Dynamic reliability analysis for planetary gear system in shearer mechanisms

Mechanism and Machine Theory 105 (2016) 244–259

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Dynamic reliability analysis for planetary gear system in shearer mechanisms Di Zhou, Xufang Zhang ⁎, Yimin Zhang ⁎ School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China

a r t i c l e

i n f o

Article history: Received 7 February 2016 Received in revised form 6 July 2016 Accepted 6 July 2016 Available online xxxx Keywords: Reliability-based sensitivity Time-variant reliability analysis Saddle-point approximation Planetary gear system Shearer

a b s t r a c t The paper considers the reliability and reliability-based sensitivity analyses of planetary gear systems in shearer mechanisms. It is important to quantify the dynamic performance of planetary gear system to aid in risk assessment and risk mitigation of the shearer mechanism. Failure of the system is defined as the dynamic contact stress exceeding the material strength within a time period. It is necessary to evaluate the cumulative failure event, which is defined as the combination of failure events at each time point. The method of saddle-point approximation (SPA) is introduced to evaluate the probability distribution of the equivalent maximum contract stress for the planetary gearbox. Additionally, reliability-based sensitivity indices are derived to investigate the parametric significance of random input variables. The gear transmission system of traction unit is used to demonstrate the engineering applications of the proposed method for dynamic reliability and reliability-based sensitivity analyses. A crude Monte Carlo simulation is performed to provide benchmark results. The results indicate that the structural parameters of the sun gear have more of an influence on system reliability than other parts of the mechanism. © 2016 Published by Elsevier Ltd.

1. Introduction A mining shearer is critical for maintaining efficiency and security in underground coal mining, for which reliability, maintainability and safety are prominent factors. As shearer size and complexity continue to increase, the implications of shearer failure have become more important. Gupta [1] proposed using maintenance information to analyze the failure logic of a long-wall shearer. Dhillon [2] summarized the statistical approaches to analyze the reliability and maintainability of shear loaders. Barabady [3] presented a case study that described the reliability and availability analysis of the crushing plant number 3 at Jajarm Bauxite Mine in Iran. Hoseinie [4] used an event-based Monte Carlo simulation to conduct a reliability analysis of the long-wall shearer mechanism. The planetary gear system is one of the most important subsystems of the shearer. The reliability analysis of the force transmission system is crucial for the safe operation of shearer loaders. Yang [5] investigated the fatigue analysis of the gear system based on the modified linear cumulative damage criteria. Zhang [6] proposed using the stochastic perturbation method for the reliability-based design of gear pairs. Zhang [7] employed the Kriging model to globally optimize the volume and reliability of large ball mill gear transmissions. Nejad [8] presented a long-term fatigue damage method for gear tooth bending in wind

⁎ Corresponding authors. E-mail addresses: [email protected] (X. Zhang), [email protected] (Y. Zhang).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.007 0094-114X/© 2016 Published by Elsevier Ltd.

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turbines. Li [9] used a logical diagram to evaluate the reliability of generic geared wind turbine systems. Guerine [10] analyzed the dynamic statistical responses of a gear system with uncertain parameters. The reliability analysis must consider the dynamic factors of gear transmission systems, including the time-variant meshing force. The saddle-point approximation (SPA) method was first introduced by Daniels [11] to estimate the probability distribution of a function of random variables. This method has been widely used in statistics, biostatistics, engineering sciences, econometrics, and applied mathematics to estimate parent distributions. SPA generally provides probability approximations with much greater accuracy than the current supporting theory would suggest [12]. Du [13] extended the SPA for the parent distribution estimation of linear and nonlinear performance functions in structural reliability analysis. Huang [14] used the SPA and first-order Taylor series expansion at the mean values to estimate probability distributions. Yuen [15] investigated random samples of the performance variable to calculate saddle points in the reliability analysis of dynamic structural systems. Zhang [16] demonstrated the utility of the saddle-point approximation (SPA) in probabilistic importance analysis of machine components. Song [17] implemented the line sampling method to analyze the influence of design parameters in the linear performance function. Xiao [18] considered epistemic and aleatory uncertainties to estimate structural reliability based on the mean-value first-order SPA method. The objective of the paper is to perform a dynamic reliability analysis on the planetary gear system used in the shearer mechanism. The safety of the system is characterized by the dynamic stress as a fraction of the material strength at each time point. The SPA is used to estimate the extreme-value distribution of the system performance function. Reliability-based sensitivity indices of input random variables are derived. The remainder of the paper is organized as follows. Section 2 presents the dynamic model and determines the dynamic responses of the planetary gear system in the shearer loader. Section 3 proposes a performance function for reliability analysis of the planetary gear system. The reliability and reliability-based sensitivity indices are derived using SPA. The planetary gear system of traction unit in the shearer mechanism is utilized as a numerical example in Section 4 to demonstrate the proposed method for dynamic reliability and reliability-based sensitivity analyses. The conclusions are presented in Section 5.

2. Dynamic analysis of planetary gear system in shearer loader The planetary gear system is one of the most important subsystems in the transmission chains of the shearer loader. Because of its ability to carry heavy torques and compact volumes, the planetary gearbox is extensively used in the design of mining equipment.

2.1. Performance of transmission system in shearer The shearer loader enables a highly mechanized production technique for underground long-wall mining. The entire structure and key units of the shearer loader are illustrated in Fig. 1. The rocker arm delivers the power to the roller for continuous mining, whereas the traction units support and drive the entire shearer mechanism. Maintenance can be performed underground to guarantee system

Fig. 1. Structure of the shearer loader.

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safety in harsh working environments. However, maintenance and testing must be regularly conducted in the factory to maintain structural reliability. The planetary gear system is the end component of the transmission chain; it links the traction unit with the rocker arm of the shearer mechanism, as illustrated in Fig. 1. To illustrate the maintenance behaviour of the shearer mechanisms, fault records for model MG300/700-WD at ATB Opencast Coal Mine were collected over two years. The ratios of maintenance time for subsystems are shown in Fig. 2. The planetary gear system was a critical part of the coal-mining progress over the data collection time. The 484 h (63%) of repair time represents considerable financial loss and use of human resources. The poor operational record and intricate structural features necessitate a reliability analysis of the planetary gearbox, one of the most vital components in the shearer loader. The dynamic model for the vibration analysis of the planetary gear system is illustrated in Fig. 3. The physical parameters are summarized in Table 1. The time-varying meshing stiffness and backlash of the planetary gearbox should be considered based on the working conditions. 2.2. Dynamic analysis for planetary gear system The planetary gear system is widely used in the drive train of automobiles, wind turbines, marine vehicles and mining equipment. A single-stage planetary gearbox consists of a sun gear, a ring gear, a carrier and several planets. Lin [19] established an analytical model of planetary gears. Ambarisha [20] compared the vibration model with the finite-element model in terms of the nonlinear characteristics. Liu [21] developed the planetary gear model and analyzed the dynamic response. The positive directions of the system are defined as the positive direction of each coordinate system and compressive direction of each tooth surface throughout this analysis. In Fig. 3, the relative displacement along the line of action for the sun-planet mesh is expressed as



 δspi ¼ −uxs sin ψspi þ uys cos ψspi −uxpi sin α spi −uypi cos α spi þ urs þ urpi −espi

ð1Þ

where ψspᵢ = φi − αspᵢ. The relative displacement is derived from the sun and planet deflections in the direction of the line of action for the sun-planet mesh. The displacements are shown as the red lines in Fig. 3. The kinetic equations of the sun gear can be described as 8 4 4
 X X > > € xs ¼ −ksx uxs þ > kspi f δspi sinψspi þ cspi δ_ spi sinψspi −csx u_ xs ms u > > > > i¼1 i¼1 > > < 4 4
 X X € ys ¼ −ksx uys − ms u kspi f δspi cosψspi − cspi δ_ spi cosψspi −csx u_ ys > > i¼1 i¼1 > > > 4 4
 > X X > Js > > € sr ¼ − kspi f δspi −kst usr − cspi δ_ spi : : 2 u r bs i¼1 i¼1

Fig. 2. Component maintenance time of shearer loader MG300/700-WD.

ð2Þ

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Fig. 3. Dynamic model of a planetary gear system.

Table 1 Physical description of parameters. Parameter s r c pi ωc ml Jl rl φi kηt δε M G C T u Kb Kg KΩ kspᵢ, krpᵢ cspᵢ, crpᵢ kηx, kηy cηx, cηy knpᵢ, ktpᵢ cnpᵢ, ctpᵢ esp, erp bspᵢ, brpᵢ αspᵢ, αrpᵢ uxl, uyl, url

SI-unit

Physical description

rad·s−1 kg kg·m2 m rad N·m−1 m kg kg·s−1 kg·s−1 N m N·m−1 N·m−1 N·m−1 N·m−1 kg·s−1 N·m−1 kg·s−1 N·m−1 kg·s−1 m m rad m

Sun gear Ring gear Carrier gear Planet gear i (i = 1, 2, 3, 4) Rotation speed of carrier Mass of the lth gear (l = s, c, r, pi) Rotation inertia of the lth gear Pitch radius of the lth gear Positional angle of the ith planet gear Tangential supporting stiffness (η = s, c, r) Projection of displacement on the line of action (ε = spi, rpi, cxi, cyi) Mass matrix Gyroscopic matrix Damping matrix Excitation vector Vibration displacement vector Supporting stiffness matrix Time-varying meshing stiffness matrix Radial stiffness matrix Time-varying meshing stiffness of the gear pair Meshing damping of the gear pair Bearing stiffness in the horizontal or vertical direction Bearing damping in the horizontal or vertical direction Bearing stiffness in the normal or tangential direction of planet gear i Bearing damping in the normal or tangential direction of planet gear i Manufacturing error Nonlinear backlash of the gear pair Mesh angle of the gear pair Responses of the lth gear in the directions (l = s, c, r, pi)

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Similarly, the expressions of the relative displacements and kinetic equations of other gears can be derived. The matrix form of the dynamic model is expressed as h i 2 _ € þCuþω _ _ Mu c Guþ Kb þKg ðt Þþωc KΩ þωc Kt ηðu; bÞ¼T ðt Þ

ð3Þ

in which symbols are introduced in Table 1. The detailed expressions of the matrix parameters are not listed here; the reader is referred to related studies in [19,20]. The nonlinear backlash model [22,23] η(u, b) can be expressed as 8 < δ−b ηðδ; bÞ ¼ 0 : δþb

δNb
 ðδ; bÞ ¼ ðδspi ; bsp Þ or ðδrpi ; brp Þ jδj ≤b δb−b

ð4Þ

where (δspi, bsp) and (δrpi, brp) are the parameters of the sun-planet mesh and ring-planet mesh, respectively. The meshing stiffness is taken as a trapezoidal form to account for the effect of transition between single-pair and double-pair tooth meshes [24]. The stiffness between the ring and the planet gear krp(t) in Fig. 3 is shown in Fig. 4. The contact process can be idealized as a spring and a damper, as illustrated in Fig. 3. The linear and nonlinear models for the contact force [25,26] are considered. The dynamic meshing force Fspi between the sun and planet gears is represented as h iρ h iq F spi ðt Þ ¼ kspi ðt Þ δspi ðt Þ þ cspi δspi ðt Þ δ_ spi ðt Þ

ð5Þ

where the models can be divided into two types: linear model (Case 1: ρ = 1, q = 0) and nonlinear model (Case 2: ρ = 3/2, q = 1/2). Both cases are energy-dissipating models for the time-varying force. The dynamic contact stress σC can be determined as

σ C ðt Þ ¼ Z H Z E Z X

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F spi ðt Þ u  1 KAKVKBKR u λd3

ð6Þ

where ZH, ZE and ZX represent the nodal field coefficient, elastic coefficient and contact ratio coefficient, respectively. Coefficients λ, d and u are the tooth width coefficient, pinion pitch diameter, and reduction rate of the gear pair, respectively. Coefficients KA, KV, KR and KB are the work condition coefficient, dynamic load coefficient, transverse load distribution coefficient and longitudinal load distribution coefficient, respectively. Dynamic responses are calculated herein to represent the movement of the planetary gearbox in the shearer mechanism for use in the reliability analysis.

Fig. 4. Meshing stiffness for ring-planet mesh.

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3. Dynamic reliability analysis The planetary gearbox is one of the driveline components of both the rocking arm and traction unit of the shearer mechanism. The dynamic responses vary with time for a system and differ among systems because of the drive characteristics and random parameters. Reliability is essential to ensure correct dynamic operation of the system. 3.1. Saddle-point approximation method The cumulative damage criteria [5,8] are based on the linear damage hypothesis to analyze the system reliability. The Kriging model [7] is a half-parameter interpolation method with strict requirements for the orthogonality and uniformity of the sample points. The stochastic perturbation method [6,10] uses statistical moments such as the mean and standard deviation to describe the dynamic responses for linear and weakly nonlinear systems. However, the SPA method [14,27] can be used to compute the probability density function (PDF) and cumulative distribution function (CDF) of random variables. The statistical moments can also be calculated using these functions. The method has the advantage of high accuracy in the tail of the distribution function. The SPA has been utilized to analyze and calculate distribution functions and the system reliability of shearer loader in this study. The SPA method [11] can estimate the output distribution of a multivariate function for a joint probability distribution of input random variables. For clarity, X represents a random variable, x denotes a realization of X, X represents a vector of random variables [X1, X2,…, Xn]T, and an observation of X is expressed as x = [x1, x2,…, xn]T. For a joint PDF of input vector X, the moment-generating function (MGF) is defined as Z MX ðsÞ ¼

þ∞ −∞

expðsxÞf X ðxÞdx

ð7Þ

and the cumulant generating function (CGF) is: K X ðsÞ ¼ ln ½MX ðsÞ:

ð8Þ

Appendix 1 summarizes the expressions of CGFs for various probability distributions. Two useful properties of CGF [28] are expressed as follows. Property I. If X1,…Xj,… are independent random variables, and the CGFs are KXj(s) (j = 1,2,…,n), the CGF of total variables is described as K Y 1 ðsÞ ¼

n X

K X j ðsÞ

if Y 1 ¼

n X

j¼1

X j:

ð9Þ

j¼1

Property II. If KX(s) is the CGF of random variable X, the CGF of Y2 is described as K Y 2 ðsÞ ¼ K X j ðξsÞ þ γs if Y 2 ¼ ξX þ γ:

ð10Þ

Considering a multivariate function Y = g(X), the first-order Taylor series approximation is expressed as Y ¼ g ðμÞ þ

 n X ∂g   ∂X j  j¼1


 X j −μ j

ð11Þ

x¼μ

where μ j is the mean value of Xj, μ = [μ1,.. μj,…μn]T. Properties I and II are used to calculate the CGF of Y, and the result is 82  < n X ∂g  K Y ðsÞ ¼ K X 4g ðμÞ−  : ∂X j  j¼1

x¼μ

9 =

3

 n X ∂g  μ j5 þ  ∂X j  j¼1

x¼μ

Xj

;

 n X ∂g  ¼ gðμÞs−  ∂X j  j¼1

x¼μ

0

 ∂g  μ js þ KXj @  ∂X j  j¼1 n X

1 sA

ð12Þ

x¼μ

which determines a close expression of the output PDF of Y as ( f Y ðyÞ ¼

)1=2 1 exp½KY ðsÞ−sy: 2πK ″Y ðsÞ

ð13Þ

Here, K′Y(•) and KY″(•) are the first- and second-order derivatives of the CGF. In particular, the saddle point s0 is the root of the following equation: 0

K Y ðsÞ ¼ y:

ð14Þ

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Given the saddle-point approximation of the PDF, the CDF of a multivariate function is further determined as  F Y ðyÞ ¼ P fY ≤yg ¼ ΦðwÞ þ φðwÞ

1 1 − w v

 ð15Þ

where Φ(•) and φ(•) are the CDF and PDF of the standard normal distribution, respectively. The variables w and v are expressed as 1=2

w ¼ sgnðsÞf2½sy−K Y ðsÞg

ð16Þ

h i1=2 ″ : v ¼ s K Y ðsÞ

ð17Þ

The saddle-point approximation method aims to evaluate the CDF and PDF of the system response Y = g(X), which can be considered as a performance function. 3.2. Dynamic reliability model The dynamic reliability analysis of a planetary gear system must evaluate the time-dependent failure probability, which implies that the reliability probability of the system at time tm (m = 0,…, N) is represented as

Rðt m Þ ¼ Pr½σ H −σ C ðt m ÞN0 ¼ Pr½g ðX; t m ÞN0 h i ¼ Pr gðZ H ; Z E ; Z X ; K A ; K V ; K B ; K R ; λ; σ H ; F sp ðt m ÞÞN0

ð18Þ

¼ Pr½g ðZ H ; Z E ; Z X ; K A ; K V ; K B ; K R ; λ; σ H ; M; C; G; K; T; u; t m ÞN0 where the material strength σH is considered a time-independent random variable. The contact stress σC in Eq. (18) is calculated using Eq. (5); the dynamic force Fsp is obtained from Eqs. (3) and (5). Both working parameters and structural parameters in Eq. (3) can affect the system performance and cause the dynamic responses to fluctuate. These random variables should be considered for the reliability analysis. However, the failure of the planetary gear system must evaluate the following system performance function: Es ¼ ½gðX; t0 ÞN0∪½gðX; t1 ÞN0⋯∪½gðX; tend ÞN0 N

N

m¼0

m¼0

¼ ∪ fσ H −σ C ðX; t m ÞN0g ¼ ∪ fgðX; tm ÞN0g

ð19Þ

where tm = t0 + mΔt and Δt = (tend − t0)/N. The envelope function [29] is used to describe the worst-case failure-safety boundary for the system reliability. The response surface model [30] is used to evaluate the minimum and maximum system responses. The response surface method [31] is used in Fig. 5 to calculate the worst case of the limit state for the dynamic reliability of the planetary gear system in the shearer loader.

Fig. 5. Extreme function of the planetary gear system.

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The extreme function min[g(X;t)] describes the hazardous situation of the planetary gear system in the shearer loader in Fig. 5. Therefore, the system reliability during the lifetime interval [t0, tend] is determined as 

 N N ∪ fgðX; tm ÞN0g ¼ Pr ∪ fσ H −σ C ðX; t m ÞN0g m¼0  m¼0 ¼ Pr σ H − max ½σ ðX; tm Þ N0 ¼ Prf min½gðX; tÞN0g; t∈½t 0 ; t end 

R ¼ Pr

ð20Þ

m¼1;2; :::

where the function g(X; t) represents the dynamic system performance in Fig. 5. The SPA is proposed in this paper to estimate the probability distribution of the system maximal contact stress. The reliability of the system is evaluated as   1 1 − R ¼ Pr½g min ðXÞ≥0 ¼ 1−Fg ð0Þ ¼ 1−Φðw0 Þ−φðw0 Þ w0 v0

ð21Þ

where w0 and v0 are the constants determined by Eqs. (16) and (14) and s0 is determined by 0

K Y ðs0 Þ ¼ 0

ð22Þ

h i1=2 ″ v0 ¼ s0 K Y ðs0 Þ

ð23Þ

1=2

w0 ¼ sgnðs0 Þf2½−K Y ðs0 Þg

:

ð24Þ

The dynamic reliability of the planetary gear system is calculated based on the SPA. The PDF of the system can also be represented by Eq. (13). 3.3. Reliability-based sensitivity analysis The reliability-based sensitivity analysis must evaluate the derivative of the reliability parameter with respect to the mean and standard deviation of the input random variables. Substituting for the reliability function in Eq. (21), one can determine the following reliability-based sensitivity of the mean value: " # ∂R ∂½1−FY ð0Þ ∂ΦðβÞ ∂Φ ∂β ∂β ∂w0 ∂β ∂v0 ¼ ¼ ¼ ¼ φðβÞ þ ∂μ j ∂μ j ∂μ j ∂β ∂μ j ∂w0 ∂μ j ∂v0 ∂μ j

ð25Þ

where β ¼ w0 þ

  1 v ln 0 ; w0 w0

∂β 1 ln ðv0 =w0 Þ ∂β 1 ¼ 1− 2 − ; ¼ ∂w0 ∂v0 w0 v0 w0 w20

ð26Þ

and ∂w0 −1=2 ∂ ¼ −2sgnðs0 Þ½−2K Y ðs0 Þ ½K ðs Þ ∂μ j Y 0 ∂μ j

ð27Þ

Table 2 Derivative of the CGF for various probability distributions. Derivative Normal Exponential Extreme value Chi-squared Uniform

K″

K′ 2

K′ = μ + σ s K′ = β/(1 − βs) K′ = a− Ψ(1 − bs)b K′ = n/(1 − 2s) K 0 ¼ ½bexpðbsÞ−aexpðasÞ= ½expðbsÞ−expðasÞ−1=s

K″ = 2σ K″ = β2/(1 − βs)2 K″ = Ψ(1, 1 − bs)b2 K″ = 2n/(1 − 2s)2 K ″ ¼ ½b expðbsÞ−a2 expðasÞ= ½expðbsÞ−expðasÞ þ 1=s2 − 2 ½bexpðbsÞ−aexpðasÞ = ½expðbsÞ−expðasÞ2 2

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h i ∂v0 −1=2 ∂ ″ ¼ s0 ½−2K Y ðs0 Þ K ðs Þ : ∂μ j Y 0 ∂μ j

ð28Þ

The reliability-based sensitivity of the standard deviation is " # ∂R ∂½1− FY ð0Þ ∂ΦðβÞ ∂Φ ∂β ∂β ∂w0 ∂β ∂ν0 ¼ ¼ ¼ ¼ φðβÞ þ ∂σ j ∂σ j ∂σ j ∂β ∂σ j ∂w0 ∂σ j ∂ν0 ∂σ j

ð29Þ

∂w0 −1=2 ∂ ¼ −2sgnðs0 Þ½−2K Y ðs0 Þ ½K ðs Þ ∂σ j Y 0 ∂σ j

ð30Þ

h i ∂v0 −1=2 ∂ ″ ¼ s0 ½−2K Y ðs0 Þ K Y ðs0 Þ : ∂σ j ∂σ j

ð31Þ

where

The derivatives of CGF with respect to the distribution parameter are summarized in Table 2 for the input random variables. The normally distributed parameters are used as an example in Appendix 2 to analyze the reliability and reliability-based sensitivity. The CGFs cannot be directly computed using Eqs. (7) and (8) for several random distributions such as the log-normal distribution and Weibull distribution [32,33]. The approximate method is utilized to calculate the CGFs. The power expression [29] is ~ ðsÞ ¼ K Y

∞ X ι¼1

κι

sι ι!

ð32Þ

where κι (ι = 1,2,…) is the ιth cumulant of Y. The first four cumulants of random variables are calculated using the first four central moments and origin moments as follows 8 κ > > > 1 < κ2 > κ3 > > : κ4

0

¼ μ 1 ¼ ΕðY Þ 0 02 2 ¼ μ 2 −μ 1 ¼ ΕðY−ΕðY ÞÞ 0 0 0 03 3 ¼ μ 3 −3μ μ 2 þ 2μ 1 ¼ ΕðY−ΕðY ÞÞ 0 0 0 02 0 02 04 4 2 ¼ μ 4 −4μ 3 μ −3μ 2 þ 12μ 2 μ 1 −6μ 1 ¼ ΕðY−ΕðY ÞÞ −3ðκ 2 Þ

ð33Þ

where μ ′ι (ι = 1,2,3,4) are the first central moments and Е is the index of mathematical expectation. Two cases are presented to calculate the approximate CGFs using Eq. (32). Case 1: Normal distribution N(μ a, σa). The approximate CGF in Eq. (32) is calculated as follows  2 3 2
  4 ~ ðsÞ ¼ μ s þ σ 2 s þ 0 s þ 3σ 4 −3 σ 2 2 s ¼ μ s þ σ 2 s : K a a a a a a 2! 3! 4! 2

ð34Þ

The result is identical to the expression in Appendix 1. Case 2: Log-normal distribution LN(μ b, σb). The first four cumulants are expressed as 8 μ b þ0:5σ 2b > > > κ1 ¼ e   > > > 2μ b þσ 2b σ2 > > e b −1 < κ2 ¼ e  2   2 2 > σ 2b > > κ 3 ¼ e3μ b þ1:5σ b eσ b −1 e þ 2 > > > > > 2 2 2 2 2 : 4μ þ8σ b 4μ þ5σ b 8μ þ3σ b 4μ þ2σ b 4μ þ4σ b −4 e b þ 8e b −4 e b −e b κ4 ¼ e b

ð35Þ

Table 3 Structural parameters of the planetary gear system. Structural parameter

Sun gear

Ring gear

Planet gear (1–4)

Carrier gear

Number of teeth Modulus/mm Width/mm Diameter of centre hole/mm Torque/N·m

16 6 – 0 6500

68 6 – 400

26 6 – 110

– – 182 – 34,125

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Table 4 Probability distribution of input random variables. Variable

Symbol

Description

μj

σj

Distribution

X1 X2 X3 X4 X5 X6 X7 X8

Bs Bc B p1 B p2 B p3 Bp4 ZH ZE

95 182 105 105 105 105 2.4946 189.8

4.75 9.10 9.10 5.25 5.25 5.25 0.1247 9.49

Normal Normal Normal Normal Normal Normal Normal Normal

X9 X10 X11 X12 X13 X14 X15

KA KV λ KB KR ZX σH

Tooth width: Sun gear/mm Tooth width: Carrier/mm Tooth width: Planet 1/mm Tooth width: Planet 2/mm Tooth width: Planet 3/mm Tooth width: Planet 4/mm Nodal field coefficient pffiffiffiffiffiffiffiffiffiffi Elastic coefficient/ MPa Work condition factor Dynamic load coefficient Width coefficient Longitudinal load factor Transverse load factor Contact ratio coefficient Structural strength/MPa

2.25 2.8577 1.20 1.16 1.10 0.9325 1400

0.113 0.15 0.06 0.058 0.055 0.0466 100

Normal Log-normal Normal Normal Normal Normal Normal

Furthermore, the approximate PDF of the log-normal distribution is obtained by substituting cumulants into Eq. (13).

4. Numerical example In this section, the reliability-based sensitivity analysis of a planetary gearbox in the traction unit of the shearer mechanism is used to illustrate the application of the proposed method. The following dimensionless indices are used to calibrate the sensitivities of the mean and standard deviation of random variables.

Sμ ¼

∂R μ j  ; ∂μ j R

Sσ ¼

∂R σ j  ∂σ j R

ð36Þ

Fig. 6. Dynamic responses of planetary gears in the traction unit (a: Dynamic meshing force of the planet-ring gear pair; b: Dynamic meshing force of the sunplanet gear pair; c: Dynamic contact stress of the planet-ring gear pair; d: Dynamic contact stress of the sun-planet gear pair; Case 1: Linear model; Case 2: Nonlinear model; T: Mesh period).

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Fig. 7. Probability distributions of the stress-strength interference model (Case 1: Linear model; Case 2: Nonlinear model; T: Mesh period).

4.1. Planetary gear model The s1-p4 NGW planetary gear system in the traction unit of shearer loader MG300/700-WD is considered. The structural parameters are listed in Table 3. Additionally, stochastic structural and environmental parameters are considered for the system. The probability distributions of random variables are listed in Table 4. The linear and nonlinear models in Eq. (5) are used to calculate the meshing force and dynamic contact stress of the planetary gear system with the mean values of the input random variables, as shown in Fig. 6(a)–(d). The results of two models change with the cycle of the mesh period, in which the linear models have larger values than the nonlinear models. The curves in Fig. 6(a)–(d) exhibit similar variation tendencies and reach their peaks simultaneously. The dynamic contact stresses of the sun-planet and planet-ring gear pairs are calculated and shown in Fig. 6(c) and (d), respectively. The engineering failure analysis indicates that the primary major failure mode of the planetary gear system is the overstress of the sun and planet meshing system in the traction unit of the shearer loader.

4.2. Dynamic reliability analysis With the proposed SPA method, the instantaneous probability distribution function of the system contact stress and material strength are determined as shown in Fig. 7. These results determine the instantaneous reliability in Table 5. The cross-field between stress distribution and strength distribution is an index to represent the system failure. Case 2 has a smaller range in Fig. 7 and higher reliability in Table 5 than case 1. Case 1 is used to calculate the maximal contact stress of the planetary gear system in the shearer loader for safety. The probability distribution of the system performance function is shown in Fig. 8. The system reliability of the dynamic planetary gear set is determined as RS ¼ Pr½g S ðXÞN0 ¼ Prf min½gðX; tÞN0g ¼ 0:995185:

ð37Þ

The reliability of worse case in traction unit is 0.995273 by Monte Carlo simulation. Structural parameters and dynamic factors should be controlled to minimize the impact indices for system reliability and ensure proper operation.

Table 5 Reliability results at t1 and t2.

R(t1) = Pr {[σH − σC(t1)] N 0} R(t2) = Pr {[σH − σC(t2)] N 0}

Case 1: Linear model

Case 2: Nonlinear model

0.999971 0.996074

0.999998 0.998355

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Fig. 8. Probability distribution of the system performance function (SPA: Saddle-point approximation method; MCS: Crude Monte Carlo simulation).

4.3. Reliability-based sensitivity analysis The reliability-based sensitivity indices of the mean and standard deviation of input random variables are determined as shown in Fig. 9. A Monte Carlo simulation with 106 samples is used to provide benchmark results for the sensitivity analysis. The consistency in Fig. 9 highlights the high accuracy of the proposed method.

Fig. 9. Reliability-based sensitivity analysis of the planetary gear system.

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D. Zhou et al. / Mechanism and Machine Theory 105 (2016) 244–259

The sensitivity analysis illustrates that the tooth width of the sun gear is the most significant random variable with respect to the system reliability of the nonlinear dynamic system. The working parameters (X7–X14) act as the demanding factors in the system reliability analysis of the planetary gear system, which implies that the increase in mean and standard deviation of these random variables causes the decrease in system reliability. The results are assessed by comparing with a crude Monte Carlo simulation with 106 samples. However, evaluating the system reliability using the crude Monte Carlo simulation is time-consuming. The MCS benchmark requires 106,202 s of CPU time with an Intel(R) Core(TM) i5-4500 CPU at 3.30 GHz with 3.5 GB RAM. The SPA method requires 134 s to analyze the dynamic reliability and reliability-based sensitivity of the shearer loader. The computational efficiency should be considered for the reliability calculation. 5. Conclusion The planetary gear system is a critical link that consumes more maintenance time than other mechanical parts in the shearer loader. Linear and nonlinear contact models are used to evaluate the time-varying responses and distribution functions according to dynamic models of the planetary gearbox. The SPA method is utilized to establish the dynamic reliability model of the timedependent system. The linear contact model is used to estimate the probability distribution of the system maximal contact stress to analyze the reliability of the planetary gear system in the shearer mechanism. The method and Monte Carlo simulation are consistent. The derivative of the SPA method is proposed to evaluate the derivative of reliability with respect to the mean value and standard deviation of input random variables. The sun gear affects the dynamic performance more than other gears in the planetary gearbox of the shearer loader. The tooth width of the sun gear is more sensitive to the system reliability than other structural parameters. Both machining precision and working accuracy of the sun gear should be controlled in the manufacturing and maintenance process. During the working process, the degradation and rapid variation of environment can decrease the system reliability, so that improvements in workplace conditions are beneficial for the system reliability. The material property also plays an important role to minimize the reliability-based sensitivity indices in the design process. Acknowledgements The research work in this paper was fully supported by the National Key Basic Research Development Plan of China (the 973 Program) under Grant No. 2014CB046303 and the National Natural Science Foundation of China under Grant No. 51405069. Appendix 1. CGFs for various probability distributions

Table A1 CGFs of random variables Distribution

PDF

CGF

Uniform

fX(x) = 1/(b −a)

Normal

f X ðxÞ ¼ ð2πÞ−1=2 σ −1 expf−1=2 ðx−μÞ2 =σ 2 g fX(x) = (1/β) exp (−x/β) f X ðxÞ ¼ 1=σexp½−ðx−μÞ=σ  expfexp½−ðx−μÞ=σ g fX(x) = βα/Γ(α)xα −1 exp (−βx)

K X ðsÞ ¼ ln½expðbsÞ−expðasÞ −lnðb−aÞ−lnðsÞ KX(s) = μs + (1/2)σ2s2

Exponential Extreme value Gamma Chi-squared

f X ðxÞ ¼ 1=ðΓðn=2Þ2n=2 Þ xn=2−n expð−1=2xÞ

KX(s) = − ln (1 − βs) KX(s) = μs + ln Γ(1 − σs) KX(t) = α KX(s) = (−1/2)n ln (1 − 2s)

Appendix 2. Reliability-based sensitivity for normal distribution In practice, because most random variables obey the normal distribution, the computational method is derived for the normal distribution. If element Xj (j = 1,2,…,n) in vector X obeys normal distribution N(μ,σ), then μ = [μ1, μ2, …, μn] and σ = [σ1, σ2,…, σn]. The reliability and reliability-based sensitivity are simplified, and the results are consistent with FORM. The system reliability is described as Rs = 1 − FY(0). The CGF is expressed by Eq. (8) and Appendix 1 as follows:  n X ∂g  K Y ðsÞ ¼ g ðμÞs−  ∂X j  j¼1

x¼μ

0

 ∂g  @ μ js þ KXj  ∂X j  j¼1 n X

1 sA: x¼μ

ðB1Þ

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257

The first-order derivative of KY(s) is derived from Eq. (14). 0

K Y ðsÞ ¼ y ¼ 0:

ðB2Þ

The second-order derivative of KY(s) is expressed as

″ K Y ðsÞ

0

 ∂g  @ ¼  ∂X j  j¼1 n X

0   A K ″X @ ∂g  j ∂X j 

1

12 x¼μ

sA ¼ x¼μ

0

  @ ∂g  ∂X j  j¼1

n X

12 σ jA :

ðB3Þ

x¼μ

The saddle-point expression is computed as

0 K Y ðs0 Þ

 n X ∂g  ¼ g ðμÞ−  ∂X j  j¼1

x¼μ

s0 ¼ −

n X j¼1

 n X ∂g  μj þ  ∂X j  j¼1

0

 ∂g   ∂X j 

1

0 Kx j @ x¼μ

g ðμÞ 12   @ ∂g  σ j A : ∂X j 

s0 A ¼ 0 x¼μ

ðB4Þ

0

x¼μ

Moreover, w0 and v0 are the constants determined by Eqs. (23) and (24):

1=2

w0 ¼ sgnðs0 Þ½−2K Y ðs0 Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12 u  uX u n @ ∂g  ¼ −g ðμÞ=t  σ jA ∂X j  j¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12 u  uX i1=2 u n @ ∂g  ″ t v0 ¼ s0 K Y ðs0 Þ ¼ −g ðμÞ=  σ jA : ∂X j  h

j¼1

ðB5Þ

x¼μ

ðB6Þ

x¼μ

The reliability by SPA is described as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12 1 u  uX u n @ ∂g  B C R ¼ 1−FY ð0Þ ¼ 1−Φðw0 Þ ¼ 1−Φ@−g ðμÞ=t  σ jA A ∂X j  j¼1 x¼μ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 12 1 u  uX u n @ ∂g  B C ¼ ΦðβÞ ¼ Φ@g ðμÞ=t  σ jA A : ∂X j  0

j¼1

ðB7Þ

x¼μ

Based on FORM, the first-order Taylor series expansion of Y = g(X) is  n X ∂g  Y ¼ g ðμÞ þ  ∂X j  j¼1


 X j −μ j :

ðB8Þ

x¼μ

The mean and standard deviation values are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12 u  uX u n @ ∂g  μ Y ¼ g ðμÞ; σ Y ¼ t  σ jA : ∂X j  j¼1

x¼μ

ðB9Þ

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The reliability by FORM is expressed as     Y−μ Y μ μ ≤− Y ¼ 1−ϕ − Y R ¼ 1−FY ð0Þ ¼ 1−PðY ≤0Þ ¼ 1−P σ Yvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σY σY 0 !2ffi1 u n  X u ∂g  σj A: ¼ 1−ΦðβÞ ¼ 1−Φ@g ðμÞ=t ∂X 

ðB10Þ

x¼μ

j¼1

The SPA in Eq. (B7) provides the identical result as that of Eq. (B10), which was calculated by FORM. Thus, the SPA method is a valid approach to analyze the system reliability. A simple form of the reliability-based sensitivity can be derived from Eqs. (25) and (29). The parameters of Eq. (26) can be expressed as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12 1 0 u   uX u n @ ∂g  ∂β ∂ B ∂g  C ¼  σ jA A ¼ @  @g ðμÞ=t ∂μ j ∂μ j ∂X j  ∂X j  j¼1 0

x¼μ



∂β ∂ 1 ¼ μY ∂σ j ∂σ j σ Y



1

x¼μ

20 0   n ∂ 6@X @ ∂g  ¼ μY 4 ∂σ j ∂x j  i¼1

x¼μ

A 1 σj

ðB11Þ

12 1−12 3 0  ∂g  7 A A σj 5 ¼ −@  ∂x j 

x¼μ

12 A

σ jμg σ 3Y

:

ðB12Þ

The mean-value sensitivity and standard deviation sensitivity can be determined as 0  ∂R ∂g  ¼ φðβÞ@  ∂μ j ∂X j 

x¼μ

1 A 1 ; σj

0  ∂R ∂g  ¼ −φðβÞ@  ∂σ j ∂X j 

12 A x¼μ

σ jμg σ 3Y

:

ðB13Þ

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