Strojniški vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 © 2016 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2016.3706 Original Scientific Paper Received for review: 2016-04-29 Received revised form: 2016-09-18 Accepted for publication: 2016-09-28 A Novel Universal Reducer Integrating a Planetary Gear Mechanism with an RCRCR Spatial Mechanism Cheng Gui2 - Xinbo ChenU* 1 Tongji University, Clean Energy Automotive Engineering Center, China 2 Tongji University, School of Automotive Studies, China A novel speed reducer with self-adaptability to variable transmission angles is proposed in this paper, in which a planetary gear mechanism with a difference of a few teeth and an RCRCR (R-revolute pair, C-cylindrical pair) spatial mechanism are integrated. The difference of a few teeth presents the advantages of high transmission ratio and compact structure, while the RCRCR spatial mechanism achieves the translational motion of the planetary gear in space. Theoretical kinematics analyses are carried out to prove the motion of each component in space. Models are established in the Automatic Dynamic Analysis of Mechanical Systems (ADAMS) to verify the performances of universal transmission and torque fluctuation tolerance. The simulation results demonstrate that the proposed speed reducer maintains a stable transmission ratio with variable transmission angles, presenting a high level of tolerance with the position perturbations. Strength analysis and manufacturing and lubrication guidelines of the prototype design are presented. Keywords: integrate, self-adaptability, planetary gear mechanism, few teeth difference, RCRCR spatial mechanism, universal reducer Highlights • A planetary gear mechanism with a difference of a few teeth presents the advantages of high transmission ratio and compact structure. • Translational motion of the planetary gear is achieved by the RCRCR spatial mechanism. • The transmission ratio of the reducer never varies with the intersection angle or the distance. • The universal reducer owns the self-adaptability to variable transmission angles in the process. 0 INTRODUCTION Speed reducers, which perform the function of speed and torque conversion, are widely utilized in modern machine applications. A common speed reducer can offer speed reduction with fixed relative position between the input shaft and output shaft. For instance, to reduce the speed between parallel shafts of bicycles, Wu and Chen proposed an eight-speed internal cylinder gear hub [1]. Instead of traditional four discs, Blagojevic et al. adopted two cycloid discs to achieve two-stage speed reduction with the advantages of compact structure, good load distribution and dynamic balance [2]. Furthermore, it is proved that efficiency is decreased with the increase of friction and contact forces [3]. By numerical simulation and experiment verification, Concli and Gorla studied the influences of parameters on the planetary speed reducer, including lubricant temperature and rotational speed [4]. Speed reduction transmission with fixed intersect shafts can be achieved with bevel gear mechanisms. Wang summarized the mathematical models of the spiral bevel gear, including the applications of thermal analysis, frictional analysis and advanced manufacturing technology [5]. For spiral bevels and hypoid gears manufactured with the duplex helical method, Zhang et al. calculated the basic machinetool settings with a new method [6]. Worm reducers or the screw gear reducer transmits two screwed vertical shafts. Dudás established the model of a new worm gear drive having point-like contact and conducted simulations to verify the better tolerance with misalignment [7]. For the case of a non-vertical intersection angle between the shafts, by defining and calculating the sliding rate, Ding et al. proposed a parameter selection criterion of the helix curve meshing wheel (HCMW) [8]. As it is transmitted between the space curves instead of space surfaces, the load capacity is limited, and the transmission ratio is unstable. However, the reducers mentioned above have no ability to adapt to variable intersection angles in the transmission process. A couple of universal joints are usually introduced, which leads to a more complicated connection relationship among the shafts and a larger size of transmission system in the axial direction. Our research team proposed a speed reducer of externally engaged gears with the RCRCR mechanism [9]. Litvin and Tan proposed a general approach in determining the singularities of the motion and displacement functions of the RCRCR mechanism [10]. Through several tests of the prototype, the phenomenon of "lock up" among the components appeared in the process of transmission, preventing it from rotating smoothly. After theoretical analyses and prototype tests, it shows that, except the factors 730 *Corr. Author's Address: Tongji University, No.4800 Cao'an highway, Shanghai, China. xinbo_chen@126.com Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 of assembly and lubrication, uneven stress plays an important role as the torque is transmitted by only one pair of eccentric shafts. In contrast, with the same transmission ratio, the speed reducer achieved by synchronous pulleys or external gear pair occupies a larger size than the internal gear pair, which obviously limits the application of the mechanism ([11] to [13]). The "few teeth difference" planetary gear mechanism, named for the small difference between the engaged gear teeth, has been widely used in mechanisms for the outstanding advantages of compact structure, large transmission ratio and high efficiency ([14] to [16]). For instance, if the teeth of the centre gear and planetary gear are 50 and 49 respectively, the transmission ratio is 50, while the eccentric distance is only m/2, in which m is the module of gear. Regarding the two problems mentioned above, a novel planetary gear reducer of internal gearing with a difference of a few teeth is proposed in this paper to achieve universal transmission and compact structure. A common application for such a universal reducer is the transmission system in the industry, as depicted in Fig. 1. In the traditional reduction configuration of Fig. 1a, two extra couplings are introduced to transmit the power, which requires high precision alignment. However, errors and tolerances caused by manufacturing, assembly, load, friction and temperature are inevitable, which requires significant efforts in installation and maintenance. In Fig. 1b, with the fault tolerance ability to compensate the relative displacement error, even for the inconsistent axes of the transmission components, there is great convenience and feasibility in the installation and operation processes. Coupling a) Load Motor Reducer r — Load 1 1 1 b) Universal reducer Fig. 1 Application of universal reducer; a) traditional reduction configuration with two extra couplings, and b) the reduction configuration with the universal reducer 1 MECHANISM THEORY The universal reducer integrates a planetary gear mechanism of internal gearing and an RCRCR spatial mechanism; the architectures are introduced in detail below. 1.1 Planetary Gear Mechanism A K-H (K-centre gear, H-planetary carrier) planetary gear is depicted in Fig. 2, which is composed of centre gear O1 with teeth z1, planetary gear O2 with teeth z2, and planetary carrier H connecting O1 and O2. According to the degree of freedom (DOF) equation, the mechanism DOF obviously is 2, which means a certain restriction is needed to guarantee a stable transmission ratio. Fig. 2. Planetary gear train Define the transmission ratio as the angular velocity ratio of the input shaft to the output shaft. Taking the planetary carrier angular velocity mH as a reference, the angular velocity transmission ratio of gear 2 to gear 1 is satisfied as Eq. (1) shows. H _ ®2 - mH (1) where mh m2 and mH are angular velocities of the centre gear, planetary gear and planetary carrier, respectively, z1 and z2 are the centre gear and planetary gear teeth respectively. The symbol "+" means that the centre gear and planetary gear rotate in the same direction (planetary carrier H acts as the reference object), while the symbol "-" means that the centre gear and planetary gear rotate in the opposite direction. This needs to be emphasized for the case of = œH . Although, according to Eq. (1): z z m, - mH z 2 A Novel Universal Reducer Integrating a Planetary Gear Mechanism with an RCRCR Spatial Mechanism 731 Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 ■H _ -mH (2) as an integral, the relative positions of the centre gear, planetary gear, and planetary carrier never change in the transmission process, as depicted in Fig. 1. Consequently, the transmission ratio is 1 for certainty as ©J = m2 = (°h . To obtain a stable transmission ratio, supposing planetary gear O2 is constrained to perform a translational motion in space by the RCRCR spatial mechanism proposed below, which means ©2 = 0, the following equations can be deduced according to Eq. (1). ■H _ ~®H = '21 œ H œ (3) (4) Consequently, taking the angular velocities of planetary carrier mH and centre gear ©J as the input and output velocities respectively, a stable transmission ratio is concluded. To obtain the transmission ratio in Eq. (4), it can also obviously be achieved by adopting two external gearing pairs with three gears at least, making one of the gears an idler gear. However, it would increase the mechanism components and make the transmission system more complicated. As a comparison, if an external gearing pair is adopted, the transmission ratio of the planetary gear would be deduced as below. (5) Comparing the results of Eqs. (4) and (5), it is proved that, with the same gear parameters, the transmission ratio of the mechanism with internal gearing is much larger than that with external gearing. 1.2 RCRCR Spatial Mechanism To achieve the translational motion of the planetary gear, the RCRCR spatial mechanism is proposed in Fig. 3. The spatial mechanism is composed of shafts "1", "2", "3", "4" and stander "0", including three revolution pairs (R) and two cylinder pairs (C). The characteristics of the RCRCR spatial mechanism are stated below in detail. The axes of the revolution pairs are denoted as "a", "d" and "f", in which, "a" and "d" are the fixed axes of the intersection shafts; the floating axes of the cylinder pairs are denoted as "b" and "c", which are parallel to the fixed axes "a" and "d" respectively. The intersection distance d0 between the connecting shafts "a" and "d" is equal to the distance "d1" between the connecting shafts "b" and "c". The distance r1 between "a" and "b" is equal to the distance rh between "c" and "d". The intersection angle a14 between "a" and "d" equals a23 between "b" and "c". a Fig. 3. RCRCR spatial mechanism The degree of freedom (DOF) of the RCRCR spatial mechanism is calculated below. F = Z f-l = Y kPk (6) in which £f is the DOF of the open chain mechanism of the space-connecting shaft, which is 7, supposing the mechanism is disconnected with the stander. X is the DOF of the end link of the open chain, which is 6. As a result, the RCRCR mechanism is of 1 DOF, which proves the transmission feasibility. F = £ f-1= 1. 2 KINEMATIC ANALYSES (7) Kinematic analysis is carried out to prove the translational motion of the connecting shafts in the RCRCR spatial mechanism. For the convenience of analysis, a right-hand rectangular coordinate system is constructed on each component depicted in Fig. 4. Z-axis is along the axis of each kinematic pair, and X-axis is along the common perpendicular of two axes of the adjacent kinematic pairs. hj is the shortest distance between z, and Zj along the axis x;-. s}- is the distance between x, and x;- along the axis z,-. a,j is the angle between z,- and Zj along the axis x;-. 6j is the rotating angle of the corresponding component i. A direction cosine matrix is employed to express the exchanging relationships between coordinate b 2 z d k=1 z 732 Gu, C. - Chen, X. Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 system x, (y) zt and x}- (y) Zj, which can be written as the equation (8). Fig. 4. RCRCR spatial mechanism C ] = cos d j - sin dj cos aj sin 6j sin aj sin d j cos d j cos a s - cos dj sin ajj 0 sina- cosa„ (8) Along axes z2 and z3, the closed loop can be separated into two parts: fixed chain 3-4-0-1-2 and floating chain 2-3. According to the geometric equivalence property of the kinematics, the direction cosine of the axes z2 and z3 in fixed chain 3-4-0-12 equals to that in floating chain 2-3, which can be defined by the following equation. cos( z2, z 3 )3_4_0-i-2 = cos( z2, Z3 )2_3 = cos a23, (9) in which, cos( z2, z 3W,_2 = [0,0,l][C34][C40][CJ[CI2][0,0lf. (10) Through calculation, Eq. (5) can be simplified as cos(01 +e2) = 0. (11) Thus, e2 = 270° -e1. (12) According to the equal geometry principle, the vector projection of the closed mechanism to coordinate axis x4 is zero. Then the following equation is satisfied. h cos(xj, x4) - sj cos(zj, x4) - s2 cos(z2, x4) + h4 - -h0cos( x0, x4) + s0cos( z0, x4) = 0, (13) in which, cos( x,, x4 ) = [1,0,0][C12][C23 ][CM ][1,0,0]r cos( z,, X4 ) = [1,0,0][Ci2 ][C23 ][C34 ][0,0,1f cos(Z2,X4) = [1,0,0][C23][C34P,0,1f . (14) cos( X0 , X4 ) = [1,0,0][C01 ][C12 ][C23 ][C34 ][1,0,0]r cos( Z0 , X4 ) = [1,0,0][C01 ][C12 ][C23 ][C34 ][0,0,1]r It is concluded that, Sj = s0 -h(1 + cosa4O)sm0j /sina40. (15) In a similar way, other motion parameters can be concluded as follows. e0 =0j -180° ,e0 =el 02 = -0i e4 = 90°-0P04 = -^1 i1 = s0 - h1(1 + cos a40 )sin 01 / sin a40 s1 = s4 - h (1 + cos a40 )sin 01 / sin a40 (16) 03 = 180 +«40 Assuming the angular velocity of connecting shaft 2 is -01 in relative to input shaft 1, and the angular velocity of input shaft 1 is in relative to the stander, the angular velocity of connecting shaft 2 is 0 in relative to stander, which means the connecting shaft only translates in space. In the same way, the angular velocity of connecting shaft 3 is 04 = — in relative to input shaft 4, and the angular velocity of input shaft 4 is set as 90 (relative to the stander). Thus, relative to the stander, connecting shaft 3 only translates in space. However, the motion features of the spatial mechanism are never related to intersection angle a40. Therefore, the transmission relationship will not change, even though the intersection angle a40 changes in the process. Consequently, translational motions of the connecting shafts "2" and "3" in space are achieved with the RCRCR mechanism. Fixing the planetary gear O2 in Fig. 1 and the connecting shaft "3" in Fig. 2 together, the planetary gear O2 would translate together with the connecting shaft "3", instead of revolving around axis "c". Furthermore, connecting the centre gear with the stander along the axis "d", the angular velocity relationship between the center gear and the carrier is determined by Eq. (4). As mH = ma, m2 = 0, the transmission ratio in Eq. (4) is concluded below. m„ m, ma m (17) z z A Novel Universal Reducer Integrating a Planetary Gear Mechanism with an RCRCR Spatial Mechanism 733 Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 From the equation above, the transmission ratio of the reducer would never vary with the intersection angle and the distance between the shafts "1" and "2" in the process of transmission. Thus, it is a universal speed reducer maintaining a stable transmission ratio, which is adaptive to different intersect angles. If intersect distance d0 = d1 = 0, then the transmission shafts axes "a" and "d" intersect with each other; and if distance d0 = d1 ^ 0, then shafts axes "a" and "d" are the spatial crossing. In conclusion, the RCRCR mechanism shown in Fig. 3 is a 1-DOF spatial mechanism adaptive to variable intersection angles and distances, achieving the required translational motion of the planetary gear in the universal reducer. In addition, the motion pairs in the mechanism are all low pairs, which is beneficial to improve wear-resisting performance. 3 SIMULATION RESULTS AND DISCUSSION To prove the function of the speed reduction transmission of the mechanism, a simulation model of the mechanism is established in Automatic Dynamic Analysis of Mechanical Systems (ADAMS) software. The simulation model is depicted in Fig. 5 in detail. To show the structure inside the speed reducer clearly, the gearbox is shown as partially transparent. Taking Figs. 3 and 4 as references, the input shaft "a" is designed with a flange, while the output shaft, which is in alignment with "d", is designed fixing with the centre gear. For the convenience of manufacturing and testing, let the intersection distance d0 = d1 = 0. The centre pair of connecting shafts corresponds to shafts "b" and "c" in Fig. 3. The offset of the input shaft "a" and the centre connecting shaft "b" is equal to the offset of the output shaft "d" and the centre connecting shaft "c", which is the rotational radius of the planetary carrier. Since the offset is so small while a large driving torque is needed, three additional connecting shafts are introduced, distributed on a larger circle equally to help the connecting shaft revolve and thus to ensure the reducer work smoothly. Furthermore, it is beneficial to improve supporting capacity and avoid the phenomenon of "lock up" mentioned above as they distribute the load more evenly. The gearbox not only performs the function of a hermetically sealed chamber but also that of a transmission component. The necessary parameters of the simulation model are listed in Table 1. Table 1. Universal reducer parameters Parameters Values Planetary gear teeth 29 Center gear teeth 34 Module of gear [mm] 2 Eccentric offset [mm] 5 Transmission ratio 6.8 Figs. 6 and 7 illustrate the angular velocities of the input shaft, output shaft, and planetary gear of the speed reduction mechanism with the intersect angles a = 135° and a = 160°, respectively, simulating with the same parameters, including simulation period and input velocity. Comparing the simulation results in Figs. 6 and 7, we can prove the translational motion of the Fig. 5. Simulation model of the mechanism 734 Gu, C. - Chen, X. Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 100.0 80.0 60.0 40.0 20.0 0.0 -I ^^»input angular velocity — —output angular velocity ■ ■ ■ ■ planet gear angular velocity \ \ / \ / \ / \ / \ / \ * m ^ ~ — I 4 'Mf' \ 0.0 1.0 2.0 3.0 4.0 Time [s] Fig. 6. Angular velocities of the input and output shafts with a=135° Fig. 7. Angular velocities of the input and output shafts with a=160° "a. 1000.0 <2J t 500 0 < - 1 ^^Hnput angular velosity — — Output angular velosity ■ ■■■Planet gear angular velosity / x i / Is 1.0 2.0 3.0 4.0 Time [s] Fig. 8. Step torque response of the universal reducer connecting shafts in the RCRCR mechanism by testing its angular velocities. It is concluded that the mechanism, which maintains the stable transmission ratio of 6.8, can be adaptive to different intersect angles, even the intersection angle changes in the process of transmission, which means the universal reducer presents brilliant robustness with the position perturbations of input and output shafts. It is also a significant advantage in the applications of the variable intersection angle between the input and 735 A Novel Universal Reducer Integrating a Planetary Gear Mechanism with an RCRCR Spatial Mechanism 39 Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 output shafts during transmission, which is common in transmission close to wheel for electric vehicles. To suppress the negative effects of a heavy unsprung mass in electric vehicles, a common practice is to arrange the motor on the vehicle framework as the sprung mass. Consequently, the universal reducer is introduced to transmit the power between two floating components perfectly without a constant velocity universal joint. The torque fluctuation is also an important performance aspect for a speed reducer. A step torque response simulation is carried out, in which the initial input torque is 100 Nm and changes to zero at the time of 1 second; the whole simulation lasts 5 seconds. From the angular velocity results shown in Fig. 8, the output angular velocity is much smoother than the input angular velocity, while the transmission ratio remains stable. It is concluded that the universal speed reducer with the few teeth difference planetary gear has an excellent performance of torque fluctuation tolerance. Note that the working condition of the reducer is complex, including the varying loads and misalignment of each element, especially the connecting shafts and cylinders. All the above may influence the transmission smoothness and reliability, so it is important to emphasize manufacture and assembly precision. 4 PROTOTYPE DESIGN 4.1 Structure Design To test the speed reducer performances with the few teeth difference planetary gear mechanism, the detailed design is carried out. The parameters of the universal reducer are the same as those used in the simulation shown in Table 1. The universal reducer prototype is designed in Fig. 9, and the components are listed in Table 2. The oil-less bearing is adopted for a compact structure. The reference diameters of the centre gear and planetary gear are 68 mm and 58 mm respectively, while the distribution diameter of the three additional connecting shafts is 52 mm. The offset of the input shaft and the centre connecting shaft is equal to the offset of the output shaft and the centre connecting shaft, which are both 5 mm. The diameters of the input shaft and output shaft are both 25 mm. The radius of the centre connecting shaft and the three additional connecting shafts are 15 mm and 10 mm, respectively. The gearbox radius is 53 mm, and the overall thickness is 84 mm. In conclusion, the designed universal reducer has the advantage of a compact structure. Table 2. Prototype components No. Name No. Name 1 Input shaft 13 Planetary gear 2 Elastic collar 14 Reducer shell 3 Deep groove ball bearing 15 Center gear 4 Bush 16 Deep groove ball bearing 5 Oil-less bearing 17 Elastic collar 6 Oil-less bearing 18 Deep groove ball bearing 7 Connecting shaft 19 Bush 8 Connecting shaft 20 Bush 9 Bolt 21 Bolt 10 Connecting shaft 22 Rubber gasket 11 Planetary carrier 23 Socket head cap screw 12 Reducer shell 736 Strojniski vestnik - Journal of Mechanical Engineering 62(2016)12, 730-739 4.2 Strength Analysis A torque Min is applied on the input shaft, while the load on the output shaft is Mout. With the transmission ratio i, Eq. (18) is concluded. Mout = i ■ Min (18) Firstly, taking the connecting shafts (marked as components 2 and 3) as a whole object, the forces and torques applied to the object are depicted in Fig. 2. The distance between B and E is L1, the same as the distance between C and E. The angle between the connecting shafts 2 and 3 is 2a, and the distance between the input shaft and the connecting shaft is r, which equals to the distance between the output shaft and the connecting shaft (component 3). Fig. 10. Dynamic analysis of component 2 and 3 In the rotating process, when the connecting shafts are located at the highest position, with the involvements of FB and FC on the shafts 2 and 3, to keep the translational motion of the connecting shafts, a torque of ME is generated on point E. Me = 2FBL sin a, (19) in which, Fb = Min / r . A torque of MC1 is applied on point C under the tangential force of the planetary gear. MC1 = P,r2 Ft = 2M0Ut / (mzi (20) in which, Ft is the tangential force, m is the module of the internal gear and z1 is the teeth of the internal gear. According to the kinematic analysis, to guarantee the translational motion of the connecting shafts, the torques should be kept balanced as follows. IMB sina + MC sina -MC1 cosa -ME = 0 [Mb cosa -MC cosa -MC1 sina = 0 (21) The torques MB and MC can be concluded according to the equations above. Clearly, the critical position of connecting shaft 3 locates C point. The connecting shaft (component 3 in Fig. 10) is torqued by MC1 and bent by MC simultaneously. 45# steel with the allowable bending stress of 60 MPa is preferred. Based on the third strength theory, the equation below should be satisfied.