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.

💡 Research Summary

The paper addresses a long‑standing difficulty in numerical simulation of musical instruments that involve contact or impact, such as strings striking a bridge, reeds hitting a mouthpiece, or percussive elements colliding with a body. Conventional approaches start from Newton’s second law and discretise the resulting ordinary differential equations with explicit finite‑difference or finite‑element schemes. Because the impact force is a non‑analytic, piecewise‑defined function of displacement (often modelled as a power‑law repulsive potential that is zero until a gap closes), explicit schemes cannot guarantee stability; the time step must be extremely small or the simulation blows up.

To overcome this, the authors adopt a Hamiltonian formulation. The total energy—kinetic plus potential, where the potential includes both the usual elastic energy and an additional contact potential (V(q)=\frac{k}{\alpha+1}\max(0,\delta-q)^{\alpha+1})—is taken as the Hamiltonian (H(p,q)). Hamilton’s equations (\dot q=\partial H/\partial p) and (\dot p=-\partial H/\partial q) are then discretised in time using a symmetric, implicit midpoint‑type rule. This yields the update equations

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