Application of modal analysis and synthesis of reed and pipe to numerical simulations of a clarinet

Abstract

A finite-element model of a complete clarinet is proposed in order to simulate the acoustical dynamics of the instrument. Following the work of Pinard and Laine on isolated reeds [unpublished reports of the Ecole Polytechnique], modal analysis of clarinet reeds performed with holographic interferometry has been compared with modal synthesis derived from the FEM. In this linear model, the reed is considered alone first, then coupled to the mouthpiece, and finally to the mouthpiece and the barrel. The good agreement between the two approaches makes the mixed solid-acoustical FEM usable in a numerical simulation of the whole clarinet. The rest of the pipe is modeled by lumped mechanical elements whose parameters are fitted with measurements of the acoustical input impedance. Eigenmodes of the complete system—reed, mouthpiece, barrel, pipe—are computed and damping is subsequently attributed to each mode. The evolution of the system subject to nonlinear boundary conditions—incoming airflow, one-sided contact of the reed against the curved lay of the mouthpiece—is computed in the time domain: at each time step, external forces and internal variables are projected onto the modal basis and the evolution of each modal amplitude is integrated. Findings of the simulations will be discussed.