Under mass production conditions, mechanical parts cannot be manufactured to exact dimensions or geometric perfection. A tolerance specifies the range of imperfections in size and shape that can be permitted for a part to be acceptable for assembly and use. Understanding the causes and effects of dimensional and geometric variations is a major concern in the design and manufacture of mechanical products. Designers are essentially concerned with the following geometric dimension and tolerance (GD&T) issues:

1. What mating conditions/clearances can achieve the intended function(s)? (Functionality/assemblability)
2. Which dimensions contribute to variations of each mating condition and how? (Tolerance Analysis)
3. How to optimally distribute the allowable net variation at mating surfaces between all the contributing dimensions and geometric variations? (Tolerance Allocation)

National and international standards (ASME Y14.5, ISO 1101) have been established to ensure proper communication of geometric dimensions and tolerances (GD&T). But these standards are based on ad-hoc conventions collected from years of engineering practice rather than on mathematical principles. Comprehensive 3D analysis of stack-ups is only possible if a mathematical model of such variations exist. The attempt to “retrofit” an “official” math model (Y14.5.1) to the tolerance standard has not gone far enough. Researchers have proposed replacing the standard completely—a proposition unacceptable to industry because the valuable empirical knowledge contained in the current standard will be lost. The research challenge is to build a math model of geometric variations that is consistent with already existing tolerance standards and capable of supporting comprehensive 3D analysis of stack-up conditions.

Current Projects: (NSF CMMI grant)

• Development of maps for complex/compound tolerance classes (2007-08)
• Development of a universal library of m-maps for manufacturing variations (2007-08)
• Development of a universal library of t-maps
• m-map based statistical analysis of tolerance accumulation (2008-09)
• Manufacturing maps for process planning (2008-09)
• Direct “fitting” of CMM inspection data into corresponding i-Maps (2009-10)

• Bi-level mathematical model of geometric tolerances
• Tolerance-maps (T-maps©) for size, form orientation of planar features
• Topological model of tolerances
• Tolerance-maps (T-maps©) for lines (cylindrical features)
• Accumulation maps: Minkowski sum algorithms
• T-maps applied to Black & Decker Miter Saw
• Automation of 1-D min/max charts for parts & assemblies
• Integrated GD&T Testbed based on global model
• Comparative studies of Tolerance Analysis Methods: 1D charts, 2D/3D Monte Carlo sim and T-maps
• Rule based Advisor for supporting GD&T specification
• Computational model for validating D&T based on entity clusters