Mathcad Software Research Articles

Using applied program packages in determining optimum cutting modes

Purpose. Comparison of the functionality of application program packages when determining optimal cutting modes during mechanical processing. Creation of a mathematical model with technological limitations when boring a shaft in one pass and carrying out calculations. The methods. Research methods were based on the principles of optimization of cutting processes, the theory of cutting and cutting tools, standard methods of determining correction factors were used when creating technological limitations. The smallest machine processing time is chosen as the objective function. Using the functionality of the mathematical package Mathcad and the "Turning optimization" program. Findings. The created mathematical model of turning optimization sufficiently fully describes the machining process. The difference between calculations by two methods is no more than 10%. The functionality of the application programs is enough to determine the optimal parameters of the cutting modes - the number of revolutions of the machine spindle and feed. In addition, when using the Mathcad package, it is possible to add or remove technological constraints. Without a suitable database of correction factors, the search and calculation process is complicated. On the other hand, the "Turning optimization" program works in automatic mode, you only need to enter the initial data. But it is impossible to change the technological limitations, the insufficient number of standard materials of cutting tools and parts. The originality. A mathematical model of optimization of cutting modes when boring a shaft to a pass with corresponding technological limitations was created and investigated. A comparison of application program packages was carried out when determining the optimal cutting modes. Advantages and disadvantages of application programs are defined. Practical implementation. The proposed mathematical model of optimization of cutting modes during turning allows to determine the optimal parameters of the number of revolutions and feed of the machine. Its use will greatly facilitate calculations carried out by designer-technologists when designing technological routes for processing parts and calculating cutting modes. This will significantly reduce production preparation time, especially when using CNC machines.

Thermal investigation into the Oldroyd-B hybrid nanofluid with the slip and Newtonian heating effect: Atangana–Baleanu fractional simulation

The significance of thermal conductivity, convection, and heat transportation of hybrid nanofluids (HNFs) based on different nanoparticles has enhanced an integral part in numerous industrial and natural processes. In this article, a fractionalized Oldroyd-B HNF along with other significant effects, such as Newtonian heating, constant concentration, and the wall slip condition on temperature close to an infinitely vertical flat plate, is examined. Aluminum oxide (Al2O3) and ferro-ferric oxide (Fe3O4) are the supposed nanoparticles, and water (H2O) and sodium alginate (C6H9NaO7) serve as the base fluids. For generalized memory effects, an innovative fractional model is developed based on the recently proposed Atangana–Baleanu time-fractional (AB) derivative through generalized Fourier and Fick’s law. This Laplace transform technique is used to solve the fractional governing equations of dimensionless temperature, velocity, and concentration profiles. The physical effects of diverse flow parameters are discussed and exhibited graphically by Mathcad software. We have considered 0.15≤α≤0.85,2≤Pr⁡≤9,5≤Gr≤20,0.2≤ϕ1,ϕ2≤0.8,3.5≤Gm≤8, 0.1≤ Sc ≤0.8, and 0.3≤λ1,λ2≤1.7. Moreover, for validation of our present results, some limiting models, such as classical Maxwell and Newtonian fluid models, are recovered from the fractional Oldroyd-B fluid model. Furthermore, comparing the results between Oldroyd-B, Maxwell, and viscous fluid models for both classical and fractional cases, Stehfest and Tzou numerical methods are also employed to secure the validity of our solutions. Moreover, it is visualized that for a short time, temperature and momentum profiles are decayed for larger values of α, and this effect is reversed for a long time. Furthermore, the energy and velocity profiles are higher for water-based HNFs than those for the sodium alginate-based HNF.

Relative magnetic field and slipping effect on Casson dusty fluid of two phase fluctuating flow over inclined parallel plate

Due to its significant applications in magnetic devices for cell separation, magnetic drugs for cancer tumor treatment, blood flow adjustment during surgery, magnetic endoscopy, and fluid pumping in industrial and engineering processes, magnetohydrodynamics (MHD) has a wide range of applications. Therefore, the aim of this study to investigate the Relative Magnetic Field and slipping effect on Casson dusty fluid of two-phase fluctuating flow over inclined parallel plate. More exactly, the relative magnetic phenomena are fixed relatively to the plate (MFFRP) or fixed relatively to the fluid (MFFRF). The mathematical modeling was established as the set of partial differential equations for dusty Casson fluid. Buckingham's pi theorem is used to find out the dimensionless variables, to make our system dimensionless. The solution in perturb form is to find out by incorporating Poincare–Lighthill perturbation technique (PLPT). To know in-depth different parameters the graphical results for both velocities are plotted with the help of software Mathcad- 15. It is clearly detected that the relative magnetic field performs a vital part in the fluid as well as particle motion. Moreover, the relative magnetic field play a vital role in the decreasing of skin friction of both plates.

Approximation Solution of Fuzzy Singular Volterra Integral Equation by Non-Polynomial Spline

A non-polynomial spline (NPS) is an approximation method that relies on the triangular and polynomial parts, so the method has infinite derivatives of the triangular part of the NPS to compensate for the loss of smoothness inherited by the polynomial. In this paper, we propose polynomial-free linear and quadratic spline types to solve fuzzy Volterra integral equations (FVIE) of the 2nd kind with the weakly singular kernel (FVIEWSK) and Abel's type kernel. The linear type algorithm gives four parameters to form a linear spline. In comparison, the quadratic type algorithm gives five parameters to create a quadratic spline, which is more of a credit for the exact solution. These algorithms process kernel singularities with a simple technique. Illustrative examples use MathCad software to deal with upper and lower-bound solutions to fuzzy problems. The method provides a reliable way to ensure that an exact solution is approximated. Also, figures and tables show the potential of the method.

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

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

Time fractional analysis of Casson fluid with application of novel hybrid fractional derivative operator

<abstract><p>A new approach is used to investigate the analytical solutions of the mathematical fractional Casson fluid model that is described by the Constant Proportional Caputo fractional operator having non-local and singular kernel near an infinitely vertical plate. The phenomenon has been expressed in terms of partial differential equations, and the governing equations were then transformed in non-dimensional form. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on the newly introduced Constant Proportional Caputo fractional derivative operator. This fractional model has been solved analytically, and exact solutions for dimensionless velocity, concentration and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. For the physical significance of various system parameters such as $ \alpha $, $ \beta $, $ Pr $, $ Gr $, $ Gm $, $ Sc $ on velocity, temperature and concentration profiles, different graphs are demonstrated by Mathcad software. The Constant Proportional Caputo fractional parameter exhibited a retardation effect on momentum and energy profile, but it is visualized that for small values of Casson fluid parameter, the velocity profile is higher. Furthermore, to validated the acquired solutions, some limiting models such as the ordinary Newtonian model are recovered from the fractionalized model. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, from the literature, it is observed that to deriving analytical results from fractional fluid models developed by the various fractional operators is difficult, and this article contributes to answering the open problem of obtaining analytical solutions for the fractionalized fluid models.</p></abstract>

НЕЧЕТКАЯ КЛАСТЕРИЗАЦИЯ КОМПЛЕКСНОЗНАЧНЫХ ДАННЫХ

The article discusses the most well-known methods of fuzzy clustering of data: the k-means method and the fuzzy algorithm of c-ellipsoids. Algorithms of these methods are implemented in the MathCAD package and an example with complex-valued data is considered.

A New Bivariate Odd Generalized Exponential Gompertz Distribution

The objective of this study was to present a novel bivariate distribution, which we denoted as the bivariate odd generalized exponential gompertz(BOGE-G) distribution. Other well-known models included in this one include the gompertz, generalized exponential, odd generalized exponential, and odd generalized exponential gompertz distribution. The model introduced here is of Marshall-Olkin type [16]. The marginals of the new bivariate distribution have odd generalized exponential gompertz distribution which proposed by[7]. Closed forms exist for both the joint probability density function and the joint cumulative distribution function. The bivariate moment generating function, marginal moment generating function, conditional distribution, joint reliability function, marginal hazard rate function, joint mean waiting time, and joint reversed hazard rate function are some of the properties of this distribution that have been discussed. The maximum likelihood approach is used to estimate the model parameters. To demonstrate empirically the significance and adaptability of the new model in fitting and evaluating real lifespan data, two sets of real data are studied using the new bivariate distribution. Using the software Mathcad, a simulation research was conducted to evaluate the bias and mean square error (MSE) characteristics of MLE. We found that the bias and MSE decrease as the sample size increases.

Автоматизированный синтез топологий интегральных сверхвысокочастотных LC-фильтров с минимизацией потерь в полосе пропускания

The paper proposes the developed technique for automated synthesis of integrated microwave LC filter layouts with passband losses minimization by minimizing interconnection lengths and searching for optimal layouts of conductors, inductors, and capacitors. Algorithms for minimizing losses and length of LC-filter interconnections are proposed. The layouts of conductors, inductors, and capacitors are optimized based on silicon-verified models that take into account the main characteristics and limitations of the process, skin effect, edge effects, substrate and via influences. The CAD is developed in MathCAD software based on the proposed technique. The use of CAD made it possible to reduce the engineering time for integrated LC-filters to tens of minutes, reduce the filters passband losses by 1,1–2,7 dB and increase the AFCs gain slope by 47–267 dB/dec.

The processing method of the parameters of the beam pattern of phased array antennas

The investigation results of the radiation pattern as one of the important parameters of the phased array antenna are presented at the article. Practical results: obtaining skills in using MathCad and researching methods for improving the directional pattern in order to increase the accuracy of determining the goal. The paper presents an algorithm for calculating the radiation pattern of antenna arrays with four modules and a method for improving it. Software and mathematical results are available in the Internet for free, that allows anyone to do similar work to confirm the calculations and conclusions. The results could be used at students practice in technical universities while studying in specialties related to radio electronic engineering in courses connected with study of the design principles of antenna systems with an equidistant arrangement of elements. Publication could be recommended as methodical manual to students for homework solving or laboratory work using specialized MathCad software.

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

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

Model of the aggregation process of fine ferromagnetic particles

The most important result of the effect of interparticle interactions of ferromagnetic particles in suspensions is the formation of aggregates and modification of their physical properties, which allows changing the kinetics of separation processes. The interest of the authors to this problem is dictated by both its theoretical content and purely applied aspects within mineral processing. Based on the block-structural approach to technological equipment modeling within the framework of the ideal mixing model, a mathematical model of processes occurring in the working volumes of apparatuses using magnetic forces to intensify the separation of mineral suspensions was created. The use of the MATHCAD software has allowed obtaining analytical solutions of the differential equations system of the model and predictions of the kinetics of aggregation of ferromagnetic particles and physical properties of the formed aggregates, such as coarseness, density, deposition velocity in the gravity field as functions of medium parameters and external magnetic fields. The analytical solutions are necessary for developing simulation models of magnetic separation processes in strong, weak, uniform and non-uniform magnetic fields, magnetic deslammers and filters.

ПОГРУЗЧИК-ШТАБЕЛЕР С РАСШИРЕННЫМИ ФУНКЦИОНАЛЬНЫМИ ВОЗМОЖНОСТЯМИ

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

USE OF NEURAL NETWORKS FOR PLANNING THE CORRECT SELECTION OF PLANT AND SOIL SAMPLES IN PRECISION AGRICULTURE TECHNOLOGIES

The article is devoted to the study of the use of neural networks to optimize the selection of plant stands in precision agriculture technologies. The study takes into account the complex aspects of sample selection, such as the speed of image acquisition, the effectiveness of assessing the state of mineral nutrition and soil moisture, etc. This data is a necessary component for precision farming technologies and, in particular, crop management. Research was conducted on production fields in 2019-2020 in Boryspil district of Kyiv region. Spectral studies were performed using the Slantrange 3p complex installed on the UAV. Data processing was performed both with the specialized software for spectral data Slantview and with the mathematical package MathCad. The assessment of the nature of the distribution of both individual spectral channels and their combination in the form of vegetation indices turned out to be unprepared for the identification of uneven water supply of areas. The red channel and its derivatives turned out to be the most promising in the direction of identifying the water supply of wheat. The use of neural networks made it possible to identify probable areas with increased water supply on the maps of the distribution of vegetation indices in the field. The duration of identification using neural networks will not interfere with the sampling procedure, so that such a procedure can be effectively implemented in agronomic practices. Therefore, the use of neural networks allows you to automate and increase the accuracy of selection, improving the quality of the analysis of plant stands, subject to compliance with soil sample evaluation technologies. The obtained results indicate the prospects of implementing this approach in modern agriculture.

Mathematical programming in Mathcad and Mathematica

An article generalizes the long-term work of authors with packages of applied mathematical programs. It discusses and demonstrates the features and methods of solution of mathematical tasks in mathematical package Mathcad and Mathematica: from the simplest ones, included in the set of typical problems of mathematical disciplines for training specialists for shipbuilding, to complex computational tasks and applied problems of professional orientation, which require the construction of a mathematical model and analysis of the results obtained. The examples show the solution of mathematical problems in symbolic form, mathematical studies in the Mathcad and Mathematica environment, and mathematical programming with these packages.

Экологиялық тізбектер және олардың математикалық-компьютерлік моделдері

The article deals with the problem of studying the structures and evolutions of ecological chains at the level of microorganisms using mathematical and computer models. Several mathematical models have been constructed based on the laws of biological kinetics. Computer experiments were developed using these models. Mathematical models describing the dependence of the concentration of bacterial biomass, substrates and protozoa on time have been created. With the help of the Mathcad package, graphical solutions of mathematical models were obtained and analyzed. Graphs of the states of ecological chains with changes in the rate of penetration of the nutrient substrate into the system were created and these states were studied. Mathematical models describing the mutual competition of 6 different bacteria were developed in Mathcad environment. The parameters characterizing the biomass concentration of each bacterium were established. This parameter included: saturation constant of bacteria, saturation constant of protozoa, maximum value of specific growth rate of bacteria, maximum value of specific growth rate of protozoa, and economic growth coefficient of protozoa. Graphical solution of mathematical models have been analyzed. These models characterize the dynamics of the concentration of bacterial biomass. A computer experiment representing competition between bacterial species was developed. The results of mutual competition of ten different bacterial colonies were obtained on the basis of numerical and graphical methods. The software has been compiled in Microsoft Visual Basic 6.0 environment.

Changes in the immunological reactivity in patients with coxarthrosis and the factors that determine them

Background. The participation of immune responses in the pathogenesis of degenerative-dystrophic diseases of the synovial joints determines the feasibility of using more rational approaches to assessing the immune status of patients with this pathology. Despite the ongoing immunological studies in patients with degenerative-dystrophic diseases of the hip joint, there are questions related to the functioning of the immune system as a whole, which require additional research.The aim of the study was to establish the features of the relationship between the indicators of the immune status and to identify the factors that determine the nature and severity of immunological disorders in patients with coxarthrosis.Material and methods. The study is based on the results of immunological parameters in 73 patients (mean age – 45.3 ± 2.56 years) with stage III dysplastic coxarthrosis. The data obtained using cluster analysis made it possible to distribute patients into two groups. The first clinical group included 46 patients, the second one – 27 patients. The control group consisted of 28 clinically healthy individuals. Mathematical processing of the results was carried out using the Statistica 6.0 software package (StatSoft Inc., USA) and the mathematical package Mathcad 2001 Pro (MathSoft Inc., USA).Results. The levels of CD4+ and CD8+ cells and IgA were determined as the classification features that make the greatest contribution to the division of the studied groups. Model experiments have shown that a specific combination of the magnitude of each of the noted factors with a certain degree of probability makes it possible to assess the state of the initial immunological reactivity and verify the nature and severity of immunological disorders.Conclusion. The information obtained is of great importance, since it allows predicting the course of the recovery period after surgery and assessing the risk of developing possible postoperative complications in patients with severe articular pathology.

DINAMIC FORCES IN THE ECCENTRICALLY COMPRESSED MEMBERS OF REINFORCED CONCRETE FRAMES UNDER ACCIDENTAL IMPACTS

Interest to solving scientific problems related to the evaluation of facilities' resistance and its protection against progressive collapse increases and attracts more and more attention of specialists in the field of structural analysis and design. Therefore, the article presents the results of a computational analysis of dynamic forces in eccentrically compressed reinforced concrete members of structures under accidental impact such as sudden removal of a load bearing member. Using relations for specific deformation energy and integrating it through the cross-section area, the analytical expressions for dynamic strains and curvatures have been obtained for physically and structurally nonlinear RC frame members under eccentric compression. These expressions in some cases allow symbolic solution, for example, in MathCAD software. In contrary, it can be solved with the approximate iterative method. To assess the reliability and effectiveness of the proposed quasi-static method, the analysis of the cast-in-situ reinforced concrete frame resistance to progressive collapse has been performed. The article also provides comparison of the simulation results of the nonlinear quasi-static analysis and the nonlinear dynamic time-history analysis.

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

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

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

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