# A Method for Optimizing the Dwell Time of Optical Components in Magnetorheological Finishing Based on Particle Swarm Optimization

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{#}and 2

^{#}), the root-mean-square (RMS) and peak–valley (PV) values of 1

^{#}converged from the initial values of 169.164 nm and 1161.69 nm to 24.79 nm and 911.53 nm. Similarly, the RMS and PV values of 2

^{#}converged from the initial values of 187.27 nm and 1694.05 nm to 31.76 nm and 1045.61 nm. The simulation results showed that compared with the general pulse iteration method, the proposed algorithm could obtain a more accurate dwell time distribution of each point under the condition of almost the same processing time, subsequently acquiring a better convergence surface and reducing mid-spatial error. Finally, the accuracy of the optimization algorithm was verified through experiments. The experimental results demonstrated that the optimized algorithm could be used to perform high-precision surface machining. Overall, this optimization method provides a solution for dwell time calculation in the process of the magnetorheological finishing of optical components.

## 1. Introduction

## 2. Calculation Method of Dwell Time

- A. Fourier transform method

- B. Iteration method (PI)

- C. Linear equations method

## 3. Dwell Time Calculation by Particle Swarm Optimization Algorithm

#### 3.1. Evaluation Criteria of Surface Error

_{x}is the height element along the outline, and n is the number of discrete elements. The RMS result is calculated as the standard deviation of the height (or depth) of the test surface relative to the reference at all data points in the dataset. The RMS result is the root-mean-square of the surface error or transmission error relative to the reference surface. The RMS result is an area-weighted statistic. In measuring the performance of optical components, RMS can be used to describe the optical performance of the component surface more accurately than PV statistics because it uses all the data in the associated calculations.

#### 3.2. Establishing the Dwell Time Optimization Algorithm

## 4. Simulation Analysis

^{#}and 2

^{#}, with a diameter of 156 mm, a curvature radius of −425.15 mm, an off-axis quantity of 121.23 mm, and a K coefficient of −1, were used as the components to be machined. The XY grating scanning path was used to carry out the simulation of magnetorheological finishing. The process flow chart is shown in Figure 2.

^{#}is shown in Figure 3a, and its surface peak–valley (PV) and root-mean-square (RMS) values were 1161.69 nm and 169.16 nm, respectively. The initial surface error distribution $Z\left(x,y\right)$ of 2

^{#}is shown in Figure 3b, and its surface peak–valley (PV) and root-mean-square (RMS) values were 1694.05 nm and 187.27 nm, respectively. The removal function used in the machining process is shown in Figure 4. The length and width of the removal function were 16 mm and 8 mm. In addition, the peak removal efficiency was 17.17 μm/min; the volume removal efficiency was 0.89 mm/min.

^{#}, the residual error of the surface after seven iterations is shown in Figure 5a. The corresponding peak–valley (PV) value of the surface was 912.14 nm, and the root-mean-square (RMS) value was 29.33 nm. Using the optimization method, the residual error of the surface was obtained after thirteen iterations (Figure 5b). The corresponding peak–valley (PV) value of the surface was 911.53 nm, and the root-mean-square (RMS) value was 24.79 nm. In the case where the optimization method was not used for 2

^{#}, the residual error of the surface after eleven iterations is shown in Figure 5c. The corresponding peak–valley (PV) value of the surface was 1187.25 nm, and the root-mean-square (RMS) value was 38.88 nm. Using the optimization method, the residual error of the surface was obtained after fourteen iterations (Figure 5d). The corresponding peak–valley (PV) value of the surface was 1045.61 nm, and the root-mean-square (RMS) value was 31.76 nm. Compared with Figure 5a,c, the surface distributions corresponding to Figure 5b,d were smoother, and the peak–valley (PV) and root-mean-square (RMS) values were also smaller, indicating that the optimization method can improve the surface accuracy of components.

^{#}obtained using the particle swarm optimization and pulse iteration methods, it was found that, after using the optimization method, the PSD values decreased in the spatial frequency band of 0.05 mm

^{−1}to 0.16 mm

^{−1}, indicating that the particle swarm optimization method can reduce the corresponding middle-and low-frequency surface errors during processing (Figure 6a). Although the spatial frequency was improved below 0.05 mm

^{−1}, the overall surface error peak–valley value (PV) and root-mean-square value (RMS) values were reduced; thus, this part could not be considered. By comparing the PSD curves of the residual error of 2

^{#}obtained using the particle swarm optimization and pulse iteration methods, it was found that, after using the optimization method, the PSD values decreased in the spatial frequency band of 0.06 mm

^{−1}to 0.16 mm

^{−1}, indicating that the particle swarm optimization method can reduce the corresponding middle-and low-frequency surface errors during processing (Figure 6b). Although the spatial frequency was improved to below 0.03 mm

^{−1}, the overall surface error PV and RMS values were reduced; thus, this part could not be considered.

^{#}, the high-frequency error of the surface is shown in Figure 7a. The corresponding peak–valley (PV) value of the surface was 156.63 nm, and the root-mean-square (RMS) value was 3.38 nm. The high-frequency error of the surface obtained using the optimization method is shown in Figure 7b. The corresponding peak–valley (PV) value of the surface was 158.60 nm, and the root-mean-square (RMS) value was 3.45 nm. For the case where the optimization method was not used for 2

^{#}, the high-frequency error of the surface is shown in Figure 7c. The corresponding peak–valley (PV) value of the surface was 101.56 nm, and the root-mean-square (RMS) value was 2.30 nm. The high-frequency error of the surface obtained using the optimization method is shown in Figure 7d. The corresponding peak–valley (PV) value of the surface was 100.82 nm, and the root-mean-square (RMS) value was 2.39 nm. Given that magnetorheological finishing is not sensitive to high-frequency information, the removal process is only applicable to low-frequency and mid-frequency errors. Therefore, the high-frequency errors obtained by filtering are similar regardless of whether they are processed using the pulse iteration method or the particle swarm optimization method.

^{#}obtained with and without the particle swarm optimization method were compared (Figure 8a,b). Specifically, when the particle swarm optimization method was not used, the total dwell time distribution $T\left(x,y\right)$ obtained after the seventh iteration was satisfied: $T\left(x,y\right)={\displaystyle {\sum}_{k=0}^{6}{T}_{k}\left(x,y\right)}$. The feed speed corresponds to a peak–valley (PV) value of 4000 mm/min and a root-mean-square (RMS) value of 507.20 mm/min. In comparison, by using the particle swarm optimization method, the total dwell time distribution ${T}^{\prime}\left(x,y\right)$ obtained after the thirteenth iteration was satisfied: ${T}^{\prime}\left(x,y\right)={\displaystyle {\sum}_{k=0}^{12}{{T}_{k}}^{\prime}\left(x,y\right)}$. The feed speed corresponds to a peak–valley (PV) value of 4000 mm/min and a root-mean-square (RMS) value of 205.85 mm/min. The feed velocity distributions of the polishing wheel for 2

^{#}obtained with and without the particle swarm optimization method were also compared (Figure 8c,d). Specifically, when the particle swarm optimization method was not used, the total dwell time distribution $T\left(x,y\right)$ obtained after the eleventh iteration was satisfied: $T\left(x,y\right)={\displaystyle {\sum}_{k=0}^{10}{T}_{k}\left(x,y\right)}$. The feed speed corresponds to a peak–valley (PV) value of 4000 mm/min and a root-mean-square (RMS) value of 350.41 mm/min. In comparison, by using the particle swarm optimization method, the total dwell time distribution ${T}^{\prime}\left(x,y\right)$ obtained after the fourteenth iteration was satisfied: ${T}^{\prime}\left(x,y\right)={\displaystyle {\sum}_{k=0}^{13}{{T}_{k}}^{\prime}\left(x,y\right)}$. The feed speed corresponds to a peak–valley (PV) value of 4000 mm/min and a root-mean-square (RMS) value of 178.44 mm/min. The above observations indicate that after using the optimization method, although the peak–valley (PV) value of the polishing wheel’s feed speed obtained via solving did not change, the root-mean-square (RMS) value of the obtained polishing wheel feed speed decreased, and the speed distribution was gentler. Therefore, it can be concluded that the optimization method can render the polishing wheel’s feed speed more uniform, reduce instantaneous acceleration and deceleration movement, and ensure the stability of the machine tool. Thus, the particle swam optimization method reduced the introduction of mid-spatial errors and ensured the high precision of the processing of the surface.

^{#}obtained with and without the particle swarm optimization method were compared (Figure 9a,b). When the particle swarm optimization method was not used, the total dwell time required was 145.56 min. Meanwhile, when particle swarm optimization was used, the total dwell time required was 168.93 min. The dwell time distributions of 2

^{#}obtained with and without the particle swarm optimization method were also compared (Figure 9c,d). When the particle swarm optimization method was not used, the total dwell time required was 208.12 min. Meanwhile, when particle swarm optimization was used, the total dwell time required was 222.46 min. Consequently, in the case where there was no significant difference in dwell time, the surface accuracy of the component was improved by using particle swarm optimization.

^{#}and 2

^{#}are shown in Table 1. It can also be concluded that with almost the same dwell time, particle swarm optimization can increase the uniformity of the feed speed of the polishing wheel in the machining process, reduce instantaneous acceleration and deceleration movement, and ensure the stability of the machine tool during machining. Therefore, particle swarm optimization reduced the introduction of mid-spatial error, subsequently improving the surface accuracy of the components after processing.

## 5. Experimental Verification

^{−1}to 0.37 mm

^{−1}were reduced compared to those obtained using the pulse iteration method (PI), indicating that the particle swarm optimization method can reduce the corresponding middle- and low-frequency surface errors in the machining process. Under the condition of a similar convergence rate and dwell time, lower PV and RMS values could be obtained after optimization. After the machining experiment, compared with the PSO-2 and PI curves, the PSD value of the spatial frequency in the band from 0.035 mm

^{−1}to 0.16 mm

^{−1}decreased, indicating that the optimization algorithm effectively reduced the middle- and low-frequency bands in the actual machining process. In addition, there was no significant difference in the PSD values between the values of the PSO-2 and PSO-1 curves in the spatial frequency band of 0.16 mm

^{−1}to 0.37 mm

^{−1}, and this finding was attributed to an error in the actual machining process.

## 6. Conclusions and Implications

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**(

**a**) Initial surface error of 1

^{#}optical element; (

**b**) initial surface error of 2

^{#}optical element.

**Figure 5.**(

**a**) The residual error of surface calculated using the pulse iteration method for 1

^{#}; (

**b**) the residual error of surface calculated via the particle swarm optimization algorithm method for 1

^{#}; (

**c**) the residual error of surface calculated using the pulse iteration method for 2

^{#}; (

**d**) the residual error of surface calculated via the particle swarm optimization algorithm method for 2

^{#}.

**Figure 6.**(

**a**) The power spectral density (PSD) curve comparison of 1

^{#}surface residual error; (

**b**) the power spectral density (PSD) curve comparison of 2

^{#}surface residual error.

**Figure 7.**(

**a**) The high-frequency error of surface calculated using pulse iteration method for 1

^{#}; (

**b**) the high-frequency error of surface calculated using the particle swarm optimization algorithm method for 1

^{#}; (

**c**) the high-frequency error of surface calculated using the pulse iteration method for 2

^{#}; (

**d**) the high-frequency error of surface calculated via particle swarm optimization algorithm method for 2

^{#}.

**Figure 8.**(

**a**) The polishing wheel moving velocity distribution determined via the pulse iteration method for 1

^{#}; (

**b**) the polishing wheel moving velocity distribution determined via particle swarm optimization algorithm method for 1

^{#}; (

**c**) the polishing wheel moving velocity distribution determined via the pulse iteration method for 2

^{#}; (

**d**) the polishing wheel moving velocity distribution determined via the particle swarm optimization algorithm method for 2

^{#}.

**Figure 9.**(

**a**) The dwell time distribution obtained via the pulse iteration method of 1

^{#}; (

**b**) the dwell time distribution obtained via the particle swarm optimization algorithm method of 1

^{#}; (

**c**) the dwell time distribution obtained via the pulse iteration method of 2

^{#}; (

**d**) the dwell time distribution obtained via the particle swarm optimization algorithm method of 2

^{#}.

**Figure 12.**(

**a**) The polishing wheel moving velocity distribution obtained using the pulse iteration method; (

**b**) the polishing wheel moving velocity distribution obtained using the particle swarm optimization algorithm method.

**Figure 13.**(

**a**) The dwell time distribution obtained using the pulse iteration method; (

**b**) the dwell time distribution obtained using the particle swarm optimization algorithm method.

**Figure 14.**(

**a**) The residual error of the surface calculated using the pulse iteration method; (

**b**) the residual error of the surface calculated using the pulse iteration method.

Index | Pulse Iteration Method of 1^{#} | Particle Swarm Optimization of 1^{#} | Pulse Iteration Method of 2^{#} | Particle Swarm Optimization of 2^{#} |
---|---|---|---|---|

Initial surface error PV/nm | 1161.69 | 1161.69 | 1694.05 | 1694.05 |

Initial surface error RMS/nm | 169.16 | 169.16 | 187.27 | 187.27 |

Surface error after machining PV/nm | 912.14 | 911.53 | 1187.25 | 1045.61 |

Surface error after machining RMS/nm | 29.33 | 24.79 | 38.88 | 31.76 |

Polishing wheel feed speed PV/mm/min | 4000 | 4000 | 4000 | 4000 |

Polishing wheel feed speed RMS/mm/min | 507.20 | 205.85 | 350.41 | 178.44 |

Total dwell time of machining/min | 145.56 | 168.93 | 208.12 | 222.46 |

Index | Pulse Iteration Method | Particle Swarm Optimization |
---|---|---|

Initial surface error PV/nm | 1545.13 | 1545.13 |

Initial surface error RMS/nm | 345.51 | 345.51 |

Surface error after simulation machining PV/nm | 1095.65 | 1095.10 |

Surface error after simulation machining RMS/nm | 301.44 | 295.27 |

Polishing wheel feed speed PV/mm/min | 4000 | 4000 |

Polishing wheel feed speed RMS/mm/min | 439.88 | 124.74 |

Total dwell time of machining/min | 153.73 | 178.93 |

Surface error after actual machining PV/nm | - | 1526.55 |

Surface error after actual machining RMS/nm | - | 312.35 |

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**MDPI and ACS Style**

Gao, B.; Fan, B.; Wang, J.; Wu, X.; Xin, Q.
A Method for Optimizing the Dwell Time of Optical Components in Magnetorheological Finishing Based on Particle Swarm Optimization. *Micromachines* **2024**, *15*, 18.
https://doi.org/10.3390/mi15010018

**AMA Style**

Gao B, Fan B, Wang J, Wu X, Xin Q.
A Method for Optimizing the Dwell Time of Optical Components in Magnetorheological Finishing Based on Particle Swarm Optimization. *Micromachines*. 2024; 15(1):18.
https://doi.org/10.3390/mi15010018

**Chicago/Turabian Style**

Gao, Bo, Bin Fan, Jia Wang, Xiang Wu, and Qiang Xin.
2024. "A Method for Optimizing the Dwell Time of Optical Components in Magnetorheological Finishing Based on Particle Swarm Optimization" *Micromachines* 15, no. 1: 18.
https://doi.org/10.3390/mi15010018