The function of fixtures is to locate and hold workpieces for machining, assembly, inspection and other operations. The essential criteria in fixture design are accuracy and repeatability of workpiece location, and lack of interference between fixture elements and tools. Other criteria of fixtures may include ease of loading and unloading workpieces, light weight, low cost and rigidity.; In this study we present a mathematical theory for automatic design of fixtures for machining of general prismatic parts. We assume that workpieces are represented by surface regions in a framework of their major orientations. We also assume that locators and clamps apply uni-sense forces.; The following four functional requirements of fixturing are formulated using both Screw Theory and statics: (1) locating stability, (2) deterministic location, (3) clamping stability and (4) total restraint. Analysis procedures for each requirement are presented.; An algebraic formalism has been developed to model the reasoning process of fixture synthesis. The synthesis problem is approached by solving two nearly decomposable problems: sub-problem XY that comprises x and y translations and z rotation, and sub-problem Z that comprises z translation and x and y rotations. The subproblems are dealt with individually in transformed spaces taking into consideration their interaction. The decomposition not only reduces the search space but also allows general polygon support bases (vs. 3-point support) to be dealt with. A mechanism is also provided for adjusting locators and supports after clamps are determined.; Cutting forces and couples of boring, milling and drilling operations are analyzed and represented as wrenches. There is an infinite number of machining wrenches in 3D space. The aggregate effects of machining wrenches over their course of action are estimated. Necessary clamping forces are then determined using Linear Programming formulations to withstand the cutting wrenches.; By this theory, the positions of locators and clamps and necessary clamping forces can be generated automatically with little search. Also, repetitive generation of similar fixture configurations can be avoided and no non-solutions are generated. The theory provides a foundation for automatic design of fixtures.
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