Mode localization and eigenvalue loci veering phenomena in disordered structures

Abstract

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.

Buckling loads and eigenfrequencies of a braced beam resting on an elastic foundation

“…The dependence of the eigenvalues upon the system parameters is often of interest in structural dynamics and design, which is frequently illustrated by a family of loci [19,20] . When two loci approach each other, they often cross (curve crossing) or abruptly diverge (curve veering) [20] .…”

Ultrasensitive mass sensing using mode localization in coupled microcantilevers

“…While numerous studies of mode localization in coupled structures [10][11][12][13][14][15] and arrays of coupled oscillators [16][17][18] have been performed, the question of whether this phenomenon can be used in a sensing capacity has not been examined. This question is addressed in this letter both theoretically and by means of experiments using coupled gold-foil microcantilevers.…”

Far-Field Earthquake Responses of Overhead Line Equipment (OHLE) Structure Considering Soil-Structure Interaction

“…As for mast structure, it was noticeable that the rotational stiffness affected the natural frequencies and mode shape of vibration in a lower mode but rarely affected the fundamental mode in a higher mode (Ngamkhanong et al, 2017). This was because the dynamic behavior was characterized by coincident eigenfrequencies, mode order change, while the eigenfunctions remain associated with the corresponding eigenvalues (Pierre, 1988;Benedettini et al, 2009;Sari et al, 2017). In addition, soil-structure interaction has a crucial influence on the seismic response of structures especially founded on soft soils (Davoodi and Sadjadi, 2015).…”