Model Reduction Techniques for Structural Dynamic Analyses
This chapter presents a model reduction technique based on substructure coupling for dynamic analysis. The dynamic behavior of the substructures is described by a set of dominant fixed-interface normal modes along with a set of interface constraint modes
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Model Reduction Techniques for Structural Dynamic Analyses
Abstract This chapter presents a model reduction technique based on substructure coupling for dynamic analysis. The dynamic behavior of the substructures is described by a set of dominant fixed-interface normal modes along with a set of interface constraint modes that account for the coupling at each interface where the substructures are connected. Based on these modes, the corresponding reducedorder matrices are derived. The internal dynamic behavior of the substructures is then enhanced by consideration of the contribution of residual fixed-interface normal modes. Next, the interface degrees of freedom are reduced by consideration of a small number of characteristic constraint modes. Pseudo-codes are provided in order to illustrate how the reduced-order matrices are constructed, by including dominant and residual fixed-interface normal modes as well as interface reduction. Finally, the dynamic response of reduced-order models is discussed.
1.1 Structural Model Attention is focused on a general class of structural dynamical systems, with localized nonlinearities characterized by multi-degrees of freedom models that satisfy the equation of motion ˙ ¨ ˙ y(t)) + f(t) Mu(t) + Cu(t) + Ku(t) = fNL (u(t), u(t),
(1.1)
˙ the velocity vector, where u(t) denotes the displacement vector of dimension n, u(t) ˙ ¨ y(t)) the vector of nonlinear restoring u(t) the acceleration vector, fNL (u(t), u(t), forces, y(t) the vector of a set of variables that describes the state of the nonlinear components, and f(t) the external force vector. The matrices M, C, and K, which are assumed to be symmetric, describe the mass, damping, and stiffness, respectively. The evolution of the set of variables y(t) is described through an appropriate nonlinear model that depends on the nature of the nonlinearity. The equation of motion for the
© Springer Nature Switzerland AG 2019 H. Jensen and C. Papadimitriou, Sub-structure Coupling for Dynamic Analysis, Lecture Notes in Applied and Computational Mechanics 89, https://doi.org/10.1007/978-3-030-12819-7_1
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1 Model Reduction Techniques for Structural Dynamic Analyses
displacement vector u(t) and the equation for the evolution of the set of variables y(t) constitute a system of coupled nonlinear equations. This characterization of the dynamical system allows the description of different types of models commonly used for nonlinearities, such as hysteresis, degradation, plasticity, and other types of nonlinearities [2, 3, 27]. In the dynamic characterization of this class of structural dynamical systems, it is often inefficient to carry out a finite element analysis of the entire model. In fact, in many dynamic analysis problems, the lower frequencies and the corresponding modes tend to dominate the dynamic behavior of the structure. It is also common for component structures to be analyzed independently in complex systems, which makes it more convenient to perform a dynamic analysis at the substructure level. In this framework, model r
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