This article focuses on the milling of spiral bevel gears and proposes a method for the collaborative optimization of tooth surface geometric accuracy. By conducting tooth surface modeling and simulation of double helical milling, and using the load gear contact analysis method (NLTCA), the data-driven relationship between the load contact mechanical performance evaluation and assembly errors is determined. Through modifying the assembly error evaluation, an adaptive data-driven collaborative optimization model is established. On one hand, when determining the target tooth surface through the multi-objective optimization (MOO) of the load contact mechanical performance evaluation, the achievement function method is used to obtain the Pareto optimal solution; on the other hand, the geometric shape of the tooth surface is optimized through assembly error correction, and the sensitivity analysis strategy is used to select the optimal design variables. Finally, a numerical example is used to verify the proposed method, and the results show that the proposed method has good tooth surface geometric accuracy and load contact mechanical performance evaluation.
1. Introduction
Spiral bevel gears and hypoid gears are the key components for power conversion between shaft transmissions, and their tooth surface geometries directly affect the load contact mechanical performance. The geometric characteristics, processing technology, and manufacturing process of spiral bevel gears make the control of geometric accuracy more complex. Through the comprehensive analysis of existing literature, it is shown that machine tool settings are the main design variables of the three main technologies: tooth surface modeling, tooth surface shape error correction, and tooth surface contact analysis (TCA). The machine tool setting correction is actually an “adaptive” optimization design, which can achieve accurate “data-driven” compensation for the geometric accuracy caused by various errors in the manufacturing process through the correction of the initial machine tool settings. Currently, the machine tool setting correction has developed into a closed-loop system of measurement – design – manufacturing. With the application of the universal motion concept (UMC), the machine tool setting has broken through the limitations of previous tooth surface designs based on different tooth systems, processing technologies, and hypoid generators.
However, in addition to the compensation or correction of tooth surface errors, the collaborative optimization of load contact mechanical performance has also become an indispensable main content. In establishing the relationship between machine tool settings and contact performance evaluation, a large amount of numerical operations and optimizations are required. In this regard, the numerical load tooth contact analysis (NLTCA) will provide a main method, but it is very complex. Based on this, this article proposes a precise double helical surface milling geometric accuracy control driven by the load contact mechanical performance to achieve collaborative optimization, considering non-orthogonal spiral bevel gears and hypoid gears, using an improved TCA to solve gear assembly problems, proposing a new load gear contact analysis method (NLTCA), establishing a data-driven relationship, considering the multi-objective optimization of load contact mechanical performance evaluation, and establishing an accurate collaborative optimization model. This method has high calculation efficiency and fast data-driven effect.
2. Double Helical Milling Tool Design and Kinematic Model
In data-driven non-orthogonal tooth surface milling, the double helical method is adopted as the free surface processing method. In the data-driven tooth surface modeling of non-orthogonal gear transmission, the gear adopts the inclined method, and the pinion adopts the double helical method. The entire kinematic simulation of double helical surface milling of non-orthogonal tooth surfaces is expressed as follows: . Here, represents the tooth point vector, represents the gear, and represents the tool. The design variables include Gaussian parameters and motion parameters . Under the UMC framework, machine tool settings include rotation angle , inclination angle , tool radial setting , rolling ratio , etc., and each parameter can be expressed as a polynomial about the basic motion parameter . Among them, is a complex kinematic chain: . Figure 1 describes the kinematic model of double helical surface milling of non-orthogonal spiral bevel gears. At each cutting point position , there is an additional helical motion, so represents the relative speed of the tool with respect to the working gear blank, which can be expressed as.
3 Adaptive Data-Driven Collaborative Optimization
4.1 Assembly Errors and Data-Driven Design
This article takes assembly errors as the basic data for collaborative optimization. Figure 4 shows the basic definition of assembly errors of spiral bevel gears and hypoid gears, where represents the displacement of in the direction of the pinion shaft, represents the displacement of in the direction of the gear shaft, represents the offset between the two gear shafts, and represents the angle between the two gear shafts. The positive direction is represented by “+”, and the negative direction is represented by “-“. There exists: . Assembly errors can replace machine tool settings in their correction, and assembly error correction is also an adaptive data-driven design.
4.2 Adaptive Data-Driven Collaborative Optimization Model
An adaptive optimization model for the accuracy driven by the load contact performance of double helical surface milling is established: . Transforming Equation (30) into a nonlinear least squares problem is:
. The goal is to obtain accurate assembly errors , which is actually a feedback and optimization about the initial assembly errors , by approaching the target tooth surface, and the target tooth surface is determined by the MOO solution.
