Fluid-structure coupling

For the fully coupled fluid-structure interaction model, the MAM unit cell from Fig. 2.1 is considered to be part of an infinite planar array of this unit cell, as indicated in Fig. 2.5. Each MAM cell is uniquely iden-tified with the indices p ∈ {. . . ,−2,−1,0,1,2, . . .} and q ∈ {. . . ,−2,

−1,0,1,2, . . .}. The index pair (p, q) refers to the MAM unit cell rang-ing overpL_x < x < (p+ 1)L_x and qL_y < y < (q+ 1)L_y. A fluid half space with density ρ₀ and speed of soundc₀ is coupled to each side of the MAM array, extending toz→ −∞andz→+∞, respectively. The acoustic pressure fields in those half spaces are given by ˆP⁻(x, y, z), for the source side of the MAM at z < 0, and ˆP⁺(x, y, z), for the trans-mission side of the MAM at z > 0. The MAM is acoustically excited by an obliquely incident plane wave with pressure amplitude ˆP_i0 and wave vector

where θi is the incidence angle and φi is the azimuth of the plane wave (see Fig. 2.5). After introducing the nondimensionalized incident pressure field as βi= ˆPiLx/Tm, the dimensionless pressure field of the incident sound wave in the source side fluid half space is given by

β_i(ξ, η, ζ) =β_i0e⁻ⁱ(^κξξ+κηη+κζζ), (2.34)

Figure 2.5:Setup for the fluid-structure interaction model between the MAM unit cell and the adjacent fluid half spaces with Bloch-Floquet periodic bound-aries.

where κξ = kixLx, κη =kiyLy, and κζ = kizLx are the dimensionless forms of the Cartesian incident wave number components. The nondi-mensional acoustic pressure fieldβ⁻ = ˆP⁻L_x/T_m can be decomposed into a blocked pressure partβbl, given by

β_bl(ξ, η, ζ) =βi(ξ, η, ζ) +βi(ξ, η,−ζ)

=βi0e⁻ⁱ(^κξξ+κηη)

e^−iκζζ+ e^iκζζ

, (2.35)

and a reradiated pressure partβrr, resulting from the radiation of sound on both sides of the MAM [14, p. 97]. Thus, the source side pressure field can be expressed as β⁻ = β_bl −β_rr. The pressure field on the other side of the MAM β⁺ = ˆP⁺L_x/T_m is equal only to β_rr, because no external sound field is present on this side and the MAM is many orders of magnitude thinner than the acoustic wavelength. Thus, the nondimensional acoustic pressure amplitude difference on the MAM surface ∆β is

∆β(ξ, η) =β⁻(ξ, η,0)−β⁺(ξ, η,0)

= 2β_i0e⁻ⁱ(^κξξ+κηη)−2β_rr(ξ, η,0). (2.36) Consequently, the acoustic pressure difference vector∆βin the system of equations (2.27) can be written as

∆β= 2β_i0b−2β_rr, (2.37) where b = (bmn) ∈ C^N is the membrane mode excitation vector de-pending only on the properties of the incident sound field with

b_mn= 2mnπ²(−1)^me^−iκξ −1 κ²_ξ−m²π²

(−1)ⁿe^−iκη−1

κ²_η−n²π² (2.38) and β_rr is obtained from the reradiated sound pressure field, which in general is a function of the membrane displacement amplitudeu.

From the nature of the incident plane wave it follows that each MAM unit cell is excited by a pressure field, which differs from the pressure excitation field of any other unit cell only by a constant phase shift.

Thus, the incident sound pressure field can be expressed in the form of a Bloch-Floquet wave as [29]

βi(ξ+p, η+q, ζ) =βi(ξ, η, ζ)e⁻ⁱ(^κξp+κηq). (2.39) Eq. (2.39) indicates that the incident sound field is – apart from a phase difference factor of e^{−i(κξp+κηq)} – periodic at the unit cell edges.

This periodicity also applies to the vibrational response of the MAM unit cells and it is therefore sufficient to consider only a single unit cell of the MAM and the coupled fluid half spaces with the Bloch-Floquet boundary conditions as shown in Fig. 2.5. Following the nondimension-alization procedure for the structural MAM model from Section 2.1.1, the reradiated acoustic field in both half-spaces is expressed in terms of the nondimensional acoustic potential functionϕ_rr, which is governed by the Helmholtz equation

ϕrr,ξξ+Λ²ϕrr,ηη+ϕrr,ζζ+κ²₀ϕrr= 0 (2.40) with the periodic boundary conditions

ϕ_rr(ξ+p, η+q, ζ) =ϕ_rr(ξ, η, ζ)e⁻ⁱ(^κξp+κηq). (2.41) In Eq. (2.40), κ₀ = k₀L_x is the dimensionless acoustic wave number, which is related to the dimensionless speed of soundς0 =c0/p

Tm/m⁰⁰_m by κ₀ = Ω/ς₀. The general solution of Eq. (2.40) under the periodic boundary conditions specified in Eq. (2.41) consists of the superposition of the fluid cavity modes, each given by the product of a lateral mode function

Ψrs= e^{−i(κrξ+κsη)}, (2.42)

where κr = κξ + 2πr, r = . . . ,−1,0,1, . . ., and κs = κη + 2πs, s = . . . ,−1,0,1, . . ., and a plane wave propagating in positive ζ-direction.

Thus, the reradiated acoustic potential function is approximated by truncating the superposition of the fluid cavity modes, i.e.

ϕrr(ξ, η, ζ)≈

Nx^(ϕ)

X

r=−Nx^(ϕ)

Ny^(ϕ)

X

s=−Ny^(ϕ)

arsΨrse^−iκrsζ

=X

rs

arsΨrse^−iκrsζ.

(2.43)

Like the expansion foru in Eq. (2.6), the approximate relationship in Eq. (2.43) becomes more accurate for higher values ofNx^(ϕ)andNy^(ϕ). In most cases it is sufficient to consider only fluid modes up to an order of ten, because the higher-order fluid modes are strongly evanescent and do not contribute much to the sound radiation of MAMs. It follows from the dispersion relation that theζ-component of the dimensionless wave number in Eq. (2.43) is given by

κ_rs =± q

κ²₀−κ²_r−Λ²κ²_s, (2.44) with the positive sign for positive radicands (i.e. realκrs) and the neg-ative sign for negneg-ative radicands (i.e. imaginary κrs) such that only physically reasonable solutions for the reradiated acoustic field are ob-tained.

The unknown constantsarsin Eq. (2.43) have to be determined from the vibro-acoustic coupling condition between the MAM and the fluid.

For this, the nondimensional particle displacement amplitude vector field urr inside the fluid half space adjacent to the MAM follows from

the gradient of the potential functionϕrr, i.e.

The continuity condition requires the MAM displacement amplitudeu and theζ-component of the fluid particle displacementu_ζ,rrto be equal at the MAM-fluid interface. With the series representations of u and ϕ_rr given in Eqs. (2.6) and (2.43), respectively, the following equation for the determination of the constantsa_rs results:

X A weak form of Eq. (2.46) is obtained by employing the Galerkin method with the complex conjugates of the lateral fluid mode func-tions Ψ_r^∗0s⁰ as weighting functions. After surface integration over the MAM, the resulting approximate relationship is given by

X

mn

hΨ_rs^∗, Φmnicmn=−1

Ωκrsars. (2.47) This can be rewritten using matrix notation as

a=−ΩK⁻¹R^Hc, (2.48)

where the elements of the vector a ∈ C^N

(ϕ) correspond to the modal amplitudesa_rsof the reradiated acoustic field withN^(ϕ)= (1+2Nx^(ϕ))· (1 + 2Ny^(ϕ)) being the total number of acoustic modes considered in the series representation in Eq. (2.43). K = (κ_rsr⁰_s⁰) ∈ C^N

(ϕ)×N^(ϕ)

is a diagonal matrix containing the ζ-components of the modal wave numbers, i.e.

κ_rsr⁰_s⁰ = (±p

κ²₀−κ²_r−Λ²κ²_s ifr=r⁰∧s=s⁰

0 else, (2.49)

and R^H denotes the conjugate transpose of R = (r_mnrs) ∈ C^N×N The nondimensional reradiated acoustic pressure field is derived from the potential functionϕrr via

βrr=−iκ₀Z₀ϕrr, (2.51) with the nondimensional characteristic impedance defined as Z₀ = Z0Lx/p

m⁰⁰_mTm. Since the elements of the reradiated pressure vector β_rr= (β_rr,mn)∈C^N in Eq. (2.37) are given by

β_rr,mn =hΦ_mn, β_rr(ξ, η,0)i, (2.52) the substitution of the series representation of ϕ_rr in Eq. (2.43) into Eq. (2.51) yields

βrr,mn =−iκ₀Z₀X

rs

hΦ_mn, Ψrsiars, (2.53) which can be written as

β_rr=−iκ₀Z₀Ra. (2.54)

Substituting Eq. (2.48) for afinally yields the following expression for β_rr, which is independent of the fluid modes amplitude vectora:

β_rr = iΩκ0Z₀RK⁻¹R^Hc. (2.55) Inserting this expression into Eq. (2.37) and rearranging the system of equations (2.27) results in a system of equations for the fully coupled vibro-acoustic problem:

where the dimensionless radiation impedance matrix of the MAM unit cell

Ξ= 2κ₀Z₀RK⁻¹R^H (2.57) has been introduced. In terms of the modal formulation from Sec-tion 2.1.2, the modal system of equaSec-tions (2.32) is thus given by

Λˆ + iΩΞˆ −Ω²I

ˆ

c= 2βi0b,ˆ (2.58) whereΞˆ =Vˆ_c^TΞ ˆVc is the modal radiation impedance matrix andbˆ = Vˆ_c^Tb.

In order to obtain the transmission coefficient τ from Eq. (1.1), the incident and transmitted sound powers W_i and W_t have to be cal-culated. Since the incident sound field is a plane acoustic wave with amplitude ˆPi0 and incidence angleθi, the former is given by

Wi= LxLy

2Z₀ Pˆ_i0²cosθi (2.59) or, written in nondimensional form, by

Πi= Wi

pT_m³/m⁰⁰_mL_x = β_i0²

2ΛZ₀cosθi. (2.60) The transmitted sound power, on the other hand, is equal to the rera-diated sound power, which can be obtained from

Wt = 1 2

Z Z

S

Re n

ˆ

v^∗(x, y) ˆPrr(x, y,0) o

dydx, (2.61)

where ˆv^∗ = −iωwˆ^∗ is the complex conjugate of the MAM velocity amplitude and ˆPrr is the reradiated acoustic pressure amplitude field

[22, p. 10]. Using the nondimensionalizations introduced above, this With Eqs. (2.6) and (2.51) as well as the series expansion of ϕrr in Eq. (2.43), this becomes

Using the solution for the coefficient vector a in Eq. (2.48) and the definition of the nondimensional radiation impedance matrix Ξ from Eq. (2.57), the final expression for the radiated power is obtained as

Πt= Ω²κ₀Z₀

2Λ Re

c^HRK⁻¹R^Hc = Ω²

4Λc^HRe{Ξ}c. (2.64) Thus, using Eqs. (2.60) and (2.64), the transmission coefficient τ_θi of the MAM unit cell is given by

τ_θi = Πt whereec=c/(2βi0) contains the membrane modal participation factors normalized by the excitation amplitude. Expressed in terms of the nor-malized MAM modal participation factors eˆc =ˆc/(2β_i0), Eq. (2.65) is given by

Im Dokument metamaterials for aircraft noise shields (Seite 54-63)