On the Meshing Efficiency of 2K-2H Type Planetary Gear Reducer

Abstract

This paper proposes 2K–2H type planetary gear reducer and analyzes its meshing efficiency. First, according to the concept of train value equation, the kinematic design of 2K–2H type planetary gear reducers is carried out. Three 2K–2H type planetary gear reducers are designed to illustrate the design algorithm. Then, based on the latent power theorem, the meshing efficiency equation of 2K–2H type planetary gear reducer is derived. According to the meshing efficiency equation, the meshing efficiencies of 2K–2H type planetary gear reducers are analyzed. The 2K–2H type planetary gear reducer has the following characteristics. (1) There is a power circulation in 2K–2H type planetary gear reducer. (2) Larger reduction ratio makes less meshing efficiency. (3) For the same reduction ratio, larger value $\left({\xi }_{42}\right)\hspace{0.17em}(\left({\xi }_{\alpha }\right))$ will get better meshing efficiency. (4) For 2K–2H type planetary gear reducer, the quality of gears is an important factor. (5) The efficiency of gears manufactured by grinding is only improved by 1.5%; however, meshing efficiency of 2K–2H type planetary gear reducer is improved by 28%~44.8%.

1. Introduction

Planetary gear trains can be designed to have high reduction ratios and can be used as the gear reducers for power machinery. However, the planetary gear train with high reduction ratio is a coupled planetary gear train with compound gear system.

Planetary gear trains were the subject of intensive research directed at kinematic analysis [1–9], kinematic design [10–14], efficiency analysis [15–24], and patents [25–30]. Few studies focused on the efficiency analysis of 2K–2H type planetary gear trains. The 2K–2H type planetary gear train has a problem of “power circulation” that makes the meshing efficiency not good. The purpose of this paper is to design 2K–2H type planetary gear reducer with better meshing efficiency.

By referring to the studies of kinematic analysis and design of planetary gear trains, this paper proposes 2K–2H type planetary gear trains. Three 2K–2H type planetary gear trains are proposed for the design of gear reducers with high reduction ratios (such as 50, 100, and 99). Based on the latent power theorem, the equation of meshing efficiency of 2K–2H type planetary gear reducer is derived. According to the efficiency equation, the 2K–2H type planetary gear reducers with better meshing efficiencies can be synthesized. Some 2K–2H type planetary gear reducers are designed and their meshing efficiencies are analyzed to illustrate the results. Based on the conclusions of this paper, 2K–2H type planetary gear reducer with better meshing efficiency can be synthesized.

2. 2K–2H Type Planetary Gear Train

Since the reduction ratio of noncoupled planetary gear trains is less than 10 (R _r < 10), 2K–2H type planetary gear train is proposed to have high reduction ratio. 2K–2H planetary gear train is a planetary gear train with two sun gears (including ring gears) and two carriers (planet arms). Figures 1(a)–1(f) show the six design concepts of 2K–2H type planetary gear trains.

Figure 1:

Six design concepts of 2K–2H type planetary gear trains.

According to Figure 1(c), in 2012, Hsieh and Chen [29] proposed a 2K–2H-type planetary gear reducer (as shown in Figure 2(a)) to have high reduction ratio and got the Taiwan patent right (M428280). According to Figure 2(a), in 2013, Hsieh and Tang [14] designed 2K–2H type planetary gear trains with reduction ratios 29, 34, and 726, respectively (shown in Figures 2(b)–2(d)).

Figure 2:

2K–2H type planetary gear reducer (Taiwan patent number M428280) [29].

3. Kinematic Design

3.1. Train Value Equation

For a planetary gear train, let us denote the first sun gear as i, the last sun gear as j, and the carrier (arm) as k, respectively. The train circuit contains sun gear i and sun gear j, and carrier k can be denoted as (i, j; k). The relationship among ω _i , ω _j , and ω _k can be expressed as

${\omega }_i-{\xi }_{ji}{\omega }_j+\left({\xi }_{ji}-1\right){\omega }_k=0.$

(1)

The 2K–2H type planetary gear reducer shown in Figure 2 has two sun gears and two carriers; it is a planetary gear train with two train circuits. The 2K–2H type planetary gear reducer is coupled with two train circuits. According to the research of Hsieh et al. [11], the coupling relationship (Figure 2(a)) can be expressed as in Figure 3.

Figure 3:

The coupling relationship of 2K–2H type planetary gear train.

Let (2, 4; 5) and (2, 4; 7) be the 1st and 2nd train circuits of the planetary gear train, respectively, and ξ₄₂ (ξ_4′2′) the train value of ring gear 4 (4′) to sun gear 2 (2′). According to (1), the two train value equations can be rewritten as

$\begin{array}{c}{\omega }_2-{\xi }_{42}{\omega }_4+\left({\xi }_{42}-1\right){\omega }_5=0,\\ {\omega }_2-{\xi }_{4^{\prime }{2}^{\prime }}{\omega }_4+\left({\xi }_{4^{\prime }{2}^{\prime }}-1\right){\omega }_7=0.\end{array}$

(2)

Based on (2), by eliminating the variable ω₄, we get

$\left({\xi }_{4^{\prime }{2}^{\prime }}-{\xi }_{42}\right){\omega }_2+{\xi }_{4^{\prime }{2}^{\prime }}\left({\xi }_{42}-1\right){\omega }_5-{\xi }_{42}\left({\xi }_{4^{\prime }{2}^{\prime }}-1\right){\omega }_7=0.$

(3)

Sun gear 2 is adjacent to input shaft (ω₂ = ω_in); ring gear 4 is free link, carrier 5 is fixed (ω₅ = 0); carrier 7 is adjacent to the output shaft (ω₇ = ω_out). For the 2K–2H type planetary gear reducer, the reduction ratio (R _r ) can be written as

$R_r=\frac{{\omega }_{\text{in}}}{{\omega }_{\text{out}}}=\left(-\right)\frac{-{\xi }_{\alpha }\left({\xi }_{\beta }-1\right)}{\left({\xi }_{\beta }-{\xi }_{\alpha }\right)}=\frac{{\xi }_{42}\left({\xi }_{4^{\prime }{2}^{\prime }}-1\right)}{\left({\xi }_{4^{\prime }{2}^{\prime }}-{\xi }_{42}\right)}.$

(4)

If Z₂ = 12, Z₃ = 38, Z₄ = 88, Z_4′ = 87, and Z₇ = 37, then ξ₄₂ = – 7.333, and ξ_4′2′ = – 7.25. According to (4), the 2K–2H type planetary gear reducer shown in Figure 2 (d) has ξ₄₂ = – 7.333 and ξ_4′2′ = – 7.2; its reduction ratio is R _r = 726.

3.2. Examples

1. If Z₂ = 26, Z_2′′ = 28, Z₃ = 21, Z₄ = 72, and Z₆ = 22, then ξ₄₂ = – 2.7692, and ξ_4′2′ = – 2.5714. According to (4), the 2K–2H type planetary gear reducer shown in Figure 4 (a) has ξ₄₂ = – 2.7692 and ξ_4′2′ = – 2.5714; its reduction ratio is R _r = 50.

2. If Z₂ = 27, Z_2′′ = 28, Z₃ = 22, Z₄ = 72, and Z₆ = 22, then ξ₄₂ = – 2.6667 and ξ_4′2′ = – 2.5714. According to (4), the 2K–2H type planetary gear reducer shown in Figure 4 (b) has ξ₄₂ = – 2.6667 and ξ_4′2′ = – 2.5714; its reduction ratio is R _r = 100.

3. If Z₂ = 18, Z₃ = 24, Z₄ = 66, Z_4′ = 63, and Z₆ = 21, then ξ₄₂ = – 3.6667 and ξ_4′2′ = – 3.5. According to (4), the 2K–2H type planetary gear reducer shown in Figure 4 (c) has ξ₄₂ = – 3.6667 and ξ_4′2′ = – 3.5; its reduction ratio is R _r = 99.

Figure 4:

2K–2H type planetary simple gear reducer.

4. Meshing Efficiency Analysis

4.1. Latent Power Theorem

For a planetary gear train with sun gear i, ring gear j, and carrier k, let T _i , T _j , and T _k be the torques of members i, j, and k, respectively. Since they rotate around the same axis, according to the torque balance, we have:

$T_i+T_j+T_k=\mathrm{0}.$

(5)

Furthermore, the latent powers of sun gear i and ring gear j relative to carrier k (P _i ^k and P _j ^k ) are defined as

$P_i^k=T_i\times \left({\omega }_i-{\omega }_k\right),$

(6)

$P_j^k=T_j\times \left({\omega }_j-{\omega }_k\right).$

(7)

Let η _ij ^k (η _ji ^k ) be the meshing efficiency of planetary gear train if the latent power is transmitted from sun gear i (ring gear j), to ring gear j (sun gear i) when carrier k is relatively fixed. Based on the concept of latent power, the relationship among P _i ^k and P _j ^k can be expressed as

$-P_j^k=P_i^k\times {\eta }_{ij}^k\hspace{1em}\text{for}\hspace{0.17em}{P}_i^k\ge \mathrm{0}\hspace{0.17em}\text{and}\hspace{0.17em}{P}_j^k\le \mathrm{0},$

(8a)

$-P_i^k=P_j^k\times {\eta }_{ji}^k\hspace{1em}\text{for}{P}_i^k\le 0\text{and}{P}_j^k\ge 0.$

(8b)

Substituting (6)-(7) into (8a)-(8b), we get

$T_j=-\left({\xi }_{ji}{\eta }_{ij}^k\right)T_i\hspace{1em}\text{for}{P}_i^k\ge 0\text{and}{P}_j^k\le 0,$

(9a)

$T_j=-\frac{{\xi }_{ji}}{{\eta }_{ji}^k}{T}_i\hspace{1em}\text{for}{P}_i^k\le 0\text{and}{P}_j^k\ge 0.$

(9b)

4.2. Equation of Meshing Efficiency

According to the kinematic analysis, if ξ_4′2′ = ξ_β < ξ₄₂ = ξ_α < 0, then the reduction ratio is R _r > 0. We get ω₂ > ω₇ > 0 > ω₄.

4.2.1. 2nd Train Circuit

For the 2nd train circuit of 2K–2H type planetary gear reducer shown in Figure 3, i₂ = 2′, j₂ = 4′, k₂ = 7, and ξ_4′2′ = ξ_β < 0, the relationship of angular velocity among ω_i2, ω_j2, and ω_k2 can be expressed as

${\omega }_{i\mathrm{2}}>{\omega }_{k\mathrm{2}}=\mathrm{0}>{\omega }_{j\mathrm{2}}.$

(10)

Since carrier k₂ is output and ω_k2 > 0, we get T_k2 < 0. According to (5), (9a), and (9b), since ξ_4′2′ = ξ_β < 0, we get T_i2 > 0 and T_j2 > 0. Then, based on (7) and (10), the latent powers of ring gear j₂ relative to carrier k₂(P_j2^k2) can be expressed as

$P_{j\mathrm{2}}^{k\mathrm{2}}=T_{j\mathrm{2}}\times ({\omega }_{j\mathrm{2}}-{\omega }_{k\mathrm{2}})<\mathrm{0}.$

(11)

According to (8a), we have

$-T_{j\mathrm{2}}\times \left({\omega }_{j\mathrm{2}}-{\omega }_{k\mathrm{2}}\right)=T_{i\mathrm{2}}\times \left({\omega }_{i\mathrm{2}}-{\omega }_{k\mathrm{2}}\right){\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}.$

(12)

According to the definition of train value and (9a), we have

$T_{j\mathrm{2}}=-\left({\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)T_{i\mathrm{2}}.$

(13)

For the 2nd train circuit, sun gear i₂, ring gear j₂, and carrier k₂ rotate around the same axis; (5) can be rewritten as

$T_{i\mathrm{2}}+T_{j\mathrm{2}}+T_{k\mathrm{2}}=\mathrm{0}.$

(14)

Then, based on (13) and (14), we have

$T_{i\mathrm{2}}=\frac{\mathrm{1}}{\left(\mathrm{1}-{\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)}{T}_{k\mathrm{2}}>\mathrm{0},$

(15)

$T_{j\mathrm{2}}=\frac{{\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}}{\left(\mathrm{1}-{\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)}{T}_{k\mathrm{2}}>\mathrm{0}.$

(16)

According to (10) and T_i2 > 0, T_j2 > 0, we get

$\begin{array}{c}{P}_{i2}=T_{i2}\times {\omega }_{i2}>0,\\ P_{j2}=T_{j2}\times {\omega }_{j2}<0.\end{array}$

(17)

Based on (17), the power flow of 2nd train circuit can be drawn as Figure 5 (b).

Figure 5:

The power flow of 2K–2H type planetary gear reducer.

4.2.2. 1st Train Circuit

Since ring gear j₁ and ring gear j₂ are coupled together to be the free link, we have

$P_{j1}+P_{j2}=0,$

(18)

$P_{j1}=-P_{j2}>0,$

(19)

$T_{j\mathrm{1}}=-T_{j\mathrm{2}}<\mathrm{0}.$

(20)

For the 1st train circuit of 2K–2H type planetary gear train shown in Figure 3, i₁ = 2, j₁ = 4, k₁ = 5, and ξ₄₂ = ξ_α < 0, the relationship of angular velocity among ω_i1, ω_j1, and ω_k1 can be expressed as

${\omega }_{i\mathrm{1}}>{\omega }_{k\mathrm{1}}=\mathrm{0}>{\omega }_{j\mathrm{1}}.$

(21)

According to (20) and (21), the latent powers of ring gear j₁ relative to carrier k₁(P_j1^k1) can be expressed as

$P_{j\mathrm{1}}^{k\mathrm{1}}=T_{j\mathrm{1}}\times \left({\omega }_{j\mathrm{1}}-{\omega }_{k\mathrm{1}}\right)>\mathrm{0}.$

(22)

According to the definition of train value and (9b), we have

$T_{j1}=-\frac{{\xi }_{\alpha }}{{\eta }_{j1i1}^{k1}}{T}_{i1},\hspace{1em}\text{or}\hspace{1em}{T}_{i1}=-\frac{{\eta }_{j1i1}^{k1}}{{\xi }_{\alpha }}{T}_{j1}.$

(23)

Based on (16), (20), and (23), we get

$T_{i\mathrm{1}}=\frac{{\xi }_{\beta }{\eta }_{j\mathrm{1}i\mathrm{1}}^{k\mathrm{1}}{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}}{{\xi }_{\alpha }\left(\mathrm{1}-{\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)}{T}_{k\mathrm{2}}<\mathrm{0}.$

(24)

According to (15) and (24), the input torque T_in = T_i1 + T_i2 can be expressed as

$T_{\text{in}}=T_{i\mathrm{1}}+T_{i\mathrm{2}}=\frac{\left({\xi }_{\beta }{\eta }_{j\mathrm{1}i\mathrm{1}}^{k\mathrm{1}}{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}-{\xi }_{\alpha }\right)}{{\xi }_{\alpha }\left(\mathrm{1}-{\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)}{T}_{k\mathrm{2}}>\mathrm{0}.$

(25)

Based on (25), the meshing efficiency of the 2K–2H type planetary gear reducer can be obtained as

${\eta }_m=-\frac{P_{\text{out}}}{P_{\text{in}}}=-\frac{P_{k\mathrm{2}}}{P_{i\mathrm{2}}}\times \frac{{\omega }_{\text{out}}}{{\omega }_{\text{in}}}=\frac{{\xi }_{\alpha }\left(\mathrm{1}-{\xi }_{\beta }{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)}{\left({\xi }_{\alpha }-{\xi }_{\beta }{\eta }_{j\mathrm{1}j\mathrm{1}}^{k\mathrm{1}}{\eta }_{i\mathrm{2}j\mathrm{2}}^{k\mathrm{2}}\right)}\times \frac{\mathrm{1}}{R_r}.$

(26)

For the gear manufacturing, if the gears are manufactured by shaving, the meshing efficiency of external (internal) gear pair is 0.98 (0.99). And if the gears are manufactured by grinding, the meshing efficiency of external (internal) gear pair is 0.99 (0.995). The relative meshing efficiencies η _ij ^k and η _ji ^k can be regarded equal to 0.98 × 0.99 = 0.9702 (0.99 × 0.995 = 0.985). Then, based on (4) and (26), the 3D drawing among ξ₄₂ (ξ_α), R _r , and η _m can be expressed as in Figure 6. For specified reduction ratio (e.g., R _r = 50 and 100), the meshing efficiency between ξ₄₂ (ξ_α) and η _m can be expressed as in Figure 7.

Figure 6:

The 3D drawing of meshing efficiency of 2K–2H type planetary gear reducer.

Figure 7:

The meshing efficiency of 2K–2H type planetary gear reducer.

Example 1 (Figure 4 (a), R _r = 50). Figure 4 (a) shows a 2K–2H type planetary gear reducer with ξ₄₂ = ξ_α = – 2.7692 and ξ_4′2′ = ξ_β = – 2.5714; according to (4), its reduction ratio is R _r = 50.

1. If η₂₄⁶ = η_2′4′⁶ = 0.9702, based on Figure 7 (a), its meshing efficiency is η _m = 55.50%.

2. If η₂₄⁶ = η_2′4′⁶ = 0.9850, based on Figure 7 (a), its meshing efficiency is η _m = 71.32%.

Example 2 (Figure 4 (b), R _r = 100). Figure 4 (b) shows a 2K–2H type planetary gear reducer with ξ₄₂ = ξ_α = – 2.6667 and ξ_4′2′ = ξ_β = – 2.5714; according to (4), its reduction ratio is R _r = 100.

1. If η₂₄⁶ = η_2′4′⁶ = 0.9702, based on Figure 7 (b), its meshing efficiency is η _m = 37.84%.

2. If η₂₄⁶ = η_2′4′⁶ = 0.9850, based on Figure 7 (b), its meshing efficiency is η _m = 54.82%.

Example 3 (Figure 4 (c), R _r = 99). Figure 4 (c) shows a 2K–2H type planetary gear reducer with ξ₄₂ = ξ_α = – 3.6667 and ξ_4′2′ = ξ_β = – 3.5; according to (4), its reduction ratio is R _r = 99.

1. If η₂₄⁶ = η_2′4′⁶ = 0.9702, based on (4) and (26), its meshing efficiency is η _m = 43.74%.

2. If η₂₄⁶ = η_2′4′⁶ = 0.9850, based on (4) and (26), its meshing efficiency is η _m = 60.80%.

The meshing efficiency of 2K–2H type planetary gear reducer with R _r = 100 (Figure 4 (b)) is less than 2K–2H type planetary gear reducer with R _r = 99 (Figure 4 (c)). Hence, when we design the 2K–2H type planetary gear reducer, we must choose the larger |ξ₄₂|(|ξ_α|) as possible. Based on the above reasoning, we conclude the following.

1. There is a power circulation in 2K–2H planetary gear reducer.

2. Basically, larger reduction ratio makes less meshing efficiency.

3. For the same reduction ratio, larger value |ξ₄₂|(|ξ_α|) will get better meshing efficiency.

4. The efficiency of gears manufactured by grinding is only improved by 1.5% (from 97.02% to 98.5%). However, the meshing efficiency of 2K–2H type planetary gear reducer is improved by 28%–44.8%. For 2K–2H type planetary gear reducer, the quality of gears is an important factor.

If the gears are manufactured by grinding, the meshing efficiency of external (internal) gear pair can be regarded as 0.99 (0.995) the meshing efficiency of 2K–2H type planetary gear reducer is good.

5. Summary

This paper proposes a design methodology for the 2K–2H planetary simple gear reducer. First, according to the concept of train value equation, the kinematic design of 2K–2H type planetary simple gear reducers is carried out. Three 2K–2H type planetary simple gear reducers with high reduction ratios (50, 100, and 99) are designed to illustrate the design algorithm. Then, based on the latent power theorem, the meshing efficiency equation of 2K–2H type planetary gear reducer is derived. According to the meshing efficiency equation, the meshing efficiency of 2K–2H type planetary simple gear reducer is analyzed. The 2K–2H type planetary gear reducer has the following characteristics.

1. There is a power circulation in 2K–2H type planetary gear reducer.

2. Basically, larger reduction ratio makes less meshing efficiency.

3. For the same reduction ratio, larger value |ξ₄₂|(|ξ_α|) will get better meshing efficiency.

4. If the gears are manufactured by grinding, the meshing efficiency of external (internal) gear pair can be regarded as 0.99 (0.995) and the efficiency of gear pairs is only improved by 1.5% (from 97.02% to 98.5%). However, the meshing efficiency of 2K–2H type planetary gear reducer is improved by 28%–44.8%.

5. For 2K–2H type planetary gear reducer, the quality of gears is an important factor.

Based on the above results, 2K–2H type planetary gear reducer with better meshing efficiency can be synthesized.