Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

Abstract

The failure types in gear systems vary, with typical ones mainly including pitting, cracking, wear, and broken teeth. Different modeling and stiffness calculation methods have been developed for various gear failure types. A unified method for typical gear failure modeling and stiffness calculation is introduced in this study by considering the deviations in the time-varying meshing stiffness (TVMS) of faulty gears resulting from the use of different methods. Specifically, a gear tooth is discretized into a large number of microelements expressed with a matrix, and unified models of typical gear failures are built by adjusting the values of the matrix microelements. The values and positions of the microelements in the tooth failure model matrix have the same physical meaning as the parameter variables in the potential energy method (PEM), so the matrix-based failure model can be perfectly matched with PEM. Afterward, a unified method for TVMS is established. Modeling of healthy and faulty gears with pitting, wear, crack, and broken tooth is performed with the matrix equation, and the corresponding TVMS values are calculated by incorporating the matrix models with PEM. On the basis of the results, the mechanism of typical fault types that affect TVMS is analyzed, and the conclusions are verified through the finite element method. The developed unified method is a promising technique for studying the dynamic response characteristics of gear systems with different failure types because of its superiority in eliminating stiffness deviations.

Graphical abstract

Keywords

gears / matrix equation / failure modeling / TVMS calculation / unified method

References

[1] Gao P, Liu H, Yan P F, Xie Y K, Xiang C L, Wang C. Research on application of dynamic optimization modification for an involute spur gear in a fixed-shaft gear transmission system. Mechanical Systems and Signal Processing, 2022, 181: 109530

[2] Guan X L, Tang J Y, Hu Z H, Wang Q S, Kong X N. A new dynamic model of light-weight spur gear transmission system considering the elasticity of the shaft and gear body. Mechanism and Machine Theory, 2022, 170: 104689

[3] Ouyang T C, Mo X Y, Lu Y C, Wang J X. CFD-vibration coupled model for predicting cavitation in gear transmissions. International Journal of Mechanical Sciences, 2022, 225: 107377

[4] Al-TubiI SLong HZhangJShawB. Experimental and analytical study of gear micropitting initiation and propagation under varying loading conditions. Wear, 2015, 328–329: 8–16

[5] Huangfu Y F, Dong X J, Chen K K, Tu G W, Long X H, Peng Z K. A tribo-dynamic based pitting evolution model of planetary gear sets: a topographical updating approach. International Journal of Mechanical Sciences, 2022, 220: 107157

[6] Wang W, Liu H J, Zhu C C, Wei P T, Wu W. Micromechanical analysis of gear fatigue-ratcheting damage considering the phase state and inclusion. Tribology International, 2019, 136: 182–195

[7] Jia P, Liu H J, Zhu C C, Wu W, Lu G C. Contact fatigue life prediction of a bevel gear under spectrum loading. Frontiers of Mechanical Engineering, 2020, 15(1): 123–132

[8] Mohammed O D, Rantatalo M. Gear fault models and dynamics-based modeling for gear fault detection––a review. Engineering Failure Analysis, 2020, 117: 104798

[9] Bonaiti L, Gorla C. Estimation of gear SN curve for tooth root bending fatigue by means of maximum likelihood method and statistic of extremes. International Journal of Fatigue, 2021, 153: 106451

[10] Zhou K, Diehl E, Tang J. Deep convolutional generative adversarial network with semi-supervised learning enabled physics elucidation for extended gear fault diagnosis under data limitations. Mechanical Systems and Signal Processing, 2023, 185: 109772

[11] Karabacak Y E, Gürsel Özmen N, Gümüşel L. Intelligent worm gearbox fault diagnosis under various working conditions using vibration, sound and thermal features. Applied Acoustics, 2022, 186: 108463

[12] Chen X X, Yao Y P, Xing J Z. Meshing stiffness property and meshing status simulation of harmonic drive under transmission loading. Frontiers of Mechanical Engineering, 2022, 17(2): 18

[13] Xiang L, An C H, Zhang Y, Hu A J. Failure dynamic modeling and analysis of planetary gearbox considering gear tooth spalling. Engineering Failure Analysis, 2021, 125: 105444

[14] Meng F S, Xia H, Zhang X, Wang J X, Jin Y L. Study on nonlinear dynamic characteristics of gear system with 3D anisotropic rough tooth surface based on fractal theory. International Journal of Non-linear Mechanics, 2023, 150: 104366

[15] Meng F S, Xia H, Zhang X, Wang J X, Jin Y L. Mechanism analysis for GDTE-based fault diagnosis of planetary gears. International Journal of Mechanical Sciences, 2023, 259: 108627

[16] Yu X L, Huangfu Y, Yang Y, Du M G, He Q B, Peng Z K. Gear fault diagnosis using gear meshing stiffness identified by gearbox housing vibration signals. Frontiers of Mechanical Engineering, 2022, 17(4): 57

[17] Liang X H, Zuo M J, Feng Z P. Dynamic modeling of gearbox faults: a review. Mechanical Systems and Signal Processing, 2018, 98: 852–876

[18] Wang Q B, Chen K K, Zhao B, Ma H, Kong X G. An analytical-finite-element method for calculating mesh stiffness of spur gear pairs with complicated foundation and crack. Engineering Failure Analysis, 2018, 94: 339–353

[19] Andary F, Heinzel C, Wischmann S, Berroth J, Jacobs G. Calculation of tooth pair stiffness by finite element analysis for the multibody simulation of flexible gear pairs with helical teeth and flank modifications. Multibody System Dynamics, 2023, 59(4): 395–428

[20] Yu X, Sun Y Y, Li H G, Wu S J. An improved meshing stiffness calculation algorithm for gear pair involving fractal contact stiffness based on dynamic contact force. European Journal of Mechanics A: Solids, 2022, 94: 104595

[21] Raghuwanshi N K, Parey A. Mesh stiffness measurement of cracked spur gear by photoelasticity technique. Measurement, 2015, 73: 439–452

[22] Raghuwanshi N K, Parey A. Experimental measurement of gear mesh stiffness of cracked spur gear by strain gauge technique. Measurement, 2016, 86: 266–275

[23] Raghuwanshi N K, Parey A. Experimental measurement of mesh stiffness by laser displacement sensor technique. Measurement, 2018, 128: 63–70

[24] Verma J G, Kumar S, Kankar P K. Crack growth modeling in spur gear tooth and its effect on mesh stiffness using extended finite element method. Engineering Failure Analysis, 2018, 94: 109–120

[25] Chaari F, Fakhfakh T, Haddar M. Analytical modeling of spur gear tooth crack and influence on gearmesh stiffness. European Journal of Mechanics A: Solids, 2009, 28(3): 461–468

[26] El Yousfi B, Soualhi A, Medjaher K, Guillet F. New approach for gear mesh stiffness evaluation of spur gears with surface defects. Engineering Failure Analysis, 2020, 116: 104740

[27] Marafona J D M, Marques P M T, Martins R C, Seabra J H O. Mesh stiffness models for cylindrical gears: a detailed review. Mechanism and Machine Theory, 2021, 166: 104472

[28] Shen Z X, Yang L H, Qiao B J, Luo W, Chen X F, Yan R Q. Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear. Frontiers of Mechanical Engineering, 2022, 17(1): 9

[29] Chaari F, Baccar W, Abbes M S, Haddar M. Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission. European Journal of Mechanics A: Solids, 2008, 27(4): 691–705

[30] Meng F S, Xia H, Zhang X, Wang J X. A new tooth pitting modeling method based on matrix equation for evaluating time-varying mesh stiffness. Engineering Failure Analysis, 2022, 142: 106799

[31] Ma H, Zeng J, Feng R J, Pang X, Wen B C. An improved analytical method for mesh stiffness calculation of spur gears with tip relief. Mechanism and Machine Theory, 2016, 98: 64–80

[32] Xie C Y, Hua L, Han X H, Lan J, Wan X J, Xiong X S. Analytical formulas for gear body-induced tooth deflections of spur gears considering structure coupling effect. International Journal of Mechanical Sciences, 2018, 148: 174–190

[33] Feng M J, Ma H, Li Z W, Wang Q B, Wen B C. An improved analytical method for calculating time-varying mesh stiffness of helical gears. Meccanica, 2018, 53(4–5): 1131–1145

[34] Huang W K, Ma H, Zhao Z F, Wang P F, Peng Z K, Zhang X X, Zhao S T. An iterative model for mesh stiffness of spur gears considering slice coupling under elastohydrodynamic lubrication. Journal of Central South University, 2023, 30(10): 3414–3434

[35] Zhao W Q, Liu J, Zhao W H, Zheng Y. An investigation on vibration features of a gear-bearing system involved pitting faults considering effect of eccentricity and friction. Engineering Failure Analysis, 2022, 131: 105837

[36] Thirumurugan R, Gnanasekar N. Influence of finite element model, load-sharing and load distribution on crack propagation path in spur gear drive. Engineering Failure Analysis, 2020, 110: 104383

[37] Brethee K F, Zhen D, Gu F S, Ball A D. Helical gear wear monitoring: modeling and experimental validation. Mechanism and Machine Theory, 2017, 117: 210–229

[38] Huangfu Y F, Zhao Z F, Ma H, Han H Z, Chen K K. Effects of tooth modifications on the dynamic characteristics of thin-rimmed gears under surface wear. Mechanism and Machine Theory, 2020, 150: 103870

[39] Huangfu Y F, Chen K K, Ma H, Li X, Han H Z, Zhao Z F. Meshing and dynamic characteristics analysis of spalled gear systems: a theoretical and experimental study. Mechanical Systems and Signal Processing, 2020, 139: 106640

[40] Liu Z M, Shang E L, Huangfu Y F, Ma H, Zhu J Z, Zhao S T, Long X H, Li Z W. Vibration characteristics analysis of flexible helical gear system with multi-tooth spalling fault: simulation and experimental study. Mechanical Systems and Signal Processing, 2023, 201: 110687

[41] Chen K K, Huangfu Y F, Ma H, Xu Z T, Li X, Wen B C. Calculation of mesh stiffness of spur gears considering complex foundation types and crack propagation paths. Mechanical Systems and Signal Processing, 2019, 130: 273–292

[42] Chen K K, Huangfu Y F, Zhao Z F, Ma H, Dong X J. Dynamic modeling of the gear-rotor systems with spatial propagation crack and complicated foundation structure. Mechanism and Machine Theory, 2022, 172: 104827

[43] Liu Z M, Chang C, Hu H D, Ma H, Yuan K G, Li X, Zhao X J, Peng Z K. Dynamic characteristics of spur gear system with tooth root crack considering gearbox flexibility. Mechanical Systems and Signal Processing, 2024, 208: 110966

[44] Pedersen N L, Jørgensen M F. On gear tooth stiffness evaluation. Computers & Structures, 2014, 135: 109–117

[45] Munro R G, Palmer D, Morrish L. An experimental method to measure gear tooth stiffness throughout and beyond the path of contact. Proceedings of the Institution of Mechanical Engineers Part C, Journal of Mechanical Engineering Science, 2001, 215: 793–803

[46] Pimsarn M, Kazerounian K. Efficient evaluation of spur gear tooth mesh load using pseudo-interference stiffness estimation method. Mechanism and Machine Theory, 2002, 37(8): 769–786

[47] Cooley C G, Liu C G, Dai X, Parker R G. Gear tooth mesh stiffness: a comparison of calculation approaches. Mechanism and Machine Theory, 2016, 105: 540–553

[48] Ma H, Song R Z, Pang X, Wen B C. Time-varying mesh stiffness calculation of cracked spur gears. Engineering Failure Analysis, 2014, 44: 179–194

[49] Düzcükoğlu H, İmrek H. A new method for preventing premature pitting formation on spur gears. Engineering Fracture Mechanics, 2008, 75(15): 4431–4438

[50] Xu X Y, Lai J B, Lohmann C, Tenberge P, Weibring M, Dong P. A model to predict initiation and propagation of micro-pitting on tooth flanks of spur gears. International Journal of Fatigue, 2019, 122: 106–115

[51] Tan C K, Irving P, Mba D. A comparative experimental study on the diagnostic and prognostic capabilities of acoustics emission, vibration and spectrometric oil analysis for spur gears. Mechanical Systems and Signal Processing, 2007, 21(1): 208–233

[52] Aslantaş K, Tasgetiren S. A study of spur gear pitting formation and life prediction. Wear, 2004, 257(11): 1167–1175

[53] Lei Y G, Liu Z Y, Wang D L, Yang X, Liu H, Lin J. A probability distribution model of tooth pits for evaluating time-varying mesh stiffness of pitting gears. Mechanical Systems and Signal Processing, 2018, 106: 355–366

[54] Mao K, Chetwynd D G, Millson M. A new method for testing polymer gear wear rate and performance. Polymer Testing, 2020, 82: 106323

[55] Shen Z X, Qiao B J, Yang L H, Luo W, Yang Z B, Chen X F. Fault mechanism and dynamic modeling of planetary gear with gear wear. Mechanism and Machine Theory, 2021, 155: 104098

[56] Flodin A, Andersson S. Simulation of mild wear in helical gears. Wear, 2000, 241(2): 123–128

[57] Wan Z G, Cao H R, Zi Y Y, He W P, He Z J. An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack. Engineering Failure Analysis, 2014, 42: 157–177

[58] Wang S Y, Zhu R P. An improved mesh stiffness calculation model for cracked helical gear pair with spatial crack propagation path. Mechanical Systems and Signal Processing, 2022, 172: 108989

[59] Jedliński Ł, Syta A, Gajewski J, Jonak J. Nonlinear analysis of cylindrical gear dynamics under varying tooth breakage. Measurement, 2022, 190: 110721

[60] Yang W G, Jiang D X, Han T. Effects of tooth breakage size and rotational speed on the vibration response of a planetary gearbox. Applied Sciences, 2017, 7(7): 678

[61] Jiang H J, Liu F H. Dynamic features of three-dimensional helical gears under sliding friction with tooth breakage. Engineering Failure Analysis, 2016, 70: 305–322

[62] Li Z W, Ma H, Feng M J, Zhu Y P, Wen B C. Meshing characteristics of spur gear pair under different crack types. Engineering Failure Analysis, 2017, 80: 123–140

[63] Han H Z, Ma H, Tian H X, Peng Z K, Zhu J Z, Li Z W. Sideband analysis of cracked planetary gear train considering output shaft radial assembly error. Mechanical Systems and Signal Processing, 2023, 200: 110618

[64] Pandya Y, Parey A. Experimental investigation of spur gear tooth mesh stiffness in the presence of crack using photoelasticity technique. Engineering Failure Analysis, 2013, 34: 488–500

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 52175122 and 52075456), the Sichuan Science and Technology Program, China (Grant No. 2023NSFSC0362), the Sichuan Province Innovative Talent Funding Project for Postdoctoral Fellows, China (Grant No. BX202214), and the China Postdoctoral Science Foundation (Grant No. 2023M732917).

Conflict of Interest

The authors declare that they have no conflict of interest.

RIGHTS & PERMISSIONS

2024 Higher Education Press