Transient dynamics of a thin isotropic spherical belt

DOI: 10.34759/trd-2023-131-05


Zuskova V. N.*, Okonechnikov A. S.**, Serdyuk D. O.***

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



Thin-walled spherical shells are common structural elements in various industries, such as aircraft, rocket and mechanical engineering. When designing the corresponding structural elements, one of the topical issues is the calculations of structures operating in non-stationary interaction modes. Such calculations are complex and time-consuming, since in such problems the desired solution is significantly heterogeneous in spatial coordinates and time. In an axisymmetric formulation, the study of the transient dynamics of a thin spherical belt with emission boundary conditions under the influence of a moving transient load. The belt material is elastic and isotropic. The Kirchhoff-Love hypothesis was implemented in the qualitative mathematical model of the spherical scenario. An approach to research based on the principle of superposition, the Green’s function method and the method of compensating influence. The essence of the results is in connection with the desired solution with the acting and compensating load using integral operations such as convolution in coordinate and time. Being this operation, is the Green’s function for spherical exploitation, which is the normal displacement in response to the core, random focusing on the load coordinate and time, mathematically described by the Dirac delta function. The compensation solution is the result of studying some specially calculated values, at which decisions are made from the acting load and the compensating sensation, satisfying the boundary conditions at the ends of the spherical belt. An example of calculating the unsteady dynamics of a spherical belt is given. The results for transient functions of normal displacements, angles of turns and bending moments are presented in the form of graphs. The method of compensating loads applied in the work allows us to study the transient dynamics of a spherical belt with intermediate axisymmetric supports, as well as spherical segments.


transitional dynamics, compensating load, Green's function, spherical layer, spherical shell


  1. Ganapathi M., Varadan T. K. Dynamic Buckling of Laminated Anisotropic Spherical Caps, ASME Journal of Applied Mechanics, 1995, vol. 62 (1), pp. 13-19. DOI: 10.1115/1.2895879
  2. Zenkour A.M. Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model, Archive of Applied Mechanics, 2004, vol. 74, pp. 262–276. DOI: 10.1007/s00419-004-0348-3
  3. Narasimhan M.C. Dynamic-response of laminated orthotropic spherical shell, Journal of the acoustical society of America, 1992, vol. 91 (5), pp. 2714-2720. DOI: 10.1121/1.402953
  4. Sundararajan N., Prakash T., Ganapathi M. Dynamic buckling of functionally graded spherical caps, AIAA Journal, 2006, vol. 44, no. 5. DOI: 10.2514/1.17320
  5. Awrejcewicz J., Krysko V.A., Shchekaturova T.V. Transitions from regular to chaotic vibrations of spherical and conical axially-symmetric shells, International journal of structural stability and dynamics, 2005, vol. 5, no. 3, pp. 359-385. DOI: 10.1142/S0219455405001623
  6. Lugovoi P.Z., Meish V.F., Orlenko S.P. Numerical simulation of the dynamics of spherical sandwich shells reinforced with discrete ribs under a shock wave, International applied mechanics, 2020, vol. 56 (5), pp. 590-598. DOI: 10.1007/s10778-020-01037-3
  7. Nath, K. Sandeep. Effect of transverse shear on static and dynamic buckling of antisymmetrically laminated polar orthotropic shallow spherical shells, Composite Structures, 1997, vol. 40, no. 1, pp. 67-72. DOI: 10.1016/s0263-8223(97)00153-0
  8. Alexey A. Semenov. Models of Deformation of Stifened Orthotropic Shells under Dynamic Loading, Journal of Siberian Federal University. Mathematics & Physics, 2016, vol. 9 (4), pp. 485-497. DOI: 10.17516/1997-1397-2016-9-4-485-497
  9. Chekhov V.N., Zakora S.V. Vestnik Donetskogo natsional’nogo universiteta. Seriya A: estestvennye nauki, 2018, no. 3-4, pp. 92-101.
  10. Ganeeva M.S., Moiseeva V.E. Ekologicheskii vestnik nauchnykh tsentrov ChES, 2012, vol. 9, no. 4, pp. 37-47.
  11. Vinogradov Yu.I. Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana, 2015, no. 3, pp. 68-84. DOI: 10.7463/0315.0760049
  12. Firsanov V.V., Fam Vin’ Tkhien. Trudy MAI, 2019, no. 105. URL:
  13. Firsanov V.V., Fam Vin’ Tkhien, Chan Ngok Doan. Trudy MAI, 2020, no. 114. URL: DOI: 10.34759/trd-2020-114-07
  14. Petrov I.I., Serdyuk D.O., Skopintsev P.D. Trudy MAI, 2022, no. 124. URL: DOI: 10.34759/trd-2022-124-11
  15. Mikhailova E.Yu., Tarlakovskii D.V., Fedotenkov G.V. Obshchaya teoriya uprugikh obolochek (General theory of elastic shells), Moscow, Izd-vo MAI, 2018, 112 p.
  16. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v
    sploshnykh sredakh
    (Waves in continuous media), Moscow, FIZMATLIT, 2004, 472 p.
  17. Ventsel’ E.S., Dzhan-Temirov K.E., Trofimov A.M., Negol’sha E.V. Metod kompensiruyushchikh nagruzok v zadachakh teorii tonkikh plastinok i obolochek (Compensation method for advanced studies of thin plates and shells), Khar’kov, B.i., 1992, 92 p.
  18. Koreneva E.B. Metod kompensiruyushchikh nagruzok dlya resheniya zadachi ob anizotropnykh sredakh, International Journal for Computational Civil and Structural Engineering, 2018, vol. 14 (1), pp. 71–77.
  19. Gobson E.V. Teoriya sfericheskikh i ellipsoidal’nykh funktsii (Theory of spherical and ellipsoidal functions), Moscow, Izd-vo Inostrannoi literatury, 1952, 476 p.
  20. Dech G. Rukovodstvo k prakticheskomu primeneniyu preobrazovaniya Laplasa i Z-preobrazovanii (Guide to the practical application of the Laplace transform and Z-transforms), Moscow, Izd-vo NAUKA, 1971, 288 p.

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