A Survey of Research in the Application of Tolerance Analysisto the Design of Mechanical AssembliesADCATS Report No. 91-1Kenneth W. ChaseAlan R. ParkinsonMechanical Engineering DepartmentBrigham Young UniversityProvo, Utah 84602Published in Research in Engineering Design (1991) 3:23-37April 5, 1991AbstractTolerance analysis is receiving renewed emphasis as industry recognizes that tolerancemanagement is a key element in their programs for improving quality, reducing overall costs andretaining market share. The specification of tolerances is being elevated from a menial task to alegitimate engineering design function. New engineering models and sophisticated analysis toolsare being developed to assist design engineers in specifying tolerances on the basis of performancerequirements and manufacturing considerations.This paper presents an overview of tolerance analysis applications to design with emphasis onrecent research that is advancing the state of the art. Major topics covered are:1) New models for tolerance accumulation in mechanical assemblies, including the MotorolaSix Sigma model.2) Algorithms for allocating the specified assembly tolerance among the components of anassembly.3) The development of 2-D and 3-D tolerance analysis models.4) Methods which account for non-Normal statistical distributions and nonlinear effects.5) Several strategies for improving designs through the application of modern analytical tools.11 IntroductionInterest in tolerance analysis is rapidly increasing in industry. The quest for quality hasfocused attention on the effects of variation on cost and performance of manufactured products.Excess cost or poor performance will eventually show up as a loss of market share. Therefore,the specification of tolerance limits on each dimension and feature of engineering drawings isconsidered by many to be a vital design function. Tolerance requirements have a far-reachinginfluence that touches nearly every aspect of the manufacturing enterprise as shown in Fig. 1.Both engineering design and manufacturing personnel are concerned about the effects oftolerances. Engineers like tight tolerances to assure fit and function of their designs.Manufacturers prefer loose tolerances which make parts easier and less expensive to produce.Therefore tolerance specifications become a critical link between engineering and manufacturing,a common meeting ground where competing requirements may be resolved. A Critical Link Between Design and ManufacturingEngineering DesignTolerances ManufacturingResultant Dimensions Fit and Function Design Limits Performance Sensitivity Robust to VariationProduction Cost Process Selection Machine Tools Operator Skills Tooling, Fixtures Inspection Precision Assemblability Scrap and ReworkCompeting RequirementsTight LooseFig. 1. The effects of assigned tolerances are far-reaching.In the last few years, numerous companies have established comprehensive programs inquality management. Noteworthy among them are the efforts of Motorola, IBM and Xerox,whohave initiated formal, corporate-wide programs for improved tolerance specification, monitoringand control. Their success in reducing waste while cutting cost and development time andreclaiming lost market share has received national praise [Placek 1989a, Placek 1990, Kendrick1991].Another indication of the growing interest in tolerancing is the Mechanical TolerancingWorkshop sponsored by NSF and ASME in 1988 which brought together an international groupof experts in tolerancing to discuss the state of the art and identify research opportunities [Paleck,21989b, ASME 1990]. This has been followed by special theme sessions at several ASMEconferences, such as the Design Technical Conference in Montreal (1989), the Design Show inChicago (1990), and the Computers in Engineering Conference in Boston (1990).2 Models for Tolerance AccumulationThe basis for rational tolerance specification is to create an analytical model to predict theaccumulation of tolerances in a mechanical assembly. Critical clearances or fits or other resultantfeatures of an assembly are generally controlled by the stackup or sum of several componenttolerances. A number of analytical models exist with varying levels of sophistication as shown inFig. 2.TOLERANCE ANALYSISWorst Case Statistical SampledRoot Sum Squares Mean Shift Six Sigma Hasofer-Lind Method of Moments IntegrationMonte CarloFig. 2. Mathematical models of tolerance accumulation.Common models for predicting how component tolerances sum are Worst Case (WC) andRoot Sum Square (RSS) as shown in Eqs. 1 and 2 [Fortini 1967].One-dimensional assemblies: Two- or three-dimensional assemblies:Worst CasedU = Σ Ti ≤ TASMdU = Σ(| ixf∂∂| Ti) ≤ TASM(1)Root Sum SquaredU = [Σ Ti2 ]1/2 ≤ TASMdU = [Σ ( ixf∂∂)2 Ti2 ]1/2 ≤ TASM(2)where Xi are the nominal component dimensions, Ti are the component tolerances, dU is thepredicted assembly variation, TASM is the specified limit for dU, and f(Xi) is the assemblyfunction describing the resulting dimension of the assembly, such as the clearance or interference.3The partial derivatives ∂ f/xi represent the sensitivity of the assembly tolerance to variations inindividual component dimensions. For a one-dimensional tolerance stack the sensitivity is ±1.0.The Worst Case model assumes all the component dimensions occur at their worst limitsimultaneously. It is used by designers to assure that all assemblies will meet the specifiedassembly limit. However, as the number of parts in the assembly sum increases, the componenttolerances must be greatly reduced in order to meet the assembly limit, requiring higherproduction costs. In the RSS model, the low probability of the worst case combination occurringis taken into account statistically, assuming a Normal or Gaussian distribution for componentvariations. Tolerances are commonly assumed to correspond to six standard deviations (6σ or±3σ). Component tolerances may be increased significantly since they add as the root sumsquared (RSS).Statistical distributions may be used to predict the yield of an assembly, that is, the number orfraction of assemblies which are likely to lie inside the spec limits. RSS analysis generally predictstoo few rejects when compared to real assembly processes. This is due to the fact that theNormal distribution is only an approximation of the true distribution, which may be flatter orskewed.