The Steady State Of The Forced Vibration Of Nanoplate Structure

With the development of high technology,the lightweight,miniaturization,microminiaturization and intelligentization of equipment and instrument have formed the development trend.The development of nanomaterials,nanostructures and nanotechnology provides the conditions and impetus for this trend.China’s "13th five-year plan" clearly puts forward the key development requirements for nanomaterials and devices,and the outline of the national medium and long-term scientific and technological development plan also clearly puts forward the research needs for controllable preparation,self-assembly and functionalization of nanomaterials.In recent years,China’s material manufacturing industry has developed rapidly,and the application of nanomaterials in the production and processing of electronic equipment provides the possibility for the miniaturization and lightweight of electronic equipment,especially the unprecedented development of micro-nano electromechanical system.At the same time,the accompanying dynamic performance and behavior problems of new material structures are more prominent.Therefore,the development of corresponding research methods and revealing their laws has important scientific significance and application value.In this dissertation,a kind of steady-state forced vibration problem of nanoplates is studied,and the analytical solution of steady-state forced vibration problem of nanoplates with different boundary conditions is solved by using the small deflection bending theory of non-local Kirchhoff plates and the Hamiltonian system method,which provides methods and basis for further study of similar problems.Specific research contents include:(1)To construct the Hamiltonian solution system for the steady-state forced vibration of nanoplate,and obtain the analytic solution expressed by the symplectic eigensolution series.Taking the simply supported opposite edges(SS)nanoplate as an example and establishing the relation between local variables and non-local variables,the governing equation of forced vibration of nanoplates is described by using local variables.By using dual variable and variational method,the regular equation of Hamiltonian system is introduced.In the Hamiltonian system,the problem is reduced to the symplectic eigenvalue and symplectic eigensolution,and the homogeneous general solution and non-homogeneous particular solution expressions are obtained.By using boundary conditions,symplectic conjugate orthogonal relations and expansion theorem,the problem is transformed into the solution of algebraic equations.Thus,the undetermined coefficients in symplectic analytic solution expressions are obtained,that is,symplectic analytic solutions for the steady state forced vibration of rectangular nanomaterials with simply supported opposite edges and analytical solutions for nanomaterials with other boundary conditions are obtained.(2)The symplectic analysis model of forced vibration of non-uniform rectangular nanoplates on elastic medium is established.On the basis of Hamiltonian system,the steady state forced vibration law of a group of rectangular nanoplates with clamped adjacent edges and simple/clamped supports(CCCC,CCCS and CCSS)is studied and analyzed,and the analytical solutions of these problems are obtained.The research method is mainly aimed at the boundary conditions of nanoplates which are difficult to be solved directly.The steady state forced vibration problem of nanoplates is transformed into several subproblems.The relation between subproblems is established by undetermined coefficients.By using the expression of the relation between the subproblems and the solution of a single subproblem,the original problem is transformed into a simple problem of algebraic equations,and the analytical solution of the steady state forced vibration problem of non-uniform rectangular nanoplates is obtained.The analytical solutions of non-uniform rectangular nanoplates with CCCC,CCCS and CCSS boundary conditions are obtained.(3)The steady state forced vibration model of ortho tropic rectangular nanoplates in viscoelastic media is established under symplectic system.The analytical solution of the nanoplate problem with four edges free(FFFF)boundary conditions is obtained by the method of boundary superposition.Specific methods include:the symplectic analytic solution for the steady state forced vibration of rectangular nanoplates with guided supported(GG)is obtained by Hamiltonian solution;two solutions of vibration problems are analyzed for the change of the motion angle of the boundary of the four edges guided supported nanoplates;by superimposing the boundary conditions of the three problems and the external excitation loads,it is exactly equivalent to the basic problem of the steady-state forced vibration of orthotropic rectangular nanoplates with four edges free(FFFF)boundary conditions.According to the equivalence relation,the undetermined coefficients are determined,and the analytical solution of the problem is obtained.The characteristics and laws of the steady state forced vibration of FFFF orthotropic rectangular nanoplates are given.(4)Using the method of boundary decomposition,the steady state forced vibration model of rectangular nanocrystals with complex boundary supports such as free-free-clamped-clamped supported(FFCC),free-free-clamped-simply supported(FFCS)and free-free-simply-simply supported(FFSS)on viscoelastic media was established.This kind of model can be decomposed by boundary in Hamiltonian system,and the problem can be divided into several series solutions which can be expressed by symplectic eigensolutions.The undetermined coefficients of series solutions can be reduced to the roots of algebraic equations by virtue of symplectic conjugate orthogonal relations of eigensolutions.The analytical solution of the problem is obtained.The numerical results show that the viscosity coefficient has a great influence on the vibration amplitude;the nonlocal parameters are inversely proportional to the resonance frequency;the supporting condition will affect the stiffness of the whole structure,so it will affect the resonance frequency.(5)A symplectic system and Hamiltonian regular equations are established to solve the steady-state forced vibration problem of double-layer cantilever(FFFC)rectangular nanoplates embedded in viscoelastic media.It is found that the sum and difference of upper and lower plate displacements satisfy the same Hamiltonian regular equations.Therefore,the problem of double-layer nanoplates can be attributed to the symplectic eigenvalue and eigen solution of the same Hamiltonian system,that is,displacement can be expressed by the combination of symplectic eigen solutions of the same family.In this study,the boundary decomposition method and the double Hamiltonian system expression technique with two different coordinate simulation time are used to establish the correlation conditions among the sub-problems,and the analytic solution of the problem is obtained by symplectic superposition method,so that a special Hamiltonian system method is formed in this kind of problems.The results show that the resonance frequency of forced vibration of double-layer plate is more than that of single-layer plate.In fact,the extra resonance frequency corresponds to the different vibration modes of the double plates.The research method provides the basis for analyzing the dynamic behavior of nanocrystals and provides a path and effective method for solving similar problems.