PART3:仮動的実験における実験誤差の累積 : 仮動的実験応答の安定と精度(梗概)

Architectural Institute of Japan

ArchitecturalInstitute of Japan

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PART3:EXPERIMENTAL

ERROR

GROWTH

IN

PSEUDO

DYNAMIC

TESTING

(Stability

and

Accuracy

Behavior

of

Pseudo

Dynamic

Response)

by

MASAYOSHI

NAKASHIMA*

and

HIROTO

KATO'",

Members

of

A.

I.

J.

1,

lntroduction

The

pseudo

dynamic

(PSD)

test

(also

referred

to

as

the

on-line computer

te$t

control method

)

is

an

experimental

technique

to

simulate

the

earthquake response

behavior

of structural systems without using a

shaking

table

device,

Since

first

devised

by

Takanashi

et al.

<Ref,

1)

of

the

Institute

of

Industrial

Science,

the

University

of

Tokyo,

this

technique

has

been

employed

by

many researchers

in

both

Japan

and overseas

(Refs.2

and

3).

One

should not overlook,

however,

that

the

PSD

test

is

nothing

else

but

an

appreximate

method with various assumptions and

simplifications,

and,

therefore,

the

obtained response

i$

not

identical

with

the

true

response,

As

the

major assumptions, one can state

_l)

the

discrete

spring-mass representat'ion of a continuum,

2)

the

discretization

with respect

to

the

tirne

domain,

and

3>

the

arbitrary selectionof viscous

damping.

Since

those

assumptions are

inevitable

in

the

course of

the

basic

formulation

of

the

PSD

test,

the

response error caused

by

those

assumptions may

be

said as

the

intrinsic

error.

Recently,

studies

have

been

made

to

investigate

the

intrinsic

eTror

behavior

in

the

PSD

test

(Refs,4

to

_9).

There

is

anotheT

type

of error sources

that

also

bring

deterioration

in

the

PSD

[esponse.

In

the

PSD

test,

the

load

applying

actuatbrs should

lead

the

test

specimen

to

the

computed

displacements

and

then

measure

the

reactional

forces.

Because

of

the

finite

accuracies

of

the

displacement

and

load

measuring

_fiensors

as well as

the

servo control

limitation,

neither

the

displacements

that

could

be

_positioned

by

the

actuators would

be

the

same as

those

commanded, nor

the

measured reactional

forces

be

the

true

reactional

forces.

Those

experimental error sources

also

appear

in

our

conventional

qttasi-static

test.

Beeause

of

the

_{postdetermined}

nature

in

the

displacement

history,

the

PSD

test

is

found

to

be

more vulnerable against

those

error sources

than

the

quasi-static

test

with a

predetermined

loading

history.

A

simple example

to

demonstrate

this

is

given

below.

Let

us suppose an undamped

linear

elastic and single

degree

of

fieedom

(SDOF)

system subjected

to

free

vibration,

and,

further,

suppose

that

the

displacement

transdueer

to

measure

the

displacement

is

not

accurate

and

has

hystersis

exhibiting acounterclockwise

loop

as shown

in

Fig.

1.

Although

the

analyzed

system

is

_purely

linear

elastic,

in

the

PSD

test,

energy eAual

to

the

area enclosed

by

the

transducer's

hysteresis

loop

is

added

to

the

system after

every

one

cycle

of vibration, and

this

added

energy

causes

a

diveTgent

response.

In

the

conventional

Force

static

test,

such

an error

in,

the

transducer

_produces

DiSP･

shifts

in

the

refeTence

displacement,

but

by

no means

make$

the

test

structure

to

behave

in

a

divergent

fashion.

In

fact,

nota

few

inve$tigators

reported

the

cases

m

which

the

PSD

test

response could

be

lead

erroneous

because

of a

divergent

error response

(Refs.2,

3,

4,

6,

10J13).

From

the

above considerations,

several

research needs seem

to

be

surfaced out regarding

the

application

of

the

PSD

test.

TheY

may

be

classified

into

1)

defining

-l[etl,!rdlXmXcDisp.

Time

Flg.

₁

PSD

Response

ef

Linear

SDOF

System

When

ducer

Has

Countercleckwise

Hysteresis

i

_Senior

_Research

_Engineer

#

Research

Engineer,

Buildlng

Resea[ch

lnstitute.

Ministfy

_of

Construction

(Manaseript

received

August

13,

l987)

-36-NII-Electronic Library Service

the

types

of

experirnental

error

sources

included

in

the

PSD

test,

2)

examining

their

effects on

the

response

(the

response

deviation

caused

by

those

experimental error sources

is

clesignated

herein

as

the

experimental

(response)

6rror),

3)

setting up

_quantitative

_guidelines

for

enabling

us

to

estimate

the

magnitude of

the

experimental error, and

4)

developing

test

control algorithms

that

could suppress

the

experimental error

_growth.

This

_paper

(Part

3>

_presents

the

experimentar studies

to

investigate

the

first

two

subjects

indicated

above, and works related

_to

_the

second

half

will

be

reported

in

the

companion

_paper

(Part

4).

2.

Experimental

Error

Sources

In

this

section, sources

that

could

_produce

the

experimentaL error' aie

defined,

and

_procedttres

to

estimate

their

effects

en･the

experimental

error

are

introduced;

Throughout

this

study.

the

PSD

test

system

developed

by

the

Building

Research

Institute

(BRI),

Ministry

of

Construction,

will

be

refe[red.

EFror

sources

included

in

the

PSD

test

are sometimes said

to

be

system

dependent.

According

to

an extensive survey on

the

existing

PSD

test

systems

developed

by

_Japanese

researchers

(Ref.

13),

however,

the

basic

test

opeTation

is

not much

different

from

one

to

another,

and,

therefore,

the

writers

believe

that

the

discussion

based

upon

the

BRI

PSD

test

system

holds

its

generality

with respect

to

the

experimental

e[ror

sources.

A

cliagram

of

the

basic

PSD

test

operation

system

is

given

in

Fig.2.

In

this

figure,

it

is

supposed

that

all

information

up

to

the

i-th

step

is

already

obtained,

and

that

(

i+1)-th

step

operation

is

to

be

performed.

Symbols

appearing

in

Fig.

₂

and

details

in

the

diagram

can

be

found

elsewhere

(Ref,

_14),

In

the

BRI

PSD

test

system,

digital

displacement

transducers'are

incorporated

to

directly

measure

the

disPlacements

of

the

test

specimen

for

the

purpose

of ensuring

high

resolutions

in

the

large

stroke measuiement.

Since

the

servo control

is

established

in

analeg

form

as

done

conventionally, a combined

digital-analog

leading

algorithm

has

been

employed.

This

algorithm can

be

explained

in

reference

to

Fig.

_3.

Here,

the

digital

displacement

transduceris

set

up

to

measure

the

displacement

of

the

test

structure, whereas

the

analbg

displacbment

transducer

is

to

measure

the

displacement

of

the

actuato[

ram.

Provided

that

tlte

test

structure

be

deformed.from

_position

xt

to

xt.i,

first

A

times

Axo

(='xt.i-xi)

of

the

displacernent

increment

is

commanded

to

the

actuator.

HeTe,

A

is

a coefficient

(siay

O.

5)

that

should

be

specified as an

initial

input

by

the

test

operator.

Because

of･the

structure

stiffness

andlor

loading

apparatus

flexibility,

the

displacement

of

the

structure after

this

actuator motion

is

most

likely

not

identical

with

the

actuator

disptacement.

At

this

_point,

the

displacement

of,the

structure

is

monitored

bY

_the

digitaL

displacement

transducer,

and

the

rernaining

displacement

increment

is

measured.

Then,

A

times

this

remaining

displacernent

increment

is

cornmanded

to

the

actuator, _and

this

_proeess

is

repeated until

the

displacement

reaches

x,.,

w.ith an allowable e'rror of

2E.

This

allowable error

is

another

initial

input.

When

multiple actuators are employed

in

one

test,

the

velocities of

the

actuators are continuously adjusted so

that

those

actuqtors can reach

their

respective

target

_positions

(in

each

incren)ental

segment

loading)

approximately

at

the

same

tirne.

Furtherrnore,

iterative

adjustment

is

made until all actuators reach

their

final

-targets

with

their

specified

allowable errors.

Looking

into

this

control mechanism

(Figs.

2

and

3),

one can examine

the

_potential

expeTimental error sources,

First,

the

achieved

displacements

after

the

necessary actuator motion are

likely

to

be

deviated

from

the

computed

target

displacements,

This

deviation

takes

_place

because

of

the

finite

accuracy of

the

displacement

sensors as well as

the

lirnitation

_of

the

_{servo control}

(i,e.

the

_{servo action"s}

inability

to

lead

the

_structure

to

the

target

_posit}'on),

Further,

the

measured

forces

at

the

_pesitioned

displacement

level

may

be

deviated

from

the

true

reactional

･forces

at

A/DCONVERSION

{t}1}Cteedieeked)1=j+1

{tFl}

: cmedsured),

_r,

'

i

N

･[m]Cg"+[c]{'ft11+{i/1=-[m]il}ild

･tXFFt]=C[M]+At12[c])-i

c2[m]{n}.CAt12[c]-[ml]txH}

-btt{{n+[m]ol'x'd})

'

-{Xd+1}

A

[meesureti)

/

s1t

{X;+1)

Ccomputed)

Fig.2

Flow

DIA CONVERSION

Diagram

of

i

VLo,ff

SERVO

,

CONTReLLERS

1

:

ottuator'

{Xi+1}

(com-rn-sp.d..e-Ql

PSD

Test

_Operation

ON-LavE

Fig.3

Comblned

ALLOWAeLE PECIMEND15P. ERRoReouND ep

ACTUATORDISP.

Digital･Ana!og

Loading

Control

-37-Architectural Institute of Japan

ArchitecturalInstitute of Japan

this

level

because

of

the

finite

accuracy of

the

load

measuring sensors.

Finally,

the

measured

forces

are

AID

converted;this

_process

includes

truncation

or round-of £

As

described

above,

there

are several experirrlental error sources

in

the

closed

loop

of

the

test

operation

(Fig.

2),

but

all

of

those

errors

accumulated

during

one

step

operation

are

finally

combined

into

one

reactional error

force

quantity.

If

the

true

reactional

forces

corresponding

te

the

computed

displacements

aJe

taken

to

be

(ljU,D,

which, although, would never

be

correctly estimated, one can see

that

the

difference

between

Gfh,,l)

and

the

feedbacked

reactional

forces

(Y,i)

serves

as

the

reactional

error

forces

applied

at

each

step,

This

can

be

understood

frem:

[m]IXtl+[c]1thil+l.fht]I=-[m]l1IX.i+GJhtE-lfsl)････-････････-･･-･･･-･･････････--･･････-･･-･･･････-････-･-･･-･-･-･･(1)

in

which

_m,

_c,

and

x.t

are

the

mass, viscous

damping

coefficient, and

input

acceleration.

The

bracket

([

]),

brace

(l

l),

and

dot

indicate

the

matrix, vector, and

differentiation,

and symbol,

i,

the

time

step.

Here,

in

the

computation,

all

associated

_quantities,

i.

e.

thedisplacements,

velocities,

and

accelerations,

havetheir

basis

on

the

computed

displacements

Jather

than

the

measured

displacements,

According

to

Kaminosono

(Ref.

Is>

and

Mahin

and

Shing

(Ref.

4),

use of

the

computed

displacements

is

better

in

achieving

more

stable amd accurate solutions.

Frem

Equation

1,

one can see

that

the

experimental error caused

by

those

experimental

error

sources

is

nothing

else

but

the

response.of

the

analyzed system subjected

to

the

reactionai error

forces

as

the

input

forces.

It

is

worth while

to

comment

on

the

relative contribution of

those

individual

error

sources

to

the

reactional

erTor

force,

The

error

_given

as

the

difference

between

the

computed

displacement

and

the

displacement

achieved after

the

actuator motion

(the

errer

is

here

defined

as

the

dispiacement

error)

has

a

dimension

of

the

disptacement,

and,

therefore,

the

resultant [eactional error

force

is

estimated as

the

displacement

error multiplied

by

the

stiffness of

the

test

strueture.

On

the

other

hand,

the

errors

_generated

because

of

the

load

measuring sensor's

inaccuracy

and

the

AID

conversion

(here

defined

as

the

force

errors)

have

adimension of

foree.

Atthis

_point,

one can readily

find

that

the

stiffness

of

the

test

structure

_plays

an

important

role

to

the

relative contribution of

those

errors

to

the

reactional

error

force.

If

the

test

structure

is

stiff,

the

contribution of

the

displacement

error

is

more significant since

the

error

force

generatecl

by

this

displacement

error

is

directly

_proportional

to

the

stiffness of

the

structure, whereas

the

force

errors

remain

unchanged

regardless

of

the

stiffness

of

the

structnre.

It

is

also

to

be

noted

that,

once

the

test

structuTe

falls

into

its

inelestic

iange,

the

displacement

error effect on

the

final

reactional error

force

decreases

since,

in

this

range,

the

structure usually

loses

its

stiffness.

3.

Experimental

Error

Growth

Behavior

(I):PSD

Test

for

2DOF

System

3.1

Description

of

Test

To

investigate

the

effects

of

the

above

defined

error

sources

on

the

experimental

(response)

error,

a

serie$

of

PSD

tests

were

carried

out.

Considering

the

observatiens

that

the

experimental

errer

eflect

is

more

significant

and

complex

in

the

PSD

test

applied

to

multi

DOF

systems

(Refs.

6,

11,

13>,

the

structure

used

in

the

tests

was a

two

story

steel

braced

frame.

The

basic

dimensions

of

the

test

structure

as

well as an overview

_Qf

the

test

setup

are

shown

in

Fig.4.

The

$tructure was

treated

as

a

two

DOF

system with each mass

assigned

at each

floor

level.

Through

preliminary

small

loading,

the

elastic

stiffnesses

of

the

test

structure

were

estimated

as

shown

in

Table

1.

This

table

also shows

the

assigned masses, and

the

natural

frequencies

and vibrational modes computed

based

on

those

properties.

If

the

error sources stated earlier were reminded,

the

_paTameteTs

that

could

be

selected

in

the

test

are

1)

the

resolutions

(accuracies)

of

the

displacement

and

load

measuring sensors,

2)

the

resolution of

the

AID

converter,

3)

the

servo contiol

_gains,

4)

the

allowable

displacement

eirois, and

5)

the

type

of

input

motion,

It

is

natural

that

we

can

obtain

more reliable results with

higher

resolutions of

the

sensors and converter,

then

the

test

operator's choice should

be

to

have

those

devices

adjust so

that

the

highest

resolutions could

be

achieved.

Table2

shows

the

[esolutions

(accuracies)

of

the

sensors and converteT employed

in

the

test.

This

table

also

includes

the

error

force

leveis

that

could

be

_generated

by

those

devices,

The

servo control

_gains

and allowable

errors

are something

to

be

examined rnore closely.

It

is

known

that

the

test

structure

is

_positioned

more accurately with

the

increase

of

the

servo

gains,

but

with

the

Sacrifice

of control stability.

According

to

the

leading

algorithm employed

in

the

test,

however,

the

final

displacement

errors are

_governed

sotely

by

the

specified allowable eTro[s

no

matter what

_gains

are selected,

because

the

loading

is

continued until

the

test

structure

is

converged

to

the

target

displacements

(measured

by

the

digital

displacement

transducers

)

within

those

allowable errors.

According

to

_previous

experiences,

if

the

allowable

-38-NII-Electronic Library Service

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l

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g

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o as

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YNIT

:rrtn {al tb)

Test

Structure

and

Setup

Used

in

2DOF

PSD'Test

((a)

Test

Structure,

(b)

Test

Setup)

Properties

of

2DOF

Test

Structure

Table

3

Designation

and

Test

Parameters

fDr2DOF

PSD

Test

Table2

FuLl

Scales

and

Resolutions

of

Sensors

Used

ln

2DOF

PSD

TestFul]ScalePossibleError

LoadCel1s

Digital'Disp.

Transducers

AID[enverter

10,OOOkg500rtrm'12bit

20kgtO.Ol"r"

s(loooo/2o4s}kg

DisptacementWeignt

rcr-a-ngd-u-c-ecxc

Actuetore

1

--=F."---･e･.-

_'

Tt

= IM

-

e

[-7

-LL.p

'..=.

--e.!

_tt

'-tt-

Jt

L

-Specimen

Straingage

Fp.2000MM--TestBed

.

_by

_Preliminary

_Calibration

_of

_Load

_Cells

errors are set

to

be

large,

the

test

control

is

easier, whereas,

if

they

are set

to

be

very srnall, much

time

is

often needed

to

cemplete

one

step

of

loading

because

of many

iteTations.

Considering

the

trade"off

between

the

comfort of

the

test

operation and

the

accuracy

in

the

displacement

control,

this

allowable error

term

should

be

a

_parameter

worth

while

to

examine,

The

type

of

input

motion also

is

a

candidate

which may affect

the

experimental error

behavior

in

the

PSD

test.

Based

on

those

examinations,

two

_parameters:O

the

amount of

the'allowable

brrors

and

2)

the

type

of

input

motion were selected, and a

total

of six

PSD

te'sts

were conducted with

two

different

allowable errors

:

_O.

_e2

mm

(representing

the

accurate

displacement

control) and

O.

15

mm

(representing

the

easy

test

operation), and

three

different

input

motions

:'

the

zero

input

(providing

the

response

by

the

ept'perimental erroT

force

only),

the

initial

impulse

input

_{(representing}

the

fundamental

form

of earthquake

loading>,

and

the

_ground

motion

inpfit.

TabLe

3

summarizes

the

basic

test

_parameters

empleyecl

in

the

tests

with

the

designatien

ef

each

test,

The

tests

were conducted so

that

the

_test

structure should

behave

elastically.

The

number of'steps･in

Table

3

indicates

the

last

step of

the

integratieti

just

before

the

test

was

terminated.

3,2

Test

Results

It

was

found

from

the

test

results

that

the

effect of

the

type

of

inpttt

motion on

the

experimental error

behavior

was minimal, whereas

the

amount of

the

allowable errors

incleed

influenced

the

error

behavior

a

_great

cleal.

The

discussions

to

follow

therefore

are only concerried with

the

tests

under

the

_ground

motion

<Tests

EQ02

and

EQ15),

'

'

Details

in

the

other

tests

can

be

founcl

_elsewhere

(Ref.l6).

_.

'

Following

diagrhms

are

_presented

for

discussions.

They

are

1)

<computed)

displacement

and shear

force

time

'

histories

and

their

Fourier

spectTa

<Figs.

5

and

_6>

:

₂₎

displacement

error

tirne

histories

and

their

Fourier

s ectra

DesignationLoadingConditionErrorLastStepof

ofTest

BoundIntegration

(":n)

ZERO02

Zerornput

O.02

103

ZEROIS

ZeroInput

O.15

36

PULSE02lnitialPulseO.02

1OO

tSOegal}

PULSE15InitialPulseO.l5

33

{500.gal}

EQ02

EarthquakeInputO.02

13]

(343.7gal}

EQ15

EarthquakeInputO.15

35

(343.7gal)

'IntegratianTimeInterval;O.O05sec.

(Figs,

7

and

8).

Here,

the

displaeement

error

is

redefinecl as

the

computecl

displacement

minus

the

measured

(by

the

digital

displacement

transducer)

displacement

and regarded

to

be

representative of

the

experimental error

sources.

Thir

representation was adopted

becauseit

was

the

onlyerror source

that

could

directly

be

estimated

from

the

test.

In

fact,

this

error was

found

to

be

the

major source of

the

experimental error as will

be

verified

late'r

:

3)

corre.lations

-39-Architectural Institute of Japan

ArchitecturalInstitute ofJapan

betweenthedisplacement

errorsand(computed)

dislacementmcrementsobtained

in

the

test

(Figs,

histograms

of

the

displacement

errors(Fig

6)11)

:

Here5}'1

numericallyobtaineddislacement and shear

force9

and

10)

:

4)

time

histories

l.llt!L!bgi!-Egy!lelsRggy!g

tththeirFouner

t(Figs.sand

PSD

test

and using

the

_properties

listed

in

Table'

numerical analysis

into

which reactional error

the

computation was made under

the

:

and

6)

displacement

and shear

force

conditions

identical

_to

_the

timehistoriesobtainedfrom

forceswereincorporated{Figs.12

and

_13).

In

_this

analysis,

the

2F 2･eo.o D[GPLACEnENT

-2.0FOVRr

SPEHA:.V

.

o. I o. ICum･s IF 2o SHEAH

-pFeU

s"At

a

(kg

FrequencyCHI)

Fig.5

2foe

-OFVV

smx

=(mm

Fig

fOH:[ IF2.o. OISPLhCE"EHT

-u･FDUHI

5PE"AX.V

-

a. [ o, tCrm･s

(a)

l; Frequency(Hi} ANALTSIS o SHEAA FrequencyCHI] FoecE

tzFOU

S"AX

-

{ltg

Frequency(Hz) EXeEHIHEHT

(b)

Time

Histories

and

Fourier

Spectra

Obtained

in

EQ02

_((a)

Displacement

and

Cb}

Shear

Force

Histories)

OlfFEHENCE O:Se. IF e[FFEHENCE olse. FrequencyCHi;

TestTime

.7

DisplacementErrorTime

(Test

EQ02)

o2FVRN,"-n-.if..'I

U./f

.-ep,'eli':Pe";eodis.,

o,.".le--:p-eO--;e.o-e-"-eD.,:"''eZ.-:

ee-es

/o1,,eN

Histories

IF

0

FrequenEyCHi)

and

Fourier

Spectra

oxe's./

n

o-oe

oeoe'"teijsoeeoo-""tsu}..b4ie..e

-2"oeeee-e--eeo"t'e".e;-s.-::ss-.1d

/

-1x

v

v

ov

o

U

:

Undershoot

,

O

:

Overshoot

Fig.9

_Co[relation

Between

Displacernent

ETrors

and

{computed)

Disptacement

Increments

<Test

EQ02)

40

2F 2o･o

-2,FSuHl

seEHAX.

m

D, o,Cm.s 2F 2 DISPLACEnENT SHEAH

-2Fou

SHAX

.

Ckg

FrequencyCHi)

Fig.6

2E O,2 FoHCE FrequencyCHI} -NALTSIS

,F

_OISPLACEnEHT zo

-2Fov

S"AX

g

[mm

FrequencytHi)

(a)

Lf SHEAn fOHCE ?o

-2Fou

sHA:

.

Ckg

Frequency(H:)

E:eEHIHE"T

(b)

Time

Histories

and

Fourier

Spectra

Obtalned

in

EQ15

_((a)

Displacernent

and

(b}

Shear

Force

Histories)

OIFFEHENCE OISe. IF D[fFERE-CE OISP.

rm O

TestTime

O,O D

-e

ne

-D.2

,}

-O

FaUR:E FaU s

Cmm,se

(nrm

o･e loD･e o

FrequencyCHz)

_{Frequency(Hz)}

Fig.

8

Dlsplacement

Error

Time

Histories

and

Fourier

Spectra

(Test

EQI5}

2F

1F

O

O

I

U

*KsL

:-U/o

Pi,,,

ep e-.eee

'1.:e.

1

eo"l

-1N

v

Fig.10U

.

-1

"

.

2

ot

b

e-.

,,･7pt

:･:-

g----lx

t

ov

o

:

Undershoot

,

O

:

Overshoot

Correlation

Between

Displace.ment

Errors

and

NII-Electronic Library Service

reactional error

forcas

were estimated as

the

_product

of

the

stiffness

(Tablel)

and

the

displacement

errors obtained at

the

corresponding step

in

the

experiment.

In

other words,

the

anal'ys'is was

_performed

using

Equation

1,

with

[k]

lx,l

as

iX,,,l

and

[k]Idx,I

as

(ifl,,,l-if,l).

Note

that

[h]

and

ldrptl

are

the

stiffness matrix

{Table

1)

and

displacement

errors measured at

the

i-th

step

in

the

corresponding

test.

3.3

Discussions

on

Test

Results

･

The

following

summarizes

the

major

findings

ex-amined

from

those

diagTams.

(

1

)

In

T.est

EQ02,

the

obtained responses weTe reasonably

close

･to

the

numerical

responses

(Fig.5),

but

the

vibration

having

the

frequency

of

19Hz

was more

_promoted

in

the

test

_particularly

in

the

shear

force

responses

(Fig.5(b)).

This

frequency

of

₁₉

Hz

structure.

(

2

)

In

Test

EQ02,

the

displacement

errors

were

(Fig.

7)

and

also with respect

to

its

amplitude

(Fig,

11).

O.

1

mm

to

O.

1

mm

for

the

second

stQry

and

from

-O,

two

to

five

times

larger

than

the

specified allowable combined effect of

inertia

in

the

loading

and aslight

time

1

measuring

the

forces.

'

(

3

)

In

Test

EQ02,

no strong correlation

displacement

increments

(Fig.9).

In

fact,

similar

'quantities

(such

as

the

shear

foices,

restoring

forpes,

'

found

in

any of

those

_plots.

(4)

In

Test

EQ15,

the

responses,

inc

were

utterly

dominated

by

the

yibration

having

th

divergent

behavior

(Fig.6).

Because

of

this,

the

test

(5)

In

Test

EQ15,

the

displacement

errors

had

increments,

and most of

those

_plots

scatterecl

in

int[oduced

If

the

absolute value of

the

cornputed

displacement

increment,

(xi.-pmtT"c),

this

situation,

it

is

defined

as

the

overshoot.

Then,

i

between

the

displacement

error

(abscissa)

and

the

7:.oD:SrrtrtbLACEHEHT

'

1;.oalsmtLAcE"EHT e.s

-?.o

2F 2.0 e.o

-2,OFig.12

88tsb2z=

20

15

10

5

o

-o

SHE-H FOBEE

lo3

kg

g･o-zalf2.0o.o-2.0

corresponded

randomly

The

amplitude

of

those

errors ranged

approximately

from

05

mm

to

_O,

₀₅

mm error

ag

between

the

instants

of

detecting

th

was examined

between

plots

were

shear

foice

increments

Luding

the

displacement,

e

frequency

of

19

Hz

was

distinct

either

the

measured

displacement

increment

condition

is

f

a

_plot

is

(computed}

e.eTine(sec,)SHEAH FeHCE

io3

kg o.eunetse[.) o.e Tire

{sec.)

AHALTSIS

Cemparison

Between

Numerical

Response

Obtained

Displacement

o,e' Tine

(sec.)

E:eEHI"ENI

ExpeTimental

Response

and

Including

Experimental]y

Errors

(Test

EQ02)

.15

-O.1

-O.05

0.0

O.05

O.1

O.15

Error

(nvn)

"-ean

IF:O.ool

mn

2

F

:-O.oo1

orn

Standard

1

F

:

O.030

"tn

Deviation

2

F

:

O.059

mm

Fig,

11

Histograms

of

Displacem.ent

ErTors

(Test

EQ02)

to

the

second mode natural

frequency

of

the

test

scattered

w.ith

respect

to

the

'frequency

domain

for

the

first

story

(Fig.

11).

Those

errors arg of

O.

02

mm.

This

discJepancy

occurred

because

of a

econvergence and of

the

displacement

errors and

the

(computed)

made

between

the

displacement

errors

and

other

, etc,

},

but

no strong

corretation

was

sheaT

force,

and

displacement

error

responses,

(the

second

mode vib[ation). and

exhibitecl

forced

to

be

terminated

in

an earlier stage.

correlation with

the

(computed)

displacement

first

or

third

_quadrant

_(Fig.

10).

Here,

a

new

term

is

,

{xim'ni.i)c),

is

smaller

than

that

of

the

defined

as

the

undershoot,

and,

in

the

reversed

in

either

the

first

or

third

_quadrant

in

the

relationship

displacement

increment

{ordinate),

it

means

the

2F DISPLACE"ENr lf D[SPLAgEHENT za xo O,O D.O

_-zo

_-zo

)

2F SHEA-FOnCE tF SHEhB FenCE

ZO _{m] kg} ZO

,]o3

_Lg o･o o-o D-e o.e Time Tine

'2･O

AHCASLeTCsl)s

-2･O

_i:pEHInENT

(SeC:)

Fig.13

Comparison

Between

Experimental

Response

and

Numerical

Response

Ineluding

Experimenta]ly

Obtained

Displacement

ErTo[s

(Test

EQ15}

･-Architectural Institute of Japan

ArchitecturalInstitute of Japan

undershoot

According

to

Fig.10,

the

displacement

error

obtained

in

Test

E915

had

the

_property

of strong

unclershoot.

This

undershoot

behayior

is

understandable

if

the

loading

algorithm

(Fig,3)

is

reminded.

As

the

test

structure

incrementally

and

gradually

approaches

the

target

displacement,

and

the

displacement

incrernent

in

each

incremental

loading

is

always

smaller

than

the

iemaining

displacement,

the

final

displacement

that

can

be

recognized

by

the

computer as

the

target

displacement

is

most

likely

undershooted.

(6)

The

tindershoot

indeed

has

the

effect

of

adding

some energy

into

the

test

structure

(similar

to

that

in

Fig.

1),

because

the

restoring

force

corresponding

to

the

displacement

_positioned

after

the

actuator motion

is

combined with

the

computed

displacement,

It

is

then

possible

to

understand

the

reason why

the

response was more

_pronounced,

but

the

reason why

the

second mode vibration

(instead

of

the

first

mode vibiation) was more significantly

_promoted,

however,

is

yet

to

be

verified.

(

7

)

Numerical

analysis

including

_the

effect

of

the

experimentally

obtained

displacement

errors

provided

the

responses

that

almost

matched

the

experimental responses

for

both

Tests

EQ02

and

EQ15

(Figs.12

and

13).

Based

upon

the

above

observations

(<

1)

to

(7

)),

the

conclusions

drawn

from

this

test

are

as

follows,

First,

the

match

in

response

between

the

test

and

the

numerical analysis

including

the

effect

of

the

experimentally

obtained

displacement

errors

(Observation

{

7

))

demonstrated

that

the

displacement

error was

the

rnajor error source as

speculated

earlier,

It

should

be

emphasized

that

this

displacement

error was

_generated

because

of

the

actuator's

inability

to

lead

the

test

structure

at

the

exact

target

position.

Second,

the

clisplacement

error

had

the

_properties

as stated

in

Observations

(2

),

(3

),

(5

),

and<6

).

and

those

_pioperties

were responsibEe

forthe

response

distortion

characterized

in

ObseTvations

(1)

and

(4).

4.

Experimental

Error

Growth

Behavior

(E)

:

PSD

Test

for

6DOF

System

4,

1

Description

of

Test

In

order

to

further

examine

the

experimental error effect on

the

PSD

response, another structure was

tested.

The

test

structure was a

full

scale six story steel

braced

frame

(treated

as a

6DOF

system), which was

fabricated

under

the

scope of

the

U.

S.

-Japan

Cooperative

Research

Program

Utilizing

Large

Scaie

Testing

Facilities,

The

conditions employed

in

the

test

are summarized

in

Tables4

to

_6.

The

allowable errors adopted

in

the

test

were

determined

through

_preliminary

small

loading

so

that

the

test

could

be

Proceeded

without

involving

too

many

iterations

for

making

the

structure

to

converge

to

the

target

_position.

The

stiffness

_properties

listed

in

Table

5

were

estimated

fiom

small unit

loading

tests,

in

which

the

flexibility

matrix was

first

estimated, and

then

that

matrix was

inverted,

Table6

shows

the

structure's natural

frequencies

and corresponding

vibration

rnodes

that

were cornputed

based

on

the

_quantities

in

Tables

4

and

5.

As

indicated

in

Table

4,

the

yiscous

damping

ratios were

set

to

be

large

for

the

fourth

to

six vibrational modes

in

order

to

intentional-ly

suppress

the

vibration

for

_those

_three

rnodes,

and

the

viscous

damping

matrix was established

from

the

assumed mass and viscous

damping

_properties

as well as

the

estimated stiffness values.

This

high

viscous

damping

was

introduced,

because,

through

prelimin-ary elastic

loading,

it

was

disclosed

that

the

vibration corresponding

to

higher

modes

was

promoted

Table4

St[ucturar

PToperties

of

6DOF

Test

Structure

Story

Mass

VisceusOampingError

2

{kg･sec/cm)

Ratioc%}

Bound(mm)

6F

77.24

6th.:90.0

O.10

5F

90.51

5th.:90.0

O.04

4F

90.51

4th.:90.0

O.02

3F

90.51

3rd.:4.23O.02

2F

90.51

2nd.:3.0

O.02

IF

95.00

lst.:3.0

O.02

Table5

E]astic

Stiffness

Matrix

of

6DOF

Test

Structure

6

5

₄

3

z

1

692.6-108.28.81.23.02.6

5-108.2243.6-148.512.8-O.91.2

48.8-148.5325.8-204.419.1-1.2

31.212.S-204.4417.8-249.424.7

23.0-O.919.1-249.4478.5-272.2

1Z61.2-1.224.7-272.2541.6

unit:tenlcm

Tabte6

Natural

Frequencies

and

Vibration

Modes

of

6DOF

Test

StructuTe

Mode

1st2nd3rd4th5th6th

Natural{Hl)

Freqvency1.6084.4257.57610.1012.6615a63

6F

1.00-1.00･1.eotO.473e.163-e.olgs 5F

O.921-O.236-O.8481.0D-O.542e,og7]

os

e.7o7O.592-1.00-O.4691.00-O.338

3F

e.solO.946O.156-O.635-O.813O,737

2F

O.306O.863O.996e.361Lo.Z65-1.00

IF

O.131O.471O.821O.]14O.934O,789

-42-NII-Electronic Library Service

ably and

that

this

vibration was caused

by

some

ex-perimental

error effects.

Details

of

the

test

structure

and

the

overall research

_p'rogram

using

this

test

struc-ture

can

be

found

elsewhere

(Refs,

17,

18),

4.2

Test

Results

and

Discussions

Figures

14

to

16

show

the

cornputed

displacement,

story shear

force,

and

disptacement

error

time

histor-ies

and

their

Fourier,spectra.

Figures

14

and

15

also

include

the

numerical respohses obtained

by

using

the

properties

Listed

in

Tables

4

and

5.

In

Fig.

17,

the

ex-perimental

story shear

force

responses are compaTed with

the

responses obtained

from

numerical

computa-tion,

into

which

the

experimentally obtained

displace-ment errors were

incorporated

(the

same

_procedure

as

that

te

_provide

Figs.12

and

13).

The

majoT

findings

that

could

be

drawn

from

those

figures

are

as

follows.

(1)

The

experimental responses were reasonably close

to

the

numerical responses

(Figs.14,

15)

_, verifying

the

suGcess of

the

test,

but

the

vibration of

7,6Hz

(the

third

mode vibration) was also

_promoted

in

the

story shear

force

responses

(Fig.

Is),

and

parti-cularly

in

the

displacement

error responses

(Fig.16).

Note

that

the

fourth

and

higher

inode vibrations were

Sf OISeL-CEnENT

'

SF D[SPLACE"E"T

FrequencyCHi) 4F OISPLAtE"ENT 3F OSSPLA:EliEHT 2D.O rm o･o 10.0 Time

.2o.o

(secJ

t.D SPECIHU"

Ht:.eV.AsLIUEg IAHA.) 1.29 dExP.)

{

(nVn.SeC.)o.o

2o.o Frequency{Hi)

Frequency(Hl}

2F D[sFLACE"ENT LF OISPLACEnENT rrm 20.0 10.0 m o-o-20.0 MAX.

₌

5. it

.EOURIEH

SPECTHUH YALUE 2e IANA,1 S7 EXf.:

(eTn-sec.)

Fig.14

t,o

'gao

Displacement

Obtained

o･o･ to,o IO.O Tane Trime

Csec,:

-2o,o

{sec.)

1.0 FUUHIEH

SPECTHUM

MtX.IY.AILIUE

g

IANA,) 2.IS IE:P,1

?o.e

(an'Sec･)

o･a po･o Frequency(Hz) FrequencyCHi} ANALTSIS EXeEHtttENT

'

Time

Histories

and

Fourier

Spectra

in

6DOF

PSD

Test

6E SHEAH FOHCE SF SHEAH fOHCE

60･O _{lo3 kg} o･n tO.O Time

-Eo.o

to

CSecJ

FOUHIEfi-,

spEcTnUHt

HA:.VALUE

-294a4.n

1-HA.; l"?2,02 dEXr.]

(ltg･sec･}o.o

lo.e Freguencv{Hl}

-F SHiAH FOeEE eF SHEaHFOHCE

so.o

3

6o.o

3

o.o o.o o,o e-o ine 1me FaUn:E FOUHLE HA:. HA:. FrequencyCHi} Frequency{Hi) lf SHEAfi FaACE Lf SHEAH FenCE

6e.n

3

e･o o-a rlme

)

-6o･o

ec-1 fOVHIE

HAx.

Frequency(Ht) FrequencytHt) ANALTS]e EXe[HI"EKT

Fig15

Sheaf

Force

Time

Histories

and

Fourier

Spectra

Obtained

in

6DOF

PSD

Test

6F elFFEHEHCE OIS- SF OtFFEHE"[E OISe.

o. O-t o･e o･o

-o.

-o.1

FauRI FOuHr SEE SPE

IH:-:x

lol

:ii"n.

';,

FreqtiencyCHz)

FrequencyCHi)

-F OIFFEeEHCE O:Se, af D[FFEHENCEOISP.

FeUHI SPE

-

o.

Cmm･s

2f O.1 e,o

-e,

Foue[ SPE NAX.V

-

o.

Cmn･s

Fig.16

Frequency[Hi)

OIFFEHENCEOISP. o.

Distra

O 50. Frequency[Hl)

placement

Err6r

Obtained

in

emec,} oT FoualEn

T

seE[THunHAX.VALUE

=

o.ele IExp.;lmm･sec,) 1:. e.

-e･Founl

SPEHA:.

.

a,(ntn･s 1rne6DOF Frequency{Hl) DIFFEHENCE'OISP.o omec.) e.o so.e Frequency{Hl)

Histories

and

Fourier

Spec-PSD

Test

43

Architectural Institute of Japan

ArchitecturalInstitute ofJapan

suppressed

intentionally

by

high

viscous

damping

ratios.

The

numerical responses,

on

the

other

hand,

did

not

show any sign of

the

third

mode vibratio]

(Fig$.

!4,

15).

(2)

The

numerical

responses

including

the

effect ef

the

experimental

displacement

error

duplicated

the

ex-perimental

responses with

_great

accuracy

(Fig.

17).

This

observation

again

supported our

early

fincling

from

the

2DOF

test

that

_the

_displacement

error was

the

major cause

that

distorted

the

experimental responses.

(3)

In

Fig.18,

the

displacement

errors are

_plotted

against

the

(computed)

displacement

increments.

The

undershootlovershoot

behavior

was

not

so

distinct

as

in

the

2DOF

test.

The

di$placement

errors

and

(computed)

displacement

increments

can

be

transformed

to

their

respective modal

clisplacement

errors and moclal

<computed)

displacement

increments

if

the

concept

of

the

modal analysis

is

employed.

Using

the

modes

listed

in

Table

_6,

those

two

terms

were

transformed

to

their

respective modal values.

In

Fig.19,

the

modal

displacement

errors

(abscissa)

are

plotted

against

the

modal

(computed)

displacement

increments

(ordinate).

This

figure

clearly

de-monstrates

that

the

modal undershoot occurred

in

the

third

mode, and

this

significant

third

mode undershoot was

believed

to

_be

the

cause

that

_promoted

the

third

mode responses

in

the

test.

Note

that

Fig.10

_{for

the

2DOF

Test)

was also

transformed

to

the

modal coordinates, and

the

significant second mode undershoot was observed.

s.

CohcludingRemarks

The

major

findings

drawn

from

this

study are $ummarized

below.

(

1

)

The

displacement

error,

defined

as

the

difference

between

the

computed

and

measured

displacements,

was

the

major

source

that

distorted

the

experimental

response.

This

displacement

error

was

generated

because

of

the

actuator's

incapability

to

lead

the

test

structure

to

the

exact

target

position,

(2)

The

displacement

error

had

the

tendency

of

promoting

the

highest

mode vibration

(the

second mode vibration

in

the

2DOF

test

and

the

third

mode vibration

in

the

6DOF

test),

and

this

_promotion

was more significant

in

the

shear

force

response

than

in

the

displacement

response.

<

3

)

In

the

2DOF

test,

if

the

allowable

error was

set

to

be

very

small,

the

displacement

errors

were

ran-domly

_generated

with respect

to

the

magnltude and

frequency,

On

the

other

hand,

if

the

allowable error was set

to

be

relatively

large

for

the

sake of easy

test

operation,

the

displacement

errors were more

systema-tically

_generated

and

had

the

_property

of undershoot.

(4>

In

the

6DOF

test,

the

displacement

errors

had

also a systematic nature, and

the

undershoot

in

i Rxy

Fig.18

Correlation

(computed}

Test)

44

-2F

1 IF 1

[:sg,l}･sii`:1.,

_-i

1'l',.lf;,'E･..;,,,

r-,i,,E:s:

,i,:'g"ijt/-l-i "

';l;i,g･,{,i'

Il,?,,lif.,i･･/･

･i

',

'1.',i･'f'iG':,tsi

''l

li'/tl.iieis

: Cerreletlon

CoefficSent

Between

Displacement

Errors

and

Displacement

Inerements

_(6DOF

PSD

6FGO.Oo･e-GO.o4FGO.Oo.o-so･o2FEO.Oo,o-SO.OSHEAn FOHCE ]o3 kg SHE-H FOntE Le.oTime[sec.; SHEAfiFeH:E Di"be

.)

o,o1ne

SF6e.oe.o-6e,o3F60.0e.o-60.0IF6o･ee.o-60.0SH[-fiFOHCE

io3

kg 5HESflEaHCE to･oTime(sec.} SHEAH FOHCE o･oime.] e･oime

Csec,)

ec.)

.

ANALTSIS

-

EXPEH]"ENT

Fig.17

_Cornparison

_Between

_{Experirnental}

_Response

ancl

Nurnerical

Respon$e

Including

Experirnentally

Obtained

Displacement

Errors

(6DOF

PSD

Test)

.

"!

-1

3rd. 1

..

!nd-,

.1

_lst.

_,r

,..-i,･l･ttlillliilli

..i,.l.･1iil.IEt･l:i:,/,,{:ii,.,,

.

ii

,l/tffr,,s.i,'･i' '

_{i･i'ii;1.S･ill,,/i/,.,3as,.,,Jlh'"}

.tttt

/[:r:,i

''

_--

D･14?

,

_;,'

_.･1

0･ID{

Rxy

;

Correletien

Coefficient

Fig.19

Correlation

Between

Modal

Modal

(computed)

Displacement

PSD

Test)

:,:/,,

'

･;･s.t;'t:'t/･i･

i･

_l･

1･1･,/

lt-l･.･za

:1

.

ttt'I't'':I

.[

.".s:,!,;=

]

t+tt

O:Oll

DispLacement

Errors

and