Basic Equations of Structural Acoustics and Vibration

Chapter 2 Basic equations of structural acoustics and vibration 2.1 INTRODUCTION Copyrighted Material – Taylor & Francis As mentioned in Chapter 1, this book addresses classical numerical tech- niques to solve various vibroacoustics problems. In practice, a typical problem involves a structural domain coupled to bounded or unbounded fluid domains and sound-absorbing materials. We then seek to predict the coupled vibratory/acoustic response of the whole system subjected to the given excitations and boundary conditions. In this chapter, we introduce the fundamental equations governing the linear sound wave propagation in fluid domains (acoustics), in solid domains (elastodynamics), and in porous sound-absorbing materials (poroelasticity). We also establish the coupling equations between these domains. All these partial differential equations will be then solved using specific numerical methods, which will be pre- sented in Chapters 3 and 4. 2.2 LINEAR ACOUSTICS The acoustic pressure disturbance px(,t) in a perfect fluid volumeΩ f at rest with sound speed c0, and density ρ0, due to an acoustic source distribution Q(x,t), satisfies the inhomogeneous wave equation: 1 2 ∇−px(,t)(2 px,)tQ=− (,xt) (2.1) c0 In the case of a mass source, Q(x,t) represents the rate of mass injection and is related to the volume velocity Qs(x,t) by dQs Qx(,t) =ρ0 (2.2) dt 9 K20584_C002.indd 9 2/9/2015 7:07:16 PM 10 Finite element and boundary methods in structural acoustics and vibration Mass injection is the typical model of monopoles. In this case, the volume velocity, also called source strength, is the product of the surface area and the normal surface velocity of the monopole. For a proof of Equation 2.1, the reader is invited to look at the excellent book of Pierce (1989). To the pressure disturbance, we can associate a particle velocity Vx(,t) given by Euler’s equation: ∂Vx(,t)(∇px,)t =− (2.3) ∂t ρ0 A particle displacement Ux(,t) written as ∂Ux(,t) Vx(,t) =Copyrighted Material – Taylor & Francis (2.4) ∂t A density fluctuationρ (,xt) px(,t) ρ(,xt) = 2 (2.5) c0 Equation 2.1 must be associated to the boundary and initial conditions (Figure 2.1): • Neumann boundary condition: Normal acoustic displacement is specified on the part∂Ω f,N of boundary ∂Ωf: Un⋅=Un⋅ (2.6) r → ∞ ∂Ω f,N n x ∂Ω ∂Ω f,N f,R Ω f ∂Ω ∂Ω ∪ ∂Ω ∪ ∂Ω f = f,D f,N f,R Figure 2.1 Fluid domain and boundary conditions. K20584_C002.indd 10 2/9/2015 7:07:18 PM Basic equations of structural acoustics and vibration 11 • Dirichlet boundary condition: Acoustic pressure is specified on the part ∂Ωf,D of boundary ∂Ωf: pp= (2.7) • Mixed, Robin, or impedance boundary condition:* Specific acoustic admittance is specified on the part∂Ω f,R of boundary ∂Ωf: ρ00c Vn. β ==ρ00c (2.8) Zn p where Zn is the specific acoustic impedance applied on∂Ω f,R. In addition, for problems involving acoustic radiation in unbounded media (exteriorCopyrighted problem, see Material Chapter – 7), Taylor we must & Francis specify a condition to ensure that the wave amplitude vanishes at infinity.† This is the Sommerfeld radiation condition, which in the 3-D case, reads ∂p 1 ∂p lim r + = 0 (2.9) r →∞ rct ∂ 0 ∂ Finally, the previous equations must be completed with initial conditions ∂px(,t) px(,t) and at all points of . t =0 Ωf ∂t t =0 In the following equation, we are interested in harmonic problems, that is, dynamic problems for which the temporal dependency of all variables is sinusoidal with circular frequency ω.‡ A convenient mathematical descrip- tion of the variables is to use the complex formalism px(,tp)[=ℜ()xiexp( ωϕtp)] =ℜ[(xi)exp( ()xi)exp( ωt)] (2.10) Solving for px(,t) is equivalent to solving for the complex-valued function pxˆ(). There is an amplitude pxˆ() and a phase ϕ()x associated to pxˆ(). In the following equation, we systematically omit the factor exp(iωt) in all the equations. However, we have to keep in mind that the physical value of the acoustic pressure at point x and time t is recovered by multiplying pxˆ() by exp(iωt) and taking the real part. * Also called radiation or absorption condition. † This condition assumes the absence of acoustic sources at infinity. ‡ Note that if the disturbance spectrum is broadband, the temporal signal can be recon- 1 +∞ structed using Fourier transform: px(,tp)(= ∫−∞ xi,)ωωexp( td),ω ω being the conjugate variable of time t. 2π K20584_C002.indd 11 2/9/2015 7:07:25 PM 12 Finite element and boundary methods in structural acoustics and vibration For harmonic temporal dependence, Equation 2.1 can be rewritten as ∇+22pxˆ() kpˆ()xQ=−ˆ ()x (2.11) where k = ω/c0 is the wave number. Equation 2.11 is known as Helmholtz equation. Dirichlet boundary condition rewrites ˆ (2.12) pp= Neumann boundary condition can be rewritten using Euler’s equation (2.3): ∂pˆ 2 =⋅ρω0 Un (2.13) ∂n Copyrighted Material – Taylor & Francis The impedance boundary condition rewrites ∂pˆ +=ikβˆ pˆ 0 (2.14) ∂n Finally, for exterior problems, Sommerfeld radiation condition becomes ∂p lim r + ikp = 0 (2.15) r →∞ ∂r 2.3 LINEAR ELASTODYNAMICS Let us consider a linear elastic solid occupying volume Ωs (see Figure 2.2). The fundamental equations governing its dynamic behavior are given by the conservation of mass equation, the conservation of momentum equation, and the behavioral law of the solid (the model defining how the solid deforms in response to an applied stress). In the framework of small dis- turbances around the solid equilibrium position, we can linearize these equations to come up with the following linear elastodynamics equations, relating the linearized displacement field u, strain tensor ε, and stress tensor σ at all points of Ωs 2u ∂ (2.16) ρss2 =∇⋅+σρFb ∂t σε= C (2.17) K20584_C002.indd 12 2/9/2015 7:07:27 PM Basic equations of structural acoustics and vibration 13 1 ∂ui ∂uj εij = + (2.18) 2 ∂xj ∂xi where ρs is the density of the solid, ∇ is the nabla operator (see Appendix 3B), Fb is the body force vector per unit volume, C is the fourth-order stiff- ness (elasticity) tensor, εij and ui, i,j = 1,2,3 are the strain tensor and displacement components along the 3 directions of a cartesian coordinate system. Equation 2.17 represents Hooke’s law. In Equation 2.16, the body force distribution per unit mass Fb can include various effects: gravity, thermal effects, initial deformation or pre- stress, etc. It is common to rewrite this equation according to d’Alembert as ∇⋅σρ+=sbF 0 where the body force ρsbF accounts for the inertial 22 pseudo-forceCopyrighted Iu=−ρs()∂∂/ tMaterial. – Taylor & Francis The boundary ∂Ωs of the solid is subjected to two types of boundary conditions (Figure 2.2): • Specified contact forces F per unit area applied on ∂Ωs,N: σ⋅nF= (2.19) • Specified displacement over∂Ω s,D = ∂Ωs/∂Ωs,N: uu= (2.20) Finally, the previous equations are supplemented by two initial conditions providing the values of ux(,t) and ((ux,)tt/ ) at all points of . t =0 ∂∂t =0 Ωs For harmonic problems, the linear elastodynamics equation writes ˆ 2 ∇⋅σρˆ ++sbFuρωs ˆ = 0 (2.21) ∂Ω s,D n F x ∂Ω s,N Ω s ∂Ω ∂Ω ∪ ∂Ω s = s,N s,D Figure 2.2 Solid domain and boundary conditions. K20584_C002.indd 13 2/9/2015 7:07:30 PM 14 Finite element and boundary methods in structural acoustics and vibration The boundary conditions of Equations 2.19 and 2.20 remain unchanged. For more details, the reader is invited to refer, for example, to Reddy’s book (Reddy 2010). 2.4 LINEAR POROELASTICITY From a qualitative point of view, a porous material is made up of a solid phase (the matrix or skeleton) and a fluid phase (pores) saturating its net- work of pores. The matrix can be continuous (e.g., plastic foams, porous ceramics) or not (fibrous or granular materials). The complexity of the microscopic geometry of such a medium makes it difficult to model it at this scale. The modeling is rather done at a macroscopic scale (defined by the wavelength in the medium), wherein this heterogeneous medium is seen as the superpositionCopyrighted in time Material and space – ofTaylor two continuous & Francis coupled media, a solid and a fluid. This is the basis of the Biot theory. An extension of this theory, the Biot–Allard theory, is dedicated to the acoustics of porous media. It establishes partial differential equations involving macroscopic solid and fluid displacements(, uUsf) averaged over a representative elemen- tary volume. Alternatively, these equations can be rewritten in terms of solid-phase displacement and interstitial pressure (,ups f ). Thus, the wave propagation in poroelastic materials is commonly described using either the classic displacement form (,uUsf) or a mixed- displacement pressure (,ups f ) form. These are the most popular forms that have been implemented in the context of the FEM over the years. It is not the purpose of this book to cover all the various forms of poroelasticity equations. The reader can refer to Allard and Atalla’s book (Allard and Atalla 2009) for details about the modeling of poroelastic materials. Instead, it has been chosen to focus on the mixed (,ups f ) displacement pressure form, which proves to be pretty efficient from the numerical point of view. Other forms can be implemented in a similar way (Allard and Atalla 2009).

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