Symplectic Method For Vibration Of Magnetically Affected Nanoplates

With the development of nano science and technology,new nanoplate structures have been widely used in the design of nano devices such as nanoelectromechanical systems.Among them,the performance indexes of some new nano devices based on nanomagnetic sensing materials such as graphene show significant improvement in the magnetic field environments.It is necessary to understand the mechanical properties and behaviors of nanoplate structures on the nanometer scale for the accurate design,manufacture and assembly of such nano devices.However,for the complex nanostructures,limitations like high experimental cost and large calculation exist for the current experimental methods and molecular dynamics simulation methods.Therefore,it is of great scientific significance to enrich and enhance the understanding of the mechanical behaviors of nanoplate structures for the development of new nano devices,by using the modified continuum mechanics method which considering the scale effect,and developing the analytical research methods and theories in this field from the theoretical point of view.In this paper,a series of vibration problems based on nonlocal thin plate model are studied analytically under the framework of Hamiltonian system.A kind of analytical solution of nano plate vibration problem with complex boundary conditions such as cantilever and corner support in the magnetic field is solved,and then its vibration characteristics are analyzed.The work in this paper can provide theoretical support for the design and analysis of nano devices based on the above nano plate structures.The specific research content of this paper includes:(1)Based on the nonlocal thin plate theory and Hamiltonian system,the Hamiltonian solution system of the vibration problem of nano plate under in the magnetic field is established,and the analytical solution with combined boundaries including opposite simple supported and sliding supported is obtained.In this paper,the displacement and rotation angle of the nanoplate are taken as the original variables,and the dual variables of generalized shear force and bending moment are derived by Legendre transformation.Based on the Hamiltonian variational principle,the Hamiltonian canonical equation for the vibration of nanoplates in the magnetic field is derived.The Hamiltonian operator matrix contains both Lorentz force and nonlocal theoretical parameters.In the Hamiltonian system,the natural frequency and the corresponding mode function of the nano plate can be expressed by the eigenvalues and eigensolutions of the Hamiltonian operator matrix.The natural frequency and mode function of the influence of the nanoplate on the boundary conditions are solved.(2)The vibration problems of four edges free,corner supported and cantilevered nanoplates in magnetic fields can be solved directly by boundary decomposition.The edges free,corner supported and cantilevered boundaries of the nanoplates can be decomposed into several subproblems with simple boundaries,and the solutions of each subproblem can be expressed by the symplectic eigensolutions of the opposite simply supported/sliding supported nanoplates.By boundary superposition,the original vibration problem of complex boundary nanoplates is transformed into a set of algebraic equations with undetermined coefficients,and the natural frequency and mode function are obtained directly.Results show that the external magnetic field can improve the natural frequency of the nanoplate and change its mode arrangement.(3)The Hamiltonian solution system of the vibration problem for a kind of double layered nanoplate system in the magnetic field environment is established,and the analytical solutions under various boundary conditions are obtained.Based on the research on the vibration of the single layered nanoplate,the vibration of the double layered nanoplate system can be divided into two subproblems:in-phase vibration and out-of-phase vibration.By introducing the adjustment coefficients,the Hamiltonian canonical control equation corresponding to the vibration problem of the double layered nanoplate in the magnetic field is established.The original problem is attributed to the eigenproblem in the Hamiltonian system,so that the analytical solution of the opposite simply supported/sliding supported double layered nanoplate in the magnetic field can be obtained directly.On this basis,the boundary superposition method is used to further obtain the analytical solution of the vibration problem for the double layered nanoplate system with complex boundary conditions in the magnetic field environment.Results show that the external magnetic field can also improve the natural frequency of the double layered nanoplate system,and the natural frequencies of the same phase and the opposite phase vibrations are similar to each other.