| With the rapid development of nanoindustry,graphene,carbon nanotubes and other new materials have been widely used,especially in micro-nano electromechanical systems(NEMS).As an important component of NEMS,nanostructures play a key role due to their excellent properties in electromagnetic,thermoelectric,mechanics and other aspects.In view of the great application potential of nanostructures,it is of great scientific significance and application value to study their mechanical properties and kinetic mechanism.Traditional theories and research methods are no longer applicable at nanoscale due to different characterization of material properties at macroscale.Therefore,based on the Eringen nonlocal theory and the classical Kirchhoff thin plate theory,which take the nanoscale effect into account,combined with the classical Maxwell electromagnetic theory and the Hamiltonian solution system,the vibration problem of three-layer nanoplates is studied and analyzed as follows:(1)In order to describe the basic problem of nanoplate vibration,the Van der Waals force and Casimir force between nanoplates are equivalent to Winkler spring force,and the Lorentz force of vertical nanoplate under magnetic field is obtained by using Maxwell's electromagnetic theory.By means of the nonlocal theoretical constitutive relation expressed by the differential operator,the governing equations and boundary conditions for the vibration of the nanoplate(generalized Kirchhoff thin plate)under the action of magnetic field are obtained,and the basic problems are described.(2)The dual variable of the original variable is introduced by the energy method,and the Hamiltonian regular equations are introduced by the variational principle of the full state vector,that is,the Hamiltonian system for the vibration problem of the nanoplate is established.Using the method of separating variables in Hamiltonian system and combining with boundary conditions,the natural frequencies and modes of vibration of nanoplates are reduced to the symplectic eigenvalues and eigensolutions.Each symplectic eigensolution represents a free vibration mode.Because of the symplectic conjugate orthogonality relation between the symplectic eigensolutions,the solution of the nanoplate vibration problem can be expressed by the series of eigensolution expansion.The comparison with other numerical results shows that the results obtained by the symplectic method are accurate.Numerical results show that the influence of the related parameters of the nanoplate is analyzed.(3)The vibration problem of the cantilever nanoplate was decomposed into two subproblems of the boundary conditions of the opposite edge sliding supports by using the boundary decomposition method of the nanoplate.After solving the expression of the subproblem,the analytical solution of the free vibration problem of the cantilever nanoplate is obtained by using the boundary superposition method.The natural frequencies and mode shapes of the problem are shown by numerical calculation.The results show that the method of boundary decomposition and boundary superposition is effective.(4)For the vibration problem of triple-layered nanoplates system under the action of magnetic field,through mathematical transformation,the coupling governing equations of the triple-layered nanoplates are summed up into a unified equation form.It is proved that the solution of the unified form equation corresponds to the solution of the original triple-layered nanoplates problem.Taking the triple-layered cantilever nanoplates as an example,the results show that the natural frequencies and modes of the system can be divided into three types.The numerical results show the characteristics of the natural frequencies and three vibration types of the triple-layered cantilever nanoplates.The method presented in this paper provides a research idea for the vibration of multilayer nanoplates. |
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