Nonuniformity in Milling

Docent A. M. Rosenberg.

It is known that the productivity of cutting work, expressed through the volume of chip removed per unit time, increases with an increase in the thickness of the chip being removed (with all other conditions constant). This circumstance must be taken into account in roughing conditions, when the principal task is to remove as large a volume of chip as possible per unit time.

In the milling process, both with a cylindrical and with a face mill, this is of no less importance than in other types of cutting work.

It is not difficult to be convinced of this from Fig. 1, which gives the experimentally obtained dependence of the specific work \(A_w\), expended on removing a unit volume of chip, on the mean chip thickness \(S_m\) (in microns) when milling cast iron \(H_B = 150\).

Fig. 1.

Similar graphs were also obtained experimentally by the author for other metals\(^{1,2}\).

All of them indicate that the specific work, as the mean chip thickness in milling increases, decreases noticeably; that is, working with a thicker chip, while expending the very same work, we remove a greater volume of chip, or, in other words, by expending the same power available to us, we obtain greater productivity.

From Fig. 1 it is evident that, in passing from \(S_m = 10\) microns to \(S_m = 100\) microns (both being possible in roughing work), we reduce the specific work by more than a factor of two, i.e., we increase productivity by the same factor, without increasing the expenditure of energy.

We shall not touch at all upon the question of tool life, assuming that tool life in all cases ensures the selected re-

Thus the rule of greatest productivity in rough milling may be formulated as follows:

The greatest possible productivity on a given machine can be obtained with full utilization of the machine power and while working, at the same time, with chips as thick as possible.

When selecting a roughing regime for milling, usually the milling depth (thickness of the layer removed) \(t\) and the milling width \(b\) are specified and, if the tool is also considered specified (number of teeth—\(z\), diameter \(D\)), then the machine power can be fully loaded by selecting various combinations of the number of revolutions \(n\) and the feed per tooth \(S_z\); moreover, it is quite natural that productivity will be greater in the case of choosing the largest possible feed \(S_z\) and the corresponding value of the number of revolutions \(n\) from the condition of using the power \(N_e\).

With an increase in feed, productivity will increase, but in parallel with this the force load on the tool and in the transmission mechanisms of the machine will increase. For each machine there exists, and can be determined, a limit to the increase in the torque on the cutter and of the total circumferential force. This limit is governed by the strength of the parts of the main spindle drive and of the feed-mechanism drive.

When selecting the regime of maximum productivity, it is necessary to take into account the actual load on the links of the machine drive, so as not to allow them to be overloaded.

The problem of taking the acting loads into account is complicated in the present case by the fact that the circumferential force and the torque on the cutter are, in the general case, not quantities constant in time, but undergo, as the tooth passes under the chip, more or less considerable fluctuations. These fluctuations, caused by the geometry of the cutting process and by the variability of the specific cutting pressure in milling, occur both in the operation of face mills and in the operation of cylindrical milling cutters. The variability of the force and torque (nonuniformity) can create unfavorable conditions when working on a milling machine, causing vibrations in individual parts of the machine, affecting the finish of the machined surface and the life of the cutter, and thereby reducing productivity. In addition, the force and torque, passing through all their instantaneous values \(zn\) times per minute, at their maxima can significantly overload individual links of the machine transmissions, thereby leading to their rapid wear and, in some cases, to breakages.

A number of works of a theoretical and experimental character\(^3\) had as their aim the study of questions of nonuniformity, but the conclusions obtained in them do not make it possible simply to determine the maximum force and torque on the cutter for the selected regime.

The average circumferential force and the average torque are not difficult to determine, having first found graphically or analytically the power at the spindle \(N_e\) for the selected regime. In this case

\[ P_{\mathrm{cp}}=\frac{4500N_e}{v}\ \text{kg} \]

\[ M_{\mathrm{cp}}=716,2\frac{N_e}{n}\ \text{kgm}. \]

But the maximum of these quantities, recurring \(zn\) times per minute, may considerably exceed their average value. Orientation in selecting the regime by the average values of these quantities is inadmissible, since this would prescribe a periodic overload of the machine links, especially considerable in face milling, where, as we shall show below,

the maximum of the force and of the torque in some cases may exceed their average values by several times.

We shall call the ratio of the maximum total circumferential force \((\Sigma P_{\mathrm{circ}})_{\max}\) to the average \(P_{\mathrm{avg}}\) the coefficient of nonuniformity \(\mu\)

\[ \mu=\frac{(\Sigma P_{\mathrm{circ}})_{\max}}{P_{\mathrm{avg}}} =\frac{M_{\max}}{M_{\mathrm{avg}}}. \]

If, for the selected operating condition, along with the easily determined \(P_{\mathrm{avg}}\) or \(M_{\mathrm{avg}}\), the value \(\mu\) is also known, we can simply determine \((\Sigma P_{\mathrm{circ}})_{\max}\) and \(M_{\max}\) and will have everything necessary for judging the admissibility of the selected condition both from the standpoint of utilization of the machine power and from the standpoint of the load on its transmission mechanisms.

I. Nonuniformity in face milling.

In face milling, when the width of the milled surface \(b\) is equal to or less than the cutter diameter \(D\) and is located symmetrically with respect to the cutter, the power equation at the cutter will be \(^{2)}\):

\[ N_e= \frac{2^{-\frac{k}{2}}\lambda n^{-\kappa}}{1000\cdot 4500} \cdot S^{\kappa+1}\cdot t^{l+1}\cdot Z^{-\kappa}\cdot D \left[ 1-\left(1-\frac{b}{D}\right)^{\frac{\kappa+2}{2}} \right] \tag{1} \]

hence the average circumferential force is

\[ P_{\mathrm{avg}}=\frac{4500N_e}{v} = \frac{2^{-\frac{k}{2}}}{\pi}\lambda\cdot S_z^{\kappa+1}\cdot t^{l+1}\cdot z \left[ 1-\left(1-\frac{b}{D}\right)^{\frac{\kappa+2}{2}} \right] \tag{2} \]

Here:

\[ \begin{aligned} \lambda &\text{ — a quantity depending on the material being machined and on the profile of the cutter tooth.}\\ n &\text{ — number of revolutions per minute.}\\ t &\text{ — depth of milling in mm.}\\ S &\text{ — feed in mm per minute.}\\ Z &\text{ — number of cutter teeth.}\\ S_z &\text{ — feed per one cutter tooth.}\\ D &\text{ — cutter diameter.}\\ b &\text{ — milling width in mm.}\\ k,l &\text{ — exponents characterizing the material being machined.} \end{aligned} \]

The circumferential force on a face mill for the given instantaneous position of its teeth is

\[ \sum P_{\mathrm{circ}}= \lambda\cdot S_z^{\kappa+1}\cdot t^{l+1}\sum \sin^{\kappa+1}\psi \tag{3} \]

hence

\[ \mu= \frac{\left(\sum P_{\mathrm{circ}}\right)_{\max}}{P_{\mathrm{avg}}} = \frac{ \pi\left(\sum \sin^{\kappa+1}\right)_{\max} }{ 2^{-\frac{k}{2}}Z \left[ 1-\left(1-\frac{b}{D}\right)^{\frac{k+2}{2}} \right] } \tag{4} \]

Thus, the coefficient of nonuniformity \(\mu\) is a function of \(\frac{b}{D}\) and \(Z\). These two quantities (for \(k=\mathrm{const}\)) completely determine the value-

of \(\mu\), since \(\left(\sum \sin^{\kappa+1}\psi\right)_{\max}\) is also a function of these two quantities.

For any ratio \(\dfrac{b}{D}\), we shall have the maximum sum of sines (and hence also the maximum total circumferential force) in the case when, over the given width \(b\), the greatest possible number of teeth is placed under the chip, and moreover they are placed symmetrically with respect to the axis of the workpiece. Such a position is shown in Fig. 2 for a twelve-tooth milling cutter \((Z=12)\) at \(b=b_1\). Calculating for this case

\[ \left(\sum \sin^{\kappa+1}\psi\right)_{\max} = \sin^{\kappa+1}\psi_1 + \sin^{\kappa+1}\psi_2 + \sin^{\kappa+1}\psi_3 + \sin^{\kappa+1}\psi_4, \]

and substituting all quantities into expression 4, we obtain the coefficient of nonuniformity \(\mu\), which will show how many times the maximum total circumferential force will be greater than the mean circumferential force for the given milling case.

Fig. 2.

If, for this case \((Z=12)\), we begin gradually to reduce the milling width \(b_1\), then the denominator of the fraction in expression 4 will gradually decrease, while the numerator will remain constant until the milling width becomes equal to \(b_2\). With this change of width from \(b_1\) to \(b_2\), the coefficient of nonuniformity \(\mu\) will gradually increase. Four teeth will be under the chip (at the instant \((\Sigma P)_{\max}\)).

If the milling width is reduced still further, then as soon as it becomes less than \(b_2\), only three teeth will be able to fit under the chip. The position corresponding to \(\left(\sum \sin^{\kappa+1}\psi\right)_{\max}\) will correspond to Fig. 3, and in the numerator of the fraction of expression 4 one must substitute

\[ \left(\sum \sin^{\kappa+1}\psi\right)_{\max} = \sin^{\kappa+1}\psi_{\mathrm{I}} + \sin^{\kappa+1}\psi_{\mathrm{II}} + \sin^{\kappa+1}\psi_{\mathrm{III}}. \]

At the moment when the milling width passes (as it is reduced) through the value \(b_2\), the numerator of expression 4, which had remained constant, immediately decreases, since

\[ \sin^{\kappa+1}\psi_{\mathrm{I}} + \sin^{\kappa+1}\psi_{\mathrm{II}} + \sin^{\kappa+1}\psi_{\mathrm{III}} < \sin^{\kappa+1}\psi_1 + \sin^{\kappa+1}\psi_2 + \sin^{\kappa+1}\psi_3 + \sin^{\kappa+1}\psi_4 \]

and the coefficient $\mu$, which had been gradually increasing, will at that moment abruptly decrease in value. With a further decrease in the milling width within the limits between $b_2$ and $b_3$, we shall again have a gradual decrease of the denominator with a constant numerator, i.e., a gradual increase of $\mu$. When the milling width reaches the value $b_3$, we shall again have a sudden decrease of $\mu$, and so on.

Such a stepwise process of variation of $\mu$ with variation of $\dfrac{b}{D}$ is presented for a number of values of $Z$ in Figs. 4–17.

From a comparison of these figures it is evident that, as $Z$ increases, the number of jumps of $\mu$ increases, but the absolute value reached by the nonuniformity coefficient at a jump decreases.

It is not difficult to observe that the jumps of the nonuniformity coefficient occur at values of $\dfrac{b}{D}$ satisfying the condition:

\[ \frac{b}{D} = \sin\left(\frac{m}{2}\cdot\frac{360^\circ}{Z}\right), \]

where $m$ is the greatest number of cutter teeth fitting under the chip at the given ratio $\dfrac{b}{D}$.

The number of jumps of $\mu$ for each $Z$ will be, in the case of an even number of teeth, equal to $\dfrac{Z}{2}-1$, and in the case of an odd number of teeth, $\dfrac{Z}{2}+\dfrac{1}{2}$.

From Figs. 4–17 it is evident that, with a small number of teeth on the cutter, the magnitude of the nonuniformity coefficient $\mu$ reaches high values; i.e., the maximum force on the cutter in this case considerably (by several times) exceeds its mean value.

From Fig. 9, for example, it is evident that the values of $\mu$ can be fairly large even at $Z=10$; thus, at $\dfrac{b}{D}=0.311$, $\mu=2$; this means that the maximum total tangential force and the maximum torque exceed the mean values of these quantities by a factor of two. At $\dfrac{b}{D}=0.59$, $\mu=1.45$. The fluctuations of the load on the cutter, i.e., the ratio of the maximum values of the forces and moments to their minimum values, will be still greater.

It is also evident from Fig. 9 that if, by changing the cutter diameter at a given milling width, we change the ratio $\dfrac{b}{D}$ from $\dfrac{b}{D}=0.311$ to $\dfrac{b}{D}=0.30$, then by this simple measure we reduce the value of $\mu$ from $\mu=2$ to $\mu=1.08$, i.e., we obtain a practically completely uniform cutter load, owing to which productivity can undoubtedly be increased by increasing the cutter feed.

If in the first case, at $\dfrac{b}{D}=0.311$, the maximum total tangential force on the cutter exceeds the minimum by several times, which at high cutting speeds will undoubtedly cause unsteady operation of the machine, then at $\dfrac{b}{D}=0.3$ we shall have a cutter load that is practically constant in magnitude.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

Fig. 16.

Fig. 17.

When considering Figs. 4–17, the question arises as to what the coefficient \(\mu\) will practically be equal to for values of \(\frac{b}{D}\) associated with a stepwise change in \(\mu\). Thus, for \(\frac{b}{D}=0.31\), from Fig. 9 we have two values: the lower one—\(\mu=1.05\), and the upper value—\(\mu=2.01\).

For \(\frac{b}{D}=0.59\), from Fig. 9 we have the values \(\mu\): 1.45 and 1.03. Thus, for each ratio \(\frac{b}{D}\) at which a jump in \(\mu\) occurs, we shall have two values of the coefficient of nonuniformity. The question is, which of these two values of \(\mu\) will we actually have during operation?

To answer this question it is necessary to turn to the real cutting process, with all the phenomena accompanying it. In Fig. 18 the case \(z=10,\ \frac{b}{D}=0.59\) is presented.

In analyzing the question posed, it is not difficult to arrive at the conclusion that the larger value of \(\eta\) will be obtained in the case where, at \(\frac{b}{D}=0.59\) (Fig. 18), the chip pressure acts simultaneously on three teeth.

Fig. 18.

According to Fig. 18, it would seem that this is precisely what happens, since at the moment when tooth 1 is at point \(A\), tooth 3 will already be at point \(B\). In reality, however (and this is confirmed by experimental verification), the chip removed by tooth 1 breaks away from the workpiece somewhat earlier than the tooth reaches point \(A\); i.e., tooth 1 loses its load before tooth 3 reaches point \(B\), enters under the chip, and is loaded by it. That is, the load can simultaneously be only on two teeth, and in practice we shall have the smaller value of the coefficient of nonuniformity \(\mu\). And in all cases where the ratio \(\frac{b}{D}\) is associated with a jump in \(\mu\), in practice we shall have in operation the smaller of the two values of \(\mu\), and by—

therefore the ratios \(\dfrac{b}{D}\) associated with jumps of \(\mu\), i.e., satisfying the condition:

\[ \frac{b}{D}=\sin\left(\frac{m}{2}\cdot\frac{360^\circ}{z}\right) \]

will be the most favorable from the point of view of uniformity of the cutting process.

From an analysis of equation 4, and also from a comparison of the figures we have presented, it may be concluded that an increase in the number of teeth of the milling cutter leads to a decrease in the values of \(\mu\).

From all that has been set forth it should be clear how important it is to know the magnitude of the coefficient of nonuniformity \(\mu\) when operating a milling machine, and how necessary it is to have an understanding of the regularities in the variation of \(\mu\) when selecting or designing a face mill for a given operation, when, by some often insignificant change in the ratio \(\dfrac{b}{D}\) or \(Z\), a substantial effect can be achieved. It is therefore necessary to be able simply to find the magnitude of \(\mu\).

Equation 4, although it makes it possible to calculate \(\mu\) exactly, is inconvenient to use, since each calculation takes a great deal of time. The graphs presented in Figs. 4–17 are considerably more convenient, but for each number of cutter teeth one would have to have a separate graph.

Sufficient accuracy for practice can be obtained from the proposed graph (see Fig. 19), on which the lines \(\mu=\mathrm{const}\) are plotted in the coordinates \(\dfrac{b}{D}\) and \(Z\). Here, for any value of \(\dfrac{b}{D}\) and \(z\), the value of \(\mu\) can simply be determined with an accuracy of up to 0.1–0.2. Some places on the graph seem unclear at first glance; for example, for \(z=3\) at \(\dfrac{b}{D}=0.85\) we do not have a value of \(\mu\) on the graph. But we see that for \(z=3\), at

\[ \frac{b}{D}=0.78 \quad\text{and at}\quad \frac{b}{D}=0.9,\quad \mu=1.3. \]

Moreover, the lines \(\mu=1.5\), \(\mu=1.7\), and \(\mu=1.1\) do not extend to \(z=3\). Hence it may be concluded that for \(z=3\), at \(\dfrac{b}{D}=0.85\), we shall have a value of \(\mu\) between 1.3 and 1.5. Likewise, for \(z=6\) and \(\dfrac{b}{D}=0.9\), it may be concluded that \(\mu\) will have a value between 1.1 and 1.3.

From the graph it is evident that, for small numbers of teeth, the values of \(\mu\) may be very large, especially at small ratios \(\dfrac{b}{D}\). In the region of larger \(z\) and \(\dfrac{b}{D}\), on the contrary, we deal with coefficients of nonuniformity more or less close to unity. At \(z>20\) and \(\dfrac{b}{D}>0.75\), we shall always have \(\mu<1.1\).

Figure 20 gives a graph of the optimal values of \(\dfrac{b}{D}\), which, for the chosen \(z\), give the minimum value of \(\mu\), i.e., which satisfy the relation

\[ \frac{b}{D}=\sin\left(\frac{m}{2}\cdot\frac{360^\circ}{z}\right). \]

Using this graph, by drawing a horizontal line through the chosen \(z\), at the intersections of this horizontal with the curves we obtain the optimal value of

Fig. 19.

\(\dfrac{b}{D}\). It should not be thought that in this case we shall have \(\mu=\mathrm{const}\). For \(z=\mathrm{const}\), the coefficient of nonuniformity will be the smaller, the larger the ratio \(\dfrac{b}{D}\). Thus, for \(z=10\), according to Fig. 20 we have 5 optimal values

$\dfrac{b}{D}$: 0.31; 0.588; 0.811; 0.95; 1.0. According to Fig. 9 the values of $\mu$ in this case will be: 1.05; 1.03; 1.02; 1.01; 1.03.

A complete picture of the relationship between $\mu$, $z$, and $\dfrac{b}{D}$ is given by Fig. 21, in which, in the spatial coordinates $z$—$\dfrac{b}{D}$—$\mu$, the dependence of the coefficient of nonuniformity $\mu$ on $z$ and $\dfrac{b}{D}$ is presented.

Fig. 20.

To construct all the graphs, we carried out all the calculations taking for the value $k$, which enters into the exponents of equation 4, the mean value $k=-0.25$.

From our experimental work with face mills it follows that, in face milling, the value of this quantity depends on the material being machined and varies within the limits 0.2–0.315.

Thus, for each material being machined, the indicated graphs should be constructed using its own value of \(k\). The error in constructing the graphs using the average value (\(k = -0.25\)) is in practice quite negligible; however, this frees us from the need to construct graphs for different materials being machined.

Fig. 21.

We have not touched here at all on the question of nonuniformity in the case of side cutting. This question will be considered by us together with the phenomenon of nonuniformity in the operation of a cylindrical milling cutter, with which it has much in common.

In conclusion, we give an example illustrating what effect can be achieved by taking into account the coefficient of nonuniformity in face milling.

Let it be given: power at the spindle, 4 HP.
The maximum torque at the spindle according to the strength of the spindle drive is \(M_{max} = 90\ \text{kgm}\).
Machining cast iron, \(H_B = 150\).

\[ \begin{aligned} b &= 120\ \text{mm}\\ t &= 10\ \text{''}\\ D &= 200\ \text{''}\\ z &= 10\ \text{''} \end{aligned} \qquad \begin{aligned} &\text{Data for calculating}\\ &\text{power:}\\ \lambda &= 85\\ k &= -0.315\\ l &= -0.09 \end{aligned} \]

\[ \frac{b}{D}=\frac{120}{200}=0.6 \]

For \(z = 10\) and \(\dfrac{b}{D}=0.6\), \(\mu = 1.41\).

If we take \(n = 32\ \text{rev/min}\), in order to obtain a cutting speed of \(20\ \text{m/min}\), then at a feed \(S = 110\ \text{mm/min}\) we obtain the average torque at the spindle

\[ M_{cp}=63.5\ \text{kgm}. \]

Taking into account the nonuniformity coefficient, we shall have

\[ M_{max}=\mu M_{cp}=1.41\cdot 63.5=90\ \text{kgm}. \]

Thus, at \(n = 32\ \text{rev/min}\) and all the other adopted conditions, according to the strength of the spindle drive, which permits \(M_{max}=90\ \text{kgm}\), a feed greater than \(S = 110\ \text{mm/min}\) cannot be used.

If, in this same case, we take the cutter diameter as \(218\ \text{mm}\) and thereby obtain the ratio

\[ \frac{b}{D}=\frac{120}{218}=0.55, \]

then the nonuniformity coefficient for this case will be \(\mu = 1.1\).

Applying, with all other conditions constant, a feed \(S = 164\ \text{mm/min}\), we shall have the average torque

\[ M_{cp}=82\ \text{kgm}. \]

Taking into account the nonuniformity coefficient,

\[ M_{max}=\eta \cdot M_{cp}=1.1\cdot 82=90.2\ \text{kgm}. \]

In other words, by a simple change in the cutter diameter, with the same maximum torque available at the machine spindle, we increase productivity by 49%, since

\[ \frac{164}{110}=1.49. \]

Moreover, in the second case, at the higher productivity, the working conditions of the machine will be more favorable, owing to the greater stability of the load.

II. Nonuniformity in the operation of a cylindrical cutter.

When a cylindrical cutter is operating, irrespective of the helix angle of the tooth, the average circumferential force may be obtained from the power equation

\[ N_e=\frac{2^{k+1}}{k+2}\cdot \frac{\lambda n^{-k}}{1000\cdot 4500}\cdot b\cdot s^{k+1}t^{\frac{k+2}{2}}Z^{-k}D^{-\frac{k}{2}} \]

\[ P_{cp.}=\frac{2^{k+1}}{(k+2)}\frac{\lambda\cdot b}{\pi}\,S_z^{k+1}Z\cdot \sin^{k+2}\left(\frac{\varphi}{2}\right) \]

Here \(\varphi\) is the contact angle.

To determine the nonuniformity coefficient \(\mu\), it is necessary to have an expression for \((\sum P_{okr})_{max}\), which will differ for cutters with straight and helical teeth.

Cylindrical cutter with straight teeth

For a cutter with straight teeth we shall have

\[ \sum P_{okr}=\sum (b S_e K_s); \]

\[ K_s=\lambda S_e^k \]

\[ S_e=S_z\sin\psi \]

where \(S_e\) is the chip thickness on the tooth at the given instant,
\(\psi\) is the angle determining the instantaneous position of the tooth,
\(S_z\) is the feed per tooth.

\[ \sum P_{okr}=\lambda b S^{{k+1}}\sum(\sin^{k+1}\psi) \]

\[ \mu=\frac{(\sum P_{okr})_{max}}{P_{sr.}} =\frac{K+2}{2^{k+1}}\cdot \frac{\pi\left(\sum \sin^{k+1}\psi\right)_{max}} {Z\cdot \sin^{k+2}\left(\frac{\varphi}{2}\right)} \tag{6} \]

Denoting the central angle between two teeth by \(\varepsilon\), we shall have

\[ \left(\sum \sin^{k+1}\psi\right)_{max} = \sin^{k+1}\varphi +\sin^{k+1}(\varphi-\varepsilon) +\sin^{k+1}(\varphi-2\varepsilon) + \]

\[ +\sin^{k+1}\{\varphi-(m-1)\varepsilon\}. \]

Putting, moreover, \(\frac{\varepsilon}{\varphi}=u\), we obtain

\[ \left(\sum \sin^{k+1}\psi\right)_{max} = \sin^{k+1}\varphi +\sin^{k+1}\left[\varphi(1-u)\right] +\sin^{k+1}\left[\varphi(1-2u)\right] + \]

\[ +\sin^{k+1}\left[\varphi\{1-(m-1)u\}\right] \tag{7} \]

In this expression \(m\) is the number of teeth simultaneously under chip load.

Substituting 7 into 6, and also replacing

\[ Z=\frac{2\pi}{\varepsilon}=\frac{2\pi}{u\varphi} \]

we obtain:

\[ \mu= \frac{K+2}{2^{k+2}}\cdot u\varphi \left\{ \frac{ \sin^{k+1}\varphi +\sin^{k+1}\left[\varphi(1-u)\right] +\sin^{k+1}\left[\varphi(1-2u)\right] +\ldots } {\sin^{k+2}\left(\frac{\varphi}{2}\right)} + \right. \]

\[ \left. \ldots+ \frac{ \sin^{k+1}\left[\varphi\{1-(m-1)u\}\right] } {\sin^{k+2}\left(\frac{\varphi}{2}\right)} \right\} \tag{8} \]

From expression 8 it may be concluded that the nonuniformity coefficient \(\mu\) depends on \(u=\frac{\varepsilon}{\varphi}\) and on \(\varphi\).

The quantity \(\mu\) is also influenced by the value \(K\), which enters into the exponent of the sine and depends on the material being machined. Guided by our work on cylindrical milling cutters, \(^{1)}\) for subsequent calculations concerning the coefficient of nonuniformity in the operation of a cylindrical cutter, for all materials being machined one may take the mean value \(k=-0.3\).

In addition, it is not difficult to show that, in the main, \(\mu\) depends only on \(u\), while the magnitude of the contact angle \(\varphi\) has a very slight influence.

If we specify the limiting values \(\varphi_{\min}=8^\circ\) and \(\varphi_{\max}=60^\circ\), which will correspond to

\[ \frac{t}{D}=0.005 \quad\text{and}\quad \frac{t}{D}=0.25 \]

(limits beyond which, in production, one never goes when working with cylindrical cutters), then the influence of the angle \(\varphi\) on the value of \(\mu\) may be seen from Table 1.

Table 1

\(U=\dfrac{\varepsilon}{\varphi}\)	\(\mu\) at \(\varphi=8^\circ\)	\(\mu\) at \(\varphi=60^\circ\)	\(\mu\) mean
0.2	1.155	1.155	1.155
0.4	1.36	1.36	1.36
0.6	1.54	1.52	1.53
0.8	1.78	1.75	1.765
1	1.68	1.60	1.64
1.5	2.52	2.40	2.46
2	3.36	3.20	3.28
3	5.04	4.8	4.92
5	8.4	8.0	8.2

Thus the influence of \(\varphi\) on the coefficient of nonuniformity \(\mu\) may be entirely neglected.

Consequently, for a straight-tooth cutter, \(\mu\) depends only on \(\dfrac{\varepsilon}{\varphi}\), i.e., on how many teeth are simultaneously under the chip.

The fact that we neglect the influence of \(\varphi\), and also take, for the value \(K\), the mean value \(K=-0.3\), introduces certain errors into all calculations; however, these errors are practically immaterial, since, no matter how precisely we determine the magnitude of the coefficient of nonuniformity \(\mu\), we still can never account for the change in nonuniformity under the influence of the runout of the cutter, which is always present. Therefore, we consider the assumptions we have adopted to be fully acceptable.

Cylindrical cutter with a helical tooth.

When a cutter with a helical tooth is operating, the mean circumferential force is expressed by the same equation (5). The maximum sum of the circumferential forces has the following dependence \(^{4)}\)

\[ \left(\sum P_{\text{circ.}}\right)_{\max} = D\,\omega\,S_z^{k+1}\operatorname{tng}\beta\, \frac{2^{k+1}}{K+2} \sum \left[ \sin^{k+2}\left(\frac{\psi_2}{2}\right) - \sin^{k+2}\left(\frac{\psi_1}{2}\right) \right]_{\max} \tag{9} \]

Here \(\beta\) is the helix angle of the tooth,

\(\psi_1\) and \(\psi_2\) are the angles determining the instantaneous position of each tooth.

The expression for \(\mu\) is obtained in the following form:

\[ \mu=\frac{\left(\sum P_{o\kappa p.}\right)_{max}}{P_{op}} = \frac{\pi D\operatorname{tg}\beta}{bz}\cdot \frac{\sum\left[\sin^{\kappa+2}\left(\frac{\psi_2}{2}\right)-\sin^{\kappa+2}\left(\frac{\psi_1}{2}\right)\right]_{max}} {\sin^{\kappa+2}\left(\frac{\varphi}{2}\right)}; \]

\[ \frac{\pi D\operatorname{tg}\beta}{Z}=h \]

\(h\)—the axial distance between two teeth.

\[ \mu=\frac{h}{b}\, \frac{\sum\left[\sin^{\kappa+2}\left(\frac{\psi_2}{2}\right)-\sin^{\kappa+1}\left(\frac{\psi_1}{2}\right)\right]_{max}} {\sin^{\kappa+2}\left(\frac{\varphi}{2}\right)} \tag{10} \]

If the position of the teeth of the milling cutter is represented on the developed surface of contact, then

\[ \sum\left[\sin^{\kappa+2}\left(\frac{\psi_2}{2}\right)-\sin^{\kappa+2}\left(\frac{\psi_1}{2}\right)\right]_{max} \]

will always correspond to such a position of the teeth when one of them passes through the point \(q\), lying on the line of exit of the teeth from under the chip \(pq\). (Fig. 22)

Fig. 22.                Fig. 23.

Moreover, on this tooth there will be the maximum possible transverse section of the chip, and on all teeth simultaneously located under the chip, the maximum total force. This is proved as follows*).

If, over the milling width \(b\), we separate out a width \(h\), equal to the axial distance between two teeth, then over the width \(h\) we shall have \(\sum P=const.\)

Over the width \((b-h)\), the depicted position of the teeth corresponds to \(\left(\sum P\right)_{max}\), since, upon their further displacement over the width supplementing \((b-h)\) to \(h\), there will be an increase in the forces.

Let us determine on what factors the value of \(\mu\) depends when a cylindrical milling cutter with helical teeth is operating. For this purpose let us consider several possible ratios

\[ \frac{\varepsilon}{\varphi}=u \quad \text{and} \quad \frac{b}{h}=c. \]

1st case: \(\varepsilon>\varphi;\ b<h\) (Fig. 23).

* The present proof, as well as the derivation following it, was proposed by the author’s collaborator B. V. Sudnishnikov, who took part in the author’s work.

For this case we have

\[ U>1;\quad C<1; \]

\[ \frac{\psi_1}{\varepsilon}=\frac{a}{h};\qquad \frac{\varepsilon}{\varphi}=\frac{-h}{a+b};\qquad a=\frac{\varphi h-b\varepsilon}{\varepsilon}; \]

\[ \psi_1=\frac{\varepsilon a}{h}=\varphi-\frac{b}{h}\cdot\varepsilon . \]

Substituting into the last relation \(\frac{b}{h}=C\) and \(\varepsilon=u\varphi\), we have

\[ \psi_1=\varphi(1-cu) \]

Substituting what has been obtained into relation (10) for \(\mu\), we obtain:

\[ \mu=\frac{1}{c}\, \frac{ \sin^{\kappa+2}\left(\frac{\varphi}{2}\right) - \sin^{\kappa+2}\left[\frac{\varphi}{2}(1-cu)\right] }{ \sin^{\kappa+2}\left(\frac{\varphi}{2}\right) } \]

\[ \mu=\frac{1}{c}-\frac{1}{c}\cdot \frac{ \sin^{\kappa+2}\left[\frac{\varphi}{2}(1-cu)\right] }{ \sin^{\kappa+2}\left(\frac{\varphi}{2}\right) } \tag{11} \]

II case: \(\varepsilon>\varphi;\ b>h\) (Fig. 24).

Let us denote by \(p\) the number of whole intervals \(h\) fitting into the width \(b\) (for Fig. 24, \(p=2\)).

Fig. 24.

Fig. 25.

\[ \frac{\varphi-\psi_1}{\varepsilon}=\frac{b_1}{h};\qquad \psi_1=\varphi-\varepsilon\frac{b_1}{h};\qquad b_1=b-ph; \]

\[ \psi_1=\varphi-\varepsilon\frac{b-ph}{h} \]

\[ \psi_1=\varphi-u\varphi(c-p) \]

\[ \psi_1=\varphi\left[1-u(c-p)\right]. \]

The expression for \(\left(\sum P_{\mathrm{okr.}}\right)_{\max}\) for this case will take the form:

\[ \left(\sum P_{\mathrm{okr.}}\right)_{\max} = D_{\lambda}S_z^{\kappa+1}\operatorname{tng}\beta\, \frac{2^{\kappa+1}}{k+2} \left\{ (p+1)\sin^{\kappa+2}\left(\frac{\varphi}{2}\right) - \right. \]

\[ \left. -\sin^{\kappa+2}\left[ \frac{\varphi}{2}\left\{1-u(c-p)\right\} \right] \right\}. \]

The expression for the coefficient of nonuniformity is obtained as

\[ \mu=\frac{p+1}{c}-\frac{1}{c}\cdot \frac{ \sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c-p)\}\right] }{ \sin^{k+2}\left(\frac{\varphi}{2}\right) }. \tag{12} \]

III case: \(\varepsilon<\varphi;\ b<h\) (Fig. 25).

\[ \frac{\varphi-\psi_1}{b}=\frac{\varepsilon}{h};\qquad \psi_1=\varphi-\frac{b\varepsilon}{h};\qquad \psi_1=\varphi(1-cu) \]

For this case we shall have:

\[ \begin{aligned} \left(\sum P_{okr}\right)_{max} &=D\Lambda S_z^{k+1}\operatorname{tng}\beta\, \frac{2^{k+1}}{k+2} \Bigg\{ \sin^{k+2}\left(\frac{\varphi}{2}\right)+ \\ &\quad+\sin^{k+2}\left[\frac{\varphi}{2}(1-u)\right] +\sin^{k+2}\left[\frac{\varphi}{2}(1-2u)\right] +\sin^{k+2}\left[\frac{\varphi}{2}(1-3u)\right]+ \\ &\quad+u\ \text{etc.} -\sin^{k+2}\left[\frac{\varphi}{2}(1-cu)\right] -\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c+1)\}\right]- \\ &\quad-\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c+2)\}\right] -\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c+3)\}\right] -\text{etc.} \Bigg\} \end{aligned} \]

\[ \begin{aligned} \mu&=\frac{1}{c}+\frac{1}{c}\cdot \frac{ \sin^{k+2}\left[\frac{\varphi}{2}(1-u)\right] +\sin^{k+2}\left[\frac{\varphi}{2}(1-2u)\right] }{ \sin^{k+2}\left(\frac{\varphi}{2}\right) }+\\ &\quad+\frac{1}{c}\cdot \frac{ \sin^{k+2}\left[\frac{\varphi}{2}(1-3u)\right]+\text{etc.} }{ \sin^{k+2}\left(\frac{\varphi}{2}\right) } -\frac{1}{c}\cdot \frac{ \sin^{k+2}\left[\frac{\varphi}{2}(1-cu)\right] }{ \sin^{k+2}\left(\frac{\varphi}{2}\right) }-\\ &\quad-\frac{1}{c} \frac{ \sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c+1)\}\right] +\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c+2)\}\right]+\text{etc.} }{ \sin^{k+2}\left(\frac{\varphi}{2}\right) }\ldots \tag{13} \end{aligned} \]

IV case: \(\varepsilon<\varphi;\ h<b\) (Fig. 26)

For this case

\[ \frac{\varphi-\psi_1}{\varepsilon}=\frac{b-ph}{h};\qquad \psi_1=\varphi\left[1-u(c-p)\right] \]

\[ \left(\sum P_{okr}\right)_{max} = D\Lambda S_z^{k+1}\operatorname{tng}\beta\, \frac{2^{k+1}}{k+2}\times \]

\[ \times \left\{ (p+1)\sin^{k+2}\left(\frac{\varphi}{2}\right) +\sin^{k+2}\left[\frac{\varphi}{2}(1-u)\right]+ \right. \]

\[ \left. +\sin^{k+2}\left[\frac{\varphi}{2}(1-2u)\right] +\text{etc.}- \right. \]

Fig. 26.

\[ -\sin^{k+1}\left[\frac{\varphi}{2}\{1-u(c-p)\}\right] -\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c-p+1)\}\right] -\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c-p+2)\}\right] -\text{etc.}\}; \]

\[ \mu=\frac{p+1}{c}+ \frac{1}{c\cdot \sin^{k+2}\left(\frac{\varphi}{2}\right)} \cdot \left\{ \sin^{k+2}\left[\frac{\varphi}{2}(1-u)\right] +\sin^{k+2}\left[\frac{\varphi}{2}(1-2u)\right] +\sin^{k+2}\left[\frac{\varphi}{2}(1-3u)\right] +\text{etc.} \right\} \]
\[ - \frac{1}{c\cdot \sin^{k+2}\left(\frac{\varphi}{2}\right)} \left\{ \sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c-p)\}\right] +\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c-p+1)\}\right] +\sin^{k+2}\left[\frac{\varphi}{2}\{1-u(c-p+2)\}\right] +\text{etc.} \right\}. \tag{14} \]

It is not difficult to observe that equation 14 is the general form of the equation of nonuniformity, of which equations 11, 12, and 13 are special cases.

From equation 14 it may be concluded that the quantity \(\mu\) depends on \(c=\dfrac{b}{h}\), \(u=\dfrac{\varepsilon}{\varphi}\), and \(\varphi\); \(p\) is a function of \(c\), cannot vary independently of \(c\), and cannot have an independent influence on \(\mu\).

The influence of the contact angle \(\varphi\) is entirely absent in those cases when \(u\geq 1\) and \(u(c-p)\geq 1\).

And in all other cases the influence of \(\varphi\) on \(\mu\) is very insignificant, as can be seen from Table 2.

Table 2.

\(u=\dfrac{\varepsilon}{\varphi}\)	\(c=\dfrac{b}{h}\)	\(\mu\), \(\varphi=8^\circ\)	\(\mu\), \(\varphi=60^\circ\)	\(\mu_{\text{average}}\)
0.2	0.1	1.14	1.11	1.125
0.2	0.5	1.11	1.07	1.09
0.2	1.5	1.036	1.023	1.03
0.5	0.1	1.34	1.25	1.29
0.5	0.5	1.195	1.176	1.185
0.5	1.5	1.068	1.059	1.063
1	0.1	1.62	1.49	1.55
1	0.5	1.405	1.35	1.37
1	1.5	1.23	1.11	1.17
3	0.1	4.6	4.3	4.45
3	0.5	2	2	2
3	1.5	1.33	1.33	1.33
10	0.1	10	—	—
10	0.5	2	—	—
10	1.5	1.33	—	—

In Table 2 the values of $\mu$ have been calculated for a number of particular values of $u$ and $c$, taken over fairly wide ranges. Within the limits $\varphi = 8^\circ$, $\varphi = 60^\circ$ the influence of $\varphi$ is so small that it may be neglected. Thus the nonuniformity coefficient $\mu$ can be expressed as a function of $\dfrac{\varepsilon}{\varphi}$ and $\dfrac{b}{h}$.

In view of the complexity of the expressions obtained for $\mu$, we set ourselves the goal of constructing a graph by which, for any values encountered in production of $\dfrac{b}{h}$ and $\dfrac{\varepsilon}{\varphi}$, it would be possible simply to find the value of the nonuniformity coefficient $\mu$. In all calculations we took $K=-0.3$, as an average for machining various materials with a cylindrical milling cutter. In addition, the value of $\mu$ was calculated by us as the mean of the values for $\varphi = 8^\circ$ and $\varphi = 60^\circ$.

In Fig. 27 a graph is presented of the variation of $\mu$ for several particular values of

\[ u=\frac{\varepsilon}{\varphi} \]

as a function of

\[ c=\frac{b}{h}. \]

From the graph it is seen that $\mu$ decreases (uniformity increases) with a decrease in $\dfrac{\varepsilon}{\varphi}$, i.e., either with an increase in the number of cutter teeth at a constant ratio $\dfrac{t}{D}$, or with an increase in $\dfrac{t}{D}$ at constant $Z$. With variation of $\dfrac{b}{h}$, the nonuniformity coefficient changes periodically, decreasing to $\mu = 1$ when $\dfrac{b}{h}$ is equal to any integer, while in the intervals between these values it increases to more or less high values. The maxima of $\mu$ are not located in the middle between integer values of $\dfrac{b}{h}$. From this it may be concluded that the maximum of the nonuniformity coefficient does not coincide with the maximum degree of nonuniformity, which, judging from the works of Granovskii and Salomon’s[^3], is located exactly midway between integer values of $\dfrac{b}{h}$.

Fig. 27.

In Fig. 28 we give a summary graph of the nonuniformity coefficient. Here, in the coordinates $\dfrac{b}{h}$ and $\dfrac{\varepsilon}{\varphi}$, curves of constant $\mu$ are shown, with an indication on each of them of the value of $\mu$ to which it corresponds.

From this graph, for any $\dfrac{b}{h}$ within the range from 0.01 to 10 and for any $\dfrac{\varepsilon}{\varphi}$ within the range from 0.2 to 10, the value is simply found:

\[ \mu=\frac{\left(\sum P_{\mathrm{okr}}\right)_{\max}}{P_{\mathrm{sr}}}= \]

\[ =\frac{M_{\max}}{M_{\mathrm{cp}}}. \]

Thus, having found from the power equation \(P_{\mathrm{cp}}\) or \(M_{\mathrm{cp}}\), and having determined from Fig. 28 the value \(\mu\), we easily obtain the maximum circumferential force on the cutter or the maximum torque for a specified cutting condition.

In order to facilitate the calculation of the values \(\dfrac{\varepsilon}{\varphi}=f(D,t_z,Z)\) and

\[ \frac{b}{h}=f(D,\beta,Z,b), \]

a combined graph is given in Fig. 29.

Fig. 28.

As we have established above, for a straight-tooth cutter \(\mu\) depends only on \(\dfrac{\varepsilon}{\varphi}\). In Fig. 28 the curves of constant \(\mu\) at \(\dfrac{b}{h}=0.01\) give, for differ-

… values of the ratio \(\dfrac{\varepsilon}{\varphi}\)—of the magnitudes of the coefficients \(\mu\) for a straight-tooth milling cutter, as is not difficult to verify if they are compared with the data of Table I.

From Fig. 28 it is seen that \(\mu\) reaches high values when \(\dfrac{b}{h}<1\) and \(\dfrac{\varepsilon}{\varphi}>1\). Such ratios are not uncommon in production, since modern milling cutters, especially those intended for heavy work, usually have a small number of teeth.

Fig. 29.

From these same graphs it follows that, in order to achieve the greatest possible uniformity, one should reduce \(\dfrac{\varepsilon}{\varphi}\) and increase \(\dfrac{b}{h}\). The former would require, for a given milling depth, either increasing the number of teeth or decreasing the diameter of the cutter.

Increasing the number of teeth is undesirable, and modern high-productivity milling cutters usually have a small number of teeth \((Z = 6—12)\).

a reduction in diameter likewise cannot be significant, since it is desirable to have a cutter diameter that permits it to be mounted on a sufficiently strong arbor.

Therefore, in order to attain the smallest possible values of $\mu$, it is necessary to strive to increase the ratio $\dfrac{b}{h}$; this is achieved by using cutters with a steep helix, which, with a small number of teeth, have small axial distances between teeth $(h)$. Modern high-productivity cutters usually do have a small number of teeth with a large helix angle relative to the cutter axis.

It is a known proposition that fluctuations of the load on the cutter will be entirely absent in the case where the milling width $b$ satisfies the equality

\[ b = C \frac{H}{Z} = C \frac{\pi D \operatorname{tng}\beta}{Z}, \]

where $C$ is any integer.

This case is the most favorable both from the standpoint of the dynamics of the process and from the standpoint of the durability of the cutter and the finish of the machined surface. One should always strive for the indicated relation by selecting an appropriate cutter.

In many practical cases, complete absence of force fluctuations can also be achieved when

\[ b = C \frac{H}{Z}. \]

For this purpose, the workpiece must be positioned relative to the cutter in such a way that the dimension of the milled surface, measured in the direction parallel to the cutter axis, is equal to $C \dfrac{H}{Z}$.

Fig. 30.

When working on a universal milling machine, this can be achieved by turning the table through an angle $\omega$, with

\[ \cos \omega = \frac{bz}{C \pi D \operatorname{tng}\beta}. \]

In the first part of the work we indicated that the question of nonuniform—

of nonuniformity in the case of peripheral milling with \(b \le \dfrac{D}{2}\), when each tooth of the face mill takes a chip of zero thickness (see Fig. 30), has much in common with the operation of a cylindrical cutter. The nonuniformity coefficient \(\mu\) for this case can be obtained from Fig. 28 in the same way as for a straight-tooth cutter \((\beta = 90^\circ)\).

In calculating the value of \(\mu\) for the operation of a cylindrical cutter we used the mean value of the exponent \(k = -0.3\), whereas for the operation of face mills one should take \(k = -0.25\); however, this deviation will not substantially affect the value of \(\mu\), and, as calculations show, will distort it by no more than \(2\)–\(2.5\%\).

Conclusion

As a result of the investigation carried out, we have presented graphs of the nonuniformity coefficient, by means of which it is possible easily to determine the ratio of the maximum load on the cutter to its mean value for a given cutting regime. From these same graphs it is not difficult to find the most favorable relationships between the width of the milled surface and the principal parameters of the cutter, satisfying the condition of greatest uniformity of the cutting process and making it possible, without increasing the force load on the tool and the machine parts, to achieve an increase in productivity.

All the conclusions presented do not take into account the influence of cutter runout on nonuniformity; this will somewhat distort the phenomenon, but the distortion will be smaller the more correctly the cutter is sharpened and the thicker the chip removed by each cutter tooth.

References

1. Rozenberg. — Operation of a cylindrical cutter. Orga—Informatsiya, 1935, No. 12.
2. Rozenberg. — Dynamics of face milling. Proceedings of the Tomsk Industrial Institute, 1937.
3. Granovskii. — On the uniformity of milling. Orga—Informatsiya, 1935, No. 6.
   Larin. — Nonuniformity in the operation of a cylindrical cutter. Orga—Informatsiya, 1935, No. 9.
   Schröder. — The significance of chip thickness in milling. Maschinenbau, 1934, No. 19/20.
   Salomon. — Inertia-free measurements in the cutting process. Leve-Notitsen, 1929.
   Kikebush and Pol. — Forces and their oscillations in milling with cylindrical cutters. Maschinenbau, 1935, No. 9/10.
   Rozenberg and Sudnishnikov. — Theory of the operation of a cylindrical milling cutter with a spiral tooth. Vestnik metallopromyshlennosti, 1933, No. 4.