On the Linear and Nonlinear Vibration Responses of Elastically End Restrained Beams Using DTM
11 Mar 2014-Mechanics Based Design of Structures and Machines (Taylor & Francis Group)-Vol. 42, Iss: 2, pp 135-150
TL;DR: In this paper, the authors apply the differential transformation method (DTM) to solve linear and nonlinear vibration problems of elastically end-restrained beams, which demonstrates many advantages such as rapid convergence, high accuracy, and computational stability.
Abstract: The objective of this paper is to apply the differential transformation method (DTM) to solve linear and nonlinear vibration problems of elastically end-restrained beams. The method demonstrates many advantages such as rapid convergence, high accuracy, and computational stability to determine linear and nonlinear natural frequencies as well as mode shapes of such beams. The mathematical models provided in this paper can be solved easily using symbolic tools in available software packages such as Maple and Matlab. An accuracy of the present solutions is confirmed by comparing with some published results in the open literature. New numerical results of nonlinear frequency ratio of beams supported by various types of elastic boundary conditions are presented and discussed in detail. The significant effects of translational and rotational springs including vibration amplitudes on linear and nonlinear vibration results are also taken into investigation. Based on the numerical exercises, it is revealed that the...
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TL;DR: In this article, the authors investigated the dynamic analysis of porous bi-directional functionally graded (FG) plates reinforced by eccentrically outside stiffeners and subjected to a moving load with a constant velocity.
Abstract: The principal purpose of this study is investigating the dynamic analysis of porous bi-directional functionally graded (FG) plates reinforced by eccentrically outside stiffeners and subjected to a moving load with a constant velocity. The materials are assumed to be graded in two directions and their effective properties are computed by the rule of mixtures. The FG plates are assumed to have both even and uneven distribution of porosities over the plate cross-section. Using appropriate kinematic relations, the displacements of the plate mid-plane are compatible with those of the stiffeners. The governing differential equations of porous bi-directional FG plates are derived through Hamilton's principle based on the first order shear deformation theory (FSDT) and Von Karman relations for large deflections. Moreover, dynamic relaxation method with kinetic damping (K-DR) coupled with Newmark integration technique are used to solve the plate's time-varying nonlinear equations. The effects of some numerical aspect ratios such as volume fraction, boundary conditions, porosity coefficients and distribution patterns and the existence of stiffeners on dynamic behaviors are investigated. The results show that the stiffness of the porous bi-directional FG plates is highly improved with the aid of eccentric stiffeners; hence, better dynamic behaviors are provided.
59 citations
TL;DR: In this paper, an exact free vibration and buckling analysis of a beam-column with general connections is presented, where all structural elements are made of functionally graded material and all the connections are made from the same material.
Abstract: An exact free vibration and buckling analysis of a tapered beam-column with general connections is calculated. All structural elements are made of functionally graded material. In this study, a pow...
39 citations
Cites methods from "On the Linear and Nonlinear Vibrati..."
...In another study, the differential transformation method (DTM) was applied to solve linear and nonlinear vibration problems of elastically restrained beams by Wattanasakulpong and Chaikittiratana (2013)....
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TL;DR: In this article, the von Karman strain-displacement relationship together with Hamilton's principle and Eringen's theory are employed to derive equations of motion for the nonlinear free vibration of a nanobeam under electro-thermo-mechanical loading with elastic medium and various boundary conditions, especially the elastic boundary condition.
Abstract: This study presents an examination of nonlinear free vibration of a nanobeam under electro-thermo-mechanical loading with elastic medium and various boundary conditions, especially the elastic boundary condition. The nanobeam is modeled as an Euler–Bernoulli beam. The von Karman strain-displacement relationship together with Hamilton's principle and Eringen's theory are employed to derive equations of motion. The nonlinear free vibration frequency is obtained for simply supported (S-S) and elastic supported (E-E) boundary conditions. E-E boundary condition is a general and actual form of boundary conditions and it is chosen because of more realistic behavior. By applying the differential transform method (DTM), the nanobeam's natural frequencies can be easily obtained for the two different boundary conditions mentioned above. Performing a precise study led to investigation of the influences of nonlocal parameter, temperature change, spring constants (either for elastic medium or boundary condition) and imposed electric potential on the nonlinear free vibration characteristics of nanobeam. The results for S-S and E-E nanobeams are compared with each other. In order to validate the results, some comparisons are presented between DTM results and open literature to show the accuracy of this new approach. It has been discovered that DTM solves the equations with minimum calculation cost.
29 citations
23 citations
References
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TL;DR: In this article, a closed-form solution for the postbuckling problem in terms of the applied axial load was obtained and the critical buckling loads and their associated mode shapes were obtained as a byproduct.
Abstract: We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.
250 citations
01 Dec 2013
185 citations
"On the Linear and Nonlinear Vibrati..." refers methods in this paper
...Using analytical and approximate methods to solve linear vibration problems of beams with general boundary conditions is well-known (Dimarogonas, 1996; Meirovitch, 2001)....
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TL;DR: In this article, the effects of material property distribution and end supports on the nonlinear dynamic behavior of functionally graded materials (FGMs) beams are discussed, and it is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.
Abstract: Nonlinear vibration of beams made of functionally graded materials (FGMs) is studied in this paper based on Euler-Bernoulli beam theory and von Karman geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. The direct numerical integration method and Runge-Kutta method are employed to find the nonlinear vibration response of FGM beams with different end supports. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FGM beams are discussed. It is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.
169 citations
"On the Linear and Nonlinear Vibrati..." refers background in this paper
..., in the study of Ke et al. (2010). As suggested by Singh et al....
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..., in the study of Ke et al. (2010). As suggested by Singh et al. (1990), the factor of 3/4 could be used to reduce the axial stretching force when the effect of axial displacement of the beams is neglected....
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TL;DR: In this article, differential transformation method was used to predict buckling behavior of single walled carbon nanotube (SWCNT) on Winkler foundation under various boundary conditions.
Abstract: In the present work differential transformation method (DTM) is used to predict the buckling behaviour of single walled carbon nanotube (SWCNT) on Winkler foundation under various boundary conditions. Four different boundary conditions namely clamped–clamped, simply supported, clamped hinged and clamped free are used to study the critical buckling loads. Effects of (i) size of SWCNT (ii) nonlocal parameter and (iii) Winkler elastic modulus on nonlocal critical buckling loads are being investigated and discussed. The DTM is implemented for the nonlocal SWCNT analyses and this yields results with high degree of accuracy. Further, present method can be applied to linear and nonlinear problems.
113 citations
"On the Linear and Nonlinear Vibrati..." refers methods in this paper
...The use of the DTM has been continued recently which can be seen, for example, in references (Pradhan and Reddy, 2011; Mao, 2012a)....
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TL;DR: In this article, the out-of-plane free vibration analysis of a double tapered Euler-Bernoulli beam, mounted on the periphery of a rotating rigid hub is performed.
Abstract: In this study, the out-of-plane free vibration analysis of a double tapered Euler–Bernoulli beam, mounted on the periphery of a rotating rigid hub is performed. An efficient and easy mathematical technique called the Differential Transform Method (DTM) is used to solve the governing differential equation of motion. Parameters for the hub radius, rotational speed and taper ratios are incorporated into the equation of motion in order to investigate their effects on the natural frequencies. Calculated results are tabulated in several tables and figures and are compared with the results of the studies in open literature where a very good agreement is observed.
105 citations