Speakers
Description
Gyroscopic systems are mechanical systems described by the equation:
$$M \ddot x(t) + G\dot x(t) + K x(t) = 0,$$
where the mass matrix $M\in\mathbb{R}^{n\times n}$ is symmetric positive definite, the gyroscopic matrix $G \in\mathbb{R}^{n\times n}$ is skew-symmetric, the stiffness matrix $K\in\mathbb{R}^{n\times n}$ is symmetric, and $x=x(t)$ is a time-dependent displacement vector. The properties of above system are determined by the algebraic properties of the quadratic matrix polynomial:
$$
\mathcal{G}(\lambda ) = \lambda^2 M + \lambda G + K,$$
and the corresponding quadratic eigenvalue problem:
$$
\mathcal{G}(\lambda)x=(\lambda^2 M+\lambda G+K)x=0, \quad x\in\mathbb{C}^{n}, x\ne 0. $$
Perturbation bounds for mechanical systems are crucial for understanding the stability and behaviour of these systems under external disturbances or variations in parameter values. Stability in this context implies that all eigenvalues of the system are purely imaginary and semi-simple. We present an upper bound for the relative change in eigenvalues, as well as a $\sin\Theta$ type bound for the corresponding eigenvectors for stable gyroscopic systems under perturbations of the system matrices. The case when $K$ is positive definite is treated separately from the case when it is negative definite. To demonstrate the effectiveness of the obtained bounds, we illustrate their performance through numerical experiments.
