Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams

Abstract

The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparison published results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beams with arbitrary boundary conditions.

Free Vibrations with Large Amplitude of Axially Loaded Beams on an Elastic Foundation Using the Adomian Modified Decomposition Method

“…e terms in equation (23) are linear except for the nonlinear Fredholm integral coefficient shown in equation (24). Before continuing with the main solution method, the nonlinear term will be treated first through the use of appropriate Cauchy products.…”

“…In the present work, the adomian modified decomposition method [21,22] is utilized to calculate free transverse vibration characteristics of axially loaded Euler-Bernoulli beams with various end restrains, resting on a Winkler one-parameter foundation. e method is chosen as it has proved efficient and accurate [23,24] for solving linear and nonlinear differential equations, and it has the advantage of computational simplicity. In addition, it does not involve linearization, discretization, perturbation, or a priori assumptions, which may alter the physics of the problem considered [21].…”

Free Vibrations of Spinning Beams Under Nonclassical Boundary Conditions Using Adomian Modified Decomposition Method

“…The AMDM has been applied to the free vibration problems for beam structures, and the method has furnished reliable results in providing analytical approximation that converges rapidly. [11][12][13][14][15] By using the AMDM, the governing di®erential equations for the spinning beam become a recursive algebraic equation system. The boundary conditions become simple algebraic frequency equations that are suitable for symbolic computations.…”

“…into Eq. (28), C m ðm ¼ 0; 1; 2; 3Þ can be obtained, then using Eq (15),. all other coe±cient vectors C m ðm !…”

Simulation of the vibrations of a non-uniform beam loaded with both a transversely and axially eccentric tip mass

“…Specific to this work, the Adomain decomposition and Adomian modified decomposition method have been used by several groups [14,15] for uniform and non-uniform beams, starting with either the Euler-Bernoulli or Timoshenko formulations. Mao [14] applied the AMDM to rotating uniform beams and included a centrifugal stiffening term while Adair and Jaeger [15] applied the AMDM to rotating non-uniform beams which also included a centrifugal stiffening term. Yaman [16] has used the Adomian decomposition method to investigate the influence of the orientation effect on the natural frequency of a cantilever beam carrying a tip mass.…”