[1995] a System Identification Approach to the Detection Of

7/24/2019 [1995] a System Identification Approach to the Detection Of

1/13

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


2/13

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.


3/13

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


4/13

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


5/13

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


6/13

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


7/13

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.


8/13

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


9/13

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


10/13

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

/;


11/13

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


12/13

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


13/13

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.

REFERENCES

1.

J.

T. P. Y ao, Safety and Reliability of Existing Structures, Pitman Advanced Publishing Program, Boston, 1985.

2. J. E. Stephens, Structural damag e assessment using response m easurements, Ph.D. Thesis, School of Civil Engineering, Purdue

3. H . G. Nathe and J . T. P. Yao, Research topics in structural identification, Proc . 3rd con$ on dynamic response ofstructu res, ASCE,

4. M.

S.

Agbabian, S.

F.

Masr i, R. K. Miller an d T. K. Caughey, System identification approa ch to detection of structural damage,

J .

5. C. H.

Loh

and S. T. Chung, A three-stage identification approach for hysteretic systems, Earthquake eng. struct. dyn. 22, 129-150

6 . Y . K. Wen and T. T. Baber, Random vibration

of

hysteretic degrading system,

J .

eng. mech. din ASC E

107,

1069-1087 (1981).

7. P. Caravani, M. L. Wa tson and W. T. Thom son, Recursive east-square time doma in identification of struc tural parameters,

J .

appl.

8. Torsten Sode rstrom and Petre Stoica,

System Identification,

Chapters 8 and 9, Prentice-Hall, U.K., 1989.

9. M. Hoshiya a nd E. Saito, Structural identification by extended Kalman filter,

J .

eng. mech. AS CE 110, 1757-1770 (1984).

10. M. Hoshiya, A mplication of the extended Kalm an filter-WGI metho d in dynamic system identification, in S. T. Ariaratnam,

G. 1. Schueller and I. Elishakoff (eds),Stochastic Structural Dynamics Progress in Theory and Application, Elsevier, Amster dam, 1988,

11.

R. G . Ghanem, H. Gavin and M. Shinoz uka, Experimental verification of a numb er of structu ral system identification algorithm,

12. K. C. Tsai, H. W. Chen, C. P. Hong and Y.

F. Su,

Design of steel triangular plate energy absorbers for seismic-resistant

University, West Lafayette, IN, 1985.

Los Angeles, 1986, pp. 542-550.

eng. mech. AS CE 117, 370-390 (1991).

(1993).

mech. ASME

44,

135-140 (1977).

pp. 103-124.

Technical Report NCEER-91-0024, National Center for Earthquake Engineering Research, Buffalo, NY, 1991.

construction, Earthquake spectra 9, 505-528 (1993).

[1995] a System Identification Approach to the Detection Of

Documents

Transcript of [1995] a System Identification Approach to the Detection Of