摘要
准双曲面齿轮用于传递空间相错轴的运动和动力,多用于汽车后桥的减速传动,同时也在工程机械、船舶和飞行器等领域具有广泛的应用。由于其几何关系与螺旋锥齿轮有相似之处,目前生产中常用的螺旋锥齿轮加工方法和切齿机床可以在计算和调整后用于加工准双曲面齿轮,并逐渐形成了以Gleason公司所采用的面铣削法和面滚切法为代表的齿轮加工体系。但是,上述两种方法加工的螺旋锥齿轮并不是理想的球面渐开线齿形,因此加工的齿面存在理论误差,瞬时速比不恒定,同时也造成齿面修形与机床调整复杂、设计制造周期较长、非同批加工的齿轮之间无法互换等问题,因此需要对机床参数和刀具进行不断地修正和改进,以减小这种误差的影响。
产形线切齿法是基于渐开线和圆锥螺旋渐开面的生成原理提出的一种新型齿轮加工方法,该方法加工的螺旋锥齿轮齿形为理想的球面渐开线,具有瞬时速比恒定、非同批加工齿轮可以互换等优点,且机床结构相对简单,加工效率较高,易于实现齿面的接触区控制。
为扩大产形线切齿法的适用范围,使该方法的诸多优点在更大范内得到发挥,本文对产形线切齿法加工准双曲面齿轮的理论与方法进行了深入研究。通过分析准双曲面齿轮的基本几何,在目前常用的描述准双曲面齿轮的几何关系基础上,构造了产形线切齿法所必需的基圆锥与基平面,提出了小轮与大轮产形线的几何关系与运动关系,推导了与基锥相关的几何参数的求解公式,从而建立了产形线切齿法加工准双曲面齿轮的基本原理。
根据该原理加工生成齿面的过程,本文建立了描述齿面的数学模型,利用空间啮合理论论证了该方法加工的一对准双曲面齿轮齿面能够进行正确的点接触啮合传动,并以此模型为基础,分别建立了已知大轮产形线形状为直线和圆弧线的情况下,加工大、小齿轮各齿面产形线的数学模型。
为了研究准双曲面齿轮各齿面的切齿加工方案,本文根据产形线切齿法加工螺旋锥齿轮的齿面切削加工原理,分别阐述了产形线为直线、圆弧线以及平面一般曲线的情况下齿面切削区的确定和避免发生过切的手段,并将之推广到准双曲面齿轮,提出了准双曲面齿轮小轮和大轮各齿面的切削区确定方法以及各齿面的切齿加工方案。在此基础上,对加工各齿面所需要的机床运动和结构进行了研究和汇总,提出基于产形线切齿法的用于加工螺旋锥齿轮和准双曲面齿轮的通用型6轴3联动数控机床的结构和运动方案。
在上述理论研究的基础上,参考一对已知设计参数的Gleason准双曲面齿轮,本文实例计算了各基锥参数和各齿面的切削区,并分别在设定大轮产形线为直线或圆弧线的情况下,计算了小轮和大轮各齿面的产形线方程。采用Matlab在直角坐标系内绘制了各条理论产形线以观察其形状,由于小轮的理论产形线近似于直线或圆弧线,故本文提出了产形线的代用方法并建立了代用误差的评价方法,以便采用产形线为直线或圆弧线的切齿原理和机床加工小轮齿面以提高切齿效率。通过实例计算,验证了本文提出的产形线代用方法能够将代用误差控制在齿面弹性变形的范围内,从而表明其可行性。
对于以圆弧线代用的小轮产形线,直接计算而来的代用圆弧半径通常不是整数,不利于机床刀具系列化以简化机床配置,故本文提出了对代用产形线半径进行圆整优化的方法。对于小轮产形线采用圆弧线代用,特别是小轮左、右齿面均采用圆弧线代用的情况下,分析齿面预设几何参数的变化对各代用产形线半径的影响,采用曲面拟合及插值求解的方法实现产形线半径的圆整优化。从实例计算结果可以看出,该方法能够将优化后的代用产形线半径残差控制在0.1%以内,且仍可根据需要作进一步的优化,从而表明该方法具有较高的可行性。
通过本文的研究,使产形线切齿法的适用范围进一步扩展到通常不探讨齿形概念的准双曲面齿轮,为加工机床的研制和进一步计划开展的切齿实验积累了重要的研究资料,同时也为产形线切齿法加工其它类型的齿轮提供了一种新的思路。纵观全文,本论文的创新性研究工作主要有以下几个方面:
1.在目前常用的准双曲面齿轮几何关系的基础上构建了基锥和基平面,并确定了小轮和大轮产形线间的平面共轭关系,从而使采用产形线切齿法加工准双曲面齿轮成为可能,并提供了基本的条件和要素。
2.提出了采用产形线切齿法加工螺旋锥齿轮的原理加工准双曲面齿轮各齿面。由于产形线切齿法在螺旋锥齿轮的加工理论和实践上具有更多的理论基础和实践经验,将准双曲面齿轮齿面转化为螺旋锥齿轮的加工方法有利于问题的简化和避免重复研究。
3.提出了产形线代用方法和代用产形线优化方法。产形线代用的提出可以避免使小轮齿面按照产形线为一般曲线的切齿方案进行加工,有利于发挥产形线切齿法的优势;代用产形线圆整优化方法的提出使得在进一步机床研制过程中系列化圆弧刃刀具半径成为了可能,有利于简化刀具设计和机床配置。
Hypoid gears are widely used to transmit crossed-axis power and motion in vehicles (such as the rear drive-axles of passenger cars and trucks), engineering machineries, ships, aircrafts and other areas. Because of the similarity of the geometrical relationship between hypoid gears and spiral bevel gears, the current methods and machines which are commonly used to manufacture spiral bevel gears can be used to manufacture hypoid gears after calculating and adjusting, and the gear cutting system, which consist of face milling and face hobbing and are used by Gleason Works, is gradually formed. However, the tooth profile curves of spiral bevel gears which are manufactured by face milling or face hobbing are not ideal spherical involutes, so there are theoretical errors on the tooth surfaces. The effects of the error are: (1) transmission ratio is not constant; (2) modification and adjustment of tooth surfaces are complex; (3) designing and manufacturing need to take a longer period; (4) gears can not be interchanged unless in the same batch. Therefore, in order to reduce to effect of the error, the parameters of machine tools and cutting tool need to be modified and improved continuously.
Based on the generating principle of spherical involute and the theory of conjugated tooth surfaces, we have proposed a new method of cutting involute gears called generating-line method. This new method can be used to process ideal spherical involute spiral bevel gears that the transmission ratio is constant and gears can be interchanged even in different batches. Moreover this method makes the calculation and adjustment of machine tools settings relatively simple, the processing efficient relatively high, and the controlling of tooth contact areas easily.
In order to expand the applying scope of the generating-line method, this paper researched the theory of manufacturing hypoid gears by this new method. Based on analyzing the basic geometry of hypoid gears, the base cones and base planes which are necessary for generating-line method were established, and the geometrical and kinematical relationships between pinion and gear generating lines were proposed, then the formulas of the geometrical parameters of the base cone were studied. Accordingly the basic principle of manufacturing hypoid gears by generating-line method was built.
On basis of the process of generating tooth surfaces by this principle, the mathematical model of tooth surfaces was established. Based on the space engagement principle, this paper proved that the pair of tooth surfaces could conjugate exactly in point contact. And then the mathematical model of each side of pinion and gear generating lines which the shape of gear is known as straight line or circular arc was built on basis of the tooth surfaces model.
In order to study the processing program of cutting each side of hypoid gears, on the basis of the principle of cutting spiral bevel gear tooth surfaces, the author researched the method of determining the cutting areas and avoiding over-cutting in the condition of the generating lines are straight line, circular arc and plane general curve respectively, then promoted this method to hypoid gears, and proposed the method of determining the cutting areas of each side of pinion and gear and the processing program of cutting. And then the author researched and gathered the necessary motion and structure of machine tools, and built a commonly used six-three axis CNC machine tools to manufacture spiral bevel and hypoid gears by generating-line method.
Based on the theoretical studies mentioned above, reference to a pair of Gleason hypoid gears which the designing parameters were known, the author calculated the base cone parameters, cutting areas of each tooth surfaces of pinion and gear, and calculated the equations of gear generating lines and the theoretical pinion generating lines in the condition of setting the gear generating lines as a straight line or circular arc respectively. Because of the theoretical pinion generating lines which were plotted in Matlab looks approximate straight lines or circular arc, in order to improve the cutting efficiency by using the cutting theory of straight line or circular arc generating lines, the methods of substituting simple curves for the complex theoretical pinion generating lines and estimating the errors of substituting were proposed. As the radius of circular arc substituted generating lines were usually not integer, in order to serialize the cutting tool radiuses and simplify the structure of machine tools, the author proposed an optimizing method of rounding the radiuses of circular arc substituted generating lines. It can be seen from the results of the calculating example: the method of substituting generating lines could control the errors under the area of elastic deformation of tooth surfaces, and the method of optimizing substituted generating lines could control the residual errors less than 1% and could make a further optimization, therefore the methods of substituting and optimizing generating lines were both feasible.
Through theses studies, the author extended the suitable range of generating-line method to hypoid gears which usually do not have the concept of tooth profile, and accumulated lots of important research data for the further machine tools research and cutting experiments, and also provided a new way to study generating-line method for manufacturing other types of gears. Throughout the full text of this paper, the innovative research works mainly shown in the following areas:
1. Built the base cones and base plane for hypoid gears based on the commonly used geometrical relationship, and determined the plane conjugated relationship between the generating lines of pinion and gear, therefore the basic conditions and elements of manufacturing hypoid gears by generating-line method were provided
2. Proposed the method of using the theory of manufacturing spiral bevel gears by generating-line method to process hypoid gears. As the theoretical basis and practical experience of manufacturing spiral bevel gears by generating-line method were much more, transforming hypoid gears cutting process to spiral bevel gears could simplify the problems and avoid redundant researches.
3. Proposed methods of substituting generating lines and optimization of substituted generating lines. The method of substituting generating lines could avoid cutting the pinion as the generating line was a plane general curve, and would help to play the strengths of generating-line method. The method of rounding optimization of substituted generating lines made the cutting tool radius serialization possible in the further studies of the machine tools, and would help to simplify the design of cutting tools and the configuration of machine tools.
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