Modeling and numerical simulation of a nonlinear system of piano strings coupled to a soundboard

Construction of a physical model for the grand piano implies complex and multidimensional phenomena. We present a model of piano strings coupled to a soundboard, and its numerical approximation. Measurements on piano strings and bridge show phantom partials and a time precursor that both cannot be explained by the linear scalar string model. A classical model of nonlinear strings has been written by Morse & Ingard, it implies to consider the longitudinal displacement as well as the standard transversal displacement of the string, in a nonlinear coupled system. Various approximate (polynomial) models have been written from this one, by expanding the nonlinearity (a square root term) around the rest position of the string. We provide a mathematical justification of the most used model. Transmission of the string motion to the rest of the structure is essential from the acoustical point of view. We use a modal approach for the soundboard, and we write a nonstandard reciprocal coupling condition between strings and soundboard at the bridge. Numerical approximation of such a nonlinear, multidimensional and coupled problem is a difficult issue. We use an energy approach to achieve stability, which leads to an innovating implicit numerical scheme. MODELING THE PIANO The work presented in this paper is the ongoing subject of the first author’s PhD, which consists in modeling a grand piano