Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant

Samuel Macharia Karimi; Mark Kimathi; Mathew Ngugi Kinyanjui

doi:doi:10.11648/j.ajam.20200805.13

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Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant

Samuel Macharia Karimi, Mark Kimathi, Mathew Ngugi Kinyanjui

Published in American Journal of Applied Mathematics (Volume 8, Issue 5)

Received: 12 August 2020 Accepted: 3 September 2020 Published: 11 September 2020

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Abstract

There is always a demand in the industry sector to increase the efficiency of machine components to reduce wear and tear. This paper presents the numerical solution to the study of Elastohydrodynamic lubrication point contact for sliding/rolling bearing where the viscosity of the lubricant is non-Newtonian. The assumption that a lubricant is Newtonian reduces validation of the model hence the Reynolds-Eyring model in this research will incorporate the non-Newtonian nature of the lubricant of the bearing. The mathematical model comprises of Reynold-Eyring equation, film thickness, load balance, lubricant viscosity and lubricant density equations together with their boundary conditions. The Reynolds-Eyring equation governing the flow is non-linear hence the finite difference method numerical technique is used to discretize it together with the other two dimensional equations. These equations are solved simultaneously and Matlab software is used simulate the results. The film thickness and pressure profiles with various loads and speeds are presented. The findings note that an increase in load lowers the pressure and film thickness while an increase in the speed results to a direct increase in pressure and film thickness. A pressure spike is also noted at the exit of the bearing.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 5)
DOI	10.11648/j.ajam.20200805.13
Page(s)	257-264
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Elastohydrodynamic, Thermal, Eyring, Non-newtonian

References

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[4]	H. Liu, K. Mao, C. Zhu, X. Xu, and M. Liu, "Parametric studies of spur gear lubrication performance considering dynamic loads”, Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, May 9, 2012.
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[6]	H. Liu, "Lubricated Contact Analysis of a Spur Gear Pair with Dynamic Loads”, PHD Thesis, School of Engineering, University of Warwick April 2013.
[7]	H. Lu, M. Berzins, C. E. Goodyer, and P. K. Jimack, "High order discontinuous Galerkin method for elastohydrodynamic lubrication line contact problems”, Communications in Numerical Methods in Engineering, vol. 21, pp. 643-650, 2005.
[8]	T. Almqvist and R. Larsson, "Thermal transient rough EHL line contact simulations by aid of computational fluid dynamics”, Tribology International, vol. 41, pp. 683-693, 2008.
[9]	M. Berzins, C. E. Goodyer and P. Jimack, " High order discontinuous Galerkin method for elastohydrodynamic lubrication line contact”, Communications in Numerical Methods in Engineering, vol. 00, pp. 1-6, 2000.
[10]	Y. Liu, Q. J. Wang, S. Bair, and P. Vergne, "A quantitative solution for the full shear thinning EHL point contact problem including traction”, Tribology Letters, vol. 28 (2):171-181, 2007.
[11]	D. V. Srikanth, A. N. Kaushal, K. B. Chaturvedi et al., "Determination of a large tilting pad thrust bearing angular stiffness”, Tribology International, 47, 69-76, 2012.
[12]	P. R. Kiogora, M. N. Kinyanjui, D. M. Theuri, "A Conservative Scheme Model of an Inclined Pad Thrust Bearing”, International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 1, January 2014.
[13]	P. R. Kiogora, M. N. Kinyanjui, D. M. Theuri, "Numerical Solution of the Momentum and Energy Equations of an Inclined Pad Thrust Bearing”, International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 3, May 2014.
[14]	V. Petrone, A. Senatore and V. D’Agostino, "Effect of an Improved Yasutomi Pressure-Viscosity Relationship on the Elastohydrodynamic Line Contact Problem”, International Scholarly Research Notices, vol. 2013, pp. 1-7, 2013.
[15]	J. S. Issa and W. Habchi, “Influence of Realistic Lubricant Density-Pressure Dependence on the Stiffness of Elastohydrodynamic Lubricated Contacts”, Journal of Tribology, vol. 142 (3), pp. 1-20, 2019.
[16]	S. Jadhav, G. D. Thakre and S. C. Sharma, “Numerical modeling of elastohydrodynamic lubrication of line contact lubricated with micropolar fluid”, J Braz. Soc. Mech. Sci. Eng., vol. 40, 326 (2018).
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APA Style

Samuel Macharia Karimi, Mark Kimathi, Mathew Ngugi Kinyanjui. (2020). Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant. American Journal of Applied Mathematics, 8(5), 257-264. https://doi.org/10.11648/j.ajam.20200805.13

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ACS Style

Samuel Macharia Karimi; Mark Kimathi; Mathew Ngugi Kinyanjui. Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant. Am. J. Appl. Math. 2020, 8(5), 257-264. doi: 10.11648/j.ajam.20200805.13

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AMA Style

Samuel Macharia Karimi, Mark Kimathi, Mathew Ngugi Kinyanjui. Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant. Am J Appl Math. 2020;8(5):257-264. doi: 10.11648/j.ajam.20200805.13

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@article{10.11648/j.ajam.20200805.13,
  author = {Samuel Macharia Karimi and Mark Kimathi and Mathew Ngugi Kinyanjui},
  title = {Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {5},
  pages = {257-264},
  doi = {10.11648/j.ajam.20200805.13},
  url = {https://doi.org/10.11648/j.ajam.20200805.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.13},
  abstract = {There is always a demand in the industry sector to increase the efficiency of machine components to reduce wear and tear. This paper presents the numerical solution to the study of Elastohydrodynamic lubrication point contact for sliding/rolling bearing where the viscosity of the lubricant is non-Newtonian. The assumption that a lubricant is Newtonian reduces validation of the model hence the Reynolds-Eyring model in this research will incorporate the non-Newtonian nature of the lubricant of the bearing. The mathematical model comprises of Reynold-Eyring equation, film thickness, load balance, lubricant viscosity and lubricant density equations together with their boundary conditions. The Reynolds-Eyring equation governing the flow is non-linear hence the finite difference method numerical technique is used to discretize it together with the other two dimensional equations. These equations are solved simultaneously and Matlab software is used simulate the results. The film thickness and pressure profiles with various loads and speeds are presented. The findings note that an increase in load lowers the pressure and film thickness while an increase in the speed results to a direct increase in pressure and film thickness. A pressure spike is also noted at the exit of the bearing.},
 year = {2020}
}

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TY  - JOUR
T1  - Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant
AU  - Samuel Macharia Karimi
AU  - Mark Kimathi
AU  - Mathew Ngugi Kinyanjui
Y1  - 2020/09/11
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200805.13
DO  - 10.11648/j.ajam.20200805.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 257
EP  - 264
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200805.13
AB  - There is always a demand in the industry sector to increase the efficiency of machine components to reduce wear and tear. This paper presents the numerical solution to the study of Elastohydrodynamic lubrication point contact for sliding/rolling bearing where the viscosity of the lubricant is non-Newtonian. The assumption that a lubricant is Newtonian reduces validation of the model hence the Reynolds-Eyring model in this research will incorporate the non-Newtonian nature of the lubricant of the bearing. The mathematical model comprises of Reynold-Eyring equation, film thickness, load balance, lubricant viscosity and lubricant density equations together with their boundary conditions. The Reynolds-Eyring equation governing the flow is non-linear hence the finite difference method numerical technique is used to discretize it together with the other two dimensional equations. These equations are solved simultaneously and Matlab software is used simulate the results. The film thickness and pressure profiles with various loads and speeds are presented. The findings note that an increase in load lowers the pressure and film thickness while an increase in the speed results to a direct increase in pressure and film thickness. A pressure spike is also noted at the exit of the bearing.
VL  - 8
IS  - 5
ER  -

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Author Information

Samuel Macharia Karimi

Pan African University, Institute of Basic Sciences, Technology and Innovation, Nairobi, Kenya
Mark Kimathi

Department of Mathematics, Statistics and Actuarial Science, Machakos University, Machakos, Kenya
Mathew Ngugi Kinyanjui

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

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APA Style

Samuel Macharia Karimi, Mark Kimathi, Mathew Ngugi Kinyanjui. (2020). Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant. American Journal of Applied Mathematics, 8(5), 257-264. https://doi.org/10.11648/j.ajam.20200805.13

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ACS Style

Samuel Macharia Karimi; Mark Kimathi; Mathew Ngugi Kinyanjui. Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant. Am. J. Appl. Math. 2020, 8(5), 257-264. doi: 10.11648/j.ajam.20200805.13

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AMA Style

Samuel Macharia Karimi, Mark Kimathi, Mathew Ngugi Kinyanjui. Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant. Am J Appl Math. 2020;8(5):257-264. doi: 10.11648/j.ajam.20200805.13

Copy | Download

@article{10.11648/j.ajam.20200805.13,
  author = {Samuel Macharia Karimi and Mark Kimathi and Mathew Ngugi Kinyanjui},
  title = {Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {5},
  pages = {257-264},
  doi = {10.11648/j.ajam.20200805.13},
  url = {https://doi.org/10.11648/j.ajam.20200805.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.13},
  abstract = {There is always a demand in the industry sector to increase the efficiency of machine components to reduce wear and tear. This paper presents the numerical solution to the study of Elastohydrodynamic lubrication point contact for sliding/rolling bearing where the viscosity of the lubricant is non-Newtonian. The assumption that a lubricant is Newtonian reduces validation of the model hence the Reynolds-Eyring model in this research will incorporate the non-Newtonian nature of the lubricant of the bearing. The mathematical model comprises of Reynold-Eyring equation, film thickness, load balance, lubricant viscosity and lubricant density equations together with their boundary conditions. The Reynolds-Eyring equation governing the flow is non-linear hence the finite difference method numerical technique is used to discretize it together with the other two dimensional equations. These equations are solved simultaneously and Matlab software is used simulate the results. The film thickness and pressure profiles with various loads and speeds are presented. The findings note that an increase in load lowers the pressure and film thickness while an increase in the speed results to a direct increase in pressure and film thickness. A pressure spike is also noted at the exit of the bearing.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solution of Elastohydrodynamic Lubrication for Sliding/Rolling Bearing for Non-newtonian Lubricant
AU  - Samuel Macharia Karimi
AU  - Mark Kimathi
AU  - Mathew Ngugi Kinyanjui
Y1  - 2020/09/11
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200805.13
DO  - 10.11648/j.ajam.20200805.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 257
EP  - 264
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200805.13
AB  - There is always a demand in the industry sector to increase the efficiency of machine components to reduce wear and tear. This paper presents the numerical solution to the study of Elastohydrodynamic lubrication point contact for sliding/rolling bearing where the viscosity of the lubricant is non-Newtonian. The assumption that a lubricant is Newtonian reduces validation of the model hence the Reynolds-Eyring model in this research will incorporate the non-Newtonian nature of the lubricant of the bearing. The mathematical model comprises of Reynold-Eyring equation, film thickness, load balance, lubricant viscosity and lubricant density equations together with their boundary conditions. The Reynolds-Eyring equation governing the flow is non-linear hence the finite difference method numerical technique is used to discretize it together with the other two dimensional equations. These equations are solved simultaneously and Matlab software is used simulate the results. The film thickness and pressure profiles with various loads and speeds are presented. The findings note that an increase in load lowers the pressure and film thickness while an increase in the speed results to a direct increase in pressure and film thickness. A pressure spike is also noted at the exit of the bearing.
VL  - 8
IS  - 5
ER  -

Copy | Download