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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 inthe construction of numerical schemes for simulation and sound synthesis.Finite difference schemes and other time-stepping algorithms used for musicalinstrument modelling purposes are normally arrived at by discretising aNewtonian description of the system. However because impact forces arenon-analytic functions of the phase space variables, algorithm stability canrarely be established this way. This paper presents a systematic approach toderiving energy conserving schemes for frictionless impact modelling. Theproposed numerical formulations follow from discretising Hamilton's equationsof motion, generally leading to an implicit system of nonlinear equations thatcan be solved with Newton's method. The approach is first outlined for pointmass collisions and then extended to distributed settings, such as vibratingstrings and beams colliding with rigid obstacles. Stability and other relevantproperties of the proposed approach are discussed and further demonstrated withsimulation examples. The methodology is exemplified through a case study ontanpura string vibration, with the results confirming the main findings ofprevious studies on the role of the bridge in sound generation with this typeof string instrument.

Energy conserving schemes for the simulation of musical instrument contact dynamics