The interrelated modelling method of the nonlinear dynamics of rigid rotors in passive and active magnetic bearings

Authors

DOI:

https://doi.org/10.15587/1729-4061.2016.65440

Keywords:

rotor dynamics, passive magnetic bearings, active magnetic bearings, magnetic energy, mathematical model, nonlinear vibrations

Abstract

A method is suggested for building mathematical models of dynamics of rotors in magnetic bearings of different types (passive and active). It is based on Lagrange-Maxwell differential equations in a form identical to that of Routh equations in mechanics. The expressions for magnetic energy and forces in active magnetic bearings with account for control laws for introducing them into the mathematical models have been found by adapting the analytical method of analysing magnetic circuits. This method is based on building equivalent circuits and using the loop flux method to account for dissipation fluxes and magnetic resistances of AMB magnetic circuit sections and ensure noncriticality of the mathematical model to emergence of “zero” gaps and currents. Besides, the mathematical models account for such nonlinearities as nonlinear dependence of magnetic forces on gaps in passive and active magnetic bearings and on currents in the coils of electromagnets, nonlinearities linked to coil inductance, a geometric link between electromagnets in one AMB and links between all AMB in one rotor, which results, among other factors, in connectedness of processes in orthogonal directions. The method’s validity has been confirmed experimentally by a laboratory setup being a prototype of a complete combined magnetic-electromagnetic suspension in small-size rotor machinery. The suggested approach has helped detect in the system and investigate different nonlinear rotor dynamics phenomena such as super- and subharmonic vibrations with determination of resonance modes.

Author Biography

Gennadii Martynenko, National Technical University «Kharkiv Polytechnic Institute» 21 Bagaliya str., Kharkiv, Ukraine, 61002

PhD, Associate professor

Dynamics and Strength of Machine Department

References

  1. Schweitzer, G., Bleuler H., Traxler A. (1994). Active magnetic bearings, ETH-Zurich, 244.
  2. Maslen, E. H. (2000). Magnetic Bearings, University of Virginia Department of Mechanical, Aerospace, and Nuclear Engineering Charlottesville, 231.
  3. Schweitzer, G.; Gupta, K. (Ed.) (2011). Applications and Research Topics for Active Magnetic Bearings. IUTAM Bookseries, 263–273. doi: 10.1007/978-94-007-0020-8_23
  4. G. Schweitzer, E. H. Maslen (Eds.) Magnetic Bearings. Theory, Design, and Application to Rotating Machinery. Berlin: Springer, 2009, 535. doi: 10.1007/978-3-642-00497-1
  5. Polajžer, B. (Ed. )(2010). Magnetic Bearings. Theory and Applications, Sciyo, Rijeka, 140.
  6. Jansen, R., DiRusso, E. (1996). Passive Magnetic Beating With Ferrofluid Stabilization. Lewes Research Center, Cleveland, 154.
  7. Earnshaw, S. (1842). On the nature of molecular forces which regulate the constitution of luminiferous ether. Transactions of Cambridge Philosophie Society, V–VII, Part I, 97–112.
  8. Braunbek, W. (1939). Freies Schweben diamagnetischer Körper im Magnetfeld. Zeitschrift fur Physik, 112 (11-12), 764–769. doi: 10.1007/bf01339980
  9. Bassani, R. (2006). Earnshaw (1805–1888) and Passive Magnetic Levitation. Meccanica, 41 (4), 375–389. doi: 10.1007/s11012-005-4503-x
  10. Simms, J. (2009). Fundamentals of the Turboexpander: Basic Theory and Design, Gas Technology Services, Santa Maria, 34.
  11. Ji, J. C., Hansen, C. H., Zander, A. C. (2008). Nonlinear Dynamics of Magnetic Bearing Systems. Journal of Intelligent Material Systems and Structures, 19 (12), 1471–1491. doi: 10.1177/1045389x08088666
  12. Ehrich, F. F. (2008). Observations of Nonlinear Phenomena in Rotordynamics. Journal of System Design and Dynamics, 2 (3), 641–651. doi: 10.1299/jsdd.2.641
  13. Peel, D. J., Bingham, C. M., Wu, Y., Howe, D. (2002). Simplified characteristics of active magnetic bearings. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 216 (5), 623–628. doi: 10.1243/0954406021525296
  14. Skricka, N., Markert, R. (2002). Improvements in the integration of active magnetic bearings. Control Engineering Practice, 10 (8), 917–922. doi: 10.1016/s0967-0661(01)00106-x
  15. Ji, J. C. (2004). Dynamics of a piecewise linear system subjected to a saturation constraint. Journal of Sound and Vibration, 271 (3-5), 905–920. doi: 10.1016/s0022-460x(03)00759-4
  16. Chinta, M., Palazzolo, A. B., Kascak, A. (1996). Quasi-periodic Vibration of a Rotor in a Magnetic Bearing with Geometric Coupling. Proceedings of the Fifth International Symposium on Magnetic Bearings. Kanazawa, Japan, 147–152.
  17. Chinta, M., Palazzolo, A. B. (1998). Stability and bifurcation of rotor motion in a magnetic bearing. Journal of Sound and Vibration, 214 (5), 793–803. doi: 10.1006/jsvi.1998.1549
  18. Ho, Y. S., Liu, H., Yu, L. (2003). Effect of Thrust Magnetic Bearing on Stability and Bifurcation of a Flexible Rotor Active Magnetic Bearing System. Journal of Vibration and Acoustics, 125 (3), 307–316. doi: 10.1115/1.1570448
  19. Zhang, W., Zhan, X. P. (2005). Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness. Nonlinear Dynamics, 41 (4), 331–359. doi: 10.1007/s11071-005-7959-2
  20. Zhang, W., Yao, M. H., Zhan, X. P. (2006). Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness. Chaos, Solitons & Fractals, 27 (1), 175–186. doi: 10.1016/j.chaos.2005.04.003
  21. Inayat-Hussain, J. I. (2007). Chaos via torus breakdown in the vibration response of a rigid rotor supported by active magnetic bearings. Chaos, Solitons & Fractals, 31 (4), 912–927. doi: 10.1016/j.chaos.2005.10.039
  22. Raus, Je.; Arhangel'skiy, Ju. A., Djomin, V. G. (Eds.) (1983). Dinamika sistemy tverdyh tel. In 2 volumes. Vol. 1. Moscow: Nauka, 464.
  23. Bekinal, S. I., Anil, T. R. R., Jana, S. (2013). Analysis of radial magnetized permanent magnet bearing characteristics. Progress In Electromagnetics Research B, 47, 87–105. doi: 10.2528/pierb12102005
  24. Martynenko, G. (2008). Modeling the Dynamics of a Rigid Rotor in Active Magnetic Bearings. Proceedings of the 6th EUROMECH Nonlinear Dynamics Conference (ENOC 2008), St. Petersburg, 1–6. Available at: http://lib.physcon.ru/doc?id=9531874f673b
  25. Martynenko, G. (2010). Method of Detuning from Resonance Modes for Rotors in Active Magnetic Bearings with Nonlinear Force Characteristics. Proceedings of the Third International Nonlinear Dynamics Conference. Kharkiv, 135–140.
  26. Rogovyj, Je. D., Buholdin, Ju. S., Levashov, V. O., Martynenko, G. Ju., Smirnov, M. M. (2007). Patent. 77665. Ukrai'na. MPK F16C 32/04. Sposib dyskretnogo keruvannja elektromagnitnym pidvisom obertovyh rotoriv. Applicant and the patentee VAT «Sum. nauk.-vyrob. ob-nja im. M.V. Frunze», Nac. tehn. un-t «Hark. politehn. in-t»; №2003076309; applicated: 08.07.03; published: 15.01.07, Bjul. №1/2007, 6.

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Published

2016-04-25

How to Cite

Martynenko, G. (2016). The interrelated modelling method of the nonlinear dynamics of rigid rotors in passive and active magnetic bearings. Eastern-European Journal of Enterprise Technologies, 2(5(80), 4–13. https://doi.org/10.15587/1729-4061.2016.65440

Issue

Vol. 2 No. 5(80) (2016): Applied physics. Materials Science

Section

Applied physics

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Copyright (c) 2016 Gennadii Martynenko

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