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A methodology is presented which extends to non-linear systems the cDncept of normal modes of motion which is well developed for linear systems. The method is constructive for weakly non-linear systems and provides the physical natUlre of the normal modes along with the non-linear differential equations which govern their dynamics. It also provides the non-linear co-ordinate transformation which relates the original system co-ordinates to the modal co-ordinates. Using this transformation, we demonstrate how an approximate nunlinear version of superposition can be employed to reconstruct the overall motion from the individual non-linear modal dynamics. The results presented herein for non-linear systems reduce to modal analysis for the linearized system when non-linearities are neglected, even though the approach is entirely different from the traditional one. The tools employed are from the theory of invariant manifolds for dynamical systc~ms and were inspired by the center manifold reduction technique. In this paper the basic ideas are outlined, a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions.

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An investigation of the effects of disorder on the modes of vibration of nearly periodic structures is presented. It is shown that, in structures with close eigenvalues, small structural irregularities result in both strong localization of the mode shapes and abrupt veering away, or mutual repulsion, of the loci of the eigenvalues when these are plotted against a parameter representing the disorder in the system. Perturbation methods for the eigenvalue problem are applied to predict the occurrence of strong localization and eigenvalue loci veering, which are shown to be two manifestations of the same phenomenon. Also, perturbation methods that handle the dramatic effects of small disorder are developed to analyze eigenvalue loci veering and strong localization. Two representative disordered nearly periodic structures are studied: a mistuned assembly of coupled oscillators and a multi-span beam with irregular spacing of the supports. 485 0022460X/88/210485+ 18 $03.00/O 0 1988 Academic Press Limited 486 C. PIERRE It has also been observed by Leissa and others [16-181 that when eigenvalue loci veering occurs, the eigenfunctions undergo dramatic, albeit continuous, changes in the veering region. Indeed, the eigenfunctions corresponding to each eigenvalue locus are interchanged during veering. However, to date, the occurrence of eigenvalue loci veering has never been associated with the occurrence of strong mode localization.Both eigenvalue loci veering and mode localization are catastrophic type phenomena, because small changes in the system parameters result in large variations in the eigenvalues and the mode shapes, respectively. Since both phenomena occur when a particular system parameter is varied (for mode localization, this parameter is a measure of disorder), one wonders whether the occurrence of these two phenomena might be related in any way.This paper describes an investigation of the effects of small disorder on the modes of free vibration of nearly periodic structures. Special attention is paid to the loci of the eigenvalues versus a parameter representing the amount of disorder in the system. It is shown that when small disorder is introduced in conservative nearly periodic structures with weak internal coupling, both strong mode localization and veering of the eigenvalue loci occur, indicating that these are two manifestations of the same phenomenon. This simultaneous occurrence is illustrated in section 2 with a simple two-degree-of-freedom (DOF) system. Section 3 presents perturbative approaches for both the prediction and analysis of mode localization and curve veering in self-adjoint disordered systems. In sections 4 and 5 the general theory is applied to two simple disordered systems, namely an assembly of coupled oscillators and a two-span beam whose localization of the modes has been previously studied by the author [12,13]. The (modified) perturbation method developed in references [12,13] for the analysis of strong mode localization is shown to apply to the analysis of curve veering as well.

The literature on reduced-order modeling, simulation, and analysis of the vibration of bladed disks found in gas-turbine engines is reviewed. Applications to system identification and design are also considered. In selectively surveying the literature, an emphasis is placed on key developments in the last decade that have enabled better prediction and understanding of the forced response of mistuned bladed disks, especially with respect to assessing and mitigating the harmful impact of mistuning on blade vibration, stress increases, and attendant high cycle fatigue. Important developments and emerging directions in this research area are highlighted. I. Introduction T URBINE engine rotors, or bladed disks, are rich dynamical systems that are known to suffer from severe vibration problems. Although a bladed disk is typically designed to have identical blades, there are always random deviations among the blades caused by manufacturing tolerances, wear, and other causes. This is called mistuning. Even though mistuning is typically small (e.g., blade natural frequency differences on the order of a few percent of the nominal values), mistuned bladed disks can have drastically larger forced response levels than the ideal, tuned design. The attendant increase in stresses can lead to premature high cycle fatigue (HCF) of the blades. HCF is a major cost, safety, and reliability issue for gas-turbine engines. For example, in 1998 it was estimated by the U.S. Air Force that about 55% of fighter jet engine safety Class A mishaps (over $1 million in damage or loss of aircraft) and 30% of all jet engine maintenance costs were due to HCF. 1 It is clearly of great interest to be able to predict-and, ultimately, to reduce-the maximum blade response as a result of mistuning. The comprehensive modeling, analysis, and understanding of bladed disk vibration is thus critical to reducing the occurrence of HCF and improving the performance and reliability of turbine engines. Bladed disk vibration first received significant attention from the research community in the late 1960s and the 1970s. Notable early work was done by Whitehead, 2 Wagner, 3 Dye and Henry, 4 and Ewins. 5−8 The bladed disk vibration literature has been surveyed by Srinivasan 9,10 and Slater et al. 11 The 1997 survey by Srinivasan 10 Matthew P. Castanier is an Associate Research Scientist in the Department of Mechanical Engineering at the University of Michigan. He received his Ph.D. in Mechanical Engineering from the University of Michigan in 1995. His research interests are in the area of structural dynamics and vibration, including reduced-order modeling, low-to mid-frequency vibration and power flow in complex structures, localization and related phenomena in periodic or cyclic structures, and vibration of mistuned bladed disks in turbine engines. Christophe Pierre is Dean of the Faculty of Engineering at McGill University in Montréal, where he is also Professor of Mechanical Engineering and holds the Canada Research Chair in Structural Dynamics and Vibration. He ...