CALCULATION OF CUTTING FORCES IN FACE MILLING

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

At the present time, for calculating cutting forces in milling, empirical power-law relationships of the following type are used:

$$ P = C B^{w} Z S_z^{y} t^{x} D^{u} V^{n} K. $$

The coefficients \(C\) and \(K\), as well as the exponents for the parameters of the milling process \(w, y, x, u, n\) that appear in these equations, have no strict physical meaning and, consequently, cannot correctly take into account the dependence of the cutting force on the parameters of the cutting process, the physicomechanical properties of the work material, the cutter material, and the tool geometry. These equations describe the cutting process only approximately and, for this reason, do not provide sufficient accuracy in the calculation results.

The main source of errors in the power-law relationship is the fact that this formula does not take into account the presence of two independent groups that make up the total cutting force, whose physical nature is entirely different.

The forces acting on the rake face of the tool are determined by the process of deforming the metal into a chip, and therefore depend primarily on the volume of the layer being removed and on the degree of its deformation.

The forces acting on the flank face of the tool do not participate in the process of chip deformation and depend neither on the volume of metal being removed nor on the degree of chip deformation. This group of forces, being the forces of elastoplastic contact of the tool flank face with the machined surface, depends primarily on the contact area and the elastic properties of the work material.

It is also necessary to take into account the presence of two independent groups of forces because, under different cutting conditions, their ratio will be different.

With thick chips and a sharply sharpened tool, the forces on the rake face are considerably greater than the forces on the flank face, and the latter may be neglected.

With thin chips, which occurs especially often in milling, the forces on the flank face are comparable with the forces on the rake face and cannot be neglected.

A change in the rake angle affects only the forces on the rake face, whereas the forces on the flank face remain constant.

Thus, in Fig. 1, the straight-line dependences of the cutting force on feed at constant temperature (galvanometer reading \(T=25\,\mu\text{A}\)) for different rake angles cut off on the ordinate axis one and the same value of the forces on the flank face—\(6\) kg. Consequently, in the equations of the cutting forces for different rake angles, only those parts that determine the forces on the rake face will be variable.

Fig. 1. Dependence of the circumferential force on feed at constant temperature (galvanometer reading \(T=25\,\mu\text{A}\)) for different rake angles.

Conversely, when the wear of the teeth of a face mill on the flank face increases, the total cutting force increases only due to the forces acting on the flank face, whereas the forces on the rake face remain unchanged. The validity of this proposition is confirmed by the graph in Fig. 2.

It is evident from the graph that the straight lines \(P=f(S_z)\) for different degrees of wear, \(T=25\,\mu\text{A}\), are parallel to one another but displaced along the vertical axis. The vertical displacement is caused by the increase of forces on the flank face. The parallelism of the straight lines is explained by the constancy of the forces on the rake face. For small feeds, when the forces on the rake face are small, the increase of forces on the flank face due to wear will lead to a significant relative increase of the total force. For large feeds, when the forces on the rake face are the principal forces, the increase of forces on the flank face due to wear will lead to a less substantial relative change of the total force.

In power-law equations, the increase of forces due to tool wear is taken into account by a common multiplier and, as is seen from Fig. 2, this multiplier must be different for different feeds.

The indicated imperfections of the power-law dependence made it necessary to seek a form of equation for the total force in milling that would separately take into account the influence of the various parameters of the milling process on each of the indicated groups of cutting forces.

Graph labels and notes: \(P\), kg; \(S_z\), mm/tooth. Material and conditions: L62; VK8; \(z=8\); \(D=207\) mm; \(B=80\) mm; \(t=3\) mm; \(T=25\ \mu\text{A}\). Curves: \(\delta=0.52\), \(\delta=0.3\), \(\delta=0.09\), \(\delta=0\). Legend: \(\blacktriangle\) — \(\tilde S_z=0.02\); \(\bullet\) — \(\tilde S_z=0.0317\); \(\triangle\) — \(\tilde S_z=0.05\); \(+\) — \(\tilde S_z=0.08\); \(\circ\) — \(\tilde S_z=0.125\); \(\nabla\) — \(\tilde S_z=0.201\); \(\times\) — \(\tilde S_z=0.317\).

Fig. 2. Dependence of the circumferential force on feed at constant temperature (galvanometer reading \(T=25\ \mu\text{A}\)) for various degrees of wear.

Below is a brief derivation of the equations for determining cutting forces in face milling, taking into account the separation of forces on the front and rear faces, and a method for determining the constant coefficients included in these equations.

The principal component of the cutting force on the front face is determined by the simple dependence [1]:

\[ P_{\mathrm{pg}}=\tau k a b, \tag{1} \]

where \(\tau\) is the shear stress in the chip shear zone, in kg/mm\(^2\);

\(a\) is the chip thickness, in mm;

\(b\) is the chip width, in mm;

\(k\) is a multiplier depending on the rake angle, the coefficient of friction, and the degree of chip deformation [1, 2] (a dimensionless quantity).

For simplicity and convenience, equation (1) may be written as

\[ P_{\mathrm{pg}}=C_{\mathrm{pg}}ab, \]

where

\[ C_{\mathrm{pg}}=\tau k. \]

The force on the rear face in the direction of the cutting speed is equal to

\[ P_{\mathrm{zg}}=C_{\mathrm{zg}}\sum l, \]

where \(\sum l\) is the total length of the cutting edges of the tool that are in contact with the cutting surface;

\(C_{zg}\) is a factor depending on the metal being machined, the cutting speed, the temperature, and the wear of the tool on the flank face.

Then the circumferential force in general form is written as

\[ P=P_{pg}+P_{zg}=C_{pg}ab+C_{zg}\sum l . \]

In expanded form, the expression for the mean circumferential force will have the following form:

\[ P_{\mathrm{avg}}= \frac{BtZ}{\pi D} \left[ C_{pg}S_z+ C_{zg}\frac{\Theta D}{2B} \left\{ \frac{1}{\sin\varphi} + \frac{l_n}{t} \left( 1-\frac{\sin\varphi_n}{\sin\varphi} \right) \right\} \right], \tag{2} \]

where \(B\) is the milling width, in mm;

\(t\) is the milling depth, in mm;

\(Z\) is the number of cutter teeth;

\(D\) is the cutter diameter, in mm;

\(S_z\) is the feed per tooth, in mm/tooth;

\(\Theta\) is the angle of contact of the cutter with the workpiece, in radians;

\(\varphi\) is the angle in the plane of the main cutting edge;

\(l_n\) is the length of the transitional cutting edge, in mm;

\(\varphi_n\) is the angle in the plane of the transitional cutting edge.

For the case of symmetrical milling with a cutter having an angle in plan \(\varphi=90^\circ\) and \(l_n=0\), equation (2) takes the form

\[ P_{\mathrm{avg}}= \frac{BtZ}{\pi D} \left[ C_{pg}S_z+ C_{zg}F_1\left(\frac{B}{D}\right) \right]. \]

The quantity \(F_1\left(\frac{B}{D}\right)\) can either be calculated

\[ F_1\left(\frac{B}{D}\right) = \frac{\arcsin\frac{B}{D}}{\frac{B}{D}}, \]

or taken from Table 1.

Table 1

The feed force is likewise composed of the force on the rake face \(P_{s\,pg}\) and the force on the flank face \(P_{s\,zg}\).

In this case

\[ P_{s\,pg}=U_{pg}ab, \]

\[ P_{s\,zg}=U_{zg}\sum l. \]

Here \(U_{pg}\) is a factor similar to \(C_{pg}\), and \(U_{zg}\) is similar to \(C_{zg}\), with

\[ U_{pg}=C_{pg}\tan(\eta-\gamma), \]

where \(\eta\) is the angle of friction, and \(\gamma\) is the rake angle. Then the feed force is expressed as

\[ P_s = P_{s\,\mathrm{pg}} + P_{s\,\mathrm{zg}} = U_{\mathrm{pg}} aB + U_{\mathrm{zg}} \sum l . \]

In expanded form, the expression for the average feed force in the case of symmetrical milling will have the following form:

\[ P_{s\,\mathrm{avg}} = \frac{Zt}{\pi} \left[ U_{\mathrm{pg}} S_z \sin \varphi_{\mathrm{red}} \cdot \frac{\Theta + \sin \Theta}{4} + U_{\mathrm{zg}} \sin \frac{\Theta}{2} \right], \]

where \(\varphi_{\mathrm{red}}\) is the reduced angle in plan [3].

In the simplest case, when \(\varphi = 90^\circ\), \(l_n = 0\), one may take \(\varphi_{\mathrm{red}} = \varphi\); then

\[ P_{s\,\mathrm{avg}} = \frac{BtZ}{\pi D} \left[ U_{\mathrm{pg}} S_z F_2\left(\frac{B}{D}\right) + U_{\mathrm{zg}} \right], \]

where \(F_2\left(\dfrac{B}{D}\right)\) can either be calculated from the formula

\[ F_2\left(\frac{B}{D}\right) = \frac{\Theta + \sin \Theta}{4 \dfrac{B}{D}}, \]

or found from Table 2.

Table 2

Numerous experiments carried out by the authors on ferrous and nonferrous metals showed that, for each metal at a constant rake angle, the coefficients \(C_{\mathrm{pg}}\), \(C_{\mathrm{zg}}\), \(U_{\mathrm{pg}}\), and \(U_{\mathrm{zg}}\) will be constant under a condition of constant deformation, which is close to the condition of constant temperature or to the condition of constant tool life.

The validity of this proposition is confirmed by the graph (Fig. 3), in which the circumferential force and the feed force vary, at constant temperature, in direct proportion to the feed.

From this graph it is possible, for each temperature, to determine the values of the coefficients \(C_{\mathrm{pg}}\), \(C_{\mathrm{zg}}\), \(U_{\mathrm{pg}}\), and \(U_{\mathrm{zg}}\) required for calculating cutting forces, and to present them as functions of temperature (Fig. 4) (for L62 brass, the values of \(C_{\mathrm{zg}}\) and \(U_{\mathrm{zg}}\) remain constant at all temperatures and are equal to \(C_{\mathrm{zg}} = U_{\mathrm{zg}} = 1.7\)).

Since, in carrying out each experiment, not only the forces but also the temperature were measured, from the experimental results and Fig. 4 one can construct a graph (Fig. 5) of the variation of the constants \(C_{\mathrm{pg}}\), \(C_{\mathrm{zg}}\), \(U_{\mathrm{pg}}\), and \(U_{\mathrm{zg}}\) as functions of speed for each feed (as an example, only one graph is given—the graph of the variation of \(C_{\mathrm{pg}}\)).

On the basis of the latter, one can obtain a graph (Fig. 6) of the variation of \(C_{\mathrm{pg}}\) as a function of speed and feed. Since each curve of this graph corresponds to a constant temperature, on the basis of Fig. 4 each of these curves will be not only a curve of constant \(C_{\mathrm{pg}}\), but also a curve of constant \(U_{\mathrm{pg}}\).

From this graph it is not difficult to determine intermediate values of \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) for any values of cutting speed and feed.

It was indicated above that the rake angle affects only the forces on the rake face; consequently, for different rake angles

Fig. 1. Dependence of the distances \(C_{\mathrm{pr}}\) and \(L_p\) on temperature for various rake angles

In-figure labels:

\[ C_{\mathrm{pr}},\ L_p \]

\[ \text{L62; VK8; } z=8;\quad D=207\ \mathrm{mm};\quad B=80\ \mathrm{mm};\quad t=3\ \mathrm{mm} \]

\[ \gamma = 0^\circ,\qquad \gamma = +10^\circ,\qquad \gamma = +18^\circ \]

\[ T\ \mathrm{m/min} \]

Fig. 3. Dependence of the circumferential force and the feed force on the feed per tooth at various temperatures

In-figure labels:

\[ P_s,\ \mathrm{kg} \]

\[ S_z,\ \mathrm{mm/tooth} \]

\[ T=15,\quad T=20,\quad T=25,\quad T=30,\quad T=35 \]

\[ P_k \]

\[ \text{L62; VK8; } \gamma=+10^\circ;\quad z=8;\quad D=207\ \mathrm{mm};\quad B=80\ \mathrm{mm};\quad t=3\ \mathrm{mm} \]

\[ S_z=0.02,\quad S_z=0.031,\quad S_z=0.05,\quad S_z=0.08,\quad S_z=0.125,\quad S_z=0.201,\quad S_z=0.317 \]

Scale of feed \(S_z\).

variables will be \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\). The dependence of \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) on the lead angle can be given only under the condition of constant temperature.

For this purpose one can use the graph (Fig. 4), which gives the dependence of the constants \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) on temperature for various lead angles.

Fig. 5. Dependence of the coefficient \(C_{\mathrm{pg}}\) on cutting speed for various feeds.

Visible labels in Fig. 5:

• Vertical axis: \(C_{\mathrm{pg}}\)
• Horizontal axis: \(v\), m/min
• Conditions: L62, VN8, \(z=8\); \(D=207\) mm; \(B=80\) mm; \(\gamma=+10^\circ\); \(t=3\) mm
• Legend:
• \(\blacktriangle\) — \(S_z=0.02\)
• \(\bullet\) — \(S_z=0.0317\)
• \(\triangle\) — \(S_z=0.05\)
• \(+\) — \(S_z=0.08\)
• \(\circ\) — \(S_z=0.125\)
• \(\nabla\) — \(S_z=0.201\)
• \(\times\) — \(S_z=0.317\)
• \(\blacktriangledown\) — \(S_z=0.50\)

Having determined from the graph in Fig. 6 the values of \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) for the principal angle (in the example given, the principal angle is the lead angle

Visible labels in Fig. 6:

• Vertical axis: \(S_z\), mm/tooth
• Horizontal axis: \(v\), m/min
• Conditions: L62; VN8; \(\gamma=+10^\circ\); \(B=80\) mm; \(t=3\) mm; \(z=8\); \(D=207\) mm
• Along the lower part of the graph:
• \(C_{\mathrm{pg}}=140,\ 130,\ 120,\ 115,\ 110,\ 105\)
• \(U_{\mathrm{pg}}=27,\ 23,\ 18,\ 14.5,\ 11.5,\ 7\)

Fig. 6. Dependence of the constants \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) on cutting speed and feed.

\(\gamma=+10^\circ\)), from the graph in Fig. 4 one can find the values of the coefficients \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) for the specified angle at the same temperature.

By an analogous procedure, the constants \(C_{\mathrm{zg}}\) and \(U_{\mathrm{zg}}\) are determined for calculating the forces on the rear face; however, for most metals their dependence on cutting speed and feed is so insignificant that it may be neglected without introducing any appreciable error into the calculation results.

As indicated above, the constants \(C_{\mathrm{zg}}\) and \(U_{\mathrm{zg}}\) depend to a considerable degree on the magnitude of tool wear along the rear face.

The procedure for determining the influence of wear on these constants is analogous to the procedure for determining the dependence of the constants \(C_{\mathrm{pg}}\) and \(U_{\mathrm{pg}}\) on the rake angle.

Experiments carried out by the authors on eleven grades of steels and four grades of nonferrous metals showed the high accuracy of calculating cutting forces on the basis of the method presented. The deviations of the actual cutting forces measured with a dynamometer differ from the calculated values by no more than 5–7%.

REFERENCES

1. A. M. Rozenberg and A. N. Eremin. Elements of the Theory of the Metal-Cutting Process. Mashgiz, 1956.

2. A. M. Rozenberg and L. A. Khvorostukhin. Hardness and stresses in a plastically deformed body. Journal of Technical Physics, Vol. XXV, No. 2, 1955.

3. L. A. Khvorostukhin. Calculation of forces in high-speed cutting on the basis of the physicomechanical characteristics of metals. Dissertation, Tomsk, 1953.