Improved independent component analysis based modal identification of higher damping structures
Introduction
To assess structural health status and develop effective maintenance strategies, researchers in the field of structural health monitoring are in full swing [27]. With the deepening of theoretical research, many structures have been equipped with SHM systems, such as Tsing Ma Bridge in Hong Kang of China, Runyang Bridge and CB32A Offshore Platform in mainland of China, Okashi Kaiko Bridge in Japan, I40 Bridge in New Mexico of United States, Confederation Bridge in Canada, Oresund Bridge in Denmark and Geumdang Bridge in Korea [18].
As one of necessary prerequisites to structural health monitoring, modal identification directly determines the effectiveness of a SHM system health monitoring system and further maintenance strategies [28]. Structural modal identification comprises the identification of frequency, damping ratio and mode shape.
Traditional structural modal parameter identification complies with the wisdom of system identification, which is based on the relationship between input and output signals. This corresponds to an ideal test situation in which excitation to the system can be controlled or measured. In the real-world, due to the large size of structures, such as long-span bridges, high rise buildings and large scale dams, it is extremely difficult and expensive to excite the structure, which limits the application of traditional structural modal identification in actual structures. Ambient excitation is usually a combination of the natural excitations such as micro-tremors, wind, traffic and other environmental effects. Related studies show that the ambient excitation basically has the characteristics of a stationary broadband excitation, which can be approximated by white noise, which makes it possible to identify structural modal parameter under ambient excitation. In the past decades, structural modal identification due to ambient excitation has attracted more and more attention for its advantages, such as no need of an input excitation. After decades of development, considerable progress has been made in output only ambient excitation methods, in which more sophisticated methods involving frequency domain methods, time domain methods and combined time–frequency domain methods have been adopted. Frequency domain methods include peak picking method, frequency domain decomposition method and least squares complex frequency domain method; time domain methods include time series analysis, random decrement method, ITD methods and natural excitation method, stochastic subspace methods, the minimum squares complex exponential method and the features of the system realization method; time–frequency domain methods include Wigner and short-time Fourier transform, wavelet transform and Hilbert–Huang transform methods [5], [19]. Most of these methods are sensitive to the measurement noise and non-stationary excitation which structures commonly confront in their service environment, which undermine the effectiveness of the existing methods.
In recent years, Blind Source Separation (BSS), due to its advantages of directly extracting sources from observed signals without known information about sources or mixing processing, have been used in many fields, such as acoustics [2], communication [16], image processing [3] and neural science [10], [14]. BSS shows prominent capacity as a new unsupervised signal processing tool [9] and has been introduced into structural dynamics [1]. Since it can recover the hidden sources and their underlying factors using only observed mixtures; it may thus be suitable to perform output-only modal identification [24].
Independent component analysis (ICA), as a popular tool to solve BSS problem, was proposed with the development of BSS at the end of 1990s [12], [26]. Based on the assumption of statistical independent sources, ICA tries to extract the sources from the observed mixture sources without knowing the information about sources and mixture matrix. According the principle of modal shape superposition of linear dynamical system, modal shapes and modal coordinate response can be separated from system outputs by ICA. Modal parameter identification technique is employed to obtain frequencies and damping rations then. Though ICA has obtained so much attention for its advantages, however, its fatal flaw of restricting to low level of damping in structures (the damping ratio less than 1%) [15], limits the actual structural application of ICA in civil structures for their higher damping. For example, the damping ratio of steel structures is always bigger than 1%. So, we give the definition of “higher damping structure” here as the structure with damping ratio more than 1%.
For using this powerful technical in structural modal identification without the limitation of low level damping, a new method of structural modal identification called “IDT + ICA” is proposed in the paper. The main contributions in the paper may be summarized as follows: (1) Analyzing the reasons why ICA is unable to identify higher damping structural outputs; (2) Introducing Inversed Damping Transfer (IDT) to turn the outputs of higher damping structure into low-damping signal. (3) Performing ICA to the low-damping signal obtained in step (2) and obtaining structural modal parameters including frequencies, damping ratios and model shapes. (4) Further treatment is performed to eliminate the impacts of IDT and obtain the actual structural modal parameters.
The layout of this paper is as follows: Section 2 provides the basic theory of BSS and ICA. Section 3 elucidates the process of structural modal identification by ICA. Inversed Damped Technique (IDT) technique is addressed in Section 4 and new method called “IDT + ICA” identifying higher damping structural modal is proposed in Section 5. Numerical simulation of 3-dof mass-spring and a simply supported concrete beam are introduced to verify the effectiveness of the presented methodology in Section 6, and Section 7 presents experimental results of a three-story steel frame to demonstrate the application of the method, followed by the discussion and concluding remarks in Section 8.
Section snippets
Blind source separation, BSS
The fundamental goal of Blind Source Separation (BSS) is to estimate unknown original sources from a set of observed mixtures without prior information about either the sources or the mixing process [1]. To limit the generality, specific restrictions are placed on the mixing model and the source signals. Although convolutive and non-linear mixtures can be considered [22], [21], the paper focuses on linear and static mixtures for which BSS is well established.
The observed mixtures are defined by
Structural modal parameters identification by ICA
Structural modal parameter identification includes extraction of structural modal frequencies, modal shapes and modal damping from inputs and outputs of structures [20].
According to structural dynamic theory, the governing equation of motion EMO of a linear time-invariant system iswhere M, C and K are the constant mass, damping, and stiffness matrices of the system, respectively; t is the time; f(t) is the external force vector applied to the system.
For small damping
Inverse Damping Transform (IDT)
According to the reason why ICA failing in modal identification of a highly damped structure, only if the signal has small enough damping, can it meet the requirement of independence, and can it be used to identify a highly damped structure. To meet the requirement of small damping, inversed damping transfer (IDT) is introduced to turn the signal of high damping into small damping and make the condition for the application of ICA.
IDT + ICA identify structures with high damping
Kerschen et al. used ICA to extract the modal responses directly from the measured system responses in the time domain. However, it failed in higher-damped structures (only within 1%) [15]. IDT + ICA is proposed in this study to address the aforementioned issues.
The proposed IDT + ICA modal identification algorithm under ambient excitation is completely unsupervised and capable of performing blind modal identification by directly extracting modal information from the measured structural responses.
Numerical simulations
To investigate the effectiveness of IDT + ICA applied in structural modal identification under free vibration and ambient excitations, two numerical simulations—a mass-spring model and a simply supported concrete beam—are introduced. The structures are subjected to two different excitations: one is free vibration and anther is ambient excitation.
Experimental verification
A three-story steel frame is borrowed to experimentally demonstrate the applicability of the proposed IDT + ICA modal identification method [25], see Fig. 14. The masses on each story are dominant and the two columns are flexible, with an extra damper between the base and the first floor. Three accelerometers are attached on top of the masses to record the system responses. Horizontal impact at the top floor (the third mass) is applied to the system, and the responses are measured and transmitted
Practical application
The seismic responses of the University of Southern California (USC) hospital building during the Northridge earthquake in 1994 is also analyzed by the proposed IDT–ICA algorithm to validate its applicability on real-world data. The USC hospital building is an eightstory base-isolated system, which is highly damped. The base and three stories (fourth, sixth, and the roof) are equipped with sensors to record the seismic responses, and the sensor/channel outline is presented in Fig. 18. A 3D
Conclusion
In this study, an improved ICA-based Blind Source Separation method was developed to perform structural modal identification under ambient excitation and overcome the fatal flaw of ICA failing in modal identification of highly damped structures. Based on the theoretical study, the fundamental reason of the flaw mentioned above is that high damping will not lead to realization of independence of signals completely, hence, the separation of ICA cannot completed either. To overcome this issue, a
Acknowledgments
This work is partially supported by Jiangsu provincial Natural Science Foundation of China (No. 20141180) and Scientific Research Project of Jiangsu Province Key Laboratory of Structure Engineering (No. ZD1405) and Science and Technology Projects of Construction Industry of Jiangsu Province (No. 2015ZD77), which is acknowledged gratefully. The author Jun Chang is supported by a grant from Jiangsu Government Scholarship for Overseas Studies, which is also gratefully acknowledged.
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