Three-dimensional tuning of idiophone bar modes via finite element analysisa

The timbre of marimba and other idiophone bars can sometimes be polluted by untuned torsional modes, leading to substandard instruments or rejected materials. Makers have complained of problems with these untuned modes over a specific range of notes. Marimba, vibraphone, and similar idiophone bars are tuned by carving one side of the bar to bring up to three flexural modes into harmonic relationships. Torsional and other mode types are commonly left untuned. The relative frequency of these untuned modes with respect to the fundamental mode will vary along the keyboard. This paper investigates tuning both torsional and flexural modes simultaneously. This tuning is achieved using sophisticated carved geometries, and without employing concentrated masses or additional materials. Bars are modeled using three-dimensional finite elements. Geometry is defined by a large number of input parameters. Algorithms are implemented to identify bar modes automatically, eliminating the need for human intervention. Tuning is performed via a Newton-Raphson approach using the Moore-Penrose generalized matrix inverse to solve systems of tuning equations. This method is found to be effective at finding satisfactory bar geometries in proximity to initial conditions. Numerous example marimba and vibraphone bar models are provided, representing both typical and atypical modal tuning ratios.