MODELLING OF ACOUSTIC POWER RADIATION FROM MOBILE SCREW COMPRESSORS


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1 11 th International Conference on Viration Prolems Z. Dimitrovová et.al. (eds.) Lison, Portugal, 9 12 Septemer 213 MODELLING OF ACOUSTIC POWER RADIATION FROM MOBILE SCREW COMPRESSORS Jan Dupal* 1, Jan Vimmr 1, Ondřej Bulík 1, Michal Hajžman 1 1 European Centre of Excellence NTIS  The New Technologies for Information Society, Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Repulic Keywords: Screw Compressor, ViroAcoustics, Helmholtz Equation, Finite Element Method. Astract. The method for a numerical solution of the viroacoustic prolem in a moile screw compressor is proposed and the inhouse 3D finite element (FE) solver is developed. In order to reduce the complexity of the prolem, an attention is paid to the numerical solution of the acoustic pressure field in the compressor cavity interacting with linear elastic compressor housing. Propagation of acoustic pressure in the cavity is mathematically descried y Helmholtz equation in the amplitude form and is induced y periodically varying surface velocity of the compressor engine which can e determined experimentally. Numerical solution of Helmholtz equation for acoustic pressure amplitude distiution inside the cavity with regards to prescried oundary conditions is performed using FE method on unstructured tetrahedral grid. For the FE discretisation of the elastic compressor housing (modelled as a thin metal plate), a sixnoded thin flat shell triangular finite element with 18 DOF ased on the Kirchhoff plate theory was developed and implemented. The resulting strong coupled system of linear algeraic equations descriing the viroacoustic prolem, i.e. the prolem of interaction etween the air inside the cavity and the screw compressor housing, is solved numerically y wellknown algorithms implemented in Matla. The developed 3D FE solver is verified against the approximate analytical solution of a specially designed enchmark test case. A construction of the enchmark test analytical solution is also presented in the paper. The verified FE solver is applied to viroacoustic prolem of the simplified model of a screw compressor and its numerical results, i.e. distriution of the acoustic pressure amplitudes in the cavity and asolute values of the compressor housing deflection amplitudes, are discussed.
2 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman 1 INTRODUCTION Considering customer requirements and the necessity to satisfy hygienic standards, the producers of moile screw compressors are compelled to minimise the emitted acoustic power and the related noise. The ongoing research is focused on a proposal of a suitale and efficient method and its implementation within an inhouse computational software for the numerical solution of the acoustic power radiation from screw compressors. The knowledge from computational results will e used to alter the moile screw compressor design so that the total emitted acoustic power is significantly decreased. The solution of the aove stated goals represents a complex prolem. This paper is primarily focused on the numerical solution for the viroacoustic prolem of the simplified model of the moile screwcompressor. Main attention is paid to the acoustic pressure distriution [1, 2, 3, 4] in the compressor cavity interacting with the linear elastic housing of the screw compressor. It is assumed that the propagation of the acoustic pressure in the cavity is induced y periodically varying surface velocity of the compressor engine, which is known a priori (e.g. experimentally measured). Since the periodic function can e expressed in the form of Fourier series, the prolem can e solved for a single frequency value and with regards to linearity of the prolem the gloal solution can e derived using a superposition of the harmonic solutions. Based on this assumption, the acoustic pressure p (x, t) and the acoustic velocity v (x, t) can e expressed in the complex harmonic form p (x, t) = p(x)e iωt and v (x, t) = v(x)e iωt, respectively, where ω is an angular frequency of compressor engine movement. The method used for numerical solution of the viroacoustic prolem is implemented into a newly developed 3D finite element (FE) solver in Matla. The FE solver is verified ased on the approximate analytical solution of a specially designed enchmark test case. Furthermore the FE solver is applied to the viroacoustic prolem in the simplified model of a screw compressor and its numerical results, i.e. distriution of the acoustic pressure amplitudes in the cavity and asolute values of the compressor housing deflection amplitudes, are discussed. 2 PROBLEM FORMULATION In the following, the computational domain is divided into two regions, the compressor cavity Ω R 3 and the elastic compressor housing Ω R 3, as displayed in Figure 1. The cavity domain Ω is ounded y the oundary Ω = Γ in Γ out Γ w Γ t where Γ in, Γ out and Γ w denote the inlet, the outlet and the rigid walls of the computational domain, respectively. The elastic housing Ω is ounded y oundary Ω = Γ f Γ u Γ t where Γ f denotes the oundary with prescried force loads and Γ u denotes the oundary with prescried displacement field. Figure 1: Computational domain The interface etween the two interacting domains, i.e. etween the cavity Ω and the housing Ω, (Ω Ω) R 3. is denoted as Γ t. In the cavity domain Ω, the distriution of acoustic pressure p(x) is descried y the Helmholtz equation in the amplitude form [2, 4, 5] k 2 p + p = (1) where k = ω is the wave numer, c is the speed of sound and ω is the angular frequency of the c engine motion. The weak solution of the Helmholtz equation is given as 2
3 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman Ω k 2 ϕ p dω Ω ϕ p dω + Ω ϕ p ds = (2) n where ϕ(x) is a well chosen test function. The momentum conservation law for the inviscid fluid yields [2, 4] p n = iωϱ av n (3) where i is the imaginary unit, ϱ a is the air density and v n = n v is the normal component of the acoustic velocity. The oundary condition v n = is prescried at the rigid wall oundary Γ w which yields Γ w ϕ p ds = n Γ w ϕ i ωϱ a v n ds =. The normal velocity v n at the anechoic outlet Γ out is given as v n = p/(ϱ a c) [1, 4] which yields Γ out ϕ p ds = n Γ w ϕ i ω p ds. For c the inlet oundary Γ in (i.e. the engine surface), the surface acoustic velocity v n is prescried. The interface Γ t etween the cavity and elastic housing holds Γ t ϕ p ds = n Γ t ϕ i ωϱ a v n ds, where the normal component of the acoustic velocity v n is an unknown velocity resulting from the interaction etween the acoustic environment in the cavity and the elastic housing of the screw compressor. The solution in the housing domain Ω is ased on the principle of virtual work (PVW) in the following form δε T σ d Ω = ϱ h δu T ü d Ω + p δu T n ds (4) Γ t Ω Ω where ε = [ε x, ε y, ε z, γ yz, γ xz, γ xy ] T is the strain vector, σ = [σ x, σ y, σ z, τ yz, τ xz, τ xy ] T is the stress vector, u = [u, v, w] T is the displacement vector and ϱ h is the material density of the housing. The surface integrals over the oundary Γ u with prescried zero displacements and Γ f with prescried zero loads are equal to zero and are therefore omitted from Eq FINITE ELEMENT METHOD DISCRETISATION The finite element discretisation of the cavity domain Ω R 3 is carried out for an unstructured tetrahedral mesh, a sample tetrahedral element is shown in Figure 2(left). The amplitudes of the acoustic pressure and the acoustic velocity are approximated linearly as p(ξ, η, ζ) = Φ(ξ, η, ζ)x e S 1 e p e, v ν (ξ, η, ζ) = Φ(ξ, η, ζ)x e S 1 e v e ν, ν {x, y, z}, (5) where Φ(ξ, η, ζ) = [1, ξ, η, ζ], p e = [p 1, p 2, p 3, p 4 ] T is the vector of the acoustic pressure amplitudes at each node of the finite element, Figure 2(left), and vν e = [v ν1, v ν2, v ν3, v ν4 ] T, ν {x, y, z} is the vector of νth components of the acoustic velocity amplitudes at each node of the finite element. The matrices S e and X e are of the following form S e = 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4, X e = 1 x 1 y 1 z 1 x 2 ȳ 2 z 2 x 3 ȳ 3 z 3 x 4 ȳ 4 z 4, (6) while x i = x i x 1, ȳ i = y i y 1, z i = z i z 1 for i = 2, 3, 4. The test function ϕ(ξ, η, ζ) is approximated linearly, similarly to the amplitudes of the acoustic pressure and the acoustic 3
4 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman Figure 2: Tetrahedral finite element Ω e (left) and normalised finite element Ω e (right). velocity. Sustituting the approximations of the test function and of the acoustic pressure and velocity amplitudes, Eq. 5, into the weak solution given y Eq. 2 and taking the oundary integrals into consideration, we otain a system of linear algeraic equations ω 2 J e c 2 S T e X T e A X e S 1 e } {{ } H e 1 p e 6 J e S T e X T e L T D T e D e LX e S 1 e } {{ } G e p e (7) iω J ej c S T e X T e ÃjX e S 1 e p e iωϱ a Jej S T e X T e A j T e P v e = iωϱ h J }{{}}{{} ej S T e X T e A j T e P }{{} F C t e e C in e where the matrices H e and G e are computed for all inner elements of the cavity domain Ω and the matrices F e, C t e and C in e are computed for the oundary elements at the oundaries Γ out, Γ [ ] t T Φ and Γ in, respectively, see Figure 1. The matrix L = T, ΦT, ΦT ξ η ζ, the matrices De and T e are of the following form ȳ 3 z 4 z 3 ȳ 4 z 2 ȳ 4 ȳ 2 z 4 ȳ 2 z 3 z 2 ȳ 3 X e S 1 e D e = z 3 x 4 x 3 z 4 x 2 z 4 z 2 x 4 z 2 x 3 x 2 z 3, T e = X e S 1 e, (8) x 3 ȳ 4 ȳ 3 x 4 ȳ 2 x 4 x 2 ȳ 4 x 2 ȳ 3 ȳ 2 x 3 X e S 1 e P is the permutation matrix that changes the sequence of the FE parameters, J e = det J e where J e = x 2 ȳ 3 z 4 x 2 ȳ 4 z 3 x 3 ȳ 2 z 4 + x 3 ȳ 4 z 2 + x 4 ȳ 2 z 3 x 4 ȳ 3 z 2, Jej expresses twice the area of 1 1 ξ 1 ξ η each face of tetrahedral element, A = Φ T (ξ, η, ζ)φ(ξ, η, ζ) dζ dη dξ, the matrices Ã j, j = 1, 2, 3, 4 are otained from the integration of the product of ase functions Φ(ξ, η, ζ) over each face of the normalised tetrahedral element, Figure 2(right), and the matrices A j, j = 1, 2, 3, 4 are otained from the integration of the product of ase function matrices for the test function and for the normal velocity over each face of the normalised tetrahedral element. After dividing Eq. 7 y the term ( iω) and after assemling the gloal matrices, the following equation is otained ( 1 iωh + 1 ) iω G + F p + C t v t = C in v in (9) ϱ a where v in is the vector of prescried acoustic velocity amplitudes in the element nodes located at the engine surface Γ in and v t is the vector of unknown acoustic velocity amplitudes in the element nodes located at the interface Γ t. v e 4
5 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman Figure 3: Sixnoded thin flat shell triangular finite element Ω e efore (left) and after (right) transformation. For the FE discretisation of the housing domain Ω R 3, a new 6noded thin flat shell triangular finite element with 18 degrees of freedom (DOF) was developed, see Figure 3(left). This thin flat shell element is ased on the Kirchhoff plate theory. Each corner node i, j and k contains three displacements û, ˆv, ŵ and two rotations ŵˆx, ŵŷ. The midside nodes l, m and n store the information aout normal derivatives of deflection ŵ n (l), ŵ n (m), ŵ n (n), where the sense of rotation is determined y the sign of difference of numers of the corresponding edge corner nodes, see Figure 3(left). Applying the PVW Eq. 4, the equation of motion for a housing element in the amplitude form is otained with regards to a local coordinate system (ˆx, ŷ, ẑ) of the element as iωm e v h e + B e v h e + 1 iω K ev h e + Q e p e =, (1) where M e and K e are the derived mass and stiffness matrices of the 6noded thin flat shell triangular finite element, B e = βk e is the proportionaldamping matrix and ve h is the vector of velocity amplitudes for the nodes located in the housing of the screw compressor. The matrix Q e expresses the distriution of acoustic pressure amplitudes at the element Ω e. In order to assemly the gloal matrices of mass M, stiffness K and proportionaldamping B and the gloal loading vector Qp, it is necessary to perform a transformation of all matrices and vectors from the local coordinate system of the element (ˆx, ŷ, ẑ) to the gloal coordinate system (x, y, z). This way, two rotations ŵˆx, ŵŷ at each corner node i, j and k of the element are transformed into three rotations ϕ, ϑ, ψ in the gloal coordinate system (x, y, z), while the rotations in the midside nodes l, m and n are not transformed, i.e. ŵ n (l) sgn(j i) α (l) sgn(j i), ŵ n (m) sgn(k j) α (m) sgn(k j) and ŵ n (n) sgn(k i) α (n) sgn(k i). The displacements û, ˆv, ŵ in the corner nodes i, j and k of the element are transformed into displacements u, v, w. Using this transformation the otained finite element has 21 DOF, as shown in Figure 3(right). The resulting matrix equation descriing the motion of the compressor s housing in the gloal coordinate system (x, y, z) yields iωmv h + Bv h + 1 iω Kvh + Qp =. (11) The vector of acoustic velocity amplitudes v h can e divided into the suvector of velocity amplitudes v t of compressor housing nodes at the interface Γ t and into the suvector of velocity amplitudes v Ω of the other nodes inside the compressor housing domain Ω. Thus the coefficient 5
6 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman matrices M, B, K can e divided into dedicated locks and Eqs. (9), (11) can e rewritten as ( 1 ϱ a iωh+ 1 G+F) C t p C in v in iω iωm 11 +B K iω 11 iωm 12 +B K iω 12 v Ω =. (12) Q iωm 21 +B K iω 21 iωm 22 +B K iω 22 v t The resulting strong coupled system of linear algeraic equations descriing the viroacoustic prolem, i.e. the prolem of interaction etween the air inside the cavity Ω and the elastic housing Ω, is solved numerically y wellknown algorithms implemented in Matla. 4 ANALYTICAL SOLUTION OF THE VIBROACOUSTIC PROBLEM In order to verify the method descried aove and to validate numerical results of the developed solver, a enchmark test case, which can e solved analytically, is suggested. For simplicity, a rectangular cavity domain Ω R 2 with dimensions a is considered. The amplitude of normal velocity vn in (x) = A sin 2πx is prescried at the oundary Γ in at the ottom of the cavity domain and normal velocity v n = is prescried at the rigid side walls Γ w. At the top of the cavity, there is a simple supported elastic eam Ω R 1. Between the cavity and the elastic eam, there is the interface oundary Γ t. The test setup is displayed in Figure 4. It is possile to find an approximate analytic solution of this simple viroacoustic prolem descriing the distriution of the acoustic pressure amplitudes in domain Ω and the deflection amplitudes in domain Ω. Figure 4: Computational domains Ω R 2 and Ω R 1 for the enchmark test case. In this case, the weak solution of Eq. 1 for acoustic pressure amplitudes is a ω 2 c ϕ p dydx 2 a ϕ p dydx + Γ in Γ w Γ t ϕ p ds = (13) n where ϕ = ϕ(x, y) are wellchosen test functions. It is assumed, that the solution for the amplitude of the acoustic pressure can e approximated y finite series p(x, y) = k=1 l=1 ( a kl sin kπx sin lπy a + kl cos kπx sin lπy a + (14) +c kl sin kπx cos lπy a + d kl cos kπx cos lπy a where N is a numer of considered harmonics in x and y directions. With regards to Eq. 3, the following conditions are applied at the rigid wall oundary Γ w p n = iωϱ av n = = p p (, y) = (, y) =. (15) x x Thus the a kl and c kl terms must e excluded from Eq. 14. The final form of the approximate solution for the acoustic pressure amplitudes is then 6 )
7 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman p(x, y) = k=1 kl cos kπx l=1 sin lπy a } {{ } ϕ kl (x,y) +d kl cos kπx cos lπy a } {{ } ϕ kl (x,y). (16) Due to the fact that Eq. 1 is of second order, the oundary conditions p are unstale and the n solution must e from a function space where a ϕ kl(x, y) dydx = a ϕ kl (x, y) dydx =, which are the functions with null mean values. The function in Eq. 16 meets this requirement. The oundary integral over Γ w from Eq. 13 is omitted with regards to Eq. 15. The velocity v n at the oundary Γ in is prescried. Thus the term p is given according to Eq. 3. At the interface Γ n t, the velocity v n is identical for oth the cavity domain Ω and the elastic eam Ω. The amplitude of the elastic eam deflection is assumed to e the harmonic function w(x) = j=1 which yields the amplitude of velocity of the eam v(x) = iω j=1 α j sin jπx (17) α j sin jπx. (18) In the following, the Galerkin method is applied. The estimated aproximate solution Eq. 16 is sustituted into Eq. 13 and the functions ϕ kl (x, y), ϕ kl (x, y) respectively, are applied as the test functions. This yield to two systems of N 2 algeraic equations for unknown coefficients kl and d kl, k, l = 1,..., N. Due to the fact that ϕ kl (x, ) = ϕ kl (x, a) = the first system holds ω 2 c 2 a The second system is given as ω 2 c 2 a ϕ kl (x, y)p(x, y) dydx ϕ kl(x, y)p(x, y) dydx +ω 2 ϱ a cos lπ j=1 a a 2j α j π j 2 k 2 jk = i ω A ϕ kl (x, y) p(x, y) dydx =. (19) ϕ kl(x, y) p(x, y) dydx + (2) [ ] 4 π(4 k 2 ) 2k where jk = 1 for (j k) even and jk = for (j k) odd. When considering the ortogonality of functions ϕ kl (x, y) and ϕ kl (x, y), Eqs. (192) ecome significantly simplified. The solution for the elastic eam Ω R 1 is given elow. With regards to Eq. 16, the acoustic pressure amplitude that is acting on the eam can e expressed in the form p(x, a) = k=1 l=1 d kl cos kπx cos lπ. (21) The amplitude form of the elastic eam equation of motion can e written as EI 4 w x 4 ω2 ϱ h S 2 w = p(x, a) (22) t2 7
8 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman where I = h3 and S =.1 h. After sustituting Eq. 17 and Eq. 21 into Eq. 22, consequent 12 multiplication y the term sin jπx, j = 1,..., N and after integration over domain Ω (i.e. for x ; ), the following system of equations is derived ) α j (EI j4 π 4 ω 2 ϱ 2 4 h S = k=1 d kl cos lπ 2j π j 2 k jk. (23) 2 Systems of Eqs. (19), (2) and (23) form a system of 2N 2 + N linear algeraic equations for 2N 2 + N unknown coefficients kl, d kl and α j where j, k, l = 1,..., N. The system can e written in a lock form A 11 A 12 A 21 A 22 A 23 A 32 A 33 l=1 d α = c 2 (24) where vectors, d and α include unknown coefficients kl, d kl and α j, respectively. Sustituting coefficients kl, d kl and α j into Eqs. (1618) yields the solution for the acoustic pressure amplitudes in the cavity and for amplitudes of elastic eam deflections and velocities. 5 ANALYTICAL AND NUMERICAL RESULTS First, results of the approximate analytical solution for the enchmark test case of section 4 are presented and a comparison with numerical results of the inhouse developed 3D FE solver is provided. In order to test the performance of the new thin flat shell triangular element, the 2D1D enchmark test case (rectangular cavity and elastic eam) is numerically solved as a 3D 2D case (cuoidal cavity and elastic plate), Figure 5(right) and the numerical results from the central plane section are compared to the aproximate analytical solution. Both the analytical and the numerical solution of the viroacoustic prolem is provided for the cavity domain with dimensions a = 1 m, = 2 m and the amplitude A =.1 m s 1 of the prescried normal velocity at oundary Γ in, Figure 4. In the numerical model, the third dimension is given as z =.1 m. The air density is ϱ a = kg m 3 and the speed of sound is c = 34 m s 1. The elastic eam with thickness h =.1 m, density ϱ h = 78 kg m 3, Young modulus E = Pa and Poisson ratio µ =.3 is applied. The angular frequency ω = 5 rad s 1 and proportional damping coefficient β = are considered. Comparison of analytically and numerically computed asolute values of elastic eam/plate deflection amplitudes is provided in Figure 5(left). A minor deviations of the two solutions can e oserved. These are likely caused y a limited numer of harmonic functions respected in the approximate analytical solution (the provided solution refers to N = 8). For higher numer of harmonics the prolem ecomes illconditioned. This is actually an advantage of numerical solution y means of FE method. Figure 5(right) displays a complete numerical solution of the viroacoustic prolem. Both distriution of acoustic pressure amplitudes in the cavity domain and asolute values of deflection amplitudes at elastic plate. For completeness, there is a comparison of acoustic pressure amplitudes in the cavity as computed using approximate analytical solution, Figure 6(left), and numerically, Figure 6(right), showing a good agreement of the two solutions. The second case studied numerically is a simplified model of moile screw compressor. The compressor cavity is represented y a cuoidal computational domain geometry with dimensions mm as depicted in Figure 7(topleft). The cavity ottom is rigid and corresponds to the oundary Γ w. All other walls of the cuoidal domain form the interface 8
9 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman Figure 5: Benchmark test case: comparison of analytical (red) and numerical (lue) solution of asolute values of elastic eam/plate deflection amplitudes (left) and numerical solution of distriution of acoustic pressure amplitudes in cavity and of asolute values of deflection amplitudes at elastic plate (right). Figure 6: Benchmark test case: distriution of acoustic pressure amplitudes in cavity  approximate analytical solution (left) and numerical results (right). oundary Γ t etween the compressor cavity and elastic housing. The elastic compressor housing is made of a thin metal plate with thickness h =.1 m and density ϱ h = 78 kg m 3. The compressor engine ody is modelled as a small lock attached to the rigid ottom and its surface represents the Γ in oundary. The engine lock dimensions are mm and the prescried amplitudes of normal velocity v n are equal to.2249 m s 1. The following two parameters are selected: angular frequency of compressor engine motion ω = 2 rad s 1 and the coefficient of proportional damping β =. All other parameter values are identical to values from the previous enchmark test case. In Figure 7(right), the resulting distriution of the acoustic pressure amplitudes in the cavity of the simplified model of the screw compressor as solved y the implemented 3D FE solver is shown. Figure 7(ottomleft) displays asolute values of deflection amplitudes at compressor housing in a direction that is normal to the compressor housing surface. The numerical results of a viroacoustic prolem solved for a real geometry of the computational domain prepared according to the technical drawings provided y compressor producer and consisting of the engine ody, the inner cavity and compressor housing will e presented during the conference ICOVP CONCLUSIONS The inhouse 3D FE solver for viroacoustic analysis of screw compressors has een developed and the new 6noded flat shell triangular finite element with 18 DOF implemented. Correctness of the proposed method and of the implemented solver has een verified using the enchmark test case, for which the analytical solution has een derived. First numerical results of viroacoustic analysis have een presented for a simplified screw compressor model. In 9
10 Jan Dupal, Jan Vimmr, Ondřej Bulík, Michal Hajžman Figure 7: Simplified screw compressor model: computational domain geometry (topleft), numerical solution of asolute values of deflection amplitudes at compressor housing (ottomlet) and numerical solution of distriution of acoustic pressure amplitudes in compressor cavity (right). order to quantify the total emitted acoustic power, the numerical solution in amplitude form has to e performed for all angular frequencies of periodically varying surface velocity of the compressor engine. Then the total emitted acoustic power is a superposition of all particular solutions. In the near future, the solution of compressor housing virations that are kinematically excited y the rotating parts of the screw compressor will e addressed and superimposed to the housing virations caused y the acoustic pressure field. The results of this numerical analysis will enale modifications of the present moile screw compressor design which will lead to significant decrease in the total emitted acoustic power. 7 ACKNOWLEDGEMENTS This work was supported y the project TA21565 of the Technology Agency of the Czech Repulic. An assistance of Lior Loovský and Alena Jonášová during the preparation of the manuscript is also acknowledged. REFERENCES [1] L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of acoustics. John Wiley & Sons, Inc., Forth eddition, 2. [2] S. Temkin, Elements of acoustics. Acoustical Society of America, 21. [3] P.M. Morse, K.U. Ingard, Theoretical acoustics. Princeton University Press, New Jersey, [4] R. Matas, J. Kňourek, J. Voldřich, Influence of the terminal muffler geometry with three chamers and two tailpipes topology on its attenuation characteristics. Applied and Computational Mechanics, 3, , 29. [5] A. Bermúdez, P. Gamallo, R. Rodríguez, Finite element methods in local active control of sound. SIAM Journal on Control and Optimization, 43, , 24. 1
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