Iteration framework for solving mixed lubrication computation problems

doi:10.1007/s11465-021-0632-8

Frontiers of Mechanical Engineering

2021, Vol. 16

Issue (3): 635-648 https://doi.org/10.1007/s11465-021-0632-8

本期目录

Iteration framework for solving mixed lubrication computation problems

Shi CHEN¹, Nian YIN¹, Xiaojiang CAI², Zhinan ZHANG¹(

)

¹. State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China
². Shanghai Aerospace Control Technology Institute, Shanghai 201109, China; Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai 201109, China

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Abstract：

The general discrete scheme of time-varying Reynolds equation loses the information of the previous step, which makes it unreasonable. A discretization formula of the Reynolds equation, which is based on the Crank–Nicolson method, is proposed considering the physical message of the previous step. Gauss–Seidel relaxation and distribution relaxation are adopted for the linear operators of pressure during the numerical solution procedure. In addition to the convergent criteria of pressure distribution and load, an estimation framework is developed to investigate the relative error of the most important term in the Reynolds equation. Smooth surface with full contacts and mixed elastohydrodynamic lubrication is tested for validation. The asperity contact and sinusoidal wavy surface are examined by the proposed discrete scheme. Results show the precipitous decline in the boundary of the contact area. The relative error suggests that the pressure distribution is reliable and reflects the accuracy and effectiveness of the developed method.

Key words： mixed lubrication discretization formula relative error Reynolds equation asperity

收稿日期: 2020-09-22 出版日期: 2021-09-24

Corresponding Author(s): Zhinan ZHANG

引用本文:

. [J]. Frontiers of Mechanical Engineering, 2021, 16(3): 635-648.
Shi CHEN, Nian YIN, Xiaojiang CAI, Zhinan ZHANG. Iteration framework for solving mixed lubrication computation problems. Front. Mech. Eng., 2021, 16(3): 635-648.

链接本文:

https://academic.hep.com.cn/fme/CN/10.1007/s11465-021-0632-8
https://academic.hep.com.cn/fme/CN/Y2021/V16/I3/635

Fig.1

Fig.2

Fig.3

Parameter	Value
Synthetic curvature radius, $R x$ , $R y$ /mm	19.05
Synthetic elastic modulus, $E ′$ /GPa	219.78
Viscosity, $η 0$ /(Pa·s)	0.096
Pressure–viscosity coefficient, $α$ /GPa⁻¹	18.2
Reference shear stress, $τ 0$ /MPa	18.0
Entrainment velocity, $U$ /(mm·s^–1)	625
Applied load, $w$ /N	800
Time step length $Δ t$ /μs	5.855
Gauss–Seidel relaxation factor, $θ G$	0.15
Distributive relaxation factor, $θ d$	0.1
Tolerance of pressure, $ζ p$	10⁻⁶
Tolerance of load, $ζ w$	10⁻⁶
Grid nodes	$257 × 257$

Tab.1

Fig.4

Fig.5

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

$a, b$	Semi-major axis and semi-minor axis of Hertz contact (m)
$E ′$	Synthetic elastic modulus (GPa)
$h$	Local film thickness (m)
$h ‾$	Average film thickness
$h 0$	Distance between surfaces at $x = 0$ without accounting for deformation (m)
$h 1$	A threshold value for checking the contacted state (m)
$h 2$	A threshold value for checking contacted boundary (m)
$h a$	Roughness amplitude (m)
$H = h R x a b$	Dimensionless film thickness
i, j	Node number
$k$	Iteration step
$k m$	Maximum iteration step
$l w x, l w y$	Wavelengths in x and y directions, respectively (m)
$p$	Hydrodynamic pressure, or pressure in general (Pa)
$p H$	Maximum Hertzian contact pressure (Pa)
$p r$	Pressure distribution error
$P = p / p H$	Dimensionless pressure
$P ˜ i, j$	Value of dimensionless pressure of previous iteration in node (i, j)
$P ‾ i, j$	Value of dimensionless pressure of current iteration in node (i, j)
$R s$	Spherical radius (m)
$R x, R y$	Synthetic curvature radius of contacted surface in x and y direction, respectively (m)
$t$	Time (s)
$T = t / a$	Dimensional time (s/m)
$U$	Entrainment speed (m/s)
$V$	Average speed in y direction (m/s)
$V e$	Surface elastic deformation (m)
$w$	Applied load (N)
$w r$	Load error
$x$	$x -coordinate$
$x L$	Leading edge of roughness surface (m)
$x s$	Initial position (m)
$X = x / a$	Dimensionless length in $x$ direction
$y$	$y -coordinate$
$Y = y / a$	Dimensionless length in $y$ direction
$α$	Pressure–viscosity exponent (Pa^–1)
$β$	Thermal diffusivity (W?m/°C)
$ε$ $= ρ * H 2 p H 12 η 0 E ′ (a R x) 3$	The coefficient of dimensionless pressure
$ξ x$ , $ξ y$	x- and y-coordinate of pressures when computing deformation, respectively
$ζ w$ , $ζ p$	Tolerance of load and pressure, respectively
$θ d$	Relaxation factor of distributive relaxation method
$θ G$	Relaxation factor of Gauss–Seidel relaxation method
$ϑ$	Temperature (°C)
$ρ$	Density (kg/m³)
$ρ 0$	Density under ambient condition (kg/m³)
$ρ ∗ = ρ / ρ 0$	Dimensionless density
$η 0$	Viscosity under ambient condition (Pa?s)
$η ∗$	Effective viscosity (Pa?s)
$λ$	Film thickness parameter
$σ$	Composite root–mean–square roughness of contact surfaces
$τ 0$	Reference shear stress (Pa)
$τ 1$	Shear stress on the lower surface (Pa)
$δ$	Absolute computation errors of governing equation
$δ p (i, j)$	Pressure change of node (i, j) due to relaxation
$δ r$	Relative error
$Ω$	Control volume
$℧$	Calculated domain

1	G E Morales-Espejel, P Rycerz, A Kadiric. Prediction of micropitting damage in gear teeth contacts considering the concurrent effects of surface fatigue and mild wear. Wear, 2018, 398–399: 99–115 https://doi.org/10.1016/j.wear.2017.11.016
2	V Brizmer, C Matta, I Nedelcu, et al.. The influence of tribolayer formation on tribological performance of rolling/sliding contacts. Tribology Letters, 2017, 65(2): 57 doi:10.1007/s11249-017-0839-3
3	V Brizmer, A Gabelli, C Vieillard, et al.. An experimental and theoretical study of hybrid bearing micropitting performance under reduced lubrication. Tribology Transactions, 2015, 58(5): 829–835 https://doi.org/10.1080/10402004.2015.1021944
4	N Patir, H S Cheng. An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. Journal of Tribology, 1978, 100(1): 12–17 https://doi.org/10.1115/1.3453103
5	K H Geng, A N Geng, X Wang, et al.. Frictional characteristics of the Vane–Chute pair in a rolling piston compressor based on the second-order motion. Tribology International, 2019, 133: 111–125 https://doi.org/10.1016/j.triboint.2018.11.030
6	C Liu, Y Lu, P Wang, et al.. Numerical analysis of the effects of compression ring wear and cylinder liner deformation on the thermal mixed lubrication performance of ring-liner system. Mechanics & Industry, 2018, 19(2): 203 doi:10.1051/meca/2018015
7	S H Cui, L Gu, M Fillon, et al.. The effects of surface roughness on the transient characteristics of hydrodynamic cylindrical bearings during startup. Tribology International, 2018, 128: 421–428 https://doi.org/10.1016/j.triboint.2018.06.010
8	S W Zhang, C H Zhang, Y Z Hu, et al.. Numerical simulation of mixed lubrication considering surface forces. Tribology International, 2019, 140: 105878 https://doi.org/10.1016/j.triboint.2019.105878
9	A Azam, A Ghanbarzadeh, A Neville, et al.. Modelling tribochemistry in the mixed lubrication regime. Tribology International, 2019, 132: 265–274 doi:10.1016/j.triboint.2018.12.024
10	X Jiang, D Y Hua, H S Cheng, et al.. A mixed elastohydrodynamic lubrication model with asperity contact. Journal of Tribology, 1999, 121(3): 481–491 https://doi.org/10.1115/1.2834093
11	Y Z Hu, D Zhu. A full numerical solution to the mixed lubrication in point contacts. Journal of Tribology, 2000, 122(1): 1–9 doi:10.1115/1.555322
12	Y Z Hu, H Wang, W Z Wang, et al.. A computer model of mixed lubrication in point contacts. Tribology International, 2001, 34(1): 65–73 https://doi.org/10.1016/S0301-679X(00)00139-0
13	X P Wang, Y C Liu, D Zhu. Numerical solution of mixed thermal elastohydrodynamic lubrication in point contacts with three-dimensional surface roughness. Journal of Tribology, 2017, 139(1): 011501 https://doi.org/10.1115/1.4032963
14	T He, D Zhu, C J Yu, et al.. Mixed elastohydrodynamic lubrication model for finite roller-coated half space interfaces. Tribology International, 2019, 134: 178–189 https://doi.org/10.1016/j.triboint.2019.02.001
15	D Zhu, J Wang, N Ren, et al.. Mixed elastohydrodynamic lubrication in finite roller contacts involving realistic geometry and surface roughness. Journal of Tribology, 2012, 134(1): 011504 https://doi.org/10.1115/1.4005952
16	Z J Xie, Q H Xue, J Q Wu, et al.. Mixed-lubrication analysis of planetary roller screw. Tribology International, 2019, 140: 105883 https://doi.org/10.1016/j.triboint.2019.105883
17	X Q Lu, Q B Dong, K Zhou, et al.. Numerical analysis of transient elastohydrodynamic lubrication during startup and shutdown processes. Journal of Tribology, 2018, 140(4): 041504 https://doi.org/10.1115/1.4039371
18	S B Xiao. Numerical analysis of transient thermal mixed lubrication in point contact. Thesis for Master’s Degree. Wuhan: Wuhan University of Science and Technology, 2019 (in Chinese)
19	Y C He. Numerical analysis for the thermal EHL in point contacts of cam-tappet pair. Thesis for Master’s Degree. Qingdao: Qingdao University of Technology, 2019 (in Chinese)
20	J Marti, P. RyzhakovAn explicit/implicit Runge–Kutta-based PFEM model for the simulation of thermally coupled incompressible flows. Computational Particle Mechanics, 2020, 7(1): 57–69 https://doi.org/10.1007/s40571-019-00229-0
21	E Ezzatneshan, K Hejranfar. Simulation of three-dimensional incompressible flows in generalized curvilinear coordinates using a high-order compact finite-difference lattice Boltzmann method. International Journal for Numerical Methods in Fluids, 2019, 89(7): 235–255 https://doi.org/10.1002/fld.4693
22	P Huang. Lubrication Numerical Calculation Methods. Beijing: Higher Education Press, 2012
23	Y Hamid, A Usman, S K Afaq, et al.. Numeric based low viscosity adiabatic thermo-tribological performance analysis of piston-skirt liner system lubrication at high engine speed. Tribology International, 2018, 126: 166–176 https://doi.org/10.1016/j.triboint.2018.05.022
24	S Z Wen, P Huang. Principles of Tribology. 2nd ed. Beijing: Tsinghua University Press, 2002
25	D Zhu. On some aspects of numerical solutions of thin-film and mixed elastohydrodynamic lubrication. Proceedings of the Institution of Mechanical Engineers. Part J, Journal of Engineering Tribology, 2007, 221(5): 561–579 doi:10.1243/13506501JET259
26	J D Anderson. Computational Fluid Dynamics: The Basics with Applications. New York: McGraw-Hill, 1995
27	S K Liao, Y Zhang, D Chen. Runge–Kutta finite element method based on the characteristic for the incompressible Navier–Stokes equations. Advances in Applied Mathematics and Mechanics, 2019, 11(6): 1415–1435 doi:10.4208/aamm.OA-2018-0150
28	M V Fischels, R G Rajagopalan. Family of Runge–Kutta-based algorithms for unsteady incompressible flows. AIAA Journal, 2017, 55(8): 2630–2644 https://doi.org/10.2514/1.J055106
29	X N Zhang, R Glovnea, G E Morales-Espejel, et al.. The effect of working parameters upon elastohydrodynamic film thickness under periodic load variation. Tribology Letters, 2020, 68(2): 62 https://doi.org/10.1007/s11249-020-01303-y
30	S B Liu, Q Wang, G Liu. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear, 2000, 243(1–2): 101–111 doi:10.1016/S0043-1648(00)00427-0
31	S Bair, Y Liu, Q J Wang. The pressure-viscosity coefficient for Newtonian EHL film thickness with general piezoviscous response. Journal of Tribology, 2006, 128(3): 624–631 https://doi.org/10.1115/1.2197846
32	C H Venner, A A Lubrecht. Multilevel Methods in Lubrication. Amsterdam: Elsevier, 2000
33	F R Lv, C X Jiao, N Ta, et al.. Mixed-lubrication analysis of misaligned bearing considering turbulence. Tribology International, 2018, 119: 19–26 https://doi.org/10.1016/j.triboint.2017.10.030
34	G Xiang, Y F Han, J X Wang, et al.. Coupling transient mixed lubrication and wear for journal bearing modeling. Tribology International, 2019, 138: 1–15 https://doi.org/10.1016/j.triboint.2019.05.011

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