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

19

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%

21

Design Under Uncertainty

Analysis /SimulationInput Output

Uncertainty (Quantified)

Uncertainty (Calculated)

1. Quantification

Propagation

2. Propagation

Design

3. Design (RBDO)

22

Feasible Region

Increased Performance

x2

x1

f(x1,x2) contours

g1(x1,x2)=0

g2(x1,x2)=0

Design Under Uncertainty (RBDO)

Reliable Optimum

23

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

24

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%