1. Introduction
Gear transmission is an important part of mechanical transmission systems. The performance of gears directly affects the efficiency and reliability of the entire system. Among various gear types, spur gear is widely used due to their simple structure and easy manufacturing. In recent years, with the development of high-speed and heavy-duty machinery, the requirements for gear performance have become higher and higher. One of the important indicators to evaluate the performance of spur gear is the loaded transmission error. The loaded transmission error reflects the deviation between the actual transmission motion of spur gear pair and the ideal transmission motion under load conditions. A smaller fluctuation amplitude of the loaded transmission error indicates better dynamic meshing performance of spur gear pair, which can reduce vibration and noise during gear operation and improve the service life of spur gear.
1.1 Research Background
In the past few decades, scholars at home and abroad have conducted a large number of studies on spur gear transmission errors. For example, Litvin et al. proposed a modified surface topology structure of involute helical gear to reduce gear loaded transmission errors in 2005. Fang Zongde et al. optimized the loaded transmission error of spiral bevel gear with high contact ratio. Sun Yuehai et al. derived the relationship between the static loaded transmission error of modified spur gear and the comprehensive modification parameters of tooth pairs under a certain load condition. These studies have made important contributions to the understanding and improvement of spur gear transmission performance. However, most of the existing research focuses on external gear or helical gear, and there is relatively less research on internal spur gear, especially internal short tooth spur gear.
1.2 Research Significance
Internal spur gear is often used in planetary transmissions, and the short tooth form is usually adopted to meet certain design requirements. The research on the loaded transmission error of internal short tooth spur gear is of great significance for improving the performance of planetary transmissions. By studying the influence of tooth profile modification and helix modification parameters on the loaded transmission error, and optimizing the modification design method, it is possible to effectively reduce the fluctuation amplitude of the loaded transmission error, improve the dynamic meshing performance of spur gear pair, and thus enhance the overall performance of the planetary transmission system.
2. Modeling of Internal Short Tooth Spur Gear
2.1 Gear Parameters
In this study, the Romax software is used for spur gear modeling. The gear set type is set as spur gear, and the spur gear are defined by parameters such as pressure angle, module, modification coefficient, tooth width, addendum coefficient, and dedendum coefficient. The parameters of the internal meshing gear pair studied in this paper are shown in Table 1.
| Items | Pinion | Gear |
|---|---|---|
| Number of teeth | 53 | 160 |
| Module (mm) | 4.0 | |
| Pressure angle (°) | ||
| Tooth height coefficient | ||
| Addendum coefficient | 0.3 | |
| Tooth width (mm) | 212 | |
| Input power (kW) | 600 | |
| Pinion rotational speed (r/min) | 1000 |
2.2 Consideration of Gear Structure
For the internal meshing gear pair, the pinion adopts a solid structure, and spur gear needs to consider the rim thickness. The rim thickness of spur gear is calculated according to the formula δ0 = (2.5-4)m (m≥8), and in this study, the rim thickness is taken as 16 mm.
3. Modification Methods of Spur Gear
3.1 Tooth Profile Modification
3.1.1 Modification Parameters and Representation
The tooth profile modification parameters include the modification amounts and lengths at the tooth tip and tooth root. In this paper, for the convenience of modification processing, the tooth profile modification diagram is represented along the tooth height direction. The tooth tip modification part and the tooth root modification part can adopt parabolic or linear forms. y1 represents the maximum tooth tip modification amount, y2 represents the tooth tip modification length in the tooth height direction, y3 represents the maximum tooth root modification amount, and y4 represents the tooth root modification length in the tooth height direction.
3.1.2 Determination of Parameter Ranges
To analyze the influence of tooth profile modification parameters on the loaded transmission error, it is necessary to determine the variation range of the modification parameters. The maximum tooth tip and tooth root modification amounts each adopt 5 values: 0.010 mm, 0.015 mm, 0.020 mm, 0.025 mm, and 0.030 mm.
3.2 Helix Modification
3.2.1 Types of Helix Modification
The helix modification of the pinion includes two types: end slope modification and drum modification. y5 represents the maximum helix modification amount. y0 represents the helix slope modification length. The slope modification curve is a straight line, and the drum modification curve is a parabola.
3.2.2 Parameter Ranges
For the slope modification, the maximum modification amount is generally 13 – 35 μm, and the modification length is 0.25b (b is the tooth width). For the drum modification of high-precision and high-reliability spur gear, the drum modification amount is 10 – 25 μm, plus a manufacturing error of 5 μm. To analyze the influence of the helix modification amount on the loaded transmission error, the maximum helix modification amount also adopts 5 values: 0.010 mm, 0.015 mm, 0.020 mm, 0.025 mm, and 0.030 mm.
4. Analysis of Modification Effects on Loaded Transmission Error
4.1 Influence of Tooth Profile Modification Parameters
4.1.1 Simulation of Unmodified Gear
First, the Romax software is used to simulate the loaded transmission error of the unmodified standard tooth surface. the fluctuation amplitude of the loaded transmission error is 6.69 μm.
4.1.2 Influence of Tooth Tip Modification Amount
When analyzing the influence of tooth profile modification parameters on the loaded transmission error, the helix modification adopts drum modification, and the drum modification amount is kept at 0.020 mm. When the tooth tip maximum modification amount changes and other modification parameters remain unchanged ( mm, mm, mm). It can be seen that with the increase of the tooth tip maximum modification amount, the fluctuation amplitude of the loaded transmission error first decreases and then increases.
4.1.3 Influence of Tooth Root Modification Amount
When the tooth root maximum modification amount changes and other modification parameters remain unchanged ( mm, , mm), for parabolic modification, with the increase of the tooth root maximum modification amount, the fluctuation amplitude of the loaded transmission error decreases; but for linear modification, the fluctuation amplitude of the loaded transmission error first decreases and then increases.
4.2 Influence of Helix Modification Parameters
When analyzing the influence of helix modification parameters, the helix modification adopts end slope modification and drum modification for comparison. When the helix maximum modification amount changes and other modification parameters remain unchanged ( mm, , mm, , ), the analysis result. It can be seen that with the increase of the helix maximum modification amount, the fluctuation amplitude of the loaded transmission error increases, and the drum modification is superior to the end slope modification.
5. Optimization Design of Modification Parameters
5.1 Optimization Algorithm
In this paper, the particle swarm optimization algorithm (PSO) is adopted. The PSO algorithm is a type of random optimization technique based on swarm intelligence. Compared with the genetic algorithm, both are based on iterative search of the swarm, but the PSO algorithm has no crossover and mutation operators and searches for the optimal solution through the cooperation among individuals, making use of the idea of information sharing in the biological swarm. The PSO algorithm is easy to implement and is more suitable for engineering applications.
5.2 Optimization Objectives and Parameters
The optimization objective is to minimize the fluctuation amplitude of the loaded transmission error. The modification optimization parameters are , , , , and , and the particle is defined as . The target function is the fluctuation amplitude of the loaded transmission error, which is calculated by the loaded tooth contact analysis (LTCA) method.
5.3 Optimization Results and Comparison
The optimization process of the PSO algorithm includes 5 steps: random initialization of particles in the particle swarm, calculation of the fitness of each particle (target function value), calculation of the best position experienced by the particle, evolution of the particle, and judgment of the end condition (whether it has evolved to the preset number of generations, otherwise return and continue). The optimization results are: μm, mm, μm, mm, μm.
In order to compare and analyze, the Romax software recommended modification design method and the optimized modification design method are used. The Romax recommended modification method is a one-way linear tooth profile modification method, and the helix is not modified without considering the installation error condition. The modification parameters are μm, mm, μm, and mm, and the fluctuation amplitude of the loaded transmission error is 6.45 μm. The Romax optimized modification method adopts parabolic tooth profile modification and helix drum modification, and the genetic algorithm is used for optimization. Under the same optimization parameters as this paper, the Romax optimized modification results are μm, mm, μm, mm, and μm, and the fluctuation amplitude of the loaded transmission error is 3.10 μm. The optimization effect of this paper, and both the tooth profile and helix adopt parabolic modification, and the fluctuation amplitude of the loaded transmission error is 2.86 μm. The comparison of modification effects is shown in Table 2.
| Modification Method | Error (μm) |
|---|---|
| Standard Gear (Unmodified) | 6.69 |
| Romax Recommended Modification | 6.45 |
| Romax Optimized Modification | 3.10 |
| This Paper’s Optimized Modification | 2.86 |
6. Conclusions
6.1 Summary of Research Results
- The influence of tooth tip and tooth root modification amounts on the fluctuation amplitude of the loaded transmission error is nonlinear. Due to different tooth profile modification methods (linear and parabolic modification), the influence of the maximum tooth tip and tooth root modification amounts on the fluctuation amplitude of the loaded transmission error is different.
- The helix modification has an impact on the fluctuation amplitude of the loaded transmission error of the internal meshing gear pair, and the drum modification effect is better than the slope modification.
- Reasonable modification can effectively reduce the fluctuation amplitude of the loaded transmission error.
- The modification parameter optimization design method proposed in this paper can effectively reduce the loaded transmission error.
6.2 Significance and Application Prospects
The research results of this paper provide a basis for the modification design of internal short tooth spur gear with high transmission performance. In the future, the research on spur gear transmission errors can be further deepened, and more accurate and efficient modification design methods can be explored to meet the higher requirements of modern machinery for spur gear performance. At the same time, the research results can also be applied to the design and optimization of planetary transmissions and other mechanical transmission systems to improve the overall performance and reliability of the systems.