# Application of Differential Transform Method in Free Vibration Analysis of Rotating Non-Prismatic Beams

01 Jan 2008-

TL;DR: In this paper, free vibration of non-prismatic rotating Euler-Bernoulli beams is studied by using differential transform method, a powerful numerical tool in solution of ordinary differential equations, for solving the governing equation of motion.

AbstractRotating beams are considerably used in different mechanical and aeronautical installations. In this paper, free vibration of non-prismatic rotating Euler-Bernoulli beams is studied. Dynamic stiffness matrix is evaluated by using differential transform method, a powerful numerical tool in solution of ordinary differential equations, for solving the governing equation of motion. The method is capable of modeling any beam whose cross-sectional area and moment of inertia vary along beam with any two arbitrary functions and any type of cross-section with just one or few elements so that it can be used in most of engineering applications. In order to verify the competency of the method, natural frequencies are obtained for two problems and the effects of rotational speed parameter and taper ratio on natural frequencies are investigated.

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TL;DR: In this article, the vibration response of non-homogenous and non-uniform microbeams is investigated in conjunction with Bernoulli-Euler beam and modified couple stress theory, where boundary conditions of the microbeam are considered as fixed at one end and free at the other end.
Abstract: In the present study, vibration response of non-homogenous and non-uniform microbeams is investigated in conjunction with Bernoulli–Euler beam and modified couple stress theory. The boundary conditions of the microbeam are considered as fixed at one end and free at the other end. It is taken into consideration that material properties and cross section of the microbeam vary continuously along the longitudinal direction. Rayleigh–Ritz solution method is utilized to obtain an approximate solution to the free transverse vibration problem. A detailed study is carried out to show the effects of material properties and taper ratios on natural frequencies of axially functionally graded tapered microbeams. In order to demonstrate the validity and accuracy of the current analysis, some of present results are compared with previous results in the literature and an excellent agreement is observed between them.

302 citations

Journal ArticleDOI
TL;DR: In this paper, the free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams were studied through solving the governing differential equations of motion. But, the convergence rate of the conventional differential transform method (DTM) does not necessarily converge to satisfactory results, and a new approach based on DTM called differential transform element method (DTEM) is introduced which considerably improves the convergence of the method.
Abstract: The free vibration and stability of axially functionally graded tapered Euler–Bernoulli beams are studied through solving the governing differential equations of motion. Observing the fact that the conventional differential transform method (DTM) does not necessarily converge to satisfactory results, a new approach based on DTM called differential transform element method (DTEM) is introduced which considerably improves the convergence rate of the method. In addition to DTEM, differential quadrature element method of lowest-order (DQEL) is used to solve the governing differential equation, as well. Carrying out several numerical examples, the competency of DQEL and DTEM in determination of free longitudinal and free transverse frequencies and critical buckling load of tapered Euler–Bernoulli beams made of axially functionally graded materials is verified.

176 citations

Journal ArticleDOI
TL;DR: In this article, the free bending vibration of rotating axially functionally graded (FG) Timoshenko tapered beams (TTB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL).
Abstract: The free bending vibration of rotating axially functionally graded (FG) Timoshenko tapered beams (TTB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL). These two methods are capable of modelling any beam whose cross sectional area, moment of inertia and material properties vary along the beam. In order to verify the competency of these two methods, natural frequencies are obtained for problems by considering the effect of material non-homogeneity, taper ratio, shear deformation parameter, rotating speed parameter, hub radius and tip mass. The results are tabulated and compared with the previous published results wherever available.

84 citations

Journal ArticleDOI
TL;DR: In this article, the free bending vibration of rotating axially functionally graded (FG) Euler-Bernoulli tapered beams (ETB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL).
Abstract: The free bending vibration of rotating axially functionally graded (FG) Euler–Bernoulli tapered beams (ETB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL). These two methods are capable of modeling any beam whose cross sectional area and moment of inertia vary along the beam. In order to verify the competency of these two methods, natural frequencies are obtained for problems by considering the effect of material non-homogeneity, taper ratio, rotating speed parameter, hub radius and tip mass. The results are tabulated and compared with the previous published results.

66 citations

Journal ArticleDOI
TL;DR: In this article, the free vibration analysis of rotating tapered beams is studied from a mechanical point of view, and exact shape functions are derived in terms of Basic Displacement Functions (BDFs).
Abstract: This paper deals with enhancing the existing Finite Element formulations through employing basic principles of structural mechanics accompanied with mathematical techniques. Introducing the concept of Basic Displacement Functions (BDFs), the free vibration analysis of rotating tapered beams is studied from a mechanical point of view. It is shown that exact shape functions could be derived in terms of BDFs. The new shape functions turn out to be dependent on the rotational speed, circular frequency, the position of element along the beam and variation of cross-sectional dimensions along the element. Dynamic BDFs are obtained by applying Adomian Modified Decomposition Method (AMDM) to the governing differential equation of motion. Carrying out numerical examples, the competency of the method is verified.

38 citations

##### References
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TL;DR: This book describes the development of a dynamic method for earthquake engineering equivolent staic lateral force method - uniform Building Code 1994 dynamic method - Uniform Building Code - 1994.
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TL;DR: In this paper, structural analysis of a single-degree-of-freedom system undamped single degree of freedom system response to haarmonic loading response to general dynamic loading Fourier analysis and response in the frequency domain generalized coordinates and Rayleigh's method nonlinear structural response response spectra.
Abstract: Part I: structures modelled as a single degree-of-freedom system undamped single degree-of-freedom system damped single degree-of-freedom system response of one-degree-of-freedom system to haarmonic loading response to general dynamic loading Fourier analysis and response in the frequency domain generalized co-ordinates and Rayleigh's method nonlinear structural response response spectra. Part II: structures modelled as shear buildings the multistory shear building free vibration of a shear building forced motion of shear buildings damped motion of shear buildings reduction of dynamic matrices. Part III: framed structures modeled as discrete multidegree-of-freedom systems dynamic analysis of beams dynamic analysis of plane frames dynamic anaylsis of grids three-dimensional frames dynamic analysis of trusses time history response of multidegree-of-freedom systems. Part IV: structures modelled with distributed properties dynamic analysis of systems with distributed properties discretization of continuous systems. Part V: introduction to finite element method dynamic analysis of plates dynamic analysis of shells dynamic analysis of three-dimensional solid structures. Part VI: random vibration. Part VII: earthquake engineering equivolent staic lateral force method - uniform Building Code 1994 dynamic method - uniform Building Code - 1994.

460 citations

Journal ArticleDOI
TL;DR: Three-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time and exact solutions of linear and non-linear systems of partial differential equations have been investigated.
Abstract: In this study, three-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time. Moreover, as an application of two and three-dimensional differential transform, exact solutions of linear and non-linear systems of partial differential equations have been investigated. The results of the present method are compared very well with those obtained by decomposition method. Differential transform method can easily be applied to linear or non-linear problems and reduces the size of computational work. With this method exact solutions may be obtained without any need of cumbersome work and it is an useful tool for analytical and numerical solutions.

361 citations

### "Application of Differential Transfo..." refers background in this paper

• ...In recent years, concept of DTM has broadened to problems involving partial differential equations and systems of differential equations [11-13]....

[...]

Journal ArticleDOI
Ming-Jyi Jang
TL;DR: It is demonstrated that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation.
Abstract: The differential transform is a numerical method for solving differential equations. In this paper, we present the definition and operation of the two-dimensional differential transform. A distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations. Partial differential equation of parabolic, hyperbolic, elliptic and nonlinear types can be solved by the differential transform. We demonstrate that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation.

255 citations

### "Application of Differential Transfo..." refers background or methods in this paper

• ...In recent years, concept of DTM has broadened to problems involving partial differential equations and systems of differential equations [11-13]....

[...]

• ...[11] state that “the differential transform is an iterative procedure for obtaining analytic Taylor series solutions of differential equations”....

[...]

Journal ArticleDOI
TL;DR: In this paper, the exact frequencies and mode shapes for rotating beams in which both the flexural rigidity and the mass distribution vary linearly were solved using the Frobenius method.
Abstract: The method of Frobenius is used to solve for the exact frequencies and mode shapes for rotating beams in which both the flexural rigidity and the mass distribution vary linearly. Results are tabulated for a variety of situations including uniform and tapered beams, with root offset and tip mass, and for both hinged root and fixed root boundary conditions. The results obtained for the case of the uniform cantilever beam are compared with other solutions, and the results of a conventional finite-element code.

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