Mathematical modeling of influence between surface roughness and thermoelectric current.

Cirstoiu, Carmen Adriana

1. INTRODUCTION

The measurement of surface state evaluates the defects of surface inevitably generated during the manufacture of parts. A good knowledge of these defects allows manufacturing the parts with the precision and quality required, in the best economic conditions.

The thermoelectric effect or thermoelectricity encompasses three separately identified effects: the Seebeck effect, the Peltier effect and the Thomson effect. The thermoelectric effect is the direct conversion of temperature differences from the cutting area to electric voltage. We can make an analogy between the phenomena appeared in thermocouple and what happens during the cutting process.

In literature there are studies showing the influence of the cutting regime on thermoelectric current. Novelty brought by this work represents the mathematical modeling of dependence between roughness and the thermoelectric current. The thermoelectric current can easily be measured during the cutting process on conventional machine tools, enabling the assessment of roughness in real time and identify the causes that lead to lower quality areas turned, in order to take the necessary measures.

For assessing roughness by indirect methods, the relationships between roughness and voltage or intensity of thermoelectric current were determined, using specific software for processing experimental data. The methods used were: the direct method for measurement by contact of the surface roughness, before and after processing and natural thermocouple method for measuring voltage and intensity of thermoelectric current. Further research will lead to automation of taken over data, by building a data acquisition unit.

2. DESIGN OF THE EXPERIMENT

We measure the values of voltage and the thermoelectric current intensity in turning of 42MoCr11 alloy, thanks to a modified cutting regime. We didn't quantify the vibrations effect on the experimental studies carried out and we also didn't use cooling liquid (dry cutting). Experimental investigations were conducted on cylindrical surfaces, separated by gorges. The work piece of 42MoCr11 alloy has 52 mm in diameter and 310 mm length.

[FIGURE 1 OMITTED]

In order to measure accurately the thermoelectric current in turning was adjusted the centering peak of the work piece by constructing a collector with copper brushes.

The processing was performed on a conventional machine tool, with changeable tool inserts--TNMG 22 04 08-P15, chosen by CoroGuide and CoroKey--PC programs ([alpha] = 7[degrees]; y = 6[degrees]; [K.sub.r] = 93[degrees]; [K.sub.r] = 27[degrees]; [[lambda].sub.s] = -6[degrees]; [r.sub.[epsilon]] = 0,8 mm).

To measure voltage U or intensity of the thermoelectric current I, was used a professional digital multimeter Metrix MX 54. The roughness of the processed surface was measured using a Diavite -11 rugosimeter.

3. RESULTS ANALYSIS

3.1 Determine the form of regression functions

Regression is the technique of processing experimental data to obtain quantitative relationships between a dependent variable y and one or more independent variables x1, x2, ..., xn, such that y = f (xl, x2,..., xn).

Functions were established basing on experimental measurement of the output variables that are correlated with the inputs into the process by geometric regression.

In accordance with literature recommendations (Liteanu & Rica, 1985; Darwish,. 1997; Puertas & Perez, 2003), the general form of regression functions in turning process, when studying the influence of three independent variables on the dependent variable, may be a function of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Multiple linear regression is chosen for the mathematical modeling of the intensity depending on the cutting regime parameters in logarithmic coordinates.

Intensity function in logarithmic coordinates, determined by the working environment MathCAD is the following:

y = [b.sub.0] + [b.sub.1][X.sub.1] + [b.sub.2][x.sub.2] + [b.sub.3][x.sub.3] + [epsilon] (2)

where: [x.sub.1] = ln([v.sub.c]); [x.sub.2] = ln(f); [x.sub.3] = ln([a.sub.p]); [epsilon] includes perturbation variables.

In order to determine the regression coefficients, was applied the method of least squares, using the MathCAD work.

With these coefficients, we obtain a function of the form (2) and then, applying a logarithmic function, a function of the form (1).

In this work was found the geometric regression function with three variables (cutting speed [v.sub.c], feed f and depth of cut [a.sub.p]), according to relationship (1).

The experimental plan used for turning 42MoCr11 steel is organized on two levels and contains a total of 18 experiments, four experiments being necessary for the calculation of systematic errors (Cirstoiu, 2007).

3.2 "Intensity of thermoelectric current" process function analysis

Expression of intensity function I, in logarithmic coordinates, determined using MathCAD program is as follows:

z = 1.665 + 0.264 x x1 + 0.104 x x2+0.079 x x3 (3)

where: z = lnI, x1 = [lnv.sub.c]; x2 = lnf; x3 = [lna.sub.p].

By applying inverse logarithmic function relationship (3) one gets to (4).

I = 5.286 x [v.sup.(0.264).sub.c] x [f.sup.(0.104)] x [a.sup.(0.079).p] [[mu]A] (4)

Figure 2 shows the 3D representation, using, "Mathematica" program, of the function determined by geometric regression, while the depth of cut is constant, [a.sub.p] = 1.5 mm.

[FIGURE 2 OMITTED]

3.3 "Voltage" process function analysis

Voltage U function in logarithmic coordinates, determined by the working environment MathCAD is the following:

z = 1.829+0.23 x x1 + 0.094 x x2 + 0.065 x x3 (5)

where: z = lnU; x1 = [lnv.sub.c]; x2 = lnf; x3 = [lna.sub.p].

By applying inverse logarithmic function relationship (5) one gets (6).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

A similar representation of the figure 2 has voltage.

3.4 Relations between roughness and termocurent

Similarly, was obtained the relationship (7) indicating the Ra roughness parameter dependence of the cutting regime parameters (Cirstoiu, 2007).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

From relations (4), (6), (7), for f = 0.208 mm/rot; ap = 1.5 mm, such relations are obtained between roughness and the thermoelectric current:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

R = 1.74872 x 1/[v.sup.0.488.sub.c] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

R = 1.46966 x U/[v.sup.0.454.sub.c]

In Figure 3 was represented in 3D roughness Ra changes depending on the speed [v.sub.c] and intensity I, while the other parameters of cutting regime are kept constant. A similar representation of the figure 3 has Ra changes depending on the speed [v.sub.c] and voltage U.

[FIGURE 3 OMITTED]

4. CONCLUSIONS

Increases of the intensity or voltage of thermoelectric current values indicate a deterioration of surface quality, reflected in the increasing values of roughness parameter Ra. Therefore it is the thermoelectric current a true indicator of the normal state of the cutting process, in terms of quality of processed surface. Further research will have as results: increasing speed of experimental data acquisition and accuracy of results, by using a microcontroller, automating data acquisition and processing, optimizing technological processes.

5. REFERENCES

Cirstoiu A. (2007). Mathematical modeling of quality of surface processed, Bibliotheca Publishing, ISBN 978-973-712-2988, Targoviste

Darwish S. M. (1997). Formulation of surface roughness models for machining Nichel super alloy with different tools, Materials and Manufacturing Processes, Vol. 12, No. 3, 395-408

Liteanu C. & Rica I. (1985). Optimization of analytical processes, Academy Publishing, Bucharest

Murzec B. & Muz F. (2003).Integral model of selection of optimal cutting conditions from different databases of tool makers. Journal of Materials Processing Technology, Vol. 133 1-2, pg. 158-165

Puertas Arbizu I. & Perez Luis C. J. (2003). Surface roughness prediction by factorial design of experiments in turning processes, Journal of Materials Processing Technology, pg. 390-396