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.
Highlights
Collisions play a key role in the operation of many musical instruments
Collision modelling represents an active field of research in musical acoustics
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
Summary
Collisions play a key role in the operation of many musical instruments. The most obvious examples are the hammer-string and mallet-membane interactions, but there are many others: fret/string interactions in instruments such as the guitar; reed-beating effects in wind instruments; the sitar and tanpura; and wire/membrane collisions in the snare drum. Some collisions may be modelled as lumped, and considered to act only over a very small portion of a system (e.g., a piano hammer). The possibility of modelling a collision via energy methods is attractive from a numerical design perspective since this passivity property can be used as a condition on stability.17 Such potential-based methods have been extended to cases involving rigid obstacles, though the interpenetration is interpreted as a penalty.. For collisions taking place in systems of finite spatial extent, approaches based on modal decompositions are impaired due to the implicit character of the update equations, and efficient solutions are only available in the case of linear barrier force.. On top of the reduction in computational cost, in this case, existence and uniqueness of solutions follow in an obvious manner Such schemes are based on the quadratisation of the collision potential energy, through the introduction of an auxiliary function treated as a new additional state variable. V and VI present applications of the proposed schemes for the cases of the snare drum and the tromba marina
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