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Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities

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Abstract

This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication (EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newton-generalized minimal residual (GMRES) from the Krylov subspace method (KSM). Acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner. Profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings. The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.

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Abbreviations

a :

amplitude of roughness, µm

a 0 :

dimensionless amplitude of roughness, a0 = aR/b2

\(\overline a\) :

normalized surface roughness amplitude, \(\overline a = {a_0}/H_{\min}^{{\rm{Dowson - Higginson}}};\)

b :

half width of Hertzian contact, \(b = 4R\sqrt {W/\left({2\pi} \right),{\rm{m}}} \)

E′ :

reduced modulus of elasticity, Pa

G :

dimensionless material parameter, G = αE′

H :

dimensionless film thickness, H = hR/b2

h :

film thickness, m

\(\overline h \) :

dimensionless film thickness, \(\overline h = h/\left({R{{\left({2U} \right)}^{- 1/2}}} \right)\)

H 00 :

dimensionless offset film thickness, m

K ij :

discrete approximation of K-logarithmic kernel

λ :

dimensionless velocity parameter

l :

dimensionless wavelength, l = λ/b

N :

number of nodes on the grid

P :

dimensionless pressure, P = p/ph

p :

pressure, Pa

α :

pressure viscosity relation

Δx :

dimensionless mesh size

Δt :

dimensionless time step

p h :

maximum Hertzian pressure, ph = E′b/(4R), Pa

R :

reduced radius of curvature, m

U :

dimensionless speed parameter, U = (η0us)/(E′R)

u s :

average rolling velocity, us = u1 + u2, m/s

u1, u2 :

velocities of lower and upper surfaces, respectively, m/s

W :

dimensionless load parameter, w/(E′R)

w :

external load per unit length, N/m

XC :

dimensionless location of a pressure spike

X :

dimensionless coordinate, x/b;

z :

pressure viscosity index

η :

Newtonian viscosity

η 0 :

viscosity at ambient pressure

\(\overline \eta \) :

dimensionless viscosity, \(\overline \eta = \eta /\eta 0\)

M :

dimensionless load parameter (Moes parameter)

L :

dimensionless material parameter (Moes parameter).

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Acknowledgements

The authors thank the reviewers for their useful suggestions for the earlier draft of the article. N. M. BUJURKE acknowledges the financial support from the Indian National Science Academy, New Delhi, India. M. H. KANTLI would like to thank Biluru Gurubasava Mahaswamiji Institute of Technology for the encouragement and support.

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Authors and Affiliations

  1. Department of Mathematics, Karnatak University, Dharwad, 580003, India

    N. M. Bujurke

  2. Department of Mathematics, Biluru Gurubasava Mahaswamiji Institute of Technology, Mudhol, 587313, India

    M. H. Kantli

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Correspondence to N. M. Bujurke.

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Citation: BUJURKE, N. M. and KANTLI, M. H. Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities. Applied Mathematics and Mechanics (English Edition), 41(6), 881–898 (2020) https://doi.org/10.1007/s10483-020-2616-8

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Bujurke, N.M., Kantli, M.H. Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities. Appl. Math. Mech.-Engl. Ed. 41, 881–898 (2020). https://doi.org/10.1007/s10483-020-2616-8

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  • DOI: https://doi.org/10.1007/s10483-020-2616-8

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