Frequency isolation for gyroscopic systems via hyperbolic quadratic eigenvalue problems

Abstract

The solutions of a forced gyroscopic system of ODEs may undergo large oscillations whenever some eigenvalues of the corresponding quadratic eigenvalue problem (QEP) (λ2M+λG+K)v=0,0≠v∈ℂn, are close to the frequency of the external force (both M,K are symmetric, M is positive definite, K is definite and G is skew-symmetric). This is the phenomenon of the so-called resonance. One way to avoid resonance is to modify some (or all) of the coefficient matrices, M, G, and K∈Rn×n in such a way that the new system has no eigenvalues close to these frequencies. This is known as the frequency isolation problem. In this paper we present frequency isolation algorithms for tridiagonal systems in which only the gyroscopic term G is modified. To derive these algorithms, the real gyroscopic QEP is first transformed into a complex hyperbolic one, which allows to translate many of the ideas in Moro and Egaña (2016) for undamped systems into the full quadratic framework. Some numerical experiments are presented.