Multi-scale Dynamical Modeling And Vibration Study Of Axially Moving Nanobeams

An axially moving nanobeam is an important simplified engineering model.Many components of micro-and nanoscale devices in MEMS/NEMS,such as the subminiature belt,nanofibre,micropump,and silicon acceleration sensor,can be modelled as axially moving nanobeams for theoretical analysis.On one hand,the size effect on mechanical properties and physical responses is not negligible at micro scales and that this effect is not represented by the classical continuum theory.On the other hand,due to the impact of axial velocity,the structures produce a large transverse vibration,which may cause system fatigue and low quality.To ensure the security and stability of the systems,it is imperative to establish and perfect new theoretical models for analysing and understanding the mechanical characteristics of nanobeam-like structures.Based on nonlocal strain gradient theory,this paper established a nonlinear dynamical equation and the corresponding boundary conditions for an axially moving nanobeam using the calculus of variations.By introducing two scale parameters,i.e.,a nonlocal parameter and a material characteristic parameter,we investigated the scale effect of multiple vibration modes of the system,including free vibration,self-excited vibration,parametrically excited vibration,forced vibration,internal resonance and combination resonance.First,we investigated the free vibration of the derived system of an axially moving nanobeam,obtained the natural frequency and the modal function of the linear system semi-analytically,and calculated an explicit representation of the critical velocity.In the range of subcritical and supercritical velocity,the vibration characteristics of the system exhibited significant differences.The divergence instability and flutter instability of the system were analysed at different velocities.This paper discussed whether the nonlocal parameter or the material characteristic parameter dominates the scale effect and investigated the effect of the two scale parameters on the natural frequency,modal function,critical(flutter)velocity and instability modes.The numerical results showed that the nonlocal parameter softens the stiffness,whereas the material characteristic parameter hardens the stiffness.The greater of the two parameters dominates.When the two parameters are equal in magnitude,the softening and hardening of the scale parameters cancel out;these results agree with the results of classical elasticity.Second,we investigated the parametric resonance of an axially moving nanobeam subjected to a velocity disturbance using the averaging method and discussed the effect of multiple parameters,particularly scale parameters,on the flutter instability regime.The viscoelastic coefficient can attenuate the system's energy and inhibit resonance vibration.Theoretical deduction showed that the nonlocal parameter and the material characteristic parameter can change the equivalent damping of the system and thus play a role similar to that of viscoelastic damping.This was verified through numerical simulation.The results showed that the nonlocal parameter enhances parametric resonance of the system and thus weakens the damping.In contrast,the material characteristic parameter inhibits parametric resonance of the system and thus enhances the damping.Third,at first order in the method of multiple scales,we analysed the nonlinear free vibration characteristics of an axially moving nanobeam based on solvability conditions.In addition,we investigated parametric vibration in response to velocity variations,discussed the stability of the response amplitude based on the Routh-Hurwitz stability criterion,and performed a detailed analysis for the multi-value and jumping phenomena.Furthermore,we analysed the effect of viscoelastic damping on the free vibration characteristics of an axially moving nanobeam at second order.The results showed that the impact of nonlinear items on the frequency occurs at first order;the nonlinear items affect the stiffness softening and hardening of the two scale parameters.The numerical size of these two parameters do not determine which is dominant;that requires further investigation.The scale parameter influences the stability boundary and the bifurcation point of the response amplitude of the nonlinear resonance vibration,enhancing or inhibiting parametric resonance.The impact of the viscoelastic coefficient on the frequency of the system is of second order.Fourth,we investigated the nonlinear forced vibration characteristics of the system using the method of multiple scales and discussed the impact of each parameter on the amplitude response curves.The results showed that the nonlocal parameter enhances resonance vibration,whereas the material characteristic parameter inhibits resonance vibration.These two parameters also influence the degree of bending and the amplitude of the peaks of the resonance vibration response curves.Fifth,we investigated fundamental harmonic resonance accompanied by internal resonance using the multidimensional L-P method when the external exciting frequency approaches the first-and second-order natural frequencies of the system and emphatically investigated the scale effect in the resonance regime and at the critical amplitude of external excitation.The results showed that the internal resonance regime is related to the amplitude of the external excitation.When the amplitude of the external excitation exceeds a certain critical value,the internal resonance disappears.The scale parameters influence the critical amplitude.Specifically,the nonlocal parameter decreases the critical amplitude,whereas the material characteristic parameter increases the critical amplitude.Finally,we investigated the bifurcation of one type of simplified nonautonomous planar system.The results showed that with a change in the nonlocal parameter and the material characteristic parameter,period doubling,and intermittent chaos appeared in the system.