# Non-Linear Regression Models with Vibration Amplitude Optimization Algorithms in a Microturbine

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Results from the GWO using a 2nd-order model: (

**a**) Regression of the 2nd-order model. (

**b**) Number of iterations in which the RMSE converged with the 2nd-order model.

**Figure 5.**Results from the GWO using a 3rd-order model: (

**a**) Regression of the 3rd-order model. (

**b**) Number of iterations in which the RMSE converged with the 3rd-order model.

**Figure 6.**Results from the GWO using a 4th-order model: (

**a**) Regression of the 4th-order model. (

**b**) Number of iterations in which the RMSE converged with the 4th-order model.

**Figure 7.**Results from the GWO using an exponential model: (

**a**) Regression of the exponential model. (

**b**) Number of iterations in which the RMSE converged with the exponential model.

**Figure 8.**Results from the GWO using an gaussian model: (

**a**) Regression of the gaussian model. (

**b**) Number of iterations in which the RMSE converged with the gaussian model.

**Figure 9.**Results from the GWO using a sinusoidal model: (

**a**) Regression of the sinusoidal model. (

**b**) Number of iterations in which the RMSE converged with the sinusoidal model.

Parameter | Description |
---|---|

Fuel | Butane/propane gas with maximum pressure of 3.5 kg/cm ^{2} |

Turbine blades outer/inner diameter | 68.6/40.5 mm |

Compressor wheel outer/inner diameter | 64.5/32.8 mm |

Turbine wheel diameter | 70 mm |

Burner hole spacing | 10 mm |

Number of gas outlet holes | 16 |

Avg. Frequency (Hz) | Standrad Deviations | Avg. Amplitude (µm) | Standard Deviations |
---|---|---|---|

26.9 | 0.5676 | 1.7891 | 0.2009 |

77 | 1.1547 | 5.2697 | 0.7028 |

125.9 | 0.5676 | 2.0426 | 0.3287 |

Parameter | 2nd Order | 3rd Order | 4th Order | Exponential | Gaussian | Sinusoidal |
---|---|---|---|---|---|---|

SearchAgent | 300 | 300 | 300 | 300 | 300 | 300 |

Iterations | 500 | 500 | 500 | 500 | 500 | 500 |

Dimension | 3 | 4 | 5 | 4 | 3 | 3 |

LowerBoundary | [−5 −5 −5] | [−5 −5 −5 −5] | [−5 −5 −5 −5 −5] | [−2 −5 −2 −5] | [0 0 0] | [−10 −5 −5] |

UpperBoundary | [5 5 5] | [5 5 5 5] | [5 5 5 5 5] | [2 5 2 5] | [10 100 100] | [10 5 5] |

Model | Coefficients | ||||
---|---|---|---|---|---|

a | b | c | d | e | |

2nd order | −0.00136 | 0.21149 | −2.90663 | · | · |

3rd order | 8.01950$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-06}$ | −0.00321 | 0.33308 | −5 | · |

4th order | 2.85$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-07}$ | −7.46$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-05}$ | 0.00502 | 0.00150 | −0.58955 |

Exponential | −0.37629 | 0.03966 | 1.30276 | 0.03008 | · |

Gaussian | 5.27487 | 77.94529 | 34.75833 | · | · |

Sinusoidal | 5.27116 | 0.02421 | 0.30448 | · | · |

Model | RMSE | ${\mathit{R}}^{2}$ | Model | RMSE | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|

2nd order | $0.19155\pm 8.0\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-07}$ | $0.92913\pm 2.9\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-07}$ | Exponential | $0.21936\pm 0.00343$ | $0.91884\pm 0.00126$ |

3rd order | $0.19175\pm 0.00206$ | $0.85524\pm 0.23333$ | Gaussian | $0.19153\pm 8.1\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-07}$ | $0.92914\pm 3.0\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-07}$ |

4th order | $0.19129\pm 0.00272$ | $0.92922\pm 0.00100$ | Sinusoidal | $0.19137\pm 2.3\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-06}$ | $0.92919\pm 8.5\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-07}$ |

Model | Time (s) | MBE |
---|---|---|

2nd order | $0.86225\pm 0.01958$ | $1.99\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-06}\pm 0.00045$ |

3rd order | $1.69912\pm 0.04212$ | $7.84\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-06}\pm 0.00085$ |

4th order | $2.61961\pm 0.13905$ | $0.00016\pm 0.00361$ |

Exponential | $1.27736\pm 0.44059$ | $0.00059\pm 0.00334$ |

Gaussian | $0.95463\pm 0.02762$ | $4.35\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-06}\pm 0.00059$ |

Sinusoidal | $0.97166\pm 0.03725$ | $7.20\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-06}\pm 0.00085$ |

Model | 26.9 Hz | 77 Hz | 125.9 |
---|---|---|---|

2nd order [27] | −0.53% | 0.03% | −12.88% |

2nd order | 0.163% | 0.010% | −0.059% |

3rd order | 0.204% | 0.004% | −0.012% |

4th order | 0.539% | 0.010% | −0.078% |

Exponential | 2.774% | −0.675% | 0.633% |

Gaussian | 0.271% | 0.060% | −0.263% |

Sinusoidal | 0.125% | 0.035% | −0.127% |

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**MDPI and ACS Style**

Rodríguez-Abreo, O.; Rodríguez-Reséndiz, J.; Montoya-Santiyanes, L.A.; Álvarez-Alvarado, J.M.
Non-Linear Regression Models with Vibration Amplitude Optimization Algorithms in a Microturbine. *Sensors* **2022**, *22*, 130.
https://doi.org/10.3390/s22010130

**AMA Style**

Rodríguez-Abreo O, Rodríguez-Reséndiz J, Montoya-Santiyanes LA, Álvarez-Alvarado JM.
Non-Linear Regression Models with Vibration Amplitude Optimization Algorithms in a Microturbine. *Sensors*. 2022; 22(1):130.
https://doi.org/10.3390/s22010130

**Chicago/Turabian Style**

Rodríguez-Abreo, Omar, Juvenal Rodríguez-Reséndiz, L. A. Montoya-Santiyanes, and José Manuel Álvarez-Alvarado.
2022. "Non-Linear Regression Models with Vibration Amplitude Optimization Algorithms in a Microturbine" *Sensors* 22, no. 1: 130.
https://doi.org/10.3390/s22010130