The approximate equations oscillations of cylindrical shells of variable thickness
DOI:
https://doi.org/10.26577/JMMCS-2019-4-m8Keywords:
oscillations, rod, rotation, strain, deformationAbstract
To date, a huge number of studies have been carried out to bring a three-dimensional problem
to a two-dimensional one by engineering and mathematical methods. But these studies do not
exhaust the problem completely. The solution of this problem for bodies with different geometries
continues today, as evidenced by the publications of domestic, Russian and foreign scientists.
Adjacent to them is the problem of studying the dynamic behavior of circular rods interacting with
the deformable medium on the basis of oscillation equations derived using a rigorous mathematical
apparatus. Approximate equations of oscillation of rod systems above the second order with respect
to the derivatives of the desired function and the theory of oscillations of a circular cylindrical
shell, in particular torsional oscillations, taking into account the inertion of rotation and the
strain of the transverse shear, are devoted to a relatively small number of scientific publications.
The approximate equations of rod systems of variable thickness presented in this paper allow us
to construct approximate theories of oscillation depending on the conditions at the ends of the
rod, on the order of the derivatives sought in the approximate equations and initial conditions.
The results obtained make it possible to formulate boundary value problems in solving particular
problems of oscillations of a cylindrical shell under various conditions at the end of the shell
References
konferencija po teorii obolochek i plastinok, (1970): 714-722.
[2] Blend D, "Teorija linejnoj vjazkouprugosti [The theory of linear viscoelasticity]" , (1965): 428.
[3] Timoshenko S.P, "Ustojchivost’ sterzhnej, plastin i obolochek [Stability of rods, plates and shells]" , (1971): 807.
[4] Filippov I.G.and Cheban V.G, "Matematicheskaja teorija kolebanij uprugih i vjazkouprugih plastin i sterzhnej [The
mathematical theory of vibrations of elastic and viscoelastic plates and rods]" , (1988): 190.
[5] Seitmuratov A.and Taimuratova L.and Zhussipbek B.and Seitkhanova А.and Kainbaeva L, "Conditions of extreme stress
state" , News of the National Academy of Sciences of the Republic of Kazakhstan, no 5(2019): 202-206.
[6] Seitmuratov A.and Tileubay S.and Toxanova S.and Ibragimova N.and Doszhanov B.and Aitimov M.Z, "The problem of
the oscillation of the elastic layer bounded by rigid bouhdaries" , News of NAS RK. Series of physico-mathematical, no
5(2018): 42 – 48.
[7] Filippov I.G.and Filippov S.I, "Dinamicheskaja teorija ustojchivosti sterzhnej [Dynamic Theory of Stability rods]" ,Trudy
Rossijsko-Pol’skogo seminara «Teoreticheskie osnovy stroitel’stva», (1995): 63-69.
[8] Filippov I.G. "Dinamicheskaya teoriya otnositel’nogo dvizheniya mnogokomponentnyh sred. [Dynamic theory of relative
motion of multicomponent media.]" , Prikl. mekhan., no 10(1974): 92-94.
[9] Filippov I.G., Egorychev O.A. "Volnovye processy v linejnyh vyazkouprugih sredah. [Wave processes in linear viscoelastic
media.]" , M: Mashinostroenie, (1983): 272.
[10] Filippov I.G. "Tochnye uravnenij poperechnyh kolebanij vyazkouprugih plit. [Exact equations of transverse oscillations
of viscoelastic plates.]" , Trudy Vsesoyuz. konf. Po dinamike osnovanij, fundamentov i podzemnyh sooruzhenij., (1995):
63-69.
[11] Filippov I.G., Dzhanmuldaev B.D., Egorychev O.O., Skhropkin S.A., Filippov S.I. "Teoriya dinamicheskogo povedeniya
ploskih elementov stroitel’nyh konstrukcij. [Theory of dynamic behavior of flat elements of building structures.]" , Tezisy
i dokl.// Rossijsko-Pol’skogo seminara ”Teoreticheskie osnovy stroitel’stva”, (1993).
[12] Filippov I.G. "Priblizhennyj metod resheniya dinamicheskih zadach dlya vyazkouprugih sred. [Approximate method for
solving dynamic problems for viscoelastic media.]" , PMM, (1979): 133-137.
[13] Filippov I.G. "K nelinejnoj teorii vyazkouprugih izotropnyh sred. [On the nonlinear theory of viscoelastic isotropic media.]", Prikl. mekhanika, no 3(1983): 3-8.
[14] Filippov I.G., Ishripkulov T.Sh., Mirzanabilov S.M. "Nestacionarnye kolebaniya linejnyh uprugih i vyazkouprugih sred.
[Unsteady oscillations of linear elastic and viscoelastic media.]" , «FAN» Uz. SSR, (1909).
[15] Filippov I.G., Filippov S.I. "Uravneniya kolebaniya kusochno-odnorodnoj plastinki peremennoj tolshchiny. [Equations of
oscillation of a piecewise homogeneous plate of variable thickness.]" , MTT, no 5(1989): 149-157.
[16] Filippov I.G., Filippov S.I., Kostin V.I. "Dinamika dvumernyh kompozitov. [Dynamics of two-dimensional composites.]" ,
Trudy Mezhdun. konferencii po mekhaniki i materialam, (1995): 75-79.
[17] Filippov I.G., Filippov S.I., Egorychev O.A. "Vliyanie sloistosti deformirovannogo osnovaniya na kolebaniya ploskih elementov.
[Influence of stratification of deformed base on vibrations of plane elements.]" , Sb. trudov Respub. konfer.
«Aktual’ny problemy mekhaniki kontaktnogo vzaimodejstviya», (1997): 70-71.
[18] Filippov A.I. "Rasprostranenie voln v uprugom sterzhne, okruzhennom sredoj tipa Vinklera. [Propagation of waves in an
elastic rod surrounded by a Winkler-type medium.]" , MGU. Ser. 1. Matematika, mekhanika, (1983): 74-78.
[19] Almagambetova A.and Tileubay S.and Taimuratova L.and Seitmuratov A.and Kanibaikyzy K, "Problem on the distribution
of the harmonic type Relay wave" , News of the National Academy of Sciences of the Republic of Kazakhstan, no
1(2019): 242-247.
[20] Seitmuratov A.and Zharmenova В.and Dauitbayeva А.and Bekmuratova A. K.and Tulegenova Е.and Ussenova G, "Numerical
analysis of the solution of some oscillation problems by the decomposition method" , News of NAS RK. Series of
physico-mathematical, no 1(2019): 28 – 37.
[21] Seitmuratov A.and Zhussipbek B.and Sydykova G.and Seithanova А.and Аitimova U, "Dynamic stability of wave processes
of a round rod" , News of NAS RK. Series of physico-mathematical, no 2(2019): 90 – 98.
[22] Seitmuratov A., Ramazanov M., Medeubaev N., Kaliev B. "Mathematical theory of vibration of elastic or viscoelastic
plates, under non-stationary external influences" , News of the National Academy of Sciences of the Republic of Kazakhstan.
Series of geology and technology sciences., no 426 (2017), 255-263.
[23] Ashirbayev N., Ashirbayeva Zh., Abzhapbarov A., and Shomanbayeva M. "The features of a non-stationary state of stress
in the elastic multisupport construction" , AIP Conference Proceedings, (2016).
[24] Ashirbayev N., Ashirbayeva Zh., Sultanbek T., and Bekmoldayeva R. "Modeling and solving the two-dimensional nonstationary
problem in an elastic body with a rectangular hole" , AIP Conference Proceedings, (2016).
[25] Ashirbayev N.K., Banas I., Bekmoldayeva R. "A Unified Approach to Some Classes of Nonlinear Integral Equations" ,
Journal of Function Spaces, (2014), 9.
