EMD-BASED STOCHASTIC SUBSPACE IDENTIFICATION OF CIVIL ENGINEERING STRUCTURES UNDER OPERATIONAL CONDITIONS

Transcription

1 EMD-BASED STOCHASTIC SUBSPACE IDENTIFICATION OF CIVIL ENGINEERING STRUCTURES UNDER OPERATIONAL CONDITIONS Wei-Xin Ren, Department of Civil Engineering, Fuzhou University, P. R. China Dan-Jiang Yu Department of Civil Engineering, Fuzhou University, P. R. China Abstract Modal parameter identification of ambient vibration system need extract information from data using signal process method. In this paper, a newly developed signal processing technique, Empirical Mode Decomposition (EMD), is implemented to decompose data set into several intrinsic mode functions (IMF) by a procedure called sifting process. By using intermittency frequency, the sifting process can overcome the difficulty of mode mixing and decompose data to modal response functions. Then the stochastic subspace identification (SSI) method is used to identify the modal parameters of civil engineering structures. The process combines the advantages of those two methods, it can process nonlinear and non-stationary signal generated by ambient vibration measurements. And the structure modal parameters are easily identified ignoring the influence of other modal frequency components and the influence of fae frequencies and unwanted noise by proposed EMD-based stochastic subspace identification. A case study Beichuan arch bridge using real operational acceleration signals excited by ambient vibration is presented to show the applicability of the present technique. 1 Introduction Modal analysis is a technique that estimates the dynamic characteristics of a structure through vibrational measurements, which is not a recent practice and many studies have been carried out in the past. It was originally developed in more advanced mechanical and aerospace engineering disciplines [1,2] where the modal parameter identification was based on both input and output measurements. There is a clear merit in trying to transfer this technology into civil engineering applications where we are dealing with problems which have a completely different scale, logistics and rationale. The encountered civil engineering structures are often complex, large in size and low in frequency. The major difference of vibration measurements in civil engineering structures is that the measurements are always carried out under operational conditions where the input forces or excitations are extremely difficult to quantify and the output only data are measured. A modal parameter identification procedure will therefore need to base itself on output only measurements. The operational modal analysis, or output only modal analysis, is associated with several advantages of being inexpensive since no equipment is needed to excite the structures. The service state need not have to be interrupted to use this technique. More realistic boundary and loading conditions are present. However, the mathematical bacground of operational modal analysis was wea (without enough input information), computer power was insufficient and equipment not

2 sensitive enough for this ind of measurements in the early time, the technology has become popular only during the last 10 years. The improvement of computing capacities and signal processing techniques led to the development of powerful algorithms for the determination of the dynamic characteristics from operational vibration measurements. There have been several modal parameter identification techniques available that were developed by different investigators for different uses such as the pea-picing method from the power spectral densities [3], natural excitation technique (NExT) [4], and stochastic subspace identification (SSI) [5,6].The mathematical bacground of these operational modal parameter identification techniques is very similar. The difference is often due to the implementation aspects such as data reduction, type of equation solvers, sequence of matrix operations, etc. The SSI algorithm is probably the most advanced method for operational vibration measurements. It identifies the state space matrices based on the measurements and by using robust numerical techniques such as QR-factorization, singular value decomposition (SVD) and least squares. SSI has been successfully applied to several types of civil engineering structures under operational conditions. Presently, the operational modal analysis methods usually assume that the unmeasured operational excitation is a white noise and output only measurement is a stationary response to simplify the algorithm. Operational vibration measurements, however, are not always stationary signals. Historically, Fourier spectral analysis has provided a general tool for the data analysis efforts. But it has some crucial restrictions: the system must be linear and the data must be stationary. One of the available non-stationary data processing methods, the wavelet approach has been available only in the last decades or so. The wavelet is essentially an adjustable windows Fourier spectral analysis. It still has the problems such as its leaage generated by the limited length of the basic wavelet function. A new signal process technique, referred to as Empirical Mode Decomposition (EMD) has been recently proposed by Huang et al [7,8]. EMD considers signals at the level of their local oscillations, EMD can decompose any data set into several intrinsic mode functions (IMF) by a procedure called the sifting process. IMF is a mono-component and can be referred to as modal response function in a sense [9]. The frequency components of each IMF only depend on the original signal. It is an adaptive method and suitable to dealing with non-stationary signals. This newly developed technique has already been used in the damage detection of civil engineering structures [10,11]. The purpose of this paper is to present an EMD-based stochastic subspace identification of civil engineering structures from operational vibration measurements. The output only measurements are first decomposed into the modal response functions by using the empirical mode decomposition technique with the specified intermittency frequencies. The stochastic subspace identification method is then applied to the decomposed signals to identify the modal parameters. The results demonstrate that the stabilization diagram of SSI becomes more clearly and the identification is easier. A case study of the output only measurements from a real bridge is presented to show the applicability of the present technique. 2 Empirical Mode Decomposition (EMD) The essence of the empirical mode decomposition is to identify the time lapses between the successive extrema in the signal empirically, and then to decompose the signal accordingly. Suppose x( is the signal to be decomposed. The sifting process is implemented as follows. (1) Identify all the local extrema, and then fit all of the local maxima and local minima by a cubic spline line as the upper and lower envelopes. The upper and lower envelopes should cover all the

3 data between them. Their mean is designated m 1, and the difference between the data ( and m is defined by 1 x 1 x 1 h = ( m (1) If h 1 satisfies two conditions: (a) in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; and (b) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero, should be the first intrinsic mode function (IMF). h 1 (2) If h1 doesn t satisfy IMF conditions, suppose h1 as the original data and repeat the sifting process (1) until the conditions achieved and the first IMF is obtained. (3) The original signal is then subtracted from the IMF and the repeated sifting process is applied to the remaining signal to obtain another IMF. The process is repeated to obtain n IMFs, i.e. n x( = = i 1 c ( i + r ( n (2) in which c i ( (i=1,2, n) is the IMFs of the measured signal x( from high frequency to low frequency components, each c i ( contains different frequency component. r n ( is the residual that is the mean trend of the signal or a constant. Such a process is so called the empirical mode decomposition (EMD). However, the above sifting process sometimes may run into difficulties and each IMF contains more than one natural frequency components, which is not the modal response. Namely, a few natural frequency components of the original signal may mix together. This mode mixing is due to the EMD algorithm and sifting process. The sample frequency, signal component and amplitude etc. will also affect the decomposition result. In order to avoid the mode mixing, a criterion f ini, called the intermittency frequency, is introduced. The intermittency frequency is defined based on the period length to separate the waves of different periods into different modes. The criterion frequency f ini can be set as the upper limit of the period that can be included in any given IMF component, so that the resulting IMF will not contain any natural frequency components smaller than the intermittency frequency. This criterion will enable us to extract a modal response as an IMF. As a result, the measured response can be decomposed into nth modal response functions and many other IMFs. The decomposed modal response functions are afterwards used for the modal parameters identification. 3 Stochastic Subspace Identification (SSI) Stochastic subspace identification (SSI) is an output-only time domain method that directly wors with time data. It is especially suitable for operational modal parameter identification. It is beyond the scope of this paper to explain all details about the stochastic subspace identification method and only the main issues are explained below. The dynamic behaviour of a vibrating structure can be described by a continuous-time state-space model: x& ( = A c y( = C c x( + B u( c x( + D u( c (3)

4 where x ( is the system state vector; u( describes the inputs as a function of time; A c is the system state matrix and B c is the system control influence coefficient matrix. Cc is the output matrix and Dc is the outputs; is the direct transmission matrix. The measurement data are always discrete in nature. After sampling the continuous-time state-space model can be converted to the discrete-time state-space model x y +1 = Ax = C x + Bu + D u in which = x( is the discrete-time state vector containing the sampled displacements and x velocities; u, y are the sampled input and output; A = exp( Ac is the discrete state matrix; 1 B = [ A I] A c B c is the discrete input matrix; C is the discrete output matrix; D is the direct transmission matrix. So far it is assumed that the structure is only driven by a deterministic input u. However, the deterministic model is not able to exactly describe real measurement data since in practice there are always system uncertainties including process and measurement noises. The process noise is due to the disturbances and modeling inaccuracies, whereas the measurement noise is due to the sensor inaccuracy. Stochastic components have to be included in the model and following discrete-time combined deterministic-stochastic state-space model can be obtained: x y +1 = Ax = C x + Bu + D u where w is the process noise and v is the measurement noise. They are both unmeasurable vector signals assumed to be zero mean, white and with covariance matrices. The vibration information is the responses of a structure excited by the operational inputs. Due to the lac of input information it is impossible to distinguish deterministic input u from the noise terms w, v in Eq (5). If the deterministic input term u is modeled by the noise terms w, v, the discrete-time purely stochastic state-space model of a vibration structure can be obtained: x y +1 = Ax = C x The structure is now purely excited by the noise terms w, v. However the white noise assumptions of these noise terms cannot be omitted. The consequence is that if this white noise assumption is violated, for instance, if the input contains also some dominant frequency components in addition to white noise, these frequency components cannot be separated from the eigenfrequencies of the system and they will appear as spurious poles of the state matrix A. The essence of stochastic subspace identification is the projection of the row space of the future outputs into the row space of the past outputs. The main difference with other algorithms is that the subspace algorithm is data driven instead of covariance driven so that the explicit formation of the covariance matrix is avoided. It is clear that the stochastic subspace identification is a time domain method that directly wors with time data, without the need to convert them to correlations or spectra. Common to all system identification methods for operational vibration measurements, the identified mode shapes are the operational ones instead of absolute scaling (mass normalized) mode shapes because the input remains unnown. + v + w + v + w y( (4) (5) (6)

5 4 Case Study The real case studied is a half-through concretefilled steel tubular arch bridge with the span of 90 m as shown in Figure 1. The bridge is named as Beichuan Bridge that lies over the Beichuan River in Xining, Qinghai Province, China. The specific feature of the bridge is that the arch is made by the concrete-filled steel tube. The ratio of rise to span of the bridge is 1/5. Measurement locations were selected on the dec near the location of connection between suspenders and dec. A total of 32 locations (16 points per side) were selected. The accelerometers were arranged in the vertical direction only. Four test setups were conceived to cover the planned measurement locations. One reference Figure 1 Beichuan arch bridge accelerometer (base station), located at the bearing, was common to each setup. As a result, each setup consisted of one reference accelerometer and eight moveable accelerometers. The sampling frequency on site is 80 Hz and data recording time is about 15 minutes. The raw measurement data are re-sampled and low-pass filtered to 16Hz. Figure 2(a) shows the re-sampled signal and corresponding Fourier spectrum at measurement location 4 (1/4 span). It can be estimated that the range of the third modal frequency f 3 is within f L3 = 3.40 f 3 f H 3 = Subsequently, the original signal is decomposed by using the empirical mode decomposition technique with the intermittency frequency f ini = f H 3 = 3.55 Hz. After the sifting process, the resulting first IMF is the time-history signal only containing the natural frequencies lower than 3.55Hz. To subtract this IMF from the original data, the empirical mode decomposition is applied again to this signal with the intermittency frequency f ini = f L3 = Hz. The resulting first IMF and its Fourier transform are shown in Figure 2(b). It is clearly shown that there is only one pea left in the spectrum. It is the third natural frequency f 3 of the bridge. Other natural frequency components of the original data are removed from this IMF and modes are decomposed. (a) Time data and spectrum of original signal (b) IMF and spectrum of decomposed 3 rd mode

6 (c) IMF and spectrum of decomposed 2 nd mode (d) IMF and spectrum of decomposed 1 st mode Figure 2 Time-histories and spectrums of original and decomposed signals The above decomposition process is repeated by using the empirical mode decomposition technique with the specified intermittency frequencies. The rest modal components can be decomposed from the signals as shown in Figures 2(c, d). After all measurements of 32 measured locations and reference location have been decomposed, the data are ready for the vibration characteristics extraction by using stochastic subspace identification. The calculated SSI stabilization diagrams of the decomposed three IMFs and original signal are compared in Figure 3. It can be clearly seen from Figure 3(d) that the SSI stabilization diagram of the original signal without the decomposition processing is mixed among modes and there are many stable poles. It is sometimes difficult to choose the right poles. Each SSI stabilization diagram of the decomposed three IMFs as shown in Figure 3(a~c), on the contrary, demonstrates only one clear stable pole. The modes of a vibrating structure are decomposed. Signal stable pole in the decomposed SSI stabilization diagram maes the modal parameter identification much easy. The identified modal parameters of the arch bridge by using the proposed EMD-based stochastic subspace identification technique are compared in Table 1 with those obtained from the stochastic subspace identification only and pea-picing method. It is demonstrated that the current results are comparable. Figure 4 shows the identified first three vertical bending mode shapes of the arch bridge from the EMD-based stochastic subspace identification. These mode shapes agree well with those obtained from the finite element analysis [12]. All these results have proven that the present EMD-based stochastic subspace identification technique can effectively identify the dynamic characteristics of a full-size bridge under operational vibration measurements. (a) SSI stabilization diagram of 1 st mode (b) SSI stabilization diagram of 2 nd mode

7 (c) SSI Stabilization diagram of 3 rd mode (d) SSI stabilization diagram without decomposition Figure 3 SSI stabilization diagrams 1st bending mode shape 2nd bending mode shape 3rd bending mode shape Figure 4 Identified first three vertical bending mode shapes of the bridge Table 1: Comparison of identified modal parameters of the bridge Mode EMD+SSI SSI [12] Pea-picing [12] Frequency (Hz) Damping ratios Frequency (Hz) Damping ratios Frequency (Hz) % % % % % % Conclusions The newly developed empirical mode decomposition (EMD) is a promising and attractive signal processing technique. It is basically an adaptive high pass filter. By using the specified intermittency frequencies the sifting process can overcome the difficulty of mode mixing. The decomposed IMFs can be referred to as the modal response functions in a sense. By using EMD as a pre-processing tool to decompose the operational vibration measurements into several modal response functions, an EMD-based stochastic subspace identification of civil engineering structures for operational vibration measurements is presented in the paper. A case study of the output only measurements from a real bridge has demonstrated that the identified frequencies and mode shapes are comparable. By using decomposed time data the stable pole in the stabilization diagram becomes sole ignoring the influence of other modal components and fae frequencies due to unwanted noise. It is convinced that the proposed technique will be a promising in dealing with the operational vibration measurements if the input contains some dominant frequency components in addition to white noise during the structural health monitoring.

8 6 References [1] Ewins, D.J. Modal testing: theory and practice, England: Research Studies Press Ltd, [2] Juang, J,N. Applied system identification, New Jersey: Prentice-Hall Inc. Englewood Cliffs, [3] Bendat, J,S. and Piersol, A.G. Engineering applications of correlation and spectral analysis, 2nd edition, John Wiley & Sons, New Yor, NY, [4] James III, G.H., Carne, T.G. and Lauffer, J.P. The natural excitation technique (NExT) for modal parameter extraction from operating structures, International Journal of Analytical and Experimental Modal Analysis, 10(4): , [5] Van Overschee, P. and De Moor, B. Subspace identification for linear systems: theory, implementation and applications, Dordrecht, Netherlands, Kluwer Academic Publishers, [6] Peeters, B. and De Roec, G. Reference based stochastic subspace identification in civil engineering, Inverse Problems in Engineering, 8(1): 47-74, [7] Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, E.H., Zheng, Q., Tung, C.C. and Liu, H,H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis, Proceedings of Royal Society of London, SeriesA, 454: , [8] Huang, N.E., Shen, Z. and Long, S.R. A new view of nonlinear water waves: The Hilbert spectrum, Annu Rev Fluid Mech., 31: , [9] Yang, J.N. and Lei, Y. Identification of natural frequencies and dampings of in situ tall buildings using ambient wind vibration data. Journal of Structural Engineering, ASCE, 130(5): , [10] Yang, J.N. and Lei, Y. Hilbert-Huang based approach for structural damage detection. Journal of Structural Engineering, ASCE, 130(1): 85-95, [11] Xu, Y.L. and Chen, J. Structural damage detection using empirical mode decomposition: experimental investigation, Journal of Structural Engineering, ASCE, 130(11): , [12] Zong, Z.H., Jaishi, B., Ge, J.P. and Ren, W.X. Dynamic behavior analysis of a CFT halfthrough arch bridge, Engineering Structures, 27(1): 3-15, 2005