Investigation of free vibration and buckling of Timoshenko nano-beam based on a general form of eringen theory using conformable fractional derivative and Galerkin method

Q2 Engineering

Engineering Transactions Pub Date : 2019-04-26 DOI:10.24423/ENGTRANS.1001.20190426

F. Mohammadi, Z. Rahimi, W. Sumelka, Xiao-jun Yang

{"title":"Investigation of free vibration and buckling of Timoshenko nano-beam based on a general form of eringen theory using conformable fractional derivative and Galerkin method","authors":"F. Mohammadi, Z. Rahimi, W. Sumelka, Xiao-jun Yang","doi":"10.24423/ENGTRANS.1001.20190426","DOIUrl":null,"url":null,"abstract":"The purpose of this paper is to study the free vibration and buckling of a Timoshenko \nnano-beam using the general form of the Eringen theory generalized based on the fractional derivatives. \n \nIn this paper, using the conformable fractional derivative (CFD) definition the generalized form of the Eringen nonlocal theory (ENT) is used to consider the effects of integer and noninteger stress gradients in the constitutive relation and also to consider small-scale effect in the vibration of a Timoshenko nano-beam. The governing equation is solved by the Galerkin method. \n \nFree vibration and buckling of a Timoshenko simply supported (S) nano-beam is investigated, and the influence of the fractional and nonlocal parameters is shown on the frequency ratio and buckling ratio. In this sense, the obtained formulation allows for an easier mapping of experimental results on nano-beams. \n \nThe new theory (fractional parameter) makes the modeling more flexible. The model can conclude all of the integer and non-integer operators and is not limited to the special operators such as ENT. In other words, it allows to use more sophisticated/flexible mathematics to model physical phenomena.","PeriodicalId":38552,"journal":{"name":"Engineering Transactions","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Engineering Transactions","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.24423/ENGTRANS.1001.20190426","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Engineering","Score":null,"Total":0}

引用次数: 6

Abstract

The purpose of this paper is to study the free vibration and buckling of a Timoshenko nano-beam using the general form of the Eringen theory generalized based on the fractional derivatives. In this paper, using the conformable fractional derivative (CFD) definition the generalized form of the Eringen nonlocal theory (ENT) is used to consider the effects of integer and noninteger stress gradients in the constitutive relation and also to consider small-scale effect in the vibration of a Timoshenko nano-beam. The governing equation is solved by the Galerkin method. Free vibration and buckling of a Timoshenko simply supported (S) nano-beam is investigated, and the influence of the fractional and nonlocal parameters is shown on the frequency ratio and buckling ratio. In this sense, the obtained formulation allows for an easier mapping of experimental results on nano-beams. The new theory (fractional parameter) makes the modeling more flexible. The model can conclude all of the integer and non-integer operators and is not limited to the special operators such as ENT. In other words, it allows to use more sophisticated/flexible mathematics to model physical phenomena.

查看原文本刊更多论文

基于广义eringen理论的Timoshenko纳米梁自由振动和屈曲的保形分数导数和Galerkin方法研究

利用基于分数阶导数的广义Eringen理论的一般形式，研究了Timoshenko纳米梁的自由振动和屈曲问题。本文利用合形分数阶导数(CFD)定义，利用广义的Eringen非局部理论(ENT)来考虑整数和非整数应力梯度在本构关系中的影响，并考虑Timoshenko纳米梁振动中的小尺度效应。用伽辽金法求解控制方程。研究了Timoshenko简支纳米梁的自由振动和屈曲，得到了分数阶参数和非局部参数对频率比和屈曲比的影响。从这个意义上说，所获得的公式允许更容易地映射纳米光束的实验结果。新的理论(分数参数)使建模更加灵活。该模型可以归纳出所有的整数和非整数运算符，并且不局限于特殊运算符，如ENT。换句话说，它允许使用更复杂/灵活的数学来模拟物理现象。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Engineering Transactions Engineering-Engineering (all)

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： Engineering Transactions (formerly Rozprawy Inżynierskie) is a refereed international journal founded in 1952. The journal promotes research and practice in engineering science and provides a forum for interdisciplinary publications combining mechanics with: Material science, Mechatronics, Biomechanics and Biotechnologies, Environmental science, Photonics, Information technologies, Other engineering applications. The journal publishes original papers covering a broad area of research activities including: experimental and hybrid techniques, analytical and numerical approaches. Review articles and special issues are also welcome. Following long tradition, all articles are peer reviewed and our expert referees ensure that the papers accepted for publication comply with high scientific standards. Engineering Transactions is a quarterly journal intended to be interesting and useful for the researchers and practitioners in academic and industrial communities.