Experimental Signal Deconvolution in Acoustic Emission Identification Setup

Transcription

1 6th NDT in Progress 20 International Workshop of NDT Experts, Prague, 0-2 Oct 20 Experimental Signal Deconvolution in Acoustic Emission Identification Setup Zuzana FAROVÁ23, Zdeněk PŘEVOROVSKÝ, Václav KŮS2, Serge DOS SANTOS 3 Institute of Thermomechanics AS CR, v. v. i., Dolejškova 402/5, Praha 8, Czech Republic; Phone: , farova@it.cas.cz, zp@it.cas.cz 2 Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 3, Prague 20 00, EU - Czech Republic; Phone: , vaclav.kus@fjfi.cvut.cz 3 Centre - Val de Loire Université, Unité Mixte de Recherche Imagerie et Cerveau, INSERM U930 - CNRS ERL 306, ENI Val de Loire, 3 Rue de la Chocolaterie, BP340, 4034 Blois cedex, FRANCE, serge.dossantos@univ-tours.fr Abstract Identification of acoustic sources is a great deal and the most important problem emerging in biomedical applications and nondestructive testing. We present the basics of time reversal Acoustic Emission (AE) principle and point out the advantages of AE nondestructive testing for localization and classification of acoustic sources. This approach provides the tool for the defect localization by means of the reversal wave focusing in nonlinearity position in the material under consideration. Also, using TR operator to AE signal, we obtain convolutional signal in the closest neighborhood of acoustic source, which gives us the pure image of the real characteristics of AE source after deconvolution process applied to the signals detected. We describe the mathematical background for backside deconvolution through the Green functions. Further, we design the laboratory settings realizing experiments for AE signal deconvolution of the reversed signals measured by the laser beam device centering to the position of the defect. Finally, we apply the recently developed classification methods to these signals. Keywords: TRA, Green function, experimental deconvolution, acoustic emission, identification. Introduction The AE sources identification techniques are of great importance in the field of nondestructive testing. Thanks to the exact determination of acoustic source position, we are able to decide whether the source represents the threat of damage in the device or material considered. Unfortunately, in most cases, the received signal is influenced by passing through the device itself and also by the AE sensors. Thus, instead of the original signal, we measure a convolution of the original signal and the Green function. Therefore, the classification of signals into the groups according to their origin is a very difficult task and sometimes equivocal. To solve the problem, we suggest employing the Time Reversal Acoustics (TRA) in order to perform the so called experimental deconvolution, i.e. using TRA to avoid the influences of AE sensors and the device itself. In this paper we deal with the proposed concept of experimental deconvolution.

2 2. Green function - the background To introduce the Green function let us consider a simple source like unit impulse at x = ξ (space position) and t = τ (time instant). Then the vector Green function G represents the displacement in general point (x,t), where we denote by G i (x, t; ξ, τ) the i-th component of this displacement. The Green function depends on both the coordinates of source and the receiver and also it must satisfy the following equations under zero initial conditions 2 G t 2 = δ(x ξ)δ(t τ) + c2 2 G, () G(x, t; ξ, τ) = 0, for t τ, G(x, t; ξ, τ) = 0, for x ξ. t The solution of () can be found in the form G(x, t; ξ, τ) = δ(t τ x ξ 4πc 2 x ξ c ). (2) If we have the source s(t) instead of δ pulse, then the Green function is a solution of the following equation 2 G t = δ(x ξ)s(t) + 2 c2 2 G, (3) again with zero initial conditions. Let us notice that s(t) = s(τ)δ(t τ)dτ. Thus, it follows that the resolution of the equation (3) is given by G(x, t; ξ, τ) = s(t τ x ξ 4πc 2 x ξ c ). (4) For more details concerning the Green function and the solution of wave equation, see []. 3. Time reversal acoustics (TRA) The theory of TRA is based on the fact that acoustic wave equation in non-dissipative heterogeneous medium is invariant with respect to TR operation. The basic TR experiment introduced in [2] can be described as follows. Let us consider point source at the position r 0 inside the volume V surrounded by the field of transducers. First step. Let the source emit a signal s(t) = δ(t) at the initial time t = 0. In an arbitrary position r V we can measure the response in the form of Green function G(r, r 0 ; t) resulting from the wave equation. Assume that we are able to measure pressure field and

3 its normal derivative at each point of transducers area V during the time interval 0, T. Moreover, T is high enough to ensure that the information loss is insignificant. Second step. The original source at the position r 0 is removed and the transducers now represent sources. TR operation is described by means of transformation t T t and the secondary sources can be expressed as follows φ s (r, t) = G(r, r 0 ; T t) with the normal derivative in the direction n n φ s (r ) = n G(r, r 0 ; T t). The resulting pressure field φ tr (r, t) propagates inside the cavity. TR pressure field can be derived by means of Helmhotz-Kirchhoff integral φ tr (r, t) = dt S (G(r, r ; t t ) n φ s (r, t ) φ s (r, t ) G(r, r ; t t )) d2 r ρ(r). By the particular calculations (see [2]) we obtain the TR field in the form φ tr (r, t) = G(r, r 0 ; T t) G(r, r 0 ; t T) which in the case of homogeneous medium leads to φ tr (r, t) = 4π r r 0 δ(t + r r 0 ) c 4π r r 0 δ(t r r 0 ). c In non-homogenous medium we use only several transmitters during the actual TR experiment, so the TR field can be rewritten through the sum over all transmitters φ tr (r = r 0, t) N G(r 0, r i ; t) G(r 0, r i ; t), i= where N is the number of transmitters. TR field propagates back into the cavity and refocuses exactly at the initial source position, see [2]. For detailed solution of the wave equation and mathematical treatment of TR, see [3]. 4. Signal deconvolution via TRA Let us consider some point source s(t) at the position r 0 and a receiver at the position r i. The acoustic signal measured at the position of receiver in time interval t 0, T arises as the result of convolution between source signal and corresponding Green function This signal s G (t) is time reversed to s G (t) = s(t) G(t, r 0, r i ), t 0, T. s(t t) G(T t, r 0, r i ), t 0, T,

4 Figure : Schematic diagram of the experimental deconvolution principle. and sent back from the position r i to the position r 0, see Figure. At the position r 0 we now receive the signal which can be expressed as follows s = s(t t) G(T t, r 0, r i ) G(t, r i, r 0 ), t 0, T. (5) We are interested in the relation between s and the original signal s(t). For this purpose we use the Fourier transform F(f) satisfying the well-known property F(f g) = F(f)F(g). Now, we can convert the signal s by means of the Fourier transform s = s(t t) G(T t, r 0, r i ) G(t, r i, r 0 ) F( s(t)) = F(s(T t))f(g(t t, r 0, r i ))F(G(t, r i, r 0 )). Further, applying the Fourier transform of the Green function ( ) δ(t x ξ ) c F = exp ik x ξ 4πc 2 x ξ 4π x ξ we transform the equation (5) into F( s(t)) = F(s(T t)) 4π x ξ exp (ik x ξ )exp (iωt) exp ( ik x ξ ) 4π x ξ = F(s(T t)) exp (iωt). (6) 6π x ξ 2 If we apply inverse Fourier transform to (6), we obtain the desired TR signal after passing through the device F(s(T t)) 6π x ξ exp (iωt) IFT 2 6π x ξ 2s(t) = a. s(t) So it can be seen that the resulting signal (after passing through the device, time reversing and sending back to the transducer) is proportional to the original signal emitted by transducer, see Figure 2.

5 a. s(t) Figure 2: The resulting schema of experimental deconvolution principle. 5. TRA experiment for deconvolution How we derived in the previous section, the resulting signal after time reversing and sending back to the initial source position is proportional to the original signal. This fact leads to a very important consequences for AE signals measuring and also for AE source localization. During a standard AE measuring we receive signals which are not exactly original source signals, but they are influenced by passing through the investigated device and also by the characteristics of AE sensors. We always measure the signal s G (t) which is convolution of the initial AE signal and the corresponding Green function s G (t) = s(t) G(t, r 0, r i ), r 0 and r i denotes the initial source position and the sensor position, respectively. To verify our theoretical results in practice, an experiment on small aircraft component has been performed. The experiment was made on a steering actuator bracket (SAB), which is a part of the aircraft nose landing gear, for more details see [4]. We recorded and monitored the AE signals by means of DAKEL XEDO AE system. The AE measuring was performed on the SAB during the laboratory fatigue tests. The bracket was loaded under stress cycles on Instron-Schenck 00-kN uniaxial loading machine to the maximal load level of 43 kn causing the maximum stress of about 870 MPa in the critical points. AE was monitored during the loading cycles by the AE system and all 07 detected AE signals were stored, and after certain loading periods, the detailed AE source location analysis was performed so as to detect crack initiation. Most AE events arose around the mounting holes of the bracket. Figure 3 shows localization results of acoustic sources in the bracket. It can be observed from Figure 3 that we obtain 2 main area of sources of AE. Most of the AE sources originated from the clearly visible damages on the interior surface of the hole caused by the friction with mounting shank. So the two main areas of acoustic sources are situated on the left hole and on the right hole. Several signals were selected originating from the right hole and several from the left hole. Figure 4, resp. Figure 5, shows AE signals from the three events on the left hole, resp. on the right hole, every event was

6 Figure 3: Bracket with localized acoustic emission sources. (a) Event (b) Event 2 (c) Event 3 Figure 4: Events originated from the left hole in bracket. measured by the four nearest AE sensors around the hole, see Figure 3. From these signals several various parameters were extracted and Q-factor analysis was applied to these parameters. Q-factor analysis distinguish from R-factor analysis. Whereas through the commonly used R-factor analysis the correlation between variables (i.e. signal parameters) is computed, in the Q-factor analysis we compute correlation between the measurements (i.e. AE events). In the Q-factor analysis we obtain information about the number of significant object groups, which have similar properties. Consequently, this number can be used in the cluster analysis, which divides objects into the mentioned groups (clusters). The Q-factor analysis revealed two groups among the signals, which are in Figures 4 and 5. The first cluster belongs to the events from the left hole and the second one belongs to the events from the right hole. Afterwards, the following process was produced for the both holes: We time reversed the signals measured on individual sensor and sent it back from respective sensor. The

7 (a) Event 4 (b) Event 5 (c) Event 6 Figure 5: Events originated from the right hole in bracket. (a) Event (b) Event 2 (c) Event 3 Figure 6: Events measured on the left hole (after time reversing). (a) Event 4 (b) Event 5 (c) Event 6 Figure 7: Events measured on the right hole (after time reversing).

8 resulting signals were measured by means of sensor placed at the position near the hole. You can see the resulting measured signals in Figure 6 for the left hole and in Figure 7 for the right hole. We applied Q-factor analysis also to these time reversed signals. The analysis determined that all signals belong to only one group (cluster). Regarding the fact that all the measured AE signals come from the same type of AE source (i.e. from damages on the interior surface of the hole caused by the friction with mounting shank), the classification based on all measured signals is more appropriate for the purpose of determining the real AE source. 6. Conclusions In this paper we have proposed the so called experimental deconvolution using TRA principle. We found theoretically as well as experimentally that we are able to obtain the signal very similar to the initial AE signal by this newly introduced measurement approach. From the above mentioned facts it follows that it is possible, by means of time reversing of the measured signals and sending them back to the initial position of AE source, to eliminate the influence of passing through the material and also the influence of AE sensors. A disadvantage is that the initial source position has to be localized beforehand because we require the measurements of the TR signal to be done at the initial source position. In the near future, we plan to use laser interferometer instead of placing AE sensor nearby the initial source position. Also, we would like to confirm the apparent advantages of the experimental deconvolution principle for the exact classification of AE signals in order to decide if the AE source can cause a damage in the material under consideration. Acknowledgements This work was supported by the grant SGS0/209/OHK4/2T/4 and by the research program of the Ministry of Education of Czech Republic under the contract MSM References [] Aki K., Richards P. G., Quantitative Seismology. University Science Books, [2] Fink M., Time-reversed Acoustics. Rep. Prog. Phys., 63, , [3] Klibanov M.V., Timonov A., On the Mathematical Treatment of time reversal, Institute of physics publishing. Inverse Problems, 9, , [4] Chlada M., Prevorovsky Z., Blahacek M., Neural Network AE Source Location Apart From Structure Size and Material Journal of Acoustic Emission, 28, 99 08, 200.