Mathcad Software Package Research Articles

ГЕОДИНАМІКА

The main goal is to determine and analyze the recent rotation poles of the main tectonic plates based on measurements of continuous GNSS stations for the period of 2002–2021. Using procedures based on the method of the least squares, we suggested an algorithm to determine recent rotation poles of tectonic plates on the basis of processing time series of daily solutions of continuous GNSS stations. The algorythm was implemented in the MathCAD software package. It uses, generalizes, and modernizes the approaches presented in previous studies. Structurally, this algorithm consists of five consecutive stages: transformation of data into an internal format; compliance check and time series filtering; determination of horizontal displacement rates; compliance check and filtering of specified velocities; determination of rotation poles. The algorithm involves the use of freely available time series of daily solutions of continuous GNSS stations, or any other data prepared in a similar format. The study has developed an algorithm to determine recent rotation poles of tectonic plates. It is based on processing time series of daily solutions of continuous GNSS stations. The algorithm was tested to define the recent rotation poles of the main tectonic plates. We determined the components of recent horizontal displacement vectors of 3169 continuous GNSS stations located on 7 large, 7 medium and 3 micro plates for the period of 2002-2021 in the ITRF2014/IGS14 reference frame. The accuracy of determining the component vectors of horizontal displacements is in the range of 0.9-6.4 mm and is on average 10–15% of the vector length. The research allowed us to construct a map scheme of the spatial distribution of the velocity field of recent horizontal movements of continuous GNSS stations. Recent rotation poles of the main tectonic plates were determined for the period 2002-2021 in ITRF2014/IGS14 reference frame. It was established that, in general, the obtained values of recent rotation poles correlate well with known models of tectonic plate movements. This confirms the correctness of the chosen method, as well as the reliability of the obtained results. Recent rotation poles of tectonic plates are the basis for modeling and analysis of global, regional and local geodynamic processes, so their accurate determination is an urgent and necessary task. GNSS data is an alternative, and recently, practically irreplaceable basis for determining such parameters. The rapid increase in the number of continuous GNSS stations, as well as the high quality of their measurements, contributes to improving the accuracy of determining the recent rotation poles of tectonic plates, but leads to the need for their constant recalculation and refinement. The presented algorithm and the obtained results can be used to develop new and refine existing models of tectonic plate movements and reference frames, as well as to forecast the movements of the Earth’s crust.

Equations of motion of masses of the Chelomey pendulum model

To verify the provisions stated by V.I. Bogomolov, B.I. Puzanov. and Linevich E.I. about the possibility of performing over-unit work by inertial forces, a closed mechanical system in the form of kinematically connected rotating masses is proposed for consideration. The research aimed, within the framework of Newtonian mechanics, to study the fulfillment of the laws of conservation of momentum, angular momentum and energy, to establish the possibility of performing work by inertial forces (centrifugal and Coriolis), to assess the change in kinetic parameters using the example of the Chelomey pendulum model. For the complex radial-circular motion of the masses of the Chelomey pendulum model, resolving equations are obtained. To verify the analytical calculations, algorithms for numerical solutions of the above problems have been developed and implemented in the MathCAD software package

Economic and mathematical model of selecting a gas supply system based on biomethane and natural gas

Introduction. A promising direction in the development of gas supply systems, which is widely used in the developed countries of the world, is the use of an alternative energy source — biomethane. The effective use of biomethane requires the development of scientific provisions and recommendations for the use of biomethane gas supply systems, taking into account technical and economic factors. Therefore, it is urgent to develop an economic and mathematical model for choosing a gasification method and determining the area of rational use of gas supply systems with biomethane.
 
 Materials and methods. To develop an economic and mathematical model for choosing a method for gasification of settlements, methods of mathematical modeling and parametric optimization were used. The implementation of the economic and mathematical model was carried out in the Mathcad software package. Mathematical and graphic processing of the results obtained was carried out.
 
 Results. An economic and mathematical model was developed for determining the rational area of application of biome­thane for gas supply to settlements. The model is based on the technical and economic calculation of the reduced costs for gas supply systems with network natural gas and biomethane. Using the developed model, the technical parameters of the biomethane gas supply system were determined: the critical length of the gas pipeline branch, the radius of the biomethane plant, the biomethane consumption, the length of the gas pipeline for the biomethane supply, and the reduced costs to the gas supply system. A graph of the dependence of the critical length of the gas pipeline branch on the population size has been obtained, which makes it possible to determine the area of rational use of biomethane for gas supply to settlements. It was found that for a rural settlement with a population of 5,000 people, the critical length of the gas pipeline branch is 24.820–28.044 km, for an urban-type settlement with a population of 15,000 people, the length of the gas pipeline branch is 50.923–64.677 km, and for a city with a population of 30,000 people — 89.617–127.279 km.
 
 Conclusions. The results obtained can be used for the development and design of gas supply systems based on biomethane for the gasification of settlements remote from the main gas pipelines. The use of biomethane makes it possible to gasify settlements in developed agricultural regions and to increase the efficiency of the gas transportation system of the Russian Federation.

Equations of motion of masses of the Chelomey pendulum model

Abstract. To verify the provisions stated by V.I. Bogomolov, B.I. Puzanov. and Linevich E.I. about the possibility of performing over-unit work by inertial forces, a closed mechanical system in the form of kinematically connected rotating masses is proposed for consideration. The research aimed, within the framework of Newtonian mechanics, to study the fulfillment of the laws of conservation of momentum, angular momentum and energy, to establish the possibility of performing work by inertial forces (centrifugal and Coriolis), to assess the change in kinetic parameters using the example of the Chelomey pendulum model. For the complex radial-circular motion of the masses of the Chelomey pendulum model, resolving equations are obtained. To verify the analytical calculations, algorithms for numerical solutions of the above problems have been developed and implemented in the MathCAD software package.

Justification of the use of square pipes in the frame of the removable roof of the open wagon

This paper reports determining the basic strength indicators for the removable roof of a railroad gondola. It has been established that the typical roof design has a significant margin of safety in the components of the supporting structure. In order to reduce the roof material intensity, the reserves of its strength have been determined and optimized based on the criterion for minimal material intensity. Pipes of square cross-section have been proposed for using as the components of the roof frame. When taking into consideration the proposed measures, it becomes possible to reduce the mass of the frame of the removable roof for a railroad gondola by almost 15 % compared to the typical design. At the same time, to apply the roof on different types of gondolas, its cantilevered parts can move in a longitudinal plane. It is possible to use deflectors on the removable roof. The roof can be attached to the body in a regular way. It is also possible to fix it using shog-connections. To substantiate the proposed solution, the strength of the improved structure of the removable roof was determined. It was established that the maximum equivalent stresses in the load-bearing structure of the removable roof did not exceed permissible ones. To define the indicators of removable roof dynamics, its dynamic loading was investigated. The calculation was performed in a flat coordinate system. The oscillations in bouncing and galloping were taken into consideration as the most common types of a railroad car oscillations when running on a rail track. The mathematical model of dynamic loading was solved in the Mathcad software package (Boston, USA). The study has shown that the acceleration of the body in the center of masses is 0.4 g and is within the permissible limits. At the same time, the ride of a railroad car is excellent. The study reported here would contribute to the improvement of the efficiency of railroad transportation.

Justifying the prolongation of the service life of the bearing structure of a tank car when using Y25 bogies

This paper substantiates the use of Y25 bogies under tank cars in order to prolong their service life. The reported study has been carried out for a tank car with rated parameters, as well as the actual ones, registered during full-scale research. Mathematical modeling was performed to determine the basic indicators of the tank car dynamics. The differential equations of motion were solved by a Runge-Kutta method using the Mathcad software package (USA). It was established that the use of Y25 bogies under a tank car with rated parameters could reduce the acceleration of its bearing structure by almost 39 % compared to the use of standard 18‒100 bogies. Applying the Y25 bogies under a tank car with the actual parameters reduces the acceleration of its load-bearing structure by almost 50 % compared to the use of standard 18‒100 bogies. The derived acceleration values were taken into consideration when calculating the bearing structure of a tank car for strength. The calculation was performed using the SolidWorks Simulation software package (France). The resulting stress values are 18 % lower than the stresses acting on the load-bearing structure of a tank car equipped with 18‒100 bogies. For the load-bearing structure of a tank car with the actual parameters, the maximum equivalent stresses are 16 % lower than the stresses when the 18‒100 bogies are used. The design service life of the load-bearing structure of a tank car was estimated taking into consideration the use of Y25 bogies. The calculations showed that the design service life of the bearing structure of a tank car equipped with Y25 bogies is more than twice as high as that obtained for 18‒100 bogies. The study reported here would contribute to compiling recommendations for prolonging the service life of the load-bearing structures of tank cars

INVESTIGATION OF THE DYNAMIC LOAD OF THE JOINT-PLATFORM OF THE JOINT TYPE FROM ROUND PIPES LOADED BY TANKER CONTAINERS

The dynamic loading of the load-bearing structure of the articulated platform car made of round pipes loaded with tank containers was determined. The research was conducted by mathematical modeling. The solution of the equations of motion is carried out in the MathCad software package. It is established that under the influence of a longitudinal force of 2.5 MN on the front stops of the autocoupling, the acceleration on the first section of the platform car is 27.7 m/s² from the side of the force application, and on the second - 24.4 m/s2. To reduce the dynamic load of the load-bearing structure of the platform car, the possibility of using the concept of a harness device on it as an alternative to a typical self-coupling device is considered. The quenching of the kinetic energy of the impact is carried out due to the viscous resistance forces arising in the concept. The proposed implementation allows to reduce the dynamic load of the load-bearing structure of the platform car by 10% compared to the use of a standard self-coupling device. The conducted researches will promote creation of innovative designs of cars-platforms and increase of efficiency of use of railway transport.

Dynamic Loading Research of the Gondola car Supporting Structure with an Elastic-Viscous Filler in a Center Sill

Purpose. The work aims to investigate dynamic loading of the supporting structure of a gondola car with an elastic-viscous filler in the center sill by means of mathematical modeling. Methodology. Mathematical modeling of the dynamic loading of the supporting structure of a gondola car with a closed center sill filled with a filler with elastic-viscous properties has been carried out. The case of the highest load on the supporting structure of the gondola car in operation is taken into account – a shunting collision, taking into account the action of a load of 3.5 MN on the rear stop of the automatic coupler. To determine the dynamic loading of a gondola car, a mathematical model formed by prof. G. I. Bohomaz was used. However, within the framework of this study, the model was refined by adapting it to the determination of the dynamic loading of a gondola car. It also takes into account the friction forces arising between the center plates of the body and the center pivots of the bogies, as well as the properties of the energy-absorbing material. The solution of the mathematical model was carried out in the MathCad software package. In this case, the differential equations of motion were reduced to the Cauchy normal form, and then integrated using the Runge-Kutta method. Initial displacements and speeds are taken equal to zero. The calculation was carried out on the example of a universal gondola car model 12-757 built by Kriukivskyi Carriage Works PJSC (Kremenchug) on standard bogies 18-100. Findings. Accelerations are obtained as components of a dynamic load acting on a gondola car with a closed center sill structure filled with an elastic-viscous filler. It was found that with the stiffness of the center sill filler of 82 kN/m, as well as the viscous resistance coefficient of -120 kN∙s/m, the maximum accelerations of the gondola car supporting structure is about 37 m/s2 (0.37g). Originality. A mathematical model is proposed for determining the dynamic loading of a gondola car with a closed structure of a center sill filled with an elastic-viscous filler. The model makes it possible to obtain accelerations as the components of the dynamic loading acting on the supporting structure of the gondola car, taking into account the improvement measures during a shunting collision. Practical value. The results of the research will help to reduce the damage to the supporting structures of gondola cars in operation, reduce the cost of their maintenance, create developments in the design of innovative structures of rolling stock, as well as increase the efficiency of its operation.

Обгрунтування впровадження SIN-балок в несучу конструкцію напіввагона

The scientific substantiation of SIN-beam installation into the load-bearing structure of a nonversatile filling car was carried out. The research was carried out with regard to the freight car model 12-757 produced by Krukivsky Carriage Works. Use of SIN-beams as the main bearing elements of the frame reduces its weight by 6% in comparison with the frame of the prototype car. The strength of the frame of the car with SIN-beams as load-bearing elements has been calculated. At that, the SolidWorks Simulation program complex was used, which implements the method of joined elements. Spatial isoparametric tetrahedrons were used as the joined elements. It was found that the maximum equivalent loads occur in the area of interaction between the trunk beam and the kingpin beam are close to 335.3 MPa and do not exceed the permissible values. The maximum displacements are centered in the central part of the frame and are about 29.0 mm. The main indices of the load-bearing structure dynamics of the freight car were determined. The calculation was carried out under the conditions of empty wagon running with steep nerves, as a case of the highest load of the load-carrying structure. When carrying out the calculations, the parameters of the suspension arm of the model 18-100 were taken into account. At the same time, the investigation was carried out in the flat coordinate system. The differential equations of motion were developed in the MathCad software package. The calculation was carried out by the Runge-Kutta method. It was found that the indicators of the dynamics of the car are within the permissible limits. The maximum acceleration of the wagon body in the center of mass is 4.8 m/s². The acceleration of the cars is 5.7 m/s². In this case, the evaluation of the movement of the wagon is «excellent». The conducted research will contribute to the creation of innovative carriage designs, reduction of costs for their production, as well as increasing the efficiency of railway transport functioning

Determination of dynamic loading of bearing structures of freight wagons with actual dimensions

The determination of the dynamic loading of the bearing structures of the main types of freight wagons with the actual dimensions under the main operating conditions is carried out. The inertial coefficients of the bearing structures of the wagons are determined by constructing their spatial models in the SolidWorks software package. Two cases of loading of the bearing structures of the wagons – in the vertical and longitudinal planes – have been taken into account. The studies were carried out in a flat coordinate system. When modeling the vertical loading of the bearing structures of wagons, it was taken into account that they move in the empty state with butt unevenness of the elastic-viscous track. The bearing structures of the wagons are supported by bogies of models 18-100. The solution of differential equations of motion was carried out by the Runge-Kutta method in the MathCad software package. When determining the longitudinal loading of the bearing structures of wagons, the calculation was made for the case of a shunting collision of wagons or a "jerk" (tank wagon). The accelerations acting on the bearing structures of the wagons are determined. The research results will help to determine the possibility of extending the operation of the bearing structures of freight wagons that have exhausted their standard service life. It has been established that the indicators of the dynamics of the load-carrying structures of freight wagons with the actual dimensions of the structural elements are within the permissible limits. So, for a gondola wagon, the vertical acceleration of the bearing structure is 4.87 m/s2, for a covered wagon – 5.5 m/s2, for a flat wagon – 5.8 m/s2, for a tank wagon – 4.25 m/s2, for a hopper wagon – 4.5 m/s2. The longitudinal acceleration acting on the bearing structure of a gondola wagon is 38.25 m/s2, for a covered wagon – 38.6 m/s2, for a flat wagon – 38.9 m/s2, for a tank wagon – 27.4 m/s2, for a hopper wagon – 38.5 m/s2. This makes it possible to develop a conceptual framework for restoring the effective functioning of outdated freight wagons. The conducted research will be useful developments for clarifying the existing methods for extending the service life of the bearing structures of freight wagons that have exhausted their standard resource

Determination of the dynamic load of the carrying structure of the hopper wagon with the actual dimensions of structural elements

The object of research is the supporting structure of the pellet wagon with the actual dimensions of the supporting elements. One of the most problematic areas is the determination of the indicators of dynamics and strength of the supporting structure of the hopper wagon with the actual dimensions of the structural elements.
 A study of the dynamic loading of the supporting structure of the hopper wagon was carried out. At the same time, the actual dimensions of the structural elements were determined by means of field studies. Mathematical modeling of the dynamic loading of the load-carrying structure of a hopper wagon with the actual dimensions of structural elements was carried out by means of mathematical modeling. The studies were carried out in a flat coordinate system. The presence of three degrees of freedom of the supporting structure of the hopper wagon was taken into account: vibrations of twitching, bouncing and galloping. Differential equations were solved in the MathCad software package. In doing so, they were reduced to the Cauchy normal form, and then integrated using the Runge-Kutta method. It was found that the maximum value of the acceleration acting on the supporting structure of the hopper wagon is 38.5 m/s2, which is 2.7% higher than the acceleration of the supporting structure with nominal dimensions.
 Computer simulation of the dynamic loading of the supporting structure of the hopper wagon was carried out. The calculation was carried out using the finite element method in the SolidWorks Simulation (CosmosWorks) software package. It was found that the maximum accelerations are concentrated in the middle part of the supporting structure of the hopper wagon and amount to 36.2 m/s2. The F-criterion was used to verify the developed model. The calculations showed that the calculated value of the criterion is Fc = 1.09 and is less than the table value Ft = 3.29. The adequacy hypothesis is not rejected.
 The natural frequencies and vibration modes of the hopper wagon supporting structure were determined. It has been established that the values of natural vibration frequencies of the hopper wagon bearing structure with the actual dimensions of the structural elements are within the permissible limits.
 The research will contribute to the creation of relevant developments to extend the service life of wagons that have exhausted their standard resource, as well as to increase the efficiency of railway transport operation.

Research of the Vertical Dynamics of the Supporting Structures of Freight Cars Made of Round Pipes

Purpose. This study is aimed at determining the vertical dynamics of supporting structures of freight cars made of round pipes. Methodology. Mathematical modeling of the dynamic loading of the supporting structures of the main types of freight cars made of round pipes (gondola car, covered car, flat car, hopper car) was carried out. The studies were carried out in a plane coordinate system – the XZ plane. At the same time, it was taken into account that the car is moving in an elastic-viscous track so that the reactions of the track are proportional to both its deformation and the rate of this deformation. The studies were carried out for the case of empty cars. The joint inequality is described by a periodic function. The calculation was performed at a speed of 80 km/h. Differential equations of motion were solved in the MathCad software package using the Runge-Kutta method. Findings. Based on the mathematical modeling of the dynamic loading of the supporting structures of cars made of round pipes, the main indicators of their dynamics were obtained: accelerations acting on the supporting structures in the mass center, forces acting in the spring suspension of bogies, dynamics coefficients. For gondola car, covered car, and hopper car, the acceleration at the mass center of the supporting structure is within 0.4 g, and for a flat car – 0.5 g. It was found that the obtained indicators of the dynamics of cars made of round pipes are within the permissible limits. The accelerations acting on the supporting structures of cars made of round pipes are almost the same as those obtained for prototype cars. At the same time, the motion of cars is assessed as "excellent" for gondola car, covered car, and hopper car and "good" for flat car. Originality. Mathematical modeling of the dynamic loading of the supporting structures of cars from round pipes was carried out and the main indicators of their dynamics were obtained. Practical value. The research carried out will contribute to the creation of recommendations for the design of supporting structures of freight cars of round pipes, and can also be useful developments in the creation of innovative car designs.

The impact effect of underground trains in motion analyzed together with other factors contributing to the stress-strain state arising along the tunnel lining boundary

Introduction. Urban expansion and compaction of urban environments trigger the need for delay-free transport communications between urban districts. An underground railway is a modern means of transport, whose construction and operation are feasible below the ground surface. However, it is a source of vibrations that negatively affect human comfort and high-tech production processes. Train speed, weight, wheel and rail defects, the geological structure of the site, subway route construction conditions, the remoteness of measuring equipment installation points are the main factors that influence the intensity of vibrations of the ground surface or building structures. The effect, produced by moving subway trains on the tunnel lining, as well as the wheel-rail junction interface are analyzed in this work.
 Materials and methods. The problem of stresses, triggered by underground trains along the boundary of the tunnel lining, can be reduced to the well-known problem of a beam on an elastic foundation. MathCAD software package was used to obtain the analytical solution and diagrams.
 Results. The co-authors identified the values of stresses arising along the boundary of the tunnel lining as a result of the interface between underground trains and rail junctions. Diagrams that illustrate deflections of a bending curve of an infinite beam, bending moments, and stresses arising along the boundary of the tunnel lining have been made. Values of regular stresses arising along the tunnel boundary are obtained by transforming external loads into Fourier series.
 Conclusions. The data obtained as a result of the analysis are further used to solve problems of identifying the wave field in a continuous elastic medium using methods of successive wave approximations and compensating loads.

Long periodicals in the position of permanent GNSS stations

The relevance of the study of the movements of base reference stations is determined by the task of improving the accuracy and reliability of the results of geodetic monitoring carried out using GNSS technologies. The paper presents the results of an analysis of the position of five permanent GNSS stations located in St. Petersburg for 2013-2020. Data processing by the method of precise positioning was performed using the online service CSRS-PPP. The time series were processed using the “Caterpillar”-SSA algorithm implemented in the MathCAD software package. The main attention is paid to the identification of periodic components in the movement of stations. The analysis of the series showed consistently identified periods of 3 and 12 months in the plane coordinates and 70 days and 12 months in altitude. The amplitudes of the periodicals were in the range from 1 to 10 millimeters. A trend was also identified, demonstrating a tendency for all stations to move to the northeast. The paper presents comparisons of the results obtained with previous studies.

Explicit Finite Difference Approximation Method for Solving the Riesz Space Fractional Percolation Equation

ABSTRACT This paper presents an efficient approximation method for solving the Riesz space fractional percolation equation mixed fractional derivative (RSFPEMFD) with the initial condition (IC) and derivative boundary conditions (DBC) using explicit finite difference method (EFDM). The shifted Grunwald estimate was used for analysis of the mixed fractional derivatives. In addition, the analysis method of consistency, stability, and convergence was used for analyzing the EFDM. To demonstrate the efficiency and validity of the proposed algorithm, some illustrative examples are given and comparing results with the exact solution. The results are displayed when necessary in tables using the MathCAD and MATLAB software package. The EFDM for solving RSFPEMFD has been demonstrated to be effective and reliable.

Застосування середовища Mathcad у загальному курсі фізики при підготовці фахівців інженерних спеціальностей

The article discusses a methodological approach to solving problems when studying the topic «Harmonic oscillations» in the course of general physics using the Mathcad software package when training specialists in engineering specialties. In the course of the research, the scientific literature was theoretically analyzed, the experience of using teaching methods was studied. The methodological basis of the research is an integrated approach that combines traditional forms of teaching and learning using the Mathcad software package. An illustrative principle of teaching and presenting information is given on the example of two tasks to demonstrate the capabilities of the Mathcad software package in the classroom in higher education institutions. The considered examples are given in general form without numerical calculations, when supplemented with numerical values and changing the initial conditions, it is possible to increase the number of possible tasks based on the given templates. The methodical approach is introduced into the educational process. As a result of observation of the lessons, the results obtained were summarized and systematized. It is proposed to supplement the solution of these problems with the help of computer modeling with an analytical solution for comparing and checking the results obtained. The introduction of the proposed methodological approach into the educational process demonstrates an increase in students' motivation to master the discipline as a whole; the acquisition by students of the skills of using computer programs to solve professionally oriented problems; presentation of educational material in a visual form; saving study time, etc. Solving problems of applied content encourages students to independently establish intersubject connections in natural-mathematical and special disciplines at different levels and contributes to the transfer of knowledge from one field of science to another. The approach used encourages students to independently model oscillating processes to study the operation of various devices and structures. Using the considered methodology for solving the above examples in the Mathcad environment, students can master not only mathematical competencies, but also additionally related to information processing; to gain the skills of analysis and search for optimal solutions, which is inherent in engineering specialties.

METHOD MANY-PARAMETRIC FUNCTIONS FOR ANALYSIS OF THE CONNECTING PROCESSES IN ELECTRIC CIRCUIT WITH SEMICONDUCTOR COMMUTATORS

One of the most challenging tasks in analyzing rozgaluzhenih electric power with switches є the task is to complete the transition process in such stake with vikoristanny high-frequency remix. A mathematical model has been invoked for the analysis of the transition processes in the on-demand transfer of installations of the electrical pulse processing of the open mediums. Indicated graphs, scho vidobrazuyayut pereh_dnі processes in electric kolakh. The protests of the analysis of the electromagnet processes in such cola for the round wrap of the method of bagatoparametric modular functions and the special MATHCAD software package are spread. Primed by the major industrial and industrial frequency retarder (NPP) frequency for the change of the FRI modulus. It is shown that such a piddid by the day of Victorian Bagatorov’s overtake of the most varied tasks at the post office and at the same time, the methods of the quick assessment of the public relations (key mode). An analysis is made of the method of bagoparametric functions for the Victoria MATHCAD package for the analysis of intermittent processes in discharge-pulse systems with high-frequency voltage switches that are primed with the highest frequency. The results of accession for the visualization of the system and frequency conversion of the mm generator are presented, as well as the minimum dispersion phase of the output signal is secured. The method of changing the uninterrupted mode of operation is monitored.

STUDY OF THE DYNAMIC LOADING OF THE LOAD-BEARING STRUCTURE OF A FLAT WAGON DURING TRANSPORTATION BY SEA

To increase the efficiency of combined transportation, the supporting structure of flat wagon has been created. A feature of the wagon is that the sections have a low center of gravity. This solution allows for the transportation of oversized cargo on a flat wagon within the established dimensions. The design of this flat wagon can be used for the carriage of goods not only by main lines, but also in rail and water traffic when transported by rail ferries. To ensure the safety of transportation of a flat wagon with containers on a railway ferry, their dynamic loading was determined. It is taken into account that a large-capacity container of 1AA standard size is placed on each section. The solution of the mathematical model was carried out in the MathCad software package. The resulting accelerations, as components of the dynamic load, were taken into account when determining the stability of a container on a flat wagon during transportation by a rail ferry. It was found that the stability of the container is ensured at tilt angles up to 25°. A computer simulation of the dynamic loading of the supporting structure of an articulated flat wagon with containers during transportation by a railway ferry has been carried out. The calculation is implemented in the CosmosWorks software package using the finite element method. The fields of distribution of accelerations relative to the supporting structure of the flat wagon and containers are determined. The maximum percentage of discrepancy between the results of mathematical and computer simulation does not exceed 11 %. The research carried out will contribute to the creation of innovative designs of flat wagons, as well as to increase the efficiency of the operation of combined transport in international traffic

Determining the dynamic loading and strength of the bearing structure of a covered wagon when firing from it

The bearing structure of a covered rail wagon has been improved to enable firing from it at motion. The covered wagon of model 11–217 was chosen as a prototype. To enable firing in the vertical plane, it has been proposed to use a sliding roof, which consists of shutters that move by means of a pneumatic or hydraulic drive. To accommodate military equipment inside the covered wagon, its frame is equipped with supporting sectors. Mathematical modeling was performed in order to determine the dynamic load on a covered rail wagon when firing from it. The mathematical model was solved in the Mathcad software package. We have established the dependence of the accelerations of the bearing structure of a covered rail wagon on the recoil force induced by the combat equipment that it hosts. It has been found that in order to maintain the dynamics indicators within acceptable limits, combat equipment should have a maximum recoil at a shot of about 3.2 kN. The maximum accelerations that act on the bearing structure of a covered wagon in a vertical plane are about 6 m/s 2 . In the zones of interaction between the body and bogies, the maximum accelerations are about 9.5 m/s 2 and the accelerations of bogies are 10 m/s 2 . To reduce the dynamic load on the bearing structure of a covered rail wagon, it has been proposed to use a viscous connection between the supporting sectors and frame. We have determined the dependence of accelerations on the coefficient of viscous resistance between the supporting sectors and the bearing structure of a wagon. It has been established that taking into consideration the use of a viscous connection between the supporting sectors and frame makes it possible to reduce the dynamic load on a wagon at least by 15 %. The basic indicators of strength for the bearing structure of a covered rail wagon when firing from it have been determined. We have derived the dependence of the maximum equivalent stresses in the bearing structure of a covered wagon on the recoil force of combat equipment. The maximum equivalent stresses at a recoil force of 3.2 kN arise in the console part of the girder of a covered wagon and are about 300 MPa. The maximum displacements were registered in the area where the front stops of the auto-coupling are arranged; they are equal to 2.9 mm. The maximum deformations amounted to 6.98∙10 -3 . Modal analysis of the bearing structures of a covered rail wagon has been carried out. It has been determined that the values of the oscillation natural frequencies are within the permissible limits. Our study will contribute to the construction of innovative rolling stock for the transportation of military equipment and for firing at motion

Оптимизация формулы расчетного сопротивления грунта

The assumptions made in the derivation of the formula for the design soil resistance are considered. It is shown that in the conditions of the plane problem, taking into account the coefficient of lateral soil pressure leads to lower values of the initial critical load and, consequently, to a decrease in the design resistance. The equation of limiting equilibrium is implemented in the MathCad software package, which makes it possible to obtain isolines of deviation angles at any depth from the applied uniformly distributed load. To determine the initial critical load, a formula is given that takes into account the coefficient of lateral soil pressure K0. In the final part of the article, the solution to the problem is given according to the formula of N. P. Puzyrevsky and according to the formula derived by the author, taking into account K0, the difference in the isolines of the deflection angles is presented, and an analysis of a similar problem in the Plaxis 2D software package is carried out.