 Proceedings of Engineering and Technology Innovation, vol. 11, 2019, pp. 46-50 The Design and Prototype Manufacture of a Planetary Gear Reducer Tzu-Hsia Chen 1,* , Long-Chang Hsieh 2 1 Department of Mechanical Engineering, Minghsin University of Science and Technology, Taiwan, ROC 2 Department of Power Mechanical Engineering, National Formosa University, Yunlin, Taiwan, ROC Received 02 May 2018; received in revised form 28 May 2018; accepted 05 August 2018 Abstract The gear reducer of the power system can be ordinary spur gear reducer, worm and worm gear reducer, or planetary gear reducer. Due to the compact size, light weight, and multi-degrees of freedom; planetary gear reducers are commonly used in various transmissions. One of its applications is used as the gear reducer for industrial purpose. The reduction ratios of 1-stage planetary gear reducers are limited to 3 ~ 10. This paper focused on the design and prototype manufacture of a 1-stage planetary gear reducer. First, according to the concept of train value equation, the planetary gear reducer with reduction ratio 4 is proposed. Based on the involute theorem, the gear data of the planetary gear reducer are obtained. Finally, based on the results of kinematic design and meshing efficiency analysis, the integrated design of the planetary gear reducer was carried out and the prototype was manufactured to verify the design theorem. The results of this paper can be used as a reference for engineers to design the gear reducers for industrial purpose. Keywords: engineering design, planetary gear reducer, prototype manufacture, train value equation 1. Introduction Traditionally, the reduction ratio of an ordinary spurs gear pair, worm and worm gear pair, and 1-stage planetary gear train is limited to 4~7, 10~100, and 3 ~ 10, respectively. Figs. 1 (a)-(c) show the ordinary spur gear reducer, worm and worm gear reducer, and 1-stage planetary gear reducer, respectively. Due to the reason of the compact size, light weight, and multi-degrees of freedom; planetary gear reducers are commonly used in various transmissions. One of its applications is used as the gear reducer for industrial purpose. (a) Ordinary spur gear reducer (b) Worm and worm gear reducer (c) 1-stage planetary gear reducer Fig. 1 Gear reducers In the past, planetary gear trains were the subjects of intensive researches directed at kinematic analysis [1-5], kinematic design [6-13], and meshing efficiency analysis [14-19]. The purpose of this paper is to propose a planetary gear reducer with reduction ratio Rr=4 for industry purpose. The results of this paper can be used as a reference for engineers to design all kinds of planetary gear reducers. The research result can also provide the application experiences for the transmission design practice course. * Corresponding author. E-mail address: summer34134@gmail.com Tel.: +886-3-559-3142 ext. 3023; Fax: +886-3-557-3797 Proceedings of Engineering and Technology Innovation, vol. 11, 2019, pp. 46-50 47 2. Train Value Equation For a planetary gear train, shown in Fig. 2, we denoted the first sun gear as i, the last sun gear as j, and the carrier (arm) as k, respectively. The train circuit can be denoted as (i, j; k); the relationship among ωi, ωj, and ωk can be expressed as: Fig. 2 Planetary gear train 0)1(  kjijjii  (1) Here ξji is the train value of sun gear j to sun gear i. The sum of the coefficients of Eq. (1) is equal to 0, i.e., 0)1(1  jiji  . For a planetary gear train with the single train circuit, there are three members (i, j, and k) can be adjacent to input shaft, output shaft, and frame. If sun gear j is adjacent to the frame (ωj=0), and sun gear i is adjacent to the input shaft (ωi=ωin), and carrier k is adjacent to the output shaft (ωk=ωout), according to Eq. (1), its reduction ratio (Rr) can be written as:  1 ( 1) 1 jiin i r ji out k R             (2) For a planetary gear train with ξji < 0, the reduction ratio Rr > 1. For the planetary gear reducer shown in Fig. 3, let Z2=24, Z5=24, and Z3=72, then there is the train value ξ32=-3. Since the sun gear 3 is adjacent to the frame (ω3=0), and the sun gear 2 is adjacent to the input shaft (ω2=ωin); the carrier 4 is adjacent to the output shaft (ω4=ωout), according to Eq. (2), its reduction ratio Rr = 4. Fig. 3 Planetary gear reducer with reduction ratio Rr = 4 3. Latent Power Theorem and Meshing Efficiency For a planetary gear train with sun gear i, ring gear j, and carrier k. Ti, Tj and Tk are the torques of members i, j and k, respectively. Since they rotate around the same axis, and according to the torque balance, we have: 0 kji TTT (3) Furthermore, the latent powers of sun gear i and sun gear j relative to carrier k ( 𝑃𝑖 𝑘 and 𝑃𝑗 𝑘 )are defined as :   kii k i TP   (4)   kjj k j TP   (5) 𝜂𝑖𝑗 𝑘 ( 𝜂𝑗𝑖 𝑘 ) is the meshing efficiency of planetary gear train; if the latent power is transmitted from sun gear i (sun gear j) to sun gear j (sun gear i), when carrier k is relatively fixed. Based on the concept of latent power [14-19], the relationship between k i P and k j P can be expressed as: Proceedings of Engineering and Technology Innovation, vol. 11, 2019, pp. 46-50 48 0P and 0 Pfor PP k j k i k ij k i k j   (6) 0P and 0 Pfor PP k j k i k ji k j k i   (7) The meshing efficiency of the planetary gear train is defined as the ratio of output power to input power, i.e., ( ) out m PGT in P P    (8) According to Eqs. (3) ~ (8), the meshing efficiency of planetary gear reducer can be expressed as follows:     ( ) 1 1 k ji ij m PGT ji        (9) where 𝜂𝑖𝑗 𝑘 ( 𝜂𝑗𝑖 𝑘 ) is the meshing efficiency of planetary gear train; if the latent power is transmitted from sun gear i (sun gear j) to sun gear j (sun gear i) when carrier k is relatively fixed. The meshing efficiency of ordinary external gear pair (m(ex)) and internal gear pair(m(in)) can be expressed as [20-22]: 2 2 ( ) 1 1 1 1 1 1 / 1 [ ] 2 2 ar r m ex A R A R fR f           (10) 2 2 ( ) 1 1 1 1 1 1 / 1 [ ] 2 2 ar r m in A R A R fR f           (11) where Rr is the reduction ratio of gear pair, βA1 and βA2 are the angles of approach of driving and driven gears, fa and fr are the average friction coefficient of approach and recess. For external (internal) gear pair, the meshing efficiencies are between 99.2% ~ 99.6% (99.7% ~ 99.9%). For the planetary gear reducer in Fig. 3, if the meshing efficiencies of external and internal gear pairs are 0.994 and 0.998, then the meshing efficiency of 𝜂𝑖𝑗 𝑘 is 0.994 0.998 0.992 k ij     . Then, according to Eq. (9), the meshing efficiency of the planetary gear reducer is 99.4%. Based on the above results, we concluded that the meshing efficiencies of planetary gear reducers are very good. 4. Engineering Design and Prototype In the following paragraph, the planetary gear reducer, shown in Fig. 3, is the example to illustrate how to design the planetary gear reducer. Table 1 shows some important gear data of this feasible design example. Table 1 Gear data of the planetary gear reducer shown in Fig. 3 External gear pair (2, 5) Internal gear pair (5, 3) Sun gear 2 Planet gear 5 Ring gear 3 Teeth Number 24 24 72 Normal module (mm) 1.25 1.25 1.25 Helical Angle 0⁰ 0⁰ 0⁰ Shift Coefficient 0 0 0.3 Pitch Dia (mm) 30 30 90 Addendum Dia. (mm) 32.5 32.5 88.25 Base Dia. (mm) 28.19078 28.19078 84.57234 Center Distance (mm) 30.2 30.2 Pressure Angle (deg.) 21D 1M 4S 21D 1M 4S Contact Ratio (Approach/ Recess) 1.44721 (0.72361 / 0.72361) 1.71049 (0.98689 / 0.72361) According to the gear data in Table 1, the engineering design drawings shown in Figs. 4(a), 4(b), 4(c),and 4(d) are sun gear 2 (Z2=24), planet gear 5 (Z5=24), ring gear 3 (Z3=72), and planetary gear set of planetary gear reducer. Figs. 4(e) and 4(f) show its corresponding 1/2 Section view and isometric view. Proceedings of Engineering and Technology Innovation, vol. 11, 2019, pp. 46-50 49 (a) Sun gear (Z2=24) (b) planet gear (Z5=24) (c) rnig gear (Z3=72) (d) planetary gear set (e) sectional view (f) isogonal view Fig. 4 Engineering drawing of planetary gear reducer Figs. 5(a), 5(b), 5(c), and 5(d) show the corresponding prototype photo of sun gear 2 (Z2=24), planet gear 5 (Z5=24), ring gear 3 (Z3=72), and planetary gear set of planetary gear reducer. Figs. 5(e) and 5(f) show its corresponding prototype photo of 1/2 Section view and isometric view. (a) Sun gear (Z2=24) (b) Planet gear (Z5=24) (c) Rnig gear (Z3=72) (d) Planetary gear set (e) Sectional view (f) Isogonal view Fig. 5 Prototype photos of planetary gear reducer 5. Conclusions The reduction ratios of 1-stage planetary gear reducers are limited to 3 ~ 10. This paper focuses on the integrated design and prototype manufacture of the planetary gear reducer (with reduction ratio Rr=4) for industry purpose. According to the kinematic design, a planetary gear reducer with Rr=4 is synthesized. According to the latent power theorem, its corresponding meshing efficiency is calculated to be 99. 4%. Finally, the engineering design of the planetary gear reducer is accomplished and its corresponding prototype is manufactured to verify the design theorem. 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