1 Zissimos P. Mourelatos, Associate Prof. Daniel N. Wehrwein, Graduate Student Mechanical 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 Study Optimize 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. 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 Simulation Input Output Design Gear Ratio Optimization Simulink Simulation Input Output Design Transmission Shift Point Optimization Stage 1 Stage 2

15 Deterministic Optimization of Axle and Transmission Ratios ConstraintDescription G1=G1=(Quarter Mile Time) G2=G2= -(0 to 30 Time) G3=G3=(0 to 30 Time) G4=G4=(0 to 60 Time) G5=G5=(0 to 90 Time) G6=G6= -(Gradeability) G7=G7= -(0 to 30 Towing Time) G8=G8=(0 to 30 Towing Time) G9=G9=(0 to 60 Towing Time) G 10 = - (Towing Gradeability) G 11 =(Max Engine RPM) Trans ratios Axle ratio Fuel Economy

16 Deterministic Optimization of Axle and Transmission Ratios Initial PointDet. Opt Design Variables N n ngear Objective f(X) Constraints G 1 =(Quarter Mile Time) G 2 = -(0 to 30 Time) G 3 =(0 to 30 Time) G 4 =(0 to 60 Time) G 5 =(0 to 90 Time) G 6 = -(Gradeability) G 7 = -(0 to 30 Towing Time) G 8 =(0 to 30 Towing Time) G 9 =(0 to 60 Towing Time) G 10 = - (Towing Gradeability) G 11 =(Max Engine RPM)

17 Optimal Ratios vs. Production Feasible Ratios Optimal Point Production Feasible Point Design Variables N n ngear Objective f(X) Constraints G 1 =(Quarter Mile Time) G 2 = -(0 to 30 Time) G 3 =(0 to 30 Time) G 4 =(0 to 60 Time) G 5 =(0 to 90 Time) G 6 = -(Gradeability) G 7 = -(0 to 30 Towing Time) G 8 =(0 to 30 Towing Time) G 9 =(0 to 60 Towing Time) G 10 = - (Towing Gradeability) G 11 =(Max Engine RPM)

18 Deterministic Optimization of WOT Transmission Shift Points ConstraintDescription G1=G1=(Quarter Mile Time) G2=G2= -(0 to 30 Time) G3=G3=(0 to 60 Time) G4=G4= -(Gradeability) G5=G5=(0 to 30 Towing Time) G6=G6=(0 to 60 Towing Time) G7=G7= - (Towing Gradeability) G8=G8=(Max Engine RPM) shift points 0 to 90 time

19 Deterministic Optimization of Transmission Shift Points Initial PointOptimized Point Design Variables One Two WOT Shift Speed Two Three WOT Shift Speed Three Four WOT Shift Speed120 One Two Trailer Shift Two Three Trailer Shift80 Three Four Trailer Shift120 Objective f(X) Constraints G 1 = (Quarter Mile Time) G 2 = -(0 to 30 Time) G 3 = (0 to 60 Time) G 4 = -(Gradeability) G 5 = (0 to 30 Towing Time) G 6 = (0 to 60 Towing Time) G 7 = - (Towing Gradeability) G 8 = (Max Engine RPM)

20 Deterministic Optimization Results

21 Design Under Uncertainty Analysis / Simulation Input Output Uncertainty (Quantified) Uncertainty (Calculated) 1. Quantification Propagation 2. Propagation Design 3. Design (RBDO)

22 Feasible Region Increased Performance x2x2 x1x1 f(x 1,x 2 ) contours g 1 (x 1,x 2 )=0 g 2 (x 1,x 2 )=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 ConstraintDescription G1=G1=(Quarter Mile Time) G2=G2= -(0 to 30 Time) G3=G3=(0 to 30 Time) G4=G4=(0 to 60 Time) G5=G5=(0 to 90 Time) G6=G6= -(Gradeability) G7=G7= -(0 to 30 Towing Time) G8=G8=(0 to 30 Towing Time) G9=G9=(0 to 60 Towing Time) G 10 = - (Towing Gradeability) G 11 =(Max Engine RPM) Viscous friction coef. Engine torque multiplier Deterministic design variables Probabilistic design parameters

25 Probabilistic Optimization of Axle and Transmission Ratios Initial PointDet. OptRBDO Design Variables N n ngear Objective f(X) Constraints G 1 =(Quarter Mile Time) G 2 = -(0 to 30 Time) G 3 =(0 to 30 Time) G 4 =(0 to 60 Time) G 5 =(0 to 90 Time) G 6 = -(Gradeability) G 7 = -(0 to 30 Towing Time) G 8 =(0 to 30 Towing Time) G 9 =(0 to 60 Towing Time) G 10 = - (Towing Gradeability) G 11 =(Max Engine RPM)

26 Probabilistic Optimization of Transmission Shift Points ConstraintDescription G1=G1=(Quarter Mile Time) G2=G2= -(0 to 30 Time) G3=G3=(0 to 60 Time) G4=G4= -(Gradeability) G5=G5=(0 to 30 Towing Time) G6=G6=(0 to 60 Towing Time) G7=G7= - (Towing Gradeability) G8=G8=(Max Engine RPM)

27 Probabilistic Optimization of Transmission Shift Points Initial PointDet. OptRBDO Design Variables One Two WOT Shift Speed Two Three WOT Shift Speed Three Four WOT Shift Speed 120 One Two Trailer Shift Two Three Trailer Shift 80 THREE FOUR TRAILER SHIFT 120 Objective f(X) Constraints G 1 = (Quarter Mile Time) G 2 = -(0 to 30 Time) G 3 = (0 to 60 Time) G 4 = -(Gradeability) G 5 = (0 to 30 Towing Time) G 6 = (0 to 60 Towing Time) G 7 = - (Towing Gradeability) G 8 = (Max Engine RPM)

28 Probabilistic Optimization Results

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%