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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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Book ChapterDOI
01 Jan 2017
TL;DR: In this paper, the authors provide an assessment of the literature dealing with the use of composite materials in relatively recent applications to marine structures. But the analysis of the behavior of the blade is complex because of the geometry, the effects of the rotation of the propeller, the interaction with the water, and the anisotropic nature of the material.
Abstract: This chapter provides an assessment of the literature dealing with the use of composite materials in relatively recent applications to marine structures. The analysis of the behavior of the blade is complex because of the geometry, the effects of the rotation of the propeller, the interaction with the water, and the anisotropic nature of the material. These issues are discussed in details, recent results from analyses of the structure-hydrodynamics problem are assessed and recommendations for future work are presented.

4 citations

Journal ArticleDOI
01 Nov 2012
TL;DR: In this article, a novel approach is introduced to derive an efficient beam element, which is derived through power series solution of the static part of the governing differential equations for rotating tapered Timoshenko beams.
Abstract: The accuracy of results predicted by finite element method considerably depends on the shape functions used to formulate the displacement field along the element. In this article, a novel approach is introduced to derive an efficient beam element. Special functions, namely Basic Displacement Functions (BDFs), are introduced and derived through power series solution of the static part of the governing differential equations for rotating tapered Timoshenko beams. It is shown that the shape functions could be derived in terms of BDFs through basic principles of structural mechanics. It is shown that the proposed shape functions have the advantage of including the effect of the rotational speed, hub radius, and varying cross-sectional dimensions. In order to verify the competency of the present element in the determination of natural frequencies, several numerical examples are carried out, and the results compared with those in the literature.

4 citations

Journal ArticleDOI
TL;DR: In this article, the effect of different parameters such as rotating speed, taper ratio, hub radius ratio and gradient index on flapwise frequency is discussed, and the authors obtained the flapwise frequencies using Differential Transform Method (DTM) and showed that the material properties of the beam change symmetrically in the direction of thickness from middle surface to outer surfaces.
Abstract: Rotating beams are the practical application for aircraft propellers, helicopter blades, wind mill propeller, turbine blades and spinning space structures. In these structures, taper beams are preferred due to optimum weight distribution with uniform strength. Vibration behavior of tapered variable section beams subjected to centrifugal forces is important in analysis of structures. The research work deals with flapwise vibration of functionally graded (FG) rotating taper beam. The material properties of the beam change symmetrically in the direction of thickness from middle surface to the outer surfaces. Mori-Tanaka method estimates these properties of FG beam. The flapwise frequencies are obtained using Differential Transform Method (DTM). The effect of different parameters such as rotating speed, taper ratio, hub radius ratio and gradient index on flapwise frequency is discussed.

4 citations

01 Jan 2010
TL;DR: Wang et al. as mentioned in this paper generalized the differential transformation method to solve higher order linear boundary value problems and provided several numerical examples and compared the results with the exact solutions, demonstrating that the proposed method has high accuracy.
Abstract: . In this paper, we propose the generalization of differential transformation method to solve higher order of linear boundary value problem. Previous studies show that the differential transformation method is a powerful method to solve several lower order linear boundary value problems. In our study, we generalized the method so that one can solve � -th order boundary value problems with � -th order linear differential equation for �> �, �< � or �= � . To illustrate the accuracy of the proposed method, we provide several numerical examples and we compare the results with the exact solutions. The comparisons demonstrate the proposed method has high accuracy. KEYWORDS. Differential transformation method; boundary value problems; linear differential equations. INTRODUCTION The differential transformation method (DTM) is one of the numerical methods in ordinary differential equations, partial differential equations and integral equations. Since proposed in (Zhou, 1986), there are tremendous interests on the applications of the DTM to solve various scientific problems. For instance, see (Arikoglu & Ozkol, 2005), (Ayaz, 2004), (Bildik

4 citations


Cites background from "Application of Differential Transfo..."

  • ...Recent study also shows that the DTM can be applied in free vibration analysis of rotating non-prismatic beams (Attarnejad & Shahba, 2008)....

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Dissertation
01 Jan 2012
TL;DR: In this article, the dynamic stability of FGM beams subjected to parametric excitation is studied using finite element method and the shape functions for the beam element are established from the differential equation of static equilibrium.
Abstract: The dynamic stability of functionally graded material (FGM) beams subjected to parametric excitation is studied using finite element method. First order shear deformation theory (Timoshenko beam theory) is used for the analysis of the beams. The shape functions for the beam element are established from the differential equation of static equilibrium. Floquet’s theory is used to establish the stability boundaries. A steel-alumina functionally graded ordinary (FGO) beam with steel-rich bottom is considered for the analysis. For the analysis of functionally graded sandwich (FGSW) beam, alumina and steel are chosen as top and bottom skin respectively and the core is FGM with steel and alumina as constituent phases. The material properties in the direction of thickness of FGM are assumed to vary as per power law and exponential law. The effect of property distribution laws on critical buckling load, natural frequencies and parametric instability of the beams is investigated. Also, the effect of variation of power law index on the critical buckling load, natural frequencies and dynamic stability of beams is determined. It is found that the property variation as per exponential law ensures better dynamic stability than property variation as per power law. Increase in the value of power law index is found to have detrimental effect on the dynamic stability of the beams. Influence of the elastic foundations on the dynamic stability of the beams is studied. Pasternak elastic foundation is found to have more enhancing effect on the dynamic stability of the beam than Winkler elastic foundation. The dynamic stability of FGO and FGSW beams used in high temperature environment is investigated. It is observed that increase in environmental temperature has an enhancing effect on the instability of the beams. The effect of beam geometry, rotary inertia, hub radius and rotational speed on natural frequencies as well as on the parametric instability of rotating FGO and FGSW cantilever beams is studied. It is observed that increase in rotational speed enhances the dynamic stability of the beams. Parametric instability of a pre-twisted FGO cantilever beam is investigated. The effect of property distribution laws and pre-twist angle on critical buckling load, natural frequencies and parametric instability of the beam is studied. The increase in the value of power law index is found to have enhancing effect on the parametric instability of the beam. The increase in pre-twisting of the beam reduces the chance of parametric instability of the beam with respect to the first principal instability region. But the increase in pre-twist angle has a detrimental effect on the stability of the beam for second principal instability region.

3 citations


Cites methods from "Application of Differential Transfo..."

  • ...Attarnejad and Shahba [12] have studied free vibration of non-prismatic rotating Euler-Bernoulli beams using differential transform method....

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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”....

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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