On the analysis of axially loaded Euler-Bernoulli beam using Galerkin, Fourier transform, and Lie symmetry approach

Abstract

The beam with axial load is one of the fundamental models in the field of physics and engineering. The applications of this model span from a guitar string to the rotating blade of a helicopter. It has an important role in vibration theory and structural mechanics. Also, it establishes a relationship between the mathematical continuum concept and continuum mechanics as it is the most basic example of a continuum physical model. For the long and slender beams, the Euler-Bernoulli beam theory is used. In this thesis, Euler-Bernoulli beam with axial load is studied. For the wide application in industry, an accurate and simple closed form solution for the general problem is highly desirable. Here, general problem implies a beam with variable stiffness, variable mass, and variable axial load. The governing equation of motion is a partial differential equation of order four in the spatial variable and of order two in the temporal variable. However, there is a deficiency in the research on exact solutions of the above mentioned model. The modern numerical methods such as finite element methods can give an approximate solution but it requires huge computer time and sometimes extensive coding. In this thesis, we look at some basic methods such as the Galerkin method and Fourier transform method. We also apply Lie symmetry method to extract the closed form solution. There are three parts in the thesis. In Part I, the Galerkin method is written; in Part II, the Fourier transform method is discussed and finally, in Part III, the Lie method is given