A Numerical Continuation Method to Compute Nonlinear Normal Modes Using Modal Reduction

Abstract

Nonlinearities in structural dynamic systems introduce behavior that cannot be described with linear vibration theory, such as frequency-energy dependence and internal resonances. The concept of nonlinear normal modes accommodates such phenomena, providing a rigorous framework to characterize and design nonlinear structures. A recently developed method has enabled the computation of nonlinear normal modes for structures with hundreds of degrees of freedom, but the formulation is not readily applicable to large scale geometrically nonlinear structures that are modeled within finite element software. This work presents a variation on that approach that can be used to extract the nonlinear normal modes of a structure using commercial finite element software. A model of the structure is created in the finite element package and the algorithm then iterates on the nonlinear transient response in a non-intrusive way to estimate the nonlinear modes. A modal coordinate transformation is used to reduce the order of the Jacobians required by the algorithm. The method is demonstrated on a fixed-fixed beam that is geometrically nonlinear due to coupling between transverse and axial displacements. An alternative procedure is also presented in which static load cases are used to compute a reduced order model of the nonlinear system and then standard continuation is used to find the nonlinear modes of the reduced order model. That approach is explored using both enforced displacements and applied loads and the results obtained are compared with those from the full-order model.