Algebraic method for shaping percussive bars

In the tuning optimization problem for percussive bars, shape profiles are parameterized in terms of some appropriate set of functions, and parameters are sought that produce desired mode frequencies. For a specific bar shape, the bending mode frequencies can be accurately determined within the Timoshenko-Ehrenfest model using a finite element analysis. Under certain conditions, a useful set of functions that reduces the differential equation to a purely algebraic problem can be constructed from solutions to the corresponding uniform system problem. These constructed functions vary on the same scale as the uniform beam bending modes and hence couple effectively to shift the mode frequencies. This algebraic methodology is applied to the percussive bar problem, and a Monte Carlo optimization is performed to determine shape functions that produce bars with the frequency ratios of the marimba (1:3:6), the xylophone (1:4:10), and an "octave bar" with ratios 1:2:4:8. Bars were constructed according to these designs, and acoustical measurements are in good agreement with model predictions.