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Anisotropic 3D rough surface reconstruction model based on fractal method and its application in contact characteristics

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Abstract

This study proposes a novel anisotropic model based on fractal theory for three-dimensional (3D) rough surface reconstruction to improve the large error associated with isotropic models. The proposed model comprises a fractal-based two-dimensional (2D) contour curve, a conversion operator and a coefficient vector ratio. The effects of the virtual random curve and fractal-based 2D contour curve in the conversion operator are discussed. The largest height ratio of the virtual contour curve and the measured surface discrete point curve of each column are used in the coefficient vector ratio. The rough surface is also smoothed using a five-point cubic smoothing algorithm. Compared with the isotropic model, the proposed model resembles the measured surface more closely with an average fitting error reduction of 49.84% and 64.7% before and after smoothing, respectively. The surface exhibits horizontal and vertical textures that can be controlled by the fractal dimension and roughness in the X and Y directions. Additionally, an asperity contact model is established and analysed. The proposed modelling method for anisotropic 3D rough surface contact condition is more consistent with the measured surface model. The transverse and longitudinal textures of the anisotropic surface can be changed by the fractal dimension and fractal roughness in the X and Y directions. This research provides insights into characterising 3D rough surfaces of high-precision gears and improving nominal contact area accuracy.

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Abbreviations

D, D Sms, Dx, Dy :

Fractal dimension

G, G Sms, Gx, Gy :

Fractal roughness (μm)

L :

Sampling length (mm)

n :

Asperity grade

φ k :

Weighting ratio

m :

Smoothing times

γ n :

Spatial frequency

ω L :

Lowest frequency

ω U :

Highest frequency

Ls :

Cut-off length (mm)

Ns :

Number of discrete points

σ :

Standard deviation (μm)

δ :

Sampling interval (mm)

Sms :

Fractal rough contour curve discrete-point height (μm)

Sx :

Virtual curve discrete-point height (μm)

Szs :

Largest height of each column of the measured surfaces’ discrete points (μm)

Szy :

Largest height of Zy (μm)

Z(x i,y j), Z(x i), Z(y j), Zx, Zy, Y, Z :

Discrete-point height (μm)

k 1, k 2, k 3 :

Double logarithmic curve slope

a :

Weight proportion of the mean value of the transverse and longitudinal roughness

b :

Weight proportion of the difference value of the transverse and longitudinal roughness

A:

Conversion operator

B:

Coefficient scale vector

r t :

Contact radius of the single asperity (μm)

r s :

Cut-off radius of the single asperity (μm)

r ′ :

Basal radius of the single asperity (μm)

h :

Asperity height (μm)

ω :

Deformation (μm)

R :

Asperity peak curvature radius (μm)

A :

Contact area (m2)

F :

Contact load (N)

K :

Pressure coefficient (Pa)

H :

Hardness

1 :

Elastic contact stage

2 :

First elastic–plastic contact stage

3 :

Second elastic–plastic contact stage

4 :

Plastic contact stage

1c :

First critical state

2c :

Second critical state

3c :

Third critical state

References

  1. Zhou C, Ning L, Wang H, Tang L (2021) Effects of centring error and angular misalignment on crack initiation life in herringbone gears. Eng Fail Anal 120:105082

    Article  Google Scholar 

  2. Wang H, Tang L, Zhou C, Shi Z (2021) Wear life prediction method of crowned double helical gear drive in point contact mixed elastohydrodynamic lubrication. Wear 484:204041

    Article  Google Scholar 

  3. Fernandez RC, Tobie T, Rommel S, Collazo J (2023) Improve wind gears bending performance by means of IGS (Improved gear surface). Int J Fatigue 172:107618

    Article  Google Scholar 

  4. Muthukumar M, Bobji MS (2018) Effect of micropillar surface texturing on friction under elastic dry reciprocating contact. Meccanica 53:2221–2235

    Article  Google Scholar 

  5. Chen Z, Zhou J, Liu B, Fu H, Meng X, Ji J, Zhang Y, Hua X, Xu X, Fu Y (2023) Meshing frictional characteristics of spur gears under dry friction and heavy loads: effects of the preset pitting-like micro-textures array. Tribol Int 180:108296

    Article  Google Scholar 

  6. Wang H, Zhou C, Wang H, Hu B, Liu Z (2021) A novel contact model for rough surfaces using piecewise linear interpolation and its application in gear wear. Wear 476:203685

    Article  CAS  Google Scholar 

  7. Zhou C, Wang H, Wang H, Hu B (2021) The physical and mechanical properties for flexible biomass particles using computer vision. Fuel 315:123278

    Google Scholar 

  8. Prajapati DK, Tiwari M (2020) Experimental analysis of contact fatigue damage using fractal methodologies. Wear 450:203262

    Article  Google Scholar 

  9. Greenwood JA, Williamson JP (1966) Contact of nominally flat surfaces. Proc R Soc Lond Ser A 295:300–319

    Article  ADS  CAS  Google Scholar 

  10. Whitehouse D, Archard J (1970) The properties of random surfaces of significance in their contact. Proc R Soc Lond Ser A 316:97–121

    Article  ADS  Google Scholar 

  11. Chang WR, Etsion I, Bogy DB (1987) An elastic–plastic model for the contact of rough surfaces. J Tribol 109:257–263

    Article  Google Scholar 

  12. Zhao Y, Maietta DM, Chang L (2000) An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow. J Tribol 122:86–93

    Article  Google Scholar 

  13. Zhao Y, Chang L (2001) A model of asperity interactions in elastic–plastic contact of rough surfaces. J Tribol 123:857–864

    Article  Google Scholar 

  14. Kogut L, Etsion I (2002) Elastic–plastic contact analysis of a sphere and a rigid flat. J Appl Mech 69:657–662

    Article  Google Scholar 

  15. Jackson RL, Green I (2005) A finite element study of elasto–plastic hemispherical contact against a rigid flat. J Tribol 127:343–354

    Article  Google Scholar 

  16. Brake MR (2012) An analytical elastic-perfectly plastic contact model. Int J Solids Struct 49:3129–3141

    Article  Google Scholar 

  17. Majumdar AA, Bhushan B (1990) Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J Tribol 112:205–216

    Article  Google Scholar 

  18. Jin Y, Liu X, Song H, Zheng J, Pan J (2019) General fractal topography: an open mathematical framework to characterize and model mono-scale-invariances. Nonlinear Dyn 96:2413–2436

    Article  Google Scholar 

  19. Panagiotopoulos PD (1992) Fractals and fractal approximation in structural mechanics. Meccanica 27:25

    Article  Google Scholar 

  20. Majumdar A, Tien CL (1990) Fractal characterization and simulation of rough surfaces. Wear 136:313–327

    Article  Google Scholar 

  21. Majumdar A, Bhushan B (1991) Fractal model of elastic–plastic contact between rough surfaces. J Tribol 131:1–11

    Article  Google Scholar 

  22. Morag Y, Etsion I (2007) Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces. Wear 262:624–629

    Article  CAS  Google Scholar 

  23. Wang S, Komvopoulos K (1994) A fractal theory of the interfaoial temperature distribution in the slow sliding regime: part I elastic contact and heat transfer analysis. J Tribol 16:812–822

    Article  Google Scholar 

  24. Yan W, Komvopoulos K (1998) Contact analysis of elastic–plastic fractal surfaces. J Appl Phys 84:3617–3624

    Article  ADS  CAS  Google Scholar 

  25. Yang J, Komvopoulos K (2005) A mechanics approach to static friction of elastic–plastic fractal surfaces. J Tribol 127:315–324

    Article  CAS  Google Scholar 

  26. Liou JL, Tsai CM, Lin J-F (2010) A microcontact model developed for sphere and cylinder based fractal bodies in contact with a rigid flat surface. Wear 268:431–442

    Article  CAS  Google Scholar 

  27. Miao J, Du X, Li C, Shan X, Chen B (2022) Multi-scale modelling of the thermally coupled planetary roller screw mechanism with curved fractal contact. Meccanica 57:2771–2795

    Article  MathSciNet  Google Scholar 

  28. Peng YF, Guo YB (2007) An adhesion model for elastic–plastic fractal surfaces. J Appl Phys 102:053510

    Article  ADS  Google Scholar 

  29. Yuan Y, Cheng Y, Liu K, Gan L (2017) A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces. Appl Surf Sci 425:1138–1157

    Article  ADS  CAS  Google Scholar 

  30. Huo L, Wang F, Li H, Song G (2017) A fractal contact theory based model for bolted connection looseness monitoring using piezoceramic transducers. Smart Mater Struct 26:104010

    Article  ADS  Google Scholar 

  31. Liou JL, Lin JF (2010) A modified fractal microcontact model developed for asperity heights with variable morphology parameters. Wear 268:133–144

    Article  CAS  Google Scholar 

  32. Zhao Z, Han H, Wang P, Ma H, Zhang S, Yang Y (2021) An improved model for meshing characteristics analysis of spur gears considering fractal surface contact and friction. Mech Mach Theory 158:104219

    Article  Google Scholar 

  33. Scaraggi M, Angerhausen J, Dorogin L, Murrenhoff H, Persson BNJ (2018) Influence of anisotropic surface roughness on lubricated rubber friction: extended theory and an application to hydraulic seals. Wear 410–411:43–62

    Article  Google Scholar 

  34. Ausloos M, Berman DH (1997) A multivariate Weierstrass-Mandelbrot function. Proc R Soc Lond Ser A 400:331–350

    ADS  MathSciNet  Google Scholar 

  35. Blackmore D, Zhou G (1998) Fractal analysis of height distributions of anisotropic rough surfaces. Fractals-complex Geometry Patterns & Scaling in Nature & Society 6:43–58

    Google Scholar 

  36. Blackmore D, Zhou G (1998) A new fractal model for anisotropic surfaces. Int J Mach Tools Manuf 38:551–557

    Article  Google Scholar 

  37. Wu J-J (2002) Analyses and simulation of anisotropic fractal surfaces. Chaos Solitons Fractals 13:1791–1806

    Article  ADS  MathSciNet  Google Scholar 

  38. Tian H (2013) Normal contact mechanics model of fixed joint interface adopting anisotropic fractal geometrical theory. J Mech Eng 49:108

    Article  Google Scholar 

  39. Mei MX, Diao HX (2016) Fractal contact model of ground guide surface. Modul Mach Tool Autom Manuf Tech 11:41–45

    Google Scholar 

  40. Feng Y, Yang P, Zhang Y, Shi L, Hang Z, Feng Y (2022) Fractal model of thermal elasto–plastic contact of rough surfaces. J Central South Univ 29:1500–1509

    Article  Google Scholar 

  41. Wu JJ (2000) Simulation of rough surface with FFT. Tribol Int 33:47–58

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support by the National Natural Science Foundation of China (Grant Nos. 52075153 and 52005051) and National Key Laboratory of Science and Technology on Helicopter Transmission (Grant No. HTL-O-21K01).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, People’s Republic of China

    Ming Zheng, Yu Han, Shengwen Hou, Changjiang Zhou & Jie Su

  2. Shanxi Key Laboratory of Gear Transmission, Shaanxi Fast Gear Co., Ltd., Xi’an, 710077, People’s Republic of China

    Yu Han & Shengwen Hou

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  1. Ming Zheng
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Correspondence to Yu Han.

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Zheng, M., Han, Y., Hou, S. et al. Anisotropic 3D rough surface reconstruction model based on fractal method and its application in contact characteristics. Meccanica 59, 235–254 (2024). https://doi.org/10.1007/s11012-023-01750-6

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  • DOI: https://doi.org/10.1007/s11012-023-01750-6

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