Title of the article VIBRATIONS OF A THREE-LAYER CIRCULAR STEP PLATE UNDER PERIODIC IMPACT
Authors

LEONENKO Denis V., D. Sc. in Phys. and Math., Prof., Professor of the Department “Structural Mechanics”, 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.

MARKOVA Marina V., Postgraduate Student of the Department “Structural Mechanics”, 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 2022
Issue 3(60)
Pages 68–76
Type of article RAR
Index UDK 539.3
DOI https://doi.org/10.46864/1995-0470-2022-3-60-68-76
Abstract Forced oscillations of a three-layer circular plate with step-variable thickness of the outer layers are analyzed. The deformation of the plate is described with the zig-zag theory. In thin border layers of plate Kirchhoff’s hypotheses are valid. In a relatively thick in thickness medium layer Timoshenko’s hypothesis on the straightness and incompressibility of the deformed normal is fulfilled. The equations of motion are derived from Hamilton’s variational principle. A special case of exposure is considered: periodic sequence of strokes with constant intensity. The problem is reduced to finding three required functions in each section, deflection, shear and radial displacement of the median plane of the filler. The solution is presented as a sum of quasi-static and dynamic components of the unknown displacements. Numerical results of the obtained solution are presented. The influence of the impact stress on the oscillatory character is analyzed.
Keywords circular three-layer plate, plate with step-variable thickness, forced vibration, periodic strokes
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Bibliography
  1. Nguyen C.H., Butukuri R.R., Chandrashekhara K., Birman V. Dynamics and buckling of sandwich panels with stepped facings. International journal of structural stability and dynamics, 2011, no. 4(11), pp. 697–716.
  2. Nguyen C.H., Chandrashekhara K., Birman V. Enhanced static response of sandwich panels with honeycomb cores through the use of stepped facings. Journal of sandwich structures & materials, 2011, no. 2(13), pp. 237–260.
  3. Lal R., Rani R. On the use of differential quadrature method in the study of free axisymmetric vibrations of circular sandwich plates of linearly varying thickness. Journal of vibration and control, 2016, no. 7(22), pp. 1729–1748.
  4. Rani R., Lal R. Axially symmetric vibrations of circular sandwich plates of linearly varying thickness. Proc. 3rd International conference on soft computing for problem solving “Advances in intelligent systems and computing”. New Delhi, 2014, no. 258, pp. 169–181.
  5. Lal R., Rani R. On radially symmetric vibrations of circular sandwich plates of non-uniform thickness. International journal of mechanical sciences, 2015, no. 99, pp. 29–39.
  6. Lal R., Rani R. On the radially symmetric vibrations of circular sandwich plates with polar orthotropic facings and isotropic core of quadratically varying thickness by harmonic differential quadrature method. Meccanica, 2016, no. 51, pp. 611–634.
  7. Rani R., Lal R. Radially symmetric vibrations of exponentially tapered clamped circular sandwich plate using harmonic differential quadrature method. Mathematical analysis and its applications, 2015, no. 143, pp. 633–643.
  8. Süsler S., Türkmeni H. Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading. International journal of mechanics and materials in design, 2017, no. 13, pp. 429–451.
  9. Jalali S.K., Heshmati M. Buckling analysis of circular sandwich plates with tapered cores and functionally graded carbon nanotubes-reinforced composite face sheets. Thin-walled structures, 2016, no. 100, pp. 14–24.
  10. Chang J.S., Chen H.C. Free vibrations of sandwich plates of variable thickness. Journal of sound and vibration, 1992, no. 2(155), pp. 195–208.
  11. Bauchau O., Craig J. Kirchhoff plate theory. Structural analysis, 2009, vol. 163, pp. 819–914.
  12. Timoshenko S.P. On the correction for shear the differential equation for transverse vibrations of the prismatic bars. Philosophical magazine and journal of science, 1921, no. 245(41), pp. 744–746.
  13. Bolotin V.V., Novichkov Yu.N. Mekhanika mnogosloynykh konstruktsiy [Mechanics of multilayer structures]. Moscow, Mashinostroenie Publ., 1980. 375 p. (in Russ.).
  14. Bolotin V.V. K teorii sloistykh plit [Towards the theory of layered slabs]. Izvestiya AN SSSR. Mekhanika i mashinostroenie, 1963, no. 3, pp. 65–72 (in Russ.).
  15. Novichkov Yu.N. Variatsionnye printsipy dinamiki i ustoychivosti mnogosloynykh obolochek [Variational principles of dynamics and stability of multilayer shells]. Trudy Moskovskogo energeticheskogo instituta. Dinamika i prochnost mashin, 1973, no. 164, pp. 14–22 (in Russ.).
  16. Grigolyuk E.I., Chulkov P.P. Ustoychivost i kolebaniya trekhsloynykh obolochek [Stability and vibrations of three-layer shells]. Moscow, Mashinostroenie Publ., 1973. 172 p. (in Russ.).
  17. Grigolyuk E.I., Chulkov P.P. Nelineynye uravneniya tonkikh mnogosloynykh obolochek regulyarnogo stroeniya [Nonlinear equations of thin multilayer shells of regular structure]. Inzhenernyy zhurnal. Mekhanika tverdogo tela, 1967, no. 1, pp. 163–169 (in Russ.).
  18. Leonenko D.V. Elastic bending of a three-layer circular plate with step-variable thickness. Mechanics of machines, mechanisms and materials, 2021, no. 1(54), pp. 25–29.
  19. Leonenko D.V. Lokalnoe nagruzhenie stupenchatoy krugovoy sendvich-plastiny [Local loading of a stepped circular sandwich plate]. Mechanics. Researches and innovations, 2021, no. 14(14), pp. 126–130 (in Russ.).
  20. Leonenko D.V. Poperechnyy izgib krugovoy sendvich-plastiny stupenchatoy tolshchiny [Transverse bending of a circular sandwich plate of stepped thickness]. Proceedings of Francisk Skorina Gomel State University, 2020, no. 6(123), pp. 151–155 (in Russ.).
  21. Parfenova V.S. Deformirovanie krugloy uprugoy trekhsloynoy plastiny so stupenchato-peremennoy granitsey [Deformation of a circular elastic three-layer plate with a step-variable boundary]. Mechanics. Researches and innovations, 2017, no. 10(10), pp. 157–163 (in Russ.).
  22. Starovoytov E.I., Tarlakovskiy D.V. Deformirovanie trekhsloynoy ortotropnoy plastiny stupenchato-peremennoy tolshchiny [Deformation of a three-layer orthotropic plate of stepwise variable thickness]. Fundamental and applied problems of technics and technology, 2014, no. 2(304), pp. 38–43 (in Russ.).
  23. Shlyakhin D.A. Vynuzhdennye osesimmetrichnye kolebaniya tonkoy krugloy plastiny stupenchato peremennoy tolshchiny i zhestkosti [Forced axisymmetric oscillations of a thin circular plate of stepwise variable thickness and stiffness]. News of higher educational institutions. Construction, 2013, no. 4(652), pp. 13–20 (in Russ.).
  24. Shlyahin D.A. Vynuzhdennye osesimmetrichnye kolebaniya tonkoy krugloy bimorfnoy plastiny stupenchato peremennoy tolshchiny i zhestkosti [Forced axisymmetric oscillations of a thin circular bimorphic plate of stepwise variable thickness and stiffness]. Engineering journal of Don, 2013, no. 1(24), pp. 36–45 (in Russ.).
  25. Hosseini-Hashemi Sh., Rezaee V., Atashipour S.R., Girhammar U.A. Accurate free vibration analysis of thick laminated circular plates with attached rigid core. Journal of sound and vibration, 2012, no. 25(331), pp. 5581–5596.
  26. Hosseini-Hashemi Sh., Derakhshani M., Fadaee M. An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates. Applied mathematical modelling, 2013, no. 6(37), pp. 4147–4164.
  27. Molla-Alipour M. Bending analysis of FG circular and annular plates with stepped thickness variations by using a new exact closed form solution. 2017. Available at: https://www.semanticscholar. org/paper/Bending-Analysis-of-FG-Circular-and-Annular- Plates-Molla-Alipour/a052dc4841b7b7f41692655a0da- 0f2abc14cfba1#paper-header.
  28. Zorich V.A. Matematicheskiy analiz. Chast 1 [Mathematical analysis. Part 1]. Moscow, MTsNMO Publ., 2012. 710 p. (in Russ.).
  29. Grigolyuk E.I., Kogan F.A. Sovremennoe sostoyanie teorii mnogosloynykh obolochek [The current status of the theory of multilayer shells]. Prikladnaya mekhanika, 1972, no. 6(8), pp. 5–17 (in Russ.).
  30. Carrera E. Historical review of zig-zag theories for multilayered plates and shells. Applied mechanics reviews, 2003, no. 3(56), рp. 287–308.
  31. Icardi U., Sola F. Assessment of recent zig-zag theories for laminated and sandwich structures. Composites Part B-engineering, 2016, no. 97, pp. 26–52.
  32. Nowacki W. Teoria sprężystości. Warszawa, Państwowe Wydawnictwo Naukowe, 1970.
  33. Zhou Z.H., Wong K.W., Xu X.S., Leung A.Y.T. Natural vibration of circular and annular thin plates by Hamiltonian approach. Journal of sound and vibration, 2011, no. 5(330), pp. 1005–1017.
  34. Markova M.V., Leonenko D.V. Postanovka nachalno-kraevoy zadachi ob osesimmetrichnykh kolebaniyakh krugovoy trekhsloynoy plastiny peremennoy tolshchiny [Definition of the initial-boundary value problem for axisymmetric vibrations of a circular three-layer plate with variable thickness]. Teoreticheskaya i prikladnaya mekhanika, 2022, iss. 36, pp. 3–10 (in Russ.).
  35. Starovoytov E.I., Pleskachevskiy Yu.M., Leonenko D.V., Tarlakovskiy D.V. Deformirovanie stupenchatoy kompozitnoy balki v temperaturnom pole [Straining of a step-variable thickness composite beam in a temperature field]. Inzhenerno-fizicheskii zhurnal, 2015, vol. 88, no. 4, pp. 987–993 (in Russ.).
  36. Starovoytov E.I. Poddubnyy A.A. Izgib trekhsloynogo sterzhnya so stupenchato-peremennoy granitsey, chastichno opertogo na uprugoe osnovanie [The bending of three-layer beam with variable border laying on the elastic basis]. Mechanics of machines, mechanisms and materials, 2011, no. 1(14), pp. 47–55 (in Russ.).
  37. Starovoytov E.I., Leonenko D.V., Rabinskiy L.N. Deformirovanie trekhsloynykh fizicheski nelineynykh sterzhney [Deformation of three-layer physically-nonlinear bars]. Moscow, MAI Publ., 2016. 184 p. (in Russ.).
  38. Tong K.N. Theory of mechanical vibration. New York, Wiley, 1960. 370 p.
  39. Aramanovich I.G., Levin V.I. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1969. 288 p. (in Russ.).
  40. Markova M.V. Sobstvennye kolebaniya krugovoy trekhsloynoy stupenchatoy plastiny [Self-oscillations of the circular three-layer staged-thickness plate]. Mechanics. Researches and innovations, 2021, no. 14(14), pp. 147–158 (in Russ.).
  41. Bateman H., Arthur E. Higher transcendental functions. New York, McGraw-Hill, 1953, 990 p.
  42. Watson G.N. A treatise on the theory of Bessel functions. Cambridge University Press, 1944. 804 p.