ACCELERATED METHOD FOR DETERMINING THE CUTTING PROPERTIES OF TOOLS AND THE MACHINABILITY OF METALS

A. M. ROZENBERG
Professor, Doctor of Technical Sciences

The various rapid methods proposed up to now for determining the cutting properties of tools and the machinability of metals have either not fully justified themselves, or have a number of shortcomings and limited ranges of application. Among the accelerated methods known at present, those deserving attention are Reichel’s method and the Van-Dongen method (face turning). The first of these has so far not been sufficiently tested, and the possibility of its application has not been definitively established. The Van-Dongen method has been tested quite reliably by the metal-cutting laboratory of the Gorky Automobile Plant named after Molotov, but it has narrow limits of application. This method is applicable for testing only cutting tools, and moreover only tools made of high-speed steel. Up to now it has been used only for this tool. It may be assumed that it is not applicable in those cases where the cutter becomes blunt through gradual wear of the flank face and a specified width of the wear land on the flank face is taken as the criterion of dulling. Such dulling is observed in the operation of cutters with carbide-tipped plates, and the indicated method is not applicable for testing them. It is self-evident that, by its very nature, it is also not applicable for testing all other tools except cutters.

Up to now, the most widely used method, both in determining the cutting properties of various types of tools and in determining the machinability of various metals, has been the “\(vT\)” method, and recently, in those cases where the tool becomes blunt on the flank face, this method has been supplemented by observation of flank-face wear. The “\(vT\)” method, being quite reliable, has the disadvantage that it requires considerable expenditures of time, metal, and tool, especially if the tests are carried out on a metal with unknown—even approximately—machinability, or with a tool whose cutting properties are unknown.

The cutting properties of a tool, as well as the machinability of a metal, are characterized by the equation

\[ v = \frac{K}{T^{\frac{1}{m}}} \tag{1} \]

where \(v\) is the cutting speed in m/min,
\(T\) is the tool life in minutes,
\(K\) is a constant depending on the cutting properties of the tool and the machinability of the metal,
\(m\) is a quantity depending mainly on the character of tool wear.

The exponent \(\frac{1}{m}\) is the smaller, the more thermal wear of the tool predominates over mechanical wear. Table 1 shows that the value of the exponent of relative tool life \(\frac{1}{m}\) varies over wide limits for different tools.

Table 1

Thus, the exponent of relative tool life for different tools varies over wide limits: from 0.05 to 1.1.

The smaller the value of \(\frac{1}{m}\), i.e., the larger \(m\) is, the narrower are the limits of cutting speeds corresponding to the limits of variation in tool life used in production or adopted in experiments.

Thus, for example, at \(m = 15\), i.e., at \(\frac{1}{m} = 0.066\), a change in cutting speed by 10% changes tool life by a factor of 4.2; a change in speed by 20% changes tool life by a factor of 15.5. Therefore, if tool-life tests of such a tool are being carried out and it is desired to obtain tool lives from 4 to 60 minutes, i.e., varying by a factor of 15, all tests must be conducted at speeds varying within only 20%. If, in this case, the cutting qualities of the tool or the machinability of the metal are not known in advance, it is very difficult to find the range of speeds that will correspond to the desired limits of tool life; and small deviations upward from this speed range will cause instantaneous blunting of the tool, while small deviations downward will make the tests very prolonged.

and requiring a considerable expenditure of metal. Thus, the experimenter is required to carry out a whole series of trial tests that would bring him into that range of speeds which corresponds to the desired tool-life dimensions; and often more time, metal, and tools are spent on these trial tests than on the main ones.

In what follows we describe a method of variable speed which makes it possible, by means of only one trial test, to determine the range of speeds in which the experiments should be conducted; and, if necessary, to determine approximately the cutting qualities of a tool or the machinability of a metal, it makes it possible to be limited to only two tests in all, including the trial one. This method, unlike Van-Dongen’s method, is a method not of continuously but of stepwise varying cutting speed, and is applicable to any cutting tools, with the indispensable condition that there be such a machine tool on which the speed can be varied in sufficiently small steps.

Method of Stepwise Varying Cutting Speed

Let the tool under test be made to work for \(\tau\) minutes at speed \(V_1\), then \(\tau\) minutes at speed \(V_2\), then \(\tau\) minutes at speed \(V_3\), and so on, increasing the speed until, at speed \(V_n\), after having worked \(\Theta\) minutes, the tool becomes dull. Here \(\Theta < \tau\). Let the speeds \(V_1, V_2, V_3\), etc., increase according to some geometric progression with ratio \(C\), that is:

\[ V_2 = CV_1;\quad V_3 = CV_2 = C^2V_1;\quad V_4 = CV_3 = C^3V_1;\ldots\ldots V_n = C^{n-1}V_1. \]

At the first speed step \(V_1\), the tool would have worked until dulling for \(T_1\) minutes,

\[ T_1=\frac{K^m}{V_1^m}. \]

After the tool has worked at the first speed step for a time \(\tau\), and \(\tau < T_1\), it has an unused remainder of tool life \(T_1-\tau\). At the speed \(V_2=CV_1\), this remainder of tool life would give the tool a life of

\[ (T_1-\tau)\cdot\left(\frac{V_1}{V_2}\right)^m=\frac{T_1-\tau}{C^m}. \]

After the tool has worked for \(\tau\) minutes at speed \(V_2\), it has the remaining tool life

\[ \frac{T_1-\tau}{C^m}-\tau, \]

which at speed \(V_3\) would give a life of

\[ \left(\frac{T_1-\tau}{C^m}-\tau\right)\cdot\left(\frac{V_2}{V_3}\right)^m = \frac{T_1-\tau}{C^{2m}}-\frac{\tau}{C^m}. \]

After working \(\tau\) minutes at speed \(V_3\), the tool has the remaining tool life

\[ \frac{T_1-\tau}{C^{2m}}-\frac{\tau}{C^m}-\tau, \]

which at speed \(V_4\) would give the tool a life of

\[ \left(\frac{T_1-\tau}{C^{2m}}-\frac{\tau}{C^m}-\tau\right)\cdot\left(\frac{V_3}{V_4}\right)^m = \frac{T_1-\tau}{C^{3m}}-\frac{\tau}{C^{2m}}-\frac{\tau}{C^m}. \]

At the speed \(V_n\), at which blunting occurs, the remaining tool life \(\Theta\), for which the following equality may be written,

\[ \Theta=\frac{T_1-\tau}{C^{(n-1)m}}-\frac{\tau}{C^{(n-2)m}}-\frac{\tau}{C^{(n-3)m}}-\ldots-\frac{\tau}{C^{2m}}-\frac{\tau}{C^m}. \]

This equation can be rewritten in the following form:

\[ \Theta=\frac{T_1}{C^{(n-1)m}}-\tau\left[ \frac{1}{C^{(n-1)m}}+\frac{1}{C^{(n-2)m}}+\frac{1}{C^{(n-3)m}}+\ldots+\frac{1}{C^{2m}}+\frac{1}{C^m} \right]. \]

Here

\[ \frac{T_1}{C^{(n-1)m}}=\frac{K^m}{V_1^m\cdot C^{(n-1)m}}=\frac{K^m}{V_n^m}. \]

Therefore

\[ \Theta=\frac{K^m}{V_n^m}-\tau\left[ \frac{1}{C^{(n-1)m}}+\frac{1}{C^{(n-2)m}}+\frac{1}{C^{(n-3)m}}+\ldots+\frac{1}{C^m} \right]. \]

For \(C>1\) and \(n>1\), the terms in square brackets form an increasing geometric progression with ratio \(C^m\), number of terms \((n-1)\), and sum of terms \(S\), where

\[ S=\frac{C^{(n-1)m}-1}{C^{(n-1)m}\cdot (C^m-1)}; \]

If \(C^{(n-1)m}\geq 40\), then the unity in the numerator may be neglected. In this case

\[ S=\frac{C^{(n-1)m}}{C^{(n-1)m}\cdot (C^m-1)}=\frac{1}{C^m-1} \]

and

\[ \Theta=\frac{K^m}{V_n^m}-\frac{\tau}{C^m-1}. \tag{2} \]

In the last expression,

\(V_n\)—the speed at which blunting of the tool occurred,
\(\tau\)—the interval of time during which the tool worked at each speed step,
\(C\)—the speed-increase coefficient,
\(\Theta\)—the interval of time for which the tool worked at the last speed step before blunting.

All these quantities are known, and in equation (2) the unknown quantities are \(k\) and \(m\), which enter into equation (1). To determine them, one more equation is needed, and to obtain it we carry out a second experiment at the constant speed \(V_n\). At this speed the tool will have a tool life \(T_n\), where

\[ T_n=\frac{K^m}{V_n^m}. \]

Having determined \(T_n\) and substituting it into equation (2), we obtain

\[ \Theta=T_n-\frac{\tau}{C^m-1}, \tag{3} \]

whence

\[ m=\frac{\lg\left\{\frac{\tau}{T_n-\Theta}+1\right\}}{\lg C}, \tag{4} \]

\[ K=V_n\cdot T_n^{\frac{1}{m}}. \tag{5} \]

Thus, by carrying out two tests, we determine the quantities of interest to us, \(K\) and \(m\).

Both of these tests will be of short duration. The first of them will take a time interval \(\tau(n-1)+\Theta\); the second, conducted at the high cutting speed \(V_n\), will take a time interval greater than \(\Theta\), but less than \(\Theta+\tau\).

The duration of work at each speed step should be taken as small, namely \(\tau=3\text{–}4\) min.; the number of speed steps should preferably be \(n=4\text{–}6\). In this case the duration of the first test will be within 12–24 minutes, and that of the second no more than 6–8 minutes.

Having the quantities \(K\) and \(m\), we can, from equation (1), determine the cutting speed for any tool life; but it must be borne in mind that, by carrying out only two tests, we obtain only approximate values of \(K\) and \(m\). To obtain exact values, two tests are insufficient, since each test inevitably gives more or less substantial errors both because of possible fluctuation in the hardness of the workpiece being machined and because of a certain difference in the cutting qualities of different cutters used in the tests (even if these cutters were made completely identically), and also because of possible slight fluctuation of the cutting speed within each test. Even if the same cutter was used in the two tests, some change in its cutting qualities is still possible as it is resharpened. All these causes, each individually exerting an insignificant influence, may together produce a noticeable deviation. Therefore, to obtain exact values of \(K\) and \(m\), two tests are insufficient.

In finding exact values of \(K\) and \(m\), it should be considered necessary to carry out 4–6 tests. In this case the first test is conducted at a stepwise-changing speed and serves to find that range of speeds in which the subsequent tests should be carried out in order to obtain convenient values of tool life (not too long and not too short). If in the first test blunting of the cutter occurred at speed \(V_n\), then the second test may be carried out at speed \(V_{n-1}\), the third at speed \(V_{n-2}\). The tool lives obtained at these speeds will not be very large and will show at what speeds the subsequent (fourth, fifth, etc.) tests should be carried out.

When carrying out the first test (at stepped speed), it is important to choose successfully the value \(C\)—the speed-increase coefficient. In deriving equation (2) we indicated that the accuracy of the derivation will be sufficient if \(C^{(n-1)m}\geq 40\). This inequality may serve for choosing the value of \(C\). The value of \(m\) depends mainly on the type of tool, and we know it approximately. It is desirable to have the number of speed steps equal to 4–6. Therefore, if we take \(n=5\) and \(C^{(n-1)m}\cong 40\), then

\[ \lg C\cong \frac{0.4}{m} \]

and the value of \(C\) for various \(m\) will take the values presented in Table 2.

Table 2

The convenience of the stepped-speed method for determining the range of speeds suitable for experiments can be illustrated by a simple example.

Suppose that the first speed step $V_1$ has been chosen by us so that, if we worked at it up to blunting, the tool life would be $T_1 = 1000$ min, which would be unacceptable for carrying out a constant-speed test because of the great duration of the test. If the test is carried out at stepped speed and the value $\tau = 3$ min is adopted for $\tau$, and the value $C$ is chosen from Table 2, then the test will proceed as follows:

\[ V_1;\quad T_1 = 1000\ \text{min};\quad \tau = 3\ \text{min};\quad T_1 - \tau = 997\ \text{min}. \]

\[ V_2 = C V_1;\quad T_2 = \frac{T_1 - \tau}{C^m} = \frac{997}{2.5} = 398.8\ \text{min};\quad T_2 - \tau = 395.8\ \text{min}. \]

\[ V_3 = C V_2;\quad T_3 = \frac{T_2 - \tau}{C^m} = \frac{395.8}{2.5} = 158.32\ \text{min};\quad T_3 - \tau = 155.32\ \text{min}. \]

\[ V_4 = C V_3;\quad T_4 = \frac{T_3 - \tau}{C^m} = \frac{155.32}{2.5} = 62.13\ \text{min};\quad T_4 - \tau = 59.13\ \text{min}. \]

\[ V_5 = C V_4;\quad T_5 = \frac{T_4 - \tau}{C^m} = \frac{59.13}{2.5} = 23.65\ \text{min};\quad T_5 - \tau = 20.65\ \text{min}. \]

\[ V_6 = C V_5;\quad T_6 = \frac{T_5 - \tau}{C^m} = \frac{20.65}{2.5} = 8.26\ \text{min};\quad T_6 - \tau = 5.26\ \text{min}. \]

\[ V_7 = C V_6;\quad T_7 = \frac{T_6 - \tau}{C^m} = \frac{5.26}{2.5} = 2.1 = \Theta. \]

Thus, the test is carried out at 7 speed steps and takes a time $(n - 1)\tau + \theta = 20.1$ min instead of 1000 min. In the example given, $C^m = 2.5$ was taken, which will always be the case if the value of $C$ is taken from Table 2.

Of course, when carrying out the test, not knowing the exact value of $m$, we shall be able only approximately to choose the value of $C$, after roughly estimating the possible value of $m$; but this will not change anything essential either in the test procedure or in its results.

If the stepped-speed test is carried out in order, together with a second test, to make it possible by equations (4) and (5) to determine the values of $K$ and $m$, then the procedure described has the disadvantage that the value $\Theta$ obtained is very small ($\Theta \ll \tau$), owing to which one cannot particularly vouch for its reliability. In addition, a test carried out at stepped speed with small values of $\tau$ is a cutting test with interruptions, which will somewhat alter the tool-life relation.

To avoid these disadvantages, the procedure for carrying out the stepped-speed test should be modified as follows. Passing from

from step to step, the operating time at each subsequent step should be increased according to some geometric progression.

Then we shall have

Let the life at the first speed step $V_1$ be $T_1$. After the tool has worked at the first step for $\tau_1$ minutes and the speed has been switched to $V_2$, the remaining life of the tool at speed $V_2$ will be

\[ (T_1-\tau_1)\cdot\left(\frac{V_1}{V_2}\right)^m = \frac{T_1-\tau_1}{C^m}. \]

After working at $V_2$ for the interval of time $\tau_2=e\cdot\tau_1$, the tool at the next speed step $(V_3)$ will have a remaining life of

\[ \left(\frac{T_1-\tau_1}{C^m}-\tau_2\right)\cdot\left(\frac{V_2}{V_3}\right)^m = \frac{T_1-\tau_1}{C^{2m}}-\frac{\tau_2}{C^m}. \]

Continuing similar reasoning, at speed $V_n$ we obtain

\[ \Theta= \frac{T}{C^{(n-1)m}} - \left[ \frac{\tau_1}{C^{(n-1)m}} + \frac{\tau_2}{C^{(n-2)m}} + \frac{\tau_3}{C^{(n-3)m}} +\ldots+ \frac{\tau_{n-2}}{C^{2m}} + \frac{\tau_{n-1}}{C^m} \right] \]

or

\[ \Theta= \frac{K^m}{V_n^m} - \tau_1 \left[ \frac{1}{C^{(n-1)m}} + \frac{e}{C^{(n-2)m}} + \frac{e^2}{C^{(n-3)m}} +\ldots+ \frac{e^{n-3}}{C^{2m}} + \frac{e^{n-2}}{C^m} \right]. \]

For $e>1;\ C>1$ and $n>1$, the expression in square brackets is an increasing geometric progression with ratio $eC^m$ and number of terms $(n-1)$. The sum of its terms will be

\[ S= \frac{e^{\,n-1}\cdot C^{m(n-1)}-1}{eC^{nm}-C^{(n-1)m}}. \]

In the numerator, unity may be neglected. In this case we shall have

\[ S=\frac{e^{\,n-1}}{eC^m-1}; \]

\[ \Theta= \frac{K^m}{V_n^m} - \frac{\tau_1\cdot e^{\,n-1}}{eC^m-1}, \tag{6} \]

or

\[ \Theta=T_n-\frac{\tau_n}{eC^m-1}. \tag{7} \]

Here

$T_n$—the tool life obtained in testing at the constant speed of the last step $V_n$,

$\tau_n$—the interval of time during which the tool would have had to work at the last speed step if blunting had not occurred, $(\tau_n=\tau_1\cdot e^{\,n-1})$,

\(\Theta\)—the actual interval of time for which the tool worked before becoming dull at the last speed stage,

\(C\)—the coefficient of increase of speed,

\(e\)—the coefficient of increase of working time at each speed stage.

Having carried out the first test at a stepwise varying speed and, from it, determined the value \(\Theta\), then having carried out a second test at the constant speed \(V_n\) and obtained the tool life \(T_n\), we determine the value \(m\)

\[ m=\frac{\lg\left\{\frac{\tau_n-1}{T_n-\Theta}+\frac{1}{e}\right\}}{\lg C}. \tag{8} \]

The time spent on the test at stepwise speed will be

\[ \tau_1+\tau_2+\tau_3+\ldots+\tau_{n-1}+\Theta = \tau_1(1+e+e^2+\ldots+e^{n-2})+\Theta, \]

that is,

\[ \tau_1\cdot\frac{e^{\,n-1}-1}{e-1}+\Theta. \]

Let, for the tool being tested, the actual relationship be

\[ V=\frac{40}{T^{0.1}}. \]

In the test the following were taken:

\(V_1=20\ \text{m/min};\)
\(\tau_1=2\ \text{min},\)
\(e=1.5;\)
\(C=1.1.\)

The speed stages in the test will be

The durations of the test at each stage

If the work to dulling were carried out at the first stage at \(V=20\ \text{m/min}\), then the tool life would be

\[ T_1=\left(\frac{40}{20}\right)^{10}=1024\ \text{min}. \]

At the first stage the duration of work is \(\tau=2\ \text{min}\), and therefore the remaining tool life, reduced to the second stage, will be

\[ \frac{T_1-\tau_1}{C^m}=393.7\ \text{min}. \]

The duration of operation at the second stage is \(\tau_2 = 3\) min, and the remaining tool life, reduced to the third stage, is 150.5 min. The duration of operation at the third stage is \(\tau_3 = 4.5\) min, and the remaining tool life, reduced to the fourth stage, is 56.6 min.

The duration of operation at the fourth stage is \(\tau_4 = 6.75\) min, and the remaining tool life, reduced to the fifth stage, is 19.26 min.

The duration of operation at the fifth stage is \(\tau_5 = 10.1\) min, and the remaining tool life, reduced to the sixth stage, is 3.50 min \(= \theta\).

Thus, blunting occurs at the sixth stage. The time expended on the first test is 30.58 minutes.

The second test is carried out at a speed \(v_6 = 32.2\), with a tool life of

\[ T_n=\left(\frac{40}{32.2}\right)^{10}=8.75\ \text{min}. \]

In view of the fact that tests at stepped speed are performed with time intervals that increase from stage to stage, almost the entire tool life of the tool is used up at the last two speed stages. This is illustrated by Table 3, in which the cutting time at each speed stage is expressed as the time, equivalent in tool life, at the speed of the first stage \((V_1)\).

Table 3

Thus, more than 84% of the tool life is used at the last two speed stages, which therefore are the principal and decisive ones. It is natural to conclude that the influence of interruptions in cutting during the transition from stage to stage cannot have any appreciable effect here, while the considerable duration of the tests at the last speed stages gives greater accuracy to the results obtained.

Conclusion

The article sets forth a method of stepped cutting speed, making it possible, by means of a single test, to determine the range of speeds in which the tool or the machinability of the metal should be tested.

When an approximate determination of the cutting properties of a tool or of the machinability of a metal is required, according to the method presented it is sufficient to carry out only two tests.

Errata

Proceedings of TPI, vol. 61, issue 3.