| The generation of reduced-order models of nonlinear systems is particularly difficult, due to the complex interactions of the system components. This work applies the invariant manifold formulation for nonlinear normal modes to create rigorous reduced-order models of a wide variety of nonlinear structures, including discrete, finite element, and continuous dynamic systems. This is accomplished through two types of expansion-based solutions for the invariant manifolds which govern the nonlinear normal modes of the structure.;The first expansion is polynomial-based and produces analytic, third-order, invariant manifolds which are asymptotically accurate. The solution obtained is applicable to a subclass of weakly nonlinear structural systems with quadratic and cubic nonlinearities in displacement. The second method uses a Galerkin projection and numerical solver to determine the invariant manifold over a chosen domain. This approach is shown to be accurate for strong nonlinear effects as well as being more adaptable than the polynomial-based approach. Both methods are applied to various nonlinear structural systems, and the results indicate that, in general, the high accuracy of the Galerkin-based solution compensates for the additional computational effort.;One field in which nonlinear interactions play a critical role, and are difficult to capture, is rotorcraft dynamics. In particular, blade simulations are cumbersome due to the large models which have been necessary to achieve accurate results. Equations of motion are developed for a uniform nonlinear Euler-Bernoulli beam, rotating at constant velocity, and constrained to move in only the transverse and axial directions. In the interest of improving the rotorcraft design process, the above reduction methods were applied to this simplified blade model. The results indicate that, although both methods capture the critical nonlinear coupling terms at low amplitudes, the Galerkin-based solution achieves excellent results, allowing accurate analysis for tip deflections as large as one meter (peak-to-peak), for a nine meter blade. However, the polynomial-based solutions remain applicable, as they allow investigations of internal resonances which are currently not possible using the present Galerkin-based formulation. |
