This paper presents a novel model order reduction technique for 3D flexible multibody systems featuring nonlinear elastic behavior. We adopt the mean-axis floating frame approach in combination with an enhanced Rubin substructuring technique for the construction of the reduction basis. The standard Rubin reduction basis is augmented with the modal derivatives of both free-interface vibration modes and attachment modes to consider the bending–stretching coupling effects for each flexible body. The mean-axis frame generally yields relative displacements and rotations of smaller magnitude when compared to the one obtained by the nodal-fixed floating frame. This positively impacts the accuracy of the reduction basis. Also, when equipped with modal derivatives, the Rubin method better considers the geometric nonlinearities than the Craig–Bampton method, as it comprises vibration modes and modal derivatives featuring free motion of the interface. The nonlinear coupling between free-interface modes and attachment modes is also considered. Numerical tests confirm that the proposed method is more accurate than Craig–Bampton’s, a nodal fixed floating frame counterpart originally proposed in Wu and Tiso (Multibody Syst. Dyn. 36(4): 405–425, [2016]), and produces significant speed-ups. However, the offline cost is increased because the mean-axis formulation produces operators with decreased sparsity patterns.

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The Hurty/Craig-Bampton method in structural dynamics represents the interior dynamics of each subcomponent in a substructured system with a truncated set of normal modes and retains all of the physical degrees of freedom at the substructure interfaces. This makes the assembly of substructures into a reduced-order system model relatively simple, but means that the reduced-order assembly will have as many interface degrees of freedom as the full model. When the full-model mesh is highly refined, and/or when the system is divided into many subcomponents, this can lead to an unacceptably large system of equations of motion. To overcome this, interface reduction methods aim to reduce the size of the Hurty/Craig-Bampton model by reducing the number of interface degrees of freedom. This research presents a survey of interface reduction methods for Hurty/Craig-Bampton models, and proposes improvements and generalizations to some of the methods. Some of these interface reductions operate on the assembled system-level matrices while others perform reduction locally by considering the uncoupled substructures. The advantages and disadvantages of these methods are highlighted and assessed through comparisons of results obtained from a variety of representative linear FE models.

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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. @en

Slender structures with high stiffness-to-weight ratio form the main bearing element of modern engineering. This renders geometrical non-linear effects a key feature to be considered throughout the whole life-time of diverse structural components, such as Wind Turbine (WT) blades. Although the Finite Element (FE) method constitutes a well-established tool for the analysis of such systems, the resulting models are often prohibitively expensive in terms of computational resources and thus cannot be implemented in the design. The problem of state estimation for condition diagnostics and control applications is therefore rendered a challenging and intricate task when it comes to systems experiencing geometrical non-linearities. This is firstly due to the computationally demanding FE models associated with such systems and, secondly, to the requirement that estimation methods must consider non-linear phenomena. The problem is further pronounced in online applications, where real-time performance is required, as is commonly the case in structural health monitoring (SHM). Within this context, the focus is on computationally efficient models that operate on subspaces of significantly smaller size as compared to the full-order problem and which can be tailored to the framework of non-linear state estimation. This study proposes the implementation of physics-based reduced-order models (ROMs) for response prediction of systems featuring geometrically non-linear effects. In so doing, the concept of modal derivatives is adopted and combined with a flexible multibody approach in order to capture secondorder effects, e.g. twist-bend coupling, that arise as the system departs from the linear regime. In identifying the vibration response of such structures, the ROMs are compounded with the unscented Kalman filter (UKF) for the non-linear state estimation. The outlined approach is tested on the real-time response prediction of a WT blade, assuming that a limited number of artificial vibration measurements is available. The effectiveness of the scheme is assessed as a tool for online SHM and vibration control.@en

Component mode synthesis is commonly used to simulate the structural behavior of complex systems. Among other component mode synthesis techniques, the Craig–Bampton method stands out for its popularity. However, for finely meshed systems featuring many components, the size of the resulting assembled system is dominated by the interface degrees of freedom. The system-level interface reduction technique aims at reducing the size of the assembled reduced model by extracting a few dominant interface modes. If the size of the interface degrees of freedom is large, the resulting problem is almost as computationally expensive as the one associated to the full model. Conversely, the local-level interface reduction technique reduces the interface of each substructure before assembly. In this case, the computational effort associated to the local eigenvalue problem is moderate, but issues arise when enforcing compatibility between interfaces. In this paper, the computational effort related to the interface reduction is significantly reduced by performing two variants of the multilevel Craig–Bampton reduction when the subsystems are assembled in subsets. This procedure localizes the interface reduction by applying a multilevel static condensation and eigenvalue analysis on each subset in parallel. The different interface reduction techniques are assessed on large-size realistic examples.

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Component Mode Synthesis is commonly used to simulate the structural behavior of complex systems with many degrees of freedom. The Craig-Bampton approach is one of the most commonly used techniques. A novel reduction method is proposed here for geometrically nonlinear models by augmenting the constraint modes and internal vibration modes with the modal derivatives. A subset of the corresponding modal derivatives can therefore be efficiently used to consider the geometric nonlinearities. This modal substructuring technique is an extension of the Craig-Bampton method without increasing the difficulty of implementation. The applicability and efficiency of the modal derivative based Craig-Bampton method for nonlinear system is demonstrated by a numerical example.

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The Floating Frame of Reference (FFR) provides a natural framework for the Model Order Reduction (MOR) of flexible multibody systems. The classical reduction carried out by a Galerkin projection on a reduced basis of Vibration Modes (VMs), however, is not applicable when the elastic deformations become finite. In this contribution, we present a MOR technique based on a quadratic manifold on which the reduced solution lives. The manifold is build by an expansion of the elastic displacements for each flexible body. The quadratic terms are formed by Modal Derivatives (MDs) that properly account for the effect of the geometric nonlinearity. As opposed to classical Galerkin projection for geometrically nonlinear systems, this approach minimizes the size of the reduced order model, at the price of a more complex nonlinear system.@en