A Novel Hybrid FEM–Dynamic Modeling Approach for Enhanced Vibration Diagnostics in a Two-Stage Spur Gearbox
Abstract
:1. Introduction
- Hybrid FEM–dynamic modeling approach: We propose a novel approach that integrates refined FEM-based stiffness estimation with a dynamic model of a two-stage gearbox. This method accurately simulates gear-body compliance and distributed load effects, effectively reproducing vibration responses under different fault scenarios.
- Parametric sensitivity analysis: We perform an extensive sensitivity analysis using the FEM approach on five critical parameters that impact TVMS estimation:
- -
- Load-dependent stiffness variations under operational torque.
- -
- Assembly tolerance-induced interaxial variations.
- -
- Tooth profile deviations and intentional geometry modifications.
- -
- Hub-radius effects on the overall stiffness response.
- -
- Mesh convergence behavior in the contact zones.
- Integrated validation framework: We validate our methodology by using a closed-loop process that combines simulation and experimental data. The dynamic gearbox model generates vibration profiles for healthy and faulty conditions, and these are benchmarked against experimental results from a gearbox test bench, demonstrating the method’s capability to replicate the experimental signal characteristics.
2. Proposed Approach
2.1. Dynamic Modeling of Gearbox
- (for ): mass and rotational inertia of gear i.
- : vertical displacement of gear i along line of action.
- : angular displacement of gear i.
- : base circle radius of gear i, converting into effective linear displacement.
- : motor inertia, angular displacement, and torque, respectively.
- : load inertia, angular displacement, and torque, respectively.
- : bearing stiffness and damping for each gear shaft, modeling radial support.
- : torsional stiffness and damping between motor and Gear 1.
- : gear mesh stiffness and damping between Gear 1 and Gear 2 (first stage).
- : torsional stiffness and damping of intermediate shaft connecting Gear 2 and Gear 3.
- : gear mesh stiffness and damping between Gear 3 and Gear 4 (second stage).
- : torsional stiffness and damping between Gear 4 and load.
2.2. Analytical Calculation of TVMS
2.3. FEM Calculation of TVMS
- 1.
- Define the initial contact position between meshing teeth as the angular reference (0°).
- 2.
- Perform a quasi-static structural analysis under a specified torque to compute the resultant gear deformations.
- 3.
- Determine the angular displacement from the FE solution, which quantifies the rotational deflection of the pinion relative to a reference point (RP) induced by the applied torque.
- 4.
- Incrementally rotate the gear pair by an angular step proportional to the speed ratio () transitioning the contact to subsequent tooth pairs.
- 5.
- Repeat steps 2–4 across successive angular positions until a full mesh cycle is completed.
2.4. R-FEM and C-FEM
2.5. TVMS Sensitivity to Parameter Variations
2.5.1. Effects of Applied Torque
2.5.2. Impact of Gear Center Distance
2.5.3. Impact of Tooth Profile
2.5.4. Impact of Gear Hub Radius
2.5.5. Impact of Mesh Size
3. Results and Discussion
3.1. Experimental Setup
3.2. Approach Validation Under Healthy Conditions
3.3. Approach Validation Under Faulty Conditions
3.3.1. Chipped Gear Fault
3.3.2. Broken Tooth Gear Fault
3.4. General Discussion
- 1.
- Conducting modal analysis to develop transfer functions that account for vibration transmission paths from gear teeth to sensors through various gearbox components, including shafts and housing structures.
- 2.
- Implementing systematic calibration protocols to develop high-fidelity digital twins capable of accurately simulating a diverse range of gear fault scenarios under industrial conditions.
- 3.
- Expanding the investigation to include other fault morphologies, particularly initiated root cracks and progressive wear patterns, to enhance diagnostic capabilities.
- 4.
- Integrating additional factors such as friction, lubrication, and tool wear effects into the simulation framework to improve its industrial applicability.
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
TVMS | Time-Varying Mesh Stiffness |
FEM | Finite Element Method |
FE | finite element |
R-FEM | Refined Finite Element Method |
C-FEM | Conventional Finite Element Method |
CBM | condition-based monitoring |
DOFs | degrees of freedom |
RP | reference point |
3D | three-dimensional |
EMP | electromagnetic particle |
GWMA | Gaussian Weighted Moving Average |
GMF | gear mesh frequency |
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Parameter | Pinion | Gear |
---|---|---|
Number of teeth (Z) | 19 | 31 |
Base circle radius (mm) | 28.3 | 46.2 |
Module (mm) | 3.2 | |
Pressure angle | 20° | |
Face width (mm) | 38.1 | |
Young’s modulus (MPa) | 2.068 × 105 | |
Poisson’s ratio | 0.3 |
Parameter | Pinion 1 | Gear 1 | Pinion 2 | Gear 2 |
---|---|---|---|---|
Number of teeth | 29 | 100 | 36 | 90 |
Bore radius (mm) | 10 | 15 | 10 | 15 |
Base circle radius (mm) | – | 70.4770 | – | 63.4293 |
Face width (mm) | 15 | |||
Modulus (mm) | 1.5 |
Parameter | First Stage | Second Stage | Bearings | Motor |
---|---|---|---|---|
Pinion mass (kg) | 0.16 | 0.294 | – | – |
Gear mass (kg) | 1.74 | 1.79 | – | – |
Pinion inertia (kg·m2) | 4.76 × 10−5 | 1.21 × 10−4 | – | – |
Gear inertia (kg·m2) | 4.82 × 10−3 | 3.89 × 10−3 | – | – |
Motor inertia (kg·m2) | – | – | – | 6.63 × 10−3 |
Bearing stiffness (N/m) | – | – | 6.56 × 108 | – |
Bearing damping (N·s/m) | – | – | 1.8 × 103 | – |
Young’s modulus (MPa) | 2.068 × 105 | |||
Poisson’s ratio | 0.3 |
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El Amli, A.; El Yousfi, B.; Soualhi, A.; Guillet, F. A Novel Hybrid FEM–Dynamic Modeling Approach for Enhanced Vibration Diagnostics in a Two-Stage Spur Gearbox. Energies 2025, 18, 2176. https://doi.org/10.3390/en18092176
El Amli A, El Yousfi B, Soualhi A, Guillet F. A Novel Hybrid FEM–Dynamic Modeling Approach for Enhanced Vibration Diagnostics in a Two-Stage Spur Gearbox. Energies. 2025; 18(9):2176. https://doi.org/10.3390/en18092176
Chicago/Turabian StyleEl Amli, Amine, Bilal El Yousfi, Abdenour Soualhi, and François Guillet. 2025. "A Novel Hybrid FEM–Dynamic Modeling Approach for Enhanced Vibration Diagnostics in a Two-Stage Spur Gearbox" Energies 18, no. 9: 2176. https://doi.org/10.3390/en18092176
APA StyleEl Amli, A., El Yousfi, B., Soualhi, A., & Guillet, F. (2025). A Novel Hybrid FEM–Dynamic Modeling Approach for Enhanced Vibration Diagnostics in a Two-Stage Spur Gearbox. Energies, 18(9), 2176. https://doi.org/10.3390/en18092176