Wavelet‐based approach of time series model for modal identification of a bridge with incomplete input

Identification of modal parameters of a bridge from its earthquake responses is crucial for performing damage assessment of the structure. However, all the input base excitations of the bridge may not be measured because of economic concerns and sensor malfunctions. Consequently, evaluating the modal parameters of a bridge under the consideration of incomplete input measurements is a challenging and important task. An approach that combines the continuous Cauchy wavelet transform with an autoregressive time‐varying moving average with exogenous input (AR‐TVMA‐X) model is proposed in this study to identify the modal parameters of a multispan bridge under multiple support earthquake excitations with incomplete measurements. The efficiency and efficacy of the proposed approach are first validated using numerically simulated responses of a three‐span continuous beam subjected to multiple support nonstationary excitations. A standard procedure of using the proposed approach to identify the modal parameters is established according to comprehensive studies on the effects of noise in the data, the number of supports whose excitations are used in the AR‐TVMA‐X model, and the orders of the AR‐TVMA‐X model on the accuracy of identifying the modal parameters. This procedure is further applied to process the earthquake responses of a two‐span cable‐stayed 510‐m‐long bridge to demonstrate the engineering applicability of the proposed approach.     

Analytic wavelet transformation‐based modal parameter identification from ambient responses

Analytic wavelet transform (AWT) based on Gabor wavelet function overcomes the deficiency of the time‐domain localization of traditional Fourier transform and the limitation of the constant resolution in the time‐frequency domain of short‐time Fourier transform. The identification of modal parameters of structures may be carried out by both the amplitude and phase frequency information revealed by resorting to matching mechanism between the wavelet function and complex‐valued signal. By applying the AWT in conjunction with the well‐known random decrement technique, this paper analyses the time‐frequency resolution of Gabor wavelet and the process of identifying structural modal parameters. The method of selecting the parameters of Gabor wavelet function and the formula determining the usable lengths of signal are thus proposed. Eventually, the efficiency of the present method is confirmed by applying it to a numerical simulation data without and with noise contamination of a three degree‐of‐freedom (3DOF) structure with the closely spaced natural frequencies and to ambient vibration full‐scale measurements of a super high‐rise building—Shanghai Jin Mao Building excited by wind. Copyright © 2010 John Wiley & Sons, Ltd.     

Vibration based structural health monitoring: Wavelet packet transform based solution

One prominent problem for vibration-based structural health monitoring is to extract condition indices which are sensitive to damage and yet insensitive to measurement noise. In this paper, a condition index extraction method based on the wavelet packet transform (WPT) is proposed. This transform leads to the formulation of a novel condition index: wavelet packet signature (WPS). The sensitivity of the WPS to the change of structural parameters is derived and validated on a five-degrees-of-freedom spring-mass system. Results show that the WPS is significantly more sensitive to the stiffness change than the natural frequencies and the mode shapes. Its sensitivity is slightly better or comparable to that of the modal flexibility matrices depending on the location of damage. A variability analysis is also performed to study the effect of measurement noise on the proposed WPS. Results show that the WPS does not show any significant variation even under the presence of 10 dB noise. To illustrate the potential of the WPS, a damage indicator is formulated and used to monitor the health condition of the structural system. An experimental study on a three-storey frame shows that when incorporated with a statistical process control approach, the WPS-based damage indicator can distinctly identify the presence of damage in the system.     

Identification of Instantaneous Modal Parameter of Time‐Varying Systems via a Wavelet‐Based Approach and Its Application

This work presents an efficient approach using time‐varying autoregressive with exogenous input (TVARX) model and a substructure technique to identify the instantaneous modal parameters of a linear time‐varying structure and its substructures. The identified instantaneous natural frequencies can be used to identify earthquake damage to a building, including the specific floors that are damaged. An appropriate TVARX model of the dynamic responses of a structure or substructure is established using a basis function expansion and regression approach combined with continuous wavelet transform. The effectiveness of the proposed approach is validated using numerically simulated earthquake responses of a five‐storey shear building with time‐varying stiffness and damping coefficients. In terms of accuracy in determining the instantaneous modal parameters of a structure from noisy responses, the proposed approach is superior to typical basis function expansion and regression approach. The proposed method is further applied to process the dynamic responses of an eight‐storey steel frame in shaking table tests to identify its instantaneous modal parameters and to locate the storeys whose columns yielded under a strong base excitation.     

A Wavelet-Based Approach to Identifying Structural Modal Parameters from Seismic Response and Free Vibration Data

Abstract:   This work presents the use of a discrete wavelet transform to determine the natural frequencies, damping ratios, and mode shapes of a structure from its free vibration or earthquake response data. The wavelet transform with orthonormal wavelets is applied to the measured acceleration responses of a structural system, and to reconstruct the discrete equations of motion in various wavelet subspaces. The accuracy of this procedure is numerically confirmed; the effects of mother wavelet functions and noise on the ability to accurately estimate the dynamic characteristics are also investigated. The feasibility of the present procedure to elucidate real structures is demonstrated through processing the measured responses of steel frames in shaking table tests and the free vibration responses of a five-span arch bridge with a total length of 440 m.     

Blind Modal Identification in Frequency Domain Using Independent Component Analysis for High Damping Structures with Classical Damping

Output‐only modal identification methods are practical for large‐scale engineering. Recently, independent component analysis (ICA) which is one of the most popular techniques of blind source separation (BSS) has been used for output‐only modal identification to directly separate the modal responses and mode shapes from vibration responses. However, this method is only accurate for undamped or lightly damped structures. To improve the performance of ICA for high damping structures, this article presents an extended ICA‐based method called ICA‐F, which establishes a BSS model in frequency domain. First, the basic idea of BSS and ICA applied in modal identification is introduced in detail. The free vibration responses and the correlation functions of ambient responses can be cast into the frequency‐domain BSS framework just by mapping the time history responses to frequency domain through fast Fourier transform (FFT). Then, an ICA‐based method in frequency domain called ICA‐F is proposed to accurately extract mode shapes and modal responses for both light and high damping structures. A simulated 3 degree of freedom mass‐spring system and a 4‐story simulated benchmark model developed by the IASC‐ASCE Task Group in Health Monitoring are employed to verify the effectiveness of the proposed method. The results show that the proposed method can perform accurate modal identification for both light and high damping structures. Finally, the IASC‐ASCE experimental benchmark structure is also utilized to illustrate the proposed method applied to practical structure.     

Wavelet-Based Structural Health Monitoring of Earthquake Excited Structures

Abstract:   The article presents a wavelet-based structural health monitoring technique for structures subjected to an earthquake excitation utilizing the instantaneous modal information. The instantaneous mode shape information is first extracted from the vibration response data collected during an earthquake event by using a wavelet packet sifting process. A confidence index (CI) is proposed to validate the results obtained. The identified normalized instantaneous mode shapes in conjunction with the corresponding CIs can be effectively used to monitor damage development in the structure. The effectiveness of the proposed approach is illustrated for two damage scenarios, sudden stiffness loss and progressive stiffness degradation, and different base excitations including three real earthquake signals and a random signal. Consistently good results were obtained in all cases. Issues related to robustness of the method in the presence of a measurement noise and sensitivity to damage severity are discussed.     

多段微差爆破振动信号频带能量分布特征的小波包分析

爆破振动分析是研究爆破振动危害控制的基础,也是控制爆破振动危害的前提。根据爆破振动信号具有短时非平稳的特点,利用小波包分析技术对满足分析要求的多段微差爆破振动信号的能量分布特征进行研究。首先,简略地介绍了小波变换与小波包分析的特点;其次,对6条多段微差爆破振动信号进行小波包分析,得到了爆破振动信号在不同频带上的能量分布图;最后,总结了多段微差爆破振动信号频带能量的分布特征。该分析手段为综合研究爆破地震效应特别是为将来构建振动速度–频率相关安全准则提供了一种有效的分析技术。     

On‐line system identification of structures using wavelet‐Hilbert transform and sparse component analysis

A new wavelet‐Hilbert transform based sparse component analysis (WHT‐SCA) method is presented for online system identification in indeterminate conditions. The instantaneous phase ratios of output signals are obtained by using a wavelet‐Hilbert transform based filter; and the out‐of‐phase data, that causes errors in identification accuracy, is detected and eliminated. Then, modal parameters of the structure are identified through existing relationships between the dispersion of filtered data in the frequency domain. Subsequently, to demonstrate the capability of the online identification, a new controller is introduced by combining the WHT‐SCA and a semi‐active tuned mass damper (STMD), resulting in creation of smart structures. The performance of the proposed method and controller is investigated through examples. The results demonstrate that, modal parameters of structures are identified accurately even with noise contamination and limited number of sensors. Also, the STMD is effectively robust against any variations in modal parameters of the structure.     

小波包分析在结构损伤监测中的应用

小波分析具有良好的时频局部化性质,特别适合于分析和处理突变信号。在获得结构的动力响应的基础上,对结构响应信号做小波包分解。根据各种响应信号对损伤的灵敏度,选择损伤特征,通过捕捉结构出现损伤的时刻,实现对结构损伤时刻监控。为模拟测试实际结构响应噪声的影响,在第一层加速响应信号中加入信噪比为5:1的白噪声,运用小波包消噪后再运用小波包分解识别结构的损伤时刻。     

A Comparative Study of Modal Parameter Identification Based on Wavelet and Hilbert–Huang Transforms

Abstract:   This article presents a comparative study of the modal parameter identification of structures based on the continuous wavelet transform (WT) using the modified complex Morlet wavelet function and the improved Hilbert–Huang transform (HHT). Special attention is given to some implementation issues, such as the modal separation and end effect in the WT, the optimal parameter selection of the wavelet function, the new stopping criterion for the empirical mode decomposition (EMD) and the end effect in the HHT. The capabilities of these two techniques are compared and assessed by using three examples, namely a numerical simulation for a damped system with two very close modes, an impact test on an experimental model with three well-separated modes, and an ambient vibration test on the Z24-bridge benchmark problem. The results demonstrate that for the system with well-separated modes both methods are applicable when the time–frequency resolutions are sufficiently taken into account, whereas for the system with very close modes, the WT method seems to be more theoretical and effective than HHT from the viewpoint of parameter design.     

延拓小波变换识别的桥梁模态参数研究

振动系统模态识别是当今桥梁结构动力特性研究的热点之一。从复模态理论的一般阻尼系统的模态参数分析入手,利用径向神经网络插值技术,对含有噪声的振动信号进行信号预测延拓降噪处理,借助连续的Morlet小波变换,识别出了振动结构系统的模态。以重庆大佛寺长江大桥为研究背景,使用模态叠加法和Morlet小波分析识别结构,二者吻合程度较高。研究结果表明,基于径向神经网络的延拓预测的信号降噪效果好;Morlet小波变换识别模态参数精度满足工程要求。     

爆心距对爆破振动信号频带能量分布的影响

根据爆破振动信号具有短时非平稳的特点,利用小波包分析技术对满足分析要求的多段微差爆破振动信号的能量分布特征进行研究。首先,简略地介绍了小波变换与小波包分析的特点。其次,基于MATLAB对爆破振动信号进行小波包分析,得到了爆破振动信号在不同频带上的能量分布图。最后,总结了爆破振动信号频带能量的分布规律,重点探讨了爆心距对爆破振动信号频带能量分布的影响。结果表明,爆破震动信号在传播过程中,其主振频带有往低频发展的趋势,且宽度增加。     

Wavelet-based iterative data enhancement for implementation in purification of modal frequency for extremely noisy ambient vibration tests in Shiraz-Iran

The main purpose of the present study is to enhance high-level noisy data by a wavelet-based iterative filtering algorithm for identification of natural frequencies during ambient wind vibrational tests on a petrochemical process tower. Most of denoising methods fail to filter such noise properly. Both the signal-to-noise ratio and the peak signal-to-noise ratio are small. Multiresolution-based one-step and variational-based filtering methods fail to denoise properly with thresholds obtained by theoretical or empirical method. Due to the fact that it is impossible to completely denoise such high-level noisy data, the enhancing approach is used to improve the data quality, which is the main novelty from the application point of view here. For this iterative method, a simple computational approach is proposed to estimate the dynamic threshold values. Hence, different thresholds can be obtained for different recorded signals in one ambient test. This is in contrast to commonly used approaches recommending one global threshold estimated mainly by an empirical method. After the enhancements, modal frequencies are directly detected by the cross wavelet transform (XWT), the spectral power density and autocorrelation of wavelet coefficients. Estimated frequencies are then compared with those of an undamaged-model, simulated by the finite element method.     

基于小波变换的爆破振动时频特征分析

应用小波变换方法对短时非平稳爆破振动过程提出了时频特征分析。根据离散小波变换的分层分解展开关系，将爆破振动时间历史信号用分层重构信号进行扫描。应用这些信号可以给出不同频率带上爆破振动的相对能量分布和振动强度的时间变化规律。一个爆破振动实测结果的分析表明，与建立在传统Fourier变换基础上的频谱分析方法相比，基于小波变换的爆破振动时频特征分析可以给出更为准确的细节信息。文中的研究结果为爆破振动结构安全性分析提供了新的途径。     

基于小波包分析的建(构)筑物爆破振动安全判据研究

基于现场实测爆破振动数据,采用小波包分析技术对爆破振动信号进行了时频特征分析。根据小波包变换的分层分解关系,推导出爆破振动信号不同频带的小波包频带能量,小波包频带能量能同时反映爆破振动三要素(振动的强度、频率和持续时间)的作用影响。从结构动力响应角度,探讨了受控建(构)筑物本身的固有特性对爆破振动动态响应的影响,获得了受控建(构)筑物在爆破振动作用下的频带响应系数。首次建立了能考虑爆破振动的强度、频率和持续时间以及受控建(构)筑物本身的动态响应特性(固有频率和阻尼比)等因素综合的安全判据--响应能量判据,并用工程实例验证了该判据的可行性和可靠性。     

New method for modal identification of super high‐rise building structures using discretized synchrosqueezed wavelet and Hilbert transforms

Measured signals obtained by sensors during dynamic events such as earthquake, wind, and wave contain nonlinear, nonstationary, and noisy properties. In this paper, a new approach is presented for modal parameter identification of structures particularly suitable for very large real‐life structures such as super high‐rise building structures based on the integration of discretized synchrosqueezed wavelet transform, the Hilbert transform, and the linear least‐square fit. Its effectiveness is demonstrated first by application to a two‐dimensional frames from the literature, and then to the 123‐story Lotte World Tower (LWT) under construction in Seoul, Korea. The LWT measurements are very low‐amplitude ambient vibrations. Extracting the natural frequencies and damping ratios from such low‐amplitude signals are known to be very challenging. Further, the new methodology was compared with the empirical mode decomposition. It is demonstrated that the new method is capable of extracting both natural frequencies and damping rations from low‐amplitude signals effectively and with a higher accuracy compared with the empirical mode decomposition approach. The results of this research indicate a super high‐rise building like LWT has a damping ratio in the range 0.7–3.4%. The new method is quite promising for practical implementations of health monitoring of large real‐life structures.     

Enhanced Hilbert–Huang transform and its application to modal identification

The well‐known Hilbert–Huang transform (HHT) consists of empirical mode decomposition to extract intrinsic mode functions (IMFs) and Hilbert spectral analysis to obtain time–frequency characteristics of IMFs through the Hilbert transform. There are two mathematical requirements that limit application of the Hilbert transform. Moreover, noise effects caused by the empirical mode decomposition procedure add a scatter to derivative‐based instantaneous frequency determined by the Hilbert transform. In this paper, a new enhanced HHT is proposed in which by avoiding mathematical limitations of the Hilbert spectral analysis, an additional parameter is employed to reduce the noise effects on the instantaneous frequencies of IMFs. To demonstrate the efficacy of the proposed method, two case studies associated with structural modal identification are selected. In the first case, through identification of a typical 3‐DOF structural model subjected to a random excitation, accuracy of the enhanced method is verified. In the second case, ambient response data recorded from a real 15‐story building are analyzed, and nine modal frequencies of the building are identified. The case studies indicate that the enhanced HHT provides more accurate and physically meaningful results than HHT and is capable to be an efficient tool in structural engineering applications. Copyright © 2012 John Wiley & Sons, Ltd.     

Wavelet Packet-Autocorrelation Function Method for Traffic Flow Pattern Analysis

Abstract:   Accurate and timely forecasting of traffic flow is of paramount importance for effective management of traffic congestion in intelligent transportation systems. A detailed understanding of the properties of traffic flow is essential for building a reliable forecasting model. The discrete wavelet packet transform (DWPT) provides more coefficients than the conventional discrete wavelet transform (DWT), representing additional subtle details of a signal. In wavelet multiresolution analysis, an important decision is the selection of the decomposition level. In this research, the statistical autocorrelation function (ACF) is proposed for the selection of the decomposition level in wavelet multiresolution analysis of traffic flow time series. A hybrid wavelet packet-ACF method is proposed for analysis of traffic flow time series and determining its self-similar, singular, and fractal properties. A DWPT-based approach combined with a wavelet coefficients penalization scheme and soft thresholding is presented for denoising the traffic flow. The proposed methodology provides a powerful tool in removing the noise and identifying singularities in the traffic flow. The methods created in this research are of value in developing accurate traffic-forecasting models .     

Hybrid Time‐Frequency Blind Source Separation Towards Ambient System Identification of Structures

Abstract: Ambient system identification in noisy environments, in the presence of low‐energy modes or closely‐spaced modes, is a challenging task. Conventional blind source separation techniques such as second‐order blind identification (SOBI) and Independent Component Analysis (ICA) do not perform satisfactorily under these conditions. Furthermore, structural system identification for flexible structures require the extraction of more modes than the available number of independent sensor measurements. This results in the estimation of a non‐square modal matrix that is spatially sparse. To overcome these challenges, methods that integrate blind identification with time‐frequency decomposition of signals have been previously presented. The basic idea of these methods is to exploit the resolution and sparsity provided by time‐frequency decomposition of signals, while retaining the advantages of second‐order source separation methods. These hybrid methods integrate two powerful time‐frequency decompositions—wavelet transforms and empirical mode decomposition—into the framework of SOBI. In the first case, the measurements are transformed into the time‐frequency domain, followed by the identification using a SOBI‐based method in the transformed domain. In the second case, a subset of the operations are performed in the transformed domain, while the remaining procedure is conducted using the traditional SOBI method. A new method to address the under‐determined case arising from sparse measurements is proposed. Each of these methods serve to address a particular situation: closely‐spaced modes or low‐energy modes. The proposed methods are verified by applying them to extract the modal information of an airport control tower structure located near Toronto in Canada.