On the derivation of inequality constraints for independent component optimization maintaining a minimum system eigenfrequency

Abstract

The lowest eigenfrequency of a structure depends on the local distributions of stiffness and inertia. When components of a structural system are designed separately, which is typical, e.g., in the aerospace industry, design targets for stiffness and mass of components support independent design work: they can be chosen such that realizing them ensures a minimum system eigenfrequency. However, meeting design targets exactly may be difficult, e.g., due to unforeseen restrictions such as requirements from other disciplines or limitations with respect to manufacturing. This paper uses Rayleigh’s Quotient to derive inequality constraints as limits on the spatial distributions of stiffness and mass, rather than target distributions without tolerance. They serve as component requirements with design freedom and help to decompose the system for independent design.A general formulation based on limit functionals for deformation energy and kinetic energy is proposed for arbitrary structures. For the special case of simply connected systems with rigid interfaces between components and mass-less support elements, limit functionals for support stiffness can be expressed using limit stiffness matrices. A particular choice for them is motivated assuming rigid body motion of the main body. This is then relaxed to produce limit stiffness matrices for arbitrary deformable main bodies.The approach is applied to the structural design of a space payload instrument, the ATHENA Wide Field Imager, enabling independent mass optimization of the so-called primary structure and the support elements. The resulting total mass is 7% larger than the one obtained by monolithic optimization. This could be further reduced to 3% by appropriately adjusting the requirements.