Discrete port-Hamiltonian system model of a single-reed woodwind instrument

Time-domain simulation of woodwind instruments typically involves the development of separate discrete-time sub-models for the excitation mechanism and the resonator. These components have largely been modeled via digital waveguide or finite-difference time-domain (FDTD) methods. We present a separate approach based on the modular and energy-based port-Hamiltonian system (PHS) framework. We recast the three main components of a woodwind instrument—the single-reed, the bore, and the tonehole—as PHS models and incorporate novel elements in each derivation. In the beating reed model, we make use of recent work on energy quadratization to formulate a linearly implicit scheme of the nonlinear Hunt-Crossley contact force coupled to a nonlinear Bernoulli flow. In the horn model, we discretize a distributed PHS representing the horn equation with a generalized symplectic Störmer-Verlet scheme, verifying previously proposed FDTD schemes. In the tonehole model, we propose a new low-frequency model of the tonehole and model note transitions with a switching PHS. The benefit of describing each element as a PHS is demonstrated by the ability to interconnect all sub-models in a modular and energy-conserving manner to simulate a complete instrument. Simulations are performed on a test instrument and the numerical stability of the overall scheme is demonstrated.