ON THE EIGENVALUES AND CRITICAL SPEED STABILITY OF GYROSCOPIC CONTINUA

In order to provide analytical eigenvalue estimates for general contin uous gyroscopic systems, this paper presents a perturbation analysis t o determine approximate eigenvalue loci and stability conclusions in t he vicinity of critical speeds and zero speed. The perturbation analys is relies on a formulation of the general continuous gyroscopic system eigenvalue problem in terms of matrix differential operators and vect or eigenfunctions. The eigenvalue lambda appears only as lambda(2) in the formulation, and the smoothness of lambda(2) at the critical speed s and zero speed is the essential feature. First-order eigenvalue pert urbations are determined at the critical speeds and zero speed. The de rived eigenvalue perturbations are simple expressions in terms of the original mass, gyroscopic, and stiffness operators and the critical-sp eed/zero-speed eigenfunctions. Prediction of whether an eigenvalue pas ses to or from a region of divergence instability at the critical spee d is determined by the sign of the eigenvalue perturbation. Additional ly, eigenvalue perturbation at th critical speeds and zero speed yield s approximations for the eigenvalue loci over a range of speeds. The r esults are limited to systems having one independent eigenfunction ass ociated with each critical speed and each stationary system eigenvalue . Examples are presented for an axially moving tensioned beam, an axia lly moving string on an elastic foundation, and a rotating rigid body. The eigenvalue perturbations agree identically with exact solutions f or the moving string and rotating rigid body.