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SN Applied Sciences

, 2:60 | Cite as

Influence of transmission errors to load sharing behaviour in Ravigneaux planetary gear sets

  • Shaobo ChengEmail author
  • Shijing Wu
  • Xiaosun Wang
Research Article
  • 108 Downloads
Part of the following topical collections:
  1. Engineering: Adaptive Model, Mesh and Resolution Techniques for Computationally-Demanding Problems in Simulation-Based Engineering

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 translational-torsional 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 formula 

1 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 torque-to-weight 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 time-varying 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 axis-symmetric 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 run-out 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, multi-stage 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. Ai-shyyab [13] used a hybrid harmonic balance method (HBM) to semi-analytically solve the torsional dynamic model of a multi-stage 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, time-varying 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 double-floating, single-floating and non-floating, found the load sharing coefficient of double-floating structure system is 9% smaller than that of the single -floating structure, which is 16% smaller than that of non-floating structure. In 2012, Kim [17] researched the influence of time-varying pressure angle and contact ratio caused by bearing deformation, found that time-varying 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 step-size 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 single-period response, quasi-periodic response, multi-periodic response and chaotic response successively. Zhu and Wu [20] developed the translational-torsional 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 uncertain-but-bounded 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 above-mentioned 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], time-varying 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. Multi-factor 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 multi-factor analysis but also quantitative analysis about the influence degree are realized.

As Fig. 1 illustrates: (1) First, according to the lumped mass method and second Lagrange equation, 30 DOF translational-torsional coupling dynamic equations of motion of Ravigneaux planetary gear sets are derived. The load sharing coefficients of gear sets under the condition of different transmission errors are calculated via Maple. (2) Then, according to the results of orthogonal experiment, this paper qualitatively analyzes the influence degree of transmission errors on the load sharing coefficients of gear sets. Key errors and key members which affect load sharing behaviour most are found out. (3) Next, according to the results of variance analysis, this paper quantitatively reveals the influence degree of transmission errors on the load sharing coefficients, and some meaningful conclusions are obtained. (4) Finally, several fitting formulas are innovatively proposed, to calculate the load sharing coefficients of Ravigneaux planetary gear sets conveniently. These achievements can promote the development of researches on load sharing characteristics of Ravigneaux planetary gear sets and have guiding significance for realizing the load sharing in practical engineering.
Fig. 1

Analysis flow chart

2 Dynamic model

The structure diagram of double-ring Ravigneaux planetary gear sets is shown as Fig. 2 It consists of one sun gear, two ring gears, one carrier and two kinds of planetary gears. Planetary gears are mounted on a single carrier c and equally spaced. It is obvious that all meshing pairs include \(sb_{n} ,r_{1} b_{n} ,r_{2} a_{n} ,b_{n} a_{n} (n = 1,2 \ldots N)\) (N is the group number of planetary gears, in Fig. 2 N = 3).
Fig. 2

Double-ring Ravigneaux planetary gear sets

In order to procure the translational-torsional dynamic model of the system (Fig. 3), some reasonable simplifications as follows are necessary:
Fig. 3

The translational-torsional model of double-ring Ravigneaux planetary gear sets

  1. 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. 2.

    All members are assumed as rigid bodies. Flexibility of members (especially the flexibility of ring gears) is neglected.

     
  3. 3.

    All mesh forces between gears are in one mesh plane, the dynamics problem of CPGS can be regarded as a two-dimensional problem rather than three-dimensional.

     
  4. 4.

    All frictions between meshing pairs are neglected, there are only elastic forces and damping forces between meshing pairs.

     
  5. 5.

    For the same kind of planetary gears, their mass as well as moment of inertia is exactly equal.

     
Nomenclature
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: = Ii/ri2

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 time-varying mesh stiffness (TVMS), backlash and transmission errors.

Generally speaking, for the sake of the solving convenience, a Cosine function as Eq. (2.1) is used to express TVMS.
$$k_{j} (t) = \bar{k}_{j} + k_{j\_fluc} \cdot \cos (\omega_{m} \cdot t + \varphi_{j} ),\quad j = sb,r_{1} b,r_{2} a,ba$$
(2.1)

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 piece-wise non-linear function.

$$f\left( {\delta_{j} } \right) = \left\{ {\begin{array}{*{20}l} {\delta_{j} - b,} \hfill & {{\text{if}}:\delta_{j} > b} \hfill \\ {0,} \hfill & {{\text{else}}\;{\text{if}}: - b < \delta_{j} < b} \hfill \\ {\delta_{j} + b,} \hfill & {{\text{else}}:\delta_{j} < - b} \hfill \\ \end{array} } \right.$$
(2.2)
here \(\delta_{j}\) is the relative mesh displacement of meshing pair, b is the half of backlash.

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.

As Fig. 4 shows,\(\theta_{j} \left( {j = sb,sa,ba} \right)\) is the relative position angle between the sun gear, the long planetary gear, and the short planetary gear. Utilizing Cosine Law, they can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {\theta_{sb} = {\text{arc}}\,\cos \left( {\frac{{L_{sb}^{2} + L_{ba}^{2} - L_{sa}^{2} }}{{2L_{sb} L_{ba} }}} \right)} \\ {\theta_{sa} = {\text{arc}}\,\cos \left( {\frac{{L_{sa}^{2} + L_{ba}^{2} - L_{sb}^{2} }}{{2L_{sa} L_{ba} }}} \right)} \\ {\theta_{ba} = {\text{arc}}\,\cos \left( {\frac{{L_{sb}^{2} + L_{aa}^{2} - L_{ba}^{2} }}{{2L_{sb} L_{sa} }}} \right)} \\ \end{array} } \right.$$
(2.3)
here \(L_{sb} = r_{ps} + r_{pb} ,L_{sa} = r_{{pr_{2} }} - r_{pa} ,L_{ba} = r_{pb} + r_{pa}\), \(r_{pi} (i = s,b,a,r_{1} ,r_{2} )\) is the radius of pitch circle of member i.
Fig. 4

Relative position angle between s, b and a

Then, position angle of planetary gears can be expressed as:
$$\left\{ \begin{aligned} &\psi_{{b_{n} }} = \frac{{2 \cdot \pi \cdot \left( {n - 1} \right)}}{N} \hfill \\ &\psi_{{a_{n} }} = \frac{{2 \cdot \pi \cdot \left( {n - 1} \right)}}{N} + \theta_{ba} \hfill \\ &\psi_{{b_{n} a_{n} }} = \theta_{sb} - \frac{\alpha \cdot \pi }{180} - \psi_{{b_{n} }} \hfill \\ \end{aligned} \right.$$
(2.4)
Here \(\psi_{{b_{n} }}\) is the position angle of long planetary gear \(b_{n}\),\(\psi_{{a_{n} }}\) is the position angle of short planetary gear \(a_{n}\), \(\psi_{{b_{n} a_{n} }}\) is the relative mesh angle between \(b_{n}\) and \(a_{n}\).
The relative mesh angle of meshing pairs can be defined as:
$$\left\{ \begin{aligned}&\psi_{{sb_{n} }} = \psi_{{b_{n} }} - \alpha_{s} \hfill \\ &\psi_{{r_{1} b_{n} }} = \psi_{{b_{n} }} + \alpha_{{r_{1} }} \hfill \\ &\psi_{{r_{2} a_{n} }} = \psi_{{a_{n} }} + \alpha_{{r_{2} }} \hfill \\ \end{aligned} \right.$$
(2.5)
The installation errors of central members (IE) are generated due to the inconsistency between the actual installation central points and theoretical central points of the central member [20]. This paper projects these errors to the mesh line as:
$$\left\{ \begin{aligned} &\bar{A}_{{s - sb_{n} }} = A_{s} \cdot \sin \left( { - \omega_{c} t + \alpha_{s} + \lambda_{s} - \psi_{{b_{n} }} } \right) \hfill \\ &\bar{A}_{{r_{1} - r_{1} b_{n} }} = A_{{r_{1} }} \cdot \sin \left( {\omega_{c} t + \alpha_{{r_{1} }} - \lambda_{{r_{1} }} + \psi_{{b_{n} }} } \right) \hfill \\ &\bar{A}_{{r_{2} - r_{2} b_{n} }} = A_{{r_{2} }} \cdot \sin \left( {\omega_{c} t + \alpha_{{r_{2} }} + \theta_{ba} - \lambda_{{r_{2} }} + \psi_{{b_{n} }} } \right) \hfill \\ \end{aligned} \right.$$
(2.6)
Here, \(A_{i} (i = s,r_{1} ,r_{2} )\) is the amplitude of installation error of central members, \(\omega_{c}\) is the angular velocity of carrier, \(\alpha_{i} (i = s,r_{1} ,r_{2} )\) is the pressure angle, \(\lambda_{i} (i = s,r_{1} ,r_{2} )\) is the initial phase angle of installation error.
The carrier planet pinhole position errors (CPE) are the inconsistency between the planetary axis on the carrier and the central hole of the planetary gear. It’s time-invariant and independent of assembly [7]. This paper projects these errors to the mesh line as:
$$\left\{ \begin{aligned} &\bar{A}_{{b_{n} - sb_{n} }} = - A_{{b_{n} }} \cdot \sin \left( {\alpha_{b} + \lambda_{{b_{n} }} } \right) \hfill \\ &\bar{A}_{{b_{n} - r_{1} b_{n} }} = A_{{b_{n} }} \cdot \sin \left( {\alpha_{{r_{1} }} - \lambda_{{b_{n} }} } \right) \hfill \\ &\bar{A}_{{b_{n} - b_{n} a_{n} }} = A_{{b_{n} }} \cdot \sin \left( {\alpha_{b} + \pi - \theta_{sb} - \lambda_{{b_{n} }} } \right) \hfill \\ &\bar{A}_{{a_{n} - b_{n} a_{n} }} = - A_{{a_{n} }} \cdot \sin \left( {\alpha_{a} + \theta_{sa} - \lambda_{{a_{n} }} } \right) \hfill \\ &\bar{A}_{{a_{n} - r_{2} a_{n} }} = A_{{a_{n} }} \cdot \sin \left( { - \alpha_{{r_{2} }} + \lambda_{{a_{n} }} } \right) \hfill \\ \end{aligned} \right.$$
(2.7)
Here,\(A_{i} (i = b_{n} ,a_{n} )\) is the amplitude of carrier planet pinhole position error, \(\alpha_{i} (i = b,a,r_{1} ,r_{2} )\) is the pressure angle, \(\lambda_{i} (i = b_{n} ,a_{n} )\) is the initial phase angle of position error.
The eccentric errors of gears (EE) is also called as “gear radial circular run-out errors”. These errors exist in the gear system when the gear rotates around its own theoretical center, its rotation center produces a certain offset at the pitch circle. It’s time-varying and dependent of assembly [7]. This paper projects these errors to the mesh line as:
$$\left\{ \begin{aligned} &\bar{E}_{{s - sb_{n} }} = E_{s} \cdot \sin \left( {\left( {\omega_{s} - \omega_{c} } \right) \cdot t + \alpha_{s} + \eta_{s} - \psi_{{b_{n} }} } \right) \hfill \\ &\bar{E}_{{b_{n} - sb_{n} }} = - E_{{b_{n} }} \cdot \sin \left( {\left( {\omega_{b} - \omega_{c} } \right) \cdot t + \alpha_{b} + \eta_{{b_{n} }} } \right) \hfill \\ &\bar{E}_{{b_{n} - r_{1} b_{n} }} = E_{{b_{n} }} \cdot \sin \left( { - \left( {\omega_{b} - \omega_{c} } \right) \cdot t + \alpha_{{r_{1} }} - \eta_{{b_{n} }} } \right) \hfill \\ &\bar{E}_{{b{}_{n} - b_{n} a_{n} }} = E_{{b_{n} }} \cdot \sin \left( { - \left( {\omega_{b} - \omega_{c} } \right) \cdot t + \alpha_{b} + \pi - \nu_{sb} - \eta_{{b_{n} }} } \right) \hfill \\ &\bar{E}_{{a_{n} - b_{n} a_{n} }} = - E_{{a_{n} }} \cdot \sin \left( { - \left( {\omega_{a} - \omega_{c} } \right) \cdot t + \alpha_{a} + \nu_{sa} - \eta_{{a_{n} }} } \right) \hfill \\ &\bar{E}_{{a_{n} - r_{2} a_{n} }} = E_{{a_{n} }} \cdot \sin \left( {\left( {\omega_{a} - \omega_{c} } \right) \cdot t - \alpha_{{r_{2} }} + \eta_{{a_{n} }} } \right) \hfill \\ &\bar{E}_{{r_{1} - r_{1} b_{n} }} = E_{{r_{1} }} \cdot \sin \left( { - \left( {\omega_{{r_{1} }} - \omega_{c} } \right) \cdot t + \alpha_{r1} - \eta_{{r_{1} }} + \psi_{{b_{n} }} } \right) \hfill \\ &\bar{E}_{{r_{2} - r_{2} a_{n} }} = E_{{r_{2} }} \cdot \sin \left( { - \left( {\omega_{{r_{2} }} - \omega_{c} } \right) \cdot t + \alpha_{r2} + \nu_{ba} - \eta_{{r_{2} }} + \psi_{{b_{n} }} } \right) \hfill \\ \end{aligned} \right.$$
(2.8)
Here \(E_{i} (i = s,r_{1} ,r_{2} ,b_{n} ,a_{n} )\) is the amplitude of eccentric error, \(\eta_{i} (i = s,r_{1} ,r_{2} ,b_{n} ,a_{n} )\) is the initial phase angle of eccentric error.
To sum up, transmission errors eij between meshing pairs can be expressed as:
$$e_{ij} = A_{i - ij} + A_{j - ij} + E_{i - ij} + E_{j - ij} \, \left( {ij = sb_{n} ,r_{1} b_{n} ,r_{2} a_{n} ,b_{n} a_{n} } \right)$$
(2.9)

2.2 System equations of motion

This paper chooses general coordinates as:
$$q = \left[ {x_{i} ,y_{i} ,u_{i} ,x_{{b_{n} }} ,y_{{b_{n} }} ,u_{{b_{n} }} ,x_{{a_{n} }} ,y_{{a_{n} }} ,u_{{b_{n} }} } \right]^{T} ,(i = s,c,r_{1} ,r_{2} ;n = 1,2,3)$$
(2.10)
By utilizing the second Lagrange equation, equations of motion of double-ring Ravigneaux planetary gear sets can be obtained. Here, considering that the floating of central members would not influence the computation results a lot and this paper is mainly focused on the influence of synthetic mesh errors on the load sharing coefficient, floating of central members are ignored. This paper selects the dimensionless nominal scale:\(b_{c} = 10\,\upmu{\text{m}}\), \(\omega_{d} = \sqrt {\bar{k}_{sb} /\left( {1/m_{s} + 1/m_{b} } \right)}\). These equations of motion can be nondimensionalized by these dimensionless nominal scales (dimensionless equations of motion are listed in “Appendix”). Load sharing coefficient (LSC) represents the ability of gear system to distribute load uniformly between different planets, the larger LSC is, the weaker this ability is. LSC of meshing pairs are defined as Ref. [22],\(B_{j} (j = sb,r_{1} b,r_{2} a,ba)\) represents the load sharing coefficient of meshing pair j. According to the practical engineering, this paper configures r1 as the fixed member, s as the input member while r2 as the output member. The input torque from s: \(T_{s} = 1000\,{\text{N}}\,{\text{m}}\), the rotational speed of s: \(n_{s} = 1000{\text{r}}/\hbox{min}\), since the transmission ratio \(i = \frac{{\omega_{s} }}{{\omega_{{r_{2} }} }} = \frac{{z_{{r_{2} }} \cdot \left( {z_{s} + z_{r1} } \right)}}{{z_{s} \cdot \left( {z_{r1} + z_{r2} } \right)}} = 1.669\), the output torque from r2 can be calculated: \(T_{{r_{2} }} = - T_{s} \cdot i = - 1669.6\,{\text{N}}\,{\text{m}}\). Mesh damping ratio: \(\xi = 0.07\), half of backlash: b = 10 μm. Based the above-mentioned parameters, we have written the Maple program to solve these equations. Basic parameters of double-ring Ravigneaux planetary gear sets to be solved are listed in Table 1.
Table 1

Basic Parameters of double-ring Ravigneaux gear sets

Parameter

Sun (s)

Small ring (r1)

Large ring (r2)

Carrier (c)

Long planetary gear (bn)

Short planetary gear (an)

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 mm2)

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 multi-factor 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

The results of orthogonal experiment about installation errors and carrier planet pinhole position errors are listed in Table 2, while Table 3 is about eccentric errors.
Table 2

Orthogonal experiment results about IE and CPE

Series No.

As

Ar1

Ar2

Ab1

Aa1

Bsb

Br1b

Br2a

Bba

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

Table 3

Orthogonal experiment results about EE

Series no.

Es

Er1

Er2

Eb1

Ea1

Bsb

Br1b

Br2a

Bba

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

In order to simplify the calculation, take a algebra substitution as Eq. (3.1) shows, (it can be easily mathematically proved that this substitution doesn’t make any difference to the result of analysis).
$$BB_{j} = 100*\left( {B_{j} - 1} \right) \, \left( {j = sb,r_{1} b,r_{2} a,ba} \right)$$
(3.1)
Take Table 2 for instance: for meshing pair sb, relevant data after substitution are listed in the following Table 4.
Table 4

Orthogonal experiment results about meshing pair sb (after substitution)

Series no.

As

Ar1

Ar2

Ab1

Aa1

Bsb

BBsb

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

K1

34.63

33.72

34.75

42.11

32.71

 

163.68

K2

37.06

37.63

38.35

41.2

37.63

  

K3

41.5

43.87

42.69

40.11

42.58

  

K4

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

  
Here \(K_{i} (i = 1,2,3,4)\) represents the sum of \(BB_{sb}\) in same level of \(A_{i}\), for example, rows from 1 to 4 are corresponding to As = 8 μm, therefore:
$$K_{1} = \sum\limits_{i = 1}^{4} {row_{i} } = 3.56 + 6.44 + 10.05 + 14.58 = 34.63$$
(3.2)
\(K_{i} (i = 2,3,4)\) is corresponding to As= 16, 25, 36 μm respectively, similarly, they can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {K_{2} = \sum\limits_{i = 5}^{8} {row_{i} } = 9.08 + 7.15 + 11.22 + 9.61 = 37.06} \\ {K_{3} = \sum\limits_{i = 9}^{12} {row_{i} } = 8.03 + 9.04 + 12.10 + 12.33 = 41.50} \\ {K_{4} = \sum\limits_{i = 13}^{16} {row_{i} } = 13.05 + 15 + 10.5 + 11.94 = 50.49} \\ \end{array} } \right.$$
(3.3)
  • Qi 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: \(P = \frac{1}{16}\left( {\sum\nolimits_{i = 1}^{16} {row_{i} } } \right)^{2} = 1674.4\)

Si is the sum of squares of deviations (SSD), which is equal to the difference between Qi and P, and S1 represents the influence degree of factor 1 (namely, As):
$$S_{ 1} = Q_{ 1} - P = 1711 - 1674.4 = 36.6$$
(3.4)

3.1.2 Qualitative analysis

The relationship between \(A_{i} (i = s,r_{1} ,r_{2} ,b_{1} ,a_{1} )\) and \(B_{j} (j = sb,r_{1} b,r_{2} a,ba)\) is as Fig. 5 shows:
Fig. 5

Influence of various Ai on different meshing pairs. a meshing pair sb, b meshing pair, r1b, c meshing pair r2a, d meshing pair ba

  1. (1)

    From Fig. 5a, we can know Bsb increases approximately linearly as As, Ar1, Ar2, Aa1 increases. For meshing pair sb, the influence degree of installation error and position error is ranked from big to small as: As= Aa1> Ar1= Ar2> Ab1≈0. Obviously, As, Aa1, Ar1, Ar2 make similar difference to Bsb. Aa1 makes a large difference to Bsb (the green line), while Ab1 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 r1) simultaneously, central members are more stable (besides this paper configures r1 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 Bsb. Nevertheless, for short planetary gear a, because it meshes with one planetary gear and only one central member (b and r2), it is more unstable relatively, resulting in that: Bsb is more sensitive to the position error of short planetary gear a.

     
  2. (2)

    From Fig. 5b, we can notice that Br1b increases approximately linearly as Ar1 or Aa1 increases, especially Ar1 (the red line). It can be seen As, Ar2, Ab1 all make a little difference to Br1b. Similarly, Aa1 still makes a large difference to Br1b while Ab1 makes a little difference. For meshing pair r1b, the influence degree is ranked from big to small as: Ar1> Aa1> As> Ar2= Ab1 0; (3) From Fig. 5c, d, we can know Br2a= Bba, this is perceptible because a only meshes with b and r2, so according to the law of force balance:\(F_{{r_{2} a}} = F_{ba}\), we can deduce the equation: Br2a= Bba. Br2a (or Bba) increases as As, Ar1, Ar2, Aa1 increases, while it decreases as Ab1 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 r2a (or ba), the influence degree is ranked from big to small as: Aa1> Ar2= As= Ar1> Ab1. Aa1 makes the biggest difference to Br2a (the green line), and As, Ar1, Ar2 all have a similar medium influence.

     
Similarly, the relationship between \(E_{i} (i = s,r_{1} ,r_{2} ,b_{1} ,a_{1} )\) and \(B_{j} (j = sb,r_{1} b,r_{2} a,ba)\) is as Fig. 6 demonstrates.
Fig. 6

Influence of various Ei on different meshing pairs. a meshing pair sb, b meshing pair r1b, c meshing pair r2a

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 Br1b. 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 Bsb increases approximately linearly as Es, Er1, Er2, Eb1, Ea1 increases (especially Es, it makes the biggest difference to Bsb (the blue line)), for meshing pair sb, the influence degree of various eccentric errors is ranked from big to small as: Es > Er2 > Er1 > Eb1 > Ea1. (2) From Fig. 6b, we can notice that Br1b is obviously bigger than Bsb and Br2a, this tells the meshing pair r1b is undergoing the worst load sharing. Br1b increases as Er1, Eb1 increases, and Er1 and Eb1 have the similar biggest influence (the red, purple line). For meshing pair r1b, the influence degree is ranked from big to small as: Er1= Eb1> Es> Er2= Ea1. Some nonlinearities appear, especially, as Es increases, the increase rate of Br1b is gradually going down (the blue line). (3) From Fig. 6c, we can see that: for meshing pair r2a, the influence degree is ranked from big to small as: Ea1= Er2> Es> Eb1> Er1. There are also some nonlinearties, and it can be seen that Ea1, Er2, Es all make a remarkable difference to Br2a (the green, yellow, blue line), while Er1 and Eb1 make a little difference.

Above-mentioned analysis is just qualitative (As Table 5 lists), in order to set up a quantitative description, variance analysis is necessary, Sect. 3.2 will discuss it.
Table 5

The influence degree of various transmission errors on various meshing pairs

Influence degree

Mesh pairs

Type of error

sb

r1b

r2a

ba

Ai

    

As

***

*

**

**

Ar1

***

***

**

**

Ar2

***

*

**

**

Ab1

*

*

*

*

Aa1

***

**

***

***

Ej

    

Es

***

**

**

**

Er1

**

***

*

*

Er2

**

*

***

***

Eb1

*

***

*

*

Ea1

*

*

***

***

*** Remarkable influence

** Medium influence

* Subtle influence

3.2 Variance analysis

3.2.1 Quantitative analysis about IE and CPE

Si, as is mentioned in Sect. 3.1.1, is the sum of squares of deviations (SSD), and represents the influence degree of various factors (As, Ar1, Ar2, Ab1, Aa1), so the proportion of Si to \(\sum\nolimits_{i = 1}^{5} {S_{i} }\) can indicate the relative influence degree of various errors quantitatively, as Table 6 displays. From it we can know that: (1) For meshing pair sb, the influence degree of Ai is ranked as: Aa1> As> Ar1> Ar2> Ab1≈ 0 (it coincides with the results in Sect. 3.1.2). Aa1 makes the most remarkable influence (32.25%), while Ab1 almost makes little difference. Aa1 makes a bigger influence to meshing pair sb than Ab1 does, a reasonable explanation has been given in Sect. 3.1.2.
Table 6

Variance analysis table of Ai about meshing pair sb

Variance source

Sum of squares of deviations

Proportion (%)

As

36.59625

26.55

Ar1

32.0545

23.25

Ar2

24.0969

17.48

Ab1

0.64655

0.47

Aa1

44.45235

32.25

Sum

137.8466

100

Similarly, variance analysis of Ai about meshing pair r1b, r2a can be calculated.We can notice that: (2) For meshing pair r1b, the influence degree of Ai is ranked as: Ar1> Aa1> As= Ab1= Ar2≈ 0. Ar1 makes a dominating influence (78.55%), Aa1 counts up a second importance (19.36%), others have little influence. (3) For meshing pairs r2a, the influence degree is ranked as: Aa1> As= Ar1= Ar2> Ab1≈ 0. Aa1 makes a remarkable influence (52%), installation errors of central members (As, Ar1, Ar2) make similar influence (14–16%), while Ab1 makes a little difference (2%).

Obviously, these quantitative conclusions are highly consistent with the qualitative conclusions of Sect. 3.1.2. Therefore, four important conclusions can be drawn:
  1. 1.

    The position errors of SHORT planetary gear (Aa) should be put more attention rather than the position errors of LONG planetary gear (Ab);

     
  2. 2.

    Ab, which makes a little difference to all meshing pairs, can be ignored;

     
  3. 3.

    For meshing pair sb and r2a, the influence degrees of installation errors of central members (As, Ar1, Ar2) are similar, they can be treated equally;

     
  4. 4.

    The installation of small ring gear (Ar1) should be paid special attention, because it makes a predominate influence to the meshing pair r1b (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: Es> Er2> Er1> Eb1> Ea1. Es makes the most remarkable influence (57.47%), Er1 and Er2 make a medium difference (12–18%), while Eb1 and Ea1 make a little difference (3–7%). (2) For meshing pair r1b, the influence degree is ranked from big to small as: Eb1= Er1> Es> Er2> Ea1. Eb1 and Er1 have the most remarkable influence (33–35%), Es makes a medium difference (23.81%), while Er2 and Ea1 make a little difference (2–4%). (3) For meshing pair r2a, the influence degree is ranked from big to small as: Ea1> Er2> Es> Eb1= Er1. Ea1 and Er2 have the most remarkable influence (34–40%), Es makes a medium difference (21%), while Eb1 and Er1 make a little difference (1–3%).

These quantitative conclusions are consistent with the qualitative conclusions of Sect. 3.1.2, so following conclusions can be drawn:
  1. 1.

    The eccentric error of SUN gear (Es) should be put most attention, because Es makes an unneglected difference to all meshing pairs

     
  2. 2.

    For meshing pair r1b and r2a, the eccentric errors of their OWN components influence the load sharing coefficient most; the eccentric error of sun gear (Es) counts up a second importance; while other kinds of eccentric errors can be ignored. (Namely, Er2 and Ea don’t affect Br1b, and Er1 and Eb don’t affect Br2a.)

     
The quantitative description about the law of influence can be summarized as follows: the three most important errors to the load sharing coefficient are: Aa1 (156.67%), Ar1 (131.21%), Es (123.25%).They have been already marked red in Fig. 7. Therefore, in practical engineering, precision of these three members should be guaranteed first, in order to improve the load sharing characteristic.
Fig. 7

The key components and key errors to the load sharing coefficient

3.3 Fitting formulas

Based on the above-mentioned qualitative and quantitative conclusions, this paper proposes several fitting formulas to calculate the LSC of meshing pairs directly. From Figs. 5 and 6, it can be seen that most lines are approximately linear, so this paper neglects the nonlinearities and presumes fitting formulas are linear. Take the meshing pair sb for instance, first, only taking into account IE&CPE, the mathematical form of its fitting formula can be expressed as Eq. (3.5),
$$B_{sb}^{A} - 1 = k \cdot \left( {0.2655 \cdot A_{s} + 0.2325 \cdot A_{{r_{1} }} + 0.1748 \cdot A_{{r_{2} }} + 0.0047 \cdot A_{{b_{1} }} + 0.3225 \cdot A_{{a_{1} }} } \right) + b$$
(3.5)
In order to solve parameters k and b, this paper defines matrices as follows,
$$A = \left[ {\begin{array}{*{20}l} \begin{aligned} 0 \hfill \\ 8 \hfill \\ \end{aligned} & \begin{aligned} 0 \hfill \\ 8 \hfill \\ \end{aligned} & \begin{aligned} 0 \hfill \\ 8 \hfill \\ \end{aligned} & \begin{aligned} 0 \hfill \\ 8 \hfill \\ \end{aligned} & \begin{aligned} 0 \hfill \\ 8 \hfill \\ \end{aligned} \\ 8 & {16} & {16} & {16} & {16} \\ {} & {} & \ldots & {} & {} \\ {36} & {36} & 8 & {25} & {16} \\ \end{array} } \right]_{17 \times 5} ,B = \left[ {\begin{array}{*{20}l} {0.2655} \\ {0.2325} \\ {0.1748} \\ {0.0047} \\ {0.3225} \\ \end{array} } \right]_{5 \times 1} ,C = A \cdot B,D = \left[ {\begin{array}{*{20}l} \begin{aligned} 0.0000 \hfill \\ 0.0356 \hfill \\ \end{aligned} \\ {0.0644} \\ \ldots \\ {0.1458} \\ \end{array} } \right]_{17 \times 1}$$
(3.6)
Then, fitting formula (3.5) can be expressed in matrix forms,\(D = k \cdot C + b\). Finally, utilizing “LeastSquares ()” function of the function package “CurveFitting” in Maple, parameters k and b can be solved: k = 0.00513, b = − 0.0062, so the fitting formula about meshing pair sb is as follows,
$$B_{sb}^{A} = 0.00513 \cdot \left( {0.2655 \cdot A_{s} + 0.2325 \cdot A_{{r_{1} }} + 0.1748 \cdot A_{{r_{2} }} + 0.0047 \cdot A_{{b_{1} }} + 0.3225 \cdot A_{{a_{1} }} } \right) - 0.0062 + 1$$
(3.7)
Then, only taking into account EE, the fitting formula can be expressed as,
$$B_{sb}^{E} = 0.00669 \cdot \left( {0.5747 \cdot E_{s} + 0.1275 \cdot E_{{r_{1} }} + 0.1873 \cdot E_{{r_{2} }} + 0.0718 \cdot E_{{b_{1} }} + 0.0387 \cdot E_{{a_{1} }} } \right) + 1.0223$$
(3.8)
Finally, formula (3.7) and (3.8) can be assembled to obtain the general fitting formula which includes IE, CPE&EE:\(B_{sb} = B_{sb}^{A} \cdot B_{sb}^{E}\), then omitting the higher-order terms, the general fitting formula can be written as Eq. (3.9), analogously, other fitting formulas are written as Eq. (3.10) and (3.11).
$$\begin{aligned} B_{sb} & = B_{sb}^{A} + B_{sb}^{E} - 1 \\ & = 0.00513 \cdot \left[ {\begin{array}{*{20}l} {A_{s} } & {A_{{r_{1} }} } & {A_{{r_{2} }} } & {A_{{b_{1} }} } & {A_{{a_{1} }} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}l} {0.2655} & {0.2325} & {0.1748} & {0.0047} & {0.3225} \\ \end{array} } \right]^{T} \\ & \quad + 0.00669 \cdot \left[ {\begin{array}{*{20}l} {E_{s} } & {E_{{r_{1} }} } & {E_{{r_{2} }} } & {E_{{b_{1} }} } & {E_{{a_{1} }} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}l} {0.5747} & {0.1275} & {0.1873} & {0.0718} & {0.0387} \\ \end{array} } \right]^{T} + 1.0161 \\ \end{aligned}$$
(3.9)
$$\begin{aligned} B_{{r_{1} b}} & = 0.00856 \cdot \left[ {\begin{array}{*{20}l} {A_{s} } & {A_{{r_{1} }} } & {A_{{r_{2} }} } & {A_{{b_{1} }} } & {A_{{a_{1} }} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}l} {0.0129} & {0.7855} & {0.0030} & {0.0049} & {0.1936} \\ \end{array} } \right]^{T} \\ & \quad + 0.01658 \cdot \left[ {\begin{array}{*{20}l} {E_{s} } & {E_{{r_{1} }} } & {E_{{r_{2} }} } & {E_{{b_{1} }} } & {E_{{a_{1} }} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}l} {0.2381} & {0.3393} & {0.0459} & {0.3512} & {0.0255} \\ \end{array} } \right]^{T} + 1.0178 \\ \end{aligned}$$
(3.10)
$$\begin{aligned} B_{{r_{2} a}} & = 0.00676 \cdot \left[ {\begin{array}{*{20}l} {A_{s} } & {A_{{r_{1} }} } & {A_{{r_{2} }} } & {A_{{b_{1} }} } & {A_{{a_{1} }} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}l} {0.1660} & {0.1475} & {0.1409} & {0.0201} & {0.5255} \\ \end{array} } \right]^{T} \\ & \quad + 0.0085 \cdot \left[ {\begin{array}{*{20}l} {E_{s} } & {E_{{r_{1} }} } & {E_{{r_{2} }} } & {E_{{b_{1} }} } & {E_{{a_{1} }} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}l} {0.2101} & {0.0110} & {0.3414} & {0.0343} & {0.4031} \\ \end{array} } \right]^{T} + 1.0106 \\ \end{aligned}$$
(3.11)
Here, we compared the results of fitting formulas (3.9)–(3.11) with the results of Maple calculation, as is shown in Fig. 8, they coincide well with each other, the fitting formula results are just slightly larger than Maple results (the maximum value of relative error is 3.0%). Therefore, fitting formulas (3.9)–(3.11) are all effective, and can be directly used to instead the complex theoretical calculation, to obtain the LSC of meshing pairs (sb, r1b, r2a) conveniently.
Fig. 8

Comparison between fitting formula (3.9)–(3.11) and Maple

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 (Aa) makes a large difference to Bsb rather than Ab (in response to this anomaly, an explanation is given in Sect. 3.1.2), furthermore, the position error of short planetary gear (Aa) and the eccentric error of sun (Es) are the largest influence factors (32%, 57%, respectively); (2) while for meshing pair r1b, the installation error of small ring gear (Ar1) and the eccentric error of long planetary gear (Eb) are the largest influence factors (78%, 35%, respectively); (3) for meshing pair r2a and ba, the position error of short planetary gear (Aa) and the eccentric error of short planetary gear (Ea) make the largest difference (52%, 40%, respectively).

The three most important errors to LSC of Ravigneaux planetary gear sets are: Aa (156.67%), Ar1 (131.21%), Es (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 (Ab) 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

In the present work, the influence of transmission errors to LSC of Ravigneaux planetary gear sets are investigated, using a combination of orthogonal experiment method and variance analysis. Some innovative discoveries are summarized as follows:
  1. 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 (Aa), the installation error of small ring gear (Ar1), the eccentric error of sun (Es).

     
  2. 2.

    For meshing pair r1b and r2a, the eccentric errors of their own components influence the load sharing coefficient most remarkably, Er2 and Ea don’t make difference to Br1b, and Er1 and Eb don’t make difference to Br2a.

     
  3. 3.

    The position errors of long planetary gear b (Ab) can be appropriately increased to improve the load sharing behaviour. The installation error of large ring gear (Ar2), the eccentric error of small ring (Er1), and the eccentric error of long planetary gear (Eb) make a small influence to LSC of meshing pairs.

     
  4. 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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The Department of Power and Mechanical EngineeringWuhan UniversityWuhanChina

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