A 3 D finite element model for sound transmission through a double plate system with isotropic elastic porous materials

Transcription

1 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA Département génie mécanique Université de Sherbrooke Sherbrooke, QC CANADA Tel.: (819) Fax: (819) A 3 D finite element model for sound transmission through a double plate system with isotropic elastic porous materials Revised title: A 3-D finite element model for the vibroacoustic response of saturated porous media by Raymond Panneton, PhD Student (GAUS, QC CANADA) Pr. Noureddine Atalla (GAUS, QC CANADA) Pr. Jean-François Allard (Université du Maine, FRANCE) 128th Meeting of the Acoustical Society of America

2 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA O U T L I N E INTRODUCTION MODELING VARIATIONAL FORMULATION FINITE ELEMENT FORMULATION RESULTS DISCUSSION

3 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA I N T R O D U C T I O N MOTIVATIONS Currently, the use of Multilayer Structures in Automobile, Aeronautics and Space Industries is increasing In Structural Acoustics and Vibration, this means an assembly of elastic, viscoelastic and porous materials Consequently, the demand of prediction tools for the acoustical and structural behaviors is increasing Analytical models are limited to simple geometries Low frequency domain

4 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA I N T R O D U C T I O N MAIN OBJECTIVES 3D-Finite Element model for the Vibroacoustic response of porous materials Evaluate the damping characteristic of porous materials SPECIFICATIONS Compatible with classical FE models for elastic and viscoelastic materials, and with FE and BE models for fluid media

5 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA M O D E L I N G METHODOLOGY Stress-Strain Relations for porous a material Kinetic, Strain and Dissipation energies for a porous material Hamilton's Function in terms of (u s, U f ) t2 H = F( us, u s, U f, U f ) dt t 1 = [ ]{ } U f = [ N f ]{ q f } us Ns qs Solution Discretized Hamilton's Function Variational Principle q s q f q s q f [ K] + [ M ] + [ C] q s q f s = Q f Q

6 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA M O D E L I N G Variational Formulation (Hamilton's Function) ( ) 2 2 Η = T U + W dt + W * dt t t 1 T = 1 us us + us U f + U f U f d 2 ρ11 ρ12 ρ 22 Ω Ω t t 1 Poroelastic element based on Biot theory: Use a displacement formulation (analogy with solid elements) { } 2 { } { } { } { } s s f f U = 1 d 2 σ ε + σ ε Ω Ω ( s f ) D = 1b u U d 2 2 Ω Ω f { s} Γ { f } s W = F u d + F U dγ Γ Γ frequency dependent

7 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA M O D E L I N G Stress-Strain Relations (M.A. BIOT) { s} [ s]{ s} [ ]{ f σ = D ε + Q ε } { f } [ f ]{ f } [ ]{ s σ = D ε + Q ε } Solid element analogy λ + 2µ λ λ λ λ + 2µ λ s λ λ λ + 2µ [ D ] = µ µ µ D hk Q = h K f ( 1 ) f [ ] = f [ ] frequency dependent

8 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA M O D E L I N G Finite Element Formulation Interpolation Schemes = [ ]{ } U [ N ]{ q } f = f f us Ns qs Discretized Hamilton's Function Variational Principle 2 M M ω Discrete equation of motion [ ss] [ sf ] T [ Msf ] [ M ff ] + jω [ Css] [ Csf ] T [ Csf ] [ C ff ] + [ Kss] [ Ksf ] T [ Ksf ] [ Kss] { q } s { q f } = s { Q } f { Q } frequency dependent

9 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA RESULTS Validation against an analytical model (J.F. Allard) Configuration Typical fibrous material Normal Incidence plane wave in AIR h=10 cm σ s zz (1) σ f zz (1) σ s z zz (2) σ f zz (2) σ s zz (3) σ f zz (3) σ s zz (4) σ f zz (4) 8 nodes - Linear Element: 6 dof / nodes (u s, v s, w s, U f, V f, W f ) z Rigid Wall

10 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA RESULTS Re(Zi)-Anal Im(Zi)-Anal Re(Zi)-FEM Im(Zi)-FEM 1500 Surface Impedance (Pa*s/m) Frequency (Hz)

11 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA RESULTS Effects of the Bondary Conditions on the edges d Absorption coefficient Laterally infinite extent : d= FEM: d=2.5cm FEM: d=5cm 2 constrained FEM: d=10cm edges multilayered poroelastic material Frequency (Hz)

12 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA DISCUSSION Modeling New variational formulation based on the Biot theory for fluid-saturated porous materials with elastic frames A 3-D Finite Element Model for the prediction of the vibroacoustic behavior of porous materials The F.E. has proved to be accurate following the comparaisons with the analytical model of Allard

13 Acoustics and Vibrations Group Université de Sherbrooke, QC CANADA DISCUSSION Results The edge effects are important for small samples On going works Sound transmission loss through multilayer structures containing porous materials Investigations on mixed formulations (u s,u f,p) and (u s,p) See : Overview of the numerical models for the vibroacoustics of multilayer panels with porous materials Raymond Panneton J. Acoust. Soc. Am. 102, 3113 (1997);

14 Groupe d acoustique et vibrations Acoustics and Vibrations Group Département de génie mécanique Université de Sherbrooke Sherbrooke, QC CANADA Tel. : (819) Fax : (819) Overview of the numerical models for the vibroacoustics of multilayer panels with porous materials Their Experimental Validations by Raymond Panneton 134th Meeting of the Acoustical Society of America Invited paper

15 OUTLINE Recall of the poroelastic models Validation # 1 : Double plate with unbonded foam Validation # 2 : Single plate with bonded foam Conclusion

16 Recall of the poroelastic models 1) Biot displacement formulation : (u,u) ( ~ ~ ) div σ s ( u, U ω 2 ) + ρ 11 u + ρ 12 U = 0 ( ~ ~ ) div σf ( u, U) + ω 2 ρ 12 u + ρ 22 U = 0 (Elastodynamic + source) (Elastodynamic + source) 2) Mixed formulation : (u,p) 2 div σ( u) + ω ρ ~ u = ~ γ p (Elastodynamic + source) p e + ω ρ ~ K p = ω ρ ~ ~ ~ 2 γ div h e u (Helmholtz + source)

17 Recall of the poroelastic models 3) Linear interpolation : wedge element (u,u) u y n U y n (u,p) u y n 1 2 e Ω p u z n Uz n u x n Ux n p u z n u x n 4) Discretized elasto-poro-acoustic equations 2 [ K ] ω [ M ] [ C ] m m ma ω ρ T [ C ] [ H] [ Q] [ C ] ma K pa a a [ ] [ ~ ] 2 C K [ ~ M ] pa p ω p only for (u,u) T { m q } { a q } { p q } = { F } m { R} { F } p

18 Validation # 1 Double plate configuration with foam y Accéléromètre Capteur de force x maillage 11x9 99 mesures Pré-amplificateur F(ω) z Pot vitrant A (ω) A(ω)/F(ω) Amplificateur 2 <V > ω nm FFT-Analyzer SMS STAR-Struct system

19 Validation # 1 1) Clamped plate Configuration y Vibration spectrum 140 (1,1) (2,1) (3,1) (1,2) (2,2) (4,1) (3,2) (1,3) (4,2) Quadratic velocity (db ref.: 5e-8 m/s) predicted measured 35 x 22 cm mm thick Materials Aluminum x Frequency (Hz)

20 Validation # 1 2) Plate-Air-Plate system y Configuration Vibration spectrum of the rear plate Quadratic velocity (db ref.: 5e-8 m/s) (1,1) (1,1 (2,1) (2,1)* predicted (3,1) (2,2) (1,2 (1,2)* (2,2)* (4,1) (1,0, measured (4,1)* (3,2) (1,3) (4,2) Cavity : 5.4 cm Materials - Aluminum x Frequency (Hz)

21 Validation # 1 3) Plate-air-foam-air-plate system y Configuration Vibration spectrum of the rear plate 140 Quadratic velocity (db ref.: 5e-8 m/s) (1,1) (2,1) predicted (3,1) Double plate (ref.) (1,2) (2,2) (4,1) measured (3,2) (1,3) (4,2) Frequency (Hz) Foam thick. : 5 cm Air gaps : 2 mm (11 x 9 x 6 f.e. mesh) Materials - Aluminum - Polyurethane x

22 Validation # 2 Single free plate configuration with foam Aluminum 30 x 35 cm mm thick 10 x 9 mesh

23 Validation # 2 1) Free plate Vibration spectrum Configuration y 140 Quadratic velocity (db ref.: 5e-8 m/s) Predicted Measured 30 x 35 cm mm thick Materials - Aluminum x Frequency (Hz)

24 Validation # 2 2) Plate with foam Polyurethane 30 x 22 cm 2 5 cm thick Plate 10 x 9 mesh Foam 10 x 7 x 6 mesh

25 Validation # 2 2) Free plate with foam Configuration y Vibration spectrum 140 Quadratic velocity (db ref.: 5e-8 m/s) Predicted Plate (ref.) Measured Frequency (Hz) Foam thick. : 5 cm Bonded Materials - Aluminum - Polyurethane x

26 Conclusion Discrepancies «due to» Errors related to the vibration measurements Errors related to the evaluation of the 9 physical properties of poroelastic material Errors related to boundary conditions on the foam (free b.c.) Anisotropy of the poroelastic material Good comparisons «any way!» Validate the f.e. poroelastic formulations Validate the coupling conditions Validate the numerical implementation (u,p) vs (u,u) The (u,p) formulation is much less computer memory and time consuming ((u,p) is 10 to 20 faster than (u,u))