https://oldena.lpnu.ua/handle/ntb/44892| Title: | Generalization and application of the Cauchy–Poisson method to elastodynamics of a layer and the Timoshenko equation |
| Other Titles: | Узагальнення та застосування метода Коші–Пуассона до еластодинаміки шару та рівняння Тимошенко |
| Authors: | Селезов, І. Selezov, I. |
| Affiliation: | Інститут гідромеханіки НАН України Institute of Hydromechanics, NASU |
| Bibliographic description (Ukraine): | Selezov 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. |
| Bibliographic description (International): | Selezov 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. |
| Is part of: | Mathematical Modeling and Computing, 1 (5), 2018 |
| Journal/Collection: | Mathematical Modeling and Computing |
| Issue: | 1 |
| Volume: | 5 |
| Issue Date: | 15-Jan-2018 |
| Publisher: | Lviv Politechnic Publishing House |
| Place of the edition/event: | Lviv |
| UDC: | 531.4 |
| Keywords: | метод Кошi–Пуассона евклiдiв простiр диференцiальне рiвнян- ня в часткових похiдних еластодинамiка шар гiперболiчнi апроксимацiї рiвняння Тимошенко Cauchy–Poisson method Euclidean space partial differential equation (PDE) elastodynamics layer hyperbolic approximations Timoshenko equation |
| Number of pages: | 10 |
| Page range: | 88-97 |
| Start page: | 88 |
| End page: | 97 |
| Abstract: | Метод 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. The 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. |
| URI: | https://ena.lpnu.ua/handle/ntb/44892 |
| Copyright owner: | © 2018 Lviv Polytechnic National University CMM IAPMM NASU © 2018 Lviv Polytechnic National University CMM IAPMM NASU |
| References (Ukraine): | [1] Maxwell J.C. A dynamical theory of the electromagnetic field. Cambridge University Press (1864). [2] Maxwell J.C. On the dynamical theory of gases. Phil. Trans. Roy. Soc. 157, 49–88 (1867). [3] EinsteinA. The meaning of relativity. Princeton University Press (1950). [4] Weber J. General relativity and gravitational waves. New York, Interscience Publishers (1961). [5] Selezov I.T., KryvonosYu.G. Wave hyperbolic models propagation of perturbations. Kiev, Naukova Dumka (2015). [6] Selezov I.T., KryvonosYu.G. Modeling medicine propagation in tissue: generalized statement. Cybernetics and Systems Analysis. 53 (4), 535–542 (2017). [7] CauchyA. L. Sur l’´equilibre et le mouvement d’une lame solide. Exercices Math. 3, 245–326 (1828). [8] Poisson S.D. M´emoire sur l’´equilibre et le mouvement des corps ´elastiques. M´em. Acad. Roy. Sci. 8,357–570 (1829). [9] Selezov I.T. Degenerated hyperbolic approximation of the wave theory of elastic plates. Ser. Operator Theory. Advances and Applications. Differential Operators and Related Topics. Proc. of Mark Krein Int. Conf., Ukraine, Odessa, 18–22 August 1997. Basel/Switzerland, Birkhauser. Vol. 117, 339–354 (2000). [10] DunfordN., Schwartz J.T. Linear operators. Part II. Spectral theory. Self adjoint operators in Hilbert space. New York, London, Interscience Publishers (1963). [11] CourantR., HilbertD. Methods of mathematical physics. Vol. 1, 2. Interscience, New York-London (1962). [12] KytheP.K. Fundamental solutions for differential operators and applications. Birkhauser Boston (1996). [13] KalashnikovA. S. The concept of a finite rate of propagation of a perturbation. Russian Math. Surveys. 34 (2), 235–236 (1979). [14] MisokhataC. The theory of partial differential equations. University Kioto (1965). [15] Timoshenko S.P. On the correction for shear of the differential equation for transverse vibrations of prismatic bar. Philosophical Magazine and Journal of Science. 41 (245), 744–746 (1921). [16] KirchhoffG. ¨ Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal f¨ur die reine und angewandte Mathematik. 40 (1), 51–58 (1850). [17] RayleighD. On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proc. London Math. Soc. 10, 225–237 (1889). [18] CosseratE.& F. Th´eorie de Corps d´eformables. Hermann, Paris (1909). [19] CattaneoC. Sulla conduzione del calore. Atti Semin. Mat. Fis. della Universit`a di Modena. 3, 3–21(1948). [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). [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). [22] MoninA.M. On diffusion with finite velocity. Izv. Academy of Sciences of the USSR, ser. geogr. 3, 234–248 (1955). [23] DaviesR.W. The connection between the Smoluchowski Equation and the Kramers-Chandrasekhar equation. Phys. Rev. 93 (6), 1169–1171 (1954). [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. [25] Selezov I. Extended models of sedimentation in coastal zone. Vibrations in Physical Systems. 26, 243–250 (2014). |
| References (International): | [1] Maxwell J.C. A dynamical theory of the electromagnetic field. Cambridge University Press (1864). [2] Maxwell J.C. On the dynamical theory of gases. Phil. Trans. Roy. Soc. 157, 49–88 (1867). [3] EinsteinA. The meaning of relativity. Princeton University Press (1950). [4] Weber J. General relativity and gravitational waves. New York, Interscience Publishers (1961). [5] Selezov I.T., KryvonosYu.G. Wave hyperbolic models propagation of perturbations. Kiev, Naukova Dumka (2015). [6] Selezov I.T., KryvonosYu.G. Modeling medicine propagation in tissue: generalized statement. Cybernetics and Systems Analysis. 53 (4), 535–542 (2017). [7] CauchyA. L. Sur l’´equilibre et le mouvement d’une lame solide. Exercices Math. 3, 245–326 (1828). [8] Poisson S.D. M´emoire sur l’´equilibre et le mouvement des corps ´elastiques. M´em. Acad. Roy. Sci. 8,357–570 (1829). [9] Selezov I.T. Degenerated hyperbolic approximation of the wave theory of elastic plates. Ser. Operator Theory. Advances and Applications. Differential Operators and Related Topics. Proc. of Mark Krein Int. Conf., Ukraine, Odessa, 18–22 August 1997. Basel/Switzerland, Birkhauser. Vol. 117, 339–354 (2000). [10] DunfordN., Schwartz J.T. Linear operators. Part II. Spectral theory. Self adjoint operators in Hilbert space. New York, London, Interscience Publishers (1963). [11] CourantR., HilbertD. Methods of mathematical physics. Vol. 1, 2. Interscience, New York-London (1962). [12] KytheP.K. Fundamental solutions for differential operators and applications. Birkhauser Boston (1996). [13] KalashnikovA. S. The concept of a finite rate of propagation of a perturbation. Russian Math. Surveys. 34 (2), 235–236 (1979). [14] MisokhataC. The theory of partial differential equations. University Kioto (1965). [15] Timoshenko S.P. On the correction for shear of the differential equation for transverse vibrations of prismatic bar. Philosophical Magazine and Journal of Science. 41 (245), 744–746 (1921). [16] KirchhoffG. ¨ Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal f¨ur die reine und angewandte Mathematik. 40 (1), 51–58 (1850). [17] RayleighD. On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proc. London Math. Soc. 10, 225–237 (1889). [18] CosseratE.& F. Th´eorie de Corps d´eformables. Hermann, Paris (1909). [19] CattaneoC. Sulla conduzione del calore. Atti Semin. Mat. Fis. della Universit`a di Modena. 3, 3–21(1948). [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). [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). [22] MoninA.M. On diffusion with finite velocity. Izv. Academy of Sciences of the USSR, ser. geogr. 3, 234–248 (1955). [23] DaviesR.W. The connection between the Smoluchowski Equation and the Kramers-Chandrasekhar equation. Phys. Rev. 93 (6), 1169–1171 (1954). [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. [25] Selezov I. Extended models of sedimentation in coastal zone. Vibrations in Physical Systems. 26, 243–250 (2014). |
| Content type: | Article |
| Appears in Collections: | Mathematical Modeling And Computing. – 2018. – Vol. 5, No. 1 |
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| 2018v5n1_Selezov_I-Generalization_and_application_88-97.pdf | 1.09 MB | Adobe PDF | View/Open | |
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