The dynamic performance, vibration, and noise characteristics of gear transmission systems are fundamentally governed by internal excitations. Among these, the time-varying meshing stiffness (TVMS) of the gear pair is a primary source of parametric excitation. Its accurate calculation is paramount for reliable dynamic simulation and fault diagnosis. While significant research has focused on calculating TVMS for healthy gears and those with classical faults like single-root cracks, the scenario of bilateral cracks—where cracks initiate on both flanks of a spur gear tooth—presents a more complex and realistic failure mode, especially in applications involving bi-directional loading or uneven stress distribution. This article presents a comprehensive analytical and numerical investigation into the influence of such bilateral asymmetric cracks on the mesh stiffness of spur gears.
The foundation for calculating gear tooth deflection lies in modeling the tooth as a non-uniform cantilever beam. According to the principles of mechanics of materials, the total elastic strain energy stored in a beam under the combined action of bending moment \(M\), axial compressive force \(F_N\), and shear force \(F_S\) can be expressed as:
$$ U_T = \int_{0}^{L} \frac{M^2}{2EI_z} \, dx + \int_{0}^{L} \frac{F_N^2}{2EA} \, dx + \int_{0}^{L} \frac{3F_S^2}{5GA} \, dx $$
where \(E\) is the Young’s modulus, \(G\) is the shear modulus (\(G = E / [2(1+\nu)]\), with \(\nu\) being Poisson’s ratio), \(A\) is the cross-sectional area, \(I_z\) is the area moment of inertia, and \(L\) is the effective length of the beam (tooth). The compliance (inverse of stiffness) for each deformation mode is derived from the relationship \(U = F^2/(2k)\). For a healthy spur gear tooth, its total compliance in mesh is the sum of the compliances from Hertzian contact, bending, axial compression, and shear. For a single tooth pair in contact, the mesh stiffness \(k\) is:
$$ k = \left( \frac{1}{k_h} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} \right)^{-1} $$
where subscripts \(h, b, s, a\) denote Hertzian, bending, shear, and axial components, and subscripts \(1, 2\) denote the driving and driven gears. For spur gears with a contact ratio between 1 and 2, two pairs of teeth share the load periodically. The effective mesh stiffness during double-tooth contact is the sum of the stiffness of the two individual mating pairs.
The accurate geometry of the tooth profile is critical. A healthy spur gear tooth profile consists of four segments: the root circle, the fillet (transition) curve, the involute curve, and the tip circle. The coordinates of any point on the involute segment can be parameterized by the pressure angle \(\alpha_i\):
$$ \begin{cases} x_2 = r_i \sin \phi \\ y_2 = r_i \cos \phi \end{cases}, \quad \text{where } r_i = \frac{r_b}{\cos \alpha_i}, \quad \phi = \theta_b – \text{inv} \alpha_i $$
and \(\theta_b\) is the base half-angle and \(\text{inv} \alpha_i = \tan \alpha_i – \alpha_i\). The fillet curve generated by a rack cutter with a tip radius is more complex and can be derived from the gear generation process. The integrals for bending, axial, and shear compliances are then evaluated by segmenting the tooth profile and integrating along the height of the tooth from the root to the point of load application \(\beta\).
Cracks in gear teeth effectively reduce the load-bearing cross-section. In modeling, the region bounded by the crack line and a limiting “dead zone” line is considered inactive. The novel contribution of this work is the model for bilateral asymmetric cracks, where independent cracks exist on both the left and right flanks of a single spur gear tooth. The complexity arises in determining the effective, intact cross-sectional width \(t_e(y)\) at any height \(y\) along the tooth.
The position relationship between the projections of the upper and lower crack fronts onto the tooth’s centerline can be classified into three distinct cases, which fundamentally alter the stiffness calculation. The key parameter is the effective thickness \(t_e\).
| Projection Relationship | Description | Impact on Effective Width |
|---|---|---|
| Separated | Projections do not overlap. | Width reduction is localized to each side independently. A central ligament remains fully intact. |
| Partially Overlapping | Projections overlap partially. | The overlapping region experiences compound reduction. The remaining intact width is complex to compute. |
| Contained | One crack’s projection lies entirely within the other’s. | The tooth is effectively deeply notched on one side, with the crack on the opposite side reducing the remaining ligament further. |
Assuming the crack path can be approximated by a parabolic function for modeling simplicity, the effective cross-sectional area \(A_{ck}\) and moment of inertia \(I_{ck}\) at a given section become:
$$ A_{ck}(y) = L \cdot t_e(y) $$
$$ I_{ck}(y) = \frac{1}{12} L \cdot [t_e(y)]^3 $$
These time- and position-variant parameters \(A_{ck}\) and \(I_{ck}\) are then substituted into the integrals for bending and shear compliances, replacing their healthy-tooth counterparts \(A\) and \(I\). The Hertzian and axial compression stiffnesses are assumed to be less affected by a root crack and are often kept unchanged in such models. The mesh stiffness for a pair involving a cracked driving gear tooth (tooth 11) and a healthy driven gear tooth (tooth 21) during single-tooth contact is:
$$ k_{ts11\_ck} = \left( \frac{1}{k_h} + \frac{1}{k_{b11\_ck}} + \frac{1}{k_{s11\_ck}} + \frac{1}{k_{a11}} + \frac{1}{k_{b21}} + \frac{1}{k_{s21}} + \frac{1}{k_{a21}} \right)^{-1} $$
where \(k_{b11\_ck}\) and \(k_{s11\_ck}\) are calculated using the reduced \(A_{ck}\) and \(I_{ck}\).
To illustrate the impact, a case study is performed on a standard spur gear pair. The parameters for the healthy gears are listed below.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 36 | 54 |
| Module (mm) | 2 | 2 |
| Pressure Angle (deg) | 20 | 20 |
| Face Width (mm) | 20 | 20 |
| Young’s Modulus (GPa) | 206 | 206 |
| Poisson’s Ratio | 0.3 | 0.3 |
The TVMS over two mesh periods for the healthy gear pair, calculated using the energy method and validated by Finite Element Analysis (FEA), shows the characteristic pattern with two stiffness plateaus (double-tooth contact) and one lower plateau (single-tooth contact) per mesh cycle.
Now, introducing a bilateral crack on one tooth of the driving gear. We analyze two primary crack parameters: depth (\(q\), as a percentage of total crack length to critical length) and location (\(d\), as a percentage along the tooth profile from the root). The following tables summarize the stiffness values at key engagement points for varying crack parameters. The percentage change is relative to the healthy gear stiffness at the same meshing position.
Table 1: Effect of Crack Location (Depth \(q = 35\%\) constant)
| Meshing Phase (Driving Gear Angle) | Healthy Stiffness (N/m) | Crack Location \(d = 0\%\) | Change | Crack Location \(d = 35\%\) | Change | Crack Location \(d = 70\%\) | Change |
|---|---|---|---|---|---|---|---|
| Start of Double-Toot Contact (\(\theta_1 = 0\)) | \(6.86 \times 10^8\) | \(6.69 \times 10^8\) | -2.5% | \(6.61 \times 10^8\) | -3.6% | \(6.44 \times 10^8\) | -6.1% |
| End of Double-Toot Contact (\(\theta_1 = \theta_d\)) | \(6.90 \times 10^8\) | \(5.56 \times 10^8\) | -19.4% | \(5.11 \times 10^8\) | -25.9% | \(4.11 \times 10^8\) | -40.4% |
| Middle of Single-Toot Contact (\(\theta_1 = \theta_d\)) | \(4.31 \times 10^8\) | \(2.95 \times 10^8\) | -31.6% | \(2.49 \times 10^8\) | -42.2% | \(1.48 \times 10^8\) | -65.7% |
| End of Single-Toot Contact (\(\theta_1 = 2\pi/N_1\)) | \(4.38 \times 10^8\) | \(2.36 \times 10^8\) | -46.1% | \(1.89 \times 10^8\) | -56.9% | \(0.93 \times 10^8\) | -78.8% |
Table 2: Effect of Crack Depth (Location \(d = 0\%\) constant)
| Meshing Phase (Driving Gear Angle) | Healthy Stiffness (N/m) | Crack Depth \(q = 35\%\) | Change | Crack Depth \(q = 55\%\) | Change | Crack Depth \(q = 75\%\) | Change |
|---|---|---|---|---|---|---|---|
| Start of Double-Toot Contact (\(\theta_1 = 0\)) | \(6.86 \times 10^8\) | \(6.69 \times 10^8\) | -2.5% | \(6.59 \times 10^8\) | -3.9% | \(6.44 \times 10^8\) | -6.1% |
| End of Double-Toot Contact (\(\theta_1 = \theta_d\)) | \(6.90 \times 10^8\) | \(5.56 \times 10^8\) | -19.4% | \(4.87 \times 10^8\) | -29.4% | \(4.03 \times 10^8\) | -41.6% |
| Middle of Single-Toot Contact (\(\theta_1 = \theta_d\)) | \(4.31 \times 10^8\) | \(2.95 \times 10^8\) | -31.6% | \(2.25 \times 10^8\) | -47.8% | \(1.41 \times 10^8\) | -67.3% |
| End of Single-Toot Contact (\(\theta_1 = 2\pi/N_1\)) | \(4.38 \times 10^8\) | \(2.36 \times 10^8\) | -46.1% | \(1.56 \times 10^8\) | -64.4% | \(0.84 \times 10^8\) | -80.8% |
The data reveals crucial trends for fault diagnosis in spur gears:
- Crack Location Sensitivity: Stiffness reduction is most severe when the crack is located such that the load application point passes over the most weakened section of the tooth (typically in the single-tooth contact region). A crack closer to the tooth tip (\(d = 70\%\)) causes a more dramatic drop in stiffness at the end of the single-tooth mesh compared to a root crack (\(d = 0\%\)).
- Crack Depth Sensitivity: Unsurprisingly, deeper cracks cause greater stiffness reduction. The relationship is non-linear; a change from 35% to 55% depth causes a much larger percentage drop in stiffness than an initial 35% depth does from healthy state, especially in single-tooth contact.
- Minimal Effect at Mesh Start: At the beginning of the double-tooth contact period, the load is shared, and the cracked tooth carries only part of it while the contact point is near its root (thicker section). Therefore, the overall mesh stiffness shows only a minor reduction regardless of crack parameters.
Finite Element Analysis (FEA) was conducted to validate the proposed analytical energy method for bilateral cracks in spur gears. 2D plane-strain models of the gear tooth with various crack configurations were created. The mesh stiffness was extracted by applying a rotational displacement, calculating the reaction torque, and deriving the stiffness from the resulting angular deflection. The FEA results for different crack depths and locations showed excellent agreement with the energy method predictions, with typical discrepancies within 10%, confirming the accuracy of the derived model.
The model presented for bilateral asymmetric cracks in spur gears offers a more generalized framework for fault modeling compared to traditional single-crack models. The analysis leads to several key conclusions:
- The energy method, based on accurate tooth profile geometry and a variable effective cross-section, provides an efficient and acceptably accurate tool for calculating the TVMS of spur gears with complex bilateral cracks, as validated by FEA.
- Crack parameters (depth and location) have a coupled, non-linear influence on mesh stiffness. The stiffness reduction is most pronounced during the single-tooth contact phase and is highly sensitive to the position of the load relative to the crack tip.
- The “dead zone” assumption and the treatment of the crack path are simplifications. Real crack propagation in spur gears may follow a curved path influenced by mixed-mode stress intensity factors, which could alter the effective stiffness reduction profile.
- Future work should focus on: a) Experimental validation using strain gauges or optical techniques to measure tooth deflection under load; b) Extension of the model to three dimensions to account for crack propagation through the gear face width; c) Investigation of the interaction between bilateral cracks and other faults like pitting; and d) Integration of this stiffness model into full nonlinear dynamic models of gearbox systems to study vibration signatures for fault diagnosis.
Ultimately, understanding the precise impact of bilateral cracks on the time-varying mesh stiffness is a critical step towards developing more robust condition monitoring and remaining useful life prediction algorithms for spur gear transmission systems operating under demanding conditions.