Stability of Continuous Gyroscopic Systems Using Perturbation Analysis

Continuous gyroscopic system eigenvalues and stability are analytically calculable in only a limited set of cases. This paper presents an eigenvalue perturbation analysis to determine approximate eigenvalue loci and stability conclusions in the vicinity of critical speeds and zero speed. The perturbation analysis relies on a formulation of the general continuous gyroscopic system eigenvalue problem in terms of matrix differential operators and vector eigenfunctions. The eigenvalue λ appears only as λ2 in the formulation, and the smoothness of λ2 at the critical speeds and zero speed is the essential feature required for the perturbatton. First order eigenvalue perturbations are determined at the critical speeds and zero speed. The derived eigenvalue perturbations are simple expressions in terms of the original mass, gyroscopic, and stiffness operators and the critical speed/stationary system eigenfunctions. Prediction of whether an eigenvalue passes to or from a region of divergence instability is determined by the sign of the eigenvalue perturbation. Additionally, perturbation of the critical speed/stationary system yields approximations for the eigenvalue loci at speeds away from these. The results provide analytical means for estimating continuous gyroscopic system eigenvalues and stability near critical speeds without numerical computation. The results are limited to systems having one independent eigenfunction associated with each critical speed and each stationary system eigenvalue. The techniques also apply to discrete gyroscopic systems. Examples are presented for an axially-moving, tensioned beam and a rotating rigid body and comparisons with known solutions are given.