MATHEMATICAL MODELING OF THE TECHNOLOGICAL PROCESS OF IMPROVING THE QUALITY OF POLYMERIC PRODUCTS OF MACHINE-BUILDING PURPOSES

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Introduction. Intensive development of modern sci- ence and technology demands the development and use of structural materials with special physic-mechanical char- acteristics. In this regard polymeric composite materials got great prospects for application in various branches of mechanical engineering and the rocket industry. With the increasing volume of polymeric products and increased performance requirements, the problems of materials quality improvement and reduction of manufacture costs have become relevant [1-6]. The main means in the solu- tion of the problem of manufactured products quality im- provement at various stages of life cycle is the use of non- destructive monitoring. One of such ways of monitoring is the high-frequency method (HF) of diagnosing imple- mented in industrial conditions on the basis of the UZP 2500 device [7-9]. This method is based on local defects identification mainly in products from polymeric, com- posites by the impact of an external electric field on the studied object, and further amplitude intensity synthesis of the complex of output, operated diagnostics parameters emergence. HF method allows to carry out high precision monitoring without mechanical influence, to define a type of defect, to restore physic-mechanical properties of a product [10]. Main problems. Research tasks statement. It should be noted that in modern production process a wide range of products and details of complex spatial shape applied in mechanical engineering and space industry creates par- ticular difficulties in the course of diagnosing using the high-frequency radiation method. The conducted theoreti- cal and pilot research results presented in works [11-15] confirm that for the process of complex configuration products monitoring it is necessary to select the mode of high-frequency diagnosing each time, to develop rather complex design devices (electrodes) and the equipment. Taking into account the complexity of industrial equip- ment manufacture, difficulty of equipment setup and also the selection of diagnostics modes for polymer products, it is obvious that the development and practical use of mathematical model operation of monitoring process, and implementation of the software on the basis of the de- signed model will allow to reduce considerably the cost of preparation for diagnostic operations and also to increase the quality of monitoring of polymeric composites prod- ucts on an industrial scale. Proceeding from the aforesaid during the research the primary tasks of the development and research of a mathematical model of high-frequency radiation impact on a polymeric product, including with “metallic inclusion” defect type and also the development of the accompanying software for testing, setting and ensuring diagnostic devices operation were set. Mathematical model of high-frequency radiation impact on a polymeric product. The mathematical model in the research is necessary for calculation of key parameters of the monitoring process of complex con- figuration machine-building products both in the manual, and in the automated mode. The calculation was carried out for a sample from 8 mm thickness polyamide-610 material with 30 mm diameter with "metallic inclusion" defect type. The defect of 1 mm thickness, of 4 mm2 is simulated between two polyamide plates of 3 mm and 4 mm thickness. When designing a mathematical model, the studied sample was considered as a set of three areas: the first area - the polyamide without a defect, the second area - the polyamide located over the defect, the third area - the polyamide located under the defect. The scheme for the studied sample is shown in fig. 1. Fig. 1. Scheme of the simulated sample with a defect type “metallic inclusion” Рис. 1. Схема смоделированного образца с дефектом типа «металлическое включение» To perform the given tasks there was also a need for the development of high-frequency impact technological scheme shown in fig. 2, which is a set of electrodes, a variable quantity of heat insulators and a product to be heated with the defect. Taking into account the presented high-frequency impact technological scheme, the calculation of dynamics of warming up in each area and temperature distribution in a multilayered plate are described by a set of equations of transient heat conduction [1; 5]. The area warming up without a metal plate is described by a set of equations (1), and the dynamics of increase in temperature of areas with metal inclusion is characterized by a set of equations (2). Boundary conditions on external borders of electrodes correspond: ï ¶τ ì¶T1 = ï cp1 λ1 (T1 )ρ1 ¶2T × 1 ¶x2 ; -λ ¶Ti ¶x x= xi = αi ΔTi ; (i = 5, 1, 12, 6), (3) ¶T λ ¶2T ï ï 2 = 2 × 2 ; where αi - heat exchange coefficient; ΔТi - difference of surface temperatures of a body and surrounding medium. ï ¶τ ï í ï¶T3 = ¶τ cp2 (T2 )ρ2 λ3 c (T )ρ ¶x2 × ¶2T3 + ¶x2 p3 (τ) ; c (T )ρ (1) Referring to rather low heating temperatures of the studied polymeric products at implementation of diagnosing and high dynamics of a warming up, heat exchange with its surrounding medium when monitoring the first ï ï¶T p3 3 3 p3 3 3 λ ¶2T and single products can be considered negligibly small: ï 4 = 4 × 4 ; ¶T ï ¶τ cp (T4 )ρ4 ¶x2 -λ i x= x = 0 (4) ï 4 ¶x i ïï ¶τ ï¶T5 = î ì¶Ti cp5 λ5 (T5 )ρ5 λi ¶2T × 5 ¶x2 ; ¶2Ti On borders of layers heat fluxes and temperatures are equal: Ti = Ti+1 ; (j = 1…4; 6…11); (5) ï 1 = 1 × 1 ; ï ¶τ ï ï¶Ti cpi1 (Ti1 )ρi1 λi ¶x2 ¶2Ti λ ¶Tj j ¶x = λ j +1 ¶Tj +1 ¶x ; (j = 2…4; 6…11); (6) ï 2 = 2 × 2 ; The specific power of P is other than zero only for ï ¶τ cp (Ti )ρi ¶x2 l polymer í ï¶Ti i2 2 2 λi ¶2Ti pi (τ) (2) Pl ¹ 0; (l = 3, 8, 10); (7) ï 3 = 3 × 3 + 3 ; i3 3 3 3 3 3 ï ¶τ cp (мi )ρi ¶x2 cp (Ti )ρi The carried out research calculations with the created ï¶T λ ¶2T mathematical model use showed that the sample is heated ï i4 = i4 × i4 ; î i4 4 4 ï ¶τ cp (Ti )ρi ¶x2 where for the second area i1 = 12; i2 = 11; i3 = 10; i4 = 9; for the third area i1 = 6; i2 = 7; i3 = 8; i4 = 9; Тj - local temperature of a layer, С°; x - current thickness of a layer, mm; λj - thermal conductivity, W / (m · K); срj - specific heat, KJ / (kg · K); ρj - material density, Pa; pj - specific power of an internal source of heat of W/m3; j - layer number: 1, 5, 6, 12 - electrodes; 2, 4, 7, 11 - insulators; 3, 8, 10 - polymer; 9 - metal inclusion. 0 Fig. 2. Technological scheme of high-frequency impact on a product with a defect of the type “metallic inclusion”: 1, 5, 6, 12 - electrodes; 2, 4, 7, 11 - insulators; 3, 8, 10 - polymer; 9 - metal inclusion Рис. 2. Технологическая схема высокочастотного воздействия на изделие с дефектом типа «металлическое включение»: 1, 5, 6, 12 - электроды; 2, 4, 7, 11 - изоляторы; 3, 8, 10 - полимер; 9 - металлическое включение non-uniformly, the required temperature on areas with metal inclusion is reached in 25 seconds. The analysis of the received results allows to draw a conclusion on a pos- sibility of mathematical model use for calculation of nec- essary diagnosing time which excess will lead to the ma- terial melting and, therefore, to product rejection. The following necessary research phase was finding of the instantaneous power consumption dependence on the heating time Pмпм(t) in each area which chart is presented in fig. 3. The received dependences of specific power on temperature Pмпм1(t), Pмпм2(t), Pмпм3(t) were approximated in the form of a polynomial function and used when deter- mining the energy spent monitoring a sample with the defect type “metallic inclusion”: 25 Эпотреб = ò Рмпм (t) × dt . (8) 0 Compliance check of the model with actual diagnos- ing processes showed that when monitoring samples of brand 610 polyamide of 8 mm thickness and with a 15 mm radius, the time is 111 sec., and the heating time of the same sizes sample with the defect type “metallic inclusion” of 4 mm3 in 25 seconds, the deviations on heat- ing time from the pilot research make no more than 2 % that confirms the mathematical model correctness. At the same time the power consumption of a standard sample is 1.3 W · h, and a sample with the defect - 0.35 W · h [1; 9; 12; 13]. On the basis of the given research it is possible to draw a conclusion that the more the amount of metal in- clusion in a polymer, the less energy and time is spent on monitoring object heating. The analysis of the results received by means of the created mathematical model allowed to reveal the additional controlled parameter having informational content - the consumed energy. Be- sides, the obtained data confirm the pilot studies pre- sented in works [1; 8]. For the use of a high-frequency control method and diagnosing process control it is necessary to provide a fixed level of power impact when monitoring different materials and products of various configuration. The pre- sented mathematical model allows not only to determine the voltage necessary for specific power control preserva- tion but also provides the opportunity for electrothermal equipment input data determination at automated control of high-frequency diagnosing process, differing in the established by mathematical dependence of operating voltage supply on the form and sizes of the controlled product [1; 4]. The Aleo-Diagnost software package. Calculations of productive parameters diagnosing were carried out for the particular sample of a specific form and size. The instantaneous power consumption directly depends on two components - the volume of a monitoring object and specific power. For implementation of the developed mathematical model of various forms and sizes products monitoring it is necessary to keep a condition of the de- pendence of the instantaneous power consumption invari- ance on time. For adaptation of calculations of diagnosing process necessary parameters with the high-frequency radiation method, the Aleo-Diagnost software was devel- oped and registered [1]. The package interface imple- mented in the programming language С++ is shown in fig. 4. Fig. 3. The dependence of the instantaneous power consumption in the areas of polyamide 610 from the heating time Рис. 3. Зависимость мгновенной потребляемой мощности на участках полиамида 610 от времени разогрева Fig. 4. Interface of the Aleo-Diagnost software package Рис. 4. Интерфейс программного комплекса Aleo-Diagnost The executable file contains the main libraries, forms of input and output. Calculation is conducted in 4 stages. The process of calculations with recommendations and explanations on the corresponding stages is displayed in “Information window”. When calculating the heating time the Aleo-HFH software package version 2.0 [5; 15] is used. After com- pletion of calculations of products from polyamide mate- rials heating time the software package displays key pa- rameters of calculation for descriptive reasons in the form of tables with the intermediate values of specific and in- stantaneous power consumption depending on tempera- ture and the time of heating at high-frequency radiation impact. The software package has an opportunity to send calculation results to the text file for their convenient transfer in the approximation module. For reduction of the obtained tabular data to a polynomial form a convenient to use program-approximator is installed in the package allowing to obtain coefficients of polynomical function of the instantaneous power consumption dependence on time for carrying out further research and calculations. There is a module of the consumed energy calculation on a heating sample which results are displayed in the bottom right corner of the program window in the program. Besides, the Aleo-Diagnost allows to solve a number of such prac- tical problems as calculation of operating voltage depend- ing on geometrical parameters of a product for process realization and also the calculation of the energy con- sumed on diagnostics of a product for the given time pe- riod. The solution of these problems was a necessary stage as the consumed energy is the key output parameter of diagnosing. This parameter is assumed as the basis at non-destructive monitoring process organization in the automated mode. Conclusion. In conclusion, it should be noted once again that the created mathematical model of high- frequency radiation impact on a polymeric product and the software on the basis of the designed model practical use significantly reduces the cost not only of diagnosing operations preparation, but also of the monitoring process. The results of the presented research allow to reduce the time and improve the quality of products non-destructive monitoring in the conditions of production, operation and repair work. Besides, during the research the sensitivity of diagnosing was defined. The developed method allows to reveal the “metallic inclusion” defects type of total volume 0.0017 % that allows to use this method commer- cially in machine-building and rocket-space industries. At the moment the software package on the basis of the created mathematical model is used in the scientific, edu- cational and production purposes.

Sobre autores

A. Larchenko

Irkutsk State Transport University

Email: Larchenkoa@inbox.ru
15, Chernyshevsky St., Irkutsk, 664074, Russian Federation

N. Filippenko

Irkutsk State Transport University

15, Chernyshevsky St., Irkutsk, 664074, Russian Federation

A. Livshits

Irkutsk State Transport University

15, Chernyshevsky St., Irkutsk, 664074, Russian Federation

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