Numerical modeling of surface texturing on automotive crankshaft central bearings : Modelagem numérica de texturização superficial em mancais centrais de virabrequim automotivo

Faculdade de Engenharia Mecânica

GABRIEL WELFANY RODRIGUES

Numerical Modeling of Surface Texturing on

Automotive Crankshaft Central Bearings

Modelagem Numérica de Texturização Superficial

em Mancais Centrais de Virabrequim Automotivo

CAMPINAS

2019

Numerical Modeling of Surface Texturing on

Automotive Crankshaft Central Bearings

Modelagem Numérica de Texturização Superficial

em Mancais Centrais de Virabrequim Automotivo

Dissertation presented to the School of Me-chanical Engineering of the University of Campinas in partial fulfillment of the require-ments for the degree of Master in Mechanical Engineering, in the area of Solid Mechanics and Mechanical Design.

Dissertação apresentada à Faculdade de Engenharia Mecânica da Universidade Estad-ual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestre em Engenharia Mecânica, na Área de Mecânica dos Sólidos e Projeto Mecânico.

Orientador: Prof. Dr. Marco Lucio Bittencourt

ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DIS-SERTAÇÃO DEFENDIDA PELO ALUNO GABRIEL WELFANY RODRIGUES, E ORIENTADA PELO PROF. DR. MARCO LUCIO BITTENCOURT.

CAMPINAS 2019

Biblioteca da Área de Engenharia e Arquitetura Rose Meire da Silva - CRB 8/5974 Rodrigues, Gabriel Welfany,

R618n RodNumerical modeling of surface texturing on automotive crankshaft central bearings / Gabriel Welfany Rodrigues. – Campinas, SP : [s.n.], 2019.

RodOrientador: Marco Lúcio Bittencourt.

RodDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.

Rod1. Lubrificação. 2. Tratamento de superfícies. 3. Mancais. 4. Atrito. 5. Análise numérica. I. Bittencourt, Marco Lúcio, 1964-. II. Universidade Estadual de Campinas. Faculdade de Engenharia Mecânica. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Modelagem numérica de texturização superficial em mancais

centrais de virabrequim automotivo

Palavras-chave em inglês: Lubrication Surface treatment Journal bearings Fricton Numerical analysis

Área de concentração: Mecânica dos Sólidos e Projeto Mecânico Titulação: Mestre em Engenharia Mecânica

Banca examinadora:

Marco Lúcio Bittencourt [Orientador] Gregory Bregion Daniel

Roberto Federico Ausas

Data de defesa: 22-02-2019

Programa de Pós-Graduação: Engenharia Mecânica

Identificação e informações acadêmicas do(a) aluno(a)

- ORCID do autor: https://orcid.org/0000-0001-8730-1217

- Currículo Lattes do autor: http://lattes.cnpq.br/9329954652334011

FACULDADE DE ENGENHARIA MECÂNICA

COMISSÃO DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA

DEPARTAMENTO DE SISTEMAS INTEGRADOS

DISSERTAÇÃO DE MESTRADO ACADÊMICO

Numerical Modeling of Surface Texturing

on Automotive Crankshaft Central

Bearings

Modelagem Numérica de Texturização

Superficial em Mancais Centrais de

Virabrequim Automotivo

Autor: Gabriel Welfany Rodrigues

Orientador: Prof. Dr. Marco Lucio Bittencourt

A Banca Examinadora composta pelos membros abaixo aprovou esta Dissertação:

Prof. Dr. Marco Lucio Bittencourt FEM/UNICAMP

Prof. Dr. Gregory Bregion Daniel FEM/UNICAMP

Prof. Dr. Roberto Federico Ausas ICMC/USP

A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.

Agradeço, primeiramente a Deus, aos meus pais, Francislene e Valmir, avós, Irene e João (in memoriam) e ao tio Jairo por me darem suporte em todos o momentos e possibilitarem que eu chegasse até aqui.

Ao professor Dr. Marco Lúcio Bittencourt por proporcionar-me a oportunidade de fazer parte desta pesquisa, por toda sua atenção e dedicação como professor, orientador, pela paciência, pelo auxilio no desenvolvimento deste trabalho e por contribuir com seu grande co-nhecimento para a minha formação como profissional, muito obrigado professor.

Não poderia deixar de mencionar o colega de laboratótio Jonatha, que mesmo dis-tante sempre foi atencioso em responder dúvidas e me ajudar no prosseguimento do projeto. Sem sua ajuda, não seria possível desenvolver a pesquisa até aqui.

Um agradecimento especial ao meu amigo de laboratório, Geovane, por me auxiliar em momentos difíceis da pesquisa. Muito obrigado pelas conversas, auxilio e dúvidas respon-didas.

Agradeço a todos do laboratorio por todo apoio, conversas, sugestões e conselhos. Por tornarem esse período mais agradável, pelas dicussões e informações compartilhadas, pelo suporte e por fazerem do laborátorio um ambiente melhor pricipalmente nos momentos de difi-culdade.

Também agradeço à UNICAMP e ao Programa de Pós-Graduação em Engenharia Mecânica pela oportunidade de ingresso.

Muito obrigado a todos que de alguma forma, direta ou indiretamente, contribuíram para a realização deste trabalho.

Embora mecanismos energeticamente mais eficientes sejam constantemente bus-cados na engenharia, leis ambientais têm estabelecido metas desafiadoras, o que aumenta a demanda por novas soluções. Uma grande quantidade de energia fornecida pelo combustível em motores de combustão interna é simplesmente perdida. O atrito tem papel importante nesse aspecto e mancais representam cerca de 25% da energia perdida devido ao atrito em motores. Abordagens em diversas áreas têm sido tomadas para evitar essas perdas, como lubrificantes e aditivos de baixa viscosidade, revestimentos de superfície avançados e texturas superficiais. Considerando isso, a presente pesquisa pretende modelar numericamente a texturização super-ficial aplicada ao mancal central do virabrequim de um motor de três cilindros para melhorar aspectos tribológicos, como a redução do coeficiente de atrito. Os regimes de lubrificação e a texturização da superfície são abordados em uma revisão de literatura. Um código computa-cional em ambiente Matlab é usado para simular a texturização superficial em situações de carregamento dinâmico, incluindo efeitos como rugosidade, contato de aspereza e deformação elástica no caso de lubrificação elastohidrodinâmica. Simulações foram realizadas para enten-der o efeito da texturização aplicada em diferentes regiões do mancal e consienten-derando condições operacionais distintas como velocidade rotacional do eixo. Os resultados mostraram que exis-tem regiões onde a textura é mais adequada e que é possível melhorar aspectos tribológicos. Contudo, a textura pode ter efeitos indesejados dependendo de onde é aplicada, por exemplo, um incremento no pico de pressão indicando a perda de sustentação hidrodinâmica.

Palavras-Chave: Lubrificação; Texturização Superficial; Lubrificação Mista; Lubrificação Elas-tohidrodinâmica; Mancais; Modelo de Cavitação com Conservação de Massa; Equação de Reynolds.

Although more energetically efficient mechanisms are constantly pursued in engineering, envi-ronmental laws have pushed the goals further, which increases the demand for new solutions. A great amount of the energy provided by the fuel in internal combustion engines is simply lost. Friction has an important role in this aspect and bearings represent about 25% of the energy lost due to friction in engines. Developments in several areas have been taken to avoid these losses such as low-viscosity lubricant and additives, advanced surface coatings and surface tex-turing. Considering this context, the present research intends to numerically model the surface texturing applied to the main crankshaft bearing of a three-cylinder engine to improve tribolog-ical aspects such as the friction coefficient. The lubrication regimes and the surface texturing are covered in the literature review. A computational code in Matlab environment is used to simulate the surface texturing for a dynamic loaded condition, including effects such as rough-ness, asperity contact, and elastic deformation in the case of elastohydrodynamic lubrication. Simulations were carried out to understand the texturing effect applied to different regions of the bearing and considering distinct operational conditions such as the journal rotational speed. The results showed that there are regions where the texture is more appropriate and that is pos-sible to improve tribological aspects. However, texture can have undesired effects depending on where it is applied, for instance, an increment on the pressure peak indicating the loss of hydrodynamic lift.

Keywords: Lubrication; Surface Texturing; Mixed-Lubrication; Elastohydrodynamic Lubrica-tion; Journal Bearings; Mass-Conserving Cavitation Model; Reynolds Equation.

Figure 1.1 – Breakdown of passenger car energy consumption (HOLMBERG et al., 2012). 19 Figure 3.1 – Hydrodynamic radial bearing. . . 31 Figure 3.2 – Lubrication regimes: (a) hydrodynamic and elastohydrodynamic

lubrica-tion; (b) mixed-lubricalubrica-tion; (c) boundary lubrication, adapted from Hamrock et al.(2004). . . 32 Figure 3.3 – Conformal and non-conformal geometry (adapted from ASM (1992)). . . . 33 Figure 3.4 – Friction coefficient versus relative velocity of the lubricated surfaces (adapted

from NORTON (2004)). . . 33 Figure 3.5 – Wedge effect auto-pressurization mechanism (adapted from Izuka et al. (2009)). 34 Figure 3.6 – Squeeze effect auto-pressurization mechanism (adapted from Izuka et al.

(2009)). . . 35 Figure 3.7 – Representation of the journal bearing coordinate system (a) isometric view;

(b) planned superior view. . . 38 Figure 3.8 – Representation of the fluid film cavitation for a generic lubricated domain

(adapted from Profito (2015)). . . 39 Figure 3.9 – Schematic representation of the journal bearing geometry (adapted from

Profito (2015)). . . 42 Figure 3.10–Representation of the film thickness hT considering roughness of surfaces 1

(δ1) and 2 (δ2) for a nominal film thickness h (adapted from Profito (2015)). 43

Figure 3.11–Longitudinal, isotropic and transversally oriented surfaces contact areas (PROF-ITO, 2015). . . 44 Figure 3.12–The main surface variables on the Williams and

Greenwood-Trip model (adapted from Profito (2015)). . . 46 Figure 3.13–The main surface variables on the Williams and

Greenwood-Trip models (adapted from Profito (2015)). . . 47 Figure 3.14–Journal bearing sliding surface domain (adapted from Reis (2017)). . . 48 Figure 3.15–Finite difference mesh for two-dimensional problems. . . 49 Figure 3.16–Representation of parallel sliding surfaces with a protrusion (a) and the

ef-fect of the protrusion on the hydrodynamic field for low and sufficiently high velocity (b) (ETSION, 2013). . . 54 Figure 3.17–Textured journal bearing (a), detail of the dimple geometry (b) and the

tex-tured surface (c) (adapted from Kango et al. (2014)). . . 55 Figure 3.18–Texture cell (adapted from Gropper et al. (2016)). . . 56 Figure 3.19–Dimple influence over the film thickness. . . 57 Figure 4.1 – Representation of the full and the reduced finite element model (adapted

from Bittencourt (2014)). . . 62

Figure 4.3 – Reduced model element face representation(adapted from Profito (2015)). . 63

Figure 4.4 – Geometry of the rotor-bearing system (LIU et al., 2010). . . 65

Figure 4.5 – Results achieved for the rotor-bearing system by Liu et al. (a) pressure dis-tribuition of the oil film, (b) vertical displacement of the bearing inner sur-face (LIU et al., 2010). . . 66

Figure 4.6 – Results achieved for the rotor-bearing system using the code developed.(a) pressure distribuition of the oil film, (b) vertical displacement of the bearing inner surface. . . 66

Figure 4.7 – Computational routine flowchart. . . 68

Figure 5.1 – Three-cylinder engine crankshaft. . . 69

Figure 5.2 – Load components generated from the pressure curve for 1750 rpm. . . 71

Figure 5.3 – Load components generated from the pressure curve for 2500 rpm. . . 71

Figure 5.4 – Load components generated from the pressure curve for 4000 rpm. . . 72

Figure 5.5 – Maximum pressure for a dynamically loaded journal bearing using different meshes. . . 73

Figure 5.6 – Pressure peak convergence curve. . . 73

Figure 5.7 – Maximum pressure peak considering different values of σr for 1750 rpm. . . 75

Figure 5.8 – Mean friction coefficient for different values of σrfor 1750 rpm. . . 75

Figure 5.9 – Journal center position for 1750 rpm (a), 2500rpm (b) and 4000 rpm (c) in polar and Cartesian coordinates. . . 76

Figure 5.10–Power losses for one engine cycle at 1750 rpm, 2500 rpm and 4000 rpm. . . 77

Figure 5.11–Fluid film maximum hydrodynamic pressure for one engine cycle at 1750 rpm, 2500 rpm and 4000 rpm. . . 77

Figure 5.12–Arrangement of dimples in the circumferential and axial directions of the journal bearing surface (KANGO et al., 2014). . . 78

Figure 5.13–Journal bearing divisions into 6 regions of 60 degrees. . . 79

Figure 5.14–Fluid film thickness distribution of a textured journal bearing surface for two simulated cases. Fully (a) and partially (b) textured in the axial direction. . . 79

Figure 5.15–Textured regions analyzed: top view. . . 80

Figure 5.16–Percentage changes of the total power loss and maximum pressure peak compared to the bearing with no texture for 1750 rpm. . . 81

Figure 5.17–Percentage changes of the total power loss and maximum pressure peak compared to the bearing with no texture for 2500 rpm. . . 82

Figure 5.18–Percentage changes of the total power loss and maximum pressure peak compared to the bearing with no texture for 4000 rpm. . . 83

for different dimple radius for 2500 rpm . . . 85 Figure 5.21–Total power loss on the dynamically loaded journal bearing surface for

dif-ferent dimple radiuses for 2500 rpm. . . 85 Figure 5.22–Mean friction coefficient on the dynamically loaded journal bearing for

dif-ferent dimple depth for 2500 rpm. . . 86 Figure 5.23–Total power loss on the dynamically loaded journal bearing surface for

dif-ferent dimple depths for 2500 rpm. . . 87 Figure 5.24–Oil film thickness distribution for the journal bearing fully textured. . . 87 Figure 5.25–Oil film thickness distribution for the journal bearing textured in the central

regions. . . 88 Figure 5.26–Power saved comparison of the non-textured with the textured journal

bear-ing surfaces for different rotational speeds and one engine cycle. . . 90 Figure 5.27–Hydrodynamic pressure profile at 200 crankshaft degree for the rigid bearing

isometric view (a), and top view (b) for 2500 rpm. . . 91 Figure 5.28–Hydrodynamic pressure profile at 200 crankshaft degree for the flexible

bearing isometric view(a), and top view (b) for 2500 rpm. . . 92 Figure 5.29–Hydrodynamic film fraction field at 200 crankshaft degree for the rigid (a)

and flexible (b) bearing for 2500 rpm. . . 92 Figure 5.30–Top view of the texture distribution along to the elastic journal bearing. . . . 95

Table 4.1 – Input parameters for the EHL calculation. . . 65

Table 5.1 – Parameters used to calculate the loads for each pressure curve. . . 70

Table 5.2 – Parameters for the dynamically loaded journal bearing simulation. . . 70

Table 5.3 – Surface parameters. . . 74

Table 5.4 – Pressure peak percentage deviation considering different roughness for 1750 rpm. . . 75

Table 5.5 – Texture geometric parameters. . . 78

Table 5.6 – Surface parameters. . . 80

Table 5.7 – Geometrical parameters for dimples considering different radiuses. . . 84

Table 5.8 – Geometrical parameter for dimples considering different radius. . . 86

Table 5.9 – Dimple geometric parameters for the fully textured journal bearing surface. . 87

Table 5.10–Maximum pressure, total power loss, mean friction for different rotation speed and full texturing. . . 88

Table 5.11–Dimple geometrical parameters for the fully textured journal bearing surface. 89 Table 5.12–Maximum pressure, total power loss, mean friction for different rotation speed and texture in the central regions. . . 89

Table 5.13–Bearing material properties. . . 90

Table 5.14–Hydrodynamic pressure peak, power loss and main friction coefficient for the rigid and elastic bearings. . . 93

Table 5.15–Hydrodynamic pressure peak, power loss, and main friction coefficient for different lubricant dynamic viscosity. . . 94

Table 5.16–Hydrodynamic pressure peaks, power losses, and main friction coefficients for the textured and non-textured elastic bearings. . . 96

Latin Letters

a, b, c Dynamic viscosity thermal constants x, y, z Cartesian axis coordinates

u_x, u_y, u_z Local velocity components rx, ry, rz Elliptical dimple radius

xc, yc, zc Elliptical dimple origin center

U,W Rotational speeds on the x and y axes

C_p Specific heat

h₀ Radial clearance

h Film thickness

h_asp Normalized mean surfaces separation h_T Film thickness considering rough surfaces h_tex Film thickness considering textures

t Time

e Eccentricity

ex, ey Eccentricity Cartesian coordinates

k Successive over relaxation iteration step q_x, qz Lubricant film flow

P_cav Cavitation pressure P_amb Ambient pressure

n_x, n_z Number of dimples in x and z direction, respectively A₀ Nominal contact area

P Hydrodynamic pressure

T Temperature in Kelvin T_◦C Temperature in Celsius E Combined elastic modulus N_x, N_z Number of integration points N_aspGT Number of Asperities

AGT_asp Real contact area pGT_asp Contact pressure L_x Bearing thickness

Lz Bearing length

L_tx, L_tz Length of dimple cell W_extx ,W_exty External loads

Fx, Fy Calculated loads

F_TotalHydro Resultant hydrodynamic force

Z_s Mean asperity height of combined rough surfaces A_cell Cell texture superficial area

A_texture Texture superficial area

x_textured, ztextured Portion occupied by the textures in direction x and z

M, D, K Mass, damping, and stiffness matrix

M_r, D_r, K_r Reduced mass, damping, and stiffness matrix

F Nodal force vector

q Hydrodynamic pressure vector f_re Load vector of each element face n_e Normal unit vector

Tr Transformation reduced matrix

p₁, p₂, p₃, p₄ Pressure at each node

A Global matrix of the reduced model p_h Reduced model global pressure vector

Greek Letters

µ Dynamic viscosity

µs Kinematic viscosity

µf riction Friction coefficient

ρ Lubricant density

ρc Lubricant density on cavitation pressure

ρtexture Texture density

τ Shear stress

˙

γ Shear stress rate

ε Eccentricity ratio

βs Mean asperity radius of combined rough surfaces

σr Standard deviation of combine roughness amplitude

σs Standard deviation of asperity heights

ηs Asperity density

Ψ Power loss

ωj, ωb Journal and bearing friction torque

ωp Over relaxation parameter for the pressure

ωθ Over relaxation parameter for the film fraction

ωEHL Under relaxation parameter for the EHL solution

δr Sum of the surface roughness amplitude

δr, ˙δr, ¨δr Reduced displacement, velocity, and acceleration vectors

ξp Pressure tolerance

ξ_θ Film fraction tolerance

ξx Tolerance for the support load on the x axis

ξy Tolerance for the support load on the y axis item[ξEHL] Tolerance for the

EHL solution

Λ Ratio between fluid film thickness and surface roughness

φp(x,z) Patir and Cheng pressure flow factors

φs(x) Patir and Cheng shear flow factors

φf p(x,z) Patir and Cheng friction pressure flow factors

φf sx Patir and Cheng friction shear flow factors

1 Introduction . . . 19 1.1 Objectives . . . 20 1.2 Dissertation Outline . . . 21 2 Literature Review . . . 22 2.1 Lubrication . . . 22 2.2 Surface Texturing . . . 26

3 Lubrication and Surface Texturing Modeling . . . 31

3.1 Hydrodynamic Radial Bearings . . . 31

3.2 Lubrication Regimes . . . 32

3.3 Pressurizing Mechanisms . . . 34

3.4 Lubricants . . . 35

3.5 Hydrodynamic Lubrication Model . . . 36

3.5.1 Fluid Film Cavitation . . . 37

3.5.2 Modified Reynolds Equation (p − θ Model) . . . 40

3.5.3 Fluid Film Thickness . . . 41

3.6 Mixed Lubrication Model . . . 42

3.6.1 Modeling of surface roughness effects . . . 43

3.6.2 Surface asperity contact modeling . . . 46

3.7 Solution of the Reynolds Equation . . . 48

3.7.1 Finite Difference Solution of the Reynolds Equation . . . 48

3.8 Calculated Characteristics . . . 52

3.9 Surface Texturing . . . 53

4 Conformal Elastohydrodynamic Lubrication (EHL) Modeling and Computa-tional Routine . . . 58

4.1 Finite Element Model for the Elastic Bearing . . . 58

4.1.1 Solution of the Reduced Model . . . 60

4.1.2 Calculating the Force Vector . . . 61

4.2 Algorithm for the EHL Solution . . . 63

4.3 EHL Validation . . . 63

4.4 Computational Routine . . . 67

5 Results . . . 69

5.1 Input Parameters . . . 70

5.2 Mesh Analysis . . . 72

5.3 Roughness and Asperity Effects . . . 74

5.4 Surface Texturing . . . 76

5.4.3 Full Texturing . . . 86

5.4.4 Textures in the Central Regions . . . 88

5.5 Elastohydrodynamic Effects . . . 90

5.5.1 Rigid and Elastic Bearings . . . 90

5.5.2 Dynamic Viscosity Effects . . . 93

5.5.3 Elastic Bearing with Textures . . . 94

6 Conclusion . . . 97

6.1 Suggestions for Future Researches . . . 100

1 INTRODUCTION

Concerning about the impact of human-being actions on the environment is not a new subject, but it is still growing since some effects are already evident. The control of greenhouse-gases production has increased worldwide, governments have established regula-tions and goals to reduce the current level of emissions. Therefore, developments in several areas of study have been taken to minimize the emission of polluting gases in the atmosphere. In this context, the automotive industry has been pushed forward to improve automobile effi-ciency since the fuel consumption represents a major source of the carbon dioxide and other polluting gases. Even though new technologies allowed the reduction of the energy lost due to friction in modern engines, the improvement of engine efficiency and the reduction of para-sitic energy losses have a great potential still unexplored regarding the conservation of energy. In this aspect, friction and lubrication play an important role. According to Holmberg et al. (2012), about 21.5% of the total fuel energy is actually used to move cars. From the fuel energy, 38% is converted into mechanical power. However, 33% of this energy is lost due to friction. Figure 1.1 shows the fuel energy dissipation of a passenger car with a speed of 60 km/h.

Fig. 1.1 – Breakdown of passenger car energy consumption (HOLMBERG et al., 2012).

Considering the search for reducing the power loss in automobiles, the enhance-ment of engine efficiency is basically achieved by improving the mechanical design, configura-tion of the major components, surface engineering and coatings, and by lubricant and additive technologies aiming the reduction of wear and friction. The tribology field comprehends the understanding of the principles of friction, wear, and lubrication, which involves the study of the interacting surfaces in relative motion. In this context, tribologists all over the world have been working in reducing power loss by developing more environmental friendly mechanisms. Components such as connecting-rod, main crankshaft bearing, piston and piston-ring liner

in-terfaces are the focus of many researches on roughness surface modification, better material properties, advanced lubricant oils based on the friction and viscosity modifiers.

The modification of the contact surface of mechanical components is made by coat-ing and/or texturcoat-ing. The first is associated with a reduction of wear due to the abrasive contact protection offered by the material used. Surface texturing, which is generally produced by in-serting regular geometric entities on the surface of a given component, is associated with a gain on the hydrodynamic lifting during the film lubrication enhancing tribological properties. Although many studies have been performed in this area, questions about the best parameters to the surface texturing still remain. Relative velocity, lubricant properties, surface profiles and roughness are some characteristics that influence the surface texturing performance. In this as-pect, researches have shown a wide range of results that diverge in some situations and converge in others, which is a result of the different models and procedures adopted.

It is known that computational simulation and mathematical modeling constitute a powerful technique for the development, research and understanding of new components and technologies. It is used to solve and reproduce the actual behavior of complex systems. More-over, numerical simulations have positive aspects such as cheaper than experimental tests and a great number of different cases can be tested to establish patterns. It is possible to optimize and select appropriate solutions for the problem in study. Consequently, considering that there are many parameters that influence the performance of surface texturing in enhancing tribological properties, numerical simulation is a valuable tool.

This work aims, through mathematical modeling and numerical simulation tools, to predict the effects of surface texturing applied on a main crankshaft bearing of a three-cylinder engine to study its performance in terms of power loss and friction coefficient for dynamic loaded condition. In addition, it is intended to understand the influence of texture geometric parameters and distribution along the journal bearing surface. Therefore, this dissertation inves-tigates the possibility to improve the vehicle efficiency by improving tribological properties of journal bearings using surface texturing.

1.1 Objectives

The main purpose of this dissertation is to study the effects of surface texturing applied to automotive engine crankshaft bearings, aiming to investigate tribological properties such as the friction coefficient and hydrodynamic lifting. The contents of the analysis here proposed are listed bellow:

• Use of a comprehensive mathematical model, with a numerical procedure solution aiming to numerically describe the behavior and tribological performance of lubricated automo-tive crankshaft bearings subjected to dynamically loaded conditions;

• Numerical solution of the Reynolds equation with a mass-conservative cavitation model; • Numerical application of textures on the journal bearing surface, investigation of its influ-ence for different texture geometric parameters and different surface texturing distribution along the journal bearing surface;

• Consider the elastic deformation caused by the hydrodynamic pressure (EHL) where a partitioned solution for conformal contacts (journal bearings) is considered. The hydro-dynamic and structural problems are solved separately;

• Analytical discussion of the results obtained.

1.2 Dissertation Outline

This dissertation is organized as follows:

• Chapter 1 gives the motivation and a brief presentation of the approaches and challenges regarding to more efficient internal combustion engines followed by the objectives of this study.

• Chapter 2 is a literature review of important works that summarizes the evolution of the lubrication theory and the surface texturing development along time.

• Chapter 3 explores the lubrication solution model, the lubrication regimes are discussed as well the cavitation phenomenon. In addition, it presents the modeling methodology to consider surface texturing.

• Chapter 4 describes the approach and methodology considered to take into account the elastic deformation. The computational procedure adopted in this work is also given in this chapter.

• Chapter 5 presents the results achieved by the numerical simulations performed and dis-cusses the outcomes.

• Chapter 6 summarizes the model and results obtained and presents the conclusions. The last part of this chapter is dedicated to present suggestions for future works.

2 LITERATURE REVIEW

The human history is made of constant knowledge evolution. Computers, mechan-ical machines, electronic components are always evolving to adapt, make life simpler or solve a problem. In this context, the automotive industry has a great interest in improving the ef-ficiency of automobiles; scientists have been pushing forward the barriers of engineering by pursuing more reliable and efficient components keeping in mind the growing environmental concerning (WONG; TUNG, 2016). Power loss reduction is a key aspect to achieve these goals. There are many paths to follow such as downsizing, working with more efficient combustion systems, friction loss reduction among others extensively explored fields. It is significant the number of papers addressing the surface modification of mechanical components as a way to improve tribological properties (MA; ZHU, 2011; TALA-IGHIL et al., 2011; SUDEEP et al., 2013; ARSLAN et al., 2015; TALA-IGHIL; FILLON, 2015).

Reducing wear and friction in mechanical components with relative movement is essential to reach an adequate lifetime for a given machine part. Lubricants is mainly used to achieve such objective. They may be liquids, solids, greases or gases applied to reduce friction and wear. It is necessary to have a good understanding of physical and chemical interactions to get satisfactory results in terms of wear and friction reduction (HAMROCK et al., 2004). The technique of using lubricants to reduce friction is called lubrication and tribology is the field that studies it.

Surface texturing is the process of modifying a given mechanical component sur-face to achieve better tribological performance. Textures can have various geometric shapes, but dimples are the most common type of surface texturing and studied in many papers. The enhancement of the tribological behavior is generally associated with the micro-dimple func-tioning as oil micro-reservoirs, providing lubricant in case of mixed or boundary lubrication; micro-traps for wear particles; and the generation of lift, also called the micro-bearing effect (ARAUJO et al., 2004; LU; KHONSARI, 2007; QIU; KHONSARI, 2011a; PROFITO, 2010; REIS, 2017).

The review here presented intends to be a synthesis of the progress over the years about lubrication and surface texturing to improve tribological properties. Some important works and recent findings related to these topics are presented.

2.1 Lubrication

The hydrodynamic lubrication starts with Tower’s studies in 1883 noticing the pres-ence of a pressure field in the lubricant by measurements of the pressure in experimental analy-ses. These investigations also reported that the viscous dissipation in the lubricant film is

propor-tional to the operapropor-tional speed (TOWER, 1883). Petrov reaffirmed this conclusion by measuring the friction and concluding that the dynamic viscosity has an important influence in friction the coefficient (PETROV, 1883). However, these first studies were basically experimental. Only in 1886, the first mathematical model to describe the fluid flow effects in lubricated contact was developed using a reduced form of the Navier-Stokes equation in association with the conti-nuity equation. Reynolds established the principles of the modern lubrication theory through a formal representation in terms of a second-order partial differential equation, which makes possible the calculation of the hydrodynamic pressures developed within the lubricant in the lubricated surfaces. The isothermal and isoviscous Reynolds equation for laminar and transient flow is (REYNOLDS, 1886) ∂ ∂ x  h3 12µ ∂ p ∂ x  + ∂ ∂ z  h3 12µ ∂ p ∂ z  =U 2 ∂ h ∂ x+ ∂ h ∂ t, (2.1)

where h is the oil film thickness, p the hydrodynamic pressure, µ and U the dynamic viscosity and velocity in x-direction, respectively.

Eight years passed to obtain the first analytical solution of the Reynolds equation, which considered an infinitely long journal bearing. Nevertheless, some results did not cor-respond to the physical reality, since the pressure in the divergent zone of the film could be negative, due to the boundary conditions not considering the film rupture in the bearing (SOM-MERFELD, 1904). Later, the solution proposed to overcome this limitation was to consider only the positive pressures to the bearing load calculation, which is known as half-Sommerfeld solution (GUMBEL, 1914). A more accurate description for the rupture considers that the pres-sure in the rupture film is equal to the saturation prespres-sure and the gradient is zero, which is known as the Reynolds boundary condition (SWIFT, 1932; STIEBER, 1933).

In 1953, neglecting circumferential pressure gradient for the Reynolds equation, an analytical solution using the boundary conditions defined by Gümbel was obtained for short bearings (DUBOIS; OCVIRK, 1953). Later, a numerical solution was introduced for the so-called finite bearings (PINKUS, 1958). A conceptual change occurred on the lubrication theory when Downson proposed a more general equation derived from the fundamental ones, which could be reduced to any form used by that time in the fluid-film-bearing analysis. He was the first to present a thermohydrodynamic equation, considering the lubricant property variations (DOWSON, 1962).

In the search for a more accurate model to describe the cavitation zone, Jakobson, Floberg and Olsson imposed the mass-conservation of the lubricant flow in the lubricated do-main using complementary boundaries conditions in the rupture and reformation boundaries (JAKOBSSON; FLOBERG, 1957). However, only a few years later, Elrod and Adams de-veloped an algorithm with the conditions proposed by Jakobson, Floberg and Olsson (JFO) considering the cavitation phenomena by defining a new parameter, the lubricant film fraction.

This parameter made possible to introduce the cavitation biphasic mixture into the Reynolds equation (ELROD; ADAMS, 1974; ELROD, 1981).

With the development of the lubrication theory and the concepts of contact mechan-ics introduced by Hertz in 1881, many models described the contact in the mixed-lubrication regime. Greenwood and Williamson published an original work treating the asperity distribution in a statistical form, with one flat and smooth surface and a parallel rough surface (GREEN-WOOD; WILLIAMSON, 1966). Later, Greenwood and Tripp considered the contact between two rough surfaces and the interaction of the asperities on them (GREENWOOD; TRIPP, 1970).

In 1978, Partir and Cheng proposed a modification of the Reynolds equation in-troducing an average flow model with flow factors obtained from the numerical flow solution including the roughness (PATIR; CHENG, 1978). Other authors proposed modifications to this model (BAYADA; CHAMBAT, 1988; BAYADA; FAURE, 1989; BUSCAGLIA et al., 2002).

Hu and Zhu proposed a full numerical solution of the mixed elastohydrodynamic lubrication in point contacts. They used the Reynolds equation with hydrodynamic lubrication regime. A method related to the mathematical formulation of dry contact is applied on the asperity contact areas where the film thickness is zero. Thus, it is possible to use the same numerical procedure for all the lubricated regions. However, no averaging procedure was used and this is the reason the model can only be applied in small regions such as point contacts (HU; ZHU, 2000).

Dobrica and partners proposed a deterministic formulation for the mixed-lubrication regime and compared this model with the stochastic Partir and Cheng method. They concluded that parameters as film thickness, attitude angle and friction torque were influenced by the roughness suggesting that they are very dependent on the roughness orientation. They also ver-ified the accuracy of Partir and Cheng model in describing the behavior for different roughness types and the calculation for the average minimum film thickness, but underestimating the fric-tion torques (DOBRICA et al., 2006).

In 2007, Ausas et. al compared the Reynolds cavitation model with the one pro-posed by Elrod and Adams (1974) applying micro textures in a cuboid form on the bearing surfaces and comparing the pressure fields. They concluded that the Reynolds model gives in-accurate results in the micro textured situation, as the friction torque, because the cavitated zone is underestimated. A mass-conserving cavitation model represents better the phenomenon with more accurate results for the micro textured bearings. Another consideration is that the numerical imposition of the film-rupture and film reformation could have great effects on the results. Later, Ausas developed a numerical algorithm for fully-dynamic lubrication problems with mass-conserving boundary condition (AUSAS et al., 2007; AUSAS et al., 2009).

with appropriate implementation, can predict cavitation in dimples more accurately than other types of boundary conditions. They verified experimentally the cavitation on the superficial dim-ples and compared the pressure profiles for the JFO, half-Sommerfeld, and Reynolds cavitation boundary condition (QIU; KHONSARI, 2009). Zhang and Meng also confirmed the previous works reaffirming Qiu and Khonsari conclusions. They once again used micro textured surfaces in hydrodynamic lubrication and compared the experimental and numerical results (ZHANG; MENG, 2012).

Sahlin and coworkers suggested a mixed lubrication model, where the flow fac-tors are calculated using a homogenization process and compared the proposed method with the Partir and Cheng method. They found that their method has similar results for longitudi-nal roughness lay. However, for a cross-patterned surface roughness, the result was divergent probably because of the method used (SAHLIN et al., 2010a; SAHLIN et al., 2010b).

Qiu and Khonsari used a model considering the JFO boundary condition to inves-tigate the behavior of thrust bearings and seals with the presence of dimples on the surface. The roughness was taken into account by considering the Partir and Cheng approach. They analyzed parameters such as film thickness, dimple depth, friction, load-carrying capacity and their impact on the tribological performance. An interesting conclusion was that roughness in the dimple-modified surfaces improved the load-carrying capacity until a certain limit. More-over, the cavitation only has a significant behavior for relatively thin film thickness. There is an optimal dimple depth and dimple density to produce the maximum load-carrying capacity according to the operation condition. Another finding is that the cavitation in dimples has the effect of reducing the friction force (QIU; KHONSARI, 2011b).

Brunetière and Tournerie presented a model to solve the isothermal Reynolds equa-tion with mas-conservative cavitaequa-tion. They considered the Hertzian asperity contact to analyze the behavior of a textured mechanical seal and the hydrodynamic lift related to the texture. In their model, it was possible to achieve a substantial friction reduction in the mixed and hydro-dynamic regimes for textured surfaces, which seems to be related with the interaction between micro-roughness and pressure generated due to the textures (BRUNETIÈRE; TOURNERIE, 2012).

Profito proposed a general discretization scheme to solve different lubrication regimes. It associates the flexibility of the finite element method to work with irregular geometry, such as surfaces with dimples, and the finite volume method for the flow to solve the isothermal Reynolds equation with the mass-conservative cavitation model proposed by Elrod-Adams (PROFITO, 2015).

Some researchers suggested modification of the Elrod-Adams cavitation algorithm. Fesanghary and Khonsari suggested a modification on the switch function in the Elrod cavita-tion model by using an exponentially decreasing or increasing funccavita-tion. Their work was based on the instabilities associated to sudden changes in the original binary switch function. With a

successive over-relaxation solver, they improved the convergence speed and numerical instabil-ities (FESANGHARY; KHONSARI, 2011). In addition, Woloszynski and coworkers modified the Elrod-Adams cavitation implementation coming out with an algorithm where the system of discretized equations is continuously differentiable. The cost required to solve the system is similar when cavitation is not considered (WOLOSZYNSKI et al., 2015). More recently, Mi-raskari proposed a modification on the pressure field calculation of a cavitated axial grooved journal bearings by means of finite volume methods (MIRASKARI et al., 2017).

2.2 Surface Texturing

The reduction of fuel consumption and emissions in combustion engine has been of interest in the academia and industry. In this aspect, the reduction of power loss due to friction represents a great challenge and many approaches to achieve this goal have been discussed. Researches have addressed the surface texturing effects regarding lubrication, load capacity and wear.

The potential of micro-asperities in enhancing tribological properties with extra hy-drodynamic lifting for film lubrication is well known. Firstly, Hamilton’s studies found that the micro-irregularities and cavities were associated with the hydrodynamic lubrication process producing hydrodynamic pressure and making possible higher load carrying capacity (HAMIL-TON et al., 1966). Although some papers were published regarding this subject, only some years later, surface modification gained interest with Etsion and Burstein findings. They de-veloped a mathematical model considering the presence of regular surface structures in form of pores in non-contacting mechanical seals, and found that, with proper selection of geo-metric pore parameters, it is possible to improve the mechanical seal performance (ETSION; BURSTEIN, 1996). Etsion expanded his studies publishing other papers addressing surface tex-turing. First, he and coworkers studied, theoretical and experimentally, the partial laser surface texturing of a mechanical seal and found a significant reduction in the friction torque (ET-SION; HALPERIN, 2002). Subsequently, an experimental study of the performance improve-ment of parallel-thrust bearing using surface laser texturing was performed (ETSION et al., 2004). Moreover, Etsion published a review of works related to surface texturing focusing on laser texturing (ETSION, 2005).

The increasing demand for more efficient mechanical components pushed forward researches related to surface texturing and interesting papers were published addressing this subject. Kovalchenko et al. investigated the lubrication regime through experiments in a pin-on-disk test ring with unidirectional sliding. Testing a series of textures and conditions, they found that the laser surface texture enhanced the loading carrying capacity pointed out that the results obtained at high speeds, high loads and high viscosities which showed better im-provement (KOVALCHENKO et al., 2005). Wang, from analytical and experimental studies in the reciprocating slider end-stroke region, obtained the reduction of asperity contact with the

surface texture application (WANG, 2009).

Later in 2012, Brizmer and Kligerman produced a theoretical study analyzing the laser textured surface applied to a journal bearing under hydrodynamic lubrication. An analysis considering the load capacity and attitude angle was performed and optimized dimple param-eters were found. Their results showed an improvement in load capacity when partial textur-ing was applied with low eccentricities. However, no significant properties improvement were achieved at high eccentricity for partial or fully textured journal bearing (BRIZMER; KLIGER-MAN, 2012). In this same year, Kango et al. studied the application of different surface texture shapes in a finite journal bearing applied on different regions to verify the influence over the load carrying capacity and friction coefficient. Through numerical solution using the finite difference method, they observed that the micro cavities were able to enhance the bearing performance, the friction coefficient and force decreased (KANGO et al., 2012).

Tang et al. investigated through experiments the effects of textured steel surface under lubricated sliding contact on reducing friction and wear analyzing the size of wear par-ticles. They conclude that the main mechanisms responsible to reduce friction and wear were the micro-reservoirs and the micro-trap for wear debris formed by the dimples (TANG et al., 2013).

Kim and colleagues verified the friction effects for textured cast iron surfaces for automotive engines. Micro dimples geometry and distribution, different loads and speeds were studied. They found the dimple aspect ratio was the most significant factor in terms of friction coefficient (KIM et al., 2014).

A study performed by Qiu et al. analyzed different texture shapes evaluating the friction coefficient and stiffness of parallel slider bearings. They optimized the geometric pa-rameters in order to obtain the lowest friction coefficient and maximum bearing stiffness. The results showed that the geometric parameters to achieve each of these goals are slightly dif-ferent. In addition, these geometries are also different from those ones to maximize the load carrying capacity. Moreover, they suggested that the elliptic dimple yielded the maximum bear-ing stiffness and lowest friction coefficient. However, spherical texture can be more advanta-geous in terms of manufacturing. In addition, the requirement of bearing performance be in-dependent of the sliding direction is an important factor to consider spherical dimples (QIU et al., 2013). In this context of understanding the influence of surface texturing in friction, Hsu and coworkers studied surface texture patterns for the hydrodynamic, elastohydrodynamic and boundary lubrication regimes for sliding components. They showed that each regime requires specific dimple parameters as well orientation to achieve friction reduction (HSU et al., 2014). Moreover, Kango made a comparative analysis between spherical textured and grooved journal bearings under hydrodynamic lubrication. According to their results, textures and grooves were able to enhance tribological properties decreasing friction coefficient and average temperature when placed in appropriate location depending on the eccentricity ratio (KANGO et al., 2014).

Pratibha varied parameters like shaft speed, bearing pressure distribution and load condition to analyze the journal bearing lubrication behavior with surface textures. It was re-ported that the percentage increase of maximum pressure is greater in the textured surface sit-uation compared to the smooth bearing for constant loads and increasing speeds (PRATIBHA; CHANDRESHKUMAR, 2014).

Adjemout studied the effects of real dimple shape in mechanical seals considering a mass conservation cavitation model and hydrodynamic lubrication. Due to the fabrication process, real dimples have imperfections. Thus, the authors examined different defect types and showed the importance of the dimple shape control. There is be a limit where the imperfections on the dimples will make the positive surface texturing effects disappear. In addition, the leakage is strongly affected by the roughness height and orientation inside the dimples. The friction and leakage increase when substantial increment in the curvature angle edge of the dimple occurs (ADJEMOUT et al., 2014).

Tala-Ighil and Fillion used a numerical approach to analyze the journal bearing char-acteristics with and without textures on the bearing surface. They studied the thermal effects as well and reached interesting conclusions. Among them, texturing the entire bearing surface does not automatically improve the tribological behavior of journal bearing contacts in terms of friction reduction and hydrodynamic lift effect. However, at low rotational speeds, the fully textured surface can have positive effects on reducing the friction coefficient. Moreover, they suggested that, in the outlet of the active pressure zone, the partial texturing has a positive effect since a better pressure distribution occurs for average and high rotational speeds. In addition, the dimple geometric characteristics and type have a great influence over the enhancement of tribological properties. The texture distribution, mainly the dimple location on the journal bear-ing surface, is the principal parameter for performance improvement (TALA-IGHIL; FILLON, 2015).

Zavos and Nikolakopoulos developed a numerical model using the Navier-Stokes equations to evaluate the fluid-structure interaction of the piston ring operating under various conditions and with some texturing patterns. They considered spherical and rectangular micro-dimples to examine the hydrodynamic friction force. Their findings suggested for the conditions analyzed that the rectangular dimples proved to be more efficient in the reduction of friction force (ZAVOS; NIKOLAKOPOULOS, 2015).

Ahmed et al. performed a study in a piston ring/cylinder assembly and in a mechan-ical seal investigating the influence of surface texturing geometric parameters such as dimple depth, diameter, shape and dimple density. They concluded that the aspect ratio is the most im-portant factor to improve tribological properties in the piston ring/cylinder assembly, while the dimple depth is the most important for mechanical seals (AHMED et al., 2016).

Kang and coworkers experimentally studied how the tribological performance of injection cam is affected by the local laser surface texturing. They examined the friction

coef-ficient and wear to determine the optimal texture designs. They verified that the laser surface texture promoted wear resistance due to the remelting process and enhance lubrication of cam surface (KANG et al., 2017).

Silva and Costa investigated a recent technique called Maskless Electrochemical Texturing applied to texture gray cast-iron diesel cylinder liners. In this technique, the texture is produced through a pulsed voltage where the workpiece is anodic and the tool is cathodic. According to the authors, this method is faster and cheaper compared to laser surface texturing (SILVA; COSTA, 2017).

Dong et. al experimentally examined the effects of texture distribution on the sur-face of the journal bearing over the vibration and rotor stability analyzing the acceleration and shaft frequency amplitudes, shaft center orbits and frequency spectrum. They used different types of texture distribution. Their results showed that there is a significant decrease in acceler-ation amplitude of textured bearing. However, one of the configuracceler-ations showed better vibracceler-ation damping effect (DONG et al., 2017).

Recently, Ma and coworkers explored the roughness and texture effects on the so-lutions of Reynolds equation considering the JFO mass-conservative boundary condition and the asperity contact. They analyzed texture parameters such as dimple depth-over-diameter ratio and area density. These parameters are influenced by the surface roughness in hydro-dynamic and mixed lubrication regimes. They concluded that the surface roughness effect on load-carrying capacity could be neglect under hydrodynamic lubrication regime. An interest-ing conclusion of their work is that a large dimple area density is positive in terms of friction coefficient reduction (MA et al., 2017).

Elo and coworkers applied surface texturing to the valve sealing surfaces of a com-bustion engine. The valve system is responsible to control the flow that enters and goes out of the combustion chamber. Their objective was to promote a tribofilm formation to avoid wear since this component operates under many operation cycles. The results suggested that the tex-ture applied to the valve is able to store oil residues improving the tribofilm stability (ELO et al., 2018).

Interesting researches have been reported from the School of Mechanical Engineer-ing of the University of Campinas. The study of surface textures applied to the dynamic load condition of a lightweight crankshaft bearing was studied by Reis using numerical simulations. His results showed that it is possible to combine the weight reduction of the crankshaft with fric-tion and power loss reducfric-tion promoting addifric-tional hydrodynamic lift by choosing appropriate geometric parameters for the textures (REIS, 2017). Ramos numerically studied the application of textures in bearings under static load condition using a full multigrid technique to solve the system of equations. The effects of partial and total surface texturing were investigated. The results showed that the operational parameters have significative effects over additional hydro-dynamic lift. Friction reduction was achieved by carefully choosing the position and geometric

characteristics of the textures (RAMOS, 2018).

Akbarzadeh and Khonsari experimentally studied different combinations of coated and textured surfaces applied to a piston ring analyzing the friction behavior. They analyzed five combinations of surfaces in the piston ring: without textures and coating, with textures only, with coating only, first textured and then coated, first coated and then textured. Their results showed that the combination of coating and laser surface texturing reduced the running-in time and produced the lowest frictional force in the piston ring tested. Moreover, the first textured and then coated surface combination as well the coated and then textured produced good results reducing the total friction. A texturing and coated surface combination was able to reduce the friction force by 15% (AKBARZADEH; KHONSARI, 2018).

A recent research presented by Wang and colleagues investigated the friction and wear behavior effects of laser surface texturing. Three types of textures were tested: circular dimples, elliptical dimples and grooves. The circular dimples offered the best results in terms of friction reduction, the elliptical dimples showed the best performance in wear resistance and the grooves the best outcomes in friction stabilization. However, they pointed out that these results will depend on the operational conditions, area material contact, contact form, and lubrication (WANG et al., 2018).

Through analyses of textured journal bearings operating in transient regime, Lin et. alobserved the performance in terms of load carrying capacity. They concluded that the texture best location is influenced by the operational conditions. In addition, textures in the pressure built-up region favors the decrement of eccentricity, while in the pressure drop region results in an increment of the eccentricity (LIN et al., 2018).

The literature review presented in the last paragraphs gives the basic support for the present contribution. The starting point of this dissertation is the work developed by Reis (2017). In this work, aspects such as mass conservation, superficial surface characteristics and the texturing effects were considered. Some analysis considering the temperature influence was also performed. In addition, he developed and implemented in Matlab® environment a code to study surface texturing in lightweight crankshaft bearings, which was used as the base for the models developed here. Considering dynamically loaded condition, the computing time can be very costly. Therefore, the code was improved and adapted in some aspects to make it more ef-ficient for the present purposes. In addition, for more realistic results it is important to consider the elastohydrodynamic lubrication and a model is presented to take into account the elastic de-formation effects caused by the hydrodynamic pressures. Another important contribution was the study of texture location effects for different operational conditions. Thus, this disserta-tion investigates the surface texturing applied to an automotive three-cylinder engine crankshaft bearing in a dynamic load condition with the goal to understand the surface texturing influence on friction and power loss.

3 LUBRICATION AND SURFACE TEXTURING MODELING

This chapter presents to the mathematical models considered for the radial hydro-dynamic bearing and the hydrohydro-dynamic and mixed-lubrication regimes. In addition, the ap-proximation of the Reynolds equation and the procedures to consider the surface texturing are detailed.

3.1 Hydrodynamic Radial Bearings

Hydrodynamic bearings are mechanical components used to separate two rigid parts using a lubricant fluid to lessening friction and consequently wear. This occurs due to the rel-ative movement of the surfaces, which produces the hydrodynamic force that balances the ex-ternal load promoting the separation between the parts. The bearing can be axial or radial. The first has circular surfaces and it is used to separate parts submitted to axial loads. The second is composed by a bearing house and a shaft in its interior. It is used when there is a radial load ap-plied to the shaft with relative angular velocity to the bearing house. In addition, there is a third classification for the bearings able to handle axial and radial loads (DUARTE, 2005). Figure 3.1 is a schematic representation of a radial bearing.

3.2 Lubrication Regimes

Lubrication is the process of separating two surfaces with relative movement us-ing lubricants. Accordus-ing to the separation of the surfaces, there are four classical lubrication regimes: hydrodynamic, elastohydrodynamic, mixed and boundary lubrication. Parameters such as loading, speed, lubricant properties and roughness of the surfaces have influence on the lu-brication condition (HAMROCK et al., 2004). Figure 3.2 exemplifies the lulu-brication regimes and a brief description of each regime is given bellow.

1. Hydrodynamic Lubrication - Generally occurs in conformal contact geometry (see Figure 3.3). In this type of lubrication, the relative movement of the surfaces generates sufficient hydrodynamic pressure to make the oil film thick enough to prevent the solids surfaces to interact with each other providing low friction and high resistance to wear.

2. Elastohydrodynamic Lubrication – It is a case of hydrodynamic lubrication that gener-ally occurs in non-conformal geometry contacts (see Figure 3.3). The pressures are high enough to cause substantial elastic deformation on the lubricated parts, which cause local increase of the oil film thickness.

3. Mixed-Lubrication – Characterized by an oil film that is not thick enough to completely avoid the contact of the surface irregularities. In this type of lubrication, the applied load is supported by the oil film and the asperity contacts.

4. Boundary Lubrication - In this regime, the solid surfaces are very close and there is a thin film of molecular magnitude on the surfaces. There is not separation of the surface asperities by the lubricant.

Fig. 3.2 – Lubrication regimes: (a) hydrodynamic and elastohydrodynamic lubrication; (b) mixed-lubrication; (c) boundary lubrication, adapted from Hamrock et al. (2004).

An interesting way to represent and understand different lubrication regimes is us-ing the Stribeck curve illustrated in Figure 3.4. At low relative speeds, the boundary lubrication regime occurs and the friction is high due to the contact of the asperities. As the speed increases,

Fig. 3.3 – Conformal and non-conformal geometry (adapted from ASM (1992)).

higher than I, the regime change to the mixed-lubrication and the hydrodynamic pressure is not sufficient to completely prevent the asperity contact. The friction in this regime is lowered as the speed increases. For speeds higher than II, the hydrodynamic pressure is sufficient to gen-erate an oil film thick enough to prevent the asperity contact characterizing the hydrodynamic lubrication regime. The friction coefficient tends to increase as the speed raises for the hydro-dynamic lubrication regime, because the viscous friction increases with the increment of the relative velocity of the lubricated surfaces.

Fig. 3.4 – Friction coefficient versus relative velocity of the lubricated surfaces (adapted from NORTON (2004)).

A dimensionless factor (Λ) can be used to characterize the lubrication regime. This factor is the ration of the lubricant film thickness and the surface roughness standard devia-tion amplitude (DUARTE, 2005; IZUKA et al., 2009). Hydrodynamic lubricadevia-tion is considered when 3 ≤ Λ ≤ 100. In this situation, the lubricant film thickness is considerably greater than the roughness dimensions, and the lubricant properties generate the friction (IZUKA et al., 2009).

The mixed-lubrication is characterized for 1 ≤ Λ < 3.

3.3 Pressurizing Mechanisms

The main purpose of journal bearings is to separate two rigid mechanical parts to reduce friction, wear and temperature by avoiding or lessening surface contact. Although there are dry journal bearings, a lubricant fluid is usually applied to promote the surface separa-tion. In addition, the relatively simple geometry and easiness of construction do not reflect the complex theory behind this component operation. Depending on the application, journal bear-ings can be hydrostatic or hydrodynamic. The first needs an external pressurization source to avoid the contact surface. In the second, due the relative movement of the parts, there is an auto-pressurization and no need of an external pressurization source (DUARTE, 2005). The mechanisms that promote the pressurization are known as squeeze and wedge effects.

The wedge effect occurs due to the relative rotational movement of the surfaces, as illustrated in Figure 3.5. The shaft rotational speed makes the fluid particles flow from a region of higher to one of lower volume. This process increases the pressure and generates a hydrodynamic pressure gradient, which is responsible to keep the surfaces separated avoiding the dry contact.

Fig. 3.5 – Wedge effect auto-pressurization mechanism (adapted from Izuka et al. (2009)).

The squeeze effect occurs when a radial velocity of the shaft compresses the fluid particles against the bearing surface and the pressure increases in that region as showed in Figure 3.6.

Fig. 3.6 – Squeeze effect auto-pressurization mechanism (adapted from Izuka et al. (2009)).

These effects generate a pressure field that separates the surfaces. In this way, the viscous friction on the lubricant fluid is predominant instead of the dry friction for the contact surfaces.

3.4 Lubricants

Considering two surfaces in relative motion, lubricants reduce wear and friction between them as defined previously. Lubricity is the name given to the property of reducing friction. Moreover, other important aspects considered in lubricants are their thermal and hy-draulic stability, corrosion of the contact surfaces and cost (IZUKA et al., 2009). Considering that there are many different mechanisms subject to the most diverse conditions, each applica-tion requires adequate lubricant properties.

One of the most relevant property is the lubricant dynamic viscosity, which is of fundamental influence on the load-carrying capacity. The viscosity is a function of the temper-ature, shear stress rate and pressure. The dynamic viscosity µ is defined by:

µ = Γ ˙

γ, (3.1)

where Γ is the shear stress and ˙γ the shear stress rate.

There are some models that relates the viscosity to the temperature such as the Reynolds, Slotte, Walther and Vogel.

The Reynolds model is the most used for initial guess and application limited to the temperature range. The Slotte model gives results with average accuracy. The Walther model is the base of the ASTM viscosity × temperature graph. The Vogel model is very useful in numeric calculations (IZUKA et al., 2009; REIS, 2017). The previous models are given respectively by the following expressions:

µ = ae−bT, (3.2)

µ = a

(b + T )c, (3.3)

log₁₀log₁₀(µs+ c) = −a log10(T ) + b, (3.4)

µ = ae

b

T°c+c_{, (3.5)}

where T and T°c are the temperatures in Kelvin and Celsius, respectively, µs the kinematic

viscosity, and a, b, c are constants dependent on the lubricant oil.

Numeric models are used to consider the temperature effect on the hydrodynamic lubrication, and known as thermohydrodynamic models (THD). When the elastic deformations are considered with the temperature, the models are referred to as thermoelastohydrodynamic (TEHD). Generally, approaches considering the temperature effects become more complex. For the scope of the present dissertation, the effects of the temperature over the viscosity are not considered. Thus, the lubrication model adopted assumes constant viscosity .

3.5 Hydrodynamic Lubrication Model

In hydrodynamic lubrication, the relative speed of the surfaces in contact produces a pressure field from the lubricant film that balances the external loads. The fluid film pres-sure distribution for an operating journal bearing is described by the Reynolds equation. The classical paper published by Reynolds not only developed the theoretical fundamentals of lu-brication, but also found good agreement with experimental results (REYNOLDS, 1886). The Reynolds equation is a second-order partial differential equation derived from the continuity (mass conservation) and Navier-Stokes equations considering the following hypotheses:

• The fluid is Newtonian; • Laminar flow;

• The lubricant fluid viscosity is constant; • Inertial and body forces are not considered;

• Curvature effects of the lubricant film are negligible;

• The fluid film thickness is very thin compared to other dimensions of the contact surface and the hydrodynamic pressure is constant on the fluid film thickness direction

• The contact surfaces are assumed to be perfectly smooth.

The classical isothermal and isoviscous Reynolds equation is defined in Cartesian coordinates as follows (REYNOLDS, 1886; GROPPER et al., 2016):

∂ ∂ x  h3 12µ ∂ p ∂ x  + ∂ ∂ z  h3 12µ ∂ p ∂ z  | {z } I =U 2 ∂ h ∂ x | {z } II + ∂ h ∂ t |{z} III , (3.6) where

h: oil film thickness, p: hydrodynamic pressure, t: time,

xand z: circumferential and axial directions of the cylindrical bearing,

µ and U : dynamic viscosity and tangencial velocity in x-direction of the lubricant, respectively.

Figure 3.7 represents the coordinate system. The term (I) in equation (3.6) is the pressure flow or Poiseuille term and describes the flow rates caused by the pressure gradients in the lubricated surfaces. The term (II) is the Couette or wedge flow term and describes the effect on the flow rates due to the relative motion of the contact surfaces, the lubricant density local variation and the surface speeds. The third term (III) is the local-expansion and represents the transient effects of the lubricant film thickness (FRENE et al., 1997; HAMROCK et al., 2004; PROFITO, 2015; REIS, 2017).

The lubricant flows and shear stresses rates can be derived using the same hypothe-ses. The lubricant flows in x and z directions can be written as (PROFITO, 2015; REIS, 2017):

q_x= −ρ h 3 12µ ∂ p ∂ x+ ρhU, q_z= −ρ h 3 12µ ∂ p ∂ z. (3.7)

The shear rate components in x and z directions are given by (PROFITO, 2015; REIS, 2017): τ_x1,2= −₊ 1 2µ ∂ p ∂ x(2y − h) + U h, τ_z1,2= −₊ 1 2µ ∂ p ∂ z(2y − h). (3.8)

3.5.1 Fluid Film Cavitation

The pressure distribution calculated from the Reynolds equation can reach negative values. This means that the fluid is under significant traction on some regions of the lubricated domain. In practice, when the fluid is submitted to high tensile stresses, it is likely to suffer rupture. In these regions, the fluid experience what is known as cavitation. The hydrodynamic

Fig. 3.7 – Representation of the journal bearing coordinate system (a) isometric view; (b) planned superior view.

pressure falls below the saturation of the gas component dissolved in the lubricant or below the vapour pressure of the fluid in a specific temperature. In this situation, the fluid film is broken and the cavities in these regions are filled with a biphasic mixture of vapour/gas (PROF-ITO, 2015). The cavitation effect is illustrated in Figure 3.8 where the blue lines represents the gas/vapour mixture. θ = 1 represents the inside region where the fluid film is not broken. The cavited region occurs for 0 ≤ θ < 1. The pressured regions are separated from the cavitation zone due to discontinuity in the lubricated domain. The boundaries between the full film and the cavitated zone can be the rupture boundary, where the fluid film rupture initiates, or reforma-tion boundary, where the fluid film begins its reconstrucreforma-tion due to the rise of the hydrodynamic pressure.

The cavitation phenomenon of lubricated parts can be of three types (BRAUN; HANNON, 2010; PROFITO, 2015):

1. Gaseous cavitation: In this type of cavitation, one or more species of gases are dissolved in the fluid. It occurs when the pressure goes below the saturation pressure of a particular gas dispersed in the fluid.

2. Pseudo-cavitation: It is a type of gaseous cavitation, the gas bubble (biphasic mixture) grows due to the lost of surrounding pressurization and expands without addition of mass due to the diffusion from the liquid to gas phase.

Fig. 3.8 – Representation of the fluid film cavitation for a generic lubricated domain (adapted from Profito (2015)).

liquid due to a thermodynamically non-equilibrium situation for a given temperature. It is important to consider the cavitation since it has an important role in lubrication systems. There are some approaches to consider this phenomenon and the most used models are: the Gümbel (or half-Sommerfeld) formulation, Reynolds (or Swift-Steiber) and the JFO. Each formulation has a different influence on the the location of the lubricant film rupture. 3.5.1.1 Half-Sommerfeld Model

Also known as Gümbel model, it is the simplest formulation to deal with cavitation. It considers that all the hydrodynamic pressures below the cavitation pressure is equal to it. This assumption can be mathematically written as:

p= (

pcav if p< pcav

p if p≥ pcav

. (3.9)

Due to its simplicity, this model is largely used, but it does not enforce the mass flow conservation on the cavitation boundaries.

3.5.1.2 Reynolds Model

To deal with the lack of mass conservation of the Half-Sommerfeld model, the Reynolds or Swift-Steiber model assumes that the pressure gradient field on the cavitation boundaries are zero. It is represented as:

p→ (

p= pcav inside of the cavitation region ∂ p

∂ n= 0 on the cavitation boundary