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Bykova T. V., Grushenkova E. D., Popov V. S., Popova A. A. Hydroelastic Response of a Sandwich Plate Possessing a Compressible Core and Interacting with a Rigid Die Via a Viscous Fluid Layer. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 3, pp. 351-366. DOI:

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Hydroelastic Response of a Sandwich Plate Possessing a Compressible Core and Interacting with a Rigid Die Via a Viscous Fluid Layer

Bykova Tatyana V., Saratov State Technical University
Grushenkova Ekaterina D., Saratov State Technical University
Popov Victor Sergeevich, Saratov State Technical University
Popova Anna A., Saratov State Technical University

The three-layered plate interaction with a rigid die through a layer of viscous fluid was investigated. The plate and rigid die formed a narrow channel with rectangular parallel walls. The channel was completely filled with a viscous incompressible fluid. The fluid movement in the channel was studied as a creeping one. The motion law of the rigid die was considered to be given as a harmonic one and the forced steady-state oscillations problem of the sandwich plate was considered. The upper and lower face sheets of the plate satisfied Kirchhoff’s hypotheses, as well as, the core was assumed a compressible one. The displacements of the channel walls were believed to be much smaller than the distance between them, and the longitudinal size of the channel was considered to be much larger than its transverse one. The plane hydroelastic problem consisting of the Navier –Stokes equations, the continuity equation and the dynamics equations of the threelayered plate with compressible core was studied. The boundary conditions of the problem were the no-slip conditions, the conditions for pressure at the channel edges and the simply supported conditions at the plate edges. In the course of study, normal and shear stresses of the fluid, acting on the upper face sheet of the plate were taken into account. The elastic displacements of the plate layers were chosen in the form of a trigonometric function series of the longitudinal coordinate. From the solution of the problem, expressions of the fluid layer hydrodynamic parameters and the plate layers elastic displacements were obtained. Also, the frequency-dependent amplitude distribution functions of the plate layers displacements and the pressure of the viscous fluid layer were constructed.

Article type: 
RAR - research article
This work was supported by the Russian Foundation for Basic Research (projects No. 18-01-00127-a, No. 19-01-00014-a).
  1. Lamb H. On the vibrations of an elastic plate in contact with water. Proc. Roy. Soc. A, 1921, vol. 98, pp. 205–216. DOI:
  2. Amabili M., Kwak M. K. Free vibrations of circular plates coupled with liquids: Revising the Lamb problem. J. Fluids Struct., 1996, vol. 10, iss. 7, pp. 743–761. DOI:
  3. Amabili M. Vibrations of Circular Plates Resting on a Sloshing Liquid: Solution of the Fully Coupled Problem. J. Sound Vib., 2001, vol. 245, iss. 2, pp. 261–283. DOI:
  4. Askari E., Jeong K.-H., Amabili M. Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface. J. Sound Vib., 2013, vol. 332, iss. 12, pp. 3064– 3085. DOI:
  5. Alekseev V. V., Indeitsev D. A., Mochalova Yu. A. Vibration of a flexible plate in contact with the free surface of a heavy liquid. Tech. Phys., 2002, vol. 47, iss. 5, pp. 529–534. DOI:
  6. Ankilov A. V., Vel’misov P. A., Tamarova Iu. A. Research on dynamics and stability of an elastic element of the flow channel. Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 2016, vol. 18, no. 1, pp. 94–107 (in Russian).
  7. Bochkarev S. A., Lekomtsev S. V., Matveenko V. P. Hydroelastic stability of a rectangular plate interacting with a layer of ideal flowing fluid. Fluid Dynamics, 2016, vol. 51, no. 6, pp. 821–833. DOI:
  8. Avramov K. V., Strel’nikova E. A. Chaotic oscillations of plates interacting on both sides with a fluid flow. Int. Appl. Mech., 2014, vol. 50, no. 3, pp. 303–309. DOI:
  9. Haddara M. R., Cao S. A Study of the Dynamic Response of Submerged Rectangular Flat Plates. Marine Struct., 1996, vol. 9, no. 10, pp. 913–933. DOI:
  10. Chapman C. J., Sorokin S. V. The forced vibration of an elastic plate under significant fluid loading. J. Sound Vib., 2005, vol. 281, iss. 3, pp. 719–741. DOI:
  11. Ergin A., Ugurlu B. Linear vibration analysis of cantilever plates partially submerged in fluid. J. Fluids Struct., 2003, vol. 17, iss. 7, pp. 927–939. DOI:
  12. Kozlovsky Y. Vibration of plates in contact with viscous fluid: Extension of Lamb’s model. J. Sound Vib., 2009, vol. 326, iss. 1–2, pp. 332–339. DOI:
  13. Onsay T. Effects of layer thickness on the vibration response of a plate-fluid layer system. J. Sound Vib., 1993, vol. 163, iss. 2, pp. 231–259. DOI:
  14. Ageev R. V., Bykova T. V., Kondratova J. N. Mathematical Modeling of Interaction Between Layer of Viscous Liquid and Elastic Walls of Channel, Which Was Installed on Vibration Foundation. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2011, vol. 11, iss. 2, pp. 48–54 (in Russian). DOI:
  15. Faria C. T., Inman D. J. Modeling energy transport in a cantilevered Euler-Bernoulli beam actively vibrating in Newtonian fluid. Mech. Syst. Signal Processing, 2014, vol. 45, iss. 2, pp. 317–329. DOI:
  16. Mogilevich L. I., Popov V. S. Investigation of the interaction between a viscous incompressible fluid layer and walls of a channel formed by coaxial vibrating discs. Fluid Dyn., 2011, vol. 46, no. 3, pp. 375–388. DOI:
  17. Alekseev V. V., Indeitsev D. A., Mochalova Yu. A. Resonant oscillations of an elastic membrane on the bottom of a tank containing a heavy liquid. Tech. Phys., 1999, vol. 44, no. 8, pp. 903–907. DOI:
  18. Kondratov D. V., Mogilevich L. I., Popov V. S., Popova A. A. Hydroelastic Oscillations of a Circular Plate, Resting on Winkler Foundation. J. Phys.: Conf. Ser., 2018, vol. 944, 012057. DOI:
  19. Mogilevich L. I., Popov V. S., Popova A. A., Christoforova A. V. Mathematical Modeling of Hydroelastic Oscillations of the Stamp and the Plate, Resting on Pasternak Foundation. J. Phys.: Conf. Ser., 2018, vol. 944, 012081. DOI:
  20. Gorshkov A. G., Starovoitov E. I., Yarovaya A. V. Mekhanika sloistykh vyazkouprugoplasticheskikh elementov konstruktsii [Mechanics of layered viscoelastoplastic structural elements]. Moscow, Fizmatlit, 2005. 576 p. (in Russian).
  21. Starovoitov E. I., Leonenko D. V. Deformation of a three-layer elastoplastic beam on an elastic foundation. Mech. Solids, 2011, vol. 46, no. 2, pp. 291–298. DOI:
  22. Leonenko D. V., Starovoitov E. I. Thermal impact on a circular sandwich plate on an elastic foundation. Mech. Solids, 2012, vol. 47, no. 1. pp. 111–118. DOI:
  23. Starovoitov E. I., Leonenko D. V. Bending of a Place Sandwich Beam by Local Loads in the Temperature Field. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 1, pp. 69–83. DOI:
  24. 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, no. 3. P. 725–739. DOI:
  25. Starovoitov E. I., Leonenko D. V. Variable Bending of a Three-layer Rod with a Compressed Filler in the Neutron Flux. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2017, vol. 17, iss. 2, pp. 196–208 (in Russian). DOI:
  26. Kramer M. R., Liu Z., Young Y. L. Free vibration of cantilevered composite plates in air and in water. Composite Structures, 2013, vol. 95, pp. 254–263. DOI:
  27. Ageev R. V., Mogilevich L. I., Popov V. S. Vibrations of the walls of a slot channel with a viscous fluid formed by three-layer and solid disks. Journal of Machinery Manufacture and Reliability, 2014, vol. 43, no. 1, pp. 1–8. DOI:
  28. Popov V. S., Mogilevich L. I., Grushenkova E. D. Hydroelastic response of three-layered plate interacting with pulsating viscous liquid layer. In: Radionov A., Kravchenko O., Guzeev V., Rozhdestvenskiy Y. (eds). Proceedings of the 4th International Conference on Industrial Engineering. ICIE 2018. Lecture Notes in Mechanical Engineering. Cham, Springer, 2019, pp. 459–467. DOI:
  29. Chernenko A., Kondratov D., Mogilevich L., Popov V., Popova E. Mathematical modeling of hydroelastic interaction between stamp and three-layered beam resting on Winkler foundation. Studies in Systems, Decision and Control, 2019, vol. 199, pp. 671–681. DOI:
  30. Loitsianskii L. G. Mekhanika zhidkosti i gaza [Mechanics of Liquids and Gases]. Moscow, Drofa, 2003. 840 p. (in Russian).
  31. Panovko Ia. G., Gubanova I. I. Ustoichivost’ i kolebaniia uprugikh sistem [Stability and Oscillations of Elastic Systems]. Moscow, Nauka, 1987. 352 p. (in Russian).
  32. Van Dyke M. Perturbation methods in fluid mechanics. Stanford, Parabolic Press, 1975. 271 p.
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