# Towards determination of critical speeds of a rotating shaft with eccentric sleeves: equations of motion

Kirk, A., Georgiadis, F. and Bingham, C. (2014) Towards determination of critical speeds of a rotating shaft with eccentric sleeves: equations of motion. In: 9th IFToMM International Conference on Rotor Dynamics, 22 - 25 September 2014, Milan; Italy.

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Item Type: Conference or Workshop contribution (Paper) Live Archive

## Abstract

A primary problem in the turbine industry is associated with the mitigation of bending vibration modes of high-speed rotating shafts. This is especially pertinent at speeds approaching the critical frequencies. Here, a shaft, complete with eccentric sleeves at the free ends, is designed and developed, with a view to passively control critical speeds and vibration induced bending. In this article, using the Extended Hamiltonâ��s principle, the equations of motion (axial, torsional, inplane and out-of-plane bending) for a rotating flexible shaft are derived; considering non-constant rotating speed, Coriolis and centrifugal forces, with the associated boundary conditions due to the eccentric sleeves and torsional springs in angular deformations of lateral vibrations in bending. The numerical dynamic analysis showed that considering the sleeves as flexible only had a small effect upon the first critical speed of the shaft. Therefore, rigid body modelling of the sleeves is sufficient to capture the essential dynamics of the system. The derived equations of motion with the associated boundary conditions show that in the case of constant rotating speed, the eccentric sleeves are coupling xy-bending with xz-bending and also torsion. Also the derived equations of motion and the associated boundary conditions in the case of non-constant rotating speed are essentially nonlinear due to inertia terms. This work is essential to the advance of linear and nonlinear dynamic analysis of the system by means of determination of normal modes and critical speeds of the shaft. Â© Springer International Publishing Switzerland 2015.