Abstract
To address the issues of efficiency and accuracy in calculating the time-varying meshing stiffness of straight bevel gears, a slicing method considering the coupling effect of tooth pairs is proposed. Based on the slicing discretization approach, the gear teeth and body are discretized into slices along the tooth width and axis directions, respectively. These slices are then unfolded using the equivalent back cone principle. When the slice width is sufficiently small, the bevel gear can be approximated as a series of spur gears, enabling the use of the energy method to calculate the meshing stiffness of each slice. To account for the coupling effect during multi-tooth meshing, a gear stiffness correction coefficient is introduced in the calculation of double-tooth meshing stiffness, thereby reducing errors. An error-state meshing stiffness model is established, and the meshing stiffness, angular displacement of slice centers, and tooth surface load distribution of straight bevel gears under ideal and assembly error conditions are analyzed. Finally, the accuracy and efficiency of the proposed method are validated by comparison with finite element analysis results.
1. Introduction
Straight bevel gears, as crucial components in gear transmission systems, are widely used in various industries, including automotive, precision machine tools, and aerospace applications. Time-varying meshing stiffness (TVMS) is a vital parameter for evaluating the performance of bevel gear transmissions. Accurate TVMS calculation is essential to ensure stable gear operation and optimal performance. However, achieving both efficiency and accuracy in TVMS calculations remains challenging.
Extensive research has been conducted on TVMS calculations for gears, particularly for cylindrical gears where the problem is relatively well-understood. However, the study of TVMS for spatial bevel gears, particularly straight bevel gears, is less developed. Most scholars have relied on finite element analysis (FEA) for TVMS calculations due to its high accuracy. However, FEA is time-consuming and computationally expensive. Alternatively, traditional analytical methods, while faster, often neglect the coupling effect between tooth pairs, leading to substantial errors in multi-tooth meshing stiffness calculations.
This paper proposes a slicing method that combines the advantages of FEA and analytical methods to efficiently and accurately calculate the TVMS of straight bevel gears. This method considers the coupling effect between tooth pairs and assesses the TVMS under ideal and assembly error conditions.
2. Meshing Stiffness Calculation Model for Straight Bevel Gears
2.1 Slicing Model of Straight Bevel Gears
The three-dimensional structure and spatial meshing characteristics of straight bevel gears necessitate a unique approach to TVMS calculation. By discretizing the gear teeth and body into slices of equal width along the tooth width and axis directions, respectively, and unfolding the slices based on the back cone equivalent principle, the bevel gear can be approximated as a series of spur gears.
The parameters related to the slicing model are defined as follows:
- db: Width of each slice
- rj,rfj,raj: Base, tip, and root circle radii of the bevel gear (where j denotes the gear, either driving or driven)
- δj,B,R,rkj: Cone angle, tooth width, outer cone distance, and hole radius
The equivalent parameters for each slice, such as the equivalent number of teeth zvj, equivalent base, tip, and root circle radii (( r’{ij}, r’{aij}, r’_{fij} )), can be calculated using Equations (1).
zvj=cosδjzj
rij′=rjsinδj
raij′=rajsinδaj
rfij′=rfjsinδfj
where i is the slice number, and δa and δf are the tooth tip and root cone angles, respectively.
2.2 Calculation of Slice Meshing Stiffness
Each slice can be approximated as a spur gear, and the overall TVMS of the bevel gear is obtained by summing the instantaneous meshing stiffness of all participating slices. The equivalent meshing stiffness kdli of slice i is determined by considering various deformation components, including Hertz contact stiffness kh, bending stiffness kb, shear stiffness ks, radial compression stiffness ka, and body stiffness kf:
kdli=(kh11+kb11+ks11+ka11+kf11+kh21+kb21+ks21+ka21+kf21)−1
where the superscripts 1 and 2 represent the driving and driven gears, respectively.
To account for the coupling effect between tooth pairs, a gear stiffness correction coefficient λ1j and λ2j are introduced in the double-tooth meshing stiffness calculation:
ksli=j=1∑2(kb1j1+ks1j1+ka1j1+kf1jλ1j+kb2j1+ks2j1+ka2j1+kf2jλ2j+kh1)−1
The torsional meshing stiffness kt can then be calculated using Equation (6):
kt=i=1∑nrbi2kli
where rbi is the base circle radius of slice i.
2.3 Calculation of Gear Body Stiffness Correction Coefficients
To determine the gear stiffness correction coefficients λ1j and λ2j, FEA models of each slice are constructed in Ansys. By setting the elastic modulus of the teeth to be significantly higher than that of the gear body, the teeth can be considered rigid relative to the body, allowing the local Hertz contact and tooth deformations to be neglected. Displacements along the line of action are measured at the midpoint of each slice, and the correction coefficients are calculated using Equations (7) to (10)
3. Meshing Stiffness Model Considering Assembly Errors
3.1 Tooth Profile Deviations Due to Assembly Errors
Assembly errors, particularly shaft intersection angle errors θ and shaft intersection point errors ϵ, significantly impact the meshing stiffness of straight bevel gears. These errors cause deviations in the tooth profiles, affecting the meshing process.
The tooth profile deviations Ei1 and Ei2 due to shaft intersection angle and point errors, respectively, are calculated using Equation (11). Similarly, the angular deviations ei1 and ei2 are determined using Equation (13).
3.2 Error-State Meshing Stiffness Model
Under assembly error conditions, the meshing stiffness calculation must account for the different participation of slices in the meshing process. The angular displacement θi of each slice under load is determined using Equation (15), and the overall torsional meshing stiffness K is calculated using Equation (17).
The load distribution among slices is evaluated, and the total applied torque T is balanced by the sum of torques carried by all engaged slices (Equation 16). The calculation flowchart for angular displacements under error conditions.
4. Case Study and Results
4.1 Finite Element Analysis Model
An FEA model of a straight bevel gear pair is constructed in Ansys. The model is analyzed for a torque of 100 N·m applied to the driving gear, while the driven gear is fully constrained. The mesh convergence is verified to ensure accuracy.
4.2 Validation of Meshing Stiffness Calculation
The TVMS calculated using the proposed method is compared with traditional analytical methods and FEA results. The proposed method shows significantly higher accuracy, with errors of approximately 2% and 3% for double-tooth and single-tooth meshing, respectively.
4.3 Analysis of Meshing Stiffness and Load Distribution under Assembly Errors
The effect of assembly errors on meshing stiffness is analyzed, and the results are compared with FEA . The proposed method accurately captures the reduction in meshing stiffness with increasing assembly errors.
The angular displacements and load distributions of slices under error conditions are analyzed , demonstrating the effect of errors on gear performance.
5. Conclusion
This paper proposes a slicing method for efficiently and accurately calculating the TVMS of straight bevel gears, considering the coupling effect between tooth pairs. The method discretizes the gear into slices, which are then approximated as spur gears for stiffness calculations. By introducing gear stiffness correction coefficients, the errors in double-tooth meshing stiffness calculations are significantly reduced.
Validation against FEA results demonstrates the accuracy and efficiency of the proposed method, with errors of approximately 2% and 3% for double-tooth and single-tooth meshing, respectively. Furthermore, the method is shown to accurately calculate meshing stiffness and load distributions under assembly error conditions.
In conclusion, the proposed slicing method offers a promising approach for the efficient and accurate TVMS calculation of straight bevel gears, paving the way for optimized gear design and improved transmission performance. Future work could explore the influence of other factors, such as surface roughness, friction, and lubrication, on the meshing stiffness of bevel gears.