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Article

A Novel Hybrid FEM–Dynamic Modeling Approach for Enhanced Vibration Diagnostics in a Two-Stage Spur Gearbox

by
Amine El Amli
1,
Bilal El Yousfi
2,
Abdenour Soualhi
1,* and
François Guillet
1
1
LASPI Laboratory, Jean Monnet University, 42300 Roanne, France
2
Ecole Militaire Polytechnique (EMP), BP 17, Bordj el Bahri, Algiers 16046, Algeria
*
Author to whom correspondence should be addressed.
Energies 2025, 18(9), 2176; https://doi.org/10.3390/en18092176
Submission received: 22 March 2025 / Revised: 14 April 2025 / Accepted: 23 April 2025 / Published: 24 April 2025
(This article belongs to the Special Issue Failure Diagnosis and Prognosis of AC Rotating Machines)

Abstract

:
The condition monitoring of gearboxes is crucial to ensuring the reliability and efficiency of modern industrial machinery. The accurate estimation of Time-Varying Mesh Stiffness (TVMS) is a key aspect of modeling gear meshing behavior and generating vibration signals used for fault diagnosis. In this study, TVMS is calculated by using the Refined Finite Element Method (R-FEM), which captures detailed gear-body compliance and distributed load effects. The dynamic model of a two-stage gearbox is then used to generate vibration responses under both healthy and faulty conditions. A comprehensive parametric sensitivity analysis is conducted on critical gear modeling parameters, including tooth profile deviations, mesh convergence in contact zones, assembly tolerance-induced interaxial variations, load-dependent stiffness variations, and hub-radius effects. Experimental validation using a gearbox test bench confirms that the proposed methodology accurately reproduces fault-specific harmonic components. These results indicate that the hybrid FEM–dynamic modeling approach effectively balances accuracy and computational efficiency, thereby providing a robust framework for advanced fault detection and maintenance strategies in gear systems.

1. Introduction

Gearbox systems are fundamental to modern industrial machinery, enabling controlled power transfer in applications ranging from wind turbines to aerospace propulsion. As essential load-bearing components, gears are exposed to extreme mechanical stresses, including fluctuating torsional loads and high rotational speeds. These demanding conditions predispose them to degradation mechanisms such as pitting, spalling, and tooth fractures (Figure 1), potentially leading to catastrophic failures and unplanned downtime [1,2,3]. To mitigate these risks, industries are increasingly adopting condition-based monitoring (CBM) strategies that leverage vibration [4], acoustic [5], and thermal data [6] for early fault detection. While many CBM implementations rely on straightforward signal processing techniques, achieving advanced diagnostic accuracy for complex gear faults often requires high-fidelity dynamic models capable of capturing subtle, fault-induced vibrational signatures. In these sophisticated approaches, the precise estimation of Time-Varying Mesh Stiffness (TVMS) is critical, as it significantly influences gear meshing behavior and the resulting vibration patterns that serve as key indicators of gear health [7,8,9,10,11].
Shen et al. [12] developed a modified TVMS model using potential energy methods to quantify how tooth wear reduces mesh stiffness in planetary gear systems, particularly affecting bending, shear, and axial stiffness components. Similarly, Saxena et al. [13] used potential energy methods to show that shaft misalignment and friction forces significantly reduce TVMS in spur gear pairs, while Luo et al. [14] proposed a gear mesh kinematic model demonstrating that center distance variations significantly affect the TVMS of spur gear pairs. For helical gear applications, Wang et al. [15] developed an analytical model incorporating axial mesh force components to improve TVMS calculations. Addressing gear faults, Ma et al. [16] introduced an Improved Analytical Model (IAM) to calculate TVMS for cracked spur gears, correcting double-tooth engagement errors, and Wang et al. [17] further investigated the influence of profile shifts on TVMS in spur gears by using potential energy methods. For more complex geometries, Li et al. [18] developed a numerical calculation method for the TVMS of spiral bevel gears with spalling defects by considering multi-tooth meshing and gear flank flexibility, validated through the FEM, while Li et al. [19] advanced TVMS calculation for shaped gears by incorporating tooth surface deviations and processing textures into a wedge-shaped contact model, also validated through the FEM. Building on these TVMS calculations, Yu et al. [20] developed a dynamic single-stage gearbox model that uses TVMS computed via potential energy methods as gear mesh input to analyze coupled vibrations with multiple localized defects, and Huangfu et al. [21] employed loaded tooth contact analysis (LTCA) to calculate TVMS for spalled gear pairs, validated through the FEM, and integrated these results into a single-stage geared rotor model for vibration analysis. El Yousfi et al. [22] extended this approach by coupling an induction machine with a two-stage gear system by using an improved potential energy method for gear tooth fault detection, while Mahapatra and Mohanty [23] proposed a modified limiting line approach for calculating TVMS in cracked spur gears by using a single-stage gearbox model for dynamic response analysis. In addition, Li et al. [24] proposed an enhanced numerical loaded tooth contact analysis (NLTCA) model for spiral bevel gears that incorporates both local contact deformation and global tooth compliance, thereby improving accuracy relative to FEA. Complementing these studies, Zeng et al. [25] proposed a 24-degree-of-freedom coupled dynamic model for gear transmission systems with localized defects, incorporating TVMS derived from Hertz contact theory to analyze the modulation effects of bearing faults on gear meshing vibrations. Addressing crack detection, Kumar and Roy [26] presented a mathematical model for detecting variable gear tooth cracks in wind turbine gearboxes under ramp-up wind speed conditions by integrating analytical TVMS computation with dynamic simulation. Extending the work, Yang et al. [27] developed a dynamic model of a wind turbine gear–bearing coupling system that incorporates tooth root cracks and slicing coupling effects through an improved analytical TVMS method. More recently, digital twin techniques have emerged as promising tools for predictive maintenance and fault diagnosis. Moghadam et al. [28] presented a digital twin model for the predictive maintenance of gearboxes in floating offshore wind turbine drivetrains by integrating real-time torsional measurements with simulation-based TVMS estimations, and Lu et al. [29] introduced a digital twin-driven framework that leverages water-wave information transmission and a recurrent acceleration network to predict the remaining useful life of gearboxes. Lastly, Habbouche et al. [30] proposed a digital twin-based gearbox fault diagnosis approach integrating variational mode decomposition with dynamic vibration modeling, while Feng et al. [31] developed a hybrid methodology for monitoring and predicting gear wear that combines a discretized spatial wear model with real-time updates of wear coefficients. For wind turbine applications, Xu et al. [32] introduced a digital twin framework for gearbox operation and maintenance through multi-source data fusion and a WOA-TCN-Attention model for real-time fault prediction.
Despite significant advances in gear dynamic modeling, critical gaps remain in current research. Integrated methodologies combining the FEM and dynamic analysis for fault diagnostics in two-stage parallel gearboxes are still largely unexplored. Moreover, while previous FEM-based TVMS studies have examined torque magnitude effects, they have not addressed how the location of torque application, such as on the gear body versus the hub, affects mesh stiffness calculations. As a result, the boundary condition implications of torque application position remain uninvestigated. Furthermore, comprehensive multi-parametric studies using the FEM to assess TVMS behavior are notably sparse in the literature.
This study addresses these research needs by developing a combined framework integrating the FEM and dynamic modeling for enhanced fault diagnosis while systematically investigating both the effects of torque application conditions and parameter sensitivity on TVMS through comprehensive analysis. Through this integrated approach, we seek to both advance the theoretical understanding of gear dynamics and provide practical tools for improved fault diagnostics in gear systems.
The key contributions of this study are as follows:
  • 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.
In this work, we introduce a hybrid methodology that seamlessly combines FEM-based stiffness estimation with a dynamic gearbox model to advance the state of the art in gearbox dynamic modeling. The paper is organized as follows: Section 1 provides the introduction, including the background and literature review. Section 2 presents the hybrid methodology, covering the gearbox dynamic model, the analytical approach for mesh stiffness calculation, and a detailed FEM approach for TVMS calculation, along with analyses of various influencing parameters. Section 3 presents the results and discussion, including the experimental setup, gear mesh stiffness calculation, and model validation under both healthy and defective gear conditions. Finally, Section 4 offers conclusions and directions for future research.

2. Proposed Approach

The primary objective of this study is to estimate vibration signals in a two-stage gearbox system through a hybrid approach (Figure 2) that leverages TVMS as a key dynamic parameter. TVMS serves as a direct indicator of vibration characteristics, enabling the simulation of both healthy and faulty operational states by modeling defect scenarios and assigning their corresponding stiffness profiles. To assess the influence of different modeling strategies on TVMS accuracy, a tripartite comparison is conducted among an analytical method and two finite element (FE) approaches. The analytical model is based on a potential energy formulation under Hertzian contact assumptions, while the R-FEM and C-FEM differ in their treatment of load distribution, using distributed load with gear-body compliance and localized load application, respectively.
Yu et al. [20] employed an approach that relies primarily on analytical formulations to model gearbox dynamics, in which TVMS is estimated by using a potential energy method. In contrast, the framework proposed in this study integrates a two-stage dynamic model with refined FEM-based TVMS estimation. This hybridization enables the modeling of complex gear faults, such as tooth cracks and their propagation, which are challenging to capture accurately by using analytical methods. Chaari et al. [7], for example, adopted a simplified gear model that neglects torsional mesh stiffness, while Verma et al. [33] utilized a 2D model with a single-tooth representation and localized force application. In the present approach, the FEM model is based on a detailed 3D gear geometry, incorporates torsional mesh stiffness, and applies a distributed load across the contact zone. These refinements contribute to enhancing the fidelity of the gearbox dynamic model. Overall, the proposed framework represents a promising step toward the development of a robust digital twin for gear fault diagnosis and prognosis.

2.1. Dynamic Modeling of Gearbox

The dynamic behavior of the two-stage spur gear system is modeled by using lumped parameter theory, which simplifies the system into concentrated masses, inertias, and spring–damper elements, as depicted in Figure 3. In this approach, gears are represented as concentrated masses m i and inertias I i , while shafts and bearings are modeled as spring–damper systems characterized by stiffness k i j , damping c i j , and radial coefficients k b and c b .
Gear meshing, which enables power transmission between rotating shafts, is defined by the engagement of gear teeth. Figure 4 illustrates the pinion–gear engagement along the line of action, representing the theoretical path for transmitting contact forces. These mesh interactions are mathematically represented through mesh stiffness k m ij and damping c m ij parameters. The kinematic states of the system are defined by linear displacements y i along the line of action and angular displacements θ i , with subscripts (1a), (1b), (1c) and (1d) denoting the gears. Additionally, the motor and load are characterized by their inertias ( I m and I l ), angular displacements ( θ m and θ l ), and applied torques ( T m and T l ), with subscripts m and l denoting motor and load parameters, respectively. Nonlinear effects, such as backlash and friction, are excluded from this model to simplify the analysis.
This study focuses on the second stage of the gearbox. Specifically, we aim to extract the vibration signal y 3 ¨ , corresponding to the vertical displacement y 3 (see Equation (1c)). We simulate both healthy and faulty TVMS for the gear pair in this stage by incorporating the stiffness parameter k m 34 .
The equations of motion are derived by applying Newton’s second law, resulting in the following system:
m 1 y 1 ¨ = k b y 1 c b y 1 ˙ + F k 12 + F c 12 ,
m 2 y 2 ¨ = k b y 2 c b y 2 ˙ F k 12 F c 12 ,
m 3 y 3 ¨ = k b y 3 c b y 3 ˙ + F k 34 + F c 34 ,
m 4 y 4 ¨ = k b y 4 c b y 4 ˙ F k 34 F c 34 ,
I 1 θ 1 ¨ = k m 1 ( θ m θ 1 ) + c m 1 ( θ m ˙ θ 1 ˙ ) r b 1 ( F k 12 + F c 12 ) ,
I 2 θ 2 ¨ = k 23 ( θ 2 θ 3 ) c 23 ( θ 2 ˙ θ 3 ˙ ) + r b 2 ( F k 12 + F c 12 ) ,
I 3 θ 3 ¨ = k 23 ( θ 2 θ 3 ) + c 23 ( θ 2 ˙ θ 3 ˙ ) r b 3 ( F k 34 + F c 34 ) ,
I 4 θ 4 ¨ = k 4 l ( θ 4 θ l ) c 4 l ( θ 4 ˙ θ l ˙ ) + r b 4 ( F k 34 + F c 34 ) ,
I m θ m ¨ = T m k m 1 ( θ m θ 1 ) c m 1 ( θ m ˙ θ 1 ˙ ) ,
I l θ l ¨ = T l + k 4 l ( θ 4 θ l ) + c 4 l ( θ 4 ˙ θ l ˙ ) .
The meshing forces, comprising elastic and damping components, are expressed as:
F k ij = k m ij ( r b i θ i r b j θ j y i + y j ) ,
F c ij = c m ij ( r b i θ i ˙ r b j θ j ˙ y i ˙ + y j ˙ ) .
In the present work, the shaft parameters, torsional stiffness ( k ij ), and shaft damping ( c ij ) are adopted from the methodology described in [22].
Below are the key parameters of the two-stage gearbox model:
  • m i , I i (for i = 1 , , 4 ): mass and rotational inertia of gear i.
  • y i : vertical displacement of gear i along line of action.
  • θ i : angular displacement of gear i.
  • r b i : base circle radius of gear i, converting θ i into effective linear displacement.
  • I m , θ m , T m : motor inertia, angular displacement, and torque, respectively.
  • I l , θ l , T l : load inertia, angular displacement, and torque, respectively.
  • k b , c b : bearing stiffness and damping for each gear shaft, modeling radial support.
  • k m 1 , c m 1 : torsional stiffness and damping between motor and Gear 1.
  • k m 12 , c m 12 : gear mesh stiffness and damping between Gear 1 and Gear 2 (first stage).
  • k 23 , c 23 : torsional stiffness and damping of intermediate shaft connecting Gear 2 and Gear 3.
  • k m 34 , c m 34 : gear mesh stiffness and damping between Gear 3 and Gear 4 (second stage).
  • k 4 l , c 4 l : torsional stiffness and damping between Gear 4 and load.

2.2. Analytical Calculation of TVMS

The potential energy method is a widely used analytical approach to evaluating gear mesh stiffness [34]. In this method, the gear tooth is modeled as a nonuniform cantilever beam, fixed at its root circle, as illustrated in Figure 5. Beam theory is then applied to derive mathematical expressions for the deformation energies stored within the gear tooth structure. These expressions are subsequently used to determine the deflection of the gear tooth at various points under the applied transmitted force F.
Gear mesh stiffness represents the resistance to elastic deformation when gear teeth engage during power transmission. As gears rotate, this stiffness varies cyclically due to changing contact conditions, creating the primary excitation source in gear systems. The total mesh stiffness ( k t ) is composed of five main components: Hertzian contact stiffness ( k h ), which quantifies local deformation at the contact interface; axial compression stiffness ( k a ), characterizing tooth deformation along its longitudinal axis; bending stiffness ( k b ), accounting for tooth deflection under transverse loading; shear stiffness ( k s ), representing deformation due to shear forces; and fillet foundation stiffness ( k f ), describing the elastic behavior at the tooth root–body junction. These components act as springs in series to determine the overall mesh compliance. The mathematical formulations of these stiffness components, expressed as functions of gear tooth geometry, material properties, and loading conditions, are provided below [36,37]:
k h = π E L 4 1 ν 2
where L is the tooth width, E is Young’s modulus, and ν is Poisson’s ratio.
1 k a = 0 x c sin 2 α E S x d x
1 k b = 0 x c x c x cos α y c sin α 2 E I x d x
1 k s = 0 x c 1.2 cos 2 α G S x d x
where x c is the contact line coordinate along the tooth, y c is the perpendicular offset from the reference axis, α is the force application angle on the tooth, and x is the integration variable along the tooth width, as illustrated in Figure 5. Meanwhile, G denotes the shear modulus, and S and I represent the cross-sectional area and moment of inertia, respectively.
1 k f = cos 2 α L E L * u f S f 2 + M * u f S f + P * 1 + Q * tan 2 α
The coefficients L * , M * , P * , and Q * , along with parameters u f , S f , and α , are provided in [37].
Finally, the overall gear mesh stiffness at a given contact point is obtained by combining the individual stiffness components through the following relationship [35]:
1 k t = 1 k h + i = p , g 1 k a i + 1 k b i + 1 k s i + 1 k f i
For the gear pair parameters listed in Table 1, the TVMS, denoted by k t in Equation (8), is calculated to validate the modeling approach. The resulting mesh stiffness curve, shown in Figure 6, illustrates the characteristic variation in TVMS throughout the meshing cycle, during which the gear teeth engage and disengage as they rotate, transitioning from single-tooth contact to double-tooth contact. The horizontal axis represents the shaft rotation angle, and the sudden drop in the curve marks the transition from double-tooth to single-tooth contact.

2.3. FEM Calculation of TVMS

As established in prior discussions, FEM delivers high-fidelity results, which explains the widespread adoption of FEM-based methodologies in gear stiffness analysis. However, the accuracy of these approaches critically depends on their fidelity to physical loading conditions and boundary constraints. For instance, the simplified methodology adopted by Chaari et al. [7] restricts the load application exclusively to the tooth profile while neglecting the mechanical coupling between the gear body and the tooth root. This simplification introduces significant inaccuracies, particularly by excluding torsional mesh stiffness contributions arising from gear-body compliance. The FE analysis involves a quasi-static study conducted by using ABAQUS/CAE 2024 software. A three-dimensional (3D) model of the gear pair, imported in the STEP (*.step) file format, is assigned linear–elastic material properties, as specified in Table 1. Boundary conditions are applied, where the pinion is fully constrained (all degrees of freedom (DOFs) fixed), while the gear is restricted to rotation exclusively about the z-axis (with all other DOFs fixed). A hexahedral mesh is generated by using C3D8R elements with reduced integration. The mesh features multi-zone refinement with a 0.2 mm mesh size for the contact zones of both the pinion and gear and coarser meshes of 2.5 mm and 3.5 mm for the non-contact regions of the pinion and gear, respectively (see Figure 7). A reference point (RP-1), defined at the geometric center of the gear hub, is kinematically coupled to the gear body to enforce rigid rotational behavior. Finally, a torque T is applied to this reference point to simulate the rotational loading conditions.
Mesh stiffness is characterized by initially computing the torsional mesh stiffness ( k tor ) and subsequently converting it into the rectilinear mesh stiffness ( k t ) through the relationship [39]
k t = k tor r b 1 2 = T Δ θ · r b 1 2
where Δ θ represents the angular displacement of the driving gear under an applied torque T and r b 1 denotes the base circle radius of the driving gear. This formulation establishes the equivalent linear stiffness by normalizing the torsional response with respect to the base radius of the gear.
In Figure 8, the gear tooth is depicted in both its original geometry (undeformed) and the geometry under load (deformed). The deflection δ quantifies how much the tooth bends or is displaced due to the applied meshing force.
The methodology for evaluating TVMS follows this iterative procedure:
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 Δ θ 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 ( Z driven Z driving ) 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

This subsection aims to compare the R-FEM and C-FEM approaches to modeling gear meshing dynamics, with a focus on their respective load application methods and their influence on stress distribution and TVMS. Figure 9 illustrates how the torque (T) is applied to the gear body. Although computationally expedient, the localized loading assumption isolates the tooth from the gear body, yielding a nonphysical stress distribution that neglects the structural interaction between the tooth and the gear hub. Consequently, the C-FEM does not account for the influence of the base radius ( r b 1 ), a critical geometric parameter that governs the conversion of torsional stiffness into rectilinear stiffness. This omission ultimately leads to systematic errors in the derived mesh stiffness values. In contrast, the refined methodology addresses these limitations by applying the torque at a reference point ( X rp ) rigidly coupled to the gear hub. This configuration ensures load distribution through the gear body, thereby capturing the torsional compliance of the entire structure.
By enforcing kinematic constraints at X rp , the model replicates the boundary conditions of a mounted gear, enabling the accurate computation of both angular displacement ( Δ θ ) and torsional mesh stiffness ( k tor ). The inclusion of gear-body dynamics in this approach preserves the mechanical coupling between the base radius and tooth deformation, yielding results that align more closely with experimental observations. This methodological distinction underscores the necessity of comprehensive load application strategies in FEM-based gear analyses to ensure physically consistent stiffness characterization.
In Figure 10, the interaction modes between the FEM model and the control surfaces are illustrated. In the R-FEM approach, the model is coupled to RP-1 as the control point and uses the internal gear hub as the control surface. This coupling is achieved via a kinematic connection that constrains six degrees of freedom, namely, the translational displacements U 1 , U 2 , U 3 and the rotational displacements U R 1 , U R 2 , U R 3 . In contrast, the C-FEM approach employs the external gear surface as the control surface, which is a key difference compared with the R-FEM.
Under identical conditions with an applied moment of 100 N·m in the counterclockwise direction, the FE analysis reveals marked differences between the two approaches. As shown in Figure 11, which illustrates the single-contact transition, the R-FEM approach demonstrates a stress distribution aligned with the line of action (Figure 4), with stress being observed at the root of both engaged teeth, reflecting their resistance to the applied torque. In contrast, the C-FEM approach exhibits a stress distribution that does not fully align with the line of action; stress is concentrated primarily at the root of the pinion tooth, while the gear tooth subjected to the applied torque shows minimal stress.
For the double-contact scenario (Figure 12), similar phenomena are observed. The stress distribution produced by the R-FEM remains well aligned with the line of action and is evident at the root of both engaged tooth pairs. In contrast, the C-FEM approach shows significantly reduced stress levels in the gear pair compared with the pinion pair; specifically, while stress is present in the pinion tooth root, the gear tooth subjected to torque exhibits negligible stress. The contrast observed in the single-contact scenario underscores the importance of selecting appropriate FEM approaches for accurate gear stress analysis.
Both R-FEM and C-FEM approaches preserve the characteristic TVMS profile across double-contact (0–10°) and single-contact (10–19°) regions, as illustrated in Figure 13. However, distinct differences emerge in stiffness magnitudes, and in the double-contact region, the C-FEM yields values higher than the R-FEM. This discrepancy aligns with the stress analysis results (see Figure 12), where reduced stress concentrations in the C-FEM, indicative of lower localized deformation, correlate with its higher computed stiffness. During single-tooth engagement, the C-FEM retains marginally higher stiffness, though the difference diminishes relative to the double-contact phase. The stress distribution in Figure 11 clarifies this trend, as the C-FEM localizes stress predominantly at the pinion tooth root, whereas the R-FEM redistributes deformation across both meshing teeth and the gear body. The error between the methods reaches 11.23%, underscoring the non-negligible impact of the torque application strategy (hub-coupled vs. surface-loaded) on TVMS accuracy. These findings emphasize the critical role of boundary condition fidelity in gear stiffness modeling, particularly for systems operating under dynamic multi-tooth contact conditions.
A comparative analysis demonstrates that while the C-FEM exhibits good correlation with the analytical method (Figure 6), thus validating the potential energy formulation, the R-FEM is implemented in this investigation due to its superior fidelity in several critical aspects. The R-FEM approach captures realistic, nonuniform load distributions across tooth contact surfaces; accurately represents the complete three-dimensional gear geometry, including root fillets and profile modifications; characterizes complex stress distributions and localized deformations under operational loading conditions; and facilitates the precise geometric modeling of tooth defects. These capabilities are essential to fault diagnosis, as the accurate representation of stress concentrations and the resulting TVMS significantly influences the fidelity of simulated vibration signatures and dynamic response characteristics.

2.5. TVMS Sensitivity to Parameter Variations

2.5.1. Effects of Applied Torque

At a first glance, Figure 14 shows that the TVMS profiles demonstrate negligible variation across the applied torque range (6–100 N·m), particularly in the double-contact region, where stiffness trends remain consistent regardless of the loading magnitude. However, subtle distinctions arise in the single-contact region, where higher torque levels yield marginally elevated stiffness magnitudes. Notably, the transition between engagement phases becomes progressively smoother with the increase in load, as illustrated in the second hand-over region between x-axis angles [16°, 20°]. This smoother transition under elevated torque conditions indicates enhanced contact rigidity and reduced displacement variability, consistent with the load-dependent behavior of gear systems. These observations highlight the nuanced influence of torque magnitude on TVMS characteristics, particularly during transitional phases of gear engagement.

2.5.2. Impact of Gear Center Distance

The gear center distance, or internal axis (see Figure 15), is a critical design parameter that governs the spatial relationship between meshing gear pairs [14]. By using 80 mm as the reference, we vary the center distance to 79.85 mm, 79.95 mm, and 80.5 mm to assess its impact on TVMS.
As shown in Figure 16, reducing the center distance increases the TVMS magnitude and shortens the single-tooth contact region. This occurs due to interpenetration between the gears, which elevates contact forces and stiffness, causing premature tooth engagement. As a result, the single-contact phase is compressed, and the double-contact region is extended. Conversely, increasing the center distance beyond 80 mm delays engagement, elongating the single-contact region and reducing double-contact duration due to decreased tooth overlap. These findings highlight the critical influence of axis distance adjustments on dynamic load distribution in gear systems.

2.5.3. Impact of Tooth Profile

The tooth profile plays a significant role in gear dynamics [40]. In this study, we examine three cases: gears without chamfers and gears with chamfers of two sizes, 0.5 mm and 1 mm.
Introducing a chamfer (see Figure 17) leads to notable changes in the TVMS profile, as illustrated in Figure 18. The 0.5 mm chamfer smooths mesh stiffness variations without significantly affecting the duration of the single- and double-tooth contact regions. In contrast, the larger 1 mm chamfer induces a more progressive shift in the contact phases and extends the single-tooth contact period by delaying the transition to double contact. This prolongs the interval during which only one tooth pair carries the load.

2.5.4. Impact of Gear Hub Radius

The R-FEM employs a hub-coupled torque strategy, enabling it to capture stiffness variations associated with gear bore radius modifications. In this study, the pinion bore radius (green) was fixed at 17.5 mm, while the gear bore radius (red) was systematically varied, as illustrated in Figure 19.
As demonstrated in Figure 20, increasing the gear bore radius significantly elevates the stiffness magnitudes, though the characteristic stiffness profile remains intact. This trend aligns with the enhanced structural rigidity provided by larger bore geometries, which reduce torsional compliance.

2.5.5. Impact of Mesh Size

A mesh sensitivity study evaluated three finite element sizes, 0.1 mm, 0.2 mm, and 0.3 mm, as shown in Figure 21.
The results in Figure 22 show that the 0.1 mm and 0.2 mm meshes produce nearly identical TVMS profiles, while the 0.3 mm mesh introduces unstable oscillations due to insufficient contact resolution. By using an Intel® Xeon® E5-2667 v4 dual-processor system (3.20 GHz, with 96 GB of RAM), the computational time for one complete mesh cycle was 54 min for the 0.1 mm mesh and 25 min for the 0.2 mm mesh. The 0.2 mm mesh was selected as optimal, offering a 54 % reduction in computational time compared with the 0.1 mm mesh while maintaining result accuracy, which is a critical balance for FE analysis.
In addition to the computational time, the pre-processing time should also be considered, as it contributes significantly to the total computational effort.

3. Results and Discussion

This section presents the experimental findings and their analysis. We begin by describing the experimental setup used for data acquisition. Next, we detail the gear mesh stiffness calculation process. We then validate our model by comparing simulated results with experimental vibration signals obtained from the two-stage gearbox under healthy operating conditions. Finally, we extend the validation to defective gear conditions, specifically examining chipped and broken tooth fault cases.

3.1. Experimental Setup

The test bench features a two-stage parallel-shaft gearbox (Figure 23), consisting of a 29/100 tooth gear pair in the input stage and a 36/90 tooth gear pair in the output stage. A torque sensor (model 8661-5020, ± 20 Nm range), manufactured by Burster Praezisionsmesstechnik GmbH & Co. KG (Gernsbach, Germany), is positioned between the motor and the gearbox to measure the torque and speed generated by the motor. An ICP® accelerometer (model 603C00, 10 mV/g sensitivity), manufactured by PCB Piezotronics, Inc. (Depew, NY, USA), is mounted along the horizontal axis of the second-stage pinion. The electromagnetic particle (EMP) brake, acting as the load, is coupled with a fan system, collectively consuming a total of 1400 W of power.
Data are collected over a period of 10 s, with a sampling frequency of 25.6 kHz. Three cases are considered; in these cases, alongside the healthy case, the faulty pinions (Figure 24) are positioned in the second stage, and the data are collected from the intermediary shaft. The experiments are conducted at a motor speed of 2900 rpm and an input load of 3.3 Nm, representing 75 % of the maximum motor load.

3.2. Approach Validation Under Healthy Conditions

The R-FEM computes TVMS by using the parameters from Table 2. First, a quasi-static analysis evaluates deformation to determine the TVMS for each stage. A healthy gear condition establishes a baseline stiffness profile, and subsequent simulations of faulty scenarios reveal TVMS degradation under defect conditions.
Figure 25 illustrates the calculated TVMS for the first and second gear stages. The first stage (3.3 Nm) exhibits lower stiffness amplitude, while the second stage (11.73 Nm) shows a more pronounced response, reflecting the distinct configurations of each stage.
The vibration signals for all cases were simulated by incorporating the TVMS into the dynamic model (Figure 3). The simulation parameters from Table 3 were applied under operating conditions matching the experimental setting. The TVMS profile was resampled to 25.6 kHz and smoothed by using the Gaussian Weighted Moving Average (GWMA) technique, as shown in Figure 26, to mitigate rigid transitions between double- and single-contact zones.
Figure 27 presents the simulated vibration responses for the healthy case. The spectral analysis in Figure 28 shows dominant peaks at the first-stage gear mesh frequency (GMF1; 1399.74 Hz) and the second-stage gear mesh frequency (GMF2; 503.921 Hz), with a clear harmonic at 1007.84 Hz. These findings confirm proper engagement dynamics in the two-stage transmission system under healthy conditions.

3.3. Approach Validation Under Faulty Conditions

3.3.1. Chipped Gear Fault

During single-tooth meshing cycles involving the chipped gear, where one-third of the tooth is broken, the reduced contact area leads to increased deformation and a significant drop in effective stiffness (Figure 29). The stiffness loss is more pronounced during single-tooth contact, as the compromised tooth bears the load alone, whereas during double-tooth contact, the neighboring intact tooth partially compensates for the defect.
Figure 30 presents the simulated vibration responses for the chipped tooth condition. The spectral analysis in Figure 31 reveals distinct gear mesh frequencies and prominent side bands, which serve as reliable indicators of localized gear faults.

3.3.2. Broken Tooth Gear Fault

In the broken tooth case, the stiffness reduction is even more pronounced (Figure 32). During the double-tooth contact phase, only one tooth is engaged, effectively halving the load-bearing capacity. In the subsequent single-tooth contact phase, the effective stiffness drops abruptly, as the broken tooth is unable to transfer load effectively.
To mitigate the high-frequency content resulting from abrupt stiffness changes, a low-pass Butterworth filter with a 1500 Hz cutoff was applied to the simulated signal. This filter attenuates transient impulses while preserving the gear mesh frequencies.
Figure 33 presents the simulated vibration responses for the broken tooth condition. The spectral analysis of Figure 34 reveals heightened side-band prominence relative to baseline conditions, confirming the model’s capability to capture defect-induced modulation effects. These findings are consistent with established contact mechanic principles governing gear fault dynamics under reduced stiffness conditions.

3.4. General Discussion

The results demonstrate the effectiveness of the hybrid FEM–dynamic model in simulating gearbox system behavior. The generated vibration signals accurately capture the principal dynamic characteristics of the physical system, particularly the gear mesh frequencies of both stages, which closely match experimental measurements (Figure 27 and Figure 28). The R-FEM proves to be a reliable method for calculating the TVMS of gear pairs and effectively capturing gear fault signatures, as evidenced in both faulty case studies. In the chipped tooth scenario, Figure 29 clearly shows the characteristic stiffness drop, while the simulated signal (Figure 31) exhibits distinct impulses in the mesh frequency period of the faulty gear in the second stage. Although both simulated and experimental signals have comparable overall amplitudes, the fault-related peaks are more pronounced in the simulation. This discrepancy can be attributed to physical phenomena such as friction, lubrication effects, and transfer path characteristics through the gearbox housing, which were not incorporated into the dynamic model. These factors modulate the experimental signal and highlight a limitation of the current approach that should be addressed in future studies.
In the frequency domain, while the principal mesh frequencies are well represented in both signals with similar amplitudes, the experimental spectrum exhibits richer side-band content due to the aforementioned physical phenomena. Despite these differences, fault signatures remain identifiable in both spectra, confirming the fundamental accuracy of the proposed method. In the broken tooth case, a significant TVMS reduction of approximately 10 8 N/m is observed (Figure 32). The application of a GWMA smoothing operation is crucial to mitigating transition effects between single- and double-tooth contact regions, resulting in more consistent vibration patterns. Nonetheless, a substantial stiffness discontinuity persists in the vibration response (Figure 33), with simulated signal amplitudes reaching ±15 g compared with the experimental measurements constrained within ±3 g. This amplitude discrepancy, along with the more prominent side bands in the simulated vibration spectrum (Figure 34), suggests that additional mechanical interactions in real systems are not fully captured by the model.
The limitations of the current study include the exclusion of certain dynamic phenomena such as friction, lubrication effects, and detailed transfer path characteristics, as well as the absence of considerations for tool wear during manufacturing. These factors play a critical role in real-world gearbox operation and impact the accuracy of fault signature reproduction.
Future research should focus on incorporating these additional phenomena into the dynamic model. Specific directions for future work include the following:
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

This study presented a hybrid approach that integrates Refined FEM-based TVMS estimation with a dynamic model of a two-stage gearbox to simulate vibration responses in fault scenarios. The method was validated against experimental data, demonstrating its ability to reproduce key vibration signatures associated with the gearbox’s dynamic characteristics. A comprehensive parametric sensitivity analysis revealed that while TVMS remains largely stable under varying torque in double-contact regions, subtle increases occur in single-contact zones with higher loads. Furthermore, adjustments to gear center distance and tooth profile chamfer significantly affect the duration of contact phases and stiffness amplitude, thereby influencing the dynamic behavior. Analyses of gear hub radius and mesh resolution further underscore the importance of optimizing model parameters to balance accuracy and computational efficiency. Overall, the proposed framework offers a robust tool for enhancing gearbox fault diagnosis, with potential for further extension to incorporate additional dynamic interactions and operating conditions. Future investigations should include additional phenomena, such as transfer function, lubrication parameters, and other influential factors, to more accurately replicate the experimental vibration signatures.

Author Contributions

Conceptualization, A.E.A. and A.S.; methodology, A.E.A. and B.E.Y.; software, A.E.A. and B.E.Y.; validation, A.E.A.; formal analysis, A.E.A. and A.S.; investigation, A.E.A.; resources, A.S. and F.G.; data curation, A.E.A.; writing—original draft preparation, A.E.A. and B.E.Y.; writing—review and editing, A.E.A. and A.S.; visualization, A.E.A.; supervision, A.S. and F.G.; project administration, A.S. and F.G.; funding acquisition, A.S. and F.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The datasets presented in this article are not readily available because the data are part of an ongoing study.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
TVMSTime-Varying Mesh Stiffness
FEMFinite Element Method
FEfinite element
R-FEMRefined Finite Element Method
C-FEMConventional Finite Element Method
CBMcondition-based monitoring
DOFsdegrees of freedom
RPreference point
3Dthree-dimensional
EMPelectromagnetic particle
GWMAGaussian Weighted Moving Average
GMFgear mesh frequency

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Figure 1. Gears fault types (LASPI).
Figure 1. Gears fault types (LASPI).
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Figure 2. Flowchart of hybrid FEM–dynamic modeling approach.
Figure 2. Flowchart of hybrid FEM–dynamic modeling approach.
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Figure 3. Gearbox lumped parameter model illustrating key components and their interactions [22].
Figure 3. Gearbox lumped parameter model illustrating key components and their interactions [22].
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Figure 4. Pinion–gear tooth engagement along the line of action. Point P indicates the mesh point, and F represents the gear mesh force.
Figure 4. Pinion–gear tooth engagement along the line of action. Point P indicates the mesh point, and F represents the gear mesh force.
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Figure 5. Gear tooth model used in potential energy method [35].
Figure 5. Gear tooth model used in potential energy method [35].
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Figure 6. Gear mesh stiffness with the potential energy method.
Figure 6. Gear mesh stiffness with the potential energy method.
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Figure 7. Three-dimensional gear model with zoom-in contact zone.
Figure 7. Three-dimensional gear model with zoom-in contact zone.
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Figure 8. A comparison of the tooth geometry before and after deformation, illustrating the deflection δ of the tooth.
Figure 8. A comparison of the tooth geometry before and after deformation, illustrating the deflection δ of the tooth.
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Figure 9. Torque applied at the external gear surface: (A) R-FEM and (B) C-FEM.
Figure 9. Torque applied at the external gear surface: (A) R-FEM and (B) C-FEM.
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Figure 10. Interaction modes: (A) the C-FEM with the external gear surface as the control surface; (B) the R-FEM with the internal gear hub as the control surface.
Figure 10. Interaction modes: (A) the C-FEM with the external gear surface as the control surface; (B) the R-FEM with the internal gear hub as the control surface.
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Figure 11. Single-contact comparison of R-FEM and C-FEM approaches.
Figure 11. Single-contact comparison of R-FEM and C-FEM approaches.
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Figure 12. Double-contact comparison of R-FEM and C-FEM approaches.
Figure 12. Double-contact comparison of R-FEM and C-FEM approaches.
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Figure 13. TVMS profiles comparing R-FEM and C-FEM.
Figure 13. TVMS profiles comparing R-FEM and C-FEM.
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Figure 14. TVMS variation with load.
Figure 14. TVMS variation with load.
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Figure 15. Distance between the centers of two gears.
Figure 15. Distance between the centers of two gears.
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Figure 16. TVMS variation with interaxial distance.
Figure 16. TVMS variation with interaxial distance.
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Figure 17. Tooth corner (dashed circle) without chamfer in (1) and with chamfer in (2).
Figure 17. Tooth corner (dashed circle) without chamfer in (1) and with chamfer in (2).
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Figure 18. Chamfer geometry impact on TVMS.
Figure 18. Chamfer geometry impact on TVMS.
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Figure 19. Bore radii of the pinion (left) and the gear (right).
Figure 19. Bore radii of the pinion (left) and the gear (right).
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Figure 20. TVMS variation with different gear bores.
Figure 20. TVMS variation with different gear bores.
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Figure 21. Mesh contact zone element size.
Figure 21. Mesh contact zone element size.
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Figure 22. Mesh size elements impact on TVMS.
Figure 22. Mesh size elements impact on TVMS.
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Figure 23. Albert test bench at LASPI.
Figure 23. Albert test bench at LASPI.
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Figure 24. Faulty pinions (LASPI).
Figure 24. Faulty pinions (LASPI).
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Figure 25. TVMS of stage 1 and stage 2, calculated by using R-FEM for healthy gears.
Figure 25. TVMS of stage 1 and stage 2, calculated by using R-FEM for healthy gears.
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Figure 26. The TVMS of the two healthy stages: original (left) and smoothed (right).
Figure 26. The TVMS of the two healthy stages: original (left) and smoothed (right).
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Figure 27. Experimental and simulated vibrations for the healthy case.
Figure 27. Experimental and simulated vibrations for the healthy case.
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Figure 28. Experimental and simulated vibration spectra for healthy case.
Figure 28. Experimental and simulated vibration spectra for healthy case.
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Figure 29. Stage 2 chipped tooth TVMS.
Figure 29. Stage 2 chipped tooth TVMS.
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Figure 30. Experimental and simulated vibrations for chipped tooth condition.
Figure 30. Experimental and simulated vibrations for chipped tooth condition.
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Figure 31. Experimental and simulated vibration spectra for chipped tooth condition.
Figure 31. Experimental and simulated vibration spectra for chipped tooth condition.
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Figure 32. Stage 2 broken tooth case.
Figure 32. Stage 2 broken tooth case.
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Figure 33. Experimental and simulated vibrations for broken tooth condition.
Figure 33. Experimental and simulated vibrations for broken tooth condition.
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Figure 34. Experimental and simulated vibration spectra for broken tooth condition.
Figure 34. Experimental and simulated vibration spectra for broken tooth condition.
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Table 1. Spur gear parameters [38].
Table 1. Spur gear parameters [38].
ParameterPinionGear
Number of teeth (Z)1931
Base circle radius (mm)28.346.2
Module (mm)3.2
Pressure angle20°
Face width (mm)38.1
Young’s modulus (MPa)2.068 × 105
Poisson’s ratio0.3
Table 2. Geometrical properties of gear pairs.
Table 2. Geometrical properties of gear pairs.
ParameterPinion 1Gear 1Pinion 2Gear 2
Number of teeth291003690
Bore radius (mm)10151015
Base circle radius (mm)70.477063.4293
Face width (mm)15
Modulus (mm)1.5
Table 3. Dynamic and simulation parameters.
Table 3. Dynamic and simulation parameters.
ParameterFirst StageSecond StageBearingsMotor
Pinion mass (kg)0.160.294
Gear mass (kg)1.741.79
Pinion inertia (kg·m2)4.76 × 10−51.21 × 10−4
Gear inertia (kg·m2)4.82 × 10−33.89 × 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 ratio0.3
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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

AMA Style

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 Style

El 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 Style

El 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

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