On eigenproblem solution of damped vibrations associated with gyroscopic moments

A new efficient approach is presented for solving the quadratic eigenvalue problem of weakly, nonproportionally damped vibration systems. In the analysis of these systems, gyroscopic moments and external damping are both considered. Traditional restriction of symmetry of inertia, damping and stiffness matrices is slightly relaxed. A second-order perturbation theory is developed such that the perturbed solution is based on the eigensolution of an unperturbed subproblem. This subproblem considers the unperturbed system in two different forms: (i) a conservative, gyroscopic part of an original problem, or (ii) a nonconservative, gyroscopic part of an original problem that is proportionally damped. To cope with asymmetry of the system matrices, a Duncanʹs like state formulation is used to bring these matrices into a suitable form for perturbations. Two numerical examples are introduced for explaining the detailed implementation of the presented approach. Additionally, a practical problem of rotor supported by two tilting pad-bearings is investigated. The eigensolutions obtained by the current approach match, to a great extent, other solutions obtained by time-consuming exact methods. The investigation procedure given here gives a framework to handle vibration problems of weakly nonproportional damping and/or weakly asymmetric inertia, damping and stiffness matrices.