Zuo-Cai Wang

Analytical mode decomposition of time series with decaying amplitudes and overlapping instantaneous frequencies

In this study, the recently developed analytical mode decomposition with Hilbert transform was extended to the decomposition of a non-stationary and nonlinear signal with two or more amplitude-decaying and frequency-changing components. The bisecting frequency in the analytical mode decomposition became time-varying, and could be selected between any two adjacent instantaneous frequencies estimated from a preliminary wavelet analysis. The mathematical foundation for this new extension was integration of the bisecting frequency over time so that the original time series is actually decomposed in the phase domain. Parametric studies indicated that the analytically derived components are insensitive to the selection of bisecting frequency and the presence of up to 20% noise, sufficiently accurate when the sampling rate meets the Nyquist–Shannon sampling criterion, and applicable to both narrowband and wideband frequency modulations even when the signal amplitude decays over time. The proposed analytical mode decomposition is superior to the empirical mode decomposition and wavelet analysis in the preservation of signal amplitude, frequency and phase relations. It can be directly applied for system identification of buildings with time-varying stiffness.

Strain modes based dynamic displacement estimation of beam structures with strain sensors

This paper presents a new technique to estimate the dynamic displacement based on strain mode shapes of beam structures with strain sensors. Strain mode shapes are first estimated from the cross-correlation function of the measured dynamic strain data. Then, the displacement mode shapes can be estimated from the strain mode shapes based on the displacement-strain relation. For an oscillating beam structure under service conditions, the dynamic response can be expressed as the superposition of their corresponding mode shapes weighed by the corresponding modal coordinates. Thus, by knowing the strain mode shapes and strain data, the corresponding modal coordinates can be estimated. The dynamic displacement can then be estimated by the obtained displacement mode shapes and modal coordinates. This method is verified by numerical simulations of a simply supported beam subjected to impulsive excitation and earthquake excitation. Experimental tests of a simply supported beam under various hammering excitations are also conducted to verify the effectiveness of the proposed method. Both the numerical and test results show that the proposed method can estimate the dynamic displacement of beam structures with high accuracy.

Time-Varying Linear and Nonlinear Structural Identification with Analytical Mode Decomposition and Hilbert Transform

AbstractAnalytical mode decomposition (AMD) of a time series concerning any preselected bisecting frequency with Hilbert transform has been developed for closely spaced multicomponent signal decomposition. For this class of structures, it is often challenging, if not impossible, to apply empirical mode decomposition. In this study, the instantaneous structural frequencies are directly derived from the decomposed modal responses for systems with single and multiple degrees of freedom with both free and force vibrations, based on AMD combined with Hilbert transform analysis. The results show that the slow varying component of the instantaneous frequency of the signal is approximately equal to the instantaneous frequency of the systems for slowly time varying linear or weakly nonlinear structures. Both numerical simulations and experimental tests show that the proposed method is capable of tracking the frequency variations with high accuracy for time varying linear structures or weakly nonlinear structures.

Deflection Estimation of Bending Beam Structures Using Fiber Bragg Grating Strain Sensors

This paper uses strains measured by fiber Bragg grating (FBG) sensors to estimate the static or dynamic deflection curve of bending beam structures. The deflection estimation method is only based upon the geometric equations of a beam structure without knowing the material information. At each cross section of a beam structure, two FBG strain sensors are installed, and the curvature at the cross section can be estimated by using the two measured strains from the geometric equation. Then the curvature function is assumed as a polynomial function and the coefficients can be estimated using least squares method. Finally, the deflection curve is estimated by integrating the curvature function twice. For dynamic deflection estimation, since only the geometric equations are used, and at each time step, the geometric equations can be also used for a dynamic system. Therefore, at each time step, the deflection can be estimated using the proposed method and the dynamic deflection can be finally obtained. The method is verified by the simulations of a continuous beam under static loads and experimental tests of a simply supported beam under various static loads. A simply support beam under moving loads is also simulated to verify the method for dynamic deformation estimation. All the numerical and test results show that the method can estimate the static and dynamic deflection curve of beam structures with a high accuracy.

Suboptimal Rayleigh damping coefficients in seismic analysis of viscously-damped structures

An optimization method for the consistent evaluation of two Rayleigh damping coefficients is proposed. By minimizing an objective function such as an error term of the peak displacement of a structure, the two coefficients can be determined with response spectral analysis. The optimization method degenerates into the conventional method used in current practices when only two modes of vibration are included in the objective function. Therefore, the proposed method with all significant modes included for simplicity in practical applications results in suboptimal damping coefficients. The effects of both spatial distribution and frequency content of excitations as well as structural dynamic characteristics on the evaluation of Rayleigh damping coefficients were investigated with a five-story building structure. Two application examples with a 62-story high-rise building and a 840 m long cable-stayed bridge under ten earthquake excitations demonstrated the accuracy and effectiveness of the proposed method to account for all of the above effects.

Hilbert low-pass filter of non-stationary time sequence using analytical mode decomposition

In this paper, the recently developed analytical mode decomposition with a constant or time-varying cutoff frequency is extended into the decomposition of a non-stationary discrete time sequence. The discretization of the signal and the selection of the cutoff frequency may cause the failure of low frequency component extraction. In this study, to eliminate the effects of the signal discretization, the one-step, two-step, and four-step low-pass filters with cutoff frequencies are proposed. Based on the theoretical derivation, the previous one-step low-pass filter is effective only when the cutoff frequency is not greater than a quarter of the sampling frequency and the maximum frequency of the signal not greater than a half of the sampling frequency. In this study, if the cutoff frequency is less than or equal to a quarter of the sampling frequency, a two-step low-pass filter is proposed to extract the low frequency component. If the cutoff frequency is greater than a quarter of the sampling frequency, a four-step low-pass filter with frequency shifting process is proposed. When the time-varying cutoff frequency is not always larger than or less than a quarter of the sampling frequency, a sufficient condition, which is the sampling frequency is greater than four times of the maximum frequency of the signal component, is provided in this study. Two numerical examples are used to validate the effectiveness of the proposed low-pass filters. Both the theoretic derivation and numerical simulations show that the proposed filters can analytical extract the discrete low frequency component with an appropriate cutoff frequency.

Time-varying system identification of high voltage switches of a power substation with slide-window least-squares parameter estimations

This paper is aimed at identifying the time-varying parameters and ultimate behavior of high voltage switch structures based on a series of full-scale shake table tests with harmonic excitations. Each structure involves a mechanical device for switch-on and switch-off, a friction-based switch, and three porcelain pillars. To identify the structural properties over time, a novel slide-window least-squares estimation method is developed. Each time-varying parameter is firstly approximately expressed by a simple polynomial or exponential function with time in a short slide-window. The time-invariant coefficients of the polynomial or exponential function are then estimated using a least-squares method. Finally, the time-varying parameters can be simply calculated from the estimated polynomial or exponential function. The proposed method is validated by simulated one- and two-story buildings with three kinds of time-varying parameters (stiffness varying abruptly, gradually, and periodically) under earthquake excitations. The application of the proposed method to the tested switch structures demonstrated that the time-varying fundamental frequency of the structures decreased from 7.5 to 6.5 Hz near resonance, which is consistent with the shake table test observations under an excitation of 1.27 and 2.54 mm in stroke. During the shake table tests, all switch structures failed at the bottom of the mechanical device under cyclic loading.

Structural Modal Parameter Identification from Forced Vibration with Analytical Mode Decomposition

In this paper, a recently developed analytical mode decomposition method is proposed for modal parameters identification of structures subjected to impulsive, harmonic, and ambient excitations. The decomposed modal response for a structure subjected to an impulsive load is free vibration, thus, the instantaneous amplitude and phase angle of each decomposed modal response can be directly used to identify the modal parameters by using the least-squares fit procedure. For a structure subjected to a harmonic load, the transient response is first extracted from the measured response. Then, the extracted transient response can be decomposed into modal responses and the modal parameters can be evaluated. The proposed method in combination with the conventional random decrement technique is proposed for modal parameter identification for structures under ambient vibration. The random decrement technique is used to extract the free vibration information from which modal parameters are evaluated. A 3 degree-of-freedom mechanical system with closely-spaced modes subjected to impulsive, harmonic, and ambient loads is simulated as a numerical example. The new method is then validated with shake table testing of a 3-story building frame subjected to white noise and earthquake excitations. Both experiments and simulations showed high accuracy and effectiveness of the new method for structural modal parameters identification.

Time–frequency analysis and applications in time-varying/nonlinear structural systems: A state-of-the-art review:

Nonlinear dynamic behaviors of civil engineering structures have been observed not only under extreme loads but also during normal operations. Characterization of the time-varying property or nonlinearity of the structures must account for temporal evolution of the frequency and amplitude contents of nonstationary vibration responses. Neither time analysis nor frequency analysis method alone can fully describe the nonstationary characteristics. In this article, an attempt is made to review the milestone developments of time–frequency analysis in the past few decades and summarize the fundamental principles and structural engineering applications of wavelet analysis and Hilbert transform analysis in system identification, damage detection, and nonlinear modeling. This article is concluded with a brief discussion on challenges and future research directions with the application of time–frequency analysis in structural engineering.

Stationarity test of vibration signals with surrogate data and time–frequency analysis:

The stationarity test of vibration signals is critical for the extraction of the signal features. In this article, the surrogate data with various time–frequency analysis methods are proposed for stationary test of vibration signals. The surrogate data are first generated from the Fourier spectrum of the original signal with keeping the magnitude of the spectrum unchanged and replacing its phase by a random sequence. The local and global spectra of the original signal and the surrogate data are then estimated by four time–frequency analysis methods, which are short-time Fourier transform, multitaper spectrograms, wavelet transform, and S-transform methods. The index of nonstationarity is then defined based on the distances between the local and global spectra. Three kinds of synthetic signals, which are stationary signals, frequency-modulated signals, and amplitude-modulated signals, are tested to compare the efficiency of the four time–frequency analysis methods as mentioned. The results show that with a certain observation scale value, the index of nonstationarity based on the short-time Fourier transform or wavelet transform method may fail to test the stationarity of the signal. The parametric studies and sensitivity analysis of the observation scale and noise-level effect are also extensively conducted. The results show that the index of nonstationarity calculated using the multitaper spectrograms’ method is more suitable for stationarity test of frequency-modulated signals, while the index of nonstationarity calculated using the S-transform method is more suitable for stationarity test of amplitude-modulated signals. The results also show that the noise has a significant effect on the stationarity test results. Finally, the stationarity of a real vibration signal measured from a cable is tested, and the results show that the proposed index of nonstationarity can effectively test the stationarity of real vibration signals.