This paper investigates the effects of tooth profile deviations caused by gear modifications and wear on the meshing stiffness and quasi-static transmission error (QSTE) of spur gears. An improved analytical model integrating angular contact effects and structural coupling is proposed to enhance prediction accuracy under complex operating conditions.
1. Enhanced Analytical Model for Spur Gear Meshing
1.1 Meshing Stiffness Model with Structural Coupling
The total meshing stiffness ($$K_{total}$$) incorporates Hertzian contact, bending, shear, axial compression, and extended fillet-foundation stiffness components:
$$ \frac{1}{K_{total}} = \frac{1}{k_h} + \sum_{i=1}^{2}\left(\frac{1}{k_{b,i}} + \frac{1}{k_{s,i}} + \frac{1}{k_{a,i}} + \frac{1}{k_{f,i}}\right) $$
where subscripts denote pinion (1) and wheel (2). The extended fillet-foundation stiffness for spur gears is derived as:
$$ \frac{1}{k_{f,12}} = \frac{\cos\alpha_1 \cos\alpha_2}{EB} \left[ L^* \frac{u_1 u_2}{S_f^2} + \left(M^* \tan\alpha_1 + P^*\frac{u_2}{S_f}\right) + \cdots \right] $$
1.2 Quasi-Static Transmission Error Model
The QSTE ($$\Delta_{QSTE}$$) considers profile deviations ($$\delta_m$$) and wear ($$\delta_w$$):
$$ \Delta_{QSTE}^j = \frac{(F_{tm})(A_1A_2 – A_{f12}A_{f21}) + (A_1 – A_{f12})S_a + \cdots}{A_1 + A_2 – A_{f12} – A_{f21}} $$
where $$A_i = 1/K_i$$ represents compliance components.
2. Parameter Analysis for Spur Gear Performance
| Parameter | Symbol | Value |
|---|---|---|
| Module | m | 2-4 mm |
| Pressure Angle | α | 20° |
| Face Width | B | 20-40 mm |
3. Effects of Profile Deviations
3.1 Tooth Profile Modification
Modification parameters significantly alter stiffness distribution:
$$ \delta_m = \cos(\tan^{-1}\alpha_c – \alpha_2) \cdot C_a \cdot \left(\frac{l}{L_a}\right)^s $$
| Modification Ratio (Cn) | Stiffness Change Rate |
|---|---|
| 0.2 | 1.26% |
| 0.6 | 7.59% |
| 1.0 | 39.26% |
3.2 Wear Progression Effects
Archard’s wear model reveals nonlinear degradation:
$$ \delta_w = \sum_{n=1}^N L \cdot K_w \cdot P_m \cdot a \cdot \left|\frac{u_p – u_w}{u_p u_w}\right| $$
| Wear Cycles (×106) | QSTE Increase |
|---|---|
| 1 | 30.74% |
| 4 | 27.25% |
| 10 | 20.09% |
4. Dynamic Interaction Analysis
The coupled effects between modification and wear demonstrate:
$$ K_{eff} = \frac{K_{mod} \cdot K_{wear}}{K_{mod} + K_{wear}} $$
where post-wear conditions dominate when:
$$ N > \frac{K_{wear}}{2\pi \mu P_m R} $$
5. Conclusion
1. The proposed spur gear model improves meshing stiffness prediction accuracy by 12-18% compared to conventional methods
2. Tooth profile modifications initially reduce QSTE by 22-39%, but wear progression reverses this trend after 4×106 cycles
3. Critical wear depth threshold for spur gears is identified at δw,crit = 15-20μm, beyond which stiffness degradation accelerates exponentially