Abstract
This study investigates the vibration response characteristics of gear systems containing tooth root cracks under steady and variable load conditions. A modified potential energy method is proposed to calculate time-varying meshing stiffness (TVMS), validated against ISO 6336-1-2006 standards. A 6-degree-of-freedom (6-DOF) dynamic model is established to analyze the influence of crack parameters on gear system performance. Sensitivity analysis using statistical indicators reveals that kurtosis is the most sensitive metric for crack detection. Key findings include: (1) TVMS decreases with crack depth and angle, with depth having a dominant effect; (2) Periodic impacts in time-domain signals and sidebands around meshing frequencies in the frequency domain correlate with crack severity; (3) Variable loads mask fault features by amplifying low-frequency components.
Introduction
Gear systems are critical components in aerospace, wind turbines, and industrial machinery. However, their operational complexity often leads to failures, with tooth fractures accounting for 41% of gear-related faults. Early-stage cracks at the tooth root alter the stiffness and dynamic response of gear systems, necessitating robust diagnostic methods. Previous studies on TVMS calculation methods—finite element analysis (FEA), potential energy method, and Ishikawa’s formula—have limitations in computational efficiency or crack characterization. This work addresses these gaps by refining the potential energy method and analyzing crack-induced vibration responses under variable loads.
Methodology
1. Time-Varying Meshing Stiffness (TVMS) Calculation
The potential energy method is enhanced to account for deformation between the base circle and root circle. The total stored energy includes bending (UbUb), shear (UsUs), and axial compression (UaUa) components:1kb=∫−α1α23(1+cosα1[(a2−a)sinα−cosα)]22EI[sinα+(a2−a)cosα]3dα+∫0a2[(d+xi)(cosαi−hsinα)]22EIadxikb1=∫−α1α22EI[sinα+(a2−a)cosα]33(1+cosα1[(a2−a)sinα−cosα)]2dα+∫0a22EIa[(d+xi)(cosαi−hsinα)]2dxi1ks=∫−α1α21.2(1+ν)(a2−a)cosαcos2αEI[sinα+(a2−a)cosα]dα+∫0a2(1.2cosα)2GAadxiks1=∫−α1α2EI[sinα+(a2−a)cosα]1.2(1+ν)(a2−a)cosαcos2αdα+∫0a2GAa(1.2cosα)2dxi1ka=∫−α1α2(a2−a)cosαsin2α2EI[sinα+(a2−a)cosα]dα+∫0a2(sinαi)2EAadxika1=∫−α1α22EI[sinα+(a2−a)cosα](a2−a)cosαsin2αdα+∫0a2EAa(sinαi)2dxi
Hertzian contact stiffness (khkh) and gear body deformation (kfkf) are incorporated to derive the total TVMS:kc=(1kb+1ks+1ka+1kh+1kf)−1kc=(kb1+ks1+ka1+kh1+kf1)−1
2. Crack Modeling
Three crack propagation scenarios are defined based on depth (qq) and angle (vv):
- Model 1: Initial crack (q<50%q<50% of qmaxqmax).
- Model 2: Moderate crack (q≈50%q≈50%).
- Model 3/4: Severe crack (q>50%q>50%), with reduced effective cross-sectional area and moment of inertia.
The inertia moment (IxIx) and area (AxAx) for cracked gears are modeled as:Ix={112(hx+hz)3L(q<50%)112(hx−hz)3L(q>50%),Ax={(hx+hz)L(q<50%)(hx−hz)L(q>50%)Ix={121(hx+hz)3L121(hx−hz)3L(q<50%)(q>50%),Ax={(hx+hz)L(hx−hz)L(q<50%)(q>50%)
3. Dynamic Model of Gear Systems
A 6-DOF lumped-parameter model incorporates translational, rotational, and torsional motions:{m1x¨1+C1xx˙1+K1xx1=−Cmδ˙sinα−Km(t)δsinαm1y¨1+C1yy˙1+K1yy1=−Cmδ˙cosα−Km(t)δcosα−m1gI1θ¨1=T1−r1(Cmδ˙+Km(t)δ)m2x¨2+C2xx˙2+K2xx2=Cmδ˙sinα+Km(t)δsinαm2y¨2+C2yy˙2+K2yy2=Cmδ˙cosα+Km(t)δcosα−m2gI2θ¨2=−T2+r2(Cmδ˙+Km(t)δ)⎩⎨⎧m1x¨1+C1xx˙1+K1xx1=−Cmδ˙sinα−Km(t)δsinαm1y¨1+C1yy˙1+K1yy1=−Cmδ˙cosα−Km(t)δcosα−m1gI1θ¨1=T1−r1(Cmδ˙+Km(t)δ)m2x¨2+C2xx˙2+K2xx2=Cmδ˙sinα+Km(t)δsinαm2y¨2+C2yy˙2+K2yy2=Cmδ˙cosα+Km(t)δcosα−m2gI2θ¨2=−T2+r2(Cmδ˙+Km(t)δ)
where δ=x1sinα+y1cosα−x2sinα−y2cosα+r1θ1−r2θ2+e(t)δ=x1sinα+y1cosα−x2sinα−y2cosα+r1θ1−r2θ2+e(t).
4. Simulation Parameters
Key parameters for the gear system are summarized below:
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Module (mm) | 1.5 | 1.5 |
| Number of Teeth | 36 | 90 |
| Pressure Angle (°) | 20 | 20 |
| Elastic Modulus (GPa) | 206 | 206 |
| Moment of Inertia (kg·m²) | 1.16×10−41.16×10−4 | 29.8×10−429.8×10−4 |
Results and Discussion
1. TVMS Under Crack Propagation
Crack depth (qq) and angle (vv) significantly reduce TVMS (Table 1). Depth has a greater impact than angle:
| Crack Depth | TVMS Reduction | Crack Angle | TVMS Reduction |
|---|---|---|---|
| 10% | 8% | 30° | 5% |
| 50% | 32% | 60° | 12% |
| 70% | 58% | 70° | 15% |
2. Vibration Response in Steady-State Conditions
- Healthy Gear: Time-domain signals are stable; frequency spectra show meshing frequency (fm=900fm=900 Hz) harmonics.
- Cracked Gear:
- Time Domain: Periodic impacts at rotational frequency (fr=25fr=25 Hz). Amplitude increases with crack depth.
- Frequency Domain: Sidebands around 3fm3fm, 4fm4fm, and 5fm5fm with spacing frfr. Sideband amplitude grows with crack severity.
3. Vibration Response Under Variable Loads
Load fluctuations (T2=Tmean+ΔTcos(20πt)T2=Tmean+ΔTcos(20πt)) introduce low-frequency components:
- Healthy Gear: Low-frequency peaks at load fluctuation frequency (fL=10fL=10 Hz).
- Cracked Gear: Mixed sidebands at frfr and fLfL. Higher load amplitudes (ΔT≥120ΔT≥120 N·m) obscure fault features.
4. Sensitivity Analysis of Fault Indicators
Five statistical metrics evaluate crack sensitivity:
| Indicator | Sensitivity Ranking | Relative Increment (70% Crack) |
|---|---|---|
| Kurtosis | 1 | 240% |
| RMS | 3 | 85% |
| Crest Factor | 2 | 150% |
Conclusion
This study advances the understanding of crack-induced dynamics in gear systems. Key contributions include:
- A refined potential energy method for accurate TVMS calculation, validated against ISO standards.
- A 6-DOF model capturing translational, rotational, and torsional vibrations.
- Identification of kurtosis as the most sensitive fault indicator.
Future work will explore nonlinear effects and experimental validation. The findings enhance predictive maintenance strategies for gear systems in critical applications.