Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods

TLDR

The first attempt to model a struck piano string using finite difference methods was made by Hiller and Ruiz (1971). This work shows how to improve the numerical approach and underlying physics to simulate piano string motion with high realism. A finite difference scheme derived from a damped, stiff string model interacting with a nonlinear hammer computes time histories of displacement and velocity at each string point, including hammer–string and bridge forces, and discusses stability and dispersion considerations. The model’s performance is demonstrated by simulated string waveforms that illustrate realistic motion. J.

Abstract

The first attempt to musical by solving the of by means of (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462–472 (1971)]. It is shown here how this numerical approach and the underlying physical can be improved in order to simulate the motion of the string with a high degree of realism. Starting from the fundamental of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included.