Dynamic analysis of a flexible rotor supported by ball bearings with damping rings based on FEM and lumped mass theory

Abstract

A dynamic model of a flexible rotor supported by ball bearings with rubber damping rings was proposed by combining the finite element and the mass-centralized method. In the proposed model, the rotor was built with the Timoshenko beam element, while the supports and bearing outer rings were modelled by the mass-centralized method. Meanwhile, the influences of the rotor’s gravity, unbalanced force and nonlinear bearing force were considered. The governing equations were solved by precise integration and the Runge-Kutta hybrid numerical algorithm. To verify the correctness of the modelling method, theoretical and experimental analysis is carried out by a rotor-bearing test platform, where the error rate between the theoretical and experimental studies is less than 10%. Besides that, the influence of the rubber damping ring on the dynamic properties of the rotor-bearing coupling system is also analyzed. The conclusions obtained are in agreement with the real-world deployment. On this basis, the bifurcation and chaos behaviors of the coupling system were carried out with rotational speed and rubber damping ring’s stiffness. The results reveal that as rotational speed increases, the system enters into chaos by routes of crisis, quasi-periodic and intermittent bifurcation. However, the paths of crisis, quasi-periodic bifurcation, and Hopf bifurcation to chaos were detected under the parameter of rubber damping ring’s stiffness. Additionally, the bearing gap affects the rotor system’s dynamic characteristics. Moreover, the excessive bearing gap will make the system’s periodic motion change into chaos, and the rubber damping ring’s stiffness has a substantial impact on the system motion.

Keywords

finite element method / Timoshenko beam / rubber damping ring / bifurcation / chaos

References

[1] FukataS, GadE H, KondouT, AyabeT, TamuraH. On the radial vibration of ball bearings (computer simulation) [J]. Bulletin of the JSME, 1985, 28: 899-904

[2] KimY B, NoahS T. Bifurcation analysis for a modified Jeffcott rotor with bearing clearances [J]. Nonlinear Dynamics, 1990, 1: 221-241

[3] TiwariM, GuptaK, PrakashO. Dynamic response of an unbalanced rotor supported on ball bearing [J]. Journal of Sound and Vibration, 2000, 238(5): 757-779

[4] HarshaS P, SandeepK, PrakashR. Non-linear dynamic behaviors of rolling element bearings due to surface waviness [J]. Journal of Sound and Vibration, 2004, 272: 557-580

[5] HarshaS P. Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect [J]. Mechanism and Machine Theory, 2006, 41(6): 688-706

[6] MevelB, GuyaderJ L. Experiments on routes to chaos in ball bearings [J]. Journal of Sound and Vibration, 2008, 318(3): 549-564

[7] BaiC Q, XuQ Y. Dynamic model of ball bearings with internal clearance and waviness [J]. Journal of Sound and Vibration, 2006, 294: 23-48

[8] ChavezP J, WiercigrochM. Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model [J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18: 2571-2580

[9] HouL, ChenY S, CaoQ J, ZhangZ Y. Turning maneuver caused response in an aircraft rotor-ball bearing system [J]. Nonlinear Dynamics, 2015, 79(1): 229-240

[10] ChenG, LiC G, WangD Y. Nonlinear dynamic analysis and experiment verification of rotor-ball bearings-support-stator coupling system for aeroengine with rubbing coupling faults [J]. Journal of Engineering for Gas Turbines and Power-transactions of the ASME, 2009, 132(2): 1-9

[11] ChenG. Vibration modelling and verifications for whole aero-engine [J]. Journal of Sound and Vibration, 2015, 349: 163-176

[12] ChenG. Simulation of casing vibration resulting from blade-casing rubbing and its verifications [J]. Journal of Sound and Vibration, 2016, 361: 190-209

[13] WangH F, GongJ J, ChenG. Characteristics analysis of aero-engine whole vibration response with rolling bearing radial clearance [J]. Journal of Mechanical Science and Technology, 2017, 37: 2129-2141

[14] LiuX J, LiuY, WangS G, YanH J, LiaoP T. Bifurcation analysis of a magnetically supported rigid rotor in auxiliary bearings [J]. Chaos, Solitons & Fractals, 2019, 118: 328-336

[15] HAN Y C, LI M. Nonlinear dynamic characteristics of marine rotor-bearing system under heaving motion [J]. Shock and Vibration, 2019: 1–16. DOI: https://doi.org/10.1155/2019/7683952.

[16] WangA M, YaoW, HeK, MengG Y, ChengX H, YangJ. Analytical modelling and numerical experiment for simultaneous identification of unbalance and rolling-bearing coefficients of the continuous single-disc and single-span rotor-bearing system with Rayleigh beam model [J]. Mechanical Systems and Signal Processing, 2019, 116: 322-346

[17] AlvesD S, WuM F, CavalcaK L. Application of gain-scheduled vibration control to nonlinear journal-bearing supported rotor [J]. Mechanical Systems and Signal Processing, 2019, 442: 714-737

[18] YangY, YangY R, CaoD Q, ChenG, JinY L. Response evaluation of imbalance-rub-pedestal looseness coupling fault on a geometrically nonlinear rotor system [J]. Mechanical Systems and Signal Processing, 2019, 118: 423-442

[19] DongJ, WangX J, ZhanG J, GuJ J. Experimental investigation of dynamic behaviour of rotor-bearing system with nitrile rubber support [J]. Proceedings of the Institution of Mechanical Engineers Part C. Journal of Mechanical Engineering Science, 2018, 232: 1502-1513

[20] SaeedN A, EissaM. Bifurcations of periodic motion of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft [J]. International Journal of Non-linear Mechanics, 2018, 101: 113-130

[21] PENG B, ZHU RP, LI M M, TANG Z Y. Bending vibration suppression of a flexible multispan shaft using smart spring support [J]. Shock and Vibration, 2017: 1–12. DOI: https://doi.org/10.1155/2017/7859164.

[22] MarainiD, NatarajC. Nonlinear analysis of a rotor-bearing system using describing functions [J]. Journal of Sound and Vibration, 2018, 420: 227-241

[23] HanB B, DingQ. Forced responses analysis of a rotor system with squeeze film damper during flight maneuvers using finite element method [J]. Mechanism and Machine Theory, 2018, 122: 233-251

[24] LiuJ. A comprehensive comparative investigation of frictional force models for dynamics of rotor-bearing systems [J]. Journal of Central South University, 2020, 27: 1770-1779

[25] ChoukseyM, DuttJ K, ModakS V. Modal analysis of rotor-shaft system under the influence of rotor-shaft material damping and fluid film forces [J]. Mechanism and Machine Theory, 2012, 48: 81-93

[26] MirtalaieS H, HajabasiM A. Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft [J]. Archive of Applied Mechanics, 2017, 87(9): 1465-1494

[27] LuX, ZhangJH, MaL, LinJ W, WangJ, WangJ, DaiH W. Effects of misalignment on the nonlinear dynamics of a two-shaft rotor-bearing-gear coupling system with rub-impact fault [J]. Journal of Vibroengineering, 2017, 19: 5960-5977

[28] NanG F, TangM, ChenE Y, YangA L. Nonlinear dynamic mechanism of rolling element bearings with an internal clearance in a rotor-bearing system [J]. Advances in Mechanical Engineering, 2016, 8(11): 1-9

[29] TadayoshiS. Nonlinear vibration of saturated water journal bearing and bifurcation analysis [J]. Journal of Vibration and Acoustics, 2019, 141(2): 1-10

[30] XU Y Y, CHEN Z B, LUO A C J. Period-1 motion to chaos in a nonlinear flexible rotor system [J]. International Journal of Bifurcation and Chaos, 30(5): 1–24. DOI: https://doi.org/10.1142/S0218127420500777.

[31] HaslamA H, SchwingshacklC W, RixA I J. A parametric study of an unbalanced Jeffcott rotor supported by a rolling-element bearing [J]. Nonlinear Dynamics, 2020, 99(4): 2571-2604

[32] ZhuH M, ChenW F, ZhuR P, GaoJ, LiaoM J. Study on the dynamic characteristics of a rotor bearing system with damping rings subjected to base vibration [J]. Journal of Vibration Engineering & Technologies, 2020, 8: 121-132

[33] FriswellM I, PennyJET, GarveyS D, LeesA WDynamics of rotating machines [M], 2010, London, Cambridge University Press

[34] YokoyamaT. Vibrations of a hanging Timoshenko beam under gravity [J]. Journal of Sound and Vibration, 1990, 141: 245-258

[35] YokoyamaT. Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations [J]. Computer and Structures, 1996, 61: 995-1007

[36] HarrisT ARolling bearing analysis [M], 2003, New York, John Wiley & Sons

[37] FeiZ X, TongS G, WeiC. Investigation of the dynamic characteristics of a dual rotor system and its start-up simulation based on finite element method [J]. Journal of Zhejiang University-Science A, 2013, 14(4): 268-280

[38] XuanH-J, HongW-R. Application of elastomeric O-rings damper in high-speed spin tester [J]. Journal of Zhejiang University(Engineering Science), 2005, 39(12): 1854-1857(in Chinese)

[39] FangX, ZhangD B, ZhangX Y, WuH C, GaoF, HeL, LvY. Chaos of flexible rotor system with critical speed in magnetic bearing based on the improved precise Runge-Kutta hybrid integration [J]. Advances in Mechanical Engineering, 2018, 10(9): 1-14

[40] ZhangS Y, DengZ C, LiW C. A precise Runge-Kutta integration and its application for solving nonlinear dynamical systems [J]. Applied and Computational Mathematics, 2007, 184(2): 496-502