Christophe Pierre

MODE LOCALIZATION AND EIGENVALUE LOCI VEERING PHENOMENA IN DISORDERED STRUCTURES

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.

Localization of vibrations by structural irregularity

An investigation of the localization phenomenon is presented for disrodered structural systems consisting of weakly coupled component systems. Emphasis is placed on the development of a perturbation method that allows one to obtain the localized modes of vibration of the disordered system from the modes of the individual subsystems. The principal feature of this perturbation method is that it not only considers the variations of the parameters as perturbations, but it also treats the small coupling between subsystems as a perturbation. Such a perturbation analysis is cost effective as compared to a global eigenvalue analysis of the entire system, and proves to be very accurate. Moreover, it provides physical insight into the localization phenomenon, and allows one to formulate a criterion that predicts the occurrence of strongly localized modes. Results are obtained for a chain of single-degree-of-freedom coupled oscillators, but these can be readilt generalized to deal with chains of multi-degree-of-freedom component systems.

Non-linear normal modes and invariant manifolds

Small-amplitude motions of dynamic systems (structural, fluid, control, etc.) about an equilibrium state are modeled by linear differential equations which have constant coefficients. These are typically obtained by a Taylor series expansion of the forces about the equilibrium point. Under quite general circumstances these equations admit a set of special solutions, called normal mode motions, in which each system component moves with the same frequency and with a fixed ratio amongst the displacements of the components (for a conservative system; for a non-conservative system all displacements and velocities are linearly related to a single displacement/velocity pair).

Natural modes of Bernoulli-Euler beams with symmetric cracks

Abstract An approximate Galerkin solution to the one-dimensional cracked beam theory developed by Christides and Barr for the free bending motion of beams with pairs of symmetric open cracks is suggested. The series of comparison functions considered in the Galerkin procedure consists of the mode shapes of corresponding uncracked beam. The number of terms in the expansion is determined by the covnergence of the natural frequencies and confirmed by studying the stress concentration profile near the crack. This approach allows the determination of the higher natural frequencies and mode shapes of the cracked beam. It is found that the Christides and Barr original solution was not fully converged and that cracks render the convergence of the Galerkins procedure very slow by affecting the continuity characteristics of the solution of the boundary value problem. To validate the theoretical results, a two-dimensional finite element approach is proposed, which also allows one to determine the parameter that controls the stress concentration profile near the crack tip in the theoretical formulation without requiring the use of experimental results. Very good agreement between the theoretical and finite element results is observed.

Modeling and analysis of mistuned bladed disk vibration : Status and emerging directions

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.

A Reduced Order Modeling Technique for Mistuned Bladed Disks

The analysis of the response statistics of mistuned turbomachinery rotors requires an expensive Monte Carlo simulation approach. Simple lumped parameter models capture basic localization effects but do not represent well actual engineering structures without a difficult parameter identification. Current component mode analysis techniques generally require a minimum number of degrees of freedom which is too large for running Monte Carlo simulations at a reasonable cost. In the present work, an order reduction method is introduced which is capable of generating reasonably accurate, very low order models of tuned or mistuned bladed disks. This technique is based on component modes of vibration found from a finite element analysis of a single disk-blade sector. It is shown that the phenomenon of mode localization is well captured by the reduced order modeling technique.

Localized vibrations of disordered multispan beams - Theory and experiment

The localization of the free modes of vibration of disordered multispan beams is investigated, both theoretically and experimentally. It is shown that small deviations of the span lengths from an ideal value may have drastic effects on the dynamics of the system. Emphasis is placed on the development of a perturbation method that allows one to obtain the strongly localized modes of vibration of the disordered system without a global eigenvalue analysis of the entire system. Such a perturbation analysis is cost-effective and accurate. More importantly, it provides physical insight into the localization phenomenon, and allows one to formulate a criterion that predicts the occurrence of strongly localized modes. Also, an experiment is described which has been carried out to verify the existence of localized modes for disordered two-span beams. Theoretical and ex- perimental results are compared in detail and excellent agreement is found, thus confirming the existence of localized modes for such weakly coupled, weakly disordered structural systems.

Component-Mode-Based Reduced Order Modeling Techniques for Mistuned Bladed Disks—Part I: Theoretical Models

Component mode synthesis (CMS) techniques are widely used for dynamic analyses of complex structures. Significant computational savings can be achieved by using CMS since a modal analysis is performed on each component structure (substructure). Mistuned bladed disks are a class of structures for which CMS is well suited. In the context of blade mistuning, it is convenient to view the blades as individual components, while the entire disk may be treated as a single component. Individual blade mistuning may then be incorporated into the CMS model in a straightforward manner. In this paper, the Craig-Bampton (CB) method of CMS is formulated specifically for mistuned bladed disks, using a cyclic disk description. Then a novel secondary modal analysis reduction technique (SMART) is presented: a secondary modal analysis is performed on a CB model, yielding significant further reduction in model size. In addition, a straightforward non-CMS method is developed in which the blade mistuning is projected onto the tuned system modes. Though similar approaches have been reported previously, here it is generalized to a form that is more useful in practical applications. The theoretical models are discussed and compared from both computational and practical perspectives. It is concluded that using SMART, based on a CB model, has tremendous potential for highly efficient, accurate modeling of the vibration of mistuned bladed disks.

Characteristic constraint modes for component mode synthesis

A technique is presented for reducing the size of a model generated by the Craig-Bampton method (Craig, R. R., and Bampton, M. C. C., Coupling of Substructures for Dynamic Analyses, AIAA Journal, Vol. 6, No. 7, 1968, pp. 1313-1319) of component mode synthesis (CMS). An eigenanalysis is performed on the partitions of the CMS mass and stiffness matrices that correspond to the so-called constraint modes. The resultant eigenvectors are referred to as characteristic constraint modes because they represent the characteristic motion of the interface between the component structures. When the characteristic constraint modes are truncated, a CMS model with a highly reduced number of degrees of freedom may be obtained. An example of a cantilever plate is considered. It is shown that relatively few characteristic constraint modes are needed to yield accurate approximations of the lower natural frequencies. Furthermore, this method yields physical insight into the mechanisms of vibration transmission in complex structures, and it provides an excellent framework for the efficient calculation of power flow.

Weak and strong vibration localization in disordered structures - A statistical investigation

A statistical investigation of the effects of disorder on the dynamics of one-dimensional nearly periodic structures is presented. The problem of vibration propagation from a local source of excitation is considered. While for the ordered infinite system there exists a frequency passband for which the vibration propagates without attenuation, the introduction of disorder results in an exponential decay of the amplitude for all excitation frequencies. Analytical expressions for the localization factors (the exponential decay constants) are obtained in the two limiting cases of weak and strong internal coupling, and the degree of localization is shown to depend upon the disorder to coupling ratio and the excitation frequency. Both modal and wave propagation descriptions are used. The perturbation results are verified by Monte Carlo simulations. The phenomena of weak and strong localization are evidenced. While the former affects little the dynamics of most engineering structures, the latter is shown to be of significant importance in structural dynamics.