Direct method to determine singular point of enveloped surface and its application to worm wheel tooth surface

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-01-31 DOI:10.1098/rspa.2023.0369
Jian Cui, Yaping Zhao, Qingxiang Meng, Gongfa Li
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Abstract

A novel methodology for determining the singular point of an enveloped surface is put forward. Unlike some existing methods, the presented method starts directly from the equation of the enveloped surface instead of that of the generating surface, and it is thus called a direct method. The calculation for the normal vector of the enveloped surface is well simplified with the help of the moving frame approach, which makes the presented method feasible. The singularity condition equation is extracted by using the theory of linear algebra. For singular points with different properties, proper solving techniques are established, including resultant elimination and simple elimination. Applying the developed method, the undercutting characteristics of the Archimedes worm wheel are investigated from the perspective of spatial meshing. The numerical results demonstrate that the worm wheel generally has one undercutting limit line, whose trend is along the tooth width of the wheel. Locating on one side of the tooth surface and near the tooth root is a dangerous part of the worm wheel undercutting. The proposed method is beneficial for the development of gear meshing science.
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确定包络面奇异点的直接方法及其在蜗轮齿面中的应用
本文提出了一种确定包络面奇异点的新方法。与现有的一些方法不同,本文提出的方法直接从包络曲面的方程而不是生成曲面的方程出发,因此被称为直接方法。借助移动框架方法,包络面法向量的计算得到了很好的简化,这使得所提出的方法是可行的。奇点条件方程是利用线性代数理论提取的。针对不同性质的奇异点,建立了适当的求解技术,包括结果消元和简单消元。应用所开发的方法,从空间网格的角度研究了阿基米德蜗轮的下切特性。数值结果表明,蜗轮一般有一条下切极限线,其趋势沿轮齿宽度方向。位于齿面一侧且靠近齿根的位置是蜗轮下切的危险部位。所提出的方法有利于齿轮啮合科学的发展。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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