Laurent Maxit
email: laurent.maxit@insa-lyon.fr

Oriol Guasch
email: oriol.guasch@salle.url.edu

A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models

Keywords: Statistical energy analysis, dual modal formulation, component modal synthesis, power flow, mid-frequency modelling, plate assembly

Statistical energy methods in vibroacoustics, like the statistical energy analysis (SEA) or the statistical modal energy distribution analysis (SmEdA), rely on specific modal coupling assumptions (MCAs) between subsystem modes. These methods assume that the behavior of subsystem mode amplitudes mimic that of oscillators, that the modes within a subsystem are uncoupled, and that the coupling between two different subsystems only takes place through the interaction of resonant modes. In the case of more than two subsystems being connected at a junction, however, it becomes difficult to establish a modal interaction scheme for them. In SEA, the problem is avoided by resorting to the travelling wave approach instead of the modal one. Nevertheless, there is a need for other energy-based methods, like SmEdA, to deal with such a situation. In this work it is proposed to extend the displacement-stress dual formulation, originally intended for two subsystems, to the case of multiple flexural waveguides connected at a junction. Numerical results are presented for a test case consisting of a floor coupled to two walls at right angle. The fulfillment of the MCAs by the dual modal formulation is examined in terms of the impedance mismatch between the floor and the walls.

Introduction

Statistical energy-based approaches are widely used to describe the dynamics of complex built-up structures at mid and high frequencies. From all methods, the most popular one is statistical energy analysis (SEA) [START_REF] Lyon | Theory and application of statistical energy analysis[END_REF][START_REF] Ungar | Fundamentals of statistical energy analysis of vibrating systems[END_REF][START_REF] Woodhouse | An approach to the theoretical background of statistical energy analysis applied to structural vibration[END_REF][START_REF] Burroughs | An introduction to statistical energy analysis[END_REF][START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF]. In SEA, the global system to be analyzed is subdivided into several subsystems (each one representing a group of resonant modes), and power balance equations are established between them. Despite their simplicity, the derivation of the SEA equations relies on several rigorous assumptions whose range of validity is not easy to establish. Many studies have addressed this subject in the past, see for example [START_REF] Sestieri | Vibroacoustics: The challenges of a mission impossible?[END_REF][START_REF] Culla | Is it possible to treat confidentially SEA the wolf in sheep's clothing?[END_REF][START_REF] Kean | Statistical energy analysis of strongly coupled systems[END_REF][START_REF] Mace | On the statistical energy analysis hypothesis of coupling power proportionality and some implications of its failure[END_REF][START_REF] Finnveden | Ensemble averaged vibration energy flows in a three-element structure[END_REF][START_REF] Yap | Investigation of damping effects on statistical energy analysis of coupled structures[END_REF], or the more recent works in [START_REF] Bot | Validity diagrams of statistical energy analysis[END_REF][START_REF] Lafont | Review of statistical energy analysis hypotheses in vibroacoustics[END_REF].

The strict hypotheses demanded by SEA essentially confine it to the high frequency range. To address the so-called mid-frequency problem and extend energy-based methods to lower frequencies than those in the range of SEA, several proposals have been made. In the framework of modal approaches, the energy distribution analysis (EDA) [START_REF] Mace | Statistical energy analysis, energy distribution models and system modes[END_REF], for instance, expresses the energy influence coefficients [START_REF] Guyader | Energy transmission in finite coupled plates, Part I: theory[END_REF] of a built-up structure in terms of the modes of the whole structure. In the Asymptotical Scaled Modal Analysis (ASMA) method [START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF][START_REF] Rosa | On the use of the asymptotic scaled modal analysis for timeharmonic structural analysis and for the prediction of coupling loss factors for similar systems[END_REF][START_REF] Rosa | Structural similitudes for the dynamic response of plats and assemblies of plates[END_REF][START_REF] Rosa | A similitude for structural acoustic enclosures[END_REF], the physical size of the solution domain is reduced based on a scaling law, while the damping loss factor is artificially increased. This leaves a scaled model that can represent the mean response of the original system, with the advantage of only needing a reduced number of modes.

Another modal approach is the statistical modal energy distribution analysis (SmEdA) method [START_REF] Maxit | Extension of SEA model to subsystem with non-uniform modal energy distribution[END_REF][START_REF] Totaro | Extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems[END_REF][START_REF] Maxit | Non-resonant transmission modelling with statistical modal energy distribution analysis[END_REF], which establishes power balance equations between individual connected subsystem modes, rather than between the subsystems themselves. This circumvents the SEA requirements of modal energy equipartition and enables applying SmEdA to cases of low modal overlap, locally excited subsystems, and to deal with complex heterogeneous subsystems. SEA is then recovered as a limiting case from SmEdA [START_REF] Maxit | Extension of SEA model to subsystem with non-uniform modal energy distribution[END_REF].

The coupling between two subsystems in SmEdA and SEA takes place according to a set of conditions that the resonant modes in the subsystems must fulfill. We may term such conditions as the modal coupling assumptions (MCAs), to be detailed below. The MCAs permit characterizing the coupling between subsystems (groups of resonant modes) in SEA, or between individual modes in SmEdA, through coupling loss factor coefficients.

The latter strongly facilitate solving SEA and SmEdA systems, and resolving problems of practical industrial interest, like the computation of energy transmission paths [START_REF] Guasch | Finding the dominant energy transmission paths in statistical energy analysis[END_REF][START_REF] Guasch | A graph cut strategy for transmission path problems in statistical energy analysis[END_REF][START_REF] Aragonès | Ranking paths in statistical energy analysis models with nondeterministic loss factors[END_REF][START_REF] Aragonès | A graph theory approach to identify resonant and non-resonant transmission paths in statistical modal energy distribution analysis[END_REF].

There is no evidence, however, indicating that the MCAs should generalize to multiple subsystems connected at a junction. It is precisely the purpose of this work to make some first steps towards this goal, by considering the case of several flexural waveguides sharing a common joint. As far as the authors know, this problem has not been tackled before.

In SEA, the latter seems to be a theoretical problem rather than a practical one. This is so because there is a way out to the situation: one can resort to the wave approach to SEA instead of the modal one. Travelling waves are considered and one can deduce the expressions for the coupling loss factors at a junction [START_REF] Crandall | On the coupling loss factor in statistical energy analysis[END_REF][START_REF] Crocker | Sound transmission using Statistical Energy Analysis[END_REF][START_REF] Langley | The wave transmission coefficients and coupling loss factors of point connected structures[END_REF][START_REF] Díaz-Cereceda | Numerical estimation of coupling loss factors in building acoustics[END_REF] from wave transmission coefficients [START_REF] Pedersen | Estimation of vibration attenuation through junctions of building structures[END_REF][START_REF] Craven | Sound transmission and mode coupling at junctions of thin plates, part 1: representation of the problem[END_REF][START_REF] Gibbs | Mode coupling and energy partition of sound in a system of plate junctions[END_REF][START_REF] Langley | Elastic wave transmission coefficients and coupling loss factors for structural junctions between curved panels[END_REF]. Finite elements [START_REF] Steel | Statistical energy analysis of structure-borne sound transmission by finite element methods[END_REF][START_REF] Simmons | Structure-borne sound transmission through plate junctions and estimated of SEA coupling loss factors using the finite element method[END_REF], spectral methods [START_REF] Poblet-Puig | Using spectral finite elements for parametric analysis of the vibration reduction index of heavy junctions oriented to flanking transmissions and EN-12354 prediction method[END_REF] and hybrid finite element (FEM)-SEA approaches [START_REF] Cotoni | Numerical and experimental validation of the hybrid finite element-statistical energy analysis method[END_REF] can also be employed to compute them. In SmEdA, however, the problem is not only of theoretical but also of practical importance. This is so because in SmEdA the subsystem modes are required to compute the coupling loss factors, and there is no alternative akin to the wave approach in SEA.

To find a modal coupling scheme for multiple subsystems connected at a junction, it is herein suggested to resort to the dual modal formulation (DMF) [START_REF] Fahy | Vibration of containing structures by sound in the contained fluid[END_REF][START_REF] Fahy | Response of a cylinder to random sound in the contained fluid[END_REF][START_REF] Karnopp | Coupled vibratory-system analysis, using the dual formulation[END_REF][START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF]. When applied to a pair of subsystems, the DMF complies with the MCAs. This formulation has been known since long ago to properly describe the dynamic behavior of a flexible structure coupled to a closed acoustic domain [START_REF] Fahy | Vibration of containing structures by sound in the contained fluid[END_REF][START_REF] Fahy | Response of a cylinder to random sound in the contained fluid[END_REF]. The structure is described by the displacement field and the uncoupled-free modes (i.e. in-vacuo modes), whereas the cavity is characterized by the acoustic pressure field and the blocked modes (i.e. cavity modes assuming rigid walls).

By analogy between the acoustic waves in a fluid medium and the longitudinal waves in a rod, the DMF was extended in [START_REF] Karnopp | Coupled vibratory-system analysis, using the dual formulation[END_REF] to coupled rods, showing that the modal series converges even when the rods have equal properties. By the end of the nineties, the DMF was generalized to continuous mechanical systems [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF]. If there is a mechanical impedance mismatch at a junction (which can be related to the weak coupling assumption), the modal series can be truncated to the resonant modes, as demanded by the MCAs [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF]. The boundary conditions of the uncoupled-subsystems can then be clearly defined; the stiffest subsystem should be free at the junction and described by its displacements, while the soft one should be blocked and represented by its stresses.

In this work, the DMF is extended to multiple subsystems sharing a junction. Still, the method requires one subsystem to be clearly stiffer than the others, the stiffest subsystem being characterized by its free modes and the remaining ones by blocked modes. Numerical tests will be presented for different impedance mismatches between subsystems, to determine under which conditions the resonant modes can represent the global behavior of the system, and therefore fulfill with the MCAs.

The paper is organized as follows. The MCAs are detailed in section 2, together with a summary of the DMF applied to a pair of subsystems. As a by-product of the paper, we also present a new derivation of the DMF equations in the Appendix, stemming from Hamilton's principle and its complementary form [START_REF] Tabarrok | Complementary variational principles in elastodynamics[END_REF], rather than from Reissner's principle. Section 3 introduces a benchmark problem consisting of four panels sharing a junction, which will be used to facilitate discussion through the remaining of the paper. Some considerations on subsystem substructuring are also outlined, followed by the derivation of the DMF for the four panel test case. Numerical simulations and a discussion on the results are provided in section 4. The conclusions and future perspectives close the paper in section 5.

Basic theory for two coupled subsystems connected at a junction

Modal coupling assumptions

The modal coupling assumptions (MCAs) characterizing the interaction between modes, or groups of modes, belonging to two different subsystems, can be summarized as follows (see e.g., [START_REF] Lyon | Theory and application of statistical energy analysis[END_REF][START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF]):

-the interaction concerns the modes of the uncoupled subsystems, -the dynamic behavior of a subsystem mode can be associated to the dynamic behavior of an oscillator (mass-spring-damper system), -the coupling between the modes in different subsystems is conservative and takes place through mass, stiffness and/or gyroscopic elements, -modes within a subsystem are uncoupled (orthogonality of modes), -the resonant modes of each subsystem suffice to account for the energy interchanged with other subsystems.

Few works seem to exist analyzing the particular fulfilment of the MCAs in mechanical systems. Crandall and Lotz [START_REF] Maxit | Extension of SEA model to subsystem with non-uniform modal energy distribution[END_REF], for instance, established the modal equations for the specific case of two flexural beams coupled by a torsion spring. The subsystem modes were chosen as the modes of the uncoupled-free subsystems and displacements were used to describe the vibration of the two beams. However, the resulting modal coupling scheme was not in accordance with the MCAs; some terms arise in the formulation which express direct coupling between modes belonging to the same subsystem. Nonetheless, if the torsion stiffness of the spring is small in comparison with the beam flexural stiffness, it was shown that the inner mode coupling can be neglected leading to fulfillment of the MCAs. Scharton and Lyon had previously obtained similar results for two weakly coupled identical beams [START_REF] Scharton | Power flow and energy sharing in random vibration[END_REF].

Despite of the MCAs being at the very core of SEA and SmEdA, as said in the Introduction it is unclear whether they could also apply to the connection of more than two subsystems. Before extending the DMF to multiple subsystems to check that issue, let us first give a summary of the method when applied to the easier situation of two coupled subsystems.

The dual modal formulation basic equations

Consider two continuous elastic subsystems 1 and 2, coupled at a junction Γ, subsystem 1 being excited by a mechanical force. The DMF [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF] allows one to calculate the vibration response of the two coupled subsystems from the uncoupled subsystem modes and the reduced modal equations. In accordance with the formulation (see [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF] and appendix A), the stiffer subsystem is described by its displacement field and its uncoupled-free modes (i.e. assuming null stresses at the connection), whereas the softer one should be characterized by a stress field (e.g., acoustic pressure) and by their uncoupled-blocked modes (i.e. contemplating null displacements at the connection).

Suppose that subsystem 1 is the stiffest one and that p u represents the displacement vector of its p-th free mode when normalized to a unit modal mass. Besides, q σ stands for the stress tensor of the q-th blocked mode of subsystem 2, when normalized to unit modal stiffness (i.e. unit modal strain energy).

The displacement vector at a point M of subsystem 1,  

         Q q q q N t t N , σ σ  , (1)
where  

  Q Q ĉard  .
Following the DMF approach, the expressions in Eq. ( 1) are to be introduced in the weak formulation of the coupled problem. Taking advantage of the orthogonality of the uncoupled modes and making the change of variables

Q q q q ,      
, the modal equations of motion can be written as (see appendix A for a derivation),

11 22 T                                            1 1 1 22 I0q 0 W q Λ 0 q f 0Iq W 0 q 0 Λ q 0 , (2)
with

1 2 12 1 1 1 2 2 2 , , , , diag , diag .
p q p pq P Q P PQ

pq P P Q Q FW                                   1 1 q q f W ΛΛ p  , q
 denote the modal angular frequencies of modes p and q, and p F the modal forces due to the external excitation. 12 pq W stands for the intermodal work between modes p and q, which is given by

12 d pq p q W    u σn . (3)
n in Eq. (3) represents the normal vector pointing outwards to the boundary of subsystem 2. Note that viscous damping could be also easily introduced in the modal equations (2).

The system of equations ( 2) corresponds well to the MCAs. In addition to the external force and the damping loss factor, these models require knowing the modal angular frequencies and the mode shapes (in term of displacements or stresses) at the coupling junction. For time harmonic excitations, the modal amplitudes can be obtained solving the matrix system [START_REF] Ungar | Fundamentals of statistical energy analysis of vibrating systems[END_REF]. The displacement at point M, or the stress at N, can then be computed from Eq. ( 1). Moreover, the total energy (kinetic energy + strain potential energy) of each subsystem can be calculated from the summation of their modal energies. The reader is referred to [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF] for details on these developments, which will be not reproduced herein.

3 Extended theory for multiple subsystems connected at a junction

Description of the benchmark model and cases

For ease of exposition, the viability of the DMF approach when applied to multiple subsystems is analyzed for a specific structure composed of four flexural panels coupled at a junction, at right angle (see Fig. 1). Panels P1 and P2 in the figure represent the floor whereas panels P3 and P4 stand for vertical walls. The plates have clamped boundary conditions on the outer edges.

This example has been chosen because it is widely met in practical applications of SEA (e.g., in the naval and building industries). However, to keep the problem amenable for analytical description we have only considered the flexural motion of the panels.

Therefore, the coupling among the plates only takes place through rotations and moments at the junction. The coupling between the out of plane motions (flexural) and the in-plane motions (longitudinal/shear motions) has been discarded in the analysis.

Two typical cases of building structures are contemplated. In the first one (test case #1), the floor is 20 cm in thickness and made of concrete, while the walls are 5 cm in thickness and made of brickwork. Therefore, there is a clear impedance mismatch at the junction, the concrete floor being much stiffer than the brickwork walls. In the second situation (test case #2), both, the floor and the walls will be made of concrete with a thickness of 20 cm. Consequently, all individual panels will have the same impedance.

A normal force excitation with unit power spectral density,

  /Hz N 1 2  f S FF
, is applied on panel P1, at the 1 kHz central frequency octave band. The point of excitation is located at coordinates (0m, 1.6m, 0.87m). The damping loss factor of the floor and walls has been fixed to 0.02 in both configurations.

The DMF calculations to be performed in subsequent sections need the subsystem modes as inputs, with appropriate boundary conditions. These modes have been computed using the finite element method (FEM) and the SDTool code [START_REF] Balmes | Structural Dynamics Toolbox & FEM link, User's guide[END_REF] in MATLAB. The computational mesh for test case #1 has been defined with a criterion of six elements per flexural wavelength at 2 kHz. The global FEM mesh is composed of 52126 nodes and 51604 quadrilateral shell elements, with 134 nodes belonging to the coupling junction. The same mesh has been considered for the second test case, which presents stiffer walls.

Clamped boundary conditions have been imposed on the nodes of the outer edges.

Moreover, the three translations remain blocked for the 134 nodes belonging to the coupling edge because, as said, only flexural motion has been taken into account. The meshes of the subsystems have been built from the global mesh. Appropriate boundary conditions at the nodes of the junction have been imposed in agreement with the requirements of the DMF method. The subsystem modes are then extracted using the Implicitly Restarted Lanczos Method (eigs command in SDTool, [START_REF] Balmes | Structural Dynamics Toolbox & FEM link, User's guide[END_REF]). All subsequent DMF computations have been carried with MATLAB.

Sub-structuring and SEA weak coupling assumption

Once defined the test cases, the next step is that of partitioning the four-plate system into subsystems. This sub-structuring [START_REF]Langley -A general derivation of the statistical energy analysis equations for coupled dynamic systems[END_REF][START_REF] Diaz-Cereceda | Automatic subsystem identification in statistical energy analysis[END_REF][START_REF] Magrans | A subsystem identification method based on the path concept with coupling strength estimation[END_REF][START_REF] Fahy | An indicator of coupling strength between SEA subsystems[END_REF][START_REF] Fahy | A study of the kinetic energy impulse response as an indicator of the strength of coupling between SEA subsystems[END_REF] should comply with the weak coupling assumption of SEA. Different definitions for the latter can be found in literature, which have always been a matter of discussion. The following ones could be useful in the present context.

In [START_REF]Langley -A general derivation of the statistical energy analysis equations for coupled dynamic systems[END_REF], Langley proposed that "… the coupling will be said to be weak if the Green function for subsystem j is approximately equal to that of the uncoupled subsystem.", while a few years later, Fahy and James [START_REF] Fahy | An indicator of coupling strength between SEA subsystems[END_REF] asserted that "Under the conditions of weak coupling, the system modes are 'localized' in the sense that they closely resemble in natural frequency and shape the modes of the uncoupled subsystems (given the appropriate boundary conditions), and that their energies reside principally within the corresponding subsystems.". Moreover, the same authors specified that "Depending on the nature of the coupling, the boundary conditions for the uncoupled system do not always correspond to free displacement at the coupling" [START_REF] Fahy | A study of the kinetic energy impulse response as an indicator of the strength of coupling between SEA subsystems[END_REF].

The above considerations suggest that the localization of the global modes could give clear indications for partitioning a system into subsystems, as well as for defining the boundary conditions of the uncoupled subsystems. For example, in the EDA formalism ( [START_REF] Mace | Statistical energy analysis, energy distribution models and system modes[END_REF][START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF]), the 𝜓 𝑗𝑘 𝑟 parameter could be used as an indicator of localization of the global modes. This parameter corresponds to the integral on subsystem r of the product of mode shapes i and j multiplied by the mass density (see Eq. ( 3) in [START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF]). As global modes are considered in EDA, the 𝜓 𝑗𝑗 𝑟 parameter approaches to the generalized mass when the mode j is localized on subsystem r.

In the case of large models, localizing global modes for sub-structuring is not very practical because it implies computing all system modes. However, it provides a helpful guideline for our small four panel benchmark problem. For instance, in Fig. 2 Altogether, this indicates that the DMF is well adapted to describe the vibration behavior of this system. The floor, which is the stiffest component, should constitute a single subsystem described in terms of displacements and by its uncoupled-free modes (null stress at the coupling junction). The vertical walls should be two additional independent subsystems characterized by their stresses and uncoupled-blocked modes (null displacement at the coupling junction). Therefore, in what follows subsystem 12 will designate the floor (panels P1 plus P2), whereas subsystems 3 and 4 will respectively correspond to the vertical walls P3 and P4.

Dual modal equations for the four panels

The DMF outlined in section 2. As mentioned in section 3.1, the modes of each uncoupled subsystem have been calculated with FEM. Remember that to compact the equations, the mode shapes of subsystem 12 are taken normalized to unit modal kinetic energy and those of subsystems 3 and 4 normalized to unit modal strain energy. For subsystem 12, the three rotations of the 134 nodes belonging to the coupling junction have been left free whilst fixed to zero for subsystems 3 and 4.

Applying the DMF and the conditions of conservation of slope and bending moment at the junction, the modal equations for the 3 coupled subsystems can be deduced. These read, in matrix form, intermodal work between the p-th mode of subsystem  and the q-th mode of subsystem  ) and can be computed from the FEM nodal variables as (see Eq. ( 80) in [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF])

                                                                               0 0 F q q q Λ 0 0 0 Λ 0 0 0 Λ q q q 0 0 W 0 0 W W W 0 q q q I 0
    i i q i p pq M W     , , . (5)
Expanding one of the rows in Eq. ( 4) one gets a particular case of the canonical problem needed to comply with the MCAs [START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF],

             q q pq p p p p q W F q q    2 , (6)
which states that each mode p of a subsystem  behaves as a mass-spring oscillator (l.h.s of Eq. ( 6)) that is coupled to oscillators (modes q) in another subsystem    through gyroscopic elements (r.h.s of Eq. ( 6)). It remains yet to be checked whether the sole consideration of the subsystem resonant modes suffices to represent the global behavior of the system. In other words, is it possible to correctly estimate the subsystem energies if one only retains the resonant modes in the DMF Eq. ( 4)? To answer this question, DMF calculations will be performed in the following section with and without non-resonant modes and compared to a reference solution using the system global modes.

To compute the modal energy of the p-th mode of subsystem  ,   f E p  , we have introduced some viscous damping in Eq. ( 4) and applied a unit harmonic excitation at frequency f. The vector of the modal energies of subsystem  ,

The mean modal energy,  p E , in response to a white noise force of unit power spectral density in the octave band of central frequency c f , is obtained by integrating (using a simple quadrature rule),

          2 1 2 1 f f p f f FF p p df f E df f S f E E    , ( 8
)
where 1

f and 2 f are the lower and upper bound frequencies of the considered octave band.

On the other hand, the energy of subsystem  at frequency f can be computed adding the energies of the modes in the subsystem,

    f E f E p p    
. Finally, the subsystem energy for a white noise force in the octave is computed as

    2 1 f f df f E E   , which should equal  p p E  .

4

Numerical simulations and discussion

Results

The total energies of subsystems 12 and 3 have been first calculated, according to the DMF, for the test case #1 of section 3.1, and plotted versus frequency in Fig. 3. This figure also contains the reference energies obtained from the global modes of the system, as described in section 2.2 of [START_REF] Mace | Energy flow models from finite element analysis[END_REF]. The calculations have been performed for the 1 kHz octave band considering 200 modes per subsystem. This frequency band has been initially chosen since it guarantees, at least, 10 resonant modes per band for each subsystem, as required by SEA [START_REF] Lyon | Theory and application of statistical energy analysis[END_REF][START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF].

A very good agreement can be observed between the different curves, even if only resonant subsystem modes are included in the DMF calculations. Some slight differences can also be appreciated at the bounds of the frequency band, which could be expected because of the stronger influence of the non-resonant modes at these frequencies. Note that the frequency matching between subsystem modes seems to have a similar effect than the one between global modes in EDA. Indeed, in the EDA formalism of [START_REF] Mace | Statistical energy analysis, energy distribution models and system modes[END_REF][START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF], the frequency cross-modal terms Γ 𝑗𝑘 are small unless modes j and k are resonant.

For better understanding the influence of neglecting the non-resonant modes, in Fig. 4 we have plotted the modal energies of each subsystem with and without considering them. The modal energies of subsystem 12 are not affected by the non-resonant modes (Fig. 4a), which is logical because subsystem 12 is directly excited with the external force.

As opposed, non-resonant modes somewhat influence the less energetic modes of subsystems 3 and 4 (Figs. 4b andc), though the higher ones are well approximated only with the resonant modes. It can be therefore concluded that resonant coupling suffices for a correct prediction of the subsystem energies at the 1 kHz octave band.

One could wonder whether the above results would remain valid for other Next, let us focus on test case #2 described in section 3.1. The same type of computations than for test #1 have been carried out. The results are summarized in Table 1, where the overall subsystem energies are presented. As observed, significant discrepancies with the global modes reference solution are found for subsystems 3 and 4, even if both, resonant and non-resonant modes are included in the computations. This means that the 200 -resonant or non-resonant -modes are not sufficient to ensure the convergence of the modal expansion and a larger number of them should be needed to achieve a better result. In fact, this is not unexpected since the considered subsystem modes do not resemble those of the system global modes as in test case #1.

To better understand this point we may compare the subsystem modal parameters considered in the DMF with those of the global modes. These parameters are the modal

frequencies p f ( 2 pp f  
) in Eq. ( 6)) and the modal angular rotations at the coupling junction, 12 , , pi i  , in Eq. ( 5). Fig. 6, shows the values of the maximum angular rotation The maximum angular rotations of subsystem 12 are systematically greater than those of the global system, whilst the modal frequencies become lower. Consequently, it is apparent that for the second test case the DMF seems not to be the appropriate approach to get a modal coupling scheme according to the MCAs.

Discussion

The basic reason for above behavior and failure of the DMF in case #2 is that the floor and the panels have the same thicknesses and material properties, and thus exhibit the same dynamic stiffness. Hence, a more pertinent partitioning scheme for #2 compliant with the DMF is needed.

One option could be that in Fig. 7a. It consists in choosing, for instance, panel P1 as a subsystem and then grouping all remaining panels into another subsystem. In such situation, there would be a significant mechanical impedance mismatch between the two subsystems and the DMF could be applied. Panel P1 (subsystem 1) could be well characterized by blocked modes, whereas the three others panels (subsystem 2) could be represented by free modes. However, this alternative sub-structuring would not be very feasible for complex built-up structures in practice, so it will be not pursued hereafter. A more manageable sub-structuring alternative for case #2 could rely on the weak coupling definition by Fahy and James [START_REF] Fahy | An indicator of coupling strength between SEA subsystems[END_REF][START_REF] Fahy | A study of the kinetic energy impulse response as an indicator of the strength of coupling between SEA subsystems[END_REF], already cited in section 3.1. Given that all panels have the same dynamic stiffness, each one perceives the other three at the junction.

One may therefore expect each panel to be blocked at the junction by the other three. The global modes could be more localized in that picture and resemble those of uncoupled panels with clamped boundary conditions at the junction. This would result in the substructuring scheme depicted in Fig. 7b, in which each panel is identified with a subsystem.

Note that this scheme corresponds to the partitioning generally assumed in the traveling wave approach to SEA [START_REF] Crandall | On the coupling loss factor in statistical energy analysis[END_REF][START_REF] Crocker | Sound transmission using Statistical Energy Analysis[END_REF][START_REF] Langley | The wave transmission coefficients and coupling loss factors of point connected structures[END_REF][START_REF] Díaz-Cereceda | Numerical estimation of coupling loss factors in building acoustics[END_REF].

However, an important question remains, namely, which could be the modal formulation amenable with the partition scheme in Fig. 7b that also satisfies the MCAs?

The DMF must be discarded because only one subsystem could be described by its uncoupled blocked modes, the other ones having to be characterized by uncoupled free modes. An appealing alternative would be to resort to the well-known Craig-Bampton (CB) method [START_REF] Craig | Substructure coupling for dynamic analysis and testing[END_REF][START_REF] Pierre | Mid-frequency dynamics of complex structural system: assessing the state of the art and defining future research directions[END_REF]. Given that the normal modes in the CB are defined with fixed-interfaces, this popular approach may be well adapted to our objective. Checking whether the CB, or some variation of it, is suitable to characterize multiple subsystems connected at a junction, with low impedance matching and according to the MCAs, will be the topic of companion paper [START_REF] Maxit | Energy-based reformulated Craig-Bampton method for multiple flexural subsystems connected at a junction with low impedance mismatch[END_REF].

Conclusions

In this paper we have studied the possibility of obtaining a modal coupling scheme for multiple flexural waveguides sharing a junction, in accordance with the required modal coupling assumptions (MCAs) of some energy-based methods, like SEA and SmEdA. In particular, the dual modal formulation (DMF) has been extended to deal with multiple connected subsystems. It has been shown that the latter can satisfy the MCAs, the modes of the different subsystems being coupled through gyroscopic elements. However, the DMF suffers from one drawback: it only works well if one of the subsystems is clearly stiffer than the others. Therefore, the approach fails when applied to subsystems with similar dynamic stiffness. This has motivated to address the problem of connected subsystems with low impedance mismatch in a companion paper [START_REF] Maxit | Energy-based reformulated Craig-Bampton method for multiple flexural subsystems connected at a junction with low impedance mismatch[END_REF].

On the other hand, we shall notice that only flexural waves have been considered in this work. Future developments may address the influence of longitudinal and shear waves on the modal coupling. Also, it would be interesting to derive coupling loss factors in SmEdA and SEA from the herein reported results.

Appendix A. Dual Modal Formulation from Hamilton's principle and its complementary form

A derivation of Eq. ( 2) in the main text steming from variational principles will be next presented. The reader is refered to [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF] for a more complete description of all involved concepts and equations regarding the DMF. The exposition will be based on Hamilton's principle and its complementary form [START_REF] Tabarrok | Complementary variational principles in elastodynamics[END_REF], while it was based on two forms of the Reissner principle in [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF].

Let us consider two elastic continuous mechanic systems rigidly coupled at the surface  (see Fig. , respectively denote the boundary surfaces with free and blocked displacements. It is assumed that each subsystem is made of an elastic, homogeneous material.

In the following, subsystem 1 is described by its displacement field and subsystem 2 by its stress field. In Lagrangian coordinates   The variational formulation of the elastodynamics of each subsystem can be introduced as follows. Consider the free vibration of the elastic conservative subsystems between time instants 1 t and 2 t . For subsystem 1, described in terms of displacements, the Hamilton functional

H  [43] reads,                        2 1 1 1 * t t i j ij V k H d u n dV U T u  , (A.1)
where

kl ijkl ij C U   2 1 
is the strain energy, with

  i j j i ij u u , , 2 1  



, and



(see [START_REF] Tabarrok | Complementary variational principles in elastodynamics[END_REF]), so that

                       2 1 2 2 * t t j ij i V ij T d n u dV T U    . (A.2) kl ijkl ij S U     2 1 * 
in Eq. (A.2) is the complementary strain energy and

dV u M V p i p   1 2 , 1  , dV C K V p kl ijkl p ij p   1 , ,   , p p p M K   . (A.3)
Analogously, the modal mass (or modal strain energy) q M , the modal stiffness (or modal kinetic energy) q K , and the modal angular frequency q  of mode q in subsystem 2, read

dV S M V q kl ijkl q ij q   2 , ,   , dV K V q j ij q   2 2 2 , ,   , q q p M K   . (A.4)
The displacement field of subsystem 1 and the stress field of subsystem 2 can then be decomposed on their respective subsystem modal basis as,

         P p p i p i M u t t M u ˆ, ,  ,          Q q q ij q ij N t t N ˆ, ,    , (A.5)
where p  and q  represent the modal amplitudes, and P ˆ, Q ˆ stand for the sets of modes of subsystems 1 and 2 retained in the modal expansion. Given that

ij ij    
, we can also decompose the impulse field as

         Q q q ij q ij N t t N ˆ, ,    with Q q q q ,       . (A.6)
Introducing the modal expansions Eqs. (A.5) and (A.6) in the functionals Eqs. (A.1) and (A.2), and making use of the orthogonality of modes (see [START_REF] Tabarrok | Complementary variational principles in elastodynamics[END_REF]), provides the modal amplitudes that render each functional stationary. This results in the Euler-Lagrange equations, which yield

                       . , 0 , , 0 2 , , 1 Q q S d n u K M P p d u n K M j q ij i q q q q p i j ij p p p p            (A.7)
To re-assemble the two subsystems, we impose displacement continuity and force equilibrium at the coupling surface  . This allows one to rewrite the prescribed displacements, l u  , and stresses, ij  , on  as,

    t M u t M u i l , ,    ,   M , (A.8)     2 1 , , j ij j ij n t M n t M     ,   M . (A.9)
From the modal expansions in Eq. (A.5) it follows

      M u t t M u P p p i p l    ˆ, ,    ,   M , (A.10)           2 ˆ, 2 ˆ, 1 , j Q q q ij q j Q q q ij q i ij n M t n M t n t M               ,   M . ( A
q q q q pq p pP M K W p P M K W q Q                            (A.12) with 12 2 ,, pq i p ij q j W u n d    
standing for the intermodal work between mode p in subsystem 1 and mode q in subsystem 2. Eq. ( 2) in the main text is nothing but Eq. (A.12) written in matrix form and normalized to unit modal mass.

  modal amplitudes. P ˆ and Q ˆ stand for the sets of modes of subsystems 1 and 2 retained in the modal expansion. The number of modes in each set are denoted by

  Figs.2g and 2h.

M

  2 is next extended to the four-plate benchmark structure considering the subsystem partitioning presented above. Hereafter the following notation will apply: l  designates the set of nodes belonging to the coupling junction, -  p stands for the angular frequency of mode p of subsystem , respectively symbolize the angular rotation and reaction moment at node l i   of mode p of subsystem  .

qW

  stand for their components, namely the p-th mode of the  are the matrices of intermodal works between subsystems 12 and 3, and between subsystems 12 and 4. Their components are designated by  pq W (i.e.,

  Figs. 3, 4 and 5 that the results from the resonant DMF are quite satisfactory. Therefore, under the premises of test case #1, the DMF offers a modal coupling scheme in agreement with the MCAs.

  A.1). 1 V and 2V represent the volumes occupied by subsystems 1 and 2, and the unit vectors 1 n and 2 n represent the outer normal to these volumes.

  . Here a comma denotes differentiation with respect to the index space coordinate. The dynamic problem related to subsystem 1 consists in finding the displacements i u satisfying the displacement boundary conditions, which render the functional H  stationary.The variational approach for subsystem 2, described in terms of stresses, involves the complementary form of Hamilton's functional,



  in the absence of volume forces. Solving the elastodynamics of subsystem 2 consists in finding the impulses ij  that satisfy the force boundary conditions and render the functional T  stationary. Next, let us consider the normal modes of the uncoupled-free subsystem 1 (i.e. with null stresses on Γ) and the normal modes of the uncoupled-blocked subsystem 2 (i.e. with null displacements on Γ). Denote by p i u , the displacement shapes (i=1,2,3) of mode p of the uncoupled-free subsystem 1, and by q ij,  the stress shapes (i,j=1,2,3) of mode q of the uncoupled-blocked subsystem 2. The modal mass p M , the modal stiffness p K , and the modal angular frequency p  of mode p in subsystem 1, are given by

Figure 1 .

 1 Figure 1. Geometry of the benchmark structure.

Figure 2 .

 2 Figure 2. Comparison of the spatial shapes of the system global modes (left column) and the subsystem modes computed with the DMF (right column) for the first test case. (a-d) examples of global modes; (e-f) blocked modes of P4: (g-h) free modes of P1+P2.

Figure 3 .

 3 Figure 3. Subsystem energy response depending on frequency for subsystem 12 (black) and subsystem 3 (grey), for test case #1. Continuous line: reference energy from global modes; Dashed line: DMF computation considering resonant and non-resonant modes; Dotted line: DMF computation with only resonant modes.

Figure 4 .Figure 5 .

 45 Figure 4. Modal energy distribution for test case #1. Cross: DMF taking into account resonant and non-resonant modes; Circle: DMF only considering resonant modes; (a) subsystem 12; (b) subsystem 3; (c), subsystem 4. Vertical dash lines indicate the bandwidth of the 1 kHz octave band;

Figure 6 .

 6 Figure 6. Maximum modal angular rotations at the coupling edge max p  versus modal frequency p f . Cross: modes of the free subsystem 12 (like in DMF); Square: global modes of case #1; Circle: global modes of case #2.

Figure 7 .

 7 Figure 7. Partitioning and boundary conditions for the second test: (a) Alternative DMF substructuring; (b) CB substructuring. (Section view of the panels).

Figure A. 1 .

 1 Figure A.1. (a) Rigid coupling of two continuous subsustems; (b) Fictive separation of the two subsystems with the associated prescribed condition on the coupling surface Γ.

Figure 1 .Figure 2 . 29 Figure 3 .Figure 4 .Figure 5 .

 1229345 Figure 1. Geometry of the benchmark structure.

Figure 6 .Figure 7 .Figure A. 1 .

 671 Figure 6. Maximum modal angular rotations at the coupling edge max p  versus modal

  One can observe how the values for subsystem 12 resemble those of the global modes for test case #1, which explains why convergence was achieved in that occasion. As opposed, the results for subsystem 12 clearly differ from the global ones for test case #2.

	max p  max i  	, p i	as a function of the modal frequencies	p f , up to 450 Hz. To facilitate
	visualization, only the free modes presenting a significant deformation of plate 1 have been
	included.

Table 1 :

 1 Comparison of subsystem energies for test cases #1 and #2. Role of resonant and non-resonant modes. Results correspond to the 1 kHz octave band (dB, ref. 10 -12 J).

	Test	Panel	1+2	3	4
	case	Energy
	#1 Reference	68.4 dB	44.7 dB	46.4 dB
	#1 DMF	68.4 dB	44.2 dB	45.9 dB
	#1 DMF (resonant modes only) 68.2 dB	44.0 dB	45.7 dB
	#2 Reference	67.9 dB	56.6 dB	57.2 dB
	#2 DMF	67.9 dB	37.7 dB	36.0 dB
	#2 DMF (resonant modes only) 67.8 dB	34.8 dB	34.3 dB

TABLE CAPTIONS Table 1 :

 CAPTIONS1 Comparison of subsystem energies for test cases #1 and #2. Role of resonant and non-resonant modes. Results correspond to the 1 kHz octave band (dB, ref. 10 -12 J).


Laurent Maxit
email: laurent.maxit@insa-lyon.fr

Oriol Guasch
email: oriol.guasch@salle.url.edu

Energy-based reformulated Craig-Bampton method for multiple flexural subsystems connected at a junction with low impedance mismatch

Keywords: Statistical energy analysis, modal formulation, component modal synthesis, power flow, mid-frequency modelling, plate assembly, Craig-Bampton method

Whereas the coupling between modes of two different subsystems is well-resolved in vibroacoustic energy-based methods, the situation becomes more intricate when several subsystems get connected at a common junction. In statistical energy analysis (SEA), the modal formulation is replaced by the travelling wave approach to solve the problem. However, this is not a viable option for other energy-based methods, like the statistical modal energy distribution analysis (SmEdA), and a modal coupling scheme is required for them. If there is a strong impedance mismatch between the multiple connected subsystems, the displacement-stress dual formulation offers a proper way to address the situation. Yet, the latter fails if all involved subsystems have similar stiffness. In this work, the feasibility of the Craig-Bampton (CB) method to address such circumstance is explored. It is shown that the original CB technique does not fulfill the modal coupling assumptions of energybased methods, so it is suggested to reformulate it to partially mitigate the problem.

Numerical tests on a benchmark problem are carried out to validate the proposal. The benchmark consists of a floor coupled with two walls at right angle, and it is analyzed for different impedance mismatch conditions.

Nomenclature

The symbols used in the main text are described in this section.

M

global coupling mass matrix between the resonant (R) normal modes and the non-resonant (NR) normal modes, plus the constraint interface (I) modes (NR+I modes) CC q amplitude vector of the characteristic constraint mode C g q amplitude vectors related to the coupling modes

g CC  q
global amplitude vector related to the subsystem normal modes and the characteristic constraint mode It is well-known that statistical energy analysis (SEA) admits two distinct formulations [START_REF] Lyon | Theory and application of statistical energy analysis[END_REF][START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF][START_REF] Crandall | On the coupling loss factor in statistical energy analysis[END_REF]. The first one arises from statistical considerations on how normal modes in a built-up structure interchange energy and is referred to as the modal approach to SEA. The second one grounds on statistical developments of disordered travelling waves propagating in a structure and is named the wave approach to SEA. The advantage of such a twofold formulation is that when difficulties are encountered when applying the modal one, those can be often overcome by switching to the wave one, and vice versa.

An example of such a type concerns the connection of several subsystems at a junction. In SEA a subsystem is identified with a group of resonant modes. Under certain conditions, which we denote as the modal coupling assumptions (MCAs) (to be specified below), the modal approach establishes the power balance equations between two distinct subsystems. However, it is not clear at all whether the MCAs generalize to multiple subsystems connected at a junction. SEA circumvents the problem by resorting to the wave approach to characterize that situation (see e.g., [START_REF] Crocker | Sound transmission using Statistical Energy Analysis[END_REF][START_REF] Langley | The wave transmission coefficients and coupling loss factors of point connected structures[END_REF][START_REF] Díaz-Cereceda | Numerical estimation of coupling loss factors in building acoustics[END_REF]). Yet in a recent work [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF], it was shown that it was also possible to establish a modal scheme satisfying the MCAs, whenever there was a strong impedance mismatch between the subsystems sharing the joint. The dual modal stress-displacement formulation was used for that purpose [START_REF] Fahy | Vibration of containing structures by sound in the contained fluid[END_REF][START_REF] Fahy | Response of a cylinder to random sound in the contained fluid[END_REF][START_REF] Karnopp | Coupled vibratory-system analysis, using the dual formulation[END_REF][START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information. Part I: Theory[END_REF][START_REF] Maxit | Reformulation and extension of SEA by taking modal energy distribution into account[END_REF]. The current work is to be viewed as a follow-up of [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF], where we now address the case of all connected subsystems having similar stiffness i.e, when there is a low impedance mismatch at the joint.

One might argue that such a problem is only of theoretical interest to SEA, given that in practice one could always avoid it by switching to the wave formulation. Even if that was the case, which is still believe interesting per se, there are other energy-based methods where there is a need for a multiple connection scheme that complies with the MCAs. The strong hypotheses underlying SEA ([1,2]) limit the theory to the high frequency range, so many methods have been proposed to extend its applicability to lower frequencies and face the so-called mid frequency problem. On the one hand, the wave approach to SEA can be derived from more general ray-tracing techniques under the assumption of diffusivity (see e.g., [START_REF] Bot | Derivation of statistical energy analysis from radiative exchanges[END_REF][START_REF] Tanner | Dynamical energy analysis-Determining wave energy distributions in vibro-acoustical structures in the high-frequency regime[END_REF]). On the other hand, in what concerns modal extensions to SEA, several methods have been proposed. For instance, in the energy distribution analysis (EDA) [START_REF] Mace | Statistical energy analysis, energy distribution models and system modes[END_REF][START_REF] Guyader | Energy transmission in finite coupled plates, Part I: theory[END_REF] energy transmission is characterized by means of energy influence coefficients, which are related to structural global modes. In the Asymptotical Scaled Modal Analysis (ASMA) method [START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF][START_REF] Rosa | On the use of the asymptotic scaled modal analysis for timeharmonic structural analysis and for the prediction of coupling loss factors for similar systems[END_REF][START_REF] Rosa | Structural similitudes for the dynamic response of plats and assemblies of plates[END_REF][START_REF] Rosa | A similitude for structural acoustic enclosures[END_REF] a scaling procedure is established to work with a reduced model that represents the average behavior of the original system. Lastly, the statistical modal energy distribution analysis (SmEdA) method [START_REF] Maxit | Extension of SEA model to subsystem with non-uniform modal energy distribution[END_REF][START_REF] Totaro | Extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems[END_REF][START_REF] Maxit | Non-resonant transmission modelling with statistical modal energy distribution analysis[END_REF][START_REF] Aragonès | A graph theory approach to identify resonant and non-resonant transmission paths in statistical modal energy distribution analysis[END_REF] proposes power balance equations between individual resonant modes belonging to different subsystems, rather than between subsystems themselves, as in SEA. In fact, the latter can be obtained as a limit case of SmEdA. It is to be noted that SmEdA fulfils the MCAs, like SEA, but has no wave approach analogue, so a modal coupling scheme for multiple connected subsystems it is not only an issue of academic interest for SmEdA, but a subject of very practical interest.

As said before, it was concluded in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF] that the DMF works well when there is a strong impedance mismatch between subsystems at the junction, but it is not a valid modal approach for low impedance mismatch values. In such situation each subsystem perceives the feedback of the other ones at the junction and may be well approximated as having clamped boundary conditions there. It was consequently suggested in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF] that one could resort to the celebrated Craig-Bampton (CB) formulation [START_REF] Craig | Substructure coupling for dynamic analysis and testing[END_REF][START_REF] Pierre | Mid-frequency dynamics of complex structural system: assessing the state of the art and defining future research directions[END_REF] instead of the DMF. In the CB method, the fixed-interface component modes (i.e., blocked modes) are computed for each subsystem and complemented with the so-called constraint modes. Thereby, the fixed-interface component modes could be -a priori -well adapted to represent the global behavior of coupled subsystems.

The organization of the paper is as follows. Section 2 reviews the MCAs and give a brief outline of the CB formulation. Section 3 presents a benchmark problem consisting of a floor and two walls at right angle, which was used to illustrate the developments in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF].

We resort to it once more to allow comparison with the formulations in this work. The modal equations for the benchmark problem are derived in section 4 for the original CB method, followed by numerical simulations and a discussion. It is shown that, unfortunately, the standard CB does not comply with the MCAs, so an alternative CB formulation is proposed and tested in section 5, to improve the results. Finally, conclusions are drawn in section 6.

2 Theoretical framework

Modal coupling assumptions

To make the present paper self-contained, let us reproduce the set of modal coupling assumptions (MCAs) listed in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF] that, for energy-based methods, describe the interaction between modes, or groups of modes, belonging to two different subsystems. The MCAs, state that (see also [START_REF] Lyon | Theory and application of statistical energy analysis[END_REF][START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF]):

-the interaction only involves the modes of the uncoupled subsystems, -the dynamics of a subsystem mode are those of an oscillator (mass-springdamper system),

-the coupling between modes in different subsystems is conservative and happens through mass, stiffness and/or gyroscopic elements, -modes within a subsystem are uncoupled (i.e., the modes conform an orthogonal set),

-the resonant modes of each subsystem account for all energy interchanged with other subsystems.

Fundamentals of the Craig-Bampton method

To facilitate the comprehension of the forthcoming sections, let us next present an outline of the most meaningful results of the original Craig-Bampton method [START_REF] Craig | Substructure coupling for dynamic analysis and testing[END_REF][START_REF] Pierre | Mid-frequency dynamics of complex structural system: assessing the state of the art and defining future research directions[END_REF]. To begin with, consider a nonspecific finite element (FEM) model of a single subsystem, its equation of motion being given by

f Ku u M     , (1)
where u stands for the nodal displacement vector and f for the external applied force vector. As usual, a dot symbol designates time derivative. M stands for the FEM mass matrix and K for the stiffness one.

Let us partition the displacements and applied forces into

       B I u u u ,        B I f f f , ( 2
)
where the vector The mass and stiffness matrices in Eq. ( 1) are partitioned accordingly as

       BB BI IB II m m m m M and        BB BI IB II k k k k K . ( 3
)
The Craig-Bampton method allows one to rewrite Eq. ( 1) with the splitting in Eqs.

(

    II i II II i II N i ,.., 2 , 1 , 0     φ m k  . ( 4
)
II N is the total number of modes retained to represent the subsystem. The eigenvectors i II φ can be ranged as the columns of a matrix

  II N II II II II φ φ φ φ ... 2 1



to finally build the normal mode matrix

        0 II φ . ( 5
)

(b) The constraint modes, ψ :

A constraint mode is defined as the static deflection induced in the subsystem when applying a unit displacement at one boundary d.o.f while keeping all remaining boundary d.o.f.s fixed. The constraint mode matrix is given by (see [START_REF] Craig | Substructure coupling for dynamic analysis and testing[END_REF][START_REF] Pierre | Mid-frequency dynamics of complex structural system: assessing the state of the art and defining future research directions[END_REF])

                    BB IB II BB IB I k k I ψ ψ 1 . ( 6
)
The constraint modes provide a complete set of deformation shapes associated with the motion of the boundary d.o.f.s.

The two type of modes defined by Eqs.

(5-6) can be concatenated into a matrix   ψ B   that can be used to transform the displacement vector u to modal coordinates

  T   q q q  , Bq u  . ( 7
)
Introducing Eq. ( 7) in Eq. ( 1) and projecting the equations on the basis B , provides

                               B I T IB I II T T f 0 f ψ f φ q k 0 0 Λ q m m m m           , (8)
with i II II T i II φ m φ m   , IB II T IB T BI T IB IB BI BB ψ m ψ m ψ ψ m m m      , II II T IB II BI φ m ψ φ m m    , IB T IB BB k ψ k k    ,   II II N N i II II T II      diag φ k φ Λ
.

Benchmark problem and sub-structuring

The developments concerning the CB approaches in the forthcoming sections will be applied to the same benchmark problem in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF]. This will ease comparison with the results reported in that previous work. The benchmark is depicted in Figure 1 and consists of a structure where four panels share a junction at right angle. The floor is made of panels P1 and P2 while panels P3 and P4 constitute vertical walls. Only flexural motions of the panels are considered to facilitate the analytical developments.

Two test cases corresponding to strong and low impedance mismatches at the junction will be contemplated. In test case #1 the floor is made of concrete and it is 20 cm in thickness, the walls being brickwork constructed and 5 cm in thickness. It was shown in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF] that in such situation the structure could be divided into three subsystems: a single subsystem for the floor (panels P1 + P2) and two more subsystems for each wall (see Figure 2a). The DMF performed well in that case and provided a modal coupling scheme in accordance with the MCAs. This was because there is a strong impedance mismatch between the floor and the walls. However, in a second test case #2, all panels were assumed to be made of concrete and of 20 cm in thickness. The DMF failed in that occasion, and it was argued in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF] that a sub-structuring scheme in which every panel was identified with a subsystem, having clamped conditions at the junction, would be more appropriate for test case #2 (see Figure 2b). This is so because all panels having the same stiffness, their motion gets blocked at the junction by the three remaining ones. Our goal in the forthcoming section will be to check whether, as suggested in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF], the CB method could perform better than the DMF for test case #2, and see if it also works well for test case #1.

The modes of the four panels needed for the CB developments have been obtained from a FEM model and the SDTool code in MATLAB [START_REF] Balmes | Structural Dynamics Toolbox & FEM link, User's guide[END_REF]. The whole benchmark structure has been meshed with 51604 quadrilateral shell elements (52126 nodes). This mesh has been used for the two test cases and fulfils the criterion of six elements per flexural wavelength at 2 kHz for test case #1. Given that results will be presented for the 1kHz octave band, the mesh is fine enough to get negligible numerical errors in the two test cases. 134 nodes have been placed at the structure junction with blocked translations in the three directions (only rotation is allowed as we are only considering flexural motion).

Besides, clamped boundary conditions have been imposed on the nodes of the outer edges.

The meshes of the subsystems have been directly extracted from the global mesh of the structure and the subsystem modes obtained from the Implicitly Restarted Lanczos Method (eigs command in SDTool, [START_REF] Balmes | Structural Dynamics Toolbox & FEM link, User's guide[END_REF]). In what concerns external excitation, a normal force with unit power spectral density (

  /Hz N 1 2  f S FF
) has been applied on panel P1, at the 1 kHz central frequency octave band. The excitation point has coordinates (0m, 1.6m, 0.87m). Besides, the damping loss factor of all panels has been set to 0.02   for the two test cases.

4 Craig-Bampton method for multiple subsystems coupled at a junction

Modal equations for four subsystems coupled at a junction

The CB developments in section 2.2 can be extended to the four-panel benchmark structure in Figure 1. To assemble the subsystem matrices, displacement continuity and force equilibrium have been imposed at the junction using the Boolean assembly matrix [START_REF] Craig | Substructure coupling for dynamic analysis and testing[END_REF][START_REF] Pierre | Mid-frequency dynamics of complex structural system: assessing the state of the art and defining future research directions[END_REF].

Given that the external excitation is only applied to an internal d.o.f. of subsystem 1, the four-panel CB modal equations become

g g g g g f q K q M     , (9)
with , ,

1 1 1 1 4 1 4 3 2 1                                    I T IB I T II g g f ψ 0 0 0 f φ f q q q q q q     
The number of constraint modes in Eq. ( 9) can be substantially diminished if one solves the generalized eigenproblem associated to its mass and stiffness matrices, (see [START_REF] Craig | Substructure coupling for dynamic analysis and testing[END_REF][START_REF] Pierre | Mid-frequency dynamics of complex structural system: assessing the state of the art and defining future research directions[END_REF]),

1 4 1 4 0 ii CC        mk ξ . ( 10
)
The resulting eigenvectors  . This matrix can be used to transform from the constraint mode amplitudes

4 1   q to the CC ones CC q , CC q q   4 1  . ( 11
)
The stiffness matrix

4 1  k diagonalizes to diag cc cc i CC cc NN      Λ
and the modal equations (9) can then be rewritten as , , ,

g CC g CC g CC g CC g CC        M q K q f , (12) with .
T II I g CC g CC g CC T T T T T T T T T T cc CC IB I                                                                        m 0 0 0 m q φf 0 m 0 0 m q 0 q f M 0 0 m 0 m q 0 0 0 0 m m q 0 m m m m I q ψf 1 2 3 4 . g CC CC                       Λ 0 0 0 0 0 Λ 0 0 0 K 0 0 Λ 0 0 0 0 0 Λ 0 0 0 0 0 Λ
The algebraic matrix system [START_REF] Maxit | Reformulation and extension of SEA by taking modal energy distribution into account[END_REF] expresses the coupling of the subsystem blocked modes to the CC ones through mass matrix elements. Expanding the system in rows allows one to recover the standard canonical problem at the basis of some energy-based methods [START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF],

      , 4 1 , 1                          q CCq pq p I T II p p p pp q q q m f φ m (13a)     . 1 1             q q pq T T p I T IB T CCp i CC CCp q q q     m f ψ (13b)
Eq. (13a) states that any normal mode of subsystem  , say the p-th mode  p q , behaves as a mass-spring oscillator (l.h.s of Eq. (13a)) that is mass-coupled to a q-th characteristic constraint CC mode (r.h.s of Eq. (13a)) but not to the normal modes of other subsystems.

Conversely, Eq. (13b) shows that any CC mode also corresponds to a mass-spring oscillator that exhibits inertial coupling with the normal modes of the four panel subsystems,

4 1   .
In comparison with the DMF approach to the problem in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF], a fifth subsystem appears in the BC formulation. It is composed by the CC modes which are not localized on one panel (see Figure 3, and in particular Figure 3c), but it rather represents the global part of the dynamic behavior of the coupled plates. This subsystem is also directly excited by the external force term 12) and (13b).

In addition to being expressed in the canonical form [START_REF] Bot | Derivation of statistical energy analysis from radiative exchanges[END_REF], to check if the above equations fulfill with the MCAs it should be verified whether resonant modes could account for the whole energy of the system at a given frequency band. This will be tested for the cases #1 and #2 described in section 3. Prior to that, however, we should depict how to compute the subsystem energies from the amplitudes of the normal modes,   q , and CC modes, .

Computation of subsystem energies

The energy of a subsystem is given by the sum of the kinetic and potential energies of the modes in the subsystem. These quantities are averaged in time though this will be not explicitly indicated hereafter.

Assuming a unit harmonic external excitation, t j e  , and introducing homogeneous structural damping into the system, the modal governing equations ( 12) transform to

          CC g CC g CC g CC g j         f q K M ) 1 ( 2 . ( 14
)
From Eq. ( 14) and the matrix definitions in Eqs. ( 11) and ( 12), the kinetic energy of subsystem α can be computed as,

                 1 k k k CC T H CC k E E E E                         q q m m m m q q , ( 15
)
with

   IB II T IB H ψ m ψ m    and   II II T IB φ m ψ m   
. The three summands in the second equality of (15) are given by, -

            q m q H k E 2 , 4 1  : kinetic energy from the normal modes, -   CC H CC k E q m q       2 , 4 1  : kinetic energy associated to the CC modes, -     CC H k E q m q         Re 2 1 2 , 
: kinetic energy arising from the cross interaction between the normal and CC modes.

Notice that

    
, k E can be either positive or negative. This term is a consequence of the fact that normal and CC modes are not orthogonal with respect to the subsystem mass matrix.

CC q

On the other hand, the time-averaged potential energies of a subsystem can be obtained from

                      , , 4 1 p p CC H CC p E E E                        q q k 0 0 Λ q q , (16)
with

   IB II T IB H ψ k ψ k   
. The two summands in the r.h.s of Eq. ( 16) correspond to, -: potential energy from the normal modes,

-   CC H CC p E q k q      4 1 ,



: potential energy associated to the CC modes.

The mean energy of a subsystem in response to a white noise force of unit power spectral density in each octave band is obtained by integrating the summation of the kinetic [START_REF] Mace | Statistical energy analysis, energy distribution models and system modes[END_REF] and potential ( 16) spectral energies over the bandwidth using a simple quadrature rule.

CB numerical simulations and discussion

The total energies of the subsystems have been computed with Eq. (15-16) for test cases #1 and #2. The energies have been integrated in the 1 kHz octave band and compared to a reference solution obtained from the global mode computation described in section 2.2 of [START_REF] Mace | Energy flow models from finite element analysis[END_REF]. 200 normal modes have been considered for the modal energy computations, which has proved more than enough to ensure convergence. As regards the CC modes, 20 of them have sufficed to achieve a good agreement with the reference solution. This is remarkable since 396 constraint modes would have been necessary to achieve similar results.

Results are given in Table 1 for the two test cases. As observed, the total energies estimated with the CB approach are compliant with the reference ones. Except for the excited subsystem, where the contribution of the normal modes dominates, the normal modes and the CC ones equally add to the energies of panels 2, 3 and 4. There are, however, some issues that prevent one identifying Eq. ( 11) as a suitable candidate to fulfill the MCAs. On the one hand the resonant modes contribute significantly to the overall energy of the normal modes, especially for subsystem 1, but hardly exceed 50-60% of the energy for subsystems 2, 3 and 4 (bear in mind that results are presented in dB in Table 1). On the other hand, the energy cross contributions are rather important (mainly for test case #1)

and attain negative values. An energy-based model like SEA or SmEdA cannot account for

           q Λ q H p E 4 1 , 
these cross contributions which arise because the subsystem normal modes and the CCmodes are not orthogonal with respect to the subsystem mass matrix. Indeed, and as seen from the MCAs in section 2.1, in those methods modes are identified with oscillators and the power flow interchanged between them is proportional to the difference between their energies. The energy of each mode can be estimated, but not the cross-contribution between them. Moreover, in the results of Table 1 the cross contributions also include the influence of non-resonant modes.

To further check the degree of compliance with the MCAs, we have made a second computation retaining only the resonant normal modes (and also the 20 CC-modes). The amount of resonant normal modes for each plate is given in Table 2. As observed, the number is much lower than that of the previous computation, where 200 normal modes per plate were considered. Yet, the number still complies with the SEA requirements of at least 10 subsystem modes for frequency band. Table 2 also includes the modal overlap factors deduced from the number of resonant modes. The modal overlap values are close to one, which indicates that the dynamic behavior of the subsystems is neither non-diffuse (as for high modal overlaps), nor dominated by a few strongly resonant modes (as for small modal overlaps). Therefore, the conditions for SEA application are fulfilled [START_REF] Bot | Foundation of statistical energy analysis in vibroacoustics[END_REF].

The results with the sole consideration of resonant modes are shown in Table 3.

The energy predictions are clearly less accurate than those from Table 1. Actually, only the energy of the excited panel is correctly estimated. For panels 2, 3 and 4, this second CB calculation underestimates the energy levels between 1 dB and 4.5 dB, depending on the subsystem and the test case. The influence of the cross terms has also noticeably decreased because of the exclusion of the non-resonant normal modes.

To better understand the reason of the discrepancies between the second and first calculations, an analysis of the modal parameters (modal injected power, modal coupling mass, modal energy) of the first CB computation has been performed. If we arbitrarily focus on subsystem 3 (equivalent results arise for the other non-excited subsystems), some hints can be grasped on what is happening. One can observe in Figure 4a that the low frequency non-resonant modes outside the 1kHz band have substantial energy. In particular, the first mode (which is clearly non-resonant) has the largest energy of all modes. This is in contrast with the global distribution of the modal energy one gets form the DMF approach (compare with Figure 4b in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF]). In that figure, the most energetic modes are mainly resonant which allows one neglecting them in the dual modal formulation without introducing much error.

We can gather more information having a look at the modal mass coupling matrix

 3  m
in Figure 4b of the current paper. The first non-resonant normal mode appears strongly coupled with the first CC-mode. Similarly, other non-resonant normal modes exhibit substantial couplings with the CC ones. This is at the origin the problem. The strong coupling makes the non-resonant modes play a significant role in the energy sharing between subsystems. Consequently, they cannot be neglected, and the MCAs may be violated if the standard CB formulation Eq. ( 11) is considered. To somewhat overcome this problem, a reformulation of the CB equations will be next proposed. 1 with the standard CB method but only considering the Resonant (R) normal modes. Results for the octave band 1 kHz.

Reformulation of the Craig-Bampton method

Reformulated CB modal equations for four subsystems coupled at a junction

Let us start by splitting the modal amplitude vector g q in Eq. ( 9) in two parts: R g q which includes the amplitude vectors of the resonant (R) normal modes, and

I NR g  q
standing for the amplitude vectors of the non-resonant (NR) normal modes, plus the constraint interface (I) modes (NR+I modes). The size of this last vector corresponds to the summation of the number of NR modes, NR N , and the number of the constraint modes (i.e. the number of boundary d.o.f.s). The algebraic system of equations ( 9) can then be rewritten as, 10), an eigen-analysis is carried out for the matrices related to the NR+I modes, 0

                                                 I NR g R g I NR g R g I NR g R g I NR g R g I NR g T I NR R g I NR R g R g f f q q K 0 0 Λ q q M M M M     / / , (17) with . , , , , , , , , 1 1 1 1 1 1
                                                                                                                                                                                               0 0 0 f φ f f ψ 0 0 0 f φ f q q q q q q q q q q q m 0
NR I i NR I i g C g      KM ζ . (18)
The eigenvectors

                   C g R g I NR g R g g q q C q q q , with        Γ 0 0 I C . (19)
Pre-and post-multiplication of Eq. ( 17) by C results in

                                        C g R g C g R g C R g C g R g C g T C R g C R g R g f f q q Λ 0 0 Λ q q M M M M     / / (20)
with

Γ M Γ M I NR g T C g   , Γ M M I NR R g C R g   / / and I NR g T C g   f Γ f
. As in Eq. ( 13), we could expand Eq. [START_REF] Rosa | A similitude for structural acoustic enclosures[END_REF] and check that this system essentially establishes an inertial coupling between the resonant subsystem modes and the C modes. The subsystem energies can be calculated following the procedure exposed in section 4.2.

Reformulated CB numerical simulations and discussion

The reformulated CB equations have been applied once more to test cases #1 and #2. 815 and 951 C modes below 20kHz have been respectively used for them. The resulting modal contributions to the subsystem energies are summarized in Table 4 and show very good agreement with the reference solution. For instance, the energy of panel P3 is correctly predicted for tests #1 and #2. This was not the case when only R normal modes were considered in the standard CB equations (see Table 3). Also, the cross contributions turn to be negligible compared with the contributions of the R normal modes and the C modes.

Hence, the latter suffice to well approximate the subsystem energies. The C modes contemplated in the computations contain both, resonant and non-resonant modes. Thus, a new calculation for test case #2 has been carried out only taking into account the six existing resonant C modes. Table 5 (second row) indicates that subsystem energies are worse estimated in this occasion. Discrepancies between 1 and 2 dB can be appreciated (compare with first row in Table 5).

To better comprehend these outcomes, the modal energy distributions of subsystems 2 and 3, obtained with the sole consideration of resonant modes, are compared in Figure 5 with those from the original CB formulation, which also include non-resonant modes (herein considered as the reference solution). As seen in the figure, the energies of most modes are correctly predicted with the reformulated CB model, even if only resonant C modes are involved. However, some large discrepancies can also be observed for a few other modes, their energy levels being generally underestimated. For instance, the calculation for the 34-th mode of subsystem 2 provides an energy of 5.0x10 -10 J, whereas the reference value is 1.9x10 -8 J. These energy differences in some resonant modes are at the origin of the slight discrepancies in dB of the subsystem energies. An analysis of the mass coupling matrix between the R normal modes and the C modes reveals that some non-resonant C modes are strongly coupled to the former. A final computation is thus presented which, along with the resonant C modes, also includes those non-resonant C modes strongly coupled with the resonant normal modes. A non-resonant C mode and a resonant normal mode are assumed to be strongly coupled if their coupling mass coefficient in C R g / M is greater than 0.1 (remember that the normal modes and the C modes are normalized to unit modal mass). Following this criterion, a total of 198 C modes (6 resonant plus 192 non-resonant) are included in the computation. This supposes neglecting a large quantity of non-resonant C-modes (753 out of 951). The subsystem energies resulting from the reformulated CB with these selected non-resonant modes are presented in Table 5 (third row), while the corresponding modal energy distributions are shown in Figure 6. A very good agreement with the reference solution is obtained.

Reference

To conclude the analysis based on the reformulated CB method, a schematic representation of the proposed modal coupling final arrangement is presented in Figure 7.

In the diagram, all modes (subsystem normal modes and C modes) can be associated to oscillators, as demanded by the MCAs. The external force directly excites the resonant normal modes of subsystem 1 and the C modes. The former are coupled to the latter through mass elements, which fulfil the conservative coupling condition of the MCAs. However, for an accurate prediction of the energy sharing between subsystems it becomes necessary to include some non-resonant C modes in the model. This point cannot be accounted by SEA and SmEdA, though it is not exclusive of the situation analyzed in this paper. For instance, non-resonant transmission is needed to model energy transfer between acoustic cavities separated by a panel, below the critical frequency (see e.g., [START_REF] Renji | Non-resonant response using statistical energy analysis[END_REF][START_REF] Craik | Non-resonant sound transmission through double walls using statistical energy analysis[END_REF]). That issue is however out of the scope of the current work.

Conclusions

In this paper we have explored the potential of the Craig-Bampton (CB) method to characterize the coupling between multiple subsystems at a junction, in accordance with the coupling modal assumptions (MCAs) of some energy-based methods. This paper constitutes a continuation of the previous work by the authors in [START_REF] Maxit | A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models[END_REF], where the dual modal formulation (DMF) was investigated to tackle the same problem. Although the DMF performs well when there is a strong impedance mismatch between subsystems at the junction, it fails when the involved subsystems have similar stiffness. This motivated resorting to the CB to address that situation.

The CB method is appealing at first sight because it considers normal modes with fixed-interfaces, which could be well adapted for subsystems with comparable stiffness (in this case every subsystem is blocked at the junction by the others). However, despite the developments based on the CB have been shown to fulfil various of the requisites in the MCAs, numerical tests have revealed that the cross contributions between the subsystem normal modes and the characteristic constraint (CC) modes are not negligible, and that non-resonant low frequency modes must be taken into account to achieve accurate energy predictions. To overcome those difficulties, a reformulation of the CB modal equations has been proposed.

The reformulated CB method has led to the definition of the coupling (C) modes for a given frequency band of excitation. C modes are obtained solving a generalized eigenvalue problem composed by the mass and stiffness matrices of the non-resonant normal modes and the constraint modes. The cross contributions have been shown to become negligible with the reformulation, thus improving the performance of the original CB method. Notwithstanding the enhancement, the influence of those non-resonant C modes strongly coupled to the normal subsystem modes cannot be neglected to get precise subsystem energies. The standard SEA or SmEdA methods cannot embed such behavior, but the situation is not so strange, and one can encounter it for instance, in panel sound radiation below the critical frequency. Developments to include non-resonant transmission have been developed in those theories and could be adapted as well to deal with the problem in this paper. Future work may also involve handling the effects of longitudinal and shear waves [START_REF] Lyon | In-plane contribution to structural noise transmission[END_REF][START_REF] Nilsson | Attenuation of structure-borne sound in super structures on ships[END_REF][START_REF] Kessissoglou | Power transmission in L-shaped plates including flexural and inplane vibration[END_REF], which have been disregarded in this paper, as well as extracting coupling loss factors from the herein reported modal coupling scheme.

1 If

 1 the normal modes of subsystem α for an unit harmonic excitation , k E   kinetic energy arising from the cross interaction between the normal and the characteristic constraint modes for an unit harmonic excitation , k E   kinetic energy of subsystem α related to the characteristic constraint modes for an unit harmonic excitation , p E   strain energy related the normal modes of subsystem α for an unit harmonic excitation , p E   strain energy of subsystem α related to the characteristic constraint modes for an unit harmonic excitation f finite element external force vector of the global system B f external force vector related to the boundary degrees of freedom C g f force vector related to the coupling modes g CC  f generalized force vector related to the subsystem normal modes and the characteristic constraint mode I NR g  f force vector related to the non-resonant (NR) normal modes, plus the constraint interface (I) modes (NR+I modes) R g f force vector related to the resonant normal modes I f external force vector related to the interior degrees of freedom external force vector related to the interior degrees of freedom applied on subsystem 1 I identity matrix BB k stiffness matrix related to the boundary degrees of freedom IB k coupling stiffness matrix between the interior degrees of freedom and the boundary degrees of freedom II k stiffness matrix related to the interior degrees of freedom  k stiffness matrix related to the constraint modes   k stiffness matrix of subsystem α related to the characteristic constraint modes K finite element stiffness matrix of the considered system g CC  K stiffness matrix related to the subsystem normal modes and the characteristic constraint mode to the non-resonant (NR) normal modes, plus the constraint interface (I) modes (NR+I modes) BB m mass matrix related to the boundary degrees of freedom IB m coupling mass matrix between the interior degrees of freedom and the boundary degrees of freedom finite element mass matrix of the considered system C g M mass matrix related to the coupling modes g CC  M global mass matrix related to the subsystem normal modes and the characteristic constraint mode I NR g  M global mass matrix related to the non-resonant (NR) normal modes, plus the constraint interface (I) modes (NR+I modes)

Λ

  related to the non-resonant (NR) normal modes, plus the constraint interface (I) modes (NR+I modes) R g q global amplitude vectors related to the resonant normal modes  q amplitude vector of the normal modes of the subsystem with fixed interfaceNR    qamplitude vectors related to the non-resonant normal modes of subsystem α to the resonant normal modes of subsystem α  q amplitude vector of the constraint modes FF S power spectral density of the force signal u displacement vector of the global system B u displacement vector related to the boundary degrees of freedom I u displacement vector related to the interior degrees of freedom Γ matrix containing the eigenvectors related to the C N first coupling modes i ζ eigenvector related to the i-th eigenvalue i C  of the generalized eigenvalue problem defined by the mass and stiffness matrices of the non-resonant normal modes, plus the constraint interface modes (i.e. eigenvector related to the i-th coupling mode)  damping loss factor i  eigenvalue of the i-th normal modes of the subsystem with fixed interface i C  eigenvalue of the i-th coupling modes i CC  eigenvalue of the i-th characteristic constraint modes eigenvalue diagonal matrix related to the coupling modes CC Λ eigenvalue diagonal matrix related to the characteristic constraint mode R g Λ eigenvalue diagonal matrix related to the resonant normal modes  Λ eigenvalue diagonal matrix related to the normal modes of the subsystem with fixed interface the i-th eigenvalue i CC  of the generalized eigenvalue problem defined by the mass and stiffness matrices related to the constraint modes (i.e. eigenvector related to the ith Characteristic Constraint (CC) mode) i II φ eigenvector related to the i-th eigenvalue i  of the generalized eigenvalue problem defined by the mass and stiffness matrices related to the interior degrees of freedom (i.e. eigenvector related the i-th normal mode of the considered subsystem with fixed interface)  matrix containing the eigenvectors related to the II N first normal modes of the considered subsystem with fixed interface ψ matrix containing the static shapes related to the interior and boundary degrees (i.e. constraint modes) IB ψ matrix containing the static shapes related to the interior degrees of freedom

Iu

  contains the displacements of the interior degrees of freedom (d.o.f.s) of the subsystem and B u encompasses the displacements of the boundary d.o.f.s. Analogously, I f stands for the forces applied on the interior d.o.f.s and B f for those at the boundary.

  )-(3) using two different sets of modes, namely, (a) The normal modes with fixed-interface,  : These modes are calculated with all boundary d.o.f.s held fixed (i.e. 0  B u ). The i-th mode is represented by the eigenvector i II φ , with eigenvalue i  , which is obtained from the generalized eigenvalue problem

iξ

  are referred to as the Characteristic Constraint (CC) modes, and their number CC N is supposed to be less than the number of constraint modes. The eigenvectors i ξ are normalized to unit generalized mass and can be organized in columns in the matrix 12 ... CC N

  similarly to what was done for the calculation of the CC modes in Eq. (

i ζ associated to the eigenvalues i C .

 C will be hereafter referred to as the Coupling (C) modes. It should be remarked that the definition of the NR modes depends on the considered frequency band. Consequently, the C modes will also depend on it. Suppose a total of C N coupling modes. Their eigenvectors can be settled in columns The modal coordinate vector g q can then be projected onto the C modes basis through the transformation

Figure 1 .

 1 Figure 1. Geometry of the benchmark structure.

Figure 2 .

 2 Figure 2. Sub-structuring of the benchmark structure (a) DMF in [7] (b) CB. (Section view of the panels).

Figure 3 .

 3 Figure 3. Illustration of the different modes shapes for the four coupled plates: (a), normal modes with fixed interface; (b), constraint modes; (c), Characteristic Constraint (CC) modes. Test case #2. (Note that constraint modes correspond to static deflections and therefore have no associated frequency).

Figure 4 .

 4 Figure 4. Test case # 2. (a), energy distribution of subsystem 3. Cross: standard CB results including non-resonant subsystem modes; Square: standard CB results with only resonant subsystem modes. Vertical dash lines indicate the bandwidth of the 1 kHz octave band; (b),

Figure 5 .

 5 Figure 5. Modal energy distributions in test case # 2 for: (a), subsystem 2; (b), subsystem 3. Cross: standard CB results considering non-resonant modes (reference); circle: reformulated CB results with only resonant subsystem modes and resonant C modes. Vertical dash lines indicate the bandwidth of the 1 kHz octave band.

Figure 6 .

 6 Figure 6. Modal energy distributions in test case # 2 for: (a) subsystem 2; (b) subsystem 3. Cross: original CB results considering non-resonant modes (reference); circle: CB results with only resonant subsystem modes and the selected C modes. Vertical dash lines indicate the bandwidth of the 1 kHz octave band.

Figure 7 .

 7 Figure 7. Schematic representation of the modal coupling scheme related to the reformulated CB method (R = resonant, NR = non-resonant).

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 1. Geometry of the benchmark structure.

Figure 7 .

 7 Figure 7. Schematic representation of the modal coupling scheme related to the reformulated CB method (R = resonant, NR = non-resonant).

CB -Total 67.9 dB 57.2 dB 56.6 dB 57.2 dBTable 1 :

 1 Subsystem energies and different contributions resulting from the standard CB method (dB, ref. 10 -12 J, (-) = negative contributions). Results for the octave band 1 kHz.

	Test	Panel	1	2	3	4
	case	Energy
	#1 #1 CB -Normal modes Reference (Resonant contrib.) #1 CB -CC modes #1 CB -Cross contributions 57.8 dB (-) 59.2 dB (-) 46.8 dB (-) 45.3 dB (-) 68.5 dB 61.4 dB 44.7 dB 68.1 dB (67.5 dB) 60.5 dB (57.7 dB) 47.1 dB (44.2 dB) 47.7 dB (46.1 dB) 60.8 dB 60.5 dB 44.2 dB 42.8 dB 46.4 dB #1 CB -Total 68.5 dB 61.5 dB 44.7 dB 46.4 dB #2 Reference 67.9 dB 57.2 dB 56.6 dB #2 CB -Normal modes (Resonant contrib.) 67.8 dB (67.4 dB) 56.1 dB (53.8 dB) 54.9 dB (52.8 dB) 54.9 dB (53.5 dB) #2 CB -CC modes 55.9 dB 55.6 dB 55.2 dB 54.9 dB #2 CB -Cross contributions 50.0 dB (-) 54.0 dB (-) 52.9 dB (-) 49.8 dB (-) #2 Test case Panel Quantity 1 2 3 4 #1 Number of resonant modes 38 27 101 78 #1 Modal overlap factor 1.1 0.8 2.8 2.2 #2 Number of resonant modes 38 27 24 17 57.2 dB #2 Modal overlap factor 1.1 0.8 0.7 0.5

Table 2 :

 2 Number of resonant modes and modal overlap factor for each panel.

	Test	Panel	1	2	3	4
	case	Energy
	#1 Reference	68.5 dB	61.4 dB	44.7 dB	46.4 dB
	#1 CB -R normal modes	67.6 dB	56.2 dB	38.8 dB	43.0 dB
	#1 CB -CC modes	54.7 dB	54.7 dB	34.2 dB	33.9 dB
	#1 CB -Cross contributions 36.9 dB (-) 39.9 dB (-) 21.9 dB (-) 26.3 dB (-)
	#1 CB -Total	67.8 dB	58.5 dB	40.1 dB	43.4 dB
	#2 Reference	67.9 dB	57.2 dB	56.6 dB	57.2 dB
	#2 CB -R normal modes	67.5 dB	54.7 dB	49.5 dB	52.3 dB
	#2 CB -CC modes	51.8 dB	51.8 dB	51.8 dB	51.7 dB
	#2 CB -Cross contributions 39.1 dB (-) 37.7 dB (-) 36.4 dB (-) 38.8 dB (-)
	#2 CB -Total	67.6 dB	56.5 dB	53.7 dB	55.1 dB

Table 3 :

 3 Same type of results than in Table

-Total 68.3 dB 61.4 dB 44.7 dB 46.4 dB #2 Reference 67.9 dB 57.2 dB 56.6 dB 57.2 dB

	Test	Panel	1	2	3	4
	case	Energy
	#1	Reference	68.5 dB 61.4 dB	44.7 dB	46.4 dB
	#1 CB -R Normal modes	67.5 dB	57.7 dB	44.2 dB	46.1 dB
	#1 CB -C modes	59.7 dB	59.2 dB	34.8 dB	34.8 dB
	#1 CB -Cross contributions 51.4 dB	46.1 dB (-) 27.3 dB (-) 29.3 dB
	#1 CB #2 CB -R Normal modes	67.4 dB	54.1 dB	52.8 dB	53.5 dB
	#2 CB -C modes	56.0 dB	54.3 dB	54.2 dB	54.2 dB
	#2 CB -Cross contributions 49.1 dB	39.9 dB (-) 39.8 dB (-) 40.6 dB
	#2	CB -Total	67.8 dB 57.1 dB	56.5 dB	57.0 dB

Table 4 :

 4 Same type of results than in Table1but with the reformulated CB calculation.The results in Table4are mainly consistent with the MCAs except for one question.

Table 5 .

 5 Subsystem energy results for the reformulated CB calculation with various sets of C modes. Test case #2.

		67.9 dB	57.2 dB	56.6 dB	57.2 dB
	CB with all the C modes	67.8 dB	57.1 dB	56.5 dB	57.0 dB
	CB with only resonant C modes 67.8 dB	55.8 dB	55.3 dB	55.7 dB
	CB with the selected C modes 67.7 dB	57.1 dB	56.4 dB	56.8 dB

TABLE CAPTIONS Table 1 :

 CAPTIONS1 Subsystem energies and different contributions resulting from the standard CB method (dB, ref. 10 -12 J, (-) = negative contributions). Results for the octave band 1 kHz.

Table 2 :

 2 Number of resonant modes and modal overlap factor each panel

Table 3 :

 3 Same type of results than in Table1with the standard CB method but only considering the Resonant (R) normal modes. Results for the octave band 1 kHz.

Table 4 :

 4 Same type of results than in Table1with the reformulated CB calculation.

Table 5 .

 5 Subsystem energy results for the reformulated CB calculation with various sets of C modes. Test case #2.

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