Design optimization of a wind turbine gear transmission based on fatigue reliability sensitivity

doi:10.1007/s11465-020-0611-5

Front. Mech. Eng.

2021, Vol. 16

Issue (1) : 61-79 https://doi.org/10.1007/s11465-020-0611-5

RESEARCH ARTICLE

Design optimization of a wind turbine gear transmission based on fatigue reliability sensitivity

Genshen LIU¹, Huaiju LIU¹(

), Caichao ZHU¹, Tianyu MAO¹, Gang HU²

¹. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, China
². School of Mechatronic Engineering, Southwest Petroleum University, Chengdu 610500, China

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Abstract

Fatigue failure of gear transmission is one of the key factors that restrict the performance and service life of wind turbines. One of the major concerns in gear transmission under random loading conditions is the disregard of dynamic fatigue reliability in conventional design methods. Various issues, such as overweight structure or insufficient fatigue reliability, require continuous improvements in the reliability-based design optimization (RBDO) methodology. In this work, a novel gear transmission optimization model based on dynamic fatigue reliability sensitivity is developed to predict the optimal structural parameters of a wind turbine gear transmission. In the model, the dynamic fatigue reliability of the gear transmission is evaluated based on stress–strength interference theory. Design variables are determined based on the reliability sensitivity and correlation coefficient of the initial design parameters. The optimal structural parameters with the minimum volume are identified using the genetic algorithm in consideration of the dynamic fatigue reliability constraints. Comparison of the initial and optimized structures shows that the volume decreases by 3.58% while ensuring fatigue reliability. This work provides new insights into the RBDO of transmission systems from the perspective of reliability sensitivity.

Keywords gear transmission fatigue reliability reliabi-lity sensitivity parameter optimization

Corresponding Author(s): Huaiju LIU

Just Accepted Date: 24 December 2020 Online First Date: 25 January 2021 Issue Date: 11 March 2021

Cite this article:

Genshen LIU,Huaiju LIU,Caichao ZHU, et al. Design optimization of a wind turbine gear transmission based on fatigue reliability sensitivity[J]. Front. Mech. Eng., 2021, 16(1): 61-79.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-020-0611-5
https://academic.hep.com.cn/fme/EN/Y2021/V16/I1/61

Fig.1 Flow chart of optimization based on dynamic reliability sensitivity. Normfit, Wblfit, and Expfit represent normal, Weibull, and exponential distributions, respectively.

Fig.2 Schematic of a 2 MW wind turbine gear transmission [²⁹,³⁰].

Gear	Teeth number	Normal module/mm	Pressure angle/(° )	Helix angle/(° )	Gear face width/mm	Shifting coefficient
$r I$	96	15	25	8	395	–0.2705
$p i I$ (input)	37	15	25	8	390	0.3924
$s I$	21	15	25	8	395	0.0000
$g 1 II$	97	11	20	10	310	0.0047
$g 2 II$	23	11	20	10	320	0.0700
$g 1 III$	103	8	20	10	180	0.0240
$g 2 III$ (output)	21	8	20	10	190	0.0200

Tab.1 Main structural parameters of the transmission

Fig.3 Input (a) torque and (b) speed history of Stage I within a test time of 600 s.

Fig.4 Changes in input torque and speed at Stage I.

Fig.5 Gear rotation direction and change in tooth contact and bending stresses at (a) Stage I and (b) Stage II.

Fig.6 Distribution fitting of tooth bending stress fitted using normal (Normfit), Weibull (Wblfit), and exponential (Expfit) distributions.

Gear	Tooth	Distribution type of bending stress	Bending stress distribution parameters/(N·mm⁻²)	Test value $p 1$	Distribution type of contact stress	Contact stress distribution parameters/(N·mm⁻²)	Test value $p 2$
$r I$	$z 92$	Normal	(231.53, 67.87)	0.87	Normal	(587.33, 56.34)	0.90
$p i I$	$z 4$	Normal	(359.42, 105.40)	0.65	Normal	(1052.25, 99.21)	0.80
$s I$	$z 1$	Normal	(319.76, 91.95)	0.76	Normal	(984.01, 93.62)	0.84
$g 1 II$	$z 44$	Normal	(342.72, 97.76)	0.93	Normal	(956.35, 123.18)	0.42
$g 2 II$	$z 19$	Normal	(311.66, 91.52)	0.54	Normal	(979.59, 100.90)	0.59
$g 1 III$	$z 78$	Normal	(278.87, 82.19)	0.60	Normal	(908.80, 94.88)	0.79
$g 2 III$	$z 2$	Normal	(275.42, 81.28)	0.02	Normal	(919.09, 113.45)	0.03

Tab.2 Stress distribution fitting conditions of representative teeth of gears

Fig.7 Flow chart of the genetic algorithm of gear transmission optimization.

Gear	$R F 0$ /%	$R H 0$ /%	$S F$ ^a)	$S H$ ^b)
$r I$	99.9785	99.9923	1.90	1.53
$p i I$	99.9934	99.9922	2.01	1.52
$s I$	99.9992	99.9988	2.10	1.61
$g 1 II$	99.9993	99.9990	2.11	1.64
$g 2 II$	99.9998	99.9986	2.14	1.60
$g 1 III$	99.9999	99.9999	2.34	1.77
$g 2 III$	99.9999	99.9999	2.37	1.75

Tab.3 Initial reliabilities and safety factors

Fig.8 Dynamic fatigue reliabilities of the three stages: (a) Dynamic bending fatigue reliabilities of the stages; (b) dynamic contact fatigue reliabilities of the stages; (c) comparison of the two types of fatigue reliability of the stages; and (d) dynamic fatigue reliabilities of the stages and gear transmission.

Fig.9 Initial fatigue reliability sensitivities in the three stages: (a) Stage I, (b) Stage II, and (c) Stage III.

Fig.10 Dynamic fatigue reliability sensitivities of the three stages.

Fig.11 Correlation coefficient matrix of the initial design parameters.

Fig.12 Cluster dendrogram of design parameters.

Tab.4 Structural parameters of the initial gear transmission structure and the optimized one

Fig.13 Dynamic fatigue reliability of the optimized gear transmission.

Tab.5 Reliability results for the initial and optimized structures

Fig.14 Optimization effect of the optimization effect of the gear transmission.

$b$	Face width, mm
CA	Convergence accuracy
$C 0$	Degradation factor
$d 1$	Diameter of the reference circle, mm
$d (X 1, X 2)$	Similarity degree between two design parameters
$F t$	Tangential tooth load, N
$g (t)$	Dynamic fatigue reliability function
$g 11$	First tooth of the pinion gear in Stage II
$g 18$	Eighth tooth of the wheel gear in Stage II
$g 1 II$	Wheel gear in Stage II
$g 2 II$	Pinion gear in Stage II
$g 1 III$	Wheel gear in Stage III
$g 2 III$	Pinion gear in Stage III
$i Stg I$	Transmission ration of Stage I
$i Stg I I$	Transmission ration of Stage II
$i Stg I I I$	Transmission ration of Stage III
$K A$	Application factor
$K Fβ$	Face load factor of bending stress calculation
$K Fα$	Transverse load factor of bending stress calculation
$K Hβ$	Face load factor of contact stress calculation
$K Hα$	Transverse load factor of contact stress calculation
$K V$	Dynamic factor
$m n$	Normal module, mm
$n$	Gear velocity
$p i I$	Planetary gear in Stage I
$R (t)$	Dynamic reliability, %
$R F 0$	Initial bending fatigue reliability, %
$R H 0$	Initial contact fatigue reliability, %
$R SF$	Initial bending fatigue reliability determined by the safety factor, %
$R SH$	Initial contact fatigue reliability determined by the safety factor, %
$R s op$	Fatigue reliability of the optimized structure
$R s ini$	Fatigue reliability of the initial structure
$R i (t)$	Dynamic fatigue reliability of Stages II and III, %
$R H i (t)$	Dynamic contact fatigue reliability of Stages II and III, %
$R F i (t)$	Dynamic bending fatigue reliability of Stages II and III, %
$R I (t)$	Dynamic fatigue reliability of Stage I, %
$R F r I (t)$	Dynamic bending fatigue reliabilities of the ring gear, %
$R H r I (t)$	Dynamic contact fatigue reliabilities of the ring gear, %
$R F s I (t)$	Dynamic bending fatigue reliabilities of the sun gear, %
$R H s I (t)$	Dynamic contact fatigue reliabilities of the sun gear, %
$R F p i I (t)$	Dynamic bending fatigue reliabilities of the planetary gear, %
$R H p i I (t)$	Dynamic contact fatigue reliabilities of the planetary gear, %
$s I$	Sun gear in Stage I
$r (n)$	Residual fatigue strength of the gear under n loading cycle numbers, N/mm²
$r (0)$	Initial fatigue strength of the gear without any damage
$S max ?$	Equivalent peak load of the gear, N/mm²
$S F$	Initial bending safety factor
$S H$	Initial contact safety factor
$r I$	Ring gear in Stage I
$T 1$	Sample torque for stress calculation
$t$	Service time, year
$u$	Gear ratio
$V op$	Volume of the optimized structure, m³
$V ini$	Volume of the initial structure, m³
$v$	Circumferential velocity, m/s
$X ¯ 1$	Basic factor set
$X ¯ 2$	Stress calculation factor set
$X ¯ 3$	Fatigue strength calculation factor set
$X ¯$	Initial design parameter set
$x ¯$	Design variable set
$Y Fa$	Tooth form factor
$Y Sa$	Dedendum stress concentration factor
$Y ε$	Contact ratio factor of bending stress calculation
$Y β$	Helix angle factor of bending stress calculation
$Y ST$	Stress correction factor of bending fatigue strength calculation
$Y NT$	Life factor of bending fatigue strength calculation
$Y X$	Size factor of bending fatigue strength calculation
$Z H$	Zone factor
$Z E$	Elasticity factor
$Z ε$	Contact ratio factor of contact stress calculation
$Z β$	Helix angle factor of contact stress calculation
$Z NT$	Life factor of contact fatigue strength calculation
$Z L$	Lubrication factor
$Z v$	Velocity factor
$Z R$	Roughness factor
$Z W$	Work hardening factor
$Z X$	Size factor of contact fatigue strength calculation
$z 1$	First tooth of the ring gear in Stage I
$z 2$	Second tooth of the ring gear in Stage I
$z 5$	Fifth tooth of the sun gear in Stage I
$α t$	Calculation factor of the zone factor, $α t = arctan ? (tan ? α n / cos ? β)$
$β$	Helix angle, º
$β (t)$	Reliability index
$β b$	Calculation factor of the zone factor, $β b = arctan ? (tan ? β cos ? α t)$
$η$	Mutation probability
$μ g (t)$	Mean value of dynamic fatigue reliability
$ρ (X 1, X 2)$	Correction coefficient between two design parameters
$σ F$	Bending stress of tooth root, N/mm²
$σ Fst$	Maximum bending static strength, N/mm²
$σ Fpst$	Maximum bending static allowable strength, N/mm²
$σ Flim$	Allowable bending stress number, N/mm²
$σ H$	Contact stress of tooth face, N/mm²
$σ Hst$	Maximum contact static strength, N/mm²
$σ Fst$	Maximum contact static allowable strength, N/mm²
$σ Hlim$	Allowable contact stress number, N/mm²
$σ g (t)$	Variance of dynamic fatigue reliability function
$Φ (?)$	Cumulative distribution function of the standard normal distribution
$ϕ (?)$	Probability density function of standard normal distribution
$⊗$	Kronecker product

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