Using planar kinematics to construct the full 4-D tolerance-map for a line-profile

Tolerances on line-profiles are used to control crosssectional shapes of parts, even mildly twisted ones such as those on turbine or compressor blades. Such tolerances limit geometric manufacturing variations to a specified twodimensional tolerance-zone, i.e. an area, the boundaries to which are curves parallel to the true profile. The single profile tolerance may be used to control position, orientation, and form of the profile. For purposes of automating the assignment of tolerances during design, a math model, called the Tolerance-Map (TMap), has been produced for most of the tolerance classes that are used by designers. Each T-Map is a hypothetical pointspace that represents the geometric variations of a feature in its tolerance-zone. Of the six tolerance classes defined in the ASME/ANSI/ISO Standards, only one attempt has been made at modeling line-profiles [1], and the method used is a kinematic description, based largely on intuition, of the allowable displacements of the middle-sized profile within its tolerance-zone. The result presented is a 4-D double pyramid having a 3-D shape for the common base. Allowable small changes in size represent the fourth dimension in the altitudedirection of the pyramids. However, that work is limited to square, rectangular, and right-triangular profile shapes for which the 3-D transverse sections (called hypersections) of the 4-D T-Map are all geometrically similar to the base because the boundaries are doubly traced. For more generally shaped profiles, [2] the hypersections are not geometrically similar to the base. The objective of this paper is to expand the kinematic description of a profile in its tolerance-zone to include the changing constraints that take place as size is incremented or decremented within the allowable tolerance-range. It provides validation of a different method that is described in a companion paper [3].