Mathematical Model to Formalize Tolerance Specifications and Enable Full 3D Tolerance Analysis

公差规定了允许的缺陷范围，因此制造的零件将能够被装配和使用。标准的存在是为了定义公差的类别，并确保工程师之间的适当沟通。本研究的目的是继续建立形位公差的数学模型。我们前期工作的结果表明，所提出的数学模型与标准是兼容的，并且它提供了装配的三维关系。在局部层次上，新模型的方法是将平面或直线的每个公差带表示为假想的点空间或地图，并将它们组合起来用于零件的装配。在全局级别，模型使用自由度将部件的所有参考系相互关联。该项目包括实施一个公差分析系统，以便将新模型与现有软件进行比较。目前分析公差的现有方法很大程度上是基于工程实践中的特殊惯例，而较少基于数学原理。因此，今天还没有对装配中的公差进行完整的三维分析，现代设计软件只与现有标准部分兼容。该项目的成果将为完成公差的全三维统计分析提供工具，从而提高质量和降低成本。这个项目将产生新的教材，这将使这门学科更容易被纳入本科生和研究生的工程课程。基于我们的模型，我们正在向美国标准委员会建议一套新的专业发展课程和认证考试，这些课程和认证考试针对的是工程师在数字时代进行公差分析。

Tolerances specify the ranges of permissible imperfections so a manufactured part will be acceptable for assembly and use. Standards exist to define classes of tolerances and to ensure proper communication among engineers. The purpose of this research is to continue creating a mathematical model of geometric tolerances. The results from our earlier work show that the proposed math model is compatible with the standards, and that it provides three-dimensional relations for assemblies. At the local level, the method of the new model is to represent each tolerance-zone for a plane or a line as a hypothetical point-space, or map, and to combine these for assemblies of parts. At the global level, the model inter-relates all reference frames of an assembly using degrees of freedom. The project includes the implementation of a tolerance analysis system for comparison of the new model with existing software. Existing methods for analysing tolerances today are based strongly on ad-hoc conventions from engineering practice and less on mathematical principles. Consequently, full three-dimensional analysis of tolerances in assemblies is not done today, and contemporary design software is only partially compatible with existing standards. The outcome of this project will provide the tools to complete full three-dimensional statistical analyses of tolerances, and, thereby, it will improve quality and lower cost. This project will result in new teaching materials, which should make the subject easier to include in undergraduate and graduate engineering curricula. Based on our model, we are proposing to the American Standards Committee a new set of professional development classes and certification exams that are directed at engineers for tolerance analysis in the digital age.