Advisory Committee Chair
Lee Moradi
Advisory Committee Members
David Littlefield
Ian Hosch
Document Type
Thesis
Date of Award
2014
Degree Name by School
Master of Mechanical Engineering (MME) School of Engineering
Abstract
Mallet percussion instruments such as the xylophone, marimba, and vibraphone have been produced and tuned since their inception by arduously grinding the keys to achieve harmonic ratios between their 1st, 2nd, and 3rd transverse modes. In consideration of this, it would be preferable to have defined mathematical models such that the keys of these instruments can be produced quickly and reliably. Additionally, physical modeling of these keys or beams provides a useful application of non-uniform beam vibrations as studied by Euler-Bernoulli and Timoshenko beam theories. This thesis work presents a literature review of previous studies regarding mallet percussion instrument design and optimization of non-uniform keys. The progression of previous research from strictly mathematical approaches to finite element methods is shown, ultimately arriving at the most current optimization techniques used by other authors. However, previous research varies slightly in the relative degree of accuracy to which a non-uniform beam can be modeled. Typically, accuracies are shown in literature as 1% to 2% error. While this seems attractive, musical tolerances require 0.25% error and beams are otherwise unsuitable. This research seeks to build on and add to the previous field research by optimizing beam topology and machining keys within tolerances that no further tuning is required. The optimization methods relied on finite element analysis and used harmonic modal frequencies as constraints rather than arguments of an error function to be optimized. Instead, the beam mass was minimized while the modal frequency constraints were required to be satisfied within 0.25% tolerance. The final optimized and machined keys of an A4 vibraphone were shown to be accurate within the required musical tolerances, with strong resonance at the designed frequencies. The findings solidify a systematic method for designing musical structures for accuracy and repeatability upon manufacture.
Recommended Citation
Kirkland, William Brandon, "Topographical Optimization of Structures for use in Musical Instruments and Other Applications" (2014). All ETDs from UAB. 2161.
https://digitalcommons.library.uab.edu/etd-collection/2161