Mathcad Software Package Research Articles

Обґрунтування вибору типу приводу стрічкового конвеєра

The designs and dynamic properties of electromechanical and hydraulic drives for a belt conveyor have been analyzed to justify their selection. The main advantages and disadvantages of their operation have been formulated. It has been established that the hydraulic drive has advantages compared to the electromechanical one for installation in a built-in drive for a belt conveyor. The design features and principles of operation of these drives have been described. Calculation schemes have been developed for the electromechanical and hydraulic drives of the belt conveyor. A mathematical model of the electromechanical drive has been constructed, taking into account the Voigt viscoelastic model for the conveyor belt, Hooke's laws for the deformation of stretching elements, equations of motion of the mechanical system, and electromagnetic phenomena in the asynchronous motor. Additionally, a mathematical model for the hydraulic drive of the belt conveyor has been constructed, which, besides the equations of motion of the mechanical system and the Voigt viscoelastic model for the conveyor belt, also includes the continuity equations of the working fluid flows in the hydraulic drive. The parameters of the belt conveyor for studying the dynamic properties of the electromechanical and hydraulic drives, which are considered in the mathematical models, have been described. The mathematical models for the electromechanical and hydraulic drives have been solved using the Mathcad software package. Theoretical curves for cases of transient processes in the mechanical system of the belt conveyor without load and with load have been constructed. Transient processes of acting torques and angular velocities have been compared by the dynamic coefficient, and it has been established that the hydraulic drive has a better indicator by 1.78 times. It has also been established that the duration of the transient process in the drive with the electric motor exceeds this parameter in the drive with the hydraulic motor by 3.5 times. It is recommended to use the built-in hydraulic drive for belt conveyors of mobile working machines because it has more advantages over the electromechanical one, and the dynamic indicators of the hydraulic drive provide better performance during operation.

Determining the influence of wheelset arrangement in the model 18-100 bogies on the level of steering efforts in the wheel-rail flange contacts

The object of this study is the process of directing the bogies of model 18-100 freight cars along a rail track, in particular, along curved sections of the track. The task to be solved was to determine the influence of wheelset arrangement on the level of steering forces in the flange contacts. A calculation diagram and a mathematical model of fitting the bogie into the curved section of the track have been built. The bogie loading scheme by external forces, including lateral rocking forces acting on the car in a curve, has been refined. In this case, the method of pseudo-statics of the mechanical system, which is a system of nonlinear algebraic equations, was applied. The calculation module Given-Find in the Mathcad software package (USA) was used to solve the mathematical model. It was established that the misalignment of wheelsets in the frames of model 18-100 bogies was of an accumulative nature. At the maximum operating angles of misalignment of wheelsets, the lateral steering forces in the flange contacts increase by 40–60 % compared to the rated setting. These angles can be up to 0.015 rad (0.85 degrees). The field of practical application of the results is railroad transportation, in particular, the system of maintenance and repair of freight cars on bogies of the 18-100 model. At the same time, the condition for the practical application of the research results is the expediency of introducing into the maintenance system the technological operation of controlling the deviation of wheelset arrangement in the bogie relative to the rated one. The current study will contribute to the construction of a measuring system for monitoring the deviation of wheelset arrangement in the bogie relative to the rated one. This proves the expediency of introducing into the trolley maintenance system the technological operation of controlling the deviation of wheelsets and designing a device for monitoring this parameter

Principle of signal-to-noise ratio direct measurement

The sensitivity of radio receivers and other systems used in all fields of practical applications of electronics is carried out at certain signal-to-noise ratio value. Examples of systems, for which this ratio is one of the important qualitative indicators, are given. The aim of the investigation is to solve the actual problem of signal-to-noise ratio measurement in a wide range of frequencies, simplify instruments and improve their technical and metrological parameters. A principle of signal-to-noise ratio measurement has been proposed, its theoretical foundation has been given. Precise relationship between the signal-to-noise ratio and the parameters of noise amplitude, phase and frequency modulation has been established, thus modulation meters can measure signal-to-noise ratio. There are three options for implementing the developed measurement principle. All of them use one existing industrial modulation meter and provide direct signal-to-noise ratio. For two options there are no restrictions on the origin and type of noise, it may be arbitrary. A pilot test of the measurement principle was conducted, and three implementation options were actually tested. Their discrepancies with the original data of the signal and noise generator lie within error limits of the instrumentation of the former. Mathematical modeling of the principle of signal-to-noise ratio measurement in Mathcad software package was also carried out. Recommendations on the use of the principle are given. The fundamental advantages of the developed principle consist of simple implementation by a single industrial instrument and solving the problem of metrological ensuring of signal-to-noise measuring instruments using the existing state verification plans. The main technical and metrological characteristics obtained by using the principle are determined by parameters of existing modulation meters and are proven to be high, the frequency range is till 26,5 GHz and the measurement error is as low as 8 %.

ЕКОНОМІКО-МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ДИНАМІКИ ЕНЕРГОСПОЖИВАННЯ ЗБАГАЧУВАЛЬНОЇ ФАБРИКИ

The purpose of the work is to establish the dependence of electricity costs on the dynamics of energy consumption from the dynamics of ore supply to the beneficiation plant bunker, namely from the loading period and its duration, and to develop, on the basis of the identified patterns of organizational, technical and economic measures to reduce the energy component of the cost of production of iron ore concentrate at the mining enrichment enterprises. Economic-mathematical methods were used to develop models of the formation of electricity costs on the dynamics of energy consumption, namely: the theory of impulse processes for the development of a model of ore supply to the bunker of the beneficiation factory; operational calculation methods, ore beneficiation theory and power supply theory for the development of a dynamic beneficiation process line model and power consumption model; methods of spectral analysis to identify the dynamic component of active power and electrical energy consumed by the ore beneficiation process; Cost analysis methods were used to calculate electricity costs; economic-mathematical modeling of energy consumption dynamics was performed by computer modeling using the Mathcad software package. An economic-mathematical model of the energy consumption dynamics of the enrichment factory was developed. The regularities of influence of ore loading parameters of beneficiation factory bunkers on the dynamics of the formation of electricity costs during the production of iron ore concentrate have been revealed. Mathematical dependencies of costs for the dynamic component of electrical energy on the time parameters of the process of loading ore into the hopper of the beneficiation factory, namely on the period and time of loading, were obtained. Organizational and economic measures have been developed to load the bunkers of the beneficiation factory, which ensure the saving of electricity. The regularities of influence of ore loading parameters of beneficiation factory bunkers on the dynamics of the formation of electricity costs during the production of iron ore concentrate have been revealed. The principles of building a system of financial incentives for operational and dispatching personnel for saving electricity based on the results of economic and mathematical modeling have been developed. For the first time, it wa s established that the energy component of the production cost of iron ore concentrate depends not only on the quantity and quality of ore, but also on the dynamic parameters of the process of loading ore into the bunkers of the beneficiation factory, while this established dependence has a non-linear nature on the period and time of loading. The proposed measures to save electricity due to the implementation of rational modes of loading bunkers of the beneficia-tion factory with ore can be used in the development of energy saving plans and programs and the development of job instructions for the dispatching staff of the ore beneficiation factory, which will allow to reduce the energy component of the production cost of iron ore concentrate. A system of financial incentives for operational dispatching staff of the beneficiation factory has been developed, which takes into account the saving of electrical energy due to the management of the ore loading of the bunkers of the beneficiation factory and the intensity of the staff’s work.

Теоретическое исследование стабилизатора вибрации с обратной связью по интегралу смещения для энергетического оборудования

Reducing the vessel hull vibration caused by the power equipment is usually obtained by the effective vibration-isolating fastening. Vibration impact on the mechanisms and automation tools greatly reduces their reliability, affects the health of passengers and crew members. Despite the modern vibration-insulating materials and compensators, the problems of eliminating the variable forces transmitted from the vibration source have not been fully solved. Increasing the operational requirements for the vibration safety of water transport requires the scientifically based and technical solutions, improving the designs of vibration isolation supports for mounting the marine power plants. Vibration reduction problems must be solved both at the design stage and at the stage of construction, repair and modernization. The conducted studies of vibrations under the influence of the harmonic force, step load and gradually increasing load have shown high efficiency of the support in all cases of load application under different simulation modes. At stabilization of the position the support was controlled by the integral of the displacement. The results of the studies conducted on the test model for the vibration protection of marine power equipment showed the effectiveness of the stabilizer. There is presented a theoretical study of a stabilizer with feedback on the displacement integral in the MathCAD software package. The numerical experiment is based on the Newton equation. The parameters of the stabilizer are determined, on the basis of which it is possible to design the actuator. It can be an electric motor, the direction of rotation of which depends on the direction of displacement of the protected object. The motor rotates a screw pair that compresses the spring. The spring acts on the clutch, the torque changes and maintains the vibration source. The simulation results helped to infer that it is necessary to stabilize the position of the unit for all load application modes.

On the issue of determining the dimensions of the rock massif cavity sufficient for the formation of a sinkhole in the overlying soil stratum

Introduction. The procedure currently recommended for karst hazard assessment includes a comparison of the cavity dimensions at the top of the karst massif with a “critical” value, sufficient for a shift of the cylindrical volume in the soil stratum. The scheme of calculating the “critical” value, proposed 40 years ago, is widely used due to its proven efficiency. However, in some cases, this procedure leads to an incorrect assessment of sinkhole development conditions, i.e., to incorrect decisions about the parameters of anti-karst measures.Aim. To adjust the procedure for calculating the “critical dimensions”, depending on the stratum structure and soil characteristics, as well as the load from the foundation.Materials and methods. The calculation formulas, transformed for computer calculations, are corrected by the introduction of a variable stability factor, the refinement of the schemes for accounting the effect of a surface load on the forces, acting along the side surface of the “shifting” volume, as well as by accounting, under the fulfillment of conditions for the general stability of a soil cylinder, the possibility of destroying the lower soil bearing layer, which is modeled as the clamped plate, loaded according to two different schemes depending on the properties of overlying soils.Results. The results of comparative calculations performed using the Mathcad software package for “standard” conditions are analyzed, and conclusions about the significance of the proposed corrections are drawn.Conclusion. When determining critical dimensions, it is necessary to introduce the values of the stability coefficient for the “shifting” soil volume, which are standard for geotechnics. In the case of using slab foundations, recommended in “karst-hazardous” areas, the effect of a surface load on the soil shift should be taken into account only when determining horizontal stresses on the side surface. Moreover, accounting can be carried out according to simple, but sufficiently justified, schemes. When the possibility of the lower layer destruction is accounted, the formation of a sinkhole is predicted at a smaller cavity span as compared to the proposed procedures.

Search for the optimal section area annular section ecentrally compressed elements of reinforcement

This article deals with the search for the optimal cross-sectional area of reinforcement of eccentrically compressed reinforced concrete elements of an annular section using the Mathcad software package. The determination of the optimal cross-sectional area of the reinforcement is carried out using the implementation of algorithms for finding the minimum that meets the requirements of the strength and stability of reinforced concrete elements. An analysis of the obtained data is given by comparison with the problem solved in the Handbook to SP 63.13330. The developed programs make it possible to save the designers time when choosing the reinforcement section and checking the strength and stability of reinforced concrete elements.

ОПТИМИЗАЦИОННЫЙ АЛГОРИТМ УПРАВЛЕНИЯ ЧАСТОТНО-РЕГУЛИРУЕМЫМ ЭЛЕКТРОПРИВОДОМ В ЭЛЕКТРОТЕХНИЧЕСКОМ КОМПЛЕКСЕ ДВУХСТУПЕНЧАТОГО ОХЛАЖДЕНИЯ ГАЗА

Relevance To cool the gas to the required temperature values in gas production and transportation in Russia, the most widespread has been implemented in the device of air-cooled gas. The installed capacity of electric drives of fans of gas air cooling devices reaches 70 % of the total installed capacity of gas field facilities. Therefore, the issue of increasing the energy efficiency of electrical complex of gas air cooling device by introducing a frequencycontrolled electric drive is relevant. Aim of research To develop a method of frequency control and to investigate the technological features of the operation of gas air cooling devices with a two-stage cooling scheme that ensures minimal electricity consumption. Research methods Mathematical methods of calculation and analysis, Mathcad software package. Results Based on the developed optimization algorithm for selecting the rotational speeds of the motors of the section of the gas air cooling apparatus, it is shown that the most effective increase in thermal efficiency can be used with separate frequency control of the fans of the section of each stage. For separate frequency control, the objective function of optimal control of the frequency-controlled electric drive of the gas air cooling apparatus according to the criterion of minimum power consumption has been developed.

ПОГРУЗЧИК-ШТАБЕЛЕР С РАСШИРЕННЫМИ ФУНКЦИОНАЛЬНЫМИ ВОЗМОЖНОСТЯМИ

The loader design with expanded functionality, in particular, an increased service area due to the use of a tripod manipulator and two guide actuators, which reduces the time of the technological operation of loading and unloading pallets in warehouse work, is proposed and developed. The kinematic parameters of the loader mechanism are calculated. Formed theoretical service area. The stability in the longitudinal and transverse planes is calculated, the maximum weight and angle of the transverse slope at which the loader will lose stability is calculated. All calculations are made in the Mathcad software package

INVESTIGATION OF LOADING STRUCTURE OF HYPER-HOPPER TWO-SECTION CARRIER DURING CARRIAGE BY RAILWAY

To increase the efficiency of operation of the hopper car, it is proposed to improve it by dividing the body into two separate sections, which facilitates the possibility of transporting different types of cargo. In order to adapt the hopper car for transportation on railway ferries in international traffic, it is proposed to install on its pivot beams knots for fastening chain ties. Determination of the dynamic load of the load-bearing structure of the hopper car was performed by mathematical modeling. The case of the on-board rocking of the railway ferry is taken into account. The solution of the mathematical model is implemented in the MathCad software package. The total amount of acceleration acting on the supporting structure of the hopper car was 2.4 m/s2 (0.24 g). The obtained value of acceleration is taken into account when calculating the strength of the load-bearing structure of the hopper car. The calculation was performed by the finite element method in the SolidWorks Simulation software package. The results of the calculations established that the maximum equivalent stresses are 312.3 MPa and occur in the node for fixing. However, they do not exceed the allowable values. The conducted researches will promote increase of efficiency of operation of hopper cars, and also creation of developments concerning designing of their perspective designs.

Modeling of the Car Traffic Air Pollution on the Territories Neighboring Multi-level Interchanges

Ensuring the proper quality of the environment is a foreground and important task for achieving sustainable development goals in contemporary urban societies. Therefore, the aim of the research is to study the air pollution on the territories neighboring multi-level interchanges and assess the additional load on the lower levels based on the analysis of physico-chemical properties of harmful substances, which are exhausts components. The scientific novelty of the paper and the authors’ contribution is in determining the level of ecological danger of territories neighboring multi-level interchanges as a result of the summation effect from emissions of vehicle flows, physical and chemical properties of pollutants, and processes of their sedimentation. To predict the pollution level of the roadside air, the model of torch approximation based on the K-theory and the equations of turbulent diffusion was used. Both mathematical modeling and visualization were performed by the Mathcad software package. Particular attention was paid to the analysis of the environmental situation neighboring multi-level interchanges and urban objects in terms of their pollution with nitrogen oxides and particulate matter PM10. Research has shown that due to the sedimentation processes a significant increase of the technogenic load on the interchange first level road will be observed. It was found that, compared to the upper road, the technogenic load in the center of the lower road increases on average by 1.4–1.6 times in terms of nitrogen oxides and by 1.5–1.6 times in terms of PM10. The territory with the harmful substances maximum permissible concentration excess is significantly increased. So, taking into account that multi-level interchanges are much more environmentally acceptable than surface sections of roads with traffic lights and pedestrian crossings, they still pose danger to the health of people who stay for a long time at the transport stops, move on the sidewalks and underground crossings, as well as for vehicle drivers and passengers. Thus, it is recommended not to place on the territories neighboring multi-level interchanges public transport stops, cafes, shops, and other city infrastructure objects, as well as to remove the existing ones.

Load of the wagon-platform for transportation of bulk cargoes

Purpose. To substantiate the improvement of the load-bearing element of the wagon-platform for the possibility of transporting bulk cargoes. Methodology. In order to be able to transport bulk cargo on the wagon-platform, it is proposed to install a composite boiler module on it. In order to determine the dynamic load of the improved load-bearing structure of the wagon-platform, mathematical modeling was performed. The mathematical model formed by professor Bohomaz H.I. was used. However, within the framework of the research this model was refined to determine the load of the wagon-platform of the proposed design. The solution of the system of differential equations is carried out in the MathCad software package. To do this, the mathematical model was reduced to the normal Cauchy form, and then integrated by the Runge-Kutta method. The obtained acceleration is taken into account when calculating the strength of the advanced load-bearing structure of the wagon-platform. The calculation is performed in the SolidWorks Simulation software package, which implements the finite element method. Also, within the research the modal analysis of a load-bearing structure of the wagon-platform is carried out. Findings. Based on the calculations, it is established that the acceleration acting on the load-bearing structure of the wagon-platform car is 0.38 g, i.e. it is within acceptable limits. The results of the calculation of the strength of the improved design of the wagon-platform showed that the maximum equivalent stresses occur in the area of interaction of the spine beam frame with the pivot and is about 340 MPa, the maximum displacement made 8.6 mm. That is, the obtained stresses do not exceed the yield strength of the structural material. The results of the modal analysis showed that the values of the natural frequencies of oscillations are within acceptable limits, because the first natural frequency has a value greater than 8 Hz. Originality. The scientific substantiation of improvement of a load-bearing structure of a universal wagon-platform to transportations of bulk cargoes is carried out. Practical value. The conducted research will promote increase in efficiency of operation of railway transport and creation of developments concerning planning of innovative designs of a rolling stock.

Loading Research of Load-Bearing Structure of Hopper Car with Composite Roof

Purpose. The work is aimed to investigate the loading of load-bearing structure with composite material roof. This will allow reducing the dead weight of the hopper car and will contribute to the possibility of increasing its carrying capacity. Methodology. Investigations were performed using the example of hoper car for grain transporting, model 19-6869, manufactured by Karpaty Experimental Mechanical Plant. It is important to say that the use of composite material reduces the roof weight by up to 40% in comparison with the metal design. That is why mathematical modeling of dynamic loading of the hopper car with composite roof was carried out. Differential equations were solved by Runge-Kutta method in MathCad software package. Initial conditions were assumed to be zero. During the calculations, the spring suspension parameters of the 18-100 bogie models were taken into account. The obtained results of calculations were used when determining the main indicators of the roof strength. The spatial model of the hopper car roof was created in SolidWorks software complex. Calculation was performed by the finite element method, which is implemented in the SolidWorks Simulation (CosmosWorks) software complex. When constructing the finite element model of the hopper car, the isoparametric tetrahedra were used. The optimum number of the model elements was determined by the grapho-analytical method. Findings. The basic indices of load-bearing structure dynamics of hopper car with composite roof were obtained. Acceleration of the body in the mass center was 5,0 m/s2. Coefficient of vertical dynamics is equal to 0.67. It was found that the maximum equivalent stresses in the roof for all the considered loading schemes do not exceed the admissible values, that is, the roof strength is ensured. Originality. The mathematical modeling of dynamic loading of the load-bearing structure of the hopper car with composite roof was carried out. The acceleration values as the components of dynamic loading acting on it during operation as well as vertical dynamics coefficient were determined. The strength indicators of the composite roof under the main operational loading modes have been found out. Practical value. The conducted research will contribute to the creation of guidelines for the design of innovative structures of the rolling stock, as well as increase the efficiency of its operation.

Evaluation of the Effect of Induction Motors on Short-Circuit Current in Electrical Installations with a Rated Voltage above 1 kV

The growth of 6(10) kV electrical networks in large cities generates the need to improve the method of calculating short-circuit (SC) currents to ensure sufficient sensitivity of the main and backup protections of cable lines, which take power supply from the generator switchgears of thermal power plants (TPP GS), which often have a galvanic link between their auxiliaries and local load. The effect an induction motor (IM) has on the SC current in electrical installations for a rated voltage above 1 kV is studied. To analyze the effect an IM has on the SC current, a computational mathematical model was developed in the MathCad software package, and variant calculations were carried out. The electrical equipment parameters and standard electric circuits of TPP GS that are applied in Russia were used as input data. It is shown that the SC current decreases depending on the IM capacity and the electrical distance of the SC from the SC current level relative to the calculations carried out without taking the IM into account. The criteria under which an IM leads to a decrease in the SC current are determined. Practical recommendations for reducing the IM influence on the SC current to ensure the necessary relay protection sensitivity are proposed.

Parametric synthesis of car suspension

Problem. When driving a vehicle with comfortable suspension settings, there is a high probability that such adjustments may, firstly, cause discomfort to the driver and passengers, and secondly, when cornering, there is a high probability that the vehicle will overturn. A stiffer suspension leads to discomfort when driving on bumps. To choose the optimal parameters of the car's suspension, it is necessary to take into account the various parameters of the car's suspension, its settings and the features of the road surface. Goal. Solving the problem of choosing the values of the varied parameters of the car suspension - the coefficients of stiffness of the elastic elements and the average values of the damping coefficients of the shock absorbers, which provide a comfortable state of the driver and passengers while driving. Methodology. The approaches adopted in the work to solve the problems are based on the algorithmic method of parametric synthesis of dynamic systems. Results. The obtained maximum values of generalized coordinates, speeds and accelerations of the sprung part of the car body, as well as the minimum values of functionalities, with the help of ratios are possible to estimate the values of weights of additive functionality based on MathCAD software package. With the help of the algorithmic method of solving the problem of parametric synthesis of a dynamic system it is possible to it automate. Analysis of the process with the specified coefficients allows to detect high efficiency of damping of vertical oscillations of the under-sprung part of the car body. The amplitude of the linear displacement of the center of mass of the body does not exceed 0,02 m, and the amplitude of the generalized veloc-ity of the centre of mass of the sprung part of the housing does not exceed 0,08 m/s-1. Originality. The obtained parameters allow to minimize the time for selection of average values when designing the vehicle suspension. Practical value. The results can be recommended when studying the design features of vehicle suspensions. Thanks to the optimization of the MATLAB software package, modeling with different suspension parameters is possible.

СРАВНИТЕЛЬНЫЙ АНАЛИЗ СПОСОБОВ ОПРЕДЕЛЕНИЯ РАСХОДА КОНДЕНСАТА В СИСТЕМЕ ВЕНТИЛЯЦИИ МАШИНОСТРОИТЕЛЬНОГО ПРЕДПРИЯТИЯ

The study objective is designing the condensate removal system of ventilation at a machine-building enterprise, eliminating the problem of the exhaust duct freezing at outdoor sub-zero temperatures due to the formation of condensate in it and, accordingly, reducing the consumption of electricity consumed by the exhaust fan.
 The problem to which the paper is devoted is to develop optimal methods for defining the condensate flow to design a condensate removal system of ventilation.
 Research methods: three methods for calculating the condensate flow were analysed: 1) taking into account changes only in the temperature of the removed air; 2) taking into account changes not only in temperature, but also in pressure using general gas equation by Clapeyron; 3) taking into account changes in temperature and pressure based on the Van der Waals equation. Using system approach the processes leading to condensate formation have been simulated, and an optimal mathematical model has been created on its basis that allows quite accurate determining the volume of condensate released with the removed air passing through the heat exchanger. 
 The novelty of the work: an optimal algorithm for calculating the amount of condensate formed in the ventilation system has been defined.
 Study results: the analysis is made using Mathcad software package and the condensate volume against the humidity and temperature of the removed air were graphed for three methods, an optimal algorithm for determining the condensate flow was developed.
 Conclusions: the analysis showed that it is possible to use a simplified first calculation method with the additional Van der Waals coefficient in the calculation dependence for designing a condensate removal system. The studies conducted allow not only to define the initial data for the design of the condensate removal system, but also to give appropriate recommendations for the construction of the entire ventilation system.

Расчет температурного поля многослойных несущих конструкций численными методами

Modern approaches to the electronic means design include modeling of thermophysical processes occurring in the apparatus being developed. At the same time, a common task in thermophysical modeling is the problem of determining the temperature field of the supporting structure of an electronic device (board, substrate). Knowledge of the quantitative indicators of the supporting structure temperature field directly affects the prediction of the entire electronic device reliability. The load-bearing structures of electronic devices are increasingly made in the form of multilayer structures where conductive, insulating and semiconductor layers can be present. These features impose new requirements on the development of mathematical models, algorithms, application software packages for calculating the temperature fields of the supporting structures of electronic means. The article considers the process of modeling the temperature field for the supporting structures of electronic means using common software systems MathCAD and SolidWorks. The whole variety of supporting structures of electronic means is divided into three categories: conditionally one-dimensional, conditionally two-dimensional and three-dimensional. It is shown that the temperature fields of conditionally one-dimensional (rods) and conditionally two-dimensional (single-layer boards and substrates) structures should be calculated by the finite difference method in the MathCAD software package. Temperature fields of three-dimensional load-bearing structures, including complex geometric configurations, should be calculated using the finite element method in the SolidWorks software package. The developed algorithms for calculating temperature fields are verified by solving a test problem. Comparative analysis has shown that the discrepancy between the calculation results relative to the test problem does not exceed 0.8%. The developed algorithms for calculating the temperature field can be practically useful in the engineering activities of the developer of electronic equipment.

Mathematical 3D-modeling in the formation process of the overhauled surfaces in the repair industry

Purpose. Mathematical representation of the geometry of the overhauled valve seat surfaces in the shape-forming process in the auto repair industry. Methodology. The research methods were based on: the theory of cutting and metalworking scientific positions, theoretical foundations of the technology of mechanical engineering and overhaul repairs; the basics of technical operation and vehicle structure; provisions of standardization, technical measurements; means of mathematical modelling, statistics and programming. Findings. The geometric 3D-model of the processing surfaces of valve seats has been presented, which is used in the overhaul process of these parts, which is basic in the general model of the shape-forming system, which allows controlling the shaping parameters with the required precision due to the structure of the model itself and introducing the results of processing particular models: interpolation (to assess the worn surfaces state) and optimization (for determining the volume of cut material and setting cutting conditions). Originality. For the first time a particular mathematical 3D-model of the general shape-forming system has been developed, consisting of other particular mathematical models (interpolation and optimization), which gives a geometric representation of the processed valve seat surfaces within their overhaul in the auto repair industry. Practical value. The proposed mathematical 3D-model allows: - implementing the shape-forming process adequately within the simultaneous boring of three inner conical surfaces by copying with a profile cutting plate, which provides the required parameters of precision and quality of processing during the valve seats overhaul; - implementing the results of the flaw detection process of worn surfaces of valve seats with a complex-variable topography due to usage of the of roundness- and profile-diagrams and method based on an interpolation geometric 3D-model; - introducing an optimization model to determine the optimal volumes of cut material and the rational cutting modes select. The MathCAD software package has been developed, which allows having a graphical performance of the formed surface based on the interpolation model.

Model of Hydrodynamic Friction in the Boundary Layer on an Inclined Flat Surface of an Aircraft

An integral method for solving the problem of longitudinal flow of an inclined flat plate located in a stream of air flow is presented. As a result of the integration of the equations of motion over the thickness of the boundary layer, the two-dimensional problem becomes one-dimensional. The problem was solved numerically in the Mathcad software package. The graphs of the dependence of the thickness of the boundary layer and friction on the longitudinal coordinate at different angles of inclination of the plate are presented. The flow mathematical model can be used to determine the characteristics of two-dimensional boundary layers with arbitrary boundary conditions.

ДЕФОРМИРОВАНИЕ КОЛЬЦЕВЫХ ОРТОТРОПНЫХ ПЛАСТИН СРЕДНЕЙ ТОЛЩИНЫ ИЗ МАТЕРИАЛОВ, ЧУВСТВИТЕЛЬНЫХ К ВИДУ НАПРЯЖЕННОГО СОСТОЯНИЯ

Discusses a model of a three-layer annular plate of medium thickness. It is assumed that the load on the plate is assumed to be uniformly distributed. The universal relations constructed in the normalized tensor stress space associated with the main axes of the anisotropy of the material are taken as the determining relations. The load was taken in such a way that the deflections of the middle surface of the plate were considered small in comparison with its thickness. The fixing of the plate is rigid along the external and internal contours. Since some orthotropic materials with different resistance exhibit a nonlinear dependence of deformations on stresses, the material characteristics are taken as functions of stress intensity. As a result of the formulation of the boundary value problem, a mathematical model was developed for the analyzed class of problems, implemented as a numerical algorithm integrated into the Mathcad software package. To solve the system of resolving differential equations of bending of annular orthotropic plates, the method of variable elasticity parameters with a finite-difference approximation of the second order of accuracy was used.