Abstract
Many rotating machines utilize a fluid-type bearing. Despite their reliability and high load capacity, these bearings often show instabilities due to the interaction between the fluid media and the rotating shaft. These instabilities, known as oil-whirl and oil-whip, occur due to a Hopf bifurcation; being the parameter the shaft speed. Identifying the type of bifurcation, either sub-critical or super-critical, is an important task to determine the safety of the machines near the instability speed, and it tells whether one experiences oil-whip or oil-whirl. This work presents an approach, based on the center manifold reduction (CMR) method, to obtain limit cycles near Hopf bifurcations of rotors supported on fluid-film bearings. The basis of the method is the obtention of the center manifold of the system, which allows one to assess the type of bifurcation at hand. To obtain the center manifold, the parameterization method for invariant manifolds is used, which is a powerful tool to obtain invariant manifolds of high-dimensional dynamical systems. The proposed method is evaluated by comparing its results with an open-source numerical continuation package (MATCONT) in two systems: a simple and a realistic rotor system. The results show that the CMR, together with the parameterization method, can be used reliably to learn if the rotor system presents sub- or super-critical bifurcations, to perform parametric studies with different bearing properties, and also to predict the amplitude of the limit cycles.
Similar content being viewed by others
Data availibility
The datasets generated during and analyzed during the current study are not publicly available but are available from the corresponding author on reasonable request
References
Newkirk, B.L.: Shaft whipping. Gen. Electr. Rev. 27, 169 (1924)
Newkirk, B.L., Taylor, H.D.: Shaft whipping due to oil action in journal bearings. Gen. Electr. Rev. 28(8), 559–568 (1925)
Kimball, A.L.: Internal friction as a cause of shaft whirling. Lond. Edinb. Dublin Philos. Mag. J. Sci. 49(292), 724–727 (1925)
Smith, D.M.: The motion of a rotor carried by a flexible shaft in flexible bearings. Proc. R. Soc. Lond. Ser. A 142(846), 92–118 (1933)
Robertson, D.: Whirling of a journal in a sleeve bearing. Lond. Edinb. Dublin Philos. Mag. J. Sci. 15(96), 113–130 (1933)
Muszynska, A.: Stability of whirl and whip in rotor/bearing systems. J. Sound Vib. 127(1), 49–64 (1988). https://doi.org/10.1016/0022-460X(88)90349-5
Muszynska, A.: Rotordynamics. Taylor & Francis, Boca Raton (2005)
Bachschmid, N., Pennacchi, P., Vania, A.: Steam-whirl analysis in a high pressure cylinder of a turbo generator. Mech. Syst. Signal Process. 22(1), 121–132 (2008). https://doi.org/10.1016/j.ymssp.2007.04.005
Vance, J.M., Laudadio, F.J.: Experimental measurement of Alford’s force in axial flow turbomachinery. J. Eng. Gas Turbines Power 106(3), 585–590 (1984). https://doi.org/10.1115/1.3239610
Untaroiu, A., Jin, H., Fu, G., Hayrapetiau, V., Elebiary, K.: The effects of fluid preswirl and swirl brakes design on the performance of labyrinth seals. J. Eng. Gas Turbines Power (2018). https://doi.org/10.1115/1.4038914
Lund, J.W.: Review of the concept of dynamic coefficients for fluid film journal bearings. J. Tribol. 109(1), 37–41 (1987). https://doi.org/10.1115/1.3261324
Tiwari, R., Lees, A.W., Friswell, M.I.: Identification of dynamic bearing parameters: a review. Shock Vib. Digest 36(2), 99–124 (2004). https://doi.org/10.1177/0583102404040173
Sawicki, J.T., Rao, T.V.V.L.N.: A nonlinear model for prediction of dynamic coefficients in a hydrodynamic journal bearing. Int. J. Rotat. Mach. 10, 507–513 (2004). https://doi.org/10.1155/S1023621X04000508
Meruane, V., Pascual, R.: Identification of nonlinear dynamic coefficients in plain journal bearings. Tribol. Int. 41(8), 743–754 (2008). https://doi.org/10.1016/j.triboint.2008.01.002
Chatterton, S., Pennacchi, P., Dang, P.V., Vania, A.: In: Pennacchi, P. (ed.) Proceedings of the 9th IFToMM International Conference on Rotor Dynamics. Mechanisms and Machine Science, pp. 931–941. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-06590-8_76
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, vol. 2. Springer, New York (2003). https://doi.org/10.1007/b97481
Hollis, P., Taylor, D.L.: Hopf bifurcation to limit cycles in fluid film bearings. J. Tribol. 108(2), 184–189 (1986). https://doi.org/10.1115/1.3261158
Wang, J.K., Khonsari, M.M.: Prediction of the stability envelope of rotor-bearing system. J. Vib. Acoust. 128(2), 197–202 (2005). https://doi.org/10.1115/1.2159035
Wang, J.K., Khonsari, M.M.: Bifurcation analysis of a flexible rotor supported by two fluid-film journal bearings. J. Tribol. 128(3), 594–603 (2006). https://doi.org/10.1115/1.2197842
Wang, J.K., Khonsari, M.M.: Application of Hopf bifurcation theory to rotor-bearing systems with consideration of turbulent effects. Tribol. Int. 39(7), 701–714 (2006). https://doi.org/10.1016/j.triboint.2005.07.031
Miraskari, M., Hemmati, F., Gadala, M.S.: Nonlinear dynamics of flexible rotors supported on journal bearings—part I: analytical bearing model. J. Tribol. (2017). https://doi.org/10.1115/1.4037730
Chasalevris, A.: Stability and Hopf bifurcations in rotor-bearing-foundation systems of turbines and generators. Tribol. Int. 145, 106,154 (2020). https://doi.org/10.1016/j.triboint.2019.106154
Troger, H., Steindl, A.: Nonlinear Stability and Bifurcation Theory. Springer, Vienna (1991). https://doi.org/10.1007/978-3-7091-9168-2
Boyaci, A., Hetzler, H., Seemann, W., Proppe, C., Wauer, J.: Analytical bifurcation analysis of a rotor supported by floating ring bearings. Nonlinear Dyn. 57(4), 497–507 (2009). https://doi.org/10.1007/s11071-008-9403-x
Kano, H., Ito, M., Inoue, T.: Order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing. Nonlinear Dyn. 95(4), 3275–3294 (2019). https://doi.org/10.1007/s11071-018-04755-z
Haro, À., Canadell, M., Figueras, J.L., Luque, A., Mondelo, J.M.: The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations. Applied Mathematical Sciences, vol. 195. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29662-3
Ponsioen, S., Pedergnana, T., Haller, G.: Automated computation of autonomous spectral submanifolds for nonlinear modal analysis. J. Sound Vib. 420, 269–295 (2018). https://doi.org/10.1016/j.jsv.2018.01.048
Vizzaccaro, A., Opreni, A., Salles, L., Frangi, A., Touzé, C.: High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point. Nonlinear Dyn. 110(1), 525–571 (2022). https://doi.org/10.1007/s11071-022-07651-9
Opreni, A., Vizzaccaro, A., Frangi, A., Touzé, C.: Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dyn. 105(2), 1237–1272 (2021). https://doi.org/10.1007/s11071-021-06641-7
Ponsioen, S., Jain, S., Haller, G.: Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems. J. Sound Vib. 488, 115,640 (2020). https://doi.org/10.1016/j.jsv.2020.115640
Opreni, A., Vizzaccaro, A., Touzé, C., Frangi, A.: High-order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to generic forcing terms and parametrically excited systems. Nonlinear Dyn. 111(6), 5401–5447 (2023). https://doi.org/10.1007/s11071-022-07978-3
Touzé, C., Vizzaccaro, A., Thomas, O.: Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dyn. 105(2), 1141–1190 (2021). https://doi.org/10.1007/s11071-021-06693-9
van den Berg, J.B., Hetebrij, W., Rink, B.: The parameterization method for center manifolds. J. Differ. Equ. 269(3), 2132–2184 (2020). https://doi.org/10.1016/j.jde.2020.01.033
Friswell, M.I., Penny, J.E.T., Seamus, D.G., Lees, A.W.: Dynamics of Rotating Machines. Cambridge University Press, New York (2010)
Vance, J.M., Murphy, B., Zeidan, F.: Machinery Vibration and Rotordynamics. Wiley, Hoboken (2010)
Someya, T. (ed.): Journal-Bearing Databook. Springer, Berlin (1989). https://doi.org/10.1007/978-3-642-52509-4
Krämer, E.: Dynamics of Rotors and Foundations. Springer, Berlin (1993)
Lee, C.W.: Vibration Analysis of Rotors, 1st edn. Springer, Dordrecht (1993)
Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dyn. 86(3), 1493–1534 (2016). https://doi.org/10.1007/s11071-016-2974-z
Meirovitch, L.: Computational Methods in Structural Dynamics, vol. 5. Sjithoff & Noordhoff International Publishers, Rockville (1980)
Jain, S., Haller, G.: How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models. Nonlinear Dyn. 107(2), 1417–1450 (2022). https://doi.org/10.1007/s11071-021-06957-4
Chouchane, M., Amamou, A.: Bifurcation of limit cycles in fluid film bearings. Int. J. Non-Linear Mech. 46(9), 1258–1264 (2011). https://doi.org/10.1016/j.ijnonlinmec.2011.06.005
Amamou, A.: Nonlinear stability analysis and numerical continuation of bifurcations of a rotor supported by floating ring bearings. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 236(5), 2172–2184 (2022). https://doi.org/10.1177/09544062211026340
Dhooge, A., Govaerts, W., Kuznetsov, Yu.A., Mestrom, W., Riet, A.M.: In: Proceedings of the 2003 ACM Symposium on Applied Computing—SAC ’03 ACM Press, Melbourne, Florida, p. 161 (2003). https://doi.org/10.1145/952532.952567
Boyaci, A.: Numerical continuation applied to nonlinear rotor dynamics. Procedia IUTAM 19, 255–265 (2016). https://doi.org/10.1016/j.piutam.2016.03.032
Anastasopoulos, L., Chasalevris, A.: Bifurcations of limit cycles in rotating shafts mounted on partial arc and lemon bore journal bearings in elastic pedestals. J. Comput. Nonlinear Dyn. (2022). https://doi.org/10.1115/1.4053593
Miura, T., Inoue, T., Kano, H.: Nonlinear analysis of bifurcation phenomenon for a simple flexible rotor system supported by a full-circular journal bearing. J. Vib. Acoust. (2017). https://doi.org/10.1115/1.4036098
Machado, T.H., Alves, D.S., Cavalca, K.L.: Discussion about nonlinear boundaries for hydrodynamic forces in journal bearing. Nonlinear Dyn. 92(4), 2005–2022 (2018). https://doi.org/10.1007/s11071-018-4177-2
El-Shafei, A., Tawfick, S.H., Raafat, M.S., Aziz, G.M.: Some experiments on oil whirl and oil whip. J. Eng. Gas Turbines Power 129(1), 144–153 (2004). https://doi.org/10.1115/1.2181185
ASTM D2270-10: Practice for Calculating Viscosity Index from Kinematic Viscosity at 40 and 100C ASTM International (2016). https://doi.org/10.1520/D2270-10R16
Mereles, A., Cavalca, K.L.: Modeling of multi-stepped rotor-bearing systems by the continuous segment method. Appl. Math. Model. 96, 402–430 (2021). https://doi.org/10.1016/j.apm.2021.03.001
Mereles, A., Alves, D.S., Cavalca, K.L.: Continuous model applied to multi-disk and multi-bearing rotors. J. Sound Vib. 537, 117,203 (2022). https://doi.org/10.1016/j.jsv.2022.117203
Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968). https://doi.org/10.2514/3.4741
Allen, M.S., Rixen, D., van der Seijs, M., Tiso, P., Abrahamsson, T., Mayes, R.L.: Substructuring in Engineering Dynamics: Emerging Numerical and Experimental Techniques, CISM International Centre for Mechanical Sciences, vol. 594. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-25532-9
Wagner, M.B., Younan, A., Allaire, P., Cogill, R.: Model reduction methods for rotor dynamic analysis: a survey and review. Int. J. Rotat. Mach. 2010 (2010)
Miraskari, M., Hemmati, F., Gadala, M.S.: Nonlinear dynamics of flexible rotors supported on journal bearings—part II: numerical bearing model. J. Tribol. (2017). https://doi.org/10.1115/1.4037731
Asgharifard-Sharabiani, P., Ahmadian, H.: Nonlinear model identification of oil-lubricated tilting pad bearings. Tribol. Int. 92, 533–543 (2015). https://doi.org/10.1016/j.triboint.2015.07.039
Alves, D.S., Cavalca, K.L.: In: Cavalca, K.L., Weber, H.I. (eds.) Proceedings of the 10th International Conference on Rotor Dynamics—IFToMM. Mechanisms and Machine Science, pp. 1–15. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-99262-4_1
Funding
The authors would like to thank CNPq (Grants #307941/2019-1 and #140275/2021-5) for the financial support of this research and the anonymous reviewers for their corrections and valuable comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mereles, A., Alves, D.S. & Cavalca, K.L. Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction. Nonlinear Dyn 111, 17749–17767 (2023). https://doi.org/10.1007/s11071-023-08788-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-08788-x