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A New Approach to Study the Effect of Complexity on an External Gear Pump Model to Generate Data Source for AI-Based Condition Monitoring Application^{ †}

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## Abstract

**:**

## 1. Introduction

- (1)
- A new approach, which relies on a detailed geometric model for simulating the EGP, is proposed.
- (2)
- The approach is assessed by comparing two models that were previously examined, with each model being validated against experimental data from the study case of an electrified reach truck.
- (3)
- A test classifier is developed to assess the data reliability from the top-performing model for automated condition monitoring of EGP.

## 2. Methods

#### 2.1. External Gear Pump Models

_{i}is the pitch diameter of each gear, z is the number of teeth, r

_{hi}is the outer diameter of the gear, and t

_{0}is the base pitch. These parameters could be either from measuring the physical components after disassembly or from CAD files.

_{l}is the arc length between two adjacent teeth on the root diameter, h

_{r}is the gap clearance between the gears and the wear plates, µ is the dynamic viscosity, l

_{r}is the difference between the root and the shaft radius, and Δp is the pressure difference between the delivery and the suction line. The total flow rate for this type of leakage is the following:

_{1}considers the fraction of the gears’ rotation where the couple of teeth is in connection with the inlet part where the pressure difference is zero, thus giving no contribution. This factor is equal to:

_{t}is the tip width, and u

_{t}is the tangential speed. To obtain a correct representation of the event, it is necessary to consider the variability of the gap distance along the sealing zone. This variability is due to the eccentricity between the gears and the housing center created using pressure forces acting on the gears. The modeled equation is again based on an equivalent rectangular orifice geometry, so, for this reason, some evaluations are needed. The first is the need to evaluate an equivalent value of the gap clearance along the sealing zone for a single rotation. To obtain this equivalent parameter, an integral average is implemented based on geometric and trigonometric techniques [23]:

_{l}is the difference between the outer and the root radius, l

_{l}is the tooth thickness on the pitch diameter. The choice of tooth thickness is an overestimation of the true value, but it permits considerable simplification and streamlines the evaluation without losing too much accuracy. The total flow rate for this leakage is the following:

#### 2.2. Case

#### 2.3. Experimental Setup

^{3}/rev.

## 3. Results and Discussion

#### 3.1. Model Validation

^{3}/rev and 14.79 cm

^{3}/rev, respectively.

#### 3.2. Test Classifier

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AI | Artificial Intelligence |

CAD | Computer Aided Drafting |

DT | Decision Tree |

EGP | External Gear Pump |

FMI | Functional Mockup Interface |

FPGA | Functional Programmable Gateway Arrays |

KNN | K-Nearest Neighbor |

LDA | Linear Discriminant Analysis |

MLP | Multi-Layer Perceptron |

MSE | Mean Squared Error |

RF | Random Forest |

RMSE | Root Mean Squared Error |

## Nomenclature

Symbol | Description | Unit |

𝑏 | Gear Width | [m] |

bl b_{l} | Root arc length | [m] |

𝐶𝑠${C}_{s}$ | Laminar Slip Coefficient | |

𝐶𝑠𝑡 ${C}_{st}$ | Turbulent Slip Coefficient | |

e $e$ | Eccentricity | |

h h | Tooth tip gap clearance | [m] |

havg${h}_{avg}$ | Mean value of Tooth tip gap clearance | [m] |

hr h_{r} | Radial gap clearance | [m] |

k1 k_{1} | Rotational Inlet fraction | |

k2 k_{2} | Sealing zone rotational factor | |

Ll l_{l} | Lateral orifice length | [m] |

lr${l}_{r}$ | Radial orifice length | [m] |

lt${l}_{t}$ | Tooth Tip orifice length | [m] |

𝑛 | Rotational Speed | [rad/s] |

𝑟1 ${r}_{1}$ | Pitch Diameter of Gear 1 | [m] |

𝑟2 ${r}_{2}$ | Pitch Diameter of Gear 2 | [m] |

𝑟ℎ | Outside Diameter of Gear | [m] |

𝑟ℎ1 ${r}_{h1}$ | Outside Diameter of Gear 1 | [m] |

𝑟ℎ2 ${r}_{h2}$ | Outside Diameter of Gear 2 | [m] |

R | Housing Diameter | [m] |

𝑄 | Volumetric Flow | [m^{3}/s] |

𝑃 | Pressure | [Pa] |

𝑡0 | Base Pitch | [m] |

ut | Tangential speed | [m/s] |

𝑉 | Volumetric Displacement | [m^{3}/rev] |

𝑉𝑔 | Geometric Displacement Volume | [m^{3}/rev] |

𝑉𝑔𝑖 | Geometric Displacement Volume of tooth pair | [m^{3}/rev] |

w | Housing Wear parameter | [m] |

z | Number of Teeth | |

Greek letter | Description | Unit |

ε | Suction angle | [rad] |

φ | Rotational angle | [rad] |

𝜇 | Dynamic Viscosity | [Kg/ms] |

𝜂𝑣 | Volumetric Efficiency | |

𝜌 | Density | [Kg/m^{3}] |

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**Figure 1.**Physical representation of the radial leakages inside an EGP (Distance exaggerated for better visual representation): (

**a**) 3D representation, (

**b**) 2D representation.

**Figure 7.**Model validation for the three levels of complexity based on cylinder displacement. The blue solid line is the measured data; the green dotted line is the level 1 model, the black dashed line is the level 2 model, and the red dash-dotted line is the level 3 model.

Model | Displacement | Volumetric Flow | System Pressure | |||
---|---|---|---|---|---|---|

MSE | RMSE | MSE | RMSE | MSE | RMSE | |

Level 1 | 0.00018 | 0.01342 | 16.0788 | 4.0098 | 127.8042 | 11.3051 |

Level 2 | 0.00645 | 0.08031 | 22.1156 | 4.7027 | 110.3343 | 10.504 |

Level 3 | 0.00037 | 0.01923 | 16.1718 | 4.0214 | 97.3894 | 9.4847 |

Classifier | Balanced Accuracy Score (%) |
---|---|

LDA | 63.43 |

DT | 85.55 |

RF | 88.95 |

KNN | 66.46 |

MLP | 71.68 |

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**MDPI and ACS Style**

Azeez, A.A.; Mazzei, P.; Minav, T.; Frosina, E.; Senatore, A.
A New Approach to Study the Effect of Complexity on an External Gear Pump Model to Generate Data Source for AI-Based Condition Monitoring Application. *Actuators* **2023**, *12*, 401.
https://doi.org/10.3390/act12110401

**AMA Style**

Azeez AA, Mazzei P, Minav T, Frosina E, Senatore A.
A New Approach to Study the Effect of Complexity on an External Gear Pump Model to Generate Data Source for AI-Based Condition Monitoring Application. *Actuators*. 2023; 12(11):401.
https://doi.org/10.3390/act12110401

**Chicago/Turabian Style**

Azeez, Abid Abdul, Pietro Mazzei, Tatiana Minav, Emma Frosina, and Adolfo Senatore.
2023. "A New Approach to Study the Effect of Complexity on an External Gear Pump Model to Generate Data Source for AI-Based Condition Monitoring Application" *Actuators* 12, no. 11: 401.
https://doi.org/10.3390/act12110401