Stability of gyroscopic systems with respect to perturbations

A linear gyroscopic system is of the form $$ M ddot x + Gdot x + K x = 0, $$ 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. In this article we are interested in evaluating robustness of a given stable gyroscopic system with respect to perturbations. In order to do this we present an ODE-based methodology which aims to compute the closest unstable gyroscopic system with respect to the Frobenius distance. A few examples illustrate the effectiveness of the methodology.