OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 2 2023 An additional option for solving the problem of meeting requirements (2) for all members of the parameter-oriented series is the formation of conditions under which the dimensions of the used cutting tools or workpieces X will be close to the maximum allowable. The logical consequence of this condition will be the inevitable decrease in the ranges of variation D of parameter X with an increase in the dimensions of the technological equipment [19, 71] 1 2 i m D D D D > > > > > . (5) In turn, the ability of the enterprise to quickly switch to the production of new products based on the existing one (or, at least, with a minimum replacement of equipment), i.e. the mobility of the machine tool holding should also be ensured in the synthesis of parameter-oriented series. This additional condition is achieved by overlapping the size ranges of individual members of the parameter-oriented series. But, as noted earlier, multiple duplication, significantly increasing the cost of manufacturing and operating machine tools, essentially eliminates all efforts aimed at improving its mobility. Numerous statistical studies in industrialized countries have shown that the size distributions of tools and machined surfaces usually have a unimodal character with positive asymmetry. It has been proven that for machines of a certain size, the parameter X usually has a log-normal distribution [19, 20, 71, 72] ( ) ( )2 2 2 1 2 i i y y i i f y e − − s = s π where y is the natural logarithm of the random variable X; i y – average value (mathematical expectation) of the value y; si – standard deviation of y from i y . From the analysis of the log-normal distribution pattern (Fig. 6), it can be seen that the middle part of the range is the main one for the ith standard size of equipment, and the side parts serve for duplicating the production functions of adjacent members of the series. It is advisable to divide the range of variation of the logarithmically normal distribution into three approximately equal intervals (subranges). Then, within the average (main) sub-range, processing with a maximum productivity of more than 2/3 of the work will be provided. The remaining third of the work will be assigned to both other duplicating intervals ai min and ai max (Fig. 7). In this case, it is possible to use equipment outside this size range, but with some loss of productivity as a result of the discrepancy between the technical characteristics of machine tools and optimal operating conditions [19, 20, 71, 72]. This principle of forming a parameter-oriented series makes it possible to process a surface of any size with maximum productivity, while there is a threefold overlap of the range. Fig. 7. Image of the proposed parameter-oriented series (in logarithmic coordinates) with a variable denominator: amax and amin – the maximum and minimum size of the interval; 1 X and m X – the average values of the parameter X for the first and mth members of the series Fig. 6. Logarithmically normal distribution of tool sizes and surfaces machined on metal-cutting machines
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