Robust methods for controlling casting processes and the quality of castings

The paper considers the ways of controlling the processes of producing good-quality castings by using robust methods based on the identification of objects under uncertainty, adaptive control methods, stabilization methods for automatic control systems and their mathematical description. The purpose of the paper is to practically elaborate on the task of creating robust technologies for casting based on the study of the stochastic dispersion models of the casting process parameters and to develop a method for stabilizing the casting processes and a method for stabilizing the quality parameters of the castings. The regular patterns of the stochastic dispersion of the casting process parameters were investigated based on the mechanical performance of the casting alloys, their chemical composition, the physical and mechanical properties of the molding mixtures, the dimensions of the castings, and the features of the casting production process such as pouring temperature, element loss during melting and mold hardness. The results of the conducted investigations have shown that to describe the stochastic dispersion of the casting process parameters, it is appropriate to use the Johnson system of distributions. A method for stabilizing the casting quality parameters was developed. It has been shown that it is more efficient to stabilize the strength and improve the dimensional accuracy of the casting simultaneously. With the existing production process, the dimensional accuracy of the casting corresponds to Grade 11 according to GOST 26645-85. With the increase of the dimensional accuracy grade, the metal content of the casting can be reduced by 16.3% after reaching Grade 10, by 20.4% after reaching Grade 9, and by 24.2% after reaching Grade 8.


Introduction
When developing the technological process of manufacturing machine parts, there is a need to set up a quality management system for products, the formation of which depends on various phenomena that significantly affect the products performance [1,2].
Among the processes that affect the quality of machine parts, an important place is taken by the processes related to the production of good-quality castings, which determine their physical and mechanical and performance properties.
Robust methods based on the identification of objects under uncertainty, adaptive control methods, stabilization methods for automatic control systems, mathematical description of the system can be used for this purpose.Currently, there are, mainly in Japanese industry, some striking examples of the use of the concept of robustness to solve practical quality management 1254 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1254/1/012007 2 problems, where the probability of failure of sophisticated products has been reduced from the rate of 1 to 5 failures per one hundred products to one failure per one million products [3,4].

Analysis of literature sources, purpose and objectives
The parameters of the casting processes and the quality of castings consist of two components, deterministic and stochastic.The deterministic component represents the nominal value of the technology parameters, while the stochastic component accounts for the variability of the casting production processes caused by variations in raw material properties, variance and drift in technology and equipment parameters.Variations in product size, structure and properties are the major challenges in creating a perfectly homogeneous industrial production.
Depending on the specific production conditions, both the deterministic and stochastic components can have a decisive effect on the quality of the casting.Therefore, when solving the problem of casting quality control, both components of the technology parameters must be considered quantitatively.However, the case is somewhat different in practice.The Generation 1 casting technologies used at an early stage of development of the casting technology involved a simplified conceptual model of the technology, whereby controlled process parameters were considered as purely deterministic values [5,6].
Subsequently, an effective solution to the problem of achieving a high level of quality of industrial products, including cast parts, has proved impossible without any quantitative consideration of the statistical component of the technology parameters.Therefore, the industry in developed countries started to use statistical methods in the field of quality control and quality management in the early 20th century.Up-to-date casting methods developed and improved in the 20th century based on a scientific and technological approach can be identified as Generation II technologies [7].
One of the most important conclusions of the statistical theory of quality management is that in order to solve the problem of creating high-quality industrial products, it is necessary not only to consider but also to manage the degree of stochasticity of technological parameters.Within this approach, the concept of new Generation 3 technologies was developed; these were called robust, and their main feature is resilience to random factors.
The stochasticity factor plays a major role in the formation of the consumer and performance properties of cast parts.A specific feature of the casting technology is that the quality parameters are affected by lots of variables, the number of which, according to experts, is estimated to be 2000 to 3000, while only a small number of them, about a few dozen, can be referred to as controllable variables.As a consequence, there is an inevitably wide variation in the quality parameters of the castings.Therefore, the development and practical use of robust technologies is one of the major scientific and practical challenges in the field of casting technology [8].
The analysis of the situation has shown that the existing pattern of technological research established in the field of casting technology should be supplemented by a fundamentally new stage related to the development of an apparatus, methods and working algorithms to address the issue of stabilizing the parameters of the casting process and the quality of castings.
Improving the quality and competitiveness of industrial products inevitably involves additional costs and an increase in their cost price.To find a rational solution to the problem of improving the quality of castings requires the development of a conceptual and mathematical model of the quality formation process, which considers all costs, risks and losses incurred by the designer, manufacturer and consumer of the products [9].
To date, neither general method nor mathematical and algorithmic apparatus for solving the problem of improving the quality and stabilizing the parameters of castings or cast parts has been developed.
The purpose of the research paper is to practically elaborate on the task of creating robust technologies for casting based on the study of the stochastic dispersion models of the casting process parameters, and to develop a method for stabilizing the casting technology, and a method for stabilizing the quality parameters of the castings.

Method
The primary objective in controlling the reliability of a cast part is to determine the function of distributing the load capacity of the casting.Therefore, the first step in addressing the issue of controlling the reliability is to define the general law of load capacity dispersion.
Engineering problems typically use uniform, normal, and exponential distribution laws.These laws can be used to solve training and elementary tasks, but they are totally inappropriate for solving practical tasks related to the management of the quality of castings [10].
The regular patterns of the stochastic dispersion of the casting process parameters were investigated as follows.A hundred empirical functions obtained by statistical sampling in foundries in automotive and tractor, agricultural, and transport mechanical engineering industries were studied [11,12].The mechanical performance of the casting alloys, their chemical composition, the physical and mechanical properties of the molding mixtures, the dimensions of the castings, and the features of the casting production process such as pouring temperature, element loss during melting and mold hardness were studied as parameters [13,14].
To examine the possibility of an adequate description of the distribution functions of the casting process parameters, two currently existing versatile Pearson and Johnson systems of distributions were reviewed.

Results and discussion
The results of the conducted investigations have shown that to describe the stochastic dispersion of the casting process parameters, it is appropriate to use the Johnson system of distributions.Its advantages can be summarized as follows: • the system consists of only three types of functions; • there are relatively simple methods for the approximate calculation of their parameters; • the scope of the system is universal.
The Johnson system of distributions includes the equation of three curves referred to as S l , S b , S u families in mathematical statistics.The functions of the S b family describe the distribution of a random variable whose limits are finite and best meet the nature of the dispersion of the casting technology parameters.
A meaningful analysis of the problem of investigating the regular patterns of random fluctuations of the casting processes was conducted as follows.All parameters of the casting processes cannot by their nature take negative and infinitely large values; therefore, they necessarily have an upper and lower limit.
In view of this, it is necessary to impose a requirement on the functions describing the dispersion of the casting parameters, by which the nature of the distribution function used must be consistent with the nature of the parameter being described.It therefore follows that all functions having at least one infinite limit of variation should be discarded to describe the regular patterns of any parameter of the casting processes and systems.
After filtering all known distribution functions through this requirement, the only remnant of these is the Johnson S b distribution.It alone has two properties at once: meaningful and mathematical adequacy to solve both technological and problems related to the reliability of cast parts.Thus, the Johnson S b distribution should become the main function when describing the dispersion patterns of the quality parameters of castings and supersede all others, including the normal distribution law.
One of the key components of the problem of improving the quality of castings is the stabilization of the process properties of molding mixtures [15].The existing approach to addressing this issue based on the data provided by online control of the properties of the molding mixture is ineffective because control over compactability, moisture contact and strength can be carried out no more often than every 30 to 40 minutes, while control over the content of active bentonite, losses during calcination, particle size distribution no more often than once per shift.In this case, deviation from the set parameters of the mixture preparation technology are detected after its application [16,17].
To stabilize the casting technology in the in-line production of cast parts for agricultural machinery, the following tasks were solved: • a method was developed, and production and statistical research was conducted to study the parameters of dispersion of the technological properties of molding mixtures; • mathematical models of the formation process of properties of molding mixtures were built; • practical recommendations for improving the technological process of production of castings were developed and implemented.
A specific feature of solving the problem of improving the stability of the process and diagnosing the casting defects is the requirement for increased reliability and accuracy of the production information used [18,19].For this purpose, a method of issuing data sheets for castings was developed, based on which industrial experiments were conducted in foundries.The point of the method of issuing data sheets for castings was reduced to perform sequentially the following operations.
The following parameters were monitored and recorded in the melting shop: the quantity, size and chemical composition of charge materials, the melting process parameters, the quantity and composition of deoxidizers and fluxes used, the temperature of the metal in the trough and in the ladle, the poring speed and temperature.In addition, samples were taken from each heat in the steel foundry to determine the gas saturation of the metal [20].
The following operations were performed on the molding and pouring conveyor during the mold-making process: • the numerical indexes in ascending order were stamped on the horizontal surface of the lower half of the mold; • each mold was sampled for rapid determination of its properties in the shops express laboratory; • the hardness of the mold surface was determined using a hardness tester at three points on the mold.
After completing the finishing operations the castings, each with an individual numerical index, were presented to the technical control department to be sorted.The information obtained was consolidated in a single table; the increased validity and reliability of the data obtained is due to the individual determination of all process variables for each casting.The data table was used to solve the tasks of stabilizing the technology parameters, determine the causes of casting defects and the degree of influence of individual variables on the quality and extent of casting rejects.
The solution to the problem of improving the stability of the technological properties of molding mixtures under production conditions was implemented as follows.As the object of research, a unified molding mixture used for the production of carbon steel castings was chosen with the following composition: sand -89 to 92%; refractory clay -7.5 to 9.5%; fuel oil -0.8 to 1.0%.The following requirements were set for the mixture: gas permeability of at least 70 GPU, ultimate strength when wet and water content in the ranges of 0.05 to 0.06 MPa and 4.6 to 5.0%, respectively.The results of 120 data obtained at the facility for the determination of the gas generation value of the mixture were processed, and samples for its determination were collected in the workshop at regular intervals for a month.
The data obtained show that the most important technological properties of the molding mixture, which determine the quality of the mold, are extremely unstable.For example, the extreme values of the dispersion range of gas permeability are 50 and 250 GPU, and those of the gas-generation value are 10 and 65 cm 3 /g, i.e. the limit values of the dispersion range differ from each other by 5 to 6.5 times.
It therefore follows that in order to solve the problem of dealing with gas-induced casting defects, it is necessary to reduce fluctuations in the gas permeability and gas-generation value of the molding mixture.The solution of this problem is particularly important for steel foundries, in view of the fact that the surface layers of the mold when in contact with the liquid steel are heated to a high temperature of about 1,100 to 1,200 • C [21,22].
The major factor reducing the quality of steel castings is that these are prone to having gasinduced casting defects such as blowholes, pinholes, gas and shrinkage porosity.Mathematical modeling methods have been used to quantitatively study the conditions for the formation of gas-and shrinkage-induced defects in castings and to develop measures to prevent these defects.
To build mathematical models of the formation of the process properties of the mixture, the regression models were used, and their parameters were determined based on the experimental data.The investigated dependencies, which were generally non-linear, were used to choose control actions [23,24].
Based on the results of the conducted research, a method for stabilizing the process of the preparation of a molding mixture was developed, which includes the following sequential procedures: • implementing 100% acceptance control over all raw materials for the preparation of the molding mixture and systematic monitoring of the gas-generation value of the mixture; • collecting the most reliable input information based on data sheets of molds and castings; • improving the accuracy of dosing water and fuel oil in the molding mixture, which are the main sources of gas generation in the mold; • determining the effect of the mixture preparation process parameters on the gas permeability of the molding mixtures and their correction; • providing a correlation and regression analysis of the dependence of gas permeability of a mixture on its composition and preparation conditions; • determining the rational limits of gas permeability and gas-generation value for the examined conditions; adjusting the mixture preparation technology.
According to the conducted investigations, it was found that the lower allowable gas permeability limit for a unified molding mixture in a steel foundry was set at 120 GPU, and the gas-generation upper limit value was set at 28 cm 3 /g.The accuracy of dosing of the source components in the preparation of clay suspension was improved; the ongoing control over its density was implemented; the accuracy of dosing when introducing fuel oil was improved, and its dosage was corrected, which was set equal to 0.85 ś 0.05% of the batch in the mixer.
A periodic monitoring of the actual clay content in the mixture was implemented, according to which the amount of clay suspension introduced in the mixture can be adjusted.As a result of this work, the stability of the properties of the molding mixture in the steel foundry has been considerably improved.Specifically, the number of cases of gas permeability below 120 GPU was reduced from 23 to 7%; the range of fluctuations of moisture content in the mixture was reduced from 4.1-5.5% to 4.4-5.2%.While the mean gas-generation value and its root mean square deviation in the initial state were 29.2 and 4.2 cm 3 /g, respectively, after implementing the stabilization measures, the values decreased to 26.2 and 2.7 cm 3 /g, respectively [25].
Implementing the method for stabilizing the process properties of molding mixtures made it possible to improve the quality of castings and reduce rejects with gas-induced defects.
The third task to be solved in this work is to develop a method for stabilizing the quality parameters of the castings.The method for solving this problem is implemented using the following algorithm.
(i) A mathematical model describing the dependence of the quality parameter Y on the independent technological variables x 1 , x 2 , , x n , is built, which can be represented as an equation: (ii) The mathematical expectation of the studied parameter is determined.If some of the variables x i or all of them are random, then the parameter Y is also a random variable, its mathematical expectation can be found by the formula: where m y , m x i are the expected values of the parameter y and the variable x i .(iii) Determined by the numerical and approximate-analytical method of parameter dispersion Y .For the numerical determination of this quantity, the simulation method can be used.The approximate-analytical method is based on the linearization of the function by expanding it in a Taylor series and preserving the first two terms.In this case, the variance of the parameter Y how the function of the variances of the independents is determined by the formula obtained on the basis of the linearization of the dependence (1): The index of the partial derivative means that its value is determined by x i = m x i .(iv) On the basis of processing the results of industrial experiments, the average values and dispersions of the studied technological variables are determined.(v) According to formula (3), the variance of the studied quality parameter is determined, the mean values found in the previous paragraph are taken as an estimate of the mathematical expectation of the variables.(vi) The contribution of each variable to the variance of parameter Y is analyzed, recommendations are developed and implemented to reduce the value of the parameter under study.
The described algorithm was used to stabilize the load capacity of the hinge body part cast of steel 45FL of a T-150K tractor.It was found that the cross-sectional area dispersion and the mechanical property dispersion account for 55% and 45%, respectively to the variation of the load capacity; therefore, the degree of dispersion of these two factors should be reduced.The following measures were developed to improve the geometric accuracy of the part: • reducing the shrinkage variation by reducing the allowable ranges of pouring temperature and carbon content; • reducing the deformation of the mold when pouring by improving the strength of the molding mixture and its packing density; • improving the accuracy of production and assembly of the model elements and centering pins; • minimizing the assembly clearances between the mold and core pins by optimizing the dimensional chains; • adjusting the dimensions of the core boxes in view of the deformation of the cores during transportation and drying.
By implementing the above technical measures, the dimensional accuracy of the hinge body casting was improved by 2 to 3 classes according to GOST 26645-85 and the coefficient of variation of the bearing cross-section was reduced from 0.0941 to 0.0402.
To determine whether it is possible to reduce the dispersion of the strength of steel 45FL, a mathematical model of its strength was developed as follows: where C, Si, M n is the percentage of carbon, silicon and manganese in cast steel.
Based on the processing of the results of the chemical analysis of production heats, it has been found that the dispersion of strength is 2,267.4,and 93% of its value is determined by the variation in the carbon content of steel, 5% by the variation in the silicon content, and 2% by the variation in the manganese content.Therefore, to reduce the dispersion of the mechanical properties of cast steel, it is sufficient to reduce the dispersion of its carbon content.The results of experimental heats showed that by improving the accuracy of dosing admixtures and by improving the express control over the carbon content in steel, the range of its variation can be reduced from 0.09 to 0.07%, and the RMS deviation from 0.015 to 0.01-0.012%.This reduction in the RMS variation made it possible to reduce the dispersion of the ultimate strength of steel and lower the strength dispersion from 2,267.4 to 1,154.7.By reducing the carbon content variation, the minimum tensile strength increased from 540 to 581 MPa, i.e. by 7.6%.
A study of the metal content of the hinge body part showed that while maintaining the guaranteed load capacity of this part equal to 31.2 tons by stabilizing the carbon content and increasing the lower strength value, the mass of the part could be reduced by 7.6%.It is more efficient to stabilize the strength and improve the dimensional accuracy of the casting simultaneously.With the existing production process, the dimensional accuracy of the casting corresponds to Grade 11 according to GOST 26645-85.With the increase of the dimensional accuracy grade, the metal content of the castings can be reduced by 16.3% after reaching Grade 10, by 20.4% after reaching Grade 9, and by 24.2% after reaching Grade 8.

Conclusions
1.In the production of good-quality parts, an important role is taken by the processes related to the production of castings, which determine their physical and mechanical and performance properties.2. Robust methods based on the identification of objects under uncertainty, adaptive control methods, stabilization methods for automatic control systems and mathematical description of the studied processes are promising to manage the quality of castings.3. The analysis of the situation has shown that the existing pattern of technological research established in the field of casting technology should be supplemented by a fundamentally new stage related to the development of an apparatus, methods and working algorithms to address the issue of stabilizing the parameters of the casting process and the quality of castings.4. Based on production and statistical research, the regular patterns of the dispersion of the casting process parameters and the quality of castings were studied.It has been shown that the Johnson Sb distribution is the most appropriate in terms of the meaningfulness and accuracy of describing the variation in the characteristics of technological processes and parameters of castings and cast parts.The obtained results can be used to improve the efficiency of problem solving in the casting quality management systems.5. To stabilize the casting technology in the in-line production of cast parts for agricultural machinery, the following tasks were solved: • a method was developed, and production and statistical research was conducted to study the parameters of dispersion of the technological properties of molding mixtures; • mathematical models of the formation process of properties of molding mixtures were built; • Practical recommendations for improving the technological process of production of castings were developed and implemented.6.A method for stabilizing the casting quality parameters was developed.It has been shown that it is more efficient to stabilize the strength and improve the dimensional accuracy of the casting simultaneously.With the existing casting production process, the dimensional accuracy of the casting corresponds to Grade 11 according to GOST 26645-85.With the increase of the dimensional accuracy grade, the metal content of the casting can be reduced by 16.3% after reaching Grade 10, by 20.4% after reaching Grade 9, and by 24.2% after reaching Grade 8.