On the vibro-acoustic modeling of panels excited by di ff use acoustic field (DAF)

Yahya AllahTavakoli 1

Univ Lyon, ENTPE, LTDS UMR 5513, 3 rue Maurice Audin, 69518 Vaulx-en-Velin, France Catherine Marquis-Favre 2

Univ Lyon, ENTPE, LTDS UMR 5513, 3 rue Maurice Audin, 69518 Vaulx-en-Velin, France Mohamed Ichchou 3

LTDS - CNRS UMR 5513, Vibroacoustics & Complex Media Research Group, Ecole Centrale de Lyon, Écully, 69134, France Nacer Hamzaoui 4

Laboratoire Vibrations Acoustique, Univ Lyon, INSA-Lyon, LVA EA677, F-69621, Villeurbanne, France

ABSTRACT Composite materials along with isotropic materials possess abundant applications to various engineering disciplines like aerospace and mechanical issues. The design optimization of composite materials such as sandwich panels essentially needs an accurate vibro-acoustic modeling. Such a modeling of these materials and a sound synthesis can lead to a design approach based on relevant psychoacoustic analyses. In this research, the mathematical / numerical and experimental steps required by the vibro-acoustic modeling of composite and isotropic plates excited by a di ff use acoustic field (DAF) are discussed. Herein, the analytical modeling was performed by 4th and 6th order problems adapted for the composite and isotropic plates, and moreover, the numerical modeling was carried out via the Finite Element Method (FEM). The experimental observations were also performed by means of an acoustic test cabin used for ideally generating a di ff use acoustic field (DAF). The various types of analytical and numerical simulations including synthesized sounds required by future psycho-acoustic analyses, as well as experimental observations including recorded sounds were investigated and compared. Finally, based on the various analyses, the research illustrates how such investigations and comparisons are necessary for designing and enhancing the mechanical and vibro-acoustic properties of materials.

1. INTRODUCTION

In recent years there has been an accelerating trend in the application of composite materials in the aerospace and transportation industries due to their outstanding mechanical properties as well

1 yahya.allahtavakoli@entpe.fr

2 catherine.marquisfavre@entpe.fr

3 mohamed.ichchou@ec-lyon.fr

4 nacer.hamzaoui@insa-lyon.fr

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

as their low weight and proper fatigue behavior. For instance, composite materials have known an increasing acceptance into aerospace construction over an evolutionary period in four decades [1], and nowadays, new airframes are mainly composites, such as Boeing 787 Dreamliner and Airbus A350 XWB in which composites possess 80% participation by volume and 50% participation by weight, e.g. see [1, 2]. Another advanced application of composite materials belongs to the modern stage of transportation industries. Composite materials has the advantage of 1) lighter weight to decrease fuel e ffi ciency, 2) acceptable tensile strength for lighter-weight durability, 3) enhanced stability for lasting performance, 4) corrosion resistance for deterioration-free operation, 5) sound ba ffl ing for a less noisy operating environment, and 6) design flexibility for usage in complex shapes [3].

The sandwich composite panels as a well-known typical type of composite materials are constituted from thin face sheets and a thick core o ff ering a diversity of material choices and architectures. In some recent research, we can find that the vibroacoustic properties of sandwich panels can be improved playing with the face sheet and core architecture. Certain ideas have been explored to valid the impact of geometric design on the vibroacoustic indicators. For instance, in 2016, Droz and his colleagues [4] examined di ff erent types of periodic cores, which can be used to increase the transition frequency and reduce the modal density in a given frequency range. The transition frequency happens in a sandwich panel when the transverse shear sti ff ness has a significant e ff ect on the flexural motion, by comparison with the bending sti ff ness.

Similarly, in 2016, Arunkumar et al. [5] carried out a study on the influence of core geometry on vibro-acoustic response characteristics of sandwich panels, which are used in aerospace industries. They performed certain numerical analyses of vibro-acoustic behavior of sandwich panels with di ff erent cores.

The work described in (Meng et al., 2017) [6] is another study which has numerically and experimentally analyzed the e ff ects of a geometrical parameter on the vibro-acoustic indicators of sandwich panels. Meng et al. (2017) performed numerical and experimental tests to assess the low frequency sound absorption coe ffi cient (SAC) and sound transmission loss (STL) of corrugated sandwich panels with diverse perforation configurations. Meng et al. (2017) showed that compared with the classical corrugated sandwich without perforation, the corrugated sandwich with perforated pores in one of its face sheets not only exhibits a larger SAC at low frequencies but also a better STL as a result of the enlarged SAC [6].

Also, Zergoune et al. (2017) [7] published a study that proposed a modeling strategy based on a two-scale method to analyze the flexural vibroacoustic behavior of sandwich panels, especially in the low-mid frequency range. The results in this research showed that the indicators mostly depend on the geometrical and mechanical parameters. For example, in this research the parametric analyses indicated that the cell angle of the hexagonal core of sandwich panel does not have influence on the sound transmission loss (STL) of the sandwich panel. It also showed that the increase of the panel thickness can result in a significant enhancement of the STL for the mid frequencies as well as an increase of the transition frequency but also it may cause a remarkable drop of the sti ff ness-weight ratio of the sandwich panel [7].

While in the above-mentioned studies, the acoustic absorption and the sound transmission loss are mainly considered as required vibro-acoustic indicators, and they do not consider the impact on auditory perception. On the other hand, in literature we can find research studies which proposed to couple vibro-acoustics and auditory perception. These studies could gain an increasing interest in examining the relationships between the physical parameters of radiating structures, and the auditory

characteristics of the radiated or transmitted sounds. For instance, Meunier et al. (2001) [8] provided a combination of vibroacoustic and psychoacoustic investigations of sounds radiated by a vibrating structure. The calculated sound field was the sound pressure radiated by a ba ffl ed thin-plate structure immersed in a fluid. Based on the vibroacoustic and psychoacoustic investigations, this research could finally confirm an interest of extending vibroacoustic studies to a more complete psycho-mechanical investigation [8]. The work presented in (Faure and Marquis-Favre, 2005) [9] is another study for the coupling of vibro-acoustic and auditory evaluations of sounds radiated from a plate when the structural parameters are varying. In this research study, which was performed for an isotropic steel plate, they computed the sound pressure radiated from the plate at the ears of a listener. The authors could present the results both in terms of vibro-acoustic indicators and in terms of judgments given by listeners and auditory attributes for structural parameter variations and also it could finally investigate the e ff ects of structural mechanical parameters of a simply supported ba ffl ed steel plate on sound perception [9]. Such a coupled approach accounting for sound perception highlighted the benefit of optimizing the acoustical properties of the structure in comparison with the classical consideration of vibroacoustic indicators. Another example of such research is described in (Trollé et al., 2008) [10], which concerned the e ff ect of the independent variation of certain structural parameters on the auditory perception of environmental noises transmitted through a window. This research set out how some structural parameters, the structural damping factor and the mounting conditions, of a glass pane could a ff ect the auditory perception of transmitted environmental noises [10]. Trolle et al.(2012) [11] also worked on the adjustment of vibro-acoustical computational cost in the framework of a sound quality evaluation of a simple plate-cavity system. In this research, authors specified simulation parameters so as to reduce computational cost with the same conclusions regarding the perceived acoustical properties of the vibroacoustic system under study. [11].

Now, continuing the above-mentioned studies, the current research aims at proposing a strategy required for vibro / psycho-acoustically designing mechanical structures. The proposed strategy starts with the mathematical analyses associated with the excitation of panels and then it is very essential to adopt an appropriate analytical model for predicting the vibration as the next step. Another step consists in computing the pressure field, caused by the sound radiated by the vibrating plate. Finally the strategy ends with synthesizing sounds required by the psycho-acoustic investigations for designing and enhancing the mechanical and vibro-acoustic properties of materials (see Fig.1).

Figure 1: an overview of the method proposed for vibro / psycho-acoustically designing mechanical structures

In this paper, the plot of the presentation is as follows: in Section 1 an introduction as well as a literature review on the ideas of this research was presented to readers and it is followed by Section 2, which presents the main theoretical concepts and methodology of this research. Section 3 discuses numerical experiments and some computational prerequisites and then Section 4 provides us with experimental results and their comparisons with the mathematical modeling and simulations. And finally, this paper will end with certain discussions and conclusions concerning the analytical, computational and experimental achievements of this research.

Auditory Excitation Vibration Radiation 5 evaluation

2. METHODOLOGY

In order to have an appropriate prediction of the vibration field and its radiated pressured field in the medium, we need to have an accurate understating of the governing equations of the motion and the solutions to the associated problems. Hence, in this Section, the concepts, theories and methodologies of mathematical predictions of vibrating panels are explained and discussed in details. For this purpose, governing partial di ff erential equations (PDEs), called 4 th and 6 th order problems, required for modeling the vibration of isotropic and composite plates are discussed and also the section presents mathematical strategies that are necessary for solving the problems.

2.1. 4 th ORDER PROBLEM First, assume a vibrating isotropic thin flat plate, which has a a × b rectangular shape. Then suppose a case that the plate can be excited by a harmonic point force such e i ω t at point ( x 0 , y 0 ) when the bending rigidity is D = Eh 3 / (12(1 − ν 2 )) , ρ is the mass density, h is the thickness, m = ρ h is the mass per unit area, E is the Young modulus, and ν is the Poisson ratio. In such case, according to the Kirchho ff -Love hypothesis and neglecting transverse shear and rotary inertia, the 4 th order partial di ff erential equation (PDE) of motion governing the transverse displacement w ( x , y , t ) of an undamped plate will be as follows, e.g. see [12–14]:

D ∇ 4 w ( x , y , t ) + m ∂ 2 w ( x , y , t )

∂ t 2 = e i ω t δ ( x − x 0 ) δ ( y − y 0 ) (1)

where ∇ 4 ( • ) = ∇ 2 ( ∇ 2 ( • )) and ∇ 2 ( • ) = ∂ 2 ( • )

∂ x 2 + ∂ 2 ( • )

∂ y 2 is the Laplacian operator. By a modal decomposition and taking into account an ad hoc damping loss factor η for dissipation, the response H to the harmonic point force such e i ω t at point ( x 0 , y 0 ), will be as follows [13]

H ( x , y ; x 0 , y 0 ; ω ) = X

n ≥ 0 H n ( ω ) ψ n ( x 0 , y 0 ) ψ n ( x , y ) (2)

where the frequency response coe ffi cients { H n ( ω ) } will be obtained via the governing di ff erential equation (i.e. Eq.1) as follows:

H n ( ω ) = 1 m ( ω 2 n − ω 2 + i ηω n ω ) (3)

Also, ψ n denotes the modal function of mode n . For example, for a simply supported boundary condition, the mode index n is a double subscript ( n 1 , n 2 ), and the expressions of the undamped natural frequencies are as follows e.g. see [15]:

D / m  n 1 π

 2  n 2 π

 2 ! (4)

ω n = p

a

b

where the mode shapes are ψ n ( x , y ) = (2 / √

ab ) sin( n 1 π x / a ) sin( n 2 π y / b ) and ( x , y ) ∈ [0 , a ] × [0 , b ]. Now, if the plate is embedded in a rigid ba ffl e, the acoustic pressure radiated from the plate in fluid half-space z can be evaluated by the well-known Rayleigh integration with time dependence e i ω t as follows (e.g. see [16]):

S v ( x ′ , y ′ ) e ikR

2 π R dx ′ dy ′ (5)

p ( x , y , z ; ω ) = i ωρ 0

where p ( x , y , z ; ω ) is the pressure field at the location ( x , y , z ) ∈ R 3 , ω = 2 π f is the angular frequency, ρ 0 is the air density, v is the vibration velocity at the point ( x ′ , y ′ ) ∈ S , S = [0 , a ] × [0 , b ] ⊂

c 0 is the wavenumber, c 0 is the sound speed in air, and R = q

( x − x ′ ) 2 + ( y − y ′ ) 2 + z 2 . Hence, according to Eq.5 and the relation between the velocity and displacement ( v = i ω w ), we have

R 2 , k = ω

S w ( x ′ , y ′ ) e ikR

2 π R dx ′ dy ′ (6)

p ( x , y , z ; ω ) = Ray ( w ) = − ω 2 ρ 0

where Ray is the Rayleigh integral operator, which converts the flexural dispacement w to the pressure field p . Accordingly, the acoustic pressure at point ( x , y , z ), caused by the response H to the harmonic point force such e i ω t at point ( x 0 , y 0 ), can be obtained as follows:

p ( x , y , z ; ω ) = X

n ≥ 0 H n ( ω ) ψ n ( x 0 , y 0 ) Ray ( ψ n ) (7)

In Section 3, we have also discussed an issue for an optimally and accurately approximation of the Rayleigh integration.

2.2. 6 th ORDER PROBLEM In practice since the shear e ff ects are not usually negligible for composite panels, their flexural vibrating motions are usually expected not to satisfy the 4 th order problem, and instead, their governing equations of motion may fulfill higher order problems like the 6 th order problem. In fact, the 6 th order problem is a partial di ff erential equation (PDE) of sixth-order that can demonstrate flexural motion of a vibrating composite plate in a spatial-time domain, and it is introduced as follows: [17–19]:

D t ∇ 6 w − D t ´ g (1 + Y ) ∇ 4 w + m ∂ 2

∂ t 2 ( ∇ 2 w − ´ gw ) = ∇ 2 p e − ´ gp e (8)

where ´ g = 2 G (1 − v 2 )(1 + i β ) / ( E 1 h 1 h 2 ) is the shear parameter of the core, G is the shear modulus of the viscoelastic core, and β is the core loss factor, Y = 3(1 + h 2 / h 1 ) 2 is the geometric parameter, D t = E 1 h 3 1 / (6(1 − v 2 )) is the total flexural rigidity of both face plates, E 1 is the Young’s modulus of the face plate, ν is the Poisson’s ratio of the plate material, m is the mass per unit area of the entire sandwich, h 1 and h 2 , respectively, are the thicknesses of the face plate and the constrained damping layer, and p e is the external pressure loading on the panel [19]. Herein, for extracting the natural frequencies and consequently for estimating the frequency response of the vibrating composite plate and then predicting the pressure field radiated by the composite plate, as discussed by Subsection 2.1, we needed to use the above 6 th order problem equation instead of Eq.1. So, according to Eq.8, the natural frequencies corresponding to the flexural vibration of the composite plate could be obtained by the following sixth-order dispersion relation:

− D t k 6 − D t ´ g (1 + Y ) k 4 + m ω 2 ( k 2 + ´ g ) = 0 (9)

and so, we have

s

1 + ´ gY k 2 n + ´ g

! (10)

D t k 4 n m

ω n =

wh ere k n is the wave number corresponding to th e mode index n , and considering the mode shapes { (2 / √

( n 1 π/ a ) 2 + ( n 2 π/ b ) 2 . By comparison of the above equations of the 6 th order problem with the equation of the 4 th order one, we can clearly see that the 6 th order problem is the generalization of the 4 th order problem, and the solutions of these problems become equal when the shear e ff ects of the composite plate are neglected (compare the solution of Eq.10 with Eq.4 when ´ g = 0). So, based on Eq.10,the frequency response associated with the 6 th order problem can similarly be achieved by Eqs. 2 and 3.

ab ) sin( n 1 π x / a ) sin( n 2 π y / b ) } , we have k n = p

2.3. DIFFUSE EXCITATION Now, consider a di ff use acoustic field (DAF) p e as a random pressure field excitation on the plate (see Fig.2). Ideally, a di ff use field is expected to be homogeneous and isotropic (e.g. see [20]). So, if we assume that the excitation is stationary with respect to time, the power spectral density of the DAF as the external excitation can be expressed as follows [18]:

S p e ( x 0 , y 0 ; ´ x 0 , ´ y 0 ; ω ) = S p e ( ω ) C ( x 0 , y 0 ; ´ x 0 , ´ y 0 ; ω ) (11)

where C ( x 0 , y 0 ; ´ x 0 , ´ y 0 ; ω ) is in the form of a correlation function and, it can be regarded as the spectrum of the correlation coe ffi cient. If the external excitation is assumed to be an ideal white noise, the function S p e will be constant. In this research the quantity S p e has been observed by the Beta cabin, which was employed to produce an ideal di ff use acoustic field (DAF) (see Section 4). In addition, herein, the theoretical formulation of the spectrum of the correlation function C ( x 0 , y 0 ; ´ x 0 , ´ y 0 ; ω ) has been considered as follows [20,21]:

C ( x 0 , y 0 ; ´ x 0 , ´ y 0 ; ω ) = sin ( k 0 r )

k 0 r (12)

( x 0 − ´ x 0 ) 2 + ( y 0 − ´ y 0 ) 2 , k 0 = ω/ c 0 is the acoustic wave number, and c 0 is the sound speed in air. So, considering Equations 7, 11, and 12, the spectrum (i.e. auto-spectral density (ASD)) of the sound pressure p at the location ( x , y , z ) will be obtained as follows (c.f. [18]):

where r = p

S p ( x , y , z ; ω ) = X

X

´ n ≥ 0 Ray ( ψ n ) Ray ( ψ ´ n ) H n ( ω ) H ´ n ( ω ) S p e ( ω ) I n ´ n ( ω ) (13)

n ≥ 0

where I n ´ n ( ω ) = X

X

( ´ x 0 , ´ y 0 ) ∈ S ψ n ( x 0 , y 0 ) ψ n ( ´ x 0 , ´ y 0 ) C ( x 0 , y 0 ; ´ x 0 , ´ y 0 ; ω ) (14)

( x 0 , y 0 ) ∈ S

and S = [0 , a ] × [0 , b ] ⊂ R 2 is the plate surface area, P ( x 0 , y 0 ) ∈ S is the discrete form of the 2D integration over the plate, and ◦ is the complex conjugate of ◦ .

Figure 2: a di ff use acoustic field (DAF) as a random pressure field excitation on the plate. The objective is to model the spectrum of the sound pressure level at an arbitrary receiver point, caused by the vibrating plate excited by the DAF

Accordingly, based on the methodology described in this Section, we could perform certain numerical and laboratory experiments for examining the ideas of the research. In the next Section, some of numerical experiments executed before laboratory experiments have been explained in details.

une 8 Sp(%6,25 0) ( aa ‘piffuse acoustic field « — (DAF) excitation xi< Sp (@) - 2 C ~ er Cf

3. NUMERICAL EXPERIMENTS

Before carrying out any laboratory experiments, it was necessary to perform certain numerical experiments in order to numerically assure ourselves of the analytical modeling and simulations, and also in order to fulfill some numerical prerequisite of the problem. Herein, a finite elements method (FEM) was employed as the numerical tool for re-validating the procedures of the proposed strategy and the numerical prerequisite were attained via certain numerical comparisons. The finite element method was performed by means of ANSYS / APDEL and applied to a 40 cm × 60 cm aluminium plate with a thickness of 4 mm . It was supposed that the mass density, young modulus, Poisson ratio, and loss damping factor of the plate are equal to 2500 kg / m 3 , 70 GPa , 0 . 22, and 0 . 001, respectively.

Additionally, the finite element of FEM was defined such that it has four nodes, is suitable for analyzing thin to moderately-thick shell structures, is well-suited for linear and large rotation, and has six degrees of freedom (three translations and three rotations) at each node. In this numerical experiment, the boundary condition was considered to be simply supported at all the edges of the plate and a mechanical nodal excitation with a white noise spectrum was assumed for the modeling (the nodal excitation was exert on a node with the location x = 0 . 12 m , y = 0 . 40 m where x ∈ [0 , a ], y ∈ [0 , b ], and a = 40 m , and b = 0 . 60 m ).

Moreover, the element mesh size was considered 8 . 6 mm that is half of the expected value of the minimum wave number. Accordingly, the numerical experiment could be performed under the aforementioned condition, and as an example Fig.3 illustrates the vibration field at the middle point of the plate, approximated via the FEM, as well as its comparison with the analytical modeling achieved via solving the 4 th order problem (i.e via Equations 1 to 4). In this figure, we can see that the numerical result coincides with the analytical prediction of the problem (see Fig.3). This step of the numerical experiments as well as its corresponding comparisons with the analytical models is essential in the strategy in order to double-check the procedures and to avoid any mistakes in the implementation of the proposed approach.

Figure 3: a comparison between the FEM and analytical approach for f ∈ [0,10kHZ]

Furthermore, in order to have accurate approximations of the Rayleigh integration and pressure

Displacement(z){m] Harmonic Response (Displacement) 109 105 10% 10% 10° —— Analytical —— ANSYS (APDL) 10! 107 eee 108

field at an arbitrary receiver point, we needed to do other numerical analyses. After modeling the vibration field over the plate, the discretization and computation of the Rayleigh integral is a tricky part and regarding the frequency range and the desired accuracy, at first we needs to have a proper understanding of the optimum value of the mesh size and the appropriate method for the discrete 2D integration. Although it is usually recommended to consider six to ten nodes per the minimum wavelength to have an accurate estimation of mode shapes (e.g. see [22]), high numbers of nodes and consequently small mesh sizes can lead to increasing the amount of computation and its processing time. In order to avoid it and find an optimum value of the mesh size, in this step, we assumed the solution obtained by the mesh size equal to λ Min / 6 as the most accurate value of the solution and then we compared other solutions with this solution as a reference ( p Re f ).

On the other hand, as mentioned above, it was required to adopt a proper approach for discretizing and approximating the Rayleigh integration. For this purpose, at the same time, we examined three di ff erent numerical integration approaches: 1) Discretization & simple summation, 2) Trapezoidal approach, and 3) Quadrature Method [23]. Fig.4 sums up these results, which illustrates how both the trapezoidal and Quadrature methods can start to have an acceptable accuracy from mesh sizes less than λ Min / 2. For this purpose, in this research for continuing other numerical computations the mesh size was assumed equal to λ Min / 2 = 8 . 6 mm and the trapezoidal approach has been used for approximating the Rayleigh integration.

Figure 4: the relative error associated with the integration approaches and di ff erent values of the mesh size

4. LABORATORY EXPERIMENTS

Herein, we had certain objectives for performing the laboratory experiments. Our objectives were 1) to experimentally check the mathematical and numerical modeling and simulations, 2) to experimentally investigate the possibility of considering di ff use acoustic field (DAF) excitation in the mathematical modeling of the vibration and pressure field, required for the proposed vibro / psycho- acoustical approach, 3) to record the sound radiated by the plate with the purpose of a perceptual validation of the obtained synthesized sounds for such an excitation, and 4) to investigate various configurations of the problems and their limitations. For this purpose, we used an acoustic test cabin, (a Mecanum Inc. product, called Beta cabin, see [24]). This Beta cabin is usually used for a quick and accurate estimation of sound absorption and transmission loss on medium-sized samples, and it is based on the concept of di ff use acoustic field (DAF). So, by means of this instrument, herein

36 Relative Error 30 Relative Error = 2—Prerl Prep = Prey = P= asin!) Eg Sis @ 10 a=) ‘A=N2 Feary AaNe Mesh size (A) The relative error of the trapezoidal and quadrature methods for A= 2ayin/2 are less than 1.3%

we could produce a di ff use field inside of the cabin, and utilize it as a di ff use acoustic excitation for testing some sample plates (see Fig 5), and then we could measure the pressure field at di ff erent distances from the vibrating plates by means of a microphone system. The geometric and mechanical parameters of the tested plates, used by these laboratory experiments, have been considered as Table1.

Figure 5: a) Beta Cabin with the plate under study, b) inside Beta Cabin with microphones, c) four uncorrelated speakers required for generating an ideal DAF, d) a 4-microphones antenna for acquiring the DAF spectrum, e) the microphone system to record the sound radiated by the plate

Table 1: the geometric and mechanical characteristics of the tested plates

No Material Type a × b [cm] h [mm] Mechanical properties

1 Steel Isotropic 61 × 61 2 E = 210 GPa , ν = 0 . 29, η = 0 . 005, ρ = 7 . 8 g / cm 3

2 Aluminium Isotropic 61 × 61 5 E = 70 GPa , ν = 0 . 33, η = 0 . 02, ρ = 2 . 5 g / cm 3

3 Steel Isotropic 33 . 9 × 29 . 2 2 E = 210 GPa , ν = 0 . 29, η = 0 . 005, ρ = 7 . 8 g / cm 3

4 Aluminium Isotropic 33 . 9 × 29 . 2 5 E = 70 GPa , ν = 0 . 33, η = 0 . 02, ρ = 2 . 5 g / cm 3

5 Aluminium composite Composite 33 . 9 × 29 . 2 2 D t = 40 GPam 3 , ν = 0 . 3, ρ = 2 . 2 g / cm 3 , η = 0 . 01, β = 0,

G = 1 MPa , h 1 , h 2 = 0 . 9 mm , 0 . 2 mm

After acquiring the sound pressure level (SPL) at di ff erent distances from the vibrating plates, the precise observations of the DAF spectrum S p e ( ω ) inside the Beta cabin, which were measured during each experiment, enabled us to straightforwardly perform the proposed method described in Section 2. Fig.6 illustrates the main shape of the measured DAF spectrum S p e ( ω ). Accordingly, Eq.14 and the measured DAF spectrum S p e ( ω ) could provide us with the modeled spectrum of pressure field, and consequently we could carry out comparisons between the modeled spectrum of pressure field and the SPL measured by the microphone system. Fig.7 presents the corresponding comparisons for the laboratory experiments of the sample plates, performed at the distance 50cm from the plates. In this figure, the spectra of the simulated and experimental results have been illustrated. Fig.7 shows how the simulated results, obtained by the proposed approach discussed in Section2, coincide with the

experiments. In the proposed approach, the 4 th and 6 th order problems are employed for the modeling and simulations associated with the isotropic and composite plates, respectively.

Figure 6: the spectrum S p e ( ω ) of the DAF excitation, measured during the experiments

SPL [dB] 110 100 90 80 70 60 50 40 30 0 1000 2000 Spectrum of DAF Excitation: S pe (w) 3000 4000 5000 6000 7000 Frequency [HZ] 8000 —S,.() 9000 10000

Figure 7: the spectra of the sound pressure level (SPL) radiated by the 5 sample plates (see Table1)

(1) Steel plate (61cm x 61cm) s |——Experiment — Simulation SPL[d BYP Res=20 HPascal] "Frequency(HZ] _(3) Steel plate (33.9cm x 29.2cm) — Experiment “ (Simulation SPL[AB][P,.=20 Pascal] 10° Frequency[HZ] (2) Aluminium plate (61cm x 61cm) — Experiment — Simulation SPL[dBI[P,.,=20 Pascal] 10 10 Frequency[HZ] (4) Aluminium plate (33.9cm x 29.2cm) 80 — Experiment — Simulation SPL[AB][P,...=20 Pascal] 10° 0 Frequency[HZ] (5) Composite plate (33.9cm x 29.2cm) SPL[dB][P,...=20 pPascall] 10 Frequency[HZ] Experiment — Simulation

5. DISCUSSION AND CONCLUSION

The main target of this research was to provide theoretical and practical steps for vibroacoustic modeling of panels excited by a Di ff use Acoustic Field (DAF), which we ultimately needed for vibro / psycho-acoustically designing mechanical structures. Based on such a target, first we presented a theoretical strategy for analytical modelings of the problems, where the 4 th and 6 th order problems were the bases of vibroacoustic modelings of isotropic and composite plates, respectively. Then we could complete the theoretical strategy by means of certain numerical and experimental experiments to practically examined our main ideas. First, certain numerical validations and prerequisites were fulfilled and then the laboratory experiments were performed to examine the proposed analytical procedures. Meanwhile, a clear understating of DAF, which herein plays a key role of the random pressure field excitation, was another important issue. We required to theoretically investigate the DAF excitation in details and precisely model it via stochastic analyses. On the other hand, it was also essential to ideally generate such DAF excitation field in practice. Hence, in the laboratory experiments, we utilized an acoustic test cabin for ideally generating the DAF excitation and exerting it to di ff erent sample plates. The experiments were performed for 4 isotropic sample plates with di ff erent sizes as well as a composite one (see Table1), and finally, the acquired experimental results were compared with the mathematical modeling and simulations. The comparisons were able to experimentally approve the main ideas of the research, and the experimental results associated with the isotropic plates and the composite one could approve the abilities of the 4 th and 6 th order problems in the proposed approach (see Fig.7). Accordingly, based on the agreements accomplished by by the various numerical and laboratory experiments with the mathematical modelings, the proposed strategy (described in Section 2) was made ready to be used for precisely synthesizing sounds required by corresponding psycho-acoustic tests and analyses, and consequently, for completing the proposed auditory evaluation step required for vibro / psycho-acoustically designing mechanical structures (see Fig.1).

ACKNOWLEDGEMENTS

This research was funded by CeLyA (Lyon Acoustics Center, ANR-10-LABX-0060).

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