In fra mework of the geometrical diffraction theory, the explicit expressions for the pre s sure in waves arbitrarily re reflected N times from a contour, boundary surfaces of the cylindrical and spherical refle c tors are obtained. Pressure expressions in the recept ion point for reflectors of the canonical form are obtained on the basis of 2D and 3D sol utions to the problems on the pressure determination in the acoustical wave re - reflected from a set of reflectors in the case of high - frequency oscillations. The pro blem in its general formulation is studied on the basis of a modified physical diffraction of Kirchhoff diffraction theory. Within the frames of the proposed modification , diffraction int egrals which leading terms for asymptotic expansions are studied throug h the multidimensional stationary phase tec h nique, are obtained. The developed analytical expressions for the pressure in the re - reflected wave conform to the GTD. For all three cases, these expressions are connected with calculating the N - th order dete r mi nant (in the 2D case), and of the 2N - th order determinants (for reflectors in 3D space). An analytical and numerical analysis of the obtained expressions versus distances between a source and a receiver from the reflecting surfac e is pe rformed. The acousti c wave focusing points are marked. The problem of replacing non - plane reflectors by plane ones in the applied problems of acoustics is discussed.

Let   H G be a space of analytic functions of one variable in simply - connected domain G of the complex plane. It is known that a linear complex convolution operator is generated by a one - variable analytic function, a multivalued one in general. A known problem when all such functions are single - valued is solved. As it turned out, the solution to the problem is connected with the geometry of G domain. Set   s G with property   s G G G   is termed re sidue of G domain. A class of simply connected regions whose residue is a connected set is described. Let the linear operator be continuous in function space, analytical in simply - connected domain G, and let it commute with diffe rentiation. Then it can be reduced to a complex convoluti on operator. It is proved that the function generating such an operator will always be single - valued for regions with a connected residue. When the residue of region G is not connected, there is always a complex convolution operator with a multivalued func tion generating a kernel.

The authors have previously studied two - dimensional Fredholm integral operators with homogeneous kernels of fiber - singular type. For this class of operators, the symbolic calculus is built using the theory of biloc al operators by V. Pilidi, and Fredholm criterion is formulated through the inversibility of t wo families: the family of one - dimensional convolution operators, and the family of one - dimensional singular integral operators with continuous coefficients. The aim of this work is to study composite two - dimensional integral operators with homogeneous ker nels of fiber singular type analogous to Simonenko’s continual convolution integral operators. This investigation is a part of a more general study of algebra of operators with homogeneous kernels which layers are singular operat ors with piecewise continuo us coefficients. For the considered operators, the symbolic calculus and the necessary and sufficient Fredholm conditions are obtained.

An axial ly symmetric quasistatic thermoelasticity problem on the indentation of a flat - ended cylindrical punch with a constant temperature at its base into the functionally - graded half - space which elasticity modulus, Poisson ratio, heat conductivity and expansion coefficients are independently continuously varying in the boundary layer, is considered. Out of the contact area, the surface is perfectly thermally - insulated and stress - free. The earlier solution, obtained through the combined numerical and analytical ap proach (using Hankel integral transform and the modulating fun ction method) to the unmixed pro b lem on the arbitrary thermomechanical effect upon the inhomogeneous in depth thermoelastic half - space, is applied to solve the problem. The original problem is r educed to the system of dual integral equations. The properties of the dual integral equations kernel transforms allow applying a well - grounded bilateral asymptotic technique which is being actively developed at present. The approximate express ions for d et ermining the thermal flux, the half - space surface displacement, and the contact stresses under the heated stamp base, are obtained with the aid of this method. The numerical values of contact stresses for various cases of the thermomechanical properties va riation in the boundary layer of the half - space are provided. The cases either when values of the thermomechanical coating prope r ties are the same as those of the substrate, or when the property value differs twice (upward or dow n ward) on the surface, and linearly decreases (or goes up) in depth to the value in the substrate, are con sidered.

A solution to the problem on describing experimentally obtained dependences is considered. The autho r’s method is based upon getting some local approximations of fragments of these relations, and their additive reduction to a single analytical expression. This effect is determined using special “allocating” functions limitin g the domain of non zero definition for each of the approximation functions. The method is called “cut - glue” according to the applied principles. The closest analogue of the proposed method is spline approximation. However, the “cut - glue” method is much more adaptable, as it is bonded to neither the number of spline - approximable points, nor the function order approximating the areas. The order of the polynomial approximant, or another approximating function, as well as its structure for each site, can be arbitrary. Another advantageous difference of “cut - glue” approximation consists in a single a nalytic notation of the whole piecewise function instead of defining a vector spline - function through a cu mbersome system of equations. This effect has been achieved using an analytical function approximatin g and par ametrically arbitrarily approaching the Heaviside step function. The analytical and numerical studies of the properties and the effects of applying the proposed method are resulted. The obtained results are illustrated w ith the specific technical sample applications of the method to practical pr oblems, tabular and graphical data.

The three - dimensional Galin’s type contact problem for a two - layered elastic base (a layer completely attached t o a half - space from another material) is investigated when an extra loading (concentrated force) is applied o utside the contact area. The contact zone is supposed to be unknown. The punch foot form is an elliptic parabol oid. The pro blem is reduced to an in tegral equation with respect to the unknown contact pressure distributed in the unknown contact zone. Galanov’s method of nonlinear boundary integral equations is used to determine the con tact pressure and the contact zone simultaneously. Calculations made for various values of elastic and geometric parameters a llow estimating an extra force input to the dependence between the punch settlement and the force app lied to the punch. The problem is important for the strength analysis of coated surfaces of variou s elastic solids subjected to contact and extra loadings. The solution can be also useful in the frame of the discrete contact the ory for bodies with rough surfaces.

The applicability of the bionic techniques of artificial bee colonies for the implementation of the classical transpos ition cipher cryptanalysis is considered. The problem is a classical optimization problem to the solution of which the known techniques of artificial bee colonies fallen within a relatively new class of bioinspired opti mization methods are applied. It is shown that this is a subproblem of allocation, and it can b e solved with an artificial bee colony algorithm, as the bee behavior principle is a self - organization delivering a collective swarm goal. А t the first stage, a set of promising areas - sources is formed with the aid of scout - bees, at the second stage, the n eighborhood of these areas is explored with the aid of foraging bees. At this, the main goal of the bee colony is to find a source with a maximum amount of nectar. Solution representation methods (positions in search space) are considered , a formula for de termining an object function value (amount of nectar) is given. It is shown that the target search i s the d etermination of an optimal symbol combination with the highest value of the objective function. Princ iple stages of the artificial bee colony algorit hm, as well as an example of its application, are given. Keywords: 

On the basis of Navier — Stokes linear equations for the case of Lame “ thin layer ”, a technique of generating an exact self - similar solution to the hydrodynamic calculation of the elastically deformable thrust bearing with the pr ofile adapted to its bearing surface operating on a three - layer lubricant is presented. The effect of the par ameters characterizing the adapted contour supporting the slider surface, the su p porting layer deformation, viscous layer ratio, and their length, on the basic performance of the thrust bearing is evaluated. These paramete r values provi ding a rational oper ation according to the carrying capacity and friction force of the relevant thrust bearing are d efined. Furthermore, the optimal variation ranges of the constructive, operating, and all the functio nal parameters dete r mining the bearing performance, are set . The obtained results allow creating the database for designing thrust bearings operating on the three - layer lubrication.

The methods of calculating the location radii of the gear chain pivot locus on the wraps of involute sprockets are suggested. The methods involve a step - by - step “placing” of the chain loops on the sprocket. To determine the chain joints location, surface equidistance of the operating and back tooth profiles which represent the c hain pivot locus upon their contact with the sprocket tooth profiles is used. The location radii of the pivot locus o n the involute sprocket wrap can be defined through the predete r mined position of the first joint, the known length values of the gear chain contact link pitches, and the slack side pretension. The study results allow presenting t he methods of calculating t he gear chain pivot l o cus on the wraps of involute sprockets and the mathematical model algorithm of the chain gearing with involute sprockets. The methods can be used to determine setting of the chain belt with toothed chain providing the minimal we ar - out of the gearing elements.

Acoustic screens are one of the means to provide sound insulation and noise reduction on its path fr om the source to the target point (TP, the workplace). The advantages of the acoustic screens are constru c tive simplicity, light weight, and relat ively low cost. A distin c tive feature of an acoustic screen as a sound insulation system is that the area of maximum attenuation is located in close proximity to the screen. The efficiency effect of th e screens is d etermined by the following factors: - a sm all size of the noise source; - the conversion of high - frequency components of the source noise spectrum; - the location of the work area outside the technological machine; - a short distance from the noise source to the operator's station. All that fully applies to the equipment for various applications, in particular, for the metal and woodworking machinery (lathe, milling, grinding), gear - boxes, and engines on the running sites. The existing d e tailed screen calculations are designed for screen installati on. At free atmosphere, the formation of the acoustic field under the screen insta llation inside the production areas differs essentially due to the multiple reflections of sound waves. The studies on the acoustic performance of screens inside the produ c ti on are resulted in the paper.

Some issues of applying the stochastic simulation modeling to the description and stud ying of DNA’s end structures (telomeres) shortening mechanisms are considered. The aim of this research is to study — on the base of the st ochastic and simulation modeling — processes in the cells leading to the telomeres’ length shortening, and as a co nse quence, to decreasing the proliferative capacity, and developing pathologies including oncologic one s. The mat hematical model is described in the semimartingale terms. The analysis of the modeling results is bas ed on their comparison to the biological expe riments data. The formed cell distribution in accordance to their telomeres’ length is compared with the e x perimental data obtained through the tests performed with human fibroblast - culture using Levy — Prokhorov metric. The study results can be used in bi omedical research on ageing, and on developing var ious pathologies, as well as in solving a number of problems in the field of gero n tology.

The possibility to increase parameter accuracy (pitch of thread, pressure angle, external and internal thread diam eter) for various kinds of thread on the seamless casing is studied. As a means of achieving the targ et goal, the v ibration amplitude contraction arising in the system ‘machine — widget — tool — part’ is considered (MWTP). To solve the problem, a model which describes tube shell oscillations in the course of its processing a nd allows defining the impact on the work end of shell oscillations arising in its loose part is offered. The study is resulted in a math ematical model b ased on the initial parameters method. The calculations found the best (of the considered in the paper) support location diagram under which the arising oscillations are the lowest. The obtained re sults can be used at any enterprise dealing with thread cut ting on the casing tubes.  The investigation results in the following: — optimum quantity of supports for the considered equipment is five; — expansion in the number of supports by no means always leads to the vibration amplitude contraction; — changing in the relative position of supports is an effective means of the vi bration amplitude contraction that acts to increa se thread parameters accuracy.

The interrelation of the chaffing process efficiency growth and the air - screen grain - cleaner perfo r mance indices is considered. A mathematical functional model of the air separator and the screen clea n er is described as a closed loop quasi - static system. Agronomical performance indices of the machine are accepted as criteria. Through vari ation of the delivery, technological properties of the basic material and of the c ompleteness of each component sep aration, the rational values of the control factor arguments are revealed, the numerical results of t he grain loss ma gnitude, completeness of the foreign and other grain separation, as well as their content in the puri fied material, are determined. The developed mathematical model of the heterogenic granular medium chaffing allows — through the known parametric and structural synthesis methods — conducting a multivariate analysis of the process and eval uating the air separat or rational parameters for the specified conditions and the restrictions to its performance ind ices. As a result of modeling, a trend of the index variation of the machine performance and their in terrelation is r e vealed. The obtained results have shown a s ignificant increase in productivity and technological parameters of the similar machine performance from the efficiency growth of their air separator.

The thermodurability of glass carbon — one of the most promising carbon materials used in the production of lum inophors and semico nductor materials, optical single crystals, etc. — is investigated. The aim of this work was to study the behavior of glass carbon in the course of the thermal - cycle testing. As a diagnostic technique of the m icroflow initiation and progress, the signal re ception of acoustic emission was used. It is shown that a heavy disloc ation motion on the thermal - cycle exposure recognized in the acoustic emission signals is observed in the glass carbon samples when heated. As far as the number of impulses is higher whe n heated than on cooling of the sa mple, it implies the damage accumulation in the structure under the thermal - cyclic handling of the glass carbon sa mples. It is found that damage acc u mulation curves reflected by the acoustic emission pulses differ on the s tage of heating and cooling. It is proposed to use a total curve of the acoustic emission pulses as a potent ial destruction sign of a glass - carbon product when heated.

The application of a new system - cybernetic approach to the power system synchronous generator control with r egard to their nonlinearity, multidimensionality, and multicoupling, is shown. The main study subject is a nonlinear model of the synchronous generator operating for a high power network. A synergistic synthesis procedure o f the nonlinear adaptive control law for power system synchronous generator excitation in accordance with the principle of the integrated adaptation of syn ergetic control theory is presented. The synthesized control law provides the implementation of technological invariants, i.e. stabilization of terminal voltage, and a synchronou s operation with the network, suppression of parametric perturbation (parametr ic robustness). The obtained results have shown their efficiency in improving power system stability problem. A comparative analysis of the stabilit y area with the synergistic law, and with the conventional synchronous generator excitation controller has i llustrated the significant advantage of the synergetic approach.

A production facility model to be used und er the deep - hole drilling of large - diameter tube plates with a heterogen eous structure made of hard - to - machine materials is presented. Dynamic properties of the object as a composite membrane with several anchorage points are analyzed. Modes of the workpie ce vibrations with account for its i nternal heterogeneous structural features are defined. The object structure transformation in the cou rse of pr ocessing (the formation of a large number of holes) effects on its elastic and dynamic properties. Su ch effect fe atures are considered. To simplify the practical application of the model, some recommendations for t he global stif fness matrix transformation, and the production facility dissipation to the local points correspondin g to the applic ation places under the hole making are given. The research results are focused on the creation of an automatic co ntrol system for the deep - hole drilling of hard - to - machine and heterogeneous materials.

Simulating a piezoelectric generator which is an element of the energy storage device is considered. A piezoelectric generator appears a cantilever - mounted plate on which the piezoelectric elements are glued, and the inertial mass is attached. Two options of the plate vibration exc itation are investigated. In the first case, the fixed side executes heave harmonic motions at a given frequency and with the desired amplitude; in the second — a ha r monic force is applied to this side. A three - dimensional boundary - value problem of the lin e ar theory of electroelasticity for the composite elastic and piezoelectric body is considered as a mathematical model of the device. The bo undary pro blem is solved through the finite - element package ANSYS. Under the numerical solution, PKR - 7M piezoceramic is taken as a piezoelectric material, whereas fiberglass, duralumin, steel are considered as a plate ma t e rial. Aluminum is used as a material of the inertial mass. Two cases are studied computationally. In the first case , the fixed side e x ecutes vertical harmonic vibrations at a given frequency and with a preset amplitude, in the second case, the force varying in a harmonic fashion is applied to this side. The device resonance frequency dependen ce upon the plate thickness for various materials and upon the value of the inertial element mass is studied. The results are pr esented graphically that allows finding the resonant frequency for certain sizes. The output potentia l dependences at the free electrodes of the piezoelectric elements on the resonant freque ncies and in the low - frequency region on the same parameters are investigated. These results are also presented graphically that allows a des igner to select reasonable feature sizes and the mix of ma terials to optimize the device.

A supplementary friction group in the clutch design increases its load capacity. It is a decided adv antage of adaptive friction clutches of the second generation with combine d feedback. The feedback gain changes depending on the friction coefficient value. The problem of establishing such patterns, and the synthesis of the comm and unit el eme n tary diagram with its parametrization based on this concept, is solved in the paper. T he identified regularity allows stating the following. Under zero thrust force and minimum friction coefficient value, changi ng the gain level does not exceed its limit value. It provides an automatic control with a theoretical “ideal” load ch aracteristic of the clutch. A curve geometric form outlining the side wall of the rolling body socket of the command uni t, and providing an automatic control in the adaptive friction clutch, is found. In this case, the goal is to obtain a theoretically “ ideal ” load cha racteristic of the coupling.

The asymptotic solution to the problem on the hydrodynamic calculation of the porous double - layer r a dial finite length bearing is provided on the eccentricity ratio levels. The solution is based on full nonlinear Navier — Stokes equations, and on Darcy’s equation. The distinction of the suggested design model is that it allows for the perme ability anisotropy of the porous layers, and availability of a greasing source. A case when greasing forcibly gets thro ugh pores of an external porous layer is considered. A velocity and pressure field in the lubricant and porous layers is found. As a result, analytical expressions for the main performance data of the bearing ar e obtained. The effect of the permeability an isotropy of porous layers, nonlinear factors, as well as the influence of a greasing source on the basic performance chara c teristics of the bearing are evaluated.

The possibilit y to increase feedback gain and operation accuracy of the adaptive friction clutch is r e vealed at the expense of the friction couple separate force closure of the basic and supplementary friction groups . The load cha racteristic of the clutch is presented i n the form of a curve monotonically i n creasing in the full range of the friction coefficient variation. The obtained results of three options of the coupling fluent param e ters variability have shown that to attain a maximum operation accuracy of the clutch with separate force closure executed according to the second option, it is necessary to take the maximum possible number of friction couples of the basic friction group, and the maximum gain le v el