1 *, and Sh r yaxshiboyev

The Mathematical Model of Transverse Vibrations of the Three-Layer Plate
Kh Khudoynazarov^{1 *}, and Sh R Yaxshiboyev
^1,2Department of Theoretical and applied mechanics, Samarkand State University, 100104, 15 University Boulevard, Samarkand, Uzbekistan.
^*Email: [e-mail]
Annotation. The article in a flat setting investigated the antisymmetric oscillations of a three-layer plate, which is infinite in plan. It is believed that the plate is not symmetrical in thickness. Based on the exact solutions of the equations of the linear theory of elasticity in transformations, a theory of unsteady transverse vibrations of a three-layer plate is developed. The oscillation equations are derived with respect to two auxiliary functions, which are the main parts of the longitudinal and transverse displacements of the points of some “intermediate” surface of the middle layer. The distance of this surface to the coordinate plane of the plate is arbitrary. All components of the stress tensors and displacement vectors at the points of the layers are expressed, like the vibration equations, through the introduced auxiliary functions. The problem of harmonic antisymmetric vibrations of an elastic three-layer plate is solved.
Key words: plate, layer, vibration, frequency, wave number, asymmetric structure
1. Introduction.
Multilayer plates and shells are widely used in various fields of technology. The sphere of usage of the three-layer plates is extremely wide. It includes such areas as construction, aircraft manufacturing, instrumentation, etc. Therefore, the calculation of such plates for the action of various dynamic loads is widely used in the design and operation of engineering structures, often working in extreme conditions on the effects of explosive, seismic and other loads [1].
2. Methods
C onsider a three-layer, infinite in plan, elastic plate. We assume that the plate consists of two bearing layers with thicknesses  and , and the middle layer with thicknesses (Fig. 1). In the case when the space between the bearing layers is filled with lighter, i.e. less rigid material, the middle layer is called aggregate. During deriving the equations of vibration, we assume that both the plate as a whole and each of its layers separately strictly obey the mathematical linear theory of elasticity and are described in exact formulations by its three-dimensional equations Fig.1 Geometry of a three-layer plate.
2.1. Formulation of the problem.
Given the unlimited size of the plate, further we will assume that it is under conditions of plane deformation. Therefore, we will consider the plate in a system of rectangular coordinates and direct the axis along the midline of the cross section, and the axis up, perpendicular to the axis . We call the supporting layers of the plate the first and second (in accordance with their thicknesses  and  ) layers, and the middle layer is zero. Therefore, we will consider the plate in a system of rectangular coordinates and direct the axis along the midline of the cross section, and the axis up, perpendicular to the axis. According to the stresses on strains at the points of isotropic layers of the plate are described by Hooke's linear law.
In the case of plane deformation, by introducing the components of displacement vectors according to the formulas
(1)
the equations of motion of the points of layers in a Cartesian system can easily be reduced to wave equations
(2)
Here  -elastic coefficients (Lame) and bulk density of the layers; and - some potential functions to be determined; -two-dimensional Laplace operator.
Transverse vibrations of the plate are excited under boundary conditions
. (3)
Where function and antisymmetric parts of the function of external dynamic loads. In addition, on the surfaces of the middle layer, at should be fulfilled by the following dynamic and kinematic conditions

and
(4)
The initial conditions of the problem are considered zero.
When defining the displacement components in the form (1), the stresses are given by the expressions


2.2. The equations of oscillation of a three-layer plate.
For solving assigned task, the functions of external influences from (3) can be represented as [20]
(5)
where ,   - функции, regular at having a finite number of poles, taking arbitrary values within a certain area , containing the spacing of the imaginary axis , decreasing when not slower at , then  , где , and such like outside their values are negligible.
In accordance with the accepted representations for the function of external influences and , we also represent potential functions in the form (3.1), substitution of which in (2) gives the ordinary Bessel differential equations with respect to the functions transformed by (5) and [21]
(6)
where , at .
The solutions of equations (6) in the case of transverse vibrations of the plate, taking into account the antisymmetric effects in the boundary conditions (3), will be
(7)

On the other hand, substituting solutions (7) into contact conditions (4) for ,we find We represent the stresses , as well as (5).Then, boundary conditions (3) can be written as

(9)
Substituting the expressions of constants , in (9) and expanding the trigonometric functions in series, on the left-hand sides of (9) by the degrees of the coordinate and by reversing the system of equations obtained in this way, we will have the general equations of the transverse vibrations of a three-layer plate. These equations have infinitely high orders in derivatives. We assume that the truncation conditions for the infinite series indicated in are satisfied and will be limited to zero or first approximations in the expansions. As a result, we obtain approximate equations of vibration of a three-layer plate for solving applied problems in which we pass to dimensionless variables by the formulas
, , , , , , ,
we obtain the equations:


(10)

= +
Where velocity of longitudinal waves in the material of the middle layer; - shear wave velocities in layer materials; - plate length.


2.3. The stress-strain state of the plate.
Along with the vibration equations, formulas are derived for all components of the stress tensors and displacement vectors at the points of all three layers of the plate. For example, the expressions for the displacement , as well as the stresses at the points of the middle layer, corresponding to the degrees of the oscillation equations (10) have the form

, (11)
.
here


2.4. Harmonic vibrations of a three-layer plate.
As an example, we consider the problem of antisymmetric (transverse) harmonic vibrations of a three-layer plate based on the obtained approximate equations of oscillation. It should be considered that the plate surfaces are free from external loads. Then the right-hand sides of the oscillation equations (10) will be equal to zero. The solution of differential equations (10) with zero right-hand sides will be sought in the form
, , (13)
here -circular frequency; k – wave number. Substituting (13) into the oscillation equations, we have a system of two homogeneous algebraic equations with respect to and
, , (14)
where
, ,
,
.
From (14) follows the frequency equation
. (15)
The last equation (15) was solved numerically using the “Maple 17” application package. In this case, the calculations were performed for steel and aluminum bearing layers of the plate. The values of their physical and mechanical material parameters are as follows: steel- E=2,010¹¹Па;ν=0,25;ρ=7850 ; aluminum -E=0,710¹¹Па;ν=0,35;ρ=2750 . The following materials and their physical and mechanical parameters are taken as a filler: polymer -E₀=5,510¹⁰Па;ν=0,4;ρ=1700 ; fiberglass -E₀=1,810¹⁰Па;ν=0,35;ρ=1400 ; wood plastic - E₀=1,210¹⁰Па;ν=0,35;ρ=1200 and tantalite - E₀=0,410¹⁰Па;ν=0,35;ρ=1300 . The geometric characteristics of the three-layer plate are as follows: thickness of the outer layersh₁ = h₂ = 0,001 м; aggregate thickness - h₀ = 0,03; 0,05; 0,1м.