ESTIMATING WITH A GIVEN ACCURACY OF THE COEFFICIENTS AT NONLINEAR TERMS OF UNIVARIATE POLYNOMIAL REGRESSION USING A SMALL NUMBER OF TESTS IN AN ARBITRARY LIMITED ACTIVE EXPERIMENT

We substantiate the structure of the efficient numerical axis segment an active experiment on which allows finding estimates of the coefficients fornonlinear terms of univariate polynomial regression with high accuracy using normalized orthogonal Forsyth polynomials with a sufficiently smallnumber of experiments. For the case when an active experiment can be executed on a numerical axis segment that does not satisfy these conditions, wesubstantiate the possibility of conducting a virtual active experiment on an efficient interval of the numerical axis. According to the results of the experiment, we find estimates for nonlinear terms of the univariate polynomial regression under research as a solution of a linear equalities system withan upper non-degenerate triangular matrix of constraints. Thus, to solve the problem of estimating the coefficients for nonlinear terms of univariatepolynomial regression, it is necessary to choose an efficient interval of the numerical axis, set the minimum required number of values of the scalarvariable which belong to this segment and guarantee a given value of the variance of estimates for nonlinear terms of univariate polynomial regressionusing normalized orthogonal polynomials of Forsythe. Next, it is necessary to find with sufficient accuracy all the coefficients of the normalized orthogonal polynomials of Forsythe for the given values of the scalar variable. The resulting set of normalized orthogonal polynomials of Forsythe allows us to estimate with a given accuracy the coefficients of nonlinear terms of univariate polynomial regression in an arbitrary limited active experiment: the range of the scalar variable values can be an arbitrary segment of the numerical axis. We propose to find an estimate of the constant and ofthe coefficient at the linear term of univariate polynomial regression by solving the linear univariate regression problem using ordinary least squaresmethod in active experiment conditions. Author and his students shown in previous publications that the estimation of the coefficients for nonlinearterms of multivariate polynomial regression is reduced to the sequential construction of univariate regressions and the solution of the correspondingsystems of linear equalities. Thus, the results of the paper qualitatively increase the efficiency of finding estimates of the coefficients for nonlinearterms of multivariate polynomial regression given by a redundant representation.

Analysis of the influence of cutting conditions on the roughness of the surface layer when machining a composite material based on thermally еxpanded graphite

The current trend of replacing metal alloys with composite materials in the manufacture of various types of plain bearings, sealing elements of Assembly units, anti-friction bushings, and other machine parts is very promising for the development of many industries. However, in the manufacture of composite material products by modern methods, such as extrusion, pressing, and injection molding, various types of defects characteristic of these types of processing occur, resulting in the required parameters of the quality of the working surfaces of parts are not provided. This leads to the need for additional mechanical processing the quality of which largely depends on the reliability and durability of the functioning of parts and mechanisms. The article analyzes the main parameters that affect the quality of the surface obtained during turning for metal alloys, as well as for composite materials. The method of obtaining blanks by pressing from the material under consideration was developed, the technological equipment necessary for the research was designed and manufactured, a plan was developed for conducting a full-factor experiment with the creation of a model in the Mathcad program, and an active experiment was carried out to determine the influence of cutting mode parameters on surface roughness. Given the results of the study on the basis of which conclusions about the dependence of various cutting parameters on the quality of the surface layer, defined by the nuances of turning the considered composite material, practical recommendations that will positively affect the timing and success of implementation of fabrication of parts from this material.

Mathematical forecasting composition of secondary carbides in the single-crystal superalloys

Purpose: Predicting the specifics of the distribution of alloying elements between secondary carbides, their topology, and morphology, as well as the composition for a single-crystal multicomponent system of the type Ni-11.5Cr-5Co-3.6Al-4.5Ti-7W-0.8Mo-0.06C using the calculated CALPHAD (passive experiment) versus scanning electron microscopy (active experiment). Design/methodology/approach: This work presents the results of studies of the distribution of chemical elements in the composition of carbides, depending on their content in the system. The studies were carried out using an electron microscope with computer analysis of images and chemical composition. Findings: It was found that the influence of alloying elements on the composition of carbides is complex and is described by complex dependencies that correlate well with the obtained experimental results. Research limitations/implications: An essential problem is the prediction of the structure and properties of superalloys without or with a minimum number of experiments. Practical implications: The obtained dependencies can be used both for designing new superalloys and for improving the compositions of industrial alloys. Originality/value: The value of this work is that the obtained dependences of the influence of alloying elements on the dissolution (precipitation) temperatures and the distribution of elements in secondary carbides in the superalloy of the Ni-11.5Cr-5Co-3.6Al-4.5Ti-7W-0.8Mo-0.06C system. It was found that changes in the course of the curves of temperature dependence on the element content closely correlate with thermodynamic processes occurring in the system, that is, the curves exhibit extrema accompanying the change in the stoichiometry of carbides or the precipitation of new phases.

Mathematical Modeling in View of Property Prediction of DD11 Steel Laminated in LBC Liberty Galati

The paper presents a mathematical model for predicting the mechanical properties of hot rolled strips. The realization of this mathematical model relied on statistical measurements of the mechanical properties (Rm, Rp0,2 and A5) for the laminated strip in the LBC rolling mill from the Liberty Steel Galati steel plant. To achieve this mathematical model, there has been used the active experiment method.By means of this mathematical model, significant time and material savings can be made in the process of testing the mechanical properties for hot rolled tape.

Mathematical Modeling of the Cold Rolling Process and Heat Treatment for DC01 Steel

The paper presents the elaboration of a mathematical model of the cold strip rolling process combined with the recrystallization annealing after the rolling at LBR Liberty Galati.The elaborated mathematical model allows the prediction of the mechanical properties of cold rolled strips subsequently subjected to a heat treatment.The realization of this mathematical model was based on statistical measurements of the mechanical properties Rm, Rp0.2 (Rc) and A5 for the rolled steel strip DC01 from Liberty Steel Galati. To achieve this mathematical model, the active experiment method was used.With the help of this mathematical model, it is possible to optimize the rolling process by significant savings of time and materials in the process of testing the mechanical properties for cold rolled tape, but also by choosing the most appropriate process parameters.

Methodology of testing common rail fuel injectors with the use of Gauss’s formulas

The paper presents the methodology of testing common rail fuel injectors, which consisted in extending the standard diagnostic proce-dures with the analysis of the resulting fuel delivery fields. The calculations were carried out in a popular spreadsheet, using Gauss's formulas (the so-called shoelace formula). In this way, the necessity to modify the test bench software was eliminated, as the analytical process took place after the end of the active experiment phase. It has been shown that the proposed solution should be particularly useful in problematic situations where clear-cut verification and assessment of the technical condition of the fuel injectors is sometimes difficult, as shown in the example. In addition, implementation in a digital environment allows the presented algorithms to be reused in research with a similar profile.

APPLICATION OF THE “JUMPING FROGS” ALGORITHM FOR RESEARCH AND OPTIMIZATION OF THE TECHNOLOGICAL PROCESS

Context. An application of the method of a “jumping frogs” search algorithm to construct optimal experiment plans for cost (time) in the study of technological processes and systems that allow the implementation of an active experiment on them is proposed. The object of study are optimization methods for cost (time) costs of experimental designs, based on the application of a “jumping frogs” search algorithm. Objective. To obtain optimization results by optimizing the search of a “jumping frogs” search algorithm for the cost (time) costs of plans for a full factorial experiment. Method. A method is proposed for constructing a cost-effective (time) implementation of an experiment planning matrix using algorithms for searching for “jumping frogs”. At the beginning, the number of factors and the cost of transitions for each factor level are entered. Then, taking into account the entered data, the initial experiment planning matrix is formed. Then, taking into account the entered data, the initial matrix of experiment planning is formed. The “jumping frogs” method determines the “successful frog” by the lowest cost of transitions between levels for each of the factors. After that, the permutations of the “frogs” are performed. The “frog” strives for the most “successful” and, provided it stays close, remains in the location. Then the gain is calculated in comparison with the initial cost (time) of the experiment. Results. Software has been developed that implements the proposed method, which was used to conduct computational experiments to study the properties of these methods in the study of technological processes and systems that allow the implementation of an active experiment on them. The experimental designs that are optimal in terms of cost (time) are obtained, and the winnings in the optimization results are compared with the initial cost of the experiment. A comparative analysis of optimization methods for the cost (time) costs of plans for a full factorial experiment is carried out. Conclusions. The conducted experiments confirmed the operability of the proposed method and the software that implements it, and also allows us to recommend it for practical use in constructing optimal experiment planning matrices.

Mathematical simulation of welded deposit layers as for adaptation of materials to failure in quasidissipative tribosystems

Purpose. To establish mathematical models for the adaptation of materials under conditions of activation of a metastable structural-phase state of quasi-dissipative tribosystems. Determination and substantiation of factors for planning an active experiment due to which it is possible to create mathematical models of stable forecasts for increasing the wear resistance of materials. Research methods. When conducting the experiments, a priori data were used in combination with our own scientific developments of mathematical models of the dependences of the influence of the chemical composition of alloys during manual and automatic electric arc surfacing on the physical and mechanical properties of the surface layer of the material, which is destroyed under tribosystem conditions. It was decided to use mathematical planning of research on the basis of an active experiment with the creation of models for the numerical description of the mathematical expectation in the form of regression equations. Obtained results. On the basis of theoretical and practical scientific research with planning a passive and active experiment, a set of relevant knowledge has been obtained, which makes it possible to determine the main criterion requirements for the sensitivity of deposited steels and alloys to adaptation under the action of external mechanical and energy influences and allows to mathematically describe the characteristics of the alloy and provide a numeri­cal estimate of the correlation parameters among themselves. On the basis of the theory of scientific mathematical planning of the experiment, a set of corresponding experiments was carried out, which made it possible to build spatial graphic models. Scientific novelty. For the first time, theoretical and practical scientific research is presented with the reproduction of a systemic multivariate analysis of the parameters of mathematical models and processes leading to the martensitic ( γ → α ) transformation and determines the substantiation of the chemical composition of the deposited materials to increase fracture resistance under conditions of quasi-dissipative and dissipative tribosystems. Practical meaning. The obtained results of the above studies allow, within the framework of technical and technological accuracy, which is necessary within the framework of practical engineering forecasts, to determine the physical and mechanical properties of wear-resistant deposited alloys under conditions of quasi-dissipative and dissipative tribosystems.

MODELING THE TORQUE OF THE HYDRAULIC VOLUME DRIVE WITH THE PARAMETERS OF THE HYDRAULIC LOADING DEVICE

Currently, manufacturers for a reliable assessment of the technical condition of volumetric hydraulic drives use the dynamic test method, which allows you to determine the torque (braking) moment on the shaft of the tested hydraulic motor. There are difficulties in reliably determining the value of the developed torque (braking) moment. (Research purpose) The research purpose is in constructing a mathematical model of the relationship between the developed torque (braking) moment of a volumetric hydraulic drive and the parameters of a hydraulic loading device. (Materials and methods) The article presents a hydraulic loading device that provides the necessary braking torque on the shaft of the tested hydraulic motor. Authors conducted one-factor and multi- factor experiments-dynamic tests of the new Sauer-Danfoss series 90 volumetric hydraulic drive using a hydraulic loading device. The torque (braking) was monitored using a non-contact digital torque sensor M 425 3-A datum electronics and a digital indicator with LCD display (Results and discussion) As a result of one-factor passive experiments, the factors influencing the optimization parameter and the ranges of their variation were determined. A mathematical model of the relationship between the developed torque (braking) moment of a volumetric hydraulic drive and the parameters of a hydraulic loading device was obtained by a multi-factor active experiment. The article presents a complete planning matrix for a multi-factor active experiment, which reflects the sequence of all possible combinations of factors that affect the optimization parameter. The most significant factors affecting the value of the developed torque (braking) moment were: the pressure drop and the temperature of the working fluid in the hydraulic lines of the hydraulic loading device. (Conclusions) The mathematical model allows us to determine with high accuracy the developed torque (braking) of the volumetric hydraulic drive.

Construction of passport static characteristics of the system using the interval approach

An approach to constructing direct and inverse static characteristics of the system is considered proceeding from the experimental data obtained using a traditional «black box» concept, whereby the results of the experiment containing measured input and output values are used which, in practice, are determined with errors. The presence of various sources and trigger interference factors leads to a significant distortion of the error estimates and the formation of an inadequate conversion response. A crucially different character of the impact on the measurement results can be observed taking into account the type and sources of errors upon formation of the static characteristics and developing a model of noise arising upon measurements in actual operation conditions and during the experiment. However, analysis of the existing method based on a statistical approach and used for determining the error of the object model revealed a number of shortcomings: focusing mainly on the random components of errors and complexity of taking into account non-statistical information, including a priori information about the systematic errors, round-off and sampling errors, and the errors of the measurement system used in upon developing a model of the system. The statistical approach does not provide theoretical substantiation of the solution of the problem of the inverse static characteristic of the system which is relevant for many applications. Moreover, the fact that the interference models under experimental and real operation conditions have different sources and are generated by different factors is ignored in the framework of the statistical approach, which can lead to a significant distortion of the error estimates of the system model and to the formation of an inadequate direct and inverse transformation characteristics. We propose to eliminate the aforementioned shortcomings using the interval approach. The problem of developing a new method for constructing the passport characteristic of the inverse static function of a system is solved on the basis of a two-stage procedure of the experimental design. A new method based on the interval approach has been developed in which the problem of constructing the inverse static characteristic of the system and the problem of determining the error of the measurement system are solved separately using different types of active experiment. The boundaries of the uncertainty interval of the measurement system are defined in the general case by spline-functions. A concept of the operating range of the system, which does not coincide with the range of changes of the measured value in an active experiment, is introduced. It is shown that outside the operating range the system error sharply increases and becomes asymmetric with respect to the inverse static characteristic.