Chapter 2 A Priori Interface Reduction for Substructuring of Multistage Bladed Disks Lukas Schwerdt, Lars Panning-von Scheidt, and Jörg Wallaschek Abstract When analyzing the dynamics of bladed disks in turbomachinery, most methods focus on a single stage at a time because of the challenges associated with multistage structures. Whereas the cyclic symmetry of individual bladed disks is commonly exploited to yield great savings of computational effort, multistage rotors lack this symmetry due to the differing number of blades in each stage. Substructuring methods can be used to overcome this problem but they still face challenges with non-conforming finite element meshes at the interface between stages. Some state of the art methods expect the nodes at the interface to be arranged in concentric rings and use a truncated Fourier series as basis for the displacement along each ring of nodes. In this paper, a reduction basis for the interface degrees of freedom between adjacent stages is proposed which uses polynomial basis functions in the radial direction in addition to a truncated Fourier series in the circumferential direction. This enables coupling the substructures of multiple stages with arbitrary meshes. Additionally, the resulting reduced order model (ROM) can be smaller while preserving accuracy. The proposed interface reduction is demonstrated in conjunction with a cyclic Craig-Bampton (CB) reduction of each stage. Different ROMs are compared to show the impact of the CB reduction as well as the interface reduction. Keywords Model order reduction · Component mode synthesis · Multistage · Interface reduction · Mistuning Nomenclature n Number of degrees of freedom x Physical degrees of freedom y Degrees of freedom in travelling wave coordinates F DFTmatrix H Fourier harmonic basis function I Identity matrix K Stiffness matrix M Mass matrix N Number of blades P Polynomial basis function T Transformation/reduction matrix V Interface reduction basis function W Matrix of interface reduction basis η Reduced/modal degrees of freedom ϕ Mode/eigenvector Matrix of fixed interface modes Matrix of constraint modes (r,α) (Polar) coordinates on the interface h Substructure harmonic index j Interface harmonic index l Mode index k Stage index L. Schwerdt ( ) · L. Panning-von Scheidt · J. Wallaschek Institute of Dynamics and Vibration Research, Faculty of Mechanical Engineering, Leibniz University Hannover, Hannover, Germany e-mail: schwerdt@ids.uni-hannover.de © Society for Experimental Mechanics, Inc. 2020 A. Linderholt et al., Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12184-6_2 13
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