Bifurcations of eigenvalues of gyroscopic systems with parameters near stability boundaries

This paper deals with stability problems of linear gyroscopic systems M(x) double over dot + G(x) over dot + Kx =0 with finite or infinite degrees-of- freedom where the system matrices or operators depend smoothly on several r eal parameters. Explicit formulas for the behavior of eigenvalues under a c hange of parameters are obtained. It is shown that the bifurcation (splitti ng) of double eigenvalues is closely related to the stability, flutter, and divergence boundaries in the parameter space. Normal vectors to these boun daries are derived using only information at a boundary point: eigenvalues, eigenvectors, and generalized eigenvectors, as well as first derivatives o f the system matrices (or operators) with respect to parameters. These resu lts provide simple and constructive stability and instability criteria. The presented theory is exemplified by two mechanical problems: a rotating ela stic shaft carrying a disk, and an axially moving tensioned beam.