On the Physical Interpretation of Proper Orthogonal Modes in Vibrations

Feeny, Brian F.; Kappagantu, Ramana

doi:10.1006/jsvi.1997.1386

Cited by 385 publications

(213 citation statements)

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“…POD provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments, and it provides energy-wise, the most efficient way (however, perhaps not from a dynamical perspective (Mezić 2005a)) of capturing the dominant components of the process (Holmes et al 1996;Feeny and Kappagantu 1998). Consider finite simulation outputs of angular positions, {δ i (nT s )} (i = 2, .…”

Section: Empirical Mode Decompositionmentioning

confidence: 99%

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Coherent Swing Instability of Power Grids

2011

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We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations-local plant, inter-machine, and inter-area modes-interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid sta- bility studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.

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Section: Empirical Mode Decompositionmentioning

confidence: 99%

“…, 10) to be orthonormal and closest in energy norm to the output, and call them Proper Orthonormal Modes (POMs). Every vector {e ij } is obtained by computing the correlation matrix R from {δ i (nT s )} and by finding the orthonormal eigenvectors of R: see Feeny and Kappagantu (1998) …”

Section: Empirical Mode Decompositionmentioning

confidence: 99%

Coherent Swing Instability of Power Grids

2011

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“…In the papers [10] and [11] of Feeny and Kappagantu a relation of the proper orthogonal modes to normal modes of vibration is investigated. Then they used the POD as they so call optimal modal reduction and exploit the benefits of the application of these modes in comparison to the linear natural modes.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Order Reduction Strategy in Earthquake Nonlinear Response Analysis of Structures

Bamer

Amiri

Bucher

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Since earthquake dynamic response analysis of large and complex structures are computationally time demanding, efficient methods that can reduce the system order are of high interest. In this sense, there are different methods available, which try to provide a proper equivalent model. However, in the presence of nonlinearities in the structural elements, most of those methods are ruled out due to their linear assumptions. Therefore, this contribution aims at providing an efficient strategy, which can reduce the order of the nonlinear structural model while retaining important structural characteristics for further earthquake dynamic response analysis. The model order reduction (MOR) strategy is developed based on the proper orthogonal decomposition (POD) method to derive a set of nonlinear deterministic POD modes according to the information of the response history (snapshots) of the full order structure under one or a set of representative earthquake excitations. Subsequently, the POD modes are utilized to create the reduced-order models of the structure subjected to different earthquake excitations. Then, the reduced order models need substantially less amount of computational time in comparison to the full order models. This study presents the application results of the introduced new strategy to a realistic building structure, which is base-isolated by means of frictional bearing elements for better seismic performance. The results demonstrate accurate approximations of the physical (full) responses by means of this new MOR strategy if the probable behavior of the structure has already been captured in the POD snapshots.

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“…The proper orthogonal decomposition (POD) (alternatively called the Karhunen-Loève decomposition) [1,2,3,4,5,6,7,8] is applied to data obtained from a series of strain gauges distributed along the rod's length. We find strong evidence that two specific nonlinear modes, as represented by empirical basis functions obtained via the POD, are present in all of the observed responses.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear modal interactions and low order modeling of a kicked flexible rod

Cusumano

2007

Proc Appl Math and Mech

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A magnetically kicked flexible-beam oscillator is studied experimentally. Spectral bifurcation diagrams of the attracting steady states are obtained, and the system is seen to exhibit periodic, quasiperiodic and chaotic responses. The proper orthogonal decomposition (POD) is applied to spatial strain gauge data, and a specific nonlinear two-mode interaction is shown to be present in all responses. Based on these experimental observations, a two degree of freedom piecewise linear autonomous model is developed, which captures much of the behavior observed in the experiments.

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On the Physical Interpretation of Proper Orthogonal Modes in Vibrations

Cited by 385 publications

References 0 publications

Coherent Swing Instability of Power Grids

Coherent Swing Instability of Power Grids

An Efficient Order Reduction Strategy in Earthquake Nonlinear Response Analysis of Structures

Nonlinear modal interactions and low order modeling of a kicked flexible rod

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