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This paper provides a tutorial and summary of the theory of circulant matrices and their application to the modeling and analysis of the free and forced vibration of mechanical structures with cyclic symmetry. Our presentation of the basic theory is distilled from the classic book of Davis (1979, Circulant Matrices, 2nd ed., Wiley, New York) with results, proofs, and examples geared specifically to vibration applications. Our aim is to collect the most relevant results of the existing theory in a single paper, couch the mathematics in a form that is accessible to the vibrations analyst, and provide examples to highlight key concepts. A nonexhaustive survey of the relevant literature is also included, which can be used for further examples and to point the reader to important extensions, applications, and generalizations of the theory.

The nonlinear performance of centrifugally-driven, order-tuned absorbers is investigated for vibration reduction of a cyclic rotating flexible structure under traveling wave (TW) engine order excitation. A key finding from previous work by the authors using a linearized model is the existence of a no-resonance zone, that is, an entire range of absorber designs that avoid system resonance for any rotation speed. Linearization is generally valid for the rotating structure but absorber motions can become large for typical loading conditions. This work generalizes the linear results to account for large-amplitude, nonlinear absorber motions. Existence and stability of the steady-state TW response to TW excitation are investigated in terms of the absorber path design, which fixes its linear and nonlinear tuning characteristics. A TW response is unique for the linearized system and is shown to exist for the weakly nonlinear model. The nonlinear model exhibits the usual characteristics of a weakly nonlinear system, including bistability and the attendant hysteresis near resonance. More significantly, no additional instabilities associated with the symmetry could be identified. Hence the desired TW response is robust to nonlinear absorber effects and can be described by an equivalent model, which is obtained by reduction using the symmetry. It is shown that good performance can be obtained by linear absorber tuning in the no-resonance zone and the absorber paths should have a slightly softening nonlinear characteristic.

This paper investigates the use of order-tuned absorbers to attenuate vibrations of flexible blades in a bladed disk assembly subjected to engine order excitation. The blades are modeled by a cyclic chain of N oscillators, and a single vibration absorber is fitted to each blade. These absorbers exploit the centrifugal field arising from rotation so that they are tuned to a given order of rotation, rather than to a fixed frequency. A standard change of coordinates based on the cyclic symmetry of the system essentially decouples the governing equations of motion, yielding a closed form solution for the steady-state response of the overall system. These results show that optimal reduction of blade vibrations is achieved by tuning the absorbers to the excitation order n, but that the resulting system is highly sensitive to small perturbations. Intentional detuning (meaning that the absorbers are slightly over- or under-tuned relative to n) can be implemented to improve the robustness of the design. It is shown that by slightly undertuning the absorbers there are no system resonances near the excitation order of interest and that the resulting system is robust to mistuning (i.e., small random uncertainties in the system parameters) of the absorbers and/or blades. These results offer a basic understanding of the dynamics of a bladed disk assembly fitted with order-tuned vibration absorbers, and serve as a first step to the investigation of more realistic models, where, for example, imperfections and nonlinear effects are considered, and multi-DOF and general-path absorbers are employed.

This paper considers the dynamic response and order-tuning of vibration absorbers fitted to a rotating flexible structure under traveling wave (TW) engine order excitation. Of specific interest is the extension of previous results on the so-called no-resonance zone, that is, a region in linear tuning parameter space in which the coupled structure/absorber system does not experience resonance over all rotation speeds. The no-resonance feature was shown to exist for cyclic rotating structures with one structural and one absorber degree-of-freedom (DOF) per sector. This work uses a higher-fidelity structural model to investigate the effects of higher modes on the cyclically-coupled system. It is shown that the no-resonance zone is replaced by a resonance-suppression zone in which one structural mode is suppressed, but higher-order resonances still exist with the addition of the absorbers. The results are general, in the sense that one vibration mode can be eliminated using a set of identically-tuned absorbers on a rotating structure with arbitrarily many DOFs per sector.