This article delves into the numerical rolling test of oversized face gears, a crucial aspect in the field of gear transmission. With the increasing demand for high – performance gear systems, especially in the context of nuclear power applications, understanding and optimizing face gear transmission is of great significance. The paper first introduces the background and significance of the research, followed by a detailed description of the face gear transmission in a reactor. Then, it elaborates on the methods of obtaining and fitting the measured tooth surface of the face gear, as well as the numerical rolling test process. Through case analysis and experimental verification, the effectiveness of the proposed approach is demonstrated, and finally, conclusions and prospects for future research are presented.
1. Introduction
In the modern industrial landscape, gear transmission is the backbone of countless mechanical systems. Among various types of gears, face gears have unique advantages such as simple support structures and wide – range selectable shaft angles. However, when it comes to oversized face gears, challenges in manufacturing and ensuring stable meshing performance emerge.
In the nuclear power sector, as the demand for nuclear energy utilization grows, the importance of (spent fuel reprocessing) cannot be overstated. A reactor in the spent fuel reprocessing system uses an oversized face gear transmission as its core technology. The large – scale face gear in this system has to operate in harsh environments, including high radioactivity and a nitric acid – containing atmosphere. Moreover, due to its large size and weight, maintenance and repair can only be carried out remotely, making it difficult to adjust the meshing of the face gear pair and ensure installation accuracy. Therefore, evaluating the actual tooth surface meshing stability of this face gear pair is crucial for guiding the manufacturing and remote inspection and repair of the equipment.
2. Face Gear Transmission in the Reactor
2.1 Structure and Working Principle of Face Gear Transmission
The face gear transmission mechanism in the reactor consists of an involute cylindrical gear and a face gear. When the shaft angle of a bevel gear is 90°, its pitch cone surface becomes a plane, and the teeth are distributed on this plane, thus forming a face gear. Compared with bevel gear transmission, face gear transmission has several advantages. For example, it does not require extremely high installation accuracy, the cylindrical gear has no axial force, and its support structure is simple. Additionally, the transmission ratio can range from 6 to 20, and the selectable shaft angle range is wide. The working principle is based on the meshing of the cylindrical gear and the face gear, which transfers power and motion between different shafts.
| Advantages of Face Gear Transmission | Details |
|---|---|
| Installation Accuracy Requirement | Low, reducing the difficulty of installation |
| Axial Force of Cylindrical Gear | None, simplifying the support structure design |
| Transmission Ratio Range | 6 – 20, providing flexibility in different applications |
| Shaft Angle Range | Wide, adaptable to various mechanical layouts |
2.2 Tooth Surfaces of the Face Gear Pair
2.2.1 Cylindrical Gear Modified Tooth Surface
The design of the modified tooth surface of a cylindrical gear is a reverse process. To establish its theoretical model, a special method is used. Usually, a mismatched (non – standard) involute – shaped grinding wheel is made to move in a parabolic motion for processing. As shown in Figure 1 (replace with the actual figure number in the original paper), first, the straight – line profile of the rack is replaced with a parabolic profile. Then, this parabolic – profile rack is used to generate a cylindrical gear, resulting in a mismatched involute tooth profile for the cylindrical gear.
The equation of the parabolic – profile rack in a certain coordinate system is expressed as: \(R_{c}=\left[\begin{array}{cccc}\cos\alpha&\sin\alpha&0&X_{c0}\\ -\sin\alpha&\cos\alpha&0&Y_{c0}\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\left[\begin{array}{c}-a_{c}(u_{1}-u_{0})^{2}\\ u_{1}\\ 0\\ 1\end{array}\right]\) where \(\alpha\) is the pressure angle, \(X_{c0}=-0.5S_{0}\cos^{2}\alpha\), \(Y_{c0}=0.5S_{0}\cos\alpha\sin\alpha\), \(S_{0}=\pi m / 2\) is the tooth space width (m is the module of the face gear pair), \(a_{c}\) is the coefficient of the parabola, \(u_{0}\) is used to control the position of the parabola vertex, and u is the parameter of the rack profile curve.
The normal vector of the rack \(n_{c}\) is: \(n_{c}=\frac{\partial R_{c}}{\partial u_{1}}\times\left[\begin{array}{lll}0&0&1\end{array}\right]\)
After a series of coordinate transformations, the two – parameter bidirectional modified tooth surface of the cylindrical gear can be obtained. The process involves using the curve family of the parabola in different coordinate systems and considering the motion of the forming milling cutter. The final model of the bidirectional modified tooth surface of the cylindrical gear is established through equations related to the coordinate transformation matrices and the motion of the milling cutter.
2.2.2 Face Gear Tooth Surface
The tooth surface equation of the face gear, as proposed by Litvin, is given by: \(R_{2}=\left[\begin{array}{c}x_{s}\cos\varphi_{2}\cos\varphi_{s}-y_{s}\cos\varphi_{2}\sin\varphi_{s}-u_{s}\sin\varphi_{2}\\ -x_{s}\sin\varphi_{2}\cos\varphi_{s}+y_{s}\sin\varphi_{2}\sin\varphi_{s}-u_{s}\cos\varphi_{2}\\ x_{s}\sin\varphi_{s}+y_{s}\cos\varphi_{s}\end{array}\right]\) where \(x_{s}=r_{bs}[\sin(\theta+\theta_{0})-\theta\cos(\theta+\theta_{0})]\), \(y_{s}=r_{bs}[\sin(\theta+\theta_{0})-\theta\cos(\theta+\theta_{0})]\), \(\theta_{0}=0.5\pi / N_{s}-\tan\alpha+\alpha\), \(r_{bs}\) is the base – circle radius of the pinion cutter for generating the face gear, \(N_{s}\) is the number of teeth of the pinion cutter, \(\theta\) is the involute development angle of the pinion cutter, \(\varphi_{2}=N_{s}/N_{2}\), \(\varphi_{s}\) are the rotation angles of the face gear and the pinion cutter during the generation of the face gear tooth surface respectively, \(N_{2}\) is the number of teeth of the face gear, and \(u_{s}\) is the tooth – direction motion parameter of the pinion cutter. The normal vector of the face gear tooth surface \(n_{2}\) can also be calculated through relevant derivative operations.
3. Fitting Surface of the Measured Tooth Surface of the Face Gear
3.1 Distribution of Measuring Points on the Tooth Surface
When measuring the tooth surface using a gear measuring center, the distribution of measuring points significantly affects the subsequent fitting accuracy. Generally, the denser the measuring points, the higher the fitting accuracy, but it also leads to longer measurement time and lower efficiency. According to the experience of Gleason Company, usually 45 measuring points are taken, with 5 points in the tooth – height direction and 9 points in the tooth – length direction. However, due to the extremely large diameter of the face gear in this study, a measuring machine of such a large size is not available in China. Therefore, during the research process, the cutting tool on the cutting machine is replaced with a probe to measure the coordinates of the tooth surface. In this case, a total of 135 measuring points are set, and their distribution along the tooth – height direction e and tooth – width direction f is shown in Figure 2 (replace with the actual figure number in the original paper).
| Tooth Surface Measuring Method | Advantages | Disadvantages |
|---|---|---|
| Using Gear Measuring Center | High – precision measurement, standard operation | Limited by the size of the measuring machine, high – cost |
| Replacing Tool with Probe on Cutting Machine | Suitable for large – scale gears, cost – effective | Measurement accuracy may be affected by the cutting machine’s accuracy |
3.2 Fitting of the Measured Tooth Surface
The measured tooth surface is fitted using a second – order continuous bi – cubic non – uniform rational B – spline surface (NURBS). The expression of the NURBS surface \(R(e,f)\) is: \(R(e,f)=\frac{\sum_{i = 1}^{m}\sum_{j = 1}^{n}N_{i,3}(e)N_{j,3}(f)W_{i,j}P_{i,j}}{\sum_{i = 1}^{m}\sum_{j = 1}^{n}N_{i,3}(e)N_{j,3}(f)W_{i,j}}\) where m and n are the number of control vertices in the e and f directions respectively (in this paper, \(m = 9\), \(n = 15\)), \(P_{i,j}\) is the control vertex of the surface, \(W_{i,j}\) is the weight factor of \(P_{i,j}\), and \(N_{i,3}\), \(N_{j,3}\) are the cubic B – spline basis functions in the e and f directions.
The fitting process involves first calculating the control vertices of each NURBS curve in the \(e(f)\) direction based on the measured point data, and then using these control vertices as new type – value points to calculate along the \(f(e)\) direction to obtain all the control vertices of the NURBS surface. Substituting these control vertices into the above formula can obtain the bi – cubic NURBS surface. Even if the measured tooth – surface coordinates are usually not on the boundary, the data can be extended to the effective boundary of the tooth surface through extrapolation methods, which has little impact on the inverse – calculated control matrix and can ensure the accuracy of tooth – surface contact analysis.
3.3 Accuracy Verification of the Fitted Tooth Surface
Since the NURBS surface is a constructed tooth surface, it is necessary to verify its fitting accuracy. The verification steps are as follows:
- Take \(9\times15\) points on the theoretical tooth surface in the same range according to the same rule, and fit to obtain the numerical tooth surface of the theoretical tooth surface.
- Find the mid – points of the surface patches formed by every 4 adjacent points among the \(9\times15\) points on the theoretical tooth surface (\(8\times14\) points in total).
- Calculate the normal distance between these mid – points on the theoretical tooth surface and the corresponding points on the numerical tooth surface. The maximum normal distance is the maximum fitting error.
- Sort these mid – points of the surface patches in the form of \((i = 1,2,\cdots,8;j = 1,2,\cdots,14)\) and plot the fitting error.
Taking an oversized face gear as an example, the basic parameters are shown in Table 1 (replace with the actual table number in the original paper). Through verification, the fitting error is within the range of – 0.1 – 0.05 μm, which is sufficient to meet the accuracy requirements for tooth – surface contact analysis. The fitting accuracy is at a similar level to that of the cycloid – tooth hypoid gear in the literature, but the two – side tooth – surface fitting errors of the cycloid – tooth hypoid gear show differences due to its asymmetric tooth surface, while the left – and right – side tooth – surface fitting errors of the straight – tooth face gear in this study are basically the same.
4. Numerical Rolling Test of the Measured Tooth Surface of the Face Gear
4.1 Contact Analysis of the Measured Tooth Surface
The coordinate system for the contact analysis of the bidirectional modified tooth surface of the cylindrical gear \(\sum_{1}\) and the NURBS – fitted surface \(\sum_{2}^{N}\) of the measured tooth surface of the face gear is shown in Figure 3 (replace with the actual figure number in the original paper). In the figure, \(L_{0}\) is the mid – diameter of the face gear, \(B = r_{ps}-r_{p1}\) is the difference between the dividing – circle radius of the pinion cutter and the dividing – circle radius of the cylindrical gear, and \(\phi_{1}\), \(\phi_{2}\) are the rotation angles of the tooth surfaces \(\sum_{1}\), \(\sum_{2}^{N}\) with the coordinate systems \(S_{1}\), \(S_{2}\) in the fixed coordinate system \(S_{f}\) respectively.
The coordinate transformation matrices from the moving coordinate systems \(S_{1}\), \(S_{2}\) to the fixed coordinate system \(S_{f}\) are: \(M_{f1}=\left[\begin{array}{cccc}\cos\phi_{1}&\sin\phi_{1}&0&0\\ -\sin\phi_{1}&\cos\phi_{1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\) \(M_{12}=\left[\begin{array}{cccc}\cos\phi_{2}&\sin\phi_{2}&0&0\\ 0&0&1&B\\ \sin\phi_{2}&-\cos\phi_{1}&0&-L_{0}\\ 0&0&0&1\end{array}\right]\)
The contact of the two tooth surfaces needs to satisfy the following equations: \(\left\{\begin{array}{l}R_{f1}(u_{1},l_{1},\phi_{1}) = M_{f1}(\phi_{1})R_{1}(u_{1},l_{1})\\ R_{f2}(\mu,\lambda,\phi_{2}) = M_{f2}(\phi_{2})R_{2}^{N}(\mu,\lambda)\\ n_{f1}(u_{1},l_{1},\phi_{1}) = M_{f1}(\phi_{1})n_{1}(u_{1},l_{1})\\ n_{f2}(\mu,\lambda,\phi_{2}) = M_{f2}(\phi_{2})n_{2}^{N}(\mu,\lambda)\end{array}\right.\) When \(\phi_{2}\) is selected as the discrete input variable, the 5 – scalar – equation system included in the vector equation can be solved to determine the contact trajectory. The contact ellipse can be determined by using the principal curvature and principal direction of the contact points of the two tooth surfaces, and the transmission error can be determined by using the values of \(\phi_{1}\) and \(\phi_{2}\).
4.2 Determination of the Tooth Surface Modification Parameters of the Cylindrical Gear
By using the above – mentioned contact – surface equations, the contact path between the measured tooth surface of the face gear and the tooth surface of the cylindrical gear is solved. Assuming that the position of the contact path is uniform and the transmission error is continuous, the modification parameters \(a_{c}\), \(u_{0}\), \(a_{l}\), \(l_{0}\) of the cylindrical gear are adjusted to obtain an ideal and stable meshing performance, and the final modification parameters are determined.
5. Case Analysis and Experimental Verification
5.1 Error Distribution of the Measured Tooth Surface of the Face Gear
The error distribution of the measured tooth surface of the face gear corresponding to the parameters in Table 1 is shown in Figure 4 (replace with the actual figure number in the original paper). The measurement coordinates are rotated based on the tooth – space coordinate measurement benchmark to convert the tooth – space tooth surface into the tooth – flank tooth surface. Then, the coordinates of the theoretical tooth surface and the measured tooth surface are compared, and the normal distance between them is calculated as the error of the measured tooth surface.
According to the tooth – profile total deviation specified in the standard GB/T10095 – 2001, for the left tooth surface, the tooth – surface error does not exceed 35 μm, approaching a machining accuracy of level 6. For the right tooth surface, especially at the tooth root, the machining tooth – surface error is relatively large, with a maximum error exceeding 323.02 μm and an average error exceeding 116.25 μm, and its machining accuracy is below level 7 – 8. Since the machining accuracy of the left tooth surface is relatively high and that of the right tooth surface is low, this paper mainly takes the right tooth surface as an example for numerical rolling test and cylindrical gear modification.
5.2 Case Analysis
Several design cases of cylindrical gear tooth surface modification are set up, and the meshing contact situations with the average tooth surface of the right tooth surface of the face gear are analyzed.
| Design Case | \(a_{c}/mm^{-2}\) | \(u_{0}/mm\) | \(a_{l}/mm^{-2}\) | \(l_{0}/mm\) | Contact Imprint Characteristics | Transmission Error Characteristics |
|---|---|---|---|---|---|---|
| Case 1 | 0 | 0 | 0 | 0 | Contact point position varies greatly, long contact ellipse, easy to cause edge contact | Linear, almost non – intersecting, indicating unsmooth meshing in and out of teeth |
| Case 2 | – 2.0×10⁻⁴ | 5 | 0 | 0 | Contact points are relatively evenly distributed above the pitch plane of the face gear tooth surface, concentrated at the tooth root, long ellipse, poor imprint stability | Continuous but with large amplitude |
| Case 3 | 0 | 0 | – 1.0×10⁻⁴ | 0 | Severe edge contact at the tooth tip, non – unique contact points between the tooth tip and tooth root, consistent ellipse length | Overlapping but with large amplitude at the gear – tooth meshing change – over, indicating large meshing – in and meshing – out impacts |
| Case 4 | – 2.0× |
5.3 Experimental Verification
The face gear pair for the experiment consists of an oversized face gear and a cylindrical gear in a reactor, with basic parameters shown in Table 1. The modified tooth surface of the cylindrical gear is processed by a disc – shaped forming milling cutter as shown in Figure 9 (replace with the actual figure number in the original paper), and its modification parameters are in accordance with Case 4 in Table 2. The face gear is processed on the self – developed machine tool QJK007 by Qinchuan. First, a vertical milling cutter is used to open the tooth groove and roughly mill the tooth surface, and then a tapered plane disc milling cutter is used to complete the semi – finishing and finishing milling of the tooth surface based on the patent CN103264198 B.
Due to the large diameter of the face gear, a suitable measuring machine is not available. Therefore, during the research, the cutting tool on the processing machine is replaced with a measuring probe to measure the tooth – surface coordinates. To ensure the measurement accuracy, a test block with a face – gear tooth surface is first measured on a standard measuring machine and then on the processing machine. The measurement error between the two is within ±5 μm. The measurement results of the face gear and the cylindrical gear are shown in Figure 10 and Figure 11 (replace with the actual figure numbers in the original paper) respectively.
Since there is no large – scale rolling test equipment available, an experimental bench is built. The cylindrical gear is slowly driven by manual loading with a torque of about 75 Nm, and the face gear is dragged manually on its outer circumference to detect the contact imprint. As shown in Figure 12 (replace with the actual figure number in the original paper), before modification, the contact imprint of the unmodified cylindrical gear and the face gear shows severe misloading, with the contact imprint length in the tooth – width and tooth – height directions accounting for only 3.8% and 65% of the tooth width and tooth height respectively, almost like edge contact. After modification with the parameters of Case 4, the contact imprint is evenly distributed in the middle of the tooth width, and the lengths in the tooth – width and tooth – height directions account for 55% and 70% respectively, meeting the requirements of gear – transmission contact imprints with an accuracy level of 7 or above. Comparing the numerical simulation results with the experimental results, it can be seen that the contact imprints are basically consistent.
6. Conclusions
In view of the large structural size and difficult – to – control machining accuracy of the face gear in a certain reactor, a method is proposed to improve the meshing stability by modifying the design of the cylindrical gear based on obtaining the measured tooth – surface coordinates of the face gear. The following conclusions can be drawn:
- After obtaining the measured tooth – surface coordinates of the face gear, the NURBS – fitted surface can accurately describe the actual machined tooth surface with a fitting accuracy of up to 0.1 μm. This provides a technical means for the numerical rolling test of the actual machined tooth surface of the face gear.
- Although the manufacturing errors of the face gear are relatively large, through reasonable modification of the cylindrical gear, the deficiency in machining accuracy can be compensated. Even when the tooth – surface error reaches 150 μm, an ideal and stable meshing contact performance can still be obtained.
- The contact imprints of the numerical rolling test and the experimental rolling test of the face gear are basically the same. Before and after modification, the proportions of the tooth – surface contact imprints in the tooth – width and tooth – height directions increase from 3.8% and 65% to 55% and 70% respectively, significantly improving the meshing stability.
7. Future Research Prospects
The research on oversized face gears has achieved certain results, but there is still room for further exploration. In future studies, more in – depth research can be carried out on the following aspects:
- Optimization of modification parameters: Although the current method can obtain good meshing performance, there may be more optimal modification parameters. Through more advanced optimization algorithms, such as genetic algorithms or particle swarm optimization algorithms, the modification parameters of the cylindrical gear can be further optimized to achieve better meshing performance and higher transmission efficiency.
- Influence of dynamic factors: In actual operation, dynamic factors such as vibration and impact will affect the meshing performance of the face gear pair. Future research can consider these dynamic factors, establish more accurate dynamic models, and study the influence of dynamic loads on the tooth – surface contact and transmission error, so as to provide a more reliable theoretical basis for the design and operation of face gear transmission systems.
- Expansion of application scenarios: The research results of this paper are mainly aimed at the face gear transmission in a reactor. In the future, it can be extended to other industrial fields, such as aerospace, large – scale mechanical equipment, etc. Different application scenarios have different requirements for gear performance, and corresponding research and improvement can be carried out according to these requirements.
- Improvement of manufacturing technology: The manufacturing of oversized face gears still faces many challenges. Future research can focus on improving the manufacturing technology, such as developing more advanced grinding and machining methods, to further improve the machining accuracy and surface quality of face gears, and reduce manufacturing costs.
