Non-Linear Normal Modes and Non-Parametric System Identification of Non-Linear Oscillators

Ma, Xianghong; Azeez, Mohammed F. Abdul; Vakakis, Alexander F.

doi:10.1006/mssp.1999.1267

Cited by 41 publications

(18 citation statements)

References 23 publications

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“…Nonetheless, the solution for the IMO (11), which is strongly driven by the forcing Λ m (t)e jω m t becauseζ m ζ m , can approximately reproduce the IMF in Eq. (18). Similar discussions can be made not only for the response at position x 9 , but also for those at all other positions along the linear beam.…”

Section: Linear Beammentioning

confidence: 61%

“…Furthermore, the solution for the IMO (11) will appear as a free damped response, which may naturally satisfy the relation in Eq. (18). However, as is the case for many other nonlinear system identification methods where it is of more interest to check whether the proposed parametric model is able to reproduce the measured (or simulated) dynamics, the damping factor in the IMO is not necessarily the same as the physical one (i.e.,ζ m = ζ m , in general).…”

Section: Linear Beammentioning

confidence: 99%

“…Use of POD has been rather popular in studying system identification and nonlinear normal modes of coupled beams [18] and rods [19], and in structural damage detection [20]. For example, the method of POD has been utilized for studying chaotic vibrations of a 10-degree-of-freedom (DOF) impact oscillator and a flexible-beam impact oscillator in [21] and [22], respectively.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

et al. 2012

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We study the dynamics of a cantilever beam with two rigid stops of certain clearances by performing nonlinear system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics. The NSI method in this work can proceed in two directions: One for the numerical data obtained from a reduced-order model by means of the assumed-mode method, and the other for the experimental data measured at the same positions as in numerical simulations. This paper focuses on the analysis of the numerical data, providing qualitative comparison with some experimental results; the latter task will be discussed in detail in a companion paper. First, we perform empirical mode decomposition (EMD) on the acceleration responses measured at ten, almost evenly-spaced, spanwise positions along the beam leading to sets of intrinsic modal oscillators governing the vibro-impact dynamics at different time scales. In particular, the EMD analysis can separate any nonsmooth effects caused by vibro-impacts of the beam and the rigid stops from the smooth (elastodynamic) response, so that nonlinear modal interactions caused by vibro-impacts can be explored only with the remaining smooth components. Then, we establish nonlinear interaction models (NIMs) for the respective intrinsic modal oscillators, where the NIMs invoke slowly-varying forcing amplitudes that can be computed from empirical slow-flows. By comparing the spatio-temporal variations of the nonlinear modal interactions for the vibro-impact beam and those of the underlying linear model (i.e., the beam with no rigid constraints), we demonstrate that vibro-impacts significantly influence the lower-frequency modes, introducing spatial modal distortions, whereas the higher frequency modes tend to retain their linear dynamics between impacts. We introduce a linear correlation coefficient as a measure for studying the linear dependency between the slowlyvarying complex forcing amplitudes for the linear and vibro-impact beams and demonstrate that only a set of lower-frequency modes are strongly influenced by vibro-impacts, capturing most of the essential nonlinear dynamics. These results demonstrate the efficacy of the proposed approach to analyze strongly nonlinear measured time series and provide physical insight for strong nonlinear dynamical interactions.

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Section: Linear Beammentioning

confidence: 61%

Section: Linear Beammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

et al. 2012

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“…POD, similar to singular value decomposition (SVD), is a tool for extracting modes that optimize the signal energy distribution in a set of measured time series. It has been used to characterize spatial coherence in turbulence and structures [20][21][22], the dimension of the dynamics [21][22][23], empirical modes for reduced order models [24,25], and in system identification [26,27]. POD, SVD, and similar tools have been compared for structural applications [28].…”

Section: Complex Mode Decompositionmentioning

confidence: 99%

Complex Orthogonal Decomposition Applied to Nematode Posturing

Feeny

Sternberg²,

Cronin

et al. 2013

Journal of Computational and Nonlinear Dynamics

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The complex orthogonal decomposition (COD), a process of extracting complex modes from complex ensemble data, is summarized, as is the use of complex modal coordinates. A brief assessment is made on how small levels of noise affect the decomposition. The decomposition is applied to the posturing of Caenorhabditis elegans, an intensively studied nematode. The decomposition indicates that the worm has a multimodal posturing behavior, involving a dominant forward locomotion mode, a secondary, steering mode, and likely a mode for reverse motion. The locomotion mode is closer to a pure traveling waveform than the steering mode. The characteristic wavelength of the primary mode is estimated in the complex plane. The frequency is obtained from the complex modal coordinate's complex whirl rate of the complex modal coordinate, and from its fast Fourier transform. Short-time decompositions indicate the variation of the wavelength and frequency through the time record.

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“…POD produces modes that optimize the signal energy distribution in a set of measured time series. POD has been applied, for example, to characterize spatial coherence in turbulence and structures [1][2][3][6][7][8][9], to evaluate the dimension of the dynamics [3,[6][7][8]10], to detect modal interactions [11,12], to produce empirical modes for reduced order models [13][14][15][16][17][18][19], and in system identification [20][21][22][23]. The POD is similar to Karhunen-Loeve decomposition, principle components analysis, or singular value decomposition.…”

Section: Introductionmentioning

confidence: 99%

A nonsymmetric state-variable decomposition for modal analysis

Feeny

Farooq

2008

Journal of Sound and Vibration

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A modal decomposition strategy based on state-variable ensembles is formulated. A nonsymmetric, generalized eigenvalue problem is constructed. The data-based eigenvalue problem is related to the generalized eigenvalue problem associated with free-vibration solutions of the state-variable formulation of linear multi-degree-of-freedom systems. For linear free-response data, the inverse-transpose of the eigenvector matrix converges to the state-variable modal eigenvectors, and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state-variable model. As such, the eigenvalues lead to estimates of frequencies and modal damping. The interpretation holds for linear systems with multi-modal free responses, whether damping is large or small, modal or nonmodal, and without the need of input data.

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Non-Linear Normal Modes and Non-Parametric System Identification of Non-Linear Oscillators

Cited by 41 publications

References 23 publications

Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

Complex Orthogonal Decomposition Applied to Nematode Posturing

A nonsymmetric state-variable decomposition for modal analysis

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