Modeling the influence of the piano hammer shank flexibility on the sound

Abstract

A nonlinear model for a vibrating Timoshenko beam in non-forced unknown rotation is derived from the virtual work principle applied to a system of beam with mass at the end. The system represents a flexible piano hammer shank coupled to a hammer head. A novel energy-based numerical scheme is then provided and coupled to a global energy-preserving numerical solution for the whole piano (strings, soundboard, and sound propagation in the air). The obtained numerical simulations show that the pianistic touch clearly influences the spectrum of the piano sound of equally loud isolated notes. These differences do not come from a possible shock excitation on the structure, nor from a changing impact point, nor a “longitudinal rubbing motion” on the string, since neither of these features are modeled in our study.