68 5. MATHEMATICAL MODELING AND TV ANALYSIS OFHYBRID ELECTRIC VEHICLES
k
0
99
D k
S2b
r
2
b
C k
ab
r
2
b
(5.44)
k
0
10;1
D k
S2b
r
S2
r
b
(5.45)
k
0
10;7
D k
ab
r
a
r
b
(5.46)
k
0
10;10
D k
S2b
r
2
b
C k
ab
r
2
b
(5.47)
J
c
D J
0
c
C n.M
a
r
2
ca
C M
b
r
2
cb
/ (5.48)
r
ca
D r
S1
C r
a
(5.49)
r
cb
D r
S2
C r
b
(5.50)
where the moment of inertia of the carrier J
c
is defined as
J
c
D J
0
c
C n.M
a
r
2
ca
C M
b
r
2
cb
/; r
ca
D r
S1
C r
a
r
cb
D r
S2
C r
b
; (5.51)
where J
0
c
is the carrier moment of inertia, n is total number of planet gears a-b in the compound
planetary gear set, M
a
and M
b
are masses of planets a and b, r
ca
and r
cb
are radiuses of revolution
of the short and long planet gears. J
S1
is the moment of inertia of small sun gear, J
S2
is the
moment of inertia of big sun gear, J
a1
, J
a2
, and J
a3
is the moment of inertia for short planet
gears, and J
a1
D J
a2
D J
a3
J
b1
, J
b2
, and J
b3
is the moment of inertia for long planetary gears,
and J
b1
D J
b2
D J
b3
J
r
is the moment of inertia for ring gear.
5.2 THE TORSIONAL DYNAMIC MODEL OF THE
POWER-SPLIT HYBRID SYSTEM
e TV model for the power-split hybrid driveline system is presented in Fig. 5.2.
And the engine dynamics equation can be described by
J
e
R
e
C k
tc
.
e
c
/
D 0; (5.52)
where J
e
is the moment of inertia of the engine,
e
is the angular displacement of engine, and
c
is the angular displacement of planetary carrier. k
tc
D
k
t
k
c
k
t
C k
c
, k
t
, and k
c
are the CPGS of the
torsional damper and the planetary carrier. e dynamics equation of the TV can be described
by
J
m
R
m
C k
rm
r
m1
.
m
r
m1
r
r
r2
/ C k
md
r
r2
.
m
r
m2
d
r
d
/ D 0; (5.53)
where J
m
is the moment of inertia of the reducer, k
rm
is meshing stiffness between the ring and
the reducer, k
md
is meshing stiffness between the reducer and the differential. r
m1
and r
m2
are
base radiuses of reducer gears, r
r2
is the base radius of the outer ring, and
m
and
r
are angular
displacements of the reducer and ring, respectively.
e equilibrium equation of TV for the differential can be derived as follows:
J
d
R
d
C k
md
r
d
.
d
r
d
m
r
m2
/ C k
lh
.
d
lw
/ C k
rh
.
d
rw
/ D 0; (5.54)
5.2. THE TORSIONAL DYNAMIC MODEL OF THE POWER-SPLIT HYBRID SYSTEM 69
k
md
k
r
h
k
lh
k
w
k
w
k
rm
k
tc
θ
S2
J
S2
T
S2
θ
e
J
e
T
e
J
m
θ
m
J
v
θ
v
J
v
θ
v
J
lw
θ
lw
J
rw
θ
rw
J
d
θ
d
θ
S1
J
S1
T
S1
ICE
DF
Compound Planetary Gear Set
Figure 5.2: TV model of hybrid driveline [49].
where J
d
is moment of inertia of the differential, k
lh
and k
rh
are TS of the left and right half
shafts,
d
,
lw
, and
rw
are angular displacements of the differential, left and right wheels, re-
spectively.
e equilibrium equation of TV for left wheel may be given as follows:
J
lw
R
lw
C k
lh
.
lw
d
/ C k
w
.
lw
v
/ D 0; (5.55)
where J
lw
is the moment of inertia of the left wheel, k
w
is TS of the wheel,
v
is angular dis-
placement of the vehicle.
e equilibrium equation of TV for the right wheel for is written as follows:
J
rw
R
rw
C k
rh
.
rw
d
/ C k
w
.
rw
v
/ D 0; (5.56)
where J
rw
is the moment of inertia of the right wheel.
e equilibrium equation of TV for the vehicle can be written as
J
v
R
v
C k
w
.
v
lw
/ C k
w
.
v
rw
/ D 0; (5.57)
where J
v
D m
v
r
2
w
, m
v
is the mass of the vehicle, r
w
is the radius of the wheel.
By combining the equations for the compound planetary gear set in Eq. (5.9) with the
equations of motion for the driveline components in Eqs. (5.52)–(5.57), the equations of motion
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