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Master's Degree Thesis ISRN: BTH-AMT-EX--2012/D-12--SE
Supervisors: Professor Roland Larsson, LTU
Department of Mechanical Engineering Blekinge Institute of Technology
Karlskrona, Sweden
2012
Mohammad Shirzadegan
Simulation of rough contact lubrication Piston-Ring
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Simulation of rough contact
lubrication Piston-Ring
Mohammad Shirzadegan
Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
Spring 2012
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Abstract
In everyday life machine have a strong role, therefore investigation in
machine elements are desired for many researchers. Piston ring is one of
important component of engine. In order to understand the lubricant oil
consumption, friction, wear, the first step is to analyze a Piston-Ring
behavior. For this reason a theoretical models are considered and
behavior of pressure in several situations are studied. This study is aimed
to survey steady state and transient analysis condition among the Piston
Ring. Also behavior of surface feature (Texture) on the lower surface is be
modeled in some time step.
In this thesis by using MATLAB, a suitable cavitation algorithm with an
iterative method is established.
Keywords:
Piston-Ring, Cavitation, theoretical model, MATLAB, dent, steady state,
Transient
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Acknowledgement
I would like to thank Professor Roland Larsson who gave me this invaluable opportunity and supporting me all the time. It is really my
pleasure to work under his supervision. Studying in machine element
division of LTU is one of the moments that I will never forget in my life.
Special thanks to Andreas Almqvist, Assistance professor on machine
element of Lule Tekniska Universitet, who helps me in MATLAB
programing and mathematical and physical concepts.
I also thank Dr.Ansel Berghuvud from Blekinge Tekniska Hgskola
(BTH) for his valuable support and guidance through my work. I have
learned many things from him during two years studying at BTH.
Finally, I would like to express my thanks to my family (Amir, Jamileh,
Shahin, Marjan) that supporting me without any hesitation.
Mohammad Shirzadegan
Spring 2012, Lule, Sweden
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Contents 1 Notation ........................................................................................................... 5
2 Introduction ..................................................................................................... 8
2.1 Piston-Ring .............................................................................................. 9
3 Literature Review .......................................................................................... 13
4 Lubrication-Governing Equations .................................................................. 17
4.1 Film thickness equation ......................................................................... 18
4.2 Pressure-viscosity .................................................................................. 19
4.3 Density-Pressure .................................................................................... 20
4.4 Force Balance ........................................................................................ 21
5 Numerical Approaches .................................................................................. 22
5.1 Jacobi method ........................................................................................ 26
5.2 Gauss-Seidel Method ............................................................................. 27
5.3 Successive over relaxation ..................................................................... 28
5.4 Deformation scheme .............................................................................. 29
5.4.1 Direct integral ................................................................................ 31
6 Cavitation ...................................................................................................... 33
6.1 Cavitation algorithm .............................................................................. 33
7 Theoretical solutions ...................................................................................... 38
8 The model problem ........................................................................................ 41
9 Results ........................................................................................................... 45
9.1 Pressure distribution among Piston-Ring model 1 ................................ 45
9.2 Pressure distribution among piston-ring model 2 .................................. 48
9.3 Effect of surface features ....................................................................... 49
9.4 Effect of deformation ............................................................................. 51
10 Discussions and Conclusions ..................................................................... 52
11 Future works .............................................................................................. 55
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12 Bibliography ............................................................................................... 56
13 Appendix ................................................................................................... 62
13.1 Evans and Hughes coefficient ............................................................... 62
13.2 Secant Method ....................................................................................... 63
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1 Notation
Real area of contact
Acceleration
Integration constant
Integration constant
Elastic modulus
Elastic modulus
Elastic modulus
Axial friction force
Asperity contact function
Axial friction force based on asperity contact
Axial Hydrodynamic friction force
Radial friction force at ring groove pivot
Gravity constant, switch function
Height function
Minimum film thickness
Integral kernel
Length
Piston-Ring mass
Number of node
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Hydrodynamic pressure
Cavitation pressure
Axial applied gas pressure force
Radial gas pressure relief force
Gas pressure behind ring
Radial applied gas pressure relief force
Radial curvature
Axial component of Hydrodynamic force
Radial for upper surface
Radial for lower surface
Crank radius
Crank radius
Right hand side
Engine Speed
Velocity
Angular velocity
External load
Pressure viscosity index
Pressure viscosity coefficient
Bulk modulus
Indexes
Fractional contact
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Deformation
Viscosity
Viscosity at ambient pressure
Density
Interval
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2 Introduction
The main function of lubricant is to decrease wear, friction, heat and to
facilitate load support of moving surfaces. There are five types of
lubrication namely, Hydrodynamic, Hydrostatic, Elastohydrodynamic,
Boundary and Solid film.
When two surfaces separated by a thick film of lubricant , pressure and load
capacity of the system can be calculated based on fluid dynamics law (No
metal contact), the lubrication is Hydrodynamic. But if the system does not
have any motion and lubricant is inserted into the system with sufficient
pressure, the lubrication is Hydrostatic (air and water are the most common
lubricants).
Lubrication is Elastohydrodynamic when lubricant between the surfaces
reduces to the specific amount and contact deformation of surfaces is
considerable. In this situation combination of contact mechanic and fluid
dynamics law are considerable. More details will be discussed in next
sections.
Whenever load increases or velocity decreases, lacking of surface area,
increase in lubricant temperature that lead to reducing of viscosity, any of
these may put full film thickness in critical situation. Then highest
asperities may be detached by few drops of lubricant. This is called
Boundary lubrication.
The selves solid lubricant, such as graphite, is necessary for operation if the
system supposed to work on extremely high temperature and the load
carried by the asperities. Mineral oil couldnt have effective efficiency in
high temperature. The following figure shows the difference between
Hydrodynamic area and boundary condition. Viscosity () and velocity (N)
have a direct relation with friction. This graph has been obtained by Mckee
brothers during friction test for bearing.
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2.1 Piston-Ring
The Piston is one of the main parts of the engine that transfer power from
combustion chamber to crank shaft. When gases burnt at top of cylinder
(combustion chamber) pressure pushes the piston to the downside. The
Piston reciprocal action move crank shaft and crank shaft moves the other
engine components. This mechanism is repeated until the engine turned off.
Piston is surrounded by cylinder. For some important reasons space
between the cylinder and piston should be sealed. This couldnt be
happened without ring. Main duties of rings are divided in three main parts:
a) Sealing combustion chamber to prevent the gas leak from piston circumstance
b) Heat transfer from high temperature (piston) to low temperature (cylinder wall)
c) Adjust the oil
Figure 2.1. The variation of the coefficient of friction with [46].
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All rings generally can be classified into two types namely compression
rings and oil control ring. Compression rings are used for sealing of
compressed gases and they sit at the top of the piston. Cross section of these
types is rectangular, barrel or tapered shape. Gas pressure moves to the
back side of the ring and forces it towards the cylinder liner.
Oil control ring is set at the bottom of piston. Their main duty is controlling
the oil and distributes it on to the cylinder wall. The scraped oil is
collected in the oil control ring groove and transported through the piston
back to the crankcase. [1]
Many factors can fail a system high pressure and high temperature system
causes a critical environment for Piston-Ring. If the system operates under
such condition it could lead to wear, friction, deformation and other bad
effects on the system.
Figure 2.2. Schematic piston ring.
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Material have important role in running condition. As was said one function
of ring is heat transfer between piston and liner. Normally applicable
material for ring production is called Grey cast iron. It shows good
performance under starved and dried conditions. Also chromium coating is
used to prevent corrosion and abrasion. Due to the engine performance
different type of coating such as aluminum-titania, tungsten carbide is used
for piston. For further information see Andersson et al [1].
Figure 2.3. Rectangular and tapered ring [43].
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The main object of this work is to simulate a Piston-Ring and study the
behavior of pressure distribution under different conditions. Also it is a
desire to understand the effect of surface roughness by adding a small dent
in a lower surface, and finally, to study if elastohydrodynamic effects play
any significant role.
Elastohydrodynamic effects are not normally taken into account in studies
of the piston ring problem. But that deformation can cause load and film
thickness improvements.
This study is based on the theoretical model for this reason some numerical
methods like Jacobi relaxation and Gauss-Seidel relaxation and SOR are
introduced. Most of references reported that multigrid method is the ideal
method for lubricant analysis but it has some sophistication in programing
and needs more expertise.
An introduction to Elastohydrodynamic parameter such as viscosity-
pressure, density-pressure and deformation will be explained. Also step-by-
step approaches for cavitation algorithm based on modify Elrods work is
implemented. Pressure distribution of piston-Ring under two different
boundary conditions (fully flooded and starved) is studied.
A piston ring is modeled with two different geometries. Model one does not
have any groove and model two has small groove in lower surfaces. Steady
state and transient condition will be surveyed for these two models.
Friction wear and oil consumption are always occurs in reality but these
concepts are beyond the aim of current survey.
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3 Literature Review
Back in 1880s an English railway engineer Beauchamp Tower was first to
discover high pressure in full film regime [2]. He did some experimental
test on railroad bearing and founded unexpectedly high pressure. After that,
on 1886 professor Osborn Reynolds [3] published his famous theorem. This
theory was extracted from the Navier-Stoks equation and determined the
pressure distribution across the arbitrary geometry. Nowadays his work is
well known as Reynolds equation.
In 1916 Martin [4] modeled the meshing of gear teeth. He considered the
lubricant as isoviscos. He extracted the relation between operation
condition and film thickness which was far from the roughness of gear in
reality. On that period it was difficult to calculate the elastic deformation
and viscosity-pressure of lubricant at the same time [5].
The viscosity- pressure relation is obtained from the effort of Barus [6]. He
deliberated the various aspect of viscosity of marine glue in various
temperature and pressure. He observed that in any temperature the rate of
viscosity increases with pressure and he deduced the exponential relation
between pressure and viscosity.
Petrusevich [7] in 1951 solved the Reynolds and deformation equations
together. He considered highly loaded effects on EHL contact and
established the pressure spike for the first time in his study.
EHL line contact was solved with different methods. Dowson and
Higginson [8] solved the problem with inverse method and Newton-
Raphson was used by Okamura [9]. Besides Hamrock and Jacobson [10]
using the Gauss-Seidel relaxation method in order to solve the low load
EHL contact.
Lubrecht [11]solved the EHL line and point contact in 1984 with
combination of nonlinear Gauss-Seidel and multigrid technic. Lubrecht and
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Brandt [12] solved the line contact EHL for the low loaded contact. He
used the multi-integration algorithm technique and increased the solvation
process for deformation [13].
Almqvist [14] applied the block Jacobi method as iterative solution for
EHL line contact problem. He investigated multigrid technique and Jacobi
method in his work.
One important issue of lubricant analysis is cavitation. Famous solutions for
cavitation were started by Sommerfeld [15]. He presented a pressure
distribution for a full film lubricant in journal bearing. As the film rupture
was not taken into consideration the negative pressure distribution was
shown in his work. Then in 1914 Gumble [16] by changing the boundary
conditions (Half-Sommerfeld) of Reynolds equation gained better pressure
distribution but the conservation of mass was not fulfilled in his work.
Then, Swift in 1932 [17] and Stieber [18]in 1933 defined other boundary
conditions for journal bearing which were known as Reynolds boundary
conditions. Based on that, the pressure is started to build up from the
beginning of the dominant and will be disappeared where the pressure
gradient is equal to zero. These conditions also have a mass conservation
problem because the film reformation was not considered. This problem
was discussed by Brewe et al [19] work.
First experiment that presented the vapor cavitation was occurred at Lule
University of technology in machine element laboratory. They did
experiment on bearing in the motion of PMMA tube and shaft. After that
NASA Lewis search center did the same experiment and gained better
result [19]. Floberg [20]also published his experimental work on cavitation
that was occurred on bearing. Mass continuity was totally considered by
work of Floberg, Jakobsson and Olsson for moving boundary conditions.
This worked is known as JFO method and the boundary condition are
varied due to the time depended load. This method was difficult to
implement.
In 1981 Elrod [21] introduced his famous algorithm that computes the
cavitation in the bearing. He defined the switch function that made the
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Reynolds equation valid among the bearing. There are two advantageous
for this method. The first one is consideration of mass conservation among
the cavitation region and the second is easiest programing due to cavitation
complexity.
After that Vijarvaraghavan and Keith [22]developed the Elrod algorithm.
For the cavitation boundary the second part of Reynolds equation (shear
induced flow) automatically changes from central to upwind difference.
This happened by applying an artificial viscosity function to shear induced
term. They analyzed their algorithm on slider and journal bearing under a
heavy loaded.
Ausas et al [23] prepared another numerical algorithm due to the mass
conversation and Elrods work. They used the Newarks scheme and relaxation process to update the fluid fraction and pressure.
Dowson et al [24] studied the tribological behavior of a piston ring in eight
four- stroke and six two-stroke diesel engines. The effects of squeeze film
and elastohydrodynamics at top dead center were established too.
Priest et al [25] presented a free body diagram for the compression ring. He
computed hydrodynamic pressure for four different cavitation theories.
Also minimum film thickness behavior was discussed for these models.
The dynamic behavior of the piston ring in a diesel engine was investigated
by Tian [26]. He studied the performance and effect of lubricant between
top two rings for different crank angles. This experimental investigation
shows the transportation of oil in three main regions of piston and cylinder.
Rahnejat et al [27] investigated the gas force action behind the piston ring.
Variation of film thickness and load at different crank angle was discussed.
Rahnejat believed that beside deformation, asperity adhesion should be
added to model to understand the behavior of piston at TDC position.
Dellis and Arcoumanis [28] developed an experimental reciprocating rig
test in order to inspect cavitation and behavior of film thickness. They
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discussed about the shape of pressure based on film thickness.
Development of cavitation in different load is shown in details.
Spencer et al [29] studied the surface texture for combustion engine
experimentally and numerically. He determined the pressure distribution
under the ring with Homogenized technique. He modeled an artificial
texture on a piston ring that could be used in for real model. To deal with
cavitation, Vijarvaraghavan algorithm was implemented in his work.
Andersson [1] compiled useful information about design, material, product,
wear and friction in the Piston Ring-Cylinder liner. In his literature survive
nearly 150 references were used.
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4 Lubrication-Governing Equations
When the pressure distribution in lubricant reaches high enough level like
0.1 to 3 GPa the shape of film thickness changes due to the surface
deformation. Then we can speak about Elastohydrodynamic lubrication
(EHL). If we want to mention two major characters of EHL, they can be
elastic deformation and piezo viscous effects. The elastic deformation was
studied by Hertz in 1881. He considers a contact between two spherical
bodies.
Mostly, EHL consider in non-conformal contacts which are line contact
(cylinder-cylinder), elliptical contact (ball-cylinder) and circular contact
(ball-ball).
The EHL problem is governed by various equations. Viscosity and density
are function of pressure. As was said deformation equation must be added
to the height function in Reynolds equation. Force balance is a parameter
that should not be forgotten that balance an applied load and pressure.
In order to simplify the contact for example between two disks, one of the
disks is replaced with plane in the x coordinate. The figure 4.1 shows this
procedure [30].
Figure 4.1. Schematic simulation of contact [30].
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For this system a reduced radius R is defined as below
( 4.1)
In order to calculate the parabolic shape, if considered as minimum film
thickness an approximation for film thickness can be estimated as below
( ( ))( )
( 4.2)
Which can be simplified to:
( 4.3)
If
4.1 Film thickness equation
By adding deformation to the film thickness equation the final relation can
be determined. A parabola carve could be plotted by this equation.
( )
( 4.4)
Where deformation can be found from:
( )
( )
Extraction of deflection will be described briefly at next section.
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4.2 Pressure-viscosity
In EHL two famous pressure-viscosity relationships are applicable. The
first one is given by Barus [6]. He introduced a simple equation that
describes behavior of viscosity under high pressure.
( 4.5)
Where is atmospheric viscosity and is a coefficient for pressure
viscosity which is varies between for oils.
The second equation that is more complicated than Barus was introduced
by Roelands in 1966 [31]
( ) ( ( ) )( (
) ))
( 4.6)
Where are constant. Comparison between these
two equations as a function of pressure is plotted on figure 4.2.
Figure 4.2. Comparison between Barus and Roelands method [13].
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Neither Barus nor Reolands are very accurate above 500 MPa. Free volume
models should be applied in such cases. The Barus equation has been used
in this study since pressure level is low.
4.3 Density-Pressure
Dowson and Higginson [8] defined a relation for density that varies due to
the pressure. It reads
( ) (
)
( 4.7)
Where is atmospheric density. In this equation pressure is given in Pascal
unit.
Figure 4.3. Relation between density and pressure due to Dowson [13].
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4.4 Force Balance
The fluid pressure must balance externally applied load:
( )
( 4.8)
To satisfy relation 4.9 some numerical methods are suggested. The famous
one is Secant method that is easy to implement. An algorithm in Matlab can
be found in appendix 13.2. A bisection method is another mathematical root
finding that could be applicable.
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5 Numerical Approaches
In this chapter finite difference method which is used to solve the Reynolds
equation and three algebraic techniques will be explained. In a boundary
value problem suppose we have a second order equation which is generally
describe by following formula
( ) ( ) ( ) ( ) ( )
( 5.1)
That
( ) ( ) ( )
Let divide the interval [a,b] to N equal space therefore
For higher order equation it is essential to use this approximation
( ) (
)( ( ) ( ) ( )
( 5.2)
And using central, backward or forward difference for first order equation
( ) (
)( ( ) ( ))
( 5.3)
( ) (
)( ( ) ( ))
( 5.4)
( ) (
)( ( ) ( ))
( 5.5)
Now we can discretize Reynolds equation due to above procedure. It reads
(
)
( )
( 5.6)
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Define ( )
and ( )
Substitute (e) and (RHS) into Reynolds equation
(
)
( )
( 5.7)
And extracting in a finite difference form
(
)
(
) (
) (
) (
)
( 5.8)
Central difference for right hand side of Reynolds equation
( 5.9)
This equation should only use for interior point.
( 5.10)
For i=1
( ) ( ) ( )
( 5.11)
For i=N
( ) ( ) ( )
( 5.12)
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Rewriting the equation in a matrix form leads to following diagonal system:
Originally, Reynolds equation has a squeeze term that could estimate the
film thickness. Since the film history adds to the simulation value of film
thickness, location of cavitation and pressure distribution are getting
changed.
(
)
( )
( )
To fulfill initial amount of film thickness and pressure are guessed and also
initial elastic deformation is estimated. After that, the pressure of Piston-
Ring is calculated by implementation of cavitation algorithm. Convergence
criteria should have satisfied. Next second, bisection and inverse quadratic
interpolation methods are applied to balance the load behind the ring [29].
The processes are repeated for every time increment. Results for each time
step kept as history for next one. The summery of implementation of the
model into software for transient condition is shown figure (5.2)
Figure 5.1. System of equation.
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Start
Initial guess of height function
Apply pressure to calculate the
Estimate the Elastic deformation
Implement the cavitation algorithm
and determine pressure distribution
Use h function of every cycle for
Does P converge?
Force Balance
Adjust
First cycle
Use h for
previous time
step & go to
next time step
Yes
No
Print pressure distribution for every
time step
End
Yes
Figure 5.2. Algorithm for Transient analysis.
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To solve this algebraic equation an iterative solution is necessary. Jacobi
and Gauss-Seidel relaxation and successive over relaxation method are
discussed briefly.
5.1 Jacobi method
This method needs an initial guess value for start the iteration loop. Then it
produces several sequences that converge to initial guess. If we have
equation system [A]{x}={b}, this technique changes the system to the
.
An algorithm for solving algebraic system with Jacobi method implement
as follow.
Suppose A is formed by three diagonal U (upper triangular), L (lower
triangular), D (diagonal A).
( 5.13)
Each loop is identical to solve for every variable one. Now equation Ax=b
can be changed to
( )
( 5.14)
( )
( 5.15)
That ( )
This method can program to Matlab by following orders. This method is
converged vary slowly.
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Due to Venner and Lubrect [32] Jacobi relaxation in matrix form is:
( 5.16)
( )
5.2 Gauss-Seidel Method
This method is popular technique for solving an algebraic system of
equation. In contrast to Jacobi method that updates values at end of iteration
this method uses the new approximation for solving the later one. This
iteration method is diverged faster than Jacobi.
An algorithm for matrix form is:
( ) ( )
( 5.17)
Figure 5.3. Matlab algorithm for Jacobi method.
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It can be programed to Matlab by following orders.
5.3 Successive over relaxation
SOR help us to gain convergence faster than Gauss-Seidel and extracted by
extrapolating Gauss-Seidel method.
( ) ( 5.18)
In a matrix form it is rewritten as:
( ) ( ( ) ( )
( 5.20)
If w is set to one the equation changes to the Gauss-Seidel method and If
w
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5.4 Deformation scheme
There are different approaches to calculate the deflection and film
thickness. The elastic deformation for line contact problem is obtained from
the following integral.
( )
( )
(5.21)
Okamara [33] discretized the deformation equation by the following
formula in dimensionless form. He used this equation throw to numerical
analysis of isothermal elastohydrodynamic lubrication.
(|
| ) (|
|)
(5.22)
Houpert and Hamrock [34] proposed a faster approach to deformation.
They improved Okumaras method and determined the integral inside the
equation analytically. The baseline in this assumption were pressure
distinguished by a polynomial of second degree in the interval [ ]
(
)
( 5.19)
Almqvist [14] used alternative approach which differentiate
with respect to . Due to his work deformation could
be calculated as
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( )
(5.24)
Where
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Evans and Hughes [35] introduced differential deflection method that
calculated the deformation of the system for any arbitrary pressure
distribution among the lubricant. They assumed that for calculation of
deflection numerically, it could be written in a quadrature form.
( )
( 5.25)
Where is a weighting function based on pressure under the area of
integration.
By two times differentiation respect to x, the equation (4.14) can be written
in the following form. (For the mathematical procedure see appendix 13.1)
(
)
( 5.20)
Which rewritten to the form
( 5.21)
Where
If we want to rearrange the equation (4.16) for any mesh point
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( )
( 5.22)
Equation 4.17 could calculate the deformation. This equation needs two
boundary conditions that are important to implement correctly.
5.4.1 Direct integral
Due to singular kernel of
evaluation of this integral seems
difficult. Integral term of deflection equation can be determined by
(5.23)
In the interval
Analytical evaluation of kernel (k) - mostly used in direct integral method -
is extracted in the following equation.
(
) ( |
| ) (
)( |
| )
( 5.30)
Therefore deformation can be approximated as
( )
( )
( )
( 5.31)
Where P is approximation for the pressure and K is called kernel
coefficient.
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Figure 5.5. Effect of deformation on Height function.
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6 Cavitation
If the pressure inside a lubricant becomes lower than the gage/ambient
value, the liquid film couldnt withstand the high tensile stress. Then the
fluid film will break-up and cavitation occurs.
Four major category of cavitation can be summarized as hydrodynamic,
Acoustic, optic and particle cavitation. Hydrodynamic cavitation takes
place due to the variation of velocity. It can be subdivided into four groups
namely, travelling, fix, vortex and vibratory cavitation.
- Travelling cavitation happens when the bubble starts to grow from inside the liquid, become larger and afterward collapse.
- Fix cavitation occurs when the liquid flow separates from the rigid boundary of dunk body and the cavity remain fix to the boundary.
- Vortex cavitation take place in cores of vortices which form in areas of high shear.(i.e. blade of ship)
- At vibratory cavitation due to the low velocity of lubricant a given element of liquid is exposed to many cycles of cavitation instead of
one. [36]
6.1 Cavitation algorithm
In 1981 Elrod [21] introduced a switch function that computes the
cavitation automatically among the lubricant regime. The regime is divided
into two parts, full film and cavity. Reynolds equation is valid only in the
complete film and in the cavity regime the equation needs certain changes.
These two regimes are named as coquette flow and Shear flow. Due to
Elrod, universal differential equation is contained fractional content ( )
which help to calculate the pressure distribution among the lubricant.
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Fractional content or nondimensional density (in the full film regime) is
defined as
and the pressure can be obtained from ( )
where is cavitation pressure and is a bulk modulus.
Based on the variation of through the regime cavitation index g is
defined. If cavitation index is zero which means the flow are in
cavity zone. So if cavitation index will set as zero.
On X direction the mass conservation for the upwind stream and downwind
stream when the velocity set positive is written as below
( )
( ( )
( ) ( ))
( 6.1)
( ) (
) ( ( )
( ( ))
( 6.2)
Due to above equations (6.1-2), in the full film zone g =1 for parabolic
system second order central is applied. In the cavitation zone the pressure
equation is omitted because g is equal to zero and upwind difference is used
for connective term.
In order to discretize the Elrod algorithm Finite Difference is adopted for
convective and pressure term.
Switch function
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{
[ ( ) ]
[
]}
{
[
]}
{
[( ) ]
[ (
) ]}
[ ( ) ( )
( ) ]
[
(
)
]
( 6.3)
Vijarvaraghavan and Keith [22] modified the Elrod algorithm. This made it
less time-consuming and the accuracy of the result is improved with a
coarser grid. In the numerical formulation an artificial viscosity term is add
to the shear flow term where is equal to zero in the full film zone and in
the cavity zone it is equal to one. Artificial viscosity has the following
relation with switch function g:
The one dimensional steady state Reynolds equation should be changed as
follow
(
)
( 6.4)
To discretize the equation in X direction, we can rewrite it in a following
order. This helps us to fill in matrices easily. For two dimensional transient
Reynolds equations see for example Spencer et al [29]
( 6.5)
Where
[ ]
[ ]
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[ ]
[ ]
[ ]
[( ) ]
[ ( ) ]
And coefficients are introduced as
(
)
To study cavitation algorithm a parabolic geometry is modeled. A slider
parabolic bearing which has been modeled by Vijarvaraghavan [22]and
Sahlin et al [37] is applied. The geometry of the slider bearing is shown
below. The minimum film thickness is constant and boundary conditions
are considered as fully flooded film.
The height function is defined with following equation
Figure 6.1. Parabolic shape [37].
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( ) ( )
( )
( 6.6)
Input parameters that used for the model is shown on table 2.
Input parameters
( m) 2.54e-5 ( m) 7.62e-2 ( m/s) 4.57 ( pa .s) 0.039 (N/ ) 6.9e7 ( ) 0
Figure 6.2. Pressure distribution for fully flooded boundary condition.
Table 6.1. Input parameter for slider bearing.
Figure 6.3. Pressure distribution for starved boundary condition.
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7 Theoretical solutions
Here just the theoretical equation for the piston ring relation is extracted.
But for analyzing the system numerical method that explained in chapter 6
is used. Simplified mathematical model for piston ring is calculated.
Several assumptions have been made. In comparison with gas pressure, ring
weight and friction are small and they set to zero. Also at z direction
friction force which made by groove is neglected. System assumed to be
fully flooded. is Axial component of Hydrodynamic force and is
Axial applied gas pressure force [25]
( 7.1)
( 7.2)
( 7.3)
( 7.4)
Where is considered as radial gas pressure relief force and is radial
applied gas pressure relief force. is radial force due to asperity contact
-
39
and is radial component of hydrodynamic force. All the parameter and
their direction are shown in following figure.
A model that presented by Greenwood and Tripp [38]was applied to
determined . Approach proposed by Ruddy et al should be implemented.
Then [25]
( )
(
)
( 7.5)
Figure 7.1. Piston Ring [25].
-
40
Which is composite root mean square surface roughness of the piston
ring and cylinder bore and is asperity radius of curvature
When the hydrodynamic pressure integrated among the inlet and outlet,
radial hydrodynamic load will be obtained.
( 7.6)
And hydrodynamic pressure can be obtained from Reynolds equation (by
two times integration)
( 7.7)
Which c1 and c2 will be calculated based on boundary conations. For more
details about calculating cavitation and friction see Priest [25]
-
41
8 The model problem
To implement all information into our differential equation, two models are
introduced. The first one is piston-ring in a steady state condition. Pressure
distributions for three different crank angles -20, -10,-2 are extracted. It is
assumed that the surface is totally smooth and it doesnt have any
roughness. Boundary conditions are variable during the analysis due to the
gas pressure behind the piston-ring and they are always set as fully flooded.
Effects of temperature on viscosity and other parameters are neglected.
All the geometry are defined in table 8.1
The force that applied behind the ring is obtained from ring tension (T) and
combustion gas pressure. It has different values on different crank angle.
The maximum value of gas pressure is near TDC. According to the figure
(8.2) this amount is close to 1.53 Mpa. Relation between ring tension and
gas pressure with force can be determined by equation (8.1) [39].
( )
( 8.1)
Figure 8.1. Schematic Piston-Ring, first model.
-
42
Parameter value
X-start (m) -7.3751 X-end (m) 7.3751 Ring tension T (MPa) 0.341
Bulk modulus (Pa) 1.721 Radius R (m) 0.0183
Cavitation pressure (MPa) 0.02
Bore diameter D (mm) 0.889
Viscosity coefficient (Pa-1) 1.8
Due to the cavitation algorithm two boundary conditions are demanded as
function of fraction content( ). Both of them can be estimated by
substituting pressure from top ring ( ) and pressure from the blow-by( ).
According to Yang et al [40]
( 8.2)
( 8.3)
and are constant and equal to .
Figure 8.2. Combustion gas pressure [39].
Table 8.1. Input parameter for first model.
-
43
Whenever the piston move in every crank angle velocity of piston is
changed therefore a relation needed to define this movement. It can be
estimated by following equation [40].
(
( )
)
( 8.4)
Where
and n is an engine speed, R is a crank radius and L is
connecting rod length.
The second model has the same geometry as the first one but lower surface
has a small dent on it. Contrary to reality it is assumed that only one surface
of the Piston-Ring considered rough. In order to study the behavior of
surface feature, a mathematical statement should add to height function
which is [41]
( (
( )
) ) ( (
( )
))
(8.5)
Where is the amplitude, is the center of dent which is varies in
each time step and is the dent wavelength.
-
44
In the second model input data for gas pressure in every crank angle is
needed here the load and velocity are assumed to be constant.
Characteristics value
Load (Mpa) 5.5
Amplitude (m) 0.8
Wavelength (mm) 0.1
Time (s) 0.000926
Velocity(m/s) 2
Figure 8.3. Schematic view of model 2.
Table 8.2. Input parameter for second model.
-
45
9 Results
9.1 Pressure distribution among Piston-Ring model 1
Pressure distribution for different crank angle is calculated. At the
velocity is set to and maximum pressure reaches to near .
When engine goes to gas pressure behind the ring make the contact
thinner also velocity reduces to and maximum pressure
reaches . Before TDC ( ) maximum pressure increases rapidly.
The amount reaches to
Whenever crank angle get close to TDC an applied load behind the ring
increased, velocity decreased to and deformation influenced more
on pressure build up. Maximum pressure value is close to
Figure 9.1. Pressure distribution- 20 and -10 crank angle.
-
46
It can be seen that position of the cavitation is shifted to the center. Also
reformation in a divergent part is started to grow. See figure (9.1 and 9.2)
Film thickness for three different crank angles is plotted. See figure (9.3).
When the Piston moves to the TDC position the value of film thickness
reaches its minimum value.
Figure 9.2. Pressure distribution -2 and TDC.
-
47
Figure 9.3. Film thickness -20,-10 and TDC.
-
48
9.2 Pressure distribution among piston-ring model 2
In the analysis of small dent the load and velocity are kept constant.
Behavior of dent during in four different steps is shown in figure 9.4.
There is no effect of dent on figure 9.4.a. Then it moves through to system.
Because of the parabolic shape of height function in the beginning the
pressure decreases but after a while it starts to build up and after that when
the dent reaches to the middle of height function once again the pressure
drops and gains its minimum values.
Finally, when the dent passes the middle the second peak pressure reduces
until it disappears from sight. See figure 9.3.c and 9.3.d. behavior of
Figure 9.4. Pressure distribution for model 2 in different time step.
(a) (b)
(c) (d)
-
49
pressure during this action doesnt change even when the amount of load
varies.
9.3 Effect of surface features
In this section maximum pressure and minimum height function in every
time step are plotted. Effects of surface feature that introduced by equation
(8.5) in details has been plotted. In order to get converged results the load
behind of Piston-Ring is set to 6.51 and the velocity 2 m/s.
Figure 9.5. Maximum pressure
in each step.
Figure 9.6. Minimum height
in each step.
Figure 9.7. Parabola shape
when the dent is in the middle.
Figure 9.8. Pressure distribution
when the dent is in the middle.
-
50
When the dent reaches to the middle of parabola shape as was expected the
pressure started to decrease. While center of dent reaches to diverge part of
parabola once again the pressure start to build up. This is shown in figure
(9.7).
-
51
9.4 Effect of deformation
In order to see what will happen if the deformation is omitted from the
analysis of the system a high constant load (5 MPa) is applied to the
system. After one complete engine cycle the results show how the amount
of pressure varies. These changes are become more important whenever the
system have roughness (dent) see figure (9.9, 9.10).
Figure 9.10. Effect of deformation on dent in high pressure.
Figure 9.9. Pressure distribution with and without deformation.
(a) (b)
(a) (b)
-
52
10 Discussions and Conclusions
- The results shows EHL have effects on film thickness and perhaps pressure when the load is high. These functions such as viscosity,
deformation, density improve the value of load capacity and
therefore it could be considerable for further calculation like
friction, wear or power loss. Elastohydrodynamic effect changes the
peak pressure. Comparison between pressure for system with and
without deformation shows that the value of pressure peak is
decreases due to implementation of deformation. By increasing
amount of applied pressure (load) to the system deformation have
active role in results. As can be seen in figures (9.9) for constant
load 5.5e6 MPa and velocity 2 m/s the peak of pressure is changed.
Also for the same position of dent in the second model the shape of
pressure peak is different.
- Pressure reformation is started to grow up as long as the piston ring get close to the TDC. As was seen on fig (8.1) in it is small but near the TDC as the velocity decreases and gas pressure increases,
the reformation started from . Another reason for this phoneme is elastohydrodynamic effects.
- It is necessary to satisfy the force balance. It has direct influence on
squeeze film-term (
). On the steady state condition, due to
elimination of squeeze term pressure peak is goes high near the top
dead center.
- Transient analysis of the system may differ from steady state. Position of cavitation, peak pressure and minimum film thickness
are varying due to the squeeze-term.
- Investigation of the surface features shows that the effects of texture may not be neglected and even a small dent into system cause
changes in output. This texture may increase the friction and wear
and the life time of the machine will be reduced.
-
53
- Due to the iterative procedure for Piston-Ring analysis the initial value of some parameter like height function should be guessed. For
gaining suitable results it is necessary to run the system at least for
one engine cycle.
- Behavior of iteration method like Jacobi and G-S shows that for the high loads system these methods are totally unstable and they never
converged.
- It can be mention that in higher load, the pressure spike will be added to the pressure distribution among the system.
- In order to analyze the extracted results from mathematic view coefficient matrix A is considered. A Is a square matrix which is
sparse and band matrix with a constant bandwidth of three (see fig
(10.1)).
At first the eigenvalue of the matrix plotted. In the spectrum of the
eigenvalues it is observed that most of the values are clustered around zero.
Furthermore, the large distance between largest and smallest eigenvalues
concludes that the condition number of the analyzed matrix should to be
large. This reveals that the coefficient matrix is an ill-conditioned matrix
which means that it is close to be singular. Therefore, conventional direct
strategies will not guarantee to maintain the desired accuracy of the results
due to the sensitivity of the system because of unavoidable rounding errors.
Alternatively, an iterative solver has been employed in order to solve the
equation of this ill-condition system, i.e. condition number=1.547e9.
Figure 10.1. Diagonal of matrix A. this figure is zoomed
-
54
In the further analysis the singular values of the coefficient matrix is
studied. Fig. 9.3 demonstrates the singular values of the coefficient matrix.
Obviously the values decade slowly as expected, and no large and
detectable gaps are observed. Hence, the evaluated system is not rank
deficient and uniqueness of the answer of the system is ensured.
Figure 10.2. Spectrum of the eigenvalues of the system.
Figure 10.3. Spectrum of the singular values of the system.
-
55
11 Future works
Effect of wear and friction will be discussed. Investigation in this area
seems challenging. The iteration solution shows that they are too slow and
the rate of convergence for high pressure is too low. Multigrid techniques
will be studied in order to spread solution to the high pressure area and
could be able to analyze the pressure spike and surface texture at same
time. This method also has more advantageous simulation speed and
accuracy.
It is interesting to adopt another cavitation algorithm and compare with
modified Elrods algorithm. FFT method for determining the deformation
will be investigated. This method is commonly used in LTU.
And finally, In order to solve Reynolds equation and evaluate film
thickness simultaneity coupled method will be investigated.
-
56
12 Bibliography
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-
62
13 Appendix
13.1 Evans and Hughes coefficient
Quadrature coefficient f split to three separate equations which is
[
( ) (
)]
[
( ) (
)]
[
( ) (
)]
To solve this equation a finite difference method is applied and extracted
the left hand side.
( )
Two boundary conditions are needed to evaluate the above equation. There
are two methods available for this reason. It can be substituted by direct
integration method or by taking an arbitrary constant and use the following
equation. [35]
( ) ( ) ( )
-
63
13.2 Secant Method
One of the numerical ways to estimate the force balance condition is secant
method. In this method the root of function is estimated by line tangent of
carve with two points. It is not necessary for the start and end points have
different signs.
The secant method is define by the following equation
For i=1,2,max iteration
( )[
( ) ( )]
In Matlab the following file can calculate the root of equation with arbitrary
tolerance and iteration. It just need too input function and interval.
function secant(f,x1,x2,tol,j)
itr=0; g1=feval(f,x1); g2=feval(f,x2); err=abs(x2-x1); disp('______________________________________________________________
') disp('itr xj f(xj) f(xj+1)-f(xj) |xj+1-xj|') disp('______________________________________________________________
') fprintf('%2.0f %12.6f %12.6f\n',iter,x0,u) fprintf('%2.0f %12.6f %12.6f %12.6f %12.6f\n',itr,x1,g2,g2-
g1,err) while (err>tol)&(itrj) disp(' Convergency problem ') end
-
School of Engineering, Department of Mechanical Engineering Blekinge Institute of Technology SE-371 79 Karlskrona, SWEDEN
Telephone: E-mail:
+46 455-38 50 00 [email protected]
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