On research of a phase synthesis time reversal focusing method for damage imaging of complex composite structures

Abstract

Time reversal focusing method has been proved to be an effective method for active Lamb wave–based structural health monitoring. In this article, aiming at developing a practical method for online localization of damage on aircraft composite structures that can take advantage of time reversal focusing and do not rely on the transfer function, a phase synthesis–based time reversal focusing method was proposed. In this method, damage images are given out directly through time reversal focusing, and the other imaging processes such as the delay-and-sum imaging method adopted in many researches of time reversal focusing are not needed. Based on the damage imaging method, a structural health monitoring demonstration system was built on a composite panel of an aircraft wing box with many bolt holes and stiffeners. The demonstrated results show that this method can estimate the positions of damages efficiently with a low sensitivity of group velocity errors and a high antijamming capability.

Keywords

Complex composite structure structural health monitoring time reversal focusing phase synthesis damage imaging wing box

Introduction

Structural health monitoring (SHM) technology utilizes the sensors/actuators network that is integrated on structures to obtain information about the healthy state of structures online and adopts advanced signal processing methods and mechanical modeling methods to extract the characteristic parameters to recognize the damage state of structures so as to realize the self-diagnosis of structures, improve their safety, reduce their maintenance cost, and extend their service life (Qing et al., 2007; Yuan et al., 2006).

Much attention has been paid to the research of time reversal focusing method in active Lamb wave–based SHM because this method shows a promising advantage to give a focusing image of the structural damage and can improve the signal-to-noise ratio of the propagating waves (Fink, 1992, 1999; Ing and Fink, 1996, 1998; Prada and Fink, 1998).

Wang investigated the feasibility of this time reversal focusing process applied in guided waves of plate-like structures and discussed a damage imaging method based on the time reversal concept (Wang et al., 2004). Based on this research, Wang and Yuan (2009) studied a time reversal focusing method for baseline-free damage imaging. However, the study objects of these researches were simple isotropic plate-like structures, and the damage imaging method was a kind of delay-and-sum imaging method (Michaels, 2008; Wilcox et al., 2006) but not time reversal focusing. Sohn et al. (2007) and Park (2007, 2009) developed a time reversal focusing based damage identification method by comparing the differences of focused signal and original input signal at the excitation point. The time reversal focusing process was realized by hardware, and this method was validated on a simple plate-like composite structure. However, high precision damage localization would require a large number of actuator–sensor channels of lead zirconate titanate (PZT) sensors to cover damage monitoring areas. Besides, reexciting PZT sensors simultaneously needs complicated hardware to support. Thus, some researchers conducted further researches to use software realization of the time reversal focusing process (Cai et al., 2011; Derveaux et al., 2007; Wang and Yuan, 2005). To realize the process, transfer functions of the propagation of the signals on structures are obtained firstly and stored in computers, and the virtual realization based on the transfer functions of the time reversal focusing process is conducted in software. A key issue in these methods is to obtain the transfer functions of the wave propagation between excitation element and the sensing element. Two typical methods are studied to obtain the transfer functions: (a) theoretic modeling of the structure (Derveaux et al., 2007; Wang and Yuan, 2005); (b) measuring the transfer functions through experiments (Cai et al., 2011). The first method is difficult to be achieved in complex composite structures. In Derveaux et al.’s (2007) research, the transfer functions were assumed and simplified to be Green’s functions, and the dispersion characteristics of Lamb wave were not considered. This is reasonable when the frequency of Lamb wave is low and the frequency band is narrow (Xu and Giurgiutiu, 2007). The method was validated only on numerical simulating signals but not on real structures. To the second method, the traveling time of Lamb waves is completely compensated in the time reversal focusing process, and the time-of-arrival information is removed. To solve this problem, Cai et al. (2011) adopted a changing-element excitation and reception mechanism to obtain the time-of-arrival information. In this research, the study object was also a simple aluminum plate, and when the time reversal focusing signal was acquired in software, the delay-and-sum imaging method was adopted to achieve the final damage imaging.

This article aims at developing a practical method for online localization of damage on aircraft composite structures that can take advantage of the time reversal focusing method, and therefore, a PZT sensor array-based phase synthesis time reversal focusing method for damage imaging was proposed and validated on a complex composite structure. This method achieves the time reversal focusing and imaging of damage depending on the phase synthesis of damage scattered signals acquired by PZT sensors and gives out the damage imaging results directly through time reversal focusing. To the practical application of complex composite structures, the implementation process of this method is very simple, and this method does not require any work on the modeling or the measuring of transfer functions.

Phase synthesis–based time reversal focusing and imaging method

According to the time reversal focusing concept, an input signal can be focused at an excitation source point if an output signal recorded at another point is reversed in the time domain and reemitted back to the original excitation source point. This time reversibility is based on the spatial reciprocity and time reversal invariance of linear wave equations (Fink, 1992). Figure 1 shows the time reversal focusing process. In Figure 1(a), an excitation signal is input to a PZT sensor, and Lamb wave is excited on a plate-like structure. When the Lamb wave arrives at a damage C, it is scattered as shown in Figure 1(b).

E_C(ω) is the frequency response of the damage scattered source. H_Ci is the frequency response of transfer function of the signal propagating from C to the PZT i, and the propagating distance is r. The output signal of PZT i can be represented as

E_{i} (ω) = H_{Ci} (ω, r) E_{C} (ω)

(1)

When the output signal of each PZT sensor is time reversed and reemitted back to the damage scattered source position, a synthesis signal at the damage scattered source C can be obtained by

E'_{C} (ω) = \sum_{i = 1}^{n} H_{Ci} (ω, r) E_{i} {(ω)}^{*}

(2)

where the superscript “*” denotes complex conjugate. It means in the frequency domain, time reversal of a signal is equivalent to phase conjugation (Wang et al., 2004). According to the spatial reciprocity of linear wave equation, there is H_iC = H_Ci. By substituting equation (1) into equation (2), the modulus value of synthesis signal at the position C can be represented as

| {E'}_{C} (ω) | = | \sum_{i = 1}^{n} H_{Ci} (ω, r) H_{Ci} {(ω, r)}^{*} E_{C} {(ω)}^{*} | = | \sum_{i = 1}^{n} {H'}_{Ci} (ω, r) E_{C} {(ω)}^{*} |

(3)

Depending on the basic time reversal theory of Lamb wave propagating on plate-like structures, the modulus value of the synthesis signal reaches the maximum at the position C because all the time reversed signals arrive at the source point at the same time and add together due to the spatial reciprocity and time reversal invariance of linear wave equations (Núñez and Negreira, 2005; Park et al., 2007, 2009; Wang et al., 2004).

To Lamb wave signal at lower frequency, only A₀ and S₀ modes exist, and the amplitude of A₀ mode is much higher than that of S₀ mode. Thus, the Lamb wave can be considered to be a frequency narrowband single-mode Lamb wave (Xu and Giurgiutiu, 2007; Yu and Giurgiutiu, 2009), and the transfer function $H (ω, r)$ can be simplified to

H (ω, r) \approx a_{A_{0}} (ω, r) e^{- j k_{A_{0}} (ω) r}

(4)

where $a_{A_{0}} (ω, r)$ and $k_{A_{0}} (ω) r$ are the amplitude term and the phase term of the transfer function, respectively. $k_{A_{0}} (ω)$ denotes the wave number of A₀ mode: $k_{A_{0}} (ω) = ω / C_{A_{0}}$ . $C_{A_{0}}$ denotes phase velocity of A₀ mode. By substituting equation (4) into equation (3), the synthesis signal at the position C can be changed to

{E^{'}}_{C} (ω) = (\sum_{i = 1}^{n} a_{A_{0}} (ω, r_{i C}) a_{A_{0}} {(ω, r_{i C})}^{*} e^{- j k_{A_{0}} (ω) r_{i C}} e^{j k_{A_{0}} (ω) r_{i C}}) E_{C} {(ω)}^{*} = \sum_{i = 1}^{n} {| a_{A_{0}} (ω, r_{i C}) |}^{2} E_{C} {(ω)}^{*}

(5)

where the distance from C to all the PZT sensors is denoted as r_iC, i = 1, 2, …, n. |···| denotes modulus value.

Depending on equation (5), if the transfer functions of A₀ mode can be obtained beforehand, the time reversal focusing process can be realized in software as mentioned in section “Introduction.” But to complex composite structure, the transfer functions of A₀ mode are difficult to be obtained.

Equation (5) shows that the focusing signal has nothing to do with the phase term of the transfer functions. It indicates that time information of the focusing signal has been also removed when the time reversal focusing happens. Thus, in Cai et al.’s (2011) research, a changing-element excitation and reception mechanism process was adopted to reobtain the time information. Once the time reversal focusing signal was acquired, other imaging technique should be adopted to give out the final damage image. Thus, a method not to rely on the transfer functions and is easy to be applied to complex composite structure should be presented for the practical application.

The transfer functions shown in equation (4) have two terms, the amplitude term and the phase term. Based on the time reversal theory, the two terms have different influences on the focusing results. When the time reversed signals travel back to the source, they arrive at the source at the same time (the same phase), which produces the focusing effect of the time reversal method. After the time reversal focusing, the similarity of the waveform shape of the time reversal focusing signal to that of the original source signal is mainly decided by the amplitude term of the transfer function. Regarding the damage imaging, what is concerned is the position of the damage. It is not necessary to obtain a signal synthesized with the same shape as the original signal. In this case, the amplitude term of the transfer function becomes not important. If a virtual synthesis is realized just according to the phase term of the transfer function and neglecting the amplitude term, the time reversed signal will still arrive at the source with the same phase, and the focusing effect will still happen, though the shape of the focusing signal will change and no longer has the same shape as the original signal. This means at the damage scattered source point, still a maximum amplitude focusing signal will be formed. This method can be called a phase synthesis method. Instead of transfer functions, this method just needs its phase term that can be calculated from the phase velocity of the A₀ mode when single-mode Lamb wave is considered. Based on this principle, a phase synthesis method is put forward to search the time reversal focusing position of the damage scattered signals and give out the damage image directly.

To introduce the proposed phase synthesis time reversal focusing and imaging method in detail, a plate-like structure with n PZT sensors arranged is adopted as an example to explain the method as shown in Figure 2.

The PZT sensors construct a damage monitoring area. A random position D is chosen whose coordinate is (x, y). The distances from the position D to all the PZT sensors are denoted as r₁, r₂, …, r_n. E_i(ω) is the frequency response of the damage scattered signal of the PZT i. The output synthesis signal at the position D can be represented as equation (6) by time reversing E_i(ω) and applying the phase term of the transfer function $e^{- j k_{A_{0}} (ω) r_{i}}$ to E_i(ω) at the same time

V_{S} = \sum_{i = 1}^{n} E_{i} {(ω)}^{*} e^{- j k_{A_{0}} (ω) r_{i}}

(6)

The modulus value of the inverse Fourier transform of equation (6) can be represented as

| v_{s} (t) | = | \frac{1}{2 π} \sum_{i = 1}^{n} \int_{- \infty}^{+ \infty} E_{i} {(ω)}^{*} e^{- j k_{A_{0}} (ω) r_{i}} e^{j ω t} d ω |

(7)

where $v_{s} (t)$ is denoted as the phase synthesis signal. The phase delay factor $e^{- j k_{A_{0}} (ω) r_{i}}$ in equation (6) can be represented as

e^{- j k_{A_{0}} (ω) r_{i}} = e^{- j r_{i} ω / C_{A_{0}}}

(8)

Considering that the E_i(ω) is frequency narrowband, equation (7) can be changed to

| v_{s} (t) | = | \frac{1}{2 π} \sum_{i = 1}^{n} \int_{- \infty}^{+ \infty} (\sum_{ω_{0} \in [ω_{l}, ω_{h}]} E_{i} {(ω_{0})}^{*} e^{- j r_{i} ω_{0} / C_{ω_{0} A_{0}}}) e^{j ω t} d ω | = \frac{1}{2 π} \sum_{i = 1}^{n} [\sum_{ω_{0} \in [ω_{l}, ω_{h}]} {e^{'}}_{i} (τ - t - r_{i} / C_{ω_{0} A_{0}})]

(9)

where $E_{i} (ω_{0})$ is a frequency component of $E_{i} (ω)$ when the frequency is ω₀. $e'_{i} (t)$ is the corresponding time domain signal. $C_{ω_{0} A_{0}}$ denotes the phase velocity of the frequency component. The ω₀ is in the frequency range of [ω_l, ω_h]. The frequency band is defined as ω_b = ω_h−ω_l and the central frequency is ω_c.

In the phase synthesis imaging process, the pixel value corresponding to the position D is calculated by equation (9) and represents the modulus value of the synthesis signal at the position D. It is known that the real focusing only happened at the original source, which means here the damage position. The pixel value at the damage position will be the largest of all the points in the image. Thus, the whole monitored area can be divided into many small areas, by an algorithm to calculate the pixel value of all these small areas; the damage position can be decided by choosing the largest pixel value position or larger pixel value area.

In the phase synthesis imaging process, the phase velocity of each frequency component in the frequency narrowband signal is needed. To complex composite structure, the phase velocity of each wave propagation direction is different, and the phase velocity of different frequency components is also different. Thus, the errors of the phase velocity of each frequency component will introduce a large accumulated error to the phase synthesis result. The measuring of lots of phase velocity is also a laborious work. In the phase synthesis imaging process, there are two Fourier transform processes and one phase modulation process. For searching the position of damage in the whole damage monitoring area, a large amount of calculation is required. Therefore, some simplification should be made to solve these problems.

Depending on the superposition principle given out by the Fourier transform, the frequency narrowband Lamb wave signal can be considered to be consisted of finite single frequency components. In equation (9), $C_{ω_{0} A_{0}}$ denotes the phase velocity of each single frequency component. Equation (9) indicates that the time reversal focusing of damage scattered signals can be realized using the phase velocity to modulate the phase of each corresponding single frequency component depending on the propagating distance.

According to the dispersion characteristics of Lamb wave, the summation of each modulated single frequency component is approximately equal to the signal obtained using the group velocity to modulate the frequency narrowband Lamb wave signal directly

\sum_{ω \in [ω_{l}, ω_{h}]} {e^{'}}_{i} (τ - t - r_{i} / C_{ω_{0} A_{0}}) \approx e_{i} (τ - t - \frac{r_{i}}{C_{g A_{0}}})

where $e_{i}$ denotes the damage scattered signal of the PZT sensor i and $C_{g A_{0}}$ denotes the group velocity of the signal.

Based on this point, equation (9) can be represented as equation (10) approximately

| v_{s} (t) | \approx | \sum_{i = 1}^{n} e_{i} (τ - t - \frac{r_{i}}{C_{g A_{0}}}) |

(10)

The error of approximation in equation (10) is related to the dispersion degree of the Lamb wave signal. For low frequencies and S₀ mode, the error of approximation is small. But in this condition, the amplitude of S₀ mode is very low. Thus, the A₀ mode should be used. According to the discussion in amplitude comparing of A₀ and S₀ modes on the composite panel of section “Damage imaging validation and demonstration,” the dispersion degree of the A₀ mode of the Lamb wave signal at low frequencies on the composite panel is also very low. Thus, the error of approximation will be small.

In this article, the phase synthesis process is realized in time domain using group velocity of the signals at the central frequency ω_c. Equation (10) can be changed to the final expression of the phase synthesis signal using the positive envelope of the signals

| v'_{s} (t) | = \sum_{i = 1}^{n} Envelope (e_{i} (τ - t - \frac{r_{i}}{C_{g A_{0}}}))

(11)

where Envelope denotes of positive envelope. There is only one group velocity of the signals at the central frequency ω_c that is needed. The amount of calculation in the phase synthesis imaging process is greatly reduced. The implementation process of the phase synthesis–based time reversal focusing damage imaging method is shown in Figure 3.

Complex Shannon wavelet transform–based group velocity measuring

In the phase synthesis process, group velocity is needed. Threshold-based method and cross-correlation–based method are often used to measure group velocity of signals. Some researchers also studied time–frequency signal processing methods to solve the problem, such as the Hilbert–Huang transform (HHT) and the Morlet wavelet transform (Apostoloudia et al., 2007; Ding et al., 2004; Leonard and Hinders, 2005; Peng and Yuan, 2005). But HHT is limited by the number of decomposition levels and the stop conditions. In the process of using the Morlet wavelet transform to calculate time-of-flight of the signals, some approximation needed to be made (Apostoloudia et al., 2007; Peng and Yuan, 2005). Overall speaking, wavelet transform is an effective tool for this application because of its high time–frequency resolution. This article adopts complex Shannon wavelet transform to solve the problem.

Equation (12) shows the complex Shannon wavelet function

Ψ (t) = \sqrt{f_{b}} \sin c (f_{b} t) e^{2 π i f_{c} t}

(12)

The terms of f_b and f_c are frequency band and central frequency of the wavelet, respectively. The function of sin c is represented as

\sin c (x) = {\begin{matrix} 1 & x = 0 \\ \frac{\sin (π x)}{π x} & x \neq 0 \end{matrix}

(13)

The Fourier transform of equation (12) can be represented as

Φ (ω) = {\begin{matrix} \sqrt{\frac{2 π}{ω_{b}}} & ω_{c} - \frac{ω_{b}}{2} < ω \leq ω_{c} + \frac{ω_{b}}{2} \\ 0 & Others \end{matrix}

(14)

where $ω_{b} = 2 π f_{b}$ , $ω_{c} = 2 π f_{c}$ , and $ω_{c} > ω_{b} / 2$ . Figure 4 gives out the real part of complex Shannon wavelet function in different central frequencies, frequency bands, time factors, and theoretical frequency responses.

It indicates that the frequency response of complex Shannon wavelet function is a kind of frequency square window. The width of the window is depending on the central frequency and frequency band. Equation (12) indicates that the center time of complex Shannon wavelet function is at t = 0.The central frequency according to equation (14) is at $ω = ω_{c}$ , which is limited in the range of $(ω_{c} - ω_{b} / 2, ω_{c} + ω_{b} / 2]$ . Therefore, the center time of complex Shannon wavelet transform $Ψ ((t - b) / 1)$ is at t = b. The central frequency is at $ω = ω_{c} / a$ . The complex Shannon wavelet transform of the signal $x (t)$ is a frequency narrowband signal that represents the time–frequency components at t = b, $ω = ω_{c} / a$ , and the frequency band is $(ω_{c} / a - ω_{b} / 2 a, ω_{c} / a + ω_{b} / 2 a]$ .

A simple waveform of Lamb wave containing two harmonic waves and propagating along X axis can be represented as

f (x, t) = e^{- i (k_{1} x - ω_{1} t)} + e^{- i (k_{2} x - ω_{2} t)}

(15)

where k₁ and k₂ denote the wave number, and ω₁ and ω₂ denote central frequency of the waveform. The complex Shannon wavelet transform of $f (x, t)$ is

WT (x, a, b) = \sqrt{a} {Ψ^{*}}_{g} (a ω_{1}) e^{- i (k_{1} x - ω_{1} b)} + \sqrt{a} {Ψ^{*}}_{g} (a ω_{2}) e^{- i (k_{2} x - ω_{2} b)}

(16)

where ${Ψ^{*}}_{g} (a ω_{1})$ is equal to ${Ψ^{*}}_{g} (a ω_{2})$ when the ω₁ and ω₂ are in the range of $(ω_{c} / a - ω_{b} / 2 a, ω_{c} / a + ω_{b} / 2 a]$ . Equation (16) can be changed into equation (18) by supposing

{\begin{matrix} {Ψ^{*}}_{g} (a ω_{c}) = {Ψ^{*}}_{g} (a ω_{1}) = {Ψ^{*}}_{g} (a ω_{2}) \\ \frac{k_{2} - k_{1}}{2} = Δ k, \frac{ω_{2} - ω_{1}}{2} = Δ ω \\ \frac{k_{2} + k_{1}}{2} = k_{c}, \frac{ω_{2} + ω_{1}}{2} = ω_{c} \end{matrix}

(17)

WT (x, a, b) = 2 \sqrt{a} A {Ψ^{*}}_{shan} (a ω_{c}) \cos (Δ kx - Δ ω b) e^{i (k_{c} x - ω_{c} b)}

(18)

The modulus value of equation (18) can be represented as

| W T (x, a, b) | = 2 \sqrt{a} | Ψ_{g}^{*} (a ω_{c}) c o s (2 Δ k x - 2 Δ ω b) |

(19)

It indicates that the modulus value will reach the maximum when $a \in (1 - ω_{b} / 2 ω_{c}, 1 + ω_{b} / 2 ω_{c}]$ and $b = Δ kx / Δ ω = x / c_{g}$ , where C_g denotes the group velocity of Lamb wave signal. Thus, the group velocity can be acquired

C_{g} = \frac{x}{b}

(20)

Figure 5 gives out an example of the group velocity measuring. The excitation signal is a five-peak wave signal, as shown in Figure 5(a). The central frequency of the excitation signal is 50 kHz. The sampling rate is 5 MHz. The modulus value is obtained by complex Shannon wavelet transform (f_c = 50 kHz and f_b = 0.4f_c) of the signals, as shown in Figure 5(b). The modulus value of the excitation signal and response signal is denoted as blue line and red line, respectively. The time of the maximum modulus value of the excitation signal relative to that of the sampling zero point is b₀, and the time of the maximum modulus value of the response signal relative to that of the sampling zero point is b₁. The time-of-flight of the signal can be denoted as b = b₁−b₀. Depending on equation (20), the group velocity can be obtained.

Damage imaging validation and demonstration

The damage imaging demonstration system shown in Figure 6 is consisted of top panel of an aircraft wing box specimen, PZT sensors that are placed on the wing box, and an integrated structural health monitoring system (ISS).

Wing box specimen and composite panel

Wing box is an important part of an aircraft structure (Grondel et al., 2004). The dimension of the wing box in this article is 1000 × 1800 × 200 mm. The top panel is made of carbon fiber composite material, and the bottom panel is made of aluminum. The thickness of the composite panel is 4 mm. The panels are fastened to the steel box frame. There are totally six T-shaped stiffeners with a distance of 130 mm between each other. Vertical to the stiffeners, there are totally five lines of bolt holes. The distance between the lines is 280 mm. The structural damage imaging demonstration system that is based on the damage imaging method is built on the top panel combined with PZT sensors and an ISS.

The top panel of the wing box is a complex structure. It is a composite material and contains lots of stiffeners, and the distance between them is very short. The stiffeners can reduce the amplitude of signals that are propagating through them and also introduce a large reflecting signals that are propagating near them but not propagating through them. There are lots of bolt holes on it that can also introduce a lot of reflecting signals and change the signals propagating through them.

PZT sensors and placement

To make it easy to arrange the PZT sensors and ensure the arrangement process of each element to be the same, a kind of smart layer (Yuan et al., 2008) is manufactured as shown in Figure 7. Each layer contains three PZT sensors. The signal interfaces are small subminiature B coaxial connectors. Eight layers are placed on the inner side of the top panel. The size of PZT sensor adopted in the smart layers is 8 mm × 0.48 mm (diameter × thickness). Twenty-four PZT sensors are used to construct the PZT sensor array, as shown in Figure 8. The 34 actuator–sensor channels are defined and shown in Figure 9. The start point of the blue arrow is the actuator and the end point of the blue arrow is the sensor. The size of the monitoring area is 460 mm × 800 mm (width × height).

ISS

An ISS (Qiu and Yuan, 2009) is adopted to fulfill the scanning task of the 34 actuator–sensor channels and the damage imaging process. The software of ISS is based on the LabVIEW platform. Based on the secondary development function of ISS, the damage imaging algorithm is integrated into it. First, the algorithm is implemented as an M-function in MATLAB. And then, the M-function is converted to a C++ shared library. Finally, the M-function converted to C++ shared library is encapsulated by standard windows dynamic library, which is called by the software of ISS.

Amplitude comparing of A ₀ and S ₀ modes on the composite panel

Depending on the discussion of section “Complex Shannon wavelet transform-based group velocity measuring,” Lamb wave of single mode is needed in the phase synthesis–based time reversal focusing process. According to the researches of Giurgiutiu (Xu and Giurgiutiu, 2007; Yu and Giurgiutiu, 2009), when the central frequency of Lamb wave is lower than 100 kHz, the amplitude of A₀ mode is much higher than that of S₀ mode. In this condition, the Lamb wave can be considered to be single mode (A₀ mode). But the amplitude of A₀ and S₀ modes mainly depends on the size of PZT sensors and mechanical properties of structure. The researches are performed on aluminum plate (1 and 3 mm thickness), and the diameter of the PZT sensors used in the research is 7 mm.

In this article, the amplitude of A₀ and S₀ modes is compared on the composite panel. Figure 10 gives out a waterfall plot of signals acquired from actuator–sensor channel 8-7. The central frequencies of excitation signal range from 20 to 200 kHz with an interval of 10 kHz. The electromechanical impedance (EMI) in the Lamb wave response signal is a crosstalk signal introduced by the ISS. The waterfall plot shows that the amplitude of S₀ mode is very low and nearly invisible when the central frequency of excitation signal is lower than 60 kHz. It also shows that the group velocity of A₀ and S₀ modes from 50 to 200 kHz is nearly the same, which indicates the low degree of dispersion of Lamb wave propagating on the composite panel.

The amplitude comparing result of A₀ and S₀ modes of actuator–sensor channel 8-7 at the central frequency range of 20–200 kHz with an interval of 10 kHz is shown in Figure 11. The results indicate that the amplitude of A₀ mode is up to the peak value when the central frequency of the excitation signal is 80 kHz, and the amplitude of A₀ mode is much higher than that of S₀ mode when the central frequency is less than 80 kHz. Depending on the comparing results, 50 kHz is selected to be the central frequency of excitation signal in the following research.

Group velocity of Lamb wave signals propagating on the composite panel

Figure 12(a) and (b) gives out the measured group velocity around the PZT 10 and PZT 8, respectively. Figure 12(a) shows that the group velocities are of great difference at each direction. Figure 12(b) shows the same.

Comparing Figure 12(a) with Figure 12(b), the group velocity is different even at the same direction. Thus, the average group velocity is adopted. According to the measured group velocity of the 34 actuator–sensor channels, the average velocity of Lamb wave signal propagation on the composite panel is 1455 m/s approximately.

Damage simulating method

Many researches have shown that the defects or damage of structure behaves like secondary acoustic source or scattered source (Derveaux et al., 2007; Sohn et al., 2007; Wang and Yuan, 2009; Yuan et al., 2005). The artificial simulating damage is adopted in some researches to simulate the scattered source. A block mass in the form of a 22 mm diameter, 3-mm-thick metallic coin is pasted on the structure to simulate the damage in Poddar et al.’s (2011) research. And in Cai et al.’s (2011) research, two identical hexagonal hollow screws are bonded on the plate to simulate damages. There are two main problems of using these methods to simulate damage in this article. The first problem is that the adhesive that is used to paste the block mass needs a time to cure. But the surface of the composite panel in this article is vertical to the ground. The second problem is that the work of removing the bonded damage is hard and may damage the surface of the composite panel.

In this article, the damages are simulated by a kind of solid adhesive tape. The damage effect of Lamb wave caused by real damage and the solid adhesive tape is compared by the following experiment to validate this damage simulating method.

An experiment of impact damage of a carbon fiber composite plate is implemented. The experimental setup is shown in Figure 13. An impact hammer is used to apply the impact on the composite plate. Four PZT sensors are placed on the surface of the plate, and three actuator–sensor channels are defined, as shown in Figure 14.

When the accumulative impact energy of the impact is up to 38.69 J, the plate generates a visible damage, as shown in Figure 15. The response signal acquired before and after the generation of the visible damage is shown in Figures 16 and 17. The impact location is on the wave propagation path of the actuator–sensor channel 1-3. Figure 16 shows that the energy of the direct wave is reduced by the damage, and the phases of the signals are changed. To the actuator–sensor channel 1-4, the impact location is not on the wave propagation path. Figure 17 shows that the damage introduces little influence to the direct wave, but it introduces lots of scattered signals to the response signals.

The solid adhesive tape with dimensions of 30 mm × 30 mm (length × width) is placed on the composite panel of the wing box, as shown in Figure 18. Figure 19 gives out the signals acquired from actuator–sensor channels 8-1 and 8-2 before and after the placement of the solid adhesive tap between PZT 8 and PZT 1, respectively. Figure 19(a) shows that the solid adhesive tape can reduce the energy of the direct wave and change the phase of the response signal. Figure 19(b) shows that the solid adhesive tape introduces lots of scattered signals to the response signal. Comparing Figure 19 with Figures 16(b) and 17(b), the effect of response signals introduced by the impact damage and the simulating damage is nearly the same.

Damage imaging demonstration

The implementation process of the damage imaging demonstration is (a) scanning the actuator–sensor channels and acquiring the response signals of the PZT sensor array when the composite panel is under healthy state, (b) placing the solid adhesive tap on the panel to simulate the damage, (c) scanning the actuator–sensor channels and acquiring the response signals of the PZT sensor array when the composite panel is under damage state, and (d) running the damage imaging program and showing the damage image to estimate the damage position.

The search intervals Δx and Δy shown in Figure 3 are set to be 4 and 8 mm, respectively. Many simulating damages are applied to the composite panel, respectively. Figure 20 shows four damage imaging results. The point with the largest pixel value of the damage image is estimated to be the center of the damage.

Table 1 gives the position comparison between the center of actual damage and the estimated center of damage depending on the damage images. It indicates that the distance errors of damage localization by the damage imaging method are less than 3.0 cm. Considering the velocity errors in each direction, the localization errors are reasonable.

Table 1.

Damage localization results.

Damage no.	Center coordinate of actual damage (mm, mm)	Estimated center coordinate of damage (mm, mm)	Error (cm)
D1	(130, 150)	(125, 136)	1.5
D2	(0, 150)	(25, 150)	2.5
D3	(0, −150)	(10, −150)	1.0
D4	(−100, 0)	(−102, 16)	1.6

The error of the group velocity should be considered. The velocity error is defined as

Erro r_{C_{g}} = \frac{max_{i = 1, 2, \dots} | C_{gi} - C_{gaverage} |}{C_{gaverage}} \times 100 %

(21)

where $Erro r_{C_{g}}$ is the velocity error, $C_{gaverage}$ is the average velocity, and $C_{gi}$ denotes the group velocity on the ith direction. The maximum, the minimum, and the average group velocities are 1844, 1247, and 1455 m/s, respectively. Thus, the error of the group velocity is 26.7%. The error of the group velocity is assumed to be 30% (C_gaverage = 1418 m/s) and 35% (C_gaverag = 1366 m/s) and applied to the damage imaging of the four damages again. The damage localization errors are compared in Table 2. It shows that if the errors of the group velocity are less than 35%, the maximum distance errors of damage localization by the damage imaging method are no more than 4.0 cm.

Table 2.

Damage localization errors of different group velocity errors.

Damage no.	Localization error (cm) of velocity error of 26.7%	Localization error (cm) of velocity error of 30%	Localization error (cm) of velocity error of 35%
D1	1.5	2.7	3.3
D2	2.5	3.1	3.5
D3	1.0	1.7	2.1
D4	1.6	1.6	2.6

These results show that the damage imaging method using the average group velocity can be applied to some more complex composite structures if the errors of the velocities are in a reasonable range. To some structures of strong anisotropy such as the study object of Mahzan et al. (2010), the velocities of each direction must be adopted.

To validate the antijamming capability of the damage imaging method, the white noises of signal-to-noise ratio of 10 and 5 dB are added to the response signals of the 34 actuator–sensor channels under the healthy and damaged states of the composite panel, respectively. Figure 21 shows an example. The damage imaging result of Figure 22(a) is nearly the same with Figure 20(c). The damage position can still be estimated from Figure 22(b).

Conclusion

This article proposes a damage imaging method for damage localization of complex composite structures. A phase synthesis time reversal focusing method is proposed to achieve the time reversal focusing of damage scattered signals. The damage imaging method does not require any work on the modeling or measuring of the transfer functions of the propagation of the signals on the complex composite structure. Damage images are given out directly through time reversal focusing. The complex Shannon wavelet transform is adopted to construct analytic signals and measure the time-of-flight of Lamb wave response signals. Based on the damage imaging method, a SHM demonstration system of a composite panel with many bolt holes and stiffeners of an aircraft wing box is built combined with eight smart layers of 24 PZT sensors and an ISS.

According to the demonstrated results, in the monitoring area of 460 mm × 800 mm, the damage localization errors of the damage imaging method are less than 3.0 cm when the error of the group velocity is 26.7%. The damage imaging method using average group velocity can be applied to some more complex composite structures if the errors of velocities in each direction are in a reasonable range. To some structures of strong anisotropy, the group velocity of each direction must be used. Due to the time reversal focusing and the complex Shannon wavelet transform, the damage imaging method performs a high antijamming capability. Further study is ongoing to apply the damage imaging method to multidamage localization and study the stability to temperature variations of this method.

Footnotes

Funding

This study was supported by the National Nature Science Foundation of China (50875132, 51110105008 and 51205189), Open Foundation of Jiangsu Provincial Key Laboratory of ASIC Design, and China Postdoctoral Science Foundation (2012M510183).

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On research of a phase synthesis time reversal focusing method for damage imaging of complex composite structures

Abstract

Keywords

Introduction

Phase synthesis–based time reversal focusing and imaging method

Complex Shannon wavelet transform–based group velocity measuring

Damage imaging validation and demonstration

Wing box specimen and composite panel

PZT sensors and placement

ISS

Amplitude comparing of A 0 and S 0 modes on the composite panel

Group velocity of Lamb wave signals propagating on the composite panel

Damage simulating method

Damage imaging demonstration

Conclusion

Footnotes

Funding

References

Amplitude comparing of A ₀ and S ₀ modes on the composite panel