Strojniški vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 © 2020 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2020.6868 Original Scientific Paper Received for review: 2020-07-20 Received revised form: 2020-10-02 Accepted for publication: 2020-10-13 A Loaded Analysis Method for RV Cycloidal-pin Transmission Based on the Minimum Energy Principle Tianxing Li12* - Hang Xu3 - Meng Tian2 1 Henan University of Science and Technology, School of Mechatronics Engineering, China 2 Collaborative Innovation Center of Machinery Equipment Advanced Manufacturing of Henan Province, China 3 Zhongyuan University of Technology, School of Mechatronics Engineering, China Due to the complexity of load distribution and contact conditions, as well as the lack of effective analysis methods, the theoretically designed rotary vector (RV) cycloidal-pin drive with good meshing characteristics shows poor loaded performance in practical applications. In this paper, an effective analysis method based on the minimum energy principle is proposed, which can accurately obtain the real loaded characteristics in line with the actual operations. In the process of loaded analysis, through the innovative introduction of the minimum energy principle, the actual number of teeth engaged simultaneously was accurately determined, which directly affects the quality of meshing. The results of simulation and measurement experiment demonstrate the correctness and practicability of the theoretical analysis method and the effectiveness of the introduction of the minimum energy principle. This study solves the problem that the actual meshing performance is inconsistent with the theoretical analysis results, and provides an effective way for the improvement and pre-control of the transmission accuracy and meshing quality of the robot RV reducer. Keywords: cycloidal-pin gear, loaded characteristics, minimum energy principle, rotary vector (RV) reducer, transmission error Highlights • An analysis method based on the minimum energy principle is proposed, which can accurately obtain the real loaded characteristics in line with the actual operations of the RV transmission. • Through the innovative introduction of the minimum energy principle, the actual number of teeth engaged simultaneously is accurately determined. • The possible meshing teeth and their numbers are quickly estimated according to the distribution of the minimum meshing clearance and the maximum deformation of the cycloidal-pin gear. • The nonlinear mathematical relationship between the meshing load and the deformation is established. 0 INTRODUCTION The rotary vector (RV) reducer, accounting for about 38 % of the total cost, is the core component of a robot [1] and [2]. The RV reducer is mainly composed of a two-stage mechanism. The cycloidal-pin gear at the low-speed stage is directly connected to the output shaft, so its transmission performance determines the output accuracy of a reducer [3] and [4], which directly affects the motion accuracy, repeated positioning accuracy and service life of the robot. One of the main features of cycloidal-pin transmission is simultaneous multi-tooth engagement. The error equalization effect generated by multi-tooth meshing is conducive to improving the transmission accuracy, which allows high meshing rigidity and large carrying capacity. However, due to the complicated load distribution and contact conditions of the cycloidal-pin gear with simultaneous multi-tooth meshing, as well as the lack of effective load analysis methods, the theoretically designed RV mechanism with good meshing characteristics shows poor load performance in practical applications. Low transmission precision, insufficient bearing capacity, and short accuracy retention are common problems to be solved in RV transmission. Many researchers have conducted in-depth studies on the meshing characteristics of cycloidal-pin drives. Lin et al. [1] analysed the kinematic characteristics of the cycloidal drive by using the tooth contact analysis. Through analysing the contact dynamics in a cycloidal-pin transmission, Xu et al. [3] concluded that multi-tooth meshing characteristics have the main influence on the kinematic accuracy and the contact force. Based on unloaded and loaded contact analyses of the cycloid speed reducer, Li et al. [5] predicted the loads on various components in the presence of clearances and eccentricity errors. Li et al. [6] proposed an unloaded meshing analysis method for RV cycloidal-pin drive considering the influence of manufacturing errors. Based on the influence coefficient method, Huang and Tsai [7] proposed a computerized approach of loaded tooth contact analysis for the contact tooth pairs of the involute stage and the cycloid stage. Demenego et al. [8] developed a computer program for tooth contact analysis, which *Corr. Author's Address: Henan University of Science and Technology, School of Mechatronics Engineering, Luoyang, China, litianxing@haust.edu.cn 655 Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 can determine the virtual pair of contacting profiles, the backlash and the transmission errors. Furthermore, the research on the meshing characteristics of ordinary gears laid the foundation for cycloidal-pin transmission. Hao et al. [9] established an analysis model to obtain the load-sharing characteristics of the aeronautical two-stage five-branching planets gear train. Tsai and Ye [10] proposed a computerized approach based on the influence coefficient method to solve the statically indeterminate problem of multiple loaded contact tooth pairs in planetary gear drives. Wang et al. [11] solved the meshing characteristic problem of orthogonal face-gear drives in real working conditions. Bodzas [12] carried out loaded tooth contact analysis of straight bevel gears and obtained the mechanical characteristics by different loads. Vivet et al. [13] proposed a multi-body meshing analysis method and analysed the transmission error and contact patterns of spiral bevel gears. Lin [14] presented the loaded contact analysis of curve-face gear pair. Nishino [15] developed a computerized tool to predict the loaded tooth contact pattern and static transmission error of hypoid gears. Kolivand and Kahraman [16] proposed a practical methodology based on ease-off topography for loaded tooth contact analysis of hypoid gears. Batsch [17] conducted tooth contact analysis on the convexo-concave helical bevel Novikov gear. Wang et al. [18] improved the transmission characteristics of equal base bevel gears from a processing perspective. Yang et al. [19] proposed a dynamic analysis method to improve the transmission performance of spiral bevel and hypoid gears. Changbin et al. [20] simulated the dynamic meshing process of the elliptic cylinder gear. The load distribution and actual contact situation of the multi-tooth meshing of the cycloidal-pin gear are extremely complicated, and its load-bearing transmission process is a multi-body contact process. The key to loaded analysis is to accurately determine the actual number of meshing teeth and the nonlinear relationship between the contact force and deformation. In order to solve the static problems of elastic components, the principle of minimum energy is often used. It avoids the complex process of integral calculation directly using the relationship between force and deformation but gives a more concise result from the stability of energy. Li et al. [21] proposed a calculation method for cutting force based on an energy method. Based on the energy balance, Wang et al. [22] predicted the contribution of the axial force of the gear. Wang and Feng [23] derived the formula of time-varying mesh stiffness of the gear train based on the energy method. Liang et al. [24] evaluated the time-varying mesh stiffness of a planetary gear set by means of the potential energy method. Wan et al. [25] proposed an accumulated integral potential energy method to calculate the mesh stiffness of helical gears. Song et al. [26] proposed a potential energy-based slice grouping method to calculate the mesh stiffness of straight bevel gears. Wang et al. [27] studied the chaos and stability of spur gear based on the energy method and floquet theory. The above work provides a novel idea for the study of the complex meshing characteristics of cycloidal-pin gears. In this paper, the nonlinear mathematical relationship between the meshing load and the deformation is established first. Next, the possible meshing teeth and their numbers are quickly estimated according to the distribution of the minimum meshing clearance and the maximum elastic deformation. More importantly, the principle of minimum energy is innovatively introduced into the loaded analysis process to accurately determine the actual number of teeth engaged simultaneously and their tooth numbers. Finally, according to the loaded analysis model of cycloid-pin gear, some important parameters of loaded characteristics are calculated and analysed. 1 TRANSMISSION CHARACTERISTICS OF CYCLOIDAL-PIN GEAR The RV reducer is a closed differential gear system composed of a two-stage transmission mechanism, as shown in Fig. 1. The sun gear (driving gear) is connected to the input shaft. The planet gear meshed with the sun gear is fixedly connected with the crankshaft. Two cycloidal gears with the phase difference of 180° are installed on the crankshaft and engage with the pin gear fixed on the pin box. If the driving gear rotates counter-clockwise at a certain speed, the planetary gear meshing with the sun gear rotates clockwise, and the crankshaft fixed to the planetary gear rotates clockwise. The rotation of the crankshaft drives the cycloidal gear to revolve clockwise, and at the same time, the cycloidal gear rotates counter-clockwise under the action of the pin gear. The rotation of the cycloidal gear, in turn, causes the crankshaft to revolve counter-clockwise around the input shaft. The angular velocity of this revolution is the output angular velocity of the RV reducer, and its direction is the same as that of the input shaft. When the crankshaft rotates 360°, the cycloidal gear rotates by a pitch angle in the opposite direction. The rotation angular velocity of the cycloidal gear is equal to the revolution angular velocity of the crankshaft 656 Li, T.X. - Xu, H. - Tian, M. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 connected to the output mechanism, so the angular velocity of the cycloidal gear can be transmitted and output by the output mechanism at a speed ratio of 1:1. It can be seen that the transmission performance of the cycloidal-pin mechanism directly connected to the output shaft has an important effect on the motion accuracy of the RV reducer. b) Fig. 1. Transmission mechanism of RV reducer; a) cycloidal-pin gear; and b) transmission mechanism The simultaneous meshing of multiple teeth of the cycloidal-pin gear not only allows the load to be distributed to each tooth pair, which improves the carrying capacity, and its error equalization effect greatly improves the transmission accuracy. However, the load distribution and actual contact conditions of the multi-tooth meshing are quite complicated, and higher requirements are also placed on the machining errors and assembly processes [6]. In practical applications, many factors, such as machining errors, assembly errors, and modification errors, have significant and complex effects on the meshing contact state of gear teeth. In fact, according to the meshing principle of the gear, the interactive influence of various error factors indirectly changes the profile shape of the meshing tooth pair, which in turn affects the meshing contact state and transmission accuracy. In this case, the characteristics of simultaneous multi- tooth engagement no longer exist. Under no-load conditions, the smaller or larger cycloid gear and the pin gear only have a pair of teeth to maintain contact. There are initial clearances of different sizes between the other gear teeth. In the loaded state, only when the deformation of the first contact tooth is greater than the initial clearance, the remaining teeth can come into contact one after another. Therefore, the modification errors, machining errors, and thermal deformations have caused the change in the contact characteristics of cycloidal-pin gear, which inevitably produce transmission errors and lost motion. The change in the shape of the tooth profile leads to uneven load distribution, which will aggravate wear and reduce the accuracy retention of RV transmission. 2 MESHING MODEL OF CYCLOIDAL-PIN GEAR Tooth contact analysis is one of the important means of studying meshing characteristics. The analysis method and model have been generated by some researchers [28] and [29]. In this section, the mathematical model of the cycloidal-pin transmission will be given for later use [6]. As shown in Fig. 2, Sf(xf,yf) is a coordinate system fixedly connected to the frame. The coordinate systems of the pin gear and cycloidal gear are defined as Sp (xp ,yp) and Sc (xc ,yc), respectively. Their rotation angles are denoted as ^ and The eccentricity is represented by a. Fig. 2. Meshing contact model of cycloidal-pin gear In the RV transmission process, the position vector and normal vector of the cycloidal gear and the pin gear at any meshing point should be consistent. Therefore, the meshing contact equation can be established [6] R Y W2) = R(/ )(P,A) (f )t ,c f) fab) = n(f ^M) (1) A Loaded Analysis Method for RV Cycloidal-pin Transmission Based on the Minimum Energy Principle 657 Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 where the superscript (f ) is indicated in the coordinate system Sf. The vector Rc and nc respectively represent the position and normal of the cycloidal gear, with q and as variables [1] and [6]. The vector R and np are the position and normal of the pin gear, with ft and as variables. There are three nonlinear equations and four unknowns in Eq. (1). By giving different driving angle a series of driven angles and contact position parameters q and ft can be solved. The process of solving the meshing model is described as follows: first, an initial value of the driving angle ^ is given, which is denoted as 0/o). In this case, Eq. (1) is a nonlinear equation system consisting of three equations and three unknowns. The driven angle 02(o) and the initial contact position parameters ft(0) and q(0) can be solved by the optimization method. Then, by continuously changing ^ in small increments, a series of meshing contact parameters $k, 02(kk, ft(k) and q(k) can be solved (superscript k = 1, 2, . .. is the index of the meshing contact point). The nonlinear relationship between ^ and determines the transmission error. The parameters q and ft determine the contact position on a tooth profile. Under no-load conditions, the transmission error is mainly determined by the actual clearance between the gear teeth. In the loaded state, the transmission error is mainly caused by the contact deformation and the meshing clearance, which is the main difference between the meshing characteristics analysis under unloaded and loaded conditions. 3 LOADED ANALYSIS BASED ON MINIMUM ENERGY PRINCIPLE In theory, the deformation quantity at a meshing position should be the sum of the contact deformation and bending deformation. However, in order to increase the rigidity of the mechanism, a pillowlike pin-tooth structure is often used. Therefore, the bending deformation is ignored, and the contact deformation is mainly considered in this paper. Also, the error equalization effect of multi-tooth engagement makes the two symmetrically installed cycloidal gears have similar meshing characteristics, so only one cycloidal gear is considered. 3.1 Meshing Force at Contact Position In the loaded transmission, the cycloidal gear bears the load torque. At the same time, due to its meshing contact with the pin gear, a torque opposite to the direction of the load torque is also imposed. According to the principle of torque balance, when the RV transmission runs stably, the algebraic sum of all torques should be zero. Therefore, the nonlinear relationship between the torque and the meshing force can be established. Tc = £f (P, Pi A A) • Li (p), (2) i=\ where i = 1 ~m, m is the number of simultaneous meshing teeth, and the meaning is the same below. The meshing force of the /'th tooth is represented by F, which is related to parameters q and ft. The distance L is the force arm at the contact point of the ith tooth, with q as the variable. By solving Eq. (1), the contact position parameters q and ft of the cycloidal gear and the pin gear can be determined. Assuming that the number of meshing teeth m is determined, the force arm Lt can be calculated. According to Eq. (2), the meshing force of the contact position can be obtained. 3.2 Meshing Contact Deformation and Coordination Relationship According to the Hertz contact theory, the elastic contact problem of the cycloidal-pin gear at the meshing position is simplified in this paper [30]. The contact is considered as the smooth contact of the elastic body without friction. The vertical pressure on the contact surface is considered to be uniformly distributed, and the contact area is a small deformation. Thus, the meshing contact of the cycloidal-pin gear can be simplified into an elastic contact between two parallel cylinders, as shown in Fig. 3. Fig. 3. Contact deformation model of cycloidal-pin gear Under no-load conditions, the contact pattern of the cycloidal-pin gear should be a straight line. After loading, the contact pattern will become rectangular. The contact half-width a, can be expressed as: 658 Li, T.X. - Xu, H. - Tian, M. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 ai (Pi, ßi Aii ,A ) = 4 • F (P, ß Ai ) -p (P ) n • b 1 , 1 2 \ (3) where b is the width of the cycloidal gear. Young's modulus of the cycloidal gear and the pin gear are denoted as Ec and Ep, respectively. Their Poisson ratio are ^c and The comprehensive curvature radius, pi, at the meshing point, which can be determined by the radius of curvature of the cycloidal gear and the pin gear. Pi 1 Pciiv) ■ ^ + ± Pciiv) ± rrp (4) PciivY where the positive sign is used for external contact and the negative sign is used for internal contact. The roller radius, which is represented by rrp , is the radius of curvature of the pin gear. The radius of curvature at the contact point of the cycloidal gear is denoted as pci , which can be calculated as follows: PciiV) = irP + Arp ) • (1 + k1 - 2k1 • cos(zc q))3 ki • (zp +1)• cos(zc •q)-(1+zp • ki ) r + Ar„ (5) where zc and zp is the number of teeth of cycloidal gear and pins; k1 is the short width coefficient; rp is the roller position; Arp and Arrp is the modification by the radius and position of the roller. According to the geometric relationship in Fig. 3, the deformation ôci and ôpi of the cycloidal-pin gear at the ith contact position can be calculated as follows Sci=Pci-4p2ci~ 8pi= rrp -J 2 2 rrp - ai . (6) (7) The normal stiffness Kci and Kpi at the ith contact position of the cycloidal gear and the pin gear can be determined respectively: K - F o„. K - Fi p' (8) (9) From this, the normal meshing stiffness Kni of a pair of teeth of the cycloid-pin gear can be obtained: K _ KciKpi K, + K„ (10) The equivalent torsional stiffness Kti of the cycloidal-pin gear at the ith contact position can be expressed as follows: K = K, ■ L ' (11) Therefore, the nonlinear relationship between the meshing force Fi and the deformation di can be determined as follows: Si {(Pi, ßi Ai A2i )= ~T¡~ . K„ (12) Additionally, the contact deformation of the cycloidal gear and the pin gear is not independent. A certain coordination relationship should be met, which ensures the integrity and continuity after contact deformation. In the paper, the ratio of the sum of di and di to Li is defined as the error rotation angle. According to the coordination relationship, the angles should be equal at all contact positions. Thus, the deformation coordination equation can be obtained ¿>j + dj S2 + d2 S, + d, L L (13) 3.3 Determination of Possible Number of Meshing Teeth Under no-load conditions, only one pin tooth maintains meshing contact with the cycloidal tooth, and there are initial clearances of different sizes between the other teeth. According to Eq. (1), the distribution of the minimum meshing clearance of the cycloidal-pin gear can be obtained [6], as shown in Fig. 4. Fig. 4. Minimum meshing clearance and maximum deformation It can be seen from the distribution of the minimum clearance shown in Fig. 4 that the clearance between the 35 th pin tooth and the cycloidal profile is rp r rp m A Loaded Analysis Method for RV Cycloidal-pin Transmission Based on the Minimum Energy Principle 659 Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 zero. Therefore, the 35th pin should be the first tooth in contact under no-load condition. After loading, the 35th pin tooth produces a certain deformation displacement under the action of the meshing force. When the next pin comes into contact, the deformation quantity reaches the maximum value. Only when the deformation value is greater than the meshing clearance of other pin teeth can they come into contact with the cycloidal tooth. Lines 1, 2, and 3 in Fig. 4 illustrate the position of the maximum deformation with respect to the clearance distribution. If the maximum deformation is at the position of line 1, the No. 30~38 pin teeth bellow line 1 will be most likely to contact after load. In this case, the maximum number of meshing teeth can be estimated to be 9. It should be noted that the estimated number of meshing teeth and their tooth numbers are only the maximum possible value. Each possibility needs to be grouped to discuss the actual number of meshing teeth. The detailed process of estimating the number of teeth that may be meshed and their tooth numbers is shown in Fig. 5. Output torque Tc * Meshing analysis model I Initial meshing position parameters 3 350 | 300 ¿■250 3 200 § 150 100 50 0 603.58 —'■— 0.02 rad \ —•— 0.04 rad \ —A— 0.06 rad \ -T- 0.08 rad ............... 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Number of pins in meshing contact Fig. 9. Contact pressure at meshing points It can be seen that the contact pressure at each meshing position increases to the peak first and then decreases with the increase of the number of pins. The maximum pressure is at the 32nd pin tooth, and its value is 603.58 MPa. Compared with the distribution of the contact force in Fig. 8, the peak position of the maximum contact pressure is different. The pressure value is related not only to the contact force Fi, but also to the comprehensive curvature radius Pi. If the contact force is larger, the corresponding comprehensive radius of curvature is also larger, but the contact pressure is not necessarily larger. Therefore, the influence of the curvature radius on contact pressure is more obvious. 4.2 Loaded Deformation Fig. 10 shows the deformation distribution at each meshing point when the rotation angle of the pin gear is different. It can be seen from Fig. 10 that the loaded deformation at the meshing position of the cycloidal-pin gear is essentially consistent with the contact force. With the increase of the pin number, the deformation value at each meshing point first increases to the peak and then decreases. The maximum deformation is 0.057 mm at the 35th pin. If the pin gear rotates counter-clockwise, the deformation before the peak increases, and the deformation after the peak decreases with the increase of the angle of the pin gear. According to the deformation distribution trend, with the rotation of the pin gear, the deformation of the 39th pin gradually decreases until it becomes zero, at which time the pair of pin teeth are disengaged. At the same time, the 23rd pin tooth is in meshing contact immediately, and its deformation quantity increases until it reaches the peak and then gradually decreases until disengaged. Fig. 10. Deformation distribution at each meshing point According to Eq. (12), the deformation d, is influenced by the contact force F, and the curvature radius pThe changes in the meshing force and deformation shown in Figs. 8 and 10 are similar. It can be concluded that the curvature radius p has little effect on the loaded deformation. 4.3 Loaded Contact Pattern At the initial meshing position of the cycloidal-pin gear, the contact parameter and the half-width a, were calculated, as shown in Table 2. 662 Li, T.X. - Xu, H. - Tian, M. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 Table 2. Position parameters and contact half-width Parameters_Number of simultaneous meshing teeth Pin number 24 25 26 27 .. . 36 37 38 39 Position [rad] -2.51 -2.36 -2.2 -2.04 .. . -0.63 -0.47 -0.31 -0.16 Half-width [mm] 0.15 0.19 0.24 0.27 .. . 0.44 0.37 0.34 0.24 In order to more clearly compare the positions of the contact patterns when multiple teeth are in contact, all positions are displayed on the same tooth surface, as shown in Fig. 11. It can be seen that the loaded contact pattern of the cycloidal-pin gear is approximately rectangular. From the 24th to 39th teeth, the contact patterns at each meshing position are gradually widened from 0.15 mm to 0.47 mm, and then gradually reduced to 0.24 mm. The change rule of the loaded contact pattern is essentially the same as that of the meshing force and contact pressure, that is, the greater the loaded contact force and the contact pressure is, the larger the area of the contact pattern is. Fig. 11. Loaded contact patterns 4.4 Loaded Transmission Error Using the loaded analysis model, continuously changing the rotation angle ^ of the pin gear, the actual rotation angles of the cycloidal gear can be obtained. Then the loaded transmission error curve was drawn, as shown in Fig. 12. It can be seen from Fig. 12 that the loaded transmission error of the cycloidal-pin gear changes periodically and is smooth. In one cycle, as the rotation angle of the pin gear increases, the transmission error gradually decreases to -3.80 arcmin and then slowly increases to -7.46 arcmin until the next cycle begins. The amplitude of the fluctuation is approximately 3.66 arcmin. Since the assembly error and machining error are not considered, the transmission error in the theoretical model mainly depends on the contact deformation, torsional deformation and meshing clearance. Therefore, the fluctuation amplitude of each meshing cycle is consistent and relatively small. Fig. 12. Loaded transmission error curve (Three cycles) It should be mentioned that the design purpose of the holes in the blank of the cycloid is mainly to reduce the weight. The prerequisite for lightweight design is that the meshing performance of the gear train cannot be changed. Therefore, the effect of the stiffness of holes in the blank on the contact pressure and transmission error can be ignored, which was proved in the comparison and analysis of ABAQUS results. 5 EXAMPLES AND DISCUSSION The parametric modelling of the RV reducer was completed through Unigraphics (UG) software. Due to the pillow-like structure of the pin gear and the symmetrical distribution of the cycloidal gear in the robot reducer, the model is simplified to the meshing contact form of a cycloidal gear and a pin gear. The simplified model was imported into ABAQUS for finite element analysis. In order to better compare the examples, two ABAQUS models have been established, as shown in Fig. 13. Model A in Fig. 13a is the "full blank". Model B in Fig. 13b is the "blank with holes". The finite element mesh has a significant impact on the analysis results. The meshing contact A Loaded Analysis Method for RV Cycloidal-pin Transmission Based on the Minimum Energy Principle 663 Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 deformation of the cycloidal-pin gear is mainly located near the contact area. Therefore, a method of mesh refinement near the contact area was adopted. The approximate size of the elements of the refined mesh is 0.2 mm. The element type is C3D8R (8-node linear brick). The total number of meshes of the cycloidal gear (without holes) and the pin gear are 203,973 and 183,792, respectively. Also, the gear axes (Z-axis) of the cycloid and pin gears were free, and other degrees of freedom were restricted. The bending moment imposed on a cycloidal gear was set to 28.5 Nm, which is half of the actual load. The contact property was defined as frictionless and "hard" contact. The pin surface was assigned as the master surface and the cycloidal surface as the slave surface. which shows the correctness of the theoretical model and the results. a) bj Fig. 13. ABAQUS model; a) model A; and b) model B 5.1 Comparison and Analysis of Contact Pressure The contact pressure distributions of the finite element simulation of the cycloidal gear and the pin gear are shown in Figs. 14 and 15, respectively. It can be seen from Fig. 14 that the maximum contact pressures of the cycloidal gear are 601.3 MPa and 600.0 MPa, respectively. The difference between the two is within 0.22 %, which shows that the stiffness of the hole in the blank has almost no effect on the contact pressure. For the pin gear in Fig. 15, the conclusions are similar. a) ^^ b) Fig. 14. Contact pressure distribution of the cycloidal gear; a) Model A; and b) Model B In order to verify the correctness of the theoretical model, the contact pressure distribution of the Model B simulation and the theoretical calculation was compared in Fig. 16. It can be seen that their trends are similar, and the maximum pressure is at the 32nd pin. The difference between the two is within 0.6 %, a) bj Fig. 15. Contact stress distribution of the pin gear; a) Model A; and b) Model B Fig. 16. Comparison of contact pressure between simulation and theoretical calculation 5.2 Comparison and Analysis of Contact Pattern According to the contact stress of the finite element analysis, the actual number of meshing teeth and the tooth numbers were determined, and the meshing patterns of the cycloidal-pin gear was obtained, as shown in Fig. 17. Fig. 17. Contact patterns of simulation From the simulation results, it was found that the contact patterns of Models A and B were the same, which shows that the stiffness of the holes in the cycloidal gear has a very small effect on the contact 664 Li, T.X. - Xu, H. - Tian, M. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 patterns. Only the results of Model A were drawn in Fig. 17. It can be seen from the comparison with Fig. 11 that the actual number of meshing teeth is also 16. The size of the meshing area is slightly different from the theoretical one, which is due to the influence of the mesh fineness in the simulation model and the rotation angle error of the theoretical model on the comparison results. 5.3 Measurement of Loaded Transmission Error In order to better compare and analyse the results of the theoretical model and the ABAQUS model, the loaded transmission errors of Models A and B in ABAQUS were respectively extracted, as shown in Fig. 18. — Results of model A — Results of model B 20 40 60 80 100 120 140 160 180 \AAAvVV\^VV\AAAAAAVW\A a1 o i -5 o J ¿-10 « 0 ■g -5 -o ,„ o" 180 200 220 240 260 280 300 320 340 360 -1 Output angle [deg] Fig. 18. Loaded transmission errors of Models A and B In Fig. 18, the fluctuation amplitudes of loaded transmission errors of models A and B are 3.69 arcmin and 3.7 arcmin, respectively. The difference in simulation results is 0.01 arcmin. Therefore, the influence of the stiffness of the holes in Model B on the loaded transmission error can be ignored. The simulation results of Models A and B are the same. In addition, compared with the theoretical results in Fig. 12, the difference between Model B and the theoretical model is within 1.1 %, which indicates the correctness of the theoretical model. Furthermore, a measurement experiment of the loaded transmission error was completed on a RV reducer comprehensive performance tester, as shown in Fig. 19. The assembled reducer and its cycloidal gear with holes are shown in Fig. 20. During the measurement process, the RV reducer is mounted on a precision bracket. The input is driven by the input motor. The output end is connected to the output assembly through a coupling, and is loaded through a load motor and a torque sensor. Through the high-precision circular grating installed at the input and output ends, the angle changes of the input and output are collected in real time to realize the transmission error measurement. The measurement accuracy of the tester is within 8.0 arcsec, which can meet the experiment requirements. Fig. 19. RV reducer comprehensive performance tester Fig. 20. a) RV reducer and b) cycloidal gear In the 360° range of the output shaft, 1800 angles of the output shaft were collected. By calculating the difference between the theoretical and actual output angle, the loaded transmission error curve was drawn, as shown in Fig. 21. Fig. 21. Loaded transmission errors of measurement It can be seen from Figs. 12 and 21 that the transmission error curves of theoretical calculation, actual measurement and simulation analysis all show a relatively consistent parabolic trend. In Fig. 21, the actual measured transmission error is larger than those of the numerical model and experimental results. Their maximum fluctuation amplitudes are 3.83 arcmin, 3.66 arcmin and 3.7 arcmin, respectively. The maximum difference is about 4.4 %. From the comparison of data and trends, it can be concluded that the transmission error of the numerical model, the finite element model and the experimental results have good consistency. The effectiveness and practicability of the proposed method and simulation analysis can also be validated. However, further improvement of the experimental results is an important topic for future work. The second-stage mechanism in the RV reducer has a certain influence on the experimental results. In addition, there are many influencing factors in the actual measurement process, such as assembly errors, profile errors and pitch errors of cycloidal gears, processing errors of other key components, bearing clearance, etc. The comprehensive influence of these A Loaded Analysis Method for RV Cycloidal-pin Transmission Based on the Minimum Energy Principle 665 Strojniski vestnik - Journal of Mechanical Engineering 66(2020)11, 655-667 factors is very complex and cannot be accurately determined [31] to [33]. 6 CONCLUSIONS This paper presents the loaded analysis method based on the minimum energy principle for RV cycloidal-pin transmission. The main conclusions include: 1. The Hertz contact theory, the principle of minimum energy and the finite element method were integrated into the loaded contact analysis of the cycloidal-pin drive, and the actual loaded characteristics were obtained. 2. The possible meshing teeth and their numbers were quickly estimated according to the distribution of the minimum meshing clearance and the maximum deformation of the cycloidal-pin gear. 3. By comparing the minimum energy of the contact tooth pair, the actual number of teeth engaged simultaneously was accurately determined, which is critical to the accuracy of the loaded analysis. 4. The comparison and verification of theoretical modelling, finite element analysis and measurement experiment show that the difference in contact pressure, the number of meshing teeth, contact pattern and transmission errors were essentially the same, which proves the effectiveness and correctness of the proposed method. This study solves the bottleneck problem in which the actual meshing performance is inconsistent with the theoretical analysis results in the loaded transmission of RV reducer, and has an important reference value for the improvement and pre-control of the kinematic accuracy of the robot RV Reducer. However, for the proposed method, some work remains to be done, including the comprehensive transmission characteristics of RV reducer in the theoretical model, the influence law of load torque on transmission error and the number of meshing teeth, and experimental research more in line with the actual working conditions. 7 ACKNOWLEDGEMENTS This work supported by National Natural Science Foundation of China (No. U1504522, 51705134, and 51675161), Key Scientific Research Projects of Higher Education Institutions in Henan Province of China (No. 20A460010) and Key Scientific and Technological Project in the Henan Province of China (No. 202102210079). 8 REFERENCES [1] Lin, W.S., Shih, Y.P., Lee, J.J. (2014). Design of a two-stage cycloidal gear reducer with tooth modifications. Mechanism and Machine Theory, vol. 79, p. 184-197, D0l:10.1016/j. mechmachtheory.2014.04.009. [2] Li, T.X., Zhou, J.X., Deng, X.Z., Li, J.B., Xing, C.R., Su, J.X., Wang, H.L. (2018). 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