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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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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We demonstrate a Model Order Reduction technique for a system of nonlinear equations arising from the Finite Element Method (FEM) discretization of the three-dimensional quasistatic equilibrium equation equipped with a Perzyna viscoplasticity constitutive model. The procedure employs the Proper Orthogonal Decomposition-Galerkin (POD-G) in conjunction with the Discrete Empirical Interpolation Method (DEIM). For this purpose, we collect samples from a standard full order FEM analysis in the offline phase and cluster them using a novel kk-means clustering algorithm. The POD and the DEIM algorithms are then employed to construct a corresponding reduced order model. In the online phase, a sample from the current state of the system is passed, at each time step, to a nearest neighbor classifier in which the cluster that best describes it is identified. The force vector and its derivative with respect to the displacement vector are approximated using DEIM, and the system of nonlinear equations is projected onto a lower dimensional subspace using the POD-G. The constructed reduced order model is applied to two typical solid mechanics problems showing strain-localization (a tensile bar and a wall under compression) and a three-dimensional square-footing problem.@en

Residual stresses are common in Micro Electro Mechanical System (MEMS) membrane structures. Experimental assessment of these stresses can provide valuable information on the production process. In general, experimental stress assessment for MEMS is challenging due to the limited possibilities for non-destructive testing. This work investigates the use of dynamic modal data to identify the residual stress state. In view of the computational feasibility, the focus is on two aspects: 1) A method is proposed that expresses the unknown stress field as a combination of few, carefully selected stress modes. An optimization algorithm is deemed to identify the amplitudes of such modes. 2) A meta-model is constructed using the empirical interpolation method (EIM), to facilitate a fast evaluation of the iterations.

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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

Keywords: FWMAV control, eigensolution sensitivity, repeated eigenvalues, structural modifications, modal basis. Abstract: Several Flapping Wing Micro Air Vehicle (FWMAV) designs use structural resonating properties to increase energy efficiency. For practical use, control of the resonating structure, i.e., the eigensolutions, is required which is complicated by constraints on weight and power consumption. Hence, systematic ways to induce substantial changes in the resonance response are of high interest. Eigensolution sensitivity is used to approximate the change in the resonance vibration, i.e., the eigenmode, due to local structural modifications. Expressing these sensitivities in a modal basis, shows that (nearly) repeated eigenvalues are an indication for effective eigenmode control. The eigenmodes corresponding to repeated eigenvalues are not uniquely defined but any linear combination of these eigenmodes is also an eigenmode. This could lead to substantial changes in the eigenmode, or vibration response, if the eigenmodes become uniquely defined again if the corresponding eigenvalues become distinct due to local structural modifications. A smart structural design could use this idea to exhibit substantial changes in the eigenmodes as introduced by relatively limited control action. The response of a simple symmetric structure, with repeated eigenvalues, shows the response change if the structure becomes asymmetric. Additionally, the influence of structural damping on the effectiveness of the response change is shown.@en