virtual texturing of lightweight crankshafts bearings ... · texturização virtual de mancais de...
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UNIVERSIDADE ESTADUAL DE CAMPINASFaculdade de Engenharia Mecnica
JONATHA OLIVEIRA DE MATOS REIS
Virtual Texturing of Lightweight CrankshaftsBearings
Texturizao Virtual de Mancais de VirabrequinsLeves
CAMPINAS
2017
JONATHA OLIVEIRA DE MATOS REIS
Virtual Texturing of Lightweight CrankshaftsBearings
Texturizao Virtual de Mancais de VirabrequinsLeves
Dissertation presented to the School of Me-chanical Engineering of the University ofCampinas in partial fulfillment of the require-ments for the Masters degree, in the field ofSolid Mechanics and Mechanical Design.
Dissertao apresentada Faculdade deEngenharia Mecnica da UniversidadeEstadual de Campinas como parte dosrequisitos exigidos para a obteno do ttulo deMestre em Engenharia Mecnica, na rea deMecnica dos Slidos e Projeto Mecnico.
Orientador: Prof. Dr. Marco Lucio Bittencourt
ESTE EXEMPLAR CORRESPONDE VERSO FINAL DA DIS-SERTAO DEFENDIDA PELO ALUNO JONATHA OLIVEIRADE MATOS REIS, E ORIENTADA PELO PROF. DR. MARCOLUCIO BITTENCOURT.
CAMPINAS
2017
Agncia(s) de fomento e n(s) de processo(s): FAPESP, 2015/16470-1; CAPES
Ficha catalogrficaUniversidade Estadual de Campinas
Biblioteca da rea de Engenharia e ArquiteturaLuciana Pietrosanto Milla - CRB 8/8129
Reis, Jonatha Oliveira de Matos, 1991- R277v ReiVirtual texturing of lightweight crankshafts / Jonatha Oliveira de Matos Reis.
Campinas, SP : [s.n.], 2017.
ReiOrientador: Marco Lucio Bittencourt. ReiDissertao (mestrado) Universidade Estadual de Campinas, Faculdade
de Engenharia Mecnica.
Rei1. Lubrificao. 2. Rolamentos. 3. Tratamento de superfcies. 4. Rugosidade
de superfcie. 5. Efeito da temperatura. I. Bittencourt, Marco Lucio,1964-. II.Universidade Estadual de Campinas. Faculdade de Engenharia Mecnica. III.Ttulo.
Informaes para Biblioteca Digital
Ttulo em outro idioma: Texturizao virtual de mancais de virabrequins levesPalavras-chave em ingls:LubricationBearingsSurface treatmentSurface roughnessTemperature effectrea de concentrao: Mecnica dos Slidos e Projeto MecnicoTitulao: Mestre em Engenharia MecnicaBanca examinadora:Marco Lucio Bittencourt [Orientador]Gregory Bregion DanielFrancisco Jos ProfitoData de defesa: 04-08-2017Programa de Ps-Graduao: Engenharia Mecnica
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UNIVERSIDADE ESTADUAL DE CAMPINASFACULDADE DE ENGENHARIA MECNICA
COMISSO DE PS-GRADUAO EM ENGENHARIA MECNICADEPARTAMENTO DE SISTEMAS INTEGRADOS
DISSERTAO DE MESTRADO ACADMICO
Virtual Texturing of LightweightCrankshafts Bearings
Texturizao Virtual de Mancais deVirabrequins Leves
Autor: Jonatha Oliveira de Matos ReisOrientador: Prof. Dr. Marco Lucio Bittencourt
A Banca Examinadora composta pelos membros abaixo aprovou esta Dissertao:
Prof. Dr. Marco Lucio BittencourtFEM/UNICAMP
Prof. Dr. Gregory Bregion DanielFEM/UNICAMP
Prof. Dr. Francisco Jos ProfitoPME/EPUSP
A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vidaacadmica do aluno.
Campinas, 04 de agosto de 2017.
minha famlia,
que mesmo longe, esteve sempre comigo.
ACKNOWLEDGEMENTS
Gostaria de agradecer, primeiramente, aos meus pais, Ana e Valter, que me apoia-ram em todos os momentos e que conhecem os caminhos que me levaram at aqui. Foi a estradaque vocs trilharam que me trouxe to longe e onde ela comea o lugar que chamarei semprede lar. Obrigado tambm minha irm Jocasta e minha prima Thais, por fazerem parte dostimos momentos que uso de recordao e inspirao.
Agradeo imensamente ao professor Marco Lcio Bittencourt, pela chance de fazerparte desse projeto. O aprendizado proporcionado por esse mestrado estar comigo sempre ese hoje dou mais um passo em direo pesquisa, porque ele me deu a oportunidade e meinspirou a continuar buscando o conhecimento sempre.
Deixo aqui tambm um agradecimento especial ao professor Francisco Jos Profito.Sem a sua ajuda, eu no teria chegado to longe nessa pesquisa. Muito obrigado pelas conversase pelas dvidas respondidas.
Muito obrigado a todos do lab, por tornarem a experincia do mestrado muito maistranquila e divertida. Alfredo, obrigado pelo template em LATEX, mas tambm por dividiros problemas e alegrias do mestrado comigo. Guilherme, muito obrigado por toda a ajuda noincio da minha pesquisa, em especial pelo suporte na utilizao dos softwares comerciais desimulao. Darla e Mari, muito obrigado pelo convvio e pela dose de humor e loucura quasediria. Eduardo, obrigado pela ajuda a cada novo documento que precisava ser entregue e pelacompanhia nas reunies de projeto. Agradeo imensamente a todos e espero o melhor paravocs.
Agradeo eternamente aos meus conterrneos e amigos, Alessandro, Breno e Lvia,que passaram pelas mesmas dificuldades e saudades que passei para virem at aqui e que esti-veram comigo durante toda essa jornada. Vocs foram essenciais para tornar essa jornada maisfcil e saibam que, aonde quer que estejamos, podero sempre contar comigo.
Meu muito obrigado quela que esteve comigo durante todos os momentos. Cintia,meu amor, obrigado por fazer parte dessa histria e dividir comigo os melhores momentos. Vocfoi e vai ser sempre minha referncia do melhor que posso ser. Seu apoio foi vital para minhasanidade durante esse mestrado. Obrigado por estar comigo na alegria e na tristeza, na sade ena doena, na pobreza e na menos pobreza, por todos os dias.
Por fim, meus agradecimentos ao professor Waldyr Luiz Ribeiro Gallo, pela gern-cia do projeto temtico, e Fundao de Amparo Pesquisa do Estado de So Paulo (FAPESP)pela concesso do auxlio pesquisa e bolsa de mestrado, sob o processo nmero 2015/16470-1.
RESUMO
Em 2009, 56 milhes de litro de combustvel foram gastos com o atrito em mo-tores de combusto, sendo os mancais responsveis por cerca de um quarto dessa perda. Essapesquisa tem o objetivo de modelar numericamente os mancais de um virabrequim com alviode peso, estudando os efeitos que a texturizao superficial desses mancais pode ter no seudesempenho e focando em reduzir o atrito deste componente. Uma reviso bibliogrfica foirealizada, tratando da texturizao superficial, dos regimes de lubrificao e aspectos de rugosi-dade, contato de superfcies e temperatura. Um programa de computador foi desenvolvido paraestudar o comportamento de um mancal carregado dinamicamente, levando em conta efeitos derugosidade, contato de asperezas, efeitos trmicos e texturizao superficial. Simulaes foramrealizados no mancal de um virabrequim com alvio de peso, aplicando diferentes designs detextura sua superfcie. Os resultados mostraram que possvel reduzir o coeficiente de atritohidrodinmico mdio e mximo destes componentes, mas alguns designs de texturas tendem aaumentar a presso no fluido lubrificante, especialmente se forem levados em conta os efeitostrmicos.
Palavras-Chave: Texturizao Superficial; Equao de Reynolds; Regime de Lubrificao Mista;Mancal de Deslizamento;
ABSTRACT
In 2009, over 56 million liters of fuel were used to overcome friction in combustion engines,being the bearings alone accountable for nearly a quarter of this loss. This research aims tonumerically model a lightweight crankshaft bearing and to study the effects that surface tex-tures can have on its performance, aiming to reduce friction in this component. A literaturereview was performed, covering the surface texture treatments, lubrication regimes and aspectsof rough contact and temperature. A computer program was developed to study the behaviorof a dynamically loaded bearing, considering roughness effects on lubrication, asperity contact,global thermal effects and surface texturing. Simulations were performed for the bearing of alightweight crankshaft, considering different texture designs on its surface. The results showedthat it is possible to reduce the mean and maximum hydrodynamic friction coefficient of thesebearings, but some texture designs may lead to an increase in the fluid pressure, specially whenthe thermal effects are take into account.
Keywords: Surface Texture; Reynolds Equation; Mixed Lubrication;Crankshaft Journal Bear-ings; Roughness Effects.
LIST OF FIGURES
Figure 2.1 Textured mechanical seal model, as present by Etsion (ETSION; HALPERIN,2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Figure 2.2 Parallel bearing with a surface pattern, as shown by Fesanghary (FESANG-HARY; KHONSARI, 2013) . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 3.1 Coordinate system and velocity components for the Navier-Stokes equation(adapted from (SANTOS, 1997)). . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 3.2 Laminar flow between the two lubricated surfaces (adapted from (SANTOS,1997)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Figure 3.3 Journal bearing geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.4 Fluid film cavitation on a lubricated domain. The inside region is a non
cavitated domain, where the fluid film is not broken and = 1. Outsidethat region is the cavitated domain, where the thin blue lines represent thegas/vapor mixture and where 0 < 1 (adapted from (PROFITO et al.,2015)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 3.5 Pressure distributions along the center line of a dimple cell, comparing thethree cavitation models, as found by Khonsari (QIU; KHONSARI, 2009). . 41
Figure 3.6 Total film thickness hT between two surfaces with roughness 1 and 2, witha nominal film thickness h (PROFITO et al., 2015). . . . . . . . . . . . . . 45
Figure 3.7 The main surface variables on the Greenwood-Tripp model (PROFITO etal., 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Figure 3.8 Relation between dynamic viscosity and temperature for different viscositymodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Figure 3.9 Cartesian system of a journal bearing, with the isometric view (left) and thefront view (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 3.10Textured journal bearing (a), with detail for the dimple geometry (b) and thetextured surface (c) (KANGO et al., 2014). . . . . . . . . . . . . . . . . . . 55
Figure 3.11Isometric view of a single dimple on the surface of a mechanical seal (top)and the center line view from the dimple (bottom). . . . . . . . . . . . . . . 56
Figure 3.12Arrangement of dimples in the circumferential and axial directions of a jour-nal bearing surface (KANGO et al., 2014). . . . . . . . . . . . . . . . . . . 57
Figure 3.13Fluid film thickness of a textured journal bearing. . . . . . . . . . . . . . . 57Figure 3.14Computational domain, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.15Computation grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.16Computational procedure flow chart. . . . . . . . . . . . . . . . . . . . . . 63Figure 4.1 Cross-section of the journal bearing geometry with two grooves, as used by
Elrod and Adams (adapted from (FESANGHARY; KHONSARI, 2011)). . . 65
Figure 4.2 Results for the pressure and film content in a journal bearing cross-sectionas achieved by Elrod (top) and by this study (bottom) (ELROD; ADAMS,1974). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Figure 4.3 Results for the pressure and film content for different mesh sizes . . . . . . 66Figure 4.4 Textured mechanical seal with the detail for the surface roughness of the
texture (adapted from (QIU; KHONSARI, 2011b)). . . . . . . . . . . . . . 67Figure 4.5 Load-carrying capacity for a single cell of a texture mechanical seal as found
by Khonsari (left) and by us (righ) (QIU; KHONSARI, 2011b). . . . . . . . 68Figure 4.6 Pressure profile for a journal bearing considering temperature variation (TALA-
IGHIL; FILLON, 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.7 Fluid film: 3D view (left) and cross-section (right) for a smooth journal bear-
ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 4.8 Fluid film: 3D view (left) and cross-section (right) for a textured journal
bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 4.9 Resulting pressure profile for the textured journal bearing (TALA-IGHIL;
FILLON, 2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.10Journal bearing schematics and computational domain, as presented by Ausas
(AUSAS et al., 2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 4.11Loads evolution with time. (Adapted from (AUSAS et al., 2009)) . . . . . . 74Figure 4.12The maximum pressure over time for a dynamically loaded journal bearing. 74Figure 5.1 Crankshaft without (top) and with (bottom) weight reduction (RODRIGUES,
2013). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 5.2 Load chart for a dynamically load journal bearing for a complete engine
cycle, from 0 to 720 degrees. . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 5.3 Numerical results of fluid film maximum pressure over time in journal bear-
ing using different mesh sizes. . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.4 Convergence curve for the fluid film peak pressure. . . . . . . . . . . . . . 78Figure 5.5 Fluid film maximum hydrodynamic pressure on a dynamically load journal
bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 5.6 Lubricant thickness on the journal bearing. . . . . . . . . . . . . . . . . . . 79Figure 5.7 Journal center position in both polar (left) and Cartesian (right) coordinate
systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 5.8 Journal eccentricity over time on the dynamically load bearing. . . . . . . . 79Figure 5.9 Power loss during the journal bearing operation. . . . . . . . . . . . . . . . 80Figure 5.10Hydrodynamic friction coefficient for the journal bearing during its operation. 80Figure 5.11Results of fluid film maximum pressure over time for different operation
speeds with detail for the maximum pressure peaks. . . . . . . . . . . . . . 81Figure 5.12Lubricant thickness on the journal bearing, highlighting the minimum film
thickness reached during the bearings operation. . . . . . . . . . . . . . . . 81
Figure 5.13The hydrodynamic peak pressure for a rough and a smooth dynamicallyloaded journal bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Figure 5.14Relative minimum fluid film thickness for a rough and a smooth dynamicallyloaded journal bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Figure 5.15The hydrodynamic coefficient of friction for a rough and a smooth dynami-cally loaded journal bearing. . . . . . . . . . . . . . . . . . . . . . . . . . 84
Figure 5.16Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.17Temperature distribution over time for a dynamically loaded journal bearing
considering thermal, roughness and asperities effects. . . . . . . . . . . . . 85Figure 5.18Lubricant maximum pressure values for a dynamically loaded journal bear-
ing with and without thermal effects. . . . . . . . . . . . . . . . . . . . . . 86Figure 5.19Fluid film thickness for a dynamically loaded journal bearing with and with-
out thermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 5.20Asperity pressure for a dynamically loaded journal bearing with and without
thermal effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 5.21Hydrodynamic coefficient of friction for a dynamically loaded journal bear-
ing with and without thermal effects. . . . . . . . . . . . . . . . . . . . . . 87Figure 5.22Lubricant film thickness for a journal bearing with dimples and different
depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 5.23Fluid maximum pressure for a journal bearing with the presence of dimples
with different depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 5.24Power loss for a journal bearing with the presence of dimples with different
depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 5.25Average power loss for a journal bearing with the presence of dimples with
different depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 5.26Hydrodynamic friction coefficient over time for a journal bearing with the
presence of dimples with different depths . . . . . . . . . . . . . . . . . . . 91Figure 5.27The maximum and mean friction coefficient on the dynamically load journal
bearing for different values of dimple depths. . . . . . . . . . . . . . . . . . 92Figure 5.28Comparison between the hydrodynamic friction coefficient and the power
loss of a journal bearing for different values of dimple depths. . . . . . . . . 92Figure 5.29Fluid maximum pressure for a journal bearing dimples and different depths,
considering thermal effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 5.30Fluid minimum thickness for a journal bearing with the presence of dimples
with different depths, considering thermal effects. . . . . . . . . . . . . . . 93Figure 5.31Power loss for a journal bearing with the presence of dimples with different
depths, considering thermal effects. . . . . . . . . . . . . . . . . . . . . . . 94Figure 5.32Average power loss for a journal bearing with the presence of dimples with
different depths, considering thermal effects. . . . . . . . . . . . . . . . . . 95
Figure 5.33Hydrodynamic friction coefficient over time for a journal bearing with thepresence of dimples with different depths and considering thermal effects . . 95
Figure 5.34The maximum and mean friction coefficient on dynamically load journalbearing for different values of dimple depth, considering thermal effects. . . 96
Figure 5.35Comparison between the hydrodynamic friction coefficient and the powerloss of a journal bearing for different values of dimple depths,consideringthermal effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Figure 5.36Lubricant film thickness for a journal bearing with dimples and differentradius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Figure 5.37Fluid maximum pressure for a journal bearing with dimples and differentradius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Figure 5.38Average power loss for a journal bearing with the presence of dimples withdifferent radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Figure 5.39The maximum and mean friction coefficient on dynamically load journalbearing for different values of dimple radius. . . . . . . . . . . . . . . . . . 99
Figure 5.40Comparison between the hydrodynamic friction coefficient and the powerloss of a journal bearing for different values of dimple depths. . . . . . . . . 100
Figure 5.41Fluid maximum pressure for a journal bearing dimples and different radius,considering thermal effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 5.42Average power loss for a journal bearing with the presence of dimples withdifferent radius, considering thermal effects. . . . . . . . . . . . . . . . . . 101
Figure 5.43The maximum and mean friction coefficient on dynamically load journalbearing for different values of dimple radius, considering thermal effects. . . 101
Figure 5.44Comparison between the hydrodynamic friction coefficient and the powerloss of a journal bearing for different values of dimple depths,consideringthermal effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Figure 5.45Lubricant film thickness top view for a journal bearing with the presence ofdifferent dimples distribution . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 5.46Fluid maximum pressure for a journal bearing with different dimples distri-bution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 5.47Pressure profile in a journal bearing with the presence of different dimplesdistribution and considering thermal effects. . . . . . . . . . . . . . . . . . 104
Figure 5.48Fluid maximum pressure for a journal bearing with the presence of differentdimples distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
LIST OF TABLES
Table 3.1 Dynamic viscosity-temperature equations: . . . . . . . . . . . . . . . . . . 51Table 3.2 Constants for the dynamic viscosity models. . . . . . . . . . . . . . . . . . 51Table 4.1 Input parameters for the Elrods hydrodynamic lubrication problem . . . . . 65Table 4.2 Input parameters for Khonsaris mechanical seal problem . . . . . . . . . . . 67Table 4.3 Tala-Ighil input parameters for the journal bearing under thermal effects . . . 69Table 4.4 Geometrical parameters for the dimples on the textured journal bearing . . . 71Table 4.5 Ausas dimensionless parameters for the dynamically load journal bearing . . 73Table 5.1 Geometrical parameters for the dynamically load journal bearing. . . . . . . 76Table 5.2 Surface properties values . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Table 5.3 Parameters for the thermal model . . . . . . . . . . . . . . . . . . . . . . . 85Table 5.4 Geometrical properties of the dimples, considering different depths. . . . . . 88Table 5.5 Geometrical properties of the dimples, considering different radius. . . . . . 97Table 5.6 Geometrical properties of the dimples, considering different distributions. . . 102Table 5.7 Hydrodynamic friction coefficient values for the different texture distribu-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Table 5.8 Hydrodynamic friction coefficient values for the different texture distribu-
tions, considering thermal effects. . . . . . . . . . . . . . . . . . . . . . . . 105
LIST OF SYMBOLS
a,b,c Constants for the dynamic viscosity thermal models
x,y,z Cartesian axis
ux,uy,uz Local velocity components
rx,ry,rz Elliptical dimple radius
xc,yc,zc Elliptical dimple origin center
h Fluid film thickness
ho Radial clearance
h Mean surface separation
hT Film thickness between two rough surfaces
t Time
e Eccentricity
k Successive over relaxation iteration step
qx,qz Lubricant film flow
nx,nz Number of dimples
P Hydrodynamic pressure
Pcav Cavitation pressure
Pamb Ambient pressure
R Bearing radius
R j Journal radius
U,W Rotational speeds on the x and y axes.
T Temperature in Kelvin
TC Temperature in Celsius
Cp Lubricant specific heat
Q Axial fluid film flux
E Combined elastic modulus
N Number of asperities
Nx,Nz Number of integration points
Lx Bearing thickness
Lz Bearing length
Ltx,Ltz Length of dimple cell
Wx,Wy Length of dimple cell
Fx,Fy Length of dimple cell
Zs Mean asperity height of combined rough surfaces
Dynamic viscosity
Lubricant density
c Lubricant density on cavitation pressure
Shear stress
Eccentricity ratio
Bulk modulus
s Mean asperity radius of combined rough surfaces
r Standard deviation of combine roughness amplitude
s Standard deviation of asperity heights
r Combined roughness amplitude
s Asperity density
Dissipated power
Friction torque
p Over relaxation parameter for the pressure
p Over relaxation parameter for the film fraction
p Pressure tolerance
Film fraction tolerance
x Tolerance for the support load on the x axis
y Tolerance for the support load on the y axis
T Temperature tolerance
Ratio between fluid film thickness and surface roughness
p(x,z) Patir and Cheng pressure flow factors
s(x) Patir and Cheng shear flow factors
f p(x,z) Patir and Cheng friction pressure flow factors
f sx Patir and Cheng friction shear flow factors
f(x,z) Friction contact factor
CONTENTS
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Surface Texturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Solution Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Hydrodynamic Lubrication Modeling and Solution . . . . . . . . . . . . . . . 323.2 Mixed Lubrication Modeling Solution . . . . . . . . . . . . . . . . . . . . . . 443.3 Global Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Fluid Film for Spherical Geometry Dimples . . . . . . . . . . . . . . . . . . . 543.5 Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.6 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Program Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1 Case 1: Cavitation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Case 2: Roughness Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Case 3: Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4 Case 4: Texture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5 Case 5: Dynamically Loaded Journal Bearing . . . . . . . . . . . . . . . . . . 73
5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Operation Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4 Roughness and Asperity Effects . . . . . . . . . . . . . . . . . . . . . . . . . 825.5 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.6 Surface Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.1 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
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1 INTRODUCTION
This work aims to develop procedures for virtual texturing of crankshaft bearingsto improve their performance in terms of friction and load carrying capacity. There is an ever-increasing demand for automobiles worldwide and enhancing the tribological properties of en-gine components is important since nearly one third of the fuel energy is used to overcomefriction (HOLMBERG et al., ). According to the author, only in 2009, 208,000 millions ofliters of fuel were used to overcome friction, and, as gasoline is a nonrenewable power sourceand, therefore, has a limited life cycle, the amount of money wasted due to friction is onlyincreasing. If would be possible to reduce the amount of friction in crankshaft bearings, it ispossible to reduce the amount of gasoline required to provide the same power to the engine and,as a result, increase the vehicle efficiency, allowing it to run with lower costs.
In a regular four-stroke engine, gasoline works in an intake-compression-combustion-exhaust cycle, performed using mechanical components. About 28% of the energy fuel, withoutconsidering braking friction, is lost due to friction. Lubrication is important to reduce frictionon these mechanical parts, but for a bearing, even when lubricated, the friction still representsabout 25% of the power loss in engines (LIGIER; NOEL, 2015).
Researches were performed in mechanical seals and concluded that it is possibleto reduce friction in seals by modifying their surfaces. In this study, performed by Etsion andBurstein, they added dimples to the seal surface and improved the components performance,for an optimum set up. With those modifications, there was a friction reduction up to 50%(ETSION; BURSTEIN, 1996). This research led to questions about whether it is possible toreproduce this results for different engine components. If so, the amount of energy lost due tofriction in engine components would fall drastically, improving the vehicles power efficiency.
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1.1 Objective
The objective of this dissertation is to evaluate the surface texturing applied tocrankshafts bearings in order to decrease friction in engines aiming to improve the vehiclesperformance.
The study will implement numerical procedures to simulate the bearing behaviorduring its operation following the steps:
Solve the Reynolds equation for a hydrodynamic lubricated journal bearing;
Solve the modified Patir and Cheng Reynolds equation to deal with the roughness effectsin the fluid;
Use the Greenwood-Trip rough contact model to deal with the asperities interaction;
Use an average temperature model to deal with the variation of lubricant viscosity withthe temperature during the bearing operation;
Numerically apply textures to the journal bearings surface.
1.2 Dissertation Outline
This dissertation is divided into the following chapters:
Literature review: where the main lubrication regimes are presented along with the sur-face texturing and the solution methods used to solve the approximate Reynolds equation;
Methodology: with a brief explanation of the equations used to model the journal bear-ings behavior;
Program validation: with the comparison between this dissertation models for the cavita-tion, roughness, temperature, texture and dynamical with papers from the literature;
Results and discussion: with application of the program to investigate lightweight crankshafttextured bearings.
Conclusion: with a summary of what was done and ideas for future works.
This project is part of So Paulo Research Foundation (FAPESP) Center of Researchin Engineering Prof. Urbano Ernesto Stumpf, under project number 2013/50238-3 and grantnumber 2015/16470-1.
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2 LITERATURE REVIEW
The increasing need of the automobile industry for more efficient cars has leadresearchers to push the boundaries of science when trying to find ways to make engine com-ponents smaller and lighter. These features are desirable without losing the engine ability tosustain high loads. One of the components that require high capability of sustain these loadsis the engine bearings. The bearings are responsible, among other things, to uphold the shaftmovement in different velocities and loads, with the least wear and friction. There has been anincreasing number of papers related with the enhancement of tribological properties of enginebearings, specially through the manipulation of their surface (MARIAN et al., 2007; MA; ZHU,2011; ARSLAN et al., 2015).
This review aims to present the current worldwide progress when dealing with themodeling of textured surfaces in order to enhance their tribological properties, particularly whenapplied to journal bearings. In order to better understand the behavior of bearings, we are goingto present aspects of lubrication and how it influences the bearings operation. The most relevantpapers related to the subject are presented here, including common designs for textures and themost common used modeling techniques for the different lubrication regimes.
2.1 Lubrication
The main purpose of a lubricant is to reduce friction and wear on machine elements,making sure that these elements run smoothly during their life cycle. The most used type oflubricant in the automobile industry is liquid, although it can also be presented in solid and gasforms. There are four main lubricant regimes, each one dealing differently with the physicaland chemical interactions between the lubricant and the contacted surface (HAMROCK et al.,2004). This section is going to discuss each of these regimes and present the most well knownworks on the field, scaling to their applications in car engine components, specific on bearings.
2.1.1 Hydrodynamic Lubrication
Hydrodynamic lubrication is characterized by a lubricant film thick enough tokeep the lubricated surfaces separated from each other. This lubrication regime is consideredideal because of the low friction and wear associated with the positive pressure created by theconverging surfaces, the relative movement and fluid viscosity, occurring specially on confor-mal surfaces such of journal and thrust bearings, although it can also happen for non-conformalcontacts (HAMROCK et al., 2004).
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2.1.1.1 Reynolds Equation
The Reynolds equation is a partial differential equation that governs the fluid filmpressure distribution for lubricated contacts. It was first derived in 1886, by Osborne Reynolds,who assumed the condition of dominance of the pressure and viscosity laminar flow terms ofthe Navier-Stokes equations. Reynolds isothermal and isoviscous equation can be written as(REYNOLDS, 1886; GROPPER et al., 2016):
x
(h3
12 px
)+
z
(h3
12 p z
)=
U2
hx
+h t
(2.1)
where:h: film thickness;p: hydrodynamic pressure;: dynamic viscosity;U : velocity in x-direction.
Later on, Dowson extended the lubrication theory, writing generalized equations,that included the thermohydrodynamic effects and the lubricant properties variation across thefilm thickness (DOWSON, 1962). Berther and Godet took into account the existence of local-ized perturbations and defects on both surfaces, creating the today known as Generalized Equa-tions of the Mechanics of Viscous Thin Films (BERTHE; GODET, 1973). Afterwards, Patirand Cheng proposed a modification to the Reynolds equation, this time including the roughnesseffects on the pressure distribution by introducing an average flow model (PATIR; CHENG,1978; GROPPER et al., 2016).
2.1.1.2 Fluid Film Cavitation
Cavitation is the phenomenon where the hydrodynamic pressure reaches valueslower than the gas saturation or vapor pressure, resulting on a rupture of the lubricant film. Thecreated cavity is filled with a mixture of liquid and gases or vapor, modifying the lubricationsystem behavior (PROFITO et al., 2015).
The first researcher to study lubricant film rupture was Gumbel, in 1914. He pro-posed that only the pressures with values equal or higher than a defined cavitation pressure limitshould be considered. All the pressures bellow this values were admitted to be equal to the as-sumed cavitation pressure. The Gumbel model is the simplest model to deal with cavitation onthe Reynolds equation and is also known as the half-Sommerfeld cavitation model (GUMBEL,1914).
A few years later, Swift and Steiber proposed a new formulation to improve thehalf-Sommerfeld model. The new formulation considered that the cavitation boundaries hada null pressure gradient and within the cavitated zones, the pressure were constant and equal
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to the cavitation limit pressure. This formulation is known as the Reynolds cavitation model(SWIFT, 1932; STIEBER, 1933).
The half-Sommerfeld and the Reynolds cavitation models both have the drawbackthat there is no enforcement of mass-conservation on the reformation boundaries. It was Jakobs-son, Floberg and Olsson who proposed complementary boundary conditions that ensured mass-conservation of all the lubricant flow domain (JAKOBSSON; FLOBERG, 1957; OLSSON,1965).
The first algorithm considering mass conservation, was developed by Elrod andAdams, who implemented the boundary conditions proposed by Jakobsson, Floberg and Ols-son (JFO model). They divided the cavitation region using a switch function, eliminating theneed for multiple equations and making sure that there was a pure Couette flow present in thecavitated areas (ELROD; ADAMS, 1974; ELROD, 1981).
Recently, the Elrod algorithm was compared with Reynolds model by Ausas andco-workers, in 2007, when studying the cavitation influence on micro-textured journal bear-ings. The conclusion they reached was that, for textured journal bearings, models without massconservation underestimate the cavitation area, resulting in higher values for the friction torque.It is important that the cavitation models used when modeling journal bearings consider mass-conservation, in order to avoid calculation errors. Ausas also made his algorithm available,making it easier for new studies (AUSAS et al., 2007; AUSAS et al., 2009).
Qiu and Khonsari used a mass conserving algorithm to predict the cavitation behav-ior on flat dimpled surfaces. They ran experimental tests to validate their model and concludedthat its possible to predict the cavitation effect using the JFO boundary conditions. The studyalso showed that using half Sommerfeld and Reynolds models leads to distinct solutions whendifferent dimple sizes are used (QIU; KHONSARI, 2009).
The stability problem of the Elrod algorithm was later studied by Fesanghary andKhonsari. They focused their effort on developing a variation for the switch function in theElrod algorithm, that would result in less instability. The solution they found was to substitutethe binary switch proposed in the JFO method by a progressive regressive exponential equation.The results showed an improvement in the solution stability and also in the computation time(FESANGHARY; KHONSARI, 2011).
2.1.2 Mixed-Lubrication
The mixed-lubrication, or partial lubrication, is the regime where both the fluidfilm and contact of asperities take place during a certain operation condition. It can occur, forexample, when lubricated components operate under high pressure loads (or too low velocities)(HAMROCK et al., 2004; ADJEMOUT et al., 2014).
Jiang and his team proposed a model to study the behavior of machine elements that
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operate under mixed-elastohydrodinamic lubrication regime with very thin film thickness. Heused a fast Fourier transform (FFT) technique to solve the forward (surface displacement) andinverse problem (asperity contact pressure), developing a code that analyzed asperity of contacton 3D surfaces with a good computational time (JIANG et al., 1999).
A deterministic model was developed later by Shi and Salant to treat the behaviorof two surfaces operating in a mixed soft elastohydrodynamic lubrication regime, a flat andan elastic rough. The model took into account the effects of inter-asperity cavitation, usingthe JFO boundary conditions through the Elrod method. They showed that it is necessary toconsider inter-asperity when modeling mixed EHL, for better understatement of the soft EHLproblems (SHI; SALANT, 2000).
Later on, in 2006, Dobrica and other researchers proposed a deterministic model andcompared it with the Patir and Cheng stochastic model. They concluded that roughness had aninfluence on all the parameters they observed (film thickness, attitude angle and friction torque)and that the stochastic model proposed by Patir and Cheng was able to predict accurately theeffect of different roughness types, giving good predictions for average minimum film thickness,but underestimating friction torque (DOBRICA et al., 2006).
Experimental tests were performed by Braun and co-workers to analyze the effi-ciency that laser surface texturing had in the reduction of friction in steel sliding pairs, con-sidering a mixed lubrication regime. The group used different sizes of dimples and performedtests using two different lubricants temperatures, 50 and 100 C They concluded that there wereno linear correlation between dimple size and friction coefficient, operation speed or temper-ature. Despite that, they managed to reduce the friction by 80% for an optimum diameter andoperation speed, being this reduction highly dependent on the oil temperature (BRAUN et al.,2014).
Profito modelled the different lubrication regimes on journal and sliding bearings,using the isothermal generalized equation of the mechanics of viscous thin films and the mass-conserving p- Elrod-Adams cavitation model. He proposed a new way to numerically calculatethe modified mass-conserving Reynolds equation based on the element based finite volumemethod. This technique gathers the finite elements conservation flexibility to work with complexgeometries and the finite volume to work with the flow, making it ideal to work with irregularmeshes, such as textured surfaces (PROFITO et al., 2015).
2.1.2.1 Roughness Effects
The interaction between two surfaces is limited by the roughness and can haveits location defined by an influence area, being this area of contact the responsible for carryingall the external load. Asperity can be understood as the highest roughness peaks, with the actualcontact area being the one generated by the deformation of these asperities (PROFITO, 2010).
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The milestone for contact mechanics was the paper published by Hertz, in 1881,where he considered two surfaces with no lubrication between them. After that, many otherstudies were published, trying to define models for the contact area between surfaces (dry con-tact). For instance, Greenwood and Williamson presented one of the first models where the ac-tual random nature of roughness asperity peaks was considered (GREENWOOD; WILLIAMSON,1966).
Harp and co-workers paper presented a variation of the Reynolds equation, includ-ing the effects of cavitation induced by inter-asperity. The equation is based on the JFO model,implemented in the Elrod algorithm, and the Patir and Cheng method. They later publishedanother paper with the application of this equation for the analyzes of a mechanical seal, show-ing that the microscopic cavitations are meaningful in seals with small film/roughness ratio.Including inter-asperity changed the predictions of previous models that neglected it, reachinghigher load supports numbers and very different leakage rates, even for hydrodynamic lubrica-tion regimes (HARP; SALANT, 2001; HARP; SALANT, 2002).
Almost a decade later, an algorithm was created by Qiu and Khonsari to analyze theperformance of seals and thrust bearings with surface texture. The algorithm considers mass-conservation, using JFO boundary conditions and applies the Patir and Cheng formulation totake into account roughness effects. Among other things, they studied the film thickness, depth-diameter ratio and dimple density necessary to improve the surface performance. The mainoutcome were that the cavitation is only relevant for thin fluid films, the roughness influenceis small when compared with the cavitation effect and there is an optimum depth and densityof dimples that improves the performance, but it depends on the operation conditions (QIU;KHONSARI, 2011b).
2.1.3 Elastohydrodynamic Lubrication
The elastohydrodynamic lubrication can be understood as a variation of the hy-drodynamic lubrication, where the elastic deformation of the contact surfaces are considered(HAMROCK et al., 2004).
Zhu and Cheng wrote a FORTRAN code to calculate point and line contacts lubrica-tion, using inlet heating and thermal reduction factor, evaluating their effects on film thickness.They also adopted a viscoelastic fluid model to calculate shear stress. The code and results wereuseful to determine the efficiency of mechanical components and predict failure on elastohy-drodynamic contacts. A year later, they improved their code in order to predict both mixed andelastohydrodynamic lubrication parameters, including friction. This new code used real 3D sur-face roughness under critical loads to predict lubrication characteristics, although it still neededimprovement in the numerical approach and the computational time (ZHU; CHENG, 1989;ZHU; HU, 2001).
Three methods to calculate the surface elastic deformation were compared by Wang
25
and his colleagues: direct summation, multilevel multi-integration (MLMI) and discrete con-volution fast Fourier transform (DC-FFT). These methods are used for contact problems andresults obtained by Wang showed that all three methods are capable of reaching the same pre-cision, but with different calculation times. Out of the three methods, the DC-FFT had the bestcomputational efficiency, specially when dealing with dense meshes. It was able to solve sur-face deformation and also temperature rise problems, being three times faster than the MLMI(WANG et al., 2003).
2.1.4 Thermohydrodynamic and Thermoelastohydrodynamic Lubrication
Thermohydrodynamic (THD) and thermoelastohydrodynamic (TEHD) lubrica-tions regimes are important when the temperature has an influence on the hydrodynamic andelastohydrodynamic behavior of the lubricants, respectively (HAMROCK et al., 2004).
Shi and Wang developed a thermoelastohydrodynamic model for journal bearingsconformal contacts where they considered the roughness, asperity contact, thermal and ther-moelastic effects, working with large eccentricity ratios. They concluded that roughness effectsshould be considered when working with thermal-tribological analysis and, afterwards, wroteanother paper where they put this model into practice. This paper simulated the performance ofjournal bearings for different arrangements of convective heat-transfer conditions. The conclu-sion was that, when considering thermal effects, the contact asperity does not decrease linearlywith the increase of operational speed, because the speed increases the hydrodynamic effect andalso the amount of heat generated. They also reported a transition of the contact asperity areas,from the edge to a global central area. This phenomenon was explained by the transition processof lubrication regimes or by the thermal expansion difference between the bearing inside andoutside surfaces (SHI; WANG, 1998; SHI et al., 1998).
A transient studied was done by Zhang considering non-Newtonian dynamicallyloaded journal bearings in mixed lubrication to analyze their TEHD behavior. He solved theproblem using the Reynolds equation, a 2D conduction equation and heat balance, demon-strating the effects, among other things, of the texture roughness and the thermal and elasticdeformations performance of bearings under dynamic load. His results showed that the bear-ing system is very sensitive to the contact conditions and temperature distribution varies withthe degree of contact asperity. He also concluded that, for better results, roughness, degree ofcontact and thermal and elastic deformation must be coupled (ZHANG, 2002).
Kucinschi and colleagues analyzed the performance of radially groove thrust washerwith the presence of thermal deformations, coupling the flow effects of the lubrication withheat transfer of both solid and fluid. They created a model to study elastic deformation andtemperature influence on this mechanical component and found higher minimum film thicknessvalues than the ones presented by rigid stators. Also, when comparing THD and TEHD regimes,they found only a small difference between results, when a small misalignment was present
26
(KUCINSCHI et al., 2004).
A new model for the thermoelastohydrodynamic lubrication was proposed by Fatuand his team to analyze a dynamically loaded journal bearing. The model considered massconservation cavitation and the effect of temperature in the viscosity, proposing a new heat fluxalgorithm to take into account the lubricant film temperature. They used a big-end connecting-rod bearing to validate their algorithm, verifying its efficiency and also concluding, from theresults, that temperature for the oil film varies significantly over space and time (FATU et al.,2006).
A 3D thermohydrodynamic analysis was performed by Cupillard and his team ona textured slider, considering laminar and steady flow. They used hot and cold lubricants inthe grooves and studied their performance on different operation conditions. They concludedthat thermal effects must be considered for better results, having a maximum performance withlong grooves and achieving an increase in the loading carry capacity of about 16% for crit-ical conditions. They also discovered that, when compared with smooth sliders, the texturedones performed better for a large range of operation conditions, specially under high speeds(CUPILLARD et al., 2009).
A micro-textured journal bearing was thermally analyzed using JFO boundary con-ditions and non-Newtonian rheology by Kango and co-workers, comparing with a model usingReynolds boundary conditions. They showed that the loading carrying capacity was similarfor flat and full textured surfaces, but the JFO provided more realistic results for the texturedbearings. Additionally, they investigated the partial texturing and found that, when located inthe convergence area, it improves the loading carrying capacity, decreasing the temperature,specially for small eccentricity ratios (KANGO et al., 2014).
2.2 Surface Texturing
The first group to associate surface texturing with the improvement of a surfacetribological properties was Hamilton and his co-workers, in 1966 (HAMILTON et al., 1966).They pointed out that the classical theory of lubrication did not predict the existence of anadditional pressure that separate two planar parallel surfaces, so there should exist a mechanismthat would explain this effect. The proposed mechanism was the cavitation generated in thesmall irregularities of the surface. Therefore, the cavitation in the micro cavities was responsiblefor creating a hydrodynamic pressure that would balance the asymmetric pressure distributionof the fluid film, producing a high carrying load capacity.
Very little attention was paid for this discovery at that time and only few paperswere published related to the subject (ANNO et al., 1968; ANNO et al., 1969; WILLIS, 1986).It was only a few years later, with the publication of Etsion and Burstein and the positive resultsthey obtained with a surface treatment on mechanical seals, that surface texturing became a
27
subject of interest in academia. They found that with a proper selection of the size and ratioof the surface pores, it was possible to improve the mechanical seals performance (ETSION;BURSTEIN, 1996). After their work, there were a huge number of publications, most of themtheoretical, exploring the surface texturing in different components, such as thrust and journalbearings, piston rings, piston pins, etc.
Etsion published many other theoretical and experimental papers about surface tex-turing. He investigated the effects of laser surface texturing on partially textured mechanicalseals in order to find an optimum texture design to reduce friction and heat generation on me-chanical components, discovering that this type of treatment enable an impressive improvementin the tribological properties. Figure 2.1 shows the textured mechanical seal model used byEtsion (ETSION; HALPERIN, 2002). Later, Brizmer and Etsion published a paper about theadvantages of using laser surface texture in parallel thrust bearings and, furthermore, the rela-tion between the porous density and influence on the pressure (BRIZMER et al., 2003; ETSIONet al., 2004). Finally, Etsion published a review showing the efforts worldwide related to lasersurface texturing, describing different models for particular machine elements (ETSION, 2005).
Fig. 2.1 Textured mechanical seal model, as present by Etsion (ETSION; HALPERIN, 2002)
Other researchers investigated the effects of surface texturing using different ap-proaches. Kucinschi and his team published a paper in 2004 describing a numerical model con-sidering the thermoelastohydrodynamics effects of a grooved thrust washer. He used Reynoldsequation to solve the fluid flux problem, the energy equation for the temperature and the equa-tion of thermo-elasticity for the deformation (KUCINSCHI et al., 2004). A year later, Ko-valchenko and co-workers performed an experimental study on laser surface texturing effect ontransitions and found out that, for a hydrodynamic lubrication regime, the loading carrying per-formance was improved, specially for high loads, velocities and viscosities (KOVALCHENKOet al., 2005).
28
It was Wang and Zhu, in 2005, who published the first known paper using the term"virtual texturing" when referencing to the act of modeling surfaces with different treatments.In their paper, they related the texture designs with the component tribological properties andillustrated the idea of what was virtual texturing. They listed these concepts as: 1. Virtuallygeneration of the surface, based on the operation requirements; 2. Analysis of the contact and thelubrication to evaluate the performance; 3. Prediction of the efficiency, life and the evolution ofthe surface; 4. Modification and optimization of the surface; 5. Validation of the model throughexperimental analysis (WANG; ZHU, 2005).
More recently, in 2013, Kango and his team made a comparison with texturedand grooved journal bearings, studying the performance of components with dimples, paral-lel grooves and transverse grooves. They conclude that both dimples and groove can improvethe average temperature, as long as they are optimally located and operating under certain ec-centricities ratios (KANGO et al., 2013).
2.2.1 Definition and Purpose
The act of creating features on a surface, like micro dimples or grooves, with theintention of improving its mechanical properties, is denominated surface texturing. There arethree main mechanism responsible for the improvement of the tribological parameters of tex-tured surfaces. Firstly, the micro-dimples are believed to function as micro reservoirs, providinglubricant for the mechanical components in case of mixed or boundary lubrication. Secondly,the dimples function as micro-traps for wear particles. Finally, the generation of lift, also calledthe micro-bearing effect (mechanism only captured by theoretical models). Even though theseeffects are addressed repeatedly in the literature, there are no researches that corroborate withthe first two assumptions (ARAUJO et al., 2004; LU; KHONSARI, 2007; QIU; KHONSARI,2011a; GROPPER et al., 2016; PROFITO, 2010).
2.2.2 Texturing Design
Texturing design depends on many parameters, but it is possible to limit themost important ones to texture density, dimple aspect ratio and dimple depth. There are manystudies focused solely on optimizing these parameters, aiming to achieve better tribologicalperformance, either for full or partial textured surfaces.
Tnder studied the influence of dimples on a full textured plane bearing, using theReynolds equation to run numerical simulations. He found that the loading carrying capacitycan be increased under specific operation conditions, even though the friction increases as well.In addition, for certain operation conditions, it is possible to improve the film stiffness anddamping, with the cost of higher values for the coefficient of friction (TONDER, 2010).
Ma and Zhu considered the optimum design for surfaces textured with elliptical-shaped dimples. They used many dimple depths and found that an increase in the optimum
29
diameter is followed by an increase in the optimum corresponding depth. Furthermore, theydiscovered that the ideal area ratio does not depend on texture or operating parameters, validat-ing these results with experimental tests (MA; ZHU, 2011).
Podgornik and his colleagues analyzed the best texture designs for different lubrica-tion regimes, using experimental tests, fluid dynamic models and 2D finite element simulations.They discovered that, under starved lubrication, friction was increased and only low dimpledensity resulted in the micro oil reservoir effect. The best results for friction were obtainedwhen the full film lubrication was reached, but a CFD simulation was needed to define the idealdensities and shapes (PODGORNIK et al., 2012).
Fesanghary and Khonsari used the Sequential Quadratic Programming optimizationalgorithm (SQP) to find the optimum groove shapes for loading carrying capacity on parallelbearings. They ran a great amount of theoretical and experimental analyzes, validating their re-sults with a good correlation and finding that the aspect ratio has a major influence in the loadingcarrying capacity, reaching an improvement of 36% when comparing with parallel bearings nor-mal spiral grooves. Figure 2.2 presents one of the texture designs used by Fesanghary on theparallel bearing (FESANGHARY; KHONSARI, 2013).
Fig. 2.2 Parallel bearing with a surface pattern, as shown by Fesanghary (FESANGHARY;KHONSARI, 2013)
Qiu and co-workers investigated the effects of many different textured shapes onthe friction coefficients and stiffness of parallel slider bearings. They concluded that there isan optimum textured shape that minimizes the friction coefficient, but that one is not neces-sarily the one that gives the higher values of loading carrying capacity. Also, they found thatcurved shaped dimples produce the lowest friction coefficient, specially ellipsoid shapes, butfor isotropic performance, an spherical shape dimple is the best option (QIU et al., 2013).
Changing the focus of the studies for a more practical sense, Adjemout and col-leagues showed the effects of real dimple shapes on the performance of textured mechanicalseals. They analyzed the shapes obtained after the fabrication process and implemented them ina hydrodynamic lubrication model, studying the influence of different dimple defects. Results
30
showed that there is a limit where the fabrication imperfection ceases the positive effects ofthe texturing, being the depth and orientation of roughness inside the dimples the most rele-vant parameters for the leakage. They concluded that an increase in the curvature angle of theedge of the dimples increases the leakage, affecting the pumping and increasing the friction(ADJEMOUT et al., 2014).
2.2.3 Journal Bearings
As a result of the circumferential converging-diverging gap of the journal bear-ings film thickness, the surface texturing on journal bearings are the most challenge ones andthere are not many publications in the field (GROPPER et al., 2016).
Tala-Ighil and co-workers studied the effects of surface texture and the influenceof texturing location on the performance of journal bearings. They considered the inertia ef-fects and found that there is a negative influence of these effects on the bearings performance,concluding that the optimum texture depends on the geometry and operation conditions of thebearing (TALA-IGHIL et al., 2011).
A paper published by Cupillard, in 2008, showed how he used computational fluiddynamics analysis on a textured journal bearing, trying to determinate the texture effect on theloading carrying capacity. The paper considered hydrodynamic lubrication and used a multi-phase flow cavitation model for the analyzes. Cupillard and his team tested the textured designin a wide range of eccentricity ratios and discovered that it is possible to improve the loadingcarrying capacity for a specific range of eccentricity ratios. They also figured that the frictioncoefficient is reduced for deep dimples located in the regions of maximum pressure when theeccentricity ratio is high, but these effects depend on the operation conditions (CUPILLARD etal., 2008).
A theoretical research was conducted by Brizmer and Kligerman in 2012 to evalu-ate the potential of laser surface texturing on journal bearings. Optimum parameters were foundfor the dimples when seeking for high values of loading carrying capacity, analyzing both par-tial and fully textured bearings. Their results showed that, for fully textured bearings, there is adetrimental in the tribological properties of the bearings and, for the partial texturing, the load-ing carrying capacity increases for low eccentricities, but no real improvements were found forhigh eccentricities (BRIZMER; KLIGERMAN, 2012).
2.3 Solution Schemes
There are many types of solution for the lubrication models commented so far,each one with its benefits, trying to solve the equations using the least amount of time. Re-searchers try to avoid too many simplifications, in such way that the model becomes too simpleor to little (or too heavy with prohibited time consuming). The most common methods to solve
31
the Reynolds equations are:
Finite Element Method (FEM): A numerical method for the solution of boundary valueproblems involving ordinary and partial differential equations. This method was first de-veloped for civil engineering applications, but, with the work of mathematicians, a solidtheoretical base was created and it was further used in many different applications (POD-GORNIK et al., 2012; JACKSON; GREEN, 2008; FATU et al., 2006; KUCINSCHI etal., 2004; ZHANG, 2002; BITTENCOURT, 2015);
Finite Difference Method (FDM): Numerical method to obtain the approximate solu-tion of a partial differential equation. This method uses the Taylors series to expand thederivated function (ADJEMOUT et al., 2014; CUPILLARD et al., 2009; DOBRICA etal., 2006);
Finite Volume Method (FVM): Consists of approximating the integral form of a dif-ferential equation into discrete regions called finite volumes. The basic assumption ofthis method is taking a physical representation of the problem through a differentialequation (KANGO et al., 2014; FESANGHARY; KHONSARI, 2013; QIU et al., 2013;BRIZMER; KLIGERMAN, 2012; MA; ZHU, 2011; TALA-IGHIL et al., 2011; PROF-ITO et al., 2015).
Each of these methods have a specific characteristic that makes them preferable forsolving lubrication problems and they were analyzed by Woloszynski and colleagues, consid-ering the difference between analytical and numerical results, size of the mesh, load carryingcapacity and friction coefficient. Woloskynski and his team came up with the following con-clusions, when analyzing the methods for hydrodynamics bearing with and without surfacetexturing (WOLOSZYNSKI et al., 2013):
The finite element method gives good results, having the second best result when dealingwith textured surfaces;
The finite volume method has the best stability when mesh size parameters were changed,maintaining small difference in the results for pressure and load carrying capacity, forexample. Unfortunately, it was the method that presented the least accurate results;
The FDM showed fairly precise results for pressure and other parameters only when usedfor the inclined surface.
32
3 METHODOLOGY
The methodology used in this work was the gathering and replication of papersfrom the literature review, coming up with a new computational program capable of accuratelydescribe the behavior of dynamically loaded journal bearing, considering effects of roughness,temperature and surface textures. The articles were selected to improve the learning process,beginning with a hydrodynamic lubrication solution of the Reynolds equation and increasingits complexity, scaling up to consider surface roughness, temperature and texture effects.
3.1 Hydrodynamic Lubrication Modeling and Solution
The Reynolds equation is a second order partial differential equation that deals withthe pressure distribution in the fluid film. The equation is named after Osborne Reynolds, whowas the first person to describe the basic differential equation of fluid film, comparing his the-oretical predictions with experimental results (REYNOLDS, 1886). The Reynolds equation isobtained from the Navier-Stokes and mass conservation equations. Consider two lubricated sur-faces with the Cartesian system described in Figure 3.1.
Fig. 3.1 Coordinate system and velocity components for the Navier-Stokes equation (adaptedfrom (SANTOS, 1997)).
33
We can write the Navier-Stokes equation for this system as (SANTOS, 1997):
ux t
= px
+(
2 2uxx2
+ 2uxy2
+ 2ux z2
)+2
x
uxx
+ y
(uyx
+uxy
)+
z
(uzx
+ux z
)
uy t
= py
+(
2 2uyx2
+ 2uyy2
+ 2uy z2
)+2
y
uyy
+ x
(uyx
+uxy
)+
z
(uzy
+uy z
)
uz t
= p z
+(
2 2uzx2
+ 2uzy2
+ 2uz z2
)+2
z
uz z
+ x
(uzx
+ux z
)+
y
(uzy
+uy z
) (3.1)
where:x, y and z: Cartesian coordinatesux, uy and uz: velocity components: fluid density: fluid dynamic viscosityp: pressure of the fluid filmt: time variable
Now consider the following hypotheses:
The lubricant oil flow is laminar and therefore, the fluid inertia effects can be neglectedand
ux t
= 0, uy t
= 0, uz t
= 0. (3.2)
The velocity variations in the x and z directions are null, as illustrated in Figure 3.1.Therefore,
2uxx2
2ux
y2
2ux z2
;
2uyx2
2uyy2
2uy z2
;
2uzx2
2uz
y2
2uz z2
.
(3.3)
The pressure gradient and lubricant properties are negligible across the film thicknessdimension, since its too small when compared to the other dimensions (106). Therefore,
py
= 0, y
= 0. (3.4)
34
The product between the derivative terms of viscosity and velocity are small and negligi-ble. Consequently,
x
(uzx
+ux z
)= 0,
x
(uyx
+uxy
)= 0;
z
(uzx
+ux z
)= 0,
z
(uzy
+uy z
)= 0.
(3.5)
Using the described assumptions in Equation (3.1), we have:
px
= 2uxy2
;
py
= 2uyy2
;
p z
= 2uzy2
.
(3.6)
We can obtain the velocity profiles by integrating Equation (3.6) twice. Performingthe integrations for the x, y and z coordinates, we have:
ux =y2
2 px
+ c1y+ c2;
uy =y2
2 py
+ c3y+ c4;
uz =y2
2 p z
+ c5y+ c6;
(3.7)
where c1 through c6 are the integration constants. In order to determine these constants, considerthe Figure 3.2 describing the flow between the lubricated surfaces.
35
Fig. 3.2 Laminar flow between the two lubricated surfaces (adapted from (SANTOS, 1997)).
From Figure 3.2, we can define the components of velocity for y = 0
ux =U1, uy = 0, uz = 0. (3.8)
And for y = hux =U2cos Vysin;
uy =U2sin +Vycos;
uz = 0;
(3.9)
where:U1: speed of surface 1U2: speed of surface 2Vy = h t
As the curvature of the solid components is much greater, when compared with thefluid film thickness, the angle can be considered very small, thus:
cos = 1, sin tan = hx
, Vyhx
= 0. (3.10)
Considering that there is no slip between the fluid and the walls and substituting
36
Equations (3.8) and (3.9) in Equation (3.7), we have:
ux =1
2 px
(y2 yh)+ h yh
U1 +yh
U2;
uy =yh
Vy;
uz =1
2 p z
(y2 yh).
(3.11)
The fluid is considered a continuum medium and the continuity equation is defineby
DDt
+ div~u = 0. (3.12)
Considering that density is constant and substituting Equation (3.11) on Equation(3.12), we have
x
[1
2 px
(y2 yh)+(
h yh
U1 +yh
U2
)]+
y
(yh
Vy)+
z
[1
2 p z
(y2 yh)]= 0. (3.13)
Integrating Equation (3.13) on the y direction for a domain [0,h], we have
x
[1
2 px
(h3
3 h
3
2
)+
h h22h
U1 +h2
2hU2
]+Vy +
z
[1
2 p z
(h3
3 h
3
2
)]= 0. (3.14)
Considering surface 1 fixed, we have velocity U1 null. Rearranging and simplifyingEquation (3.14), we then have:
x
(h3
12 px
)+
z
(h3
12 p z
)
I
=U2
hx
II
+h tIII
. (3.15)
where U is the velocity of surface 2. Equation (3.15) is known as the Reynolds equation. Thefirst term (I) is the Poiseuille, or the pressure flow term, and deals with the lubricant flow due tothe pressure gradients. The second term (II) is the Couette term, also known as the wedge flowterm, and is responsible for the relative motion between the surfaces in contact and variations ofthe fluid density and surface velocities. Finally, the third term (III) is the normal squeeze termand deals with the transient effects of the film thickness.
37
We can now use the same assumptions to define the lubricant flows and shear stress.The lubricant flows for the x and z components can be defined as:
qx =h3
12 px
+hU ;
qz =h3
12 p z
.
(3.16)
The shear stresses in the lubricant are defined as:
1,2x =+12
px
(2yh)+ Uh
;
1,2z =+12
p z
(2yh).(3.17)
3.1.1 Film Thickness
The film thickness geometry of a journal bearing is a function of the nominal clear-ance (ho) and the axis eccentricity (e). The equation for the film thickness can be obtainedgeometrically from Figure 3.3, where R and R j are the bearing and journal radius respectively.
Fig. 3.3 Journal bearing geometry.
We know that for a typical journal bearing, the nominal clearance ho is much smallerthan R and R j and
AB OA, EF CE. (3.18)
38
Hence,
OB CF . (3.19)
We can now say that the film thickness h is
h() = EF = AB. (3.20)
Thus,
h = OBDE
= OB (CE CD)
= OBCE + ecos()
= RR j + ecos()
= ho + ecos().
(3.21)
We can use the following eccentricity relation:
= e/ho. (3.22)
Substituting Equation (3.22) in Equation (3.21) we have the film thickness equationfor a journal bearing i.e.,
h = ho(1+ cos). (3.23)
3.1.2 Film Cavitation
The Reynolds equation can provide the pressure profile on a journal bearing, butdepending on the operational conditions, these pressures can reach negative values in someregions of the fluid film. These negative values mean that the fluid film is under tension andis likely to be dissolved in that region. This cavity is filled with a biphasic mixture of liquidand gas/vapor (PROFITO et al., 2015). This process is known as cavitation or film rupture andaffects directly the performance of lubricated journal bearings. There are many models that tryto predict this phenomenon numerically, but this dissertation is going to focus in only three ofthem: Gmbel, Swift-Steiber and JFO models. Figure 3.4 represents a lubricated domain underthe cavitation effect, according to JFO approach, where the boundary between the full film andthe broken film is the rupture boundary (left boundary) and the boundary between the brokenfilm and the reconstructed film is the reformation boundary (right boundary).
39
Fig. 3.4 Fluid film cavitation on a lubricated domain. The inside region is a non cavitateddomain, where the fluid film is not broken and = 1. Outside that region is thecavitated domain, where the thin blue lines represent the gas/vapor mixture and where0 < 1 (adapted from (PROFITO et al., 2015)).
3.1.2.1 Gmbel
The Gmbel model was the first proposed procedure to deal with the cavitation inhydrodynamic pressure problems. The proposed model, also known as half-Sommerfeld model,consists of excluding all pressures values lower than a specific cavitation pressure, setting themto be equal to the cavitation pressure, as
p =
{pcav i f p < pcavp i f p pcav
(3.24)
The Gmbel model is probably the simplest way to deal with the cavitation problem,but it fails to enforce the mass conservation principle in the cavitation boundaries. Because ofthe abruptly break of the pressure film, the Reynolds conservative nature is not maintained.
3.1.2.2 Swift-Steiber
The Swift-Steiber model is an extension of the Gmbel model, assuming that thepressure gradient on the cavitation boundaries are null. Therefore,
{p = pcav (cavitation region)
p~n = 0 (cavitation boundary)
(3.25)
The Swift-Steiber model, also known as the Reynolds cavitation model, predictsaccurately the film rupture, since it enforces the mass conservation principle in that region. Themain issue with the Swift-Steiber cavitation model is that it is not capable of predicting the re-
40
formulation boundary with precision, due to the lack of mass conservation on the reformulationboundary of the fluid film.
3.1.2.3 JFO
The cavitation model proposed by Floberg and Jakobsson and later on improved byOlsson is a more accurate model to predict cavitation, specially for practical approaches, sinceit deals with the mass conservation on both rupture and reformulation boundaries. The mainassumptions proposed by them are (PROFITO et al., 2015):
The cavitation pressure inside the broken film region is constant and equal to the limitpressure (pcav);
Inside the cavitated region, the mixture liquid-gas/vapor flows in thin lines that are com-pletely separated by the vapor/gas phase;
On the reformulation and rupture boundaries, the mass conservation flow is enforcedusing the following complementary conditions:
h( 1)(
Un2
Wn)+
h3
12 p~n
= 0 (3.26)
where is the film fraction, ~n is the unit vector normal to the cavitation boundaries andUn and Wn are the components of the sliding and moving boundary velocities for a localdirection~n, respectively.
Figure 3.5 shows the difference between the cavitation zones for the three differentmodels, as presented by Khonsari (QIU; KHONSARI, 2009). There are other types of cavitationmodels, but the ones described here are the most common used. This dissertation is going touse the JFO complementary boundary conditions to solve the cavitation problem on journalbearings. The JFO boundary conditions are not simple to implement due to the fact that therupture and reformulation boundaries are not fixed and their location are not known beforehand.In order to implement the JFO boundary conditions, we are going to use the Elrod and Adamsalgorithm, proposing two ways to implement it, the gfunction and the p approaches.
41
Fig. 3.5 Pressure distributions along the center line of a dimple cell, comparing the three cav-itation models, as found by Khonsari (QIU; KHONSARI, 2009).
3.1.2.3.1 gFunction Model
Vijayaraghavan and Keith modified the Elrod algorithm in a way that it automat-ically changed the form of differentiation (central or upwind) of the shear flow terms in bothregions. Their method avoids the trial and error step used during the development of the El-rod algorithm. Consider the following form of the Reynolds equation (VIJAYARAGHAVAN;KEITH, 1989):
x
(h3
12 px
)+
z
(h3
12 p z
)=
U2
(h)x
+ (h)
t(3.27)
We can relate the fluid density with the film pressure using the definition of the bulkmodulus. The bulk modulus can be defined as:
= p
(3.28)
Elrod proposed a variable called fraction film content that made it possible to im-plement an universal partial differential equation that covers both the cavitated and full-filmregions. The film fraction variable proposed by Elrod is:
=c
(3.29)
42
where: : fraction film contentc: fluid density at the cavitation pressure
The fraction film removes the need to distinguish the boundaries of the cavitatedregion, working as an auxiliary variable that represents the proportion of lubricant at everypoint of the solution domain. However, it is necessary to implement a switch function, capableof making the partial differential equation consistent with the uniform pressure assumptionwithin the cavitated region. Elrods switch function is defined as (ELROD; ADAMS, 1974;ELROD, 1981; FESANGHARY; KHONSARI, 2011):
g =
{0 < 11 1
(3.30)
Using Equations (3.28), (3.29) and (3.30) in Equation (3.27), we have:
x
(h3g()
12x
)+
z
(h3g()
12 z
)=
U2
hx
+h t
(3.31)
Now solving Equation (3.31) for , we can obtain the pressure profile in the fluidfilm using the following equation:
p = pc + ln (3.32)
with pc is the cavitation pressure. Elrods algorithm is highly nonlinear, making it susceptible tonumerical instabilities and often causes results to diverge. Fesanghary and Khonsari proposeda modification to the Elrods switch function, aiming to improve the stability and convergencespeed of the Elrod algorithm (FESANGHARY; KHONSARI, 2011). Instead of using the binaryswitch function, they proposed the a continuous function implement in Algorithm 3.1:
43
Algorithm 3.1 Switch Function (FESANGHARY; KHONSARI, 2011)1: procedure SWITCH CODE2: if ((i, j) 1) then3: if gFactor > 0 then4: g(i, j) = g(i, j)/gFactor5: else6: g(i, j) = 1
7: end if8: else9: g(i, j) = g(i, j) x gFactor
10: end if11: if (g(i, j) > 1) then12: g(i, j) = 1
13: end if14: if (g(i, j) < 106) then15: g(i, j) = 0
16: end if
The gFactor is a constant that varies between zero and one and is predefined bythe user. The problem with the switch function approach is that, even if the stability can beimproved using the Fesanghary modified switch function, it still requires that the user defines avalue for the gFactor. For a dynamic problem this is not viable, since the optimum gFactor mayvary through the solution and possible instabilities may occur.
3.1.2.3.2 p Model
Similar with the gFunction model, the p model defines a variable as thefraction film content, which represents the proportion of lubricant oil present in the solutiondomain. The difference between the two models is that the p fixes a range of values for depending on the solution region, as follows:
with
{0 < 1, cavited region = 1, pressured region
(3.33)
Equation (3.33) denotes that the fluid density reduces to a percentage of insidethe cavitated regions (0 < 1). For the pressures regions, the film is complete and is equalto one. From these assumptions, substituting Equation (3.33) on equation (3.27) we have:
x
(h3
12 px
)+
z
(h3
12 p z
)=
U2
(h)x
+ (h)
t(3.34)
44
with the complementary conditions:
(p pcav)(1) = 0
{p > pcav = 1,p = pcav 0 < 1,
(3.35)
Different from the switch function model, the p model requires two sets ofpartial differential equation to be solved, one for the pressure profile and one for the fractionfilm content. Using two equations makes the solution more time consuming, but the advantageof this method is that it is more stable than using a single function to solve the entire problemand no longer requires a predefinition of a variable for every time period (gFactor). The pmodel is going to be used here for all the cases presented. The equations for the lubricant flowsand the shear stress need to be changed to take into account the fraction film . Assumingthat there is a linear correlation between the viscosity and the film fraction within the cavitatedregions, the equations for the fluid flows are:
qx =h3
12 px
+hU ;
qz =h3
12 p z
.
(3.36)
The shear stress components in the lubricant are now defined as:
1,2x =+ 12
px
(2yh)+ Uh
;
1,2z =+ 12
p z
(2yh).(3.37)
3.2 Mixed Lubrication Modeling Solution
The journal bearing can be submitted to high loads or reach low speeds during itsoperation. When this happens, the lubricant can start swifting from a hydrodynamic lubricationregime to a mixed lubrication one, where the surface topography have a much higher influenceon the film. When dealing with micro dimples, its important that the surface micro geometry aretaken into consideration, since they will change the solution of the film inside the dimples. Wecan separate the interactions between these topography features in two: the interaction betweenthe fluid and the surface of the bearing, which is the roughness influence on the lubricant flow,and the interaction between the imperfections on the bearing surface and the shell surface, whichis the roughness influence in terms of asperity contact.
3.2.1 Roughness Effect on Lubrication
Patir and Cheng proposed a modification in the Reynolds equation to consider gen-eral roughness patterns in its solution, dealing with the roughness effects statistically (PATIR;
45
CHENG, 1978). They proposed an average flow model, introducing flow factors that are ob-tained by solving the flow numerically, for different roughness configurations (PATIR; CHENG,1978; PROFITO et al., 2015).
Fig. 3.6 Total film thickness hT between two surfaces with roughness 1 and 2, with a nomi-nal film thickness h (PROFITO et al., 2015).
Figure 3.6 shows the definition of film thickness between two rough surfaces (hT )as
hT = h+R (3.38)
where R is the sum of the roughness amplitude of surfaces 1 and 2, with a standard deviationR. The influence of the surface roughness on lubrication is defined by:
=h
R(3.39)
For values of lower than 3, it generally means that the roughness from the twosurfaces are starting to interact with each other. When has values greater than 3, it means thatthe fluid film is thickness enough to separate the two surfaces without causing the roughnessto affect the lubrication. The Patir and Cheng model writes the unit lubricant flows in terms ofpressure p(x,z) and shear sx flow factors as (PATIR; CHENG, 1978; PROFITO et al., 2015):
qx =pxh3
12 px
+hTU sxUr;
qz =pzh3
12 p z
.
(3.40)
46
The first term in both expression of Equation (3.40) is the flow due to the pressuregradients, where the pressure flow factors p(x,z) are correction factors relating the Poiseuilleflow in rough surfaces with that of smooth ones. An additional term is introduced to deal withthe flow due to the relative motion between the rough surfaces contact, which can be related withthe combined perturbation effect due to the superficial roughness and sliding onto the fluid flow.We can now write the modified Patir and Cheng Reynolds equation with the mass-conservingcavitation model as (PROFITO et al., 2015):
x
(pxh3
12 px
)+
z
(pzh3
12 p z
)=
U2
(h)x
Ur2
(sx)x
+ (h)
t(3.41)
The pressure and shear flow factors are obtained numerally, performing simulationsin small rectangle domains which have a microscopic scale. The results are collected in curvesas functions of the fluid film thickness ratio and the surface pattern . Here we are going toassume that the parameter has a fixed value of 1/9, which relates to a transversely orientedroughness pattern. Therefore, the pressure flow factors and shear flow factor are calculated asfollowing (PROFITO et al., 2015):
px() =
{11.48e0.42, 11+1.480.42, > 1
(3.42)
py() =
{10.87e1.5, 11+0.871.5, > 1
(3.43)
sx() =
{2.051.12e0.78+0.032 , 51.86e0.25 , > 5
(3.44)
We can write the friction pressure f p(x,z) flow factor and friction shear f sx flowfactor using the following equation (PROFITO et al., 2015):
f px() = 11.51e0.52 (3.45)
f py() = 10.73e0.91 (3.46)
f sx() = 14.12.45e2.3+0.1
2(3.47)
47
The hydrodynamic shear stresses acting in each rough surface can be defined interms of friction pressure and shear flow factors as:
1,2x =+2
px
(2yh) f px +Uh
( f + f sx);
1,2z =+2
p z
(2yh) f pz(3.48)
where f is the friction contact factor that comes from the average sliding velocities in the shearstress definitions for rough contacts. Assuming a normalized probability function, we can definethe friction contact factor as
f =2
es22+ s
ds (3.49)where s is the normalized roughness. We can find the solution of Equation (3.49) using thefollowing curved-fitted expression (PROFITO et al., 2015):
f () =
3532z[(1 z2)3 ln
(z+1109
)+ za
], 3
3532z[(1 z2)3 ln
( z+1z1)+ zb
], < 3
(3.50)
where:z = 3za = 160 [55+ z(132+ z(345+ z(160+ z(405+ z(60+147z)))))]
zb = z15[66+ z2(30z2 80)
]3.2.2 Asperity Contact
The contact between the microscopic irregularities on the bearing surfaces are calledasperity contact. This type of contact happens because of the natural roughness present in eachsurface. One of the first researchers to work with the asperity contact problem were Greenwoodand Williamson. They extended the Hertz theory of contact for a single asperity using a Gaus-sian distribution, spreading the contact in the whole surface. A few years later, Greenwood andTripp improved the previous model, considering two rough surfaces and the misalignment be-tween asperities. This dissertation is going to use the Greenwood-Tripp model to determine thepressure due to the contact of the asperities on non smooth bearings because of its simplicity,robustness and easiness to implement.
3.2.2.1 Greenwood-Williamson
The following assumptions are considered when working with the Greenwood Williamsonasperity contact model (PROFITO et al., 2015; GREENWOOD; WILLIAMSON, 1966):
48
The contact between two flat parallel rough surfaces is assumed as the contact between arigid smooth plane and a combined rough topography;
The rough surface has an asperity height distribution randomly placed on it, describedaccording to their respective probability density functions;
The asperities have a spherical shape with constant mean radius;
The asperities deformations and displacements are small, so their influence on the neigh-bors asperities is neglected;
The contact of each asperity is modeled as a Hertzian contact.
Fig. 3.7 The main surface variables on the Greenwood-Tripp model (PROFITO et al., 2015).
The smooth and rough surfaces with asperities in contact are illustrated in Figure3.7. The probability of a given asperity to be in contact with the rigid plane can be defined as(PROFITO et al., 2015):
prob(z > h) =
h(z)dz (3.51)
where h is the mean surfaces separation defined as
h = hZs (3.52)
49
For a N number of asperities on the rough surface, the total number of asperities incontact (nasp) can be expressed as a function of h as
nasp(h) = N
h(z)dz (3.53)
Assuming that the real contact area is the sum of all micro areas resulted by thelocal deformations at every asperity interaction, the total real contact area (Aasp ) as a functionof h is
Aasp(h) = sN
h[(zh)(z)]dz (3.54)
where s is the mean asperity radius of the combined rough surface. Just like the real contactarea, we can write the total contact load as the sum of all micro loads acting on every asperityinteraction. We can then write the total contact load Wasp as function of h. Therefore,
Wasp(h) =43
E 1/2s N
h[(zh)3/2(z)]dz (3.55)
where E is the combined elastic modulus. We can generalize Equations (3.53) through (3.55)by considering the following variables:
s =NA0
(3.56)
hasp =hZs
s(3.57)
where A0 is the nominal contact area, hasp is the normalized mean surfaces separation, s theasperity density of the combined rough surface, Z is the mean asperity height of the combinedrough surface and s is the standard deviation of the asperity heights of the combined roughsurface. We can now write the normalized form of Equations (3.53) through (3.55) as:
nGWasp (hasp) = sA0F0(hasp) (3.58)
AGWasp (hasp)