The technique enabling to measurement the
dispersion influence of the phase velocity Lamb wave A 0 mode
Lina Draudviliene
Kaunas University of Technology, Ultrasound Institute, Barsausko st. 59,Kaunas LT-51423, Lithuania
Hacène Ait-Aider
Mouloud Mammeri University of Tizi-Ouzou, Algeria
Abstract— The application of ultrasonic guided waves (UGW) is one of the latest methods of non-destructive testing (NDT) and structural health monitoring (SHM). Nowadays, it is widely used in various industrial fields to inspect and screen many structures by exploring the properties of its waves which can propagate long distances with high sensitivity to structural changes.
In this paper a technique based on the zero – crossing approach combined with the spectrum decomposition method is proposed. In general, according to the spectrum decomposition technique, the frequency spectrum of the signal is multiplied by special filter and the signal is reconstructed using inverse Fourier transform. Using this filtered signal, further signal processing is performed applying the zero- crossing method. This technique has been investigated using simulated and modelled signals of the Lamb wave asymmetric A0mode propagating in the higher dispersion zone. In order to obtain Lamb wave signals for analysis aluminium plates having thickness of 2 mm are used. The excitation signal was 3 periods, 300 kHz burst with the Gaussian envelope. The obtained measurement results are compared with the theoretical dispersion curves obtained by SAFE method. The results have shown that the proposed measurement technique enables not only to reconstruct the phase velocity dispersion curve of the Lamb wave A0mode, but also to choose in what precision limits and in what extent of the frequency ranges the dispersion curve of the phase velocity can be reconstructed.
Keywords — Lamb waves, phase velocity, dispersion, frequency spectrum
I. INTRODUCTION
The ultrasonic guided waves (UGW) are widely used in ultrasonic aplications of non-destructive testing (NDT) and structural health monitoring (SHM) to inspect of large engineering constructions, pipes, plates, rails, for these waves propagate a long distance with high sensitivity to structural changes [1-4]. According to propagating signal amplitude or
velocity changes defects, delaminating, corrosions, cracks and non-homogeneous can therefore be detected in objects [2-5].
However, UGW possess a dispersion phenomenon which leads to two different propagation velocities phase and group both are frequency depending and characterized by dispersion curves [6]. This UGW feature is complicated to determine accurately signal amplitudes and/or velocities. Therefore, to use UGW in investigations the question which can arise is related not only with velocities but also with frequency to which it corresponds.
In previous article [7], the phase velocity method based on the analysis of the zero-crossing point in the signal was presented. However, when using this method the phase velocity dispersion curves of the Lamb waves A0 and S0
modes‘ are reconstructed only in a narrow frequency range.
Using modelled signals obtained the Lamb wave propagating in 2 mm thikness aluminium plate A0 mode phase velocity dispersion curve is reconstructed only from 284 kHz up to 316 kHz (Δf=32 kHz) [6], using experimantal signals in range from 250 up to 305 kHz (Δf=55 kHz) [8]. Therefore, in order to restore the phase velocity dispersive curves in a wider frequency range it was conceived at first to perform the filtering of the Lamb wave signals and then apply the zerro- crossing technique. To achieve this goal using simulated and modeled signals of the A0mode of Lamb waves were filtered with different bandwidths of the filter having different central frequencies.
The objective of the work is to propose and investigate a technique based on the zero – crossing approach combined with the spectrum decomposition method for the measurement of the phase velocity of A0 mode of Lamb waves and to reconstruct the dispersion curve in the analyze frequency range.
II. MEASUREMENTMETHOD
The proposed phase velocity measurement technique which is based on the spectrum decomposition and the zero- crossing methods is briefly explained in Fig. 1 and can be described by the following steps which are presented below.
Fig.1. The algorithm of the proposed phase velocity dispersion curve reconstruction technique
According to Fig. 1:
1. the two signals at different distances are selectedx1and x2,
2. the frequency spectra of these two signals are calculated using the Fourier transform,
3. the filter bandwidth is selected,
4. the different central frequencies of the filter are selected,
5. the frequency spectra of these two signals are filtered using the selected filter with different central frequencies,
6. the filtered signals are reconstructed to the time domain using the inverse Fourier transform,
7. the phase velocity is calculated using the zero-crossing function,
8. the durations of the half–periods of first signal are calculated,
9. the equivalent frequencies that correspond to calculated duration of the half–period are estimated,
10. the phase velocity values and the corresponding number of the equivalent frequency are related to each other and corresponding set of pairs are obtained.
This obtained set defines estimation of the segment of phase velocity dispersion curve. For the verification of the proposed measurement method, investigations were carried out using simulated and modeled signals of the Lamb waves propagating in a 2 mm thickness aluminium plate.
III. INVESTIGATIONOFTHEPROPOSEDMETHODUSING THESIMULATEDSIGNALS
At first the proposed technique is investigated at theoretical level. For the obtained simulated signals of the asymmetric A0 Lamb wave mode the reference signal with known frequency bandwidth and the simplified complex transfer function of the object under investigation was exploited [9]. The 2 mm thickness aluminium plate was used for the investigation and 300 kHz frequency range is chosen for the analysis where the asymmetric A0 mode possesses a big dispersion nature [9]. The parameters of the aluminium alloy plate are: ρ the density (ρ = 2780 kg/m3), E the Young modulus (E = 71.78 GPa), ν the Poisson‘s ratio (ν = 0.3435).
For the investigation the signals of Lamb waves were calculated from distances 0 mm up to 140 mm with 0.1 mm step away from the excitation point. In the overall, 1401 signals were obtained for analysis. The obtained B-scan image of the A0mode signals is presented in (Fig. 2).
Fig.2.The B – scan image of the Lamb wave simulated signals measured on thed= 2 mm thickness aluminium plate with selected A0mode According to the proposed algorithm the two signalsx1,x2
at different distances are obtained and the frequency spectra of these two signals are calculated using the Fourier transform (Fig.3). The spectrum magnitude of the signals at – 6 dB is 160 kHz and the frequency spectra of the signals possess frequency bandwidth from 100 up to 500 kHz at very low – 40 dB level (Fig.3).
Signal No.2 Signal No.1
The spectrum of
the signal 2 The spectrum of
the signal 1 The filter with
different central frequencies
The filtered spectrums of the
signal 2
The filtered spectrums of the
signal 1
The reconstructed
signal 2 The reconstructed
signal 1
The zero-crossings of the signal 1
Estimated frequency for
the signal 1
Reconstructed dispersion curve
x1
x2
Transmitter Receiver 1 Receiver 2 Direction of waves propagating
∆x
Estimated phase velocity The zero-crossings
of the signal 2
Fig.3.The frequency spectrum of the A0mode of Lamb wave signals and filter bandwidth 80 kHz with different central frequencies: 1-150 kHz, 2 – 200 kHz, 3 - 250 kHz, 4 – 300 kHz, 5 – 350 kHz, 6 – 400 kHz, 7 – 450 kHz
Therefore, the width of the filter bandwidth is selected the half past narrower it is 80 kHz and the central frequencies of the filter: 150 kHz, 200 kHz, 250 kHz, 300 kHz, 350 kHz, 400 kHz, 450 kHz are selected according of the frequency spectrum of the analyse signals (Fig.3).
According the algorithm Fig.1, calculations were performed. The separate parts of the presented algorithm Fig.1 have been previously proposed by Mazeika et al. [6] and Draudviliene et al. [10]. To compare the obtained results of the proposed measurement method, the dispersion curves of the A0mode of Lamb waves propagating in 2 mm thickness aluminium plate were calculated using the semi-analytical finite element method (SAFE) [11]. According proposed measurement technique using different central frequencies of the 80 kHz bandwidth of the filter the values of the phase velocity dispersion curve are obtained and displayed on the theoretical ones (Fig.4).
Fig.4.The phase velocities of the simulated signals of the A0mode (dots) obtained using the80 kHz filter bandwidth with different central frequencies
The obtained results of the A0mode on the phase velocity dispersion curve are distributed in the frequency range from 173 kHz up to 419 kHz (Δf=246 kHz). Whereas, the frequency bandwidth of the filter is 80 kHz and the central frequencies are selected with 50 kHz step, therefore, the dispersion curve is reconstructed in a segments.
Selecting the 120 kHz frequency bandwidth of the filter it is ¾ spectrum magnitude of the analysis signalsand central frequencies of the filter are left the same: 150 kHz, 200 kHz, 250 kHz, 300 kHz, 350 kHz, 400 kHz, 450 kHzthe obtained values of the A0 mode phase velocity are distributed on the theoretical dispersion curve in the whole analized frequency range Fig.5.
Fig.5.The phase velocities of the simulated signals of the A0mode (dots) obtained using the 120 kHz filter bandwidth with different central frequencies
Using the 120 kHz frequency bandwidth of the filter the A0
mode results on the dispersion curve are distributed in the frequency range from 172 kHz up to 394 kHz (Δf=222 kHz).
However, the restore of the dispersive curve is reduced 24 kHz in the frequency range compared with the results obtained using the 80 kHz frequency bandwidth of the filter. It leads to missing in information.
IV. INVESTIGATIONOFTHEPROPOSEDMETHODUSING THEMODELLEDSIGNALS
The 2D finite element model of the aluminum plate was created and used for the verification of proposed method [6].
The geometry parameters and the elastic constants of the aluminum plate were used the same like in simulated case.
The asymmetric A0 mode was excited by applying the tangential force to one of the plate edges Fig.6. The waveform of the excitation force was a 3 period, 300 kHz burst with the Gaussian envelope and the central frequency 300 kHz. For the reflections of the end of plate the Lamb wave signals were calculated at the distances from 0 mm up to 140 mm with 0.1 mm step away from the excitation point. Totally, 1401 signals were obtained.
Fig.6.The finite element model for investigation of the A0mode of Lamb waves propagating in 2 mm thickness aluminium plate
The 80 kHz and 120 kHz width of the filter bandwidths are selected. The central frequencies of the both filters: 150 kHz, 200 kHz, 250 kHz, 300 kHz, 350 kHz, 400 kHz, 450kHz.
The carried out investigation is showed that applying proposed measurement technique for the Lamb waves A0
2 mm
l=200 mm Aluminium
100 150 200 250 300 350 400 1500
1600 1700 1800 1900 2000 2100 2200
f, kHz cph,
m/s
Theoretical dispersion curve Reconstructed segment of the dispersion curve
100 150 200 250 300 350 400 1500
1600 1700 1800 1900 2000 2100 2200 cph, m/s
f, kHz Theoretical
dispersion curve 150 kHz
200 kHz 250 kHz
300 kHz 350 kHz
400 kHz
450 kHz 0 100 200 300 400 500 600 0.1
0.3 0.5 0.7 0.9
160 kHz The central
frequency of the signal U(f),
n.v.
f, kHz 80 kHz
1 2 3
4 5
6 7
Excitation force
mode modelled signals the segments of the phase velocity dispersion curve are obtained. The obtained results of the investigation using different 80 kHz and 120 kHz filter bandwidths are displayed on the theoretical once and presented in Fig. 7 and Fig. 8 respectively.
Fig.7.The phase velocities of the modelled signals of the A0mode (dots) obtained using the80 kHz filter bandwidth with different central frequencies
Fig.8.The phase velocities of the modeled signals of the A0mode (dots) obtained using the120 kHz filter bandwidth with different central frequencies
The obtained results of the A0 mode using 80 kHz filter bandwidth on the dispersion curve are distributed in the frequency range from 167 kHz up to 420 kHz (Δf=253 kHz).
Using 120 kHz filter bandwidth the results are distributed from 179 kHz up to 397 kHz (Δf=218 kHz). The comparison of the simulated and modeled results is presented in Table 1.
TABLE I. FREQUENCIESOBTAINEDUSINGSIMULATEDAND
MODELLEDSIGNALS
Values of frequency at distance of 140 mm, kHz
filter
bandwidths, kHz Interval Covers
Simulated signals 80 173 - 419 246
Simulated signals 120 172 - 394 222
Modelled signals 80 167 - 420 253
Modelled signals 120 179 - 397 218
According to the presented results in Table 1 it can be seen that using 120 kHz filter bandwidth the phase velocity dispersion curve is restorted in a narrow frequency range comparing with those obtained using 80 kHz filter bandwidth.
The restorted A0phase velocity frequency range is narrowed 24 kHz using simulated signals and 35 kHz in a modeling case.
V. CONCLUSIONS
The obtained simulated and modeled results indicate that the proposed technique based on the zero – crossing approach combined with the spectrum decomposition method enable to reconstruct the segment of the A0 mode phase velocity dispersion curve. The investigations are carred out in 300 kHz frequency range and the proposed tecnique is investigated using different 80 kHz and 120 kHz filter bandwidths with different central frequencies. The theoretical and modeling investigations indicate that using narrower 80 kHz filter bandwidth, the A0mode phase velocity dispersion curves are reconstructed in wider frequency range compering with results obtained using 120 kHz filter bandwidth, however is needed to apply the smaller step of the central frequencies of the filter.
Using wider 120 kHz filter bandwidth, the part information is lost, so that the dispersion curve is restorted in a narrower frequency range, in a simulated case 24 kHz, in modeling case 35 kHz.
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Ultrasonics.2003, 41(3), 175–183.
100 150 200 250 300 350 400 1500
1600 1700 1800 1900 2000 2100 2200 cph, m/s
f, kHz Theoretical dispersion curve 150 kHz
200 kHz 250 kHz
300 kHz 350 kHz
400 kHz
450 kHz
100 150 200 250 300 350 400 1500
1600 1700 1800 1900 2000 2100 2200
Theoretical dispersion curve
f, kHz cph,
m/s
Reconstructed segment of the dispersion curve
