Physical Modeling of Piano Sound

Abstract

This paper aims to develop a comprehensive physical model and numerical simulation schemes for a grand piano. The model encompasses various subsystems, including hammer felt, hammer shank, string, soundboard, air and room barriers, each modeled in three dimensions to approach their realistic dynamics. A general framework for 3D elastic solids accounting for prestress and prestrain is introduced, particularly addressing the the nonlinearities arising from the large deformation of piano strings and the one-sided nature of hammer felt-string contact. The study also examines coupling between subsystem through mechanisms of surface force transmission and displacement/velocity continuity. To facilitate numerical simulations, strong PDEs are translated into weak ODEs via a flexible space discretization approach. Modal transformation of system ODEs is then employed to decouple and reduce DOFs, and an explicit time discretization scheme is customized for generating digital audio in the time domain. The study concludes with a discussion of the piano models capabilities, limitations, and potential future enhancements.