An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets  

作　　者：Xinsheng Xu Dalun Rong C.W.Lim Changyu Yang Zhenhuan Zhou 

机构地区：[1]State Key Laboratory of Structure Analysis of Industrial Equipment, Department of Engineering Mechanics,International Research Center for Computational Mechanics,Dalian University of Technology [2]Department of Architecture and Civil Engineering, City University of Hong Kong

出　　处：《Acta Mechanica Sinica》2017年第5期912-925,共14页力学学报（英文版）

基　　金：support of the National Natural Science Foundation of China (Grant 11672054);the Research Grant Council of Hong Kong (11215415);the National Basic Research Program of China (973 Program) (Grant 2014CB046803)

摘　　要：A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach. A Hamiltonian system is established by introducing a total unknown vector consisting of the displacement amplitude, rotation angle, shear force, and bending moment. The high-order governing differential equation of the vibration of SLGSs is transformed into a set of ordinary differential equations in symplectic space. Exact solutions for free vibration are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary conditions. Vibration modes are expressed in terms of the symplectic eigenfunctions. In the numerical examples, comparison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given. A parametric study of the natural frequency is also included.A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach. A Hamiltonian system is established by introducing a total unknown vector consisting of the displacement amplitude, rotation angle, shear force, and bending moment. The high-order governing differential equation of the vibration of SLGSs is transformed into a set of ordinary differential equations in symplectic space. Exact solutions for free vibration are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary conditions. Vibration modes are expressed in terms of the symplectic eigenfunctions. In the numerical examples, comparison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given. A parametric study of the natural frequency is also included.

关 键 词：Hamiltonian system Analytical method Nonlocal elasticity theory Orthotropic graphene sheet Natural frequency 

分 类 号：O327[理学—一般力学与力学基础]