The application of Modal Derivatives (MDs) in conventional reduction methods is an effective method to capture geometric nonlinearities in Finite Element Models (FEMs). Reduction methods, in general, are used to effectively reduce the number of unknowns in FEMs for the sake of computational efficiency. We investigated the applicability of three MDs-based reduction methods in parametrically excited and parametric resonating structures, the focus lays on the steady-state responses. Benchmark in this project is a mechanical Frequency Divider (FD) that consists of cascading mechanical components, each of which is excited by the preceding one by means of parametric resonance. After full activation of the FD, a frequency division along the array of resonators is achieved with a factor 2^i at resonator i. The modularity of the FD makes it an excellent candidate for testing the application of component mode synthesis, where each substructure of the cascade is independently reduced and connected to other members via common interface.

Dynamic analysis of large-size finite element models has been commonly applied by mechanical engineers to simulate the dynamic behavior of complex structures. The ever-increasing demand for both detailed and accurate simulation of complex structures forces mechanical engineers to pursue a balance between two conflicting goals during the simulations: low computational cost and high accuracy. These goals become extremely difficult for geometric nonlinear structural dynamical problems. When geometrical nonlinearities are introduced, the internal force vector and Jacobians are configuration dependent, and the corresponding updates are computationally expensive. This thesis presents nonlinear model order reduction techniques that aim to perform detailed dynamic analysis of multi-component structures with reduced computational cost, without degrading the accuracy too much. Special attention is given to flexible multibody system dynamics.

For multi-component structures featuring many interface degrees of freedom, standard substructuring dynamics can be combined with interface reduction techniques to obtain compact reduced order models. Chapter~2 summarized a variety of interface reduction techniques for the well-known Craig-Bampton substructuring method. These approaches are reviewed and compared in terms of both computational cost and accuracy. A multilevel interface reduction method is presented as a more generalized approach, where a secondary Craig-Bampton reduction is performed when the subsystems are assembled within localized subsets. The multilevel interface reduction method provides an accurate representation of the full linear model with significantly lower computational cost.

In Chapter~3, we extend the Craig-Bampton method to geometric nonlinear problems by augmenting the system-level interface modes and internal vibration modes of each substructure with their corresponding modal derivatives. The modal derivatives are capable of describing the bending-stretching coupling effects exhibited by geometric nonlinear structures. Once the reduced order model is constructed by Galerkin projection, the upcoming challenge is the computation of the reduced nonlinear internal force vectors and tangent matrices during the time integration. The evaluation of these objects scales with the size of the full order model, and it is therefore expensive, as it needs to be repeated multiple time within every time step of the time integration. To address this problem, we directly express the reduced nonlinear vectors and matrices as a polynomial function of the modal coordinates, using substructure-level higher-order tensors with much smaller size. This enhanced Craig-Bampton method offers flexibility for reduced modal basis construction, as modal derivatives need to be computed only for substructures actually featuring geometrical nonlinearities, and do not need the prior knowledge of the nonlinear response of the full system with training load cases.

For flexible multibody systems, each body undergoes both overall rigid body motion and flexible behavior. To describe the dynamic behavior of each body accurately, the floating frame of reference is commonly applied. In Chapter~4, the enhanced Craig-Bampton method, as proposed in Chapter~3, is embedded in the floating frame of reference. We consider here structures modeled with von-Karman beam elements. Interface reduction methods are in this context unnecessary since the adjacent bodies are connected through a single node. The proposed reduction method constitutes a natural and effective extension of the classical linear modal reduction in the floating frame.

For more complex geometries, like wind turbine blades, extremely simplified beam models can not capture the complexity of the real three-dimensional structure, and therefore the dynamic behavior might not be accurately modeled. In Chapter~5, we present an enhanced Rubin substructuring method for three-dimensional nonlinear multibody systems. The standard Rubin reduction basis is augmented with the modal derivatives of both the free-interface vibration modes and the attachment modes to include bending-stretching coupling effects triggered by the nonlinear vibrations. When compared to the enhanced Craig-Bampton method proposed in Chapter~4, the enhanced Rubin method better reproduces the geometrical nonlinearities occurring at the interface, and, as a consequence, higher accuracy can be achieved.

In Chapter~6, the overall conclusions are drawn and recommendations for further study are provided.