1. Introduction
Helical gear systems are widely used in various industries, such as aerospace, marine, and automotive, due to their advantages like reliable transmission, high efficiency, and compact structure. However, the vibration and noise generated by these systems have always been significant concerns. The time – varying mesh stiffness of gear pairs is one of the major internal excitation sources, which can lead to vibrations that transmit from the gear teeth to the entire system, causing discomfort in applications like automotive interiors and reducing the reliability of high – speed aerospace transmissions.
In the past, many methods have been proposed to address the vibration issue in gear systems. These include optimizing the vibration transfer path by modifying components like gears, shafts, and bearings, and adding dampers or using gear tooth profile modification. However, the impact of mesh stiffness fluctuation on the vibration characteristics of the system has not been thoroughly studied. This article focuses on a novel low – fluctuation mesh stiffness design method for helical gear systems to improve their vibration performance.
2. Understanding Helical Gear Meshing and Stiffness Fluctuation
2.1 Meshing Characteristics of Helical Gears
Helical gears have a unique meshing process compared to spur gears. The contact line on the tooth surface changes during meshing. As shown in Figure 1, the meshing of helical gears involves different contact line lengths at the entry and exit of meshing. This is a crucial difference from spur gears.
| Meshing Stage | Contact Line Length Feature |
|---|---|
| Entering Meshing | Contact line length starts to increase |
| Mid – meshing | May reach a relatively stable length in some cases |
| Exiting Meshing | Contact line length decreases |
The contact line length variation is related to parameters such as the helix angle (), face width (), and 重合度 (contact ratio). The contact ratio of helical gears includes the transverse contact ratio () and the longitudinal contact ratio (). The total contact ratio () is the sum of these two, i.e., .
2.2 Causes of Mesh Stiffness Fluctuation
The change in contact line length during helical gear meshing is the main cause of mesh stiffness fluctuation. When the contact line length changes, the load – bearing capacity and deformation characteristics of the gear teeth also change, resulting in a fluctuating mesh stiffness. For example, when more teeth are in contact (higher contact line length), the mesh stiffness is relatively higher.
The mesh stiffness fluctuation can be quantitatively described by parameters. Let’s assume the mesh stiffness at a certain moment is , and its maximum value is and minimum value is . The stiffness fluctuation coefficient is defined as . A smaller indicates less mesh stiffness fluctuation.
3. Low – Fluctuation Mesh Stiffness Design Method
3.1 Design Idea
The core of the low – fluctuation mesh stiffness design method is to minimize the change in the total contact line length during the meshing process of helical gears. According to the meshing principle, we can deduce the conditions for achieving this.
To make the contact line length change the least, we can design either the transverse contact ratio or the longitudinal contact ratio to be an integer. When or is an integer, the number of teeth in contact at different meshing positions is more stable, which helps to reduce the contact line length fluctuation and thus the mesh stiffness fluctuation.
3.2 Parameter Optimization
Changing the basic parameters of helical gears, such as the number of teeth (, ), normal module (), normal pressure angle (), helix angle (), addendum coefficient (), and face width (), can adjust the contact ratio. In this study, we focus on optimizing the helix angle () and face width () because they have a significant impact on .
The relationship between the face width (), helix angle (), and normal module () for achieving a specific contact ratio can be expressed as , where is a positive integer representing the target value of the contact ratio component.
For example, consider a helical gear pair with a certain normal module . When we want to make or , we can calculate the corresponding face width values for different helix angles according to the above formula. The following table shows some sample data:
| Helix Angle () | Face Width for () | Face Width for () |
|---|---|---|
| mm | mm | |
| mm | mm |
4. Establishment of Meshing Stiffness Model
4.1 “Slice Method” and “Offset Method”
The “slice method” and “offset method” are commonly used to calculate the meshing stiffness of helical gear pairs. For a helical gear pair with a contact ratio in the range of 2 – 3, the meshing area can be divided into different regions, such as three – tooth meshing areas and two – tooth meshing areas, as shown in Figure 2.
When calculating the meshing stiffness, we consider the stiffness of each tooth slice. The stiffness of a tooth slice is composed of axial compression stiffness, bending stiffness, and shear stiffness. The overall meshing stiffness of the helical gear pair can be calculated based on the stiffness of these slices. The formula for calculating the meshing stiffness is , where is the stiffness of the -th tooth slice and is the maximum stiffness of a certain component.
4.2 Calculation of Related Parameters
In the process of calculating the meshing stiffness, we need to determine various parameters. For example, the position of the meshing point, the pressure angle at the meshing point, and the load – sharing coefficient of each tooth slice.
The pressure angle at the meshing point can be calculated by , where is the base circle radius and is the radius at the meshing point. The load – sharing coefficient of each tooth slice is related to the load distribution among different teeth and can be obtained through relevant research or calculation methods.
5. Model Verification
5.1 Optimization Schemes
To verify the effectiveness of the proposed design method and the meshing stiffness model, two optimization schemes were designed.
| Optimization Scheme | Optimization Target | Parameter Changes |
|---|---|---|
| Scheme 1 | Optimize helix angle while keeping bearing capacity constant | Increase or decrease helix angle |
| Scheme 2 | Optimize face width while keeping center distance constant | Adjust face width |
The basic parameters of the helical gear pairs before and after optimization are shown in Table 1.
| Parameter | Gear Pair 1 (Before Optimization) | Gear Pair 2 (Optimized Helix Angle) | Gear Pair 3 (Optimized Face Width) | Gear Pair 4 (Another Example) |
|---|---|---|---|---|
| Number of Teeth , | 29/49 | 29/49 | 29/49 | 29/49 |
| Normal Module (mm) | 1.75 | 1.75 | 1.75 | 1.75 |
| Pressure Angle () | 25 | 25 | 25 | 25 |
| Helix Angle () | 10 | 15.95 | 10 | 21.84 |
| Face Width (mm) | 20 | 20 | 31.66 | 20 |
| Addendum Coefficient | 1 | 1 | 1 | 1 |
| Dedendum Coefficient | 0.25 | 0.25 | 0.25 | 0.25 |
| Elastic Modulus (GPa) | 210 | 210 | 210 | 210 |
| Poisson’s Ratio | 0.3 | 0.3 | 0.3 | 0.3 |
| Inner Hole Radius (mm) | 15/25 | 15/25 | 15/25 | 15/25 |
| Mass (kg) | 0.2165/0.6266 | 0.2326/0.6725 | 0.3427/0.9919 | 0.2576/0.7441 |
| Moment of Inertia ( kg·m²) | 0.96/7.90 | 1.07/8.78 | 1.52/12.5 | 1.25/10.3 |
| Transverse Contact Ratio | 1.470 | 1.422 | 1.470 | 1.355 |
| Longitudinal Contact Ratio | 0.632 | 1.000 | 1.000 | 1.353 |
| Total Contact Ratio | 2.102 | 2.422 | 2.470 | 2.708 |
5.2 Finite Element Model and Results
A finite – element model of the helical gear pair was established, as shown in Figure 3. In the model, the load torque N·m was applied to the driven gear, and the angular displacement was applied to the pinion to simulate gear rotation.
The results of the finite – element method (FE) and the analytical model (AM) for the meshing stiffness of different gear pairs are compared in Figure 4. The following conclusions can be drawn:
- After optimization, the stiffness fluctuation is significantly reduced. Before optimization, the stiffness fluctuation coefficient of the FE method is 18.97%. After optimizing the helix angle and face width, the fluctuation coefficients are reduced to 2.02% and 2.96% respectively.
- The calculation results of the AM method are in good agreement with those of the FE method. The maximum calculation error is 4.7%, which verifies the effectiveness of the model.
- When other parameters remain unchanged, the meshing stiffness after optimizing the helix angle is close to the average meshing stiffness before optimization, while the meshing stiffness after optimizing the face width increases significantly.
The impact of mesh stiffness fluctuation on the loaded static transfer error () was also analyzed. The comparison results of of different gear pairs are shown in Figure 5. The optimization design can significantly reduce the fluctuation of . For example, for Gear Pair 1, the average is a certain value, and the peak – to – peak value is 1.80 m. After optimizing the helix angle (Gear Pair 2), the mean value is 11.66 m, and the peak – to – peak value is 0.08 m, with a 95.6% reduction in the peak – to – peak value.
6. Results and Discussion
6.1 Dynamic Modeling of Helical Gear System
A 8 – degree – of – freedom helical gear system dynamic model was established using the lumped – mass method, as shown in Figure 6. In this model, the meshing effect of the gear pair is equivalent to an elastic element with time – varying meshing stiffness and time – varying damping . The influence of unloaded static transfer error and tooth profile clearance is also considered.
The displacement vector of the helical gear system is . The dynamic transfer error () along the meshing line is . The dynamic meshing force along the meshing line is , where .
According to Newton’s second law, the motion equations of the helical gear pair can be obtained, which involve parameters such as bearing damping , bearing stiffness , input load , and output load .
6.2 Vibration Characteristics Analysis
6.2.1 Dynamic Response
The dynamic characteristics of the gear system before and after optimization, including , vibration displacement, and vibration velocity, were compared and analyzed. The root – mean – square value () of displacement and vibration velocity are important parameters to characterize the system vibration.
The of displacement is defined as , where can be , , , . The vibration intensity can be reflected by the of the vibration velocity at a specified point on the mechanical device, , where can be , , , .
The of and vibration displacement in the , , directions with the change of rotational speed are shown in Figure 7. The results show that optimizing the face width has a significant impact on , which can reduce by about a certain value in most rotational speed cases. Optimizing the helix angle has little impact on . In the and directions, the vibration amplitude change trends of the two optimization schemes are basically the same. In some rotational speeds after optimizing the face width, the of displacement slightly decreases. After optimizing the helix angle, the system vibration is reduced by about 2%. In the direction, the results of the two optimization schemes are slightly different. After optimizing the face width, the – direction vibration basically remains unchanged, while after optimizing the helix angle, the – direction vibration increases by about a certain value due to the increase in the axial force caused by the increase in the helix angle.
The of the vibration velocity is shown in Figure 8. Ignoring the change in resonance velocity caused by stiffness and mass changes, the optimization can reduce the and the vibration response in the and directions, and also reduce the system energy. For example, in the third peak region of Figure 8(b), the reduction amplitudes of the after optimizing the helix angle and face width are a certain value and 21.99% respectively. In Figure 8(d), the axial vibration of Gear Pair 2 after optimization is much larger than that of Gear Pair 3 before optimization due to the increase in the helix angle.
6.2.2 Comparison of Two Optimization Schemes
For Gear Pair 1, when the helix angle is increased (positive optimization), the mesh stiffness fluctuation is significantly reduced, but the axial vibration is also significantly intensified. If the helix angle is decreased (negative optimization), this problem can be alleviated.
The contact line fluctuation coefficient is defined to describe the contact line fluctuation. The change of the contact line fluctuation coefficient with the helix angle in the range of is shown in Figure 9. When the helix angle is in the range of , the contact line fluctuation coefficient first decreases and then increases, showing a “V” – shaped curve.
There are two optimization methods. Method 1 is positive optimization, such as optimizing from Gear Pair 1 to Gear Pair 2 by increasing the helix angle to make the contact ratio an integer. Method 2 is negative optimization, such as optimizing from Gear Pair 4 to Gear Pair 2 by decreasing the helix angle to make the contact ratio an integer.
To further compare the differences between the two optimization methods, the time – varying mesh stiffness (TVMS) curves of Gear Pairs 1, 2, and 4 are shown in Figure 10. It can be seen that the stiffness fluctuation coefficient of Gear Pair 4 is 7.01%, and that of Gear Pair 2 is 0.82%.
After negative optimization, the comparison results of the of displacement and velocity of Gear Pairs 2 and 4 are shown in Figure 11. The and axial vibration are both suppressed after negative optimization. The axial vibration is reduced by 26.7%, the derivative of remains basically unchanged, and the axial vibration velocity is reduced by 14.3%. Therefore, compared with Gear Pair 4, the vibration of Gear Pair 2 is effectively suppressed. In general, the proposed parameter optimization design method can effectively reduce the stiffness excitation fluctuation of the gear system. However, it should be noted that modifying parameters may also introduce other excitations, such as an increase in axial force and axial vibration when the helix angle is positively optimized.
7. Conclusion
This article presents a new method for reducing the mesh stiffness fluctuation of helical gear systems through parameter optimization design. The main conclusions are as follows:
- By optimizing design parameters to make the transverse contact ratio or longitudinal contact ratio of the helical gear pair an integer, the contact line fluctuation can be minimized, achieving the goal of low – fluctuation mesh stiffness design. This is based on the principle that a stable number of contacting teeth can lead to a more consistent contact line length and reduced stiffness variation during meshing.
- The finite – element method was used to verify the optimization design method and the meshing stiffness analytical calculation model. Optimizing the face width and helix angle can achieve low – fluctuation stiffness design, proving the accuracy and effectiveness of the calculation model. The significant reduction in mesh stiffness fluctuation after optimization compared to before optimization indicates the practical value of this approach.
- The system can significantly reduce the loaded static transfer error (LSTE) and dynamic response through parameter optimization design. Both optimizing the face width and helix angle can reduce the peak – to – peak value of LSTE. Positively optimizing the helix angle will decrease the radial dynamic response but increase the axial dynamic response. Optimizing the face width and negatively optimizing the helix angle can reduce both the axial and radial dynamic responses of the system. This shows that the proposed parameter optimization design method can effectively achieve vibration reduction, providing valuable guidance for the design of gear systems.
8. Future Research Directions
Although this study has achieved certain results in reducing the mesh stiffness fluctuation of helical gear systems, there are still some aspects that can be further explored.
- Multi – parameter Coupling Optimization: In this research, only the helix angle and face width were optimized. Future studies can consider coupling multiple parameters simultaneously, such as jointly optimizing the number of teeth, modulus, and pressure angle in addition to the helix angle and face width. This comprehensive approach may lead to more significant improvements in vibration reduction and better – optimized gear performance.
- Considering actual working condition factors: Real – world operating conditions of gear systems are complex, involving factors like temperature changes, lubrication conditions, and variable loads. Future research should incorporate these factors into the model. For example, temperature can affect the material properties of gears, and lubrication can change the friction and contact characteristics between teeth. Considering these aspects will make the research results more applicable to practical engineering applications.
- Expand to complex gear systems: This study focused on simple helical gear pairs. Future work can extend this research to more complex gear systems, such as multi – stage gear trains and planetary gear systems. These complex systems have their own unique vibration characteristics, and the low – fluctuation mesh stiffness design method needs to be adjusted and optimized accordingly to meet the requirements of different applications.
