Abstract
This paper delves into the optimal design of meshing contact for a helical gear pair in a cutting reducer. Focusing on the effects of shaft system deformations on gear meshing performance, the paper calculates the load capacity of the gears and verifies the simulation results. Based on tooth profile and tooth orientation modification principles, various modification schemes are analyzed to optimize the distribution of tooth contact stress and reduce transmission errors. The most suitable modification scheme is identified, resulting in a significant improvement in gear pair transmission performance.
Keywords: helical gear, gear modification, contact stress, stress distribution, shaft system deformation
1. Introduction
Gear transmission is one of the most widely used forms of mechanical transmission, and its performance significantly impacts the overall operation of mechanical systems. In the context of mining machinery, such as cutting reducers, gear pairs are subjected to high loads and harsh operating conditions, making their performance crucial for reliable operation. Research has been conducted on various aspects of gear transmission, including the adhesive state of tooth surfaces under dry friction, root cracks, and vibration characteristics of mining reducers. However, the impact of shaft system deformations on gear meshing contact has been largely overlooked.
This paper aims to address this gap by analyzing the effects of shaft system deformations on helical gear meshing contact and proposing optimization strategies. Specifically, we calculate the load capacity of the gears, verify simulation results, and employ gear modification techniques to optimize the distribution of tooth contact stress and transmission errors.
2. Gear Basic Parameters and Load Capacity Calculation
2.1 Gear Parameters
The study focuses on a helical gear pair from a cutting reducer, with the following specifications:
- Active Gear (Z1): Number of teeth = 20, module = 6 mm, helix angle = 10°, pressure angle = 20°, face width = 75 mm, modification coefficient = 0.5593
- Driven Gear (Z2): Number of teeth = 32, module = 6 mm, helix angle = 10°, pressure angle = 20°, face width = 75 mm, modification coefficient = 0.5834
- Material quality grade: 6 (ISO 1328)
2.2 Load Capacity Calculation
The nominal tooth surface contact stress (σH0) is calculated using the following formula:
sigmaH0=ZHZEZϵZβd1b⋅u+1Ft
where:
- ZH, ZE, Zϵ, and Zβ are coefficients related to node region, elasticity, contact ratio, and helix angle, respectively.
- Ft is the nominal tangential force on the reference pitch circle.
- d1 is the pitch circle diameter of the pinion.
- b is the face width.
- u is the gear ratio (( z_2/z_1 )).
Considering various factors such as use coefficient (K_A \)), dynamic load factor (\( K_V \)), and contact intensity-related coefficients (\( K_{H\beta} and ( K_{H\alpha} )), the actual contact stress (( \sigma_H )) is determined as:
sigmaH=ZB(ZD)σH0KAKVKHβKHα
For the given parameters, the calculated contact stresses are:
- σH1=1175.01 MPa for the pinion
- σH2=1140.79 MPa for the gear
The material’s allowable contact stresses are 1455 MPa and 1476 MPa for the pinion and gear, respectively, with corresponding safety factors of 1.25 and 1.30.
3. Simulation Analysis of Gear Meshing Contact
3.1 Simulation without Considering Shaft System
A simulation of the gear pair’s meshing contact, excluding the effects of the shaft system, was conducted. The results are presented showing the transmission error curve and stress distribution on the tooth surfaces of both gears.
3.2 Simulation Considering Shaft System
A subsequent simulation, this time incorporating the effects of the shaft system, was performed. The results reveal a wider range of transmission errors and a more severe stress distribution skew.
4. Comparison and Analysis
Comparing the results from the two simulations (Figures 2 and 3), the following key differences are observed:
- Transmission Error Range: Without the shaft system, the transmission error range is [-69, -43] μm, whereas with the shaft system, it widens to [-74, -49] μm.
- Stress Distribution: Without the shaft system, stress is distributed uniformly across the tooth surfaces. However, with the shaft system, stress distribution becomes skewed, indicating a more severe load on one side of the teeth.
5. Optimal Design of Tooth Contact through Gear Modification
To address the issues observed in the simulations, gear modification techniques are employed, focusing on tooth profile and tooth orientation modifications.
5.1 Tooth Profile Modification
To reduce elastic deformations, manufacturing errors, and improve lubrication, tooth profile modifications (elastic and crowning) are applied.
- Elastic Modification: Following the recommendations in the Mechanical Design Handbook, an elastic modification with a depth of 0.025 to 0.040 mm and a rounding radius of 0.75 mm is implemented.
- Crowning Modification: Different crowning amounts (10, 20, 30, 40, and 50 μm) are analyzed. depicts the effect of varying crowning amounts on stress distribution.
Based on the analysis, a crowning amount of 30 μm is selected, as it effectively reduces stress concentrations while keeping the maximum contact stress within acceptable limits.
5.2 Tooth Orientation Modification
To achieve a more uniform load distribution and improve contact patterns, tooth orientation modifications are applied. Considering the gear pair’s operating conditions, only elastic deformations are considered for modification calculations, with recommended values ranging from 13 to 35 μm.
Simulations are conducted with different crowning amounts (13, 18, 23, 28, 33, and 35 μm). The results indicate improved stress distributions as the crowning amount increases.
A crowning amount of 28 μm is selected, resulting in a more stable transmission with a reduced transmission error range ([-75, -61] μm) and improved stress distribution.
6. Optimized Results and Discussion
After implementing the optimal modification scheme, significant improvements are observed:
- Transmission Error: The transmission error range is reduced from [-74, -49] μm to [-75, -61] μm, indicating smoother operation.
- Stress Distribution: Stress concentrations are mitigated, and the contact pattern is optimized.
7. Conclusion
This paper presents an optimal design approach for helical gear meshing contact in a cutting reducer, considering the effects of shaft system deformations. By calculating the load capacity and verifying simulation results, various gear modification schemes were analyzed to optimize tooth contact stress distribution and reduce transmission errors. The most suitable modification scheme was identified, resulting in significant improvements in gear pair transmission performance. This study demonstrates the importance of considering shaft system deformations in gear design and optimization.