Formulation of Tolerance Analysis

The formulation of tolerance analysis can be stated as follows. Given a set of tolerances

{T} = {T₁, T₂, . . . , T_n} on a set of dimensions {D} = {D₁, D₂, . . . . , D_n} and given a set of design constraints {C} = {C₁, C₂, . . . , C_m}, is {T} satisfactory? Constraints could be functional requirements of a manufacturing, assembly costs etc. The dimensions in the set {D} include both the nominal dimensions {DN} and their tolerances {T}, that means,

 {D} = {D_N} + {T}. To assess tolerance appropriateness, we formulate a Resultant dimension in terms of {D}, that is,

RD = f ({D}) = f (D₁, D₂, . . . , D_n)            . . . (1)

The variability of RD because of variability in {D} is determined (by using methods described below). If RD satisfies {C} all the time, {T} is satisfactory and assembly is accepted. If not, {T} is unsatisfactory and assembly is discarded. Design functions are frequently complex and their formulation form the hardest part of tolerance analysis and can be time- consuming.

Tolerance analysis methods might be divided into two types. In the simpler type, dimensions have conventional tolerances, and the consequence of tolerance analysis is the nominal value of the design function (RDN) and its upper (RD_max) and lower (RD_min) limits. This type of analysis is sometimes called as worst-case analysis. It means that all possible combinations of in-tolerance parts should result in an assembly that satisfies the design constraints. The upper & lower limits of the design function represent the worst possible combination of the tolerances of the design function variables. Though, the likelihood of worst-case combination of these tolerances in any particular product is very low. Therefore, worst-case tolerance analysis is very conservative.

The other type of tolerance analysis is performed on a statistical basis. Tolerance analysis methods of this type allow statistical tolerances and output a statistical distribution for the design function. This allows for more realistic analysis. Manufacturing costs are reduced by loosening up the tolerances, and accepting a calculated risk that the design constraints {C} may not be satisfied 100 percent of the time. By assuming a probability distribution for each toleranced dimension, it is possible to determine the likelihood that the specified design limits will be exceeded. Effectively, a reject rate is determined for the assembly. A nonzero reject rate may be preferable to an increase in individual part manufacturing costs due to tight tolerances. Both the worst-case and statistical approaches are important in practice.