Vibrations in engineering and technology

DUBCHAK Viktor – Candidate of Technical Sciences, Associate Professor of the Department of Mathematics, Physics and Computer Technologies, Vinnitsa National Agrarian University (21008, 3, Sonyachna Str., Vinnytsia, e-mail: viktor_dubchak@rambler.ru).

In this paper we investigate the question of the optimal, extreme (minimum) location of one plane figure, namely, the square in the first case, as well as the equilateral triangle in the second case relative to the circle with the common center of the said figures. In comparison with previous works, the analysis of studies of such a mutual effective location continues. But if in the previous results a function based on determining the value of an area over which the geometric figures did not match was calculated and investigated, then in these studies another numerical characteristic, such as a function, is proposed as such a criterion for the basic working function. which determines the length of the total polygon (straight or arc, etc.). For certain ratios of the radius of a circle with respect to the length of the side of a square or triangle, the length of the arc of the total line along which the discrepancy between the specified geometric objects is observed will take on an extreme (minimum) value. The optimality of such mutual arrangement of one figure relative to another is evaluated by such a criterion as the effective estimation of the length of the set of lines by which the divergence of these figures occurs. The value of a certain function, which determines the length of such sum of lines of divergence of figures in both cases, is obtained, the study of the extremality of this function is performed, it is shown that at the found extremum point, the function that determines the total length of the line of divergence of figures acquires a minimum value. A study of a function that establishes the entire length of the line over which the discrepancy of the figures is observed indicates, at the extreme, that the function acquires minimal values in the case of such arrangement of figures, when the circle is inscribed squarely or an equilateral triangle, and when described around these figures when such a function is maximized. Drawings are given for a better understanding of the formulation and solution of the task, conclusions are drawn, which summarize the values of the arguments sought when the corresponding function acquires minimum values.

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