Straight bevel gear is widely used in various mechanical products such as differentials in automotive and agricultural machinery transmissions, and machine tools to transmit power between two intersecting shafts, with an intersection angle of mostly 90 degrees. In theory, the tooth profile curve of a straight bevel gear is usually approximately a spherical involute. Due to the straight tooth line of the straight bevel gear, there is contact along the entire tooth length during meshing. Compared with other bevel gears, it is more sensitive to installation errors and load-bearing contacts, so it is suitable for low-speed and heavy-duty applications.

In recent years, with the rapid development of computer technology, computer modeling and simulation analysis technology have become important tools for current product development and manufacturing. This requires the establishment of a mathematical model that is completely consistent with the real product.

This article describes the method of establishing an accurate mathematical model for straight bevel gear by taking straight bevel gear as the research object and using the gear meshing principle and the generating method of gear shaping. The contact area is dynamically simulated in the software. This method has promotional value in the gear industry.

**1. Establishment of the Three-Dimensional Mathematical Model**

**1.1 Solution of Tooth Surface Points**

Straight bevel gear is usually processed by the generating method on a gear shaping machine. Based on the establishment of the meshing coordinate system of the straight bevel gear with installation errors, a method is proposed to use the Gaussian curvature of the differential surface to represent the sensitivity of the installation error of the modified tooth surface of the straight bevel gear on the two tooth surfaces in contact at one point. The penalty function method is used to optimize the major axis of the contact ellipse to obtain the imprint diagram of the modified straight bevel gear with low sensitivity to installation errors. Studies have shown that the major axis of the contact ellipse has the greatest influence on the sensitivity coefficient of the tooth surface. After optimizing the major axis of the contact ellipse, a better contact trace can be obtained. When the contact trace is perpendicular to the root cone, the sensitivity of the installation error of the modified tooth surface of the straight bevel gear can be reduced. By establishing the meshing coordinate system of the straight bevel gear pair with the installation error of the modified tooth surface of the straight bevel gear, the Gaussian curvature at the meshing point of the straight bevel gear pair is derived, and the value of the Gaussian curvature is used as an indicator to evaluate the sensitivity of the installation error of the modified tooth surface of the straight bevel gear. After analyzing the influence of the position of the reference point and the second-order contact parameters on the Gaussian curvature, the imprint diagram with low sensitivity to the installation error can be obtained by optimizing the major axis of the contact ellipse.

To improve the processing accuracy of bevel gears, a new digital rolling inspection method is proposed. Firstly, the cubic NURBS curve is studied, and the linear equations for solving the control vertices of the NURBS curve are derived. Then, from the curve to the surface, a pair of straight bevel gear parameters is given, and the tooth surface is fitted using the bicubic NURBS surface. The influence of the data points, node vectors, and boundary conditions on the fitting accuracy during the reconstruction of the tooth surface is analyzed. Finally, the fitting error is calculated. After analysis, it can be seen that the error is small, and the fitted tooth surface can replace the real tooth surface. This method provides convenience for the digital design and manufacturing of gears.

Based on the above theoretical research and the derivation of the theoretical contact area, using the Visual Basic language, a software for calculating the tooth surface points of straight bevel gear is developed, as shown in Figure 1. By simply inputting a few basic parameters of the straight bevel gear, the spatial coordinates of the tooth surface points of the gear can be calculated, and then imported into the UG software to complete the establishment of the three-dimensional mathematical model.

Take the straight bevel gear of the differential of the Dongfanghong 58.9 – 66.2 kW wheeled tractor as an example for modeling. The differential of this power segment is composed of two axle gears and two planetary gears. The main parameters of the gear pair are shown in Table 1.

Table 1 Main Parameters of Straight Bevel Gear

Item | Axle Gear | Planetary Gear |
---|---|---|

Number of Teeth | 19 | 11 |

Large End Module (mm) | 6.17 | 6.17 |

Cone Angle (°) | 59.93 | 30.07 |

Top Cone Angle (°) | 63.83 | 36.52 |

Pitch Distance (mm) | 67.73 | 67.73 |

Normal Backlash (mm) | 0.18 | 0.18 |

Pressure Angle (°) | 22.5 | 22.5 |

Tooth Width (mm) | 23 | 26 |

Chordal Tooth Thickness at Large End (mm) | 8.640s | 10.51 – |

Using the written tooth surface point calculation software, by inputting the basic parameters of the axle gear, the tooth surface point data can be obtained, as shown in Figure 2.

**1.2 Establishment of the Three-Dimensional Mathematical Model**

According to the above tooth surface point calculation software, the three-dimensional accurate mathematical model of the straight bevel gear can be established in the UG software as follows:

(1) Create a single tooth surface sheet. Import the tooth surface point data into the UG software in the form of a (.dat) file, and use the “Through Points – Points in File” command in the surface modeling toolbar to establish the tooth surface sheet, as shown in Figure 3.

(2) Create the other two sheet surfaces of the tooth groove. Use the mirror feature to mirror the tooth surface sheet in the first step to establish a symmetrical tooth groove surface, as shown in Figure 4. Use the “Ruled Surface” feature to establish the bottom surface of the tooth groove, as shown in Figure 5.

(3) Create the entire tooth groove. Sew the two sheet surfaces of the tooth groove and the bottom surface of the tooth groove to form a complete tooth groove sheet surface, as shown in Figure 6.

(4) Create the gear blank. According to the requirements of the drawing, create the 804 – 601 left axle gear blank model, as shown in Figure 7.

(5) Create the final digital model. Use the tooth groove sheet surface created in the third step to cut out a tooth groove, as shown in Figure 8. Through the “Circular Array” command, create the final modified digital model, as shown in Figure 9.

Using the same method, establish the mathematical model of the 804 – 603 planetary gear, as shown in Figure 10. The planetary gear mathematical model is not modified in the tooth shape and tooth length direction.

**2. Simulation Analysis of the Contact Area**

In the motion simulation module of the UG software, a motion simulation model of the meshing between the axle gear and the planetary gear is established using the transmission ratio and motion relationship to simulate the cold contact area at multiple moments. Then, by judging the size and position of the contact area based on the simulation situation of the contact area, the modification amount is continuously adjusted and fed back. In this way, the product quality is controlled at the design stage, thereby avoiding the production of unqualified workpieces at the processing stage.

Import the three-dimensional digital models of the established axle gear and planetary gear into the motion module of the UG software, overlap the nodes of the two gears, assemble the two gears according to the actual motion situation, set the tooth surface interference amount to 0.005 mm, set the motion time to 30 s, set the rotation speed to 50 r/min, and the simulation situation of the contact area is shown in Figure 11.

It can be seen from Figure 11 that the contact area situation is good at any intercepted moment of motion. Under no-load conditions, the contact area is close to the small end and occupies about 50% of the tooth length in the tooth length direction, which preliminarily verifies the correctness of the digital model design.

**3. Processing and Testing of the Standard Gear**

**3.1 Processing of the Standard Gear**

The gear manufacturing needs to be completed on the DMG five-axis machining center. Since the processing software equipped with this machine tool can automatically generate the numerical control processing program based on the digital model and the selected tool after importing the three-dimensional mathematical model, the parts completely consistent with the three-dimensional digital model can be manufactured. As shown in Figure 12, it is the five-axis machining center used for processing the standard gear.

To verify the correctness of the three-dimensional digital model, import the established three-dimensional digital model into the software of the DMG five-axis machining center, and then generate the corresponding numerical control processing program. The tool uses a finger milling cutter, and the 45 steel material is used to produce the corresponding standard gear according to the generated processing program, as shown in Figure 13.

**3.2 Testing of the Standard Gear**

Import the three-dimensional digital models of the established axle gear and planetary gear into the software of the gear measuring machine, and compare and test them with the actually processed standard gear. A total of 45 tooth surface points near the pitch circle of the tooth surface are tested. The test results show that the accuracy of the standard gear reaches level 4 accuracy, and the actually processed standard gear has a maximum tooth surface deviation of 0.007 mm compared with the three-dimensional digital model, which meets the processing accuracy required for the standard gear during design.

To verify the contact area, vibration, and noise situation, the standard gear is further tested on the rolling inspection machine, as shown in Figure 14.

It can be seen from Figure 14 that the actual rolling inspection effect is close to the contact area simulation effect in the UG software in Figure 11. Under no-load conditions, the contact area is close to the small end and occupies about 50% of the tooth length and about 60% of the tooth height, close to the middle of the tooth height, without edge contact and other situations. The vibration and noise situation is good, which is the most ideal contact area. This proves the correctness of the established cold mathematical model. This modeling method and contact area simulation analysis method are universal in the straight bevel gear industry.

**4. Conclusion**

(1) Based on the principle of gear shaping by the generating method, a software for calculating the tooth surface points of the modified straight bevel gear is written using the Visual Basic language, and this software is universal.

(2) According to the calculated spatial tooth surface points, a mathematical model of the straight bevel gear is established in the UG software, and the dynamic simulation analysis of the contact area is carried out.

(3) The standard gear is processed according to the established three-dimensional digital model, and the actual contact area is tested, which is consistent with the contact area simulated in the software. This indicates the correctness of the established mathematical model. This modeling and simulation analysis method can effectively shorten the trial production cycle and cost.