# Predictive Models of Double-Vibropolishing in Bowl System Using Artificial Intelligence Methods

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## Abstract

**:**

## 1. Introduction

#### 1.1. Double Vibro-Polishing

#### 1.2. Analytical and Mathematical Modelling

#### 1.3. Computational Intelligence Methods

## 2. Materials and Methods

#### 2.1. Regression

#### 2.2. Artificial Neural Network

#### 2.3. Genetic Programming

#### 2.4. Model Conditioning

#### 2.5. Acquisition and Pre-Processing of Data

- time subjected to finishing, $t$,
- initial surface roughness, ${R}_{i}$,
- frequency of vibratory bowl, ${f}_{b}$,
- state of vibratory fixture, ${s}_{v}$

#### 2.6. Feature Generation

#### 2.7. Data Extrapolation Using Exponential Function

#### 2.8. Prediction Models

#### 2.8.1. Regression Model

_{b}, ${R}_{i}$, and ${S}_{v}$ affect the model through interaction with $t$, including 3rd and 4th order interactions between the variables.

#### 2.8.2. Artificial Neural Network

#### 2.8.3. Genetic Programming

## 3. Discussion of Results

#### 3.1. Model Selection

#### 3.2. Family of Curves Generation

- time, $t$, are 15 min intervals from 0 to 180 min
- bowl frequencies, ${b}_{f}$, are 3000, and 4500 rpm
- secondary vibratory fixture state of 0 for “off” and 1 for “on”, and
- initial roughness, ${R}_{i}$ of 5 equally spaced points from 0.6 to 1.5 $\mathsf{\mu}\mathrm{m}$

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${R}_{a}$ | Arithmetic average of the roughness profile (µm) |

${R}_{i}$ | Initial roughness before subjecting to vibropolishing (µm) |

${R}_{s}$ | Roughness saturation (µm); the minimum roughness that a material achieves when subjected to a vibropolishing process |

$\widehat{{R}_{a}}$ | Normalized roughness average (dimensionless) |

${f}_{b}$ | frequency of vibratory bowl (rpm) |

${s}_{v}$ | state of vibratory fixture (0:off or 1:on) |

MAPE | Mean absolute percentage error |

## References

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**Figure 3.**Data for vibratory bowl, ${S}_{v}$ is the secondary vibratory fixture state, ${f}_{b}$ is the bowl frequency.

**Figure 5.**Artificial Neural Network (ANN) predicted vs. measured (normalized values) for bowl system.

**Figure 8.**Program tree generated through genetic programming for bowl system, continued from Figure 7.

**Table 1.**Ti-6Al-4V chemical composition [10].

Carbon (Maximum) | 0.10 | % | Titanium | Balance | |
---|---|---|---|---|---|

Aluminum | 5.50 to 6.75 | % | Vanadium | 3.50 to 4.50 | % |

Nitrogen | 0.05 | % | Iron (Maximum) | 0.40 | % |

Oxygen (Maximum) | 0.020 | % | Hydrogen (Maximum) | 0.015 | % |

Other, Total (Maximum) | 0.40 | % |

Time (t) | Bowl Frequency (f_{b}) | Initial Roughness (R_{i}) | Vibratory Fixture State (s_{v}) | Combined Terms |
---|---|---|---|---|

0 | 0 | 0 | 1 | (s_{v}) |

0 | 0 | 1 | 0 | (R_{i}) |

0 | 0 | 1 | 1 | (R_{i})(s_{v}) |

0 | 1 | 0 | 0 | (f_{b}) |

0 | 1 | 0 | 1 | (f_{b})(s_{v}) |

0 | 1 | 1 | 0 | (f_{b})(R_{i}) |

0 | 1 | 1 | 1 | (f_{b})(R_{i})(s_{v}) |

1 | 0 | 0 | 0 | (t) |

1 | 0 | 0 | 1 | (t)(s_{v}) |

1 | 0 | 1 | 0 | (t)(R_{i}) |

1 | 0 | 1 | 1 | (t)(R_{i})(s_{v}) |

1 | 1 | 0 | 0 | (t)(f_{b}) |

1 | 1 | 0 | 1 | (t)(f_{b})(s_{v}) |

1 | 1 | 1 | 0 | (t)(f_{b})(R_{i}) |

1 | 1 | 1 | 1 | (t)(f_{b})(R_{i})(s_{v}) |

Terms | Coefficient | Standard Error | t | P > |t| |
---|---|---|---|---|

(t) | −13.15 | 0.13 | −103.41 | 0 |

(t)(f) | 11.65 | 0.14 | 81.12 | 0 |

(t)(R_{i}) | 24.55 | 0.24 | 103.03 | 0 |

(t)(S_{v}) | −45.29 | 0.27 | −167.16 | 0 |

(t)(f_{b})(R_{i}) | −32.42 | 0.26 | −125.05 | 0 |

(t)(f_{b})(S_{v}) | 34.33 | 0.3 | 113.14 | 0 |

(t)(R_{i})(S_{v}) | 40.71 | 0.33 | 122.31 | 0 |

(t)(f_{b})(R_{i})(S_{v}) | −32.95 | 0.37 | −90.14 | 0 |

Parameter | Value |
---|---|

Number of hidden layers | 1 |

Number of nodes in hidden layer | 7 |

Learning rate | 0.03 |

Regularization constant | 0.0 |

Total Epochs | 5320 |

Parameter | Value |
---|---|

Parsimony coefficient | 4 × 10^{−5} |

Generations | 20 |

Crossover probability | 0.9 |

Subtree-mutation probability | 0.05 |

Population size | 5000 |

Stopping criterion | $\left|\overline{\mathrm{e}}\right|<0.01$ |

Function set | (add, subtract, multiply, divide, negate, exponential) |

Model Type | Train MAPE (%) | Train R-Squared | Test MAPE (%) | Test R-Squared |
---|---|---|---|---|

Exponential regression | 8.0 | 0.98 | 6.1 | 0.99 |

ANN | 9.8 | 0.97 | 15.4 | 0.96 |

GP | 7.4 | 0.97 | 5.3 | 0.98 |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Alcaraz, J.Y.I.; Ahluwalia, K.; Yeo, S.-H. Predictive Models of Double-Vibropolishing in Bowl System Using Artificial Intelligence Methods. *J. Manuf. Mater. Process.* **2019**, *3*, 27.
https://doi.org/10.3390/jmmp3010027

**AMA Style**

Alcaraz JYI, Ahluwalia K, Yeo S-H. Predictive Models of Double-Vibropolishing in Bowl System Using Artificial Intelligence Methods. *Journal of Manufacturing and Materials Processing*. 2019; 3(1):27.
https://doi.org/10.3390/jmmp3010027

**Chicago/Turabian Style**

Alcaraz, Joselito Yam II, Kunal Ahluwalia, and Swee-Hock Yeo. 2019. "Predictive Models of Double-Vibropolishing in Bowl System Using Artificial Intelligence Methods" *Journal of Manufacturing and Materials Processing* 3, no. 1: 27.
https://doi.org/10.3390/jmmp3010027