Frequency Analysis of an Arbitrarily Supported, Exponentially Tapered Beam

• Published in: International journal of mechanical engineering education 2013-07, Vol.41 (3), p.252-268
• Main Author: Cha, Philip D.
• Format: Article
• Language: English
• Subjects: Analysis; Frequencies; Structural engineering; Vibration; Vibration analysis
• ISSN: 0306-4190; 2050-4586
• DOI: 10.7227/IJMEE.41.3.7

In most introductory vibration textbooks, the vibration of non-uniform beams is not covered. In this paper, the free vibration of an arbitrarily supported Euler–Bernoulli beam of constant thickness and exponentially varying width is investigated. The ‘assumed modes‘ method is utilized to obtain the approximate natural frequencies of the tapered beam, where the basis functions used in the expansion correspond to the eigenfunctions of the uniform beam, including its rigid-body modes. Expressing the eigenfunctions in terms of exponentials, the generalized masses and stiffnesses for the non-uniform beam can all be obtained in closed form, which substantially simplifies the analysis. The proposed scheme is simple to code, efficient to apply and can be easily modified to accommodate a tapered beam with various boundary conditions. Additionally, with slight modifications, the proposed method can also be used to analyze the free vibration of a combined system consisting of an exponentially varying beam carrying lumped attachments. Numerical experiments validate the present approach, and show that the approximate natural frequencies converge very quickly to the exact solution.