Sinc-Galerkin solution to the clamped plate eigenvalue problem

El-Gamel, Mohamed; Mohsen, A.; Abdrabou, Amgad

doi:10.1007/s40324-016-0086-9

Cited by 10 publications

(8 citation statements)

References 42 publications

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“…Of course, this is not a big difference, but it is interesting to know which results are more exact. So, a resent work of El -Gamel M. at all [21], where approximately 200 N = terms were used (this means N 2 equations), can shed light on the value of "exact" values. They gave only first four frequencies.…”

Section: Static Centric Loadingmentioning

confidence: 99%

“…Except the trial functions described in [14] many other different functions are used in literature. For example, the four side clamped plate is analyzed by set of translated on small distance each from other the Sink functions [21], which are the ratio of sin(x) to x. There also many other presentations of trial functions in literature, most of which are combinations of trigonometric functions.…”

Section: Introductionmentioning

confidence: 99%

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Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

Bai¹,

Оrynyak²

2021

M&AT

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The paper is continuation of our efforts on application of the properly constructed sets of exponential functions as the trial (basic) functions in weighted residuals method, WRM, on example of classical tasks of structural mechanics. The purpose of this paper is justification of new method’s efficiency as opposed to getting new results. So, static deformation and free vibration of isotropic thin – walled plate are considered here. Another peculiarity of paper is choice of weight (test) functions, where three options are investigated: it is the same as trial one (Galerkin method); it is taken as results of application of differential operator to trial function (least square method); it equals to the second derivative of trial function with respect to both x and y coordinate (moment method). Solution is considered as product of two independent sets of functions with respect to x or y coordinates. Each set is the combination of five consequent exponential functions, where coefficient at first function is equal to one, and four other coefficients are to satisfy two boundary conditions at each opposite boundary. The only arbitrary value in this method is the scaling factor at exponents, the reasonable range of which was carefully investigated and was shown to have a negligible impact on results. Static deformation was investigated on example of simple supported plate when outer loading is either symmetrical and concentrated near the center or is shifted to any corner point. It was demonstrated that results converge to correct solution much quickly than in classical Navier method, while moment method seems to be a best choice. Then method was applied to free vibration analysis, and again the accuracy of results on frequencies and mode shape were excellent even at small number of terms. At last the vibration of relatively complicated case of clamped – clamped plate was analyzed and very encouraged results as to efficiency and accuracy were achieved.

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Section: Static Centric Loadingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

Bai¹,

Оrynyak²

2021

M&AT

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“…The accuracy of proposed method is verified on example of clamped rectangular plate by comparison with results of two variants of Galerkin method. Note, that chosen geometry is very popular in practical verifications of various methods, because it usually gives the maximal discrepancies between results [5,25].…”

Section: Introductionmentioning

confidence: 99%

Applications of randomly selected sets of exact Voight's solutions for vibration of thin plates

Оrynyak¹,

Bai

Kostiushko³

2022

Mech. Adv. Technol.

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The principally new method of selected exact solutions, SES, for plate vibration based on fundamental solutions of Voigt is suggested. In contrast to similar known methods, it employs the frequency dependent functions for both space coordinates. The sets of exact solutions which depends on some arbitrary chosen parameters are constructed. This allows to choose any number of exact solutions, while the required number of them depends on the boundary conditions which should satisfy in considered collocation points. The efficiency of method is demonstrated for the most unfavorable case of all sides clamped rectangular plate. Nevertheless, the accuracy is quite satisfactory for first six natural frequencies even for relatively small number of collocation boundary points, and testify about big prospects as to application for complex structures, different geometries, various boundary conditions. Additionally two variants of the Galerkin method are realized and compared. First one, employs the exponential functions, while the second one –the very popular beam functions. The calculation results show the superiority of first variant as in technical realization as in accuracy, and in further applications in structural mechanics.

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“…Sinc methods were used for solving a wide range of linear and nonlinear problems arising from scientific and engineering applications including Blasius equation [16], Hallens integral equation [17], inverse problem [18], oceanographic problems with boundary layers [19], Troeschs problem [20]and the fourth-order parabolic equation [21]. Very recently, El-Gamel used the sinc procedure to solve time-dependent partial differential equations and the clamped plate eigenvalue problem [22,23] and Babaei et al developed the sinc spectral collocation technique for time-fractional multi-dimensional partial integro-differential equation [24].…”

Section: Introductionmentioning

confidence: 99%

Novel efficient collocation method for Sturm–Liouville problems with nonlocal integral boundary conditions

El-Gamel

El-Hady

2020

SeMA

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Sinc-Galerkin solution to the clamped plate eigenvalue problem

Cited by 10 publications

References 42 publications

Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

Applications of randomly selected sets of exact Voight's solutions for vibration of thin plates

Novel efficient collocation method for Sturm–Liouville problems with nonlocal integral boundary conditions

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