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61 ISSN 1392 - 1207. MECHANIKA. 2008. Nr.2(70) Mathematical modeling of deep drawing force with double reduction of wall thickness I. Karabegović*, E. Husak** *University of Bihać, Dr. Irfana Ljubijankića bb, 77 000 Bihać, Bosnia and Herzegovina, E-mail: tfb@bih.net.ba **University of Bihać, Dr. Irfana Ljubijankića bb, 77 000 Bihać, Bosnia and Herzegovina, E-mail: erminhusak@yahoo.com 1. Introduction tion, revitalization and controlling of processes and sys- tems. Experimental research is used for inspection, cor- Regarding that, the main aim of process and sys- rection and verification of numerical results and at stochas- tem modeling is the construction of mathematical models. tic modeling which describes the most realistically machin- The main aim of experimental research is to get exact, ap- ing processes and systems. Technological construction and proximately correct data which will serve as relationships, projecting of the modern machining processes demand to necessary for mathematical model. Mathematical model is analyze all technical and technological parameters of the necessary to start optimization of process. Regarding that, process and to apply scientific methods for modeling and the main aim of modeling and optimization of machining defining of optimal conditions of machining processes and process and systems is cheaper and higher quality produc- systems. tion [1]. The main goal of modeling and optimization of Mathematical model of deep drawing force with machining process is to increase productivity, economy, double reduction of wall thickness is presented in this pa- total quality of the product or partial segments (machined per. surfaces, tool durability, etc.) also to decrease material costs, energy, machining time, and machining costs per 2. Election of significant factors one piece of the product. Using theoretical analytical models it is hard to For significant factors electing it is used criteria define precisely parameters of machining processes like: that elected factors are not related and not depends from wasting tool, optimal geometric shape, deformation ap- outside factors. Outside factors are unelected factors which pearance at tool or press die, limitation level of deforma- also belong to the process of deep drawing but are not sig- tion, tribologic (friction) processes, tool loading. In each of nificant for it. Significant factors values have to be appli- the mentioned machining processes a lot of significant cable for the measuring process. For the process of deep factors and theirs interactions were applied. Therefore the drawing are elected three significant factors deformation φ, application of experiments and analysis of their results is diameter ratio d1/d2 and friction coefficient μ. unchangeable in developing the new and improve existing Variation of the factors limits are shown in Ta- machining processes and systems. ble 1 [1 - 4]. The experiment was conducted with the varia- The main goal of modeling is to define mathe- tions factors of two levels. The experiment was repeated matical model which is necessary in optimization, simula- three times for each sample. Table 1 Significant factors of machining EXPERIMENT LEVEL Significant factors Deformation φ Diameter ratio d1/d2 Friction coefficient μ Maximal 1.22 1.055 0.20 Minimal 0.95 1.017 0.10 3. Equipment for researching d0=26.4 mm (Fig. 3). Force of this process is measured on the same hydraulic testing machine. In experiment planning it was decided to use three Force measuring equipment in this machining independent changeable input values with two levels which process are presented in Fig. 4 [6]: 1 - hidraulic testing make eight samples. For each sample measuring was re- machine, 2 - signal acquisition unit, 3 - monitor for presen- peated three times which makes twenty four measuring tation of data and 4 - unit for data processing. times. This experiment of deep drawing process with 4. Experiment results double reduction of wall thickness is executed on hydraulic testing machine Amsler 300 kN (Fig. 1). Sensors for meas- After decision which significant factors would be uring friction coefficient (Fig. 2) were installed [5, 6]. The used in this process, the number of variation levels and material of raw specimen is CuZn28 which before deep decision about the number of repeating for each sample, drawing process was prepared by heating, washing and start phase of the experiment process which are presented lubricating. Figs. 1 - 4. Outer diameter of the raw specimen is 62 a b Fig. 1 Testing equipment: a - hydraulic testing machine, b - specimen grips Plug d0=Ø 26.4 12º Ø 22 d1 Die 12º α Sensor for measuring Piece normal contact tension pn Sensor for measuring d2 tangential contact tension τn Fig. 2 Process of deep drawing with the system for measuring of contact tensions pn and τn d0=Ø26.4 1 Ø22 2 3 h0=40 4 Fig. 3 Raw specimen Fig. 4 Measuring system of force 63 Table 2 Experiment results Significant factors data Results after measuring of deep Arith- Square Degree of Varia- No of sample drawing force metic sum freedom tion value ∑ ( y ji − y j ) n 2 Fj1 , kN F j 2 , kN F j 3 , kN yj = F fj S2 φ d1/d2 μ i =1 j 1. 0.95 1.017 0.10 34 33 34.2 33.7 0.83 2 0.415 2. 1.22 1.017 0.10 46 43 45.2 44.7 4.83 2 2.415 3. 0.95 1.055 0.10 28.1 27.4 27.9 27.8 0.26 2 0.13 4. 1.22 1.055 0.10 42.6 43.1 42.3 42.7 0.33 2 0.165 5. 0.95 1.017 0.20 34.8 34.8 36.1 35.3 1.14 2 0.57 6. 1.22 1.017 0.20 47.1 47.2 46.1 46.8 0.74 2 0.37 7. 0.95 1.055 0.20 29.5 28.1 29.8 29.2 1.66 2 0.83 8. 1.22 1.055 0.20 48 48 47.5 47.8 0.17 2 0.085 Changing the value of all or just some input pa- 5. Processing of experimental results rameters and repeating the experiment the matrix of output result values is completed. Repeating of experiment in After inspection of the homogeneity of experi- same sample with the same values of significant factors mental results, the next step would be the calculation of was founded three different results. Final experimental regression coefficients. Regression coefficients are calcu- results are given in Table 2. For further procedure it is im- lated using the following formulas (models) [1, 7] portant to find arithmetical mean value of three repeated experiments. On these bases it is possible to finish inspec- 1 N tion of homogeneity of result dispersion. bi = N ∑ X ij y j , i = 1, 2, ...., k (4) Inspection of homogeneity of dispersion for j =1 sorted level of reliability (P=0.95) has been done by Cohran’s criteria [1, 7] and 1 N max S 2 bim = ∑ X ij X mj y j (5) Kh = N j ( ≤ Kt f j , N ) (1) N j =1 ∑ S 2j where X ij is the value of X i in j-th experiment presented j =1 in Table 3, y j is measured value in j-th experiment and where K t is value by Cohran’s criteria. N is the number of experiments (samples). Dispersion (variation) is calculated by the next For correct structuring of mathematical model of model (2) (Table 2) any process in machining it is very important to choose a correct approximate mathematical model. If the first is not S2 = j 1 3 n j − 1 i =1 ( ∑ y ji − y j ) 2 j = 1, 2 ,...,8 (2) correct it is necessary to repeat all the steps for new ap- proximate model construction. In this paper the function of deep drawing force f j = n j −1 (3) correlates with the mathematical model presented in for- mula (6) where is k=3 where f j is degree of freedom, n j = 3 is the number of Y = b0 + b1 X 1 + b2 X 2 + b3 X 3 + b12 X 1 X 2 + repeating for one sample By Cohran’s criteria: + b23 X 2 X 3 + b13 X 1 X 3 + b123 X 1 X 2 X 3 (6) - after calculation process with the results in Table 2 After calculation of regression coefficients results K h = 0.4849 are presented below - using data from Table 1 founded in reference [1] it is b0 = 38.5, b1 = 7 , b2 = −1.625, b3 = 1.275 obtained b12 = 1.375, b13 = 0.525, b23 = 0.35, b123 = 0.4 ( ) K t f j , N = K t (2,8) The next step is testing of significances of regres- sion coefficients. Each coefficient should be tested for im- K t = 0.516 portance in our mathematical model of deep drawing force (function) (6). In this calculation were use coded values. Regarding that K h = 0.4849 < K t =0.516, process Evaluation of significances of regression coeffi- of getting mathematical model to be continued. Cohran’s cients is going to be done separately one by one. Regres- criteria confirms that homogeneity of experimental results sion coefficients which are not significant should be re- dispersion satisfy. moved from the mathematical model. It is not necessary to do value correction for significant regression coefficients. 64 Table 3 Code value of process No of Variables of physical process Code variable of process (y ) 2 sample y jE yR j E j − yR j φ d1/d2 μ X1 X2 X3 1. 0.95 1.017 0.10 -1 -1 -1 33.7 33.7 0 2. 1.22 1.017 0.10 +1 -1 -1 44.7 44.7 0 3. 0.95 1.055 0.10 -1 +1 -1 27.8 27.8 0 4. 1.22 1.055 0.10 +1 +1 -1 42.7 42.7 0 5. 0.95 1.017 0.20 -1 -1 +1 35.3 35.3 0 6. 1.22 1.017 0.20 +1 -1 +1 46.8 46.8 0 7. 0.95 1.055 0.20 -1 +1 +1 29.2 29.2 0 8. 1.22 1.055 0.20 +1 +1 +1 47.8 47.8 0 Two known criteria for the evaluation of signifi- b3 = 1.275 > 0.28183 significant cances of regression coefficients can be used: t – Student’s criteria or Fisher’s criteria and correlation [1, 7] existing b12 = 1.375 > 0.28183 significant between them b13 = 0.525 > 0.28183 significant b23 = 0.35 > 0.28183 significant F (1, f ) = t 2 ( f ) (7) b123 = 0.4 > 0.28183 significant For the evaluation of significances of regression coefficients of model bi , it would be used t’s criteria or where Student’s test. t ( f 1 ,ε ) = t ( f E ,α ) = t (16,0.05) = 1.75 (13) The formula for significance testing of regres- sions coefficients bi with t - criteria is After significance testing of regression coeffi- cients in the mathematic model, the conclusion is that all regression coefficients are significant. The next step is to bi bi N n return original values for all regression coefficients in tri = = ≥ tt (8) Sbi Sy ( f y ,α ) model (6). After this model will have the structure pre- sented in formula for i = 0, 1, 2, ...., k or Y = 38.5 + 7 X 1 − 1.625 X 2 + 1.275 X 3 + Sy bi ≥ Δbi = ±tt Sbi = tt (9) + 1.375 X 1 X 2 + 0.525 X 1 X 3 + 0.35 X 2 X 3 + ( f y ,α ) ( f y ,α ) N n + 0.4 X 1 X 2 X 3 (14) for i = 0, 1, 2, ..., k The variation of experimental error can be de- Model Eq. (14) presents deep drawing force as the scribed by the model function of significant parameters but in the code value. Used those coded values the values of deep drawing force are calculated. They are presented in Table 3. ∑ ∑ (y ji − y j ) N n 2 Sy = 2 j =1 i =1 (10) 6. Structuring of final mathematical model fy or After calculating and writing down the results of 2 deep drawing force, the next step is to test this mathemati- S y Sbi = 2 , i = 0, 1, 2, ..., k (11) cal model (14) for adequacy. For this purpose, Fisher’s N n criteria presented below are used where Fa < Ft (15) ( ) N f y = ∑ n j − 1 = N ( n − 1) (12) where Ft is value founded in Table 3 in reference [1] j =1 where is determinate by level of significance and f y is total number of degree of freedom, n j is the p (Fa > Ft ) = α = 0.05, or (1 − α ) = 0.95 , it is 95% reli- number of experiment repeating in j-th line of matrix, ability. Where is when is the same repeating number n = n j . 2 S b ⋅ t = 0.28183 , in the condition for being significant. The Fa = Sa 2 ( ≤ Ft ( f 1 , f 2 ) = Ft f a , f y ) (16) Sy coefficients which satisfy this condition bi > S b ⋅ t , are: b0 = 38.5 > 0.28183 significant for Sa > S y 2 2 b1 = 7 > 0.28183 significant 2 Sy b2 = − 1.625 > 0.28183 significant Fa = 2 ( ≤ Ft ( f 1 , f 2 ) = Ft f a , f y ) (17) Sa 65 for S y > Sa . 2 2 F, kN The value of dispersion of adequacy is determined by the following model 55 d1/d2=1.017 50 µ = 0.20 ∑ n(y E − y R ) N 2 j j 45 d1/d2=1.017 j =1 Sa = 2 (18) 40 µ = 0.10 fa 35 d1/d2=1.055 30 µ = 0.20 where f a = N − k − 1 is the number of degree of freedom 25 which is related to the dispersion of adequacy. d1/d2=1.055 The corresponding mathematical relation is ob- 20 µ = 0.10 tained using the model (16) when S y > Sa 2 2 15 10 2 5 Sa Fa = 2 =0 0 Sy 0.7 0.8 0.9 1.0 1.1 1.2 1.3 φ Fig. 5 Theoretical correlation between deep drawings force From this it can be concluded that the mathemati- F, kN and deformation degree φ cal model 100 % describes finished experiment. That is F, kN Fa = 0 < Ft = 3 φ=1.22 φ=1.22 ∑ n(y ) N µ = 0.10 R 2 µ = 0.20 E j −y j is a part of model (19). Model (19) pre- 50 j =1 sents coefficient of multiple regression. Result of 45 ∑ n(y E − y R ) N 2 40 j j equals zero (Table 3). That shows coeffi- j =1 35 cient of multiple regression will equals 1. 30 φ=0.95 φ=0.95 When regression model correctly describes the 25 µ = 0.20 µ = 0.10 process R → 1 . Coefficient of multiple regression is de- 20 scribed by the model (19) 15 10 ∑ (y E − y R ) N 2 j =1 j j 5 R = 1− (19) 0 ∑ (y E − y E ) N 2 j 1 1.01 1.02 1.03 1.04 1.05 1.06 d1/d2 j =1 Fig. 6 Theoretical correlation between deep drawing force As the model (14) adequately describes mean F, kN and diameter ratio d1/d2 value of deep drawing force F in relation with significant parameters: φ, d1/d2 and μ, then the next proceeding is the F, kN d1/d2=1.017 d1/d2=1.055 conception of final mathematical model. In order to get φ = 0.20 φ = 0.20 mathematical model with real coefficients it is necessary to finish decoding of the model using transformation equation 50 [1]. 45 Mathematical model for deep drawing the force 40 where force is related with significant parameters is de- scribed in the next model 35 30 F = 213.4 − 30.46ϕ − 213.58 d1 d 2 + 3049.14 μ + 25 d1/d2=1.017 d1/d2=1.055 + 68ϕ ( d1 d 2 ) − 3149.88ϕ μ − 2999.9 μ ( d1 d 2 ) + 20 φ = 0.10 φ = 0.10 + 3115.69ϕ μ ( d1 / d 2 ) (20) 15 10 The correlation between the force and parameters 5 can be shown graphically Fig. 5, 6 and 7. Diagrams (Fig. 5, 0 6 and 7) present the correlation of deep drawing force with 0 0.05 0.10 0.15 0.20 0.25 μ double reduction of wall thickness with: deformation de- Fig. 7 Theoretical correlation between deep drawings force gree φ, diameter ratio of press die d1/d2 and friction coeffi- F, kN and friction coefficient μ cient μ. 66 7. Conclusions tinis modeliavimas įgalina teisingai analitiškai aprašyti deformacijos procesą. Turint analitiškai aprašytą deforma- Experiments of deep drawing process with double cijos procesą, galima skaičiuoti giliojo ištempimo jėgą ir reduction in press die of wall thickness were performed. optimizuoti visą deformacijos procesą bei ištempimo jėgą. The following conclusions were made: Straipsnyje aprašoma planuojamo eksperimento eiga, pra- - final mathematical model is adequate for describ- dedant svarbiausių parametrų įvertinimu, sienelės storio ing the process of deep drawing what is confirmed tolygumo kitimo, eksperimente naudojamos įrangos, gautų by Fisher’s test; rezultatų palyginimu su žinomais analiziniais modeliais bei - final mathematical model is absolutely correct for galutinio matematinio modelio giliojo ištempimo jėgai describing the process of deep drawing what is con- nustatyti sudarymu. firmed by multiple regression coefficient. Which is equal 1; - deep drawing force is in line function of independ- I. Karabegović, E. Husak ent input parameters: deformation degree φ, diame- ter ratio of press die d1/d2 and friction coefficient μ; MATHEMATICAL MODELING OF DEEP DRAWING - final mathematical model should be used for opti- FORCE WITH DOUBLE REDUCTION OF WALL mization of deep drawing process. THICKNESS The main goal of the optimization of this mathe- Summary matical model is to get minimal deep drawing force with the reduction of wall thickness with optimal input parame- The main goal of this paper is to get a mathema- ters (factors). The importance of this force minimizing tical model for the process of deep drawing force with the would be multiple: from minimizing energy consumption reduction of wall thickness. Mathematical model should to decreasing of intensity of wasting on press die guides have a correct analytic description of this deformation and other pieces of press die. process and along with that a possibility to calculative deep drawing force and after that a possibility of optimization of References the whole deformation process and force. In this paper experiment planning is presented, from defining significant 1. Jurković M. Matematičko modeliranje inženjerskih factors (parameters), levels of variation, equipment for procesa i sistema.-Mašinski fakultet Bihać, 1999, p.43- experiment performing, processing of receiving results 127. with known analytic models and on the end structuring 2. Karabegović I. Jurković M. Mustafić E. Matematič- final mathematical model for deep drawing force. ko modeliranje sile izvlačenja žice. -5th International Scientific Conferenece RIM 2005, p.205-210. 3. Karabegović E. Karabegović, I. Jurković, M. Mathe- I. Karabegović, E. Husak matical modelling of hard surfacing influence on tool stability. -Mechanika. -Kaunas: Technologija, 2003, МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ СИЛЫ Nr.1(39), p.61-66. ГЛУБОКОЙ РАСТЯЖКИ, УМЕНЬШАЮЩЕЙ 4. Karabegović I. Jurković M. Bejdić M. Mathematical ТОЛЩИНУ СТЕНКИ В ДВА РАЗА modelling of the main cutting force at turning, -Mecha- nika. -Kaunas: Technologija, 2004, Nr.3(47), p.59-63. Рeзюме 5. Musafia, B. Obrada metala plastičnom deformacijom, -Zavod za udžbenike i nastavna sredstva. -Svjetlost. Статья посвящена математическому модели- -Sarajevo, 1988, p.493-500. рованию силы пластической деформации уменьшаю- 6. Zaimović Uzunović, N. Mjerna tehnika.-Zenica: щей толщину стенки образца в два раза при глубокой Univerzitet u Sarajevu, Mašinski fakultet u Zenici, вытяжке. Математическое моделирование использова- 1997, p.15-74. но с целью аналитического описания процесса дефор- 7. Silvio E. Planiranje eksperimenata. I izdanje.-Sarajevo: мации образца. При аналитической оценке процесса Zavod za izdavanje udžbenika, 1979, p.134-136. деформации возможен подсчет силы глубокой вытяж- ки, оптимизации ее и всего процесса вытяжки. Описан ход планируемого эксперимента, начиная с оценки I. Karabegović, E. Husak основных параметров, изменения равномерности ши- рины стенки, в эксперименте используемого оборудо- GILIOJO IŠTEMPIMO JĖGOS, PERPUS вания. Полученные результаты сравнены с результата- SUMAŽINANČIOS SIENOS STORĮ, MATEMATINIS ми, полученными при использовании известных ана- MODELIAVIMAS литических моделей и в конечном счете создана мате- матическая модель для определения силы глубокой Reziumė вытяжки. Straipsnis skirtas plastinio deformavimo jėgos, Received January 21, 2008 susidarančios giliojo ištempimo metu ir perpus sumažinan- čios sienelės storį, matematiniam modeliavimui. Matema-

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