Sensitivities of multiple singular values for optimal geometries of precision structures

This paper presents the sensitivities of a repeated singular value of a mat rix with respect to perturbations in that matrix. The difficulty in computi ng the sensitivities of a repeated singular value is linked to the fact tha t the multiplicity of the singular value may change during the perturbation , The derivative is developed based on an approach used for repeated eigenv alues of self adjoint systems, by constraining the singular values to remai n bundled during the perturbation. The need for the sensitivities of singul ar values arose when optimizing the geometry of precision structures under a family of disturbances characterized by a disturbance influence matrix. T he aim was to modify the geometry of the structure in a way which enhances its performance. Since the structure is subjected to a multitude of loading cases the objective is to minimize the worst possible distortion. It is sh own that this is equivalent to minimizing the first singular value of the d isturbance influence matrix. Consequently, in the mathematical programming formulation the objective function is the first singular value under the co nstraints inherent to the method for computing the sensitivities. This is t hen solved by a Lagrangian method. It is shown that the technique is very r eliable as visualized in two typical truss examples. (C) 1999 Elsevier Scie nce Ltd. All rights reserved.