Numerical and Experimental Investigation on Parameter Identification of Time-Varying Dynamical System Using Hilbert Transform and Empirical Mode Decomposition

This paper proposes an approach to identifying time-varying structural modal parameters using the Hilbert transform and empirical mode decomposition. Definition of instantaneous frequency and instantaneous damping ratio based on Hilbert transform for single-degree-of-freedom (SDOF) system is first introduced. The following is the definition of Hilbert damping spectrum from which the time-varying damping ratio of multi-degree-of-freedom (MDOF) system can be calculated. Identification procedures for both instantaneous frequency and damping ratios based on their definition are then introduced. Applicability of the proposed identification algorithm has been validated through several numerical examples. The instantaneous frequency and damping ratios of SDOF system under free vibration and under sinusoidal and white noise excitation have been identified. The proposed method is also applied to MDOF system with slow and sudden changing structural parameters. The results demonstrate that when the system modal parameters are slowly changing, the instantaneous frequency could be easily and well identified with satisfied accuracy for all cases. However, the instantaneous damping ratio could be extracted only when the system is lightly damped. The damping results are better for free vibration situation than for the forced vibration cases. It is also shown that the suggested method can easily track the abrupt change of system modal parameter under free vibration. The proposed method is then applied to a 12-story short-lag shear wall structure model tested on a shaking table. The instantaneous dynamic properties of the structure were identified and were then introduced as known parameters into a finite element model. Comparisons with the numerical results using constant structural parameters demonstrate that the calculated structural responses using the identified time-varying parameters are much closer to the experimental results.

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A posteriori verification methodology for astrochronology: a step further to improve the falsifiability of cyclostratigraphy

10.5194/egusphere-egu2020-10874 ◽

2020 ◽

Author(s):

Sébastien Wouters ◽

Michel Crucifix ◽

Matthias Sinnesael ◽

Anne-Christine Da Silva ◽

Christian Zeeden ◽

...

Keyword(s):

Empirical Mode Decomposition ◽

Hilbert Transform ◽

Instantaneous Frequency ◽

Linear Interpolation ◽

Time Interval ◽

A Posteriori ◽

Intrinsic Mode Functions ◽

Zero Crossing ◽

Mode Decomposition ◽

The Hilbert Transform

Cyclostratigraphy is increasingly used to improve the Geologic Time Scale and our understanding of past climatic systems. However, except in a few existing methodologies, the quality of the results is often not evaluated.We propose a new methodology to document this quality, through a decomposition of a signal into a set of narrow band components from which instantaneous frequency and amplitude can be computed, using the Hilbert transform. The components can be obtained by Empirical Mode Decomposition (EMD), but also by filtering a signal (be it tuned or not) in any relevant way, and by subsequently performing EMD on the signal minus its filtered parts.From that decomposition, verification is performed to estimate the pertinence of the results, based on different concepts that we introduce:<ul><li>&#160;Integrity quantifies to what extent the sum of the components is equal to the signal. It is defined as the cumulated difference between (1) the signal, and (2) the summed components of the decomposition. EMD fulfils integrity by design, except for errors caused by floating-decimal arithmetic. Ensemble Empirical Mode Decomposition (EEMD) may fail to satisfy integrity unless noisy realisations are carefully chosen in the algorithm to cancel each other when averaging the realisations. We present such an algorithm implemented in R: &#8220;extricate&#8221;, which performs EEMD in a few seconds.</li> <li>&#160;Parsimony checks that the decomposition does not generate components that heavily cancel out. We propose to quantify it as the ratio between (1) the cumulated absolute values of each component (except the trend), and (2) the cumulated absolute values of the signal (minus the trend). The trend should be ignored in the calculation, because an added trend decreases the parsimony estimation of a similar decomposition.</li> <li>&#160;IMF departure (IMFD) quantifies the departure of each component to the definition of intrinsic mode functions (IMF), from which instantaneous frequency can reliably be computed. We define it as the mean of the absolute differences of the base 2 logarithms of frequencies obtained using (1) a robust generalized zero-crossing method (GZC, which simplifies the components into extrema separated by zero-crossings) and (2) a more local method such as the Hilbert Transform.</li> <li>&#160;Reversibility is the concept that all initial data points are preserved, even after linear interpolation and tuning. This allows to revert back to the original signal and discuss the significance of each data point. To facilitate reversibility we introduce the concept of quanta (smallest depth or time interval having significance for a given sampling) and an algorithm computing the highest rational common divisor of given values in R: &#8220;divisor&#8221;.</li> </ul>This new methodology allows to check the final result of cyclostratigraphic analysis independently of how it was performed (i.e. a posteriori). Once the above-mentioned concepts are taken into account, the instantaneous frequencies, ratios of frequencies and amplitudes of the components can be computed and used to interpret the pertinence of the analysis in a geologically meaningful way. The instantaneity and independence of frequency and amplitude so obtained open a new way of performing time-series analysis.

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Identification of Linear Time-Varying Dynamical Systems Using Hilbert Transform and Empirical Mode Decomposition Method

Journal of Applied Mechanics ◽

10.1115/1.2188538 ◽

2006 ◽

Vol 74 (2) ◽

pp. 223-230 ◽

Cited By ~ 36

Author(s):

Z. Y. Shi ◽

S. S. Law

Keyword(s):

Dynamical Systems ◽

Empirical Mode Decomposition ◽

Hilbert Transform ◽

Decomposition Method ◽

Degrees Of Freedom ◽

Linear Time ◽

Time Varying ◽

Mode Decomposition ◽

Empirical Mode Decomposition Method ◽

The Hilbert Transform

This paper addresses the identification of linear time-varying multi-degrees-of-freedom systems. The identification approach is based on the Hilbert transform and the empirical mode decomposition method with free vibration response signals. Three-different types of time-varying systems, i.e., smoothly varying, periodically varying, and abruptly varying stiffness and damping of a linear time-varying system, are studied. Numerical simulations demonstrate the effectiveness and accuracy of the proposed method with single- and multi-degrees-of-freedom dynamical systems.

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Empirical Assay of the Use of the Hilbert-Huang Transform for the Spectral Analysis of Storm Waves

Volume 2: Structures, Safety and Reliability ◽

10.1115/omae2008-57461 ◽

2008 ◽

Author(s):

Joaqui´n Ortega ◽

George H. Smith

Keyword(s):

Empirical Mode Decomposition ◽

Hilbert Transform ◽

Instantaneous Frequency ◽

Sampling Rate ◽

Stationary Time Series ◽

Hilbert Spectrum ◽

Hilbert Huang Transform ◽

Storm Waves ◽

Mode Decomposition ◽

The Hilbert Transform

The Hilbert-Huang Transform (HHT) was proposed by Huang et al. [2] as a method for the analysis of non-linear, non-stationary time series. This procedure requires the decomposition of the signal into intrinsic mode functions using a method called empirical mode decomposition. These functions represent the essential oscillatory modes contained in the original signal. Their characteristics ensure that a meaningful instantaneous frequency is obtained through the application of the Hilbert Transform. The Hilbert Transform is applied to each intrinsic mode function and the amplitude and instantaneous frequency for every time-step is computed. The resulting representation of the energy in terms of time and frequency is defined as the Hilbert Spectrum. In previous work [7] using the HHT for the analysis of storm waves it has been observed that the number of IMFs needed for the decomposition and the amount of energy associated to different IMFs differ from what has been observed for the analysis of waves under ‘normal’ sea conditions by other authors. In this work we explore in detail the effect that the sampling rate has in the empirical mode decomposition and in the Hilbert Spectrum for storm waves. The results show that the amount of energy associated to different IMFs varies with the sampling rate and also that the number of IMFs needed for the empirical mode decomposition changes with record length.

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Output-Only Modal Analysis Based on Improved Empirical Mode Decomposition Method

Advances in Materials Science and Engineering ◽

10.1155/2015/945862 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12 ◽

Cited By ~ 3

Author(s):

Shiqiang Qin ◽

Qiuping Wang ◽

Juntao Kang

Keyword(s):

Modal Analysis ◽

Empirical Mode Decomposition ◽

Modal Parameters ◽

Band Pass Filter ◽

Pass Filter ◽

Random Decrement Technique ◽

Mode Decomposition ◽

Frequency Components ◽

Output Only ◽

Emd Method

The output-only modal analysis for bridge structures based on improved empirical mode decomposition (EMD) is investigated in this study. First, a bandwidth restricted EMD is proposed for decomposing nonstationary output measurements with close frequency components. The advantage of bandwidth restricted EMD to standard EMD is illustrated by a numerical simulation. Next, the modal parameters are extracted from intrinsic mode function obtained from the improved EMD by both random decrement technique and stochastic subspace identification. Finally, output-only modal analysis of a railway bridge is presented. The study demonstrates the mode mixing issues of standard EMD can be restrained by introducing bandwidth restricted signal. Further, with the improved EMD method, band-pass filter is no longer needed for separating the closely spaced frequency components. The modal parameters extracted based on the improved EMD method show good agreement with those extracted by conventional modal identification algorithms.

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