Underdetermined blind modal identification of structures by earthquake and ambient vibration measurements via sparse component analysis
Introduction
Modal identification of structural systems when only system responses are available is a challenging issue [1], [2]. Blind source separation (BSS) technique [3] is a statistical signal processing tool that has been widely used to deal with this problem [4], [5], [6], [7]. Introducing modal displacements as virtual sources paved the way for application of BSS in modal identification problems [8], [9]. Independent component analysis (ICA) [10] and second order blind identification (SOBI) [11] are the mostly used techniques in BSS, which are based on fourth order and second order statistics, respectively. Independence of the sources is the basic assumption of the ICA-based methods; however SOBI-based ones only require that the source signals should be uncorrelated.
Sparse component analysis (SCA) is another approach which is well suited to underdetermined problems. This approach comprises two major stages: mixing (modal) matrix estimation and source (modal displacements) recovery. SCA-based methods require no condition on independence or uncorrelatedness of the sources and sparsity is the only requirement that can be satisfied in transformed domain [12]. The choice of underlying transform plays a pivotal role in determining the type of excitation that any identification algorithm can undertake. Trying to exploit sparsity of structural responses in transformed domain, different transforms have been used in the literature. To cite a few, Discrete Cosine Transform (DCT) [13], linear time–frequency analysis [14], [15] and quadratic time–frequency analysis [16], [17] were used successfully in blind modal identification field.
In first stage, clustering algorithms [18] can be used to estimate modal matrix of structural systems. However, considering the fact that in practical situations the sources may have overlap in transformed domain, a criterion should be set to discard these points. Different researchers have investigated this issue exclusively and have reported a number of criteria [19], [20], [21], [22]. Of those, comparing the direction between real and imaginary part of the mixture signals was used in this study to distinguish Single Source Points (SSPs) from Multiple Source Points (MSPs), where only one and multiple sources make contribution in the responses, respectively [21]. Although this criterion was originally developed for speech signals, it showed promising results for vibration problems as well. Using this preprocessing step not only enhanced the estimation accuracy significantly, but it also reduced the computational cost while conducting clustering analysis.
Once the incomplete mixing matrix has been obtained, an underdetermined system of linear equations is formed. Solving this equation, especially when the source signals are complex-valued, is a challenging task [15]. In attempt to avoid complex elements, the authors in Ref. [13] chose DCT as underlying transform and used -magic algorithm [23] in recovery stage. But regarding the fact that DCT transform is in frequency domain, their proposed method was incapable of handling non-stationary excitations, like earthquake. Using Smoothed zero norm (SL-0) algorithm [24], [25], [26] in this paper enabled us to use any kind of transform, regardless of obtaining real-valued or complex-valued signals in transformed domain.
In this study, Short Time Fourier Transform (STFT) was chosen to fulfill sparsity requirement and to provide a general framework for tackling wide variety of excitations like earthquake and ambient vibration. In addition, constituent elements of the first and second stages were designed in such a way to render the proposed algorithm robust against noise, computationally efficient and less dependent on type of sparsifying transform that might yield real or complex coefficients. Whereas, constant resolution of the STFT transform, which requires a predefined window length, and restriction of the proposed technique to proportionally damped structures remained as the major limitations of the proposed technique.
The rest of this paper is organized as follows. The modal identification problem and it’s affinity to BSS problem are explored in Section 2. Then, two main stages of the SCA-based approach are presented in 3 First stage: modal matrix estimation, 4 Second stage: modal displacements estimation, independently. Numerical simulations using a synthetic example and a benchmark structure are carried out to investigate the performance of the proposed method in Section 5. Finally, a brief conclusion is given in Section 6 as a closing section.
Section snippets
Modal identification and BSS
A Linear Time Invariant (LTI) N-DOF structural system subjected to an excitation force is governed by the following differential equation [27]:where M, C and K are the N×N mass, proportional damping (diagonalizable) and stiffness matrices, respectively, and is a column-vector that contains excitation force.
is the absolute nodal displacement vector, which is transformed into modal coordinate as follows:where is the square real-valued modal
First stage: modal matrix estimation
It is aimed to estimate modal matrix of an LTI structural system which is subjected to earthquake and ambient excitation. The key idea is that by scattering all of the responses with respect to each other in sparse domain, the main components (contributing sources) align in certain directions representing the eigenvectors of the system. Then, they are grouped by means of a clustering algorithm to yield columns of the mixing matrix. However, in real life examples source signals may not be fully
Second stage: modal displacements estimation
Modal displacements are recovered with the knowledge of mixing matrix and responses. For determined case, modal displacements could be recovered by using a simple inversion technique (Eq. (14)).
However, in underdetermined problems complete modal matrix is no longer available. In this case, system responses and its partial modal matrix form an underdetermined system of linear equations.where , and every sub-matrix of A is of full rank [12].
In general, the Eq.
Performance evaluation and verification
The proposed algorithm was applied to a number of structural models to render the verification process. The capability of the proposed method in handling earthquake and ambient vibrations were all examined through the following simulations.
Modal assurance criterion was used to quantify the accuracy of the obtained mode shapes, where is the estimated mode shape and is the theoretical one.
Conclusions
A novel blind modal identification technique—which requires a limited number of sensors—was proposed in this paper. It is capable of identifying modal parameters of structures subjected to earthquake and ambient excitations. Time–frequency nature of the proposed method laid the foundation of such capability in dealing with different sorts of vibration. Time–frequency representation of each response was refined through the SSPs selection process. This preprocessing step paved the way for
Acknowledgment
The authors would like to thank the editor and reviewers for their invaluable comments that made significant improvement to our paper.
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