Journal of Applied Mathematics and Physics, 2013, 1, 93-97
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15014
Open Access JAMP
Approximation Solution for TEHL of
Bevel Gears in PSD
Yunhui Zhang, Zuoxin Li, Shuangshi Feng, Yongjiang Ma, Lei Tang
College of Mechanical Science and Engineering, Jilin University, Changchun, China
Received October 6, 2013; revised November 6, 2013; accepted November 15, 2013
Copyright © 2013 Yunhui Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
To study the lubrication of the contact zone between half-shaft gears and planet gears in the power-split device (PSD),
TEHL line contact of bevel gears in PSD is approximately computed based on a theory in which a bevel gear is equal-
ized to an equivalent spur gear. In the calculation, the housing is taken as the reference system and the influence of the
housing’s rotating on the lubrication is ignored. Film pressure, film thickness and temperature rise are analyzed under
maximum load condition. This research provides some approximate reference data for the design of lubrication and
cooling system of PSD.
Keywords: HEV; PSD; Equivalent Bevel Gears; TEHL
1. Introduction
The hybrid electric vehicle (HEV) is a cutting-edge tech-
nology in automobile field [1]. The power coupling de-
vice is the core component of the transmission system for
series-parallel HEV. Based on the working principle of
automobile differential, Ref [2] invented a power-split
device (PSD) used in HEV, as shown in Figure 1. Al-
though the kinetics principle of PSD is similar to the
automobile differential, the working time and the speed
difference between the left and right half-shaft gears in
PSD are far greater than that in the automobile differen-
tial [3]. This leads to a completely different lubrication
Engine
Shaft
Half-Shaft
Generator Half-Shaft Motor
Right half-shaft gear
Planet gear
Left half-shaft gear
Housing
A
B
Figure 1. Arrangement of differential-based PSD in HEV.
status. In this paper, the lubrication properties of PSD are
approximately calculated by applying thermal elastohy-
drodynamic lubrication (TEHL) at the contact zone of
planet gears and half-shaft gears. The research to TEHL
involves mathematic model, numerical method and lu-
brication parameters [4].
In this paper, the influence of the housing’s rotation on
the lubrication is ignored, and the housing is chosen as
the reference system. The mathematical model of TEHL
is established based on the non-Newtonian Ree-Ryring
model [5], and multi-grid method is used to get the com-
plete numerical solution for the mathematical model [6].
Planet gears and half-shaft gears are bevel gears in PSD.
And TEHL line contact of the gears is approximately
analyzed by using equivalent spur gears, which simpli-
fies TEHL of bevel gears in PSD greatly. Finally lubri-
cating properties of planet gears and half-shaft gears,
namely film pressure, film thickness and temperature rise,
are solved under maximum load condition.
2. Theory Model
2.1. Equivalent Model of Bevel Gear
Since TEHL of bevel gears is complicated [7], within the
allowed error, an equivalent spur gear model is intro-
duced to simplify the analysis of the lubrication status of
bevel gears, as shown in Figure 2. Based on the principle

Y. H. ZHANG ET AL. 94

b 
2v
d

1v
d

1m
d 
1
d 
2m
d
2
d
b
R

1

2


2b
M

Figure 2. Equivalent spur gears model of bevel gears.
of equivalent gears [7] and theory of lubrication of spur
gears [8], the linear load, the relative speed between tooth
surfaces and the equivalent radius of the contact area are
determined when equivalent gears mesh at the pitch point.
Thus, TEHL of bevel gears in PSD is easy to analyze.
1) Equivalent radius of contact surfaces
The meshing of the equivalent spur gear pair is shown
in Figure 3. The intersection point between N1N2 (line of
action) and O1O2 (line of centers), namely pitch point M,
is chosen as the research point for TEHL.
Considering the geometrical relationship of bevel gears
and the equivalent radius formulas of TEHL, the equiva-
lent radius at point M can be expressed as:
121 2
1212 2
sin
1
2cos cos
vv mm
vv mm
RRd d
RRR dd

1




(1)
2) Normal force at unit length
According to the force analysis of bevel gears, the
normal force at unit length of the contact line at point M
is expressed as:
2
cos
n
K
FZ
Wb

 (2)
where
K
is a coefficient related to equivalent gears,
F
represents peripheral force of bevel gears, and
Z
is
a coefficient expressing the contact ratio of equivalent
gears.
3) Equivalent speed at tooth contact point
Based on the geometrical relationship of bevel gears
and the equivalent speed equation of TEHL, the equiva-
lent speedat point M can be expressed as:
12 1122
12
2120 coscos
vvm m
e
uund nd
U


 
 


(3)
O
2
O
1
r
vb1
r
vb2
r
v2
r
v1
N
1
N
2
R
v1
R
v2
M
Figure 3. Meshing in an equivalent spur gear pair.
2.2. TEHL Model
According to TEHL line contact model of spur gears [8],
the mathematical model of bevel gears is built, which
includes Reynolds equation, film thickness equation, en-
ergy equation, thermal interface equations and load bal-
ance equation.
1) Reynolds equation
The generalized Reynolds equation, which allows the
viscosity and the density of the oil to vary along the film
thickness direction, can be expressed as:

312 e
e
h
p
hu
x
xx






 




 


(4)
where

12 ee ee
e


 

  





ee baeae
uu uu  






2
000
11
d, d
hh
ee
hh
 







d
300 0
1d11
d
hh
eeh
h
 
d





 


 

20
11 d
h
eh





00
sinh





 
 
 
Boundary conditions for Equation (4) are as follows:



 
in out
in out
0
0
px px
pxxx x





(5)
Open Access JAMP

Y. H. ZHANG ET AL. 95
2) Film thickness equation
The film thickness is composed of rigid displacement,
geometric film thickness and elastic deformation, whose
equation can be expressed as:
  
out
in
22
00
2ln d
2
x
x
x
hxhpxx xx
RE 
 
 
(6)
3) Viscosity equation
The equation about viscosity-pressure-temperature re-
lationship proposed by Roelands is chosen in this paper,
which can be expressed as:


0
0
9
00
0
exp9.67ln115.1 10
138
138
z
H
S
Pp
T
T
 












 





(7)
4) Energy equation
Ignoring influence of heat radiation, thermal conduc-
tivity and gravity, the energy equation can be expressed
as:
2
2
2
pTTTT pu
Cu qKu
xTx




 
 
 
 
  
 
 




(8)
where
0dqu
x


 
5) Thermal interface equation
To determine the temperature on contact surfaces, the
gear is considered as a half-space body with a moving
heat source. Therefore, the heat conduction equations of
gears are built, as shown in Equation (9).


0
111 1,0
0
222 2,h
d
,0
d
,
x
ax
x
bx
KTs
Tx T
cuKx s
KTs
Txh T
cuKx s


















(9)
where, is the temperature on the contact surface
between planet gear and oil film, is the tem-
perature on the contact surface between half-shaft gear
and oil film.

,0Tx


,Txh
The continuous condition of the heat flow of the con-
tact surface is given as:
1
00
2
0
a
b
a
hb
TT
KK
TT
KK














6) Load balance equation
The load equation is given as
out
in
d
x
x
Wpx (10)
3. Numerical Solutions
A complete numerical solution to TEHL line contact of a
spur bevel gear is obtained by combining multi-grid
method, Jacobi iteration method and scanning method [9].
The pressure loop and the temperature loop are included
in the numerical analysis of TEHL, which form a whole
loop, and the flowchart is shown in Figure 4. Jacobi it-
eration method is chosen for the pressure loop, scanning
method is chosen for the temperature loop and multi-grid
method is chosen for the whole loop. In order to simplify
the calculation, all the variables related are dimensionless,
so are the outputs of the calculation, except those of
temperature rise. They are transformed to the actual
value to express temperature rise evidently.
4. Results
As the housing is considered as the reference system and
the influence of the housing’s rotation on the lubrication
is ignored, the lubrication of meshing gears in PSD is
Calculate normal force at unit length,
equivalent speed and radius
Calculate the pressure and
film thickness
Results converge?
Calculate the temperature rise
Temperature rise
converge?
Load converges?
Pressure
loop

Temperature
loop

No
Yes
Yes
No
No
Yes
Output results
Revise pressure
Revise temperature
Revise pressure
and temperature
Input torque and rotational speed Input
torque and rotational speed of bevel
gears and lubrication parameters
Figure 4. Flow chart of TEHL of bevel gears.
Open Access JAMP

Y. H. ZHANG ET AL.
Open Access JAMP
96
TEHL of bevel gears, as shown in Figure 1. Point A is
the meshing point between planet gear and left half-shaft
gear, and point B is the meshing point between planet
gear and right half-shaft gear. The rotational speed and
torque of bevel gears and the housing under maximum
load condition are shown in Table 2.
simplified as a steady-state issue about a pair of bevel
gears. In such case, when THEL of gears in PSD is ana-
lyzed, the autorotation speed of planet gears and the rota-
tional speed relative to the housing of half-shaft gears are
adopted during calculating.
The parameters of bevel gears in PSD are shown in
Table 1. Figures 5-7 show the distributions of film pressure,
film thickness and temperature rise at point A and B Point A and B are chosen to study the steady-state
Table 1. Parameters of gears in PSD.
Modulus (mm) Pressure angle (˚) Number of teeth Density (kg/m3)
m = 5.4  = 22.5 z1 = 10 z2 = 14  = 7800
Equivalent elastic modulus (Pa) Coefficient of tooth width Tooth width at pitch circle (mm) Reference cone angle (˚)
E = 2.06e11 
R = 0.2846 b1 = 15.4 b2 = 15.4

1 = 35.53

2 = 54.47
Notice: Subscript 1 and 2 represent planet gear and half-shaft gear respectively.
Table 2. Parameters of working condition of PSD.
Output torque (Nm) Absolute rotational speed (r/min) Rotational speed relative to housing (r/min)
Planet gear - 1110.2 1110.2
Left half-shaft gear (A) −30 1816 793
Right half-shaft gear (B) −35 230 −793
Housing 65 1023 0
-1.5 -1 -0.500.5 11.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
X/dimensionless
P/dimensionless
A
B
Figure 5. Distributions of film pressure.
-1.5 -1 -0.5 00.5 11.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X/dimensionless
H/dimensi onless
A
B
Figure 6. Distributions of film thickness.

Y. H. ZHANG ET AL. 97
Figure 7. Distributions of temperature rise.
when PSD is under maximum load condition.
Figures 5-7 show that, under maximum load condition,
the pressure distribution of point A agrees with that of
point B, film thickness at point A is bigger than that at
point B evidently and temperature rise at point A is lower
than that at point B. This mainly results from the differ-
ence of output torques from left and right half-shaft. Ac-
cording to the indoor experiment [10], the reason for the
difference is frictions of two parts: one is that between
planet gears and their shafts, and the other is that be-
tween back cone of gears and the housing.
5. Conclusion
The lubrication properties of planet gears and half-shaft
gears are obtained by the approximate solution of TEHL
line contact of bevel gears in PSD. According to the re-
sults, lubrication properties at point A are different from
those at point B, which result from the frictions in PSD.
The research provides some approximate reference data
for the design of lubrication and cooling system of PSD.
However, as the solution of TEHL line contact of bevel
gears in PSD is approximate, the equivalent model needs
to be improved in the future.
6. Acknowledgements
Authors of the present paper gratefully acknowledge the
National Natural Science Foundation of China (NSFC no.
51075179) for supporting this research.
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