Physical Modelling of Piano Sound

This paper develops a comprehensive physical model and numerical implementation schemes for a grand piano, building upon the prior works of Chabassier et al. The model encompasses various subsystems, including hammer felt, hammer shank, string, soundboard, air and room barriers, each modelled 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 complexities of prestressed piano strings. 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.