Anderson Localization Phenomenon in Periodic Structures

2013 ◽  
Author(s):  
Reyolando M. Brasil
Keyword(s):  
Anderson Localization ◽  
Periodic Structures ◽  
Small Variation ◽  
Slight Variation ◽  
Vibration Modes ◽  
Axial Loads ◽  
Mode Localization ◽  
Localization Phenomenon ◽  
Stiffness Matrices ◽  
Light Coupling

We study the occurrence in structures of a phenomenon similar to Anderson localization. This the vibration modes localization in structures composed of several nominally identical lightly coupled modular substructures. In an ideal perfect model, the vibration modes are global in nature, spreading to the whole structure. In real structures there are no two completely identical segments. Constructive or loading imperfections generate slight variation of the dynamic characteristics of each module. As the level of disorder grows and coupling between modules becomes lighter, the resulting vibration modes change considerably. Vibration energy may become confined to a few segments. This is the Mode Localization Phenomenon. We present models of long modular planar trussed structures. Light coupling is considered between the initially identical modules. A certain degree of imperfection is introduced by adopting a slight variation in the loading of the modules. This will generate a small variation in the global stiffness of the system as the axial loads in the bars affect their Geometric Stiffness Matrices.

Influence of Loading on Mode Localization in Periodic Structures

1995 ◽  
Vol 48 (11S) ◽  
pp. S132-S137 ◽  
Author(s):  
Reyolando M. L. R. F. Brasil ◽  
Carlos E. N. Mazzilli
Keyword(s):  
Small Parameter ◽  
Periodic Structures ◽  
Computer Code ◽  
Vibration Modes ◽  
Finite Element Program ◽  
Mode Localization ◽  
Element Program ◽  
The Masses ◽  
Lumped Masses ◽  
Two Degree Of Freedom

This paper addresses the problem of the presence of a slightly disordered loading in otherwise ordered periodic structures as an element to trigger the phenomenon of vibration mode localization due to its effect in their stiffness. The sample structures are basically cantilever columns supporting the loads due to large lumped masses in their top which may vary according to a disorder related small parameter. They are connected by very flexible springs. A first series of results deals with a two-degree-of-freedom model where localization is achieved due to loading disorder. The frequencies and displacements show very sharp and nonlinear variation when the small parameter changes slightly around its zero value. The results for this simple model compare well with those of a finite element program developed by the authors. For a more complex example, another model of a rank of six columns is analyzed by the same computer code. A pseudo-random variation of the masses is considered and the resulting vibration modes are compared to those of the ordered structure, which are global in nature and present a sinusoidal spatial distribution. Again, due to mode localization, motions in the perturbed structure are found to be restricted largely to one of the masses.

Localization of Natural Modes of Vibration in Bladed Disks

1987 ◽  
Author(s):  
O. O. Bendiksen ◽  
N. A. Valero
Keyword(s):  
State Physics ◽  
Periodic Structures ◽  
Solid State Physics ◽  
Mode Localization ◽  
Natural Modes ◽  
Localization Phenomenon ◽  
Modes Of Vibration ◽  
Manufacturing Tolerances ◽  
Trigger Mode ◽  
Disk Mode

A study is presented of the mode localization phenomenon in imperfect blade-disk and blade-shroud-disk assemblies. This surprising phenomenon, first discovered in solid state physics, occurs in a number of periodic structures when the periodicity is broken by disorder or imperfections. Localization is signalled by the amplitude of a global blade-disk mode becoming confined to one or more local regions consisting of just a few blades, and could have serious implications for blade fatigue life. The results of this investigation indicate that unshrouded blades mounted on stiff disks are especially susceptible, and even small blade imperfections within manufacturing tolerances are likely to trigger mode localization. Increasing the interblade coupling by adding shrouds or reducing the disk stiffness greatly reduces the localization susceptibility, although certain modes may still become localized if the shrouds are free to slip.

Effect of a local crack on the dynamic characteristics of a rotating grouped blade disc

2001 ◽  
Vol 216 (4) ◽  
pp. 447-457 ◽  
Author(s):  
B W Huang ◽  
J H Kuang
Keyword(s):  
Dynamic Characteristics ◽  
Rotational Speed ◽  
Crack Depth ◽  
Numerical Results ◽  
Galerkin’S Method ◽  
Mode Localization ◽  
Torsion Spring ◽  
Crack Distribution ◽  
Localization Phenomenon ◽  
Blade Crack

The effects of a local blade crack and the group arrangement on the mode localization in a rotating turbodisc are studied in this paper. Periodically coupled Euler—Bernoulli beams are used to approximate the grouped and shrouded turboblades. A two-span beam with a torsion spring is used to model the cracked blade. The crack depth characterizes the stiffness of the assumed torsion spring. Galerkin's method is applied to formulate the localization equations of the grouped turbodisc system. Numerical results indicate that the crack depth, crack distribution and rotational speed in a rotating grouped blade disc may significantly affect the localization phenomenon.

Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures

1997 ◽  
Author(s):  
Mintae Kim ◽  
Joonho Moon ◽  
Jonathan A. Wickert
Keyword(s):  
Mode Shape ◽  
Natural Frequencies ◽  
Periodic Structures ◽  
Spatial Modulation ◽  
Vibration Modes ◽  
Algebraic Relation ◽  
Natural Frequencies And Modes ◽  
Axisymmetric Structure ◽  
Technical Interest ◽  
Model Features

Abstract When a structure deviates from axisymmetry because of circumferentially varying model features, significant changes can occur to its natural frequencies and modes, particularly for the doublet modes that have non-zero nodal diameters and repeated natural frequencies in the limit of axisymmetry. Of technical interest are configurations in which inertia, dissipation, stiffness, or domain features are evenly distributed around the structure. Aside from the well-studied phenomenon of eigenvalue splitting, whereby the natural frequencies of certain doublets split into distinct values, modes of the axisymmetric structure that are precisely harmonic become contaminated by certain additional wavenumbers in the presence of periodically spaced model features. From analytical, numerical, and experimental perspectives, this paper investigates spatial modulation of the doublet modes, particularly those retaining repeated natural frequencies for which modulation is most acute. In some cases, modulation can be sufficiently severe that a mode shape will beat spatially as harmonics with commensurate wavenumbers combine, just as the superposition of time records having nearly equal frequencies leads to classic temporal beating. A straightforward algebraic relation and a graphical checkerboard diagram are discussed with a view towards predicting the wavenumbers present in modulated eigenfunctions given the number of nodal diameters in the base mode and the number of equally spaced model features.

Geometric Mistuning Reduced-Order Model Development Utilizing Bayesian Surrogate Models for Component Mode Calculations

2019 ◽  
Vol 141 (10) ◽  
Author(s):  
Joseph A. Beck ◽  
Jeffrey M. Brown ◽  
Alex A. Kaszynski ◽  
Emily B. Carper ◽  
Daniel L. Gillaugh
Keyword(s):  
Periodic Structures ◽  
Model Development ◽  
Element Model ◽  
Forced Response ◽  
Mode Shapes ◽  
Order Model ◽  
Mode Localization ◽  
Reduced Order ◽  
Structural Periodicity ◽  
Computationally Intensive

AbstractIntegrally bladed rotors (IBRs) are meant to be rotationally periodic structures. However, unique variations in geometries and material properties from sector-to-sector, called mistuning, destroy the structural periodicity. This results in mode localization that can induce forced response levels greater than what is predicted with a tuned analysis. Furthermore, mistuning and mode localization are random processes that require stochastic treatments when analyzing the distribution of fleet responses. Generating this distribution can be computationally intensive when using the full finite element model (FEM). To overcome this expense, reduced-order models (ROMs) have been developed to accommodate fast calculations of mistuned forced response levels for a fleet of random IBRs. Usually, ROMs can be classified by two main families: frequency-based and geometry-based methods. Frequency-based ROMs assume mode shapes do not change due to mistuning. However, this assumption has been shown to cause errors that propagate to the fleet distribution. To circumvent these errors, geometry-based ROMs have been developed to provide accurate predictions. However, these methods require recalculating modal data during ROM formulations. This increases the computational expense in computing fleet distributions. A new geometry-based ROM is presented to reduce this cost. The developed ROM utilizes a Bayesian surrogate model in place of sector modal calculations required in ROM formulations. The method, surrogate modal analysis for geometry mistuning assessments (SMAGMA), will propagate uncertainties of the surrogate prediction to forced response. ROM accuracies are compared to the true forced response levels and results computed by a frequency-based ROM.

Dynamic Characteristics of a Mistuned Heat Exchanger in Cross-Flow

2007 ◽  
Author(s):  
Bo-Wun Huang ◽  
Huang-Kuang Kung ◽  
Jao-Hwa Kuang
Keyword(s):  
Heat Exchanger ◽  
Shock Waves ◽  
Dynamic Characteristics ◽  
Tube Bundle ◽  
Cooling Water ◽  
Cross Flow ◽  
Water Film ◽  
Local Defect ◽  
Mode Localization ◽  
Localization Phenomenon

The dynamic behaviors of tubes of a heat exchanger are frequently affected by the existence of local flaw. These tubes are worn from the hot-cold fluid shock waves. This local defect may alter the tube dynamics and introduce mode localization in the periodically arranged tube array. The variation of the dynamic characteristics of a component cooling water heat exchanger with wear tubes in cross-flow is investigated in this study. Periodically coupled cooling tubes are used to approximate a heat exchanger. Each tube is considered to be coupled to adjacent tubes through the squeezed water film in the gaps. This work addresses the probability of mode localization is occurring in a heat exchanger in cross-flow. A dynamic model of the coupled tube bundle is proposed. The numerical results reveal that the local defect in a tube array may introduce the so-called mode localization phenomenon in a periodically coupled tube bundle.

Numerical Investigations of Localized Vibrations of Mistuned Blade Integrated Disks (Blisks)

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
T. Klauke ◽  
A. Kühhorn ◽  
B. Beirow ◽  
M. Golze
Keyword(s):  
Strain Energy ◽  
Vibration Modes ◽  
Blade Vibration ◽  
Nodal Diameter ◽  
Mode Localization ◽  
Bladed Disk ◽  
High Pressure Compressor ◽  
Localized Vibrations ◽  
Pressure Compressor ◽  
Higher Sensitivity

Blade-to-blade variations of bladed disk assemblies result in local zoning of vibration modes as well as amplitude magnifications, which primarily reduces the high cycle fatigue life of aeroengines. Criteria were introduced to determine the level of these mode localization effects depending on various parameters of a real high pressure compressor blisk rotor. The investigations show that blade vibration modes with lower interblade coupling, e.g., torsion modes or modes with high numbers of nodal diameter lines, have a significantly higher sensitivity to blade mistuning, which can be characterized by the higher percentage of blades on the total blisk strain energy.

Mode Localization in Coupled Circular Plates

1994 ◽  
Vol 116 (3) ◽  
pp. 286-294 ◽  
Author(s):  
C. O. Orgun ◽  
B. H. Tongue
Keyword(s):  
Periodic Structures ◽  
Qualitative Change ◽  
Disk Drive ◽  
Circular Plates ◽  
Mode Localization ◽  
Infinite Dimensional ◽  
Write Heads ◽  
The Individual ◽  
Support Motion ◽  
Computer Disk

When analyzing structures that are comprised of many similar pieces (periodic structures), it is common practice to assume perfect periodicity. Such an assumption will lead to the existence of eigenmodes that are global in character, i.e., the structural deflections will occur throughout the system. However, research in structural mechanics has shown that, when only weak coupling is present between the individual pieces of the system, small amounts of disorder can produce a qualitative change in the character of the eigenmodes. A typical eigenmode of such a system will support motion only over a limited extend of the structure. Often only one or two of the smaller pieces that make up the structure show any motion, the rest remain quiescent. This phenomenon is known as “mode localization”, since the modes become localized at particular locations on the overall structure. This paper will examine the behavior of several circular plates that are coupled together through springs, a system that models a multiple disk computer disk drive. These drives typically consist of several disks mounted on a single spindle, coupled by read/write heads, which act as weak springs, thus leading one to suspect the possibility of localization. Since such an effect would impact accurate read/write operations at small fly heights, the problem deserves attention. Although computer disk drives contain space fixed read/write heads, this paper will consider springs that are fixed to the plates in order to understand the effect of localization on a set of infinite dimensional structures (the circular plates). Later work will extend the model to the case of space fixed springs and the wave behavior and destabilizing effects that such a configuration will induce.

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