# Modelling of Lubricated Electrical Contacts

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## Abstract

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## 1. Introduction

## 2. Numerical Methods

#### 2.1. Assumptions

- In the absence of accurate wear and corrosion prediction techniques for lubricated electrical contact conditions, wear and corrosion of the surfaces are not included. The effects of lubricant chemistry at the interfaces, such as tribofilm formation, are not directly considered. In other words, the surface properties, resistivity and roughness topography remain unchanged throughout the simulation, although deflections are considered at the asperity scale and large macro-scales.
- The current quasi-static model neglects time dependent effects such as vibrations and squeeze film effects, but as noted in previous work, this could be important [27].
- Thermal effects, such as heat generation due to Joule heating and friction are ignored, along with temperature dependent properties.

#### 2.2. Hydrodynamic and Mixed Lubrication

_{x}, and ϕ

_{y}, are mathematical flow factors that describe the obstruction of the fluid flow from the surface roughness. Patir and Cheng provide flow factor equations which depend on the film thickness, surface roughness and asperity orientation (longitudinal or lateral). The roughness is assumed to be isotropic here and thus the flow factors are independent of the direction of flow. The isotropic flow factor equations given by Patir and Cheng are:

_{x}and ϕ

_{y}approach the value of one.

_{o}is the initial film thickness that includes the undeformed geometry of the cylinder with radius, R, or

#### 2.3. Elastic-Plastic Rough Surface Contact

_{c}is derived independently of the hardness, to be:

_{peak}is the average radius of curvature of the peaks and C is derived to be

_{s}is given as [50]

_{peak}, σ) can be approximated from profilometer measurements using the methods outlined in McCool [51]. The average contact force, P, is predicted by Equation (11) as a function of surface separation, h. Therefore, at each node the contact force is predicted between the surfaces of the electrical contact. The contact forces on each node are summed together to give the total solid contact force between the surfaces, F

_{cont}.

#### 2.4. Statistical Electrical Contact Resistance

#### 2.5. Force Balance and Numerical Solution

_{cont}. The modified Reynolds equation (Equation (1)) is solved for the fluid pressures which are integrated to provide the total fluid force, F

_{fluid}using Equation (5). The numerical solution is complete when the forces balance as given by:

#### 2.6. Mesh Convergence

## 3. Results and Discussion

_{y}) are for a typical tin material, which is commonly used in electrical contacts. Both surfaces of the contact are considered to be tin, and that is considered using the effective elastic modulus:

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

d | distance between the mean of the surface asperities or peaks |

E | elastic modulus |

F | applied force |

f | friction coefficient |

G | Gaussian asperity height distribution |

h | film thickness, separation of mean surface height |

p | fluid pressure |

P | contact force |

R_{peak} | asperity radius of curvature |

S_{y} | yield strength |

U | sliding velocity |

x, y, z | Cartesian coordinate system |

δ | surface deformation |

η | areal asperity density |

ϕ | flow factor for modified Reynolds equation |

μ | dynamic fluid viscosity |

σ | RMS roughness |

ν | Poisson’s ratio |

## References

- Chudnovsky, B.H. Lubrication of Electrical and Mechanical Components in Electric Power Equipment; CRC Press: Boca Raton, FL, USA, 2019. [Google Scholar]
- Johnson, K.L. Contact Mechanics; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Jackson, R.L. Chapter 14: Lubrication. In Handbook of Lubrication and Tribology, Volume II: Theory and Design; Bruce, R.W., Ed.; CRC Press: Boca Raton, FL, USA, 2012; pp. 14.1–14.14. [Google Scholar]
- Williams, J.A.; Kennedy, F.E. Engineering Tribology. J. Tribol.
**1998**, 120, 644. [Google Scholar] [CrossRef] [Green Version] - Campbell, W. The lubrication of electrical contacts. IEEE Trans. Compon. Hybrids Manuf. Technol.
**1978**, 1, 4–16. [Google Scholar] [CrossRef] - Chudnovsky, B.H. Lubrication of electrical contacts. In Proceedings of the Fifty-First IEEE Holm Conference on Electrical Contacts, Chicago, IL, USA, 26–28 September 2005; pp. 107–114. [Google Scholar] [CrossRef]
- Sawa, K.; Watanabe, Y.; Ueno, T. Effect of lubricant on sliding conditions in Au-plated slip-ring system for small electric power transfer. In Proceedings of the 2014 IEEE 60th Holm Conference on Electrical Contacts (Holm), New Orleans, LA, USA, 12–15 October 2014; pp. 1–6. [Google Scholar]
- Sawa, K.; Takemasa, Y.; Watanabe, Y.; Ueno, T.; Yamanoi, M. Fluctuation components of contact voltage at AgPd brush and Au-plated slip-ring system with lubricant. In Proceedings of the 2015 IEEE 61st Holm Conference on Electrical Contacts (Holm), San Diego, CA, USA, 11–14 October 2015; pp. 250–255. [Google Scholar]
- Jackson, R.L.; Coker, A.B.; Tucker, Z.; Hossain, M.S.; Mills, G. An Investigation of Silver-Nanoparticle-Laden Lubricants for Electrical Contacts. IEEE Trans. Compon. Packag. Manuf. Technol.
**2018**, 9, 193–200. [Google Scholar] [CrossRef] - Crilly, L.; Jackson, R.L.; Bond, S.; Mills, G.; Bhargava, S. An Investigation of the Electrical Contact Resistance Change, Lubrication, and Wear Properties of a Nanolubricant. In Proceedings of the 2020 IEEE 66th Holm Conference on Electrical Contacts and Intensive Course (HLM), San Antonio, TX, USA, 30 September–7 October 2020; pp. 1–7. [Google Scholar]
- Cao, Z.; Xia, Y.; Liu, L.; Feng, X. Study on the conductive and tribological properties of copper sliding electrical contacts lubricated by ionic liquids. Tribol. Int.
**2019**, 130, 27–35. [Google Scholar] [CrossRef] - Ko, S.-D.; Seo, M.-H.; Yoon, Y.-H.; Han, C.-H.; Lim, K.-S.; Kim, C.-K.; Yoon, J.-B. Investigation of the Nanoparticle Electrical Contact Lubrication in MEMS Switches. J. Microelectromech. Syst.
**2017**, 26, 1417–1427. [Google Scholar] [CrossRef] - Berman, D.; Erdemir, A.; Sumant, A.V. Graphene as a protective coating and superior lubricant for electrical contacts. Appl. Phys. Lett.
**2014**, 105, 231907. [Google Scholar] [CrossRef] - Lang, H.; Xu, Y.; Zhu, P.; Peng, Y.; Zou, K.; Yu, K.; Huang, Y. Superior lubrication and electrical stability of graphene as highly effective solid lubricant at sliding electrical contact interface. Carbon
**2021**, 183, 53–61. [Google Scholar] [CrossRef] - Achanta, S.; Drees, D. Effect of lubrication on fretting wear and durability of gold coated electrical contacts under high frequency vibrations. Tribol.-Mater. Surf. Interfaces
**2008**, 2, 57–63. [Google Scholar] [CrossRef] - Kaushik, L.; Azarian, M.H.; Pecht, M. Fretting Performance Comparison between PFPE and PAO Based Lubricants for Lightly-Loaded Gold-Plated Electrical Contacts. In Proceedings of the 2019 IEEE Holm Conference on Electrical Contacts, Milwaukee, WI, USA, 14–18 September 2019. [Google Scholar]
- Chaudhary, R.; Kaushik, L.; Azarian, M.H.; Pecht, M. Comparison between Synthetic Oil Lubricants for Reducing Fretting Degradation in Lightly Loaded Gold-Plated Contacts. 2021. Available online: https://www.researchsquare.com/article/rs-476954/v1 (accessed on 15 April 2021).
- Swingler, J. The automotive connector: The influence of powering and lubricating a fretting contact interface. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2000**, 214, 615–623. [Google Scholar] [CrossRef] - Graton, O.; Fouvry, S.; Enquebecq, R.; Petit, L. Effect of Lubrication on DC and RF Electrical Endurance of Gold Plated Contacts Subjected to Fretting Wear. In Proceedings of the 2018 IEEE Holm Conference on Electrical Contacts, Albuquerque, NM, USA, 14–18 October 2018; pp. 426–434. [Google Scholar]
- Fu, Y.; Qin, H.; Xu, X.; Zhang, X.; Guo, Z. The effect of surface texture and conductive grease filling on the tribological properties and electrical conductivity of carbon brushes. Tribol. Int.
**2021**, 153, 106637. [Google Scholar] [CrossRef] - Noel, S.; Brezard-Oudot, A.; Chretien, P.; Alamarguy, D. Fretting behaviour of tinned connectors under grease lubrication. In Proceedings of the 2017 IEEE Holm Conference on Electrical Contacts, Denver, CO, USA, 10–13 September 2017; pp. 109–116. [Google Scholar]
- Larsson, E.; Andersson, A.M.; Rudolphi, Å.K. Grease lubricated fretting of silver coated copper electrical contacts. Wear
**2017**, 376–377, 634–642. [Google Scholar] [CrossRef] - Amada, Y.; Sawa, K.; Ueno, T. Effects of lubricant oil on sliding contact phenomena in carbon brush-slip ring system. In Proceedings of the 2017 IEEE Holm Conference on Electrical Contacts, Denver, CO, USA, 10–13 September 2017; pp. 182–186. [Google Scholar]
- Gohar, R. Elastohydrodynamics; World Scientific: Singapore, 2001. [Google Scholar]
- Gohar, R.; Cameron, A. Optical Measurement of Oil Film Thickness under Elasto-hydrodynamic Lubrication. Nature
**1963**, 200, 458–459. [Google Scholar] [CrossRef] - Jackson, A.; Cameron, A. An Interferometric Study of the EHL of Rough Surfaces. ASLE Trans.
**1976**, 19, 50–60. [Google Scholar] [CrossRef] - Mehdigoli, H.; Rahnejat, H.; Gohar, R. Vibration response of wavy surfaced disc in elastohydrodynamic rolling contact. Wear
**1990**, 139, 1–15. [Google Scholar] [CrossRef] - Teodorescu, M.; Rahnejat, H.; Gohar, R.; Dowson, D. Harmonic decomposition analysis of contact mechanics of bonded layered elastic solids. Appl. Math. Model.
**2007**, 33, 467–485. [Google Scholar] [CrossRef] - Patir, N.; Cheng, H.S. An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication. J. Lubr. Technol.
**1978**, 100, 12–17. [Google Scholar] [CrossRef] - Patir, N.; Cheng, H.S. Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces. ASME J. Tribol.
**1979**, 101, 220–230. [Google Scholar] [CrossRef] - Dmochowski, W.M.; Dadouche, A.; Fillon, M. Finite Difference Method for Fluid-Film Bearings. In Encyclopedia of Tribology; Springer Science and Business Media LLC: Berlin, Germany, 2013; pp. 1137–1143. [Google Scholar]
- Szeri, A.Z. Fluid Film Lubrication; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Booser, E.R. CRC Handbook of Lubrication. Theory and Practice of Tribology. Volume II: Theory and Design; CRC Press: Boca Raton, FL, USA, 1984. [Google Scholar]
- Zhao, B.; Dai, X.-D.; Zhang, Z.-N.; Xie, Y.-B. A new numerical method for piston dynamics and lubrication analysis. Tribol. Int.
**2016**, 94, 395–408. [Google Scholar] [CrossRef] - Zhang, X.; Xu, Y.; Jackson, R.L. A mixed lubrication analysis of a thrust bearing with fractal rough surfaces. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2020**, 234, 608–621. [Google Scholar] [CrossRef] - Zhang, X.; Jackson, R.L. A mixed lubrication analysis of a flat-land thrust bearing with a surface optimisation method. Lubr. Sci.
**2021**, 33, 335–346. [Google Scholar] [CrossRef] - Jackson, R.L.; Green, I. The Behavior of Thrust Washer Bearings Considering Mixed Lubrication and Asperity Contact. Tribol. Trans.
**2006**, 49, 233–247. [Google Scholar] [CrossRef] - Jackson, R.L.; Green, I. The Thermoelastic Behavior of Thrust Washer Bearings Considering Mixed Lubrication, Asperity Contact, and Thermoviscous Effects. Tribol. Trans.
**2008**, 51, 19–32. [Google Scholar] [CrossRef] - Angadi, S.; Jackson, R.L.; Choe, S.-Y.; Flowers, G.T.; Lee, B.-Y.; Zhong, L. A Multiphysics Finite Element Model of a 35A Automotive Connector Including Multiscale Rough Surface Contact. J. Electron. Packag.
**2012**, 134, 011001. [Google Scholar] [CrossRef] - Angadi, S.V.; Jackson, R.L.; Pujar, V.; Tushar, M.R. A Comprehensive Review of the Finite Element Modeling of Electrical Connectors Including Their Contacts. IEEE Trans. Compon. Packag. Manuf. Technol.
**2020**, 10, 836–844. [Google Scholar] [CrossRef] - Polchow, J.R.; Angadi, S.; Jackson, R.; Choe, S.-Y.; Flowers, G.T.; Lee, B.-Y.; Zhong, L. A Multi-Physics Finite Element Analysis of Round Pin High Power Connectors. In Proceedings of the 2010 Proceedings of the 56th IEEE Holm Conference on Electrical Contacts, Charleston, SC, USA, 4–7 October 2010; pp. 1–9. [Google Scholar]
- Kogut, L.; Etsion, I. Electrical Conductivity and Friction Force Estimation in Compliant Electrical Connectors. Tribol. Trans.
**2000**, 43, 816–822. [Google Scholar] [CrossRef] - Leidner, M.; Schmidt, H.; Myers, M.; Schlaak, H.F. A new simulation approach to characterizing the mechanical and electrical qualities of a connector contact. Eur. Phys. J. Appl. Phys.
**2010**, 49, 22909. [Google Scholar] [CrossRef] - Liu, H.; Leray, D.; Colin, S.; Pons, P.; Broué, A. Finite Element Based Surface Roughness Study for Ohmic Contact of Microswitches. In Proceedings of the 2012 IEEE 58th Holm Conference on Electrical Contacts (Holm), Portland, OR, USA, 23–26 September 2012; pp. 1–10. [Google Scholar]
- Israel, T.; Gatzsche, M.; Schlegel, S.; Grosmann, S.; Kufner, T.; Freudiger, G. The impact of short circuits on contact elements in high power applications. In Proceedings of the 2017 IEEE Holm Conference on Electrical Contacts, Denver, CO, USA, 10–13 September 2017; pp. 40–49. [Google Scholar]
- Jackson, R.L.; Green, I. A Statistical Model of Elasto-plastic Asperity Contact between Rough Surfaces. Tribol. Int.
**2006**, 39, 906–914. [Google Scholar] [CrossRef] - An, B.; Wang, X.; Xu, Y.; Jackson, R.L. Deterministic elastic-plastic modelling of rough surface contact including spectral interpolation and comparison to theoretical models. Tribol. Int.
**2019**, 135, 246–258. [Google Scholar] [CrossRef] - Jackson, R.L.; Green, I.; Marghitu, D.B. Predicting the coefficient of restitution of impacting elastic-perfectly plastic spheres. Nonlinear Dyn.
**2010**, 60, 217–229. [Google Scholar] [CrossRef] - Greenwood, J.A.; Williamson, J.B.P. Contact of nominally flat surfaces. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1966**, 295, 300–319. [Google Scholar] [CrossRef] - Front, I. The Effects of Closing Force and Surface Roughness on Leakage in Radial Face Seals; Technion, Israel Institute of Technology: Haifa, Israel, 1990. [Google Scholar]
- McCool, J.I. Relating Profile Instrument Measurements to the Functional Performance of Rough Surfaces. J. Tribol.
**1987**, 109, 264–270. [Google Scholar] [CrossRef] - Etsion, I.; Kligerman, Y.; Kadin, Y. Unloading of an elastic–plastic loaded spherical contact. Int. J. Solids Struct.
**2005**, 42, 3716–3729. [Google Scholar] [CrossRef] - Jackson, R.L.; Chusoipin, I.; Green, I. A Finite Element Study of the Residual Stress and Strain Formation in Spherical Contacts. ASME J. Tribol.
**2005**, 127, 484–493. [Google Scholar] [CrossRef] [Green Version] - Yang, H.; Green, I. Analysis of displacement-controlled fretting between a hemisphere and a flat block in elasto-plastic contacts. J. Tribol.
**2019**, 141, 031401. [Google Scholar] [CrossRef] - Holm, R. Electric Contacts; Springer: New York, NY, USA, 1967. [Google Scholar]
- Cooper, M.G.; Mikic. B.B. Yovanovich, M.M. Thermal contact conductance. Int. J. Heat Mass Transf.
**1969**, 12, 279–300. [Google Scholar] [CrossRef] - Siddaiah, A.; Kasar, A.K.; Khosla, V.; Menezes, P.L. In-Situ Fretting Wear Analysis of Electrical Connectors for Real System Applications. J. Manuf. Mater. Process.
**2019**, 3, 47. [Google Scholar] [CrossRef] [Green Version] - Sharma, A.; Ahn, B. Dry Sliding Wear Behavior of Sn and NiSn Overlays on Cu Connectors. Tribol. Lett.
**2018**, 66, 136. [Google Scholar] [CrossRef] - Song, J.; Yuan, H.; Schinow, V. Fretting corrosion behavior of electrical contacts with tin coating in atmosphere and vacuum. Wear
**2019**, 426–427, 1439–1445. [Google Scholar] [CrossRef] - Jackson, R.; Ashurst, W.R.; Flowers, G.T.; Angadi, S.; Choe, S.-Y.; Bozack, M.J. The Effect of Initial Connector Insertions on Electrical Contact Resistance. In Proceedings of the Electrical Contacts-2007 Proceedings of the 53rd IEEE Holm Conference on Electrical Contacts, Pittsburgh, PA, USA, 16–19 September 2007; pp. 17–24. [Google Scholar]
- Angadi, S.; Wilson, W.E.; Jackson, R.L.; Flowers, G.T.; Rickett, B.I. A Multi-Physics Finite Element Model of an Electrical Connector Considering Rough Surface Contact. In Proceedings of the 2008 Proceedings of the 54th IEEE Holm Conference on Electrical Contacts, Orlando, FL, USA, 27–29 October 2008; pp. 168–177. [Google Scholar]
- Zhang, X.; Jackson, R.L. The influence of multiscale roughness on the real contact area and contact resistance between real reference surfaces. In Proceedings of the ICEC 2014—The 27th International Conference on Electrical Contacts, Dresden, Germany, 22–26 June 2014; VDE: Berlin, Germany. [Google Scholar]

**Figure 3.**Predicted contact resistance as a function of sliding velocity using varying mesh densities.

**Figure 4.**Predicted contact resistance as a function of sliding velocity also considering various values of effective surface roughness.

**Figure 6.**Predicted contact resistance as a function of sliding speed for various macro-scale radii.

R | 0.01 m |

Width | 0.001 m |

σ | 0.5 µm |

η | 10^{9} m^{2} |

Rpeak | 100 µm |

E | 41.4 GPa |

Sy | 14 MPa |

ν | 0.36 |

µ | 0.1 Pa·s |

ρ | 1.15 × 10^{−7} Ω/m |

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**MDPI and ACS Style**

Jackson, R.L.; Angadi, S.
Modelling of Lubricated Electrical Contacts. *Lubricants* **2022**, *10*, 32.
https://doi.org/10.3390/lubricants10030032

**AMA Style**

Jackson RL, Angadi S.
Modelling of Lubricated Electrical Contacts. *Lubricants*. 2022; 10(3):32.
https://doi.org/10.3390/lubricants10030032

**Chicago/Turabian Style**

Jackson, Robert L., and Santosh Angadi.
2022. "Modelling of Lubricated Electrical Contacts" *Lubricants* 10, no. 3: 32.
https://doi.org/10.3390/lubricants10030032