Mathematic modelling and the deep drawing force simulation with the wall thickness thinning experiment application

The principal technological process parameters are, amongst others, made of: speed, deformation rate (dependent of weight and of matrix cone angle), the states of material in contact (area topography, tribological conditions, physical and chemical material characteristics), tool geometry and working part. The process projecting of deep-drawing with wall-thickness thinning for its natural processes and many influential parameters demands a detailed analysis of all influential parameters where the basic aim is the cheaper, more qualitative and profitable fabrication [1-3]. Hence, with the election of optimal values of the influential parameters the demanded product quality can be obtained by the mi-nimum energy consumption. Theoretically, with the application of the analytical models it is difficult to determine the optimal processing conditions whereas in every processing process more influential factors and their interactions are present. For the process known as the stochastic processes, when experiment performing, certain approximations which are the constituent part of analytical modelling are eliminated. With the application of the statistic methods and obtained experimental results, the more accurate data that determines the real process parameters are obtained. The mathematical modelling and optimization methods represent the basic methods in analysis of projecting process with the basic aim to innovate the existing processes, their modernisation and rising to a higher technological level. Modelling of the analysis process is the foundation of the optimization and defining of optimal analysis conditions which is impossible without preliminary definition of the reliable mathematical model. In this respect, as a subject of this experimental process research of deep-drawing with wall-thickness thinning is made. The main goal is to obtain the exact and accurate data to serve defining the mathematical model for the deep-drawing force. Thereafter, with the simulation of the obtained mathematical model, the determination of the dimension and the character of the drawing force, depending on input independent variables of the process parameters is included in experiment.


Abstract
Deep drawing with wall-thickness thinning has a broad application in product fabrication with the height higher than diameter and the base thickness higher than the product wall thickness.Some of them are given in Fig. 1.The principal technological process parameters are, amongst others, made of: speed, deformation rate (dependent of weight and of matrix cone angle), the states of material in contact (area topography, tribological conditions, physical and chemical material characteristics), tool geometry and working part.
The process projecting of deep-drawing with wall-thickness thinning for its natural processes and many influential parameters demands a detailed analysis of all influential parameters where the basic aim is the cheaper, more qualitative and profitable fabrication [1][2][3].Hence, with the election of optimal values of the influential parameters the demanded product quality can be obtained by the mi-nimum energy consumption.
Theoretically, with the application of the analytical models it is difficult to determine the optimal processing conditions whereas in every processing process more influential factors and their interactions are present.
For the process known as the stochastic processes, when experiment performing, certain approximations which are the constituent part of analytical modelling are eliminated.With the application of the statistic methods and obtained experimental results, the more accurate data that determines the real process parameters are obtained.
The mathematical modelling and optimization methods represent the basic methods in analysis of project-ing process with the basic aim to innovate the existing processes, their modernisation and rising to a higher technological level.
Modelling of the analysis process is the foundation of the optimization and defining of optimal analysis conditions which is impossible without preliminary definition of the reliable mathematical model.
In this respect, as a subject of this experimental process research of deep-drawing with wall-thickness thinning is made.The main goal is to obtain the exact and accurate data to serve defining the mathematical model for the deep-drawing force.Thereafter, with the simulation of the obtained mathematical model, the determination of the dimension and the character of the drawing force, depending on input independent variables of the process parameters is included in experiment.

Election of the input-output process parameters
A successful performance of the experiment demands the identification and limitation of certain influential process parameters to a concrete number.It refers to defining of only a certain input variables, as the independent variables x i , as an input into the process , and function defining of output process y i that are variable dependent dimensions.Such approach enables qualitative managing of the process and development towards modelling achievement [1,3].
The input-output process parameters included in the experiment are shown in Fig. 2 where are: ψ is deformation, s 1 is wall-thickness after drawing, mm, μ is abrasion coefficient and F i is drawing force, kN, while the other variables were observed as the constants.Fig. 2 Election of the input-output process parameters included into the experiment [1,3,4] During the experiment, the variable-independent process parameters, ψ, s

Description of the research equipment
With the experiment plan it was estimated the performance of entirely 12 tests, i.e., measuring of the drawing force F i , the test of which eight tests were performed with different influential parameter values and four with the medium values Table 2.
The scheme of the deep-drawing process with wall-thickness thinning is shown in Fig. 3.The working parts of dimensions D 0 , s 0, and h 0 are set up in the tool matrix, then operating of the lifting piece the working part draws through the matrix where the complete part of the new dimensions is obtained: The structure of the research equipment utilized for the experiment is shown in Fig. 4. The experiment performance consists of: on the hydraulic strainer ¨1¨, on the tool place, a specially produced tool for deep-drawing with wall-thickness thinning ¨2¨ is fitted, then sensors for drawing force measurement and contact constrains are connected with the connector cables (which are placed o the tool) with the measuring apparatus ¨3¨.After that, the previously prepared working parts are fitted in the tool, and the process of drawing is preformed according to the defined experiment plan (number of tests, working part dimension, greasing manner, etc.).

Homogeneity evaluation of experimental results
According to already defined input and output process variables (Fig. 2), as well as their values variations (Table 1.), 12 tests are performed with the repetition only in the central dot of the orthogonal plan (n 0 = 4) [1].The test repetition is necessary in order to perform dispersion analyses of the experimental results (experiment homogeneity, experiment error estimation, model adequacy, etc.).The obtained experimental results for the drawing force F i, with physical and coded values of the selected influential parameters x i are given in the complete experiment matrix plan, Table 2.
The elected mathematical model for the drawing force modelling is linear mathematical model, and in the coded form along with where X i are the influential parameters in coded form, b i are the regression model coefficients.After the performed experiment it is necessary to evaluate the single generic dispersions, i.e. to evaluate the experiment homogeneity in order to determine the difference of the obtained numerical values [1,4].In this regard, Cohran's criteria for dispersion homogeneity evaluation is applied (specification in the literature [1]) with a form With the measuring repetition performed was only in dots 9, 10, 11, and 12 tests, and after that obtained values for the variables 2 j S , i, According to the form (3), Cohran's criteria condition given in the form (2) is achieved.The experimental result dispersion is homogenous and further modelling with data obtained by the experiment can be carried out.

Processing of the experiment results
The mathematical model of the process is given in the form (1), in order to be accepted, it is necessary to calculate and evaluate the regression coefficient signification of the model (b i ) and to examine the model adequacy.For the conditions of test repetitions in the central dot of the orthogonal plan, the regression coefficient values of the model are obtained by applying the following forms where n 0 is number of repeated experiments in the central dot, y 0j is experiment results in the central plan dot, y j is experiment results.According to this, applying the forms (4), ( 5), ( 6) and a Table 2, the values of regression coefficient b i are: .In addition to the obtained coefficient values, it is necessary to determine their signification in function model in the form (1), i. e. their single influence on model accuracy [1].For the estimation of the coefficient signification Fisher's criteria is applied, i. e. F-test with the form where are error evaluation of coefficient: fn    -degree of freedom in cen- tral dot plan.
With applying the specified forms from (7) to (11), and with the additional value calculating, the estimation of the coefficient signification is shown in Table 3, where the 'Verification' is in the line, and by applying the symbol ¨*¨, their signification is determined, and those are the coefficients 0 , The coefficients b i , not significant for further process are excluded from the mathematical model in the form (1), and the new model form is obtained After defining the model function F i , the next step in the experiment result processing is adequacy determination of the obtained model and calculation of the multiple regression coefficient R as an additional adequacy criteria.
In general case, the adequacy of the obtained mathematical model is evaluated by the comparison of the experimentally obtained values E j y and calculating values R j y obtained from the model, where the adequacy condi- tion is determined by the F-criteria where are adequacy dispersion: When the form values are calculated (13), ( 15) and ( 16), F-test criteria has the value   which presents that the obtained mathematical model ( 12) adequately describes the deep drawing force F i .
The second criteria for the mathematical model election is the multiple regression coefficient R, the value of is determined according to the form The coefficient value R = 0.99 is obtained by applying data from Table 4 and the term (18).
Hence, in both cases of the adequacy model evaluation (Fisher's criteria and multiple regression coefficient R), the results show that the obtained mathematical model in coded form (12), adequately describes the deep drawing process with wall-thickness thinning, i.e the deep drawing force F i .

The experimental results presentation
After achieved evaluation of the mathematical model adequacy, it is necessary to compile the term (12) compile from the coded form with variables (X i ) to the physical model, with the real process parameters (ψ, s 1 , μ), instead of the coded.
With the applied forms for transformation according to [1], and with the additional recalculating the equations are obtained The form (20) presents the final mathematical model in physical form for the deep drawing force with wall-thickness thinning F i , which was the main goal of this paper.
By the model simulation in the graphic package ¨Graphic 2. 9¨ with the parameter value variation ψ, s 1 , μ, the 3D diagrams are obtained: presented in Figs.5-7.
Thus, the insight into the character and the drawing force values F i enables not only defining the gauge of the input parameter values (ψ, s 1 , μ), but also out of them.

Conclusion
The research results and analysis showed:  the obtained mathematical model for the drawing force F i adequately describes the process, which is explic-itly confirmed with the adequacy criteria (Fisher's criteria and criteria of the multiple coefficient regression R);  the drawing-force function is completely linear;  with the simulation of the obtained mathematical model for the drawing force F i in the programme package ¨Graphic 2.9¨, the monitoring of the character and drawing force values is enabled, not only for the defined gauge values of the input parameters ψ, s 1 , μ, but also out of them.
The applied modelling and simulation methods in this paper, can also be used for defining the optimal values of bigger number of the technological process parameters of deep drawing, with wall-thickness thinning, by the applied higher order mathematical model.Principally, it would relate to the reduction of the drawing force with the obtained products, according to the demanded quality along with the minimum energy consumption.

Fig. 1
Fig. 1 Product selection obtained by deep drawing with wall-thickness thinning

Fig. 3
Fig. 3 Scheme process presentation with the elected statistic value (P = 0.99) and with the applied condition given in the form (2) it is 9)error evaluation in central dot of the test plan:

Final
mathematical model with the significant coefficient values b i , now has the form

Table 4
Calculating values for R