Determination of the parameters of typical multi-link dimensional chains (closing link is tension) using recurrent dependencies

The existing approach to the use of the group interchangeability method during assembly, which are concentrated, has a number of significant drawbacks: 1) there are no formulas for calculating the upper and lower deviations of the component parts of the dimensional chain; 2) deviations are selected without taking into account the design of the component links (the method of equal tolerances is used); 3) the method is applicable for the number of links no more than three with a small number of groups. For other types of problems there is no algorithm, mathematical apparatus, and examples of calculation are not considered sequentially; 4) the calculation by the group interchangeability method is limited by the condition of equal tolerances, i.e. excluding compensators and compensating properties of materials as a part of mechanisms; 5) the calculations are based on assumptions about the economic feasibility of increasing the closing link by a factor of n without taking into account the feasibility of such an increase in the component links that is not permissible in complex mechanisms; 6) the size of the batch of parts and the volume of work in progress are not reflected, as one of the factors of economic feasibility of establishing production tolerance. In this regard, it is necessary to improve the method of achieving accuracy by group interchangeability. This article discusses typical problems that rise when solving dimensional chains in which the closing link is a tension.


Introduction
This article discusses typical problems that rise when solving dimensional chains in which the closing link is a tension. We accept that in the case of tension compared with the gap joints, the diameter of the sleeve is a reducing link, since with increasing it the closing link (tension) reduces. The diameter of the shaft in the joint with the tension is an increasing link.
The first problem concerns a three-link dimensional chain. The next two problems concern dimensional chains with one reducing link and two increasing links .
When solving dimensional chains, the rules set forth in section 1 are applied. Let us consider the solutions with specific examples.

Three-link dimensional chain
Given: a three-link dimensional chain ( Figure 1) and layout of tolerance fields ( Figure 2). The method of equal tolerances is applied, the number of groups is n. The tolerances of the increasing and reducing links are TD = Td. Closing link is a tension. Determine: upper and lower deviations of component parts when applying the method of group interchangeability and the method of equal tolerances.  For reducing links (Eqs 1,2): where i=1,2…n-the current variable characterizing the number of groups, Td is the tolerance of the reducing link, since the tension decreases with increasing hole diameter. Upper and lower deviation of the increasing links (Eq 3,4):: where Nmin is minimal tension, TD-the tolerance of the increasing link (the shaft), since with an increase in the diameter of the shaft the tension (closing link) increases. The calculation was made with the following initial data: TD = Td = 40 microns, n = 4. The calculation results for the recurrent formulas are given in Table 1

A multi-link dimensional chain with a tension. The method of equal tolerances
Given  (Figure 4), we derive the recurrent Eqs. 5, 6 by the induction method:   In Eqs. 5 and 6 the product with mred = 1, m = 3, j=2,3 is determined as follows: The calculation was made with the following initial data: n = 4, Td = 80 microns, Nmin = 20 microns, Nmax = 60 microns. The calculation results are given in the Table 2.
The formulas (5) and (6) are applied when the nominal sizes of the links are in the same interval of the normal series of numbers, i.e. tolerances for all increasing links are equal.

Lower EI A (i,j) ES A (i,j) EI A (i,j) es A (i,j) ei A (i,j)
Max Мin   1  80  70  40  30  80  60  60  20  40   2  70  60  30  20  60  40  60  20  40   3  60  50  20  10  40  20  60  20  40   4  50  40  10  0  20  0  60  20  40 It should be noted that in the manufacture brackets are used for mounting tires and boxes with wires. If the tension is greater than permissible, then there may be a breakdown of the bracket. Therefore, the sizes are sorted according to the Table 2.

A multi-link dimensional chain with a tension. The method of equal accuracy is applied
Given: multi-link dimensional chain. Two increasing links minc = 2 and one reducing link ( Figure 5). The layout of tolerance fields is given in Figure 6. The sums of the tolerances of the increasing and reducing links are equal. The method of equal accuracy is applied. The number of groups is n.
Determine: upper and lower deviations of the component parts when applying the method of group interchangeability and the method of equal accuracy.  First, we derive formulas for determining the tolerances of the increasing links (Eq.8): T (i,j)-the tolerance for the size of the increasing link; a -the number of tolerance units (Eq.9): ) ...
where ej is the tolerance unit determined using the Eq. 10.
Then, by induction using the layout in Figure 6, we determine the recurrent dependencies to determine the upper and lower deviations of the increasing links (Eqs. 11,12): where m is the number of component links. The calculation was made with the following initial data: A2 = 27 mm; A3 = 8 mm; A1 = 35 mm; Td = TD = 45 microns; Nmax = 20 microns, Nmin = 5 microns; n = 3, minc = 2 is the number of increasing links sorted into groups; Tgr = 15 microns -group tolerance of the reducing link.
The results of solvation of the problem are given in table 3.

Conclusion
The presented typical problems using the obtained recurrent dependencies reveal dimensional bonds in multilink chains in which the closing link is the tension when applying the group interchangeability method. The tables with the calculation results are the main part of the technological document for sorting parts into groups during assembly.