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Research on the Possibility of Lowering the Manufacturing Accuracy of Cycloid Transmission Wheels with Intermediate Rolling Elements and a Free Cage

Research on the Possibility of Lowering the Manufacturing Accuracy of Cycloid Transmission Wheels... applied sciences Article Research on the Possibility of Lowering the Manufacturing Accuracy of Cycloid Transmission Wheels with Intermediate Rolling Elements and a Free Cage 1 , 2 , 3 4 5 Egor A. Efremenkov *, Nikita V. Martyushev , Vadim Yu Skeeba , Maria V. Grechneva , 6 7 Andrey V. Olisov and Anatoly D. Ens Mechanical Engineering Depurtment, Tomsk Polytechnic University, 30, Lenin Ave., 634050 Tomsk, Russia Innovation Management Department, Tomsk State University of Control Systems and Radioelectronics, 40, Lenin Ave., 634050 Tomsk, Russia Materials Science Depurtment, Tomsk Polytechnic University, 30, Lenin Ave., 634050 Tomsk, Russia; martjushev@tpu.ru Department of Industrial Machinery Design, Novosibirsk State Technical University, 20, K. Marks Ave., 630073 Novosibirsk, Russia; skeeba_vadim@mail.ru Department of Materials Science, Welding and Additive Technologies, Irkutsk National Research Technical University, 83, Lermontov Str., 664074 Irkutsk, Russia; mgrech@irk.ru Scientific and Educational Center “Additive Technologies”, National Research Tomsk State University, 36, Lenin Ave., 634050 Tomsk, Russia; a.v.olisov@mail.tsu.ru Faculty of Electrical Engineering and Computer Science, Technical University of Ostrava, 17, listopadu 2172/15, 70800 Ostrava, Czech Republic; ens.t@enspg.ru * Correspondence: egorefr@tpu.ru; Tel.: +7-(3822)-606392 Abstract: Purpose: In the present work, different combinations of fits and accuracies, in relation to the profiles of mating parts, have been analysed in order to assess the degree of the engagement of Citation: Efremenkov, E.A.; transmissions that contain intermediate rolling elements. The aim of this work is to determine which Martyushev, N.V.; Skeeba, V.Y.; fits have decreased accuracy, but nevertheless provide a minimum manufacturing clearance for the Grechneva, M.V.; Olisov, A.V.; Ens, transmission engagement in order to reduce the cost of parts production. Methods and materials: A.D. Research on the Possibility of Considering the normal probabilistic distribution law in relation to the obtained dimensions of the Lowering the Manufacturing manufacturing equipment, a combination of fits were selected using the incomplete interchangeability Accuracy of Cycloid Transmission method, taking into account the peculiarities of the cycloid engagement in transmissions with inter- Wheels with Intermediate Rolling mediate rolling elements (IRE). Results: Having studied various combinations of fits of parts that are Elements and a Free Cage. Appl. Sci. engaged in transmissions with intermediate rolling elements and a free cage (IREFC), a combination 2022, 12, 5. https://doi.org/ of fits for a “ring, rolling-element cam” were determined, in which a technological clearance of 3 m 10.3390/app12010005 is formed in the engagement. At the same time, cycloid disk profiles are manufactured according Academic Editor: Alberto Boschetto to the 9th tolerance grade, which reduces the laboriousness and cost of the production. Discussion. Received: 19 November 2021 When reducing the manufacturing accuracy of cycloid disks, it is possible to obtain both very ample Accepted: 16 December 2021 clearances and significant negative allowances. For example, having manufactured a ring with the Published: 21 December 2021 H9 fit, rolling elements with h6 and a cam with js9, the maximum manufacturing clearance can reach 0.086 mm, while the clearance limits vary from 0.025 mm to 0.061 mm. Additionally, if mating parts Publisher’s Note: MDPI stays neutral are manufactured using a combination of K9-h6-js9 fits, a negative allowance varying from 0.014 mm with regard to jurisdictional claims in published maps and institutional affil- to 0.026 mm will emerge in the engagement. Both described cases are unacceptable because both iations. ample clearances and large negative allowances will negatively influence the working capacity of the mechanism. However, it is possible to select a combination of fits using the 9th tolerance grade of the basic parts, by which the parts will contact in the range from a small negative allowance of 1 m to a clearance of 3–4 m. Furthermore, if this is considered, taking into account the machine settings, it is Copyright: © 2021 by the authors. possible to obtain parts according to the 9th accuracy tolerance grade and, at the same time, provide Licensee MDPI, Basel, Switzerland. a clearance in the engagement that is almost equal to zero. Moreover, such a combination of fits is This article is an open access article relevant for any transmission with IRE. This is a positive result because it reduces the laboriousness distributed under the terms and when manufacturing parts and, at the same time, provides high accuracy of the mechanism. Conclu- conditions of the Creative Commons sions: It has been established that when lowering the accuracy of manufacturing transmission parts Attribution (CC BY) license (https:// with IRE, both clearances and negative allowances may occur in the engagement, depending on the creativecommons.org/licenses/by/ 4.0/). combination of fits. At the same time, it is possible to select such a combination of fits, by which the Appl. Sci. 2022, 12, 5. https://doi.org/10.3390/app12010005 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 5 2 of 10 parts manufactured according to the 9th tolerance grade, will provide almost zero clearance of the engagement of the transmission. In this way, it is possible to reduce the cost of manufacturing the parts for gears with intermediate rolling elements and, at the same time, maintain a high accuracy of the transmission mechanism. Keywords: manufacturing accuracy; cycloidal transmission; kinematic parameters; tolerances; deviations; technological gap 1. Introduction When manufacturing mechanical transmissions, special attention is paid to the accu- racy and quality of the surfaces of the parts directly engaged in the transmission of speed and efforts. The accuracy of these parts must be high and the roughness of their contact surfaces must be low [1–6]. In practice, in general engineering, these structural elements are manufactured with the accuracy of contact surfaces according to the 7th tolerance grade, which ranges in the interval from 0.025 to 0.057 mm (depending on their size). At the same time, the roughness of the contact surfaces is brought to a level in the order of Ra 0.8–1.6 m. The shape of the cycloid profile of the tooth also influences the working capacity of the mechanism in both longitudinal [7] and transverse [8,9] directions. The cycloid profile modification allows errors to be compensated for in the geometry of the cycloid engage- ment and the power characteristics to be enhanced [7,10]. In particular, authors [7] note that, when correcting the cycloid profile, the contact voltage changes on average by several tens of megapascals. These design requirements cause certain technological challenges and increase the manufacturing time of such parts. At the same time, a decrease in the accuracy of manufacturing parts leads to a quality deterioration of the mechanism and loss of its accuracy characteristics due to emerging manufacturing clearances [11]. Therefore, the study of the possibility for lowering the manufacturing accuracy of the contacting surfaces of disks engaged in a cycloid transmission, while maintaining the accuracy of the transmission itself, is a vital task. The solution to this will allow a decrease in the manufacturing time of such parts and a reduction in the cost of the entire mechanism. The accuracy of the transmission mechanism, among other things, is influenced by such a characteristic as the clearance in the engagement (angular play) [8]. In [8], a method for modifying the cycloid tooth profile is considered, taking into account the clearances in the engagement, but the influence of the manufacturing accuracy of the cycloid profile has not been analysed. The clearance in the engagement can be designed in, when designing the transmission and it also definitely emerges when manufacturing the transmission links, being directly in the engagement. The structurally laid clearance is calculated and can be adjusted when assembling the transmission. Additionally, the manufacturing clearance, that is emerging due to dimension deviations from the absolute value when manufacturing parts, is difficult to take into account owing to the probabilistic obtainment of dimensions during manufacture. At the same time, there are transmissions, for which the clearance adjustment in the engagement is not provided because the design does not allow it. Such transmis- sions are cycloid pinwheel [8,9] transmissions with intermediate rolling elements (IRE), in general [12–15], and transmissions with intermediate rolling elements and a free cage (IREFC) [16], in particular (Figure 1). In IREFC transmissions, the loaded parts are cam 1, rolling elements 2 and ring 3 (Figure 1). The accuracy of the manufacturing of these parts influences the engagement quality. There is also a separator 4 between the cycloidal profiles of cam 1 and wheel 3. However, the manufacturing accuracy will not affect the accuracy of the engagement of transmissions with IRE. This is explained by the fact that a separator 4 is only used for the distribution of the rolling elements 2 on a mutual angular placement and it keeps them from moving into the root of the cycloidal profile (for example, the cam) when they are not under load. The standard technology for manufacturing the Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 10 the engagement of transmissions with IRE. This is explained by the fact that a separator 4 Appl. Sci. 2022, 12, 5 3 of 10 is only used for the distribution of the rolling elements 2 on a mutual angular placement and it keeps them from moving into the root of the cycloidal profile (for example, the cam) when they are not under load. The standard technology for manufacturing the separator separator allows the ob allows tainm the ent obtainment of such a th of ickne such ss o a thickness f the part tha of the t it wi part ll n that ot wors it will en the not engag worsen e- the ment in engagement any case. Cyclo in any icase. d pinwheel Cycloid transmissi pinwheel ons transmissions are designed so are that the c designedle so ara that nce be- the tween the cycloid profiles and the pin is not regulated. The situation with transmissions clearance between the cycloid profiles and the pin is not regulated. The situation with transmissions having IRE is even more complicated having IRE is even mor , w e complicated, here the rolling elemen where thet is rolling able to move free element is able ly [2– to move 4]. Moreov freelyer [2 , in som –4]. Mor e t eover ransm , in iss some ions transmissions this contacts sthis imucontacts ltaneoussimultaneously ly with two cycwith loid p two ro- cycloid profiles [12,13]. Transmissions with a cycloid wheel are widely used in the industry files [12,13]. Transmissions with a cycloid wheel are widely used in the industry in Russia in Russia (“SIMACO” Company, Tomsk, Russia), Europe [17,18] and abroad [19–21]. The (“SIMACO” Company, Tomsk, Russia), Europe [17,18] and abroad [19–21]. The problems problems with improving the manufacturability of the cycloid disk profile are acute for with improving the manufacturability of the cycloid disk profile are acute for manufac- manufacturers. Therefore, the purpose of this work is to determine the influence of the turers. Therefore, the purpose of this work is to determine the influence of the manufac- manufacturing accuracy of the contact surfaces of profile disks on the clearance size in the turing accuracy of the contact surfaces of profile disks on the clearance size in the engage- engagement for transmissions with IREFC. ment for transmissions with IREFC. Figure Figure 1 1.. C Cr ros oss-section s-section of of a a t transmission ransmission wi with th interm intermediate ediate rol rolling ling e elements lements and and a a free free cag cage: e: H— H— generator; 1—cam; 2—intermediate rolling elements; 3—ring; 4—cage; 5—bearing. generator; 1—cam; 2—intermediate rolling elements; 3—ring; 4—cage; 5—bearing. 2. Materials and Methods 2. Materials and Methods After measuring the root diameters of the cycloid profiles of wheels manufactured 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 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 the obtainment of cycloid profiles with a normal distribution of dimensions that were ap- approximately within the tolerance zone, as shown in Figure 2. The distribution of the proximately within the tolerance zone, as shown in Figure 2. The distribution of the results results of measuring the diameter of the disk roots (Figure 2) does not exactly conform to the 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 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 deviation of the presented distribution from the normal law. The cycloid profile with a diameter of Ø175 mm was measured (the accuracy was according to the 7th tolerance grade). root diameter of Ø175 mm was measured (the accuracy was according to the 7th tolerance The diameter was measured using the contact method with a clock-type indicator. The data grade). The diameter was measured using the contact method with a clock-type indicator. on five pivot points were obtained in the root of the profile. Among these points, the point The data on five pivot points were obtained in the root of the profile. Among these points, with the largest modulus coordinate was found. The measurements were also performed in the point with the largest modulus coordinate was found. The measurements were also the diametrically opposite root. The obtained diameter of the roots of the cycloidal profile performed in the diametrically opposite root. The obtained diameter of the roots of the was determined by the difference in the obtained coordinates. The diagram shows that cycloidal profile was determined by the difference in the obtained coordinates. The dia- the equipment processes the maximum number of parts according to the dimensions, the gram shows that the equipment processes the maximum number of parts according to the accuracy of which shifts to the upper limit of the tolerance zone from its centre. In this way, dimensions, the accuracy of which shifts to the upper limit of the tolerance zone from its it is possible to determine the limits within the tolerance zone, to which more than 80% centre. In this way, it is possible to determine the limits within the tolerance zone, to which of the dimensions fall. When determining the permissible manufacturing clearance, one more than 80% of the dimensions fall. When determining the permissible manufacturing should focus on this range. Based on the measurement results (Figure 2), the following clearance, one should focus on this range. Based on the measurement results (Figure 2), limits can be accepted within the tolerance zone: from 0.262 mm to 0.288 mm. Overall, 65% the following limits can be accepted within the tolerance zone: from 0.262 mm to 0.288 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. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 10 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. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 10 mm. Overall, 65% of the entire tolerance zone, that is 80% of the dimensions, obtained on Appl. Sci. 2022, 12, 5 4 of 10 the equipment are distributed approximately within two thirds of the tolerance zone per dimension. Figure 2. Diagram of deviation distribution of the root diameter of the cycloid profile within the tolerance zone, TD = 0.04 mm. Figure 2. Figure 2.Diag Diagram ram of de of dev viati iation on di distribution stribution of of the root the root ddiameter iameter of the of the cycloid cycloid profil profile e within the within the We considered the normal law of the probability distribution for the obtained dimen- tolerance zone, TD = 0.04 mm. tolerance zone, TD = 0.04 mm. sions and the method of incomplete interchangeability when selecting fits for parts in the We considered the normal law of the probability distribution for the obtained dimen- We considered the normal law of the probability distribution for the obtained dimen- cycloid engagement. sions and the method of incomplete interchangeability when selecting fits for parts in the sions and the method of incomplete interchangeability when selecting fits for parts in the In considering the engagement of a transmission with intermediate rolling elements cycloid engagement. cycloid engagement. In considering the engagement of a transmission with intermediate rolling elements 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— and free cage (IREFC) (Figure 3), this consists of a wheel with an external cycloid profile— 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 cam 1, cylindrical rolling elements 2 and a wheel with an internal cycloid profile—ring cam 1, cylindrical rolling elements 2 and a wheel with an internal cycloid profile—ring 3 [16,22]. 3 [16,22]. [16,22]. Figure 3. Engagement of a transmission with IREFC and tolerance zones of dimensions of link con- tacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. As previously noted, when manufacturing contacting profiles of transmission links, Figure 3. Engagement of a transmission with IREFC and tolerance zones of dimensions of link Figure 3. Engagement of a transmission with IREFC and tolerance zones of dimensions of link con- the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and contacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they tacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. are made larger than the nominal dimension. In fact, the dimensions of these three links As previously noted, when manufacturing contacting profiles of transmission links, can be manufactured in any combination but there is the possibility of manufacturing the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and As previously noted, when manufacturing contacting profiles of transmission links, parts with dimensions using extreme tolerance values (Figure 4). rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and are made larger than the nominal dimension. In fact, the dimensions of these three links rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they can be manufactured in any combination but there is the possibility of manufacturing parts with dimensions using extreme tolerance values (Figure 4). 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). Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 10 Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 10 Appl. Sci. 2022, 12, 5 5 of 10 Figure 4. Diagram of the link position in the cycloid tooth system of a transmission with IREFC Figure 4. Diagram of the link position in the cycloid tooth system of a transmission with IREFC Figure when m 4.aDiagram nufactured with the m of the link position aximum in po thessib cycloi le cd learanc toothe: 1—cam system of; 2— a transmission intermediate r with olling IREFC ele- when manufactured with the maximum possible clearance: 1—cam; 2—intermediate rolling ele- ments; 3—ring. when manufactured with the maximum possible clearance: 1—cam; 2—intermediate rolling elements; ments; 3—ring. 3—ring. Having considered the case presented in Figure 4, let us determine the maximum Having considered the case presented in Figure 4, let us determine the maximum 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 possible manufacturing clearance in the engagement of a transmission with IREFC using possible manufacturing clearance in the engagement of a transmission with IREFC using the following expression: the following expression: the following expression: v k T D Td rb (1) Δ = −𝑇𝑑 − , D = Td , (1) m ax (1) 2 2 Δ = 2 −𝑇𝑑 − 2 , 2 2 where TD is the deviation tolerance of the diametric dimension of the ring profile; where TD is the deviation tolerance of the diametric dimension of the ring profile; rb where TD is the deviation tolerance of the diametric dimension of the ring profile; Td rb is the deviation tolerance of the diametric dimension of the rolling elements; Td is the deviation tolerance of the diametric dimension of the rolling elements; rb Td is the deviation tolerance of the diametric dimension of the rolling elements; Td k is the deviation tolerance of the diametric dimension of the cam profile. Td is the deviation tolerance of the diametric dimension of the cam profile. Td is the deviation tolerance of the diametric dimension of the cam profile. Formula (1) describes the case where the contact surfaces of the ring are carried out Formula (1) describes the case where the contact surfaces of the ring are carried out Formula (1) describes the case where the contact surfaces of the ring are carried out according to the upper deviation and the contact surfaces of the rolling elements and the according to the upper deviation and the contact surfaces of the rolling elements and the according to the upper deviation and the contact surfaces of the rolling elements and the cam are carried out according to the lower one. However, the probability of such an event cam are carried out according to the lower one. However, the probability of such an event cam are carried out according to the lower one. However, the probability of such an event is extremely small and it is more likely that all parts will be manufactured according to the is extremely small and it is more likely that all parts will be manufactured according to the lim lim is extremely small and it is more likely that all parts will be manufactured according to the upper (D ) tolerance value (Figure 5a) or according to the lower (D ) value (Figure 5b). u p dn upper (Δ ) tolerance value (Figure 5a) or according to the lower (Δ ) value (Figure 5b). upper (Δ ) tolerance value (Figure 5a) or according to the lower (Δ ) value (Figure 5b). (a) (b) (a) (b) Figure 5. Cases of emergence of maximum manufacturing clearances in the engagement of a trans- Figure 5. Cases of emergence of maximum manufacturing clearances in the engagement of a transmis- Figure 5. Cases of emergence of maximum manufacturing clearances in the engagement of a trans- mission with IREFC: 1—cam; 2—intermediate rolling elements; 3—ring; (a) maximum upper clear- sion with IREFC: 1—cam; 2—intermediate rolling elements; 3—ring; (a) maximum upper clearance; mission with IREFC: 1—cam; 2—intermediate rolling elements; 3—ring; (a) maximum upper clear- ance; (b) maximum lower clearance. (b) maximum lower clearance. ance; (b) maximum lower clearance. In this case, the maximum manufacturing clearance in the engagement, using the In this case, the maximum manufacturing clearance in the engagement, using the In this case, the maximum manufacturing clearance in the engagement, using the agreed notations of deviations [23], can be determined as follows: agreed notations of deviations [23], can be determined as follows: agreed notations of deviations [23], can be determined as follows: To manufacture all contacting parts according to the upper deviation To manufacture all contacting parts according to the upper deviation To manufacture all contacting parts according to the upper deviation v k (2) Δ = −𝑒𝑠 − ; ES es lim rb (2) D Δ= = −𝑒 es𝑠 − ; ; (2) u p 2 2 𝑇𝑑 𝑇𝐷 𝑇𝑑 𝑇𝐷 Appl. Sci. 2022, 12, 5 6 of 10 To manufacture all contacting parts according to the lower deviation v k E I ei lim rb D = ei . (3) dn 2 2 The tolerance value is substituted into Formula (1) with a sign corresponding to the area of its location relative to the zero line. For example, if a hole with a tolerance by H is considered, then the tolerance in Formula (1) is substituted with the sign “+” and for the shaft by the same tolerance grade, with the sign “”. Formulas (1)–(3) can be used to determine the clearance in the engagement of any cycloidal gearing with IRE. These equations take into account the actual shape of the contacting surfaces of the transmission links, i.e., the shape that is obtained after manufacturing these surfaces using the manufacturing equipment. In Formulas (2) and (3), deviations are substituted with their own signs. Expressions (1)–(3) are applicable for any fits; the exceptions are fits J and Js because their deviations are located on both sides of the zero line. When using these fits, the upper deviation for the hole and the lower deviation for the shaft should be substituted in Formula (1). 3. Results At the manufacturing site, the contacting profile surfaces of the ring and cam are cur- rently manufactured using the 7th tolerance grade with a fit of H (h), and the intermediate rolling elements with h6. For the engagement with a rolling element diameter of 12 mm and the root diameters of the ring of 175 mm and the cam diameter of 151 m by the specified fits and tolerance grades, the deviations will be as follows: Ø175H7 TD = 40 m; Ø151h7 TD = 40 m; Ø12h6 TD = 11 m. Having performed the calculation according to Formula (1), let us determine the maximum manufacturing clearance for such engagement as being D = 51 m. According max to Formulas (2) and (3), the clearance limits for the engagement under consideration will lim lim be equal to D = 20 m and D = 31 m. In this way, the transmissions with IRE that u p dn are currently manufactured at enterprises have a clearance in the engagement changing from 0.02 to 0.05 mm. Therefore, investigating combinations of fits with coarser tolerance grades lets us determine such combinations in which the manufacturing clearances will be in a mentioned range or less. It should be noted that during manufacturing, the dimension distribution within the tolerance zone is uneven. Let us study the combination of fits on the contacting parts of transmissions with intermediate rolling elements and free cage (IREFC) to determine the minimum clearance limits in the selected accuracy range. When considering this further, for notational con- venience of the engagement with a combination of fits and tolerance grades, let us make the fit and the tolerance grade of the ring a priority. Then, those of the rolling elements follow and finally, those of the cam. For example, for the case under consideration, let us designate the engagement as H7-h6-h7. The analysis of Formulas (1)–(3) shows that for a combination of H-h-h fits, when increasing the tolerance grade (substituting for a coarser one), the manufacturing clearances in the engagement will only increase. Therefore, it is worth considering other combinations of fits. It should also be noted that changing the accuracy of the diameter of the rolling elements is not always justified. Currently, the manufacture of cylindrical or spherical rolling elements with an accuracy of h6 has been launched into mass production and such parts can be purchased for less than the expenditures for manufacturing a small lot at one’s own manufacturing site, even with less accuracy. Further, in this way, let us use the h6 fit for the rolling elements and consider the combinations of fits with coarser tolerances, only for profile cycloid disks. Let us take into consideration several combinations of fits: H-h-k; K-h-h; Js-h-h; H-h-js; K-h-js; Js-h-k. The deviation values will be used for the following diameters: outer profile Appl. Sci. 2022, 12, 5 7 of 10 disk—Ø127.8 mm; rolling elements—Ø18 mm; internal profile disk—Ø82.5 mm. For the considered combinations of fits, the change in the tolerance grade, varying from the 7th to the 9th for cycloid disk fits, has been studied (Table 1). Table 1. Values of manufacturing clearances when changing the tolerance grade and combination of fits. Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 10 Maximum Combination of Fits and Upper Clearance Lower Clearance Manufacturing lim lim Tolerance Grades. Limit D , mm Limit D , mm up dn Clearance D , mm max Table 1. Values of manufacturing clearances when changing the tolerance grade and combination of fits. H7-h6-k7 0.013 0.001 0.008 Combinat K7-h6-h7 ion of Fits Maximum M 0.008 anufactur- Upper Clearance 0.001 Lower Cl 0.007 earance Limit 𝚫 , mm and Tolerance Grades. ing Clearance ∆max, mm Limit 𝚫 , mm H8-h6-k8 0.015 0.003 𝒖 𝒑 0.008 H7-h6-k7 0.013 0.001 0.008 K8-h6-h8 0.006 0.001 0.005 K7-h6-h7 0.008 −0.001 0.007 H9-h6-k9 0.017 0.005 0.008 H8-h6-k8 0.015 0.003 0.008 K9-h6-h9 0.004 0.001 0.003 K8-h6-h8 0.006 −0.001 0.005 Js7-h6-h7 0.038 0.01 0.018 H9-h6-k9 0.017 0.005 0.008 H7-h6-js7 0.041 0.01 0.031 K9-h6-h9 0.004 −0.001 0.003 Js7-h Js8-h6-h86-h7 0.03 0.0538 0.0.01501 0.01 0.0228 H7-h6-js7 0.041 0.01 0.031 H8-h6-js8 0.058 0.016 0.042 Js8-h6-h8 0.053 0.015 0.022 Js9-h6-h9 0.079 0.025 0.029 H8-h6-js8 0.058 0.016 0.042 H9-h6-js9 0.086 0.025 0.061 Js9-h6-h9 0.079 0.025 0.029 K7-h6-js7 0.001 0.011 0.001 H9-h6-js9 0.086 0.025 0.061 Js7-h6-k7 0.003 0.009 0.001 K7-h6-js7 0.001 −0.011 −0.001 K8-h6-js8 0.005 0.017 0.006 Js7-h6-k7 0.003 0.009 −0.001 K8-h Js8-h6-k8 6-js8 −0.005 0.001 −0.017 0.013 −0.006 0.006 Js8-h6-k8 −0.001 0.013 −0.006 K9-h6-js9 0.014 0.026 0.015 K9-h6-js9 −0.014 −0.026 −0.015 Js9-h6-k9 0.007 0.020 0.015 Js9-h6-k9 −0.007 0.020 −0.015 The results, obtained using expressions (1)–(3), are conveniently presented in graphic The results, obtained using expressions (1)–(3), are conveniently presented in graphic form (Figures 6–8). form (Figures 6–8). Figure 6. Change of the clearances in the engagement when manufacturing the external profile disk Figure 6. Change of the clearances in the engagement when manufacturing the external profile disk by the H fit. by the H fit. 𝒅𝒏 𝒍𝒊𝒎 𝒍𝒊𝒎 Appl. Sci. 2022, 12, 5 8 of 10 Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 10 Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 10 Figure 7. Change of the clearances in the engagement when manufacturing the external profile disk Figure 7. Change of the clearances in the engagement when manufacturing the external profile disk Figure 7. Change of the clearances in the engagement when manufacturing the external profile disk by the K fit. by the K fit. by the K fit. Figure 8. Figure 8. Change of cl Change of clearanc earances es i in n the engagement when the engagement when manufacturing the external profi manufacturing the external profi le d le d isk by isk by Figure 8. Change of clearances in the engagement when manufacturing the external profile disk by the the Js Js fit fit.. the Js fit. 4. Discussion 4. Discussion 4. Discussion The diagrams (Figure 6) demonstrate that when manufacturing the ring using the H9 The diagrams (Figure 6) demonstrate that when manufacturing the ring using the H9 The diagrams (Figure 6) demonstrate that when manufacturing the ring using the H9 fit and the cam using k9, the manufacturing clearance for transmissions with intermediate fit and the cam using k9, the manufacturing clearance for transmissions with intermediate fit and the cam using k9, the manufacturing clearance for transmissions with intermediate rolling elements and free cage (IREFC) varies from 0.005 mm to 0.017 mm. This is less than rol rolling ling element elements s aand nd frfr ee ca ee cage ge (IR (IREFC) EFC) vavaries ries from from 0.00 0.005 5 mm mtm o 0.017 to 0.017 mm. This i mm. This s less is tless han than the clearance that emerges in the engagement when manufacturing profile disks, accord- the clear the clearance ance that emer that emer ges gesin the en in the engagement gagement when when manufac manufacturing turing profile d profileidisks, sks, acco accor rd-ding ing to the 7th tolerance grade, as is carried out at enterprises. Such fits can be assigned to ing to the 7th tolerance grade, as is carried out at enterprises. Such fits can be assigned to to the 7th tolerance grade, as is carried out at enterprises. Such fits can be assigned to the cycloidal profiles of the disks if other options are not considered. Manufacturing the the cycloidal profiles of the disks if other options are not considered. Manufacturing the the cycloidal profiles of the disks if other options are not considered. Manufacturing cam profile with the js9 fit results in the obtainment of a manufacturing clearance ranging cam profile with the js9 fit results in the obtainment of a manufacturing clearance ranging the cam profile with the js9 fit results in the obtainment of a manufacturing clearance from 0.025 mm to 0.086 mm. This exceeds the parameters currently provided at the man- from 0.025 mm to 0.086 mm. This exceeds the parameters currently provided at the man- ranging from 0.025 mm to 0.086 mm. This exceeds the parameters currently provided at ufacturing site and is not acceptable. ufacturing site and is not acceptable. the manufacturing site and is not acceptable. In this case, it is important to bear in mind that the obtained clearance ranges repre- In this case, it is important to bear in mind that the obtained clearance ranges repre- In this case, it is important to bear in mind that the obtained clearance ranges represent sent the field of a possible location of real profiles of cycloidal wheels and the rolling sent the field of a possible location of real profiles of cycloidal wheels and the rolling the field of a possible location of real profiles of cycloidal wheels and the rolling elements. At the same time, the cycloidal surfaces of the disks and the cylindrical surface of the rolling elements are not equidistant from their theoretical profile. Appl. Sci. 2022, 12, 5 9 of 10 When manufacturing the ring with the K9 fit and the cam with h9 (Figure 7), the man- ufacturing clearance for transmissions with IREFC changes from 0.001 mm to 0.004 mm. This is also less than the clearance currently obtained at enterprises when manufacturing profile disks according to the 7th tolerance grade. However, the minimum value of the clearance has turned out to have a “” sign, that is a negative allowance may form in- stead of a clearance in the engagement. However, for the combination of fits under study, this is admissible since the negative allowance is 1 m and can be eliminated during the mechanism roll-on due to roughness crumpling. If the cam is manufactured with the js9 fit, then instead of the clearance, it is possible to obtain only a negative allowance that can be sufficiently large, varying from 0.014 mm to 0.026 mm, which is not acceptable. Figure 8 shows that the manufacture of the ring with the Js fit does meet the purpose of the research. First, in this case, a large negative allowance of up to 0.015 mm can arise. Second, a very large clearance of up to 0.079 mm may appear. In this way, to design and manufacture cycloid profile disks, it is possible to recom- mend K9 fits for the ring, h9 for the cam and h6 for the rolling elements. In case of such a combination of fits, the transmission accuracy increases since there is the possibility of the emergence of an insignificant negative allowance. Such combination of fits decreases the clearance in the engagement, even compared to the more precise manufacture of profile disks. However, if the dimension distribution law within the tolerance zone is considered when manufacturing these disks, then 90% of the parts will be produced without a negative allowance and this will provide a more precise engagement of transmissions with IREFC. The considered results are related to the accuracy of the manufacturing of the surfaces of parts that contact and create the engagement of transmissions with intermediate rolling elements (IRE). At the same time, the mutual bracing of these surfaces can be different. The cycloidal surfaces of the disks must theoretically be parallel to the cylindrical surface of the rolling elements. In fact, these surfaces may not be parallel to each other. The problem of studying the mutual bracing of the surfaces of parts in the engagement of transmissions with IRE seems to be an interesting task for further research. 5. Conclusions It has been shown that in the case of different combinations of fits of the mating parts in transmissions with intermediate rolling elements and free cage (IREFC), both a clearance and a negative allowance can occur in the engagement. At the same time, when manufacturing parts according to the 9th accuracy tolerance grade, clearances of up to 0.086 mm (H9-h6-js9) and negative allowances of up to 0.026 mm (K9-h6-js9) may occur in the engagement. It is obvious that it is not expedient to assign the Js (js) fit to the cycloidal profiles of parts. However, it is possible to select such a combination of fits according to the 9th tolerance grade (K9-h6-h9), by which the clearance in the engagement changes from 0.001 mm to 0.003 mm. Furthermore, taking into account the machine settings, this clearance will vary from 0 mm to 0.003 mm. This fact will allow the provision of high transmission accuracy, in the case of low accuracy of the manufacturing of the relevant parts. This combination of fits should be used for any transmission with intermediate rolling elements (IRE). Moreover, in general, for complex combinations of structural joints, having three or more elements in simultaneous contact, it is possible to recommend the designation of fits with K for mated holes and with h for mated shafts. Further development of the presented scientific results will allow qualitative im- provement in transmission design and significantly facilitate the process of technological preparation for the production of gearing, which will subsequently allow the achievement of high technical and economic indicators in general. Also, the study of the problem of the mutual bracing of the surfaces of parts in the engagement of transmissions with IRE is promising. Appl. Sci. 2022, 12, 5 10 of 10 Author Contributions: Conceptualization and methodology, E.A.E.; validation and formal analysis, A.D.E. and E.A.E.; investigation, V.Y.S.; resources, M.V.G.; data curation, A.V.O.; writing—original draft preparation, V.Y.S., M.V.G.; writing—review and editing, V.Y.S., N.V.M.; visualization, A.V.O.; supervision, E.A.E.; project administration, N.V.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the TPU development program. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Belyaev, A.E. Mechanical Transmissions with Ball Intermediate Elements; Tomsk Polytechnic University: Tomsk, Russia, 1992. 2. Pankratov, E.N. Design of Mechanical Systems of Automated Complexes in the Machining Production: Workshop of the Leader-Designer; Tomsk State University: Tomsk, Russia, 1998. 3. Lustenkov, M.E. Criteria for the strength of mechanical transmissions with constituent intermediate rolling elements. Bull. Belarusian-Russ. Univ. 2015, 49, 33–41. 4. Chen, B.; Fang, T.; Li, C.; Wang, S. Gear geometry of cycloid drives. Sci. China Ser. E Technol. Sci. 2008, 51, 598–610. [CrossRef] 5. Mihailidis, A.; Athanasopoulos, E.; Agouridas, K. EHL film thickness and load dependent power loss of cycloid reducers. Mech. Eng. Sci. 2016, 230, 1303–1317. [CrossRef] 6. Pokatilov, D.A.; Efremenkov, E.A. Analysis of the technological process of manufacturing the cycloid profile of transmission parts with intermediate rolling elements. Proc. Samara Sci. Cent. RAS 2015, 17, 868–873. 7. Zhang, T.; Li, X.; Wang, Y.; Sun, L. A Semi-Analytical Load Distribution Model for Cycloid Drives with Tooth Profile and Longitudinal Modifications. Appl. Sci. 2020, 10, 4859. [CrossRef] 8. Li, T.; An, X.; Deng, X.; Li, J.; Li, Y. A New Tooth Profile Modification Method of Cycloid Gears in Precision Reducers for Robots. Appl. Sci. 2020, 10, 1266. [CrossRef] 9. Wang, H.; Shi, Z.-Y.; Yu, B.; Xu, H. Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load. Appl. Sci. 2019, 9, 4099. [CrossRef] 10. Hu, Y.; Li, G.; Zhu, W.; Cui, J. An Elastic Transmission Error Compensation Method for Rotary Vector Speed Reducers Based on Error Sensitivity Analysis. Appl. Sci. 2020, 10, 481. [CrossRef] 11. Kociško, ˇ M.; Pollák, M.; Töröková, M.; Baron, P.; Paulišin, D.; Kundrík, J. Determination of Methodology and Research of the Influence of the Trial Run of High-Precision Reducers on the Change of Their Characterizing Properties. Appl. Sci. 2021, 11, 3859. [CrossRef] 12. Lustenkov, M.E. Transmissions with Intermediate Rolling Elements: Definition and Minimization of Power Losses; Belarusian-Russian University: Mogilev, Belarus, 2010. 13. Kan, A. Distribution of efforts between the links of the planetary gearing of the k-h-v type. Bull. Mech. Eng. 2016, 5, 60–63. 14. Kan, A.; Il’In, A.S.; Lazurkevich, A.V. Load analysis of the planetary gear train with intermediate rollers. Part 2. IOP Conf. Ser. Mater. Sci. Eng. 2016, 124, 012004. [CrossRef] 15. Lustenkov, M.E. Strength calculations for cylindrical transmissions with compound intermediate rolling elements. Int. J. Mech. Robot. Syst. 2015, 2, 111–121. [CrossRef] 16. Efremenkov, E.A.; Kobza, E.E.; Efremenkova, S.K. Force Analysis of Double Pitch Point Cycloid Drive with Intermediate Rolling Elements and Free Retainer. Appl. Mech. Mater. Sci. J. 2015, 756, 29–34. [CrossRef] 17. Li, T.; Tian, M.; Xu, H.; Deng, X.; An, X.; Su, J. Meshing contact analysis of cycloidal-pin gear in RV reducer considering the influence of manufacturing error. J. Braz. Soc. Mech. Sci. Eng. 2020, 42, 1–14. [CrossRef] 18. Lustenkov, M.E. Evaluation of the resource and the load capacity of transmissions with constituent intermediate elements. Top. Quest. Mach. Sci. Collect. Rescuers Paper 2014, 3, 189–191. 19. Chen, B.; Zhong, H.; Liu, J.; Li, C.; Fang, T. Generation and investigation of a new cycloid drive with double contact. Mech. Mach. Theory 2012, 49, 270–283. [CrossRef] 20. Bao, J.; He, W. Parametric Design and Efficiency Analysis of the Output-Pin Wheel Cycloid Transmission. International. J. Control Autom. 2015, 8, 349–362. [CrossRef] 21. Meng, Y.; Wu, C.; Ling, L. Mathematical modeling of the transmission performance of 2K–H pin cycloid planetary mechanism. Mech. Mach. Theory 2007, 42, 776–790. [CrossRef] 22. Efremenkov, E.A.; I-Kan, A. Euler-Savari Determination of Radii of Curvature of Cycloid Profiles. Russ. Eng. Res. 2010, 30, 1001–1004. [CrossRef] 23. Myagkov, V.D. Tolerances and Fits; Reference book. In 2 parts. 6th rev.; Mechanical Engineering: Leningrad, Russia, 1982. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Research on the Possibility of Lowering the Manufacturing Accuracy of Cycloid Transmission Wheels with Intermediate Rolling Elements and a Free Cage

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applied sciences Article Research on the Possibility of Lowering the Manufacturing Accuracy of Cycloid Transmission Wheels with Intermediate Rolling Elements and a Free Cage 1 , 2 , 3 4 5 Egor A. Efremenkov *, Nikita V. Martyushev , Vadim Yu Skeeba , Maria V. Grechneva , 6 7 Andrey V. Olisov and Anatoly D. Ens Mechanical Engineering Depurtment, Tomsk Polytechnic University, 30, Lenin Ave., 634050 Tomsk, Russia Innovation Management Department, Tomsk State University of Control Systems and Radioelectronics, 40, Lenin Ave., 634050 Tomsk, Russia Materials Science Depurtment, Tomsk Polytechnic University, 30, Lenin Ave., 634050 Tomsk, Russia; martjushev@tpu.ru Department of Industrial Machinery Design, Novosibirsk State Technical University, 20, K. Marks Ave., 630073 Novosibirsk, Russia; skeeba_vadim@mail.ru Department of Materials Science, Welding and Additive Technologies, Irkutsk National Research Technical University, 83, Lermontov Str., 664074 Irkutsk, Russia; mgrech@irk.ru Scientific and Educational Center “Additive Technologies”, National Research Tomsk State University, 36, Lenin Ave., 634050 Tomsk, Russia; a.v.olisov@mail.tsu.ru Faculty of Electrical Engineering and Computer Science, Technical University of Ostrava, 17, listopadu 2172/15, 70800 Ostrava, Czech Republic; ens.t@enspg.ru * Correspondence: egorefr@tpu.ru; Tel.: +7-(3822)-606392 Abstract: Purpose: In the present work, different combinations of fits and accuracies, in relation to the profiles of mating parts, have been analysed in order to assess the degree of the engagement of Citation: Efremenkov, E.A.; transmissions that contain intermediate rolling elements. The aim of this work is to determine which Martyushev, N.V.; Skeeba, V.Y.; fits have decreased accuracy, but nevertheless provide a minimum manufacturing clearance for the Grechneva, M.V.; Olisov, A.V.; Ens, transmission engagement in order to reduce the cost of parts production. Methods and materials: A.D. Research on the Possibility of Considering the normal probabilistic distribution law in relation to the obtained dimensions of the Lowering the Manufacturing manufacturing equipment, a combination of fits were selected using the incomplete interchangeability Accuracy of Cycloid Transmission method, taking into account the peculiarities of the cycloid engagement in transmissions with inter- Wheels with Intermediate Rolling mediate rolling elements (IRE). Results: Having studied various combinations of fits of parts that are Elements and a Free Cage. Appl. Sci. engaged in transmissions with intermediate rolling elements and a free cage (IREFC), a combination 2022, 12, 5. https://doi.org/ of fits for a “ring, rolling-element cam” were determined, in which a technological clearance of 3 m 10.3390/app12010005 is formed in the engagement. At the same time, cycloid disk profiles are manufactured according Academic Editor: Alberto Boschetto to the 9th tolerance grade, which reduces the laboriousness and cost of the production. Discussion. Received: 19 November 2021 When reducing the manufacturing accuracy of cycloid disks, it is possible to obtain both very ample Accepted: 16 December 2021 clearances and significant negative allowances. For example, having manufactured a ring with the Published: 21 December 2021 H9 fit, rolling elements with h6 and a cam with js9, the maximum manufacturing clearance can reach 0.086 mm, while the clearance limits vary from 0.025 mm to 0.061 mm. Additionally, if mating parts Publisher’s Note: MDPI stays neutral are manufactured using a combination of K9-h6-js9 fits, a negative allowance varying from 0.014 mm with regard to jurisdictional claims in published maps and institutional affil- to 0.026 mm will emerge in the engagement. Both described cases are unacceptable because both iations. ample clearances and large negative allowances will negatively influence the working capacity of the mechanism. However, it is possible to select a combination of fits using the 9th tolerance grade of the basic parts, by which the parts will contact in the range from a small negative allowance of 1 m to a clearance of 3–4 m. Furthermore, if this is considered, taking into account the machine settings, it is Copyright: © 2021 by the authors. possible to obtain parts according to the 9th accuracy tolerance grade and, at the same time, provide Licensee MDPI, Basel, Switzerland. a clearance in the engagement that is almost equal to zero. Moreover, such a combination of fits is This article is an open access article relevant for any transmission with IRE. This is a positive result because it reduces the laboriousness distributed under the terms and when manufacturing parts and, at the same time, provides high accuracy of the mechanism. Conclu- conditions of the Creative Commons sions: It has been established that when lowering the accuracy of manufacturing transmission parts Attribution (CC BY) license (https:// with IRE, both clearances and negative allowances may occur in the engagement, depending on the creativecommons.org/licenses/by/ 4.0/). combination of fits. At the same time, it is possible to select such a combination of fits, by which the Appl. Sci. 2022, 12, 5. https://doi.org/10.3390/app12010005 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 5 2 of 10 parts manufactured according to the 9th tolerance grade, will provide almost zero clearance of the engagement of the transmission. In this way, it is possible to reduce the cost of manufacturing the parts for gears with intermediate rolling elements and, at the same time, maintain a high accuracy of the transmission mechanism. Keywords: manufacturing accuracy; cycloidal transmission; kinematic parameters; tolerances; deviations; technological gap 1. Introduction When manufacturing mechanical transmissions, special attention is paid to the accu- racy and quality of the surfaces of the parts directly engaged in the transmission of speed and efforts. The accuracy of these parts must be high and the roughness of their contact surfaces must be low [1–6]. In practice, in general engineering, these structural elements are manufactured with the accuracy of contact surfaces according to the 7th tolerance grade, which ranges in the interval from 0.025 to 0.057 mm (depending on their size). At the same time, the roughness of the contact surfaces is brought to a level in the order of Ra 0.8–1.6 m. The shape of the cycloid profile of the tooth also influences the working capacity of the mechanism in both longitudinal [7] and transverse [8,9] directions. The cycloid profile modification allows errors to be compensated for in the geometry of the cycloid engage- ment and the power characteristics to be enhanced [7,10]. In particular, authors [7] note that, when correcting the cycloid profile, the contact voltage changes on average by several tens of megapascals. These design requirements cause certain technological challenges and increase the manufacturing time of such parts. At the same time, a decrease in the accuracy of manufacturing parts leads to a quality deterioration of the mechanism and loss of its accuracy characteristics due to emerging manufacturing clearances [11]. Therefore, the study of the possibility for lowering the manufacturing accuracy of the contacting surfaces of disks engaged in a cycloid transmission, while maintaining the accuracy of the transmission itself, is a vital task. The solution to this will allow a decrease in the manufacturing time of such parts and a reduction in the cost of the entire mechanism. The accuracy of the transmission mechanism, among other things, is influenced by such a characteristic as the clearance in the engagement (angular play) [8]. In [8], a method for modifying the cycloid tooth profile is considered, taking into account the clearances in the engagement, but the influence of the manufacturing accuracy of the cycloid profile has not been analysed. The clearance in the engagement can be designed in, when designing the transmission and it also definitely emerges when manufacturing the transmission links, being directly in the engagement. The structurally laid clearance is calculated and can be adjusted when assembling the transmission. Additionally, the manufacturing clearance, that is emerging due to dimension deviations from the absolute value when manufacturing parts, is difficult to take into account owing to the probabilistic obtainment of dimensions during manufacture. At the same time, there are transmissions, for which the clearance adjustment in the engagement is not provided because the design does not allow it. Such transmis- sions are cycloid pinwheel [8,9] transmissions with intermediate rolling elements (IRE), in general [12–15], and transmissions with intermediate rolling elements and a free cage (IREFC) [16], in particular (Figure 1). In IREFC transmissions, the loaded parts are cam 1, rolling elements 2 and ring 3 (Figure 1). The accuracy of the manufacturing of these parts influences the engagement quality. There is also a separator 4 between the cycloidal profiles of cam 1 and wheel 3. However, the manufacturing accuracy will not affect the accuracy of the engagement of transmissions with IRE. This is explained by the fact that a separator 4 is only used for the distribution of the rolling elements 2 on a mutual angular placement and it keeps them from moving into the root of the cycloidal profile (for example, the cam) when they are not under load. The standard technology for manufacturing the Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 10 the engagement of transmissions with IRE. This is explained by the fact that a separator 4 Appl. Sci. 2022, 12, 5 3 of 10 is only used for the distribution of the rolling elements 2 on a mutual angular placement and it keeps them from moving into the root of the cycloidal profile (for example, the cam) when they are not under load. The standard technology for manufacturing the separator separator allows the ob allows tainm the ent obtainment of such a th of ickne such ss o a thickness f the part tha of the t it wi part ll n that ot wors it will en the not engag worsen e- the ment in engagement any case. Cyclo in any icase. d pinwheel Cycloid transmissi pinwheel ons transmissions are designed so are that the c designedle so ara that nce be- the tween the cycloid profiles and the pin is not regulated. The situation with transmissions clearance between the cycloid profiles and the pin is not regulated. The situation with transmissions having IRE is even more complicated having IRE is even mor , w e complicated, here the rolling elemen where thet is rolling able to move free element is able ly [2– to move 4]. Moreov freelyer [2 , in som –4]. Mor e t eover ransm , in iss some ions transmissions this contacts sthis imucontacts ltaneoussimultaneously ly with two cycwith loid p two ro- cycloid profiles [12,13]. Transmissions with a cycloid wheel are widely used in the industry files [12,13]. Transmissions with a cycloid wheel are widely used in the industry in Russia in Russia (“SIMACO” Company, Tomsk, Russia), Europe [17,18] and abroad [19–21]. The (“SIMACO” Company, Tomsk, Russia), Europe [17,18] and abroad [19–21]. The problems problems with improving the manufacturability of the cycloid disk profile are acute for with improving the manufacturability of the cycloid disk profile are acute for manufac- manufacturers. Therefore, the purpose of this work is to determine the influence of the turers. Therefore, the purpose of this work is to determine the influence of the manufac- manufacturing accuracy of the contact surfaces of profile disks on the clearance size in the turing accuracy of the contact surfaces of profile disks on the clearance size in the engage- engagement for transmissions with IREFC. ment for transmissions with IREFC. Figure Figure 1 1.. C Cr ros oss-section s-section of of a a t transmission ransmission wi with th interm intermediate ediate rol rolling ling e elements lements and and a a free free cag cage: e: H— H— generator; 1—cam; 2—intermediate rolling elements; 3—ring; 4—cage; 5—bearing. generator; 1—cam; 2—intermediate rolling elements; 3—ring; 4—cage; 5—bearing. 2. Materials and Methods 2. Materials and Methods After measuring the root diameters of the cycloid profiles of wheels manufactured 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 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 the obtainment of cycloid profiles with a normal distribution of dimensions that were ap- approximately within the tolerance zone, as shown in Figure 2. The distribution of the proximately within the tolerance zone, as shown in Figure 2. The distribution of the results results of measuring the diameter of the disk roots (Figure 2) does not exactly conform to the 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 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 deviation of the presented distribution from the normal law. The cycloid profile with a diameter of Ø175 mm was measured (the accuracy was according to the 7th tolerance grade). root diameter of Ø175 mm was measured (the accuracy was according to the 7th tolerance The diameter was measured using the contact method with a clock-type indicator. The data grade). The diameter was measured using the contact method with a clock-type indicator. on five pivot points were obtained in the root of the profile. Among these points, the point The data on five pivot points were obtained in the root of the profile. Among these points, with the largest modulus coordinate was found. The measurements were also performed in the point with the largest modulus coordinate was found. The measurements were also the diametrically opposite root. The obtained diameter of the roots of the cycloidal profile performed in the diametrically opposite root. The obtained diameter of the roots of the was determined by the difference in the obtained coordinates. The diagram shows that cycloidal profile was determined by the difference in the obtained coordinates. The dia- the equipment processes the maximum number of parts according to the dimensions, the gram shows that the equipment processes the maximum number of parts according to the accuracy of which shifts to the upper limit of the tolerance zone from its centre. In this way, dimensions, the accuracy of which shifts to the upper limit of the tolerance zone from its it is possible to determine the limits within the tolerance zone, to which more than 80% centre. In this way, it is possible to determine the limits within the tolerance zone, to which of the dimensions fall. When determining the permissible manufacturing clearance, one more than 80% of the dimensions fall. When determining the permissible manufacturing should focus on this range. Based on the measurement results (Figure 2), the following clearance, one should focus on this range. Based on the measurement results (Figure 2), limits can be accepted within the tolerance zone: from 0.262 mm to 0.288 mm. Overall, 65% the following limits can be accepted within the tolerance zone: from 0.262 mm to 0.288 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. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 10 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. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 10 mm. Overall, 65% of the entire tolerance zone, that is 80% of the dimensions, obtained on Appl. Sci. 2022, 12, 5 4 of 10 the equipment are distributed approximately within two thirds of the tolerance zone per dimension. Figure 2. Diagram of deviation distribution of the root diameter of the cycloid profile within the tolerance zone, TD = 0.04 mm. Figure 2. Figure 2.Diag Diagram ram of de of dev viati iation on di distribution stribution of of the root the root ddiameter iameter of the of the cycloid cycloid profil profile e within the within the We considered the normal law of the probability distribution for the obtained dimen- tolerance zone, TD = 0.04 mm. tolerance zone, TD = 0.04 mm. sions and the method of incomplete interchangeability when selecting fits for parts in the We considered the normal law of the probability distribution for the obtained dimen- We considered the normal law of the probability distribution for the obtained dimen- cycloid engagement. sions and the method of incomplete interchangeability when selecting fits for parts in the sions and the method of incomplete interchangeability when selecting fits for parts in the In considering the engagement of a transmission with intermediate rolling elements cycloid engagement. cycloid engagement. In considering the engagement of a transmission with intermediate rolling elements 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— and free cage (IREFC) (Figure 3), this consists of a wheel with an external cycloid profile— 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 cam 1, cylindrical rolling elements 2 and a wheel with an internal cycloid profile—ring cam 1, cylindrical rolling elements 2 and a wheel with an internal cycloid profile—ring 3 [16,22]. 3 [16,22]. [16,22]. Figure 3. Engagement of a transmission with IREFC and tolerance zones of dimensions of link con- tacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. As previously noted, when manufacturing contacting profiles of transmission links, Figure 3. Engagement of a transmission with IREFC and tolerance zones of dimensions of link Figure 3. Engagement of a transmission with IREFC and tolerance zones of dimensions of link con- the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and contacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they tacting surfaces: 1—cam; 2—intermediate rolling elements; 3—ring. are made larger than the nominal dimension. In fact, the dimensions of these three links As previously noted, when manufacturing contacting profiles of transmission links, can be manufactured in any combination but there is the possibility of manufacturing the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and As previously noted, when manufacturing contacting profiles of transmission links, parts with dimensions using extreme tolerance values (Figure 4). rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they the resulting dimensions are in the range of the tolerance zones (Figure 3). For cam 1 and are made larger than the nominal dimension. In fact, the dimensions of these three links rolling elements 2, they are made smaller than the nominal dimension and for ring 3, they can be manufactured in any combination but there is the possibility of manufacturing parts with dimensions using extreme tolerance values (Figure 4). 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). Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 10 Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 10 Appl. Sci. 2022, 12, 5 5 of 10 Figure 4. Diagram of the link position in the cycloid tooth system of a transmission with IREFC Figure 4. Diagram of the link position in the cycloid tooth system of a transmission with IREFC Figure when m 4.aDiagram nufactured with the m of the link position aximum in po thessib cycloi le cd learanc toothe: 1—cam system of; 2— a transmission intermediate r with olling IREFC ele- when manufactured with the maximum possible clearance: 1—cam; 2—intermediate rolling ele- ments; 3—ring. when manufactured with the maximum possible clearance: 1—cam; 2—intermediate rolling elements; ments; 3—ring. 3—ring. Having considered the case presented in Figure 4, let us determine the maximum Having considered the case presented in Figure 4, let us determine the maximum 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 possible manufacturing clearance in the engagement of a transmission with IREFC using possible manufacturing clearance in the engagement of a transmission with IREFC using the following expression: the following expression: the following expression: v k T D Td rb (1) Δ = −𝑇𝑑 − , D = Td , (1) m ax (1) 2 2 Δ = 2 −𝑇𝑑 − 2 , 2 2 where TD is the deviation tolerance of the diametric dimension of the ring profile; where TD is the deviation tolerance of the diametric dimension of the ring profile; rb where TD is the deviation tolerance of the diametric dimension of the ring profile; Td rb is the deviation tolerance of the diametric dimension of the rolling elements; Td is the deviation tolerance of the diametric dimension of the rolling elements; rb Td is the deviation tolerance of the diametric dimension of the rolling elements; Td k is the deviation tolerance of the diametric dimension of the cam profile. Td is the deviation tolerance of the diametric dimension of the cam profile. Td is the deviation tolerance of the diametric dimension of the cam profile. Formula (1) describes the case where the contact surfaces of the ring are carried out Formula (1) describes the case where the contact surfaces of the ring are carried out Formula (1) describes the case where the contact surfaces of the ring are carried out according to the upper deviation and the contact surfaces of the rolling elements and the according to the upper deviation and the contact surfaces of the rolling elements and the according to the upper deviation and the contact surfaces of the rolling elements and the cam are carried out according to the lower one. However, the probability of such an event cam are carried out according to the lower one. However, the probability of such an event cam are carried out according to the lower one. However, the probability of such an event is extremely small and it is more likely that all parts will be manufactured according to the is extremely small and it is more likely that all parts will be manufactured according to the lim lim is extremely small and it is more likely that all parts will be manufactured according to the upper (D ) tolerance value (Figure 5a) or according to the lower (D ) value (Figure 5b). u p dn upper (Δ ) tolerance value (Figure 5a) or according to the lower (Δ ) value (Figure 5b). upper (Δ ) tolerance value (Figure 5a) or according to the lower (Δ ) value (Figure 5b). (a) (b) (a) (b) Figure 5. Cases of emergence of maximum manufacturing clearances in the engagement of a trans- Figure 5. Cases of emergence of maximum manufacturing clearances in the engagement of a transmis- Figure 5. Cases of emergence of maximum manufacturing clearances in the engagement of a trans- mission with IREFC: 1—cam; 2—intermediate rolling elements; 3—ring; (a) maximum upper clear- sion with IREFC: 1—cam; 2—intermediate rolling elements; 3—ring; (a) maximum upper clearance; mission with IREFC: 1—cam; 2—intermediate rolling elements; 3—ring; (a) maximum upper clear- ance; (b) maximum lower clearance. (b) maximum lower clearance. ance; (b) maximum lower clearance. In this case, the maximum manufacturing clearance in the engagement, using the In this case, the maximum manufacturing clearance in the engagement, using the In this case, the maximum manufacturing clearance in the engagement, using the agreed notations of deviations [23], can be determined as follows: agreed notations of deviations [23], can be determined as follows: agreed notations of deviations [23], can be determined as follows: To manufacture all contacting parts according to the upper deviation To manufacture all contacting parts according to the upper deviation To manufacture all contacting parts according to the upper deviation v k (2) Δ = −𝑒𝑠 − ; ES es lim rb (2) D Δ= = −𝑒 es𝑠 − ; ; (2) u p 2 2 𝑇𝑑 𝑇𝐷 𝑇𝑑 𝑇𝐷 Appl. Sci. 2022, 12, 5 6 of 10 To manufacture all contacting parts according to the lower deviation v k E I ei lim rb D = ei . (3) dn 2 2 The tolerance value is substituted into Formula (1) with a sign corresponding to the area of its location relative to the zero line. For example, if a hole with a tolerance by H is considered, then the tolerance in Formula (1) is substituted with the sign “+” and for the shaft by the same tolerance grade, with the sign “”. Formulas (1)–(3) can be used to determine the clearance in the engagement of any cycloidal gearing with IRE. These equations take into account the actual shape of the contacting surfaces of the transmission links, i.e., the shape that is obtained after manufacturing these surfaces using the manufacturing equipment. In Formulas (2) and (3), deviations are substituted with their own signs. Expressions (1)–(3) are applicable for any fits; the exceptions are fits J and Js because their deviations are located on both sides of the zero line. When using these fits, the upper deviation for the hole and the lower deviation for the shaft should be substituted in Formula (1). 3. Results At the manufacturing site, the contacting profile surfaces of the ring and cam are cur- rently manufactured using the 7th tolerance grade with a fit of H (h), and the intermediate rolling elements with h6. For the engagement with a rolling element diameter of 12 mm and the root diameters of the ring of 175 mm and the cam diameter of 151 m by the specified fits and tolerance grades, the deviations will be as follows: Ø175H7 TD = 40 m; Ø151h7 TD = 40 m; Ø12h6 TD = 11 m. Having performed the calculation according to Formula (1), let us determine the maximum manufacturing clearance for such engagement as being D = 51 m. According max to Formulas (2) and (3), the clearance limits for the engagement under consideration will lim lim be equal to D = 20 m and D = 31 m. In this way, the transmissions with IRE that u p dn are currently manufactured at enterprises have a clearance in the engagement changing from 0.02 to 0.05 mm. Therefore, investigating combinations of fits with coarser tolerance grades lets us determine such combinations in which the manufacturing clearances will be in a mentioned range or less. It should be noted that during manufacturing, the dimension distribution within the tolerance zone is uneven. Let us study the combination of fits on the contacting parts of transmissions with intermediate rolling elements and free cage (IREFC) to determine the minimum clearance limits in the selected accuracy range. When considering this further, for notational con- venience of the engagement with a combination of fits and tolerance grades, let us make the fit and the tolerance grade of the ring a priority. Then, those of the rolling elements follow and finally, those of the cam. For example, for the case under consideration, let us designate the engagement as H7-h6-h7. The analysis of Formulas (1)–(3) shows that for a combination of H-h-h fits, when increasing the tolerance grade (substituting for a coarser one), the manufacturing clearances in the engagement will only increase. Therefore, it is worth considering other combinations of fits. It should also be noted that changing the accuracy of the diameter of the rolling elements is not always justified. Currently, the manufacture of cylindrical or spherical rolling elements with an accuracy of h6 has been launched into mass production and such parts can be purchased for less than the expenditures for manufacturing a small lot at one’s own manufacturing site, even with less accuracy. Further, in this way, let us use the h6 fit for the rolling elements and consider the combinations of fits with coarser tolerances, only for profile cycloid disks. Let us take into consideration several combinations of fits: H-h-k; K-h-h; Js-h-h; H-h-js; K-h-js; Js-h-k. The deviation values will be used for the following diameters: outer profile Appl. Sci. 2022, 12, 5 7 of 10 disk—Ø127.8 mm; rolling elements—Ø18 mm; internal profile disk—Ø82.5 mm. For the considered combinations of fits, the change in the tolerance grade, varying from the 7th to the 9th for cycloid disk fits, has been studied (Table 1). Table 1. Values of manufacturing clearances when changing the tolerance grade and combination of fits. Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 10 Maximum Combination of Fits and Upper Clearance Lower Clearance Manufacturing lim lim Tolerance Grades. Limit D , mm Limit D , mm up dn Clearance D , mm max Table 1. Values of manufacturing clearances when changing the tolerance grade and combination of fits. H7-h6-k7 0.013 0.001 0.008 Combinat K7-h6-h7 ion of Fits Maximum M 0.008 anufactur- Upper Clearance 0.001 Lower Cl 0.007 earance Limit 𝚫 , mm and Tolerance Grades. ing Clearance ∆max, mm Limit 𝚫 , mm H8-h6-k8 0.015 0.003 𝒖 𝒑 0.008 H7-h6-k7 0.013 0.001 0.008 K8-h6-h8 0.006 0.001 0.005 K7-h6-h7 0.008 −0.001 0.007 H9-h6-k9 0.017 0.005 0.008 H8-h6-k8 0.015 0.003 0.008 K9-h6-h9 0.004 0.001 0.003 K8-h6-h8 0.006 −0.001 0.005 Js7-h6-h7 0.038 0.01 0.018 H9-h6-k9 0.017 0.005 0.008 H7-h6-js7 0.041 0.01 0.031 K9-h6-h9 0.004 −0.001 0.003 Js7-h Js8-h6-h86-h7 0.03 0.0538 0.0.01501 0.01 0.0228 H7-h6-js7 0.041 0.01 0.031 H8-h6-js8 0.058 0.016 0.042 Js8-h6-h8 0.053 0.015 0.022 Js9-h6-h9 0.079 0.025 0.029 H8-h6-js8 0.058 0.016 0.042 H9-h6-js9 0.086 0.025 0.061 Js9-h6-h9 0.079 0.025 0.029 K7-h6-js7 0.001 0.011 0.001 H9-h6-js9 0.086 0.025 0.061 Js7-h6-k7 0.003 0.009 0.001 K7-h6-js7 0.001 −0.011 −0.001 K8-h6-js8 0.005 0.017 0.006 Js7-h6-k7 0.003 0.009 −0.001 K8-h Js8-h6-k8 6-js8 −0.005 0.001 −0.017 0.013 −0.006 0.006 Js8-h6-k8 −0.001 0.013 −0.006 K9-h6-js9 0.014 0.026 0.015 K9-h6-js9 −0.014 −0.026 −0.015 Js9-h6-k9 0.007 0.020 0.015 Js9-h6-k9 −0.007 0.020 −0.015 The results, obtained using expressions (1)–(3), are conveniently presented in graphic The results, obtained using expressions (1)–(3), are conveniently presented in graphic form (Figures 6–8). form (Figures 6–8). Figure 6. Change of the clearances in the engagement when manufacturing the external profile disk Figure 6. Change of the clearances in the engagement when manufacturing the external profile disk by the H fit. by the H fit. 𝒅𝒏 𝒍𝒊𝒎 𝒍𝒊𝒎 Appl. Sci. 2022, 12, 5 8 of 10 Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 10 Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 10 Figure 7. Change of the clearances in the engagement when manufacturing the external profile disk Figure 7. Change of the clearances in the engagement when manufacturing the external profile disk Figure 7. Change of the clearances in the engagement when manufacturing the external profile disk by the K fit. by the K fit. by the K fit. Figure 8. Figure 8. Change of cl Change of clearanc earances es i in n the engagement when the engagement when manufacturing the external profi manufacturing the external profi le d le d isk by isk by Figure 8. Change of clearances in the engagement when manufacturing the external profile disk by the the Js Js fit fit.. the Js fit. 4. Discussion 4. Discussion 4. Discussion The diagrams (Figure 6) demonstrate that when manufacturing the ring using the H9 The diagrams (Figure 6) demonstrate that when manufacturing the ring using the H9 The diagrams (Figure 6) demonstrate that when manufacturing the ring using the H9 fit and the cam using k9, the manufacturing clearance for transmissions with intermediate fit and the cam using k9, the manufacturing clearance for transmissions with intermediate fit and the cam using k9, the manufacturing clearance for transmissions with intermediate rolling elements and free cage (IREFC) varies from 0.005 mm to 0.017 mm. This is less than rol rolling ling element elements s aand nd frfr ee ca ee cage ge (IR (IREFC) EFC) vavaries ries from from 0.00 0.005 5 mm mtm o 0.017 to 0.017 mm. This i mm. This s less is tless han than the clearance that emerges in the engagement when manufacturing profile disks, accord- the clear the clearance ance that emer that emer ges gesin the en in the engagement gagement when when manufac manufacturing turing profile d profileidisks, sks, acco accor rd-ding ing to the 7th tolerance grade, as is carried out at enterprises. Such fits can be assigned to ing to the 7th tolerance grade, as is carried out at enterprises. Such fits can be assigned to to the 7th tolerance grade, as is carried out at enterprises. Such fits can be assigned to the cycloidal profiles of the disks if other options are not considered. Manufacturing the the cycloidal profiles of the disks if other options are not considered. Manufacturing the the cycloidal profiles of the disks if other options are not considered. Manufacturing cam profile with the js9 fit results in the obtainment of a manufacturing clearance ranging cam profile with the js9 fit results in the obtainment of a manufacturing clearance ranging the cam profile with the js9 fit results in the obtainment of a manufacturing clearance from 0.025 mm to 0.086 mm. This exceeds the parameters currently provided at the man- from 0.025 mm to 0.086 mm. This exceeds the parameters currently provided at the man- ranging from 0.025 mm to 0.086 mm. This exceeds the parameters currently provided at ufacturing site and is not acceptable. ufacturing site and is not acceptable. the manufacturing site and is not acceptable. In this case, it is important to bear in mind that the obtained clearance ranges repre- In this case, it is important to bear in mind that the obtained clearance ranges repre- In this case, it is important to bear in mind that the obtained clearance ranges represent sent the field of a possible location of real profiles of cycloidal wheels and the rolling sent the field of a possible location of real profiles of cycloidal wheels and the rolling the field of a possible location of real profiles of cycloidal wheels and the rolling elements. At the same time, the cycloidal surfaces of the disks and the cylindrical surface of the rolling elements are not equidistant from their theoretical profile. Appl. Sci. 2022, 12, 5 9 of 10 When manufacturing the ring with the K9 fit and the cam with h9 (Figure 7), the man- ufacturing clearance for transmissions with IREFC changes from 0.001 mm to 0.004 mm. This is also less than the clearance currently obtained at enterprises when manufacturing profile disks according to the 7th tolerance grade. However, the minimum value of the clearance has turned out to have a “” sign, that is a negative allowance may form in- stead of a clearance in the engagement. However, for the combination of fits under study, this is admissible since the negative allowance is 1 m and can be eliminated during the mechanism roll-on due to roughness crumpling. If the cam is manufactured with the js9 fit, then instead of the clearance, it is possible to obtain only a negative allowance that can be sufficiently large, varying from 0.014 mm to 0.026 mm, which is not acceptable. Figure 8 shows that the manufacture of the ring with the Js fit does meet the purpose of the research. First, in this case, a large negative allowance of up to 0.015 mm can arise. Second, a very large clearance of up to 0.079 mm may appear. In this way, to design and manufacture cycloid profile disks, it is possible to recom- mend K9 fits for the ring, h9 for the cam and h6 for the rolling elements. In case of such a combination of fits, the transmission accuracy increases since there is the possibility of the emergence of an insignificant negative allowance. Such combination of fits decreases the clearance in the engagement, even compared to the more precise manufacture of profile disks. However, if the dimension distribution law within the tolerance zone is considered when manufacturing these disks, then 90% of the parts will be produced without a negative allowance and this will provide a more precise engagement of transmissions with IREFC. The considered results are related to the accuracy of the manufacturing of the surfaces of parts that contact and create the engagement of transmissions with intermediate rolling elements (IRE). At the same time, the mutual bracing of these surfaces can be different. The cycloidal surfaces of the disks must theoretically be parallel to the cylindrical surface of the rolling elements. In fact, these surfaces may not be parallel to each other. The problem of studying the mutual bracing of the surfaces of parts in the engagement of transmissions with IRE seems to be an interesting task for further research. 5. Conclusions It has been shown that in the case of different combinations of fits of the mating parts in transmissions with intermediate rolling elements and free cage (IREFC), both a clearance and a negative allowance can occur in the engagement. At the same time, when manufacturing parts according to the 9th accuracy tolerance grade, clearances of up to 0.086 mm (H9-h6-js9) and negative allowances of up to 0.026 mm (K9-h6-js9) may occur in the engagement. It is obvious that it is not expedient to assign the Js (js) fit to the cycloidal profiles of parts. However, it is possible to select such a combination of fits according to the 9th tolerance grade (K9-h6-h9), by which the clearance in the engagement changes from 0.001 mm to 0.003 mm. Furthermore, taking into account the machine settings, this clearance will vary from 0 mm to 0.003 mm. This fact will allow the provision of high transmission accuracy, in the case of low accuracy of the manufacturing of the relevant parts. This combination of fits should be used for any transmission with intermediate rolling elements (IRE). Moreover, in general, for complex combinations of structural joints, having three or more elements in simultaneous contact, it is possible to recommend the designation of fits with K for mated holes and with h for mated shafts. Further development of the presented scientific results will allow qualitative im- provement in transmission design and significantly facilitate the process of technological preparation for the production of gearing, which will subsequently allow the achievement of high technical and economic indicators in general. Also, the study of the problem of the mutual bracing of the surfaces of parts in the engagement of transmissions with IRE is promising. Appl. Sci. 2022, 12, 5 10 of 10 Author Contributions: Conceptualization and methodology, E.A.E.; validation and formal analysis, A.D.E. and E.A.E.; investigation, V.Y.S.; resources, M.V.G.; data curation, A.V.O.; writing—original draft preparation, V.Y.S., M.V.G.; writing—review and editing, V.Y.S., N.V.M.; visualization, A.V.O.; supervision, E.A.E.; project administration, N.V.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the TPU development program. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Dec 21, 2021

Keywords: manufacturing accuracy; cycloidal transmission; kinematic parameters; tolerances; deviations; technological gap

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