Dynamic Analysis of High Contact Ratio Spur Gear Considering Surface Wear

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This study investigates the dynamic behavior of high contact ratio spur gears under tooth surface wear. By integrating Archard’s wear model with energy-based stiffness calculations, we establish a comprehensive framework for analyzing wear-induced changes in gear meshing characteristics.

1. Dynamic Model Formulation

The single-degree-of-freedom dynamic equation for spur gear systems is expressed as:

$$m_e\ddot{x}(t) + c\dot{x}(t) + k(t)f(x) = F_m + F_h(t)$$

Where:

  • $m_e$ = equivalent mass
  • $c$ = damping coefficient
  • $k(t)$ = time-varying mesh stiffness
  • $F_m$ = mean load component

The dimensionless form becomes:
$$\ddot{\bar{x}} + 2\xi\dot{\bar{x}} + \bar{K}(\tau)\bar{f}(\bar{x}) = \bar{F}_m + \bar{F}_h(\tau)$$
with dimensionless parameters:
$$\xi = \frac{c}{2\sqrt{K_m m_e}},\ \bar{K}(\tau) = \frac{k(t)}{K_m},\ \tau = \omega_n t$$

2. Wear Characterization

Using Archard’s wear model for spur gear analysis:

$$h = \frac{2a\lambda n t}{\varepsilon_\alpha}I_h$$

Table 1: Spur Gear Parameters
Parameter Value
Module 3.25 mm
Pressure Angle 20°
Contact Ratio 2.23
Young’s Modulus 206 GPa

Wear distribution characteristics:

Table 2: Wear Depth vs. Operating Cycles
Cycles (×10⁶) Max Wear (μm)
2 12.4
6 37.8
10 63.2

3. Mesh Stiffness Calculation

Total mesh stiffness for high contact ratio spur gears:

$$K(t) = \begin{cases}
\sum_{i=1}^3 k_i(t) & 0 \leq \tau \leq (\varepsilon-2)T \\
\sum_{i=1}^2 k_i(t) & (\varepsilon-2)T \leq \tau \leq T
\end{cases}$$

Component stiffness terms include:
$$k = \left(\frac{1}{k_h} + \frac{1}{k_{b1}+k_{s1}+k_{a1}} + \frac{1}{k_{b2}+k_{s2}+k_{a2}}\right)^{-1}$$

4. Dynamic Response Analysis

Key findings for spur gear dynamics under wear:

Table 3: Dynamic Behavior Under Different Loads
Condition Wear State System Behavior
Light Load N=2×10⁶ Chaotic
N=6×10⁶ Quasi-periodic
N=10⁷ Chaotic
Heavy Load All Cases Periodic

Characteristic frequency relationships:
$$\omega_h/\omega_n = \begin{cases}
1.0 & \text{Primary resonance} \\
0.2 & \text{Subharmonic}
\end{cases}$$

5. Conclusion

This investigation of spur gear dynamics reveals:

  1. Light-load systems show sensitivity to wear progression (quasi-periodic ↔ chaotic transitions)
  2. Heavy-load conditions maintain periodic behavior despite wear
  3. Contact ratio significantly influences stability thresholds

The developed model enables accurate prediction of spur gear performance degradation, particularly valuable for high contact ratio applications requiring prolonged service life.

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