An analytical model taking feed rate effect into consideration for scallop height calculation in milling with torus-end cutter

Abstract

Feed rate effect on scallop height in complex surface milling by torus-end mill is rarely studied. In a previous paper, an analytical predictive model of scallop height based on transverse step over distance has been established. However, this model doesn’t take feed rate effect into consideration. In the present work an analytical expression of scallop height, including feed rate effect, is detailed in order to quantify feed rate effect and thus to estimate more precisely the surface quality. Then, an experimental validation is conducted, comparing the presented model predictions with experimental results. Actually, the share of the scallop height due to feed effect is highly dependent on the machining configuration. However, most of time, the feed effect on total scallop height values is far from being negligible.

Appendix: Scallop height approximation

Considering $\varrho$ the curvature of the surface locally constant, the scallop height calculation is based on the equation:

$$\begin{aligned} R_{eff}^2=(\varrho +h_1)^2+(\varrho +R_{eff})^2-2\,\cos (\beta )(\varrho +R_{eff})(\varrho +h_1) \end{aligned}$$

From this equation the following can be calculated:

$$\begin{aligned} \cos (\beta ) = \frac{2\,\varrho ^2 + 2\,\varrho \,R_{eff} + 2\,\varrho \,h_1 + h_1^2}{2(\varrho +R_{eff})(\varrho +h_1)} \end{aligned}$$

$h_1^2$ may be considered negligible in comparison with other quantities. Thus:

$$\begin{aligned} \cos (\beta ) = \frac{2\,\varrho ^2 + 2\,\varrho \,R_{eff} + 2\,\varrho \,h_1}{2(\varrho +R_{eff})(\varrho +h_1)} \end{aligned}$$

Considering $s_{od}$ the distance between the two contact points (see Figs. 4, 5), the following can be stated:

$$\begin{aligned} \sin (\beta ) = \frac{s_{od}}{2\,\varrho } \end{aligned}$$

Using the fact that $\sin ^2(\beta ) = 1-\cos ^2(\beta )$ it can be deduced that:

$$\begin{aligned} \left( \frac{s_{od}}{2\,\varrho }\right) ^2 = 1-\left( \frac{2\,\varrho ^2 + 2\,\varrho \,R_{eff} + 2\,\varrho \,h_1}{2(\varrho +R_{eff})(\varrho +h_1)}\right) ^2 \end{aligned}$$

whence

$$\begin{aligned} \frac{s_{od}^2}{4\,\varrho ^2}&= 1-\frac{(\varrho ^2 + \varrho \,R_{eff} + \varrho \,h_1)^2}{\left( (\varrho +R_{eff})(\varrho +h_1)\right) ^2} \\&= \frac{\left( (\varrho +R_{eff})(\varrho +h_1)\right) ^2 - (\varrho ^2 + \varrho \,R_{eff} + \varrho \,h_1)^2}{\left( (\varrho +R_{eff})(\varrho +h_1)\right) ^2} \\&= \frac{R_{eff}\,h_1(2\,\varrho ^2 + 2\,\varrho \,R_{eff} + 2\,\varrho \,h_1+R_{eff}\,h_1)}{(\varrho +R_{eff})^2(\varrho +h_1)^2} \\&= \frac{R_{eff}\,h_1(2\,\varrho (\varrho + R_{eff} + h_1)+R_{eff}\,h_1)}{(\varrho +R_{eff})^2(\varrho +h_1)^2} \end{aligned}$$

$h_1$ may be considered negligible when added with $\varrho$. Thus:

$$\begin{aligned} \frac{s_{od}^2}{4\,\varrho ^2}&= \frac{R_{eff}\,h_1(2\,\varrho ^2 + 2\,\varrho \,R_{eff}+R_{eff}\,h_1)}{\varrho ^2\,(\varrho +R_{eff})^2}\\&= \frac{R_{eff}\,h_1(2\,\varrho ^2 + R_{eff}(2\,\varrho +h_1))}{\varrho ^2\,(\varrho +R_{eff})^2}\\ \end{aligned}$$

again $h_1$ may be considered negligible when added with $\varrho$. Thus:

$$\begin{aligned} \frac{s_{od}^2}{4\,\varrho ^2}&= \frac{R_{eff}\,h_1(2\,\varrho ^2 + 2\,\varrho \,R_{eff})}{\varrho ^2\,(\varrho +R_{eff})^2} \\&= \frac{2\,\varrho \,R_{eff}\,h_1(\varrho + R_{eff})}{\varrho ^2\,(\varrho +R_{eff})^2} \\&= \frac{2\,R_{eff}\,h_1}{\varrho \,(\varrho +R_{eff})} \end{aligned}$$

Finally the expression of $h_1$ is:

$$\begin{aligned} h_1 = \frac{s_{od}^2(\varrho +R_{eff})}{8\,\varrho \,R_{eff}} \end{aligned}$$

Calculation of the scallop height $h_{1p}$ for a plane surface is directly derived of the equation:

$$\begin{aligned} R_{eff}^2 = \left( \frac{s_{od}}{2}\right) ^2 + (R_{eff}-h_{1p})^2 \end{aligned}$$

which, considering $h_{1p}^2$ negligible in comparison with other quantities, leads to:

$$\begin{aligned} h_{1p} = \frac{s_{od}^2}{8\,R_{eff}} \end{aligned}$$

Thus, considering that

$$\begin{aligned} \lim _{\varrho \rightarrow \infty } \frac{s_{od}^2(\varrho +R_{eff})}{8\,\varrho \,R_{eff}} = \frac{s_{od}^2}{8\,R_{eff}} \end{aligned}$$

the following can be stated:

$$\begin{aligned} \lim _{\varrho \rightarrow \infty } h_1 = h_{1p} \end{aligned}$$