Antony Kirk

Theoretical investigation into balancing high-speed flexible shafts, by the use of a novel compensating balancing sleeve

Traditional techniques for balancing long, flexible, high-speed rotating shafts are inadequate over a full range of shaft speeds. This problem is compounded by limitations within the manufacturing process, which have resulted in increasing problems with lateral vibrations and hence increased the failure rates of bearings in practical applications. There is a need to develop a novel strategy for balancing these coupling shafts that is low cost, robust under typically long-term operating conditions and amenable to on-site remediation. This paper proposes a new method of balancing long, flexible couplings by means of a pair of balancing sleeve arms that are integrally attached to each end of the coupling shaft. Balance corrections are applied to the free ends of the arms in order to apply a corrective centrifugal force to the coupling shaft in order to limit shaft-end reaction forces and to impart a corrective bending moment to the drive shaft that limits shaft deflection. The aim of this paper is to demonstrate the potential of this method, via the mathematical analysis of a plain, simply supported tube with uniform eccentricity and to show that any drive shaft, even with irregular geometry and/or imbalance, can be converted to an equivalent encastre case. This allows for the theoretical possibility of eliminating the first simply supported critical speed, thereby reducing the need for very large lateral critical speed margins, as this requirement constrains design flexibility. Although the analysis is performed on a sub 15 MW gas turbine, it is anticipated that this mechanism would be beneficial on any shaft system with high-flexibility/shaft deflection.

Generalised analysis of compensating balancing sleeves with experimental results from a scaled industrial turbine coupling shaft

The paper furthers the analysis of a recently proposed balancing methodology for high-speed, flexible shafts. This mechanism imparts corrective balancing moments, having the effect of simulating the fixing moments of equivalent double or single encastre mounted shafts. This is shown to theoretically eliminate/nullify the first lateral critical speed, and thereby facilitate safe operation with reduced lateral critical speed margins. The paper extends previously reported research to encompass a more generalised case of multiple, concentrated, residual imbalances, thereby facilitating analysis of any imbalance distribution along the shaft. Solutions provide greater insight of the behaviour of the balancing sleeve concept, and the beneficial implications for engineering design. Specifically, (1) a series of concentrated imbalances can be regarded as an equivalent level of uniform eccentricity, and balance sleeve compensation is equally applicable to a generalised unbalanced distribution, (2) compensation depends on the sum of the applied balancing sleeve moments and can therefore be achieved using a single balancing sleeve (thereby simulating a single encastre shaft), (3) compensation of the second critical speed, and to a lesser extent higher orders, is possible by use of two balancing sleeves, positioned at shaft ends, (4) the concept facilitates on-site commissioning of trim balance, which requires a means of adjustment at only one end of the shaft, (5) the Reaction Ratio, RR, (simply supported/encastre), is independent of residual eccentricity, so that the implied benefits resulting from the ratio (possible reductions in the equivalent level of eccentricity) are additional to any balancing procedures undertaken prior to encastre simulation. Analysis shows that equivalent reductions in the order of 1/25th, are possible. Experimental measurements from a scaled model of a typical drive coupling employed on an industrial gas turbine package, loaded asymmetrically with a concentrated point of imbalance, are used to support the analysis and conclusions.

Passive Control of Critical Speeds of a Rotating Shaft Using Eccentric Sleeves: Model Development

This paper considers the passive control of lateral critical speeds in high-speed rotating shafts through application of eccentric balancing sleeves. Equations of motion for a rotating flexible shaft with eccentric sleeves at the free ends are derived using the extended Hamilton Principle, considering inertial, non-constant rotating speed, Coriolis and centrifugal effects. A detailed analysis of the passive control characteristics of the eccentric sleeve mechanism and its impact on the shaft dynamics, is presented. Results of the analysis are compared with those from three-dimensional finite element simulations for 3 practical case studies. Through a comparison and evaluation of the relative differences in critical speeds from both approaches it is shown that consideration of eccentric sleeve flexibility becomes progressively more important with increasing sleeve length. The study shows that the critical speed of high-speed rotating shafts can be effectively controlled through implementation of variable mass/stiffness eccentric sleeve systems.

Towards Determination of Critical Speeds of a Rotating Shaft with Eccentric Sleeves: Equations of Motion

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 Hamilton’s principle, the equations of motion (axial, torsional, in-plane 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.