Analytic Geometry Method Research Articles

STUDY OF THE DESIGN OF RAILWAY TRACKS OF INDUSTRIAL ENTERPRISES WITHIN THE AREAS OF SMOOTH TRACK EXPANSION

The goal. Railway sidings of industrial enterprises are the shippers and receivers of goods, which determine their main nomenclature for transportation by mainline rail transport. The length of railway sidings of industrial enterprises varies widely, from a few hundred meters to hundreds of kilometers, including the length of the siding of PJSC ArcelorMittal Kryvyi Rih exceeding 700 km. The combination of factors such as the presence of technological transportation on the sidings, the development of the industrial enterprises' transportation system over time, and the master plan of the main production facilities leads to the need to apply railway transport design standards for severe and especially severe conditions. This is mainly expressed in reducing the radii of curved track sections to the minimum possible values. When using reinforced concrete sleepers of the Sh6 type, it is possible to arrange a track gauge of 1539 mm in curved sections. But currently, there is no approved design for smooth track widening sections on reinforced concrete sleepers in front of curves with radii of less than 300 meters. There is also a reasonable need to increase the track gauge to at least 1540 mm in curves of such radii. Together with the limited service life of wooden sleepers when they are laid in areas of smooth track widening compared to reinforced concrete sleepers, this necessitates the development of a track structure on reinforced concrete sleepers within the area of smooth track widening. Methods. Methods of scientific analysis and synthesis of the track development structure, methods of analytical geometry. Results. Based on the analysis of the design of intermediate fasteners of the KB50 type, the design of a section of smooth track widening using reinforced concrete sleepers of types Sh1-1 and Sh6 and rails R50 is proposed. The track widening from 1520 to 1545 mm is achieved by developing the order of laying sleepers of different types and turning the KB50 substrates. In this case, the length of the section of smooth track widening from 1520 mm to 1545 mm is 2800 mm, and the actual deviation is 8.9‰. Practical significance. The use of the considered structure within the areas of smooth track widening, provided that the results of trial operation and testing of the established procedure are positive, will reduce the operating costs associated with the current maintenance of industrial enterprises' tracks, the parameter of which correspond to severe and especially severe design conditions.

CONSTRUCTION OF THE RAILWAY TRACK OF UKRAINE AND PROSPECTS FOR THEIR IMPROVEMENT

The goal. The process of improving the design of the railway track requires a detailed study of the history of the development of its structural elements, methods of combination, interconnections and limits of mutual influence. Modern track designs are the result of many years of hard work by scientists and long operational tests. Some designs were inventions at the time, some were breakthrough technologies, some remained just ideas, and some have existed for more than a century. The study of the history of the development of railway track structures and modern existing structures in operation made it possible to identify a promising direction for improvement and further development, taking into account the existing problems on the mainline transport tracks and industrial enterprise tracks. Reinforced concrete sleepers and intermediate fasteners that allow for track gauge adjustment are the railroad track structure that should be improved in the coming years. Improving railroad track design is a continuous process in the development of rail transport. Recently, new track designs have been introduced on Ukrainian railways. As a rule, the problems of track facilities are first considered and technical tasks are formulated. Then, a public discussion is held in the scientific and practical environment and an appropriate development strategy is born. Such a strategy is reviewed and approved at scientific and technical meetings and specialized commissions. This article aims to identify promising areas for improving the track design and solving the problems of installing and operating a continuous track by analyzing existing designs and world experience of railways. Methods. Methods of scientific analysis and synthesis of track development design, methods of analytical geometry. Results. Based on the analysis of the history of the development of the railroad track structure and modern designs, a promising direction of improvement is proposed. A promising railroad track structure is defined as a reinforced concrete sleeper and fasteners capable of forming an adjustable track gauge of up to 1545 mm. At the same time, it is necessary that the new design has sufficient stability of the jointless track plates. Practical significance. The application of the proposed structure within small radius curves, provided that the results of trial operation and testing of the established procedure are positive, will reduce the operating costs associated with the current maintenance of the tracks, expand the scope of laying of the jointless track and use reinforced concrete sleepers in small radius curves at industrial enterprises.

BASIC IMPERATIVES FOR MODELING PENSION PAYMENTS IN THE PENSION SECURITY SYSTEM OF UKRAINE

The purpose of the article is to research and analysis of the amount of indexation of pension payments related to inflationary processes. The object of the study is the state of pension provision in Ukraine. The subject of the study is the process of indexation of pension payments as a result of the implementation of social policy in Ukraine. An analysis of the history of the development and reform of pension provision in Ukraine was carried out. This allowed for a deeper understanding of the content of events related to the establishment of pension provision in Ukraine. The research methodology involves the synergy of mathematical methods of analytical geometry and economic-mathematical modelling. It is shown that the presence of inflationary processes in the national economy leads to the need to increase the pension provision of citizens. At the same time, the size of the increase in pension payments should be differentiated in order to ensure social "justice". The scientific task set in the article is to demonstrate a modern approach to choosing a strategy for implementing the state pension policy in Ukraine. As a result of the research, two models of appropriate indexation of pension payments in Ukraine are presented. The calculation made it possible to choose the most significant model. In the future, it is advisable to direct the research to the search for other more effective ways of indexing pensions. It is assumed that for this it is necessary to formalize the mathematical formulation of the problem of finding the optimal value of the index of pension payments as extreme, taking into account the necessary restrictions. A scientific novelty in solving this issue is the receipt of practically significant recommendations regarding the indexation of pensions. The practical significance of the obtained results lies in the elimination of social defects in the indexation of pension payments. Further research should be directed to the search for rational solutions that will stimulate the creation of an effective pension system in Ukraine.

Mathematical modeling of the regional budget formation procedure

The regional budget is a key element of the regional financial system. According to budget legislation, the formation of the budget of the subject of the Russian Federation is carried out by state authorities of the constituent entity of the Russian Federation in accordance with the requirements established by the Budget Code of the Russian Federation, as well as the laws of the subject of the Russian Federation and other regulatory documents adopted in accordance with it. The procedure for forming a regional budget is quite well regulated, but poorly formalized. It should be noted that budget planning tactics and strategy differ significantly, since long-term goals are set on the basis of the development of strategic plans. The purpose of this study is to implement an approach to the formation of the regional budget according to the optimal structure. The uniqueness of the concept lies in the development of the target vector and positioning of budget planning as a Bayesian process. The criterion of optimality is the location of a multidimensional point expressing the composition of the budget on a straight line reflecting the target vector. Such an approach in relation to the procedure for the formation of the regional budget has not been considered before. The solution to this problem involves the formation of a target vector for both its revenue and expenditure parts. The methodological tools used in the work are classical methods of analytical geometry and an algorithm for forming a regional budget. To increase the adequacy of the results when modeling the regional budget, it is advisable to add the condition that the random process belongs to the Bayesian type, then, in addition to the obtained values, the share of the item in the regional budget is also taken into account. The Bayesian process takes into account conditional probabilities. In this work, the relative measure of achievement of each budget item of the target state in the base period acts as conditional probabilities. The article uses an illustrative example to show the use of the proposed methodological approach for the formation of an optimal regional budget within the framework of revenue and expenditure parts for items approved by Russian legislation. The construction and analysis of a regional budget allows us to take a new approach to solving this problem, and most importantly, to create an optimal budget while saving intellectual and time costs for participants in the budget process. The results of mathematical modeling of the regional budget are presented in the form of an algorithm, which makes it possible to consolidate the relevant procedures for calculating the revenue and expenditure parts of the budget in regulatory documents (at the federal and/or regional level) for their further use in the budget process.

Research on Measurement of Multi-beam Transducer Based on Analytic Geometry Method

The ocean is inseparable from the survival and development of human beings. It not only provides rich resources for human beings, but also provides a large number of samples for human scientific research. For the efficient development of the ocean, detailed and accurate seabed topography and geomorphology information is needed as a prerequisite. In this paper, aiming at the problem of multi-beam detection system detecting full coverage water depth strip in waterway, the measurement model of multi-beam transducer under different survey lines is established. Based on the combination of numbers and shapes, firstly, this paper considering the special situation that the survey line is perpendicular to the horizontal component of the normal of the seabed slope, it is concluded that the depth of seawater is positively correlated with the width of the survey line. Secondly, when the angle of the survey line is arbitrary, and the spatial rectangular coordinate system model is established, this paper considering the law of the width of the survey line at different angles and distances. Finally, through analytic geometry method, the measuring line equations at different positions and angles are expressed in this paper, and the measuring line width is quantified according to the distance formula between two points. The results show that the coverage width is unchanged at the center of the sea area or when the direction of the survey line is perpendicular to the horizontal component of the normal of the submarine slope. The coverage width is symmetrically distributed at the survey line angle. The results show that the multi-beam transducer measurement model is effective.

A Method for Calculating the Optical Efficiency of Solar Tower Heliostats

Solar tower heliostat is one of the highlights of China's new energy development in recent years, the Optical Efficiency is an important index for evaluating the field distribution of heliostat, but the calculation of Shadow Blocking Efficiency and Truncation Efficiency is more complicated. In this paper, through the ray tracing method, based on the idea of Monte Carlo algorithm, combined with the calculation method of spatial analytic geometry, a more complete and operable method of calculating Optical Efficiency is proposed.

Dynamic effects of rough elements in fractures on coal permeability during different stages of methane extraction

Fractures in coal seams are channels for gas migration, and their geometric characteristics can affect permeability, thereby affecting the effectiveness of coalbed methane (CBM) extraction and carbon dioxide sequestration. The surface of natural fractures is rough and composed of minerals with complex shapes. At present, research on rough fracture is mainly focused on surface characteristics, with less attention given to the dynamic effects of rough structure in fractures on coal permeability considering effective stress and matrix adsorption expansion during different stages of methane extraction. In this work, we construct the complex geometric structure in the fracture surface into simplified rough elements. First, a physical model of fracture rough elements was constructed by using the analytical geometry method; meanwhile, the roughness applicable to this model was proposed, and the characterization of roughness degree of different-shaped rough elements was unified. Subsequently, a definition and an approach for selecting and computing effective fracture aperture were proposed. By taking into account the competition between effective stress and matrix adsorption expansion, the change of effective fracture aperture related to pore pressure was determined. Additionally, a novel permeability evolution model was formulated according to the cubic law and compared with the numerical simulation of other researchers for verification. The variation patterns of permeability, gas pressure and fracture aperture under diverse characteristic fracture roughness elements during CBM extraction were obtained by utilizing the proposed model. The findings indicate that the fractures with smaller roughness, higher stagger degree and gentler inclination angle of rough elements are more favorable for CBM extraction. This has inspired a new understanding of the mechanism of CO2-enhanced coal bed methane recovery (CO2-ECBM) technology in deep coal seams with regard to fracture rough element characteristics, which offers valuable theoretical support for the practical implementation of this technology.

Underwater Cable Localization Method Based on Beetle Swarm Optimization Algorithm

Dear Editor, This letter concerns the localization of three-core underwater cables which are widely deployed for offshore energy transmission. It is a non-trivial problem, since the external magnetic field of three-core underwater cables is variable which reduces the accuracy of localization. To solve this problem, in this letter, an approximate equation is firstly derived to formulate the external magnetic field of a three-core armored underwater cable by considering the seafloor environments and the structure of three-core cables. Then, a new underwater cable localization method is proposed based on dual three-axis magnetic sensor array and the beetle swarm optimization (BSO) algorithm. The method constructs a fitness function based on the magnetic flux density amplitude and replaces the existing analytical geometry method with an optimization algorithm to achieve underwater cable localization with higher accuracy.

AUTOMATION OF CALCULATING THE POSITION OF THE BOUNDARY COLUMN BETWEEN ADJACENT TRACKS OF DIFFERENT DESIGNS

The purpose of the study is to develop appropriate algorithms and procedures used in simulating the movement of a model car along adjacent tracks. These algorithms will automate the process of calculating the position of the boundary column between adjacent tracks of different designs with a given accuracy and compliance with traffic safety requirements. Scientific methods. In the course of the study, simulation methods were used to model the movement of the design car along the adjacent tracks. Analytical geometry methods were used to determine the main geometric characteristics of the car in terms of its relative position relative to the track axis. Results. The following conclusions were obtained as a result of the research. The multivariability of the design of adjacent tracks in the plan does not make it possible to create a universal analytical expression for determining the position of the boundary column. Existing methods for calculating the position of the boundary post provide for the maximum spread of the dimension in curved sections, which results in an excessive distance between the boundary post and the switch and inefficient use of the useful track length. The developed algorithms and procedures for determining the position of the boundary column are universal and allow automating the process of calculating the position of the boundary column between adjacent tracks of different designs in the plan, even when technical standards or parameters of the design cars change. In this case, only the initial data for the calculation need to be changed. Scientific novelty. The scientific novelty lies in the development of algorithms and procedures for formalizing the design of adjacent tracks and simulating the movement of design rolling stock on them. Practical significance. The developed algorithms and procedures can be implemented in the form of a software package for automated calculation of the position of the boundary column with high accuracy. Also, the obtained results of calculations can be used in the analysis and creation of an analytical description of the position of the boundary column for complex structures of adjacent tracks.

Mutual Inductance Calculation of Circular Coils for an Arbitrary Position With a Finite Magnetic Core in Wireless Power Transfer Systems

Mutual inductance can be increased, and magnetic leakage can be reduced by adding finite magnetic core materials to the coil structure. However, the problem of the calculation of the mutual inductance between two arbitrarily positioned circular coils with a finite magnetic core has not been resolved. In this article, an analytical formula for calculating the mutual inductance between two circular coaxial coils on a finite magnetic core is obtained based on regional hierarchical expansion analysis (RHEA), the Maxwell equation, the Bessel function, and the boundary conditions for electromagnetic fields. The mutual inductance between two arbitrarily positioned coils is obtained by using the analytical geometry method. A series of experiments are carried out to verify the analytic calculations. The experimental results for each misalignment case are compared with the calculated results and simulated results. The theoretical results and simulated results are in good agreement with the experimental data, which verifies the validity of the proposed method.

Complex numbers in the school course of Mathematics and in the perspective model of the final exam

The article is devoted to the use of the concept of a complex number in a school Mathematics course. Until recently, the topic of complex numbers had not been included in the state final exam, however studied in classes with in-depth study of mathematics. In connection with the introduction of tasks related to this concept into the prospective model of the unified state exam, it is now becoming topical. The article primarily touches upon the analysis of the presentation of material in school textbooks, in particular the concept of a complex number. Then there are several examples of problem solving, which was presented in a promising model of the unified state exam in Mathematics, using both analytical and geometric interpretation of a complex number. In addition, various approaches to solving the problem are shown, both algebraic (estimation method) and geometric (finding the altitude in a triangle, similarity method, methods of analytical geometry) methods, as well as methods of mathematical analysis (finding the largest or smallest value using the derivative). The article also contains some methodological recommendations for preparing on this topic and a system of tasks that can be offered both to teachers in preparing for the exam, and to pupils to consolidate the skill of solving tasks of this type.

ANALYSIS OF EXISTING METHODS FOR CALCULATING THE POSITION OF BOUNDARY COLUMNS

The purpose of the research is to improve the methods of calculating the position of limit posts on the railways of Ukraine. To achieve the established goal, a comprehensive analysis of the existing methods of determining the position of boundary posts was performed, taking into account the multivariate design features of the railway track connection and compliance with traffic safety requirements. Methodology. In the process of research, methods of analysis and synthesis were used to study the content and basic provisions of existing methods for determining the position of limit posts on railways; methods of analytical geometry to describe the position of the limit column relative to adjacent tracks, taking into account the geometric parameters of the rolling stock. The results. As a result of the conducted research, the following conclusions were obtained. The basis of the current methods of determining the position of the limit posts are the requirements for the parameters of the calculation car, which is not operated on the railways of Ukraine, however, the recalculation of the position of the boundary columns on the existing tracks of the main and industrial railway transport using the characteristics of the car that will replace the existing calculation car is currently impractical and irrelevant Taking into account the multivariate constructions of the connection of railway tracks, there is no universal analytical solution for determining the position of the limit column. New approaches to determining the distribution of the overall distance when calculating the position of the limit posts in the conditions of automated design have inaccuracies, and also do not take into account the variety of structural features of the connection of railway tracks. The results of the performed analysis are the basis for continuing research on improving the methods of calculating the position of limit posts on the railways of Ukraine. Scientific novelty. The obtained results make it possible to summarize the scientific basis for the imperfection of the existing methods of calculating the position of the limit posts on the railways of Ukraine. Practical significance. From a practical point of view, the results of the analysis create conditions for further research on the improvement of the above methods. The result of the research should be the development of a universal method of theoretically accurate calculation of the position of the limit post or other objects on railway tracks for any design of track development. The use of this method in design practice is possible by developing a software product for determining the position of the boundary column, including on tracks with a width of 1435 mm, for cases where tracks with a width of 1435 and 1520 mm are combined on the same ground surface and under the condition of using calculated wagons with any parameters.

Mathematical modeling of the process of vacuum drying of corn cobs

Purpose. To determine the effect of vacuum on the process of drying corn seeds on the cob using mathematical modeling. Methods. Differential calculus methods were used, in particular, methods for solving differential diffusion equations with time-varying boundary conditions, as well as methods of differential and analytical geometry. The mathematical model was analyzed using graphs constructed by the computer program MathCad 13 from the solutions of differential equations and methods of technical analysis. Results. On the basis of the differential diffusion equation with variable boundary conditions in time, the processes of moisture removal from corn seeds in the cob were described. These processes are described by analytical dependencies that describe the kinetics of drying corn seeds and the entire cob, while taking into account diffusion processes both inside the seed and in the cob core. They are used to construct graphs that describe the dynamics of the moisture field inside a single seed and all the seeds along the cob. They show that the part of the seed that is outside the cob takes the longest to dry. The graphs also show the unevenness of drying along the cob: the ends of the cob dry the fastest, and the middle dries the slowest. Conclusions. The vacuum drying of corn cobs consists of 3 processes that take place in parallel: evaporation of moisture from the end surface of the cob, diffusion of moisture to the outer surface of the cob, and diffusion of moisture from the seed to the cob. The outer part of the seed takes the longest to dry, resulting in a moisture gradient that moves toward the core. Reducing the pressure inside the drying chamber reduces the time for moisture equalization, both over the volume of the seed and along the length of the cob, and increases the drying speed of the cob. To increase the uniformity of drying, the distribution of the heating temperature of the seeds along the length of the cob should correspond to a parabolic relationship with the highest temperature in the middle of the cob length, or it is necessary to divide the cob into parts to reduce the length of the moisture path. Keywords: seeds, moisture diffusion, drying time, moisture evaporation, vacuum, humidity, corn cob, cob, drying kinetics, drying chamber.

Розробка системи кріплення мікрохвильового радара для виявлення перешкод

The subject of scientific research is the consideration of layout options and the possibility of placing a microwave radar on board a helicopter. The goal is to develop a methodology for creating a three-dimensional parametric model of the radar mounting system. The task is to develop an algorithm (sequence of steps) for creating a three-dimensional parametric radar mount model without making changes to the basic design and layout of an existing aircraft. The methods used are: methods of analytical geometry; methods of scanning three-dimensional objects; methods of three-dimensional parametric modeling using modern computer-integrated CAD\CAM\CAE systems in the context of existing assemblies. The following results were obtained. On the basis of the scanned point cloud, a parametric model of the nose of the aircraft was created, the area of additional equipment installed on the aircraft was determined, a microwave radar mounting system was developed, a three-dimensional model of the mounting system was made, and design documentation was developed for it. Conclusions. The scientific novelty of the obtained results is as follows: a general algorithm, stages of design and three-dimensional parametric modeling of the aircraft unit using the methods and principles of reverse engineering have been developed. The implementation of this algorithm, methods and principles of reverse engineering is presented on en example of the creation of a microwave radar mounting system on the F1 compartment of the Mi8-MSB1 helicopter. The work presents the stages of designing a platform for attaching a microwave obstacle detection radar to the frame of the F1 compartment Mi-8MSB1 helicopter, as well as the radar scanning zone during the flight.

FEATURES OF PROCESSING AND ASSEMBLY OF COMPOSITE PRODUCTS

The paper examines the assembling of cylindrical products consisting of sector parts with dihedral angles. A crucial issue in the assembly of such parts is the accuracy of dihedral angles during processing. Based on methods of analytical geometry, we proved that in order to achieve gapless assembly of three sectors, the sum of all dihedral angles of the sectors should be less than or equal to 360°. Since it is impossible to ensure the exact equality of the sum of all dihedral angles of 360°, characteristic “flatness deviation” is always formed during assembly. In order t obtain a gapless connection, we propose a new method for the selective assembly of sectors and developed a scheme of the dihedral angle controller. This selective assembly method will allow sectors to be selected in such a way that their “twist” angle in the assembled conditions is minimal.

Justification and calculation of design and strength parameters of screw loaders

Existing designs of screw working bodies of transport and processing machines cannot always provide the required performance and avoid overloading, which causes deformation and damage to parts of the mechanisms. Thus, it is necessary to establish new efficient screw working bodies that would ensure the execution of the technological process. The purpose of the study was to substantiate and explore the design and methodology for calculating the main structural-power parameters of the developed screw loader, to synthesise screw loaders using the method of morphological analysis to obtain designs of mechanisms with better technical and economic characteristics. The methods of surface theory, analytical and differential geometry, computer graphics and mathematics were used to conduct the research. As a result of theoretical and experimental studies, the designs of screw conveyors with hopper loading and screw conveyors with nozzle loading are presented. In addition, a classification of loaders for screw transport and technological mechanisms is presented. The dependencies for calculating the design of screw conveyor loaders are determined. The synthesis of loaders and mixers by the method of morphological analysis was performed to obtain designs of mechanisms with the best technical and economic characteristics and competitive designs of mechanisms were established. A system of coding and synthesis of structural elements and mechanisms of screw implements with improved technological properties is proposed, and competitive designs of screw implement that perform technological processes efficiently and effectively are selected. A methodology for calculating the main structural and power parameters of a screw loader has been developed, which allows for determining the parameters of a screw loader. The materials of the research can be used in further studies of screw conveyors and, in particular, screw loaders to determine their design, technological, energy and power parameters

Power Arcing Fault Detection and Location Algorithm Solution in Two and Three Dimensions Euclidean Planes Using an Analytic Geometry Method Instead of an Iterative or Numerical Method

The relationship between the electric current and the magnetism is well demonstrated by the Faraday's induction and conductivity laws. Under this concept, it follows that, through intensive laboratory experiments, it has been established that the arcing fault current path related sparks are indeed source of ElectroMagnetic Radiated (EMR) signals waves. These EMR can be detected and located at a relatively remote distance, with the aid of directional antennas arrays, at considerable and reasonable accuracy. Under this novel concept of power arcing fault source point triangulation, the arc EMR data play a critical role in providing essential timing information onto which the arc fault detection and location algorithm is based. Thus, from these arc EMR recorded and gathered data, the signals Time Differences Of Arrival (TDOA) information or signatures can be easily extracted from a pair of antennas, with the aid of an appropriate signal processing tool, namely the cross-correlation (Xcorr) function. Since the electric arc phenomenon is unpredictable practically, therefore, the use of the arc signal Time Of Arrival (TOA), is impossible for the fault source point triangulation solution. In contrast, the values of TDOA seem to be efficient for solving the system of hyperbolic equations which has been imposed by this particular field measurement structure and constraints. It is of interest to mention that, the arc fault EMR field measurement structure and constraints are given and determined by the distributed antennas arrays locations or positions only. In other words, with the knowledge of both the antennas arrays locations Cartesian Coordinates (CC), together with the TDOA values, it is possible to pinpoint the unknown target power arcing fault exact source point position or location, not only in two-dimensions (2D), but also in three-dimensions (3D) Euclidean plane. Thus, this kind of arc fault source point position or location solution finding is generally referred to as a source point triangulation method. This source point triangulation method consists of solving a system of hyperbolic equations as aforementioned. In fact, under a geometry analysis process, it follows that, if only if the arcing fault hazard formation time is known in advance, therefore, the solution of a system of hyperbolic equations consists of finding the intersection point of at least two or more circles. Such that, each of these circles, will be centered at each antenna in such a way that each circle will have a radius given by the distance derived from the value of TOA together with the power arcing fault hazard ignition time stamp or signature. In contrast, since the arcing fault hazard formation stage time is unknown in practice, therefore the solution of the system of hyperbolic equations consists of finding the intersection point of at least two or more hyperbolas. Such that each of these hyperbolas will have a focal point determined by the location of each antenna and the distance between pair of foci will be given by the values of TDOA. However, generally, a system of hyperbolic equations can be solved either with the aid of an analytic method or with that of a numerical method (also known as iterative method). Unlike the majority of this research field related relevant published literature into which the iterative methods have been used intensively, this paper presents an accurate analytic geometry method in order to solve the electric arc faults EMR source point exact location or position. As a result, it appears that, this novel proposed analytic geometry method has shown its potential in clarifying the power arcing fault source position or location, at a reasonable and considerable accuracy.

Study on propagation mechanisms of the actuations generated by plasma synthetic jet actuator in a supersonic flow

The plasma synthetic jet actuator (PSJA) has a potential application in the flow control, especially for the supersonic and hypersonic flow. In order to make a deeper understanding on the propagation processes of the actuation generated by the PSJA in a supersonic flow, both the schlieren experiment and the numerical simulation are adopted for the analyses. Based on the analytic geometry method, the propagation of the diffracted wave, the synthetic jet and the dis-charged heat are discussed in detail. The effects of the mainstream Mach number and the discharge strength on the propagation processes are also deeply analyzed. Besides, an analytic model and relevant motion equations are established for the forecast of the actuations. The results suggest that the 1st diffracted wave expands around and propagates along the mainstream direction. The synthetic jet will change the start point of the oblique wave. Further, the propagation of the dis-charged heat is also related to the mainstream.

Surfaces of congruent sections of pendulum type on cylinders with generatrix superellipses

In 1972, I.I. Kotov proposed to separate the surfaces of congruent sections into a separate class and to include the surfaces of plane-and-parallel translation, surfaces of revolution, carved surfaces of Monge, cyclic surfaces with a generatrix circle of constant radius, rotative, spiroidal, and helical surfaces in it. The aim of the research is to obtain generalized parametric equations of surfaces of congruent sections of the pendulum type on right cylinders with plane-and-parallel translation of movable rigid superellipses. Analytical geometry methods are used. Computer systems MathCad and AutoCAD are applied to visualize surfaces. The results consist in the derivation of parametric equations of the studied surfaces in a general form convenient for the use of computer modeling methods. The technique is demonstrated on five examples with congruent mobile superellipses. The possibility of using obtained surface shapes in parametric architecture, free-form architecture, and in shaping of the surfaces of some technical products is noted.

ДВИЖЕНИЕ ИСПОЛНИТЕЛЬНЫХ ОРГАНОВ МЕХАНИЗМОВ ПО ЭЛЛИПТИЧЕСКИМ И КРУГОВЫМ ТРАЕКТОРИЯМ

The work objective is to generalize the principle of combining movements into circular movements.
 The problem to which the article is devoted is the analytical description of the trajectories of combined movements.
 Research methods. Analytical geometry methods are used. The coordinate system x'0'y' is considered, which rotates in the coordinate system x0y without angular acceleration with ω velocity. The rotation radius is equal to ρ1. At the same time 0x || 0'x', 0y || 0'y'. Object a rotates in the coordinate system x'0'y' without angular acceleration at the velocity ± ω. The rotation radius is equal to ρ2.
 The novelty of the work realises in ellipse formulas expressed in terms of radii of opposite directions.
 The results of the study: it is established that during rotations in opposite directions, the trajectory of the total motion is an ellipse; all the standard characteristics of the ellipse are determined in relation to the case under consideration; the inclination of the elliptical trajectory is established; it is shown that if the trajectory of the total motion is elliptical and the semi-axes are equal to (ρ1 + ρ2) and |ρ1 – ρ2|, then object a moves along a circular in the coordinate system x'0'y' without angular acceleration with velocity - ω; just as the result of the superposition of two non-accelerated movements is also non-accelerated, i.e. uniform and rectilinear motion, with rotations in one direction, the trajectory of the total motion is a circle; with circular movements with multiple velocities, the trajectory of the total motion is snail. 
 Conclusions: the practical aspect of the study is determined by the fact that the formulas obtained can be directly used in CAD when performing design work.