2012 - Response-Only Modal Identification of Structures Using Strong Motion Data

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    Response-only modal identification of structures using strongmotion data

    S. F. Ghahari1, F. Abazarsa2, M. A. Ghannad1 and E. Taciroglu3,*,

    1Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iran2Structural Engineering Department, Int. Institute of Earthquake Eng. and Seismology, P.O. Box 3913/19395, Tehran, Iran

    3Civil and Environmental Engineering Department, 5731E Boelter Hall, University of California, Los Angeles,

    CA 90095, USA

    SUMMARY

    Dynamic characteristics of structures viz. natural frequencies, damping ratios, and mode shapes are

    central to earthquake-resistant design. These values identied fromeld measurements are useful for modelvalidation and health-monitoring. Most system identication methods require input excitations motions to bemeasured and the structural response; however, the true input motions are seldom recordable. For example,when soilstructure interaction effects are non-negligible, neither the free-eld motions nor the recordedresponses of the foundations may be assumed as input. Even in the absence of soilstructure interaction, inmany instances, the foundation responses are not recorded (or are recorded with a low signal-to-noise ratio).Unfortunately, existing output-only methods are limited to free vibration data, or weak stationary ambientexcitations. However, it is well-known that the dynamic characteristics of most civil structures are amplitude-dependent; thus, parameters identied from low-amplitude responses do not match well with those from strongexcitations, which arguably are more pertinent to seismic design. In this study, we present a new identicationmethod through which a structures dynamic characteristics can be extracted using only seismic response(output) signals. In this method, rst, the response signals spatial time-frequency distributions are used forblindly identifying the classical mode shapes and the modal coordinate signals. Second, cross-relations amongthe modal coordinates are employed to determine the systems natural frequencies and damping ratios on thepremise of linear behavior for the system. We use simulated (but realistic) data to verify the method, and also

    apply it to a real-life data set to demonstrate its utility. Copyright 2012 John Wiley & Sons, Ltd.

    Received 7 November 2011; Revised 24 September 2012; Accepted 1 October 2012

    KEY WORDS: blind system identication; modal identication; output-only techniques; strong groundmotions; soilstructure interaction; spatial time-frequency distributions

    1. INTRODUCTION

    Identication of dynamic characteristics of civil structures from response recorded during strong

    ground shaking has been a subject of research for more than three decades [14]. System

    identication techniques are not only useful for condition assessment, but also for improving futuredesigns, validating predictive models, and verifying retrot procedures [5]. Active and semi-active

    control systems that are now being frequently employed in important structures throughout the

    world also make use of system identication techniques, which provide intelligent feedback from

    the structure so that the active and passive (damping) forces supplied to the system can be properly

    controlled [6]. Although the diversity of applications continue to ourish, a major part of system

    identication research remains focused on model validation and damage assessment. Numerous

    *Correspondence to: E. Taciroglu, Department of Civil and Environmental Engineering, 5731E Boelter Hall, Universityof California, Los Angeles, CA 90095-1593, USA.

    E-mail: [email protected]

    Copyright 2012 John Wiley & Sons, Ltd.

    EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2268

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    studies have been undertaken in recent years to develop robust methods by which structural damage

    can be detected and quantied (see, e.g., [7, 8]). In most of these studies, changes in dynamic

    properties are used as indices for detecting the location and severity of damage (see, e.g., [9, 10]

    and references therein).

    Many of the existing system identication methods are common to mechanical, electrical, and

    structural engineering elds. Quite a number of those methods have been developed for output-only

    modal identication, and they are especially pertinent for civil structures, for which input excitationsare not always readily available. However, most of these techniques are designed to use free or

    ambient vibration data as input [1113] wherein the identied modal properties are related typically

    to small-amplitude vibrations. Moreover, in techniques devised for ambient vibrations, it is usually

    assumed that the input excitation is a broadband stochastic process, which is modeled as stationary

    Gaussian white noise [14]. On account of these facts, available output-only identication techniques

    cannot be employed for nonstationary response signals recorded during strong ground motions.

    Consequently, apart from a few notable exceptions [15, 16], nearly all of the existing techniques

    employed in the identication of civil structures from their seismic responses require the input

    motions to be known/measured. Yet, in most practical cases, the true foundation input motion (FIM)

    is not recorded at all, or recorded with a low signal-to-noise ratio (SNR), or cannot be recorded

    because of, for example, soilstructure interaction (SSI) effects. When there is signicant SSI,

    the FIM is different from both the free-eld motion recorded in the structures vicinity, and

    the responses recorded at the foundation level [17]. The usual contrast between the stiffness of

    a (nearly rigid) foundation and the surrounding soil causes the motion experienced by the foundation

    (i.e., FIM) to differ from the free-eld motion; an effect coined as kinematic interaction. Inertial

    interaction, which is caused by the soils exibility and attenuation, and the structures and

    its foundations masses, compounds the said difference between the foundation response and

    the FIM.

    Even in the absence of SSI, for many real cases, foundation responses are not always recorded, or

    are recorded with low SNR. As such, recorded foundation responses are usually assumed to be input

    motions in dynamic analyses, and in system identication studies. For example, Skolniket al. [18]

    used signals recorded at the ground level of a 15-story, steel-frame building (i.e., the UCLA Factor

    Building) as input motions for their nite element model updating studies. However, the Fourier

    spectra of those signals (Figure 1) indicate that they are affected by the building s own dynamic

    response: in the Fourier amplitude spectrum of the eastwest (EW) acceleration (cf., Figure 1(a)),there are two dominant peaks occurring at frequencies 0.42 and 0.65 Hz, which are very close to

    the reported rst natural frequency of the EW mode (0.47 Hz) and the natural frequency of the

    rst torsional mode (0.68 Hz), respectively. This similarity is also seen in the northsouth (NS)

    direction spectrum, wherein a dominant peak corresponding to rst natural frequency in the NS

    direction (i.e., 0.51 Hz) is observed (cf., Figure 1(b)). Clearly, the motion used as input is inuenced

    by the structures response. Examples such as these are not uncommon [19].

    Figure 1. Fourier spectra of the ground level responses of the UCLA Factor Building during the 2004 Parkeldearthquake.

    S. F. GHAHARIET AL.

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

    DOI: 10.1002/eqe

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    In this paper, we present a new system identication method through which the modal properties of

    civil structures are identied from their seismic responses without having measurements of input

    excitations. The method works in two steps:

    i. Mode shapes and modal coordinates are extracted by applying a blind source separation (BSS)

    technique to the time-frequency representation of recorded responses; and

    ii. Concurrent analyses of modal coordinates are carried out using a cross-relation (CR) technique to

    identify natural frequencies and damping ratios from modal coordinates extracted in the rst step.

    The theoretical background needed for the rst step is presented in the next section, where the time-

    frequency distribution (TFD) of a signal and also spatial TFD (STFD) of signals are briey introduced.

    This is followed by detailed descriptions of the method s two steps (i.e., BSS and CR). Performance of

    this new method is evaluated subsequently through simulated and real data examples.

    2. THEORETICAL BACKGROUND

    2.1. Time-frequency distribution of a signal

    Signals are classically represented in either the time-domain or the frequency-domain. In both forms, the

    time and frequency variables are treated as mutually exclusive. Consequently, these representations arenonlocal with respect to the excluded variable [20]. As such, they are only suitable for signals with

    time-invariant or frequency-invariant properties. TFDs are needed for nonstationary signals, or signals

    from a nonlinear system, for which the signal characteristics are changing. The theoretical background

    of TFDs is beyond the scope of this paper; and interested readers are referred to Boashashs textbook [20]

    and the references cited therein. For brevity, only the pertinent fundamental formulae are presented

    below.

    The discrete-time form of the Cohen-class of TFDs is given by [21]

    Dxx t;f X1

    l1

    X1m1

    m; l x t m l x t m l ej4pfl (1)

    where x(t) is a time signal, and tand frepresent the time and frequency variables, respectively. Thekernel function, (m,l), depends on both the time (t) and the lag (l) variables. Different choices ofthe kernel function lead to different TFD realizations. TFDs dened through Equation (1) are

    categorized as nonlinear or quadratic TFDs. Contrary to linear TFDs, such as the short-time Fourier

    transform, or the wavelet transform, the signals product with its complex conjugate is used.

    Choosing the most suitable TFD for a given situation depends on which of its characteristics are

    desired. Linear TFDs are sometimes preferred, because they are real-valued, simpler, and free of

    interference-terms; however, their simultaneous time-frequency resolutions are limited [22].

    Quadratic TFDs have higher time-frequency resolutions, but suffer from interference. Interference

    terms, which are also dubbed as outer artifacts or cross-terms [20], are spurious features that

    appear when representing a multicomponent signal in the time-frequency domain using one of the

    quadratic methods. To wit, consider the signal x(t) =x1(t) +x2(t) in which both x1(t) and x2(t) are

    analytic and monocomponent signals. Substituting x(t) into Equation (1), its TFD will be,

    Dxx t;f Dx1x1 t;f Dx2x2 t;f 2Re Dx1x2 t;f f g (2)

    in whichDx1x1 t;f andDx2x2 t;f are the auto-terms, which represent energy concentration in the time-frequency plane, whereas 2Re Dx1x2 t;f f gis a cross-term that occurs in (t,f) points in which no energyis expected at all. These cross-terms have large oscillating amplitudes on average times and frequencies

    of the true components, and can make the TFD difcult to interpret. This is especially true if the

    components are numerous (or they are close to each other), and also in the presence of noise,

    because they can be produced between the signal and the noise components. Therefore, in most

    practical applications, quadratic TFDs are not used despite their higher resolution. There are three

    RESPONSE-ONLY MODAL IDENTIFICATION OF STRUCTURES USING STRONG MOTION DATA

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

    DOI: 10.1002/eqe

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    approaches to suppress the cross-term effects: (i) by multiplying the TFD with the spectrogram, which

    is a cross-term free TFD [23]; (ii) by using one of the existing components as a reference signal [24];

    and (iii) by attenuating the cross-terms through a properly selected kernel function, which results in a

    TFD family that is referred to as reduced interference distribution. Herein, we adopt the smoothed

    pseudo-WignerVille distribution (SPWVD) [25], which is an enhanced version of WignerVille

    distribution (WVD) [26] and belongs to the said reduced interference distribution family.

    In all TFD methods, the analytic associate of the real signal is used to eliminate the unnecessarynegative frequencies without losing information. This removal has two benecial effects: rst, it

    halves the total bandwidth, allowing the signal to be sampled at half the usual Nyquist rate without

    aliasing [27, 28]; second, it avoids the appearance of various interference terms generated by the

    interaction of positive and negative components in quadratic TFDs [20]. The analytic associate,xa(t), of

    a signalx(t) is dened as

    xa t x t jx t ; x t 1

    p

    Z11

    x t

    t tdt (3)

    wherex t denotes the Hilbert Transform ofx(t).

    2.2. Spatial time-frequency distribution

    The TFD dened in the previous section conveys the energy distribution of a single signal in the time-

    frequency plane, and thus, it is called an auto-TFD. In many cases, it is necessary to have the joint

    energy distribution of two signals, for which the cross-TFD is used. The cross-TFD of two signals

    x1(t) and x2(t) is dened in similar fashion to Equation (1),

    Dx1x2 t;f X1

    l1

    X1m1

    m; l x1 t m l x2 t m l e

    4pjfl: (4)

    On the basis of Equation (4), the STFD of a vectorxcontaining n signals is dened as

    Dxx t;f X1

    l1

    X1m1

    m; l x t m l xH t m l e4pjfl (5)

    where Dxx t;f ij Dxixj t;f fori,j2 {1, . . .,n}; and the superscript H denotes a Hermitian transpose.

    Although the same kernel function is used in Equation (5) for all pairs, it is possible to use a speci c

    kernel for each pair.

    In the previous section, auto-terms and cross-terms were introduced as points with true and ghost

    energy concentrations, that is, nonzero TFD, respectively. For an STFD, two extra terms are

    introduced: The point (ta,fa) is dubbed an auto-source TF point of a source (signal) xi(t), if its auto-

    TFD at this point, that is, Dxixi t;f , exhibits an energy concentration. This energy concentration canbe true if xi(t) is monocomponent, or a ghost if the source is multicomponent. Additionally, the

    point (tc,fc) is dubbed a cross-source TF point between signals xi(t) and xj(t), if their cross-TFD atthis point, that is, Dxixj t;f , exhibits an energy concentration [29].

    3. THE PROPOSED IDENTIFICATION METHOD

    In this section, we present a new system identication method with which modal properties, that is,

    natural frequencies, damping ratios, and mode shapes, can be extracted from a buildings response

    recorded during strong ground motions. This method works without the knowledge of the input

    motions, so it may be categorized as an output-only identication method. At the present time, there

    appears to be no other robust technique proposed in the open literature that claims the same feat.

    S. F. GHAHARIET AL.

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

    DOI: 10.1002/eqe

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    The governing equations of motion for an MDOF system with ndDOFs, which is excited by a

    unidirectional (scalar) ground acceleration, can be expressed as follows:

    Mx t Cx t Kx t Mlxg t (6)

    in which M, C, and K are the constantnd ndmass, damping, and stiffness matrices of the system,

    respectively. The vector x

    (t) contains relative displacement responses of the system at all DOFs;xg t is a scalar time-signal, which represents the (unknown) ground acceleration; and l is theinuence vector [30]. In practical cases, the absolute acceleration of structure is recorded, which is

    xt t x t lxg t (7)

    By assuming a proportional damping matrix, the absolute acceleration response can be expressed in

    modal space as

    xt t fq t (8)

    where f is annd ndreal-valued mode shape matrix whosei-th column (fi) is thei-th mode shape; andq t is a vector that contains the absolute acceleration modal coordinates. The i-th modal coordinate isthe absolute acceleration response of an SDOF system that corresponds to thei-th mode, given by

    qi t hi t bi xg t ; i 1; . . . ; nd (9)

    where the operator * indicates linear convolution; bi fiTMl=fTi Mfi is the modal contribution

    factor [30]; and hi(t) is the SDOFs impulse response function, which is calculated as [31],

    h t 1

    odexont od

    2 x2on2

    sin odt 2xonodcos odt

    (10)

    where x, on, and od onffiffiffiffiffiffiffiffiffiffiffiffiffi

    1 x2p are the damping ratio, and the undamped and damped naturalfrequencies of the SDOF system, respectively. As such, the identi

    cation of a system may be dividedinto two stages: (i) modal decomposition and (ii) SDOF system identication. The following sections

    describe the said two stages of the proposed output-only identication method.

    3.1. Blind source separation

    Blind source separation is a well-established methodology in the sound separationeld [3235]. BSS

    techniques are used for recovering the source signals and the unknown mixing matrix using only the

    recorded signals (output). Here, we apply BSS to estimate the modal coordinates (source signal) and

    the mode shape matrix (mixing matrix).

    Consider again Equation (8) in which nm contributing modal coordinates are linearly combined to

    produce nn response signals, with nm nn nd. Because Equation (8) is a single equation with two

    unknowns, restricting assumptions must be considered for the source signals and the mixing matrix.

    If we assume that the source signals have distinct structures and localization properties in the time-frequency domain, and that the mixing matrix has full-column rank, then the mixing matrix can be

    identied, and the source signals can be recovered up to an arbitrary scaling factor and permutation

    via BSS [3638].

    Before describing the details of the time-frequency domain BSS method, we present a simple

    conceptual example to illustrate the implication of being localized in the time-frequency domain.

    Figure 2 displays the time-frequency distributions of two synthetic signals, schematically. In

    Figures 2(a) and 3(b), the types of signals that are mostly encountered in the electrical engineering

    eld are depicted [29, 39, 40]. Having a distinct localization in the time-frequency domain means

    that the signals have disjoint time-frequency signatures, as in Figure 2(a). However, BSS techniques

    can also be applied to signals that are quasi-disjoint, that is, they have overlapping regions in the

    RESPONSE-ONLY MODAL IDENTIFICATION OF STRUCTURES USING STRONG MOTION DATA

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

    DOI: 10.1002/eqe

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    Figure 2. A schematic diagram of time-frequency distributions of disjoint and quasi-disjoint signals intypical electrical (a, b) and earthquake (c, d) engineering applications.

    Figure 3. TFDs of the rstoor dynamic response of the 5-DOF system under the El Centro Array #9 accel-erogram recorded in Imperial Valley earthquake, 1940 [50]: (a, c) WVD, and (b, d) SPWVD for systems

    with stiffness-proportional and mass-proportional damping, respectively.

    S. F. GHAHARIET AL.

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

    DOI: 10.1002/eqe

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    time-frequency domain, as in Figure 2(b). Herein, the modal coordinates are taken as the source

    signals. Thus, they must have distinct time-frequency signatures so that BSS techniques are

    applicable. Fortunately, in most cases, the time-frequency distributions of modal coordinates are

    similar to the schematic TFD distribution shown in Figure 2(c). In the worst-case scenario, the input

    excitation also has a dominant frequency that contributes in all modal coordinates as shown in

    Figure 2(d) in which the modal coordinates can be assumed as quasi-disjoint sources. Hence, time-

    frequency-based BSS methods can be applied to separate these source signals blindly. This methodis described next.

    Calculating the STFD of both sides of Equation (8) and neglecting the noise effects yields,

    Dx tx t t;f fDqq t;f fH (11)

    whereDx tx tandDqq t;f are, respectively,nn nnand nm nmmatrices whose elements are the auto-TFDs and cross-TFDs of the recorded signals and the modal coordinates. Equation (11) is similar to the

    well-known BSS equation in the sound-separation eld (cf., [41]). For simplicity, consider an ideal

    TFD, that is, one without any cross-terms in any time-frequency point that corresponds to an auto-

    term of only a source signal. In this case, Dqq t;f is diagonal with only one nonzero diagonalelement. Thus, at each auto-term, Equation (11) is converted to,

    Dx tx t ti;fi fiDq iq i ti;fi fiH (12)

    where fiis i-th column off, andDq iq i ti;fi isi-th modes auto-TFD. Knowing the proper auto-term

    of each modal coordinate, an eigenvalue decomposition (EVD) can be used to extract fi and

    Dq iq i ti;fi . However, Dqq ti;fi is rank-decient, and thus, f cannot be uniquely determined.

    Moreover, external criteria are needed to specify the true auto-terms of each source signal and to

    decide which modes auto-term should be used for EVD. The difference between the number of

    modes and the number of sensors poses further difculties for EVD. Belouchrani et al. [37] showed

    that it is necessary and sufcient to use all auto-terms simultaneously, without knowing to which

    source they belong; hence, a joint approximate diagonalization (JAD) is employed instead of

    EVD [42, 43]. In this method, several Jacobi rotations are applied on a set of matrices to diagonalize

    them (details are omitted here for brevity, but may be found, for example, in [44]).

    Because the JAD algorithm is restricted to nding a unitary diagonalizing matrix, a preprocessingstep that converts the problem ofndingf (which is not always unitary) to nding a unitary matrix

    U is needed. This step, dubbed whitening, renders Jacobi rotations applicable, and also solves the

    dimensional problem mentioned earlier. As the term whitening implies, this step converts the

    sensor signals to white signals, that is, their zero-lag correlation matrix becomes equal to the identity

    matrix I. The whitening process reduces the determination of the nn nm mode shape matrix f tothat of a unitary nm nm matrix (U) as follows. On the basis of the aforementioned denition, annm nn whitening matrix (W) converts the recorded signals, x

    t t , toz t Wxt t , such that,

    Rz 0 I (13)

    whereRz(0) is a zero-lag correlation matrix of the new version of recorded signals; and is computed as

    Rz 0 Ez t z t (14)

    where E[.] denotes the expected value and the superscript * indicates the complex conjugate. For

    discrete deterministic signals, Equation (14) can be written as,

    Rz 0 1

    N

    XNk1

    zk z k (15)

    whereNis the number of time samples, and kis a discrete index. Insertingx t fq t intoz t Wxt t ,and applying the condition presented in Equation (13) yields

    RESPONSE-ONLY MODAL IDENTIFICATION OF STRUCTURES USING STRONG MOTION DATA

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

    DOI: 10.1002/eqe

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    Rz 0 Ez t z t E Wfq t q t fHWH

    WfRq 0 f

    HWH I (16)

    Because of the intrinsic ambiguity embedded in BSS techniques, and without any loss of generality,

    it is assumed that Rq 0 I . In other words, it is implicitly assumed that the source signals are

    uncorrelated at zero-lag with unit variance; thus,

    Wf Wf H I (17)

    Equation (17) implies thatW f= U, in whichU is a unitary matrix. If the whitening matrix is known,

    then f may be identied fromU by a simple MoorePenrose pseudo-inverse [41]. For determinate

    cases in which the number of sensors is equal to the number of modes, it can be easily shown that

    the whitening matrix can be obtained as follows [41]:

    W Rxt 0 0:5

    (18)

    whereRxt 0 is the correlation matrix of recorded signals, and Rxt 0 0:5

    is its principal square root.

    For cases in which nm< nn, the rstnm largest eigenvalues and eigenvectors of Rxt 0 are used asfollows [41]:

    W 0:5nm HHnm

    (19)

    where nm is annm nm diagonal matrix in which diagonal entities are the nmlargest eigenvalues ofRxt 0 , and Hnm is an nm nm matrix that contains the corresponding eigenvectors. The whiteningprocess must be applied on a noise-free portion of the recorded signals, which is clearly not

    possible for real data. Hence, for the under-determined case, the average of (nn nm) the remainingeigenvalues of Rxt 0 is used as noise variance, and is subtracted from the diagonal elements ofnm to reduce the noise effects [41]. The aforementioned whitening process may be carried out

    with other approaches, for example, Robust Whitening [45, 46], which may improve the results in

    exchange for additional computation.The STFD of the, now whitened, signals can be expressed as

    Dzz t;f UDqq t;f UH (20)

    Therefore, any whitened STFD-matrix is diagonal when stated in the basis of the columns of the

    matrixU. As mentioned previously, this unknown unitary matrix can be identied through a JAD of

    Dzz(t,f) at the chosen time-frequency points that are true auto-terms. After the identication of U,

    the mode shape matrix f, and the modal coordinates q t , may be recovered as follows:

    f W#U; (21)

    q t UHWxt t (22)

    where the superscript # denotes a MoorePenrose pseudo-inverse.

    As mentioned above, the JAD procedure is to be applied on a set ofDzz(t,f) matrices calculated at

    time-frequency points that are auto-terms. Several researchers have proposed various criteria to

    determine/select the said auto-terms [29, 37, 38, 47]. However, before attempting to identify the

    auto-terms, it is expedient to increase the odds of choosing proper points (and also to decrease the

    computational effort involved) by removing the noise effects and also by discarding the low-energy

    points. To remove the time-windows with no signicant energy (e.g., those portions that usually

    comprise the beginning and the end of strong motion records) the following criterion may be used:

    S. F. GHAHARIET AL.

    Copyright 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2012)

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    maxf

    Dxtxt ti;f k kF

    maxt;f

    Dxtxt t;f k kF>E1 (23)

    where . Fdenotes the Frobenius norm. The criterion includes time slices ti, in which the ratio of the

    maximum energy along the frequency axis to the maximum energy in the time-frequency plane is

    greater than some scalarE1. The value ofE1 should be chosen based on the signal-to-noise ratio. Our

    experiences with real-life records suggest that values greater than 1% will work well for most cases.

    However, for weak earthquake motions with noisy data, it may have to be increased up to 10%.

    Next, an energy criterion can be used to remove unnecessary points in each time slice as in,

    Dxtxt ti;f k kFmax

    fDxtxt ti;f k kF

    > E2: (24)

    Here, in each time sliceti, the auto-term points are selected if the ratio of their energy to the maximum

    energy at that time window is greater than some scalarE2. The value ofE2is chosen based on earthquake

    energy; for weak input motions, larger values ofE2should be used (typically greater than 1%).

    To identify the auto-term points, Belouchrani et al. [38] suggested exploiting the off-diagonal

    structure of the STFD matrices at cross-terms; that is, for time-frequency points that correspond tocross-terms, the following condition can be written:

    traceDzz t;f trace UDqq t;f UH

    trace Dqq t;f

    0 (25)

    This equation is valid because the trace of a matrix is invariant under a unitary transform. Thus, they

    suggested using the criterion, traceDzz t;f = Dzz t;f k kF< E3, to detect the cross-terms and to excludethem. HereE3is a positive scalar less than 1 (typically,E3 = 0.8) [29]. Although this criterion works well

    for excluding the cross-terms (produced because of deciencies inherent to quadratic TFD), it cannot

    detect cross-sources. In other words, at common points (i.e., points at which several sources are

    contributing) the value of trace Dzz t;f = Dzz t;f k kF< E3 may be greater than 1; and these pointswill be selected as auto-source points for the diagonalization process. This scenario is highly likely

    for a system that is subjected to strong ground shaking the scenario in which the input excitationcontains several dominant frequencies. If the dominant frequencies of input are observed in several

    modes, then the aforementioned criterion cannot lter out the time-frequency points corresponding

    to those frequencies; and another criterion would have to be used to select the best auto-sources.

    Under the source time-frequency disjointness assumption, each auto-source STFD matrix is of rank

    one; or at least, each matrix has one signicantly large eigenvalue compared with its other

    eigenvalues. Therefore, the following criterion may be used to deselect the cross-source points [29]:

    lmaxDxtxt t;f

    Dxtxt t;f k kF 1

    >E4 (26)

    whereE4is a small positive scalar (typically, E4 = 0.001) andlmax[.] represents the largest eigenvalue of

    its argument matrix. Note that in this new criterion, the STFD matrix of the original data (as opposed tothat of the whitened data) is used.

    Remark 1

    To reduce modal interference in systems with closely spaced modes in two directions, signals of those

    two directions should be analyzed separately for symmetric buildings. For asymmetric cases, a stricter

    auto-source point selection criterion (i.e., smaller values forE4) must be used.

    By employing the aforementioned process for selecting the auto-source points, and by applying

    JAD for the diagonalization of whitened auto-source STFD matrices, the mode shapes and modal

    coordinates in absolute acceleration forms will have been identied. The second step is to extract

    the natural frequencies and damping ratios, which is described next.

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    3.2. Simultaneous identication of the modal coordinates

    On the basis of Equation (9), the i-th absolute acceleration modal coordinate qi t , is the response of anSDOF system to the translational acceleration bixg t . Applying a Z-transform to both sides ofEquation (9), the response of the system in theZ-plane can be represented by the following equation:

    q i z

    hi z bi

    xg z (27)

    Here, q i z ,xg z , and hi z denote theZ-transforms of the modal coordinate, the input motion, and theimpulse response function, respectively. UnilateralZ-transform of a discrete-time signalx[k] is dened

    as [48],

    x z X1k0

    x k zk (28)

    wherekis an integer time index and z is, in general, a complex number. The term hi z is calculated indiscrete form based on Equations (10) and (29) as

    hi z 1

    T

    Ci Diz1

    1 Aiz1 Biz2

    (29)

    where Tis sampling time and the coefcients are given by

    Ai 2exiwniT cos wdi T ;Bi e

    2xiwni T; Ci 2xiwni T;

    Di oniTexioniT

    oni

    odi1 2xi

    2

    sin odiT 2xicos odiT

    (30)

    Substituting these expressions into Equation (27) yields

    T

    biCi1 Aiz

    1 Biz2

    q i z 1 D

    iz1

    xg z (31)

    where D i Di=Ci, that is,

    D i exioniT

    oni

    odi

    1 2xi2

    2xi

    sin odiT cos odiT

    (32)

    Transforming Equation (31) back to the time-domain, a new representation is obtained,

    1biCi

    qi k Aiqi k 1 Biqi k 2 xg k D

    ixg k 1 ; 1inm; 1kN (33)

    with N denoting the number of samples used for discretization. This equation indicates that the

    absolute acceleration modal coordinate at discrete time k, that is, q k , may be written as a lineardifference equation involving previous responses, and previous and current input excitations. In the

    literature, such models are termed as auto-regressive models with eXternal/eXogeneous input. Using

    a nonlinear least-squares technique, the nm linear equations corresponding to all of the extracted

    modes may be solved simultaneously to determine the natural frequencies, damping ratios, and modal

    contribution factors of the system, and the input motion. However, this approach is very expensive

    computationally. An alternative is presented below that alleviates this burden.

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    Equation (33) may be restated for all time instants in matrix form as,

    Qii xgDi for 1inm (34)

    where

    Qi qi 2 qi 1 qi 0

    qi N 1 qi N 2 qi N 3

    24 35 (35)

    i gi0 g

    i1 g

    i2

    T; (36)

    xgDi xg 2 D

    ixg 1

    xg N 1 D

    ixg N 2

    24

    35; (37)

    where gi0 1= biCi , gi1 g

    i0Ai , g

    i2 g

    i0Bi and T denotes matrix transposition. On the basis of

    Equation (34),nm (nm 1)/2 possible relations can be established among all modes as follows:

    Qii Qjj D

    i D

    j

    xg for 1i;jnm (38)

    where

    xg xg 1 xg N 2 T: (39)

    Repeating Equation (38) for a different pairs and merging those results, we may obtain a new

    relation that is free from input excitation. To wit,

    Qi Qj Qk Ql i

    D i D

    j

    j

    D i D

    j

    k

    D k D

    l

    l

    D k D

    l

    T 0 for i;j 6 k; l : (40)

    Using any four modes in Equation (40), we may write Lnm2

    2

    0@

    1A relations provided that

    nm3. These relations can be combined in a matrix form as follows:

    Q L nm nm1 nm nm1 1 0 (41)

    where

    nm

    D nm D

    nm1

    nm 1

    D nm D

    nm1

    nm

    D nm D

    nm2

    nm2

    D nm D

    nm 2

    2

    D 2 D

    1

    1

    D 2 D

    1

    " # (42)

    in which

    Q LLQ nm (43)

    where

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    L2 I I ; Ll L11l L

    12l

    0 Ll1

    for 3lL (45)

    and

    L11l I I I T; L12l

    I 0

    0 I

    24

    35 for 3lL (46)

    where I is an identity matrix with size N 2.In the presence of noise, Equation (41) can be solved in a least-squares sense through the

    constrained minimization problem,

    CR arg mink k1

    HQHQ (47)

    as described in [49]. Alternatively, the eigenvector associated with the smallest eigenvalue of the

    matrix QHQ can also be used as a solution of Equation (47). The present equations would yield

    (nm 1) answers for each i, each of which containing a different denominator term. Nevertheless,the (nm 1) answers for the damping ratio and natural frequency of each mode can be identied from

    the following equations, wherein gi0 , gi1 , and g

    i2 related to each answer have the same denominator,

    which cancel out,

    xioni 1

    2T ln

    gi2gi0

    ; oni

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 xi

    2

    q

    1

    T cos1

    gi1exioniT

    2gi0

    for 1inm; (48)

    xi xioni

    oni; oni

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixioni

    2 oni

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 xi

    2

    q 2s for 1inm: (49)

    4. VERIFICATION OF THE METHOD AND ITS APPLICATION TO REAL DATA

    4.1. A simulated MDOF system for method verication

    Consider a 5-DOF linear shear-building model, in which the oor mass is taken to be 3 105 for eachstory. Interstory stiffnesses, from the rst to the fth story, are 7k, 5k, 3k, 2k, andk, respectively, where

    k= 5 107. This structure mimics the dynamic characteristics of a typical 5-story building fairly well.Two separate cases are investigated: in the rst case, the damping ratio, xm(%), is set to be mass-

    proportional with a 10% rst mode damping ratio; in the second case, the damping ratio, xs(%), is

    set to be stiffness-proportional with a 0.5% rst mode damping ratio. Horizontal accelerogram

    recorded in El Centro Array #9 during Imperial Valley earthquake, 1940 [50] is used as input

    motion for generating the dynamic response of the system. This dynamic analysis is carried out

    using the lsim command in MATLAB [51] with a 100 Hz sampling frequency. Figure 3 displays the

    rst oors absolute acceleration responses using two different kinds of TFDs, viz. WVD and

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    displays the said time variation of MAC values for both systems (markers in these plots are located at the

    end-time of each time window). As this gure indicates, the 4th and 5th modes can be more accurately

    identied only during the rst time window for the system with stiffness-proportional damping; whereas,

    for the system with mass-proportional damping, the proposed method yields nearly the same results at alltime windows.

    Remark 3

    It is worth mentioning here that the use of all of the data can increase the MAC values, as it happened

    for the 4th mode of the system with stiffness-proportional damping. The primary reason for this

    phenomenon is related to the JAD procedure, which seeks orthogonal directions. Thus, it extracts the

    estimates of inactive modes even at time-frequency points at which they are not contributing.

    The application of the second step of the identication process yields the natural frequencies and

    damping ratios of the two systems as shown in Table I. The exact/analytical values are shown here

    also for comparison. As can be seen, for both systems, the identied modal values are nearly the same,

    especially for the lower modes. It is important to note here that the inaccuracies in the identied frequency

    and damping ratio values primarily belong to the previous step, which produced inaccurate modalcoordinates. The natural frequencies and damping ratios presented in Table I are the average of (n 1)

    solutions (their variations are not presented here for brevity). Results indicate that variations of the identied

    natural frequencies are negligible, while the identied damping ratios for the mass-proportional system

    display some variations. However, these variations are mostly due to errors accrued during modal

    decomposition, because the second step of the proposed method works perfectly for exact modal coordinates.

    To prove this statement, the identication technique was applied to the exact modal coordinates,

    which resulted in exact natural frequencies and damping ratios (again, omitted here for brevity).

    4.2. A simulated soilstructure system for method verication

    The 5-DOF structure introduced in the previous section is now placed on a

    exible foundation tomodel a new 7-DOF soilstructure system with two additional DOFs (foundation sway and system

    rocking). The foundation mass, and the horizontal and rocking soil stiffnesses are as mf= 3 105,

    kh = 9 108, and kr= 3 10

    10, respectively. Mass moment of inertia of all stories and the foundation

    are set as Ii = 25mi, wherein mi denotes story mass. Also, a constant story height equal to 4 m is

    considered for all stories. For the sake of simplicity, only mass-proportional damping is considered

    so that the rst mode has a 10% damping ratio. Similar to the xed-base example, dynamic analysis

    with the same horizontal input motion is carried out in MATLAB [51] with a 100 Hz sampling

    frequency. Figure 6 displays the SPWVD graphs of foundation s and roofs absolute acceleration

    responses along with rocking response of the system. As can be seen, all modes are not detectable

    in all DOFs. For example, the highest mode is only observed in foundations response, while the

    Figure 5. Time variation of MAC for systems with (a) stiffness-proportional and (b) mass-proportionaldamping.

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    roofs response is composed of the rst three modes. This gure also shows that rocking response is

    mainly affected by the fourth mode.

    Using all DOFs absolute accelerations, mode shapes are identied by applying the BSS time-

    frequency decomposition method with E4 = 0.001. The calculated MAC indices for all modes are

    1.000, 0.993, 0.997, 0.737, 0.981, 0.998, and 0.996, which indicates that the 4th mode is not

    identied accurately. However, this mode is a rocking mode, and the MAC index is low because the

    order of rotation is small compared with the horizontal displacement. For this reason, the

    COordinate Modal Assurance Criterion [52] index is calculated for all DOFs, which is dened as

    COMACl

    X7k1

    faklfikl

    2X7k1

    fakl2 X7

    k1fikl2

    (51)

    wherefaklandfikldenote the analytical and the identied mode shapes, respectively, at thel-th DOF in

    the k-th mode. This index for foundation sway and rocking and the other ve structural DOFs are

    0.8271, 0.9226, 0.8795, 0.9201, 0.9207, 0.9361, and 0.9505, respectively. As seen, the rocking

    DOFs deformation, which is the dominant deformation in the fourth mode, is identied well.

    Repeating the decomposition process with only ve signals (excluding the foundations horizontal

    and rocking response signals), ve modes are identied. A comparison of the identied and analyticalmode shapes indicates that the identied modes are the 1st, 2nd, 3rd, 5th, and 6th modes. MAC indices

    for the identied mode shapes are 1.000, 0.996, 0.998, 0.978, and 0.996. Thus, although the exclusion

    of two response signals converts this case to an under-determined problem, ve modes are still

    identiable. Figure 7 displays the time variation of MAC values for both cases. As this gure

    indicates, the MAC index for mode 4 is high between 25 to 30 s, because during this time window,

    the contribution of mode 4 relative to other modes is signicant (cf., Figure 6). Moreover, as can be

    seen in Figure 7(b), all identied modes are nearly accurate in all time segments, even with only

    Figure 6. SPWVDs of foundation (a) translation and (b) rocking, and (c) roof translation responses under theEl Centro Array #9 accelerogram recorded during the 1940 Imperial Valley earthquake [50].

    Figure 7. Time variation of MAC index for two cases: (a) using all DOFs response signals and (b) excludingfoundation horizontal and rocking response.

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    ve sensors. The averages of the identied natural frequencies and damping ratios using two sets of

    sensors are presented in Table II. This table shows that the identied values using ve sensors are

    fairly as accurate. Nevertheless, excluding the foundation and rocking sensors results in identication

    errors, especially for the rst mode damping ratio.

    4.3. Application of proposed method to data recorded at the Van Nuys Hotel

    Herein, we apply the proposed method to real-life data recorded at the Van Nuys Hotel. Data are

    provided by the Centre for Engineering Strong Motion Data website (http://strongmotioncenter.org).

    This building is ideally suited for the present study, because there are several structural vibration

    records from damaged states with extents that range from moderate to severe [19]. Not surprisingly,

    quite a number of studies have been conducted on data from this building. A brief summary of

    those studies along with a description and history of the building are presented in what follows.

    The Van Nuys Hotel is a 7-story building with a 6200 m2 oor plan, located in the San Fernando

    Valley of Los Angeles County, California (34.2203 N, 118.4713 W). It was designed in 1965 and

    built during 1966 [53]. The structure is essentially symmetric in both directions. Spandrel beams and

    exterior columns form the primary frame that resists the lateral loads in both directions. The oors

    are reinforced concrete at slabs. The structure sits atop a group of pile foundations on recentalluvial soils that primarily consist of ne silty sands [54]. The building was rst lightly, and then

    severely damaged during the 1971 San Fernando (M6.6), and the 1994 Northridge earthquakes

    (M6.7), respectively. After the 1994 earthquake, the building was retrotted with new reinforced

    concrete shear walls in the eastwest direction at exterior frames [53].

    The building was instrumented in 1980 through the California Strong Motion Instrumentation

    Program (CSMIP Station No. 24386). Of the 16-channel recording system, which comprises uni-

    axial, bi-axial, and tri-axial instruments, only ve channels are devoted to the eastwest direction.

    These ve channels are mounted on the ground, second, third, sixth, and roof levels. Several

    researchers have used acceleration data provided by these instruments to determine the natural

    frequencies and damping ratios of the building. Ambient tests conducted soon after construction in

    1967, and following the 1971 San Fernando earthquake before and after a seismic retrot revealed

    that the

    rst natural frequency of the east

    west response decreased from 1.89 Hz before theearthquake to 1.39 Hz after the earthquake, and then increased to 1.56 Hz following repair [55]. Two

    detailed ambient vibration tests were also conducted by Ivanovic et al. [54] after the 1994

    Northridge earthquake. They placed sensors on one of the interior eastwest frame at all oors to

    determine the lateral frequencies. They reported the frequencies of the system as 1.0, 3.5, and 5.7 Hz

    for three translational modes in the EW direction. Note that all of the aforementioned frequencies

    are apparent (or pseudo-exible base) frequencies, because they are calculated from relative

    spectra of the response signals to the ground oor signals, while rocking response is not excluded [56].

    Owing to the existence of large sets of recorded earthquake data, there were also numerous attempts

    to estimate apparent natural frequencies from recorded seismic responses. Trifunac et al. [57, 58]

    studied the changes in natural frequencies of the building from earthquake to earthquake, using

    Table II. Comparison of the identied and analytical modal properties.

    Identied

    5 Sensors 7 Sensors Analytical

    Mode no. fn(Hz) x(%) fn(Hz) x(%) fn(Hz) x(%)

    1 0.99 28.00 0.90 8.02 0.91 10.002 2.42 3.42 2.43 3.76 2.45 3.703 3.87 2.34 3.86 2.29 3.87 2.344 4.81 1.83 4.91 1.855 5.44 1.53 5.30 1.59 5.43 1.676 7.72 1.17 7.59 1.21 7.57 1.207 10.96 0.83 10.97 0.83

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    Fourier analyses and short-time Fourier transform techniques. Table III displays some of their results,

    where the rst apparent frequency of the building in the EW direction at the beginning and end ofve

    earthquakes, along with its minimum value obtained during shaking, is presented. On the basis of these

    results, Trifunacet al. concluded that the system frequencies changed from one earthquake to another,

    and with the intensity of shaking.

    In another study, Alimoradi and Naeim [19] used ground oor responses recorded during the

    1992 Big Bear and 1994 Northridge earthquakes as input. They identied the rst (EW) natural

    frequency as 0.87 Hz from Big Bear data. They went on to apply their method to three segments of

    the Northridge data, and estimated the natural frequencies of the rst three modes along the EW

    direction as 0.61, 1.90, 3.41 (rst segment), 0.56, 1.74, 3.11 (second segment), and 0.48, 1.82,

    4.20 Hz (third segment). They also found the damping ratios to be approximately 10% for all modes

    for segment 1, 15% for all modes for segment 2, and 6.7%, 8.4%, and 15% for three modes for

    segment 3 of the Northridge data. However, if SSI is signicant for this building, as reported by

    Trifunac et al. [57, 58], then neither the xed-base nor the exible-base natural frequencies were

    identied in their study. Indeed, they have extracted pseudo-exible-base parameters, that is, rocking

    motion is included, but sway is excluded.

    Recently, Todorovska and Trifunac [59] used wave travel times of vertically propagating waves in

    the Van Nuys building between the ground oor and roof to estimate the rst natural frequency of the

    xed-base system in the EW direction. They found that the xed-base system vibrated at a frequency

    of around 1.2 Hz during the Landers and the Big Bear earthquakes (both occurred in 1992). They also

    found that the rst frequency varied from 0.85 to 0.55 Hz during the 1994 Northridge earthquake.

    As the summaries of previous studies indicate, there appear to be several unresolved issuesregarding the interpretation of data recorded at the Van Nuys Hotel. To wit,

    Apparent natural frequencies of the system identied from ambient tests are different from those

    expected during earthquakes (we also note that the damping ratios have not been reported in [54]).

    In calculating the apparent natural frequencies, rocking response is included, but sway is discarded.

    To identify the dynamic characteristics of the system from strong motion data, two approaches

    were used: (i) assuming the ground oor response as the input motion [19], and (ii) extracting

    information from the relative response spectra [57, 58]. The rst approach is not applicable for

    buildings that interact with the surrounding soil (Van Nuys Hotel appears to have this attribute),

    and the second approach cannot be used to identify the natural frequencies and damping ratios of

    soilstructure systems.

    As demonstrated through the simulated (veri

    cation) problem earlier, the identi

    cation methodproposed herein is able to accurately estimate the modal characteristics from strong motion data

    without the need to know Foundation Input Motions. As such, aforementioned problems regarding

    the Van Nuys buildings data can be overcome with the proposed method, which is applied here to

    data recorded at the four eastwest channels. The ground oor data are excluded because of its

    evidently low signal-to-noise ratio. Because of the limited number of sensors along the height of the

    building, we seek only the rst three translational modes along the EW direction, and use the data

    recorded during four relatively strong earthquakes, which are summarized in Table IV. In addition

    to the high level of recorded peak amplitudes (except for the Big Bear earthquake), these

    earthquakes are specically chosen so that the identied results can be compared with the previous

    studies mentioned earlier. We use the full lengths of recorded signals for the Whittier, Landers, and

    Table III. The rst EW apparent natural frequency of Van Nuys Hotel reported by Trifunacet al. [57, 58].

    Number Earthquake Date M fbeg(Hz) fend(Hz) fmin(Hz)

    1 San Fernando 02/09/1971 6.6 1.05 0.85 0.702 Whittier 10/01/1987 5.9 1.00 0.75 0.803 Landers 06/28/1992 7.5 1.00 1.30 0.704 Big Bear 06/28/1992 6.5 0.80 0.80 0.80

    5 Northridge 01/17/1994 6.4 0.95 0.60 0.45

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    Big Bear earthquakes, and divide the Northridge data into four 15-s segments, because signicant

    damages have been reported during this earthquake.

    Figure 8 displays the mode shapes identied through the proposed method with E2 = 0.01 and

    E4 = 0.001. The identied rst modes for all earthquake data (Figure 8(a)) appear nearly the same;

    although a small discrepancy is observed at the lower stories during the latter portions of Northridge

    earthquake, which is arguably related to the reported damages. This discrepancy is also observed for

    the Big Bear earthquake, which may be related to measurement noise, because the level of shaking

    is fairly low in this earthquake.

    Remark 4

    It is useful to note here that the sensors are limited to points shown by markers in Figure 2, and linear

    interpolation is used for the mode shape displacements of other stories. Additionally, the contribution

    of torsional modes may cause further inaccuracies in estimating the translational modes.

    Remark 5

    Figures 8(b) and (c) imply that changes are occurring during the nal 45 s of the Northridge earth-

    quake: the second and especially the third mode shapes identied from these data windows are differ-

    ent from their earlier versions. Because of the limited number of sensors, it is not possible to exactly

    identify the location of these changes (a potential extension of the technique proposed here for local-

    izing damage is deferred to a future study). Nevertheless, as a comparison, the mode shapes identied

    from the last segment of the Northridge earthquake are redrawn in Figure 9, along with mode shapes

    identied from ambient tests conducted one month after this earthquake [54]. This gure reveals thatthe proposed method can extract the mode shapes of this structure by using only four signals recorded

    during the Northridge earthquake (note that the identied mode shapes from ambient data are related to

    a pseudo-exible-base system).

    Table IV. Earthquake data used in this study.

    Number Earthquake Date M Dist. (km) PGA* (g) PSA** (g)

    1 Whittier 10/01/1987 5.9 41 0.17 0.202 Landers 06/28/1992 7.5 187 0.04 0.193 Big Bear 06/28/1992 6.5 152 0.03 0.064 Northridge 01/17/1994 6.4 7 0.47 0.59

    *PGA, peak ground acceleration.**PSA, peak structure acceleration.

    Figure 8. The identied mode shapes. (a) rst mode, (b) second mode, and (c) third mode.

    S. F. GHAHARIET AL.

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    In the second step, natural frequencies and damping ratios are extracted from the identied modal

    coordinates. The obtained results are displayed in Table V. As observed, natural frequencies, especially

    in the rst mode, are decreasing during successive earthquakes, which may be an indication of damage.

    The said trend is only violated during the nal segment of the Northridge earthquake. More specically,

    the minimumrst natural frequency is observed during the third segment of the Northridge earthquake,

    potentially because nonlinearities/damage occurred during that segment, and then it is seen recovering

    during the last segment, which incidentally is consistent with the results reported in Figure 10 of Ref. [59].

    Remark 6

    Damping ratios higher than 20% are not reported here, because they are unusual for civil structures. It

    is also useful to note that the natural frequencies and damping ratios identied here may be question-

    able because of potential contributions from other (undetected torsional or other higher) modes.

    5. CONCLUSIONS

    A new system identication method was presented. This method can identify modal properties of civil

    structures from their recorded responses when the input motion is unknown. Contrary to currently

    available output-only techniques, which are only applicable to ambient vibration data, the proposed

    method is able to extract dynamic characteristics of structures using strong motion responses. This is

    quite attractive, because the actual input motions exciting the structures are rarely recordable during

    earthquakes. This is especially true for the case of soil structure systems where neither the free-eld

    motion nor the recorded response of foundation may be assumed as input to the system. This

    Figure 9. Comparison between the mode shapes identied from the last 15 s of the 1994 NorthridgeEarthquake data (black), and from ambient test one month after that [54] (gray).

    Table V. Identied natural frequencies and damping ratios from earthquake response.

    Mode No. Whittier Landers Big Bear Northridge-1 Northridge-2 Northridge-3 Northridge-4

    fn(Hz) 1 0.87 0.86 0.77 0.63 0.50 0.43 0.502 2.92 3.31 2.29 2.00 1.66 1.46 1.133 5.41 5.50 4.54 4.24 3.43 3.07 2.81

    x(%) 1 8.96 7.51 9.76 2.72 16.67 11.38 12.622 7.34 5.01 1.71 9.41 7.25 3 13.74 7.89 3.83 1.67 16.76

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    method comprises two steps. First, a time-frequency blind source separation technique is employed to

    decompose the recorded response into a linear combination of modal coordinate signals for which the

    combination factors are the mode shapes. In the second step, modal cross-relations are employed to

    extract the natural frequencies and damping ratios. To verify the proposed method, two numerical

    simulations were presented: (i) identication of axed-base system without having the input motion

    (i.e., the foundation response); and (ii) identication of a exible-base system without having the

    foundation input motion (i.e., the result of kinematic interaction between the foundation and thesurrounding soil). Moreover, response signals recorded at the Van Nuys Hotel during four strong

    earthquakes were used as a case study. Identication results for both scenarios, that is, simulated

    and real-life data, indicate that the proposed method can be successfully employed to identify

    dynamic properties of civil structures.

    ACKNOWLEDGEMENTS

    The work presented in this manuscript was funded, in part, by the NSF Grant CMMI-0755333. Anyopinions, ndings, conclusions or recommendations expressed in this material are those of the authorsand do not necessarily reect the views of the sponsoring agency. The authors also would like to thankProfessor Boashash for providing MATLAB codes for calculating the time-frequency distributions.

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