Bifurcations of Eigenvalues of Gyroscopic Systems With Parameters Near Stability Boundaries

Abstract

This paper deals with stability problems of linear gyroscopic systems Mẍ+Gẋ+Kx=0 with finite or infinite degrees-of-freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulas for the behavior of eigenvalues under a change of parameters are obtained. It is shown that the bifurcation (splitting) of double eigenvalues is closely related to the stability, flutter, and divergence boundaries in the parameter space. Normal vectors to these boundaries are derived using only information at a boundary point: eigenvalues, eigenvectors, and generalized eigenvectors, as well as first derivatives of the system matrices (or operators) with respect to parameters. These results provide simple and constructive stability and instability criteria. The presented theory is exemplified by two mechanical problems: a rotating elastic shaft carrying a disk, and an axially moving tensioned beam.