A Spectral Taylor Polynomial Solution of Euler-Bernoulli Beam Equation by a Matrix Approach

Year 2025, Volume: 1 Issue: 1, 1 - 12, 28.05.2025

Özer Tatar , Muhammet Mustafa Bahşı , Mehmet Çevik

Abstract

The Euler-Bernoulli beam theory, widely applied in structural engineering, describes the relationship between applied loads and resulting deformations in beams. This study addresses the transverse vibration of beams with simply supported, cantilever, and fixed-fixed boundary conditions using the Taylor matrix method. The governing differential equation, which includes both boundary and initial conditions, is transformed into a matrix form through Taylor series expansion. This matrix approach simplifies the process of solving the Euler-Bernoulli equation, providing an efficient method for analyzing beam vibrations under various support conditions. The accuracy of the Taylor matrix method is validated by comparing it with exact solutions derived through the separation of variables. Numerical examples illustrate that the method yields results closely aligning with the exact solutions, with minimal discrepancies that decrease as the number of terms N in the Taylor series expansion increases. This method shows promise as an accessible and accurate approach for studying mechanical vibrations, especially in engineering applications requiring efficient computational techniques. The findings contribute to the literature on beam theory and demonstrate the applicability of the Taylor matrix method in structural analysis.

Keywords

Taylor matrix method , vibration , Euler-Bernoulli beam , spectral method

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