Introduction
Gear transmission systems are critical components in locomotive operations, responsible for transferring torque from traction motors to wheels. However, under complex wheel-rail excitations—such as wheel polygonal wear and rail corrugation—cracks often initiate and propagate at gear tooth roots, leading to premature failure. This study investigates the dynamic contact characteristics and stress intensity factors (SIFs) of cracked gears under such excitations. A numerical approach is employed to calculate time-varying meshing stiffness, establish a six-degree-of-freedom (6-DOF) gear dynamics model, and simulate crack propagation using finite element analysis (FEA). The results reveal how wheel-rail excitations exacerbate crack growth rates and compromise gear longevity.
Time-Varying Meshing Stiffness of Cracked Gears
The meshing stiffness of gears with root cracks is calculated using an energy-based numerical method. The total potential energy U during gear meshing comprises Hertzian contact energy Uh, bending energy Ub, shear energy Us, and axial compression energy Ua. These components relate to stiffness parameters as:U=2pF2,
where F is the meshing force, and p represents stiffness. The total time-varying meshing stiffness pt is expressed as:pt=ph1+pb11+ps11+pf11+pa11+pb21+ps21+pf21+pa211.
Here, subscripts 1 and 2 denote driving and driven gears, respectively.
Key Findings:
- Meshing stiffness decreases by 18% and 32% for crack depths of 1 mm and 3 mm, respectively.
- Crack propagation angles beyond 30° accelerate stiffness reduction (Table 1).
Table 1: Meshing Stiffness Reduction vs. Crack Depth
| Crack Depth (mm) | Stiffness Reduction (%) |
|---|---|
| 1 | 18 |
| 2 | 25 |
| 3 | 32 |
6-DOF Gear Dynamics Model
A 6-DOF lumped-parameter model incorporates translational and rotational motions of driving/driven gears. The governing equations are:⎩⎨⎧m1j¨1+cj1j˙1+pj1j1−Ff=0,m2j¨2+cj2j˙2+pj2j2+Ff=0,m1z¨1+cz1z˙1+pz1z1+(ptz+ctz˙)=0,m2z¨2+cz2z˙2+pz2z2−(ptz+ctz˙)=0,I1θ¨1−Rb1(ptz+ctz˙)−M1=T1,I2θ¨2+Rb2(ptz+ctz˙)−M2=−T2,
where m, c, p, I, Rb, M, and T denote mass, damping, stiffness, inertia, base radius, friction torque, and external torque, respectively.
Dynamic Response Analysis:
- Newmark-β simulations show that crack depths >2 mm induce violent vibration shocks (Figure 1).
- Pulse factor (PF) is the most sensitive statistical indicator for crack detection (Table 2).
Table 2: Sensitivity of Statistical Indicators to Crack Depth
| Indicator | Sensitivity (1 mm) | Sensitivity (3 mm) |
|---|---|---|
| RMS | 0.12 | 0.28 |
| Kurtosis | 0.18 | 0.35 |
| Pulse Factor | 0.24 | 0.42 |
Finite Element Analysis of Cracked Gears
A locomotive gear pair (Table 3) is modeled in ABAQUS to analyze contact stresses and SIFs under wheel-rail excitations.
Table 3: Gear Parameters
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Module (mm) | 8 | 8 |
| Pressure Angle (°) | 20 | 20 |
| Number of Teeth | 23 | 120 |
Key Observations:
- Stress concentration occurs at the tooth root (Figure 2).
- Maximum contact stress increases by 240% under 24th-order wheel polygon excitation.
Influence of Wheel-Rail Excitations
Wheel Polygon Excitation
- 18th/19th-Order Polygons: Contact force peaks at 1,100 N.
- 24th-Order Polygon: Contact force surges to 4,125 N (3× baseline).
Table 4: Peak Contact Forces Under Wheel Polygon Excitation
| Polygon Order | Peak Force (N) |
|---|---|
| 18 | 1,100 |
| 19 | 1,200 |
| 24 | 4,125 |
Rail Corrugation Excitation
- Corrugation depth >1.0 mm increases single-tooth contact forces by 6.4× (Figure 3).
- Crack-tip SIFs for Mode-I cracks dominate over Mode-II (Table 5).
Table 5: Stress Intensity Factors (SIFs) vs. Load
| Load (N) | KI (MPa√m) | KII (MPa√m) |
|---|---|---|
| 800 | 12.4 | 0.8 |
| 1,000 | 15.6 | 1.2 |
| 1,200 | 18.9 | 1.5 |
Crack Propagation Mechanism
- Mode-I Dominance: KI exceeds KII by 12–15×, confirming tensile-driven crack growth.
- J-Integral Analysis: J-values correlate with SIFs, validating fracture energy release rates (Figure 4).
Critical Insights:
- Wheel-rail excitations elevate SIFs by 52%, accelerating crack propagation.
- A 3 mm crack reduces gear lifespan by 40% under 24th-order polygon excitation.
Conclusion
This study quantifies the detrimental effects of wheel-rail excitations on cracked gears. Key outcomes include:
- Time-varying meshing stiffness declines nonlinearly with crack depth and propagation angle.
- Pulse factor (PF) is optimal for early crack detection.
- Wheel polygon and rail corrugation excitations amplify contact forces by 3–6×, exacerbating SIFs.
- Mode-I cracks dominate failure mechanisms, with KI being critical for lifespan prediction.
These findings underscore the need for condition-monitoring strategies to mitigate crack-induced failures in locomotive gears.
Equations and Variables
- Hertzian Contact Stiffness:
ph=4(1−ν2)πE,
where E is Young’s modulus and ν is Poisson’s ratio.
- Stress Intensity Factor:
KI=Yσπa,
where Y is geometry factor, σ is applied stress, and a is crack length.
- J-Integral:
J=∫Γ(Wdy−Ti∂x∂uids),
where W is strain energy density, Ti is traction vector, and ui is displacement.