Decomposition of governing equations in the analysis of resonant response of a nonlinear and non-ideal vibrating system

Jan Awrejcewicz,Grażyna Sypniewska-Kamińska,Roman Starosta

doi:10.1007/s11071-015-2158-2

Abstract

The dynamic response of a nonlinear system with three degrees of freedom in resonance that is loaded, inter alia, with a non-ideal excitation is investigated. A direct current motor (DC motor) with an eccentrically mounted rotor serves as a non-ideal source of energy. The general coordinate corresponding to the rotor dynamics steadily increases as a result of rotational motion. The decomposition of the equations of motion proposed in the paper allows us to separate the vibration of rotor from its rotations. The presented approach can be used to separate the vibration from rotations in many other mechanical and mechatronic systems. The behaviour of the considered non-ideal system near two simultaneously occurring resonances is examined using the Krylov–Bogolyubov averaging method. The stability analysis of the resonant response is also carried out.

Highlights

In modelling the excitation of vibrating systems, one can distinguish two approaches
When the source of excitation is non-ideal, its dynamic behaviour and the reciprocal interactions between the vibrating system and the source should be taken into account in the mathematical modelling of the problem
All that causes dynamics of the non-ideal vibrating system requires a more sophisticated mathematical description, especially when nonlinearities are taken into account

Summary

Introduction

In modelling the excitation of vibrating systems, one can distinguish two approaches. When the power supplied to the system is reduced during the passage through the resonance, the jump phenomena are observed, but the threshold values are not the same as in the case described previously Such a behaviour, characteristic only in the case of a non-ideal system, is known as the Sommerfeld effect. The procedure, carried out in such a way, leads to the model equations containing the unknown functions, which describe only the vibration of the whole system Such a form of equations allows us to apply the Krylov–Bogolyubov averaging method in order to investigate the behaviour of the non-ideal system in the resonance case. This is the main novelty of the approach presented in the paper.

Formulation of the problem

Decomposition of the governing equations

Vibration near resonance

B10 B20c12υ2

Conclusions