64 5. MATHEMATICAL MODELING AND TV ANALYSIS OFHYBRID ELECTRIC VEHICLES
e full hybrid vehicle has multiple power sources and is a complex electromechanical
coupling system. e establishment of the TV model is more complicated than the traditional
automobile, especially the establishment of the dynamic coupling device-composite planetary
TV model. To solve this problem, the dynamic equation of the compound planetary gear train
is deduced by Lagrangian equation, and the dynamic equations of the components of the hybrid
powertrain are listed to establish the TV model of hybrid powertrain, and the TV mode of the
powertrain are calculated.
5.1 DYNAMIC MODELING OF THE COMPOUND
PLANETARY GEAR SET
Figure 5.1 shows the dynamic model of the compound planetary gear set [49].
k
ra
k
ab
k
S2b
k
S1a
θ
r
θ
aci
θ
S1
θ
bci
θ
c
θ
S
2
a
r
b
c
Figure 5.1: Dynamic model of the compound planetary gear set.
In Fig. 5.1 k
S2b
is the meshing stiffness between big sun gear and long planet, k
ab
repre-
sents the meshing stiffness between short planet and long planet, k
S1a
is the meshing stiffness
between small sun gear and short planet, k
ar
is the meshing stiffness between ring and short
planet,
S1
is the angular displacement of small sun gear,
S2
is the angular displacement of big
sun gear,
r
is the angular displacement of ring gear,
c
is the angular displacement of carrier,
aci
and
bci
are the angular displacement of short and long planet, respectively.
5.1. DYNAMIC MODELING OF THE COMPOUND PLANETARY GEAR SET 65
e Lagrangian equation of the torsional dynamic system can be described by [72]:
L D
1
2
I
c
P
2
c
C
1
2
nI
a
P
aci
C
P
c
2
C
1
2
nI
a
P
bci
C
P
c
2
C
1
2
I
S1
P
2
S1
C
1
2
I
S2
P
2
S2
C
1
2
I
r
P
2
r
1
2
k
S1a
3
X
iD1
Œ
r
a
.
aci
C
c
/
r
ca
c
C r
S1
S1
2
1
2
k
S2b
3
X
iD1
Œ
r
b
.
bci
C
c
/
r
cb
c
C r
S2
S2
2
1
2
k
ar
3
X
iD1
Œ
r
a
.
aci
C
c
/
C r
cb
c
r
r
r
2
1
2
k
ab
3
X
iD1
Œ
r
a
aci
C r
b
bci
2
: (5.1)
e equations can be expressed by the generalized coordinates as follows:
d
dt
@L
@
P
q
!
@L
@
q
D 0; q D c; S1; S 2; r; ac1; ac2; ac3; bc1; bc2; bc3; (5.2)
where the derivate of a variable represents differentiation w.r.t. time. By combing Eqs. (5.1) and
(5.2), we can rewrite the coupled homogeneous ordinary differential equations as follows:
.
I
c
C nI
a
C nI
b
/
R
c
C
3
X
j D1
I
a
cai
C
3
X
j D1
I
b
R
cbi
k
S1a
r
S1
3
X
j D1
Œ
r
a
cai
C r
S1
S1
r
S1
c
k
S2b
r
S2
3
X
iD1
Œ
r
b
cbi
C r
S2
S2
r
S2
c
C k
ar
r
r
3
X
iD1
Œ
r
a
cai
C r
r
c
r
r
r
D 0 (5.3)
I
S1
R
S1
C k
S1a
r
S1
3
X
iD1
Œ
r
a
cai
r
S1
c
C r
S1
S1
D 0 (5.4)
I
S2
R
S2
C k
S2b
r
S2
3
X
iD1
Œ
r
b
cbi
r
S2
c
C r
S2
S2
D 0 (5.5)
I
r
R
r
k
ar
r
r
3
X
iD1
Œ
r
a
cai
C r
r
c
r
r
r
D 0 (5.6)
I
a
R
ai
C I
a
R
c
C k
S1a
r
a
Œ
r
a
cai
r
S1
c
C r
S1
S1
C k
ab
r
a
Œ
r
a
cai
C r
b
cbi
C k
ab
r
a
Œ
r
a
cai
C r
r
c
r
r
r
D 0; i D 1; 2; 3 (5.7)
66 5. MATHEMATICAL MODELING AND TV ANALYSIS OFHYBRID ELECTRIC VEHICLES
I
b
R
bi
C I
b
R
c
C k
S2b
r
b
Œ
r
b
cbi
r
S2
c
C r
S2
S2
C k
ab
r
b
Œ
r
a
cai
C r
b
cbi
D 0; i D 1; 2; 3: (5.8)
e equilibrium equation of the compound planetary gear set TV can be described by:
M
0
Rq C K
0
q D 0; (5.9)
where the displacement vector fqg, the mass matrice ŒM
0
and stiffness matrice ŒK
0
are given as
follows:
fqg D
c
S1
S2
r
ac1
ac2
ac3
bc1
bc2
bc3
T
(5.10)
M
0
D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
J
c
C 3J
a
C 3J
b
0 0 0 J
a1
J
a2
J
a3
J
b1
J
b2
J
b3
0 J
S1
0 0 0 0 0 0 0 0
0 0 J
S2
0 0 0 0 0 0 0
0 0 0 J
r
0 0 0 0 0 0
J
a1
0 0 0 J
a1
0 0 0 0 0
J
a2
0 0 0 0 J
a2
0 0 0 0
J
a3
0 0 0 0 0 J
a3
0 0 0
J
b1
0 0 0 0 0 0 J
b1
0 0
J
b2
0 0 0 0 0 0 0 J
b2
0
J
b3
0 0 0 0 0 0 0 0 J
b3
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(5.11)
K
0
D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
k
0
11
k
0
12
k
0
13
k
0
14
k
0
15
k
0
16
k
0
17
k
0
18
k
0
19
k
0
1;10
k
0
21
k
0
22
0 0 0 0 0 0 0 0
k
0
31
0 k
0
33
0 0 0 0 0 0 0
k
0
41
0 0 k
0
44
0 0 0 0 0 0
k
0
51
0 0 0 k
0
55
0 0 k
0
58
0 0
k
0
61
0 0 0 0 k
0
66
0 0 k
0
69
0
k
0
71
0 0 0 0 0 k
0
77
0 0 k
0
7;10
k
0
81
0 0 0 k
0
85
0 0 k
0
88
0 0
k
0
91
0 0 0 0 k
0
96
0 0 k
0
99
0
k
0
10;1
0 0 0 0 0 k
0
10;7
0 0 k
0
10;10
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(5.12)
5.1. DYNAMIC MODELING OF THE COMPOUND PLANETARY GEAR SET 67
where
k
0
11
D 3k
S1a
r
2
S1
C 3k
S2b
r
2
S2
C 3k
ar
r
2
r
(5.13)
k
0
12
D 3k
S1a
r
2
S1
(5.14)
k
0
13
D 3k
S2b
r
2
S2
(5.15)
k
0
14
D 3k
ar
r
2
r
(5.16)
k
0
15
D k
ar
r
r
r
a
k
S1a
r
S1
r
a
(5.17)
k
0
16
D k
ar
r
r
r
a
k
S1a
r
S1
r (5.18)
k
0
17
D k
ar
r
r
r
a
k
S1a
r
S1
r
a
(5.19)
k
0
18
D k
S2b
r
S2
r
b
(5.20)
k
0
19
D k
S2b
r
S2
r (5.21)
k
0
1;10
D k
S2b
r
S2
r
b
(5.22)
k
0
21
D 3k
S1a
r
2
S1
(5.23)
k
0
22
D 3k
S1a
r
2
S1
(5.24)
k
0
31
D 3k
S2b
r
2
S2
(5.25)
k
0
33
D 3k
S2b
r
2
S2
(5.26)
k
0
41
D 3k
S2b
r
2
S2
(5.27)
k
0
44
D 3k
ar
r
2
r
(5.28)
k
0
51
D k
ar
r
r
r
a
k
S1a
r
S1
r
a
(5.29)
k
0
55
D r
2
a
.k
S1a
C k
ab
C k
ar
(5.30)
k
0
58
D k
ab
r
a
r
b
(5.31)
k
0
61
D k
ar
r
r
r
a
k
S1a
r
S1
r
a
(5.32)
k
0
66
D r
2
a
.k
S1a
C k
ab
C k
ar
/ (5.33)
k
0
69
D k
ab
r
a
r
b
(5.34)
k
0
71
D k
ar
r
r
r
a
k
S1a
r
S1
r
a
(5.35)
k
0
77
D r
2
a
.k
S1a
C k
ab
C k
ar
/ (5.36)
k
0
7;10
D k
ab
r
a
r
b
(5.37)
k
0
81
D k
S2b
r
S2
r
b
(5.38)
k
0
85
D k
ab
r
a
r
b
(5.39)
k
0
58
D k
ab
r
a
r
b
(5.40)
k
0
88
D k
S2b
r
2
b
C k
ab
r
2
b
(5.41)
k
0
91
D k
S2b
r
S2
r
b
(5.42)
k
0
96
D k
ab
r
a
r
b
(5.43)
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