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Article(id=1263881621460103860, tenantId=1146029695717560320, journalId=1263187878914834467, issueId=1263881604263437054, articleNumber=null, orderNo=null, doi=10.16578/j.issn.1004.2539.2026.01.021, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1727280000000, receivedDateStr=2024-09-26, revisedDate=1730217600000, revisedDateStr=2024-10-30, acceptedDate=null, acceptedDateStr=null, onlineDate=1779262460341, onlineDateStr=2026-05-20, pubDate=1768406400000, pubDateStr=2026-01-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1779262460341, onlineIssueDateStr=2026-05-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1779262460341, creator=13041195026, updateTime=1779262460341, updator=13041195026, issue=Issue{id=1263881604263437054, tenantId=1146029695717560320, journalId=1263187878914834467, year='2026', volume='50', issue='1', pageStart='1', pageEnd='191', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1779262456241, creator=13041195026, updateTime=1779263107607, updator=13041195026, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1263884336399528882, tenantId=1146029695717560320, journalId=1263187878914834467, issueId=1263881604263437054, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1263884336403723187, tenantId=1146029695717560320, journalId=1263187878914834467, issueId=1263881604263437054, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=174, endPage=183, ext={EN=ArticleExt(id=1263881628175184587, articleId=1263881621460103860, tenantId=1146029695717560320, journalId=1263187878914834467, language=EN, title=Visual modeling and simulation analysis method of spiral bevel gear transmission system, columnId=1263881623129416517, journalTitle=Journal of Mechanical Transmission, columnName=Development·Application, runingTitle=null, highlight=null, articleAbstract=
Objective

In response to the visualization requirements of the models and analysis results necessary for the development of simulation analysis software for spiral bevel gear transmission systems, based on the structural characteristics of the bevel gear transmission system, the topological structure of the transmission system and the complete expression of the interrelationships among components were achieved by applying graph theory and object-oriented data structures.

Methods

The generation method of regularized point sets on the geometric surfaces of key heterogeneous components in the transmission system was investigated, and the precise construction and rapid assembly of the geometric models of bevel gears, transmission shafts, and bearings were accomplished using the open-source 3D computer graphics tool VTK. On this basis, the mapping between the geometric model of the transmission system and the mechanical model of loading contact analysis was established, and the visualization methods for the analysis results such as system deformation under loading and tooth surface meshing state were studied. Finally, the visualization effects of modeling and simulation analysis results of the bevel gear transmission system were verified through examples.

Results

Research has shown that the comprehensive application of graph theory and object-oriented data structures can achieve a complete expression of the topological configuration and geometric correlation properties of bevel gear transmission systems. By applying the parameter expression modeling, sweep modeling, and triangulation modeling methods of the VTK library, accurate modeling of heterogeneous components can be achieved. Based on this, a mapping between the system geometry model and the mechanical analysis model can be established to complete the loading contact analysis of the bevel gear transmission system, and visualize the analysis results. The above research results provide technical support for the development of modeling and simulation analysis software for bevel gear transmission systems.

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目的

针对弧齿锥齿轮传动系统仿真分析软件开发所需模型与分析结果的可视化需求，基于锥齿轮传动系统的结构特点，应用图论及面向对象的数据结构，实现了传动系统拓扑结构及零部件关联关系的完整表达。

方法

首先，研究了传动系统关键异构元件几何表面规则化点集的生成方法，并应用开源的三维计算机图形学工具VTK实现了锥齿轮、传动轴和轴承几何模型的精确构建及快速装配；以此为基础，建立了传动系统几何模型与加载接触分析力学模型之间的映射，研究了加载状态下系统变形、齿面啮合状态等分析结果的可视化方法；最后，通过实例对锥齿轮传动系统建模及仿真分析结果的可视化效果进行了验证。

结果

研究表明，综合应用图论及面向对象的数据结构，可实现弧齿锥齿轮传动系统拓扑构型及其几何关联属性的完整表达；应用VTK库的参数表达式造型、扫掠造型以及三角剖分造型方法，可实现异构零部件的精确建模；在此基础上，建立系统几何模型与力学分析模型的映射，完成锥齿轮传动系统加载接触分析，并实现了分析结果的可视化显示。上述研究结果为锥齿轮传动系统建模仿真分析软件的开发提供了技术支持。

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张卫青（通信作者），男，1983年生，江西吉安人，博士，教授；主要研究方向为复杂零件的数字化及智能化制造技术；。
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李名扬，男，1999年生，重庆璧山人，硕士研究生；主要研究方向为齿轮仿真分析软件开发；。

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李名扬，男，1999年生，重庆璧山人，硕士研究生；主要研究方向为齿轮仿真分析软件开发；。

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李名扬，男，1999年生，重庆璧山人，硕士研究生；主要研究方向为齿轮仿真分析软件开发；。

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pageStart=62, pageEnd=67, url=null, language=null, rfNumber=1, rfOrder=0, authorNames=孙悦桐, 田国富, 傅雁博, journalName=机械设计, refType=null, unstructuredReference=孙悦桐，田国富，傅雁博. 基于有限元法的弧齿锥齿轮接触性能研究［J］. 机械设计，2023，40（增刊2）：62-67., articleTitle=基于有限元法的弧齿锥齿轮接触性能研究, refAbstract=null), Reference(id=1263881775080682125, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, doi=null, pmid=null, pmcid=null, year=2023, volume=40, issue=Suppl.2, pageStart=62, pageEnd=67, url=null, language=null, rfNumber=1, rfOrder=1, authorNames=SUN Yuetong, TIAN Guofu, FU Yanbo, journalName=Journal of Machine Design, refType=null, unstructuredReference=SUN Yuetong，TIAN Guofu，FU Yanbo. 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Journal of Central South University，2022，29（1）：157-172., articleTitle=Mathematic model and tooth contact analysis of a new spiral non-circular bevel gear, refAbstract=null)], funds=null, companyList=[AuthorCompany(id=1263881698278781073, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, xref=null, ext=[AuthorCompanyExt(id=1263881698375250066, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, companyId=1263881698278781073, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=School of Mechanical Engineering, Chongqing University of Technology, Chongqing400054, China), AuthorCompanyExt(id=1263881698392027283, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, companyId=1263881698278781073, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, 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segment structure, figureFileSmall=IQFhhqoxQ2gOOI4v6ixVYQ==, figureFileBig=sf7H/EE074NBwdpY/VyUYw==, tableContent=null), ArticleFig(id=1263881742000206219, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图4, caption=轴段结构体定义, figureFileSmall=IQFhhqoxQ2gOOI4v6ixVYQ==, figureFileBig=sf7H/EE074NBwdpY/VyUYw==, tableContent=null), ArticleFig(id=1263881742302196108, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.5, caption=Modeling process of the drive shaft, figureFileSmall=83Mp7qsF6qI7hgj19rmEMw==, figureFileBig=8z61Bk/sFIuOUs0LSJ8ffQ==, tableContent=null), ArticleFig(id=1263881742750986641, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图5, caption=传动轴建模过程, figureFileSmall=83Mp7qsF6qI7hgj19rmEMw==, figureFileBig=8z61Bk/sFIuOUs0LSJ8ffQ==, tableContent=null), ArticleFig(id=1263881743157834134, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.6, caption=Definition of the bearing structure, figureFileSmall=oB+jLon2He8TEIhkxQ1uAg==, figureFileBig=YKgVzZcLWPo7an185/nhnA==, tableContent=null), ArticleFig(id=1263881744017666462, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图6, caption=轴承结构体定义, figureFileSmall=oB+jLon2He8TEIhkxQ1uAg==, figureFileBig=YKgVzZcLWPo7an185/nhnA==, tableContent=null), ArticleFig(id=1263881744806195626, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.7, caption=Modeling process of the angular contact ball bearing, figureFileSmall=W3Q85zd5JTQUUQZZqudGjQ==, figureFileBig=rmMrFWA3Dj630LVAL67YNw==, tableContent=null), ArticleFig(id=1263881748304245165, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图7, caption=角接触球轴承建模过程, figureFileSmall=W3Q85zd5JTQUUQZZqudGjQ==, figureFileBig=rmMrFWA3Dj630LVAL67YNw==, tableContent=null), ArticleFig(id=1263881749105357234, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.8, caption=Definition of the spiral bevel gear structure, figureFileSmall=Sepv9l1oZfnEAqSVqRpzeg==, figureFileBig=xInyN4VbySPpYwDheeAbcQ==, tableContent=null), ArticleFig(id=1263881751110234557, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图8, caption=弧齿锥齿轮结构体定义, figureFileSmall=Sepv9l1oZfnEAqSVqRpzeg==, figureFileBig=xInyN4VbySPpYwDheeAbcQ==, tableContent=null), ArticleFig(id=1263881751563219393, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.9, caption=Theoretical tooth surface points of driving and driven gears, figureFileSmall=BUij7HDe9ZkPe8FOvLoTug==, figureFileBig=gE5e8b/H3OSuRCkBEV4uhw==, tableContent=null), ArticleFig(id=1263881755690414540, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图9, caption=主、从动轮的理论齿面点, figureFileSmall=BUij7HDe9ZkPe8FOvLoTug==, figureFileBig=gE5e8b/H3OSuRCkBEV4uhw==, tableContent=null), ArticleFig(id=1263881756126622162, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.10, caption=Modeling process of the spiral bevel gear single tooth, figureFileSmall=BySav3CvklXUCxL5pOdBfw==, figureFileBig=CaJqJVDy6MkHUfeQx3jGqw==, tableContent=null), ArticleFig(id=1263881756554441178, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图10, caption=弧齿锥齿轮单齿建模过程, figureFileSmall=BySav3CvklXUCxL5pOdBfw==, figureFileBig=CaJqJVDy6MkHUfeQx3jGqw==, tableContent=null), ArticleFig(id=1263881757015814623, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.11, caption=Driving and driven gear models of spiral bevel gears, figureFileSmall=bsF9btlpmFDYE8rtZ6byZg==, figureFileBig=NI37optbuR7v1noOK7ip6w==, tableContent=null), ArticleFig(id=1263881757384913380, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图11, caption=弧齿锥齿轮的主、从动轮模型, figureFileSmall=bsF9btlpmFDYE8rtZ6byZg==, figureFileBig=NI37optbuR7v1noOK7ip6w==, tableContent=null), ArticleFig(id=1263881758244745711, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.12, caption=Principle of coordinate transformation for the shafting assembly, figureFileSmall=tXjlfTAcMG6xMOWFUtqeDQ==, figureFileBig=jfRY/Mfi9OuDHxEQNcogCg==, tableContent=null), ArticleFig(id=1263881760044102134, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图12, caption=轴系装配的坐标变换原理, figureFileSmall=tXjlfTAcMG6xMOWFUtqeDQ==, figureFileBig=jfRY/Mfi9OuDHxEQNcogCg==, tableContent=null), ArticleFig(id=1263881760354480636, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.13, caption=Assembly of the input shaft and the output shaft, figureFileSmall=7fE29tOLsjcT2TTSUO2qhQ==, figureFileBig=X2wB77lzZB6mU20i36KL9Q==, tableContent=null), ArticleFig(id=1263881760731968001, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图13, caption=输入轴、输出轴的装配体, figureFileSmall=7fE29tOLsjcT2TTSUO2qhQ==, figureFileBig=X2wB77lzZB6mU20i36KL9Q==, tableContent=null), ArticleFig(id=1263881760996209159, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.14, caption=Assembly of the bevel gear transmission system, figureFileSmall=kvYj153dBbqbLeBY3RJYIA==, 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journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.16, caption=Input shafting mechanical analysis model based on beam element, figureFileSmall=RNAeOxo2RLEM6GU/NVxuiw==, figureFileBig=l2dSw7LVao3TT6GusoMXaw==, tableContent=null), ArticleFig(id=1263881762170614307, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图16, caption=基于梁单元的输入轴系力学分析模型, figureFileSmall=RNAeOxo2RLEM6GU/NVxuiw==, figureFileBig=l2dSw7LVao3TT6GusoMXaw==, tableContent=null), ArticleFig(id=1263881762493575720, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.17, caption=Principle of the contact imprinting, figureFileSmall=ouZn8cPeFL1f3K+NutJb3A==, figureFileBig=aZ/XCpyX2KFIJYCqd2JwEQ==, tableContent=null), ArticleFig(id=1263881762585850413, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图17, caption=接触印痕生成原理, figureFileSmall=ouZn8cPeFL1f3K+NutJb3A==, figureFileBig=aZ/XCpyX2KFIJYCqd2JwEQ==, tableContent=null), ArticleFig(id=1263881764259377713, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.18, caption=Process of the contact stress nephogram visualization, figureFileSmall=/ARnl+yfN7D4L2DmD/P/EA==, figureFileBig=TZqB8xcVaw8B8GHZSZbNag==, tableContent=null), ArticleFig(id=1263881764607504952, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图18, caption=接触应力云图可视化流程, figureFileSmall=/ARnl+yfN7D4L2DmD/P/EA==, figureFileBig=TZqB8xcVaw8B8GHZSZbNag==, tableContent=null), ArticleFig(id=1263881764989186622, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.19, caption=Contact stress nephogram of the tooth surface, figureFileSmall=MA8ExSwdsl5fiR9EOaRnhQ==, figureFileBig=G7VLn+UEL73SeofC2dxUUQ==, tableContent=null), ArticleFig(id=1263881765563806273, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图19, caption=齿面接触应力云图, figureFileSmall=MA8ExSwdsl5fiR9EOaRnhQ==, figureFileBig=G7VLn+UEL73SeofC2dxUUQ==, tableContent=null), ArticleFig(id=1263881765907739206, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.20, caption=Assembly of the spiral bevel gear transmission system, figureFileSmall=j4VbGaVNJijEeOiDpgxd9w==, figureFileBig=KUkhXdYK/ZO8uI67dBmtiA==, tableContent=null), ArticleFig(id=1263881766612382285, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图20, caption=弧齿锥齿轮传动系统装配体, figureFileSmall=j4VbGaVNJijEeOiDpgxd9w==, figureFileBig=KUkhXdYK/ZO8uI67dBmtiA==, tableContent=null), ArticleFig(id=1263881767006646864, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.21, caption=Comparison of the deformation results of the two analysis methods, figureFileSmall=zq1vtDuME/23ppsjseUszQ==, figureFileBig=oXeU8/x4kuZUB5uHDlz5ZQ==, tableContent=null), ArticleFig(id=1263881767195390548, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图21, caption=两种分析方法的变形结果对比, figureFileSmall=zq1vtDuME/23ppsjseUszQ==, figureFileBig=oXeU8/x4kuZUB5uHDlz5ZQ==, tableContent=null), ArticleFig(id=1263881769107993177, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.22, caption=Comparison of the contact area of the initial design tooth surface, figureFileSmall=F5MyHV2xe0R3cHy+OSY/Jw==, figureFileBig=cwT0BiUkERiBjnqRMOMO+w==, tableContent=null), ArticleFig(id=1263881769510646366, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图22, caption=初始设计齿面的接触区对比, figureFileSmall=F5MyHV2xe0R3cHy+OSY/Jw==, figureFileBig=cwT0BiUkERiBjnqRMOMO+w==, tableContent=null), ArticleFig(id=1263881769670029921, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.23, caption=Comparison of the contact area after the tooth surface modification, figureFileSmall=HSKHtDoXvewBaiheg9/43w==, figureFileBig=rONqrl5LNQjde7yyDz4EZw==, tableContent=null), ArticleFig(id=1263881770013962852, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图23, caption=齿面修形后的接触区对比, figureFileSmall=HSKHtDoXvewBaiheg9/43w==, figureFileBig=rONqrl5LNQjde7yyDz4EZw==, tableContent=null), ArticleFig(id=1263881770458559080, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Fig.24, caption=Nephogram of the adjusted loading contact stress, figureFileSmall=jBTc+CVQfLCz2nZxbF/0qw==, figureFileBig=zf7aDB8LNXRFkTaQ/PwIWA==, tableContent=null), ArticleFig(id=1263881770957681259, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=图24, caption=调整后的加载接触应力云图, figureFileSmall=jBTc+CVQfLCz2nZxbF/0qw==, figureFileBig=zf7aDB8LNXRFkTaQ/PwIWA==, tableContent=null), ArticleFig(id=1263881771125453422, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Tab. 1, caption=

Parameters of the spiral bevel gear pair

, figureFileSmall=null, figureFileBig=null, tableContent=
基本参数小齿轮（主动轮）大齿轮（从动轮）
齿数2841
模数/mm3.75
轴交角/（°）90
偏置距/mm0
压力角/（°）20
齿面宽/mm2828
外锥距/mm93.0993.09
螺旋角/（°）3535
节锥角/（°）34.3355.67
), ArticleFig(id=1263881771347751536, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=表1, caption=

弧齿锥齿轮副参数

, figureFileSmall=null, figureFileBig=null, tableContent=
基本参数小齿轮（主动轮）大齿轮（从动轮）
齿数2841
模数/mm3.75
轴交角/（°）90
偏置距/mm0
压力角/（°）20
齿面宽/mm2828
外锥距/mm93.0993.09
螺旋角/（°）3535
节锥角/（°）34.3355.67
), ArticleFig(id=1263881771511329396, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Tab. 2, caption=

Parameters of the drive shaft

, figureFileSmall=null, figureFileBig=null, tableContent=
类型类别轴段一轴段二轴段三
输入轴长度/mm359075
外径/mm605550
内径/mm252525
输出轴长度/mm3010070
外径/mm908580
内径/mm404040
), ArticleFig(id=1263881773289714296, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=表2, caption=

传动轴参数

, figureFileSmall=null, figureFileBig=null, tableContent=
类型类别轴段一轴段二轴段三
输入轴长度/mm359075
外径/mm605550
内径/mm252525
输出轴长度/mm3010070
外径/mm908580
内径/mm404040
), ArticleFig(id=1263881773423932027, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Tab. 3, caption=

Bearing designation

, figureFileSmall=null, figureFileBig=null, tableContent=
轴承编号轴承型号轴承编号轴承型号
①7011C③7017C
②7010C④7016C
), ArticleFig(id=1263881773507818110, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=表3, caption=

轴承型号

, figureFileSmall=null, figureFileBig=null, tableContent=
轴承编号轴承型号轴承编号轴承型号
①7011C③7017C
②7010C④7016C
), ArticleFig(id=1263881773751087744, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Tab. 4, caption=

Input axis deformation calculated by FEM

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节点编号A法-X/µmA法-Y/µmA法-Z/µm
1-47.6336.3110.73
2-42.9531.3310.73
3-30.3919.6610.4
4-25.0515.1110.23
5-10.562.799.72
61.02-5.999.21
73.55-7.809.04
811.87-14.239.04
920.20-20.969.04
), ArticleFig(id=1263881773990163074, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=表4, caption=

有限元方法计算的输入轴变形量

, figureFileSmall=null, figureFileBig=null, tableContent=
节点编号A法-X/µmA法-Y/µmA法-Z/µm
1-47.6336.3110.73
2-42.9531.3310.73
3-30.3919.6610.4
4-25.0515.1110.23
5-10.562.799.72
61.02-5.999.21
73.55-7.809.04
811.87-14.239.04
920.20-20.969.04
), ArticleFig(id=1263881774308930179, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Tab. 5, caption=

Input axis deformation calculated by the proposed method

, figureFileSmall=null, figureFileBig=null, tableContent=
节点编号B法-X/µmB法-Y/µmB法-Z/µm
1-48.05（0.9%）36.51（0.6%）10.73（0.0%）
2-43.25（0.7%）31.49（0.5%）10.73（0.0%）
3-30.57（0.6%）19.63（0.2%）10.40（0.0%）
4-25.00（0.2%）15.00（0.7%）10.22（0.1%）
5-10.54（0.2%）2.78（0.4%）9.72（0.0%）
61.01（1.0%）-6.00（0.2%）9.21（0.0%）
73.52（1.0%）-7.73（0.9%）9.04（0.0%）
811.92（0.4%）-14.35（0.8%）9.04（0.0%）
920.33（0.6%）-20.81（0.7%）9.04（0.0%）
), ArticleFig(id=1263881774392816261, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=表5, caption=

本方法计算的输入轴变形量

, figureFileSmall=null, figureFileBig=null, tableContent=
节点编号B法-X/µmB法-Y/µmB法-Z/µm
1-48.05（0.9%）36.51（0.6%）10.73（0.0%）
2-43.25（0.7%）31.49（0.5%）10.73（0.0%）
3-30.57（0.6%）19.63（0.2%）10.40（0.0%）
4-25.00（0.2%）15.00（0.7%）10.22（0.1%）
5-10.54（0.2%）2.78（0.4%）9.72（0.0%）
61.01（1.0%）-6.00（0.2%）9.21（0.0%）
73.52（1.0%）-7.73（0.9%）9.04（0.0%）
811.92（0.4%）-14.35（0.8%）9.04（0.0%）
920.33（0.6%）-20.81（0.7%）9.04（0.0%）
), ArticleFig(id=1263881774506062471, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=EN, label=Tab. 6, caption=

Results of the meshing misalignment

, figureFileSmall=null, figureFileBig=null, tableContent=
参数ΔP/μmΔW/μmΔE/μmΔΣ/mrad
数值65.02-54.84-122.80.262
), ArticleFig(id=1263881774644474505, tenantId=1146029695717560320, journalId=1263187878914834467, articleId=1263881621460103860, language=CN, label=表6, caption=

啮合错位量结果

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参数ΔP/μmΔW/μmΔE/μmΔΣ/mrad
数值65.02-54.84-122.80.262
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弧齿锥齿轮传动系统可视化建模与仿真分析方法
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李名扬 , 张卫青 , 谭儒龙 , 傅涛
机械传动 | 开发应用 2026,50(1): 174-183
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机械传动 | 开发应用 2026, 50(1): 174-183
弧齿锥齿轮传动系统可视化建模与仿真分析方法
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李名扬 , 张卫青 , 谭儒龙, 傅涛
作者信息
  • 重庆理工大学 机械工程学院，重庆400054

    • 李名扬，男，1999年生，重庆璧山人，硕士研究生；主要研究方向为齿轮仿真分析软件开发；。

通讯作者:

张卫青（通信作者），男，1983年生，江西吉安人，博士，教授；主要研究方向为复杂零件的数字化及智能化制造技术；。
Visual modeling and simulation analysis method of spiral bevel gear transmission system
Mingyang LI , Weiqing ZHANG , Rulong TAN, Tao FU
Affiliations
  • School of Mechanical Engineering, Chongqing University of Technology, Chongqing400054, China
出版时间: 2026-01-15 doi: 10.16578/j.issn.1004.2539.2026.01.021
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摘要
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目的

针对弧齿锥齿轮传动系统仿真分析软件开发所需模型与分析结果的可视化需求，基于锥齿轮传动系统的结构特点，应用图论及面向对象的数据结构，实现了传动系统拓扑结构及零部件关联关系的完整表达。

方法

首先，研究了传动系统关键异构元件几何表面规则化点集的生成方法，并应用开源的三维计算机图形学工具VTK实现了锥齿轮、传动轴和轴承几何模型的精确构建及快速装配；以此为基础，建立了传动系统几何模型与加载接触分析力学模型之间的映射，研究了加载状态下系统变形、齿面啮合状态等分析结果的可视化方法；最后，通过实例对锥齿轮传动系统建模及仿真分析结果的可视化效果进行了验证。

结果

研究表明，综合应用图论及面向对象的数据结构，可实现弧齿锥齿轮传动系统拓扑构型及其几何关联属性的完整表达；应用VTK库的参数表达式造型、扫掠造型以及三角剖分造型方法，可实现异构零部件的精确建模；在此基础上，建立系统几何模型与力学分析模型的映射，完成锥齿轮传动系统加载接触分析，并实现了分析结果的可视化显示。上述研究结果为锥齿轮传动系统建模仿真分析软件的开发提供了技术支持。

锥齿轮  /  传动系统  /  可视化  /  啮合特性分析
Objective

In response to the visualization requirements of the models and analysis results necessary for the development of simulation analysis software for spiral bevel gear transmission systems, based on the structural characteristics of the bevel gear transmission system, the topological structure of the transmission system and the complete expression of the interrelationships among components were achieved by applying graph theory and object-oriented data structures.

Methods

The generation method of regularized point sets on the geometric surfaces of key heterogeneous components in the transmission system was investigated, and the precise construction and rapid assembly of the geometric models of bevel gears, transmission shafts, and bearings were accomplished using the open-source 3D computer graphics tool VTK. On this basis, the mapping between the geometric model of the transmission system and the mechanical model of loading contact analysis was established, and the visualization methods for the analysis results such as system deformation under loading and tooth surface meshing state were studied. Finally, the visualization effects of modeling and simulation analysis results of the bevel gear transmission system were verified through examples.

Results

Research has shown that the comprehensive application of graph theory and object-oriented data structures can achieve a complete expression of the topological configuration and geometric correlation properties of bevel gear transmission systems. By applying the parameter expression modeling, sweep modeling, and triangulation modeling methods of the VTK library, accurate modeling of heterogeneous components can be achieved. Based on this, a mapping between the system geometry model and the mechanical analysis model can be established to complete the loading contact analysis of the bevel gear transmission system, and visualize the analysis results. The above research results provide technical support for the development of modeling and simulation analysis software for bevel gear transmission systems.

Bevel gear  /  Transmission system  /  Visualization  /  Analysis of meshing characteristics （编辑：刘欢欢）
李名扬, 张卫青, 谭儒龙, 傅涛. 弧齿锥齿轮传动系统可视化建模与仿真分析方法. 机械传动, 2026 , 50 (1) : 174 -183 . DOI: 10.16578/j.issn.1004.2539.2026.01.021
Mingyang LI, Weiqing ZHANG, Rulong TAN, Tao FU. Visual modeling and simulation analysis method of spiral bevel gear transmission system[J]. Journal of Mechanical Transmission, 2026 , 50 (1) : 174 -183 . DOI: 10.16578/j.issn.1004.2539.2026.01.021
正文
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0　引言
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弧齿锥齿轮传动系统被广泛应用于航空航天、船舶工业和汽车工业中，对其进行仿真分析是设计优化的重要环节。在锥齿轮传动系统仿真分析方面，国内外学者开展了许多研究。孙悦桐等[1-2]基于弧齿锥齿轮齿面数学模型，采用Pro/E软件构建齿轮副三维模型，并利用Abaqus软件进行了齿面接触分析。喻子豪等[3-4]依据齿轮啮合原理，利用Matlab软件计算理论齿面点，结合Pro/E、AutoCAD等建模软件，实现了弧齿锥齿轮的精确建模。高洁等[5]考虑直升机主减速器机匣、轴系及轴承的支撑变形，利用Ansys软件对减速器进行了静力学分析。MOSLEM等[6]基于FEM软件研究了传动轴轴向和径向不对中两种情况下弧齿锥齿轮的加载接触分析。NAMBOOTHIRI等[7]利用Abaqus软件对直齿轮进行断裂行为分析，确定其最大弯曲应力区域。上述研究多依赖通用CAD、CAE软件进行齿轮传动系统的建模分析，整个建模过程烦琐、分析流程复杂，不利于推广应用。也有学者进行了齿轮专用软件的研究开发。刘桂源等[8]基于C++和Qt开发了同向平双齿轮设计分析软件，对传动系统进行了迭代设计优化。刘怀举等[9]开发高性能齿轮数据库软件，实现了工况分析、寿命预测、结构优化等功能。聂少武等[10]利用VC编程软件和Matlab软件联合开发摆线锥齿轮分析计算软件，可对其进行切齿计算与啮合仿真。上述开发软件中的分析模型多采用二维图显示，分析结果无法直观展示。国外的Romax、Masta、KISSsoft等齿轮专用分析软件可实现模型及分析结果的可视化，便于用户交互，但其具体的实现方法和技术尚未公开[11-12]。
针对上述问题，本文研究了弧齿锥齿轮传动系统的数据结构表达、零部件建模、传动系统力学分析模型构建及分析结果可视化的方法，并应用VTK库开发了其实现算法。研究可为锥齿轮传动系统三维建模仿真分析专用软件的开发提供支撑，从而提高其设计分析效率。
1　传动系统的拓扑结构表达
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1.1　弧齿锥齿轮传动系统的拓扑结构
弧齿锥齿轮传动系统一般由输入轴系和输出轴系组成。轴系主要包括4部分：传动轴、齿轮、轴承及外部载荷，轴系之间通过齿轮副进行连接。弧齿锥齿轮传动系统模型如图1所示。
本节根据传动系统各部件拓扑连接组成特点，设计具体的数据结构用于各部件参数的管理。传动系统结构体System变量由各轴系结构体ShaftSystem变量组成；同时，各轴系结构体变量中包含各部件结构体Shaft、Gear、Bearing、Load变量，齿轮之间构建Pair齿轮副结构体变量。各零部件的材料属性、位置信息等参数均包含在各自结构体中，最终整合得到其面向对象的数据结构形式，如图2所示。
1.2　部件间的连接关系
本节通过明确部件之间的连接关系，确定各结构体变量的所属关系，从而方便获取对应部件参数。图论的邻接矩阵能够有效地描述轴系及整个传动系统的连接情况，且便于编程实现[13]。本文采用图论的邻接矩阵进行分析。
以图1中的输入轴系为例，形成对应的邻接矩阵A1，即
               abcdeA1=abcde0111110000100001100000000
式中，a、b、c、d、e分别表示输入轴、轴承1、轴承2、主动轮、输入转矩。当两个部件有连接关系时，对应位置的元素为1，反之则为0。同理，可形成输出轴系的邻接矩阵A2。
轴系间的连接关系用齿轮副邻接矩阵进行描述。在图1中，输入轴系上的主动轮与输出轴系上的从动轮构成一对齿轮副，形成邻接矩阵B12，即
                  fghijB12=abcde0000000000000000000001000
式中，f、g、h、i、j分别表示输出轴、轴承3、轴承4、从动轮、输出转矩。矩阵B12中，（d，i）位置元素为1，表明主、从动轮所在轴系之间构成一级齿轮传动系统。
对于多级齿轮传动系统，将所有轴系邻接矩阵An、齿轮副邻接矩阵Bin进行组合，可得到整个传动系统邻接矩阵的一般形式，如式（3）所示。其组合方式为：轴系邻接矩阵依次呈对角线排列，Bin在系统邻接矩阵中的位置根据其具体表示某两个轴系之间的连接关系来确定。
A1B120…00B12TA2B23…000B23TA3…00⋮⋮⋮⋮⋮000…An-1Bin000…BinTAn
其中，i表示序号为i的轴；n表示序号为n的轴；Bin表示序号i的轴和序号n的轴通过齿轮副连接构成的齿轮副邻接矩阵。其中，i，n=1，2，3，…；i≠n；同时序号i和n对应齿轮副邻接矩阵在系统邻接矩阵中的元素位置。
通过遍历传动系统邻接矩阵中某行元素，可确定与该行对应部件有连接关系的部件。
2　传动系统部件的可视化建模
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基于传动系统的数据结构获取相应部件参数，可建立各零部件的三维模型及进行传动系统的装配。本节针对不同部件模型，研究了轴、轴承等异构元件的扫掠造型方法、参数表达式造型方法以及弧齿锥齿轮复杂曲面模型的三角剖分造型方法。根据这些方法，利用VTK进行零部件的精确建模。
2.1　VTK可视化流程
利用VTK库，可实现弧齿锥齿轮传动系统各部件模型的三维可视化。VTK全称Visualization Toolkit（可视化工具包），以其开源、跨平台、面向对象等优点，被广泛应用于计算机图形、图像处理及可视化软件开发中。VTK中包含两个重要概念：可视化管线和渲染引擎。可视化管线负责生成或读取数据、处理数据、数据映射以及将数据传递给渲染引擎；渲染引擎负责将映射后的图元数据进行可视化。VTK的可视化流程如图3所示。
2.2　轴和轴承的模型构建
传动轴和轴承部件均为回转体结构，建模时可采用旋转拉伸的方式。可通过输入各轴段的长度Li、外径Di、内径di等几何参数，利用Shaft结构体变量进行存储，其结构体定义如图4所示。将所有轴段数据存放在vector容器中，可得到整个传动轴的几何参数。
首先，以传动轴左端轴截面几何中心为原点建立坐标系，根据传动轴参数确定各个轴段矩形截面P1、P2、P3、P4等4个点的坐标值，构建矩形截面单元；然后，利用VTK中的vtkRotationalExtrusionFilter类对截面单元进行旋转拉伸，生成相应的轴段模型。按此方法构建所有轴段模型并进行拼接，最终可得到完整的传动轴模型，过程如图5所示。构成传动轴模型的点矢量集合用Vs（i）表示。
轴承模型通常由内圈、外圈、滚动体、保持架等4部分组成，建模过程与传动轴相似。以角接触球轴承建模为例，将输入的轴承相关参数（如内径d、外径D、宽度B、接触角α等）通过Bearing结构体变量进行储存，其结构体定义如图6所示。
首先，以轴承内圈几何中心为原点设立坐标系，构建轴承一般参数（如外径、内径、宽度、接触角等）与其余参数之间的参数表达式；接着，根据轴承一般几何参数推导出内外圈截面各轮廓点的坐标值，构建截面多边形单元，将截面单元旋转拉伸，形成规则化点集进行模型显示；最后，根据滚动体直径和数目，利用vtkSphereSource类建立球形滚动体模型并阵列得到所有滚动体。轴承各部分可在同一坐标系下构建完成，其内外圈截面轮廓尺寸共同约束了滚动体的相对位置。建立完成的轴承模型如图7所示。构成轴承模型的点矢量集合用Vb（i）表示。
2.3　弧齿锥齿轮的模型构建
构建弧齿锥齿轮结构体Gear，用以存储齿轮几何参数、齿坯参数、刀具参数等，结构体定义如图8所示。根据这些参数并结合齿轮啮合原理，对齿面在拓扑截面上进行规则化网格划分并导入到啮合方程中，可得到规则化齿面点集[14]。通过上述方式可计算出主从动轮凹凸齿面点，齿长方向取M个均匀点，齿高方向取N个均匀点，整个齿面具有M×N个齿面点。本例中，M=67，N=101。所有理论齿面点呈均匀矩阵式分布，如图9所示。
研究齿面点的分布特点和齿面的三维曲面特征，利用三角剖分法构建整个齿面模型。首先，将齿面点数据存储到vtkPoints中，并为每个点附加索引编号；接着，从0号开始遍历齿面点，将当前点与其右方、下方和右下方的3个点组成P5P6P7和P6P7P8两个vtkTriangle三角形单元，通过循环，完成整个齿面的三角形单元构建；然后，根据上述方法构建单齿模型凹凸齿面、大小端面、齿顶齿底面等6个面所有的三角形单元，并存储到vtkCellArray中；最后，利用vtkPolyData整合模型的齿面点数据和三角形单元拓扑数据，实现单齿模型的构建。建模过程如图10所示。
进一步，将单齿模型旋转阵列，获得所有轮齿；在此基础上，构建齿轮基底模型，则建立的完整的弧齿锥齿轮模型如图11所示。构成齿轮模型的点矢量集合用Vg（i）表示。
2.4　传动系统的部件装配
基于构建完成的部件模型，进行传动系统装配。先完成轴系的装配，进而完成整个传动系统的装配。轴系装配时，将传动轴坐标系作为基准坐标系，按照各部件安装位置及朝向，通过齐次坐标变换到基准坐标系中，完成轴系装配。其原理如图12所示。
图12中，O1X1Y1Z1和O2X2Y2Z2坐标系为传动轴坐标系，其余为部件的局部坐标系。以图12中输出轴系为例，锥齿轮安装在基准坐标系Z1轴S2位置；两轴承各自绕Y1轴旋180°确定朝向，然后分别安装在Z1轴S1与S3位置。
以输出轴系中的齿轮装配为例，Vg(s)（i）计算式为
Vg(s)(i)=0MY1(β)0Sj0001•Vg(g)(i)
式中，Vg(s)(i)为变换后在传动轴坐标系下的齿轮模型点矢量集合；Vg(g)(i)为齿轮坐标系下齿轮模型点矢量集合；MY1(β)为模型绕Y1轴的旋转矩阵，β为传动轴轴向与齿轮安装朝向之间的夹角；Sj为部件在传动轴上的位置。其余部件的装配原理和计算方法类似。最后得到的输入轴系和输出轴系的装配体模型如图13所示。
传动系统装配时，将箱体坐标系作为基准坐标系。先将输入轴系置于箱体坐标系中，基于弧齿锥齿轮副的安装条件，推导出输出轴系在箱体中的相对位置关系；在此基础上，根据坐标变换计算式完成输出轴系的装配，计算式为
Vss(box)(i)=txM(θ)tytz0001•Vss(s)(i)
式中，Vss(box)(i)、Vss(s)(i)分别为在齿轮箱坐标系下和轴系坐标系下的轴系模型矢量集合；M（θ）为轴系的正交旋转矩阵，θ为轴系模型轴向与理论安装朝向之间的夹角；tx、ty、tz分别为模型平移分量。
最终完成的传动系统装配体模型如图14所示。
3　传动系统分析模型构建及结果可视化
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3.1　弧齿锥齿轮传动系统的静力学分析
将轴系作为传动系统分析的基本单元，构建其力学分析模型，其流程如图15所示。根据系统数据结构中存储的轴系几何、装配、载荷等相关数据，构建轴系的梁单元力学分析模型，从而可建立轴系几何模型与力学分析模型的映射关系，实现各轴系力学分析模型的搭建。
以图1中输入轴系为例，构建其梁单元力学分析模型。根据轴系的几何参数和装配参数，将单元节点分别设置在轴段变截面位置Li、部件安装位置Si、载荷施加位置及边界条件位置。对于非实心轴，需根据内孔轴段长度li添加相应节点。建立的分析模型如图16所示。
接着，根据轴段参数求解分析模型中相邻节点之间梁单元刚度矩阵Kij，将共节点的梁单元进行刚度矩阵耦合；同时，计算出轴承刚度矩阵Kb并在轴承节点位置进行约束刚度矩阵耦合[15]，实现边界条件的添加。最后得到的轴系的总刚度矩阵为
K=K12+K23+K34+K45+K56+K67=A1112A12120⋯0A2112A2212+A2223A2323⋯00A3223A3323+A3334+Kb1⋯0⋮⋮⋮⋮000⋯A7767
矩阵K中元素的下标表示在总刚度矩阵中的位置；上标表示归属于哪两个节点。根据输入轴施加的转矩T，结合齿轮参数计算出齿轮副的啮合力F，并将载荷施加到相应节点位置。输出轴系采用同样的方式计算出总刚度矩阵和外部载荷。根据力与变形的关系，构建的整个传动系统的受力平衡方程组为
[δ1,δ2,⋯,δn]T=K-1·[F1,F2,⋯,Fn]T
式中，Fn为节点n处6个自由度上的外载荷向量；δn为节点n处6个自由度上的变形量向量。通过式（7）可以计算出轴系分析模型上各个节点的变形量。
3.2　考虑系统变形的齿面接触分析及结果可视化
根据齿轮安装位置的节点变形量，计算出输入轴和输出轴相对于轴交错点的位移量，可求出齿轮副的啮合错位量[16-17]。基于共轭差曲面理论，结合啮合错位量ΔP、ΔW、ΔE、ΔΣ，可计算出弧齿锥齿轮副受载变形后的接触迹线（ΔP为齿轮副沿着小齿轮轴线方向的相对位移；ΔW为齿轮副沿着大齿轮轴线方向的相对位移；ΔE为齿轮副沿着偏置距方向的相对位移；ΔΣ为齿轮副沿着轴交角方向的相对角位移），讨论接触迹线上每一共轭接触点处的接触特性。首先，基于切片法与刚度串、并联模型推导出单齿刚度计算式；然后，考虑多齿啮合状态下的载荷平衡原理，在单齿刚度计算式中引入力矩平衡方程和变形协调方程，求解出载荷分配；最后，将齿面分配载荷代入赫兹接触理论模型中，求解出加载接触椭圆。接触椭圆内应力计算式[18-19]为
P=3F'2πabP(x,y)=P1-x2a2-y2b2
式中，P为最大赫兹接触应力；F'为分配的法向接触载荷；a、b分别为接触椭圆长、短轴；P（x，y）为接触椭圆范围内任意点处的接触应力。
沿接触迹线的所有接触椭圆共同构成了齿面的接触印痕区域，其原理如图17所示。图17中，大齿轮为从动轮，小齿轮为主动轮。
根据上述方式可计算出各齿面点的最大接触应力值，利用VTK中的vtkLookupTable类构建一个颜色查找表，将接触应力值从小到大依次与不同的颜色相对应；然后，利用vtkPolyData，根据各齿面点应力值的大小赋予相应的颜色属性，可绘制出接触应力云图。图18、图19分别为其可视化流程和应力云图。
4　应用实例
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为了验证传动系统建模及分析结果的可视化效果，以某弧齿锥齿轮传动系统为例进行仿真分析。齿轮副参数、传动轴参数及轴承型号分别如表1~表3所示，其输入转矩为400 N∙m。
采用上述参数，应用第2节所述方法进行各零部件模型构建及传动系统的装配，结果如图20所示（表3中轴承编号与图20对应）。图20中模型准确反映了各部件几何特征且装配规范无干涉。
进一步，根据第3.1节所述方法，基于构建的传动系统几何模型形成系统的力学分析模型，进行传动轴受力变形分析，计算出传动轴梁单元模型各节点的变形量。以输入轴为例，提取X、Y、Z三个方向上的节点位移变形结果，并与有限元方法（Finite Element Method, FEM）的分析结果进行对比。
两种方法的计算结果分别如表4、表5所示。表中，A法-X表示采用有限元方法计算的X方向变形量，B法-X表示采用第3.1节方法计算的X方向变形量，其余依此类推。两种方法计算结果的对比如图21所示。
由表5可知，本方法计算结果与有限元方法计算的轴变形最大误差不超过1%，验证了本方法计算结果的正确性。根据传动轴变形量计算出齿轮副的啮合错位量，结果如表6所示。
图22（a）所示为齿轮副初始设计的齿面接触状态。可见在轻载作用情况下，齿轮副的理论接触区位于齿面中部，呈菱形。图22（b）所示为考虑系统在载荷作用下发生变形时齿轮副的理论接触区形状。可见啮合错位状态下，齿轮副的理论接触区朝大端齿顶移动，出现了明显的边缘接触。
基于啮合错位状态下齿轮副的理论接触区偏移情况，对齿轮副初始的齿面形状进行调整，使其理论接触区朝小端齿根偏移；同时，为了降低齿轮副啮合状态对安装误差的敏感性，适当加大齿面在齿高方向的修形量，使其更加细长。优化后齿面不考虑啮合错位影响的接触区形态如图23（a）所示。考虑载荷作用下系统变形的影响，其啮合错位情况下的理论接触区如图23（b）所示。可见接触区由小端略偏齿根回到齿面中部位置，且在齿根和齿顶均未出现边缘接触状态。
进一步，对优化后的齿面进行加载接触分析，同时，考虑系统和轮齿变形下的齿面啮合接触状态，图24为加载后的齿面接触应力云图。可见整个啮合过程中接触区域呈椭圆分布；最大应力位于椭圆中部，向四周扩展时应力逐渐减小；其轻载和重载时在齿顶、齿根均未发生接触。这说明，通过分析结果指导后的优化齿面获得了更加理想的啮合效果。
5　结论
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针对弧齿锥齿轮仿真分析软件的开发需求，研究了锥齿轮传动系统关键元件的建模及仿真分析结果的可视化方法，并基于VTK平台探索了其具体的实现技术手段。得出以下结论：
1）综合应用图论及面向对象的数据结构，可实现弧齿锥齿轮传动系统拓扑构型及其几何关联属性的完整表达，为系统几何及分析模型的构建提供了数据交互基础。
2）根据传动系统各元件的结构特点，分别应用VTK库的参数表达式造型法、扫掠造型法以及三角剖分造型法，可实现异构零部件的精确建模。
3）通过系统建模流程，对系统邻接矩阵的控制数据结构实施逐级赋值，能够构建系统几何模型与力学分析模型之间的映射关系，进而完成锥齿轮传动系统的加载接触分析，并实现分析结果的可视化呈现。
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2026年第50卷第1期
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doi: 10.16578/j.issn.1004.2539.2026.01.021
  • 接收时间：2024-09-26
  • 首发时间：2026-05-20
  • 出版时间：2026-01-15
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  • 收稿日期：2024-09-26
  • 修回日期：2024-10-30
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    重庆理工大学 机械工程学院，重庆400054

通讯作者:

张卫青（通信作者），男，1983年生，江西吉安人，博士，教授；主要研究方向为复杂零件的数字化及智能化制造技术；。
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2种不同金属材料的力学参数
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Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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