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Combined spectral estimator for phase velocities of multimode Lamb waves in multilayer plates
Ultrasonics 44 (2006) e1145–e1150 www.elsevier.com/locate/ultras
Combined spectral estimator for phase velocities of multimode Lamb waves in multilayer plates De-an Ta a
a,*
, Zhen-qing Liu b, Xiao Liu
b
Department of Electronic Engineering, Fudan University, Shanghai 200433, PR China b Institute of Acoustics, Tongji University, Shanghai 200092, PR China Available online 9 June 2006
Abstract A novel combined spectral estimate (CSE) method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates is presented in this paper, which based on auto-regressive (AR) model and 2-D FFT. Simulated signals in brass plate were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The multimode Lamb waves are excited in these laminates, and the received signal is processed by 2-D FFT and CSE, respectively. The results showed that the phase velocities of multimode signals whose wave numbers are much closed cannot be differentiated by 2-D FFT, but CSE has strong spatial resolution. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole. It promises to be a useful method in experimental signals processing of multimode Lamb waves. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Combined spectral estimator; Lamb waves; Phase velocity; Multiplayer plates
1. Introduction Lamb waves, which are dispersive and contain multiple modes, have received extensive attention [1–6] since the study by Worlton [7]. The key problem in Lamb wave testing is the measurement of the amplitudes and the phase velocity of the individual modes present in a dispersive multimode signal. If this could be achieved, the relative amplitudes of the different modes generated by mode conversion at a defect could be measured, leading to possibility of defect detection. The difficulty of phase unwrapping, which occurs in the Fourier phase dispersion measurement technique [8], is not present in time–frequency based analysis. The Fourier phase technique also cannot be used when multiple modes or reflections are superimposed in time. However, another technique that overcomes these limitations is the 2-D FFT *
Corresponding author. Tel.: +86 21 5566 4473; fax: +86 21 5566 4473. E-mail address: [email protected] (D.-a. Ta).
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.156
[9], which was used to operate on multiple, equally spaced waveforms [10]. Unfortunately, the need for exact, spatially sampled data restricts the practicality of the 2-D FFT for some inspection applications. Liu and Huang [4] used phase unwrapping algorithm to obtain the velocity dispersion curves of A0 and S0 Lamb wave modes of the received waves experimentally. However, when the plates are comparatively thicker, the components of high-order modes become greater in the acousto–ultrasonic waves, and various kinds of modes overlap one another. It makes difficult to separate different Lamb modes in the time domain. In contrast, time–frequency representations require only a single signal. The analyses such as the short-time Fourier transform (STFT), have been used to characterize dispersion by Kwun and Bartels [11]. They evaluated the group velocity dispersion of the first and second modes of the axisymmetric longitudinal wave in cylindrical steel shells. The pseudo-Wigner–Ville distribution (PWVD) [6] can offer several advantages for velocity dispersion measurements in comparison with the more traditional techniques. For
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example, they can be applied to broadband signals so that one measurement may be required to determine the velocity of multiple modes over a wide range of frequencies. Another important advantage is the ability to analyze signals containing multiple propagation modes and/or reflections, which superimpose and interfere in the time domain. Although, in theory, the group velocity can be obtained from phase velocity dispersion measurements, small measurement errors in the phase velocity come into being significant errors in the calculated group velocity. Hayashi et al. [12] determined the thickness and the elastic properties of thin metallic foils (thickness of less than 40 lm) by calculating the group velocity of a single mode (the A0 up to 3.5 MHz) using the wavelet transform of laser-generated and detected Lamb waves. However, if it is necessary to measure the phase velocities of multimode signals whose wave numbers are much close, it may also not be possible to separate the modes by using wavelet transform, since its spatial resolution is not strong enough. In this paper, we developed a combined spectral estimate (CSE) method based on AR model and 2-D FFT in order to differentiate and estimate the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates. Simulated signals in brass plate were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The multimode Lamb waves are excited in these laminates, and the received signal is processed by 2-D FFT and CSE, respectively. 2. Material and methods 2.1. Principle of combined spectral estimator The phase velocity is obtained commonly from dividing the angular frequency by the wavenumber. This method gives satisfactory results in the case of single-mode propagation, but does not in the case of the superposition of several modes. The measured velocities correspond to a function of the various velocities of all the propagating modes. So the experimental signals are transformed from time-domain into frequency domain, and 2-D FFT is employed to process the data. The discrete expression of 2-D FFT can be given as: N1 X N2 X X ðejx1 ; ejx2 Þ ¼ xðn1 ; n2 Þejx1 n1 ejx2 n2 ð1Þ n1 ¼1 n2 ¼1
Eqs. (1) can be rewritten as: X 1 ðn1 ; ejx2 Þ ¼
N2 X
xðn1 ; n2 Þejx2 n2
ð2Þ
n2 ¼1
X ðejx1 ; ejx2 Þ ¼
N1 X n1 ¼1
X 1 ðn1 ; ejx2 Þejx1 n1
ð3Þ
where n1 is discrete temporal sequence number, n2 is discrete spatial sequence number, N1 and N2 is discrete temporal sequence length and discrete spatial sequence length, respectively. The spatial frequency spectrum is obtained by doing Fourier transform of experimental signal gotten at every sampling point. So the received amplitude–time records are transformed into amplitude–wavenumber records under discrete frequency. 2-D FFT overcomes some problems caused by multimode propagation and dispersive nature. The spatial frequency spectrum of experimental data and the amplitude–wavenumber–frequency information were obtained by it. Using the information, we can gain Lamb waves’ amplitudes and phase velocities. However, the spatial sampling points in the experimental investigation cannot be very large, so the spatial resolution of 2-D FFT is not very powerful. Its difficult for it to differentiate the modes whose wavenumbers are much close. To improve the spatial resolution, a novel combined spectral estimate (CSE) method is proposed to combine FFT with AR method for exact determination of the ultrasonic phase velocity. From Eqs. (1) to (3), we can seen that 2-D FFT has two steps. First, 1-D FFT of every line (or row) in x(n1, n2) can be executed to obtain X 1 ðn1 ; ejx2 Þ. Then, 1-D FFT of X 1 ðn1 ; ejx2 Þ can be executed at every row (or line). Since the 2-D FFT’s temporal resolution is suitable and its spatial resolution is weak, we propose a novel two-dimensional combined-spectral estimating method, which is Fourier transform combined with modern spectral estimation. To deal with the experimental data, it also has two steps: firstly, we executing 1-D FFT of every spatial sample and getting intermediate production, then getting spectral estimation value on every frequency point using AR model. In other words, spectral estimating values of Eqs. (2) production can be obtained by executing AR model spectral estimating method. The combined spectral estimation values can be written as r2 Dt 2 X AR ðejx2 Þ ¼ jX 1 ðn1 ; ejx2 Þj ¼ P 1 þ p ak ejx2 k 2
ð4Þ
k¼1
where r2 is the variance of exciting white noise, Dt the sampling interval and ak the model coefficient got by Burg algorithm [13]. p is the order of AR model. AR model [13] is widely used in modern spectral estimation. It is very suitable to be used to analyze the peak value in the spectrum. One of the key problems in AR model’s application is the choice of order. If AR model’s order is too low, the spectrum resolution is not powerful enough; if the order is too high, illusive peaks will appear in the spectrum. There is an experiential formula (N/3) < p < (N/2) (where N is the spatial sample length). At the same time, we estimate it by using trial method and the order is selected as 64 finally.
D.-a. Ta et al. / Ultrasonics 44 (2006) e1145–e1150
Fig. 1. Experimental set-up for measurement.
2.2. Experimental testing Fig. 1 shows the schematic diagrams of experimental setup. A pulser/receiver unit (Panametrics 5052 PR, Waltham, MA) was used to excite transducer. Normal angle transducers with central frequency 2.5 MHz were used to transmitting and receiving Lamb waves. The initial distance between two transducers is 40 mm and the spatial sampling interval is 1 mm, and total number of samples is 150. The received signals were amplified (Panametrics 5052 PR) and digitized to 8 bits (HP 54642A, HewlettPackard, Palo Alto, CA) at 20 mega-samples/s. Each 2000-point waveform was averaged 128 times in the time domain and then stored on a computer for offline analysis using MATLABÒ software. The whole sampling process is monitored by PC. In the experiment, we made two three-layered aluminum/xpoxy/aluminum plates (symmetrical plate: 1.44 mm/ 0.67 mm/1.44 mm; unsymmetrical plate: 1.44 mm/0.65 mm/ 0.94 mm). The longitudinal wave and the transverse wave velocity of aluminum plate is cl = 6350 m/s and ct = 3050 m/s, density is q = 2700 kg/m3. In epoxy layer, the longitudinal wave and transverse wave velocity is cl = 2450 m/s and ct = 1590 m/s, density q = 1259 kg/m3. Matrix method [14] was employed to calculate the dispersion curves and wavenumber–frequency curves of Lamb waves in multi-layered plate. 3. Results and discussion 3.1. The simulated results of brass plate In order to examine the performance of CSE, The signal time waveform was simulated for monolayer brass plate in vacuum and they were processed by 2-D FFT and CSE.
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Fig. 2 shows the simulated signals time waveform (a) and disperse curve of wavenumber–frequency (b) in monolayer brass plate. Where the thickness of brass plate is 1 mm, the density is 8400 kg/m3, longitudinal and shear velocity is 4.4 m/ms and 2.2 m/ms, respectively. The number of cycles in the excitation signal is 5 with 3 MHz central frequency. The excitation signal is amplitude modulated by Gaussian window. The propagation distance is 400 mm. Lamb wave A0, S0, A1 and S1 modes generated from transmission point and overlap one another. But it cannot be differentiate the modes, especially A0, S0 and A1 modes from time waveform. However, Lamb wave modes can be differentiated by process of 2-FFT and CSE. Fig. 3 shows the results of 2D FFT (a) and CSE (b) for simulated signals of monolayer brass plate in vacuum. From Fig. 3, it can be seen that two methods can differentiate Lamb wave modes. But the spatial resolution of 2-D FFT is poor, especially for between A0 and S0, A1 and S1 modes, respectively. Those simulated results showed that CSE could improve the spatial resolution of 2-D FFT, the Lamb wave A0, S0, A1 and S1 modes can be separated. Therefore, the CSE may be an effective method for differentiation of Lamb wave modes whose wave numbers are much close or overlap one another in plate. 3.2. The results of symmetrical plate In the experiment of symmetrical plate, the central frequency of transducer is 2.5 MHz and the incident angle of transducer is 57° and 28°, respectively. As an example, when excited-angle is 57°, there have two modes A2 and S2. Fig. 4 is the results of 2-D FFT and it contraposed with wave number in experiment of symmetrical plate under this excited-angle, where Fig. 4(a) is the process result of 2-D FFT. In order to make analysis easier, Fig. 4(a) is overlapped with wave number to get Fig. 4(b). Fig. 4(b) shows that the generated Lamb waves were basically A2 and S2 near 2.4 MHz. But it is too difficult to discern individual modes. Fig. 5(a) is the process result of CSE for experimental data of symmetrical plate, where the AR model’s order was selected as 64 by experiential formula and trial method. In order to make analysis easier, Fig. 5(a) is also overlapped with wave number to get Fig. 5(b). Fig. 5(b)
Fig. 2. Simulated signals (a) time waveform and (b) disperse curve of wavenumber–frequency in monolayer brass plate.
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Fig. 3. Results of (a) 2-D FFT and (b) CSE for simulated signals of monolayer brass plate.
Fig. 4. Result of (a) 2-D FFT and (b) it contraposes with wavenumber in experiment of symmetrical plate.
Fig. 5. Result of (a) CSE and (b) it contraposes with wavenumber in experiment of symmetrical plate.
shows that the generated Lamb waves were basically A2 and S2 near 2.4 MHz. It is clear enough to discern exact modes to which the Lamb waves belong in this frequency range. The spatial resolution was improved notably. The estimative values of spatial spectrum of 2-D FFT and CSE at 2.39 MHz (where the peak value is) are shown as Fig. 6(a) and (b), respectively. It is thus evident that 2-D
FFT cannot discern the Lamb wave modes. From Fig. 6(b), it can be seen that CSE got two peak values, corresponding to A2 and S2. Table 1 is the comparison of measured value and calculated values. From this table, it can be seen that CSE can differentiate A2 and S2 modes. The new method has strong spatial resolution.
Fig. 6. Results of (a) 2-D FFT and (b) CSE in experiment of symmetrical plate.
D.-a. Ta et al. / Ultrasonics 44 (2006) e1145–e1150 Table 1 Comparison of measured value and calculated values (2.5 MHz, 57°)
Calculated value Measured value Error
Table 2 Comparison of measured value and calculated values (2.5 MHz, 35°)
S2
A2
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S1
A1
Wavenumber (1/m)
Phase velocity (m/s)
Wavenumber (1/m)
Phase velocity (m/s)
4996.3 4900.9 1.91%
3005.6 3064.1 1.95%
4686.2 4733.3 1.01%
3204.5 3172.6 1.00%
3.3. The results of unsymmetrical plate In the experiment of unsymmetrical plate, the incident angle of transducer is 35°, and the central frequency of transducer is 1 MHz and 2.5 MHz, respectively. As an example, when central frequency is 2.5 MHz and in that excited-angle, there have two modes A1 and S1. That the results of 2-D FFT and it contraposed with wave number in experiment of unsymmetrical plate under this excitedangle is also obtained. And the generated Lamb waves were basically A1 and S1 at near 2.4 MHz. But it is too difficult to discern exact modes to which the Lamb waves belonged. The process result of CSE for experimental data of unsymmetrical plate is also obtained. It is clear enough to discern exact modes to which the Lamb waves belong in this frequency range. The spatial resolution was improved notably. Fig. 7(a) and (b) are the estimative values of spatial spectrum at 2.42 MHz (where the peak value is) gotten by 2-D FFT and CSE, respectively. And the AR model’s order was also selected as 64 by experiential formula and trial method. It was thus evident that 2-D FFT cannot discern the Lamb wave modes. From Fig. 7b, it can be seen that CSE got two peak values, corresponding to A1 and S1. Table 2 is the comparison of measured value and calculated values. From this table, it can be seen that CSE can differentiate A1 and S1 modes. The spatial resolution of 2-D FFT is not very powerful due to the restriction of the spatial sampling points in the experimental investigation. And it is difficult to separate the modes whose wavenumbers are much close or various kinds of modes overlap one another (it is also can be seen from the Fig. 3(a), Fig. 4, Fig. 6(a) and Fig. 7(a)). From the comparison of measured value and calculated values, we can be seen that CSE can improve the spatial
Calculated value Measured value Error
Wavenumber (1/m)
Phase velocity (m/s)
Wavenumber (1/m)
Phase velocity (m/s)
5475.5 5403.5 1.31%
2777.0 2814.0 1.33%
5287.4 5194.1 1.76%
2875.8 2927.4 1.79%
resolution of 2-D FFT. The method can differentiate Lamb wave modes whose wave numbers are much close or overlap one another, which were difficult to separate by 2-D FFT. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole (it is can be seen from Tables 1 and 2). The results showed that the CSE may be an effective method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates. This result also was approved by experiments, in which the central frequency of transducer is 2.5 MHz with 28° incident angle and 1 MHz with 35°, respectively. However, there have some errors between the result of CSE and theoretical value. The errors may be conduced by the choice of AR model’s order. The AR model’s order was selected as 64 by experiential formula (N/3) < p < (N/2) and trial method in the paper, in some cases it may be inadequate. On the other hand, effects of noise, measurement and control of step length (spatial sampling interval) and uniformity of plate thickness also are the roots of error. In our experiment, the precision of the step length of transducer not so higher and the step length may not be able to too little. Those situations also have effects on the process result of signal. When the plates are comparatively very thicker, the components of high-order Lamb wave modes become greater, and many kinds of modes overlap one another. It makes may be difficult to separate different Lamb modes. In order to improve on the precision of 2-D combined spectral estimation, it is need to mend the algorithm. The improving of the precision of the step length may also assist the performance of 2-D combined spectral estimation. Those ameliorations and applying CSE to other structures is just our further works.
Fig. 7. Results of (a) 2-D FFT and (b) CSE in experiment unsymmetrical plate.
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4. Conclusion
References
A combined spectral estimate (CSE) method is proposed to combine FFT with AR model for exact detection of the Lamb waves phase velocity. And it was used to differentiate and estimate the phase velocities of multimode Lamb waves propagation whose wave numbers are much close or overlap one another in multiplayer plates. Simulated signals from monolayer brass plate in vacuum were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The results showed that 2-D FFT could not discern the exact modes of Lamb waves that propagated in the plate simultaneously and its spatial resolution is weak. The prerequisites for estimating the phase velocity accurately of Lamb waves are to discern the modes exactly. The novel method is proposed for exact detection of multimode Lamb waves in multiplayer plates. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole. The results showed that the CSE may be an effective method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates.
[1] F. Simonetti, P. Cawley, M.J.S. Lowe, Long range inspection of lossy bilayers, AIP Conf. Proc. 700 (1) (2004) 222–229. [2] D.N. Alleyne, P. Cawley, The interaction of Lamb waves with defects, IEEE Trans. Ultrason. Ferr. Freq. Control 39 (1992) 381–397. [3] P. Cawley, D. Alleyne, The use of Lamb waves for the long range inspection of large structures, Ultrasonics 34 (1996) 287–290. [4] Z.Q. Liu, R.J. Huang, Experimental analysis of acousto–ultrasonic wave propagation mode in thin plate, Chin. J. Acoust. 19 (2000) 277–284. [5] M. Ech-Cherif El-Kettani, F. Luppe´, A. Guillet, Guided waves in a plate with linearly varying thickness: experimental and numerical results, Ultrasonics 42 (2004) 807–812. [6] W.H. Prosser, M.D. Seale, B.T. Smith, Time–frequency analysis of the dispersion of Lamb modes, J. Acoust. Soc. Am. 105 (1999) 2669– 2676. [7] D.C. Worlton, Ultrasonic testing with Lamb waves, Non-Destructive Test. 15 (1957) 218–222. [8] W.H. Prosser, M.R. Gorman, Plate mode velocities in graphite/epoxy plates, J. Acoust. Soc. Am. 96 (1994) 902–907. [9] D. Alleyne, P. Cawley, A two-dimensional Fourier transform method for measurement of propagating multimode signals, J. Acoust. Soc. Am. 89 (1991) 1159–1168. [10] C. Eisenhardt, L.J. Jacobs, J. Qu, Application of laser ultrasonics to develop dispersion curves for elastic plates, J. Appl. Mech. 66 (1999) 1043–1045. [11] H. Kwun, K.A. Bartels, Experimental observation of wave dispersion in cylindrical shells via time–frequency, J. Acoust. Soc. Am. 97 (1995) 3095–3097. [12] Y. Hayashi, S. Ogawa, H. Cho, M. Takemoto, Non-contact estimation of thickness and elastic properties of metallic foils by wavelet transform of laser-generated Lamb waves, NDT & E Int. 32 (1999) 21–27. [13] S.M. Kay, Modern Spectral Estimation—Theory and Application, Prentice-Hall, 1988. [14] M.J.S. Lowe, Matrix techniques for modeling ultrasonic waves in multilayered media, IEEE Trans. Ultrason. Ferr. Freq. Control 42 (1995) 525–542.
Acknowledgements This project was supported by the National Natural Science Foundation of China (Grant No. 10074050 and 10304003) and the Ph.D. Programs Foundation of Ministry of Education of China (No. 20040246017).
Combined spectral estimator for phase velocities of multimode Lamb waves in multilayer plates De-an Ta a
a,*
, Zhen-qing Liu b, Xiao Liu
b
Department of Electronic Engineering, Fudan University, Shanghai 200433, PR China b Institute of Acoustics, Tongji University, Shanghai 200092, PR China Available online 9 June 2006
Abstract A novel combined spectral estimate (CSE) method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates is presented in this paper, which based on auto-regressive (AR) model and 2-D FFT. Simulated signals in brass plate were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The multimode Lamb waves are excited in these laminates, and the received signal is processed by 2-D FFT and CSE, respectively. The results showed that the phase velocities of multimode signals whose wave numbers are much closed cannot be differentiated by 2-D FFT, but CSE has strong spatial resolution. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole. It promises to be a useful method in experimental signals processing of multimode Lamb waves. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Combined spectral estimator; Lamb waves; Phase velocity; Multiplayer plates
1. Introduction Lamb waves, which are dispersive and contain multiple modes, have received extensive attention [1–6] since the study by Worlton [7]. The key problem in Lamb wave testing is the measurement of the amplitudes and the phase velocity of the individual modes present in a dispersive multimode signal. If this could be achieved, the relative amplitudes of the different modes generated by mode conversion at a defect could be measured, leading to possibility of defect detection. The difficulty of phase unwrapping, which occurs in the Fourier phase dispersion measurement technique [8], is not present in time–frequency based analysis. The Fourier phase technique also cannot be used when multiple modes or reflections are superimposed in time. However, another technique that overcomes these limitations is the 2-D FFT *
Corresponding author. Tel.: +86 21 5566 4473; fax: +86 21 5566 4473. E-mail address: [email protected] (D.-a. Ta).
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.156
[9], which was used to operate on multiple, equally spaced waveforms [10]. Unfortunately, the need for exact, spatially sampled data restricts the practicality of the 2-D FFT for some inspection applications. Liu and Huang [4] used phase unwrapping algorithm to obtain the velocity dispersion curves of A0 and S0 Lamb wave modes of the received waves experimentally. However, when the plates are comparatively thicker, the components of high-order modes become greater in the acousto–ultrasonic waves, and various kinds of modes overlap one another. It makes difficult to separate different Lamb modes in the time domain. In contrast, time–frequency representations require only a single signal. The analyses such as the short-time Fourier transform (STFT), have been used to characterize dispersion by Kwun and Bartels [11]. They evaluated the group velocity dispersion of the first and second modes of the axisymmetric longitudinal wave in cylindrical steel shells. The pseudo-Wigner–Ville distribution (PWVD) [6] can offer several advantages for velocity dispersion measurements in comparison with the more traditional techniques. For
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D.-a. Ta et al. / Ultrasonics 44 (2006) e1145–e1150
example, they can be applied to broadband signals so that one measurement may be required to determine the velocity of multiple modes over a wide range of frequencies. Another important advantage is the ability to analyze signals containing multiple propagation modes and/or reflections, which superimpose and interfere in the time domain. Although, in theory, the group velocity can be obtained from phase velocity dispersion measurements, small measurement errors in the phase velocity come into being significant errors in the calculated group velocity. Hayashi et al. [12] determined the thickness and the elastic properties of thin metallic foils (thickness of less than 40 lm) by calculating the group velocity of a single mode (the A0 up to 3.5 MHz) using the wavelet transform of laser-generated and detected Lamb waves. However, if it is necessary to measure the phase velocities of multimode signals whose wave numbers are much close, it may also not be possible to separate the modes by using wavelet transform, since its spatial resolution is not strong enough. In this paper, we developed a combined spectral estimate (CSE) method based on AR model and 2-D FFT in order to differentiate and estimate the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates. Simulated signals in brass plate were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The multimode Lamb waves are excited in these laminates, and the received signal is processed by 2-D FFT and CSE, respectively. 2. Material and methods 2.1. Principle of combined spectral estimator The phase velocity is obtained commonly from dividing the angular frequency by the wavenumber. This method gives satisfactory results in the case of single-mode propagation, but does not in the case of the superposition of several modes. The measured velocities correspond to a function of the various velocities of all the propagating modes. So the experimental signals are transformed from time-domain into frequency domain, and 2-D FFT is employed to process the data. The discrete expression of 2-D FFT can be given as: N1 X N2 X X ðejx1 ; ejx2 Þ ¼ xðn1 ; n2 Þejx1 n1 ejx2 n2 ð1Þ n1 ¼1 n2 ¼1
Eqs. (1) can be rewritten as: X 1 ðn1 ; ejx2 Þ ¼
N2 X
xðn1 ; n2 Þejx2 n2
ð2Þ
n2 ¼1
X ðejx1 ; ejx2 Þ ¼
N1 X n1 ¼1
X 1 ðn1 ; ejx2 Þejx1 n1
ð3Þ
where n1 is discrete temporal sequence number, n2 is discrete spatial sequence number, N1 and N2 is discrete temporal sequence length and discrete spatial sequence length, respectively. The spatial frequency spectrum is obtained by doing Fourier transform of experimental signal gotten at every sampling point. So the received amplitude–time records are transformed into amplitude–wavenumber records under discrete frequency. 2-D FFT overcomes some problems caused by multimode propagation and dispersive nature. The spatial frequency spectrum of experimental data and the amplitude–wavenumber–frequency information were obtained by it. Using the information, we can gain Lamb waves’ amplitudes and phase velocities. However, the spatial sampling points in the experimental investigation cannot be very large, so the spatial resolution of 2-D FFT is not very powerful. Its difficult for it to differentiate the modes whose wavenumbers are much close. To improve the spatial resolution, a novel combined spectral estimate (CSE) method is proposed to combine FFT with AR method for exact determination of the ultrasonic phase velocity. From Eqs. (1) to (3), we can seen that 2-D FFT has two steps. First, 1-D FFT of every line (or row) in x(n1, n2) can be executed to obtain X 1 ðn1 ; ejx2 Þ. Then, 1-D FFT of X 1 ðn1 ; ejx2 Þ can be executed at every row (or line). Since the 2-D FFT’s temporal resolution is suitable and its spatial resolution is weak, we propose a novel two-dimensional combined-spectral estimating method, which is Fourier transform combined with modern spectral estimation. To deal with the experimental data, it also has two steps: firstly, we executing 1-D FFT of every spatial sample and getting intermediate production, then getting spectral estimation value on every frequency point using AR model. In other words, spectral estimating values of Eqs. (2) production can be obtained by executing AR model spectral estimating method. The combined spectral estimation values can be written as r2 Dt 2 X AR ðejx2 Þ ¼ jX 1 ðn1 ; ejx2 Þj ¼ P 1 þ p ak ejx2 k 2
ð4Þ
k¼1
where r2 is the variance of exciting white noise, Dt the sampling interval and ak the model coefficient got by Burg algorithm [13]. p is the order of AR model. AR model [13] is widely used in modern spectral estimation. It is very suitable to be used to analyze the peak value in the spectrum. One of the key problems in AR model’s application is the choice of order. If AR model’s order is too low, the spectrum resolution is not powerful enough; if the order is too high, illusive peaks will appear in the spectrum. There is an experiential formula (N/3) < p < (N/2) (where N is the spatial sample length). At the same time, we estimate it by using trial method and the order is selected as 64 finally.
D.-a. Ta et al. / Ultrasonics 44 (2006) e1145–e1150
Fig. 1. Experimental set-up for measurement.
2.2. Experimental testing Fig. 1 shows the schematic diagrams of experimental setup. A pulser/receiver unit (Panametrics 5052 PR, Waltham, MA) was used to excite transducer. Normal angle transducers with central frequency 2.5 MHz were used to transmitting and receiving Lamb waves. The initial distance between two transducers is 40 mm and the spatial sampling interval is 1 mm, and total number of samples is 150. The received signals were amplified (Panametrics 5052 PR) and digitized to 8 bits (HP 54642A, HewlettPackard, Palo Alto, CA) at 20 mega-samples/s. Each 2000-point waveform was averaged 128 times in the time domain and then stored on a computer for offline analysis using MATLABÒ software. The whole sampling process is monitored by PC. In the experiment, we made two three-layered aluminum/xpoxy/aluminum plates (symmetrical plate: 1.44 mm/ 0.67 mm/1.44 mm; unsymmetrical plate: 1.44 mm/0.65 mm/ 0.94 mm). The longitudinal wave and the transverse wave velocity of aluminum plate is cl = 6350 m/s and ct = 3050 m/s, density is q = 2700 kg/m3. In epoxy layer, the longitudinal wave and transverse wave velocity is cl = 2450 m/s and ct = 1590 m/s, density q = 1259 kg/m3. Matrix method [14] was employed to calculate the dispersion curves and wavenumber–frequency curves of Lamb waves in multi-layered plate. 3. Results and discussion 3.1. The simulated results of brass plate In order to examine the performance of CSE, The signal time waveform was simulated for monolayer brass plate in vacuum and they were processed by 2-D FFT and CSE.
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Fig. 2 shows the simulated signals time waveform (a) and disperse curve of wavenumber–frequency (b) in monolayer brass plate. Where the thickness of brass plate is 1 mm, the density is 8400 kg/m3, longitudinal and shear velocity is 4.4 m/ms and 2.2 m/ms, respectively. The number of cycles in the excitation signal is 5 with 3 MHz central frequency. The excitation signal is amplitude modulated by Gaussian window. The propagation distance is 400 mm. Lamb wave A0, S0, A1 and S1 modes generated from transmission point and overlap one another. But it cannot be differentiate the modes, especially A0, S0 and A1 modes from time waveform. However, Lamb wave modes can be differentiated by process of 2-FFT and CSE. Fig. 3 shows the results of 2D FFT (a) and CSE (b) for simulated signals of monolayer brass plate in vacuum. From Fig. 3, it can be seen that two methods can differentiate Lamb wave modes. But the spatial resolution of 2-D FFT is poor, especially for between A0 and S0, A1 and S1 modes, respectively. Those simulated results showed that CSE could improve the spatial resolution of 2-D FFT, the Lamb wave A0, S0, A1 and S1 modes can be separated. Therefore, the CSE may be an effective method for differentiation of Lamb wave modes whose wave numbers are much close or overlap one another in plate. 3.2. The results of symmetrical plate In the experiment of symmetrical plate, the central frequency of transducer is 2.5 MHz and the incident angle of transducer is 57° and 28°, respectively. As an example, when excited-angle is 57°, there have two modes A2 and S2. Fig. 4 is the results of 2-D FFT and it contraposed with wave number in experiment of symmetrical plate under this excited-angle, where Fig. 4(a) is the process result of 2-D FFT. In order to make analysis easier, Fig. 4(a) is overlapped with wave number to get Fig. 4(b). Fig. 4(b) shows that the generated Lamb waves were basically A2 and S2 near 2.4 MHz. But it is too difficult to discern individual modes. Fig. 5(a) is the process result of CSE for experimental data of symmetrical plate, where the AR model’s order was selected as 64 by experiential formula and trial method. In order to make analysis easier, Fig. 5(a) is also overlapped with wave number to get Fig. 5(b). Fig. 5(b)
Fig. 2. Simulated signals (a) time waveform and (b) disperse curve of wavenumber–frequency in monolayer brass plate.
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Fig. 3. Results of (a) 2-D FFT and (b) CSE for simulated signals of monolayer brass plate.
Fig. 4. Result of (a) 2-D FFT and (b) it contraposes with wavenumber in experiment of symmetrical plate.
Fig. 5. Result of (a) CSE and (b) it contraposes with wavenumber in experiment of symmetrical plate.
shows that the generated Lamb waves were basically A2 and S2 near 2.4 MHz. It is clear enough to discern exact modes to which the Lamb waves belong in this frequency range. The spatial resolution was improved notably. The estimative values of spatial spectrum of 2-D FFT and CSE at 2.39 MHz (where the peak value is) are shown as Fig. 6(a) and (b), respectively. It is thus evident that 2-D
FFT cannot discern the Lamb wave modes. From Fig. 6(b), it can be seen that CSE got two peak values, corresponding to A2 and S2. Table 1 is the comparison of measured value and calculated values. From this table, it can be seen that CSE can differentiate A2 and S2 modes. The new method has strong spatial resolution.
Fig. 6. Results of (a) 2-D FFT and (b) CSE in experiment of symmetrical plate.
D.-a. Ta et al. / Ultrasonics 44 (2006) e1145–e1150 Table 1 Comparison of measured value and calculated values (2.5 MHz, 57°)
Calculated value Measured value Error
Table 2 Comparison of measured value and calculated values (2.5 MHz, 35°)
S2
A2
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S1
A1
Wavenumber (1/m)
Phase velocity (m/s)
Wavenumber (1/m)
Phase velocity (m/s)
4996.3 4900.9 1.91%
3005.6 3064.1 1.95%
4686.2 4733.3 1.01%
3204.5 3172.6 1.00%
3.3. The results of unsymmetrical plate In the experiment of unsymmetrical plate, the incident angle of transducer is 35°, and the central frequency of transducer is 1 MHz and 2.5 MHz, respectively. As an example, when central frequency is 2.5 MHz and in that excited-angle, there have two modes A1 and S1. That the results of 2-D FFT and it contraposed with wave number in experiment of unsymmetrical plate under this excitedangle is also obtained. And the generated Lamb waves were basically A1 and S1 at near 2.4 MHz. But it is too difficult to discern exact modes to which the Lamb waves belonged. The process result of CSE for experimental data of unsymmetrical plate is also obtained. It is clear enough to discern exact modes to which the Lamb waves belong in this frequency range. The spatial resolution was improved notably. Fig. 7(a) and (b) are the estimative values of spatial spectrum at 2.42 MHz (where the peak value is) gotten by 2-D FFT and CSE, respectively. And the AR model’s order was also selected as 64 by experiential formula and trial method. It was thus evident that 2-D FFT cannot discern the Lamb wave modes. From Fig. 7b, it can be seen that CSE got two peak values, corresponding to A1 and S1. Table 2 is the comparison of measured value and calculated values. From this table, it can be seen that CSE can differentiate A1 and S1 modes. The spatial resolution of 2-D FFT is not very powerful due to the restriction of the spatial sampling points in the experimental investigation. And it is difficult to separate the modes whose wavenumbers are much close or various kinds of modes overlap one another (it is also can be seen from the Fig. 3(a), Fig. 4, Fig. 6(a) and Fig. 7(a)). From the comparison of measured value and calculated values, we can be seen that CSE can improve the spatial
Calculated value Measured value Error
Wavenumber (1/m)
Phase velocity (m/s)
Wavenumber (1/m)
Phase velocity (m/s)
5475.5 5403.5 1.31%
2777.0 2814.0 1.33%
5287.4 5194.1 1.76%
2875.8 2927.4 1.79%
resolution of 2-D FFT. The method can differentiate Lamb wave modes whose wave numbers are much close or overlap one another, which were difficult to separate by 2-D FFT. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole (it is can be seen from Tables 1 and 2). The results showed that the CSE may be an effective method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates. This result also was approved by experiments, in which the central frequency of transducer is 2.5 MHz with 28° incident angle and 1 MHz with 35°, respectively. However, there have some errors between the result of CSE and theoretical value. The errors may be conduced by the choice of AR model’s order. The AR model’s order was selected as 64 by experiential formula (N/3) < p < (N/2) and trial method in the paper, in some cases it may be inadequate. On the other hand, effects of noise, measurement and control of step length (spatial sampling interval) and uniformity of plate thickness also are the roots of error. In our experiment, the precision of the step length of transducer not so higher and the step length may not be able to too little. Those situations also have effects on the process result of signal. When the plates are comparatively very thicker, the components of high-order Lamb wave modes become greater, and many kinds of modes overlap one another. It makes may be difficult to separate different Lamb modes. In order to improve on the precision of 2-D combined spectral estimation, it is need to mend the algorithm. The improving of the precision of the step length may also assist the performance of 2-D combined spectral estimation. Those ameliorations and applying CSE to other structures is just our further works.
Fig. 7. Results of (a) 2-D FFT and (b) CSE in experiment unsymmetrical plate.
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4. Conclusion
References
A combined spectral estimate (CSE) method is proposed to combine FFT with AR model for exact detection of the Lamb waves phase velocity. And it was used to differentiate and estimate the phase velocities of multimode Lamb waves propagation whose wave numbers are much close or overlap one another in multiplayer plates. Simulated signals from monolayer brass plate in vacuum were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The results showed that 2-D FFT could not discern the exact modes of Lamb waves that propagated in the plate simultaneously and its spatial resolution is weak. The prerequisites for estimating the phase velocity accurately of Lamb waves are to discern the modes exactly. The novel method is proposed for exact detection of multimode Lamb waves in multiplayer plates. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole. The results showed that the CSE may be an effective method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates.
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Acknowledgements This project was supported by the National Natural Science Foundation of China (Grant No. 10074050 and 10304003) and the Ph.D. Programs Foundation of Ministry of Education of China (No. 20040246017).