Modeling of Contact between Liner Finish and Piston Ring in Internal
Combustion Engines Based on 3D Measured Surface
by
Qing Zhao
MASSACHUSETTS ITITUTE
OF TECHNWOLOGY
AUG 152014
B.Sc., Mechanical Engineering
Purdue University, 2012
Shanghai Jiao Tong University, 2012
LIBRARIES
SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE, 2014
@2014 Massachusetts Institute of Technology. All rights Reserved.
Signature redacted
Signature of Author:
Department of Mechanical Wngineering
May 9, 2014
Certified by:
Signature redacted,
Dr. Tian Tian
Principle Research Scientist, Department of Mechanical Engineering
,
Accepted by:
The.supervisor
________Signature redacted
David E.Hardt
Professor of Mechanical Engineering
Chairman, Committee on Graduate Students
Modeling of Contact between Liner Finish and Piston Ring in Internal
Combustion Engines Based on 3D Measured Surface
by
Qing Zhao
Submitted to the Department of Mechanical Engineering on May 9, 2014 in Partial Fulfillment
of the Requirements for the Degree of Master of Science in Mechanical Engineering
Abstract
When decreasing of fossil fuel supplies and air pollution are two major society problems in the
2 1 st century, rapid growth of internal combustion (IC) engines serves as a main producer of
these two problems. In order to increase fuel efficiency, mechanical loss should be controlled in
internal combustion engines. Interaction between piston ring pack and cylinder liner finish
accounts for nearly 20 percent of the mechanical losses within an internal combustion engine,
and is an important factor that affects the lubricant oil consumption. Among the total friction
between piston ring pack and cylinder liner, boundary friction occurs when piston is at low
speed and there is direct contact between rings and liners. This work focuses on prediction of
contact between piston ring and liner finish based on 3D measured surface and different
methods are compared.
In previous twin-land oil control ring (TLOCR) deterministic model, Greenwood-Tripp
correlation function was used to determine contact. The practical challenge for this single
equation is that real plateau roughness makes it unreliable. As a result, micro geometry of liner
surface needs to be obtained through white light interferometry device or confocal equipment
to conduct contact model. Based on real geometry of liner finish and the assumption that ring
surface is ideally smooth, contact can be predicted by three different models which were
developed by using statistical Greenwood-Williamson model, Hertzian contact and revised
deterministic dry contact model by Professor A.A. Lubrecht.
The predicted contact between liner finish and piston ring is then combined with hydrodynamic
pressure caused by lubricant which was examined using TLOCR deterministic model by Chen. et
al to get total friction resulted on the surface of liner finish. Finally, contact model is used to
examine friction of different liners in an actual engine running cycle.
Thesis Supervisor:
Dr. Tian Tian, Department of Mechanical Engineering
3
Acknowledgements
There are many people who I would like to thank for their contributions to this research, and to
my past two years' study at MIT. These contributions have given me many opportunities for
developments on both personal and professional level.
First and foremost, I would like to thank my supervisor, Dr. Tian Tian, for his support and
guidance throughout my research and the course of my work. I have learnt a great deal through
my exposure to his depth of knowledge, insight, experience and logical approach to problem
solving. I would like to thank Professor Ton Lubrecht, for his guidance and help throughout my
research at MIT. I couldn't have finished the work without his help.
I would also like to thank my peer worker, Dallwoo Kim, who built the experiment aspect of
contact model. I would also like to thank Yang Liu and Renze Wang, who made the part to
predict hydrodynamic pressure between liner finish and piston ring and generated ideal rough
surfaces based on measured liner surface. I absorbed numerous knowledge and ideas through
the intense and inspiring discussions with them. I couldn't have finished the work without their
continuous help in these two years.
This work is sponsored by the consortium on lubrication in internal combustion engines with
additional support by Argonne National Laboratory and the US department of energy. The
current consortium members are Daimler, Mahle, PSA Peugeot Citroen, Renault, Shell, Toyota,
Volkswagen, Volve Cars, and Volve Truck. I would like to thank them, for their financial support,
and more specifically, their representatives and others for their continued encouragement over
the years, and for sharing their extensive experience with me. Our regular meetings provided
not only a motivation for completing work, but also an invaluable opportunity to share
knowledge and obtain constructive feedback. Without their help and guidance this research
would not be possible.
I would also like to thank the members of the Sloan Automotive Laboratory for their support
and friendship. In particular I would like to thank students of the Lubrication Consortium and
my office mates, Eric Zanghi, Pasquale Totaro, Renze Wang, Yang Liu, Camille Baelden, Mathieu
Picard, Tianshi Fang, Kai Liao, Dallwoo Kim for their help to make the stressful time relieving.
Finally, I would like to thank my parents and boyfriend for their great support and love
throughout my stay here and all the friends I met at MIT that made my stay here more colorful.
4
Table of Contents
Abstract.......................................................................................................................................................
3
Acknow ledges.............................................................................................................................................4
Table of Contents.......................................................................................................................................5
List of Figures..............................................................................................................................................7
List of Tables..............................................................................................................................................10
Introduction....................................................................................................................................11
1.1 Project M otivation......................................................................................................................11
1.2 Piston Ring Pack..........................................................................................................................12
1.3 Cylinder Liner Finish............................................................................................................... 13
1.4 Surface Roughness M easurem ent Techniques................................................................... 14
1.5 Previous Work on Modeling Contact between Liner Finish and Piston Ring Pack........16
1.6 Scope of Thesis W ork............................................................................................................. 19
2
M easured Liner Processing M ethod...................................................................................... 20
2.1 M easured Liner............................................................................................................................20
2.1.1 M easured Liner Geom etry Profile.................................................................................... 20
2.1.2 M ean Plateau Height..........................................................................................................
21
2.1.3 Asperity Height Distribution and Mean Plateau Height of Different Sample Liners...22
1
2.1.4 Plateau Surface Roughness - ..........................................................................................
24
2.2 M easurem ent Errors on M easured Liner............................................................................ 24
2.3 M easured Liner Processing M ethod.................................................................................... 26
2.3.1 Rem oving Unexpected Large Spikes................................................................................. 26
2.3.2 Plateau Surface Roughness u- of Different Sam ple Liners............................................ 27
2.4 M easured Liner M odeling M ethod..........................................................................................28
2.5 Conclusion...........................................................................................................................
........ 30
3
Statistical M odel and Hertzian Contact M odel..................................................................... 31
3.1 Challenges of Applying Previous Contact M odel............................................................... .... 31
3.1.1 Assum ptions and Equations of Previous Contact M odel............................................... 31
3.1.2 Correlation Function based on Gaussian Distribution.................................................... 33
3.1.3 Comparison of Previous Contact Model with Real Situation.........................................34
3.2 Statistical M odel..........................................................................................................................35
3.2.1 Assum ptions and Equations of Statistical M odel........................................................... 35
3.2.2 Application of Statistical M odel........................................................................................ 38
3.2.3 Results of Different Sam ple Liners by Statistical M odel................................................. 39
3.2.4 Application of Statistical Model based on Gaussian Distribution................................ 42
3.3 Hertzian Contact M odel.......................................................................................................
43
3.3.1 Assum ptions and Equations of Hertzian Contact M odel.............................................. 44
5
3.3.2 Application of Hertzian Contact M odel............................................................................ 45
3.3.3 Results of Different Sample Liners by Hertzian Contact M odel.................................... 46
3.4 Discussions of Results by Statistical Model and Hertzian Contact Model for Different
Sample Liners..............................................................................................................................48
3.4.1 Comparisons of Statistical Model and Hertzian Contact M odel.................................. 48
3.4.2 Factors Influence Contact Pressure................................................................................... 53
3.4.3 Comparison of Previous Contact Model and Statistical Model based on Gaussian
Distribution...............................................................................................................................53
3.5
4
Conclusion....................................................................................................................................54
Deterministic Contact M odel.....................................................................................................56
Approach of Deterministic Contact M odel.........................................................................
4.1.1 Assumptions and Formulas in Deterministic Model......................................................
4.1
56
56
4.1.2 M ulti-level M ethod.............................................................................................................. 58
4.1.3 Application of Deterministic Contact Model.................................................................. 59
4.2 Results of Different Sample Liners by Deterministic Contact M odel............................. 60
Discussions of Results by Deterministic Contact M odel................................................... 62
4.3.1 Comparisons of Deterministic Contact Model and Hertzian Contact Model, Statistical
4.3
M odel........................................................................................................................................63
4.3.2 Com parisons of M odeled Liner Surface and Original Liner Surface............................. 66
4.3.3 Boundary Effect in Deterministic Contact M odel.......................................................... 68
4.3.4 Another Choice of Normalized Process............................................................................ 70
4.3.5 Influence of Level Size..........................................................................................................72
4.3.6 Relation between Contact Area and Contact Pressure................................................. 73
4.4
Conclusion....................................................................................................................................74
Evaluation of Contact M odel....................................................................................................
5
5.1
5.2
75
Correlation Functions of Contact Pressure......................................................................... 75
Test Results in Cycle Model and Comparison with Experimental Results for Three
76
Different Contact Model.......................................................................................................
5.2.1
5.2.2
5.2.3
Calculation Results of Sample Liner #1 1....................................................................... 76
Calculation Results of Sample Liner #2 2....................................................................... 79
Calculation Results of Sample Liner #4 3...................................................................... 81
5.3
Discussion.....................................................................................................................................82
5.4
Conclusion....................................................................................................................................83
Conclusion.......................................................................................................................................85
6
6.1
6.2
Summary and Conclusion.......................................................................................................
Potential Future W ork............................................................................................................
References................................................................................................................................................87
6
85
86
List of Figures
Figure 1.1: Breakdown of Total Diesel Engine Energy, Mechanical Friction and Ring Pack Friction
[1]............................................................................................................................................1
1
Figure 1.2: Position of Piston Ring Pack in Combustion Chamber of an Internal Combustion
E ng in e.....................................................................................................................................12
Figure 1.3: New Cylinder Liner Finish Geometry Profile................................................................13
Figure 1.4: Worn Cylinder Liner Finish Geometry Profile..............................................................14
Figure 1.5: Schematic Drawing of Stylus Profiler Method [10]......................................................15
Figure 1.6: Schematic Drawing of Confocal Microscope [13]...................................................... 15
Figure 1.7: Schem atic Draw ing of W LI [17]......................................................................................
16
Figure 2.1: Sample Liner Geometry Profile in 2D and 3D View....................................................21
Figure 2.2: Sam ple Liner Surface Height Distribution..................................................................... 21
Figure 2.3: Plateau of Sample Liner Geometry Profile in 2D and 3D View..................................22
Figure 2.4: Sample Liners and Mean Plateau Height of Them......................................................24
Figure 2.5: Unexpected Large Spikes on Measured Liner..............................................................25
Figure 2.6: Unexpected Spikes along Border of Plateau and Valley............................................25
Figure 2.7: Flow Chart of Iteration to Remove Unexpected Large Spikes...................................26
Figure 2.8: Original Surface and Processed Surface without Large Spikes..................................27
Figure 2.9: Unexpected Spikes along Border of Plateau and Valley on Filtered Surface...........27
Figure 2.10: Original ap and Filtered up of Sample Liners.............................................................28
Figure 2.11: M odeled O ne Asperity...................................................................................................
29
Figure 2.12: M odeled a Sm all Patch of Surface...............................................................................
30
Figure 3.1: Ring in Contact with Liner Finish at Clearance Height h............................................32
Figure 3.2: Comparison of F 2.s based on Gaussian Distribution and Hu et al Correlation Function
and New Correlation Function........................................................................................ 34
Figure 3.3: Comparison between Plateau Height Distribution of Liner Surface #1 with Normal
D istrib utio n ............................................................................................................................
35
Figure 3.4: Contact between One Rough Surface and One Smooth Surface............................. 36
Figure 3.5: Contact of One Smooth Surface and One Rough Surface..........................................37
Figure 3.6: Measured Liner Surface and Modeled Liner Surface................................................. 38
Figure 3.7: Relation between Contact Pressure and Clearance Height.......................................39
Figure 3.8: Contact Pressure of Different Sample Liners by Statistical Method.........................40
Figure 3.9: Comparison of F 1.5 with Gaussian Distribution and Correlation Function.............43
Figure 3.10: One Asperity and One Smooth Surface in Contact....................................................45
7
Figure 3.11: Measured Liner Surface and Modeled Liner Surface...............................................45
Figure 3.12: Relation between Contact Pressure and Clearance Height....................................46
Figure 3.13: Contact Pressure of Different Sample Liners by Hertzian Contact Model............48
Figure 3.14: Comparison of Statistical Model and Hertzian Contact Model for Different Sample
Lin e rs......................................................................................................................................50
Figure 3.15: Asperity Size Distribution of Sample Liner #2.......................................................... 52
Figure 3.16: Comparison of Statistical Model and Hertzian Contact Model for Generated Surface
w ith Identical Asperity Size.............................................................................................
52
Figure 3.17: Comparison of Previous Contact Model and Statistical Model based on Gaussian
54
D istributio n ............................................................................................................................
Figure 4.1: Original M easured Liner Surface..................................................................................
59
Figure 4.2: Deform ed Liner Surface.........................................................................................
60
Figure 4.3: Relation between Contact Pressure and Clearance Height.......................................60
Figure 4.4: Contact Pressure of Different Sample Liners by Deterministic Contact Model...........62
Figure 4.5: Contact Pressure of Different Sample Liners by Deterministic Contact Model,
Hertzian Contact Model and Statistical Model............................................................65
Figure 4.6: Comparison of Original Liner Surface and Modeled Liner Surface...........................68
Figure 4.7: Constructed Liner Surface without Boundary Effect and Original Liner Surface with
Boundary Effect.....................................................................................................................69
Figure 4.8: Comparison of Contact Pressure with Boundary Effect and Contact Pressure without
Boundary Effect.....................................................................................................................69
Figure 4.9: Contact between Punch and Smooth Surface.............................................................70
Figure 4.10: Comparison between Different Normalized Process, Hertzian Contact and Punch
1
Co ntact...................................................................................................................................7
Figure 4.11: Comparison of Contact Pressure by Different Level Size.........................................72
Figure 4.12: Relation between Contact Area and Clearance Height of Sample Liner #2..........73
Figure 4.13: Relation between Contact Area and Contact Load of Sample Liner #2............... ...... 74
Figure 5.1: Comparison of Friction between Sample Liner #1 and Oil Control Ring by Different
Contact Models and Experimental Results at 100*C..................................................77
Figure 5.2: Comparison of Friction between Sample Liner #1 and Oil Control Ring by Different
Contact Models and Experimental Results at 40*C....................................................78
Figure 5.3: Comparison of Friction between Sample Liner #2 and Oil Control Ring by Different
Contact Models and Experimental Results at 100*C..................................................80
Figure 5.4: Comparison of Friction between Sample Liner #2 and Oil Control Ring by Different
Contact Models and Experimental Results at 40C.....................................................80
8
Figure 5.5: Comparison of Friction between Sample Liner #4 and Oil Control Ring by Different
Contact Models and Experimental Results at 100*C.................................................81
Figure 5.6: Comparison of Friction between Sample Liner #4 and Oil Control Ring by Different
Contact Models and Experimental Results at 40*C....................................................82
Figure 5.7: Discontinuous Spikes along Plateau/Deep Valley......................................................83
9
List of Tables
Table 3.1: Statistical Data of Different Sample Liners....................................................................39
Table 3.2: Maximum Difference and Average Difference of Statistical Model and Hertzian
Contact Model for Different Sample Liners.................................................................. 51
Table 4.1: Maximum Difference and Average Difference of Deterministic Contact Model and
Hertzian Contact Model for Different Sample Liners..................................................65
Table 4.2: Com parison of Different Level Size............................................................................... 72
10
1. Introduction
1.1 Project Motivation
When decreasing of fossil fuel supplies and air pollution are two major society problems in the
2 1st
century, rapid growth of internal combustion (IC) engines serves as a main contributor of
these two problems. The challenge of energy demand and environmental protection can be
alleviated by increasing the engine's efficiency and reducing its CO 2 emissions. These are the
two demanding goals for the whole automotive industry.
Among total consumed energy in a typical diesel automotive, mechanical friction loss accounts
for approximately 10% of the total fuel energy, and of which around 20% is dissipated into
friction between piston rings and liner finish, as illustrated in Figure 1.1 [1]. Meanwhile, oil
control ring is responsible for more than half of the piston ring pack friction loss. As a result,
there is still a space for automotive industry to increase energy efficiency by reducing piston
ring pack friction.
Total Engurgy Brmiadmwn
Msc d
Iechaneal Pricion Breakdown
Pdc~m
Ring Friction Breakdown
Rigs
r
Top Rhig
(13-40%1
.d
Second R
Rods 110-.22%)
(0ing
(19-44%)
Figure 1.1: Breakdown of Total Diesel Engine Energy, Mechanical Friction and Ring Pack
Friction [1]
Asperity contact between liner finish and rings can occur due to a combination of limited oil
supply and low piston sliding speed at top dead center (TDC) and bottom dead center (BDC) of
the stroke among piston ring pack friction. Asperity contact occurs in a boundary lubrication
regime, when asperities carry the entire ring load, and in a mixed lubrication regime, when the
11
ring load is shared by asperity contact and hydrodynamic pressure. Friction due to asperity
contact has been identified as an important contributor to total ring pack friction [2].
1.2 Piston Ring Pack
In a combustion chamber of an internal combustion engine, a piston ring is a split ring that fits
into a grove on the outer diameter of a piston and the main functions of piston rings are sealing
the combustion chamber so that there is no transfer of gases and oil from the combustion
chamber to the crank case and regulating engine oil consumption [3].
The piston ring pack consists of three different rings (from top to bottom): top ring
(compression ring), second ring (scraper ring) and oil control ring (OCR) in an internal
combustion chamber, as illustrated in Figure 1.2.
Combustion Chamber
Top Ring
Second Rine
Twin Land Oil
Control Ring
Figure 1.2: Position of Piston Ring Pack in Combustion Chamber of an Internal Combustion
Engine
Twin land oil control ring (TLOCR) is widely used in automotive diesel engines, and it was
focused in this work. In order to seal oil in crank case from combustion chamber, a high normal
force is exerted by the oil control ring spring to conform the ring onto the cylinder bore, and
consequently it results in a larger portion of the entire ring pack friction loss. Moreover,
another function of oil control ring is to limit oil film thickness left on the liner which is the
source of oil supply to top two rings. If the controlled film thickness by oil control ring is thicker,
12
it will increase oil consumption while result in less contact friction. The trade-off between the
contact friction of the top two rings and oil consumption makes the design of oil control ring
complicated.
1.3 Cylinder Liner Finish
To guarantee reproducibility with efficient productivity in mass production, cylinder liners of
internal combustion engines are finished using an interrupted multi-stage honing process,
known as plateau-honing process. This process is a succession of three honing stages. The first
stage categorized as a rough honing establishes the form of the bore. The second operation
creates the basic surface texture and the third honing operation serves for removing surface
peaks [4]. The whole honing process gives cylinder liner the desired finish, dimensional
accuracy, form, and a surface with characteristic cross-hatch groove pattern [5].
A typical new cylinder liner finish geometry profile is shown in Figure 1.3. The plateau area is
formed by the third honing operation and the deep valley part comes from second honing
process. When piston runs in a cylinder, ring land surface slides over liner finish and it is in the
plateau part that all asperity contact occurs. Consequently, surface roughness of the plateau
part is the most important factor to determine asperity contact and o- which is root mean
square (RMS) of the plateau area is used to define liner finish surface roughness.
color scale length unit (um)
1400
1
1200
0.5
0
"0.5
10
0
600
1500
1000
Axial Cirecd on (prn)
2000
Figure 1.3: New Cylinder Liner Finish Geometry Profile
13
-
600
Surface topology of liner finish changes with time due to asperity contact between liner and
piston ring pack in both break-in and wear process. Figure 1.4 shows a worn cylinder liner finish
geometry profile. During break-in period, some asperity peaks due to honing process can be
removed and thus reducing contact friction between liner and piston ring pack [6] [7].
color scale length unit (jpm)
1400
,-%1200
6
Si1ooo
5100
Boo
-0.5
c 600
E400
-1-
0200
ON
0
600
1000
1600
2000
Axial direction (pm)
Figure 1.4: Worn Cylinder Liner Finish Geometry Profile
1.4 Surface Roughness Measurement Techniques
There are different methods to measure surface texture and among them, stylus profiler
method, white light interferometry (WLI) microscopy and confocal microscopy are widely used
and provide higher accuracy. Stylus Profiler uses contacting method, while WLI microscopy and
confocal microscopy are based on optical techniques [8].
The stylus profiler senses surface height through mechanical contact where a stylus traverses
peaks and valleys of the surface with a small contacting force, as illustrated in Figure 1. 5.
Vertical motion of the stylus is converted to an electrical signal by a transducer, which
represents the surface profile. Vertical resolution of the stylus profiler can be very high, while
the lateral resolution is limited by size of the stylus tip. One disadvantage of the stylus
instrument, however, is that stylus may damage the surface, depending on the hardness of the
surface relative to the stylus and tip size [8] [9].
14
up
I
Ntemiurrc
tolic
."in1m
Figure 1.5: Schematic Drawing of Stylus Profiler Method [10]
Confocal microscope uses aperture (pinhole) to scan surface relative to a finely focused spot of
laser light. The transmitted or reflected light is then collected and focused onto a point detector,
as illustrated in Figure 1.6. The resulting signal strength can be used to modulate brightness of
the spot which can tell the height on the surface spot by spot [11]. Confocal microscope has the
unique capability of creating a bright image of the in-focus region of the specimen while causing
all out-of-focus regions to appear dark [12].
ronrs
laser
screen with
pinhole
detector (PMT)
microscope
fluorescent
specimen
Figure 1.6: Schematic Drawing of Confocal Microscope [13]
WLI technique, as illustrated in Figure 1.7 is an established optical method to measure surface
roughness. A Michelson interferometer is usually used to generate interferometry. The
15
interferometer is illuminated by a broadband light source such as a light-emitting diode, a
super-luminescent diode, or an incandescent lamp. In the Michelson interferometer, light
source is split into two parts through a beam splitter. One goes directly to the surface and the
other travels onto a smooth reference mirror. The reflected two beams can produce
interference fringes around the equal path condition. At the output of the interferometer, a
CCD camera serves as a detector to record the fringe pattern. Scanning the surface vertically
with respect to microscope and detecting the optimum equal path condition at every pixel in
the camera result in a topographic image [14] [15] [16].
Detector
Reference
Beam
mirror
splitter
White light
z
Reference mirror
position
lw
tu
h(xy)
y
z
Surface
x or y axis
Figure 1.7: Schematic Drawing of WII [17]
Optical Methods, including confocal method and WLI method, have the advantages that they
are non-contacting and hence, non-destructive. Optical methods based on imaging and
microscopy also have a higher measurement speed than contacting technique, stylus profiler
method, which rely on mechanical scanning of a contacting probe [8]. However, accuracy of the
optical methods is limited to moderate surface slopes. Sharp edges, inclusions, defects, and
other peculiarities of the surface can scatter light away from objective and cause outliers and
dropouts of data points in the topographic images measured with optical microscopes [8] [18].
1.5 Previous Work on Modeling Contact between Liner Finish and Piston Ring
Pack
In previous TLOCR deterministic model, asperity contact is based on Hertzian contact,
Greenwood-Tripp model, and Hu et al asperity contact equation [19] [20] [21].
16
Hertzian suggests contact force between two elastic solids has the following relation [19],
4
1 3
P = -E'Rfwf
3
1+
E'
El
2
E2
In the above equation, P is contact force between two elastic spheres, El and E2 are young
moduli of the two spheres, v, and v 2 are Poisson ratios of the two spheres, E' is combined
modulus of two bodies in contact, R is combined radius of the two spheres, w is interference of
the two spheres [19].
According to Greenwood-Tipp model, asperity contact force between two rough surfaces reads
[20],
P(d) = 2N2Af
f
Z1
Z2
P(w, r)0(z,)0(z 2 )rdrdzdz 2
In the above equation, P(d) represents contact force at nominal separation distance d between
two rough surfaces, N is number of asperities on each surface, A is surface area (nominal
contact area), P(w, r) is contact force on each asperity which depends on interference of two
asperities and misalignment r, 0(zl) and O(z 2 ) are asperity height distribution on two surfaces
[20].
Then based on the following assumptions,
a.
b.
c.
d.
e.
Two surfaces are covered with spherical asperities.
Shape of each asperity is identical, at least the summit part.
Deformation is constrained to elastic deformation and no plastic deformation exists.
Asperity height distribution on each surface is Gaussian distribution.
Interaction between asperities on same surface is neglected.
Equation of contact friction between two rough surfaces has the form of [20]:
P(h) =
1 f 0
C)
16AF
J(NR-)2E' -/A
!<
15R is
N17r _-
17
h
s - ->
0
s
S2
e Tds
In the above equation, a is standard deviation of asperity peak height which represents surface
roughness.
Greenwood-Tripp model is based on statistical analysis of all asperities on the surface. The real
shape and height of each asperity is not important to apply this model, and as long as statistical
radius of all asperities, number of asperities and roughness of the surface are available, contact
friction can be obtained.
Hu et al gave a correlation function to simplify Greenwood-Tripp model and suggested the
approximated number for a rough liner surface. In Hu et al paper, he used contact pressure
instead of contact friction [21],
P = KE'F2 .5
K= 16,2T(NR)2_
(
F2.s h)
1
S2
f *c < s - h >2se-2ds=
h
- )z
A(&)
=
_<
->
0
where w
h
4.0, A = 4.4068 x 10- 5 , Z = 6.804, K = 2.396 x 10-4 [21].
According to experiment results of friction between liner finish and piston ring pack, a constant
parameter, cfct = 20 [22] [23], has been added before the following equation. Then previous
contact model can be expressed as
h
P = cfctKE'
A(co - -)z
a
0
h
_
o
->to
The previous contact model can be easily used due to its simplified form, but unfortunately it is
not sensitive to surface geometry profile and asperity peak height distribution. The only
parameter related to real condition is root mean square of plateau area on liner finish. As a
result, root mean square of plateau area sometimes needs to be changed to match with
experimental data in Tian cycle model.
18
1.6 Scope of Thesis Work
The objective of this thesis is to model contact between cylinder liner finish and piston ring
pack in an internal combustion engine. Three different approaches have been evaluated. The
first one is based on Greenwood-Williamson statistical model [24]. The second one applies
Hertzian contact to entire measured surface [19]. The third one is a deterministic contact model.
All three models are based on measured liner finish and the assumption that ring surface is
ideally smooth.
Second chapter introduces measured surface of different sample liners and unexpected large
spikes shown on measured surfaces due to measurement errors. It also discusses the methods
to numerically remove unexpected large spikes which have a large impact on contact part. In
order to apply Greenwood-Williamson statistical model [24] and Hertzian contact model,
asperities should be in regular shape, such as spherical shape and ellipsoidal shape. An
approach to fit irregular asperities to regular shape has also been included in this part.
Third chapter introduces Greenwood-Williamson statistical model and Hertzian contact model
which are based on modeled liner surfaces and neglect interactions between asperities on liner
finish. Applications of the two models on different sample liners are demonstrated, as well as
the comparisons between them.
Fourth chapter presents deterministic contact model which is based on original measured liners
instead of modeled liner surfaces. In addition, it doesn't neglect interaction between asperities
on liner surface. Application of this model is also shown for sample liners and the results are
compared with the results by Greenwood-Williamson statistical model and Hertzian contact
model.
Fifth chapter discusses applications of three contact models in cycle model to predict total
friction between liner finish and piston ring pack. The first step is to fit contact pressure and
clearance height into a correlation function, and then test it in cycle model. Comparisons
between testing results by different contact models and experimental data are also shown in
this chapter.
Sixth chapter summarizes and concludes the thesis work and suggests potential future work on
this topic.
19
2. Measured Liner Processing Method
In this chapter, small patches of different sample liner finish measured by confocal microscope
are used. Contact between liner and piston ring pack is dependent on surface roughness of
plateau area, and thus the first step is to define mean plateau height (from where plateau area
starts) according to asperity height distribution. Besides that the unexpected large spikes on
measured surfaces caused by measurement errors are pointed out. Therefore, a measured liner
finish processing method is introduced to numerically remove unexpected large spikes based
on root mean square of plateau area.
In order to apply Greenwood-Williamson statistical model [24] and Hertzian contact model,
asperities should be in regular shape, such as spherical shape and ellipsoidal shape. An
approach to fit irregular asperities to regular shape is also included in this chapter.
2.1 Measured Liner
In this section, measured liner is demonstrated to clearly show surface geometry profile.
Contact is highly dependent on plateau part on measured liners, and thus mean plateau height
(which plane separates plateau and valley) needs to be calculated for each liner finish before
applying contact model. In this section, a method to define mean plateau height is introduced
and that of different sample liners are shown. An approach to calculate plateau surface
roughness which is the most important factor influences contact is introduced after that.
2.1.1 Measured Liner Geometry Profile
Sample Liner finish has been measured by confocal microscope. The resolutions of the confocal
microscope are 0.37 micrometer in both axial and circumferential directions, and thus the
height of every spot which has the area of 0.37 micrometer by 0.37 micrometer is recorded and
represented by a number. Size of the small patch of measured liner surface shown below is
0.185 millimeter by 0.185 millimeter (500 spot by 500 spot).
20
Color scale length unit (micrometer)
Color scale length unit (micrometer)
500
400
2,2
.t
0)
.2
S200
400
100
C./ 200
7
0
100
200
300
400
0
....
/Oj. 0
'fee
500
C400t7
0
Axial direction
Figure 2.1: Sample Liner Geometry Profile in 2D and 3D View
2.1.2 Mean Plateau Height
In order to conduct contact model, height of each spot is important because contact pressure is
dependent on compressed height of each asperity at different clearance height. Mean plateau
height is the plane that separates area of plateau and valley on the surface. When height of
each spot on liner is measured, a reference plane has been chosen and the height of each spot
is relative to this chosen reference plane. This is not the real mean plateau height, and thus we
need to find mean plateau height according to asperity height distribution. In this work and
previous work by Chen [22], height on liner finish with the maximum asperity height
distribution is defined to be the mean plateau height, which is shown below. For the following
sample liner, mean plateau height is 0.063 micrometer.
x 104 sample liner asperity height distribution
L
12,
10k
8
.0
6
CL
4
2
06
-6
-4
-2
0
height (m)
2
4
6
X 10-
Figure 2.2: Sample Liner Surface Height Distribution
21
Asperity contact highly depends on plateau of liner finish and the figure below shows plateau of
the sample liner surface shown in Figure 2.1 (height of valley is set to zero).
Color scale length unit (micrometer)
500
.6
0.5
C:
0
.5
C.)
0)
.4
:05
.3
0..2
2)
Q
.2
6400
Qr'ec
40
017f'ee,20020
0
100
200
300
Axial direction
400
500
Figure 2.3: Plateau of Sample Liner Geometry Profile in 2D and 3D View
2.1.3 Asperity Height Distribution and Mean Plateau Height of Different Sample
Liners
In figure 2.4 below, five different sample liners and their relative asperity height distribution are
shown, as well as tabulated mean plateau height.
.1
10
2
I:
I
x 10 sample liner 1 asperity height distribution
L
L
L
L
1.5
-0
E
1
CL
2
0
Sbdng Dimcon (urn)
0.5 F
-6
200
-4
-2
0
height (m)
Sample Liner #1: 0.026micro//mean plateau height
22
2
4
6
x 10-
.7
12
12
160
x 104 sample liner 2 asperity height distribution
-
10
L
10-
140
8
E
100
80
0
~60
91)
100
I
6
4
3
2
-5
-6
-4
-2
2
0
4
height (m)
Sling Dirction (urn)
6
x 10-7
Sample Liner #2: 0.066micro//mean plateau height
10.7
200
16
X 104
sample liner 3 asperity height distribution
L
L
L
L
14
160
12
140
10
12D
E
100
.5
0
180
CL
0
2
6D
K
8
6
4
3
2
0
-6
-4
-2
Shing DcvIon (um)
0
height (m)
2
4
6
x 10-
Sample Liner #3: 0.022micro//mean plateau height
x 104 sample liner 4 asperity height distribution
.10
8
I
(D
.0
E
6
M
j
0
4
2
2
3
4
-5
-1.5
-1
-0.5
0
height
Sliding DO ction (urn)
0.5
(m)
Sample Liner #4: 0.010micro//mean plateau height
23
1.5
1
x 10
10
-7
15.
IGO
1
X 104 sample liner 5 asperity height distribution
0 10
0
m
-1
20-1.5
Shdai Dwhon urn)height
-0.5
0
0.5
(in)
1
1.5
x 106
Sample Liner #5: O.O63micro//mean plateau height
Figure 2.4: Sample Liners and Mean Plateau Height of Them
2.1.4 Plateau Surface Roughness crp,
Contact is only dependent on plateau of liner surface and the influence of valley can be
neglected because they don't have direct contact with piston ring pack, and thus surface
roughness can be defined by Root Mean Square (RMS) of each spot in plateau area on liner
surface and represented by o-,.
2.2 Measurement Errors on Measured Liner
Different sample liners have been measured by confocal microscope. All surfaces are worn ones
which are after break-in process, and thus large spikes cannot exist on surfaces and should be
removed by contact with ring surfaces in break-in process. In addition, for many surfaces after
interrupted multi-stage honing process, the height distribution tends to be Gaussian
distribution and nearly no asperity is larger than 4a, [24]. But unexpected large spikes are still
shown on some measured liners, as illustrated in Figure 2.5. Such large spikes are not on real
liners and they will highly affect contact between liners and piston rings. Dusts on measured
surfaces and dusts in the air through which light path travels in the confocal microscope can be
reasons of this kind of measurement errors [11] [12]. Consequently, they should be removed
before applying contact model.
24
100
Color scale length unit (micrometer)
Color scale length unit (micrometer)
80
00
.60
2
2
40
E4
-10.
0 2010
00
20
0
40
60
80
100
6
Ccfir
tiaent
0 0
' ; tio
-dtC
Maa
\
Axial direction
Figure 2.5: Unexpected Large Spikes on Measured Liner
Second type of measurement error is due to large slopes on measured surface, which means
two adjacent spots on measured surface have a large height difference and one is in plateau
part while the other is in deep valley part. If there is a large slope on measured surface, light of
confocal microscope cannot be reflected vertically back to detector and it will cause
measurement error. Some discontinuous spikes are shown along the border of valley and
plateau and in reality they are not on the surface, as shown in Figure 2.6.
Color scale length unit (micrometer)
.6
1
.4
06..2
.2
.4
.6
s
200
c 0 0
100C
0
10
.8
pad
Figure 2.6: Unexpected Spikes along Border of Plateau and Valley
25
2.3 Measured Liner Processing Method
An approach to remove unexpected large spikes is introduced in this section based on plateau
surface roughness up. Plateau surface roughness up of different sample liners after the process
of numerically removing large spikes is tabulated.
2.3.1 Removing Unexpected Large Spikes
Unexpected large spikes on measured liners should be numerically removed before applying
contact model to predict contact friction between piston ring pack and liner finish because they
will highly increase contact. The method is based on the assumption that, asperity height
distribution tends to be Gaussian distribution and no asperity is larger than 4U, for surfaces
after interrupted multi-stage honing [24]. First step is to remove obvious measurement errors
on original surface which have much larger height than the other spots. Average height of the
whole surface will be given for such spots. Second step is to calculate surface roughness up of
plateau area without the obvious measurement errors and remove all the spikes larger than
4%p. Such kind of spots will be given a new value of the local average height around them. Then
a new up is calculated based on new surface without spikes larger than 4%p. After that all the
spikes larger than new up will be removed. Such iteration needs to be done for ten times to get
final surface without large spikes. The process of removing unexpected large spikes is illustrated
in Figure 2.7. Figure 2.8 shows the comparison between original measured surface and
processed surface without unexpected large spikes.
calculate ap of
liner surface
get the processed
surface
check height of each
spot on liner surface
10 times
I
If it is larger than 10
replace height of it by
total average height
Calculate new a,
of liner surface
check height of each
spot on liner surface
replace height of it by
local average height
If it is larger than 4 ap
Figure 2.7: Flow Chart of Iteration to Remove Unexpected Large Spikes
26
Color scale
length
AAJ
unit (micrometer)
Color scale
length
unit (micrometer)
200
150
150
10-
100
2
0
50
100
150
2
0
200
Axial direction
50
100
Axial direction
150
200
Figure 2.8: Original Surface and Processed Surface without Large Spikes
Unfortunately spikes along borders of plateau and valley cannot be filtered by this method. This
is because large slopes on measured surface may cause measurement and interpretation errors
of spikes between 2r, and 4o, which cannot be numerically removed by the above method,
but have a large impact on contact. Figure 2.9 shows a small patch of filtered surface after
removing unexpected large spikes, but discontinuous spikes are still shown along borders of
plateau and valley part.
Color scale length unit (micrometer)
2.5
2.5
2
51.
10
0 0 0 0 -5(3
15%*
Figure 2.9: Unexpected Spikes along Border of Plateau and Valley on Filtered Surface
2.3.2 Plateau Surface Roughness up of Different Sample Liners
Contact is only dependent on plateau of liners, and thus surface roughness is defined by Root
Mean Square (RMS) of plateau on liner surface and represented by up, as introduced in 2.1.4.
27
up can be obtained based on original measured surfaces, represented by original u-,, and
filtered surfaces without unexpected large spikes, represented by filtered op, as compared in
Figure 2.10. Surface roughness of original surface is larger than that of filtered surface and
more difference, more unexpected large spikes on the measured liner surface.
1
10
30r.
M:86
153
10
Io
so
S4
1se
II
2
3
I2t
so
S"u D-0-~M
100
SWg DrntO-m~
"un
153
2W
0
S"u DWecuu (%04
Sample Liner #3
0.072micro//original
0.057micro//filtered
Sample Liner #2
0.091micro//original up
0.055micro//filtered up
Sample Liner #1
0.049micro//original ap
0.038micro//filtered UP
10
1154
2
Ix
1
-1
10C
.2
-3
so
100
1W3
2
I
Sb"g Due-n(u
Sample Liner #5
0.162micro//original %p
0.120micro//filtered up
Sample Liner #4
0.381micro//original ap
0.310micro//filtered up
Figure 2.10: Original up and Filtered a, of Sample Liners
2.4 Measured Liner Modeling Method
One attempt in this thesis work is to apply Hertzian ellipsoidal contact model to the measured
surface. However, measured liner surface taken by confocal microscope reflects real geometry
on the surface and the shape of each asperity is very random and irregular. There is no theory
28
Up
predicting force between two asperities with random shapes, so before applying contact model,
asperities with irregular shapes should be modeled into new asperities with regular shapes. By
examining real asperity shape, it was found that ellipsoidal shape is a good approximation of
real asperity and there is formula for moderately ellipsoidal Hertzian contact [25].
The first step of modeling real surface to new surface with ellipsoidal asperities is to define
border of an asperity, i.e. the area covered by one asperity. The method is to find the maximum
rectangle in which all of the spots are in plateau area (heights of all the spots in the area are
larger than zero). Then width and length of the rectangle is known and half of them can be
defined as semi-major axis and semi-minor axis of the ellipsoid. The maximum height in the
rectangle area can be defined as height of the ellipsoid. By applying this method, each asperity
can be explored and modeled into ellipsoidal shape by using width and length of the asperity
area and maximum height in the asperity area, as illustrated in Figure 2.11. The real surface can
be modeled into new surface with ellipsoidal asperities by checking each asperity, as illustrated
in Figure 2.12.
Contact between Liner finish and piston rings is only dependent on plateau part of liner finish,
so only plateau part on measured liner surface is modeled. Because only if all spots in a
rectangle area are larger than zero on real measured liner surface, it will be defined as an
asperity, some area which is former plateau part will not be plateau anymore. After modeling
real measured surface, area of the plateau part decreases. In addition, o of modeled surface
becomes larger than that of real surface as a result of maximum height in a real asperity being
set to height of the modeled ellipsoidal new asperity and the average height of the plateau part
is becoming larger.
color scale length unit (micrometer)
0.30
0
Fgctur
color scale length unit (micrometer)
2
dA
0
Figure 2.11: Modeled One Asperity
29
0
Color scale length unit (micrometer)
Color scale length unit (micrometer)
1.2
.1
Qv.
1.0
.6
.4
06
1000
so~
c~c
01>
s
ecPIS-
Figure 2.12: Modeled a Small Patch of Surface
2.5 Conclusion
In this chapter, different sample liners are shown and the method to find mean plateau height,
as well as mean plateau height of different sample liner surfaces, has been demonstrated.
However, real measured liner surfaces cannot be fully trusted and some apparant
measurement errors due to dusts on measured surfaces and large slopes on measured surfaces
have been pointed out. In order to remove measurement errors, a measured surface
processing method has been introduced to numerically filter unexpected large spikes on
measured surfaces. After applying measured processing method, plateau surface roughness up
has been illustrated for different sample liners and compared with o-, of original measured
surfaces. An approach to model irregular asperities into regular ellipsoidal shapes for applying
Hertzian contact model is introduced in last section of this chapter and the disadvantages of
using this approach are also indicated.
30
3. Statistical Model and Hertzian Contact Model
In this chapter, challenges of using previous contact model based on Greenwood-Tripp model
and Hu et al correlation function have been demonstrated [20] [21]. It also introduces
Statistical Model based on Greenwood-Williamson Model and Hertzian Contact Model [19] [24].
Both models can be used to directly calculate pressure between liner finish and piston ring pack
at certain clearance height based on measured liner surface on which unexpected large spikes
have been numerically removed and asperities have been modeled to regular ellipsoidal shape,
as introduced in Chapter 2. In addition, results of relation between contact pressure and
clearance height by using both Statistical Model and Hertzian Contact Model have been given
for different sample liners.
3.1 Challenges of Applying Previous Contact Model
Details of previous contact model, including assumptions and equations are given in this section.
Based on the assumption that asperity height distribution is Gaussian distribution, discussions
about the inaccuracy caused by it are presented in two aspects. One is from Hu et al numerical
correlation function to simplify relation between contact pressure and clearance height, and
the other is from difference between Gaussian distribution and real asperity height distribution.
3.1.1 Assumptions and Equations of Previous Contact Model
Previous contact model is based on Greenwood-Tripp Contact Model and Hu et al correlation
function [20] [21]. It predicts relation of contact pressure P and clearance height h between
liner surface and ring surface, as illustrated in Figure 3.1. Asperities which are above ring
surface will be in contact with the ring surface and thus deformed. There are several
assumptions of this model including:
a.
b.
c.
d.
e.
f.
Deformation is constrained to purely elastic deformation
Contact is between two equally rough surfaces
There is no interaction between asperities on same surface
Asperity height on both rough surfaces is Gaussian distribution
Contact is based on statistical properties of asperities on rough surface, i.e. average
asperity size, number of asperity and asperity height distribution.
Asperities are in spherical shape [19] [20] [21]
31
Color scale length unit (micrometer)
Color scale length unit (micrometer)
151
05
I]
Sos
05
3-
I
D
45
06
1.6.
I
OcnfrentIa dir a
trction
3M0
1ectilOO
Aildf
I
I
i
2W
2W0
IM
i
I
so
10M
III
WO
.1
4s
L
Axial direction
Figure 3.1: Ring in Contact with Liner Finish at Clearance Height h
Equations of the previous contact model are demonstrated below. Hu et al gave fitting values
of A, w and Z based on Gaussian distribution and assumed value of K in the correlation
function: w = 4.0, A = 4.4068 x 10- 5 , Z = 6.804, K = 2.396 x 104 [21]. In the equation
which is used in Tian's cycle model, a coefficient cfct = 20 is added before Hu et al correlation
function in order to match with experimental data [22] [23] [26].
P
2 .5
=KEF
1
1-vi
E'
El
(h
1-v
E2
16Kf
rr(NRa) 2
15R
K =
F2.5
((h) =
__
f 0 <-
h
-
5
s
2
>2 e2ds=
a
_.
Aw
h
a
h
h
->0
a*
h
h
)P
P = cfct K E'
h
->)
10
UP
32
In above equations, E1 and E2 are young moduli of liner finish and piston ring, v, and v 2 are
relatively Poisson ratios of them, E' is combined modulus, N is asperity density on liner surface,
R is average radius of all asperities on liner surface, a is plateau surface roughness of liner
surface [20].
3.1.2 Correlation Function based on Gaussian Distribution
The part F2 .5
in Greenwood-Tripp model which relates the probability distribution of
asperity height has been fit into a correlation formula for convenience of numerical calculation.
In this function: o = 4.0, A = 4.4068 x 10- 5 , Z = 6.804 [21]
1
F2.s=5-
By comparing the real value of F2.5
A(-)z _a
h 5 s
< s - - >2 e--ids=
h
h
2
()
and Hu et al correlation formula, there is still a
difference which is not negligible, as illustrated in Figure 3.2. The red star indicates real value of
F2 .5 (s) with Gaussian distribution and black cycle indicates Hu et al correlation function. As a
result, a new correlation function represented by blue plus marker can be attained and it has
the same form as Hu et al correlation function, but different values for o, A and Z. In the new
correlation function, w = 4.0, A = 1.101 x 10-4, Z = 5.529. From the figure shown below,
with Gaussian
distribution than Hu et al correlation function. In the figure below, lambda =
33
.
one can conclude that new correlation function fits better to F2.s
6x
10
e
v
Gaussian Distribution
Hu et al Correlation Function
New Correlation Function
r
4
U-
3
p
p
1
r
r
2
2.2
r- -Ilnnq=tsx
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
lambda
Figure 3.2: Comparison of F2 .5 h based on Gaussian Distribution and Hu et al Correlation
Function and New Correlation Function
3.1.3 Comparison of Previous Contact Model with Real Situation
For the real situation of contact between liner finish and piston ring pack in an internal
combustion engine, piston ring surface can be assumed to be a purely smooth surface due to its
manufacturing processing method. Therefore, it is more reasonable to model the situation into
contact between a smooth surface and a rough surface, while the previous contact model is for
two rough surfaces in contact with each other.
The other ambiguity of applying previous contact model is related to asperity height
distribution. In previous contact model, the assumption that asperity height distribution is
Gaussian distribution was made, but real asperity height distribution is not normal distribution,
as illustrated in Figure 3.3 for sample liner #1, especially in the area of 2a-, to 4%p which highly
affects contact.
34
__
-
-
-
-
-
-
-
__
-
---
Sample liner surface #1 plateau height distribution
x 104
4.5:-
Sample liner surface #1 plateau height distribution
4000
Liner Surface #1
Normal Distribution -r
-
4
Liner Surface #1
Normal Distribution
3500
3.5--
3000-
32500
E
-
2.5
.
2
2-
*1
1.5 r
2000
1500
10001.
1
0.50o
0
-- _1
0.5
-
1
-
1.5
-_
--I --
2
lambda
- -r
2.5
_
3
___
_
.-
3.5
.
------------ Z__
2
4
2.2
2.4
2.6
2.8
3
lambda
3.2
3.4
3.6
3.8
4
Figure 3.3: Comparison between Plateau Height Distribution of Liner Surface #1 with Normal
Distribution
3.2 Statistical Model
Measured liners with real geometry can be obtained by confocal microscope. Therefore, the
assumption that asperity height distribution is Gaussian distribution can be discarded and more
accurate and reasonable results based on real geometry can be obtained. In this section,
Statistical Model which is based on Greenwood-Williamson Model and for the situation of
contact between one rough surface and one smooth surface has been applied to measured
surface, and relation between contact pressure and clearance height is demonstrated for
different sample liners.
3.2.1 Assumptions and Equations of Statistical Model
Statistical Model considers the situation of contact between one rough surface and one smooth
surface, as illustrated in Figure 3.4, and predicts relation between contact pressure P and
clearance height h. Asperities which are beyond ring surface will be in contact with ring surface
and thus deformed. It is based on real asperity height distribution and not Gaussian distribution
anymore. There are several assumptions of this model including:
a. Deformation is constrained to purely elastic deformation
b. Contact is between one rough surface for liner surface and one smooth surface for ring
surface
c. No interaction between asperities on same surface
35
d. Contact is based on statistical property of asperities on whole surface, i.e. average
asperity size, number of asperity and asperity height distribution.
e. Shape of asperity is ellipsoidal [24].
Color scale length unit (micrometer)
0%
Crc,
200
100
Ccirect/
0
0
pa\ 6ifection
Figure 3.4: Contac between One Rough Surface and One Smooth Surface
A rough surface is represented by an array of identical asperities (with average size of asperities)
differing only in their heights above a reference plane, which is the zero datum plane on
measured liner surface. The situation of one rough surface which is measured liner surface and
one smooth surface which is piston ring surface has been considered, as shown in Figure 3.5.
Suppose O(z) is distribution of asperity heights, N is surface density of asperity peaks on rough
surface, h is clearance height between reference plane of smooth surface and reference plane
of rough surface, A is nominal rough surface area. For contact, asperity height z above
reference plane must be larger than h:
z> h
Interference, deformation height of asperity, can be defined as:
w = z - h
Number of asperities with heights in the range z to z+dz situated on rough surface is:
ANO(z)dz
36
Therefore, expected contact force on rough surface due to compression of smooth surface is:
P(w)0(z)dz
P(h) = AN
In the above equation, P(w) is contact force due to deformation of one asperity and based on
Hertzian contact [19]:
1 3
4
P(w) = - E'Riwf
3
1
1 E22
R is radius of the deformed asperity. El and E 2 are Young Moduli of rough surface and smooth
surface. v, and v 2 are Poisson ratio of rough surface and smooth surface.
Thus, the expected total force between rough surface and smooth surface is,
1
4
P(h) = -NE'RA
3
f
3
fm(z
- h)-f O(z)dz
The expected pressure between rough surface and smooth surface will is [24],
4
1
Pressure(h) = -NE'R
3
3
(z - h)2 O(z)dz
fh
-Y
clearance
height h
,z
I
I
\ I
(N
reference plane of
smooth surface
reference plane of
rough surface
Figure 3.5: Contact of One Smooth Surface and One Rough Surface
37
3.2.2 Application of Statistical Model
In order to apply Statistical Model, modeled surface based on original measured liner is needed,
as shown in Figure 3.6. The procedure to generate modeled surface is described in Chapter 2
and it just took the plateau into account. Based on modeled surface, asperity number and
average asperity radius and asperity height distribution need to be calculated to apply in the
statistical formula.
Color scale length unit (micrometer)
Color scale length unit (micrometer)
JA
6
0
. 31W
20MW
~ISD
3W3
31
12W
Sol
em
&e0 0
0
W
Li
Figure 3.6: Measured Liner Surface and Modeled Liner Surface
Based on the modeled liner surface above, average asperity radius is 1.2505 micrometer and
asperity number is 108. Height of each asperity can also be obtained, as well as asperity height
distribution. After plugging into the equation relates contact pressure and clearance height
shown below, results of contact pressure can be obtained in Figure 3.7.
4
1
Pressure(h) = 3NE'Ri
38
3
(z - h)f O(z)dz
-
0.4
0.35
0.3
-
-
2 0.12
-
a- 0.25
0.1
0.05
0
-2
-
-
0.1
2.5
3
lambda
3.5
4
Figure 3.7: Relation between Contact Pressure and Clearance Height
3.2.3 Results of Different Sample Liners by Statistical Model
In Table 3.1 below, average asperity radius, asperity number and surface roughness of plateau
have been tabulated for different sample liners. In Figure 3.8 below, relation between contact
pressure and clearance height has been demonstrated for different sample liners.
Statistical Method
1.40E+04
1.3763
0.038
1.08E+04
1.2569
0.055
1.15E+04
1.3434
0.057
2.42E+04
1.7362
0.31
1.97E+04
1.3313
0.12
Table 3.1: Statistical Data of Different Sample Liners
39
Sample Liner #1
10
5
4
I;
I
co
2o
4)
CL
I
3
2
1
0
1w
ISO
Mo
3
lambda
2.5
2
Sh mg
Dvsw (Urn)
3.5
Sample Liner #1
Sample Liner #2
07
5
I
I
-
-
---
-
-
4
,D
3
CL
2
1
0
2
2.5
Sbd1g Drctif (n)
3
lambda
3.5
Sample Liner #2
Sample Liner #3
10
8
6
a 4
I
-- -...
------2
0
2
2.5
3
lambda
Srg DOeton (um)
Sample Liner #3
40
3.5
4
Sample Liner #4
o60
50
0
1601
11
30
-
20
OD0
2
cc 40
-
300
0
so
100
ISO
2
00
Sldng Dved on (um)
2.5
3
3.5
4
lambda
Sample Liner #4
ern
F g0r .. : C.n.c P r ss r..D Sample Linerr #5
30
140~
a.
1210
400
Sk~noOndm(UM)lambda
Sample Liner #5
Figure 3.8: Contact Pressure of Different Sample Liners by Statistical Method
Contact pressure of sample liner 4 and sample liner 5 are much stronger than that of the other
three sample liners, as shown in Figure 3.8. The first reason is asperity density of these two
liners is larger, which means there are more asperities on these two liners. However, this is not
the most important factor because contact pressure is proportional to asperity density in
Statistical Model as shown in the equation above and asperity density of liner 4 and liner 5 is
just around two times larger than that of the other three liners, as demonstrated in Table 3.1.
The second reason leading-to the large difference is higher plateau surface roughness up of
liner 4 and liner 5. Though the range of clearance height is the same for all sample liners from
2%p to 4-p, the one with a larger op has more deformed height on its asperities, which can
cause much stronger contact. This is the main reason leading to the larger difference and it
41
gives the trend that contact increases with larger a.. Other factors, such as average asperity
size and asperity height distribution also influence contact pressure.
3.2.4 Application of Statistical Model based on Gaussian Distribution
In some situation when surface measurement techniques are not available and real liner
surface geometry cannot be obtained, Statistical Model for contact of one rough surface and
one smooth surface can still be used based on the assumption that asperity height distribution
is Gaussian distribution. Therefore, contact pressure can be represented by,
w1
4
Pressure(h) = -E'NRa
3
Pressure(h)
h
3
s2
1-\~ir
< s - - >2 e Tds
<s
=
a
hEF.
g
4
K = -NRa43R
F1;
< s --
=---
h
3
s2
>2 e-2ds
a
For convenience of numerical calculation, the part related to Gaussian distribution in the
pressure function can also be fit into a correlation function like that of contact for two rough
surfaces.
h
A(o -)z
h 3 s2
a
< s- ->2 e~ 2ds ==-d
F1s (-)
=,-- f C
h
0
->0
where A = 1.844 x 10-4, w = 4, z = 5.133, as shown in Figure 3.9.
42
X 103
Gaussian Distribution
- -
6
_- _ Correlation Function
4
U?
2-
u-3
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
lambda
Figure 3.9: Comparison of F1 .5
with Gaussian Distribution and Correlation Function
The value of K can be assumed based on statistical data of five sample liners. The average
asperity density of different sample liners is 1.6 x 10 4 /mm 2 , the average asperity radius is
1.4088 micrometer and the average plateau surface roughness is 0.116 micrometer. Therefore,
the assumed K value can be 1 x 10-3
3.3 Hertzian Contact Model
Hertzian contact model is based on measured liner surface with real geometry, and the
assumption that asperity height distribution is Gaussian distribution can be discarded like
Statistical Method. After modeling the measured liner surface, the shape of asperities on the
modeled surface is regular ellipsoid. Based on Hertzian contact theory, if deformation of an
ellipsoid is known, the contact force caused by the deformation can be calculated based on
deformation height, ellipsoid size and material property. All the asperities on the modeled
surface can be regarded as separated ellipsoids. At certain clearance height, the deformation of
each asperity is known if asperity height is larger than clearance height, and thus the contact
force due to the deformation. Contact force on the whole surface is the sum of contact force at
each asperity. In this section, Hertzian Contact Model has been introduced and applied to
sample liners. The relation between contact pressure and clearance height by Hertzian Contact
Model is also demonstrated.
43
3.3.1 Assumptions and Equations of Hertzian Contact Model
Hertzian Contact Model predicts contact between one rough surface and one smooth surface.
The difference between Hertzian Contact Model and previous contact model is that it is based
on real asperity height distribution and not the assumption of Gaussian distribution. It also
differs from Statistical Model because it calculates contact force of each asperity and sums
them up to get contact force on whole surface. Statistical data, such as average asperity radius
and asperity density are not necessary. There are several assumptions of applying Hertzian
Contact Model:
a. Deformation is constrained to purely elastic deformation
b. Contact is between one rough surface for liner surface and one smooth surface for ring
surface
c.
No interaction between asperities on same surface
d. Shape of asperities is ellipsoid
When one ellipsoid is in contact with a smooth surface at a clearance height h, as illustrated in
Figure 3.10, it means the height of the asperity is larger than clearance height and the asperity
is deformed. Assume semi-major radius of the ellipsoid is A, and semi-minor radius of the
ellipsoid is B. The height of the ellipsoid above reference plane is L and clearance height is h.
Therefore, the equivalent radius of the ellipsoid which can be used in formula for circular
contacts is [27],
R= (A x Bx (
2 B)
Elastic deformation of the ellipsoid is,
hdeform =
L- h
Thus, contact force caused by the deformation is,
4
1
3
F = 3 E'Rhdeform2
Therefore, contact due to deformation of this ellipsoid is obtained by the above equation.
When there are lots of ellipsoids on a surface like the condition of modeled liner surface, each
ellipsoid can be regarded as separated ones. For each ellipsoid, equivalent contact radius can
44
41
be calculated by semi-major radius and semi-minor radius of the ellipsoid and deformation is
from the height of the ellipsoid and clearance height between two reference planes on liner
surface and piston ring surface. By plugging into the above equation, contact induced by each
ellipsoidal asperity is known and thus the total contact between the two surfaces.
Color scale length unit (micrometer)
0.51
.2
0
10
C41
"Mis
Ile'
6
.1
o
4
2
C
/06
Figure 3.10: One Asperity and One Smooth Surface in Contact
3.3.2 Application of Hertzian Contact Model
The first step of applying Hertzian contact model is modeling irregular asperities into regular
ellipsoidal asperities, as described in Chapter 2. A small patch of original measured surface and
modeled surface based on it are shown in Figure 3.11.
Color scale length unit (micrometer)
Color scale length unit (micrometer)
1.5
0
(%liJ
5-0
(9^5O
4'.5-
U
20
60
r 40 W""'-
X&te
PA'a
Figure 3.11: Measured Liner Surface and Modeled Liner Surface
45
80
There are twenty ellipsoids on modeled surface shown above and when modeling measured
liner surface, data of semi-major radius, semi-minor radius and asperity height of each asperity
have been saved for further calculation. Results of contact pressure at different clearance
height are shown below in Figure 3.12.
12
-
10 -- ------ ------- -- -------- ------ --- ---------- ------ ---- - -------------- -----------
02
- --- ----- - - ---- - ---- - ---- -
-
C - 8 _- - -
2.5
3
3.5
4
lambda
Figure 3.12: Relation between Contact Pressure and Clearance Height
o.7.
ISOA
3.3.3 Results of Different Sample Liners by Hertzian Contact Model
In Figure 3.13 below, relation between
2-2
contact pressure and clearance height has been
demonstrated for different sample liners.
200
Sample Liner #1
1
....
1
6
20
""----
3
40
0
W0
1W
Sh&dg Dnoln
1W
20
2
amb
Sample Liner #1
46
2.5
3
lambda
3.5
4
Sample Liner #2
10
2w3
10 r
180
8
160
140
3
120
Cal
6
--K 41
4
80
so
CL
0
2
zu
1i
Slidw
Oircton (uM)
150
-
0
u
r
2
2.5
3
lambda
-
___________________
3.5
4
3.5
4
3.5
4
Sample Liner #2
Sample Liner #3
12
200
180
10
160
I 140
I
I'I
8
IOD
820
CL
6
4
80
(-)
3
40
2
2DI
n
-5
2
2.5
Slidg DWcwon (um)
3
lambda
Sample Liner #3
Sample Liner #4
107
IUU
F-
80
CL
I
60
(D
40
10
j
2
20
02
2
Sb" Netion (urn)
Sample Liner #4
47
2.5
3
lambda
Sample Liner #5
10
60
50
40
30
I IS MO2
2.53 3.4
-
20
10
0-
n
_0
o
Sbdn Dwcten (um)
____
____
______
-
_
_I
_
_
_
_
_
_
lambda
Sample Liner #5
Figure 3.13: Contact Pressure of Different Sample Liners by Hertzian Contact Model
3.4 Discussions of Results by Statistical Model and Hertzian Contact Model for
Different Sample Liners
Relation between contact pressure and reference height has been procured for different
sample liners by both Statistical Model and Hertzian Contact Model. Though the results are
based on same modeled surface of measured sample liners, Statistical Method requires
asperity average radius, asperity density on surface and asperity height distribution to calculate
contact pressure, while Hertzian Contact Model computes contact force on each asperity and
uses total force to get contact pressure. Therefore, contact pressure is different for same
sample liner by different models. Difference of contact pressure for same liner surface has been
tabulated and reasons are given to explain the difference. Influence of asperity size, plateau
surface roughness on contact pressure is also discussed. In addition, previous contact model
and Statistical Model based on Gaussian distribution have been compared for application in
case that measured technique is unavailable to get measured liner surface.
3.4.1 Comparisons of Statistical Model and Hertzian Contact Model
Results of contact pressure in relation to clearance height by Statistical Model and Hertzian
Contact Model have been displayed in same figure of each sample liner to compare the two
different methods, as illustrated in Figure 3.14. In addition, maximum difference and average
difference of the results by different method are calculated in percentage of the result by
48
Hertzian Contact Model in clearance range of 2cr to 4%p to numerically compare the difference,
as demonstrated in Table 3.2.
Sample Liner #1
.7
7
Statistical Model
Hertzian Contact Model
'
6
-
10
5
0M
3,
4
-D
3
2
2
3
0
0D
2
2.5
Sliding D~irm (urn)
3
lambda
3. 5
Sample Liner #1
Sample Liner #2
10
200
10
' Statistical Model
180
160
I120
Hertzian Contact Model
""" '
8-
140
Cu
0~
6
_
11)
100
U)
Cl)
so0
w6
4-
2
2
3
S"dg Dircson (uM)
ISO20
0
..
2
2.5
3
lambda
Sample Liner #2
49
3.5
4
Sample Liner #3
.7
10
Statistical Model
Hertzian Contact Model
-
10
S1
10
CL
2
3
D
S"in
In
28
---
--
}
0
-- O
2
a-
Dircbinm)
2.5
3
lambda
4
r #3
Sample
10
C,
Sample Liner #4
00
""
80
(,
0)
I'U
3.
" Statistical Model
Hertzian Contact Model
40
2
20--
3
0
1(0
SdM D#Wm (um)
-
(.7
2
2.5
3
lambda
3.5
4
Sample Line
Line r #4
Sample Liner #5
10
30
=---Statistical Model
-
50
II
'U
*
i"
-" Hertzian Contact Model
1
1430--'~
*2
0.)
*
3
10
0
200
SWing Diciioan (urn)
Sample
2
2.5
3
lambda
3.5
4
r #5
Figure 3.14: Comparison of Statistical Model and Hertzian Contact Model for Different Sample
Liners
50
Compare of Statistical Method (SM) and Hertzian Contact Method (HCM)
29.94
29.67
30.47
30.13
30.53
30.22
39.64
37.9
34.96
34.22
Table 3.2: Maximum Difference and Average Difference of Statistical Model and Hertzian
Contact Model for Different Sample Liners
From the figures above, Hertzian Contact Model predicts larger contact pressure than Statistical
Model for all sample liner surfaces. This is mostly caused by the part of asperity height
distribution in statistical model. When numerically taking asperity height probability
distribution, the asperities with similar height will be in the same small height range. When
clearance height is classified, some asperities will be neglected for contact with liner surface,
but they are actually in contact with liner surface.
Another factor inducing the difference is average asperity size in Statistical Model. When
average asperity size is taken in replace of exact size of each asperity, it will make contact force
a little bit larger. But in real situation, there are not many large asperities due to existence of
honing angle, and thus the influence of asperity size is negligible. Figure 3.15 shows asperity
size distribution of sample liner #2 and the radius of most asperities are in the range of 1
micrometer to 2 micrometer. In order to ensure the difference by Statistical Model and
Hertzian Contact Model is from statistical asperity height distribution on sample liner instead of
statistical asperity size, a surface with identical asperity size is generated. The results of the
generated surface by two different methods are shown in Figure 3.16. Statistical Model still
predicts smaller contact than Hertzian Contact Model.
51
x 10
3
3
-_-_
_
2.5
----
2
E
1.5
1
0.501
3
2
2.5
asperity radius (micro)
1.5
3.5
4
Figure 3.15: Asperity Size Distribution of Sample Liner #2
_
12
4
""" "Statistical
Model
--"
Hertzian Contact Model
10
c~8'
26
CL,
2
n
2
2.5
3
lambda
3.5
4
Figure 3.16: Comparison of Statistical Model and Hertzian Contact Model for Generated
Surface with Identical Asperity Size
The differences between Hertzian Contact Model and Statistical Model of sample liner #1, #2,
#3 are around 30 percent of the results by Hertzian Contact Model. Plateau surface roughness
of such sample liners is around 0.04micro and 0.05micro. For sample liner #4 and #5 whose
plateau surface roughness are larger and around 0.1 micro and 0.3 micro, differences between
the two methods become larger. As a result, difference between Hertzian Contact Model and
Statistical Model increases with plateau surface roughness.
52
Additionally, average difference and maximum difference for same sample liner are nearly the
same which means the trends of Hertzian Contact Model and Statistical Model are similar. If the
relation between contact pressure and clearance height has been fit into a correlation function,
the difference of the correlation function by the two methods is just a parameter.
3.4.2 Factors Influence Contact Pressure
Plateau surface roughness of liner surface highly influences contact pressure between liner
surface and piston rings. From Table 3.1 and Figure 3.14, it is found that sample liner #1 has the
smallest plateau surface roughness, and then sample liner #2, sample liner #3, sample liner #5,
and sample liner #4 has the largest plateau surface roughness. Contact pressure predicts the
same ranking no matter by Statistical Model or Hertzian Contact Model.
Another observation is that trend of the plot of contact pressure and clearance height is not
similar for different sample liners and this is due to different asperity height distribution. When
contact pressure is calculated by Statistical Method, integration of asperity height distribution
results in power of the trend between contact pressure and clearance height.
3.4.3 Comparison of Previous Contact Model and Statistical Model based on
Gaussian Distribution
When measured liner surface is unavailable due to lack of surface measurement techniques,
real asperity height distribution is unknown and Gaussian distribution for that is a good
assumption [20]. Previous contact model is based on the assumption of Gaussian distribution
and Hu et al assumed values for asperity radius, asperity density and an experienced parameter
added in Tian cycle model [22] [23]. In section 3.2.4, a statistical model based on Gaussian
distribution has also been introduced, as well as the assumed values for asperity radius and
asperity density based on sample liner surfaces. In Figure 3.17, the two different models are
compared.
53
2.5
C
-
Previous Model Gaussian Distribution
"" Statistical Model Gaussian Distribution
-
2
.5
0.5
0
--
2
2.5
_
3
3.5
04
4
lambda
Figure 3.17: Comparison of Previous Contact Model and Statistical Model based on Gaussian
Distribution
Statistical Model based on Gaussian distribution and average value of asperity radius, asperity
density of sample liner surfaces predicts less contact than previous contact model. The power
of trend of statistical model based on Gaussian distribution is smaller than that of previous
model based on Gaussian distribution because the previous one is for two rough surfaces and
new statistical one is for one rough surface and one smooth surface.
3.5 Conclusion
In this chapter, the calculations by Statistical Model and Hertzian Contact Model are based on
the assumption of no asperity interaction on same surface, which is the same as all Greenwood
based models, including previous contact model.
The first section evidently demonstrated limitations of the previous contact model when
applied to different honing surfaces. Firstly, previous contact model is for contact between two
rough surfaces while the real situation is contact between one rough surface and one smooth
surface. Secondly, previous contact model is based on the assumption that asperity height
distribution is Gaussian distribution, while real asperity height distribution is available through
confocal microscope and the assumption is not necessary. Additionally Hu et al correlation
function is not accurate.
54
In second part and third part of this chapter, Statistical Model and Hertzian Contact Model were
introduced. The details of assumptions and equations in each model were clarified. Results of
different sample liner surfaces by Statistical Method and Hertzian Contact Model were
demonstrated.
Comparisons between Statistical Model and Hertzian Contact Model were also given. Statistical
Model always predicts less contact than Hertzian Contact Model due to integral of asperity
height distribution in statistical model. The difference of two models depends on plateau
surface roughness of sample liner and rougher surface will give larger difference. It was also
identified that surface roughness and asperity height distribution highly influence relation
between contact pressure and clearance height. In addition, statistical model based on
Gaussian distribution is also given in case that measured liner surface is not available.
55
4. Deterministic Contact Model
This chapter introduces Deterministic Contact Model which is originally proposed by A.A.
Lubrecht [26]. In this model, modeled liners with regular asperities are not necessary for
application and the effect of interaction between asperities is considered. Therefore, this model
is more related to real situation and practical application. In this chapter, assumptions and
equations of deterministic model have been demonstrated, as well as multi-level method which
can save calculation time and simplify computation procedure. Results of contact pressure at
certain clearance height are shown for different sample liners. Additionally, comparison
between Deterministic Contact Model and Statistical Model, Hertzian Contact Model is
discussed, as well as boundary effects, different dimensionless process, contact area, accuracy
of modeled surface, convergence speed and influence of level size in deterministic contact
model.
4.1 Approach of Deterministic Contact Model
Measured liner surface is necessary for applying Deterministic Contact Model and serves as an
input. Modeled liner surface described in Chapter 2 is not needed for application and thus
deterministic contact model preserves real surface geometry and is more accurate than
Statistical Model and Hertzian Contact Model introduced in Chapter 3. Additionally, asperities
on measured surface cannot be regarded as separated ones because they will interact with
each other. Contact pressure caused by deformation of one asperity will also deform other
asperities on the surface. In Deterministic Contact Model, interaction between asperities has
been considered which makes it closer to reality than the other two methods in Chapter 3.
4.1.1 Assumptions and Formulas in Deterministic Model
There is an assumption of Deterministic Contact Model:
a. Deformation is constrained to purely elastic deformation
Because interaction between each asperity is identified in this model, elastic deformation
w(x, y) of one point due to pressure distribution on whole surface needs to be calculated and it
can be approximated by [28]:
w(x'y)
2
TE'
+ +0oo00
_O
p(x',y')dx'dy'
_c, V(x - x') 2 + (y
56
-
yf)2
where
2
_
E'
V2
1+
1-_V2
2
El
E2
and v, and v 2 represent Poisson ratio of liner finish and piston ring. E1 and
moduli of liner finish and piston ring respectively.
E2
are elastic
Defining height of point (x, y) above reference plane of liner finish surface as 1(x, y), finally
gives gap between liner finish and ring surface as
2
+oo
+oo_______________
rE' _00
_cO V(X _ XI2 + (y - y')2
h(x, y) = ho - l(x, y) +-
I,
where ho represents clearance height between liner and piston ring surface.
When two surfaces are loaded together, the gap between them should become zero (contact,
positive pressure: domain to), or remain positive (no contact, zero local pressure: domain w 2 ).
The complementarity problem can be expressed as:
h(x,y) =0 ,p(x,y) > 0
(x,y) E a)
h(x,y) >0,p(x,y) = 0
(x,y) E
2
wE'
h(xy) = ho - I(x,y) + 2I
+oo
_ 0
+o
(
2
p(x',y')dx'dy'
_O V(x
X')2
+ (y _y)
2
In mathematical terms this is a complementarity problem. The two equations are valid on the
sub domains w, and w 2 respectively, but the division of the domain w 1 and w 2 into the two sub
domains is a priori unknown.
A numerical approach is needed when liner surface is not smooth and geometry is random. For
the numerical solution it is convenient to rewrite the equations introducing dimensionless
variables [28]. The first choice is to use parameters of Hertzian contact solution.
When a sphere with known radius or curvature R pressed together with a smooth surface, the
contact area is a disc of radius a, so h(x, y) = 0 for (x)2 + (Z)2 < 1, and the pressure is given
a
by a semi-elliptical pressure distribution:
57
a
p(x,y) = f(x)= fPh
2
2
2
0,
otherwise
and deformed distance is given by w, where variable ph is given by
P =
E'w
7ra
Therefore, to normalizing the equation related gap between liner finish and ring surface and
pressure distribution, introducing:
-
x
y
p
h
X = -; Y = -; P = -; H =
a
a
Ph
w
The problem can be written as:
H(x, y) = 0, P(x, y) > 0
H(x,y) > 0,P(x,y) = 0
H(x,y) = HO -L(x,y)
+
V2+w f+c
ff
contact
no contact
P(X',Y')dX'dY'
(X'Y')dX+y'
The objective of deterministic contact model is to solve the above equations numerically and
find the pressure distribution on whole surface. An initial guess of pressure distribution is given
and gap distribution can be obtained based on that. But for the points where pressure is larger
than zero, gap has a positive or negative value instead of being exactly zero on them, and thus
pressure needs to be adjusted. The core theory of numerically calculation is to achieve the
balance, when there is contact pressure, the gap is zero and when there is no contact pressure,
gap is larger than zero.
4.1.2 Multi-level Method
In deterministic contact model, a multi-level method is used to find balance of the above
equations and calculate contact pressure distribution. The approach is based on discretizing
liner surface using a square grid of uniform mesh size in each direction, but it is not real grid on
58
measured liner surface. Grid size is dependent on user's indicated level size. For example, no
matter what the size of the input surface is, level 8 gives a discretization of 2048 grids by 2048
grids. Therefore, when a liner surface is input, rearrangement of liner surface geometry will give
a new surface with 2048 grids by 2048 grids if level size is 8. Then, a V-cycle multi-grid
technique is employed to find contact pressure at each grid. In this particular case of the
calculation of multi-integrals, the aim is to utilize coarser grids to decrease computing time
without significantly reducing accuracy of the integrals. Larger levels will give more grids and
more accurate results, while requires longer calculation time. Meantime more cycle results in
more accurate results and also longer computation time. Results tabulated in 4.2 are all run at
level 8 and cycle 8. This is because input surface has 3243 grids by 3243 grids on original surface
and level 8 results in the smallest grid size which is larger than original grid size.
4.1.3 Application of Deterministic Contact Model
In Deterministic Contact Model, original measured liner, shown in Figure 4.1, and clearance
height are inputs, as well as number of level, number of cycle which decide accuracy and
convergence. The output will be contact force on input surface at designated clearance height
and deformed surface, as demonstrated in Figure 4.2.
Color scale length unit (micrometer)
02
20
12
Ci150
enirci
F
agu
r 1 a 0rc
p
Figure 4.1: Original Measured Liner Surface
59
Color scale length unit (micrometer)
0
2
-15
10
200
150
-0
r'CO
12
e/ -*frect 00 s
1%ctio
d01
Figure 4.2: Deformed Liner Surface
Result of contact pressure at different clearance height by deterministic contact model for
2.5
-____
___
-
-
2
-
above measured surface is shown below in Figure 4.3.
-
-
1.5
U,
0
2
2.5
3
lambda
3.5
"
0.5
Figure 4.3: Relation between Contact Pressure and Clearance Height
4.2 Results of Different Sample Liners by Deterministic Contact Model
In Figure 4.4 below, relation between contact pressure and clearance height has been
demonstrated for different sample liners by Deterministic Contact Model.
60
Sample Liner #1
10
200 r
2.5;--
2
1.51
I
CD
1
2
0.5
3
A
2.5
2
SW"ing Dclion (um)
3
lambda
3.5
Sample Liner #1
Sample Liner #2
.0
3
2.5
10
cc
0
I
Go)
2
F
1.5
U)
1
2
(-)
0.5
-
3
0
.5
2
2.5
SM"ing Oection (urn)
3
lambda
i 'v
vI
Sample Liner #2
Sample Liner #3
10
3.5
3
3
2.5
2
Cu
0.
I
.1
2
U)
U)
1.5
0.
1
-2
-3
0.5
-4
0
2
SWing Dirction (un)
Sample Liner #3
61
2.5
3
lambda
3.5
v
Sample Liner #4
10
10
8
0M
6
(n
cn
10
Ia
4
CL
2
2
4
0
Si"
(
-5
2
25
"edon
3
lambda
3.5
Sample Liner #4
Sample Liner #5
.0
6
2W
160
5
10
4
160
100
02
~60
2
1
00
60
ndf
1
200
0
-6
2
23
lambda
Sample Liner #5
Figure 4.4: Contact Pressure of Different Sample Liners by Deterministic Contact Model
4.3 Discussions of Results by Deterministic Contact Model
Relation between contact pressure and clearance height has been procured by Deterministic
Contact Model for different sample liners. In this model, input is original measured liner surface
which is different from that in Hertzian Contact Model and Statistical Model as described in
Chapter 3. Another difference is interaction between asperities has been considered in
Deterministic Contact Model. In this part, comparisons between Deterministic Contact Model
and Hertzian Contact Model, Statistical Model have been discussed to see the influence of
interaction. Modeled liner surfaces which are used in Hertzian contact model and Statistical
62
Model are also examined by Deterministic Contact Model to test the accuracy of modeling.
Boundary effect in Deterministic Contact Model is also discussed, as well as different
dimensionless process which is based on contact of punch with smooth surface. Influence of
level size and relative calculation time are also presented.
4.3.1 Comparisons of Deterministic Contact Model and Hertzian Contact Model,
Statistical Model
Results of contact pressure in relation to clearance height by Deterministic Contact Model,
Statistical Model and Hertzian Contact Model have been displayed in same figure of each
sample liner to compare the three different methods, as illustrated in Figure 4.5. In addition,
maximum difference and average difference of the results by Deterministic Model and Hertzian
Contact Model are calculated in percentage of the result by Hertzian Contact Model in
clearance range of 2%- to 4%p to numerically compare the difference, as demonstrated in Table
4.1.
Sample Liner #1
10
""
6
"
-
Statistical Model
Hertzian Contact Model
" Deterministic Contact Model
0.
1122D
2
slidig
onetwa(um)lambda
Sample Liner #1
63
2.4
.
C-)U
Sample Liner #2
10
Statistical Model
Hertzian Contact Model
Deterministic Contact Model
10
a)
0~
~1)
C.)
(a
(a
a)
a-
1P
4
2
2""4
0
2
2 5
3
.5
lambda
SMing Dwon (r)
Sample Liner #2
10
Sample Liner #3
.7
12.
Statistical Model
" Hertzian Contact Model
"
Deterministic Contact Model
"
10
18
1"
6
10
Ca
a)
8
4-
is
2
2
3
>
n
Sing Dcbon
2
6
10
---2.5
3
35
lambda
)
83
"
i
8
U
Sample Liner #3
Sample Liner #4
10
100
"
160
Statistical Model
Hertzian Contact Model
80
"
Deterministic Contact Model
-too
60
10
160
ISO
10
Ca
ca
40
2
40
3
0-
45
20
0
2
Sbdmg Diretion (un)
25
"3
lambda
Sample Liner #4
64
v
vN'
Sample Liner #5
60
0
21
1
50
-
160
Statistical Model
"
Hertzian Contact Model
"""m"" Deterministic Contact Model
-C 404-
140
120
30
-
(,
100
CL 20
0
100
0
S""n
DIf~h
ISO
2
200
(Ur)
2.5
3
.6v V
lambda
Sample Liner #5
Figure 4.5: Contact Pressure of Different Sample Liners by Deterministic Contact Model,
Hertzian Contact Model and Statistical Model
Compare of Deterministic Contact Model (DCM) and Hertzian Contact Method (HCM)
93.09
64.31
99.09
68.97
92.73
65.96
97.50
86.57
99.80
90.18
Table 4.1: Maximum Difference and Average Difference of Deterministic Contact Model and
Hertzian Contact Model for Different Sample Liners
From Figure 4.5, it is noticeable that Deterministic Contact Model predicts much less contact
than Hertzian Contact Model and Statistical Model. The large difference results from interaction
between asperities on liner surface and it displays the significant influence of interaction.
Meanwhile, Hertzian Contact Model and Statistical Model overestimate contact pressure
because modeled surface increasing plateau surface roughness which serves as another
important factor inducing the large difference.
65
As shown in Table 4.1, results by Deterministic Contact Model and Hertzian Contact Model
show larger gap for sample liner #4 and sample liner #5 which are relatively rougher than the
other three sample liners. This is due to more asperities and higher asperities on rougher liners
which result in more interaction between asperities. Another behavior is that maximum
difference and average difference between results by deterministic model and Hertzian contact
model is not consistent. It indicates different trend for the results by two methods.
4.3.2 Comparisons of Modeled Surface and Original Surface
As mentioned above, modeled liner surface serves as an important factor inducing difference of
results by deterministic contact model and Hertzian contact model. Modeled liner surface only
preserves roughness morphology on original surface and changes real geometry. Some small
plateau area has been neglected due to modeling procedure and average height of an asperity
is also changed because of shape limit. However, modeled liner surface is necessary for
application of Hertzian Contact Model and Statistical Model which require less calculation time
than Deterministic Contact Model. In order to examine accuracy of modeled surface,
Deterministic Contact Model was applied to both original measured liner surface and modeled
liner surface to compare the difference, as illustrated in Figure 4.6.
2
Sample Liner #1
3.
Original Liner Surface
ISO"
2.
Modeled Liner Surface
1
1.5
-
1,
100
AIL
200
0
0020
2
2.5
3
lambda
S"n DMIon (UM)
Sample Liner #1
66
3 .5
V
Sample Liner #2
3
10
Original Liner Surface
IS'
Modeled Liner Surface
14D2.5-
- -------
2 -
2D-
too
1.5
GCL
.
0
40
001-
0
15
0
2 2.53 3.5
lambda
swng Oc6on ()
Sample Liner #2
o.
Sample Liner #3
Original Liner Surface
Modeled Liner Surface
3-
10
120
2
2-
sio
-V-
2
60
20
0
100
150
3
2002
3.5
.
1010
So
1W1
00
lambda
Sample Liner #3
400
Sample Liner #4
10
200
~2
252
10
v
Original Liner Surface
Modeled Liner Surface
10
u120
100
a
1a
5
40
0
0
100
1M
200
S
e
e
2
2.5
3
lambda
Sample Liner #4
67
.5
Sample Liner #5
10
10
F
Original Liner Surface
"""Modeled Liner Surface
I"V
1
122
so
4-
1
4
100_
0
Wo
100
shmg D"scion ()
ISO
2
200
2.5
3
3.5
lambda
Sample Liner #5
Figure 4.6: Comparison of Original Liner Surface and Modeled Liner Surface
In above figures, it suggests that modeled liner surface have larger contact pressure than
original liner surface at same clearance height. It is consistent with the guess that modeling
process gives modeled liner surface a comparatively large plateau surface roughness. This is
also a reason of large difference between results by Deterministic Contact Model and Hertzian
Contact Model.
Meanwhile, it is observed that the above modeling procedure gives more accurate modeled
liner surface for relatively smoother surfaces, such as sample liner #1, sample liner #2 and
sample liner #3, while it is not accurate for rougher surfaces, sample liner #4 and sample liner
#5. This is because for sample liner #4 and sample liner #5, there are more asperities with large
height shown on surface. When modeling one asperity to ellipsoidal shape, the height of
ellipsoid takes the maximum height on whole asperity region. In original asperity area, it has a
high possibility that just one spot has very large height and all the other points are relatively
low compared with that spot. However, lots of points have large heights which are similar to
the maximum height on the modeled asperity and it makes the rough surface rougher.
Therefore, smoother modeled liner surface, plateau surface roughness on the order of 0.01
micrometer, preserves better original surface geometry.
4.3.3 Boundary Effect in Deterministic Contact Model
In deterministic calculation of contact between liner finish and piston ring surface, the
boundary of measured liner surface is assumed to be purely smooth and has no contact with
piston ring surface. However, boundary of measured liner surface has similar surface geometry
as that inside of liner surface and contact pressure at boundary will also deform asperities
68
inside measured liner surface area, and thus boundary effect cannot be neglected. A surface
with center area being real measured liner surface and the outside part being artificially smooth
surface is constructed to see the influence of boundary effect, as indicated in Figure 4.7.
Contact pressure on centered rough area of constructed surface neglects boundary effect. The
other surface is the real measured liner surface and contact pressure is calculated just in the
center part as indicated in Figure 4.7. This condition considers boundary effect which arises
from the outer rough area in the second surface shown in Figure 4.7. Contact Pressure on
centered part in constructed surface is compared with contact pressure on cycled part in
original liner surface, as indicated in Figure 4.8.
x 10
x 10
200
150
2
2
100
150
5o
10
12
10
0
50
0
100
150
20
Figure 4.7: Constructed Liner Surface without Boundary Effect and Original Liner Surface with
Boundary Effect
'
2.5
-
No Boundary Effect
"'" With Boundary Effect
1.5
--
-
2
C,,
ca-
1
- -- ---- - - - -
-
0.5
n
2
2.5
3
lambda
.
3.5
V
v
v
v
Figure 4.8: Comparison of Contact Pressure with Boundary Effect and Contact Pressure
without Boundary Effect
69
In Figure 4.8, it indicates that boundary effect in Deterministic Contact Model is not negligible.
Contact pressure due to asperities outside calculation area also deforms asperities inside
calculation area and results in smaller contact pressure. By calculating the difference of contact
pressure without boundary effect and difference of contact pressure with boundary effect
numerically, the maximum difference is around 6 percent which is not small enough to be
neglected. In addition, difference depends on area size outside calculation area and it increases
with larger area outside calculation area.
4.3.4 Another Choice of Normalized Process
The above dimensionless function relating gap between liner surface and ring surface and
pressure distribution in Deterministic Contact Model is based on Hertzian contact solution
between a sphere and a smooth plane. Another choice is contact between a punch and a
smooth surface, as shown in Figure 4.9.
Figure 4.9: Contact between Punch and Smooth Surface
When a punch with radius a is in contact with a smooth surface, minimum pressure takes place
in the center of punch area and it reads [19],
E'w
Pra
where E' is combined Young's modulus of two materials and w is deformed distance on smooth
surface. Therefore, to normalizing the equation related gap between liner finish and ring
surface and pressure distribution, introducing:
70
x
y
p
X=-;Y =-;P =-;H
a
a
Ph
=-
h
w
The problem can be written as:
H(x, y) = 0, P(x, y) > 0
contact
H(x,y) > 0,P(x,y) = 0
P(X',Y')dX'dY'
+x+
f
f-+-0
(X - X') 2 + (y
-x + -X0 +0 -Y)
-
2
H(x,y) = HO - L(x,y) +
no contact
Contact pressure by different normalized process, with Hertzian contact and punch contact, is
compared in Figure 4.10 for sample liner surface #2.
Sample Liner #2
3
2.5
Hertzian contact
Punch contact
7
2
Cu
0n
-7
1.5
'7
----- " -7----
C.
9)
1
7'7
I
-
0.5
-7
0 '2
2
,~' ,~,'~-
--
2.5
3
lambda
7
3.5
Figure 4.10: Comparison between Different Normalized Process, Hertzian Contact and Punch
Contact
Contact pressure given by defined dimensionless parameters of Hertzian contact and
dimensionless parameters from punch contact is nearly the same and difference between two
normalized processes is negligible.
71
4.3.5 Influence of Level Size
Different level size discretizes liner surface differently. Larger level size gives more grids and
smaller grid size, and thus more accurate results. However, it requires more calculation time.
Level size 8 which gives 2048 grid by 2048 grid is used to obtain the results shown above, and
this is because there are 3243 grid by 3243 gird on original input surface and level size 8 gives
most grids which are less than original grid number on input surface. In order to compare
results by different level size, contact pressure of sample liner surface #2 has been calculated at
level size 6, level size 7, level size 8 and level size 9 to show the difference, as illustrated in
Figure 4.11, as well as calculation time, as tabulated in Table 4.2. However, it is found that in
smaller level size, more cycles are needed for convergence. 15 cycles operate at level 6, 10
cycles operate at level 7, 8 cycles operate at level 8 and 8 levels operate at level 9.
3.5
3
- --
Level size 6
I size 7
Level Size 8
Size 9
- -Le
-
2.5
- - -Leel
CU
2-
0.5
02
2.5
3
lambda
3.5
Figure 4.11: Comparison of Contact Pressure by Different Level Size
Compare of Different Level Size
15
30
10
95
8
1215
8
5415
Table 4.2: Comparison of Different Level Size
72
4.3.6 Relation between Contact Area and Contact Pressure
Contact area on measure liner surface at certain clearance height is another important
phenomenon to examine. Deterministic Contact Model can also calculate contact area at each
clearance height. Relation between contact area and clearance height ranging
from 2c- , to 4rp
for sample liner #2 is shown in Figure 4.12 and it indicates very small contact area in this range
of separation.
A A
0.12
I--
0.1
0.08
0
0.06
C,
0.04
0.02
0
2
2.5
3
lambda
3.5
Figure 4.12: Relation between Contact Area and Clearance Height of Sample Liner #2
The relation between contact load and contact area for liner surface #2 is plotted in Figure 4.13,
as well as linear fitting line between contact area and contact load. One can conclude that the
relation between them is almost straight and this is consistent with Greenwood's theory that
contact area is proportional to contact load [20].
73
0.2
contact area
--
S
0 .05
linear fitting line
------- -- ----- -- -------------- --------- - ----------
0.5
1
1.5
2
-----------
2.5
3
contact load (N)
Figure 4.13: Relation between Contact Area and Contact Load of Sample Liner #2
4.4 Conclusion
This chapter introduces Deterministic Contact Model to evaluate contact pressure between
liner finish and piston ring pack. This model is an improvement from Hertzian Contact Model
and Statistical Model. It considers interaction between asperities on liner finish and doesn't
require modeled surface which changes real geometry of liner surface. Therefore, it predicts
less contact than the other two models and thus more accurate.
Contact pressure of different sample liner surfaces are demonstrated by Deterministic Contact
Model and compared with that by Hertzian Contact Model and Statistical Model. Some other
issues regarding Deterministic Contact Model are examined. It is found that boundary effect
influences the results and cannot be neglected. Relation between contact load and contact area
is consistent with suggestions in literature. Evaluation of modeled surface is also presented by
using Deterministic Contact Model to compare with original liner surface. Influence of level size
in the aspect of accuracy and calculation time is discussed. More level size can generate more
accurate results while require more calculation time.
74
5. Evaluation of Contact Model
In this chapter, different contact models are evaluated in Tian's cycle model and compared with
experimental results for different sample liners [26]. Sample Liner #1, sample liner #2 and
sample liner #4 are selected according to their plateau surface roughness. The first step is to
correlate relation between contact pressure and clearance height into a function with same
form as previous contact model. The correlation function serves as an input in cycle model. And
then contact friction is combined with hydrodynamic friction due to lubricant between liner and
piston ring to obtain total friction. The results of total friction between liner finish and piston
rings are compared with experimental results at different temperature and piston speed. Other
issues causing inaccuracy are discussed in this chapter.
5.1 Correlation Functions of Contact Pressure
In order to compare computational results by contact model and experimental data, contact
friction and hydrodynamic friction should be combined because experimental results are for
total friction including contact part and hydrodynamic part. In cycle model, a function relating
contact pressure and clearance height is required to predict contact friction and it should be in
the same form as previous contact model,
h
P = a(b -)c
By using least squares method, correlation functions of different sample liners by three
different contact models can be obtained. Correlation functions of sample liner #1 by Statistical
Contact Model, Hertzian Contact Model and Deterministic Contact Model are respectively,
P(MPa) = 0.4512 * (4.157
h )2.993
-
UP
P(MPa) = 0.6404 * (4.158
h )2.998
-
UP
P(MPa) = 0.4687 * (4
h
-
)2.19
Up
Correlation function of sample liner #2 by Statistical Contact Model, Hertzian Contact Model
and Deterministic Contact Model are respectively,
75
P(MPa) = 0.4873 * (4.183
-
P(MPa) = 0.7108 * (4.179
-
P(MPa) = 0.2492 * (4.297
h
-)3.171
h
h)3.157
h
-
)2.92
UP
Correlation function of sample liner #4 by Statistical Contact Model, Hertzian Contact Model
and Deterministic Contact Model are respectively,
h
P(MPa) = 1.55 * (4.07
P(MPa) = 2.71 * (4.12
-
-
P(MPa) = 0.7361 * (4.04
)4.786
h )4.667
h
--
).p
5.2 Test Results in Cycle Model and Comparison with Experimental Results for
Three Different Contact Model
Correlation function of contact pressure and clearance height can be input into cycle model to
get prediction of contact friction. Contact friction is then combined with hydrodynamic friction
to present total friction between liner and piston ring. In this section, prediction results of
friction between liner and oil control ring are compared with experimental results at two
different temperatures 400C and 1000C. Piston runs at four different speeds ranging from 100
RPM to 700 RPM.
5.2.1 Calculation Results of Sample Liner #1
Figure 5.1 and Figure 5.2 show the results of friction between liner and oil control ring at each
crank angle (degree). The liner is a relatively smooth liner (, is 0.038 pum). Oil control ring
76
tension is 19.5N. OCR ring-land-width is 0.15 mm. Two different temperatures have been
considered. The first one is 100*C, as indicated in Figure 5.1 and the second one is 40C, as
shown in Figure 5.2. Different temperature will have impact on friction and higher temperature
results in more contact friction. For each temperature, piston runs at four different speeds,
100RPM, 300RPM, 500RPM and 700RPM. Contact friction dominates at low piston speed and
faster piston speed generates more hydrodynamic friction.
Statistical Contact Model
. ......... Hertzian Contact Model
- Deterministic Contact Model
------ Experiment Results
_
Sample Liner 1 OCR 100C 300RPM
Sample Liner 1 OCR 100C 100RPM
20
20
10 K
i,
_
t
10'-7..1~
'
I-.
J/
z
C
0
0
0-
02
I~ 4
.10 r
-200
0
crank angle
200
-20
-400
400
K-
20-
10
-
-
10
li
'
.
0
0
200
ii
4
-10
-
-10-
-20 --400
0
-
z
400
200
0
crank angle
Sample Liner 1 OCR 100C 700RPM
Sample Liner 1 OCR 100C 500RPM
20
-200
-
-20'-400
-10-
-200
0
crank angle
200
-20
-400
400
-200
0
crank angle
200
400
Figure 5.1: Comparison of Friction between Sample Liner #1 and Oil Control Ring by Different
Contact Models and Experimental Results at 100*C
77
Sample Liner 1 OCR 40C 300RPM
Sampl e Liner 1 OCR 40C 100RPM
20
20
10
10
*
I
C
0
-
0
C
.2
.
p
-
I
"
~..d
I
.-. i--
0
0
-0
. iI
-10 H
-10
1~~~
-200
-400
-200
0
crank angle
200
-20'-400
400
Sample Liner 1 OCR 40C 500RPM
20
10
10
C:
0-
200
400
V
N
0
-10V
-
-10
-20
-400
0
crank angle
'1
Sample Liner 1 OCR 40C 700RPM
20
-
C:
02
-200
I
I~
I
0
-200
crank
200
-20 K-400
400
angle
-200
0
crank angle
200
400
Figure 5.2: Comparison of Friction between Sample Liner #1 and Oil Control Ring by Different
Contact Models and Experimental Results at 40*C
For sample liner #1, Deterministic Contact Model predicts well for contact friction between
liner finish and oil control ring. When piston is at top side or bottom side in combustion
chamber where piston speed is very low, contact friction dominates and it is in the region of
boundary friction. With increasing of piston speed, it goes to the region of mixed friction where
contact friction and hydrodynamic friction are equally important. When piston speed is very
high and temperature is relatively low, hydrodynamic friction plays more important role.
As shown in Figure 5.1, contact friction by Deterministic Contact Model matches well with
experimental data in the region of both boundary friction and mixed friction, especially for the
conditions of 100RPM and 300RPM. Another phenomenon is that Deterministic Contact Model
predicts less contact than Hertzian Contact Model and Statistical Model and this is consistent
with the conclusions in Chapter 4. When piston runs at relatively high speed, difference of total
friction by different contact model reduces because hydrodynamic pressure is more important
78
and contact friction part is getting less. In Figure 5.2, it shows more hydrodynamic friction
because temperature is reduced to 40C from 100C. Results from cycle model also predicts
well compared with experimental data. When piston runs at speed equal to or larger than
300RPM, hydrodynamic friction plays an important role and contact friction falls at nearly same
line by different contact models.
5.2.2 Calculation Results of Sample Liner #2
In order to test the accuracy of contact models, friction prediction and comparison with
experimental data are conducted to another sample liner #2. Figure 5.3 and Figure 5.4 show the
results of friction between liner and oil control ring at each crank angle (degree) for this sample
liner. The liner is also a relatively smooth liner (o-, is 0.055 pm), but rougher than sample liner
#1. Oil control ring tension is 19.5N. OCR ring-land-width is 0.15 mm. Two different
temperatures have been considered. The first one is 100'C, as indicated in Figure 5.3 and the
second one is 40*C, as shown in Figure 5.4. For each temperature, piston also runs at four
different speeds, 100RPM, 300RPM, 500RPM and 700RPM.
Statistical Contact Model
Hertzian Contact Model
Deterministic Contact Model
----- Experiment Results
........
Sample Liner 2 OCR 100C 300RPM
Sample Liner 2 OCR 1OOC 100RPM
20
2010
o-
I
10
-
z
I
-
0
-10
-20
-400
-10\
-200
0
200
-20
-400
400
crank angle
-200
0
crank angle
79
200
400
Sample Liner 2 OCR 100C 500RPM
Sample Liner 2 OCR 100C 700RPM
20
20
101-
101
i~*-~~
I
I
I
~I/
I
i
1 ~%...
~.#
3
I
1
1
I
z
z
0
0
I%
I
I
-101
101-
9.
-.
-200
0
crank angle
-20
-400
400
200
-200
.%.
~I
I
~
I
-.-
-20-400
-
I
4,
'I
~
'I
-
0
crank angle
*%
400
200
Figure 5.3: Comparison of Friction between Sample Liner #2 and Oil Control Ring by Different
Contact Models and Experimental Results at 1000C
Sample Liner 2 OCR 40C 100RPM
Sample Liner 2 OCR 40C 300RPM
20
20
1
I1
10r
10
C
z
0
0
.0
-10k-
0
-10p
-2-00
1-
0
-200
-400
-200
400
200
-200
-400
crank angle
Sample Liner 2 OCR 40C 500RPM
400
200
Sample Liner 2 OCR 40C 700RPM
20
20r
I'
~J
10 h
I
1
1
p.J
-
10
0
crank angle
~~
Il
C
C
0
0
I'
ji
1'
0
F=)
%
'9.
-101-
-20 --
-400
3
-10-
-200
0
crank angle
200
20
400
-4C
-200
0
crank angle
/A' 9.
I
1
1
9.11~'-'i--I 1
I
200
/
.2
400
Figure 5.4: Comparison of Friction between Sample Liner #2 and Oil Control Ring by Different
Contact Models and Experimental Results at 400C
80
For sample liner #2, though Deterministic Contact Model gives smaller contact than the other
two contact models, all three contact models predict much stronger contact compared with
experimental data.
5.2.3 Calculation Results of Sample Liner #4
The third input is a relatively rougher liner (up is 0.31 pm). Results of total friction by different
contact models are plotted at each crank angle, as illustrated in Figure 5.5 and Figure 5.6. They
run at same condition as that of sample liner #1 and sample liner #2. All contact models predict
larger contact than experimental data.
-- Statistical Contact Model
......... Hertzian Contact Model
--- '--- Deterministic Contact Model
----- Experiment Results
Sample Liner 4 OCR 100C 100RPM
Sample Liner 4 OCR 1OOC 300RPM
30
30 r-
20
A
20
10-
10
0
I
-10
I
-
-10-
0
-
U
-
0
0-
-20
-400
-200
0
crank angle
200
-20
-4 00
400
0
-200
200
400
crank angle
Sample Liner 4 OCR 100C 700RPM
Sample Line r 4 OCR 100C 500RPM
20
20.
10
z
0
I
101,
'I
-20'
-400
-200
0
crank angle
200
-20
-400
400
\
-
10F
-
I
-
0
0I
*
C
0
-200
0
crank angle
200
400
Figure 5.5: Comparison of Friction between Sample Liner #4 and Oil Control Ring by Different
Contact Models and Experimental Results at 100*C
81
Sample Liner 4 OCR 40C 100RPM
Sample Liner 4 OCR 40C 30ORPM
30
30
20
20
----------
10-
10
C
0
0
0
0
-10 F
-10
-20
-400
-200
0
200
-20
400
-400
0
-200
Sample Liner 4 OCR 40C 500RPM
Sample Liner 4 OCR 40C 700RPM
20
207
Lf
%
ft
1oF
1oo
-- ]IN
0-10
1by
-200
0
crank angle
200
-20 L
400
lj
6
-
0
-20
-400
N/
V~%.
c
z
0)
400
200
crank angle
crank angle
-400
-200
0
crank angle
200
400
Figure 5.6: Comparison of Friction between Sample Liner #4 and Oil Control Ring by Different
Contact Models and Experimental Results at 400 C
5.3 Discussion
Deterministic Contact Model predicts less contact than Hertzian Contact Model and Statistical
Model, and is closer to experimental data. For sample liner 1, it gives good prediction, while for
the other two sample liners, it results in stronger contact. By examining surface geometry of
the three liners, more discontinuous spikes along deep valley are found for sample liner 2 and 4
which show larger contact compared with experimental data. Such discontinuous spikes are not
real parts on surfaces and are from measurement errors, as indicated in Figure 5.7. In chapter 2,
it mentions that optical equipment cannot give accurate measured results when there are large
slopes on surfaces. But for sample liner 1, there are not many discontinuous spikes along
borders of plateau and valley and the results by Deterministic Contact Model match well with
82
experimental data. As a result, one can conclude that using deterministic hydrodynamic and
contact models match experiment fairly well for the friction of the TLOCR when surface
measurement is free of errors, as sample liner #1.
Color scale length unit (micrometer)
0.4
3
Figure 5.7: Discontinuous Spikes along Plateau/Deep Valley
5.4 Conclusion
In this chapter, different contact models are evaluated in cycle model. In order to be easily used
in cycle model, trend between contact pressure and clearance height is correlated to a formula
which has the same form as function of previous contact model. Contact friction is then
combined with hydrodynamic friction to generate total friction between liner finish and piston
ring (oil control ring) as an output of cycle model. It can be compared with experimental data to
see accuracy of different contact models.
Three sample liners have been tested at different piston speeds and different temperatures.
Results of one sample liner match well with experimental data, especially with Deterministic
Contact Model, while the other two liners show stronger contact by simulation. Deterministic
Contact Model predicts less contact than the other two models because it considers interaction
between asperities and applies real surface geometry. Therefore, Deterministic Contact Model
is more accurate than Hertzian Contact Model and Statistical Model which overestimate
contact. By reexamining surface geometry of the three liners, more discontinuous spikes along
83
plateau and deep valley are observed on the two surfaces which predict much more contact
than experimental data. Such kind of discontinuous spikes cannot be filtered by the method
introduced in Chapter 2 because they are not apparently large spikes that can be numerically
removed. However, they highly influence the prediction of contact. As a result, one can
conclude that using deterministic hydrodynamic and contact models match experiment fairly
well for the friction of the TLOCR when surface measurement is free of errors, as sample liner
#1.
84
6. Conclusion
6.1 Summary and Conclusion
The objective of this thesis is to develop a contact model based on 3D measured liner surface to
simulate contact friction between piston ring pack and liner. Two different kinds of contact
models are introduced. The first type requires modeled surface with regular asperities and
neglect interaction between asperities on liner. The second type which uses deterministic
method is dependent on real surface geometry and considers influence of interaction.
Hertzian Contact Model and Statistical Model are developed based on Hertzian theory of
ellipsoidal asperities, and thus liner surface with regular ellipsoidal asperities is a desired input.
Chapter 2 introduces the procedure to model liner surfaces. Chapter 3 presents application of
Hertzian contact model and Statistical model to different liner surfaces and makes comparisons
between them.
Deterministic Contact Model introduced by Lubrecht is introduced in Chapter 4 which is an
improvement of Hertzian Contact Model and Statistical Model. It considers the effect of
interaction and can be applied to any surface with randomly irregular asperities, and thus
preserves the original surface geometry. Application of Deterministic Contact Model to
different liners is also shown in this chapter. It is found that interaction between asperities
highly influences contact pressure in comparison with results by Hertzian Contact Model.
Boundary effects, influence of level size are also discussed in chapter 4. It seems that boundary
effect is not negligible and larger level size is required for deterministic model. Relation
between contact area and contact pressure, as well as accuracy of modeled surface, is
examined in chapter 4. It is found that contact area is nearly proportional to contact pressure as
suggested in literature and modeled surface is more accurate for smooth surfaces.
In order to test the accuracy of three different contact models, they are evaluated in Tian's
cycle model and compared with experimental data. It is shown that Deterministic Contact
Model is better than Hertzian Contact Model and Statistical Contact Model and predicts less
contact. When compared with experiment data, one liner surface demonstrates good match,
while the other two liners display stronger contact. By examining surface geometry of the three
liners, discontinuous spikes along plateau/deep valley on surfaces is a factor causing large and
inaccurate contact. Such discontinuous spikes do not exist on real surface and are
measurement errors due to optical measurement techniques. As a result, one can conclude
that using deterministic hydrodynamic and contact models match experiment fairly well for the
friction of the TLOCR when surface measurement is free of errors. Errors introduced by either
85
measurements or interpolation for the locations with large slope can greatly inflate the
magnitude of contact prediction while hydrodynamic prediction is less affected [23].
6.2 Potential Future Work
One potential future work of this project is to verify accuracy of measured liner surface. Contact
pressure is highly dependent on spikes on surfaces which occupy a small portion of area, and
thus contact is very sensitive to spikes. If some spikes are from measurement errors, they will
significantly change the results and predict much larger contact. However, if measurement
errors are inevitable, especially in the region along deep valley, a reasonable method to filter
such spikes without changing other surface geometry is needed before applying contact model.
Another potential future work is to consider plastic deformation. All three contact models are
based on the assumption of purely elastic deformation, while plastic deformation exists in
reality. In addition, evolution of surface geometry can be another potential project because
when piston ring slides on liner surface, it scratches liner surface and takes away some peak
asperities.
86
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