SCIENCE & EDUCATION

The authors consider different mathematical models of thermal interaction of ball inclusions and the matrix of a composite, used for estimating the effective heat conduction coefficient of this kind of composite. Application of a dual formulation of the variation problem of stationary thermal conductivity in a nonhomogeneous solid body allowed to get double-sided estimations of possible values of the coefficient. It was determined that transition from ball inclusions to cubic ones influenced slightly the effective heat conduction coefficient of a composite. The authors derived an evaluation formula that allowed to obtain a reliable estimation of this coefficient within the whole range of possible variations of the volume concentration of inclusions. Due to the electrothermal analogy it was possible to apply the obtained results in order to estimate characteristics of electrical conductivity and of dielectric capacitance of composites modified with ball inclusions (considering nanostructured elements).

On the basis of a developed mathematical model of heat transfer in composite with anisotropic inclusions of ellipsoidal shape, a procedure for calculating components of an effective heat-conduction tensor of the textured composite was proposed. A variant of inclusions in the form of ellipsoids of rotation was considered. The main axes of the heat-conduction tensor of such inclusions coincide with symmetric axes of an ellipsoid. In case of conic texture of a composite its possible dispersion was taken into account. The obtained results could be used for estimating effective thermal conductivity of composites modified with nanostructured elements (including carbon nanotubes). Due to electrothermal analogy these results are applicable when consideringcharacteristics of electrical conductivity and dielectric capacitance of textured composites.

Works of Lode V., Yagna Yu. I. and other researchers point to the presence of deviations from the laws of plasticity that found wide application in scientific literature. Analysis of the results of studies with such materials as steel, nickel, copper, and duralumin shows the possibility of studying and description of marked deviations from the standpoint of modern representations of the tensor of nonlinearity. In this work, on the basis of the results of the mentioned works, it was shown that consideration of this non-linearity leads to increases in accuracy of determination of both stresses and deformations, if we reject a single relation between intensity of stress and intensity of deformations. For this purpose, derivation of determining equations of the deformation theory of plasticity, which takes into account the effect of loosening, a form of hardening curves and their dependence on the kind of stressed state was performed using one of the variants of the theory of V. V. Novozhilova.

Nowadays small plants for tire recycling based on milling and with an annual consumption of raw materials up to 5000 tons are widespread in Russia. The authors research the problem of extreme vibrations in such equipment when used tires are disposed. Analysis is based on the developed setup and its prototype. This article describes the main causes of these vibrations, their adverse effects, as well as an example of a mathematical model of the process, which takes into account parameters of the technological system, mutual influence of the cutting tool kinematics and dynamics of the cutting force formation.

The authors carried out a comparative quantitative analysis of mathematical models for estimating effective thermal conductivity coefficient of polycrystalline material which consists of chaotically oriented anisotropic crystal grains with identical crystal grid. Along with well-known methods for generating mathematical models, approaches which allowed to obtain new calculating dependences were also considered. The upper and lower bounds for accepted values of this coefficient were obtained with the use of a dual variational formulation of a stationary thermal conductivity problem for a non-homogeneous solid body. Formulas for calculating best-attested estimations were recommended. These formulas could be used for estimating electric conductivity of polycrystalline materials due to electro thermal analogy.

In this paper the authors propose improved formulas for calculating coefficients of transfer matrix in the presence of damping. The authors consider a classical damping which means the damping matrix is proportional to the stiffness matrix. These formulas are designed for large finite-element models. Preliminary determination of the lower bound for natural frequencies and oscillation modes is expected. Approximate formulas are used only to calculate coefficients of obtained bending modes. In the revised formula contribution of higher vibration tones is taken into account; corrections for both real and imaginary parts of coefficients of the transfer matrix were also found. Contribution of rejected vibration tones is considered to be static, but within a given range of frequencies in the lower part of the spectrum accuracy is quite acceptable, and the effect of correction is significant.

For mechanical systems with geometric constraints application of the Shulgin method for obtaining motion equations without joining factors, which is based on a simple derivation of constraint equations, is considered in this article. Obtained motion equations for holonomic system in redundant coordinates are a special case of the Voronetz equations for non-holonomic systems in case of integrable constraints. The method for using these equations is developed for stability and stabilization problems of the equilibrium position of systems with redundant coordinates. The proposed approach allows to use previously obtained results based on the theory of critical cases. The problem of non-asymptotic stability was tackled by reducing it to the Lyapunov-Malkin theorem of stability for a special case of several zero roots. In this paper, the authors prove that consideration of constraints on initial perturbations asymptotically stabilizes the equilibrium state despite formal reduction to a special case, if the number of zero roots of the characteristic equation is equal to the number of constraints and real parts of other roots are negative. Efficiency of this approach was verified by investigating the stabilization problem of the equilibrium state for real mechatronic workbench GBB1005 Ball&Beam Educational Control System.

A mathematical model of heat transfer in a composite material with identically oriented anisotropic ellipsoidal inclusions was created. Formulae for calculating effective thermal conductivity of composite materials, which are anisotropic with respect to the property of thermal conductivity, were obtained on the basis of a developed mathematical model. To estimate the possible error of the results, a dual formulation of the variational problem of stationary heat conduction in a heterogeneous solid body was applied. These results can be used to predict effective thermal conductivity of composite materials with ellipsoidal inclusions, which are among other nanostructured elements (including carbon nanotubes). The obtained formulas can be used for estimating electric conductivity of composite materials with identically oriented anisotropic ellipsoidal inclusions due to electro thermal analogy.

The paper describes reduced boundary conditions for an elastic half-plane covered with a thin viscoelastic coating in case of normal force applied to the surface. A long-wave approximation was constructed with the use of the method of asymptotic integration for motion equations and constitutive relations of the viscoelastic coating.

In this work a method of microstructural finite element analysis for calculation of strength surface of composite materials with periodic fabric structure of reinforcement on the basis of a developed software was proposed. This method is a method of asymptotic averaging adapted for multi-scale periodic structures developed by Prof. Y. Dimitrienko at the "Computational Mathematics and Mathematical Physics" department. The method is based on solving a special class of elasticity problems on 1/8th of the periodicity cell of the composite in order to determine effective elastic properties of the material. The results of calculation of ultimate strength for disperse reinforced and textile composites are presented.

 The authors propose a technique of creating various mathematical models of mechanical oscillatory systems with two degrees of freedom with elastic supports and a rigid body for protection against vibrations. The problem is solved by selection of corresponding motion coordinates. It is supposed that after realization of a system of joints between the rigid body and the base of the system, choice of corresponding coordinates is made. Those coordinates describe relative movement between two selected points. After transformation of a mathematical model and the setting coordinate of relative movement equal to zero, the system loses one degree of freedom. The mathematical model of the system is simplified; it allows one to estimate dynamical properties of the system from the required point of view. The proposed method also allows one to search for and estimate new constructional and technical solutions in the field of protection against vibration and vibration isolation of technical systems.

The authors propose a technology for creating mathematical models of mechanical oscillatory systems with a chain structure and several degrees of freedom. Various descriptions of movement in different coordinates are considered. Some coordinates characterize relative motion. Nulling of such coordinates results in combining corresponding mass-inertia elements; hence, the system on the whole reduces the number of degrees of freedom by one. Generalization allowing to set several coordinates equal to zero simultaneously is also considered. Some examples are provided.

 Analyses of Novozhilov equations show that existence of tensor nonlinearity is due to the difference of values ​​of the shear moduli in the direction of the principal shear stresses. It is shown that materials which have different degrees of non-linear charts at simple stress state develop tensor nonlinearity where, with increasing stress state, the deformed state is significantly different from the decisions made ​​by models which do not take into account this effect. 

The authors consider a mechanical system that has an object of protection in the form of a solid body on two elastic supports. The object of protection has a mass and a moment of inertia and can oscillate around the center of gravity. The dynamic lever vibration dampener has a lever construction and a support point on the object as a rotary joint. Movement of the hinge or choosing a seat for the dampener can significantly change the properties of the system as a whole. Features of the dampener include expression of dynamic properties which depend on the choice of the generalized coordinate system. The authors propose a method of constructing mathematical models and evaluating dynamic properties. They also give results of numerical experiments.

The article presents an analytical study of hydrodynamic modes for a model of magnetic fluid with internal rotation. The author shows that there are three hydrodynamic modes in the ferrohydrodynamics model with internal rotation. The author obtained phase velocity and absorption coefficients for fast and slow magnetosonic waves, as well as for the modified Alfven wave type. It is shown that in the absence of an external magnetic field the ferrofluid with internal rotation does not have anisotropy of sound and has the properties of a homogeneous fluid. The article also presents a comparison of hydrodynamic modes of ferrohydrodynamics with internal rotation with hydrodynamic modes of ferrohydrodynamics with frozen magnetization.

This paper considers the uniqueness of solution to the variational problem in the method of density functionals. The authors investigated the scaling relations that arise in systems containing two types of particles, as well as in systems of charged particles in an electric field created by an outward charge. The possibility of using the virial theorem was studied. It is shown that for quantum systems of charged particles in the field of distributed charge the well-known relation is not executed; this relation connects average values of the kinetic energy of the system, of potential energy of particle interaction with the external field and with each other.

The author considers nanoscale systems of noninteracting fermions with cylindrical symmetry (quantum wires) which are in the simplest rectangular confining potential. It was shown that for low-dimensional quantum systems, the degeneracy of levels is primarily determined by the symmetry of the problem, and - depending on the type of building -  not only degeneration caused by the presence of axial symmetry is possible, but also accidental degeneracy. Thus, the state with the same energy corresponds to different spatial distributions of electrons, which leads to divergence of results for electron density, which determines, in particular, system response to external stimuli. It follows that the usual method for estimating reliability of theoretical analysis results - comparison of energy performance characteristics – does not to allow judge how close to reality the result of calculating the electron density is.

The authors developed a method for numerical evaluation of the mechanical momentum transferred to the wall by the gas-bubble cavitation at the moment when it collapses near the wall of the channel. For the first time the impact of the cavitation bubble on the wall is considered as a point explosion in an incompressible fluid. The impact of the empty cavity formed on the spot of the explosion of the bubble on the wall of the channel. The authors obtained the dependence of the momentum transferred to the channel wall on the initial bubble radius and the distance to the wall on which the bubbles collapses. The cavitation bubble is considered spherical, and the process of its evolution is adiabatic. In the calculations the authors considered water as the liquid.

The authors formulate a mathematical model for relaxation resistance of a spring working under the conditions of neutron irradiation and high temperature; a method of its analysis was developed. The model and the analysis method are based on the fundamental equations of the creep theory constructed with the use of the finite-difference algorithm. This model is capable of creep and relaxation analysis of a spring based on various initial spring loads and geometrical parameters. Comparison of results of the proposed model with the well-established industrial analysis computer program “Ansys 12” is included in the article.

The author proposes a finite-element quartz resonator model which allows to take into account the quality of the quartz plate surface layer for calculating its characteristics. For modeling the development of a damaged plate layer a modified law of crack growth in geological materials is proposed. Results of calculation of the resonator Q-factor change as a function of time are presented. In the calculations the proposed law of damaged layer development was taken into consideration. The results of the experiments confirming the proposed resonator model and development of the quartz plate damaged surface layer during its vibrations in quartz oscillator scheme are included in the article.