Free Vibration Analysis of Non-Uniform Timoshenko Beams Using Haar Wavelets

Abstract

In this paper, we present a method using Haar wavelets for solving axially functionally graded (FG) Timoshenko beam equations with non-uniform cross-sections. We compare two different approaches to the solution. The first approach involves approximating the resulting function (i.e., rotation) of the differential equation with a polynomial function using Haar wavelets, which is a classical application in the Haar wavelet method. The second method employs an auxiliary function that uses Haar wavelets, but the rotation or deflection do not directly equal the sought function. The rotation and deflection are derived from this auxiliary function. In both methods, the coupled governing equations are transformed into a single governing equation. Using the Haar wavelet method, this single differential equation transforms into a system of linear algebraic equations. For different boundary conditions, the roots of the characteristic polynomial equation obtained from the system of linear algebraic equations are solved to determine the lowest to highest-order natural frequencies. Our results show that the use of auxiliary function is faster and more consistent with the results available in the literature. We presented several natural frequency predictions of Timoshenko beams with different taper ratios and support conditions that have not been detailed in the literature before.