{"title":"Direct method to determine singular point of enveloped surface and its application to worm wheel tooth surface","authors":"Jian Cui, Yaping Zhao, Qingxiang Meng, Gongfa Li","doi":"10.1098/rspa.2023.0369","DOIUrl":null,"url":null,"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.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"227 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","FirstCategoryId":"103","ListUrlMain":"https://doi.org/10.1098/rspa.2023.0369","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
引用次数: 0
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.
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.