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Hydroelastic response of a circular sandwich plate interacting with a liquid layer

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.


Introduction
Beams and plates are the main design elements in the schemes of actual engineering structures. Nowadays, composite structures, and in particular, sandwich plates, are widely used in order to reduce weight and size parameters, as well as to protect against various aggressive factors. The historical review of developing the model for the multilayered structural elements deformation based on the zigzag kinematic theories is given in [1]. The problems of studying the statics and dynamics of threelayered structural elements were considered in the monograph [2]. In reference [3], the problem of bending a three-layered beam under concentrated and distributed load action, placed in a temperature field, was solved. In reference [4], the deformation of an elastic-plastic three-layered circular plate in a temperature field and under the distributed load influence was studied. In reference [5], deformations of an annular sandwich plate under local loads were studied. The axisymmetric vibrations of a threelayered circular plate resting on the Winkler foundation were investigated in [6]. Considering the interaction of elastic structure with dissimilar bodies, we can specify hydroelasticity problems. Historically, one of the first problems of the kind was considered in [7]: it reviewed free oscillations of a circular plate contacting with unlimited volume of ideal liquid. In reference [8], the above-mentioned problem was solved on the basis of the coupled hydroelasticity problem, and in addition, in [9], the fluid viscosity was estimated. The circular plate oscillations immersed in ideal fluid located in a rigid cylinder and possessing a free surface were studied in [10]. The stability problems of rectangular plates interacting with ideal and viscous fluid were studied in [11,12]. In reference [13], the hydroelastic vibrations of a disk and a circular plate with viscous liquid in between them, caused by the foundation vibration were considered. The oscillations of a circular plate resting on a Winkler foundation and interacting with a viscous fluid layer were studied in [14]. The hydroelastic vibrations of the sensor elastic element in the form of a circular plate were investigated in [15]. However, there were fewer studies of the hydroelasticity problems for composite plates. For example, in [16][17][18], free oscillations and stability of the multilayer composite cantilevered beams and plates in air and water were studied both analytically and numerically. In references [19,20], the hydroelastic vibrations problems of three-layered beams and plates forming the wall of a narrow channel filled with viscous liquid were considered, taking into account the opposite wall vibration and the fluid pressure pulsation. However, the shear stresses of the liquid, as well as the inertia forces of the sandwich plate in the longitudinal direction, were excluded from consideration in above-mention publications.

Statement of the Problem
Let us consider a narrow channel formed by a three-layered circular plate and a rigid disk parallel and coaxial to it, as shown in Fig. 1. The radii of the plate and disk are R. We assume that the channel walls are clamped along the contour and consider the axisymmetric problem taking into account the channel axial symmetry. The sandwich circular plate is formed by the upper and lower sheets with a thickness of h 1 , h 2 , and the core with a thickness of 2c. We assume the plate core is lightweight and incompressible, and take into account the core work in tangential direction. Let us associate the cylindrical coordinate system center with the middle surface centre of the plate core. The distance between the channel walls in the unperturbed state is h 0 . The channel is filled with some viscous incompressible fluid. Along the contour, the fluid flows freely into the same fluid having a constant pressure of p 0 and a pulsating component p 1 (ωt). Furthermore, we assume that h 0 << R. We obtained the circular three-layered plate dynamics equations from the known equations of its equilibrium [2] using d'Alembert principle, i.e. taking into account the inertia forces of the upper and lower plate sheets in the radial and normal directions. These equations are: where q zr , q zz are shear and normal fluid stresses, respectively, acting on the plate surface; G k is the shear modulus of the k-th layer; K k is the bulk modulus of the k-th layer; ρ k is the density of the k-th layer material; u is the radial three-layered circular plate displacement; w is the three-layered circular plate deflection; φ is the rotation angle of the deformed normal in the three-layered circular plate core. The accepted notation for a 1 -a 6 : The stresses q zr , q zz on the plate surface according to [21] are written as ( The boundary conditions of Eq. (1) are: ( We assume the creeping fluid motion in a narrow channel, i.e. viscous fluid dynamics equations have the form [21]: where V r , V z are the fluid vector velocity projections on the axes of the cylindrical coordinate system, ν is the kinematic viscosity coefficient of the fluid, ρ is the fluid density. The boundary conditions of Eq. (4) are non-slip ones: In addition, we formulate the conditions for pressure at the channel contour and the symmetry axis:

Determining Circular Sandwich Plate Response
Let us introduce dimensionless variables and small parameters of the problem where ψ, λ are small parameters characterizing the problem. By substituting variables (7) into Eqs. (1)-(6), and neglecting the small terms [22], we obtain: dynamics equations of viscous incompressible fluid layer Solving the fluid dynamics equations (8) with boundary conditions (10), we obtain: The plate elastic displacements and the angle rotation of the plate core normal are determined from the solution of Eq. (9). Taking into account the boundary conditions (11) Here J 0 is the zero-order Bessel function of the first kind; I 0 is a modified zero-order Bessel function; β k is the root of the transcendental equation I 1 (β k )/I 0 (β k ) = -J 1 (β k )/J 0 (β k ), k = 1, 2,..., where J 1 (β k ), I 1 (β k ) are the corresponding first-order Bessel functions [2]. The coefficients 0 k R , 0 k Q , 0 k T correspond to static plate movements at a constant pressure level p 0 , while harmonic time functions k R , k Q , k T correspond to plate movements due to pressure pulsation. Substituting (13) in (12) and (9) and re-performing representation of the remaining constants and functions as the series (13), we obtain the following equations for the main mode of oscillations:     (   11  21  21  11  11  12  21  22  11  12   11  21  21  11  12  12  21  22  11 12 where the following symbols are entered
These amplitude frequency responses allow us determining the resonant frequencies of the main mode of the radial and bending vibrations, and to estimate the vibrations amplitude at those frequencies. The results of the calculations are shown in Fig.2 and Fig.3.

Summary and Conclusion
Thus, we constructed the equations for the radial and bending vibrations of the sandwich plate and amplitude-frequency responses for the main mode of the plate's oscillations. Computations have shown that two resonant frequencies were observed on the main oscillation mode of the plate, which could be explained by the cross-influence of inertia and stiffness forces in the radial and normal directions. Therefore we conclude that, in contrast to the hydroelasticity problems of homogeneous plates, for three-layered plates, it is important to take into account the inertia forces and the fluid stresses in the radial direction.