Eigensensitivity Analysis for Space Structures with Applications in Design Optimization

Eigensensitivity analysis is concerned with the rates of changes of eigenvalues and eigenvectors with respect to design variables. Eigensolutions of space structures are found to display four distinct characteristics, three of which are related to various types of repeated eigenvalues. This study will address three cases of the real symmetric structural eigenvalue problem. The three cases that will be considered are: distinct eigenvalues, repeated eigenvalues with distinct first eigenvalue derivatives, and repeated eigenvalues with infinitely repeating eigenvalue derivatives. Analytic formulations for eigenvalue/ vector derivatives will be presented for each of the three cases. The directional differentiability of eigenvalues for the case of repeated eigenvalues with distinct eigenvalue derivatives will be presented. Also, a theory to determine the set of design variable dependent differentiable eigenvectors will be introduced. Particular emphasis will be placed on the applications of eigensensitivity analysis to the integrated design of large finite element based flexible space structures. The simultaneous control/structure optimization of the Earth Pointing Satellite structure will be used as an example to demonstrate the usefulness of eigensensitivity analysis in an integrated design environment.