双功能梯度纳米梁系统振动分析的辛方法  

A Symplectic Approach for Free Vibration of Functionally Graded Double-Nanobeam Systems Embedded in Viscoelastic Medium

作　　者：周震寰[1] 李月杰 范俊海 隋国浩 张俊霖 徐新生[1] ZHOU Zhenhuan;LI Yuejie;FAN Junhai;SUI Guohao;ZHANG Junlin;XU Xinsheng(State Key Laboratory of Structural Analysis for Industrial Equipment（Dalian University of Technology;Department of Engineering Mechanics,International Research Center for Computational Mechanics,Dalian University of Technology Dalian,Liaoning 116024,P.R.China)

机构地区：[1]大连理工大学国际计算力学中心工程力学系工业装备结构分析国家重点实验室(大连理工大学),辽宁大连116024

出　　处：《应用数学和力学》2018年第10期1159-1171,共13页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11672054);国家重点基础研究发展计划(973计划)(2014CB046803);国家重点研发计划(2016YFB0201600);辽宁省自然科学基金(20470540186);中央高校基本科研业务费(DUT17LK57)~~

摘　　要：在辛力学与非局部Timoshenko(铁木辛柯)梁理论的基础上,针对黏弹性介质中的双功能梯度纳米梁系统的自由振动问题,提出了一种全新的解析求解方法.在Hamilton(哈密顿)体系下,位移与广义剪力、转角与广义弯矩互为对偶变量.以对偶变量为基本未知量,Lagrange(拉格朗日)体系下的高阶偏微分控制方程简化为一系列常微分方程.该纳米梁系统的振动问题归结为辛空间下的本征问题,解析频率方程和振动模态可以通过辛本征解和边界条件直接获得.数值结果验证了该方法的正确性与有效性,并针对纳米梁系统的小尺度效应、纳米梁间的相互作用以及黏弹性地基的影响进行了系统的参数分析.A new analytical approach was proposed for free vibration of functionally graded（ FG） doublenanobeam systems（ DNBSs） embedded in viscoelastic medium under the framework of symplectic mechanics and the nonlocal Timoshenko beam theory. In the Hamiltonian system,the dual variables of the displacement and the rotation angle are the generalized shear force and bending moment,respectively. The high-order governing partial differential equations in the classical Lagrangian system were simplified into a set of ordinary differential equations through introduction of an unknown vector composed of the fundamental variables and their dual variables. The free vibration of DNBSs was finally reduced to an eigenproblem in the symplectic space. Analytical frequency equations and vibration mode functions were directly obtained with the symplectic eigensolutions and boundary conditions. Numerical results verify the accuracy and efficiency of the presented method. A systematic parametric study on the small size effect,the interaction between the double nanobeams and the viscoelastic foundation influence,was also provided.

关 键 词：HAMILTON体系 辛方法 双功能梯度纳米梁系统 自由振动 解析解 

分 类 号：O326[理学—一般力学与力学基础]