20 L. Schwerdt et al. maximum harmonic 0 14 Number of degrees of freedom Max relative eigenfrequency error among the first 15 modes 0 100 200 300 400 500 10−9 10−7 10−5 10−3 10−1 101 Fig. 2.6 Non-dominated ROMs for a fixed value of the maximum interface harmonic 2.6 Conclusions In this paper, a model order reduction method for multistage bladed disks is presented. It is based on a cyclic Craig-Bampton reduction of each stage. The interfaces are reduced beforehand using basis functions that are a product of Fourier harmonics in the circumferential direction and polynomials in radial direction. It is a strict improvement over the approach of using only Fourier constraint modes, as the requirements for the interface meshes are removed and ROMs with less DOF are possible. The method is evaluated on an academic two-stage rotor. The possible trade-offs between ROM size and accuracy are demonstrated for different choices of number of interface harmonics and polynomial degrees as well as number of fixed interface modes. The benefits of the proposed method are even greater for larger models with finely meshed interfaces but decrease with higher accuracy requirements for the ROM. All calculations can be performed on single sector sized matrices. Using prior basis functions for the interface ensures that no intermediate reduction step potentially requires extreme amounts ofmemory. For mistuning analyses the presented ROM can be extended using an additional modal analysis [23], or by implementing the interface reduction into more advanced methods such as MMDA [5] and PRIME [6]. Acknowledgement The authors kindly thank the German Research Foundation (DFG) for enabling this publication by funding the research project Influence of Regeneration-induced Mistuning on the Aeroelasticity of Multi-Stage Axial Compressors as part of the Collaborative Research Center 871 Regeneration of Complex Capital Goods. References 1. Bladh, R., Castanier, M.P., Pierre, C.: Effects of multistage coupling and disk flexibility on mistuned bladed disk dynamics. J. Eng. Gas Turbines Power 125(1), 121 (2003) 2. Song, S.H., Castanier, M.P., Pierre, C.: Multi-stage modeling of turbine engine rotor vibration. In: Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, pp. 1533–1543. ASME, Long Beach (2005) 3. Laxalde, D., Thouverez, F., Lombard, J.P.: Dynamical analysis of multi-stage cyclic structures. Mech. Res. Commun. 34(4), 379–384 (2007) 4. Sternchüss, A., Balmes, E., Jean, P., Lombard, J.P.: Model reduction applied to multi-stage assemblies of bladed disks. In: ISMA, 15p (2008) 5. Bhartiya, Y., Sinha, A.: Reduced order model of a multistage bladed rotor with geometric mistuning via modal analyses of finite element sectors. J. Turbomach. 134(4), 041001 (2012) 6. Kurstak, E., D’Souza, K.: Multistage blisk and large mistuning modeling using Fourier Constraint Modes and PRIME. In: Volume 7B: Structures and Dynamics, p. V07BT35A012. ASME, Long Beach (2017) 7. Klerk, D.D., Rixen, D.J., Voormeeren, S.N.: General framework for dynamic substructuring: history, review and classification of techniques. AIAAJ. 46(5), 1169–1181 (2008) 8. Craig, J.R., Roy, R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968) 9. Schwerdt, L., Willeke, S., Panning-von Scheidt, L., Wallaschek, J.: Reduced-order modeling of bladed disks considering small mistuning the disk sectors. J. Eng. Gas Turbines Power. 141(5), 052502 (2018). Advance online publication https://doi.org/10.1115/1.4041071
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