Introduction
High contact ratio spur gears are defined as gears with a contact ratio greater than 2. These gears are known for their enhanced dynamic characteristics, such as reduced noise and improved performance. The study focuses on analyzing the total meshing stiffness of such gears, factoring in the wear on their tooth surfaces. The research is crucial for understanding how wear influences the overall gear system, especially under different load conditions.
Key Points:
- The high contact ratio leads to better dynamic behavior and lower operational noise.
- Wear calculations based on Archard’s model and energy methods are integrated into the analysis.
Gear System Dynamic Model
The dynamic model for the gear system is expressed through a single-degree-of-freedom differential equation, capturing the system’s response to time-varying meshing stiffness and various external and internal forces.
Equation:
mx¨(t)+cx˙(t)+k(t)f(x)=Fm+Fh(t)m\ddot{x}(t) + c\dot{x}(t) + k(t)f(x) = F_m + F_h(t)mx¨(t)+cx˙(t)+k(t)f(x)=Fm+Fh(t)
Where:
- mmm = equivalent mass of the system
- ccc = damping coefficient
- k(t)k(t)k(t) = time-varying mesh stiffness
- FmF_mFm = equivalent external excitation
- Fh(t)F_h(t)Fh(t) = internal excitation
The model is further non-dimensionalized to simplify the analysis, leading to the equation:x¨+2ξx˙+Kˉ(τ)f(xˉ)=Fˉm+Fˉh(τ)\ddot{x} + 2\xi\dot{x} + \bar{K}(\tau)f(\bar{x}) = \bar{F}_m + \bar{F}_h(\tau)x¨+2ξx˙+Kˉ(τ)f(xˉ)=Fˉm+Fˉh(τ)
Where:
- ξ\xiξ = damping factor
- Kˉ(τ)\bar{K}(\tau)Kˉ(τ) = non-dimensional mesh stiffness
Wear Characterization and Impact on Stiffness
Surface wear significantly influences the gear system’s meshing stiffness. In particular, the wear is calculated based on the Archard model, considering the sliding coefficient, rotational speed, and other parameters.
Wear Calculation Formula:
h=2aλntϵαIhh = 2a\lambda nt \epsilon_{\alpha} I_hh=2aλntϵαIh
Where:
- aaa = equivalent contact width at the meshing point
- λ\lambdaλ = sliding coefficient
- nnn = rotational speed
- ttt = operating time
- ϵα\epsilon_{\alpha}ϵα = overlap ratio
- IhI_hIh = wear rate
The wear is found to affect the mesh stiffness, especially under lighter load conditions, where the system undergoes complex dynamic behavior such as quasi-periodic and chaotic motions.
Tables for Gear Parameters
The following table summarizes key gear parameters used in the analysis.
| Parameter | Driving Gear (Active) | Driven Gear (Passive) |
|---|---|---|
| Number of Teeth | 32 | 25 |
| Module (mm) | 3.25 | 3.25 |
| Pressure Angle | 20° | 20° |
| Addendum Coefficient | 1.35 | 1.35 |
| Base Circle Radius | 43.3 mm | 36.8 mm |
Dynamic Response Analysis
The study then investigates how surface wear influences the dynamic behavior of the gear system. The analysis shows that under light load conditions, the system’s dynamic response becomes more sensitive to wear, with the system transitioning through various states of motion from quasi-periodic to chaotic and back. In contrast, under heavy load conditions, the effect of wear on the dynamic behavior is minimal, and the system maintains periodic motion for extended periods.
Key Findings:
- Light Load Condition: Significant impact of wear, leading to transitions between quasi-periodic and chaotic motion.
- Heavy Load Condition: Minimal effect of wear, with the system predominantly exhibiting periodic motion.
Conclusion
The study concludes that surface wear plays a critical role in the dynamic response of high contact ratio spur gears, particularly under light loading conditions. The inclusion of wear in the total meshing stiffness reveals the gear system’s vulnerability to wear-induced changes in dynamic behavior. Under heavy loads, however, the influence of wear is less pronounced, and the system remains stable.