## 2 Materials and Methods  
After measuring the root diameters of the cycloid profiles of wheels manufactured by the OOO \"Siberian Engineering Company\", it was found that the equipment allowed the obtainment of cycloid profiles with a normal distribution of dimensions that were approximately within the tolerance zone, as shown in Figure 2. The distribution of the results of measuring the diameter of the disk roots (Figure 2) does not exactly conform to the normal distribution law but can be approximately taken as such, owing to the insignificant deviation of the presented distribution from the normal law. The cycloid profile with a root diameter of \(\mathcal{O}\)175 mm was measured (the accuracy was according to the 7 th tolerance grade). The diameter was measured using the contact method with a clock-type indicator. The data on five pivot points were obtained in the root of the profile. Among these points, the point with the largest modulus coordinate was found. The measurements were also performed in the diametrically opposite root. The obtained diameter of the roots of the cycloidal profile was determined by the difference in the obtained coordinates. The diagram shows that the equipment processes the maximum number of parts according to the dimensions, the accuracy of which shifts to the upper limit of the tolerance zone from its centre. In this way, it is possible to determine the limits within the tolerance zone, to which more than 80% of the dimensions fall. When determining the permissible manufacturing clearance, one should focus on this range. Based on the measurement results (Figure 2), the following limits can be accepted within the tolerance zone: from 0.262 mm to 0.288 mm. Overall, 65% of the entire tolerance zone, that is 80% of the dimensions, obtained on the equipment are distributed approximately within two thirds of the tolerance zone per dimension. Figure 1: Cross-section of a transmission with intermediate rolling elements and a free cage: H—generator; 1—cam; 2—intermediate rolling elements; 3—ring; 4—cage; 5—bearing. We considered the normal law of the probability distribution for the obtained dimensions and the method of incomplete interchangeability when selecting fits for parts in the cycloid engagement. In considering the engagement of a transmission with intermediate rolling elements and free cage (IREFC) (Figure 3), this consists of a wheel with an external cycloid profile--cam 1, cylindrical rolling elements 2 and a wheel with an internal cycloid profile--ring 3 [16; 22]. As previously noted, when manufacturing contacting profiles of transmission links, the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they are made larger than the nominal dimension. In fact, the dimensions of these three links can be manufactured in any combination but there is the possibility of manufacturing parts with dimensions using extreme tolerance values (Figure 4). Figure 3: Engagement of a transmission with IREFC and tolerance zones of dimensions of link contacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. Figure 2: Diagram of deviation distribution of the root diameter of the cycloid profile within the tolerance zone, TD = 0.04 mm. Having considered the case presented in Figure 4, let us determine the maximum possible manufacturing clearance in the engagement of a transmission with IREFC using the following expression:  
\[\Delta_{max}=\frac{TD^{v}}{2}-Td^{\text{rb}}-\frac{Td^{\text{k}}}{2}, \tag{1}\]  
where \(TD^{v}\) is the deviation tolerance of the diametric dimension of the ring profile;  
\(Td^{\text{rb}}\) is the deviation tolerance of the diametric dimension of the rolling elements;  
\(Td^{\text{k}}\) is the deviation tolerance of the diametric dimension of the cam profile.