Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II)
Abstract
:1. Introduction
2. Diaphragm Spring Multi-Objective Mathematical Model
2.1. Multiple Objective Functions
2.2. Design Variables
2.3. Constraints
- (1)
- (2)
- To ensure the stable operation after the friction plate, the pressing force of the spring work after the damaged F1A should not be less than the new corresponding parameters F1B [3]:
- (3)
- Considering the depth-thickness ratio H/h has impactions on the load-deformation curve of the diaphragm, it should be met within a certain range. At the same time, the diaphragm spring initial cone angle α0 should be controlled within a certain range [8]. That is:
- (4)
- For the friction plate on the pressing force distribution, the radius of the big end of the friction plate R1 should be taken between the mean radius and the outside diameter of the friction plate, according to engineering experience [9]:
- (5)
- In order to meet the structural arrangement of the diaphragm spring actual situation, the big end of the radius R, the support ring radius R1, load radius r1 and inner diameter r should be in a certain range [9]:
- (6)
- To ensure the working point, the wear point and separation point should be distributed relatively reasonabley the new location λ1B should meet the following conditions [8]:
- (7)
- The work pressing force of the new designed diaphragm FB should be not less than the force FC in the separation process [9]:
- (8)
- In order to make diaphragm spring satisfy a certain leverage ratio during the separation, the ratio of the outer diameter to inner diameter should be met [3]:
- (9)
- To take advantage of spring material, part size should meet certain requirements, according to engineering experience [8]:
- (10)
- The highest point of the tensile stress σAmax(σCmax) in the bottom of the A (or C) dangerous parts of the diaphragm spring separating finger holes should meet a strength condition [9]:
- (11)
- During the diaphragm spring manufacturing process, there are some major dimensions machining errors, the error during assembly process should meet certain requirements [9]:
3. NSGA-II Algorithm and Multi-Objective Solution
4. Results and Discussion
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Plan | Objective Function | Algorithm |
---|---|---|
A | Penalty function method | |
B | genetic algorithm | |
C | NSGA-II |
Population Size | Stop Algebra | Fitness Function Value Deviation | Optimal Front End Individual Coefficient | Maximum Iterative Algebra |
---|---|---|---|---|
100 | 200 | 1 × 10−100 | 0.3 | 200 |
Plan | H (mm) | h (mm) | R (mm) | r (mm) | R1 (mm) | r1 (mm) | |
---|---|---|---|---|---|---|---|
Original | 5.8 | 2.93 | 145.7 | 116.8 | 143.66 | 116.1 | 4.80 |
A | 5.24 | 2.80 | 140.00 | 115.00 | 138.68 | 115.00 | 4.21 |
B | 5.20 | 2.80 | 140.04 | 115.18 | 138.80 | 114.00 | 4.02 |
C | 5.21 | 2.81 | 140.35 | 115.48 | 140.66 | 114.50 | 4.01 |
Plan | Fb (N) | Fa (N) | Fc (N) | |Fb − Fa| (N) | |Fb − Fa|/|Fb| (%) |
---|---|---|---|---|---|
Original | 5226 | 5925 | 3817 | 699 | 13.37 |
A | 4834 | 5185 | 3709 | 351 | 7.23 |
B | 4757 | 5016 | 3715 | 259 | 5.44 |
C | 4422 | 4603 | 3567 | 181 | 4.09 |
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Zhou, J.; Wang, C.; Zhu, J. Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II). Math. Comput. Appl. 2016, 21, 47. https://doi.org/10.3390/mca21040047
Zhou J, Wang C, Zhu J. Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II). Mathematical and Computational Applications. 2016; 21(4):47. https://doi.org/10.3390/mca21040047
Chicago/Turabian StyleZhou, Junchao, Chun Wang, and Junjun Zhu. 2016. "Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II)" Mathematical and Computational Applications 21, no. 4: 47. https://doi.org/10.3390/mca21040047
APA StyleZhou, J., Wang, C., & Zhu, J. (2016). Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II). Mathematical and Computational Applications, 21(4), 47. https://doi.org/10.3390/mca21040047