2014. № 77
Mathematics. Physics. Mechanics  
The method of numerical analysis of the mechanical fields in the deformable body, based on graph model of an elastic medium in the form of directed graph, is considered. A new type of elementary cell is introduced to describe the features that appear near a crack tip in isotropic elastic material. An elementary cell is considered as a subgraph that corresponds to a single element of continuum. The configuration of a cell is defined by installing hypothetical devices upon an element of a solid. The equations of elementary cell are derived using the deformation energy as invariant that remains constant under the transformation of an element into a cell. A procedure determining the parameters of the elementary cell is described. The graph of a whole body is constructed the same way as for an elementary cell. Equations of state of the original solid body are derived by using the transformation of generalized coordinates of a decomposed solid body elements into a system of generalized coordinates of entire solid body. This transformation is performed using nonsingular and mutually inverse matrices. The specific nature of the graph discrete solid body model is such that allow you to construct nonsingular matrices without numerical inversion. Kirchhoff graph laws were used for derivation of a system of defining equations. Graph rules (apex and contour) have mechanical interpretation and their application cause the equations of equilibrium and strain compatibility to be satisfied when the net dimensions are reduced. The results of numerical calculation of compliance and stress intensity factor in a centrally cracked tensile plate obtained by graph method are presented. Keywords: mathematical modeling, elasticity, directed graph, crack, stress singularity, stress intensity factor, compliance 

A statement of the bilevel problem of stochastic programming with quantile criterion and several followers is suggested. Random parameters of the problem assumed to have a discrete distribution. In this problem, there are several decision makers: leader and several followers. These decision makers may have different own purposes. The followers choose strategies solving their optimization problems when a leader’s decision is already known. The leader chooses his own strategy taking into account the followers’ optimal strategies and solving his own optimization problem. The followers’ problems are assumed to be linear in follower’s decision variables. Due to this fact, the followers’ problems may be replaced by nonlinear equilibrium constraints using the slackness complementary conditions. The following theorem is proved: the original problem can be reduced to a mixed integer mathematical programming problem. This theorem is used in solving the problem of optimization of energy saving projects. In this problem, the leader is a transport company and the followers are companies carrying out the energy saving projects. The aim of the transport company is to minimize investment costs and expenses for the purchase of energy resources. The aim of a followers is to maximize own profit carrying out their project. In this application, the random parameters are the demands for different energy resources in different planning periods. It is proved that this applied problem can be reduced to a mixed integer linear programming problem. The results of a numerical experiment concerning typical section of the railway are presented. Three projects for saving two energy resources: electrical energy and diesel oil are considered in this experiment. Keywords: stochastic programming, bilevel problem, quantile criterion, energy saving 

Bending of elastic rectangular orthotropic threelayer plate with a hard core and the composite layers under the influence of local load is considered.
Keywords: threelayer circular plate, oscillations, local load, stressstrain behavior, composites 

Aviation technics and technology  
Physicalchemical processes, which take place in the gas and on the surface that is streamlined by this gas, depend on the energy of thermal motion of molecules. In nonequilibrium flows, the energy of thermal motion is distributed unevenly between translational and internal degrees of freedom of molecules. Under these conditions it is necessary to consider the energy of the translational degrees of freedom (translational temperature T_{t}) separately from the energy of the internal degrees of freedom (internal temperature T_{Ω}). The physicalmathematical models of the first approximation are most common in practical applications. These models can be obtained from the system of moment equations of polyatomic gases [1]. A method for constructing models of the first approximation is described in [2]. Two models of the first approximation are considered. The first model is written down by using the thermodynamic variables. Its system of equations contains 5 scalar equations (5moment model). This model is the NavierStokesFourier model. The coefficient of bulk viscosity is adduced in an explicit form. The system of equations does not contain T_{t} and T_{Ω} temperatures. These parameters are defined by a special dependency through the coefficient of bulk viscosity. The second model contains 6 scalar equations. The T_{t} and T_{Ω} temperatures are defined by independent moment equations (twotemperature model). The heat flow is represented by two components. The first component is the energy flow of the translational degrees of freedom. The second component is the energy flow of the internal degrees of freedom. The coefficient of bulk viscosity is not used. The exchange of energy between translational and internal degrees of freedom is described explicitly. Numerical testing of the models was conducted on the sample problem of the shock wave shape. The tests have shown that 5moment model produces qualitatively incorrect values of T_{t} and T_{Ω} temperatures. The second law of thermodynamics was violated in some profile areas. The twotemperature models only produce errors of quantitative nature. These errors can be reduced by the adjustment of free parameters of the model. The twotemperature model has significant advantages over the 5moment model during the modeling of the nonequilibrium flows. These advantages are essentially important for the description of physicalchemical processes. Keywords: moment equations, first approximation, bulk viscosity, twotemperature model, shock wave shape 

The paper is devoted to the development of a program, which is aimed at automating the design and calculation of parameters of the pressure control system valves within the airplane air conditioning systems. Thus the program should accelerate the development of new valves with the help of computers. A generalized classification of valves has been proposed for the development of this program. This classification unites the whole diversity of types of valves within one scheme. The classification system divides the valves into the following types according to their purpose: exhaust valves, emergency valves, and combined valves. Three categories of actuator valves are allocated according to the type of their closure members: valves with solid, elastic and aerodynamic closure members. Three types of valves can be distinguished by the kind of energy they consume (actuation types): pneumatic, hydraulic and electric. This paper considers the actuator valves, which are classified according to the type of consumed energy (actuation).
Keywords: automatic pressure control system, design automation, programming, valves, calculation 

Space technics and technology  
Flight safety of spacecraft (SC) in the space debris environment has become significantly important.
Keywords: spacecraft, space debris, orbit parameters, onboard measurement facilities, analytical method, transformation of equation system, computational errors 

Development of space solar power station technology shows the necessity of demonstration space solar power station development based both on the available spacerocket equipment and on electrooptical technology. Laser emission is chosen as energy transmission channel, because it gives the possibility to receive focal spot of acceptable size.
Keywords: spacecraft, satellite, laser 

Aerospace propulsion engineering  
The design of lithiumion battery for space application is developed. The lithiumion battery contains prismatic lithiumion accumulators in series connected into the electric circuit with the electronics module and the device for active equalizing voltage battery charge and discharge. The full cycle of ground tests, including electrical, mechanical, climatic tests are carried out with prototypes of battery. The program and methodology of testing, including cyclic tests simulated mode in geostationary orbit (charge current: 20 A within 120 minutes; discharge current: 40 A within 60 minutes; discharge depth  80%) are developed for carrying out electrical tests. Battery tests were carried out in both normal climatic conditions and under vacuum (3·10^{6} mm Hg). The test results showed that the actual discharge capacity at the beginning of the testing was approximately 81 Ah (the nominal capacity is 50 Ah), the actual powerintensity was approximately 2300 Wh. When cyclic battery tests on the appropriate battery modes operation on geostationary and low earth orbits in normal climatic conditions were conducting the energy efficiency was approximately 0,96, the maximum surface temperature of the batteries wasn’t greater than 22 ºС while the temperature of the cooling system plate surface was about 13,5 ºС. When cycling of the battery in a vacuum were conducting the temperature increased to 3234 ºC while discharge current was 40 A and tendency to a slight further growth with cycling continuation was observed. Keywords: lithiumion battery, transport properties, discharge capacity, discharge velocity, temperature regime conditions, powerintensity 

A continuum thermomechanical model for densely perforated electrodes of the ionextraction system (IES) was developed. It is based on representing the emissive, accelerating and decelerating electrodes as structurally orthotropic sloping thinwalled spherical (profiled in general case) panels. An uneven heating of the electrodes along the radius and through their thickness was considered as the basic loading factor. As a consequence, the coefficients of electrode filling with materials, which are simultaneously the coefficients of constructive orthotropy in the considered model, were directly taken into account in the initial equations of thermoelasticity.
Keywords: ion thruster, ionoptical system, perforation, deformation, constructive orthotropy, thermal load, thermal deformation, thermomechanical model 

In this article a physicalmathematical model, which gives the possibility to compute features of individual and group regimes of evaporation and combustion of dense turbulent sprays, consisting of fuel droplets, is considered. Investigation results of operational process peculiarities inside combustion chambers by varying the regimes of evaporation and combustion of fuel droplets are presented. Working process development inside liquid rocket engine combustion chamber is in a significant extent determined by an interaction of liquid sprayed flows of fuel and oxidizer. Combustion of propellant droplet arrays can be profoundly affected by the collective interaction between droplets that impedes large scale intermixing of fuel and oxidizer vapor thus forming oxidizer or fuel reach regions with unflammable conditions inside sprays. Additionally, the presence of large number of comparatively cold liquid droplets tends to decrease the temperature inside spray. Three spray combustion regimes can be proposed: 1) individual regime of combustion, typical for sparse sprays, when every droplet is surrounded by an individual combustion front; 2) regime, when droplets in internal regions of the spray have no individual combustion fronts, and at the same time in the outer regions of spray individual droplet combustion regime takes place; 3) regime, when droplets inside the spray only evaporate, and combustion front surrounds all the spray on the outside. The mathematical model includes equations of the NavierStokes type and the k — ε turbulence model. The combustion rate is defined by the model of turbulent vortex dissipation and from the generalized equation of chemical kinetics of the Arrhenius type. Calculation of motion, warming up, evaporation and breaking of droplets is conducted in the Lagrange approach. Thereat, the interchange with mass, momentum and energy between phases of gas and droplets with regard to turbulent flow characteristics are taken into account. The equations are digitized with the controlvolume method and solved with an algorithm of the SIMPLER type. Additional relations, which give the possibility to identify realization of individual or group spray combustion regimes, are being proposed. This technique is widely used at Keldysh Research Centre for numerical modeling of the working process in subscale and fullscale combustion chambers and gas generators that operate with a variety of injectors and using different propellants. Possibilities of the method have been demonstrated by an investigation of features of the working process inside a combustion chamber with coaxial jet injectors under different regimes of fuel spray combustion. Keywords: evaporation and combustion of fuel droplets, combustion chamber, numerical simulation 

Control and navigation systems  
This paper presents an approach to the problem of aircraft systems technical state estimation method. This method uses information stored by regular onboard flight information monitoring systems. The principle of the method is based on reference pattern formed by way of finite set of points within the space of state variables of controlled onboard system for every generic flight mode upon condition of its normal operation. To estimate current aircraft system condition considering noncontrolled factors action including factors of accidental action, we use probability criterions that characterize probability of current controlled parameters set membership to reference pattern. In this work we suggest computational algorithms for such probability criterions. These algorithms intend to apply Mahalanobis distance from reference region points to its geometric center as a variability measure of controlled system parameters values. Estimation of the probabilistic criteria is performed using Mahalanobis distance distribution function. This function is obtained by approximating Mahalanobis distance cumulative distribution function which can be calculated based on the information about previous flights of the exact aircraft. With the help of distribution function we can calculate the probability of occurrence Mahalanobis distance value exceeding the argument of the function in case of normal operating of controlled system. We define this probability as a main probabilistic criteria. We also define an inversed probabilistic criteria as a quantile of the distribution function for given probability value. In this case quantile is a Mahalanobis distance value that will not be exceeded with a given probability value. Developed algorithms allow us to identify the moment of abnormal operation mode occurrence. To solve this task we use inverse probabilistic criteria. This moment occurs when the current Mahalanobis distance calculated using the current flight data exceeds predetermined quantile value. Developed methods and algorithms were used to estimate Su35 aircraft conditioning system state. The analyzed period encompassed 50 flights, beginning from 80^{th} to 129^{th}, carried out during two years of exploitation. By using the proposed method an abnormal performance in the temperature channel of air conditioning system was detected in 2 out of 50 flights (105^{th} and 106^{th}). Obtained results fully correspond to the state of the system fairly confirmed by technical inspection report. According to the report the breakdown of turbocompressor due to inadmissible overheating of a bearing was registered during 106th flight. Keywords: aircraft system, aircraft system reference region, probabilistic criteria, Mahalanobis distance, environment control system 

The article describes existing methods and systems of monitoring of GLONASS radio navigation field and highlights their weak side i.e. low precision of monitoring of user accuracy. The presented example of current accuracies for highprecision positioning methods demonstrates that the monitoring of radio navigation field must perform in high precision mode. Technique combining two highprecision methods (relative positioning by phase measurements and precise point positioning) was overviewed and main results of its implementation were presented. The results of relative positioning for two couples of Russian receivers (manufactured in «Navis» and «Izhevsk radio factory») showed that using of two similar receivers on a base station and rover can result to fixed solution with accuracies not worse that for GPSonly solution. It was also obtained that optimal solution of high precision GLONASS monitoring is autonomous monitoring in a receiver because of high influence of local error sources like multipath, loss of visibility in city canyon, radio noises and systematic errors of phase measurements. The necessity of global network monitoring of continuously reference stations with using various high precision ephemeris and clock information is approved in real time mode and in postprocessing by receivers of various manufactures as well as high precision monitoring in various software. Keywords: GLONASS, accuracy, radionavigation field monitoring, relative method, precise point positioning, highprecision ephemeris & clock information 

This paper considers the possibility of assessing influence of the reflected signal on the parameters of the GNSS signal from the simulator satellite navigation signals Spirent, so that let getting measurements of the selected satellite when exposed only multipath errors. The research was depended of the multipath error of a GPS C/A signal in the presence of one of the reflected signal from the delay of given amplitude of the reflected signal and the comparison of experimental results with theoretical calculations. A result of comparison was revealed that the value of the envelope multipath errors are obtained in the real experiment, in a good agreement with the value of the envelope multipath errors are obtained by the theoretical calculation for the navigation signal receiver. This fact allow to use theoretical calculation at the design stage of a new receiver, in order to assess the influence of multipath errors on the accuracy of the measured navigation parameters at the selected frequency plan and the filter in a highfrequency part of the receiver. The results of field experiments can be used to analyze the ability of the equipment to withstand the multipath errors at a stage of mass production. A theoretical calculation and methodology are in the experimental study are universal and can be used to estimate the multipath error for any GNSS signal. Keywords: global navigation satellite system, multipath error, strobe method, navigation field simulator, GPS, pseudorandom number 

Technical cybernetics. Information technology. Computer facilities  
The purpose of this work is to develop the technique, by which one can effectively develop software to realize mathematical tool for aerodynamic forces simulator with FPGA. To develop this technique we carry out system analysis of the aerodynamic equation, which constitutes a part of the aerodynamic forces (AFS) simulator. This analysis resulted in selection of integral form of the equation for further implementation with FPGA. We performed discretization of the equations, and reduced them to fixedpoint arithmetic. To increase the speed of algorithm operation, we replace division operator with binary shift. While developing the technique we used existing recommendations for FPGA programming in LabVIEW. After implementation of the developed technique a list of typical mathematical operations and methods of their implementation, based on the analysis of the equations included into the algorithm are formed. In addition the separation of slowly varying and rapidly varying parameters calculations is performed. Implementation of this technique during aerodynamic forces simulator design allowed to increase the speed of the algorithm for eight interconnected channels from 17.7kHz to 416.6 kHz, which corresponds to 24 times increase in performance. To obtain the results we used National Instruments equipment: eightchannel analogtodigital converter (ADC) with 16 bit resolution and sampling rate of 750 kHz as well as eightchannel digitaltoanalog converter (DAC) with 16 bit resolution and sampling rate of 1 MHz. The proposed technique can be implemented during various measurement and information as well as management systems design, with severe requirements for operation speed, accuracy and channeling with explicit synchronization of execution. Keywords: programmable logic integrated circuit, digitaltoanalog convertor, aerodynamic forces simulator, analogtodigital convertor, electromechanical modeling method 

Material authority  
The disadvantages of the majority of wideband electromagnetic shields with geometrically nonregular surface areas follows: their large thickness, weight and sufficient cost. The electromagnetic shielding constructions of a linear corrugated shape are suggested, which are based on compressed cellulose filled with a hygroscopic solution. The EMR attenuation and reflection characteristics are investigated in the frequency range of 0.717 GHz for the different orientation of corrugations with reference to the electromagnetic waves polarization. The electromagnetic shielding constructions based on liquidcontaining compressed cellulose of a linear corrugated shape and flat shape are suggested. In order to stabilize the liquidcontent, the hygroscopic metal salt solution in equilibrium concentration was synthesized. The solution is characterized by high sorption and, as a result, prevents liquid content variation when the ambient temperature and humidity alter. The sample surface microstructure was studied using metallographic microscope МЕТАМ—Р1 with amplification up to 507 times. The obtained microphotographs show, that there are local volumes of the solution preserved within the cellulose fiber structure during the whole period of the investigations. The experiments show, that the samples of flat shape 5 mm in thickness possess the reflection coefficient as low as −12 dB. The reflection characteristics for the flatshaped sample on a metal base are quite flat in a wide frequency band. The EMR attenuation increases from 2 to 14 dB with the frequency increase due to the radioabsorbing properties of the liquidcontaining filler. Keywords: electromagnetic radiation shield, composite materials, liquidcontaining matrrials 

Economics and management  
Russian production companies engaged in space industry are controlled by the Government through a special federal authority, ROSKOSMOS (Federal Space Agency), and another authorized organization, jointstock company “ORKK”, as stated in Decree of the President № 874 of 2nd December, 2013. The revenue secured by the stateguaranteed order in the total income of such companies exceeds 80%. Strong governmental support gives the companies significant advantages in terms of stable guaranteed orders and loan capital. However, in these circumstances they become exposed to regional and country risks.
Keywords: management cost, mechanism of budgeting 

While production order making, one usually has little information about cost calculating, so that it is difficult to decide whether order profitability is sufficient or not. This problem is especially important if production life cycle is rather long, which is common considering aviation industry. The article considers the method of gage cost preliminary evaluation. The gages mentioned are used for rotation solids diameter measurement at the industrial companies. A set of recently designed gages was chosen and a number of different parameters were subjected to analysis for giving preliminary evaluation of material and labor costs. One who makes tool set order has his requirements and the designer while developing the gage fixes a set of variables. While investigation it is proved that connection does exist between the first and the latter mentioned, which is shown in the form of correlation coefficients matrix. It is stated that gage parameters required determine tool set material and labor costs. The cost evaluation method is proposed based on econometrics equations. The method gives the description of the parameters connection in the form of linear regression equations, in which order requirements are used as independent variables and the cost as the dependent one, so that it can be calculated at the moment the order is transferred to the tool set department. The equation gives a preliminary sum of costs providing rather high degree of its accuracy due to strong parameter correlation. Production profitability can be easily evaluated using such method that is really important for industrial companies. Keywords: cost evaluation, econometrical equations, instrumental manufacturing, gages 

Purpose
Keywords: restricting, aero engine building, efficiency, competitiveness, cumulative potential 