Investigation On Nonlinear Dynamic Behavior Of Energy Harvesting Of Magnetic Levitation

In recent years,with the increasing development of very large scale integration circuit,the level of power consumption of wireless electronic devices decreases from milliwatts to microwatts order of magnitude.Vibration energy harvesting for powering such wireless device becomes possible.However,when the frequency of the environment is far away from the base frequency,the performance of linear energy harvester rapidly declines.To increase the energy capture bandwidth,the nonlinearity is an effective way to improve the energy harvesting effect.Magnetic technology is a kind of typical nonlinear factors,therefore they are widely used in energy harvesting field.The energy harvester of magnetic levitation has caught the attention of many researchers.To improve the energy harvesting effect,this dissertation studies a type of vibration energy harvester of magnetic levitation.The two magnets of energy harvester are fixed on both ends of a pillar that inserts into a middle magnet that can move up and down.A spring oscillator is suspended under the middle magnet.The poles of the magnets are placed in such a way that the middle magnet is levitated.While this system is excited by the external environment,the middle of the magnet and the spring oscillator nonlinear vibration is generated and the mechanical energy is converted to electrical energy through the energy converter on the middle.magnet.The magnetic levitation force of the middle magnet is expressed approximately with cubic nonlinearities and the nonlinear vibration of the energy harvester can be described Duffing equations.When the energy harvester is subjected to a harmonic excitation,the mathematical model equations of electricity-force coupling can be derived based on the Newton’s second law and Faraday’s law.Analytical solution of the equations is obtained by the harmonic balance method.The effects of the gravity,damping,inductance,length of the coil,and magnetic flux density of the system parameters on the displacement and the average output power of the middle magnet are discussed,respectively.The results show that,compared with the linear energy harvester,the response amplitude of the displacement and average power frequency increases,and the response frequency bandwidth is widened.Moreover,using a linear transformation,the Duffing energy harvesting system is transformed into the quadratic nonlinearities energy harvesting system.The displacement amplitude and average power amplitude frequency responses of the middle magnet are obtained by the harmonic balance method,Newton iteration method,and arc length method,and the effect of the nonlinear coefficient on the amplitude-frequency response of the displacement and average power is given.Under the multi-frequency excitation,by the use of the harmonic balance method,Newton iteration method and arc-length method,the approximate analysis of the displacement and average power is obtained.Also,the impact of the two frequency ratio on the middle magnet displacement and average power is investigated.In order to enlarge the area of magnetic levitation nonlinear energy harvesting,the paper researches the nonlinear dynamics of a two-degree-of-freedom nonlinear energy harvester of magnetic levitation.A one-degree-of-freedom nonlinear energy harvester of magnetic levitation is expanded by introducing a linear oscillator.The nonlinear ordinary differential equations are approximately analyzed by using the harmonic balance method.Under the harmonic excitation,the characteristics of the steady-state amplitude-frequency response of primary resonance are studied.The results show that the resonance area of the two degree-of-freedom system is significantly expanded.Moreover,by comparing the various parameters on the amplitude-frequency response and the resonance area,the effects of the mass ratio,frequency ratio and nonlinear coefficient ratio enhanced are investigated.Therefore,the influences of the intensity of energy acquisition and the bandwidth are determined.The steady-state amplitude response of the energy harvester under the multi-frequency harmonic is further studied.The results have shown that adding a linear spring vibrator and under multi-frequency excitation,the amplitude of amplitude-frequency response and the bandwidth of the amplitude frequency response of the energy acquisition system can be increased.Two resonance peak of the middle magnet is close in the range of low frequency by adjusting the the mass ratio of the two degrees of freedom.The influence of the system parameter change on the displacement and average power of the middle magnet is investigated as well.Under simple harmonic excitation and multi-frequency harmonic excitation,the approximate analytic results of the steady-state response of the energy harvester of magnetic levitation are verified through a direct runge kutta numerical simulation.Also,the approximately analytic results of the energy harvester of magnetic levitation appending the linear oscillator are validated under simple harmonic excitation and multi-frequency harmonic excitation.In conclusion,the proposed methodology is not only in-depth study of the energy harvester of magnetic levitation,can also extended to tackle other complex beam or plate vibratory energy harvesters.These findings provide a reference and help for the design of the future vibratory energy harvester.