ACOUSTIC FILTERS CHARACTERIZATION BASED ON FINITE ELEMENTS MODELS

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1 ACOUSTIC FILTERS CHARACTERIZATION BASED ON FINITE ELEMENTS MODELS Cartaxo, António José Ferreira Mourão SUMMARY: In this paper, acoustic filters characterization is made using finite elements models. These models were built to verify the results of sound wave propagation through repetitive structures obtained with Bloch wave techniques and to study the effect of some geometric modifications in a simple expansion chamber muffler in sound attenuation. Results obtained include an example of acoustic attenuation through an array of metallic cylinders, and others of one expansion chamber muffler. KEYWORDS: wave propagation, passive attenuation, periodic filters, mufflers, band gap, finite elements. 1. INTRODUCTION The first problem investigated here is the sound transmission through a periodic structure. This subject continues receiving interest from researchers in areas dealing with the existence of absolute acoustic band gaps and, also, in areas dealing with the sound attenuation by an array of rigid cylinders (see [1, 2]).Bloch wave technique allows, for this cases, the prediction of frequency ranges where no wave propagates or where attenuation is considerable. The first aim of the present work is to check the validity of such predictions when a finite repetition of the medium is considered instead of the infinite repetition. For it, a numerical model and also a physical prototype were developed by Cartaxo and Neves (see [3, 4]) to investigate the presence of such acoustic attenuation for specific frequency ranges. A Fluid-Structure Interaction (FSI) modeled by a finite element technique is used to analyze harmonically the case of a wave propagating in the air through a finite repetition of base cells, e. g. periodic distribution of cylinders perpendicular to the direction of the wave propagation. The model assumes no friction dissipation in the solid material model, no rotational fluid, perfect reflection in the material surfaces and material continuity between all elements (tubes and metallic sheets). The fluid (air) was modeled without friction losses. The second problem investigated here work is to predict the acoustic behavior of expansion muffler, referring to finite element models and comparing some results with those predicted by theory. This subject continues receiving interest from researchers due the demand of mufflers with complex geometries. Although not studied here, the response of complex geometry muffler can hardly be study by conventional methodologies. We had reviewed briefly the response of mufflers with simple geometries to check the potential and limitations of finite element methods.

2 This article is organized as follows: Section 2 contains a brief description of the theoretical formulation. In section 3 a brief comparison of sound pressure distribution in a uniform section tube model is compared with theory. In section 4, acoustic attenuation results from Bloch wave technique (infinite medium) are compared with the ones obtained for finite repetitive medium through a fluid-structure interaction medium. In section 5 some muffler models of one expansion chamber are compared in terms of attenuated frequency range and sound attenuation and, finally, in section 6, conclusions are presented. The experimental results obtained with the built prototype in an anechoic chamber compare reasonable well with the numerical results, although some laboratory difficulties found are also presented here. 2. TEORICAL FORMULATION, MODELLING AND NUMERICAL PROCEDURES 2.1. Wave propagation model To model the wave propagation trough the periodic solid, a finite-element analysis is used for the linear acoustic jωt harmonic analysis. The acoustic pressure p is assumed to be harmonic, p = p0e, and the solution is obtained from the following differential equation (vd. [18]): ω c 2 2 p0 = p 2 0, (1) where ω is the excitation frequency, c the speed of sound in the propagation medium, and p 0 is the pressure magnitude. Fluid-structure interaction (FSI) is also considered through the relations between K, C and M, i. e. the stiffness, damping, and mass-matrix, respectively, and F the force vector. [ R ] is a coupling matrix that appears in equation (2) in the form K FSI = [ R] and also in M FSI = ρ0[ R]. By placing the unknown load on the left hand side of the equations one gets the following system of equations (vd. [29]): M M S FSI 0 M f.. u u C.. + p 0 0 C P. S u K. + p 0 K K FSI f u F = p F This system of equations requires that nodes on a fluid-structure interface have displacement and pressure degrees of freedom. S f (2) 2.2. Transmission loss in a simple expansion chamber muffler Transmission loss through a simple expansion chamber muffler is predicted by expression (3). This result is obtained through the continuity of pressure and fluid flow in both section transition of the muffler (vd. [21]): TL log m sin ( kl ) (3) 4 m = e where m is the areas ratio in the section transition, length. ω k = wave number, c sound speed and l e the camera c 3. COMPARISON OF RESULTS IN UNIFORM SECTION TUBE MODELS A finite model of sound wave propagation in a Ø40x500mm constant section tube was calculated by theory and compared with finite element model. The model didn t consider fluid-structure interaction, viscosity losses in the fluid and fluid flow, and a plane sound wave of 1Pa (94dB) was imposed at the left top (Figure 1, a)). At the right top a rigid steel top of 4mm thickness was modeled to reflect all acoustic incident sound waves (Figure 1 a), red volume). 2

3 Figure 1 Uniform section tube model, a), and respective finite element mesh, b). For a tube with absolute reflection at the right top, sound pressure distribution of a plane wave propagating in the air for the excitation of 1500Hz is presented in Figure 2 b) (red line). Figure 2 Sound pressure (Pa) distribution at 1500Hz obtained by FEM, a), and comparison of sound pressure values along the axis (black line) with theory (red line), b); the green line represents the relative error, for a model with 36 elements per wave length. The results presented at Figure 2 b) were obtained for a model with 36 elements per wavelength. It is shown a good concordance with theory and, ignoring the singular points around 0Pa, less than 5% errors are obtained. 4. RESULTS FROM THE ACOUSTIC ATTENUATION THROUGH A FINITE ARRAY OF METALLIC CYLINDERS D Numerical Results From 2D models implemented, it was found that the results did not change with the fluid-structure interaction (due to the right in plane rigidity of the 2D metal tubes compared with the air). For this reason, the 2D studies presented next assume air holes (vacuum) in place of the tubes and still give rigorous results. The model was built with 6 parallel lines of 4 cylinders of diameter of 40mm, with a filling fraction of (as described ). A single layer of 2D acoustic fluid elements is loaded with a harmonic pressure wave of 1Pa at the left side of the array, and an anechoic termination is imposed at the opposite top. The resulting sound pressure distribution for a frequency of 1560 Hz is the shown at Figure 3 a). 3

4 Figure 3 Sound pressure distribution for a plane wave from left to right excited at a frequency of 1560 Hz, a), and sound pressure attenuation obtained with the same model for an interval of frequencies from 500Hz up to near 2500Hz (red points). The black line was obtained by Sánchéz Pérez et al.[2]. The curve for the attenuation obtained with this model is presented at Figure 3 b). The presence of a considerable attenuation between 1250Hz and 1750Hz is evident. These results are much close to that obtained [14]. A question remains. Can one expect that the results are still valid for the corresponding 3D prototype? The main differences expected are the higher flexibility of the tubes and its fixation at extremes D Numerical Results Tubes Fixed at One End Introducing the metallic tubes and activating a fluid-structure flag were obtained the 3D results of Figure 4 a). An excitation frequency of 1500Hz is used. It can be observed an irregular sound pressure distribution especially after the first tubes. By studying a continuous decrease of the tubes free length, it was observed that the shorter the tubes, the more similar to 2D were the results and the more the sound wave tended to stay plane, as illustrated by Figure 4 b). Figure 4 Sound pressure distribution (db) obtained for fix-free tubes with a 500mm free length, a), and with a 125mm free length, b). In Figure 5 are shown the vibration modes of the cylinders in the frequency range, evidencing that these are responsible for the bad distribution of sound field represented in Figure 4 a). 4

5 Figure 5 Vibration modes of the cylinders in the excited frequency range. First mode at 1153Hz, a), and third mode at 1795Hz, b). This FEM model was modified to keep studying the sound attenuation, increasing the rigidity of the tubes as shown in next chapter Tubes Fixed at Both Ends An immediate idea is to require the tubes to be fixed at both ends. The results obtained (see Figure 6 a)) are very similar to those obtained earlier and presented in Figure 3 a). For the frequencies between 1000Hz and 2000Hz, the attenuation values are presented in Figure 6 b). Figure 6 Sound pressure (db) distribution obtained for a model with tubes fixed at both ends, a), and sound pressure attenuation comparison between 2D (red line) and 3D (green line) numerical models. To verify this apparent concordance between the FEM models and Sánchéz Pérez et al. results, a physical prototype was built and tested as shown in next chapter Test of the Prototype in an Anechoic Chamber A physical model was built by connecting the tubes to two metal sheets with fasteners (nut and bolt) and was positioned on the floor, as illustrated by Figure 7 a), and it was tested in an anechoic chamber, with the setup indicated at Figure 7 b). The sound attenuation measured with the prototype showed some agreement with the numerical results and with those presented in [2] (see Figure 8). 5

6 Channel 1 Oscilloscope Channel 2 Amplifier 1 Amplifier 2 Signal Generator Pre amplifier 2 Pre amplifier 1 Microphone 2 Microphone 1 Speaker Figure 7 Final prototype, a, and speaker and microphone set-up used in the measurements, b). Figure 8 Comparison of sound attenuation results obtained by measurement (green line, for values obtained with a sound meter, and red line, with two microphones) and by Sánchéz Pérez et al. [2] (black line). During the experiments in the anechoic chamber, some difficulties were observed, mainly due to the available material. 5. RESULTS FROM THE ACOUSTIC ATTENUATION THROUGH A SIMPLE EXPANSION CAMERA MUFFLER Let us change to the next problem studied at this work. For a simple expansion chamber muffler the presented a satisfactory concordance with the expected theory results, namely the peaks and valleys of the attenuation, Figure 9 b). The muffler was modeled with the following dimensions: tubes Ø40x240mm and chamber Ø180x240mm. The models didn t include fluid-structure interaction and it was considered fluid without friction losses. It was applied a plane wave of 1 Pa (94dB) at the left top and the opposite top was modeled as anechoic. It can be seen that the attenuation obtained is about 30% bigger that that predicted by theory using (3). This fact can be explained due to the number of elements used per wave length, for the case, 24 (a result obtained by Cartaxo is that for the study of acoustic wave propagation should have a minimum of 36 elements per wave length should be used, to get a relative error of 10% [3]). A model with more finite elements per wavelength should be use. That wasn t done because the frequency ranges studied and the modified models were enough to take a long time to get the comparison of results. 6

7 Figure 9 Sound level (db) distribution in a simple expansion chamber muffler, a), and results of sound attenuation (black points), compared with theory (red points), b). Some modifications were made to the initial model, Figure 10, to study the effects of geometric changes in the global attenuation. It is shown that the best modification is to include interior divisions in the chamber for the studied frequencies range, Figure 11. Figure 10 Results of sound pressure level distribution at 2230Hz for: two simple mufflers is serie, a); one expansion chamber muffler with extended tubes, b); a simple expansion chamber muffler with a internal division, c), and a simple expansion chamber with two internal divisions, d). 7

8 Figure 11 Comparison of attenuation results for one expansion muffler model and its modifications of Figure 10. Figure 11 shows that while increasing the number of internal divisions in the expansion chamber, the attenuated frequency range is increased, as well as the attenuation values. These results should, in future, be verified with models using more elements per wavelength (minimum of 36 elements as studied in section 3) and should be compared with theory presented in literature [21]. 6. CONCLUSIONS The first numerical model described gives the possibility of modeling acoustic attenuation in a media composed by an array of metallic cylinders. In this paper, the actual status of the work dealing with wave propagation through a finite 3D periodic material was presented. A good agreement between the results from the infinite repetition model and from the finite repetition model and from the prototype tested in anechoic chamber was observed. The experimental work is not concluded and will require more experiments to acquire better results. The second numerical model and it modifications shows that while increasing the number of internal divisions in the expansion chamber, larger are the attenuated frequency range an bigger are the attenuations. FEM works well with conventional muffler geometries and some comparison at no conventional geometries should require some experimental verification. 7. ACKNOWLEDGMENTS This work received support from FCT - through the project POCTI/EME/ 44728/2002 (MMN), FEDER - and from IDMEC-IST. The authors are grateful to Mr. Onofre Moreira and Rafael Serrenho from Centro de Análise e Processamento de Sinais - CAPS (IST, Lisbon), Technical University of Lisbon, for providing their support and time during the tests at the CAPS s anechoic chamber. 8. REFERENCES [1] C. Barbarosie, M. M. Neves, Periodic Structures for Frequency Filtering: Analysis and Optimization, Computers and Structures, , , [2] J. V. Sánchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-Dehesa, F. Sánchez-Pérez, J. Llinares, F. Gálvez, Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders, Physical Review Letters, The American Physical Society, USA, [3] A. J. Cartaxo, M. M. Neves, Análise da Propagação de Ondas em Estruturas Periódicas Finitas e Validação de Resultados Obtidos com Técnicas de Ondas de Bloch, Relatório 001/2006, IDMEC, Lisbon, Portugal, [4] A. Cartaxo, M. M. Neves, Validation of Band Gap Results Obtained with a 3D Numerical Model, Departamento de Engenharia Mecânica, Instituto Superior Técnico, Lisbon,

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