Algorithmization of the solution of dynamic boundary value problems of the theory of flexible plates taking into account shift and rotation inertia

A. Yuldashev   ; Sh. T. Pirmatov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 75 (2022), pp. 150-165

Résumé

Most of the problems on flexible plates are solved in the Föppl-von Karman formulation, which is Love's special case. The constructed algorithms are not economical in terms of implementation on a computer. Therefore, construction of algorithms for the complete calculation of flexible plates with a given degree of accuracy with allowance for the shear and inertia of rotation is becoming a topical issue. The problem of creating an automated inference system and solving the equations of the theory of elasticity and plasticity were first posed in the monograph by V.K. Kabulov. In this work, for the first time, the main problems of algorithmization are formulated and ways of their machine solution are outlined. The problem of algorithmization is solved as follows: depending on geometric characteristics of the object and physical properties of the material, a design scheme of this model is selected; derivation of the initial differential equations and the corresponding boundary and initial conditions; selection of a computational algorithm and numerical solution of the obtained equations; analysis of the obtained numerical results describing the stress-strain state of the structure under consideration. This work consists of an introduction, three sections and a conclusion. In the first paragraph, the equations of motion of rectangular plates are given. Substituting the expression for the force of moments and shearing forces and introducing a dimensionless value, a system of equations in displacements is obtained. In the second section, using the central difference formulas, a system of quasilinear ordinary differential equations is obtained. Taking into account the boundary and initial conditions, the system of equations is reduced to matrix form, which can be solved by the Runge-Kutta method. In the third paragraph, an analysis of the results obtained is presented.

Détail

Keywords: algorithm, plates, shear, inertia of rotation, theory of elasticity, difference schemes.

Bibliographie

[1] Volmir A. S., Nelineinaya dinamika plastin i obolochek, Nauka, M., 1972, 432 pp. | MR

[2] Kabulov V. K., Algoritmizatsiya v teorii uprugosti i deformatsionnoi teorii plastichnosti, Fan, Tashkent, 1966, 394 pp.

[3] Demidovich B. P., Maron I. A., Chislennye metody analiza, ed. B.P. Demidovich, Fizmatgiz, M., 1962

[4] Lyav A., Matematicheskaya teoriya uprugosti, ONTI-NKTL SSR, M.-L., 1935, 674 pp.

[5] Yuldashev A., Aliboev A., “Integrirovanie uravneniya dvizheniya gibkikh obolochek s uchetom sdviga i inertsii vrascheniya”, Sb. nauch. tr. TashPI «Chislennye metody», Tashkent, 1978, 50–60

[6] Berezin I. S.,Zhidkov N. P., Metody vychislenii, v. I-II, Fizmatgiz, 1959 | MR

[7] Aliboev A., Yuldashev A., “Integrirovanie uravneniya dvizheniya pryamougolnykh plastin metodom setok”, Sb. nauch. tr. TashPI «Avtomatizatsiya proektirovaniya», 237, Tashkent, 1978, 89–93

[8] Bate K. Yu., Metody konechnykh elementov, Fizmatlit, M., 2010, 1024 pp.

[9] Kornishin M. S., “Nekotorye voprosy primeneniya metoda konechnykh raznostei dlya resheniya kraevykh zadach teorii plastin”, Prikladnaya mekhanika, 9:3 (1963)

[10] Yuldashev A., Pirmatov Sh. T., Minarova N., “Uravnenie ravnovesiya gibkikh kruglykh plastin”, Austrian J. Technical and Natural Sciences, 2015, no. 3–4, 32–35

[11] Yuldashev A., Pirmatov Sh.T., “Algorithmization of solving dynamic edge problems of the theory of flexible rectangular plates”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2020, no. 66, 143–157 | DOI | MR

[12] Korobeinikov S. N., Nelineinoe deformirovanie tverdykh tel, Izd-vo SORAN, Novosibirsk, 2000, 262 pp.

[13] Rabotnov Yu. N., Mekhanika deformiruemogo tverdogo tela, Nauka, M., 1988, 712 pp.

[14] Berikkhanova G. E., Zhumagulov B. T., Kanguzhin B. E., “Matematicheskaya model kolebanii paketa pryamougolnykh plastin s uchetom tochechnykh svyazei”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2010, no. 1(9), 72–86

[15] Bazarov M. B., Safarov I. I., Shokin Yu. I., Chislennoe modelirovanie kolebanii dissipativno odnorodnykh i neodnorodnykh mekhanicheskikh sistem, Izd-vo SO RAN, Novosibirsk, 1996, 189 pp.

[16] Vyachkin E. S., Kaledin V. O., Reshetnikova E. V., Vyachkina E. A., Gileva A. E., “Razrabotka matematicheskoi modeli staticheskogo deformirovaniya sloistykh konstruktsii s neszhimaemymi sloyami”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 55, 72–83 | DOI | MR