The study of gear dynamics is fundamentally linked to the accurate characterization of internal excitations, with time-varying mesh stiffness (TVMS) standing as a primary parameter. This is particularly critical for herringbone gears, which are renowned for their high load capacity, compact structure, and ability to balance axial forces, making them indispensable in heavy-duty applications such as naval propulsion, turbine drives, and heavy vehicle transmissions. The operational integrity of these systems is continuously challenged by surface and structural degradation mechanisms. Among these, pitting corrosion—a surface-initiated fatigue failure—and root cracks—a structural fault—are prevalent. While significant research exists on the individual impact of cracks or pitting on the TVMS of spur and helical gears, the coupled effect of these compound faults on herringbone gears remains insufficiently explored. The complex three-dimensional geometry of herringbone gears, involving dual helical sections, necessitates a specialized analytical approach to model stiffness degradation under combined faults accurately. This work, therefore, presents a comprehensive methodology to calculate the TVMS of herringbone gears subjected to concurrent pitting and crack faults, integrating modified potential energy principles with a slicing technique to account for the unique gear geometry and fault interactions.
The foundation of the TVMS calculation lies in the potential energy method, which considers the elastic energy stored in the gear teeth during meshing. For a healthy gear tooth, the total stiffness is derived from several components: Hertzian contact stiffness ($k_h$), bending stiffness ($k_b$), shear stiffness ($k_s$), and axial compressive stiffness ($k_a$). For a herringbone gear, the model is applied to individual thin slices along the face width, and the total stiffness is the sum of the stiffness contributions from all slices in contact. The mesh stiffness for a single slice $k$ is given by:
$$
\frac{1}{k_{km}} = \frac{1}{k_{kh}} + \frac{1}{k_{kb1}+k_{ks1}+k_{ka1}} + \frac{1}{k_{kb2}+k_{ks2}+k_{ka2}}
$$
where subscripts 1 and 2 denote the pinion and gear slices, respectively. The total mesh stiffness $K_m$ over the complete meshing cycle is the summation across all engaged slices $j$:
$$
K_m = \sum_{j} k_{km}
$$
The stiffness components for a healthy tooth slice are calculated as follows. The Hertzian contact stiffness, representing the localized contact deformation, is constant for a given slice and material:
$$
k_h = \frac{\pi E l}{4(1-\nu^2)}
$$
where $E$ is Young’s modulus, $l$ is the slice contact length, and $\nu$ is Poisson’s ratio. The bending, shear, and axial stiffnesses are derived from strain energy. For a force $F$ applied at an angle $\alpha_1$ to the tooth centerline, with components $F_b = F\cos\alpha_1$ and $F_a = F\sin\alpha_1$, the energy expressions and corresponding stiffnesses are found by integration along the tooth height $d$:
$$
\frac{1}{k_b} = \int_{0}^{d} \frac{[F_b (d-x) – F_a h_x]^2}{2E I_x} dx, \quad \frac{1}{k_s} = \int_{0}^{d} \frac{1.2 F_b^2}{2G A_x} dx, \quad \frac{1}{k_a} = \int_{0}^{d} \frac{F_a^2}{2E A_x} dx
$$
Here, $I_x$ is the area moment of inertia, $A_x$ is the cross-sectional area, $h_x$ is the distance from the load point to the neutral axis at section $x$, and $G$ is the shear modulus.
Modeling faults requires modifying these geometric and integral parameters. A pitting fault is modeled as a cylindrical void on the tooth flank. Its influence is characterized by a local reduction in the effective contact length ($\Delta l_x$) and cross-sectional area ($\Delta A_x$), which in turn affects the Hertzian stiffness and the moment of inertia ($\Delta I_x$) in the energy integrals over the pitted region. For a single pit with radius $R_{sp}$ and depth $d_{sp}$, centered at a distance $u$ from the root, these reductions are:
$$
\Delta l_x = \begin{cases}
2\sqrt{R_{sp}^2 – (u-x)^2}, & x \in [u-R_{sp}, u+R_{sp}] \\
0, & \text{otherwise}
\end{cases}
$$
$$
\Delta A_x = \begin{cases}
\Delta l_x \cdot d_{sp}, & x \in [u-R_{sp}, u+R_{sp}] \\
0, & \text{otherwise}
\end{cases}
$$
The reduction in the moment of inertia is more complex, accounting for the shift in the neutral axis due to the missing material. A root crack is modeled as a straight-line propagation from the root fillet at an angle $\xi$. The primary effect of a crack is to reduce the effective area $A_x$ and moment of inertia $I_x$ along a significant portion of the tooth root, drastically altering the integrals for $k_b$ and $k_s$. The critical parameter is the crack depth $q_1$. When both a crack and pitting are present, the modifications to $A_x$ and $I_x$ must account for their overlapping or separate domains. In severe cases where the crack-damaged zone envelops the pitted zone, the fault model becomes dominated by the crack parameters, with pitting causing an additional, secondary reduction in contact length. The generalized integrals for the bending stiffness under a compound fault become segmented, evaluating different expressions for the healthy, solely cracked, coupled crack-pit, and solely pitted sections along the tooth. The shear stiffness calculation follows a similar segmented integration approach.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth, $Z$ | 25 | 60 |
| Module, $m_n$ (mm) | 2 | 2 |
| Helix Angle, $\beta$ (°) | 15 | 15 |
| Pressure Angle, $\alpha_n$ (°) | 20 | 20 |
| Face Width, $L$ (mm) | 40 | 40 |
| Young’s Modulus, $E$ (GPa) | 210 | 210 |
| Poisson’s Ratio, $\nu$ | 0.3 | 0.3 |
The proposed analytical model was applied to the herringbone gear pair detailed in Table 1. Various fault scenarios were analyzed: individual crack faults (10% mild, 70% severe crack depth ratio), individual pitting faults (mild: one pit of R=0.3mm, moderate: one pit of R=0.4mm, severe: two pits of R=0.4mm), and their compound combinations. The TVMS over one mesh cycle was computed for each case. The results for healthy gears show the characteristic periodic fluctuation due to changing number of tooth pairs in contact. The introduction of any fault reduces the average TVMS and alters its waveform.
| Fault Type | Severity Level | TVMS (% of Healthy) | Dominant Mechanism |
|---|---|---|---|
| Crack Only | Mild (10%) | ~94% | Reduced root bending section |
| Severe (70%) | ~85.5% | Severely compromised tooth integrity | |
| Pitting Only | Mild | ~98% | Local contact length reduction |
| Moderate | ~96% | Increased area/contact length loss | |
| Severe | ~94.9% | Significant flank material loss | |
| Compound (Mild Crack + Pitting) | + Mild Pitting | ~92% | Combined root and surface effect |
| + Severe Pitting | ~90% | Pitting initially more influential in mid-mesh | |
| Compound (Severe Crack + Pitting) | + Mild Pitting | < ~85.5% | Crack dominates, pitting effect marginal |
| + Severe Pitting | < ~85.5% | Crack fully dominates, waveform distortion |
Analysis of the compound fault scenarios reveals intricate interactions. For a mild crack coupled with pitting, the TVMS curve shows characteristics of both faults. The pitting causes a distinct local dip in stiffness when the meshing occurs over the pitted region, superimposed on the overall reduced stiffness baseline set by the crack. As pitting severity increases in this case, its influence on the TVMS reduction becomes more pronounced relative to the mild crack. The formula for the combined stiffness in such interactive zones must account for the simultaneous reduction in $A_x$ and $I_x$ from the crack and the additional loss from $\Delta A_x$ and $\Delta I_x$ due to pitting within the same segment.
$$
I_{x,\text{compound}} = \frac{1}{12}(2h_x – d_{sp})^3 l_1 \quad \text{for } x \text{ in coupled region}
$$
In contrast, for a severe crack coupled with any level of pitting, the TVMS is overwhelmingly dominated by the crack. The severe crack so drastically reduces the load-bearing cross-section that the additional material loss from pitting has a negligible further impact on bending and shear stiffness. However, the pitting still contributes by reducing the effective contact length, which affects the Hertzian stiffness component $k_h$. This leads to a TVMS that is even lower than that from the severe crack alone, but the waveform’s major fluctuations are dictated by the crack’s geometry. The transition point where the crack zone subsumes the pitting zone is a key feature of the model, mathematically handled by conditional statements in the integrals for $A_x$ and $I_x$.
To validate the analytical model, a three-dimensional finite element analysis (FEA) was conducted for both a healthy gear pair and one with a mild compound fault. The FEA model applied realistic boundary conditions and contact definitions. The TVMS was extracted from the FEA results by relating the applied torque to the resulting angular deflection. The comparison showed excellent agreement between the analytical and FEA results for the healthy gear. For the compound fault case, the analytical model predicted the TVMS with a high degree of accuracy, with a maximum discrepancy of approximately 2.5% over the meshing cycle. This close correlation validates the underlying assumptions of the analytical model, including the slicing technique, the fault geometry approximations, and the energy method formulation for herringbone gears.
In conclusion, this study establishes a robust analytical framework for calculating the time-varying mesh stiffness of herringbone gears with compound pitting and crack faults. The model effectively captures the individual and coupled effects of these common gear failures. The key findings are that pitting primarily reduces stiffness through loss of contact area and localized flank weakening, while cracks cause a more severe and global reduction by compromising the tooth root structure. In compound faults, a severe crack dominates the TVMS response, marginalizing the additional effect of pitting on bending stiffness, though pitting still influences contact stiffness. The validated model provides a critical tool for predicting the dynamic excitation in herringbone gear systems undergoing progressive degradation, which is essential for condition monitoring, fault diagnosis, and remaining life prediction in high-performance mechanical drives. Future work could integrate this TVMS model into a full nonlinear dynamic model of a herringbone gear transmission system to simulate vibration responses under compound faults.