Finite Element Method Applied to the Eigenvalue Analysis of Flexible Rotors Supported by Journal Bearings

Abstract

This work deals with a finite element\rprocedure developed to perform the eigenvalue analysis of damped gyroscopic\rsystems, represented by flexible rotors supported on fluid film journal bearings.\rThe rotor finite element model is based on the Timoshenko beam theory,\raccounting for the shaft rotary inertia and\rgyroscopic moments. The governing equations for the hydrodynamic journal\rbearing are obtained through the Galerkin weighted residual method applied to\rthe classical Reynolds equation. A perturbation scheme on the fluid film\rgoverning equation permits to obtain the zero-th and first order lubrication\requations for the bearings, which allow the computation of the dynamic force\rcoefficients associated with the bearing stiffness and damping. The\rrotor-bearing system equation, which consists of a case of damped gyroscopic\requation, is rewritten on state form to compute the complex eigenvalues. The\rnatural frequencies at several operating conditions are obtained and compared\rto the technical literature data. The influence of the effective damping on the\reigenvalue real part sign is analyzed for some examples of rotor-bearing\rsystems, showing how the stability conditions can be predicted by the\reigenvalue analysis. The procedure implemented in this work can provide useful\rguidelines and technical data about the selection of the more appropriate set of bearing parameters for\rrotating machines operating at stringent conditions.