The Symplectic Method On Dynamic Behaviors Of Nanostructures Using The Nonlocal Theory

With the development of materials science in China, the carbon nanomaterials such as carbon nanotube, graphene sheet, and so on, with the excellent performance in light, electricity, magnetic, thermal, mechanical physical and chemical features, had been widely used in manufacturing the Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical Systems (NEMS). Comparing with the macroscopic materials, the carbon nanomatericals have better dynamic characteristics, for instance the higher natural frequency and buckling critical load and so on. However, due to the size effects, the dynamical behaviors of the carbon nanomatricals can not be analyzed using the traditional macroscale structures analysis method precisely, and the complex or larger structures can not be computed with the existing experiments and computing methods. So there are important theoretical and practical significance studying the dynamical behaviors of carbon nanomatericals. The present dissertation investigates the vibration behaviors of two important components (nanobeam and nanoplate) in NEMS, and suggests an analytic method for solving the free vibration frequencies and modes based on classical continuum mechanics, nonlocal theory and Hamiltonian mechanics symplectic method.Firstly, combining with the classical elasticity and the nonlocal theory, the current paper describes carbon nanostructures as containing continuum mechanics model with scale effect. Secondly, the issues are convert to new systems — Hamiltonian solution systems from the traditional Lagrange system by introducing the displacements variables of the dual variables, and the Hamilton canonical equation (governing equation) with corresponding boundary conditions are obtained. Finally, the governing equation is figured out using the separation variable method in completed solution space, and the original problems are attributed to eigenvalue problem in symplectic space, and the analytic frequency equations and vibration frequencies are acquired. The work finds that the free vibtration frequencies and modal with definite physical meanings can be shown by symplectic eigenvalue and eigensolution in Hamiltonian systems. Main content of our works are as follows:(1) The free vibration frequecy equations of nanbeam under the common boundary conditions using the Euler-Bernoulli namobeam theory. In this article, The high-order differential governing equations for the free vibration of nanobeams are transformed into the low-order Hamiltonian canonical equation by using the original variables (the displacement and the angle of rotation) and dual variables (the bending moment and the nonlocal equivalent shear force). According to the characteristics of Hamiltonian matrix in which the eigenvalues and eigenvectors are gained using the separation variable method, and eventually frequency equations that satisfies the boundary conditions are obtained. The study find that the vibration frequency of the nanobeam reduce with the incraesing nonlocal parameter, it shows that the whole stiffness matrix of nanomatricals have a softening effect with nonlocal effect. It can be concluded that the nonlocal parameter and boundary condition have significant influences on the natural frequency in nanoscale mechanics. The nonlocal effect decrease with the increasing of the length of the nanobeam. And considering corresponding boundary conditions, the nanobeam in clamped-clamped end has the highest natural frequency whereas the clamped-free corresponds to the lowest ones.(2) The symplectic method is investigated free vibration of the Levy-type isotropic, orthotropic and circluar/annular nanoplate according to the Kirchhoff plate theory. In these issues, the nonlocal Lagrange function is constucted, and the canonical equation described by original variables and dual variables is gained, then the free vibration problems is come down to symplectic eigenvalues in Hamiltonian system using the symplectic method. The analytical solution for the free vibration of the Levy-type rectangular nanoplate is obtained by the method of separation variables based on symplectic geometry, and the natural frequency equations under different boundary conditions are given.The first six non-dimensional natural frequencies of different edge end (clamped, simply supported, free and guided boundary conditions) with the nonlocal parameters, aspect ratio are given in isotropic and orthoprotic nanoplate in this thesis. It is indicates that the natural frequency by the influence of the boundary condition, and the lowest and highest effects of the nonlocal parameter correspond to SFSF and SCSC rectangular nanoplates, respectively. Furthermore, the effect of the nonlocal parameter becomes more pronounced for higher-mode numbers and higher aspect ratios. According to the existing literature and using the orthotropic nanoplate model, we discuss the inherent frequency of single-layered graphene sheet with the change of the thickness, length and nonlocal parameter. The different boundary conditions, the ratio between the internal and external radius (R1/R2), nonlocal parameter are detailed investigations in free vibration preblems of circular/annular nanoplate, It implies that the effect of nonlocal parameter strongly depends on the boundary conditions, only the higher order natural frequencies exists with R1/R2=0.8 and nonlocal parameter lagerer than 1 nm. For a better understanding, the lowest eight dimensionless frequency parameters for the annular nanopaltes with R1/R2=0.8 are presented.(3) The free vibration of rectangular double-nanoplate systems (DNPSs) and circular double-nanoplate systems are established in Hamiltonian system and then the frequency equations are obtained. In this study, we explicitly consider three different cases of free vibration on the double-nanoplate systems. The cases studied will be nanoplates free vibration with out-of-phase (asynchronous) sequence, in-phase (synchronous) sequence, and when one of the nanoplates is considered to be fixed. According to different situations, different Hamiltonian systems are built, and the analytic solution for vibration problems of Levy-type rectangle double-nanoplate system are obtained. Similar methods could be adopted for circular double nanoplate system, and the frequency equations are obtained in term of Hamlitonian canonical equation consisting of original variables and their dual variables. The exhaustive numerical results show that the natural frequency of the asynchronous situations are the highest, while the synchronous sequence give the lowest ones. And the nonlocal parameters have little influence on the fundamental frequency, whileas has great influence on the high frequency.The results above show that the symplectic method is a effective solution to dynamic behavior of nanostructure with high precision, convenient calculation and high efficiency. The present method is suitable for the accurate analysis of the mechanical behavior of nanomaterials for strong universality.