Structural Engineering and Mechanics

  • 제81권4호
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  • Pages.411-428
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  • 2022
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Model order reduction for Campbell diagram analysis of shaft-disc-blade system in 3D finite elements

  • Phuor, Ty (School of Mechanical Engineering, Hanyang University) ;
  • Yoon, GilHo (School of Mechanical Engineering, Hanyang University)
  • 투고 : 2021.08.08
  • 심사 : 2021.11.07
  • 발행 : 2022.02.25
⟨ 이전 논문 다음 논문 ⟩

초록

This paper presents the Campbell diagram analysis of the rotordynamic system using the full order model (FOM) and the reduced order model (ROM) techniques to determine the critical speeds, identify the stability and reduce the computational time. Due to the spin-speed-dependent matrices (e.g., centrifugal stiffening matrix), several model order reduction (MOR) techniques may be considered, such as the modal superposition (MS) method and the Krylov subspace-based MOR techniques (e.g., Ritz vector (RV), quasi-static Ritz vector (QSRV), multifrequency quasi-static Ritz vector (MQSRV), multifrequency/ multi-spin-speed quasi-static Ritz vector (MMQSRV) and the combined Ritz vector & modal superposition (RV+MS) methods). The proposed MMQSRV method in this study is extended from the MQSRV method by incorporating the rotational-speed-dependent stiffness matrices into the Krylov subspace during the MOR process. Thus, the objective of this note is to respond to the question of whether to use the MS method or the Krylov subspace-based MOR technique in establishing the Campbell diagram of the shaft-disc-blade assembly systems in three-dimensional (3D) finite element analysis (FEA). The Campbell diagrams produced by the FOM and various MOR methods are presented and discussed thoroughly by computing the norm of relative errors (ER). It is found that the RV and the MS methods are dominant at low and high rotating speeds, respectively. More precisely, as the spinning velocity becomes large, the calculated ER produced by the RV method is significantly increased; in contrast, the ER produced by the MS method is smaller and more consistent. From a computational point of view, the MORs have substantially reduced the time computing considerably compared to the FOM. Additionally, the verification of the 3D FE rotordynamic model is also provided and found to be in close agreement with the existing solutions.

키워드

과제정보

This work was supported by Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government (MOTIE) (2021202080026D, Development of platform technology and operation management system for design and operating condition diagnosis of fluid machinery with variable devices based on AI/ICT).

참고문헌

  1. Bladh, R., Castanier, M.P. and Pierre, C. (2001), "Component-mode-based reduced order modeling techniques for mistuned bladed disks-Part 1: Theoretical models", J. Eng. Gas Turbin. Power, 123(1), 89-99. https://doi.org/10.1115/1.1338947.
  2. Castanier, M.P. and Pierre, C. (2002), "Using intentional mistuning in the design of turbomachinery rotors", AIAA J., 40(10), 2077-2086. https://doi.org/10.2514/2.1542.
  3. Chatelet, E., D'Ambrosio, F. and Jacquet-Richardet, G. (2005), "Toward global modelling approaches for dynamic analyses of rotating assemblies of turbomachines", J. Sound Vib., 282(1-2), 163-178. https://doi.org/10.1016/j.jsv.2004.02.035.
  4. Crawely, E.F., Ducharme, E.H. and Mokadam, D.R. (1986), "Analytical and experimental investigation of the coupled bladed disk/shaft whirl of a cantilevered turbofan", J. Eng. Gas Turbin. Power, 108, 567-575. https://doi.org/10.1115/1.3239948.
  5. Feiner, D.M. and Griffin, J.H. (2002), "A fundamental model of mistuning for a single family of modes", J. Turbomach., 124(4), 597-605. https://doi.org/10.1115/1.1508384.
  6. Genta, G. (2005), Dynamics of Rotating System, Springer Science & Business Media, New York, USA.
  7. Genta, G. and Gugliotta, A. (1988), "A conical element for finite element rotor dynamics", J. Sound Vib., 120(1), 175-182. https://doi.org/10.1016/0022-460X(88)90342-2.
  8. Gruttmann, F. and Wagner, W. (2004), "A stabilized one-point integrated quadrilateral Reissner-Mindlin plate element", Int. J. Numer. Meth. Eng., 61, 2273-2295. https://doi.org/10.1002/nme.1148.
  9. Gu, J., Ma, Z.D. and Hulbert, M.G. (2000), "New load-dependent Ritz vector method for structural dynamics analyses: Quasi-static Ritz vectors", Finite Elem. Anal. Des., 36, 261-278. https://doi.org/10.1016/S0168-874X(00)00036-6.
  10. Han, J.S. (2014), "Krylov subspace-based model order reduction for Campbell diagram analysis of large-scale rotordynamic systems", Struct. Eng. Mech., 50(1), 19-36. https://doi.org/10.12989/sem.2014.50.1.019.
  11. Huang, S.C. and Ho, K.B. (1996), "Coupled shaft-torsion and blade-bending vibrations of a rotating shaft-disk-blade unit", J. Eng. Gas Turbin. Power, 118, 100-106. https://doi.org/10.1115/1.2816524.
  12. Khulief, Y.A. and Mohiuddin, M.A. (1997), "On the dynamic analysis of rotors using modal reduction", Finite Elem. Anal. Des., 26(1), 41-55. https://doi.org/10.1016/S0168-874X(96)00070-4.
  13. Kirchgassner, B. (2016), "Finite elements in rotordynamics", Procedia Eng., 144, 736-750. https://doi.org/10.1016/j.proeng.2016.05.079.
  14. Koh, H.S., Kim, J.H. and Yoon, G.H. (2020), "Efficient topology optimization of multicomponent structure using substructuring-based model order reduction method", Comput. Struct., 228, 106146. https://doi.org/10.1016/j.compstruc.2019.106146.
  15. Lazarus, A., Prabel, B. and Combescure, D. (2010), "A 3D finite element model for the vibration analysis of asymmetric rotating machines", J. Sound Vib., 329(18), 3780-3797. https://doi.org/10.1016/j.jsv.2010.03.029.
  16. Li, C., She, H., Tang, Q. and Wen, B. (2019), "The coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features", Appl. Math. Model., 67, 557-572. https://doi.org/10.1016/j.apm.2018.09.041.
  17. Lim, S.H., Bladh, R., Castanier, M.P. and Pierre, C. (2007), "Compact, generalized component mode mistuning representation for modeling bladed disk vibration", AIAA J., 45(9), 2285-2298. https://doi.org/10.2514/1.13172.
  18. Ma, O. and Wang, J.G. (2007), "Model order reduction for impact-contact dynamics simulations of flexible manipulators", Robotica, 25, 397-407. https://doi.org/10.1017/S026357470600316X.
  19. Martel, C. and Sanchez-Alvarez, J.J. (2018), "Intentional mistuning effect in the forced response of rotors with aerodynamic damping", J. Sound Vib., 433, 212-229. https://doi.org/10.1016/j.jsv.2018.07.020.
  20. Mogenier, G., Baranger, T., Ferraris, G., Dufour, R. and Durantay, L. (2014), "A criterion for mode shape tracking: Application to Campbell diagrams", J. Vib. Control, 20(2), 179-190. https://doi.org/10.1177/1077546312463714.
  21. Nandi, A. (2004), "Reduction of finite element equations for a rotor model on non-isotropic spring support in a rotating frame", Finite Elem. Anal. Des., 40(9-10), 935-952. https://doi.org/10.1016/S0168-874X(03)00121-5.
  22. Nandi, A. and Neogy, S. (2001), "Modelling of rotors with three-dimensional solid finite elements", J. Strain Anal. Eng. Des., 6(4), 359-371. https://doi.org/10.1243/0309324011514539.
  23. Nelson, F.C. (2007), "Rotor dynamics without equations", Int. J. COMADEM, 10(3), 2-10.
  24. Nelson, H.D. (1980), "A finite rotating shaft element using Timoshenko beam theory", J. Mech. Des., Trans., ASME, 102(4), 793-803. https://doi.org/10.1115/1.3254824.
  25. Nelson, H.D. and McVaugh, J.M. (1976), "The dynamics of rotor-bearing systems using finite elements", J. Mech. Des., Trans., ASME, 98(2), 593-600. https://doi.org/10.1115/1.3438942.
  26. Okabe, A., Otawara, Y., Kaneko, R., Matsushita, O. and Namura, K. (1991), "An equivalent reduced modelling method and its application to shaft-blade coupled torsional vibration analysis of a turbine-generator set", Proc. Inst. Mech. Eng., Part A: J. Pow. Energy, 205, 173-181. https://doi.org/10.1243/PIME_PROC_1991_205_026_02.
  27. Petrov, E.P., Zachariadis, Z.I., Beretta, A. and Elliott, R. (2013), "A study of nonlinear vibrations in a frictionally damped turbine bladed disk with comprehensive modeling of aerodynamic effects", J. Eng. for Gas Turbin. Power, 135(3), 032504-11. https://doi.org/10.1115/1.4007871.
  28. Phuor, T. (2020), "Development and application of three-dimensional finite element interface model for soil-jack-up interaction during preloading", PhD Thesis, Universiti Teknologi Petronas.
  29. Phuor, T., Harahap, I.S., Ng, C.Y. and Al-Bared, M.A.M. (2021a), "Development and of the skew boundary condition for soil-structure interaction in three-dimensional finite element analysis", Comput. Geotech., 137, 104264. https://doi.org/10.1016/j.compgeo.2021.104264.
  30. Phuor, T., Harahap, I.S.H. and Ng, C.Y. (2021b), "Bearing capacity factors for rough conical footing by viscoplasticity finite element analysis", ASCE Int. J. Geomech., 22(1), 04021266. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002256.
  31. Rahman, T. and Valdman, J. (2013), "Fast MATLAB assembly of FEM matrices in 2D and 3D: Nodal elements", Appl. Math. Comput., 219(13), 7151-7158. https://doi.org/10.1016/j.amc.2011.08.043.
  32. Saito, A., Castanier, M.P. and Pierre, C. (2009), "Effects of a cracked blade on mistuned turbine engine rotor vibration", J. Vib. Acoust., Trans., ASME, 131(6), 061006-9. https://doi.org/10.1115/1.4000458.
  33. She, H., Li, C., Tang, Q. and Wen, B. (2018), "The investigation of the coupled vibration in a flexible-disk blades system considering the influence of shaft bending vibration", Mech. Syst. Signal Pr., 111, 545-569. https://doi.org/10.1016/j.ymssp.2018.03.044.
  34. Stephenson, R.W. and Rouch, K.E. (1993), "Modeling rotating shafts using axisymmetric solid finite elements with matrix reduction", J. Vib. Acoust., Trans., ASME, 115(4), 484-489. https://doi.org/10.1115/1.2930376.
  35. Sternchuss, A. (2009), "Multi-level parametric reduced models of rotating bladed disk assemblies", Ecole Centrale Paris.
  36. Vest, T.A. and Darlow, M.S. (1990), "A modified conical beam element based on finite element analysis: Experimental correlations", J. Vib. Acoust., Trans., ASME, 112(3), 350-354. https://doi.org/10.1115/1.2930515.
  37. Vollan, A. and Komzsik, L. (2012), Computational Techniques of Rotor Dynamics with the Finite Element Method, CRC Press.
  38. Wagner, M.B., Younan, A., Allaire, P. and Cogill, R. (2010), "Model reduction methods for rotor dynamic analysis: A survey and review", Int. J. Rotat. Mach., 2010, Article ID 273716. https://doi.org/10.1155/2010/273716.
  39. Wang, S., Bi, C.X. and Zheng, C.J. (2019), "A reduced-order model for the vibration analysis of mistuned blade-disc-shaft assembly", Appl. Sci., 9(22), 4762. https://doi.org/10.3390/app9224762.
  40. Wang, S., Wang, Y., Zi, Y. and He, Z. (2015), "A 3D finite element-based model order reduction method for parametric resonance and whirling analysis of anisotropic rotor-bearing systems", J. Sound Vib., 359, 115-135. https://doi.org/10.1016/j.jsv.2015.08.027.
  41. Wang, S., Wang, Y., Zi, Y., Li, B. and He, Z. (2015), "Reduced-order modeling for rotating rotor-bearing systems with cracked impellers using three-dimensional finite element models", J. Sound Vib., 355, 305-321. https://doi.org/10.1016/j.jsv.2015.06.037.
  42. Wang, S., Zi, Y., Li, B., Zhang, C. and He, Z. (2014), "Reduced-order modeling for mistuned centrifugal impellers with crack damages", J. Sound Vib., 333(25), 6979-6995. https://doi.org/10.1016/j.jsv.2014.07.009.
  43. Wilson, E.L., Yuan, M.W. and Dickens, J.M. (1982), "Dynamic analysis by direct superposition of Ritz Vectors", Earthq. Eng. Struct. Dyn., 10(6), 813-821. https://doi.org/10.1002/eqe.4290100606.
  44. Yang, M.T. and Griffin, J.H. (2001), "A reduced-order model of mistuning using a subset of nominal system modes", J. Eng. Gas Turbin. Power, 123(4), 893-900. https://doi.org/10.1115/1.1385197.
  45. Yoon, G.H. (2010), "Structural topology optimization for frequency response problem using model reduction schemes", Comput. Meth. Appl. Mech. Eng., 199(25-28), 1744-1763. https://doi.org/10.1016/j.cma.2010.02.002.
  46. Yoon, G.H. (2012), "Toward a multifrequency quasi-static Ritz vector method for frequency-dependent acoustic system application", Int. J. Numer. Meth. Eng., 89, 1451-1470. https://doi.org/10.1002/nme.3301.
  47. Yoon, G.H., Kim, J.H., Jung, K.O. and Jung, J.W. (2015), "Transient quasi-static Ritz vector (TQSRV) method by Krylov subspaces and eigenvectors for efficient contact dynamic finite element simulation", Appl. Math. Model., 39(9), 2740-2762. https://doi.org/10.1016/j.apm.2014.10.059.
  48. Yuan, J., Scarpa, F., Allegri, G., Titurus, B., Patsias, S. and Rajasekaran, R. (2017), "Efficient computational techniques for mistuning analysis of bladed discs: A review", Mech. Syst. Signal Pr., 87, 71-90. https://doi.org/10.1016/j.ymssp.2016.09.041.