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We investigated the natural oscillations of dissipative inhomogeneous plate mechanical systems with point connections. Based on the principle of virtual displacements, we equate to zero the sum of all active work force, including the force of inertia which obtain equations vibrations of mechanical systems. Frequency equation is solved numerically by the method of Muller. According to the result of numerical analysis we established nonmonotonic dependence damping coefficients of the system parameters.

Studies related to the definition of inherent characteristics of plates with attached masses are discussed in [

We will consider the mechanical system consisting of N isotropic viscoelastic plates, occupying volume

Mathematically, the viscoelastic problem is as follows. Let all the n-th point of the body are subject to a harmonic law fluctuations, i.e.

where

For rectangular plates

where x, y?coordinate. Proceeding from the principle of possible movements, we will equate to zero sum of works of all active forces, including inertia forces on possible movements

where

where

We will determine physical ratios for n-th viscoelastic body of system by equality [

where

here

Considering (1), time function in equality (5) will be with slowly

where

Time allows to exclude it from the variation equation integrated members and, finally. In a symbolical look it can be presented in a look

We will write out concrete representation of functionality of G, for example, for a package of rectangular plates with dot communications:

where

where

instant rigidity according to of n-th plate, of l-th shock-absorber, of l'-th support. In an elastic case

The similar functionality can be written down for system of covers of rotation.

Components of a vector of movements

It was necessary to impose rigid dot communications which don’t make work at fluctuations on system. Terms of hard hinged support the n-th body in

where

If the part of support has jamming, conditions will be added

where

Existence of rigid racks between n-th and (n + 1)-th body at

where

Thus, on a vector of movements restrictions of types (8)-(11) are in addition imposed. On system dot communications we will consider imposing by means the method of Lagrange multipliers. Then the variational Equation (10) will correspond in a look

where

It is necessary to find a range of complex own frequencies

Approach the solution of the variational Equation (12), as well as in case of an elastic task, we look for in the form of the approximating form made of fundamental functions, satisfying both to the equation, and the set geometrical boundary conditions on surfaces of

where

Previously

Without providing concrete calculations, we will write down this system in a matrix look:

where^{0} of dimension represents

Structurally matrixes A and B are similar described in work [

The degeneracy of the matrix B as a resilient problem is caused in the introduction of additional point connections (rigid supports and pillars). The transformed matrixes will have dimension

which, unlike a case of an elastic task, will be complex. The most effective way of the solution of the similar equations, apparently, is the method of Müller [

Systems of rectangular plates and with dot communications are considered. We will consider the design representing a package from two parallel square elastic plates with the shock-absorber and the attached weight. The relaxation kernel for the shock-absorber is chosen in a look

where_{0} = 0.05. Plates square(a = b = 1), supported on a contour, thickness of the bottom plate h_{1} = 0.1, and of the top plate h_{2} = 0.046, on the bottom plate in the center dot weight is attached. Viscosity of the shock-absorber is accepted such that its deformation of creep at quasistatic process made a small share from the general (~12%). For this case kernel parameters following:^{−4} to 10^{−1}. On the right this range is limited by the size since at C = C_{2} there is a change of the second form. On

Real illustration of this effect is existence of a point of intersection of schedules of damping coefficients _{H} and A_{b} can be transferred to the following matrix) and, secondly, all functions are identical. Then the system of the Equation (14) in a matrix look can be copied so:

where A^{n}?a numerical matrix of the total instantaneous stiffness of viscoelastic elements of the system.After an exception linearly dependent the component from system also can be written down (15) transformed matrixes of the generalized instant

where^{*} point * thereisenergy “transfer”. Consequently, for free oscillations between forms of energy is exchanged. This is especially true if the forms have similar natural frequencies. At the point of intersection of the graphs of damping factors ^{*}, the “pumping” of energy from the second form to the first, so the latter most intensively dissipates energy. After the point of intersection of the difference between the first natural frequency increases, the appropriate forms of interaction decreases and their dissipative properties take a regular character. The practical conclusion is that the damping capacity of the structure is mainly determined by the absolute value of the minimum damping factor (in this case the latest damped oscillations precisely this form); global (determining) the damping factor of the system is the first ^{*}, when the damping factor of the global maximum.

The second example provides the mechanical system consisting of two parallel identical mechanical properties and geometry of the elastic plat in

connected by a weightless viscoelastic damper. The parameters of its core relaxation

In this problem, we investigate the following variant: shock absorber is located in the center of the plates, the masses are the central axis of the structure, and М_{1} is fixed to the first (bottom) plate at a distance of 0.04 m from the absorber, and the position of the mass М_{2} to the second (upper) plate was varied along the central axis of the structure.

_{2}. Calculation showed a strong dependence of the damping of the first global form _{2}. Eigenfrequencies _{2} of the plate at those points become identical as to the rigidity, and on the inertial characteristics. If the mass М_{2} to mix in other areas on the top plate, then there is also the point at which the damping factor _{2} there are an infinite number of positions (at fixed positions of the shock absorber, and М_{1}, in which no attenuation of the first global shape. These points form a closed curve (close to a circle) on the second plate. Analysis of

suitable arrangement of added mass. Symmetry of the left and right branches of the graph provides an opportunity to obtain the maximum (and equal) the effectiveness of the shock absorber arrangement as cargo on either side of him (left leg), and on one side (right branch).

Described effect shows that the energy of the system depends not only on the rheological properties of the damping material, but also on the geometry of the structure as a whole. Similar effect was observed for the system with finite number of degrees of freedom [

Safarov Ismail Ibrahimovich,Teshaev Muhsin Hudoyberdievich,Madjidov Maqsud, (2014) Natural Oscillations of Viscoelastic Lamellar Mechanical Systems with Point Communications. Applied Mathematics,05,3018-3025. doi: 10.4236/am.2014.519289