Full text of DOI:10.1177/107754630100700507

Journal of Vibration and Control
http://jvc.sagepub.com/
Identification of Two Cracks in a Simply Supported Beam from Minimal Frequency
Measurements
Antonino Morassi and Matteo Rollo
Journal of Vibration and Control 2001 7: 729
DOI: 10.1177/107754630100700507
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ofTwoCracks in
a
Identification
SupportedBeam
Simply
Measurements
Minimal
From
Frequency
MORASSI
ANTONINO
MATTEOROLLO
DepartmentofCivil
30March
Via delle Scienze
33100
Udine,
208,
Udine,
Engineering,
University of
2000)
Italy
5
December
2000;
(Received
accepted
a
Abstract: This
for the identification of
two cracks
of
in
paperpresents diagnostic technique
equal
severity
beamunder flexural vibrations. Thecrack
Theanalysis
a
is simulated
a
rotational
simply supported
by
expression
ofthe
spring
connecting
ofthebeam.
is basedonan
thetwo
of the
sensi-
adjacent segments
explicit
frequency
the damagedsystem
perturbation
to
and
is considered as a
damage,
virgin system.
Bymeasuring
tivity
the
the first three natural
it is
to
the
inverse
of
frequencies,
possible
study
namely,
changes
problem—identification
and
The inverse
ofcrack location
tions cracks
is
even
severity.
problem
ill-posed;
by leaving symmetrical
posi-
with different
in two sets
ofdifferent locations can
identical
in
aside,
severity
produce
changes
Numerical results show
small with
if the
the
the first three natural
tification are affected
that
to
natural
used as
in
iden-
data
then
frequencies.
errors
frequencies
by
relatively
respect
frequency-induced
changes,
damage
leads to
results.
identification
satisfactory
Words:
inverse
cracks
detection,
beams,
Damage
bendingvibrating
problems,
Key
INTRODUCTION
1.
in
a
paper, weseekto detecttwo
cracks of
beam
In this
equal severity
simply supported
open
aminimumnumberof
measurements.
from
Despite
1997
frequency
extensive literature on
identification
most ofthe
the
Ruotolo, 1997;
(see
very
for recent and
damage
work
Salawu,
complete state-of-the-art),
previous
dealing
present.
multicracked
if
a
crack is
cracked beamsconsiders
andlocation
the damagesizing
with
just
single
in
haveturnedtheir attention
assessment
to damage
researchers
Only recently,
was considered in Ostachowicz and Krawczuk
and in
(1991)
The direct
beams.
problem
(1999).
the natural
and Krawczuk
studied the effect oftwo
and
Ostachowicz
frequencies
Shifrin
Ruotolo
(1991)
of the flexural vibration
for
an efficient
in
a
cantilever beam.
the
cracks
upon
open
and Shifrin
Ruotolo
technique
solving
eigenvalue
(1999) presented
The
multicracked beam.
inverse
vibration of
a
related to the free
problem
bending
identification in multicracked beams
problem
of
in
and
hasbeenconsidered
and Vestroni and
Hu,
Liang,
Choy
damage
and
and Surace
a
Ruotolo
Hu,
(1997);
diagnostic technique
Capecchi (1996). Liang,
(1992);
based on
to localized
frequency sensitivity
beamstructures. Ruotolo and Surace
developed
(1992)
Choy
in
and
cracks
for
detecting
assessing multiple
damage
Downloaded from jvc.sagepub.com at CLEMSON UNIV on November 19, 2014

730
terms and used
a
the inverse
in
formulated
solution
optimization
in cantileverbeam.
Withthe
diagnostic
(1997)
problem
to
a
twocracks
algorithms
ofthe
procedureemployinggenetic
identify
aim of
the
Vestroni and
reducing
indeterminacy
diagnostic problem,
Capecchi (1996)
a
identification
of variational
that is based on the
a
presented damage
information that the
procedure
type
priori
in
is located
a
few sections of the beam.
damage
only
deals withthe identification oftwo
uniformbeamfromthe
The
small
cracks of
open
presentpaper
simplysupported
equal severity
in
first
knowledgeof the
in
a
shifts
damage-induced
As in Freund and Herrmann
lower
crack is simulated
every
(1976),
bending frequencies.
rotational
ofstiffness
an
by
massless
the two
of
K,
spring,
cross section.
equivalent
connecting
segments
the beam
to the
that the
is
damaged
Assuming
adjacent
undamaged system
three
need to be
the
K
stiffness
of
as
known,
determined,
namely,
completely
only
parameters
S2
the
andthe abscissas
of the crackedcross sections.
weconsidered
this set of
Therefore,
Sl,
spring
a
minimal set ofdatathe first threenatural
of the beam.
data,
frequencies
Byusing
by leaving
symmetrical
the
even
is
solutions
namely,
diagnostic problem
generally ill-posed,
andofdifferent
In
can
identical
still produce
ill-posedeness,
cracks in different locations
first three natural
aside,
in the
severity
changes
of this
the effects
of the
K
and
frequencies.
spite
dramatic,
cos
are notso
becauseit is foundthat the
of the solution
stiffness
ofthe
nonuniqueness
(whereL
the location variables x
cos
is the
length
27rs11 L,y
27rS2/L
beam)
in
all
cases,
of
ofsecond
are solutions
closed form
Therefore,
degree.
in
terms ofthe data. Our
polynomial equations
K
for
and
is
are deduced
ofthe
based on
an
x, y,
analysis
expressions
to
derived
Morassi
and
by
sensitivity
frequency
damage
(1993),
expression
explicit
Narkis
and
Morassi
extension of
ofthose obtained
our results are an
by
(1994)
part
crack identification
(1999)
Someof the results are also valid forcracked
rods
forthe
problem.
single
vibration.
in axial
the
onsimulatedcrackedbeams
methodfor
tests
supported
proposed
Dynamic
performed
diagnostic
in
Numerical
situations.
results show
or
if
that
ofthe
the solution
the natural
errors
problem
practical
usedas data in identification are affected
(model
by
measurement)
the
frequencies
induced
small with
tothevariationsof the
frequencies
by
relatively
respect
leads to
damage,
results.
identification
then
satisfactory
damage
DAMAGE
SENSITIVITYTO
2.
FREQUENCY
in
is
a
this
which will be
The
model,
paper,
located
mainly investigated
simply supported
physical
at
cracks
cross sections
beam with two
of
Euler-Bernoulli
uniform
of
equal severity
L
where
is the
ofthe
length
and
We assume that
cracks remain
0
<
<
S2
s,
< L,
the flexural
abscissa sl
s2.
crack is
every
beam.
Assumingthat
vibration,
during
spring,
alwaysopen
rotational
massless
as in FreundandHerrmann
As
a
(1976).
the
inserting
represented
by
in
a
K
the stiffness
ofthe
can berelated
precise wayto
geometry
Denotingby
it is
well-known,
spring
example, by
the material and
and
of the
for
as
Paipetis
Dimarogonas
(1983).
suggested,
damage,
Young’s
the volume
y
the mth
E
the
modulus of
C02 dm m
mass-density,
eigenpair
by
bending
the
vibrations of the cracked beamsatisfies
of
Vdm -
1, 2, ... ,
(s) ,
(wm
the
value
boundary
problem:
following
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731
where
thejumpconditions
cross
of
hold at the
sections
abscissa s =
and s
s,
wherecracks occur. In
the
section of
denotes
S2
momentofinertia andthe
equations
I
A
the
and
area ofthe cross
the
above,
represent
equations (3) through (4),
at s. The
beam,
In
of
(~(s)~ - (~(s+) - ~(s-))
to K ~
the jump
respectively.
the
00 or E m
0.
function 0
undamagedsystem corresponds
1/K ~
Ifcracks are
is small
thenwe
the
mayfind
first-order
variation
small,
namely, E
enough,
ofthe natural
with E as shown in Morassi
in
Ruotolo
or
(1993)
frequencies
(5.7)). Bytaking
(1997) (Section
5.2,
equation
that the first variation ofthe mth
is
wefind
eigenvalue
given by
yAwm(s) ds
1
been
has taken into
where the
that the
account. Note
normalizing
condition fo
in
a
natural
a
crack
be
may
andthe secondofwhich
as the
change
frequency produced by
single
expressed
of
the first
location
is
two
of which
ofthe
tothe
terms,
onthe
proportional
severity
this second term is the of
square
product
In
only
bending
depends
damage.
particular,
moment
the
the
of
beam evaluated at the cracked cross section. We
for
mth mode
in the
shape
undamaged
expression
thedamage
sensitivity ofnatural
will see inthe nextsectionthatthe
(6)
explicit
we observe
role in our
of
that the
of
a
crucial
analysis. Finally,
assumption
plays
damages
frequencies
small
of the methodto cracked
the
that
confines
ofthe
configurations
range
application
one.
this is not
to
a
severe limitation because in
a
are
However,
undamaged
perturbation
practical
in
position
the damageright
situations it is crucial to be
as it arises.
most
identify
THEORETICALRESULTS
3.
of the
cracksandtheir
now
the
the
Wecan
s2
sl,
natural
of identifying
changes
severity
pose
problem
positions
in
ofthe
the lower
of
beam. Since
the
K
the
from
knowledge
frequencies
Downloaded from jvc.sagepub.com at CLEMSON UNIV on November 19, 2014

732
~F
to
three
needto be
it is reasonable
to what
extent the
determined,
investigate
only
parameters
for
measurement ofthe first three natural
can be useful
the
identifying
crack
located
at
frequencies
damage.
any one
a
is
with
to s
therefore
The
and
symmetrical
system
setof
respect
L/2,
in
will
natural
of
a
identical
changes
ofthe
symmetricallyplacedpoints
produce
frequencies.
assume
follows
without
the character of
we
It
can
that,
analysis,
affecting
generality
to
that the two cracks are located on the interval
the left
halfof
the
corresponding
Cm
by
(0, L/2)
for
Let us denote
the
0
<
<
s,
s2 <
beam,
example,
quantity
L/2.
wherem >
1
is an
and B is the constant
integer
Theeigenpairs
the
vibrations are
simply supported uniform beam in bending
of
m = 1, 2, ....
Inserting
following system
of
for m =
into
the
we obtain
the
1, 2, 3,
equation (6),
expression
wm(s),
three nonlinear
of
equations:
with
to
Notethat
since 0
>
<
<
to be solved
Cm
s,
s2 <
0, m
1, 2, 3,
respect
(Sl, s2, K).
can rewrite the
we
standard
identities,
system ( 11 a)
through
By using
trigonometric
L/2.
(lie)
the
form:
in
following equivalent
where
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733
one-to-one
are
Since
and
functions x
x
0
<
<
s2 <_
s,
correspondences
y
= y
L/2,
the
(~i)
(s2 )
identities X3
and
+
+
By using
algebraic
+ y3 =
y2 -
x # y.
(x
y) (x2
xy)
+ y2 )
in
and
and
( 12c),
by
expressing
2xy
means
equation
(x + y)
(x2
by
(x + y) 2 - (x2 + y2 )
we can deduce the
of
(12a) through (12b),
following polynomial equation
equations
in
the unknownK:
-i
L
thetrivial solution
K
0
does not
to the
By
(which
clearly
neglecting
correspond
assumption
ofsmall
we now
that the
ofsecond
solutions
enclosed in
for
every
damage),
prove
polynomial
degree
square
brackets in
has tworeal
set of
data
KZ
(14) always
equation
positive
showing
positive.
Kl ,
that
we
start
>
that the coefficient of Kis
C1
0,
Recalling
C2,
(Ci,
C3).
negative
coefficient ofthetermoforderzero
In
andthe
is
from
fact,
equations
(12a) through
wehave
( 12c),
a
shows
that
E
c
for x E
Moreover,
and y
and y
simple computation
~-1,1 ) .
[- 1, 1).
~-1,1 )
for x
weshowthat the discriminant A ofthe
c
Finally,
polynomial
following:
~-1, 1 )
1
is
in the variable K
nonnegative:
for
wehavethe
the
In
C3,
fact,
( 12c)
Cl , C2,
I-
byusing
expressions (12a) through
~2
where
with
to the variablex:
Weorder f (x, y)
respect
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734
that
Observing
as follows:
wecan
rewrite f (x, y)
that
is,
is
a
which
nonnegative quantity.
conditions
clearly
and(17), wecan concludethat there exist two
into account
(15), (16),
Taking
in
roots of the
thevariable
that
there existtwo
real
K,
is,
equal)
polynomial
positive (possibly
the
ofthe stiffness
Kofthe
values
simulating
damage:
spring
and -
1
2
to
+
where indexes and
sign
of Kinto
sign, respectively.
correspond
(24)
we
canlocalize
the
equations ( 12a) through( 12b),
inserting
expression
therole of variables
By
x
in
and y
system( 12a)through( 12c) is completely
Notethat
the
damage.
a
the
then
is
solution of
is
diagnostic problem,
(~ y, x)
By
using equations
interchangeable, namely, if (I~ x, y)
variable x.
to determine the
the
solution.
a
it is
also
Then,
position
enough
of
second
in
wecan deduce
equation
degree
following polynomial
through (12b),
(12a)
the variable
.
x:
real
has two distinct
in
the discriminant A is
fact,
in
The
roots;
(25)
equation
polynomial
strictly positive:
thattworoots
to
of
to the
interval
To
because
and then
X2
(25) belong
damage
prove
xl,
equation
x #y.
(-1, 1),
reasonable
locations
of
definition
(see
they correspond
physically
(13)),
ofthe
that the value
it is
to
evaluated at
polynomial
equation
(25)
(i)
enough
verify
g(x)
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735
x
= -1
andx
and
1- is nonnegative
and
hasminimum
positive, respectively,
(ii)
thatg(x)
wehavethe
atxm;n E
(12a)
Recalling equations
through (12c),
(-1, 1).
following:
and
the condition
is
satisfied.
for
A
c
E
direct
( 12a)
(ii)
Then,
at xmin
every x
and y
(i)
(-1,
1
)
(-1, 1 ) .
1 -
shows
has minimum
E
KCI. Then,
computation
using equation
and
that g(x)
and
that
x
C
it turns out that
xmin E
is
recalling
and y
(-1, 1)
[-1, 1),
(-1,1)
wecan evaluate
Hence,
proved.
1
2
and -
before the
square
where indexes and
+
root,
respectively.
sign
represent
sign
set ofsolutions ofthe
through( 12c) with reference to
the
( 12a)
Finally,
complete
system
is
located onthe left halfofthe beam
cracks
given by
the stiffness of the rotational
the crack
In
once the value of
is
fact,
spring simulating
possible damage
let’s
wecan evaluate the
locations
determined
K,
say
(24)),
expression
(via expression
let’s
say
xl, wecan evaluate
It turns outthat
Given one
crack via
ofone
x-unknown,
via
X2
(29).
xl,
the
value ofthe
let’s
( 12a).
equation
say yl,
y-unknown,
corresponding
is
a
of the stiffness
there
ofsolutions
onefixedvalue
for
K,
xi . Then,
problem:
pair
and,
similarly,Y2
ofthe inverse
=
=
are one-to-one
weobtain
for
cos
relations
x
and
y
the
Since
correspondences
2Jrf
cos 2Jrf
with
<
the
to
0
< s
s-variable,
respect
solving
by
L/2,
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736
andthenthere
exists
one
two
These
solution of
s2
coincide,
given
configurations clearly
corresponding
damage
problem
only
to
a
value of
for
inverse
the
That
K,
example,
(I~ sl (K) ,
(K)).
shown that two cracks of same
have
via
we
Kl
(evaluated
is,
severity
expression
(evaluated
identical
located at the cross sections of
abscissa sl
s2
sign)
plus
(24}-with
via
(Kl ) ,
(K¡)
and
in the first three natural
(29)
(13)) producechanges
frequencies
expressions
two cracks ofthe same
induced
minus
to those
located
K2
(expression (24)-with
by
severity
sign)
to
cross sections ofabscissa
and
the
at
s,
S2
(expressions (29)
present
(K2 )
( 13)).
(K2 ) ,
ofthe
it shouldbenotedthatthe
methodcanbe
In conclusion
section,
in an
adapted
small cracks
the mth
vibratingbeamwith free ends. To show
two
identify
observe that
suffice to
this,
axially
of
a
free-free uniform
eigenpair,
are
,11m
0,1, 2, ... ,
(um (s)
(02 m M
beamin axial vibration
.
I
&dquo;
the
vibration
and
same
used for
case to formulate the
crack is
then
the
bending
procedure
repeat
Freund and Herrmann
to
here
a
inverse
(1976),
represented
According
massless
by
problem.
translational
ofstiffness
at
cross
the damaged
on
the insertion of
a
section. If
K,
spring,
andof
namely,Kis
as in
Morassi
cracks are small
large enough,
equal severity,
proceeding
andwith the above
the first-order variation ofthe mth
with
notation,
(1993)
eigenvalue
1/K
is
given by
m
axial force at the cross section ofabscissa
s
in
the mth
is the
the
where
Nm (s)
EA d&dquo;~S
of
beam. In
and
denote
the
axial mode
s,
S2
undamaged
equation (35),
(normalized)
cross sections.
the
of
modes
of the
abscissas
damaged
and
Taking
expression (34)
vibrating
as data the variations of the first three natural
into account
considering
frequencies
m
we obtain
a
ofthree nonlinear
mode is
omitted,
coincident
1, 2, 3),
system
equations
e.g.,
withthe
(rigid
formally
procedure
nowthe
same
( 12a) through( 12c). By
previoussystem
repeating
problem.
case, wecan solve the inverse
used
the
vibration
for
bending
4. APPLICATIONS
the measurementofthe first
it waselucidated howto
three
In
the
section,
employ
beam with two cracks of
preceding
bending frequencies
aswellasthe
so as to
of
a
equal severity
toaccountforthe
simply supported
magnitudeof the damage.
assessthe location
Aiming
practical
the
section is devoted to
within the
ofreal
cases,
use ofthe results above
some
present
analysis
ofnumerical character.
outlining
applications
detection is solved for different
The inverse
of
cases,
using pseudo-
problem
damage
from
that
the
are obtained
the direct
the
three
in
undamaged
problem
data,
is,
frequencies
experimental
defined
conditions
conditions andin some
s2.
K,Sl,
parameters
by
damaged
study
damage
is
a
double
T
steel beam
IPE
the series
with the
stiffness
the case
of
300
beam for
The
and mechanical characteristics:
5
m,
following geometrical
1.721 ~ 107
length L =
42.2
bending
linear mass
and
EI
Nm2,
density p - yA
kg/m.
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737
1.
for
the
Table
beamandtheirvaluesassociated
to thecasesof
Frequencies
(cases
undamaged
Values
in Hz.
free of
/,/undamaged ~100.
0% _
damage
error).
~,fundamaged -,/damaged ~
Table 2.
of
Results
identification
free of
Determination of
locations
the
S2
damage
(cases
and of
error).
stiffness
spring
K (Nm/rad, equation (24))
corresponding
damage
Sl,
and
equations (29)
(13)).
(m,
cases of
several studied
are
Two main different
are
damage among
presented: they
ofthe main features ofthe inverse
and ofthe identification
illustrative
problem
technique.
&dquo;small&dquo;
that
the value ofthe stiffness K
is such
The first case is characterized
is,
by
damage,
frequencies
of the first three
are about
to of the initial values for
1.7%
the variations
0.5%
that
set of
locations. For the other
&dquo;moderate&dquo;
a
different
case,
damage
damage corresponding
considered. In
variations of the same
about 1.7% to 6% is
both
S2
to
cases,
frequencies
for
results are
a
set oftwo
locations:
s,
identification
presented
damage
L14,
L13
Case
ands,
Case
cracks,
(separate
S2
cracks,
(close
S).
C)
L/5,
2L/5
andtheir values associated withthe cases
values for
beam
The
frequency
are shown
the undamaged
in
obtained
in
exactwaythe
Table
as
the latter are
1;
of
bysolving
eigenvalue
damage
in Ostachowicz and Krawczuk
for
shown,
example,
( 1991 ).
( 1 ) through (4),
problem
to
in Table 2. It is
observe that in the
results of identification are
The
possible
the
presented
for the
oftwosolutions
estimate
mathematical
The
of
the
absence
errors,
theory
(30)
by
predicted
pair
real solution ofthe
the
contains
of)
damage problem.
(a satisfactory
problem
which are
dueto the
from the exact
deviations
damageparameters,
exclusively
perturbation
locations and
of
for damage
(6~arenegligible
and
are
order
and this
equations (5)
assumption-see
smaller for less severe
are
for damage severity. Discrepancies
to 15%
4%
behavior
damages,
because
is formulatedonthe
the inverse
is
diagnosticproblem
assumption
expected
is
of the
a
&dquo;small&dquo;
the
that
virgin system.
system
perturbation
analysis
damaged
of
in absence
errors
so
the
as
have
it is well-
We
far,
but,
developed
results
damage
of mostidentification
on
measurement
the
strictly depend
known,
techniques
possible
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738
3. Results of
identification for Case
stiffness
C
with
Table
data.
errors on
the
(24)) andof
corresponding
damage
(small
damage)
Determination ofthe
locations
spring
K (Nm/rad, equation
and
(~,., -~~) //~~100.
(13)). err% f =
equations
SI, S2 (m,
(29)
damage
The
means
solution.
imaginary
symbol
(*)
Table4. Results of
data. Determination
locations
identification forCaseC(moderate
stiffness
errorsonthe
damage
damage)with
of the
andof
equation
err% f =
spring
K (Nm/rad,
(24))
corresponding
and
100.
sl, S2 (m, equations (29)
(13)).
damage
(ferror -,lexact)
oferrors in the
data into
we
account,
somenoise.
To take the effect
errors.
and
experimental
were
modeling
cases in whichthe natural
of
series
considered
a
corrupted by
frequencies
Case C-withsmall and moderate
To
an
Tables
of
3 and 4 refer to the
previous
give
example,
damage-inpresence
orof an alternate
Asa
errors of the same
sign
frequencies
sign.
general
increasing
usedas datain identification are
observethat if the natural
to
it is
remark,
affected
possible
errors
the
small with
to thevariations of
induced
by
by
damage,
relatively
respect
frequencies
results.
in
identification leads to
the inverse
ofthe
the
then
However,
satisfactory
damage
noise in the data is
and
the estimates
the
solution,
usually amplified strongly
problem
errors.
seemto berather sensitive to
input
damageparameters
Downloaded from jvc.sagepub.com at CLEMSON UNIV on November 19, 2014

739
of
5.
In
CONCLUSIONS
this paper, wehavefocusedon
two
cracks
fromthe
detecting
of equal severity
frequencies
knowledge
the
in
the first threenatural
of
a
beam
damage-inducedchanges
simplysupported
under
vibration. The
is based on an
of
the
bending
analysis
explicit expression
frequency
to
and the
is
a
considered as
ofthe
sensitivity
system.
severity
natural
damage
damaged system
by leaving symmetrical positions
perturbation
virgin
It was found
even
cracks
with
different
that,
aside,
in
two sets ofdifferent locations can
identical
in the first
three
with numerical
produce
changes
by comparison
The theoretical
results are confirmed
a
frequencies.
performed
tests
oncracked beams.
NOTE
This
was
out to us
Michele Di
Lena.
1.
argument
pointed
by
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