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Robust polarization filter for separation of Lamb wave modes acquired using a 3D laser vibrometer
Mechanical Systems and Signal Processing 93 (2017) 368–378
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Robust polarization filter for separation of Lamb wave modes acquired using a 3D laser vibrometer Łukasz Ambrozin´ski ⇑, Tadeusz Stepinski Department of Robotics and Mechatronics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 12 April 2016 Received in revised form 11 January 2017 Accepted 4 February 2017
Keywords: Lamb waves Polarization Modes separation 3D laser scanning vibrometer
a b s t r a c t Interpretation of signals related to Lamb waves propagation and scattering can rise serious difficulties due to the multi-modal nature of these waves. Different modes propagating with different velocities can be mixed up and hinder extraction of damage reflected components. As a feasible solution to this problem we propose a technique for separation of the propagating modes using a new type of polarization filter. The proposed directional polarization filter (DPF) can be applied if two components of particle movement, the in-plane and the out-of-plane, are available, for instance, from the measurement performed using laser vibrometer. The DPF is robust in the sense that it does not need a complete amplitude information of both components. Operation principle of the DPF is presented and illustrated by the simulated results in the form of B-scans obtained for an aluminum plate. The simulated results are verified by the experimental data obtained by processing the signals captured using a 3D laser vibrometer. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The main difficulty hindering applications of Lamb waves for non-destructive evaluation (NDE) of planar structures is their multi-modal and dispersive nature. An infinite number of modes can propagate in a plate with velocities depending on the frequency-plate thickness product [1–4]. Although, it is possible to limit the low frequency band where only two fundamental modes can exist, the responses can be complex even for an intact structure. Therefore, special techniques have been developed to obtain pure single-mode excitation or to process the acquired data to separate the wave modes. A common approach, that can be used for selective generation and reception of Lamb waves is based on the Snell’s law of refraction. Refraction of ultrasonic waves at an interface between two different media enables selective tuning of wavelength. Transducer’s angle of incidence can be adjusted to obtain the desired phase velocity corresponding to the selected Lamb mode in the investigated medium. This principle can be realized by means of contact transducers with perspex variable-angle-wedge [5] as well as immersion or air-coupled setups [6–8]. Another solution, applied mostly in structural health monitoring (SHM) applications is the use of interdigital transducers (IDT) with comb-shaped electrodes spaced with pitch corresponding to the desired wavelength [9]. Similarly, theoretical solution proposed by Xu et al. [10] permits tuning the length of piezoelectric wafer active sensor (PWAS) to the wave mode
⇑ Corresponding author. E-mail addresses: [email protected] (Ł. Ambrozin´ski), [email protected] (T. Stepinski). http://dx.doi.org/10.1016/j.ymssp.2017.02.002 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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that has be to excited. The main drawback of those techniques is that the transducers have to be perfectly matched to the application by taking into account thickness of the inspected plate and the desired frequency band. Since exciting and sensing a single wave mode is a complicated task in practical applications a variety of signal postprocessing techniques for wave mode identification and separation has been proposed. One of the most straightforward approaches relies on the time-frequency representation of the acquired ultrasonic signals. Due to the dispersion different spectral components of an excited burst propagate with different velocities. It results in deformed waveforms of the received signals that can be observed as elongated long-tailed tone-burst. Although the time-frequency representation allows to reveal the dispersion-related delays it requires a broadband excitation with sufficient energy within the whole frequency band where the dispersion characteristics are to be estimated. Various time-frequency representations, including reassigned spectrogram, reassigned scalogram, smoothed Wigner–Ville distribution as well as Hilbert spectrum, have been used for the identification of Lamb modes [11]. Moreover, it was shown that selected features of the time-frequency representations can be used for damage detection [12]. Since the frequency-wavenumber space enables separation of different Lamb modes, two-dimensional Fourier transform (2DFT) is one of the most commonly used techniques to identify and quantify the existent modes [13–17]. However, a relatively dense network of sensing points is required to apply this technique and therefore laser interferometers are often used to collect data with sufficient spatial sampling [18–22]. These measurements are, however, sensitive to the surface quality and often reflective sprays or foils have to be used. Availability of laser vibrometers has opened completely new possibilities for sensing and processing Lamb waves; new ideas could be realized by means of a contactless measurement of vibrations at different points at the investigated surface or a free plate edge. A specific application of laser vibrometer for mode separation if the plate surface is inaccessible for testing was presented in [23]. The proposed orthogonality-based method is based on measurements by means of a 3D vibrometer (an instrument that combines 3 different laser beams for simultaneous measurement of both vibration components) of the wave field scattered from the plate’s free edge at a dense grid of points. Thus, the method requires access to a free edge, which makes it impractical in many applications. As illustrated in [19], elliptical particle motion trajectory resulting from Lamb wave propagation can be acquired by means of a single point laser interferometer, however, the measurements should be repeated using a laser beam incident at different angles with respect to the surface. A different approach was presented in [24] where the in-plane vibration component was estimated based on the out-of-plane motion. To apply this method, however, both plate’s thickness and bulk wave velocities have to be known. More recently we showed how the Lamb waves modes can be separated based on their polarization parameters [25] using oblique polarization filter (OPF), originally developed for seismic waves processing [26,27]. Both S0 and A0 Lamb modes exhibit elliptical polarization, however, their polarization parameters, i.e., the ratios of the in-plane and out-ofplane displacements and the phase-shifts between these components are different. Therefore, if the vertical and horizontal components of the wave motion are available it is possible to apply signal processing permitting projection of individual modes on orthogonal axes. Contrary to the frequency-wavenumber domain filtering techniques, single-spatial point measurement is sufficient for the polarization-based method. The in-plane and out-of-plane signals can be acquired using a 3D laser scanning vibrometer. The main issue with the OPF implementation is that precise information on the investigated structure is needed, i.e., plate thickness and bulk wave velocities are required to calculate the polarization parameters that are used to set the OPF. These parameters are valid only for a narrow frequency band, therefore, the filter may require adjustment for a wave propagating over a significant distance. The attenuation characteristics differs significantly for both modes and can therefore shift the spectrum of subsequent modes. Moreover, misalignment of the laser beams and poor signal to noise ratio can significantly influence the acquired data and reduce the performance of OPF. In this paper we present a new robust directional polarization filter (DPF) for efficient separation of Lamb waves modes. The proposed method is based on the observation that the symmetrical and anti-symmetrical modes have different polarization directions. This feature is pronounced by a phase difference between the in-plane and out-of-plane motion components, which is observed for a broad range of frequency-thickness products. Therefore, no additional prior information on the modes’ polarization parameters, i.e., the amplitude ratio and phase shift between the modes, is required. Thanks to those features, the proposed DPF algorithm becomes both significantly more robust and simplified in comparison with the classical OPF. This paper is organized as follows: we start from presenting theory related to motion trajectories of Lamb waves, which is used for explaining the concept of oblique and directional polarization filters. In the next section, operation of the proposed directional polarization filtered is explained using simulated results. Subsequently, experimental results are presented that verify the simulations and demonstrate the filter performance. The final section includes conclusions and digests future works. 2. Theory This section provides a brief introduction to polarization mechanisms. The basic polarization kinds and the signal parameters related to them are explained followed by a description of the OPF. Finally, the proposed technique is presented in details based on signals simulated using a finite difference method.
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2.1. Lamb wave motion trajectory In this paper we consider an elastic isotropic homogeneous medium, with free boundaries as presented in Fig. 1. We assume here that the plate is a wave-guide for plane harmonic Lamb waves propagating in the xdirection. Following [28] we recall two characteristic equations, determining the eigenvalues of the wave number ks of the symmetric 2
2
2
ðks þ s2s Þ cothðqs d=2Þ 4ks qs ss cothðss d=2Þ ¼ 0;
ð1Þ
and the wavenumber ka of the antisymmetric Lamb modes 2
2
2
ðka þ s2a Þ tanhðqa d=2Þ 4ka qa sa cothðsa d=2Þ ¼ 0; where qs;a
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ ks;a kl ; ss;a ¼ ks;a kt ; kl and kt are velocities of longitudinal and transverse waves, respectively. The sub-
scripts a and s denote coefficients related respectively to the antisymmetric and symmetric modes. Numerical solution of Eqs. (1) and (2) leads to the dispersion curves, i.e., the frequency-wave number pairs satisfying these formulas. In the assumed case, the total Lamb waves motion remains in the x z plane. The displacements uðzÞ and wðzÞ, occurring along the xand z axes respectively, can be found as functions of the coordinate zusing the following equations [28]:
"
# chðqs zÞ 2qs ss chðss zÞ p us ðzÞ ¼ Aks eiðks xxt2Þ ; shðqs d=2Þ ðk2s þ s2s Þ shðss d=2ÞÞ
ð3Þ
"
# 2 shðqs zÞ 2k shðss zÞ 2 s eiðks xxtÞ ; shðqs d=2Þ ðks þ s2s Þ shðss d=2ÞÞ
ð4Þ
# shðqa zÞ 2qa sa shðsa zÞ p ua ðzÞ ¼ Bka eiðka xxt2Þ ; chðqa d=2Þ ðk2a þ s2a Þ chðsa d=2Þ
ð5Þ
ws ðzÞ ¼ Aqs "
" wa ðzÞ ¼ Bqa
# 2 chðqa zÞ 2k chðsa zÞ eiðka xxtÞ ; 2 a chðqa d=2Þ ðka þ s2a Þ chðsa d=2Þ
ð6Þ
where A and B are arbitrary constants, x is angular frequency and tis time. Omitting the exponential terms in Eqs. (3)–(6) allows for finding the displacement profiles over the plate thickness. For illustration, particle displacements of the S0 and A0 modes obtained for f d ¼ 1 MHz mm, are presented in Fig. 2a and b, respectively. Additional plots for other modes and f d products can be found in [1,28,29]. From Fig. 2a it can be seen that the displacements in normal direction wðzÞ have opposite signs for the upper and lower part of the plate, the motion is thus symmetrical with respect to the neutral plane at z ¼ 0. An opposite behavior can be seen in Fig. 2b, where the wðzÞ is positive within the whole plate thickness, which means that the upper and lower plate surfaces have in phase out-of-plane vibrations for antisymmetric modes. Further analysis of the profiles reveals trajectory of particle motion at the plate surface. Since the displacements are real, using the real parts of Eqs. (3)–(6) yields the following equations
^ s;a ðtÞ ¼ us;a ðz ¼ d=2Þ sinðxtÞ u ; ^ s;a ðtÞ ¼ ws;a ðz ¼ d=2Þ cosðxtÞ w
ð7Þ
where us;a ðz ¼ d=2Þ and ws;a ðz ¼ d=2Þ are the in-plane and out-of-plane displacements, respectively, evaluated for z ¼ d=2 at the upper and z ¼ d=2 at the lower surface of the plate. It can be easily observed that Eq. (7) is a formula of ellipse in a parametric form with tas a parameter. From Eq. (7) it is possible to find not only the elliptical shapes of the trajectories, presented in Figs. 3, but also the directions of particle movement, denoted in these figs. by arrows. To illustrate this relation more thoroughly, assume the S0 mode with displacement profiles presented in Fig. 2a. At the upper surface, at z ¼ d=2,
Fig. 1. Geometry of the considered problem – a wave propagates along the x-axis with a clockwise elliptical polarization.
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Fig. 2. Displacement profiles of the S0 (a) and A0 (b) Lamb modes for f d ¼ 1 MHz mm.
Fig. 3. Particle motion trajectory at z ¼ d=2 – the S0 (a) and the A0 (b) modes. Particle motion is indicated by horizontal arrows at t ¼ 0 and by vertical arrows for xt ¼ p=2.
the displacements ws < 0 and us > 0. Inserting these features to Eq. (7) facilitates tracking the trajectory over time – for t ¼ 0 ^ s ð0Þ ¼ 0. From the properties of the cosineit is evident that the sinfunction is 0 and thus the in-plane displacement u ^ s ð0Þ ¼ ws ðz ¼ d=2Þ for t ¼ 0. To find the direction of motion, note again that us ðz ¼ d=2Þ > 0, which multiplied by negative w values of sinfor small values of the argument xt, results in a clockwise direction denoted by arrows in Fig. 3a. Similar analysis can be also conducted for the trajectory of A0 mode acquired at the upper surface. From the resulting ellipses, presented in Fig. 3b, it can be seen that the A0 mode has greater out-of-plane displacement than the in-plane displacements. Moreover, on the contrary to the S0 mode, the A0 mode exhibits counterclockwise trajectory. 2.2. The oblique polarization filter Before introducing details of the proposed technique, we provide a brief theoretical overview of polarization states and applicable signal processing. Three basic polarization states are illustrated by means of a pure tone-burst signal consisting of 9 cycles of sine modulated with Hanning window, as shown in Fig. 4. In a general case, when the Uand Wmotion components (see Fig. 1) differ in phase, as presented in Fig. 4a, the resulting trajectory, and hence polarization, is elliptical, as shown in Fig. 4d. In some special cases, when the amplitudes of the Uand Wcomponents are equal and their phase shift is p=2, the resulting polarization is circular. When the particle motion exists along both directions and both vibrations are in phase, as illustrated in Fig. 4b, the polarization becomes linear, as shown in Fig. 4e. Finally, let us assume that the particle motion occurs only along one component, for instance, along a horizontal line aligned with the direction of wave propagation. The considered cases are illustrated in Fig. 4c and f. It is not surprising that the polarization presented in the Lissajous plot in Fig. 4e and f are linear.
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Fig. 4. Examples of polarized waves (a–c) and the corresponding Lissajous plots (d and e). Elliptical polarization (a) and (d). Linear polarization, with different amplitude ratio (b) and (e). Linear polarization with horizontal particle motion (c) and (f).
In our case the task of the polarization filter is to manipulate the signals of the in-plane and out-of-plane motion to bring elliptically polarized modes to linear polarization and then to project them on separate axes. Assume that two analytic signals sU ðtÞ and sW ðtÞ, representing the horizontal and vertical vibration components, respectively, are available. In the general case of elliptical polarization these signals are phase-shifted and have different amplitudes, as shown in Fig. 4a and d. Matching the components’ phase yields linear polarization. The phase-shift h can be applied to the signals sU ðtÞ and sW ðtÞ using the following operation
1
0
0 eih
sU ðtÞ
sW ðtÞ
:
ð8Þ
The phase-shift operation leads to a slanted line in the Lissajous plot. The angle of this line, as presented in Fig. 4e, depends on the ratio between the amplitudes of sH ðtÞ and sV ðtÞ. Applying rotation to these components, or in other words, modifying the signals’ amplitude, allows for projection of the vibration components onto the selected axis, as presented in Fig. 4f. Rotation of the Lissajous plot with an angle c can be described as
cosðcÞ
sinðcÞ
sinðcÞ cosðcÞ
sU ðtÞ sW ðtÞ
:
ð9Þ
Since below the so called cut-of frequency in the frequency-wavenumber product range only two Lamb modes exist, they can be separated using a set of time-shifts and rotations subsequently applied for the signals to project one mode on the U and the other on the W axis. This can be accomplished using the oblique polarization filter originally developed in [27]. The OPF procedure consists of the following steps: 1. 2. 3. 4. 5. 6.
Phase-shift the W component to obtain linear polarization of the A0 mode. Apply rotation to project the A0 mode on the horizontal axis. Phase-shift the W component to obtain linear polarization of the S0 mode. Apply rotation to obtain vertical axis as an axis of symmetry. Modify amplitudes to make the S0 and A0 modes orthogonal. Rotate the signals to generate the separated H and V components.
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The main problem in application of the OPF is obtaining the information on the modes polarization parameters that is needed to calculate the required phase-shifts and rotation angles. It can be evaluated theoretically, however, the discrepancy between the model and the real structure can prohibit proper separation. In some cases, when the two modes are timeseparated, the parameters can be easily obtained analyzing the signals in short windows [27]. However, the noise and misalignment in 3D measurements can influence the phase and amplitude of the acquired signals. Therefore, there is a need for a robust OPF technique that can perform well without a model, i.e., using features of the acquired signal only. 3. The directional polarization filter To illustrate principles of the proposed DPF, we will use the in-plane and out-of-plane signals from a simulated propagation of Lamb waves. The simulation of wave propagation was performed using local interaction simulation approach with sharp interface model (LISA/SIM) [30]. This technique features 3D analysis, therefore, all motion components could be easily resolved from a single simulation. A model of an aluminum plate with dimensions 600 600 2 mm and with a grid cell size of 0.5 mm was build. In the analysis a sine consisting of 5 cycles at 600 kHz, modulated with Hanning window was used for excitation of the plate’s central node in the out-of-plane direction. The responses – the in-plane and out-of plane motion components were captured at a set of nodes located at a line along the wave propagation. Two motion components simulated at a distance of 140 mm from the source were used to illustrate the direction of Lamb modes polarization. At this point the modes were already well separated in time and the corresponding parts of the signals were used to plot the normalized motion trajectories of the S0 and A0 modes. From Fig. 5a, where the simulation results are presented, it can be seen that the S0 mode is dominated by the in-plane motion. On the other hand, Fig. 5b shows that the motion of the A0 mode is dominated by the out-of-plane displacements. Moreover, the polarization direction of the S0 mode is clockwise as opposed to the A0 mode with the counterclockwise motion, which is in good agreement with the theoretical results presented in Fig. 3. In order to apply the OPF, described in the previous subsections, a complete information on modes polarization is required in the form of phase-shifts and amplitude ratios of the in-plane and out-of-plane modes’ components. The proposed solution, however, is based on the fact that two modes measured at the same, either upper or lower, surface have opposite polarization directions. Furthermore, to compensate the lack of information on the amplitude ratio between the modes, the instantaneous amplitudes of both components are matched. This is done by replacing the envelope of the sW ðtÞ signal (i.e., its absolute value) with the envelope of sU ðtÞ signal, which can be easily performed for analytic signals using the following operation
^sW ðtÞ ¼ sU ðtÞeiargðsW ðtÞ=sU ðtÞÞ :
ð10Þ
To illustrate the first step of the DPF algorithm the simulated in-plane and out-of plane responses are presented in Fig. 6a in the form of the time waveforms and Lissajous plots. As it can be seen in 6b, after step 1 the instantaneous amplitudes of both components are equal and both signals differ only in phase. In perfect conditions for Lamb waves the phase difference is 90 , which means that the polarization is circular. Due to some measurement errors, however, the phase shift can be slightly different, therefore in the next step of the algorithm the exact phase difference between the maximums of both components is found. Next, using Eq. (8) the phase shift is applied to component U, which brings both modes to linear polarization. From time waveforms in Fig. 6c it can be seen that after step 2 the phase difference for the first mode signal is 0 while for the second it is p. Since the in-plane and out-of-plane amplitudes are equal it means that 45 rotation is required as the last
Fig. 5. Elliptical motion trajectories in the nodes at the upper surface of the simulated model: the S0 mode (a), and the A0 mode (b).
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Fig. 6. Directional polarization filtration of the simulated data. Time waveforms and the corresponding Lissajous plots: raw data (a); signals after instantaneous amplitude agreement – step 1 (b); phase-shifted signals – step 2 (c), and the signals after 45 rotation – step 3 (d).
step (step 3) to project these signals on both axes, as presented in Fig. 6d. Note that in the effect of 45° rotation, the signal pffiffiffi amplitudes shown in Fig. 6d became 2 times greater than the amplitudes of motion components shown in Fig. 6c. Summarizing, the DPF filtration can be performed using the following steps: 1. Replace the envelope of the signal W with envelope of the signal U (this makes polarization of both modes circular). 2. Find phase shifts between the components at maximum of the out-of-plane component. Apply phase shift to align the dominant mode (this makes polarization of both modes linear). The components of A0 mode are in phase and the components of the S0 mode are out of phase. 3. Apply 45 rotation to project the modes on the U and W axes. All simulated in-plane and out-of-plane responses are presented in the form of B-scan images in Fig. 7a and b, respectively. The subsequent A-scans were normalized by their maximum amplitude. The modes can be identified based on their angles (velocity) in the B-scan images, for instance, both fast S0 and slower A0 incident modes can be seen in the figures; reflections from the plate edge are pronounced also. For illustration of the DPF operation, the algorithm was applied to the time-signals composing the B-scans presented in Fig. 7a and b and the filtering results are presented in Fig. 7c and d, respectively. After processing a clear incident S0 mode can be seen in Fig. 7c, while the incident A0 mode has been suppressed. Additionally, a reflected A0 mode can be seen in this Bscan. It can be explained by the fact that the polarization direction is related to the direction of wave propagation. Therefore, the A0 and S0 modes traveling at opposite directions will appear simultaneously at the DPF output. Similarly to the previous case, a clear incident A0 mode can be observed in Fig. 7d. Also here the reflected S0 mode can be seen.
4. Experimental results The experiments were performed to validate the simulated results using setup schematically illustrated in Fig. 8. A 4 mm thick aluminum plate of dimensions of 1 1 m was instrumented using a multilayer PZT transducer of dimensions of 2 2 2 mm, NAC2011 form NoliacÒ, Denmark, coupled to the plate using cyjanoacrylate adhesive. The transducer was placed 100 mm from the edge of the plate. Ton-burst signals consisting of 5 cycles of a sine at 300 kHz, modulated with Hanning window were generated and amplified using PAQ-G device from EC-SystemsÒ, Poland, which means that the experiments were performed for f d ¼ 1:2 MHz mm. The measurements were performed using a 3D laser scanning vibrometer, PSV-400-3D, from PolytecÒ, Germany, which allowed to capture full 3D vibration velocity of the signals. Antiglare tape was used to improve light scattering in various directions and hence to improve measurement conditions of the 3D components. The measurements were conducted along a line starting at 50 mm from the source, in 150 points spaced with a distance of 1 mm. In the first step, two responses were selected from a point where the modes were separated in time to confirm the predicted polarization direction. The normalized motion trajectories of the S0 and A0 modes are presented in Fig. 9a and b, respectively. As expected, the S0 mode is dominated by the in-plane components whereas for the A0 mode the out-ofplane motion component is higher than the in-plane one. These modes differ also in the polarization direction, i.e., the S0
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Fig. 7. Simulated Bscan images: the in-plane (a), and the out-of-plane displacements (b). The same B-scans after the DPF filtration: the S0 mode (c), and the A0 mode (d).
Fig. 8. Schematic of the experimental setup.
has clockwise and the A0 has counterclockwise rotation direction. Thus, these results are in good agreement with the theoretical and simulated profiles presented in Figs. 3 and 5. The complete set of the in-plane and out-of-plane vibration data is presented in the form of B-scans in Fig. 10a and b. Note that each A-scan in the B-scans was normalized by its maximum amplitude to compensate for the attenuation. Multiple modes can be seen in both B-scan images. The signals were processed subsequently with the DPF and the resulting filtered S0 and A0 modes can be seen in Fig. 10c and d, respectively. In Fig. 10c it can be seen that the incident A0 mode
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Fig. 9. Particle motion trajectories: the S0 mode (a) and the A0 mode (b).
Fig. 10. B-scan images of the in-plane (a) and the out-of-plane displacements (b); filtration results of the S0 (c) and the A0 (d) modes. Yellow lines at (a) and (b) indicate spatial points where the waveforms presented in Fig. 11 were extracted.
was suppressed and only the incident S0 mode remained. Additionally, the reflected S0 mode can be seen in the image. The PZT transmitter in the experimental setup was placed close to the plate’s edge and therefore, the reflected S0 mode is delayed and propagates in the same direction as the incident mode. Analogically, in Fig. 10d the incident S0 mode has been attenuated by the DPF while the two A0 modes, the incident and reflected are apparent. The normalization performed on the B-scans, permitted displaying the images using full dynamic range of the color scales, however it amplified also noise. Moreover it brought up points incorrectly measured due to laser sensing issues. Signals from these points were not correctly resolved as it can be seen for instance in Fig. 10c for distance below 50 mm, where a discontinuity in A0 mode can be observed.
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Fig. 11. Time waveforms and corresponding Lissajous plots of the signals captured at a distance of 30 mm (a,c) and 120 mm (b,d). Raw data (a,b) and the corresponding signals processed using DPF (c,d).
To provide a more profound explanation of the proposed technique two signals corresponding to the dashed lines in Fig. 10 are presented in Fig. 11a and b. The first example, shown in Fig. 11a, presents signals acquired in a proximity of the source where the modes were not separated yet. Since the out-of-plane component was slightly higher than the in-plane vibrations these signals may suggest that a single A0 mode was recorded. However, the time plot presented in Fig. 11c reveals that after the DPF processing both modes appear, the faster S0 and slower A0 . In the second example, presented in Fig. 11b, the modes were already well separated in the time domain. It should be possible to separate the modes using simple gating based on the ratio of the in-plane and out-of-plane vibrations, however, the amplitude ratio for A0 mode does not agree with the theoretical predictions – the in-plane motion is bigger than out-ofplane. This fact can be explained mainly by difficulties related to the 3D measurements performed using the vibrometer used in the experiments. Precise calibration of the vibrometer is performed only in few points, therefore, misalignment of laser beams may occur during scanning. Moreover, the measured ratio can be affected by features of the anti-glare tape. In the considered case the classical OPF would fail due to the inconsistent polarization parameters. The DPF, however, can work without information on the amplitude ratio of both components and, as it can be seen from Fig. 11d, the modes have been correctly projected on the respective axes. 5. Conclusions and future work A new processing technique for the separation of Lamb wave (LW) modes using signals captured by a laser vibrometer was presented in this paper. The proposed directional polarization filter (DPF) is a robust version of the known from geophysics oblique polarization filter (OPF) using different polarization directions of the LW modes. We demonstrated both on simulated and experimental signals that contrary to the OPF, the DPF can operate without full information on polarization parameters of the processed modes, i.e., the amplitude ratio and phase shift between the inplane and out-of-plane components. Therefore, the method can be used without a priori knowledge of material properties and thickness of the inspected plates. This can be a significant practical advantage over the existing techniques since these properties are often unknown prior the non-destructive evaluation. The DPF performed well, even for distorted experimental signals, thus it is expected to operate correctly in the case of slight signal distortion resulting from the presence of noise or wave attenuation. On the other hand, the distorted experimental results indicate that additional improvements are required concerning the measurement system to increase accuracy of the acquired data and to enable practical applications of the method. Note, however that here we performed the scanning to obtain a set of signals that were processed independently. During the automatic scanning it is difficult to ensure good light reflection conditions over the whole area. The expected practical application would involve however only few measurements taken over a given area. These points should be selected to ensure good wave sensing using the interferometer. We also showed using numerical data that the filtration results are related to the direction of wave propagation, i.e., if the DPF is set to suppress the incident S0 mode it will also bring up the A0 mode, which propagates in the opposite direction. To our best knowledge this is the first publication on the method that uses direction of polarization for SHM applications, therefore this article was limited to a comprehensive theoretical, numerical and experimental illustration of the proposed technique. Our future work will be concerned with an extension of the proposed method to enable simultaneous processing
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of all 3D vibration components, which is expected to provide information on the wave’s direction of arrival; another direction to explore is the issue of broad-band signal processing. These steps are essential to achieve the ultimate goal, that is damage detection and localization using a 3D laser sensing Lamb wave propagation. Acknowledgments The first author would like to acknowledge Foundation for Polish Science for ”START”– stipend for young talented researchers. References [1] J.L. Rose, Ultrasonic Waves in Solid Media, Cambridge University Press, 2004. [2] Z. Su, L. Ye, Y. Lu, Guided lamb waves for identification of damage in composite structures: a review, J. Sound Vib. 295 (3–5) (2006) 753–780, http://dx. doi.org/10.1016/j.jsv.2006.01.020.. [3] V. Giurgiutiu, Structural Health Monitoring With Piezoelectric Wafer Active Sensors, Academic Press, New York, 2008. [4] L. Ambrozinski, B. Piwakowski, T. Stepinski, T. Uhl, Evaluation of dispersion characteristics of multimodal guided waves using slant stack transform, NDT&E 68 (2014) 88–97, http://dx.doi.org/10.1016/j.ndteint.2014.08.006. [5] D. Alleyne, P. Cawley, The interaction of lamb waves with defects, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (3) (1992) 381–397. [6] M. Castaings, P. Cawley, The generation, propagation, and detection of Lamb waves in plates using air-coupled ultrasonic transducers, J. Acoust. Soc. Am. 100 (5) (1996) 3070–3077, http://dx.doi.org/10.1121/1.417193. . [7] D. Chimenti, Review of air-coupled ultrasonic materials characterization, Ultrasonics 54 (7) (2014) 1804–1816, http://dx.doi.org/10.1016/j. ultras.2014.02.006. . [8] P.-Y. Le Bas, M. Remillieux, L. Pieczonka, J. Ten Cate, B. Anderson, T. Ulrich, Damage imaging in a laminated composite plate using an air-coupled time reversal mirror, Appl. Phys. Lett. 107 (18). doi:http://dx.doi.org/10.1063/1.4935210. . [9] M. Manka, M. Rosiek, A. Martowicz, T. Stepinski, T. Uhl, PZT based tunable interdigital transducer for Lamb waves based NDT and SHM, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2015.12.013. . [10] B. Xu, V. Giurgiutiu, Single mode tuning effects on Lamb wave time reversal with piezoelectric wafer active sensors for structural health monitoring, J. Nondestr. Eval. 26 (2007) 123–134, http://dx.doi.org/10.1007/s10921-007-0027-8. [11] M. Niethammer, L.J. Jacobs, J. Qu, J. Jarzynski, Time-frequency representations of lamb waves, J. Acoust. Soc. Am. 109 (5) (2001) 1841–1847, http://dx. doi.org/10.1121/1.1357813. [12] C. Valle, J.W. Littles Jr., Flaw localization using the reassigned spectrogram on laser-generated and detected lamb modes, Ultrasonics 39 (8) (2002) 535–542, http://dx.doi.org/10.1016/S0041-624X(02)00249-4. . [13] Z. Tian, L. Yu, Single mode Lamb wave phased array beamforming with hybrid PZT-SLDV sensing, in: Proc. SPIE 9061, 2014. [14] D. Alleyne, P. Cawley, A two-dimensional Fourier transform method for the measurement of propagating multimode signals, J. Acoust. Soc. Am. 89 (3) (1991) 1159–1168, http://dx.doi.org/10.1121/1.400530. . [15] T. Hayashi, K. Kawashima, Single mode extraction from multiple modes of lamb wave and its application to defect detection, JSME Int. J. Ser. A Solid Mech. Mater. Eng. 46 (4) (2003) 620–626, http://dx.doi.org/10.1299/jsmea.46.620. [16] M. Ruzzene, Frequency wavenumber domain filtering for improved damage visualization, Smart Mater. Struct. 16 (6) (2007) 2116–2119. . [17] T.E. Michaels, J.E. Michaels, M. Ruzzene, Frequency-wavenumber domain analysis of guided wavefields, Ultrasonics 51 (4) (2011) 452–466, http://dx. doi.org/10.1016/j.ultras.2010.11.011. . [18] W.J. Staszewski, B.C. Lee, L. Mallet, F. Scarpa, Structural health monitoring using scanning laser vibrometry: I. Lamb wave sensing, Smart Mater. Struct. 13 (2) (2004) 251. . [19] J. Ayers, N. Apetre, M. Ruzzene, K. Sabra, Measurement of lamb wave polarization using a one-dimensional scanning laser vibrometer (L), J. Acoust. Soc. Am. 129 (2) (2011) 585–588, http://dx.doi.org/10.1121/1.3523429. . [20] L. Yu, Z. Tian, Lamb wave structural health monitoring using a hybrid PZT-laser vibrometer approach, Struct. Health Monit. 12 (5–6) (2013) 469–483, http://dx.doi.org/10.1177/1475921713501108, arXiv:http://shm.sagepub.com/content/12/5-6/469.full.pdf+html URL . [21] M. Radzieñski, L. Doliñski, M. Krawczuk, M. Palacz, Damage localisation in a stiffened plate structure using a propagating wave, Mech. Syst. Signal Process. 39 (1–2) (2013) 388–395, http://dx.doi.org/10.1016/j.ymssp.2013.02.014. . [22] L. Pieczonka, L. Ambrozinski, W.J. Staszewski, D. Barnoncel, P. Peres, Damage detection in composite panels based on mode-converted lamb waves sensed using 3d laser scanning vibrometer, Opt. Lasers Eng. (2017), http://dx.doi.org/10.1016/j.optlaseng.2016.12.017. . [23] M. Ratassepp, A. Klauson, F. Chati, F. Léon, D. Décultot, G. Maze, M. Fritzsche, Application of orthogonality-relation for the separation of lamb modes at a plate edge: numerical and experimental predictions, Ultrasonics 57 (2015) 90–95, http://dx.doi.org/10.1016/j.ultras.2014.10.022. [24] N. Rajic, C. Rosalie, P. Norman, C. Davis, Determination of the in-plane components of motion in a lamb wave from single-axis laser vibrometry, J. Acoust. Soc. Am. 135 (6) (2014) 3446–3454, http://dx.doi.org/10.1121/1.4876181. . [25] L. Ambrozinski, T. Stepinski, T. Uhl, Separation of Lamb waves modes using polarization filter of 3d laser measured signals, Proc. SPIE 9437, Structural Health Monitoring and Inspection of Advanced Materials, Aerospace, and Civil Infrastructure, Vol. 9437, 2015, http://dx.doi.org/10.1117/12.2084047, pp. 94371E–94371E–8. [26] J. Mars, F. Glangeaud, J. Vanpe, J. Boelle, Wave separation by an oblique polarization filter, in: Physics in Signal and Image Processing, 1999. [27] A. Roueff, J. Chanussot, J. Mars, Estimation of polarization parameters using time-frequency representations and its application to waves separation, Signal Process. 86 (2006) 3714–3731. [28] I.A. Viktorov, Rayleigh and Lamb Waves Physical Theory and applications, Plenum Press, New York, 1967. [29] P.F. Pai, H. Deng, M.J. Sundaresan, Time-frequency characterization of lamb waves for material evaluation and damage inspection of plates, Mech. Syst. Signal Process. 62–63 (2015) 183–206, http://dx.doi.org/10.1016/j.ymssp.2015.03.011. . [30] P. Packo, T. Bielak, A.B. Spencer, W.J. Staszewski, T. Uhl, K. Worden, Lamb wave propagation modelling and simulation using parallel processing architecture and graphical cards, Smart Mater. Struct. 21 (7) (2012) 075001. .
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Robust polarization filter for separation of Lamb wave modes acquired using a 3D laser vibrometer Łukasz Ambrozin´ski ⇑, Tadeusz Stepinski Department of Robotics and Mechatronics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 12 April 2016 Received in revised form 11 January 2017 Accepted 4 February 2017
Keywords: Lamb waves Polarization Modes separation 3D laser scanning vibrometer
a b s t r a c t Interpretation of signals related to Lamb waves propagation and scattering can rise serious difficulties due to the multi-modal nature of these waves. Different modes propagating with different velocities can be mixed up and hinder extraction of damage reflected components. As a feasible solution to this problem we propose a technique for separation of the propagating modes using a new type of polarization filter. The proposed directional polarization filter (DPF) can be applied if two components of particle movement, the in-plane and the out-of-plane, are available, for instance, from the measurement performed using laser vibrometer. The DPF is robust in the sense that it does not need a complete amplitude information of both components. Operation principle of the DPF is presented and illustrated by the simulated results in the form of B-scans obtained for an aluminum plate. The simulated results are verified by the experimental data obtained by processing the signals captured using a 3D laser vibrometer. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The main difficulty hindering applications of Lamb waves for non-destructive evaluation (NDE) of planar structures is their multi-modal and dispersive nature. An infinite number of modes can propagate in a plate with velocities depending on the frequency-plate thickness product [1–4]. Although, it is possible to limit the low frequency band where only two fundamental modes can exist, the responses can be complex even for an intact structure. Therefore, special techniques have been developed to obtain pure single-mode excitation or to process the acquired data to separate the wave modes. A common approach, that can be used for selective generation and reception of Lamb waves is based on the Snell’s law of refraction. Refraction of ultrasonic waves at an interface between two different media enables selective tuning of wavelength. Transducer’s angle of incidence can be adjusted to obtain the desired phase velocity corresponding to the selected Lamb mode in the investigated medium. This principle can be realized by means of contact transducers with perspex variable-angle-wedge [5] as well as immersion or air-coupled setups [6–8]. Another solution, applied mostly in structural health monitoring (SHM) applications is the use of interdigital transducers (IDT) with comb-shaped electrodes spaced with pitch corresponding to the desired wavelength [9]. Similarly, theoretical solution proposed by Xu et al. [10] permits tuning the length of piezoelectric wafer active sensor (PWAS) to the wave mode
⇑ Corresponding author. E-mail addresses: [email protected] (Ł. Ambrozin´ski), [email protected] (T. Stepinski). http://dx.doi.org/10.1016/j.ymssp.2017.02.002 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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that has be to excited. The main drawback of those techniques is that the transducers have to be perfectly matched to the application by taking into account thickness of the inspected plate and the desired frequency band. Since exciting and sensing a single wave mode is a complicated task in practical applications a variety of signal postprocessing techniques for wave mode identification and separation has been proposed. One of the most straightforward approaches relies on the time-frequency representation of the acquired ultrasonic signals. Due to the dispersion different spectral components of an excited burst propagate with different velocities. It results in deformed waveforms of the received signals that can be observed as elongated long-tailed tone-burst. Although the time-frequency representation allows to reveal the dispersion-related delays it requires a broadband excitation with sufficient energy within the whole frequency band where the dispersion characteristics are to be estimated. Various time-frequency representations, including reassigned spectrogram, reassigned scalogram, smoothed Wigner–Ville distribution as well as Hilbert spectrum, have been used for the identification of Lamb modes [11]. Moreover, it was shown that selected features of the time-frequency representations can be used for damage detection [12]. Since the frequency-wavenumber space enables separation of different Lamb modes, two-dimensional Fourier transform (2DFT) is one of the most commonly used techniques to identify and quantify the existent modes [13–17]. However, a relatively dense network of sensing points is required to apply this technique and therefore laser interferometers are often used to collect data with sufficient spatial sampling [18–22]. These measurements are, however, sensitive to the surface quality and often reflective sprays or foils have to be used. Availability of laser vibrometers has opened completely new possibilities for sensing and processing Lamb waves; new ideas could be realized by means of a contactless measurement of vibrations at different points at the investigated surface or a free plate edge. A specific application of laser vibrometer for mode separation if the plate surface is inaccessible for testing was presented in [23]. The proposed orthogonality-based method is based on measurements by means of a 3D vibrometer (an instrument that combines 3 different laser beams for simultaneous measurement of both vibration components) of the wave field scattered from the plate’s free edge at a dense grid of points. Thus, the method requires access to a free edge, which makes it impractical in many applications. As illustrated in [19], elliptical particle motion trajectory resulting from Lamb wave propagation can be acquired by means of a single point laser interferometer, however, the measurements should be repeated using a laser beam incident at different angles with respect to the surface. A different approach was presented in [24] where the in-plane vibration component was estimated based on the out-of-plane motion. To apply this method, however, both plate’s thickness and bulk wave velocities have to be known. More recently we showed how the Lamb waves modes can be separated based on their polarization parameters [25] using oblique polarization filter (OPF), originally developed for seismic waves processing [26,27]. Both S0 and A0 Lamb modes exhibit elliptical polarization, however, their polarization parameters, i.e., the ratios of the in-plane and out-ofplane displacements and the phase-shifts between these components are different. Therefore, if the vertical and horizontal components of the wave motion are available it is possible to apply signal processing permitting projection of individual modes on orthogonal axes. Contrary to the frequency-wavenumber domain filtering techniques, single-spatial point measurement is sufficient for the polarization-based method. The in-plane and out-of-plane signals can be acquired using a 3D laser scanning vibrometer. The main issue with the OPF implementation is that precise information on the investigated structure is needed, i.e., plate thickness and bulk wave velocities are required to calculate the polarization parameters that are used to set the OPF. These parameters are valid only for a narrow frequency band, therefore, the filter may require adjustment for a wave propagating over a significant distance. The attenuation characteristics differs significantly for both modes and can therefore shift the spectrum of subsequent modes. Moreover, misalignment of the laser beams and poor signal to noise ratio can significantly influence the acquired data and reduce the performance of OPF. In this paper we present a new robust directional polarization filter (DPF) for efficient separation of Lamb waves modes. The proposed method is based on the observation that the symmetrical and anti-symmetrical modes have different polarization directions. This feature is pronounced by a phase difference between the in-plane and out-of-plane motion components, which is observed for a broad range of frequency-thickness products. Therefore, no additional prior information on the modes’ polarization parameters, i.e., the amplitude ratio and phase shift between the modes, is required. Thanks to those features, the proposed DPF algorithm becomes both significantly more robust and simplified in comparison with the classical OPF. This paper is organized as follows: we start from presenting theory related to motion trajectories of Lamb waves, which is used for explaining the concept of oblique and directional polarization filters. In the next section, operation of the proposed directional polarization filtered is explained using simulated results. Subsequently, experimental results are presented that verify the simulations and demonstrate the filter performance. The final section includes conclusions and digests future works. 2. Theory This section provides a brief introduction to polarization mechanisms. The basic polarization kinds and the signal parameters related to them are explained followed by a description of the OPF. Finally, the proposed technique is presented in details based on signals simulated using a finite difference method.
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2.1. Lamb wave motion trajectory In this paper we consider an elastic isotropic homogeneous medium, with free boundaries as presented in Fig. 1. We assume here that the plate is a wave-guide for plane harmonic Lamb waves propagating in the xdirection. Following [28] we recall two characteristic equations, determining the eigenvalues of the wave number ks of the symmetric 2
2
2
ðks þ s2s Þ cothðqs d=2Þ 4ks qs ss cothðss d=2Þ ¼ 0;
ð1Þ
and the wavenumber ka of the antisymmetric Lamb modes 2
2
2
ðka þ s2a Þ tanhðqa d=2Þ 4ka qa sa cothðsa d=2Þ ¼ 0; where qs;a
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ ks;a kl ; ss;a ¼ ks;a kt ; kl and kt are velocities of longitudinal and transverse waves, respectively. The sub-
scripts a and s denote coefficients related respectively to the antisymmetric and symmetric modes. Numerical solution of Eqs. (1) and (2) leads to the dispersion curves, i.e., the frequency-wave number pairs satisfying these formulas. In the assumed case, the total Lamb waves motion remains in the x z plane. The displacements uðzÞ and wðzÞ, occurring along the xand z axes respectively, can be found as functions of the coordinate zusing the following equations [28]:
"
# chðqs zÞ 2qs ss chðss zÞ p us ðzÞ ¼ Aks eiðks xxt2Þ ; shðqs d=2Þ ðk2s þ s2s Þ shðss d=2ÞÞ
ð3Þ
"
# 2 shðqs zÞ 2k shðss zÞ 2 s eiðks xxtÞ ; shðqs d=2Þ ðks þ s2s Þ shðss d=2ÞÞ
ð4Þ
# shðqa zÞ 2qa sa shðsa zÞ p ua ðzÞ ¼ Bka eiðka xxt2Þ ; chðqa d=2Þ ðk2a þ s2a Þ chðsa d=2Þ
ð5Þ
ws ðzÞ ¼ Aqs "
" wa ðzÞ ¼ Bqa
# 2 chðqa zÞ 2k chðsa zÞ eiðka xxtÞ ; 2 a chðqa d=2Þ ðka þ s2a Þ chðsa d=2Þ
ð6Þ
where A and B are arbitrary constants, x is angular frequency and tis time. Omitting the exponential terms in Eqs. (3)–(6) allows for finding the displacement profiles over the plate thickness. For illustration, particle displacements of the S0 and A0 modes obtained for f d ¼ 1 MHz mm, are presented in Fig. 2a and b, respectively. Additional plots for other modes and f d products can be found in [1,28,29]. From Fig. 2a it can be seen that the displacements in normal direction wðzÞ have opposite signs for the upper and lower part of the plate, the motion is thus symmetrical with respect to the neutral plane at z ¼ 0. An opposite behavior can be seen in Fig. 2b, where the wðzÞ is positive within the whole plate thickness, which means that the upper and lower plate surfaces have in phase out-of-plane vibrations for antisymmetric modes. Further analysis of the profiles reveals trajectory of particle motion at the plate surface. Since the displacements are real, using the real parts of Eqs. (3)–(6) yields the following equations
^ s;a ðtÞ ¼ us;a ðz ¼ d=2Þ sinðxtÞ u ; ^ s;a ðtÞ ¼ ws;a ðz ¼ d=2Þ cosðxtÞ w
ð7Þ
where us;a ðz ¼ d=2Þ and ws;a ðz ¼ d=2Þ are the in-plane and out-of-plane displacements, respectively, evaluated for z ¼ d=2 at the upper and z ¼ d=2 at the lower surface of the plate. It can be easily observed that Eq. (7) is a formula of ellipse in a parametric form with tas a parameter. From Eq. (7) it is possible to find not only the elliptical shapes of the trajectories, presented in Figs. 3, but also the directions of particle movement, denoted in these figs. by arrows. To illustrate this relation more thoroughly, assume the S0 mode with displacement profiles presented in Fig. 2a. At the upper surface, at z ¼ d=2,
Fig. 1. Geometry of the considered problem – a wave propagates along the x-axis with a clockwise elliptical polarization.
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Fig. 2. Displacement profiles of the S0 (a) and A0 (b) Lamb modes for f d ¼ 1 MHz mm.
Fig. 3. Particle motion trajectory at z ¼ d=2 – the S0 (a) and the A0 (b) modes. Particle motion is indicated by horizontal arrows at t ¼ 0 and by vertical arrows for xt ¼ p=2.
the displacements ws < 0 and us > 0. Inserting these features to Eq. (7) facilitates tracking the trajectory over time – for t ¼ 0 ^ s ð0Þ ¼ 0. From the properties of the cosineit is evident that the sinfunction is 0 and thus the in-plane displacement u ^ s ð0Þ ¼ ws ðz ¼ d=2Þ for t ¼ 0. To find the direction of motion, note again that us ðz ¼ d=2Þ > 0, which multiplied by negative w values of sinfor small values of the argument xt, results in a clockwise direction denoted by arrows in Fig. 3a. Similar analysis can be also conducted for the trajectory of A0 mode acquired at the upper surface. From the resulting ellipses, presented in Fig. 3b, it can be seen that the A0 mode has greater out-of-plane displacement than the in-plane displacements. Moreover, on the contrary to the S0 mode, the A0 mode exhibits counterclockwise trajectory. 2.2. The oblique polarization filter Before introducing details of the proposed technique, we provide a brief theoretical overview of polarization states and applicable signal processing. Three basic polarization states are illustrated by means of a pure tone-burst signal consisting of 9 cycles of sine modulated with Hanning window, as shown in Fig. 4. In a general case, when the Uand Wmotion components (see Fig. 1) differ in phase, as presented in Fig. 4a, the resulting trajectory, and hence polarization, is elliptical, as shown in Fig. 4d. In some special cases, when the amplitudes of the Uand Wcomponents are equal and their phase shift is p=2, the resulting polarization is circular. When the particle motion exists along both directions and both vibrations are in phase, as illustrated in Fig. 4b, the polarization becomes linear, as shown in Fig. 4e. Finally, let us assume that the particle motion occurs only along one component, for instance, along a horizontal line aligned with the direction of wave propagation. The considered cases are illustrated in Fig. 4c and f. It is not surprising that the polarization presented in the Lissajous plot in Fig. 4e and f are linear.
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Fig. 4. Examples of polarized waves (a–c) and the corresponding Lissajous plots (d and e). Elliptical polarization (a) and (d). Linear polarization, with different amplitude ratio (b) and (e). Linear polarization with horizontal particle motion (c) and (f).
In our case the task of the polarization filter is to manipulate the signals of the in-plane and out-of-plane motion to bring elliptically polarized modes to linear polarization and then to project them on separate axes. Assume that two analytic signals sU ðtÞ and sW ðtÞ, representing the horizontal and vertical vibration components, respectively, are available. In the general case of elliptical polarization these signals are phase-shifted and have different amplitudes, as shown in Fig. 4a and d. Matching the components’ phase yields linear polarization. The phase-shift h can be applied to the signals sU ðtÞ and sW ðtÞ using the following operation
1
0
0 eih
sU ðtÞ
sW ðtÞ
:
ð8Þ
The phase-shift operation leads to a slanted line in the Lissajous plot. The angle of this line, as presented in Fig. 4e, depends on the ratio between the amplitudes of sH ðtÞ and sV ðtÞ. Applying rotation to these components, or in other words, modifying the signals’ amplitude, allows for projection of the vibration components onto the selected axis, as presented in Fig. 4f. Rotation of the Lissajous plot with an angle c can be described as
cosðcÞ
sinðcÞ
sinðcÞ cosðcÞ
sU ðtÞ sW ðtÞ
:
ð9Þ
Since below the so called cut-of frequency in the frequency-wavenumber product range only two Lamb modes exist, they can be separated using a set of time-shifts and rotations subsequently applied for the signals to project one mode on the U and the other on the W axis. This can be accomplished using the oblique polarization filter originally developed in [27]. The OPF procedure consists of the following steps: 1. 2. 3. 4. 5. 6.
Phase-shift the W component to obtain linear polarization of the A0 mode. Apply rotation to project the A0 mode on the horizontal axis. Phase-shift the W component to obtain linear polarization of the S0 mode. Apply rotation to obtain vertical axis as an axis of symmetry. Modify amplitudes to make the S0 and A0 modes orthogonal. Rotate the signals to generate the separated H and V components.
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The main problem in application of the OPF is obtaining the information on the modes polarization parameters that is needed to calculate the required phase-shifts and rotation angles. It can be evaluated theoretically, however, the discrepancy between the model and the real structure can prohibit proper separation. In some cases, when the two modes are timeseparated, the parameters can be easily obtained analyzing the signals in short windows [27]. However, the noise and misalignment in 3D measurements can influence the phase and amplitude of the acquired signals. Therefore, there is a need for a robust OPF technique that can perform well without a model, i.e., using features of the acquired signal only. 3. The directional polarization filter To illustrate principles of the proposed DPF, we will use the in-plane and out-of-plane signals from a simulated propagation of Lamb waves. The simulation of wave propagation was performed using local interaction simulation approach with sharp interface model (LISA/SIM) [30]. This technique features 3D analysis, therefore, all motion components could be easily resolved from a single simulation. A model of an aluminum plate with dimensions 600 600 2 mm and with a grid cell size of 0.5 mm was build. In the analysis a sine consisting of 5 cycles at 600 kHz, modulated with Hanning window was used for excitation of the plate’s central node in the out-of-plane direction. The responses – the in-plane and out-of plane motion components were captured at a set of nodes located at a line along the wave propagation. Two motion components simulated at a distance of 140 mm from the source were used to illustrate the direction of Lamb modes polarization. At this point the modes were already well separated in time and the corresponding parts of the signals were used to plot the normalized motion trajectories of the S0 and A0 modes. From Fig. 5a, where the simulation results are presented, it can be seen that the S0 mode is dominated by the in-plane motion. On the other hand, Fig. 5b shows that the motion of the A0 mode is dominated by the out-of-plane displacements. Moreover, the polarization direction of the S0 mode is clockwise as opposed to the A0 mode with the counterclockwise motion, which is in good agreement with the theoretical results presented in Fig. 3. In order to apply the OPF, described in the previous subsections, a complete information on modes polarization is required in the form of phase-shifts and amplitude ratios of the in-plane and out-of-plane modes’ components. The proposed solution, however, is based on the fact that two modes measured at the same, either upper or lower, surface have opposite polarization directions. Furthermore, to compensate the lack of information on the amplitude ratio between the modes, the instantaneous amplitudes of both components are matched. This is done by replacing the envelope of the sW ðtÞ signal (i.e., its absolute value) with the envelope of sU ðtÞ signal, which can be easily performed for analytic signals using the following operation
^sW ðtÞ ¼ sU ðtÞeiargðsW ðtÞ=sU ðtÞÞ :
ð10Þ
To illustrate the first step of the DPF algorithm the simulated in-plane and out-of plane responses are presented in Fig. 6a in the form of the time waveforms and Lissajous plots. As it can be seen in 6b, after step 1 the instantaneous amplitudes of both components are equal and both signals differ only in phase. In perfect conditions for Lamb waves the phase difference is 90 , which means that the polarization is circular. Due to some measurement errors, however, the phase shift can be slightly different, therefore in the next step of the algorithm the exact phase difference between the maximums of both components is found. Next, using Eq. (8) the phase shift is applied to component U, which brings both modes to linear polarization. From time waveforms in Fig. 6c it can be seen that after step 2 the phase difference for the first mode signal is 0 while for the second it is p. Since the in-plane and out-of-plane amplitudes are equal it means that 45 rotation is required as the last
Fig. 5. Elliptical motion trajectories in the nodes at the upper surface of the simulated model: the S0 mode (a), and the A0 mode (b).
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Fig. 6. Directional polarization filtration of the simulated data. Time waveforms and the corresponding Lissajous plots: raw data (a); signals after instantaneous amplitude agreement – step 1 (b); phase-shifted signals – step 2 (c), and the signals after 45 rotation – step 3 (d).
step (step 3) to project these signals on both axes, as presented in Fig. 6d. Note that in the effect of 45° rotation, the signal pffiffiffi amplitudes shown in Fig. 6d became 2 times greater than the amplitudes of motion components shown in Fig. 6c. Summarizing, the DPF filtration can be performed using the following steps: 1. Replace the envelope of the signal W with envelope of the signal U (this makes polarization of both modes circular). 2. Find phase shifts between the components at maximum of the out-of-plane component. Apply phase shift to align the dominant mode (this makes polarization of both modes linear). The components of A0 mode are in phase and the components of the S0 mode are out of phase. 3. Apply 45 rotation to project the modes on the U and W axes. All simulated in-plane and out-of-plane responses are presented in the form of B-scan images in Fig. 7a and b, respectively. The subsequent A-scans were normalized by their maximum amplitude. The modes can be identified based on their angles (velocity) in the B-scan images, for instance, both fast S0 and slower A0 incident modes can be seen in the figures; reflections from the plate edge are pronounced also. For illustration of the DPF operation, the algorithm was applied to the time-signals composing the B-scans presented in Fig. 7a and b and the filtering results are presented in Fig. 7c and d, respectively. After processing a clear incident S0 mode can be seen in Fig. 7c, while the incident A0 mode has been suppressed. Additionally, a reflected A0 mode can be seen in this Bscan. It can be explained by the fact that the polarization direction is related to the direction of wave propagation. Therefore, the A0 and S0 modes traveling at opposite directions will appear simultaneously at the DPF output. Similarly to the previous case, a clear incident A0 mode can be observed in Fig. 7d. Also here the reflected S0 mode can be seen.
4. Experimental results The experiments were performed to validate the simulated results using setup schematically illustrated in Fig. 8. A 4 mm thick aluminum plate of dimensions of 1 1 m was instrumented using a multilayer PZT transducer of dimensions of 2 2 2 mm, NAC2011 form NoliacÒ, Denmark, coupled to the plate using cyjanoacrylate adhesive. The transducer was placed 100 mm from the edge of the plate. Ton-burst signals consisting of 5 cycles of a sine at 300 kHz, modulated with Hanning window were generated and amplified using PAQ-G device from EC-SystemsÒ, Poland, which means that the experiments were performed for f d ¼ 1:2 MHz mm. The measurements were performed using a 3D laser scanning vibrometer, PSV-400-3D, from PolytecÒ, Germany, which allowed to capture full 3D vibration velocity of the signals. Antiglare tape was used to improve light scattering in various directions and hence to improve measurement conditions of the 3D components. The measurements were conducted along a line starting at 50 mm from the source, in 150 points spaced with a distance of 1 mm. In the first step, two responses were selected from a point where the modes were separated in time to confirm the predicted polarization direction. The normalized motion trajectories of the S0 and A0 modes are presented in Fig. 9a and b, respectively. As expected, the S0 mode is dominated by the in-plane components whereas for the A0 mode the out-ofplane motion component is higher than the in-plane one. These modes differ also in the polarization direction, i.e., the S0
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Fig. 7. Simulated Bscan images: the in-plane (a), and the out-of-plane displacements (b). The same B-scans after the DPF filtration: the S0 mode (c), and the A0 mode (d).
Fig. 8. Schematic of the experimental setup.
has clockwise and the A0 has counterclockwise rotation direction. Thus, these results are in good agreement with the theoretical and simulated profiles presented in Figs. 3 and 5. The complete set of the in-plane and out-of-plane vibration data is presented in the form of B-scans in Fig. 10a and b. Note that each A-scan in the B-scans was normalized by its maximum amplitude to compensate for the attenuation. Multiple modes can be seen in both B-scan images. The signals were processed subsequently with the DPF and the resulting filtered S0 and A0 modes can be seen in Fig. 10c and d, respectively. In Fig. 10c it can be seen that the incident A0 mode
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Fig. 9. Particle motion trajectories: the S0 mode (a) and the A0 mode (b).
Fig. 10. B-scan images of the in-plane (a) and the out-of-plane displacements (b); filtration results of the S0 (c) and the A0 (d) modes. Yellow lines at (a) and (b) indicate spatial points where the waveforms presented in Fig. 11 were extracted.
was suppressed and only the incident S0 mode remained. Additionally, the reflected S0 mode can be seen in the image. The PZT transmitter in the experimental setup was placed close to the plate’s edge and therefore, the reflected S0 mode is delayed and propagates in the same direction as the incident mode. Analogically, in Fig. 10d the incident S0 mode has been attenuated by the DPF while the two A0 modes, the incident and reflected are apparent. The normalization performed on the B-scans, permitted displaying the images using full dynamic range of the color scales, however it amplified also noise. Moreover it brought up points incorrectly measured due to laser sensing issues. Signals from these points were not correctly resolved as it can be seen for instance in Fig. 10c for distance below 50 mm, where a discontinuity in A0 mode can be observed.
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Fig. 11. Time waveforms and corresponding Lissajous plots of the signals captured at a distance of 30 mm (a,c) and 120 mm (b,d). Raw data (a,b) and the corresponding signals processed using DPF (c,d).
To provide a more profound explanation of the proposed technique two signals corresponding to the dashed lines in Fig. 10 are presented in Fig. 11a and b. The first example, shown in Fig. 11a, presents signals acquired in a proximity of the source where the modes were not separated yet. Since the out-of-plane component was slightly higher than the in-plane vibrations these signals may suggest that a single A0 mode was recorded. However, the time plot presented in Fig. 11c reveals that after the DPF processing both modes appear, the faster S0 and slower A0 . In the second example, presented in Fig. 11b, the modes were already well separated in the time domain. It should be possible to separate the modes using simple gating based on the ratio of the in-plane and out-of-plane vibrations, however, the amplitude ratio for A0 mode does not agree with the theoretical predictions – the in-plane motion is bigger than out-ofplane. This fact can be explained mainly by difficulties related to the 3D measurements performed using the vibrometer used in the experiments. Precise calibration of the vibrometer is performed only in few points, therefore, misalignment of laser beams may occur during scanning. Moreover, the measured ratio can be affected by features of the anti-glare tape. In the considered case the classical OPF would fail due to the inconsistent polarization parameters. The DPF, however, can work without information on the amplitude ratio of both components and, as it can be seen from Fig. 11d, the modes have been correctly projected on the respective axes. 5. Conclusions and future work A new processing technique for the separation of Lamb wave (LW) modes using signals captured by a laser vibrometer was presented in this paper. The proposed directional polarization filter (DPF) is a robust version of the known from geophysics oblique polarization filter (OPF) using different polarization directions of the LW modes. We demonstrated both on simulated and experimental signals that contrary to the OPF, the DPF can operate without full information on polarization parameters of the processed modes, i.e., the amplitude ratio and phase shift between the inplane and out-of-plane components. Therefore, the method can be used without a priori knowledge of material properties and thickness of the inspected plates. This can be a significant practical advantage over the existing techniques since these properties are often unknown prior the non-destructive evaluation. The DPF performed well, even for distorted experimental signals, thus it is expected to operate correctly in the case of slight signal distortion resulting from the presence of noise or wave attenuation. On the other hand, the distorted experimental results indicate that additional improvements are required concerning the measurement system to increase accuracy of the acquired data and to enable practical applications of the method. Note, however that here we performed the scanning to obtain a set of signals that were processed independently. During the automatic scanning it is difficult to ensure good light reflection conditions over the whole area. The expected practical application would involve however only few measurements taken over a given area. These points should be selected to ensure good wave sensing using the interferometer. We also showed using numerical data that the filtration results are related to the direction of wave propagation, i.e., if the DPF is set to suppress the incident S0 mode it will also bring up the A0 mode, which propagates in the opposite direction. To our best knowledge this is the first publication on the method that uses direction of polarization for SHM applications, therefore this article was limited to a comprehensive theoretical, numerical and experimental illustration of the proposed technique. Our future work will be concerned with an extension of the proposed method to enable simultaneous processing
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of all 3D vibration components, which is expected to provide information on the wave’s direction of arrival; another direction to explore is the issue of broad-band signal processing. These steps are essential to achieve the ultimate goal, that is damage detection and localization using a 3D laser sensing Lamb wave propagation. Acknowledgments The first author would like to acknowledge Foundation for Polish Science for ”START”– stipend for young talented researchers. References [1] J.L. Rose, Ultrasonic Waves in Solid Media, Cambridge University Press, 2004. [2] Z. Su, L. Ye, Y. Lu, Guided lamb waves for identification of damage in composite structures: a review, J. Sound Vib. 295 (3–5) (2006) 753–780, http://dx. doi.org/10.1016/j.jsv.2006.01.020.