14 L. Schwerdt et al. m Master/interface degrees of freedom s Slave/inner degrees of freedom ˜ Reduced model 2.1 Introduction Finite element models are a vital part of the product development process in turbomachinery to predict the vibratory response, but detailed models of multistage rotors are prohibitively large. Therefore models of single stages are used predominantly although the importance of multistage effects is known [1]. Increasingly more methods are developed to generate reduced order models for multistage assemblies [2–6]. Most use a variant of Component Mode Synthesis (CMS) to split the rotor into substructures, which are reduced separately. This can reduce the amount of memory required for an analysis and allows to perform multiple analyses efficiently using the ROM. In this paper, a model order reduction method for multistage bladed disks is presented, that uses CMS with the CraigBampton reduction (CB-CMS) with a priori interface reduction. As basis functions for the interface motion, a product of Fourier harmonics in circumferential direction and polynomials in radial direction is used. First, the CB-CMS method is presented by itself and then with interface reduction. afterwards, an overview of the existing model reduction methods for multistage bladed disks is given before the proposed method is presented. Finally, the new method is demonstrated on an academic two stage rotor. 2.2 Craig-Bampton Method and Interface Reduction One method to generate reduced order models is the Component Mode Synthesis. It is widely used in turbomachinery and other applications, by itself and as a basis for more advanced model order reduction methods. In the CMS method, the structure is split up in to multiple components, also known as substructures. Each of the components is then reduced individually and the reduced components are assembled to yield the reduced model of the complete structure. Different methods for the reduction and assembly are available, cf. [7]. 2.2.1 Craig-Bampton Method The most popular of these methods is the Craig-Bampton reduction[8]. Here the degrees of freedom (DOF) of each substructure are split intomaster (xm) and slave (xs) DOF and the stiffness and mass matrices are partitioned accordingly: K= Kmm Kms Ksm Kss M= Mmm Mms Msm Mss (2.1) The master DOF are kept in the reduced model while a truncated set of modal DOF (ηs) represent the slave DOF in the reduced system: xm xs = I 0 Ψ Φ xm ηs =T xm ηs (2.2) The constraint modes Ψ and fixed-interface normal modes Φ= ϕ1, ϕ2, . . . are obtained by: Ψ =−K−1 ss Ksm Kss −ω 2 i Mss ϕi =0 (2.3) The reduced stiffness and mass matrices are ˜K=T H KT ˜M=T H MT (2.4)
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