Model reduction of nonlinear structural systems using nonlinear normal modes and component mode synthesis

This work addresses the general problem of model size reduction for describing the nonlinear vibration of structural elements and systems. The aim is to provide computational tools that allow one to accurately capture nonlinear dynamic behavior using a minimal number of degrees of freedom. In typical applications the finite element (FE) method is used to generate structural dynamic models, and model size reduction is carried out using linear modal analysis with truncation. However, in some cases one must retain many modes in order to accurately capture essential nonlinear coupling between the linear modes. In this work we utilize nonlinear normal modes (NNMs) defined in terms of invariant manifolds for the purposes of model size reduction, since it directly addresses modal coupling. This approach, which makes use of master and slave modes, along with the concept of dynamic invariance, allows one to generate accurate reduced order models (ROM) with only a few DOF, while capturing the effects of all modeled linear modes without directly simulating them. There are three main contributions of the present effort: (1) Two new numerical approaches for solving the invariant manifold equations are introduced. Both approaches employ master modal displacement and velocity coordinates and are based on weighted-residual techniques. When compared with previous methods that utilize amplitude and phase variables, the new methods are found to be superior in terms of computational time but inferior in terms of accuracy. (2) A specific application is considered: the finite amplitude vibrations of a rotating beam, which is a crude model for a rotorcraft blade. This system is known to possess essential nonlinear coupling between axial and transverse displacements, thereby leading to slow modal convergence. The proposed method systematically captures this coupling and provides an accurate single degree of freedom ROM. These results demonstrate the utility of NNM-based ROM, since they combine the versatility of the finite element method with the accurate NNM model reduction technique. (3) A model reduction technique suitable for structures that can be partitioned into substructures is developed. This allows one to build ROMs using NNMs at the substructure level and to assemble these using a component mode synthesis (CMS) technique. It is found that the proposed nonlinear CMS technique generally provides an accurate model only when the couplings between substructures are weak.