Inversion of first-order perturbation theory and its application to structural design

Abstract

Present methods of structural design, such as the finite-element method, NASTRAN, are capable of accurately determining the shape and frequencies of the normal modes of complicated structures, if the stiffness and mass matrices are specified. The inverse problem, that of determining the stiffness and mass matrices of a structure in order to arrive at a desired set of normal mode shapes and frequencies, in general is unsolved. A mathematical method is presented here which addresses the inverse problem by the use of first-order perturbation theory. In this approach, the usual perturbation calculation is inverted. That is, the structural changes necessary to effect a given change in vibration modes are computed, rather than computing the effect on the vibration modes resulting from a small change in the design of that structure. The application of this technique requires a specification of a number of the normal modes of vibration of a base line structural design. This information can be obtained by a conventional finite-element calculation or by measurement of the normal modes of an existing prototype. In the former case, the inverse perturbation calculation can be used to proceed from the baseline design to an optimum design by an iterative procedure. In the latter case, the inverse perturbation calculation employs normal mode data from an existing structure to determine the changes in that structure necessary to improve its vibratory response. The basic mathematical structure of the inverse perturbation technique is presented herein, together with two illustrative numerical examples, the first employing four springs and three masses, and the second treating the modification of the modes of a cantilever beam.