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Model Correlation

and Damage Location for

Large Space Truss Structures:

L_

Secant Method Development

arid Evaluation

L |7

11 :

2

L

SuzanneWeaver Smith Departmentof EngineeringScience and Mechanics Christopher

t _

A. Beattie

Departmentof Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061

Rnal Report NASA Grant NAG-I-960 Period Covered:

February I, 1989 - January

(NASA-CR-188ZO2) MO_L CORRELATION AND DAMAGE LOCATION FOR LARGE SPACE TRUSS STRUCTURES: S_C&NT METHOD DFVSLOPMENT AND EVALUATION Final Report, [ _eb. I969 - 31 Jan. I99I (Virginia Polytechnic Inst. end

\

3I, 1991

_91-21213

Unclas G3/18

00075_9

--_,_-

z

:

w

z

Model Correlation

and Damage

Location

for

___'.

Large Space Truss Structures:

I

Secant Method

I

Development

Suzanne Department

Weaver

of Engineering

and Evaluation

Smith

Science

I

and Mechanics

jw

Christopher A. Beattie Department of Mathematics r w

Virginia °

Polytechnic Institute and State University Blacksburg, Virginia 24061

i

Final Report I

NASA Grant NAG-I-960 Period

L_

I_

° Currently,

Covered:

Department

February

of Engineering

1, 1989 - January

Mechanics,

31, 1991

University

of Kentucky

w

Abstract

=

On-orbit testing of a large space structure, like Space Station Freedom, will be required to complete the certification of any mathematical model for the structure dynamic response. Prior to assembly in orbit, model verification will occur only at the substructure level or with scale model technology. Ground tests of the complete structure are not possible.

V

The process of establishing a mathematical model that matches measured structure response is referred to as "model correlation." Most model correlation approaches have, as their essence, an identification technique to determine structural characteristics from measurements of the structure response.

t_

The current research program, under Grant NAG-1-960, approached this problem with one particular class of identification techniques - matrix adjustment methods - which use measured data to produce an optimal update of the structure property matrix, often the stiffness matrix. Previous research had led to new methods for identification to handle problems of the size and complexity expected for large space structures. Further development and refinement of these "secant-method" identification algorithms was undertaken. Also, evaluation of these techniques in an approach for model correlation and damage location was initiated.

w

Efforts for development and evaluation of secant methods for structure identification have produced understanding in a broad area now te_ed optimal-update identification or optimal matrix adjustment. This is the major result of this grant effort. Constrained optimization formulations for updating the property matrices of a model of a structure using measured dynamic response data have been developed and examined. In additon, these techniques are applicable to other problems in which multiple constraints can be specified for a matrix updating problem, for example, control gain calculations.

L m

Evaluation of these techniques for the problem of damage detection is still in progress, however. Improved understanding resulted from the development efforts and evaluation studies under this grant but additional efforts were def'med as a result. In particular, attention to the problem of sensor placement, among others, is needed to produce the best mode shape information for the identification process. As a result of this research effort, ten publications have, or will, appear. In the summary section, these publications are listed. Abstracts are reproduced where they are available.

L 1E-

w

Table of Contents P

1.0 Introduction

.................................................................................................................................

1

2.0 Background

.................................................................................................................................

6

3.0 Optimal-Update

w

Identification

4.0 New Developments

..................................................................................................

in Optimal Matrix Adjustment

.................................................................

17

24

5.0 Evaluation Studies ....................................................................................................................

28

6.0 Summary ...................................................................................................................................

33

Bibliography

39

m

...................................................................................................................................

w

_T

L_

m

L w

i

ii

w

1.0 Introduction w

On-orbit

testing

of a large space structure,

complete

the certification

assembly

in orbit, model verification

technology.

of establishing

to as "model

Freedom,

model for the structure

structure

a mathematical

correlation."

Station

will be required

dynamic

will occur only at the substructure

Ground tests of the complete

The process is referred

of any mathematical

like Space

response.

to

Prior to

level or with scale model

are not possible.

model that matches

Most model correlation

measured

approaches

structure

response

have, as their essence,

m

an identification =

technique

to determine

structural

characteristics

from

measurements

of the

:

structure

response.

Once

the

mathematical

model

is updated

and

accepted

as the

correlated

model

of the

.._4

undamaged

m

structure,

monitoring

problem

just completed.

response

measurements.

undamaged

structure

the structure

Again, This

to locate

for damage

a mathematical

current

model

model

is equivalent

of the structure

is compared

areas of disagreement,

to the model correlation

which

to the indicate

is developed

correlated

model

that damage

using for the

has occurred

and its location. A research

program,

class of identification

under Grant NAG-I-960, techniques

- matrix

approached

adjustment

these problems

methods

- which

with one particular

use

measured

data to

U m w

produce

an optimal

research

[R-177]

and complexity

w

of the structure

had developed expected

algorithms

Also, evaluation

of these techniques

matrix,

often the stiffness

for identification

for large space structures.

identification

was initiated.

property

new methods

"secant-method"

r

w

update

Further development

was undertaken

in an approach

to handle

as a part

matrix.

problems

Previous of the size

and refinement of the research

for model correlation

and damage

of these program. location

m

This report January

presents

31, 1991.

Results

in the form of journal

w

referenced

the research

program

results for a period

that have appeared,

papers or conference

in the following,

or will appear,

from February

as a part of the technical

papers [R-107 to R-ll0,

but not reproduced

I, 1989 to

in this text.

R-ll2,

literature

R-190 to R-193]

Primarily,

the report

are

includes

an

of the findings which are have been published

in

_ffi=:_

extensive

review and bibliography.

Summaries

other forms are included as well. wm_

The rl

space

following structure

identification literature

introduction identification

techniques

of other

adjustment

sections and

methods.

damage

and damage

engineering New

provide

a brief

location.

location

disciplines. developments,

Chapter

in the

Chapter

discussion

aerospace

new

developments

2 is an extensive

3 presents

including

of recent

literature,

review

as well

a summary applications

of

as in the

of optimal for

in

matrix

optimal-update

u

identification,

w

w

are discussed

for performance

for model

with simple spring-mass

is the summary

and conclusions.

5 summarizes

correlation

models

results of the evaluation

and damage

location.

and using a laboratory

These

studies

evaluations

truss structure.

Chapter

6

1.1 Space Structure Identification model correlation

characteristics. task

w

of the methods

were conducted

Successful m

in Chapter 4. Chapter

A review

committee

whose

of on-orbit stated

goal

identification

of large structures

experimental

evaluations

More

recently,

identification. experimental damage

assessment

1.2 Damage

Location

to discussions are included,

of large

[R-178].

from

was

held

reviews

of nonlinearities

as well.

on the ability to identify space structures

a state-of-the-art

to discuss

on methods

included

by a for

a call for

methods. current

of identification and model

structural

was prepared

report

Their recommendations

studies of identification

[R-179]

ranged

depends

was to "develop

in space"

a workshop

location

identification

and comparison

Presentations results

or damage

programs

techniques

uncertainties.

in on-orbit to reports

Some

of

talks on

w

This year, a second

w

space

structures.

an assessment

In identification,

by establishing

response.

Optimal

Second,

results in an inteUigent

ground

identification

learning

tests.

experiment

programs

These

methods

identification

monitoring"

for

was adopted.

is possible.

First, uncertainties models

in the structure

model may be

that best represent

identification

may

use

the model identification.

the previous

techniques.

on-orbit

the structure

to be robust with respect to the entire set of models

system

way to complete

to supplement

Several

of the state-of-the-art

are designed

addressed

and on "health

the set, or sets, of system

controllers

for the structure.

[R-180],

programs

two ideas were prominent.

addressed

tailored

Workshop

A focus on experimental

From this workshop

_z

USAF/NASA

compared

the

use measured

all previous

Additional

performance

structure

experimental

experiments

can be

of

modal

response

various

to produce

modal

w

parameters;

frequencies,

damping

factors,

Eigensystem

Realization

Algorithm

(ERA)

with the strongest were presented, Primarily, structures,

E

performance

shapes

and modal

participation

factors.

The

[R-181, R-77, R-76, R-78] was cited as the technique programs.

Refinements

of and improvements

to ERA

also. two types of structures

either deployable

are different

in several

mode

were examined

or erectable,

and reflector

for the two types of structures,

but overall

in the various structures.

research

Characteristics

they present

programs

- truss

of the response

many of the same challenges:

m

closely-spaced, i w

m

modes,

low

damping,

possible

nonlinearities,

and

symmetry,

among others. Various including

!

low-frequency

sensing devices

optical

truss structures "Health structural

sensing,

are under investigation laser

sensing

and imbedded

their capabilities

fiber-optic

for identification

sensing.

Active

members

system

failures

along

of

may also be used for identification. monitoring"

degradation

encompasses

and failure.

sensor

Results

of this report were the only experimental location.

to explore

failure

and

other

from the experimental results

presented

program

[R-109]

which

detailed focussed

with

in Chapter on damage

5

ira#

w_

On-orbit

!1=.

experiments

which

are underway

or which

are planned

were

highlights

of the

workshop: E

1.

Recent

parabolic

flight profiles of a KC- 135 aircraft

tions and free-free

tests of two truss structures.

were used to achieve

free-fall condi-

With results of the flight tests compared

v

to results ground

of ground

test techniques

tures Division The program 2.

tests designed is anticipated.

of the Hight is focussed

Flight dynamics

to simulate

This program

Dynamics

Laboratory

on system identification

experiments

free-free

conditions,

is a continuing

an evaluation effort

at Wright-Patterson

of

of the Struc-

Air Force Base.

and active control.

for modal identification

of the Low Altitude

Compensation

W

Experiment

(LACE)

satellite

(which

is currendy on-orbi0

Three

flexible,

are being

conducted

by the

tlw

Naval

Research

Laboratory.

deployable

truss booms

are the primary

I;;;

structure E

of the satellite.

identification

Wi-

Using ground-based

laser measurements

of all six of the first flexible modes is feasible.

and on-board

sensors,

The actual experiments

will

occur later this year. 3.

The Space Station Freedom Structural under development

Characterization

by NASA Langley

Research

testing can be applied to various build-up

Experiment

Center.

configurations

is a flight experiment

Techniques

for on-orbit

modal

of the Space Station for identifi-

F

cation of the properties currendy

underway

of this large flexible

to support

space structure.

the development

Analytical

effort and to evaluate

studies are contechniques

for on-

orbit identification. 4.

The MIT Space Engineering Gravity

z

Experiment)

of contained middeck

Research

experiment

Center

to identify

fluids and of very flexibIe

of the space shuttle.

may be evaluated

Model

the MODE

linear and nonlinear

structures.

coupling,

as a result of this experiment

pared to flight experimental

is designing

(Middeck

dynamic

The experiment

as well as nonlinear

Zero-

characteristics

is designed model

for the

techniques,

in which ground test results will be com-

data.

4

The

field

of structure

addressed

issues.

to reduce

the uncertainty,

uncertainties location

Foremost

identification

contains

many

among these is the issue of uncertainty

how to quantify

are all questions

or "health

still

monitoring"

the uncertainty,

to be explored. is another

unaddressed

in the identified

how to use identified

Using system identification

major

issue.

more detail in this final report, but the first is considered

or incompletely model.

How

models even with

techniques

Of these two, the second

for damage is explored

in

as well.

F

h

E

5

w

=

=

rL_ V

2.0 Background E

=

In the subsequent

sections,

background

material is presented

on structure

identification

and

w

damage location.

First, optimal matrix updates

These techniques niques

led to the secant methods

for structure

reviews

identification

of test/analysis

assessment

along with damage

2.1 Optimal

research,

location

presented

location

which employs

structural

which uses dynamic

matrices

General

for modal

and damage

research

property

herein.

and on techniques

correlation

(or location)

for structure

will be reviewed.

background

on other tech-

identification

research

follows.

close the chapter.

parameter

identification,

Then Damage

is presented,

testing, but not identification.

Matrix updates

2.1.1 Early Work Methods

L_

based on matrix

approximation

ideas have been in use in structural

dynamics

since

m_

the late-1960's. structural

Rodden

influence

[R-132] used results from ground

coefficients.

using the generalized

inverse,

The resulting

vibration

underdetermined

then further adjustments

helped

tests to derive the matrix

system of equations to satisfy physical

of

was solved

constraint

condi-

J

tions. Brock m

measured matrix

[R-133]

examined

conditions

and

is found

the problem

which

of determining

may also

which most closely

satisfies

possess

an optimal

matrix

to satisfy

matrix

properties.

The

certain

conditions

with the measured

data.

a set of

symmetric

In the current

B

research, definiteness

these ideas

may be imposed.

can be incorporated close as possible

m

are extended.

by finding

In addition

Initial information the symmetric

to symmetry,

properties

such as positive

about the matrix (for example,

an initial model)

property

matrix

matrix

as above,

subject

to having

it as

to the initial model as well.

6

The problem

of incorporating

with the updated

matrix

initial model information

as close as possible

can be framed

in a different

to the initial matrix and also satisfying

way,

conditions

L_

with the measured r,

data and symmetry.

with the measured lows in Chapter

data (which

assuming W

formulation errors).

problem

the stiffness

the f'mite-element

which

orthogonalization

and mode

to be an accurate

matrix must satisfy conditions

might be from the finite-element

formulation

started

model,

shapes,

representation

with the measured

and

of the

modes.

An ini-

with the initial mass matrix as

of measured

modes,

stiffness

[R-65] adopted

test data, addressing

of above is the basis of a sequence

with the work of Baruch

for the stxucture

man and Nagy

frequencies

techniques.

Updates

The latter problem

adjustment

of this point fol-

mass matrix.

2.1.2 Recent Optimal

identification

a more rigid constraint

Further discussion

with measured

an initial model of the mass matrix is known

mass distribution,

imposes

more detail on these two views of optimal-update

identification

tial model of the stiffness

m l E

is likely to contain

3 which includes

For the structural

This alternate

Baruch

matrix,

and Bar

Itzhack

using measured

a similar approach

the possibility

and Bar Itzhack

[R-54].

developed

frequencies

in an effort to improve

of both mass and stiffness

work in optimal-update In their work on a minimal

matrix

and mode

shapes.

Bet-

analytical

models

using

matrix updates.

W

Baruch m

[R-56

with an exchange further

to R-62]

and Berman

[R-55,R-64,R-66

of ideas and extensions

clarifications

and extensions,

to R-74]

of the techniques.

most

recently

followed

their initial works

Wei JR-137,R-139,R-141]

combining

the mass

provided

update

and stiffness

l

qr

update

in a single constrained

minimization

With each of the preceding

methods,

formulation the sparsity

[R-141]. (the zero/nonzero

pattern)

of the original

ilm stiffness mm

[]

matrix

is not preserved

sponds to a reduced

model

in the adjusted

with few zero elements,

the similar adjustment

of an original

the nonzero

created

elements

finite-element

in the adjusted

that do not exist in the actual structure.

stiffness

matrix.

If the original

this might not be a difficulty. stiffness

stiffness

model

However,

matrix having considerable

matrix

This reduces confidence

correspond

to physical

in the adjusted

correfor

sparsity, load paths

model.

"Mr

7 m

Kabe [R-85,R-86]

v

cedure

which

matrix.

preserves

An updated

straints imposed found.

addressed

the physical

stiffness

of the stiffness

and presented

connectivity

matrix

by the measured

The measure

Kammer

these concerns

of the original

that is "closest"

JR-90] presented work.

a stiffness

w

as in Kabe's

Producing

and the sparsity

t

method

is an important

reformulation

weights

in the objective

function.

results

with similar

of Kabe's method

yet satisfies

is

element

techniques.

with the same objectives

computational

that permits

con-

matrix,

for each nonzero

method

pro-

stiffness

of the original

norm used in previous

matrix identification

equivalent

in the updated matrix,

is a sum of the change percentages

matrix rather than the matrix Frobenius

matrix adjusmaent

model

to the original

modal data, symmetry

of "closeness"

a stiffness

effort,

more flexibility

Kammer's in defining

m

I

z

Prior to the current research was

initiated

[R-187, which

effort, a new technique

by the authors

R-188]

update

preserves

[R-177].

A multiple-secant

of computational

the sparsity

for optimal

linear

of the original

algebra

model,

stiffness

generalization produced

without

matrix identification of the Marwil-Toint

an identification

excessive

storage

technique

requirements

or

!

computational m

update

effort.

techniques

Details

of the development

are presented

of this new method

and other related

optimal-

in Chapter 3.

2.2 Mode Shape Expansion and Orthogonalization !

Throughout i

that all modeled costs, limited

m

ation.

_===

of freedom

data handling

The

eigenvalue

[R-183]

capabilities

unmeasured

presented

mathematical problem,

Rearranging E

degrees

discussions

it has been

were accessible

the premiere

model

corresponding

the second

for each of the observed

to measurement.

work on model reduction

of the structure to the measured

dynamic

response

and unmeasured

matrix

equations

modes,

Due to instrumentation

the actual number

10% of the total number

of two resulting

degrees of freedom

assumed

and inaccessibility,

dom is far less in practice - rarely exceeding Guyan

w l

the previous

of degrees

of free-

that may be modeled. to accommodate is a reordered, degrees

produces

this situpartitioned

of freedom. an expression

of the mode shape which can be used to reduce

for the

the mathematical

w model

representation

unmeasured

to one with only the measured

degrees-of-freedom

the measured Guyan

degrees-of-freedom reduction

are significant,

i.e., significant

representation efficient

computations

An alternate w

(SEREP)"

and more accurate technique

[R-89,R-91]

by O'Callahan,

in cases where

to address

degrees-of-freedom reduced

Avitabile

the neglected

all

in terms

of

dynamic

Conti [R-188]

these shortcomings.

with a converging

terms

presented

He expanded

power

a the

series to allow

models.

was presented

and separately

them

words,

equations.

high frequencies.

of the Guyan method

reduction

ment by Kammer

models

mass or relatively

of the unmeasured

in other

from the model by expressing

using static equilibrium

may lead to inaccurate

higher order generalization

W

are eliminated

degrees-of-freedom,

for "test-analysis-model

for "system

and Riemer

equivalent

[R-100,R-102

(TAM)"

reduction

to R-105].

develop-

and expansion

A vector

of physical

E

displacements, m _gw

bination

i.e., a mode shape vector in the physical

of analytical

and partitioning

sented in terms of the measured

the measured

reduction, values,

as before, the unmeasured

degrees-of-freedom

tions for the modal coordinates For model

is expressed

as a linear com-

model mode shapes.

After reordering

i

coordinates,

using a generalized

once

the

substitution

degrees-of-freedom

by solving

can be repre-

the first of the partitioned

equa-

inverse.

unmeasured

degrees-of-freedom

into the original

eigenproblem

are expressed produces

in terms

a new model

of

with

m

"reduced" IgJ

stiffness

and mass matrices.

with r as the number of measured

The new stiffness

and mass matrices

are dimension

r x r,

degrees-of-freedom.

It is often useful to extrapolate

values

for the unmeasured

degrees

of freedom

based on both

i m m

modeled !

U

dynamic

a reduced

information

mathematical

tion process,

model.

thus, "expanding"

and the available

measured

degrees-of-freedom,

This extrapolation

can be accomplished

one mode at a time.

The set of expanded

without

by reversing

producing the reduc-

mode shapes is not like-

I E

ly to be orthogonal zation

E

procedure

with respect to the analytical is necessary

to introduce

model mass matrix, so a subsequent

this property

to the set of expanded

orthogonali-

modes.

Several

w

optimal-update

identification

respect to the analytical IP"

Baruch

and

f

nique.

assume orthogonality

[R-54]

presented

an optimal

set of modes to the closest

This orthogonalization The weighted

of the measured

mode

data with

mass matrix.

Bar Itzhack

adjusts the measured matrix.

techniques

problem

optimization

technique

orthogonalization

set which is orthogonal

to the structure

can also be framed as an optimal matrix

technique

performs

symmetric

which

adjustment

mass

adjustment

tech-

of the mode shape

vectors. m

Prior to the work of Baruch and Bar ltzhack, Targoff [R-127,R-129] cussed orthogonality

of measured

modes and methods

to "correct"

and Rodden

the measured

[R-128]

modes.

dis-

Gravitz

v

IR-130]

also presented

Chu, et.

E

shapes

al.

derived

determine

an approach

JR-114] from

for orthogonalization

discussed

measured

the consequences

data.

the pitfalls Many

of using

of measured a mass

of the previously

of their application

in circumstances

modes.

orthogonality

check

cited techniques leading

for mode

were tested

to structure

to

identifica-

tion. Finally, measured

Kabe [R-87] proposed a procedure

mode shapes." Different excitation

lation of the modes and therefore

2.3 Structural

tion is appropriate

_mB

updates

orthogonality

of

levels and locations can be used to improve the iso-

improve the result.

Identification

A discussion

U

"to improve the mass weighted

of techniques

for structural

at this point.

Ultimately

identification

which do not employ

a hybrid of these alternate

matrix optimiza-

techniques

and optimal

may be the most effective.

2.3.1 Sensitivity Adelman

Methods

and Haftka [R-165]

recently presented a review of research

in the area of sensitivity

m analysis

for structures.

in this area.

Optimal

A variety control

of applications

and structural

for sensitivity

optimization

derivatives

use derivatives

has encouraged to establish

work

effective

10 z

z

V

search

directions

improve

for optimal solutions.

computationally

ment using sensitivity

intensive

Efficient

optimization

derivatives

computation

procedures.

has received

attention,

of derivatives

Recently,

is also studied

analytical

to

model improve-

along with applications

in design meth-

odology. For structural identification usingsensitivity derivatives, a setof physicalparameters(densities, moduli of elasticity, areas,lengths, masses)ischosen to represent thestructure. Elemcntsof the dynamic model property of the

physical

expressed

matrices,

parameters.

as functions

mass, stiffness

In turn,

or damping,

eigenvalues

of the parameter

are functions

and eigenvectors

of the

set. Using local linearization,

(often nonlinear) response

the functional

can be

relationship

W

of system properties Adjustments E i

structure

to the parameter

in the physical

and the sensitivity

depending

on the number

which set whether

set is determined.

parameter

set are identified

derivatives. of physical

The update parameters

the system of equations

using the dynamic

is accomplished

and on the number

if overdetermined

response

iteratively

of the

or optimally,

of measured

properties,

or underdetermined.

m_

Creamer structure ficulties

and Hendricks

using measured encountered

[R-80]

natural

presented

frequencies

with symmetry

a technique and sensitivity

to identify

physical

derivatives.

parameters

They examined

of a

the dif-

in the structure.

E

Calculation Lim, Junkins

of eigenvalue

and eigenvector

and Wang [R-79] recently

derivatives

re-examined

has received

the problem

considerable

of eigenvector

attention.

derivatives

pre-

E m

senting an improved Property

matrices

ined for sensitivity

method for non-self-adjoint and modal quantities with respect

systems.

are not the only structure

to the physical

parameter

set.

models which can be exam-

Chon

[R-166]

discussed

strain

lm energy sensitivity Applications

to joint stiffness. of structural

identification

using sensitivity

derivatives

are available

in the liter-

m !

ature.

BriUhart, et. al. [R-174,R-175]

for a transfer

orbit stage structure.

reported

Additional

results of modal tests and test/analysis applications

are cited in the sections

correlation

which follow.

w

2.3.2 Eigenstructure

Assignment 11

=

Another

class of structure identification

eigenstructure which

assignment.

determined

with a particular

Srinathkumar

the appropriate

grew

[R-163] presented

control

system response

techniques

(an adjustment

(or eigenstructure).

out of the controls

a method

for optimal

to the control/plant

Andry, Shapiro

research control

system)

on

design to result

and Chung IR- 164] extend-

ed these results. m

L

For structure identification, tious control model

eigenstructure

assignment

which would be required

to produce

This fictitious control

is translated

system.

ate for the observed

structure

response.

theory is applied to determine

the measured dynamic into adjustments

Minas and Inman

the ficti-

response with the initial

to the initial model appropri-

[R-75]

and Zimmerman

and Widen-

m

gren [R-39] presented

variations

of this idea.

z

2.3.3 Other Methods While this review is extensive, tification

techniques

mention,

though.

Rajaram, elements

which

it is not comprehensive

have been presented

due to the numbers

over the last 20 years.

of structural

Several

iden-

ideas deserve

Some of them are covered in this section.

et. al. [R-83,R-84]

of the stiffness,

presented

damping

an identification

and mass matrices

technique

which

is notable in that all

of a linear model can be determined

the size of the model as the only initial information.

Measurements

acceleration

to provide

of displacement,

velocity

with and

u_t

u z

determined nique

E W

are required

at each degree-of-freedom

system of equations

is not practical

to solve for the elements.

with current

measurement

enough

Symmetry

technology,

data to produce

is not presumed.

but demonstrates

an overThis tech-

one extreme

in

identification. C_n

and Garba

[R-121]

used a perturbation

analysis

to formulate

a structure

identification

E procedure. ments,

White and Maytum

to adjust

with the frequency

the response response

[R-146] used scaled submatrices, model.

function

Creamer

finite elements

JR-81] used techniques

and extended

them to determine

or groups of ele-

for modal

structure

identification

property

matrices.

L E

IB

12

Nonlinear oped w

systems

have been considered

only for simple

nonlinear

single

degree-of-freedom

systems were considered

by Gifford

Kuo and Wada [R-154,R-156] boundary

conditions

employed

nonlinear

to enable sensitivities

Modal identification,

Juang

and Pappa [R-184]

systems

ground

and Tomlinson

testing

for identification

Multi degree-of-freedom

structures

with multiple,

identification.

They

different also have

[R- 161 ].

of modal parameters

an extensive

are devel-

JR-162].

and to enhance

is a related area of research prepared

the techniques

at this time.

have studied testing flexible

determination

ing and modal amplitudes,

by Mook [R-98], although

- frequencies,

mode

shapes,

damp-

which is, in turn, related to modal testing.

review of modal testing

and identification

with a

focus of control of structures. Additional summary. L_

references

Identification

out using a spatial identification

on modal identification with statistical

discretization

methods

and modal testing have been omitted and identification

are also excluded.

with optimal matrix updates,

While

of continuous

these topics

they are not essential

from this

structures

with-

are related to structure

to the subsequent

discussions.

_4

2.4 Test/Analysis m i

v_

m

Correlation

Many applications tion) are presented sensitivity

with Structure

of structure

in the literature.

Identification

identification

for test/analysis

Olximal-update

correlation

identification

(or model verifica-

is employed

in some cases;

techniques are used in others.

mmm m m

Matzen [P,-23,R-24] truss structures. elements

His approach for system identification

of mass, damping

than for physical m

iu

While damage

and stiffness

parameters

location

play a key role in damage Detection

L m

has studies structural identification

for shear buildings

uses sensitivity

matrices of a discretized

of substructures

(i.e., elastic

was not addressed specifically

and indeterminate

techniques

to solve for the

model of the structure

modulus,

cross-section

in this work, structural

rather

area, etc.).

identification

can

detection.

of changes in structural parameters using system identification

gated by Agbabian and Masri, with others [R-20,R-41].

is the topic investi-

A system identification

technique,

simi-

l

13

E

w

lar to that used by Matzen,

is evaluated

in the presence

of noisy measurements.

Simulated

data

for an idealized bridge structure is used in the most recent studies. Douglas five-span System

w

and Reid JR-182]

four-hundred-foot identification

long, reinforced

techniques

of the structure.

The authors

highway

provide

bridges

ture by Lapierre

interaction

correlation =

approach.

investigates

optimal-update

box girder

carefully

[R-115].

of a bridge

bridge

stiffness

conducted

about the structure

structure.

was tested

properties, dynamic

A

extensively.

among

others,

tests of full-scale

(and) ... shed much experimen-

of the problem..."

was used for test/analysis

attempts

Simulated

identification

to determine

"Thus,

aspects

Hashemi-Kia

characteristics

Caesar [R-116,R-117]

concrete

information

identification

modal tests to investigate

i

concluded,

and Ostiguy

system

were employed

detailed

tal light on the soil-structure Optimal-update

also investigated

correlation

[R-38]

of a welded

plate struc-

used optimal-matrix

updates

and

of an elastic wing model.

to unify the works of Berman and Baruch in a comprehensive

studies with a 27-dof structure

methods

for structural

are presented.

model matrices

Link [R-142]

for test/analysis

also

correlation.

2.5 Damage Location The subjects in recent years.

of damage detection

and damage

location

have received

Much work has been done in the Civil Engineering

considerable

community,

attention

although

several

m

researchers

in the aerospace

2.5.1 Damage

Location

community

with Structure

are active as well.

Identification

In a series of work, Yao and coauthors lem of damage m

to examine structural

changes verification

requirements m w

detection

in buildings.

in the damping

[R-10,R-12,R-13,R-15,R-16]

Vibration

testing and identification

of the structure.

was discussed.

and, thus, may be verified

A damaged

A conflict or flawed

as acceptable.

between

structure

The complexity

have explored

the prob-

techniques

were used

damage may

detection

still satisfy

of the damage

and

system location

problem prompted Yao to suggest artificial intelligence as an approach for damage assessment.

B

14

An excellent

w

ambient

series of work focussed

excitation

success

with deducing

"The laboratory testing) !,

platforms

using

Rubin

and Coppolino

[R-28,R-29,R-17]

reported

of the platform

were damaged

for key structural

elements.

and field testing to date strongly to become

in fixed offshore

viable

suggest

that the method

for cost-effective

contribution

(of ambient

vibration

to an overall

structural

program."

Hajela

and Soeiro [R-48,R-49,R-50]

identification. structures

proposed

approaches.

Stubbs

and on composite

and coauthors

damage.

have recently

Their work is focussed

truss-type

locating

identification.

which members

can be developed

inspection

matrix

and system

on locating damage

Simulation

on several

materials.

[R-37,R-152]

studied damage

using optimal-

types of structures,

Simulation

have

assessment

but primarily

studies were conducted

also considered

using

system

on

to evaluate

identification

studies on simple models have been used to demonstrate

for

the tech-

niques.

2.5.2 Damage Location r

Kircher determine

i[5

different "identical"

[R-176]

with Dynamic studied

if similarities floors

identical

vibration

existed

were tested

design was comparable,

power

could indicate

changes

density

functions

in the structure,

Tracy and coauthors

was then investigated

response.

as well.

Similar Similar

The dynamic

period.

are obtained at different

perhaps

times."

occurring

Kircher

consider

damage

the changing

dynamic

response the damage

tested

for buildings

of built

that "Virtually

measurements changes

to

(in this case, of buildings

also reports

Therefore,

due to damaging

were

response

from acceleration

[R-34,R-35]

as a tool to determine

structures substructures

with the most similar response

in the same building

were used to examine

of buildings.

during the same construction

spectral

the same location

testing

in the frequency

of the buildings)

by the same contractor

Testing

taken at

in the response

earthquakes.

in composite

materials.

in delaminated

First, experiments

structures.

by using the dynamic

Modal

testing

response.

15

ira,

-_

z

Complex

aerospace

[R-30,R-33,R-46,R-47] aged aircraft testing

structures,

smactures

can be tested

and Pacia [R-43,R-45] including

was used to detect damage

to evaluate

structural

have been involved

a small plane,

integrity,

as well.

with experiments

with dam-

an F-16 wing, and the shuttle orbiter.

in each of these studies

by examining

changing

West

Modal

frequency

response.

u

r_

L_

16

:

.

3.0 Optimal-Update

Identification

In this section the developments

we present can be found

developments

in secant-method

in References

R-177,

R-108,

identification.

Further

R-110 and R-190.

details

of

This grant sup-

ported the second, third and fourth of these publications.

3.1 Optimal-Update

Identification

Optimal-update

identification

optimization matrices) w

problem,

adjusted

that more closely

ured modal data, qualifies preservation

of structure

In optimal-update minimization Baruch L_

ment.

or updated

of symmetry, connectivity

and Bar Itzhack

matrices

the solution

(i.e., mass,

modal response.

Along

closeness

to the initial model,

are imposed

in various techniques.

one of two views may be adopted

which produces [R-54]

to produce, through

property

match the structure

identification

problem

Generally,

is an approach

the adjusted

and by Kabe

this view can be formulated

property

matrix.

[R-85] in their methods

of a constrained

damping

or stiffness

with matching positive

the meas-

definiteness,

to establish

and

the constrained

The first view was used by for stiffness

matrix

adjust-

as

min IIA- All F subject to AS=Y and A = At,

(3.1)

where A is the nxn adjusted property matrix, A, is the nxn initial-model property matrix, S, Y are nxp matrices that define constraints with the measured n is the number of degrees-of-freedom of the model, p is the number of matrix constraint equations, t indicates the matrix transpose, and II E IIF= (Y--_,i=_1%12) _a is the Frobenius

norm.

Here a matrix closest to the original

model matrix is found.

is symmetric

the constraints

and that also satisfies

data,

It is, however,

of the measured

data.

the closest

matrix that

Additional

constraints

L h

17

2__:_

may

be

imposed,

Weighting Specific

including

matrices

structure

algorithms

identification.

Minimal

However,

to model

exhibit

for actual

indicates

that

with the other constraints

data inconsistent

with the desired

may occur [R-85]. in the objective

several

function

and

positive

of the minimization

vary in the measure

in terms

structures

constraints

definiteness.

problem

of closeness

sparsity

that make

of the number

have

them

as well. used and

with

attractive

of modes

been accomplished

imposed

of the technique

[R-66,

measured

data

can lead to poor results.

the measured-data

thus finding the closest

constraints

for

[R-85] and R-115].

which

are

In the case of

pattern, an updated model which is not positive

One option is to introduce

function,

characteristics

data is required

refinement

experience

inconsistent

pattem)

are included.

these

applications

zero/nonzero

that stem from this first viewpoint

in which constraints

w,,

(the

may be added in the objective

techniques

In general,

sparsity

def'mite

as a "soft" constraint

matrix which most nearly

matches

the dynamic

response. The

L C

second

view

This view was

adopted

linear systems

minimizers

Generally,

the optimal-update

by Brock

and by Kammer

with symmetric there.

for framing

[R-133]

problem

in his development

[R-90] in his projector

for the measured-data

this view is formulated

follows

of optimal

matrix method.

constraint

a different

and imposes

matrices

The second

philosophy. describing view starts

further constraints

from

as

rain IIAS - YI]v subject to A = Atand other constraints.

l

Although stiffness next

optimal-update

property

section

developed

3.2 Stiffness

matrices,

we present

identification stiffness

a series

with an optimal-update

(3.2)

algorithms

can be developed

matrix identification of stiffness

identification

matrix

for mass, damping

has been most widely explored.

adjustment

techniques

that have

and In the

all been

approach.

Matrix Adjustment

18

r'q'.:

A review

w

approach

and

summary

of optimal

allows a discussion are included.

of stiffness

updates serves of performance

Further

matrix

to illustrate

identification

the two viewpoints

characteristics.

discussion

and proofs

techniques

General

are presented

under

described

observations

the

general

previously

and

about these algorithms

in the references

supported

by this

=

grant. =

:

For this approach, undamped

+ [K]x

model

of the free,

(3.3) positive definite, original-model original-model stiffness matrix,

x is the nxl vector of displacements, is the nxl vector of accelerations. Baruch presented

structural

= 0

M° is the nxn symmetric, K° is the nxn symmetric,



at first an n degree-of-freedom

vibration,

[M]i

L

consider

and Bar ltzhack a technique

[R-54],

for stiffness

mass matrix,

and

in their work on orthogonalization

matrix adjustment

which exemplifies

of measured

modes,

the first viewpoint.

The

W

stiffness

matrix

for a structure

similar technique models,

was adopted by Berman

but they considered

matrix adjustment

using

measured

is formulated

frequencies

and Nagy [R-66] in their efforts

both mass matrix updates

problem

and stiffness

matrix

and mode

shapes.

to improve updates,

A

analytical

The stiffness

as

-IF K)Mo -1121

min IIM_tn(K-

subject

is improved

to KS=M

SD 2 and

K = K t,

(3.4)

where E

K is the nxn adjusted stiffness matrix, D 2 is the pxp matrix of measured eigenvalues, S is the nxp matrix of p corresponding measured IIE IIF= (_.Jelil2) A closed-form

112is the Frobenius

solution involving

mode shapes, and

norm.

only matrix multiplications

exists and was given by Baruch

and Bar Itzhack [R-54]: K = K + M SD2StM •

II

- K SS"M - M SStK + M SStK SStM. •

II

II

II

II

II

11

(3.5) II

19

Positive

w

definiteness

is positive

definite.

is guaranteed However,

in the adjusted

the sparsity

stiffness

matrix

pattem of the original

if the original

stiffness

matrix

model is lost in the update.

For

=

dense models which arise from reduced system models, this may not be an issue. Sparsity problem.

is preserved lObe

[R-85]

expressed to compare

i-I

in the adjusted presented

with

jG adj(i)

(K_)

property

matrix by adding

a technique

the formulations

for

stiffness

of this work,

an additional

matrix

is stated

constraint

adjustment

which,

research

sponsored

when

as follows:

2

(3.6)

subject to KS=MaSt_2 and K = Kt'

The

to the

by this grant

further

included

development

of a sparsity

preserving

update which is an analog to Kabe's,

= min IID'I(K-

subject to KS=MaSf_ 2, K =

-1

K)D

2

IIF

K t,

(3.7)

and sparse(K) = sparse(K), where D = diag(di) = diag(Nf_._.). These

sparsity-preserving

optimal-update

techniques

require

the

solution

of an auxiliary

z

problem r

_

to produce

the update.

Lagrange

objective

function.

Minimization

equations

to solve

for the Lagrange

multipliers

of the

incorporate

Lagrangian

function

the constraints produces

into an extended

a system

of linear

Z

The auxiliary

problem

multipliers

has the dimension

and an update

equation

of the number of measured

for the stiffness

modes,

matrix.

p, times the number

m Ires

of original

w

E

model degrees-of-freedom,

matrix adjustment the second storage

method

of Kabe [R-85]

method is symmetric

requirement

n, or np x np. While the auxiliary is symmetric

and positive

and with less computational

and indefinite,

semi-definite,

permitting

problem

in the stiffness

the auxiliary solution

problem

without

of

a large

effort.

20

With

w

the solution

of the auxiliary

formed from the original stiffness

problem,

elements

of the adjusted

stiffness

matrix

are

matrix as

w

Kij = K_j+ L

didj[(PiDSFi)

j +

(PjDSFj)i]

for ij = 1,2,...,n.

(3.8)

_

The diagonal matrix Pi contains sparsity

pattern

only ones and zeros to mask the mode shape vectors

of the ith row of K,. The sparsity

pattern

of K, does not consider

in S with the

zeros produced

w

by structural w

structure

symmetry

between

(force

cancellation),

degrees of freedom

The partitioned

vector

but rather

represents

physical

connections

in the

of the model.

{F} is the solution

of the auxiliary

system of equations,

constructed

w

from the original

model and measured

modal data as

r_

Fag + I-0F

=

.

(3.9)

..,

Lr.J F is a block diagonal tJ

matrix of weighted

masked vectors

using the same projection

matrix P_,

F = diag [PtDS, P2DS ..... PDS]

which

is combined

with the identity

(3.10)

matrix

I and a reordering matrix

H to form the auxiliary

=_ m

problem

coefficient

matrix.

The right-hand-side

vector

{A} is another

partitioned

vector

formed

with the given data

Ai = _i

,

(3.11)

L;:J

--= r_

f,i

weighting

each row of Y = MSt22

- K,S.

u

Implementation L_

positive

semidefinite

of the method system

can be accomplished

and solving

for {F}.

However

by

assembling

the dimension

the full

symmetric

of the system (np x

21

np) can exceed for solving assemble F

computer

Equation the

preconditioning

of the repetitive

explicitly.

for

the

requirements.

Experience

condition

method

and never

with

diagonal

no more storage

than

problem

is

much

In the case of stiffness

orthogonal

to the original

noise

imperfect

and inconsistent

has indicated

even if the original

in

this

second

data have been shown to

that inconsistent

data may lead to an

model matrix is positive accurate,

measured

matrix

data

updates,

is necessary

measured

definite.

but even then, noisy

make

it inevitable

that

for the existence

mode

model mass matrix (which is presumed

modeling

reduced

performance.

on the

solutions.

and

patterns

and required

if the initial model matrix is sufficiently

poor algorithm

underlying

gradient

problem

of sparsity

model which is not positive definite

data may produce

substructure

A conjugate

auxiliary

method, but preservation

be incompatible

Iterative methods

matrix [R-110].

effort

Often this is not a problem

m

advantage

matrix

stiffness

sparsity-preserving

An

if a large structure is considered.

was used for solution of the auxiliary

Computational

updated

(3.8) take

coefficient

that for the original

W

storage capabilities

shapes

are presumed

to be accurate).

this

will

not

of these

be

to be

Environmental exactly

satisfied.

_=

Modifications

to the measured

modes to recover

A second option is to introduce function,

rather than requiring

We explored m:!

Formulated

this option

mass orthogonality

the measured

constraints

that the data be matched

in an error-compensating

[R-54] is an option.

as a soft constraint

exactly

version

in the objective

as in the preceding of the sparsity

formulations.

preserving

update.

as

rainIIDt(K- Ko)Dil_+ lICKS - M.SflZ)[cc]tl_ w

subject to K =

where Here

[t_] is a diagonal a matrix

consistent

is found

Kt

and sparse(K) = sparse(K ),

weighting which

with the measured

(3.12)

matrix with oq as a weighting is close

data, subject

value for the ith measured

to the original

stiffness

to symmetry

and sparsity

matrix

and

constraints.

which

mode.

is nearly

The weighting

22

===

values,

% are intended

measured w

data by providing

problem

must be solved,

def'mite,

allowing

values to express Kammer problem



L

_.

a confidence

although

[R-90] presented

with the philosophy

the structure iterative

nonzero

elements

to minimize

data

algorithms.

is a difficult

of the previously

in the stiffness

the measured

of the linear

a sparsity-preserving

by creating

from the initial model

value for each measured

the mode shape confidence

is preserved

constraint

the information

the use of efficient

sparsity

using

L

to balance

matrix.

and the original

the difference

model

between

is symmetric

and positive

of appropriate

matrix second

stiffness

update

weighting

that approaches

viewpoint. elements

from the unknown,

with subsequent

and corrected

the

In his method,

the norm of the error vector

is minimized,

the original

Here too an auxiliary

task, though.

presented

In his method,

system Selection

stiffness

a vector of unknown

mode.

with that from the

stiffness

generated

addition

of a

matrices.

i1¢

L

.

w

E_

23

4.0 New Developments

in Optimal Matrix Adiustment

w

In this chapter,

the second

viewpoint

is adopted

for the development

of new techniques

for

w

property

matrix

adjustment.

R-191 and R-192,

both supported

These techniques However,

using

constraints.

data

ideas

follow

These

where the measured

in a general

for specific

modal

of the developments

can be found

in References

by this grant.

are presented

they can be applied

adjustment

situation

Further details

format, using A to represent the property

property

(frequencies

and

matrices, mode

shapes)

the early ideas of Brock

data presents

as with the case of stiffness to form

[R-133],

a set of constraints

the

matrix. matrix

measured-data

but here we consider

that number

the

less than the model

w

order (p < n). w

The matrices

second

viewpoint

for in the optimal-update

which solve the problem

of Equation

approach

starts

with a set of symmetric

(3.2), repeated here for reference,

min IIAS - YI_ subject to A = At. S and Y are nxp matrices A. The columns --=

A symmetric

that use the test data to define the p constraints

on the property

matrix

of S are independent. matrix A which solves the problem

above satisfies

the Sylvester

equation

w

ASS t + SStA = ys t + Sy t,

(4.1)

w

and can be determined

as

A = UBIf,

(4.2)

where m m

U is a unitary matrix of the left singular S = LIZV t, and B is an nxn symmetric B is defined

vectors

of S, from the singular

value decomposition

matrix.

by

r E

24

C,,

b.. = -..-..K..--

(4.3)

for each id with min(i,j__p,

w

where

{oi}iP 1 are the ordered nonzero

singular values

of S and Cij are the elements

of

C = Ut(YS t + SY_U.

Details of the proofs An alternate

(4.4)

associated

expression

with this result are included in R-192.

of the solution follows

from partitioning

the matrix B as

(4.5)

using B = U'AU and a partitioned Bm_

range of S. With respective

U, U = [U_ I U2], with U_ as the nxp submatrix

partitioning

that spans the

of C,

B lldiag(oj z) + diag(°i 2)Bn = Cl1'

(4.6)

or elementwise, 2 (Bll)ij_

2

j + Oi (Bli)ij

= (el

l)ij.

In addition, diag(oiZ)Bt2 = Ca2. Several

observations

the stated

problem

information

from

imposition

(4.7)

stem from this development.

are nonunlque, the measured-data

def'med

constraints.

The matrix

constraints,

such as closeness

Note that the conditions

for positive

definiteness

identification

problem.

The desired

is provided

property

matrices

Bal and

to an original

model,

positive definite for another

which

B12 which

B22 is arbitrary

of A are that B u is positive

is chosen so that Bzz = D + Bt_2BiatB_2for an arbitrary understanding

property

only by the matrices

of additional

With this observation,

Updated

solve include

at this point,

but

fix B22 as well as A. definite

and that Bz2

matrix D.

very different

approach

matrix A and the measured data, represented

to the in S

and Y should satisfy 25

AS= V.

(4.8)

=

so a set of linear equations can be constructed

m

imposed).

Listing these unknowns

for the n(n+l)/2

unknowns

in A (once symmetry

in a long vector, _i,the linear system can be represented

is

as

Xi = b,

(4.9)

=

where X is an n(n+l)/2 b is an n(n+l)/2

x n(n+l)/2 matrix, and column vector,

w==_ m

both assembled

with appropriate

entries

which satisfy this equation

becomes

a least

Even

to replicate

computationally

Equation extensive

(4.8).

Finding

the values

of

for large models and may require

m

L

_

squares

characteristics available

from

that results

definite

this

generalized

arbitrary

or even positive

model can he presented

If

(4.2),

is obtained,

definiteness

the

solution

inverse, ii = X&dagger.

and of

this

no information

closeness system

B22 precludes

is obtained

b, then the assembled

semidefinite.

For applications

is an important

formulation, as an additional

the property

additional model,

through

is the

A would he the same A choice

matrix

to structure

about

to an original

(4.6) and (4.7) with the particular

submatrix

or semidefiniteness

In the optimal-update

positive

approach.

from Equations

for the

definiteness

if the solution

of A, including

Moore-Penrose

choice

solution.

of B22 = 0.

A from being

property

matrices,

This

positive positive

characteristic.

the addition of a priori information

in the form of an initial

constraint

rain IIAS - YIIF subject to A = At and IIA- A, IIF minimal.

n

Now, the solution is unique

and given by A = UBU _ for B as defned

for B_ and B_2, respectively, 1

t

(4.10)

by Equations

(4.6) and (4.7)

and with

t

B22 = _-U2(Z ' + Z)U 2,

(4.11)

m

m

26

1 w

+ z'o).

whereA.= Finally,

adding

a condition

for positive

semidefinitcness

results

in a formulation

for the

i

optimal-update

problem

as

rain IIAS- YIIF subject to A = At, A positive scmidefinite, w

and IIA- A°IIF minimal.

_mm

Here, A = UBIf

(4.12)

for Bll and B12 as defined by Equations

(4.6) and (4.7), respectively,

while

t -i 1 t t -1 B_ = BI2BnBt2+ pos(_-lf2(Z ' + Z)U 2- BI2BnB12),

(4.13)

whcrc pos(W) isdefinedas the positivcpartof thenxn matrixW.

factorization of 2(W

If UwAU w is the spectral

+ W _)withA = diag(_),then

w

= with 1]k---max(O,Xk) fork=l .....n. Final observations matrices

include

which result from this second

the ability to estimate

in the measured

(4.14)

the sensitivity

viewpoint

on optimal

of the solution

data S and Y. The closer the columns

updates

matrices

of S arc to being

for property

A to perturbations

an orthogonal

set, the

v

more stable thc solution is to perturbations The requirement L

LJ n

but for stiffness sparsity

for a specific

matrix adjustment

preserving

sparsity Kammer

in the data [R- 192]. pattem

was not included

in these new developments,

JR-90] has adopted this viewpoint

and incorporated

a

technique.

L_

w

27

W

---2

5.0 Evaluation Studies

w

In this section secant-method

we present

and optimal-update

all of the references arc summarized

5.10rthogonal

supported

identification. by the grant.

Further

studies that have been conducted details of these studies

In particular,

References

are included

R-i 12, R-107

for in

and R-109

Expansion

Mode shape expansion truss to enable damage the unmeasured

of the evaluation

below.

Procrustes

particularly

a summary

is necessary

location

degrees

to a specific

of freedom

to optimal-update

to preserve

the structure

truss element.

is critical

identification

of the finite element

The process

to the performance techniques

which

model of a

used for the estimation

of the identification

are sensitive

of

algorithm,

to inconsistencies

in

w

the mode shape data. Optimal

w

related w

mode

shape

to the problem

updates,

new expansion

grant as well. evaluation

expansion,

Reference

R-112

Orthogonal-Procrustes

for

structure

techniques 1.

presents

techniques

the details

Within

identification.

with the following

This

is

a

technique

which

compared

are

of optimal

under

this research

and the results

of the

problem

were performed. simultaneously

the mass-orthogonal,

was

framework

studies with a small spring-mass

technique

to produce

the

orthogonalization

were developed

data from a laboratory truss structure expansion

shape

of this development

Both simulated

the mode shape vectors

Orthogonal

and mode

identification.

and orthogonalization

studies of these techniques.

orthogonalizes

reduction

of optimal-update

and studies with experimental

w

model

expands

full mode shapes

to several

previously

and needed

published

results:

Procrustes

expansion

which were calculated

efficiently,

produced

estimates

of the

full mode

shape

vectors

!

simultaneously

and ultimately

orthogonal

with respect

to the model mass matrix.

28

t_lui,"

Dynamic

o

expansion

nalization

r

using the technique

and Bar Itzhack

tions are required

to produce

expansion

subsequent

orthogonalization.

preserving

a selection

The first problem

is a widely-used

structure

identification.

updating

for a local discrepancy,

results

Extensive

that

computa-

of freedom

that resulted

versions analogous

from Orthogonal-

with dynamic

exapnsion

of inconsistent

and

data to

Problem algorithms

algorithms spring-mass

on a small problem, Alternate

orthogo-

techniques.

of identification

of investigating,

produced

expansion.

were not as frequent

of the identification

purposes

[R-54],

These sign errors present a problem

optimal-update

characteristics

for comparison,

degrees

in some cases

Studies with a Spring-Mass

To illustrate

Procrustes

with subsequent

a large matrix inverse, though.

in the small-displacement

Procrustes

5.2 Evaluation

!

of Baruch

to those using Orthogonal

sparsity

lems.

by Berman and Nagy, R-66)

were comparable

Sign errors

o

(as described

and to provide

are applied example

[R-85].

original

to a damage

prob-

It was used here for the

problem

detection

examples

to two demonstartion

ideas for mode shape expansion of Kabe's

common

and techniques

were used,

for

to examine

situation.

i

Kabe's eight degree-of-freedom

spring-mass

problem

is shown in Figure

5.1, which includes

6_

the stiffness

and mass values

for structure

identification

for the exact model.

This problem

presents

a chalianging

situation

are included.

Closely-

w

m

spaced frequencies Variations

u

___.

are exhibited

of Kabe's

which had incorrect

in that stiffness

values

studies were only incorrect

original

values

of various

magnitudes

as well. problem

were used.

Rather

for all of the connecting

springs,

for the spring between

two masses,

than the initial

initial models

model

used for most of the

i.e. the spring between

and 5, for example.

A value of 500, five times that of the exact spring, was assumed

The large difference

in the initial value is more representative

because

often the initial system experiences

of a situation

a large, local reduction

he used,

of damage

of stiffness

masses

3

in this trial. detection,

when damaged.

w

29 L

w

L

w

m

E T_

==

i

F=;

k, = I000

m l = 0.001

k z - I0

m s - 0.002

k 3 = 900

m i = 1.0

]=

2,7'

k4 = I00

w

k 5 = 1.5 i

k 6 = 2.0

Figure

5.1

Kabe's

Spring-Mass

Problem

3O

Z

]brat

Several

ltt¢

Expansion

different

studies were performed

techniques

were evaluated

with the various

and reported

trial situations

in reference

R-112

of Kabe's

as one.

problem.

Mode

selection

r

studies for best identification publications 1.

supported

Mode

performance

by this grant.

selection

A summary

was investigated

data in the identification

t-_

ing judgement

respect to those predicted

that exhibited

w

correlation

as a means

of available

was accomplished

the largest

frequency

as

by engineerdifference

Modes were also selected

by frequency

difference

But often they are not. of selection

through

and selected

with the use

is indicated

as a promising

option

modes that contain extreme errors also may be poorly correlated.

approach

which

frequency

differences

In

by mode shape

In the smaU studies, use of mode shape

However,

uses

modes

actually lack of it, with respect to the initial model modes.

many cases, these two sets - selected - are identical.

combinations

At first mode selection

modes

in several

of some of the results follows:

by the intial model.

of mode shape correlation,

difference

Results of these studies appear

by using various

process.

- selecting

was another.

and

mode

shape

in many

cases.

A combined

differences

should

be

explored. 2.

Related

to the idea of mode

selection

degrees

of freedom

to have the best data for identification.

ies, arbitrary

w

to measure

selection

of measured

issue is one of major

importance

is the problem

degrees

of freedom

for identification

of sensor

placement,

or which

In the small stud-

led to the conclusion

performance.

that this

Optimal-update

tech-

B m

niques

are sensitive

unwise

measurement

to mode shape information locations seems

common.

and poor mode Research

shape data as a result of

and more practical

experience

is needed in this area. 3.

Initial model

correlation

there was no indication exception r

of the sparsity

the application

was studied

as well.

For these small

that the initial model of the stiffness pattem.

of fairly accurate,

identification

matrix

problems

is critical,

with the

Large and varied intial model errors are identified full modes.

However,

use of the initial model

with

to esti-

31

mate the full mode placement

A

F'maUy,

,

shape vector

does lead to poor performance

in some cases.

Sensor

is a critical factor in this result as well. data accuracy

algorithms

tolerate

sparse updates,

was examined

up to 10 percent

one symptom

in several

studies.

Generally,

the identification

random noise added to the mode shape vectors.

of poor data is a non-positive-definite

For

result.

5.3 Evaluation Studies with a Laboratory Truss A focus of the research

effort was to apply these techniques

to the problem

of identification

w

and damage described

detection

of a laboratory truss structure.

in detail in references

tion to be a journal

R-107 and R-109.

Experiments

A final summary

and preliminary

results

are

of these results is in prepara-

publication.

w

32

zJ

6.0 Summary

Efforts

for development

produced

understanding

matrix

adjustmenL

lations

for updating

response niques

w

the property

of these techniques understanding

damage

problem

in simulated

of a model

and examined. in which

resulted

from

problems,

the development as a result.

inconsistent

modal data.

placement,

among others, is needed to produce

dynamic

result is that these tech-

constraints

detection

can be specified

for a

is still in progress,

efforts

In particular,

and

Inconclusive

in comparison

has shown the need for further

for handling,

measured

formu-

for example.

truss structure,

techniques

have

or optimal

optimization

using

important

of damage

efforts were defined

for the laboratory

Constrained

multiple

identification

identification

of a structure

Another

for the problem

for structure

optimal-update

i.e. control gain calculations,

under this grant but additional age location

methods

we termed

to other problems

problem,

Improved

area

matrices

been developed

are also applicable

Evaluation ever.

in a broad

of secant

This is the major result of this grant effort.

data have

matrix updating

and evaluation

evaluation results

howstudies

for the dam-

with the ability to locate

understanding attention

of, and improved

to the problem

the best mode shape information

of sensor

for the identifi-

cation process. With the understanding

and experience

developed

in this research

effort, further

advances

for

U

the application knowledge

of damage

location

base is experience

are possible.

with various

At this time, the most important

real-data

situations.

This, in concert

addition

to the

with understand-

m

ing of other approaches fication

i

algorithms

for structure

for damage

identification,

will enable eventual

implementation

of identi-

location.

B

During

the course

the evaluation

studies.

of this research, Cameron

an undergraduate

A. Bryant will graduate

researcher

was employed

with a BSCE,

May 1991.

to help with

j--

W

33

w

As a result of this research

effort, ten publications

have, or will, appear.

In the following

chronological

Abstracts

sec-

V

tion, these publications

are listed

by type in reverse

order.

are repro-

duced where they are available.

=

w

r_

v

! w

E

34

r

Publications

v

under Grant NAG- 1-960

ip--_ v

Journal Articles Smith, S.W. and P.E. McGowan, "Locating Damage submitted for consideration to the AIAA Journal



in a Truss Using Modal Tests," to be

2,

On-orbit assessment of large flexible space truss structures can be accomplished, in principle, with dynamic response information, structural identification and model correlation techniques which produce an adjusted mathematical model. In this approach for damage location, an optimal update of the structure model is formed using the response data, then examined to locate damage members, An experiment designed to demonstrate the performance of the on-orbit assessment approach uses a laboratory truss structure which exhibits characteristics expected for large space truss structures. Vibration experiments were performed to generate response data for the damaged truss. This paper describes the damage location approach, analytical work performed in support of the vibration tests, the measured response of the test article and damage location results.

_--L'

.

Identifica-

Problems of model correlation and system identification are central in the design, analysis and control of large space structures. Of the numerous methods that have been proposed, many are based on finding minimal adjustments to a model matrix sufficient to introduce some desirable quality into that matrix. In this work, several of these methods are reviewed, placed in a modern framework, and linked to other previously known ideas in computational linear algebra and optimization. This new framework provides a point of departure for a number of new methods which are introduced here. Significant among these is a method for stiffness matrix adjustment which preserves the sparsity pattern of an original matrix, requires comparatively modest computational resources, and allows robust handling of noisy modal data. Numerical examples are included to illustrate methods presented herein.

r

v

w

Beat-tie, C.A. and S.W. Smith, "Optimal Matrix Approximants in Structural tion," Journal of Optimization Theory and Applications, to appear.

.

Smith, S.W. and C.A. Beatfie, "Secant-Method Adjustment AIAA Journal, Vol. 29, No. 1, January 1991, pp. 119-126.

for Structural

Models,"

h m w

7=-

On-orbit testing will be required for final tuning and validation of any mathematical model of large space structures. Identification methods using limited response data to produce optimally adjusted property matrices seem ideal for this purpose, but difficulties exist in that application of previously published methods to large space truss structures. This article presents new stiffness matrix adjustment methods that generalize optimalupdate secant methods found in quasi-Newton approaches for nonlinear optimization. Many aspects of previously published methods of stiffness matrix adjustment may be better understood within this new framework of secant methods. One of the new methods

v

preserves realistic structural connectivity with minimal storage requirements and computational effort. A method for systematic compensation for errors in measured data is introduced that also preserves structu_ connectivity. Two demonstrations are presented to compare the new methods' results to those of previously published techniques.

35

x.-.

It

:

Conference .

Proceedings

(* denotes presenter)

Smith, S.W.* and C.A. Beattie, "Optimal Identification Using Inconsistent Modal Data," to be presented at the AIAAJASMFJASCE/AHS/ASC 32nd Structures, Structural Dynamics and Materials Conference, Baltimore, Maryland, April 8-10, 1991. Structure identification is the _ss oTusing measured response to produce a mathematical model that accurately represents the structure's dynamic characteristics. This capability is essential for large space structures which can not be fully assembled and ground tested before their assembly 0n-orbit. Consequently, various approaches have been adopted to address this need and within each of these general approaches, numerous techniques have been developed. This work examines techniques under the general approach of optimal-update identification which produce optimally adjusted, or updated, property matrices (i.e., mass, stiffness and damping matrices) to more closely match the structure modal response. For practical applications, the techniques must perform when the modal response is inconsistent with other constraints on the desired model. Here we present an alternate view of the optimal' update problem that leads to new techniques for addressing inconsistent data. Viewpoints Used for previously published technique, s are also examined to explore in optimal-update identification.

w

W

.

r

Beattie, C.A.* and S.W. Smith, "Matrix Approximation Structures," Proceedings of the International Conference tural Mechanics, Catania, Italy, July 2-5, i_.

in the Identification of Vibrating on New Developments in Struc-

w

Numerous methods have been proposed f, _: the adjustment of analytical models of vibrating structures in order to inc0_rate experimentally obtained information. Often this involves a modification to a model property matrix that is minimal in some appropriate sense, yet sufficient to induce consistency with experimental data. A naive formulation leads to the solution of a large unsXmctured linear system requiring (.potentially) an enormous amount of computation and_ding no insight into other qualities one may wish to preserve in the material property matrix. We provide a path of development here that maintains the matrix theoretic aspects of the approximation problem and permits a substantial reduction in computational complexity while providing insight into circumstances that yield positive definite approximants. We show how these techniques can be applied to identify structural damping matrices and provide a simple illustrative example.

i-

ra

m W

E

i m

.

r

E

r

Smith, S.W.* and C.A. Beattie, "Simultaneous Expansion and Orthogonalization of Measured Modes for Structure Identification," Proceedings of the AIAA SDM Dynamics Specialist Conference, Long Beach, California, April 5-6, i990, pp. 261-270. Tests of large structures on-orbit will be performed with measurements at a relatively few structure points. Values for the unmeasured degrees of freedom (dofs) can be estimated based on measured dofs and analytical model dynamic information. These "expanded" mode shapes are useful for optimal-update identification and damage location as well as test/analysis correlation. A new method of expansion for test mode shape vectors is developed from the orthogonal Procrustes problem from computational linear algebra. A subspace defined by the set of measured dofs is compared to a subspace defined by modes shapes from an analytical model of the structure. The method simultaneously expands and orthogonalizes the mode shape vectors. Two demonstration problems are used to compare the new method to current expansion techniques. One demonstration uses test data from a laboratory scale-model truss structure. Performance of the

i

36

k..

new method is comparable or superiorm that of the previous expansion require separate orthogonalization.

methods

which

F_ w ,

McGowan, P.E.*, S.W. Smith, and M. Javeed, "Experiments for Locating Damaged Members in a Truss Structure," Proceedings of the 2nd USAF[NASA Workshop on System Identification and Health Monitoring of Precision Space Structures, Pasadena, California, March 27-29, 1990. This paper summarizes experiments performed to demonstrate and evaluate a previously developed method for locating damage in truss structures. A review of the damage location method and its application to large space truss structures is presented. Two focus test articles, which have been developed for the Dynamic Scale Model Technology (DSM'D project at NASA Langley, are described. Analytical studies show the sensitivity of a complex, dynamically scaled model Of space station to damaged members. Vibration experiments were performed on a scale model truss section with and without damaged members to generate modal data required for the damage detection algorithm. Experimental results from different cases are included to validate the damage location method.

w

L

W

i w .

On-orbit assessment of large flexible space truss structures can be accomplished, in principle, with dynamic response information, structural identification and model correlation techniques which produce an adjusted mathematical model. In a previously developed approach for damage location, an optimal update of the structure model is formed using the response data, then examined to locate damage members. An experiment designed to demonstrate and verify the performance of the on-orbit assessment approach uses a laboratory scale model truss structure which exhibits characteristics expected for large space truss structures. Vibration experiments were performed to generate response data for the damaged truss. This paper describes the damage location approach, analytical work performed in support of the vibration tests, the measured response of the test article and some preliminary results.

m

.

m

m

:

W

u

:"N

Smith, S.W.* and P.E. McC-owan, "Locating Damaged Members in a Truss Structure Using Modal Test Data: A Demonstration Experiment," NASA-TM-101595, April 1989. (also presented at the 30th Structures, Structural Dynamics and Materials Conference, Mobile Alabama, April 1989)

Smith, S.W.* and C.A. Beattie, "Secant-Method Matrix Adjustment for Structural Models," Proceedings of the AiAA/ASME/ASCE/AHS/ASC 30th Structures, Structural Dynamics, and Materials (SDM) Conference, Mobile, AL, April 1989, pp. 1041-1051. On-orbit testing will be required for final tuning and validation of any mathematical model of large space structures. Identification methods using limited response data to produce an optimally adjusted stiffness matrix seem ideal for this purpose, but difficulties arise in their application to large space truss structures. Typically, either unrealistic physical connectivity is produced in the adjusted model or massive storage and computational effort is required to preserve realistic connectivity. This paper presents new methods of stiffness matrix adjustment generalizing optimal-update secant methods from quasiNewton approaches for nonlinear optimization. One of the new methods, the MSMT method, preserves realistic structural connectivity with minimal storage and computational effort, while an extension of the MSMT method allows compensation for errors in

37

measured data. Within the new framework of secant updates, aspects of previously used methods of stiffness matrix adjustment are better understood as well. Two demonstrations are presented to compare the new methods' results to those of previously published techniques. r

Monograph Chapter °

Zirnmerman, D.C. and S.W. Smith, "Model Refinement and Damage Location for Intelligent Structures," to appear in Intelligent Structural Systems, Ed. H. S. Tzou., 1991.

w

Re_ v

= =

= = IE5

38

Bibliokrraphy

R-|°

Henneke, E. G. "Nondestructive Evaluation of Fiber Reinforced Composite Laminates," Proceedings of the l lth World Conference on Nondestructive Testing (WCNIYD, Las Vegas, Nevada, November 3-8, 1985, pp. 1332-1343.

R-2.

Wagdi, M. N., "An Adaptive Control Approach to Sensor Failure Detection and Isolation," Journal of Guidance, Control and Dynamics., Vol. 5, No. 2, March-April 1982, pp. 118-123.

R-3.

Adelman, H. M. and R. T. Haftka, "Sensitivity Analysis of Discrete tems," AIAA Journal, Vol. 2, No. 5, May 1986, pp. 823-832.

R-4.

Arora, J. S., D. F. HaskeU and A. K. Govil, "Optimal Design of Large Structures for Damage Tolerance," AIAA Journal, Vol. 18, No. 5, May 1980, pp. 563-570.

R-5.

Batla, F. A., "Methodology of Design and Analysis of External Walls of Space Station for Hypervelocity Impacts by Meteoroids and Space Debris," Marshall Space Flight Center Research Reports, N86-24512, August 1985.

R-6°

Dorsey, J. T., "Structural Performance bers," NASA Technical Memorandum

R°7°

Salama, M., T. Rose, and J. Garba, "Optimal Placement of Excitations and Sensors for Verification of Large Dynamical Systems," Proceedings of the AIAA/ASME/ASCE/AHS 28th Structures, Structural Dynamics and Materials Conference, Monterey, California, April 1987, pp. 1024-1031.

R-8.

Gran, R. and M. Rossi, "On Orbit Testing for Large Space Structures," Proceedings of the AIAA 17th Aerospace Sciences Meeting, New Orleans, Louisiana, January 15-17, 1979.

R-9.

Smith, S. W. and S. L. Hendricks, "Evaluation of Two Identification Methods for Damage Detection in Large Space Trusses," Proceedings of the 6th VPI&SU/AIAA Symposium on Dynamics and Controls for Large Structures, Blacksburg. Virginia, June 29 - July 1, 1987, pp. 127-142.

R-10.

Yao, J. T. P., "Damage Detection using System Identification," SEM Spring Conference on Experimental Mechanics, Houston, 1987 (extended summary).

z

z

Structural

Sys-

[] t

of Space Station Trusses with Missing Mem87715.

E

i

w

Proceedings of the Texas, June 14-19,

39

r

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