Quantitative Rectangular Notch Detection of Laser－induced Lamb Waves in Aluminium Plates with Wavenumber Analysis

， ， ，

1．College of Mechanical Engineering and Applied Electronics Technology，Beijing University of Technology，Beijing 100124，P．R．China；

2．Beijing Engineering Research Center of Precision Measurement Technology and Instruments，Beijing University of Technology，Beijing 100124，P．R．China）

0 Introduction

Notch is one of the most common defect types in aluminium aircraft structures．Lamb waves have shown great potential for non－destructive testing and structural health monitoring purposes in plate－like structures due to their ability to propagate long distances with low energy loss and their reasonable sensitivity for defects of different types［1－4］．Defect detection technology based on Lamb waves can effectively identify and locate defects by using transducer array，but there are still many challenges in the quantitative detection of defects［5］．The recent progresses on wavefield analysis technique have provided a new way to achieve quantitative detection of defects［6－7］．Some scholars have also proposed the phased array imaging method for defects detection and location in plates，as well as the wavefield analysis and other techniques for defect quantification［8－9］．Recently，with the development of laser ultrasonic technology，ultrasonic optical testing equipment，represented by laser Dopper vibrometer and two－wave mixing interferometer，has been widely used in the wavefield measurement．The two－wave mixing interferometer provides a high spatial resolution measurement of the out－ofplane displacement component at both one－dimensional（1D）and two－dimensional（2D）spaces，and wavefield signal is further analyzed by multidimensional Fourier transform（FT）to reveal the characterization of the defect［10］．In the existing wavefield analysis based on multi－dimensional FT，the wavenumber analysis exhibits great potential in the quantitative detection of defects［11－16］．

This paper illustrates the process of the wavenumber analysis to realize the quantitative detection of the rectanguar notch depth，and provides a feasible non－contact quantitative defect detection method．Firstly，the simulation models of different notches with different depths and widths are established to obtain the effects of notch depth，notch width and window width on the quantitative detection accuracy of wavenumbr analysis．Then，the finite element analysis is performed on experimental plates and the simulated data are analyzed by wavenumber analysis．Further，the experimental system is establised，and the aluminum plate with different notch depths are quantitatively detected．In the experiment，the Nd：YGA pulsed laser is used as the actuator of Lamb waves，and the signal is obtained by two－wave mixing interferometer．Finally，the results of experiment and simulation verify the effectiveness of wavenumber and analysis method．

1 Wavenumber Analysis

Wavenumber is one of the intrinsical parameters of Lamb waves．The wavenumber of excited Lamb wave mode is a fixed value for a given frequency，agiven thickness and material properties of the plate．Thus，for the determined excitation frequency and plate properties，the corresponding plate thickness can be calculated according to wavenumber．The defects in plate－like structure will cause the thickness of the plate change．Therefore，the depth of the defect can be quantitatively evaluated according to the wavenumber corresponding to the defect area．

1．1 Lamb wave dispersion characteristics

Lamb waves are dispersive and multi－modal．From Rayleigh－Lamb equation，the relationship between frequency and wavenumber can be calculated in plates．Fig．1depicts Lamb waves dispersion curves of a 2mm－thick aluminium plate．As shown in Figs．1（a，b），when the values of frequency－thickness product is lower than 1．5MHz·mm，only the fundamental Lamb waves modes，A0andS0modes，exist．As the product of frequency and thickness increases，more Lamb wave modes（such asA1，A2，S1，andS2）appear．

Fig．1 Lamb waves dispersion curves of a 2mm－thick aluminium plate

The dispersion characteristics can be represented by Fig．1（b）for aluminium plate with different thicknesses．Compared with theS0mode，the phase velocity of theA0mode changes more greatly with plate thickness duo to its highly dis－persive nature at low frequency－thickness product（fd）．Thicker plates will cause greater product of frequency and thickness for a given frequencyf，which will cause greater phase velocity eventually．According to the equationk＝2πf／vp，greater phase velocityvpresults in smaller wavenumberk．Thinner plates will cause greater wavenumber conversely．Therefore，theA0mode at low frequency is selected to evaluate the thickness of aluminium plate for the detection of defect depth．

1．2 Two－dimensional Fourier transform

Mmeasured points are taken with a spatial resolution ofΔx，and the number of samples per point isN，thus a spatial－temporal data matrixu（x，t）with dimension ofM×Ncan be obtained．The concept of Fourier analysis is straightforwardly extended to multi－dimensional signals，such as the spatial－temporal wavefield datau（x，t）．The frequency－wavenumber representation of a spatial－temporal signal can be interpreted as an alternative representation of the signal in terms of the familiar temporal frequency variableωand the wavenumberk．For 1Dwave propagation，its representation in frequency－wavenumber domain is obtained by 2DFT．Spatial－temporal signalu（x，t）can be transformed to frequency－wavenumber domain by using 2DFT，which is defined as

wherexis the distance between a actuator and a receiver，tthe time variable，ωthe frequency variable，andkthe wavenumber．According to Eq．（1），the amplitude distribution of frequencywavenumber spectrum depends on wavenumber component contained in wavefieldu（x，t）at an interested frequencyω0．From the dispersion characteristics of Lamb waves，it can be found that the change of the plate thickness at the defect will produce new wavenumber components，which implies that the amplitude distribution of wavenumber spectrum will change at an interested frequencyω0in the frequency－wavenumber domain．

1．3 Space－frequency－wavenumber analysis

The wavefield data is transformed from the time－space domain to the frequency－wavenumber domain through 2DFT which reveals the wave propagation characteristics that can not be clearly seen in the time－space domain．However，this approach cannot retain the space or time information．Therefore，according to the change of wavenumber spectrum distribution at a certain center frequency，only whether there exists a defect in the signal propagation path can be determined，while the specific location of the defect remains unknown．Similar to the idea of the short－time FT using a windowing technique，a short－space 2D FT is adopted in this paper to perform the spacefrequency－wavenumber analysis where the spatial information is retained．Mathematically，this process can be expressed as

wherexiis the space index，W（x，t）the 2Dwin－dow function．Ifthe function ofW（x，t）can be given

Otherwise，W（t，x）＝0．Here，Dxis the window width in space domain．Previous research has shown that the window width should be greater than at least two wave periods in order to accurately represent the spectral peak［17］．The phase velocity of Lamb waves is calculated according to the wavenumber of the defect region，and the frequency－thickness product corresponding to the phase velocity in the theoretical dispersion curves is selected to calculate the thickness of the aluminium plate at the defect region，so as to evaluate the defect depth．

2 Rectangular Notch Simulation

2．1 Simulation methodology

2Dsimulation has been implemented using the ABAQUS finite element package in this study．An aluminium plate of 2mm thickness and 1 000mm length was modelled for obtaining the effects of notch depth，notch width and window width on the quantitative detection accuracy of wavenumbr analysis．Material properties of the aluminium plate are listed in Table 1．

Table 1 Material properties of the aluminium plate

A schematic illustration of the simulation model is shown in Fig．2．A0mode was input according to the displacement mode shape of A0in the left end of plate．Here，the three－cycle excitation signal modulated by Hanning window at the center frequency of 200kHz was used．The simulation was solved using a fixed time stepping solver with 1ns time step and 0．1mm maximum quadratic element size in order to satisfy the requirements of the stability and accuracy of the solution．Notch was 100mm away from the left end of the model．

Fig．2 Schematic illustration of simulation model

2．2 Simulated wavefield analysis

As illustrated in Fig．2，taking the left end of the model as the starting point，the out－of－plane components were taken from left to right side of the model with a spatial resolution of 1mm，and 281points were extracted．Take the defected model as an example，the notch widthwis 20 mm and the depthdis 1．5mm，and the simulation data were analyzed by space－frequency－wavenumber analysis method．A 2DHanning window with a width of 20mm （equal to notch width）was selected and the moving step size was 1mm．The resulted time－space wavefield and its frequency－wavenumber spectrum are shown in Fig．3（a）and Fig．3（b），respectively．Fig．3（a）shows transmitted，reflected and multiple reflected waves caused by notch，as well as the incident wave．Furthermore，when the signal propagates inside the notch，the wave velocity decreases（the slope of the incident waves decreases in the distance－time coordinate system），which is consistent with the propagation characteristics of A0mode shown in Fig．1（b）．The frequency－wavenumber spectrum is plotted as amplitude versus frequency and wavenumber in Fig．3（b），together with the theoretical dispersion curves．As shown in Fig．3（b），the A0mode can be clearly observed，and the simulation results match well with the theoretical dispersion curves．It can be also observed that the new wavenumber component induced by notch matches well with the theoretical dispersion curves of aluminium plate with 0．5mm thickness．Note that in the frequencywavenumber spectrum，negative wavenumbers represent waves that propagate opposite to the incident wave．

Fig．3（c）presents the space－wavenumber spectrum at 200kHz．As shown in Fig．3（c），the spectrum corresponding to waves propagating in the defect－free zone is centered about the wavenumber at 0．72rad／mm，which is the same as the theoretical wavenumber value of A0mode at 200 kHz in the pristine region of the model．Furthermore，the spectrum corresponding to multiple reflected waves inside the notch increases distinctly and it is centered about the wavenumber at 1．32 rad／mm，which is the same as the theoretical wavenumber value of A0mode at 200kHz in the defect region of the model．Fig．3（d）shows spacewavenumber curves plotted by calculating the weighted average of the wavenumber at each location．Therefore，the location of the notch and the wavenumber of the notch region can be determined using the peak coordinates of the spacewavenumber curve．

2．3 Parameters analysis by wavenumber analysis

Fig．3 Space－frequency－wavenumber analysis of simulation data for the model with notch defect

The analysis procedure outlined in Section 2．3was applied to simulation data which were obtained through a range of notch depths and widths．Notch width ranged from 3mm to 13mm in 1mm increments with a constant 1．8mm notch depth．Window width ranged from 8mm to 13 mm for each width．The theoretical and calculated wavenumber are shown in Fig．4（a）．The accuracy increases along with the notch width increase．This can be explained by considering that when the notch is smaller than the window，the windowed dataset will contain the signals of Lamb waves propagating within and adjacent to the notch．Therefore，the calculated wavenumber will be reduced by the presence of the lower wavenumber components presented in the pristine region of the plate．Fig．4（b）shows wavenumber percent error versus the ratio of notch width to window widthWd／Ww．As the notch width increases and approaches the window width，the error is dramatically reduced．

Fig．4 Effect of notch width on calculated wavenumber at the frequency of 200kHz and with a 1．8mm deep notch

Simulations were also performed when notch depth varied from 0．1mm to 1．8mm in 0．1mm increments with constant 30mm notch width，in order to study the effect of notch depth on calculated wavenumber．Window width ranged from 8 mm to 13mm．The theoretical wavenumber and calculated wavenumber are shown in Fig．5（a）．Theoretical wavenumber was determined by solving the antisymmetric Rayeigh－Lamb equation at 200kHz for propagation in a plate with a thickness equal to that of the region inside the notch．Agreement between the theoretical and the calculated wavenumbers was achieved，particularly for the notch depth more than 1．2mm where maximum percent error is 3．6%．For each notch depth，a larger window results in greater accuracy，particularly for more shallow notch．For the shorter wavelengths corresponding to thinner plate or deeper notch，more cycles of the wave were contained within the window．As shown in Fig．5（b），calculated wavenumber percent error is presented versus a normalized window size equal to the product of wavenumber and window width．The normalized window size indicates the number of wavelengths contained in the window．Measurement error is less than 1．7%for normalized window sizes greater than two，which corresponds to two wave periods within the window．

Fig．5 Effect of notch depth on calculated wavenumber at the frequency of 200kHz and with a 30 mm notch depth

The simulation results show that when determining the window width，a balance must be struck between increasing window width to ensure multiple wavelengths within the window while reducing window width to ensure the notch width larger than the window width．Therefore，in order to minimize error in wavenumber measurement，the window width must be at least as large as two wavelengths and the notch must be at least as large as the window width．

2．4 Simulation analysis for experimental plate

Using the same modeling method，an aluminium plate of 1．7mm thickness and 1 000mm length was modelled．Taking the left end of the model as the starting point，the out－of－plane com－ponents were taken from the left to the right side of model with a spatial resolution of 0．5mm，and 561points were extracted．The center frequency of excitation signal was 140kHz，and other simulation parameters remained unchanged．Both pristine and defected simulated data were analyzed using space－frequency－wavenumber analysis．Two notches with different depths were detected：20 mm×0．9mm and 20mm×1．4mm．Hanning window with a width of 20mm （equal to notch width）was selected and moving step size was 0．5mm．

Fig．6shows analysis results of the pristine plate．As shown in Fig．6（a），there is only incident signal in the time－space wavefield due to the absence of defects．There is no signal attenuation in the simulation，so it can be seen from Fig．6（b）that the energy of the signal does not drop clearly as the propagation distance increases．Furthermore，it can be clearly observed from Fig．6（c）that the wavenumber did not change significantly in the propagation path，and the predicted and calculated wavenumbers well agrees．

Fig．6 Space－frequency－wavenumber analysis of simulation data for the pristine model

Simulated signals obtained from defected models were analyzed，and the results are shown in Figs．7，8．Compared with the time－space wavefield，it can be seen that transmitted signal components decrease and multiple reflection phenomenon within the notch becomes more obvious as the notch depth increases．Furthermore，as the notch depth increases，the proportion of wavefield energy increases in the defect region as shown in Fig．7（b）and Fig．8（b）．As the notch depth increases，more portion of Lamb waves are trapped inside the notch due to multiple reflections from the notch boundary，while less Lamb waves are transmitted through notch．As shown in Fig．7（c）and Fig．8（c），the wavenumber significantly increases in the defect region．Evaluation results of notch depth obtained from simulation data are listed in Table 2．

Fig．7 Space－frequency－wavenumber analysis of simulate data for the model with a 0．9mm deep notch

Fig．8 Space－frequency－wavenumber analysis of simulation data for the model with a 1．4mm deep notch

Table 2 Evaluation results of notch depth obtained from simulation data

3 Experiment

3．1 Test specimens

Three aluminium plates with dimension of 1 000mm×1 000mm×1．7mm labelled as No．1，No．2and No．3were used for testing．No．1aluminum plate had no defect．The notch defect of 30mm×20mm×0．9mm was machined on No．2 aluminium plate and the other notch defect of 30 mm×20mm×1．4mm was machined on No．3aluminium plate ．From the back side of the defect，the schematic of the test aluminium plate is shown in Fig．9．Taking the location of the pulsed laser spot as the origin，a total of 401scan points along the positive direction ofxaxis from thex＝0mm to 200mm were taken with a spatial resolution ofΔd＝0．5mm．Note the sampling intervalΔdwas selected first by meeting the Nyquist sampling theorem spatially，i．e．Δd≤λ／2，whereλis the wavelength of the interrogating wave．To improve the signal－to－noise ratio，the measurement at each scan point was averaged for 30 times．To further improve the measurement quality，reflective tape （producted by 3Mcorporation，8850model，approximately 0．1mm thick）is used to ensure the consistency of smoothness of the surface of the aluminium plates．

Fig．9 Schematic of the test aluminium plate

3．2 Experimental setup

The diagram of non－contact laser ultrasonic testing system is shown in Fig．10．The excitation was implemented by a Nd：YAG pulsed laser．Data acquisition was completed with two－wave mixing interferometer．Then，the data was further collected by a digital oscilloscope（Tektronix DPO 4054B）at a time sampling rate of 50MHz．

During excitation，the laser controller provided energy for Nd：YAG pulse laser to cause energy transitions in its internal crystal to generate a pulsed laser，which was emitted by the FHY excitation head and acted on the surface of the aluminium plate．At the same time，the laser controller provided the oscilloscope with a synchronous trigger signal to ensure the synchronous excitation and reception of the laser ultrasonic signals．

On the receiving side，the continuous laser was output by a continuous－wave fiber laser，which was split into two beams after passing through the laser splitter drawer．One of them was directly input to the demodulator as a reference light，the other was irradiated to the surface of the aluminium plate via a FHY measurement head，and then the laser （signal light）reflected by the surface of the aluminium plate and carrying the ultrasonic signal received by the FHY measurement head again，and eventually transmitted to the demodulated．In demodulator，the ultrasonic signal was demodulated by reference light and signal light．Finally，it was received and displayed by an oscilloscope．The FHY measurement head was mounted on the 2D control platform，and the position of the meas－urement head can be accurately controlled by the computer．In order to avoid the influence of the air shock wave generated by thermoelastic mechanism on the received signal，the excitation and reception probes were divided into two sides of the aluminium plate．

Fig．10 Diagram of the non－contact laser ultrasonic testing system

The wavelength of the laser produced by the Nd：YAG pulse laser in the system was 1 064 nm，the effective duration was 10ns，the energy of single laser pulse was 0—50mJ，and the diameter of the spot was about 1．2mm when the spot was focused．Laser single－pulse energy adjustment and excitation mode selection can be achieved through the laser controller．The laser ultrasonic receiver adopted the AIR－1550－TWM two－wave mixing interferometer produced by the American IOS company．The continuous laser wavelength was 1 550nm，the spot radius was at least 200μm，and the detection bandwidth was 125MHz．In the experiment，in order to ensure that the laser excites the ultrasonic signal under the thermoelastic mechanism to avoid damage to the surface of the aluminium plate，the laser single－pulse energy was set to 10mJ．To increase the signal－to－noise ratio，the laser excitation and receiving beams were kept perpendicular to the surface of aluminium plate．The experimental system of non－contact laser ultrasonic testing system is shown in Fig．11．

Fig．11 Experimental system of non－contact laser ultrasonic testing system

3．3 Analysis of received wavefield by laser－laser method

Fig．12 Time－domain signals and frequency spectra of laser－induced Lamb waves at 50mm distance from the receiving spot

On the basis of band－pass filtering，continuous wavelet transform was used to get wavefield signals with center frequency of 140kHz．According to the Shannon entropy minimum theory［18］of wavelet coefficients，gaus9was selected as the mother wavelet．In No．3aluminium plate，the time－domain signals of Lamb waves and its frequency spectra received by the receiving spot at 50 mm away from the excitation spot are shown in Fig．12．As shown in Fig．12（a） ，the time－domain signal of the original laser－induced Lamb waves contain a lot of high－frequency noises，the direct arrival signal and defect－reflected signal can not be clearly seen．However，according to the phenomenon that the amplitude of the signal has many irregular oscillations after 0．05ms，it can be inferred that after this time point，the signal includes components other than the direct arrival signal．From Fig．12（b），it can be seen that the laser－induced Lamb wave signal has a wide frequency band and is dominated by low－frequency components，and the energy of high－frequency components is relatively low due to attenuation during the propagation process．After the wavelet denoising of the original laser－induced Lamb wave signal，bandpass filter and continuous wavelet transform were used to extract Lamb wave signal with the center frequency of 140kHz．The direct arrival signal and the defect－reflected echo can be clearly seen，as Fig．12（c）shows．The frequency spectrum of the extracted signal as shown in Fig．12（d），shows that the frequency band of the signal has been concentrated near the center frequency of 140kHz．

Band－pass filtering combined with continuous wavelet transform was used to extract the wavefield signals of laser－induced Lamb waves collect－ed from the three test plates，and the wavefield signals with the center frequency of 140kHz were obtained．The analysis results of extracted experimental wavefield are shown in Figs．13—15．A－greement between the simulated and experimental results is relatively poor．The wavenumber component in the experimental results is more complicated，which has a certain impact on the evaluation of the depth of defects．Evaluation results of notch depth using experimental data obtained by laser－laser method are listed in Table 3．As shown in Table 3，the accuracy of defect depth evaluation in No．3test plate is lower than that in No．2test plate．The experimental results are not consistent with the conclusions obtained in Section 2．3and the simulation results in Section 2．4．This is mainly due to the noise contained in the experimental data，which reduces the signal－tonoise ratio．The source of noise mainly comes from the poor consistency of experimental signals．The poor consistency of the experimental signal will affect the quality of the wavefield signal［19］，and further affect the accuracy of the final defect depth assessment．The Gauss white noise contained in the experimental data is also one of the factors that reduce the signal－to－noise ratio of the experimental data．

Table 3 Evaluation results of notch depth using experimental data obtained by laser－laser method

Fig．13 Space－frequency－wavenumber analysis of experimental data for No．1plate by using laser－laser method

Fig．14 Space－frequency－wavenumber analysis of experimental data for No．2plate by using laser－laser method

Fig．15 Space－frequency－wavenumber analysis of experimental data for No．3plate by using laser－laser method

4 Conclusions

In this paper，the quantitative detection of notch defects in aluminium plates has been carried out by means of wavenumber analysis．Firstly，the simulated signals were analyzed to obtain the effects of notch depth，notch width and window width on the quantitative detection accuracy of wavenumber analysis．The results show that in order to minimize error in wavenumber measurement，the window width must be at least as large as two wavelengths and the notch must be at least as large as the window width．This implies that higher frequency excitations should perform better due to shorter wavelength and therefore smaller available window width and more accurate characterization of small defects．Laser－laser method was used in experimental investigation for signal excitation and reception．The simulation results showed that as the notch depth increase，the notch depth evaluation error decreases．However，in the experimental results，the detection error of notch with 1．4mm depth increased．The poor consistency of experimental signals and Gaussian white noise in experimental data may be responsible for this phenomenon．We only briefly illustrate some factors that influence the detection ac－curacy of detecting the existance of some wavenumber，and have not considered other factors．The simulation and experimental results verify the feasibility of the wavenumber analysis method on evaluating the depth of the rectangular notch，which implies the method′s potential in the quantitative detection of defects．

Acknowledgements

This work was supported by the National Natural Science Foundation of China（Nos．51475012，11772014，and 11272021）．

［1］ WILCOX P D，LOWE M J S，CAWLEY P．The effect of dispersion on long－range inspection using ultrasonic guided waves ［J］．NDT＆E International，2001，34（1）：1－9．

［2］ PRADO V T，HIGUTI R T，KITANO C，et al．Lamb mode diversity imaging for non－destructive testing of plate－like structures［J］．NDT＆E International，2013，59：86－95．

［3］ LIU Z H，SUN K M，SONG G R，et al．Damage localization in aluminum plate with compact rectangular phased piezoelectric transducer array［J］．Mechanical Systems and Signal Processing，2016，70：625－636．

［4］ AN Y K，PARK B，SOHN H．Complete noncontact laser ultrasonic imaging for automated crack visualization in a plate ［J］．Smart Materials and Structures，2013，22（2）：025022．

［5］ YU L Y，LECKEY C A C．Lamb wave－based quantitative crack detection using a focusing array algorithm ［J］．Journal of Intelligent Material Systems and Structures，2013，24（9）：1138－1152．

［6］ LEE J R，JEONG H，CHIA C C，et al．Application of ultrasonic wave propagation imaging method to automatic damage visualization of nuclear power plant pipeline［J］．Nuclear Engineering and Design，2010，240（10）：3513－3520．

［7］ LEE J R，CHIA C C，PARK C Y，et al．Laser ultrasonic anomalous wave propagation imaging method with adjacent wave subtraction：algorithm ［J］．Optics and Laser Technology，2012，44（5）：1507－1515．

［8］ MICHAELS J，DAWSON A，MICHAELS T，et al．Approaches to hybrid SHM and NDE of composite aerospace structures［C］∥Proceedings of SPIE－The International Society for Optical Engineering．［S．l］；［s．n．］，2014，9064：906427．

［9］ TIAN Z H，YU L Y，LECKEY C A C．Rapid guided wave delamination detection and quantification in composites using global－local sensing［J］．Smart Materials and Structures，2016，25（8）：1－11．

［10］ALLEYNE D N，CAWLEY P．A two－dimensional Fourier transform method for the measurement of propagating multimode signals［J］．Journal of the Acoustical Society of America，1991，89（3）：1159－1168．

［11］YU L Y，TIAN Z H．Lamb wave structural health monitoring using a hybrid PZT－laser vibrometer approach［J］．Structural Health Monitoring，2013，12（5－6）：469－483．

［12］RUZZENE M．Frequency－wavenumber domain filtering for improved damage visualization ［J］．Smart Materials and Structures，2007，16（6）：2116－2129．

［13］KUDELA P，RADZIENSKI M，OSTACHOWICZ W．Identification of cracks in thin－walled structures by means of wavenumber filtering ［J］．Mechanical Systems and Signal Processing，2015，50：456－466．

［14］SOHN H，DUTTA D，YANG J，et al．Automated detection of delamination and disbond from wavefield images obtained using a scanning laser vibrometer［J］．Smart Materials and Structures，2011，20（4）：045017．

［15］FLYNN E B，CHONG S Y，JARMER G J，et al．Structural imaging through local wavenumber estimation of guided waves ［J］．NDT＆E International，2013，59：1－10．

［16］ROGGE M D，LECKEY C A C．Characterization of impact damage in composite laminates using guided wavefield imaging and local wavenumber domain analysis［J］．Ultrasonics，2013，53：1217－1226．

［17］MCAULAY R J，QUATIERI T F．Speech analysis／synthesis based on sinusoidal representation ［J］．IEEE Transactions on Acoustics，Speech，and Signal Processing．1986，34（4）：744－754．

［18］LI F C，MENG G，KAGEYAMA K，et al．Optimal mother wavelet selection for Lamb wave analyses［J］．Journal of Intelligent Material Systems and Structures，2009，20（10）：1147－1161．

［19］MALINOWSKI P，WANDOWSKI T，KUDELA P，et al．Laser vibrometry for guided wave propagation phenomena visualisation and damage detection［C］∥AIP Conference Proceedings．［S．l．］：［s．n．］，2010，1253：140－149．