M. J. A. Holmes
H. P. Evans
R. W. Snidle
Cardiff School of Engineering,
Cardiff CF24 0YF, UK
Analysis of Mixed Lubrication
Effects in Simulated Gear Tooth
Contacts
The paper presents results obtained using a transient analysis technique for point contact
elastohydrodynamic lubrication (EHL) problems based on a formulation that couples the
elastic and hydrodynamic equations. Results are presented for transverse ground surfaces
in elliptical point contact that show severe film thinning and asperity contact at the
transverse limits of the contact area. This thinning is caused by transverse leakage of the
lubricant from the contact in the remaining deep valley features between the surfaces. A
comparison is also made between the point contact results on the entrainment center line
and the equivalent line contact analysis. The extent of asperity contact is shown to be
dependent on the Hertzian contact aspect ratio. It is also shown that transverse waviness
(superimposed on the roughness) of even relatively small amplitude can lead to large
increases in asperity contact rates over all waviness peaks in the contact.
关DOI: 10.1115/1.1828452兴
Introduction
The lubrication mechanism primarily responsible for the pro-
tection of gear tooth surfaces from wear and surface distress is
elastohydrodynamic lubrication 共EHL兲. In the case of very smooth
surfaces 共such as those found in rolling element bearings, for ex-
ample兲 the EHL mechanism can generate oil films which are thick
compared to the height of roughness features present on the sur-
faces. Under these conditions the thickness of the oil film may be
calculated, with reasonable accuracy, using the well-known for-
mula of Dowson and Higginson 关1兴. An important feature of gear
tooth contacts, however, is that the surfaces produced by present
day manufacturing methods have roughness features that are sig-
nificantly greater than the oil film predicted by this formula. Con-
sequently gears tend to operate in a regime described as ‘‘mixed’’
or ‘‘micro’’ EHL in which there is a significant interaction of
roughness asperities on the two surfaces. In theoretical solutions
of both the dry and micro EHL situations the presence of rough-
ness leads to significant rippling of the contact pressure distribu-
tion with maximum values far in excess of the Hertzian values
expected when the surfaces are perfectly smooth. Micro EHL so-
lutions also indicate the presence of very thin films at asperity
encounters within the overall rolling/sliding contact. Two practical
problems associated with roughness effects and film thinning in
gears are micropitting 共rolling contact fatigue on the scale of sur-
face asperities兲 and scuffing 共scoring兲 which is related to the fail-
ure of the elastohydrodynamic system. In order to gain a much
clearer understanding of these failure mechanisms it is necessary
to develop a full theoretical model of lubrication of gear contacts
under rough surface/thin film conditions. Such a model must take
account of the real operating conditions of gears in terms of loads,
speeds, surface roughness and lubricant properties, and be able to
predict pressures, local film thickness, temperatures and friction
between the teeth. A further important feature that must be con-
sidered when roughness is present is the time-dependent effect of
roughness: this occurs when roughness features move relative to
the overall contact.
The paper presents results from the numerical modeling of tran-
sient rough surface point contact problems obtained using a new
coupled numerical formulation for solving the elastohydrody-
namic 共EHL兲 point contact problem described and validated else-
where 关2,3兴. The paper focuses attention on the strong side leak-
age effects that take place at the edges of contacts that have a
surface finish transverse to the direction of rolling/sliding such as
that in conventional involute gears. The configuration chosen for
analysis is that of a gear simulation disk rig which gives rise to an
elliptical contact. In the smooth surface case the EHL contact
adopts a self-sealing configuration by developing side constric-
tions in the form of the familiar horseshoe shape seen in optical
interferometry experiments. When transverse roughness features
are present, however, this mechanism is unable to seal the
pressure-driven transverse flow in the valley features because the
closest that the surfaces can be brought together is determined by
the physical contact of asperity tip features. Even in this extreme
configuration the composite valley features on the surfaces remain
open and unsealed, and lubricant can easily escape from the con-
tact area in the transverse direction along these valleys. When oil
is lost from the contact due to this sideways leakage mechanism
the entrainment of lubricant under the downstream micro contacts
is progressively weakened at each successive following contact.
This model for EHL failure was proposed earlier by the authors
关4兴 in response to their experimental observation that initial scuff-
ing failure invariably occurs at the transverse edges of such con-
tacts 关5兴. The detailed results of micro-EHL analysis presented
here add further evidence in support of the model. It is of interest
to note that the transverse edges, where failure appears to origi-
nate in the scuffing experiments, are not subject to extreme tem-
perature or extreme pressure behavior. The identification of con-
tact edges as the location of initial scuffing failure is thus a
significant observation indicating that failure of the physical
mechanism of EHL is a primary underlying cause of scuffing
in gear tooth contacts. In real gears having a finite facewidth
effective ‘‘contact edges’’ which behave in the way suggested
are not limited to the actual face edges of the gears because of
the inevitable ‘‘waviness’’ present on the surfaces. Results pre-
sented in the paper suggest that such waviness, of even small
amplitude, can lead to a significantly increased occurrence of film
breakdown.
Contact Analysis
The EHL problem is specified by the elastic deflection equation
written in differential form 关2兴,
Manuscript received February 23, 2004; revision received August 16, 2004. Re-
view conducted by: M. Lovell.
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2
h
共
x
i
,y
j
兲
x
2
⫹
2
h
共
x
i
,y
j
兲
y
2
⫽
2
共
x
i
,y
j
兲
x
2
⫹
2
共
x
i
,y
j
兲
y
2
⫹
1
R
x
⫹
1
R
y
⫹
2
E
⬘
兺
all k,l
f
k⫺ i,l⫺ j
p
k,l
(1)
and the non-Newtonian Reynolds equation
x
冉
x
p
x
冊
⫹
y
冉
y
p
y
冊
⫺
共
U
¯
h
兲
x
⫺
共
h
兲
t
⫽ 0 (2)
The nonlinear dependence of viscosity and density on pressure are
taken to be given by the well known isothermal Roelands, and
Dowson and Higginson relationships, respectively,
⫽
0
exp
兵
ln
共
0
/
兲
共共
1⫹
p
兲
Z
⫺ 1
兲
其
(3)
⫽
0
再
1⫹
␥
p
1⫹ p
冎
(4)
For non-Newtonian situations
x
and
y
(⫽
x
) are determined
from the lubricant’s pressure, pressure gradients, film thickness,
and surface velocities as discussed in 关6兴. Equation 共2兲 is dis-
cretized using linear quadrilateral finite elements with an implicit
共Crank–Nicolson兲 time formulation, and Eq. 共1兲 using finite dif-
ferences for the Laplacian with the pressure coefficients, f
i,j
,
given by the analysis in 关7兴. The equations are thus written as
兺
k⫽ 0
n
c
A
k
p
k
⫹
兺
k⫽ 0
n
c
B
k
h
k
⫽ R
i,j
(5)
兺
k⫽ 0
n
c
C
k
p
k
⫹
兺
k⫽ 0
n
c
D
k
h
k
⫽ E
i,j
(6)
where suffix k represents the nodes contributing to the assembled
equation at node (i,j) and k⫽ 0 denotes that node. A
k
and B
k
are
the pressure and film variable coefficients for the Reynolds equa-
tion 共2兲, and n
c
is the number of neighboring nodes involved in
the formulation. Similarly C
k
and D
k
are the pressure and film
variable coefficients for the differential deflection equation 共1兲.
Expression R
i,j
contains information from the previous timestep,
and E
i,j
contains all the contributions to the pressure summation
of Eq. 共1兲 that are not explicitly contained on the left hand side of
Eq. 共6兲.
Equations 共5兲 and 共6兲 are expressed as a pair of simultaneous
equations in the variables p
0
and h
0
A
0
p
0
⫹ B
0
h
0
⫽ R
ˆ
i,j
再
⫽ R
i,j
⫺
兺
k⫽ 1
n
c
A
k
p
k
⫺
兺
k⫽ 1
n
c
B
k
h
k
冎
C
0
p
0
⫹ D
0
h
0
⫽ E
ˆ
i,j
再
⫽ E
i,j
⫺
兺
k⫽ 1
n
c
C
k
p
k
⫺
兺
k⫽ 1
n
c
D
k
h
k
冎
which are solved to give a coupled iterative scheme to update the
values of all the unknown nodal values of p and h in turn, i.e.,
p
i,j
new
⫽
R
ˆ
i,j
D
0
⫺ E
ˆ
i,j
B
0
A
0
D
0
⫺ B
0
C
0
, h
i,j
new
⫽
E
ˆ
i,j
A
0
⫺ R
ˆ
i,j
C
0
A
0
D
0
⫺ B
0
C
0
(7)
This method is found to be both effective in obtaining rapid con-
vergence in low ⌳ situations with rough surfaces, and extremely
robust. The rapidity with which the influence coefficients f
i,j
in
Eq. 共1兲 fall, as the indices i and j increase from zero 关7兴,isakey
advantage of this differential formulation of the deflection equa-
tion. This property allows the recalculation of pressure contribu-
tions to E
i,j
to be limited to those that are close to the n
c
points
used in the iteration sweep 关2,8兴.
During rough surface transient analyses fluid film breakdown
can occur resulting in contact between the micro asperities. Where
contact occurs between the two surfaces the hydrodynamic film
thickness is zero, although in practice there will typically be a
boundary film that controls the local friction coefficient. Equation
共2兲 arises from mass flow continuity of the fluid film, and at lo-
cations where the film thickness is zero there is no such mass
flow. The physical principle on which Eq. 共2兲 is founded is thus
not applicable at micro contact locations. Equation 共1兲, however,
is always applicable as it relates the pressure acting on the sur-
faces with their deflection irrespective of whether the pressure
arises from a hydrodynamic film or from direct contact of the
surfaces.
Contact situations in the iterative scheme are dealt with as fol-
lows. If the iterating equations 共7兲 result in a negative value for
h
i,j
new
, its value is set to zero and Eqs. 共7兲 are thus replaced by
h
i,j
new
⫽ 0, p
i,j
new
⫽
E
ˆ
i,j
C
0
(8)
This effectively replaces the Reynolds equation with the boundary
condition h⫽ 0, and applies the deflection equation subject to that
boundary condition.
Thus Eqs. 共7兲 are used at each mesh point during each iterative
sweep and are replaced by Eqs. 共8兲 only at mesh points where the
current evaluation of Eqs. 共7兲 yield a negative value for h
i,j
new
. The
ease with which contact conditions can be incorporated using this
approach is a further advantage of the coupled differential deflec-
tion technique.
This method can be used to solve the dry, elastic contact prob-
lem using Eq. 共2兲 as can be seen from the results for contact
start-up analysis presented in 关9兴. This problem involves a simul-
taneous solution of full film and dry contact areas as liquid is first
entrained into a dry contact by motion of the surfaces. The itera-
tive approach described above deals effectively with this situation,
maintaining a dry contact pressure that remains essentially Hert-
zian away from the area where the contact shape is distorted by
the entrained fluid. The comparison made in 关9兴 with the elegant
experimental work of Glovnea and Spikes 关10兴 for this situation
provides confirmation of the validity of the approach adopted.
Results
Behavior at the Transverse Edge of the Contact. The re-
sults presented in the current paper are based on an isothermal
analysis of elliptical contacts finished in a transverse direction. In
each case the contact is between two ellipsoidal bodies whose
surface finish is given by one of the three experimental profiles
shown in Fig. 1 共solid metal below the profile兲. Trace 共A兲 is a
Fig. 1 Profiles adopted for the surfaces used in the numerical
investigation. Profiles are offset for clarity and oriented with
metal below the curves.
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profile taken from a well run-in transverse ground disk used in
scuffing experiments by Patching 关5兴. Traces 共B兲 and 共C兲 are taken
from micropitting tests on gears and have been run for several
load stages and as a result have become run-in to some extent, but
a close examination shows that they clearly have larger asperities
than profile 共A兲. Traces 共A兲, 共B兲 and 共C兲 have R
a
values of 0.32
m, 0.22
m and 0.31
m, respectively. Intermediate heights that
are required as the surfaces move through the contact are obtained
using cubic spline interpolation, which ensures slope continuity at
the measured points. A comparison of the line contact behavior of
these profiles has been undertaken previously in 关11兴 where it was
clearly seen that Profile 共C兲 was the most aggressive of the three
profiles in terms of its tendency to produce high pressure ripples
and severe film thinning.
The lubricant modeled is Mobil Jet 2, a synthetic gas turbine
lubricant used in earlier scuffing experiments 关5兴. The operating
conditions and lubricant parameters adopted are specified in Table
1 and result in a contact whose Hertzian dimension in the trans-
verse direction is four times that in the rolling/sliding direction.
The computing mesh covers the area ⫺ 2.5a⬍ x⬍ 1.5a; ⫺ 2b
⬍ y⬍ 2b, with mesh spacing ⌬x⫽ a/200; ⌬y⫽ b/50. The
timestep adopted was ⌬t⫽ ⌬x/2u
max
so that the faster moving
surface moves through one mesh spacing over two timesteps. The
transient analysis is started from the smooth steady state result,
shown in Fig. 2 which illustrates the pressure and film thickness
contours for the operating conditions chosen for analysis. The
Moes and Bosma dimensionless groups for the contact are M
⫽ 270 and L⫽ 6. These conditions could be expected to generate
an appreciable pressure spike with a Newtonian analysis, but this
is diminished into the rudimentary shoulder feature seen in non-
Newtonian circumstances; this can be discerned in the very
closely spaced 共monotonic兲 pressure contours near the exit of the
Hertzian contact area. The maximum pressure developed is 1.03
GPa which is very close to the corresponding Hertzian contact
value of 1.05 GPa. The central film thickness ‘‘plateau’’ value is
0.48
m with a minimum value on the longitudinal center line of
0.42
m, and transverse edge constrictions where a similar mini-
mum film value of 0.43
m is developed. When the steady state
solution has been established the rough surface features, which
make the problem time dependent, are fed in with the moving
surfaces from the inlet boundary position. Because of the different
speeds of the two surfaces the time taken for both surfaces in the
contact to become fully rough is that required for the slowest of
the two surfaces to move from the inlet boundary to the exit
boundary. Once this has occurred the computation is carried on for
a further 2370 timesteps, i.e. further analysis times of 0.076 ms,
0.070 ms and 0.063 ms, respectively, for the three slide/roll ratios
adopted of 0.1, 0.25 and 0.5. The distance moved by the faster
moving surface during this further calculation is thus the same in
each case, and is equal to 5.9a.
The presence of roughness on both surfaces causes a significant
variation in both pressure and film thickness within the contact
area, and Fig. 3 illustrates this effect for one particular timestep in
the analysis for two rough surfaces having profile 共C兲 with a slide/
roll ratio of
⫽ 0.25. 共This profile was shown in 关11兴 to have the
most aggressive EHL response in rough on rough line contact of
the three profiles considered.兲 The figure shows the pressure and
film thickness along the entrainment centreline, y⫽ 0, and also
shows the orientation of the two rough surfaces offset below for
clarity. The deviations of pressure from the smooth surface result
for this example can be seen to be significant: maximum pressures
Table 1 Operating conditions for the point and line contact
comparisons
Point
contact Line contact
R
x
0.0191 m 0.0191 m
R
y
0.151 m ⬀
w,w
⬘
962 N 527 kN/m
a 0.335 mm 0.335 mm
b 1.31 mm ⬀
p
hz
1.05 GPa 1.0 GPa
E
⬘
227 GPa
␣
11.1 GPa
⫺ 1
5.1 GPa
⫺ 1
␥
2.27 GPa
⫺ 1
0
0.005 Pa•s 0.0048 Pa•s
63.2⫻ 10
⫺ 6
Pa•s
1.68 GPa
⫺ 1
U
¯
25 m/s
0.1, 0.25, 0.5
0
10 MPa
Fig. 2 PressureÕGPa „upper figure… and film thicknessÕ
m
contours of the smooth surface result for the conditions ana-
lyzed. Central and minimum film thicknesses are 0.48 and
0.42
m, respectively. The heavy curve indicates a Hertzian dry
contact area.
Fig. 3 Pressure „heavy curve… and film thickness on the en-
trainment axis,
y
Ä0, at one timestep in the analysis of contact
between two surfaces having Profile „C… with
Ä0.25. Also
shown are the two rough surfaces in their contact configura-
tion offset for clarity.
Journal of Tribology JANUARY 2005, Vol. 127 Õ 63
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of the order of 3 GPa can be seen to develop on some asperity/
asperity contacts with corresponding extremely small film thick-
ness values. As the analysis progresses contact 共as defined in the
previous section兲 occurs occasionally as asperities move past each
other. The number of timesteps in which contact occurs is calcu-
lated for each mesh point during the fully rough analysis period,
and we refer to this as the contact count, denoted Q. Figure 4
shows contours of the contact count for the case where both sur-
faces have Profile 共C兲 for the three different slide/roll ratios con-
sidered. To facilitate comparisons between cases having different
sliding speeds, and thus different timesteps, the values of Q ob-
tained are normalized with respect to the total analysis time. High
contact count values indicate repeated contact instances at that
particular location as the rough surfaces pass through the corre-
sponding smooth surface Hertzian contact area indicated by the
semi ellipse. It is clear from the contour values that contact occurs
predominantly at the transverse edges of the Hertzian contact area
downstream (x⬎ 0) of the contact centerline. It is worth noting
that the profiles used for the analysis are taken from experimental
test disks and that the surface finish has been modified by the
action of plastic deformation as the contact has run from its as-
manufactured surface finish to the current state. As-manufactured
surfaces are considerably rougher, and when finished by grinding
have a near Gaussian distribution of surface heights. The height
distribution of the current profiles, shown in Fig. 1, have a degree
of skewness introduced by the running-in process, and the asperity
tips are more rounded as a result. This is the surface configuration
that corresponds to contact failure, and as such its EHL behavior
is likely to be of more engineering relevance than the more com-
putationally challenging freshly manufactured finish.
In Fig. 4 we see that calculated contact is a relatively frequent
occurrence at the transverse margins of the Hertzian area. The
mesh point for which contact occurs most frequently experiences
contact in 42 of the 2370 timesteps for which the count is carried
out, i.e., about 2% of the calculation time. This is for the interme-
diate case where
⫽ 0.25. The number of timesteps for which
contact occurs at one or more mesh points during the calculation
is, however, a high proportion of the total. Contact occurs for
98%, 95% and 86% of the timesteps for the three analyses as
shown in Table 2.
There is little difference between the contact count pattern ob-
tained for the three slide/roll ratios. The highest occurrence is with
⫽ 0.25, and the area experiencing the higher rates of asperity
contact is more pronounced for this and the lower sliding case of
⫽ 0.1. The higher sliding speed of
⫽ 0.5 seems to lead to a
reduced rate of asperity contact, and with the area over which high
asperity contact rates occur also reduced in comparison with the
other cases. This feature of the results follows from the fact that
the entrainment effect for asperity/asperity collisions within the
Hertzian region is effectively given by 0.5 times the sliding ve-
locity as was demonstrated for line contacts in 关12兴. Thus the
asperity/asperity entrainment effect for
⫽ 0.5 is five times higher
than for
⫽ 0.1. This observation would seem to suggest that
higher sliding may be advantageous in preventing asperity/
asperity contact. However, we hasten to add that the analysis pre-
sented here is isothermal so that the detrimental effects of local-
ized heat generation due to thin film or dry contact sliding
between the asperities are not included at this stage.
Figure 5 shows contours of contact count per unit time for the
case of contact between two surfaces having Profile 共B兲. The con-
tact behavior of these surfaces is quite different as can be seen by
a comparison with Fig. 4. The mesh points experiencing the high-
est contact counts in Fig. 5 are seen to be located in a very limited
area around the boundary of the Hertzian contact region. The con-
tact counts at these locations are about 25% of the peak values
seen in Fig. 4. In addition there are almost no contact occurrences
in the remainder of the Hertzian area. This is in marked contrast to
Figure 4 where contact is seen to occur over the whole width of
the Hertzian area, which suggests strongly that contact occurs
across the whole Hertzian contact area with some particular as-
perity collisions. There are bands in Fig. 4 where no contact has
occurred but these probably result from a lack of asperity colli-
sions at these locations during the analysis time.
Figure 6 shows contours of contact count per unit time for the
case of contact between two surfaces having Profile 共A兲. The peak
contact count level is similar for each of the sliding speeds, but
this case illustrates the strong effect of the asperity/asperity en-
trainment due to the sliding velocity. For the highest sliding speed
the contact count rate is close to zero over most of the Hertzian
region and high values are concentrated at the transverse contact
boundary. For the lower sliding speed contact conditions also oc-
cur on the centerline and in bands over the exit half of the contact.
Figure 7 shows contact count contours for a contact consisting
of Profile 共A兲 running against Profile 共C兲. Comparing this figure
with Figs. 4 and 6 shows contact incidences that are intermediate
between those of the individual surfaces in contact with them-
selves. Contact between these two surfaces takes place approxi-
mately half as frequently as that between two surfaces having
profile 共C兲, and with a similar pattern of contact intensity/location.
Figure 8 examines the contact count obtained for Profile 共C兲 for
⫽ 0.25 in three different cases. In Fig. 8共a兲 the surface is in
contact with a smooth surface. In Fig. 8共b兲 it is again in contact
Fig. 4 Contours of contact count rate
Q
Õms for the transient
analysis of two surfaces each having Profile „C…. The heavy
curve indicates a Hertzian dry contact area.
Table 2 Percentage of transient analysis time for which con-
tact is calculated to occur at one or more mesh points
Surfaces
⫽ 0.1
⫽ 0.25
⫽ 0.5
AandA 382634
B and B 19 20 4.3
CandC 989586
AandC 967755
C and smooth 9.2
&⫻ C and smooth 48
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with a smooth surface, but the scale of the roughness of Profile
共C兲 has been increased by a factor of &. This is so that the
composite roughness is of the same order as that for contact be-
tween two surfaces having Profile 共C兲, which is the case illus-
trated in Fig. 8共c兲. In Fig. 8共a兲 contact hardly takes place and the
little that does occur is limited to the transverse contact boundary.
In Fig. 8共b兲 there is a six-fold increase in the contact count due to
the higher roughness and this is again concentrated at the trans-
verse contact boundary. For the case of the two rough surfaces, the
maximum contact count at the transverse boundary has increased
by a further factor of three, and bands of contact occurrence
spread across the entire contact width. Although the composite
roughness in the cases shown in Figs. 8共b兲 and 共c兲 are the same,
the relative radius of curvature at the tips of the most aggressive
asperities is smaller in the case of Fig. 8共b兲 than in the case of Fig.
8共c兲 where they will generally be in contact with surface features
having a larger radius of curvature. However contact is less preva-
lent in Fig. 8共b兲 than in Fig. 8共c兲 which supports the view that
asperity collision is an important factor in causing contact to oc-
cur. This feature of the results, and the percentage contact times
given in Table 2, point towards the possibility of experimental
verification of the predicted contact effects by measuring frac-
tional contact time using electrical contact resistance. Experiments
to investigate this effect are planned for future work.
Figure 9 shows a photograph of part of the surface of a test disk
taken from the scuffing program reported in 关5兴. The disk is from
an experiment where the contact load was removed at the first
indication of scuffing. The disks are crowned and the Hertzian
contact area is illustrated by the ellipse superimposed on the pho-
tograph. Grinding marks can be clearly seen extending across the
width of the disk but the surface finish has been totally changed in
the scuffed part of the running track. The width of the scuffing
mark is about 25% that of the running track, and its outer edge
corresponds to the transverse limit of the Hertzian contact area.
This pattern has been observed in all the scuffing experiments
utilizing transverse-ground crowned disks. 共The width of the
scuffing mark is dependent on the rapidity with which load is
removed when scuffing is detected by its characteristic sudden
increase in friction.兲 We suggest that the correspondence between
the position of the scuffing track in Fig. 9 and the location of
predicted high contact counts typically shown in Figs. 4 to 7 is
striking. This indicates strongly that the primary cause of scuffing
in these experiments was the breakdown of the EHL film as a
result of direct contact between the surfaces.
Behavior in the Center of the Contact. Although the results
given above concentrate on the features of mixed lubrication situ-
ations brought about by side flow at the transverse edges of the
elliptical contact, it is interesting to compare the center line (y
⫽ 0) behavior with that of the corresponding line contacts. Figure
10 shows one such example of the film thickness and pressure
distribution at a particular timestep. The figure includes both the
elliptical contact and equivalent line contact results at the same
timestep. It can be seen that the film thickness behavior is identi-
cal between the two methods and the minor differences in pres-
sure are no greater than inevitably exist in the comparison of these
equivalent smooth surface solutions. This equivalence is found to
be generally the case 关3兴 for low ⌳ conditions with contacts of this
aspect ratio. This gives a clear demonstration that line contact
transient analyses are a suitable tool for investigating micro-
pitting and scuffing in involute gears, failure occurrences which
are not limited to the edges of the gear face width.
The aspect ratio of the contacts considered up to this point are
4:1, i.e. a/b⫽ 0.25. For elliptical contacts that have more adverse
aspect ratios the proximity of the transverse boundaries exerts a
greater influence over the main part of the contact area as might
be expected. Figure 11 compares the contact count contours for
the case of two rough surfaces each having profile C with
Fig. 5 Contours of contact count rate
Q
Õms for the transient
analysis of two surfaces each having Profile „B…. The heavy
curve indicates a Hertzian dry contact area.
Fig. 6 Contours of contact count rate
Q
Õms for the transient
analysis of two surfaces each having Profile „A…. The heavy
curve indicates a Hertzian dry contact area.
Journal of Tribology JANUARY 2005, Vol. 127 Õ 65
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