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Title of the article NATURAL OSCILLATIONS OF THE FIVE-LAYER ROD CAUSED BY THE INITIAL DEFLECTION
Authors

STAROVOITOV Eduard I., D. Sc. in Phys. and Math., Prof., Professor of the Department “Structural Mechanics, Geotechnical and Structural Engineering”, Belarusian State University of Transport, Gomel, Republic of Belarus, This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.

BUDNIKOVA Darya A., Graduate Student of the Department “Structural Mechanics, Geotechnical and Structural Engineering”, Belarusian State University of Transport, Gomel, Republic of Belarus, This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.
In the section MECHANICS OF DEFORMED SOLIDS
Year 2025
Issue 2(71)
Pages 70–77
Type of article RAR
Index UDK 539.3
DOI https://doi.org/10.46864/1995-0470-2025-2-71-70-77
Abstract The problem of natural oscillations of a five-layer rod symmetrical in thickness, resulting from the initial deflection, is considered. The three load-bearing layers are assumed to be thin and high-strength. Bernoulli’s hypotheses are accepted for them. In two relatively thick, light fillers, the Timoshenko hypothesis is fulfilled, i. e. a displacement in the filler is taken into account. The system of differential equations of natural oscillations is obtained by the variational method, taking into account the transverse inertia forces. For the rod that is symmetrical in thickness, the system is reduced to two partial differential equations with respect to deflection and relative displacement in the fillers. The analytical solution of the corresponding initial boundary value problem is obtained by decomposing the desired displacements into a series according to the constructed system of eigenfunctions. An algebraic equation for determining eigenvalues is given. Graphs of the dependence of the first three frequencies on the thickness of the central bearing layer and fillers are presented. An example of the occurrence of natural oscillations due to the initial deflection is considered. A numerical analysis of the obtained solutions is carried out.
Keywords symmetrical five-layer rod, initial deflection, analytical solution, eigenvalues and frequencies, numerical results
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Bibliography
  1. Reddy J.N. Mechanics of laminated composite plates and shells. Theory and analysis. Boca Raton, CRC Press, 2003. 858 p. DOI: https://doi.org/10.1201/b12409.
  2. Carrera E., Fazzolari F.A., Cinefra M. Thermal stress analysis of composite beams, plates and shells: computational modelling and applications. Academic Press, 2016. 440 p.
  3. Aghalovyan L., Prikazchikov D. Asymptotic theory of anisotropic plates and shells. Singapore, World Scientific Publishing Co., 2015. 360 p.
  4. Yarovaya, A.V. Stroitelnaya mekhanika. Statika sterzhnevykh sistem [Construction mechanics. Statics of rod systems]. Gomel, Belorusskiy gosudarstvennyy universitet transporta Publ., 2013. 447 p. (in Russ.).
  5. Gorshkov A.G., Starovoitov E.I., Yarovaya A.V. Mekhanika sloistykh vyazkouprugoplasticheskikh elementov konstruktsiy [Mechanics of layered viscoelastic-plastic structural elements]. Moscow, Fizmatlit Publ., 2005. 576 p. (in Russ).
  6. Zhuravkov M.A., Starovoitov E.I. Matematicheskie modeli mekhaniki tverdykh tel [Mathematical models of solid mechanics]. Minsk, Belorusskiy gosudarstvennyy universitet Publ., 2021. 535 p. (in Russ.).
  7. Zhuravkov M., Lyu Y., Starovoitov E. Mechanics of solid deformable body. Singapore, Springer Verlag, 2023. 317 p. DOI: https://doi.org/10.1007/978-981-19-8410-5.
  8. Abdusattarov A., Starovoitov E.I., Ruzieva N.B. Deformirovanie i povrezhdaemost uprugoplasticheskikh elementov konstruktsiy pri tsiklicheskikh nagruzheniyakh [Deformation and damage of elastic-plastic structural elements under cyclic loads]. Tashkent, Tashkentskiy gosudarstvennyy transportnyy universitet Publ., 2023. 381 p. (in Russ.).
  9. Starovoitov E.I., Shafieva Yu.V., Nesterovich A.V., Kozel A.G. Deformirovanie trekhsloynykh plastin pri termosilovykh nagruzkakh [Deformation of three-layer plates under thermal force loads]. Gomel, Belorusskiy gosudarstvennyy universitet transporta Publ., 2024. 395 p. (in Russ.).
  10. Starovoitov E.I., Zhuravkov M.A., Leonenko D.V., Lyu Y. Deformation of three-layer structural elements in thermal radiation fields. Singapore, Springer Singapore, 2024. 384 p. DOI: https://doi.org/10.1007/978-981-97-7217-9.
  11. Mikhasev G.I., Altenbach H. Free vibrations of elastic laminated beams, plates and cylindrical shells. Thin-walled laminated structures, 2019, pp. 157–198. DOI: https://doi.org/10.1007/978-3-030-12761-9_4.
  12. Pleskachevsky Yu.V., Starovoitov E.I., Leonenko D.V. Sobstvennye kolebaniya krugovoy sendvich-plastiny v temperaturnom pole [Natural oscillations of circular sandwich plates in the temperature field]. Mechanics of machines, mechanisms and materials, 2024, no. 4(69), pp. 70–77. DOI: https://doi.org/10.46864/1995-0470-2024-4-69-70-77 (in Russ.).
  13. Lachugina E.A. Poperechnye kolebaniya pyatisloynoy uprugoy krugovoy plastiny s zhestkimi zapolnitelyami [Transverse vibrations of the five-layer elastic circular plate with rigid fillers]. Mechanics. Researches and innovations, 2022, iss. 15, pp. 116–122 (in Russ.).
  14. Lachugina E.A. Svobodnye kolebaniya pyatisloynoy krugovoy plastiny s legkimi zapolnitelyami [Free vibrations of a five-layer circular plate with lightweight fillers]. Mechanics. Researches and innovations, 2023, iss. 16, pp. 111–116 (in Russ.).
  15. Budnikova D.A. Uravneniya sobstvennykh kolebaniy pyatisloynogo uprugogo sterzhnya [Equations of natural vibrations of a five-layer elastic rod]. Materialy 13 Mezhdunarodnoy nauchno-prakticheskoy konferentsii “Problemy bezopasnosti na transporte” [Proc. 13th international scientific and practical conference “Transport safety issues”]. Gomel, 2024, part 2, pp. 121–123 (in Russ.).
  16. Starovoitov E.I., Leonenko D.V. Issledovanie spektra chastot trekhsloynoy tsilindricheskoy obolochki s uprugim napolnitelem [Investigation of the frequency spectrum of a three-layered cylindrical shell with an elastic filler]. Mechanics of composite materials and structures, 2015, vol. 21, no. 2, pp. 162–169 (in Russ.).
  17. Bakulin V.N., Boitsova D.A., Nedbai A.Ya. Parametric resonance of a three-layered cylindrical composite rib-stiffened shell. Mechanics of composite materials, 2021, vol. 57, iss. 5, pp. 623–634. DOI: https://doi.org/10.1007/s11029-021-09984-9.
  18. Leonenko D.V., Starovoitov E.I. Vibrations of cylindrical sandwich shells with elastic core under local loads. International applied mechanics, 2016, vol. 52, iss. 4, pp. 359–367. DOI: https://doi.org/10.1007/s10778-016-0760-8.
  19. Tarlakovskii D.V., Fedotenkov G.V. Two-dimensional nonstationary contact of elastic cylindrical or spherical shells. Journal of machinery manufacture and reliability, 2014, vol. 43, iss. 2, pp. 145–152. DOI: https://doi.org/10.3103/S1052618814010178.
  20. Fedotenkov G.V., Tarlakovsky D.V., Vahterova Y.А. Identification of non-stationary load upon Timoshenko beam. Lobachevskii journal of mathematics, 2019, vol. 40, iss. 4, pp. 439–447. DOI: https://doi.org/10.1134/S1995080219040061.
  21. Vestyak V.A., Sadkov A.S., Tarlakovskiy D.V. Rasprostranenie nestatsionarnykh obemnykh vozmushcheniy v uprugoy poluploskosti [Propagation of nonstationary volumetric perturbations in an elastic half-plane]. Izvestiya Rossiyskoy akademii nauk. Mekhanika tverdogo tela, 2011, vol. 46, no. 2, pp. 130–140 (in Russ.).
  22. Paimushin V.N., Gazizullin R.K. Static and monoharmonic acoustic impact on a laminated plate. Mechanics of composite materials, 2017, vol. 53, iss. 3, pp. 283–304. DOI: https://doi.org/10.1007/s11029-017-9662-z.
  23. Grover N., Singh B.N., Maiti D.K. An inverse trigonometric shear deformation theory for supersonic flutter characteristics of multilayered composite plates. Aerospace science and technology, 2016, vol. 52, pp. 41–51. DOI: https://doi.org/10.1016/j.ast.2016.02.017.
  24. Leonenko D.V., Markova M.V. Kolebaniya krugovoy trekhsloynoy stupenchatoy plastiny pri udarnom periodicheskom vozdeystvii [Vibrations of a three-layer circular step plate under periodic impact]. Mechanics of machines, mechanisms and materials, 2022, no. 3(60), pp. 68–76. DOI: https://doi.org/10.46864/1995-0470-2022-3-60-68-76 (in Russ.).
  25. Leonenko D.V., Markova M.V. Kolebaniya krugovoy trekhsloynoy plastiny pod deystviem vneshney nagruzki [Oscillations of a circular three-layer plate under external linear in time load]. Journal of the Belarusian State University. Mathematics and informatics, 2023, no. 1, pp. 49–63. DOI: https://doi.org/10.33581/2520-6508-2023-1-49-63 (in Russ.).
  26. Pradhan M., Dash P.R., Pradhan P.K. Static and dynamic stability analysis of an asymmetric sandwich beam resting on a variable Pasternak foundation subjected to thermal gradient. Meccanica, 2016, vol. 51, iss. 3, pp. 725–739. DOI: https://doi.org/10.1007/s11012-015-0229-6.
  27. Leonenko D.V. Kolebaniya krugovykh trekhsloynykh plastin na uprugom osnovanii Pasternaka [Vibrations of circular three-layer plates on Pasternak elastic foundation]. Ecological bulletin of research centers of the Black Sea economic cooperation, 2014, vol. 11, no. 1, pp. 59–63 (in Russ.).
  28. Ageev R.V., Mogilevich L.I., Popov V.S. Kolebaniya stenok shchelevogo kanala s vyazkoy zhidkostyu, obrazovannogo trekhsloynym i tverdym diskami [Vibrations of the walls of a slit channel with a viscous liquid formed by three-layer and solid disks]. Problemy mashinostroeniya i nadezhnosti mashin, 2014, no. 1, pp. 3–11 (in Russ.).
  29. Mogilevich L.I., Popov V.S., Kondratov D.V., Rabinskiy L.N. Bending oscillations of a cylinder, surrounded by an elastic medium and containing a viscous liquid and an oscillator. Journal of vibroengineering, 2017, vol. 19, iss. 8, pp. 5758–5766. DOI: https://doi.org/10.21595/jve.2017.18179.
  30. Tratsevskaya E.Yu. Dinamicheskaya neustoychivost kvazitiksotropnykh morennykh gruntov [Dynamic instability of quasi-thixotropic moraine soils]. Litosfera, 2017, no. 1(46), pp. 107–112 (in Russ.).
  31. Tratsevskaya E.Yu. Dempfiruyushchie svoystva slabosvyaznykh trekhfaznykh gruntov [Damping properties of loosely coupled three-phase soils]. Litosfera, 2019, no. 2(51), pp. 115–121 (in Russ.).
  32. Babaytsev A.V., Kalyagin M.Yu., Rabinskiy L.N. Razvitie defektov v mnogosloynykh kompozitakh pri staticheskikh nagruzkakh [The development of defects in multilayer composites under static loads]. Rossiyskie inzhenernye issledovaniya, 2024, no. 1(44), pp. 112–115 (in Russ.).
  33. Starovoitov E.I., Kozel A.G. Izgib uprugoy krugovoy trekhsloynoy plastiny na osnovanii Pasternaka [The bending of an elastic circular sandwich plate on the Pasternak foundation]. Mechanics of composite materials and structures, 2018, vol. 24, no. 3, pp. 392–406 (in Russ.).
  34. Kozel A.G. Sravnenie resheniy zadach izgiba trekhsloynykh plastin na osnovaniyakh Vinklera i Pasternaka [Comparison of solutions to the bending problems of three-layer plates on the Winkler and Pasternak foundations]. Mechanics of machines, mechanisms and materials, 2021, no. 1(54), pp. 30–37. DOI: https://doi.org/10.46864/1995-0470-2021-1-54-30-37 (in Russ.).
  35. Gorshkov A.G., Starovoitov E.I., Yarovaya A.V., Leonenko D.V. Deformirovanie krugovoy trekhsloynoy plastiny na uprugom osnovanii [Strain of a circle sandwich plate on elastic foundation]. Ecological bulletin of research centers of the Black Sea economic cooperation, 2005, vol. 2, no. 1, pp. 16–19 (in Russ.).
  36. Starovoitov E.I., Leonenko D.V., Suleyman M. Termouprugiy izgib koltsevoy trekhsloynoy plastiny na uprugom osnovanii [Thermoelastic bending of a ring sandwich plate on the elastic foundation]. Ecological bulletin of research centers of the Black Sea economic cooperation, 2006, vol. 3, no. 4, pp. 55–62 (in Russ.).
  37. Starovoitov E.I., Leonenko D.V. Deformirovanie trekhsloynogo sterzhnya v temperaturnom pole [Deformation of three-layer beam in a temperature field]. Mechanics of machines, mechanisms and materials, 2013, no. 1(22), pp. 31–35 (in Russ.).
  38. Zakharchuk Yu.V. Napryazhenno-deformirovannoe sostoyanie krugovoy trekhsloynoy plastiny so szhimaemym zapolnitelem [Stress-strain state of a circular three layer plate with a compressible filler]. Mechanics. Researches and innovations, 2019, iss. 12, pp. 66–75 (in Russ.).
  39. Nestsiarovich A.V. Osesimmetrichnoe nagruzhenie krugloy fizicheski nelineynoy trekhsloynoy plastiny v svoey ploskosti [Axisymmetric loading of a circular physically nonlinear three-layer plate in its plane]. Problems of physics, mathematics and technics, 2021, no. 3(48), pp. 24–29. DOI: https://doi.org/10.54341/20778708_2021_3_48_24 (in Russ.).