Spacecraft shape optimization theoretical guidelines for fundamental frequency regulation

In this paper, the possibility of efficiently optimizing the first natural frequency of a spacecraft structure is anal- ysed. The mathematical problem of optimizing the fundamental frequency was first addressed in the 19th century and has since been extensively studied by many authors. The value of this sensitive parameter, under specific constraints, is extremely important, particularly for spacecraft structures, where the fundamental frequency must meet a minimum requirement to ensure suitability for launch. It is theoretically known, via the Faber-Krahn theorem, that in R^N, an open ball has the lowest value for its first eigenvalue with respect to the Laplacian operator. Later, the same prob- lem was analysed for regular polygons. The expected results were proven for triangles and quadrilaterals, while the problem remains open for N-gons, with N > 5. This investigation has been extended here to the biharmonic opera- tor, about which very little is currently known. This operator is representative of the general structural free vibration problem. The results obtained for the biharmonic operator have been confirmed and corroborated by numerical and experimental studies. Starting from a purely mathematical approach, the theoretical results were used to derive shape optimization guidelines to increase the fundamental frequency, which can be significantly affected from appropriate design modifications suggested from those very mathematical tools. By changing the shape in a smart and efficient way, it is possible to increase the first natural frequency while maintaining the weight unchanged. Therefore, this work is suitable for practical application to real spacecraft. The design criteria were applied to various case studies, demonstrating an alternative way to optimize the fundamental frequency without the need to stiffen the structure.