# Equilibrium Points with Their Associated Normal Modes Describing Nonlinear Dynamics of a Spinning Shaft with Non-constant Rotating Speed

Georgiades, Fotios (2018) Equilibrium Points with Their Associated Normal Modes Describing Nonlinear Dynamics of a Spinning Shaft with Non-constant Rotating Speed. Journal of Vibration Testing and System Dynamics, 2 (4). pp. 327-373. ISSN 2475-4811

Full content URL: https://doi.org/10.5890/JVTSD.2018.12.003

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## Abstract

In this article, the dynamics of a spinning shaft during spin-up/down operation, is examined analytically, around the equilibrium points. The system of equations of motion of a spinning shaft with non-constant rotating speed has no linear part therefore the equilibrium points are rather essential. In the first instance, the equilibrium points of the original system are determined. A restricted system is obtained by neglecting the rigid body angular position of the shaft, and the equilibrium manifold with its’ bifurcations is defined. It is shown that this manifold, is formed by the backbone curve of the nonlinear normal modes which are associated with the rigid body angular motions of the spinning shaft. It is shown that all the equilibrium points of the original and the restricted system are degenerate. The original and restricted systems have been linearized around the equilibrium points, and examination of their stability through the eigenvalues of the Jacobian matrix showed that there are centres and unstable regions. Then, the frequencies and initial conditions of the normal modes of the linearized system around the equilibrium points are determined with the associated analytical solutions. The comparison of analytical with numerical results shows very good agreement, noting that the linearization around equilibrium is valid only for very small perturbations. This work is essential for understanding critical situations in the dynamics of the spinning shaft during spin-up/down operation, based on the associated normal modes. The stability analysis of the spinning shaft can be used further to identify regions with chaotic attractors, which should be considered for normal operation. Finally, considering other rotating structures with non-constant rotating speed, the equations of motion are similar to those of a spinning shaft. Therefore, the approach that is followed in this article can be considered as more general, and could be applied in all rotating structures during spin-up/down, and expecting similar results.

Keywords: Non-constant rotating speed, Nonlinear normal modes, Spinning shaft, Campbell diagram H Engineering > H142 Solid MechanicsH Engineering > H310 DynamicsH Engineering > H300 Mechanical EngineeringH Engineering > H342 VibrationH Engineering > H140 Mechanics College of Science > School of Engineering 33960 12 Nov 2018 11:43

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