A structured method for the design-for-frequency of a brace-soundboard system using a scalloped brace

Design-for-frequency of mechanical systems has long been a practice of iterative procedures in order to construct systems having desired natural frequencies. Especially problematic is achieving acoustic consistency in systems using natural materials such as wood. Inverse eigenvalue problem theory seeks to rectify these shortfalls by creating system matrices of the mechanical systems directly from the desired natural frequencies. In this paper, the Cayley–Hamilton and determinant methods for solving inverse eigenvalue problems are applied to the problem of the scalloped braced plate. Both methods are shown to be effective tools in calculating the dimensions of the brace necessary for achieving a desired fundamental natural frequency and one of its higher partials. These methods use the physical parameters and mechanical properties of the material in order to frame the discrete problem in contrast to standard approaches that specify the structure of the matrix itself. They also demonstrate the ability to find multiple solutions to the same problem. The determinant method is found to be computationally more efficient for partially described inverse problems due to the reduced number of equations and parameters that need to be solved. The two methods show great promise for techniques which could lead to the design of complex mechanical systems, including musical instrument soundboards, directly from knowledge of the desired natural frequencies.