Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods
Antoine J. Chaigne, Anders G. Askenfelt · The Journal of the Acoustical Society of America · 1994
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.