A new nonlinear force model to replace the Hertzian contact model in a rigid-rotor ball bearing system

doi:10.1007/s10483-018-2308-9

Abstract

Abstract: A new nonlinear force model based on experimental data is proposed to replace the classical Hertzian contact model to solve the fractional index nonlinearity in a ball bearing system. Firstly, the radial force and the radial deformation are measured by statics experiments, and the data are fitted respectively by using the Hertzian contact model and the cubic polynomial model. Then, the two models are compared with the approximation formula appearing in Aeroengine Design Manual. In consequence, the two models are equivalent in an allowable deformation range. After that, the relationship of contact force and contact deformation for single rolling element between the races is calculated based on statics equilibrium to obtain the two kinds of nonlinear dynamic models in a rigid-rotor ball bearing system. Finally, the displacement response and frequency spectrum for the two system models are compared quantitatively at different rotational speeds, and then the structures of frequency-amplitude curves over a wide speed range are compared qualitatively under different levels of radial clearance, amplitude of excitation, and mass of supporting rotor. The results demonstrate that the cubic polynomial model can take place of the Hertzian contact model in a range of deformation.

Key words: cubic polynomial, thermohaline double-diffusive system, periodic solution, stability, fractional index, Hertzian contact, rotor ball bearing system, rolling element bearing

2010 MSC Number:

74H45

Yulin JIN, Zhenyong LU, Rui YANG, Lei HOU, Yushu CHEN. A new nonlinear force model to replace the Hertzian contact model in a rigid-rotor ball bearing system. Applied Mathematics and Mechanics (English Edition), 2018, 39(3): 365-378.

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References

[1] Harris, T. A. and Kotzalas, M. N. Advanced Concepts of Bearing Technology:Rolling Bearing Analysis, 5th ed., Taylor & Francis, London (2006)
[2] Bai, C. Q., Xu, Q. Y., and Zhang, X. L. Nonlinear stability of balanced rotor due to effect of ball bearing internal clearance. Applied Mathematics and Mechanics (English Edition), 27(2), 175-186(2006) https://doi.org/10.1007/s10483-006-0205-1
[3] Hou, L., Chen, Y. S., Fu, Y. Q., Chen, H. Z., Lu, Z. Y., and Liu, Z. S. Application of the HBAFT method to the primary resonance analysis of a dual-rotor system. Nonlinear Dynamics, 88, 2531-2551(2017)
[4] Hou, L., Chen, Y. S., Cao, Q. J., and Lu, Z. Y. Nonlinear vibration analysis of a cracked rotor-ball bearing system during flight maneuvers. Mechanism and Machine Theory, 105, 515-528(2016)
[5] Zhang, Z. Y. Bifurcations and Hysteresis of Varying Compliance Vibrations of a Ball BearingRotor System, Ph. D. dissertation, Harbin Institute of Technology (2015)
[6] Zhang, Z. Y. and Chen, Y. S. Harmonic balance method with alternating frequency/time domain technique for nonlinear dynamical system with fractional exponential. Applied Mathematics and Mechanics (English Edition), 35(4), 423-436(2014) https://doi.org/10.1007/s10483-014-1802-9
[7] Zhang, Z. Y., Chen, Y. S., and Li, Z. G. Influencing factors of the dynamic hysteresis in varying compliance vibrations of a ball bearing. Science China Technological Sciences, 58, 775-782(2015)
[8] Zhang, Z. Y., Chen, Y. S., and Cao, Q. J. Bifurcations and hysteresis of varying compliance vibrations in the primary parametric resonance for a ball bearing. Journal of Sound and Vibration, 350, 171-184(2015)
[9] Harris, T. A. and Mindel, M. H. Rolling element bearing dynamics.Wear, 23, 311-337(1973)
[10] Sunnersjö, C. S. Varying compliance vibrations of rolling bearing. Journal of Sound and Vibration, 58, 363-373(1978)
[11] Fukata, S., Gad, E. H., Kondou, T., Ayabe, T., and Tamura, H. On the radial vibrations of ballbearing-computer-simulation. Bulletin of JSME, 28, 899-904(1985)
[12] Mevel, B. and Guyader, J. L. Routes to chaos in ball bearings. Journal of Sound and Vibration, 162, 471-487(1993)
[13] Mevel, B. and Guyader, J. L. Experiments on routes to chaos in ball bearings. Journal of Sound and Vibration, 318, 549-564(2008)
[14] Tiwari, T. and Gupta, K. Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. Journal of Sound and Vibration, 238, 723-756(2000)
[15] Tiwari, M., Gupta, K., and Prakash, O. Dynamic response of an unbalanced rotor supported on ball bearings. Journal of Sound and Vibration, 238, 757-779(2000)
[16] Harsha, S. P. Non-linear dynamic response of a balanced rotor supported on rolling element bearings. Mechanical Systems and Signal Processing, 19, 551-578(2005)
[17] Harsha, S. P. Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect. Mechanism and Machine Theory, 41, 688-706(2006)
[18] Harsha, S. P. Nonlinear dynamic analysis of an unbalanced rotor supported by roller bearing. Chaos Solitons and Fractals, 26, 47-66(2005)
[19] Aktürk, N., Uneeb, M., and Gohar, R. The effects of number of balls and preload on vibrations associated with ball bearings. ASME Journal of Tribology, 119, 747-753(1997)
[20] Liew, A., Feng, N., and Hahn, E. Transient rotordynamic modeling of rolling element bearing systems. ASME Journal of Engineering for Gas Turbines and Power, 124, 984-991(2002)
[21] Jang, G. H. and Jeong, S. W. Nonlinear excitation model of ball bearing waviness in a rigid rotor supported by two or more ball bearings considering five degrees of freedom. ASME Journal of Tribology, 124, 82-90(2002)
[22] Jang, G. H. and Jeong, S. W. Analysis of a ball bearing with waviness considering the centrifugal force and gyroscopic moment of the ball. ASME Journal of Tribology, 125, 487-498(2003)
[23] Ghafari, S. H., Abdel-Rahman, E. M., Golnaraghi, F., and Ismail, F. Vibrations of balanced fault-free ball bearings. Journal of Sound and Vibration, 329, 1332-1347(2010)
[24] Bai, C. Q., Zhang, H. Y., and Xu, Q. Y. Subharmonic resonance of a symmetric ball bearing-rotor system. International Journal of Non-Linear Mechanics, 50, 1-10(2013)
[25] Zhang, X., Han, Q. K., Peng, Z. K., and Chu, F. L. Stability analysis of a rotor-bearing system with time-varying bearing stiffness due to finite number of balls and unbalanced force. Journal of Sound and Vibration, 332, 6768-6784(2013)
[26] Zhang, X., Han, Q. K., Peng, Z. K., and Chu, F. L. A comprehensive dynamic model to investigate the stability problems of the rotor-bearing system due to multiple excitations. Mechanical Systems and Signal Processing, 70-71, 1171-1192(2016)
[27] Gunduz, A. and Singh, R. Stiffness matrix formulation for double row angular contact ball bearings:analytical development and validation. Journal of Sound and Vibration, 332, 5898-5916(2013)
[28] Nonato, F. and Cavalca, K. L. An approach for including the stiffness and damping of elastohydrodynamic point contacts in deep groove ball bearing equilibrium models. Journal of Sound and Vibration, 333, 6960-6978(2014)
[29] Xu, L. X. and Li, Y. G. An approach for calculating the dynamic load of deep groove ball bearing joints in planar multibody systems. Nonlinear Dynamics, 70, 2145-2161(2012)
[30] Xu, L. X. A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint. Nonlinear Dynamics, 78, 857-879(2014)
[31] Razpotnik, M., Bischof, T., and Boltežar, M. The influence of bearing stiffness on the vibration properties of statically overdetermined gearboxes. Journal of Sound and Vibration, 351, 221-235(2015)
[32] Hou, L., Chen, Y. S., Cao, Q. J., and Zhang, Z. Y. Turning maneuver caused response in an aircraft rotor-ball bearing system. Nonlinear Dynamics, 79, 229-240(2015)
[33] Zhao, C. J., Yu, X. K., Huang, Q. X., Ge, S. D., and Gao, X. Analysis on the load characteristics and coefficient of friction of angular contact ball bearing at high speed. Tribology International, 87, 50-56(2015)
[34] Jin, Y. L., Yang, R., Hou, L., Chen, Y. S., and Zhang, Z. Y. Experiments and numerical results for varying compliance vibrations in a rigid-rotor ball bearing system. ASME Journal of Tribology, 139, 041103(2017)
[35] Nayak, R. Contact vibrations. Journal of Sound and Vibration, 22, 297-322(1972)
[36] Hess, D. and Soom, A. Normal vibrations and friction under harmonic loads, part 1:Hertzian contact. ASME Journal of Tribology, 113, 80-86(1991)
[37] Rigaud, E. and Perret-Liaudet, J. Experiments and numerical results on non-linear vibrations of an impacting hertzian contact, part 1:harmonic excitation. Journal of Sound and Vibration, 265, 289-307(2003)
[38] Perret-Liaudet, J. and Rigaud, E. Response of an impacting Hertzian contact to an order-2 subharmonic excitation:theory and experiments. Journal of Sound and Vibration, 296, 319-333(2006)
[39] Fu, C. G. Aeroengine Design Manual, Rotor Dynamics and Engine Body Vibration, Vol. 19, Aviation Industry Press, Beijing (1999)
[40] Lazovic, T., Ristivojevic, M., and Mitrovic, R. Mathematical model of load distribution in rolling bearing. FME Transactions, 36, 189-196(2008)
[41] Marconnet, P., Pottier, B., Rasolofondraibe, L., Dron, J. P., and Kerroumi, S. Measuring load distribution on the outer raceways of rotating machines. Mechanical Systems and Signal Processing, 66-67, 582-596(2016)