Vehicle Suspension Optimization. J. Michael Gray Jacob Wronski Friday May 9 th, 2003

Transcription

1 Vehicle Suspension Optimization J. Michael Gray Jacob Wronski Friday May 9 th, 2003

2 Presentation Outline Project introduction Simulation model description Trade space exploration Single objective optimization Sensitivity analysis Multi-objective optimization Summary Future work

3 Project Introduction Ford F-350 rear suspension system Goal to maximize passenger comfort Objectives were vertical acceleration and settling time of the passenger cabin Modeled as 6 modules encompassing a range of disciplines

4 Simulation Modules m c xc Dynamics module fourth order system spring, mass, damper system road disturbance (F t ) modeled as a step input K s K t m s b s C s xs Leaf spring stiffness module modeled as a simply supported steel beam F t linear, obeys Hooke s Law F s L s /2 L s

5 Simulation Modules Axle stress module axle modeled as a simply supported beam passenger cabin is modeled as two point loads on the axle bending, shear and torsion stress are calculated Passenger Cabin A L a F a D a B F a L a T a L C

6 Suspension Modules Suspension mass module mass obtained from a proprietary CAD model composed of the leaf spring, shock absorber and hollow shaft Cost module priced to compare with suspension parts available on the market Gas mileage module heavily depends on mass of suspension

7 Numerical Simulation

8 Tradespace Exploration & Scaling Explored design space with L 18 orthogonal array Only one point was feasible Scaled by computing Hessian = = s a a a s s s o C L d t b h L X Before Scaling = = s a a a s s s os C L d t b h L X After Scaling

9 Suspension Optimization Gradient-based optimization SQP gradient search method used on scaled and un-scaled design vectors Design vector before and after optimization Run Description a max (m/s 2 ) Elapsed Time (s) Unscaled Scaled % decrease in a max t a and d a limited by bounds L a remains unchanged Scaling L s h s b s t a d a L a Cs J(x)=a max Before After

10 Suspension Optimization SQP Results Acceleration (m/s 2 ) = Before Optimization = After Optimization Time (s)

11 Suspension Optimization GA and SA Used Provided similar results Great disparity in run times ( times longer!) Run Type L s h s b s t a d a L a Cs J(x)=a max SQP SA GA Run Type Ω max (MPa) GM (mpg) S t (s) ω ratio C ($) xc max (m) Elapsed Time (s) SQP SA GA

12 Sensitivity Analysis Sensitivity of objective function to design variables Quarter Car Suspension Normalized Sensitivities Damping Coefficient Design Variable Width of Leaf Spring Distance To "Attach Point" Outer Diameter of Axle Tube Thickness of Axle Tube Thickness of Leaf Spring Length of Leaf Spring

13 Sensitivity Analysis Sensitivity of design variables with respect to parameters looked at: (i) material density, (ii) mass of car, (iii) tire stiffness Design Variables Parameter L s / p b s / p h s / p t a / p d a / p L a / p C s / p ρ (Material Density) -6.20E E E E E m c (Mass of Car) 2.00E E E-04 K t (Tire Stiffness) 1.50E E

14 Sensitivity Analysis Sensitivity of X* with respect to parameters 3 active constraints at X* Changing max S t from 2.5 to 5.0 changes J(X*) from m/s 2 to 6.87 m/s 2 Constraint λ GM (gas mileage) S t (settling time) ω ratio (natural frequency ratio) 1.551

15 Multi-Objective Optimization Passenger comfort also depends on settling time, S t min [ J ( x) ] i i where J 0 < w w < 1 w i+ 1 i i ( x) = = 0.1 (1 w )*( a i max ) + w i *( S t )

16 Multi-Objective Optimization Settling Time (s) Single Objecive Optimum Pareto Front Max Acceleration (m/s) Initial Design Vector

17 Multi-Objective Optimization Acceleration (m/s 2 ) = Min [Max Acceleration] = Min[Settling Time] Time (s)

18 Multi-Objective Optimization Vertical Displacement (m) = Min [Max Acceleration] = Min[Settling Time] Time (s)

19 Summary Simulation models representative but simple Orthogonal array found a good starting point SQP performed much better than both heuristic optimization algorithms used a max most sensitive to the damping coefficient Constraints such as S t, bounds, very important S t and a max are competing objectives both related to passenger comfort

20 Future Work Increase the fidelity and accuracy of the modules Look at multi-objective problems across different disciplines (ex. cost or weight vs. comfort) Integration of simulation within the DOME framework

21 References [1 ] Hyniova, Katerina, A. Stribrsky, and J. Honcu, Fuzzy Control of Mechanical Vibrating Systems, Proceedings of International Carpathian Control Conference, [2] Norton, Robert L., Machine Design: An Integrated Approach, New Jersey, Prentice-Hall, [3] Wong, Jo Yung, Theory of Ground Vehicles, New York, John Wiley, 2001.

22 Questions??????

23 Appendix-Orthogonal Array C s La d a t a h s h s L s Exp. No. Factor Levels Factors

24 Appendix-Orthogonal Array Results Contraints Exp. No. a max Ω max <= 2x10 8 x max <=.10 S t <= >= ω ratio >= 15 C <= 2500 GM >=

25 Appendix-SA Start Parameters # of Convergence Checks Rate of parameter annealing Rate of cost annealing Rate of parameter quenching Rate of cost quenching Run # # of Designs ε SA E SA E SA E SA E

26 Appendix-SA Run Results Run # Ls hs bs ta Da La Cs J(x) SA SA SA SA Gradient GA

27 Appendix-SA Run Results Run # Max Stress (Mpa) Mileage (mpg) Settling Time (seconds) Wn Ratio Cost (Dollars) Max Displacement (meters) Elapsed Time (seconds) SA SA SA SA Gradient GA

28 Appendix-SA History

29 Appendix-GA Start Parameters GA Algorithm Parameters Values Subpopulation Size 20 Number of Populations 1 Generations 50 Gene Bits 32 Crossover 1 Rate of Mutation.01 Rate of Migration.5 Interval of MIgration 5 Elite Size 1 Relative Tournament Size.5

30 Appendix-GA History

31 Appendix-Multiobjective Results Weighting Factor MaxAxcel Settling Time Ls hs bs ta da La Cs