I. S. Chizhov.

On the Sorting Method in Interchangeable Production

It is well known that the inaccuracy of machining of mating parts has a very noticeable influence on the character of the fit. This influence is expressed in a greater or lesser variation in the magnitude of clearances and interferences that determine the character of the fit. Moreover, sometimes the permissible variations in the dimensions of the manufactured parts lead to a change in the sign of the clearance or interference. We encounter the latter circumstance in the so-called “transition fits”¹, of the 2nd and \(2\frac{1}{2}\) accuracy classes. Meanwhile, the operating conditions of the parts, the strength of the initial materials used for manufacturing the parts, the magnitude of the transmitted forces and moments, the operating conditions of the lubricating layer, etc.—design considerations—may set limits on such variations in the character of the joints. In this case the natural course is to pass to higher accuracy classes when assigning fits, since in a higher accuracy class the variations of clearances (or interferences) are not so large and the fit itself thereby acquires greater definiteness. But such a transition to a higher accuracy class—1) is always accompanied by a considerable increase in the cost of machining, since any reduction of the machining tolerance progressively increases the required manufacturing time; 2) requires labor of higher qualification, increases the percentage of scrap in manufacture, requires additional equipment, tools, and other conditions that may not always be available; and 3) such a transition to a higher accuracy class is not always possible for formal reasons, since in that class there may be no analogous fits approved by the OST; to “create” independently the required fits for each designer or even plant would mean consciously renouncing the advantages provided by standardization.

Foreign, in particular American, practice has found a way out of the indicated difficulties in applying the method of so-called selective assembly. More precisely, this method may be called assembly with preliminary sorting, or machining followed by sorting.

The principle of work by this method is as follows. As is known, a fit is determined by: 1) the mutual position of the tolerance zones of the mating parts, and 2) the magnitude of the tolerance zones. The mutual position of the tolerance zones determines a certain value of the mean clearance (or, correspondingly, interference) between the mating parts (see distance \(AB\), drawing 1).

Because it is impossible to manufacture parts absolutely accurately, it is necessary in manufacturing them to assign certain tolerances, the magnitude of which characterizes the size of the tolerance zones of the indicated parts. As a result, during assembly such cases are possible when a hole of minimum

diameter meets a shaft of maximum diameter, which gives the minimum clearance (see the distance \(S^\circ_{\min}\), Fig. 1), or, conversely, a shaft of minimum diameter meets a hole of maximum diameter, which gives the maximum clearance (see the distance \(S^\circ_{\max}\), Fig. 1). Analogous reasoning may also be repeated for parts assembled with interference.

Desiring to reduce the difference between the maximum and minimum clearances or interferences (let us call it the fit tolerance and denote it by \(\Delta_s\)), without creating additional difficulties in machining, the parts \(A\) and \(B\) are sorted into several classes according to the dimensions obtained, and larger shafts are assembled with larger holes, smaller shafts with smaller holes, etc. Naturally, such sorting will lead to a change in the difference between the maximum and minimum clearances (or interferences) and will give the joints greater definiteness and sharpness, without changing the character of the fit, which is determined by the magnitude of the mean clearance or interference.

A small variation in clearances and interferences naturally leads to a sharp improvement in the quality of products; thus, for example, after the introduction of sorting of piston pins and connecting rods at Ford plants, cases of knocking in engines (too large a clearance) sharply decreased, cases of overheating of the connecting-rod-head bearing (too small a clearance) decreased, and the service life of the joint before the first repair increased sharply.

When working by the ordinary method, the magnitude of the tolerance field depended to a certain degree on the arrangement of the tolerance fields, since, if it was desired to obtain small clearances or interferences, excessively large tolerance fields could sharply change the character of the fit. When working with sorting, the magnitude of the tolerance field is chosen independently of the mutual arrangement of the tolerance fields and, in cases of difficulty in machining to the given accuracy class, may even be increased by establishing additional standard sizes for the shaft and the hole (Fig. 2). Thus, for example, if machining to tolerances \(\delta_a\) and \(\delta_b\) is difficult or gives a considerable amount of rejects, the tolerance fields may be increased respectively by the amounts \(\delta'_a\) and \(\delta'_b\) (the latter procedure has received in America the name of the over-size method).

Fig. 1.

Fig. 2.

It should be noted that in the majority of cases the advantages obtained from

machining to a coarser class of accuracy are so great that they more than cover the costs of sorting the parts, all the more since the sorting process itself can be carried out in the course of inspection. The inspection-and-rejection apparatus thereby partially turns into a sorting organ of the plant.

As a result, the application of the method of machining “with subsequent sorting,” in combination with the method of “oversizes” or without it, promises very great technical and economic advantages.

Unfortunately, a large number of questions concerning the theory and practice of applying these methods have not received sufficient treatment in the technical literature. In any case, the share of attention given to these questions by the engineering and technical circles of our machine-building industry does not correspond to the significance and importance of the issues raised.

In the present article the author proposes an approach to determining the number of intervals into which tolerance fields should be divided for the case of the so-called normal distribution of parts over the tolerance field. Along the way, a whole series of theoretical questions is clarified concerning the possibilities inherent in the method of assembly with sorting, since sometimes this method is recommended in cases where its application cannot yield the expected results.

For the sake of generality of the conclusions, in what follows by \(\delta_a\) we shall understand the larger tolerance, whether it is the tolerance for manufacturing a shaft or the tolerance for manufacturing a hole, and by \(\delta_b\) we shall understand the smaller tolerance, i.e.

\[ \delta_a \geq \delta_b \tag{1} \]

In exactly the same way, we shall express interferences as negative clearances. Thus, for example, having a maximum interference, we shall have to express it as a minimum clearance by substituting its numerical value with a minus sign.

Let us divide the tolerance field \(\delta_a\) into \(n\) equal intervals. Generally speaking, division into equal intervals is by no means necessary. Indeed, since most parts are located near the middle of the tolerance field, it is therefore more expedient, with one and the same number of intervals, to place the latter more densely near the middle of the tolerance field, thereby ensuring a higher quality of fits for the majority of parts and a lower quality for a minority of parts. But in the practice of American, as well as of our, plants applying the method of assembly with sorting, only a uniform division of the tolerance field into intervals has become established. It is obvious that the number of intervals for the tolerance field \(\delta_b\) cannot be different from that for the tolerance \(\delta_a\). Division of the tolerance \(\delta_b\) according to the same law as \(\delta_a\) (uniform) presupposes the so-called normal distribution of parts over the tolerance field, i.e. according to the Gaussian curve (Fig. 3).

Such a distribution does not always occur. The data of investigations carried out in this direction allow the following conclusion to be drawn:

Fig. 3.

Proceedings of TII, vol. 2, 5.

1. Where the manufacturing process of a part, whether a shaft or a hole, is arranged so that the final size of the part is determined by the corresponding preliminary setting of the machine tool, tool, or fixture—there we may expect a symmetrical distribution of parts over the tolerance field according to the Gaussian normal distribution curve. This includes machining on automatic machines, machining on lathes and turret lathes to stops, reaming and drilling of holes, centerless grinding.

2. Where, however, the manufacturing process of the part is arranged in such a way that bringing it to the final size is left to the skill of the operator, the distribution curve turns out to be asymmetrical. Owing to the worker’s natural desire to keep closer to the “go” side of the gauge, the peak of the distribution curve is shifted by some distance toward the beginning of the tolerance field.

Naturally, all our subsequent conclusions, based on a uniform division of the tolerance field into intervals, have only limited application, assuming, as indicated above, a Gaussian-curve distribution. Uniform division into intervals under an asymmetrical distribution of parts over the tolerance field would lead to a considerable increase in unfinished production.

For mass and large-lot production, where work with limit gauges on the basis of the principle of interchangeability is exceptionally widespread, it is precisely the first type of arrangement of technological processes that is most characteristic. This considerably increases the practical applicability of the conclusions set out below.

For the practical implementation of division into \(n\) intervals, it is necessary additionally to manufacture \((n-1)\) sets of limit gauges. Thus, when dividing into 2 intervals, it is necessary additionally to manufacture snap gauges \(\beta'\) and \(\gamma'\), and plug gauges \(\gamma\) and \(\beta\), or else to manufacture one snap gauge and one plug gauge whose go dimensions will be \(\beta'\) and \(\beta\), and whose no-go dimensions will be \(\gamma'\) and \(\gamma\).

Shafts accepted by the normal snap gauge \(PR=\alpha'\) and \(NE=\varepsilon'\) are, during sorting, checked with an additional snap gauge. If this snap gauge passes with its no-go side, the shaft is assigned to group \(c\); if even the go side of the snap gauge does not pass, the shaft is assigned to group \(a\). If the go side passes and the no-go side does not pass, the shaft is assigned to group \(b\). Gauges for holes are constructed on the same basis.

Fig. 4.

After sorting, holes of group \(A\) are assembled with shafts of group \(a\), holes of group \(B\) with shafts of group \(b\), and so on. In this case, generally speaking, each group obtains its own minimum clearances, different from the analogous clearances of the other groups.

Thus, denoting by \(S^0\) the clearances of a given fit before sorting, after sorting we obtain for the combination of holes \(A\) with shafts \(a\) (Fig. 4):

\[ \begin{gathered} S_{a\min}=S^\circ_{\min}+\frac{\delta_a}{n}(n-1)=S^\circ_{\min}+\left(1-\frac{1}{n}\right)\delta_a,\\ S_{a\max}=S^\circ_{\max}-\left(1-\frac{1}{n}\right)\delta_b,\\ S_{a\,\mathrm{avg}}=S^\circ_{\mathrm{avg}}+\left(1-\frac{1}{n}\right)\frac{\delta_a-\delta_b}{2}. \end{gathered} \tag{2} \]

for group \(X_x\)

\[ \begin{gathered} S_{x\min}=S^\circ_{\min}+\left(1-\frac{1}{n}\right)\delta_b,\\ S_{x\max}=S^\circ_{\max}-\left(1-\frac{1}{n}\right)\delta_a,\\ S_{x\,\mathrm{avg}}=S^\circ_{\mathrm{avg}}-\left(1-\frac{1}{n}\right)\frac{\delta_a-\delta_b}{2}. \end{gathered} \tag{3} \]

i.e., if \(\delta_a>\delta_b\), then larger clearances are obtained in group \(Aa\) and smaller clearances are obtained in group \(Xx\), i.e. the larger clearances are shifted toward the location of the larger tolerance field, and the smaller clearances are shifted toward the location of the smaller tolerance field.

The fit tolerance \(\Delta\) for some combination, for example \(Aa\), will be

\[ \begin{aligned} \Delta_a&=S_{a\max}-S_{a\min} =S^\circ_{\max}-\delta_b\left(1-\frac{1}{n}\right)-S^\circ_{\min}-\delta_a\left(1-\frac{1}{n}\right)=\\ &=(S^\circ_{\max}-S^\circ_{\min})-\left(1-\frac{1}{n}\right)(\delta_a-\delta_b), \end{aligned} \]

but \(S^\circ_{\max}-S^\circ_{\min}=\delta_a+\delta_b\), i.e. the fit tolerance is equal to the sum of the tolerances for the shaft and the holes.

After substitution we obtain

\[ \Delta_a=(\delta_a+\delta_b)-\left(1-\frac{1}{n}\right)(\delta_a+\delta_b)=\frac{\delta_a+\delta_b}{n}. \]

Denoting the fit tolerance before sorting by \(\Delta^\circ\), we obtain

\[ \Delta_a=\frac{\Delta^\circ}{n}. \tag{4} \]

We shall obtain an analogous result for any other group \(Bb, Cc\), etc., up to group \(Xx\).

Thus, the inaccuracies of mating (clearance fluctuations) in each group, when the tolerance fields are divided into \(n\) intervals, are reduced \(n\)-fold.

The maximum clearance among all groups must, according to the above, be sought in group \(Aa\). Denote this maximum of the maximum clearances by \(S'_{\max}\)

\[ S'_{\max}=S_{a\max}=S^\circ_{\max}-\delta_b\left(1-\frac{1}{n}\right) \tag{5} \]

The minimum clearance among all the minima will be in group \(X_x\)

\[ S'_{\min}=S_{x\min}=S^\circ_{\min}+\delta_b\left(1-\frac{1}{n}\right) \tag{6} \]

The total variation of the clearance in the fit after sorting, which we shall call the intergroup tolerance of the fit, is determined as the difference between the quantities \(S'_{\max}\) and \(S'_{\min}\) reduced above. Denote it by \(\Delta'\); then

\[ \Delta' = S'_{\max} - S'_{\min} = S^\circ_{\max} - \delta_b \left[1 - \frac{1}{n}\right] - S^\circ_{\min} - \delta_b \left[1 - \frac{1}{n}\right] = \]

\[ = (S^\circ_{\max} - S^\circ_{\min}) - 2\delta_b \left[1 - \frac{1}{n}\right] = \Delta^\circ - 2\delta_b \left[1 - \frac{1}{n}\right] \tag{7} \]

Thus, with an increase in the number of intervals, the maximum clearance decreases and the minimum clearance increases. The total (intergroup) tolerance of the fit, as well as the ratio \(\dfrac{S'_{\max}}{S'_{\min}}\), which we shall denote by \(K\), decreases with an increase in the number of groups (the number of intervals).¹ The nature of the variation of the indicated quantities with increasing \(n\) is shown in the diagram of Fig. 5.

It should be noted that the indicated diagram is constructed for the particular case of fit \(X_3\) in the shaft system for a nominal diameter of 50–80 mm. For other fits the position of the curves will, of course, be different, but

Fig. 5.

the character of the change of the curves will remain approximately the same, with the exception of the change in the quantity \(K\), which may either decrease or increase.

As may be concluded from the form of the curves, and also from an analytical investigation of the corresponding formulas, \(S'_{\max}\), \(S'_{\min}\), \(\Delta'\), and \(K\), as the number \(n\) increases to \(\infty\), tend to certain limits, asymptotically approaching the values of these limits.

Thus, for infinitely large \(n\), for group \(Aa\) we obtain from equations (2), upon substituting \(n \to \infty\), in the limit

\[ S_{a\min} = S_{a\max} = S_{a\,\mathrm{cp}} = S^\circ_{\min} + \delta_a = S^\circ_{\max} - \delta_b = S_{\mathrm{cp}} + \frac{\delta_a - \delta_b}{2}, \tag{8} \]

for group \(X\), from equations (3), upon substituting \(n \to \infty\), we obtain in the limit

\[ S_{x\min}=S_{x\max}=S_{x\mathrm{av}}=S^\circ_{\min}+\delta_b =S^\circ_{\max}-\delta_a =S_{\mathrm{av}}-\frac{\delta_a-\delta_b}{2} \tag{9} \]

The partial tolerance of the fit, as is evident from equations 8 and 9, will be equal to zero

\[ \Delta_{a\min}=\Delta_{x\min}=0 \tag{10} \]

The intergroup (overall) tolerance of the fit will be, from equations (8) and (9),

\[ \Delta'_{\mathrm{win}}=\Delta^\circ-2\delta_b=\delta_a-\delta_b \tag{11} \]

and the limiting ratio of the maximum clearance to the minimum clearance

\[ K'_{\mathrm{lim}}= \frac{S'_{\max}}{S'_{\min}} = \frac{S_{\max}-\delta_b}{S_{\min}+\delta_b} = \frac{S_{\min}+\delta_a}{S_{\max}-\delta_a} = \frac{S_{\max}-\delta_b}{S_{\max}-\delta_a} = \frac{S_{\min}+\delta_a}{S_{\min}+\delta_b} \tag{12} \]

In formulas 8, 9, 10, 11, and 12 each of the lines contains several equations that are entirely identical, so that the choice of any one of them leads to an identical result.

Equations (8), (9), and (10) show that, within a given group, as the number \(n\) increases, the minimum, maximum, and average clearances come closer together, and in the limit, as \(n \to \infty\), become equal to one another, so that the partial tolerance of the fit becomes equal to zero. The intergroup (overall) tolerance of the fit, however, decreasing with increasing \(n\), tends toward its limit, equal to the difference between the smaller and the larger tolerance. The ratio \(K=S'_{\max}/S'_{\min}\) in the limit also cannot be equal to unity if \(\delta_a \ne \delta_b\).

Of course, the case \(n \to \infty\) is of interest to us chiefly from the theoretical point of view, for clarifying the tendencies of change of the separate quantities characterizing the mating of parts, and of those theoretical limiting possibilities that can be obtained in machining with subsequent sorting before assembly.

As the character of the curves shows (Fig. 5), the quantities \(S'_{\max}\), \(S'_{\min}\), \(\Delta'\), and \(K'\) undergo sharp changes at \(n=2,3,4\). A further increase in the number of intervals does not introduce substantial changes in the values of the indicated characteristics. Therefore, in practice, in dividing the tolerance field, one rarely goes beyond 4 intervals, so that \(n=4\), and at most 5, may be accepted as the practical limit for the number of groups in sorting.

When dividing tolerance fields into intervals, the designer may be guided by various considerations. Let us examine the various cases that may occur in design practice.

1. Suppose we are given the condition that the maximum clearance (among all groups) must not exceed a prescribed value \(a\), or must be less than it.

Such a requirement arises, for example, in the case of running fits. The magnitude of the clearance, depending on the operating conditions of the bearing—specific pressure, number of revolutions, oil viscosity, length and diameter of the bearing—is known not to exceed a certain value, in order to avoid transition from fluid friction to semifluid, or even to semidry friction.^1

Likewise, in the case, for example, of press fits, the minimum interference must not be less than a prescribed value,^2 since otherwise the transmission of torque without the shaft turning in the bushing cannot be guaranteed. Thus, according to the technical specifications of the NKPS, when mounting wheel centers on wagon and locomotive axles, it is prescribed

^1 See Falz, Fundamentals of Lubrication Engineering. Lesokhin, Tolerances in Mechanical Engineering.

^2 It must not be forgotten that we have agreed to regard a decrease in interference as equivalent to an increase in clearance.

certain limits for the magnitude of the pressing-in force, which, as is known, is directly proportional to the interference.

For the case of transition fits, where usually, for one and the same fit, both interferences and clearances may occur, sometimes, by dividing the tolerance field into intervals, one tries to ensure such a situation that the possibility of obtaining assemblies with clearances is either absolutely excluded, or that the maximum clearance obtained is no greater than a prescribed value, or else that the minimum interference obtained is no less than a prescribed value.

For all the above-mentioned cases, the equation determining the number of groups will be

\[ S'_{\max} \leq a, \tag{13} \]

or

\[ a \geq S^0_{\max} - \left(1 - \frac{1}{n}\right)\delta_b \]

Hence

\[ n \geq \frac{\delta_b}{\delta_b - \left(S^0_{\max} - a\right)} . \tag{14} \]

From the last equation it is evident that the problem posed is not always solvable. The condition for obtaining a finite and positive \(n\) will be

\[ a > S^0_{\max} - \delta_b \tag{15} \]

For \(a = S^0_{\max} - \delta_b\) we obtain \(n = \infty\), and for \(a < S^0_{\max} - \delta_b\) the problem proves impossible.

Putting \(a = 0\), we obtain the condition for the possibility of assembly without clearances when sorting is introduced:

\[ S^0_{\max} \leq \delta_b \tag{16} \]

Eng. A. V. Lesokhin, in the book Tolerances in Machine Building, p. 81, 4th ed., writes with regard to tight fits of the 2nd accuracy class that, in view of the fact that clearances may be obtained instead of minimum interferences, “complete reliability of the fit in continuous assembly without sorting is not ensured.” But here it should be pointed out that even with sorting this problem proves practically feasible only for a transition fit of the 2nd class\(^{1}\) and a tight fit of the 1st class. For the fits \(H_8/n_7\) and \(H_7/N_8\) OST—1026 and OST—1016 (the so-called transition fits of \(2\frac{1}{2}\) accuracy class), it is possible to avoid clearances completely only with a very considerable number of sorting groups (\(n \simeq 7 \div 8\)).

All other fits of the 1st class (\(H_1 P_1\)), 2nd class (\(T,H,P\)) and \(2\frac{1}{2}\) class (\(M_8/h_7,\ K_8/h_7,\ I_8/h_7,\ H_8/n_7,\ H_8/m_7,\ H_8/k_7,\ H_8/j_7\)) will not be able to ensure the absence of assemblies with clearances even in the case of sorting. Nevertheless, by means of sorting, for these fits it will be possible to reduce the possible clearances, although at the cost of a corresponding reduction of the maximum interferences.

The latter circumstance must never be forgotten, since when sorting is introduced any reduction of the maximum clearance is accompanied by a reduction also of the maximum interference.

Sometimes, by introducing sorting, one tries to ensure such a condition that the minimum clearance (respectively, the maximum interference) be greater (for interference, respectively, less) than some previously prescribed value \(b\),

\(^{1}\) A transition fit of the 1st accuracy class \(\Gamma\) ensures assembly without clearances without any sorting.

In the case of running fits, according to Gumbel’s hydrodynamic theory of friction, when the clearance is reduced below the optimum, the coefficient of friction increases, and consequently so does the temperature of the bearing. Therefore one should not go below a certain minimum clearance, which, according to Lesokhin’s proposal made on the basis of Gumbel’s theory, is equal to 0.644 of the optimum clearance.^1) Similarly, in the case of press fits the designer endeavors to ensure that the maximum possible interference does not exceed that permitted by the strength of the materials from which the shaft and the bushing are made.

Thus

\[ S'_{\min} \ge b, \tag{17} \]

or

\[ S^0_{\min}+\delta_b\left(1-\frac{1}{n}\right)\ge b. \]

Hence, determining \(n\), we obtain the formula for determining the minimum number of groups

\[ n \ge \frac{\delta_b}{\delta_b+(S^0_{\min}-b)} \tag{18} \]

This problem likewise is not always feasible. The condition for obtaining a finite and positive \(n\) will be

\[ b<\delta_b+S^0_{\min} \tag{19} \]

As follows from formula (18), the minimum clearance (maximum interference) will be equal to zero when the number of groups is

\[ n=\frac{\delta_b}{\delta_b+S^0_{\min}} \tag{20} \]

Sometimes the problem confronting the designer is to reduce the tolerance of the fit during sorting to some value \(c\), which constitutes, relative to the old fit tolerance \(\Delta^0\), a certain fraction \(\gamma\), where \(\gamma<1\), i.e.

\[ \Delta' \le \gamma \Delta^0=c. \tag{21} \]

It is not difficult to derive from formula (7) the number of groups required in sorting in order to satisfy this condition:

\[ n \ge \frac{2\delta_b}{2\delta_b-(\Delta^0-c)}, \tag{22} \]

or otherwise

\[ n \ge \frac{2\delta_b}{2\delta_b-\Delta^0(1-\gamma)}. \tag{22} \]

It is of interest to trace how the ratio

\[ \frac{S'_{\max}}{S'_{\min}} \]

changes as the number of sorting groups increases. Here two cases must be distinguished. If the absolute value of the maximum clearance is greater than the absolute value of the minimum clearance, then the relative value of the indicated ratio decreases, and conversely.

Thus, for example, for fit \(X_3\),

\[ S^0_{\max}=180\mu \]

\[ S^0_{\min}=40\mu \]

\[ |S_{\max}|>|S_{\min}|; \]

therefore, the ratio

\[ \frac{S'_{\max}}{S'_{\min}} \]

decreases as the number of groups increases.

^1) For more detail see Lesokhin, Tolerances in Machine Building.

For the fit \(H_8/m_7\) according to ISA OST—1016, \(S_{\max}=+35\mu\), \(S_{\min}=+45\mu\), \([S_{\max}]<[S_{\min}]\); consequently, the ratio \(\dfrac{S'_{\max}}{S'_{\min}}\) increases as the number of groups increases, namely for \(n=2\) it is equal to \(-0.77\), and for \(n=4\) it is equal to \(-0.67\).

In the first case, when dividing into groups, the requirement is often imposed that the ratio \(\dfrac{S'_{\max}}{S'_{\min}}\) be less than a prescribed value \(K\). In the second case, the requirement is imposed that the ratio \(\dfrac{S'_{\max}}{S'_{\min}}\) be greater than the prescribed value \(K\).

The necessary number of groups for both cases is found according to the following considerations:

\[ \frac{S'_{\max}}{S'_{\min}} = \frac{S^0_{\max}-\left(1-\frac{1}{n}\right)\delta_b} {S^0_{\min}+\left(1-\frac{1}{n}\right)\delta_b} = k \tag{23} \]

From this equation we find

\[ \frac{1}{n}\lesseqgtr 1-\frac{S^0_{\max}-kS^0_{\min}}{\delta_b(1+k)} \tag{24} \]

Hence it is easy to determine \(n\).

The condition for obtaining a finite and positive value of \(n\), i.e. the condition for the possibility of a real solution of the problem posed, is obviously written in the form

\[ \left|S^0_{\max}-KS^0_{\min}\right|<\left|\delta_b(1+k)\right| \tag{25} \]

To illustrate the application of the formulas given above, let us solve several concrete examples.

Example 1. Suppose machining is carried out according to the tolerances of the fit \(X_3\) in the shaft system OST—1023, in the diameter interval \(50\)–\(80\) mm. From OST—1023 we have
\(\delta_a=80\mu;\ \delta_b=60\mu;\ S^0_{\max}=+180\mu;\ S^0_{\min}=+40\mu.\)

Consequently,

\[ K_0=\frac{S^0_{\max}}{S^0_{\min}}=\frac{180}{40}=4.5. \]

a) If we wished to divide the parts, for the fit, into such a number of groups that the value of the ratio \(\dfrac{S'_{\max}}{S'_{\min}}<2\), then it is first necessary to check the possibility of a real solution of the problem posed. According to formula (25) we have

\[ \left|S^0_{\max}-KS_{\min}\right|<\left|\delta_b(1+k)\right| \]

or

\[ |180-2.40|<|60(1+2)| \]

\[ 100<180, \]

i.e. the problem is possible.

The number of groups is determined from equation (24)

\[ \frac{1}{n}\lesseqgtr 1-\frac{S^0_{\max}-kS^0_{\min}}{\delta_b(1+k)} \]

\[ \frac{1}{n}\lesseqgtr 1-\frac{180-2,40}{60(1+2)}. \]

Whence

\[ n\geq 2.25. \]

We take

\[ n=3. \]

Then

\[ k=\frac{S^0_{\max}-\left(1-\frac{1}{n}\right)\delta_b} {S^0_{\min}+\left(1-\frac{1}{n}\right)\delta_b} = \frac{180-\left(1-\frac{1}{3}\right)60} {40+\left(1-\frac{1}{3}\right)60} \div 1.75. \]

The maximum that can be achieved in the ideal case by the sorting method, i.e. for \(n=\infty\), is, by formula (12),

\[ k_{\text{lim}}=\frac{S^0_{\max}-\delta_b}{S^0_{\min}+\delta_b} =\frac{180-60}{40+60}=1.20; \]

with the above-adopted \(n=3\) we shall have

\[ S'_{\max}=S^0_{\max}-\delta_b\left(1-\frac{1}{n}\right) =180-\left(1-\frac{1}{3}\right)60=140\mu, \]

\[ S'_{\min}=S^0_{\min}+\delta_b\left(1-\frac{1}{n}\right) =40+60\left(1-\frac{1}{3}\right)=80\mu. \]

b) If, under the same machining conditions, another problem is posed: to divide the parts into such a number of groups that the maximum clearance be less than the prescribed value \(a=150\mu\), the condition for a finite positive \(n\) will, by formula (15), be

\[ a>S^0_{\max}-\delta_b, \]

\[ 150>180-60, \]

i.e. it will be satisfied.

The required number of groups is determined by formula (14)

\[ n\ge \frac{\delta_b}{\delta_b-\left(S^0_{\max}-a\right)} = \frac{60}{60-(180-150)}. \]

\[ n\ge 2. \]

c) If the sorting is to proceed from the condition that \(S'_{\min}>b=80\mu\), then the condition for a finite positive \(n\) is written, by formula (19),

or

\[ b<\delta_b+S^0_{\min} \]

\[ 80<60+40 \quad \text{or} \quad 80<100 \]

i.e. the condition is satisfied.

The required number of groups is determined by formula (18)

\[ n\ge \frac{\delta_b}{\delta_b+\left(S^0_{\min}-b\right)} = \frac{60}{60+(40-80)}=3. \]

As was computed above, for \(n=3\), \(S'_{\min}=80\mu\).

d) Suppose that in sorting the problem is posed: to reduce the fit tolerance by a factor of 2, i.e.

\[ \Delta'\le \gamma\Delta=\frac{1}{2}\Delta. \]

By formula (22) we find the required number of groups

\[ n\ge \frac{2\delta_b}{2\delta_b-\Delta^0(1-\gamma)} = \frac{2.60}{2.60-140\left[1-\frac{1}{2}\right]} =2.4. \]

For \(n=3\) we have \(\Delta' = 140 - 80 = 60\mu\), which, relative to the initial fit tolerance, is

\[ \gamma=\frac{\Delta'}{\Delta_0}=\frac{60}{140}=0.43<0.50. \]

Example 2. Machining is carried out according to the tolerances of the \(Pr1_3\) fit of the hole system (OST--1069) for diameter \(d=60\) mm.

According to OST--1069 we have \(\delta_a=60\mu,\ \delta_b=60\mu\). Hence \(\Delta^0=120\mu\), the maximum interference \(H_{\max}=135\mu\). The minimum interference \(H_{\min}=15\mu\).

In accordance with the adopted notation we have \(S^0_{\max}=15\mu,\ S^0_{\min}=-135\mu\), i.e. the ratio

\[ \frac{S^0_{\max}}{S^0_{\min}}=\frac{15}{-135}=0.111. \]

When the tolerance fields are divided into intervals, the value

\[ \frac{S^0_{\max}}{S^0_{\min}} \]

will increase, since, according to formula (12), for \(n=\infty\)

\[ k_{\mathrm{lim}}=\frac{S_{\max}-\delta_b}{S_{\min}+\delta_b} =\frac{-15-60}{-135+60} =\frac{-75}{-75}=1.00 \]

a) Sorting from the condition \(k\geq 0.333=\dfrac{1}{3}\).

The condition for a finite positive \(n\) is written as follows:

\[ \left|S^0_{\max}-kS_{\min}\right|<\left|\delta_b(1+k)\right| \]

\[ \left|-15-\frac{1}{3}(-135)\right|< \left|60\left(1+\frac{1}{3}\right)\right| \]

\[ |30|<|80| \]

i.e. the condition is satisfied.

The number of groups

\[ \frac{1}{n}\leq 1-\frac{S^0_{\max}-kS^0_{\min}}{\delta_b(1+k)} \]

\[ \frac{1}{n}\leq 1-\frac{-15-0.333(-135)}{60(1+0.333)}. \]

Whence

\[ n\geq \frac{8}{5}. \]

We take

\[ n=2. \]

Then

\[ k= \frac{S^0_{\max}-\left[1-\frac{1}{n}\right]\delta_b} {S^0_{\min}+\left[1-\frac{1}{n}\right]\delta_b} = \frac{-15-\left[1-\frac{1}{2}\right]60} {-135+\left[1-\frac{1}{2}\right]60}, \]

\[ k\simeq 0.43. \]

b) Sorting from the condition \(S'_{\max}\leq a\). Let \(a=-50\mu\). The condition for a positive finite \(n\): \(a\geq S^0_{\max}-\delta_b\)

\[ -50\geq -15-50, \]

i.e. the condition is satisfied

\[ -50\geq -65 \]

Number of groups

\[ n \geq \frac{\delta_b}{\delta_b - \left(S_{\max}-a\right)} = \frac{60}{60-[-15-50]}=2.4. \]

Let us take \(n=3\). Then

\[ S'_{\max}=S^0_{\max}-\left[1-\frac{1}{n}\right]\delta_b =-15-\left[1-\frac{1}{3}\right]60=-55\mu. \]

This problem is posed when one wishes to preserve a certain minimum interference. In the present case the interference obtained is \(55\mu\), greater than the specified minimum \(50\mu\), as was required.

c) Determination of the number of groups from the condition \(S'_{\min}\geq b\), correspondingly, in the present problem, the condition of establishing the number of groups \(n\) from the limiting maximum interference. It is solved analogously, and therefore is omitted.

Example 3. Machining is carried out according to the tolerances for the fit \(H_8 j_7\) according to ISA; see OST—1016, which corresponds to a tight fit of \(2\frac{1}{2}\)-th accuracy class in the hole system. Diameter \(50\text{—}80\) mm, \(\delta_a=46\mu\).

\[ \delta_b=30\mu;\qquad S^0_{\max}=58\mu;\qquad S^0_{\min}=-18\mu. \]

In this fit it is of interest to trace the change of \(S'_{\max}\), \(S'_{\min}\), \(\Delta'\), and \(K\) with increasing number \(n\). As is seen from the appended diagram (Fig. 6),

Fig. 6.

\(S'_{\min}\), with the introduction of sorting, changes from interference to clearance. The quantity

\[ K=\frac{S'_{\max}}{S'_{\min}} \]

at first decreases from \(-3.25\) to \(-14\) and \(-\infty\), then decreases from \(-\infty\), \(19\), and to \(2.33\), when \(n=\infty\).

The same picture must also be expected in fits \(P_1\) and \(P\). Consequently, the introduction of sorting in tight fits not only cannot reduce the number of assemblies with clearances to zero or decrease them; on the contrary, with

with the introduction of sorting we shall obtain fits exclusively with clearances if only we take a sufficient number of groups.

Let us determine this number of groups. By formula (17)

\[ S_{\min} \geq b. \quad \text{Here } b=0. \]

The condition for a finite positive \(n\) by formula (19)

\[ b < \delta_b + S^0_{\min}\quad 0 < 30 - 18 \]

is satisfied.

Then the number of groups by formula (18)

\[ n \geq \frac{\delta_b}{\delta_b + (S^0_{\min}-b)} = \frac{30}{30+(-18-0)} = 2.5. \]

Thus, beginning with \(n=3\), from the assembly of sorted parts we shall obtain exclusively fits with clearances.

Example 4. Machining is carried out according to the tolerances of fit \(\Gamma\) OST—1012. (A blind fit of the 2nd accuracy class in the hole system.) Diameter \(50\text{—}80\ \mathrm{mm}\). According to OST—1012 we have

\[ \delta_a = 30\mu,\quad \delta_b = 20\mu,\quad S^0_{\max}=10,\quad S^0_{\min}=-40\mu,\quad K=\frac{S^0_{\max}}{S^0_{\min}}=-0.250. \]

The tolerance picture given by this fit is entirely different.

As we have already said, when the number of sorting groups is increased, \(S'_{\max}\) decreases, while \(S'_{\min}\) increases. Since the quantity \(S'_{\max}\) is close to zero, it must be expected that, when sorting is introduced, \(S'_{\max}\) will pass through zero and become a negative quantity.

Let us determine the number of groups from the condition \(S'_{\max}\geq 0\).

The condition for a finite positive \(n\)

\[ a \geq S^0_{\max}-\delta_b \]

\[ 0 \geq 10-20,\ \text{i.e. it is satisfied.} \]

The required number of groups is

\[ n \geq \frac{\delta_b}{\delta_b-(S^0_{\max}-a)} = \frac{20}{20-(10-0)} = 2, \]

i.e., when the tolerance fields are divided into 2 or more intervals, the possibility of the appearance of assemblies with clearances will be eliminated.

The conclusions presented above, as well as the examples solved, show by what route, proceeding from a definite, previously specified condition, one may find the necessary number of intervals or groups in sorting for the fulfillment of the indicated condition. In addition, the formulas given above make it possible to draw a conclusion as to the feasibility of the problem posed in each concrete case.

References

1. Lesokhin A., Tolerances in Mechanical Engineering, 4th ed.
2. Pivovar, Eng., “Distribution of Parts over the Tolerance Field.” Vestnik Metallopromyshlennosti, 1935, No. 9.

1. As will be shown below (see example 3), the quantity \(K = \dfrac{S'_{\max}}{S'_{\min}}\) is a discontinuous function of \(n\), and under certain conditions undergoes changes from \(-\infty\) to \(+\infty\). ↩↩