On one class of initial-boundary-value problems in aerohydroelasticity

P. A. Vel'misov ; J. А. Tamarova ; Yu. V. Pokladova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings – XXXI". Voronezh, May 3-9, 2020, Tome 204 (2022), pp. 16-26.

Résumé

In this paper, we consider initial-boundary problems for systems of differential equations, which are mathematical models of the mechanical system “pipeline-pressure sensor” intended for controlling pressure in gas-liquid media. Based on the models proposed, we examine the joint dynamics of the sensitive element of the pressure sensor and the medium in the pipeline. To describe the dynamics of the medium and the dynamics of the sensitive element, we use linear models of fluid and gas mechanics and mechanics of solid deformable bodies. We obtain differential equations with deviating arguments that relate the displacement (deformation) of the sensitive element of the sensor with the pressure law of the medium in the engine. Also, we develop analytical and numerical methods for solving these initial-boundary problems.

Détail

Keywords: differential equation, aerohydroelasticity, pipeline, pressure sensor, dynamics, finite difference method, Galerkin method.

Bibliographie

[1] Ageikin D. I., Kostina E. N., Kuznetsova N. N., Datchiki kontrolya i regulirovaniya, Mashinostroenie, M., 1965

[2] Andreeva L. E., Uprugie elementy priborov, Mashinostroenie, M., 1981

[3] Andreeva L. E., Bogdanova Yu. A., Metody proektirovaniya membrannykh uprugikh elementov, Mashinostroenie, M., 1972

[4] Ankilov A. V., Velmisov P. A., Gorbokonenko V. D., Pokladova Yu. V., Matematicheskoe modelirovanie mekhanicheskoi sistemy «truboprovod—datchik davleniya», UlGTU, Ulyanovsk, 2008

[5] Ash Zh. i dr., Datchiki izmeritelnykh sistem. Kn. 1, Mir, M., 1992

[6] Ash Zh. i dr., Datchiki izmeritelnykh sistem. Kn. 2, Mir, M., 1992

[7] Bellman R., Kuk K. L., Differentsialno-raznostnye uravneniya, Mir, M., 1967

[8] Belozubov E. M., Vasilev V. A., Zapevalin A. I., Chernov P. S., “Proektirovanie uprugikh elementov nano- i mikroelektromekhanicheskikh sistem”, Izmerit. tekhn., 2011, no. 1, 17–19

[9] Belozubov E. M., Mokrov E. A., Tikhomirov D. V., “Minimizatsiya pogreshnosti tonkoplenochnykh tenzorezistornykh datchikov davleniya pri vozdeistvii nestatsionarnoi temperatury”, Datchiki i sistemy., 2004, no. 1, 26–29

[10] Velmisov P. A., Gorbokonenko V. D., “Ob odnoi matematicheskoi modeli sistemy «truboprovod–datchik davleniya»”, Tr. 4 Mezhdunar. nauch.-tekhn. konf. «Matematicheskoe modelirovanie fizicheskikh, ekonomicheskikh, tekhnicheskikh, sotsialnykh sistem i protsessov. Matematika», UlGU, Ulyanovsk, 2001, 43–44

[11] Velmisov P. A., Gorbokonenko V. D., “Ob odnoi matematicheskoi modeli sistemy «truboprovod–datchik davleniya»”, Mat. 11 Mezhvuz. konf. «Matematicheskoe modelirovanie i kraevye zadachi», SamGTU, Samara, 2001, 15–17

[12] Velmisov P. A., Gorbokonenko V. D., Reshetnikov Yu. A., “Matematicheskaya model sistemy «truboprovod-datchik davleniya»”, Mekhanika i protsessy upravleniya, UlGTU, Ulyanovsk, 2002, 9–15

[13] Velmisov P. A., Gorbokonenko V. D., Reshetnikov Yu. A., “Matematicheskoe modelirovanie mekhanicheskoi sistemy «truboprovod-datchik davleniya»”, Datchiki i sistemy., 2003, no. 6 (49), 12–15

[14] Velmisov P. A., Pokladova Yu. V., Issledovanie dinamiki deformiruemykh elementov nekotorykh aerogidrouprugikh sistem, UlGTU, Ulyanovsk, 2018

[15] Velmisov P. A., Pokladova Yu. V., Serebryannikova E. S., “Matematicheskoe modelirovanie sistemy «truboprovod-datchik davleniya»”, Zh. Srednevolzh. mat. o-va., 12:4 (2010), 85–-93

[16] Gerasimov V. K., Tyzhnov G. I., “K voprosu vybora uprugogo chuvstvitelnogo elementa dlya izmereniya davleniya”, Izv. vuzov. Priborostroenie., 1973, no. 6, 80–83

[17] Godunov S. K., Ryabenkii V. S., Raznostnye skhemy. Vvedenie v teoriyu, Nauka, M., 1977 | MR

[18] Kazaryan A. A., Groshev G. P., “Universalnyi datchik davleniya”, Izmerit. tekhn., 2008, no. 3, 26–-30

[19] Kalitkin N. N., Chislennye metody, Nauka, M., 1978

[20] Korsunov V. P., Uprugie chuvstvitelnye elementy, Izd-vo Saratov. un-ta, Saratov, 1980

[21] Mikhailov P. G., Mokrov E. A., Mitrokhin S. V., Sergeev D. A., “Osobennosti metrologicheskogo obespecheniya sovremennykh datchikov pulsatsii davlenii”, Izv. Yuzhn. feder. un-ta. Tekhn. nauki., 130:5 (2012), 174–179

[22] Mokrov E. A., Lebedev D. V., Bazaev V. P., Efremov E. V., Semina I. A., Kolchin P. A., “O konstruktivno-tekhnologicheskom sovershenstvovanii tenzorezistornykh tonkoplenochnykh datchikov davlenii”, Datchiki i sistemy., 2008, no. 6, 2–7

[23] Myshkis A. D., Lineinye differentsialnye uravneniya s zapazdyvayuschim argumentom, Nauka, M., 1972 | MR

[24] Norkin S. B., Differentsialnye uravneniya vtorogo poryadka s zapazdyvayuschim argumentom, Nauka, M., 1965

[25] Pirogov S. P., Manometricheskie trubchatye pruzhiny, Nedra, SPb., 2009

[26] Etkin L. G., Vibrochastotnye datchiki. Teoriya i praktika, Izd-vo MGTU im. N. E. Baumana, M., 2004

[27] Bellman R., Cooke K. L., Differential-Difference Equations, Academic Press, New York–London, 1963 | MR | Zbl

[28] Velmisov P. A., Garnefska L. V., Gorbokonenko V. D., “An investigation of mathematical models “pipeline-pressure sensor””, Proc. XXVII Summer School “Applications of Mathematics in Engineering and Economics”, Heron Press, Sofia, Bolgaria, 2002, 542–548

[29] Velmisov P. A., Pokladova Yu. V., “Mathematical modelling of the “pipeline-pressure sensor” system”, J. Phys. Conf. Ser., 1353 (2019), 012085 | DOI

[30] Velmisov P. A., Pokladova Yu. V., Mizher U. J., “Mathematical modeling of the mechanical system “pipeline-pressure sensor””, AIP Conf. Proc., 2172:1 (2019), 030006 | DOI