1
Zissimos P. Mourelatos, Associate Prof.
Daniel N. Wehrwein, Graduate StudentMechanical Engineering Department
Oakland University
Modeling and Optimization of Vehicle Drivetrain Dynamic Performance
Considering Uncertainty
2
Outline
Purpose of study
Dynamic Vehicle Model
Bond Graph Modeling
Optimization Process
Deterministic Optimization
Probabilistic Optimization (RBDO)
Summary and Conclusions
3
Purpose of StudyOptimize drivetrain performance under uncertainty
Transmission Gear Ratios
Final Drive Ratio (axle ratio)
Transmission Shift Points
Acceleration Performance
Fuel Economy
Trailer Towing Acceleration and Gradability
Design Variables
Performance Measures
4
Vehicle Model
5
Bond Graph Modeling Graphical method for system modeling Energy based and multidisciplinary Modular; components can be modeled separately and
assembled Bond graphs and block diagrams are interchangeable.
Simulink can be used
6
Engine Model Engine is modeled as a rigid body with friction Torque input is a look up table of engine speed and
throttle position and is based off a steady state torque map
Bond Graph Block Diagram
7
Torque Converter Model
A complete model would require complex CDF modeling Dynamometer data is used instead to model torque ratio and converter efficiency
8
Transmission Description GM 4L60E, four speed automatic Two planetary gear sets in series Clutch actuation determines gear state Ratio of sun and ring gear on each planetary gear set
determines transmission ratios
9
Transmission Model Planetary gear sets, clutches, and a controller to actuate each clutch are
modeled Each planetary gear set has a sun, a ring, and a planetary gear Each clutch is actuated through a controller using a shift table.
Gear State Based on Vehicle Speed and Throttle Position
0%
20%
40%
60%
80%
100%
0 20 40 60 80 100
Vehicle Speed (mph)%
Thro
ttle
Are
a .
1-2 Upshift2-1 Dow nshift
2-3 Upshift3-2 Dow nshift
3-4 Upshift4-3 Dow nshift
Planetary Gear Set Shift Table
10
Driveline and Vehicle Model Two inertia elements connected by a spring model each shaft Tire is assumed to be in constant contact with the road The tire is modeled as a lump inertia with a discrete spring between the tire and
the road The vehicle is modeled as a rigid body with standard rolling resistance and
aerodynamic drag
2WD Driveline
11
Vehicle Performance Targets
Common performance targets for full size trucks: Acceleration Performance Gradeability Trailer Towing Performance Fuel Economy Cost, weight, and packaging (not used)
12
Drivetrain Optimization Process
Optimization design variables : Transmission planetary ratios Axle ratio Transmission shift points
Ratios of integers
Depend on ratios
13
Drivetrain Optimization Process (cont.)
In order to avoid integer programming, the optimization is done in two stages:
Optimize axle and transmission ratios for maximum highway fuel economy
Optimize transmission shift points for minimum 0 to 90 acceleration time
14
Drivetrain Optimization Process (Cont.)
Simulink SimulationInput Output
Design
Gear Ratio Optimization
Simulink SimulationInput Output
Design
Transmission Shift Point Optimization
Stage 1
Stage 2
15
Deterministic Optimization of Axle and Transmission Ratios
ngearnN
Where
4.560.680.68
2.730.350.35
11~1jfor0G
s.t.
)]([min
u
l
ul
j
X
X
X
XXX
XX
fConstraint Description
G1= (Quarter Mile Time) - 16.10
G2= -(0 to 30 Time) + 2.4
G3= (0 to 30 Time) - 2.5
G4= (0 to 60 Time) - 7.89
G5= (0 to 90 Time) -18.41
G6= -(Gradeability) + 22.256
G7= -(0 to 30 Towing Time) + 5.29
G8= (0 to 30 Towing Time) + 5.39
G9= (0 to 60 Towing Time) - 17.01
G10= - (Towing Gradeability) + 9.57
G11= (Max Engine RPM) - 6000
Trans ratios
Axle ratio
Fuel Economy
16
Deterministic Optimization of Axle and Transmission Ratios
Initial Point Det. Opt
Design Variables
N 0.4857 0.3645
n 0.4359 0.5925
ngear 3.7273 2.9956
Objective
f(X) 22.1865 25.759
Constraints
G1=(Quarter Mile Time) - 16.10 -0.05 -0.4704
G2= -(0 to 30 Time) + 2.4 -0.05 -0.06
G3=(0 to 30 Time) - 2.5 -0.05 -0.0379
G4=(0 to 60 Time) - 7.89 -0.05 -0.2
G5=(0 to 90 Time) -18.41 -0.05 -0.695
G6= -(Gradeability) + 22.256 -0.05 -0.02
G7= -(0 to 30 Towing Time) + 5.29 -0.05 -0.17
G8=(0 to 30 Towing Time) + 5.39 -0.05 -0.01
G9=(0 to 60 Towing Time) - 17.01 -0.05 -0.1327
G10= - (Towing Gradeability) + 9.57 -0.05 -0.374
G11=(Max Engine RPM) - 6000 -876 -942
17
Optimal Ratios vs. Production Feasible Ratios
Optimal PointProduction Feasible
Point
Design Variables
N 0.3645 0.3636
n 0.5925 0.5970
ngear 2.9956 3.0000
Objective
f(X) 25.7593 25.7329
Constraints
G1=(Quarter Mile Time) - 16.10 -0.4704 -0.4782
G2= -(0 to 30 Time) + 2.4 -0.06 -0.0576
G3=(0 to 30 Time) - 2.5 -0.0379 -0.0424
G4=(0 to 60 Time) - 7.89 -0.2 -0.212
G5=(0 to 90 Time) -18.41 -0.695 -0.7119
G6= -(Gradeability) + 22.256 -0.02 -0.03
G7= -(0 to 30 Towing Time) + 5.29 -0.17 -0.1593
G8=(0 to 30 Towing Time) + 5.39 -0.01 -0.015
G9=(0 to 60 Towing Time) - 17.01 -0.1327 -0.1583
G10= - (Towing Gradeability) + 9.57 -0.374 -0.8
G11=(Max Engine RPM) - 6000 -876 -881
18
Deterministic Optimization of WOT Transmission Shift Points
34towing]23towing12towing....
34shift23shift[12shift=
Where
120] 115 70 120 115 [70 =u
20] 10 5 20 10 [5 =l
ul
8~1=jfor0j
G
s.t.
)(min
X
X
X
XXX
XX
fConstraint Description
G1= (Quarter Mile Time) - 16.10
G2= -(0 to 30 Time) + 2.4
G3= (0 to 60 Time) - 7.89
G4= -(Gradeability) + 22.256
G5= (0 to 30 Towing Time) + 5.39
G6= (0 to 60 Towing Time) - 17.01
G7= - (Towing Gradeability) + 9.57
G8= (Max Engine RPM) - 6000
shift points
0 to 90 time
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Deterministic Optimization of Transmission Shift Points
Initial Point Optimized Point
Design Variables
One Two WOT Shift Speed 38.5 47.8109
Two Three WOT Shift Speed 72.5 81.3541
Three Four WOT Shift Speed 120 120
One Two Trailer Shift 40 49.84
Two Three Trailer Shift 80 80
Three Four Trailer Shift 120 120
Objective
f(X) 18.155 17.71
Constraints
G1=(Quarter Mile Time) - 16.10 -0.05 -0.097
G2=-(0 to 30 Time) + 2.4 -0.05 -0.05
G3=(0 to 60 Time) - 7.89 -0.05 -0.42
G4=-(Gradeability) + 22.256 -0.05 -0.05
G5=(0 to 30 Towing Time) + 5.39 -0.05 -0.05
G6=(0 to 60 Towing Time) - 17.01 -0.05 -0.05
G7=- (Towing Gradeability) + 9.57 -0.05 -0.01758
G8=(Max Engine RPM) - 6000 -876 -123
20
Deterministic Optimization Results
Performance Measures for Initial and Optimized Driveline Parameters
7.84
18.50
22.19
7.47
17.71
25.76
0
5
10
15
20
25
300
-60
tim
e
0-9
0 t
ime
MP
G
Tim
e (s
) an
d M
PG
Initial Point OptimizedDirection of Increasing
Performance
Direction of Increasing
Performance +16%
+4.5%
+4.7%
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Design Under Uncertainty
Analysis /SimulationInput Output
Uncertainty (Quantified)
Uncertainty (Calculated)
1. Quantification
Propagation
2. Propagation
Design
3. Design (RBDO)
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Feasible Region
Increased Performance
x2
x1
f(x1,x2) contours
g1(x1,x2)=0
g2(x1,x2)=0
Design Under Uncertainty (RBDO)
Reliable Optimum
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Viscous friction at the transmission, ring gear, and pinion gear
The engine output torque
Uncertainty in Our Model
Gear ratios are ratios of integers.
Transmission shift points are not sensitive to small errors in vehicle speed and throttle position.
Deterministic
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Probabilistic Optimization of Axle and Transmission Ratios
]05.0008.0013.002.0[
]0.108.013.02.0[
11~1jfor3j
β
11~1jR0)),(j
P(G
)],([min
j
P
P
d
d
ddd
Pd
Pdd
4.560.680.68u
2.730.350.35l
ul
s.t.
f
][
ngearnN
Where
ETMRRRtranspinring
P
d
Constraint Description
G1= (Quarter Mile Time) - 16.10
G2= -(0 to 30 Time) + 2.4
G3= (0 to 30 Time) - 2.5
G4= (0 to 60 Time) - 7.89
G5= (0 to 90 Time) -18.41
G6= -(Gradeability) + 22.256
G7= -(0 to 30 Towing Time) + 5.29
G8= (0 to 30 Towing Time) + 5.39
G9= (0 to 60 Towing Time) - 17.01
G10= - (Towing Gradeability) + 9.57
G11= (Max Engine RPM) - 6000
Viscous friction coef.
Engine torque multiplier
Deterministic design variables
Probabilistic design parameters
25
Probabilistic Optimization of Axle and Transmission Ratios
Initial Point Det. Opt RBDO
Design Variables
N 0.4857 0.3645 0.395
n 0.4359 0.5925 0.621
ngear 3.7273 2.9956 3.274
Objective
f(X) 22.1865 25.759 24.6144
Constraints
G1=(Quarter Mile Time) - 16.10 -0.05 -0.4704 -0.0972
G2= -(0 to 30 Time) + 2.4 -0.05 -0.06 -0.0513
G3=(0 to 30 Time) - 2.5 -0.05 -0.0379 -0.0487
G4=(0 to 60 Time) - 7.89 -0.05 -0.2 -0.1451
G5=(0 to 90 Time) -18.41 -0.05 -0.695 -0.0324
G6= -(Gradeability) + 22.256 -0.05 -0.02 -1.494
G7= -(0 to 30 Towing Time) + 5.29 -0.05 -0.17 -0.0651
G8=(0 to 30 Towing Time) + 5.39 -0.05 -0.01 -0.0349
G9=(0 to 60 Towing Time) - 17.01 -0.05 -0.1327 -0.1431
G10= - (Towing Gradeability) + 9.57 -0.05 -0.374 -0.25
G11=(Max Engine RPM) - 6000 -876 -942 -1026
26
Probabilistic Optimization of Transmission Shift Points
]05.0008.0013.002.0[
]0.108.013.02.0[
120] 115 70 120 115 [70 =u
20] 10 5 20 10 [5 =l
8~13
8~1)0),((
..
),(min
P
P
d
d
ddd
Pd
Pdd
ul
jforj
jRj
GP
ts
f
j
][
]342312....
342312[
ETMRRR
towingtowingtowing
shiftshiftshift
Where
transpinring
P
d
Constraint Description
G1= (Quarter Mile Time) - 16.10
G2= -(0 to 30 Time) + 2.4
G3= (0 to 60 Time) - 7.89
G4= -(Gradeability) + 22.256
G5= (0 to 30 Towing Time) + 5.39
G6= (0 to 60 Towing Time) - 17.01
G7= - (Towing Gradeability) + 9.57
G8= (Max Engine RPM) - 6000
27
Probabilistic Optimization of Transmission Shift Points
Initial Point Det. Opt RBDO
Design Variables
One Two WOT Shift Speed 38.5 47.8109 45.42
Two Three WOT Shift Speed 72.5 81.3541 87.7814
Three Four WOT Shift Speed 120 120 120
One Two Trailer Shift 40 49.84 46.1091
Two Three Trailer Shift 80 80 80
THREE FOUR TRAILER SHIFT 120 120 120
Objective
f(X) 18.155 17.71 17.78
Constraints
G1=(Quarter Mile Time) - 16.10 -0.05 -0.097 -0.0467
G2=-(0 to 30 Time) + 2.4 -0.05 -0.05 -0.05
G3=(0 to 60 Time) - 7.89 -0.05 -0.42 -0.396
G4=-(Gradeability) + 22.256 -0.05 -0.05 -0.05
G5=(0 to 30 Towing Time) + 5.39 -0.05 -0.05 -0.05
G6=(0 to 60 Towing Time) - 17.01 -0.05 -0.05 -0.05
G7=- (Towing Gradeability) + 9.57 -0.05 -0.0176 -0.0107
G8=(Max Engine RPM) - 6000 -876 -52 -49
28
Probabilistic Optimization Results
Performance Measures for Initial and Optimized Driveline Parameters
7.84
18.50
22.19
7.47
17.71
7.49
17.78
25.7624.61
0
5
10
15
20
25
300
-60
tim
e
0-9
0 t
ime
MP
G
Tim
e (s
) an
d M
PG
Initial Point Optimized RBDO Direction of Increasing
PerformanceDirection of Increasing
Performance+16%
+11%
+4.5% +3.9%
+4.7% +4.5%
29
Summary & Conclusions
A vehicle drivetrain dynamic model is developed using bond graphs.
Transmission ratios, axle ratio, and WOT shift points were optimized using a two-step optimization process.
Both deterministic and probabilistic optimization was performed.
Highway fuel economy was improved by 11%
0 to 90 time was improved by 3.9%
0 to 60 time was improved by 4.5%
