On the study of Kirchhoff plates vibration on an acoustic base
UDC
539.3DOI:
https://doi.org/10.31429/vestnik-19-4-91-99Abstract
In seismology, the study of the wave field on the surface of a medium makes it possible to build models of natural tectonic processes in the Earth's crust and upper mantle. On the scale of the Earth's structure, lithospheric plates can be considered as coatings of relatively small thickness; therefore, a plate can serve as the simplest model of an extended lithospheric structure, which leads to modeling their interactions using separated deformable plates. The problems of geodynamic interaction of deformable plates with acoustic and elastic media can be studied by methods of the contact problems theory. In this paper, we consider a plane problem of steady-state oscillations for a system of two contacting semi-bounded Kirchhoff plates on a layer of an ideal fluid located on a non-deformable foundation. Oscillations are excited by the action of a concentrated harmonic source located in an acoustic medium. The solution is constructed using the integral approach, the eigenfunction method, and the factorization method developed for similar problems arising in geophysics, seismoacoustics, and ecology. We also obtained integral representations of the amplitude values of the coating deflections (the surface displacements of the structure under consideration) and the pressure at the interface between the elastic plates and the liquid medium. The presented approach makes it possible to study the features of the propagation for waves generated by oscillating loads in acoustic media with a coating.
Keywords:
Kirchhoff plates, steady-state oscillations, acoustic medium, concentrated source, eigen function method, factorization methodFunding information
The study was supported by a grant from the Russian Science Foundation and the Kuban Science Foundation (project No. 22-21-20032, https://rscf.ru/project/22-21-20032/).
References
- Гамбурцев, Г.А., Перспективный план исследований по проблеме. Изыскание и развитие прогноза землетрясений. В сб. Развитие идей Г.А. Гамбурцева в геофизике. Наука, Москва, 1982, c. 304–311. [Gamburtsev, G.A., Perspective plan of research on the problem. Research and Development of Earthquake Forecast. In: Razvitie idey G.A. Gamburtseva v geofizike = Development of G.A. Gamburtsev in geophysics. Nauka, Moscow, 1982, pp. 304–311. (in Russian)]
- Садовский, М.А., Болховитинов, Л.Г., Писаренко, В.Ф., Деформирование геофизической среды и сейсмический процесс. Наука, Москва, 1987. [Sadovsky, M.A., Bolkhovitinov, L.G., Pisarenko, V.F., Deformirovanie geofizicheskoy sredy i seysmicheskiy protsess = Deformation of the geophysical environment and the seismic process. Nauka, Moscow, 1987. (in Russian)]
- Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., On the possibility of predicting some types of earthquake by a mechanical approach. Acta Mechanica, 2018, vol. 229, iss. 5. pp. 2163–2175. DOI 10.1007/s00707-017-2092-0
- Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., On a mechanical approach to theprediction of earthquakes during horizontalmotion of litospheric plates. Acta Mechanica, 2018, vol. 229, iss. 10, pp. 4727–4739. DOI 10.1007/s00707-018-2255-7
- Бабешко, В.А., Евдокимова, О.В., Бабешко, О.М., Хрипков, Д.А., Телятников, И.С., Евдокимов, В.С., Шестопалов, В.Л., О стартовых землетрясениях в прибрежной зоне. Экологический вестник научных центров Черноморского экономического сотрудничества, 2018, т. 15, № 3., с. 12–18. [Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., Khripkov, D.A., Telyatnikov, I.S., Evdokimov, V.S., Shestopalov, V.L., O starting earthquakes in the coastal zone. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 3., pp. 12–18. (in Russian)]
- Altenbach, J., Altenbach, H., Eremeyev, V.A., On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech (Special Issue), 2010, vol. 80, pp. 73–92. DOI 10.1007/s00419-009-0365-3
- Chandrashekhara, K., Theory of Plates. Himayat Nagar Universities Press, 2001.
- Еремеев, В.А., Зубов, Л.М., Механика упругих оболочек. Наука, Москва, 2008. [Eremeev, V.A., Zubov, L.M., Mekhanika uprugikh obolochek = Mechanics of Elastic Shells. Nauka, Moscow, 2008. (in Russian)]
- Pavlova, A.V., Rubtsov, S.E., Telyatnikov, I.S., On models of wave processes in geological structures with the presence of a interlayer water. IOP Conference Series: Earth and Environmental Science, 2022, vol. 1070. art. 012013. DOI 10.1088/1755-1315/1070/1/012013
- Вольмир, А.С., Нелинейная динамика пластинок и оболочек. Наука, Москва, 1972. [Volmir, A.S., Nelineynaya dinamika plastinok i obolochek = Nonlinear dynamics of plates and shells. Nauka, Moscow, 1972. (in Russian)]
- Гольденвейзер, А.Л., Теория упругих тонких оболочек. Наука, Москва, 1976. [Goldenweiser, A.L., Teoriya uprugikh tonkikh obolochek = Theory of elastic thin shells. Nauka, Moscow, 1976. (in Russian)]
- Исакович, М.А., Общая акустика. Наука, Москва, 1973. [Isakovich, M.A., Obshchaya akustika = General acoustics. Nauka, Moscow, 1973. (in Russian)]
- Евдокимова, О.В., Бабешко, В.А., Бабешко, О.М., Уафа, С.Б., Коваленко, М.М., Бушуева, О.А., О некоторых приложениях покрытий с жидкостью. Экологический вестник научных центров Черноморского экономического сотрудничества, 2019, т. 16, №3, с. 40–45. [Evdokimova, O.V., Babeshko, V.A., Babeshko, O.M., Uafa, S.B., Kovalenko, M.M., Bushueva, O.A., On some applications of liquid coatings. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2019, vol. 16, no. 3, pp. 40–45. (in Russian)]
- Тихонов, А.Н., Самарский, А.А., Уравнения математической физики. Наука, Москва, 2004. [Tikhonov, A.N., Samarskii, A.A., Uravneniya matematicheskoy fiziki = Equations of Mathematical Physics. Nauka, Moscow, 2004. (in Russian)]
- Телятников, И.С., К моделям и методам изучения взаимодействия литосферных структур в области разломов. Экологический вестник научных центров Черноморского экономического сотрудничества, 2016, № 2, с. 78–89. [Telyatnikov, I.S., On models and methods for studying the interaction of lithospheric structures in the fault area. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2016, no. 2, pp. 78–89. (in Russian)]
- Noble, B., Methods based on the Wiener–Hopf technique for the solution of partial differential equations. Pergamon Press, New York, 1958.
- Лаврентьев, М.А., Шабат, Б.В., Методы теории функций комплексного переменного. Наука, Москва, 1973. [Lavrentiev, M.A., Shabat, B.V., Metody teorii funktsiy kompleksnogo peremennogo = Methods of the theory of functions of a complex variable. Nauka, Moscow, 1973. (in Russian)]
- Колесников, М.Н., Телятников, И.С., О методах изучения динамики контактирующих литосферных структур. Экологический вестник научных центров Черноморского экономического сотрудничества, 2017, № 4, ч. 1, с. 50–61. [Kolesnikov, M.N., Telyatnikov, I.S., On methods for studying the dynamics of contacting lithospheric structures. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2017, no. 4, pt. 1, pp. 50–61. (in Russian)]
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Copyright (c) 2022 Телятников И.С., Колесников М.Н., Павлова А.В., Рубцов С.Е.
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