INFLUENCE OF THE CUTTING CONDITIONS ON THE ELEMENTS OF THE MICROGEOMETRY OF THE SURFACE PROFILE

A. N. EREMIN
Candidate of Technical Sciences

The microgeometry of the profile of a machined surface (Fig. 2) is characterized by various elements. The principal ones are: the greatest height of the irregularities \(H\), their mean value \(h_m\), the mean height of the depressions \(h\), and the bearing surface of the irregularities \(F_i\), measured along the line of section of the ridges at an arbitrary height \(h_i\) from the base of the profile \(cd\). In all experimental and theoretical investigations carried out in the field of finish turning, the quality of the machined surface has usually been determined by the greatest height of the irregularities \(H\). A number of relationships have been established expressing the interrelation between \(H\) and various factors of the cutting process. Less elucidated up to the present has been the question of the influence of the cutting conditions (cutting speed—\(v\), feed—\(S\), depth of cut \(t\)) on the elements of the surface profile: \(h_m\), \(h\), and \(F\).

The mean height of the irregularities \(h_m\) (or of the depressions \(h\)) is of interest because, from its magnitude, one can judge the extent to which the microgeometry of the profile of a given surface ensures the reliability of the fit in fixed joints.

According to the experiments of Schmalz [1], it has been established that, when products are pressed together, plastic deformation (flattening) of the largest irregularities occurs in the zone of surface roughness, and the material of the latter is uniformly distributed over the height \(h_m\). Lowering the microgeometric layer of the surface by the height \(H - h_m = h\) leads to a decrease in the calculated value of the interference, and consequently to a weakening of the reliability of the given fit. The smaller \(h_m\) is, the greater \(h\) is, the stronger this weakening, and vice versa.

The magnitude of the bearing surface of the irregularities of the profile \(F\) (Fig. 2) characterizes the quality of the machined surface chiefly from the standpoint of its suitability for running fits. The smaller the value of \(F\), the greater the initial wear of the surface, and the sooner the required magnitude of clearance is violated during operation of the joint.

Consequently, for the accuracy of calculating fits (running and tight fits), it is necessary to know not only the height of the irregularities \(H\), but also the magnitudes of the other elements of the surface profile: \(h_m\), \(h\), \(F\), etc.

In the proposed work the author provided for:

1. Determination of the relationships:

\[ h_m \text{ and } h = f_1(v, S, t), \]

\[ h_m \text{ and } h = f_2(H), \]

\[ F_m = f_3(v, S, t), \]

where \(F_m\) is the bearing surface of the ridges of the profile along the line of section at the height \(h_m\) (Fig. 2);

2. To establish to what extent the profile elements \(h_m\), \(h\), \(F_m\) can be objective indicators of the quality of the machined surface.

The reason for this was the work of Shmalts [2], Berman [3], and others, in which the authors recommended taking \(h_m\), \(h\), and \(F_m\) as component parts of the coefficients proposed by them for evaluating surface quality. For example, according to Shmalts, the quality coefficient (filling coefficient) is

\[ K=\frac{h_m}{H}. \]

According to Berman:

\[ K=\frac{F_m}{h}\ \%/\text{micron} \]

Here: \(F_m\) is expressed as a percentage of \(l\) (Fig. 1),

\(h\) is the mean height of the depressions in microns.

The present investigation was based on data from experiments carried out by the author on finish turning of steel 5120 with cutters of constant geometry \((\delta=70^\circ;\ \varphi=45^\circ;\ \varphi_1=10^\circ;\ r=1.5\text{ mm};\ \alpha=8^\circ)\), without the use of coolant. More than 400 profilograms of surfaces obtained under different cutting conditions were used.

In each profilogram the value \(h_m\) was determined as the quotient of the division of the total area of the profile irregularities \(f\) by the length of the profilogram \(l\). The area \(f\), indicated by hatching in Fig. 2, was in all cases determined by planimetry.

Fig. 3b

The magnitude of the bearing surface of the ridges \(F_m\), along the line of section at the mean height \(h_m\), was found as a percentage of \(l\).

In Fig. 2 the linear quantity \(F_m\) is equal to the sum of the segments \(1—1'\), \(2—2'\), \(3—3'\), etc.

To determine the dependence of the quantity \(h_m\) (or \(h\)) on the cutting-condition factors \(v\), \(S\), and \(t\), 410 profilograms were used. The investigation was carried out in three variants. In the first variant the dependence \(h_m=f(t)\) was determined. The result of the investigation is shown in Fig. 3b in the coordinates \(\frac{h_m}{H}\) and \(t\). The straight line drawn through the points corresponds to the averaged value

\[ \frac{h_m}{H}\approx 0.57=\text{const}, \]

whence

\[ h_m\approx 0.57\,H. \]

From Fig. 36 it is evident that, irrespective of the change in \(t\) within the range from \(0.1\) to \(3\) mm, the ratio \(\frac{h_m}{H}\) remains constant, even at different \(v\). The observed variation of the values of \(\frac{h_m}{H}\) over the entire range of \(t\)—from 0.46 to 0.68—amounts, relative to the averaged value 0.57, to approximately \(\pm 19\%\).

In constructing Fig. 36, experimental data for \(v = 15;\ 36;\ 51;\ 61\) and \(95\) m/min were included, at \(S = 0.22\) mm/rev. Therefore, in order to separately account for the influence of \(v\) on the magnitude \(h_m\), a second version of the investigation of the dependence \(h_m = f(v)\) was carried out at the same feed \(S = 0.22\) mm/rev and \(t_1 = 1 \div 1.1\) mm,

Fig. 37

\(t_2 = 0.4 \div 0.5\) mm (Fig. 37). Here, as in the preceding case, the arrangement of the points over the entire range of \(v\) from 2 to 200 m/min makes it possible to draw a line of averaged values \(\frac{h_m}{H} \simeq 0.57\). The greatest scatter of the points (excluding the 2 lower points) falls within the limits from 0.46 to

Fig. 38

0.68, which, relative to the mean value \(\frac{h_m}{H} \simeq 0.57\), is approximately \(\pm 19\%\).

Conclusion from the analysis of the diagrams in Figs. 36 and 37.
Irrespective of the change in \(v\) from 2 to 200 m/min and \(t\) within the range from 0.1 to 3 mm, the mean value \(\frac{h_m}{H} \simeq 0.57\) remains a constant quantity (at \(S = \mathrm{const}\)). Consequently, the mean height of the irregularities

\[ h_m \simeq 0.57 H, \]

with a variation of \(\pm 19\%\) from this value, is not a function of \(v\) and \(t\).

On this basis, in determining (by the 3rd variant) the dependence \(h_m = f(S)\), all available experimental data were used, except—

experiments with a feed of \(S-0.011\) mm/rev. In Fig. 38, in the coordinates \(\frac{h_m}{H}\) and \(S\), individual measurements of \(\frac{h_m}{H}\) are indicated by dots, while the figures mark the total number of measurements concentrated at a given place for a definite feed. The symmetrical arrangement of the boundaries of the scatter fields of the points, for each value of \(S\), makes it possible to draw a straight line for the averaged value

\[ \frac{h_m}{H} \approx 0.57 = \mathrm{const}. \]

From the diagram in Fig. 33, which includes the data of experiments with a wide variation of cutting conditions \((v = 2 \div 220\ \text{m/min};\ t = 0.1—3\ \text{mm},\ S = 0.055—1.0\ \text{mm/rev})\), it is seen that the limits of scatter of the values of \(\frac{h_m}{H}\) are bounded by the following quantities:

upper value—0.68—0.69, lower—0.45—0.46.

Consequently, the deviation from the mean value \(\frac{h_m}{H} \approx 0.57\) is equal to \(\pm 0.11 \div 0.12\), which in percent amounts to \(\approx 19 \div 21\%\).

As the results of the investigation carried out for all three variants show, the mean value and the limits of scatter of the quantity \(\frac{h_m}{H}\) do not depend on the cutting conditions, i.e., on \(v\), \(S\), and \(t\).

This circumstance permits one to assume that, for the specified material being machined (steel 5120) and a definite tool geometry \((\delta = 70^\circ,\ \varphi_1 = 10^\circ;\ \varphi = 45^\circ,\ r = 1.5\ \text{mm})\), the observed phenomenon of scatter of the values of \(\frac{h_m}{H}\) depends only on random factors of the cutting process, for example, on errors in the tool geometry in plan \((r,\ \varphi_1)\) during sharpening and setting \((\varphi_1)\), etc.

The question of according to what law the values of \(\frac{h_m}{H}\) are distributed within the scatter field is of practical importance. Knowing this law, it is possible to predict in advance what the most probable value of \(\frac{h_m}{H}\) and \(h_m\) will be in each particular machining case.

If the deviations of \(\frac{h_m}{H}\) are the result of the influence of random errors, then, on the basis of probability theory, it may be assumed that the distribution of the values of \(\frac{h_m}{H}\) in the scatter field will take place according to the law of the normal distribution curve, called the Gaussian curve.

The equation of the Gaussian curve may, in the present case, be represented in the following form:

\[ y = \frac{N}{\sigma \sqrt{2\pi}} \cdot e^{-\frac{(x-M)^2}{2\sigma^2}} \tag{1} \]

where \(\pi = 3.1416\),

\(e = 2.71828\)—the base of the system of natural logarithms.

\(N\)—the totality of all experimental data for \(\frac{h_m}{H}\).

Here \(N = 410\).

\(M\)—the arithmetic mean of the given statistical population, i.e., the mean value \(\frac{h_m}{H}\) of all \(N\) quantities.

\(x\)—the current value of the variate corresponding to the value \(\frac{h_m}{H}\) in the variational series.

\(\sigma\)—the mean square deviation of an individual variate from the arithmetic mean \(M\).

\(y\)—the frequency of occurrence of each value \(x\) (variate).

In Fig. 39, in the coordinates \(y\) and \(x\), two curves are shown, of which one (the dotted one) corresponds to the actual distribution

\[ x=\frac{h_m}{H}, \]

while the other (the solid one) represents the theoretical curve of the normal distribution of \(x\) according to Gauss’s equation (1).

Fig. 39

On the basis of calculations carried out from \(N=410\) values of \(\frac{h_m}{H}\), the Gaussian curve has the following parameters:

arithmetic mean . . . . . . . . . . \(M=0.565\),

mean square deviation \(\sigma=\pm 0.061\).

In the theoretical Gaussian curve both branches (Fig. 39) have \(y=0\) at \(x=\pm\infty\), i.e., both branches asymptotically approach the \(x\)-axis. The area enclosed between the Gaussian curve and the \(x\)-axis graphically represents 100% of all \(N\) values \(\frac{h_m}{H}=x\). In practice, both branches of the curve may be bounded along the \(x\)-axis by the value \(y\) at \(\pm 3\sigma\). As calculations show, the area enclosed between the Gaussian curve and the \(x\)-axis within \(\pm 3\sigma\) contains 99.74% of all values of \(x\), i.e., is almost equal to 100%; within \(\pm 2\sigma\) are contained 95.5% of all values of \(x\), and within \(\pm\sigma\) are found 68.5% of all values of \(x\).

In the normal distribution curve, the greatest frequency of occurrence, \(y_{\max}\), has the mean value \(x=M=0.565\).

Let us denote in equation (1) of the Gaussian curve

\[ \frac{N}{\sigma\sqrt{2\pi}}=A;\qquad e^{-\frac{(x-M)^2}{2\sigma^2}}=e^{-a}. \]

Then:

\[ y=A\cdot e^{-a}. \tag{2} \]

For \(X=M=0.565\) and \(\sigma=\pm 0.061\), from equation (2) we have:

\[ y_{\max}=26.8. \]

The coordinates of the points of the Gaussian curve, for different limits of its restriction along the \(x\)-axis, are given in Fig. 39 (table). From Fig. 39 it is seen that the curve of the actual distribution of the experimental values

\[ \frac{h_m}{H} \]

(dashed) almost exactly coincides with the theoretical Gaussian curve of the normal distribution (solid). From this the following conclusions follow:

1. The mean value

\[ \frac{h_m}{H}=0.565 \]

does not depend on the limits of variation of the components of the cutting conditions, i.e., on \(v\), \(S\), \(t\), and is governed by the law of random errors. This conclusion fully confirms what was set forth above with respect to Figs. 36, 37, 38.

1. For a given workpiece material and cutter geometry, the actual value

\[ \frac{h_m}{H} \]

may vary within the limits:

\[ \text{for }68.5\%\text{ of }N\text{ values of }\frac{h_m}{H};\qquad \frac{h_m}{H}=0.565\pm0.061 \tag{3a} \]

\[ \text{for }27.0\%\text{ of }N\text{ values of }\frac{h_m}{H};\qquad \frac{h_m}{H}=0.565\pm0.122 \tag{3б} \]

\[ \text{for }4.24\%\text{ of }N\text{ values of }\frac{h_m}{H};\qquad \frac{h_m}{H}=0.565\pm0.183 \tag{3в} \]

The dependence (3a) is of practical importance, since it satisfies 68.5% of the cases out of the total number \(N=100\%\) of values of \(\frac{h_m}{H}\).

Consequently, the most probable value of the mean height of the irregularities of the actual surface profile is

\[ h_m=(0.565\pm0.061)H, \]

or

\[ h_m=(0.89\div1.11)\,0.565\,H. \tag{4} \]

With the same cutter geometry in plan, within the limits \(S\le 1\ \text{mm/rev}\), the mean height of the irregularities of the theoretical surface profile (Fig. 30)

\[ h_m \simeq 0.30\,H, \tag{5} \]

i.e., it is on average 47% smaller than the actual one.

In general form, dependences (4) and (5) are analogous.

From the geometry of the surface profile (Fig. 2)

\[ H=h_m+h, \]

whence the mean height of the depressions is

\[ h=\left(1-\frac{h_m}{H}\right)H. \]

This shows that the actual value of \(h\) obeys the same law of dispersion as \(h_m\) and is numerically equal to:

\[ h=(0.89 \div 1.11)0.435\,H. \tag{6} \]

The method for investigating the influence of the cutting conditions on the magnitude of the bearing surface of the ridges \(F_m\) by height \(h_m\) (Fig. 2) is analogous to the determination of the dependence \(h_m=f(v,s,t)\). The results of the investigation are presented in the diagrams of Figs. 40–43.

Figure 40 shows the dependence of \(F_m\) on the depth of cut \(t\) and on the length of the machined surface \(L\), at \(S=0.22\ \text{mm/rev}\) and \(v=61\ \text{m/min}\). The character—

Fig. 40

—of the arrangement of most of the points makes it possible to draw the line \(F=\text{const}\) through the mean value \(F_m=54\%\). The scatter of the experimental data does not exceed \(\pm 10\%\) from the mean value 54.

In Fig. 41 the dependence \(F_m=f(t)\) is indicated from experiments on cutting depth, carried out at \(S=0.22\ \text{mm/rev}\), but for 5 different values of \(v\). Here, as in the preceding case, it is possible to draw the averaging line \(F_m=54\%=\text{const}\). All points, with the exception of the experiments at \(v=14.7\ \text{m/min}\), are distributed uniformly relative to the straight line \(F_m=\text{const}\).

Fig. 41

This gives grounds to suppose that the mean value \(F_m\) is influenced neither by changing \(t\) from 0.1 to 3 mm, nor by changing \(v\) by a factor of 6.5 (from 15 to 95 m/min).

To prove that the cutting speed indeed does not influence the value of \(F_m\), Fig. 42 gives experimental data for \(v\) from 2 to 200 m/min at \(S=0.22\ \text{mm/rev}\) and \(t=1 \div 1.1\) and \(0.4 \div 0.5\ \text{mm}\). From Fig. 42 it is evident that over the entire range of \(v\) the mean value \(F_m=\text{const}\).

This circumstance allowed the author, in determining the dependence \(F_m=f(S)\), to use all experimental data obtained at variable values of \(v\) and \(t\).

The result of the investigation is presented in Fig. 43 in the coordinates \(F_m\) and \(S\). The black points denote the mean values of measurements of \(F_m\), concentrated in the intervals between the points connected by a dashed line (without taking individual points into account) at definite \(S\). The figures indicate the total number of measurements concentrated at the given place. The arrangement of the points

allows one to draw a straight line of the averaged values of \(F_m\) with an inclination to the \(S\) axis at an angle \(\alpha\).

If the deviations of individual measurements are not taken into account, then the scatter limits of the latter may be bounded by dashed lines,

Fig. 42

indicating an average deviation from the mean \(F_m\) of the order of \(\pm 13 \div 14\%\).

The equation of the straight line \(F_m=f(S)\) in Fig. 43 is

\[ F_m = 49 S^{-0.065}\%, \tag{7} \]

where

\[ F_m \text{— in } \%; \qquad S \text{— mm/rev} \]

For

\[ S=1; \qquad F_m = 49\%. \]

The slope of the line \(F_m=f(s)\) is confirmed by data from partial experiments, given as an example in Fig. 44 for several values of \(v\).

Fig. 43

According to Berman [3], the quantity \(F_m\) enters as a component in the surface-quality coefficient:

\[ K=\frac{F_m}{h}\ \%/\text{microns} \tag{a} \]

As a result of a mathematical transformation, Berman gives the value of \(K\) in the form:

\[ K=100\left(\frac{1}{h}-\frac{1}{H}\right) \]

or

\[ K=100\left(\frac{H-h}{hH}\right), \tag{8} \]

where \(h\) is the average height of the depressions (Fig. 2). From equations (a) and (8) one may write the equality

\[ \frac{F_m}{h}=100\left(\frac{H-h}{hH}\right). \tag{6} \]

To verify how much this equality corresponds to reality, we make the following transformation. Substituting into equality (6), in place of \(F_m\) and \(h\), their mean values from equations (7) and (6), we have:

\[ \frac{112.5}{H S^{0.065}}=\frac{130}{H}. \]

It follows from this that the given equality can satisfy only one particular case, when

\[ S^{0.065}=0.943,\quad \text{i.e., at } S=2.51\ \text{mm/rev}. \]

For other values \(S \lesseqgtr 2.51\), the equality is violated. This proves that the expression for the coefficient \(K\) adopted by Berman according to equation (8)

Fig. 44

corresponds to the value of \(K\) according to equation (a) only in one case and therefore cannot have practical application.

In reality, according to equations (a), (6), and (7),

\[ K=\frac{112.5}{H S^{0.065}}\%\ \text{micron} \tag{9} \]

Conclusion

1. Regardless of the variation of the cutting-regime factors over wide ranges (for a given cutter profile), the mean value of the ratio

\[ \frac{h_m}{H}=0.565=\text{const.} \]

* From Proceedings of TPI, 61—III.

1. The ratio $\frac{h_m}{H}$ may vary within the limits from 0.38 to 0.75 according to the law of the Gaussian curve (Fig. 39), and for this reason $\frac{h_m}{H}$ cannot serve as a criterion for assessing surface quality for the given method of machining.

2. The area of the bearing surface of the irregularities, $F_m$, at the mean height $h_m$ (Fig. 2), does not depend on changes in the depth or speed of cutting. The value of $F_m$ changes with a change in feed. The greater $S$ is, the smaller $F_m$ is (Equation 7).

3. Berman’s surface-quality index (Equation 8) does not correspond to reality (Equation 9), and therefore cannot have practical significance.

REFERENCES

1. Schmalz, G. — Werkstattstechnik, No. 1, 1936, p. 1.
2. Schmalz, G. — Werkstattstechnik, No. 11, 1937, p. 256.
3. Berman, S. R. — Mashinostroitel’, No. 7, 1938, p. 9.