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