Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear
Abstract
1. Introduction
2. Materials and Methods
2.1. Kriging Surrogate Model of Mechanical Structural Response Based on FEM
2.2. Wear Reliability Model of Rolling Ball Bearing
2.3. Sensitivity Analysis and Reliability-Based Robust Design
3. Results
- The wear reliability target for ball bearings is set to , which means that the optimized reliability result obtained at the moment should be no less than . Therefore, the reliability constraint is expressed as follows:
- In order to ensure the point contact between the inner/outer raceway and the balls, the inner/outer raceway radius should be larger than the ball radius , and the inner raceway radius is usually larger than the outer raceway radius . According to the above principles to establish constraints as follows:
- Based on the structural design requirements of ball bearings and with reference to the various dimensional standards for ball bearings [46], the initial value constraints are established for the input parameters as follows:
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
- Weibull, W. Efficient Methods for Estimating Fatigue Life Distribution of Rolling Bearings; Elsevier: New York, NY, USA, 1962; pp. 252–265. [Google Scholar]
- Lundberg, G.; Palmgren, A. Dynamic capacity of rolling bearings. J. Appl. Mech. 1949, 16, 165–172. [Google Scholar] [CrossRef]
- Organización Internacional de Normalización. Rolling Bearings: Dynamic Load Ratings and Rating Life; ISO: Geneva, Switzerland, 2007. [Google Scholar]
- Ding, N.; Yan, S.; Ma, G.; Li, H.; Jiang, D. Mixed thermo-elasto-hydrodynamic lubrication and wear coupling simulation analysis for dynamical load journal bearings. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2023, 237, 6020–6041. [Google Scholar] [CrossRef]
- Wang, X.; Wang, B.; Chang, M.; Li, L. Reliability and sensitivity analysis for bearings considering the correlation of multiple failure modes by mixed Copula function. Proc. Inst. Mech. Eng. Part O J. Risk Reliab. 2019, 234, 15–26. [Google Scholar] [CrossRef]
- Li, X.; Yan, K.; Lv, Y.; Yan, B.; Dong, L.; Hong, J. Study on the influence of machine tool spindle radial error motion resulted from bearing outer ring tilting assembly. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2019, 233, 3246–3258. [Google Scholar] [CrossRef]
- Zhang, T.; He, D. An improved high-order statistical moment method for structural reliability analysis with insufficient data. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2017, 232, 1050–1056. [Google Scholar] [CrossRef]
- Li, Z.; Tian, G.; Cheng, G.; Liu, H.; Cheng, Z. An integrated cultural particle swarm algorithm for multi-objective reliability-based design optimization. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2013, 228, 1185–1196. [Google Scholar] [CrossRef]
- Gao, Y.; Zhang, F.; Li, Y. Reliability optimization design of a planar multi-body system with two clearance joints based on reliability sensitivity analysis. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2018, 233, 1369–1382. [Google Scholar] [CrossRef]
- E, S.; Wang, Y.; Xie, B.; Lu, F. A Reliability-Based Robust Design Optimization Method for Rolling Bearing Fatigue under Cyclic Load Spectrum. Mathematics 2023, 11, 2843. [Google Scholar] [CrossRef]
- Winkler, A.; Marian, M.; Tremmel, S.; Wartzack, S. Numerical modeling of wear in a thrust roller bearing under mixed elastohydrodynamic lubrication. Lubricants 2020, 8, 58. [Google Scholar] [CrossRef]
- Pozzebon, M.L.; Lin, C.; Meehan, P.A. On the modeling of wear in grease-lubricated spherical roller bearings. Tribol. Trans. 2020, 63, 806–819. [Google Scholar] [CrossRef]
- Zhao, Y.; Lu, Z.; Ono, T. A simple third-moment method for structural reliability. J. Asian Archit. Build. Eng. 2006, 5, 129–136. [Google Scholar] [CrossRef]
- Zhao, Y.; Ono, T. Moment methods for structural reliability. Struct. Saf. 2001, 23, 47–75. [Google Scholar] [CrossRef]
- Clark, D.L., Jr.; Bae, H.; Forster, E.E. Gaussian Surrogate Dimension Reduction for Efficient Reliability-Based Design Optimization. AIAA J. 2020, 58, 4736–4750. [Google Scholar] [CrossRef]
- Lee, I.; Choi, K.K.; Du, L.; Gorsich, D. Dimension reduction method for reliability-based robust design optimization. Comput. Struct. 2008, 86, 1550–1562. [Google Scholar] [CrossRef]
- Dauparas, J.; Anishchenko, I.; Bennett, N.; Bai, H.; Ragotte, R.J.; Milles, L.F.; Wicky, B.I.; Courbet, A.; de Haas, R.J.; Bethel, N.; et al. Robust deep learning--based protein sequence design using ProteinMPNN. Science 2022, 378, 49–56. [Google Scholar] [CrossRef] [PubMed]
- Park, G.; Lee, T.; Lee, K.H.; Hwang, K. Robust design: An overview. AIAA J. 2006, 44, 181–191. [Google Scholar] [CrossRef]
- Lee, D.; Jahanbin, R.; Rahman, S. Robust design optimization by spline dimensional decomposition. Probabilistic Eng. Mech. 2022, 68, 103218. [Google Scholar] [CrossRef]
- Park, J.; Yoo, D.; Moon, J.; Yoon, J.; Park, J.; Lee, S.; Lee, D.; Kim, C. Reliability-Based Robust Design Optimization of Lithium-Ion Battery Cells for Maximizing the Energy Density by Increasing Reliability and Robustness. Energies 2021, 14, 6236. [Google Scholar] [CrossRef]
- Yu, S.; Wang, Z.; Wang, Z. Time-dependent reliability-based robust design optimization using evolutionary algorithm. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part B Mech. Eng. 2019, 5, 20911. [Google Scholar] [CrossRef]
- Gong, C.; Frangopol, D.M. An efficient time-dependent reliability method. Struct. Saf. 2019, 81, 101864. [Google Scholar] [CrossRef]
- Yang, W.; Zhang, B.; Wang, W.; Li, C. Time-dependent structural reliability under nonstationary and non-Gaussian processes. Struct. Saf. 2023, 100, 102286. [Google Scholar] [CrossRef]
- Doltsinis, I.; Kang, Z. Robust design of structures using optimization methods. Comput. Meth. Appl. Mech. Eng. 2004, 193, 2221–2237. [Google Scholar] [CrossRef]
- Zhang, T. An improved high-moment method for reliability analysis. Struct. Multidiscip. Optim. 2017, 56, 1225–1232. [Google Scholar] [CrossRef]
- Zhang, T. Robust reliability-based optimization with a moment method for hydraulic pump sealing design. Struct. Multidiscip. Optim. 2018, 58, 1737–1750. [Google Scholar] [CrossRef]
- Zhang, T.; He, D. A Reliability-Based Robust Design Method for the Sealing of Slipper-Swash Plate Friction Pair in Hydraulic Piston Pump. IEEE Trans. Reliab. 2018, 67, 459–469. [Google Scholar] [CrossRef]
- Ant O Nio, C.C.C.C.; Hoffbauer, L.I.S.N. An approach for reliability-based robust design optimisation of angle-ply composites. Compos. Struct. 2009, 90, 53–59. [Google Scholar] [CrossRef]
- Dammak, K.; El Hami, A. Thermal reliability-based design optimization using Kriging model of PCM based pin fin heat sink. Int. J. Heat Mass Transf. 2021, 166, 120745. [Google Scholar] [CrossRef]
- Tripathy, R.K.; Bilionis, I. Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification. J. Comput. Phys. 2018, 375, 565–588. [Google Scholar] [CrossRef]
- White, D.A.; Arrighi, W.J.; Kudo, J.; Watts, S.E. Multiscale topology optimization using neural network surrogate models. Comput. Meth. Appl. Mech. Eng. 2019, 346, 1118–1135. [Google Scholar] [CrossRef]
- Allaix, D.L.; Carbone, V.I. An improvement of the response surface method. Struct. Saf. 2011, 33, 165–172. [Google Scholar] [CrossRef]
- Zhang, D.; Han, X.; Jiang, C.; Liu, J.; Li, Q. Time-dependent reliability analysis through response surface method. J. Mech. Des. 2017, 139, 41404. [Google Scholar] [CrossRef]
- Jeong, S.; Murayama, M.; Yamamoto, K. Efficient optimization design method using kriging model. J. Aircr. 2005, 42, 413–420. [Google Scholar] [CrossRef]
- Lu, C.; Feng, Y.; Fei, C.; Bu, S. Improved decomposed-coordinated kriging modeling strategy for dynamic probabilistic analysis of multicomponent structures. IEEE Trans. Reliab. 2019, 69, 440–457. [Google Scholar] [CrossRef]
- Keshtegar, B.; Mert, C.; Kisi, O. Comparison of four heuristic regression techniques in solar radiation modeling: Kriging method vs RSM, MARS and M5 model tree. Renew. Sustain. Energy Rev. 2018, 81, 330–341. [Google Scholar] [CrossRef]
- Jiang, Z.; Wu, J.; Huang, F.; Lv, Y.; Wan, L. A novel adaptive Kriging method: Time-dependent reliability-based robust design optimization and case study. Comput. Ind. Eng. 2021, 162, 107692. [Google Scholar] [CrossRef]
- Echard, B.; Gayton, N.; Lemaire, M. AK-MCS: An active learning reliability method combining Kriging and Monte Carlo simulation. Struct. Saf. 2011, 33, 145–154. [Google Scholar] [CrossRef]
- Liu, H.; Li, S.; Huang, X. Adaptive surrogate model coupled with stochastic configuration network strategies for time-dependent reliability assessment. Probabilistic Eng. Mech. 2023, 71, 103406. [Google Scholar] [CrossRef]
- Archard, J.F.; Hirst, W. The wear of metals under unlubricated conditions. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 1956, 236, 397–410. [Google Scholar]
- GB/T 25769-2010; Rolling Bearings—Measuring Methods for Radial Internal Clearance. Standards Press of China: Beijing, China, 2010.
- GB/T 4604.1-2012; Rolling Bearings—Internal Clearance—Part 1: Radial Internal Clearance for Radial Bearings. Standards Press of China: Beijing, China, 2012.
- Chakraborty, S.; Das, S.; Tesfamariam, S. Robust design optimization of nonlinear energy sink under random system parameters. Probabilistic Eng. Mech. 2021, 65, 103139. [Google Scholar] [CrossRef]
- Pellizzari, F.; Marano, G.C.; Palmeri, A.; Greco, R.; Domaneschi, M. Robust optimization of MTMD systems for the control of vibrations. Probabilistic Eng. Mech. 2022, 70, 103347. [Google Scholar] [CrossRef]
- Zhang, T. Matrix description of differential relations of moment functions in structural reliability sensitivity analysis. Appl. Math. Mech. 2017, 38, 57–72. [Google Scholar] [CrossRef]
- Zhang, S. Latest Bearing Manuals; Publishing House of Electronics Industry: Beijing, China, 2007. [Google Scholar]
Inner Diameter (mm) | Radial Clearance (mm) | Axial Clearance (mm) |
---|---|---|
<30 | 4D/1000 | 0.2 |
35~70 | 3.5D/1000 | 0.3 |
75~100 | 3D/1000 | 0.3 |
>100 | <0.3 | 0.3 |
Bearing Parameters | Mean | Variance | Skewness | Kurtosis |
---|---|---|---|---|
Center circle diameter (mm) | 102.4874 | 2.5 × 10−5 | 9.96 × 10−2 | 2.87 |
Ball diameter (mm) | 13.4940 | 4.0 × 10−6 | 4.09 × 10−2 | 2.96 |
Inner ring groove radius | 7.1010 | 1.0 × 10−6 | −7.37 × 10−2 | 3.03 |
Outer ring groove radius (mm) | 6.9960 | 1.0 × 10−6 | 1.29 × 10−1 | 2.69 |
Elastic modulus (GPa) | 208 | 1.0 × 10−4 | −1.88 × 10−4 | 3.47 |
Poisson’s ratio | 0.3 | 1.0 × 10−8 | −1.53 × 10−1 | 2.85 |
Hardness | 200 | 1.0 × 10−4 | 8.35 × 10−2 | 2.88 |
Initial radial clearance | 0.0240 | 2.5 × 10−5 | 6.46 × 10−2 | 2.92 |
Methods | Relative Error | |||
---|---|---|---|---|
FOSM method | 635 h | −1.6567 | 95.12% | 0.7521% |
The proposed method | 635 h | −1.5985 | 94.50% | 0.0953% |
MCS method | 635 h | \ | 94.41% | \ |
Input Parameters | Reliability Sensitivity | Sensitivity Gradient | |
---|---|---|---|
Center circle diameter | 0.0037 | −1.06 × 10−6 | 40.4176 |
Ball diameter | 2.2782 | −0.1616 | |
Inner ring groove radius | −1.1763 | −0.0215 | |
Outer ring groove radius | −4.4164 | −0.3036 | |
Elastic modulus | −0.0028 | −1.21 × 10−6 | |
Poisson’s ratio | −0.3828 | −2.28 × 10−4 | |
Hardness | −0.0283 | −1.25 × 10−4 | |
Initial radial clearance | 20.9554 | −34.1776 |
Input Parameters | Before Optimization | After Optimization |
---|---|---|
Center circle diameter (mm) | 102.4874 | 102.4306 |
Ball diameter (mm) | 13.4940 | 13.4501 |
Inner ring groove radius | 7.1010 | 7.1499 |
Outer ring groove radius (mm) | 6.9960 | 7.05 |
Elastic modulus (GPa) | 208 | 208.0212 |
Poisson’s ratio | 0.3 | 0.3497 |
Hardness | 200 | 201.9952 |
Initial radial clearance | 0.0240 | 0.0235 |
Input Parameters | ||||
---|---|---|---|---|
Before optimization | 635 h | −1.5985 | 94.50% | 40.4176 |
After optimization | 635 h | −5.5483 | 99.99% | 8.4755 × 10−5 |
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Wang, Y.; E, S.; Yang, K.; Xie, B.; Lu, F. Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear. Lubricants 2024, 12, 293. https://doi.org/10.3390/lubricants12080293
Wang Y, E S, Yang K, Xie B, Lu F. Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear. Lubricants. 2024; 12(8):293. https://doi.org/10.3390/lubricants12080293
Chicago/Turabian StyleWang, Yanzhong, Shiyuan E, Kai Yang, Bin Xie, and Fengxia Lu. 2024. "Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear" Lubricants 12, no. 8: 293. https://doi.org/10.3390/lubricants12080293
APA StyleWang, Y., E, S., Yang, K., Xie, B., & Lu, F. (2024). Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear. Lubricants, 12(8), 293. https://doi.org/10.3390/lubricants12080293