Acoustic modification prediction using FRF: application of Helmholtz resonators on a rocket engine combustion chamber

Transcription

1 Acoustic modificatio predictio usig FRF: applicatio of Helmholtz resoators o a rocket egie combustio chamber G. P. Guimarães 1,, R. Pirk 1,, C. D A. Souto 1,, L. C. S. Góes 1 Istitute of Aeroautics ad Space (IAE), Divisio of Itegratio ad Testig (AIE) Praça Marechal Eduardo Gomes, 50, São José dos Campos, São Paulo, Brazil gustavo.paulielli@gmail.com Techological Istitute of Aeroautics (ITA), Departmet of Mechaical Egieerig Praça Marechal Eduardo Gomes, 50, São José dos Campos, São Paulo, Brazil Abstract This paper presets the applicatio of a method that uses acoustic Frequecy Respose Fuctios (FRF) to predict modificatios i a origial acoustic system: i this case, a cavity of a rocket egie combustio chamber. As a cavity modificatio, the isertio of Helmholtz Resoators (HR) was applied. The use of HR i such chambers atteuates the effect of combustio istability, which ca seriously damage the egie. The Acoustic Modificatio Predictio (AMP) method usig FRF is based o the Structural Modificatio Usig Respose Fuctios (SMURF), which is a well-kow structural reaalysis techique. The AMP uses the FRF matrix from the origial cavity ad the aalytical model of the HR to predict the behavior of the ew cavity, avoidig the use of large models. I order to validate the predictio results, experimetal data were used. The first results preseted differeces i magitudes, but it was possible to idetify the modificatio behavior, as well as parameters to be ehaced to have a fully validated method. 1 Itroductio Durig the desig process of structures, it may be ecessary to evaluate a variety of cofiguratios i order to choose oe that fits the desired dyamic respose. This challege is ehaced whe complex structures are desiged, oce time cosumptio reaalysis with high computatio cost becomes a sigificat issue to be cosidered. I this way, reliable ad efficiet aalysis tools are usually adopted i order to simulate the dyamical behavior of a ew (modified) structure through local modificatios, usig the modal model or Frequecy Respose Fuctios (FRF) of a origial structure. There is a vast literature presetig differet techiques to perform this type of dyamic reaalysis, such as Structural Dyamic Modificatio (SDM) [1] [] [3] [4], also kow as Structural Modificatio Predictio [5] ad Structural Modificatio Usig Respose Fuctios (SMURF) [6] [7] [8]. By aalogy, acoustical bouded cavities ca be modeled as well as it is doe i structural dyamics: the acoustical FRF are calculated or measured ad a modal model is obtaied through experimetal or umerical modal aalysis. Therefore, acoustical modificatios ca be simulated. I the space idustry, high-frequecy combustio istability is a serious problem i rocket egie developmets, oce this pheomeo ca severely impair the operatio of the egie [9]. The couplig betwee the combustio process ad the acoustic modes of combustio chambers ca yield low efficiecy (or eve cause a explosio) of a rocket egie [10]. I view of atteuatig acoustic pressure fluctuatios iside combustio chambers, amog other passive techiques, Helmholtz resoators are widely used [9]. I this work, a method origially used to perform structural modificatios was applied to predict the acoustic effect of itroducig a modificatio i a origial acoustic cavity. The Acoustic Modificatio

2 Predictio (AMP) techique was applied by the itroductio of modificatios (Helmholtz resoators) i a liquid propellat rocket egie combustio chamber. The acoustic atural frequecies ad mode shapes of the origial chamber were umerically calculated by a Fiite Elemet (FE) model ad validated through acoustic experimetal modal aalysis performed i a mock-up without combustio. Afterwards, the resoators were desiged to atteuate the respose amplitude of oe specific atural frequecy ad a FRF was calculated cosiderig the dyamic behavior of the resoators. The umerical FRFs of the origial chamber were processed i combiatio with those of the resoator, i order to apply the acoustic modificatio method. I view of validatig the process, the acoustic FRF iside the chamber cavity are compared with experimetal results, cosiderig the istallatio of the referred Helmholtz resoators. Theory Review.1 Acoustics of Combustio Chambers Egie acoustic cavity characterizatio becomes a importat issue to be ivestigated, due to the combustio istability caused by the couplig betwee the combustio process ad the acoustic modes of the cavity. Durig burig tests [11], it was verified that the power spectrum of pressure levels has preseted soud pressure peaks i very well-defied frequecies with high magitude levels above the average pressure levels. These peaks are correlated with the resoace frequecies of the combustio chamber cavities. Such cavities, i geeral, preset logitudial ad trasversal (tagetial ad radial) acoustic modes, as well as, coupled modes, i.e., combiatios of those types. It is possible to describe the combustio chamber acoustics usig cold test procedures. Acoustic dyamics i combustio eviromets are obtaied by shiftig the cold test resoat frequecies by a scalar factor defied by the ratio of soud velocity at the cold test temperature ad at real operatio temperature [1]. Helmholtz resoators have bee applied as combustio stabilizatio devices for solid motors ad liquid rocket egies, with relative success. It could be oted i literature that they are used i a set of dozes or eve hudreds i each chamber cavity, distributed alog the walls or i a sigle row alog the ijector periphery [10].. Acoustic Modal Aalysis I classical structural modal aalysis theory, it is possible to trasform a ifiite degree-of-freedom system ito a limited umber of systems of oe degree-of-freedom. This limit is based o the umber of discrete poits, either for experimetal or umerical approaches. For acoustic cavity systems, oe observes the same hypothesis, cosiderig a elastic ad isotropic fluid medium, ad that modal reciprocity is valid. Other cosideratios were made i this work: the acoustic system is liear with light dampig ad light modal overlappig. The acoustic modal aalysis has bee developed sice the first studies [13], ad the iterest i develop acoustic sources [14] [15] has cotributed i this way. The primary study [16] showed that it is possible to perform experimetal ad umerical acoustic modal aalysis by usig the same tools as those used i classical structural modal aalysis.

3 ..1 Numerical Acoustic Modal Aalysis The Fiite Elemet Method (FEM) is a well-kow umerical predictio techique for solvig egieerig problems, which cosist of fidig the distributio of oe (or several) field variable(s) i a cotiuum domai, govered by a appropriate (set of) partial differetial equatio(s) ad boudary coditios. The most commoly used fiite elemet implemetatio for time-harmoic acoustic problems is based o a odal approximatio of the pressure field [17]. The fluid domai is discretized ito a umber of fiite elemets ad odes. Provided that the fiite elemet discretizatio ( mesh ) of the fluid domai cotais a total of odes, the acoustic pressure field i the domai ca be approximated, usig a set of odal shape fuctios. Based o a weighted residual formulatio of the Helmholtz equatio, a fiite elemet model i the ukow odal pressure values p i is obtaied M j a C a K a p i q i (1) The matrices [K a ] ad [M a ] are deoted as, respectively, the acoustic stiffess matrix ad the acoustic mass matrix. The acoustic dampig matrix [C a ] results from the ormal impedace boudary coditio. q is cosidered. The mode shape ad I this case, oly exteral acoustic source of volume acceleratio i atural frequecy predictios are obtaied from the followig eigevalue problem [17]: K. M ( m 1.. ) () a m m a m a where each eigevector m represets a mode shape ad where the associated eigevalue correspods to the square of the atural frequecy ω m of that mode. is calculated by usig the matrix iversio i equatio (3), which could imply i a large amout of computer resources. This process ca be simplified usig the modal superpositio method. The umerical FRF matrix FRF 1 a a a FRF M j C K (3).. Experimetal Acoustic Modal Aalysis Experimetal Modal Aalysis is a well-applied techique i structure dyamics. However, due to the developmet of commercial acoustic sources, Experimetal Acoustic Modal Aalysis ca be a suitable choice i view of extractig the acoustical FRF. I additio, the mathematical approach of the modal parameters extractio of structures ca be applied to acoustic systems, cosiderig their liear ature. There are may methods applied o experimetal data curve fittig, i order to match a parametric model. Amog them, the PolyMAX cosists of a least-squares complex frequecy-domai estimatio method, which uses the complex FRF data ito the bad of iterest. To costruct the modal model, it assembles a stabilizatio diagram icludig all possible modes [5]. The partial fractio expasio of the FRF matrix is writte i terms of modal parameters, as described i equatio (4). mod es * Rm R m LR * UR (4) 1 j Pm j P m e FRF m m m

4 where FRF is the FRF matrix, [R(m)] is the residue matrix, P(m) is the pole locatio for mode m, [LR] ad [UR] are, respectively, the lower ad upper residuals modelig the ifluece of the out-of-bad modes. The relatio betwee residues ad mode shapes is described by equatio (5). T R m A m where A(m) cotais the modal participatio factors. m m (5) The Modal Assurace Criterio (MAC) idicates the degree of liear depedece betwee two eigevectors ad ca be described as equatio (6). I this work, MAC was used to (i) validate the modes chose i the diagram, through the check of mutual orthogoality betwee modes from the same model; ad to (ii) check the orthogoality amog modes from differet modal models, comparig the experimetal ad umerical solutios. MAC ijm T * ìm jm T * T * ìrm im jm jm 100 (6).3 Helmholtz Resoators There is a wide rage of applicatios of a Helmholtz Resoator (HR) i order to suppress or atteuate the acoustic pressure iside cavities, rooms ad other volumes. There has bee still cosiderable effort ad iterest i the use of this kid of acoustic device to cotrol udesirable combustio oscillatios [10]. HR cosists of a small volume coected to a bigger cavity (a combustio chamber, i this case) through a orifice by a flaged eck. The dimesios of the resoator eed to be much less tha the acoustic wavelegth of iterest i order to cosider the resoator as lumped elemets coupled at a geometric discotiuity. The couplig coditio is that the oscillatory volume flow i the eck is equal to that imposed o the fluid i the cavity [18], eglectig the elastic property of the air i the eck. A typical HR is showed i Figure 1, beig d the eck diameter, D the cavity diameter, V c the volume cavity, l the eck legth ad L the cavity legth. P 1 is the icidet acoustic pressure ad P is the cavity pressure. Figure 1: Helmholtz resoator schematic Cosiderig that the gas beyod the ed of the eck moves as a uit with the gas iside the eck, it is ecessary to use a effective legth l eff which is bigger tha the true legth l of the eck [19]. The effective legth l eff is obtaied by addig a mass ed correctio δ, which is defied [10] by:

5 AR 0.85d for AR < 0.16 (7) where AR is the Area Ratio (A / A c ), beig A ad A c the eck cross-sectioal area ad the cavity crosssectioal area, respectively. The effective legth is calculated as l eff l. The acoustic iertace M defiitio applied to the Helmholtz resoator gives: where ρ is the air desity ad m is the effective mass. m l A l M (8) A A A eff eff The acoustic compliace C is defied as the volume displacemet that is produced by the applicatio of uit pressure [19]. A applicatio of this defiitio to HR produces: where c is the soud velocity. V c C (9) c The acoustic resistace R i the eck was approximated to the dissipatio associated with viscous forces, cosiderig the dyamic viscosity μ [10]: 8l R (10) A.4 Structural Modificatio Usig Respose Fuctios (SMURF) The first studies to estimate structural modificatios derived directly from experimetal Frequecy Respose Fuctios (FRF) were performed o the begiig of the 70s [6]. However, oly after the sigal acquisitio techologies were motivated by the advace of computig power that the iterest has itesified i this matter. I additio, the developmet ad verificatio of accurate modal models for structural modificatio, cosidered a complex ad legthy process, could be optimized usig SMURF. The SMURF provides, i these circumstaces, a excellet tool ot oly for the predictio of modificatios, but also for the validatio of a modal model. The positive poits of the techique come from the direct use of the FRFs. Thus, besides ot havig the eed for a curve fittig, the FRF already have the effect of the modes that are outside the frequecy rage cosidered, avoidig the estimatio of the residual terms which ormally accompay the modal model. The egative poits are related to the use of icosistet dataset [6]. A extesive list of works o modificatio techiques, also called reaalysis techiques, is preseted i referece [8]. Based o a aalysis of the list, it is possible to ote that the SMURF techique presets high degree of flexibility, sice it ca use directly umerical, experimetal or aalytical (i some cases) data to perform the calculatios. I additio, i cases where there is o eed to add degree of freedom to the structure, it is possible to avoid matrix iversio process. I order to calculate the FRF from umerical model, it is ecessary to cosider a system represeted by a stiffess matrix [K], a mass matrix [M] ad a structural dampig matrix [C]. Assumig the harmoic excitatio {F}, the respose {x} ca be preseted as: 1 x M jc K F The receptace matrix [α] of the origial system is defied [8] as (11)

6 1 M j C K The receptace matrix [γ] of the modified system is defied as: (1) 1 M M j C C K K (13) The modificatio matrices [Δ_ ] are icluded i the dyamic structural modificatio matrix [D], ad yields: 1 1 D (14) Cosiderig a structural modificatio with additioal degrees of freedom, the receptace matrices ca be partitioed as follows: the coordiates from origial structure oly (a), the coectio betwee the origial ad modified structure (b) ad the coordiates from the modified structure oly (c). The, the followig equatios ca be assumed. 1 aa ab ba bb 1 (15) 1 1 aa ab ac ba bb bc 00 ca cb cc D The developed method [8] perform the matrix maipulatios to show that the modified structure ca be calculated as fuctio of the dyamic structural modificatio matrix [D] ad the receptace matrix of the origial structure [α]. (16) 1 ba I 0 bb 0 ba D ca I 0 0D ba aa aa ab ca (17) (18) 3 Methodology The liquid propellat rocket egie mock-up [0] showed i Figure 3 faithfully represets the iteral acoustic cavity of the origial egie. The mai dimesios are: 18 mm of diameter ad 335 mm legth. The cavity is surrouded by alumium surface of approximately 6 mm thickess. I the left extremity of the origial egie, the face plate ad ijectors are located. I this mock-up, the face plate was simulated by a plae surface, where a acoustic source ad, i a secod step, HR were istalled. O the other had, the ozzle, which should be istalled i the right extremity of the mock-up, was eglected for the purpose of this study, oce the ozzle has o effect i the acoustic behavior of the mai chamber cavity [0]. The origial acoustic cavity was modeled usig the Fiite Elemet Method (FEM), which provided the atural frequecies ad mode shapes of such mid-complexity cavity form. After that, the Experimetal Acoustic Modal Aalysis was performed to validate the umerical model. This procedure starts from the measuremet of the Frequecy Respose Fuctios (FRF) ad, as a secod step, performs the dyamical parameters idetificatio: atural frequecies ad mode shapes, by usig the theory preseted i item..

7 The experimetal setup was performed to idetify oly the logitudial modes. Ad, it was desiged a set of three Helmholtz Resoators, which were istalled at the mock-up plae surface. It is importat to highlight that the HR were coceived to have its atural frequecy coicidet to the secod logitudial mode of the cavity. Usig the validated FE model or the origial cavity, a set of FRF were calculated by cosiderig the odes coicidet to the poits of the experimetal setup (item 3.) ad the odes where the HR will be istalled. Cosiderig the istallatio of the Helmholtz resoators as the modificatios i the origial cavity, the acoustic behavior of the modified cavity was estimated by the equatios i item 4 that are based o the set of umerical FRFs of the origial cavity ad the calculated FRF of the resoator. 3.1 Fiite Elemet Model The acoustic volume was modeled usig 11,136 liear solid hexahedral elemets, which yielded 1,510 odes (1,093 degrees of freedom), as ca be see i Figure.The fluid properties were characterized as the air, with desity of 1.5 Kg/m 3 ad the soud speed of 340 m/s. The acoustic umerical modal aalysis was calculated, from 0 to,400 Hz. Figure : Cavity s mesh 3. Experimetal Setup The mid-high frequecy volume acceleratio source [15] produces a voltage sigal proportioal to the volume acceleratio [m 3 /s ] variatio, with a omial frequecy rage of 00 up to 8,000 Hz. This source ozzle was istalled i the mock-up plae surface as show i Figure 3. The chamber excitatio was provided by a white oise fuctio geerator, a power amplifier ad the referred source. The microphoe was supported by a thi rod to be positioed i each measuremet poit iside the cavity. The pressure oscillatio iside the cavity was captured, ad registered ito the data acquisitio system. The samplig rate of 8,19 Samples/s (frequecy rage of 3,00 Hz) ad a legth of 6,400 poits assured a frequecy resolutio of 0.5 Hz. The acoustic cavity was dicretized i 7 poits of microphoe measuremets alog the cetral logitudial axis (Figure 3). The poit 0 represets the drivig poit FRF, due to the fact that it is the positio of a ideal omidirectioal poit source, called acoustic cetre [15]. The other poits (1 to 6) are placed 50 mm each other, from the face plate, i the same axis lie. The measuremets were doe by usig the free ru method, ad stopped automatically after 100 averages, to each oe of the 7 FRF. At each ew measuremet poit, the auto-rage process was applied to equalize the iput amplifier gai ad, the, to optimize the sigal-to-oise ratio.

8 Figure 3: Source ad microphoe positios (distaces i mm) The preseted measuremet methodology was repeated i order to acquire the same FRF, cosiderig the ew cofiguratio with the resoators, as ca be see i Figure Helmholtz Resoator Desig The Helmholtz Resoators were maufactured i a ylo material ad were divided i two parts: the cover ad the body (Figure 4). These parts, whe coupled, perform a cavity of 11 mm legth. The three HR covers were glued i the HR body, which were, fially, glued i the mock-up plae surface. Figure 4: Chamber with the HR istalled (left) ad HR dimesios i mm (right) It is possible to otice i Table 1 that oe of the dimesios is bigger tha 6% of the wavelegth of the resoat frequecy. The area ratio was AR = The parameters used i the desig were updated cosiderig the room temperature observed durig the experimet: approximately 8 C. Soud speed: c = m/s; desity of the air: ρ = kg/m 3 ; dyamic viscosity of the air: μ = 1.983x10-5 kg/ms. The tued resoace frequecy is 76 Hz (coicidet with the d mode of the chamber cavity). The acoustic parameters were calculated M = 1019 kg/m 4, C = 4.7x10-11 s m 4 /kg, R = 1468 kg/sm 4, usig the equatios (8), (9) ad (10), respectively. It was possible to obtai a trasfer fuctio (equatio (19)) of the desiged HR through the relatio betwee pressure iside the HR cavity (P ) ad the icidet pressure (P 1 ). The respective bode diagram is show i Figure 5.

9 P 1 P1 MCs RCs 1 (19) Dimesios (mm) Percetage of λ (%) d l D L λ Table 1: Dimesios of desiged HR Figure 5: Bode diagram of the desiged HR 4 Acoustic Modificatio Predictio Usig FRFs A set of Frequecy Respose Fuctios FRF were calculated from the umerical model of the origial cavity, as described i the item..1. The equatio (0) shows the procedure to obtai FRF usig the cocept of structural receptace matrix of equatio (1) i the formulatio of acoustic matrices i equatio (3). 1 a a a FRF M j C K (0) The estimatio of the acoustic FRFs from the modified cavity FRF ca, the, be calculated, as show i equatio (1). By aalogy, equatios (17) ad (18) are used to fid the behavior of the modified cavity. 1 1 FRF FRF D act (1)

10 The acoustic modificatio matrix D act cotais the FRF of the resoators that are calculated usig equatio (19), which uit is pressure/pressure, or Pa/Pa. I order to equalize the uits ad the gai of the trasfer fuctio, a ormalizatio was performed usig the drivig poit fuctio FRF (uit: Pa/(m 3 /s )) which is located i the iterface of the origial cavity with the resoator. Additioally, a iversio of the resoator FRF is eeded to guaratee the uits compatibility of equatio (1) (uits: (m 3 /s )/Pa). bb 5 Results The idetified atural frequecies are summarized i Table. The frequecies related to trasversal modes were idetified oly by the umerical process. Cosiderig the first four logitudial modes, the maximum error comparig the umerical ad experimetal estimatio of atural frequecies was 6.6%. Frequecy / Hz Error / % Mode Experimetal Numerical Logitudial (1) Logitudial () Trasversal (1) Trasversal () Logitudial (3) Trasversal (3) Trasversal (4) Logitudial (4) Table : Compariso betwee experimetal ad umerical atural frequecies Visualizig acoustic mode shapes is a hard task i experimetal aalysis. Figure 6 presets a parallel aalysis: the two first logitudial umerical modes i the left colum ad a compariso betwee experimetal ad umerical modes (the same showed i left colum) cosiderig the ormalized amplitude i the measured poits (0 to 6), i the right colum. To perform the compariso it was selected the relative amplitudes from the eigevectors of the umerical results i the odes that coicide with the measured poit locatios. I this way, the right colum i Figure 6 presets poor umerical results; however, they are compatible with the estimated experimetal mode shapes. The ormalized amplitudes were plotted together with its egative form to icrease the visualizatio of odes (itersectio of lies) ad ehace the comprehesio whe compare with the left colum figure. Additioally, it was possible to calculate the MAC (equatio (6)) betwee experimetal ad umerical modes. These correlatio idicators, MAC, are bigger tha 98%, cosiderig the first two logitudial modes. I the first mode, the agreemet betwee umerical ad experimetal results reaches high levels. The ode i the secod mode is almost at the same poit. The modes relative amplitudes are compatible whe comparig the umerical ad experimetal curves (right colum) ad presets good agreemet with the color scale o the left colum. The experimetal results (Figure 7) of the origial ad modified cavities, regardig the FRF from poits ad 3 with excitatio i poit 0 shows the atteuatio aroud the secod mode, due to the modificatio (HR istallatio). It was perceived a atteuatio of approximately 9 db i that frequecy rage. O the other had, the measuremet process was well-established, oce the FRF of origial ad modified cavities are almost exactly the same, except for the regio of secod mode, where the HR is tued.

11 Numerical Experimetal x Numerical Logitudial (1) MAC: 99.1 % Logitudial () MAC: 98.1 % Figure 6: Numerical logitudial modes (left) ad compariso betwee experimetal (blue solid lie) ad umerical (red dashed lie) logitudial mode shapes (right) Figure 7: Experimetal FRF: poit (left) ad poit 3 (right) with referece i poit 0

12 By comparig the results of umerical FRF ad experimetal FRF (Figure 8 (left)) from the origial cavity, it is possible to verify that the cocordace is satisfied i a good level, maily aroud the secod atural frequecy, where the resoators were tued. The applicatio results of the Acoustic Modificatio Predictio (AMP) method ca be verified i Figure 8 (right). Aroud the frequecy of iterest (730 Hz), it ca be oted two ew atural frequecies: before ad after the secod atural frequecy of origial cavity. This behavior is expected, as could be see i the experimetal results (Figure 7). However, the magitudes are differet, with peaks ad the ati-resoace beig magified. Additioally, a extra atural frequecy is oted aroud 500 Hz, probably, a calculatio residue, or ifluece of other FRF. Below 500 Hz, the AMP method maitais the origial FRF, although above 900 Hz, the method results are distorted regardig the origial FRF. Figure 8: Poit 3: umerical x experimetal results of origial cavity (left); ad umerical x acoustic modificatio method cosiderig HR (right) A set of parameters were slightly shifted i order to idetify their ifluece i the method s result. Two of them appear to be the more ifluetial: DC gai ad dampig of resoator model. About the dampig, the model cosiders oly the effect of viscous resistace of the gas movemet i the eck. Probably, it will be ecessary to iclude (i the ext steps of the work) the ope-ed soud radiatio resistace of the eck, i order to icrease the dampig. The model DC gai demads a deeper aalysis, but it could be observed that a chage i gai ca improve the results of the AMP method, as ca be see i Figure 9: i left, comparig the AMP result to the origial umerical FRF; i right, comparig to the experimetal modified cavity. I additio, the peak ear 500 Hz was reduced. This effect will be ivestigated i the ext steps.

13 Figure 9: Poit 3 with gai modificatio: umerical FRF from origial cavity x acoustic modificatio method (left) ad experimetal, cavity with HR x acoustic modificatio (right) 6 Coclusios The adopted Experimetal Acoustic Modal Aalysis proved to be a appropriate tool to characterize a acoustic cavity. The use of a volumetric source plays a importat role i the process, oce allows the accurate measuremet of acoustic FRF. The umerical model results were validated ad used as the basis for the HR desig, i a first momet. I additio, umerical ad experimetal results were used to idetify ad localize the ode regios ad the maximum amplitude poits regios of each mode. The desig of the resoators seemed to be adequate, oce it was verified a atteuatio of 9 db or greater, depedig o its locatio iside the chamber. The experimetal validatio could be doe due to the good quality of the FRFs. The AMP method was applied to the case of Helmholtz resoators isertio o a rocket egie combustio chamber. The first results were uexpected, oce the behavior aroud the frequecy of iterest was distorted. The strategy of assurig the couplig betwee the cavity ad resoator was based o the drivig poit FRF. However the HR model gai does ot seem to be appropriated i this way. It was observed that by adjustig the gai of the model; the calculated FRF agrees better with the expected FRF. It will be the aim of ivestigatio i the ext steps of this work Aother issue to be ivestigated is the dampig icreasig. A optio that could be adopted is to update the acoustic resistace of the resoator model usig the experimetal results. I this way, the radiatio resistace could be cosidered i the model. Ackowledgemets The authors would like to thak the Istitute of Aeroautics ad Space (IAE) ad Techological Istitute of Aeroautics (ITA) for the fiacial support. The authors ackowledge the support from the Natioal Istitute of Sciece ad Techology o Smart Structures for Egieerig (INCT-EIE-CNPq).

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