Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears S 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-speed spur gear contacts is investigated under dynamic conditions. A non-linear time-varying vibratory model of spur gear pairs is introduced to predict the instantaneous tooth forces under dynamic conditions 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 excitations and 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 dynamic loading conditions, considering the variations of the basic contact parameters, such as radii of curvature, sliding and rolling velocities, and measured roughness profiles, as the contact moves along the tooth from the start of active profile to the tip. The EHL predictions under dynamic loading conditions are compared to those assuming quasi-static contact loads for gear sets having smooth and rough surfaces to demonstrate the important influence of dynamic loading on gear lubrication. The unique, transient EHL behaviour under the non-linear (intermittent contact 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
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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.
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
Influence of dynamic behaviour on elastohydrodynamic lubrication of spur gears 753
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