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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Kononenko, V. O.: Vibrating systems with a limited power supply. Iliffe (1969)
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). https://doi.org/10.1023/A:1025877308510
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). https://doi.org/10.1007/s11071-018-4126-0
Kong, L. B., Li, T., Hng, H. H., Boey, F., Zhang, T., Li, S.: Waste Energy Harvesting, pp. 405–480. Springer, Berlin (2014). https://doi.org/10.1007/978-3-642-54634-1
Shafiee, S., Topal, E.: An econometrics view of worldwide fossil fuel consumption and the role of US. Energy Policy 36(2), 775–786 (2008). https://doi.org/10.1016/j.enpol.2007.11.002
Shafiee, S., Topal, E.: When will fossil fuel reserves be diminished? Energy policy 37(1), 181–189 (2009). https://doi.org/10.1016/j.enpol.2008.08.016
Mescia, L.: Innovative materials and systems for energy harvesting applications. IGI Global, 496p (2015). https://doi.org/10.4018/978-1-4666-8254-2
Mason, J.E.: World energy analysis: \(H_2\) now or later? Energy Policy 35(2), 1315–1329 (2007). https://doi.org/10.1016/j.enpol.2006.03.024
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 . https://doi.org/10.1109/ISIE.2010.5637866
Harb, A.: Energy harvesting: state-of-the-art. Renew. Energy 36(10), 2641–2654 (2011). https://doi.org/10.1016/j.renene.2010.06.014
Priya, S., Inman, D. J.: Energy Harvesting Technologies, vol. 21. Springer, New York (2009). https://doi.org/10.1007/978-0-387-76464-1
Priya, S.: Advances in energy harvesting using low profile piezoelectric transducers. J. Electroceramics 19(1), 167–184 (2007). https://doi.org/10.1007/s10832-007-9043-4
Roundy, S., Wright, P. K., Rabaey, J. M.: Energy scavenging for wireless sensor networks. Norwell, 45–47 (2003) https://doi.org/10.1007/978-1-4615-0485-6
Beeby, S.P., Tudor, M.J., White, N.M.: Energy harvesting vibration sources for microsystems applications. Meas. Sci. Technol. 17(12), R175 (2006). https://doi.org/10.1088/0957-0233/17/12/R01
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). https://doi.org/10.1109/TVT.2012.2229308
Litak, G., Friswell, M.I., Adhikari, S.: Magnetopiezoelastic energy harvesting driven by random excitations. Appl. Phys. Lett. 96(21), 214103 (2010). https://doi.org/10.1063/1.3436553
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). https://doi.org/10.1243/09544062JMES304
Crawley, E.F., Anderson, E.H.: Detailed models of piezoceramic actuation of beams. J. Intell. Mater. Syst. Struct. 1(1), 4–25 (1990). https://doi.org/10.1177/1045389X9000100102
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). https://doi.org/10.1177/1045389X09343218
Mann, B.P., Sims, N.D.: Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319(1–2), 515–530 (2009). https://doi.org/10.1016/j.jsv.2008.06.011
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). https://doi.org/10.1063/1.1138539
Wong, T.H.: Design of a magnetic levitation control system??? An undergraduate project. IEEE Trans. Educ. 4, 196–200 (1986). https://doi.org/10.1109/TE.1986.5570565
Moon, F.C.: Superconducting Levitation: Applications to Bearing and Magnetic Transportation. Wiley, New york (2008)
Braunbeck, W.: Free suspension of bodies in electric and magnetic fields. Z. für Phys. 112(11), 753–763 (1939)
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). https://doi.org/10.1007/s40430-014-0233-0
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)
Nayfeh, A.H.: Nonlinear Interactions: Analytical, Computational and Experimental Methods. Wiley, New York (2000)
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). https://doi.org/10.1016/j.physa.2016.09.020
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)
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)
The authors acknowledge support by CNPq (GRANT: 447539/2014-0) and CAPES, all Brazilian research funding agencies.
Conflict of interest
The authors declare there is no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Rocha, R.T., Balthazar, J.M., Tusset, A.M. et al. On a non-ideal magnetic levitation system: nonlinear dynamical behavior and energy harvesting analyses. Nonlinear Dyn 95, 3423–3438 (2019). https://doi.org/10.1007/s11071-019-04765-5
- Magnetic levitation
- Chaotic behavior
- Method of multiple scales
- Nonlinear dynamics
- Energy harvesting
- Non-ideal excitation