Stability And Nonlinear Dynamics Of Micro/Nano Beam Structure Immersed In A Magnetic Field

With the vigorous development and application of micro-electro-mechanical systems(MEMS)and nanotechnology,the research on Micro/Nano slender structures has attracted much attention.The study of mechanical-electro-magnetic coupling dynamic responses of slender structures such as nanobeams,fluid-conveying nanotubes and current-carrying microwires is of great significance for the design of related mirco/nano components and the mechanical analysis and security design of nanofluidic devices.By considering the above application requirements,the researches on the stability and nonlinear dynamics of micro/nano beam structures in magnetic field are carried out in this dissertation.The influence of magnetic field on the stability and buckling behavior of three typical micro/nano beam-type structures are studied by means of theoretical modeling,analytical analysis and numerical calculation.The main work and research results of this paper include:1.Nonlinear governing equation of nonlocal nanobeam in a longitudinal magnetic field is established.Linear frequencies,nonlinear frequencies and buckling displacement of the nanobeam are solved by analytic analysis.Results showed that a longitudinal magnetic field can significantly increase the critical axial pressure and makes the nanobeam more stable.For free vibrations of pinned-pinned nanobeams,with the increase of applied axial force,the evolution of nonlinear frequencies is much different from that of linear frequencies,for both pre-buckling and post-buckling problems.2.The concept of using longitudinal magnetic field to improve the stability of cantilevered nanotubes conveying fluid is proposed.Based on nonlocal elasticity theory and Euler-Bernoulli beam model,the partial equation of motion of cantilevered nanotubes conveying fluid with consideration of magnetic field effect is derived.This partial differential equation is then discretized using the differential quadrature method(DQM).Our numerical results show that the nonlocal parameters reduce the stability and the critical flow velocity of the system.The increase of nonlocal parameter may change the system from a second-mode flutter to a third-mode oscillatory instability.In addition,it is found that the magnetic field not only effectively improves the stability of the cantilevered nanotube conveying fluid,but also changes the order of instability modes of the system.More interesting dynamical behavior is obtained if the dimensionless magnetic field intensity is equal to or greater than the dimensionless flow velocity.In such case,the cantilevered nanotube becomes unconditionally stable,and the system would never lose stability,indicating that the longitudinal magnetic field can be used to control the stability of some nanofluidic devices.3.By introducing geometric nonlinearities,the stability and nonplanar buckling of a current-carrying microwire in a longitudinal magnetic field are researched,based on modified couple stress theory and Euler-Bernoulli beam model.Results revealed that the product of 1.5 power of dimensionless slenderness ratio parameter and dimensionless Lorentz force determines whether the system is unstable or not.This product is then defined as a dimensionless magnetic field force.It is found that the dimensionless critical magnetic field force only dependents on boundary conditions and dimensionless smallscale parameters.And the relationship curves between dimensionless axial pressure and dimensionless critical magnetic field force under different small scale parameters are plotted.It is found that the post-buckling configurations of current-carrying microwires are sensitive to initial conditions employed.4.The influence of three-dimensional magnetic field on the stability and buckling configurations of a current-carrying microwire is discussed.According to the modified couple stress theory and Euler-Bernoulli beam model,the nonlinear governing equations of the current-carrying microwire in a three-dimensional magnetic field are established,and the fourth-order Runge-Kutta method is used to solve the differential equations.It is found that the buckling configurations of the current-carrying microwire in a three dimensional magnetic field are independent of initial conditions employed.Compared with the effect of dimensionless longitudinal magnetic field force,a smaller dimensionless transverse magnetic field force produces larger lateral displacement.Also,it is found that the transverse magnetic field may enlarge the axial displacement of the microwire,while a larger longitudinal magnetic field would suppress the axial displacement of the microwire to a certain extent.5.The mechanism of the effect of alternating magnetic field on the stability of a currentcarrying microwire is revealed.Based on modified couple stress theory,the coupled hyperbolic nonlinear governing equations with time-varying coefficients for the currentcarrying microwire in an alternating magnetic field are derived.The stability of the system is analyzed using either a fourth-order Runge-Kutta method or a Floqute theory.The stability region diagrams about the alternating magnetic field frequency and alternating magnetic field intensity are plotted in the case of zero average magnetic field force,and the time-displacement response characteristics for different stability regions are analyzed in detail.In addition,the influence of perturbed alternating magnetic field on the stability of the current-carrying microwire is studied when the average magnetic field force is slightly larger than or slightly smaller than the critical magnetic field force.