Linearly-implicit schemes for collisions in musical acoustics based on energy quadratisation
Michele Ducceschi, Stefan Bilbao, Silvin Willemsen, Stefania Serafin
Abstract
Collision modelling represents an active field of research in musical acoustics. Common examples of collisions include the hammer-string interaction in the piano, the interaction of strings with fretboards and fingers, the membrane-wire interaction in the snare drum, reed-beating effects in wind instruments, and others. At the modelling level, many current approaches make use of conservative potentials in the form of power-laws, and discretisations proposed for such models rely in all cases on iterative root-finding routines. Here, a method based on energy quadratisation of the nonlinear collision potential is proposed. It is shown that there exists a suitable discretisation of such a model that may be resolved in a single iteration, while guaranteeing stability via energy conservation. Applications to the case of lumped as well as fully distributed systems will be given, using both finite-difference and modal methods.