Free vibration analysis of non-uniform Bernoulli beam by using Laplace Adomian decomposition method

Abstract

This paper discusses an investigation into the influence of the cone ratio and axial force on the vibration problem in a non-uniform cantilever Euler–Bernoulli beam. In the analysis, the governing equation for the non-uniform cantilever Euler–Bernoulli beam is solved using the Laplace Adomian decomposition method (LADM). The LADM is used to convert the governing equation into a characteristic equation of a non-uniform Euler–Bernoulli beam, and a simple calculation is performed to obtain the natural frequencies and corresponding modals. The obtained numerical results are verified using a comparison with the results reported in previous studies. The present study speeds up the convergent rate and the accuracy of calculation by comparing the results using the modified Adomian decomposition method (MADM) and differential transformation method (DTM). The main power and advantage of the LADM are providing an analytical approximation to a nonlinear differential equation without linearization, perturbation, approximation, and discretization, all of which lead to huge numerical computation. The numerical methods demonstrated that the natural frequency increases with increasing the rotating spring modulus and moving spring modulus, and the moving spring modulus has a greater influence on the natural frequency. The effects of the cone ratio and axial force are presented for non-uniform Euler–Bernoulli beams. The numerical results show that the LADM is a suitable technique for analyzing the behavioral characteristics of beams.