Gear loaded transmission error (LTE) serves as a critical indicator for evaluating dynamic meshing performance. Minimizing LTE fluctuation amplitude directly correlates with improved gear vibration characteristics. This study investigates the influence of modification parameters on LTE for internal short tooth spur gears using Romax software, proposes an optimization framework, and validates its effectiveness through comparative analysis.
1. Modeling of Internal Short Tooth Spur Gears
The internal spur gear pair parameters are specified in Table 1. The gear modeling considers critical design parameters including pressure angle, modulus, and tooth profile coefficients. For internal spur gears, the rim thickness of the internal gear follows:
$$ \delta_0 = (2.5 \sim 4)m \quad \text{where } m \geq 8 $$
This study adopts a rim thickness of 16 mm for the internal gear to ensure structural integrity.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 53 | 160 |
| Module (mm) | 4.0 | 4.0 |
| Pressure angle (°) | 20 | 20 |
| Addendum coefficient | 0.8 | 0.8 |
| Dedendum coefficient | 1.0 | 1.0 |
| Face width (mm) | 212 | 212 |
2. Modification Methodology for Spur Gears
2.1 Tooth Profile Modification
The parabolic tooth profile modification scheme for spur gears includes four key parameters:
$$ y_1: \text{Maximum tip relief} $$
$$ y_2: \text{Relief length at tip} $$
$$ y_3: \text{Maximum root relief} $$
$$ y_4: \text{Relief length at root} $$
2.2 Helix Modification
Two helix modification types are analyzed:
$$ \text{End relief: } y_5 = k_1 \cdot b $$
$$ \text{Crowning: } y_5 = k_2 \cdot \delta_{\text{max}} $$
where \( k_1 = 0.25 \), \( b \) represents face width, and \( \delta_{\text{max}} \) denotes maximum manufacturing error.
3. Parametric Analysis of LTE Characteristics
3.1 Profile Modification Effects
The LTE fluctuation amplitude shows non-linear responses to profile modifications:
$$ \Delta LTE = f(y_1,y_3) = a_1y_1^2 + a_2y_3^2 + b_1y_1 + b_2y_3 + c $$
Typical parametric relationships are summarized in Table 2.
| Modification Type | Parameter Range (μm) | LTE Reduction (%) |
|---|---|---|
| Tip relief (parabolic) | 10-30 | 18-42 |
| Root relief (linear) | 15-25 | 12-28 |
3.2 Helix Modification Effects
Crowning demonstrates superior LTE reduction compared to end relief:
$$ \frac{\Delta LTE_{\text{crown}}}{\Delta LTE_{\text{relief}}} = 0.78 \sim 0.85 $$
4. Optimization of Spur Gear Modifications
A particle swarm optimization (PSO) algorithm minimizes LTE fluctuation through multi-parameter coordination:
$$ \text{Minimize: } f(Y) = \max(LTE(\theta)) – \min(LTE(\theta)) $$
$$ \text{Subject to: } Y = [y_1,y_2,y_3,y_4,y_5]^T $$
The optimization results demonstrate significant improvement over conventional methods:
| Method | LTE Amplitude (μm) | Reduction (%) |
|---|---|---|
| Unmodified | 6.69 | – |
| Romax Standard | 6.45 | 3.6 |
| Romax Optimized | 3.10 | 53.7 |
| PSO Optimized | 2.86 | 57.2 |
5. Conclusion
This investigation establishes comprehensive guidelines for internal short tooth spur gear modification design:
- Parabolic profile modification achieves 42% LTE reduction versus linear modification
- Optimal crowning magnitude ranges between 13-15 μm for 212mm face width spur gears
- PSO algorithm demonstrates 57.2% LTE reduction through coordinated multi-parameter optimization
The proposed methodology provides practical solutions for enhancing dynamic performance of high-speed spur gear transmissions while maintaining manufacturing feasibility.