Three-dimensional Modeling and Simulation of a Tuning Fork

Abstract

The mathematical characterization of the sound of a musical instrument still follows Schumann’s laws [1]. According to this theory, the resonances of the instrument body, “the formants”, filter the oscillations of the sound generator (e.g., strings) and produce the characteristic “timbre” of an instrument. This is a strong simplification of the actual situation. It applies to a point source and does not distinguish between a loudspeaker and a three-dimensional instrument. In this work we investigate Finite-Element-based numerical simulations of eigenfrequencies and eigenmodes of a tuning fork in order to capture the oscillation behavior of its eigenfrequencies. We model the tuning fork as an elastic solid body and solve an eigenvalue equation derived from a system of coupled equations from linear elasticity theory on an unstructured three-dimensional grid. The eigenvalue problem is solved using the preconditioned inverse iteration (PINVIT) method with an efficient geometric multigrid (GMG) preconditioner. The latter allows us to resolve the tuning fork with a high resolution grid, which is required to capture fine modes of the simulated eigenfrequencies. To verify our results, we compare them with measurement data obtained from an experimental modal analyses of a real reference tuning fork. It turns out that our model is sufficient to capture the first eight eigenmodes of a reference tuning fork, whose identification and reproduction by simulation is novel to the knowledge of the author.