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Multi-Objective Optimization Design of Cycloid Pin Gear Planetary Reducer

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Multi-Objective Optimization Design of Cycloid Pin Gear Planetary Reducer Research Article Advances in Mechanical Engineering 2017, Vol. 9(9) 1–10 Ó The Author(s) 2017 Multi-objective optimization design of DOI: 10.1177/1687814017720053 cycloid pin gear planetary reducer journals.sagepub.com/home/ade Yaliang Wang, Qijing Qian, Guoda Chen, Shousong Jin and Yong Chen Abstract A multi-objective optimal model of a K-H-V cycloid pin gear planetary reducer is presented in this article. The optimal model is established by taking the objective functions of the reducer volume, the force of the turning arm bearing, and the maximum bending stress of the pin. The optimization aims to decrease these objectives and obtains a set of Pareto optimal solutions. In order to improve the spread of the Pareto front, the density estimation metric (crowding distance) of non-dominated sorting genetic algorithm II is replaced by the k nearest neighbor distance. Then, the improved algo- rithm is used to solve this optimal model. The results indicate that the modified algorithm can obtain the better Pareto optimal solutions than the solution by the routine design. Keywords Cycloid speed reducer, planetary transmission, multi-objective optimization, evolutionary algorithm, density estimation Date received: 11 January 2017; accepted: 15 June 2017 Academic Editor: Ismet Baran Introduction between cycloid disk and housing rollers affected the contact force, friction torque, and transmission effi- Cycloid speed reducers have some excellent characteris- ciency. Hsieh6 proved the nonpinwheel design of tics, such as compact structure, wide scope of ratios, cycloid speed reducer could effectively reduce vibra- high transmission efficiency, low noise, and smooth- tion, stress value, and stress fluctuation. ness. Thus, they are widely used in all kinds of mechan- In the engineering design, many multi-objective opti- ical transmission. However, many design parameters mization problems (MOOPs) can be found. Some or all and complex constraints in the reducer design bring objectives often conflict with each other; in other more difficulty. At present, a lot of researches are avail- 1 words, all objectives cannot achieve its own best value able in this area. Chen et al. established the equation simultaneously.7 Thus, the policymakers need to choose of meshing for small teeth difference planetary gearing the compromise design parameters according to the and a universal equation of cycloid gear tooth profile based on cylindrical pin tooth and given motion. He et al.2 carried out optimum design and experiment on Key Laboratory of Special Purpose Equipment and Advanced Processing the double crank ring-plate-type pin-cycloid planetary Technology, Ministry of Education, Zhejiang University of Technology, 3 drive to reduce its noise and vibration. Sensinger pre- Hangzhou, China sented a unified design method to optimize cycloidal drive profile, efficiency, and stress. Blagojevic et al.4 Corresponding author: Shousong Jin, Key Laboratory of Special Purpose Equipment and introduced a new two-stage cycloid speed reducer, Advanced Processing Technology, Ministry of Education, Zhejiang which was characterized by good load distribution and University of Technology, 18 Chaowang Road, Hangzhou 310014, China. 5 dynamic balance. Blagojevic et al. found the friction Email: [email protected] Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 Advances in Mechanical Engineering reality. For the last decade and more, many classical 2. Pin gear (also called pinwheel). It is uniformly multi-objective evolutionary algorithms (MOEAs) were distributed over the circumferential direction. studied, such as the non-dominated sorting genetic The pin gear consists of the pin and pin sleeve. algorithm II (NSGA-II),8 the improved strength Pareto 3. Cycloid gear. To keep the static balance of the evolutionary algorithm (SPEA2),9 and the multi- input shaft and increase the loading capacity of objective particle swarm optimization (MOPSO).10 As the reducer, two identical cycloid gear structures an efficient algorithm, NSGA-II has been widely used are usually adopted. The cycloid gears are to solve MOOPs.11–13 Some researchers used NSGA-II installed on the dual eccentric sleeves, and their to optimize gear reducers. Deb and Jain14 proved position differs 180°. In order to reduce the fric- MOEAs can solve multi-speed gearbox design optimi- tion between eccentric sleeve and cycloid gear, zation problem with more than one optimal goal and the turning arm bearing is installed between the different kinds of design parameters. Tripathi and two parts. Chauhan15 applied NSGA-II to optimize the volume of 4. Output mechanism. This reducer often uses the gearbox and the surface fatigue life factor simultane- output mechanism that called pin axle type. ously. Sanghvi et al.16 used NSGA-II to solve optimiza- There are some cylindrical pin holes on the tion design problem of a two-stage helical gear train, in cycloid gear for inserting the cylindrical pins which the bearing force and volume were optimized. (the cylindrical pin sleeve is mounted on the Three optimization methods (MATLAB optimum cylindrical pin). Thus, the rotation motion of tools, genetic algorithm, and NSGA-II) were com- cycloid gears can be output by cylindrical pins. pared. It was shown that NSGA-II obtained better optimization objective values than other methods. Li et al.17 combined NSGA-II and fuzzy set theory to Theoretical tooth profile of cycloid gear optimize dynamic model of steering mechanism and got As shown in Figure 2, rolling circle 2 is the circum- the better result than the original design. However, far scribed circle of base circle 1, and the center of base cir- too little attention has been paid to the research about cle 1 (Og) is also the origin of rectangular coordinate multi-objective optimal design of cycloid speed redu- system. According to the forming principle of cycloid cers. Yu et al.18 and Yu and Xu19 selected volume as gear profile, supposing the base circle 1 is fixed, the the optimization objective. Xi et al.20 selected volume locus of point M in circle 2 is a epicycloid when circle 2 and efficiency as optimization objectives, and the multi- scrolls from tangent point A to tangent point B. The objective optimization model was solved by transform- rotation angle of the circle 2 (with the center O) around ing it into a single-objective problem, which ignored the the circle 1 (with the center Og) is denoted by u. The 21 relationship between objectives. Wang et al. focused rotation angle of circle 2 is denoted by ub and the abso- on reducing the volume and improving the efficiency of lute angle of circle 2 is denoted by uh. The coordinates reducers, and the optimal model was solved by the of a point on the theoretic profile of cycloidal gear can MATLAB genetic algorithm toolbox. be expressed as In this article, a multi-objective optimization model of reducer is established to minimize the volume of the x = R sin u À esinu 0 z h ð1Þ reducer, force of the turning arm bearing, and maximal y0 = Rz cos u À ecosuh bending stress of the pin. It is expected to provide a new way for multi-objective optimization design of cycloid reducer and similar structural optimization problems. Moreover, in order to improve the spread of the Pareto front, the density estimation metric of NSGA-II is replaced by the k nearest neighbor distance. Optimization design model of the K-H-V cycloid driver Figure 1 shows the typical structure of the cycloidal pinwheel transmission, which mainly consists of the fol- lowing parts: 1. Planet carrier. It is composed of two parts, namely, the input shaft and the dual eccentric sleeves. Figure 1. Typical structure of cycloidal pinwheel transmission. Wang et al. 3 Figure 2. Theoretical tooth profile of cycloid gear. Figure 3. Meshing force of pin gear and cycloid gear. Fi is proportional to component of Du in the direction where Rz is the radius of the pin gear distributed circle, of Fi. That is e is the eccentric distance (e = ObOg =(K1Rz)=Zb), K1 is the short amplitude coefficient (K1 =(rb=Rz), rb is the li pitch radius of the pinwheel), and Zb is the tooth num- Fi}Ducosai = Du ð4Þ RZ ber of the pin gears (Zb =(uh=u)). Therefore, equation (1) can be expressed as where li is the vertical distance between Ob and the 9 direction of Fi. When li is equal to rb, and Fi is equal to K1 = x0 = Rz sin u À Z sin Zbu Fmax. That is b ð2Þ K1 ; y0 = Rz cos u À Z cos Zbu rb b Fmax } Du ð5Þ Rz According to the radius of curvature formula, the radius of theoretical tooth profile of cycloid gear can In the triangle ObBP, sin ui =(li=rb). According to be expressed as equations (4) and (5), Fi can be expressed as 3 li 2 2 (1 + K À 2K cosu ) R Fi = Fmax = Fmax sin ui ð6Þ 1 1 b Z r r0 = 2 ð3Þ b K1(1 + Zb) cos ub À (1 + ZbK1 ) The maximum engaging force22 is shown as Meshing force of pin gear and cycloid gear 4Tg Fmax = ð7Þ K1ZgRz As shown in Figure 3, the angular velocity (vg)of cycloid gear has the opposite direction on output tor- where Zg is the tooth number of the cycloid gear. In que when pin gears are fixed. In the switching mechan- general, Tg = 0:55Tv, and Tv is the output torque. ism, the rotation direction of cycloid gear and pinwheel Therefore, Fmax can be expressed as are same.
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