Nonlinear normal modes and their application in structural dynamics

This research aims at the development and implementation of new model reduction methods for nonlinear structural systems, based on a nonlinear modal analysis methodology. Invariant manifolds in the system’s phase space are used to define and construct nonlinear normal modes of motion for a wide class of nonlinear vibratory systems. The approach reduces to the well-known results in the linearized case, although it offers a formulation that is entirely different from the traditional one. A numerical Galerkin technique is utilized to solve for the invariant manifolds, which allows one to construct nonlinear normal modes and carry out nonlinear mode-based model reduction for motions in strongly nonlinear regions of the phase space. This method seamlessly interfaces with finite element models of engineering structures, and it allows the user to specify the vibration amplitude range and the accuracy of the model over that range. In this presentation, the nonlinear modal analysis methodology is generalized to multi-nonlinear normal mode systems, including those with internal resonances. The approach is also successfully extended to systems with piecewise linear restoring forces, which model structural components with clearance, pre-load, or different elastic materials. Furthermore, nonlinear modal analysis is developed for systems that are subjected to periodic forces, thereby providing a useful tool for attacking the important problem of obtaining the frequency response of complex nonlinear structures. Finally, the invariant-manifold-based model reduction methodology is applied to a complex engineering structure, namely a rotating rotorcraft blade with nonlinear bending-axial coupling and undergoing large-amplitude motion. While discretized blade models typically feature large numbers of degrees of freedom, the proposed approach is shown to yield a small, accurate reduced-order model.