Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 4, pp. 1381-1395, Warsaw 2017 DOI: 10.15632/jtam-pl.55.4.1381 POROUS MATERIAL EFFECT ON GEARBOX VIBRATION AND ACOUSTIC BEHAVIOR Mohamed Riadh Letaief, Lassaad Walha, Mohamed Taktak, Fakher Chaari, Mohamed Haddar Mechanical, Modeling and Manufacturing Laboratory LA2MP, National School of Engineers of Sfax, Sfax, Tunisia e-mail: walhalassaad@yahoo.fr In this paper, we define a resolution method to study the effect of a porous material on vibro-acoustic behavior of a geared transmission. A porous plate is coupled with the gear- box housing cover. The developed model depends on the gearbox characteristic and poro- elastic parameters of the porousmaterial. To study the acoustic effect of the housing cover, the acoustic transmission loss is computed by simulating numerically the elastic-porous co- upled platemodel, and the numerical implementation is performedbydirectly programming the mixed displacement-pressure formulation. To study the vibration effect, the bearing di- splacement is computed using a two-stage gear system dynamical model and used as the gearbox cover excitation. Numerical implementation is performed by direct programming of the Leclaire formulation. Keywords: porous material, gearbox, vibro-acoustic behavior 1. Introduction Controlling the vibro-acoustic behavior of rotating machinery has become a quality factor to improve the comfort by reducingnoise andvibration levels.Oneof themajor noise andvibration sources are geared transmissions (gears, shafts, roller bearings and the housing).The generalized forces which generate the vibration response of the gearbox housing are multiple, as expressed byRemond et al. (1993). Sources of vibration excitations generated by geared transmissions can be divided into two categories, first the internal excitation sources like the static transmission error under load, elastic deformations of teeth, fluctuation in the frictional force developed by Houser (1991), Aziz and Seirg (1994), schock phenomenon and the projection or flows of the lubricant on walls of the housing according to Houser (1991) andHoujoh andUmezawa (1992). External sources of excitation can be associated with the fluctuations in engine torque and load inertia. Regardless of directivity of the source, larger walls of the housing are more flexible and contribute most to noise radiation. A parametric study performed by Sibe (1997) shows that the more walls are heavy, stiff and thick, the higher is the acoustic transmission loss of the housing. An increase in the thickness of the housing is unfortunately contrary to the desire of manufacturers who always want to increase the specific power of their transmissions. Note that in the majority of gearboxes, their housings covers are more flexible than other parts body of the housing and have the largest surface of acoustic radiation while looking for a method how to decrease their acoustic emission, some research work as that carried out by Guezzen (2004), confirmed effects of structure of the gearbox cover on noise radiation. In this context, we study a housing cover of a gearbox coupled with a porous material plate to isolate sources of noise radiation. Various models have been developed to describe the acoustic propagation in porous media. One of the best known and the easiest to implement is the model of Delany and Bazley (1970). 1382 M.R. Letaief et al. However, this model is limited because it represents only tested materials and does not express the phenomenon related to skeleton vibrations. Tomodelmore accurately the dissipative effects, unlike in themodel developed by Johnson et al. (1987), onemay introduce a function of viscous formwhich is not limitedby thegeometric nature of the skeleton.Modeling of thevariation of the viscous dissipationmodulusmay require introduction of the viscous characteristic length which is an intrinsic parameter of the material that can be obtained through experience. Similarly, ChampouxandAllard (1991) defined the thermal characteristic length as an intrinsic parameter expressing thermal effects. Lafarge et al. (1997) introduced thermal permeability to improve thermal effects at low frequencies. However, the model with a rigid structure is not suitable when the skeleton of the material is deformed or mobile: this is the case in many applications where aporousmaterial is directly subjected to amechanical or acousticwave excitationwhich is the subject of our paper. Allard (1993) adapted amodel for acoustic applications by integrating various contributions previously cited, see Johnson et al. (1987), Champoux and Allard (1991) and Lafarge et al. (1997). This model, commonly called the Biot-Allard model is used in our study since porousmaterials are subjected to the imposed displacement or acoustic pressure. In Section 2, we describe equations of motion for the dynamic model of gearbox and the housing cover (elastic and porous coupled plate) implementing porousmodels. In Section 3, we present the resolutionmethod (input and output, geometry, implemented porous and boundary conditions). In Section 4, we describe the porous plate effect on vibration and the acoustic transmission loss of our gearbox housing cover by a study case. 2. Gearbox modelling In most gearboxes, especially those having reduced sizes, the wheel axis is in the same plane between the two parts of the gearbox (Fig. 1) that enables easy assembling of the wheels. Fig. 1. Plane configuration of a two-stage gear system and the porous housing cover We defined a fixed reference frame (O,X0,Y0) in the model. αi are pressure angles of two gearmesh contact. In this paper, these angles are equal to 20◦ in the case of the gearings with right teeth. 2.1. Dynamic model of a two-stage gear system A two-stage gear system is composed of two trains of gearings. Every train links two blocks. So, the gear system has in total three blocks (j=1,2,3). Every block is supported by a flexible bearingwhose bending stiffness is kxj and the traction compression stiffness is kyj. The dynamic Porous material effect on gearbox vibration and acoustic behavior 1383 model developed has twelve degrees of freedom: six angular movements γji and six linear move- ments xj and yj (Fig. 2). Themotor and receiving wheels are introduced by inertias Im and Ir as expressedbyMiller (1999) with the assumption thatwe use short shafts. The other spur gears constitute the gearbox. The gearmeshes are modeled by a linear spring ks(t) (s = 1,2) along the lines of action represented in Fig. 2. αi are pressure angles of two gearmesh contact. The angular displacements of every wheel are noticed by γji with the indices j=1 to 3 designating the number of the block, and i = 1,2 designating the two wheels of each block. Besides, the linear displacements of the bearing denoted by xj and yj are measured in the plane which is orthogonal to the axis of wheel rotation. Fig. 2. Model of the two-stage gear system developed by Walha et al. (2009) 2.2. Modeling of the mesh stiffness Generally, we canmodel variation of the gearmesh stiffness ki(t) by a squarewave whichwas developed by Velex (1988). The variation in stiffness comes from the fact that during meshing there is a change in the number of contacting pairs. For spur gears, there is a change for twopairs of teeth in contact for a period of meshing. The square wave variation is the best representative of the real phenomenon, and is represented in Fig. 3. Fig. 3. Modeling of the mesh stiffness variation The gearmesh stiffness variation can be decomposed into two components: an average com- ponent denoted by kci, and a time-dependent one denoted by kvi(t). The extreme values of the mesh stiffness are defined by kmini =− kc 2εαi kmaxi =−kmini 2−εαi εαi−1 (2.1) The terms εαi are the contact ratio corresponding to the two gearmesh contacts. 1384 M.R. Letaief et al. 2.3. Equations of motion Applying the Lagrange equations, we obtain a system of differential equations governing the dynamic behavior. It can be written in the following usual matrix form Mq̈+[Ks+K(t)]q=F0 (2.2) where q is the generalized coordinate vector, M is the mass matrix expressed by M= diag(m1,m1,m2,m2,m3,m3,Im,I12,I21,I22,I31,Ir) mj is mass of the block j, Im is the polar inertia of the motor wheel, Ir is the polar inertia of the receiving wheel. Thematrix of average stiffness of the structure is defined by Ks = [ Kp 0 0 Kθ ] Kp =    kx1 0 0 0 0 0 0 ky1 0 0 0 0 0 0 kx2 0 0 0 0 0 0 ky2 0 0 0 0 0 0 kx3 0 0 0 0 0 0 ky3    Kθ =   kθ1 −kθ1 0 0 0 0 −kθ1 kθ1 0 0 0 0 0 0 kθ2 −kθ2 0 0 0 0 −kθ2 kθ2 0 0 0 0 0 0 kθ3 −kθ3 0 0 0 0 −kθ3 kθ3   whereKp is the bearing stiffness matrix andKθ is the shaft torsional stiffness matrix. K(t) is the stiffness matrix of the engagement which is variable over time K(t)= [ K1(t) K12(t) KT12(t) K2(t) ] where K1(t)=   k1s 2 1 −k1sc1 −k1s 2 1 k1sc1 0 0 −k1sc1 k1c 2 1 k1sc1 −k1c 2 1 0 0 −k1s 2 1 k1sc1 k1s 2 1+k2s 2 2 −k1sc1−k2sc2 −k2s 2 2 k2sc2 k1sc1 −k1c 2 1 −k1sc1−k2sc2 k1c 2 1+k2c 2 2 k2sc2 −k2c 2 2 0 0 −k2s 2 2 k2sc2 k2s 2 2 −k2sc2 0 0 k2sc2 −k2c 2 2 −k2sc2 k2c 2 2   K12(t)=    0 −k1rb12s1 −k1rb21s1 k1sc1 0 0 0 k1rb12c1 k1rb21c1 −k1c 2 1 0 0 0 k1rb12s1 k1rb12s1 −k1sc1−k2sc2 −k2s 2 2 k2sc2 0 −k1rb12c1 −k1sc1−k2sc2 k1c 2 1+k2c 2 2 k2sc2 −k2c 2 2 0 0 −k2s 2 2 k2sc2 k2s 2 2 −k2sc2 0 0 k2sc2 −k2c 2 2 −k2sc2 k2c 2 2    Porous material effect on gearbox vibration and acoustic behavior 1385 K2(t)=   0 0 0 0 0 0 0 k1r 2 b12 k1rb12rb21 0 0 0 0 k1rb12rb21 k1r 2 b21 0 −k2s 2 2 0 0 0 0 k2r 2 b22 k2rb22rb31 0 0 0 0 k2rb22rb31 k2r 2 b31 0 0 0 0 0 0 0   where rb is the base radius; si, sci and c 2 i are simplifications of the functions: si = sin 2φi, sci = sinφicosφi and c 2 i = cos 2φi, respectively. F0 is the vector of external static forces and can be expressed as F0 = [0,0,0,0,0,0,Cm,0,0,0,0,−Cr] T Cm andCr are the motor and receiving wheel torques, respectively. 3. Modelling of the housing cover In our study, the housing cover is modeled as an elastic and porous coupled plate. In fact, two porousmodels are implemented. 3.1. Leclaire’s formulation Leclaire’s formulation is based on the classical theory of homogeneous plates and on the Biot stress-strain relations in an isotropic porousmediumwith a uniform porosity. The vibrations of a rectangular porous plate can be described by two coupled dynamic equations of equilibrium relating the plate deflection ws and the fluid/solid relative displacement w. In the case of a plate with thickness h and subjected to a load q, these two equations can be expressed as ( D+ φ2λ̃fh3 12φ2 ) ∇4ws+h(ρ1ẅs+ρ0ẅ)= q λ̃fh φ ∇2ws−h(ρ0ẅs+mẅ)= 0 (3.1) whereD is the flexural rigidity, ρ0 – density of the fluid, ρ1 – density of the frame, φ – porosity, λ̃f –material expansion coefficient andm is themass parameter introduced byBiot (1962) given by m(ω)= τ(ω) φ ρ0 (3.2) where ω is the pulsation, τ(ω) is the dynamic tortuosity expressed as folows τ(ω)= τ∞− j σφ ρ0 F(ω) √ 1+ 4ηα2 ∞ ρ0 σ2Λ2φ2 jω F(ω)= √ 1− i 4τ2 ∞ κ2ρ0ω ηΛ2φ2 (3.3) where F(ω) is the viscosity correction function introduced by Johnson et al. (1987), α∞ is the tortuosity of pores, η is the damping coefficient, Λ is the characteristic dimension of pores, σ is the flow resistivity. The space derivatives are written with the help of the operators ∇4 = ∇2(∇2) and ∇2 = ∂2/prtx2 + ∂2/∂y2 of the system of co-ordinates (x,y) while the double dots denote the second time derivative. 1386 M.R. Letaief et al. In the first equation of equilibrium (or plate equation) [D+φ2λ̃fh3/(12φ2)]∇4ws represents the internal potential force (per unit surface) within the fluid-saturated plate, while the inertia terms hρ1ẅs and hρ0ẅ and the load q are considered as external forces. Similarly, the inter- nal force associated with the fluid-solid relative displacement may be defined, and is given by (λ̃fh/φ)∇2ws while the external forces can be taken as hmẅ and hρ0ẅs. We note that the Leclaire formulation is a 2D one and the unknown variables arews andw. All terms used in this formulation are based on poroelastic material characteristics. 3.2. The mixed formulation In order to reduce the computation time enlarged by complexity of the problem, mixed formulations (u,p) have been implemented. This formulation was developed by Atalla et al. (1998) using the classical equations of Biot where u represents displacement field of the solid phase and p is the pore pressure. Replacing the displacement of the fluid phase by its pressure allows us to reduce degrees of freedom from 6 to 4 per node, valid only for harmonicmotion. It is also accurate in the classical formulation (u,U). The modified equations of equilibrium (for small harmonic oscillations) are expressed as follows σ̂sij/jS+ω 2ρ̃ui+ γ̃p/i =0 −ω 2 ρ̃22γ̃ φ2 ui/i+ω 2 ρ̃22 λ̃f p+p/ii =0 (3.4) where σ̂sij is the stress tensor of the material “in vacuo” (does not depend on the fluid phase). It is written by σ̂sij = ˆ̃ λsεskkδij +2µ sεsij ε s ij = 1 2 (ui/j +uj/i) (3.5) where εsij is the strain tensor of the skeleton, µ s is the shear modulus of the porous material. The above equations depend on certain factors: ˆ̃ λs, ρ̃, γ̃ and λ̃f. These are based on intrin- sic poroelastic characteristics introduced by Horoshenkov and Swift (2001) and Umnova et al. (2001). 4. Resolution method Fig. 4. SADT diagram For the two cases of study (acoustic andvibration behavior), the implemented porousmodels are analysed by the finite element software COMSOL and MATLAB. The equations of motion are introduced by the EDP module of COMSOL software. Porous material effect on gearbox vibration and acoustic behavior 1387 4.1. Porous models In COMSOL, the general form of PDE (for a temporal analysis) must be expressed in the following matrix form Γ ·∇=F (4.1) where Γ is the matrix of the flux vectors and F is the right part of the vector. In Cartesian coordinates, the gradient/divergence operator vector ∇ is defined as follows ∇=   ∂ ∂x ∂ ∂y   (4.2) 4.1.1. Leclaire’s formulation If we adapt Leclaire’s formulation, Eqs. (3.1), to the EDP form in COMSOL, we obtain the following equations Γ=    ∂z ∂x ∂z ∂y ∂ws ∂x ∂ws ∂y ∂w ∂x ∂w ∂y    F=   1 D+α2Mh3/12 ( q+hω2(ρws+ρfw) ) 1 αMh ( ∆P −hω2(ρfws+mw) ) z   (4.3) 4.1.2. The mixed formulation If we adapt „the mixed formulation”, equations (3.4),to the EDP form of COMSOL, we obtain the following equations Γ= [ Γij Γ4i ] = [ µS(ui/j +uj/i)+ λ̃ Suk/kδij p/i ] F= [ Fi F4 ] =   −ω2ρeui−γp/i −ω2 ρ̃22 λ̃f p+ω2 ρ̃22γ̃ φ2 ui/i   (4.4) 4.2. Geometry The geometry of the structure used in the numerical simulation is represented by a coupled porous plate (Fig. 5) with dimensions a= b. Thickness of the porous plate is hp, of the elastic plate hs. The system is loaded by the imposed displacement. 4.3. Input parameters The input parameters are the gear system parameters: motor torque Cm and speed Nm, bearing and shaft stiffnesses kxs, kys, kθs, teeth number, width and module Z, b, m, average mesh stiffness kc1, contact ratio εα1, pressure angle α and 9 poroelastic parameters: porosity φ, tortuosityα∞, flow resistivityσ, thermal andviscous characteristic dimensionsof pores,modulus of elasticity Λ and Λ′, density of the skeleton ρ1, skeleton Poisson’s coefficient ν, damping coefficient η and the skeleton elasticity modulusE. 1388 M.R. Letaief et al. Fig. 5. System of co-ordinates in the plate 4.4. Output parameters The first output is the normal incidence transmission loss TL, as introduced by Rossing (2007) TL =10log 1 |Ta| 2 (4.5) where |Ta| 2 is thenormal incidence power transmission coefficient for an anechoically-terminated sample, that is the ratio of the sound power transmitted by the sample to the sound power incident on the sample. In the case of perfectly anechoic termination Ta =C/A A= j(P1e jkx2 −P2e jkx1) 2sin[k(x1−x2)] C = j(P3e jkx4 −P4e jkx3) 2sin[k(x3−x4)] (4.6) with P1 to P4 are complex sound pressures at x1 to x4, and k is the wave number. The second output is the bearing block load Fb =Kx3x3+Ky3y3 (4.7) x and y are bearing displacements,K is the bearing stiffness and Fb is the bearing block load. 4.5. Boundary conditions The boundary conditions for EDP in COMSOL in their general form are as follows 0=R −Γn=G+ [∂R ∂u ]T µ (4.8) The vectorR andmatrixΓmaybe functions of the spatial co-ordinates withnbeing the normal unit vector leaving the boundary surface. These are the boundary conditions of Dirichlet and Neumann, respectively. The term µ in the Neumann boundary conditions is synonymous with the Lagrange multiplier. There are several boundary conditions to be respected since there are two clamped coupled plates with four sides and poroelastic/acoustic as well as poroelastic/elastic coupling zones. Using the Biot-Allard formulation, the boundary conditions are discussed below. Porous material effect on gearbox vibration and acoustic behavior 1389 • Imposed pressure field The imposed pressure field p on the boundary of the porous medium allows us to write the following relations σtijnj =−pni p= p (4.9) which express the continuity of the total normal stress and continuity of pressure across the interface of the border. The total stress is equal to σtij =σ S ij +σ f ij =σ S ij −φpδij = σ̂ S ij −φ ( 1+ λ̃fS λ̃f ) pδij =µS(ui/j +uj/i)+ λ̃ Suk/kδij −φ ( 1+ λ̃fS λ̃f ) pδij (4.10) Using the second boundary condition of Eq. (4.9), the first one can be expressed as follows −[µS(ui/j +uj/i)+ λ̃ Suk/kδij]nj = [ 1−φ ( 1+ λ̃fS λ̃f )] pni (4.11) After identification, the terms R andG are as follows R= [ Ri R4 ] = [ 0 p−p ] G= [ Gi G4 ] =   [ 1−φ ( 1+ λ̃fS λ̃f )] pni 0   (4.12) When a portion of the surface of the porousmedium is coupled to an infinite acoustic medium, the condition of a free edge can be applied. This is assuming that p=0. • Imposed displacement field In the case of the imposed displacement field ui, the boundary conditions can be expressed by ui =ui vini−uini =0 (4.13) The first term in Eq. (4.13) expresses the continuity between the imposed displacements and the solid phase displacements, while the second term describes the continuity of the normal displacement between the fluid and solid phase. In this second condition, it is necessary to replace the displacement of the fluid phase by the fluid pressure vi = φ ω2ρ̃22 p/i− ρ̃12 ρ̃22 ui (4.14) which yields p/ini = ω2 φ (ρ̃12+ ρ̃22)uini (4.15) such as ω2 φ (ρ̃12+ ρ̃22)= ω2 φ (ρ12+ρ22)=ω 2ρ0 (4.16) After identification, the terms R andG are as follows Ri =ui−ui R4 =0 Gi =0 G4 =− ω2 φ (ρ̃12+ ρ̃22)uini (4.17) Applying that ui =0 implies the fact that our porous domain is embedded to a rigid wall. 1390 M.R. Letaief et al. • Acoustic – poroelastic coupling In this case, the equations for continuity of the total normal stresses, acoustic pressure and fluid flow are as follows σtijnj =−p anj p= p a (1−φ)uini+φvini = 1 ρ0ω 2 ∇pani (4.18) where pa is pressure in the acoustic medium, ρ0 its density and σ t the total stress tensor in the poroelastic material. The vectors G andRwill have the following components Ri =0 R4 = p−p a Gi = [ 1−φ ( 1+ λ̃fS λ̃f )] pani G4 =0 (4.19) In addition, the continuity of the fluid flow at the coupling interface can be expressed as an imposed acceleration on the fluid in the acoustic environment. Replacing vi by its expression, the normal acceleration can be obtained by 1 ρ0 ∇pani =ω 2 [ uini ( 1−φ ( 1+ ρ̃12 ρ̃22 ))] +ω2 [ ∇pni ( φ2 ω2ρ̃22 )] (4.20) For the Leclaire formulaion, a boundary condition can be considered. It is discussed below • Clamped plate At the boundary conditions, an embedding condition is introcuced ws =0 Uf =0 (4.21) The relative solid-fluid displacement is defined as follows w=φ ( Uf −ws) Uf = 1 φ w+ws (4.22) wherews is the solid displacement andUf is the fluid displacement. Subsequently,R andG are expressed by R=   ws 1 φ w+ws 0   G=   0 0 0   (4.23) The loading conditions q and ∆P are fixed according to the type of solicitation (pressure, force,...). For the surface pressure, a value of 0.1 is assumed ∆P = q=0.1bars (4.24) 5. Study case The numerical parameters of the two-stage gear system are summarized in Table 1. Table 3 describes numerical values of parameters of the poroelastic materials. Porous material effect on gearbox vibration and acoustic behavior 1391 Table 1.Geared transmission parameters External inputs Motor torque and speed Cm =1000Nm,Nm =3000tr/mn Structure Bearing and shaft stiffness xs = kys =10 9N/m, characteristics kθs =10 5Nm/rad Gear characteristics material: 42CrMo4, ρ=7860kg/m3 First stage Second stage Teeth width andmodule [mm] b=20, m=4 b=20,m=4 Teeth number Z(12)= 26, Z(21)= 39 Z(12)= 26, Z(21)= 39 Average mesh stiffness kc1 =1.4 ·10 8N/m kc2 =1.4 ·10 8N/m Contact ratio and pressure angle εα1 =1.57, α=20 ◦ εα2 =1.53, α=20 ◦ Table 2. First eigenfrequency of the geared transmission ωi [rad] 1823 4095 6016 16063 17353 27365 fi [Hz] 290 652 957 2557 2763 4357 Table 3.Poroelastic parameters for validation of the models Parameter Unity Porous material ρ1 kg/m 3 90 φ – 0.7 σ Ns/m4 22250 α∞ – 1.3 ν – 0.05 Λ µm 75 Λ′ µm 87 E N/m2 2980000 η – 0.12 5.1. Porous plate effect on vibration level Figure 6 shows the displacement along the axis x of the output bearing at the housing cover. Thedisplacement amplitude is about 2·10−6. Theperiodicity of the bearing displacement comes from domination of the gearmesh frequency. Figure 7 shows that the RMSbearing displacement increases with themeshing frequency as it is shown in Fig. 8. The results show that the gearmesh frequency and its harmonics dominate the RMS bearing displacement with higher amplitudes when the gearmesh frequency or one of its harmonics is close to the eigenfrequency. The first peak is close to the first eigenfrequency (290Hz) the second one is close to the third eigenfrequency (957Hz). The third peak is close to the sum of the first and the third eigenfrequency (1608Hz). Figure 9 shows that the gearmesh frequency and its harmonics dominate the point plate displacement. The absence of a negative displacement is due to the elastic effect of the plate at the measurement point. Due to the same reason, there are no positive displacements in the other half of the plate. As it is shown in Fig. 10, the gearmesh frequency dominates the point plate displacement. Theabsenceof anegativedisplacement is due to the elastic effect of theplate at themeasurement point. Due to the same cause, there are no positive displacements in the other half of the plate. 1392 M.R. Letaief et al. Fig. 6. Output bearing displacement in the x direction Fig. 7. Output bearing displacement in the x direction for three gearmesh frequencies Fig. 8. RMS bearing displacement Porous material effect on gearbox vibration and acoustic behavior 1393 Fig. 9. Displacement along the axis x at a point with coordinates (0.15,0.24) on the elastic plate Fig. 10. Point displacements (solid line: elastic plate, dashed: elastic and porous coupled plate) 5.2. Acoustic effect of the porous plate Figure 11 shows the transmission lossTL of the elastic-porous coupled plate. The calculation is conducted for the porous plate with a characteristic defined in Table 3 and thickness 10mm. TL increases along the frequency axis and is dominated by the resonance frequency of the plate where TL decreases with the frequency converging to 67dB, 54dB and 83dB at, respectively, natural frequencies 620Hz, 1240Hz and 1900Hz. Figure 11 shows the dependence of the sound transmission loss on the flow resistivity which is one of the characteristic of the porousmaterial but is still dominated by the natural frequencies. 6. Conclusion A resolutionmethod to determine the porous plate effect on a gearbox hosing cover is discusseg in the paper. The developed model depends on several parameters: gearbox and porous plates parameters. It is found that coupling of the porous plate to the housing cover reduces the 1394 M.R. Letaief et al. Fig. 11. Sound transmission loss TL for different flow resistivity σ vibration level and is dominated by the gearmesh frequency. For the acoustic effect, poroelastic materials have major capacity to mitigate the noise level caused by the geared transmission. The vibration and the acoustic behavior are heavily dependent on poroelastic characteristics. These results were validated byTewes (2005), who computed the transmission loss of an infinite double wall partition for various angles of incidence and for various mass ratios. The developed method helps one to make decisions in the robust design and lessens the enormous computing time. References 1. 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