We considered the formulation and solution of the forced oscillations hydroelasticity problem for a three-layered circular plate contacting with a viscous incompressible fluid layer, the pressure in which varies according to the harmonic law. The plate is the bottom wall of a narrow channel completely filled with a viscous fluid. The axisymmetric coupled hydroelasticity problem consisting of the plate dynamics equation, the viscous fluid layer dynamics equation, and their corresponding boundary conditions was investigated. We obtained the plate dynamics equations taking into account inertia forces in the radial and normal directions in the framework of zigzag kinematic theory. In these equations, the load was expressed by the stresses of the viscous fluid contacting with the three-layered circular plate. The fluid dynamics equations were represented by the Navier-Stokes equations and continuity equation written for the case of creeping fluid flow in a channel. We obtained the forced radial and bending hydroelastic oscillations equations of the circular three-layered plate using the perturbation method. The solution of these equations was represented by a series of eigenfunctions of the corresponding Sturm-Liouville problem. We have also presented the numerical study results of the radial and bending vibrations amplitude dependence on the frequency for the main steady oscillations mode of the plate. © Published under licence by IOP Publishing Ltd.

The mathematical modelling of a vibrating stamp interaction with a flexible restrained end wall (bellows) of a narrow channel via a viscous incompressible fluid is carried out. The narrow channel formed by two parallel walls and filled with a viscous liquid is investigated. The liquid motion in the channel is considered as a laminar one. The bottom channel wall is absolutely rigid, and the top one is the absolutely rigid vibrating stamp. At the right edge of the channel, the end wall with flexible restraint (bellows) is installed, and at the left one, there is a free liquid flow. The problem of longitudinal hydroelastic vibrations of the channel end wall is formulated and solved analytically. The distribution laws of velocities and pressure in the fluid layer along the channel are found. The motion law of the channel end wall is determined. The amplitude frequency and phase frequency responses of the end wall for steady-state harmonic oscillations are constructed. The mathematical modelling has shown the possibility of damping the channel end wall (bellows) vibrations by changing the distance between the channel walls or by changing the liquid viscosity in the channel. © Published under licence by IOP Publishing Ltd.

The hydroelastic oscillations problem of the sandwich beam resting on elastic foundation and interacting with the pulsating fluid layer was investigated. The three-layered beam with an incompressible lightweight filler was considered. The broken normal hypothesis for the three-layered beam and the model of viscous incompressible fluid, as well as Winkler model for elastic foundation, was chosen to study hyroelastic problem. A mathematical model of the considered mechanical system consists of dynamic equations of the three-layered beam with incompressible lightweight filler and the Navier-Stokes equations with continuity equation for the pulsating viscous fluid layer. The no-slip conditions and pressure coincidence at the edges with the given pressure into edges cavities were selected as boundary ones. The plane problem of hydroelastic beam bending oscillations for the regime of steady harmonic ones were considered. The solution of the hydroelastic problem was carried out by the perturbations method using proposed small parameters of the hydroelastic problem. As a result, the laws of three-layered beam elastic deflections and the hydrodynamic parameters of pulsating viscous fluid layer were obtained. © 2019 IOP Publishing Ltd. All rights reserved.

The problem of mathematical modeling of longitudinal and transverse oscillations of an elastically fixed wall of a narrow wedge-shaped channel filled with a viscous incompressible liquid and installed on vibrating foundation was stated and solved analytically. The problem was considered in a flat definition for the mode of steady-state forced harmonic oscillations. The mathematical model developed includes the Navier-Stokes equations and the equation of continuity for a thin layer of viscous incompressible liquid and the equations of dynamics of an elastically fixed channel wall. The conditions of adhesion of the liquid to the channel walls and the conditions of a free outflow of liquid at the ends were chosen as the boundary conditions. For the stated problem, the system of dimensionless variables was proposed. Analytical expressions for the wall displacements and the hydrodynamic parameter of the layer of liquid were obtained. The calculations, which made it possible to estimate the influence of inclination of the channel wall and the viscosity of the liquid to damping of oscillations of the elastically fixed wall, were performed. © 2018, Allerton Press, Inc.

Longitudinal and bending oscillations of a three-layer plate interacting with vibrating stamp through a thin layer of a viscous incompressible fluid are investigated. We study the plane problem. The plate is considered as a three-layer package consisting of two bearing layers with an incompressible filler between them. The broken normal hypothesis is used to cinematically describe a three-layer package. A mathematical model including Navier-Stokes equations, continuity equation and three-layer plate dynamics equations with the corresponding boundary conditions is developed. No-slip conditions, pressure conditions at the channel edges and simply supported plate edges conditions are selected as boundary ones. The hydrodynamic parameters distribution laws of the liquid layer and the elastic displacements of the three-layer plate are found by mines of the perturbation method. © Published under licence by IOP Publishing Ltd.