Acoustic and vibration response of a structure with added noise control treatment under various excitations Dilal Rhazia) and Noureddine Atalla G.A.U.S., Mechanical Engineering Department, Universit e de Sherbrooke, 2500 Boulevard Universit e, Sherbrooke, Quebec J1K 2R1, Canada
(Received 3 August 2012; revised 16 December 2013; accepted 23 December 2013) The evaluation of the acoustic performance of noise control treatments is of great importance in many engineering applications, e.g., aircraft, automotive, and building acoustics applications. Numerical methods such as finite- and boundary elements allow for the study of complex structures with added noise control treatment. However, these methods are computationally expensive when used for complex structures. At an early stage of the acoustic trim design process, many industries look for simple and easy to use tools that provide sufficient physical insight that can help to formulate design criteria. The paper presents a simple and tractable approach for the acoustic design of noise control treatments. It presents and compares two transfer matrix-based methods to investigate the vibroacoustic behavior of noise control treatments. The first is based on a modal approach, while the second is based on wave-number space decomposition. In addition to the classical rainon-the-roof and diffuse acoustic field excitations, the paper also addresses turbulent boundary layer and point source (monopole) excitations. Various examples are presented and compared to a finite element calculation to validate the methodology and to confirm its relevance along with its limitaC 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4861361] tions. V PACS number(s): 43.40.Rj, 43.55.Ti, 43.20.Tb, 43.55.Rg [FCS]
I. INTRODUCTION
Predicting noise transmission through multilayer structures at an early design stage is still a challenging task. A simple and reliable tool that allows for a fast computation of vibroacoustic response of a structure subjected to different kinds of excitations over the audible frequency spectrum is of great interest for industry. For example, laboratory and flight tests aiming at simulating turbulent boundary layer (TBL) excitation are difficult to conduct and could only be done at prohibitive costs.1 Numerical simulation is a practical approach to analyze the response of structures to such excitations to suggest design modifications. One main drawback of classical numerical methods such as finite element (FE) and boundary element (BE) is the computational time and hardware cost, especially for predictions over a wide frequency spectrum. Hence, their use is limited to the low- and mid-frequency domain. Most of the previous work on the vibroacoustic response of structures under TBL excitation addresses bare panel configurations without sound packages.2–5 For instance, Maury et al. (Refs. 4 and 5) presented a wavenumber-frequency approach to predict both the vibration response and the acoustic radiation of a thin baffled plate excited by a large class of random fields. The response of structures under TBL, diffuse acoustic field (DAF), and purely random field were compared. The authors in Ref. 4 have presented and used a model to study the boundary layer noise transmitted through aircraft sidewalls6 that consisted of double wall panel partitions with light insulation material. Legault et al.7 a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (2), February 2014
Pages: 693–704
analyzed the sound transmission through ribbed panels excited by TBL and DAF using the periodic theory. A parameters study was done to show the effect of frame spacing in the presence of both excitations. The authors suggested that the noise control treatment (NCT), especially constituted of absorbing materials, can be included in the model using a transfer matrix method (TMM) without details. The influence of the excitation type on add-on damping efficiency was studied by Collery et al.8 Strong and varying damping effects were observed depending on the excitation, TBL or DAF. A very important component, namely the NCT, was neglected in the preceding studies. To account for the influence of the NCT and trim panels, the wave propagation in the insulating material has to be accounted for. This complicated phenomenon has been the focus of a research by Graham.9 A NCT was applied to a structural panel and was shown to have a direct effect on the structure. According to Cimerman et al.,10 the acoustic material interacts with the non-resonant response of the panel affecting the mass law sound transmission and acts as a multilayer damping treatment on the panel. Many conclusions drawn from bare panel studies have to be revised. For example, the efficiency of a structural damping treatment is likely to be limited in the presence of a sound package. As suggested by Graham,9 it will be much more productive to concentrate on optimizing the acoustic characteristics of the interior treatment. Cooper11 has shown that structural damping treatments had a negligible effect on sound levels when the cabin was trimmed. Coyette et al.12 studied the acoustic transmission characteristics of structures with porous material treatment using refined finite element models. The TBL excitation was modeled using equivalent stochastic
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C 2014 Acoustical Society of America V
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nodal forces. Due to the computational cost of the study of double wall structures with various configurations and under TBL and DAF excitations, the frequency range of interest was limited to 500 Hz. Graham13 defined a simple model for a flat panel under TBL excitation. This model is based on the standard modal analysis (modal synthesis approach). Many physical phenomena are studied such as structural damping, aerodynamic coincidence, critical frequency, and structural inhomogeneity. The author acknowledges that his conclusions are tentative as it is known that the sound package has a significant effect on the behavior of the plate. Thus the model was extended to take into account the effect of a sound package.9 The latter consists of a flat plate under TBL excitation. The panel had the internal surface covered by two dissipative layers representing insulation and an elastic plate, the trim panel. The main objective of this paper is not to discuss the effects of a sound package on the vibroacoustic response of panels with attached sound packages but rather to present and validate two simple methodologies for a quick assessment of these effects. It is an attempt to alleviate the limitations of the existing methodologies (computational time; limited frequency range, limited excitation types, etc.). The paper presents a simple TMM based methodology to predict the effect of a sound package in multilayered flat structures under various types of excitations: TBL, monopole, DAF and rain-on-the-roof (ROF). The methodology is based on two approaches. The first is a modal approach using the Rayleigh Ritz technique where the above four excitations were taken into account using the joint acceptance concept and the global indicators were computed by the summation of each mode’s contribution. The approach uses simply supported boundary conditions and accounts for the effect of the sound package through modal impedances computed using the TMM. In the latter, the modal wavenumbers are propagated, independently, through the sound package and used to calculate the added (seen by the panel) and transfer (seen by the transmission face of the sound package) modal impedances. This approach is approximate because the TMM assumes the sound package of infinite lateral extent. Moreover, cross modal coupling and fluid loading are neglected. The latter can be easily accounted for in the presented methodology but was not deemed necessary due to its expected low effect for the considered geometry and fluids (air) and high computational cost. On the other hand, accounting for the effect of cross modal coupling is not feasible in the presented methodology. However, its effect is expected to be small because for bare homogenous flat panels, it is mainly observed at the resonances of the transmission loss (TL) curve (anti-resonances of the radiated power). Moreover, this effect will be smoothed out when frequency band averaging of the TL is considered as is practically done in the use of the TMM method and in engineering TL data presentations. A more important limitation of this first method is that the propagating wavenumbers into the sound package are limited to the modal wavenumbers of the panel. This is a filter that may affect the accuracy of the method as discussed in the presented numerical examples (see the case of the foam with a heavy layer). The second approach is 694
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
more natural for the TMM. It is referred to here by the wave approach. In this approach, the excitation is decomposed in the wave-number space, and a numerical integration in the wave-number space is performed to compute the response of the structure. Note that for ROF excitation, a similar work, using TMM and a decomposition of the point load in wavenumber space, has already been published by the authors.14 However, below the critical frequency of the structure, this approach was not able to accurately capture the acoustic response of the system. To overcome this problem, this present paper proposes to use a radiation efficiency algorithm presented by Rhazi and Atalla15 to account for the panel’s size and thus correct for the radiation efficiency at low frequencies. The paper is divided in two main sections. The first section presents the theory behind the two approaches. The second section presents a systematic comparison with FE for a bare panel and two added treatments, foam and foam with a mass (heavy) layer, to corroborate the validity of the present methods and assess its accuracy and range of validity. Because the ROF case was already studied,14 the comparison will be limited to the “equivalent” TL for TBL, monopole and DAF excitations. II. THEORY
The basic model consists of a flat multilayered structure of length Lx and width Ly, set in an infinite rigid baffle and driven by various excitations, that is: TBL, monopole, ROF, and DAF. Figure 1 shows a schematic of the studied problem. The baffled structure separates two fluid domains, the source side X2 (y < 0) and the receiver side X1 (y > 0), characterized by densities q1 and q2 with the corresponding sound velocities c1 and c2, respectively. The methodology is based on TMM wherein each layer is represented by a matrix relating a set of variables at the two faces of the layer. The nature of the variables and associated matrix is a function of the properties and nature of waves propagating in the layer. For example, for an elastic porous material, the matrix is of dimension 6 6, representing three waves propagating in the layer (2 compression waves and one shear wave).16 In this section, transmission and radiation problems will be solved for a given wave-number kt with components
FIG. 1. Multilayer structure, set in an infinite rigid baffle, and driven by various excitations. Acoustic energy radiated to the interior (y > 0). D. Rhazi and N. Atalla: Sound package study with various excitations
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(kx, ky). Once the associated various indicators (pressure, velocity, surface impedance, radiated power, etc.) are computed using the TMM, the global indicators are calculated depending on the used approach. Two approaches are presented. The first is a modal approach wherein the global indicators are computed by the summation of the contribution of a number of kept modes (modal synthesis approach). In the second approach, a numerical integration in wave-number space is used. In the following, subscript A is used for the excitation side (front of structure) and B for the receiver side (rear of structure). The excitation pressure Pexc ðx; y; xÞ and normal velocity VA ðx; y; xÞ at the excited and VB ð x; y; xÞ at the receiver sides of the multilayer are related by TMM Z~A VA ð x; y; xÞ ¼ Pexc ð x; y; xÞ
(1)
TMM Here Nmn is the norm of mode shape umn and Z~A;mn is the associated modal structural impedance, accounting for the effect of the sound package. In consequence, Eq. (3) can be written
VA ðx; y; xÞ ¼
X Pmn ðxÞ exc umn ð x; yÞ; ~TMM N m;n mn Z
(6)
A;mn
where the modal excitation term is defined by Pmn exc ðxÞ ¼
ð S0
Pexc x0 ; y0 ; x umn x0 ; y0 dS0 :
(7)
Similarly, the velocity at a point B on the receiver side can be expanded in the form X vfy;mn ðBÞumn ðx; yÞ (8) VB ð x; y; xÞ ¼ m;n
and TMM Z~B!A VB ðx; y; xÞ ¼ Pexc ðx; y; xÞ:
(2)
TMM Z~A is the impedance of the multilayer seen from the exciTMM tation side and Z~B!A is a transfer impedance relating the excitation and response seen at the receiver face. These impedances are computed, for a fixed wavenumber, using the TMM and thus are independent of the location. In the following, a general form of vibroacoustic indicators and associated assumptions will be given for each approach. Three indicators will be used: Space averaged quadratic velocity, power radiated in the receiver side, and the input power.
with vfy;mn ð BÞ ¼
ð Þ Pmn exc A : TMM Nmn Z~B!A;mn
(9)
TMM Z~B!A;mn the transfer modal impedance calculated using the TMM with the trace wavenumber kt;mn . In consequence, Eq. (8) becomes
VB ð x; y; xÞ ¼
X m;n
ð Þ Pmn exc A umn ðx; yÞ: TMM Nmn Z~
(10)
B!A;mn
Finally, the radiated pressure at point B is given by A. Modal based approach
The solution of Eq. (1) can be obtained using an expansion of the panel’s displacement w(x, y, t) in terms of its normal modes. The latter can be calculated analytically or using FE. In this paper, it is done analytically assuming the panel flat and simply supported and thus simply supported modes umn , defined by the classic product of sine functions, are used. The velocity at a point A on the source side can be expanded in the form VA ð x; y; xÞ ¼
X
vfy;mn ð AÞumn ð x; yÞ
(3)
PB ð x; y; xÞ ¼
X m;n
ð Þ Pmn exc A TMM Nmn Z~
ZB;mn umn ðx; yÞ:
(11)
B!A;mn
In classical TMM, the panel is assumed of infinite size. In consequence, if Z0 is the characteristic impedance of the receiving semi-infinite fluid, the modal radiation impedance at point B is given by Z0 k0 ZB;mn ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 k02 kt;mn
(12)
m;n
with vfy;mn the y modal component of the velocity. It is computed for each mode using the TMM with a trace wavenumber kt;mn given by
kt;mn
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 mp np ¼ þ : Lx Ly
(4)
Accordingly, vfy;mn ð AÞ ¼
ð Þ Pmn exc A TMM : Nmn Z~ A;mn
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
(5)
By computing pressure and velocity on both sides for each mode, various vibroacoustic indicators can be calculated. Note at this stage that two assumptions are implicit in using Eqs. (5) and (9). First, cross modal coupling is neglected. Second, the wavenumbers propagating in the sound package are limited to the components given by Eq. (4). The presented numerical examples will corroborate the validity of the weak cross-modal coupling assumption. However, they will show that the approximation resulting in limiting the propagating wavenumbers in the sound package to the modal wavenumbers is more limiting.
D. Rhazi and N. Atalla: Sound package study with various excitations
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1. Space averaged quadratic velocity
The space averaged quadratic velocity, at the excited face, is defined by ð 1 2 hV i ¼ V ð x; yÞVA ð x; yÞdS: 2S S A
(13)
Using Eq. (6), it becomes 1 X hV i ¼ 2S mnpq 2
ðð ð S S0
Pexc ðx0 ; y0 ; xÞ/mn ð x0 ; y0 ÞPexc ðx00 ; y00 ; xÞ/mn ðx00 ; y00 Þ 0 00 dS dS /mn ðx; yÞ/pq ð x; yÞdS: TMM TMM S00 Nmn Npq Z~A;mn Z~A;pq
(14)
Using the weak cross-modal coupling assumption, orthogonality of modes, and noting that for the simply supported boundary conditions, the modal norm is given by: Nmn ¼ S/4, Eq. (14) becomes 1X 1 hV i ¼ 2 8 m;n Nmn Z~TMM A;mn 2
ð ð S0 S00
Pexc x0 ; y0 ; x Pexc x00 ; y00 ; x umn x0 ; y0 umn x00 ; y00 dS0 dS00 :
Finally 3 2 S SAuto;pp ðxÞ X Jmn ðx Þ 2 hV i ¼ 2 7 ; 6 8 4 m;n TMM 5 ~ Nmn Z A;mn 2
ZB;mn
2
(16)
2 with Jmn the modal joint acceptance defined by ð ð 1 2 Pexc x0 ; y0 ; x Jmn ðx Þ ¼ 2 SAuto;pp S S0 S00 Pexc x00 ; y00 ; x umn x0 ; y0 umn x00 ; y00 dS0 dS00 :
(17)
(15)
ðð
jq2 x ¼ exp jkt;mn cos /x0 þ sin /y0 S S S ejkR exp½ jkt ð cos /x þ sin /yÞdxdydx0 dy0 : 2pR (20)
equation, cos u ¼ mp=k Lx; sin u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t;mn 2 ¼ np=kt;mn Ly and R ¼ ðx x0 Þ þ ð y y0 Þ2 is the distance between two locations on the surface. An efficient semi-analytical implementation of Eq. (20) is given by TMM TMM by Z~ , Eq. (19) Rhazi and Atalla.15 Replacing Z~ In
the
preceding
B!A;mn
A;mn
gives the power radiated into the emission side. SAuto;pp ðxÞ is the point power spectral (auto-power) density of the excitation.
2. Radiated power
The sound power radiated into the receiver face is defined by ð 1 PB ðx; y; xÞVB ð x; yÞdS : (18) Prad ¼ Re 2 S Using Eqs. (8) and (11) and an approach similar to the calculation of the space averaged velocity, Prad is given by Prad ¼
2 S2 SAuto;pp ðxÞ X Jmn ðx Þ ; ðq2 c2 SrB;mn Þ TMM 8 ~ jNmn Z B!A;mn j2 m;n
(19) with rB;mn the modal radiation efficiency. In classical TMM, the panel is assumed of infinite size and rB;mn is computed using Eq. (12). In this work, a finite size correction is used. The principle is to replace ZB;mn of Eq. (12) by 696
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
3. Input power
Similarly, the input power is defined by ð 1 Pexc ðx; y; xÞVA ð x; yÞdS : Pinput ðxÞ ¼ < 2 S
(21)
According to Eqs. (6) and (7), the input power is given by the following equation: ! S2 SAuto;pp ðxÞ X 2 1 Jmn < : (22) Pinput ¼ TMM 2 Nmn Z~A;mn m;n The preceding expressions of the vibroacoustic indicators are the same for the four excitations in this study. The difference is in the generalized force or joint acceptance expression. In the following, and for each excitation, the joint acceptance expression is recalled. a. Turbulent boundary layer excitation. Distributed random excitations are frequently encountered in vibroacoustic applications. Common examples are TBL and DAF. Such D. Rhazi and N. Atalla: Sound package study with various excitations
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excitations are usually modeled as weakly stationary random processes and are characterized, in the homogeneous case, by a reference power spectrum and a spatial correlation function Spp x x0 ; y y0 ; x ¼ SAuto;pp ðxÞC x x0 ; y y0 ; x : (23) In the preceding, we have used the fact that the wall pressure fluctuation is homogeneous and isotropic. In space domain, Spp ð x x0 ; y y0 ; xÞ represents the cross-spectral density between the pressure at two points, M and M0 . Cð x x0 ; y y0 ; xÞ is the correlation function of the pressure field, in space-frequency domain. Many authors have developed analytical formulations of the spatial correlation function of TBL excitation. This paper
ð ð
1 S2
2 Jmn ðx Þ ¼
sin
S0 S00
sinð k0 RÞ : (26) C x x0 ; y y0 ; x ¼ k0 R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R ¼ ðx x0 Þ þ ð y y0 Þ2 is the distance between the two measurement points, M and M0 . The joint acceptance expression is given by mpx npy sin Lx Ly 0 0 0 0 0 0 sinðk0 RÞ mpx npy sin sin dxdx0 dydy0 : Lx Ly k0 R (27) ð Lx ð Ly ð Lx ð Ly
sin
This integration can be reduced to a double integral given by the following equation: 2 Jmn ðx Þ ¼
1 S2
ð Lx ð Ly a1 ¼0 b1 ¼0
(24) with dx ¼ 1=ax kc ; dy ¼ 1=ay kc , and kc ¼ x=Uc : Uc represents the convection velocity, ax and ay the coherence decay coefficients in the x (streamwise) and y (spanwise) directions, respectively, and dx and dy the associated correlation lengths. The joint acceptance given by Eq. (17) takes the following form for TBL excitation
(25)
mpa1 mpa1 a1 cos Lx Lx Lx mpa1 sin ; þ mp Lx
Amn ða1 Þ ¼ Lx cos
b. Diffuse acoustic field excitation. The incident diffuse field is classically represented by an infinite sum of uncorrelated plane waves the incidence angles of which are uniformly distributed over a half-space. Alternatively, its correlation function in space-frequency domain may be used
1 S2
0 0 0Þ ð C x x0 ; y y0 ; x ¼ ejxx j=dx ejyy j=dy ejkc xx ;
mpx mpx0 ax jxx0 jkc ay jyy0 jkc jðxx0 Þkc mpy mpy0 sin e e sin e dSdS0 : sin Lx Lx Lx Lx
This integral can be evaluated in closed form.
2 Jmn ðx Þ ¼
uses the Corcos model to demonstrate the proposed methodology. It is given by17
Fða1 ; b1 ÞAmn ða1 ÞBmn ðb1 Þda1 db1 ; (28)
npb1 npb1 Bmn ða1 Þ ¼ Ly cos b1 cos Ly Ly Ly npb1 : þ sin np Ly The latter integral is evaluated numerically. c. Rain on the roof excitation. Here the correlation function is a Dirac function: Cðx x0 ; y y0 ; xÞ ¼ dðx x0 Þdð y y0 Þ. In consequence, the joint acceptance 2 reduces to Jmn ðxÞ ¼ 1=4S. d. Monopole excitation. The monopole is represented by an ideal unit point source located at ðx0 ; y0 ; d Þ with d the height of the monopole measured from the excited side of the panel (Fig. 2). Consider a point source with unit source amplitude, the associated pressure field in a point ðx; y; 0Þ located in the plane of the panel is given by
Pð x; y; zÞ ¼
ejk0 R ; R
(29)
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin k0 a21 þ b21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Fða1 ; b1 Þ ¼ k0 a21 þ b21 J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
where R ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x x 0 Þ 2 þ ð y y 0 Þ2 þ d 2 .
The point source being a deterministic excitation, the associated generalized force is used rather than Eq. (17). It is given by D. Rhazi and N. Atalla: Sound package study with various excitations
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The space averaged quadratic velocity is defined by ð ð 1 þ1 þ1
2 hV i ¼ 2 j Vðkx ; ky Þ z¼0 j2 dkx dky : 8p S 1 1
(33)
Into the receiver domain, the radiated power is given by "ð þ1 ð þ1 1 Prad ¼ 2 Z0 rfinite ðkx ; ky Þ 8p 1 1
2 (34) Vðkx ; ky Þ z¼0 dkx dky ; where
Vðkx ; ky Þ z¼0 ¼ FIG. 2. Plate excited by a monopole.
Pmn ðx; yÞ ¼
ð
ejk0 R umn ð x; yÞdxdy: R S
(30)
This expression is evaluated numerically.
F ðk x ; k y Þ TMM Z~A
:
(35)
þ ZB;1
ZB;1 is the radiation impedance, seen from the receiver side. Fðkx ; ky Þ is the excitation in the wavenumber space that depends on the nature of the excitation. For the sake of conciseness, only the TBL and monopole excitations will be discussed. The ROF case has been presented in Ref. 14 while the DAF excitation is classically handled using superposition of plane waves.16
B. Wave based approach
This approach is based on the decomposition of the excitation field in wavenumber space ðkx ; ky Þ and assumes the structure to be infinite. The load f ðx; yÞ can be represented by an infinite number of plane waves using the spatial Fourier integral transform 8 ð ð > 1 þ1 þ1 > > Fðkx ; ky Þejðkx xþky yÞ dkx dky > < f ðx; yÞ ¼ 4p2 1 1 ð þ1 ð þ1 > > > > Fðk ; k Þ ¼ f ð x; yÞejðkx xþky yÞ dxdy: x y : 1
1
(31) For each wave-number k with components ðkx ; ky Þ, the transmission and radiation problems can be solved using the classic TMM. The TMM can be extended easily to take into account the panel’s size and corrections for the radiation efficiency at low frequencies. The approach proposed by Rhazi and Atalla,15 known under the name of FTMM (finite transfer matrix method), is used in this paper for all kind of excitations. The basic idea of this approach is to replace the radiation efficiency in the receiving medium rinfinite by the theoretical baffled radiation efficiency of the window under vibration due to forced propagating waves rfinite . In the following, the expressions of vibroacoustic indicators are given without proof for sake of conciseness. The input power is defined by ð ð 1 þ1 þ1
< Fðkx ; ky ÞV ðkx ; ky Þz¼0 dkx dky : Pin ¼ 2 8p 1 1 (32) 698
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
a. Turbulent boundary layer excitation. In the wavenumber domain, Eq. (24) takes the following form where the flow is assumed along the x direction ð þ1 ð þ1 C x x0 ; y y0 ; x C ðk x ; k y ; x Þ ¼ 1
1
exp½jðkx x þ ky yÞdxdy:
(36)
This integration leads to the following expression: Cðkx ; ky ; xÞ ¼
4dx dy : 1 þ ðkx kc Þ2 d2x 1 þ ky2 d2y
(37)
b. Monopole excitation. The spherical wave, Eq. (29), can be expanded into plane waves using a two-dimensional Fourier transform.18 In the plane z ¼ 0, 8 ð ð 1 þ1 þ1 > > > P k ; k ejðkx xþky yÞ dkx dky P x; y ¼ > < ð Þ 4p2 1 1 ð x y Þ ! ð þ1 ð þ1 jk0 pffiffiffiffiffiffiffiffiffi x2 þy2 > e > > Pðkx ; ky Þ ¼ > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ejðkx xþky yÞ dxdy: : x2 þ y2 1 1 (38)
Using polar coordinates 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < k ¼ n cos w; k ¼ n sin w; n ¼ k2 þ k2 x y x y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : x ¼ r cos u; y ¼ r sin u; r ¼ x2 þ y2 ;
(39)
the excitation term in wavenumber space becomes D. Rhazi and N. Atalla: Sound package study with various excitations
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P ðk x k y Þ ¼
ð 2p ð þ1
ejr½k0 ncosðwuÞ drdu
1
0
ð 2p ¼ j 0
dw1 1 ¼ 2pj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k0 þ n cos w1 k 2 n2
(40)
0
where w1 ¼ u w: Hence, ð ð j þ1 þ1 ejðkx xþky yÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidkx dky : Pð x; yÞ ¼ 2p 1 1 k2 n2 0
ð þ1 ð þ1 1 ð2pÞ V ðnÞndn 2S 0 0 ! ! ð þ1 J0 ðnaÞJ0 ðn0 aÞada: V ðn0 Þn0 dn0
hV 2 i ¼
(41)
This expression describes the field in the plane z ¼ 0. To extend this expression to the whole space,18 one can add the term 6jlz in the exponential qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi term of Eq. (41).
Setting l ¼ k02 n2 and using the preceding polar coordinates kx x þ ky y ¼ rn cosðu wÞ (42) dkx dky ¼ ndndw; Pðr Þ ¼
j 2p
ð 2p ð þ1 0
ejrncosðuwÞ eþjld ndndw: l 1
Using the following property: ð 2p ejucosð/wÞ dw ¼ 2pJ0 ðuÞ:
Note that only J0 ðnr Þ and J0 ðn0 rÞ depend on x and y because pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 . The integration is performed first over x and y. Using polar coordinates x ¼ a cos h and y ¼ a sin h, Eq. (48) becomes
(43)
Using the following orthogonally property of Bessel functions: ð þ1 dðn n0 Þ ; (50) J0 ðn0 aÞJ0 ðnaÞada ¼ n 0 the final expression of quadratic velocity is obtained, ð p þ1 2 hV i ¼ ndnjV ðnÞj2 : S 0
(44)
One obtains the expression of the incident pressure (the excitation term is twice this term assuming blocked pressure assumption), ð1 PðnÞJ0 ðnr Þndn; (45) Pðr Þ ¼ 0
where PðnÞ ¼ jejld =l and =mðlÞ 0,
(Received 3 August 2012; revised 16 December 2013; accepted 23 December 2013) The evaluation of the acoustic performance of noise control treatments is of great importance in many engineering applications, e.g., aircraft, automotive, and building acoustics applications. Numerical methods such as finite- and boundary elements allow for the study of complex structures with added noise control treatment. However, these methods are computationally expensive when used for complex structures. At an early stage of the acoustic trim design process, many industries look for simple and easy to use tools that provide sufficient physical insight that can help to formulate design criteria. The paper presents a simple and tractable approach for the acoustic design of noise control treatments. It presents and compares two transfer matrix-based methods to investigate the vibroacoustic behavior of noise control treatments. The first is based on a modal approach, while the second is based on wave-number space decomposition. In addition to the classical rainon-the-roof and diffuse acoustic field excitations, the paper also addresses turbulent boundary layer and point source (monopole) excitations. Various examples are presented and compared to a finite element calculation to validate the methodology and to confirm its relevance along with its limitaC 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4861361] tions. V PACS number(s): 43.40.Rj, 43.55.Ti, 43.20.Tb, 43.55.Rg [FCS]
I. INTRODUCTION
Predicting noise transmission through multilayer structures at an early design stage is still a challenging task. A simple and reliable tool that allows for a fast computation of vibroacoustic response of a structure subjected to different kinds of excitations over the audible frequency spectrum is of great interest for industry. For example, laboratory and flight tests aiming at simulating turbulent boundary layer (TBL) excitation are difficult to conduct and could only be done at prohibitive costs.1 Numerical simulation is a practical approach to analyze the response of structures to such excitations to suggest design modifications. One main drawback of classical numerical methods such as finite element (FE) and boundary element (BE) is the computational time and hardware cost, especially for predictions over a wide frequency spectrum. Hence, their use is limited to the low- and mid-frequency domain. Most of the previous work on the vibroacoustic response of structures under TBL excitation addresses bare panel configurations without sound packages.2–5 For instance, Maury et al. (Refs. 4 and 5) presented a wavenumber-frequency approach to predict both the vibration response and the acoustic radiation of a thin baffled plate excited by a large class of random fields. The response of structures under TBL, diffuse acoustic field (DAF), and purely random field were compared. The authors in Ref. 4 have presented and used a model to study the boundary layer noise transmitted through aircraft sidewalls6 that consisted of double wall panel partitions with light insulation material. Legault et al.7 a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (2), February 2014
Pages: 693–704
analyzed the sound transmission through ribbed panels excited by TBL and DAF using the periodic theory. A parameters study was done to show the effect of frame spacing in the presence of both excitations. The authors suggested that the noise control treatment (NCT), especially constituted of absorbing materials, can be included in the model using a transfer matrix method (TMM) without details. The influence of the excitation type on add-on damping efficiency was studied by Collery et al.8 Strong and varying damping effects were observed depending on the excitation, TBL or DAF. A very important component, namely the NCT, was neglected in the preceding studies. To account for the influence of the NCT and trim panels, the wave propagation in the insulating material has to be accounted for. This complicated phenomenon has been the focus of a research by Graham.9 A NCT was applied to a structural panel and was shown to have a direct effect on the structure. According to Cimerman et al.,10 the acoustic material interacts with the non-resonant response of the panel affecting the mass law sound transmission and acts as a multilayer damping treatment on the panel. Many conclusions drawn from bare panel studies have to be revised. For example, the efficiency of a structural damping treatment is likely to be limited in the presence of a sound package. As suggested by Graham,9 it will be much more productive to concentrate on optimizing the acoustic characteristics of the interior treatment. Cooper11 has shown that structural damping treatments had a negligible effect on sound levels when the cabin was trimmed. Coyette et al.12 studied the acoustic transmission characteristics of structures with porous material treatment using refined finite element models. The TBL excitation was modeled using equivalent stochastic
0001-4966/2014/135(2)/693/12/$30.00
C 2014 Acoustical Society of America V
693
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nodal forces. Due to the computational cost of the study of double wall structures with various configurations and under TBL and DAF excitations, the frequency range of interest was limited to 500 Hz. Graham13 defined a simple model for a flat panel under TBL excitation. This model is based on the standard modal analysis (modal synthesis approach). Many physical phenomena are studied such as structural damping, aerodynamic coincidence, critical frequency, and structural inhomogeneity. The author acknowledges that his conclusions are tentative as it is known that the sound package has a significant effect on the behavior of the plate. Thus the model was extended to take into account the effect of a sound package.9 The latter consists of a flat plate under TBL excitation. The panel had the internal surface covered by two dissipative layers representing insulation and an elastic plate, the trim panel. The main objective of this paper is not to discuss the effects of a sound package on the vibroacoustic response of panels with attached sound packages but rather to present and validate two simple methodologies for a quick assessment of these effects. It is an attempt to alleviate the limitations of the existing methodologies (computational time; limited frequency range, limited excitation types, etc.). The paper presents a simple TMM based methodology to predict the effect of a sound package in multilayered flat structures under various types of excitations: TBL, monopole, DAF and rain-on-the-roof (ROF). The methodology is based on two approaches. The first is a modal approach using the Rayleigh Ritz technique where the above four excitations were taken into account using the joint acceptance concept and the global indicators were computed by the summation of each mode’s contribution. The approach uses simply supported boundary conditions and accounts for the effect of the sound package through modal impedances computed using the TMM. In the latter, the modal wavenumbers are propagated, independently, through the sound package and used to calculate the added (seen by the panel) and transfer (seen by the transmission face of the sound package) modal impedances. This approach is approximate because the TMM assumes the sound package of infinite lateral extent. Moreover, cross modal coupling and fluid loading are neglected. The latter can be easily accounted for in the presented methodology but was not deemed necessary due to its expected low effect for the considered geometry and fluids (air) and high computational cost. On the other hand, accounting for the effect of cross modal coupling is not feasible in the presented methodology. However, its effect is expected to be small because for bare homogenous flat panels, it is mainly observed at the resonances of the transmission loss (TL) curve (anti-resonances of the radiated power). Moreover, this effect will be smoothed out when frequency band averaging of the TL is considered as is practically done in the use of the TMM method and in engineering TL data presentations. A more important limitation of this first method is that the propagating wavenumbers into the sound package are limited to the modal wavenumbers of the panel. This is a filter that may affect the accuracy of the method as discussed in the presented numerical examples (see the case of the foam with a heavy layer). The second approach is 694
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
more natural for the TMM. It is referred to here by the wave approach. In this approach, the excitation is decomposed in the wave-number space, and a numerical integration in the wave-number space is performed to compute the response of the structure. Note that for ROF excitation, a similar work, using TMM and a decomposition of the point load in wavenumber space, has already been published by the authors.14 However, below the critical frequency of the structure, this approach was not able to accurately capture the acoustic response of the system. To overcome this problem, this present paper proposes to use a radiation efficiency algorithm presented by Rhazi and Atalla15 to account for the panel’s size and thus correct for the radiation efficiency at low frequencies. The paper is divided in two main sections. The first section presents the theory behind the two approaches. The second section presents a systematic comparison with FE for a bare panel and two added treatments, foam and foam with a mass (heavy) layer, to corroborate the validity of the present methods and assess its accuracy and range of validity. Because the ROF case was already studied,14 the comparison will be limited to the “equivalent” TL for TBL, monopole and DAF excitations. II. THEORY
The basic model consists of a flat multilayered structure of length Lx and width Ly, set in an infinite rigid baffle and driven by various excitations, that is: TBL, monopole, ROF, and DAF. Figure 1 shows a schematic of the studied problem. The baffled structure separates two fluid domains, the source side X2 (y < 0) and the receiver side X1 (y > 0), characterized by densities q1 and q2 with the corresponding sound velocities c1 and c2, respectively. The methodology is based on TMM wherein each layer is represented by a matrix relating a set of variables at the two faces of the layer. The nature of the variables and associated matrix is a function of the properties and nature of waves propagating in the layer. For example, for an elastic porous material, the matrix is of dimension 6 6, representing three waves propagating in the layer (2 compression waves and one shear wave).16 In this section, transmission and radiation problems will be solved for a given wave-number kt with components
FIG. 1. Multilayer structure, set in an infinite rigid baffle, and driven by various excitations. Acoustic energy radiated to the interior (y > 0). D. Rhazi and N. Atalla: Sound package study with various excitations
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(kx, ky). Once the associated various indicators (pressure, velocity, surface impedance, radiated power, etc.) are computed using the TMM, the global indicators are calculated depending on the used approach. Two approaches are presented. The first is a modal approach wherein the global indicators are computed by the summation of the contribution of a number of kept modes (modal synthesis approach). In the second approach, a numerical integration in wave-number space is used. In the following, subscript A is used for the excitation side (front of structure) and B for the receiver side (rear of structure). The excitation pressure Pexc ðx; y; xÞ and normal velocity VA ðx; y; xÞ at the excited and VB ð x; y; xÞ at the receiver sides of the multilayer are related by TMM Z~A VA ð x; y; xÞ ¼ Pexc ð x; y; xÞ
(1)
TMM Here Nmn is the norm of mode shape umn and Z~A;mn is the associated modal structural impedance, accounting for the effect of the sound package. In consequence, Eq. (3) can be written
VA ðx; y; xÞ ¼
X Pmn ðxÞ exc umn ð x; yÞ; ~TMM N m;n mn Z
(6)
A;mn
where the modal excitation term is defined by Pmn exc ðxÞ ¼
ð S0
Pexc x0 ; y0 ; x umn x0 ; y0 dS0 :
(7)
Similarly, the velocity at a point B on the receiver side can be expanded in the form X vfy;mn ðBÞumn ðx; yÞ (8) VB ð x; y; xÞ ¼ m;n
and TMM Z~B!A VB ðx; y; xÞ ¼ Pexc ðx; y; xÞ:
(2)
TMM Z~A is the impedance of the multilayer seen from the exciTMM tation side and Z~B!A is a transfer impedance relating the excitation and response seen at the receiver face. These impedances are computed, for a fixed wavenumber, using the TMM and thus are independent of the location. In the following, a general form of vibroacoustic indicators and associated assumptions will be given for each approach. Three indicators will be used: Space averaged quadratic velocity, power radiated in the receiver side, and the input power.
with vfy;mn ð BÞ ¼
ð Þ Pmn exc A : TMM Nmn Z~B!A;mn
(9)
TMM Z~B!A;mn the transfer modal impedance calculated using the TMM with the trace wavenumber kt;mn . In consequence, Eq. (8) becomes
VB ð x; y; xÞ ¼
X m;n
ð Þ Pmn exc A umn ðx; yÞ: TMM Nmn Z~
(10)
B!A;mn
Finally, the radiated pressure at point B is given by A. Modal based approach
The solution of Eq. (1) can be obtained using an expansion of the panel’s displacement w(x, y, t) in terms of its normal modes. The latter can be calculated analytically or using FE. In this paper, it is done analytically assuming the panel flat and simply supported and thus simply supported modes umn , defined by the classic product of sine functions, are used. The velocity at a point A on the source side can be expanded in the form VA ð x; y; xÞ ¼
X
vfy;mn ð AÞumn ð x; yÞ
(3)
PB ð x; y; xÞ ¼
X m;n
ð Þ Pmn exc A TMM Nmn Z~
ZB;mn umn ðx; yÞ:
(11)
B!A;mn
In classical TMM, the panel is assumed of infinite size. In consequence, if Z0 is the characteristic impedance of the receiving semi-infinite fluid, the modal radiation impedance at point B is given by Z0 k0 ZB;mn ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 k02 kt;mn
(12)
m;n
with vfy;mn the y modal component of the velocity. It is computed for each mode using the TMM with a trace wavenumber kt;mn given by
kt;mn
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 mp np ¼ þ : Lx Ly
(4)
Accordingly, vfy;mn ð AÞ ¼
ð Þ Pmn exc A TMM : Nmn Z~ A;mn
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
(5)
By computing pressure and velocity on both sides for each mode, various vibroacoustic indicators can be calculated. Note at this stage that two assumptions are implicit in using Eqs. (5) and (9). First, cross modal coupling is neglected. Second, the wavenumbers propagating in the sound package are limited to the components given by Eq. (4). The presented numerical examples will corroborate the validity of the weak cross-modal coupling assumption. However, they will show that the approximation resulting in limiting the propagating wavenumbers in the sound package to the modal wavenumbers is more limiting.
D. Rhazi and N. Atalla: Sound package study with various excitations
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1. Space averaged quadratic velocity
The space averaged quadratic velocity, at the excited face, is defined by ð 1 2 hV i ¼ V ð x; yÞVA ð x; yÞdS: 2S S A
(13)
Using Eq. (6), it becomes 1 X hV i ¼ 2S mnpq 2
ðð ð S S0
Pexc ðx0 ; y0 ; xÞ/mn ð x0 ; y0 ÞPexc ðx00 ; y00 ; xÞ/mn ðx00 ; y00 Þ 0 00 dS dS /mn ðx; yÞ/pq ð x; yÞdS: TMM TMM S00 Nmn Npq Z~A;mn Z~A;pq
(14)
Using the weak cross-modal coupling assumption, orthogonality of modes, and noting that for the simply supported boundary conditions, the modal norm is given by: Nmn ¼ S/4, Eq. (14) becomes 1X 1 hV i ¼ 2 8 m;n Nmn Z~TMM A;mn 2
ð ð S0 S00
Pexc x0 ; y0 ; x Pexc x00 ; y00 ; x umn x0 ; y0 umn x00 ; y00 dS0 dS00 :
Finally 3 2 S SAuto;pp ðxÞ X Jmn ðx Þ 2 hV i ¼ 2 7 ; 6 8 4 m;n TMM 5 ~ Nmn Z A;mn 2
ZB;mn
2
(16)
2 with Jmn the modal joint acceptance defined by ð ð 1 2 Pexc x0 ; y0 ; x Jmn ðx Þ ¼ 2 SAuto;pp S S0 S00 Pexc x00 ; y00 ; x umn x0 ; y0 umn x00 ; y00 dS0 dS00 :
(17)
(15)
ðð
jq2 x ¼ exp jkt;mn cos /x0 þ sin /y0 S S S ejkR exp½ jkt ð cos /x þ sin /yÞdxdydx0 dy0 : 2pR (20)
equation, cos u ¼ mp=k Lx; sin u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t;mn 2 ¼ np=kt;mn Ly and R ¼ ðx x0 Þ þ ð y y0 Þ2 is the distance between two locations on the surface. An efficient semi-analytical implementation of Eq. (20) is given by TMM TMM by Z~ , Eq. (19) Rhazi and Atalla.15 Replacing Z~ In
the
preceding
B!A;mn
A;mn
gives the power radiated into the emission side. SAuto;pp ðxÞ is the point power spectral (auto-power) density of the excitation.
2. Radiated power
The sound power radiated into the receiver face is defined by ð 1 PB ðx; y; xÞVB ð x; yÞdS : (18) Prad ¼ Re 2 S Using Eqs. (8) and (11) and an approach similar to the calculation of the space averaged velocity, Prad is given by Prad ¼
2 S2 SAuto;pp ðxÞ X Jmn ðx Þ ; ðq2 c2 SrB;mn Þ TMM 8 ~ jNmn Z B!A;mn j2 m;n
(19) with rB;mn the modal radiation efficiency. In classical TMM, the panel is assumed of infinite size and rB;mn is computed using Eq. (12). In this work, a finite size correction is used. The principle is to replace ZB;mn of Eq. (12) by 696
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
3. Input power
Similarly, the input power is defined by ð 1 Pexc ðx; y; xÞVA ð x; yÞdS : Pinput ðxÞ ¼ < 2 S
(21)
According to Eqs. (6) and (7), the input power is given by the following equation: ! S2 SAuto;pp ðxÞ X 2 1 Jmn < : (22) Pinput ¼ TMM 2 Nmn Z~A;mn m;n The preceding expressions of the vibroacoustic indicators are the same for the four excitations in this study. The difference is in the generalized force or joint acceptance expression. In the following, and for each excitation, the joint acceptance expression is recalled. a. Turbulent boundary layer excitation. Distributed random excitations are frequently encountered in vibroacoustic applications. Common examples are TBL and DAF. Such D. Rhazi and N. Atalla: Sound package study with various excitations
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excitations are usually modeled as weakly stationary random processes and are characterized, in the homogeneous case, by a reference power spectrum and a spatial correlation function Spp x x0 ; y y0 ; x ¼ SAuto;pp ðxÞC x x0 ; y y0 ; x : (23) In the preceding, we have used the fact that the wall pressure fluctuation is homogeneous and isotropic. In space domain, Spp ð x x0 ; y y0 ; xÞ represents the cross-spectral density between the pressure at two points, M and M0 . Cð x x0 ; y y0 ; xÞ is the correlation function of the pressure field, in space-frequency domain. Many authors have developed analytical formulations of the spatial correlation function of TBL excitation. This paper
ð ð
1 S2
2 Jmn ðx Þ ¼
sin
S0 S00
sinð k0 RÞ : (26) C x x0 ; y y0 ; x ¼ k0 R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R ¼ ðx x0 Þ þ ð y y0 Þ2 is the distance between the two measurement points, M and M0 . The joint acceptance expression is given by mpx npy sin Lx Ly 0 0 0 0 0 0 sinðk0 RÞ mpx npy sin sin dxdx0 dydy0 : Lx Ly k0 R (27) ð Lx ð Ly ð Lx ð Ly
sin
This integration can be reduced to a double integral given by the following equation: 2 Jmn ðx Þ ¼
1 S2
ð Lx ð Ly a1 ¼0 b1 ¼0
(24) with dx ¼ 1=ax kc ; dy ¼ 1=ay kc , and kc ¼ x=Uc : Uc represents the convection velocity, ax and ay the coherence decay coefficients in the x (streamwise) and y (spanwise) directions, respectively, and dx and dy the associated correlation lengths. The joint acceptance given by Eq. (17) takes the following form for TBL excitation
(25)
mpa1 mpa1 a1 cos Lx Lx Lx mpa1 sin ; þ mp Lx
Amn ða1 Þ ¼ Lx cos
b. Diffuse acoustic field excitation. The incident diffuse field is classically represented by an infinite sum of uncorrelated plane waves the incidence angles of which are uniformly distributed over a half-space. Alternatively, its correlation function in space-frequency domain may be used
1 S2
0 0 0Þ ð C x x0 ; y y0 ; x ¼ ejxx j=dx ejyy j=dy ejkc xx ;
mpx mpx0 ax jxx0 jkc ay jyy0 jkc jðxx0 Þkc mpy mpy0 sin e e sin e dSdS0 : sin Lx Lx Lx Lx
This integral can be evaluated in closed form.
2 Jmn ðx Þ ¼
uses the Corcos model to demonstrate the proposed methodology. It is given by17
Fða1 ; b1 ÞAmn ða1 ÞBmn ðb1 Þda1 db1 ; (28)
npb1 npb1 Bmn ða1 Þ ¼ Ly cos b1 cos Ly Ly Ly npb1 : þ sin np Ly The latter integral is evaluated numerically. c. Rain on the roof excitation. Here the correlation function is a Dirac function: Cðx x0 ; y y0 ; xÞ ¼ dðx x0 Þdð y y0 Þ. In consequence, the joint acceptance 2 reduces to Jmn ðxÞ ¼ 1=4S. d. Monopole excitation. The monopole is represented by an ideal unit point source located at ðx0 ; y0 ; d Þ with d the height of the monopole measured from the excited side of the panel (Fig. 2). Consider a point source with unit source amplitude, the associated pressure field in a point ðx; y; 0Þ located in the plane of the panel is given by
Pð x; y; zÞ ¼
ejk0 R ; R
(29)
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin k0 a21 þ b21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Fða1 ; b1 Þ ¼ k0 a21 þ b21 J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
where R ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x x 0 Þ 2 þ ð y y 0 Þ2 þ d 2 .
The point source being a deterministic excitation, the associated generalized force is used rather than Eq. (17). It is given by D. Rhazi and N. Atalla: Sound package study with various excitations
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The space averaged quadratic velocity is defined by ð ð 1 þ1 þ1
2 hV i ¼ 2 j Vðkx ; ky Þ z¼0 j2 dkx dky : 8p S 1 1
(33)
Into the receiver domain, the radiated power is given by "ð þ1 ð þ1 1 Prad ¼ 2 Z0 rfinite ðkx ; ky Þ 8p 1 1
2 (34) Vðkx ; ky Þ z¼0 dkx dky ; where
Vðkx ; ky Þ z¼0 ¼ FIG. 2. Plate excited by a monopole.
Pmn ðx; yÞ ¼
ð
ejk0 R umn ð x; yÞdxdy: R S
(30)
This expression is evaluated numerically.
F ðk x ; k y Þ TMM Z~A
:
(35)
þ ZB;1
ZB;1 is the radiation impedance, seen from the receiver side. Fðkx ; ky Þ is the excitation in the wavenumber space that depends on the nature of the excitation. For the sake of conciseness, only the TBL and monopole excitations will be discussed. The ROF case has been presented in Ref. 14 while the DAF excitation is classically handled using superposition of plane waves.16
B. Wave based approach
This approach is based on the decomposition of the excitation field in wavenumber space ðkx ; ky Þ and assumes the structure to be infinite. The load f ðx; yÞ can be represented by an infinite number of plane waves using the spatial Fourier integral transform 8 ð ð > 1 þ1 þ1 > > Fðkx ; ky Þejðkx xþky yÞ dkx dky > < f ðx; yÞ ¼ 4p2 1 1 ð þ1 ð þ1 > > > > Fðk ; k Þ ¼ f ð x; yÞejðkx xþky yÞ dxdy: x y : 1
1
(31) For each wave-number k with components ðkx ; ky Þ, the transmission and radiation problems can be solved using the classic TMM. The TMM can be extended easily to take into account the panel’s size and corrections for the radiation efficiency at low frequencies. The approach proposed by Rhazi and Atalla,15 known under the name of FTMM (finite transfer matrix method), is used in this paper for all kind of excitations. The basic idea of this approach is to replace the radiation efficiency in the receiving medium rinfinite by the theoretical baffled radiation efficiency of the window under vibration due to forced propagating waves rfinite . In the following, the expressions of vibroacoustic indicators are given without proof for sake of conciseness. The input power is defined by ð ð 1 þ1 þ1
< Fðkx ; ky ÞV ðkx ; ky Þz¼0 dkx dky : Pin ¼ 2 8p 1 1 (32) 698
J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014
a. Turbulent boundary layer excitation. In the wavenumber domain, Eq. (24) takes the following form where the flow is assumed along the x direction ð þ1 ð þ1 C x x0 ; y y0 ; x C ðk x ; k y ; x Þ ¼ 1
1
exp½jðkx x þ ky yÞdxdy:
(36)
This integration leads to the following expression: Cðkx ; ky ; xÞ ¼
4dx dy : 1 þ ðkx kc Þ2 d2x 1 þ ky2 d2y
(37)
b. Monopole excitation. The spherical wave, Eq. (29), can be expanded into plane waves using a two-dimensional Fourier transform.18 In the plane z ¼ 0, 8 ð ð 1 þ1 þ1 > > > P k ; k ejðkx xþky yÞ dkx dky P x; y ¼ > < ð Þ 4p2 1 1 ð x y Þ ! ð þ1 ð þ1 jk0 pffiffiffiffiffiffiffiffiffi x2 þy2 > e > > Pðkx ; ky Þ ¼ > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ejðkx xþky yÞ dxdy: : x2 þ y2 1 1 (38)
Using polar coordinates 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < k ¼ n cos w; k ¼ n sin w; n ¼ k2 þ k2 x y x y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : x ¼ r cos u; y ¼ r sin u; r ¼ x2 þ y2 ;
(39)
the excitation term in wavenumber space becomes D. Rhazi and N. Atalla: Sound package study with various excitations
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P ðk x k y Þ ¼
ð 2p ð þ1
ejr½k0 ncosðwuÞ drdu
1
0
ð 2p ¼ j 0
dw1 1 ¼ 2pj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k0 þ n cos w1 k 2 n2
(40)
0
where w1 ¼ u w: Hence, ð ð j þ1 þ1 ejðkx xþky yÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidkx dky : Pð x; yÞ ¼ 2p 1 1 k2 n2 0
ð þ1 ð þ1 1 ð2pÞ V ðnÞndn 2S 0 0 ! ! ð þ1 J0 ðnaÞJ0 ðn0 aÞada: V ðn0 Þn0 dn0
hV 2 i ¼
(41)
This expression describes the field in the plane z ¼ 0. To extend this expression to the whole space,18 one can add the term 6jlz in the exponential qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi term of Eq. (41).
Setting l ¼ k02 n2 and using the preceding polar coordinates kx x þ ky y ¼ rn cosðu wÞ (42) dkx dky ¼ ndndw; Pðr Þ ¼
j 2p
ð 2p ð þ1 0
ejrncosðuwÞ eþjld ndndw: l 1
Using the following property: ð 2p ejucosð/wÞ dw ¼ 2pJ0 ðuÞ:
Note that only J0 ðnr Þ and J0 ðn0 rÞ depend on x and y because pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 . The integration is performed first over x and y. Using polar coordinates x ¼ a cos h and y ¼ a sin h, Eq. (48) becomes
(43)
Using the following orthogonally property of Bessel functions: ð þ1 dðn n0 Þ ; (50) J0 ðn0 aÞJ0 ðnaÞada ¼ n 0 the final expression of quadratic velocity is obtained, ð p þ1 2 hV i ¼ ndnjV ðnÞj2 : S 0
(44)
One obtains the expression of the incident pressure (the excitation term is twice this term assuming blocked pressure assumption), ð1 PðnÞJ0 ðnr Þndn; (45) Pðr Þ ¼ 0
where PðnÞ ¼ jejld =l and =mðlÞ 0,
