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

本项目的目标是开发一种新的几何公差的双层数学模型，即零件在使用时可接受的尺寸和形状的变化范围。 在局部层面上，新模型将把平面或轴（线）的每个公差带表示为多维区域或地图，它将包括将这些区域和其中的子区域之间的相互依赖关系联系起来的计算技术。 待开发的数学模型将包括尺寸、形状、方向、位置和跳动以及这些变化的组合的公式，这些公式可以使用点、线和平面进行建模。 研究的主要方法是用传统的数学方法表示公差带中的平面、直线和点。 在全局级别上，建议的模型将零件或组件上的所有参考框架相互关联。 它将包括一个符号代数，该代数将特征（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相关联。 为了验证集成的双层数学模型，并证明其可行性和实用性，一套原型软件模块将被设计和实现。 如果成功，这项研究的结果将影响几何公差的最新技术水平和使用结果的设计师的生产力。 在局部层面上，该模型将以三维的方式处理尺寸和几何公差，并与从业人员普遍接受的公差标准保持一致，而现有的公差模型在与标准保持一致方面仅部分令人满意。 在全球一级，预计新的模型将提供一个独特的桥梁之间的数学特性（平面，线，点）和现有的参数计算机辅助设计模型和统计公差软件包。 预计该项目的成功完成将为设计师提供一个工具，以便更好地决定如何分配几何公差。 完成的模型还应允许设计人员确定权衡和优化公差分配。 其影响应该是缩短设计时间，减少设计原型阶段的迭代次数，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.