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 the numerical simulation of musical instruments that involve collisions: the nonlinear, non‑analytic nature of contact forces (typically modeled by a one‑sided power law f(χ)=k_c⟨χ⟩^α). Traditional time‑stepping schemes—finite‑difference, trapezoidal, Newmark‑β, Verlet—are derived from Newton’s second law and, while they work well for linear or weakly nonlinear systems, they provide little theoretical guarantee of stability when strong, discontinuous contact forces are present.

To overcome this, the authors start from a variational description. They write the Lagrangian L = T − V for a simple mass‑spring‑gravity system with an added contact potential V_c(y)=k_c⟨y_c−y⟩^{α+1}/(α+1). A Legendre transform yields the Hamiltonian H(y,p)=p²/(2m)+k y²/2+V_c(y)−m g y, which is the total mechanical energy and is conserved in the absence of damping. The key idea is to discretise Hamilton’s equations directly, using a midpoint (mid‑point) rule for both the position and momentum updates.

With the notation y_n, p_n for the state at time step n and Δt the sampling interval, the discrete equations read

(y_{n+1}−y_n)/Δt = (T(p_{n+1})−T(p_n))/Δp,
(p_{n+1}−p_n)/Δt = −(V(y_{n+1})−V(y_n))/Δy.

Introducing the auxiliary variable s = y_{n+1}−y_n and the scaled momentum q_n = p_n Δt/(2m), the scheme collapses to a single scalar nonlinear equation

F(s)= ξ