Tuning of bending and torsional modes of bars used in mallet percussion instruments

The present paper proposes a generic design-optimization procedure for the concomitant tuning of bending and torsional modes of bars used in mallet percussion instruments. The undercut model uses a series of discontinuous cuts aimed to facilitate the manufacturing process. Compared to one-dimensional beam models, the use of three-dimensional (3-D) finite element modeling not only allows for the calculation of torsional modes but also provides an increased accuracy in the prediction of modal frequencies, an important aspect when dealing with the demanding tuning tolerances required in a musical context. A global optimization problem is formulated and solved using a surrogate function algorithm, which enables fast computations even with the expensive function evaluations associated with 3-D finite element models. Modal identification of experimental bars demonstrates the potential of the proposed procedure, leading to bars with demanding tuning ratios (six target frequencies) at absolute tuning deviations typically below 15 cents. Measurements of the radiated sound from the experimental bars illustrate the benefits of the improved designs compared to those without torsional tuning. The proposed framework, aside from dealing with the comprehensive tuning of percussive musical bars, also accounts for important practical considerations regarding efficient optimization, modeling accuracy, and manufacturing complexity.