Numerical simulations of xylophones. I. Time-domain modeling of the vibrating bars

Abstract

A time-domain modeling of xylophone bars excited by the blow of a mallet is presented. The flexural vibrations of the bar, with nonuniform cross section, are modeled by a one-dimensional Euler–Bernoulli equation, modified by the addition of two damping terms for the modeling of losses and a restoring force for the modeling of the stiffness of the suspending cord. The action of the mallet against the bar is described by Hertz’s law of contact for linear elastic bodies. This action appears as a force density term on the right-hand side of the bending wave equation. The model is completed by the equation of motion for the mallet, and by free–free boundary conditions for the bar. The bending wave equation of the bar is put into a numerical form by means of an implicit finite-difference scheme, which ensures a sufficient spatial resolution for an accurate tuning of the bar. The geometrical, elastic, and damping parameters of the model are derived from experiments carried out on actual xylophones and mallets. The validity of the numerical model is confirmed by three different procedures: First, a comparison is made between numerical results and analytical solutions. The second series of tests consist of examining the effects of the number of spatial steps on the convergence of the solution. Finally, various comparisons are made between measured and simulated impact forces and bar accelerations. The present model reproduces adequately the main features of a real instrument. The most significant physical parameters of bars, mallets and players’ actions can be controlled independently for producing a remarkable variety of tones.