# Reliability Analysis and Optimization Method of a Mechanical System Based on the Response Surface Method and Sensitivity Analysis Method

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## Abstract

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## 1. Introduction

## 2. Basic Theories

#### 2.1. Reliability State Function

#### 2.2. First-Order Second-Moment (FOSM) Method

#### 2.3. Response Surface Method (RSM)

#### 2.4. Sensitivity Analysis of Errors

## 3. Random Errors of Transport System

## 4. Reliability Analysis and Optimization of Transport Systems

#### 4.1. Reliability Assessment of Transport Systems under Shock Effects

#### 4.2. Reliability Optimization of a Transport System Based on Sensitivity Analysis

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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No. | Error | Error Item | No. | Error | Error Item |
---|---|---|---|---|---|

1 | $\mathsf{\Delta}\eta $ | Azimuth angle error | 7 | $\delta y\left({z}_{1}\right)$ | Rotational error around y-axis |

2 | $\delta x\left(z\right)$ | Rotational error around x-axis | 8 | $\delta y\left({y}_{2}\right)$ | Rotary error around y-axis |

3 | $\mathrm{d}y\left(z\right)$ | Displacement error in y-direction | 9 | $\delta z\left({y}_{2}\right)$ | Slewing error around z-axis |

5 | $\delta x\left({z}_{1}\right)$ | Rotational error around x-axis | 10 | $\mathrm{d}z\left({y}_{2}\right)$ | Displacement error in z-direction |

6 | $\mathsf{\Delta}h$ | Travel Error of vertical conveyor system | 11 | $\mathsf{\Delta}\theta $ | Up-swing angle error of oscillating conveyor |

4 | $\Delta \phi $ | Storage silos with rotary motion rotation error | 12 | $\mathsf{\Delta}\varphi $ | Angle error of rotating conveyors rotating |

No. | Symbol of Mean Square Deviation | Numerical Value | No. | Symbol of Mean Square Deviation | Numerical Value |
---|---|---|---|---|---|

1 | ${\sigma}_{\mathsf{\Delta}\eta}$ | 0.75 mrad | 7 | ${\sigma}_{\mathsf{\Delta}h}$ | 1.5 mm |

2 | ${\sigma}_{\delta x\left(z\right)}$ | 0.05 mrad | 8 | ${\sigma}_{\delta y\left({y}_{2}\right)}$ | 0.15 mrad |

3 | ${\sigma}_{\mathrm{d}y\left(z\right)}$ | 0.5 mm | 9 | ${\sigma}_{\delta z\left({y}_{2}\right)}$ | 0.15 mrad |

4 | ${\sigma}_{\mathsf{\Delta}\phi}$ | 3.5 mrad | 10 | ${\sigma}_{\mathrm{d}z\left({y}_{2}\right)}$ | 0.25 mm |

5 | ${\sigma}_{\delta x\left({z}_{1}\right)}$ | 0.25 mrad | 11 | ${\sigma}_{\mathsf{\Delta}\theta}$ | 2.5 mrad |

6 | ${\sigma}_{\delta y\left({z}_{1}\right)}$ | 0.25 mrad | 12 | ${\sigma}_{\mathsf{\Delta}\varphi}$ | 1.5 mrad |

No. | Azimuth Angle/rad | Elevation Angle/rad | Inertial Frame | Impact Stimulus | ||
---|---|---|---|---|---|---|

β | R | β | R | |||

1 | −2.0944 | −0.1222 | 1.7923 | 96.35% | 1.3716 | 91.49% |

2 | −2.0944 | 0 | 2.1205 | 98.30% | 1.6254 | 94.80% |

3 | −2.0944 | 0.5236 | 1.5854 | 94.36% | 1.4156 | 92.15% |

4 | −2.0944 | 0.7854 | 1.8649 | 96.89% | 1.7815 | 96.26% |

5 | −2.0944 | 1.0472 | 1.9955 | 97.70% | 1.4874 | 93.15% |

6 | −1.0472 | −0.1222 | 1.6290 | 94.83% | 1.5594 | 94.05% |

7 | −1.0472 | 0 | 1.8101 | 96.49% | 1.5904 | 94.41% |

8 | −1.0472 | 0.5236 | 2.2733 | 98.85% | 1.8031 | 96.43% |

9 | −1.0472 | 0.7854 | 2.2643 | 98.82% | 1.9943 | 97.69% |

10 | −1.0472 | 1.0472 | 2.5762 | 99.50% | 1.4770 | 93.02% |

11 | 0 | −0.1222 | 1.6437 | 94.98% | 1.4992 | 93.31% |

12 | 0 | 0 | 2.2865 | 98.89% | 1.3476 | 91.11% |

13 | 0 | 0.5236 | 1.8808 | 97.01% | 1.3820 | 91.65% |

14 | 0 | 0.7854 | 2.0945 | 98.19% | 1.8872 | 97.04% |

15 | 0 | 1.0472 | 2.2766 | 98.86% | 1.5993 | 94.51% |

16 | 1.0472 | −0.1222 | 1.6290 | 94.83% | 1.5594 | 94.05% |

17 | 1.0472 | 0 | 1.8101 | 96.49% | 1.5904 | 94.41% |

18 | 1.0472 | 0.5236 | 2.2733 | 98.85% | 1.4696 | 92.92% |

19 | 1.0472 | 0.7854 | 2.2643 | 98.82% | 1.9943 | 97.69% |

20 | 1.0472 | 1.0472 | 2.5762 | 99.50% | 1.4770 | 93.02% |

21 | 2.0944 | −0.1222 | 1.7923 | 96.35% | 1.3716 | 91.49% |

22 | 2.0944 | 0 | 2.1205 | 98.30% | 1.6254 | 94.80% |

23 | 2.0944 | 0.5236 | 1.5854 | 94.36% | 1.4156 | 92.15% |

24 | 2.0944 | 0.7854 | 1.8649 | 96.89% | 1.7815 | 96.26% |

25 | 2.0944 | 1.0472 | 1.9955 | 97.70% | 1.4874 | 93.15% |

Min | — | — | 1.5854 | 94.36% | 1.3476 | 91.11% |

Average | — | — | 2.0017 | 97.73% | 1.5837 | 94.34% |

No. | Error Symbol | Sensitivity/% | No. | Error Symbol | Sensitivity/% |
---|---|---|---|---|---|

1 | $\mathsf{\Delta}\eta $ | 0.1360 | 7 | $\mathsf{\Delta}h$ | 0.1884 |

2 | $\delta x\left(z\right)$ | 0.0042 | 8 | $\delta y\left({y}_{2}\right)$ | 0.0166 |

3 | $\mathrm{d}y\left(z\right)$ | 0.1331 | 9 | $\delta z\left({y}_{2}\right)$ | 0.0972 |

4 | $\mathsf{\Delta}\phi $ | 0.0522 | 10 | $\mathrm{d}z\left({y}_{2}\right)$ | 0.1296 |

5 | $\delta x\left({z}_{1}\right)$ | 0.0497 | 11 | $\mathsf{\Delta}\theta $ | 0.0995 |

6 | $\delta y\left({z}_{1}\right)$ | 0.0508 | 12 | $\mathsf{\Delta}\varphi $ | 0.0427 |

Name | Adjusted Parameters | Before Adjustment | After Adjustment | $\overline{\mathit{\beta}}$ | $\overline{\mathit{R}}$ |
---|---|---|---|---|---|

First optimization | ${\mu}_{\mathsf{\Delta}h}$ | 5.23 mm | 3.66 mm | 1.6839 | 95.39% |

Second optimization | ${\mu}_{\mathsf{\Delta}\theta}$ | 6.75 mrad | 4.73 mrad | 1.7708 | 96.17% |

Third optimization | ${\sigma}_{\mathsf{\Delta}h}$ | 1.5 mm | 1.0 mm | 1.8132 | 96.51% |

Fourth optimization | ${\sigma}_{\mathsf{\Delta}\eta}$ | 0.75 mrad | 0.50 mrad | 1.8494 | 96.78% |

Fifth optimization | ${\sigma}_{\mathrm{d}y\left(z\right)}$ | 0.5 mm | 0.3 mm | 1.8735 | 96.95% |

Sixth optimization | ${\sigma}_{\mathrm{d}z\left({y}_{2}\right)}$ | 0.25 mm | 0.17 mm | 1.8912 | 97.07% |

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## Share and Cite

**MDPI and ACS Style**

Zhao, L.; Yue, P.; Zhao, Y.; Sun, S.
Reliability Analysis and Optimization Method of a Mechanical System Based on the Response Surface Method and Sensitivity Analysis Method. *Actuators* **2023**, *12*, 465.
https://doi.org/10.3390/act12120465

**AMA Style**

Zhao L, Yue P, Zhao Y, Sun S.
Reliability Analysis and Optimization Method of a Mechanical System Based on the Response Surface Method and Sensitivity Analysis Method. *Actuators*. 2023; 12(12):465.
https://doi.org/10.3390/act12120465

**Chicago/Turabian Style**

Zhao, Lei, Pengfei Yue, Yang Zhao, and Shiyan Sun.
2023. "Reliability Analysis and Optimization Method of a Mechanical System Based on the Response Surface Method and Sensitivity Analysis Method" *Actuators* 12, no. 12: 465.
https://doi.org/10.3390/act12120465