Part 4 : 仮動的実験における実験誤差の制御 : 仮動的実験応答の安定と精度

(1)

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PART4:CONTROL

_'

OF

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. Introduction

The

pseudo

dynamic

(PSD)

test

(also

referred

to

as

the

on-line computer

test

control method)

is

a combined

experiment

and

numerical

analysis

th

at

simulates

the

earthquake

response

behavioT

of structural

systems

with

respect

to

the

time

dornain.

Since

devised

by

Takanashi

et al.

(Ref.

1),

this

test

has

been

employed

by

many researchers

in

both

_Japan

and overseas

(Refs.

2,

3).

Through

_previous

applications of

the,PSD

test,

it

has

been

disclosed

that

the

PSD

fesponse

is

_yery

sensitive

to

the

_perfo[mance

ol

the

hardware

employed and often

distorted

if

the

ha[dware

is

not sufficient

in

controHing

the

load

applying actuators and measuring

the

displacements

and reactional

forces.

This

distortion

in

response

has

been

designated

as

the

experimental

(response)

error.

The

writers examined

the

sources

that'would

deteriorate

the

PSD

response and

their

effects on

the

respon$e

(Ref.4).

The

exaininations

indicated

that

;

<

1 )

the

displacement

error

(definecl

as

the

displacement

commandecl

minus

the

displacement

achieved after

the

actuator motion)

has

the

most significant effect

on

the

PSD

fesponse

_;

(

2 )

the

displacement

error

tends

to

show "undershoot"

(here,

the

undershoot

is

a conclition

in

which

the

displacement

increment

achieved after

Lhe

actuator motion

is

smaller

in

it$

absotute value

than

the

displacement

increment

commanded)

_;

and

(

3 )

the

displacement

error

has

a

nature of

promoting

the

highest

mocle

vibration

if

the

PSD

test

is

applied

to

a

multiple

degree

ef

freedom

(MDOF)

system.

This

paper

is

an extension of

Ref.

4

and

presents

analytical examinations

for

quantifying

the

characteristics of

the

displacement

error.

This

_paper

also

proposes

_procedures

to

suppress

the

_groWth

of

tfie

experimental error.

2,

Effect

of

Displacement

Error

on

PSD

Response

The

experimental error can

be

regardecl as

the

response

of

the

system against

the

error

force,

and, under

the

displacement

error

condition,

this

error

force

is

_given

as

the

displacement

error

times

the

sti

ffness

of

the

system.

For

a

linear-elastic

single

degree

of

lreedorn

(SDOF)

system,

the

equation of motion

that

includes

the

dlsplacement

error

effect can

be

formulated

as:

mX(iAt)+cth(iAt)+hx(iAt)==-mX,(iAt}-+'hda:(iAt)･-･-･･･-･--l･-･･-･-････-･･-･･-･-････--(1)

where, m, c, and

k

afe

the

mass, viscousdamping, and stiffness

_;

_X.the

_ground

acceleration

_;

dx

the

displacement

error;At

the

integration

time

interval;and

i

denotes

that

the

equation

belongs

to

the

i-th

step cDmputation.

2.1 Effect'of

Random

Displacement

Error

As

indicated

in.

Ref.

4,

displacemenc

errors

scattered

randomiy

if

the

allowable error

(specified

prior

to

the

test)

was

set

to

_be

very

small.

Shing

and

Mahin

(Ref.

₅₎

conducted a comprehensive study on

the

charactefistics

of

experimental

errors,

and,

here,

referring

to

their

findings,

effects of random

displacement

errors

are

discussed.

If

the

viscous

damping

is

taken

zero

and

further

the

acceleration

term

is

neglected

(1.e.

_the

error

force

only),

the

displacement

obtained

from

Equation

1 is

:

'

iTl

･

x(idt)=

£

A･sin-w(i-n)At･dx(nAt)･････-･-･･--･･･-''-･---･･-･･-''-'HHH''H''H'･･-･････-･･-････-････(2)

nLo

･

'

where

A=wAtl

lr(a)At)!/4

t

_Associate

_Professor,

_Kobe

_Universlty

"

_Research

_Engineer,

_Buildlng

_{Research'institute,}

_MinisLry

_ef

_Construction

(Manuscript

receiyed F'ebruHry

9,

]989/Paper

Accepted

May

8,

]989}

(2)

-129-NII-Electronic Library Service

where to

is

slEhE', and

-tu

the

numerical natural

circular

frequency

of

the

system.

For

details

of

a,

see

Ref.

6. Let

us

suppose

that

dx

follows

the

Gaussian

distribution

with

the

mean

of

zero

and

the

stanclard

deviation

of a. and

is

uncor[elated with respect

to

the

time,

By

some

algebraic wo[k

into

Equation

2,

the

mean of

the

response /lsalso

found

zero,

and

its

standard

deviation,

o.

is1

i-1

a.=A'a.'

Z[sint(i-n)thAt)･---･-･--･---･･---･･-･-･---･---･--･-･---･<3)

n=o

The

time

versus standard

deviation

re]ationship obtained

from

Equation

3 is

_plotted

in

Fig.

1,

in

which

the

abscissa and

ordinate

are normalized respectively

by

the

natural

_period

of

the

system and

the

standard

deviation

of

the

displacement

e[ror.

To

be

remarked

in

this

figure

is

that

the

experirnental error

{the

ordinate)

is

more

_pronounced

with

_the

increase

of

the

inLegration

time

interval,･attd

this

is

the

key

to

explain

the

[eason why,

in

an

MDOF

systeml random

displacenient

errors

tend

to

_promote

the

response corresponding

to

the

highest

niode,

If,

i[L

an

MDOF

system, random

displacement

errors are assumed

to

be

uncorrelated

with respect

_to

_the

degree,

each

vibrational

mode

(afte[

medal

decoupling)

also

sustains

random

displacement

errors,

Since

the

relative

integration

time

inLerval

(expressed

as _toAt)

is

largest

in

the

highest

mode,

the

vibration of

this

mode

is

likely

to

be

most

_promoted.

2,2

Characteristics

of

Undershoot

A]though

displacement

errors

were

found

randomly scattered

if

we set

the

allowable error very small,

it

is

by

no means an easy

task

unless

the

structuTe

tested

is

significantly more

flexible

than

the

loading

system.

(Here,

the

stiffness of

the

loading

system should

be

expressed as

the

accuracy of

the

load

applying actuators

in

_positiening

the

structure.

)

In

most

cases,

the

undershoot

_prevails

in

the

displacement

error mechanism.

If

aconstant undersheot

(fi)

is

included

in

a

PSD

_test

of

a

linear-elastic

SDOF

system,

its

response

can

be

obtained

by

solving

Equation

1

with

dx

as

a

(Here,

a

is

taken

_positive

when

the

displacement

is

ascending

and

negative

when

descending).

Even

if

the

Fig.1

17tw

Reiationship

Between

Time

ancl

Response

Error

Caused

by

Random

Displacement

Errois

(a)

Viscous

Ratie

=Dampingo.o

I(b)

Time5.0tsec)

Viscous

Damping

Lhlit=MM

Ratio

v

O.02Respanse

Undersneot

ErrorFerce

xTime

L

l

9oex

Cc)

Steady

State

Dfisplacement

Respense

and

Undersheot

Error

Force

Fig.2

Response

of

SDOF

System

Subjected

to

{

to=50 rad.

lsec

:

a==

O. ODI

mm

)

-130-Undershoot

150 to.o

5.0 F-,..

tt'de

Hdm

Ffva

Tx/

gr;rH

(a)

SDOF

System

Subject

to

Periodic:

Rectangular

Force

(b)

Fig.3

Vfe

-L-tO,O

Displacement

Amplttude

ef

SDOF

System

in

Steady

State

(h

=

Viscous

Damping

Ratio)

Respense

of

SDOF

System

Subjected

to

Periodic

RectanguLar

Force

(3)

ground

acceleration

is

takeR

zero,

the

response still

diverges

as shown

in

Fig.

2<a}

if

the

viscous

damping

is

zero.

With

nonzero viscous

damping,

the

response

falls

into

a sLeady state

(Fig.2(b)).

The

relationship

between

the

steady

state response and

the

eiror

force

(caused

by

the

undershoot}

is

shown

in

Fig.

2<c).

Two

remarks should

be

given

from

this

figure.

First,

the

error

force

is

a

_periodic

rectangutar

force

and

lagged

by

₉₀

degrees

in

_phase

angle

from

the

response.

Second,

the

_peTiod

of

the

response equals

the

natuTal

_perlod

of

the

system;it

is

true

because,

a

phase

lag

bY

90 degrees

between

a sinusoidal externa]

force

and

its

respense can

be

achieved only when

the

system

is

in

the

resonant condition.

(The

periodic

rectangular

force

can

be

approximated reasonably as a sinusoidal

force

haying

the

sarne

frequency.

)

Next,

let

us suppose

the

response when

the

SDOF

system

is

subjected

to

a

_periodic

rectangular

forge

(Fig.

_3(a)).

This

force

is

to

represent

the

error

fo[ce

caused

by

the

undershoot.

The

amplitude of

this

syst6m

in

the

steady state

is

plotted

in

Fig.3(b)

against

the

_period

of

the

rectangular

force

ancl

foT

various viscous

clamping

ratios.

In

this

figure,

the

abscissa

ancl

ordinate

are normaliied respectively

by

the

natural

_period

of

the

system and

its

static

displacement.

Figure

3{b)

indicates

that,

in

the

range where

the

_period

of

the

rectangtilar

force

is

smaller

than

the

natufal

_period

of

the

system:{,e.

the

range

_given

by

T/T,

less

than

1,O,

the

response

decreases

drastically

with

the

dec,reafie

in

TlT..

In

the

fange

_where

TfT.

_greater

than

1,O,

the

_response

is

_never

negligible

and

_promotecl

significantly when

the

_period

of

the

rectangular

force

equals an odd number

times

the

natural

period

of

the

system.

Based

on

the

findings

obtained

in

Figs.

2

and

3,

finally

considered

is

an

MDOF

$ystem

that

undertakes undeJshoot.

Here,

the

magnitude of undershoot

is

assumed constant with respect

to

both

the

time

and

degree.

Supposed

that,

in

asteady state,

the

system responcls

in

its

highest

mode,

the

error

force

applied

to

each mode

(after

modal

decoupling)

is

a

periodic

rectangular

force

whose

period

equals

the

natural

period

of

the

highest

mode.

It

means

that,

in

Fig.3{b),

TlT,

is

_positioned

_at

1.0 for

the

highest

mode and

less

than

].O

lor

all of

the

lower

modes

(because

of

larger

_T.'s

in

these

modes).

Since

the

response

is

macle minimal

in

the

region where

Tl

T. is

smaller

than

1. 0,

these

lower

modes are moSt

likely

in

inaction,

and,

therefore,

little

contradiction arises as

to

assuming

that

the

MDOF

systeth responds with

its

highest

mode.

2,3

Magnitude

of

Respon$e

Error･Caused

by

Undershoot

･

.

The

magnitude of

the

experimental error caused

by

the

undershoot can

be

estimated

through

energy

consideration.

In

a

linear-elastic

SDOF

system sustaining aconstant magnitude of undershoot, considered

is

a

half

cycle

from

the

maximum

to

minimum

displacements

as shown

in

Fig,

_4,

in

which

the

absolute va]ue$

of

these

_{di-splacements}

are

x,

and x2,

and

xi<x!,

because

the

response

gro'ws

with

time.

The

energy added

to

the

system

during

this

ha]f

cycle

is

glven

as

₁

dE=112･k･x;-112+k･xl-･･-･･-･･-･･-･-･-･---･･---･･--･'-''-H--'---"---h-･･･-･{4)

This

energy

is

to

equal

the

energy

generated

by

the

undershoot

(dEa)

minus

the

energy

dissipated

by

the

viscous

damping

(dEv)

in

this.

half

cycle.

They

can

be

expressed as:

d.Ea=a･k･{xi+xt)---･･･---･-･･-･-･---･･･-･--･･････-･･･--･･･--･-･･-･-･-･---･----･-･(5)･

dEvin-c+o･(xi+x:}218H･"''-'H'"''"''"''HHhHh"''"'--'''""'H-'''--''H-'-'''H-'-････----･--(6)

The

energy

balance

requires:

dE=dEa-dEv･･･--･---･---･--･･-･-･-･･･-･---････-･･･--･･-･-･･-･-･---･-･---･-･-･---(7)

Substituting

Equations

4 to

6 into

Equation

7,

we

obtain

for

the

relationship

between

_x,

and

x,

as

:

ForeeIks

k(stittness}

::)plDisp.

t

''

' 1 t

:

' ₁ t 1 / ' 1 1 1 X2 Xl--)T ,

':

Fig,4

Grevvth

in

DispLacement

Amplitude

by

Undershoot

30

NUMERICAL

T7TeJO

DISPLACEMENT

ENVOLePE

(d

=Displacement

Aiplitude}

Fig.5

_G[owth

of

Respense

Error

Caused

by

Undershoot

(h=o.

e2)

'

(4)

-NII-Electronic Library Service

xtla=C2+(1-n･h12)･x,16)1(1+ff･h12)･･･････････-････････････････････････････---･--･･････････････'-････-････････････････(8)

in

which

2he)=

c!m.

This

recursive equation specifies

the

growth

ef

the

experimental error causec[

by

the

undershoot.

(Remember

that

the

time

difference

between

x,

and

xi

is

half

the

natuTal

_period

of

the

syste/m.

)

Figu[e

s

shows an numerical response

of

an

SDOF

system under a constant undershoot

(obtained

by

the

direct

time

integration)

and also

_plots

the

envelopes

_given

by

Equation

8,

which

indeed

traces

the

_peak

values of

the

numerical response.

Using

EquatLon

8,

we can estirnate

the

growth

of

the

experimental error wlth

time,

and

it

is

shown

im

Fig.

6 for

various viscous

damping

ratios.

Further,

by

equating

xt

with

x,

in

Equation

8,

we

obtain

for

the

amplitude

in

the

steady state

(d):

d=2･al(rr･h}･-･･-･---･･-･-･--･･---･--･-･-･--･----･･--･-･-･･･----･---･---･･---･-･---･(9)

Comparing

Fig.

6

with

Fig.

_1,

we

find

that

the

undershoot

_promotes

the

experimental error _more _{significantly}

than

the

random

displacement

error as

tong

as

the

system

is

Lightly

damped

and

the

integratLon

tirne

interval

is

kept

reasonably small.

Using

the

concept

of modal analysis and

_particularly

looking

into

the

highest

mode, we can also estimate

the

_growth

of

the

experimental

error

in

MDOF

systems.

2.4 Comments

on

Error

Growth

by

Displacement

Errors

Two

cemments should

be

given

about

the

value of

the

above

examinations.

First,

the

magnitude of

disp!acement

errors

(either

random

or

undershoot)

was assumed constant with

respect

to

the

time.

but,

in

real

PSD

tests,

they

should vary with

the

time

as

'well

as

the

degree.

Choosing,

for

the

magnitude of

the

dispiacement

error, one of

the

largest

values expected

in

the

te$t,

we

can

estimate an upper

bound

for

the

experLmental

error.

Second,

it

was

assumed

that

the

system

behaves

linear-elastically.

When

the

system starts

behaving

inelastically,

it

nermally

loses

the

stiffness.

This

]oss

in

stiffness

decreases

the

magnitude of

the

error

fo[ce

and acco[dingly

the

_growth

of

the

experimental

error.

It

should

be

true,

because

the

error

force

is

_given

as

the

_product

of

the

displacemente[ror

and

the

stiffness.

Thus,

estimate

based

on

the

elastic

_properties

also

_gives

an upper

bound

for

the

experiinental error.

3,

Control

of

Experimental

Error

Growth

in

PSD

Test

The

most

ideaLway

to

reduce

the

experimental erroris

to

improve

the

_performance

of

the

hardware

employed

in

the

test

and reduce

the

clisplacement

error as

much

as

possible.

However,

we always

have

to

deal

with [eal/Lty and accept

some

imperfectness

in

our

hardware.

Here,

consideration

is

_given

to

how

we

can

minimize

the

expe/iimental

error ttsing a

PSD

test

system

that

inevitably

includes

displa

¢ement errors.

3.1 Effect

of

Viscous

Damping

on

Experimental

Error

Growth

Since,

as shown

in

Fig.6,

the

magnitude

of

the

experimental error

becomes

smaller with

the

irLcrease

in

the

viscous

damping

ratio, we may

be

able

to

reduce

the

experimental error

by

adding

artificially

high

viscous

damping

into

the

equations of motion.

In

fact,

in

some

of

the

_previous

applLcations

(Refs.

7,

8).

high

viscous

damping

_ratios were

introduced

for

highe[

modes

in

order

to

suppTess

_possible

promotion

of

these

modes.

However,

such

_procedure

involves

two

drawbacks.

As,

in

this

_procedure,

the

damping

rnatrix

is

constructed

in

ieference

to

the

initial

elastic

stiffnesses,

this

matrix no

longer

_guarantee

the

desired

viscous

damping

ratios once

the

system starts

behaving

inelastically.

The

larger

are

the

damping

ratios assiglled originally

for

the

higher

modes,

the

more significantly

30 od/S

h=

O.02 "---Undersneot

d=Oisp.Amptitude

Te=NaturatPeriod

h=ViseeusDamping

Ratie

O.05O.1O.2O.5

o

40 T/Te

Fig,6.

Relatianship

Between

Tirne

and

Response

Errer

Caused

by

Unde[shoot

-132-Table1Designation

and

Test

Cond[itions

for

PSD

Tests

with

I-Modification

Designation

of

Test

Eqo2I

EQIOI

EQ02IM

EqlOIM

EQIOPIM

-

_As

_input

recorded

Loading

Cenditien

Earthquake,

Elastic

Response

Elastic

Response

Elastic

Response

Elastic

Respense

Inelastic

Response

accelerations,

part

of at

Tohoku

Univ.

during

with the rnaxfimum

rror'I-1tod"'ication oundmm)-O.02

No

O.10 No

a.o2

Yes

O.10 Yes

theacceleretions

the197BMfiyagi-oki celerationsc/aled.

(5)

affected are

the

effective viscous

damping

ratios

in

the

important

lower

modes

during

the

inelastic

response.

Second,

as shown

･in

Fig.

4,

the

essence of

the

undershoot

is

negative

Coulomb

damping,

in

which

the

energy created

_per

a cycle

of

loading

is

_proportional

to

the

disp1acement

amplitude

(if

the

magnitude of

the

undershoot

is

taken

constant

),

'On

the

6ther

hand,

the

velocity

_proportional

viscous

damping

dissipates

energy

in

_propertion

to

the

square

of

the

displacement

amplitude.

If

the

system responds steadilY' with aconstant amplitude,

it

is

_possible

to

allocate aunique

'

viscous

damping

.ratio

with which

the

energy added

by

the

undershoot can

be

canceled,

but,

in

nen$tationary responses s'uch as

those

induced

by

earthquake

lo'ading,

no such

damping

ratio can

be

assigned.

A]though

adjusting

the

viscous

damping

term

is

seemingly one of

the

handiest

ways

to

suppress

the

experimentat error,

it

is

believed

to

make us unduly

difficult

to

evaluate

the

viscgus

damPing

effect on

the

response obtained.

3.2 Algorithms

to

Suppress

Experirnental

Error

Growth

'

As

a means,to suppress

the

experimental error more effectively

than

adjusting viscous

damping,

an algorithm

vras

devisecl.

This

algorithm

had

its

basis

on our

_previous

findings

Tegarding

the

chaiacteristics of experimental e.rrors,

i.

e.

that

the

err6r source

is

clearly

defined

as

the

displaceme.nt

error and can

be

measured accurately

but

that

the

error

_,

stilloccurs

because

of

insufficiency

of

the

load

applying actuators

in

_positioning

the

test

stTucture at

the

exact

target

pos'ition.

In

the

algorithm

de'vised,

the

actuator

forces

measured

by

load

eells were modified, anyd

the

modified

forces

were used

in

_place

ot

the

measured

forces

for

solving

the

equations of motLon.

The

modification

_procedure

adopted was,

ifl=ifE+[h]ixc-xmi-''H-"'""''･-･--'-H-H'･-･--･"'-HH-･･-'"'H-HH-･-''H-'''-････----･-･--(10)

Here,

_ifl

and

VI

are

the

measured and modified

force

vectors and,

[ls]

the

stiffness mat[ix,

for

which

the

initially

estimated

{elastic)

stiffness matrix was employecl,

Further,

lx,I

and

lx.l

are

the

computed and measured

displacement

vectors.

As

long

as

the

system responds elasticatly,

in

this

algorithm

<designated

as

I-Modification},

the

modified

forces

should equal

the

reactional

fotces

corresponding

to

the

exact

target

_position.

To

evaluate

the

effectiveness of

this

algorithm,

a

steel

braced

ftame

tested

_previously

_(Fig.

₄

of

Ref.

4)

was

tested

again,

but-with

a new _{set of}

braces,

Table

]

_summarizes a

tQtal

of

five

tests

conducted

in

this

test

program

and

their

test

conditions.

The

[esults

of

Tests

:

EQ02I,

EQIOI,

EQ02IM,

EQIOIM

_(in

which

the

structure

behaved

elastically) are shown

in

Figs.

7 to

lo,

together

with

the

responses obtained nnme[ically.

In

Test

EQ02I,

the

response

did

not

diverge,

but

the

second mode vibration was more

promoted

in

the

experirnental response, wherea$

the

response of

Test

EQIOI

diverged

within a very

short

period

of

time.

On

the

other

hand,

the

results with

I-Modification

(i,

e.

Tests

EQ02IM

2F DISPLACE"ENT xoo.o ma o me(sec.) tF OISPLAZEnE"T 2.0

-XD

i.o FaVRIER

-SPEETnUH"AX.VALUE

a

.

o.og

E

CAHA.] e-lo [ExP.)

{mm-sec.}

o.oFrequency(Hi) 2F S}IEA- FOnCE

{

so.o e

)

O.D

-2.0FOURI

spEHA:.V

-

a [ e CCmm,s tF SHEAH zOa.o omec.}

.o

FrequencyCHz)

foncE 2F nlSPLACEMENT 2,O mm

Fig,7

o,o 1.0Time(sec.) IF OTSPLhCE"EHI za mm

-zo

FouHI 5PEntl.V

-I

l(kg"s

-2.0

1.0 fOURIEF

-spEtTnu"MAx.vALuE

d

'

_9iRk,G

?t:s.,(mm･sec･}o.b

tr,o 0

FreguencyCHi}

-NALTSIS

Time

Histeries

and

Test

EQ02I

Fourier

]

Freguency(Hl} 2F SHEAH FOHCE

2.0o.o

io3k

50.0

-ze

L.O FOUHIEn

-5PECTfiUHMAx,vALtrE

d

-9L:fl.,fi

gt:3.,{mm,sec.)

o:o omec.) 1.0Time{sec,)

.0

Frequency(Hl) EXPEHI"EptJSpectra

Obtainecl

in

FrequencyCHi) IF SHEAH FOH:E

zo lo3kg

-zoFOURIEH

-SPEeTEUHnA:.VALUE

.

e.3s IANA.1 S!.15 tEXP.]

Ckg･sec.)

uaae o.o se.a o me(sec.} e,e so.a FrequencyCHI-} AHALTSIS

Histories

and

EQIOI

-xeFOURIEfi

-spEcTRunHAX.YALUE

.

iL12 tANA.) SS.Sl (EXP.)

Ckg･sec,)

I,OaE

Fig8

Tinie

Test

l.OIime(sec.)

Fourier

o.q so.o Frequency(Hi} E:?EHIMEHT

Spectra

Obtaincd

in

(6)

-133-NII-Electronic Library Service

2FDTSPLACE"ENT

C

FtrequencyCHz)

0imeec.

.a

)

lf xDISPLACE"ENT

-2FOU

SMt:

-

(mm

amee.)

FreguencyCHi}o

2F ZDe,o OISPLA[E"EHT mm

-2,OFounlEn

5FECIHUnnAK,VALUE

-

O,31 tANA.] O.Sl [Exp.]

(mm･sec.)

uoaE 1.0

"meCsec.)

o.oFrequency(Hz)SD.O IF DTSeLA:E"EHI

i･ii]!4-vnvevCVSiAii,i,s,

FOURIEA

SPECTfiUNHAX.VALUE

-

O.11 [AHA.) O.IG [EXP.ICmn,sec.} n

Frequen[y(Hz)

t.odEo.o 5D

}

lfSHEAH FOHCE

C

.o

Frequency(Hi) ANALTS:S

Fig.9

Time

Histories

and

Test

EQ02IM

imeec.) tF 2.D, S"EAn EeHCE

-2.

faUH SPnhX.

'

Ckg･

Fourier

Frequency(Hl)

EXPEnl"ENTSpectra

Obtained

in

2F 2.0O.D SHEAB ₁₀

3

FOHCEk9

-2,OFOURIEH

SPECTRUNHA:.VALVE - 24S.09 iANA.1

l2:b?P

(kg･sec.)

t.oaE 1.0

Time(sec.)

Fig.10

a.o Frequency(Hl) AH-LISIS

Time

Histories'

and

Test

EQIOIM

tf SHEAH FOHCE

iiFFOURIEH

SPECTHUMnsx-v-LuE-{IalFrs

{gRpPF

Ckg-set.)

Fourier

]kg

ll!iille,e.)

o Frectuenc)'CHz) EXPEHIttENJSpectra

Obtalned

in

L,O.ns D.O 50

and

EQIOIM)

did

not show any $ign of

diverging

behavior

and clo$eLy mat ¢

hed

the

numeiical responses,

demonstrating

the

validity of

this

algorithm.

Figure

11

shows

the

responses obtained

form

Test

EQ10PIM

(in

which

the

structure

behaved

ine]astical]y),

together

with

the

responses obtained nurnerically.

In

the

numerical analysis,

the

restoring

force

behavior

was simulated

by

a combined

Ramberg-Osgood

and slip model, and

_the

coefficients

included

in

this

model

were

cleterinined

using

the

technique

of

system

identification,

See

Ref.

9 for

this

technique.

2F?oOISeLACE"EHT

-2oFOU

SHA:

-

Crm

2F

-se

SHEAH Emec.)

.e

Fpequency[H!) FOBCE

･o310kg.o

1.sTim

･e

{sec

vonlEH.PE:TRUM''NALUEnlss.esfiIANA.)6Dl,46tExP.)･sec.]O.D 30.

Fig.11

FrequencyCHz)

TimeTest

J

ANALTSIS

Histories

and

EQIOPIM

lf?oo OISPLACE"[NT

-20FaU

SnA:

-

Crm

lf s.oGHEAfi mec.]

.0

Frequency(Hl)

EOHCE o･ me

-6.

c.) FaUHI 5PEliAl.V

-

1 E 9 t(kg･s

.o

Frequeney(Hz)

EXPEHIMENT

Fourier

Speetra

Obtained

in

tlio

Force

'1:,r1t11'd'11,g1d::ddd,1

Tmsec

,,te,,,,,l xm

XJM

x-oDisp.

dtJ

_r

tl

(a)Meesurement

t-'ro3co'rPUdieor

Fig

Fig.12ef

Restoring Ferces

Procedure

to

Estimate

in

T-Modification

1 1 1 , t , , 1 d 1

1. DSSpt

Last

Meosured Pelnt

inPrev[ous Stelp

{b)

Estimate

of Tangent

Stiffness

Tangent

_Stiffness

.13

Test

Specimen

T-ModificationandSetup

X

Men

8-Used

o"or

Unit:mm

in

PSD

Tests

with

134

(7)

Figure

]1

indicates

that

the

experimental responses were ctose

to

the

numerical responses, again'

indicating

the

effectiveness

of

the

propgsed

algorithm.

One

may

question

why

this

algorithm

provided

an accurate result,

because

the

stiffness

_properties

should

･have

changed

during

the

inelastic

respo'nse, while

the

modification was still

based

upon

the

initial

_stiffnesses.

It

_{was speculated}

that,

duling

the

inelastic

response, energy

dissipated

by

the

hysteresis

'

of

the

structure completely overshadowed

possible

errors

induced

by

incorrect

esti.mation of

the

modified

forces.

To

incorporate

m'ore

directly

the

change

in

stiffness

during

the

inelastic

response

into

the

force

modification

procedure,

another algorithm,

designated

as

T-Modification,

was also

devised.

In

this

algorithm,

for

the

stiffness

Table3

Table2StTuctural

PropeTties

of

Structure

Used

in

PSD

T-Modification

Two

DOF

Tests

with

Hass

StiffnessNatrix

{kgXcm)

ckg･sec2tcm)as

_F

2FIF85.0

52.280 85.0-53,840-63.e40126.9oo

VibretienalbodeNeturalFrequencyOaupingRatio(x)

(Hi)

lst2nd1.0001.coOil.7362.19

-O.5T66.992.0s.o

Designatien

and

Test

with

T-ModificationConditions

fer

PSD

Tests

DesignationLoadingConditipnErrorT-Modfification

ofTest

BoundC"m)

EQ03TElasticResponseO.03

No

EQ15TElasticResponseO.15

Ne

EQ03TMElasticResponseO.03

Ves

EQ15TMElasticRespenseO.15

Yes

EQ03PTMInelasticResponseO.03

Yes

ST03TMquasi-Static

O.03 Yes

leAsinputeccelerations,partoftheacceleratfiens recordedatTohokuUnriv.duringthe1978Miyagi-oki

Earthquake,with･themaxfimumacceleratienscaled.

2F OISPLACE"ENT LD･ o.-lo. FOUH SPnAX.

'

{mm.

2F

'

SHEAB 30･D

_lo3kg

O.D-30.0 1.n FOUH[EH

-SPECTRU"HAX.VALVE

a

"

_litRfi?iE

l2R;S(kg･sec.)

o.o

)

Frequency(Hz}

FOHCE 6.0TimeCsec,) 20.0 Frequenty{Hi) ANALTSIS

Time

Histeries

and

Test

EQ15TM

lf DISPLACEMENT LO.O o.o-to.o FOURT SPE"tX･i

ICmm･s

2.5 -2.S

Force(103

kg}

s.e(sec)

U

est

Cal.

(a)

Second

Forceoo3kg)

e

Test-Cal.

Flg.14

0mec.)

.O

Frequency(Hz)

:F SHEAB FOHCE so-a o.o

-so.o

)

FDUHT SPEMAX.Y'

?

e(kg･s

Frequency(Hz)

-

EXFEHTMEHT

Fourier

Spectra

Obtainecl

in

2.S

-2.5

Eig.15

uxY=O.953･

'

-1,

r:'

C

(b)

First

Corre]ation

Forces

to

'be

ShenrForce(le3Sheekg)

-30-15

--50

-12-7Disp.

2 (mm1

-40.

L-.TanEst

Story

Fig,

16 Story

Betweem

Cgrrected

Corrected

_(Test

oo3kg}

'Fbrces

_arrd

EQIsTM)

Fercefio Disp-EO-{"vn)50

Est?::tedStfffness

Correlatlon

Between

Stiffnesses

_(Test

ShearForceoo3kg)t

55

t'''t'xE,.J50'4S.--r't-H-'

20304nnisp'("lnl

Estirnated

ST03TM)and

Experimental

(8)

-135-NII-Electronic Library Service

matrix

[k]

used

in

the

force

modification

(Equation

10),

the

tangent

stiffness matrix

between

the

_present

and next

steps

was used, and

the

_pfocedure

to

estimate

the

tangent

stiffness matrix

fol!owed

;

1)

in

each

time

step,

the

test

structure was

loaded

in

accordance with

the

standaTd

_procedure

of

loading;2)

during

the

loading,

the

fc/rce

and

displacement

values were

measured

continually

every after a small

tirne

interval

as shown

in

Fig.

12(a)

and

;

3)

using

the

data

collected.

the

tangent

stiffness was estimated using

the

least

square method

(Fig.

IZ(b)),

Te

vc/rify

the

validity of

this

algorithm, a

two

DOF

structure shown

in

Fig.13

was

tested.

Tabtes2

and

3 list

the

vibrational

pToperties

of

the

structure

and

the

test

program.

Figure

14 $how$

the

results obtained

from

Test

EQ15TM

(in

whLch

the

stTucture responded eLastically),

demonstrating

that

the

response obtained was veTy accurate, and

thus

the

algorithm effective

(The

response of

Test

EQ15T.

in

which

T-Modifi

¢ation was not employed,

diverged

qLiick!y).

FiguTe15

illustrates

the

forces

corrected

in

this

test

:

i.e.

the

estimated

tangent

stiffness matrix

times

the

displacernent

errers.

Correlation

be.tween

the

corrected

forces

and

the

forces

that

shouLd

have

been

be

corrected

:

i.e,

the

elastic stiffness

times

the

di$placement

errors, was

found

excellent, verifying

that

the

tangent

stiffnesses estimated were accurate.

Figure

16 illustrates

closer

looks

of

the

estimatecl

tangent

stiffnesses against

the

story shear

force

versus

deflection

curves obtained

from

Test

ST03TM.

In

this

test,

clisplacements

were applied

quasi-statically

to

the

structure, and

the

tangent

stiffnesses were estimated

for

each small

incremental

loading

by

using

the

a!gorithm

devised.

When

the

structure

be.haved

linearly,

the

estimated

stiffnesses

(dashed

lines)

matched

the

experimental

stiffnesses,

but,

in

the

inelastic

range,

they

were constantty

larger

than

the

corresponding experimental stiffnesses.

This

overestimate

was

believed

to

have

been

caused

because

the

restoring

forces

measurecl

during

the

].oading

were

larger

than

the

forces

measured when

the

actuator

motion

was

stopped,

and

this

has

to

do

with

the

effect

of

loading

rate on

the

restoring

forces.

3.3 Values

of

Algorithrns

Devised

Comments

rega;ding

the

values of

the

algorithms

proposed

follow.

The

absolute

prerequisite

for

applying

either

I-or

T-Modification

is

that

the

displacement

tltal

can

be

measured

in

the

test

is

significantly accurate relative

to

the

displacement

that

can

be

controlled.

It

is

so

because,

if

this

conclition

does

not meet,

the

displacement

error amd eventually

the

force

to

be

modified cannot

be

estimated correctly.

This

limitation,

however,

does

not seem

to

impair

the

app}icability of

these

algorithms.

The

PSD

test

after all

ls

a

test

with

dispLacement

control, and,

therefore,

no mattef what

_procedures

are ernployed, we cannot ensure reliabte results

if

the

test

requires control

beyond

the

measurable

displacement.

T-Modification

is

to

provide

more

accurate

results

than

I-Modification,

because

it

takes

into

account

the

change

in

stiffnesis

during

the

nenlinear response,

but

this

algori,thm

requires adiditional

haidware

capacity

in

order

to

continually measure and collect

the

forces

and

displacements

during

the

loading.

On

the

other

hand,

I-Moclification

is

simpler ancl

_perfectly

adaptable

to

the

basic

Loading

_procedure

developed

for

the

PSD

test.

If

we are reminded

that

the

error effect no[mally

decreases

during

the

inelastic

response,

I-Modification

is

believed

still effective

for

most of

_practical

_purposes.

4,

Conclusion

The

major

findings

obtained

from

this

study

follow

:

1. The

reason why,

in

the

clisplacement

error condition,

the

highest

mecle vibration was more

_piomoted

was verified

2. Relationship

between

the

magnitude of

displacement

errors and

the

growth

of experimental eT[oTs was

quantified.

3,

Algerithms

to

suppress

the

experimental

error

_growth

were

devised,

and

their

effectivenes,s was

demonstratecl

by

PSD

tests.

ACKNOWLEDGMENT

The

writers wish

to

express

their

_gTatitude

to

Drs,

S,

Okamoto

and

Y. Yamazaki,

Building

Reseaich

]nstitute,

Ministry

ef

Con$truction,

for

their

continuous

encouragement and valuable comments

throughout

the

study

presentecl.

References

1)

Takanashi,

K.,

et al,, "Seismic

_Fai]ure

_Analysls

of

Structures

by

Cornputer･Pulsator

On-Line

System",

JournaL

of the

(9)

-136-　 -136-　

lnstitute

　Df　

Industrial

Science

，

University

　Qf　

Tokyo

，

VoL

26 ，

No

．

11 ，

_PP

．

13 −

25 ，

December

1974

_（

in

Japanese

_）

．

2

｝

Ma

卜

in

_，

S

．

A

，

and 　

Shing

，

P

．

B

．

，

”

PseudOdynamiC

Me

亡

hOd

fOr

Se

且SmC 　

TeStlng

”

，

_」OUrnal 　Of　the　

StrUCtUral

Engineering

．

　　

ASCE

_，

VoL

ll1

，

No

．

7 ，

_pp

、

1482

−

1503

，

July

1985

．

3

｝

Takanashi

，

K

．

　and　

Nakashma

，

M

．

，

“

Japanese

ActiVities

　on　

On

−

Llne

Testing

”

，

Journal

　Qf　

Lhe

Engmeering

Mechanics

，

ASCE

，

vo1

．

113 ，

NQ

．

7 ，

pp

．

1014

一

ユ

032 ，

July

l987

、

4

｝

Nakashima

，

M

．

　and　

Kato

，

H

．

，

“

Part

3

_：

Experimental

Error

Growth

in

Pseudo

Dynamic

Testing

”

，

Journal

　of 　

Structura

［and

　　

Construction

Engineertng

，

Transactions

　oE 　

AU

，

No

．

386 ，

_pp

，

36 −

48 ，

Apri

［

1988

．

5

｝

Shing，

P

．

B

．

　 and 　

Mahin

，

S

．

A

，

」

‘

Experimental

Error

Propagahon

in

Pseudodynamic

Testi

“

9 「

’

，

UCB

_／

EERC

−

83

_／

12 ，

　　

E

ヨrthquake 　

Engineerlng

R

已search 　

Center

，

University

　of　

California

，

Berkeley

．

June

l983

．

6

）

Nakashima

，

M

．

、

“

Part

1

：

Relatienship

BeIween

Integratien

Time

Interval

　and　

Response

Stability

　ln 　

Pseudo

Dy

冂amic

　　

Testing

”

_，

Journal

　of 　

Structural

　and 　

Construc

し

ion

Engineerlng

，

Transactions

　of　

AIJ

，

No

．

353 ，

pp

．

29 −

36 ，

July

l985

，

7

）

　

YamanQuchi

，

H

．

，

et　al

、

，

’

‘

Fun

．

Sca

［e　

Seismic

Tests

　Qn　a　

Six

−

Story

CQncentrically

K

．

Braced

Steel

Building

−

US

．

japan

　　

CooperatLve

Research

Program

− ”

，

Proceedings

　Qf　the　

Fifth

Engineering

Mechanlcs

Division

Speciahy

Conference

，

　　

pp

．

603 −

606 ，

　

August

　

1984

，

8

）

Seki

，

M

．

，

e吐al

．

_，

“

S

亡udy　on　

Ear

し

hquake

Response

　of　

Two

・

Stori

巳s　

Stecl

Frames

　wtth 　

Y

・

Shaped

Brace

’

Journal

d

Structural

　　

Engineering

，

Vo

【

．

33

B

，

_PP

．

259 −

271 ，

March

l987

（

in

Japanese

）

．

g

）

Yamazaki

，

Y

．

，

et 　al

．

，

“

Accuracy

Evaluation

　of　

Pseudo

Dynamic

Response

，

Earthquake

ResPonse

Sirnulation

Capacity

　of

　　

Pseudo

Dynamic

TestLng

”

，

Journa

［of 　

Structural

　and 　

Construction

Engineerlng

，

TransactiQns

　of 　

AlJ

，

No

．

370 ，

_pp

．

40 −

49 ，

　　

Deoember

］

986

（

in

JaPanese

）

．

【

論

　

文

】

UDC ：624

．

042

．

7 ：

620 ．

1 EI

本建築学会構造系論文報告築第 40 工号

・

1989 年 7 月

Part

4 _{仮動的}

実験

_{における}

実

験

_誤

差

_の

制御

（

仮

動

的

実験応答

の

安定と精度）（

梗概）

正会員正会員

中

加

島

藤

正

博

愛

＊

人

＊＊

　

1．

序

　

仮動的実験

では

，

加力装置

や

計測

装

置

の

精度

が十分でないと

，

得

られた

応答

が

真

の応

答

から大きくずれてしま

う傾向（

実験誤差

と

呼

ぶ

）

が

あ

る。

筆者

らは

文献

4 ）

において_，この

実験誤差

の

特性を実験的

に

調

べ

，

】

）変位

誤差

（

目

標

とする

_変位

（

計

算変

位

）

と

加力後実際

に

到達

しえた

変位（

計測変位〉

の

差｝

が

実験誤差

に

最も大

きな

影響

を及ぼす

誤差

因子であること

，

2 ）変位誤菱

はアンダ

ー

シュ

ー

ト

（

計測変位

が

計算変位

によりその

増

分の絶

対値

において

小

さい

現象）

という

性質

を

持

ちやすいこと，

3 ）多

自由度系

に

対

する

仮動的実験

においては

，

変位誤

差は

系

の

_最

高次のモ

ー

ドを

励起

する

傾向

にあること

，

を

明

らかにした

。

本論

では，

変

位誤差

による

実験誤差

の

特

性を解析的

に

定量化

し

，

ま

た

実験誤差

の

累積を抑制す

る

手順

を

提案

する

。

ホ _{神戸大学}

_{助教授}

・

Ph

．

D

．

＊

_建_設_{省建築}_研_{究所　研究員} 　

C

］

98gijz

月

9

日原稿受理

，

1989

年

5

月

8

日採用決定｝

　

2，

変位誤差

の

応答

に

及ぼ

す

影

響

　

線形

1 自由度

系

においては

，

実験誤差

は

，

変位誤差

と剛

性

の

積

として

表

される誤差力による

系

の

応答

と

解

釈

できる

（

式

1 ）

。

　

2．

1 　

ラン

ダ

ムな

変位誤差

の

影響

　Shing

と

Mahin

にょる

詳細

な

検

討（

文献

5 ｝

を

参考

にして

，

ラン

ダ

ムな

変位誤差

を

含

む

仮動的実験

における実験誤

差

を

考察

する

。

式

1

において

，

粘性減衰

が

0

でありまた

地動加速度

がないとし

（

誤

差力

だけが

作用

する

状

態）

，

さらに

変位誤

差が正

規

分

布

に

従

い

，

またその

_平

均

値

が

0

で

_{標準}

偏差

が σ

，

で

あ

るとすると

，

実

験誤差

の

平

均値

は

0

となり

，

その

標準偏差（

σ。

）

は式

3

で

表

される

。

この

関

係

を図

一1

に

示

す

。

図

一

ユによると，

積

分

時

間

刻

み

（

At

）

が

大

きくなるほど

実験誤差

も

大

きくなる

。

多自

由度

系

では

，

高次

モ

ー

ドほど

相対

的

な

積分時間刻

み

（

ω

A

t

_）

が

大

きいので

，

高次

モ

ー

ドの

_応

_答

がより

励起

されやすいことが分かる

。

　

2 ．

2 　

アン

ダ

ー

シュ

ー

トの