Aalborg Universitet
Theory of Covariance Equivalent ARMAV Models of Civil Engineering Structures
Andersen, P.; Brincker, Rune; Kirkegaard, Poul Henning
Publication date:
1995
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Andersen, P., Brincker, R., & Kirkegaard, P. H. (1995). Theory of Covariance Equivalent ARMAV Models of Civil
Engineering Structures. Aalborg: Dept. of Building Technology and Structural Engineering, Aalborg University.
(Fracture and Dynamics; No. 71, Vol. R9536).
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(~~
•·INSTITUTTET
.U3
FOR BYGNINGSTEKNIK
DEJ?T. OF BUILDING TECHNOLOGY AND
AALBORG UNIVERSITET
•
AUC
•
FRACTURE & DYNAMICS
PAPER NO. 71 ·
STRUCTURAL ENGIN~ERING
AALBORG
•
DANMARK
Aalborg Universitetsbibliotek
l
530005198401
'
1111111111111111111111111111111111111111111111111111111
l
f
j
To be presented at the 14th International Modal Analysis Conference,
Dearborn, Michigan, USA,. February 12-15, 1996
:P. ANDERSEN, R. BRINCKER & P. H. KIRKEGAARD
THEORY OF COVARIANCE EQUIVALENT ARMAV MODELS OF CIVIL
ENGINEBRING STRUCTURES
DECEMBER 1995
ISSN 1395-7953 R9536
The FRAGTURE AND DYNAMICS papers are issued for early dissemirration of research results from the Structural Fradure and Dynamics Group at the Depa~tment of
Building Technology and Structural Engineering, University of Aalborg. These papers
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.
.1Printed at Aalborg University l
INSTITUTTET FOR BYGNINGSTEKNIK
DEPT. OF BUILDING TECHNOLOGY
AALBORG UNIVERSITET
•
AUC
AND
•
STRUCTURAL ENGINEERING
AALBORG
•
DANMARK
FRACTURE & DYNAMICS
PAPER NO. 71
To be presented at the 14th International Modal Analysis Conference,
Dearborn, Michigan, USA, February 12-15, 1996
P. ANDERSEN, R. BRINCKER & P. H. KIRKEGAARD
THEORY OF COVARIANCE EQUIVALENT ARMAV MODELS OF CIVIL
ENGINEERING STRUCTURES
DECEMBER 1995
ISSN 1395-7953 R9536
.--
----------------------------------------------------------------------------------------------------
THEORY OF COVARIANCE EQUIVALENT ARMA V MODELS
OF CIVIL ENGINEERING STRUCTURES
P. Andersen, R. Brincker & P.H. Kirkegaard
Department of Building Technology and Structural Engineering
Aalborg University
Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
ABSTRACT
l. INTRODUCTION
In thispaper the theoretical background for using covariance
equivalent ARMA V models in modal analysis is discussed. l t is
shown how to obtain a covariance equivalent ARMA model for
a univariate linear second arder continuous-time system excited
by Gaussian white noise. This re sult is generalized for multivariate systems to an ARMA V model. The covariance equivalent model structure is also considered when the number af
channels are different from the number af degrees offreedom
to be modelled. Finally, it is reviewed how to estimate an
ARMAV modelfrom sampled data.
The use of non-parametric FFT-based methods has for many
years been one of the most popular tools in modal analysis, but
recently the interest in using parametrical models as the basis
for modal analysis has increased. Since, the usual way of
obtaining information about a structure is through sampling, all
parametrical models are in some sense discrete equivalents to
the continuous system. There are several methods for discretization. Some of these are approximation using pole-zero
mapping, see Åstri:im et al. [1], and approximation by hold
equivalence techniques, see Safak [2]. But perhaps the most
used approximation is the covariance equivalence technique, see
Bartlett [3], Kozin et al. [4] and Pandit et al. [5] . In thispaper
the theoretical background for using covariance equivalent autoregressive moving-average vector (ARMAV) models in modal
analysis of civil engineering structures will be discussed. This
is done by showing that a second order linear continuous-time
system can be modelled by an ARMA V model. The results can
irnroidiately be used as an effective simulation tool in case of
white noise excitation, but can also be used for identification o f
structural systems from sampled data.
NOMENCLATURE
m
c
k
T
g
d
y
a/
y(t)
u(t)
x(t)
h( r)
A
B
b
m;
M
11
Il
Y,
a,
X,
Gk
Q>
e
L
l;
Å
Å
Q>;
e;
Diagonal mass matrix
Syrnroetric damping matrix
Syrnroetric stiffness matrix
Sampling period
Modal weight of impulse response
Modal weight of the covariance matrix
Lagged covariance matrix of response process
Covariance of a Gaussian white noise process x
Continuous-time system response
Continuous-time Gaussian white noise
Continuous-time state vector
Impulse response fimetion
Continous-time state space matrix
Continuous-time excitation matrix
Continuous-time excitation vector
Scaled modeshape
Eigenvectors of A
Eigenvalues of A
Diagonal matrix of eigenvalues J..l;
Discrete-time system response
Discrete-time Gaussian white noise
Discrete-time state vector
Green ' s function
Discrete-time state space matrix
Discrete-time excitation matrix
Eigenvectors of Q>
Scaled modeshapes
Eigenvalues of Q>
Diagonal matrix of eigenvalues Å;
Auto-regressive polynomial coefficients
Moving-average polynomial coefficients
The theoretical considerations of thispaper have been used in
Kirkegaard et al. [6] for identification of the skirt piled Gullfaks
C gravity platform, and in Brincker et al. [7] for identification
of a multi-pile offshore platform.
In section two it will be shown how to obtain a covariance
equivalent univariate (single channel) ARMA model of a
single-degree of freedom system. Section three generalizes these
results to a multivariate ARMA V model for a multi-degree o f
freedom system. The fourth section explains how to obtain a
covariance equivalent univariate ARMA model for a multidegree of freedom system. Finally, in the fifth section, it is
deseribed how these models can be calibrated to sampled data.
2. UNIV ARlA TE MODEL - SDOF SYSTEM
There are two eriterions that must be satisfied in order to make
an ARMA model covariance equivalent to an SDOF
continuous-time linear system. Firstly, the modal properties
must be equal, i.e. eigenfrequency and damping ratio must be
the same. Secondly, the discrete-time autocovariance function
of the system response must in some sense be equal to the
continuous-time autocovariance function. The derivation
therefore starts by considering a second order continuous-tirne
systern deseribed by the differential equation
x( t) =M e 11'M- 1+f M e JJ(r -.' >M- 1b u(s)ds
o
mji(t) + C)i(t) + ky(t)
= u(t)
(6)
(1)
=Me 11'M- 1 +fh(t-s)u(s)ds
o
where m, c and k are the mass, damping and stiffness terms. y(t)
is the response of the system, and u(t) is an independent
distributed Gaussian white noise excitation with zero-rnean and
the variance a/, It is realized that the white noise approxirnation rnay not provide a very good approxirnation of non-white
excitation, but it sirnplifies the autocovariance function and
thereby also the resulting ARMA model. In section six, it will
be explained how to deal with non-white excitation.
where h( -r)= M e 11 .' M- 1b is the irnpulse response function o f
the state space system. The first part of (6) is the deterministic
or transient part, and the last is the stochastic part.
From h( r) the irnpulse response of y( t) can be extracted .
Den oting this irnpulse response by h''( r) yields
In continuous-tirne state space (l) is deseribed by
(7)
= Ax(t)
i(t)
+
(2)
bu(t)
where the g c;' s aremodal weights. Assuming stationary conditions, i.e. initial values x( O) = O and l,ul < O for both eigenvalues,
the autocovariance function of (l) is defined as
where x(t)=[y(t), y(t)f , and
o
A
k
m
o
(3}
b
c
m
m
Yc('t')
a: fhY(t)h Y(t+'t')dt
o
The solution of (2) is given by
(8}
= d e flt' + d e fl2'
cl
x(t)
= eA 1x(O)
+ f eAU -s>bu(s)ds
c2
'
.,. >
' -
O
(4)
o
where the dc,'s also aremodal weights, defined as
'.l
·:
2
2 gel
and the response is the top half of x (t). In order to simpli fy (4)
the continuous-tirne modal matrix A is decornposed
M
"[~,
(9}
(5)
:,]
fJ = diag [f.!;] ,
2 gel ge2
- a - - - au--u 2 f.l,
f.l, + fl2
i= 1, 2
where fJ is a diagonal matrix of distinct eigenvalues, and M is
a Vandermande matrix containing the corresponding eigenvectors. It is the eigenvectors that control the structure of A ,
whereas the eigenvalues contro l the values of the battom ro w of
A.
Inserting the decornposed modal matrix in to (4) yields
Now tuming to discrete time, it will now be shown that an
ARMA(2,1) model with two auto-regressive pararneters and one
moving-average pararneter is an adequate model. Defining the
discrete response, Y< as y(kT), where T is the sampling period,
the ARMA(2,1) model is given by
where <j> 1, <!> 2 are the auto-regressive parameters, and ~ is the
moving-average parameter. a, is an independent distributed
Gaussian white noise with zero-mean and variance a} . Representing (10) in discrete-tirne state space yields
X,
Q>X, _1
+
ea,
(11)
(12)
However, the present auto-regressive model is not covariance
equivalent, which is why it is necessary to add the movingaverage. The reason for adding only one moving-average
parameter can be seen by looking at the autocovariance fimetion
o f (l 0). In orde r to calculate the autocovariance function, it is
assumed that the initial values, Xm and a,, for t< O, are zero. By
applying the modal deearnposition o f ( 14) to (13), the respons e
c an be expressed in terms o f the scalar Green ' s function , see
Pandit [8] as
t
The solution o f (Il) is given by
Y, =
+
L qiea,_j
(16)
Gjar-j
j =O
t
x, = Q>'X0
L
(13)
j=O
where the Green's function is defined as
). I
and the response Y, is the top half of X, The discrete-time modal
matrix, Q>, can be decomposed in the foliowing way
Q>
+
eI
.
--=--~).l +
).! - ).2 I
).2
+
).!
2
~
o
j<
o
).2 -
= LÅC 1
eI ).!.
j
(14)
where Å is a diagonal matrix of distinct eigenvalues, and L is a
Vandermande matrix containing the corresponding eigenvectors . Again, it is the eigenvectors that control the struerure
of Q>, and the eigenvalues that contro l the values of the top row
of Q>.
= O,
(17)
and the g;' s are modal weights . Based o n ( 16) and (17) the
autocovariance function o f ( 10), see Pand i t et al. [5) , is
~
Y.r = a~ L
GjGj .,·
j =O
(18)
Inserting the decomposed modal matrix in to ( 13) yields
The modal weights in ( 18) are defined as
Similar to the continuous-time case the frrst part of (15) is the
deterrninistic or transient part, and the last is the stochastic part.
(19)
2
2
In order to make the ARMA model covariance equivalent, the
continuous-time system and the discrete-time system must be
equal at all discrete time steps k, such that t= kT, for k =O, .. , ""·
The deterrninistic parts in (6) and (15) must therefore be equal.
This is accomplished if ). k = e pkT, or equivalently if Å. = e~'r
for all eigenvalues. The result that can be drawn from this is that
a stable underdamped continuous-time SDOF system with two
complex conjugated eigenvalues, also has two complex conjugated eigenvalues in discrete time. This is the reason for why it
is necessary to have two auto-regressive parameters. So, at this
point, the auto-regressive part of (lO) cail be constructed on the
basis o f the continuous-time modal matrix A in (3 ). Deterrnine
the eigenvalues, f.l; and the eigenvectors M. Calculate the
discrete eigenvalues, Å;, by using the identity Å. = e11 r. Finally,
calculate the discrete-time modal matrix, Q>, using the sirnilarity
transformation in (14) . The auto-regressive parameters will then
be given by Q> 1 = -Å. 1 - Å. 2 and Q> 2 = Å. 1 Å. 2•
g2
+a-a l - ).2
2
To make the ARMA model covariance equivalent, the d; 's must
be specified. This is done by using two initial conditions.
However, one condition must always hold namely by the
autocovariance at time lag zero, given by
(20)
Hence, only one initial condition remains to be specified, which
is done by requiring one moving-average parameter. A method
for calculation of and a 2 that easily conform to multivariate
systems, is to exp:ess th; autocovariance fimetion implicitly
using (10) and (17) as, see Pandit [8] ,
e
(26)
o$
O,
k $l
(21)
k > l
In (26) I is an n x n identity matrix. The modal deearnposition
of A is given by
Using (21) for k=l and k=2, and that G 1 =8 1 -<!> 1 gives the
foliowing second degree polynornial in e,
0
e2l
( -k
k +<!> l J e+
l
1
=o
(22)
A
M
l
J.1
= MJ.IM- 1
[
=
m,
m,
f.l,m,
= diag
l
f.l m::
211
[ f.l;] '
(27)
i= 1,2, ... ,2n
where
k0
= Yc( O) + <!> 1 Y/T) + <!>2 Y/2T)
(23)
k1
= Y/T)
+ <!> 1 Y/0) + <!> 2 Y/T)
In (23) it is required that yk = Y/k1), and at the same time
used that y k = y -k . For each of the solutions of e 1 in (22)
corresponds a variance a}. This variance is deterrnined by the
foliowing expression
'l
'
where J.l is a diagonal matrix of 2n distinct eigenvalues. M is a
matrix containing the corresponding eigenvectors. Them;' s are
scaled modeshapes. It is again the eigenvector and thereby the
scaled modeshapes that control the structure of A. The eigenvalues controls the values of the last n rows of A.
Assurning zero initial conditions, the response y(t) o f (25) is
y(t)
[h Y(t -s)u(s)ds
o
(24)
2n
T
"m
.m j ~,
_J_ e J
= L.-
(28)
sj
j=,
From (22) it is seen that there is in faet two covariance equivalent ARMA(2,1) models possessing the same modal properties.
l
i
i
l
3. MULTIVARrATE MODEL· MDOF SYSTEM
where hY(t-s) is the n x n impulse response function, the matrices
g ejaremodal weights, and Sj are the corresponding scalar modal
Now consider an MDOF continuous-time linear system. Such
a system can be modelled by an ARMA V model. In order to
make this model covariance equivalent, the same requirements
as for the ARMA model must be imposed on it. In this section
the procedure of the previous section will be generalized.
Consider a system with n degrees of freedom, deseribed by an
n x n diagonal mass matrix m, an n x n symmetric damping
matrix c, and an n x n symmetric stiffness matrix k. It is
assumed that the system is excited in all degrees of freedom by
an independent distributed Gaussian white noise u(t) with
covariance a/, Denoting the n x l response vector by y( t), the
continuous-time state space description of this system is
masses, see Meirovitch [9]. Assurning stationary conditions the
n x n lagged covariance function of the response is given by
:i(t)
= Ax(t)
where x( t)= [y( t), y( t) f, and
+ Bu(t)
(25)
f
y c("t) = h Y( 't) a~hYT(t +'t )dt
o
(29)
where the modal weights d ej are defined as
(30)
The resemblance between (8) and (29) is obvious. Turning to
discrete time, it will now be shown that in the multivariate case,
the ARMA(2,l) model expands to an ARMAV(2,1) model.
Denoting Yk as y(kn, the ARMA V(2, l) model is defined as
------------------------- --
-----
-
2n
(31)
=L
g!.!
i=
Gi
l
l'
j <:
o
j <
o
l
=
O,
=
l.Li(IJ...
l
l
e,
where <j>,, <1>2 are the n x n auto-regressive matrices, and
is an
n x n moving-average matrix. The n x l vector a 1 is an independent distributed Gaussian white noise with zero-mean and
covariance matrix a/. Representing (31) in discrete-time state
space yields
(32)
gi
(35)
e,)
+
where Li is the ith row of the left 2n x p submatrix of i . By
using (35) the lagged covariance matrix can be defined in a
similar manner as in the scalar case as
~
Y.,. = L G1 a~GJ~s
j =O
2n
="d
J...·~·
L..J J J
(33)
s=O, l , 2, ...
(36)
}=I
The modal deearnposition of <j> yields
<j>
L
2n
g .a2gr
i=I
[-)};
J
a
t
= LJ...C 1
(34)
Å = dia g [ !.-;]
, i = l, 2, ... ,2n
where Å is a diagonal matrix of 2n distinct eigenvalues. L is a
matrix containing the corresponding eigenvectors, and the l;' s
are the scaled modeshapes. As for the continuous-time case, the
only task for the eigenvectors and thereby the scaled modeshapes are to control the structure of <j>. If the first n rows of <!>
were interchanged with the last n rows the strueture of <j> would
be similar to the structure of A, i.e. the auto-regressive matrices
would be in the n last rows and flipped in left and right direction . Because this is so, it can be verified that the scaled
modeshapes o f <j> and A are equivalent.
On the basis of (35) and (36) it can be shown again that
covariance equivalence can be obtained using only one movingaverage matrix
Foliowing the approach in the scalar case for
calculation of e, and a/, the multivariate equivalent to (21) is
given by
e,.
o~
=O,
k ~l
(37)
k> l
which provides the foliowing second degree matrix polynomial
in
e,
As in the scalar case i t is possible at this stage to determine the
auto-regressive part of (31) o n the basis of the continuous modal
matrix A, defined in (26). The calculations follow the scalar
case. Convert eigenvalues using the identity A. = e~'r. Calculate
<j> using a similarity transformation, keeping in mind that the
scaled modeshapes m; and l; are equivalent.
where
The lagged covariance matrix Ys of (31) can be expressed using
the n x n matrix Green's fimetion G. At a given time step j, Gi
is defined using 2n modal weights gi as
In (39) it is required that y k = y JkT). Purther it is used that
y [ = y -k , and that the lagged covariance matrix is real. The
solutions of e, in (38) can be found using matrix polynomial
techniques. It can be shown that there exist K(2n,n) solutions.
For each of these solutions correspond a covariance matrix aa 2 ,
which is deterrnined by
ko
Yc( O)
+
<!>,Y J Tl
k1
Yc( T)
+
<l> 1 Y c( O) + <!>2 Yc( Tl
+
<!>2 YJ2T)T
(39)
2
aa =
v,a-lk
l
(40)
So again there are several covariance equivalent ARMA V
models possesing the same modal properties .
and excitation vector u(t)=[u(t) Du ( t) ... D 2" - 2u(t)
and
4. UNIV ARlA TE MODEL - MDOF SYSTEM
A
l
In both cases considered until now the number of channels have
been equal to the number of degrees of freedom . This approach
has provided the maximum modal information, i.e. eigenfrequencies, damping and scaled modeshapes. In this section the
covariance equivalence between a univariate ARMA model and
a continuous-time multi-degree of freedom system, will be
considered. The !imitations of this approach are, that only
eigenfrequencies and damping ratios can be deterrnined.
If only one forcing function and output response for an MDOF
system are considered, then (28) can be written as
y(t) =Ja ThY(t-s)bu(s)ds
(41)
o
l
where y(t) and u(t) are scalars. The n x l vector b is filled with
zeroes except at the element that corresponds to the forcing
fimetion u(t). The n x l vector a is also filled with zeroes except
the element that corresponds to the output response y(t).
Laplace transforming (41) yields
Y( z)
=
H( z) U( z)
=L a Tg .b
2n
C)
j=l
z -f.lj
2n
2n
(42)
L a Tgci k=l,hj
II ( z - J.lk)
j=l
where Y(z) and U(z) are the Laplace transformed of y(t) and
u(t), respectively, and z is any complex number. The last
equation in (42) is in faet a scalar rational polynornial. The
order of the de norninator polynornial is 2n, whereas the order of
the numerator polynornial is 2n-2. Applying the inverse Laplace
transform to this rational polynornial yields a differential
equation of 2n ord er of the foliowing form
(D zn+ azn- l D zn - l
+ .. . + ao) y( t)
(43)
where D is a differential operator. The coefficients a; and /3;can
be calculated explicitly from the last equation in (42). The
differential equation of (43) can be reduced to state spaceform
by defining the state vector x (t) =[y( t) D y (t) .. . D 2"- 1y(t) f,
o
o
B
o o
o o
Po Pz
o
o
o
-ao -az
Of,
- a2n- I
o o
o o
Pzn - 2
(44)
o
The resemblance with the SDOF state space formulation in (2)
should be noted. By foliowing the procedure used for univariate
SDOF systems, the modal matrix A can easily be converted to
a discrete modal matrix <j> . This matrix will also be of the
dimension 2n x 2n, i.e. corresponds to an ARMA model with 2n
auto-regressive parameters.
The multivariate model (25) does not have derivatives of the
forcing function on the right-hand side. Looking at the univariate model in (43), which is restricted to only one of the
elements of the multivariate vector, it is seen that i t does have
derivatives of the forcing fimetion on the right-hand side.
Because the basic model is of second order, it can be verified
that the derivative of the right-hand side never will exceed 2n-2.
Now, this result, of course, also holds for a univariate SDOF
system, i.e. for n= I. In section two, it was shown that the
covariance equivalent ARMA model of an SDOF system was an
ARMA(2,1) . So, by extending this result, the covariance
equivalent model for an n-degrees of freedom univariate system
is evidently an ARMA(2n,2n-l).
The result is not restricted for multivariate to univariate models.
Consider an n/m-variate system with n degrees of freedom. The
covariance equivalent ARMAV model of such a system will
then be an ARMAV(2m,2m-l). As an ex-ample, consider a
system with two channels and four degrees of freedom. This
system can be modelled by an ARMAV(4,3) model.
6. IDENTIFICA TION
In the previous sections it has been shown that it is possible to
model any linear second order continuous-time structural system
excited by Gaussian white noise using the ARMA V model. This
result is usefull by itself, because it provides a very fast and
easy simulation tool knowing m, c, k, a/ and T. The result can 1
on the other hand also be used to model a discretely sampled
system. Consider a structural system without disturbance.
Knowing that the sampled system contains n degrees of freedom ,
in p channels the covariance equivalent ARMA V model is an
ARMAV(2nlp,2nlp-l). In the case of disturbance, e.g. non- I
white excitation of the structural system i t may be necessary te
l
1
increase the arder of the model. By doing so, the non-white
excitation is modelled as a part of the resulting model. The
actual physical system can then be extracted from the model
afterwards. This can be done using e.g. partial fraction expansion.
A wetl-known method for identification of a structural system
is by applying the Ieas t square method to the ARMA V model.
Consider a p-variate ARMAV(n,m) and a p x N matrix o f
samples y,. A least-square eriterion for such a model is typical
o f the foliowing form, see Lj ung [l O]
N
V(e)
=_!__!_L e;(e)A- 1e,(e)
N21=I
7. CONCLUSION
In t~is paper the theoretical background for using covariance
e_qmvalent ARMA V models as a discrete equivalent of the
ltnear second order continuous-time system, excited b
Gaussian white noise, has been discussed. The correspondenc~
between the number of channels of response, the number of
degrees of freedom in the system, and the ord er o f the ARMA v
mod~! has been shown. The results have shown that it is actually
poss1ble
a continuous-time system explicitl Y m
·
.
.to model
.
d 1screte time m a reasonable manner. It has also been considered ho_w to identify structural system from sampled data using
a non-hnear least-square criterion.
(45)
8. ACKNOWLEDGEMENT
where V(e) is the loss function, and e, is the prediction error.
The p x p matrix A 1 weights together the relative importance o f
the components o f e,. j l( e) is the predictor o f the model,
defined as
f (e) = <t>;e
9. REFERENCES
[l]
1
e = col(<l>,<l>z···<l>n,eiez· ··em)
The Danish Technical Research Council is gratefulty acknowledged
(46)
[2]
[3]
e
1
is an (n+m)p x l parameter vector obtained by
where
stacking all columns of the auto-regressive and moving-average
matrices on top of each other.
The (n+m)p 1 x p regression matrix <!>, is obtained as the
Kronecker produet between a p x p identity matrix IP' and the
(n+m)p x l regression vector q>,, defined as
[4]
[5]
[6]
[7]
(47)
[8]
The parameters of the estimated model are the ones that
minimize the loss function V(e). In arder to perform this
minimization a numerical search procedure like the GaussNewton or the Levenberg-Marquardt is needed. In anycase the
numerical minimization will be non-linear because the prediction error e, depends on the estimated parameters. However, it
is also possible to use linear multi-stage search procedure, see
e.g. Piombo et al. [Il].
[9]
[lO]
[ 11]
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Dearborn, 1996.
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Taltav6: Modal Analysis of an Offshore Platform
using Two Different ARMA Approaches, 14th
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FRACTURE AND DYNAMICS PAPERS
PAPER NO. 41: P. H. Kirkegaard & A. Rytter: Use of a Neural Network for Damage
Det€ction and Looation in a Steel Member. ISSN 0902-7513 R9245
PAPER NO. 42: L. Gansted: Analysis and Description of High-Cycl€ Stochastic Fatigue
in Steel. Ph. D.-Thesis. ISSN 0902-7513 R9135.
PAPER NO. 43: M. Krawczuk: A New Finite Element for Static and Dynamic Analysis
of Cracked Composite l!Jeams. IS$N 0902-7513 R9305.
PAPER NO. 44: A. Rytter: Vibrational Based Insp€ction of Civil Engingering Structures. Ph. D.-Thesis. ISSN 0902-7513 R9-314.
PAPER NO. 45: P. H. Kirkegaard & A. Rytter: An Experimental Study of the Modal
Parameters of fk Damaged Steel Mast. ISSN 0902-7513 R9320.
PAPER NO. 46: P. H. Kirkegaard & A. Rytter: An Experimental Study of a Steel
Lattice Mast under Natural Excitation. ISSN 0902-7513 R9326.
PAPER NO . 47: P. H. Kirkegaard & A. Rytter: l!se of Neural Networks for Damage
Assessment in a Steel M as t. ISSN 0902-7513 R9340.
PAPER NO. 48: R. Brincker, M. Demøsthenous & G. C. Manos: Es·i imation of ih€
Coefficient of Re3titution of Roeking Systems by the Random Decrement Technique .
ISSN 0902-7513 R9341.
PAPER NO. 49: L. Gansted: Fatigue of Steel: Con"siant-Amplitude Load on CCT- ·
Specimens. ISSN 0902-7513 R9344.
PAPER NO. 50: P. H. Kirkegaard & A. Rytter: Vibration Based Damage Assessment
of a Cantilever using a Neural Network. ISSN 0902-7513 R9345.
PAPER NO. 51: J. P. Ulfkjær, O. Hededal, L B. Kroon & R. Brincker: Simple Application of Fictitiou·s Crack Model in Reinforeed Concrete Beams. ISSN 0902-7513
R9349.
PAPER NO. 52: J: P. Ulfkjær, O. Hededal, L B. Kroon & R. Brincker: Simple Application of Fictitious Crack M odel in Reinforced Concrete Beams. A nalysis and Experiments. ISSN 0902-7513 R9350.
PAPER NO. 53: P. H. Kirkegaard & A. Rytter: Vibration Based Damage Assessment
of Civil Engineering S truetures using N eural N etworks. ISSN 0902-7513 R9408.
PAPER NO. 54: L. Gansted, R. Brincker & L. PilegaaJ,"d Hansen: The Fracture Mechanical M arkov Chain Fatigue Model Compared with Empirical Data. ISSN 0902-7513
R9431.
.
PAPER NO. 55: P. H. Kirkegaard, S. R. K. Nielsen & H. I. Hansen: Identification af
Non-Linear Structures using Recurrent Neural Networks. ISSN 0902-7513 R9432.
PAPER NO. 56: R. Brincker, P. H. Kirkegaard, P. Andersen & M. E. Martinez: Damage
Detection in an Offshore Structure. ISSN 0902-7513 R9434.
PAPER NO. 57: P. H. Kirkegaard, S. R. K. Nielsen & H. I. Hansen: Structural Identification by Extended Kalman Filtering and a Recurrent Neural Network. ISSN 0902-7513
R9433.
.
l
l
FRACTURE AND DYNAMICS PAPERS
PAPER NO. 58: P. Andersen, R. Brincker, P. H. Ki-rkegaard: On the Uncertainty øf
Identification of Civil Engineering Structures using ARMA Models. ISSN 0902-7513
R9437. .
.
PAPER NO. 59: P. H. Kirkegaard & A. Rytter: A Comparative Study of Three Vibration
Based Damage Ass~s-sment Techniques . ISSN 0902-7513 R9435.
PAPER NO . 60: P. H. Kirkegaa-r d, J. C. Asmussen, P. Andersen & R . Brincker: An
Expe.r imental Study of an Offshore Platfo r-m. ISSN 0902-7513 R9441.
PAPER NO. 61 : R . Bri:ncker, P. Andersen, P. H. Kirkegaard, J. P. Ulfkjær: Damage
Detection in Labaratory Concrete Beams. ISSN 0902-7513 R9458.
PAPER NO. 62: R. Brincker, J: Simonsen; W. Hansen: Some Aspecis of Formation of
Cracks in FRC with Main Reinforcement. ISSN 0902-:7513 R9506.
PAPER NO. 63: R. Brincker, J. P. Ulfkjrer, P. Adamsen, L. Langvad, R. Toft: Analytical
Model for Hook An~hor PuU-out. ISSN 0902-7513 R9511.
~· -Qakmak: Assessment of
Damage in Seismically Excited RC-Structures from a Single Measured Response. ISSN
PAPER NO. 64: P. S. Skjærbæk, S. R. K. Nielsen, A.
1395-7953 R9528.
Asm1,1.ss~n,
S. R. Ibrahim, R. Brincker: Random Decrement and ·
Regression Ana.Zysis of Tmffic Responses of Bridges. ISSN 1395-7953 R9529.
PAPER NO. 65: J. C.
PAPER NO. 66: R. Brincker, P. Andersen, M. E. Martinez, F. Tallav6: Modal Analysis of an Offshore Platform using Two Different ARMA Approaches. ISSN 1395-7953
R9531 .
PAPER NO. 67: J. C. Asmussen, R. Brinaker: Estimation of Frequency Response Fun-_
ctions by Random Deerem en t. ISSN 1395-7953 R9532.
PAPER NO. 68: P. H. Kirkegaard, P. Andersen, R. Brincker: Identification of an
Eq-uivalent Linear Motf,el for a Non-Linear Time- Varia:z,t RC-Structure . · ISBN 13957953 R9533.
PAPER NO. 69: P. H. Kirkegaard , P. Andersen, R. Brincker: Identification of the Skirt
Piled Gullfaks C Gravity Platform using ARMA V Models. ISSN 1395-7953 R9534.
PAPER NO. 70: . P. H. Kirkegaard, P. Andersen, R. Brincker: Identification of Civil
Engineering Structures using Multivariat-.e ARMAV and RARMAV Models . ISSN 13957953 R~535.
PAPER NO. 71: P. Andersen, R. Brincker, P. H. Kirkegaard: Theory of Covariance
Equivalent ARMAV Models of Civil Engineering Structures. ISSN 1395-7953 R9536.
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