70 5. MATHEMATICAL MODELING AND TV ANALYSIS OFHYBRID ELECTRIC VEHICLES
for the power-split hybrid system can be written in matrix form as
M Rx C Kx D 0; (5.58)
where M and K are mass and stiffness matrices of 16 order, and fxg is 16 order vector of gen-
eralized displacement, and presented for details as follows:
X D Œ
e
c
S1
S2
r
a1
a2
a3
b1
b2
b3
m
d
lw
rw
v
T
:
(5.59)
5.3 NUMERICAL ANALYSIS OF NATURAL FREQUENCIES
AND MODES
A harmonic solution to Eq. (5.58) is supposed to be the following form as:
f
x
g
D sin !t
f
u
g
;
f
Rx
g
D !
2
sin !t
f
u
g
: (5.60)
By replacing Eq. (5.60) into Eq. (5.58), the following standard eigenvalue equation can be ob-
tained:
Œ
K
i
f
u
g
r
D !
2
r
Œ
M
i
f
u
g
r
; r D 1; 2 : : : N: (5.61)
With Eq. (5.61), numerical simulation results for natural frequencies and eigenvectors of
the power-split hybrid driveline system are obtained.
e frequencies corresponding to the pure electric drive mode and the hybrid drive mode
are presented in Tables 5.1 and 5.2.
Table 5.1: Natural frequencies of the hybrid system in the pure electric driving condition
Mode order 1
Fr
equency (Hz) 5.37 26.12 26.64 410.3 1584.0 2890.0
Mode order 7 8 9 10 11 12
Frequency (Hz) 4264.0 10023.6 10023.6 10168.7 16461.60 16461.60
Mode order 13
Frequency (Hz) 16566.20
2 3 4 5 6
Figures 5.3–5.6 describe the hybrid driveline mode shapes in pure electric driv-
ing mode, and Figs. 6.7–6.11 depict the mode shapes in hybrid driving mode. Where
e; S1; S2; a1; a2; a3; b1; b2; b3; r; red; diff ; lw; rw, and v denote the engine, sun gear 1, sun gear
2, three short planets, three long planets, ring, reducer, differential, left wheel, right wheel and
vehicle, respectively [36].
Figure 5.3 shows that the first-order eigen mode is the rigid mode, the second order is
related to the TV of driving wheels with respect to the half shafts, the third order is relevant
5.3. NUMERICAL ANALYSIS OF NATURAL FREQUENCIES AND MODES 71
Table 5.2: Natural frequencies of the hybrid system in the hybrid driving condition
Mode order 1 2 3 4 5 6
Fr
equency (Hz) 5.6 17.3 26.1 27.1 836.1 2076.0
Mode order 7 8 9 10 11 12
Frequency (Hz) 3722.0 4441.0 10023.6 10023.6 10508.6 16461.5
Mode order 13 14
Frequency (Hz) 16461.5 16646.5
f1 = 5.38
f2 = 26.1
f3 = 26.6
1.0
0.5
0.0
-0.5
-1.0
?2 ?1 ? ??? ???? ?? ?? ??1 ?2 ?3 ?1 ?2 ?3
Components of Vibration
Relative Amplitude of Eigenvectors
Figure 5.3: e first, second, and third eigen modes of TVs of the drivetrain in the pure electric
driving condition.
to the TV of the wheels in opposite directions, and the fourth mode represents the TV of the
wheels in the same direction. As shown in Fig. 5.4, the fifth-order refers to the sun gears, the
sixth order is the vibration of the differential, the seventh order is a coupled TV of the ring, the
reducer gear and the differential, and the vibration of the reducer gear plays a leading role in the
eighth order mode. As shown in Figs. 5.5 and 5.6, the ninth and tenth orders are corresponding
to the TVs of the long planets, the eleventh order reflects the TV of the long and short planets
72 5. MATHEMATICAL MODELING AND TV ANALYSIS OFHYBRID ELECTRIC VEHICLES
f4 = 410.3
f5 = 1584
f6 = 2890
1.0
0.5
0.0
-0.5
-1.0
Components of Vibration
Relative Amplitude of Eigenvectors
?2 ?1 ? ??? ???? ?? ?? ??1 ?2 ?3 ?1 ?2 ?3
Figure 5.4: e fourth through sixth eigen modes of TVs of the drivetrain in pure electric driving
condition.
5.3. NUMERICAL ANALYSIS OF NATURAL FREQUENCIES AND MODES 73
f7 = 4264
f8 = 10023.6
f9 = 10023.6
1.0
0.5
0.0
-0.5
-1.0
Co
mponents of Vibration
Relative Amplitude of Eigenvectors
?2 ?1 ? ??? ???? ?? ?? ??1 ?2 ?3 ?1 ?2 ?3
Figure 5.5: e seventh through ninth eigen modes of TVs of the drivetrain in the pure electric
driving condition.
74 5. MATHEMATICAL MODELING AND TV ANALYSIS OFHYBRID ELECTRIC VEHICLES
f10 = 10168.7
f11 = 16461.5
f12 = 16461.5
f13 = 16566.2
1.0
0.5
0.0
-0.5
-1.0
Components of Vibration
Relative Amplitude of Eigenvectors
?2 ?1 ? ??? ???? ?? ?? ??1 ?2 ?3 ?1 ?2 ?3
Figure 5.6: e tenth through thirteenth eigen modes of TVs of drivetrain in the pure electric
driving condition.
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