PTC Mathcad Research Articles

Analysis of 2D Beam Structure Using the Matrix Method Using the PTC Mathcad and SAP2000 Applications Article Sidebar Published: Jun 30, 2024 DOI: https://doi.org/10.37253/jcep.v5i1.9208 Keywords: Struktur Balok, Matriks kekakuan, PTC Mathcad, SAP2000 Main Article Content

Concrete beam structure that functions as an element to support bending loads and shear forces. The PTC Mathcad application software has a built-in matrix method, which makes it very easy to use to create simple structural analysis programs using the stiffness matrix method. The aim of this research is to create a simple analysis program using the PTC Mathcad application to analyze 2D beam structures using the matrix method and compare the analysis results with the SAP2000 application. PTC Mathcad and SAP2000 analysis will produce joint reaction values ​​for force, moment, and joint displacement for each beam model. Based on the results of the comparative analysis of internal force results produced from the PTC Mathcad application and the SAP2000 application, it was found that the

Application of Thermodynamic Models to Predict the Solubility of Biologically Active Substances

Introduction. The choice of solvents for the processes of extraction of biologically active substances from natural raw materials, the processes of purification of natural and synthesized substances by extraction, crystallization and dissolution methods is an important problem of the modern pharmaceutical industry, because a large number of experiments must be performed to determine the optimal solvent or mixture of solvents. To reduce the cost of developing and optimizing the extraction and purification of substances stages, it is proposed to use thermodynamic models at the stage of preliminary solubility assessment. The article investigates predicting the solubility of pharmaceutical substances issue on the example of the technology for isolating hesperidin and glycyrrhizic acid from plant materials.Aim. Theoretical determination of the dissolving power of various solvents with respect to hesperidin and glycyrrhizic acid.Materials and methods. The PMUNIFAC and NRTL-SAC thermodynamic models were used to predict solubility. The solubility calculation for the NRTL-SAC model was performed using Aspen Properties V14 software, and for the PMUNIFAC model using PTC Mathcad Prime V6. To evaluate the results obtained using thermodynamic models, a number of experiments were carried out, the object of which was the peel of an orange (dried flavedo and albedo, the degree of grinding is 0.2–0.5 mm, the moisture content is 8 %). The quantitative content of hesperidin was determined by direct spectrophotometry at a wavelength of 290 nm. Statistical data processing was performed using Minitab v20 software (Minitab Inc., USA), differences were considered statistically significant at p < 0.05.Results and discussion. On the basis of thermodynamic models, the prediction of the solubility of hesperidin and glycyrrhizic acid was made. It has been shown that the solubility can be assessed both in the presence of solubility data according to the NRTL-SAC model, and in their complete absence according to PMUNIFAC. The correspondence of the theoretically calculated data to the experimental data confirms the correctness of the calculations of thermodynamic models. The results of the calculations are evaluated and solvents are recommended that can be used in the technology of isolating hesperidin and glycyrrhizic for the stages of degreasing, extraction and crystallization with the antisolvent.Conclusion. Using the NRTL-SAC and PMUNIFAC thermodynamic models, the dissolving power of various solvents with respect to hesperidin and glycyrrhizic acid was determined. From the calculation results, a list of solvents was built, ranked by the solubility of the studied substances in them. The resulting list can be used in the development of an industrial technology for the isolation and purification of hesperidin and glycyrrhizic acid. It is shown that the NRTL-SAC and PMUNIFAC models have good prospects for quantitative prediction of the solubility of active substances.

RESEARCH OF OPTIMAL TECHNOLOGICAL SEQUENCES IN THE ORGANIZATION OF CONSTRUCTION PRODUCTION

The article discusses some problems of economics and organization of construction, which can be solved by linear programming using PTC Mathcad Prime, Excel Microsoft365 and data ranking. The authors analyzed the literature, which presents the existing optimization methods in various industries and the field of construction production. Shows the basic principles of using linear programming in various software tools. The authors analyzed in detail the comparison of different methods for the selection of optimal technological sequences. Examples of solving optimization problems in construction production by means of linear programming and data ranking method are presented. To solve the problem, the technology of applying the mixture to a vertical surface was used. Namely: cleaning the facade, plastering the surface, painting the surface. Three technologies were considered. For applying to a vertical surface using standard equipment are scaffolding, ladders and cradles. The process flow consists of the following processes: This is surface preparation. It can be performed in three ways: manual cleaning, sandblasting apparatus, hydrojetting. The next process is the technology of plastering the surface. At this stage, two options are considered: manual plastering and mechanized plaster application. Next, you perform a surface grouping process. This process can be performed either manually or using a pulvelizer. And the last process is painting the surface. Considered manual painting or mechanized device. The next step was to analyze the possibilities of each of the optimization search methods and provide recommendations for their possible application in solving similar problems. Created a mathematical model that can be scaled to implement optimization search in groups of technological sequences with different methods of work. Conclusions were also made regarding the use of the presented methods in solving similar problems of construction production.This article may be an analog for calculating not only the mechanization processes, but it may become a meeting for automating the construction processes.

Динаміка коливань очисника головок коренеплідних культур від залишків гички на корені

Goal. The study of the influence of the design parameters of the cleaner of the heads of root crops from the remains of the chaff on the basis of the analysis of a mathematical model that describes its plane-parallel movement in the longitudinal-vertical plane under the influence of external disturbance in the form of fluctuations of the ordinates of the longitudinal profile of the field. Methods. Theoretical calculations of the mathematical model of the operation of the rear-mounted cleaner of the heads of root crops were performed on a computer using the PTC Mathcad 15 software environment and the initial numerical data generated for this purpose. The optimal value of the parameter was considered, for which the actual amplitude-frequency characteristics of the root crop cleaner were as close as possible to zero, and the phase-frequency characteristics were as large as possible. Results. By means of mathematical modeling, the influence of the fluctuations of the longitudinal profile of the field on the angular oscillations of the cleaner of root crop heads in the longitudinal-vertical plane of projections for different values of the stiffness and damping coefficients of its support-copying wheels, as well as the longitudinal coordinates of their placement on the cleaner frame, was determined. Conclusions. The value of the stiffness coefficient of the tires (7.50R16) of the supporting and copying wheels of the root crop cleaner should be at the level of 315 kN ⋅ m–1, which is ensured by setting the air pressure in the indicated tires at the level of 135 kPa. A change in the value of the damping coefficient of the tires of the support-copying wheels of the root crop head cleaner in the range of 350...1350 N ⋅ s ⋅ m–1 practically does not affect the amplitude and phase frequency characteristics of its angular oscillations under the condition of the effect of oscillations of the ordinates of the longitudinal profile of the field in the determining frequency range 0...24 s–1 depends little on the change in the value of the longitudinal coordinate of the location of its supporting and copying wheels. Taking into account the nature of the operation of the head cleaner of root crops under the influence of fluctuations of the ordinates of the longitudinal profile of the field in the defining frequency range of 0...24 s–1, the place of actual placement of its supporting and copying wheels should be determined taking into account the features of the machine design.

Mathematical modeling of snacks drying process from minced fish in a fluidized bed

A mathematical model of snacks drying process from minced fish in a fluidized bed with distributed parameters of the heat exchange process between the surface of an anisotropic body and the environment is proposed. The solution of the problem of non-stationary heat transfer by heat conduction using the Galerkin method is considered. The trial and verification functions of the method used, implemented in the PTC MathCAD engineering calculation environment, are linearly independent, represent the first elements of a complete system of polynomial functions and satisfy the boundary and initial conditions. Theoretical and experimental studies have been carried out, which allow considering the process of drying snacks from minced fish in a fluidized bed and scientifically substantiate options for its improvement. According to the results of experimental studies, the adequacy of the obtained mathematical model is shown. It is proved that with a uniform initial temperature distribution during preheating, the temperature inhomogeneity increases up to the moment of a phase transition on the surface of dried object. The importance of taking into account the preheating phase of wet material is established, since at this stage a temperature profile is formed, which is characterized by significant heterogeneity. This is especially important, since temperature heterogeneity directly affects the quality of the food product. The possibility of taking into account anisotropy in heat transfer processes using a three-dimensional mathematical model of transport with distributed parameters is confirmed. The developed technique allows significantly increasing the accuracy of an anisotropic boundary value problem solving by replacing the operation of integrating the stiffness matrix elements with a system of differential equations by algebraic formulas.

A REFINED METHOD FOR ANALYTICAL MODELING OF THE PROCESSES OF SPREADING VIRUS SOFTWARE TO ASSESS THE SECURITY OF INFORMATIZATION OBJECTS

Task. The purpose of writing this article is to improve the security of information in information systems. Model. The article proposes an improved method for analytical modeling of the process of spreading virus software in information systems in relation to the features of a special class of attacks implemented by the family of virus software - ransomware. Conclusions. The proposed method makes it possible to analytically calculate the intensity of infection of hosts of a protected information system with various types of virus software and use the obtained characteristics as initial data for the final assessment of the information security risk on the protected informatization object. Value. Modeling in the environment of the mathematical package PTC Mathcad Prime 7 showed the applicability of the proposed approach for analytical threat modeling.

Calculation of aerial contact wire volume heated by a moving electric ARC

The article is devoted to the study of the problem of electrified railways electric locomotive current collectors’ interaction violation with an aerial contact wire, accompanied by the occurrence of an electric arc. Current collection disorders, accompanied by arcing, have a destructive effect on the contacting elements, causing their thermal erosion. Places where current collection violations occur should be registered and diagnosed in a timely manner in order to prevent an emergency situation associated with burnout or breakage of the aerial contact wire. In order to further develop the system of technical diagnostics for current collection disorders accompanied by arcing, it is necessary to study the nature and parameters of the processes occurring during these violations. The main part of the article is devoted to the study of the area of heating the aerial contact wire by a moving electric arc. The characteristics describing the heated area are geometric parameters, including the area in the cross section of the wire and the volume of the area. To obtain the necessary data, the method of heat sources was applied, which is a mathematical model that describes the process of heat propagation in an aerial contact wire. The electric arc arising when the current collector breaks off is considered as a mobile heat source distributed over the aerial contact wire surface in the heating zone and having a limited radius of the heating spot. Based on the heat propagation process peculiarities inside the aerial contact wire, a method for calculating the volume of the heated area bounded by an isothermal surface of a certain temperature is presented. The heated area was visualized using the PTC MathCAD.

Improvement of method for estimation of parameters of contact interaction of billet with roll tool in lines of continuous electric pipe-welded mills

A technique for calculating the parameters of the shape of the boundary and areas of contact indentations of a pipe billet with a roller tool of a forming mill, in which the boundaries of the shape of the contact indentation are described by rectilinear lines, and the shape is modeled by triangles on the plane of the pipe billet, is investigated. The results of an experiment on fixing the contact sections of a tubular billet with initial dimensions of 160×1.5×1800 mm from steel 09G2S with rolls in the section of open pass are presented. As a result of the experiment, the contact areas of the pipe billet with the lower and upper rolls were fixed. The boundary of the experimental contact areas are curved arcs. Comparison of the experimental data with the technique showed that the parameters of the boundaries of the contact area according to the technique lead to an overestimation of the values by 30-40 % compared to the experimental data. On the basis of experimental data, the development of the existing technique for determining contact areas is presented, in which the boundaries are defined in the form of arcuate curves. The boundaries of the entry contact sections for the lower and upper rolls are determined by the equations of exponential approximation. Taking into account the equations of the boundaries of the sections, a method is presented for determining the areas of contact sections of a pipe billet on the surfaces of the lower and upper rolls, the results of which do not exceed 6 % of the experimental data. The engineering mathematical software PTC Mathcad Prime 6.0 was used to calculate and construct the areas of contact areas.

Design and Analysis of Rotor Assembly of Hammer Mill Machine`

The project deals with the Manufacturing with Design and Analysis of Hammer Mill Machine and Rotor Assembly of Machine of Capacity of 200 Kg/hr. Which is due to providing the transmission power of 5 HP to the machine.
 In this project, the Hammer mill machine body structure, Angle Frame and foundation frame for machine is designed using Catia. Also each and every part or component is required for machine is designed. In the present work by using standard design procedures, Diameter of the rotor of shaft of machine has been designed. Theoretical calculations done by using PTC Mathcad software for new learning experience and ease. When the shaft of the rotor is rotated at the given speed (rpm i.e. 1728 rpm) and the load applied to the shaft it should not bend during rotation. When the shaft is rotated under free conditions deflections will be created due to critical speed of the shaft.
 After Designing process, some of required drawings are converted into Ansys supported format i.e. drawings are imported into Ansys for further analysis. Meshing of the shaft model was done and the loads, stresses that were applied for the shaft to be checked out that the design should be safe one. The design should be safe when the values obtained from the design procedure were compared with the standard values and result obtained from the analysis using Ansys.
 As per the designing, the required parts are fabricated such as side plates, bearing support, doors, hinge supports by using conventional methods like gas cutting, welding, drilling, shaft turning, slotting, milling, etc. and some hardware materials buy from stores which is used for further assembly process. At the final step all parts are assembled according as per requirements. At the end actual capacity of machine is calculated by using conventional method.

SELF-TIMED LOOK UP TABLE FOR ULAs AND FPGAs

Context. Self-Timed Circuits, proposed by D. Muller on the rise of the digital era, continues to excite researchers’ minds. These circuits started with the task of improving performance by taking into account real delays. Then Self-Timed Circuits have moved into the field of green computing. At last, they are currently positioned mainly in the field of fault tolerance. There is much redundancy in Self-Timed Circuits. It is believed that Self-Timed Circuits approaches will be in demand in the nano-circuitry when a synchronous approach becomes impossible. Strictly Self-Timed Circuits check transition process completion for each gate’s output. For this, they use so-called D. Muller elements (C-elements, hysteresis flip-flops, G-flip-flops). Usually, Self-Timed Circuits are designed on Uncommitted Logic Array. Now an extensive base of Uncommitted Logic Array Self-Timed gates exists. It is believed that SelfTimed Circuits are not compatible with FPGA technology. However, attempts to create self-timed FPGAs do not stop. The article proposes a Self-Timed Lookup Table for the Self-Timed Uncommitted Logic Array and the Self-Timed FPGA, carried out either by constants or utilizing additional memory cells. Authors proposed 1,2 – Self-Timed Lookup Table and described simulation results.
 Objective. The work’s goal is the analysis and design of the Strictly Self-Timed universal logic element based on Uncommitted Logic Array cells and pass-transistors circuits.
 Methods. Analysis and synthesis of the Strictly Self-Timed circuits with Boolean algebra. Simulation of the proposed element in the CAD “ARC”, TRANAL program, system NI Multisim by National Instruments Electronics Workbench Group, and layout design by Microwind. The reliability theory and reliability calculations in PTC Mathcad.
 Results. Authors designed, analyzed, and proved the Self-Timed Lookup Table’s workability for the Uncommitted Logic Arrays and FPGAs. Layouts of the novel logic gates are ready for manufacturing.
 Conclusions. The conducted studies allow us to use proposed circuits in perspective digital devices.

Design and Analysis of Rotor Assembly of Hammer Mill Machine

The project deals with the Manufacturing with Design and Analysis of Hammer Mill Machine and Rotor Assembly of Machine of Capacity of 200 Kg/hr. Which is due to providing the transmission power of 5 HP to the machine. In this project, the Hammer mill machine body structure, Angle Frame and foundation frame for machine is designed using Catia. Also each and every part or component is required for machine is designed. In the present work by using standard design procedures, Diameter of the rotor of shaft of machine has been designed. Theoretical calculations done by using PTC Mathcad software for new learning experience and ease. When the shaft of the rotor is rotated at the given speed (rpm i.e. 1728 rpm) and the load applied to the shaft it should not bend during rotation. When the shaft is rotated under free conditions deflections will be created due to critical speed of the shaft. After Designing process, some of required drawings are converted into Ansys supported format i.e. drawings are imported into Ansys for further analysis. Meshing of the shaft model was done and the loads, stresses that were applied for the shaft to be checked out that the design should be safe one. The design should be safe when the values obtained from the design procedure were compared with the standard values and result obtained from the analysis using Ansys. As per the designing, the required parts are fabricated such as side plates, bearing support, doors, hinge supports by using conventional methods like gas cutting, welding, drilling, shaft turning, slotting, milling, etc. and some hardware materials buy from stores which is used for further assembly process. At the final step all parts are assembled according as per requirements. At the end actual capacity of machine is calculated by using conventional method.

Ensuring the operating modes of the seeding apparatus specialized equipment

The subjects of the research are the process of sowing seeds with a seeding apparatus and the indicators of quality of their distribution in a row. The goal is to substantiate a rational technological process of distributing row crops seeds in a row with a pneumatic seeding apparatus. Special equipment was used in the system of ensuring the revolutions of the seeding disc, sucking seeds to its holes, pneumatic transport of seeds after separation from the opening of the seeding disc for contact interaction with the receiving platform of the sensor of the number of seeds and its settling in the collector. The novelty of the research lies in the use of an acoustic sensor for a single registration of the sowed seeds with a short time interval, which makes it possible to assess the quality of the operation of the sowing device by the standard deviation of seeds from a given seeding rate in their normal distribution in accordance with the law of numbers. The operating mode of the pneumatic seeding apparatus from specialized equipment as part of the test bench was carried out as an example of sowing sunflower seeds with a seeding rate of 3 seeds/linear meter for the conventional seeder speed of 9 km/h. The data obtained with different time intervals between the registered seeds in WAV format by the PTC Mathcad Prime software was represented by classical distributed seeds in graphical form and data in tabular form. The quality indicators of seeds for sowing and their class distribution in a row by rational operating modes of the seeding apparatus, assessed by specialized equipment when applying the law of large numbers, are the basis of scientific research on predicting the yield of row crops.

CYCLICITY ANALYSIS OF THE BANK PERFORMANCE INDICATORS IN TERMS OF THE IMPLEMENTATION OF INNOVATIVE CUSTOMER SERVICE TECHNOLOGIES

Technological development, especially in the field of information and communication technologies, are changing the banking business. The implementation of innovative information and communication technologies for customer service in the banking sector allows banks to increase the number of customers and reduce their costs. The study of indicators that characterize changes in bank costs and customer number, their trends and seasonal or cyclical changes, will predict the development of the banking sector and the introduction of modern technologies in their activities, which, in turn, reduce the impact of inefficient management and investment decisions when implementing the latest technologies in their activities. The purpose of the study is to identify the trend and cyclical changes in the values of banking indicators and their relationship with the implementation of innovative technologies by banks. This paper has analyzed the indicators of general administrative expenses and commission income of Ukrainian banks for the period from January 2009 to December 2020, as these indicators indirectly characterize the implementation and use of innovative technologies by banks. This paper constructs indicators change models of general administrative expenses and commission incomes which include trend and cyclical components. The models' accuracy is within 10%. The analysis of indicators' cyclicity was performed by methods of harmonic analysis using the package PTC Mathcad 6.0. Both indicators show an increasing trend. The identified periods of fluctuations in the components of the total administrative expenses are 3, 4 and 72 months. The identified periods of fluctuations in the components of the commission income are 14.4, 28.8 and 72 months. The paper has made the forecast of indicators of general administrative expenses and commission incomes for 2021. The results of this study can be considered at planning and forecasting of development of banking system of Ukraine and influence of innovative technologies on this development.

Analysis of the noise load from the manufacturing enterprise on the territory of residential buildings based on complex mathematical model

Noise is known to be one of the most common negative factors that can have detrimental effect on human and the environment. In the article the expected noise load on the territory of a residential development located in the immediate vicinity of a metallurgical plant is analyzed. A steel arc furnace is taken as an active source of noise. Account has been taken of the main techno-economic, structural and social indicators of the situation in which housing development has been organized in close proximity to existing industrial plants. All the data obtained from studying the propagation of noise from the source are processed on the basis of the complex mathematical model in the PTC Mathcad Application. The model makes it possible to carry out studies of distribution of the air noise of an active industrial source in multifunctional areas, to carry out calculations in accordance with the established normative methods, and to analyse the CPD over the whole frequency range. It is also possible to analyse the application of the most effective ways of reducing acoustic load. The most effective engineering solution is a complex screening scheme which makes it possible to reduce the sound pressure levels of the frequency range from 20Hz to 10 000Hz by 20-25 dB, thus satisfying the modern requirements of sanitary standards for residential development.

Improving the Energy Efficiency of the Use of Borehole Jet Pumps

The article presents rationales for choosing the depth of installation of an oil jet pump in the borehole that which ensures maximum efficiency of its operation. The operating parameters of the ejection system are determined by the joint solution of the characteristic equations of the high-pressure jet pump and the hydraulic system. In the process of solving the system of equations, the method of successive approximations, the Delphi software environment and PTC Mathcad resources were used. The equation of the characteristics of the jet pump hydraulic system was obtained by determining the pressures in its distinctive cross-sections and then presenting their values as the relative (dimensionless) head of the ejection system. Alteration the installation depth of the jet pump changes the characteristics of its hydraulic system, the parameters of the operating point of the pumping unit and its efficiency. In this case, the minimum permissible installation depth of the jet pump is determined by the value of the minimum pressure in the elements of the ejection system, which must exceed the value of the elastic pressure of saturated vapors of the oil and gas flow and ensure its operation in pre-cavitation mode. The probability of operation of a jet pump in cavitation mode was studied using the Bernoulli, Darcy – Weisbach and flow continity equations. The inversely dependence of the ejection coefficient and efficiency of the jet pump on the depth of its installation in the borehole has been revealed. If the jet pump is installed in the borehole at the optimal depth, its efficiency is increased by 30 %.

Effect of concreting defects on the uptime of reinforced concrete during the construction of new Arctic settlements by the criterion of exposure to air carbon dioxide

The development of Arctic territories and waters requires the construction of new settlements with industrial infrastructure. Modified concrete is used for the construction of these facilities. However, during erection of monolithic reinforced concrete structures defects and damages are formed, which affect their durability and reliability. The results of probabilistic simulation of failure-free operation of reinforced concrete according to the criterion of carbon dioxide gas effect on concrete for conditions of Sabetta settlement are given in the work. Concretes with a water-cement ratio of 0.35 and 0.50 per CEM III/A 42.5N were taken as design values. Design thickness of the protective layer is 45 mm, depth of defects and damages is 5, 10, 15, 25 mm. Probability modeling was performed using the Duracret model using Excel and PTC Mathcad. The results showed that without defects and damages the period of normal operation as a whole corresponds to the design value of 50 and 100 years. In case of defects, the first local areas may appear in 15-40 years.

Determination of the hardness of the weld metal and the metal of the heat affected zone (HAZ) during low-frequency oscillations of the weld pool

Work rele vance. The using of mechanical harmonic oscillations of welding pool or tool are inexpensive and quite simple procedure formation desirable degree of dispersion structure it means weld metal strength. Possibility of oscillations differently applied relative of the surfacing (welding) process axis lets welding seam or surfacing bead formation with certain anizotropy of mechanical properties in given direction.Many works have dedicated of researches of microstructure formation by conditions of periodic action on weld melt. However, the most part of it are devoted mostly researching to influence oscillations with frequency much superior value of 5 Hz and amplitude value range from some microns to 2–3 mm.Purpose of work is getting quantity dependencies which defining hardness value of weld metal and metal of HAZ as a function of technological and oscillations modes parameters.Research methods. For researchers the samples have had prepared and polished to #8 Mirror Finish by ISO (ГОСТ 25593 – 83). The surfacing process in CO2 protection environment had has carried out by alloy wire named G3Si1 by EN 13479 ISO 143341 (ТУ У 322-4-392-96) which diameter1.2 mm. Base metal was common quality carbon steel type Е 235-С by ISO 630:1995, ISO 1052:1982 (ДСТУ 2651:2005). Measurement of hardness (by Vickers) had carried out on hardness tester named LECO M-400 under indenter load1 kgby ISO 6507-1:2007 (ДСТУ ISO 6507-1:2007). Research resultants had treatment by regression analysis methods with application of analytics software package STATISTICA 12 and engineering math software PTC Mathcad Prime 3.1.Results. Regressive models of hardness value as function magnitudes arc current, the surfacing process velocity, amplitude and frequency oscillations have received. Contour plots of depend of hardness value from amplitude and frequency magnitudes were created. Variant of predict of the receiving maximum value hardness weld metal and metal of HAZ at arc current125 Aand the surfacing process speed 14 m/h are presented.Scientific novelty. The experiments were carried out with different amplitude and frequency values which were given through programming step motor.Empirical depends of influence of technological and oscillation modes parameters of certain range on weld metal hardness and metal HAZ hardness values are created.Practical value. An increase possibility of hardness values of weld metal and metal of HAZ due to use weld pool oscillations with frequency value range 2.5–4.5 Hz and amplitude value range 3–7 mm was proven. The magnitudes of amplitude and frequency permitting receives maximum hardness values at optimal values of arc current and the surfacing velocity were obtained.

Solution of the Burgers Shock Wave Equation in PTC MathCAD

A technique for solving the Burgers equation by the Galerkin method as applied to the problem of a propagating shock wave is considered. The solution is based on the reduction of a parabolic partial differential equation to a system of ordinary differential equations, depending on time, considering the nonlinear term included into the differential equation. The system of ordinary differential equations is integrated using the Euler computational scheme. Furthermore, the step size is maintained small enough so that the errors associated with numerical integration are negligible in comparison with the approximation error. It is established that the mean square error with an increase in the order of the approximate solution decreases raPIDly and the trial solution converges to the exact solution.

Mastery of the foundations of mathematical modeling in a technical university

One of the key components of engineering education is the possession of a basic tool for obtaining new fundamental knowledge, scientific research, and designing technical systems – mathematical modeling. The use of tools and capabilities of the engineering application PTC Mathcad Prime 3.1 in mastering the basics of the scientific method of mathematical modeling is a necessary component of the educational process, largely determining its effectiveness, providing a significant increase in students' motivated interest in research work, stimulating their creative self-realization.

Calculation of sealing joints with elastic edge in PTC MathCAD

The article touches upon the issues of the development of engineering methods of calculating a thin-walled shell-plate seat for valve and flange sealing joints, followed by the formulation of tasks using modern automated calculation software. To achieve the purpose, the authors used Parametric Technology Corporation MathCAD (PTC MathCAD) with subroutines for the search for extremum of functions and the computational block ‘Given - minimize’. The calculation is presented in the form of algorithms consisting of three logically interrelated parts: 1) stiffness calculation; 2) strength calculation; 3) the calculation of rational dimensions. The obtained results of the calculation of geometric dimensions are tested, which eliminates a probable error in the calculated block ‘Given’ of PTC MathCAD. Among the developed engineering methodologies, the article describes the following ones: a method for calculating the rational geometrical dimensions of a shell-plate valve seat, operating under shock loading at an uncertain maximum dynamic load. It ensures that the allowable stresses in a thin-walled shell-plate seat are not exceeded. Another is a method for testing the stress-strain state of a shell-plate valve seat under static loading with a force of sealing and pressure of the working medium. The last one is the method of calculating rational geometrical dimensions of the shell-plate seat of a flange connection, which minimizes the unit stiffness (minimizing sealing force), provided that the shell-plate seat strength is ensured.