Hydroelastic response of a sandwich plate possessing a compressible core and interacting with a rigid die via a viscous fluid layer

T. V. Bykova ; E. D. Grushenkova ; V. S. Popov ; A. A. Popova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 3, pp. 351-366

Résumé

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 three-layered 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.