Longitudinal Waves Modelling in a Shell with Physically Quadratic Nonlinearity Filled with Liquid and Enveloped by Elastic MediumLongitudinal Waves Modelling in a Shell with Physically Quadratic Nonlinearity Filled with Liquid and Enveloped by Elastic Medium


DOI: 10.34759/trd-2020-111-3

Аuthors

Bykova T. V.*, Evdokimova E. V.**, Mogilevich L. I.***, Popov V. S.****

Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

*e-mail: tbykova69@mail.ru
**e-mail: eev2106@mail.ru
***e-mail: mogilevichli@gmail.com
****e-mail: vic_p@bk.ru

Abstract

A perturbation technique for deformation waves simulation based on considering the coupled problem of hydroelasticity in elastic cylinder shell with quadratic physical non-linearity is developing in the presented article. The shell is encompassed by the elastic medium and filled with viscous uncompressing liquid, which inertia of movement is accounted fort while considering its dynamics. The article demonstrates that the presence of encompassing medium leads to integral-differential equation, generalizing Korteweg-de Vries equation, possessing solution in the form of a solitary wave, called solitron. It does not hold an arbitrary constant wave number, in contrast to the Korteweg-de Vries equation solution. The uncompressing liquid behavior inside the shell are being described by the Navier- Stokes equations and continuity equation. They are being solved together with boundary conditions of adhesion to the shell wall. Solution is being presented by direct expansion of the sought functions by the small parameter of the hydroelesticity problem and reduced to liquid dynamics in the framework of the hydrodynamic lubrication theory. As the result, tensions from the liquid side, acting on the shell in longitudinal direction and normally, are being determined. The presence of liquid in the shell adds into the longitudinal waves equation a term, which does not allow finding the exact solution. Thus, numerical study is being realized using up-to-date approach, based on universal algorithm of commutative algebra. As the result of Gröbner differential basis construction, difference schemes of Crank-Nicolson type, obtained using basic integral difference relations, approximating the initial system of equations were generated. Numerical experiment revealed that liquid movement inertia reduced the wave velocity, and liquid viscous friction decreases the wave amplitude.

Keywords:

nonlinear waves, viscous incompressible liquid, cylindrical physically nonlinear shell, elastic medium

References

  1. Kligman E.P., Kligman I.E., Matveenko V.P. Prikladnaya mekhanika i tekhnicheskaya fizika, 2005, vol. 46, no. 6, pp. 128 - 135.

  2. Bochkarev S.A., Matveenko V.P. Prikladnaya mekhanika i tekhnicheskaya fizika, 2012, vol. 53, no. 5, pp. 155 - 165.

  3. Gromeka I.S. K teorii dvizheniya zhidkosti v uzkikh tsilindricheskikh trubakh / Sobranie sochinenii (On the fluid motion theory in narrow cylindrical pipes. Collected works), Moscow, Izd-vo AN SSSR, 1952, pp. 149 - 171.

  4. Mogilevich L.I., Popova A.A., Popov V.S. Nauka i tekhnika transporta, 2007, no. 2, pp. 64 - 72.

  5. Kondratov D.V., Mogilevich L.I. Vestnik Saratovskogo gosudarstvennogo tekhnicheskogo universiteta, 2007, vol. 3, no. 2 (27), pp. 15 - 23.

  6. Kondratov D.V., Kondratova Yu.N., Mogilevich L.I. Problemy mashinostroeniya i nadezhnosti mashin, 2009, no. 3, pp. 15 - 21.

  7. Païdoussis M.P., Nguyen V.B., Misra A.K. A theoretical study of the stability of cantilevered coaxial cylindrical shells conveying fluid, Journal of Fluids and Structures, 1991, vol. 5, no. 2, pp. 127 - 164. DOI:10.1016/0889-9746(91)90454-W

  8. Amabili M., Garziera R. Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass; Part III: Steady viscous effects on shells conveying fluid, Journal of Fluids and Structures, 2002, vol. 16, no. 6, pp. 795 - 809. DOI: 10.1006/jfls.2002.0446

  9. Amabili M. Nonlinear vibrations and stability of shells and plates, Cambridge University Press, 2008, 374 p. DOI: 10.1017/CBO9780511619694

  10. Mogilevich L.I., Popov V.S. Izvestiya RAN. Mekhanika tverdogo tela, 2004, no. 5, pp. 179 - 190.

  11. Bochkarev S.A. Vychislitel'naya mekhanika sploshnykh sred, 2010, vol. 3, no. 2, pp. 24 - 33. DOI: 10.7242/1999-6691/2010.3.2.14

  12. Lekomtsev S.V. Vychislitel'naya mekhanika sploshnykh sred, 2012, vol. 5, no. 2, pp. 233 - 243. DOI: 10.7242/1999-6691/2012.5.2.28

  13. Bochkarev S.A., Matveenko V.P. Vychislitel'naya mekhanika sploshnykh sred, 2013, vol. 6, no. 1, pp. 94 - 102. DOI: 10.7242/1999-6691/2013.6.1.12

  14. Nariboli G.A. Nonlinear longitudinal dispersive waves in elastic rods, Journal of Mathematical Physics, 1970, vol. 4. pp. 64 - 73.

  15. Nariboli G.A., Sedov A. Burger’s-Korteweg-De Vries equation for viscoelastic rods and plates, Journal of Mathematical Analysis and Applications, 1970, vol. 32, pp. 661 - 667.

  16. Erofeev V.I., Kazhaev V.V., Pavlov I.S. Inelastic interaction and splitting of strain solitons propagating in a rod, Journal of Sound and Vibration, 2018, vol. 419, pp. 173 - 182. DOI: 10.1016/j.jsv.2017.12.040

  17. Erofeev V.I., Morozov A.N., Nikitina E.A. Trudy MAI, 2010, no. 40, available at: http://trudymai.ru/eng/published.php?ID=22861

  18. Lai T.T., Tarlakovskii D.V. Trudy MAI, 2012, no. 53, available at: http://trudymai.ru/eng/published.php?ID=29267

  19. Erofeev V.I., Mal'khanov A.O., Morozov A.N. Trudy MAI, 2010, no. 40, available at: http://trudymai.ru/eng/published.php?ID=22860

  20. Nguen T.T., Tarlakovskii D.V. Trudy MAI, 2019, no. 105, available at: http://trudymai.ru/eng/published.php?ID=104123

  21. Zemlyanukhin A.I., Mogilevich L.I. Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 1995, vol. 3, no. 1, pp. 52 - 58.

  22. Erofeev V.I., Klyueva.N.V. Akusticheskii zhurnal, 2002, vol. 48, no. 6, pp. 725 - 740.

  23. Erofeev V.I., Zemlyanukhin A.I., Katson V.M., Sheshenin S.F. Vychislitel'naya mekhanika sploshnykh sred, 2009, vol. 2, no. 4, pp. 67 - 75. DOI: 10.7242/1999-6691/2009.2.4.32

  24. Bagdoev A.G., Erofeev V.I., Shekoyan A.V. Lineinye i nelineinye volny v dispergiruyushchikh sploshnykh sredakh (Linear and nonlinear waves in dispersive continuous media), Moscow, Fizmatlit, 2009, 320 p.

  25. Erofeev V.I., Kazhaev V.V., Pavlov I.S. Vychislitel'naya mekhanika sploshnykh sred, 2013, vol. 6, no. 2, pp. 140 - 150. DOI: 10.7242/1999-6691/2013.6.2.17

  26. Zemlyanukhin A.I., Bochkarev A.V. Vychislitel'naya mekhanika sploshnykh sred, 2016, vol. 9, no. 2, pp. 182 - 191. DOI: 10.7242/1999-6691/2016.9.2.16

  27. Zemlyanukhin A.I., Bochkarev A.V. Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 2016, vol. 24, no. 4, pp. 71 - 85. DOI: 10.18500/0869-6632-2016-24-4-71-85

  28. Zemlyanukhin A.I., Bochkarev A.V. Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 2017, vol. 25, no. 1, pp. 64 - 83. DOI: 10.18500/0869-6632-2017-25-1-64-83

  29. Blinkova A.Yu., Ivanov S.V., Kovalev A.D., Mogilevich L.I. Izvestiya Saratovskogo universiteta. Novaya seriya: Fizika, 2012, vol. 12, no. 2, pp. 12 - 18. DOI: 10.18500/1816-9791-2016-16-2-184-197

  30. Blinkova A.Yu., Blinkov Yu.A., Mogilevich L.I. Vychislitel'naya mekhanika sploshnykh sred, 2013, vol. 6, no. 3, pp. 336 - 345. DOI: 10.7242/1999-6691/2013.6.3.38

  31. Blinkova A.Yu., Ivanov S.V., Kuznetsova E.L., Mogilevich L.I. Trudy MAI, 2014, no. 78, URL: http://trudymai.ru/eng/published.php?ID=53486

  32. Blinkova A.Yu., Blinkov Yu.A., Ivanov S.V., Mogilevich L.I. Izvestiya Saratovskogo universiteta. Novaya seriya: Matematika. Mekhanika. Informatika, 2015, vol. 15, no. 2, pp. 193 - 202. DOI: 10.18500/1816-9791-2015-15-2-193-202

  33. Ivanov S.V., Mogilevich L.I., Popov V.S. Trudy MAI, 2019, no. 105, available at: http://trudymai.ru/eng/published.php?ID=104003

  34. Loitsyanskii L.G. Mekhanika zhidkosti i gaza (Liquids and gases mechanics), Moscow, Drofa, 2003, 840 p.

  35. Vol'mir A.S. Obolochki v potoke zhidkosti i gaza: zadachi gidrouprugosti (Shells in a fluid and gas flow: hydroelasticity problems), Moscow, Nauka, 1979, 320 p.

  36. Vlasov V.Z., Leont'ev N.N. Balki, plity i obolochki na uprugom osnovanii (Beams, Plates and Shells on Elastic Base), Moscow, Fizmatgiz, 1960, 490 p.

  37. Ageev R.V., Evdokimova E.V., Kovaleva I.A., Mogilevich L.I. Matematicheskoe modelirovanie, komp'yuternyi i naturnyi eksperiment v estestvennykh naukakh, 2017, no. 3. pp. 33 – 41.


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