Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes*

Abstract

A method for determining reduced-order models for rotating beams is presented. The approach is based on the construction of nonlinear normal modes that are defined in terms of invariant manifolds that exist for the system equations of motion. The beam considered is an idealized model for a rotor blade whose motions are dominated by transverse vibrations in the direction perpendicular to the plane of rotation (known as flapping). The mathematical model for the rotating beam is relatively simple, but contains the nonlinear coupling that exists between transverse and axial deflections. When one employs standard modal expansion or finite element techniques to this system, this nonlinearity causes slow convergence, leading to models that require many degrees of freedom in order to achieve accurate dynamical representations. In contrast, the invariant manifold approach systematically accounts for the nonlinear coupling between linear modes, thereby providing models with very few degrees of freedom that accurately capture the essential dynamics of the system. Models with one and two nonlinear modes are considered, the latter being able to handle systems with internal resonances. Simulation results are used to demonstrate the validity of the approach and to exhibit features of the nonlinear modal responses.