Influence of transmission errors to load sharing behaviour in Ravigneaux planetary gear sets
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Abstract
Ravigneaux planetary gear sets are widely applied in automotive, national defense, shipbuilding industry, etc.In this paper, orthogonal experiment method and variance analysis are utilized, to quantitatively analyze the influence of three typical transmission errors on the load sharing behaviour of Ravigneaux planetary gear sets. A 30 DOF translationaltorsional coupling dynamic model of Ravigneaux planetary gear sets has been derived. Both qualitative and quantitative analysis indicate that load sharing coefficients of different meshing pairs are sensitive to different kinds of errors. As a result, key errors and key members which affect load sharing behaviour most are found out: the position error of short planetary gear, the installation error of small ring gear, and the eccentric error of sun gear. At the end, several fitting formulas are innovatively proposed in order to obtain load sharing coefficient, the results of proposed fitting formulas show great consistency with the calculation results of software Maple.
Keywords
Ravigneaux planetary gear sets Transmission errors Load sharing behaviour Orthogonal experiment Variance analysis Fitting formula1 Introduction
Ravigneaux planetary gear sets have been widely used in many areas over the years, such as automobile transmission [1], wind power generation [2], aerospace [3], due to its outstanding advantages including: large transmission ratio, high ability to sustain large load, small space consumption, large torquetoweight ratio and good ability to reduce the radial bearing loads. By changing the assignments to input, output and stationary members, different speed and torque ratios can be realized [4], obviously, this flexibility makes the application prospect of them very extensive.Ravigneaux planetary gear sets are a strong nonlinear system, which incorporates timevarying mesh stiffness, backlash, and transmission errors. In practical engineering, the load is usually not equally shared among the planets due to the inevitable errors and tolerances. Load sharing characteristics of gear sets are remarkably influenced by many factors, such as: mesh stiffness, bearing supportive stiffness, damping ratio, installation errors, eccentric errors, tooth thickness errors, members floating, the number of planetary gears and working condition. So far many papers about planetary gear sets have been published.
In 2001, Kahraman [4] listed ten structures of Ravigneaux planetary gear sets, then derived their free torsional dynamic models and researched the vibration characteristics, finally he found the nature modes can be classified into three groups: a rigid body mode, asymmetric planet modes and axissymmetric overall modes. He [5] also investigated the influence of internal gear flexibility, and found that a flexible internal gear can improve the load sharing behaviour of gear sets. In 2003, Sun [6] researched the influence of mesh stiffness and static transmission errors on the frequency response of the gear system. In 2004, Bodas [7] developed a deformable body contact model of planetary gear sets to research the influence of three kinds of typical errors on the load sharing behaviour of planetary gear sets and proposed the design formulas for calculating the planets loads as a function of manufacturing errors. In 2006, Fakher [8] investigated the effect of runout error and tooth profile error on the dynamic response of gear system, and pointed out: tooth profile errors would increase the gear mesh frequency amplitude. In 2007, Kiracole [9] set up dynamic models of stepped planet gear sets, multistage gear sets and meshed planet gear sets, then analyzed their vibration modes. Guo and Parker [10] developed a purely torsional model of Ravigneaux planetary gear sets, which was suitable to general configurations, in another paper [14], they analyzed the sensitivity of planetary gear natural frequency and vibration modes to system inertia and stiffness parameters. In 2008, Dhouib [11] analyzed the mode characteristics of Ravigneaux planetary gear sets in case of the different number of planetary gears, besides he pointed out that gyroscopic effect engendered by high speed working condition would separate the repeated translational modes into distinct ones. Ligata [12] designed the gear transmission system experiment to investigate the influence of the number of planetary gears and the amplitude of errors on the load sharing characteristics of gear sets. Aishyyab [13] used a hybrid harmonic balance method (HBM) to semianalytically solve the torsional dynamic model of a multistage gear sets and provided a case study to verify its accuracy.
In 2011, Wu and Liu [15] used HBM to solve the steady state response of fundamental frequency of Ravigneaux planetary gear sets then researched the influence of backlash, timevarying mesh stiffness and synthetic mesh errors on the dynamic behavior of gear system. Shang [16] compared the load sharing characteristics of gear sets under the condition of doublefloating, singlefloating and nonfloating, found the load sharing coefficient of doublefloating structure system is 9% smaller than that of the single floating structure, which is 16% smaller than that of nonfloating structure. In 2012, Kim [17] researched the influence of timevarying pressure angle and contact ratio caused by bearing deformation, found that timevarying pressure angle and contact ratio would make the mesh deformation exhibit more frequency components. In 2015, Peng [18] set up a virtual prototype of gear sets based on software Adams, and investigated the influence of the eccentric error, input speed and the load on the loading sharing characteristics of gear sets. Wang [19] used the variable stepsize Gill numerical integration method to solve the purely torsional model of gear sets, found that with dimensionless excitation frequency increasing, the system would go through singleperiod response, quasiperiodic response, multiperiodic response and chaotic response successively. Zhu and Wu [20] developed the translationaltorsional coupling dynamic model of Ravigneaux planetary gear sets considering the variable bearing stiffness coefficient, found that when the phase angel of position error existing on the planetary gears is 90 degrees, the load sharing coefficient is largest. In another paper [3], they researched the bifurcation phenomenon of Ravigneaux planetary gear sets caused by backlash, mesh stiffness and transmission errors, furthermore, they revealed the effect of nonlinearities (backlash, mesh stiffness and amplitude of error) on the frequency response characteristic. In 2016, Zhou and Wu [21] analyzed the influence of installation errors, eccentric errors and their phase angles on the load sharing characteristics of gear sets, found that the eccentric error of planet gear would cause sustained partial load. Zhang [22] investigated the effects of the mesh stiffness, backlash and bearing clearance on the load sharing coefficient of gear sets, and found that the bearing clearance of central members can improve the load sharing behavior largely. In 2017, Guerine and Hami [23] proposed the “interval analysis” to investigate the dynamic response of gear system with uncertainbutbounded parameters, discovered the interval analysis method would yield larger bounds than the probabilistic approach. Lglesias [24] investigated the effect of planet pinhole position errors and eccentric errors on the load sharing behavior of gear sets under the condition of fixed sun and floating sun. In 2018, Li [25] put forward a calculation method of the maximum phase difference of the meshing points, in order to calculate load sharing coefficient. Besides, Wang [26] analyzed influence of some key parameters (such as load ratio, mesh frequency, damping ratio, backlash and mesh stiffness), on the dynamic response of the gear system in his academic publication.
From the abovementioned literature reviews, it can be seen that: research hotspots are gradually transferring from natural frequency computation and modal analysis, to dynamic response analysis and load sharing characteristics analysis. Besides, most researchers are focused on how a single factor (including the number of planetary gears [7, 11], working conditions [17], mesh errors [7, 8, 15], backlash [2, 15], timevarying mesh stiffness [15], bearing supportive stiffness [20], bearing clearance [22], member floating [16], pressure angle [17], gyroscopic effect [11], internal gear flexibility [5], etc.) affects the dynamic response or load sharing characteristics of gear sets. Multifactor analysis is less introduced, and it has still not been explored “how the load sharing characteristics of Ravigneaux planetary gear sets are affected, when multiple transmission errors are involved simultaneously?”. Besides, it has still not been revealed “which kind of transmission error exerts the most influence on load sharing behaviour?” as well as “which member of gear sets is the key one?”. Researches about the load sharing characteristics of Ravigneaux planetary gear sets are still limited.
In this study, the orthogonal experiment method and variance analysis are applied, to explore the influence degree of three main transmission errors (including installation errors of central members, carrier planet pinhole position errors, and eccentric errors of central members) on the load sharing coefficient of Ravigneaux planetary gear sets. Not only multifactor analysis but also quantitative analysis about the influence degree are realized.
2 Dynamic model
 1.
The mesh gears are standard spur profile gears, and they can be simplified as cylinders which are connected by linear springs and damping.
 2.
All members are assumed as rigid bodies. Flexibility of members (especially the flexibility of ring gears) is neglected.
 3.
All mesh forces between gears are in one mesh plane, the dynamics problem of CPGS can be regarded as a twodimensional problem rather than threedimensional.
 4.
All frictions between meshing pairs are neglected, there are only elastic forces and damping forces between meshing pairs.
 5.
For the same kind of planetary gears, their mass as well as moment of inertia is exactly equal.
 A

\(\left\{ {\begin{array}{*{20}l} {{\text{Installation}}\;{\text{error}}\;{\text{of}}\;{\text{central}}\;{\text{member}},\;i = s,r_{1} ,r_{2} } \\ {{\text{Carrier}}\;{\text{planet}}\;{\text{pinhole}}\;{\text{position}}\;{\text{error}},\;i = b,a} \\ \end{array} } \right.\)
 b _{ ij }

Half of backlash between gear i and gear j
 c _{ ij }

Mesh damping between gear i and gear j:
\(c_{ij} = 2 \cdot \xi_{ij} \cdot \sqrt {\bar{k}_{ij} /\left( {1/m_{i} + 1/m_{j} } \right)}\), \(\xi_{ij}\) is mesh damping ratio
 E _{ i }

Eccentric error of member i
 e _{ ij }

Transmission errors between gear i and gear j
 I _{ i }

Moment of inertia of member i
 I _{ c }

Moment of inertia of the carrier (without planets)
 k _{ i }

Radial support stiffness of member i
 k _{ it }

Torsional stiffness of member i
 k _{ ij }

Mesh stiffness between gear i and gear j
 M _{ i }

Equivalent mass of member i: = I_{i}/r_{i}^{2}
 M _{c}

Equivalent mass of the carrier.
2.1 Excitation factors
Gear transmission system is a strongly nonlinear system, which includes many internal and external excitation factors, such as timevarying mesh stiffness (TVMS), backlash and transmission errors.
Here,\(\bar{k}_{j}\) is the mean value of TVMS, \(k_{j\_fluc}\) is the fluctuation value (generally, it is assumed that:\(k_{j\_fluc} = 0.25 \cdot \bar{k}_{j}\)), \(\omega_{m}\) is the mesh angular frequency,\(\omega_{m} = (z_{{r_{1} }} \cdot z_{s} )/(z_{s} + z_{{r_{1} }} ) \cdot \omega_{s}\),\(\varphi_{j}\) is the initial phase.
Backlash is usually expressed by the following piecewise nonlinear function.
This paper only assumes the following transmission errors between meshing pairs:(1) Installation errors of central members (IE); (2) Carrier planet pinhole position errors (CPE);(3) Eccentric errors of members (EE). These three kinds of errors are predominant in practical engineering, other errors (such as tooth thickness error and tooth profile error) are neglected here.
2.2 System equations of motion
Basic Parameters of doublering Ravigneaux gear sets
Parameter  Sun (s)  Small ring (r_{1})  Large ring (r_{2})  Carrier (c)  Long planetary gear (b_{n})  Short planetary gear (a_{n}) 

Number of teeth  32  70  77  –  19  20 
Mass (kg)  3.0781  8.4022  8.3331  23.0951  1.0967  1.9435 
Moment of inertia (kg mm^{2})  0.0078  0.2588  0.1901  0.2276  0.0011  0.0019 
Module (mm)  4  
Pressure angle (°)  25  
Tooth thickness (mm)  40  35  39  –  72  38 
Mesh stiffness (N/m)  \(\begin{aligned} k_{sb} = 6.99 \times 10^{8} ,k_{{r_{1} b}} = 5.21 \times 10^{8} , \hfill \\ k_{ba} = 6.92 \times 10^{8} ,k_{{r_{2} a}} = 5.56 \times 10^{8} \hfill \\ \end{aligned}\)  
Radial supportive stiffness (N/m)  \(k_{s} = k_{r} = k_{c} = k_{p} = 1 \times 10^{8}\)  
Torsional stiffness (N/m)  \(k_{{r_{1} t}} = 10^{12} ,k_{st} = k_{{r_{2} t}} = k_{ct} = k_{bt} = k_{at} = 0\) 
3 Orthogonal experiment and variance analysis
3.1 Orthogonal experiment
In industrial production, orthogonal experiment method (also called as “Taguchi Method”) is universally adopted by engineers to find out the optimal technological parameters. For multifactor analysis, it can remarkably decrease the number of experiments without weaken the effectiveness of the analysis results by choosing the part uniform experiment parameter points to represent the whole experiment parameter points. Orthogonal experiment table is designed based on the following two principles: 1. Each factor has the same number of tests at different levels. All two factors are cross group comprehensive trials; 2. All possible combinations of various levels between any two columns occur in the orthogonal experiment table and occur the same number of times.
In terms of IE and CPE, there are five different errors:\(A_{s} ,A_{{r_{1} }} ,A_{{r_{2} }} ,A_{b} ,A_{a}\), following two assumptions are adopted here: 1. The position angles of errors are all zero, only magnitudes are researched; 2. The position errors of planetary gears only exist in ONE set of planetary gears (without loss of generality, assuming they only exist in the first set of planetary gears, namely,\(A_{bn} ,A_{an} \left\{ {\begin{array}{*{20}l} { \ne 0, \, while \, n = 1} \\ { = 0, \, while \, n = 2,3} \\ \end{array} } \right.\)). For each \(A_{i} (i = s,r_{1} ,r_{2} ,b,a)\), there are four different levels: 8 μm, 16 μm, 25 μm, 36 μm (corresponding to different accuracy grades from accuracy grade 4 to 7). These error values cover a typical tolerance range for the gear sets of this size, so this paper chooses the orthogonal experiment table: \(L_{16} \left( {4^{5} } \right)\).
3.1.1 Results of orthogonal experiment
Orthogonal experiment results about IE and CPE
Series No.  A_{s}  A_{r1}  A_{r2}  A_{b1}  A_{a1}  B_{sb}  B_{r1b}  B_{r2a}  B_{ba} 

1  8  8  8  8  8  1.0356  1.0578  1.0513  1.0517 
2  8  16  16  16  16  1.0644  1.1244  1.0944  1.0951 
3  8  25  25  25  25  1.1005  1.2047  1.1450  1.1459 
4  8  36  36  36  36  1.1458  1.3043  1.2077  1.2089 
5  16  8  16  25  36  1.0908  1.1247  1.1489  1.1499 
6  16  16  8  36  25  1.0715  1.1411  1.1064  1.1073 
7  16  25  36  8  16  1.1122  1.1827  1.1643  1.1654 
8  16  36  25  16  8  1.0961  1.2173  1.1186  1.1194 
9  25  8  25  36  16  1.0803  1.0581  1.1076  1.1085 
10  25  16  36  25  8  1.0904  1.0750  1.1247  1.1256 
11  25  25  8  16  36  1.1210  1.2526  1.1887  1.1898 
12  25  36  16  8  25  1.1233  1.2773  1.1801  1.1811 
13  36  8  36  16  25  1.1305  1.1085  1.1867  1.1881 
14  36  16  25  8  36  1.1500  1.2042  1.2169  1.2184 
15  36  25  16  36  8  1.1050  1.1722  1.1220  1.1231 
16  36  36  8  25  16  1.1194  1.2642  1.1612  1.1622 
Orthogonal experiment results about EE
Series no.  E_{s}  E_{r1}  E_{r2}  E_{b1}  E_{a1}  B_{sb}  B_{r1b}  B_{r2a}  B_{ba} 

1  8  8  8  8  8  1.0605  1.1339  1.0658  1.0663 
2  8  16  16  16  16  1.1000  1.2343  1.1224  1.1233 
3  8  25  25  25  25  1.1444  1.3588  1.1862  1.1873 
4  8  36  36  36  36  1.1987  1.5095  1.2643  1.2655 
5  16  8  16  25  36  1.1377  1.3297  1.2145  1.2157 
6  16  16  8  36  25  1.1370  1.4081  1.1657  1.1667 
7  16  25  36  8  16  1.1628  1.3488  1.1907  1.1915 
8  16  36  25  16  8  1.1616  1.4190  1.1657  1.1663 
9  25  8  25  36  16  1.1784  1.4139  1.1926  1.1932 
10  25  16  36  25  8  1.1908  1.4106  1.1979  1.1988 
11  25  25  8  16  36  1.1675  1.4131  1.198  1.1992 
12  25  36  16  8  25  1.1793  1.3933  1.1792  1.1799 
13  36  8  36  16  25  1.2076  1.3637  1.2675  1.2684 
14  36  16  25  8  36  1.2106  1.3599  1.2660  1.2676 
15  36  25  16  36  8  1.2159  1.4957  1.1751  1.1757 
16  36  36  8  25  16  1.2028  1.4854  1.1854  1.1866 
Orthogonal experiment results about meshing pair sb (after substitution)
Series no.  A_{s}  A_{r1}  A_{r2}  A_{b1}  A_{a1}  B_{sb}  BB_{sb} 

1  8  8  8  8  8  1.0356  3.56 
2  8  16  16  16  16  1.0644  6.44 
3  8  25  25  25  25  1.1005  10.05 
4  8  36  36  36  36  1.1458  14.58 
5  16  8  16  25  36  1.0908  9.08 
6  16  16  8  36  25  1.0715  7.15 
7  16  25  36  8  16  1.1122  11.22 
8  16  36  25  16  8  1.0961  9.61 
9  25  8  25  36  16  1.0803  8.03 
10  25  16  36  25  8  1.0904  9.04 
11  25  25  8  16  36  1.1210  12.10 
12  25  36  16  8  25  1.1233  12.33 
13  36  8  36  16  25  1.1305  13.05 
14  36  16  25  8  36  1.1500  15.00 
15  36  25  16  36  8  1.1050  10.50 
16  36  36  8  25  16  1.1194  11.94 
K_{1}  34.63  33.72  34.75  42.11  32.71  163.68  
K_{2}  37.06  37.63  38.35  41.2  37.63  
K_{3}  41.5  43.87  42.69  40.11  42.58  
K_{4}  50.49  48.46  47.89  40.26  50.76  
Q  1711.04  1706.50  1698.54  1675.09  1718.90  P = 1674.4  
S  36.60  32.05  24.10  0.65  44.45 
Q_{i} is the average of the sum of squares of \(K_{i} (i = 1,2,3,4)\):\(Q_{ 1} = \frac{1}{4}\sum\nolimits_{i = 1}^{4} {\left( {K_{i}^{2} } \right)} = 1711\)
P is the average of the square of the sum of all data: \(P = \frac{1}{16}\left( {\sum\nolimits_{i = 1}^{16} {row_{i} } } \right)^{2} = 1674.4\)
3.1.2 Qualitative analysis
 (1)
From Fig. 5a, we can know B_{sb} increases approximately linearly as A_{s}, A_{r1}, A_{r2}, A_{a1} increases. For meshing pair sb, the influence degree of installation error and position error is ranked from big to small as: A_{s}= A_{a1}> A_{r1}= A_{r2}> A_{b1}≈0. Obviously, A_{s}, A_{a1}, A_{r1}, A_{r2} make similar difference to B_{sb}. A_{a1} makes a large difference to B_{sb} (the green line), while A_{b1} makes a little difference (the purple line). It’s somewhat surprising, considering that meshing pair sb doesn’t include short planetary gear a.
We think this abnormal phenomenon may be explained by the following hypothesis: long planetary gear b meshes with two central members (s and r_{1}) simultaneously, central members are more stable (besides this paper configures r_{1} as the fixed member), so they can partly compensate the effect of position errors of long planetary gear b, resulting in that: the position error of b makes little difference to B_{sb}. Nevertheless, for short planetary gear a, because it meshes with one planetary gear and only one central member (b and r_{2}), it is more unstable relatively, resulting in that: B_{sb} is more sensitive to the position error of short planetary gear a.
 (2)
From Fig. 5b, we can notice that B_{r1b} increases approximately linearly as A_{r1} or A_{a1} increases, especially A_{r1} (the red line). It can be seen A_{s}, A_{r2}, A_{b1} all make a little difference to B_{r1b}. Similarly, A_{a1} still makes a large difference to B_{r1b} while A_{b1} makes a little difference. For meshing pair r_{1}b, the influence degree is ranked from big to small as: A_{r1}> A_{a1}> A_{s}> A_{r2}= A_{b1}≈ 0; (3) From Fig. 5c, d, we can know B_{r2a}= B_{ba}, this is perceptible because a only meshes with b and r_{2}, so according to the law of force balance:\(F_{{r_{2} a}} = F_{ba}\), we can deduce the equation: B_{r2a}= B_{ba}. B_{r2a} (or B_{ba}) increases as A_{s}, A_{r1}, A_{r2}, A_{a1} increases, while it decreases as A_{b1} increases (the purple line), this discovery provides a possible method to improve the load sharing characteristics of gear sets in practical engineering, namely, we can appropriately increase the carrier planet pinhole position errors of long planetary gear b. For meshing pair r_{2}a (or ba), the influence degree is ranked from big to small as: A_{a1}> A_{r2}= A_{s}= A_{r1}> A_{b1}. A_{a1} makes the biggest difference to B_{r2a} (the green line), and A_{s}, A_{r1}, A_{r2} all have a similar medium influence.
Comparing with Fig. 5, more nonlinearities appear here, and the amplitude of \(B_{j} (j = sb,r_{1} b,r_{2} a,ba)\) is obviously larger, especially B_{r1b}. This indicates the load sharing behaviour of Ravigneaux planetary gear sets is more sensitive to eccentric errors than installation errors of central members and position errors of planetary gears (this phenomenon has been stated in Ref. [21]).
(1) From Fig. 6a, we can know that B_{sb} increases approximately linearly as E_{s}, E_{r1}, E_{r2}, E_{b1}, E_{a1} increases (especially E_{s}, it makes the biggest difference to B_{sb} (the blue line)), for meshing pair sb, the influence degree of various eccentric errors is ranked from big to small as: E_{s} > E_{r2} > E_{r1} > E_{b1} > E_{a1}. (2) From Fig. 6b, we can notice that B_{r1b} is obviously bigger than B_{sb} and B_{r2a}, this tells the meshing pair r_{1}b is undergoing the worst load sharing. B_{r1b} increases as E_{r1}, E_{b1} increases, and E_{r1} and E_{b1} have the similar biggest influence (the red, purple line). For meshing pair r_{1}b, the influence degree is ranked from big to small as: E_{r1}= E_{b1}> E_{s}> E_{r2}= E_{a1}. Some nonlinearities appear, especially, as E_{s} increases, the increase rate of B_{r1b} is gradually going down (the blue line). (3) From Fig. 6c, we can see that: for meshing pair r_{2}a, the influence degree is ranked from big to small as: E_{a1}= E_{r2}> E_{s}> E_{b1}> E_{r1}. There are also some nonlinearties, and it can be seen that E_{a1}, E_{r2}, E_{s} all make a remarkable difference to B_{r2a} (the green, yellow, blue line), while E_{r1} and E_{b1} make a little difference.
The influence degree of various transmission errors on various meshing pairs
Influence degree  Mesh pairs  

Type of error  sb  r_{1}b  r_{2}a  ba 
A_{i}  
A_{s}  ***  *  **  ** 
A_{r1}  ***  ***  **  ** 
A_{r2}  ***  *  **  ** 
A_{b1}  *  *  *  * 
A_{a1}  ***  **  ***  *** 
E_{j}  
E_{s}  ***  **  **  ** 
E_{r1}  **  ***  *  * 
E_{r2}  **  *  ***  *** 
E_{b1}  *  ***  *  * 
E_{a1}  *  *  ***  *** 
3.2 Variance analysis
3.2.1 Quantitative analysis about IE and CPE
Variance analysis table of A_{i} about meshing pair sb
Variance source  Sum of squares of deviations  Proportion (%) 

A_{s}  36.59625  26.55 
A_{r1}  32.0545  23.25 
A_{r2}  24.0969  17.48 
A_{b1}  0.64655  0.47 
A_{a1}  44.45235  32.25 
Sum  137.8466  100 
Similarly, variance analysis of A_{i} about meshing pair r_{1}b, r_{2}a can be calculated.We can notice that: (2) For meshing pair r_{1}b, the influence degree of A_{i} is ranked as: A_{r1}> A_{a1}> A_{s}= A_{b1}= A_{r2}≈ 0. A_{r1} makes a dominating influence (78.55%), A_{a1} counts up a second importance (19.36%), others have little influence. (3) For meshing pairs r_{2}a, the influence degree is ranked as: A_{a1}> A_{s}= A_{r1}= A_{r2}> A_{b1}≈ 0. A_{a1} makes a remarkable influence (52%), installation errors of central members (A_{s}, A_{r1}, A_{r2}) make similar influence (14–16%), while A_{b1} makes a little difference (2%).
 1.
The position errors of SHORT planetary gear (A_{a}) should be put more attention rather than the position errors of LONG planetary gear (A_{b});
 2.
A_{b}, which makes a little difference to all meshing pairs, can be ignored;
 3.
For meshing pair sb and r_{2}a, the influence degrees of installation errors of central members (A_{s}, A_{r1}, A_{r2}) are similar, they can be treated equally;
 4.
The installation of small ring gear (A_{r1}) should be paid special attention, because it makes a predominate influence to the meshing pair r_{1}b (78.55%), at the same time it has an unneglected influence to all other meshing pairs.
3.2.2 Quantitative analysis about EE
In terms of eccentric errors (EE), similarly, from the data we found:(1)For meshing pair sb, the influence degree of eccentric errors is ranked from big to small as: E_{s}> E_{r2}> E_{r1}> E_{b1}> E_{a1}. E_{s} makes the most remarkable influence (57.47%), E_{r1} and E_{r2} make a medium difference (12–18%), while E_{b1} and E_{a1} make a little difference (3–7%). (2) For meshing pair r_{1}b, the influence degree is ranked from big to small as: E_{b1}= E_{r1}> E_{s}> E_{r2}> E_{a1}. E_{b1} and E_{r1} have the most remarkable influence (33–35%), E_{s} makes a medium difference (23.81%), while E_{r2} and E_{a1} make a little difference (2–4%). (3) For meshing pair r_{2}a, the influence degree is ranked from big to small as: E_{a1}> E_{r2}> E_{s}> E_{b1}= E_{r1}. E_{a1} and E_{r2} have the most remarkable influence (34–40%), E_{s} makes a medium difference (21%), while E_{b1} and E_{r1} make a little difference (1–3%).
 1.
The eccentric error of SUN gear (E_{s}) should be put most attention, because E_{s} makes an unneglected difference to all meshing pairs
 2.
For meshing pair r_{1}b and r_{2}a, the eccentric errors of their OWN components influence the load sharing coefficient most; the eccentric error of sun gear (E_{s}) counts up a second importance; while other kinds of eccentric errors can be ignored. (Namely, E_{r2} and E_{a} don’t affect B_{r1b}, and E_{r1} and E_{b} don’t affect B_{r2a}.)
3.3 Fitting formulas
3.4 Discussion of results
In general, for different meshing pairs, the influence degree of various transmission errors are different: (1) for meshing pair sb, the position error of short planetary gear (A_{a}) makes a large difference to B_{sb} rather than A_{b} (in response to this anomaly, an explanation is given in Sect. 3.1.2), furthermore, the position error of short planetary gear (A_{a}) and the eccentric error of sun (E_{s}) are the largest influence factors (32%, 57%, respectively); (2) while for meshing pair r_{1}b, the installation error of small ring gear (A_{r1}) and the eccentric error of long planetary gear (E_{b}) are the largest influence factors (78%, 35%, respectively); (3) for meshing pair r_{2}a and ba, the position error of short planetary gear (A_{a}) and the eccentric error of short planetary gear (E_{a}) make the largest difference (52%, 40%, respectively).
The three most important errors to LSC of Ravigneaux planetary gear sets are: A_{a} (156.67%), A_{r1} (131.21%), E_{s} (123.25%). In practical engineering application, the precision of these three pivotal members are more deserved to be ensured, under the limitation of production cost and manufacture technology, pertinently decreasing errors of these pivotal members would improve the load sharing behaviour efficiently. Besides, the carrier planet pinhole position errors of long planetary gear (A_{b}) can be appropriately increased, in order to improve the load sharing behaviour. The validity of proposed fitting formulas (3.9)–(3.11) has been testified by comparing fitting formula results with Maple results.
4 Conclusions
 1.
Load sharing coefficients of different meshing pairs are sensitive to different kinds of errors. The following errors have a more remarkable influence to LSC: the position error of short planetary gear (A_{a}), the installation error of small ring gear (A_{r1}), the eccentric error of sun (E_{s}).
 2.
For meshing pair r_{1}b and r_{2}a, the eccentric errors of their own components influence the load sharing coefficient most remarkably, E_{r2} and E_{a} don’t make difference to B_{r1b}, and E_{r1} and E_{b} don’t make difference to B_{r2a}.
 3.
The position errors of long planetary gear b (A_{b}) can be appropriately increased to improve the load sharing behaviour. The installation error of large ring gear (A_{r2}), the eccentric error of small ring (E_{r1}), and the eccentric error of long planetary gear (E_{b}) make a small influence to LSC of meshing pairs.
 4.
Several effective fitting formulas about the relationship between transmission errors and LSC are proposed, and they can be used to instead the complex theoretical computation to obtain LSC conveniently.
Notes
Acknowledgements
This research is supported by the Electromechanical Transmission and Control Laboratory, the Power and Mechanical School, Wuhan University, Wuhan city, Hubei province, P.R.China. The author thanks Pro. Wu Shijing and Pro. Wang Xiaosun of the Power and Mechanical school for their valuable instructions.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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