Order-tuned vibration absorbers for systems with cyclic symmetry with applications to turbomachinery

This work investigates the performance of centrifugally-driven, order-tuned absorbers for vibration reduction in a class of systems with cyclic symmetry. The rotating flexible structures of interest are bladed disk assemblies, such as the fans, compressors and turbines in a jet engine, which consist of a nominally cyclic array of interconnected substructures. Under steady operation; these assemblies rotate at a constant speed and are subjected to traveling wave dynamic loading (the so-called engine order excitation), which is characterized by excitation frequencies that are proportional to the mean rotational speed of the rotor. Such excitations result in component vibrations and can lead to high cycle fatigue failure, noise, reduced performance, and other undesirable effects. Since order-tuned absorbers feature natural frequencies that scale directly with the rotor speed, they are ideally suited to address these vibrations. However, at the time of writing, there has been no systematic analytical treatment of absorber systems applied to cyclic rotating flexible structures under engine order excitation. This thesis reports the first such study.; The aim of this investigation is threefold: to quantify and understand the underlying linear resonance structure of a cyclically-coupled bladed disk assembly fitted with order-tuned absorbers; based on these findings, to design the absorbers to eliminate or otherwise reduce blade motions relative to the rotating hub; and to generalize the linear theory, methodology; and design to include the basic, first-order effects of nonlinearity.; The analysis is carried out assuming identical, identically-coupled substructures, which gives rise to a linearized model with block circulant matrices. A standard change of coordinates based on this cyclic structure essentially decouples the governing equations, and it gives rise to closed form expressions from which analytical results can be gleaned. The linear resonance structure is found to be surprisingly rich, a feature that arises from the order-nature of the absorbers. One of the main findings of the linear analysis, and indeed of this entire thesis, is the existence of a "no-resonance zone." that is, an entire spectrum of absorber designs for which there are no system resonances over the full range of possible rotor speeds. By designing the absorbers within this small, but finite spectrum, system resonances are avoided altogether and there is at least some level of robustness to parameter uncertainties.; In the presence of weak nonlinearity, which is introduced via the absorber path geometry, the underlying linear resonance structure is shown to qualitatively persist---including the no-resonance zone---provided that the excitation strength is sufficiently small. There does exist a nonlinear design strategy in which relative blade motions can be eliminated completely, but it depends on both the rotor speed and force amplitude. The design is thus effective for only a single set of operating conditions, which suggests that nonlinearity cannot be exploited to further improve absorber performance in the systems of interest. When nonlinearity cannot be avoided it is shown that softening characteristics are more desirable than hardening; the former simply sets an upper limit on the range of speeds over which the absorbers are effective while the latter may give rise to problematic resonances. Finally, for the weakly coupled and lightly damped systems under consideration, there may be a host of symmetry-breaking instabilities involving the desired traveling wave response. However, none could be identified. This is a very promising finding since bifurcations of this kind are highly undesirable from an applications viewpoint.