Generation of Localization in a Discrete Chain with Periodic Boundary Conditions: Numerical and Analytical Results

Vakakis, Alexander F.; Salenger, Gary

doi:10.1137/s0036139997316355

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…where the columns of the matricesˆ. j / are .j / l s with l D i s , : : : , i s C k s 1 in Equation (6). Similarly,´.t / can be represented in a truncated chaos representation as Substituting these expressions into Equation (21) and then projecting on the chaos bases yields…”

Section: Invariant Subspaces In Response Predictionmentioning

confidence: 99%

“…Variability in mechanical properties destroys the symmetry of periodic structures, resulting in what is typically referred to as mistuning in the literature on bladed disks. Mistuning leads to some interesting phenomena such as mode localization , which, in the context of wave propagation, creates frequency filters often described as stop and pass bands . Although in some special cases mistuning can be beneficial , for example, bringing stability in the presence of flutter , in most cases, it leads to practical problems such as vibration localization in bladed disks, which causes blade fatigue and failure.…”

Section: Introductionmentioning

confidence: 99%

“…In the PCE-based approach, first, the random eigenvalues and eigenvectors are expressed into PCE (6). The chaos coefficients .i / l and .i / l are then estimated using weak characterization, that is, by solving the system of nonlinear deterministic Equations (10) and (11).…”

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confidence: 99%

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An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

Ghosh

Ghanem

2012

Numerical Meth Engineering

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SUMMARY The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non‐unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal‐based reduced‐order model of the system. Copyright © 2012 John Wiley & Sons, Ltd.

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Section: Invariant Subspaces In Response Predictionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

Ghosh

Ghanem

2012

Numerical Meth Engineering

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“…Passive nonlinear localization was extensively studied in previous works for discrete 3,4] and continuous systems 3,5,6]. In these works it was shown that weakly coupled nonlinear oscillators can possess strong and weak localized modes corresponding to energy con nement in one or more substructures of the system.…”

Section: Introductionmentioning

confidence: 99%

Mode Localization Induced by a Nonlinear Control Loop

M’Closkey

Vakakis

Gutierrez

2001

Normal Modes and Localization in Nonlinear Systems

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We present the analysis of a nonlinear control system that is used to excite and maintain a speci ed amplitude of oscillation in the Jet Propulsion Laboratory vibratory gyroscope. This experimental application shows that nonlinear localization through active means can be implemented in a practical system when it is desirable to con ne the response to a favorable mode. The closed-loop system response predicted by the model shows very close agreement w i t h t h e experimental results for a signi cant range of controller parameters. We also experimentally demonstrate that the actively localized motion is eliminated through bifurcation, similar to what was observed in previous passive localization studies applied to extended exible oscillators.

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Experimental nonlinear localization in a periodically forced repetitive system of coupled magnetoelastic beams

Emad

Vakakis

Miller

2000

Physica D: Nonlinear Phenomena

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Generation of Localization in a Discrete Chain with Periodic Boundary Conditions: Numerical and Analytical Results

Cited by 6 publications

References 19 publications

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

Mode Localization Induced by a Nonlinear Control Loop

Experimental nonlinear localization in a periodically forced repetitive system of coupled magnetoelastic beams

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