Stability of Gyroscopic Systems with Respect to Perturbations

Abstract

A linear gyroscopic system is of the form: $$\displaystyle M \ddot x + G\dot x + K x = 0, $$ where the mass matrix M is a symmetric positive definite real matrix, the gyroscopic matrix G is real and skew symmetric, and the stiffness matrix K is real and symmetric. The system is stable if and only if the quadratic eigenvalue problem \(\det (\lambda ^2 M+\lambda G + K)=0\) has all eigenvalues on the imaginary axis.