{"title":"A fibonacci polynomial-based numerical approach for modal analyses of Euler–Bernoulli, Rayleigh, and Timoshenko Beams","authors":"Seda Çayan, B. Burak Özhan, 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.
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.