Introduction. Designing motion control systems for mobile robots requires the construction of mathematical models. Researchers have repeatedly addressed this topic. In particular, works have been published on the calculations of multiphysical processes, modeling the movement of various types of wheels under certain conditions. In addition, the dynamics of deformable contacting bodies during sliding, rolling and rotation, issues of autonomy and controllability of mobile robots were considered. Note, however, that the dynamics and positioning accuracy of wheeled robots is largely determined by friction. The literature does not present studies on the dynamics of a robot with a differential drive taking into account the interrelationships of sliding, spinning and rolling friction effects based on the theory of multicomponent friction. Research in this area can reveal new dynamic effects. Based on the data obtained in this way, it is possible to improve the accuracy of positioning in building mathematical control models. The presented work aims at investigating the movement of an automatic device with a differential drive taking into account three contact models: nonholonomic, Coulomb friction, and multicomponent models.

Materials and Methods. The scheme of a two-wheeled robot with differential drive and continuous movement on the support surface was adopted as the basic one. The movement of the device was provided through software control. The dynamics was described in the form of Appel equations. Mathematical models were used for calculations, taking into account friction in different ways. Coordination of the actions of the mechanism was formed at a dynamic level. The control actions were the moments of the wheel motors. When visualizing the models under study, the built-in numerical methods of the Wolfram Mathematica system were used with a minimum accuracy of 10^-6.

Results. When building a mathematical model, the equations for the angular velocities of the wheels were determined. The authors took into account the presence of a contact site and derived the equations of dynamics of a differential drive robot. The elements of the system were force and moment projections, indicators of platform spin, masses, angular accelerations, and inertia of the wheels. It was shown how control actions were formed within the framework of nonholonomic mechanics. The model of engines that created a moment of control on the driving wheels was described. The solution was derived as the relationship between the inductance of the conductors of electric motors and the operation of the power supply. Three models describing the dynamics of a differential drive robot were examined in detail. The first model was nonholonomic. The second and third included a system of equations for the dynamics of a differential drive robot for a general case with a contact platform. At the same time, in the second model, the switching time in the engine was ignored and the Coulomb friction was involved. In the third model, a parameter to determine the speed of transients in the engine was introduced, and Pade decomposition was involved. This was a model with multicomponent friction. The calculation results were shown in the form of graphs. On them, the studied models were visualized in the form of curves of different colors. Comparison of the graphs showed in which cases, after the completion of transients, the control provided the required accuracy. These were models 1 and 2. In model 3, the software control generated an error in the angular velocity of rotation of the platform. This error could not be predicted within the framework of the 1st and 2nd models. In all the systems considered, the sliding speed of the wheels in the transverse direction dropped to zero. The condition of continuous motion of the support wheel was obtained and validated.

Discussion and Conclusion. Software control is acceptable in models that do not take into account wheel friction during simultaneous sliding, spinning and rolling (general case of spatial motion). However, it is important to consider the relationship between these processes and multicomponent friction. This is required for the robot to perform program movements more accurately. It was established that software control in a model that takes into account the friction of spinning and rolling caused deviations from the program values of the angular velocity of the platform. The results obtained can be used in the building of a control system with predictive models.

Introduction. Mobile robots capable of omnidirectional movement are widely used in various fields of human activity. To provide high accuracy of positioning of omnidirectional platforms with mecanum wheels, it is required to develop their detailed mathematical models used in the construction of a motion control system. Due to the complicated design of the mecanum wheels, various errors may occur during the construction of omnidirectional platforms, including the error of installing such wheels on the platform. Its effect on the accuracy of the platform movement has not been studied before. This work aims at assessing the positioning errors that arise due to the presence of design errors in the installation of mecanum wheels, and analyzing the effect of these errors on the accuracy of program motion testing when using control at the kinematic level.

Materials and Methods. The analysis of positioning accuracy was based on mathematical modeling of the platform kinematics, taking into account structural errors in the installation of mecanum wheels. To describe the relationship between the angular speeds of rotation of the wheels and the speeds of the platform, the conditions of nonslip of the contact points on the support surface were used. Numerical calculations were carried out in the Wolfram Mathematica package.

Results. A formula was obtained for estimating errors in platform pseudovelocities under program control formed at the kinematic level. The estimation of the errors of the platform speeds for simple movements was carried out. According to the calculation results, it has been shown that the speed errors are significant for robots with mecanum wheels operating autonomously.

Discussion and Conclusion. The calculation results demonstrated the significant impact of wheel installation errors on the positioning accuracy of the mecanum-platform, and confirmed the need to take into account these design errors when creating autonomous mecanum-platforms. The constructed model of the robot's kinematics makes it possible to predict errors in platform speeds that arise under program control, as well as deviations of the coordinates of the geometric center of the platform from the program motion. The proposed kinematic model can be used to improve the positioning accuracy through forming a platform motion control that compensates for the influence of wheel installation errors.

Introduction. When numerically solving problems of elasticity theory in a three-dimensional formulation by the finite element method, finite elements (FE) in the form of parallelepipeds, prisms and tetrahedra are used. Regularly, the construction of stiffness matrices of volumetric FE is based on the principle of isoparametricity, which involves the Lagrange polynomials to approximate the geometry and displacements. In computational practice, the most widespread FE are the so-called multilinear isoparametric FE with a linear law of approximation of displacements. The main disadvantage of these elements lies in the “locking” effect when modulating bending deformations. Moreover, the error of the numerical solution increases drastically in the case when the structure, in comparison to conventional deformations, undergoes significant displacements as a rigid whole. Long-term experience in solving problems of deformable solid mechanics by the finite element method has shown that existing volumetric FE have slow convergence, specifically, when modeling bending deformations of plates and shells. This study aims at constructing stiffness matrices of multilinear volumetric FE of increased accuracy allowing for rigid displacements based on the double approximation method.

Materials and Methods. The mathematical apparatus of the double approximation method based on the principle of a separate representation of the distribution functions of displacements and deformations inside the element, was used to construct the stiffness matrices of volumetric FE. The storage and processing of the resulting system of equations was implemented in algorithmic terms of sparse matrices. Software development and computational experiments were carried out using the Microsoft Visual Studio 2013 64-bit computing platform and the Intel ® Parallel Studio XE 2019 compiler with the integrated Intel ® Visual Fortran Composer XE 2019 text editor. Visualization of the calculation results was performed using the descriptor graphics of the MATLAB computer mathematics package. A large eight-node SOLID185 CE of the ANSYS Mechanical software complex was used as a test sample.

Results. Mathematical tool and software were developed to study the stress-strain state of massive structures under various types of external actions. The authorized application software package was verified on test examples with known analytical solutions. It has been shown that the constructed FE accurately satisfy the basic requirements for finite element modeling of spatial problems of elasticity theory.

Discussion and Conclusion. The performed testing of the developed mathematical and program toolkit has shown that the finite elements constructed on the basis of the double approximation method can successfully compete with similar SOLID185 volumetric elements of the ANSYS Mechanical software complex. The proposed elements can be integrated into domestic import-substituting software systems that implement the finite element method in the form of the displacement method.

Introduction. As numerous production tests show, the use of manual arc welding (MAW) of thick-walled fittings of small diameter (up to 80 mm) does not provide a high-quality weld joint that meets the requirements of regulatory and technical documents of nuclear power plants (NPP). The solution to this problem is possible on the basis of research and development of welding technology with optimal heat input instead of MAW. Existing fusion welding technologies do not allow for optimal regulated heat input. However, this can be realized during the development and further use of the friction welding (FW) method. Therefore, this work aimed at developing a technology based on an automated technique of friction welding, which could provide the enhancing of the quality of weld joints of small diameter fittings of power equipment to the level of regulatory requirements.

Materials and Methods. Small diameter fittings with a conical contact surface made of low-alloy steel 10GN2MFA were used. The experimental study was performed on a friction welding machine MST–41. Methods of non-destructive and destructive quality control were used in accordance with the regulatory and technical documentation of nuclear power engineering.

Results. A methodology was developed, and the optimal dimensions of the conical contacting surface under welding were determined. It was shown that optimal heat input during friction welding was achieved by preparing a conical contacting surface in the angle range α = 30º–40º.  The methodology and parameters of the friction welding mode for models of small diameter fittings were experimentally tested. In the course of the research, a cyclogram of the friction welding process was obtained and described, which confirmed the stage-by-stage formation of the weld joint due to the sequential inclusion of annular sections of the conical surface of the connected parts in the heating stage. The results of non-destructive and destructive testing were obtained, confirming the presence of a high-quality weld joint at the level of the requirements of the regulatory and technical documents of the NPP.

Discussion and Conclusion. The obtained research results can be used to develop the technology of friction welding of pipes, as well as products made of carbon and low-alloy steels.

Introduction. Glass fibers significantly improve the quality of composite materials, make them lighter, stronger, more corrosion resistant and thermally stable. Strengths and weaknesses of specific composites are actively discussed in the scientific and applied literature. At the same time, the effect of the ratio of fibers and matrix material on the mechanical characteristics of composites has not been sufficiently investigated. The presented study is intended to fill this gap. The work is aimed at manufacturing a composite material on a polymer basis reinforced with glass fiber, and investigating the influence of weight ratios of elements on the mechanical characteristics of the composite. For the first time, a report on the comparison of the characteristics of composites (with different fiber content) to each other and to steel is published.

Materials and Methods. Fiberglass and polyester were used as starting materials with the addition of a mediator to speed up the molding process. The samples were made manually and tested for tensile strength, hardness, and impact strength using standard equipment. The results were summarized in the form of tables, visualized in the form of graphs, and processed by comparative analysis.

Results. The technique of creating samples and methods of their testing were described. The research showed that hardness, tensile strength and impact resistance increased with a growth in the percentage of randomly distributed fiberglass to 50% with 50% unsaturated polyester. In this case, the maximum values of tensile strength — 175.4 MPa, hardness — 38 HV and impact resistance — 1.56 J/mm² were obtained. The inexpediency of exceeding the proportion of fiberglass by more than 50% was experimentally proven, since mechanical properties deteriorated. This was due, in particular, to the fragility of the glass, which, if the proportions were violated, was transmitted to the entire composite. In addition, with an excessively high volume of reinforcing fibers, the resin was not enough for high-quality bonding of the elements, which significantly reduced the strength of the material. Most of the mechanical characteristics of a composite made of 50% polyester and 50% fiberglass are better than those of steel.

Discussion and Conclusion. It has been proved that the properties of the composite material depend significantly on the glass fiber content. The resulting composite was compared to steel. It turned out that it had better mechanical characteristics and less weight. This allows us to recommend the material for boat hulls.

Introduction. The basis for research, analysis and mathematical optimization of any chemical process is an adequate mathematical model that takes into account the kinetics of the object. Kinetic analysis is a challenge in chemical technology, since it allows for optimizing synthesis processes and predicting their efficiency. Numerous chemical processes involve several stage reactions. For successful design and optimization, a mathematical model that describes each stage is needed. Creating such a model manually can be time-consuming and costly, since it requires processing a large amount of information. The modern level of automation makes it possible to accelerate the obtaining of a mathematical formulation of the kinetics of multistage reactions. In this case, working with data is greatly simplified, and the probability of making mistakes is reduced. The resulting mathematical model can be applied for further analysis and optimization of the process. The paper considers the industrial reaction of catalytic reforming of gasoline, which occupies an important place in the modern scheme of oil refining, since it is a source of high-octane components of commercial gasolines and individual aromatic hydrocarbons. This process is characterized by the participation of a large number (up to 300) of various hydrocarbons, a change in the number of moles, and non-isothermality in it. Mathematical modeling of such processes involves detailing the stages to the required level. The detailing of up to 173 stages is considered. In this setting, automation of the formation of a mathematical formulation of kinetics for catalytic reforming of gasoline has not been carried out before. Therefore, the presented work aimed at implementing effective numerical methods and algorithms for automating the building of a mathematical model taking into account kinetics, thermodynamics, and changes in the number of moles.

Materials and Methods. The mathematical formulation of the kinetics of multistage reactions was developed on the basis of the mass action law. The kinetic parameters values were taken from literary sources. The direct kinetics problem was solved using algorithms: the Gear method, the Runge-Kutta method of the 4th order, and the scipy.odeint() method of the Python language. The automation concept was implemented using the IDEF0 methodology. The software was written in the Python programming language.

Results. A new software was created to automate the process of forming a mathematical model, taking into account the kinetics, thermodynamics, and the volume of the reaction mixture. The program results were presented by the example of catalytic reforming of gasoline. The model implemented the possibility of taking into account the intermediate heating of the mixture in the reactor cascade. Numerical values of temperature changes corresponding to industrial data were obtained.

Discussion and Conclusion. The results obtained through modeling chemical transformations in the cascade of gasoline catalytic reforming reactors confirmed the exothermic nature of the reaction. The developed software product provides displaying changes in the concentrations of reactants, as well as temperature variations in the reactor, and it can be used in scientific research organizations for the analysis of multistage catalytic processes. The results of the reaction kinetics modeling will be used in the subsequent optimization of the process conditions in production.

Introduction. Research and applied works on the placement of virtual objects in real space most often focus on issues of interactivity, integration of reality and virtuality, physical properties of virtual elements. However, the task of simultaneously free and optimal placement of objects, taking into account their size and the surrounding comfort zone, has not been sufficiently worked out. In the literature, you can find a description of a similar task — about packing in a rectangular container. In our case, the goal is not limited to the greatest possible dense placement. Two conditions should be taken into account: rigid dimensions of the objects (it is forbidden to violate them) and additional areas — comfort zones (it is undesirable to occupy them). The work aims at creating and implementing such a 2D algorithm for placing objects in physical space, which takes into account the above limitations.

Materials and Methods. Using a set of numerical methods, the authors applied the previously created 1D algorithm for the placement of objects. Calculations were based on a system of linear equations. In the one-dimensional case, the optimal placement of virtual objects was reduced to a task that did not depend on the type of comfort function. The elements of such a system were the dimensions of objects, the distances between them, as well as the distances to the edge of the embedding area, a comfort zone. The proposed 2D algorithm for optimal placement of virtual objects was implemented in the form of a program code in C# using the well-known Unity game engine. The solution was tested on gadgets in peak load mode for 5, 10, 15, 20, 25, 35, 40, 45, and 50 objects. 1.8 thousand devices were used for experiments. About 77 thousand events were analyzed. To exclude unrepresentative values, each calculation was repeated 10 times, and a z-score was performed for each value. Abnormal events (more than 3 and less than -3) were excluded.

Results. In this paper, a 2D placement algorithm that implements filling a rectangular area with virtual objects has been created. Each of the objects had a size and another characteristic — a comfort zone. The authors compiled a flowchart for the implementation of this algorithm in a given two-dimensional left-hand coordinate system. It was shown, in particular, at what stage objects were sorted by length, when their batches were formed, and arrangements were made along two axes. The first axis was horizontal, the second was directed forward from the user (this is the depth vector, or frontal measurement). The 1D-placement algorithm for the generated row provided optimal positioning the objects along X-axis based on the calculated comfort coefficient К. Calculations were made and schemes were drawn up to obtain certain comfort indicators. For each object of the first string, the displacement along Z-axis from the edge of the plane was determined so that the comfort in front was equal to the comfort along X. Starting from the 2nd row, to calculate the displacement, the presence of potential neighbors who were a row higher and had common areas along X with the object being processed, was checked. Each element of the string was set along Z-axis so that its comfort from above was the maximum of the one-sided horizontal comfort in this and the previous strings. The principle of calculating Z coordinate for a string object was presented in the form of a flowchart. The initial data for the implementation of this algorithm were 7 objects with 14 different sizes and 28 comfort zones. After the software implementation, the operation of the described 2D algorithm was tested in practice — in an augmented reality mobile application. Analytical data of user sessions was recorded. The average execution time was calculated. The hypothesis of quadratic dependence that arose during the work was tested on a personal computer. For this purpose, a similar experiment was conducted for a range of [10-10,000] objects. The hypothesis was confirmed. The algorithm can be assigned a complexity of O(n²). To compare the calculation speed, 10 of the most popular models of user devices were utilized. The results were presented in the form of a diagram. The minimum registered execution time was 0.093 ms, the maximum — 0.146 ms. Calculations showed high efficiency of the two-dimensional algorithm. Additionally, the placement schemes for different numbers and parameters of objects were visualized.

Discussion and Conclusion. The proposed algorithm of two-dimensional placement enables the user to work with a set of virtual objects with different sizes and comfort zones. Sufficiently high performance and stability are shown. On average, the algorithm is implemented in fractions of a millisecond, even with large batches of objects. Possible future focus areas:

– expansion of the approach for building 3D models and algorithms;

– inclusion of objects in the rotation algorithm for greater flexibility of their location and better use of space.

The research results can be of interest to engineers and interface designers. In the future, it is required to study the user experience and the possibilities of including additional restrictions on positioning.

Introduction. The analysis and interpretation of hydrodynamic research data are based on theoretical models and computational algorithms. Despite the demand for this topic, numerous issues related to unsteady fluid flows in oil reservoirs still require solutions. Therefore, mathematical setting of problems related to the account of unsteady fluid flow, the development of effective numerical methods and algorithms, their solution using modern web technologies are pressing. This study is aimed at developing a web application for mathematical modeling of the process of fluid filtration in single- porosity reservoirs when conducting a hydrodynamic study at a production well.

Materials and Methods. To solve the problem, the methods of continuum mechanics and computational mathematics were applied. A model of oil flow in a single-porosity reservoir was presented. Python and JavaScript programming languages were used in the development of the application. The calculation results were stored in a relational database implemented using PostgreSQL tools.

Results. A new web application has been developed for modeling the oil filtration process in single- porosity reservoirs. It is applicable for studying fluid dynamic processes and can be used to predict flowrates, production and calculation of optimal well operation modes.

Discussion and Conclusion. The developed web application provides for building pressure and temperature fields in the reservoir near a flowing and shut-in production well and at various distances from it. This information makes it possible to urgently assess the duration of hydrodynamic studies, as well as to regulate the operation of wells. The application can be deployed in an existing network infrastructure and use all the functionality by connecting to a remote server. It is optimized for use on various platforms and has broad prospects for further development.

Introduction. Inverse problems are a specific type of tasks where the consequences of phenomena are studied to identify their causes. They are widely used in scientific studies, specifically, those dealing with large amounts of experimental data. In the presented paper, inverse problems in mechanical engineering and structural diagnostics are considered. These areas require precise methods to identify internal defects in various materials, which can be critical to ensure the safety and efficiency of technical structures. Despite the many flaw detection methods available, there is a need for innovative developments that can provide higher accuracy and efficiency. This study integrates different scientific methods and technologies. It opens up new perspectives in nondestructive testing for the detection of internal defects in various materials and structures. Its objective is to develop and implement nondestructive testing methods based on a neural network device to improve the accuracy of defect identification, as well as to build a neural network model and evaluate its effectiveness for the refinement of ultrasonic visualization of internal defects in solid materials. In this regard, the task to be solved is to create a reliable tool for accurate visualization of sizes, shapes, location and orientation of internal defects in various materials.

Materials and Methods. The technique of determining the geometric parameters of defects in materials through nondestructive testing is used. The approach combining modeling of ultrasonic wave propagation in acoustic medium and artificial neural network technologies is applied. This approach identifies nonlinear relationships between the geometry of defects and the amplitude-frequency and amplitude-time data obtained during signal analysis. Artificial neural networks are a model that can be trained on examples, which provides for an effective solution to problems that are difficult to express in traditional forms. The study uses the finite difference method in the time domain. It is applied to identify and visualize internal defects in materials using ultrasonic nondestructive testing and convolutional generative neural networks.

Results. A convolutional neural network has been developed to visualize internal defects using ultrasonic nondestructive testing techniques. This neural network successfully determines the size of defects, their location, shape and orientation with high accuracy and reliability.

Discussion and Conclusion. The authors highlight the key influence of defect size on the accuracy of ultrasonic imaging in various scenarios. The validation of the model for three different cases of defects with different mechanical parameters has shown that for successful visualization of defects, the wavelength of the ultrasonic pulse must be ten times smaller than the size of the defect. When analyzing the impact of defect size on the accuracy of the neural network, it is found that the visualization error increases for defects of smaller size. It has also been found that the relative speed of sound in materials has a greater effect on the accuracy of the method than the relative density of the material. Based on the results obtained by the authors, it can be argued that the developed methods and technical solutions are of great importance for future research in the field of flaw detection. They have significant potential for scientific and practical applications.