Basic equations of structural acoustics and vibration

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1 Chater Baic equation of tructural acoutic and vibration.1 Introduction Coyrighted Material Taylor & Franci A mentioned in Chater 1, thi book addree claical numerical technique to olve variou vibroacoutic roblem. In ractice, a tyical roblem involve a tructural domain couled to bounded or unbounded fluid domain and ound-aborbing material. We then eek to redict the couled vibratory/acoutic reone of the whole ytem ubjected to the given excitation and boundary condition. In thi chater, we introduce the fundamental equation governing the linear ound wave roagation in fluid domain (acoutic), in olid domain (elatodynamic), and in orou ound-aborbing material (oroelaticity). We alo etablih the couling equation between thee domain. All thee artial differential equation will be then olved uing ecific numerical method, which will be reented in Chater 3 and 4.. Linear acoutic The acoutic reure diturbance ( xt, ) in a erfect fluid volume Ω f at ret with ound eed c, and denity ρ, due to an acoutic ource ditribution Q(x,t), atifie the inhomogeneou wave equation: 1 xt (, ) = c xt (,) Q( xt, ) (.1) In the cae of a ma ource, Q(x,t) rereent the rate of ma injection and i related to the volume velocity Q (x,t) by dq Qxt (, ) =ρ (.) dt 9

2 1 Finite element and boundary method in tructural acoutic and vibration Ma injection i the tyical model of monoole. In thi cae, the volume velocity, alo called ource trength, i the roduct of the urface area and the normal urface velocity of the monoole. For a roof of Equation.1, the reader i invited to look at the excellent book of Pierce (1989). To the reure diturbance, we can aociate a article velocity V( xt, ) given by Euler equation: Vxt (, ) = xt (,) (.3) t ρ A article dilacement U( xt, ) written a Uxt (, ) Vxt (, ) = t (.4) A denity fluctuation ρ( xt, ) ρ( xt, ) = xt (, ) c (.5) Coyrighted Material Taylor & Franci Equation.1 mut be aociated to the boundary and initial condition (Figure.1): Neumann boundary condition: Normal acoutic dilacement i ecified on the art Ω f,n of boundary Ω f : U n = U n (.6) Ω f,n n r x Ωf,R Ω f Ω f,n Ω f = Ω f,d Ω f,n Ω f,r Figure.1 Fluid domain and boundary condition.

3 Baic equation of tructural acoutic and vibration 11 Dirichlet boundary condition: Acoutic reure i ecified on the art Ω f,d of boundary Ω f : = (.7) Mixed, Robin, or imedance boundary condition: * Secific acoutic admittance i ecified on the art Ω f,r of boundary Ω f : β ρc = = ρc V. n Z n (.8) where Z n i the ecific acoutic imedance alied on Ω f,r. In addition, for roblem involving acoutic radiation in unbounded media (exterior roblem, ee Chater 7), we mut ecify a condition to enure that the wave amlitude vanihe at infinity. Thi i the Sommerfeld radiation condition, which in the 3-D cae, read r r + 1 lim c t = r (.9) Coyrighted Material Taylor & Franci Finally, the reviou equation mut be comleted with initial condition xt (, ) t = and xt (, ) at all oint of Ω f. t t = In the following equation, we are intereted in harmonic roblem, that i, dynamic roblem for which the temoral deendency of all variable i inuoidal with circular frequency ω. A convenient mathematical decrition of the variable i to ue the comlex formalim xt (, ) =R [ ( x)ex( iωt)] =R[ ( x)ex( iϕ( x))ex( iω t)] (.1) Solving for ( xt, ) i equivalent to olving for the comlex-valued function x ˆ( ). There i an amlitude x ˆ( ) and a hae ϕ( x ) aociated to x ˆ( ). In the following equation, we ytematically omit the factor ex(iωt) in all the equation. However, we have to kee in mind that the hyical value of the acoutic reure at oint x and time t i recovered by multilying x ˆ( ) by ex(iωt) and taking the real art. * Alo called radiation or abortion condition. Thi condition aume the abence of acoutic ource at infinity. Note that if the diturbance ectrum i broadband, the temoral ignal can be recontructed uing Fourier tranform: xt (, ) = 1 + ( x, ω)ex( iωtd ) ω, ω being the conjugate variable of time t. π

4 1 Finite element and boundary method in tructural acoutic and vibration For harmonic temoral deendence, Equation.1 can be rewritten a x ˆ( ) + kx ˆ( ) = Q ˆ ( x ) (.11) where k = ω/c i the wave number. Equation.11 i known a Helmholtz equation. Dirichlet boundary condition rewrite ˆ = (.1) Neumann boundary condition can be rewritten uing Euler equation (.3): ˆ n = ρωun (.13) The imedance boundary condition rewrite ˆ n + ikβˆ ˆ = (.14) Finally, for exterior roblem, Sommerfeld radiation condition become lim r + ik r = (.15) r Coyrighted Material Taylor & Franci.3 Linear elatodynamic Let u conider a linear elatic olid occuying volume Ω (ee Figure.). The fundamental equation governing it dynamic behavior are given by the conervation of ma equation, the conervation of momentum equation, and the behavioral law of the olid (the model defining how the olid deform in reone to an alied tre). In the framework of mall diturbance around the olid equilibrium oition, we can linearize thee equation to come u with the following linear elatodynamic equation, relating the linearized dilacement field u, train tenor ε, and tre tenor σ at all oint of Ω ρ u σ ρf b t = + (.16) σ = Cε (.17)

5 Baic equation of tructural acoutic and vibration 13 ε ij = 1 ui xj uj + x i (.18) where ρ i the denity of the olid, i the nabla oerator (ee Aendix 3B), F b i the body force vector er unit volume, C i the fourth-order tiffne (elaticity) tenor, ε ij and u i, i,j = 1,,3 are the train tenor and dilacement comonent along the 3 direction of a carteian coordinate ytem. Equation.17 rereent Hooke law. In Equation.16, the body force ditribution er unit ma F b can include variou effect: gravity, thermal effect, initial deformation or retre, etc. It i common to rewrite thi equation according to d Alembert a σ + ρ F b = where the body force ρ F b account for the inertial eudo-force I = ρ ( u/ t ). The boundary Ω of the olid i ubjected to two tye of boundary condition (Figure.): Secified contact force F er unit area alied on Ω,N : σ n = F (.19) Secified dilacement over Ω,D = Ω / Ω,N : u Coyrighted Material Taylor & Franci = u (.) Finally, the reviou equation are ulemented by two initial condition roviding the value of u( xt, ) t = and ( uxt (,)/ t) at all oint of Ω t =. For harmonic roblem, the linear elatodynamic equation write σˆ + ρ Fˆ + ρω uˆ = (.1) b Ω,D n F x Ω,N Ω Ω = Ω,N Ω,D Figure. Solid domain and boundary condition.

6 14 Finite element and boundary method in tructural acoutic and vibration The boundary condition of Equation.19 and. remain unchanged. For more detail, the reader i invited to refer, for examle, to Reddy book (Reddy 1)..4 Linear oroelaticity From a qualitative oint of view, a orou material i made u of a olid hae (the matrix or keleton) and a fluid hae (ore) aturating it network of ore. The matrix can be continuou (e.g., latic foam, orou ceramic) or not (fibrou or granular material). The comlexity of the microcoic geometry of uch a medium make it difficult to model it at thi cale. The modeling i rather done at a macrocoic cale (defined by the wavelength in the medium), wherein thi heterogeneou medium i een a the ueroition in time and ace of two continuou couled media, a olid and a fluid. Thi i the bai of the Biot theory. An extenion of thi theory, the Biot Allard theory, i dedicated to the acoutic of orou media. It etablihe artial differential equation involving macrocoic olid and fluid dilacement ( u, U f ) averaged over a rereentative elementary volume. Alternatively, thee equation can be rewritten in term of f olid-hae dilacement and intertitial reure ( u, ). Thu, the wave roagation in oroelatic material i commonly decribed uing either the claic dilacement form ( u, U f ) or a mixeddilacement reure ( u, ) form. Thee are the mot oular form f that have been imlemented in the context of the FEM over the year. It i not the uroe of thi book to cover all the variou form of oroelaticity equation. The reader can refer to Allard and Atalla book (Allard and Atalla 9) for detail about the modeling of oroelatic material. f Intead, it ha been choen to focu on the mixed ( u, ) dilacement reure form, which rove to be retty efficient from the numerical oint of view. Other form can be imlemented in a imilar way (Allard and Atalla 9). In addition, thee equation will only be written in the frequency domain. Equation in the time domain can be found in Gorog et al. (1997) and Fellah et al. (13). The governing equation * of a oroelatic material in the framework of f the mixed ( u, ) read a (Allard and Atalla 9) Coyrighted Material Taylor & Franci σ + ω uˆ + γ ˆ f = (.) * Thee equation can be written in vector form σ + ω u + γ f = and f f ˆ + ω ˆ uˆ =. R φ γ

7 Baic equation of tructural acoutic and vibration 15 ˆf + ˆf uˆ = R ω φ γ (.3) where σ i the in vacuo olid-hae tre tenor σ ν = N( 1 + iη) ui + N( 1 + iη) ε (.4) 1 ν where N, η, and ν are the olid-hae hear modulu, daming lo factor, and Poion ratio, reectively. ε i the olid-hae train tenor. γ i a couling coefficient given by γ 1 Q = φ R (.5) Coyrighted Material Taylor & Franci where ϕ denote the oroity, Q can be interreted a a couling coefficient between the deformation of the olid hae and the fluid hae, and R i the dynamic bulk modulu of the fluid hae occuying a fraction ϕ of a unit volume of the orou material. Q and R are related to the dynamic bulk modulu of the air in the ore K e. K e account for the diiation due to the thermal exchange between the two hae. ρ i a dynamic denity given by = 11 1 (.6) where ρ 11, ρ 1, and ρ are Biot comlex-valued dynamic denitie given by 11 = ( 1 φ) ρk + φρ( α ( ω) 1) = φ ρ ( α ( ω) 1) 1 = φ ρα ( ω) (.7) where ρ i the denity of the fluid in the ore, ρ k i the denity of the material of the keleton, and α ( ω) i the dynamic tortuoity. Thi coefficient account for the diiation due to the vicou effect in the fluid hae. Note that (1 ϕ )ρ k rereent the aarent ma of the orou material. Generally, the Johnon Chamoux Allard (JCA) model (Johnon et al. 1987; Chamoux and Allard 1991) i choen to decribe the fluid hae of the orou material. In thi cae, five inut arameter are required, namely the oroity ϕ, the flow reitivity σ, the tortuoity α, the vicou characteritic

8 16 Finite element and boundary method in tructural acoutic and vibration length Λ, and the thermal characteritic length Λ. In addition, if the vibration of the keleton i accounted for and if the olid hae i aumed iotroic, Young modulu E, lo factor η, and Poion ratio ν are required. Uing the JCA model, the dynamic tortuoity can be written a (Allard and Atalla 9) φσ αω ( ) = α i G( ω) ωρ (.8) where G( ω ) i a vicou correction factor given by G( ω) = 1 + i 4 σλ φ α ηρω (.9) and the dynamic bulk modulu K e ( ω ) read K e( ω) = γ γp 8η ( γ ) G iλ B ωρ 1 (.3) where G ( ω ) i a thermal correction factor G ( ω) = 1 + i Λ B ρω 16η (.31) Coyrighted Material Taylor & Franci where B = ηc /k tc i Prandtl number, η i the fluid dynamic vicoity, C i the heat caacity at contant reure, k tc i the thermal conductivity, and γ = C /C v i the heat caacity ratio. Other exreion of dynamic tortuoity and dynamic bulk modulu can be ued if additional orou material arameter are available (Pride et al. 1993; Wilon 1993; Lafarge and Lemarinier 1997). Note that the fluid dilacement i obtained from the intertitial reure and the olid-hae dilacement by f ˆ φ U ˆ 1 = uˆ ω f (.3) The normal flux i given by f φ ˆ ( ˆ ) U u n (.33)

9 Baic equation of tructural acoutic and vibration 17 and the total dilacement of the orou material i written a ( 1 φ ) u + φ U f (.34) Alo note that the total tre tenor of the orou material i t σˆ = σ Q φ + ˆ f 1 R I (.35) where φ ( 1 + ( QR / )) i referred to a Biot Willi coefficient. An imortant cae of interet i when the orou material keleton can be conidered rigid and motionle or lim. In the cae of a rigid motionle frame, û = and the orou material i comletely decribed by it intertitial reure ˆ f. Then, Equation.3 reduce to ˆf + ˆf = R ω (.36) which reemble Helmholtz equation. Uing the analogy with a fluid, Equation.36 can be rewritten a ˆf e + ˆ = K f ω e (.37) Coyrighted Material Taylor & Franci where e = / φ and K R e = /φ correond, reectively, to an effective comlex dynamic denity and an effective dynamic comlex bulk modulu of an equivalent fluid occuying the totality of a unit volume of orou material. Another imortant cae i the one where the elaticity modulu of the matrix i weak. Then, the elatic force in the olid hae i negligible comared to the inertial and the reure force. The equation of motion of the olid hae reduce to ωuˆ γ ˆ f + = (.38) Taking the divergence of Equation.38, the olid-hae dilatation can be ubtituted in Equation.3 to give ˆf ρ e + ˆ = K f ω e (.39)

10 18 Finite element and boundary method in tructural acoutic and vibration where ρ 1 γ = + φ e e 1 (.4) where ρ e i an aarent dynamic comlex denity of the fluid hae of the oft material. Thi equation i imilar to the one of a rigid frame motionle orou material but account for the ma and the daming added by the olid hae..5 Elato-acoutic couling Coyrighted Material Taylor & Franci When an elatic olid vibrate in the reence of a fluid, there i an interaction between the elatic and the acoutic wave. In thi cae, we mut imultaneouly olve the tructural and the fluid equation ubjected to the couling condition at the interface between the two domain. For harmonic roblem, the condition of tree-continuity at the interface i written a σn ˆ + n ˆ = (.41) In addition, the continuity of normal dilacement at the interface give 1 ρω ˆ n = uˆ n (.4) Finally, if the olid domain i couled to an unbounded fluid domain (exterior roblem, ee Chater 7), Sommerfeld condition Equation.15 mut alo be fulfilled..6 Poro-elato-acoutic couling Let u again conider the harmonic roblem. At the interface between an elatic domain and a oroelatic domain, there i continuity of both the dilacement vector and the total tre vector. Moreover, there i no relative dilacement between the two hae (the flux i null). Thu uˆ n = uˆ n t σˆ n = σˆ n ( U ˆ f φ uˆ ) n = (.43)

11 Baic equation of tructural acoutic and vibration 19 At the interface between a fluid domain and a oroelatic domain, there i continuity of the total normal dilacement, the total tre vector, and the reure. Thu f ( 1 φ) uˆ n + φuˆ n = t σ ˆ n = n ˆ f ˆ = ˆ 1 ρω ˆ n (.44) Finally, at the interface between two oroelatic domain (decribed by uercrit (1) and ()), there i continuity of the total tre vector, the intertitial reure, the olid-hae dilacement, and the fluxe. Thi can be written a ˆf,( 1) ˆf,( ) = t,( 1) t,( ) σˆ n = σˆ n,( 1),( ) uˆ n = uˆ n f,( 1) () 1 φ ˆ,( 1) f,( ) ( ) ˆ ˆ,( ) ( U u ) n = ( U uˆ ) n φ (.45).7 Concluion Coyrighted Material Taylor & Franci Thi chater decribed the fundamental governing equation for three claic roblem of mechanic: linear acoutic, linear elatodynamic, and linear oroelaticity. The next chater will introduce the aociated integral form neceary for their reolution uing the finite and BE method. REFerence Allard, J. F. and N. Atalla. 9. Proagation of Sound in Porou Media, Modelling Sound Aborbing Material. nd ed. Chicheter, UK: Wiley-Blackwell. Chamoux, Y. and J.-F. Allard Dynamic tortuoity and bulk modulu in airaturated orou media. Journal of Alied Phyic 7 (4): Fellah, Z. E. A., M. Fellah, and C. Deollier. 13. Tranient acoutic wave roagation in orou media:chater 6. In Modeling and Meaurement Method for Acoutic Wave and for Acoutic Microdevice. Vol. 61. InTech. htt://hal. archive-ouverte.fr/doc//86/8/18/pdf/intech-tranient_acoutic_wave_ roagation_in_orou_media.df.

12 Finite element and boundary method in tructural acoutic and vibration Gorog, S., R. Panneton, and N. Atalla Mixed dilacement reure formulation for acoutic aniotroic oen orou media. Journal of Alied Phyic 8 (9): 419. Johnon, D. L., J. Kolik, and R. Dahen Theory of dynamic ermeability and tortuoity in fluid-aturated orou media. Journal of Fluid Mechanic 176: Lafarge, D. and P. Lemarinier Dynamic comreibility of air in orou tructure at audible frequencie. Journal of the Acoutical Society of America 1: Pierce, A. D Acoutic, an Introduction to It Phyical Princile and Alication. New York, USA: McGraw-Hill. Pride, S. R., F. D. Morgan, and A. F. Gangi Drag force of orou-medium acoutic. Phyical Review B, Condened Matter 47 (9): Reddy, J. N. 1. Princile of Continuum Mechanic: A tudy of Conervation Princile with Alication. Cambridge, UK: Cambridge Univerity Pre. Wilon, D. K Relaxation-matched modeling of roagation through orou media, including fractal ore tructure. The Journal of the Acoutical Society of America 94 (): Coyrighted Material Taylor & Franci