Приближённые уравнения колебаний цилиндрических оболочек переменной толщины
DOI:
https://doi.org/10.26577/JMMCS-2019-4-m8Кілттік сөздер:
колебания, стержень, деформация, вращения, напряжениеАннотация
К настоящему времени выполнено огромное число исследований по приведению трехмерной
задачи к двухмерной инженерными и математическими методами. Но эти исследования не
исчерпывают проблему полностью. Решение этой проблемы для тел с различной геометрией
продолжается и в наши дни, о чем свидетельствуют публикации отечественных, российских и
зарубежных ученых. К ним примыкает и проблема изучения динамического поведения кру-
говых стрежней, взаимодействующих с деформируемой среды на основе уравнений колеба-
ний, выведенных с помощью строгого математического аппарата. Приближённые уравнения
колебаний стержневых систем выше второго порядка по производным от искомой функ-
ции и теории колебаний круглой цилиндрической оболочки, в частности крутильных колеба-
ний, с учетом инерции вращения и деформации поперечного сдвига посвящено сравнительно
небольшое число научных публикаций. Приведённые в данной работе приближённые урав-
нения стержневых систем переменной толщины позволяют строить приближённые теории
колебания в зависимости от условий на торцах стержня, от порядка производных, искомых в
приближённых уравнениях и начальных условиях. Полученные результаты позволяют фор-
мулировать краевые задачи при решении частных задач колебания цилиндрической оболочки
при различных условиях на торце оболочки.
Библиографиялық сілтемелер
[1] Aleksandrov A.Ja.and Kurshin L.M, "Mnogoslojnye plastinki i obolochki [Multilayer plates and shells]" ,VII Vsesojuznaja
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.
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.
Жүктелулер
Как цитировать
Ramazanov, M. I., Seitmuratov, A. Z., Taimuratova, L. U., Medeubaev, N. K., & Mukeeva, G. I. (2019). Приближённые уравнения колебаний цилиндрических оболочек переменной толщины. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 104(4), 71–84. https://doi.org/10.26577/JMMCS-2019-4-m8
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