Modelling the Mechanical Response of the Reed-Mouthpiece-Lip System of a Clarinet. Part II: A Lumped Model Approximation

A non-linear lumped model of the reed-mouthpiece-lip system of a clarinet is formulated, in which the lumped parameters are derived from numerical experiments with a finite-difference simulation based on a distributed reed model. The effective stiffness per unit area is formulated as a function of the pressure signal driving the reed, in order to simulate the effects of the reed bending against the lay, and mass and damping terms are added as a first approximation to the dynamic behaviour of the reed. A discrete-time formulation is presented, and its response is compared to that of the distributed model. In addition, the lumped model is applied in the simulation of clarinet tones, enabling the analysis of the effects of using a pressure-dependent stiffness per unit area on sustained oscillations. The analysed effects and features are in qualitative agreement with players' experiences and experimental results obtained in prior studies.