双功能梯度纳米梁系统振动分析的辛方法

周震寰; 李月杰; 范俊海; 隋国浩; 张俊霖; 徐新生

doi:10.21656/1000-0887.390130

双功能梯度纳米梁系统振动分析的辛方法

doi: 10.21656/1000-0887.390130

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

基金项目: 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）

详细信息

作者简介:
周震寰（1983—），男，副教授，博士(通讯作者. E-mail: zhouzh@dlut.edu.cn）.

中图分类号: O326
计量
- 文章访问数: 904
- HTML全文浏览量: 108
- PDF下载量: 659
- 被引次数: 0
出版历程
- 收稿日期: 2018-04-23
- 修回日期: 2018-05-18
- 刊出日期: 2018-10-01

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

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

Funds: The National Natural Science Foundation of China(11672054); The National Basic Research Program of China(973 Program)(2014CB046803); The National Key R&D Program of China(2016YFB0201600)

摘要

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.
- Hamilton体系 /
- 辛方法 /
- 双功能梯度纳米梁系统 /
- 自由振动 /
- 解析解
Abstract: A new analytical approach was proposed for free vibration of functionally graded (FG) double-nanobeam 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.
- Hamiltonian system /
- symplectic method /
- functionally graded double-nanobeam system /
- free vibration /
- analytical solution

HTML全文

参考文献(25)

[1]	EICHENFIELD M, CAMACHO R, CHAN J, et al. A picogram and nanometre-scale photonic-crystal optomechanical cavity[J]. Nature,2009,459(7246): 550-555.
[2]	LIN Q, ROSENBERG J, CHANG D, et al. Coherent mixing of mechanical excitations in nano-optomechanical structures[J]. Nature Photonics,2010,4(4): 236-242
[3]	FU Y, DU H, HUANG W, et al. TiNi-based thin films in MEMS applications: a review[J]. Sensors and Actuators A: Physical,2004,112(2): 395-408.
[4]	KAHROBAIYAN M H, ASGHARI M, RAHAEIFARD M, et al. Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory[J]. International Journal of Engineering Science,2010,48(12): 1985-1994.
[5]	ERINGEN A C. On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface-waves[J]. Journal of Applied Physics,1983,54(9): 4703-4710.
[6]	ERINGEN A C. Nonlocal Continuum Field Theories [M]. New York: Springer, 2002.
[7]	RAFII-TABAR H, GHAVANLOO E, FAZELZADEH S A. Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures[J]. Physics Reports,2016,638: 1-97.
[8]	林 C W. 基于非局部弹性应力场理论的纳米尺度效应研究: 纳米梁的平衡条件、控制方程以及静态挠度[J]. 应用数学和力学,2010,31(1): 35-50.(LIM C W. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection[J]. Applied Mathematics and Mechanics,2010,31(1): 35-50. (in Chinese))
[9]	尹春松, 杨洋. 考虑非局部剪切效应的碳纳米管弯曲特性研究[J]. 应用数学和力学, 2015,36(6): 600-606.(YIN Chunsong, YANG Yang. Shear deformable bending of carbon nanotubes based on a new analytical nonlocal Timoshenko beam model[J]. Applied Mathematics and Mechanics,2015,36(6): 600-606.(in Chinese))
[10]	ELTAHER M A, EMAM S A, MAHMOUD F F. Free vibration analysis of functionally graded size-dependent nanobeams[J]. Applied Mathematics and Computation,2012,218(14): 7406-7420.
[11]	LI L, LI X, HU Y. Free vibration analysis of nonlocal strain gradient beams made of functionally graded material[J]. International Journal of Engineering Science,2016,102: 77-92.
[12]	RAHMANI O, PEDRAM O. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory[J]. International Journal of Engineering Science,2014,77(7): 55-70.
[13]	ELTAHER M A, ALSHORBAGY A E, MAHMOUD F F. Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams[J]. Composite Structures,2013,99(5): 193-201.
[14]	NIKNAM H, AGHDAM M M. A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation[J]. Composite Structures,2015,119: 452-462.
[15]	EL-BORGI S, FERNANDES R, REDDY J N. Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation[J]. International Journal of Non-Linear Mechanics,2015,77: 348-363.
[16]	NAZEMNEZHAD R, HOSSEINI-HASHEMI S. Nonlocal nonlinear free vibration of functionally graded nanobeams[J]. Composite Structures,2014,110: 192-199.
[17]	ELTAHER M A, KHAIRY A, SADOUN A M, et al. Static and buckling analysis of functionally graded Timoshenko nanobeams[J]. Applied Mathematics and Computation,2014,229: 283-295.
[18]	LI L, HU Y. Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material[J]. International Journal of Engineering Science,2016,107: 77-97.
[19]	KE L L, WANG Y S, YANG J, et al. Nonlinear free vibration of size-dependent functionally graded microbeams[J]. International Journal of Engineering Science,2012,50(1): 256-267.
[20]	杨晓东, 林志华. 利用多尺度方法分析基于非局部效应纳米梁的非线性振动[J]. 中国科学: 技术科学, 2010,40(2): 152-156.(YANG Xiaodong, LIN Zhihua. The nonlinear vibration of nanoscale beam based on nonlocal effect by multiscale method[J]. Chinese Science: Technical Science,2010,40(2): 152-156. (in Chinese))
[21]	YAO W A, ZHONG W X, LIM C W. Symplectic Elasticity [M]. Hackensack: World Scientific Pubinshing Company, 2009.
[22]	LIM C W, XU X S. Symplectic elasticity: theory and applications[J]. Applied Mechanics Reviews,2011,63(5): 050802.
[23]	ZHOU Z, RONG D, YANG C, et al. Rigorous vibration analysis of double-layered orthotropic nanoplate system[J]. International Journal of Mechanical Sciences,2017,123: 84-93.
[24]	AYDOGDU M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration[J].Physica E: Low-Dimensional Systems and Nanostructures,2009,41(9): 1651-1655.
[25]	ARASH B, WANG Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes[J]. Computational Materials Science,2012,51(1): 303-313.