AN APPLICATION OF DIFFERENTIAL TRANSFORMATION TO STABILITY ANALYSIS OF HEAVY COLUMNS

This paper uses a recently developed technique, known as the differential transformation, to determine the critical buckling load of axially compressed heavy columns of various support conditions. In solving the problem, it is shown that the differential transformation technique converts the governing differential equation into an algebraic recursive equation, which must be solved together with the differential transformation of the boundary conditions. Although a fairly large number of terms are required for convergence of the solution, the differential transformation method is nonetheless efficient and fairly easy to implement. The method is also shown to be very accurate when compared with a known analytical solution. The stability of heavy columns is further examined using approximate formulae currently available in the literature. In this case, the differential transformation method offers a reference for assessing the accuracy of the approximate buckling formulae.

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Axial Vibration Analysis of Stepped Bar by Differential Transformation Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.419.273 ◽

2013 ◽

Vol 419 ◽

pp. 273-279 ◽

Cited By ~ 5

Author(s):

Zhen Gang Wang

Keyword(s):

Closed Form Solution ◽

Transformation Method ◽

Form Solution ◽

Differential Transformation Method ◽

Algebraic Equations ◽

Axial Vibration ◽

Differential Transformation ◽

Characteristics Analysis ◽

Governing Differential Equation ◽

Dynamic Characteristics Analysis

Stepped distributed dynamic systems are widely used in the engineering fields, and the dynamic characteristics analysis of them is very important. In this paper, the axial vibration of a stepped bar consisting of two uniform sections is studied, in order to solve the dynamic equation, the differential transformation method is used, the governing differential equation and the boundary conditions of the bar become simple algebraic equations. Doing some simple algebraic operations for these equations, the closed form solution of natural frequency, mode shape and the dynamic response can be obtained. Comparison the results obtained by the differential transformation method and finite element method, excellent agreement is achieved, and the effects of the stiffness of spring is discussed in this paper.

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Lateral-Torsional Buckling of Nonuniformly Loaded Beam Using Differential Transformation Method

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455415500340 ◽

2016 ◽

Vol 16 (07) ◽

pp. 1550034 ◽

Cited By ~ 6

Author(s):

Ryszard Hołubowski ◽

Kamila Jarczewska

Keyword(s):

Critical Loads ◽

Transformation Method ◽

Variable Coefficients ◽

Differential Transformation Method ◽

Lateral Torsional Buckling ◽

Torsional Buckling ◽

Differential Transformation ◽

Distributed Load ◽

The Difference ◽

The Stability

The paper presents the application of the differential transformation method (DTM) to the stability analysis of nonuniformly loaded beams under bending. The main advantage of the method is the possibility to obtain the semi-analytical solution for the critical loads of the beam undergoing lateral-torsional buckling. To determine the critical load, the system of two coupled ordinary differential equations with variable coefficients and parameters have to be solved. Numerical analyses were carried out for three different types of load functions: (i) Uniformly distributed load, (ii) linearly varying load and (iii) sinusoidally distributed load. The results were compared with the solutions computed by the finite element method (FEM) and those obtained by the authors using the variational iterative method (VIM). In each case, it was found that the difference with reference to the existing one does not exceed 3%, which testifies the effectiveness of the DTM used. Nevertheless, it should be emphasized that the number of terms of the approximation series used is fairly large and therefore the calculation of higher critical loads can be very time-consuming.

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Large Deflection of Tip Loaded Beam with Differential Transformation Method

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.250-253.1232 ◽

2011 ◽

Vol 250-253 ◽

pp. 1232-1235 ◽

Cited By ~ 3

Author(s):

Yi Xiao

Keyword(s):

Differential Equation ◽

Finite Difference ◽

Finite Difference Method ◽

Large Deflection ◽

Transformation Method ◽

Differential Transformation Method ◽

Differential Transformation ◽

The Finite Difference Method ◽

Large Deflection Problem ◽

Constant Section

This paper deals with large deflection problem of a cantilever beam with a constant section under the action of a transverse tip load. The differential transformation method (DTM) is used to solve the nonlinear differential equation governing the problem. An approach treats trigonometric nonlinearity is used in DTM. The results obtained from DTM are compared with those results obtained by the finite difference method and they agree well.

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Differential Transformation Method

Advances in Mechatronics and Mechanical Engineering - Numerical and Analytical Solutions for Solving Nonlinear Equations in Heat Transfer ◽

10.4018/978-1-5225-2713-8.ch006 ◽

2017 ◽

pp. 140-172

Keyword(s):

Heat Transfer ◽

Differential Equation ◽

Nonlinear Equation ◽

Nonlinear Differential Equation ◽

Transformation Method ◽

Differential Transformation Method ◽

Differential Transformation

In this chapter, a new linearization procedure based on Differential Transformation Method (DTM) will be presented. The procedure begins with solving nonlinear differential equation by DTM. The effectiveness of the procedure is verified using a heat transfer nonlinear equation. The simulation result shows the significance of the proposed technique.

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ON A NEW DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500575 ◽

2013 ◽

Vol 06 (04) ◽

pp. 1350057 ◽

Cited By ~ 1

Author(s):

Abdelhalim Ebaid

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Nonlinear Term ◽

Nonlinear Differential Equations ◽

Mathematical Structure ◽

Transformation Method ◽

Differential Transformation Method ◽

Differential Transformation ◽

Computational Budget ◽

Symbolic Formula

The main difficulty in solving nonlinear differential equations by the differential transformation method (DTM) is how to treat complex nonlinear terms. This method can be easily applied to simple nonlinearities, e.g. polynomials, however obstacles exist for treating complex nonlinearities. In the latter case, a technique has been recently proposed to overcome this difficulty, which is based on obtaining a differential equation satisfied by this nonlinear term and then applying the DTM to this obtained differential equation. Accordingly, if a differential equation has n-nonlinear terms, then this technique must be separately repeated for each nonlinear term, i.e. n-times, consequently a system of n-recursive relations is required. This significantly increases the computational budget. We instead propose a general symbolic formula to treat any analytic nonlinearity. The new formula can be easily applied when compared with the only other available technique. We also show that this formula has the same mathematical structure as the Adomian polynomials but with constants instead of variable components. Several nonlinear ordinary differential equations are solved to demonstrate the reliability and efficiency of the improved DTM method, which increases its applicability.

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Study of Properties of Differential Transform Method for Solving the Linear Differential Equation

Asian Journal of Engineering and Applied Technology ◽

10.51983/ajeat-2019.8.2.1138 ◽

2019 ◽

Vol 8 (2) ◽

pp. 50-56

Author(s):

Nandita Das

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Linear Equations ◽

Transformation Method ◽

Differential Transformation Method ◽

Transform Method ◽

Differential Transformation ◽

Differential Transform ◽

Linear Differential ◽

Non Linear Equations

The differential transformation method (DTM) is an alternative procedure for obtaining an analytic Taylor series solution of differential linear and non-linear equations. However, the proofs of the properties of equation have been long ignored in the DTM literature. In this paper, we present an analytical solution for linear properties of differential equations by using the differential transformation method. This method has been discussed showing the proof of the equation which are presented to show the ability of the method for linear systems of differential equations. Most authors assume the knowledge of these properties, so they do not bother to prove the properties. The properties are therefore proved to serve as a reference for any work that would want to use the properties without proofs. This work argues that we can obtain the solution of differential equation through these proofs by using the DTM. The result also show that the technique introduced here is accurate and easy to apply.

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