Identifying the Modal Parameters of a Structure from Ambient Vibration Data via the Stationary Wavelet Packet

Identifying the Modal Parameters of a Structure

from Ambient Vibration Data via the Stationary

Wavelet Packet

W. C. Su

National Center for High Performance Computing, National Applied Research Laboratories, Hsinchu 30050, Taiwan

C. S. Huang*

Department of Civil Engineering, National Chiao Tung University, Hsinchu 30050, Taiwan

C. H. Chen

Department of Civil and Environment Engineering, National University of Kaohsiung, Kaohsiung 81148, Taiwan

C. Y. Liu, H. C. Huang & Q. T. Le

Department of Civil Engineering, National Chiao Tung University, Hsinchu 30050, Taiwan

Abstract: Ambient vibration tests are conducted widely to estimate the modal parameters of a structure. The work proposes an efficient wavelet-based approach to determine the modal parameters of a structure from its ambient vibration responses. The proposed approach integrates the time series autoregressive (AR) model with the stationary wavelet packet transform. In addi-tion to providing a richer decomposiaddi-tion and allow-ing for an improved time–frequency localization of signals over that of the discrete wavelet transform, the stationary wavelet packet transform also has significantly higher computational efficiency than the wavelet packet transform in terms of decomposing time-shifted signals because the former has a time-invariance property. The correlation matrices needed in determining the coefficient matrices in an AR model are established in subspaces expanded by stationary wavelet packets. The formula-tion for estimating the correlaformula-tion matrices is shown for the first time. Because different subspaces contain sig-nals with different frequency subbands, the fine filtering property enhances the ability of the proposed approach to identify not only the modes with strong modal in-terference, but also many modes from the responses of *To whom correspondence should be addressed. E-mail: cshuang@ mail.nctu.edu.tw.

very few measured degrees of freedom. The proposed ap-proach is validated by processing the numerically sim-ulated responses of a seven-floor shear building, which has closely spaced modes, with considering the effects of noise and incomplete measurements. Furthermore, the present approach is employed to process the velocity re-sponses of an eight-storey steel frame subjected to white noise input in a shaking table test and ambient vibration responses of a cable-stayed bridge.

1 INTRODUCTION

Capable of reflecting the dynamic characteristics of a structure, modal parameters identified from its mea-sured dynamic responses can offer a valuable reference to efforts to update structural design models (Garc´ıa-Palencia and Santini-Bell, 2013; Lozano-Galant et al., 2013) and perform vibration control (Adeli and Saleh, 1997; Kang et al., 2012), health monitoring (Yuen and Katafygiotis, 2006; Soyoz and Feng, 2009; Hu et al., 2013), and damage assessment (Jiang and Adeli, 2007; Osornio-Rios et al., 2012) of a particular structure. Iden-tification of modal parameters from dynamic responses can be carried out in either the frequency, time or time–frequency domains. The development of typical

C

2014 Computer-Aided Civil and Infrastructure Engineering.

approaches of system identification in the frequency and time domains can be found in many text books (i.e., Ljung, 1999; Schoukens and Pintelon, 1991). Recently, Sirca and Adeli (2012) provided an excellent review of representative works on structural system identification published in journals since 1995. The paper includes 191 references and classifies the methods of system identification as conventional model-based approaches, biologically inspired approaches, signal processing based approaches, chaos theory, and multiparadigm approaches.

Owing to its flexibility, ambient vibration test is often undertaken to attain the required dynamic responses of a structure. Because the excitation for ambient vi-brations includes natural forces (e.g., wind, tidal waves, and earthquakes tremors), the input forces are too com-plicated to measure. Numerous algorithms have been developed to identify modal parameters of a structure from its ambient vibrations. These algorithms work in either the frequency, time or time-frequency domains. Although generally simple, the methods in the fre-quency domain (Bendat and Piersol, 1993; Yan and Ren, 2012) do not work well for structural systems with heavy damping and strong modal interference (Lardies and Ta, 2011).

In the time domain, the random decrement tech-nique (Cole, 1971; Vandiver et al., 1982; Huang and Yeh, 1999) is typically applied to convert ambient vi-bration responses into free decay responses, which are also called Randomdec signatures. The modal parame-ters from the free decay responses are then identified by using either the Ibrahim time-domain scheme (Ibrahim and Mikulcik, 1973; Ibrahim and Pappa, 1982) or the least-squares complex exponential method (Brown et al., 1976; Vold and Russell, 1983). Although the in-put forces are assumed to be white noise, Huang and Yeh (1999) demonstrated theoretically that the Ran-domdec signatures converted from velocity or displace-ment responses of a system are equivalent to the free decay responses of the system, whereas Randomdec signatures converted from acceleration responses are not. The ambient vibration responses are also of-ten processed directly using the time series models, autoregressive (AR) (Wang and Fang, 1986; Huang, 2001; Liu et al., 2011) and autoregressive moving av-erage (ARMA) (Loh and Wu, 1996; Lardies, 2010), stochastic subspace approaches (Huang and Lin, 2001; Ali and Okabayashi, 2011; He et al., 2008), natural excitation technique with the eigensystem realization algorithm (NExT-ERA) (Dionysius and Yozo, 2007; Caicedo, 2011), and blind source separation meth-ods (Zhou and Chelidze, 2007; Mcneil, 2011; Hazra et al., 2012). The input forces for ambient vibration re-sponses of structures are assumed to be white noise

in the AR model, stochastic subspace approaches and NExT-ERA; this assumption is quite acceptable in civil engineering applications. The ARMA model and blind source separation methods do not make such a strict assumption and are more complicated than the AR model, stochastic subspace approaches and NExT-ERA.

Wavelet transformations provide time–frequency analyses and offer alternative signal analysis methods by presenting information in the time and frequency domains simultaneously. Wavelet transformations were first applied to structural system identification about two decades ago. Discrete wavelet transforms (Robert-son et al.,1998; Huang et al., 2005), wavelet packet trans-forms (Jiang and Adeli, 2004; Jiang et al., 2007; Chen et al., 2013), continuous wavelet transforms (Schoen-wald, 1993; Ruzzene et al., 1997; Gouttebroze and Lardies, 2001; Kougioumtzoglou and Spanos, 2013; Hsu, 2013), and wavelet neural networks (Hung et al., 2003; Jiang and Adeli, 2005, 2008; Adeli and Jiang, 2006, 2009; Lin et al., 2012; Kodogiannis et al., 2013) were em-ployed to system identifications using the dynamic re-sponses and inputs of structures or the free vibrations of structures.

Wavelet-based approaches and Hilbert transform methods are also particularly used for identifying modal parameters of a structure from its ambient vibration re-sponses. Most of these approaches collaborate with the random decrement technique. To determine the modal parameters from the Randomdec signatures, Yang et al. (2004) employed Hilbert–Huang transform (Huang et al., 1998), whereas Lardies and Goutte-broze (2002) proposed a method based on the wavelet ridges of continuous Morlet wavelet transform. Yang and Miyamoto (2006) presented a comparative study of estimating modal parameters from Randomdec signa-tures by using the approach of Lardies and Gouttebroze (2002) and the improved Hilbert–Huang transform. In another comparative study, Lardies and Ta (2011) de-termined the modal parameters of a cable-stayed bridge by using a modified approach of Lardies and Goutte-broze (2002) and a subspace algorithm. Based on the approach of Huang and Su (2007), which combines the autoregressive with exogenous input (ARX) model with the continuous wavelet transform and was developed to identify the modal parameters of a structure un-der an earthquake, Chen and Ou (2011) processed the ambient vibration responses of a cable-stayed bridge. Without using the random decrement technique, Hazra and Narasimhan (2010) directly determined the modal parameters of a building from its ambient vibration responses by using an approach that combines second-order blind identification technique with the station-ary wavelet transform. Notably, the approaches, which

were developed for processing the forced and free vibration responses, are theoretically applicable to pro-cess the Randomdec signatures. Nevertheless, the ref-erence degree of freedom chosen to construct the Randomdec signatures may significantly affect the ac-curacy of modal parameters identified from the Ran-domdec signatures. In addition to that, the ambient vibrations are converted into free decay responses with a relatively short duration, and the edge effect in the wavelet transform can have considerable effects on ac-curately determining the modal parameters.

This work develops a novel approach in the time– frequency domain without the assistance of the random decrement procedure. The proposed approach com-bines the AR model and the stationary wavelet packet transform. The measured ambient vibration responses of a structure are expanded by the stationary wavelet packets, and the correlation functions among the mea-sured degrees of freedom are estimated in the needed subspaces expanded by stationary wavelet packets. No-tably, the correlation matrices in the stationary wavelet packet space is formulated herein for the first time in lit-erature. Then, the coefficient matrices in the AR model are simply estimated using a traditional least-squares scheme. Modal parameters of the structure are directly determined from those coefficient matrices. The fine fil-tering property in the stationary wavelet packet trans-form enhances the ability of the proposed approach to identify not only the modes with strong modal in-terference, but also many modes from the responses of very few measured degrees of freedom. Validity of the proposed approach is verified by processing the nu-merically simulated responses of a seven-degrees-of-freedom system, which has closely spaced modes, with considering the effects of noise and incomplete mea-surements. Furthermore, the proposed approach is ap-plied to process the velocity responses of an eight-storey steel frame subjected to white noise input in the shak-ing table tests and the ambient vibration responses of a cable-stayed bridge with a main span of 330 m and a side span of 180 m.

2 METHODOLOGY 2.1 AR model

As is well known, the ambient vibration responses of a system can be described by the AR model with the order I, denoted by AR(I), which is mathematically ex-pressed as y(t)= I  i₌₁ iy(t − i) + a(t) (1)

where y(t− i) is the measured velocity response vector at time t− it; 1/t is the sample rate of mea-surement; i is coefficient matrix; and a(t) belongs to a vector white-noise process with the following proper-ties

E[a(t)]= 0 and E [a(t − i) aT(t− j)] = δi jW (2) where E[] denotes the mean-value operation,δi j repre-sents the Kronecker symbol, and W is the covariance matrix of a(t).

Multiplying yT_{(t− k) by both sides of Equation (1),} taking the mean values and using the fact that

E[a(t)yT_{(t− k)] = 0 yield the following relationship}

R (−k) = I 

i₌₁

iR (i− k) (3)

where R(k)= E[y(t)yT_{(t+ k)]. Applying R(−k) =} RT_{(k) to Equation (3) yields} RT(k)= I  i₌₁ iRT(k− i) (4)

By using different k in Equation (4), the following equation can be derived from Equation (4):

ˆ R= ˜R ˜ (5) where ˆ R=RT_(k i+ 1) RT(ki+ 2) · · · RT(ki+ m) T (6a) ˜ R= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ R(ki) R(ki− 1) · · · R(ki− I + 1) R(ki+ 1) R(ki) · · · R(ki− I + 2) .. . ... ... ... R(ki+m−1) R(ki+m−2) · · · R(ki+m−I +2) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6b) ˜  =12· · · I T (6c) Notably, ki ≈ I + (4 ∼ 14) was suggested by Huang (2001) for a sample rate for measurement between 50 and 200 Hz. Such setting of ki makes ˜R exclude R(0), which is heavily affected by noise. Equation (5) nor-mally represents a set of overdeterminant algebraic lin-ear equations. Consequently, the coefficient matrices in the AR model are determined in a least-squares error sense,

˜

 =_R˜T _˜

Fig. 1. Decomposition of subspaces.

2.2 Estimation of correlation functions in wavelet domain

The correlation functions are estimated from the mea-sured responses by using the stationary wavelet packet transform. Owing to that the wavelet transform pro-vides information in the time and frequency domains simultaneously, the correlation functions correspond-ing to the frequency range of interest can be con-structed by selecting appropriate wavelets. Notably, the wavelet packet transform provides a richer decompo-sition and allows for better time–frequency localiza-tion of signals than the decomposilocaliza-tion of signals by the discrete wavelet transform. The stationary wavelet packet transform is superior to the wavelet packet method in that the former is a translation–invariance transform, explaining its excellent computational ef-ficiency for functions with time shift. Moreover, the stationary wavelet packet transform is implemented without down-sampling and maintains all of the ele-ments in the coefficients across all of the decomposition levels.

Let the discrete signal f(t) belong to the space U₀(0), it is decomposed into as many subspaces as needed and expressed as (Figure 1; Walden and Cristan, 1998; Fan and Zuo, 2006; Zhong and Oyadiji, 2007; Jiang and Adeli, 2004) U₀(0) = U₁(0)⊕ U₁(1) = U(0) 2 ⊕ U (1) 2 ⊕ U (2) 2 ⊕ U (3) 2 = · · · = U(0) k ⊕ U (1) k ⊕ U (2) k ⊕ · · · ⊕ U (2k₋₂₎ k ⊕ U (2k₋₁₎ k (8)

where ⊕ denotes the direct sum of two subspaces; subscript k denotes the level of decomposition; and

U_k(2₋₁(k−1)−n) is decomposed into U_k(2k−2n+1) and U_k(2k−2n)

with n≤ k, and U_k(i )⊥U_k( j )for i= j. Let

Uk(m)=closespan μm,k,l(t)= 2−k/2μm  2−k(t− tl)  ; l∈ Z  (9)

wheretl= ltand{μm,k,l(t), l ∈ Z}is a set of orthonormal basis functions for Uk(m). Because the measured signals are discrete, functions are expressed in discrete form. The basis functions satisfy

 μm,k,l, μm,k, j= t  ¯n∈Zμm,k,l(t)μm,k, j(t) =  1 when l= j 0 when l= j (10a)  μm,k,l(t), μn,i, j(t)= 0 when n = m (10b)

Constructing the basis functions, begins with choos-ing a mother wavelet functionψ(t) and the correspond-ing scale functionφ(t). Let μ0(t¯n)= φ(t¯n) andμ1(t¯n)= ψ (t¯n), then μ2m,k+1,l(t)= √ 2 n∈Z hk[n]μm_,k,l(2t− tn) (11a) μ2m+1,k+1,l(t)= √ 2 n∈Z gk[n]μm_,k,l(2t− tn) (11b) where h0[n]= φ(t), φ(2t − tn)/ √ 2 and g0[n]= ψ(t), ψ(2t − tn)/ √

2 are the low-pass filter and high-pass filter, respectively. The stationary wavelet packet transform is calculated in a similar manner as the wavelet packet transform, except that no decimation is performed after each filtering step (Fowler, 2005; Jose et al., 2008). The low-pass filter and high-pass at level k+ 1 are defined recursively as

hk+1[n]= ⎧ ⎨ ⎩ hk  n2 n even 0 nodd ; gk+1[n]=  gk  n2 n even 0 n odd

Decomposing f(t) for l ∈ Z into subspaces U_k(m) with m= 0, 1, 2, . . . , 2k_{− 1, yields} f(t¯n)= 2k₋₁  m₌₀  l₌₀ am_,k,lμm_,k,l(t¯n) (12)

where am,k,l refers to the stationary wavelet packet co-efficients in subspace U_k(m)and

am,k,l = 

f(t), μm,k,l(t) 

If f_k(m)(t) denotes the signals of f(t) decomposed into subspace U_k(m), Equation (12) reveals that the frequency range of fk(m)(t) depends only on the frequency range of

μm_,k,l(t). Notably, the stationary wavelet packet satisfies the translation–invariance property, leading to

f (t− tn)= 2k₋₁  m=0  l=0 ¯am,k,lμm,k,l(t) and ¯am_,k,l = am_,k,l−n (14)

The correlation functions can be estimated by (Bendat and Piersol, 1993)

R(k)= 1 L

 ¯n∈Z

y(t¯n)yT(t¯n+ tk) (15) where L= ¯T /t denotes the record length of measure-ment; y(t)= 0 when t < 0 and t > ¯T , which occurs in the measured data. Using Equation (12),

y(t)= 2j₋₁  m=0 L  l=0 {am, j,l}μm, j,l(t) (16) Based on Equations (14) and (15), Equation (16) is rewritten as R(k)= _L1  ¯n∈Z  2_j₋₁ m₌₀ L−k l₌₀ {am, j,l}μm, j,l(t¯n) ×2 j₋₁  m=0 L−k l=0 {am, j,l+k}Tμm, j,l(t¯n)  (17)

By using the orthogonal properties given in Equation (10), Equation (17) is further simplified as

R(k)= 1 Lt 2j₋₁  m₌₀ L_−k  l₌₀ {am_{, j,l}}{am_{, j,l+k}}T = 1_¯ TRw(k) (18)

where R_w(k)=2_mj₌₀−1_lL₌₀−k{am, j,l}{am, j,l+k}T. To deter-mine the coefficient matrices in the AR model from Equation (7), R(k) is replaced by Rw(k).

The stationary wavelet packet transform has fine fil-tering properties in frequency, which is highly effective in identifying the modal parameters of a structure. Us-ing the signals in a specific subspace, for example, U( ¯_jm), to estimate Rw(k) yields R_w(k)= L−k  l=0 {am¯, j,l}{am¯, j,l+k}T (19)

Fig. 2. Plots for Meyer mother waveletψ(t) and

corresponding scale functionφ(t).

Fig. 3. Fourier spectra of Meyer wavelet and scale function.

If the Fourier transform of μm¯, j,l(t) has a compact support with the frequency range [ fa, fb], subspace U

( ¯m) j corresponds to the frequency range [ fa, fb]. Also, Rw(k) in Equation (19) is applied to Equation (7) to deter-mine the coefficient matrices in the AR model and iden-tify the natural frequencies of a structure by following the procedure given in Section 2.4. Doing so leads to the identified modal frequencies lying in the frequency range [ fa, fb].

2.3 Meyer stationary wavelet packets

The orthonormal basis functions in each subspace are constructed using Meyer mother wavelet ψ(t) and its corresponding scale function φ(t) (Mallat, 1999) (Figure 2). Figure 3 shows the Fourier spectra of these two functions,| ˆψ(ω)| and | ˆφ(ω)|, indicating that ψ(t) is a band-pass filter andφ(t) is a low-pass filter.

Fig. 4. Plots for Meyer mother waveletψ(t) and corresponding scale function φ(t).

No overlapping frequency ranges among subspaces U_k(m)with different m allow for the convenient selection of an appropriate subspace to estimate the correlation function matrices through Equation (19) and identify the modal parameters of a structure in the frequency range of interest. In this work, the frequency ranges of ψ(t) and φ(t) are defined by the frequency ranges corre-sponding to 50√2% of| ˆψ(ω)| and | ˆφ(ω)|, respectively, that is, [0.5, 1] Hz and [0, 0.5] Hz (Figure 3). Conse-quently, the frequency range of subspace Uk(m)is deter-mined by the following procedure:

1. Express m in a vector of binary, v= (c1, c2, ..., c¯n), where ciis either 0 or 1.

2. Define a vector ¯v= ¯c¯n⊗ ¯c¯n−1⊗ · · · ⊗ ¯c2⊗ ¯c1, where ⊗ is the Kronecker tensor product

operator, and ¯ci are set to (1, 1) and (1, −1) for ci = 0 and 1, respectively. The components of

¯v are either−1 or 1.

3. Compute the number of zero crossing in the se-quence of the components of ¯v, and denote the number byη.

4. The frequency range of Uk(m) is

[η fs/.2k+1, (η + 1) fs/.2k+1], where fs is the sampling frequency.

For example, to find the frequency range of U₃(6), the binary expression of 6 is 110, v= (1, 1, 0), ¯v = [1, 1] ⊗ [1, − 1] ⊗ [1, − 1] = [1, − 1, − 1, 1, 1, − 1, − 1, 1], and η = 4. Thus, the frequency range of U₃(6) is [4 fs/23, 5 fs/23]. Figure 4 shows the frequency ranges of subspaces that may be used later.

Fig. 5. Fourier spectra of simulated responses of first, third,

and seventh floors.

2.4 Determination of modal parameters

By adopting the concept behind the Ibrahim time do-main system identification approach, Yang et al. (1994) and Huang (1999) proved that the modal parameters of a structure can be directly determined from the eigen-values and eigenvectors of the following matrix,

G= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 I 0 0 · · · 0 0 0 I 0 · · · 0 .. . ... ... ... · · · ... I I−1 I−2 I−3 · · · 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ¯ N_{× ¯N} (20)

where I is an ¯l× ¯l unit matrix; ¯l is the dimension of y(t), and ¯N = I × ¯l.

Let λk and ϕk denote the kth eigenvalue and eigen-vector of G, respectively. The eigenvalueλkis normally a complex number, and is set to ˜ak + i ˜bk. The natural frequency and modal damping ratio of a structure are calculated by ˜ βk=  α2 k+ βk2, ξk = −αk/ ˜βk (21) where βk= 1 ttan−1 _˜b k ˜ak  , αk= 1 2t ln ( ˜a 2 k+ ˜bk2) (22) The eigenvector ϕk can be expressed as (ϕT k1ϕ T k2...ϕ T k I) T _{where ϕ}

ki has ¯l components and satisfies ϕki = λkϕk(i−1), and ϕk1 corresponds to a mode shape of the measured degrees of freedom of a structure.

It is worthwhile to mention that spurious modes other than real structural modes occur when ¯N > 2n where n is the total number of degrees of freedom of the structural system under consideration and is usually un-known. The real structural modes should consistently arise as I, that is, the order of the AR model, in-creases. Hence, the modal parameters of a structure can be determined based on the stabilization diagrams (Kullaa, 2000), which display the variations of the iden-tified modal parameters with I. Good engineering judg-ment based on the knowledge of the structural system can be very helpful in identifying the structural modes.

3 NUMERICAL VERIFICATION 3.1 Determination of modal parameters

Numerically simulated velocity responses of a seven-storey shear building under base excitation of white noise input were processed to demonstrate the accuracy and effectiveness of the proposed approach in identify-ing the modal parameters. The theoretical natural fre-quencies of the shear building were 0.72, 2.21, 3.20, 4.11, 5.01, 5.26, and 8.01 Hz, and the corresponding modal damping ratios were 0.4%, 1.4%, 2.0%, 2.6%, 3.1%, 3.3%, and 5.0%. The responses of the shear building with a duration of 300 seconds were determined using the Runge–Kutta method witht = 0.005 seconds. No-tably, the frequencies of the fifth and sixth modes are too close to be distinguished from the Fourier spectra in Figure 5.

The agreement between the identified and theoretical mode shapes is indicated by the modal assurance crite-rion (MAC) (Allemang and Brown, 1983),

MAC( ¯ϕi I, ¯ϕi A)= ϕ¯T i Iϕ¯i A 2 ¯ ϕT i Iϕ¯i Iϕ¯Ti Aϕ¯i A (23)

where ¯ϕi I and ¯ϕi A are the identified and analytical ith mode shapes, respectively. The value of MAC is be-tween zero and unity, and the closer to one of MAC indicates two mode shapes more similar.

For convenience, the identified results are consid-ered as “accurate” when the identified frequencies and modal damping ratios are within 2% and 20% of the

Table 1

Identified modal parameters of a seven-floor shear building

Noise Subspace Orders Modal Damping

NSR (frequency range) of AR I frequencies fn(Hz) ratiosξ (%) MAC

0% U₂(0)(0 to 25 Hz) 2 0.72 (0%) 0.4 (0%) 1.00 2.21 (0%) 1.4 (0%) 1.00 3.20 (0%) 2.0 (0%) 1.00 4.11 (0%) 2.5 (−3.85%) 1.00 5.01 (0%) 3.2 (3.22%) 1.00 5.26 (0%) 3.2 (−3.03%) 1.00 8.00 (−0.12%) 5.0 (0%) 1.00 U₃(0)(0 to 12.5 Hz) 2 0.72 (0%) 0.4 (0%) 1.00 2.21 (0%) 1.4 (0%) 1.00 3.20 (0%) 2.0 (0%) 1.00 4.11 (0%) 2.5 (−3.85%) 1.00 5.01 (0%) 3.4 (9.68%) 1.00 5.25 (−0.19%) 3.3 (0%) 1.00 8.03 (0.25%) 5.0 (0%) 1.00 20% U₃(0)(0 to 12.5 Hz) 7 0.72 (0%) 0.4 (0%) 1.00 2.21 (0%) 1.4 (0%) 1.00 3.20 (0%) 2.0 (0%) 1.00 4.12 (0.24%) 2.5 (−3.85%) 1.00 5.01 (0%) 3.2 (3.23%) 1.00 5.26 (0%) 3.5 (6.06%) 1.00 7.99 (−0.25%) 4.8 (0%) 0.99 U₃(0)(0 to 12.5 Hz) 2 0.72 (0%) 0.4 (0%) 1.00 2.21 (0%) 1.4 (0%) 1.00 3.20 (0%) 2.0 (0%) 1.00 4.12 (0.24%) 2.5 (−3.84%) 1.00 U₃(0)(4.70 to 6.25 Hz) 2 5.01 (0%) 2.6 (−16.13%) 1.00 5.25 (−0.19%) 2.8 (−15.15%) 1.00 U₃(0)(0 to 12.5 Hz) 2 7.97 (−0.50%) 5.2 (4%) 0.98

theoretical ones, respectively, and the MAC values ex-ceed 0.9 as well. The identified natural frequencies are typically much more accurate than the identified modal damping ratios. Therefore, smaller errors are required for the identified natural frequencies than for identified modal damping ratios when defining accurate results.

Table 1 lists the identified modal parameters obtained using the responses of all floors. The relative differ-ences between the identified results and true values are given in parentheses. The responses are transformed into different subspaces by the stationary wavelet pack-ets. Highly accurate results are obtained using AR(2) and the signals in U₂(0)and U₃(0), whose frequency ranges are [0, 25] Hz and [0, 12.5] Hz, respectively.

3.2 Effects of noise and incomplete measurements In real applications, the measured data always contain a certain level of corrupted noise. To simulate the noise

effect, 20% variance of the noise-to-signal ratio (NSR) was randomly added to the computed responses. When the signals of all floors in subspace U₃(0) are employed to find the modal parameters, AR(I) models with I ≥ 7 are needed to have accurate results, and the results us-ing AR(7) are given in Table 1. The difficulties in accu-rately identifying modal parameters of the closed modes (fifth and sixth modes) result in using such large order of AR model. If the signals in subspace U₆(2), whose fre-quency range is [4.70, 6.25] Hz, are used for identifying the modes whose frequencies are inside the frequency range, Table 1 reveals that AR(2) yields accurate re-sults, which are given in Table 1.

In practice, a highly dense sensor network is sel-dom used for measurement even with the increasingly cheap price of a monitoring system. Correspondingly, Table 2 summarizes the results obtained using the noisy responses of the first floor. The relative differences be-tween the identified results and true values are given in

Table 2 Identified modal parameters of a seven-floor shear building using the noisy responses of the first floor Present Randomdec Huang and S u (2007) Scale factor in Subspace Orders wavelet transform a Orders (frequency range) o f A R I fn (Hz) ξ (%) (frequency range) o f A R I fn (Hz) ξ (%) U (0) 3 (0 to 12.5 Hz) 5 0 / / 1 , 0.4, 0.3, 0.2 and 0.1 (0.5 to 10 Hz) 5 0 / / 2.41 (9.05%) 3 .5 (150%) 2.32 (4.98%) 4.1 (193%) // // 4.06 (− 1.22%) 4 .0 (53.84%) 4.05 (− 1.46%) 4.9 (88.46%) // // // // 8.01 (0%) 4.5 (− 10%) 8.13 (1.50%) 5.8 (16%) U (0) 5 (0 to 3.1 Hz) 1 0 0.72 (0%) 0.4 (0%) 1 (0.5 to 1 H z) 8 0.72 (0%) 0.4 (0%) 2.20 (− 0.45%) 1 .4 (0%) 0.4 (1.25 to 2.5 Hz) 1 1 2.21 (0%) 1.4 (0%) U (3) 6 (3.1 to 4.7 Hz) 1 0 3.20 (0%) 1.9 (− 5%) 0.3 (1.66 to 3.33 Hz) 1 5 3.2 (0%) 1.9 (− 5%) 4.11 (0%) 2.5 (− 3.84%) 0.2 (0.25 to 5 H z) 12 4.12 (0.24%) 2.5 (− 3.84%) U (2) 6 (4.7 to 6.25 Hz) 1 2 4.98 (− 0.59%) 2 .6 (− 16.12%) 0.11 (4.54 to 9.09 Hz) 100 4.98 (− 0.60%) 2.2 (− 29.03%) 5.24 (− 0.38%) 3 .5 (6.06%) / / U (3) 5 (6.25 to 9.4 Hz) 5 8.02 (0.12%) 4 .5 (− 10%) 0.1 (5 to 10 Hz) 9 8.08 (0.87%) 5.1 (2%) Note: “/” d enotes that the m odes a re not identified.

Fig. 6. Stabilization diagrams of identified frequencies for a

seven-floor shear building: (a) using signals in U₃(0)only,

(b) using signals in different subspaces for different modes.

parentheses. When the signals in subspace U₃(0)are used, even the AR(50) model does not yield accurate results for all of the modes. The stabilization diagram on fre-quencies given in Figure 6a shows that the frefre-quencies of the first, third, fifth, and sixth modes are not identified when the order of AR model is less than 50 and demon-strates the difficulties of identifying accurate results of all modes from the signals in the subspace that covers the frequency range of all modes.

To resolve the above difficulties, different modes are identified using the signals in different subspaces. Table 2 demonstrates the feasibility of obtaining the ac-curate results of the first and second modes by using the AR(10) model in cooperation with the signals in U₅(0) that has a frequency range of [0, 3.1] Hz. Mean-while, the AR(10) model along with the signals in U₆(3) with a frequency range of [3.1, 4.7] Hz yields accurate

results of the third and fourth modes. Accurate results for the fifth and sixth modes can be obtained using the AR(12) model along with the signals in U₆(2)with a fre-quency range of [4.7, 6.25] Hz. Additionally, the AR(5) model along with the signals in U₅(3) with a frequency range of [6.25, 9.4] Hz yields accurate results for the seventh mode. According to the stabilization diagram in Figure 6, the real modes occur stably when increas-ing the orders of AR model, and no spurious modes are found in the frequency ranges of interest.

To compare the present approach with an existing ap-proach, Table 2 also shows the results obtained using the approach of Huang and Su (2007), which combines the ARX model with the continuous wavelet trans-form, to process the Randomdec signatures extracted from the noisy responses using the random decrement technique. In processing the Randomdec signatures, which are equivalent to free decayed responses, the AR model, instead of the ARX model, should be used. Meyer wavelets were used in the wavelet transform, and a simple method proposed by Sol´ıs et al. (2013) was em-ployed to avoid wavelet transform edge effects. When several values of scale parameter (a) in the continuous wavelet transform are applied to identify the modal pa-rameters of all modes, similar to the present approach, the AR(50) model does not yield accurate results for all of the modes. When different values of a are used for identifying modal parameters of different modes, the AR(100) model does not give accurate result for the close modes (the fifth and sixth modes).

4 APPLICATIONS

Following its verification based on numerical simula-tion, the proposed approach is used to process the mea-sured responses of an eight-storey, symmetrical steel frame under shaking table tests and ambient vibration responses of a cable-stayed bridge. In practical appli-cations, the true modal parameters are unknown and the criteria given in the previous section for judging accurately identified results are inapplicable. However, real mechanical modes occur stably when increasing the order of AR model, whereas the spurious modes do not. The stabilization diagrams contribute to the eval-uation of stable results. Modes identified at the current AR(I) are compared with the results obtained from the next six consecutive orders of AR (i.e., AR(I+ k) and k = 1, 2, . . . ,6). Importantly, the results are consid-ered stable if (a) the maximum relative differences in frequency and damping ratio for each mode are less than 2% and 20%, respectively, and (b) the values of MAC are higher than 95%. Engineering judgment is

Fig. 7. A photo for an eight-storey steel frame under shaking

table tests.

also highly effective in distinguishing real mechanical modes from spurious modes.

4.1 Eight-storey frame under shaking table tests The eight-storey steel frame analyzed in this test was 1.5 m in length, 1.1 m in width, and 9.44 m in height (Figure 7). Lead plates of 250 kg were piled on each floor, such that the total mass of each floor was ap-proximately 325 kg. The columns had H-shaped sections (H100×100×7.5×7.5). The frame was subjected to base excitation of white noise in long-span direction. A ve-locity sensor was placed at the center of each floor. The responses were recorded for 250 seconds with a sam-pling rate of 200 Hz.

Figure 8 displays the Fourier spectra of the responses at the first, third, and eighth floors, in which the peaks of the first seven modes under 40 Hz are clearly shown. Frequency of the eighth mode looks like that between 40 and 45 Hz (see the Fourier spectrum for the third floor). Table 3 lists the identified results obtained by using the signals of all floors in U₁(0), whose frequency range is [0, 50] Hz. The identified modal parameters ap-pear stably when AR(I) with I ≥ 5 are used. The paren-thesized results in Table 3 are the natural frequencies

Fig. 8. Fourier spectra of the responses of an eight-storey

steel frame at first, third, and eighth floors.

obtained from the commercial finite element package, ETABS (2013), via the designed data. Following a typi-cal way of applying ETABS to construct a finite element model for a real building, a 3D beam element was uti-lized to model a column in each storey, and rigid floor assumption was made. Apparently, the identified natu-ral frequencies and those from ETABS significantly dif-fer from each other, especially for the higher modes. Notably, the identified natural frequencies correspond very well with those associated with the peaks of the Fourier spectra in Figure 8. The values of MAC indicate the excellent correlation between the identified mode shapes and those of ETABS.

Table 3 also summarizes the identified results using the responses of the first floor only, whereas Figure 9 depicts the stabilization diagrams on frequencies ob-tained from signals in different subspaces. When the signals in U₁(0) are used, Figure 9a reveals that AR(I) with I ≥ 32 are needed to obtain stable results for all of the eight modes. Some spurious modes occur in the frequency range of interest, causing difficulty in de-termining the modal parameters precisely. The iden-tified natural frequencies and modal damping ratios closely correspond to those obtained from complete

Table 3

Identified modal parameters of an eight-storey steel frame under shaking table tests

Subspace

Responses (frequency range) fn(Hz) ξ (%) MAC

All floors U₁(0)(0 to 50 Hz) 2.11 (2.01) 1.4 0.99 7.00 (6.49) 0.9 0.99 12.86 (12.38) 0.7 0.99 19.15 (19.80) 0.4 0.99 25.89 (26.46) 0.9 0.97 33.42 (34.36) 1.0 0.97 39.36 (44.84) 1.3 0.96 41.63 (56.80) 2.1 0.96

First floor only U₁(0)(0 to 50 Hz) 2.09 4.0 /

6.97 0.8 / 12.85 0.7 / 19.15 0.4 / 25.89 0.9 / 33.42 1.0 / 39.32 1.0 / 42.34 2.5 / U₃(0)(0 to 12.5 Hz) 2.06 1.3 / 6.98 0.9 / U₃(1)(12.5 to 25.5 Hz) 12.85 0.7 / 19.15 0.4 / U₃(3)(25.5 to 37.5 Hz) 25.89 0.9 / 33.42 0.9 / U₃(2)(37.5 to 50 Hz) 39.35 0.9 / 41.60 2.1 /

Note: “/” denotes data not available.

measurements, except for the damping ratios of the first mode.

When the modal parameters of different modes are identified using the signals in different subspaces,

Figure 9b reveals that a markedly smaller order of AR model than that for using the signals in U₁(0) (see Figure 9a) is needed to achieve stable results; no spurious modes occur in the frequency ranges of

Fig. 9. Stabilization diagrams of identified frequencies for an eight-storey steel frame using the responses of the first floor only:

Fig. 10. (a) A photo of the cable-stayed bridge (provided by Taiwan Area National Expressway Engineering Bureau), (b) sensor

layout for ambient vibration measurement.

Fig. 11. Fourier spectra of vertical responses at x= 25, 260, and 420 m.

Fig. 12. Fourier spectra of transverse responses at x= 25, 260, and 420 m.

interest. The identified modal parameters are consis-tent with those obtained from complete measurements (Table 3).

4.2 Ambient vibration of a cable-stayed bridge

In a second example, the proposed approach was used to process the ambient vibrations of a highway cable-stayed bridge, which is the largest spanned bridge in Taiwan (Figure 10). The bridge has a deck of 34.5 m in width, a main span of 330 m with a steel box girder type of deck, a side span of 180 m with a prestressed

concrete box girder type of deck, and an inverted Y-shaped reinforced concrete pylon of 183.5 m in height. Each side of the pylon has 14 pairs of cables. The back stays consist of four cables, and the other stays have two cables. Thirty-three measuring stations were arranged along the centerline of the bridge deck to measure the ambient vibrations. Most of the stations were located near the connections of cables to the deck. Owing to the limited number of sensitive servo-velocity sensors, the tested bridge was divided into three seg-ments (Figure 10). Two overlapping stations for any two adjacent segments were utilized to correlate the

Table 4

Identified modal parameters of a cable-stayed bridge from ambient vibration responses

Present Chen & Ou (2011)

Direction Subspace (frequency range) fn(Hz) ξ (%) MAC fn(Hz) ξ (%) MAC

Vertical U₃(0)(0 to 6.25 Hz) 0.27 (0.29) 2.6 1 0.28 2.9 1 0.59 (0.56) 2.5 1 0.57 3.7 0.98 0.90 (0.93) 2.6 1 0.92 4.4 0.97 1.58 (1.52) 2.7 1 1.54 3.9 0.97 1.86 (1.79) 2.6 0.97 1.81 3 0.97 2.36 (2.52) 2.5 1 / / / 3.11 (3.27) 2.9 0.99 / / / 3.93 (4.02) 2.7 0.98 / / / Transverse U₃(0)(0 to 6.25 Hz) 0.65 (0.65) 2.6 1 0.64 3.3 0.99 1.64 (1.68) 2.7 1 1.64 2.9 0.99 2.19 (2.15) 2.4 1 2.17 3.2 0.99 2.57 (2.49) 2.9 1 2.51 2.5 0.99 3.17 (3.15) 2.6 0.99 3.13 3.9 0.99 3.89 (3.95) 2.8 1 / / /

Note: “/” denotes data not available.

mode shapes identified for each segment. Responses in the vertical and transverse directions were measured separately. The ambient vibration responses of the bridge were recorded for 10 minutes with a sampling rate of 100 Hz.

Figures 11 and 12 display the Fourier spectra of the measured ambient vibrations at some stations in vertical and transverse directions, respectively. Because most of the peaks in the Fourier spectra occur in frequen-cies smaller than 5 Hz, the modes with frequenfrequen-cies un-der 4 Hz were identified using the proposed approach. Because the different segments were not measured si-multaneously, the measured responses in different seg-ments were processed separately.

Table 4 lists the average values of natural frequencies and modal damping ratios obtained from measured re-sponses in three segments. The modal parameters were identified using the signals in subspace U₃(0) with a fre-quency range of [0, 6.25] Hz. Figures 13 and 14 show the identified mode shapes in vertical and transverse direc-tions, respectively. Cumulatively, eight modes in verti-cal direction and six modes in transverse direction were identified. The stable results were determined by using AR(14). The parenthesized results in Table 4 are the natural frequencies obtained from the commercial fi-nite element package, SAP2000 (2013), via the designed data with some modifications in supports and connec-tions suggested by Chen and Ou (2011). The finite ele-ment model consisted of 33 and 38 beam eleele-ments for the deck and the pylon, respectively, and 28 cable el-ements for the stay cables. The MAC values refer to

the correlation between the identified mode shapes and those from the finite element analysis. Figures 13 and 14 also show the vertical and transverse mode shapes obtained from the finite element method, respectively. The identified natural frequencies correlate well with the finite element results, whereas the identified mode shapes closely correspond to those from the finite ele-ment analysis with MAC values larger than 0.97.

For comparison, Table 4 also lists the identified re-sults of Chen and Ou (2011), who applied the approach of Huang and Su (2007) to process the Randomdec sig-natures extracted from the ambient vibration responses. They identified five modes in both of the vertical and transverse directions. The mode shapes are also given in Figures 13 and 14. The MAC values refer to the correlation between the mode shapes identified by the present approach and those of Chen and Ou (2011). The natural frequencies and mode shapes of Chen and Ou (2011) show excellent agreement with those ob-tained from the present approach, whereas reasonable agreement is found for most of the modal damping ratios.

For a situation in which only the responses of a sta-tion were processed, Table 5 lists the identified mode parameters using the responses at x = 260 m in the second segment. Again, the results were obtained us-ing the signals in different subspaces in wavelet domain cooperating with AR modes having different orders. Figures 15 and 16 show the stabilization diagrams of identified frequencies in vertical and transverse direc-tions, respectively. The excellent agreement between

Fig. 13. Mode shapes in vertical direction.

the modal parameters in Tables 4 and 5 again indi-cates the ability of the proposed approach to accurately find the natural frequencies and damping ratios of many modes from responses of very few degrees of freedom. Figures 15a and 16a reveal some spurious modes whose frequencies are near the natural frequencies. Addition-ally, when the signals in subspace U₃(0)are used to deter-mine the modal parameters listed in Table 5, the AR(I) model with I ≥ 31 is required to obtain stable results. Again, using signals in different subspaces to identify different modes eliminates the spurious modes and sub-stantially reduces the needed orders of AR model to ob-tain stable results (Figures 15b and 16b).

5 CONCLUDING REMARKS

This work has developed a wavelet-based approach for identifying the modal parameters of a structure from its ambient vibration responses. The stationary wavelet packet transform is applied to the ambient vibra-tion responses, and the correlavibra-tion funcvibra-tions between measured degrees of freedom can be estimated us-ing the signals in the subspaces in the wavelet do-main. Modal parameters of the structure are then deter-mined from the coefficient matrices of the AR model, which are established using those correlation func-tions. Compared to the discrete wavelet transform and

Fig. 14. Mode shapes in transverse direction. Table 5

Identified modal parameters of a cable-stayed bridge using ambient vibration responses of a station only Direction Subspace (frequency range) fn(Hz) ξ (%)

Vertical U₅(0)(0 to 1.55 Hz) 0.27 2.5 0.58 2.7 0.89 2.4 U₅(1)(1.55 to 3.13 Hz) 1.56 2.9 1.86 2.8 2.35 2.7 3.10 2.9 U₅(3)(3.13 to 4.7 Hz) 3.90 2.9 Transverse U₄(0)(0 to 3.13 Hz) 0.66 2.5 1.61 2.6 2.18 2.7 2.58 2.8 U₄(1)(3.13 to 6.25 Hz) 3.17 2.8 3.89 2.6

wavelet packet transform, the stationary wavelet packet transform satisfies the translation invariance property and saves the computational time in constructing the

correlation functions. The Meyer wavelets were used in the proposed approach because Meyer mother wavelet is a band-pass filter, and the corresponding scale func-tion is a low-pass filter. The fine filtering properties in the stationary wavelet packet transform enhance the ability of proposed approach to identify both the closely spaced modes and many modes from the responses of very few measured degrees of freedom.

The present approach was validated first by suc-cessfully processing the numerically simulated veloc-ity responses of a seven-storey shear building with closely spaced modes of the fifth and sixth modes under base excitation of white noise. Natural frequencies and modal damping ratios of the seven modes were identi-fied easily and accurately even when the responses of only one degree of freedom with 20% noise were used.

To demonstrate the applicability of the proposed ap-proach to process the real measured data, this work also analyzed the velocity responses of an eight-storey steel frame under shaking table tests and the ambient vibra-tion responses of a highway cable-stayed bridge sub-jected to traffic flow. Eight modes of the steel frame and

Fig. 15. Stabilization diagrams of identified frequencies for a cable-stayed bridge using the vertical responses at x= 260 m:

(a) using signals in U₃(0)only, (b) using signals in different subspaces for different modes.

Fig. 16. Stabilization diagrams of identified frequencies for a cable-stayed bridge using the transverse responses at x= 260 m:

(a) using signals in U₃(0)only, (b) using signals in different subspaces for different modes.

eight modes in vertical direction and six modes in trans-verse direction of the bridge were identified even when the measured responses of only one degree of freedom were used.

During the processing of the numerically simulated responses and measured responses, our results indi-cate two main advantages of using signals in differ-ent subspaces in wavelet domain to iddiffer-entify differdiffer-ent modes over using signals in a subspace to identify all of the modes. Those advantages are that the AR mode with markedly lower order is needed and that spuri-ous modes occur to a lesser extent. Importantly, the proposed approach can identify the natural frequen-cies and modal damping ratios of many modes from the

responses of only one degree of freedom. The capacity is highly useful for long-term structural health monitor-ing usmonitor-ing vibration-based approaches in a cost-effective manner.

ACKNOWLEDGMENTS

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for finan-cially supporting this research under Contract No. NSC 101–2625-M-009–007. The appreciation is also extended to the National Center for Research on Earthquake En-gineering for providing shaking table test data.

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