Research on lamb wave wavenumber response and PZT 2D crossshaped array based damage imaging method of aircraft composite structures
Lei Qiu^{1} , Shenfang Yuan^{2} , Zhongqing Su^{3} , Bin Liu^{4}
^{1, 2}State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China
^{1, 3}Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong
^{4}Department of Air Force Military Transportation, Air Force Service College, Xuzhou, China
^{2, 3}Corresponding authors
Journal of Vibroengineering, Vol. 18, Issue 1, 2016, p. 190201.
Received 3 September 2015; accepted 18 October 2015; published 15 February 2016
JVE Conferences
In recent decay, composite materials have widely applied to important loadbearing structures of advanced aircrafts. Online damage monitoring of composite structure is an important research topic in the field of Structural Health Monitoring (SHM). Among the existing SHM techniques for composite structures, Lamb wave and Piezoelectric Transducer (PZT) array based damage imaging methods have been widely studied because they have the advantages of high signaltonoise ratio and high damage sensitivity, and they can be applied to large scale structure monitoring. In this paper, a new online damage imaging method of composite structure which is based on Lamb wave wavenumber response and PZT 2D crossshaped array is proposed. In this method, the PZT 2D crossshaped array constructed by two linear PZT arrays is placed on composite structure to acquire Lamb wave damage scattering signal in spatial domain online. For each linear PZT array, a wavenumbertime image of the damage scattering signal can be obtained by using spatial Fourier transform and a time scanning process. Based on the two wavenumbertime images, the wavenumbers of the damage scattering signal projecting at the two arrays can be obtained. By combining with the two projection wavenumbers, the damage can be localized without blind angle. The method validation performed on a carbon fiber composite laminate plate shows a good damage localization accuracy. This method is simple but efficiency and it can be applied to online damage monitoring of composite structure easily.
Keywords: structural health monitoring, composite structure, damage imaging, lamb wave, spatial Fourier transform, PZT 2D crossshaped array.
1. Introduction
Modern structures on advanced aircrafts make increasing use of largescale composite structures due to their superior stiffness and weight characteristics, such as B787 and A350XWB, which are both composed of over 50 % composite materials [1]. However, the inner damage can reduce the strength and the stiffness of the structures, and even cause air crash [2, 3]. Therefore, to ensure the safety and reduce the maintenance cost, online damage monitoring of composite structures is a hot topic in the research field of Structural Health Monitoring (SHM) [45]. Among the existing SHM methods, much attention has been paid to piezoelectric transducer (PZT) and Lamb wave based online damage monitoring method because it is sensitive to small damage and it has a long detection range [6].
Damage imaging methodology has been widely studied in recent decade. It has utilized large number of actuatorsensor channels from a network of PZTs to map the structure being interrogated based on the measurement of damage reflections and damage induced features differences, producing a visual indication of damage location and size, which result in high signaltonoise ratio, high damage sensitivity and large scale structure monitoring. The damage imaging methodology can be divided into two branches called sparse PZT array based method and dense PZT array based method respectively. For sparse PZT array, the most straightforward damage imaging method is the delayandsum imaging method [78] whose main idea is to enhance the real damage point by overlapping time domain damage scattering signals. Another typical sparse PZT array based damage imaging method is time reversal focusing method [911] which makes use of Lamb wave spatial reciprocity and time reversal invariance. The time reversal focusing process is realized by using measured or modeled frequency domain transfer functions or by phase synthesis of time domain signals directly. Damage path probability imaging method has also been studied [12]. Different from those sparse PZT array based damage imaging methods, dense PZT array based damage imaging method, such as ultrasonic phased array [1314], has a unique feature of structural beam scanning based on phase modulation and synthesis of Lamb wave signals in time domain. It is able to enhance a desired signal component at a specific direction while eliminating other signal components from other directions. Multiple signal classification [15] is another kind of dense PZT array based damage imaging method.
In recent years, Lamb wave wavenumber processing technique for damage estimation has been gradually studied. Compared with the time domain and frequency domain analysis of Lamb wave signals, the wavenumber domain analysis is an effective approach to distinguish wave propagating direction and wave modes. Michaels [16] used a frequencywavenumber ﬁlter to analyze the direction and various wave modes of damage scattering signal on an aluminum plate. Sohn [17] created a signal processing technique in frequencywavenumber domain to isolate standing wave components on a composite plate for delamination and disband evaluation. Rogge [18] built the relationship between wavenumber and wave spatial location to estimate delamination size and depth of a composite laminate. Yu [19] presented short timespatialfrequencywavenumber analysis for obtaining a wavenumber spectrum at a selected frequency and time to decompose Lamb wave modes. In the above research, the accurate wavenumber response of Lamb wave signal was needed and measured by using a scanning Laser Doppler Vibrometer (LDV) of high spatial resolution. However, these methods researched so far still have their own limitations for online damage monitoring in practical application. For online damage monitoring, Purekar [20] and Qiu [2122] studied a spatial filtering technique based damage imaging method.
In this paper, a new online damage imaging method of composite structure based on Lamb wave wavenumber response and dense PZT array is proposed. In the method, the dense PZT array is a PZT 2D crossshaped array which is constructed by two linear PZT arrays. For each linear PZT array, the online spatial domain sampling of damage scattering signal is realized. Based on the spatial Fourier transform and combining with a time scanning process, a wavenumbertime image of damage scattering signal can be obtained. According to the two wavenumbertime images obtained by using the PZT 2D crossshaped array, the wavenumbers of damage scattering signal projecting at the two arrays can be obtained and the damage can be localized without blind angle. Finally, the method function is validated on a carbon fiber composite laminate plate. Comparing with the other dense PZT array based damage imaging method, the proposed method is simple and high computational efficiency, and it can be applied to online damage monitoring of composite structure easily.
2. The new damage imaging method
2.1. Wavenumber response based on spatial Fourier transform
A linear PZT sensor array consists of $M$ PZT sensors is placed on a platelike structure shown in Fig. 1. The PZT sensors are numbered as $m=$1, 2,..., $M$. A Cartesian coordinate is constructed on the structure. The center point of the linear PZT array is set to be the origin point of the Cartesian coordinate. The distance between the centers of each two adjacent PZT sensors is $\mathrm{\Delta}x$.
There is a damage on the structure. The position of the damage is unknown and supposed to be (${x}_{a}$, ${y}_{a}$). The direction and distance of the damage relative to the linear PZT array are supposed to be ${\theta}_{a}$ and ${l}_{a}$ respectively.
To obtain the damage scattering signal, a frequency narrowband excitation of central frequency $\omega $ is input to the PZT at the original point to excite frequency narrowband Lamb wave. Based on the studies [6, 1314], the amplitude of Lamb wave ${A}_{0}$ mode is dominant at low excitation frequency. Thus, the damage scattering signal can be approximated to be singlemode when the frequency is low. Based on this point, the damage scattering signal acquired by the PZT located at (${x}_{m}$, 0) can be expressed as Eq. (1). The wavenumber of the damage scattering signal is denoted as ${k}_{a}$. It is wavenumber narrowband and it can be considered as two components. The first is the wavenumber projecting at the array direction ($X$axis) ${k}_{x}={k}_{a}\mathrm{c}\mathrm{o}\mathrm{s}{\theta}_{a}$ and the second is the Yaxis projection wavenumber ${k}_{y}={k}_{a}\mathrm{s}\mathrm{i}\mathrm{n}{\theta}_{a}$:
where $u\left({t}_{r}\right)$ is the normalized amplitude of the damage scattering signal. ${\overrightarrow{\mathbf{L}}}_{a}$ and ${\overrightarrow{\mathbf{X}}}_{m}$ represent the distance vector of ${l}_{a}$ and ${x}_{m}$. ${x}_{m}=\left(\left(2m1\right)M\right)\mathrm{\Delta}x/2$. ${t}_{r}=r/{f}_{s}$ and $r=$1,…, $L$.
Fig. 1. Schematic diagram of Lamb wave spatial sampling by a linear PZT array
Based on Fraunhofer approximation, Eq. (1) can be approximated to be Eq. (2), in which, ${\widehat{\mathbf{L}}}_{a}$ denotes the unit direction vector of the ${\overrightarrow{\mathbf{L}}}_{a}$. In farfield situation, the damage scattering signal can be regarded as a planar wave received by the linear PZT array [13, 14]. Thus, Eq. (2) can be approximated to be Eq. (3). The amplitude term of Eq. (3) is expressed as Eq. (4):
Eq. (3) gives out the damage scattering signal of time domain sampling by using one PZT sensor. The linear PZT array can be also seemed to be a spatial sampling device. The array response at time ${t}_{r}$ is the damage scattering signal of spatial sampling and it can be expressed as Eq. (5). The spatial sampling rate is $2\pi /\mathrm{\Delta}x$.
By using spatial Fourier transform, the spatial response at time ${t}_{r}$ shown in Eq. (5) can be transformed to wavenumber response shown in Eq. (6), in which, $\delta $ is the Dirac function as shown in Eq. (7):
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}=2\pi u\left({t}_{r}\right){e}^{i\omega {t}_{r}}{e}^{i{k}_{a}{l}_{a}}\delta \left(k{k}_{a}\mathrm{c}\mathrm{o}\mathrm{s}{\theta}_{a}\right).$
Eq. (6) and Eq. (7) show that if $k={k}_{a}cos\left({\theta}_{a}\right)$ which is corresponding to the $X$axis projection wavenumber of the damage scattering signal, the wavenumber response obtained by using spatial Fourier transform will reach to the maximum value. Thus, the wavenumber of damage scattering signal projecting at the $X$axis can be easily obtained based on the spatial Fourier transform and the array spatial response.
However, only small amount of PZT sensors are allowed to be used in practical application especially in aviation field. Thus, the length of the linear PZT array is so limited as to the limited spatial sampling length of the damage scattering signal. It will lead to low wavenumber spatial resolution. But one thing should be noted that the goal of this method is to realize damage localization not to measure the accurate wavenumber. If the wavenumber which is corresponding to the maximum amplitude can be obtained correctly, this method still can be used. This goal can be realized if the length of the linear PZT array is long enough to acquire a full spatial cycle of the signal. In this paper, the method of padding zeros is adopted to increase the wavenumber resolution of spatial Fourier transform. Supposing that ${L}_{array}$ is the length of the linear PZT array and the wavenumber resolution $\mathrm{\Delta}k$ is set beforehand, the length of the padded zeros can be calculated by Eq. (8), in which, ${L}_{zeros}$ is the length of the zeros which must be padded:
To validate the function of the wavenumber response calculation method based on spatial Fourier transform mentioned above. An experimental study is performed on an aluminum plate, as shown in Fig. 2. The experimental system is constructed by a 2024T3 aluminum plate, a linear PZT array and a Lamb wave based SHM system [23]. The dimension of the 2024T3 aircraft aluminum alloy specimen is 1200 mm×1200 mm×2 mm (length×width×thickness). The mechanical properties of the 2024T3 aluminum plate are shown in Table 1.
Fig. 2. Experimental setup
a) Experiment system
b) The linear PZT sensor array
Table 1. Mechanical properties of the 2024T3 aluminum
Parameter

Value

Density (kg·m^{3})

2.78×10^{3}

Elastic modulus (GPa)

73.1

Shear modulus (GPa)

28

Poisson ratio $\mu $

0.33

As shown in Fig. 2, the linear PZT array is constructed by 21 PZT5A sensors. The diameter and thickness of the PZT5A sensor is 8 mm and 0.48 mm respectively. The distance between the centers of each two adjacent PZT sensors is $\mathrm{\Delta}x=$0.09 m. The corresponding maximum cutoff wavenumber for the array is 349 rad/m. Another PZT sensor is used as a Lamb wave excitation. The direction and the distance of excitation PZT relative to the linear PZT sensor array are 0° and 400 mm respectively. The excitation signal is a fivecycle sine burst modulated by Hanning window [8] and the center frequency is 20 kHz. The frequency used in this experiment is low. Thus, ${A}_{0}$ mode is dominant in the guided wave response signal and the theoretic central wavenumber of the signal is 212.2 rad/m.
Fig. 3 gives out an example of the spatial Fourier transform of a spatial sampling signal on the aluminum plate by using only 7 PZT sensors (Number 5, 7, ,9, 11, 13, 15 and 17) in the linear PZT array. Thus, the length of the PZT array is 7, $\mathrm{\Delta}x=$0.09 m, and $\mathrm{\Delta}k=$0.1 rad/m. Based on Eq. (8), 6974 zeros are padded to the signal and 3487 zeros on each side. As it can be seen from Fig. 3 that the wavenumber which is corresponding to the maximum amplitude of the wavenumber response is very close to the theoretic wavenumber. The range of the wavenumber response is from –${k}_{max}$ to $+{k}_{max}$. ${k}_{max}$ is the maximum cutoff wavenumber of the spatial sampling and ${k}_{max}=\pi /\mathrm{\Delta}x$. The length of the wavenumber response is $N=(2{k}_{max}/\mathrm{\Delta}k)+1$.
Fig. 3. An example of spatial Fourier transform based wavenumber response calculation
2.2. Damage imaging and localization
The damage scattering signal acquired by the linear PZT array can be considered a spatial sampling signal matrix which contains $L$ times spatial sampling. As shown in Fig. 4(a), the wavenumber response of damage scattering signal of one time spatial sampling at time ${t}_{r}$ can be obtained by the spatial Fourier transform based on the above method. This method can be also applied to the damage scattering signal of $L$ times spatial sampling from ${t}_{1}$ to ${t}_{L}$. It is considered to a time scanning process based on the spatial Fourier transform. Based on the time scanning process, a matrix of wavenumber response from ${t}_{1}$ to ${t}_{L}$ can be obtained as shown in Eq. (9):
Fig. 4. Demonstration of the time scanning process and damage imaging result
a) Demonstration of damage imaging process
b) Example of damage imaging result
Based on Eq. (9), a wavenumbertime image shown in Fig. 4(b) can be obtained by imaging the matrix $\mathbf{F}$. According to the wavenumbertime image, the wavenumber corresponding to the point of the highest pixel value can be estimated to be the $X$axis projection wavenumber ${k}_{x}={k}_{a}\mathrm{c}\mathrm{o}\mathrm{s}{\theta}_{a}$ and the corresponding time can be estimated to be the absolute arrival time ${t}_{R}$ of the damage scattering signal. But it should be noted that the absolute arrival time ${t}_{R}$ is not the actual timeofflight of damage scattering signal.
As it can be seen from the above imaging method, the axis projection wavenumber and the absolute arrival time of the damage scattering signal can be obtained by using a linear PZT array. Thus, a PZT 2D crossshaped array which is constructed by two linear PZT arrays (No. 1 PZT array and No. 2 PZT array) is used to localize the damage as shown in Fig. 5. A Cartesian coordinate is constructed on the PZT 2D crossshaped array. The Lamb wave of wavenumber narrow band is excited at the origin point at time ${t}_{e}$. When the Lamb wave passes through the damage, the damage scattering signal is generated and it can be acquired by the PZT 2D crossshaped array. Based on the imaging method, the axis projection wavenumber and the absolute arrival time (${k}_{x}$, ${t}_{R1}$) and the (${k}_{y}$, ${t}_{R2}$) of the No. 1 PZT array and the No. 2 PZT array can be obtained respectively.
For damage direction estimation as shown in Fig. 5(a), based on the two projection wavenumbers ${k}_{x}$ and ${k}_{y}$, the direction ${\theta}_{a}$ of the damage direction relative to the center point of the array can be calculated by Eq. (10). It shows that ${\theta}_{a}$ can be calculated from 0° to 360° without blind angle. Generally speaking, the wavenumber of the damage scattering signal depends on the position of the damage. It means that the wavenumbers of damage scattering signals introduced by two damages at different positions may have the same coordinateaxis projection wavenumber, even if the two damages are at different directions relative to the PZT 2D crossshaped array. But this problem can be overcome by using the ratio between the $X$axis projection wavenumber and $Y$axis projection wave number, as shown in Eq. (10):
For damage distance estimation as shown in Fig. 5(b), the arrival time ${t}_{R}$ obtained from the damage imaging result contains two parts. The first part is the Lamb wave excitation time ${t}_{e}$ which can be determined by the excitation signal. The second part is the actual timeofflight ${t}_{a}$ including the time of Lamb wave signal propagating from the excitation position to the damage position and the time of the damage scattering signal propagating from the damage position to the PZT 2D crossshaped array. The damage distance ${l}_{a}$ relative to the center point of the PZT 2D crossshaped array can be calculated by Eq. (11) combing with Lamb wave group velocity ${c}_{g}$:
Finally, the damage localization result can be obtained by using Eq. (12):
Fig. 5. Damage localization based on PZT 2D crossshaped array
a) Damage direction estimation
b) Damage distance estimation
3. Validation on a carbon fiber composite laminate plate
3.1. Validation setup
The validation system shown in Fig. 6 is constructed by a carbon fiber composite laminate plate, a PZT 2D crossshaped array and the Lamb wave based SHM system. The dimension of the composite plate is 600 mm×600 mm×2.25 mm (length×width×thickness). The composite plate is stacked by 18 single layers and the ply sequences is [45/0/–45/90/0/45/0/–45/0]s. The material property of each layer is shown in Table 2. Each linear PZT array of the PZT 2D crossshaped array is constructed by 7 PZTs as shown in Fig. 6(b) and (c). The distance between the centers of each two adjacent PZTs is 10 mm. The maximum cutoff wavenumber of the linear PZT array is ${k}_{max}=$314 rad/m. The PZTs in No.1 PZT array are labeled as PZT 11, …, PZT 17 and the PZTs in No. 2 PZT array are labeled as PZT21, …, PZT 27. A PZT used as an actuator for Lamb wave excitation is placed at the center point of the array on the opposite side of the plate.
Table 2. Material parameters of single layer of the carbon fiber composite laminate plate
Parameter

Value

0° tensile modulus / GPa

135

90° tensile modulus / GPa

8.8

±45° inplane shearing modulus / GPa

4.47

Poisson ratio $\mu $

0.328

Density / (kg·m^{3})

1.61×10^{3}

Table 3. The position of the simulated damages
Damage

Cartesian coordinates (mm, mm)

Polar coordinates (°, mm)

A

(100, 300)

(71.57, 316.2)

B

(100, 200)

(63.43, 223.6)

C

(0, 200)

(90.00, 200.0)

D

(100, 200)

(116.57, 223.6)

E

(200, 200)

(135.00, 282.8)

F

(200, 100)

(153.43, 223.6)

G

(200, 0)

(180.00, 200)

Another 3 reference PZTs shown in Fig. 6(c) are labeled as Ref 1, Ref 2 and Ref 3. They are used to measure Lamb wave group velocity. The method of applying damage on the structure is bonding a mass block shown in Fig. 6(a) on damage position to change the local stiffness of the structure at the bonding area. 7 damages labeled A to G are applied on the structure by bonding a mass block onebyone respectively as shown in Fig. 6(c). The position of these damages is shown in Table 3. The SHM system is adopted to excite and acquire Lamb wave signals. The excitation signal is a fivecycle sine burst modulated by Hanning window. The center frequency of the excitation signal is 35 kHz and the amplitude is ±70 volts. The sampling rate is 10 MS/s and the sampling length is 8000 samples including 1000 presamples.
Fig. 6. The validation system of the damage imaging method on the composite plate
a) Experiment system
b) The PZT 2D crossshaped array
c) The PZT 2D crossshaped array placement and damage position
3.2. Validation results
The validation process is 1) In the health state of the structure, the Lamb wave signals of the PZT 2D crossshaped array are acquired as the health reference signals, ${f}_{HR}$. 2) Damage is applied at each position shown in Table 2 and the corresponding Lamb wave signals of the PZT 2D crossshaped array are acquired as the online monitoring signals, ${f}_{OM}$.
To show the damage imaging and localization process, the damage $E$ is selected to be an example. The ${f}_{HR}$ and ${f}_{OM}$ of the PZT 2D crossshaped array are shown in Fig. 7 and Fig. 8 respectively. The damage scattering signal is extracted by subtracting ${f}_{HR}$ from ${f}_{OM}$ and it is shown in Fig. 9. For damage imaging based on the damage scattering signal. The wavenumber resolution of spatial Fourier transform is set to be $\mathrm{\Delta}k=$1 rad/m. Fig. 10 gives out the damage imaging results of the PZT 2D crossshaped array.
For damage direction estimation, the wavenumbers ${k}_{x}=$–203 rad/m and ${k}_{y}=$192 rad/m can be obtained from the two wavenumbertime images. According to Eq. (10), the damage direction ${\theta}_{a}=$136.6° is obtained.
Fig. 7. Health reference signal of damage $E$
a) No. 1 array
b) No. 2 array
Fig. 8. Online damage monitoring signal of damage $E$
a) No. 1 array
b) No. 2 array
Fig. 9. Damage scattering signal of damage $E$
a) No. 1 array
b) No. 2 array
For damage distance estimation, the group velocity of the signal propagating on the structure should be measured beforehand. The group velocity measuring method based on continuous complex wavelet transform [8] is used to achieve this goal. The group velocity measuring results by using the 3 reference PZTs are ${c}_{gRef1}=$ 1485.20 m/s, ${c}_{gRef2}=$ 1439.87 m/s and ${c}_{gRef3}=$ 1352.27 m/s respectively. The average group velocity ${c}_{g}=$ 1425.8 m/s is used to damage distance estimation. From Fig. 10, the absolute arrival time of the damage scattering signal ${t}_{R1}=$ 0.4957 ms and t_{R}_{2}=0.4993 ms can be obtained. But the absolute arrival time is not the actual timeofflight of the damage scattering signal. By using the continuous complex wavelet transform [8], the envelope of the excitation signal can be obtained as shown in Fig. 11. The time corresponding to the envelope maximum value of is regarded to be the excitation time, ${t}_{e}=$ 0.1059 ms. Thus, based on Eq. (11), the timeofflight is obtained to be ${t}_{a}=$ 0.3916 ms and the damage distance is ${l}_{a}=$ 279.2 mm. According to Eq. (12), the damage position is obtained to be (–202.8 mm, 191.8 mm) and the damage localization error is $\mathrm{\Delta}l=$ 8.6 mm based on Eq. (13):
where, $\left({x}_{a},{y}_{a}\right)$ is damage localization result and $\left({x}_{D},{y}_{D}\right)$ is the actual position of damage.
Fig. 10. Wavenumbertime images of damage $E$
a) No. 1 array
b) No. 2 array
The whole abovementioned damage imaging and localization process for one damage can be finished in less than 5 seconds on a common laptop computer by using MATLAB 2009a.
According to the damage imaging and location process of the damage $E$ discussed above, the damage localization results and the localization errors of the 7 damages are listed in Table 4. It indicates that the damage localization results are in good agreement with the actual damage positions, and the maximum damage localization error is less than 1 cm.
Fig. 11. The envelope of the excitation signal
Table 4. Damage localization results of the composite plate
Damage label

(${k}_{x}$, ${t}_{R1}$)
(rad/m, ms)

(${k}_{y}$, ${t}_{R2}$)
(rad/m, ms)

Localized position (°, mm)

Actual position (°, mm)

Direction and distance error (°, mm)

$\mathrm{\Delta}l$
(mm)

A

(92, 0.5415)

(251, 0.5479)

(69.9, 313.7)

(71.6, 316.2)

(–1.7, –2.5)

9.7

B

(128, 0.4274)

(237, 0.4240)

(61.6, 228.0)

(63.4, 223.6)

(–1.8, 4.4)

8.4

C

(4, 0.3785)

(264, 0.3758)

(89.1, 193.4)

(90.0, 200.0)

(–0.9, –6.6)

7.3

D

(–132, 0.4121)

(247, 0.4189)

(118.1, 220.7)

(116.6, 223.6)

(1.6, –2.9)

6.7

E

(–203, 0.4957)

(192, 0.4993)

(136.6, 279.2)

(135.0, 282.8)

(1.6, –3.6)

8.6

F

(–253, 0.4330)

(119, 0.4272)

(154.8, 232.0)

(153.4, 223.6)

(1.4, 8.4)

10.0

G

(–284, 0.3943)

(5, 0.3974)

(179.0, 207.6)

(180.0, 200.0)

(–1.0, 7.6)

8.4

4. Conclusion
This paper proposes a new online damage imaging method based on Lamb wave wavenumber response and PZT 2D crossshaped array. The method function has been validated on a composite structure. The main contributions of this method are summarized as follows.
1) It does not rely on any Lamb wave propagation model of composite structure and it is high efficiency. Thus it can be applied to online damage monitoring of aircraft composite structures easily.
2) By using the PZT 2D crossshaped array, the damage can be localized without blind angle.
However, the method has two main limitations which need to be further studied.
1) The group velocity is needed to achieve damage distance estimation. For complex composite structures, the isotropic feature will increase the damage localization error.
2) The method is based on the farfield situation at the current stage.
In addition, more validations should be performed on complex composite structures.
Acknowledgements
This work is supported by the National Science Fund for Distinguished Young Scholars (Grant No. 51225502), the Natural Science Foundation of China (Grant Nos. 51205189 and 51575263), State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant Nos. MCMS0513K01 and 0515Y01), the Priority Academic Program Development of Jiangsu Higher Education Institutions and Hong Kong Scholars Program.
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