Generalization and application of the Cauchy–Poisson method to elastodynamics of a layer and the Timoshenko equation

dc.citation.epage97
dc.citation.issue1
dc.citation.journalTitleMathematical Modeling and Computing
dc.citation.spage88
dc.citation.volume5
dc.contributor.affiliationІнститут гідромеханіки НАН України
dc.contributor.affiliationInstitute of Hydromechanics, NASU
dc.contributor.authorСелезов, І.
dc.contributor.authorSelezov, I.
dc.coverage.placenameLviv
dc.date.accessioned2019-05-07T14:01:55Z
dc.date.available2019-05-07T14:01:55Z
dc.date.created2018-01-15
dc.date.issued2018-01-15
dc.description.abstractМетод Кошi–Пуассона узагальнено на n-вимiрний евклiдiв простiр так, щоб отримати диференцiальнi рiвняння в часткових похiдних вищого порядку. Наведено застосуван- ня до побудови гiперболiчних апроксимацiй, що узагальнюють та доповнюють попе- реднi дослiдження. В евклiдовому просторi вводять обмеження на похiднi. Розглянуто гiперболiчне виродження за параметрами та його реалiзацiя у виглядi необхiдних i достатнiх умов. Як окремий випадок 4-вимiрного евклiдового простору, зберiгаючи оператори до 6-го порядку, отримано узагальнене гiперболiчне рiвняння поперечних (згинних) коливань пластин з коефiцiєнтами, залежними тiльки вiд числа Пуассона. Це рiвняння мiстить як окремi випадки всi вiдомi рiвняння Бернулi–Ейлера, Кiрх- гофа, Релея, Тимошенкo. Зазначено, що уточнене рiвняння згинних коливань балки, вперше представлене Тимошенко, потрiбно розглядати як розвиток дослiджень Макс- велла i Ейнштейна про поширення збурень зi скiнченою швидкiстю в середовищi. Вперше вiдзначено вiдповiднiсть з теорiєю Коссера.
dc.description.abstractThe Cauchy–Poisson method is extended to n-dimensional Euclidean space so that to obtain partial differential equations (PDEs) of a higher order. The application in the construction of hyperbolic approximations is presented, generalizing and supplementing the previous investigations. Restrictions on derivatives in Euclidean space are introduced. The hyperbolic degeneracy by parameters and its realization in the form of necessary and sufficient conditions are considered. As a particular case of 4-dimensional Euclidean space, keeping operators up to the 6th order, we obtain a generalized hyperbolic equation of transverse (bending) vibrations of plates with coefficients depending only on the Poisson number. Numerical calculations are carried out and presented. This equation includes, as special cases, all the known equations of Bernoulli–Euler, Kirchhoff, Rayleigh, Timoshenko. It should be noted that the refined equation of bending oscillations of a beam, firstly presented by Timoshenko, must be considered as the development of Maxwell’s and Einstein’s investigations on the perturbation propagation with finite velocity in media. For the first time, the conformity with the Cosserat theory is noted.
dc.format.extent88-97
dc.format.pages10
dc.identifier.citationSelezov I. Generalization and application of the Cauchy–Poisson method to elastodynamics of a layer and the Timoshenko equation / I. Selezov // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2018. — Vol 5. — No 1. — P. 88–97.
dc.identifier.citationenSelezov I. Generalization and application of the Cauchy–Poisson method to elastodynamics of a layer and the Timoshenko equation / I. Selezov // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2018. — Vol 5. — No 1. — P. 88–97.
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/44892
dc.language.isoen
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 1 (5), 2018
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dc.relation.referencesen[3] EinsteinA. The meaning of relativity. Princeton University Press (1950).
dc.relation.referencesen[4] Weber J. General relativity and gravitational waves. New York, Interscience Publishers (1961).
dc.relation.referencesen[5] Selezov I.T., KryvonosYu.G. Wave hyperbolic models propagation of perturbations. Kiev, Naukova Dumka (2015).
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dc.relation.referencesen[14] MisokhataC. The theory of partial differential equations. University Kioto (1965).
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dc.relation.referencesen[16] KirchhoffG. ¨ Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal f¨ur die reine und angewandte Mathematik. 40 (1), 51–58 (1850).
dc.relation.referencesen[17] RayleighD. On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proc. London Math. Soc. 10, 225–237 (1889).
dc.relation.referencesen[18] CosseratE.& F. Th´eorie de Corps d´eformables. Hermann, Paris (1909).
dc.relation.referencesen[19] CattaneoC. Sulla conduzione del calore. Atti Semin. Mat. Fis. della Universit`a di Modena. 3, 3–21(1948).
dc.relation.referencesen[20] LuikovA.V. Application of irreversible thermodynamics methods to investigation of heat and mass transfer. Int. J. Heat Mass Transfer. 9 (2), 139–152 (1966).
dc.relation.referencesen[21] DavydovB. I. The diffusion equation with allowance for the molecular velocity. Reports of the Academy of Sciences of the USSR. 2 (7), 474–475 (1935).
dc.relation.referencesen[22] MoninA.M. On diffusion with finite velocity. Izv. Academy of Sciences of the USSR, ser. geogr. 3, 234–248 (1955).
dc.relation.referencesen[23] DaviesR.W. The connection between the Smoluchowski Equation and the Kramers-Chandrasekhar equation. Phys. Rev. 93 (6), 1169–1171 (1954).
dc.relation.referencesen[24] FockV.A. Solution of a problem in the theory of diffusion by the method of finite differences and its application to diffusion of light. Proceedings of the State Optical Institute. Vol. 4, Issue 34, 1–32. §13 (1926). Connection with differential equations and an expression for diffusion. 29–31.
dc.relation.referencesen[25] Selezov I. Extended models of sedimentation in coastal zone. Vibrations in Physical Systems. 26, 243–250 (2014).
dc.rights.holder© 2018 Lviv Polytechnic National University CMM IAPMM NASU
dc.rights.holder© 2018 Lviv Polytechnic National University CMM IAPMM NASU
dc.subjectметод Кошi–Пуассона
dc.subjectевклiдiв простiр
dc.subjectдиференцiальне рiвнян- ня в часткових похiдних
dc.subjectеластодинамiка
dc.subjectшар
dc.subjectгiперболiчнi апроксимацiї
dc.subjectрiвняння Тимошенко
dc.subjectCauchy–Poisson method
dc.subjectEuclidean space
dc.subjectpartial differential equation (PDE)
dc.subjectelastodynamics
dc.subjectlayer
dc.subjecthyperbolic approximations
dc.subjectTimoshenko equation
dc.subject.udc531.4
dc.titleGeneralization and application of the Cauchy–Poisson method to elastodynamics of a layer and the Timoshenko equation
dc.title.alternativeУзагальнення та застосування метода Коші–Пуассона до еластодинаміки шару та рівняння Тимошенко
dc.typeArticle

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