhere is the complex conjugate of .

#### 3. Principle of the Presented Algorithm

As mentioned above, SPWVD has a promising time-frequency resolution. Therefore, we obtain frequencies and durations of modes from SPWVD distributions of testing guided waves. Furthermore, VKF_OT is adapted to realize isolation of different wave packages with obtained mode frequencies. Finally, the final mode waveforms are cut out from the wave packages of modes by durations of modes. The processing steps of the extension algorithm are shown in Figure 1 and are as follows.