Dynamics of the pianoforte string and narrow hammers

Abstract

A new method of solution to the equation of a perfectly flexible string, when struck at any point by hard or soft narrow hammers, is developed. The hard hammer is modeled as a point mass, and the soft hammer is modeled as a point mass backed by a linear or nonlinear spring and a dashpot, representing the compliance and dissipation of the felt. For each hammer the equation describing the string–hammer contact force during the periods of contact, and of recontacts, is derived explicitly. The Green’s function for a fixed-fixed string is represented by an infinite series of Heaviside functions. The analysis is exact in the spatial domain thereby eliminating modal truncation error and capturing all discontinuities in the contact force. The method predicts the contact force and system response under impacts given any hammer to string mass ratio that experiences any number of multiple contacts during the hammer strike.