On a non-ideal magnetic levitation system: nonlinear dynamical behavior and energy harvesting analyses

  • Rodrigo Tumolin RochaEmail author
  • Jose Manoel Balthazar
  • Angelo Marcelo Tusset
  • Silvio Luiz Thomaz de Souza
  • Frederic Conrad Janzen
  • Hassan Costa Arbex
Original Paper


Nowadays, a novelty of devices that use magnetic restoring forces to generate oscillations has increased substantially. These kinds of devices have been commonly used to energy harvesting area. Therefore, in this paper, numerical and analytical analyses of a non-ideal magnetic levitation system are carried out. The mathematical modeling of the magnetic levitation device is developed and examined considering an electrodynamical shaker to base-excite the main system, which is a non-ideal excitation. The magnetic levitation system has the form of a Duffing oscillator; thus, the nonlinear analysis is required to investigate the energy harvesting potential of this nonlinear system. The novelty here is the use of the shaker to the excitation which is non-ideal. The method of multiple scales is applied to investigate the modes of vibration of the coupled system, which will remark the non-ideality and nonlinear phenomena of the system. The average harvested power is described by through expressions related to the coupling between the mechanical and electrical domains. Moreover, it was developed an expression for the excitation frequency where the maximum harvested power is obtained. The results were obtained based on the numerical method of Runge–Kutta of fourth order with fixed step whose results are shown through phase planes, Poincare maps and parametrical variation. Such results showed multiple existence of behaviors (periodic, quasiperiodic and chaos), coexistence of attractors in a high sensibility of the initial conditions and interesting results of the maximum average power, obtaining high and continuous amount of energy in periodic and chaotic regions.


Magnetic levitation Chaotic behavior Method of multiple scales Nonlinear dynamics Energy harvesting Non-ideal excitation 



The authors acknowledge support by CNPq (GRANT: 447539/2014-0) and CAPES, all Brazilian research funding agencies.

Compliance with ethical standards

Conflict of interest

The authors declare there is no conflict of interest.


  1. 1.
    Kononenko, V. O.: Vibrating systems with a limited power supply. Iliffe (1969)Google Scholar
  2. 2.
    Balthazar, J.M., Mook, D.T., Weber, H.I., Brasil, R.M., Fenili, A., Belato, D., Felix, J.L.P.: An overview on non-ideal vibrations. Meccanica 38(6), 613–621 (2003). CrossRefzbMATHGoogle Scholar
  3. 3.
    Balthazar, J.M., Tusset, A.M., Brasil, R.M., Felix, J.L., Rocha, R.T., Janzen, F.C., Nabarrete, A., Oliveira, C.: An overview on the appearance of the Sommerfeld effect and saturation phenomenon in non-ideal vibrating systems (NIS) in macro and MEMS scales. Nonlinear Dyn. 93(1), 19–40 (2018). CrossRefGoogle Scholar
  4. 4.
    Kong, L. B., Li, T., Hng, H. H., Boey, F., Zhang, T., Li, S.: Waste Energy Harvesting, pp. 405–480. Springer, Berlin (2014).
  5. 5.
    Shafiee, S., Topal, E.: An econometrics view of worldwide fossil fuel consumption and the role of US. Energy Policy 36(2), 775–786 (2008). CrossRefGoogle Scholar
  6. 6.
    Shafiee, S., Topal, E.: When will fossil fuel reserves be diminished? Energy policy 37(1), 181–189 (2009). CrossRefGoogle Scholar
  7. 7.
    Mescia, L.: Innovative materials and systems for energy harvesting applications. IGI Global, 496p (2015).
  8. 8.
    Mason, J.E.: World energy analysis: \(H_2\) now or later? Energy Policy 35(2), 1315–1329 (2007). CrossRefGoogle Scholar
  9. 9.
    Pinna, L., Dahiya, R. S., De Nisi, F., Valle, M.: Analysis of self-powered vibration-based energy scavenging system. In Industrial Electronics (ISIE), 2010 IEEE International Symposium on IEEE, (July 2010), pp. 402–408 .
  10. 10.
    Harb, A.: Energy harvesting: state-of-the-art. Renew. Energy 36(10), 2641–2654 (2011). CrossRefGoogle Scholar
  11. 11.
    Priya, S., Inman, D. J.: Energy Harvesting Technologies, vol. 21. Springer, New York (2009).
  12. 12.
    Priya, S.: Advances in energy harvesting using low profile piezoelectric transducers. J. Electroceramics 19(1), 167–184 (2007). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Roundy, S., Wright, P. K., Rabaey, J. M.: Energy scavenging for wireless sensor networks. Norwell, 45–47 (2003)
  14. 14.
    Beeby, S.P., Tudor, M.J., White, N.M.: Energy harvesting vibration sources for microsystems applications. Meas. Sci. Technol. 17(12), R175 (2006). CrossRefGoogle Scholar
  15. 15.
    Li, Z., Zuo, L., Luhrs, G., Lin, L., Qin, Y.X.: Electromagnetic energy-harvesting shock absorbers: design, modeling, and road tests. IEEE Trans. Veh. Technol. 62(3), 1065–1074 (2013). CrossRefGoogle Scholar
  16. 16.
    Litak, G., Friswell, M.I., Adhikari, S.: Magnetopiezoelastic energy harvesting driven by random excitations. Appl. Phys. Lett. 96(21), 214103 (2010). CrossRefGoogle Scholar
  17. 17.
    Stephen, N.G.: On the maximum power transfer theorem within electromechanical systems. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 220(8), 1261–1267 (2006). CrossRefGoogle Scholar
  18. 18.
    Crawley, E.F., Anderson, E.H.: Detailed models of piezoceramic actuation of beams. J. Intell. Mater. Syst. Struct. 1(1), 4–25 (1990). CrossRefGoogle Scholar
  19. 19.
    Triplett, A., Quinn, D.D.: The effect of non-linear piezoelectric coupling on vibration-based energy harvesting. J. Intell. Mater. Syst. Struct. 20(16), 1959–1967 (2009). CrossRefGoogle Scholar
  20. 20.
    Mann, B.P., Sims, N.D.: Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319(1–2), 515–530 (2009). CrossRefGoogle Scholar
  21. 21.
    Carmichael, A.T., Hinchliffe, S., Murgatroyd, P.N., Williams, I.D.: Magnetic suspension systems with digital controllers. Rev. Sci. Inst. 57(8), 1611–1615 (1986). CrossRefGoogle Scholar
  22. 22.
    Wong, T.H.: Design of a magnetic levitation control system??? An undergraduate project. IEEE Trans. Educ. 4, 196–200 (1986). CrossRefGoogle Scholar
  23. 23.
    Moon, F.C.: Superconducting Levitation: Applications to Bearing and Magnetic Transportation. Wiley, New york (2008)Google Scholar
  24. 24.
    Braunbeck, W.: Free suspension of bodies in electric and magnetic fields. Z. für Phys. 112(11), 753–763 (1939)CrossRefGoogle Scholar
  25. 25.
    Arbex, H.C., Balthazar, J.M., de Pontes Junior, B.R., da Fonseca, R.M.L.R., Felix, J.L.P., Tusset, A.M., Bueno, A.M.: On nonlinear dynamics behavior and control of a new model of a magnetically levitated vibrating system, excited by an unbalanced DC motor of limited power supply. J. Braz. Soc. Mech. Sci. Eng. 37(4), 1139–1150 (2015). CrossRefGoogle Scholar
  26. 26.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)zbMATHGoogle Scholar
  27. 27.
    Nayfeh, A.H.: Nonlinear Interactions: Analytical, Computational and Experimental Methods. Wiley, New York (2000)zbMATHGoogle Scholar
  28. 28.
    de Souza, S.L.T., Batista, A.M., Baptista, M.S., Caldas, I.L., Balthazar, J.M.: Characterization in bi-parameter space of a non-ideal oscillator. Phys. A Stat. Mech. Appl. 466, 224–231 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Serajian, R.: Parameters’ changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie. J. Meas. Eng. 1(4), 195–206 (2013)Google Scholar
  30. 30.
    Younesian, D., Jafari, A.A., Serajian, R.: Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory. Int. J. Autom. Eng. 1(3), 186–196 (2011)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of ElectronicsFederal University of Technology - ParanaPonta GrossaBrazil
  2. 2.Faculty of Mechanical Engineering of BauruSao Paulo State UniversityBauruBrazil
  3. 3.Physics and Mathematics DepartmentFederal University of Sao Joao del-ReiOuro BrancoBrazil

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