Numerical methods for checking the stability of gyroscopic systems

Abstract

Gyroscopic mechanical systems are modeled by the second-order differential equation
\begin{equation*}\displaystyle M \ddot x(t) + G\dot x(t) + K x(t) = 0,
\end{equation*} where \(M\in\mathbb{R}^{n\times n}\) is a symmetric and positive definite matrix, $G \in\mathbb{R}^{n\times n}$ is a skew-symmetric ($G^T=-G$) matrix, and $K\in\mathbb{R}^{n\times n}$ is a symmetric matrix, representing the mass, gyroscopic, and stiffness matrices, respectively. The stability of such systems, which is the primary topic of this paper, is determined by the properties of the associated quadratic eigenvalue problem (QEP)\begin{equation*}
{\mathcal G}(\lambda)x=(\lambda^2M+\lambda G+K)x=0, \quad x\in\mathbb{C}^{n},\ x\not=0.
\end{equation*} In this paper, we provide an overview of various linearizations of the QEP and propose numerical methods for checking the stability of gyroscopic systems based on solving the linearized problem. We present examples that demonstrate how the use of numerical methods provides a significantly larger stability region, which cannot be detected using the considered non-spectral criteria, or verify stability in cases where non-spectral criteria are not applicable, highlighting the advantages of numerical methods.