Prediction of chip flow angle in orthogonal turning of mild steel by neural network approach

Abstract

Improvement of chip control is a necessity for automated machining. Chip control is closely related to chip flow and it plays also a predominant role in the effective control of chip formation and chip breaking for the easy and safe disposal of chips, as well as for protecting the surface-integrity of the workpiece. Although several ways to predict the chip flow angle (CFA) have been subjected in some researches, a good approximation has not been achieved yet. In this study, using different indexable inserts and cutting conditions for turning of mild steel, the chip flow angles were measured and some of the collected data from this experimental study were used for training with a two hidden layered backpropagation neural network algorithm. A group was formed from randomly selected data for testing. The chip flow angle values found from multiple regression, neural network (NN) and studies of previous researchers under the same turning conditions of the present study were compared. It has been seen that the best prediction was obtained by neural network approach.

Appendix

1.1 APPENDIX A-Approximation of Okushima and Minato

In this section, the approximation of Okushima and Minato on chip flow angle is given. They presented some equations, as follow, in six cases. In the present study, some of these equations have been used for comparing their approximation with the others.

Case 1 f ≤ rsinχ₁ and a ≥ r:

$$\upsilon = \frac{{{\left( {\frac{\pi } {2} + \sin ^{{ - 1}} \frac{f} {{2r}}} \right)}{\left( {45 + \frac{{90}} {\pi }\sin ^{{ - 1}} \frac{f} {{2r}}} \right)}}} {{{\left( {\frac{a} {r} - 1} \right)} + {\left( {\frac{\pi } {2} + \sin ^{{ - 1}} \frac{f} {{2r}}} \right)}}}$$

Case 2 f ≤ rsin χ₁ and a< r:

$$\upsilon = 90 - \frac{{90}} {\pi }{\left[ {\cos ^{{ - 1}} {\left( {1 - \frac{a} {r}} \right)} - \sin ^{{ - 1}} \frac{f} {{2r}}} \right]}$$

Case 3 $r{\left( {1 + \frac{1} {{\sin \chi _{1} }}} \right)}$ ≥f>2rsinχ₁ and a≥r:

$$\upsilon = \frac{{{\left( {\frac{\pi } {2} + \frac{\pi } {{180}}\chi _{1} } \right)}{\left( {45 + \frac{{\chi _{1} }} {2}} \right)} + {\left[ {\frac{f} {r}\cos \chi _{1} - {\sqrt {\frac{{2f}} {r}\sin \chi _{1} - {\left( {\frac{f} {r}} \right)}^{2} \sin ^{2} \chi _{1} } }} \right]}{\left( {90 + \chi _{1} } \right)}}} {{{\left( {\frac{a} {r} - 1} \right)} + {\left( {\frac{\pi } {2} + \frac{\pi } {{180}}\chi _{1} } \right)} + {\left[ {\frac{f} {r}\cos \chi _{1} - {\sqrt {\frac{{2f}} {r}\sin \chi ^{{}}_{1} - {\left( {\frac{f} {r}} \right)}^{2} \sin ^{2} \chi _{1} } }} \right]}}}$$

Case 4 $r{\left( {1 + \frac{1} {{\sin \chi _{1} }}} \right)}$ ≥f>2rsinχ₁ and a<r:

$$\upsilon = \frac{{{\left[ {\cos ^{{ - 1}} {\left( {1 - \frac{t} {r}} \right)} + \frac{\pi } {{180}}\chi _{1} } \right]}{\left[ {90 - \frac{{90}} {\pi }\cos ^{{ - 1}} {\left( {1 - \frac{a} {r}} \right)} + \frac{{\chi _{1} }} {2}} \right]} + {\left[ {\frac{f} {r}\cos \chi _{1} - {\sqrt {\frac{{2f}} {r}\sin \chi _{1} - {\left( {\frac{f} {r}} \right)}^{2} \sin ^{2} \chi _{1} } }} \right]}{\left( {90 + \chi _{1} } \right)}}} {{{\left[ {\cos ^{{ - 1}} {\left( {1 - \frac{a} {r}} \right)} + \frac{\pi } {{180}}\chi _{1} } \right]} + {\left[ {\frac{f} {r}\cos \chi _{1} - {\sqrt {\frac{{2f}} {r}\sin \chi _{1} - {\left( {\frac{f} {r}} \right)}^{2} \sin ^{2} \chi _{1} } }} \right]}}}$$

Case 5 f> $r{\left( {1 + \frac{1} {{\sin \chi _{1} }}} \right)}$ and a>f . $\tan \chi _{1} - \frac{r} {{\cos \chi _{1} }}$:

$$\upsilon = \frac{{{\left( {\frac{\pi } {2} + \frac{\pi } {{180}}\chi _{1} } \right)}{\left( {45 + \frac{{\chi _{1} }} {2}} \right)} + \frac{{\frac{f} {r} - {\left( {1 + \sin \chi _{1} } \right)}}} {{\cos \chi _{1} }}.{\left( {90 + \chi _{1} } \right)}}} {{{\left( {\frac{a} {r} - 1} \right)} + {\left( {\frac{\pi } {2} + \frac{\pi } {{180}}\chi _{1} } \right)} + \frac{{\frac{f} {r} - {\left( {1 + \sin \chi _{1} } \right)}}} {{\cos \chi _{1} }}}}$$

Case 6 r=0:

$$\upsilon = \frac{{\frac{f} {{\cos \chi _{1} }}.{\left( {90 + \chi _{1} } \right)}}} {{a + \frac{f} {{\cos \chi _{1} }}}}$$

1.2 APPENDIX B-Approximation of Young et al.

In this section the approximation of Young et al. on chip flow angle is given. They derived an equation for predicting the chip flow angle as follows. The general form of Young’s equation is:

$$\upsilon = \tan ^{{ - 1}} {\left( {\frac{{NUM}} {{DEN}}} \right)}$$

NUM and DEN are given for two different cases.

Case 1 $a \leqslant r{\left( {1 - \sin \chi } \right)}$

$$\begin{array}{*{20}l} {{{\text{NUM}} = {\left[ { - r.\sin \theta } \right]}^{{\theta _{3} }}_{{\theta _{1} }} + \frac{1} {2}{\left[ {\sin \theta {\left( {r^{2} - f^{2} .\sin ^{2} \theta } \right)}^{{\frac{1} {2}}} + \frac{{r^{2} }} {f}\sin ^{{ - 1}} {\left( {\frac{r} {f}\sin \theta } \right)}} \right]}^{{\theta _{2} }}_{{\theta _{1} }} + {\left[ {f\frac{{\sin {\left( {2\theta } \right)}}} {4} + \frac{\theta } {2}} \right]}^{{\theta _{2} }}_{{\theta _{1} }} + {\left[ {{\left( {r - d} \right)}\log {\left( {\sin \theta } \right)}} \right]}^{{\theta _{3} }}_{{\theta _{2} }} } \hfill} \\ {{{\text{DEN}} = {\left[ { - r.\cos \theta } \right]}^{{\theta _{3} }}_{{\theta _{1} }} + \frac{1} {2}{\left\{ {\cos \theta {\left( {r^{2} - f^{2} .\sin ^{2} \theta } \right)}^{{\frac{1} {2}}} + \frac{{r^{2} - f^{2} }} {f}\log {\left[ {{\left( {f.\cos \theta } \right)} + {\left( {r^{2} - f^{2} .\sin ^{2} \theta } \right)}^{{\frac{1} {2}}} } \right]}} \right\}}^{{\theta _{2} }}_{{\theta _{1} }} + \frac{f} {4}{\left[ {\cos {\left( {2\theta } \right)}} \right]}^{{\theta _{2} }}_{{\theta _{1} }} + {\left[ { - {\left( {r - d} \right)}\theta } \right]}^{{\theta _{3} }}_{{\theta _{2} }} } \hfill} \\\end{array}$$

where the limits of integration are:

$$\begin{array}{*{20}l} {{\theta _{1} = \cos ^{{ - 1}} {\left( {\frac{f}{{2.r}}} \right)}{\text{,}}} \hfill} \\ {{\theta _{2} = \pi - \tan ^{{ - 1}} {\left[ {\frac{{r - a}}{{{\left( {2.r.a - a^{2} } \right)}^{{\frac{1}{2}}} - f}}} \right]}} \hfill} \\ {{\theta _{3} = \pi - \sin ^{{ - 1}} {\left( {\frac{{r - a}}{r}} \right)}} \hfill} \\ \end{array}$$

Case 2 $d > r{\left( {1 - \sin \chi } \right)}$

$$\begin{array}{*{20}l} {{NUM = {\left[ { - r^{2} .\sin \theta } \right]}^{{\theta _{2} }}_{{\theta _{1} }} + \frac{r} {2}{\left[ {\sin \theta {\left( {r^{2} - f^{2} .\sin ^{2} \theta } \right)}^{{\frac{1} {2}}} + \frac{{r^{2} }} {f}\sin ^{{ - 1}} {\left( {\frac{r} {f}\sin \theta } \right)}} \right]}^{{\theta _{2} }}_{{\theta _{1} }} + {\left[ {f\frac{{\sin {\left( {2\theta } \right)}}} {4} + \frac{\theta } {2}} \right]}^{{\theta _{2} }}_{{\theta _{1} }} + {\left[ {f{\left\{ {d - r{\left( {1 - \sin C_{s} } \right)}} \right\}} - \frac{{f^{2} }} {4}\sin {\left( {2\chi } \right)}} \right]}\cos \chi } \hfill} \\ {{DEN = {\left[ { - r^{2} .\cos \theta } \right]}^{{\theta _{2} }}_{{\theta _{1} }} + \frac{r} {2}{\left\{ {\cos \theta {\left( {r^{2} - f^{2} .\sin ^{2} \theta } \right)}^{{\frac{1} {2}}} + \frac{{r^{2} - f^{2} }} {f}\log {\left[ {{\left( {f.\cos \theta } \right)} + {\left( {r^{2} - f^{2} .\sin ^{2} \theta } \right)}^{{\frac{1} {2}}} } \right]}} \right\}}^{{\theta _{2} }}_{{\theta _{1} }} + {\left[ {f{\left\{ {d - r{\left( {1 - \sin \chi } \right)}} \right\}} - \frac{{f^{2} }} {4}\sin {\left( {2\chi } \right)}} \right]}\sin \chi } \hfill} \\\end{array}$$

where the corresponding limits of integration are:

$$\begin{array}{*{20}l} {{\theta _{1} = \cos ^{{ - 1}} {\left( {\frac{f} {{2r}}} \right)}} \hfill} \\ {{\theta _{2} = \pi - \chi } \hfill} \\\end{array}$$