Proceedings

of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 133 1965

Study of Cutting Forces in the Milling of Non-Ferrous Metals

A. M. Rozenberg, G. L. Kufarev, Yu. A. Rozenberg,
A. A. Kozlov, S. I. Takhman

No research work has thus far been carried out on questions of the dynamics of milling non-ferrous metals.

The creation of high-speed machine tools of the 6M12B type, with a maximum spindle speed of \(n = 5000 \div 6000\) rpm, intended mainly for machining non-ferrous metals, requires precise information on the cutting forces in milling these metals, since otherwise it does not appear possible to determine the necessary power, which at high cutting speeds will reach considerable values.

In the present work, questions of the dynamics of milling seven grades of non-ferrous metals are considered. Their mechanical properties are given in Table 1.

Table 1

The investigations were carried out on a 6N83 horizontal milling machine and a 6N13P vertical milling machine, equipped with elastoelectric apparatus with inductive sensors, which made it possible during the experiment to measure simultaneously the torque on the spindle and the feed force.

The temperature in the cutting zone was measured by means of a natural thermocouple. Cylindrical milling cutters, both high-speed steel (R18) and carbide-tipped (T15K6), were used as the cutting tool, as well as face mills, both high-speed steel (R18) and carbide-tipped (VK8).

The variation of the principal geometric parameters of the cutters is given in Table 2.

The work studied the influence on cutting forces of the following factors: cutting speed, feed per tooth, depth of milling, width of milling, rake angle, wear of the cutter teeth, plan angle, and the relative position of the face mill and the workpiece.

Table 2

It is customary to express the dependence of cutting forces in milling on various parameters in the form of power equations.

The most complete and theoretically substantiated equations were derived by A. M. Rozenberg [1] and have the following form¹.

Mean total circumferential force on a cylindrical cutter

\[ p = \frac{2\kappa + 1}{k + 2} \cdot c_\beta \cdot \frac{\lambda}{\pi} \cdot B \cdot z s_z^{\kappa+1} \left(\frac{t}{D}\right)^{\frac{\kappa+2}{2}} v^u k\Gamma . \tag{1} \]

Mean total circumferential force on a face mill

\[ p = 2^{-\frac{\kappa}{2}} \frac{\lambda}{\pi} \cdot t^{l+1} \cdot z s_z^{\kappa+1} v^u \left\{1-\left(1-\frac{B}{D}\right)^{\frac{\kappa+2}{2}}\right\} k\Gamma , \tag{2} \]

where

\[ \begin{aligned} c_\beta &\text{ is a coefficient showing the influence of the helix angle of the cutting edge on the cutting force;}\\ t &\text{ is the milling depth, in mm;}\\ B &\text{ is the milling width, in mm;}\\ s_z &\text{ is the feed per 1 cutter tooth, in mm;}\\ z &\text{ is the number of cutter teeth;}\\ D &\text{ is the cutter diameter, in mm;}\\ v &\text{ is the cutting speed, in m/min;}\\ k,\ l,\ u &\text{ are exponents;}\\ \lambda &\text{ is a constant coefficient depending on the material being machined and the machining conditions.} \end{aligned} \]

In carrying out the investigations, the task was set of checking the possibility of applying these equations to calculate cutting forces in milling non-ferrous metals over a wide range of variation of all the parameters entering into the equations, and of determining the constant coefficients \(\lambda\) and the exponents \(k, l, u\) for the metals under investigation.

As numerous experiments showed (more than 8000 tests were carried out in the work), the difference in the force values obtained from instruments and from equations (1) and (2) lies within 10%, and only for

at small feeds, when the absolute values of the forces are small, this difference reaches 15–20%. It should be noted that a power-law binomial equation cannot provide higher calculation accuracy, as will be discussed below.

Effect of Cutting Speed and Feed on Cutting Forces

One of the main factors affecting cutting forces is the cutting speed. Owing to the considerable difference in the properties of the metals studied, cutting speed affects the cutting forces in different ways.

When milling brass LS59-1 and duralumin D-1, the influence of speed is so insignificant that it may be neglected for all types of milling. For brass L62 and red copper, as the cutting speed increases the forces decrease, especially in the low-speed range \(v \le 100\ \mathrm{m/min}\) (Fig. 1).

Fig. 1.

For silumin Al-10 and Al-4, the dependence of force on speed is expressed by curves characteristic of all metals in the cutting of which built-up edge is formed (Fig. 2).

The study of the effect of speed on cutting forces was carried out at eight feeds lying within the range \(S_z = 0.0196 \div 0.503\) mm/tooth. On the basis of these experiments, values were obtained for the constant coefficients \(\lambda\) and for the exponents “K” and “I,” needed for calculating the forces by equations (1) and (2). The values of these quantities are given in Tables 3–6.

Fig. 2.

Effect of the rake angle on cutting forces

When the rake angle is changed, the conditions of chip formation, chip shrinkage, and consequently the cutting forces on the rake face change. The forces on the flank face remain constant in this case.

For small feeds, when the forces on the rake face are relatively small in comparison with the forces on the flank face, a change in the rake angle, affecting only the forces on the rake face, leads to a very slight change in the total circumferential force. In milling with large feeds, the forces on the rake face prove to be considerably greater than on the flank face, and their change

Table 3

Cylindrical milling with high-speed-steel cutters

Table 4

Cylindrical milling with carbide cutters

\[ {}^{*)}\quad v_{\mathrm{razd}} = 47\,\frac{D^{0,2}}{S_z^{0,4}t^{0,2}}\ \mathrm{m/min}, \tag{2} \]

where
$D$ is the cutter diameter, in mm;
$S_z$ is the feed per tooth, in mm/tooth;
$t$ is the depth of cut, in mm.

Table 5

Face milling with high-speed-steel cutters

Table 6

Face milling with carbide cutters

at constant forces on the flank face leads to a substantial decrease in the mean circumferential force as the rake angle increases.

Table 7 gives the values of the correction coefficients \(k_\gamma\) for various feeds when milling with cylindrical carbide cutters.

Table 7

For the remaining types of milling, Table 8 gives the mean values of the correction coefficients \(k_\gamma\) for all feeds.

For \(\gamma = 10^\circ\), the correction coefficient for all types of milling is \(k_\gamma = 1\). As is evident from the tables, the greatest influence of the rake angle is observed when milling the most ductile metal, red copper, which fully corresponds to the premises stated above.

Influence of Wear on Cutting Forces

Wear of the cutter teeth has a significant influence on the cutting forces when milling non-ferrous metals. Wear proceeds along

Table 8

| Metal grade | \multicolumn{3}{c}{Cylindrical high-speed cutter} | \multicolumn{2}{c}{Face high-speed cutter} | \multicolumn{2}{c}{Face carbide cutter} |
|---|---:|---:|---:|---:|---:|---:|---:|
| | \(\gamma = 0\) | \(\gamma = 20^\circ\) | \(\gamma = 30^\circ\) | \(\gamma = 20^\circ\) | \(\gamma = 30^\circ\) | \(\gamma = 0\) | \(\gamma = 20^\circ\) |
| L62 | 1,2 | 0,88 | 0,8 | 0,9 | 0,82 | 1,14 | 0,87 |
| LS59-1 | 1,15 | 0,9 | 0,85 | 0,92 | 0,83 | 1,1 | 0,92 |
| BrAZh9-4 | 1,2 | 0,84 | 0,72 | 0,88 | 0,78 | 1,14 | 0,87 |
| D-1 | 1,14 | 0,9 | 0,85 | 0,88 | 0,76 | 1,26 | 0,75 |
| AL-10 | 1,15 | 0,85 | 0,79 | 0,89 | 0,78 | 1,14 | 0,85 |
| AL-4 | 1,18 | 0,89 | 0,82 | 0,87 | 0,76 | 1,37 | 0,65 |
| Copper | 1,42 | 0,76 | 0,56 | 0,75 | 0,57 | 1,25 | 0,74 |

back faces. In this case the area of contact of the cutter teeth with the cutting surface increases, which leads to an increase in the forces on the back face, whereas the forces on the front face remain practically unchanged.

Since the forces on the back face do not depend on feed, the addition to the circumferential force must be constant, for a given wear, for all feeds. This is well confirmed by the graph in Fig. 3, where

Fig. 3.

Figure annotations:

• Vertical axis: \(P\), kg
• Horizontal axis: \(\delta\), mm
• Conditions: L62; VN8; \(\delta = 10^\circ\); \(D = 208\) mm; \(t = 3\) mm; \(B = 80\) mm; \(v = 200\) m/min
• Legend:
• \(+\): \(S_z = 0,0795\)
• \(\Delta\): \(S_z = 0,05\)
• \(\bullet\): \(S_z = 0,0317\)
• \(\blacktriangle\): \(S_z = 0,0196\)
• Curves marked: \(S_z = 0,503\), \(S_z = 0,317\), \(S_z = 0,201\)

the change in force due to wear is expressed by parallel lines for all feeds.

Similar results were also obtained under other conditions of milling with face cutters.

Power equations do not reflect this characteristic feature of the change in forces due to wear, since in them wear is taken into account by the factor \(k_z\), whereas it would be more correct to take account of the increase in forces due to wear as a second term in the equation, i.e., the cutting-force equation should consist of two terms; this is what is intended to be done in the further development of this question.

In cylindrical milling, when the chip thickness is measured in hundredths of a millimeter and the share of the forces on the flank face in the total circumferential force is large, the influence of wear on the cutting forces proves still more significant (Fig. 4). Analogous results—

Fig. 4.

—were also obtained when milling other metals with cylindrical cutters.

It should also be noted that, when milling with worn cutters, the influence of speed on the cutting forces, especially on the feed force, is considerably stronger.

Thus, in cylindrical milling of brass LS59-1 with a sharp carbide cutter, the cutting forces over a wide range of speed variation \((v = 20 \div 450\ \text{m/min})\) remain practically constant, whereas when milling the same brass with a worn cutter, at a certain speed a sharp increase in the cutting forces is observed (Fig. 5). In this case the cutting process practically ceases, the temperature rises sharply, and plastic “extrusion” of the metal from the cutting zone occurs. A similar phenomenon is observed for all non-ferrous metals except copper, but under different cutting conditions. An analogous process also occurs in face milling, especially with small feeds. Simultaneously with the sharp increase in the cutting forces, the quality of the machined surface deteriorates considerably. All this

forces the permissible wear on the flank face to be limited to the value \(\delta = 0.25 \div 0.3\) mm, which is considerably less than the permissible wear in the milling of ferrous metals.

On the basis of experimental data, the value of the correction factors for the circumferential force \(k_\delta\) for the maximum permissible wear varies within the range \(1.15 \div 5\), with the smaller values of the correction factors corresponding to maximum feeds, and the larger ones to minimum feeds.

Fig. 5.

The wide limits of variation in the value of \(k_\delta\) once again emphasize the need to calculate forces with allowance for the division of the forces between the rake and flank faces.

Influence of the Plan Angle on Cutting Forces

The influence of the plan angle on cutting forces was studied in milling with face carbide cutters on brasses L62 and LS59-1. The investigations showed that a decrease in the plan angle leads to an increase in the length of the cutting edge, which, in turn, leads to an increase in the forces on the flank face and, consequently, to an increase in the total circumferential cutting force.

The average values of the correction factor \(k_\varphi\) are given in Table 9.

Table 9

Influence of the Depth and Width of Milling on Cutting Forces

The study of the influence of the depth and width of milling was carried out both for cylindrical and for face milling. The experiments showed that, when milling non-ferrous metals with face cutters, the cutting forces are directly proportional to the depth of milling, and the exponent for the depth “\(l\)” may be taken as equal to zero. The width of milling in face milling, as well as the width and depth of cut in cylindrical milling, are correctly accounted for in equations (1) and (2).

In addition to the above-listed factors affecting the cutting forces, the work investigated the influence of the relative position of the face mill and the workpiece. In symmetrical milling, the circumferential force has the smallest value.

When the axis of the mill is displaced relative to the axis of the workpiece, the forces on the front face do not change, while the forces on the back face increase owing to the increase in the number of teeth simultaneously engaged under the chip, which leads to an increase in the circumferential force (Fig. 6).

Fig. 6.

Calculation of the Feed Force

The feed force can be found either from power equations similar to (1) and (2), or through the ratio \(\dfrac{p_s}{p}\). It is considerably simpler to use the second method of determining the feed force, which was done in the present work.

In up cylindrical milling, the ratio \(\dfrac{p_s}{p}\) for all non-ferrous metals under any milling conditions lies within the limits

\[ \frac{p_s}{p} = 1 \pm 10\%. \tag{3} \]

In down milling, the ratio is

\[ \frac{p_s}{p} = 0.8 \pm 10\%. \tag{4} \]

In face milling, the ratio \(\dfrac{p_s}{p}\) has a more complex dependence than in cylindrical milling. For symmetrical milling with face mills of high-speed steel and carbide, the ratio \(\dfrac{p_s}{p}\) for all the metals studied, as a function of feed, is shown in Figs. 7 and 8.

Fig. 7.

Fig. 8.

For asymmetric milling with face mills, the dependence of the ratio \(\frac{p_s}{p}\) on the relative position of the cutter and the workpiece was studied in milling brass L62 (Fig. 9). Analysis of this study showed that, for both high-speed-steel and carbide face mills, the ratio \(\frac{p_s}{p}\) is expressed by equation (5)

\[ \frac{p_s}{p}=\frac{p_s}{p}(s_z)+2i, \tag{5} \]

where \(\frac{p_s}{p}(s_z)=\frac{p_s}{p}\) for symmetric milling and is determined from the graphs in Figs. 7 and 8.

\[ i=\pm\frac{e}{D} \]

is determined as a function of the relative position of the cutter and the workpiece; here \(l\) is the distance between the axes of the cutter and the workpiece, and \(D\) is the cutter diameter.

Fig. 9.

The “\(+\)” sign is taken when cutting begins with a smaller thickness and ends with a larger one.

The “\(-\)” sign is taken when cutting begins with a larger thickness and ends with a smaller one.

In view of the fact that the influence of cutting speed and milling width on the ratio \(\frac{p_s}{p}\) is small, it may be neglected.

LITERATURE

1. A. M. Rosenberg, Dynamics of Milling. Moscow, State Publishing House “Soviet Science,” 1945.

1. In A. M. Rozenberg’s equations the influence of speed was not taken into account, since all the investigations were carried out at low speeds, where the influence of speed is insignificant; in equations 1 and 2 the influence of speed is taken into account. ↩