16 L. Schwerdt et al. 2.3.1 Single Stage Bladed Disks Single bladed disks are cyclic symmetric structures. This results in system matrices that are block circulant. By introducing travelling wave coordinates the matrices become block diagonal. This is only possible for identical disk sectors and blades. When deviations among the blades (mistuning) are considered, the matrices in travelling wave coordinates are fully populated in general. Nonetheless, some reduction methods for mistuned bladed disks make use of this representation as well (cf. [19]). The popular SNM method for example directly uses the modes of the tuned (i.e. cyclic symmetric) structure as a reduction basis[20]. The transformation from the physical coordinates x into travelling wave coordinates y is performed according to x =(F ⊗I)y (2.7) where F is the complex DFT matrix with a size of the number of sectors n and I is the identity matrix with a size of the number of DOF per sector. In travelling wave coordinates all analyses can be performed on the sector level i.e. one can don calculations where the matrices have the size of the number of DOF of a single sector. 2.3.2 Multistage Bladed Disks Due to the variable number of blades among the different stages, a multistage rotor does not in general exhibit cyclic symmetry. Therefore, other methods have to be applied to generate a reduced order model of the rotor. These methods rely on CMS and some exploit the cyclic symmetry of individual stages on the substructure level. Sternchüss uses individual sectors as components and couples them through intermediate rings which are only one layer of elements thick [21]. Cyclic symmetry is not exploited. Laxalde presented the multi-stage cyclic symmetry approach [3]. Here the coupling between adjacent stages is reduced to enable exploiting the computational advantages of cyclic symmetry. When representing each stage in travelling wave coordinates, in general all harmonic indices of adjacent stages are coupled together [4]. There can be exceptions depending on the number of blades. The multi-stage cyclic symmetry approach discards most of these connections, but retains the one belonging to the same number of nodal diameters. Thereby the complete rotor model is decomposed into systems with a size of one sector from every stage each. The resulting modes are similar to the actual system modes, but not identical. Nonetheless, they represent a good reduction basis to project the full system into. Disadvantages are the difficulty of connecting adjacent stages with non-matching meshes at the interface and the fact that the substructures are relatively big, as they have a size of multiple sectors, one from each stage. Song uses a CB-CMS approach where each harmonic index of each stage is a single substructure [2]. A Fourier basis is used to reduce the interfaces. To facilitate this, the interface is assumed to be ring shaped and meshed with concentric rings of nodes. For each of these rings, truncated Fourier series are used to represent the displacement along each direction. Thereby the number of interface DOF is reduced significantly, while retaining the substructure size of a single sector from single stage analyses. Using this prior basis function method adjacent sectors do not need matching meshes at the interface in circumferential direction. This is advantageous because the number of elements in circumferential direction for each stage needed to achieve a matching mesh is the least common multiple of the number of blades of the stage and its neighbors, e.g. 10,100 for a two stage design with 100 and 101 blades. Currently, Song’s method for the interface reduction is used with more advanced model order reduction methods for multistage assemblies [6, 22]. The interface reduction was later called Fourier Constraint Modes (FCM). 2.4 Proposed Model Order Reduction Method The proposed reduction method for multistage bladed disks uses CB-CMS with an a priori interface reduction based on the FCM method. The rotor is split into individual stages. The ring shaped interfaces between the stages are reduced using basis functions defined a priori. These functions consist of Fourier harmonics in the circumferential direction and polynomials in the radial direction. A cyclic Craig-Bampton reduction is then applied to each stage.
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