With the development of industrial equipment, non-standard cylindrical gears have broader application prospects. This article focuses on the construction of a parametric model for non-standard cylindrical gears. By deducing the machining process and applying relevant theories and methods, the accurate tooth profile equations of non-standard gears are established, and the parametric finite element model is constructed. The model is verified by examples, and its high precision and efficiency are demonstrated. It provides important technical support for the design, manufacturing and performance analysis of non-standard cylindrical gears.
1. Introduction
Gear transmission is widely used in various fields of industry. Standard gears can no longer meet the diverse requirements of actual production. Non-standard cylindrical gears, with their variable parameters such as pressure angle, addendum, dedendum and displacement coefficient, have unique advantages in some special working conditions. However, the analysis and design of non-standard gears face challenges due to the complexity of their tooth profile changes and the influence on gear performance.
The traditional methods for evaluating non-standard gears have obvious disadvantages. The method based on ISO standard gear equivalent empirical formula has low accuracy in calculating results and cannot adapt to the changes of non-standard parameters. The method of using three-dimensional software modeling analysis is cumbersome in the modeling process, difficult to solve, and needs to be remodeled when the parameters change. Therefore, it is necessary to establish a more accurate and efficient parametric model for non-standard cylindrical gears.
2. Tooth Profile Equation Derivation of Non-standard Cylindrical Gears
2.1 Tooth Profile Equation of Hobbing External Gears
The hobbing process of external gears involves the movement and meshing of the rack cutter and the gear blank. By establishing a coordinate system and analyzing the geometric relationship, the tooth profile equation can be derived. As shown in Figure 1, in the fixed rectangular coordinate system XPY and the coordinate system X₁O₁Y₁ on the rack, the movement of the rack cutter during hobbing is considered. The equations for calculating the coordinates of the cutting points on the tooth profile are as follows:
where m is the rack modulus, ξ is the displacement coefficient, α₀ is the rack pressure angle, and S is the distance between the O₁Y₁ axis of the rack and the PY axis of the fixed coordinate.
For the rounded corner part of the cutter, the coordinates of the cutting point K can also be obtained through a series of geometric calculations. Finally, by transforming the coordinates of the cutting points B and K of the cutter tooth profile to the coordinate system of the processed gear, the coordinates of the tooth profile and the transition curve can be determined.
2.2 Tooth Profile Equation of Shaping Internal Gears
The tooth profile of the internal gear is formed by the shaping cutter. First, the tooth profile equation of the shaping cutter is derived. Assuming that the shaping cutter tooth profile is processed by a rack, through the meshing relationship and coordinate transformation, the tooth profile curve equation of the shaping cutter can be obtained:
where rp1 is the pitch circle radius of the shaping cutter, θ₁ is the rotation angle of the shaping cutter when the rack moves a certain distance, α₁ is the pressure angle at the pitch circle of the shaping cutter, m is the modulus of the shaping cutter, and xc is the displacement coefficient of the shaping cutter.
Considering the stress concentration problem at the tooth root of the internal gear caused by the sharp corner of the shaping cutter tooth top, the rounded corner is used to replace it. By determining the center and cutting point coordinates of the rounded corner, the tooth profile equation and transition curve equation of the internal gear can be further derived through coordinate transformation.
3. Parametric Model Construction of Non-standard Cylindrical Gears
3.1 Construction of Parametric Units
Based on the derived tooth profile curve equations, a program is compiled to calculate the coordinates of discrete points on the end faces of non-standard internal and external cylindrical gears. As shown in Figure 4, the discrete nodes of the single teeth of the internal and external gears are obtained. According to the natural point sorting method of the unit in the ABAQUS manual, the finite element mesh models of the single teeth of the non-standard internal and external gears are constructed. By rotating and arraying along the circumferential direction and merging the tolerances, the full tooth finite element models are obtained. In the process of model construction, the linear reduced integration unit (C3D8R) is selected considering the influence of the unit type on the calculation accuracy and efficiency. After the model is assembled accurately, the material properties, analysis steps, reference points, contact pairs, field variables, history variables, boundary conditions and loads are set, and the solution is carried out to obtain the contact stress and relative rotation angle of the gear teeth at different meshing positions.
3.2 Setting of Parametric Finite Element Model
ABAQUS provides two methods for parameterization. One is to use a script file and Python language. Python language can realize the automatic reproduction of operations through a series of commands. The other is to use the Inp file based on Fortran language, which contains the data information, material properties, contact properties, analysis step settings, loads and boundary conditions of the model. The first method has stronger automation ability, and the author finally chooses this method to realize the parameterization of the model. The specific process of parameterization is shown in Figure 8.
4. Model Verification
4.1 Verification Example Introduction
Taking a non-standard planetary transmission system as an example, the parameters of the sun gear, planetary gear and internal gear ring are shown in Table 1. The planetary transmission system contains internal and external meshing pairs, which can be used to verify the analysis model of non-standard cylindrical gears.
| Parameter | Sun Gear | Planetary Gear | Internal Gear Ring |
|---|---|---|---|
| Number of Teeth | 48 | 55 | 162 |
| Module (mm) | 3.8 | ||
| Pressure Angle (°) | 22.5 | ||
| Tooth Width (mm) | 90 | 88 | 88 |
| Rotational Speed of Sun Gear (r/min) | 1128.0 | ||
| Input Power (kW) | 2985.8 | ||
| Input Torque (N·m) | 25278.7 | ||
| Single Path Meshing Force (N) | 60003.2 |
4.2 Verification Results and Analysis
By comparing the tooth surface contact stress before and after parabolic micro-modification of the gears and the tooth surface contact stress before and after modification when there is an axis angle error of 1′, it is found that the parametric model established in this paper is accurate and efficient. The comparison results of the tooth surface contact stress of the external and internal meshing gears before and after modification are shown in Figures 9-12. These figures clearly show the changes in contact stress before and after modification, which verifies the effectiveness of the model in analyzing and predicting the performance of non-standard gears.
5. Conclusion and Prospect
This paper successfully establishes the tooth profile curve equations of non-standard internal and external gears through simulation and analysis, and constructs the parametric finite element model. The verification results show that the model has high precision and efficiency, realizes the consistency between the gear tooth surface modeling method and the actual processing technology, and provides strong technical support for the parameter design, process adjustment and tooth modification of non-standard cylindrical gears.
In the future, with the continuous development of manufacturing technology and the increasing demand for high-performance gears, the research on non-standard cylindrical gears can be further deepened. For example, more accurate models can be established considering more factors such as material properties and manufacturing errors, and the optimization design method of non-standard gears can be explored to improve the performance and reliability of gears in different application scenarios.
In addition, in the process of model construction and application, the combination of advanced software and algorithms can be further strengthened to improve the automation and intelligence level of gear design and analysis. At the same time, more experimental research can be carried out to continuously verify and improve the theoretical model to promote the development and application of non-standard cylindrical gears in the field of mechanical engineering.
