S Besset

L J Éz

LTDS Équel D2s

A MODAL ANALYSIS METHOD TO STUDY FLUID-STRUCTURE COUPLING IN HOLLOW PARTS OF A STRUCTURE

Keywords: Modal analysis, Substructuring, Fluid-structure coupling

Complex structures often include hollow parts. These hollow parts constitute the skeleton of the structure and are largely responsible for its global behavior, hence the importance of analyzing them precisely. The method we propose in this paper is a modal method based on the modal analysis of elements of hollow parts. This method does not require nodal degrees of freedom on the boundaries between the elements: "modal" elements are created, and these elements can be assembled through modal mass and stiffness matrices in the same way as the finite element method. Thus, it is possible to choose the precision of the analysis by choosing the quantity of modes used in the modal analysis of the elements. We will study not only structural systems but also coupled fluid-structure systems, and our results will be compared with experimental tests.

Introduction

Complex structures often include hollow parts. The behavior of these hollow parts is quite important for it is greatly responsible for the behavior of the whole structure. Indeed, hollow parts constitute the skeleton of many complex structures, such as cars. Moreover, they are also usually associated with plates. In this case, they are considered as stiffeners. These stiffeners can be studied using a finite element model based on the real geometry of the structure. This is a quite expensive method because it requires many degrees of freedom to be sufficiently precise. Hollow parts can also be considered as beams. Several beam models do exist, but they are not always capable of representing the real characteristics of the geometry.

A modal substructuring method has already been proposed to describe hollow parts [START_REF] Besset | Modal analysis of hollow parts of a structure[END_REF]. This method uses modes corresponding to each side of a stiffener element of length δz. The hollow parts of the structure can be described by these modes. Assembly of the stiffener elements is done through the generalized degrees of freedom. Thus, no nodal degrees of freedom are required on the boundaries. However, nodal degrees of freedom have to be kept in order to assemble the hollow part to the rest of the structure.

Acoustical coupling inside hollow parts is also very important. Hollow parts can be considered as wave guides and propagate noise. In order to predict the acoustical behavior of a structure, it is necessary to take coupling into account. Modal analysis of coupled fluid-structure systems has already been studied [START_REF] Sandberg | Domain decomposition in acoustic and structure-acoustic analysis[END_REF][START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF][START_REF] Ohayon | Reduced symmetric models for modal analysis of internal structuralacoustic and hydroelastic-sloshing systems[END_REF]. The method we propose here will use acoustical and structural modes to describe the hollow parts.

The hollow part will be divided into several elements. Each element will be analyzed as a coupled fluid-structure system. Particular modes will be chosen in order to allow assembly of these elements without the use of nodal degrees of freedom.

We will first explain the method examining a hollow part in vacuo. Through several examples, we will compare our results with finite element methods. The case of an infinite waveguide will be reviewed. A hollow part will then be coupled to a plate, in order to show how different structures can be assembled. In this case, the hollow part is considered as a stiffener. To assemble a hollow part with another structure, we will use the "Double Modal Synthesis" method proposed by Jezequel [START_REF] Jezequel | Procedure to reduce the effects of modal truncation in eigensolution reanalysis[END_REF][START_REF] Jezequel | Hybrid method of modal synthesis using vibrations tests[END_REF][START_REF] Jezequel | A method of damping synthesis from substructures tests[END_REF]. Thus, the stiffeners coupled to the plate will be represented by "branch modes", which are generalized degrees of freedom corresponding to the boundaries between the structures.

We will then study the case of an acoustical coupling inside the hollow part. Results will be compared with a finite element model and a test. This last example will show that the method proposed in this paper is able to take into account acoustics problems. Indeed, hollow parts constituting the skeleton of a complex structure can be responsible for noise propagation, because these hollow parts can be considered as wave guides. It is important to be able to predict this noise. To do this, the acoustical aspects of the hollow parts must be quite precisely represented.

Analysis of hollow parts of a structure in vacuo

The hollow part we wish to study is shown on figure 1. This structure will be divided into several elements. Each element is meshed with boundary degrees of freedom only, as shown in figure 2. It is not actually necessary to remove the internal degrees of freedom, but it will facilitate the explanation of the method. Note that internal degrees of freedom should be expressed as a function of internal modes, using the Craig & Bampton method for example [START_REF] Craig | Coupling of substructures for dynamic analysis[END_REF].

Modal analysis of an element

Figure 2 shows an element used to analyze the hollow part. When the modal analysis of this element is complete, two nodal degrees of freedom will remain, in order to couple the hollow part with another structure. The behavior of an element can be described with the equation of motion: u L and u R are then split into degrees of freedom that will be expressed as a function of generalized degrees of freedom u g L and u g R , and the others u k L and u k R -marked "retained" on figure 2:

-ω 2 M LL M LR M RL M RR + K LL K LR K RL K RR u L u R = f L f R (1)
u L = u k L u g L , u R = u k R u g R (2)
Let φ R be the matrix of the modes of the element when one node of the right side is fixed, as shown in figure 3.

Φ R is the modal matrix corresponding to right nodes of the element -it is a part of matrix φ R . According to the Craig & Bampton theory [START_REF] Craig | Coupling of substructures for dynamic analysis[END_REF], displacements of these right nodes can be expressed as follows:

u g R = Φ R q R + Ψ R u k R (3)
where Ψ R is the matrix of the static modes, corresponding to the rigid body modes of the right side of the element.

Analogous matrices are defined for the left side of the element. Let φ L be the matrix of the modes of the element when one node of the left side is fixed. Φ L is the modal matrix corresponding to left nodes of the element. Displacements of these left nodes can be expressed as follows:

u g L = Φ L q L + Ψ L u k L (4)
Thus, displacements u can be expressed as a function of generalized degrees of freedom q L and q R :

       u k L u g L u k R u g R        =     I 0 0 0 Ψ L Φ L 0 0 0 0 I 0 0 0 Ψ R Φ R            u k L q L u k R q R        (5)
Notice that as left degrees of freedom do not depend on right modes, and right degrees of freedom do not depend on left modes, assembling modal elements will be possible.

Assembling elements

In order to assemble two elements, let write the relations between the nodal degrees of freedom of the left and right side of two elements. Degrees of freedom of the right side of the first element, called u k R1 and u g R1 , must match with degrees of freedom of the left side of the second element, called u k L2 and u g L2 :

u k R1 = u k L2 u g R1 = u g L2 (6)
Equation 6 can be written for elements n and n + 1:

u k Rn = u k Ln+1 u g Rn = u g Ln+1 (7)
Equations 6 leads to the expression of degrees of freedom of the left part of the second element as a function of degrees of freedom of the first element:

u k L2 = u k R1 q L2 = ΦL Φ R q R1 + (Ψ R -Ψ L ) u k R1 (8)
which can be written for elements n and n + 1:

u k Ln+1 = u k Rn q Ln+1 = ΦL Φ R q Rn + (Ψ R -Ψ L ) u k Rn ( 9
)
where ΦL is a pseudo-inverse matrix of Φ L :

ΦL = Φ T L Φ L -1 Φ T L (10)
Equations 8 allow the assembling of N elements, using transfer matrices T n :

K tot = N n=1 T T n KT n (11)
M tot = N n=1 T T n MT n (12)
Notice that assembling matrices is as simple as in the finite element method. For two elements, matrices T n are built as follows:

T 1 =     I 0 0 0 0 0 0 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I 0 0     (13)
T 2 =     0 0 I 0 0 0 0 0 ΦL (Ψ R -Ψ L ) ΦL Φ R 0 0 0 0 0 0 I 0 0 0 0 0 0 I     (14)
Notice that it is possible to use equation 14 for matrix T 1 . Matrix given in equation 11 is more simple, but it may be easier to use the same kind of matrix for each element. Thus, for N elements, matrices T n can be written:

T n =     0 • • • 0 I 0 0 0 0 • • • 0 0 • • • 0 ΦL (Ψ R -Ψ L ) ΦL Φ R 0 0 0 • • • 0 0 • • • 0 0 0 I 0 0 • • • 0 0 • • • 0 0 0 0 I 0 • • • 0     (15) 2(n -1) columns 2(N -n) columns ( 16
)
Equation 15 is valid for n = 1 . . . N . For n = 1, it is also possible to use matrix T 1 defined as follows:

T 1 =     I 0 0 0 0 • • • 0 0 I 0 0 0 • • • 0 0 0 I 0 0 • • • 0 0 0 0 I 0 • • • 0     (17) 2(N -1) columns (18)
Notice that if the hollow part is a periodic structure, there is a relation between matrices Φ L and Φ R :

Φ R = AΦ L ( 19
)
where A is a diagonal matrix defined as follows:

A(i, i) = -1 if i is associated with : -the z translation -the x rotation -the y rotation A(i, i) = 1 else (20)
Matrices Ψ L and Ψ R correspond to rigid body modes of the element, since only one node is fixed. Hence the relation:

Ψ L = Ψ R (21)
Equations 19 and 21 allow the transfer matrices T n to be simplified.

Coupling structures using "double modal synthesis"

The "hollow parts" that have been studied in the previous sections are parts of a complex structure including plates and stiffeners. During the modal analysis of the elements constituting these hollow parts, nodal degrees of freedom have been retained. The nodes kept fixed in the modal analysis of the elements allow us to couple the hollow part with another structure -a plate, for example. Thus, the second structure is studied using a classic modal analysis method -the Craig & Bampton method for example. Coupling is made through the nodal degrees of freedom remaining from the modal analysis.

In order to use only generalized degrees of freedom, it is possible to express degrees of freedom of the boundary between the two structures as a function of "branch modes", using the "double modal synthesis" method [START_REF] Jezequel | Procedure to reduce the effects of modal truncation in eigensolution reanalysis[END_REF][START_REF] Jezequel | Hybrid method of modal synthesis using vibrations tests[END_REF][START_REF] Jezequel | A method of damping synthesis from substructures tests[END_REF]. Looking at the coupled problem, the motion equation can be written as follows:

  -ω 2   M HH M HB M HS M BH M BB M BS M SH M SB M SS   +   K HH K HB K HS K BH K BB K BS K SH K SB K SS        q H u B q S    = f (22)
We denote H the generalized degrees of freedom concerning the hollow part, B the nodal degrees of the boundary, and S the generalized degrees of freedom of the second structure. f is the vector of the generalized forces applied on the system.

Φ B is the matrix of the branch modes, which satisfy the following equation:

-

ω 2 i M BB + K BB Φ B = 0 ( 23
)
where

ω 2 i (1 < i < N
) is a diagonal matrix of N eigenfrequencies. Boundary degrees of freedom u B can be expressed as follows:

u B = Φ B q B ( 24
)
which allow the whole system to be fully described with generalized degrees of freedom: "Section modes" for the hollow parts, "surface modes for the plates", and "branch modes" for the boundaries between the hollow parts and the plates.

Results in vacuo

Analysis of an infinite waveguide

The first comparison was made for an infinite waveguide. We consider the hollow part as an infinite and periodic structure. Mass and stiffness matrices are divided into left and right sides. We compute the dispersion curves for this infinite waveguide using finite element matrices (denoted M and K) and modal matrices (computed using our proposed method and denoted M and K).

Considering equation 27, which is the motion equation of an element (properties in table 1 and 2), we use the following notation:

Z(ω) = -ω 2 M + K (25) Z(ω) = -ω 2 M + K (26) Z LL (ω) Z LR (ω) Z RL (ω) Z RR (ω) u L u R = f L f R (27)
The modal equation corresponding to equation 27 can be written: The structure is periodic, hence the relation:

Z LL (ω) Z LR (ω) Z RL (ω) Z RR (ω) q L q R = f L f R (
u R = αu L (29) f R = -αf L ( 30
)
where α = e -ik∆l . Real and imaginary parts of k = k R + ik I can be written as a function of α as follows:

     k R = ℜ(k) = - arg(α) ∆l k I = ℑ(k) = ln (|α|) ∆l (31)
Equation 27 becomes:

Z LL (ω) -I Z RL (ω) 0 u L f L = α -Z LR (ω) 0 -Z RR (ω) -I u L f L (32)
Considering modal matrices, the equation can be written:

Z LL (ω) -I Z RL (ω) 0 q L f L = α -Z LR (ω)B 0 -Z RR (ω)B -C q L f L (33)
where B and C are matrices coming from the modal synthesis of an element. These matrices allow us to write the equations between q R and q L , and f R and f L :

q R = αBq L (34) f R = -αCf L (35)
f R and f L are the modal forces associated with the modal matrices M and K. Propagation curves give k as a function of f = ω 2π (k must be real). The curves obtained from the finite element matrices are given in figure 4. The curves obtained from the modal matrices are given in figures 5, 6 and 7.

Figure 5 is a plot using a modal element of 10 modes. The results are quite poor, which is why we plot the same curves using 30 and 60 modes. Figure 6 shows the

f (Hz) k (m -1 )
Figure 4: Propagation curves -FEM model results obtained using 30 modes for an element. Dispersion curves are somewhat better. Figure 7 gives the results obtained using 60 modes for an element. The results are good -at low frequencies, curves are even exactly the same -, but many modes were required.

The need for many generalized degrees of freedom to represent an infinite wave guide can be explained by the fact that the boundaries between elements is the only parameter used in the calculations. In the case of a finite hollow part, stationary waves are generated thanks to the boundary conditions at the beginning and the end of the hollow part. Results should then be better for such cases.

Modal and finite element analysis of a tube

We consider the tube shown in figure 1 split into elements shown in figure 2. This tube may be a hollow part of a structure.

The material used for this comparison is structural steel, whose properties are given in table 3. Properties of the studied structure are given in table 4. The structure is represented in figure 8.

Table 5 shows the eigenfrequencies found by the modal analysis using the method we propose and a finite element method. The finite element model uses 144 degrees of freedom for each section, whereas the modal model uses only 41 degrees of freedom (35 section modes and 6 nodal d.o.f. for each section). The results are quite good.

Figure 9 shows the evolution of the accuracy of the method as a function of the 4: Geometry of the structure number of section modes retained. In this figure, one can see that the results are much more accurate when the number of section modes retained is high. However, the number of section modes need not be as high as in the example given in section 2.4.1. Indeed, the stationary waves make the results more accurate thanks to the boundary conditions.

f (Hz) k (m -1 )

Stiffener coupled with a plate

We also made calculations using a stiffener coupled with a plate. The whole structure is given in figure 10. The properties of the material and the geometry of the structure are given in tables 6 and 7. This case is very interesting to study because hollow parts are often used as stiffeners. Coupled to a plate, such stiffeners are useful to modify the characteristics of the structure and the values of its eigenfrequencies. Table 7: Geometry of the structure reference), the modal method we propose, and a method using beam modeling. The results show that the method we propose is quite close to the finite element method. Figures 11 and 12 shows the error between our method and the finite element method, and between the method using Timoshenko beam and the finite element method.

f (Hz) k (m -1 )
f (Hz) k (m -1 )
In this example, we first used 10 generalized degrees of freedom (figure 11), and then 50 generalized degrees of freedom (figure 12) for each element of the stiffener. Using 50 modes provides better results, but the difference between 10 and 50 section modes is quite small. In both cases, the modal method we propose gives better results than a method using Timoshenko beam modeling. Notice that for the case of coupled structures, the number of section modes retained is not as important as for the single beam studied in section 2.4.2. Indeed, the behavior of the structure does not depend only on the stiffener, but also on the plate, which is why the accuracy of the analysis of the stiffener is not as important as for the first example (given in section 2.4.2). Figure 11: Error compared to finite element method (black: using 10 section modes for each element of the stiffener, white: using beam modeling) Figure 12: Error compared to finite element method (black: using 50 section modes for each element of the stiffener, white: using a beam modeling)

Analysis of a hollow part including fluid-structure coupling

In this section, the hollow part considered in section 2 will be studied considering the acoustical aspects.

Modal analysis of an element

The hollow parts of the structure are split into several elements, as for the in vacuo case.

Fluid-structure problems have often been studied and equations concerning acoustical coupling are well known [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF][START_REF] Ohayon | Reduced symmetric models for modal analysis of internal structuralacoustic and hydroelastic-sloshing systems[END_REF][START_REF] Junger | Acoustic fluid-elastic structure interactions: Basic concepts[END_REF]. Using a finite element formulation, the motion equation of an element can be written as follows:

-

ω 2 M S 0 C M A + K S -C T 0 K A u p = f 0 (36)
The method we use to study the coupled system is the same as for the in vacuo problem. Modes of the fluid part will first be calculated and then an analysis of the structural part will be performed. Figure 19 shows the results obtained in this section. The behavior of the hollow part has been predicted quite well. The measurement curve breaks down at 350 Hz, because of the cut-off frequency of the shock hammer. Nevertheless, the two curves under 350 Hz are not so different.

Under 350 Hz, differences between the two curves may have various reasons. First, the hollow part has been suspended through a rigid bar -in order to get a punctual contact -and a sandow. This sandow induces a very low stiffness that could influence the results. Secondly, though the extremities of the hollow part have been blocked, there may be leakage because of the wire of the microphone. There may also have

Conclusion

The method we proposed in this paper allows us to study hollow parts of a structure with good precision, using generalized degrees of freedom. This modal method does not use nodal degrees of freedom on the boundaries between elements, which is quite interesting. Modal matrices created by the modal analysis of an element can be assembled the same way as finite element matrices.

Numerical testing on a waveguide and a finite element model have shown that the method can produce quite good results. Of course, results obtained for the infinite waveguide are not as good as the results obtained with finite hollow parts. The case of a stiffener coupled to a plate has been processed. It has been shown that the method proposed in the paper was able to produce good results in this case. Experimental testing has shown that the fluid-structure coupling was correctly modelized, which will allow 20

Figure 1 :Figure 2 :

 12 Figure 1: Hollow part to study

Figure 3 :

 3 Figure 3: Element of a hollow part

Figure 5 :

 5 Figure 5: Propagation curves -10 modes

Figure 6 :

 6 Figure 6: Propagation curves -30 modes

Figure 7 :Figure 8

 78 Figure 7: Propagation curves -60 modes

Figure 9 :Figure 10 :

 910 Figure 9: Error on the first modes as a function of the number of section modes retained (1: 10 modes, 2: 15 modes, 3: 20 modes, 4: 25 modes, 5: 30 modes, 6: 35 modes, 7: 40 modes)

Figure 13 :Table 8 :

 138 Figure 13: Verification of the method

Figure 17 :

 17 Figure 17: Element used for the modal analysis

Figure 19 :

 19 Figure 19: Measurements and modal analysis

Table 1 :

 1 Properties of the material

	28)

Table 2 :

 2 Geometry of the structure

Table 3 :

 3 Properties of the material

	Thickness	Section	Length
	0.001 m	0.1 × 0.1 m 2	0.2 m
	Table

Table 5 :

 5 Eigenfrequencies of the hollow part

	(Hz)	(%)

Table 6 :

 6 Properties of the material

Acknowledgement

The authors thank Mr. Stéphane Lemahieu for his help during the experimental works.

Analysis of the fluid part

In order to find modes allowing us to easily assemble elements, let us split the acoustical mass and stiffness matrices into left and right sides of an element:

The pressure vector p is split the same way:

First, the modal matrix φ A R is introduced. It is the matrix of the acoustical modes of an element when one node of the right side is fixed (p k R = 0) (see figure 3 for notations). Φ A R is the part of matrix φ A R concerning the nodes of the right side of the element.

Secondly, matrix Φ A L is the matrix of the acoustical modes of an element when one node of the left side is fixed (p k L = 0). Φ A L is the part of matrix Φ A L concerning the nodes of the left side of the element.

Pressure vector p can be written as a function of generalized degrees of freedom q A L and q A R and pressure on fixed nodes p k R and p k L :

Notice that generalized degrees of freedom of each side of the element only depend on nodal degrees of freedom of the same side, which will allow us to easily assemble elements.

Analysis of the structural part

The modal analysis of the structural part of a element has already been given in section 2.1. Using similar notations, displacements of the two sides of an element can be expressed as follow:

As for the acoustical analysis, generalized degrees of freedom of each side of the element depend only on nodal degrees of freedom of the same side.

Assembling elements

In order to assemble two elements, let us write the relations between the right side of the n th element and the left side of the (n + 1) th element:

Using equations 40, 41, 42 and 43, equation 44 leads to:

These relations lead to transfer matrices that allow us to assemble several elements, as explain in section 2.2 (equations 11 and 12).

Results

In this section, we compare the results obtained with our proposed method and experimental tests conducted on a hollow part.

Verification of proposed method

The verification of the modal synthesis method we propose is made using elements shown on figure 17. Our modal synthesis method is now compared with a finite element model. Figure 13 shows the error between these two methods for one elements. This error is quite small and shows the efficiency of the model we propose.

Validation by experimental testing

The hollow part used for the tests is shown in picture 14. We used a shock hammer for the excitation and a microphone to get the pressure inside the hollow part, as shown in picture 15. Experimental material is shown in picture 16. Of course, the acoustical cavity has been blocked up. It is open in the picture in order to show the microphone.

Modal analysis and comparison

The structure has been studied using the method proposed in the paper. A different element has been used for the curved part. The two kind of elements are shown on figure 17 and18.

The properties of the material used for this example are given in table 8. The Young's modulus has been matched in order to get the best results possible. The


S Besset

L Jézéquel Ltds

Équipe D2s

Modal Criteria for Optimizing the Acoustic Behaviour of a Coupled Fluid-Structure System Based on a Systemic Approach

Keywords: Modal analysis, Optimization, Vibroacoustic coupling

This paper proposes modal criteria to represent various noise sources within a complex structure, such as an automobile. By optimizing a complex system using criteria linked to modal mass and stiffness matrices, different modes of noise propagation can be investigated separately. Several criteria are thus suggested, each related to a vibrational propagation path. Since the system is studied using modal analysis, criteria can be found based on modes associated with the structure's hollow parts, plates, and cavities. These different criteria are analyzed based on the assumption of a complex vibroacoustic system. It is shown that by analyzing such criteria, one can determine which part of the structure needs to be optimized. The optimization of such a system could constitute a research topic in its own right, and is beyond the scope of the present paper.

Introduction

The use of a modal approach to describe a structure in order to optimize its dynamic behaviour has multiple advantages. Once modal matrices have been computed, optimization criteria can readily be defined. The dynamic amplfication phenomena and dynamic coupling between substructures can then be described using just a small number of degrees of freedom. Furthermore, it is possible to link the criteria to the modal parameters used in the systemic procedure. The criteria proposed in this paper are derived from those developed by P. Lemerle [START_REF] Lemerle | Optimisation des structures selon des critères imposés par la discrétion acoustique des navires[END_REF], using Craig & Bampton's method to describe the dynamic behaviour of a structure.

The modal analysis technique described here is based on the "double modal synthesis" proposed by Jézéquel [START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part i : Theory of hybrid models and modal truncation methods[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part ii : Numerical tests and experimental identification of hybrid model[END_REF]. Complex structures often include hollow parts and stiffeners, which must be analyzed very accurately to obtain satisfactory results. In this paper, the term "hollow parts" denotes the formed steel and stiffeners making up the skeleton of a structure. In complex structures such as automobiles, stiffeners and formed steel parts which make up the skeleton of the structure are those most responsible for the behaviour of the structure as a whole. To analyze these elements, a method will be used, which was proposed in [START_REF] Besset | Modal analysis of hollow parts of a structure[END_REF][START_REF] Besset | Optimization of structural dynamic behaviour based on effective modal parameters[END_REF]. We will study the acoustic parts of the coupled system using acoustic modes based on a "triple modal synthesis method". This method resorts to a coupling formulation which has been previously investigated [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF][START_REF] Ohayon | Reduced symetric models for modal analysis of internal structuralacoustic and hydroelastic-sloshing systems[END_REF]. An example of modal analysis of a coupled system can be found in [START_REF] Sandberg | Domain decomposition in acoustic and structure-acoustic analysis[END_REF].

Modal analysis of the structure will lead to modal mass and stiffness matrices, which are then used to obtain effective modal parameters, which in turn lead to criteria allowing the structure to be optimized. These criteria will depend on the pressure values at points located in the acoustic parts of the system -inside a car for example -as a function of an excitation point located on a hollow part of the structure -e.g. a spar near the car engine. The criteria proposed here allow various vibrational propagation paths to be considered. It is thus possible to investigate separately the different noise sources within the structure.

The method proposed in this paper may be used for any vibroacoustic system. In fact, the ultimate goal of this approach is to define modal criteria which allow the vibroacoustic system to be optimized. These are related to the coupling terms between the systems different substructures, and are expressed as functions of the terms contained in the modal matrices. Thus, the number of criteria depends on the number of substructures within the vibroacoustic system. Our method is described for the case of a specific system, but can readily be adapted to other vibroacoustic systems.

The structure considered in this paper is a complex one, comprised of acoustic cavities, hollow parts and plates. The geometry of this structure is similar to that of a car. Two cavities will be considered. Thus, the method proposed here can be used to study the different paths through which vibrations propagate and generate noise within the car. Paths can exist through the hollow parts of the structure, through the plates which bound the cavity, and through plates which partition the two cavities. Such a structure has already been used in vacuo in [START_REF] Besset | Optimization of structural dynamic behaviour based on effective modal parameters[END_REF].

Analysis of the vibroacoustic system

The structure considered here is a complex structure including hollow parts and plates. It is built using formed steels which make up its skeleton, as shown in figure 1. Plates are fixed to this skeleton, and form two cavities inside the structure, as shown. The structure's geometry is similar to that of a car, in order to show that the proposed methods can be applied in an industrial context.

[Figure 1 about here.]

The vibroacoustic problem

Figure 1 shows the geometry of the vibroacoustic system under consideration. The approach used in this paper to study the vibroacoustic behaviour is a (u, p) formulation:

σ ij,j (u) + ω 2 ρ s u i = 0 over Ω s (1) ∆p + ω 2 c 2 p = 0 over Ω f (2) σ ij (u)n s j = f i over ∂Ω s \Σ (3)
σ ij (u)n s j = -pn s i = pn f i over Σ (4) ∂p ∂n f = ω 2 ρ f u.n f over Σ (5)
Au = 0 over Σ l (6)
where Ω s is the structural part, Ω f is the acoustic part, and Σ is the boundary between the structural and acoustic parts. Σ l is that part of the structure which is submitted to boundary conditions. A is an operator that allows these boundary conditions to be applied. The notations used in the equations describing the system are given below:

• σ ij is the stress tensor of the structure;

• ω is the angular frequency of the system;

• ρ s is the density of the structure;

• f represents the forces applied to the structure;

• u is the displacement of the structure;

• p is the pressure field of the acoustic part.

The dissipation model (modal damping) will be introduced through the modal synthesis of the system in section 2.3. The next section addresses the discretization of the equations when several structural parts (plates and hollow parts) and two acoustic cavities are assumed. This model represents a car which is fixed to four points corresponding to the wheels; therefore, no rigid body modes are involved in the case studied here. Moreover, no internal acoustic source is assumed to be present.

Finite element analysis of the system

The behaviour of the vibroacoustic system studied in this paper can be approximated by the finite element model according to the following equilibrium motion equation:

      -ω 2       M HH M HP1 M HP2 M P1H M P1P1 M P1P2 M P2H M P2P1 M P2P2 0 C F M F1F1 M F1F2 M F2F1 M F2F2       +       K HH K HP1 K HP2 K P1H K P1P1 K P1P2 K P2H K P2P1 K P2P2 -C T F 0 K F1F1 K F1F2 K F2F1 K F2F2                        u H u P1 u P2 p F1 p F2            =            f H f P1 f P2 0 0            (7)
The behaviour of the plates, the hollow parts and the cavities is included within this equation. Degrees of freedom denoted u H correspond to the hollow parts; degrees of freedom denoted u P1 correspond to the plates -with the exclusion of plates denoted P in figure 1 -; degrees of freedom denoted u P2 correspond to the plate denoted P in figure 1. Finally, degrees of freedom denoted p F1 -respectively p F2 -correspond to the pressure in the first cavity denoted F 1 in figure 1 -respectively the second cavity denoted F 2 in figure 1.

Thus, equation 7 is a classical interior non-symetric vibroacoustic equation. The degrees of freedom are displacements for the structural part, and pressure for the acoustic part.

The vibroacoustic coupling appears in matrice C F defined as follows:

C F = M F1H M F1P1 M F1P2 M F2H M F2P1 M F2P2 = -   K P2F1 K P2F2 K F1F1 K F1F2 K F2F1 K F2F2   T (8)

Modal analysis of the system

The acoustic part of the system is analyzed with a modal synthesis method using cavity modes. Hollow parts of the structure are analyzed using a model already proposed in [START_REF] Besset | Modal analysis of hollow parts of a structure[END_REF]. These elements comprise the skeleton of the structure. This modeling leads to modal matrices which can be assembled in the same way as finite elements matrices.

The main feature of this approach is that it produces matrices which have only generalized degrees of freedom. Nodal degrees of freedom may remain, in order to assemble the hollow parts to other structures, but boundaries between the substructures of the hollow part have only generalized degrees of freedom. Assembly of these substructures is made possible by the way in which the modes, used for the modal analysis of the substructures, are chosen. Plates are assembled to the structure's skeleton according to the nodal degrees of freedom remaining from the modal analysis of the hollow parts in the structure. The complete structure is then analyzed according to the "double modal synthesis" method proposed by Jézéquel [START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part i : Theory of hybrid models and modal truncation methods[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part ii : Numerical tests and experimental identification of hybrid model[END_REF]. This method uses "branch modes" to describe the behaviour of the boundaries between substructures. In this paper, these "branch modes" will be used to describe the behaviour of the structure's skeleton. The use of generalized degrees of freedom allows the size of the problem to be reduced. The degrees of freedom of a hollow part are comprised of generalized degrees of freedom resulting from the modal analysis of the substructures which make up the skeleton, denoted q Hc , generalized degrees of freedom resulting from the double modal synthesis, which are denoted q Hb , and excitation points, which are nodal degrees of freedom, denoted u E . We denote

q H =    u E q Hc q Hb    .
Generalized degrees of freedom are linked to nodal degrees of freedom through the following equations:

               u Hc = Φ Hc q Hc + Ψ Hc u Hb + Ψ He u E u Hb = Φ Hb q Hb u P1 = Φ P 1 q P 1 + Ψ P 1 u Hb + Ψ P 1e u E u P2 = Φ P 2 q P 2 + Ψ P 2 u Hb + Ψ P 2e u E p F1 = Φ F 1 q F 1 p F2 = Φ F 2 q F 2 (9)
Matrices Φ F 1 and Φ F 2 are the modal matrices of the acoustic modes of the system. Matrices Φ P 1 and Φ P 2 are the modal matrices of the plates' fixed modes. Ψ P 1 and Ψ P 2 are the matrices of the plates' static modes, as in the case of the method used by Craig & Bampton [START_REF] Craig | Coupling of substructures for dynamic analysis[END_REF].

Matrices Φ Hc and Ψ Hc are modal matrices resulting from the analysis of the hollow part described in [START_REF] Besset | Modal analysis of hollow parts of a structure[END_REF] and [START_REF] Besset | Optimization of structural dynamic behaviour based on effective modal parameters[END_REF]. Although this approach to the study of hollow parts has already been described in former papers, the reader will find a summary in the following.

Matrix Φ Hb is the matrix of the "branch" modes of the structure [START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part i : Theory of hybrid models and modal truncation methods[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part ii : Numerical tests and experimental identification of hybrid model[END_REF]. Excitation points, denoted u E in the following, remain nodal degrees of freedom. Modal synthesis of the structure is not applied to these points.

The modal synthesis method used to study the hollow parts involves "section modes": The hollow part as a whole is split into several elements (substructures), as shown in figures 2 and 3. An element of a hollow part is studied with respect to the modes of the "left" and "right" sections. Elements are then assembled. This approach is used because it would be too costly, in terms of computing power, to study the structure's hollow parts using a conventional modal synthesis method.

[Figure 2 The equation of motion thus becomes:

          -ω 2           M EE M EHc 0 M EP 1 M EP 2 0 0 M HcE m Hc M HcHb 0 0 0 0 0 M HbHc m Hb M HbP 1 M HbP 2 0 0 M P 1E 0 M P 1Hb m P 1 0 0 0 M P 2E 0 M P 2Hb 0 m P 2 0 0 M F 1E 0 M F 1Hb M F 1P 1 M F 1P 2 m F 1 0 M F 2E 0 M F 2Hb M F 2P 1 M F 2P 2 0 m F 2           + iω           0 0 0 0 0 0 0 0 c Hc 0 0 0 0 0 0 0 c Hb 0 0 0 0 0 0 0 c P 1 0 0 0 0 0 0 0 c P 2 0 0 0 0 0 0 0 c F 1 0 0 0 0 0 0 0 c F 2           +           K EE 0 0 0 0 K EF 1 K EF 2 0 k Hc 0 0 0 K HcF 1 K HcF 2 0 0 k Hb 0 0 K HbF 1 K HbF 2 0 0 0 k P 1 0 K P 1F 1 K P 1F 2 0 0 0 0 k P 2 K P 2F 1 K P 2F 2 0 0 0 0 0 k F 1 0 0 0 0 0 0 0 k F 2                                        u E q Hc q Hb q P 1 q P 2 q F 1 q F 2                    =                    f E f Hc f Hb f P 1 f P 2 0 0                    (10)
where matrices in lowercase characters are diagonal matrices. At this step of the modal synthesis, modal damping is introduced. Although it involves a small degree of approximation, the damping matrix is assumed to be diagonal. This property is called Basile's hypothesis is the french terminology. This has no impact on the results provided dissipation does not need to be analysed.

Optimization criteria used in the paper

As explained in the introduction, although the methodology described here can be adapted to any generalised vibroacoustic system, a specific example has been chosen to illustrate our methodology for deriving the optimization criteria (see figure 1).

Analysis of the motion equation

The aim of this section is to express the pressure p F2 in the second cavity as a function of the structural displacements. This expression will allow the various vibrational paths to be discriminated from one another. The problem must be studied using a systemic approach. Certain criteria related to the different vibrational paths will thus be defined.

Using the last line of equation 10, the following expression is obtained:

-ω 2 M k F 2E u E + M k F 2Hb q Hb +M k F 2P 1 q P 1 + M k F 2P 2 q P 2 + m k F 2 q k F 2 + iωc k F 2 q k F 2 + k k F 2 q k F 2 = 0 (11)
where

M k F 2E , M k F 2Hb , M k F 2P 1 and M k F 2P 2 are the k th lines of matrices M F 2E , M F 2Hc , M F 2Hb , M F 2P 1 and M F 2P 2 . q k F 2
is the k th component of vector q F 2 . Equation 11 leads to:

q k F 2 = ω 2 M k F 2E u E + M k F 2Hb q Hb + M k F 2P 1 q P 1 + M k F 2P 2 q P 2 -ω 2 m k F 2 + iωc k F 2 + k k F 2 (12)
Let Φ k F 2 be the k th column of Φ F 2 . Equations 9 and 12 yield:

p F2 = k Φ k F 2 q k F 2 = k ω 2 Φ k F 2 M k F 2E -ω 2 m k F 2 + iωc k F 2 + k k F 2 u E + k ω 2 Φ k F 2 M k F 2Hb -ω 2 m k F 2 + iωc k F 2 + k k F 2 ΦHb u Hb + k ω 2 Φ k F 2 M k F 2P 1 -ω 2 m k F 2 + iωc k F 2 + k k F 2 ΦP 1 (u P 1 -Ψ P 1 u Hb -Ψ P 1e u E ) + k ω 2 Φ k F 2 M k F 2P 2 -ω 2 m k F 2 + iωc k F 2 + k k F 2 ΦP 2 (u P 2 -Ψ P 2 u Hb -Ψ P 2e u E ) (13)
The quantities with a "tilde" are left pseudo-inverse matrices. It is possible to define several pseudo-inverse matrices. In the case of singular systems, pseudo-inverse matrices allow solutions to be found [10]. In this paper, matrices are not square and the solution obtained is a least squares approximation. The reason for this is that modal synthesis does not involve fewer modes than the number of physical degrees of freedom of the system.

Equation 13 provides an approximation to the pressure field p F2 within the second cavity as a function of structural displacements. A superposition of the substructural modes clearly appears in the sums k ( ).

Modal parameters

Equation 13 allows modal parameters to be defined, which can be used to optimize the coupled system. These modal parameters correspond to each of the different vi-brational paths: a direct path, a path through the hollow parts of the structure, a path through the plates bounding the cavity, and a path through the plate located between the two cavities.

Direct path

The direct path is directly obtained through equation 13. It corresponds to an excitation point on a plate located next to cavity F 2 . It is recalled that the excitations considered here are displacement excitations.

The modal parameter corresponding to this direct path is denoted G E (ω):

G E (ω) = k ω ω k 2 1 -ω ω k 2 + 2iξ k ω ω k Gk E (14)
where:

Gk E = Φ k F 2 M k F 2E -M k F 2P 1 ΦP 1 Ψ P 1e -M k F 2P 2 ΦP 2 Ψ P 2e m k P 2 (15) with the notations c k F 2 = 2ξ k k k F 2 m k F 2 and ω k = k k F 2 m k F 2 .

Path through the hollow parts

The path through the hollow parts is given by component u Hb of equation 13. q k Hb is now expressed according to the third line of equation 10:

-

ω 2 M k HbHc q Hc + m k Hb q k Hb + M k HbP 1 q P 1 + M k HbP 2 q P 2 + iωc k Hb q k Hb + k k Hb q k Hb + K k HbF 1 q F 1 + K k HbF 2 q F 2 = f k Hb (16)
The modal parameter corresponding to the path through the hollow parts is denoted G H (ω). Using equations 16 and 11, it is possible to write:

G H (ω) =    k ω ω 1k 2 1 -ω ω 1k 2 + 2iξ 1k ω ω 1k Gk H1       k ω ω 2k 2 1 -ω ω 2k 2 + 2iξ 2k ω ω 2k Gk H2    (17)
where:

Gk H1 = Φ k F 2 M k F 2Hb ΦHb m k F 2 (18)
Gk H2 = Φ k Hb M k HbHc ΦHc Ψ He -M k HbP 1 ΦP 1 Ψ P 1e -M k HbP 2 ΦP 2 Ψ P 2e m k Hb (19) with the notations c k F 2 = 2ξ 1k k k F 2 m k F 2 , ω 1k = k k F 2 m k F 2

Path through the plates

The path through the plates is given by component u P 1 of equation 13. u P 1 is then written as a function of u E according to the fourth line of equation 10:

-

ω 2 M k P 1E u E + M k P 1Hb q Hb + m k P 1 q k P 1 + iωc k P 1 q k P 1 + k k P 1 q k P 1 + K P 1F 1 q F 1 + K P 1F 2 q F 2 = f P 1 (20)
The modal parameter corresponding to the path through the plates is denoted G P 1 (ω). Using equations 20 and 11, it is possible to write:

G 1 P 1 (ω) =    k ω ω 1k 2 1 -ω ω 1k 2 + 2iξ 1k ω ω 1k Gk P 11       k ω ω 2k 2 1 -ω ω 2k 2 + 2iξ 2k ω ω 2k Gk P 12    (21)
G 2 P 1 (ω) =    k ω ω 1k 2 1 -ω ω 1k 2 + 2iξ 1k ω ω 1k Gk P 11    ×    k ω ω 2k 2 1 -ω ω 2k 2 + 2iξ 2k ω ω 2k Gk P 13    ×    k ω ω 3k 2 1 -ω ω 3k 2 + 2iξ 3k ω ω 3k Gk P 14    (22)
where:

Gk P 11 = Φ k F 2 M k F 2P 1 ΦP 1 m k F 2 (23)
Gk P 12 = Φ k P 1 M k P 1E m k P 1
(24)

Gk P 13 = Φ k P 1 M k P 1Hb ΦHb m k P 1
(25)

Gk P 14 = Gk H2 (see equation 19) (26) with the notations c k F 2 = 2ξ 1k k k F 2 m k F 2 and ω 1k = k k F 2 m k F 2 , c k P 1 = 2ξ 2k k k P 1 m k P 1
and ω 2k =

k k P 1 m k P 1 , c k Hb = 2ξ 3k k k Hb m k Hb and ω 3k = k k Hb m k Hb .

Path through the first cavity

The path through the first cavity is related to the plate forming the boundary between the two cavities.

The modal parameter corresponding to this path can be written according to the fifth and sixth lines of equation 10:

-ω 2 M k P 2E u E + M k P 2Hb q Hb + m k P 2 q k P 2 + iωc k P 2 q k P 2 + k k P 2 q k P 2 + K k P 2F 1 q F 1 + K k P 2F 2 q F 2 = f P 2 (27) -ω 2 M k F 1E u E + M k F 1Hb q Hb + M k F 1P 1 q P 1 + M k F 1P 2 q k P 2 + m k F 1 q F 1 + iωc k F 1 q k F 1 + k k F 1 q k F 1 = 0 (28)
The modal parameter corresponding to the path through the plates is denoted G P 2 (ω). Using equations 27, 28 and 11, it is possible to write:

G P 2 (ω) =    k ω ω 1k 2 1 -ω ω 1k 2 + 2iξ 1k ω ω 1k Gk P 21    ×    k 1 1 -ω ω 2k 2 + 2iξ 2k ω ω 2k Gk P 22    ×    k ω ω 3k 2 1 -ω ω 3k 2 + 2iξ 3k ω ω 3k Gk P 23    (29)
where:

Gk P 21 = Φ k F 2 M k F 2P 2 ΦP 2 m k F 2
(30)

Gk P 22 = Φ k P 2 K k P 2F 1 ΦF 1 ω 2 2k m k P 2
(31)

Gk P 23 = Φ k F 1 M k F 1E -M k F 1P 1 ΦP 1 Ψ P 1e -M k F 1P 2 ΦP 2 Ψ P 2e m k F 1 (32) with the notations c k F 2 = 2ξ 1k k k F 2 m k F 2 and ω 1k = k k F 2 m k F 2 , c k P 2 = 2ξ 2k k k P 2 m k P 2
and ω 2k =

k k P 2 m k P 2 , c k F 1 = 2ξ 3k k k F 1 m k F 1 and ω 3k = k k F 1 m k F 1 .

Modal criteria

The modal parameters defined in the previous sections lead to criteria that are defined as follows:

C E = max k ω ω k 2 1 -ω ω k 2 + 2iξ k ω ω k Gk E (33)
C n = max k ω ω 1k 2 1 -ω ω 1k 2 + 2iξ 1k ω ω 1k
Gk n where n can be H1, P 11 or P 21 (34)

C n = max k ω ω 2k 2 1 -ω ω 2k 2 + 2iξ 2k ω ω 2k
Gk n where n can be H2, P 12, or P 13 (35)

C P 14 = max k ω ω 3k 2 1 -ω ω 3k 2 + 2iξ 3k ω ω 3k
Gk P 14

(36)

C P 22 = max k 1 1 -ω ω 2k 2 + 2iξ 2k ω ω 2k Gk P 22
(37)

C P 23 = max k ω ω 3k 2 1 -ω ω 3k 2 + 2iξ 3k ω ω 3k Gk P 23 (38)
Within the framework of an optimization problem, it is possible to use these criteria, for example, in order to optimize a structure's geometry. However, the criteria we propose do not allow derivation with respect to any one of the parameters of the structure (geometry of the hollow parts, thickness of the plates. . . ), although many optimization methods require derivating these criteria. For that reason other criteria, related to the original ones, are defined in the next section.

Derivable criteria

As mentioned below, criteria C n (where n can be E, H1, H2, P 11, P 12, P 13, P 14, P 21, P 22 or P 23) cannot be derived with respect to any parameter, although this would be useful in most optimization problems. To remedy this problem, we introduce criteria Cn , defined as follows:

Cn = 1 4 log k λ (ω, k) Gk n 4 ( 39
)
where λ (ω) is a coefficient depending on ω. This is the case, for example, of Gk E ,

λ (ω) = ω ω k 2 1-ω ω k 2 +2iξ k ω ω k .
It can be shown that these criteria are very similar to the first ones, and have almost the same minima and maxima. To prove this, let us consider the following function f p :

f p : x -→ p a 1 (x) p + a 2 (x) p + • • • + a n (x) p ( 40
)
where a q (x) > 0 ∀q ≤ n ∀x. It is possible to write:

lim p→+∞ f p (x) = max [a 1 (x), a 2 (x), . . . , a n (x)] (41)
Specifically, since a q (x) > 0 ∀q ≤ n ∀x:

p (max [a 1 (x), a 2 (x), . . . , a n (x)]) p < f p (x) < p n (max [a 1 (x), a 2 (x), . . . , a n (x)]) p ∀p ∈ N. ( 42
)
Equation 42 leads to:

max [a 1 (x), a 2 (x), . . . , a n (x)] < f p (x) < p √ n max [a 1 (x), a 2 (x), . . . , a n (x)] ∀p ∈ N. (43) Since lim p→+∞ p √ n = 1, it can be deduced that lim p→+∞ f p (x) = max [a 1 (x), a 2 (x), . . . , a n (x)].
Mathematically, this can be written as:

∀ǫ > 0 ∃η > 0 tq x > η ⇒ |f p (x) -max [a 1 (x), a 2 (x), . . . , a n (x)] | < ǫ (44)
Thus, if p is large enough, criteria Cn have nearly the same extrema as C n .

4 Optimizing a structure using the modal criteria

Analysis of the criteria

In this section, the criteria developed in the previous sections are analyzed. In order to simplify the analysis, only one excitation point placed on a hollow part bounding the first cavity F 1 is considered.

Values of criteria C n may change with the excitation frequency because of coefficient λ (ω). For example, figures 4, 5, 6, 7, 9 and 10 show the values of some parameters as a function of the excitation frequency f . It can be seen that the strength of each criterion depends on the excitation frequency. For example, it is interesting to note that criterion CP 22 increases more strongly than the others with frequency f . Therefore, the vibrational path through plate P becomes very significant at higher frequencies.

Notice that criteria CP 13 and CH2 show peaks which correspond to the global modes of the structure, because the modal matrices of the structure's hollow parts are involved in these criteria.

[Figure 4

Results for the modal parameters

In this section, the modal parameters Gk H1 , Gk H2 , Gk P 21 , Gk P 22 , Gk P 23 , Gk P 11 , Gk P 13

and Gk P 14 are analyzed. This will allow each of the modes responsible for the values of criteria C n to be defined. The modal parameters must be weighted with the previously defined coefficients λ (ω), in order to take the excitation frequency into account. Two cases of excitation are presented, at 50 Hz and 300 Hz.

Path through the first cavity

Considering the vibrational path through the first cavity and the plate P, three modal parameters need to be analyzed. Figure 12 shows that only one mode of the plate P is responsible for the transmission of vibrations between the first and second cavities. The criterion associated with this figure is C P 22 , and is related to the influence of pressure within the first cavity on plate P. Given that the action of an acoustic fluid on a structure is not very significant when compared with that of a structure on a fluid, the parameter values given in figure 12 are very small. For an influence to be exerted on the path through the first cavity, which is the object of this section, one could, for example, restrict the influence of the mode shown in figure 12.

Path through the hollow parts

For the vibrational path through the hollow parts, two modal parameters have to be analyzed. Figures 17 and18 show the values of modal parameters Gk H1 and Gk H2 as a function of the mode number k for an excitation frequency of 50 Hz. Figures 17 and18 show the same parameters for an excitation frequency of 300 Hz.

[Figure 17 It can be seen in figures 17 and 18 that many of the modes are strong. The values of these parameters increase with increasing excitation frequency. However, the values for criterion C H2 are much smaller than the values for criterion C H1 . It is then possible to choose to optimize the structure using C H1 , which seems to have a stronger influence on the transmission of vibrations. Although the influence of criterion CH2 appears to be smaller, its use in the optimization of the structure allows the vibrational path through the plates to be studied, as described in the following section.

Path through the hollow parts and the plates

For the vibrational path through the hollow parts and the plates, three modal parameters should be analyzed. Figures 21 and22 show the values of modal parameters Gk P 11 and Gk P 13 as a function of mode number k for an excitation frequency of 50 Hz. It should be noted that Gk P 14 = Gk H2 , as plotted in figure 18. Figures 23 and24 show the values of modal parameters Gk P 11 and Gk P 13 as a function of the mode number k for an excitation frequency of 300 Hz.

[Figure 21 As shown if figures 21 and 22 many modes are strong. Just as in the previous section, it is then possible to optimize the structure using criterion C P 14 = C k H2 , so as to minimize the transmission of vibrations through two different paths. It is also possible to minimize criterion C P 11 , which is larger than C P 13 .

Results and validation on a simple example

Since many criteria can be found, it is very difficult to show how efficient the method is. This is the reason for which it was decided to validate the derived criteria with a simple example.

The system shown in figure 25 is considered, in which a cavity is bounded by rigid walls and a single plate.

[Figure 25 about here.] For such a system, there are only two criteria, which can be expressed as follows:

C D = Φ k f -ω 2 m f + k f ω 2 M k F E + M k F S Φs Ψ s (45)
C P = Φ k f -ω 2 m f + k f -M k F S Φs (46)
where Φ f is the matrix of the acoustic modes, Φ s is the matrix of the modes of the plate, and Ψ s is the matrix of the static modes:

u f = Φ f q f (47) u s = Φ s q s + Ψ s u E (48)
In order to validate our methodology for the determination of suitable criteria, it is proposed to compute the latter for two values of plate thickness. Since plate thickness does not change criterion C D , which is directly related to the excitation point u E , only one criterion remains. It is then possible to study the variation of this criterion and the pressure level within the cavity.

Figures 26, 27 and 28 show how the criteria change as a function of the pressure level within the cavity for two different thickness values. The pressure level has been normalized with respect to its value for the initial thickness.

[Figure 26 These figures show that criterion C P decreases as a function of pressure level. Criterion C D remains constant because the plate thickness is not taken into account in this criterion. This simple example shows that the criteria proposed in the previous sections can be very useful.

Conclusion

In the present paper several criteria, corresponding to different vibrational propagation paths, and based on modal analysis of a coupled system, have been proposed. These illustrate the relative influence of each mode on noise propagation, and show which part of the system has the strongest influence on the generation of noise inside the structure.

Initially, a complex system was considered in order to provide an overview of the various criteria which may exist. A simple example was then introduced in order to validate the efficiency of the method. Although the criteria were found by assuming a specific structure, it is possible to apply the same method to any vibroacoustic system.

The goal of this paper was to derive criteria which can allow a vibroacoustic system to be optimized. The optimization of such a system lies beyond the scope of the present paper.

[10] C. Farhat, M. Géradin, On the general solution by a direct method on a largescale singular system of linear equations: Application to the analysis of floating structures, International journal for numerical methods in engineering 41 (1998) 675-696.

P F 1 F 2 2, 5 m

  about here.] [Figure 3 about here.]

, and c k Hb = 2ξ 2k k k Hb m

  about here.] [Figure 5 about here.] [Figure 6 about here.] [Figure 7 about here.] [Figure 8 about here.] [Figure 9 about here.] [Figure 10 about here.]

  Figures 11, 12 and 13 show the values of modal parameters Gk P 21 , Gk P 22 and Gk P 23 as a function of the mode number k, for an excitation frequency of 50 Hz. Figures 14, 15 and 16 show the same parameters for an excitation frequency of 300 Hz. [Figure 11 about here.] [Figure 12 about here.] [Figure 13 about here.] [Figure 14 about here.] [Figure 15 about here.] [Figure 16 about here.]

  about here.] [Figure 18 about here.] [Figure 19 about here.] [Figure 20 about here.]

  about here.] [Figure 22 about here.] [Figure 23 about here.] [Figure 24 about here.]

  about here.] [Figure 27 about here.] [Figure 28 about here.]

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Structure to optimize

Figure 4 :

 4 Figure 4: Values of parameter CH1 as a function of f

Figure 5 :

 5 Figure 5: Values of parameter CH2 as a function of f

Figure 6 :

 6 Figure 6: Values of parameter CP 21 as a function of f

Figure 7 :

 7 Figure 7: Values of parameter CP 22 as a function of f

Figure 8 :

 8 Figure 8: Values of parameter CP 23 as a function of f

Figure 9 :Figure 10 :446

 910 Figure 9: Values of parameter CP 11 as a function of f

Figure 20 :4Figure 26 :Figure 27 :Figure 28 :

 20262728 Figure 20: Values of parameter λ Gk H2 as a function of k -300 Hz

List of Figures


S Besset

L Jézéquel Ltds

Équipe D2s

Optimization of characteristics of porous materials based on a modal synthesis method

Keywords: Modal analysis, Optimization, Porous media

Complex structures used in the automotive industry often include porous materials, in order to reduce the noise in acoustic cavities. The method proposed in this paper aims at optimizing the characteristics of these porous materials using modal criteria based on the modal analysis of the structure. The use of a generalize modal synthesis method allow to describe the structure and the cavities with sets of modes; the size of the resulting system is smaller, for only generalized degrees of freedom are used for each part of the coupled fluid-structure system. First, a modelization of the porous media is proposed, and the generalized modal analysis method is explained. An optimization is then processed on the thickness of the porous materials.

Introduction

Acoustic problems in the automotive industry has long been studied, for the noise in acoustic cavities is a very important criterion for the customer. In order to reduce this noise, poroelastic materials are applied on the structure. In this paper, we optimize the thickness of these porous materials using optimization criteria based on a generalized modal analysis method.

The modal analysis method we use has been described in a previous paper [START_REF] Besset | Dynamic substructuring based on a double modal analysis[END_REF] for a structural problem. It has then been extended to the case of a coupled fluid-structure problem [START_REF] Besset | Triple modal synthesis for complex structures analysis[END_REF]. The principle of this modal synthesis method is summarized in the paper. It is based on the modal description of each substructure and boundary using a "double" or "triple" modal synthesis, which have first been proposed by Jézéquel [START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part i : Theory of hybrid models and modal truncation methods[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part ii : Numerical tests and experimental identification of hybrid model[END_REF].

It is possible to define optimization criteria based on the modal description of a coupled fluid-structure system. It has already been done in the case of a structural problem [5] and has been extended to the case of a coupled fluid-structure system [START_REF] Besset | Optimization of a coupled fluid-structure system using a modal approach[END_REF].

1 Such criteria correspond to a systemic approach of vibroacoustic problems. The relation between the excitation and the pressure in the cavities can be expressed through modal matrices coming from the generalized modal analysis of the structure. Several terms appear in the expression of the pressure, which correspond to several "vibration paths". The expressions of these paths constitute the criteria we use for the optimization.

Porous materials have long been studied. Biot [START_REF] Biot | Theory of propagation of elastic waves in a fluid-filled-saturated porous solid[END_REF][START_REF] Biot | Theory of propagation of elastic waves in a fluid-filled-saturated porous solid -ii. higher frequency range[END_REF][START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF][START_REF] Biot | Acoustics, elasticity, and thermodynamics of porous media, Twenty one papers[END_REF] has first proposed a method to describe these materials. Later, Allard [START_REF] Allard | Propagation of sound in porous media, Modeling sound absorbing materials[END_REF]12] proposed a more precise modelization, introducing for example frequency dependant terms in the expression of the densities. Finite element models have also been proposed [13]. Our approach aims at describing porous materials through boundary impedances; some papers have already been written on the subject [14, 15], but the method we propose here uses these impedances to construct coupling terms of finite element matrices.

Modal analysis of the structure

The complex structure we consider in this paper include plates and two acoustic cavities. It has the same geometry of the structure considered in [5]. The geometry of the structure is shown in the figure 1: • F 1 is the first acoustic cavity, where the pressure level has to be minimized;

• F 2 is the second acoustic cavity;

• P is the plate between the two cavities.

For figure 1 to be more understandable, the plates have not been drawn. The structure will be excited by a displacement applied on its skeleton.

Plate P F 1 F 2 u E 2, 5 m
      -ω 2       M EE M EB M EP M BE M BB M BP M P E M P B M P P 0 C F M A1A1 M A1A2 M A2A1 M A2A2       +       K EE K EB K EP K BE K BB K BP K P E K P B K P P -C T F 0 K A1A1 K A1A2 K A2A1 K A2A2                        u E u B u P p 1 p 2            =            f E f B f P 0 0            (1)
C F is a matrix coming from the fluid-structure coupling. In order to define the criteria for the optimization of the structure, we need to express this matrix more precisely:

C F = M A1E M A1B M A1P M A2E M A2B M A2P = -   K EA1 K EA2 K BA1 K BA2 K P A1 K P A2   T (2)
Notice that M XiYj (i=1,2 and j=1,2) are not mass and stiffness terms. These terms are related to the discretized acoustic equation. That is why equation 2 can be written without units problems.

u B , u P and p are the degrees of freedom of the system:

• u E corresponds to the excited points of the structure;

• u B corresponds to the boundaries between the plates;

• u P corresponds to the plates;

• p 1 are the degrees of freedom of the acoustical cavity F 1 ;

• p 2 are the degrees of freedom of the acoustical cavity F 2 .

The modal analysis method used to describe the system is based on the "double" and "triple" modal synthesis proposed in previous papers [START_REF] Besset | Dynamic substructuring based on a double modal analysis[END_REF][START_REF] Besset | Triple modal synthesis for complex structures analysis[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part i : Theory of hybrid models and modal truncation methods[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part ii : Numerical tests and experimental identification of hybrid model[END_REF]. Modes are used to describe the acoustic cavities, the boundaries and the plates. This method leads to the modal equation:

      -ω 2       M EE M EB M EP M BE m BB 0 M P E 0 m P P 0 C F m A1A1 0 0 m A2A2       +iω       c E 0 0 0 0 c B 0 0 0 0 c P 0 0 0 0 0 0 0 0 0       +       K EE K EB K EP K BE k BB 0 K P E 0 k P P -C T F 0 k A1A1 0 0 k A2A2                        u E q B q P q A1 q A2            =            f E f B f P 0 0            (3)
The modal damping matrix is assumed to be diagonal. The modal analysis described in equation 6 and 7 allows k AiAi , k P P , k BB , m P P , m BB and m AiAi (i = 1 or 2) to be diagonal matrices too. This equation will be used in section 4 to introduce optimization criteria. The modal matrices used for obtaining the generalized degrees of freedom are defined as follows:

u P = Φ P q P + Ψ P B u B + Ψ P E u E (4) u B = Φ B q B + Ψ BE u E (5) p 1 = Φ A1 q A1 (6) p 2 = Φ A2 q A2 (7)
Matrices Φ P , Φ B , Φ A1 and Φ A2 are the matrices of the modes of the structure:

• Φ P is the matrix corresponding to the mode of the plates when boundaries are fixed;

• Φ B is the matrix corresponding to the mode of the hollow parts constituting the skeleton of the structure, considering a static condensation of the plates on these hollow parts;

• Φ A1 is the matrix corresponding to the mode of the acoustical cavity F 1 (cavity modes);

• Φ A2 is the matrix corresponding to the mode of the acoustical cavity F 2 (cavity modes);

• Ψ P B , Ψ P E and Ψ BE are static modes of the structure.

Analysis of the poroelastic materials

The coupling terms recalled in equation 2 comes from the classical acoustical coupling equation:

∂p ∂n = ρ f ω 2 u n (8)
where ρ f is the density of the fluid, p is the pressure on the fluide-structure boundary, and u n is the normal displacement. It is possible to introduce a boundary impedance Z that allows to describe the behaviour of the boundary if we consider a poroelastic material on this boundary. In equation 8, Z is considered to be infinite. Taking into account the impedance Z, the coupling equation becomes:

∂p ∂n - iωρ f Z p = ρ f ω 2 u n (9)
The aim of this section is to determinate Z in function of the characteristics of a poroelastic material. The figure 2 shows the notations we use:

• u 0 is the displacement of the boundary between the porous material and the wall (a plate of the structure);

• u l is the displacement of the porous material at the boundary with the fluid cavity;

• l is the thickness of the porous material.

1 -h -h Q R u l -γu l - h 2 ρ22 ω 2 ∂p ∂z (z = l) = 1 ρ0 ω 2 ∂p ∂z (z = l) (10)
In order to take into account the thickness of the porous material and the structure it is fixed on, we have to express u l in function of u 0 . To do this, we consider the equation of the fluid part of the porous material [13], which is recalled here:

∆p + ρ22 R ω 2 p - ρ22 h 2 γω 2 divu = 0 (11)
h is the porosity of the porous material, b is a coefficient depending on the characteristics of the pores of the porous material and α ∞ is the tortuosity of the porous material. γ, ρ22 and ρ12 are defined as follows:

ρ22 = ρ 22 -i b ω (12)
ρ 22 = hρ f α ∞ (13) γ = h ρ12 ρ22 - Q R (14) ρ12 = ρ 12 -i b ω (15)
ρ 12 = -hρ f (α ∞ -1) (16)
Q and R are coefficients that depend on the characteristics of the porous material. Their expression can be found in the book of Allard [START_REF] Allard | Propagation of sound in porous media, Modeling sound absorbing materials[END_REF]. They take into account the thermal effects in the pore, so they have a frequency dependent complex amplitude that we recall:

Q = (1 -h) K f (17) R = hK f (18) K f = γP 0 γ -(γ -1) 1 + 8η iΛ 2 B 2 ωρ f 1 + iρ f ωB 2 Λ 2 16η 1/2 -1 (19) (20)
where B and P 0 depend on the temperature. For 18 • , we have B = √ 0, 71 and P 0 = 1, 0132.10 5 (see [START_REF] Allard | Propagation of sound in porous media, Modeling sound absorbing materials[END_REF]).

Several expressions do exist to evaluate b which depends on the frequency. We choose to use an expression proposed in [START_REF] Allard | Propagation of sound in porous media, Modeling sound absorbing materials[END_REF]:

b = σh 2 G(ω) ω (21)
where:

G(ω) = 1 + 4iα 2 ∞ ηρ f ω σ 2 Λ 2 h 2 (22)
with the notations:

• Λ is a characteristic dimension of the porous material;

• σ is the flow resistivity (intrinsic property of the material);

• η is the viscosity;

Considering only the z displacement, equation 11 can be written:

∂ 2 p ∂z 2 + ω 2 ρ22 R p -ω 2 ρ22 h 2 γ ∂u ∂z = 0 (23)
Considering that the thickness l is small (for example l < 2πc0 ω ), it is possible to write:

∂u ∂z = u l -u 0 l ( 24
)
∂ 2 p ∂z 2 = ∂p ∂z (z = l) -∂p ∂z (z = 0) l ( 25
)
If the boundary between the porous material and the structure is rigid, ∂p ∂z (z = 0) can be expressed as follows:

∂p ∂z (z = 0) = ρ 0 ω 2 u 0 (26)
The equation 23 becomes then:

1 l ∂p ∂z (z = l) + ω 2 ρ22 R p -ω 2 ρ22 h 2 l γ u l -1 - h 2 ρ 0 ρ22 γ u 0 = 0 (27)
This equation allows to express u l in function of u 0 :

u l = 1 - h 2 ρ 0 ρ22 γ u 0 + h 2 ω 2 ρ22 γ ∂p ∂z + h 2 l Rγ p (28)
The coupling equation becomes:

1 ρ 0 -1 -h -h Q R h 2 ρ22 γ ∂p ∂z -ω 2 1 -h -h Q R -γ h 2 l Rγ p = ω 2 1 -h -h Q R -γ h 2 l Rγ 1 - h 2 ρ 0 ρ22 γ u 0 (29)
which is close to the classical equation of coupling:

∂p ∂z - jωρ 0 Z p p = ρ 0 ω 2 ξ 2 u 0 ( 30
)
with the notations:

Z p = j Rγ ωh 2 lξ 1 (31) ξ 1 = 1 -h -h Q R -γ 1 -ρ 0 1 -h -h Q R h 2 ρ22γ
(32)

ξ 2 = h 2 l Rγ 1 -h -h Q R -γ 1 -h 2 ρ0 ρ22γ 1 -ρ 0 1 -h -h Q R h 2 ρ22γ (33)
The modelization retained for the porous materials is a simplified modelization. It is possible to study the influence of the simplifications by comparing the impedance Z p we propose with the classical acoustic impedance, given by the relation:

Z = p(l) iωu(l) (34)
where p(l) is the pressure and iωu(l) is the speed of the fluid at z = l (see figure 2). The characteristics of the porous material used for the comparison between Z p and the impedance proposed by Allard [START_REF] Allard | Propagation of sound in porous media, Modeling sound absorbing materials[END_REF] The results of the comparison are given in the figures 3(a), 3(b), 4(a), 4(b), 5(a) and 5(b), corresponding to thicknesses of 1 cm, 5 mm, and 2 mm. The results are good if the thickness l is small. Notice that the imaginary part of the impedance is always good, even if the thickness is not small. This imaginary part corresponds to the damping due to the porous material.

The equation 30 is a boundary condition between the fluid and the plates that generates coupling terms depending on the frequency, for the impedance Z p depends on the frequency. This coupling term can be expressed in the base of the modes of the structure defined in section 2, hence the equation:

-ω 2 M + iωC(ω) + K u = f (35)
where C(ω) is a complex matrix, depending on ω. This matrix corresponds to the impedance Z p defined in equation 31. M, K, u and f correspond to the matrices and vectors defined in equation 3. Using the poroelastic modelization we propose, equation 3 becomes: where the q i are generalized degrees of freedom, whereas u E are nodal degrees of freedom. The modal analysis is proceeded as proposed in section 2 -ie. without the porous material. The motion equation of the structure including porous materials is then written by projection on the modes calculated without it. c A1 (ω) and c A2 (ω) are assumed to be diagonal (Basile's hypothesis).

      -ω 2       M EE M EB M EP M BE m BB 0 M P E 0 m P P 0 C F (ω) m A1 0 0 m A2       + iω       c E 0 0 0 c B 0 0 0 c P 0 0 c A1 (ω) 0 0 c A2 (ω)       +       K EE K EB K EP K BE k BB 0 K P E 0 k P P -C T F (ω) 0 k A1 0 0 k A2                        u E q B q P q A1 q A2            =            f E f B f P 0 0            (

Optimization criteria

The optimization criteria we propose in this paper are based on the relationship between the pressure in the cavities and an excitation in displacement on the structure. The excited degrees of freedom are denoted u E . We are interested in the pressure level in the second cavity F 2 . The last line of equation 36 allows to express the pressure p 2 in the second cavity F 2 . First, it is necessary to express q A2 :

q A2 = ω 2 M k A2E u E + M k A2B q B + M k A2P q P -ω 2 m k A2 + k k A2 + iωc k A2 (ω) (37)
We denote Φ A2 the matrix of the modes of the second cavity. Φ k A2 is the k th column of Φ A2 . The pressure p 2 in F 2 can then be written:

p 2 = Φ A2 q A2 = k Φ k A2 q A2 (38)
Replacing q A2 in equation 38 with the expression given in equation 37, we obtain the relationship:

p 2 = k   ω 2 Φ k A2 M k A2E -ω 2 m k A2 + k k A2 + iωc k A2 (ω)   u E (39)
+ k   ω 2 Φ k A2 M k A2B -ω 2 m k A2 + k k A2 + iωc k A2 (ω)   q B ( 40
)
+ k   ω 2 Φ k A2 M k A2P -ω 2 m k A2 + k k A2 + iωc k A2 (ω)   q P (41)
q Bm and q P are now expressed in function of the nodal degrees of freedom, thanks to the relations coming from the modal analysis of the system and defined in equations 4, 5, 6 and 7:

q B = ΦB (u B -Ψ BE u E ) (42) q P = ΦP (u P -Ψ P B u B -Ψ P E u E ) (43)
where ΦB and ΦP are defined as follows:

ΦB = Φ T B MB (44) ΦP = Φ T P MP (45)
which can be written thanks to the eigenvectors property:

Φ T B MB Φ B = I (46) Φ T P MP Φ P = I ( 47
)
where MB is the condensed mass matrix used to find Φ B , and MP is the condensed mass matrix used to find Φ P .

We can now define a criterion Cd linked to the direct path between the displacement u E and the pressure p 2 :

Cd = k ω 2 Φ k A2 M k A2E -M k A2B ΦB Ψ BE -M k A2P ΦP Ψ P E -ω 2 m k A2 + k k A2 + iωc k A2 (ω) (48)
In order to reduce the pressure level for the maxima, we consider criterion Cd :

C d = max k ω k Φ k A2 M k A2E -M k A2B ΦB Ψ BE -M k A2P ΦP Ψ P E ic k A2 (ω k ) (49)
It is possible to define other criteria linked to other paths, as shown on figure 6, but this is not the purpose of the paper. We will only deal with the criterion C d to optimize the thickness of the porous media of the structure, for it is the main criterion, directly linked to the direct path between the pressure field and the excitation point (black arrow on figure 6). The optimization we use in this paper is based on two criteria. Criterion C d has been defined in section 4. Criterion C m is linked to the mass of the poroelastic material, which must not be too high. Four parts of porous materials are considered here:

• porous material 1: foam put on the plates around the second cavity F 2 ;

• porous material 2: foam put on the plate P situated between the two cavities, on the side situated in the second cavity;

• porous material 3: foam put on the plate P situated between the two cavities, on the side situated in the first cavity;

• porous material 4: foam put on the plates around the first cavity F 1 ;

The optimization algorithm considered here is a classical genetic algorithm that is given on figure 7.

The results of the optimization are given on the Pareto diagram of the figure 8. This diagram shows the optimal points: some of these points are better according to We observe that the thicknesses of the foams corresponding to the first cavity are first increased. Figure 13 shows the pressure level in the cavity for the different steps of the optimization. Notice that other paths should be taken into account to get better results. Nevertheless, multicriteria optimization is not the purpose of the paper.

Conclusion

A modelization of porous materials has been proposed, in order to be able to study the dynamic behaviour of a complex structure. A vibroacoustic criterion has been proposed, which allows to optimize the thicknesses of the porous materials used in the modeling of the structure. A genetic optimization has been processed, and we observe that the thicknesses of the foams corresponding to the first cavity are first increased.

The method allows to optimize complex structures including poroelastic components:

• The modes of the structure are first computed;

• Porous media are then analyzed through a frequency dependent and complex impedance;

• Generalized degrees of freedom of the structure are expressed in function of the modes of the structure; Notice that the example given in the paper show how to use the method on a given structure. Nevertheless, other structures may be considered, for the method can easily be generalized to other situations.

Figure 1 :

 1 Figure 1: Structure to optimize

Figure 2 :

 2 Figure 2: Notations on the purous material

Figure 3 :

 3 Figure 3: Boundary impedance for a porous material -thickness: 1 cm (-: Reference, --: Proposed model Z)

Figure 4 :

 4 Figure 4: Boundary impedance for a porous material -thickness: 5 mm (-: Reference, --: Proposed model Z)

Figure 5 :

 5 Figure 5: Boundary impedance for a porous material -thickness: 2 mm (-: Reference, --: Proposed model Z)

Figure 6 :

 6 Figure 6: Direct path (in black) taken into account for optimization

Figure 7 :

 7 Figure 7: Genetic algorithm used for optimization

  , 11 and 12 show the variation of the thicknesses of the foams in function of the mass of the structure -various optimized points are represented in these figures.

Figure 8 :

 8 Figure 8: Pareto diagram of the optimization

Figure 9 :

 9 Figure 9: Best thickness of foam 1 in function of the mass of the foam

Figure 10 :Figure 13 :

 1013 Figure 10: Best thickness of foam 2 in function of the mass of the foam

Table 1 :

 1 are recalled in table 1. Characteristics of the porous material used in the comparison

	Tortuosity	Density	Porosity Poisson's Lamé Modulus
	α ∞	of the skeleton	h	ratio	N
	1, 06	130 kgm -3	0, 94	0	220(1 + 0, 1i)
	Λ	σ	η		ρ f
	1, 1.10 -4 m 40000 Nm -4 s 1, 84.10 -5 Pl 1,213 kgm -3


S Besset

L Jézéquel Ltds

Équipe D2s

Modal criteria for optimization of structures through active structural acoustic control

Keywords: Modal synthesis, Active control, Vibroacoustic coupling

This paper deals with modal criteria allowing optimization of structures through structural acoustic control (ASAC) based on the independent modal strategy control (IMSC). The aim of the paper is to propose efficient criteria allowing to process optimization of structures by determining the most important modes to control considering only the location of the excitation on the structure. A generalized modal synthesis is used to study the coupled fluid-structure system. Modal criteria are proposed in order to determine the best choice for the controlled modes used in IMSC method. The way the ASAC will be applied is not discussed in the paper. Numerical simulations are computed to show the efficiency of the method, considering a simple vibroacoustic structure.

Introduction

In order to study vibroacoustic problems without considering too large matrices, many numerical simulation strategies can be found in the literature. Among these techniques, energy methods like SEA or MES are efficient for high frequency range [START_REF] Lyon | Statistical Energy Analysis of Dynamical Systems: Theory and Application[END_REF][START_REF] Wang | Development of an energy boundary element formulation for computing high-frequency sound radiation from incoherent intensity boundary conditions[END_REF][START_REF] Cotoni | High frequency radiation of 1-shaped plates by a local energy flow approach[END_REF]. In low and medium frequency domains, modal methods can also be used [START_REF] Besset | Vibroacoustical analysis based on a multimodal strategy: Triple modal synthesis[END_REF]. These modal methods have many advantages. Once modal matrices of the problem have been computed, modal criteria linked to vibroacoustic coupling can easily be defined. The criteria proposed in this paper are derived from those developed by P. Lemerle [START_REF] Lemerle | Optimization of structures to reduce the transmission of vibrations : on approach based on kuhn-tucker optimality conditions with constraints on modal effective parameters[END_REF], using Craig & Bampton's method to describe the dynamic behaviour of a structure. They have already been used to optimize coupled fluid-structure systems in [START_REF] Besset | Optimization of structural dynamic behaviour based on effective modal parameters[END_REF]7].

The modal synthesis technique described here to compute the criteria is based on the "double modal synthesis" proposed by Jézéquel [START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part i : Theory of hybrid models and modal truncation methods[END_REF][START_REF] Jézéquel | Component modal synthesis methods based on hybrid models, part ii : Numerical tests and experimental identification of hybrid model[END_REF] and extended to the case of fluid-structure systems [START_REF] Besset | Vibroacoustical analysis based on a multimodal strategy: Triple modal synthesis[END_REF]. This method resorts to a coupling formulation which has been previously investigated [START_REF] Ohayon | Reduced models for fluid-structure interaction problems[END_REF]. An example of modal analysis of a coupled system can be found in [START_REF] Sandberg | Domain decomposition in acoustic and structure-acoustic analysis[END_REF].

Modal analysis of the structure will lead to modal mass and stiffness matrices, which are then used to obtain effective modal parameters, which in turn lead to criteria allowing the structure to be optimized. The method proposed in this paper may be used for any vibroacoustic system. In fact, the ultimate goal of this approach is to define modal criteria which allow to find the modes that most contribute to the noise level in the cavity. The obtained criteria are related to the coupling terms between the systems different substructures, and are expressed as functions of the terms contained in the modal matrices. Our method is described for the case of a specific system, but can readily be adapted to any vibroacoustic system.

Analysis of the vibroacoustic system

The structure considered here is a simple structure including plates and an acoustic cavity. This structure is shown on figure 1. The aim of the paper is to optimize the sound level in the cavity by controlling a few modes. The example we give in this section aims at optimizing the pressure level of a few points of the cavity. The method we propose leads to optimization criteria that do not depend on external excitations.

External force

The vibroacoustic problem

Figure 1 shows the geometry of the vibroacoustic system under consideration. The approach used in this paper to study the vibroacoustic behaviour is a (u, p) formulation. Suxh kind of formulation can be found in [START_REF] Morand | Fluid Structure Interaction[END_REF]. The fluid (air) is considered to be inviscid and leads to a classical linear acoustic problem. The matrix formulation of the problem can be written as follows:

M ξ + C ξ + Kξ = A w w(t) (1)
A w w(t) is the external excitation, located on a plate of the structure. C is the damping matrix. The aim of the paper is not to deal with the influence of the damping, that is why we consider a very simple viscous modelisation for the structural part of the system. Matrices M and K and vector ξ are defined as follows:

M =   M bb M bs M sb M ss 0 C F M p   K =   K bb K bs K sb K ss -C T F 0 K pp   ξ =    u b u s p    (2)
where u b are the degrees of freedom related to the boundaries, u s are the degrees of freedom related to the surfaces, and p are the degrees of freedom related to the pressure in the acoustic cavity. Matrices M and K defined in equation 2 are mass and stiffness matrices classically used in coupled fluidstructure problems. Degrees of freedom concerning the structure are split into internal degrees of freedom denoted u s and boundary degrees of freedom denoted u b . This kind of splitting is quite classical in component mode synthesis methods that we will use in the paper.

Equation 1 is a classical interior non-symmetric vibroacoustic equation. The degrees of freedom are displacements for the structural part, and pressure for the acoustic part. The vibroacoustic coupling appears in matrix C F defined as follows:

C F = M pb M ps = - K bp K sp T (3)

Control forces

The aim of the paper is to define the best criteria for the control law. Thus, a term e(t) is added in equation 1.

M ξ(t) + C ξ(t) + Kξ(t) = A w w(t) + A e e(t) (4)
The control force A e e(t) can be provided through many ways (piezoelectric patches for example). Nevertheless, the purpose of the paper is not to propose a method to apply the control forces.

Modal synthesis of the system

The coupled system is studied thanks to component mode synthesis methods that can be found in [START_REF] Besset | Dynamic substructuring based on a double modal analysis[END_REF][START_REF] Besset | Vibroacoustical analysis based on a multimodal strategy: Triple modal synthesis[END_REF]. Although classical component mode synthesis methods are often reliable in the low frequency range, it has been shown in these papers that the component mode synthesis considered here is efficient and allows to deal with medium frequency range. This modal synthesis method aims at modelling the system through cavity modes, plate modes and branch modes. Branch modes are used to describe singularities of the system such as boundaries or stiffeners. As for classical component mode synthesis methods, all the modes are not taken into account and resulting matrices are quite smaller than initial matrices. Thus, generalized degrees of freedom are obtained as follows:

   p = Φ p q p u s = Φ s q s + Ψ s u b u b = Φ b q b =⇒ ξ = φη (5)
where

η =    q b q s q p  
 is much smaller than ξ. Φ s and Φ p are the matrices of the modes corresponding respectively to the plates and the pressure in the acoustic cavity. Ψ s is the matrix of the static boundary modes. Φ b is the matrix of the "branch modes" of the structure as defined in [START_REF] Besset | Vibroacoustical analysis based on a multimodal strategy: Triple modal synthesis[END_REF].

In fact, three modal synthesis are successively processed. First, considering acoustic modes Φ p , a transfer matrix T 1 is defined as follows:

T 1 =   Id 0 0 0 Id 0 0 0 Φ p   ( 6
)
where Id is the identity matrix. Thus, mass and stiffness matrices can be expressed as follows:

M 1 = T T 1 MT 1 and K 1 = T T 1 KT 1 (7)
Matrices M 1 and K 1 can be split as follows:

M 1 =   M 1bb M 1bs 0 M 1sb M 1ss 0 M 1pb M 1ps M 1pp   and K 1 =   K 1bb K 1bs K 1bp M 1sb M 1ss K 1sp 0 0 K 1pp   (8)
A second modal synthesis is then processed considering transfer matrix T 2 :

T 2 =   Id 0 0 Ψ s Φ s 0 0 0 Id   (9)
where Ψ s = K 1ss -1 K 1sb (Craig & Bampton theory) and Φ s are the modes of the structure given by the following equation:

K 1ss -ω 2 M 1ss Φ s = 0 ( 10
)
The last modal synthesis aims at expressing u b in function of branch modes Φ b as explained in [START_REF] Besset | Dynamic substructuring based on a double modal analysis[END_REF][START_REF] Besset | Vibroacoustical analysis based on a multimodal strategy: Triple modal synthesis[END_REF].

Thanks to relations 5, we can write the modal equation of motion as follows:

Mη + C η + Kη = A w w(t) + A e e(t) (11)
where:

K = φ T Kφ =                     ω b 1 2 . . . ω b n b 2 (ω s 1 ) 2 . . . ω s ns 2 -C T F (ω p 1 ) 2 . . . ω p np 2                     (12)
M = φ T Mφ =   I n b I ns C F I np   (13)
A w = φ T A w (14
)

A e = φ T A e (15)
Matrix C = φ T Cφ is considered to be diagonal thanks to Basile's hypothesis. Equation 11 can be rewritten using vector X = η η :

Ẋ = DX + Ew(t) + Fe(t) (16)
where D, E and F are defined as follows:

D = 0 M -K -C , E = 0 A e and F = 0 A w (17)
3 Vibroacoustic criteria

The aim of the method proposed in this paper is to use vibroacoustic criteria to determine which modes should be controlled. Thus, let define these criteria from equation 11. As there is no noise source inside the cavity, the n e line of equation 11 leads to:

-

ω 2 M n pb q b -ω 2 M n ps q s + ω n p 2 -ω 2 + iωc n q n p = 0 (18)
The assumption "there is no noise source in the cavity" is not only made for the sake of brevity and clarity. The problem we consider in this paper deals with the noise in acoustic cavities and should be applied to aircrafts or automotive industry. Thus, the only excitation sources we consider are external excitations on the boundaries of the structure.

The noise level is linked to the pressure in the acoustic cavity. The pressure can be expressed as follows:

p = Np n=1 Φ n p q n p ( 19
)
where N p is the number of acoustic modes computed in the modal synthesis. Substituting equation 18 in equation 19, we obtain the following relation linking the pressure in the cavity with the generalized degrees of freedom q b and q s :

p = Np n=1 ω 2 Φ n p M n pb q b + M n ps q s ω n p 2 -ω 2 + iωc n ( 20
)
The aim is to express the pressure p in function of degrees of freedom u b and u s in order to define vibroacoustic paths as defined in [7]. These various "vibration paths" are linked to the different ways of propagation of the vibration that cause the noise into the cavity. q b and q s can then be expressed thanks to relation 5:

q b = Φb u b q s = Φs (u s -Ψ s u b ) (21)
where Φb and Φs are pseudo-inverse of matrices Φ b and Φ s . Thus, two criteria can be defined, corresponding to the relation between the pressure and the degrees of freedom related to the surfaces and the boundaries:

C b = Np n=1 ω n p Φ n p M n pb Φb -M n ps Φs Ψ s c n (22) C s = Np n=1 ω n p Φ n p M n ps Φs c n (23)
In the following, we will only consider criterion C s for the control forces that will be applied on the structure. This criterion is linking the pressure p and the degrees of freedom of the structure u s . Thus, this criterion is not directly linked to the applied excitation forces appearing in equation 4. This should be a problem since the control we consider in the paper is applied through forces. Nevertheless, the control force f = A e e(t) can be linked to the displacement of the structure u s through the following equation:

f = H(ω)u s ⇒ u s = H(ω)f (24)
where H(ω) corresponds to the dynamic response of the structure. Thus, the link between f and u s should be expressed through criterion Cs :

Cs = Np n=1 ω n p Φ n p M n ps Φs H(ω) c n = H(ω) C s (25)
In the following, we consider that considering criterion C s is enough to define the right control law.

Criterion C s can be split into N s criteria denoted C j s corresponding to the N s modes of the structure:

C j s = Np n=1 ω n p Φ n p M n ps Φj s c n 1 ≤ j ≤ N s (26)
where Φj s is the j e column of matrix Φs . Considering these N s criteria, it is possible to evaluate the most important modes of the structure for the noise level. Figure 2 shows the values of the criteria for N s = 40. The aim of the paper is to use the modal criteria defined in section 3 to control the modes of the structure. The active control method we consider here can be found in [START_REF] Jézéquel | Optimal preview semiactive suspension[END_REF][START_REF] Roberti | Adaptive semiactive isolation of seismic buildings[END_REF][START_REF] Collet | Independant modal space control of a thin plate with piezoelectrics as sensors and actuators[END_REF].

We recall that the proposed method allows to define criteria that do not depend on the excitation. Thus, we don't expect better results than other control methods for the results would depend of the considered excitation, but we expect noise levels to be reduced in order to validate the criteria.

In order to be able to control each mode separately, it is useful to define a matrix L e so that:

A e L e = I =⇒ e(t) = L e r(t) (27)
Equation 16 can then be rewritten as follows:

Ẋ = DX + Ew(t) + 0 I r(t) ( 28
)
where I is the identity matrix. The dimension of vector r(t) corresponds to the number of control forces. The control excitation is then expressed in function of the generalized degrees of freedom η and η:

r(t) = -Gη -H η ( 29
)
G and H are diagonal matrices that have to be found so that matrix 0 I can be expressed as follows:

0 I r(t) = 0 0 -G -H η η ( 30
)
In order to find values for matrices G and H, we will consider m criteria J m to control the structure, corresponding to the m modes to attenuate. These modes have been found using the vibroacoustic criteria defined in section 3. Criteria J m can be expressed as follows:

J m = lim t→∞ E    1 t t 0   η m ηm r m   T   ω 2 m 0 0 0 I 0 0 0 R m     η m ηm r m   dt    (31)
where R m is a value that is arbitrarily chosen (the efficiency of the control will depend on this value). This relation leads to the following expression of r m (t), that can be found in [START_REF] Collet | Independant modal space control of a thin plate with piezoelectrics as sensors and actuators[END_REF]:

r m (t) = ω m ω m -ω 2 m + R -1 m 1 2 η m (t) + c m ω m -c 2 m ω 2 m + 2ω m -ω m + ω 2 m + R -1 m 1 2 + R -1 m 1 2 ηm (t) (32)
Which allows to find values for matrices G and H.

Results

Considering figure 2, we chose to attenuate four modes of the structure. The results of the control are given on figures 3 and 4. The structure considered was presented in figure 1. A force have been applied on this structure (see figure 1), and we computed the frequency response (see figure 3) and the temporal response (see figure 4) to obtain the pressure in the cavity. The calculations have been made considering the average pressure in a small volume in the cavity. Figure 4 shows the noise level, that is linked to |p| 2 , is quite attenuated with the active control, even if only four modes are controlled. Moreover, another calculation has been made, considering 20 modes to be attenuated (see figures 5 and 6). This seems to be too much for the control to be efficient. The frequency response function (see figure 5) shows that the control is effective around 170 Hz, but this is not the case in lower frequency range, for example around 65 Hz. Moreover, figure 6 shows the result with control is worth than without control. This effect can be explained by the fact that the control law has to deal with It is also possible to see the influence of the control of each of the 4 controlled modes. Figures 7 and8 show the frequency response functions of the system for each case. Figures 9 and10 show the efficiency of the control. Controlling these modes allows the pressure level in the cavity to decrease: criterion C s is efficient to detect the modes to control.

It is interesting to observe that although the value of the criteria shown in figure 2 for the 15th mode is much higher than for the 25th mode. However, it seems contradictory withe the results shown in figures 9 and 10. In fact, figure 7 shows that controlling the 15th mode leads to higher pressure levels around 65 Hz, which explains that the control is less efficient that for the 25th mode.

Conclusion

Vibroacoustic criteria have been defined and used in the case of active control of a structure. We show that these criteria allow to find the most important modes in fluid-structure coupling. Thus, treating only the four main modes, it is possible to significantly reduce the noise level in a cavity.

Moreover, we show that considering all the modes does not produce better results. Finding the most important modes is thus very important. The control should be processed thanks to piezoelectric patches for example, which could be made in future works.

At last, the results presented in the paper are related to a single example. It

Figure 1

 1 Figure 1: Studied structure

Figure 2 :

 2 Figure 2: Values of criteria C j s

Figure 3 :

 3 Figure 3: Square pressure in the acoustic cavity -FRF

Figure 4 :

 4 Figure 4: Square pressure in the acoustic cavity -4 modes controlled

Figure 5 :

 5 Figure 5: Square pressure in the acoustic cavity -FRF -20 modes controlled

Figure 6 :

 6 Figure 6: Square pressure in the acoustic cavity -20 modes controlled

Figure 7 :

 7 Figure 7: Square pressure in the acoustic cavity -FRF -15 th mode controlled

Figure 8 :

 8 Figure 8: Square pressure in the acoustic cavity -FRF -25 th mode controlled

Figure 9 :

 9 Figure 9: Square pressure in the acoustic cavity -15 th mode controlled

Figure 10 :

 10 Figure 10: Square pressure in the acoustic cavity -25 th mode controlled