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International Journal of Rotating Machinery
Volume 10, Issue 4, Pages 319-335

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

1GeM, Equipe Structures et Simulations, Ecole Centrale de Nantes, Nantes, France
2Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA
3Department of Mechanical Engineering University of Michigan, Ann Arbor 48109-2447, Michigan, USA
4Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan, USA

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. 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 (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and nonconservative 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 the simulations from the reference Craig–Bampton model.