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.
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.
Citations
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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.
321 citations
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
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
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
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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TL;DR: In this paper, free vibration differential equations of motion of one end fixed, the other simply supported and axial loaded beams on elastic soil are solved using differential transform method (DTM), analytical solution and frequency factors are obtained.
81 citations
"Application of Differential Transfo..." refers background in this paper
...Some researchers have lately applied DTM for analysis of uniform and non-uniform beams [14-18]....
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TL;DR: In this article, a relatively new approach called differential transformation is applied in analyzing free lateral vibrations of a centrifugally stiffened rotating Euler-Bernoulli beam, and both natural frequencies and modeshapes are obtained using differential transformation technique.
61 citations
"Application of Differential Transfo..." refers background or methods in this paper
...Mei [18] employed DTM for rotating uniform beams....
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...Some researchers have lately applied DTM for analysis of uniform and non-uniform beams [14-18]....
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TL;DR: In this article, a dynamic investigation method for the analysis of Timoshenko beams is proposed, which takes into account the shearing deformation and the rotating inertia, and the solution of the problem is obtained through the iterative variational Rayleigh-Ritz method and assuming as test functions an appropriate class of orthogonal polynomials which respect the essential conditions only.
61 citations
"Application of Differential Transfo..." refers methods in this paper
...Some researchers have recently used numerical techniques such as Frobenius and Rayleigh-Ritz methods [4-7]....
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01 Jan 2003
TL;DR: In this article, a method based on a spectral finite element technique was proposed to develop a low degree-of-freedom dynamic model directly and without resorting to a condensation procedure.
Abstract: The determination of model frequencies and mode shapes of rotating blades, which are either straight or tapered, is usually accomplished via application of conventional assumed displacement finite element or assumed modes methodologies. To ensure accuracy of modal information at low mode num bers, a large number of elements or admissible functions must be utilized in the model. Modal frequencies converge to the exact solutions from above with the increase of number of elements or assumed admissible functions. To apply the resulting large degree-of-freedom model in a dynamic simulation (forced or transient), or to embed such a model in a real-time model-based control problem, can be impractical. As a result, the model order must be reduced via static or dynamic condensation procedures to a practical number of degrees-of-freedom, which is constrained by simulation time, or the control interval in a digital control system. Therefore, a method is proposed, based on a spectral finite element technique, to develop such a low degree-of-freedom dynamic model directly and without resorting to a condensation procedure. The method exploits semi-analytical progressive wave solutions of the governing partial differential equations. We have calculated such results for a number of examples such as a straight beam and beams with uniform taper or compound tapers. Only one single spectral finite element is needed to obtain any modal frequency or mode shape, which is as accurate or better than other approaches reported in the literature for a straight or uniformly tapered beams. The minimum number of such spectral finite element correspond to the number of substructures (i.e., beam sections with different uniform tapers) in a rotating beam in order to capture the complete system dynamic characteristics. The element assembly procedure is accomplished in the same fashion as the conventional finite element approach. Overall, for a rotating blade system, our spectral finite element method provides highly accurate predictions for any modal frequency using a single element or very few elements corresponding to the number of uniform taper changes in the blade system.Copyright © 2003 by ASME
21 citations
"Application of Differential Transfo..." refers methods in this paper
...Some researchers have recently used numerical techniques such as Frobenius and Rayleigh-Ritz methods [4-7]....
[...]
6 citations
"Application of Differential Transfo..." refers background in this paper
...Except for some special cases [1-3], there exists no exact explicit solution....
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