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Gaer M. A., Kuzmina E. Yu. Tendentsii razvitiya tekhnologii formoobrazovaniya mnogosloinykh tsilindricheskikh konstruktsii i metody otsenki ostatochnykh tekhnologicheskikh napryazhenii [Configuration varieties of square forms of surfaces details and assemblies]. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie [Modern Technologies. System Analysis. Modeling], 2019. Vol. 62, No. 2, pp. 59–66. DOI: 10.26731/1813-9108.2019.2(62).59–66

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In modern conditions of computerized production, a whole complex of problems arises associated with the spatial permissible deviations of the shapes of the surfaces of parts and assemblies. One of these problems is the generalized mathematical representation of assemblies with allowances. Despite the significant successes achieved in the process of the improvement of CAD systems in recent years, the methods of modeling and representing the structure of assemblies remain insufficiently developed. Most popular models of assemblies take into account only the nominal dimensions and, as a result, the task and support of tolerance values, as well as the implementation of a full-sized dimensional analysis are essentially difficult. Last but not least, this is due to the fact that initially, the modeling was mainly used analytical geometry. Over time, this device was not enough. And now, with automated design of assemblies, taking into account tolerances, the approaches associated with the use of differential geometry and topology are increasingly used. This article is devoted to the description of geometric modeling of spatial deviations of parts and assemblies. A model is described for representing the configuration spaces of tolerances for the distortion of the metrics of the shapes of the surfaces of parts and assemblies using differential geometry and topology. Such an approach allows one to sufficiently successfully describe small deviations from the nominal surface shape by the magnitude of the curvature and the length of the corresponding lines. In this case, the tolerances can be characterized by a change in the coefficients of the first quadratic form with the invariability of the second. For these coefficients, configuration spaces are constructed, which make it possible in the context of an automated dimensional analysis system to represent the specified types of tolerances as a parametric model.

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