A Novel Computational Method to Calculate Nonlinear Normal Modes of Complex Structures

Abstract

In this study, a simple and efficient computational approach to obtain nonlinear normal modes (NNMs) of nonlinear structures is presented. Describing function method (DFM) is used to capture the nonlinear internal forces under periodic motion. DFM has the advantage of expressing the nonlinear internal force as a nonlinear stiffness matrix multiplied by a displacement vector, where the off-diagonal terms of the nonlinear stiffness matrix can provide a comprehensive knowledge about the coupling between the modes. Nonlinear differential equations of motion are converted into a set of nonlinear algebraic equations using DFM under harmonic motion assumption. A matrix manipulation based on dynamic stiffness concept was used to localize nonlinearities and reduce the number of nonlinear equations improving the efficiency of the approach, which becomes important in solving large complex structures. The nonlinear algebraic equations are solved numerically by using Newton’s method with Arc-Length continuation. The efficiency of proposed computational approach is demonstrated using a two-degree-of-freedom nonlinear system. The proposed approach has the potential to be applied to large-scale engineering structures with multiple nonlinear elements and strong nonlinearities.