Characterizing Slop in Mechanical Assemblies Via Differential Geometry

Abstract

Slop (or backlash) in mechanical assemblies is often present and is usually undesirable from both craftsmanship and performance points of view. It is our belief that this phenomenon is not that well understood and that current methods of assessment are based largely on only qualitative, common-sense approaches. The focus of this paper is on developing an analytical theory for accurately characterizing slop, and on presenting an illustrative example. As one might expect, in principle, with a better understanding of slop, CAD (computer-aided-design) software package designers can create more refined software tools, mechanical engineers can design better products, and manufacturing engineers can be prepared to measure and improve craftsmanship levels. The underlying theory is based on combining concepts from differential geometry, including envelopes, constrained piecewise-smooth sweeps, and sweep vector fields (SVFs), along with basic configuration space (C-space) methods. In essence, the volumetric (or areal) error, which is generated as the movable part in an assembly is swept throughout its complete constrained volume (or area), may be viewed as a quantitative manifestation of craftsmanship errors. A 2-dimensional (2D) idealization of a common assembly that often suffers from poor craftsmanship due to slop, i.e., a doorknob assembly with exaggerated slop, is analyzed. The swept area is calculated using both traditional and SVF methods with the aid of Mathematica™. High quality Mathematica™ visualization of interesting sweeps along the bounding edges of the nonlinear slop constraint region, including generation of all of the envelope curves, is done. Finally, this work attempts to serve as a paradigm for characterizing slop based on engineering criteria.