Mathematical Modeling of Geometric Variations to Integrate Parametric Computer Aided Design with Tolerance Analyses and Optimization

该项目的目标是为几何公差开发一种新的双层数学模型，即零件在使用时可接受的尺寸和形状的变化范围。在地方一级，新模型将把平面或轴（线）的每个容忍区表示为多维区域或地图，它将包括将这些区域及其内的分区域之间的相互依赖关系联系起来的计算技术。要开发的数学模型将包括大小、形式、方向、位置和跳动变化的公式，以及这些变化的组合，可以使用点、线和面来建模。研究的主要方法是使用传统数学来表示容差区内的面、线和点。在全局层面上，所提出的模型将零件或装配上的所有参考框架相互关联。它将包括一个符号代数，将一个特征（如孔）的自由度和约束与指定尺寸和公差时使用的参考框架联系起来。为了验证集成的双层数学模型，并证明其可行性和实用性，将设计和实现一套原型软件模块。如果成功，这项研究的结果将影响最先进的几何公差和使用结果的设计师的生产力。在地方一级，该模型将以与从业人员普遍接受的公差标准一致的方式以三维方式处理尺寸和几何公差，而现有的公差模型在与标准一致方面只是部分令人满意。在全球一级，预计新模型将在特征（面、线和点）的数学特征与现有的参数化计算机辅助设计模型和统计公差软件包之间提供一个独特的桥梁。预计该项目的成功完成将为设计人员提供更好的几何公差分配决策工具。完成的模型还应该允许设计人员确定权衡并优化公差的分配。其影响应该是缩短设计时间，减少设计原型阶段的迭代，从而降低成本。

The objective of this project is to develop a new bi-level math model for geometric tolerances, that is, the range of variation in size and shape that can be permitted for a part to be acceptable when in use. At the local level, the new model would represent each tolerance zone for a plane or an axis (line) as a multidimensional region or map, and it would include the computational techniques that relate interdependencies between these regions and subregions within them. The mathematical model to be developed will include formulations for variations in size, form, orientation, position, and runout, and combinations of these, which can be modeled using points, lines, and planes. The primary method for the research is to use traditional mathematics for representing planes, lines, and points in a tolerance zone. At the global level, the proposed model interrelates all frames of reference on a part or an assembly. It will include a symbolic algebra that relates the degrees of freedom and constraints for a feature, such as a hole, to reference frames used when specifying dimensions and tolerances. In order to validate the integrated bi-level mathematical model and to demonstrate its feasibility and usefulness, a set of prototype software modules will be designed and implemented. If successful, the results of this research will impact the state-of-the-art for geometric tolerances and the productivity of designers who use the results. At the local level, the model will treat dimensional and geometric tolerances in three dimensions and in a manner consistent with the standard on tolerances that is universally accepted by practitioners, whereas existing models for tolerances are only partially satisfactory in being consistent with the standard. At the global level it is projected that the new model will provide a unique bridge between mathematical characterizations of features (planes, lines, and points) and existing parametric computer-aided design models and statistical tolerance software packages. It is projected that successful completion of the project will provide the designer with a tool for making better decisions about allocating geometric tolerances. The completed model also should permit designers to identify trade-offs and to optimize the allocation of tolerances. The impact should be a shorter design time, fewer iterations in the prototype phase of design, and hence lower cost.