A modal approach to piano soundboard vibroacoustic behavior

Abstract

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 the complexity of rendering high-pitched notes homogeneous.

Physically-based sound synthesis software for Computer-Aided-Design of piano soundboards

“…The reader may find such comparisons in the original papers. For instance, in [11], the mode shapes obtained using the semi-analytical model by Trevisan et al have been compared with those obtained experimentally on an upright piano (Pleyel P131), and have been shown to be in good agreement. Chabassier et al [12] compared the simulation of the string motion of a virtual and a real Steinway D. The simulated string motion exhibits the same inharmonicity profile as the measured string of the Steinway D. Additionnally, the model has also been proven in [12] to be able to accurately simulate phenomena observed in real pianos, such as the dependance of frequency with amplitude, the presence of soundboard modes in the initial transient of the tones, and the presence of phantom partials.…”

“…Currently, the software proposes two different methods to compute the modal basis of the soundboard: a semianalytical method derived from Trevisan et al [11], which requires strong simplifications of the soundboard geometry, and a finite-element method, derived from the models proposed by Chabassier et al [12].…”

“…As a consequence, the simulation frequency, which is the sampling frequency of the output audio file, is set to 2.1F max , where F max is the maximal modal frequency. Note that the semi-analytical method to compute the modal basis (see Trevisan et al [11]) is much faster (it takes around 700 s to compute the modal basis up to 5000 Hz) at the cost of the loss of fine geometric details [25].…”

“…In the case of the piano, which is the subject of this paper, many studies have been made in the past in order to fully comprehend all of the physical phenomena that are involved in the tone production. For instance, many authors have studied the vibro-acoustic behavior of the piano soundboard, either by experimental characterization [2][3][4][5][6][7], or by proposing simplified models [8][9][10][11] or finite element models [6,12]. The string behavior, including the hammer-string interaction and the string-bridge coupling, has also been studied by several authors [12][13][14][15][16][17][18].…”

“…The normalized data about the geometry and the materials of the virtual soundboard can be defined with the help of a specifically designed GUI. The modal basis of the virtual soundboard is computed with either a semi-analytical model [11], or a finite-element method [12]. Both models use a light fluid approximation.…”