A fibonacci polynomial-based numerical approach for modal analyses of Euler–Bernoulli, Rayleigh, and Timoshenko Beams

IF 2.5 3区 工程技术 Q2 MECHANICS
Seda Çayan, B. Burak Özhan, Mehmet Sezer
{"title":"A fibonacci polynomial-based numerical approach for modal analyses of Euler–Bernoulli, Rayleigh, and Timoshenko Beams","authors":"Seda Çayan,&nbsp;B. Burak Özhan,&nbsp;Mehmet Sezer","doi":"10.1007/s00419-025-02915-3","DOIUrl":null,"url":null,"abstract":"<div><p>This study presents an enhanced matrix collocation method based on Fibonacci polynomials for free vibration problems of Euler–Bernoulli, Rayleigh, and Timoshenko beams. Firstly, governing equations of the beams are reduced to fourth-order ordinary differential equations in spatial coordinates. Then, these equations are transformed into a fundamental matrix equation through the equally spaced collocation points and operational matrices. Thereby, using the Fibonacci matrix collocation method along with the eigenvalue problem, the approximate solutions are obtained in terms of the truncated Fibonacci series. These solutions correspond to the natural frequencies and modal shape functions. Also, some examples, together with relative error, are performed to illustrate the validity and applicability of the presented method. Solving the eigenvalue problem, the natural frequencies are obtained for simple–simple and clamped-free support conditions for each beam model. In addition, normalized modal shape functions corresponding to the natural frequencies are plotted. The obtained results are compared with the existing results in the literature. Moreover, the obtained numerical results are scrutinized by using tables and figures.</p></div>","PeriodicalId":477,"journal":{"name":"Archive of Applied Mechanics","volume":"95 9","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive of Applied Mechanics","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s00419-025-02915-3","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0

Abstract

This study presents an enhanced matrix collocation method based on Fibonacci polynomials for free vibration problems of Euler–Bernoulli, Rayleigh, and Timoshenko beams. Firstly, governing equations of the beams are reduced to fourth-order ordinary differential equations in spatial coordinates. Then, these equations are transformed into a fundamental matrix equation through the equally spaced collocation points and operational matrices. Thereby, using the Fibonacci matrix collocation method along with the eigenvalue problem, the approximate solutions are obtained in terms of the truncated Fibonacci series. These solutions correspond to the natural frequencies and modal shape functions. Also, some examples, together with relative error, are performed to illustrate the validity and applicability of the presented method. Solving the eigenvalue problem, the natural frequencies are obtained for simple–simple and clamped-free support conditions for each beam model. In addition, normalized modal shape functions corresponding to the natural frequencies are plotted. The obtained results are compared with the existing results in the literature. Moreover, the obtained numerical results are scrutinized by using tables and figures.

基于fibonacci多项式的Euler-Bernoulli, Rayleigh和Timoshenko梁模态分析数值方法
针对Euler-Bernoulli、Rayleigh和Timoshenko梁的自由振动问题,提出了一种基于Fibonacci多项式的改进矩阵配置方法。首先,将梁的控制方程简化为空间坐标系下的四阶常微分方程。然后,通过等间距配点和运算矩阵将这些方程转化为基本矩阵方程。因此,利用Fibonacci矩阵搭配法结合特征值问题,得到了截断Fibonacci级数的近似解。这些解对应于固有频率和模态振型函数。通过算例和相对误差说明了所提方法的有效性和适用性。通过求解特征值问题,得到了各梁模型在简简支承和无夹固支承条件下的固有频率。此外,绘制了相应于固有频率的归一化模态振型函数。所得结果与已有文献结果进行了比较。此外,还用表格和图表对得到的数值结果进行了检验。
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
CiteScore
4.40
自引率
10.70%
发文量
234
审稿时长
4-8 weeks
期刊介绍: Archive of Applied Mechanics serves as a platform to communicate original research of scholarly value in all branches of theoretical and applied mechanics, i.e., in solid and fluid mechanics, dynamics and vibrations. It focuses on continuum mechanics in general, structural mechanics, biomechanics, micro- and nano-mechanics as well as hydrodynamics. In particular, the following topics are emphasised: thermodynamics of materials, material modeling, multi-physics, mechanical properties of materials, homogenisation, phase transitions, fracture and damage mechanics, vibration, wave propagation experimental mechanics as well as machine learning techniques in the context of applied mechanics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信