Meccanica

, Volume 48, Issue 5, pp 1071–1080

The analysis of power circulation and the simplified expression of the transmission efficiency of 2K-H closed epicyclic gear trains

Authors

    • School of Mechanical EngineeringUniversity of Jinan
  • Huan-yong Cui
    • School of Mechanical EngineeringUniversity of Jinan
Article

DOI: 10.1007/s11012-012-9652-0

Cite this article as:
Wang, C. & Cui, H. Meccanica (2013) 48: 1071. doi:10.1007/s11012-012-9652-0

Abstract

Transmission efficiency is a principal index for estimating the performance of 2K-H closed epicyclical gear trains. To calculate transmission efficiency, it must confirm the power flow direction in transformation gear train and verify if there is power circulation in closed epicyclical gear train firstly. In this paper, a simple algorithm is proposed to confirm the power flow direction and estimate if there exist the power circulation based on transmission ratio of basic links in differential gear train and transformation gear train. The simplified equations are deduced to calculate the transmission efficiency and some examples are given finally.

Keywords

2K-H closed epicyclical gear trainsTransmission efficiencyTransformation gear trainDirection of power flowPower circulation

1 Introduction

Now, numerous gear trains have been used in important and practical modern mechanical transmissions [13]. Among them, the closed epicyclical gear trains are more broad in applications due to their wide speed ratio range, big transmission power and compact structure. The closed epicyclical gear trains which base on 2K-H differential gear train are most useful in projects. When two of the three basic links in 2K-H differential gear train are closed by two fixed axis gear trains or planetary gear train, the freedom of gear trains will be reduced to 1 and the trains are named 2K-H closed epicyclical gear trains. (The paper names them as closed epicyclical gear trains for short.) The closed epicyclical gear trains can be divided into two categories with different closed links. The first one is called “non H closed epicyclical gear trains” (Fig. 1A), in which the two sun gears a and b are closed while the planet carrier H is not. The other one is called “H closed epicyclical gear trains” (Fig. 1B), in which one of two sun gears (a or b) and planet carrier H are closed while the other sun gear (b or a) is not.
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig1_HTML.gif
Fig. 1

2K-H closed epicyclic gear trains

Beside from a high load capacity, one of the main requirements is to reduce the losses in gearing [4]. Therefore, the most important steps in the design of a planetary gear train (PGT) is to estimate its efficiency. Some early studies on efficiency of planetary gear trains include those of MacMillan [5, 6], Ranzimovsky [7] and Glover [8]. The methods are far from being systematic. Instead, they are conceived for being applied manually by one engineer. In contrast to the manual methods, the systematic methods are conceived for computer implementation. Especially, these methods proposed by Lei and Lu [9], Mathis and Remond [10], Del Castillo [11] and Hedman [12] belong to this category. These methods yield only numerical values, not analytical expressions for the efficiency. Del Castillo [13] proposed two procedures for obtaining the analytical expression for the efficiency of any planetary gear train, which utilized the characteristic functional form of any PGT’s transmission ratio, and of the relationships between the gearing powers and the transmission ratio. Salgado [14] proposed a method to analyze and compare PGTs on the basis of been done in the efficiency calculation of planetary and epicyclic gear train [1522]. However, the efficiency of 2K-H closed epicyclic gear train has not been systematically investigated. And for the complicated power flow directions, power circulation will be generated when the irrational directions are chosen. In this condition, it will generate the friction losses and affect the link strength. So, judging the power flow direction correctly and avoiding power flow backward are the problems which need to be solved firstly.

2 Fundamental theories

2.1 Torque and power balance equations

In the epicyclic gear train, Ma, Mb and MH are the additional torque imposed on the sun gear a, the sun gear b and the link H, respectively. When gear train rotates at uniform speed, the additional torques are in balance state, i.e.
$$ M_{a} + M_{b} + M_{H} = 0 $$
(1)
And, the transmission efficiency in transformation gear train can be expressed by the following equation, i.e. [23]
$$ M_{a}i_{ab}^{H}\bigl(\eta^{H}\bigr)^{\alpha } + M{}_{b} = 0 $$
(2)
Then obtain from (1) and (2):
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ3_HTML.gif
(3)
where ηH is the efficiency of transformation gear train. And, α depends on the power flow direction in transformation gear train, where α is equal to 1 at the condition which the sun gear a is driving gear and the sun gear b is follower gear. And α is equal to −1 at the inverse condition.

2.2 The angular velocity relationship between basic links

The equation based on the transmission ratio in transformation gear train and two sun gears transmission ratio, i.e.
$$ i_{ab}^{H} = \frac{w_{a} - w_{H}}{w_{b} - w_{H}} $$
(4)
and
$$ i_{ab} = \frac{w_{a}}{w_{b}} $$
(5)
Then obtain from (4) and (5):
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ6_HTML.gif
(6)

2.3 The transmission power calculation of basic links

$$ P = Mw $$
(7)
where P is the transmission power of certain basic link; M is the additional torque imposed on the link; w is the angular velocity of the link.

3 Confirming the power flow direction in transformation gear train

Whether in the epicyclic gear train, or the transformation gear train, the additional torques Ma, Mb and MH are unchanged. The form of driver or driven between the sun gear a and the sun gear b is obvious in the epicyclic gear train. By comparing the angular velocity of the sun gear a or the sun gear b in the epicyclic gear train with those in the transformation gear train, the power flow direction in the transformation gear train can be confirmed.

3.1 Confirming the power flow direction for non H closed epicyclic gear train

By calculating \(w_{b}^{H}\) and comparing with wb, we can confirm the power flow direction in the transformation gear train.

Considering (6) and \(w_{b}^{H} = w_{b} - w_{H}\), we have:
$$ w_{b}^{H} = \frac{i_{aH} - 1}{i_{ab}^{H} + i_{aH} - 1}w_{b} $$
(8)

The form of driver or driven between the sun gear a and the sun gear b is obvious in the epicyclic gear train. Therefore, if the calculation of \(w_{b}^{H}\) and wb are the same sign, the form of driver or driven of the gear b in the transformation gear train is the same as in the epicyclic gear train. Otherwise, they are contrary. Then, we can confirm the power flow direction in transformation gear train, i.e., the value of α in Eqs. (3).

3.2 Confirming the power flow direction of H closed epicyclic gear train

Here we assume gear b as non closed link. Obviously, the calculating equation of \(w_{b}^{H}\) is the same as Eq. (8).

Therefore, the unified efficiency calculation expressions can be written as following:
$$ w_{j}^{H} = \frac{i_{kH} - 1}{i_{kj}^{H} + i_{kH} - 1}w_{j} $$
(9)
where j represents the sun gear of calculating angular velocity, k represents another sun gear.

4 The judgment of power circulation

4.1 The form of power circulation

When the inappropriate form or irrational parameters of closed epicyclical gear train are chose, it can cause the power circulation which has severe disadvantages to transmission. It will increase the power losses of friction and decrease the efficiency or strength of gear train. It is extremely unfavorable to transmission.

4.2 The criterion principle of power circulation

When the sign of input/output power of closed links are same, the power circulation will not generate. In the other cases, the power circulation will generate.

4.3 The judgment of power circulation in non H closed epicyclical gear train

According to the judgment principle of power circulation, when there is no power circulation in non H closed epicyclical gear train and the close links a and b are the same as input/output links, their transmission power should be the same sign, that is:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ10_HTML.gif
(10)

As Eq. (10), we can see that there is no power circulation when the sign of iab and \(i_{ab}^{H}\) are opposite. Otherwise, there is power circulation.

If there is power circulation, the judgment criterion of power circulation can express like,
$$ \frac{P_{a}}{P_{H}} = \frac{M_{a}w_{a}}{M{}_{H}w_{H}} = \frac{i_{aH}}{i_{ab}^{H}(\eta^{H})^{\alpha } - 1} > 0 $$
(11)
In Eq. (11),
  1. (1)

    when \(i_{ab}^{H} < (\eta^{H})^{ - \alpha }\) and iaH<0,a is the backwards link; otherwise, b is backwards link,

     
  2. (2)

    when \(i_{ab}^{H} > (\eta^{H})^{ - \alpha }\) and iaH>0,a is backwards link; otherwise, b is backwards link.

     

4.4 The judgment of power circulation in H closed epicyclical gear train

Assuming the sun gear a and H closed, if there is no power circulation, the sign of input/output power in closed link a or H are same. It satisfies Eq. (11).

From Eq. (11), we can see that
  1. (1)

    When \(i_{ab}^{H} < (\eta^{H})^{ - \alpha }\) and the direction of wa and wH are opposite, there is no power circulation. Otherwise, there is power circulation.

     
  2. (2)

    When \(i_{ab}^{H} > (\eta^{H})^{ - \alpha }\) and the direction of wa and wH are the same, there is no power circulation. Otherwise, there is power circulation.

     

If there is power circulation, the judgment criterion of power circulation can be expressed according to Eq. (10).

In Eq. (10), we can see that the sun gear a is backwards link when the sign of iab and \(i_{ab}^{H}\) are opposite. Otherwise, H is backwards link. The judgments of power flow direction of closed epicyclical gear train are expressed in Table 1. Moreover, the concrete expression of power flow direction also refers to Refs. [21, 22].
Table 1

The judgments of power flow direction

https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Tab1_HTML.gif

5 Transmission efficiency calculation of closed epicyclical gear train

The types of input and output in closed epicyclical gear train are classified into four categories (H closed epicyclical gear train, here, assumes the sun gear b as non closed link), as follows:
  1. (1)

    For non H closed epicyclical gear train, the planet carrier H is output link;

     
  2. (2)

    For non H closed epicyclical gear train, the planet carrier H is input link;

     
  3. (3)

    For H closed epicyclical gear train, the sun gear b is output link;

     
  4. (4)

    For H closed epicyclical gear train, the sun gear b is input link.

     
Backwards coefficients γa, γb, and γH are introduced. When power circulation exists, the backwards coefficients of corresponding backwards link are 1; otherwise are 2.

5.1 Non H closed epicyclical gear train, the planet carrier H is output link

We assume that the output power of the planet carrier H is PH; the input power of the gear d is Pd; transmission efficiency which transfer from d to the sun gear a, b are ηa and ηb respectively. Then:
$$ \eta= - \frac{P_{H}}{P_{d}} $$
(12)
and
$$ P_{d} = ( - 1)^{\gamma _{a}}P_{a}/\eta_{a} + ( - 1)^{\gamma _{b}}P_{b}/\eta_{b} $$
(13)
Substituting (3), (6) and (13) with (12), we obtain:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ14_HTML.gif
(14)

5.2 Non H closed epicyclical gear train, the H is input link

We assume that the output power of H is PH; the input power of d is Pd; transmission efficiency which transfer from the sun gear a, b to d are ηa and ηb respectively. Then:
$$ \eta= - \frac{P_{d}}{P_{H}} $$
(15)
and
$$ P_{d} = ( - 1)^{\gamma _{a}}P_{a}\eta_{a} + ( - 1)^{\gamma _{b}}P_{b}\eta_{b} $$
(16)
Substituting (3), (6) and (16) with (15), we obtain:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ17_HTML.gif
(17)

5.3 H closed epicyclical gear train, the sun gear b is output link

We assume that output power of b is Pb; the input power of d is Pd; transmission efficiency which transfer from d to a, H are ηa and ηH respectively. Then:
$$ \eta= - \frac{P_{b}}{P_{d}} $$
(18)
and
$$ P_{d} = ( - 1)^{\gamma _{a}}P_{a}/\eta_{a} + ( - 1)^{\gamma _{H}}P_{H}/\eta_{H} $$
(19)
Substituting (3), (6) and (19) with (18), we obtain:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ20_HTML.gif
(20)

5.4 H closed epicyclical gear train, the sun gear b is input link

We assume that output power of b is Pb; the input power of d is Pd; transmission efficiency which transfer from a, H to d are ηa and ηH respectively. Then:
$$ \eta= - \frac{P_{d}}{P_{b}} $$
(21)
and
$$ P_{d} = ( - 1)^{\gamma _{a}}P_{a}\eta_{a} + ( - 1)^{\gamma _{H}}P_{H}\eta_{H} $$
(22)
Substituting (3), (6) and (22) with (21), we obtain:
$$ \eta= \frac{( - 1)^{\gamma _{a}}i_{aH}\eta_{a} + ( - 1)^{\gamma _{H}}[i_{ab}^{H}(\eta^{H})^{\alpha } - 1]\eta_{H}}{(i_{aH} + i_{ab}^{H} - 1)(\eta^{H})^{\alpha }} $$
(23)

5.5 Unified efficiency calculation expressions

According to the type of closed epicyclical gear trains and the form of non closed link input or output, unified efficiency calculation expressions can be written as following:

5.5.1 Non H closed epicyclical gear train

According to (14) and (17), we obtain:
$$ \eta= \biggl\{ \frac{[( - 1)^{\gamma _{a}}i_{ab}(\eta_{a})^{\beta } - ( - 1)^{\gamma _{b}}i_{ab}^{H}(\eta^{H})^{\alpha }(\eta _{b})^{\beta }](1 - i_{ab}^{H})}{[1 - i_{ab}^{H}(\eta^{H})^{\alpha }](i_{ab} - i_{ab}^{H})} \biggr\}^{\beta } $$
(24)

5.5.2 H closed epicyclical gear train

According to (20) and (23), we obtain:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ25_HTML.gif
(25)

In the transformation gear train, when the power flow direction is from the sun gear a to the sun gear b, the value of α is 1; otherwise is −1. When there is power circulation in gear train, the value of corresponding backwards link coefficients γ is 1; otherwise is 2. When the non closed link is input one, the value of β is 1; otherwise is −1.

For H closed epicyclical gear train, confirming the power flow direction in transformation gear train, verifying if there is power circulation in closed epicyclical gear train and deducing the simplified equations of the transmission efficiency, we assumes the sun gear b as non closed link. Therefore, in actual calculation, we may regard any non closed link as gear b for H closed epicyclical gear train.

6 Illustrative examples

6.1 Example 1

Structural sketch of the electric winch reducer is shown in Fig. 2, where za=76, zb=24, zc=28, zd=24, ze=20, zf=30, zH=80, and each of tooth meshing efficiency is 0.97. Calculation of the transmission efficiency of the closed epicyclical gear train is followed.
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig2_HTML.gif
Fig. 2

Electric winch reducer

As can be seen in Fig. 2, the gear train is H closed epicyclic gear train. The sun gear b is non closed link.

(1) Correlative calculation

The calculations of basic links transmission ratio in transformation gear train are as following:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ26_HTML.gif
(26)
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ27_HTML.gif
(27)
The calculation of basic links transmission ratio in differential gear train is as following:
$$ i_{aH} = - \frac{z_{H}}{z_{e}} = - 4 $$
(28)
The calculation of transmission efficiency of differential gear train is as following:
$$ \eta^{H} = 0.97^{2} = 0.9409 $$
(29)
The calculations of transmission efficiency of a pair of teeth in differential gear train are as following:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ30_HTML.gif
(30)
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ31_HTML.gif
(31)

(2) Confirmation of power flow direction with transformation gear train

Substituting (27) and (28) with (9), the following equation can be obtained:
$$ w_{b}^{H} = \frac{ - 4 - 1}{ - 0.27 - 4 - 1}w_{b} $$
(32)
For the sun gear b is input link, i.e. wb>0, then: \(w_{b}^{H} > 0\). Therefore, the power flow direction in transformation gear train is from the sun gear b to the sun gear a, i.e. α=−1.

(3) Judgment of the power circulation

For
$$ \frac{1}{\eta^{H}} = \frac{1}{0.9409} > 0 $$
(33)
and
$$ i_{ab}^{H} = - 0.27 < 0 $$
(34)
we obtain
$$ i_{ab}^{H} < 1/\eta^{H} $$
(35)
It can be seen in Fig. 2 that the direction of wa and wH is opposite. As a result, according to the judgment principle of power circulation in H closed epicyclical gear train, there is no power circulation, i.e. the backwards coefficients γa=2 and γH=2. According to Table 1 and Refs. [21, 22], power flow direction is shown in Fig. 3.
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig3_HTML.gif
Fig. 3

Power flow direction

Here, L=(1−η)P=0.03P, while P is the power into the gear mesh.

(3) The transmission efficiency calculation

The gear train is H closed epicyclical gear train and the sun gear b is input link, we can deduce β=1. Substituting β=1,α=−1, γa=2 and γH=2 with (25), we obtain:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ36_HTML.gif
(36)

6.2 Example 2

Here za=21, zb=20, zf=20, zg=21, z1=25, z2=45, \(z_{2}' = 20\), z3=50 and each of tooth meshing efficiency η is 0.97. Gear 1 is input. Analysis of whether there is power circulation is followed.

As can be seen in Fig. 4, the gear train is H closed epicyclical gear train. The sun gear b is non closed link.
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig4_HTML.gif
Fig. 4

Closed epicyclical gear train

(1) Correlative calculation

The calculations of basic links transmission ratio in transformation gear train are as following:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ37_HTML.gif
(37)
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ38_HTML.gif
(38)
The calculation of basic links transmission ratio in differential gear train is as following:
$$ i_{aH} = \frac{z_{2}z_{3}}{z_{1}z_{2}'} = 4.5 $$
(39)
The calculation of transmission efficiency of differential gear train is as following:
$$ \eta^{H} = 0.97^{2} = 0.9409 $$
(40)
The calculations of transmission efficiency of a pair of teeth in differential gear train are as following:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ41_HTML.gif
(41)
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ42_HTML.gif
(42)

(2) Confirmation of power flow direction in transformation gear train

Substituting (38) and (39) with (9), the following equation can be obtained:
$$ w_{b}^{H} = \frac{4.5 - 1}{0.9 + 4.5 - 1}w_{b} $$
(43)

For the sun gear b is output link, i.e. wb<0, then: \(w_{b}^{H} < 0\). Therefore, the power flow direction in transformation gear train is from the sun gear a to the sun gear b, i.e. α=1.

(3) Judgment of the power circulation

For
$$ \frac{1}{\eta^{H}} = \frac{1}{0.9409} $$
(44)
and
$$ i_{ab}^{H} = 0.9 $$
(45)
we obtain
$$ i_{ab}^{H} < 1/\eta^{H} $$
(46)
It can be seen in Fig. 4 that the direction of wa and wH is the same. As a result, according to the judgment principle of power circulation in H closed epicyclical gear train, there is power circulation (Fig. 5).
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig5_HTML.gif
Fig. 5

Power flow direction

For
$$ i_{ba}^{H} = \frac{w_{b} - w_{H}}{w_{a} - w_{H}} = \frac{i_{ba} - i_{Ha}}{1 - i_{Ha}} = \frac{1/i_{ab} - 1/i_{aH}}{1 - 1/i_{aH}} $$
(47)
we obtain
$$ i_{ab} = 0.93 > 0 $$
(48)
For
$$ i_{ab}^{H} = 0.9 $$
(49)
and (48) we obtain
$$ i_{ba}^{H}/i_{ba} > 0 $$
(50)

Therefore, H is the link of power circulation. The power flow direction is shown in Fig. 5.

Here, L=(1−η)P=0.03P, while P is the power into the gear mesh.

Two advanced cases:

Case 1: change tooth numbers of differential gear train,
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equa_HTML.gif
Solve:
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ51_HTML.gif
(51)
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Equ52_HTML.gif
(52)

(1) Confirm power flow direction in transmission gear train

According to Eq. (9), we obtain:
$$ w_{b}^{H} = \frac{4.5 - 1}{1.1 + 4.5 - 1}w_{b} $$
(53)

For the sun gear b is output link, i.e. wb<0, then: \(w_{b}^{H} < 0\). Therefore, the power flow direction in transformation gear train is from the sun gear a to the sun gear b, i.e. α=1.

(2) Judgment of power circulation

For \(i_{ab}^{H} > 1/\eta^{H}\) and the direction of wa and wH is the same, there is no power circulation (Fig. 6).
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig6_HTML.gif
Fig. 6

Power flow direction of advanced gear train

Here, L=(1−η)P=0.03P, while P is the power into the gear mesh.

Case 2: change 1, 2 as internal gear (Fig. 7), z1=120
https://static-content.springer.com/image/art%3A10.1007%2Fs11012-012-9652-0/MediaObjects/11012_2012_9652_Fig7_HTML.gif
Fig. 7

Advanced gear train

For \(i_{ab}^{H} < 1/\eta^{H}\) and the direction of wa and wH is the opposite, there is no power circulation.

7 Summary and conclusion

By calculating the basic links transmission ratios in the differential gear train and the transformation gear train (other relevant parameters may be determined by the gear train concrete structure),

(1) The power flow direction in transformation gear train is confirmed and then the value of α can be determined;

(2) α obtained in step (1) and the basic links transmission ratios are together used to evaluate whether the power circulation exists. If there is power circulation, the value of corresponding backwards link coefficients γ is −1, otherwise is 1;

(3) With α and γ obtained from step (1) and step (2), the expression of transmission efficiency of the closed epicyclical gear train can be selected.

Acknowledgements

The authors wish to acknowledge the financial support of the Shangdong Province Young and Middle-Aged Scientists Research Awards Fund (BS2011ZZ002). The authors would also like to thank the editor and anonymous reviewers for their suggestions for improving the paper.

Copyright information

© Springer Science+Business Media Dordrecht 2012