Research paperA topological flank modification method based on contact trace evaluated genetic algorithm in continuous generating grinding
Introduction
Modification of gear flanks has been widely used to reach goals of better residual stresses characteristics, running noise minimization, mesh impact reduction, and assembly error elimination of gears by manufacturing tolerances [1]. As one of the common continuous generating grinding methods, worm wheel gear grinding has great advantages in the mass production of automobile gears [2]. Gear pair meshing performance is affected by tooth deformation due to bearing load, while unexpected flank twist error exists when lead modifications are implemented to solve the problem [3]. Therefore, achieving precise tooth flank modification is one of the key research topics in the field of gear grinding.
The Traditional lead modification method implemented by adjusting the center distance between the tool and the work gear leads to non-homogeneous stock removal along the direction of tooth profile in the transverse section. The essential characteristic can be explained as unparallel between contact trace on the flank and transverse section of helical gear. The phenomenon that the profile slope deviations in the middle of face width gradually increased and reversed along the direction of two end faces of gears is called the “flank twist” [4]. Therefore, the manufacturing of cylindrical helical gear on continuous generating grinding machines will cause the above flank deviations as long as a certain type of lead modification was designed. Furthermore, the meshing performance of the gear pair will be affected. Modern CNC generating grinding machines are equipped with a dressing device, which was designed for adjusting surface roughness of tooth flank, implementing dressing cycles and changing the pressure angle of worm wheel profile [5]. However, differences in the profile of dressing tools and principles of the dressing process do not affect the microgeometry of the worm wheel when a worm with a regular profile is used [6]. To address the above issues, this paper proposed a topological flank modification method of helical gears implemented by setting additional X, Y, and Z axis movement on continuous generating grinding machines, which does not require a special dressing process towards the worm wheel in the dressing stage.
Researches on gear flank modification method and its application have been studied by many researchers from different aspects, such as simulation of meshing, tool profile modification method, and higher-order flank correction technology. In earlier work, the differences in helical overlap contact ratio of tooth surface had been outlined by Hoyashita [3] through the proposed gear profile calculation method. Litvin [7] discussed the necessary and sufficient conditions of the envelope to a two-parameter family of the tool surface and applied to the generation of a helical gear by worm wheel. Shortly after, Litvin et al [8]. proposed a method of generation of asymmetric spur gear drive for reduction of contact and bending stresses. Guo et al [9]. employed a dressing wheel designed with parabolic-shaped modification to produce face gears by a six-axis CNC worm wheel machine. Fong and Chen [10] present a tooth crowning method for helical gears by variable lead grinding worm. The wheel profile forward and gear profile backward calculation process using the second envelope method of point-vector were explained by He et al [11]..
Higher-order flank correction technology was first applied to process spiral bevel and hypoid gears, and then extended to modern CNC gear cutting machines for more types of gears including spur and helical gears [12], [13], [14], [15]. To further reduce form grinding errors, Shih and Chen [16] proposed a wheel profile defined by five B-spline curves and formulated each axis as a six-degree polynomial. More recently, Han et al [17]. proposed a method for lead and profile crowning tooth flank of gear with standard involute diamond dressing gear on internal gearing power honing gear machine. Tran and Wu [18] demonstrated that gear with double-crowned tooth flanks is better than with longitudinal-crowned tooth flanks and produced the surface by a closed-loop topology modification method.
Within such investigation, researchers have investigated a variety of approaches on the implementation of flank modification by high-order flank correction technology. But when it comes to continuous generating grinding, the main research strategy is to control flank error by modifying the profile of the worm wheel. Though it can achieve mass production of work gears with flank modifications, this strategy is not suitable for the situation where the modification parameters of target tooth surface change because process parameters are required to adjust both in the dressing stage for ideal wheel profile and in the grinding stage for necessary machine tool axes movement. For example, worm-shaped tools with variable pitch and with quadratic parabola type crowning over the worm length were implemented by machine tool manufacturers Gleason-Pfauter to compensate the flank twist in continuous generating grinding of helical gears [19,20]. The involute worm form tool with variable pressure angle of left or right flanks was formerly used by Liebherr to avoid twist in gear hobbing [21]. To date, only a few manufacturers operate flank error correction by regular worm wheel, for example, the wheel with regular grinding and finish grinding zones by Reishauer [22]. This method of regular profile worm wheel requires no additional calculation of process parameters or no special design of the dresser. Therefore, the application value of this method is remarkably high considering the generating grinding time, which is almost the same as compared to the industrial production time generally required.
This paper aims to propose a method for topological modification of gear flanks in the continuous generating grinding process considering the location and movement of contact traces. Firstly, a mathematical model calculating the generating surfaces of the worm wheel from involute helicoid of cylindrical helical gear is established. The wheel can then be represented as a two-parametric form spiral surface. Subsequently, a mathematical model of generating gear grinding process is established where the axis linkage relationship in CNC continuous generating grinding machine is intentionally changed by representing the radial, axial, and tangential feed of worm wheel in forms of polynomials of the axial feed of worm wheel. Then, the polynomial coefficients of the X1, Y1, and Z1 axis of the continuous generating grinding machine are optimized by the proposed contact trace evaluated genetic algorithm (CTEGA). Finally, the differences between the CTEGA and sensitivity matrix based least squares estimation algorithm (hereinafter referred to as SM algorithm) and the validity of CTEGA are clearly demonstrated by three types of target surfaces numerically.
Section snippets
Mathematical model of the worm wheel
The characteristic of point contact between the grinding wheel and cylindrical helical gear under ideal conditions is used to establish the mathematical model of the worm wheel in this paper. According to the theory of conjugate surface, the relative motion relationship between generated surface of work gear and generating surface of tool can be expressed as functions of two independent sets of parameters [23]. Regular grinding wheel profile can be obtained by regarding the standard involute
Continuous generating grinding model
Compared with the instant line contact between the standard work gear and the tool in internal gear honing, form gear grinding, and other processes, worm wheel has point contact with standard gear in a mathematical ideal condition, as shown in Fig. 3. The continuous generating grinding movement between worm wheel and work gear makes the contact point move on the tooth flank so that the contact trace is formed. In the meantime, the relative movement of worm wheel along work gear axis is also
Flank deviations correction based on CTEGA
From the perspective of numerical calculation process, calculation of the position information of linear and rotational axes by only using contact points on the single flank of work gear is difficult to achieve as well as time-consuming. The reason is that the contact traces are spirally moved along all the tooth surfaces. A new selection method of grinding points different from the conventional method [18, 26] is required to improve the calculation efficiency of the continuous generating
Numerical examples and discussion
Three numerical examples that use YW7232 CNC gear grinding machines (CHMTI, Chongqing, China) are presented to show the validity of the proposed topological modification method. The CNC controller of the machine is SINUMERIK 840Dsl (SIEMENS, Germany). Additional motions of the machine tool axes can be realized by building virtual axis linkages in the electronic gear box [27]. The work gear data, worm wheel data, and machine setting parameters are given in Table 1. The setting angle of the wheel
Conclusion
A method for topological modification of tooth flank in continuous generating grinding is proposed by considering the characteristics of contact trace between worm wheel and work gear. The continuous generating grinding model used for flank modification is established by representing the X1, Y1, and Z1 axis of continuous generating grinding machine in forms of polynomials of the axial feed of the worm wheel. The above three numerical examples allow us to draw the following conclusions:
- (1)
The
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This paper was supported by the National Natural Science Foundation of China (Nos. 51875161, 52075142, and 51805135), the National key research and development plan "strategic science and technology innovation cooperation" (No. 2020YFE0201000), the Key Science and Technology Special Project of Anhui Province (No. 202003A05020042), and the Fundamental Research Funds for the Central Universities of China (No. PA2021KCPY0034).
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