An empirical relationship between cutting forces and length in dry drilling in aluminium alloys.

Domingo, Rosario ; Alvarez, Roberto ; Sebastian, Miguel Angel 等

1. INTRODUCTION

Dry drilling continues being an important process in aeronautical industry, and also in the machining of aluminium alloys, in particular in UNS A97050-T7 and UNS A92024-T3.

The trust cutting forces can be calculated by dynamometers, some of them of high precision. However these forces are different respect to theoretical forces (Hamade, et al., 2006; Tansel, et al., 2000). For this reason it is possible to affirm that the prediction of forces is a question still does not resolve. Other studies have demonstrated that cutting forces for the first holes, being different, are similar statistically (Domingo et al., 2008a; Domingo et al, 2008b). This paper intends to clarify some relationships between cutting forces and length that allow predicting forces in dry drilling.

Cutting forces have been calculated by a piezoelectric dynamometer, and afterwards the results have been statistically analysed by Box-Cox transformations (Box & Cox, 1964) by SPSS software.

2. EMPIRICAL RESULTS

The tests have been realised in aluminium alloys, UNS A7050-T7 and UNS A2024-T3. Dry drilling has been processed to cutting speed of 83 m/min and 60 m/min for the first alloy, and for 50 m/min for the second one. Feed rate takes the value 0.175 mm/rev always. Drills used in the tests have the characteristics showed in Table 1.

[FIGURE 1 OMITTED]

Drilling process has been performed with three drills. Each drill has machined different holes, according to increment the cutting length. Cutting forces are expressed in Newton and cutting length expressed in mm. Data from UNS A97050-T7 and UNS A92024-T3 are shown in Fig. 1.

3. BOX-COX TRANSFORMATIONS

Box-Cox transformations have been used to determine whether, in this case, a significant relationship exists between maximum cutting forces, Fzmax (dependent variable) and cutting length (independent variable), L. This transformation with power allows minimizing the mean squared error.

Moreover, an analysis of variance (ANOVA) has been realised to determine the P-value. Thus the effect of various power transformations of Fzmax can be compared on the linear regression between L and Fzmax.

For Vc83 (Box-Cox Transformations - Power = 20.11 Shift = 0.0), Vc60 (Box-Cox Transformations - Power = 10.2112 Shift = 0.0) and Vc50 (Box-Cox Transformations - Power = 7.46592 Shift = 0.0), the equations of the fitted model, expressed in Fig. 2, Fig. 3 and Fig. 4 are the following,

BoxCox ([F.sub.zmax_Vc83]) = 11.23 + 0.82 x L (1)

BoxCox ([F.sub.zmax_Vc60]) = 29.79 + 0.94 x L (2)

BoxCox ([F.sub.zmax_Vc50]) = 21.36 + 1.37 x L (3)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

In the ANOVA analysis (see Table 2), the P-value is less than 0.01, there is a statistically significant relationship between the transformed values of Fzmax_Vc83 and L at the 99% confidence level, for Fzmax_Vc60 and L at the 90% confidence level (P-value is less than 0.10), and for Fzmax_Vc50 and L at the 95% confidence level (P-value less than 0.05).

For Fzmax_Vc83, the R-squared statistic points to that the fitted model gives explanation of 68.40% of the variability in Fzmax_Vc83, of 41.24% of the variability in Fzmax_Vc60, and of 56.64% of the variability in Fzmax_Vc50.

Moreover, in Table 2 can be appreciated the correlation coefficient (0.83 for Fzmaz_V83) that it involves a moderately strong connection between the variables, and the standard error of the estimate that it shows the standard deviation of the residuals to be 12.73.

For Fzmax_V60, the correlation coefficient (0.64) describes a moderately strong relationship between the cutting forces and the length, and a standard deviation of the residuals of 24.35.

Finally, in the case of Fzmax_V50, the correlation coefficient (0.75) points to a moderately strong link between the cutting variables, and a standard deviation of the residuals of 24.89.

Thus, in the three cases, significant statistical relationships have been found, and the obtained value of cutting forces can be utilised to generate prediction limits for new observations.

4. PREDICTED VALUES

In accordance with Section 3, the Table 3 exposes the predicted values for Fzmax according the minimum and maximum cutting length (Lm and LM) that it has been drilled, at the 95% confidence level, the prediction limits and the limits of confidence. Lower (Low) and upper (Upp) limits are indicated.

The Table 3 reveals the best forecasts, at 95% of confidence level, in the prediction intervals for new observations and for the mean of the observations (confidence intervals). These predictions can be distinguished in the Fig. 2, Fig. 3 and Fig. 4, by means of discontinuous curves on the fitted models.

5. CONCLUSIONS

This paper provides a study that it permits to categorise, by sectors, the prediction of cutting forces in aluminium alloys, considering the influence of the cutting length on the maximum force. Thus, a statistical relationship has been found between cutting forces and length, after to accomplish an ANOVA analysis and Box-Cox transformations. The results allow establishing a prediction limits at 95% confidence level.

These preliminary results can suppose an important element in the experimental tests due to different values of thrust forces that dynamometers provide.

Future researches could determine the connection between cutting forces and higher lengths, and also the influence of different cutting parameters and of the drill wear on the forces variability. Analogous advances will consent to complete and verify the model.

6. ACKNOWLEDGEMENTS

This work has received financial support from the MCYT (Spanish Government), by means of project DPI2005-09325 C02-02.

7. REFERENCES

Box, G.E.P. & Cox, D.R. (1964). An analysis of transformations. Journal of the Royal Statistical Society, Series B Vol. 26, 211-246, ISSN 1369-7412.

Domingo, R.; Alvarez, R.; Rubio, E.M. & Sebastian, M.A. (2008a). Experimental analysis of cutting forces in dry drilling of UNS A92024 alloy. Journal of Machine Engineering, Vol. 8, No. 2 (2008), 73-78, ISSN 1895-7595.

Domingo, R.; Arenas, J.M.; Rubio, E.M. & Marcos, M. (2008b). Experimental analysis of cutting forces in dry drilling of UNS A97050-T7 alloy, Proceedings of Intelligent Computation in Manufacturing Engineering, Teti, R. (Ed.), 3p Naples, July 2008, University of Naples Federico II, Naples.

Hamade, R.F.; Seif, C.Y. & Ismail, F. (2006). Extracting cutting force coefficients from drilling experiments. International Journal of Machine Tools & Manufacture, Vol. 46, No. 3-4, 387-396, ISSN 0890-6955.

Tansel, I.N.; Arkan, T.T.; Bao, W.Y.; Mahendrakar, N.; Shisler, B.; Smith, D. & McCool, M. (2000). Tool wear estimation in micro-machining. Part I: tool usage--cutting force relationship. International Journal of Machine Tools & Manufacture, Vol. 40, 599-608, ISSN 0890-6955.

Tab. 1. Drill description.

Drill type Tool material Point angle Coating

NH VHM 140 [degrees] TiAlN

Tab. 2. Data from ANOVA analysis.

 P- Corr. R- Standard
 Value Coeff. squared Error Est.

V83 0.006 0.83 68.40% 12.73
V60 0.019 0.64 41.24% 24.35
V50 0.019 0.75 56.64% 24.89

Tab. 3. Predicted values.

 Predicted Prediction Confidence
 Limits (95%) Limits (95%)

L Fzmax Low Upp Low Upp
 _V83

Lm 551.12 583 572

LM 591.02 569 603 581 598

L Fzmax Low Upp Low Upp
 _V60

Lm 383.02 425 245 409

LM 419.93 368 443 400 433

L Fzmax Low Upp Low Upp
 _V50

Lm 349.59 410 388

LM 414.79 361 443 392 431