Selection of parameters in cost-tolerance functions: review and approach

Abstract

The paper deals with the cost-tolerance functions used in minimum-cost tolerance allocation on mechanical assemblies. The consistency of the cost-tolerance functions associated with different dimensions of a tolerance chain is a necessary condition for a correct optimization of tolerance values. This requires a careful selection of function parameters, in order to take into account all the variables influencing the cost, and thus to compare the costs of the different dimensions in a common scale. The cost-tolerance functions proposed in the literature are reviewed, revealing that the parameters are often selected without explicit reference to empirical data or objective rules. The development of procedures for parameter selection has been addressed in several studies, which are also reviewed along with the needed cost data published in textbooks and reports. Finally, an original approach is proposed to build cost-tolerance functions for combined processes, for which fewer results or methods are available. The parameters of a reciprocal power function, chosen for the favorable balance between accuracy and ease of use, are expressed through an empirical relationship with some design variables; these include the nominal dimensions, and the type and size of the part features involved in the allocation problem. An application example shows that the proposed procedure results in optimal tolerance values consistent with common design criteria.

Concurrent optimization of dimensions and tolerances on structures and mechanisms

“…An aspect that has been relatively neglected, and which will be discussed below, is that treating dimensions as additional optimization variables emphasizes the need of cost models including the influence of dimensions as well as tolerances. Such a model has been proposed in a recent study [110] as an extended formulation of one of the available cost-tolerance functions, and will be used in this study.…”

“…The reciprocal power function has been widely used in literature assuming values for b i and k from application-dependent cost data. In an expanded form of the same function proposed in [110], the exponent is constant and the factor is empirically related to the nominal dimension:…”

“…Therefore, the variable cost of a feature depends on its nominal dimension through a power-law relationship with an exponent equal to one-third of that of its tolerance. This result was obtained in [110] through the regression of data from several sources on the costs of machining processes as a function of the tolerance grades on the machined features. The same analysis led to the estimate of a value k = 0.55 for the exponent of the cost-tolerance function, which should be verified on real cost data in specific production contexts.…”

“…where: Compared to the basic reciprocal power function, the expanded form makes explicit the influence of the nominal dimension in the cost of a feature. As discussed in [110], this should improve the consistency of the results of tolerance allocation, as the costs of all features involved in a tolerance chain are more easily estimated on a common scale. The function is even more useful in the present work, where the nominal dimensions are treated as additional variables of the optimization problem.…”

“…Ai is the area of the feature in cm2 ; & f Fi depends on the shape of the feature; possible values on rotational parts are 0.75 and 1.25 respectively for external and internal features; on prismatic parts, 1.5 for cylindrical features, 1-1.5 for non-cylindrical features, and 0.5-1 for planar features (higher values for smaller features); & f Mi is inversely related to the machinability of the material, with base 1 for mild steel, in a range from about 0.3 for aluminum alloys to about 2 for stainless steels; & b 0 is a constant that does not influence the allocation results (b 0 = 1.0 • 10 −3 from[110]). …”