In modern mechanical transmission systems, helical gears are widely used due to their smooth operation, high load capacity, and reduced noise compared to spur gears. However, under complex and variable loading conditions, helical gears are prone to developing cracks, which significantly affect the meshing stiffness and overall system dynamics. As a key parameter in gear vibration analysis, the time-varying meshing stiffness (TVMS) must be accurately calculated to predict system behavior and prevent failures. In this study, I propose an improved analytical model for calculating the meshing stiffness of helical gear pairs under crack conditions, considering both transverse and axial influences on tooth stiffness and foundation stiffness. This model addresses two typical crack propagation paths: tooth tip propagation cracks and end face propagation cracks. By integrating these factors, I aim to provide a more precise representation of how cracks degrade meshing stiffness in helical gears, and validate the results through finite element simulations.
The importance of helical gears in applications such as wind turbines, automotive transmissions, and industrial machinery cannot be overstated. Their helical tooth design allows for gradual engagement, resulting in smoother torque transmission and higher durability. However, this complexity also makes them susceptible to crack initiation and propagation under dynamic loads. Cracks in helical gears can lead to reduced meshing stiffness, altered vibration characteristics, and eventual system failure if not detected early. Traditional models often overlook the combined effects of transverse and axial stiffness components or simplify crack propagation paths, leading to inaccuracies. My approach builds on existing energy-based methods and slice theory to develop a comprehensive model that accounts for non-uniform crack depths and paths along the gear width. This enables a detailed analysis of how different crack types and extents influence the meshing stiffness of helical gears.
To model crack propagation in helical gears, I consider two primary types: tooth tip propagation cracks and end face propagation cracks. For tooth tip cracks, the crack initiates at the tooth root and propagates towards the tip, while for end face cracks, it starts at one end of the gear face and extends across the width. The crack path is assumed to follow a parabolic curve to simulate real-world scenarios where cracks do not propagate uniformly. In the coordinate system o’xyz, with the origin at the root circle, the x-axis along the tooth centerline, the y-axis tangent to the root circle, and the z-axis parallel to the tooth width, the crack propagation equation can be expressed as:
$$ x = \begin{cases}
l_1 + l_2 \times z^2 / L^2 & z \in [0, L] \\
l_1 + (r_a – r_f – l_1 – \Delta h) \times z^2 / l_3^2 & z \in [0, l_3]
\end{cases} $$
Here, $l_1$ is the distance from the initial crack point to the tooth root, $l_2$ and $l_3$ represent the crack extension along the tooth depth and face width, respectively, and $\Delta h$ accounts for geometric changes due to rotation. For non-penetrating cracks at the tooth tip, the depth $q_{n1}(z)$ varies linearly with $z$, while for penetrating cracks, $q_{c1}(z)$ follows a quadratic function. Similarly, for end face cracks, non-penetrating depth $q_{n2}(z)$ and penetrating depth $q_{c2}(z)$ are defined based on the crack length and depth parameters. These equations allow me to model the effective crack area, which directly impacts the meshing stiffness of helical gears.
The meshing stiffness of a helical gear pair is calculated using the slice method, where the gear is divided into multiple thin slices along the tooth width. Each slice is treated as a spur gear pair with a specific offset angle, and the total meshing stiffness is obtained by summing the contributions from all slices. The transverse meshing stiffness includes tooth stiffness components such as bending, shear, and axial compression stiffness, as well as foundation stiffness. The axial meshing stiffness considers bending and torsional effects due to the helical angle. The overall meshing stiffness $K_{\text{total}}(\tau)$ is derived by combining the transverse and axial components as follows:
$$ \frac{1}{K_{\text{total}}(\tau)} = \frac{1}{\sum_{n=1}^{N} \sum_{i=1}^{\text{cell}(e)} k_t^i(\tau + \Delta T)} + \frac{1}{\sum_{n=1}^{N} \sum_{i=1}^{\text{cell}(e)} k_a^i(\tau + \Delta T)} $$
where $k_t^i$ and $k_a^i$ are the transverse and axial stiffnesses of the i-th tooth pair in the n-th slice, $\tau$ is the normalized time, and $\Delta T$ is the offset time due to the helical angle. The transverse tooth stiffness $k_t^i$ is computed using energy methods, considering the potential energy stored in bending, shear, and axial deformation under load. For a cracked tooth, the effective area moment of inertia and cross-sectional area are modified based on the crack depth, leading to reduced stiffness. The foundation stiffness accounts for the deformation of the gear body under load, and it is adjusted for cracks using a correction factor based on the crack length ratio $X = q / L$.
For transverse tooth stiffness, the bending, shear, and axial compression stiffnesses are given by:
$$ \frac{1}{k_{tb}} = \int_0^d \frac{[\cos\alpha_1 (d – x_1) – \sin\alpha_1 h]^2 \cos^2\beta}{E I_{x1}} dx_1 + \int_0^{x_C – x_D} \frac{[\cos\alpha_1 (d + x_2) – \sin\alpha_1 h]^2 \cos^2\beta}{E I_{x2}} dx_2 $$
$$ \frac{1}{k_{ts}} = \int_0^d \frac{1.2 \cos^2\alpha_1 \cos^2\beta}{G A_{x1}} dx_1 + \int_0^{x_C – x_D} \frac{1.2 \cos^2\alpha_1 \cos^2\beta}{G A_{x2}} dx_2 $$
$$ \frac{1}{k_{ta}} = \int_0^d \frac{\sin^2\alpha_1 \cos^2\beta}{E A_{x1}} dx_1 + \int_0^{x_C – x_D} \frac{\sin^2\alpha_1 \cos^2\beta}{E A_{x2}} dx_2 $$
where $E$ is the elastic modulus, $G$ is the shear modulus, $\beta$ is the helix angle, $\alpha_1$ is the pressure angle at the contact point, and $I_{x1}$, $I_{x2}$, $A_{x1}$, $A_{x2}$ are the moment of inertia and cross-sectional area in the transition and involute regions, respectively, modified for cracks. The transverse foundation stiffness $k_{tf}$ is calculated as:
$$ \frac{1}{dk_{tf}} = \frac{\cos^2\beta \cos^2\alpha’}{E \cdot dL} \left[ L^* (u_f’ / S_f’)^2 + M^* (u_f’ / S_f’) + P^* (1 + Q^* \tan^2\alpha’) \right] $$
where $u_f’$ and $S_f’$ are corrected parameters based on the crack, and $L^*$, $M^*$, $P^*$, $Q^*$ are coefficients derived from gear geometry.
For axial stiffness, the bending and torsional components are considered. The axial bending stiffness $k_{ab}$ and axial torsional stiffness $k_{at}$ are expressed as:
$$ \frac{1}{k_{ab}} = \int_0^d \frac{\sin^2\beta (d – x_1)^2}{E I_{ax1}} dx_1 + \int_0^{x_C – x_D} \frac{\sin^2\beta (d + x_2)^2}{E I_{ax2}} dx_2 $$
$$ \frac{1}{k_{at}} = \int_0^d \frac{h^2 \sin^2\beta}{G I_{px1}} dx_1 + \int_0^{x_C – x_D} \frac{h^2 \sin^2\beta}{G I_{px2}} dx_2 $$
where $I_{ax1}$, $I_{ax2}$ are the area moments of inertia, and $I_{px1}$, $I_{px2}$ are the polar moments of inertia for the cracked sections. The axial foundation stiffness $k_{af}$ is derived from the deformation of a semi-circular cantilever beam under friction torque:
$$ \frac{1}{k_{af}} = \int_0^{x_D} \frac{\sin^2\beta (d + x_C – x_d)}{E I_{af}} dx_d $$
with $I_{af}$ being the moment of inertia of the cracked foundation section.
To validate the model, I applied it to a double-helical gear pair with parameters listed in Table 1. The gears are subjected to various crack conditions, and the meshing stiffness is computed analytically and compared with finite element analysis (FEA) results. The FEA models are built in ANSYS, considering the same crack geometries and loading conditions.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 42 | 43 |
| Face Width (mm) | 30 | 30 |
| Module (mm) | 3.5 | 3.5 |
| Pressure Angle (°) | 22.5 | 22.5 |
| Helix Angle (°) | 17 | 17 |
| Elastic Modulus E (Pa) | 2.068e11 | 2.068e11 |
| Poisson’s Ratio ν | 0.3 | 0.3 |
| Input Torque (N·m) | 119 | – |
| Input Speed (r/min) | 1500 | – |
For tooth tip propagation cracks, I analyzed both penetrating and non-penetrating cracks with different parameters, as summarized in Table 2. The meshing stiffness results show that as the crack extends along the tooth width, the stiffness reduction becomes more pronounced. For example, in penetrating cracks with different end positions, a longer crack length $L_c$ leads to a greater decrease in stiffness. Similarly, for non-penetrating cracks, increasing the effective crack length reduces stiffness, but the effect is less severe due to the smaller cracked area. Moreover, deeper cracks cause a more significant stiffness reduction, especially when the initial crack depth $q_s$ increases from 0.6$q_{s-total}$ to 0.9$q_{s-total}$.
| Crack Type | Case | Crack Length $L_c$ | Initial Depth $q_s$ | Terminal Depth $q_e$ |
|---|---|---|---|---|
| Penetrating Cracks at Different Locations | 1 | 0.3L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ |
| 2 | 0.6L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| 3 | 0.9L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| Non-Penetrating Cracks of Different Lengths | 1 | 0.3L | 0.3$q_{s-total}$ | 0.15$q_{e-total}$ |
| 2 | 0.6L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| 3 | 0.9L | 0.9$q_{s-total}$ | 0.45$q_{e-total}$ | |
| Penetrating Cracks with Different Depths | 1 | 0.6L | 0.3$q_{s-total}$ | 0.15$q_{e-total}$ |
| 2 | 0.6L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| 3 | 0.6L | 0.9$q_{s-total}$ | 0.45$q_{e-total}$ |
The analytical results for tooth tip cracks are presented in Figure 1, which illustrates the meshing stiffness over one engagement cycle for different crack cases. The stiffness reduction is most evident when the crack penetrates deeply and extends across a large portion of the tooth width. The FEA simulations, shown in Figure 2, confirm these trends, with good agreement between the analytical and numerical results. This validates the accuracy of my model in capturing the effects of tooth tip cracks on the meshing stiffness of helical gears.
For end face propagation cracks, I examined non-penetrating and penetrating cracks with varying lengths and depths, as detailed in Table 3. The meshing stiffness decreases more significantly for end face cracks compared to tooth tip cracks, due to the larger effective cracked area. For non-penetrating cracks, longer crack lengths lead to greater stiffness reductions, as shown in Figure 3. For penetrating cracks, increasing the crack depth from 0.3$q_{s-total}$ to 0.9$q_{s-total}$ results in a dramatic drop in stiffness, highlighting the sensitivity of meshing stiffness to crack depth in helical gears.
| Crack Type | Case | Crack Length $L_c$ | Initial Depth $q_s$ | Terminal Depth $q_e$ |
|---|---|---|---|---|
| Non-Penetrating End Face Cracks | 1 | 0.3L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ |
| 2 | 0.6L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| 3 | 0.9L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| Penetrating End Face Cracks | 1 | L | 0.3$q_{s-total}$ | 0.15$q_{e-total}$ |
| 2 | L | 0.6$q_{s-total}$ | 0.3$q_{e-total}$ | |
| 3 | L | 0.9$q_{s-total}$ | 0.45$q_{e-total}$ |
The FEA results for end face cracks, depicted in Figure 4, align closely with my analytical calculations, further supporting the model’s reliability. The discussion of these results emphasizes that the reduction in meshing stiffness is primarily governed by the effective crack area. Larger crack areas, whether due to increased length or depth, lead to more substantial stiffness degradation. Additionally, end face cracks have a more pronounced effect than tooth tip cracks because they affect a larger portion of the gear tooth, disrupting both transverse and axial stiffness components more severely.
In conclusion, my improved meshing stiffness calculation model for cracked helical gears provides a comprehensive framework that integrates transverse and axial stiffness influences. By modeling crack propagation paths as parabolic curves and applying the slice method, I accurately capture the time-varying nature of meshing stiffness under different crack conditions. The results demonstrate that crack depth is the most critical parameter affecting stiffness reduction, and end face cracks pose a greater risk than tooth tip cracks. The strong agreement with FEA simulations validates the model’s accuracy and effectiveness, making it a valuable tool for fault diagnosis and dynamic analysis of helical gear systems. Future work could extend this model to include other fault types or more complex loading conditions to further enhance its applicability in real-world engineering scenarios.