# Multi-Objective Optimization of a Spring Diaphragm Clutch on an Automobile Based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II)

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## Abstract

**:**

## 1. Introduction

_{1B}is the diaphragm spring deformation of the point B. However, there are some problems with the clutch, such as frequent usage, high labor intensity of operators, incomplete separation during long term operation and unstable connection with easiness of slipping [2]. Because of the above results, the operating pressure of the clutch pedal stroke and many other parameters in the design are often unreasonable. This maymean that clutch separation is not complete and the clutch slips. Crucially, the performance of the clutch diaphragm spring directly affects the above parameters and it has important practical engineering optimal design implications.

## 2. Diaphragm Spring Multi-Objective Mathematical Model

#### 2.1. Multiple Objective Functions

_{b}− F

_{a}| is the lowest force to ensure a reliable pressing force in the range of the wear limit. At the same time, the average value of the separation force of bearings in the separation process should be as small as possible to ensure light manipulation. Therefore, the absolute value of the pressing force change to separate the spring is selected as the first optimization target F

_{1}(X) to transfer torque reliably. The minimum average driven disc spring-loaded force is chosen as the second optimization target F

_{2}(X). Therefore, the objective function can be expressed as [8]:

_{b}is the diaphragm spring’s working pressure of working point B; F

_{a}is the working pressure of diaphragm spring when the wear reaches the limit point, and F

_{bi}is the pressing force corresponding to point N.

#### 2.2. Design Variables

_{1B}has a greater impact on the entire characteristic curve. So the main design parameters are seven parameters according to the objective function [8], namely, h, R, r, R

_{1}, r

_{1}and λ

_{1}

_{B}. The diaphragm spring design variables matrix X is determined as:

#### 2.3. Constraints

- (1)
- The new pressing force of the spring F
_{a}should be equal to the required clamping force F_{Y}[3]:$${F}_{a}={F}_{Y}$$ - (2)
- To ensure the stable operation after the friction plate, the pressing force of the spring work after the damaged F
_{1A}should not be less than the new corresponding parameters F_{1B}[3]:$${F}_{1A}\ge {F}_{1B}$$ - (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 α
_{0}should be controlled within a certain range [8]. That is:$$\{\begin{array}{l}1.7\le H/h\le 2.2\\ {9}^{\xb0}\le {\alpha}_{0}\le {15}^{\xb0}\end{array}$$ - (4)
- For the friction plate on the pressing force distribution, the radius of the big end of the friction plate R
_{1}should be taken between the mean radius and the outside diameter of the friction plate, according to engineering experience [9]:$$\{\begin{array}{l}pull:\text{\hspace{0.17em}\hspace{0.17em}}\left(D+d\right)/4\text{\hspace{0.17em}\hspace{0.17em}}\le \text{\hspace{0.17em}\hspace{0.17em}}{r}_{1}\text{\hspace{0.17em}\hspace{0.17em}}\le \text{\hspace{0.17em}}D/2-3\\ push:\text{\hspace{0.17em}\hspace{0.17em}}\left(D+d\right)/4\text{\hspace{0.17em}\hspace{0.17em}}\le \text{\hspace{0.17em}\hspace{0.17em}}{R}_{1}\text{\hspace{0.17em}\hspace{0.17em}}\le \text{\hspace{0.17em}\hspace{0.17em}}D/2-3\end{array}$$ - (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 R
_{1}, load radius r_{1}and inner diameter r should be in a certain range [9]:$$\{\begin{array}{l}1\le R\uff0d{R}_{1}\le 7\\ 0\le {r}_{1}\uff0dr\le 6\\ 0\le {r}_{1}\uff0d{r}_{0}\le 6\end{array}$$ - (6)
- To ensure the working point, the wear point and separation point should be distributed relatively reasonabley the new location λ
_{1B}should meet the following conditions [8]:$$0.8\le \frac{{\lambda}_{1B}}{H}\left(\frac{R-r}{{R}_{1}-{r}_{1}}\right)\le 1.0$$ - (7)
- The work pressing force of the new designed diaphragm F
_{B}should be not less than the force F_{C}in the separation process [9]:$${F}_{B}\ge {F}_{C}$$ - (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]:$$\{\begin{array}{l}1.2\le R/r\le 1.3\\ 3.0\le \left({r}_{1}-{r}_{f}\right)/\left({R}_{1}-{r}_{1}\right)\le 4.0\end{array}$$
- (9)
- To take advantage of spring material, part size should meet certain requirements, according to engineering experience [8]:$$\{\begin{array}{l}1.2\le R\text{}/\text{}r\le 1.35\\ 70\le 2R\text{}/\text{}h\le 100\\ 3.5\le R\text{}/\text{}{r}_{0}\le 5.0\end{array}$$
- (10)
- The highest point of the tensile stress σ
_{Amax}(σ_{Cmax}) in the bottom of the A (or C) dangerous parts of the diaphragm spring separating finger holes should meet a strength condition [9]:$$\{\begin{array}{l}{\sigma}_{A\mathrm{max}}\le \left[{\sigma}_{A}\right]\\ {\sigma}_{C\mathrm{max}}\le \left[{\sigma}_{C}\right]\end{array}$$ - (11)
- During the diaphragm spring manufacturing process, there are some major dimensions machining errors, the error during assembly process should meet certain requirements [9]:$$\left(\Delta {F}_{h}+\Delta {F}_{t}+\Delta {F}_{R}+\Delta {F}_{r}\right)/{F}_{b}\le 0.05$$

## 3. NSGA-II Algorithm and Multi-Objective Solution

_{b}− F

_{a}| is the lowest value that ensures a reliable pressing force in the range of the wear limit but, in addition, the average value of the separation force of bearings in the separation process should be as small as possible to ensure light manipulation. These two or more design specifications achieve an optimal value that is termed multi-objective optimization. In the diaphragm spring multi-objective optimization problem, each target cannot achieve the optimal simultaneously. In the conventional processing multi-objective optimization problem, the normalization method is usually used, such as the weighting coefficient method and the hierarchical sequence method, by building an evaluation function and eventually transforming the multi-objective optimization problem into a single objective optimization solution. However, the normalization method often relies on the experience of policy makers. Since each target objection of diaphragm spring is interrelated and constrained, even a target is improved by reducing another target as the cost [17,18]. Many objective solutions are typically non-inferior solution sets. Therefore, it is important to search for the Pareto set in the multi-objective optimization of the diaphragm spring. Figure 3 shows the flow of the NSGA-II algorithm. The main steps of NSGA-II algorithm used in this paper are as follows: (Initial population) create a random population of N chromosome in the population. (Calculate the objective function) calculate the multi-objective function of spring diaphragm. (Satisfied termination criteria) check whether the termination criterion is satisfied or not. If yes, the population is terminated, otherwise, adjust the fitness distribution. (Fitness distribution) evaluate the multi-objective fitness of each chromosome in the population. (Genetic operation and insert parent) generate a new population.

## 4. Results and Discussion

_{1}, the longer the lifespan of the clutch. So the lifespan of the diaphragm spring by the NSGA-II is better than the penalty function method and the genetic algorithm. The wear range |F

_{b}

**−**F

_{a}| = 181 N and relative rate of change |F

_{b}

**−**F

_{a}|/|F

_{b}| = 4.09%, which is far less than the 10% [9], improving the compaction force stability. In addition, the steering separation force of the clutch after optimization is the smallest. The relative rate of change of the steering separation force is |3817 − 3567|/3817 = 6.55%, so the operating flexibility is improved.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Characteristics of loading-deflection diaphragm spring. A: wear limit point; B: new working point; H: working point; C: separating point; N: lowest point; λ

_{1B}: diaphragm spring deformation of point B; λ

_{1H}: diaphragm spring deformation of point H; λ

_{1N}: diaphragm spring deformation of point N; F

_{1A}: diaphragm spring’s working pressure of working point A; F

_{1B}: diaphragm spring’s working pressure of working point B.

**Figure 2.**Main structural parameters of diaphragm spring. H: cone height of the disc spring; h: spring diaphragm thickness; R

_{1}: radius of the pressure plate’s loading point; r

_{1}: radius of loading point of the support ring; r

_{0}: diameter of spring diaphragm’s small end; r

_{f}: action radius of the separating bearing force; δ

_{1}: width of the small end; δ

_{2}: window width; r

_{e}: radius of the small end; R: big end radius of the disc spring; r: small end radius of the disc spring.

Plan | Objective Function | Algorithm |
---|---|---|

A | $\alpha {F}_{1}\left(X\right)+\beta {F}_{2}\left(X\right)$ | Penalty function method |

B | $\alpha {F}_{1}\left(X\right)+\beta {F}_{2}\left(X\right)$ | genetic algorithm |

C | ${F}_{1}\left(X\right),{F}_{2}\left(X\right)$ | NSGA-II |

_{1}(X): target 1; F

_{2}(X): target 2.

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) | R_{1} (mm) | r_{1} (mm) | ${\mathbf{\lambda}}_{1\mathit{B}}$ |
---|---|---|---|---|---|---|---|

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 |

_{1}: radius of the pressure plate’s loading point; r

_{1}: radius of loading point of the support ring; λ

_{1B}: new deformation of point B.

Plan | F_{b} (N) | F_{a} (N) | F_{c} (N) | |F_{b} − F_{a}| (N) | |F_{b} − F_{a}|/|F_{b}| (%) |
---|---|---|---|---|---|

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 |

_{b}: working pressure of working point B; F

_{a}: working pressure of diaphragm spring when the wear reaches the limit point; F

_{c}: separation force; |F

_{b}

**−**F

_{a}|/|F

_{b}|: relative rate of change of the steering separation force.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Zhou, 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