Nonuniform beams with harmonically related overtones for use in percussion instruments

Abstract

Beams used in certain percussion instruments with definite pitch such as marimba, xylophone, and vibraphone are undercut in order to bring the frequencies of the first few overtones into a harmonic relationship with the fundamental frequency. This paper addresses the problem of determining the optimal dimensions of the undercut so that the frequencies of modes 2 and 3 of transverse motion are harmonically related with that of the fundamental (mode 1). The undercut is assumed to have a parabolic shape described by two variable parameters related to its depth and width. The rest of the beam on each side of the undercut is assumed to be uniform. Seven pairs of optimal undercut parameters are found that bring the frequencies of the first three transverse modes into the harmonic (integer) ratios 1:3:6, 1:4:8–9, and 1:5:10–13. Calculations performed with and without taking into account the effects of rotary inertia and shear stress are compared against measurements taken from a set of experimental beams. The comparison shows that including the effects of rotary inertia and shear stress results in a better prediction of the optimal parameters of the undercut.