Linearly-implicit schemes for collisions in musical acoustics based on energy quadratisation

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.

Application of a Musical Robot for Adjusting Guitar String Re-Excitation Parameters in Sound Synthesis

“…Moreover, such model predicted the fact that, even after a full fundamental string period, the signal spectrum did not return to its previous state, and some changes remained (Figure 11). Further improvements to the model are possible [20][21][22][23].…”

A Physical Model of the Trombone Using Dynamic Grids for Finite-Difference Schemes

“…with displacement from the equilibrium y = y(t), lip mass Mr (in kg), externally supplied (angular) frequency of oscillation ωr = ωr(t) = Kr/Mr (in rad/s) and stiffness Kr = Kr(t) (in N/m). We extend the existing models of lip reeds [1] by introducing a nonlinear collision between the lips based on potential quadratisation proposed by [14]. The collision potential is defined as…”

“…Notice, that if η ≥ 0, the lips are closed and the collision potential will be non-zero. This quadratic form of a collision potential allows for a non-iterative implementation [14]. This will be explained further in Section 3.…”

“…where κ = 1 if ψ n ≥ 0, otherwise κ = −1. It should be noted that condition (19c) has been added to the definition of g from [14] to prevent a division by 0 in (19b). Finally, η ⋆ = −y ⋆ − H0 where y ⋆ is the value of y n+3/2 calculated using system (18) (after expansion) without the collision potential.…”

“…We can split a tube with time-varying length L n into two smaller sections with lengths L n p and L n q (in m) such that L n = L n p + L n q . Splitting the schemes in (14) in this way yields two sets of firstorder systems. The pressure and particle velocity of the first (left) system p n lp and v n+1/2 lp+1/2 are both defined over discrete domain lp ∈ {0, .…”