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

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.
Abstract: Rotating 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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Citations
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Journal ArticleDOI
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.

321 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.

189 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).

89 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).

72 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.

39 citations

References
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Journal ArticleDOI
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.
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.

599 citations

Book
01 Jan 1985
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.

383 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
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.

269 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.

264 citations