Abstract
Spiral bevel gears, as core components in mechanical transmission systems, require optimization and adjustment of parameters through trial cutting before mass production to ensure the final gear meets design parameters. To reduce the actual trial cutting process, simulation technology has been employed to achieve machining simulation of spiral bevel gears. However, current simulation methods often rely on expensive and complex 3D software or require tedious secondary development. Additionally, some 3D simulation algorithms suffer from issues such as slow simulation speeds and inaccurate models. In the actual process of optimizing machining process parameters, complex machine tool adjustments are often necessary to verify the effectiveness of parameter optimization, which is inefficient and wasteful of materials and time. Therefore, it is essential to develop a method that can achieve rapid machining simulation to improve simulation efficiency and provide an effective way to optimize machining parameters.
This paper addresses the above problems by studying the structure and motion principles of traditional mechanical milling machines and CNC milling machines based on the principle of tooth cutting. A coordinate system for tooth cutting is established, and the calculation relationships for each axis during CNC milling are derived based on the conversion relationship of the cutting motion of each axis of the two types of machines. Simultaneously, STL models of the gear blank and cutter head are constructed, and a topology reconstruction method is proposed to reduce memory usage while maintaining the original shape of the models, thereby improving the operational efficiency of the simulation method. The VTK visualization tool function library is introduced to realize the visualization of models during the machining simulation process. A machining simulation method for the tooth surface forming process of spiral bevel gears is proposed based on the Boolean operation principle and intersection checking method using the CGAL geometric algorithm library. During the simulation, data points of the simulated tooth surface are extracted, and a mathematical model of the cutterhead cone surface is established. The VTK surface reconstruction method is utilized to reconstruct the tooth surface and cutterhead cone surface. The finishing cutting alignment is completed according to the alignment process, realizing the finishing machining simulation of spiral bevel gears. Based on the principle of oblique cutting, a mathematical model of cutting force is established to calculate the cutting force during the machining process. An AABB tree fast slicing algorithm is proposed to calculate the cutting areas of the inner and outer cutters. The calculation process of theoretical tooth surface points is derived, and theoretical mathematical models for calculating tooth surface, tooth height, and tooth root errors are established to verify the accuracy of the simulation method. Based on the dynamic relationship between cutting force and feed speed, the feed speed is optimized and adjusted for different cutting methods, improving processing efficiency. Finally, the rapidity and accuracy of the machining simulation method are verified through relevant simulation examples, and the effectiveness of optimizing the feed speed to improve machining efficiency is validated through actual cutting experiments.
This paper realizes a rapid machining simulation method for spiral bevel gears and, during the simulation process, accurately predicts the dynamic trend of cutting forces in the actual machining process with the aid of the established cutting force mathematical model. According to the relationship between cutting force and feed speed, the feed speed is rationally optimized, and the machining efficiency is effectively improved. The paper provides a valuable approach for research on the machining simulation method and machining process parameter optimization of spiral bevel gears and has good practical application value.
Keywords: spiral bevel gear; boolean operation; machining simulation; cutting force; feed speed optimization
1. Source of the Research
This research is supported by the major special project of new-generation artificial intelligence technology in Tianjin – “Research and Development of Key Technologies and Applications of High-end Precision Intelligent Bevel Gear Complete Machining Equipment” (No. 21ZXJBGX00020) and the National Natural Science Foundation of China project “Research on the Tooth Profile Design and Manufacturing Method of Spiral Bevel Gears for High-speed Transmission System Vibration Control” (No. 52275052).
2. Research Background and Significance
Spiral bevel gears, with their complexity and multifunctionality, are indispensable key components in mechanical transmission systems due to their high load-bearing capacity, stable transmission process, and low vibration. As manufacturing technology advances, the processing technology and manufacturing processes of spiral bevel gears have been significantly improved and optimized to meet the increasing demands for efficiency, precision, and reliability in modern mechanical systems.
The unique spiral shape of spiral bevel gears赋予 them exceptional transmission performance. If a straight bevel gear is considered the basic model, the formation of a spiral bevel gear can be imagined as numerous slices of straight bevel gears being precisely cut along the central rotation axis and then subtly twisted and inclined around the pitch cone generatrix, thereby shaping the spiral tooth profile. This enables smooth meshing during the transmission process, resulting in low noise and high tooth surface contact ratios. Since the tooth profile of spiral bevel gears is not a standardized design, it is often difficult to accurately predict root cutting phenomena during the machining process solely through theoretical calculations. Therefore, trial cutting on the machine tool is necessary to analyze and determine the reasonable range of machine tool adjustment parameters to ensure that the final machined gear pair meets the design requirements. This process is complex, time-consuming, and increases production costs and resource waste, significantly reducing manufacturing efficiency. Thus, improving production efficiency and reducing costs have become urgent issues in the mechanical field.
With the deep integration of virtual simulation technology into gear processing, strong support is provided for solving the problems that are difficult to accurately determine in traditional machining. In a virtual 3D environment, models of the machine tool, cutter head, and gear blank are established, and the machine tool’s motion drives the cutter head to cut the gear blank, simulating the actual machining process and replacing traditional trial cutting methods. This not only verifies the correctness of relevant parameters but also enhances the production efficiency of spiral bevel gears and reduces production costs.
However, current simulations of the cutting process of spiral bevel gears mostly rely on 3D software developed by foreign companies or require secondary development based on such software to achieve 3D simulations. This makes simulations dependent on the software, increasing usage costs. Additionally, some machining simulation algorithms that do not rely on existing 3D simulation software have issues such as large computational loads, slow operation speeds, low efficiency, and unstable computational processes, making it difficult to achieve real-time simulation. Therefore, research on a method that can achieve rapid machining simulation of spiral bevel gears is particularly important, not only holding academic significance but also possessing important practical value.
Furthermore, current machining simulation methods primarily focus on simulating the geometric motion between the tool and workpiece, rarely considering the influence of factors such as cutting force and feed speed on machining quality and efficiency during the actual machining process. To minimize adverse effects, it is crucial to consider the optimization of machining process parameters during the actual machining process. Process parameters not only affect the surface quality of the product but also determine the machining efficiency to a certain extent. Selecting appropriate process parameters can enhance machining efficiency, reduce production costs, and ensure product quality. In-depth research and analysis of cutting forces during the cutting process to improve machining efficiency and ensure machining quality are extremely important. Therefore, research on the optimization of machining process parameters is particularly necessary in the current field of mechanical processing.
This research focuses on the long computation time and low efficiency of the tooth surface forming process in spiral bevel gear machining simulations. It conducts in-depth research on the machining simulation method of spiral bevel gears. Based on the Boolean principle and existing geometric algorithm libraries, a precise, stable, and rapid machining simulation method is proposed to improve the efficiency of spiral bevel gear machining simulations. Addressing the current lack of application of simulation processes in considering the optimization of actual machining process parameters, a cutting force calculation model is established based on the oblique cutting principle. The simulation is used to calculate the cutting area during the machining process, accurately predicting the trend of cutting force changes during different cutting methods. Based on the relationship between cutting force and feed speed, the feed speed during the cutting process is optimized, improving cutting efficiency. This method provides a feasible reference for the optimization of machining process parameters and offers strong scientific support for selecting appropriate machining methods and parameters in the actual production process of spiral bevel gears.
3. Current Research Status
3.1 Research Status of Spiral Bevel Gear Machining Technology
The machining technology and machining tools of spiral bevel gears are closely related and mutually supportive. Over the past two centuries, the machining technology of spiral bevel gears has made significant progress, gradually developing into a mature and well-established field. The machining tools for spiral bevel gears have evolved from traditional mechanical types to partially numerically controlled types and, ultimately, to modern fully numerically controlled types.
In the early 1960s, Gleason Corporation introduced the No. 116 machine tool, a peak achievement in traditional mechanical machine tools and a pivotal transition point between traditional mechanical machine tools and numerical control machine tools. By the mid-1980s, PLC control technology was applied to the S17 machine tool by Oerlikon, marking the end of the traditional mechanical machine tool era and the advent of the numerical control era, significantly enhancing milling efficiency. Subsequently, Gleason introduced two types of machine tools that fully abandoned the complex mechanisms of traditional mechanical machine tools and fully realized parametric numerical control adjustment: the CNC spiral bevel gear milling machine and grinding machine of the Phoenix series. Shortly after, Gleason launched the Gleason Expert Manufacturing System for Conical Gears (GEMS), specifically designed for the Phoenix series CNC spiral bevel gear milling machines. This system aims to achieve information exchange and sharing between engineering workstations and Gleason CNC machine tools. At the beginning of the 21st century, Gleason Corporation introduced a column-type CNC bevel gear milling machine, represented by the Phoenix II generation machine tool. The birth of this revolutionary machine tool signifies a new era in the field of bevel gear machining tools.
Research on spiral bevel gears in China began in the 1970s, relatively late compared to foreign countries. However, significant research achievements have been made by many renowned domestic experts and scholars. Among them, Zeng Tao, Zheng Changqi, Wu Xutang, and others have conducted in-depth research on the design and machining theory of spiral bevel gears, forming their unique theoretical achievements based on foreign research. Regarding the research on machining equipment, the initial stage primarily relied on imports and imitation. After understanding the theory and machining principles of spiral bevel gears, Tianjin First Machine Tool Works produced the Y2250 and Y2280 mechanical spiral bevel gear milling machines. After 1990, with the development of numerical control technology, Chinese universities and enterprises began cooperating on the research of numerical control machine tools. For example, the Qinchuan Machine Tool Group and Xi’an Jiaotong University jointly developed the YH2240 CNC milling machine, and the Gear Research Institute of Central South University developed the YK221 and YK2245 CNC milling machines capable of six-axis and five-linkage. Additionally, Changsha Hongli Precision Machinery Co., Ltd., developed the YK2045 spiral bevel gear CNC milling machine, and Tianjin Jingcheng Machine Tool Manufacturing Co., Ltd., developed the YH6012 four-axis linkage CNC spiral bevel gear milling machine in 2005. In the same year, China also successfully established the first digital production line capable of closed-loop production of spiral bevel gears, realizing mass production. This marks the realization of full CNC machine tool production for spiral bevel gears in China in the early 21st century.
As the technology of spiral bevel gear machining tools continues to mature, their application range has gradually expanded, providing equipment foundations for subsequent research related to the spiral bevel gear machining process. The widespread use of milling machines provides important support for in-depth research on the milling process and related fields, further promoting the development of this field.
3.2 Research Status of Milling Process Parameter Optimization
With the increasing maturity of gear design theory and gear machining technology, the mechanical field is no longer satisfied solely with the technology for machining gear tooth surfaces. To meet the needs of modern production, scholars at home and abroad have shifted their attention to the machining process. Among the factors in the machining process, cutting force is a crucial one to consider. Irregularly varying cutting forces significantly impact the precision, surface quality of the workpiece, and the service life of the cutter head. Therefore, domestic and foreign scholars have conducted in-depth research on cutting forces during different machining processes and actively explored methods to optimize process parameters, aiming to enhance the stability and efficiency of the machining process.
Currently, foreign scholars such as Antoniadis et al. have established a cutting force model during the hobbing process of spur gears; Zhang et al. have proposed a cutting force prediction model combining SVM and ACO algorithms to improve the prediction accuracy of cutting forces during grinding; Habibi et al. have utilized a semi-analytical method to solve the boundary of undeformed chips and predict cutting forces during the face hobbing of spiral bevel gears; Andersson et al. have established a cutting force model for the multi-tooth cutting process and studied the factors influencing cutting forces during end milling, concluding that single-tooth cutting is affected by the adjacent two blades; Sabkhi has studied the geometry of undeformed chips during the hobbing process and calculated the cutting force coefficients through numerical models, analyzing the variation of cutting forces during the machining process and demonstrating the influence of machining parameters on material removal.
Domestically, Chen Bin et al. have established cutting force models for the down milling and up milling of spur gears using disc-shaped milling cutters and derived the calculation expressions for cutting forces; Xin Zhijie has studied the cutting thickness, cutting width, and dynamic cutting displacement caused by external excitation during the machining process, establishing a dynamic cutting force model for spiral end mills; Shi Rui et al. have proposed a method for calculating the instantaneous undeformed cutting width and thickness, calculated the instantaneous cutting area, and established a cutting force model during the generating process based on the principle of oblique cutting; Jia Xinjie has calculated the undeformed chip thickness and width during the cutting process, calibrated the cutting force coefficients through experiments, and established dynamic cutting force models for form milling and hobbing.
Simultaneously, based on their research on cutting forces, domestic and foreign scholars have studied the optimization of process parameters. Li et al. have established a regression prediction model for cutting force and surface roughness, aiming to optimize them. This model can effectively improve the surface quality of machined workpieces; Subramanian et al. have designed cutting experiments using the response surface algorithm and optimized cutting forces using a genetic algorithm, obtaining the optimal combination of parameters; Brecher et al. have optimized the gear grinding process and established an optimization model for cutting forces during gear grinding; Asokan et al. have used cutting force, spindle power, etc., as constraints and established an optimization model with production cost as the objective using the simulated annealing algorithm to obtain the best cutting parameters; Yang F et al. have adopted a particle swarm optimization algorithm, taking workpiece efficiency and cost as objectives and cutting force and stability during cutting as constraints, to optimize machining process parameters and improve machining efficiency.
Hao Hongyan et al. have established a process optimization model with cutting speed, feed rate, and axial depth of cut as variables, constraints such as cutting force, stability, and roughness, and maximum production efficiency as the objective; Li Zhizhong has calculated cutting forces during the cutting process by discretizing the tool and optimized the feed rate based on the cutting forces, ultimately optimizing the cutting tool path; Zhang Chen has used the cutting force and spindle speed as constraints and the machining efficiency as the optimization objective, utilizing the discrete method to achieve the optimization of machining process parameters.
From the above research, it can be seen that domestic and foreign scholars have successfully pointed out practical directions for the research on cutting forces and the optimization of process parameters during the machining process. Meanwhile, these studies provide a reliable theoretical basis for calculating the cutting area during the cutting process in virtual simulation technology for spiral bevel gear machining.
3.3 Research Status of Machining Simulation Technology for Spiral Bevel Gears
With the gradual maturity of computer technology, the manufacturing industry has developed rapidly. The machining technology and machining processes of spiral bevel gears have also made good progress. Attempts have been made to replace the trial cutting process for determining the rationality of machine tool adjustment parameters with virtual simulation technology to reduce costs and improve production efficiency.
Foreign scholars such as Litvin et al. have utilized computer technology and the local synthesis method to realize gear meshing motion simulation and simulated gear models with optimized machine tool adjustment parameters; Huston et al. have taken bevel gears and spiral bevel gears as research objects, written solid modeling programs on computers, and used them to simulate tooth slot cutting processes; Mohan et al. have used C language programming to control the relative positions of the cutter head and gear blank based on NC files, realizing the machining simulation of spiral bevel gears; Wang et al. have written NC codes based on the generation method of spiral bevel gears and performed virtual cutting simulations of spiral bevel gears in VERICUT 6.0. By comparing with the actual machining results, they verified the correctness of the written codes and the feasibility of virtual cutting; Efstathiou et al. have successfully developed a virtual cutting platform for spiral bevel gears using AutoCAD software, which realizes the motion simulation of face milling and hobbing of spiral bevel gears and simultaneously generates solid models of gears.
Domestically, Liang Wei from Chongqing University has studied the Gleason gear machining process and developed a software system capable of simulating the cutting process and analyzing data using C++ language; Deng Xiaozhong et al. have developed a computer-aided design system for the processing parameter design, meshing simulation, and tooth root stress analysis of aerospace spiral bevel gears, which is easy to use and operate; Xiong Yuedong et al. have realized the NC machining simulation of spiral bevel gears in the Visual C++ and AutoCAD environments using the secondary development package ObjectARX; Lu Mingwen has developed a CNC bevel gear machining simulation software under the 3D software UG platform using C++ programming tools. This software has successfully machined automotive rear axle bevel gears, verifying the correctness and reliability of the system; Wang Zhonghou et al. have utilized CATIA for programming and successfully realized the simulation of spiral bevel gear tooth cutting; Han Jiaying has proposed a cutting simulation algorithm based on analytical calculation, realizing the visualization simulation of multiple cuttings of spiral bevel gears; Li Dongying has utilized the professional NC simulation software VERICUT to realize the machining simulation of helical straight bevel gears; Li Jingcai et al. have proposed a slice segmentation algorithm, effectively solving the problems of speed and accuracy in traditional simulations. Based on this algorithm, they have successfully developed a five-axis NC machining center simulation system, which has cut quasi-hypoid solid models and verified the correctness of the system.
From the above research, it can be seen that domestic and foreign scholars have made significant achievements in machining simulation technology. However, existing simulation methods still suffer from issues such as slow simulation speeds and low model accuracy.
In summary, China’s research on the theory and manufacturing technology of spiral bevel gears has gradually improved, laying a solid foundation for in-depth research on the NC machining process of spiral bevel gears. Research on cutting forces and the optimization of machining process parameters during the machining process provides practical references for the subsequent optimization