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Volume , Issue 4. Previous Article Next Article. Technical Papers. This Site. Google Scholar. Department of Civil Engineering, K. Author and Article Information. Bart Peeters. Dec , 4 : 9 pages. Published Online: February 7, Article history Received:. Views Icon Views. Issue Section:. Heylen, W. Maia, N. Allemang, R. Ewins, D. Comparison study of subspace identification methods applied to flexible structures. Petsounis, K. Signal Process. Modal testing and analysis of structures under operational conditions: industrial applications. One-year monitoring of the ZBridge: environmental effects versus damage events.
Ljung, L. Reference-based stochastic subspace identification for output-only modal analysis. Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes. Basseville, M. Caines, P. Bendat, J. Stoica, P. Felber, A. Cunha, A. Prevosto, M. Complex mode indication function and its application to spatial domain parameter estimation. Brincker, R. Golub, G. Schoukens J.
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Moreover, a new algorithm based on virtual excitation assumption is presented to remove the harmonic components in modal parameters estimation. Finally, the quality of the proposed method is compared with that of the conventional method using a numerical simulation and a practical experiment. Keywords: operational modal analysis, enhanced spectral kurtosis, harmonic detection, harmonic removal. During the last decade, the modal analysis becomes a key technology in structural dynamics behavior study [1, 2].
Operational modal analysis OMA is a technology for the identification of the linear structural modal parameters by using only the output responses [3, 4]. It is used to derive an experimental dynamics model from vibration measurements on a structure in operational conditions. In many operational cases, it is impossible to measure the excitation force, and the structural responses are the only information that can be collected for the system identification algorithms.
A new method based on the transmissibility function is proposed to process the parameters in OMA with unrestricted types load [5, 6], but it needs twice tests with different loads and may generate fake resonance peaks. Therefore, in the OMA, the system is generally assumed to be excited by a broadband random signal [7, 8]. However, in many applications, the presence of periodic excitation in the random loads is unavoidable.
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The periodic forces generated by unbalanced masses of rotating parts or aerodynamic or electric actuators, may cause harmonic components in the responses . These harmonic components will cause uncertainty in extraction of modal parameters , and need to be detected and removed before modal identification. A direct approach for processing the harmonic components in outputs is to take them as zero-damping modes. In practice, this method would not be effective when the structural modes have very low damping, or the harmonic frequencies are very close to the structural modal frequencies .
It argues that the harmonic components are un-attenuated. The harmonic components, in other words, are also assumed to be zero-damping modes in this method.
Another method based on the probability density function PDF of the output for distinguishing a harmonic response from narrowband stochastic responses is widely used [9, 14, 15]. This method, however, is effective when the harmonic frequencies are well separated from the structural frequencies, since it would imply that temporal filter must be used. Furthermore, it will require tedious calculations and be restricted by bandwidth of the filter.
A kurtosis-based method was initially proposed in time domain for distinguishing the random and period signals , and processed in a way similar to the PDF method mentioned above. In order to avoid tedious data calculation, the spectral kurtosis SK of a signal is defined as the kurtosis of its frequency component, and has been applied to harmonic components detection in the OMA .
Another advantage of this method is that the information of each frequency component can be indicated promptly in the frequency band of interest. However, the method is not perfect, since only one channel signal can be dealt with at a time. It brings a time-consuming problem and may lead to a failure of detection. In order to improve the quality and efficiency of the harmonic components detection, an enhanced spectral kurtosis ESK based on the enhanced PSD function is proposed in this paper. Unlike the original SK method, the ESK calculation utilizes all multiple-channel signals at each resonance frequency of interest.
Moreover, due to the application of a PSD enhancement technique, the band-pass filter is not required any more. Then, a new method named the virtual excitation technique VET is presented for the removal of harmonic components even when they are close to the natural frequencies. The result from the proposed method is also compared with that from the previous method by a numerical simulation. In practice, the discrete sequence b n is divided into M un-overlapped blocks to obtain an unbiased estimator of the SK by using the k -statistics:.
According to the statistical characteristic , the SK of a random process equals to zero. By contrast, the SK of a harmonic process always equals to —1 at the harmonic frequency. Supposing b n be mixed with a harmonic signal, the SK can be comprehensively described as:. Consequently, the harmonic components will be detected by computing the value of SK at each spectral line. Assume that there are n o channels of output responses:.
By applying the modal theory, the output responses in the physical space can be transformed into a modal subspace:. The superscript H denotes the Hermitian transposition.
Therefore, the k th single degree of freedom SDOF response can be derived as:. It is noticed that Eq.
Operational Modal Analysis of Civil Engineering Structures
By inserting Eq. According to the Eq. Therefore, the ESK of a mixed-signal can also be derived as:. The calculation becomes complicated if n o is a large number, which is common in the OMA. By contrast, the ESK is as an optimized method for multiple-channel signal processing.
Supposing that the k th of ESK is close to zero, the k th modal parameter can be extracted by using the enhanced PSD directly. The component, however, should be removed first in the case where its value of ESK is close to —1, particularly when it is close to a structural mode.
The unknown mixed signal x can be considered as the linear superposition of a random signal and a harmonic signal:. The DFT of the mixed signal x can be expressed with the virtual excitations as:. Indeed, the spectrum of the random input is not always flat in practice, but can be generally considered as constant spectrum in a narrow band . The MIMO model can be written at each frequency in the narrow band as:.
Using the Least Squares algorithm, the proportional virtual responses of this system can be obtained as:. The simulation technique based on the FRF modes is employed to generate six responses with five uncorrelated random noise signals and a mixed signal with a periodic frequency at The PSD matrix of responses will be computed by the Welch method, using Hanning windowing and resolution of 0.
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The original SK method is used to detect a harmonic component and the results are shown in Fig. Mode indicator curves of two-storey building. SK curves of responses two-storey building. In Fig. Six structural modes and a harmonic component at the frequency of Moreover, the harmonic component is very close, actually only three spectral lines are far, to the 3rd natural frequency. Therefore, the SK of all responses should be calculated for a good harmonic detection. Moreover, although the harmonic is indicated in several SK curves, like Fig.
Since the frequency band is too narrow to separate harmonic frequency from nearby structural frequency by band-pass filtering, the computational accuracy is influenced. It is clearly observed that the 4th harmonic component ESK value at In contrast with the SK method, the ESK method is more reliable to detect the harmonic components, because all of the measured responses are completely considered by the calculation of ESK from the enhanced PSD.
ESK curves at each peak two-storey building.