^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

The paper solves the problem of the variation formulation of the steady-linear oscillations of structurally inhomogeneous viscoelastic plate system with point connections. Under the influence of surface forces, range of motion and effort varies harmonically. The problem is reduced to solving a system of algebraic equations with complex parameters. The system of inhomogeneous linear equations is solved by the Gauss method with the release of the main elements in columns and rows of the matrix. For some specific problems, the amplitude-frequency characteristics are obtained.

Currently, in many technical designs there are widely used shell and plate structures. Thin-walled tubes and panels in real conditions usually interact with other structures and bodies, which are based on resilient supports and also have hinge supports and associated masses. As in [

Consider a homogeneous isotropic a resilient plate of constant thickness h, limited to the size of a rectangular contour a, b. Suppose on the plate is Q dot added mass

where ω―given real frequency of the disturbing force,

Here z―coordinate of a point in a direction perpendicular to the middle surface, ε_{x}, ε_{y}, ε_{xy}―components of the strain tensor of the plate. To describe the relaxation processes occurring in the viscoelastic elements or point connections system, we adopt a linear Boltzmann theory of heredity:

where_{n}―instantaneous modulus of elasticity. To stress was periodic function of time, in a ratio heredity (2) the lower limit of integration taken to be negative infinity. If the lower limit is zero, the voltage will contain a periodic additive which decreases with time. On the influence function

In contrast to the problems of the natural oscillations [

If

where E―Young’s modulus, and v-Poisson’s ratio, which is assumed to be constant. The normal components G_{z} transverse rupture is small compared to G_{x} and G_{y}, therefore believe G_{z} = 0. The potential energy stored during the elastic deformation of the plate is given by:

where V―the volume of the plate. Substituting in (4) the values of the components of deformation and stress (2), (3) and taking into account the potential energy of elastic supports, we obtain

here

where ρ―the density of the plate material, x^{q}, y^{q}―coordinates q-th associated mass. We formulate the problem in terms of the method of virtual displacements, according to which the sum of the work of all active forces in the possible displacement δU, satisfies the boundary conditions, is equal to zero:

here

where_{q}-q-I added mass, L'―number-the elastic supports, Q―the number of additional masses, V,

where

where

transforms relaxation kernel material. This will eliminate the function of time

where G―total virtual works of the system, and F―kinematic conditions of rigid point constraints imposed on the system.

The task is now formulated as follows:

・ Let the driving force

・ Required depending on the frequency of the driving force to find a module of the displacement vector

In this paper we do not address the question of convergence of the method with a rigorous mathematical point of view, since it is not crucial for the following reasons. Energy approach used in the formulation of the problem is essentially the Ritz method, the convergence is strictly proved, for example, in [

The solution of the variation Equation (10) is sought in the form of a superposition of orthogonal basis functions. It is proposed that the elements are free from localized masses and all point connections (poles, posts) are known. Then as the desired displacement field satisfies the variation Equation (10) and specify the homogeneous boundary conditions, we assume a finite sum of these fundamental functions:

where

After substituting the sum (11) into Equation (10) coefficients

Symbols of all the quantities in the left part of the system of equations (12) coincide with the notation of [

the same routines that are used in the problem of natural oscillations [

sists of two sub vectors. If equation (10) is differentiated with respect Lagrange multiplier first

them the upper subvector vector

the disturbing force is distributed nature. Merits will not change if the applied force is concentrated. Then the virtual work space element is replaced by the virtual work of the concentrated force. Ultimately, the change shall

be subject only vector amplitudes

(12) is solved by the Gauss method with the release of the main elements in columns and rows. Note that the initial system of equations (8) has complex coefficients, so the program that implements the algorithm is written for the general case, i.e., for systems with complex numerical coefficients and complex unknowns. Right side of

the system, i.e. vector

where

The calculated components of the vector of unknowns, are complex quantities, i.e. vector

the for

sary to take

This part presents the solution of several problems, which are received and analyzed by the frequency-amplitude characteristics for the displacements of individual points of structurally inhomogeneous viscoelastic systems. The purpose of research is to confirm (or refute) the mechanical effects described in [_{1}―some fixed positive integer. The process ends after the transition from a K + 1 results in a change of the resonance curve, which is not more than 1% of the maximum in the vicinity of the resonance, or in any other region of interest.

Problem 1. Consider the design is a package of two parallel square plates with elastic shock absorber and the associated mass. This task determines the frequency-amplitude characteristics of the mechanical system depending on its geometrical parameters. The system is a package of two elastic square plates connected at the center of a weightless viscoelastic damper. Kernel for relaxation absorber selected as

where A, β, α―Kernel parameters.

This corresponds to approximately 60% surge creep contribution to the overall deformation of the viscoelastic body under quasi-static loading process. With the damper stiffness is fixed and taken to be 10.

For this case, the kernel parameters as follows: A = 0.078; α = 0.1; β = 0.05 . In both plates has one attached mass. Plate simply supported along a contour similar to mechanical and geometrical parameters, Е = 2 × 10^{11} н/м^{2}; ρ = 7.8 × 10^{3} кг/м^{3}; v = 0.3; а = b = 0.2 м; h = 0.001 м . Weights are equal to each other (М_{1} = М_{2} = 0.05 кг), one of them (М_{1}) fixed on the bottom plate at a point_{2}) can move through the central axis of the structure_{2} were the amplitude of forced oscillations of the system of plates. The amplitudes have been constructed for the central points of the two plates (x = y = 0.1 m). _{2} located as the point_{2} frequency-amplitude characteristics of the upper and lower plates are similar and therefore show on the same graph. Given in [_{2} one and the same. Thus, coincidence plots frequency amplitude characteristics at various locations weight М_{2} _{1}, ω_{2} a jump in amplitude as that lower and the upper plates, wherein the absolute value of these resonant amplitude lower and upper plates are different. Here and further the equality of natural and forced frequencies relative, since due to the dissipative properties of the structure of the jump amplitude maximum shifted somewhat to the left of their own (and hence disturbing) frequency. Let us analyze the behavior of the graphs in _{1} и ω = ω_{2} differ in terms of quantity, i.e. in the resonance amplitude. This fact is also consistent with _{2}

located at

The behavior of the resonant amplitudes for the top and bottom plates is qualitatively different from _{2} damping coefficients of the first global forms are different. It should be noted that here, as in the previous embodiments, the task number for the resonance amplitude plates plays a determinant role oscillation phase relative to each other, the first resonance occurs when the two plates oscillations in phase (with different or equal amplitudes), the second resonance occurs when vibrations of plates are in antiphase (shift for the period). Displacement field at forced oscillations separate plate within these resonant frequencies qualitatively unchanged―it is close to its own form.

Task 2. Two identical mechanical properties elastic plate (E = 28; ρ = 4; v = 0.3) are connected in the center of one weightless viscoelastic damper (spring). The mass of the spring M_{0} = 0.05, square plate (a = b = 1), supported along the contour, the thickness of the lower plate h_{1} = 0.1; and upper h_{2} = 0.046 on the bottom plate is attached at the center point mass. Recall that we consider two parallel hinged plates connected at the center of a viscoelastic damper. Plates are elastic square and thicknesses. On one of them (thicker) is fixed in the center of the concentrated mass. Parameters his relaxation kernel absorber (spring) determines the value A = 0.078; α = 0.1; β = 0.05 (higher viscosity). The design is distributed over the area of the two plates of the disturbing nature of the harmonic load. Vector amplitudes of these forces have components equal to one. For the case of forced oscillations is required to evaluate the dissipative properties of structurally inhomogeneous viscoelastic system in general, depending on the magnitude of the instantaneous stiffness of the shock absorber. Assessment methodology is as follows. Selected characteristic point of the system and for her with fixed parameter instantaneous damping built―frequency amplitude characteristic. In this case, the damping capacity of the system is determined by the maximum of the resonance amplitude. Then the parameter varies, and with all the defining (maximum) resonance amplitude is selected minimum. This parameter value and will fit most case dissipation system. As the feature points are selected as two of the central (x = y = 0.5) point on the bottom and top plates. Point to other coordinates gives qualitatively the same results. In [^{*} = 3.4 × 10^{−}^{3} и С^{*} = 5.4 × 10^{−}^{3}) explains the difference between the viscosities of shock absorbers.