Benjamin Trévisan
email: benjamin.trevisan@insa-lyon.fr

Kerem Ege

Bernard Laulagnet

A modal approach to piano soundboard vibroacoustic behavior

This paper presents an analytical method for modeling the vibro-acoustic behavior of ribbed non-rectangular orthotropic clamped plates. To do this, the non-rectangular plate is embedded in an extended rectangular simply supported plate on which a spring distribution is added, blocking the extended part of the surface, and allowing the description of any inner surface shapes. The acoustical radiation of the embedded plate is ensured using the radiation impedances of the extended rectangular simply supported plate. This method is applied to an upright piano soundboard: a non-rectangular orthotropic plate ribbed in both directions by several straight stiffeners.

A modal decomposition is adopted on the basis of the rectangular extended simply supported plate modes, making it possible to calculate the modes of a piano soundboard in the frequency range 0; 3000 Hz, showing the different associated mode families. Likewise, the acoustical radiation is calculated using the radiation impedances of a simply supported baffled plate, demonstrating the influence of the string coupling point positions on the acoustic radiated power. The paper ends with the introduction of indicators taking into account spatial and spectral variations of the excitation depending on the notes, which add to the accuracy of the study from the musical standpoint. A parametrical study, which includes several variations of soundboard design, highlights 1 the complexity of rendering high-pitched notes homogeneous.

I. INTRODUCTION

Traditionally designed using empirical approaches, musical instruments are now being studied regarding the perceptive and subjective aspects of the sounds they produce. Indeed, many parameters influence their timbres, ranging from the wood used [START_REF] Brémaud | Acoustical properties of wood in string instruments soundboards and tuned idiophones: Biological and cultural diversity[END_REF] to specifications linked to their design, which greatly determines their vibratory behavior.

In the case of the piano, the soundboard plays an essential role in the functioning of the instrument. Indeed, string sections are too thin to radiate on their own. Thus their vibrations are transmitted to the soundboard through bridges that serve as effective acoustic radiators. The geometry of a soundboard is complex and composed of a non-rectangular plate, traditionally made of spruce (orthotropic material), ribbed on one side by several beams placed perpendicular to the wood fibers, and by one or two bridges almost parallel to the fibers, on the opposite side. Fig. 1 shows an example of such a structure.

The vibro-acoustic broadband behavior of this structure has been studied by several authors including Fletcher [START_REF] Fletcher | Normal vibration frequencies of a stiff piano string[END_REF], Weinrech [START_REF] Weinreich | Coupled piano strings[END_REF], and Kindel [START_REF] Kindel | Vibrations of a piano soundboard: Modal analysis and finite element analysis[END_REF], then by Suzuki [START_REF] Suzuki | Vibration and sound radiation of a piano soundboard[END_REF], [START_REF] Suzuki | Acoustics of Pianos[END_REF],

Conklin [START_REF] Conklin | Design and tone in the mechanoacoustic piano. Part I. Piano hammers and tonal effects[END_REF]- [START_REF] Conklin | Design and tone in the mechanoacoustic piano. Part III. Piano strings and scale design[END_REF], Giordano [START_REF] Giordano | Simple model of a piano soundboard[END_REF]- [START_REF] Giordano | Sound production by a vibrating piano soundboard: Experiment[END_REF], Dérogis [START_REF] Dérogis | Computation and modelisation of the sound radiation of an upright piano using modal formalism and integral equations[END_REF], Berthaut [START_REF] Berthaut | Piano soundboard: structural behavior, numerical and experimental study in the modal range[END_REF], Bensa [START_REF] Bensa | The simulation of piano string vibration: From physical models to finite difference schemes and digital waveguides[END_REF], Stulov [START_REF] Stulov | Physical modelling of the piano string scale[END_REF], and more recently by Ege and Boutillon [START_REF] Ege | Vibroacoustics of the piano soundboard:(Non) linearity and modal properties in the low-and mid-frequency ranges[END_REF], [START_REF] Boutillon | Vibroacoustics of the piano soundboard: Reduced models, mobility synthesis, and acoustical radiation regime[END_REF], Chaigne [START_REF] Chaigne | Dynamical properties of piano soundboards[END_REF], Chabassier [START_REF] Chabassier | Modeling and simulation of a grand piano[END_REF],

Rigaud [START_REF] Rigaud | Models of music signals informed by physics. Application to piano music analysis by non-negative matrix factorization[END_REF], and Etcheniquel [START_REF] Etchenique | Coupling of transverse and longitudinal waves in piano strings[END_REF]. It is now the subject of software development for piano sound synthesis [START_REF] Guillaume | Device for producing signals representative of sounds of a keyboard and stringed instrument[END_REF].

Many other works already exist on the subject of ribbed structures and not only in the area of musical instruments. Indeed, nowadays simple structures like beams and isotropic plates are described well [START_REF] Guyader | Vibration in Continuous Media[END_REF]- [START_REF] Wang | Modal sound transmission loss of a single leaf panel: Effects of intermodal coupling[END_REF], but the design and modeling of ribbed structures nonetheless remains a major topic of research. Most of these studies have focused on periodically ribbed isotropic flat plates [START_REF] Maidanik | Response of Ribbed Panels to Reverberant Acoustic Fields[END_REF]- [START_REF] Xin | An exact elasticity model for rib-stiffened plates covered by decoupling acoustic coating layers[END_REF], curved panels [START_REF] Chang | Noise transmission into semicylindrical enclosures through discretely stiffened curved panels[END_REF], stiffened cylindrical shells [START_REF] Guyader | Sound radiation by finite cylindrical ring stiffened shells[END_REF]- [START_REF] Burroughs | Acoustic radiation from fluid-loaded infinite circular cylinders with doubly periodic ring supports[END_REF], and more recently laminated composite panels [START_REF] Mejdi | Vibroacoustic analysis of laminated composite panels stiffened by complex laminated composite stiffeners[END_REF], [START_REF] Mejdi | Transmission loss of periodically stiffened laminate composite panels: Shear deformation and in-plane interaction effects[END_REF] and sandwich structures [START_REF] Xin | Analytical modeling of fluid loaded orthogonally ribstiffened sandwich structures: Sound transmission[END_REF]- [START_REF] Xin | Transmission loss of orthogonally rib-stiffened doublepanel structures with cavity absorption[END_REF].

Problems linked to the piano soundboard have been studied from the angle of purely musical issues. These included the quest for a good compromise between "sustain / radiated power" and poor color in high frequencies (i.e poor spectrum of last notes), both of which appear to be the main difficulties confronting piano makers.

Although modelling flat ribbed structures and their baffled radiation has long been an area of research, taking into account non-rectangular geometry, non-special orthotropy (i.e. the axes of orthotropy are not parallel to the boundary if rectangular) and attached bars in orthogonal directions simultaneously is a real challenge.

With a view to solving these issues, modelling is performed using a variational approach in which the judicious use of simply supported rectangular extended plate modes and radiation impedances is introduced. The aim of this model of an upright piano soundboard, which avoids any object discretization, is to provide an interesting alternative to a complete numerical method such as the FEM-BEM method (with long computing time), which could obviously be used.

Consequently, this model facilitates parametrical studies and, for example, highlights the influence of ribs and bridges (numbers, width, length, height) on the acoustical radiation of the piano soundboard and, more generally, on orthotropic ribbed panels. This article is the continuation of works presented in [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF]. The first part of the article presents how the non-rectangular edges and orientation of the materials composing the piano soundboard are formulated by the addition of springs. Then, in the second part, the model of the soundboard of a Pleyel P131 is described. It comprises the addition of ten ribs, two bridges and two "cut-off" corner beams. The modal aspects are presented in the frequency range 0; 3000 Hz. In particular we discuss the different families of modes due to stiffening. Then, several numerical results including mobility and acoustic radiated power are presented for an excitation moving along the main bridge at the position of the string couplings. We show how the vibro-acoustical behavior of the soundboard depends on the positions of the strings, and how the discrete frequencies of the notes (the sources) make analysis difficult.

Drawing its inspiration from [START_REF] Waltham | Vibrational characteristics of harp soundboards[END_REF], [START_REF] Le Carrou | Some characteristics of the concert harp's acoustic radiation[END_REF], a parametrical study shows the complexity of obtaining homogeneous high-pitched notes and how it is related to the soundboard design (main bridge discontinuity, ribs in the high frequency domain, etc.). This is done by introducing a new indicator applied to the mobility and the acoustic radiated power, taking into account the discrete spacing aspects (discontinuous excitation points) and the discrete frequencies of the keys at the same.

II. ORTHOTROPY WITH UNSPECIFIED ANGLE

In traditional piano soundboard design, the direction of the wood fibers is not parallel to the edges: we define this as the angle of orthotropy 𝜃. See Fig. 1. In this part we focus on a method of modeling the orientation of the fibers and describing the non-rectangular contours. We demonstrate the originality of this method which allows us to describe, from the analytical standpoint, any shapes embedded within a rectangular shape, and to take into account non-specified orthotropy, i.e. when the main axes of orthotropy are not parallel to the soundboard shapes.

II.A. THEORETICAL FORMULATION

In order to describe both the non-rectangular contours and the unspecified angle of orthotropy, the following model uses the simple basis of a special rectangular orthotropic plate, i.e. the Young's moduli are parallel to the plate edges. See Fig. 2 Thus it is possible to describe any edges the user wishes (not limited to only soundboards) as suggested in Fig. 3. Obviously, it is possible to model a grand piano but we chose to study a Pleyel P131 upright soundboard available at the lab (kindly given by ITEMM, European Technological Institution for Music Professions).

II.A.1. HAMILTONIAN

The model is based on a variational approach. We express the Hamiltonian of the whole plate-spring system to determine the eigenvalue problem (no external works

E x E y E x E y !! L x L y L x L y
are considered). An additional external force will be introduced in II.A.3 for the forced response calculation.

The starting point of our modeling is a simply supported rectangular plate with specific orthotropy. It follows the thin plate Love-Kirchhoff hypothesis for which the Hamiltonian can be easily found in the literature [START_REF] Leissa | Vibration of plates[END_REF]. Additional information on the kinetic hypothesis can be found in a previous article [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF].

We consider several springs coupled to the plate at the coordinates 𝑥 & , 𝑦 & . We give the following Hamiltonian for a plate with a number of springs N s :

𝐻 *+ = 1 2 𝜌ℎ𝑤 ˙3 -(𝐷
? ;@A > B 8@A 𝑑𝑥𝑑𝑦 D E D F 𝑑𝑡 + - 1 2 𝑘𝑤 3 𝑥, 𝑦 𝛿(𝑥 -𝑥 & )𝛿(𝑦 -𝑦 & ) > ? ;@A > B 8@A 𝑑𝑥𝑑𝑦 D E D F 𝑑𝑡 J K &@7 (1)
with 𝑤 being the plate transverse displacement; 𝑘 the spring rigidity; 𝑡 A ; 𝑡 The first line of the equation is related to the plate and the second to the springs where the Dirac distributions render them localized at the coordinates 𝑥 & , 𝑦 & .

Table 1 gives the properties of the extended rectangular orthotropic spruce plate (Fig. 2).

II.A.2. MODAL DECOMPOSITION

As mentioned previously, the starting point of our model is a rectangular simply supported plate. This basis, currently used in the area of vibrations, is particularly adapted for an analytical approach and has been studied several times [START_REF] Laulagnet | Sound radiation by a simply supported unbaffled plate[END_REF], [START_REF] Guyader | Sound radiation by finite cylindrical ring stiffened shells[END_REF], [START_REF] Guyader | Structural acoustic radiation prediction: Expending the vibratory response on a functional basis[END_REF]. Of course, this basis is not the real basis of the embedded plate but only a convenient basis for our model, as will be shown in the following. Using modal decomposition, the transverse displacement is written as a linear combination of simply supported plate modes weighted by modal amplitudes 𝑎 \] 𝑡 :

𝑤 𝑥, 𝑦, 𝑡 = 𝑎 \] 𝑡 𝜙 \] 𝑥, 𝑦 J ]@7 _ \@7
∀𝑥 ∈ 0; 𝐿 8 and 𝑦 ∈ 0; 𝐿 ;

where 𝜙 \] 𝑥, 𝑦 = sin 𝑚𝜋 𝐿 8 𝑥 sin 𝑛𝜋 𝐿 ; 𝑦 .

By injecting the modal decomposition (2) into Eq. ( 1) and using the orthogonal properties of eigenvectors, it is possible to analytically calculate the surface integral in the Hamiltonian. Thus the functional depends on the two variables 𝑎 \] 𝑡 , 𝑎 \] (𝑡) and no longer on the transverse displacement 𝑤 𝑥, 𝑦, 𝑡 and its space and temporal derivatives. Thus we have:

𝐻 *+ (𝑎 \] (𝑡), 𝑎 ˙\] (𝑡)) = ℒ D E D F (𝑎 \] (𝑡), 𝑎 ˙\] (𝑡)𝑑𝑡 (3)
where ℒ(𝑎 \] (𝑡), 𝑎 ˙\] (𝑡)) is termed the Lagrangian of the system.

By applying the differential form of the principle of least action and using Euler-Lagrange equations, the action of the system is minimized, leading to the following homogeneous linear system:

𝑀 +ijDk + 𝑀 &+lm]n& 𝑎 + 𝐾 +ijDk + 𝐾 &+lm]n& 𝑎 = 0 (4)
where 𝑀 and 𝐾 represent mass and rigidity matrices. These matrices are constituted by the sum of diagonal matrices coming from the plate and full and symmetrical matrices from the springs.

Solving the corresponding eigenvalue problem Eq. ( 4) allows calculating the modes of the embedded plate: eigen-frequencies and modal shapes, the matrix of eigenvectors and thus the transfer matrix 𝑇 between the initial simply supported basis and the system basis, which is the orthonormed basis of the eigenvectors [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF]. We give the relation between the two bases with 𝑎 and 𝑏 being the vectors of modal amplitudes in the initial basis and in the new basis, respectively:

𝑎 = 𝑇𝑏 (5)
Consequently, the real modal shapes of the embedded plate are built as a linear combination of the simply supported shapes of the extended plate:

𝛷 m 𝑥, 𝑦 = 𝑎 \] (m) 𝜙 \] 𝑥, 𝑦 J ]@7 _ \@7
∀𝑥 ∈ 0; 𝐿 8 and 𝑦 ∈ 0; 𝐿 ;

Where 𝑖 indicates the column number in the matrix of eigenvectors.

Note that the size of these matrices, which is governed by the truncation on the orders M and N in the finite modal decomposition Eq. ( 2), determines the precision of the results.

II.A.3. FORCED RESPONSE

We assume that the fluid loading is negligible, so the light fluid assumption is made (for a full fluid loading coupling, see [START_REF] Laulagnet | Sound radiation by a simply supported unbaffled plate[END_REF]). Here, we consider a sinusoidal and harmonic external force applied at point (𝑥 k , 𝑦 k ) with an amplitude of 1N.

Following the same approach as before, we determine the vector of generalized force 𝐹 nk] (second member of Eq. ( 4)) for which the components are defined by:

𝐹 +u = 𝜙 +u (𝑥 k , 𝑦 k ) (7)
To determine the forced response of the system we choose to express the problem in the system basis. Using Eq. ( 5), the generalized matrix problem Eq. ( 4)

with the second member becomes diagonal:

-𝜔 3 𝑀 *+ 𝑏(𝜔) + 𝑗𝜔𝐶𝑏(𝜔) + 𝐾 *+ 𝑏(𝜔) = 𝑇 D 𝐹 nk] (8)
With 𝑀 *+ = 𝑇 D 𝑀 +ijDk + 𝑀 &+lm]n& 𝑇 and 𝐾 *+ = 𝑇 D 𝐾 +ijDk + 𝐾 &+lm]n& 𝑇.

The modal damping matrix 𝐶 is also introduced. The structure is considered as a weakly dissipative system for which the equations of generalized displacements are uncoupled [START_REF] Géradin | Theory of vibrations: dynamic of structures[END_REF]- [START_REF] Ege | The piano soundboard: Modal studies in low and mid frequencies[END_REF] so the matrix 𝐶 is also diagonal and defined by:

𝐶 = ⋱ 𝜂 m 𝑚 m 𝜔 m ⋱ (9)
With 𝑚 m , 𝜔 m and 𝜂 m being the modal mass, angular frequency and damping loss factor of the i th system mode.

In the case of a piano soundboard, the modal loss factor was evaluated between 1% and 3%, depending on the modes in our frequency range of interest [START_REF] Ege | La table d'harmonie du piano-Études modales en basses et moyennes fréquences[END_REF]. In the following, the modal loss factors of all the soundboard modes are fixed at 2%.

II.B. SPRING CALIBRATION

Minimal spring rigidity and spacing are necessary to block transverse displacement.

For the frequency range of interest 0; 3000 Hz , the spacing between springs depends on the smallest wavelength in the plate. The latter is in the direction of the weak Young's modulus [START_REF] Ege | La table d'harmonie du piano-Études modales en basses et moyennes fréquences[END_REF]:

𝜆 = 2𝜋 𝑓 𝐷 : 𝜌ℎ 7/< (10)
A criterion of 𝜆 10 is chosen to overestimate the number of springs. In the present case, it leads us to a minimal distance between the springs of around 7.3 mm (for f = 3 kHz).

Minimal spring rigidity is also important: the ratio between the rigidity of the spring and that of the plate must be sufficiently high. When the rigidity is insufficient, vibrations appear at high frequencies in the "blocked" area (where the springs are localized).

We define the ratio between the average quadratic velocity in the free area and the whole plate as a criterion in order to validate sufficient spring rigidity:

𝐿 € ratio = 10 𝑙𝑜𝑔 𝑣(𝑥, 𝑦) 3 ˆ‰Š‰ 𝑑𝑆 𝑣(𝑥, 𝑦) 3 ˆOE•‰ 𝑑𝑆 (11)
For a sufficient value of spring rigidity the velocity on the blocked area will tend to zero and thus this ratio will also tend to zero.

Fig. 4-a presents this ratio for two spring rigidities. In both cases, the truncation order is fixed at 𝑃, 𝑄 = (82,63) for which the convergence of the solution is guaranteed.

The excitation point is arbitrarily chosen near an edge. When the springs are strong enough compared to the plate, the ratio is lower than 0.06 dB for the entire frequency range (solid line in Fig. 4-a). On the contrary, insufficient springs imply a higher velocity ratio (dashed line in Fig. 4-a) from 1 kHz with a peak around 3.2 dB at 2.5 kHz in the present case. This lack of rigidity can also be detected on the modal density (Fig. 4-b); see next paragraph. Fig. 5 (on the left and center) presents modal shapes for these two cases. The modal shapes are similar at low frequency. See top left and center of Fig. 5. Indeed, at low frequency the ratios are equivalent. We can also note a small frequency difference.

From 1 kHz, three kinds of modes exist if the rigidity is insufficient: internal modes (in the free area), external modes (in the density of springs) and hybrid modes (in the two areas). The bottom left of Fig. 5 shows one of these undesired modes. On the other hand, sufficient rigidity ensures only one kind of mode: the internal mode (bottom center, Fig. 5).

As mentioned previously, the coexistence of these multi-type modes can also be detected on the modal density. Fig. 4-b shows a comparison between the modal density of the two cases of spring rigidity and the analytical formulation from [START_REF] Ege | Vibroacoustics of the piano soundboard:(Non) linearity and modal properties in the low-and mid-frequency ranges[END_REF], [START_REF] Ege | La table d'harmonie du piano-Études modales en basses et moyennes fréquences[END_REF]. This comparison shows good agreement of the modal density with the theoretical plot (solid line in Fig. 4-b) when the rigidity is sufficient (circle markers in Fig. 4-b) and tends toward the same value. On the contrary, when the spring rigidity is insufficient, the presence of external modes implies a considerable increase of modal density (square markers in Fig. 4-b). This effect is very interesting and appears even if the order of truncation is not adapted. It can facilitate the detection of external modes by solving only the eigenvalue problem. We also compared our modelling with a NASTRAN FEM model. This is a 2D model with a non-rectangular clamped plate meshed with orthotropic PSHELL elements.

The size of the elements is around 5mm to ensure the convergence of the calculations for the whole frequency range of interest [0;3000] Hz.

The comparison between our analytical model (Fig. 5, center) and numerical FEM (Fig. 5, right) shows that our solution is relevant. Indeed, the clamped boundary conditions, frequencies and modal shapes are well-described by our method: the MAC criterion is 0.9997 and 0.948 for the modes presented in Fig. 5, respectively the first modes around 30 Hz and another around 1.8 kHz. Concerning the frequencies the difference is less than 4%, which is perfectly acceptable.

III. MODELING AN UPRIGHT PIANO SOUNDBOARD

In the following, we focus on the upright piano soundboard of a Pleyel P131. This soundboard is made of spruce stiffened on one face by ten ribs also made of spruce placed in the direction of the weak Young's modulus and on the other face by a bass bridge and a main bridge (medium and high pitched notes), made of beech, placed in a direction nearly parallel to the strong Young's modulus (wood fibers). There are also two "cut-off" corners delimited by two beams made of beech. See Fig. 1.

III.A. SUPERSTRUCTURES: RIBS AND BRIDGES

The model of the superstructures is an extension of the model presented in [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF].

Indeed, it now allows having linear superstructures of different lengths in addition to different materials, widths, heights and offsets from the middle plane of the plate. Fig. 6 describes the geometry modeled. The model developed allows having only linear superstructures parallel to the entire plate edges, i.e. in the direction of the wood fibers or perpendicular to them. Taking into account a curved main bridge or a stiffener that is not parallel to an extended plate edge will imply expressing variable 𝑥 as a function of 𝑦 and vice-versa. Therefore it will be impossible to carry out simplifications useful for maintaining an analytical approach as long as possible. Thus the bridges and "cut-off" corner beams become exactly parallel to the strongest Young's modulus E x . The main bridge is cut into two parts and shifted between ribs 3 and 4. Each has a constant height and width. They are placed at position y c in the coordinates of the extended rectangular plate. All the dimensions were measured on a real Pleyel P131 piano soundboard.

Table 2 gives the dimensions and properties of the superstructures.

The ribs are oriented perpendicular to the direction of the wood fibers and perpendicular to the bridges to compensate for the weakness of the wood (weak Young's modulus E y ). Thus the ribs have a predominantly static effect. Their heights are not constant along their lengths. Indeed, they are tapered at their extremities to obtain flexibility near the edges of the soundboard. See Fig. 7-a. In our model, this progressive decrease of height is simplified. We consider a sudden change of the height, as shown in Fig. 7-b. All the ribs have a height of 5mm at their extremities. Table 3 gives the dimensions and properties of the ribs.

The boundary conditions are applied to the middle plane of the plate. The ribs and bridges are bound to it through a double continuity in displacement and rotation at each interface. Moreover, their heights are sufficiently low to neglect the warping phenomenon. Consequently, the plate controls the global motion of the whole system and its motion field can be extended to the superstructures. In addition to the continuity at the interface, the motion fields in the plate / stiffener sections are considered linear [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF]. We also take into account bending and torsion phenomena.

This leads us to express the Hamiltonians of each superstructure (for more information on this calculations, see Appendix A). We give the Hamiltonian for the middle part of a rib (in direction 𝑦) and of a bridge (in direction 𝑥):

(c) (b) (a)
𝐻 lm' = 1 2

Table 4 summarizes all the constants used in the previous equations. As in Eq. ( 1), the entire Hamiltonian for the whole system (with N r being the number of ribs and extremities and N b the number of bridges, for which all the constants and positions are different) becomes:

𝐻 &*•]š'*jlš = 𝐻 *+ + 𝐻 lm' (l) J ž l@7 + 𝐻 'lmšnk (') J Ÿ '@7 (14)
and the eigenvalue matrix problem as described in section II.A.2:

𝑀 𝑝𝑙𝑎𝑡𝑒 + 𝑀 𝑠𝑝𝑟𝑖𝑛𝑔𝑠 + 𝑀 lm'& + 𝑀 'lmšnk& 𝑎 + 𝐾 𝑝𝑙𝑎𝑡𝑒 + 𝐾 𝑠𝑝𝑟𝑖𝑛𝑔𝑠 + 𝐾 lm'& + 𝐾 'lmšnk& 𝑎 = 0 (15)
Therefore we determine the new transfer matrix 𝑇 of the Pleyel P131 and the generalized matrix formulation with a second member on the basis of the piano modes:

-𝜔 3 𝑀 &*•]š'*jlš 𝑏(𝜔) + 𝑗𝜔𝐶𝑏(𝜔) + 𝐾 &*•]š'*jlš 𝑏(𝜔) = 𝑇 D 𝐹 nk] (16)

III.B. ACOUSTIC RADIATION OF A NON-RECTANGULAR RIBBED STRUCTURE

The method we propose for acoustic radiation is an alternative to a purely numerical method such as the finite element boundary method, Rayleigh integral or Perfectly Matched Layers. Thus we calculate the acoustic radiation of the structure using the radiation impedances of the extended rectangular un-ribbed plate, which has been studied several times by Wallace, Maidanik and Stepanishen [START_REF] Maidanik | Response of Ribbed Panels to Reverberant Acoustic Fields[END_REF], [START_REF] Wallace | Radiation Resistance of a Rectangular Panel[END_REF], [START_REF] Stepanishen | Modal coupling in the vibration of fluid-loaded cylindrical shells[END_REF], for example. This approach is currently limited to a rectangular ribbed plate [START_REF] Guyader | Structural acoustic radiation prediction: Expending the vibratory response on a functional basis[END_REF] but we propose to extend it to the case of a non-rectangular structure. In addition, this approach allows comparing the radiation modal impedances of a non-rectangular ribbed structure to those of an equivalent simple plate, as was done in our previous article [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF]. This could be particularly interesting in the frequency range where the modes are similar to un-ribbed plate modes (i.e. at low frequencies).

We can consider that the piano case is an obstacle to acoustic short circuiting past the first modes. At low frequency, this results in a decrease of radiation [START_REF] Laulagnet | Sound radiation by a simply supported unbaffled plate[END_REF], [START_REF] Putra | Sound radiation from rectangular baffled and unbaffled plates[END_REF]. On the contrary, when focusing on frequencies higher than the first octave, as will be the case in the following, the wavelengths quickly become small enough to consider a baffled hypothesis. Note that this hypothesis does not change the tendencies of the results and has no repercussion on the parametric studies.

The light fluid assumption is also made as it implies omitting inter-modal coupling brought by the action of a fluid on the structure. Several authors have shown that these couplings are negligible at the first order [START_REF] Wallace | Radiation Resistance of a Rectangular Panel[END_REF], [START_REF] Wallace | The acoustic radiation damping of the modes of a rectangular panel[END_REF], [START_REF] Lesueur | Acoustic radiation of structures: vibroacoustic and fluid / structure interactions[END_REF] or do not change the general tendencies [START_REF] Laulagnet | C-valor: benchmarks for vibroacoustic software[END_REF], [START_REF] Mejdi | Dynamic and acoustic response of bidirectionally stiffened plates with eccentric stiffeners subject to airborne and structure-borne excitations[END_REF], [START_REF] Mejdi | Vibroacoustic analysis of laminated composite panels stiffened by complex laminated composite stiffeners[END_REF]. However, other authors have shown that some changes can occur in the case of heterogeneous boundary conditions, high

superstructures or at resonances [START_REF] Berry | Vibration and sound radiation of complex plates: A variational approach[END_REF], [START_REF] Keltie | The Effects of Modal Coupling on the Acoustic Power Radiation From Panels[END_REF]. We assume that they are neglected and thus assign the acoustic radiated power 𝑊 𝜔 of the piano soundboard in the frequency domain as [START_REF] Trévisan | Vibroacoustics of orthotropic plates ribbed in both directions: Application to stiffened rectangular wood panels[END_REF], [START_REF] Lesueur | Acoustic radiation of structures: vibroacoustic and fluid / structure interactions[END_REF]:

𝑊 𝜔 = 𝜔 3 2 𝑏 D * 𝑇 D 𝑅(𝜔)𝑇𝑏 (17)
with 𝑅 𝜔 being the real part of the inter-modal acoustical impedance matrix.

Due to the light fluid assumption this matrix is diagonal, i.e. inter-modal couplings are neglected. Thus the diagonal terms are calculated numerically and defined as [START_REF] Laulagnet | Sound radiation by a simply supported unbaffled plate[END_REF],

[29]:

𝑅 \]\] (𝜔) = 𝜌 A 𝜔 𝜋 3 |𝜙 \] (𝑘 8 , 𝑘 ; )| 3 𝑘 3 -𝑘 8 3 -𝑘 ; 3 Ä 𝑑𝑘 8 𝑑𝑘 ; (18)
where 𝜙 \] (𝑘 8 , 𝑘 ; ) refers to the bi-dimensional Fourier transform of a simply supported rectangular baffled plate mode.

III.C. NUMERICAL RESULTS OF THE PLEYEL P131 MODEL

The vibro-acoustical behavior of the piano soundboard can be split into different frequency ranges in which the vibration and radiation are different, due to the design of the instrument. In the following, we will first focus on the modal aspects of the instrument and then on the vibro-acoustical response of the soundboard.

To ensure the convergence of the calculations, the truncation order on the modal basis is overestimated and, as with the unribbed plate, fixed at (M,N)=(82,63).

Moreover in order to validate the blocking condition along the non-rectangular contours, the same studies as shown in Figs 4 and5 were performed (spring rigidity and number).

III.C.1. SAMPLES OF UPRIGHT PIANO SOUNDBOARD MODES

In terms of modal shapes, many phenomena appear as a function of the frequency range considered. Indeed, the more the frequency increases, the higher the rigidity of the superstructures is. Therefore the superstructures will strongly limit the transverse displacement of the soundboard when the frequency increases. This begins with the strongest of superstructures, i.e. the bridges, and then the ribs.

Some studies have also shown that the vibration is localized in certain frequency ranges. Waveguide effects occur when the wavelengths reach the same order of magnitude as the inter-rib spaces [START_REF] Ege | Vibroacoustics of the piano soundboard:(Non) linearity and modal properties in the low-and mid-frequency ranges[END_REF], [START_REF] Boutillon | Vibroacoustics of the piano soundboard: Reduced models, mobility synthesis, and acoustical radiation regime[END_REF]. Fig. 8 shows a classification of modal shapes for the 200 first modes (below 3 kHz). We define four families of modes:

• "Similar" to plate modes, i.e. unribbed modes that can be named with the number of half wavelengths in the different directions (mode (3,1) for example);

• Blocking bridges only, when the two bridges strongly minimize the plate transverse displacement;

• Blocking bridges and ribs, when all the superstructures strongly minimize the plate transverse displacement;

• Localized vibrations, when the vibration is localized in a small area delimited by the superstructures. To classify the modal shapes into these different families, we use an average quadratic linear velocity to evaluate numerically when the vibration is reduced at the stiffener locations. This allows determining the points at which the transverse displacement is strongly limited. To do this, we define a velocity threshold that allows separating a sample of modes classified visually previously (around 20 modes). Regarding the "localized vibrations" family, which is a sub-family of "blocking bridges and ribs", a criterion of spatial distribution is added. We first detect the maximum velocity magnitude of the structure and then compare the entire space average quadratic velocity with the space average quadratic velocity on a smaller surface around the maximum magnitude determined previously. We gradually increase the surface so that the local average velocity becomes higher than a certain percentage of the entire space average velocity. Similarly, this percentage is determined with a sample of modal shapes classified visually. In this way, we can classify around 73% of the modal shapes that are checked visually as a final control. The unclassified modes are determined visually and we finally classify 96% of the modes into the four families.

We can see that the ratios between them are very different. Indeed, for a large majority of modes all the superstructures strongly minimize the transverse displacement (sum of triangle markers around 90%), whereas the modes "blind" to the superstructures amount to only around 3% (see the square markers on Fig. 8).

They match with low frequencies when the wavelengths are high compared to the dimensions of the soundboard. Fig. 9 shows two examples of these low frequency modes, with one of them similar to mode (1,1) and the second similar to mode [START_REF] Weinreich | Coupled piano strings[END_REF][START_REF] Brémaud | Acoustical properties of wood in string instruments soundboards and tuned idiophones: Biological and cultural diversity[END_REF]. This phenomenon is limited to the five first modes of the soundboard. For the majority of modes (higher than 1 kHz), all the superstructures strongly minimize the transverse displacement. For around 66% of them, the vibrations are spread over the whole plate or at least over a large area (see the triangle markers pointing to the right in Fig. 8). Fig. 11 shows two of these modes, the first around 1 kHz and the second around 3 kHz: the location of the superstructures can be easily guessed. In addition, the vibrations are also localized in areas delimited by superstructures for the last 24%. It is important to split these 24% into two sub-parts: 10% of "cutoff" corner modes (upward triangle markers in Fig. 8) and 14% of vibrations localized between the ribs (downwards triangle markers in Fig. 8). Fig. 12 shows two examples of this 14% of modes. This phenomenon of localization between the ribs seems to begin at around 1600 Hz, thus when the wavelengths are around the same order of magnitude as the inter-rib spaces [START_REF] Ege | Vibroacoustics of the piano soundboard:(Non) linearity and modal properties in the low-and mid-frequency ranges[END_REF], [START_REF] Boutillon | Vibroacoustics of the piano soundboard: Reduced models, mobility synthesis, and acoustical radiation regime[END_REF]. But as shown in Fig. 12, the percentage of this family is quiet low compared to the "blocked modes" (only 14% vs. 67%): the wavelength condition seems to be a necessary but not sufficient to localize the vibrations in a reduced area of the soundboard.

Finally, the last 4% are unclassified modes: the bridges begin to minimize the transverse displacement but not completely.

III.C.2. MOBILITY, SPACE AVERAGE QUADRATIC VELOCITY AND ACOUSTIC RADIATED POWER ALONG THE BRIDGE

The mechanical mobility at the bridge is an essential quantity in musical acoustics because it characterizes the transfer between the source and the radiator. Several piano soundboard studies exist on the subject [START_REF] Giordano | Mechanical impedance of a piano soundboard[END_REF], [START_REF] Boutillon | Vibroacoustics of the piano soundboard: Reduced models, mobility synthesis, and acoustical radiation regime[END_REF]. In the case of a harp soundboard, maps of mobility along the soundboard (where the points chosen for 2913 Hz 1607,3 Hz the excitation correspond to the coupling points between the soundboard and the strings) as function of the frequency have been performed by [START_REF] Waltham | Vibrational characteristics of harp soundboards[END_REF], [START_REF] Le Carrou | Some characteristics of the concert harp's acoustic radiation[END_REF].

However, the utility of mobility is limited to the local vibration at the excitation point.

In addition, we think it is worthwhile completing the analysis with other indicators, such as space average quadratic velocity and acoustic radiated power, which take into account the entire vibration of the soundboard. In this way, we consider an excitation located on different points of the main bridge, corresponding to the string coupling points ranging from note 32 (E3 with a fundamental frequencyof f 0 =164.8 Hz) to 76 (C7 with f 0 =2093 Hz), so from the left to the right of the entire main bridge (parts 1 and 2) in Fig. 6. Fig. 13 shows, from top to bottom: the mobility, the space average quadratic velocity and the acoustic radiated power. Each of them is expressed in dB scale, plotted with the same dynamic of 60 dB.

Note that in the case of a harmonic problem when considering the modal decomposition Eq. ( 2) and the relation between the initial and piano basis Eq. ( 5), the space average quadratic velocity in the soundboard area is given by the following expression [START_REF] Lesueur | Acoustic radiation of structures: vibroacoustic and fluid / structure interactions[END_REF]:

𝑣 3 (𝜔) = 𝐿 8 𝐿 ; 𝑆 m]D 𝜔 3 8 𝑎 \] 𝜔 3 J ]@7 _ \@7 = 𝐿 8 𝐿 ; 𝑆 m]D 𝜔 3 8 𝑏 D * (𝜔)𝑇 D 𝑇𝑏(𝜔) (19)
where t refers to the transpose matrix or vector and * to the complex conjugate. S int is the surface of the soundboard, i.e. the unblocked surface. Regarding the radiated acoustic power, its expression was given in Eq. ( 17). Globally, the three indicators follow the same trends. Firstly, dB levels tend to decrease when the frequency increases. Moreover, the strings coupled near the extremity and near the cut-off of the main bridge are sensitive to the local and sudden change of the bridge height. Thus, there is an increase of mobility, space average quadratic velocity and acoustic radiated power around 0.3 m and 1.2 m in Fig. 13,i.e. near the beginning of the bridge and near its break.

Secondly, in the frequency range [400;3000] Hz the dynamic level is around 1/3 lower than 400 Hz. This becomes more apparent at high frequency when the modal overlap masks the mode resonances.

𝐿 _ 𝑘𝑒𝑦 = 20 𝑙𝑜𝑔 1 𝑁 +jlDmji& 𝑤(𝑥 ¨k; , 𝑦 ¨k; , 𝑛 𝑓 ¨k; ) 𝐹

J «¬ž‰OE¬-K ]@7 (20)
𝐿 ® 𝑘𝑒𝑦 = 10 𝑙𝑜𝑔 1 𝑁 +jlDmji& 𝑊(𝑥 ¨k; , 𝑦 ¨k; , 𝑛 𝑓 ¨k; ) 𝑊 lk"

J «¬ž‰OE¬-K ]@7 (21)
Obviously, we are still far from a realistic force because the different partials do not have the same amplitudes in reality. However, the advantage of these indicators is that they present "pictures" of pianos, making it possible to compare different instrument designs.

In all the following figures, the X-axis limits are the same but the labels are different. That of the radiated power corresponds to the note numbers with the related note names whereas that of the mobility corresponds to the fundamental frequencies of each note. Moreover, all these figures are plotted with the same dynamic in order to compare the variation of each indicator. Up to now, we have only focused on the results relating to the P131 soundboard (Fig. 14). A major tendency is the inhomogeneity of mobility and power levels along the bridge (left Fig. 14), particularly at the bridge break between notes 58 and 59, where both increase considerably. The same applies for the notes near the beginning of the main bridge (near notes 32) where dB levels have the same orders of magnitude as at the break.

The inhomogeneity of the response along the bridge is not limited to its beginning and break. Regardless of the indicators, focusing on discrete frequencies also shows that the higher variations appear after the bridge break, which is not obvious on the mobility map view in Fig. 13. Above note 59, the wavelengths become small and the waves no longer pass through the bridge break, contrary to waves with lower frequencies. Therefore the second part of the main bridge (almost half the size of the first part of the main bridge, see Fig. 6) cannot distribute the vibration over the whole soundboard.

Moreover, the latter notes suffer from poor harmonic richness that maximizes the variations. Indeed, in the case of the first notes, the large number of harmonics will average the soundboard response, partly masking this phenomenon. Thus our model shows that the higher the order of notes is, the greater the variability may be.

This raises the question of the "killer-octave" which is, according to piano makers, around the penultimate octave (C6 to C7). Fandrich [START_REF] Fandrich | Soundboard Technology & Manufacture[END_REF] describes it as a region in the fifth to sixth octave (C5 to C6) of the keyboard. In both cases, it is characterized by low sustain and is present in most pianos regardless of construction method or soundboard design. Our results show two areas that could correspond to this phenomenon: the first with a large increase of the mobility and the acoustic radiated power; the second their decrease.

Focusing only on the average values would lead to an incomplete study. Indeed, in a playing situation, a major variation between two consecutive notes implies considerable perceptive inhomogeneity for the pianist and the listener. With this in mind, the gradients between successive notes provide much interesting information (right, Fig. 14). They show considerable variability between two successive notes and thus heterogeneous playing: the average mobility and acoustic radiated power gradients are around 1.33 and 1.23 dB, respectively.

The soundboard response can also be split into two ranges of notes. For the third and fourth octaves, the note-to-note gradients are low and the responses are globally homogeneous, which is also the case for both the mobility and the acoustic radiated power. Above the beginning of the fifth octave at around 659 Hz, the gradients increase, with higher values of around 4 dB. Thus, for high-pitched notes, it is common to have a perceived acoustic power division or multiplication by 2.5 compared to the previous notes. The same effect occurs near the beginning of the bridge but to a lesser extent. Note that a significant mobility gradient does not systematically mean the same for the acoustic radiated power and vice-versa. Moreover, we do not find a regular relationship between a decrease of the indicators and the location of the ribs (represented by dashed lines in Fig. 14): sometimes a coupling close to a rib tends to decrease the three indicators and sometimes it tends to increase them.

To review, the behavior can be split into two main ranges: after the bridge break, where the note-to-note gradients are higher and where there are major global variations of mobility and acoustic radiated power; before the bridge break where global variations of mobility and radiated power also exist but are lower. Highpitched notes (and the first notes) seem to be particularly difficult to control by piano makers, contrary to medium notes.

Applying a force at the bridge tends to distribute the force over the entire plate and thus homogenize the piano soundboard response. In the case of discontinuities, this distribution effect is broken if the discontinuity is significant enough compared to the wavelength of the wave propagated: at low frequencies, the wavelengths are long and the discontinuity has no impact on the propagation; at high frequencies, the wavelengths are short and a discontinuity will be an obstacle to the propagation of waves.

Finally, as with all string instruments, piano strings have multiple modal frequencies called partials or harmonics. For 32 th note, there are 18 frequencies below 3 kHz, whereas there is only one for the last note studied, number 76. Thus a discontinuity could have a higher influence on high-pitched notes and on the high order harmonics of the first notes. Consequently, it seems that if a bridge break is necessary in the design of the instrument it could be placed at low frequency to minimize its effect. On the contrary, the removal of any break at the bridge would homogenize the response of the instrument.

III.D. EXPERIMENTAL AND NUMERICAL COMPARISONS

To confirm the hypothesis made in our model, some comparisons were made with the Nastran FEM method and experimental results. The measurements were performed in the framework of an internship Master degree [START_REF] Margerit | Vibroacoustic behavior of an upright piano soundboard: comparison between experiment and analytical modelling[END_REF]. We present several results in this paper to validate the hypothesis made in our model. The Pleyel P131 soundboard used was separated from the case and clamped to a solid wall made of concrete to recreate baffled conditions. The structure was excited at the bridge in the treble zone with a white noise and the vibration measured on the rib side with a POLYTEC PSV-400 vibrometer laser with a mesh of 4600 points. Despite the geometrical simplifications in our model, the experimental results showed good agreement between the modal shapes and those of our model, as shown by the images in the top row in Fig. 15, for low frequency modes. In addition, the lower images in Fig. 15 give examples of high order modes around 2 kHz, where the superstructures strongly minimize the vibrations. However the first eigen-frequencies were higher in our model: the experimental eigen-frequency was around 66 Hz whereas that of the analytical model was around 100 Hz. Therefore the acoustic radiation was overestimated around the first modes without, however, changing the general trends. These differences highlight that the structure in our model was too rigid. This was not due to geometrical simplifications (superstructures parallel to plate edges, rib heights, etc.) since the spatial aspects (modal shapes) were well-described and compared with the measurements.

Experimental results

Analytical modelling

However, membrane effects may have had an impact on the eigen-frequencies of the first modes and quickly disappeared. To determine whether this was the case, we compared the experimental and our own results with a Nastran FEM model with the same geometrical simplifications. In order to confirm the geometrical simplifications, they were also simulated in the FEM model. The plate was meshed with PSHELL elements and the superstructures with PBEAM elements deported on each face. The FEM model is shown in Fig. 16 in which we consider two cases: the first with a blocked membrane imposing no displacements of the plate elements; the second with a membrane effect. To ensure the convergence of the calculations, the size of the finite elements must be at least 1mm due to the addition of stiffeners (see Fig. 17). On the contrary, the same figure shows that it is not useful to increase the order of truncation in our model even if it slightly increases the precision of our calculations. This result also shows that both models converge to the same solution via two different paths: our model converged from the top (increasing the order of truncation provides flexibility) whereas the FEM model converged from the bottom. The comparison of the three modelling and experimental results shows that the modal shapes are very similar, as shown in Fig. 18. This main result confirms the geometrical simplifications made in our model. On the contrary, the frequencies are different and also confirm our hypothesis relating to the membrane effect. Indeed, the frequencies of the FEM model without membrane were close to the frequencies of our model, whereas those of the FEM with membrane were close the experimental frequencies.

These comparisons show that the geometrical simplifications were relevant.

Membrane effects have an influence on the first eigen-frequencies and thus imply an overestimation of the acoustic radiation at low frequencies. However, it is important to keep in mind that the aim of this model is to perform parametrical studies. Therefore, even if the first frequencies were higher than those of the real soundboard, the Obviously, these comparisons assume a flat soundboard and not one in a playing situation. In practice the soundboard is manufactured with an initial crown in order to ensure the contact between the strings and the bridges. Applying and tuning the strings have two consequences: the initial crown lowers and compression stresses are applied to the plate edges. This phenomenon is known as "downbearing" and implies an increase of the first eigen-frequencies, as was shown in [START_REF] Moore | Interferometric studies of a piano soundboard[END_REF] and [START_REF] Ege | La table d'harmonie du piano-Études modales en basses et moyennes fréquences[END_REF] where we can find crossed comparisons of [START_REF] Kindel | Vibrations of a piano soundboard: Modal analysis and finite element analysis[END_REF], [START_REF] Conklin | Design and tone in the mechanoacoustic piano.Part II. Piano structure[END_REF], [START_REF] Moore | Interferometric studies of a piano soundboard[END_REF]- [START_REF] Nakamura | The vibrational character of the piano soundboard[END_REF].

However, simple reasoning leads to conclusions to the contrary. Indeed, lowering the crown implies decreasing the eigen-frequencies [START_REF] Elishakoff | Distribution of natural frequencies in certain structural elements[END_REF]. The same effect occurs when compression stresses are applied to the plate [START_REF] Busch-Vishniac | Drive point impedance of an infinite orthotropic plate under tension[END_REF], [START_REF] Géradin | Theory of vibrations: dynamic of structures[END_REF]- [START_REF] Ege | The piano soundboard: Modal studies in low and mid frequencies[END_REF]. In both cases, it quickly becomes negligible when the frequency increases. To explain this, it seems that the initial crown produced by manufacturing implies a non-linear evolution of the first eigen-frequencies of the soundboard. Thus the latter can increase or decrease depending on the initial crown: the consequences depend on the soundboard [START_REF] Mamou-Mani | Numerical simulation of a piano soundboard under downbearing[END_REF].

Nevertheless, neglecting downbearing effects does hinder parametrical studies in the same as membrane effects: the frequency shift will be around the same order of magnitude in all the following studies.

IV. SENSITIVITY TO STRUCTURAL CHANGES

From subsection III.C.3, we know that structural heterogeneities imply variations of the piano soundboard response, probably leading to a discontinuous timbre or perceived sound, which is a difficulty expressed by piano makers. Although the components of the force injected at the coupling point with the bridge do not have the same magnitudes, we think that ensuring a linear or progressive evolution of each indicator presented in III.C.3 will be interesting from the musical viewpoint. At least, their variations and especially the gradients should be greatly reduced in order to obtain more homogeneous playing.

Therefore we wanted to see how the vibro-acoustic behavior is modified through structural changes. Hence we now consider two additional cases illustrated in Fig. 19:

Pleyel P131 soundboard with a continuous main bridge; P131 soundboard with ribs 1 and 2 removed. These cases are compared to the reference case presented previously in III.C.3. In order to simplify the reading, solid lines are used instead of markers although we continue to focus on a spectrum of discrete frequencies.

IV.A. CONTINUOUS MAIN BRIDGE

Obviously, a continuous main bridge implies eliminating the defect between note numbers 58 and 59 because there is no longer any abrupt and local change of stiffness (Fig. 20, left). Moreover, the waves are better spread all over the soundboard because small wavelengths are no longer blocked by the bridge break.

Thus the global variations above note 56 are now lower than in the reference case (left Fig. 14), regardless of the indicators. Concerning the note-to-note gradients, their average values are lower than in the reference case, especially that of mobility (see average values in Fig. 20, right).

However, we can see that the gradients of high-pitched notes are lower but still remain high for both indicators. In the case of the acoustic radiated power, it leads to a perceived division or multiplication of this power by two from the previous notes (3 dB). For the notes originally placed before the bridge break, few changes can be noted on the mobility gradients. On the contrary, a continuous bridge increases the acoustic radiated power gradients in the frequency band [165;650] Hz. Therefore, as the average radiated power gradients are around the same order of magnitude for both cases, it appears that it is to the detriment of mid-frequency notes (fundamental Average values of acoustic radiated power gradients frequency lower than 650 Hz) and to the advantage of notes with a fundamental frequency higher than 650 Hz.

IV.B. REMOVAL OF RIBS IN THE TREBLE ZONE

Contrary to the previous case, the removal of two ribs has no effect on the first part of the main bridge (below note number 56 in Fig. 21). In this range, the changes are negligible for both indicators, as much for the average values as for the gradients.

Obviously, the effect of the main bridge break remains, with a pronounced increase of the two indicators around note number 58 (Fig. 21, left), and also significant local gradients (Fig. 21, right). For high-pitched notes, the two indicators do not follow the same trends, contrary to the previous cases. For high-pitched notes, the mobility level increases because the stiffness becomes lower locally due to the removal of the stiffeners (top left Fig. 21). The loss of mobility from notes 62 to 67 in the reference case (900 to 1250 Hz) has now been eliminated. At higher frequencies, mobility increases until note number 71 (1550 Hz) and finally decreases. Thus the removal of ribs will lead to a lower sustain due to the quicker injection of string energies in highpitched notes.

However, this solution presents the advantage of substantially decreasing the gradient between notes: globally the ratio of average gradients is around 3/4 between the two cases (Fig. 21, top right). Focusing above note number 62, this ratio falls to around 1/2 with average gradients of 1.5 dB and 0.8 dB, respectively, for the reference case and the P131 soundboard without two ribs. As for the mobility, the acoustic radiated power is impacted after the bridge break and seems to approximately follow a decreasing linear tendency, passing in the middle of the reference case (Fig. 21, bottom right). However, some variations of radiated power moving to high frequency remain. Consequently, the maximum gradient is only slightly lower but its average gradient is the same.

Finally, for the high-pitched notes, removing the ribs implies higher mobility and thus a lower sustain. By way of compensation, the variations of the two indicators are lower than in the reference case. From the musical viewpoint, removing the ribs allows more homogeneous playing with lower variations between consecutive notes for this frequency range.

These parametrical studies show the difficulty of controlling the piano soundboard response, particularly in the treble zone, which can be considered as a sensitive range. For these notes, a structural change has a considerable effect on their behavior, as shown in this article. Moreover, the bridge break homogenizes the behavior of the third and fourth octaves. Regarding the bridge length, it could be interesting to increase it at the beginning without changing the string coupling points, in order to homogenize the behavior of the first notes. We also suggest several improvements in order to reduce global and note-to-note variations in the ranges where the response is inhomogeneous.

Gradients between two successive notes [dB]

Considering the design constraints (addition of stiffeners over the whole soundboard due to the stress applied by the strings and breaking the bridge to let the cast iron plate pass through), the reference case demonstrates the skill and knowledge of piano makers. Indeed, their empirical method of design allows them to control the sound produced without scientific prediction tools.

V. CONCLUSION

This article presented how to model the piano soundboard, that is to say a complex structure with non-rectangular contours, composed of a material with an angle of orthotropy and several stiffeners in positioned in different direction. Here, we limited our study to the case of an upright piano soundboard in the frequency domain [0;3000] Hz.

The model developed allowed us to calculate the modes of this complex ribbed structure and classify them into different families: those for which modal shapes are similar to unribbed plate modes, those for which only strong superstructures inhibit the vibration, those for which all superstructures block vibrations, and those for which vibration is localized in different areas of the soundboard. It was shown that these families depend on the frequency ranges and the wavelengths of modes.

Moreover, we also presented maps of the mobility, the space average quadratic velocity and the acoustic radiated power as a function of the excitation points (string coupling positions). Moreover, we showed that focusing on these maps is not fully relevant and introduced indicators that are more accurate from the musical standpoint.

We highlighted a frequency range that seems to be difficult to control for piano makers, due to high global and note-to-note variability (high-pitched notes), whereas the low and mid-range frequencies are well mastered by them. Mainly guided by technical constraints, the traditional design appears to be a good solution considering that piano makers do not necessarily use scientific prediction tools.

To conclude, although the model presented in this paper suffers from limitations (membrane effects, downbearing and geometrical simplifications), and so does not describe a soundboard in a playing situation, it appears that these limitations do not affect the general trends, meaning that the method is relevant to parametrical studies. Some of the proposals, such as a continuous main bridge and removing ribs for highpitched notes, provide paths of improvement to make playing more homogenous.

However, they imply changing the design of the instrument to make it possible. that we define as the torsion dynamic rigidity of an orthotropic beam in the plane 𝑦, 𝑧 .

For the sake of simplicity, we consider the beam as isotropic in the following. 𝐷 n becomes 𝐷 n = 𝐺 › 𝐼 n› and we define 𝐸 › as the Young modulus of the bridge. Thus we express the Hamiltonian of the bridge: On the left: the initial plate with specific orthotropy; on the right: the addition of springs. The non-rectangular contours and the non-specified angle of orthotropy are well described.

Table of figures

Fig. 1 -

 1 Fig. 1 -(Color online) Upright piano soundboard of Pleyel P131 with Ex and Ey being the strong and the weak Young's moduli of the wood. View from front side (bridge side) with ribs and cut-off corner beams added in transparency.

Fig. 2 -

 2 Fig. 2 on the right. The boundary conditions will tend to clamped conditions.

Fig. 3 -

 3 Fig. 3 -(Color online) Examples of possible geometries with different orthotropy angles and edges.

Fig. 4 -

 4 Fig. 4 -(Color online) Influence of spring rigidity. a) Average quadratic velocity ratio for two spring rigidities. Solid line: sufficient rigidity; dashed line: insufficient rigidity. The excitation is placed at (xe,ye)=(1.26,0.76). b) Modal density per octave for two spring rigidities and analytical plot from [11], [17], [67] Eq. (A.13): circle markers: sufficient spring rigidity; square markers: insufficient spring rigidity; solid line: analytical expression.

- 1 ]

 1 Analytical modal density of a clamped plate with an angle of orthotropy of -34.804 °Sufficient spring rigidity (k=2.5E6) Insufficient spring rigidity (k=2.5E4)

Fig. 5 -

 5 Fig. 5 -(Color online) Modal shapes of a clamped non-rectangular plate with an angle of orthotropy of 34.8°. Left: insufficient spring rigidity; Center: sufficient spring rigidity; Right: FEM-NASTRAN with a mesh of 5mm.

Fig. 6 -

 6 Fig. 6 -(Color online) Geometry modelled for the Pleyel P131.

Fig. 7 -

 7 Fig. 7 -(Color online) Rib geometry. a) Section view of a real soundboard rib. b) Simplified rib in the present model. c) Partial front view of the soundboard.

Fig. 8 -

 8 Fig. 8 -(Color online) Classification of the first 200 modes of a Pleyel P131 soundboard.

Fig. 9 -

 9 Fig. 9 -(Color online) Low frequency modes similar to unribbed plate modes.

Fig. 10 -

 10 Fig. 10 -(Color online) Modes where the transverse displacement is strongly minimized by the bridges.

Fig. 11 -

 11 Fig. 11 -(Color online) Modes where the transverse displacement is strongly minimized by all the superstructures.

Fig. 12 -

 12 Fig. 12 -(Color online) Modes where the vibrations are localized into areas delimited by superstructures.

Fig. 13 -

 13 Fig. 13 -(Color online) Evolution of local and general indicators at string/main bridge coupling points: Top: mobility; Center: space average quadratic velocity; Bottom: Acoustic radiated power.

Fig. 14 -

 14 Fig. 14 -(Color online) Evolution of mobility (top) and acoustic radiated power (bottom) of a P131 soundboard. Left: dB values of indicators; Right: gradients (absolute values) between two successive notes in dB.

Fig. 15 -

 15 Fig. 15 -(Color online) Comparison of Pleyel P131 modal shapes between our analytical modelling and experimental results.

Fig. 16 -

 16 Fig. 16 -(Color online) Nastran FEM model of Pleyel P131 soundboard with the same geometrical simplifications as in our model. a) and c) view of ribs view with cut-off corner beams; b) and d) bridge view.

Fig. 17 -

 17 Fig. 17 -(Color online) Eigen-frequency convergences for our model and the Nastran FEM model (without membrane) as a function of truncation order and the mesh size.

  membrane effects do not change the conclusions resulting from the comparison made with our model.

Fig. 18 -

 18 Fig. 18 -(Color online) Comparison of numerical and experimental modal shapes using four methods. From left to right: analytical modelling, Nastran FEM without membrane, Nastran FEM with membrane and experimental results.

Fig. 19 -

 19 Fig. 19 -(Color online) Different sets of upright piano soundboards based on the Pleyel P131 configuration. a) Reference case: Pleyel P131; b) Continuous main bridge; c) Removal of ribs for highpitched notes.

Fig. 20 -

 20 Fig. 20 -(Color online) Comparison between the P131 soundboard analyzed in III.C.3 and the P131 soundboard with a continuous main bridge.

Fig. 21 -

 21 Fig. 21 -(Color online) Comparison between the P131 soundboard analyzed in III.C.3 and the P131 soundboard with ribs 1 and 2 removed.

Fig. 1 -

 1 Fig. 1 -(Color online) Upright piano soundboard of Pleyel P131 with E x and E y being the strong and the weak Young's moduli of the wood. View from front side (bridge side) with ribs and cut-off corner beams added in transparency. Fig. 2 -(Color online) The addition of springs on a rectangular simply supported plate.On the left: the initial plate with specific orthotropy; on the right: the addition of springs. The non-rectangular contours and the non-specified angle of orthotropy are well described. Fig. 3 -(Color online) Examples of possible geometries with different orthotropy angles and edges. Fig. 4 -(Color online) Influence of spring rigidity. a) Average quadratic velocity ratio for two spring rigidities. Solid line: sufficient rigidity; dashed line: insufficient rigidity. The excitation is placed at (x e ,y e )=(1.26,0.76). b) Modal density per octave for two spring rigidities and analytical plot from [11], [17], [67] Eq. (A.13): circle markers: sufficient spring rigidity; square markers: insufficient spring rigidity; solid line: analytical expression. Fig. 5 -(Color online) Modal shapes of a clamped non-rectangular plate with an angle of orthotropy of 34.8°. Left: insufficient spring rigidity; Center: sufficient spring rigidity; Right: FEM-NASTRAN with a mesh of 5mm. Fig. 6 -(Color online) Geometry modelled for the Pleyel P131. Fig. 7 -(Color online) Rib geometry. a) Section view of a real soundboard rib. b) Simplified rib in the present model. c) Partial front view of the soundboard. Fig. 8 -(Color online) Classification of the first 200 modes of a Pleyel P131 soundboard. Fig. 9 -(Color online) Low frequency modes similar to unribbed plate modes. Fig. 10 -(Color online) Modes where the transverse displacement is strongly minimized by the bridges. Fig. 11 -(Color online) Modes where the transverse displacement is strongly minimized by all the superstructures. Fig. 12 -(Color online) Modes where the vibrations are localized into areas delimited by superstructures. Fig. 13 -(Color online) Evolution of local and general indicators at string/main bridge coupling points: Top: mobility; Center: space average quadratic velocity; Bottom: Acoustic radiated power. Fig. 14 -(Color online) Evolution of mobility (top) and acoustic radiated power (bottom) of a P131 soundboard. Left: dB values of indicators; Right: gradients (absolute values) between two successive notes in dB. Fig. 15 -(Color online) Comparison of Pleyel P131 modal shapes between our analytical modelling and experimental results.Fig. 16 -(Color online) Nastran FEM model of Pleyel P131 soundboard with the same geometrical simplifications as in our model. a) and c) view of ribs view with cut-off corner beams; b) and d) bridge view.

Fig. 3 -

 3 Fig. 1 -(Color online) Upright piano soundboard of Pleyel P131 with E x and E y being the strong and the weak Young's moduli of the wood. View from front side (bridge side) with ribs and cut-off corner beams added in transparency. Fig. 2 -(Color online) The addition of springs on a rectangular simply supported plate.On the left: the initial plate with specific orthotropy; on the right: the addition of springs. The non-rectangular contours and the non-specified angle of orthotropy are well described. Fig. 3 -(Color online) Examples of possible geometries with different orthotropy angles and edges. Fig. 4 -(Color online) Influence of spring rigidity. a) Average quadratic velocity ratio for two spring rigidities. Solid line: sufficient rigidity; dashed line: insufficient rigidity. The excitation is placed at (x e ,y e )=(1.26,0.76). b) Modal density per octave for two spring rigidities and analytical plot from [11], [17], [67] Eq. (A.13): circle markers: sufficient spring rigidity; square markers: insufficient spring rigidity; solid line: analytical expression. Fig. 5 -(Color online) Modal shapes of a clamped non-rectangular plate with an angle of orthotropy of 34.8°. Left: insufficient spring rigidity; Center: sufficient spring rigidity; Right: FEM-NASTRAN with a mesh of 5mm. Fig. 6 -(Color online) Geometry modelled for the Pleyel P131. Fig. 7 -(Color online) Rib geometry. a) Section view of a real soundboard rib. b) Simplified rib in the present model. c) Partial front view of the soundboard. Fig. 8 -(Color online) Classification of the first 200 modes of a Pleyel P131 soundboard. Fig. 9 -(Color online) Low frequency modes similar to unribbed plate modes. Fig. 10 -(Color online) Modes where the transverse displacement is strongly minimized by the bridges. Fig. 11 -(Color online) Modes where the transverse displacement is strongly minimized by all the superstructures. Fig. 12 -(Color online) Modes where the vibrations are localized into areas delimited by superstructures. Fig. 13 -(Color online) Evolution of local and general indicators at string/main bridge coupling points: Top: mobility; Center: space average quadratic velocity; Bottom: Acoustic radiated power. Fig. 14 -(Color online) Evolution of mobility (top) and acoustic radiated power (bottom) of a P131 soundboard. Left: dB values of indicators; Right: gradients (absolute values) between two successive notes in dB. Fig. 15 -(Color online) Comparison of Pleyel P131 modal shapes between our analytical modelling and experimental results.Fig. 16 -(Color online) Nastran FEM model of Pleyel P131 soundboard with the same geometrical simplifications as in our model. a) and c) view of ribs view with cut-off corner beams; b) and d) bridge view.

Fig. 7 -

 7 Fig. 1 -(Color online) Upright piano soundboard of Pleyel P131 with E x and E y being the strong and the weak Young's moduli of the wood. View from front side (bridge side) with ribs and cut-off corner beams added in transparency. Fig. 2 -(Color online) The addition of springs on a rectangular simply supported plate.On the left: the initial plate with specific orthotropy; on the right: the addition of springs. The non-rectangular contours and the non-specified angle of orthotropy are well described. Fig. 3 -(Color online) Examples of possible geometries with different orthotropy angles and edges. Fig. 4 -(Color online) Influence of spring rigidity. a) Average quadratic velocity ratio for two spring rigidities. Solid line: sufficient rigidity; dashed line: insufficient rigidity. The excitation is placed at (x e ,y e )=(1.26,0.76). b) Modal density per octave for two spring rigidities and analytical plot from [11], [17], [67] Eq. (A.13): circle markers: sufficient spring rigidity; square markers: insufficient spring rigidity; solid line: analytical expression. Fig. 5 -(Color online) Modal shapes of a clamped non-rectangular plate with an angle of orthotropy of 34.8°. Left: insufficient spring rigidity; Center: sufficient spring rigidity; Right: FEM-NASTRAN with a mesh of 5mm. Fig. 6 -(Color online) Geometry modelled for the Pleyel P131. Fig. 7 -(Color online) Rib geometry. a) Section view of a real soundboard rib. b) Simplified rib in the present model. c) Partial front view of the soundboard. Fig. 8 -(Color online) Classification of the first 200 modes of a Pleyel P131 soundboard. Fig. 9 -(Color online) Low frequency modes similar to unribbed plate modes. Fig. 10 -(Color online) Modes where the transverse displacement is strongly minimized by the bridges. Fig. 11 -(Color online) Modes where the transverse displacement is strongly minimized by all the superstructures. Fig. 12 -(Color online) Modes where the vibrations are localized into areas delimited by superstructures. Fig. 13 -(Color online) Evolution of local and general indicators at string/main bridge coupling points: Top: mobility; Center: space average quadratic velocity; Bottom: Acoustic radiated power. Fig. 14 -(Color online) Evolution of mobility (top) and acoustic radiated power (bottom) of a P131 soundboard. Left: dB values of indicators; Right: gradients (absolute values) between two successive notes in dB. Fig. 15 -(Color online) Comparison of Pleyel P131 modal shapes between our analytical modelling and experimental results.Fig. 16 -(Color online) Nastran FEM model of Pleyel P131 soundboard with the same geometrical simplifications as in our model. a) and c) view of ribs view with cut-off corner beams; b) and d) bridge view.

Fig. 16 -

 16 Fig. 1 -(Color online) Upright piano soundboard of Pleyel P131 with E x and E y being the strong and the weak Young's moduli of the wood. View from front side (bridge side) with ribs and cut-off corner beams added in transparency. Fig. 2 -(Color online) The addition of springs on a rectangular simply supported plate.On the left: the initial plate with specific orthotropy; on the right: the addition of springs. The non-rectangular contours and the non-specified angle of orthotropy are well described. Fig. 3 -(Color online) Examples of possible geometries with different orthotropy angles and edges. Fig. 4 -(Color online) Influence of spring rigidity. a) Average quadratic velocity ratio for two spring rigidities. Solid line: sufficient rigidity; dashed line: insufficient rigidity. The excitation is placed at (x e ,y e )=(1.26,0.76). b) Modal density per octave for two spring rigidities and analytical plot from [11], [17], [67] Eq. (A.13): circle markers: sufficient spring rigidity; square markers: insufficient spring rigidity; solid line: analytical expression. Fig. 5 -(Color online) Modal shapes of a clamped non-rectangular plate with an angle of orthotropy of 34.8°. Left: insufficient spring rigidity; Center: sufficient spring rigidity; Right: FEM-NASTRAN with a mesh of 5mm. Fig. 6 -(Color online) Geometry modelled for the Pleyel P131. Fig. 7 -(Color online) Rib geometry. a) Section view of a real soundboard rib. b) Simplified rib in the present model. c) Partial front view of the soundboard. Fig. 8 -(Color online) Classification of the first 200 modes of a Pleyel P131 soundboard. Fig. 9 -(Color online) Low frequency modes similar to unribbed plate modes. Fig. 10 -(Color online) Modes where the transverse displacement is strongly minimized by the bridges. Fig. 11 -(Color online) Modes where the transverse displacement is strongly minimized by all the superstructures. Fig. 12 -(Color online) Modes where the vibrations are localized into areas delimited by superstructures. Fig. 13 -(Color online) Evolution of local and general indicators at string/main bridge coupling points: Top: mobility; Center: space average quadratic velocity; Bottom: Acoustic radiated power. Fig. 14 -(Color online) Evolution of mobility (top) and acoustic radiated power (bottom) of a P131 soundboard. Left: dB values of indicators; Right: gradients (absolute values) between two successive notes in dB. Fig. 15 -(Color online) Comparison of Pleyel P131 modal shapes between our analytical modelling and experimental results.Fig. 16 -(Color online) Nastran FEM model of Pleyel P131 soundboard with the same geometrical simplifications as in our model. a) and c) view of ribs view with cut-off corner beams; b) and d) bridge view.

Fig. 17 -

 17 Fig. 17 -(Color online) Eigen-frequency convergences for our model and the Nastran FEM model (without membrane) as a function of truncation order and the mesh size. Fig. 18 -(Color online) Comparison of numerical and experimental modal shapes using four methods. From left to right: analytical modelling, Nastran FEM without membrane, Nastran FEM with membrane and experimental results. Fig. 19 -(Color online) Different sets of upright piano soundboards based on the Pleyel P131 configuration. a) Reference case: Pleyel P131; b) Continuous main bridge; c) Removal of ribs for high-pitched notes. Fig. 20 -(Color online) Comparison between the P131 soundboard analyzed in III.C.3 and the P131 soundboard with a continuous main bridge. Fig. 21 -(Color online) Comparison between the P131 soundboard analyzed in III.C.3 and the P131 soundboard with ribs 1 and 2 removed.

  𝐸 8 and 𝐸 ; the two Young moduli of the plate.

	Q B R S ET EUV B? V ?B	, 𝐷 : =	Q ? R S ET EUV B? V ?B	, 𝐷 3 =	V ?B Q B R S W EUV B? V ?B	and 𝐷 < =	X B? R S S
	the plate dynamic stiffness; 𝜈 8; and 𝜈 ;8 = 𝜈 8;	𝐸 ;	𝐸 8	its Poisson coefficients;

7 an arbitrary time interval; 𝐿 8 and 𝐿 ; the plate dimensions; 𝜌 the plate mass density; ℎ the plate thickness; 𝐷 7 =

  With 𝑀 &*•]š'*jlš = 𝑇 D 𝑀 +ijDk + 𝑀 &+lm]n& + 𝑀

			𝑟𝑖𝑏𝑠	+ 𝑀	𝑏𝑟𝑖𝑑𝑔𝑒𝑠 𝑇 and 𝐾 &*•]š'*jlš =
	𝑇 D 𝐾 +ijDk + 𝐾 &+lm]n& + 𝐾	𝑟𝑖𝑏𝑠	+ 𝐾

𝑏𝑟𝑖𝑑𝑔𝑒𝑠 𝑇 . Damping matrix 𝐶 is the same as that presented in Sec. II.A.3.

Table 1 -Properties of the extended orthotropic spruce plate.

 1

		Main bridge	Main bridge	Bass	Cut-off corner	Cut-off corner
		part 1	part 2	bridge	beam 1	beam 2
	𝜌 › [kg.m -3 ]			674
	H c [m]	0.033	0.038	0.0185	0.0192
	b c [m]	0.035	0.0325	0.0345
	E c [Pa]			1.40E+10
	G c [Pa]			4.70E+008
	y c [m]	0.78	0.72	0.48	0.26	1.27
	L [m]	0.88	0.41	0.47	0.51	0.52

Table 2 -Dimensions and properties of bridges and "cut-off" corner beams.

 2

	Rib n°	1		2	3	4	5		6	7	8	9	10
	𝜌 l [kg.m -3 ]							440
	H [m]	0.01 5	0.019	0.02	0.02	0.02	0.02	0.02	0.022	0.019	0.016
	b [m]							0.025
	E r [Pa]						1.59E+10
	G r [Pa]						6.20E+008
	L 1 [m]	0.04	0.05	0.07	0.08	0.15	0.16	0.17	0.18	0.18	0.10
	L 2 [m]	0.21	0.41	0.57	0.69	0.76	0.76	0.65	0.56	0.46	0.31
	L 3 [m]	0.11	0.16	0.14	0.11	0.08	0.10	0.11	0.10	0.10	0.10
	Rib spacing [m]	0.117 0.123 0.127	0.147	0.143 0.144 0.134	0.132 0.123

Table 3 -Dimensions and properties of ribs.

 3

	1
	L x	Plate length	L y	Plate width
	𝜌	Plate mass density	h	Plate thickness
	D i	Plate dynamic stiffness	𝜌 l	Rib mass density
	E r	Rib Young's modulus	H	Rib height
	b	Rib width	I f	Rib momentum of bending inertia
	G r	Rib shear modulus	I g	Rib momentum of torsion inertia
	E c	Bridge Young's modulus	H c	Bridge height
				Bridge momentum of bending
	b c	Bridge width	I fc
				inertia
				Bridge momentum of torsion
	G c	Bridge shear modulus	I gc
				inertia
	𝜌 ›	Bridge mass density	H plate	Plate Hamiltonian
	H rib	Rib Hamiltonian	H bridge	Bridge Hamiltonian

Table 4 -Description of the different constants used in the Hamiltonians.

 4

Acknowledgments

This work was performed in the framework of the Labex CeLyA of Université de Lyon, supervised by the French National Research Agency (Nos. ANR-10-LABX-0060/ ANR-11-IDEX-0007).

The authors thank Pierre Margerit for the measurements obtained during his Master degree internship at LVA. We are also very grateful to ITEMM (European Technological

However, the dynamic level of the mobility is higher than the others, regardless of the frequency range (see Fig. 13). This is consistent because mobility is a local indicator whereas the others take into account the entire behavior of the soundboard. Below 400 Hz, the mobility fluctuates around 65 dB whereas the space average quadratic velocity and the radiated power only around 40 dB, so around 1/3 lower. Above 400 Hz, the ratio is around the same order of magnitude.

III.C.3. MOBILITY AND ACOUSTIC RADIATED POWER FROM THE MUSICAL

VIEWPOINT

From the musical viewpoint, the large quantity of information in this kind of presentation makes it difficult to interpret the results. Indeed, the response of an instrument is not continuous in frequency but discrete, in addition to the discrete variation of the excitation position. In the case of a harp soundboard, [START_REF] Waltham | Vibrational characteristics of harp soundboards[END_REF], [START_REF] Le Carrou | Some characteristics of the concert harp's acoustic radiation[END_REF] focused on admittance at discrete frequencies corresponding to string fundamentals. These authors also did the same for the first and the second partials, neglecting the inharmonicity brought about by mobility, i.e. by simply multiplying the fundamental frequencies. In the following, we extend this idea to the acoustic radiated power. This study is limited to partials due to string transverse waves below 3 kHz. We present the average values of each indicator and the gradients between successive notes (in absolute values) from the 32 th to 76 th . All the partials have the same magnitudes. We give the following expressions for the average values of mobility and acoustic radiated power: Institution for Music Professions) and particularly to Marthe Curtit and Patrick Sinigaglia 1 for having kindly given the Pleyel P131 soundboard used in this study to LVA.

Appendix A Hamiltonian calculations of a superstructure

In this appendix, we give the main steps of the calculations for expressing the Hamiltonian of an arbitrary superstructure. To make reading simpler, we have chosen a bridge. At the end of the appendix, we give the function for a rib including the change of height.

The partial Hamiltonian is defined as:

Where 𝐸 › refers to kinetic energy and 𝐸 š refers to strain energy. 𝑡 A ; 𝑡 7 is an arbitrary time interval.

We first begin by defining the displacement field of the superstructure in direction 𝑥 at position y=y c : Where 𝑦 ∈ 𝑦 › -𝑏 › 2 ; 𝑦 › + 𝑏 › 2 and 𝑧 ∈ ℎ 2 ; ℎ 2 + 𝐻 › where ℎ and 𝐻 are respectively the plate thickness and the rib height.

The kinetic and strain energies are calculated on the entire bridge volume placed at y=y c . The superstructure is defined for 𝑥 ∈ 𝑥 m]m ; 𝑥 "m] . Let the kinetic energy be:

By replacing 𝑢 › 3 , 𝑣 › 3 and 𝑤 › 3 by their expression from A.2 and after making the integrals of variables y and z, we obtain:

Where the bending inertia is equal to 𝐼 "› = ' µ : ℎ 2 + 𝐻 › : -ℎ 2 : and torsion inertia is

.

For the strain energy, we must calculate the small deformation tensor 𝝐 = The strains are linked to deformation through a flexibility matrix. The matrix 𝑀 & for an orthotropic material is: The same approach is followed for the ribs. Beyond the change of orientation, the only change is that the height of the ribs is different at the extremities (around 5mm). See