Modeling of Piston Pin Lubrication in Internal Combustion ...

Modeling of Piston Pin Lubrication in Internal Combustion Engines

by

Zhen Meng

B. Eng., Vehicle Engineering Tsinghua University, 2015

S.M., Mechanical Engineering

Massachusetts Institute of Technology, 2017

Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the Requirement for the Degree of

Doctor of Philosophy in Mechanical Engineering and Computation

at the

MASSACHUSETTS INSTITUTE of TECHNOLOGY

September 2020

© 2020 Massachusetts Institute of Technology. All rights reserved.

Signature of Author: _____________________________________________________________ Department of Mechanical Engineering

August 15, 2020

Certified by: _ _________________________________________________________________ Tian Tian

Principal Research Engineer, Department of Mechanical Engineering, MIT Thesis Supervisor

Accepted by: ___________________________________________________________________

Nicolas Hadjiconstantinou Chairman, Department Committee on Graduate Students, MIT

Co-Director, Computational Science and Engineering, MIT

3

Modeling of Piston Pin Lubrication in Internal Combustion Engines

by

Zhen Meng

Submitted to the Department of Mechanical Engineering on August 15, 2020

in Partial Fulfillment of the Requirements of the Degree of

Doctor of Philosophy in Mechanical Engineering and Computation

Abstract

The piston pin joins the piston and the connecting rod to transfer the linear force on the

piston to rotate the crankshaft that is the eventual power outlet of the engine. The interfaces

between the piston pin and the pin bore as well as the connecting rod small end are one of the

most heavily loaded tribo pairs in engines. Piston pin seizure still occurs often in the engine

development and the solution often comes from applying expensive coatings. Furthermore, it

has been found that the friction loss associated with the pin can be a significant contributor to

the total engine mechanical loss. Yet, there lacks a basic understanding of the lubrication

behavior of the pin interfaces.

This work is aimed to develop a piston pin lubrication model with consideration of all

the important mechanical processes. The model predicts the dynamics of the pin and the

lubrication of the interfaces between the pin and pin bore as well as small end. The model

couples the dynamics of the pin with the structural deformation of the mating parts, the

hydrodynamic and boundary lubrication of all the interfaces, and oil transport. The model is

successfully implemented with an efficient and robust numerical solver with the second order

accuracy to compute this highly stiff system. The preliminary results applying the model to a

gasoline engine show that the boundary lubrication is the predominant contributor to the total

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friction. As a result, the interface with more asperity contact tends to hold the pin with it. Thus,

the pin friction loss is coming from the interface with less contact. Solely from friction reduction

point of view, ensuring efficient hydrodynamics lubrication in one interface is sufficient.

Furthermore, as the heavy load is supported in several small areas, mechanical and thermal

deformation of all the parts are critical to load distribution, oil transport, and the generation of

hydrodynamic and asperity contact pressure, providing the necessity of the elements

integrated in the model.

This work represents the first step to establishing a more comprehensive engineering

model that helps the industry understand the pin lubrication and find cost-effective solutions to

overcome the existing challenges.

Thesis Supervisor: Dr. Tian Tian

Title: Principle Research Engineer, Department of Mechanical Engineering, MIT

5

Acknowledgements

This work was sponsored by Daimler and the Consortium on Lubrication in Internal

Combustion Engines in the Sloan Automotive Laboratory, Massachusetts Institute of

Technology. The consortium members were Mahle, MTU, Shell, Toyota, Volkswagen, Volvo

Trucks, and Weichai Power.

First of all, I would like to express my sincere gratitude for my advisor, Dr. Tian Tian, for

his inspiration and guidance during my time at MIT. He showed and lead me to a higher level of

thinking and working in terms of solving real-world problems as an engineer. He has been not

only the best source for me to seek advice in research, but also a valuable friend as I’ve learned

a lot from his experience and insights of life.

I would also like to thank the other members in my thesis committee, Professor Ton

Lubrecht, Professor Wai Cheng, and Professor Irmgard Bischofberger. They have given me a lot

of advice from various perspectives to help me make my work more comprehensive.

I would like to thank the members of the Sloan Automotive Laboratory at MIT for their

help and exchange of ideas and knowledge. Especially, I’ve been really enjoying the time

working together with fellow students in my group, Yang Liu, Tianshi Fang, Sebastian Ahling,

Chongjie Gu, Qin Zhang, Wang Zhang, Jerome Sacherer, Zhiyuan Shu, and Koji Kikuhara. Special

thanks to Janet Maslow for assisting our research in the lab.

Finally, I would like to thank my parents, Xuemin Zhao and Qiang Meng, and my

girlfriend, Jingru Cao, for everything they’ve done for me.

Zhen Meng, Cambridge, MA, August 2020

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Table of Contents

ABSTRACT ............................................................................................................................................... 3

ACKNOWLEDGEMENTS ........................................................................................................................... 5

LIST OF FIGURES ................................................................................................................................... 11

LIST OF TABLES ..................................................................................................................................... 15

CHAPTER 1 INTRODUCTION .............................................................................................................. 17

1.1 BACKGROUND AND MOTIVATION ...................................................................................................... 17 1.1.1 Power cylinder system .......................................................................................................... 18 1.1.2 Piston pin system .................................................................................................................. 20

1.2 EXISTING WORK ............................................................................................................................. 21 1.3 OBJECTIVES ................................................................................................................................... 23 1.4 THESIS SCOPE ................................................................................................................................ 24

CHAPTER 2 MAJOR FACTORS OF THE MODEL ................................................................................... 26

2.1 KINEMATICS AND DYNAMICS ............................................................................................................. 26 2.1.1 Definitions and simplifications .............................................................................................. 26 2.1.2 System kinematics ................................................................................................................ 27 2.1.3 System dynamics ................................................................................................................... 29

2.2 GEOMETRY OF THE INTERFACES ......................................................................................................... 30 2.2.1 Contours of the interfaces ..................................................................................................... 30 2.2.2 Clearance profile between the surfaces ................................................................................ 32

2.3 ASPERITY CONTACT SUB-MODEL ........................................................................................................ 33 2.4 HYDRODYNAMIC SUB-MODEL ........................................................................................................... 33

2.4.1 Governing equations ............................................................................................................. 33 2.4.2 Boundary conditions ............................................................................................................. 34

2.5 CALCULATION OF FRICTION ............................................................................................................... 35 2.5.1 Calculation of friction power loss .......................................................................................... 35 2.5.2 Friction in boundary lubrication regime ................................................................................ 38 2.5.3 Friction from hydrodynamic shear stress .............................................................................. 39 2.5.4 Summary ............................................................................................................................... 39

2.6 STRUCTURAL DEFORMATION ............................................................................................................. 39 2.7 SUMMARY .................................................................................................................................... 40

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CHAPTER 3 SOLUTION METHOD ....................................................................................................... 42

3.1 STRUCTURE OF THE SOLVER .............................................................................................................. 42 3.1.1 Main architecture ................................................................................................................. 42 3.1.2 Hydrodynamic module with P-q algorithm ........................................................................... 44

3.2 NUMERICAL SCHEMES ..................................................................................................................... 46 3.2.1 Reynolds equation ................................................................................................................. 46 3.2.2 Equations of system dynamics .............................................................................................. 47

3.3 JACOBIAN MATRIX FOR NEWTON’S METHOD ....................................................................................... 48 3.4 ACCURACY OF THE SCHEMES ............................................................................................................. 51 3.5 SUMMARY .................................................................................................................................... 54

CHAPTER 4 MODEL RESULTS WITH BASIC INPUT ............................................................................... 55

4.1 DYNAMICS OF THE SYSTEM ............................................................................................................... 56 4.1.1 Reciprocation of the piston ................................................................................................... 56 4.1.2 Normal forces on the pin ...................................................................................................... 57 4.1.3 Rotation of the pin ................................................................................................................ 60 4.1.4 Friction force and power loss ................................................................................................ 61

4.2 OIL TRANSPORT PATTERN ................................................................................................................ 64 4.2.1 Intake stroke ......................................................................................................................... 67 4.2.2 Compression stroke ............................................................................................................... 69 4.2.3 Expansion stroke ................................................................................................................... 70 4.2.4 Exhaust stroke ....................................................................................................................... 72 4.2.5 Summary ............................................................................................................................... 73

4.3 PARAMETRIC STUDY ........................................................................................................................ 74 4.3.1 Installation clearance ............................................................................................................ 74

4.3.1.1 Between the pin and the pin boss ................................................................................................... 74 4.3.1.2 Between the pin and the small end bearing .................................................................................... 77

4.3.2 Roughness of the surfaces .................................................................................................... 79 4.3.2.1 Roughness of the pin boss ............................................................................................................... 80 4.3.2.2 Roughness of the small end bearing ............................................................................................... 83

4.3.3 Coefficient of friction ............................................................................................................ 85 4.3.3.1 Coefficient of friction between pin and pin boss ............................................................................ 85 4.3.3.2 Coefficient of friction between pin and small end bearing ............................................................. 89

4.3.4 Engine speed ......................................................................................................................... 92 4.3.5 Cavitation pressure ............................................................................................................... 93

4.4 EFFECTS OF DEFORMATION OF THE PIN ............................................................................................... 95

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4.5 SUMMARY .................................................................................................................................. 102

CHAPTER 5 APPLICATION TO A MODERN GASOLINE ENGINE .......................................................... 104

5.1 INPUT SETUP ............................................................................................................................... 104 5.2 RESULTS WITH RIGID AND ELASTIC COMPONENTS ............................................................................... 107 5.3 EFFECTS OF DESIGN FEATURES ON THE PIN BOSS ................................................................................ 113

CHAPTER 6 CONCLUSIONS .............................................................................................................. 117

6.1 CONCLUSIONS .............................................................................................................................. 117 6.2 FUTURE WORK ............................................................................................................................ 118

REFERENCES ....................................................................................................................................... 120

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List of Figures

FIGURE 1.1 POWER CYLINDER SYSTEM AND THE MOTIONS OF THE PISTON ................................................................................... 19 FIGURE 1.2 FORCES ON THE PISTON EXCLUDING FRICTION FORCES AT EARLY EXPANSION STROKE ...................................................... 20 FIGURE 1.3 CONNECTION OF PISTON AND CON ROD THROUGH THE PIN ....................................................................................... 21 FIGURE 1.4 EXPANDED VIEW OF THE INTERFACES WITH THE PIN ................................................................................................ 21 FIGURE 2.1 IMPORTANT GEOMETRICAL PARAMETERS OF THE PISTON PIN LUBRICATION MODEL ........................................................ 26 FIGURE 2.2 AXIAL OIL SLOTS ON THE PISTON PIN BOSS AND THE CORRESPONDING INTERFACES ........................................................ 30 FIGURE 2.3 CIRCUMFERENTIAL OIL GROOVES ON THE PISTON PIN BOSS AND THE CORRESPONDING INTERFACES ................................... 31 FIGURE 2.4 FORCES ON THE PISTON, PIN, CON ROD, AND CRANKSHAFT WITHOUT FRICTION ............................................................ 35 FIGURE 2.5 RELATION BETWEEN BOUNDARY FRICTION COEFFICIENT AND RELATIVE VELOCITY ........................................................... 38 FIGURE 3.1 OVERALL STRUCTURE OF THE NUMERICAL SOLVER .................................................................................................. 43 FIGURE 3.2 DIFFERENCE IN THE PRESSURES IN THE MIDDLE OF THE PIN AT PEAK CYLINDER PRESSURE WITH DIFFERENT GRIDS AND FIRST

ORDER ACCURACY SCHEME ....................................................................................................................................... 53 FIGURE 3.3 DIFFERENCE IN THE PRESSURES IN THE MIDDLE OF THE PIN AT PEAK CYLINDER PRESSURE WITH DIFFERENT GRIDS AND SECOND

ORDER ACCURACY SCHEME ....................................................................................................................................... 53 FIGURE 4.1 AXIAL ACCELERATION OF THE PISTON AND PIN IN THE BASELINE CALCULATION .............................................................. 57 FIGURE 4.2 HYDRODYNAMIC NORMAL FORCES ON THE PIN IN THE BASELINE CALCULATION ............................................................. 57 FIGURE 4.3 ASPERITY CONTACT NORMAL FORCES ON THE PIN IN THE BASELINE CALCULATION .......................................................... 58 FIGURE 4.4 MAGNITUDE OF HYDRODYNAMIC AXIAL FORCES ON THE PIN IN THE BASELINE CALCULATION ............................................ 58 FIGURE 4.5 MAGNITUDE OF ASPERITY CONTACT AXIAL FORCES ON THE PIN IN THE BASELINE CALCULATION ........................................ 59 FIGURE 4.6 NET AXIAL FORCE ON THE PIN IN THE BASELINE CALCULATION .................................................................................... 59 FIGURE 4.7 NET ASPERITY CONTACT NORMAL FORCE ON THE PIN IN THE BASELINE CALCULATION ..................................................... 60 FIGURE 4.8 ANGULAR VELOCITIES OF CON ROD AND PIN IN THE BASELINE CALCULATION ................................................................. 61 FIGURE 4.9 FRICTION FORCES ON THE PIN IN THE BASELINE CALCULATION ................................................................................... 62 FIGURE 4.10 NET FRICTION FORCE ON THE PIN IN THE BASELINE CALCULATION ............................................................................. 62 FIGURE 4.11 EFFECTIVE COEFFICIENT OF FRICTION IN BOUNDARY LUBRICATION IN THE BASELINE CALCULATION ................................... 63 FIGURE 4.12 FRICTION POWER LOSS IN THE BASELINE CALCULATION .......................................................................................... 64 FIGURE 4.13 TYPICAL TRACES OF THE PISTON SLIDING VELOCITY AND THE FRICTION FORCE BETWEEN THE PISTON AND THE CYLINDER FROM

THE PISTON SKIRT LUBRICATION MODEL ....................................................................................................................... 64 FIGURE 4.14 DISTRIBUTION OF ASPERITY CONTACT PRESSURE AT 384 CA IN THE BASELINE CALCULATION ......................................... 65 FIGURE 4.15 DISTRIBUTION OF HYDRODYNAMIC PRESSURE AT 384 CA IN THE BASELINE CALCULATION ............................................. 66 FIGURE 4.16 DISTRIBUTION OF VOID RATIO AT 384 CA IN THE BASELINE CALCULATION ................................................................. 66 FIGURE 4.17 DISTRIBUTION OF PRESSURES AND OIL EXCHANGE AT THE BOUNDARIES AT 90 CA IN THE BASELINE CALCULATION .............. 68

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FIGURE 4.18 DISTRIBUTION OF PRESSURES AND OIL EXCHANGE AT THE BOUNDARIES AT 135 CA IN THE BASELINE CALCULATION ............ 69 FIGURE 4.19 DISTRIBUTION OF PRESSURES AND OIL EXCHANGE AT THE BOUNDARIES AT 315 CA IN THE BASELINE CALCULATION ............ 70 FIGURE 4.20 DISTRIBUTION OF PRESSURES AND OIL EXCHANGE AT THE BOUNDARIES AT 380 CA IN THE BASELINE CALCULATION ............ 71 FIGURE 4.21 CUMULATIVE OIL VOLUME ABSORBED FROM AMBIENT IN THE BASELINE CALCULATION ................................................. 72 FIGURE 4.22 DISTRIBUTION OF PRESSURES AND OIL EXCHANGE AT THE BOUNDARIES AT 675 CA IN THE BASELINE CALCULATION ............ 73 FIGURE 4.23 NORMAL FORCES ON THE PIN WITH DIFFERENT PIN-PIN BOSS INSTALLATION CLEARANCES ............................................. 75 FIGURE 4.24 AVERAGE OIL FILM THICKNESS BETWEEN PIN AND PIN BOSS WITH DIFFERENT PIN-PIN BOSS INSTALLATION CLEARANCES ....... 76 FIGURE 4.25 HYDRODYNAMIC PRESSURE ON A CROSS-SECTION WITH THE SAME BOUNDARY CONDITIONS AND DIFFERENT INSTALLATION

CLEARANCE. .......................................................................................................................................................... 76 FIGURE 4.26 ANGULAR VELOCITIES OF THE PIN WITH DIFFERENT PIN-PIN BOSS INSTALLATION CLEARANCES ........................................ 77 FIGURE 4.27 NORMAL FORCES ON THE PIN WITH DIFFERENT PIN-SMALL END INSTALLATION CLEARANCES .......................................... 78 FIGURE 4.28 ANGULAR VELOCITIES OF THE PIN WITH DIFFERENT PIN-SMALL END INSTALLATION CLEARANCES ..................................... 78 FIGURE 4.29 NORMAL FORCES ON THE PIN WITH DIFFERENT ROUGHNESS ON THE PIN BOSS AND THE SMALL END BEARING .................... 80 FIGURE 4.30 NORMAL FORCES ON THE PIN WITH DIFFERENT PIN BOSS ROUGHNESS ....................................................................... 81 FIGURE 4.31 ANGULAR VELOCITIES OF THE PIN WITH DIFFERENT PIN BOSS ROUGHNESS .................................................................. 82 FIGURE 4.32 FRICTION FORCE BETWEEN THE PIN AND THE SMALL END BEARING WITH DIFFERENT PIN BOSS ROUGHNESS ....................... 82 FIGURE 4.33 FRICTION FORCE BETWEEN THE PIN AND THE PIN BOSS WITH DIFFERENT PIN BOSS ROUGHNESS ...................................... 83 FIGURE 4.34 FRICTION POWER LOSS AND FMEP WITH DIFFERENT PIN BOSS ROUGHNESS ............................................................... 83 FIGURE 4.35 ANGULAR VELOCITIES OF THE PIN WITH DIFFERENT SMALL END BEARING ROUGHNESS .................................................. 84 FIGURE 4.36 NORMAL FORCES ON THE PIN WITH DIFFERENT SMALL END BEARING ROUGHNESS ....................................................... 84 FIGURE 4.37 ANGULAR VELOCITIES OF THE PIN WITH DIFFERENT COEFFICIENT OF FRICTION BETWEEN PIN AND PIN BORE ...................... 85 FIGURE 4.38 NORMAL FORCES ON THE PIN WITH DIFFERENT COEFFICIENT OF FRICTION BETWEEN PIN AND PIN BORE ............................ 86 FIGURE 4.39 SLIDING OIL FLOW RATE INTO THE CRITICAL REGION OF THE SMALL END WITH DIFFERENT COEFFICIENT OF FRICTION BETWEEN

PIN AND PIN BORE ................................................................................................................................................... 86 FIGURE 4.40 PRESSURE DISTRIBUTIONS AND OIL FLOW THROUGH THE BOUNDARIES AT 135 CA WITH DIFFERENT COEFFICIENT OF FRICTION

BETWEEN PIN AND PIN BORE ..................................................................................................................................... 88 FIGURE 4.41 PRESSURE DISTRIBUTIONS AND OIL FLOW THROUGH THE BOUNDARIES AT 315 CA WITH DIFFERENT COEFFICIENT OF FRICTION

BETWEEN PIN AND PIN BORE ..................................................................................................................................... 88 FIGURE 4.42 NORMAL FORCES ON THE PIN WITH DIFFERENT COEFFICIENT OF FRICTION BETWEEN PIN AND SMALL END BEARING ............. 89 FIGURE 4.43 ANGULAR VELOCITIES OF THE PIN WITH DIFFERENT COEFFICIENT OF FRICTION BETWEEN PIN AND SMALL END BEARING ....... 90 FIGURE 4.44 FRICTION FORCE BETWEEN THE PIN AND THE SMALL END BEARING WITH DIFFERENT COEFFICIENT OF FRICTION .................. 91 FIGURE 4.45 EFFECTIVE COEFFICIENT OF FRICTION BETWEEN THE PIN AND THE SMALL END BEARING WITH DIFFERENT SLIDING COEFFICIENT

OF FRICTION .......................................................................................................................................................... 91 FIGURE 4.46 FRICTION POWER LOSS AND FMEP WITH DIFFERENT COEFFICIENT OF FRICTION BETWEEN PIN AND SMALL END BEARING ..... 91 FIGURE 4.47 NORMAL FORCES ON THE PIN WITH DIFFERENT ENGINE SPEEDS ............................................................................... 92

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FIGURE 4.48 SLIDING OIL FLOW RATE INTO THE CRITICAL REGION OF THE SMALL END WITH DIFFERENT ENGINE SPEEDS ......................... 93 FIGURE 4.49 NORMAL FORCES ON THE PIN WITH DIFFERENT CAVITATION PRESSURES .................................................................... 94 FIGURE 4.50 HYDRODYNAMIC PRESSURE [BAR] AT 380 CA WITH DIFFERENT CAVITATION PRESSURES ............................................... 95 FIGURE 4.51 AVERAGE OIL FILM THICKNESS IN THE CRITICAL REGION OF THE PIN-SMALL END INTERFACE ........................................... 95 FIGURE 4.52 DEFORMATION OF THE PIN WITH TWO 1000N FORCES AT THE ENDS AND A 2000N FORCE IN THE MIDDLE ...................... 96 FIGURE 4.53 NORMAL FORCES ON THE INTERFACES WITH RIGID AND ELASTIC PINS ........................................................................ 97 FIGURE 4.54 ANGULAR VELOCITIES OF RIGID AND ELASTIC PINS ................................................................................................. 97 FIGURE 4.55 CLEARANCE, DEFORMATION, OIL FLOW RATE, AND PRESSURES ON THE INTERFACES AT 120 CA ..................................... 99 FIGURE 4.56 CLEARANCE, DEFORMATION, OIL FLOW RATE, AND PRESSURES ON THE INTERFACES AT 300 CA ................................... 100 FIGURE 4.57 CLEARANCE, DEFORMATION, OIL FLOW RATE, AND PRESSURES ON THE INTERFACES AT 500 CA ................................... 101 FIGURE 4.58 FRICTION POWER LOSS WITH RIGID AND ELASTIC PINS .......................................................................................... 102 FIGURE 5.1 COLD RADIAL VARIANCE (MICROMETER) OF THE PIN .............................................................................................. 105 FIGURE 5.2 ADDITIONAL RADIAL VARIANCE (MICROMETER) OF THE PIN BOSS DUE TO THERMAL LOAD ............................................. 105 FIGURE 5.3 TOTAL RADIAL VARIANCE (MICROMETER) OF THE PIN ............................................................................................ 106 FIGURE 5.4 COMBUSTION PRESSURE TRACE USED IN THE CALCULATION .................................................................................... 106 FIGURE 5.5 NORMAL FORCES ON THE PIN WITH DIFFERENT CONSIDERATIONS OF DEFORMATION .................................................... 107 FIGURE 5.6 FRICTION FORCE BETWEEN THE PIN AND THE PIN BOSS WITH DIFFERENT CONSIDERATIONS OF DEFORMATION ................... 108 FIGURE 5.7 FRICTION POWER LOSS AND FMEP BETWEEN THE PIN AND THE SMALL END WITH DIFFERENT CONSIDERATIONS OF

DEFORMATION ..................................................................................................................................................... 108 FIGURE 5.8 ANGULAR VELOCITY OF THE PIN WITH DIFFERENT CONSIDERATIONS OF DEFORMATION ................................................. 109 FIGURE 5.9 ASPERITY CONTACT PRESSURE [BAR] AT 385 CA IN THE RIGID CASE ......................................................................... 109 FIGURE 5.10 HYDRODYNAMIC PRESSURE [BAR] AT 385 CA IN THE RIGID CASE ........................................................................... 110 FIGURE 5.11 MAGNIFIED DEFORMATION [𝝁M] OF THE PIN AT 385 CA IN THE CASE WITH ELASTIC PIN ........................................... 110 FIGURE 5.12 CLEARANCE BETWEEN PIN AND SMALL END AT 385 CA ....................................................................................... 111 FIGURE 5.13 ASPERITY CONTACT PRESSURE [BAR] AT 385 CA IN THE CASE WITH ELASTIC PIN ....................................................... 111 FIGURE 5.14 HYDRODYNAMIC PRESSURE [BAR] AT 385 CA IN THE CASE WITH ELASTIC PIN ........................................................... 111 FIGURE 5.15 MAGNIFIED DEFORMATION [𝝁M] OF THE PIN BOSS AT 385 CA IN THE CASE WITH ELASTIC PIN AND PIN BOSS ................. 112 FIGURE 5.16 HYDRODYNAMIC PRESSURE [BAR] AT 385 CA IN THE CASE WITH ELASTIC PIN AND PIN BOSS ....................................... 112 FIGURE 5.17 NORMAL FORCES ON THE PIN WITH DIFFERENT PIN BOSS DESIGNS .......................................................................... 114 FIGURE 5.18 ANGULAR VELOCITY OF THE PIN WITH DIFFERENT PIN BOSS DESIGNS ....................................................................... 115 FIGURE 5.19 HYDRODYNAMIC PRESSURE DISTRIBUTION AT 340 CA WITH DIFFERENT PIN BOSS DESIGNS ......................................... 115 FIGURE 5.20 FRICTION FORCE BETWEEN PIN AND PIN BOSS AND FMEP WITH DIFFERENT PIN BOSS DESIGNS .................................... 116

15

List of Tables TABLE 3.1 MAJOR PARAMETERS FOR TESTING THE ACCURACY OF THE NUMERICAL SCHEMES ........................................................... 52 TABLE 4.1 ENGINE SPECIFICATION OF THE BASELINE CALCULATION ............................................................................................. 55 TABLE 4.2 OTHER IMPORTANT PARAMETERS IN THE BASELINE CALCULATION ................................................................................ 55 TABLE 5.1 SPECIFICATION AND RUNNING CONDITION OF THE GASOLINE ENGINE STUDIED IN SECTION 5.1 ........................................ 104

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Chapter 1 Introduction

1.1 Background and Motivation

Internal combustion engines have been widely used in the propulsion systems of

automotive vehicles and are expected to keep their predominance in the market in the

following decades [1]. Yet, continuous increase of the consumption of motor fuels with the

expansion of the car ownership has imposed to the world a significant challenge in terms of

environmental disruption and resource depletion. In order to address this challenge, extensive

effort has been made to improve the efficiency of the engines.

The energy loss due to the friction in the piston assembly is estimated to account for

15% of the total loss in typical engines [2], and 55-72% of it can be attributed to the piston and

the connecting rod (con rod) [3]. As the connection between these two components, the piston

pin is crucial to the performance of the entire engine.

Because of the limited surface area, the pressure taken by the pin during engine

operation can attain 100 MPa, and a large proportion of it is supported by boundary

lubrication. As a result, the pin still experiences severe wear and sometimes seizure in the

development of modern engines. Quiet often, standard protocols to change the mechanical and

geometrical designs of the pin fail to resolve the issue and costly DLC coating has to be applied.

In an experimental study on the Friction Mean Effective Pressure (FMEP) of a turbo-

charged gasoline engine, it was observed that modifications to the piston pin boss, such as axial

oil slots, circumferential oil grooves, and larger diameters, are able to reduce the FMEP

significantly [4]. However, the exact mechanisms contributing to the observed friction change

are largely unclear although the evidences suggested that the amount of oil lubricating the

pin/pin-bore interface may play a critical role in pin friction. Therefore, an adequate analytical

tool is needed in the industry to help develop new designs systematically and cost-effectively.

While maximizing the hydrodynamic pressure generation and minimizing direct asperity

contact are the goal to reduce wear, friction, and risk for seizure, there is limited understanding

how the lubricant outside of the interfaces of the pin and its mating parts interacts with the

18

primary and secondary motion of the pin, and how oil is transported to the area bearing the

thrust force. Furthermore, the lubrication depends highly on the rotation of the pin, the

geometry of the pin and the bearings, the structural deformation of the components, and the

surface roughness. On the other hand, the rotation of the pin is also dependent on the

lubrication. This introduces a unique challenge for modeling the pin lubrication problem, as the

primary motion of the system needs to be predicted as well. In addition, although the

interfaces, namely, pin and piston and pin and con rod, are completely separated geometrically,

they are affecting each other through their influence on the rotation of the pin.

1.1.1 Power cylinder system

Figure 1.1 shows the main components of the power cylinder system in a four-stroke

internal combustion engine at early expansion stroke, when the combustion of the fuel

generates enormous pressure in the combustion chamber and pushes the piston to move

downwards. Guided by the cylinder liner, the primary motion of the piston is the reciprocation

along the axial direction. This motion is transferred to the rotation of the crankshaft through

the piston pin and the connecting rod.

Due to the gap between the piston and the liner, which is usually in the order of 10

micrometers, the piston also translates in the lateral direction and tilts around the pin. The

lateral motion and tilting are referred to as the piston secondary motion.

Figure 1.2 shows the forces acting on the piston from other components with the

friction neglected. In the axial direction, the force from the pin is balancing the pressure from

the combustion chamber. In the lateral direction, it balances the forces from the liner. Both the

forces from the pin and the liner are contributed by hydrodynamic pressure and solid-solid

contact pressure.

Bai [5], Totaro [6], and Zhen [7-9] developed successively a comprehensive model to

study the interaction between the piston and the liner. The model integrates the dynamics of

the components, structural deformation, friction generation on the interface, lubrication and

oil transport between the piston skirt and the liner, and the breaking-in of the piston-skirt

19

surfaces. The model is also able to calculate the piston secondary motion, which will be directly

associated with this study.

Figure 1.1 Power cylinder system and the motions of the piston

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Figure 1.2 Forces on the piston excluding friction forces at early expansion stroke

1.1.2 Piston pin system

Figure 1.3 [10] shows how the piston pin is installed inside the piston and the con rod.

The directions of the axes are the same as in Figure 1.1. In most of the modern engines, the pin

has a fully floating configuration where it is able to rotate around its axis (𝑦). The rotation will

be referred to as the primary motion of the pin.

The pin has a basic shape of a hollow cylinder, with the outer diameter more than one

tenth of the length. Therefore, in the expanded view of its side surface, as shown in Figure 1.4,

the length in both axial and circumferential directions have the same order of magnitude of 10s

of millimeters. The interface between the pin and the con-rod small-end is in the center,

whereas the interfaces between the pin and the piston pin boss are on the two ends.

Although the centers of the pin, the con rod small end, and the piston pin boss are

overlapping at 𝑃 in Figure 1.3, their locations will be different because of the clearance among

the components, which are usually in the order of 10 micrometers. The relative movement will

be referred to as the secondary motion of the system.

The secondary motion will determine the interaction on the interfaces, such as the

generation of hydrodynamic pressure and asperity contact pressure. The distribution of the

pressures, in turn, will affect the dynamics. This makes it more challenging to simulate the

system.

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Figure 1.3 Connection of piston and con rod through the pin

Figure 1.4 Expanded view of the interfaces with the pin

1.2 Existing Work

There have been both experimental and computational studies on the piston pin

system. In 1997, Suhara et al. [11] designed an original device to measure the friction on the

piston pin boss bearing from the phase difference between the rotational oscillations of the

bearing and the semi-floating pin. Although the oscillation signals were affected by the

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intermediate spring, the device was able to help generalize the pattern of friction during an

engine cycle and study the effects of different design parameters. Iwasaki et al. [12] developed

a measurement device for a gasoline engine that collected bending stress of the pin as the

friction on the small end bearing. It was found that at higher engine speeds, the edges of the

bearing would experience severe asperity contact, which can be alleviated with the addition of

side relief to the bearing. The device was also able to measure the secondary motion of the pin,

showing that the pin would stay with the upper side of the small end bearing during most of the

engine cycle.

Clark et al. [13] tested the dynamics of the floating piston pin in a diesel engine and

identified the outer edges of the small end and the inner edges of piston pin bore as the places

with highest dry contact. Pin rotation was also measured and was shown to have a clockwise

cumulative rotation angle as defined in Figure 1.1. Abed et al [14] developed a motion data

acquisition system to capture the rotation angle of the pin against the piston pin bore, and

received a similar pattern in terms of net rotation.

Miura et al. [15] used thin-film sensors to measure the oil pressure in the fully flooded

interface between the pin and the piston pin boss. It was found that the highest hydrodynamic

pressure occurs near the inner edges of the pin boss and around 𝜑 = ±45° in the

circumference as shown in Figure 1.3.

On the modeling side, Ligier and Ragot [16] developed a simple hydrodynamic

lubrication program with a refined contact model to simulate the general behavior of the piston

pin, including the motion and the wear. The surface of the pin is assumed to be fully flooded

with lubricating oil, and the load carrying capacity is provided only by the squeeze of the oil.

Fridman, Piraner, and Clark [17] developed a model for the pin in heavy-duty diesel

engines. The results showed that the pin would slightly rotate in the direction of the crankshaft

rotation, which is consistent with the experimental results discussed above. It also showed a

contact pressure distribution that is similar to the wear pattern observed in the pin bushing.

The model does not include the structural deformation of the bearings.

Wang et al. [18] carried out an EHD lubrication analysis between the pin and the small

end bush, assuming the clearance is entirely filled with lubricant. Bertocchi, Giacopini, and Dini

23

[19] developed an algorithm to study the small end and big end based on the complementarity

relation between the hydrodynamic pressure and void fraction of the clearance of the

interfaces to guarantee the conservation of mass while considering cavitation. However, the

components are simplified as rigid.

Shi [20] designed a more comprehensive analytical model for the floating piston pin that

coupled the pin rotation, elastic deformation, lubrication, cavitation, and asperity contact. It

showed that the contact pressure became more concentrated with the deflection of the pin,

which can be improved with higher stiffness. Because the model calculates the rotation of the

pin with explicit time-marching integration, some of the results are unstable and oscillating

during the expansion stroke of the engine.

Ba et al. [21] built a multi-body dynamic model to analyze the lubrication for the piston

pin joint. It mainly consisted of an EHD solver that included film cavitation and asperity contact.

Different modifications to the geometries of the pin bore were tested to provide guidance for

the design. The structural deformation of the pin is simplified as bending deflection, whereas

the deformation of the pin boss is calculated from structural analysis rather than implicitly

computed in the model.

1.3 Objectives

The existing models for piston pin lubrication have at least one of the following

simplifications

• The interfaces between the pin and its mating part are fully filled with lubricant.

• The engine components are rigid.

• The engine components are not rigid, but their mechanical deformation is pre-

determined.

• Asperity contact pressure is not considered.

• The rotation of the pin is calculated explicitly.

The main objective of this study is to develop a comprehensive piston pin lubrication

model that has the following features

24

• To integrate the major processes critical to the lubrication of the pin, including

primary (pin rotation) and secondary motion of the pin, structural deformation of

the components, oil transport at the interfaces, hydrodynamic lubrication with mass

conservation between full and partial film regions, asperity contact and friction of

boundary lubrication considering stick and slip of the pin with its interfaces.

• To couple the two interfaces, which are indirectly affecting each other as both the

small end and the pin boss have forces on the pin.

• To apply an implicit numerical scheme that solves the system with robustness and

efficiency. The system is not only highly non-linear, but also very stiff as the net

force/inertia force of the pin is much smaller than the forces at each interface.

• To establish a relation among the major factors and understand how the system

works.

1.4 Thesis Scope

The second chapter introduces the major mechanical processes considered in the

model. It identifies the degrees of freedom in the system dynamics, as well as the equations

regarding their equilibrium. It also specifies the other unknowns and the related correlations,

including the geometry of the interfaces, the hydrodynamic and asperity contact pressures,

generation of friction force, and structural deformation.

The third chapter discusses the solution method of the system. It first introduces the

overall structure of the solver, including the algorithm in the hydrodynamic module that

incorporates full film and partial film regions while guaranteeing mass conservation. It also

explains the finite difference scheme with second-order accuracy to discretize the equations for

the Jacobian matrix of the Newton’s method.

The fourth chapter shows the results of the model with input data derived from a turbo-

charged gasoline engine. It starts with analyzing the general patterns of the dynamics and oil

transport during the engine cycle, followed by the parametric study on the installation

clearance, roughness, coefficient of friction, and engine speed. Finally, the effects of the

structural deformation of the pin is investigated.

25

The fifth chapter discusses the application of the model to a modern gasoline engine

with the profile and structural deformation of the pin boss considered.

The last chapter summarizes the thesis work with conclusions and remarks and suggests

several aspects on this topic to investigate in this future.

26

Chapter 2 Major Factors of the Model

2.1 Kinematics and Dynamics

2.1.1 Definitions and simplifications

Figure 2.1 shows some important geometrical parameters involved in the model. As

explained in the first chapter, the centers of the pin (𝑃), the con rod small end (𝑃!), and the

piston pin boss (𝑃") are three different points, and the relative locations among them are

critical to the system. 𝑄 denotes the center of the con rod big end and 𝐶 denotes the center of

gravity of the con rod. The center of the crank shaft (𝑂) is defined as the origin of the

coordinate system.

Figure 2.1 Important geometrical parameters of the piston pin lubrication model

27

The model has a few simplifications regarding the kinematics and dynamics. First, the

angular velocity of the crank shaft, �̇�, is assumed to be constant throughout the entire engine

cycle.

The forces between the con rod big end and the crank shaft are neglected when

calculating the torque equilibrium of the con rod around the big end because of their limited

arms compared to the forces at the small end.

Finally, the friction forces between the pin and the pin boss and between the pin and

the small end are neglected in the equations of force balance as their magnitudes are much

smaller than the normal forces.

2.1.2 System kinematics

There are five degrees of freedom in the system. The first three are the location of the

pin in the lateral (𝑥) and axial (𝑧) directions and the rotation of the pin along the 𝑦 axis, which

are denoted by 𝑥#, 𝑧#, and 𝛼#. The fourth degree of freedom is the rotation of the con rod, 𝛼#!.

Both 𝛼# and 𝛼#! are defined as positive when they are clockwise from the opposite direction of

𝑧 (𝛼#! is negative in Figure 2.1). The last degree of freedom is the axial location of the piston pin

boss, 𝑧#".

The locations of 𝑃! are dependent on 𝛼#!, whereas the lateral location of 𝑃" can be

imported from the result of the piston skirt lubrication model [5]. This input can be substituted

with no secondary motion if the secondary motion is not available and the results should not

have appreciable difference with the ones with secondary motion provided.

In the model, every time step is represented by the angle 𝛽 of the crankshaft, defined as

clockwise from 𝑧 axis. With 𝛽 known, the location, velocity, and acceleration of the con rod big

end are

𝑥$ = −𝑟% sin 𝛽 (2.1)

𝑧$ = 𝑟% cos 𝛽 (2.2)

28

�̇�$ = −𝑟%𝜔 cos 𝛽 (2.3)

�̇�$ = −𝑟%𝜔 sin 𝛽 (2.4)

�̈�$ = 𝑟%𝜔" sin 𝛽 (2.5)

�̈�$ = −𝑟%𝜔" cos 𝛽 (2.6)

Here 𝑟% is the radius of the crankshaft, which is the fixed distance between 𝑂 and 𝑄.

𝜔 = �̇� is the constant angular velocity of the crankshaft.

The movement of the center of the small end is

𝑥#! = −𝑟% sin 𝛽 − 𝑙! sin 𝛼#! (2.7)

𝑧#! = 𝑟% cos 𝛽 + 𝑙! cos 𝛼#! (2.8)

�̇�#! = −𝑟%𝜔 cos 𝛽 − 𝑙!�̇�#! cos 𝛼#! (2.9)

�̇�#! = −𝑟%𝜔 sin 𝛽 − 𝑙!�̇�#! sin 𝛼#! (2.10)

�̈�#! = 𝑟%𝜔" sin 𝛽 + 𝑙!H�̇�#!I" sin 𝛼#! − 𝑙!�̈�#! cos 𝛼#! (2.11)

�̈�#! = −𝑟%𝜔" cos 𝛽 − 𝑙!H�̇�#!I" cos 𝛼#! − 𝑙!�̈�#! sin 𝛼#! (2.12)

where 𝑙! is the distance between the centers of the small end and the big end.

The distance of the con rod’s center of gravity from the big end is

𝑥&/$ = 𝑥&( cos 𝛼#! − 𝑧&( sin 𝛼#! (2.13)

𝑧&/$ = 𝑥&( sin 𝛼#! + 𝑧&( cos 𝛼#! (2.14)

with 𝑥&( and 𝑧&( the original distance when the con rod is vertical.

Therefore, the motion of 𝐶 can be derived through the motion of 𝑄

𝑥& = 𝑥&/$ + 𝑥&/$ (2.15)

29

𝑧& = 𝑧$ + 𝑧&/$ (2.16)

�̈�& = �̈�$ − H�̇�#!I"𝑥&/$ − �̈�#!𝑧&/$ (2.17)

�̈�& = �̈�$ − H�̇�#!I"𝑧&/$ + �̈�#!𝑥&/$ (2.18)

2.1.3 System dynamics

Each degree of freedom can be described by an equilibrium equation. For the pin, it

takes a distributed force from the small end bearing and another from the piston pin boss, both

of which can be decomposed into the lateral direction and the axial direction. Therefore, the

force balance for the pin is

𝐹)*"+,- − 𝐹+,-"+ −𝑚�̈�# = 0 (2.19)

𝑁)*"+,- − 𝑁+,-"+ −𝑚�̈�# = 0 (2.20)

Here, 𝑚 is the mass of the pin, 𝐹+,-"+ and 𝑁+,-"+ are the lateral and axial forces from

the pin to the piston, and 𝐹)*"+,- and 𝑁)*"+,- are the forces from the con rod to the pin. The

forces are the summation of hydrodynamic and asperity contact pressures projected to the

corresponding directions.

The torque on the pin comes from the friction forces, and determines the angular

acceleration of the pin

𝑓𝑟 − 𝐼�̈�# = 0 (2.21)

Here, 𝑟 is the radius of the pin, 𝐼 is the moment of inertia around its center of gravity.

The net friction force 𝑓 on the pin will be analyzed in detail in the following subsections.

The equilibrium of torques of the con rod around its big end is

𝑚!H�̈�&𝑧&/$ − �̈�&𝑥&/$I + 𝑙!H𝐹)*"+,- cos 𝛼#! + 𝑁)*"+,- sin 𝛼#!I − 𝐼!,/�̈�#! = 0 (2.22)

30

Here, 𝑚! is the mass of the con rod. 𝐼!,/ is the moment of inertia around 𝑄. The first

term on the left-hand side of the equation is the contribution from the inertia, and the second

term comes from the forces at the small end.

Finally, the axial forces on the piston, including the force from the pin and the

combustion force, are balanced in the 𝑧 direction

𝑁+,-"+ − 𝜋𝑟""𝑝)01/ −𝑚"�̈�#" = 0 (2.23)

𝑟" and 𝑚" are the radius and mass of the piston, 𝑝)01/ is the gauge pressure in the

combustion chamber.

2.2 Geometry of the Interfaces

2.2.1 Contours of the interfaces

As shown in Figure 1.4, in a typical pin system, the side surface of the pin can be divided

into 5 sections: the two interfaces between the pin and the pin boss at the two ends of the pin,

the interface between the pin and the small end in the middle, and two areas in ambient

separating the interfaces.

In some other designs, however, the surface will be more complicated. For example,

when there are oil slots along the axial direction or oil grooves in the circumference of the pin

boss, the interfaces will be further divided if the features are considered as boundary, as shown

in Figures 2.2 and 2.3 [4].

Figure 2.2 Axial oil slots on the piston pin boss and the corresponding interfaces

31

Figure 2.3 Circumferential oil grooves on the piston pin boss and the corresponding interfaces

It is important to identify accurately the contours of each section of the interface and

the ambient areas for arbitrary designs of the components. The first main reason is that the

interpolation of clearance needs to be performed within the same section. In addition, the

ambient areas should not have any asperity contact pressure or more hydrodynamic pressure

than atmospheric.

In order to assign the nodes to correct sections, the model performs a simple cluster

analysis. Suppose that there are 𝑛 nodes in total, the algorithm works as follows

• Assign node 1 to section 1.

• If the distance from node 2 to node 1 is smaller than a threshold, assign node 2

to section 1. Otherwise, assign it to a new section 2.

• For each new node 𝑖, use 𝒢 to denote the set of previous nodes whose distances

from 𝑖 are smaller than the threshold.

o If 𝒢 is empty, assign node 𝑖 to a new section.

o If all the nodes in 𝒢 are in the same section 𝑗, assign 𝑖 to this section.

o If the nodes in 𝒢 are from more than one sections 𝑗!, … , 𝑗2, assign node 𝑖

as well as all the nodes in sections 𝑗", … , 𝑗2 to section 𝑗!.

32

2.2.2 Clearance profile between the surfaces

The base shape of the outer surface of the pin, as well as the inner surfaces of the small

end and pin boss, are cylindrical with constant nominal radii. The difference between the

nominal radii of the mating surfaces is named installation clearance and determines the initial

clearance between the components.

According to the exact design of the bearings in the pin boss and small end, the local

radius might be different from the nominal one. In addition, thermal load also alters the shapes

of the surfaces. Since the effects on the clearance by the installation clearance, designed

profile, and thermal load are not changing within an engine cycle, they will be treated as input

to the model and denoted by ℎ(.

Next, the relative locations among 𝑃, 𝑃!, and 𝑃" will affect the geometry of the

interfaces. As their influence changes with crank angle, they will be part of the solution. As a

result, the clearance between the pin and the small end becomes

ℎ! = ℎ( + H𝑧#! − 𝑧#I cosH𝜑 − 𝛼#!I + H𝑥#! − 𝑥#I sinH𝜑 − 𝛼#!I (2.24)

where 𝜑 is the circumferential location on the side surface of the pin, as shown in Figures 1.3

and 1.4.

The clearance between the pin and the pin boss becomes

ℎ" = ℎ( + H𝑧#" − 𝑧#I cos𝜑 + H𝑥#" − 𝑥#I sin𝜑 (2.25)

Finally, the structural deformation of the surfaces, 𝑑, contributes directly to the

geometry. The final clearance profile in the model is a function of motion and deformation.

ℎ = ℎ( + 𝑓3(𝜉) + 𝑑 (2.26)

Here 𝜉 represents the collection of the five degrees of freedom,𝑓3(𝜉) refers to the

relations in equations 2.24 and 2.25.

33

2.3 Asperity Contact Sub-model

The formula developed by Hu et al. [22] based on the Greenwood-Tripp model [23] is

applied to calculate the asperity contact pressure. If the roughness of the pin and the small end

are 𝜎 and 𝜎!, the effective roughness of their interface will be [24]

𝜎!,455 = [𝜎" + 𝜎!" (2.27)

When the clearance between them is smaller than 4 times of the effective roughness,

asperity contact pressure will arise with a magnitude of

𝑝6 = 𝑐! ]4 −ℎ

𝜎!,455^6"

∙ 𝕀3789!,$%% (2.28)

𝑐! and 𝑐" are constant parameters derived from the material properties. 𝕀 is the

Boolean variable indicating whether or not ℎ is lower than 4𝜎455.

Therefore, the asperity contact is only a function of the clearance and does not need to

be treated as an independent variable in the system.

2.4 Hydrodynamic Sub-model

2.4.1 Governing equations

As the clearances of the interfaces are in the order of 10s of micrometers, which is much

smaller than the length and width, it suffices to use the Reynolds equation to describe the flow

of the lubricating oil.

In order to consider the partial film regions, where there is not enough oil to fill the

entire clearance, the present model applies the algorithm designed by Biboulet and Lubrecht

[25] to guarantee mass conservation. In addition to the hydrodynamic pressure 𝑝, a void

fraction 𝜃 is defined as the ratio between the volume of air and the total volume of the local

cell in the finite difference mesh. Thus, the effective oil film thickness is (1 − 𝜃)ℎ, and the

34

original Reynolds equation that only applies to full film region is generalized to the following

form with constant oil density and dynamic viscosity 𝜂

𝜕𝑟"𝜕𝜑

]ℎ:

12𝜂𝜕𝑝𝜕𝜑^ +

𝜕𝜕𝑦]ℎ:

12𝜂𝜕𝑝𝜕𝑦^ =

𝑢2𝜕(ℎ − 𝜃ℎ)𝑟𝜕𝜑

+𝜕(ℎ − 𝜃ℎ)

𝜕𝑡(2.29)

In the full film regions, where the entire clearance is filled with oil and 𝜃 = 0, the

hydrodynamic pressure will be positive. In the partial film regions, where 𝜃 is larger than 0,

there will be no hydrodynamic pressure. Thus, there exists a complementarity relation between

𝑝 and 𝜃

f𝑝 ≥ 0𝜃 ≥ 0𝑝 ∙ 𝜃 = 0

(2.30)

which is equivalent to

𝑝 + 𝜃 − h𝑝" + 𝜃" = 0 (2.31)

The Reynolds equation and the complementarity equation will be associated with 𝑝 and

𝜃 in the system equations.

2.4.2 Boundary conditions

In addition to the distribution and transport of lubricating oil within the interfaces, there

will be oil exchange with the ambient through the boundaries of the interface. The boundary

conditions for the film thickness and pressure of the oil, therefore, have direct impact on the

lubrication condition of the system. Since related measurement data are hardly available, it is

necessary to make reasonable assumptions regarding the boundary conditions.

The baseline setup of the model assumes that the boundary is filled with lubricant at

ambient pressure. This represents an upper limit of the natural oil supply to the system.

The partial film regions on the interfaces are assumed to be at a constant cavitation

pressure that is lower than ambient. Thus, oil will be absorbed into the interface if the area

near the boundary is not fully flooded.

35

2.5 Calculation of Friction

2.5.1 Calculation of friction power loss

Unlike the calculation of the friction power loss between the piston skirt and the

cylinder liner, which is simply the multiplication of the friction force on the interface and the

sliding velocity of the piston, the power loss associated with the pin is not as straightforward.

Figure 2.4 shows the forces on the components of the piston assembly. Note that 𝛼#!

here is negative. In order to calculate the power loss, the power output to the crankshaft

without any friction is needed. The torque exerted on the crankshaft is

𝑇( = 𝐹);𝑟% cos 𝛽 + 𝑁);𝑟% sin β (2.32)

Figure 2.4 Forces on the piston, pin, con rod, and crankshaft without friction

36

The force on the piston from the combustion chamber is the driving force of the system.

Its magnitude is

𝑁)01/ = 𝜋𝑟""𝑝)01/ (2.33)

The difference between𝑁)01/ and 𝑁);, the axial force from the crankshaft to the con

rod big end, provides the acceleration of the pin, the con rod, and the piston

𝑁); = 𝑁)01/ +𝑚�̈�# +𝑚!�̈�& +𝑚"�̈�#" (2.34)

In order to get an expression for 𝐹); that does not contain any other forces except for

𝑁);, the torque balance of the con rod needs to be based on the center of the small end, 𝑃!

𝑚!H�̈�&𝑧&/#! − �̈�&𝑥&/#!I + 𝑙!H𝐹); cos 𝛼#! + 𝑁); sin 𝛼#!I − 𝐼!,;�̈�#! = 0 (2.35)

Here 𝑥&/#! and 𝑧&/#! are the relative location between the center of gravity of the con

rod and 𝑃!, which is only a function of 𝛼#!. 𝐼!,; is the moment of inertia around 𝑃!.

Therefore,

𝐹); =𝐼!,;�̈�#! +𝑚!H�̈�&𝑥&/#! − �̈�&𝑧&/#!I

𝑙! cos 𝛼#!− 𝑁); tan 𝛼#! (2.36)

The torque output with frictionless condition is

𝑇( =𝑟%

cos 𝛼#!m𝐼!,;�̈�#! +𝑚!H�̈�&𝑥&/#! − �̈�&𝑧&/#!I

𝑙!cos 𝛽 + n𝑁)01/ +o𝑚�̈�p sinH𝛽 − 𝛼#!Iq (2.37)

As the rotational speed of the crankshaft is assumed to be constant, the primary motion

of the piston is almost fixed. Therefore, even if there is friction on the interfaces, the force

balance in the axial direction will be the same as equation (2.34) because the friction forces can

be viewed as internal forces and will cancel off each other. This means that 𝑁); will be the same

as well.

The torque balance of the con rod around 𝑃! will become

37

𝑚!H�̈�&𝑧&/#! − �̈�&𝑥&/#!I + 𝑙!H𝐹); cos 𝛼#! + 𝑁); sin 𝛼#!I − 𝐼!,;�̈�#! − 𝑓!𝑟! = 0 (2.38)

with 𝑓! being the total friction force between the pin and the small end and 𝑟! the radius of the

small end bearing.

Therefore,

𝐹); =𝐼!,;�̈�#! +𝑚!H�̈�&𝑥&/#! − �̈�&𝑧&/#!I + 𝑓!𝑟!

𝑙! cos 𝛼#!− 𝑁); tan 𝛼#! (2.36)

The torque output will decrease from 𝑇( by

∆𝑇 = ∆𝐹);𝑟% cos 𝛽 =𝑟!𝑟% cos 𝛽𝑙! cos 𝛼#!

𝑓! (2.37)

The power output is the multiplication of the torque and the angular velocity of the

crankshaft

∆𝒫 = �̇�∆𝑇 (2.38)

Neglecting the secondary motion of the pin, the rotation of the con rod and crankshaft

can be related as

𝑙! sin 𝛼#! = 𝑟% sin 𝛽 (2.39)

𝑙!�̇�#! cos 𝛼#! = 𝑟%�̇� cos 𝛽 (2.40)

From equations (2.37), (2.38), and (2.40), the final friction power loss is

∆𝒫 = 𝑟!𝑓!�̇�#! (2.41)

This indicates that the friction between the pin and the piston pin boss does not directly

generate loss of power. However, as the interfaces are integrated by the rotation of the pin, the

friction will alter the behavior of the pin/small end interface. Detailed analysis will be provided

in the following chapters.

38

2.5.2 Friction in boundary lubrication regime

In the pin system, if the friction force from the small end bearing is larger than the

friction force from the pin boss, the pin will tend to follow the rotation of the con rod and

rotate against the pin boss. Otherwise, the pin will stay with the pin boss and have relative

sliding with the small end. The stick-slip model [26-27], where the coefficient of friction for

boundary lubrication changes with the sliding velocity, is suitable to describe this process.

However, the function for the coefficient is not always differentiable, which can compromise

the robustness of numerical solver.

The model uses an approximation to the stick-slip model to determine the coefficient of

friction. The coefficient between the pin and the small end bearing is

𝜇! = 𝜇!,( m1 −2

1 + 𝑒2<=̇&!?@̇Aq (2.42)

Here 𝜇!,( is the reference coefficient when the sliding velocity is large enough, 𝑘 is a

pre-determined parameter that alters the shape of the curve.

The coefficient between the pin and the pin boss is

𝜇" = 𝜇",( w1 −2

1 + 𝑒2=̇&!x (2.43)

Figure 2.5 shows the curves with different values of 𝑘.

Figure 2.5 Relation between boundary friction coefficient and relative velocity

39

2.5.3 Friction from hydrodynamic shear stress

The total drag on the pin from the hydrodynamic effects is equivalent to the shear stress

on the bearings.

𝜏 = −𝜂𝑢ℎ+ℎ2𝜕𝑝𝑟𝜕𝜑

(2.44)

Here 𝑢 is sliding velocity of the interface.

2.5.4 Summary

The total friction forces on the interface of the pin and the small end bearing (𝑆!) and

the interface between the pin and pin boss (𝑆") are

𝑓! = 𝜇!,( m1 −2

1 + 𝑒2<=̇&!?@̇Aq ∙o𝑝6

B!

+om−𝜂𝑟H�̇�#! − �̇�I

ℎ +ℎ2𝜕𝑝𝑟𝜕𝜑

qB!

(2.45)

𝑓" = 𝜇",( w1 −2

1 + 𝑒2=̇&!x ∙o𝑝6

B"

+ow−𝜂𝑟�̇�#!ℎ

+ℎ2𝜕𝑝𝑟𝜕𝜑

xB"

(2.46)

They are functions of the motion 𝜉, clearance ℎ, and hydrodynamic pressure 𝑝.

2.6 Structural Deformation

The components will deform accordingly as they are loaded with hydrodynamic and

asperity contact pressures. In the model, elastic deformation will be considered with a linear

relation with the pressure distribution.

𝑑 = 𝑀)01+ ∙ (𝑝 + 𝑝6) (2.47)

Here 𝑀)01+ is the sum of compliance matrices of the components in contact. It will be

calculated prior to the model calculation as an input. 𝑑, 𝑝, and 𝑝6 are vectors with the number

of cells in the mesh as their length.

As the hydrodynamic module requires the size of the cells in the mesh to be small to get

accurate pressure distributions, the structural deformation is generally smoother. On the other

40

hand, while the other parts of the Jacobian matrix are very sparse, the compliance matrix is

dense and will take up considerable computation time.

In order to reduce the size of the compliance matrix so that the calculation can be more

efficient and less memory consuming, two different sets of meshes are defined on the pin. The

calculation of hydrodynamic pressure and clearance is performed on the fine mesh, whereas

the deformation is calculated on the coarse mesh. Since it is straightforward to get the matrices

that interpolate and transform the variables between the meshes, the final form of the

deformation is

𝑑 = 𝑀)"5 ∙ 𝑀)01+ ∙ [𝑀5")(𝑝 + 𝑝6)] (2.48)

Suppose that there are 𝑛5 cells in the fine mesh and 𝑛) cells in the coarse mesh, 𝑀5")

will be a matrix with size 𝑛) × 𝑛5 that transforms the pressures to the coarse mesh, 𝑀)"5 will be

a matrix with size 𝑛5 × 𝑛) that transforms the deformation back to the fine mesh.

2.7 Summary

The pin lubrication model developed in this study integrates most of the important

features in the pin system. The dynamics of the engine components are calculated, including

the rotation of the pin.

To calculate the clearance profile of the interfaces, the model considers the fixed

geometry due to thermal load, the effects of the secondary motions among the components,

and the contribution from the structural deformation.

The model considers the effects of roughness on the asperity contact pressure.

Although the roughness is not included in the calculation of hydrodynamic pressure, the

lubrication part of the model incorporates both full film and partial film regions while

guaranteeing mass conservation of oil.

In the boundary lubrication regime, the model uses an approximation to the stick-slip

model to determine the coefficient of friction, which is differentiable and will not complicate

the solution method.

41

The model uses constant viscosity for the lubricant without considering the effects of

temperature and shear thinning.

42

Chapter 3 Solution Method

As described in the previous chapter, the variables to be solved at eath time step are

• Motion of the components 𝜉 = [𝑥# 𝑧# 𝛼# 𝛼#! 𝑧#"]

• Clearance between the surfaces ℎ

• Hydrodynamic pressure 𝑝

• Void fraction 𝜃

• Deformation 𝑑

As will be shown in the next chapter, the system is very stiff in the sense that the net

forces on the pin are much smaller than the corresponding forces on each interface. Therefore,

all the unknowns will be solved implicitly in order to ensure a stable solution.

The model applies the Newton-Raphson method to solve the unknowns by linearizing

the system equations with regards to the differentials of the unknowns. The details will be

introduced in the following subsections.

3.1 Structure of the Solver

3.1.1 Main architecture

Figure 3.1 shows the flow chart of the numerical solver. At the beginning of each time

step 𝑡 represented by the angle of the crankshaft, the results at the previous steps are

extrapolated to get the initial guess of motion, clearance, and deformation

𝜉(D) = 2𝜉(D?!) − 𝜉(D?") (3.1)

ℎ(D) = 2ℎ(D?!) − ℎ(D?") (3.2)

𝑑(D) = 2𝑑(D?!) − 𝑑(D?") (3.3)

43

Then, the initial guess is imported to the hydrodynamic module to calculate the

hydrodynamic pressure 𝑝(D) and void fraction𝜃(D). The asperity contact pressure 𝑝6(D) can be

determined by ℎ(D).

Figure 3.1 Overall structure of the numerical solver

After getting the pressures, the forces can be computed. The equations for motion,

clearance, and deformation are likely to have non-negligible errors instead of being balanced.

The errors are

𝑒F =

⎣⎢⎢⎢⎢⎡

𝐹)*"+,- − 𝐹+,-"+ −𝑚�̈�#𝑁)*"+,- − 𝑁+,-"+ −𝑚�̈�#

𝑓𝑟 − 𝐼�̈�#𝑚!H�̈�&𝑧&/$ − �̈�&𝑥&/$I + 𝑙!H𝐹)*"+,- cos 𝛼#! + 𝑁)*"+,- sin 𝛼#!I − 𝐼!,/�̈�#!

𝑁+,-"+ − 𝜋𝑟""𝑝)01/ −𝑚"�̈�#" ⎦⎥⎥⎥⎥⎤

(3.4)

𝑒3 = ℎ( + 𝑓3H𝜉(D)I + 𝑑(D) − ℎ(D) (3.5)

𝑒G = 𝑀)01+ ∙ n𝑝(D) + 𝑝6(D)p − 𝑑(D) (3.6)

44

Equations (3.5) and (3.6) correspond respectively to equations (2.26) and (2.47).

Next, the Jacobian matrix 𝐽 regarding the system equations is assembled. The details

about the Jacobian matrix can be found in section 3.3.

The amount of adjustment to the variables can be calculated by

⎣⎢⎢⎢⎡∆F∆3∆G∆H∆I⎦⎥⎥⎥⎤

= 𝐽?! ∙

⎣⎢⎢⎢⎡𝑒F𝑒3𝑒G00 ⎦⎥⎥⎥⎤

(3.7)

There are no errors for the Reynolds equation and complementarity equation because

they have been balanced in the hydrodynamic module, which will be discussed in the next

subsection.

The variables are then updated as

𝜉(D) = 𝜉(D) − ∆F (3.8)

ℎ(D) = ℎ(D) − ∆3 (3.9)

𝑑(D) = 𝑑(D) − ∆G (3.10)

The process will be iterated until all the errors are lower than their corresponding

threshold.

In the model, all the variables are non-dimensionalized, although their original values

are shown in the equations above.

3.1.2 Hydrodynamic module with P-q algorithm

In the hydrodynamic module, 𝑃 and 𝜃 are the only unknowns. The dimensionless forms

of the Reynolds equation and the complementarity equation are

𝐹 =𝜕𝜕𝜑 �

𝐻: 𝜕𝑃𝜕𝜑�

+𝜕𝜕𝑌 �

𝐻: 𝜕𝑃𝜕𝑌�

−𝑈2𝜕(𝐻 − 𝜃𝐻)

𝜕𝜑−𝜕(𝐻 − 𝜃𝐻)

𝜕𝑇= 0 (3.11)

45

𝐺 = 𝑃 + 𝜃 − h𝑃" + 𝜃" = 0 (3.12)

with

𝐻 =ℎ𝛿

𝑌 =𝑦𝑟

𝑈 =𝑢𝜔𝑟

𝑃 =𝑝

12𝜂𝜔𝑟"𝛿?"

𝑇 =𝑡𝜔?!

Here, 𝛿 is the average installation clearance between the components, 𝜔 is the angular

velocity of the crankshaft.

Denoting 𝐹# and 𝐹I as the partial derivatives of 𝐹 in equation (3.11) regarding 𝑃 and 𝜃,

respectively, and 𝐺# and 𝐺I are the partial derivatives of 𝐺 in equation (3.12). Using finite

differences, the equations can be discretized with local linearization. If the residuals of 𝐹 and 𝐺

are 𝑟J and 𝑟K , the correction terms for 𝑃 and 𝜃, ∆# and ∆I, can be computed by

w𝐹# 𝐹I𝐺# 𝐺I

x w∆#∆Ix = − �

𝑟J 𝑟K �

(3.13)

Note that 𝐺# + 𝐺I is the identity matrix 𝐼. Thus, by defining a new variable 𝛾 = 𝑃 − 𝜃,

equation (3.13) can be transformed into

(𝐹#𝐺I − 𝐹I𝐺#)∆L= −𝑟J + (𝐹# + 𝐹I)𝑟K (3.14)

Because of the complementarity relation between 𝑃 and 𝜃, the positive entries of 𝛾

indicate full film regions and give the hydrodynamic pressure, whereas the negative entries give

the void ratio in the partial film regions. After the iteration with equation (3.14) has led to a

converging result for 𝛾, the original unknown can be calculated by

𝑃 = max(𝛾, 0) (3.15)

46

𝜃 = max(−𝛾, 0) (3.16)

Compared with equation (3.13), equation (3.14) has half the number of variables,

therefore is more efficient. More detailed information can be found in [25].

3.2 Numerical Schemes

The model applies finite difference schemes. The scaling factors for non-

dimensionalization are

• Clearance ℎM: average installation clearance of the surfaces

• Angular velocity 𝜔M: velocity of the crankshaft

• Distance along the axis of the pin 𝑦M: radius of the pin

• Sliding velocity of the interface: 𝑢M = 𝜔M𝑦M

• Velocity in the axial direction of the piston (𝑧): 𝑣M = 𝜔M𝑟%

• Velocity in the lateral direction (𝑥): 𝑤M = 𝜔MℎM

• Time: 𝑢M = 𝜔M?!

• Pressure: 𝑝M = 12𝜂𝑦M"ℎM?"𝜔M

• Normal force: 𝐹M = max𝐹)01/

• Moment on the con rod: 𝑀M = 𝐹M𝑙!

3.2.1 Reynolds equation

In the dimensionless Reynolds equation

𝜕𝜕𝜑 �

𝐻: 𝜕𝑃𝜕𝜑�

+𝜕𝜕𝑌 �

𝐻: 𝜕𝑃𝜕𝑌�

−𝑈2𝜕(𝐻 − 𝜃𝐻)

𝜕𝜑−𝜕(𝐻 − 𝜃𝐻)

𝜕𝑇= 0 (3.17)

the Poiseuille terms use a central difference scheme, which has second-order accuracy. With a

rectangular mesh defined on the pin, and the indices in the circumferential and axial directions

denoted as 𝑖 and 𝑗, the discretization is

w𝜕𝜕𝜑 �

𝐻: 𝜕𝑃𝜕𝜑�

xN,O=𝐻N,O: + 𝐻NP!,O:

2∆𝜑"H𝑃NP!,O − 𝑃N,OI +

𝐻N,O: + 𝐻N?!,O:

2∆𝜑"H𝑃N?!,O − 𝑃N,OI (3.18)

47

w𝜕𝜕𝑋 �

𝐻: 𝜕𝑃𝜕𝑋�

xN,O=𝐻N,O: + 𝐻N,OP!:

2∆𝑋"H𝑃N,OP! − 𝑃N,OI +

𝐻N,O: + 𝐻N,O?!:

2∆𝑋"H𝑃N,O?! − 𝑃N,OI (3.19)

The Couette term uses upwind scheme. For first order accuracy, the term is

w𝑈2𝜕𝛬𝜕𝜑xN,O

=𝑈2∆𝜑 ∙ �

𝛬N,O − 𝛬N?!,O 𝑈 ≥ 0𝛬NP!,O − 𝛬N,O 𝑈 < 0 (3.20)

with Λ = 𝐻 − 𝜃𝐻. For second order accuracy, the term becomes

w𝑈2𝜕𝛬𝜕𝜑xN,O

=𝑈4∆𝜑 ∙ �

3𝛬N,O − 4𝛬N?!,O + 𝛬N?",O 𝑈 ≥ 0−𝛬NP",O + 4𝛬NP!,O − 3𝛬N,O 𝑈 < 0 (3.21)

The squeeze term uses backward scheme. For first and second order accuracy, it is

discretized as

w𝜕𝛬𝜕𝑇xN,O

(D)

=𝛬N,O(D) − 𝛬N,O

(D?!)

∆𝑇(3.22)

w𝜕𝛬𝜕𝑇xN,O

(D)

=3𝛬N,O

(D) − 4𝛬N,O(D?!) + 𝛬N,O

(D?")

2∆𝑇(3.23)

3.2.2 Equations of system dynamics

In the equations for system dynamics, the acceleration of the motions is discretized with

either second-order or third-order backward scheme

m𝜕"𝜉𝜕𝑡"

q(D)

=𝜉(D) − 2𝜉(D?!) + 𝜉(D?")

∆𝑡"(3.24)

m𝜕"𝜉𝜕𝑡"

q(D)

=2𝜉(D) − 5𝜉(D?!) + 4𝜉(D?") − 𝜉(D?:)

∆𝑡"(3.25)

The schemes provide, respectively, first-order accuracy and second-order accuracy.

48

3.3 Jacobian Matrix for Newton’s Method

Suppose that the fine rectangular mesh on the pin has 𝑛Q cells in the circumferential

direction, 𝑛R cells in the circumferential direction, and 𝑛 = 𝑛Q𝑛R in total.

The forces on the interfaces are the summation of the forces on the cells projected to

the corresponding directions.

𝐹)*"+,- = −𝑎oH𝑝N + 𝑝6NI sinH𝜑N − 𝛼#!IN∈B!

(3.26)

𝑁)*"+,- = −𝑎oH𝑝N + 𝑝6NI cosH𝜑N − 𝛼#!IN∈B!

(3.27)

𝐹+,-"+ = 𝑎oH𝑝N + 𝑝6NI sin𝜑NN∈B"

(3.28)

𝑁+,-"+ = 𝑎oH𝑝N + 𝑝6NI cos𝜑NN∈B"

(3.29)

Here 𝑎 is the area of the cells, 𝑆! and 𝑆" represent respectively the interfaces between

the pin and the small end and between the pin and the pin boss.

According to section 2.3, the asperity contact pressure is

𝑝6N = 𝑐! �4 −ℎN𝜎455

�6"∙ 𝕀3'789$%% (3.30)

Therefore, the equation for the force balance of the pin in the lateral direction

𝐹)*"+,- − 𝐹+,-"+ −𝑚�̈�# = 0 (3.31)

can be discretized as

𝑎 o m𝑝N + 𝑐! �4 −ℎN𝜎455

�6"∙ 𝕀3'789$%%q 𝑠𝑖𝑛H𝜑N − 𝛼#! ∙ 𝕀N∈B!I

N∈B!∪B"

+𝑚2𝑥# − 𝑓H𝑥#,0UVI

∆𝑡" = 0 (3.32)

49

Here 𝑓H𝑥#,0UVI = 5𝑥#(D?!) − 4𝑥#

(D?") + 𝑥#(D?:).

Denoting the left-hand side of the equation as 𝑓, the related entries in the Jacobian

matrix before scaling are

𝛿𝑓𝛿𝑥#

= 2𝑚∆𝑡?" (3.33)

𝛿𝑓𝛿𝛼#!

= −𝑎oH𝑝N + 𝑝6NI cosH𝜑N − 𝛼#!IN∈B!

(3.34)

𝛿𝑓𝛿𝑝N

= 𝑎 sinH𝜑N − 𝛼#! ∙ 𝕀N∈B!I (3.35)

𝛿𝑓𝛿ℎN

= −𝑎 sinH𝜑N − 𝛼#! ∙ 𝕀N∈B!I ∙𝑝6N𝑐"

4𝜎455 − ℎN(3.36)

The terms related to the other equilibriums, as well as the equations for clearance and

deformations, are straightforward and can be derived in a similar fashion.

The Reynolds equation with the second-order accuracy scheme has the following terms

in the Jacobian matrix

• 𝑃N,O

−2𝐻!,#$ +𝐻!%&,#$ +𝐻!'&,#$

2∆𝜑( −2𝐻!,#$ +𝐻!,#%&$ +𝐻!,#'&$

2∆𝑋(

• 𝑃N±!,O and 𝑃N,O±!

𝐻!,#$ +𝐻!±&,#$

2∆𝜑( and𝐻!,#$ +𝐻!,#±&$

2∆𝑋(

• 𝜃N,O (+ if 𝑈 ≥ 0; − if 𝑈 < 0)

3𝐻!,#2∆𝑇 ±

3𝑈𝐻!,#4∆𝜑

• 𝜃N?!,O (only if 𝑈 ≥ 0) and 𝜃NP!,O (only if 𝑈 < 0)

50

−𝑈𝐻!'&,#∆𝜑 and

𝑈𝐻!%&,#∆𝜑

• 𝜃N?",O (only if 𝑈 ≥ 0) and 𝜃NP",O (only if 𝑈 < 0)

𝑈𝐻!'(,#4∆𝜑 and

𝑈𝐻!%(,#4∆𝜑

• 𝐻N,O (− if 𝑈 ≥ 0; + if 𝑈 < 0)

30𝑃!%&,# + 𝑃!'&,# − 2𝑃!,#2𝐻!,#(

2∆𝜑( +30𝑃!,#%& + 𝑃!,#'& − 2𝑃!,#2𝐻!,#(

2∆𝑋( −01 − 𝛷!,#2

∆𝑇 ∓3𝑈01 − 𝜃!,#2

4∆𝜑

• 𝐻N?!,O 3#𝑃()*,+ − 𝑃(,+&𝐻()*,+,

2∆𝜑, + 𝕝-./𝑈#1 − 𝛷()*,+&

∆𝜑

• 𝐻NP!,O 3#𝑃(0*,+ − 𝑃(,+&𝐻(0*,+,

2∆𝜑, − 𝕝-1/𝑈#1 − 𝛷(0*,+&

∆𝜑

• 𝐻N?",O (only if 𝑈 ≥ 0) and 𝐻NP",O (only if 𝑈 < 0)

−𝑈01 − 𝛷𝑖−1,𝑗2

4∆𝜑and

𝑈01 − 𝛷𝑖+1,𝑗24∆𝜑

• 𝐻N,O±! 3#𝑃(,+±* − 𝑃(,+&𝐻(,+±*,

2∆𝑋,

In addition, the sliding velocity 𝑈 is a function of the angular velocities. The velocities

between the pin and the small end bearing and the pin and the pin boss are

𝑈! =1𝑢M𝑟H�̇�#! − �̇�#I = Α̇#! − Α̇# (3.37)

𝑈" =1𝑢M𝑟(−�̇�#) = −Α̇# (3.38)

Here the reference sliding velocity 𝑢M is the multiplication of the radius of the pin 𝑟 and

the angular velocity of the crankshaft.Α̇ represents the dimensionless angular velocity.

Therefore, the Reynolds equation also has entries in the Jacobian matrix regarding 𝛼#!

and 𝛼#.

51

3.4 Accuracy of the Schemes

Suppose that a discretization scheme with first-order accuracy is applied, the calculated

value of a variable at some point 𝑥 will be the actual value of the variable plus an error term,

which is proportional to the step size of the mesh, ℎ

𝛾!(𝑥) = 𝛾(𝑥) + 𝑂(ℎ) (3.39)

If the grid is refined by a factor of 2 in each dimension in space and time, the calculated

value will give

𝛾" = 𝛾 + 𝑂 �ℎ2�

(3.40)

When the grids are sufficiently fine and higher order errors can be neglected, the

discretization error of the coarse grid, O(h), is roughly twice as much as the difference between

the two results, γ! − γ", whereas the discretization error of the fine grid is around the same

size as the difference. If the grid is further refined to get a new result γ:, it will satisfy the

following relation

𝛾" − 𝛾: =12(𝛾! − 𝛾") (3.41)

If a second-order accuracy scheme is applied, the results will become

𝛾! = 𝛾 + 𝑂(ℎ") (3.42)

𝛾" = 𝛾 + 𝑂(ℎ" 4⁄ ) (3.43)

Similarly,

𝛾" − 𝛾: =14(𝛾! − 𝛾") (3.44)

In order to test and compare the accuracy of the schemes, the model is applied to a

simple working example. The motions of the con rod and the piston are pre-determined at each

52

time step, and the asperity contact pressure is not included. The only unknowns are the

location of the pin, the clearance, the hydrodynamic pressure, and the void ratio.

Some important parameters are listed in Table 3.1.

Table 3.1 Major parameters for testing the accuracy of the numerical schemes

Installation clearances 10 µm

Radius of pin 10 mm

Mass of pin 70 g

Mass of piston 200 g

Crankshaft radius 43 mm

Con rod length 138 mm

Engine speed 2000 rpm

Maximum combustion pressure 100 bar

Four cases with different grid sizes are calculated. The first case has a grid with 30 cells

in the circumferential direction of the pin and 25 cells in the axial direction, and a time step of

1-degree crank angle (1 CA). In the next cases, the number of cells is doubled constantly in both

directions, while the time step is decreased by half. The finest grid has 240 × 200 cells and a

time step of 1/8 CA.

The metric for comparison is the hydrodynamic pressure along the middle cross-section

of the pin at the peak cylinder pressure. Figure 3.2 shows the relative difference between the

pressures of two consecutive grids for the scheme with first order accuracy, where equations

(3.17), (3.18), (3.20), (3.22), and (3.24) are used. The difference is scaled with the result of the

coarsest grid.

When the grids are refined, the difference in the pressures is reduced approximately by

half, which is consistent with the previous analysis.

Figure 3.3 shows the similar results for the second order accurate schemes. Every time

the grid is refined by a factor of 2, the difference will decrease roughly by a factor of 4. In

addition, the magnitude of the differences is smaller than that in Figure 3.2. This shows the

53

advantage of the second order accurate scheme, which will be used in the full version of the

model.

Figure 3.2 Difference in the pressures in the middle of the pin at peak cylinder pressure with

different grids and first order accuracy scheme

Figure 3.3 Difference in the pressures in the middle of the pin at peak cylinder pressure with

different grids and second order accuracy scheme

54

3.5 Summary

The model solves the system equations based on the Newton-Raphson method, but

with the hydrodynamic part independent from the main solver to facilitate future modification.

Finite difference schemes with second-order accuracy are applied to discretize the equations in

order to guarantee the efficiency of the model.

55

Chapter 4 Model Results with Basic Input

In this chapter, the model is applied to a turbo-charged gasoline engine to study the

interaction among different factors.

In order to analyze the results systematically, some input data in this chapter are

simplified. First, on all the surfaces, the radial variances from the nominal diameters due to

designs of profiles and thermal expansion are neglected. The structural deformation of the

components is also excluded until the last section of this chapter. With all these simplifications,

the clearance in the interfaces is uniform in the axial direction.

The first two sections will be focused on the overall pattern of system dynamics and oil

transport on the interfaces using a baseline setup. Then, the effects of different parameters will

be analyzed. Finally, the elastic deformation of the pin will be introduced.

The engine specification and other important parameters for the baseline calculation

are listed, respectively, in Table 4.1 and Table 4.2.

Table 4.1 Engine specification of the baseline calculation

Stroke 86 mm

Cylinder bore diameter 77 mm



Peak cylinder pressure 101 bar

Table 4.2 Other important parameters in the baseline calculation

Installation clearance between pin and small end bearing 20 µm

Installation clearance between pin and piston pin boss 20 µm

Pin diameter 23 mm

Pin length 50 mm

56

Roughness of pin 0.1 µm

Roughness of small end bearing 0.8 µm

Roughness of piston pin boss 0.8 µm

Boundary friction coefficient between pin and small end bearing 0.10

Boundary friction coefficient between pin and piston pin boss 0.10

Cavitation pressure 0.7 bar

Initial oil film thickness on the interfaces 5 µm

4.1 Dynamics of the System

4.1.1 Reciprocation of the piston

Figure 4.1 shows the axial acceleration of the piston during a complete engine cycle. 0-

CA is when the piston is at the top dead center (TDC) before the intake stroke.

The acceleration pattern is mostly determined by the rotation of the crankshaft. In the

first half of the downstrokes, the piston is accelerating downwards; in the last half of the

upstrokes, it is decelerating. During these periods of time, the piston is closer to the TDC. When

the piston is closer to the bottom dead center (BDC), it is either accelerating in the upstrokes or

decelerating in the downstrokes.

Since the distance between the centers of the pin and the pin boss is only in the order of

10 micrometers, the axial acceleration of the pin is almost the same as the piston.

57

Figure 4.1 Axial acceleration of the piston and pin in the baseline calculation

4.1.2 Normal forces on the pin

Figures 4.2-4.3 show the normal forces on the pin, which are divided into the

contribution from the hydrodynamic pressure and the asperity contact pressure. A normal

force is defined as the direct integration of corresponding pressure over the interfaces. Since all

the cells on the mesh have the same area, it is equivalent to the multiplication of the area and

the summation of the pressure.

Figure 4.2 Hydrodynamic normal forces on the pin in the baseline calculation

58

Figure 4.3 Asperity contact normal forces on the pin in the baseline calculation

Figure 4.4 Magnitude of hydrodynamic axial forces on the pin in the baseline calculation

59

Figure 4.5 Magnitude of asperity contact axial forces on the pin in the baseline calculation

Figures 4.4-4.5 show the magnitude of the axial forces on the pin. An axial force is the

integration of pressure after projecting to the axial direction of the piston (𝑧). The forces are

similar with the normal forces, indicating that the pressures are concentrated on the top and

bottom of the interfaces. Note that the axial forces from the small end and the pin boss are in

opposite directions.

The net axial force on the pin, which is shown in Figure 4.6, provides its acceleration and

is consistent with Figure 4.1.

Figure 4.6 Net axial force on the pin in the baseline calculation

60

4.1.3 Rotation of the pin

In general, the friction force between the pin and the small end drives the pin to rotate

with the con rod, whereas the friction force from the pin boss keeps the pin from rotating. On

the other hand, the coefficient of friction for boundary lubrication is the same on the two

interfaces in the baseline calculation and is much higher than the coefficient in the

hydrodynamic lubrication regime.

As a result, the pin has the tendency to follow whoever has the higher asperity contact

normal force with it. Figure 4.7 shows the net asperity contact normal force on the pin. It is

defined as the contact normal force from the small end bearing subtracting the force from the

pin boss. Figure 4.8 shows the angular velocities of the con rod and the pin.

Figure 4.7 Net asperity contact normal force on the pin in the baseline calculation

61

Figure 4.8 Angular velocities of con rod and pin in the baseline calculation

There is a connection between the patterns of the rotation and the axial acceleration of

the pin. When the acceleration of the pin is positive in Figure 4.1, there is more contact force

from the small end, which is pushing it upwards, and vice versa. Thus, when the piston is closer

to the TDC (around 0/720 CA and 360 CA), the pin stays with the pin bore and does not rotate.

When the piston is closer to the BDC (around 180 and 540 CA), the pin rotates with the con rod.

There is an exception between 360 and 450 CA, when the pin is accelerating downwards

but rotating with the con rod. This is the most critical period of time during the entire engine

cycle, as the combustion pressure above the piston is so high that both of the interfaces are

generating tremendous amount of pressures. The reason of this exception will be analyzed in

section 4.2.

4.1.4 Friction force and power loss

Figures 4.9 shows the friction forces from the small end and the pin boss, which can

reach 4000 Newtons. Figure 4.10 shows the net friction force on the pin, which determines its

rotation in Figure 4.8. The magnitude of the net friction force is much smaller than the

magnitude on individual interface, indicating a very stiff system.

62

Figure 4.11 shows the effective coefficient of friction in the boundary lubrication regime

at each interface. As analyzed before, the interface without relative rotation has a coefficient

less than the nominal value.

Figure 4.9 Friction forces on the pin in the baseline calculation

Figure 4.10 Net friction force on the pin in the baseline calculation

63

Figure 4.11 Effective coefficient of friction in boundary lubrication in the baseline calculation

As shown in Figure 4.8, the maximum angular velocity of the con rod is around 100 rad

per second, which gives a velocity of 1m/s as the radius of the pin is roughly 0.01m. Although

the speed is low, the maximum friction force on the pin occurs at the same time as the

maximum velocity of the con rod at the beginning of the expansion stroke. As a result, the

friction generates high power loss. Figure 4.12 shows the friction power loss of the pin system,

which can attain several thousand Watts.

By contrast, the sliding speed of the piston is low when the friction between the piston

and the cylinder liner is high. Figure 4.13 shows a typical pattern of the sliding velocity of the

piston and the friction force with the liner. It is calculated with the piston skirt lubrication

model [9] using the data from a different engine for qualitative comparison.

64

Figure 4.12 Friction power loss in the baseline calculation

Figure 4.13 Typical traces of the piston sliding velocity and the friction force between the

piston and the cylinder from the piston skirt lubrication model

4.2 Oil Transport Pattern

In this section, the overall pattern of the distribution and transport of lubricating oil will

be investigated.

Figures 4.14-4.16 show the distribution of asperity contact pressure, hydrodynamic

pressure, and void ratio at 384 CA, where the peak cylinder pressure (PCP) attained in the

combustion chamber takes place. The expanded view of the pin surface is the same as Figure

65

1.4, where 𝑦 is the axial direction of the pin and 𝜑 is the circumferential direction with 0/360

degrees being the top of the original surface and 180 degree as the bottom. The areas circled

by the red lines are the interfaces with the small end and the pin boss.

In order to balance the combustion pressure on the piston, the pin is in contact with the

piston pin boss at the top of their interface. The consequent force is in turn balanced by the

force from small end bearing at the bottom of the other interface. These areas will be referred

to as critical regions.

In the following subsections, some adjustments will be made to the plots of

distributions for simplicity and illustration

• Only half of the interfaces will be shown because of symmetry.

• The interface between the pin and the pin boss will be shifted by 180 degree in

the circumferential direction so that the critical regions on the interfaces are

aligned.

Figure 4.14 Distribution of asperity contact pressure at 384 CA in the baseline calculation

66

Figure 4.15 Distribution of hydrodynamic pressure at 384 CA in the baseline calculation

Figure 4.16 Distribution of void ratio at 384 CA in the baseline calculation

67

4.2.1 Intake stroke

Figure 4.17 shows the distribution of pressures at 90 CA, when neither the pin nor the

con rod is rotating. The red arrows represent the oil flow rate into/out of the boundaries of the

interfaces in the pin axial (𝑦) direction, and the black arrows show the oil flow rate in the

circumferential (𝜑) direction. The left and right boundaries of the two interfaces are not aligned

because of the angle of the con rod.

Since there is no relative rotation, the circumferential oil flow is driven by pressure. On

one hand, oil is squeezed out of the interface in the areas where hydrodynamic pressure builds

up. On the other hand, in the partial film regions, oil is drawn into the interfaces.

After 90 CA, the piston and the pin start to decelerate, and the pin rotates against the

pin boss. As shown in Figure 4.18, oil will be brought from the right portion of the interface to

the center portion, which is the critical region. The large curved arrows in yellow illustrate the

overall path of oil transport.

68

Figure 4.17 Distribution of pressures and oil exchange at the boundaries at 90 CA in the

baseline calculation

69



4.2.2 Compression stroke

In the late compression stroke, the acceleration of the piston becomes negative, and the

pin stops rotating. As shown in Figure 4.19, the relative rotation between the pin and the small

end has the similar effect on the oil transport. Lubricant will be brought to the critical region of

the interface to build hydrodynamic pressure.

70



4.2.3 Expansion stroke

Figure 4.20 shows the pressure distributions and oil flow rate at the boundaries in the

early expansion stroke. Because of the large hydrodynamic pressure, oil will be pushed from

the critical regions back to the right portion of the interfaces.

Although the small end has lower total normal force than the pin bore, it has higher

asperity contact force and drags the pin to rotate. This is mainly because the interfaces

between the pin and the pin boss have around twice as much boundaries as the pin-small end

interface. Because of the assumption that ambient pressure is higher than the constant

71

cavitation pressure, more oil will be sucked into the pin-pin boss interfaces. Figure 4.21 shows

the cumulative amount of oil transport absorbed from ambient into the interfaces through the

boundaries. It is scaled without dimension.



72

Figure 4.21 Cumulative oil volume absorbed from ambient in the baseline calculation

4.2.4 Exhaust stroke

According to Figure 4.8, the rotation of the pin during the exhaust stroke follows a

similar pattern with the compression stroke. The difference is that the pressure in the

combustion chamber is close to the ambient pressure.

In the later part of the exhaust stroke, the piston decelerates, so it is in contact with the

pin at the bottom of the interface. The small end, on the other hand, interacts with the pin at

the top of the interface. The pressure distribution and oil transport are shown in Figure 4.22.

73



4.2.5 Summary

The oil transport pattern on the interfaces is significantly affected by the axial

acceleration of the piston and the pin. In general, the pin will rotate with the con rod small end

when it has positive acceleration and stay with the piston pin boss when it has negative

acceleration.

74

Relative rotation is beneficial for the lubrication condition of the interfaces as it brings

oil to the critical regions. Therefore, it helps that the pin has relative rotation with both of its

mating components during an engine cycle.

Because the interface between the pin and the piston pin boss is divided into two

separate sections, it benefits from larger perimeter of boundary across which oil is absorbed

from ambient.

Although the results discussed in this section is based on an ideal setup where no

structural deformation is considered and some important parameters (roughness, sliding

coefficient of friction, installation clearance, etc.) are the same for both interfaces, it is helpful

to establish a basic oil transport pattern. It provides a powerful perspective that facilitates

further investigation in the effects of individual parameters.

4.3 Parametric Study

A parametric study is conducted to examine the effects of some critical parameters. This

study is to demonstrate the consistency of the model rather than identifying the trend in real

cases as the geometry is over-simplified.

4.3.1 Installation clearance

In the baseline calculation, both the installation clearances of the interfaces are 10

micrometers in radius. In this section, the effects of increasing each clearance are studied and

compared.

4.3.1.1 Between the pin and the pin boss

Figure 4.23 shows the normal forces in the calculation with different clearances

between the pin and the pin boss. The other clearance is 10 micrometers in all cases.

Between 90 CA and 360 CA, the cases with larger clearances have higher hydrodynamic

forces between the pin and the pin boss. This is mainly due to the amount of lubricant on the

interface. Figure 4.24 shows the average oil film thickness between the pin and the pin boss.

When the clearance is larger, the interface is able to gain more lubricant from the ambient. For

75

example, the curve for the 20-micrometer clearance not only grows faster than the other

curves, but also maintains more oil throughout the entire engine cycle.

After 360 CA, the load is so high that the minimum clearance between the pin and the

pin boss becomes very small for all the cases. As a result, the ability of the profiles to generate

hydrodynamic pressure is important. Figure 4.25 shows some artificial one-dimensional profiles

representing a cross-section of the pin-pin boss interface with different clearances. The black

curves represent the surfaces and the blue curves indicate the sliding velocity. The

hydrodynamic pressure was solved with the same minimum clearance, sliding velocity, and

boundary oil film thickness. It shows that a smaller clearance is able to generate higher

hydrodynamic force, which is the reason why there is more asperity contact force on the pin

boss at early expansion stroke when the clearance is larger.

Figure 4.23 Normal forces on the pin with different pin-pin boss installation clearances

76

Figure 4.24 Average oil film thickness between pin and pin boss with different pin-pin boss

installation clearances

Figure 4.25 Hydrodynamic pressure on a cross-section with the same boundary conditions

and different installation clearance.

Figure 4.26 shows the angular velocities of the pin. The most significant difference

occurs in the late compression stroke, when the pin in the 20-micrometer case rotates with the

small end. This is because it has higher hydrodynamic force and therefore lower asperity

contact between the pin and the pin boss.

77

Figure 4.26 Angular velocities of the pin with different pin-pin boss installation clearances

4.3.1.2 Between the pin and the small end bearing

Figure 4.27 shows the normal forces in the calculation with different clearances

between the pin and the small end. The other clearance is 10 micrometers in all cases.

Both interfaces will have more contact when the clearance gets larger. The changes on

the small end are because of the same reason discussed above, whereas the higher asperity

contact pressure on the pin boss can be attributed to the rotation of the pin.

During the late intake stroke and the early compression stroke (90-270 CA), the load is

not high, and the minimum oil film thickness is not as low as PCP timing. Compared to the

baseline calculation, the relative sliding between pin and small end helps build hydrodynamic

pressure. Therefore, although the small end has larger total force than the pin boss, the

contribution from asperity contact is actually less, and the pin stays with pin boss, as shown in

Figure 4.28. As a result, less amount of oil will be brought to the critical region to be used later.

78

Figure 4.27 Normal forces on the pin with different pin-small end installation clearances

Figure 4.28 Angular velocities of the pin with different pin-small end installation clearances

79

4.3.2 Roughness of the surfaces

According to the asperity contact sub-model introduced in Chapter 2, with the same

minimum clearance on an interface, the asperity contact pressure will be much lower when the

roughness is smaller. This means that in the case with lower roughness, the two surfaces can be

closer to each other to generate higher hydrodynamic pressure without producing excessive

contact in the meantime.

In the baseline calculation, both the pin boss and the small end bearing have a surface

roughness of 0.8 micrometer. In this section, the effects of decreasing each roughness are

studied and compared.

Figure 4.29 shows the normal forces in the calculation with different roughness on both

the pin boss and the small end bearing. It is obvious that smaller roughness leads to lower

asperity contact pressure and higher hydrodynamic pressure.

80

Figure 4.29 Normal forces on the pin with different roughness on the pin boss and the small

end bearing

4.3.2.1 Roughness of the pin boss

Figure 4.30 shows the normal forces in the calculation with different roughness only on

the pin boss. The roughness on the small end bearing is 0.8 micrometer in all cases.

The asperity contact force between the pin and the pin boss decreases with the

roughness. However, the contact on the other interface becomes larger.

This is a direct illustration of the subtle connection between the two interfaces. When

the pin-pin boss interface has lower asperity contact, it will have less friction and therefore

more relative rotation. Consequently, the pin-small end interface will have less relative

rotation, which results in less lubricant brought to the critical region and more asperity contact.

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However, because of the limited relative rotation, the friction force between the pin and

the small end bearing is decreased as well, as shown in Figures 4.32. Figure 4.33 shows the

friction force between the pin and the pin boss, and Figure 4.34 shows the total friction power

loss and friction mean effective pressure (FMEP) of the system.

Figure 4.30 Normal forces on the pin with different pin boss roughness

82

Figure 4.31 Angular velocities of the pin with different pin boss roughness

Figure 4.32 Friction force between the pin and the small end bearing with different pin boss roughness

83

Figure 4.33 Friction force between the pin and the pin boss with different pin boss roughness

Figure 4.34 Friction power loss and FMEP with different pin boss roughness

4.3.2.2 Roughness of the small end bearing

Figures 4.35 and 4.36 show the angular velocities and normal forces in the calculation

with different roughness only on the small end bearing. The roughness on the pin boss is 0.8

micrometer in all cases.

84

The situation is similar to the previous subsection. The surface with lower roughness has

less asperity contact while the other surface has more. The pin is rotating less closely with the

con rod.

Figure 4.35 Angular velocities of the pin with different small end bearing roughness

Figure 4.36 Normal forces on the pin with different small end bearing roughness

85

4.3.3 Coefficient of friction

The role of the coefficient of friction in the boundary lubrication regime is

straightforward. With the similar magnitudes of asperity contact force, the interface with a

higher friction coefficient will have higher potential in generating friction force and prevent

relative rotation.

4.3.3.1 Coefficient of friction between pin and pin boss

Figures 4.37 and 4.3 show the rotation of the pin and the normal forces with different

coefficients of friction between the pin and the pin boss. The coefficient of the other interface

is 0.10.

When the friction coefficient between the pin and the pin boss is larger, it will be

difficult for the pin to rotate with the con rod. This is because once the pin starts to rotate

against the pin boss, a much higher friction force will arise and stop the rotation. Therefore, the

lubrication condition on the interface is deteriorated because less oil can be redistributed to

the critical region. As a result, there will be higher asperity contact between the pin and the pin

boss.

Figure 4.37 Angular velocities of the pin with different coefficient of friction between pin and pin bore

86

Figure 4.38 Normal forces on the pin with different coefficient of friction between pin and pin

bore

Figure 4.39 Sliding oil flow rate into the critical region of the small end with different coefficient of friction between pin and pin bore

87

On the other hand, although the pin-small end interface has relative relation during the

entire engine cycle, it has more contact as well. This is exactly due to the non-stopping sliding

between the surfaces.

Figure 4.39 shows the net oil flow rate into the critical region of the small end bearing

due to sliding motion. Here the ‘critical region’ is defined as the area centered around the very

bottom of the small end bearing with a circumferential range of 0.6 rad, as shown by the green

box in Figure 4.40. The flow rate pattern with modified coefficient of friction is significantly

different from the baseline case.

As shown in Figure 4.37, in the baseline case, the angular velocity of the con rod is

almost always smaller than or equal to the pin (the blue curve is always above or overlap with

the black dotted curve). This means that the pin can rotate against the small end only in one

direction, which is from right to the left in Figure 4.41. On the other hand, the clearance around

the critical region is usually the lowest in the entire circumference. This leads to the

accumulation of lubricant to the right of the critical region as discussed in section 4.2.

In the other cases, the pin can rotate against the small end in both directions. Thus, the

oil will not be accumulated as a long-term effect. The only exception is the extruded part of the

interface, as shown in the red circle in Figure 4.40, which receives Couette flow from outside

the boundaries. Therefore, while the baseline case can build hydrodynamic pressure along the

entire pin axis, the other cases only have hydrodynamic pressure in limited area.

88

Figure 4.40 Pressure distributions and oil flow through the boundaries at 135 CA with

different coefficient of friction between pin and pin bore

Figure 4.41 Pressure distributions and oil flow through the boundaries at 315 CA with

different coefficient of friction between pin and pin bore

89

4.3.3.2 Coefficient of friction between pin and small end bearing

Figure 4.42 shows the normal forces on the pin with different coefficients of friction

between the pin and the small end bearing. The coefficient of the other interface is 0.10.

The effects are similar to the previous subsection. Both interfaces will have higher

asperity contact if the friction coefficient is larger.

As shown in Figure 4.43, with larger friction coefficient between the pin and the small

end bearing, the pin will follow the rotation of the con rod during most of the engine cycle.

Figure 4.42 Normal forces on the pin with different coefficient of friction between pin and

small end bearing

90

Figure 4.43 Angular velocities of the pin with different coefficient of friction between pin and small end bearing

Figures 4.44 and 4.45 show the friction force and effective friction coefficient between

the pin and the small end bearing. In the baseline calculation, the pin stays with the pin boss in

the late compression stroke (270-360 CA) and its effective friction coefficient with the small end

bearing reaches the upper bound. As the higher upper bound is increased from 0.10 to 0.15,

more friction force can be generated so that the pin instead rotates with the small end.

However, when the upper bound further increases to 0.20, the extra potential for friction

forces will not be utilized. Therefore, both the friction force and the effective coefficient do not

change anymore. This is also the situation in the early expansion stroke (360-450 CA).

As a result, as long as the other interface stays the same, the overall friction loss will not

increase by much from 0.15 to 0.20, as shown in Figure 4.46.

91

Figure 4.44 Friction force between the pin and the small end bearing with different coefficient

of friction

Figure 4.45 Effective coefficient of friction between the pin and the small end bearing with

different sliding coefficient of friction

Figure 4.46 Friction power loss and FMEP with different coefficient of friction between pin

and small end bearing

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4.3.4 Engine speed

Figure 4.47 shows the normal forces on the pin in the cases with different engine

speeds. When the speed is lower, there is less hydrodynamic pressure builds up in the early

expansion stroke. This is because less amount of oil is brought to the critical region by the

sliding motion, as shown in Figure 4.48.

Figure 4.47 Normal forces on the pin with different engine speeds

93

Figure 4.48 Sliding oil flow rate into the critical region of the small end with different engine

speeds

4.3.5 Cavitation pressure

In the baseline calculation, the cavitation pressure in the partial film region is assumed

to be constant at 0.7 bar. Figure 4.50 shows the hydrodynamic and contact normal forces on

the pin with different cavitation pressures. Compared with the baseline, more oil can be

absorbed into the interfaces if the cavitation pressure is lower because the boundary ambient

pressure is fixed at 1 bar. As a result, there will be more hydrodynamic normal force and less

asperity contact force on both interfaces. However, the difference is limited. Even when the

cavitation pressure is at 1 bar as well such that the suction does not exist, there is still

hydrodynamic pressure built up, as shown in Figure 4.50. This indicates that the relative

rotation plays an important role in bringing the lubricant to the critical regions from the

boundaries unparallel to the sliding direction, as shown in Figures 4.18-20. Figure 4.51 shows

the average oil film thickness in the critical region of the pin-small end interface, which is the

boxed area in Figure 4.50. Before PCP (380 CA), there is an increase in the oil film thickness to

provide hydrodynamic pressure.

Thus, the geometry of the boundary and the clearance profile of the enlarged area on

the bottom of the small end and top of the pin boss can play a role in determining the oil supply

94

to and pressure generation of the contact area. This model can be used to help optimize related

design features.

Figure 4.49 Normal forces on the pin with different cavitation pressures

95

Figure 4.50 Hydrodynamic pressure [bar] at 380 CA with different cavitation pressures

Figure 4.51 Average oil film thickness in the critical region of the pin-small end interface

4.4 Effects of Deformation of the Pin

As stated in Chapter 2, only the component of deformations in the radial direction of

the pin will be included, and all of them are assumed to be elastic.

Figure 4.51 shows the magnified radial deformation on the pin with a simple loaded

condition. Each end of pin takes a normal force of 1000 Newtons on the top, while another

normal force of 2000 Newtons is applied to the middle of the pin on the bottom.

The total deformation can be roughly decomposed into the following parts.

• Bending

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• Ovality of the cross-section

• Local distortion near the loaded places

In Figure 4.51, the contribution of bending at the two ends is around 1 micrometer.

Since the actual loads on the pin can attain an order of magnitude of 10 thousand Newtons, the

effect of bending will be significant.

Figures 4.52 and 4.53 show the normal forces on the interfaces and the angular

velocities of the pin. The ‘no deform.’ label represents the baseline calculation discussed in the

previous chapter, while ‘deform.’ stands for the case with the deformation of the pin included.

Figure 4.52 Deformation of the pin with two 1000N forces at the ends and a 2000N force in

the middle

97

Figure 4.53 Normal forces on the interfaces with rigid and elastic pins

Figure 4.54 Angular velocities of rigid and elastic pins

98

Compared with the baseline, the most pronounced difference of the case with an elastic

pin is that there is much less pin rotation during the engine cycle. The direct reason is the

higher asperity contact force between the pin and the pin boss, as shown in the bottom right of

Figure 4.52.

Figures 4.54 shows the distributions of hydrodynamic pressure, asperity contact

pressure, clearance, and deformation of the cases at 120 CA. The oil flow rate is illustrated by

the red arrows. During the late intake stroke, because the pin bends away from the outer edge

of the pin boss on the top of the interface, the large clearance makes it difficult to generate

either hydrodynamic pressure or contact pressure. In order to compensate it, the inner edge

has to be much closer to the pin. As a result, the contact on the pin boss is higher than the small

end bearing, and the pin will stay with the pin boss and rotate against the small end. Thus, the

asperity contact force between the pin and the small end bearing is smaller than that in the

baseline case.

In the late compression stroke, the load rises significantly due to the pressure in the

combustion chamber. As shown in Figure 4.55, the pressures are concentrated not only on the

inner edge of the pin boss, but also on the outer edge of the small end bearing. The edges,

therefore, have limited clearances between the surfaces. On the other hand, the pin is rotating

relatively with the small end and transporting oil to the critical region. The arrows in Figure 4.55

show that in the baseline case, some of the oil transport will be squeezed out of the interface

when it meets the existing hydrodynamic pressure that has been developed in almost the

entire critical region. However, when the pin is able to deform, the center of the critical region

will have higher clearance because of bending. Therefore, lubricant can be accumulated and

available in the following strokes.

As shown in Figure 4.56, at 500 CA, the accumulated oil will provide hydrodynamic

pressure in the center of the small end bearing, whereas in the baseline case the load is entirely

supported by asperity contact.

99

Figure 4.57 shows the friction power loss of the two cases. With deformation, there is

more power loss around the TDC of expansion stroke (360 CA) and less power loss in the later

expansion stroke. The overall loss is slightly higher than the baseline case.

Figure 4.55 Clearance, deformation, oil flow rate, and pressures on the interfaces at 120 CA

100


101


102

Figure 4.58 Friction power loss with rigid and elastic pins

4.5 Summary

In this chapter, the model was applied to a real gasoline engine to understand the

relation between the pin rotation and the lubrication of the two interfaces. For this purpose,

the profiles of small end and pin bore bearings and their thermal and mechanical deformations

are not considered to ease the analysis. The basic pattern in the interaction of the pin with the

small end and the pin boss was identified, and the effects of various design and operation

parameters were analyzed. The model was found to generate consistent results for the

mechanisms considered.

When the two interfaces are in the same condition in terms of installation clearance,

surface roughness, coefficient of friction, and oil supply at the boundaries, the system will

follow a certain kind of symmetry, which originally comes from the axial acceleration pattern of

the pin. During approximately one half of the engine cycle, there is higher load between the pin

and the small end bearing, and the pin rotates with the small end, bringing oil to the top of the

pin-pin boss interface. During the other half of the cycle, the pin stays with the pin boss and

brings oil to the bottom of the pin-small end interface. Since the amounts of cumulative

rotation of the pin against both pin boss and small end are similar, both interfaces are to some

extent equally lubricated.

103

On the other hand, the system is very stiff in the sense that the forces on the pin from

each interface are more than two orders of magnitude larger than the corresponding net

forces. Therefore, a slight change in the condition of one interface will alter the behavior on the

other as well.

On each interface, the product of the normal asperity contact force and the sliding

coefficient of friction determines an upper bound of the friction force or friction potential,

while the lower bound is zero. As the angular inertia of the pin is much less than the friction

moment of either interface, the friction torque from two interfaces are approximately

balanced. If the pin stays with one interface, it means that this interface has higher friction

potential than the other interface. For instance, if the interface of pin and small end is

frictionless, the friction between the pin and pin boss will be zero no matter how large its

friction potential is. This concept of friction potential and effective friction coefficient is helpful

in understanding the rotation of the pin and its relationship with the lubrication of two

interfaces. Therefore, in order to reduce friction loss, only one of the interfaces need to be

improved. This has been shown by the parametric study where the friction power is smaller

when one of the interfaces has lower roughness or sliding coefficient of friction. The model’s

ability to reflect this interesting process shows the adequacy of the approximation of stick-slip

friction used in this work.

The opposite is also true. If the asperity contact or friction force on one interface is

increased, the other interface will be negatively affected as well, and the overall friction loss of

the system will be higher. An example will be shown in the next Chapter, where the

deformation of the pin leads to concentrated contact pressure on the edge of the pin boss.

104

Chapter 5 Application to a Modern Gasoline Engine

5.1 Input Setup

In this section, the model is applied to a gasoline engine. The important parameters are

shown in Table 5.1.

The profile of the pin boss is included in the form of the variance from the nominal

radius in the radial direction. It consists of the initial cold shape and the expansion due to

thermal load, which are shown respectively in Figures 5.1 and 5.2. Figure 5.3 shows total

variance. The pin boss features axial oil slots, which are treated in the same way as the

boundary regions, as shown in Figure 2.2.

Table 5.1 Specification and running condition of the gasoline engine studied in section 5.1

Stroke 92.8 mm

Cylinder bore diameter 82.5 mm



Peak cylinder pressure 15.6 bar

Installation clearance between pin and small end bearing 20 µm

Installation clearance between pin and piston pin boss 20 µm

Pin diameter 21 mm

Roughness of pin 0.1 µm

Roughness of small end bearing 0.8 µm

Roughness of piston pin boss 0.8 µm

Boundary friction coefficient between pin and small end bearing 0.10

Boundary friction coefficient between pin and piston pin boss 0.10

Cavitation pressure 0.7 bar

Initial oil film thickness on the interfaces 5 µm

105

Figure 5.1 Cold radial variance (micrometer) of the pin

Figure 5.2 Additional radial variance (micrometer) of the pin boss due to thermal load

106

Figure 5.3 Total radial variance (micrometer) of the pin

The total radial variances for the pin and the con rod small end bearing are assumed to

be uniform at 20 micrometers.

The structural deformations of both the pin and the pin boss are included in the

calculation, while the small end bearing is assumed to be rigid.

The pressure in the combustion chamber is shown in Figure 5.4.

Figure 5.4 Combustion pressure trace used in the calculation

107

5.2 Results with Rigid and Elastic Components

Figures 5.5 shows the normal forces on the pin from the calculation of the following

cases

• No deformation is considered. All the components are rigid.

• Elastic deformation of the pin is considered.

• Elastic deformation of both the pin and the pin boss is considered.

Figures 5.6 and 5.7 show the friction forces, friction power loss, and FMEP for these

cases. When only the pin is deforming, there will be less asperity contact and friction between

the pin and the pin boss around expansion TDC. When the deformation of the pin boss is also

considered, the situation will be slightly further improved.

Figure 5.5 Normal forces on the pin with different considerations of deformation

108

Figure 5.6 Friction force between the pin and the pin boss with different considerations of

deformation

Figure 5.7 Friction power loss and FMEP between the pin and the small end with different

considerations of deformation

Figure 5.8 shows the rotation of the pin. The patterns are similar in the cases, especially

before the early expansion stroke. Therefore, the difference in the friction loss should be

mainly attributed to the direct effect of deformation on the clearance.

109

Figure 5.8 Angular velocity of the pin with different considerations of deformation

Figures 5.9 and 5.10 show the distributions of asperity contact pressure and

hydrodynamic pressure at early expansion stroke in the rigid case. Because the asperity contact

is higher when the clearance is lower, the contact on the pin boss will be very concentrated

around the minimum clearance place, which is shown inside the dotted circle in Figure 5.3. On

the small end, the contact is uniform along the axial direction because of the uniform clearance.

In addition, it is difficult to generate hydrodynamic pressure away from the minimum

clearance.

Figure 5.9 Asperity contact pressure [bar] at 385 CA in the rigid case

110

Figure 5.10 Hydrodynamic pressure [bar] at 385 CA in the rigid case

When the pin is elastic, the deformation at peak cylinder pressure is shown in Figure

5.11. It not only has local distortion in the areas with the highest pressures, but also bends

downwards. Because of the bending, the clearance at the edges of the pin-small end interface

is lower than the middle, as shown in Figure 5.12. Consequently, the pressures are higher near

the edges, as shown in Figures 5.13 and 5.14.

On the other hand, the deformation makes the pin and pin boss conform more to each

other, so the asperity contact at their interface is more spread, and more hydrodynamic

pressure can be generated compared to the rigid case.

Figure 5.11 Magnified deformation [𝝁m] of the pin at 385 CA in the case with elastic pin

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Figure 5.12 Clearance between pin and small end at 385 CA

Figure 5.13 Asperity contact pressure [bar] at 385 CA in the case with elastic pin

Figure 5.14 Hydrodynamic pressure [bar] at 385 CA in the case with elastic pin

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Figure 5.15 shows the deformation of the pin boss at 385 CA when it is elastic as well. In

this case, it will conform even better with the pin, and the lubrication condition will be further

improved. Figure 5.16 shows the hydrodynamic deformation.

Figure 5.15 Magnified deformation [𝝁m] of the pin boss at 385 CA in the case with elastic pin

and pin boss

Figure 5.16 Hydrodynamic pressure [bar] at 385 CA in the case with elastic pin and pin boss

The change in the asperity contact due to deformation is in the different direction from

what was discussed in the previous parametric study. This is because in the previous chapter,

all the surfaces are originally cylindrical and fit each other well. When deformation occurs,

there will be places with lower local clearance so that the contact becomes more concentrated.

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In this case, where the original profile of the pin boss introduces a natural minimum clearance

point, deformation is able to improve the situation.

5.3 Effects of Design Features on the Pin Boss

In this section, three cases will be calculated with different pin boss designs.

• Baseline: no oil grooves or slots.

• Grooves: with circumferential oil grooves, as shown in Figure 2.3.

• Slots: with axial oil slots, as discussed in the previous sections in this chapter.

In order to study whether the design features can reduce friction loss, a higher

combustion load is applied with a PCP of 102 bar.

Figure 5.17 shows the hydrodynamic and asperity contact normal forces on the pin. The

most noticeable difference is on the pin boss during the late compression stroke and early

expansion stroke, when the case with grooves has lower hydrodynamic force and the case with

slots has slightly higher hydrodynamic force.

For the grooves, it is mainly because they are treated as ambient, and there is less area

near the top of the pin boss to generate hydrodynamic pressure. As a result, there is more

asperity contact between pin and pin bore and the pin sticks with the pin bore after mud

compression stroke – much earlier than the other two cases.

As for the slots, because they are along the axial direction and assumed to be fully

flooded with oil, they are helpful in bringing lubricant to the critical region (top) of the pin boss.

However, the benefit is limited because the oil supply for the baseline case is sufficient already.

Figure 5.19 shows the distributions of hydrodynamic pressure around the end of compression

stroke. For the baseline case, there is already full film and hydrodynamic pressure built on the

two sides of the pin boss.

The oil slots might be more useful if the boundaries of the interfaces do not have

sufficient oil, as assumed in the model. As mentioned in the first chapter, it was discovered by

measurement [4] that the slots are useful. Therefore, the assumption of fully-flooded

boundaries in reality may not be met all the time and needs to be relaxed in future works.

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Figure 5.20 shows the friction forces between the pin and the pin boss and the FMEP

associated with the pin. The effects on FMEP are not significant from both variants. The FMEP

increases from the baseline case by around 4% with grooves added, and decreases by 2.5% with

slots added.

Figure 5.17 Normal forces on the pin with different pin boss designs

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Figure 5.18 Angular velocity of the pin with different pin boss designs

Figure 5.19 Hydrodynamic pressure distribution at 340 CA with different pin boss designs

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Figure 5.20 Friction force between pin and pin boss and FMEP with different pin boss designs

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Chapter 6 Conclusions

6.1 Conclusions

The main challenge of the pin lubrication is that its primary motion or pin rotation is

unknown. The pin rotation is determined by the competition of the forces generated in the

small end and piston bearing interfaces, where the pin rotation plays a central role in the

outcome of oil transport, pressure, and friction. Thus, understanding the lubrication of the

piston pin needs to resolve the coupling of the primary motion of the pin, oil transport on the

interfaces, and oil exchange with the surroundings. Additionally, both hydrodynamic and

boundary lubrication are highly dependent on the exact geometry of the mating parts during

operation. Therefore, coupling the thermal and mechanical deformation of the pin, pin bore,

and small end is essential for the model to reflect the reality. Furthermore, the problem is

numerically challenging as the magnitude of the two competing forces at the two interfaces is

much greater than the inertia of the pin rotation. The present work shows that a stable model

with consistent results can be realized by employing adequate lubrication sub-models with oil

conservation and a stable algorithm.

The present work has been limited to model development and examination of the

consistency of the model in reflecting the built-in physics and the effects of the design and

operating parameters. Comparing the model prediction and the real measurement requires

accurate data on temperature, thermal and structural deformation, and geometry change due

to wear, and is beyond the scope of this work.

Preliminary results do show that substantial friction power loss can come from this

system despite its relatively slow sliding speed. This is because high magnitude of contact force

and the alignment of the timing between highest friction force and highest sliding speed.

Furthermore, the friction is mainly contributed by the boundary lubrication while hydrodynamic

friction is almost negligible. As pin is free, the friction can thus be effectively reduced by

enhancing the hydrodynamic effect in only one interface, which includes ensuing oil supply to

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the contact regions and favorable clearance shape during engine operation. The current model

has the capability to help make necessary design changes toward that direction.

The friction potential developed in this work is a useful concept to help understand the

rotation of the pin. When the coefficient of friction of the boundary lubrication of the two

interfaces are the same, it has been found that the reciprocating nature of the piston leads to a

pattern where the pin rotates with the connecting rod during half of the engine cycle, and stays

with the piston during the other half. Because of the larger width on the bottom of the small

end and the top of the pin boss, this pattern is favorable for bringing oil to the regions with the

highest load. More friction will be generated if the pin rotates more with either of the

components. This might be able to provide some guidance for new designs.

As the first step to establish a comprehensive engineering model to understand the pin

lubrication, this work not only lays a foundation but also helps gain a great deal of insights to

the complex interactions involved in the pin lubrication.

6.2 Future Work

A number of simplifications in the present model can be improved in the future. First of

all, more factors need to be added for more accurate simulation, such as the structural

deformation of the con rod small end, and the effects of temperature and shear stress on the

viscosity of the lubricant.

On the other hand, there are special design features for different engines in the market.

For instance, in some commercial vehicle diesel engines, there is an oil bore in the shank of the

connecting rod through which oil can be transported to lubricate the small end bearing; there

are also cases where the hole on the piston pin has an offset such that the inner and outer

surfaces of the pin are not concentric. The current model can be improved accordingly in order

to consider such design features.

The oil supply to the boundaries of the interfaces, if found not necessarily sufficient

temporally and spatially, will be a subject of further work as well. While relaxing the presently

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fully-flooded boundary condition in the model is straightforward, where and when the

boundaries may become starved need more work assisted with experimental findings.

Finally, the results of the model, especially the friction on the pin, can be used to

improve the other models. As an example, the piston skirt lubrication model [9] neglects the

friction force between the pin and the pin boss when calculating the torque of the piston. This

assumption can be dropped with information provided by the pin lubrication model.

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