9th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 7 PIANO SOUNDBOARD UNDER PRESTRESS: A NUMERICAL APPROACH PACS: 43.75.Mn Mamou-Mani, Adrien ; Frelat, Joël ; Besnainou, Charles Institut Jean Le Rond d Alembert, Université Pierre et Marie Curie; 4 place Jussieu 755 Paris, France; mamou-mani@lmm.jussieu.fr frelat@lmm.jussieu.fr chbesnai@ccr.jussieu.fr ABSTRACT The piano soundboard is subjected to different sources of prestress, due to its assembling with the ribs and the bridges, and the downbearing due to the strings tension. A numerical simulation of the soundboard with its different compounds and an initial crown gives effects of the initial stresses on the shape and the vibrations of the soundboard. Linear and nonlinear approaches of prestress are evaluated in term of modal geometries and frequencies. The simulation drives to appreciable differences in modal frequencies between the cases of a soundboard without prestress, with linear evaluation and nonlinear evaluations of prestresses. INTRODUCTION The piano soundboard vibrations have been investigated experimentally several times, see [] [3][4] [5] for examples. [4],[5] and [7] have studied the effects of the strings tension on modal properties. They observed changes in input impedance and modal frequencies with the the strings tension. In fact, the strings apply a downward force called "downbearing". The instrument makers rely the adjustment of this downbearing to the initial crown they give to the soundboard [].The combination of these procedures achieves the final quality of the instrument. Using the prestress theory [6], we present here a model demonstrating the effects of the crown and the downbearing on the modal behavior of the soundboard. PRESTRESS THEORY The Hamilton principle for a prestress structure is (see [6]) δ t t (T V int V g V ext)dt = (Eq. ) δu i (t ) = δu i (t ) = (Eq. ) where V g = Vext = Vint = V σ ijε () V C ijkl ( V ε () ij T = ρ u i u i dv the kinetic energy (Eq. 3) V ε () kl )dv the linear strain energy (Eq. 4) ij dv the geometric strain energy due to prestress (Eq. 5) X i u i dv t i u i ds the potential of the external loading S σ (Eq. 6) u i the displacement (Eq. 7)
u m x j ) where ρ is the density, σij is the initial stress field, ε () ij = ( u i x j + u j x i ) and ε () ij = ( u m x i are respectively the linear part and the quadratic part of the strain field, C ijkl is the elasticity tensor, X i and t i are respectively the volume and surface forces. Using a finite element method, the discretized Hamilton principle is t δ ( t qt M q qt Kq qt K g q + q T g)dt = (Eq. 8) where M is the mass matrix, K is the stiffness matrix K g is the geometric stiffness matrix,g is the load vector and q are the generalzed coordinates of the problem. The discretized equations of motion are: (K + K g )q + M q = g(t) (Eq. 9) and the eigenvalues problem: (K + K g )q = ω Mq (Eq. ) where ω are the eigenpulsations of the structure. There are two ways to evaluate the geometric rigidity K g, a linearized and a nonlinear one. One of the goal of this paper is to determine what is the more appropriate procedure to the piano soundboards modelling. Here is the descriptions of these two approaches. Linearized geometric rigidity approximation If the initial deformations are sufficiently small, K g = λk g (Eq. ) where K g is the geometric rigidity of an infinitesimal transformation and λ the loading parameter, and the eigenvalue problem becomes: (K + λk g)q = ω Mq (Eq. ) Nonlinear model Incremental iterative schemes are necessary in case of much larger displacements. They are particularly satisfying when the system is sufficiently far from a critical point (snap-through or bucking mode), which is the case here. We have chosen the classical Newton-Raphson method. FINITE-ELEMENT MODEL Figure : the finite-element model of the soundboard 9th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID
Finite-element model for the soundboard of a.8m IBACH piano presented in figure has been carried out using CAST3M software. The complete mesh is composed of 858 nodes for elements of Love-Kirchhoff shell type (see [8]) (even for the ribs and the bridges). geometrical properties: a simplified geometry is assumed: a constant thickness of 8mm for the plate and a completely clamped bass bridge (much more complex in reality). The crown has been achieved with a regular form obtained by a continuous deformation of the plate (Figure ). mechanical properties: The elastic parameters and density were chosen to correspond to typical values of spruce: density: 5kg/m 3, young modulus E = P a, E = 9 P a, shear modulus G = 9 P a, Poisson ratio ν =.3, i = : direction along the grain of wood; i =, 3: directions across the grain of wood. boundary conditions and loadings: we assume a simply supported plate to take into account the decrease of the thickness close to the edges. The downbearing is modeled by a constant vertical distributed force applied on the two bridges, varying from to 5N/m. Figure : Crown profile (on the left) and final geometry with the additional transverse load (on the right) NUMERICAL RESULTS Modal shapes Figure 3: First five modes of the soundboard without crown and downbearing In figure 3, the modal shapes and frequencies of the first five eigenmodes are presented for a flat soundboard without crown and downbearing. But the modal shapes are not modified by the crown and the downbearing. Then the effects of prestress on the only eigenfrequencies will be discussed below. The downbearing: linear or nonlinear evaluation of prestresses? The choice of a linear or a nonlinear prestress approach of the downbearing is driven by the magnitude of the induced displacements. A significant difference is observed for the first five eigenfrequencies when displacements are close to the thickness of the plate. The linearized theory gives appreciable smaller values (see figure 4 for five different crown values as multiples of the thickness). The initial crown has a non negligible influence on these differences. The example of the first mode in figure 5 shows the more the crown is, the more the nonlinear approach is preferable. 3 9th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID
3 3 5 5 frequencies 5 frequencies 5 nonlinear model geo rigidity 5 nonlinear model geo rigidity 5..4.6.8...4...3.4.5.6.7.8.9. Figure 4: Evolution of the first five eigenfrequencies with a nonlinear theory and a geometrical rigidity approximation for a soundboard with no crown (left) and with a crown equal to two times the thickness (right)..8.6.4...8.6.5.5.4..5..5 Figure 5: Relative difference of frequency between the first modal frequency obtained with geometrical rigidity approximation and with a complete nonlinear model, for five different crown values (multiples of the thickness) Effects of the crown and the downbearing on modal properties The nonlinear approach is now adopted because it describes better the effects of the downbearing. The calculus shows no appreciable differences between the modal shapes of the soundboard without and with crown and downbearing. This result is in contradiction with the experimental observation of [7]. This difference probably results from the direction of the downbearing that we choose simply vertical. A strong horizontal component of the tensions of strings potentially induces modifications in modal geometries. Concerning the frequencies, the numerical model predicts a significant evolution, particularly for the first mode, as shown in figure 9. The increasing or decreasing of the frequencies depends of the crown value. For a non crowning soundboard, all modes increase with downbearing. This result is in good agreement with [4], specially for the first mode presented in figure 6,7,8. But when the crown increases, a minimum in the frequencies evolutions, occurs. However, this result has never been observed experimentally, probably because the initial crown was not considered as influencing the downbearing effects. Finally, the qualitative evolutions of the five eigenfrequencies are extremely similar, but quantitatively different. CONCLUSIONS The downbearing has considerable effects on the modal frequencies of the presented numerical soundboard. The modeling of these effects with a nonlinear approach gives results close to experimental measurements for a flate soundboard and shows significant differences with a linear approximation. Moreover, the crown modifies the behavior of the soundboard under the downbearing. The crowning process has to be evaluate initially, before to predict the influence of the downbearing. 4 9th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID
.6.5.4.3...9.8.7.5.5.5..5 Figure 6: Evolution of the modal frequencies with the transverse load for five different crown values for the first mode.5..5..5.95.5.5.5..5.95.5.5.9.5..5.9.5..5 Figure 7: Evolution of the modal frequencies with the transverse load for five different crown values for the second mode (left) and the third mode (right) ACKNOWLEDGMENTS The authors want to thank C. Barrois, R. Bresson and F. Conti, students of the Ecole Nationale des Ponts et Chaussées, for their contributions to the finite-element model. References [] Fenner, K., La Table d Harmonie du Piano: Construction, EuroPiano 998 [] Suzuki, H., Vibrations and sound radiation of a piano soundboard, J. Acoust. Soc. Am., 8 (986), 573-58 [3] Nakamura, I., The vibrational character of the piano soundboard, Proc. th ICA, Paris (983), vol. 4, p.385 [4] Wogram, K., in Five Lectures on the Acoustics of the Piano, edited by A. Askenfelt, Royal Swedish Academy of Music, Stockholm, 99 [5] Conklin, H. A. Jr, in Five Lectures on the Acoustics of the Piano, edited by A. Askenfelt, Royal Swedish Academy of Music, Stockholm, 99 [6] Géradin, M. and Rixen, D., Mechanical Vibrations. Theory and Application to Structural Dynamics. John Wiley & Sons, 994 [7] Moore, T. R. and Zietlow, S. A., Interferometric studies of a piano soundboard, J. Acoust. Soc. Am. 9 (6), 783-793 [8] Batoz, J.-L., Dhatt G., Modélisation des structures par élements finis, Hermes, 99 5 9th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID
.5..5.5.5.5..5.5.5.95.95 5 5 x 3.9.5..5 Figure 8: Evolution of the modal frequencies with the transverse load for five different crown values for the fourth mode (left) and the fifth mode (right).5 relative difference of frequencies.4.3.. 3 4 5 number of mode Figure 9: Increasing of the first five mode for a flat soundboard with a.38cm transverse loading displacement 6 9th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID