# Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. A New Fractal Characterization Approach for a Rough Surface Texture

_{n}. φ

_{m,n}is the random phase. γ

_{n}is the spatial frequency of the rough surface. M indicates the number of overlapping elevated parts of the rough surface. When M = 1, the rough surface morphological characteristics show isotropic, when M ≠ 1, the rough surface morphological characteristics show anisotropic. α

_{m}is the three-dimensional rough surface cosine direction, α

_{m}= πm/M. f(x,y) and g(x,y) are the functions of independent variables related to the morphological characteristics of the rough surface, respectively.

_{x}, D

_{y}are the fractal dimensions in the direction x, y, G

_{x}, G

_{y}are the fractal roughness in the direction x, y, respectively. n

_{1}, n

_{2}are the number of sampling points within a finite length of the rough surface in the direction x, y, respectively.

_{x}and D

_{y}and the fractal roughness G

_{x}and G

_{y}in the fractal characterization method (Equation (3)), the fractal characterization of the rough surface texture can be obtained, as illustrated in Figure 2a,b; it is not difficult to discover that a rough surface with a certain texture may be efficiently simulated by modifying the fractal parameters in the x and y dimensions.

## 3. Contact Mechanics Model of Elliptical Asperity with Rough Tooth Surface

#### 3.1. Geometric Model of Single Elliptical Asperity

_{n}is the actual deformation of the elliptical asperity, and its value is between 0 and δ. The profile curve z(x, y) of a single elliptical asperity with the major diameter l

_{x}and the minor diameter l

_{y}of the elliptical region as the base can be deduced from Equation (3):

_{n}between the tip of the elliptical asperity and the base can be presented as:

_{n}of the elliptical asperity can be expressed as:

_{n}is significantly smaller than the curvature radius of the elliptical asperity.

_{x}= R

_{y}= R

_{m}. When the eccentricity e is not 0, the equivalent radius of curvature R

_{m}corresponding to the elliptical asperity can be expressed as:

#### 3.2. Contact Mechanics Model of Single Elliptical Asperity

#### 3.2.1. Elastic Contact of Elliptical Asperity

_{n}≤ ω

_{nec}, it merely experiences elastic contact deformation. The maximum contact pressure P

_{m}, the semimajor radius of the contact ellipse r

_{x}, and the elastic deformation ω can be expressed, respectively, as:

_{1}, ν

_{2}and E

_{1}, E

_{2}are Poisson’s ratio and the elastic modulus of the two rough surface materials in contact with each other, respectively.

_{e}can be expressed as:

_{m}is equal to KH, and the equation is:

_{nec}between elastic and inelastic deformation can be deduced as:

#### 3.2.2. Elastic–Plastic Contact of Elliptical Asperity

_{n}of the asperities on the rough surface is greater than the critical elastic deformation ω

_{nec}and less than or equal to 110 ω

_{nec}. In addition, a large number of simulation results have proven that the elastic–plastic deformation is divided into two stages: the asperities are in the first elastic–plastic deformation stage when the actual deformation ω

_{n}is greater than the critical elastic deformation ω

_{nec}and less than or equal to 6 ω

_{nec}, and the asperities are in the second elastic–plastic deformation stage when the actual deformation ω

_{n}is greater than the critical elastic deformation 6 ω

_{nec}and less than or equal to 110 ω

_{nec}. 6 ω

_{nec}and 110 ω

_{nec}are defined as the critical elastic–plastic deformation and critical plastic deformation, respectively, and their expressions can be expressed as follows, respectively:

_{nec}≥ a

_{n}≥ a

_{nepc}, and the elliptical asperity is in the second elastic–plastic deformation when the real contact area satisfies a

_{nepc}≥ a

_{n}≥ a

_{npc}. The critical elastic–plastic contact area a

_{nepc}and the critical plastic contact area a

_{npc}can be expressed as:

_{ep}

_{1}= 0.93, n

_{ep}

_{1}= 1.136, c

_{ep}

_{1}= 1.03, d

_{ep}

_{1}= 1.425.

_{ep}

_{2}= 0.94, n

_{ep}

_{2}= 1.146, c

_{ep}

_{2}= 1.40, d

_{ep}

_{2}= 1.263.

#### 3.2.3. Plastic Contact of Elliptical Asperity

_{n}≥ 110 ω

_{nec}, the asperity is in full plastic deformation, and the contact area a

_{np}and the normal contact pressure F

_{np}can be expressed as follows:

#### 3.3. Modified Model of the Island Area Distribution Function for a Point Contact

#### 3.3.1. Contact Area Distribution Function

_{r}can be obtained as:

#### 3.3.2. Elliptical Contact Area Distribution Function

_{x}and a semimajor radius r

_{y}.

_{l}of any elliptical asperities can be deduced as:

_{1}, D

_{2}are the fractal dimensions of the rough surface in the x-axis and y-axis directions, respectively. ζ is the contact coefficient of the elliptical asperity, and the contact coefficient ζ can be expressed as:

_{r}can be obtained by integrating the contact area distribution function described in Equation (44) as:

#### 3.3.3. Contact Coefficient of the Elliptical Area of a Beveloid Gear

_{11}, ρ

_{12}, ρ

_{21}, and ρ

_{22}, respectively, the contact area distribution functions of the beveloid gear should satisfy the following relationship:

_{m}is the integrated curvature coefficient, whose expression is:

_{x}of the contact ellipse can be expressed as:

_{n}

_{1}and ρ

_{n}

_{2}are the radii of curvature at the nodes of the gear and pinion, respectively, in the form of:

_{n}= 20°, and the modulus m = 4 mm. As seen in Figure 8a, the contact coefficient reduces as the integrated curvature of the elastic surface increases. When the integrated curvature of the elastic surface tends to 0, the two elastic surfaces are approximately in-plane contact and the surface contact coefficient λ tends to 1. As seen in Figure 8b, the contact coefficient rises when the contact load increases. This is because as the contact load increases, the contact area of the elliptical asperities grows proportionately, resulting in an increase in the surface contact coefficient.

## 4. The Fractal Contact Model for Rough Curved Surfaces with Elliptical Asperities

#### 4.1. Real Contact Area and Contact Load

#### 4.1.1. Real Contact Area

_{l}> a

_{nec}, the asperities may deform totally elastically, totally plastically, or both elastically and plastically. The real contact area A

_{r}is the sum of the plastic contact area A

_{rp}, the second plastic contact area is A

_{rep}

_{2}, the first plastic contact area is A

_{rep}

_{1}, and the elastic contact area is A

_{re}, namely:

#### 4.1.2. Real Contact Load

_{l}> a

_{nec}, the real contact load on the rough surface can be expressed as:

#### 4.1.3. The Relationship between the Real Contact Area and the Real Contact Load

_{l}> a

_{nec}, the elliptical asperities in the first elastic–plastic deformation, the second elastic–plastic deformation, plastic deformation, and elastic deformation are significant.

#### 4.2. Calculation of the Contact Stiffness of Rough Tooth Surfaces

#### 4.2.1. Contact Stiffness Model of a Single Elliptical Asperity

_{l}< a

_{npc}, the elliptical asperity is only in plastic deformation, and the normal contact stiffness k

_{np}can be expressed as:

_{nepc}> a

_{l}> a

_{npc}, the elliptical asperity is in the second elastic–plastic deformation, and the normal contact stiffness k

_{nep}

_{2}can be expressed as:

_{nec}> a

_{l}> a

_{nep}, the elliptical asperity is in the first elastic–plastic deformation, and the normal contact stiffness k

_{nep}

_{1}can be expressed as:

_{l}> a

_{nec}, the elliptical asperity is only in elastic deformation, and the normal contact stiffness k

_{ne}can be expressed as:

#### 4.2.2. Contact Stiffness Model of a Rough Surface

_{l}> a

_{nec}, the normal contact stiffness K of the rough surface can be expressed as:

_{nep}

_{2}can be presented as:

_{nep}

_{1}can be presented as:

_{ne}can be presented as:

## 5. Numerical Analysis and Discussion of Results

#### 5.1. Effect of Fractal Parameters and Eccentricity on Contact Area

_{1}= 1.56, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−10}m, and the value range of fractal dimension D

_{2}is 1.3–1.8. Figure 9 shows the relationship curves between different fractal dimensions D

_{2}and the real contact area of the rough surface.

#### 5.2. Effect of Fractal Parameters on the Contact Load

_{y}= 235 × 10

^{6}Pa, the elastic modulus E = 2.06 × 10

^{11}Pa, and the Poisson’s ratio ν = 0.26. The model in this paper and the revised MB model take the same fractal parameters, namely the fractal dimension D = D

_{1}= D

_{2}= 1.46, the fractal roughness G = G

_{1}= G

_{2}= 1.0 × 10

^{−10}m, and the eccentricity of the model in this paper e = 0.2. As can be seen, the mechanical curves of the model in this paper deviate from those of the MB model, and the real contact load calculated by the model in this paper is less than the real contact load calculated by the MB model at the same real contact area. This is because the eccentricity consequences of the elliptical asperity are accounted for in the calculation model in this paper; hence, the dimensionless contact load of the model in this paper is reduced at the same real contact area.

_{1}= 1.36, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−11}m, and we set the fractal dimension D

_{2}as 1.36, 1.46, 1.56, 1.66, and 1.76, respectively. The relationship between the real contact area and the real contact load for different fractal dimensions can be seen in Figure 12a; it can be observed that the real contact load grows as the fractal dimension increases. The reason for this is that the fractal dimension D is positively associated with the rough surface’s smoothness, and as the fractal dimension rises, the rough surface’s topology performs more finely, decreasing the real contact load on the rough surface.

_{1}= D

_{2}= 1.46, the fractal roughness G

_{1}= 1.0 × 10

^{−10}m, and we set the fractal roughness G

_{2}as 1.0 × 10

^{−10}m, 3.0 × 10

^{−10}m, 5.0 × 10

^{−10}m, 7.0 × 10

^{−10}m, and 9.0 × 10

^{−10}m, respectively. Figure 12b illustrates the relationship between the real contact area and the real contact load for various fractal roughness values, demonstrating that the real contact load reduces as the fractal roughness increases. This is because the fractal roughness G is inversely related to the rough surface’s smoothness. When the fractal roughness G is raised, the projections and depressions in the rough surface topology increase, resulting in a reduction in the rough surface’s real contact load.

_{1}= D

_{2}= 1.46, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−11}m, and we set the eccentricity e of the elliptical asperity to 0.1, 0.3, 0.5, and 0.7, respectively. The relationship between the real contact area and the real contact load with increasing eccentricity e can be seen in Figure 13a. As seen in the figure, the real contact load reduces as the eccentricity e grows. This is because whenever the eccentricity e grows, the contact area of a single elliptical asperity diminishes, reducing the total contact area of the rough surface and, hence, reducing the rough surface’s real contact load.

_{1}= D

_{2}= 1.46, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−11}m, and we set the contact coefficient to 0.9, 0.7, 0.5, and 0.3, respectively. The relationship between the real contact area and the real contact load for different contact coefficients is shown in Figure 13b. As seen in the figure, the real contact load reduces as the contact coefficients drop. The reason for this is that when the contact coefficients are reduced, the real contact area of the rough surfaces decreases, resulting in a decrease in the real contact load.

#### 5.3. Effect of Fractal Parameters on Normal Contact Stiffness

_{1}= 1.36, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−11}m, and we set the fractal dimension D

_{2}as 1.36, 1.46, 1.56, 1.66, and 1.76, respectively. The influence of different fractal dimensions on the relationship between the normal contact stiffness and the real contact area can be seen in Figure 14a. As a consequence, when the real contact area is known, the normal contact stiffness is proportional to the fractal dimension, which suggests that the normal contact stiffness rises as the fractal dimension grows. This is because the fractal dimension does have a physical significance that pertains to the smoothness of the rough surface. On a macroscopic level, the larger the fractal dimension, the higher the roughness value of the rough surface, and the more asperities are present in the contact on the rough surface, enhancing the rough surface’s normal contact stiffness.

_{1}= D

_{2}= 1.46, the fractal roughness G

_{1}= 1.0 × 10

^{−10}m, and we set the fractal roughness G

_{2}as 1.0 × 10

^{−10}m, 3.0 × 10

^{−10}m, 5.0 × 10

^{−10}m, 7.0 × 10

^{−10}m, and 9.0 × 10

^{−10}m, respectively. The influence of the specific fractal roughness values on the relationship between the normal contact stiffness and the real contact area is depicted in Figure 14b. As can be observed, the normal contact stiffness is inversely related to the fractal roughness for a given real contact area, namely, the normal contact stiffness decreases as the fractal roughness grows. This is because when the fractal roughness G value improves, the rough surface morphology becomes rougher and the number of asperities in the contact diminishes, decreasing the normal contact stiffness of the rough surface.

_{1}= D

_{2}= 1.46, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−11}m, and we set the eccentricity e of the elliptical asperity to 0.1, 0.3, 0.5, and 0.7, respectively. The influence of varying the eccentricity e on the relationship between the normal contact stiffness and the real contact area is illustrated in Figure 15a. As shown in the figure, the normal contact stiffness is inversely proportional to the eccentricity for a given real contact area, which indicates that the normal contact stiffness diminishes as the eccentricity increases. This is because when the eccentricity value grows, the contact area of the single elliptical asperity decreases, resulting in a drop in the rough surface’s total actual contact area and consequently a decrease in the rough surface’s normal contact stiffness.

_{1}= D

_{2}= 1.46, the fractal roughness G

_{1}= G

_{2}= 1.0 × 10

^{−11}m, and we set the contact coefficient to 0.9, 0.7, 0.5, and 0.3, respectively. Figure 15b illustrates the effect of varying the contact coefficients on the relationship between the normal contact stiffness and real contact area. As shown in the graphic, the normal contact stiffness is proportional to the contact coefficients for a given real contact area, which suggests that normal contact stiffness reduces as contact coefficients decrease. This is because the contact coefficients’ physical significance is to correct the contact area between two surfaces, and when the integrated curvature of the elastic surface approaches 0, namely, when the two elastic surfaces are in intimate interaction, the contact coefficient value approaches 1 at this time. As the integrated curvature of the elastic surface grows, the contact coefficients and contact area between the elastic surfaces decrease, reducing normal contact stiffness.

## 6. Conclusions

_{1}and D

_{2}and the eccentricity e of the elliptical asperities, as well as the contact coefficient. D has a direct influence on the real contact area. The real contact area grows concerning the fractal dimension D. Conversely, when the value of fractal dimensions is given, the real contact area declines as the eccentricity e increase, and the real contact area reduces as the contact coefficients decrease;

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Greenwood, J.A.; Williamson, J.B.P. Contact of nominally flat surfaces. Proc. R. Soc. Lond.
**1966**, 295, 300–319. [Google Scholar] - Majumdar, A.; Bhushan, B. Fractal model of elastic-plastic contact between rough surface. ASME J. Tribol.
**1991**, 113, 1–11. [Google Scholar] [CrossRef] - Bhushan, B.; Majumdar, A. Elastic-plastic contact model for bifractal surfaces. Wear
**1992**, 153, 53–64. [Google Scholar] [CrossRef] - Chang, W.R.; Etsion, I.; Bogy, D.B. An elastic-plastic model for the contact of rough surfaces. ASME J Tribol.
**1987**, 109, 257–263. [Google Scholar] [CrossRef] - Zhao, Y.; Maietta, D.M.; Chang, L. An asperity micro-contact model incorporating the transition from elastic deformation to fully plastic flow. J. Tribol.
**2000**, 122, 86–93. [Google Scholar] [CrossRef] - Kogut, L.; Etsion, I. Elastic–plastic contact analysis of a sphere and a rigid flat. J. Appl. Mech.
**2002**, 69, 657–662. [Google Scholar] [CrossRef] [Green Version] - Kogut, L.; Etsion, I. A static friction model for elastic-plastic contacting rough surfaces. J. Tribol.
**2004**, 126, 34–40. [Google Scholar] [CrossRef] - Jackson, R.L.; Green, I. A finite element study of elasto-plastic hemispherical contact against a rigid flat. J. Tribol. Trans. ASME
**2005**, 127, 343–354. [Google Scholar] [CrossRef] [Green Version] - Jackson, R.L.; Streator, J.L. A multi-scale model for contact between rough surfaces. Wear
**2006**, 261, 1337–1347. [Google Scholar] [CrossRef] - Sun, J.J.; Ji, Z.B.; Zhang, Y.Y.; Yu, Q.P.; Ma, C.B. A Contact Mechanics Model for Rough Surfaces Based on a New Fractal Characterization Method. Int. J. Appl. Mech.
**2018**, 10, 1850069. [Google Scholar] [CrossRef] - Morag, Y.; Etsion, I. Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces. Wear
**2007**, 262, 624–629. [Google Scholar] [CrossRef] - Yuan, Y.; Cheng, Y.; Liu, K.; Gan, L. A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces. Appl. Surf. Sci.
**2017**, 425, 1138–1157. [Google Scholar] [CrossRef] - Yuan, Y.; Gan, L.; Liu, K.; Yang, X.H. Elastoplastic contact mechanics model of rough surface based on fractal theory. Chin. J. Mech. Eng.
**2017**, 30, 207–215. [Google Scholar] [CrossRef] - Yu, Q.; Sun, J.; Ji, Z. Mechanics Analysis of Rough Surface Based on Shoulder-Shoulder Contact. Appl. Sci.
**2021**, 11, 8048. [Google Scholar] [CrossRef] - Pan, W.J.; Li, X.P.; Wang, L.L.; Guo, N.; Mu, J.X. A normal contact stiffness fractal prediction model of dry-friction rough surface and experimental verification. Eur. J. Mech. A-Solids
**2017**, 66, 94–102. [Google Scholar] [CrossRef] - Pan, W.J.; Song, C.X.; Ling, L.Y.; Qu, H.Y.; Wang, M.H. Unloading contact mechanics analysis of elastic-plastic fractal surface. Arch. Appl. Mech.
**2021**, 91, 2697–2712. [Google Scholar] [CrossRef] - Zhou, H.; Long, X.H.; Meng, G.; Liu, X.B. A stiffness model for bolted joints considering asperity interactions of rough surface contact. J. Tribol. Trans. ASME
**2021**, 114, 011501. [Google Scholar] [CrossRef] - Wang, R.Q.; Zhu, L.D.; Zhu, C.X. Research on fractal model of normal contact stiffness for mechanical joint considering asperity interaction. Int. J. Mech. Sci.
**2017**, 134, 357–369. [Google Scholar] [CrossRef] - Li, L.; Wang, J.J.; Shi, X.H.; Ma, S.L.; Cai, A.J. Contact Stiffness Model of Joint Surface Considering Continuous Smooth Characteristics and Asperity Interaction. Tribol. Lett.
**2021**, 69, 43. [Google Scholar] [CrossRef] - Cohen, D.; Kligerman, Y.; Etsion, I. The Effect of Surface Roughness on Static Friction and Junction Growth of an Elastic-Plastic Spherical Contact. J. Tribol. Trans. ASME
**2009**, 131, 021404. [Google Scholar] [CrossRef] - Cohen, D.; Kligerman, Y.; Etsion, I. A model for contact and static friction of nominally flat rough surfaces under full stick contact condition. J. Tribol. -Trans. ASME
**2008**, 130, 031401. [Google Scholar] [CrossRef] - Wang, D.; Zhang, Z.S.; Jin, F.; Fan, X.H. Normal Contact Model for Elastic and Plastic Mechanics of Rough Surfaces. ACTA Mech. Solida Sin.
**2019**, 32, 148–159. [Google Scholar] [CrossRef] - Xiao, H.F.; Sun, Y.Y.; Chen, Z.G. Fractal modeling of normal contact stiffness for rough surface contact considering the elastic-plastic deformation. J. Braz. Soc. Mech. Sci. Eng.
**2019**, 41, 11. [Google Scholar] [CrossRef] - Yu, X.; Sun, Y.Y.; Zhao, D.; Wu, S.J. A revised contact stiffness model of rough curved surfaces based on the length scale. Tribol. Int.
**2021**, 164, 107206. [Google Scholar] [CrossRef] - Kragelskii, I.V.; Mikhin, N.M. Handbook of Friction Units of Machines; ASME Press: New York, NY, USA, 2002. [Google Scholar]
- Horng, J.H. An elliptic elastic-plastic asperity microcontact model for rough surfaces. ASME J. Tribol.
**1998**, 120, 82–88. [Google Scholar] [CrossRef] - Jamari, J. An Elliptic Elastic-Plastic Asperity Micro-Contact Model. Rotasi J. Tek. Mesin
**2006**, 8, 1–5. [Google Scholar] - Jeng, Y.R.; Wang, P.Y. An elliptic microcontact model considering elastic, elastoplastic, and plastic deformation. ASME J.
**2003**, 125, 232–240. [Google Scholar] - Wen, Y.Q.; Tang, J.Y.; Zhou, W.; Zhu, C.C. A new elliptical microcontact model considering elastoplastic deformation. Proc. Inst. Mech. Eng. Part J. J. Eng. Tribol.
**2018**, 232, 1352–1364. [Google Scholar] [CrossRef] - Jamari, J.; Schipper, D.J. An elastic-plastic contact model of ellipsoid bodies. Tribol. Lett.
**2006**, 21, 262–271. [Google Scholar] [CrossRef] - Lan, W.; Fan, S.; Fan, S. A fractal model of elastic-plastic contact between rough surfaces for a low-velocity impact process. Int. J. Comput. Methods
**2021**, 18, 2150039. [Google Scholar] [CrossRef] - 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.
**2016**, 87, 253–264. [Google Scholar] [CrossRef] - Liu, Y.; Wang, Y.S.; Yu, H.C. A spherical conformal contact model considering frictional and microscopic factors based on fractal theory. Chaos Solitons Fractals
**2018**, 111, 96–107. [Google Scholar] [CrossRef] - Wang, H.H.; Jia, P.; Wang, L.Q.; Yun, F.H.; Wang, G.; Liu, M.; Wang, X.Y. Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction. Appl. Sci.
**2020**, 10, 742. [Google Scholar] [CrossRef] [Green Version] - Yang, W.; Li, H.; Dengqiu, M.; Yongqiao, W.; Jian, C. Sliding Friction Contact Stiffness Model of Involute Arc Cylindrical Gear Based on Fractal Theory. Int. J. Eng.
**2017**, 30, 109–119. [Google Scholar] - Zhao, Z.F.; Han, H.Z.; Wang, P.F.; Ma, H.; Zhang, S.H.; Yang, Y. An improved model for meshing characteristics analysis of spur gears considering fractal surface contact and friction. Mech. Mach. Theory
**2021**, 158, 104219. [Google Scholar] [CrossRef] - Mao, H.C.; Sun, Y.G.; Xu, T.T.; Yu, G.B. Numerical Calculation Method of Meshing Stiffness for the Beveloid Gear considering the Effect of Surface Topography. Math. Probl. Eng.
**2021**, 2021, 8886792. [Google Scholar] [CrossRef] - Ausloos, M.; Berman, D.H. A Multivariate Weierstrass-Mandelbrot Function. Proc. R. Soc. A Math. Phys. Eng. Sci.
**1985**, 400, 331–350. [Google Scholar] - Mandelbrot, B. The Fractal Geometry of Nature; W.H. Freeman and Company: New York, NY, USA, 1982; pp. 35–45. [Google Scholar]

**Figure 1.**Roughness measurements of a beveloid gear tooth surface. (

**a**) The measurement experiment of tooth surface topography; (

**b**) The three-dimensional morphological characteristics of the rough surface.

**Figure 2.**Fractal characterization of rough surfaces with different processing texture features: (

**a**) G

_{x}= 1.0 × 10

^{−11}m, G

_{y}= 1.0 × 10

^{−9}m, D

_{x}= 1.6, D

_{y}= 1.3; (

**b**) G

_{x}= 1.0 × 10

^{−9}m, G

_{y}= 1.0 × 10

^{−11}m, D

_{x}= 1.3, D

_{y}= 1.6.

**Figure 8.**Fractal characterization of rough surfaces with different processing texture features: (

**a**) G

_{x}= 1.0 × 10

^{−11}m, G

_{y}= 1.0 × 10

^{−9}m, D

_{x}= 1.6, D

_{y}= 1.3; (

**b**) G

_{x}= 1.0 × 10

^{−9}m, G

_{y}= 1.0 × 10

^{−11}m, D

_{x}= 1.3, D

_{y}= 1.6.

**Figure 9.**Variations of the dimensionless contact area ${A}_{r}^{*}$ with fractal dimension D: (

**a**) different eccentricity e; (

**b**) different contact coefficient λ.

**Figure 10.**Relationship between the ratio of the elastic contact area to the real contact area and fractal dimension D at different eccentricity e.

**Figure 11.**Comparison of the mechanical properties between the model in this paper and the MB contact model.

**Figure 12.**Variations in the dimensionless contact area ${A}_{r}^{*}$ with fractal dimension D: (

**a**) different fractal dimensions D; (

**b**) different fractal roughness G.

**Figure 13.**Variations in the dimensionless contact area ${A}_{r}^{*}$ with fractal dimension D: (

**a**) different eccentricity e; (

**b**) different contact coefficient λ.

**Figure 14.**Variations in the dimensionless contact stiffness ${K}_{n}^{*}$ with the dimensionless contact area ${A}_{r}^{*}$: (

**a**) different fractal dimensions D; (

**b**) different fractal roughness G.

**Figure 15.**Variations in the dimensionless contact stiffness ${K}_{n}^{*}$ with the dimensionless contact area ${A}_{r}^{*}$: (

**a**) different eccentricity e; (

**b**) different contact coefficient λ.

Parameters | Profile 1 | Profile 2 |
---|---|---|

Fractal dimension D | 1.3–1.8 | 1.3–1.8 |

Characteristic scale G (m) | 1.0 × 10^{−10} | 1.0 × 10^{−10} |

Young’s modulus E (Pa) | 2.06 × 10^{11} | |

Poisson’s ratio ν | 0.26 | |

Plastic yield stress σ_{y} (Pa) | 235 × 10^{6} |

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**MDPI and ACS Style**

Yu, G.; Mao, H.; Jiang, L.; Liu, W.; Valerii, T.
Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities. *Appl. Sci.* **2022**, *12*, 4071.
https://doi.org/10.3390/app12084071

**AMA Style**

Yu G, Mao H, Jiang L, Liu W, Valerii T.
Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities. *Applied Sciences*. 2022; 12(8):4071.
https://doi.org/10.3390/app12084071

**Chicago/Turabian Style**

Yu, Guangbin, Hancheng Mao, Lidong Jiang, Wei Liu, and Tupolev Valerii.
2022. "Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities" *Applied Sciences* 12, no. 8: 4071.
https://doi.org/10.3390/app12084071