Optimal undercuts for the tuning of percussive beams

Abstract

The paper investigates the use of optimization techniques for the design of percussive beams. Profiles were generated to satisfy predetermined overtone tunings and specified optimization criteria. It is shown that there is no unique solution to the problem, and that the generated profiles need not be the same as traditionally used profiles. The vibration of the beams was modeled using a thick beam theory to take into account the possible effects of shear deformation on the natural frequencies. A number of beams were manufactured with the generated profiles, and their frequency characteristics were obtained experimentally. The experimental results confirm the validity of the numerically generated profiles.

Designing musical structures using a constrained optimization approach

“…Hence, a onedimensional model that can account for shear deformation is usually considered suitable. 2 However, when the profile of the bar entails sudden jumps in height, the one-dimensional model fails to accurately account for the complex stress system set up around the sharp corners. In this case, a twodimensional model is more accurate.…”

“…2 Figure 2͑a͒ shows the beam discretized into N sections each of length, l i , and height, h i , iϭ1,2,...,N. The response of the beam is thus a function of h i , with constraints imposed on the heights. For ease of construction, the height variations are restricted so that no material is added, and hence a maximum height h max is stipulated.…”

“…For example, the limitation of the onedimensional model used to describe the bar is such that sudden jumps in height between adjacent sections are undesirable. 2 The choice is either to use a more complex two-dimensional model 13 or to use the one-dimensional model and choose an optimizing function that produces a smoother profile. For example, the optimization criterion can be adapted to generate smooth profiles by stipulating that the differences between two adjacent heights be minimized.…”

“…For instance, in recent years the study of percussive bars, such as the xylophone bar, has been performed using finiteelement techniques, [1][2][3][4][5][6][7] whereby the bar is discretized into many small parts to approximate the profile. Similarly, finiteelement analysis has been used to study a diverse set of musical problems, such as the design of a pentangle, 8 the vibrational patterns of a violin-shaped air cavity, 9 and the design of a bell profile.…”

The design of a harmonic percussion ensemble (L)

“…stiffness at the bar centre both lower the frequency of these modes. Shape optimization has been applied to FEA models of xylophone bars to alter the undercut arch and tune the first three transverse modes to ratios of 1:4:10 (Petrolito and Legge, 1997;Petrolito and Legge, 2005) and the ratios 1:3:6, 1:4:8-9, and 1:5:10-13 (Orduña-Bustamente, 1991). Similarly up to five transverse modes have been tuned to a variety of frequency ratios for vibraphone bars with complex undercut surface shapes (Henrique and Antunes, 2003).…”