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    EARTHQUAKE ENGINEERING A ND STRUCTURAL DYNAMICS, VOL.

    24,

    85- 97 199 5)

    A SYSTEM IDENTIFICATION APPROACH TO THE DE TECT ION OF

    PARAMETERS

    CHANGES

    IN

    BOTH LINEAR AND NON-LINEAR STRUCTURAL

    CHIN-HSIUNG LOH* AND IAT-CHUN TOU

    Department of Civil Engineering, National Taiwan University, Taipei, Taiwun.

    R . 0 . C

    SUMMARY

    This paper explores the potential

    of

    a new time domain identification procedure to detect changes in structural dyna mic

    characteristics on the basis of measurements. This procedure is verified using mathematical models simulated on the

    computer. The experiments involve two eight-storey steel structures with and without energy devices, and a 47-storey

    building at San Francisco during the Lom a Prieta earthquake. Th e recursive instrumen tal variable method a nd extended

    Kalm an filter algorithm are used as identification algorithms. An exp loratory investigation is m ade

    of

    the usefulness

    of

    various indices, such as mode shape and storey drift, that can be extracted accurately from identification to quantify

    changes in the characteristics

    of

    the physical system. It is concluded th at the change

    of

    storey drift is the key information

    to the detection

    of

    changes in structural parameters, from which the proposed system identification algorithm can be

    applied with an appropriate inelastic model to simulate the dynamic behaviour of real structures undergoing strong

    ground motion excitations.

    I N T R O D U C T I O N

    System identification is a branch of numerical analysis that utilizes experimental input and output data to

    create o r improve mathe matic al models of systems. It is an effective too l for developing analy tical models

    from experimental dat a. In the field of earthqu ake engineering, analytical modelling of structures subjected t o

    ground motions is an important aspect

    of

    fully dynamic earthquake-resistant design. In general, linear

    models are only sufficient to represent structural responses resulting from earthquake motions of small

    amplitudes. However, the response of structures during strong ground motions is highly non-linear and

    hysteretic, and the use of system identification techniques for both non-destructive and damage evaluation is

    an im portant an d challenging problem. Recent efforts to apply form al system identification concepts to the

    problem of detection of changes in structural param eters include the work of Yao, Stephens,* Na the an d

    Yao3 and Agbabian

    et aL4

    In general, identification methods can be categorized as those in the time domain and the frequency

    dom ain. In the frequency dom ain metho d, modal q uantities such as natural frequencies, dam ping ratios an d

    modal shapes are identified using measurement in the frequency domain. O n the othe r hand , in the time

    dom ain method, system parameters are determined from observational da ta sampled in time. In the present

    study, mainly pa rametric identification techn iques in the time dom ain are investigated.

    A

    new identification

    scheme to detect the changes in structural dynamic characteristics on the basis of measurements is

    investigated. Based on the three-stage identification app roach proposed by Loh an d Chung, an explorato ry

    investigation is undertaken of the usefulness of various iden tified indices, such a s mod e shap es, storey drifts,

    elastic forces and base shear distributions, which can

    be

    extracted to quantify the changes of dynamic

    characteristics of structural systems during strong g round motions.

    *Professor.

    +Graduate student.

    CCC Oo98-8847/95/0 10085- 13

    995

    by Jo hn Wiley Sons, Ltd.

    Received 6 August 1993

    Revised

    25

    April

    1994

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    86

    C.-H. LOH AND 1.-C. TOU

    MATHEMATICAL MODEL FOR TH E STRUCTURA L SYSTEM

    The class of structures tha t fall within the scop e of the present investigation can be adequately modelled by

    the following N-dimensional system

    of

    equations which describes the motion of the structure,

    (1)

    ii +

    CU

    +

    KU

    +

    g[u,

    63

    = f(t)

    Here, M denotes the inertia matrix associated with the structure,

    C

    enotes the viscous damping matrix a nd

    K is the stiffness matrix . The vector f(t) denotes the externally applied forces, an d g[u, u] is a vector whose

    components are non-linear functions of the structural displacement u and its first derivative

    8.

    The

    formulation of the earthquak e response analysis of a lumped M D O F system can be carried ou t in matrix

    notation. Using a modal synthesis approach, one may express the calculated response

    u i ( t )

    n terms of the

    modal response xl through the following equation:

    where 4i l s the mode shape comp onen t of the Ith mode a t location i . The mo dal responses are defined by the

    second-order differential equation for a classically damped system:

    Xj(t) + 2tjUji i ( t )

    + WfXj t)

    - pjiig(t)

    (3)

    where

    p j

    is the participation factor of the jt h mode. If the m ultiple input acceleration time h istories are

    considered ugk(t) , =

    1,2,

    .

    ,

    m quation (3) can be replaced by

    in which

    j i k

    = 4ijpjkand yij(t) is the response of the jt h mode at location i

    stiffness ma trix

    The elastic force associated with the relative displacement can be obtained by premultiplying by the

    {m)

    CK1

    {Y t,> = CKI

    @I

    {x t)>

    = [MI

    [@ICfi21 x t))

    5 )

    in which [Qz] is a diagonal matrix of the squared modal frequencies 0: he base shear force V ( t )of the

    system is given by the sum of

    of

    all the storey forces, that is,

    The value of any desired force resultant at the same time can be com puted by summ ing the elastic force from

    the to p floor to the desired level. Besides the elastic force an d base shear an oth er impo rtant index is the storey

    drift. The storey drift between two stories can be expressed a s

    Fro m the proposed system identification a pproac h, the nat ura l frequencies, dam ping ratios an d mod e shapes

    of

    the first few m odes of a M D O F system can

    be

    identified; then the storey drift, elastic force and base she ar

    force can be recovered from the identified ea rth qu ak e response da ta.

    For a non-linear SDOF system, the model is represented according to Baber and Wen6 as follows:

    where to s the fraction of critical viscous damping for small amplitudes, and o s the pre-yielding natural

    frequency (small amp litude response). Z is the hysteretic comp onen t

    of

    the displacement, a A,

    B,

    S and n are

    parameters describing the shape of the hysteresis loop, the elastic and inelastic deformation, and the

    maxim um stre ngth for softening springs. A large num ber of hysteresis shape s can be described by varying th e

    parameters.

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    A

    SYSTEM IDENTIFICATION APPROACH 87

    To extend the non-linear SDOF system to a non-linear MDOF system, the structural model to be

    considered herein can also be represented as the shear type. The equations of equilibrium may be written as

    follows:

    m l i i l + f i g ) +fD(al,ul,t)-fD(U2,U2,t) = o

    m2(ii2

    +

    iil

    +

    i,)

    +fD(u2,

    u 2 , ) - j D ( t i 3 , u 3 , )

    =

    0

    / N \

    in which

    ii

    is the acceleration of the base, ui is the relative displacement of the ith floor relative to the

    i

    - 1)th

    floor; andfD(ui,u i , ) is the restoring force at the ith floor originated by the relative displacement,ui, between

    the ith and i - 1)thflook. The restoring forces,fD,and the relative displacement,u i ,must be constructed to

    reflect the type of behaviour (elastic or inelastic). The model shown in equation

    (8)

    can be used to represent

    restoring force.

    THE IDENTIFICATION ALGORITHMS

    Many approaches to system identification are described in the literature. However, to detect the changes of

    structural parameters a systematic way of identification must be used. In this study, recursive least-squares

    analy~is,~ecursive instrumental variable method and extended Kalman filtering techniqueg-

    '

    are used.

    Recursive least-squares time domain identijication'

    A

    method

    of

    identifying structuraI parameters such as damping and stiffness of a building from its time

    response under dynamic excitation is used. A least-squares recursive algorithm which requires no matrix

    inversion was developed from one time point and updated at the next point. These identification algorithms

    are considered to be the initial phase of this study in the time domain. Since measurements equal to the

    number of floors are required, and also the manner of adding noise in the excitation leads to biased estimates,

    the present method becomes impractical for buildings of many stories. If only the predominant mode is

    considered (single DOF systems), the time variation of equivalent modal stiffness and damping can be

    identified. In the identification procedures, this method is applied first to estimate the elastic or inelastic

    behaviour of structural response roughly.

    Recursive instrumental variable method'

    A continuous Lth-order autoregressive AR) process is defined as the output from a linear system which is

    excited by random impulses. In continuous time the relationship between input u ( t ) and output y(t) is

    described by the Lth-order differential equation

    dL- y(t)

    dtL-

    + ..- + aoy(t)= u ( t )

    LY 0

    at

    dt .

    E L -

    1

    A discrete representation of an Lth-order AR process is

    This equation can be rewritten as

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    88

    in which

    C.-H.

    LOH AND I.-C. TOU

    $ ( t )

    = -

    t - ),

    ..., - ( t

    - na),

    u t

    -

    , ..., u t

    -

    nb))'

    e = ( A ~ ,..,

    A,,,

    B, ...,

    B , ~ ) T

    (13)

    0

    is the vector of estimated parameters and

    $ ( t )

    is the measurement vector. Taking an example of

    a single-degree-of-freedom system with mu ltiple inputs, the equa tion of mo tion can be expressed a s

    n

    my + cy + k y = - m 1 iU g i

    i = l

    Following equation (12), the vector of estimated parameters O t) and the measurement vector $ ( t ) can be

    expressed as follows:

    e t)= (elm, k l m , P I ~ 2 r..* n

    $ ( t ) =

    -

    0 ,

    y t ) ,

    Ugl(t), U g 2 t ) ,

    ...

    )

    In order to find the vector 0, the following equatio n must be satisfied:

    1 '

    1

    c Z(s)44 = c Z(S)CY(S)-

    ( 4 T e ( S ) l

    = 0

    N,=1 N , = l

    in which Z(s) is a vector which is uncorrelated with disturbance ~ ( t ) ,rom which one can obtain

    (17)

    The resulting recursive algorithm is quite sim ilar to the one derived for the recursive least squares, an d is

    given by the following equations:

    P ( t )

    =

    P ( t - I )

    - P ( t

    -

    ) Z ( t ) $ T ( t ) P ( t- 1)[1 +

    $ T ( t ) P ( t -

    )z(t)]-

    K ( t )= P ( t ) Z ( t )= P ( t

    -

    )Z(t)[l

    +

    ICIT(t)P(t

    -

    )Z(t)]-

    &t)

    = 6(t

    - 1) +

    K ( t )

    ~ ( t )

    T(t)6(t)]

    with initial condition

    P

    = sol, in which a0 is a sufficiently large real number.

    Extended Kalman

    filter

    It is based o n considering an extended state vector which includes, in addition to the response vector an d

    its derivative, all the param eters to be identified. Starting from a n initial guess, this extended state space is

    recursively updated as new observations are made available. The update is based on the Kalman filter

    formalism. The algo rithm

    is

    started with a n initial guess for the parameters a nd the e rror covariance matrix.

    Th e convergence

    of

    the algorithm a s well as the final values are known to depend, t o

    a

    great extent, on this

    initial guess. This technique has already been applied to system identification problems of seismic structu ral

    systems with inelastic behaviour. -

    P R OP O S E D I DE NT I F I C AT ION S C HE M E S O N S I M UL AT I ON DAT A

    Two sets of experiments provided acceleration time histories of floor responses for the verification of the

    identification algorithm. The experiments involved an eight-storey steel frame structure and a 47-storey

    building. The identification approach to the detection of changes in structural parameters is proposed.

    Under the assumption of a linear N-degree of freedom system with well-separated modes, identification

    can be performed for each single mode in the time domain. A band pass filter is used to isolate one single

    mode (basically starts from the fundamental mode). The recursive instrum ental variable me thod is then used

    to identify the mod al parameters from time do main analysis. This identification algorithm can simplify the

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    A

    SYSTEM IDENTIFICATION APPROACH 89

    multiple input/multiple output problem to a multiple input/single output problem. Follow the results of

    identification on fundamental mode, remove the identified mode from the response data, and then perform

    the identification on the next higher modes. After identifying the total rn-modes and if there are only

    m

    modes

    to

    be

    identified, then the error between measured and identified data can be calculated. The error function is

    defined as the root-mean-square value of acceleration records.

    If

    iteration is needed, then the identification

    can return to the starting point to identify the modal parametefs .of the fundamental mode. The previously

    identified modal parameters can be used as initial values. Then follow the same procedures described above

    to identify the modal parameters. The flow chart of this sweep modal identification procedure is shown in

    Figure

    1.

    The response quantities, such as storey drift, base shear, etc., can also be recovered from the

    analysis.

    Input/Output

    ?ta

    Fourier Transform

    m

    Identify i-th mode

    for analysis

    Using

    band-pass filter

    to

    isolate i-th mode data

    Fourier Transform

    i-th mode model parameters

    w s o i

    I

    Time response

    of i-th mode

    parameters

    of

    i-th mode

    (remove modes

    other than

    i-th mode)

    l

    I=

    1

    Remove i-th mode

    from records

    j=m

    c

    Figure 1.

    Flow

    chart

    of

    sweep modal identification procedure

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    C.-H.

    LOH

    AND

    1 . C TOU

    The identification on the non-linear response of a structural system, especially on the h ysteresis restoring

    force in each floor, can be established from the following equatio n:

    The restoring force diagram can be constructed from the top floor to the bottom level sequentially. The

    identification problem is then formulated as an optimal state determination problem whose solution is

    sought by extended Kalman filtering methods. Finally, the p aramete rs of the non-linear hysteresis model can

    be identified starting from the to p floor to the basement.

    I DE NT I F IC AT I ON S T UDY O N

    T W O

    S T R UC T UR E S

    The above-mentioned system identification approach was applied to a 47-storey building and a simulated

    structure. First, the earthquake response of a 47-storey office building at San Francisco was used. Earth-

    quak e data recorded from this building durin g the Lom a Prieta ear thq ua ke were used to identify the system

    dynamic ch aracteristics. Figure 2 shows the plan view

    of

    the building and the location of accelerometers.

    Second, two eight-storey full-scale steel structures with and without Added Damping and Added Stiffness

    (ADA S) devices were simulated. Figu re

    3

    shows the geometry of the eight-storey steel frame with a nd w ithout

    energy devices. DR AIN -2D inelastic plane structural analysis was performed on these two steel structures for

    a wide range of acceleration records. The circles in this figure denote the angle of rotation at both ends of

    a column. The larger circle denotes a larger rotational angle. The angle of rotation can be measured as

    compared to the reference circle. Fo r on e steel structure energy dissipation devices were placed between the

    Figure 2. Location of accelerometers at a 47-storey office

    building

    at San Francisco

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    A SYSTEM IDENTIFICATION APPRO ACH

    91

    E

    m

    '9

    +

    n

    +

    E

    *7

    '9

    En

    9

    3

    Figure 3. Geometry of eight-storey steel frames. The angle

    of

    rotation at column is also shown by circle (the reference circle is shown

    with number). (a) Steel frame without A DAS devices; (b) steel frame with ADAS devices

    Table I. Identified modal parameters for the 47-storey San Francisco building during the Loma Prieta

    earthquake north-south direction)

    Storey two) f ( H 4

    P N S

    PEW pUD Error%

    Mode 1 1.40 0.1900

    1.6659 - 0.0092

    Mode 2

    1.74 0.5730 - 09898 om42

    44th Mode 3

    1.89 0.9852 06296

    - 0.0721

    Mode 4

    2-87 1.3167 - 05601

    0.0685

    Mode 5 1.81 1.6956

    02227 - 0.1140

    Mode

    1

    1.39 0.1900 1.4700 0.0063

    Mode 2 1.79 0.5699 - 05440 - 0.0062

    39th Mode 3 1.88 0.9849 00628 00192

    Mode 4 2.84 1.3164

    02032

    0.0645

    Mode 5

    1.73 1.687

    1

    - 1420 - 0933

    Mode 1 1.40 0.1898 05086 0.3010

    Mode 2 1.70 05680 04936 0.024 1

    16th Mode 3 2.02 0.9841

    02022 01236

    Mode 4

    - - - -

    Mode

    5

    2.02 1.7168 - 00408

    -

    0.0162

    Mode 6

    207 2.1077 - 01865

    - 0.0932

    - 0.0882

    0.0139

    0.1057

    - 0.1511 1.4532

    -

    0.2007

    -

    0.0744

    00052

    -

    0.1560 0.8216

    - .0260

    0.1257

    - 0.1313

    - .0630

    - 0.1378

    -

    0.0183

    0.0252

    Note:

    P,

    denotes the participation factor

    of

    the north-south direction, and

    PEW

    nd

    P,,

    are for the the east-west and

    up do w n directions.

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    9 2

    C.-H. LOH

    AND

    I.-C.

    TOU

    + 1 s t MODE

    a::::

    rc- 0.24

    0.111

    30

    40 60 110 70

    no

    100

    TIME

    Sec)

    o , L

    , I I I I

    I I 1 ,

    I

    I I

    I

    lb do 30

    40 60

    6 0 70 110 90

    100

    TIME S e c )

    y

    2nd

    MODE

    0.60 I

    2 .46

    U

    0 30

    0 . 1 5 : ,

    I

    , I , ,

    I 1 , I I I I

    0 10

    20 30 40

    60 110

    70 110 90 100

    TIME

    S e c )

    19

    a4

    0 10

    90

    30

    40

    60

    110 70

    110

    90

    100

    TIME S e c )

    +

    3rd MODE

    3 0.00

    0.76

    0.00

    U

    0.411

    . 0 6 6

    I0

    PO

    SO

    4 0

    110

    60 70

    110 90 100

    TIME

    Sec)

    94 tclo

    TIME

    S e c )

    Figure 4. Identified modal damping and frequencies from sequential regressional analysis by using data from the 39th

    floor

    of the

    47-storey building during the Loma Prieta earthquake (north-south direction)

    basement and the first floor an d between the fourth an d fifth floors. The scaled El Centro acceleration records

    with different intensity levels (from

    50

    to

    300

    gal) were specified as input mo tions and produced responses for

    identification. In the analysis, yielding may ta ke place only in concen trated plastic hinges located at the ends

    of column elements. It is assumed that the horizontal floor responses at each floor level can be recorded for

    identification.l 2

    Identijication on the earthquake response

    of

    a 47-storey building

    The recursive least-squares method was applied first to identify the natura l frequency an d dam ping ratio of

    the first three modes of the building (identification was performed for each single mod e th roug h b and pass

    filtering of the response data). The natura l frequencies an d da mp ing ratio s of this building a re almost time

    invariant (after 9 s of recorded motion), as shown in F igure 4. Some negative damp ing values were identified

    at the beginning of the records. This is because the recursive least-squares method takes two or three time

    window data to converge the identified parameters, and the initial guess of modal parameters will also

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    A SYSTEM IDENTIFICATION APPROACH 93

    influence the convergence. Since the recursive least-squares meth od was applied only for a n individual m ode,

    the linear identification technique (sweep mo dal identification and recursive instrum ental v ariable meth od)

    as discussed before can be ap plied. Table

    I

    shows the results

    of

    identification

    of

    this 47-storey office build ing

    at San Francisco. In this study multiple inputs (three components of basement mo tion) and single outp ut

    200

    100

    m

    z

    -100

    4

    -200

    a

    2

    40

    6a

    i aa

    TIME

    ( S c c )

    TIME

    ( S e c )

    Record

    Idontrllostlon

    aeord

    Idmtiliomllm

    6

    24

    16

    -

    V

    2 8

    f

    g 4

    t

    k

    8

    -8

    P Z

    I

    -16

    a -24

    0 0

    a.s

    1.0

    1 s 2.0 2.6 3.0 0

    FREQUENCY Hz) TIME (Sec)

    0

    Figure

    5.

    Comp arison between recorded an d identified time series (acceleration, velocity an d displacemen t) as well as Fou rier am plitude

    for the 39th floor response

    of

    the 47-storey San Francisco office building during the Loma Prieta earthq uak e (north-south direction)

    8

    7

    6

    2

    1

    0

    f a )

    C )

    0.0 1.0

    -1.0 0.0 1.0 0.0

    2.0 4.0

    S t o r y D r i f t

    Figure

    6.

    Comparison between the identified mode shapes and storey drift from the response data of an eight-storey steel frame with

    ADAS subjected to different input intensities:

    (

    100

    gal;

    (-

    -

    - -

    -)

    200 gal;

    -

    -)

    300 gal). (a) First mode shape, (b) second

    mode shape, (c) storey drift

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    94 C.-H. LOH AN D

    1 . C

    TOU

    6 i

    0.52

    -

    0.50

    2I

    4 -

    /*

    ,.' 0 . 4 8

    az

    3 -

    f 0.46

    ( w i t h

    ADAS )

    r 0.44

    z 1

    2 0 .4 2

    r . , . , L

    /

    H 5 -

    ( w i t h o u t

    AOA S ) / '

    ,.- u

    m

    / /

    u

    /;

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    a

    )

    R

    e

    o

    n

    o

    r

    c

    p

    U

    n

    i

    m

    a

    G

    a

    )

    s

    o

    r

    y

    S

    t

    e

    a

    m

    e

    w

    i

    h

    o

    u

    t

    D

    S

    d

    c

    M

    a

    m

    u

    m

    n

    p

    u

    t

    n

    t

    e

    n

    s

    i

    y

    ,

    P

    G

    A

    =

    )

    b

    )

    R

    e

    o

    n

    o

    r

    c

    p

    u

    n

    i

    m

    a

    G

    a

    )

    s

    o

    r

    y

    f

    a

    m

    e

    w

    i

    h

    O

    S

    d

    c

    M

    a

    m

    u

    m

    n

    p

    u

    t

    n

    t

    e

    n

    s

    i

    y

    ,

    P

    G

    A

    =

    )

    SYSTEM IDENTIFICATION APPROAC H 9

    also shown

    in

    Figure

    7.

    Comparison

    between

    the identified fundamental frequency and damping ratio

    of

    the

    equivalent linear system from data of different excitation intensity is also shown in Figure 8

    t

    is pointed out

    that the

    steel

    structure

    with

    energy dissipation devices will increase the system natural frequency about

    12

    per

    cent

    and

    decrease

    1

    per cent of the

    system

    damping ratio from

    the

    system without devices.

    Since

    the

    structural system behavesas an inelastic system during strong ground motion especially when the excitation

    Figure

    9.

    Plot

    of

    restoring orce with

    respect

    to relative displacementbetween

    floors

    a) eight-storey steel frame without

    ADAS

    devices;

    b)

    eight-storey steel frame with D S devices

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    9

    C.-H. LOH

    AND I.-C. TOU

    l a )

    Figure 10. Comparison between the simulated and identified response data of the fifth floor using a non-linear hysteresis model

    subjected to input PGA = 300 gal: (a) w ithout ADAS devices; (b) with ADAS devices

    I d e n t i f i e d

    WITH-

    O i s p l

    acement.cm

    Figure 11. The identified modal parameters of hysteretic restoring force of equation (8) from the response between the fifth and sixth

    floors

    of an eight-storey steel frame with and without

    ADAS

    devices subjected

    to

    excitation with

    PGA

    =

    300 gal

    intensity level is 300 gal), comparisons between predicted a nd simulated da ta using the linear model a re no t

    quite consistent. A non-linear hysteretic model must be used to revise the linear model.

    As

    discussed in the previous section, the Bouc-Wen non-linear hysteretic mod el was used and adde d

    between each floor level to consider the hysteretic behaviour of yielding at a column. Figure 9 shows the

    hysteretic loops between floors. The extended Kalman filtering technique is used to identify the modal

    parameters shown in equation (8). Figure 10 shows the comparison

    of

    relative displacement, velocity and

    acceleration between the two systems by using the non-linear hysteretic m odel. The m odal param eters a s well

    as the respective hysteretic loop are also shown in Figure

    11.

    The viscous damping ratio

    to

    nd n atural

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    A S YS TE M IDE NT IF IC AT ION AP P R OAC H

    97

    frequencyf, of the mod al pa rameters w ere identified first from the d at a of low intensity excitation (for

    example: groun d intensity 50 gal).

    C ONC L US I ONS

    The purpo se of this study is to use the identification app roach to detect the changes of structural param eters

    during ground excitations. With this approach the damage detection can be obtained. First, under the

    assum ption of linear response, identification is performed on the stiffness-reduced struc ture , an d the dyna mic

    characteristics of the structural system are observed. These dynamic characteristics include mode shapes,

    storey drift, elastic force and base shear distribu tion. Second, applica tion of this technique to a no n-linear

    system is also discussed. The following conclusions are drawn:

    1. For

    a building structure, with little modal interference, a sweep model analysis is proposed. This

    approa ch can reduce the modal parameters from the case of multiple input/multiple ou tpu t to the case

    of multiple input/single output without loss

    of

    accuracy in system identification.

    2. The recursive instrum ental v ariable method, which is insensitive to noise conta mina tion of the da ta, can

    provide accurate estimates of modal parameters under sweep modal analysis.

    3. Recovery of mode shapes from the results of modal identification by using polynomials can be

    extrapolated to estimate the storey drift, elastic force an d base shear.

    4. For

    a system where the stiffness is changed, the identified mod e shap es and natural frequencies did not

    provide strong evidence

    of

    the chan ges of stiffness. How ever, the storey drift index did p rovide a good

    indication for the detection of changes in structural parameters. Large storey drift may be caused by the

    reduction of stiffness or the structu ral response goes into the inelastic range.

    AC KNOW L E DGE M E NT S

    The sup port of the N ational Science Council, R.O.C., through grant NSC83-0414-P-002-005-B during

    this research is gratefully acknowledged.

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