Study On The Nonlinear Dynamics Of Electrostatically Driven Non-ideal Micro/Nano Beam Resonator

Micro-electro-mechanical system(MEMS)and nano-electro-mechanical system(NEMS)involve interdisciplinary,which can be applied and developed in almost all natural and engineering fields.Electrostatically driven double end clamped micro /nano beam resonator is a typical micro / nano device.In the machining process of micro/ nano beam,machining error makes micro / nano beam in nonideal state,such as support elasticity and initial curvature.These nonideal states have significant impacts on the dynamic performance of micro / nano systems.Therefore,in order to predict the performance of the system more accurately,in addition to considering the inherent nonlinearity of electrostatic drive,the nonideal state of micro / nano structure is also a factor that can’t be ignored.In this paper,a typical micro / nano beam resonator with fixed support at both ends driven by electrostatic monopole is taken as the research object.Considering two manufacturing imperfects of flexible support and initial curvature,the static bifurcation and nonlinear vibration are studied and discussed in detail.The specific research contents and achievements are as follows.(1)Based on the generalized Hamiltonian principle,the lateral vibration governing equation of micro / nano beam system with flexible support was derived.The influence of flexible boundary and fringing field on the static bifurcation and resonance frequency was studied by using the high-order Galerkin discrete method.Then,based on the fifth order discrete static solution,the first order discrete dynamic model was introduced and the third-order nonlinear stiffness was considered.The method of multiple scales was used to study the influence of flexible boundary and fringing field on the primary resonance under small amplitude vibration.The fringing effect of electrostatic force was described by Palmer model.The flexible boundary was simulated by torsion spring and translational spring.It was assumed that the spring stiffness at both ends was equal.The numerical results agree well with the analytical results with small amplitude.The results show that the electrostatic fringing effect and flexible boundary not only reduce the static pull-in voltage and natural frequency,but also increase the dynamic response amplitude,and also change the nonlinear softening and hardening characteristics by adjusting DC voltage;the flexibility of boundary makes the system obtain a larger displacement movement under a smaller electrostatic force,which provides a theoretical basis for the realization of energy-saving resonator.(2)The influence of boundary spring coefficient was studied on the static bifurcation and resonance frequency of the system by using the ninth order discrete static model,and compared with the first order discrete static model.Based on the ninth order static solution,the first order discrete dynamic model considering the fifth order nonlinear stiffness was introduced;the multi-scale method was used to study the influence of boundary spring coefficient on the primary resonance.The judgment conditions of hard soft transition and soft hard transition are deduced.The frequency response curves under different DC and AC voltage were used to verify the correctness of the method.The results show that the DC voltage and AC voltage can change the nonlinear characteristics.(3)Using generalized Hamiltonian principle and consistent couple stress theory,the transversal vibration control differential equation of micro / nano beam resonators considering the scale effect and flexible boundary was derived;the law of the natural frequency with DC voltage was obtained;the influence of scale effect and Casimir force on the performance of micro / nano system with flexible boundary was analyzed.The results show that the scale effect is not enough to affect the judgment of the nonlinear characteristics of the MEMS with flexible boundary,but it affects this feature of the NEMS;Casimir force reduces the static pull-in voltage and critical deflection of the NEMS,increases the natural frequency and the dynamic response amplitude,which can’t be ignored.(4)The transversal vibration control differential equation of the system was established when the scale effect and Casimir force were regarded.Considering the quintic nonlinear stiffness,the influence of initial deformation on the snap-through,static pull-in instability and dynamic response of the system is studied.The results show that the initial curvature increases the pull-in displacement and the snap-through voltage compared with the straight beam,which leads to the decrease of the difference between the snap-through voltage and the static pull-in voltage;the initial curvature also reduces the natural frequency and dynamic response amplitude,and changes the nonlinear characteristics.The results of this paper predict the dynamic performance of the nonideal micro /nano beam resonator system more accurately,and reveal the transformation mechanism of the nonlinear characteristics,which has significant theoretical and engineering significance for guiding the dynamic optimization design of the resonator.