Nonlinear Dynamics

, Volume 96, Issue 4, pp 2581–2600 | Cite as

Asymmetric effect of static radial eccentricity on the vibration characteristics of the rotor system of permanent magnet synchronous motors in electric vehicles

  • Feng Liu
  • Changle XiangEmail author
  • Hui LiuEmail author
  • Lijin Han
  • Yunhao Wu
  • Xiaojie Wang
  • Pu Gao
Original Paper


Considering static radial eccentricity, a Jeffcott rotor model is established for the rotor system of the permanent magnet synchronous motors in electric vehicles. The system conservative force, including unbalanced magnetic pull, which results in nonlinearity is analyzed, and center manifold theorem and Lyapunov method are used to determine the stabilities of multiple equilibrium points. This analysis shows that static eccentricity spoils the symmetry of the equilibrium points, although they are distributed in the line along the direction of the static eccentricity. This asymmetry leads to the pitchfork bifurcation of equilibrium points to a generic bifurcation with a defect. This analysis provides two stability conditions for the rotor system. Furthermore, the effect of the asymmetry on the dynamic characteristics that can induce backward whirling motion coupled with forward whirling motion is quite different from the case without static eccentricity. These characteristics are investigated by multi-scale method. As a result, the analytical solution of the system at steady state is obtained. The frequency characteristics of the main resonance are analyzed, and the stability of the solution is determined using Routh–Hurwitz criterion and the geometric constraint of the rotor whirling motion. The characteristics reveal that a globally unstable frequency band appears due to the geometric constraint. However, this frequency band narrows and even vanishes with increases in damping and electromagnetic stiffness and decreases in mass imbalance, mechanical stiffness and static eccentricity. The analysis by multi-scale method is based on the assumption of the time invariance of the forward and backward whirling amplitudes, which is validated by the numerical method. The results of the two methods agree well, which indicates that this assumption and the analysis are reasonable.


Static eccentricity Unbalanced magnetic pull Static characteristics Frequency characteristics Stability 



This work was supported by the National Natural Science Foundations of China (Grant No. U1564210, Grant No. 51775040).

Compliance with ethical standards

Conflict of interest

We declared that we have no conflicts of interest to this work.


  1. 1.
    Pennacchi, P., Frosini, L.: Dynamical behaviour of a three-phase generator due to unbalanced magnetic pull. IEE Proc.-Electr. Power Appl. 152(6), 1389–1400 (2005)CrossRefGoogle Scholar
  2. 2.
    Smith, A.C., Dorrell, D.G.: Calculation and measurement of unbalanced magnetic pull in cage induction motors with eccentric rotors. Part 1: analytical model. IEE Proc.-Electr. Power Appl. 143(3), 193–201 (1996)CrossRefGoogle Scholar
  3. 3.
    Burakov, A., Arkkio, A.: Comparison of the unbalanced magnetic pull mitigation by the parallel paths in the stator and rotor windings. IEEE Trans. Magn. 43(12), 4083–4088 (2007)CrossRefGoogle Scholar
  4. 4.
    Dorrell, D.G.: Sources and characteristics of unbalanced magnetic pull in three-phase cage induction motors with axial-varying rotor eccentricity. IEEE Trans. Ind. Appl. 47(1), 12–24 (2011)CrossRefGoogle Scholar
  5. 5.
    Donát, M.: Computational modelling of the unbalanced magnetic pull by finite element method. Procedia Eng. 48, 83–89 (2012)CrossRefGoogle Scholar
  6. 6.
    Yang, H.D., Chen, Y.S.: Influence of radial force harmonics with low mode number on electromagnetic vibration of PMSM. IEEE Trans. Energy Convers. 29(1), 38–45 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Žarko, D., Ban, D., Vazdar, I., Jarić, V.: Calculation of unbalanced magnetic pull in a salient-pole synchronous generator using finite-element method and measured shaft orbit. IEEE Trans. Ind. Electron. 59(6), 2536–2549 (2012)CrossRefGoogle Scholar
  8. 8.
    Wang, L., Cheung, R.W., Ma, Z.Y., Ruan, J.J., Peng, Y.: Finite-element analysis of unbalanced magnetic pull in a large hydro-generator under practical operations. IEEE Trans. Magn. 44(6), 1558–1561 (2008)CrossRefGoogle Scholar
  9. 9.
    Han, B., Zheng, S., Liu, X.: Unbalanced magnetic pull effect on stiffness models of active magnetic bearing due to rotor eccentricity in brushless dc motor using finite element method. Math. Probl. Eng. 2013(1), 147–170 (2013)Google Scholar
  10. 10.
    Kawase, Y., Mimura, N., Ida, K.: 3-D electromagnetic force analysis of effects of off-center of rotor in interior permanent magnet synchronous motor. IEEE Trans. Magn. 36(4), 1858–1862 (2000)CrossRefGoogle Scholar
  11. 11.
    Kim, U., Lieu, D.K.: Effects of magnetically induced vibration force in brushless permanent-magnet motors. IEEE Trans. Magn. 41(6), 2164–2172 (2005)CrossRefGoogle Scholar
  12. 12.
    Lee, S.K., Kang, G.H., Jin, H.: Analysis of radial forces in 100 kW IPM machines for ship considering stator and rotor eccentricity. In: IEEE International Conference on Power Electronics & Ecce Asia, pp. 2457–2461 (2011)Google Scholar
  13. 13.
    Dorrell, D.G.: Calculation of unbalanced magnetic pull in small cage induction motors with skewed rotors and dynamic rotor eccentricity. IEEE Trans. Energy Convers. 11(3), 483–488 (1996)CrossRefGoogle Scholar
  14. 14.
    Guo, D., Chu, F., Chen, D.: The unbalanced magnetic pull and its effects on vibration in a three-phase generator with eccentric rotor. J. Sound Vib. 254, 297–312 (2002)CrossRefGoogle Scholar
  15. 15.
    Perers, R., Lundin, U., Leijon, M.: Saturation effects on unbalanced magnetic pull in a hydroelectric generator with an eccentric rotor. IEEE Trans. Magn. 43(10), 3884–3890 (2007)CrossRefGoogle Scholar
  16. 16.
    Calleecharan, Y., Aidanpää, J.-O.: Stability analysis of a hydropower generator subjected to unbalanced magnetic pull. IET Sci. Meas. Technol. 5, 231–243 (2011)CrossRefGoogle Scholar
  17. 17.
    Ebrahimi, B.M., Javan Roshtkhari, M., Faiz, J., Khatami, S.V.: Advanced eccentricity fault recognition in permanent magnet synchronous motors using stator current signature analysis. IEEE Trans. Ind. Electron. 61(4), 2041–2052 (2014)CrossRefGoogle Scholar
  18. 18.
    Ghoggal, A., Zouzou, S.E., Razik, H., Sahraoui, M., Khezzar, A.: An improved model of induction motors for diagnosis purposes-slot skewing effect and air-gap eccentricity faults, Energy Convers. Manage 50(5), 1336–1347 (2009)Google Scholar
  19. 19.
    Ebrahimi, B.M., Faiz, J.: Configuration impacts on eccentricity fault detection in permanent magnet synchronous motors. IEEE Trans. Magn. 48(2), 903–906 (2012)CrossRefGoogle Scholar
  20. 20.
    Ebrahimi, B.M., Roshtkhari, M.J., Faiz, J., Khatami, S.V.: Advanced eccentricity fault recognition in permanent magnet synchronous motors using stator current signature analysis. IEEE Trans. Ind. Electron. 61(4), 2041–2052 (2014)CrossRefGoogle Scholar
  21. 21.
    Roux, W.L., Harley, R.G., Habetler, T.G.: Detecting rotor faults in low power permanent magnet synchronous machines. IEEE Trans. Power Electron. 22(1), 322–328 (2007)CrossRefGoogle Scholar
  22. 22.
    Ebrahimi, B.M., Faiz, J., Roshtkhari, M.J., Nejhad, A.Z.: Static eccentricity fault diagnosis in permanent magnet synchronous motor using time stepping finite element method. IEEE Trans. Magn. 44(11), 4297–4300 (2008)CrossRefGoogle Scholar
  23. 23.
    Ebrahimi, B.M., Faiz, J., Roshtkhari, M.J.: Static-, dynamic-, and mixed-eccentricity fault diagnoses in permanent-magnet synchronous motors. IEEE Trans. Ind. Electron. 56(11), 4727–4739 (2009)CrossRefGoogle Scholar
  24. 24.
    Ebrahimi, B.M., Faiz, J.: Diagnosis and performance analysis of three-phase permanent magnet synchronous motors with static, dynamic and mixed eccentricity. IET Electr. Power Appl. 4(1), 53–66 (2010)CrossRefGoogle Scholar
  25. 25.
    Ebrahimi, B.M., Faiz, J.: Magnetic field and vibration monitoring in permanent magnet synchronous motors under eccentricity fault. IET Electr. Power Appl. 6(1), 35–45 (2012)CrossRefGoogle Scholar
  26. 26.
    Akar, M., Taşkin, S., Şeker, S., Çankaya, İ.: Detection of static eccentricity for permanent magnet synchronous motors using the coherence analysis. Turk. J. Electr. Eng. Comput. Sci. 18(6), 963–974 (2010)Google Scholar
  27. 27.
    Akar, M., Hekim, M., Orban, U.: Mechanical fault detection in permanent magnet synchronous motors using equal width discretization-based probability distribution and a neural network model. Turk. J. Electr. Eng. Comput. Sci. 23, 813–823 (2015)CrossRefGoogle Scholar
  28. 28.
    Mirimani, S.M., Vahedi, A., Marignetti, F., Santis, E.D.: Static eccentricity fault detection in single-stator-single-rotor axial-flux permanent-magnet machines. IEEE Trans. Ind. Appl. 48(6), 1838–1845 (2012)CrossRefGoogle Scholar
  29. 29.
    Mirimani, S.M., Vahedi, A., Marignetti, F., Stefano, R.D.: An online method for static eccentricity fault detection in axial flux machines. IEEE Trans. Ind. Electron. 62(3), 1931–1942 (2015)CrossRefGoogle Scholar
  30. 30.
    Hong, J., Lee, S.B., Kral, C., Haumer, A.: Detection of airgap eccentricity for permanent magnet synchronous motors based on the d-axis inductance. IEEE Trans. Power Electron. 27(5), 2605–2612 (2012)CrossRefGoogle Scholar
  31. 31.
    Da, Y., Shi, X.D., Krishnamurthy, M.: A new approach to fault diagnostics for permanent magnet synchronous machines using electromagnetic signature analysis. IEEE Trans. Power Electron. 28(8), 4104–4112 (2013)CrossRefGoogle Scholar
  32. 32.
    Karami, M., Mariun, N., Mehrjou, M.R., Ab Kadir, M.Z.A., Misron, N., Radzi, M.A.M.: Static eccentricity fault recognition in three-phase line start permanent magnet synchronous motor using finite element method. Math. Probl. Eng. 2014, 1–12 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Huang, Y.K., Guo, B.C., Hemeida, A., Sergeant, P.: Analytical modeling of static eccentricities in axial flux permanent-magnet machines with concentrated windings. Energies 9(11), 1–19 (2016)Google Scholar
  34. 34.
    Goktas, T., Zafarani, M., Akin, B.: Discernment of broken magnet and static eccentricity faults in permanent magnet synchronous motors. IEEE Trans. Energy Convers. 31(2), 578–587 (2016)CrossRefGoogle Scholar
  35. 35.
    Naderi, P., Fallahi, F.: Eccentricity fault diagnosis in three-phase-wound-rotor induction machine using numerical discrete modeling method. Int. J. Numer. Model.-Electron. Netw. Devices Fields 29(5), 982–997 (2016)CrossRefGoogle Scholar
  36. 36.
    Naderi, P.: Torque/current spectral analysis for healthy and eccentricity faulty synchronous reluctance machine using mathematical modeling method. Int. Trans. Electr. Energy Syst. 26(8), 1625–1645 (2016)CrossRefGoogle Scholar
  37. 37.
    Oumaamar, M.E., Maouche, Y., Boucherma, M., Khezzar, A.: Static air-gap eccentricity fault diagnosis using rotor slot harmonics in line neutral voltage of three-phase squirrel cage induction motor. Mech. Syst. Signal Process. 84, 584–597 (2017)CrossRefGoogle Scholar
  38. 38.
    Kim, T.J., Hwang, S.M., Park, N.G.: Analysis of vibration for permanent magnet motors considering mechanical and magnetic coupling effects. IEEE Trans. Magn. 36(4), 1346–1350 (2000)CrossRefGoogle Scholar
  39. 39.
    He, G.H., Huang, Z.Y., Qin, R., Chen, D.Y.: Numerical prediction of electromagnetic vibration and noise of permanent-magnet direct current commutator motors with rotor eccentricities and glue effects. IEEE Trans. Magn. 48(5), 1924–1931 (2012)CrossRefGoogle Scholar
  40. 40.
    Shin, H.J., Choi, J.Y., Park, H.I., Jang, S.M.: Vibration analysis and measurements through prediction of electromagnetic vibration sources of permanent magnet synchronous motor based on analytical magnetic field calculations. IEEE Trans. Magn. 48(11), 4216–4219 (2012)CrossRefGoogle Scholar
  41. 41.
    Xu, Y., Li, Z.H.: Computational model for investigating the influence of unbalanced magnetic pull on the radial vibration of large hydro-turbine generators. J. Vib. Acoust.-Trans. ASME 134(5), 1–9 (2012)CrossRefGoogle Scholar
  42. 42.
    Wu, B.S., Sun, W.P., Li, Z.G., Li, Z.H.: Circular whirling and stability due to unbalanced magnetic pull and eccentric force. J. Sound Vib. 330(21), 4949–4954 (2011)CrossRefGoogle Scholar
  43. 43.
    Lundström, N.L.P., Aidanpää, J.O.: Dynamic consequences of electromagnetic pull due to deviations in generator shape. J. Sound Vib. 301(1–2), 207–225 (2007)CrossRefGoogle Scholar
  44. 44.
    Xiang, C.L., Liu, F., Liu, H., Han, L.J., Zhang, X.: Nonlinear dynamic behaviors of permanent magnet synchronous motors in electric vehicles caused by unbalanced magnetic pull. J. Sound Vib. 371, 277–294 (2016)CrossRefGoogle Scholar
  45. 45.
    Xu, X.P., Han, Q.K., Chu, F.L.: Nonlinear vibration of a generator rotor with unbalanced magnetic pull considering both dynamic and static eccentricities. Arch. App. Mech. 86(8), 1521–1536 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Mechanical EngineeringAcademy of Armored Force EngineeringBeijingChina

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