Vibration of a circular beam with variable cross sections using differential transformation method

Abdelghany, Seif Mahmoud; Ewis, Karem Mahmoud; Mahmoud, A.A.; Nassar, May Fouad

doi:10.1016/j.bjbas.2015.05.006

Cited by 16 publications

(13 citation statements)

References 13 publications

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“…This is in contrast to the traditional approach, where a zero centre of approximation is usually used for restoring the original, e.g. [36][37][38][39][40][41][42].…”

Section: Dtm Applied To Free Vibrations Of Inhomogeneous and Non-unifmentioning

confidence: 98%

“…In [35] and [36] the authors have utilized the differential transformations to analyse free lateral vibrations of rotating tapered Euler-Bernoulli beams. Abdelghany et al in [37] have used the DTM to compute natural frequencies and appropriate mode shapes of homogeneous beams with smooth and continuous variations of non-uniform cross-sections. Free vibrations of stepped FGM beams having abrupt changes of geometrical characteristics have been studied by means the DTM as well.…”

Section: Introductionmentioning

confidence: 99%

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Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

et al. 2017

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“…This is in contrast to the traditional approach, where a zero centre of approximation is usually used for restoring the original, e.g. [36][37][38][39][40][41][42].…”

Section: Dtm Applied To Free Vibrations Of Inhomogeneous and Non-unifmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

et al. 2017

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“…Wang [69] proposed the differential quadrature element method for the natural frequencies of multiple-stepped beams with an aligned neutral axis. Abdelghany [70] utilized the differential transformation method to examine the free vibration of nonuniform circular beam.…”

Section: Introductionmentioning

confidence: 99%

Free Vibration of Axially Functionally Graded Beam

Cao¹,

Wang²,

Hu³

et al. 2020

Mechanics of Functionally Graded Materials and Structures

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Axially functionally graded (AFG) beam is a special kind of nonhomogeneous functionally gradient material structure, whose material properties vary continuously along the axial direction of the beam by a given distribution form. There are several numerical methods that have been used to analyze the vibration characteristics of AFG beams, but it is difficult to obtain precise solutions for AFG beams because of the variable coefficients of the governing equation. In this topic, the free vibration of AFG beam using analytical method based on the perturbation theory and Meijer G-Function are studied, respectively. First, a detailed review of the existing literatures is summarized. Then, based on the governing equation of the AFG Euler-Bernoulli beam, the detailed analytic equations are derived on basis of the perturbation theory and Meijer G-function, where the nature frequencies are demonstrated. Subsequently, the numerical results are calculated and compared, meanwhile, the analytical results are also confirmed by finite element method and the published references. The results show that the proposed two analytical methods are simple and efficient and can be used to conveniently analyze free vibration of AFG beam.

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“…The investigation of the dynamics of geometrical structures is of big interest because these structures are found in a lot of practical applications. As a consequence in the literature, there are many scientific works [1][2][3][4][5] dealing with this subject. A particular and important case that has been subject of many studies is the dynamics of beams [1][2][3][6][7][8], with applications ranging from stress tension in mechanical/structural systems [9] to vibrations of microtubules in biological models [10].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Euler–Bernoulli beam on nonlinear viscoelastic foundations: a parameter space analysis

Soares

Ladeira

Oliveira

2020

J Braz. Soc. Mech. Sci. Eng.

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The dynamics and spectrum of vibrations of an uniform Euler-Bernoulli beam on nonlinear viscoelastic foundations are investigated in parameter space. An alternative method of approach based on mean-field approximation is proposed. Spatial and temporal solution are obtained in an interactive way. The spectrum is reached directly from spatial solutions. It was shown that the system exhibits sudden birth and death of chaotic attractors (crises) and that there are periodic windows immersed in chaotic regions. It was determined the regions in the parameter space where chaotic and regular attractors occur. Moreover, the model presents the coexistence of many attractors. The numerical simulations found geometrical structures known as shrimps, and these periodic structures are involved by chaotic regions. In this work it was investigated the dynamics of a beam of Euler-Bernoulli with viscous and dynamic support. It is presented, then, a model to treat this type of problem, which is nonlinear and non-autonomous. The dynamics of the model presents a complex behavior, being in some cases sensitive to the initial conditions. Changing properly the parameters the dynamics goes from chaotic behavior to periodic regime. As the system is forced and dissipative, complex structures in the parameter space, known as shrimps, arise. Also, it was found that the spatial solution is not sensitive to the dynamics, although they are coupled. However, the dynamics is very sensitive to the mean quadratic deformation of the beam and, therefore, it is related to the vibration modes obtained in the spatial solution.

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Vibration of a circular beam with variable cross sections using differential transformation method

Cited by 16 publications

References 13 publications

Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

Free Vibration of Axially Functionally Graded Beam

Dynamics of a Euler–Bernoulli beam on nonlinear viscoelastic foundations: a parameter space analysis

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