Mathematical analysis of rotor shaft displacements in asynchronous machines; a critical speed or just a rotation of the orbit axis?

TLDR: The aim of this paper is to show the mathematical coherence concerning typical rotor eccentricities in asynchronous machines and to demonstrate the necessity to focus not only on the amplitudes relative to two fixed sensor positions, but to also consider the semi-major axis of the calculated orbit and its angular position.

Abstract: This paper presents a mathematical analysis of rotor shaft displacements in asynchronous machines caused by different types of rotor eccentricity. Based on a simplified rotor model, the theoretical coherence between electromagnetic, rotor dynamic, and the specific characteristics of sleeve bearings is shown. The orbits of the rotor mass and the shaft journal are mathematically described for each kind of eccentricity and the shaft displacement with respect to two virtual fixed sensors is derived. Based on this theoretical description and on a numerical example, the paper shows that focusing in a theoretical rotor dynamic analysis only on the calculated amplitudes in direction of two fixed sensor positions, may lead to wrong conclusions concerning the evaluation of resonances. The aim of this paper is, based on a simplified rotor model, to show the mathematical coherence concerning typical rotor eccentricities in asynchronous machines and to demonstrate the necessity to focus not only on the amplitudes relative to two fixed sensor positions, but to also consider the semi-major axis of the calculated orbit and its angular position. The aim of the paper is not to replace a detailed finite element rotor dynamic analysis by a simplified analytical rotor model for predicting the real shaft vibrations. The intension is to prepare the basis for adopting the conclusions, derived from a simplified analytical model, into a more detailed rotor dynamic model – e.g. a finite element rotor dynamic model – and therefore to derive a more precise theoretical analysis of the real shaft displacements.