Parametric Optimization of Structures Under Combined Base Motion Direct Forces and Static Loading

Abstract

This paper deals with the optimization of vibrating structures as a mean for minimizing unwanted vibration. Presented in this work is a method for automatic determination of a set of preselected design parameters affecting the geometrical layout or shape of the structure. The parameters are selected to minimize the dynamic response to external forcing or base motion. The presented method adjusts the structural parameters by solving an optimization problem in which the constraints are dictated by engineering considerations. Several constraints are defined so that the static deflection, the stress levels and the total weight of the structure are kept within bounds. The dynamic loading acting upon the structure is described in this work by its power spectral density, with this representation the structure can be tailored to specific operating conditions. The uncertain nature of the excitation is overcome by combining all possible spectra into one PSD encompassing all possible loading patterns. An important feature of the presented method is its numerical efficiency. This feature is essential for any reasonably sized problem as such problems are usually described by thousands of degrees of freedom arising from a finite-element idealization of the structure. In this paper, efficient, closed form expressions, for the cost function and its gradients are derived. Those are computed with a partial set of eigenvectors and eigenvalues thus increasing the efficiency further. Several numerical examples are presented where both shape optimization and the selection of discrete components are illustrated.