Time-varying system identification by enhanced Empirical Wavelet Transform based on Synchroextracting Transform
Introduction
Dynamic behaviors of engineering structures often change over time due to the environmental condition changes, mass and stiffness changes due to the material loss or strength degradation, and the effects of extreme loads, etc. These time-varying system effects can be widely observed in many engineering fields, i.e. civil and mechanical engineering. For example, the friction mechanisms used in industry can introduce the changes in the stiffness and damping of a structure under normal operations. Civil structures may also exhibit time varying vibration characteristics under earthquake, tornados and hurricanes, because of the nonlinearities in the structures, and the changes in the stiffness and boundary conditions [1]. Therefore, identifying the vibration characteristics of time-variant structures is vital for researchers and engineers to understand and assess the operational conditions of structures.
Over the past decades, system identification of time-variant structures based on the measured vibration responses (i.e. acceleration, displacement responses) has obtained a broad attention. Various techniques have been developed and reported in the literatures [2], [3]. Generally, these methods could be classified into two categories: (1) time-varying system identification based on adaptive algorithms [4], [5], [6]; (2) Time-varying system identification by using time-frequency analysis techniques [7], [8], [9]. For example, Wang et al. [3] used a least-squares (LS) parameter estimation method with slide-window function to track the real-time frequency of a high-voltage switch structure under the cyclic loading excitations in the laboratory. Yang et al. [6] proposed a novel LS parameter estimation approach to adaptively track the system parameters of a time-variant structure. In addition, in the literature [10], an improved LS strategy is developed to identify the hysteretic parameters of a nonlinear system under arbitrary external excitations.
In recent years, time-frequency analysis techniques have been widely conducted for system identification of time-variant structures, i.e. by using Hilbert Transform (HT) [11], [12] and Wavelet Transform (WT) [13], [14], [15]. Shi et al. [16] applied Empirical Mode Decomposition (EMD) with HT for the modal parameter identification of a time-varying multi-degree-of-freedom (MDOF) system. Bao et al. [2] developed an improved Hilbert-Huang Transform (HHT) method for time-varying system identification by using the autocorrelation functions of structural dynamic responses as the input to HHT, and therefore reduced the noise effect and improved the accuracy of identification. Wang et al. [17] proposed a recursive HT system identification approach, which have been successfully used to track the real-time structural characteristics of linear shear-type buildings under the forced vibration. WT is an alternative time-frequency analysis approach, which has been widely used for the system identification of linear and non-linear structures. Hou et al. [18] developed a novel approach for instantaneous modal identification of a time-varying structure subjected to an earthquake excitation based on continuous wavelet transform (CWT). Wang et al. [7] used the extracted wavelet ridges to effectively identify the instantaneous frequency (IF) of a cable structure with different tension forces under the stochastic excitations.
More recently, a new time-frequency analysis technique, named Synchrosqueezing Transform (SST), has been developed by Daubechies et al. [19], and has been applied for IF identification [20], [21]. The main advantage of SST is that it squeezes the time-frequency coefficients into the IF trajectory, which can be approximated to an ideal tine-frequency analysis representation. However, SST has a lower time-frequency resolution when it is used to reconstruct the interested components of a non-stationary signal. Based on the theory of SST, a novel time-frequency analysis method, namely Synchroextracting Transform (SET), have been developed by Yu et al. [22], which can generate a more energy-concentrated analysis result than using SST.
In this study, time-frequency representation based on SET is employed to detect the filtering boundaries of the Empirical Wavelet Transform (EWT) process [23] for non-stationary signal analysis. In the past studies, several modified EWT methods have been successfully applied for operational modal identification [24], [25]. However, to the authors’ best knowledge, there has been no study yet on using or improving EWT method for IF identification of time-varying structures. With the vibration responses measured from a time-varying structure, time-frequency analysis based on SET is first performed to determine the filtering boundaries of EWT instead of using the ordinary Fourier Spectrum. Then EWT is applied to extract the individual modes from the vibration response signals. Each mode is an amplitude-modulation and frequency-modulation signal with a narrow-band property with a varying IF. The IF of each time-variant component can be identified by using HT. A synthetic signal which consists of two time-varying frequency components is first used to verify the feasibility and accuracy of the proposed approach. Then the proposed method is employed to identify the IF of a two-storey shear-type building under the forced vibration. Experimental studies on a real bridge under the heavy traffic loads are conducted to further validate the effectiveness of the proposed method.
The remainder of this paper is organized as follows. Section 2 briefly explains the principle of EWT and SET, and provides a fundamental process of time-varying system identification based on the proposed approach. In Section 3, numerical studies on a synthetic signal and a two-storey time-varying structure are conducted to investigate the accuracy and effectiveness of the proposed approach. In Section 4, Experimental verifications on a highway bridge under the traffic loads are performed to identify the instantaneous frequencies. Section 5 provides the discussions and conclusions on the obtained results.
Section snippets
Empirical Wavelet Transform (EWT)
Using the traditional EWT method for vibration signal decomposition consists of two main steps: (1) Segmenting the Fourier spectrum of the target vibration signal; (2) Constructing the filtering bank, and processing each segmental part of the signal. To determine the filtering banks of the EWT method, the local peaks of the Fourier spectrum are firstly identified. The lowest local minima between the two sequential peaks are detected, which are defined as the boundaries of each filtering bank.
A simulation signal
In this section, a simulated signal, as defined in Eq. (24), is used to investigate the effectiveness of using SET to determine the boundaries for EWT analysis. It consists of two time-variant frequency components and which are described in Eqs. (25), (26), respectively.
To further validate the feasibility of using SET to improve the performance of EWT, a high-level noise, that is, 20% Gaussian white noise, is added to
A highway bridge
To further validate the performance of using the improved EWT process for structural time-varying dynamic characteristic identification, experimental studies on an operational highway bridge are conducted. The target bridge consists of three spans, which is shown in Fig. 15. The beams are 17.10 m long in the 1st and 3rd spans, and the central-span beam is 16.96 m long with two half joints at the ends. The half joints shown in Fig. 15(b) have been strengthened by using external vertical steel
Conclusions
This paper proposes an enhanced EWT approach based on SET for the time varying system identification. The time-frequency analysis of a vibration signal is performed by using SET to determine the filtering boundaries of EWT analysis instead of using Fourier Spectrum. An enhanced EWT method is developed to separate the vibration signal into several IMFs based on the predefined filtering boundaries. When IMFs of a vibration signal are obtained, HT can be conducted to identify and extract IF of
Acknowledgements
The work described in this paper was financially supported by China Scholarship Council Postgraduate Scholarship No. 201606690031, and Australia Research Council Linkage project LP160100528. The support and permission granted by Main Roads Western Australia (MRWA) to use the data and publish the information in this study are thankfully acknowledged. Conclusions drawn from the data provided by MRWA are those of the authors based on the conducted analyses/evaluations and do not necessarily
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