Influence of dynamic behaviour onelastohydrodynamic lubrication of spur gearsS Li* and A Kahraman
Gleason Gear and Power Transmission Research Laboratory, The Ohio State University, Columbus, OH, USA
The manuscript was received on 1 November 2010 and was accepted after revision for publication on 14 April 2011.
DOI: 10.1177/1350650111409517
Abstract: In this study, the elastohydrodynamic lubrication (EHL) behaviour of high-speedspur gear contacts is investigated under dynamic conditions. A non-linear time-varying vibratorymodel of spur gear pairs is introduced to predict the instantaneous tooth forces under dynamicconditions within both the linear and non-linear operating regimes. In this model, the periodi-cally time-varying gear mesh stiffness and the motion transmission error are used as excitationsand a constant damping ratio is employed. This model allows the prediction of steady-state non-linear response in the form of tooth separation (contact loss). An earlier gear mixed EHL model [1]is adapted to simulate the lubrication behaviour of spur gear contacts under these dynamicloading conditions, considering the variations of the basic contact parameters, such as radii ofcurvature, sliding and rolling velocities, and measured roughness profiles, as the contact movesalong the tooth from the start of active profile to the tip. The EHL predictions under dynamicloading conditions are compared to those assuming quasi-static contact loads for gear setshaving smooth and rough surfaces to demonstrate the important influence of dynamic loadingon gear lubrication. The unique, transient EHL behaviour under the non-linear (intermittentcontact loss) condition is also illustrated.
Keywords: dynamics, non-linear, gear, mixed EHL, roughness
1 INTRODUCTION
Dynamic behaviour of gear systems has been studied
extensively for two primary reasons. One is that the
noise generated by a gear system is a direct conse-
quence of its dynamic behaviour. Any effort to reduce
gear noise of a transmission must focus on the reduc-
tion of the vibration levels of the gears. The second
reason is the durability concern. As the gear and bear-
ing force and stress amplitudes are often amplified
under dynamic conditions, such dynamic effects
must be taken into account in the design of gear
pairs. A large number of dynamics models have
been developed over the past 50 years. Most of
these models are for spur or helical gears as summar-
ized in the review papers by Ozguven and Houser [2]
and Wang et al. [3]. Based on the measured non-
linear dynamic behaviour documented in the
literature for spur gears [4–9], several non-linear
time-varying models of spur gears were proposed
with reasonable success in describing the published
experiments. These mostly torsional spur gear
dynamics models [10–15] used a clearance type
non-linear gear mesh function to take into account
the tooth separations in the presence of gear backlash
and considered the periodic time variation of the gear
mesh stiffness due to the fluctuation of the number of
loaded tooth pairs as the gears rotate. Likewise, in
line with the linear behaviour observed experimen-
tally [16], most helical gear pair models have
*Corresponding author: Gleason Gear and Power Transmission
Research Laboratory, The Ohio State University, 201 West 19th
Avenue, Columbus, OH 43210, USA.
email: [email protected]
740
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
neglected backlash while including additional rota-
tional, translational, and axial motions [16–20].
The tribology literature contains a wide spectrum
of elastohydrodynamic lubrication (EHL) models.
Sophisticated isothermal or thermal, Newtonian or
non-Newtonian, and line or point EHL models con-
sidering smooth or rough surface contacts have been
proposed to simulate the very complicated lubrica-
tion behaviour in a more comprehensive and accu-
rate way [21–26]. However, the contact parameters of
a gear pair, including the normal tooth force, radii of
curvature, and surface velocities, all vary as the gears
roll in mesh. Such a transient contact behaviour
might not be fully represented through the analyses
of certain discrete gear mesh positions of interest
using these EHL models with constant contact
parameters.
There are a limited number of published studies
that attempted to include the geometric, kinematic,
and load variations of gear contacts. Approximating
the elastic deformation by that of a smooth dry
Hertzian contact, Wang and Cheng [27, 28] predicted
the minimum film thickness and thermal character-
istics of spur gears having smooth tooth surfaces.
Larsson [29] and Wang et al. [30] proposed involute
spur gear EHL models for isothermal non-Newtonian
and thermal Newtonian fluids, respectively, employ-
ing assumed time-varying normal tooth force as the
contact moves along the line of action. These three
studies, while establishing the need for a specialized
EHL model for spur gears, lacked the ability to handle
the rough surface condition. In addition, these stu-
dies limited their treatments of the gear mesh defor-
mation to Hertzian effect. However, other effects due
to tooth bending, base rotation, and shear deforma-
tion were shown to be equally important in defining
gear mesh compliance [31]. Incorporating a gear load
distribution model considering all these essential
components of the gear tooth compliance, Li and
Kahraman [1] proposed a transient mixed EHL
model for spur gear pairs that is capable of handling
extreme asperity contact condition robustly by also
including the gear contact transient effects. Any man-
ufacturing errors or intentional tooth profile modifi-
cations were also taken into account.
In view of the above review of the gear dynamics
and gear tribology literature, one can point to an
apparent gap between these two disciplines. Tribo-
dynamic models for gear systems are not readily
available. Although there have been a few studies on
the gear mesh damping mechanism (induced by the
gear EHL contacts) [32] and on the influence of fric-
tion of lubricated contacts on gear dynamics [33, 34],
the impact of dynamic tooth force on gear EHL con-
tact is yet to be fully understood. The research by
Wang and Cheng [27, 28] incorporated a torsional
vibratory model into the EHL analysis to solve for
the minimum film thickness. However, the transient
gear parametric effects were not fully included.
Neither was the surface roughness. Accordingly, the
main objective of this study is to investigate the influ-
ence of gear dynamics on gear EHL behaviour.
In this study, a non-linear, time-varying dynamic
model of a spur gear pair is incorporated with a gear
mixed EHL model to predict the gear contact beha-
viour under both the linear and non-linear dynamic
conditions. The dynamic model includes the gear
mesh stiffness fluctuation, displacement excitation
due to the manufacturing errors and intentional
tooth corrections (also known as the transmission
error), and gear backlash non-linearity in a manner
similar to the model of Tamminana et al. [15]. The
dynamic model uses the quasi-static gear mesh stiff-
ness and the transmission error excitation predicted
using a gear load distribution model [31] to deter-
mine the instantaneous tooth contact force. The
EHL model used here is based on an earlier spur
gear EHL model [1] where the instantaneous contact
radii and surface velocities are defined using the invo-
lute geometry. With the dynamic tooth force provided
by the dynamic model, instantaneous film thickness
and normal pressure distributions of the tooth con-
tact are predicted by the EHL model as the contact
moves along the tooth surface.
In this study, the effect of the lubrication character-
istics on the dynamic response in terms of lubricant
stiffness and damping is not considered, such that the
dynamic solver and the EHL solver are uncoupled.
The investigation is also kept limited to spur gear
pairs with no significant lead modifications to allow
a line contact formulation. Other types of gear pairs
such as helical and hypoid gears were beyond the
scope of this study.
2 MODEL FORMULATIONS
2.1 Prediction of dynamic gear tooth
contact forces
In order to predict the dynamic loading on the teeth
of a spur gear pair, a single-degree-of-freedom (DOF)
discrete model similar to that of Tamminana et al.
[15] is employed. This torsional dynamic model,
shown in Fig. 1, consists of two rigid discs of radii
rb1 and rb2 (to represent the base circle radii of gears
1 and 2) and polar mass moments of inertia I1 and I2,
respectively. The gear mesh interface model consists
of (a) a parametrically time-varying gear mesh stiff-
ness kðt Þ, (b) a constant viscous damper c, and (c) an
externally applied gear mesh displacement excitation
Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 741
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
eðt Þ, all of which are applied along the line of action
(line tangent to the base circles of the gears). Here, the
least known parameter is c since there are several
power loss mechanisms present, including the vis-
cous losses at the gear mesh interface, viscous
losses on the bearings, and windage, churning, and
pocketing losses due to fluid–gear interactions within
the gear box. The damping element c is intended to
represent all these mechanisms. However, some of
them are not feasible to easily quantify. The time var-
iance of kðt Þ is mainly due to the fluctuation of the
number of loaded tooth pairs between two integers
(typically 1 and 2) as the gears roll in mesh. The dis-
placement excitation eðt Þ represents the motion
transmission deviation caused by intentional tooth
profile modifications as well as the manufacturing
errors under unloaded condition.
Bulk of the non-linear behaviour observed in spur
gear pairs occurs at instances when the dynamic force
amplitude exceeds the static load (preload) trans-
mitted by the gear pair. With the presence of gear
backlash, the gear teeth lose contact at such instances
and the gear mesh stiffness drops to zero instanta-
neously. As proposed earlier [12, 15], the mesh stiff-
ness kðt Þ is subjected to a piecewise linear clearance
function, d as illustrated in Fig. 1. This function is
composed of a dead zone (backlash) of size 2b
bounded by two unity slope regions representing
the linear and back contact conditions (no contact
loss).
With the positive directions of the alternating rota-
tional displacements, #1 and #2, and the applied
torque, T1 and T2, defined as shown in Fig. 1, the
dynamic equations read
I1€#1ðt Þ þ rb1kðt Þ�ðt Þ þ crb1½_sðt Þ � _eðt Þ� ¼ T1 ð1aÞ
I2€#2ðt Þ � rb2kðt Þ�ðt Þ � crb2½_sðt Þ � _eðt Þ� ¼ �T2 ð1bÞ
where sðt Þ ¼ rb1#1ðt Þ � rb2#2ðt Þ is the dynamic trans-
mission error and
�ðt Þ ¼sðt Þ � eðt Þ � b, sðt Þ � eðt Þ4b0, sðt Þ � eðt Þ
�� �� � bsðt Þ � eðt Þ þ b, sðt Þ � eðt Þ5�b
8<: ð1cÞ
Since the generalized parameters of the two-DOF
model of Fig. 1 are semi-definite with a rigid body
mode at zero natural frequency, a new relative dis-
placement parameter is defined as �ðt Þ ¼ sðt Þ � eðt Þ.
With this, the equation of motion of the resultant
definite single-DOF model is derived as
me€�ðt Þ þ c _�ðt Þ þ kðt Þ�ðt Þ ¼ �F �me €eðt Þ ð2aÞ
where me is the equivalent mass defined as
me ¼ I1I2
�ðI1r2
b2 þ I2r2b1Þ. The constant force trans-
mitted by the gear mesh �F ¼ me rb1T1=I1ð þrb2T2=I2Þ,
where T1 and T2 are the constant external torques
applied to gears 1 and 2, respectively. The overdot
denotes the differentiation with respect to time t. The
non-linear restoring function � in Fig. 1 has the form of
�ðt Þ ¼�ðt Þ � b, �ðt Þ4b0, �ðt Þ
�� �� � b�ðt Þ þ b, �ðt Þ5�b
8<: ð2bÞ
It is noted that the mesh stiffness kðt Þ consists of a
mean component, �k, and an alternating component,
kaðt Þ, i.e. kðt Þ ¼ �k þ kaðt Þ. With this, a set of dimen-
sionless parameters can be defined as �ðt Þ ¼ �ðt Þ�
b,
eðt Þ ¼ eðt Þ�
b, �ðt Þ ¼ �ðt Þt=b, � ¼ c=ð2
ffiffiffiffiffiffiffiffiffiffime
�k
qÞ and
F ¼ �F=ðb �kÞ, leading to a dimensionless form of equa-
tion (2a) as
d2�ðt Þ
dt 2þ 2�!n
d�ðt Þ
dtþ !2
n 1þkaðt Þ
�k
� ��ðt Þ
¼ !2nF �
d2eðt Þ
dt 2ð3Þ
where !n ¼
ffiffiffiffiffiffiffiffiffiffiffiffi�k=me
qis the undamped natural fre-
quency. The mesh stiffness kðt Þ of a gear pair can be
determined using a gear load distribution model [31]
or any finite element based gear contact model. In
the latter case, the difference of the gear static trans-
mission error between the loaded (~eðt Þ) and unloaded
(eðt Þ) conditions can be used to estimate the
mesh stiffness as kðt Þ ¼ ðT1
�rb1Þ
�½~eðt Þ � eðt Þ� [15].
Fig. 1 A purely torsional dynamic model of a spur gearpair
742 S Li and A Kahraman
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
Defining the dynamic mesh force from equation (3)
as
DMFðt Þ ¼ b �k2�
!n
d�ðt Þ
dtþ 1þ
kaðt Þ�k
� ��ðt Þ
( )ð4Þ
The individual dynamic tooth force, Wdðt Þ, can be
obtained approximately from the quasi-static tooth
force, Wsðt Þ, as [15]
Wdðt Þ ¼Wsðt ÞDMFðt Þ
T1=rb1
� �ð5Þ
2.2 Time-varying gear tooth contact parameters
Unlike the simple contact of cylinder pair, the gear
contact parameters, including the radii of curvature,
tangential surface velocities, and normal tooth force,
all vary as the contact moves along the line of action
when the gears rotate. According to the basic involute
gear geometry and kinematics, the contact radii of
curvature r1 and r2 are given as
r1ðt Þ ¼ rb1�1ðt Þ ð6aÞ
r2ðt Þ ¼ rb2�2ðt Þ ð6bÞ
where �1ðt Þ and �2ðt Þ are the roll angles of gears 1 and
2, respectively. As a result, the corresponding surface
velocities tangent to the tooth surfaces become vari-
able as well such that
u1ðt Þ ¼ !1r1ðt Þ ð7aÞ
u2ðt Þ ¼N1
N2!1r2ðt Þ ð7bÞ
where N1 and N2 are the number of teeth of gears 1
and 2, respectively, and !1 the angular velocity of
gear 1. Here, u1ðt Þ5u2ðt Þwhen the contact point tra-
vels long the dedendum of the driving gear (gear 1)
from the start of active profile (SAP) to the pitch
point such that the sliding velocity is negative, i.e.
usðt Þ ¼ u1ðt Þ � u2ðt Þ5 0. When the contact reaches
the pitch point, usðt Þ ¼ 0 (pure rolling condition).
Within the addendum range from the pitch point to
the tooth tip of the driving gear, usðt Þ4 0. In the pro-
cess, the rolling velocity urðt Þ ¼12 u1ðt Þ þ u2ðt Þ½ � varies
with the gear rotation as well. The resultant slide-
to-roll ratio of the tooth contact, defined as
SRðt Þ ¼ usðt Þ�
ur ðt Þ, varies from a negative limit to a
positive one with SRðt Þ ¼ 0 at the pitch point.
As the driving and driven gears roll in mesh, the
number of loaded tooth pairs alternates between
two integers that bound the profile contact ratio of
the gear pair (average number of tooth pairs in con-
tact), resulting in the periodic quasi-static tooth force
Ws ¼Wsðt Þ.
The gear load distribution model [31] referred to
earlier for the predictions of kðt Þ and eðt Þ is also
used to predict Wsðt Þ. This model includes the flex-
ibilities associated with tooth bending, shear defor-
mation, base rotation, as well as the contact
deformation. It also considers any deviation of the
tooth profile from the perfect involute primarily due
to the intentional modifications such as tip relief and
profile crown. In the absence of such modifications,
Wsðt Þ experiences sudden and drastic changes at the
instants when the number of loaded tooth pairs
changes from two to one (the lowest point of single
tooth contact, LPSTC) and from one to two (the high-
est point of single tooth contact, HPSTC).
2.3 Transient mixed EHL model for spur
gear contacts
Assuming negligible shaft misalignment and a rea-
sonable level of lead modification, the contact of a
spur gear pair can be characterized as a line contact.
The one-dimensional transient Reynolds equation
governs the fluid flow in the contact areas where suf-
ficient lubricant film is established
@
@xf ðt Þ
@pðx, t Þ
@x
� �¼@ ur ðt Þ�ðx, t Þhðx, t Þ½ �
@x
þ@ �ðx, t Þhðx, t Þ½ �
@tð8aÞ
where pðx, t Þ, hðx, t Þ, and �ðx, t Þ are the transient pres-
sure, film thickness, and density distributions along
the rolling direction x. The Ree–Eyring flow coeffi-
cient, f ðt Þ, is approximated as f ðt Þ ¼ ½�h3�ð12�Þ� cos
h½�mðt Þ��0� [1], where � is the lubricant viscosity, �0
the lubricant reference stress, and �m the mean vis-
cous shear stress; �mðt Þ ¼ �0 sinh�1 �usðt Þ�ð�0hÞ
. For
any local area where the fluid film is extremely thin,
say less than two layers of lubricant molecules,
hydrodynamic lubrication becomes impossible and
the reduced Reynolds equation [1, 25, 26] is used to
describe the contact
@ ur ðt Þ�ðx, t Þhðx, t Þ½ �
@xþ@ �ðx, t Þhðx, t Þ½ �
@t¼ 0 ð8bÞ
Once a smooth transition from the fluid film region
to the asperity contact region is assumed, this unified
Reynolds equation system of equations (8a) and (8b)
govern the EHL behaviour of the contact, considering
the hydrodynamic and asperity contact pressures
simultaneously.
Assuming only elastic deformation, the local film
thickness can be defined as
hðx, t Þ ¼ h0 tð Þ þ g0ðx, t Þ þ V ðx, t Þ
� R1ðx, t Þ � R2ðx, t Þ ð9Þ
Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 743
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
where h0 is the reference film thickness, V the surface
elastic deflection, and g0 the unloaded geometric gap
between the mating tooth surfaces, which is time
dependent through the variable equivalent radius
of curvature reqðt Þ ¼ r1ðt Þr2ðt Þ�½r1ðt Þ þ r2ðt Þ� for
g0ðt Þ ¼ x2�½2reqðt Þ�. The terms R1ðx, t Þ and R2ðx, t Þ
represent the tooth surface roughness profiles mea-
sured in the profile (rolling and sliding) direction x.
Here, it is assumed that these roughness profiles
remain constant along the tooth face direction to
allow the use of this line contact formulation. This
is a reasonable assumption for certain finishing pro-
cesses such as grinding and shaving.
The next equation of interest is the load balance
equation, which states that the total contact force
due to the instantaneous pressure distribution over
the entire contact zone must balance the dynamic
tooth force at the same instant, i.e.
W 0dðt Þ ¼
Zpðx, t Þ dx ð10Þ
where W 0dðt Þ is the dynamic tooth force intensity
along the contact line (dynamic tooth force per unit
face width). The value of h0 in equation (9) must be
adjusted iteratively until the pressure distribution
pðx, t Þ satisfies equation (10).
Various forms of viscosity–pressure relationships
have been used in the past including the Barus’ expo-
nential relationship, the Roeland’s equation, as well
as the two-slope exponential relationship. However,
these relationships might not be accurate within very
wide pressure ranges experienced in rough gear con-
tacts [35]. The Doolittle–Tait relationship was pro-
posed as a potential remedy [35]. While any of the
other viscosity–pressure relationships can be used
as long as they represent the measured behaviour of
the lubricant accurately within the operating pressure
range, the Doolittle equation [36] used in this study is
� ¼ �0 exp B�Voccð1� �V Þ
ð �V � �VoccÞð1� �VoccÞ
� �ð11aÞ
where �0 the ambient viscosity, B the Doolittle para-
meter, and �Vocc the normalized occupied volume. The
normalized volume �V is a pressure-dependent para-
meter that can be modelled through the empirical
Tait equation of state [37] as
�V ¼ 1�1
ð1þ K 00Þln 1þ
p
K0ð1þ K 00Þ
� �ð11bÞ
where K0 and K 00 are the bulk modulus and the deri-
vative of bulk modulus with respect to pressure,
respectively, when p ¼ 0. The density–pressure rela-
tionship of the lubricant is modelled the same way as
described in Li and Kahraman [1], in which the details
of the discretization and linearization of the govern-
ing equations can be found.
3 RESULTS OF AN EXAMPLE ANALYSIS
In this section, the transient contact pressure and film
thickness distributions between the mating teeth of
an example spur gear pair under the dynamic loading
condition are presented at different mesh frequencies
to demonstrate the substantial impact of the dynamic
response on the EHL behaviour. The design para-
meters of the example spur gear pair are listed in
Table 1. The mass and inertia values of this gear
pair are also listed. This gear pair design was used
in several earlier experimental studies on the non-
linear dynamic behaviour of spur gear pairs [7–9,
12, 13]. The lubricant used in this study is Mil-
L23699, whose viscosity–pressure relationship at an
inlet temperature of 100�C is plotted in Fig. 2. The
black dots in the figure denote the measured viscosity
values at different pressures extracted from Fig. 2
of Bair et al. [35]. The solid line represents the
viscosity computed from equation (11) with the
required parameters regressed from the measured
data as �0 = 0.004 573 Pa s, B = 3.3054, �Vocc ¼ 0:6132,
K0 = 1.0747 GPa, and K 00 ¼ 10:076. The viscosity esti-
mates of equation (11) with these parameter values
are in good agreement with the measurement of Bair
et al. [35]. The density of this lubricant at the same
temperature and ambient pressure is �0 = 947.8 kg/
m3. The computational domain with the dimension
of �2:5amax � x � 1:5amax (amax is the maximum
Hertzian half-width of the contacts along the line of
action) is discretized into 512 elements with the grid
size �x ¼ 1:3 mm, which reasonably represents the
measured roughness resolution. The entire analysis
from the SAP to the tip is discretized into 1000 time
steps, resulting in the time resolution of 8� 10�5 s at
fm = 1250 Hz, 5� 10�5 s at fm = 2085 Hz and 4� 10�5 s at
fm = 2375 Hz. This represent a more refined time incre-
ment compared to those used in Larsson [29] and
Wang et al. [30]. To start the simulation at the SAP,
the Hertzian pressure is used as the initial guess and
iterated until the converged stationary EHL solution is
obtained. During the analysis from the SAP to the tip,
whenever the tooth force reaches zero (due to the
dynamic effect), a very small loading value of 5 N was
applied artificially, that is negligibly small compared to
typical tooth force levels at several kilo-Newtons.
Employing the experimentally determined damp-
ing ratio of � ¼ 0:01 (1 per cent) in equation (3) [7–9],
the dynamic model proposed above is used to predict
the steady-state dynamic response of the gear pair at
T1 = 250 Nm within the gear mesh (tooth passing) fre-
quency range of fm = 500–3500 Hz (fm ¼1
2�N1!1 where
744 S Li and A Kahraman
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
!1 is in rad/s). The gear load distribution model is
implemented to determine kðt Þ of this gear pair at
T1 = 250 Nm, as shown in Fig. 3(a). Although the
load distribution model is capable of including any
profile modification, the example gear pair used here
has perfect involute profile such that eðt Þ ¼ 0. The
Fourier spectrum of kðt Þ, shown in Fig. 3(b), indicates
that �k ¼ 3:26� 108 N/m with the first three harmonic
amplitudes of k1 = 3.46� 107, k2 = 2.16� 107, and
k3 = 9.583� 106 N/m. With this, the natural frequency
of the corresponding linear, undamped system is !n
¼
ffiffiffiffiffiffiffiffiffiffiffiffi�k=me
q¼ 20 375 rad/s (fn ¼ !n=ð2�Þ ¼ 3242 Hz).
Here, the lubricant stiffness is not included sinceits magnitude, which ranges 2.5� 1010 to5.1� 1011 N/m for the operating conditions consid-ered (estimated from the Hamrock–Dowson for-mula), is several orders larger than that of thegear mesh stiffness. The predicted root-mean-squared (r.m.s) value of sðt Þ versus fm plot ofFig. 3(c) reveals one primary resonance peak at
fm ¼ fn � 3240 Hz caused by the first harmonic ofthe excitation as well as two super-harmonic reso-nances at fm ¼
12 fn � 1620 Hz and fm ¼
13 fn � 1080
Hz caused by the second and third harmonicterms of the excitation. In the vicinity of these reso-nance peaks, two stable motions coexist: the lowerbranch linear motion without tooth separation andthe upper branch non-linear motion with toothseparation which exhibits a softening type non-linear behaviour. It is the initial condition that dic-tates which motion should be exhibited by the gearpair. This predicted response of Fig. 3(c) agrees wellwith the published measurement using the sameexample gear pair [7–9].
Three representative operating conditions marked
in Fig. 3(c) are considered here to demonstrate the
impact of the dynamic response on the EHL beha-
viour of the gear set. They include two off-resonance
conditions at fm = 1250 and 2085 Hz (marked as points
I and II in Fig. 3(c)) and a non-linear resonant condi-
tion at fm = 2375 Hz (marked as point III in Fig. 3(c)).
The variations of the contact geometry parameters
r1, r2, and req, and the speed parameters u1, u2, us, ur,
and SR of the example gear pair operating at
fm = 1250 Hz as the contact moves from the SAP at
�1 ¼ 14:5� to the tip at �1 ¼ 27:2� are shown in Fig. 4.
As seen, the variations of the radii of curvature, sur-
face velocities, and sliding velocity are evident, while
ur is constant in this case since the gear pair has unity
ratio (i.e. the driving and driven gears are identical).
In Fig. 5(a), the quasi-static tooth force Ws predicted
by the gear load distribution model at T1 = 250 Nm is
compared to its dynamic counterpart Wd when the
gears are operated under the condition of point I
(Fig. 3(c)). Here, Ws is shown to increase linearly
from the SAP to the LPSTC at �1 ¼ 20:0�, and then
almost double at the LPSTC where one of the two
loaded tooth pairs loses contact. After staying rela-
tively constant till the HPSTC at �1 ¼ 21:7�, Ws experi-
ences a sudden drop as the gear mesh transmits to
two loaded tooth pairs. As shown in Li and Kahraman
[1], these drastic changes in Ws impact the EHL beha-
viour significantly. The corresponding dynamic tooth
force Wd (in the same figure) has a substantially dif-
ferent shape from that of Ws. As this frequency is near
the resonance peak of fm ¼13fn caused by the third
harmonic of the excitation, nearly three cycles of fluc-
tuation are observed in Fig. 5(a) for the Wd curve
within a base pitch. In Figs 5(b) and (c), the Wd
curves for points II and III (Fig. 3(c)) are compared
to the same Ws. It is seen that both the amplitude and
shape of Wd change significantly with fm and show
little resemblance to those of Ws, suggesting that per-
forming any EHL analysis using Ws is inaccurate in
the case of high-speed gearing where dynamic effects
Table 1 Design parameters of the example unity-ratio
spur gear pair used in this study
Number of teeth 50Module (mm) 3.0Pressure angle (degrees) 20.0Outside diameter (mm) 156.0Pitch diameter (mm) 150.0Root diameter (mm) 140.0Centre distance (mm) 150.0Face width (mm) 20.0Backlash (mm) 0.14Polar mass moment of Inertia (kg �m2) 0.0078Equivalent mass, me (kg) 0.785
Fig. 2 Viscosity–pressure relationship of lubricantMil-L23699 at 100�C
Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 745
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
are prominent. It is also noted in Fig. 5(c) that the
dynamic tooth force becomes intermittent in the
non-linear region of operation. The Wd amplitude
for the upper branch motion of point III is amplified
to more than twice the maximum Ws value, while
sizable portions of the mesh cycle are spent under
zero contact load.
Starting with the case of perfectly smooth surfaces
(i.e. R1ðx, t Þ ¼ R2ðx, t Þ ¼ 0 in equation (8)), the mini-
mum film thickness hmin predictions of the EHL
simulations under Wdðt Þ and Wsðt Þ are compared in
Fig. 6 at the points I–III defined in Fig. 3(c). The hmin
curves that correspond to Ws (dashed lines) oscillate
between the LPSTC and somewhere after the HPSTC,
which is due to the strong squeezing effect at the
LPSTC induced by the sudden load jump-up and
the pumping effect right after the HPSTC, induced
by the sudden load jump-down as it was described
in Li and Kahraman [1]. Such film thickness spikes
were also reported in Larsson [29] and Wang et al.
[30] while the severity of these jumps varied, perhaps
due to the different operating conditions and coarser
time resolution used in those studies. Such quasi-
static behaviour, however, becomes irrelevant as the
Fig. 3 Gear mesh stiffness: time history kðt Þ (a) and the corresponding frequency spectrum ki ofthe example spur gear pair (b), and the r.m.s DTE amplitudes as a function of gear meshfrequency fm (c). Note: T1 = 250 Nm
Fig. 4 The variations of contact radii (a) and surfacevelocities (b) with �1 during a single toothengagement cycle of the example gear pair atthe mesh frequency of fmesh = 1250 Hz (point Iin Fig. 3)
746 S Li and A Kahraman
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
hmin behaviour corresponding to Wd (solid lines) is
drastically different from that for Ws at all three fm
values of operation. The wavy shape of this dynamic
condition hmin with �1 is observed to be dictated by
the shape of Wd that itself is defined by the harmonic
term(s) of the excitation effective at that fm value. For
instance, at point I where fm is near the third super-
harmonic resonance, three cycles of the fluctuation of
Wd (in Fig. 5(a)) result in a similar qualitative shape of
hmin as shown in Fig. 6(a). Similarly, the effect of the
second harmonic of the excitation is evident in Fig.
6(b) at point II. Finally, under the non-linear condi-
tion which exhibits the loss of contact (point III), it is
difficult to point to the formation of fluid film as large
portions of the mesh cycle are spent with the contact-
ing surfaces away from each other. The resultant
dynamic film thickness variation in Fig. 6(c) is
accordingly very different from that using Ws as the
normal contact load.
The lubrication behaviour summarized in the form
of hmin predicted by the EHL model in Fig. 6 is com-
plemented by Figs 7 to 9 that provide instantaneous
pressure and film thickness distributions at various �1
values along the mesh cycle. In Fig. 7, the hðx, t Þ and
pðx, t Þ are shown at ten �1 values (denoted by points A
to J) under the quasi-static loading of Ws and rota-
tional speed of 1500 r/min (fm = 1250 Hz). The effects
of the sudden change in Ws at the LPSTC and the
variations of the other contact parameters (radii and
sliding velocity) are evident in this figure. When the
dynamic load Wd is considered (at point I,
fm = 1250 Hz), completely different hðx, t Þ and pðx, t Þ
are obtained in Fig. 8 at the same �1 values as in Fig. 7.
At the SAP (mesh position A) where Wd peaks, typical
smooth surface EHL pressure and film thickness
Fig. 6 Comparisons of the predicted EHL minimumfilm thickness along the driving gear tooth sur-face with Wsðt Þ and Wdðt Þ as the tooth force:(a) fm = 1250 Hz (point I in Fig. 3(c)); (b)fm = 2085 Hz (point II in Fig. 3(c)); and (c)fmesh = 2375 Hz (point III in Fig. 3(c)). Note:T1 = 250 Nm
Fig. 5 Comparisons of the tooth force variations alongthe driving gear roll angle between the quasi-static loading condition and the dynamic load-ing condition: (a) fm = 1250 Hz (point I in Fig.3(c)); (b) fm = 2085 Hz (point II in Fig. 3(c));and (c) fmesh = 2375 Hz (point III in Fig. 3(c)).Note: T1 = 250 Nm
Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 747
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
distributions are observed. As Wd decreases with the
increasing �1, the hydrodynamic fluid film becomes
less pressurized and thicker. Beyond the mesh posi-
tion B, Wd starts to increase with the consequence of
wider fluid film and larger thickness, as shown for the
mesh position C. The resultant hðx, t Þ and pðx, t Þ,
shown in Fig. 8, point to the influence of Wd. A com-
parison between Figs 7 and 8 indicates that the hðx, t Þ
and pðx, t Þ solutions obtained using Ws or Wd are sub-
stantially different. It is also noted that the pressure
ripples at both the inlet and outlet zones introduced
by the sudden load change at the LPSTC (position G)
under the quasi-static condition are absent in Fig. 8
with Wd as the normal load. Next, the hðx, t Þ and
pðx, t Þ at ten different �1 values are shown in Fig. 9
for point III. It is evident that the EHL behaviour
under such non-linear dynamic condition has abso-
lutely no resemblance to the corresponding quasi-
static condition shown in Fig. 7. The fluid film is
formed between the mesh positions A and E to certain
extent during the first loaded segment of the contact
that dissolves completely during the period between
E and F where there is no tooth load. Afterwards, the
transient effort to form the fluid film is observed start-
ing from the mesh position F where the tooth contact
is reestablished. The hðx, t Þ and pðx, t Þ distributions
Fig. 7 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair ata series of mesh positions as defined in the top figure under the quasi-static loading condi-tion with a rotational speed of 1500 r/min and T1 = 250 Nm
748 S Li and A Kahraman
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
are highly transient for the rest of the load cycle (from
F to J).
At the end, an analysis using the ground tooth
surface profiles, shown in Fig. 10, is presented to
illustrate the EHL contacts of rough spur gear sur-
faces under the dynamic condition. These mea-
sured R1 and R2 profiles have the Rq values of
0.54 and 0.53 mm, respectively. Considering these
roughness profiles, the transient hðx, t Þ and pðx, t Þ
distributions of the example gear pair at the same
mesh positions and under the same dynamic con-
tact condition as in Fig. 8 are shown in Fig. 11.
Comparing Fig. 11 with Fig. 8, it can be seen that
the size of the contact zone of the dynamic rough
contact varies in the same way as that under the
dynamic smooth condition when the Wd fluctu-
ates. Due to the surface irregularities, however,
the local contact pressures in Fig. 11 can easily
exceed 1 GPa. At various instantaneous local con-
tact points, hðx, t Þ is zero, indicating actual asperity
contacts with the corresponding spikes displaying
in the pressure distributions. Meanwhile, the deep
roughness valleys introduce much larger local
film thickness values. Neither the pressure distri-
bution nor the film thickness distribution is
smooth and continuous. The influences of the vari-
able load Wd on hðx, t Þ and pðx, t Þ are also evident
in Fig. 11.
Fig. 8 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair ata series of mesh positions as defined in the top figure under the dynamic loading conditionwith fmesh = 1250 Hz and T1 = 250 Nm
Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 749
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
Fig. 9 Instantaneous p (solid line) and h (dashed line) distributions of the example spur gear pair ata series of mesh positions as defined in the top figure under the dynamic loading conditionwith fmesh = 2375 Hz and T1 = 250 Nm
Fig. 10 Measured tooth surface roughness profiles in the direction of relative sliding and rolling
750 S Li and A Kahraman
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
4 CONCLUSIONS
In this study, an EHL model of a gear pair was used in
conjunction with a dynamic model to predict the
lubrication behaviour under various dynamic speed
conditions. The non-linear time-varying dynamic
model of a spur gear pair was used to predict the
instantaneous tooth contact force under the dynamic
condition within both the linear and non-linear oper-
ating regimes. The predicted instantaneous tooth
force was fed into the gear EHL model to simulate
the lubrication behaviour of the spur gear contact
under the dynamic loading condition. The EHL
model included any asperity interaction activity as
well as the variations of radii of curvature, sliding,
and rolling velocities and measured roughness pro-
files as the contact moves along the gear tooth from
the SAP to the tip. The EHL results presented under
various dynamic loading conditions were shown to
differ from those under static tooth load conditions,
suggesting that the dynamic behaviour of the gear
pair must be included in describing the high-speed
gear tribology. The tooth separation that takes place
due to the backlash non-linearity was also shown to
impact the transient EHL behaviour in a unique
manner.
� Authors 2011
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Appendix
Notation
amax maximum Hertzian half-width along the
line of action
b half backlash
B Doolittle parameter
c viscous damping
e, ~e gear static transmission error under
unloaded and loaded conditions,
respectively
f flow coefficient
fm gear mesh frequency (Hz)
fn natural frequency (Hz)
g0 geometry gap before deformation
h film thickness
h0 reference film thickness
I1, I2 polar mass moments of inertia of gears 1
and 2, respectively
752 S Li and A Kahraman
Proc. IMechE Vol. 225 Part J: J. Engineering Tribology
k gear mesh stiffness�k, ka mean and alternating components of gear
mesh stiffness
K0 bulk modulus at p ¼ 0
K 00 derivative of bulk modulus with respect to
pressure at p ¼ 0
me equivalent mass
N1, N2 number of teeth of gears 1 and 2,
respectively
p pressure
rb1, rb2 base circle radii of gears 1 and 2, respectively
req equivalent radius of curvature,
req ¼ r1r2
�ðr1 þ r2Þ
r1, r2 contact radii of curvature of gears 1 and 2,
respectively
R1, R2 surface roughness profiles of gears 1 and 2,
respectively
s dynamic transmission error
SR slide-to-roll ratio, SR ¼ us=ur
t time
T1, T2 torques applied to gears 1 and 2,
respectively
u1, u2 surface velocities in the direction of rolling
of gears 1 and 2, respectively
ur rolling velocity, ur ¼12ðu1 þ u2Þ
us sliding velocity, us ¼ u1 � u2
V surface elastic deformation�V normalized volume�Vocc normalized occupied volume
Wd dynamic tooth force
W 0d dynamic tooth force per unit face width
Ws quasi-static tooth force
x coordinate along the rolling direction
� non-linear restoring function
� damping ratio
� lubricant viscosity
�0 lubricant viscosity at ambient pressure
�1, �2 roll angles of gears 1 and 2, respectively
#1, #2 alternating rotational displacements of
gears 1 and 2, respectively
1, 2 Poisson’s ratios of gear 1 and 2, respectively
� lubricant density
�0 lubricant density at ambient pressure
�0 reference shear stress of the lubricant
!n natural frequency (rad/s)
!1, !2 angular velocities of gear 1 and 2,
respectively
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