Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

Abstract

The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, the forces arising from the supporting hydraulic bearings are nonlinear, even when the shaft deformation is in the linear range. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig-Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (DOF). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-DOF reduced-order model for each nonlinear mode. A generalized formulation for the manifolds is required, since the system features damping as well as gyroscopic and non conservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well simulations from the reference Craig-Bampton model.