Tolerance Design Methods for Mechanical Systems General

The aim of tolerance design is to minimize the effects of the deviation on the functional suitability of the product and to achieve an economically optimal distribution of the tolerances, taking into account the functional requirements (Ref. 12). For a cylindrical gearbox, for example, a function-orientated tolerancing of all components should be carried out in such a way that all units reliably pass the noise limit values in the end-of-line test at minimum cost. No additional costs are then incurred for reworking or scrapping. A distinction must be made between arithmetic and statistical tolerancing, whereby the latter represents the tolerance as a probability distribution or its characteristic values (Ref. 13). Tolerance field-based micro geometry optimization has become established for the micro-geometric design of gears. After creating a full-factorial variant space in the FE-based tooth contact analysis, each variant is evaluated according to pre-defined criteria for the application behavior (Ref. 14). Hallmann et al. presented a general method for tolerance optimization with regard to manufacturing costs and functionality. Initially selected tolerances were evaluated for the fulfilment of functionality and costs using a tolerance-cost function. Adjusted tolerances were assigned based on the results of the evaluation (Ref. 15). Schleich et al. presented a procedure for analyzing the geometric functional suitability of an assembly, based on finite surface models of the individual parts, in order to be able to take shape tolerances in particular into account (Ref. 16). Concrete approaches to tolerance optimization were not described. Whitney developed a general systematic approach for the function-oriented design and mathematical modelling of assemblies (Ref. 17). DIN 7186 presents mathematical relationships for statistical tolerancing that allow the superimposition of statistical distributions and their parameters (Ref. 18). This is applicable for dimensional chains that are based on a Gauss normal distribution and in which there is a linear relationship between individual tolerance and overall tolerance. In real manufacturing and assembly processes, however, complex frequency density functions are often superimposed (Refs. 19, 20). Goetz et al. developed a general approach for assemblies with which preliminary tolerances can be defined economically at an early design stage using tolerance graphs (Ref. 21). However, this approach tends not to be suitable for complex systems such as gearboxes, including the quality parameters. Schleich et al. investigated the influence of manufacturing-related deviations of the microgeometry on a pair of cylindrical gears on the transmission error by means of a regression adapted to the results of the tooth contact analysis (TCA). Interactions between different profile and lead line deviations were analyzed with the aim of achieving an appropriate tolerance of the microgeometric deviations, but not the influence of other component deviations (Ref. 22).