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Coupling failure dynamics of tooth surface morphology and wear based on fractal theory

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Abstract

This paper aims to reveal the coupling relationship between surface topography and wear and introduce a new coupling failure dynamic model of tooth surface topography and wear. Based on the fractal theory, the theoretical model of gear rough surface and the prediction model of wear are established and reveal the relationship between tooth surface roughness, wear depth, and meshing position. In particular, we analyze rough tooth surfaces’ time-varying meshing stiffness and system dynamic characteristics. The greater the contact surface roughness level, the more severe the wear behavior, and the greater the contact deformation of the gear; the vibration response and amplitude growth rate are highly related to the surface roughness. The results of coupled vibration characteristics show that the topography of the contact surface causes the system noise and shock to increase from the two aspects of stiffness excitation and error excitation. The multi-state coupled dynamic model of the contact surface can provide a theoretical basis for optimizing gear systems.

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Data availability

The authors declare that the data supporting the 11 findings of this study are available within the article.

Abbreviations

D :

Fractal dimension

G :

The characteristic scale parameter

L r :

The length scale of the asperity

Ф :

The random phase

ν :

Poisson’s ratio

z :

Number of teeth

m :

Module

L :

Tooth width

E :

Elastic modulus

T :

The torque applied to the gearyhjhj

N :

Meshing times

k a :

Axial compression stiffness

k b :

Bending stiffness

k s :

Shear stiffness

k h :

Hertzian stiffness

k f :

Flexible base stiffness

k re :

The stiffness of the elastic deformation

k rep 1 :

The stiffness of the first elastic–plastic deformation

k rep 2 :

The stiffness of the second elastic–plastic deformation

k mesh :

Time-varying meshing stiffness under interdental coupling effects

α 1 :

The base circle arc is half of the center angle of the circle

α 2 :

The root arc of the tooth is half of the center angle of the circle

α f :

The difference between the root circle pressure angle and α1 of the internal gear

θ 0 :

The angle between the centerline of a tooth and the centerline of the adjacent tooth slot

r b :

Base circle radius

r f :

Root fillet radius

h x :

Half the thickness of the teeth at the meshing point

d :

The effective tooth length of integral

V :

Wear volume

S :

The relative sliding distance

H :

Material hardness

G w :

Viscosity coefficient

G s :

The material shear deformation-related parameters

P w :

The contact load changes with the meshing angle

S w :

The tooth surface contact parameters

R e :

The comprehensive radius of curvature

a L :

The contact area of the largest asperity

ψ :

The scale expansion coefficient

K v :

Hardness coefficient

h p :

Wear depth of contact point P

References

  1. Sayles, R., Thomas, T.: Surface topography as a nonstationary random process. Nature 271(5644), 431–434 (1978). https://doi.org/10.1038/271431a0

    Article  Google Scholar 

  2. Li, Z.X., Yan, X., Yuan, C., et al.: Virtual prototype and experimental research on gear multi-fault diagnosis using wavelet-autoregressive model and principal component analysis method. Mech. Syst. Signal Process. 25(7), 2589–2607 (2021). https://doi.org/10.1016/j.ymssp.2011.02.017

    Article  Google Scholar 

  3. Liang, X., Zhang, H., Liu, L., et al.: The influence of teeth pitting on the mesh stiffness of a pair of external spur gears. Mech. Mach. Theory 106(1), 1–15 (2016). https://doi.org/10.1016/j.mechmachtheory.2016.08.005

    Article  Google Scholar 

  4. Wu, B., Sun, Y.: Normal contact analysis between two self-affine fractal surfaces at the nanoscale by molecular dynamics simulations. Tribol. Lett. 71, 30 (2023). https://doi.org/10.1007/s11249-023-01705-8

    Article  Google Scholar 

  5. Janakiraman, V., Li, S., Kahraman, A.: An investigation of the impacts of contact parameters on wear coefficient. J. Tribol. 136(3), 031602 (2014). https://doi.org/10.1115/1.4027440

    Article  Google Scholar 

  6. Kogut, L., Etsion, I.: Elastic-plastic contact analysis of a sphere and a rigid flat. J. Appl. Mech. 69(5), 657–662 (2002). https://doi.org/10.1115/1.1490373

    Article  Google Scholar 

  7. Amarnath, M., Chandramohan, S., Seetharaman, S., et al.: Experimental investigations of surface wear assessment of spur gear teeth. J. Vib. Control 18(7), 1009–1024 (2012). https://doi.org/10.1177/1077546311399947

    Article  Google Scholar 

  8. Yu, G.B., Mao, H.H., Jiang, L.D., et al.: Fractal contact mechanics model for the rough surface of a beveloid gear with elliptical asperities. Appl. Sci. 12(8), 4071 (2022). https://doi.org/10.3390/app12084071

    Article  Google Scholar 

  9. Chen, Z.G., Zhang, J., Zhai, W., et al.: Improved analytical methods for calculation of gear teeth fillet-foundation stiffness with teeth root crack. Eng. Fail. Anal. 82, 72–81 (2017). https://doi.org/10.1016/j.engfailanal.2017.08.028

    Article  Google Scholar 

  10. Zhao, Z., Yang, Y., Han, H.Z., et al.: Meshing characteristics of spur gears considering three-dimensional fractal rough surface under elastohydrodynamic lubrication. Machines 10(8), 705 (2022). https://doi.org/10.3390/machines10080705

    Article  Google Scholar 

  11. Liu, Z., Zhang, T., Zhao, Y., et al.: Time-varying stiffness model of spur gear considering the effect of surface morphology characteristics. Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng. 233(2), 242–253 (2019). https://doi.org/10.1177/0954408918775955

    Article  Google Scholar 

  12. Pan, W.J., Li, X.P., Wang, L.L., et al.: A normal contact stiffness fractal prediction model of dry-friction rough surface and experimental verification. Eur. J. Mech. A. Solids 66, 94–102 (2017). https://doi.org/10.1016/j.euromechsol.2017.06.010

    Article  MathSciNet  Google Scholar 

  13. Chen, H., Yin, Q., Dong, G., et al.: Stiffness model of fixed joint considering self- affinity and elastoplasticity of asperities. Ind. Lubr. Tribol. 72, 128–135 (2020). https://doi.org/10.1108/ILT-05-2019-0192

    Article  Google Scholar 

  14. Guan, D., Jing, L., Gong, J., et al.: Normal contact analysis for spherical pump based on fractal theory. Tribol. Int. 124, 117–123 (2019). https://doi.org/10.1016/j.triboint.2018.04.002

    Article  Google Scholar 

  15. Chen, Q., Xu, F., Liu, P., Fan, H.: Research on fractal model of normal contact stiffness between two spheroidal joint surfaces considering friction factor. Tribol. Int. 97, 253–264 (2016). https://doi.org/10.1016/j.triboint.2016.01.023

    Article  Google Scholar 

  16. Yang, W., Li, H., Ma, D.Q., Yongqiao, W., Jian, C., et al.: Sliding friction contact stiffness model of involute arc cylindrical gear based on fractal theory. Int. J. Eng. 30, 109–119 (2017)

    Google Scholar 

  17. Xia, H., Meng, F.S., Zhang, X., et al.: Nonlinear dynamics analysis of gear system considering time-varying meshing stiffness and backlash with fractal characteristics. Nonlinear Dyn. 111, 14851–14877 (2023). https://doi.org/10.1007/s11071-023-08649-7

    Article  Google Scholar 

  18. Gao, S., Han, Q., Zhou, N., et al.: Dynamic and wear characteristics of self-lubricating bearing cage: effects of cage pocket shape. Nonlinear Dyn. 110, 177–200 (2022). https://doi.org/10.1007/s11071-022-07611-3

    Article  Google Scholar 

  19. Chang, W.R., Etsion, I., Bogy, D.B.: Static friction coeÿcient model for metallic rough surfaces. J. Tribol. 110(1), 57–63 (1988). https://doi.org/10.1115/1.3261575

    Article  Google Scholar 

  20. Kim, J.Y., Baltazar, A., Rokhlin, S.I.: Ultrasonic assessment of rough surface contact between solids from elastoplastic loading–unloading hysteresis cycle. J. Mech. Phys. Solid. 52, 1911–1934 (2004). https://doi.org/10.1016/j.jmps.2004.01.006

    Article  Google Scholar 

  21. Bajpai, P., Kahraman, A., Anderson, N.E.: A surface wear prediction methodology for parallel-axis gear pairs. J. Tribol. 126(3), 597–605 (2004). https://doi.org/10.1115/1.1691433

    Article  Google Scholar 

  22. Morag, Y., Etsion, I.: Resolving the contradiction of asperities plastic to elasticmode transition in current contact models of fractal rough surfaces. Wear 262, 624–629 (2007). https://doi.org/10.1016/j.wear.2006.07.007

    Article  Google Scholar 

  23. Wang, S., Komvopoulos, K.: A fractal theory of the interfacial temperature distribution in the slow sliding regime: part I—elastic contact and heat transfer analysis. J. Tribol. 116, 812–823 (1994). https://doi.org/10.1115/1.2927341

    Article  Google Scholar 

  24. Brizmer, V., Kligerman, Y., Etsion, I.: Elastic–plastic spherical contact under combined normal and tangential loading in full stick. Tribol. Lett. 25(1), 61–70 (2007). https://doi.org/10.1007/s11249-006-9156-y

    Article  Google Scholar 

  25. Liu, Y., Shangguan, B., Xu, Z.: A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade. Nonlinear Dyn. 70, 2247–2257 (2012). https://doi.org/10.1007/s11071-012-0615-8

    Article  MathSciNet  Google Scholar 

  26. Liu, P., Zhao, H., Huang, K., et al.: Research on normal contact stiffness of rough surface considering friction based on fractal theory. Appl. Surf. Sci. 349, 43–48 (2015). https://doi.org/10.1016/j.apsusc.2015.04.174

    Article  Google Scholar 

  27. Zhai, C., Gan, Y.X., Proust, G., et al.: The role of surface structure in normal contact stiffness. Exp. Mach. 56, 359–368 (2016). https://doi.org/10.1007/s11340-015-0107-0

    Article  Google Scholar 

  28. Yousfi, B.E., Soualhi, A., Medjaher, K., et al.: Electromechanical modeling of a motor–gearbox system for local gear tooth faults detection. Mech. Syst. Signal Process. 166, 108435 (2022). https://doi.org/10.1016/j.ymssp.2021.108435

    Article  Google Scholar 

  29. Huang, K., Cheng, Z.B., Xiong, Y.S., et al.: Bifurcation and chaos analysis of a spur gear pair system with fractal gear backlash. Chaos Solitons Fractals 142, 110387 (2021). https://doi.org/10.1016/j.chaos.2020.110387

    Article  MathSciNet  Google Scholar 

  30. Kumar, M., Bharti, R.K., Das, M.: Study of surface finishing mechanism in arotational-magnetorheological miniature gear profile polishing using novel flow restrictor. Wear 488–489, 204120 (2022). https://doi.org/10.1016/j.wear.2021.204120

    Article  Google Scholar 

  31. Wang, S.Y., Zhu, R.P.: An improved mesh stiffness model for double-helical gear pair with spalling defects considering time-varying friction coefficient under mixed EHL. Eng. Fail. Anal. 121, 105174 (2021). https://doi.org/10.1016/j.engfailanal.2020.105174

    Article  Google Scholar 

  32. Wei, D., Zhai, C., Hanaor, D.A., et al.: Contact behaviour of simulated rough spheres generated with spherical harmonics. Int. J. Solids Struct. 193–194, 54–68 (2020). https://doi.org/10.1016/j.ijsolstr.2020.02.009

    Article  Google Scholar 

  33. Sainsot, P., Velex, P., Duverger, O.: Contribution of gear body to tooth deflections - a new bidimensional analytical formula. J. Mech. Des. 126, 748–752 (2004). https://doi.org/10.1115/1.1758252

    Article  Google Scholar 

  34. Zhou, W., Tang, J., He, Y., et al.: Modeling of rough surfaces with given roughness parameters. J. Central South Univ. 24, 127–136 (2017). https://doi.org/10.1007/S11771-017-3415-Y

    Article  Google Scholar 

  35. Zhao, B., Zhang, S., Wang, P., et al.: Loading–unloading normal stiffness model for power-law hardening surfaces considering actual surface topography. Tribol. Int. 90, 332–342 (2015). https://doi.org/10.1016/j.triboint.2015.04.045

    Article  Google Scholar 

  36. Liao, J.P., Zhang, J.F., Feng, P.F., et al.: Identification of contact stiffness of shrink-fit tool-holder joint based on fractal theory. Int. J. Adv. Manuf. Technol. 90, 2173–2184 (2017). https://doi.org/10.1007/s00170-016-9506-3

    Article  Google Scholar 

  37. Wang, D., Wang, R., Wang, B., et al.: Effect of vibration on emergency braking tribological behaviors of brake shoe of deep coal mine hoist. Appl. Sci. 11, 6441 (2021). https://doi.org/10.3390/app11146441

    Article  Google Scholar 

  38. Thanh, C.L., Khuong, D.N., Minh, H.L., et al.: Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate. Physica B 631, 413726 (2022). https://doi.org/10.1016/j.physb.2022.413726

    Article  Google Scholar 

  39. Tran, V.T., Nguyen, T.K., Hung, X.N., et al.: Vibration and buckling optimization of functionally graded porous microplates using BCMO-ANN algorithm. The-Walled Structures. 182, 110267 (2022). https://doi.org/10.1016/j.tws.2022.110267

    Article  Google Scholar 

  40. Dai, H., Long, X.H., Chen, F., et al.: An improved analytical model for gear mesh stiffness calculation. Mech. Mach. Theory 159, 104262 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104262

    Article  Google Scholar 

  41. Ma, H., Song, R.Z., Pang, X., et al.: Time-varying mesh stiffness calculation of cracked spur gears. Eng. Fail. Anal. 44, 179–194 (2014). https://doi.org/10.1016/j.engfailanal.2014.05.018

    Article  Google Scholar 

  42. Xie, C.Y., Hua, L., Lan, J., et al.: Improved analytical models for mesh stiffness and load sharing ratio of spur gears considering structure coupling effect. Mech. Syst. Signal Process. 111, 331–347 (2018). https://doi.org/10.1016/j.ymssp.2018.03.037

    Article  Google Scholar 

  43. Xie, C.Y., Hua, L., Han, X.H., et al.: Analytical formulas for gear body-induced teeth deflections of spur gears considering structure coupling effect. Int. J. Mech. Sci. 148, 174–190 (2018). https://doi.org/10.1016/j.ijmecsci.2018.08.022

    Article  Google Scholar 

  44. Chen, Z.G., Zhou, Z.W., Zhai, W.M., et al.: Improved analytical calculation model of spur gear mesh excitations with teeth profile deviations. Mech. Mach. Theory 149, 103838 (2022). https://doi.org/10.1016/j.mechmachtheory.2020.103838

    Article  Google Scholar 

  45. Shen, Z.X., Qiao, B.J., Yang, L.H., et al.: Evaluating the influence of teeth surface wear on TVMS of planetary gear set. Mech. Mach. Theory 136, 206–223 (2019). https://doi.org/10.1016/j.mechmachtheory.2019.03.014

    Article  Google Scholar 

  46. Wang, H.B., Zhou, C.J., Wang, H.H., et al.: A novel contact model for rough surfaces using piecewise linear interpolation and its application in gear wear. Wear 476, 203685 (2021). https://doi.org/10.1016/j.wear.2021.203685

    Article  Google Scholar 

  47. Yu, X., Sun, Y.Y., Zhao, D., et al.: A revised contact stiffness model of rough curved surfaces based on the length scale. Tribol. Int. 164, 107206 (2021). https://doi.org/10.1016/j.triboint.2021.107206

    Article  Google Scholar 

  48. Xiao, H.F., Sun, Y.Y.: On the normal contact stiffness and contact resonance frequency of rough surface contact based on asperity micro-contact statistical models. Eur. J. Mech. A. Solids 75, 450–460 (2019). https://doi.org/10.1016/j.euromechsol.2019.03.004

    Article  MathSciNet  Google Scholar 

  49. Yang, L., Wang, L., Yu, W., et al.: Investigation of tooth crack opening state on time varying meshing stiffness and dynamic response of spur gear pair. Eng. Fail. Anal. 121, 1–13 (2021). https://doi.org/10.1016/j.engfailanal.2020.105181

    Article  Google Scholar 

  50. Raghuwanshi, N.K., Parey, A.: Experimental measurement of gear mesh stiffness of cracked spur gear by strain gauge technique. Measurement 86, 266–275 (2016). https://doi.org/10.1016/j.measurement.2016.03.001

    Article  Google Scholar 

  51. Mo, S., Luo, B.R., Song, W.H., et al.: Geometry design and teeth contact analysis of non-orthogonal asymmetric helical face gear drives. Mech. Mach. Theory 173, 104831 (2022). https://doi.org/10.1016/j.mechmachtheory.2022.104831

    Article  Google Scholar 

  52. Mo, S., Zhang, Y.X., Song, Y.L., et al.: Nonlinear vibration and primary resonance analysis of non-orthogonal face gear-rotor-bearing system. Nonlinear Dyn. 108, 3367–3389 (2022). https://doi.org/10.1007/s11071-022-07432-4

    Article  Google Scholar 

  53. Mo, S., Wang, L., Liu, M., et al.: Study of the time-varying mesh stiffness of two-stage planetary gear train considering tooth surface wear. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. (2023). https://doi.org/10.1177/09544062231170633

    Article  Google Scholar 

  54. Mo, S., Zhang, T., Jin, G.G., et al.: Analytical investigation on load sharing characteristics of herringbone planetary gear train with flexible support and floating sun gear. Mech. Mach. Theory 144, 103670 (2022). https://doi.org/10.1016/j.mechmachtheory.2019.103670

    Article  Google Scholar 

  55. Mo, S., Li, Y.H., Wang, D.D., et al.: An analytical method for the meshing characteristics of asymmetric helical gears with tooth modifications. Mech. Mach. Theory 185, 105321 (2023). https://doi.org/10.1016/j.mechmachtheory.2023.105321

    Article  Google Scholar 

  56. Mo, S., Li, Y.H., Lou, B.R., et al.: Research on the meshing characteristics of asymmetric gears considering the tooth profile deviation. Mech. Mach. Theory 175, 104926 (2022). https://doi.org/10.1016/j.mechmachtheory.2022.104926

    Article  Google Scholar 

  57. Litak, G., Friswell, M.I.: Dynamics of a gear system with faults in meshing stiffness. Nonlinear Dyn. 41, 415–421 (2005). https://doi.org/10.1007/s11071-005-1398-y

    Article  Google Scholar 

  58. Chen, Z., Ning, J., Wang, K., et al.: An improved dynamic model of spur gear transmission considering coupling effect between gear neighboring teeth. Nonlinear Dyn. 106, 339–357 (2021). https://doi.org/10.1007/s11071-021-06852-y

    Article  Google Scholar 

  59. Zhang, Y., Zhu, L.Y., Gou, X.F.: Calculation methods of load distribution ratio for spiral bevel gear. Int. J. Mech. Sci. 257C, 108531 (2023). https://doi.org/10.1016/j.ijmecsci.2023.108531

    Article  Google Scholar 

  60. Mu, X., Sun, W., Liu, C., Yuan, B., et al.: Numerical simulation and accuracy verification of surface morphology of metal materials based on fractal theory. Materials 13, 4158 (2022). https://doi.org/10.3390/ma13184158

    Article  Google Scholar 

  61. Yu, X., Sun, Y.Y., Liu, S., et al.: Fractal-based dynamic response of a pair of spur gears considering microscopic surface morphology. Int. J. Mech. Syst. Dyn. 1(2), 194–206 (2021). https://doi.org/10.1002/msd2.12004

    Article  Google Scholar 

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Acknowledgements

This research is financially supported by National Natural Science Foundation of China (No. 52265004), National Key Laboratory of Science and Technology on Helicopter Transmission (No. HTL-0-21G07), Guangxi Science and Technology Major Program (No. 2023AA19005), Open Fund of State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology (No. DMETKF2021017), Interdisciplinary Scientific Research Foundation of Guang Xi University (No. 2022JCC022), and Entrepreneurship and Innovation Talent Program of Taizhou City, Jiangsu Province (No. 20212333).

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Author notes

  1. Shuai Mo and Lei Wang have contributed equally to this work and they are co-first authors of the article.

Authors and Affiliations

  1. State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, Guangxi University, Nanning, 530004, China

    Shuai Mo

  2. School of Mechanical Engineering, Guangxi University, Nanning, 530004, China

    Shuai Mo

  3. School of Mechanical Engineering, Tiangong University, Tianjin, 300387, China

    Lei Wang & Qingsen Hu

  4. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    Shuai Mo

  5. Jiangsu Wanji Transmission Technology Co., Ltd., Taizhou, 225400, China

    Shuai Mo

  6. Ningbo Zhongda Leader Transmission Equipment Co., Ltd., Ningbo, 315301, China

    Guojian Cen

  7. Shenzhen Hefa Gear Machinery Co., Ltd., Shenzhen, 518109, China

    Yunsheng Huang

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  1. Shuai Mo
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Shuai Mo and Lei Wang contributed equally to this manuscript, Shuai Mo and Lei Wang are co-first authors of the article. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Mo, S., Wang, L., Hu, Q. et al. Coupling failure dynamics of tooth surface morphology and wear based on fractal theory. Nonlinear Dyn 112, 175–195 (2024). https://doi.org/10.1007/s11071-023-09038-w

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