Reconstruction of impact force on curved panel using piezoelectric sensors

Transcription

1 Avalable onlne at Proceda Engneerng 48 (0 ) MMaMS 0 Reconstructon of mpact force on curved panel usng pezoelectrc sensors Vladslav Laš a *, Robert Zemík a, Tomáš Kroupa a, Jan Bartošek a a Unversty of West Bohema, Unverztní 8, 306 4, Plsen, Czech Republc Abstract Ths paper s focused on reconstructon of mpact force n an unknown locaton on a general thn-walled curved structure wth a hole. Ten rregularly dstrbuted pezoelectrc sensors are attached to the panel for the measurement of response sgnals. A correspondng fnte element model s created n MSC.Marc. The methodology for reconstructon s based on the transfer functon approach. The tme dependence of mpact force s reconstructed and the locaton of mpact s dentfed on a dfferent coarse mesh and t s compared to the soluton on the orgnal mesh for all possble mpact locatons. Moreover, the nfluence of nterpolaton of the transfer functons wthn elements (or sectors) of the coarse mesh on ncrease of the accuracy of mpact locaton dentfcaton s nvestgated. 0 The Authors. Publshed by Elsever Ltd. 0 Publshed by Elsever Ltd.Selecton and/or peer-revew under responsblty of the Branch Offce of Slovak Metallurgcal Socety at Faculty Selecton of Metallurgy and/or peer-revew and Faculty under of Mechancal responsblty Engneerng, of the Branch Techncal Offce Unversty of Slovak of Košce Metallurgcal Open access Socety under at CC Faculty BY-NC-ND of Metallurgy lcense. and Faculty of Mechancal Engneerng, Techncal Unversty of Košce. Keywords: fnte element method; deconvoluton; transfer functon; nverse problem; regularzaton Nomenclature d dstance between real and reconstructed locaton e pezoelectrc constant f loadng sgnal (nput) g transfer functon h response sgnal (output) t tme E Young s modulus F matrx of loadng sgnal G matrx of transfer functons H matrx of response sgnals K number of mpact locatons L number of sensors M number of mpacts at one locaton N number of sgnal samples Greek symbols δ response reconstructon error ε delectrc permttvty constant * Correspondng author. Tel.: ; fax: E-mal address: las@kme.zcu.cz Publshed by Elsever Ltd.Selecton and/or peer-revew under responsblty of the Branch Offce of Slovak Metallurgcal Socety at Faculty of Metallurgy and Faculty of Mechancal Engneerng, Techncal Unversty of Košce Open access under CC BY-NC-ND lcense. do:0.06/j.proeng

2 368 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) λ regularzaton parameter ν Posson s rato ρ materal densty ξ, η local coordnate system Subscrpts mpact locaton j sensor. Introducton The safety or functonalty of every structure can be sgnfcantly affected by defects. These defects can be nvsble to surface nspecton. Therefore, they must be found pror to any catastrophc scenaro. Currently, they are detected by nondestructve technques lke ultrasonc, X-ray, con tappng or other methods [9], whch are tme and cost consumng and requre the constructon to be taken out of servce. In contrary, the condton of constructon can be evaluated durng operaton from measurements of sensors placed over the structure. Ths prncple s so-called structural health montorng. The dentfcaton of mpact force and mpact locaton s an mportant task of such systems and the deal dentfcaton method should dentfy the mpact force, or even the combnaton of mpact forces, on complex structures n real tme wth low dependence of operatng nose. The hdden defects are very common especally n modern composte materals, such as carbon fber renforced epoxy lamnates, that are wdely used thanks to ther hgh strength and stffness to weght ratos. Not only s the desgn process of structures whch contan parts from composte materals complcated due to effects such as non-lnear behavor [3], specfc damage behavor [9], and drectonal dependence of velocty of propagaton of stress waves [5], but, furthermore, compostes are hghly susceptble to transverse loadng, whch can cause delamnaton and cracks n matrx and thus sgnfcantly reduce the stffness or strength of the constructon. The mpact dentfcaton problems have been studed by many researchers n recent years and several methods were proposed. The often used one s the nverson of forward problem, whch can be performed n tme, frequency or spectral doman. Drect deconvoluton s a well-known ll-condtoned problem and ts results are strongly nfluenced by qualty of expermental data, approprateness of the mechancal model and robustness of employed algorthm. Many researches defne the problem rather as a mnmzaton of the dfference between measured and modeled responses of the mpacted structure. Addtonal terms and constrants are added to mnmzaton to regulate oscllatons n results. Jacqueln et al. [6] analyzed the deconvoluton n tme doman. The nfluence of sensors locaton and dfferent regularzaton methods were nvestgated. Smlarly, Gunawan et al. [8] used the tme doman. The mpact force was approxmated by cubc splne and the two-step B-splne regularzaton method was developed. On the other hand Yan and Zhou [3] used Chebyshev polynomals to represent the mpact force and the modfed genetc algorthm to solve the mnmzaton problem. Park and Chang [7] determned the system expermentally and nvestgated several types of mpacts. Martn and Doyle [] used the Fast Fourer Transform to swtch nto frequency doman and solved the deconvoluton drectly. Furthermore, Doyle [] employed the wavelet deconvoluton and modelng wth FEA. Other researches preferred to work n spectral doman. Hu et al. [] formulated the mnmzaton wth regularzaton parameter and constrant, whch was solved by quadratc programmng method. Moreover, dfferent types of sensors were compared and Chebyshev polynomals were employed to reduce the number of unknowns. Atobe et al. [4] used the gradent projecton method to solve the mnmzaton problem and compared the determnaton of the system by experment or by FEA. Fnally Sekne and Atobe [6] formulated the mnmzaton where multple mpacts can be dentfed. Another possblty s to defne the mnmzaton n recursve form n tme doman and to use flterng method to solve the nvestgated problem. Seydel and Chang [4, 5] used smoothng-flter method and nvestgated the nfluence of sensor locatons and boundary condtons. Smlarly, Zhang et al. [] mplemented smoothng-flter algorthm wth the possblty of real-tme computatons. The locaton of mpact wthn these methods s often estmated from the mnmzaton of the error between measured and modeled responses along the structure. Ths can be done by drect search methods [] or by some other optmzaton technques [4]. Another possblty s to use the technques derved from methods used n acoustc emsson (AE) [0, 4], where the dfference n arrval tme of sgnal s determned and the locaton of mpact s estmated from velocty of waves. Unfortunately, the determnaton of exact tme of arrval n composte materal or complex structures s lmted because of the dsperson and reflecton of waves on boundares. The alternatve s calculaton of dstrbuton of energy n defned tme step and the determnaton of ts maxmum [7]. Totally dfferent approach s the determnaton of mpact force and force locaton from models based on neural networks [7]. The model s composed of parallel elements connected by defned relatons and traned by prelmnary tests.

3 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) The output of the model s then set by learned behavor. The weakness of such approach s the necessty of learnng perod and uncertan reacton of model to not learned mpacts. The above cted papers dffer n several features lke complexness of geometry, determnaton of the system model or the type and number of sensors. The mpact force was nvestgated on metal beam [], metal plate [, 6, 8], composte plate [0, ] or stffened composte plate [5, 7,,, 3, 4, 6]. The model of the system s defned analytcally [4,, 3] or determned by FEA [8,, 4, 6] or by experment [7, 4]. Sgnal s mostly obtaned from stran gauges [4, 6], accelerometers [,, ], smple pezoelectrc sensors [] or from sensor network [7, 3, 0].. Dscrete convoluton and nverse problem The methodology used n ths work s based on the transfer functon approach. For a lnear system, ts response h to an nput f can be expressed by convoluton f * g = h () where g s so-called transfer functon and t represents the characterstcs of the system. In order to fnd the locaton of mpact and to reconstruct the tme dependence of the mpact force, t s necessary to perform to consecutve steps; a) a calbraton procedure,.e., to perform expermental measurements whle recordng the correspondng nput and response, and to calculate the transfer functons for all combnatons of mpact locatons and sensors, and b) a reconstructon procedure,.e. to reconstruct the force n each possble locaton for expermentally measured response for unknown mpact and to seek the mpact locaton by mnmzng the error of response reconstructon. Ths approach can be carred out ether expermentally or numercally. In ths work, a vrtual experment s performed usng fnte element analyss (FEA). Hence, a dscrete deconvoluton method s used. All pre- and post-processng algorthms are performed usng Matlab scrpts... Transfer functon calculaton Let us consder a system (a structure) wth K mpact locatons and L sensors. Frst, we measure the force n locaton and the correspondng response n sensor j. For dscrete system the nput and response sgnals, each consstng of N samples (assumng constant tme ncrement Δt), can be wrtten as T f = [ f, f,, fn ] () and T h j = [ h, h,, hn ], (3) respectvely. The force vector can be rearranged to matrx [N N] for each performed mpact (or experment) m = M as 0 0 f g h f f g h (4) = 0 f f fn gn hn m F and then a global matrx system can be assembled for up to M subsequent mpacts as F H j. (5) gj = M M F H j F The system s then represented by all solutons for each combnaton of and j. However, each system of algebrac equatons n (5) s overdetermned and ll-posed. If we rewrte (5) concsely as Fg H (6) the soluton can be obtaned by varous methods, for example by smple pseudonverson g F T T = ( F) F H, (7) by mnmzng the resduum usng least squares method by quadratc programmng technques mn g j H j m H j mn { Fg H }, (8) T T T T { g Pg Q g}, P = F F, Q = H F +, (9) or others. Nonetheless, to avod unrealstc oscllatons of the soluton, t s advsable to use adequate regularzaton technque that mposes addtonal condton on the soluton. In ths work, the Tkhonov regularzaton [8]

4 370 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) mn { Fg H + λ g } (0) s used, where the addtonal term, compared to least square method, means that the norm of the soluton wll be mnmzed too. A proper choce of the parameter λ s needed to balance the rato between the standard resduum and the oscllatons... Impact force reconstructon When all transfer functons g j are known, we can attempt to reconstruct the unknown nput sgnal from measured responses only. The response agan wth N samples as n (3) s obtaned n all L sensors or only n a subset of sensors. Now, the transfer functon for each combnaton of and j must be rearranged to matrx [N N] so that 0 0 g f h g g f h () = 0 g g gn fn hn G j and then a global system for all selected sensors can be assembled as G H. () f = G H L L G The soluton can be performed for each possble (suspected) mpact locaton. Agan, the problem (), wrtten concsely as Gf H, (3) s overdetermned and ll-posed. Moreover, as the mpact force s always non-negatve (assumng non-stckng mpact), addtonal nequalty constrant f 0 (4) mght be advantegous [0]. In ths work, the Tkhonov regularzaton s used agan wthout the nequalty constrant. Hence, the soluton f s found from mn Gf H + λ f. (5).3. Impact locaton search strategy f H { } To fnd the real locaton (or at least a good estmate) of the unknown mpact, t s necessary to seek the locaton whch produces the smallest error δ between the measured response (H) and the response (H ) reconstructed usng the correspondng soluton of (5) as G f H H = Gf. (6) Therefore, the goal s to solve mn { δ }, δ = H H. (7) The soluton f whch mnmzes (7) can be sought by varous methods, however, n ths work, a brute-force search n all locatons was conducted to ensure that the global mnmum s found. 3. Fnte element model of panel wth pezoelectrc transducers The accuracy of the presented approach s tested on a vrtual experment n ths work. A curved thn-walled steel panel wth a hole s equpped wth L = 0 rregularly placed pezoelectrc transducers (so-called patches). The panel s loaded by known mpact force n the drecton of normal to the outer surface (see Fg. and Fg. ). The force s represented by Gauss functon as shown n Fg. 3. Only one mpact per locaton s used (M = ). The transducers are used to measure the response (Fg. 3) of the panel, whereas the resultng sgnal s electrc potental between ts electrodes (voltage). A smplfed FEA model of the panel s created n MSC.Marc usng 8-node sold elements (ths s necessary for the pezoelectrc sensors). Only one layer of elements through the thckness of the panel was used, therefore, the assumed stran opton was engaged. The transent analyss used sngle-step Houbolt tme ntegraton scheme wth N = 500 tme ncrements. Each ncrement was Δt = 0 μs. Nodal lnks were used on nodes on the outer electrodes of sensors to nduce constant electrc potental across the electrodes, whle the latter nodes were grounded. The mesh conssted of 800 nodes (380 defnng the outer surface nodes mpact locatons), 34 elements (33 for steel and 0 for patches). The materal characterstcs are shown n Tab.. H j

5 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) Table. Materal propertes used n fnte element analyss. Materal ρ E ν e 3 ε [kg/m 3 ] [GPa] [-] [C/m ] [C/m ] [C/m ] [F/m] Steel Pezoelectrc e 3 e 33 Fg. FEA model of panel wth sensors (left) and detal of one sensor wth nodal lnks (rght). Fg. Surface of the panel wth normals n all mpact locatons (left) and surface wth reduced number of elements (rght). Fg. 3 Example of nput (left) and correspondng output (rght).

6 37 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) Identfcaton of mpact locaton and reconstructon of force on coarse mesh The correspondng nput and output sgnals were calculated on the orgnal mesh and these were consdered as the exact soluton wthn the vrtual experment. Then, a smplfed coarse mesh was created (Fg. ) and the reconstructon was performed usng only the responses and transfer functons correspondng to the smplfed mesh. Ths mesh conssted of 55 nodes (mpact locatons) and 39 elements (or sectors to avod confuson wth fnte elements). The dstance d between the exact mpact locaton and the locaton found usng (7) on the coarse mesh data s plotted n Fg. 4. The maxmum dstance dfference s d max =.353 m (at node [ 638, 98, 777] mm) and the mean dfference s d mean = 0.53 m, whle the mean dameter (sze) of the sectors s m. For mpacts at nodes that belong to the coarse mesh, the error was neglgble. Fg. 4 Values of dstance between real mpact locaton and that calculated usng transfer functons of coarse mesh only. 5. Interpolaton of transfer functons An attempt was made to ncrease the accuracy of the reconstructon on the coarse mesh by nterpolatng the neghborng transfer functons. Each sector of the coarse mesh was subdvded nto 5 5=5 subsectors, thus creatng new nternal nodes. The transfer functons at nternal nodes q are calculated usng standard approxmaton functons for soparametrc quadrlaterals as g = N g, j = a, b, c (8) ( ) d q j j, j where the numbers a, b, c, and d correspond to nodes defnng the gven sector and N = ( ξ )( η), = ( + ξ )( η), = a N ( + ξ )( + η), = ( ξ )( + η) 4 b N 4 c N. (9) 4 d 4 Usng ths technque, the locaton error for the worst case at node [ 638, 98, 777] mm decreased to d = Two other examples are presented heren. If the mpact locaton ([ 96, 687, 986] mm) was ncorrectly found n a neghborng sector (d = 0.38), the locaton error decreased to d = m usng nterpolaton. Last, f the mpact locaton ([ 87, 5, 00] mm) was ncorrectly found wthn the same sector (d = 0.5), the error decreased to d = m. The varaton of dstrbuton of the reconstructon error (7) for real mpact locaton at node [ 96, 687, 986] mm s llustrated n Fg. 5 for the three mesh confguratons. The cross symbol represents the exact mpact locaton and the crcle symbol denotes the dentfed mpact locaton (mnmum response error).

7 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) (a) (b) (c) Fg. 5 Examples of error surfaces calculated wth transfer functons of orgnal mesh (a), coarse mesh (b), and extrapolated (c) data.

8 374 Vladslav Laš et al. / Proceda Engneerng 48 ( 0 ) Conclusons The methodology for dentfcaton of mpact locaton and reconstructon of mpact force tme dependence was successfully demonstrated on vrtual experment usng fnte element model. The relablty of the fnte element model s to be valdated by comparson wth real experment. Of course, secondary dffcultes are lkely to emerge durng the real experment. For example, t wll be dffcult to perform mpacts wth drecton normal to surface and to mpact the same locaton f repeated experments are used (e.g. to measure mpacts wth dfferent ampltudes, veloctes). Also, the real system needs not to be perfectly lnear. The non-lnearty can be caused by varous effects, such as non-neglgble dampng, plastcty, and structural or sensor damage. Other problems may occur when selectng proper samplng frequency, sgnal trggerng algorthms, number of samples, and also the numercal parameters, such as number of teratons, tolerance values, or the regularzaton parameter. These effects could be partally suppressed by optmzed placement and numbers of both the sensors and tranng mpact locatons. Acknowledgements Ths work was supported by the European Regonal Development Fund (ERDF), project NTIS New Technologes for Informaton Socety, European Centre of Excellence, CZ..05/..00/ and by grant projects GA P0//088 and SGS References [] Martn, M. T., Doyle, J. F., 996. Impact force dentfcaton from wave propagaton responses, Internatonal Journal of Impact Engneerng 8, pp [] Doyle, J. F., 997. A wavelet deconvoluton method for mpact force dentfcaton, Expermental Mechancs 37, pp [3] Kroupa, T., Laš, V., Zemík, R., 0. Improved nonlnear stress-stran relaton for carbon-epoxy compostes and dentfcaton of materal parameters. Journal of composte materals, vol. 45, n. 9, pp Prnt ISSN: , Onlne ISSN: X [4] Seydel, R., Chang, F. K., 00. Impact dentfcaton of stffened composte panels: I. System development, Smart Materals and Structures 0, pp [5] Seydel, R., Chang, F., K., 00. Impact dentfcaton of stffened composte panels: II. Implementaton studes, Smart Materals and Structures 0, pp [6] Jacqueln, E., Bennan, A., Hameln, P., 003. Force reconstructon: analyss and regularzaton of a deconvoluton problem, Journal of Sound and Vbraton 65, pp [7] Park, J., Chang, F. K., 005. System dentfcaton method for montorng mpact events, Proc. SPIE 5758, pp [8] Gunawan, F. E., Homma, H., Kanto, Y., 006. Two-step B-splnes regularzaton method for solvng an ll-posed problem of mpact-force reconstructon, Journal of Sound and Vbraton 97, pp [9] Laš, V., Zemík, R., 008. Progressve Damage of undrectonal Composte Panels, Journal of Composte Materals, 4(), pp [0] Kudu, T., Das, S., Martn, S. A., Jata, K. V., 008. Locatng pont of mpact n ansotropc fber renforced composte plates, Ultrasoncs 48, pp [] Zhang, B., Zhang, J., Wu, Z., Du, S., 008. A load reconstructon model for advanced grd-stffened composte plates, Composte Structures 8, pp [] Hu, N., Fukunaga, H., Matsumoto, S., Yan, B., Peng, X. H., 007. An effcent approach for dentfyng mpact force usng embedded pezoelectrc sensors, Internatonal Journal of Impact Engneerng 34, pp [3] Yan, G., Zhou, L., 009. Impact load dentfcaton of composte structure usng genetc algorthms, Journal of Sound and Vbraton 39, pp [4] Atobe, S., Kuno, S., Hu, N., Fukunaga, H., 009. Identfcaton of Impact Force on Stffened Composte Panels, Transactons of Space Technology Japan 7, pp. 5 [5] erv, J., Kroupa, T., Trnka, J., 00. Influence of prncpal materal drectons of thn orthotropc structures on Raylegh-edge wave velocty. Composte structures 9, pp , ISSN: [6] Sekne, H., Atobe, S., 009. Identfcaton of locatons and force hstores of multple pont mpacts on composte sogrd-stffened panels, Composte Structures 89, pp. 7 [7] LeClerc, J. R., Worden, K., Staszewsk, W. J., Haywood, J., 007. Impact detecton n an arcraft composte panel A neural-network approach, Journal of Sound and Vbraton 99, pp [8] Hansen, P. C., 007. Regularzaton Tools, Numercal Algorthms 46, pp [9] Trebua, F. et al., 007. Analyss of causes of defects n mechancal components of mechatronc systems usng methods of expermental mechancs (n Slovak). AT&P Journal plus,, pp [0] Laš, V., Kroupa, T., Bartošek, J., Zemík, R., 0. Impact force reconstructon for structural health montorng of composte beam, Acta Mechanca Slovaca 5 (), pp. 6 3.