A Generalized Framework for Probabilistic Design With Application to Rotors

Abstract

This paper generalizes an existing method for deriving probability models of manufacturing quality metrics. We specifically consider the problem of deriving probability models for the inertia tensor of a rotor. The inertia tensor is a 3 /spl times/ 3 matrix that determines various dynamical properties of the rotor as it spins, affecting its reliability. The key contribution of this paper is that the quality metric of interest is a matrix or a second-order tensor, and the various manufacturing imperfections that cause deviations in the inertia tensor may be vectors. Existing methods, by contrast, assume that the quality metric, as well as the manufacturing imperfections, are scalar quantities. By using rotational properties of matrices & vectors, we show that the relationship between the inertia tensor, and the manufacturing errors must have a specific form, when the errors are small. This structure significantly restricts the class of allowable distributions for the inertia tensor. For example, we show that the multivariate s-normal distribution is not a physically appropriate distribution for the inertia tensor. The results in this paper, while applied specifically to the inertia tensor, are general, and depend only on transformational properties of vectors & matrices. Thus, the framework is applicable to modeling other engineering systems involving second-order tensors, such as the stress tensor, or strain tensor.