OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 ( ) max ln ln ln 3 i i i X X = + d − s ; 2 exp ln 2 i i i X X s = + , where i s is a standard deviation of a random variable X; ln i X is an average value of the logarithm of a random variable X; i X is a mean value of X. A rather visually simulated system is shown in Figure 9, where it is depicted as a pattern of distributions. Here, the characteristic values are shown in logarithmic coordinates: horizontally – spindle speed n, vertically – torque M, and diagonally – effective power N. The figure shows lines of equal probabilities of work performed on the machine and the optimal boundaries of the limit values of the simulated performance are plotted. It corresponds to the extreme maximum values of the second derivative of the differential functions f(y) of the final (resulting) distributions of machine characteristics y 1 ( ) ( ), q q q f y p f y w = = ∑ where pq is a probability of realization of processing conditions q; fq(y) is a differential function of the elementary (private) distribution of the characteristic for the processing condition q (y is the natural logarithm of n, M or N); w is a total number of machining conditions on the machine. The proposed methodology differs from the traditional approach in that it allows introducing restrictions on the minimum used power values. This feature significantly increases the average motor power factor (cos ϕ) and reduces energy consumption. The characteristic values obtained in this way are subsequently corrected, if necessary, based on standard series of preferred numbers, typical motor power values, etc. Taking into account the fact that the optimization of technical characteristics is carried out according to the productivity of the machine, the choice of the values of these characteristics at the modeling stage for greater convenience is carried out in accordance with the results of calculating the percentage component of work performed on the machine at maximum productivity, taking into account the established limitations of operational parameters. Conclusions An original technique for conducting structural-kinematic analysis for pre-project studies of hybrid metal-cutting equipment is presented. The research results showed that in most cases, parameter-oriented (dimension) series of general-purpose machine tools with constant denominators, built in accordance with the law of geometric progression, lead to the need to duplicate individual size ranges on machines of the same series, and, as a result, to an inappropriate increase in the number its members. In this regard, the costs of design work, manufacturing and operation of equipment are growing. In the case of using a parameteroriented series, built using a variable denominator, the maximum efficiency of hybrid metal-cutting equipment is achieved, which has been theoretically proven. Thanks to the use of this principle of forming a parameter-oriented series, it becomes possible to provide an almost equal probability of processing surfaces of Fig. 9. The distribution pattern of the hybrid process equipment operational characteristics
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