In this paper, we present an efficient numerical algorithm for approximate solutions of higher-order boundary value problems with two-point boundary conditions A modified form of the Adomian decomposition method will be implemented to construct such solutions The approach provides the solution in the form of a rapidly convergent series The analysis is accompanied by numerical examples The obtained results demonstrate reliability and efficiency of the proposed scheme

Approximate solutions to boundary value problems of higher order by the modified decomposition method

In this paper, we apply the variational iteration method using He's polynomials (VIMHP) for solving the higher-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed VIMHP solves nonlinear problems without using the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

https://www.degruyter.com/view/j/ijnsns.2008.9.2/ijnsns.2008.9.2.141/ijnsns.2008.9.2.141.pdf

Variational Iteration Method for Solving Higher-order Nonlinear Boundary Value Problems Using He's Polynomials

Coupled bending-torsional dynamic stiffness matrix for timoshenko beam elements

Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping

This paper presents a free-vibration analysis of simply supported, cross-ply beams via several higher-order as well as classical theories. The three-dimensional displacement field is approximated along the beam cross-section in a compact form as a generic N-order polynomial expansion. Several higher-order displacements-based theories accounting for non-classical effects can be, therefore, formulated straightforwardly. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the principle of virtual displacements. Thanks to the compact form of the displacement field approximation, governing equations are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations. Besides the fundamental natural frequency, natural frequencies associated to higher modes (such as torsional, axial, shear and mixed ones) are investigated. A half waves number equal to one is considered. The effect of the length-to-thickness ratio, lamination, aspect ratio and material properties on: (1) the accuracy of the proposed theories and (2) the natural frequencies and modes is presented and discussed. For the latter case, the modes change in order of appearance (modes swapping) and in shape (modes mutation) is investigated. Results are assessed towards three-dimensional FEM solutions. Numerical results show that, upon the choice of the appropriate approximation order, very accurate results can be obtained for all the considered modes.

S.M. Cannon

Papers

On coupled bending and torsional vibration of uniform beams