machine design, Vol.6(04) No.3, ISSN 8-59 pp. 79-84 ANALYSIS OF CLEARANCES AND DEFORMATIONS AT CYCLOID DISC Mirko BLAGOJEVIĆ * University of Kragujevac, Faculty of Engineering, Kragujevac, Serbia Preliminary note Received (5.05.04); Revised (6.08.04); Accepted (8.08.04) Abstract: Cycloidal speed reducer belong to the new generation of mechanical gear train. Thanks to a lot of favorable working characteristics, they have a wide application in industry. This paper presents a method for calculating the deformations of cycloid disc teeth and clearances which occur between the cycloid disc teeth and ring gear rollers. The influence of profile correction on deformations and clearances, and on the number of teeth which simultaneously transmit the load is analyzed in the paper. Based on the analysis of the obtained results it can be concluded that the number of teeth which transmit the load decreases with increasing the size of profile correction and the size of the gear tooth deformation is a periodic function of time. Key words: cycloidal speed reducer, cycloid disc, clearance, deformation. INTRODUCTION Gears with cycloid profile (cycloid discs) have the great application at cycloidal speed reducer, Fig.. Cycloidal speed reducer belongs to a group of new generation planetary gears, [,, 3]. They have a number of excellent characteristics such as: long and reliable work life, large range of possible transmission ratios, extremely reliable functioning in dynamical load conditions, compact design, and high efficiency coefficient, [4, 5, 6, 7, 8]. Internal non - involute roller gearing is applied at cycloidal speed reducer, [9]. Typical internal gear has been replaced by rollers placed between two discs, while the cycloid disc has the usual appearance (external gear). The exploded view of one - stage cycloidal speed reducer is presented on Fig.. At cycloid disc with non corrected profile, Fig. 3, all ring gear rollers are in the contact with the corresponding cycloid disc tooth and half of them transfer the load However, at the real cycloidal speed reducer, it is not the case. Between the ring gear rollers and the cycloid disc teeth, there is the adequate clearances, [0,,, 3]. Fig.. Exploded view of one - stage cycloidal speed reducer ( - input shaft, - eccentric, 3 - needle bearings, 4 - cycloid discs, 5 - body of the ring gear, 6 - ring gear rollers, 7 - output rollers, 8 - carrier, 9 - output shaft Fig.. One - stage cycloidal speed reducer Fig.3. Cycloid disc with non corrected profile *Correspondence Author s Address: University of Kragujevac, Faculty of Engineering, Sestre Janjic 6, 34000 Kragujevac, Serbia, mirkob@kg.ac.rs
Machine Design, Vol.6(04) No.3, ISSN 8-59; pp. 79-84 These clearances are necessary for several reasons, Fig. 4, in order to: The starting point for analysis is Fig. 5 which presents the cycloid disc in contact with the ring gear rollers. 80 Fig.4. Cycloid disc with corrected profile. compensate the errors made in the process of production of cycloidal speed reducer elements;. provide better conditions for lubricating surfaces of teeth and central gear rollers; 3. achieve the process of assembly and disassembly. At cycloidal speed reducer there are a lot of locations where clearances may occur. In this paper were analyzed only clearances between cycloid disc teeth and ring gear rollers. The above-mentioned clearances can be achieved in several ways as follows: correcting the profile of cycloid disc tooth, increasing the diameter of a stationary ring gear in which are arranged the rollers, reducing the diameter of the ring gear rollers. Regardless of the method by which the mentioned clearances are achieved, cycloidal speed reducer should have: constant transmission ratio, quiet operation, uniform load distribution,.... CALCULATING OF DEFORMATIONS AND CLEARANCES AT CYCLOID DISC Unlike the cycloid discs with theoretical (non corrected) profile in which all the teeth are in the contact with the relevant ring gear rollers, and half of them are involved in the process of transmission of loads, at cycloid discs with corrected profile it is not the case. At cycloid discs with corrected profile the number of teeth in contact depend from profile correction value (from clearance between cycloid disc teeth and ring gear rollers). Based on the mathematical models from papers [4, 5], this paper presents the procedure for calculation of deformations and clearances between the cycloid disc teeth and ring gear rollers, as well as procedure for determining the number of teeth that are also in contact and carry the load. Fig.5. Calculating the deformations and clearances at cycloidal speed reducer elements Clearance between the "i" ring gear roller and appropriate cycloid disc tooth (Fig. 5), is calculating, [4, 5]: k i q sin i ez ez cosi r r q - profile correction value, r - radius of the ring gear where the rollers are arranged, r - radius of the fixed circle, r - radius of the moving circle, e - eccentricity value, i - position angle of "i" ring gear roller (the angle between the "i" ring gear roller axis and the direction of eccentricity CR) z - number of ring gear rollers. Total deformation in contact point of cycloid disc tooth and ring gear roller can calculate as, [4, 5]: sin i wi w () max ez ez cosi r r w max - the maximum value of the total deformation. The maximum value of the total deformation can calculate as: w w w (3) max max max w max - the maximum contact deformation, w max - the maximum bending deformation of the ring gear pin at the acting point. ()
Machine Design, Vol.6(04) No.3, ISSN 8-59; pp. 79-84 The maximum contact deformation w max is calculated from the expression: w F 6 q Nmax max ln b 3 C - Poisson`s ratio of cycloid disc material ( for steel, ), E - elasticity modulus of cycloid disc material (for steel 5 E =,0 MPa ), F - the maximum value of the normal force in contact Nmax of cycloid disc tooth and the ring gear roller, b - width of cycloid disc, q - radius of the ring gear roller, - curvature radius of cycloid disc at the point where the force acting on the tooth is maximum, C - constant. Curvature radius of cycloid disc at the point where the force acting on the tooth is maximum is calculated from the expression: ez ez z r cos r r z q ez z ez z z cos r z r z - number of cycloid disc teeth, - drive angle. Constant C is calculated as: 3 FNmax q C 4,99 0 b q 3/ (4) (5) (6) The main parameters of the cycloidal speed reducer are shown in Table. Table. Parameters of the cycloidal speed reducer Power Transmission ratio Input speed Efficiency Number of cycloid disc teeth Number of ring gear rollers Radius of the ring gear where the rollers are arranged Radius of the ring gear roller Radius of the fixed circle Radius of the moving circle Eccentricity value Width of cycloid disc The maximum value of the normal force in contact of cycloid disc tooth and the ring gear roller P CR n in kw u CR min CR z z r mm q mm r mm r mm e mm b mm F Nmax N On the basis of expressions () - (8), and the parameters of cycloidal speed reducer given in the Table, calculation of clearances and deformations into a function of drive angle was realized. Clearances and deformations are calculated for different values of cycloid disc profile correction q. The results of calculation are presented on Figures from (6) to (9). The maximum bending deformation of the ring gear pin at the acting point: w max 3 FNmax l (7) I x where is: l- distance between the supports of the ring gear roller, I - moment of inertia of the ring gear pin. x I x 4 0 r r - radius of ring gear pin. 0 (8) Fig.6. Distribution curves of deformations and clearances for values of profile correction q = 0,005 mm 3. CLEARANCES AND DEFORMATIONS AT THE PROJECTED ONE - STAGE CYCLOIDAL SPEED REDUCER Calculation of clearances and deformations for the concrete one - stage cycloidal speed reducer is presented in this paper. By analyzing the expressions for calculating the clearances and deformations of cycloidal speed reducer elements, it is easy to conclude that the value of profile correction q does not effect on the size of the deformations, and directly effects on the size of the clearances. In Figures from (6) to (9) it could be seen two curves: curve and clearance and curve of deformation. 8
Machine Design, Vol.6(04) No.3, ISSN 8-59; pp. 79-84 They intersect at two points, and these points represent the boundaries of contact cycloid disc teeth and ring gear rollers. All rollers of the ring gear and the corresponding cycloid disc teeth which position angles is located between the x coordinates of intersection points, respecting the rollers (teeth) whose the deformation is larger than clearance, take part in the transfer of loads number of rollers (teeth) which simultaneously transmits the load reduces. Fig.0. Dependence of clearance curve from profile correction value q Fig.7. Distribution curves of deformations and clearances for values of profile correction q = 0,0 mm Dependence of clearance from profile correction value q is presented in Figure 0. As it was expected, the higher values of profile correction q provide greater clearances, and therefore smaller number of ring gear rollers and cycloid disc teeth which simultaneously transmit the load. Since the normal force in the contact between the ring gear roller and cycloid disc tooth F N and deformation are directly in proportion, with increasing the value of the normal force, increasing the value of deformation, Figure. Fig.8. Distribution curves of deformations and clearances for values of profile correction q = 0,05 mm Fig.. Dependence of deformation curve from normal force F N Dependence of deformation from width of cycloid disc is presented in Figure. As shown in the Figure, with increasing the cycloid disc width, the deformations are smaller. Fig.9. Distribution curves of deformations and clearances for values of profile correction q = 0, mm Ring gear rollers with position angle in front of the first intersection point or behind the second intersection point (rollers where the size of the deformation is less than the size of the clearance), do not participate in the transfer of loads. From the above - mentioned Figures, it is easy to see that with increasing the value of profile correction q, the 8 Fig.. Dependence of deformation curve from cycloid disc width b
Machine Design, Vol.6(04) No.3, ISSN 8-59; pp. 79-84 Since the process of meshing between the cycloid disc teeth and ring gear rollers is periodic function of time, the same can be said for the deformations of mentioned elements. Deformation curve for profile correction q = 0,05 mm, maximum value of normal contact force F N = 588,6 N and cycloid disc width b = 3,3 mm is presented in Figure 3. This is explained in a way that is at smaller value of profile correction q, a wider range of meshing (larger number of rollers and teeth is in simultaneously contact). In those cases, the deformation does not remain only in the zone of high values. Dependence of number of ring gear rollers which are in simultaneously contact with corresponding cycloid disc teeth and transmit the load from profile correction q is presented in Figure 5. By increasing the profile correction q, the number of ring gear roller and cycloid disc teeth in simultaneously contact is smaller. 4. CONCLUSION Fig.3. Deformation curve as periodic function of time Dependence of deformation from drive angle, for different values of profile correction q from q = 0,00 mm do q = 0, mm is presented in Figure 4. Fig.4. Dependence of deformation from drive angle (q = 0,00 mm to q = 0, mm) From Figure 4 it can be concluded that the value of maximum deformation is the same for all values of the profile correction q. However, the minimum values of deformations are significantly different. Namely, by reducing the value of the profile correction q, the minimum value of the deformation of a cycloid disc tooth and the ring gear roller in contact is reducing, too. Fig.5. Dependence of number of ring gear rollers and cycloid disc teeth in simultaneously contact from profile correction Internal non - involute roller gearing is applied at cycloidal speed reducer. To achieve the necessary clearances at cycloidal speed reducer, correcting of cycloid disc tooth profile is done. As there is a clearance between the cycloid disc teeth and the ring gear rollers, the number of the above mentioned cycloidal speed reducer elements which are in simultaneously contact and transfer the load is reducing. Based on the results presented in this paper, it can be concluded that: Number of cycloid disc teeth that simultaneously transmit the load depends on the size of the profile correction; Size of profile correction does not affect on the size of the deformation, and directly affects on the size of the clearance; With increasing the size of the profile correction, reduces the number of the cycloid disc teeth that simultaneously transmit the load; With increasing the normal contact force, increases the size of the deformation; The value of deformation is inversely proportional to the cycloid disc width; The value of deformation is a periodic function of time. In further research it would be interesting to describe the clearances that occur at other locations in cycloidal speed reducer. Also, it is necessary to analyze the impact of these clearances on the load distribution. REFERENCES [] Kudrijavcev, V.N. (966). Planetary Gear Train (in Russian), Mech. Eng., Leningrad, Russia [] Malhotra, S.K. & Parameswaran, M.A. (983). Analysis of a Cycloid Speed Reducer. Mechanism and Machine Theory, Vol. 8, No. 6. pp. 49-499, ISSN 0094-4X [3] Chen, C.; Zhang, X. & Angeles, J. (007). Kinematics and Geometric Analysis of a Pure - Rolling Epicyclic Train. Journal of Mechanical Design, Vol. 9, No. 8, (August 007) ISSN 050-047 [4] Gorla, C.; Davoli, P.; Rosa, F.; Longoni, C.; Chiozzi, F. & Samarani, A. (008). Theoretical and 83
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