Calculation of mechanical oscillations in the field of complex numbers


DOI: 10.34759/trd-2020-115-01

Аuthors

Popov I. P.

Kurgan State University, 63/4, Sovetskaya str., Kurgan, 640020, Russia

e-mail: ip.popow@yandex.ru

Abstract

It was noted that traditional calculation of complex mechanical devices at forced sinusoidal oscillations is, as a rule, a rather difficult task. Most often, calculators, designers and technologists are interested in steady-state oscillation modes. The goal of this study consists in significant calculations simplifying by replacing the need to solve differential equations with simpler algebraic methods. The author employs complex representation of harmonic values and the values associated with them. Such approach is widely used in theoretical electrical engineering. The main research methods within the framework of this work are methods of mathematical modeling and analysis. With this, a mathematical model, i.e. the object “equivalent”, reflecting its basic properties, namely the laws it obeys, bonds peculiar to its constituent parts, etc., is being studied, rather than the physical object itself. The article considers the scheme of parallel connection, in which the speeds of all mechanical system elements iare equal, while the forces are different, as well as the scheme of series connection, in the forces applied to the mechanical system elements are equal, while speeds differ. Application of symbolic (complex) description of mechanical systems at forced harmonic oscillations (in the steady-state mode) allowed abnegate the extremely cumbersome and laborious computational algorithm, associated with differential equations solving, and replace it by the much simpler algebraic operations. Due to this fact, the computation time was reduced manifold. Being an unnecessary component of the mechanical systems studying at harmonic impact, vector diagrams are of significant methodological meaning, since they demonstrate quantitative and phase relationships between the system parameters.

Keywords:

consumers of mechanical power, forced oscillations, parallel, series connection, resonance of forces, resonance of speeds

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