1. Introduction
1.1 Research Background and Significance
Spur gear is widely used in robotics, automotive systems, and industrial machinery due to their high transmission accuracy, stability, and load-bearing capacity. However, tooth surface wear is an inevitable issue during gear operation, accounting for over 50% of mechanical component failures. High contact ratio (HCR) spur gear, defined as gears with a contact ratio greater than 2, offer advantages such as reduced noise, higher load capacity, and smoother transmission compared to standard gears. Despite these benefits, the dynamic response of HCR spur gear is significantly affected by tooth surface wear. This study investigates the impact of wear on the dynamic behavior of HCR spur gear under varying operational conditions, providing theoretical insights for optimizing gear design and predicting system longevity.
1.2 Current Research Status
- Gear Design: Traditional methods for selecting modification coefficients (e.g., line diagram method) lack flexibility for negative modifications. The closed graph method, based on constraints like anti-undercutting and anti-interference, offers a more intuitive approach.
- Wear Calculation: The Archard wear model is widely adopted for predicting surface wear. However, existing studies often neglect the influence of contact ratio on wear distribution.
- Gear Dynamics: Single-degree-of-freedom torsional vibration models are commonly used to analyze dynamic responses, but the coupling effects of wear and time-varying meshing stiffness require deeper exploration.
2. Establishment of Tooth Profile Equations for HCR Spur Gear
2.1 Closed Graph Method for Modification Coefficient Selection
The closed graph method ensures gear design parameters satisfy constraints such as no undercutting, no interference, sufficient tooth tip thickness, and a contact ratio ≥2. Key equations include:
- Anti-undercutting condition:x≥ha∗−zsin2α2x≥ha∗−2zsin2α
- Anti-interference condition:tanα′−i(tanαa2−tanα′)≥tanα−4(ha∗−x1)z1sin2αtanα′−i(tanαa2−tanα′)≥tanα−z1sin2α4(ha∗−x1)
- Tooth tip thickness constraint:sa1=ra1[4x1tanα+πz1−2(invαa1−invα)]≥[sa]minsa1=ra1[z14x1tanα+π−2(invαa1−invα)]≥[sa]min
A closed graph for a gear pair (Table 1) is illustrated in Figure 2.5 of the original thesis, delineating valid regions for modification coefficients.
Table 1: Basic Parameters of Example Gear Pair
| Parameter | Pinion | Gear |
|---|---|---|
| Teeth (zz) | 25 | 32 |
| Module (mm, mm) | 3.25 | 3.25 |
| Pressure angle (αα) | 20° | 20° |
| Addendum coefficient (ha∗ha∗) | 1.35 | 1.35 |
2.2 Tooth Profile Modeling
Using the generating method, the tooth profile equations for HCR spur gear is derived based on the hob cutter geometry. The coordinate transformation between the hob and workpiece is expressed as:{x=(ri−x0)cosφ+(riφ−y0)sinφy=(ri−x0)sinφ−(riφ−y0)cosφ{x=(ri−x0)cosφ+(riφ−y0)sinφy=(ri−x0)sinφ−(riφ−y0)cosφ
where (x0,y0)(x0,y0) are hob coordinates, and φφ is the rotation angle.
3. Impact of Tooth Surface Wear on Meshing Stiffness
3.1 Archard Wear Model
The non-uniform wear depth hh is calculated as:h=2aηntϵIhh=2aηntϵIh
where aa is the Hertzian contact half-width, ηη is the sliding coefficient, nn is rotational speed, tt is time, ϵϵ is contact ratio, and IhIh is the wear rate.
3.2 Influence of Gear Parameters on Wear
- Number of Teeth: Smaller gears experience higher wear due to increased sliding velocity.
- Module: Larger modules reduce wear by distributing loads over broader contact areas.
- Modification Coefficient: Positive modification reduces wear by optimizing load distribution.
Table 2: Wear Characteristics Under Different Parameters
| Parameter | Wear Trend | Key Observation |
|---|---|---|
| Teeth (zz) | ↓ with ↑ zz | Small gears wear faster. |
| Module (mm) | ↓ with ↑ mm | Larger modules enhance durability. |
| Modification (xx) | ↓ with ↑ xx | Positive xx reduces wear. |
3.3 Time-Varying Meshing Stiffness
Using the potential energy method, the total meshing stiffness ktotalktotal is derived as:1ktotal=1kh+∑i=12(1kai+1kbi+1ksi+1kfi)ktotal1=kh1+i=1∑2(kai1+kbi1+ksi1+kfi1)
where khkh, kaka, kbkb, ksks, and kfkf represent Hertzian, axial, bending, shear, and fillet stiffness, respectively.
3.4 Effect of Wear on Stiffness
Long-term wear reduces meshing stiffness, particularly in triple-tooth contact regions. For example, after 107107 cycles, stiffness decreases by 1.26% on average, with triple-contact zones showing 2.1% reduction compared to 0.8% in double-contact zones.
4. Dynamic Response of Spur Gear Systems Under Wear
4.1 Torsional Vibration Model
The dimensionless dynamic equation is:x‾¨+2ζx‾˙+K(τ)f(x‾)=Fm+Fh(τ)x¨+2ζx˙+K(τ)f(x)=Fm+Fh(τ)
where x‾x is dimensionless displacement, ζζ is damping ratio, K(τ)K(τ) is time-varying stiffness, and Fh(τ)Fh(τ) represents internal excitation.
4.2 Bifurcation and Chaos Analysis
- Load Variation: At medium speeds, heavy loads stabilize periodic motion, while light loads induce chaotic behavior.
- Speed Variation: High speeds under heavy loads promote chaos; low speeds favor periodic motion.
Table 3: Dynamic Responses Under Different Conditions
| Condition | Dynamic Behavior | Key Observation |
|---|---|---|
| Heavy load, low speed | Periodic motion | Minimal backlash冲击. |
| Light load, high speed | Chaos | Bidirectional冲击加剧. |
| Heavy load, high speed | Chaos | Severe单向冲击. |
4.3 Impact of Wear on Dynamics
- Light Load: Wear alters bifurcation patterns (e.g., 4-periodic to 2-periodic motion) and reduces resonance frequencies.
- Heavy Load: Wear has negligible impact due to dominant inertial forces.
Table 4: Wear Impact Under Different Loads
| Load Condition | Wear Effect | Example (Ω=2.5) |
|---|---|---|
| Light load, high speed | Changes motion patterns | 4-periodic → 2-periodic. |
| Heavy load | Insignificant | Stiffness reduction < 0.5%. |
5. Conclusions and Future Work
5.1 Conclusions
- Tooth Profile Design: The closed graph method effectively selects modification coefficients for HCR spur gear, balancing anti-wear and load capacity.
- Wear Analysis: Non-uniform wear peaks at the tooth root and tip. Gear parameters (module, teeth, modification) significantly influence wear distribution.
- Dynamics: Heavy loads stabilize periodic motion, while light loads induce chaos. Wear impacts light-load systems more profoundly.
5.2 Future Work
- Develop interactive tools for closed graph-based gear design.
- Incorporate dynamic load and lubrication effects into wear models.
- Validate theoretical predictions through experimental testing.