Research On Nonlinear Dynamics And Control Of A Bistable Oscillator With Elastic Boundary

Improving the probability of a bistable oscillator exhibiting large-amplitude inter-well oscillation under a small excitation and enhancing the ability of converting external excitation into kinetic energy are great challenges for the design of bistable oscillator and related application fields(such as vibration energy harvester,actuator,and vibration attenuation).To solve these problems,a novel oscillator called the bistable oscillator with elastic boundary(BOEB)is proposed in this dissertation,and it has high-efficiency kinetic energy output.Compared with the conventional fixed-boundary bistable oscillator,the BOEB has a shallower potential barrier,it is easier for the BOEB to realize large-amplitude inter-well oscillation,thus,generate more kinetic energy.In this dissertation,the dynamic behaviors of different structures(frame,bistable oscillator,and elastic boundary oscillator)under forced vibration are studied in detail.According to the forced vibration of the BOEB under different vibration conditions,combined with Faraday law of electromagnetic induction,Lorentz force law,and the principle of locally resonant,new types of vibration energy harvester,electromagnetic actuator,and locally resonant phononic crystal are proposed,respectively.The occurring possibilities of large-amplitude inter-well oscillations in different types of forced vibrations are studied,and the application prospects of the BOEB in vibration energy harvester,electromagnetic actuator,and locally resonant phononic crystal are proved.According to different structural models,the specific research contents of this dissertation are as follows:The dynamic behaviors of the BOEB when excited by base acceleration excitation are studied.Combined with Faraday law of electromagnetic induction,a bistable vibration energy harvester with elastic boundary(BEH＿EB)is conceived.The BEH,which has no elastic boundary,is employed as a benchmark.The parametric studies are presented to uncover the merits of the proposed BEH＿EB.The bifurcation analyses as well as the basinof-attraction maps are conducted to develop insights into the nonlinear behaviors of both the BEH＿EB and the BEH.Several intriguing nonlinear phenomena are discovered,such as the periodic inter-well oscillation and the chaotic inter-well oscillation.Subsequently,the internal resonance characteristic is introduced into the BEH＿EB to study the effects of internal resonance on dynamic behaviors.The multi-scale method is employed to calculate the first primary resonance amplitude-frequency response of the BEH＿EB under small excitation.The double-jump phenomenon is observed and it could extend the bandwidth of the BEH＿EB.In addition,the time-domain responses and spectral responses of the BEH＿EB under the condition of internal resonance are analyzed,and the energy exchange phenomenon in internal resonance is observed.Finally,the effect of internal resonance on the large-amplitude inter-well oscillation of the BEH＿EB is quantitatively analyzed through the bifurcation analysis and the basin-of-attraction map.The dynamic behaviors of the BOEB when the bistable oscillator is excited by harmonic excitation are studied.Taking electromagnetic actuator as the research object,and the application prospect of the BOEB in the field of actuator is explored.The BOEB is applied to the electromagnetic actuator and the bistable electromagnetic actuator with elastic boundary(BEMA-EB)is conceived in combination with the Lorentz force law.Besides,the fixed boundary bistable electromagnetic actuator(BEMA)and equivalent linear electromagnetic actuator(LEMA)are introduced as two counterparts.The effects of bistable characteristics on the actuation performance are proved,and the effects of elastic boundary on the bistability are analyzed.In addition,the internal resonance characteristic is introduced into the electromagnetic actuator for the first time,and the first and second primary resonances of the BEMA-EB are studied in detail.The amplitude-frequency responses of the BEMA-EB are derived by the multi-scale method under small input signal amplitude,and the double-jumping phenomenon and saturation phenomenon are observed.Afterward,the time-domain responses,spectral responses,and the effects of internal resonance on the bistability of the BEMA-EB are studied numerically.Finally,the comparison of actuation performance for the BEMA-EB and an actuator without internal resonance are conducted,and the internal resonance is proved to be an effective method to improve the actuation performance.Based on the forced vibration of the BOEB elastic boundary and combined with the principle of locally resonant,a novel phononic crystal beam is conceived.BOEB plays the role of a scatterer(locally resonant element).Firstly,according to the dynamic behaviors of the BOEB(intra-well oscillation and inter-well oscillation),the new phononic crystal is divided into a linear model and a nonlinear model.Afterward,the stopbands of an infinite length linear model are calculated by two theoretical methods,and the influences of different parameters on the stopbands are studied.It is determined that the negative effective mass model is one of the reasons for the stopband characteristic of the linear model.Besides,the transmission coefficient of the finite length nonlinear model is calculated by the finite element method,and the comparisons with the stopband characteristics obtained above are conducted.It is found that the large-amplitude inter-well oscillation could be activated in the forced vibration of the elastic boundary,and the inter-well oscillation is more conducive to hinder the propagation of elastic wave than the intra-well oscillation.