1. Introduction
1.1 Background and Significance
In mechanical transmission systems, spur gears play a crucial role. The accurate analysis of the meshing stiffness of spur gear pairs is essential for understanding the dynamic characteristics and transmission performance of the gear system. Tooth length modification and axial misalignment are common phenomena in gear manufacturing and assembly processes, which have a significant impact on the meshing stiffness and overall performance of the gear pair. Therefore, studying the stiffness calculation method of spur gear pairs with tooth length modification and misalignment is of great theoretical and practical significance for improving the design and performance of gear transmissions.
1.2 Research Status at Home and Abroad
Many scholars have conducted in-depth research on the calculation method of the time-varying meshing stiffness of spur gears. The research methods mainly include the finite element method, the analytical method, and the analytical-finite element method. The finite element method can accurately simulate the actual tooth profile and analyze the time-varying meshing stiffness with high precision, but it consumes a large amount of resources. The analytical method has the advantage of high computational efficiency and is widely used in the rapid calculation of time-varying meshing stiffness. However, previous studies have not fully considered the influence of tooth length modification and axial misalignment on the meshing stiffness of spur gear pairs.
1.3 Research Objectives and Contents
This paper aims to propose an improved analytical method for calculating the meshing stiffness of spur gear pairs with tooth length modification and misalignment. The specific research contents include: establishing the mathematical model of spur gear errors, analyzing the meshing characteristics of spur gear pairs with tooth length modification and misalignment using tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA), deriving the improved analytical formula of time-varying meshing stiffness based on the potential energy method and slice theory, verifying the proposed analytical model by comparing with the finite element results, and analyzing the influence of different tooth length modification and misalignment amounts on the meshing stiffness of the gear pair.
2. Mathematical Model of Spur Gear Errors
2.1 Drum Tooth Model
The drum tooth is a common form of tooth length modification. According to the polynomial function, the drum tooth profile can be expressed as Cci=Cc•(bi-b0/B/2)^s, where is the curve bending coefficient, and , are related parameters..
| Parameter | Description |
|---|---|
| Cci | Tooth profile modification amount at an arbitrary point on the tooth profile |
| Cc | Maximum tooth width modification amount |
| bi | Width coordinate of an arbitrary point on the tooth profile |
| b0 | Reference width coordinate |
| B | Tooth width |
| s | Curve bending coefficient |
2.2 Axial Misalignment Model
Axial misalignment occurs when the two parallel axes of the spur gear pair are misaligned by an angle. The misalignment amount at an arbitrary section tooth profile Cmi can be defined by the helix angles of the pinion and gear.
| Parameter | Description |
|---|---|
| Cmi | Misalignment amount at an arbitrary section tooth profile |
| bi | Width coordinate of an arbitrary point on the tooth profile |
| βb | Base helix angle |
| θx1, θx2, θy1, θy2 | Helix angles of the pinion and gear in different directions |
| ψ12 | Installation angle between the pinion and gear |
2.3 Total Error of Tooth Surface
The total error of the tooth surface of the gear pair Ei is the sum of the drum tooth profile modification amount and the axial misalignment amount, i.e., Ei=Cci^(p)+Cci^(g)+Cmi, where the superscripts (p) and (g) represent the pinion and gear, respectively.
3. Tooth Contact Analysis and Loaded Tooth Contact Analysis
3.1 Tooth Contact Analysis (TCA)
For spur gear pairs with tooth length modification and axial misalignment, the meshing state changes from line contact to local conjugate contact. TCA is used to calculate the instantaneous contact points and meshing traces. The tooth surface of the gear is represented by a B-spline surface, and the position vector and normal vector of the tooth surface in different coordinate systems are established. By satisfying the conditions of the position vector and normal vector at the contact point, the contact points can be obtained by iterative solution of nonlinear equations.
3.2 Loaded Tooth Contact Analysis (LTCA)
LTCA is used to analyze the contact characteristics of the gear pair under load. After obtaining the contact points through TCA, the deformation at the contact points under load is considered. The coordinate system at the contact point is established, and the relationship between the elliptical axis length and the principal curvature and direction of the tooth surface at the contact point is analyzed. According to Euler’s formula and Hertz theory, the length of the elliptical major and minor axes is calculated.
4. Time-Varying Meshing Stiffness Model
4.1 Slice Theory and Cantilever Beam Model
The gear is sliced along the tooth width direction, and each slice is regarded as an equivalent variable cross-section cantilever beam. The bending strain energy, shear strain energy, and axial compressive strain energy of the slice under load are calculated using the potential energy method. The expressions of the equivalent stiffness of bending, shear, and axial compression of the slice are derived.
4.2 Hertz Contact Stiffness
The Hertz contact stiffness between the teeth is related to the nonlinear contact force and also shows nonlinear characteristics. The expression of the Hertz stiffness of the slice is given as dKh=Ee^0.9db^0.8Fi^0.1/1.275, where Ee is the equivalent elastic modulus and Fi is the slice load.
4.3 Gear Foundation Stiffness
The gear foundation stiffness Kf is an important part of the meshing stiffness of the gear pair. The expression of the gear foundation stiffness is given, and the relevant parameters are defined.
4.4 Coupling Effect and Spring Model
Due to the nonlinear contact between the tooth surfaces of the spur gear pair with tooth length modification and axial misalignment, there is a coupling effect between the slices. An equivalent spring model is established to represent the coupling effect between the slices. The expression of the coupling spring equivalent stiffness is derived.
4.5 Calculation of Single Tooth Meshing Stiffness
Based on the above models and theories, the single tooth meshing stiffness is calculated. The relationship between the slice deformation and the meshing stiffness is established, and the expression of the single tooth meshing stiffness is derived.
4.6 Calculation of Multi-Tooth Meshing Stiffness
For multi-tooth meshing, the traditional calculation method and the improved calculation method are introduced. The improved method takes into account the sharing of the gear foundation by multiple teeth and corrects the calculation of the gear foundation stiffness. The expression of the improved multi-tooth meshing stiffness is derived.
5. Model Verification and Analysis
5.1 Model Verification
5.1.1 Comparison of Ideal Tooth Profile Stiffness
The proposed method is used to calculate the time-varying meshing stiffness of the ideal tooth profile spur gear pair (when the tooth length modification amount and axial misalignment amount are 0), and the results are compared with those of the cumulative integral potential energy method and the finite element method. The comparison results show that the three methods have similar calculation results and consistent curve trends. The mean value of the time-varying meshing stiffness calculated by the proposed method is 483 kN/mm, and the error compared with the finite element method result (473 kN/mm) is 2.1%, indicating that the proposed method has high calculation accuracy for the ideal tooth profile spur gear pair.
5.1.2 Comparison of Error Spur Gear Pair Stiffness
The contact trace and contact area of the spur gear pair with an axial misalignment amount of 0.01° and a drum tooth amount of 0.01 mm are calculated using the proposed TCA and LTCA techniques, and the results are compared with the finite element results. The two methods have similar calculation results, laying a foundation for the subsequent establishment of the meshing stiffness model. Then, the time-varying meshing stiffness of the spur gear pair under this misalignment and drum tooth amount is calculated using the proposed method, the cumulative integral potential energy method, and the finite element method. The results show that the meshing stiffness calculated by the proposed method is slightly higher than the finite element result, and the calculation result of the cumulative integral potential energy method is slightly lower. The mean value of the time-varying meshing stiffness calculated by the proposed method is 339 kN/mm, and the errors compared with the finite element result (329 kN/mm) and the cumulative integral potential energy method result (315 kN/mm) are 3% and 4.2%, respectively, indicating that the proposed method has higher calculation accuracy.
5.2 Analysis of the Influence of Errors on Meshing Stiffness
5.2.1 Influence of Misalignment Amount on Meshing Stiffness
The axial misalignment amount is set to 0.01°, 0.02°, 0.03°, and 0.04°, respectively, and the meshing stiffness of the spur gear pair under different misalignment amounts is calculated. The results show that as the misalignment amount increases, the meshing stiffness decreases significantly. This is because the increase in the misalignment amount causes the contact area to shift from the middle of the tooth surface to the tooth end, reducing the range of the meshing teeth and the contact long axis, resulting in a decrease in the geometric stiffness, nonlinear Hertz contact stiffness, and comprehensive meshing stiffness. The relationship between the stiffness change and the misalignment amount is not linear, and the influence of the misalignment amount cannot be ignored in the analysis of the meshing stiffness and dynamic characteristics of the gear pair.
5.2.2 Influence of Drum Tooth Amount on Meshing Stiffness
When the drum tooth amount is 10 μm, 15 μm, 20 μm, and 25 μm, respectively, the meshing stiffness of the spur gear pair is calculated. The results show that the time-varying meshing stiffness decreases with the increase in the drum tooth amount. This is because the increase in the drum tooth amount leads to an increase in curvature and a decrease in the radius of curvature, reducing the range of the elliptical contact area under load and the meshing part of the teeth, resulting in a decrease in the geometric stiffness, nonlinear Hertz contact stiffness, and comprehensive meshing stiffness. The relationship between the meshing stiffness change and the drum tooth amount is not linear, and the influence of the drum tooth amount cannot be ignored in the analysis of the time-varying meshing stiffness and dynamic characteristics of the spur gear pair.
6. Conclusion
6.1 Research Achievements
- An improved analytical method for calculating the meshing stiffness of spur gear pairs with tooth length modification and misalignment is proposed. The method combines the potential energy method and slice theory, considering the coupling effect between slices, and can accurately calculate the meshing stiffness of the gear pair.
- The TCA and LTCA techniques are used to accurately calculate the meshing characteristics of the spur gear pair with tooth length modification and misalignment, providing a basis for the accurate construction of the meshing stiffness model.
- The influence of tooth length modification and axial misalignment on the meshing stiffness of the spur gear pair is analyzed. The results show that both axial misalignment and drum teeth have a significant impact on the meshing stiffness, and the influence is nonlinear.
6.2 Research Limitations and Future Research Directions
- In this study, only the influence of tooth length modification and axial misalignment on the meshing stiffness of spur gear pairs is considered, and other factors such as tooth profile modification and surface roughness are not included. Future research can further expand the scope of research to consider more factors affecting the meshing stiffness.
- The proposed method is verified by comparing with the finite element method and experimental results. However, the experimental verification is relatively simple, and more accurate experimental verification is needed in the future.
- In practical applications, the dynamic characteristics of the gear system are also affected by factors such as lubrication and temperature. Future research can focus on the influence of these factors on the dynamic characteristics of the gear system and further improve the design and performance of the gear transmission.
In summary, this research provides a theoretical basis and practical method for the design and analysis of spur gear transmissions with tooth length modification and misalignment. Future research can continue to explore and improve in related fields to promote the development and application of gear technology.