Skip to main content
SpringerLink
Log in

Nonlinear normal modes and primary resonance for permanent magnet synchronous motors with a nonlinear restoring force and an unbalanced magnetic pull

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The unbalanced magnetic pull (UMP) in permanent magnet synchronous motors (PMSMs) leads to the generation of vibrations and noise because of mechanical and magnetic coupling effects. This paper investigates the nonlinear oscillations of a PMSM based on a Jeffcott rotor-bearing system, and the effects of the UMP, nonlinear restoring forces due to the Hertz contact force and bearing clearance, rotor weight and rotor mass eccentricity are considered. A second-order approximate solution in the primary resonance case is obtained by applying the multiple-scale perturbation method. The Routh–Hurwitz criterion is utilized to investigate the stability of the obtained solution. The linear natural frequencies of the horizontal and vertical modes show little difference based on the constructed model. The localized oscillations and nonlocalized oscillations of the rotor-bearing system are analyzed through its frequency response curves by introducing different values of system parameters. The response curves exhibit hard spring characteristics with unstable regions and jump phenomena. Detailed numerical research including Poincare map and bifurcation diagram reveals the effect of excitation amplitude on the system. Additionally, this work provides some theoretical and practical guidance for the design of PMSMs with strong dynamic behaviors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Abbreviations

\(F_{\mathrm{r}x}, F_{\mathrm{r}y}\) :

Restoring forces in the horizontal and vertical directions, respectively

\(F_{\mathrm{u}x}, F_{\mathrm{u}y}\) :

Unbalanced magnetic pull in the horizontal and vertical directions, respectively

\(X, \dot{X}, \ddot{X}\) :

Displacement, velocity and acceleration, respectively, of the horizontal oscillation mode

\(Y, \dot{Y}, \ddot{Y}\) :

Displacement, velocity and acceleration, respectively, of the vertical oscillation mode

\(u, \dot{u}, \ddot{u}\) :

Dimensionless displacement, velocity and acceleration, respectively, of the horizontal oscillation mode

\(v, \dot{v}, \ddot{v}\) :

Dimensionless displacement, velocity and acceleration, respectively, of the vertical oscillation mode

\({\omega }_{1}, {\omega }_{2}\) :

Dimensionless linear natural frequencies of the horizontal and vertical oscillation modes, respectively

a, b :

Dimensionless rotation amplitudes of the rotor geometric center in the horizontal and vertical directions, respectively

\(\xi \) :

Dimensionless linear damping coefficient in the horizontal and vertical directions

\(\lambda \) :

Dimensionless cubic and quadratic nonlinearity stiffness coefficient of the PMSM rotor-bearing system

\({\varOmega } \) :

Dimensionless PMSM rotor spinning speed

\(\beta \) :

Dimensionless PMSM rotor eccentricity magnitude

\(\alpha \) :

Dimensionless electromagnetic stiffness coefficient of the PMSM rotor-bearing system

References

  1. Mehrjou, M.R., Mariun, N., Misron, N., Radzi, M.A.M., Musa, S.: Broken rotor bar detection in LS-PMSM based on startup current analysis using wavelet entropy features. Appl. Sci. 7(8), 845 (2017)

    Article  Google Scholar 

  2. Yang, H., Chen, Y.: Influence of radial force harmonics with low mode number on electromagnetic vibration of PMSM. IEEE Trans. Energy Convers. 29(1), 38–45 (2014)

    Article  MathSciNet  Google Scholar 

  3. Liu, Z.J., Li, J.T.: Accurate prediction of magnetic field and magnetic forces in permanent magnet motors using an analytical solution. IEEE Trans. Energy Convers. 23(3), 717–726 (2008)

    Article  MathSciNet  Google Scholar 

  4. Kuo, C., Liu, C., Chang, H., Lin, K.: Implementation of a motor diagnosis system for rotor failure using genetic algorithm and fuzzy classification. Appl. Sci. 7(1), 31 (2017)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. Gustavsson, R.K., Aidanpää, J.O.: The influence of nonlinear magnetic pull on hydropower generator rotors. J. Sound Vib. 297(3), 551–562 (2006)

    Article  Google Scholar 

  7. Lundström, N.L.P., Aidanpä, J.O.: Dynamic consequences of electromagnetic pull due to deviations in generator shape. J. Sound Vib. 301(1–2), 207–225 (2007)

    Article  Google Scholar 

  8. Wu, B.S., Sun, W.P., Li, Z.G.: Circular whirling and stability due to unbalanced magnetic pull and eccentric force. J. Sound Vib. 330(21), 4949–4954 (2011)

    Article  Google Scholar 

  9. Calleecharan, Y., Aidanpää, J.O.: Stability analysis of a hydropower generator subjected to unbalanced magnetic pull. IET Sci. Meas. Technol. 5, 231–243 (2011)

    Article  Google Scholar 

  10. He, G.H., Huang, Z.Y., Qin, R.: 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)

    Article  Google Scholar 

  11. Shin, H.J., Choi, J.Y., Park, H.I.: 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)

    Article  Google Scholar 

  12. Xu, Y., Li, Z.: Computational model for investigating the influence of unbalanced magnetic pull on the radial vibration of large hydro-turbine generators. J. Vib. Acoust. 134(5), 1–9 (2012)

    Article  Google Scholar 

  13. Werner, U.: Rotor-dynamic model for electromagnetic excitation caused by an eccentric and angular rotor core in an induction motor. Arch. Appl. Mech. 83(8), 1215–1238 (2013)

    Article  Google Scholar 

  14. Werner, U.: Mathematical rotor-dynamic model for lateral vibration analysis of induction motors with dynamic eccentricities regarding start-up. Forsch. im Ing. 78(1–2), 27–43 (2014)

    Article  Google Scholar 

  15. Zhang, L., Ma, Z.: Dynamic characteristics of a rub-impact rotor-bearing system for hydraulic generating set under unbalanced magnetic pull. Arch. Appl. Mech. 83(6), 817–830 (2013)

    Article  MATH  Google Scholar 

  16. Zhang, L., Ma, Z., Wu, Q.: Vibration analysis of coupled bending-torsional rotor-bearing system for hydraulic generating set with rub-impact under electromagnetic excitation. Arch. Appl. Mech. 86(9), 1665–1679 (2016)

    Article  Google Scholar 

  17. Xu, X., Han, Q., Chu, F.: A four degrees-of-freedom model for a misaligned electrical rotor. J. Sound Vib. 358, 356–374 (2015)

    Article  Google Scholar 

  18. Xu, X., Han, Q., Chu, F.: Nonlinear vibration of a generator rotor with unbalanced magnetic pull considering both dynamic and static eccentricities. Arch. Appl. Mech. 86(8), 1521–1536 (2016)

    Article  Google Scholar 

  19. Xiang, C.L., Liu, F., Liu, H.: Nonlinear dynamic behaviors of permanent magnet synchronous motors in electric vehicles caused by unbalanced magnetic pull. J. Sound Vib. 371, 277–294 (2016)

    Article  Google Scholar 

  20. Xu, H., Palazzolo, A.: Unstable force analysis for induction motor eccentricity. J. Sound Vib. 370, 230–258 (2016)

    Article  Google Scholar 

  21. Liu, F., Xiang, C., Liu, H.: Nonlinear vibration of permanent magnet synchronous motors in electric vehicles influenced by static angle eccentricity. Nonlinear Dyn. 90(3), 1851–1872 (2017)

    Article  Google Scholar 

  22. Xia, Y., Wang, S., Sun, W.: Analytical estimation on divergence and flutter vibrations of symmetrical three-phase induction stator via field-synchronous coordinates. J. Sound Vib. 386, 407–420 (2017)

    Article  Google Scholar 

  23. Chen, X., Yuan, S., Peng, Z.: Nonlinear vibration for PMSM used in HEV considering mechanical and magnetic coupling effects. Nonlinear Dyn. 80(1–2), 541–552 (2015)

    Article  Google Scholar 

  24. Yabuno, H., Kashimura, T., Inoue, T., Ishida, Y.: Nonlinear normal modes and primary resonance of horizontally supported Jeffcott rotor. Nonlinear Dyn. 66(3), 377–387 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Saeed, N.A., El-Gohary, H.A.: On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force. Nonlinear Dyn. 88(1), 293–314 (2017)

    Article  Google Scholar 

  26. Ishida, Y., Yamamoto, T.: Linear and Nonlinear Rotor-dynamics. Wiley, New York (2012)

    Book  Google Scholar 

  27. Yabuno, H., Kunitho, Y., Inoue, T.: Nonlinear analysis of rotor dynamics by using the method of multiple scales. In: Iutam Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty. Springer, Netherlands (2007)

  28. 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(2), 297–312 (2002)

    Article  Google Scholar 

  29. Nayfeh, A.H., Mook, D.: Nonlinear Oscillations. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  30. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  31. Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40(1), 61–102 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, A., Bai, Y., Yang, B.: Analysis of nonlinear vibration in permanent magnet synchronous motors under unbalanced magnetic pull. Appl. Sci. Basel 8(1), 113 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 51775040.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Hui Liu, Yunhao Wu, Xiaojie Wang, Pengfei Yan & Xun Zhang

Authors
  1. Hui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunhao Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaojie Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pengfei Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Liu.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflicts of interest concerning the publication of this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

$$\begin{aligned} R_{11}= & {} -\frac{1}{2}\xi -\frac{R_3}{2\omega _1} r_{20}^2 \sin (4\varphi _{20} -4\varphi _{10} ) \\ R_{12}= & {} \frac{2R_3}{\omega _1} r_{10} r_{20}^2 \cos (4\varphi _{20} -4\varphi _{10} )-\frac{1}{2}\beta \omega _1 \cos 2\varphi _{10} \\ R_{13}= & {} -\frac{R_3}{\omega _1} r_{10} r_{20} \sin (4\varphi _{20} -4\varphi _{10} ) \\ R_{14}= & {} -\frac{2R_3}{\omega _1} r_{10} r_{20}^2 \cos (4\varphi _{20} -4\varphi _{10} ) \\ R_{21}= & {} \frac{R_1}{2\omega _1} r_{10} +\frac{1}{8r_{10}^2} \beta \omega _1 \cos 2\varphi _{10} \\ R_{22}= & {} \frac{R_3}{\omega _1} r_{20}^2 \sin (4\varphi _{20} -4\varphi _{10} )+\frac{1}{4r_{10}} \beta \omega _1 \sin 2\varphi _{10} \\ R_{23}= & {} \frac{R_2}{2\omega _1} r_{20} +\frac{R_3}{2\omega _1 }r_{20} \cos (4\varphi _{20} -4\varphi _{10} ) \\ R_{24}= & {} -\frac{R_3}{\omega _1} r_{20}^2 \sin (4\varphi _{20} -4\varphi _{10} ) \\ R_{31}= & {} -\frac{R_3}{\omega _1} r_{10} r_{20} \sin (4\varphi _{10} -4\varphi _{20} ) \\ R_{32}= & {} -\frac{2R_3}{\omega _1} r_{10}^2 r_{20} \cos (4\varphi _{20} -4\varphi _{10} ) \\ R_{33}= & {} -\frac{1}{2}\xi -\frac{R_3}{2\omega _1} r_{10}^2 \sin (4\varphi _{20} -4\varphi _{10} ) \\ R_{34}= & {} \frac{2R_3}{\omega _1} r_{10}^2 r_{20} \cos (4\varphi _{10} -4\varphi _{20} )+\frac{1}{2}\beta \omega _1 \sin 2\varphi _{20} \\ R_{41}= & {} \frac{R_2}{2\omega _1} r_{10} +\frac{R_3}{2\omega _1 }r_{10} \cos (4\varphi _{10} -4\varphi _{20} ) \\ R_{42}= & {} -\frac{R_3}{\omega _1} r_{10}^2 \sin (4\varphi _{10} -4\varphi _{20} ) \\ R_{43}= & {} \frac{R_4}{2\omega _1} r_{20} -\frac{1}{8r_{20}^2} \beta \omega _1 \sin 2\varphi _{20} \\ R_{44}= & {} \frac{R_3}{\omega _1} r_{10}^2 \sin (4\varphi _{10} -4\varphi _{20} )+\frac{1}{4r_{20}} \beta \omega _1 \cos 2\varphi _{20} \end{aligned}$$

Appendix B

$$\begin{aligned} \rho _1= & {} -R_{44} -R_{33} -R_{22} -R_{11} \\ \rho _2= & {} R_{11} R_{44} +R_{11} R_{33} +R_{11} R_{22} +R_{22} R_{44} +R_{22} R_{33} \\&+R_{33} R_{44} -R_{21} R_{12} -R_{31} R_{13} -R_{41} R_{14} \\&-R_{34} R_{43} -R_{32} R_{23} -R_{42} R_{24} \\ \rho _3= & {} R_{11} R_{32} R_{23} -R_{31} R_{12} R_{23} -R_{11} R_{22} R_{44} \\&-R_{42} R_{23} R_{34} +R_{31} R_{22} R_{13} +R_{32} R_{23} R_{44} \\&-R_{11} R_{22} R_{33} -R_{32} R_{43} R_{24} -R_{21} R_{32} R_{13}\\&+R_{11} R_{42} R_{24} +R_{31} R_{13} R_{44} +R_{21} R_{12} R_{44} \\&-R_{21} R_{42} R_{14} +R_{42} R_{24} R_{33} +R_{21} R_{12} R_{33} \\&-R_{11} R_{33} R_{44} -R_{22} R_{33} R_{44} -R_{41} R_{12} R_{24} \\&-R_{31} R_{43} R_{14} +R_{41} R_{22} R_{14} +R_{22} R_{34} R_{44}\\&+R_{41} R_{14} R_{33} +R_{11} R_{34} R_{43} -R_{41} R_{13} R_{34} \\ \rho _4= & {} -R_{41} R_{12} R_{23} R_{34} -R_{21} R_{12} R_{33} R_{44}\\&+R_{11} R_{22} R_{33} R_{44} -R_{11} R_{42} R_{24} R_{33} \\&+R_{21} R_{32} R_{13} R_{44} +R_{11} R_{32} R_{43} R_{24} \\&-R_{41} R_{32} R_{13} R_{24} +R_{41} R_{32} R_{23} R_{14} -R_{11} R_{22} R_{34} R_{43} \\&+R_{21} R_{12} R_{34} R_{43} -R_{31} R_{12} R_{43} R_{24} -R_{21} R_{42} R_{13} R_{34} \\&-R_{21} R_{32} R_{43} R_{14} +R_{31} R_{12} R_{14} R_{44} +R_{31} R_{42} R_{13} R_{24} \\&-R_{31} R_{42} R_{23} R_{14} +R_{21} R_{42} R_{14} R_{33} +R_{11} R_{42} R_{23} R_{34} \\&-R_{11} R_{32} R_{23} R_{44} +R_{31} R_{22} R_{43} R_{14} -R_{31} R_{22} R_{13} R_{44} \\&+R_{41} R_{12} R_{24} R_{33} -R_{41} R_{22} R_{14} R_{33} +R_{41} R_{22} R_{13} R_{34} \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Wu, Y., Wang, X. et al. Nonlinear normal modes and primary resonance for permanent magnet synchronous motors with a nonlinear restoring force and an unbalanced magnetic pull. Nonlinear Dyn 97, 1197–1213 (2019). https://doi.org/10.1007/s11071-019-05040-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05040-3

Keywords

Search

Navigation