An Improved Lower Order Method of Modal Parameter Estimation for Offshore Structures Using Reconstructed Signals

LIU Fushun, QIN Junfei, LI Huajun, LU Hongchao, and WANG Shuqing

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An Improved Lower Order Method of Modal Parameter Estimation for Offshore Structures Using Reconstructed Signals

LIU Fushun*, QIN Junfei, LI Huajun, LU Hongchao, and WANG Shuqing

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For modal parameter estimation of offshore structures, one has to deal with two challenges: 1) identify the interested frequencies, and 2) reduce the number of false modes. In this article, we propose an improved method of modal parameter estimation by reconstructing a new signal only with interested frequencies. The approach consists of three steps: 1) isolation and reconstruction of interested frequencies using FFT filtering, 2) smoothness of reconstructed signals, and 3) extraction of interested modal parameters in time domain. The theoretical improvement is that the frequency response function (FRF) of filtered signals is smoothed based on singular value decomposition technique. The elimination of false modes is realized by reconstructing a block data matrix of the eigensystem realization algorithm (ERA) using the filtered and smoothed signals. The advantage is that the efficiency of the identification process of modal parameters will be improved greatly without introducing any false modes. A five-DOF mass-spring system is chosen to illustrate the procedure and demonstrate the performance of the proposed scheme. Numerical results indicate that interested frequencies can be isolated successfully using FFT filtering, and unexpected peaks in auto spectral density can be removed effectively. In addition, interested modal parameters, such as frequencies and damping ratios, can be identified properly by reconstructing the Hankel matrix with a small dimension of ERA, even the original signal has measurement noises.

modal parameter estimation; interested frequency; frequency response function (FRF)

1 Introduction

Accurate modal parameters such as modal frequencies and mode shapes are essential to model update and damage detection of offshore structures (Li, 2008; Liu, 2012), especially for poor spatial coverage of measured modes (Liu, 2011; Liu and Li, 2012, 2013). Only lower order modes could be excited by environmental forces for real structures and measured signals are inevitably contaminated with noise, Thus, one has to deal with two challenges: 1) how to identify genuine (interested) modal frequencies from false ones, and 2) how to reduce the influence of measurement noise and improve the efficiency of identification procedure of modal parameters.

Modal parameter identification involves estimating the modal parameters of a structural system from measured input-output data (Skingle and Urgueira, 1997; Ewins, 2009). Over the past thirty years, many algorithms have been developed to estimate modal parameters based upon the measured frequency response function (FRF) or equivalent impulse response function (IRF). The problem is that the measured FRF and IRF may not be clear and accurate enough because of measurement noises. Tradi-tional method of identifying genuine physical modes is that the computational model order is set higher than the true system order with the so-called ‘noise modes’ being absorbed (Allemang and Brown, 1998). Among many researchers proposing the usage of an over-determined system (, one with the order higher than that of a physical system) for improving the accuracy of parameter identification in noisy situations, Braun and Ram (1987) described the strategy to determine the necessary extent of over determination and the procedure for distinguishing true system modes. By presenting an efficient perturbation method based on the singular value decomposition, they also demonstrated that such an over determination is both possible and useful in the frequency domain.

In contrast to the classical method, setting a higher model order to absorb noise in the modal parameter estimate, the present proposed procedure is to isolate interested frequencies by performing signal filtering, reconstructing a new signal, and applying a time domain technique for modal parameter estimation. A low-pass filter can be readily applied to perform signal filtering, but can only be used for noise cancelation with a priori known bandwidth. Another important element in the modern theory of statistical signal extraction was proposed by Kalman (1960) and Kalman and Bucy (1961), which describes the filtering and forecasting of time-varying linear stochastic systems in discrete and continuous times. But the problem is how to adjust the Kalman filter if there is no information about the accuracy of measurements and no possibility to determine and select a better system model (Jwo and Chang, 2008). Pickrel (1996) focused on the assessment of data quality and used the singular value decomposition (SVD) technique to estimate the effects of frequency band, number of measurement locations and signal-to-noise ratio on measured response. Sanliturk and Cakar (2005) presented a method based on singular value decomposition (SVD) to eliminate noise from measured FRFs and to improve the quality of measured data. Hu(2010) proposed a different approach from the tradi- tional methods for estimating modal parameters from noisy IRF. But no discussion was presented about how to reconstruct a signal only with components of interested frequencies.

In this study, the Eigensystem Realization Algorithm (ERA) (Huang and Pappa, 1985) is used for the modal parameter identification of a system and special attention is paid to the construction of block data matrix aiming at improving computational efficiency. Synthesized measurements of a 5 degree-of-freedom (DOF) mass-spring-dashpot system are used to demonstrate the performance, and illustrate the procedure of the proposed scheme.

2 Signal Reconstruction with Interested Frequencies

Assume a noisy signal vector in which the noise is created and correlated with the signal,,

Convert the signal to the frequency domain using the Fourier transformation,

whereis the frequency in radian, and

. (3)

Assume that the original signal() consists of a number of individual components in the frequency domain,(1,),(2,), L, then Eq. (2) can be rewritten as

whereis the number of components of the original signal in the frequency domain.

Define a window, such as Hanning window with 50% overlap, in the frequency domain as(). Multiplying()with the FRF, the filtered signal can be found as

and the filtered signal can be expressed as

, (6)

where |()| is the absolute value of(), and() is the phase angle.

Theoretically, the Hankel matrix, corresponding to the noisy but band filtered signaly() in Eq. (6), can be partitioned into two parts

, (8)

in whichy,srepresents theth measured signal.

The SVD technique is also utilized for the estimation of the rank of the matrixin the process of noise elimination from measured FRFs,,

whereandare the orthogonal matrices,is the real diagonal matrix whose diagonalelementsσare called the singular values of, and the superscript ‘T’ denotes the transpose of a matrix. For measured data, singular values can become very small but will never be zero due to random errors and implemented band filtering; thus Eq. (9) can be changed to

. (10)

, (11)

with,being the number of rows and columns, respectively.

After a band-filtered and noise-removed signal is obtained, then move the window to a new position which covers the other interested-order frequency one by one; thus one can construct a new signal with only interested frequencies.

3 Improved Eigensystem Realization Algorithm

Using the traditional Eigensystem Realization Algorithm to obtain expected modal parameters,andhave to be assumed very large. But the assumption will generate unexpected false modes and reduce the efficiency of ERA. Therefore, the Hankel matrix with proper dimensions and desired modal information is constructed based on Eq. (12),

. (14)

When several sensors are used for modal identification simultaneously, the nearest matrixshould be changed to be

. (16)

, (18)

, (19)

is a minimum realization, where ? represents a quantity estimate. The desired modal damping rates and damped natural frequencies are simply the real and imaginary parts of eigenvalues after the transformation from the discrete-time domain to the continuous-time domain using the relationship.

4 Numerical Test: A Five-DOF Mass-Stiffness System

A five-DOF mass-spring system, as shown in Fig.1, was chosen for a numerical test to illustrate the procedure and demonstrate the performance of the proposed scheme. Uniform mass and stiffness coefficients were taken to bem=60kg andk=2.3×107Nm?1. In addition, dashpots were provided and the uniform damping coefficient wasc=100Nsm?1. The coordinates of the five-DOF model are denoted byxwith1at the fixed end and5at the freeend. Performing the eigenvalue analysis of the system, five modal frequencies are obtained: 13.139, 63.756, 116.97, 159.88, 187.54Hz, and the corresponding damping ratios: 1.7947×10?4, 8.7085×10?4, 15.977×10?4, 21.838×10?4, 25.616× 10?4.

In this study, first whether all components could be isolated is investigated. Then the signal is reconstructed to verify the approach. Finally the proposed method is applied on the isolation of some interested frequencies, including the usage of measured data corrupted with noise.

Fig.1 A five-DOF system.

4.1 Signal Reconstruction by All Frequency Components

To demonstrate the proposed scheme, initial investigation was conducted on whether all the frequencies could be isolated and a proper new signal could be reconstructed. In this test, first the five frequencies were isolated step by step, and then a new signal was constructed and compared with the simulated signal. A total of 2000 simulated points were used to represent the signal with the 500Hz sampling rate.

Using FFT band filtering, it is first investigated that whether the first frequency component of signal at1=13.139Hz could be isolated. To reflect the fact that the exact frequency of a system is never known in practice, the filter is designed with a center frequency that is different from the known value1=13.139Hz. Here it is assumed to be 13.270Hz, which is 1% higher than the true value. The pass band has a frequency range from 11.77 to 14.77Hz, and thus the first frequency component is inside the band. The requirement for determining the center frequency is that the chosen window should cover the interested frequency. Considering the interested frequency is usually unknown in practice, an alternative is to use a wider window initially.

Fig.2 shows the isolated signal in the frequency domain (auto spectral density with frequency). One can conclude from Fig.2 that the FFT filter can isolate the first frequency component while unexpected peaks appear because of the roll-off of the filter around a cut-off frequency. Likewise, one can isolate signals at four other frequencies. Combining the isolated signals, one can get a new reconstructed signal. Fig.3 shows the auto spectral density for the signal reconstructed by the five isolated components, which displays the consistency between the exact and filtered signals but the opposite in phase angle. Besides these, Fig.3 also shows unexpected peaks corresponding to other frequencies of the auto spectral density.

Fig.2 Auto spectral density of the isolated component f1=13.139Hz.

To smooth FRFs of filtered signals in the frequency domain, it is first needed to determine the model order of smoothed signals, which is by estimating the rank of a related Hankel matrix as shown in Eq. (8) if the singular values of the matrix are ordered sequentially from the largest to the smallest. The ordered singular values associated with the Hankel matrix of size=750 are plotted in Fig.4, where each singular value is normalized by the first (largest) singular value. For a system with five significant frequencies, the rank of the Hankel matrix associated with the filtered signal must be equal to 10, which is twice the number of modes of the system. Applying Eqs. (8) to (14) for the band filtered signal in Fig.3, one can obtain the smoothed FRFs. Fig.5 clearly demonstrates the effectiveness of the proposed method.

Fig.4 Normalized singular values of Hankel matrix.

4.2 Signal Reconstruction by Interested-Order Frequency Components

As an example, the first two frequencies are used for damage detection and model updating. A new signal is reconstructed only with these two frequencies. Based on the same consideration on isolating frequency1=13.139Hz, the second frequency2=63.756Hz can be isolated as discussed above. The only difference is that the model order in this example should be equal to 4. Implementing the proposed approach, one can reconstruct a new signal as shown in the frequency domain in Fig.6. Based on Eq. (14), the Hankel matrix with a dimension of 4×4can be used as the ERA block data matrix. Numerical results indicate that the first two interested frequencies and damping ratios can be calculated accurately.

4.3 Modal Parameter Identification

In reality, measurements always contain errors. The remaining numerical study focuses on implementing the proposed method with corrupted signals which are generated by adding a Gaussian white noise to a noise-free signal. The level of the white noise is quantified by a stated percentage, defined as the ratio of the standard deviation of the white noise to that of the no noise signal. In this example, it is assumed that the white noise level is 5% and the interested order of frequencies is the first two frequencies as discussed above. Fig.7 is the comparisons of magnitudes and phase angles between the assumed noisy signal, and the filtered and smoothed signal after 10 iterations. One can also find that the interested-frequencies could be isolated and smoothed effectively even the measured signal contains a noise level of 5%. The estimated frequencies are 13.140Hz and 63.756Hz, which are very close to their true values; and the estimated damping ratios are 4.1656×10?4and 8.635×10?4, respectively. Though the second damping ratio is estimated properly, the first one shows clear influence by measurement noise.

Further investigation is conducted to evaluate the performance of damping ratio estimation with different noise levels varying from 1% to 10% with an interval of 1%. Fig.8 shows the comparison of the first damping ratio between the preset and the estimated value and the estimated first damping ratio displays clear correspondence to increased noise levels.

Fig.8 Estimation of the first damping ratio and change in noise level.

5 Conclusions

Because of measurement noise and limitation of excitation techniques, accurately estimating model parameters has been a challenging task. An improved estimate method is proposed by reconstructing a new signal only with interested frequencies. The approach includes three steps: 1) isolation and reconstruction of interested frequencies using FFT filtering by defining a series of windows and pass bands, 2) FRF smoothness of the reconstructed signal by implementing the Cadzow’s algorithm for the structured low rank approximation (SLRA) on the Hankel matrix, and 3) extraction of interested model parameters using filtered and smoothed signals in the time domain. In implementing the method, it is found that choosing the bandwidth of a filter is not so strict comparing to the traditional FFT filtering. The only requirement is that the center frequency lies in the selected pass band. Numerical results for a 5-DOF mass-spring system indicate that even the true values of the model parameters of a test structure are not known, the interested frequencies can be isolated successfully using the FFT filtering, and unexpected peaks in auto spectral density can be removed effectively. In addition, interested model parameters, both frequencies and damping ratios, can be identified properly by reconstructing a small dimension Hankel matrix even the original signal has measurement noises.

Acknowledgements

The authors wish to acknowledge the financial support of the Excellent Youth Foundation of Shandong Scientific Committee (Grant no. JQ201512) and the National Natural Science Foundation of China (Grant nos. 51279188, 51479184, 51522906).

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(Edited by Xie Jun)

DOI 10.1007/s11802-015-2438-y

ISSN 1672-5182, 2015 14 (6): 969-974

? Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2015

(July 7, 2013; revised October 25, 2013; accepted August 24, 2015)

* Corresponding author. Tel: 0086-532-66781672 E-mail: percyliu@ouc.edu.cn