Shaped sensor for material agnostic Lamb waves direction of arrival (DoA) estimation

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1 Shaped sensor for material agnostic Lamb waves direction of arrival (DoA) estimation More info about this article: Abstract Luca De Marchi, Marco Dibiase, Nicola Testoni and Alessandro Marzani Advanced Research Center on Electronic Systems (ARCES), University of Bologna Bologna, 4036, Italy In this work, a sensor formed by clustering three ad-hoc shaped piezoelectric patches is proposed for guided waves direction of arrival (DoA) estimation in laminate composite and metallic structures. The irregular shaping of the transducer electrodes allows to simplify the signal processing procedures which are necessary to detect the wave DoA. The cluster is conceived so that there is a linear dependence between the difference in time of arrival (DToA) of the wavefront at two of the sensor patches and the DoA of the wavefront itself. The third piezoelectric patch is shaped so that the estimation of the DoA from the DToA can be performed without knowing the actual wave velocity. The transducer performance in terms of DoA estimation accuracy are evaluated through numerical simulations, in which the plate response to a point source is evaluated in the frequency domain using the Green's function approach. Results show that the standard deviation of the error in the estimation of the DoA is less than. This device is meant as a basic building block for the development of passive sensor networks in smart structure applications.. Introduction Passive-only networks of piezoelectric sensors can be exploited to detect and localize impacts or anomalous acoustic emissions for the early identification of crack initiation and growth from the analysis of the emitted ultrasonic guided waves [][]. The actual approaches for localization are based on: - hyperbolic positioning, in which the wave time different of arrivals is measured at various plate positions with single piezosensors and next used to feed triangulation procedures, - DoA intersection, in which the wave DoA, estimated at least at two clusters of sensors, is combined to pinpoint the wave source as schematically shown in figure.

2 Figure. Two cluster can be used to determine the impact position (a) Figure. (a) Sensor with a pair of point-like electrodes P and P spaced apart by a distance d. Upper and lower bound errors in the DoA estimation for the sensor in (a) considering d = 0 mm and d = 50 mm (as from Ref. [6]). In this work, a novel cluster of shaped piezo patches is designed with the aim of minimizing the number of clusters per monitored area and reducing the uncertainty in the DoA estimation. W.r.t. what already presented in Ref. [6], the cluster here proposed operates without knowing the waveguide s geometrical and mechanical properties (dispersion curves) as well as without the need of a calibration procedure (i.e. without knowing the actual wave velocity). The proposed cluster design strategy is based on the Radon Transform and its Inverse. The approach accuracy is evaluated numerically, by estimating DoA of a fundamental antisymmetric mode (A 0 ) wave propagating in an aluminum plate.. Propagation of uncertainty in DoA estimation Sensor clusters commonly used in literature for DoA estimation consist of couples of circular piezo, e.g. the piezopatches P and P as depicted in figure (a). Assuming a planar wavefront impinging the cluster, the difference in time of arrival (DToA or Δ t ) at the two patches is related to the wave DoA (θ ) by this formula:

3 d cosθ Δ t( θ ) = (.) v where d is the distance among the patches P and P and v is wave velocity. (a) Figure 3. (a) Cluster with a point-like electrode P and an Archimedean Spiral shaped electrode P. Upper and lower bound errors in the DoA estimation for the sensor in (a) (as from Ref. [6]). Consequently, the DoA can be evaluated as: θ est v Δtest = arccos (.) d It is interesting to investigate how the errors in the measurements of the DToA affect the estimation of the DoA. This can be done by exploiting the Propagation of Uncertainty theory. In particular, in figure, the upper and lower bounds of the worst-case error in estimating the actual DoA (θ ) are represented. The Matlab tool by Ridder [3] was exploited for the computation of such quantity for two values of d, considering an uncertainty in Δ t equal to just µs and an arbitrary wave velocity est equal to v = 000 m/s. It can be seen in figure that the error in the estimation can be very large when θ 40 deg. The error can be reduced by enlarging the spacing between the two transducers d. However, a larger d implies a quadratically larger Fraunhofer distance, i.e. the source location distance beyond which the plane wave approximation can be considered valid. In Ref. [6], it was also shown that by properly reshaping the piezopatch P as an arch of Archimedean spiral a linear dependence between the DToA and the DoA can be achieved : αθ Δ t( θ) = +Δ t(0) (.3) v 3

4 0 0 and that the worst-case error is constant over a range of angles ( θ [0,90 ]) and considerably lower than the error for low values of θ that can be achieved for the conventional circular shaped patches. It is worth noting that since the sensors are shaped differently, they generally have different frequency responses. This may hamper the possibility of using the DToA estimation method based on the tracking of the peak of the cross-correlation envelope, which is the optimal method in noise-affected measurements. In the next section, we will summarize the method presented in Ref. [6] which allows to achieve equal frequency responses and a linear dependence between DToA and DoA. 3 The Radon Transform as sensor design tool To achieve the desired targets, it is necessary to investigate the frequency response of a generic piezo patch impinged by a Lamb wave. According to formulation presented in Ref. [4], the voltage VP ( ω ) generated by a patch of arbitrary shape Ω P(, xy ) in presence of a plane Lamb wave propagating at angle θ can be expressed as: V ( ω) = ju( ω) k ( ω) H( θ) D ( ω, θ) (3.) P 0 where U ( ω ) denotes the amplitude and the polarization of the wave component relevant to the piezo properties of the patch at the considered frequency ω, k ( ) 0 ω is the wavenumber which characterizes the propagation, H ( θ ) is a quantity related to the material properties of the piezo-structure system, and finally DP ( ωθ, ) is the patch directivity function which can be computed by the following integral: P D = e x y dxdy (3.) jk0 ( ω)( xcosθ+ ysin θ) P( ωθ, ) φp(, ) ΩP where φ ( xy P, ) is referred to as shape function. φ ( xy P, ) is a step function which is equal to if the point of coordinates (x, y) belongs to the area of the piezoelectric patch Ω P(, xy ), and 0 elsewhere. Two different patches have different frequency responses V ( ) P ω e V ( ) P ω due to the relative H ( θ ) e H ( ) θ functions, which do not depend on the frequency, and two directivity functions D (, ) P ωθ and D (, ) P ωθ. When the directivity functions are equal, ik0( ω)( ρ0 αθ) a part from a linear phase-factor in θ, i.e. D ( ωθ, ) = D ( ωθ, ) e +, and in the P P ω ω / vg case of a non-dispersive propagation, i.e. k ( ) 0 =, we have that: the frequency responses V ( ) P ω e V ( ) P ω are equal a part from a scaling factor ω ρ αθ ( 0 ) VP( ω ) VP ( ω + ) v and phase-shift, i.e. = e g. Then, it is possible to use the H( θ) H( θ) cross-correlation method for estimating the DoA; 4

5 antitrasforming the frequency responses in the time-domain with the inverse Fourier transform we get the time responses v () P t e v () P t which are equal except for a scale factor and a linear time-shift in θ, as desired: H ( θ ) v t v t ( ρ + αθ) 0 P() = P( ) H( θ ) vg ( ρ0 + αθ) Δ t = v g (3.3) 3. Sensor directivity analysis and synthesis From equation (3.) it follows that DP ( ωθ, ) can be computed from the coefficients in the direction θ of a bidimensional (spatial) Fourier Transform (FT) of φ ( xy P, ). It is worth noting that the Projection-slice theorem [5] states that the bidimensional FT of the initial function along a line at inclination angle θ is equal to the mono-dimensional FT of the Radon Transform (acquired at angle θ) of that function. The Radon Transform at angle θ of a generic shape function φ ( xy, ) defined in domain Ω can be computed by the following formula: Rθ ( ρ)[ ϕ( x, y)] = φ( x, y) δ( ρ xcosθ ysin θ) dxdy (3.4) and it consists of multiple line-integrals. From equation (3.) and the Projection-slice theorem we can conclude immediately that if two piezoelectric patches φ and φ have the same Radon Transform along a given direction θ apart from a linear spatial-shift in θ, i.e. Rθ( ρ)[ ϕ] = Rθ( ρ ρ0 αθ)[ ϕ], their directivity functions D and P D (when the patches are solicited by the same P plane wave) differs only by a phase shift directly related to the spatial shift, i.e. ik0( ω)( ρ0 αθ) DP( ωθ, ) = DP ( ωθ, ) e +. The shape synthesis procedure exploits the fact that the Radon Transform can be inverted with suitable algorithms, and consists of the following steps: (i) define the geometry of the piezo patch P, by designing its shape function φ ( xy, ) (a regular geometry such as that of a disc can be selected); (ii) compute the Radon Transform R( θ ρ ) of the shape φ ( xy, ) associated to the first piezo patch; (iii) design the desired Radon Transform of the shape φ ( xy, ) associated to the second piezo patch so that: 0 R( ρ) = R( ρ ρ αθ) (3.5) θ θ 5

6 (a) Figure 4. (a) Circular patch P (diameter 0mm). Radon Transform of φ ( xy, ) of the circular patch P. (a) (c) (d) Figure 5. Sample design procedure: the desired Radon transform is depicted in subplot (a); its inverse transform (IRT) is depicted in subplot ; the sensor shape results from the binary quantization procedure applied on the IRT, and corresponds to the red patch in subplot (c); the RT of the sensor shape function is depicted in subplot (d). 6

7 in a predetermined interval [ θ, θ ], it is worth noting that selection of the parameter α directly influences the sensitivity of the sensor to the variation of the DoA θ ; (iv) compute the inverse Radon transform (IRT) of R( θ ρ ) to obtain ; (v) apply a binary quantization procedure which transforms function φ ( xy, ). in a step It is worth noting that the sensor design strategy just presented is not based on the knowledge of the waveguide geometrical and mechanical properties (dispersion curves), i.e. it is material agnostic, and can be applied both to isotropic and anisotropic waveguides. An example of the results which can be achieved with the shape synthesis procedure is represented in figure 5(c). (a) (c) (d) Figure 6. Sample design procedure: the desired Radon transform R3( θ ρ ) is depicted in subplot (a); IRT [ R3( θ ρ )] is depicted in subplot ; φ ( xy, ) results 3 from the binary quantization procedure applied on IRT [ R3( θ ρ )], and corresponds to the green electrode in subplot (c); the actual RT of φ ( xy, ) is 3 depicted in subplot (d). 7

8 In particular, P was assumed to be a piezoelectric circular disc (diameter equal to 0 mm), then the shape of the second patch P was designed following steps to 5, and using the following parameters: ρ 0 = 5mm, α = 0.3 mm / deg, θ = 0 o and θ = 90 o. Such parameters generate a patch whose geometrical dimensions are similar those of the Archimedean spiral depicted in figure 3(a). The resulting R( θ ρ ) is depicted in figure 5(a). The IRT of R( θ ρ ) is then depicted in figure 5. Finally, the piezopatch shape shown in fig. 5(c) is obtained by selecting φ ( xy, ) as the region of the domain in which is greater than the 0% of its maximum value. By comparing subplots (a) and (d), it can be observed that the quantization procedure somehow degrades the desired RT. It is worth noting that the second of equations (3.) (or rather its inverse) indicates that it is necessary to know the group velocity for the estimate θ. In the next paragraph we will see how to design a third sensor to perform the estimation of DoA without knowing the actual wave group velocity. 4 Design of third sensor (P3) of cluster To design a third patch P3 for the estimation of the group velocity, we must make sure that the directivity functions D (, ) P ωθ and D (, ) P3 ωθ are equal apart from a constant ik0( ω) ρ3 phase shift in θ, i.e. D ( ωθ, ) = D ( ωθ, ) e. In this way, for a non-dispersive P3 P mode, i.e. k ( ) 0 ω = ω v, we immediately get that v () t and v () 3 t are equal except for / g a scale factor and a constant time-shift in θ, obtaining the desired method of estimation of v g : P P H ( θ) ρ v () t = v ( t ) 3 3 P3 P H( θ ) vg ρ3 Δ t3 = v g (3.6) For the considerations made in paragraph 3., to obtain the aforementioned relationship between the directivity functions, it is sufficient to impose a relation between the Radon transforms of the shape functions φ and φ 3 as follows: R ( ρ)[ ϕ ] = R ( ρ ρ )[ ϕ ] (3.7) θ 3 θ 3 The steps of shape synthesis procedure are the same as those seen in the previous paragraph except for the third step in which we must impose equation (3.7). A graphical illustration of the procedure for generating the shape function is given in Figure 6 (a) - (c), where the following parameters: ρ 3 = 0mm, θ = 0 o and θ = 90 o, have been used. In particular, P was assumed to be a piezoelectric circular disc (diameter equal to 0 mm), the resulting R3( θ ρ ) is depicted in figure 6(a). The IRT of 3( ρ ) is then R θ 8

9 depicted in figure 6. Finally, the piezopatch shape shown in figure 6(c) is obtained by selecting φ ( xy, ) as the region of the domain in which the IRT of R 3 3( θ ρ ) is greater than the 0% of its maximum value. By comparing subplots (a) and (d), it can be observed that the quantization procedure somehow degrades the desired RT. We will see how much this difference distorts the results from the expected ones. 5 Numerical validation To validate the proposed sensor design strategy and the sensor perfoemances, impact occurring on an aluminium plate 3 mm thick were simulated. In particular, the response of a shaped piezo-patch to impacts generating the fundamental antisymmetric mode (A 0 ) was computed using the Green function formalism adopted in Ref. [6]. Signals detected by the three patches of the designed cluster were computed for different impact locations obtained by varying the cluster-impact distance (400mm, 800mm) and the DoA ( θ = 0,5,...,90 o ). The sensors P, P and P3 and the impact locations, schematically depicted as circles, are represented in figure 7(a). Then, the DToA between P and P ( Δ t ), and the one between the P and P3 ( Δ t ) 3 were computed by applying the cross-correlation procedure, and their ratio Δt was used to estimate DoA (θ ). No calibration procedure is required. Δ t / 3 The estimated DoA ( θ est ) is represented in figure 7 as a function of its actual value (i.e. θ ). The estimation error θ -θ est is mainly due to the effects of the quantization procedure. (a) Figure 7. (a) Circles indicate the considered impact positions placed at 400 mm and 800 mm and angles θ=0, 5, 0, 5,, 90. The circular piezodisk P, the spirally shaped electrode P, and arch-shaped sensor P3, are located at the bottom left corner (the center of P in position x=y=0). Estimated DoA ( θ est ) with respect to the actual DoA (θ ). The solid lines represent the upper and lower bound limits computed according to the Propagation of Uncertainty theory on the basis of the ideal sensor geometries. 9

10 The estimated DoA is between the upper and lower bounds derived with the Propagation of Uncertainty for the ideal patches (as those shown in figure 3(a)) and the standard deviation of θ θ is below.4 o degrees. 6 Conclusions est In this work, a novel procedure to design piezoelectric sensors for Lamb waves direction of arrival (DoA) estimation in proposed. The designed cluster is composed by three piezo patches. The first one can be conventionally shaped (e.g. as a disk). The second and third pacthes are derived from the first one with a procedure which is based on the direct and inverse Radon Transform. This method is conceived so that a linear dependence is imposed between the difference in time of arrival (DToA, Δ t ) of the wavefront at P and P and the direction of arrival (DoA, θ ) of the wavefront itself, minimizing the uncertainty in the DoA estimation. Furthermore, it s possible to use the cross-correlation procedure to estimate Δ t, and then θ. The third patch P3 is designed using again the direct and inverse Radon Transform as a mean to estimate the wave group velocity, necessary for DoA (θ ) estimation. A numerical validation on synthetically generated A0 guide wave signals shows excellent performance of the designed cluster in the estimation of the DoA. References [] T. Kundu, S. Das, and K. Jata. Detection of the point of impact on a sti ened plate by the acoustic emission technique. Smart Materials and Structures, 8(035006), 009. [] H. Matt and F. Lanza di Scalea. Macro-fiber composite piezoelectric rosettes for acoustic source location in complex structures. Smart Mater. Struct., (6):489{499, 007. [3] Ridder B 04 [4] Senesi M and Ruzzene M 0 A frequency selective acoustic transducer for directional lamb wave sensing The Journal of the Acoustical Society of America [5] Gaskill J D 978 Linear Systems, Fourier Transforms, and Optics (New York: Wiley). [6] L. De Marchi, N. Testoni, A. Marzani, Spiral-shaped piezoelectric sensors for Lamb waves direction of arrival (DoA) estimation, Smart Mater. Struct. 7 (08) (0pp) 0