# Vibrations computing for ramified mechanical systems in the field of complex numbers

DOI: 10.34759/trd-2021-116-01

### Аuthors

Popov I. P.

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

e-mail: ip.popow@yandex.ru

### Abstract

The article considers both parallel-series and series-parallel connection of consumers of mechanical power. The purpose of the work consists in developing compact algebraic methods for ramified mechanical systems computing at forced vibrations in the steady-state modes. Speeds of mechanical systems’ elements and forces applied to them are being determined algebraically through the known systems parameters and disturbing harmonic impact. A complex representation of harmonic and related mechanical quantities is used. This approach is widely used in electrical engineering. The main research methods within the framework of this work are methods of mathematical modeling and analysis. It is not the physical object itself herewith, which is being studied, but its mathematical model, namely the object “equivalent” reflecting its major properties, i.e. the laws it follows, connections peculiar to its constituent parts etc. As for the considered ramified mechanical systems, classical methods based on solving the second order differential equations are being multiply complicated and require solving the systems of equations, which are being reduced to the systems of higher orders. Symbolic (complex) description employing for the mechanical processes and systems allows apply instead simple and compact algebraic methods, which labor intensity is tenfold less.

A relation between mechanical values for various types of elements connection of mechanical systems was established. Being an unnecessary component of mechanical systems studying, vector diagrams are of great methodological value, since they demonstrate quantitative and phase relationships between systems’ parameters.

### Keywords:

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

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