Generalization and application of the Cauchy–Poisson method to elastodynamics of a layer and the Timoshenko equation
Date
2018-01-15
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Lviv Politechnic Publishing House
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.
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.
Description
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
Citation
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.