- Original Article
- Open Access
Life Distribution Transformation Model of Planetary Gear System
- Ming Li^{1},
- Li-Yang Xie^{1}Email author,
- Hai-Yang Li^{1} and
- Jun-Gang Ren^{1}
https://doi.org/10.1186/s10033-018-0221-x
© The Author(s) 2018
- Received: 17 July 2017
- Accepted: 9 March 2018
- Published: 10 April 2018
Abstract
Planetary gear systems have been widely used in transportation, construction, metallurgy, petroleum, aviation and other industrial fields. Under the same condition of power transmission, they have a more compact structure than ordinary gear train. However, some critical parts, such as sun gear, planet gear and ring gear often suffer from fatigue and wear under the conditions of high speed and heavy load. For reliability research, in order to predict the fatigue probability life of planetary gear system, detailed kinematic and mechanical analysis for a planetary gear system is firstly completed. Meanwhile, a gear bending fatigue test is carried out at a stress level to obtain the strength information of specific gears. Then, a life distribution transformation model is established according to the order statistics theory. Transformation process is that, the life distribution of test gear is transformed to that of single tooth, and then the life distribution of single tooth can be effectively transformed to that of the planetary gear system. In addition, the effectiveness of the transformation model is finally verified by a processing method with random censoring data.
Keywords
- Planetary gear system
- Reliability modeling
- Probabilistic life
- Random censoring data
1 Introduction
Planetary gear system has the advantages of light weight, small size, large transmission ratio, powerful bearing capacity and high transmission efficiency [1–3]. Therefore, it has been widely used in a variety of mechanical equipment. Actual situation and experience data have showed that the reliability of key gears (such as sun gear, planet gear and ring gear) has a significant impact on the reliability of the whole transmission system [4]. Therefore, some scholars have predicted the life or reliability of the gear system consisting of these gears. On the basis of considering time-varying meshing stiffness of gears, Qin et al. [5] studied the dynamic reliability of wind turbine gear system. Hu et al. [6] established a reliability model for closed planetary gear system, which considered the effects of load, tooth width and load sharing on reliability of the gear system. Zhou et al. [7] considered reliability-based sensitive factors to conduct the reliability analysis for the planetary gear system in shearer mechanism. Li et al. [8] studied the influence of unequal load sharing on the reliability of planetary gear system. Wu et al. [9] established a typical two stage planetary gear transmission reliability model based on the product rule of system reliability. Huang et al. [10] presented a novel method to evaluate the reliability of the kinematic accuracy of gear mechanisms with truncated random variables. Nejad et al. [11] presented a long-term fatigue damage reliability analysis method for tooth root bending in wind turbine drive trains. Shang et al. [12] combined fuzzy mathematics with reliability theory, and adopted multi-objective optimization design to study the design technology of high reliability and high power density for planetary gear transmission in large energy installations. Meanwhile, Hao [13] carried out a multi-objective fuzzy reliability optimization design for the multistage planetary gear system. Zhang et al. [14] applied an improved genetic algorithm to the reliability optimization design of NGW planetary gear transmission to improve the stability and transmission efficiency of the transmission system. In addition, many other researchers have conducted relevant studies [15–18].
In planetary gear system, movement state of each gear is very complicated, and their working environment is generally worse [19]. Therefore, for the validity and simplicity of forecasting method, how to establish model, and how to get input variable for the model should be a focus. A large number of scholars have established the reliability model of planetary gear system based on dynamics theory, which makes the form of the models too complex or difficult to ensure the accuracy. Field or accelerated gear system tests can effectively obtain the life or reliability information, but for planetary gear system, long time and high cost make it difficult to realize [20]. In order to predict the probability life of planetary gear system simply and effectively, this paper combines a simple gear pair meshing test with a life distribution transformation model. Specifically, test result of the special gears (whose parameters are the same as those of service gears) is used as input variables for the transformation model in order to fully reflect the influence of service gear performance on the gear system life. The gear pair meshing test can effectively simulate the running state of gears in planetary gear system to a certain extent, so that the test data will contain a large number of the gear system life information. Therefore, prediction accuracy of the model is guaranteed while its complexity is reduced.
2 Kinematic and Mechanical Analysis
2.1 Kinematic Analysis
Computational results of motion relations
Parts | Absolute velocity | Relative velocity | Load quantity |
---|---|---|---|
Sun gear | \(\omega_{\text{s}}\) | \(\frac{{R_{\text{r}} }}{{R_{\text{s}} + R_{\text{r}} }}\omega_{\text{s}}\) | \(\frac{{R_{\text{r}} \omega_{\text{s}} n_{\text{p}} t}}{{R_{\text{s}} + R_{\text{r}} }}\) |
Planet gear | \(\frac{{R_{\text{s}} }}{{R_{\text{s}} - R_{\text{r}} }}\omega_{\text{s}}\) | \(\frac{{2R_{\text{s}} R_{\text{r}} }}{{R_{\text{s}}^{2} - R_{\text{r}}^{2} }}\omega_{\text{s}}\) | \(\frac{{4R_{\text{s}} R_{\text{r}} \omega_{\text{s}} n_{\text{p}} t}}{{R_{\text{r}}^{2} - R_{\text{s}}^{2} }}\) |
Ring gear | 0 | \(- \frac{{R_{\text{s}} }}{{R_{\text{s}} + R_{\text{r}} }}\omega_{\text{s}}\) | \(\frac{{R_{\text{s}} \omega_{\text{s}} n_{\text{p}} t}}{{R_{\text{s}} + R_{\text{r}} }}\) |
Planet carrier | \(\frac{{R_{\text{s}} }}{{R_{\text{s}} + R_{\text{r}} }}\omega_{\text{s}}\) | 0 | ‒ |
2.2 Calculation of Root Bending Stress
System parameters and stress calculation results
Parameters | Sun gear | Planet gear | Ring gear |
---|---|---|---|
Normal modulus (mm) | 1 | 1 | 1 |
Number of teeth | 51 | 21 | 93 |
Pressure angle (°) | 20 | 20 | 20 |
Helix angle (°) | 26 | 26 | 26 |
Face width (mm) | 22 | 21.5 | 22 |
Tooth thickness (mm) | 1.49 | 1.55 | 1.57 |
Profile shift coefficient | −0.1 | 0 | 0 |
Root fillet radius (mm) | 0.41 | 0.42 | 0.15 |
Root roughness (μm) | R_{z} = 10 | R_{z} = 10 | R_{z} = 10 |
ISO quality grade | 6 | 6 | 6 |
Material | 20CrMnTi | 20CrMnTi | 20CrMnTi |
Root stress (MPa) | 388 | 362 | 420 |
3 Gear Fatigue Test
Bending fatigue fracture of teeth is the most severe failure type for gears [22, 23], which may lead to a direct collapse of the power transmission system. More seriously, in the field of aviation, fracture of the teeth can lead to serious accidents [24]. Therefore, a gear bending fatigue test is carried out to collect the life data of specific gears and to use the statistical result as input variable for the transformation model.
3.1 Gear Sample
Parameters of gear sample
Parameters | Values | Parameters | Values |
---|---|---|---|
Normal modulus (mm) | 3.7 | Root radius (mm) | 2 |
Number of teeth | 27 | Root roughness (μm) | R_{z} = 10 |
Pressure angle (°) | 20 | ISO quality grade | 6 |
Helix angle (°) | 26 | Material | 20CrMnTi |
Face width (mm) | 20 | Case depth (mm) | 0.8 ± 0.13 |
Tooth thickness (mm) | 6.6 | Surface hardness | HRC59–63 |
Base pitch (mm) | 11.8 | Core hardness | HRC35–48 |
Modification coefficient | 0.1 | Precision machining | Grinding |
3.2 Experimental Equipment
3.3 Test Method and Result
4 Life Distribution Transformation Model
For the probabilistic life prediction of gear, we consider a single gear as a sequential system, with each tooth as a component in the system. If any teeth failed, the gear would not be able to fulfill its function of power or motion transmission, thus resulting in failure of the sequential system. According to the definition of order statistics, the gear probabilistic life equals the minimal order statistics of the single tooth probabilistic life [26].
In the planetary gear system, the sun gear, planet gear, and ring gear engage with each other, so they have the same modulus and thus have the same carrying capacity. If material properties of all the gears are also the same, the strength of the teeth can be considered the same. Therefore, the tooth life under a given load level can be considered as the random variable with independent and identical distribution. Furthermore, based on the probabilistic life of tooth and the concept of minimal order statistics, a probabilistic life transformation model of planetary gear system is established.
5 Model Validation
5.1 Random Censoring Data
In life-cycle testing some products fail, so an accurate failure time (i.e., product life failure data) can be obtained; while other products may exit the test before failure due to some reasons, thereby obtaining a higher life than the testing time, namely the life censoring data. The failure data and censoring data are generally called random censoring data.
In order to verify the transformation model, the processing method with random censoring data is used. The gear test will stop when any tooth breaks, thereby deriving failure data by the tooth and censoring data of the other teeth. By processing the random censoring data of the teeth, the distribution of tooth life can be obtained, and it can be compared with the results of the transformation model.
5.2 Estimating Distribution Parameters of Random Censoring Data
5.3 Parameters Comparison and Model Validation
The concept of minimal order statistics is used to create the probabilistic life transformation model. First, the probabilistic life of the tooth is obtained based on that of the specific gear. Then, the probabilistic life of the planetary gear system can be obtained based on the probabilistic life of the tooth. Since the two transformations used the same principle, only the first transformation is actually validated. From Eq. (8), we can know the estimate of the distribution parameters of tooth life. From the process of random censoring data, we can also derive the estimate of distribution parameters of tooth life. The results of the two methods can be compared to validate the rationality of the transformation model.
In the gear test, the gear pairs are engaged to transfer power. The drive gear has a higher dynamic torque than the driven gear [29]. It was also found in the test that all the teeth breaking occurred on the drive gears; thus, the life information of the drive gears is used for the validation of the transformation model.
The ten gear life data in Figure 6 can be used. If the number of teeth in a drive gear is z, failure of any tooth will cause the test to stop, and the circulating life is recorded. For the gear, the life data are comprised of the failure data for one tooth and censoring data for the z − 1 teeth. Given j as the number of “circulating life” obtained during the test, then the failure data of the j teeth and censoring data of the j(z − 1) teeth can be obtained.
6 Discussion
- (1)
The life information of specific gear is taken as the input variables for the transformation model, thereby enabling the model to reflect the large amount of factors that influence system life, which not only improved the prediction precision, but greatly simplified the model as well.
- (2)
Through the fatigue test, the P-S-N curves of the specific gear can be obtained. Based on the transformation model, the probabilistic life of the planetary gear system at an arbitrary stress level can be derived.
- (3)
The actual load of the planetary gear system can be first statistically processed, and then the results are applied to the fatigue test with the specific gear, then this method can improve the life prediction precision of the gear system and is more effective for the life prediction of service gear system.
- (4)
The model does not consider any failure correlation among gears. In practice, the load of gears are random, thus, the results of the transformation model tend to be conservative.
7 Conclusions
- (1)
Based on the concept of minimal order statistics, the model for the probabilistic life transformation for planetary gear system is established. In the model the probabilistic life of single tooth is first calculated based on that of the specific gear. Further, the probabilistic life of the planetary gear system can be estimated based on that of the single tooth.
- (2)
A gear bending fatigue test is performed at a constant stress level, and the bending fatigue life data of ten 20CrMnTi carbonized gears are obtained.
- (3)
The validity of the transformation model is illustrated by the method of random censored data, and it also indicates that the model has the ability to process small sample data. A large number of calculations, analyses and comparisons show that the model is also valid for data with different scatter.
Declarations
Authors’ Contributions
L-YX was in charge of the whole trial; ML wrote the manuscript; H-YL and J-GR assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authors’ Information
Ming Li, born in 1986, is currently a PhD candidate at School of Mechanical Engineering and Automation, Northeastern University, China. His research interests include mechanical transmission design and fatigue reliability. Tel: +86-13940397486; E-mail: a15941660820@163.com.
Li-Yang Xie, born in 1962, PhD, is currently a professor and a doctorial supervisor at Northeastern University, China. His research interest is reliability theory. Tel: +86-13804011565; E-mail: lyxie@mail.neu.edu.cn.
Hai-Yang Li, born in 1988, is currently a PhD candidate at School of Mechanical Engineering and Automation, Northeastern University, China. His research interests include mechanical transmission design and fatigue reliability. Tel: +86-15940426127; E-mail: naxiaozi_a@126.com.
Jun-Gang Ren, born in 1982, is currently a PhD candidate at School of Mechanical Engineering and Automation, Northeastern University, China. His research interests include mechanical transmission design and fatigue reliability. Tel: +86-18640345458; E-mail: 289188724@qq.com.
Competing Interests
The authors declare that they no competing interests.
Ethics Approval and Consent to Participate
Not applicable
Funding
Supported by National Key Technology Research and Development Program of China (Grant No. 2014BAF08B01), and Natural Science Foundation of China (Grant No. 51335003), and Collaborative Innovation Center of Major Machine Manufacturing in Liaoning Province of China.
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Authors’ Affiliations
References
- S Mo, Y D Zhang, Q Wu, et al. Load sharing behavior of star gearing reducer for geared turbofan engine. Chinese Journal of Mechanical Engineering, 2017, 30(4): 796-803.View ArticleGoogle Scholar
- J Zhang, F Guo. Statistical modification analysis of helical planetary gears based on response surface method and Monte Carlo simulation. Chinese Journal of Mechanical Engineering, 2015, 28(6): 1194-1203.View ArticleGoogle Scholar
- I Miguel. Planetary gear profile modification design based on load sharing modelling. Chinese Journal of Mechanical Engineering, 2015, 28(4): 810-820.View ArticleGoogle Scholar
- M Savage, C A Paridon, J J Coy. Reliability model for planetary gear trains. Transactions of the ASME, Journal of Mechanical, Transmissions, and Automation in Design, 1983, 105: 291-297.View ArticleGoogle Scholar
- D T Qin, Z G Zhou, J Yang, et al. Time-dependent reliability analysis of gear transmission system of wind turbine under stochastic wind load. Journal of Mechanical Engineering, 2012, 48(3): 1-8. (in Chinese)View ArticleGoogle Scholar
- Q C Hu, F H Duan, S S Wu. Research on reliability of closed planetary transmission systems. Chinese Mechanical Engineering, 2007, 18(2): 146-149. (in Chinese)Google Scholar
- D Zhou, X F Zhang, Y M Zhang. Dynamic reliability analysis for planetary gear system in shearer mechanisms. Mechanism and Machine Theory, 2016, 105: 244-259.View ArticleGoogle Scholar
- M Li, L Y Xie, L J Ding. Reliability analysis and calculation for planetary mechanism. Chinese Journal of Aeronautics, 2017, 38(8): 1-14. (in Chinese)Google Scholar
- S S Wu, F H Duan, Q C Hu. The influence of allocation of systematic parameters on the reliability of multi staged planet gear transmission. Journal of Machine Design, 2007, 24(10): 43-45. (in Chinese)Google Scholar
- X Z Huang, S Hu, Y M Zhang, et al. A method to determine kinematic accuracy reliability of gear mechanisms with truncated random variables. Mechanism and Machine Theory, 2015, 92: 200-212.View ArticleGoogle Scholar
- A R Nejad, Z Gao, T Moan. On long-term fatigue damage and reliability analysis of gears under wind loads in offshore wind turbine drivetrains. International Journal of Fatigue, 2014, 61: 116-128.View ArticleGoogle Scholar
- Z Shang, Z M Liu, C M Wang. Research on design technology of high reliability and high power density for planetary gear transmission. Journal of Machine Design, 2010, 27(6): 48-51. (in Chinese)Google Scholar
- L P Hao. Fuzzy reliability optimization design with multi-objective for multi-stage planetary gear train. Journal of Mechanical Transmission, 2015, 39(1): 87-91. (in Chinese)Google Scholar
- D H Zhang, F L Wu, S T Zhang, et al. Application of improved genetic algorithm in reliability optimization design of NGW planetary gear transmission. Journal of Mechanical Transmission, 2013, 37(2): 44-46. (in Chinese)Google Scholar
- G Zhang, G Wang, X Li. Global optimization of reliability design for large ball mill gear transmission based on the kriging model and genetic algorithm. Mechanism and Machine Theory, 2013, 69: 321-336.View ArticleGoogle Scholar
- Y F Li, S Valla, E Zio. Reliability assessment of ge-neric geared wind turbines by GTST-MLD model and Monte Carlo simulation. Renewable Energy, 2015, 83: 222-233.View ArticleGoogle Scholar
- A Guerine, A Elhami, L Walha. A perturbation approach for the dynamic analysis of one stage gear system with uncertain parameters. Mechanism and Machine Theory, 2015, 92: 113-126.View ArticleGoogle Scholar
- H J Ren, H Zhang, X G Yu, et al. Nonlinear dynamics of the gear system in five shaft integrally geared centrifugal compressor. Journal of Mechanical Engineering, 2017, 53(9): 1–7 (in Chinese).View ArticleGoogle Scholar
- L Xiang, N Gao, A J Hu. Dynamic analysis of a planetary gear system with multiple nonlinear parameters. Journal of Computational and Applied Mathematics, 2017, 327(2018): 325-340.MathSciNetView ArticleMATHGoogle Scholar
- Z G Zhou, D T Qin, J Yang, et al. Fatigue life prediction of gear transmission system of wind turbine under stochastic wind load. Acta Energiae Solaris Sinica, 2014, 35(7): 1183-1190. (in Chinese)Google Scholar
- M Li, L Y Xie, L J Ding. Load sharing analysis and reliability prediction for planetary gear train of helicopter. Mechanism and Machine Theory, 2017, 115: 97-113.View ArticleGoogle Scholar
- D Y Zhang, S G Liu. Investigation on bending fatigue failure of a micro-gear through finite element analysis. Engineering Failure Analysis, 2013, 31: 225-235.View ArticleGoogle Scholar
- A Osman. Fatigue failure of a helical gear in a gearbox. Engineering Failure Analysis, 2006, 13: 1116-1125.View ArticleGoogle Scholar
- A S Nauman, K M Deen. Investigating the failure of bevel gears in an aircraft engine. Case Studies in Engineering Failure Analysis, 2013, 1: 24-31.View ArticleGoogle Scholar
- L Y Xie. Fatigue reliability evaluation method for gearbox component and system of wind turbine. Journal of Mechanical Engineering, 2014, 50: 1-8. (in Chinese)View ArticleGoogle Scholar
- L Y Xie. Issues and commentary on mechanical reliability theories, methods and models. Journal of Mechanical Engineering, 2014, 50(14): 27-35. (in Chinese)View ArticleGoogle Scholar
- Z W An, Y Zhang, B Liu. A method to determine the life distribution function of components for wind turbine gearbox. Journal of University of Electronic Science and Technology of China, 2014, 43(6): 950-954. (in Chinese)Google Scholar
- M Kateri. On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data. Statistics and Probability Letters, 2008, 78: 2971-2975.MathSciNetView ArticleMATHGoogle Scholar
- X L Zhu. Gear test technology and equipment. Beijing: China Machine Press, 1988.Google Scholar