Energy conserving schemes for the simulation of musical instrument contact dynamics

Collisions are an innate part of the function of many musical instruments. Due to the nonlinear nature of contact forces, special care has to be taken in the construction of numerical schemes for simulation and sound synthesis. Finite difference schemes and other time-stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a Newtonian description of the system. However because impact forces are non-analytic functions of the phase space variables, algorithm stability can rarely be established this way. This paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling. The proposed numerical formulations follow from discretising Hamilton's equations of motion, generally leading to an implicit system of nonlinear equations that can be solved with Newton's method. The approach is first outlined for point mass collisions and then extended to distributed settings, such as vibrating strings and beams colliding with rigid obstacles. Stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples. The methodology is exemplified through a case study on tanpura string vibration, with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument

Virtual-Acoustic Instrument Design: Exploring the Parameter Space of a String-Plate Model

Exploration is an intrinsic element of designing and engaging with acoustic as well as digital musical instruments. This paper reports on the ongoing development of a virtual-acoustic instrument based on a physical model of a string coupled nonlinearly to a plate. The performer drives the model by tactile interaction with a string-board controller fitted with piezo-electric sensors. The string-plate model is formulated in a way that prioritises its parametric explorability. Where the roles of creating performance gestures and designing instruments are traditionally separated, such a design provides a continuum across these domains. The string-plate model, its real-time implementation, and the control interface are described, and the system is preliminarily evaluated through informal observations of how musicians engage with the system