Xin Q. Diesel Engine System Design
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700 Diesel engine system design
© Woodhead Publishing Limited, 2011
compression stroke increases as EGR rate, or essentially the soot amount in
the EGR gas, increases, while the piston friction in the mid of each stroke
reduces at high EGR rate (Urabe et al., 1998). The reason was believed that
the soot deposited in the top ring groove made the piston rings slide on the
soot particles attached to the cylinder bore surface and caused an increase
in the boundary friction coefcient and wear on the components.
Piston ring axial dynamic motions (ring lifting, twisting, and uttering) and
groove design signicantly affect blow-by and oil consumption. They may
also impact ring friction and wear via the changes in inter-ring pressure and
ring tilting. The ring motion is affected by the ring gap. Too large a ring gap
results in excessively large blow-by, power loss, and emissions. Too small a
ring gap leads to ring butting problems at high operating temperatures. The
gap in the second ring can be properly designed to increase the top ring’s
sealing ability by preventing the inter-ring pressure from building up and
lifting the top ring off the bottom of the groove.
Ring thickness signicantly affects oil lm thickness and ring friction.
Furuhama et al. (1981) conducted experiments and theoretical calculations
to nd that the decrease of the ring thickness may not necessarily reduce
friction because it decreases the oil lm thickness at the same time. The
reduced lm thickness can increase the friction force and wear around the
TDC and the BDC. Off-centered barrel shape prole has been found as the
optimum for the top compression ring. The optimum choice of the degree of
curvature of the barrel prole is a trade-off between the mid-stroke and end-
stroke performance. A highly curved prole gives a large oil lm thickness
in mid-stroke due to the strong ‘wedge effect’, resulting in low friction.
However, the oil lm thickness falls rapidly around the TDC and the BDC,
and the ‘squeeze lm effect’ is weak there, resulting in high wear. A atter
ring face prole behaves oppositely.
Piston ring tension is another important parameter for friction. A piston
ring design procedure to dene a free shape producing circumferentially
uniform contact force distribution was provided by Ma et al. (1996). Oil
ring tension is an important design parameter since it determines whether
the oil ring operates in the hydrodynamic or boundary lubrication regime.
It also controls the amount of oil that is available for other rings. Uras and
Patterson (1985) used the instantaneous IMEP experimental method to nd
that the friction of a 70 N high-tension oil ring decreased signicantly after
break-in, but still higher than the friction of a 17.8 N low-tension oil ring.
When the surface roughness increases, the ratio of the piston ring oil lm
thickness to roughness decreases. When the ratio is less than a threshold value,
mixed lubrication occurs. A higher roughness results in a larger proportion
of mixed lubrication and a higher friction power loss.
Oil viscosity is also important for ring friction. Friction modiers in the
oil may reduce boundary lubrication friction, with the degree of friction
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reduction varying based on the oil formulation. Oil starvation usually
occurs more severely at full load ring than motoring. This contributes to
thinner oil lm thickness and higher friction at full load (Sanda et al., 1997).
Uras and Patterson (1984, 1985) found that the piston assembly friction
force increased when the oil viscosity became either too high or too low.
As viscosity increases, the hydrodynamic lubrication regime is promoted
and the boundary lubrication is depressed. Glidewell and Korcek (1998)
conducted experimental work to nd that oil starvation increased the friction
coefcient, most noticeably in the mid-stroke region. They also found that
the effectiveness of the friction modier signicantly degraded with aged
engine oil. Richardson and Borman (1992) found that the temperature rise
in the oil lm due to viscous heating was negligibly small. They indicated
that the cylinder inner wall temperature should be used to determine the oil
viscosity for modeling.
10.5.2 Piston ring lubrication dynamics
Piston ring pack lubrication modeling was overviewed by Dowson (1993).
The calculations of inter-ring gas pressure and blow-by (Kuo et al., 1989)
provide key input data of gas pressure for ring dynamics and friction
calculations. The gas mass ow rate between the adjacent volumes (at the
ring gap, circumferential gap, and the ring side clearance) is calculated
by using the orice ow equations. Ring motion (e.g., ring radial collapse
and axial uttering) is very sensitive to the inter-ring pressures. There is a
strong coupling between the inter-ring pressure and the ring motion. The
characteristics of inter-ring gas pressures and ring lift motion were researched
by Furuhama et al. (1979), Curtis (1981), Dursunkaya et al. (1993), Chen
and Richardson (2000), and Herbst and Priebsch (2000) using experimental
and numerical investigations.
The piston ring lubrication dynamics has several levels of complexity,
from low to high as follows:
∑ one-dimensional Reynolds equation for a ring pack with the closed-form
analytical solution with various lubrication cavitation boundary conditions
(e.g., half-Sommerfeld, the mass-conservation Reynolds or JFO condition)
and under fully or partially ooded boundary condition
∑ one-dimensional Reynolds equation with the closed-form analytical
solution including surface asperity and mixed lubrication models
∑ one-dimensional Reynolds equation with numerical solutions
∑ one-dimensional Reynolds equation coupled with ring axial and twist
dynamics
∑ two-dimensional Reynolds equation with numerical solutions
∑ two-dimensional Reynolds equation including surface asperity and
complex mixed lubrication models
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∑ two-dimensional mixed lubrication coupled with ring axial and twist
dynamics
∑ including the features of elastohydrodynamic lubrication
∑ including some of the models of bore distortion, ring conformability,
ring groove deformation, ring–groove contact pressure distribution, and
lubricant rheological properties
∑ including some of the models of blow-by, oil consumption, ring face–liner
wear, and ring groove wear.
For engine system design to estimate piston ring friction, usually the rst
and second levels of model mentioned above are suf cient. They also offer
a real-time fast computation for an instantaneous crank-angle-resolution
solution. Other higher level models are suitable for component designs.
A simple one-dimensional Reynolds equation for a piston ring can be
written as
d
d
d
d
= –
d
d
+ 12
d
d
y
dyd
h
p
dpd
y
dyd
h
y
dyd
h
t
o
lu
b
o
v
o
3
6
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
mm
mm
d
mm
d
+ 12
mm
+ 12
v
mm
v
y
mm
y
dyd
mm
dyd
vP
mm
vP
v
vP
v
mm
v
vP
v
o
mm
o
10.27
where the instantaneous oil lm thickness h
o
= h
o,min
+ h
profi le
(y), h
o,min
is
the minimum oil lm thickness, y refers to the axial direction of the ring,
h
profi le
(y) is the ring face pro le as a function of the axial distance of the
ring, and p
lub
is the lubricating oil lm pressure. The detailed derivation of
the Reynolds equation was provided by Richardson and Borman (1992).
The lubricating oil lm force per unit length of the ring is given as:
Fp
A
Fp
lu
Fp
Fp
br
Fp
Fp
in
Fp
gl
Fp
gl
Fp
ub
c
inle
t
outle
t
,
Fp
,
Fp
Fp
br
Fp
,
Fp
br
Fp
Fp =Fp
Fp
gl
Fp =Fp
gl
Fp
d
Fp dFp
gl
d
gl
Fp
gl
Fp dFp
gl
Fp
ub
d
ub
Ú
gl
Ú
gl
Fp
gl
Fp
Ú
Fp
gl
Fp
inle
Ú
inle
d
Ú
d
Fp dFp
Ú
Fp dFp
Fp
gl
Fp dFp
gl
Fp
Ú
Fp
gl
Fp dFp
gl
Fp
Ú
gl
Ú
gl
Fp
gl
Fp
Ú
Fp
gl
Fp
d
Ú
d
Fp dFp
Ú
Fp dFp
Fp
gl
Fp dFp
gl
Fp
Ú
Fp
gl
Fp dFp
gl
Fp
10.28
The ring diametrical dynamics is given by a force balance as:
mx
FF
F
ri
mx
ri
mx
ng
mx
ng
mx
ri
ng
gas
FF
gas
FF
tensio
FF
tensio
FF
nl
F
nl
F
ub
ri
ng
=
FF FF
FF
gas
FF FF
gas
FF
FF+ FF
nl
–
nl
,
–
–– –
F
gr
F
gr
F
oov
e
10.29
where x
ring
is the diametrical displacement of the ring at each moment of
time, and F
groove
is the lateral friction force between the ring and the ring
groove per unit length of the ring. With the mass inertia term
mx
ri
mx
ri
mx
ng
mx
ng
mx
ri
ng
included in the dynamic formulation, equations 10.27 and 10.29 generate
a stiff ODE system which requires a higher stiff-order implicit numerical
integration scheme with strong B-stability (for B-stability, see Hairer and
Warner (1996)).
Usually, the inertia term and the ring groove force are small and can be
neglected for simplicity. Then, with the load balanced by the lubricant force
in the normal (diametrical) direction, equation 10.29 becomes
F
~
gas
+ F
~
tension
= F
~
lub,ring
10.30
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which determines the required oil lm thickness and the ‘squeeze lm’
term to satisfy this force balance. The friction force of the piston ring can
be calculated as
FF
F
fr
FF
fr
FF
in
FF
in
FF
gf
FF
gf
FF
ri
ng
hf
F
hf
F
ri
ng
b
,,
fr,,fr
in,,in
gf,,gf
,,
ng,,ng
hf,,hf
,
ng,ng
FF =FF
FF
gf
FF =FF
gf
FF
gf
gf
FF
gf
FF FF
gf
FF
ri
ri
ng
ng
hf
hf
+
hf
+
hf
=
∂
pA
cf
Af
cf
Af
ri
cfricf
ba
pA
ba
pA
sp
pA
sp
pA
erity
pA
erity
pA
A
c
c
Ê
Ë
ˆ
¯
ÚÚ
ÚÚ
v
ÚÚ
v
h
ÚÚ
h
h
ÚÚ
h
v
ÚÚ
v
P
ÚÚ
P
o
ÚÚ
o
o
ÚÚ
o
+
ÚÚ
+
m
ÚÚ
m
2
ÚÚ
2
∂
ÚÚ
∂
ÚÚ
ppp
ÚÚ
ppp
∂p∂p∂p∂
ÚÚ
∂p∂p∂p∂
y
ÚÚ
y
Af
ÚÚ
Af
lu
ÚÚ
lu
b
ÚÚ
b
cf
ÚÚ
cf
Af
cf
Af
ÚÚ
Af
cf
Af
A
ÚÚ
A
∂
ÚÚ
∂
y∂y
ÚÚ
y∂y
Ê
ÚÚ
Ê
Ë
ÚÚ
Ë
Ê
Ë
Ê
ÚÚ
Ê
Ë
Ê
Ê
Á
Ê
ÚÚ
Ê
Á
Ê
Ë
Á
Ë
ÚÚ
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ÚÚ
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
ÚÚ
ˆ
¯
ÚÚ
¯
ˆ
¯
ˆ
ÚÚ
ˆ
¯
ˆ
ˆ
˜
ˆ
ÚÚ
ˆ
˜
ˆ
¯
˜
¯
ÚÚ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
ÚÚ
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
ÚÚÚÚ
dd
AfddAf
pAddpA
cf
dd
cf
Af
cf
AfddAf
cf
Af
ri
dd
ri
cfricf
dd
cfricf
ba
dd
ba
pA
ba
pAddpA
ba
pA
pA
sp
pAddpA
sp
pA
pA
erity
pAddpA
erity
pA
ÚÚ
dd
ÚÚ
Af
ÚÚ
AfddAf
ÚÚ
Af
Af
cf
Af
ÚÚ
Af
cf
AfddAf
cf
Af
ÚÚ
Af
cf
Af
Af+Af
ÚÚ
Af+AfddAf+Af
ÚÚ
Af+Af
Af
cf
Af+Af
cf
Af
ÚÚ
Af
cf
Af+Af
cf
AfddAf
cf
Af+Af
cf
Af
ÚÚ
Af
cf
Af+Af
cf
Af
,
10.31
where p
asperity
is the asperity contact pressure of the ring. The rst term in
the hydrodynamic friction force is the viscous shear term due to the Couette
ow, and the second term is the hydrodynamic pressure term due to the
Poisseuille ow. The friction power loss due to the translation or squeeze
term in the lubrication is neglected here.
The closed-form analytical solution for the one-dimensional Reynolds
equation 10.27 was provided by Ting and Mayer (1974a, 1974b), Dowson
et al. (1979), Wakuri et al. (1981), Furuhama et al. (1981), Jeng (1992a,
1992b, 1992c), and Sawicki and Yu (2000).
Oil starvation of the piston ring means the oil does not cover the entire
ring face. Oil starvation modeling at the leading edge (i.e., inlet) of the
ring can be as simple as assuming an effective ring thickness as input (e.g.,
assuming 50% of the ring face covered by oil), or can be as complex as
predicting the oil starvation boundary with oil transport and ow continuity
models. Oil starvation happens often on piston rings. Modeling the
starvation is important for the prediction of oil lm thickness and friction
loss. The oil supplied to each ring is dependent upon the amount of the
oil left on the cylinder wall by the preceding ring. In the downstroke the
leading ring can be assumed fully ooded. Usually, fully ooded inlet
condition can be assumed for the second rail of the oil control ring during
the downstroke. Starved lubrication should be assumed for all other rings
during the downstroke. Partially ooded lubrication should be assumed for
all the rings during the upstroke. The compression rings have more severe
oil starvation conditions during ring than motoring. During ring in the
upward strokes, the inlet of the two compression rings and the upper rail
of the oil control ring have oil starvation. In the downward strokes, the
inlet of the two compression rings has oil starvation. Ma and Smith (1996)
suggested that the bottom ring in a ring pack can be treated as fully ooded
on the downstroke in the model. Oil starvation with the one-dimensional
Reynolds equation in mixed lubrication was modeled by Sanda et al. (1997).
They found the predicted oil lm regions with starvation were as small as
half of the total thickness of the compression rings, and the predicted oil
lm thickness with starvation was less than half of the oil lm thickness
predicted by the fully ooded condition for the compression rings, resulting
in much greater friction force. They also found that the starved lubrication
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simulation for the oil control ring matched the measured data much better
than the fully ooded simulation, and the starved simulation gave about
35% higher friction force than the fully ooded simulation data.
Cavitation and separation modeling at the trailing edge of the piston ring
is important for the oil pressure distribution. The assumption of the cavitation
boundary condition at the trailing edge of the ring in the modeling signicantly
affects the calculated hydrodynamic lubrication pressure distribution prole,
load capacity, oil lm thickness, and friction force of the ring. When negative
pressures in the lubricating oil lm occur, gas bubbles are formed as cavities
from the dissolved gases or ventilation to the surroundings. The oil lm ruptures
to lose any hydrodynamic lubrication pressure. The cavitated oil lm has the
atmospheric pressure. The viscous shear force in a cavitated/ruptured oil lm,
although not as high as in a non-cavitated oil lm, cannot be neglected. The
full-Sommerfeld boundary condition unrealistically requires the lubricating oil
lm to continuously sustain large negative pressures. The half-Sommerfeld
condition simply discards the negative pressures hence violates ow continuity
and mass conservation. The Reynolds (Swift–Stieber) cavitation condition
obeys mass conservation by requiring the pressure gradient equal to zero
at the cavitation boundary. The Reynolds condition is the most commonly
used condition for piston rings although other cavitation assumptions have
been proposed in tribology (e.g., the ow separation boundary condition,
the Floberg condition, the Coyne and Elrod separation condition, and the
JFO boundary condition).
Unlike the piston skirt, a large gas pressure may exist at the outlet of the
lubricated region for piston rings. This results in the oil lm reforming after
the cavitation by generating a hydrodynamic pressure gradually rising to the
outlet gas pressure at the trailing edge of the ring. This cavitation–reformation
sequence forms a cavitated ‘pocket’ within the oil lm near the outlet.
Pressure reformation may have strong effects on the peak oil lm pressure
in the full-lm region and on the cavitation location (Yang and Keith, 1995).
Some authors believe that when a negative pressure happens the oil lm
completely separates from the ring face and does not reform at the outlet.
As a result, the outlet gas pressure is applied in the entire original negative
pressure region all the way back to the separation boundary. Richardson and
Borman (1992) reported that there was no indication of oil lm reforming
at the rear of the ring in their measurement. This indicated that separation
occurred rather than ventilation cavitation. It seems that the JFO or Elrod
cavitation model used in the journal bearings cannot be used in the piston
ring and skirt. Ma and Smith (1996) proposed that it is more reasonable to
assume an ‘open cavitation’ without oil lm reformation. Piston ring lubrication
cavitation modeling with different boundary conditions was reviewed by
Priest et al. (1996, Fig. 10.18). They showed that the most commonly used
Reynolds cavitation and reformation boundary condition produced thinner
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oil lm thickness and higher shear friction force than the modied Reynolds
reformation condition of separation without reformation. They showed that
the Reynolds condition gave 34% higher FMEP than the modied one.
Arcoumanis et al. (1995) and Sawicki and Yu (2000) compared different
cavitation conditions for piston ring lubrication. Although the calculation
results from these authors are very sensitive to the choice of cavitation
boundary condition, there is neither consensus nor solid experimental evidence
on which condition is more appropriate. The half-Sommerfeld condition
predicts thinner oil lm thickness than the Reynolds condition does. It seems
the Reynolds cavitation and reformation boundary condition is still the most
effective to date for piston ring lubrication. Moreover, it should be noted
that the difference in the friction force caused by the different cavitation
boundary conditions is usually much smaller for the piston skirt than the
piston rings. This is partly because the two sides (thrust and anti-thrust) of
the skirt tend to offset or balance each other on the overall oil lm thickness
and shear friction around the piston when one side produces thinner lm
thickness than the other side.
U
p
p
x
x
U
p
1
p
1
p
2
p
2
x
1
x
1
x
2
x
2
x
3
x
4
Cavitated region
Crank angle (degree)
(a) Hydrodynamic pressure profile and
film shape with Reynolds cavitation and
reformation boundary conditions
(b) Hydrodynamic pressure profile and
film shape with modified Reynolds
separation boundary conditions
F (N)
100
50
0
–50
–100
0 180 360 540 720
Reynolds cavitation and reformation
Modified Reynolds separation
(c) Predicted cyclic variation of friction force
10.18 Piston ring lubrication dynamics, cavitation, and friction
modeling (from Priest et al., 1996).
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Surface topography plays a dominant role in mixed lubrication. Both
surface roughness pattern (oriented in the transverse, isotropic, or longitudinal
direction) and roughness magnitude have signicant inuence on piston
ring oil lm thickness, friction and wear. A rougher surface increases the
proportion of boundary friction, in the mixed lubrication. The piston ring
surface topography effect in the mixed lubrication was modeled by Sui and
Ariga (1993) and Michail and Barber (1995). Sui and Ariga (1993) concluded
that the friction of the oil ring and the second compression ring is the most
sensitive to surface roughness variations, while the top compression ring is
less affected by surface roughness. Tian et al. (1996b) studied the effect of
surface roughness on oil transport in the top liner region. Arcoumanis et al.
(1997) developed mixed lubrication models for Newtonian and non-Newtonian
shear thinning uids on rough surfaces. Gulwadi (2000) introduced a model
to calculate the ring–liner wear.
The forces and moments due to gas pressures, axial inertia, hydrodynamic
normal and shear forces and the reaction and friction forces at the ring–groove
pivot positions cause the ring to move axially and twist in the groove. Ruddy
et al. (1979), Keribar et al. (1991), Tian et al. (1996a, 1997, 1998), and
Gulwadi (2000) extended the axisymmetrical one-dimensional Reynolds
equation lubrication analysis by including the ring dynamics of the radial,
axial, and twist motions within the groove so that blow-by, oil consumption
and ring–groove wear can be analyzed in addition to a more accurate prediction
of ring friction. Tian et al. (1996a) also introduced a lubrication and asperity
contact model for the oil lm pressure between the ring and its groove.
Piston ring hydrodynamic lubrication and friction are signicantly affected
by the dynamic twist of the ring and the inter-ring gas pressure loading that
is inuenced by the ring axial motion. The inuences of particles on the
tribological performance of piston ring packs were numerically studied by
Meng et al. (2007b, 2010). Piston ring dynamics modeling can be conducted
with commercial software packages such as Ricardo’s RINGPAK (Keribar
et al., 1991; Gulwadi, 2000) and AVL’s EXCITE Piston&Rings and GLIDE
(Herbst and Priebsch, 2000).
The lateral dynamic friction force between the piston ring and the ring groove
is believed to be signicant, and this partly contributes to a circumferential
variation of the oil lm thickness of the ring. Non-axisymmetrical oil lm
distribution is also caused by other factors such as circumferentially non-
uniform ring elastic pressure (e.g., caused by improper design of the ring
free shape), bore distortion, the circumferential variation of the ring face
prole, dynamic ring twist, the non-uniform static twist caused by ring
groove deformation, circumferential ring gap position, and the different/
asymmetrical inter-ring gas pressures at the thrust and anti-thrust sides due
to the piston secondary motions. The modeling work by Das (1976) was
one of the earliest efforts to solve the two-dimensional Reynolds equation
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for piston rings. Two-dimensional piston ring lubrication was modeled in
detail by Hu et al. (1994), Yang and Keith (1996a, 1996b), and in a series
of research conducted by Ma and Smith (1996) and Ma et al. (1995a, 1995b,
1997), particularly modeling the effects of bore out-of-roundness and ring
eccentricity on ring friction and oil transport. The non-axisymmetrical
modeling gives circumferentially non-uniform hydrodynamic lubricating
oil lm pressure distribution and uneven oil lm thickness. Yang and Keith
(1996b) found that the circumferential ow lowers the load-carrying capacity
of the ring, hence the minimum oil lm thickness in the non-axisymmetrical
modeling is smaller than that in the axisymmetrical modeling. Ma et al.
(1995b) found that better overall ring performance can be achieved when
the ring face barrel prole has a small offset (i.e., asymmetric barrel).
Piston ring elastohydrodynamic lubrication was modeled by Yang and
Keith (1995, 1996b) with one- and two-dimensional Reynolds equations,
respectively, by considering the pressure–viscosity relation and ring elastic
deection and deformation. They found that the elastohydrodynamic effect
is a strong factor in piston ring lubrication because their calculated minimum
oil lm thickness around the TDC is thicker than that in the rigid ring
case (Fig. 10.19). They also found that under high cylinder pressures the
elastohydrodynamic oil lm thickness tends to be uniform circumferentially
because the elastic deformation of the ring tends to reduce the gap caused
by the noncircular bore.
Two-dimensional elastic ring
Two-dimensional rigid ring
One-dimensional EHL
–360 –180 0 180 360
Crank angle (degree)
Minimum film thickness (mm)
3.5 ¥ 10
–3
3 ¥ 10
–3
2.5 ¥ 10
–3
2 ¥ 10
–3
1.5 ¥ 10
–3
1 ¥ 10
–3
5 ¥ 10
–4
0
10.19 Comparisons of predicted oil film thickness of a piston ring
(from Yang and Keith, 1996b).
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10.6 Engine bearing lubrication dynamics
Engine bearing tribology has been extensively researched and there is a large
body of literature available. A broad discussion in this highly specialized eld
is outside the scope of this book. Excellent reviews have been provided by
Campbell et al. (1967), Martin (1983, 1985), Goenka and Paranjpe (1992),
Taylor (1993a), and Tanaka (1999). The following discussion focuses on the
key characteristics of engine bearing lubrication dynamics, general analysis
methods, and bearing friction calculation.
10.6.1 Characteristics of engine bearing lubrication
dynamics
The engine bearings are dynamically loaded bearings, including the crankshaft
main bearings, the connecting rod bearings, the piston pin bearings, the
camshaft bearings, and the balancer shaft bearings. The analytical solutions
of the Reynolds equation for bearing lubrication were usually based on the
‘short bearing’ approximation. Booker’s mobility method (Booker, 1965)
to calculate the journal trajectory and the oil lm thickness of dynamically
loaded bearings was very popular and is still being used in many commercial
software packages for the calculations of oil lm thickness and friction. The
mobility method determines the journal eccentricity and the attitude angle
for a given dynamic force.
A more accurate solution of the journal trajectory and friction loss requires
a numerical algorithm such as nite difference or nite element to solve the
two-dimensional Reynolds equation. The mass inertia effect of the journal
within the bearing clearance may become important for some bearings. It may
impact the journal trajectory, for example in the crankshaft main bearings
near the ywheel. The journal bearing dynamics including the mass inertia
or the moment of inertia term (or the acceleration term) coupled with the
Reynolds equation possesses a fundamental characteristic of stiff ODE. Stiff
ODE requires a time-marching implicit numerical integration algorithm with
superior numerical stability in order to avoid (1) the round-off error going out
of control in any explicit integration algorithm, or (2) an arti cial oscillation
of the lubricant force with respect to time caused by a less stable implicit
integration algorithm. The stiffer the ODE system, the more dif cult the
numerical integration. A more detailed discussion of the stiff ODE feature
is summarized in Section 10.4.4. One important indicator of the stiffness
of the ODE system is the largest eigenvalue of the Jacobian matrix of the
ODE, which is related to the lubrication and dynamic parameters of journal
bearings as follows (Xin, 1999):
l
m
ODE
l
ODE
l
,m
ODE,mODE
ax
vB
m
vB
m
B
JB
rL
vB
rL
vB
mc
JB
mc
JB
µ
3
rL
3
rL
3
10.32
Diesel-Xin-10.indd 708 5/5/11 12:00:36 PM
709Friction and lubrication in diesel engine system design
© Woodhead Publishing Limited, 2011
where m
v
is the lubricant dynamic viscosity, r
B
is the bearing radius, L
B
is
the bearing length, m
J
is the journal mass, c
B
is the bearing radial clearance.
Equation 10.32 is derived using the short-bearing assumption. A larger
l
ODE,max
means higher stiffness of the ODE. Equation 10.32 shows the major
parameters affecting the stiffness. For example, a large value of oil viscosity,
a very light journal mass or a very small bearing clearance can all increase
the severity of the numerical instability in the time-marching integration,
especially when an explicit integrator is used.
Stiff ODE is one of the most important fundamental characteristics of any
lubrication dynamics problem when the mass inertia or moment of inertia
term is included in the dynamics equation of the motion coupled with the
Reynolds equation. Correctly handling stiff ODE is critical for the robustness
and the computational efciency of the model. Fortunately, in many practical
applications of lubrication dynamics, the mass inertia effect is small hence
the inertia term may be neglected for simplicity. However, the inertia term
should be included if a rigorous dynamic formulation is adopted. In this case
the stiff ODE feature is inevitable.
The complexity of the bearing models can vary from a simple level of
the rigid isothermal two-dimensional hydrodynamic lubrication with the
half-Sommerfeld (non-mass-conserving) cavitation boundary condition
to a sophisticated level of thermo-elastohydrodynamic three-dimensional
lubrication including the effects of journal tilting, surface topography, and
non-Newtonian uid with the JFO mass-conserving cavitation boundary
condition, and the Elrod algorithm to solve the boundary condition. Although
a true mass-conserving cavitation boundary condition has a secondary effect
on the calculations of oil lm thickness, oil lm pressure, and friction loss,
it is important for the oil ow and temperature predictions (Goenka and
Paranjpe, 1992). The half-Sommerfeld and Reynolds cavitation conditions
do not satisfy the condition that the oil inow should equal the outow,
whereas the JFO condition does. For engine system design calculations,
usually a simplied short-bearing approximation is sufcient to estimate
the bearing friction. Paranjpe et al. (2000) provided a comparison between
the theoretical calculations and the oil lm thickness measurements for the
engine crankshaft main bearings and the connecting rod big-end bearings.
The relationship between the typical instantaneous bearing load and the oil
lm thickness behavior of those bearings was illustrated. An example of the
instantaneous bearing friction torque simulation with a mixed lubrication model
for dynamically loaded journal bearings was provided by Ai et al. (1998).
Bearing oil operating temperature and operating viscosity have a large
impact on the accuracy of friction calculations. The friction generated inside
the bearing in turn heats the oil as it ows through the bearing. Therefore,
the estimation of the bearing oil temperature and viscosity needs to account
for the viscous heating due to the uid shear friction. The rate of the viscous
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