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A methodological method based on the use of the theory of preferred numbers has been proposed in order to improve the most important parameters of the working bodies of noise mufflers. As a result of many years of scientific research, the authors have established a previously unknown theoretical relationship between the main series of preferred numbers, golden ratio and Fibonacci series numbers. A new direction in the development of number theory has been considered in the paper, its classification has been compiled, including the geometric theory of numbers, preferred numbers, containing a new basic series of preferred numbers using the Fibonacci sequence. New formulas have been obtained to determine the denominators of geometric progressions for the series of preferred numbers and the area of a circle. Determining the area of a circle using the new formula allows to get more accurate values. A new formula for determining the circumference of a circle has also been derived. The designs of perforated partitions have been developed, in which the laws of the new basic series of preferred numbers are used. Determining the area of a circle using the new formula allows you to get more accurate values. A new formula for determining the circumference of a circle is also obtained. The designs of perforated partitions have been developed, in which the regularities of the new basic series of preferred numbers have been used. A calculated substantiation of the main geometric and structural dimensions of noise mufflers is given using a mathematical model of a perforated golden partition and new basic series of preferred numbers, which allow to obtain a noise muffler design that has the lowest possible aerodynamic resistance with the maximum possible reduction in the noise level of exhaust gases from internal combustion engines. An innovative model of a noise muffler for reciprocating internal combustion engines with improved hydraulic and acoustic characteristics based on the theory of numbers is proposed in the paper. The theory of preferred numbers applies to any technical device.

Abstract. Objectives The purpose of the study is to create a methodology for computing the design of the primary drive mechanism of metal cutting machines, making it possible to reduce error when realising a set of preferred output criteria. Methods The accuracy of established modes of operation is a key determinant on which the efficiency of cutting operations depends. In order to determine and evaluate drive errors in metal-cutting machines, three methods were proposed: direct measurement of the frequencies of the series, calculations using kinematic balance equations and summation of individual components. It is recommended that an unloaded machine be used to check the accuracy of the series gained by the machine's drive. Results By measuring the actual spindle rotation frequencies and kinematic calculation – as well as summing the components of the total error – the developed methodology makes it possible to calculate and estimate the error of a series gained by the drive of a metal-cutting machine at the design stage with a sufficiently high accuracy. The presented complex helps to reveal the role of the rounding error of the output criteria in the formation of the general relative error and provides a basis for its possible reduction. Conclusion The use of the proposed methodology expands the scientific basis for the development of algorithms and programs facilitating the selection of the optimal number of gear teeth for multiplying groups and structures and guaranteeing high accuracy of the gained series.