Optimization of Vehicle Powertrain Mounting System Based on Generalized Inverse Cascade Method under Uncertainty
Abstract
:1. Introduction
2. Multi-Objective Two-Level Model Construction
3. Introduction to the Generalized Inverse Cascade Method
Algorithm 1: generalized inverse cascade method |
Input: initial interval of design variables xi,1: =(ai,1, bi,1), mapping relationship fn between design variables x and design objectives Y, confidence threshold H0 Ouput: optimal interval of design variables (ai,3, bi,3) 1. Adjust the design variable feasible domain xi,2: =(ai,2, bi,2). 2. Calculate the optimal solution xi,opt for the design variables according to the intelligent optimization algorithm. 3. (ai,3, bi,3): =(xi,opt − ε, xi,opt + ε)//choose the optimal solution neighbourhood (ai,3, bi,3): =(xi,opt − ε, xi,opt + ε) 4. While Hc > H0, do 5. ai,3: =ai,3 × (90% to 95%)//Reduce the left-hand side of all design variables by 5% to 10%. 6. bi,3: =bi,3 × (105% to 110%)//Increase the right-hand side of all design variables by 5% to 10%. 7. Calculate Hc from fn. 8. end while 9. ai,3: =ai,3 × (105% to 110%)//Increase the left-hand side of all design variables by 5% to 10%. 10. bi,3: =bi,3 × (90% to 95%)//Reduce the right-hand side of all design variables by 5% to 10%. 11. While Hc > H0, do 12. ai,3: =ai,3 × (98% to 99%)//Reduce the left-hand side of all design variables by 1% to 2%. 13. bi,3: =bi,3 × (101% to 102%)//Increase the right-hand side of all design variables by 1% to 2%. 14. Calculate Hc from fn 15. end while 16. ai,3: =ai,3 × (101% to 102%)//Increase the left-hand side of all design variables by 1% to 2%. 17. bi,3: =bi,3 × (98% to 99%)//Reduce the right-hand side of all design variables by 1% to 2%. 18. for i: =1 to imax do//from the 1st design variable to the imax design variable 19. While Hc > H0, do 20. ai,3: =ai,3 × (99% to 99.9%)//0.1% to 1% reduction in the left-hand side of the ith design variable 21. Calculate Hc from fn 22. end while 23. ai,3: =ai,3 × (100.1% to 101%)//0.1% to 1% increase in the left-hand side of the ith design variable 24. While Hc >H0, do 25. bi,3: =bi,3 × (100.1% to 101%)//0.1% to 1% increase in the right-hand side of the ith design variable 26. Calculate Hc from fn 27. end while 28. bi,3: =bi,3 × (99% to 99.9%)//0.1% to 1% reduce in the right-hand side of the ith design variable 29. end for 30. Return (ai,3, bi,3) 31. End the generalized inverse cascade method. |
4. Example
4.1. Identify Design Objectives and Design Variables
4.2. A Platform for Acquiring Basic Sample Data
4.3. The Mapping Function for Design Variables and Design Objectives
4.4. Optimal Intervals for Design Variables Based on Generalized Inverse Calculation Solving
5. Experimental Verification
5.1. Idle Condition
5.2. Third Gear Full Throttle Acceleration Condition
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Mount Direction | Mount Stiffness (N/mm) |
---|---|
Front left longitudinal mount (X-direction) | 28 ≤ SFL,X ≤ 52 |
Front left transversal mount (Y-direction) | 35 ≤ SFL,Y ≤ 65 |
Front left vertical mount (Z-direction) | 210 ≤ SFL,Z ≤ 390 |
Front right longitudinal mount (X-direction) | 28 ≤ SFR,X ≤ 52 |
Front right transversal mount (Y-direction) | 35 ≤ SFR,Y ≤ 65 |
Front right vertical mount (Z-direction) | 210 ≤ SFR,Z ≤ 390 |
Rear longitudinal mount (X-direction) | 36.4 ≤ SRR,X ≤ 67.6 |
Rear transversal mount (Y-direction) | 117.6 ≤ SRR,Y ≤ 218.4 |
Rear vertical mount (Z-direction) | 176.4 ≤ SRR,Z ≤ 327.6 |
Sample | SFL,X (N/mm) | SFL,Y (N/mm) | SFL,Z (N/mm) | SFR,X (N/mm) | SFR,Y (N/mm) | SFR,Z (N/mm) | SRR,X (N/mm) | SRR,Y (N/mm) | SRR,Z (N/mm) |
---|---|---|---|---|---|---|---|---|---|
1 | 48 | 41 | 358 | 44.1 | 62.7 | 260 | 53.7 | 126 | 224 |
2 | 28.3 | 49 | 364 | 32.8 | 47.5 | 378 | 53.3 | 160 | 306 |
… | … | … | … | … | … | … | … | … | … |
59 | 37.7 | 53.2 | 328 | 51 | 37.6 | 360 | 62.4 | 178 | 298 |
60 | 52 | 61.2 | 314 | 34 | 60.4 | 323 | 51 | 138 | 293 |
Design Variable | Lower Limit of the Optimization Interval (N/mm) | Upper Limit of the Optimization Interval (N/mm) |
---|---|---|
SFL,X | 28 | 38 |
SFL,Y | 36 | 49 |
SFL,Z | 233 | 271 |
SFR,X | 28 | 35 |
SFR,Y | 35 | 48 |
SFR,Z | 229 | 282 |
SRR,X | 51 | 62 |
SRR,Y | 172 | 207 |
SRR,Z | 178 | 327 |
Design Variable | Lower Limit of the Optimization Interval (N/mm) | Upper Limit of the Optimization Interval (N/mm) |
---|---|---|
SFL,X | 28 | 36 |
SFL,Y | 39 | 51 |
SFL,Z | 256 | 311 |
SFR,X | 28 | 41 |
SFR,Y | 40 | 50 |
SFR,Z | 265 | 304 |
SRR,X | 57 | 66 |
SRR,Y | 203 | 213 |
SRR,Z | 199 | 325 |
Mount Direction | Mount Element Stiffness Value (N/mm) | ||||
---|---|---|---|---|---|
Original Stiffness Value | Median Value of Pareto Optimal Solution Set #1 | Median Value of Pareto Optimal Solution Set #2 | Minimum Value of Pareto Optimal Solution Set #1 | Maximum Value of Pareto Optimal Solution Set #1 | |
SFL,X | 40 | 33 | 32 | 28 | 38 |
SFL,Y | 50 | 42.5 | 45 | 36 | 49 |
SFL,Z | 300 | 252 | 283.5 | 233 | 271 |
SFR,X | 40 | 31.5 | 34.5 | 28 | 35 |
SFR,Y | 50 | 41.5 | 45 | 35 | 48 |
SFR,Z | 300 | 255.5 | 284.5 | 229 | 282 |
SRR,X | 52 | 56.5 | 61.5 | 51 | 62 |
SRR,Y | 168 | 189.5 | 208 | 172 | 207 |
SRR,Z | 252 | 252.5 | 262 | 178 | 327 |
Mount Direction | Force Transmission Rate (N·s2/m) | ||||
---|---|---|---|---|---|
Original Stiffness Value | Median Value of Pareto Optimal Solution Set #1 | Median Value of Pareto Optimal Solution Set #2 | Minimum Value of Pareto Optimal Solution Set #1 | Maximum Value of Pareto Optimal Solution Set #1 | |
SFL,X | 700 | 599 | 638 | 615 | 668 |
SFL,Y | 4560 | 3872 | 4053 | 4096 | 3989 |
SFL,Z | 956 | 860 | 816 | 890 | 877 |
SFR,X | 1400 | 1182 | 1226 | 1207 | 1266 |
SFR,Y | 4606 | 4179 | 4110 | 4197 | 4050 |
SFR,Z | 923 | 825 | 807 | 807 | 798 |
SRR,X | 1200 | 1004 | 1128 | 1120 | 1156 |
SRR,Y | 800 | 716 | 681 | 696 | 749 |
SRR,Z | 736 | 618 | 695 | 688 | 638 |
Weighted force transmission rate | 1385 | 1212 | 1233 | 1253 | 1235 |
Mount Direction | Decoupling Rate of Main Vibration Mode (%) | ||||
---|---|---|---|---|---|
Original Stiffness Value | Median Value of Pareto Optimal Solution Set #1 | Median Value of Pareto Optimal Solution Set #2 | Minimum Value of Pareto Optimal Solution Set #1 | Maximum Value of Pareto Optimal Solution Set #1 | |
X | 70.71 | 83.28 | 79.76 | 80.22 | 81.46 |
Y | 74.51 | 81.75 | 83.68 | 80.11 | 78.60 |
Z | 84.43 | 91.91 | 90.90 | 86.08 | 89.46 |
θx | 76.80 | 83.05 | 80.63 | 88.57 | 77.69 |
θy | 72.27 | 79.45 | 81.46 | 77.00 | 86.42 |
θz | 63.06 | 83.70 | 78.54 | 81.11 | 79.58 |
Weighted decoupling rate | 72.26 | 83.31 | 82.20 | 81.50 | 80.91 |
Mount Direction | Vibration Isolation Rate before Optimization (dB) | Vibration Isolation Rate after Optimization (dB) |
---|---|---|
SFL,X | 22.72 | 26.83 |
SFL,Y | 15.15 | 18.24 |
SFL,Z | 23.34 | 24.50 |
SFR,X | 20.45 | 20.51 |
SFR,Y | 14.84 | 17.62 |
SFR,Z | 23.75 | 24.67 |
SRR,X | 20.38 | 22.75 |
SRR,Y | 24.72 | 23.42 |
SRR,Z | 23.89 | 26.53 |
Position | X-Direction (m/s2) | Y-Direction (m/s2) | Z-Direction (m/s2) | RSS Test Value (m/s2) | RSS Limit Value (m/s2) |
---|---|---|---|---|---|
Seat rail | 0.0261 | 0.0134 | 0.0254 | 0.039 | 0.05 |
Steering wheel | 0.1 | 0.1 | 0.0532 | 0.15 | 0.5 |
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Shui, Y.; Wen, H.; Zhao, J.; Wu, Y.; Huang, H. Optimization of Vehicle Powertrain Mounting System Based on Generalized Inverse Cascade Method under Uncertainty. Appl. Sci. 2023, 13, 7615. https://doi.org/10.3390/app13137615
Shui Y, Wen H, Zhao J, Wu Y, Huang H. Optimization of Vehicle Powertrain Mounting System Based on Generalized Inverse Cascade Method under Uncertainty. Applied Sciences. 2023; 13(13):7615. https://doi.org/10.3390/app13137615
Chicago/Turabian StyleShui, Yongbo, Hansheng Wen, Jian Zhao, Yudong Wu, and Haibo Huang. 2023. "Optimization of Vehicle Powertrain Mounting System Based on Generalized Inverse Cascade Method under Uncertainty" Applied Sciences 13, no. 13: 7615. https://doi.org/10.3390/app13137615