Empirical modelling of the electron beam hardening.

Munteanu, Adriana ; Coteata, Margareta ; Neagu, Dumitru 等

1. INTRODUCTION

Due to present industrial technological requirements, the nonconventional technologies were applied more and more often. The non-conventional technology manufacturing today face greater challenges than ever and offers new and intelligent solution in case of some manufacturing process. One of the nonconventional techniques is the electron beam process. Electron beam machining is a nonconventional method of machining which has been used since 1879 when W. Croockes melt his own platinum anode with electron beam.

It is well known that the thermal process using the electron beam technology is based on the changing the kinetic energy of an electron beam with high speed into heat. The electron beam can be considered as thermal tool of a concentrated energy that melts the workpiece material, but in case of few methods (electronolithography, irradiation etc) didn't works like a thermal tool (Rykalin et al., 1988).

Electron beam techniques have developed in many areas. Trends on activities carried out by researchers depend on the interest of the researchers and the availability of the technology. Among the processing methods using the electron beam, we could mention: drilling, welding, cutting, surface cleaning and degasification, the superficial heat-treating, surface micro-alloying and coating (or plating) by melting, doping, electron beam metalizing, electronolithography etc.

2. EMPIRICAL MODELLING

In this paper we are focused on electron beam hardening, that use extremely high-energy input to austenitize a very thin surface layer. The bulk of the substrate remains cool and provides an adequate heat sink for "self-quenching".

Their advantages includes: a) minimal workpiece distortion, b) the ability to selectively harden zones of a surface and better process control, c) ability to harden areas inaccessible to conventional induction techniques, d) repeatability and e) high speed (Neagu, 2001).

The question is what method we can apply, using the same values for the input parameter, to obtain empirical models to express an output parameter.

If we try to realize some experiments of the electron beam hardening, in classical manner, a complete set will suppose a number of 3125 experimental data ([5.sup.5]).

Similar research was carrying on by the Dr. Eng. D. Neagu. The model of the process function, in case of the electron beam hardening, obtain by him is build upon this classical experimental plan. For example, when the hardening bandwidth [L.sub.HV] is considered as output parameter, the input parameters are the following: the working distance ([L.sub.l]), transversal deflecting angle ([beta]), the electron beam current ([I.sub.FE]), accelerating voltage ([U.sub.a]) and linear speed of the workpiece ([V.sub.m]) (Neagu, 2001).

The solution obtained in this condition was:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

So that, in order to obtain similar results we can use the Taguchi method as thus, to reduce the number of experimental data.

The Taguchi method supposes that if we modify a limited number of parameters the chances to optimize the system results are increased when the varying parameters are carefully selected, with the condition that this factor is relevant reported to optimization criterion (Goupy, 1990). The high energy machining processes as the electron beam machining process are interrelated directly with a system built on a plurality of input parameters. Taking into consideration such parameters as the material chemical composition, the electron beam energy and the electron hardening phenomena and their action on the working system, they are transformed in relevant outputs, respectively in the products characteristics (Dagnelie, 2000).

The matrix model of the system includes the effects of the main factors: working distance, transversal deflecting angle, the electron beam current, accelerating voltage, and linear speed of the workpiece, on the output parameter the hardening band width--[L.sub.HV].

A general model can be written as in equation 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Were: [L.sub.HV.sup.t] is the theoretical answer of the system; M is the general average; [E.sub.Ll1] [E.sub.Ll2] [E.sub.Ll3] [E.sub.Ll4] [E.sub.Ll5]] x [[L.sub.l]] is the effect of the working distance; [[E.sub.IFE1] [E.sub.IFE2] [E.sub.IFE3] [E.sub.IFE4] [E.sub.IFE5]] x [[I.sub.FE]] is the effect of the electron beam current; [[E.sub.b1] [E.sub.b2] [E.sub.b3] [E.sub.b4] [E.sub.b5]] x [[beta]] is the effect of the transversal deflecting angle; [[E.sub.Ua1] [E.sub.Ua2] [E.sub.Ua3] [E.sub.Ua4] [E.sub.Ua5]] x [Ua] is the effect of the accelerating voltage; [[E.sub.Vm1] [E.sub.Vm2] [E.sub.Vm3] [E.sub.Vm4] [E.sub.Vm5]] x [Vm] is the effect of the workpiece linear speed;

For our specific output parameter--the hardening band width--we have the following Taguchi model (in this case we neglect the interferences between input parameters):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The other case is the least-squares method. We must to consider the fact that sometimes the values of the output parameters of a process can be affected by the aleatory errors, this would make senseless the selection of one function y=y(x) that describe all the n values obtained in the experimental program (Munteanu & Ilii, 2008).

The complete set of the experimental data can be described like a plurality of different type of function y(x). The least-squares method indicates the best function y(x) for our set of data. This method permits to determine the most likely values of the coefficients for the prior function based on the theoretical consideration regarding the phenomena of the process for which we obtain the experimental data.

The software program used in present paper permit to determine a relation of conection between the independent output parameter y and some liniar input parameter [x.sub.1], [x.sub.2], ... [x.sub.n], a relation writen as:

y = f([x.sub.1], [x.sub.2], [x.sub.n]) (4)

The software gives a best function to express the process influeces, but also offers the possibility to choose between more forms of mathematical expression that can model the answer.

For our output parameter [L.sub.HV] (hardening band width) we shall have five distinct functions. The control criterion is the Gauss sum. So that, the best function is polynomial type (the 2nd degree polynomial function) because the Gauss sum is the smallest (Gauss sum: S = 146973,1).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

But to obtain a better image of the influence of the each input parameter and an easier ordering of them, we prefer to choose the power function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

In this case, the GWBASIC language program takes in consideration only five distinct functions like: the exponential function, 1nd and 2nd degree polynomial function, power function and hyperbolic function. Related to the experimental condition we can obtain the regression functions in the conditions to accomplish the request imposed by the law of the minimum Gauss sum (Cretu, 1992).

Using software Datafit we obtain another dependence relation, considered by the software as best expression, an exponential function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

If we modified the input parameter of the electron beam hardening and using appropriate software of experimental data processing (Datafit) we can see the variation of the output parameter depending on the considered input parameters. A graphical representation related to the mathematical model (7), but without the variation of the all input parameters.

[FIGURE 1 OMITTED]

When we analyse the different models we can see that the easiest way to represent the influence of different parameters regarding the one or another output parameter is the mathematical modelling. In this way we can emphasize the influence of each input parameter on the hardening band-width that we considered for our study.

3. CONCLUSION

The optimization of the system answer supposes to find the maximum of all possible answers, since we are concerned to obtain a bigger hardness of the superficial layer, a good sized of the deep penetration and a grater hardening band-width.

The other mathematical modelling methods cannot consider the interaction between the main variables, while the Taguchi matrix model permits besides the interaction between the input parameters, also some other aspects. In the future we will considered to extend the theoretical and experimental research in order to improve the mathematical model so that to obtain the best solution.

The poupose of this paper was to find the best relation to expres the hardening band width using different methods and, in the same time, to compare these dependence relations.

4. REFERENCES

Cretu, Gh. (1992). Base of the experimental data (in Romanian). Publishing House Rotaprint, Iassy

Dagnelie, P. (2000). The planification of the experiments: the chose of the experimental divice and method (in Franch). Journal of the Statistics French Societe, No. 141 (1-2), (5-29)

Goupy, J. L. (1990). Comparative analysis of the different experimental plans (in Franch). Journal of the applied statistics, Vol. 38, No. 4, (5 - 44), Available from: http://books.google.com/books?hl=ro&lr=&id=yiHjQlbN8 A8C&oi=fnd&pg=PA9&ots=Q I28Dz8Xx&sig=S6sb2Srs9xV23kuiqC8S7wRlQew#PPA10,M1 Accessed: 2007.05.10

Munteanu, A. & Ilii, S.M. (2008). Taguchi model versus mathematic model in the electron beam process. Scientific Bulletin of Politechnica Iasi. Tomul LIV (LVIII), Fasc.1-3 (May), (83-89)

Neagu, D. (2001). Electron beam hardening (in Romanian). Publishing House Printech, ISBN 973-652-377-2, Bucharest

Rykalin, N.; Uglov, A.; Zuev, I. & Kokora, A. (1988). Laser and electron beam material processing, handbook, Publishing House Mir Publishers, ISBN 5-03-0000-23-2, Moscow