Abstract
This article delves into the impact of center distance error on the dynamic characteristics of helical gear. Utilizing a pair of subway helical gear as the research object, the study employs the slicing method and potential energy method to calculate the time-varying meshing stiffness and dynamic transmission error in the presence of center distance error. A bending-torsional-axial eight-degree-of-freedom helical gear dynamics model is established using the lumped mass approach to analyze the dynamic meshing characteristics of the gear system. The results indicate that the time-varying meshing stiffness of helical gear decreases with increasing center distance error, while the average dynamic meshing force remains relatively stable. However, the error leads to reduced gear meshing stability, providing valuable insights for vibration and noise reduction in gear systems.
1. Introduction
Helical gear is widely utilized in various industries, including vehicles, machine tools, shipping, and power generation, due to their compact structure, high load-bearing capacity, and efficient power transmission. The meshing characteristics of gears significantly impact the vibration, noise, and lifespan of gear transmission systems. Therefore, studying the meshing behavior and internal excitations of gear systems is crucial for improving their performance.
While scholars have conducted extensive research on the meshing characteristics of gear systems and the effects of various internal excitations, limited studies have focused specifically on the influence of center distance error on helical gear. This article aims to fill this gap by investigating the dynamic characteristics of helical gear under different center distance errors.
2. Theoretical Background and Methodology
2.1 Meshing Theory of Helical Gear
Helical gear differ from spur gears in that their teeth are inclined at an angle to the axis of rotation. This inclination results in a smoother and quieter operation, as well as increased load-carrying capacity. The meshing process involves a continuous line of contact along the gear teeth, which changes over time.
2.2 Center Distance Error Mechanism
In an ideal scenario, helical gear is installed with no center distance error, ensuring that their pitch circles are tangent and the meshing angle equals the pressure angle at the pitch circle. However, in practice, center distance errors can occur, altering the distance between the center axes of the two gears. This change affects the gear engagement, causing deviations in the meshing angle and reducing the contact line length .
The end-face meshing angle ((\alpha’_t)) in the presence of center distance error ((\Delta a)) can be calculated using the formula:
cosαt′=cosαt⋅a′a
where αt is the theoretical end-face meshing angle, a is the ideal center distance, and a′=a±Δa is the actual center distance.
2.3 Time-Varying Meshing Stiffness Calculation
The slicing method is employed to approximate the helical gear by uniform slices along its tooth width, transforming it into a series of spur gears for analysis. The potential energy method is then used to compute the meshing stiffness of each slice. The overall time-varying meshing stiffness ((k_m(t))) is obtained by integrating the stiffness values of all slices:
km(t)=i=1∑Nki(t)
where ki(t) is the stiffness of the i-th slice at time t, and N is the total number of slices.
2.4 Dynamic Model of Helical Gear
A bending-torsional-axial eight-degree-of-freedom dynamic model is constructed using the lumped mass method. This model considers the transverse, torsional, and axial vibrations of the gears.
The dynamic equations of motion are derived based on Newton’s second law and D’Alembert’s principle, incorporating the time-varying meshing stiffness, damping, and external loads.
3. Experimental Setup and Results
3.1 Geometric Parameters and Operating Conditions
The study utilizes a pair of subway helical gear with specific geometric parameters outlined in Table 1. The gears operate at a speed of 1800 rpm and an input power of 138 kW.
| Parameter | Active Gear | Driven Gear |
|---|---|---|
| Number of Teeth (Z) | 16 | 101 |
| Module (m) | 6 mm | 6 mm |
| Pressure Angle (α) | 25° | 25° |
| Face Width (b) | 97 mm | 90 mm |
| Helix Angle (β) | 12.5° | 12.5° |
Table 1: Geometric Parameters of Subway Helical Gear
3.2 Time-Varying Meshing Stiffness and Dynamic Transmission Error
The time-varying meshing stiffness and dynamic transmission error are calculated for various center distance errors (−0.4 mm, −0.2 mm, 0 mm, +0.2 mm, +0.4 mm). As the center distance error increases, the contact line length and meshing stiffness decrease .
3.3 Dynamic Meshing Force and Vibration Characteristics
The dynamic meshing force and y-axis vibration acceleration are simulated under different center distance errors. the meshing force amplitude fluctuates more significantly with negative center distance errors, indicating increased vibration.
Table 2 summarizes the mean and standard deviation of the dynamic meshing force and y-axis vibration acceleration for various center distance errors.
| Center Distance Error (mm) | Mean Dynamic Meshing Force (N) | Std. Dev. Dynamic Meshing Force (N) | Mean Vibration Acceleration (μm/s²) | Std. Dev. Vibration Acceleration (μm/s²) |
|---|---|---|---|---|
| -0.4 | 20,410.57 | 537.09 | 3,633.74 | 91.52 |
| -0.2 | 20,410.57 | 536.95 | 3,622.85 | 91.49 |
| 0 | 20,410.57 | 543.33 | 3,610.44 | 90.19 |
| +0.2 | 20,410.57 | 545.56 | 3,582.56 | 92.58 |
| +0.4 | 20,410.57 | 545.56 | 3,555.82 | 92.96 |
Table 2: Summary of Dynamic Meshing Force and Vibration Acceleration
4. Discussion
4.1 Effect on Meshing Stiffness
The results demonstrate that the time-varying meshing stiffness of helical gear decreases with increasing center distance error. This is attributed to the reduction in the contact line length caused by the error.
4.2 Impact on Dynamic Meshing Force
While the average dynamic meshing force remains relatively stable across different center distance errors, the amplitude fluctuations increase significantly with negative errors, indicating reduced meshing stability.
4.3 Vibration Characteristics
The y-axis vibration acceleration decreases with increasing positive center distance errors, indicating improved meshing smoothness. However, negative errors exacerbate vibration, posing a challenge for noise reduction and system reliability.
5. Conclusion
This study comprehensively analyzed the impact of center distance error on the dynamic characteristics of helical gear. The key findings are:
- The time-varying meshing stiffness decreases with increasing center distance error due to the reduced contact line length.
- While the average dynamic meshing force remains stable, negative center distance errors significantly increase the amplitude fluctuations.
- Vibration characteristics are adversely affected by negative errors, emphasizing the importance of precise gear alignment for improved system performance.
The results provide valuable insights for designers and engineers working with helical gear, particularly in applications requiring low vibration and noise levels, such as subway systems. Future research could explore advanced control strategies and gear modifications to mitigate the effects of center distance errors.