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COMPDYN 2009
ECCOMAS Thematic Conference on
Computational Methods in Structural Dynamics and Earthquake Engineering
M. Papadrakakis, N.D. Lagaros, M. Fragiadakis (eds.)
Rhodes, Greece, 22–24 June 2009
FATIGUE LIFETIME ESTIMATION IN STRUCTURES USING
AMBIENT VIBRATION MEASUREMENTS
Costas Papadimitriou1, Claus-Peter Fritzen2, Peter Kraemer2, and Evangelos Ntotsios1
1University of Thessaly
Department of Mechanical Engineering, Volos 38334, Greece
e-mail: {costasp,entotsio}@uth.gr
2University of Siegen
Institute of Mechanics and Control Engineering, Siegen 57076, Germany
{fritzen,kraemer}@imr.mb.uni-siegen.de
Keywords: Stochastic Fatigue, Damage Accumulation, Prognosis, Kalman Filter, Stochastic
Dynamics, Spectral Methods.
Abstract. A methodology is proposed for estimating damage accumulation due to fatigue in
the entire body of a metallic structure using output-only vibration measurements from a sen-
sor network installed at a limited number of structural locations. This method can also be
seen as a tool for a life-time prognosis within SHM concepts. Available frequency domain
stochastic fatigue methods based on Miner’s damage rule, S-N fatigue cycle curves and
Dirlik’s probability distribution of the stress range are used to predict the expected fatigue
accumulation of the structure in terms of the power spectral density of the stress processes. In
predicting the damage and fatigue lifetime, it is assumed that the unmeasured excitations can
be modeled by stationary stochastic processes. The power spectral densities of stresses at
unmeasured locations are estimated from the response time history measurements available
at the limited measured locations using Kalman filter and a model of the structure. The pro-
posed formulation is demonstrated using a MDOF spring-mass chain model arising from
structures that consist of members with uni-axial stress states.
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
2
1 INTRODUCTION
Damage accumulation due to fatigue is an important safety-related issue in metallic struc-
tures. The linear damage accumulation law introduced by [1] is often used to evaluate fatigue
damage using available methods for cycle counting per stress level in measured stress re-
sponse time histories. Available S-N fatigue curves [2] are used to predict the fatigue damage
at different stress levels for harmonic stress time histories. The damage accumulation predic-
tions are based on measurements taken from a sensor network, consisting usually of strain ro-
settes, attached to the structure and are only applicable for the locations where measurements
are available. Due to practical and economical considerations, the number of sensors placed in
a structure during operation is very limited and in most cases they do not cover all critical lo-
cations. Moreover, there are locations in the structure that one cannot install sensors such as
underwater locations in offshore structures (oil refinery structures, offshore wind turbines,
offshore steel jackets, etc.), within the body of solid structures, and non-approachable areas of
extended structures. Available fatigue prediction methods based only on measurements cannot
be used to predict fatigue damage accumulation at the locations where measurements are not
available. In order to infer damage due to fatigue at structural members where measurements
are not available, one needs to predict the stress response time histories in these structural
members using the available measurements obtained from the sensory system. In limited cir-
cumstances, such predictions can be possible if one combines the available measurements
with the information obtained from a model (e.g. a finite element model) of the structure.
The methods for fatigue damage accumulation have been extended to treat the case the ex-
citations can be represented by a stochastic vector process with known correlation characteris-
tics. Assuming that the structure behaves linearly and the excitation is modeled by a Gaussian
stochastic vector process, the stress response at any point is a stochastic process that can be
completely defined using the correlation characteristics of the stochastic excitations [3]. The
fatigue damage accumulation at a structural location can then be computed using the charac-
teristics of the stochastic processes of the components of the stress sensor at such location.
Methods for fatigue accumulation for Gaussian narrow band stress processes have been first
introduced (e.g. [4]) and then extended to the case of wide-band Gaussian stress processes
(e.g. [5]). A review and comparison of spectral methods for stochastic fatigue analysis based
on wide-band Gaussian stochastic processes can be found in the work by Benasciutti and
Tovo [6]. The formulations depend on the probability distribution of stress cycles correspond-
ing to different stress levels in a stress response time history signal and the expected number
of peaks per unit time of a stress process. Results for the expected fatigue damage accumula-
tion predicted by the Miner linear rule have been presented in terms of the spectral moments
of the stress process which are readily obtained from the power spectral density of the stress
components involved. For the important case of wide band processes encountered often in
applications, simulation-inspired Dirlik approximation [5] is widely used and is considered to
be the most accurate formula for modeling the probability of stress cycles in terms of the
spectral moments of the stress process. It is worth noting that the aforementioned frequency
domain methods based the power spectral densities use no information available from a sen-
sor network. Instead, their predictive accuracy depend on the assumptions employed for the
stochastic excitation characteristics and the models representing structural behavior. However,
these prediction fail to integrate the information provided by a network of sensors. Such sen-
sor information is expected to update and improve the fatigue predictions, making them con-
sistent with the available measurements.
This work addresses the problem of estimating the expected damage accumulation due to
fatigue in the entire body of a structure using output-only vibration measurements at a limited
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
3
number of locations provided by a sensor network installed in the structure. The measure-
ments may consist of response time histories such as e.g. strain, acceleration, velocity, dis-
placement, etc. The proposed methodology is applicable for the case where the responses can
be modeled by stationary processes and the measured response time histories are long enough
so that they can be considered to approach stationary processes. The excitation in the structure
is considered to be unknown. However, for most operational condition of structures for which
the measured response is approaching a stationary process, the excitation can be considered to
be a Gaussian stationary stochastic process with unknown intensity and frequency content.
The expected fatigue damage accumulation in the entire structure is obtained by integrating
(a) methods for predicting strain/stress response time histories and their correlation/spectral
characteristics in the entire structure from output-only measured response time histories avail-
able at limited locations in the structure, and (b) frequency domain methods for estimating
fatigue damage accumulation using the spectral characteristics of the predicted strain/stress
response time histories. The idea is to use prediction methods such as Kalman filter [7] meth-
ods to predict the strain/stress response time histories at various locations within structural
components using the measurements available at a limited number of locations. Such predic-
tions are restricted here to the case of excitations that can be adequately represented by Gauss-
ian stationary stochastic processes.
This work is organized as follows. In Section 2, the frequency domain formulation for pre-
dicting damage due to fatigue in structural elements subjected to uniaxial stress state in linear
structures under Gaussian stochastic excitations is reviewed. The formulation is applicable to
Gaussian wide-band stress processes, often encountered in engineering applications, and it
depends on the spectral moments of the power spectral densities of the stress process at the
location of a structure. Section 3 presents the formulation for predicting the strain/stress re-
sponse time histories and the associated power spectral densities at the corresponding loca-
tions of the structure using Kalman filter and the measured time histories at a limited number
of locations in the structure. Section 4 demonstrates the effectiveness of the proposed meth-
odology using a chain-like mass-spring MDOF structure and measured data that are simulated
using white noise excitations. Conclusions are summarized in Section 5.
2 SPECTRAL METHOD FOR FATIGUE ESTIMATION
The Miner rule [1] is used to predict the damage accumulation due to fatigue. According to
this rule, a linear damage accumulation law at a point in the structure is defined by the formu-
la
mi
ii
n
DN
(1)
where
i
n
is the number of cycles at a stress level
i
,
i
N
is the number of cycles required for
failure at a stress level
i
, and
m
is the number of stress levels identified in a stress time his-
tory. Available S-N fatigue curves [2] are used to describe the number of cycles
i
N
required
for failure given a stress level
i
. The number of cycles
i
n
at a stress level
i
are usually
obtained using available stress cycle count methods, provided that the stress time histories are
available through measurements. Alternatively, frequency domain methods based on spectral
methods can be used to predict the expected damage due to fatigue using the linear damage
law (1). The methodology assumes that the power spectral density of the stress process is
available. For linear systems excited by time-varying loads that can be modeled by stationary
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
4
stochastic processes, these power spectral densities can be straightforward computed using
available random vibration results [3].
The methodology for estimating the fatigue accumulation at a point assuming a uni-axial
stochastic stress state is next reviewed. For multi-axial stress states one can refer to the work
by You and Lee [8] and Pitoiset and Preumont [9] to extend the applicability of the present
methodology. Let
()t
be the uni-axial stress at a point in a structural element. The stress is
considered to be a stationary Gaussian stochastic process. This is the case encountered in line-
ar structures that are subjected to stationary Gaussian stochastic processes. Let
()S
be the
power spectral density of the stationary Gaussian stochastic process
()t
and
()
i
iSd
(2)
be the spectral moments of the stress process. Using frequency domain methods for fatigue
estimation under stochastic excitations and the linear damage accumulation law (1), the ex-
pected fatigue damage accumulation for a uni-axial stochastic stress process is given by [6]
1
00
()
[ ] [ ] ( )
()
a
n
E D d c TE P p d
N
(3)
where
( ) [ ] ( )n TE P p d
is the number of cycles at stress levels within the stress interval
[ , ]d
,
()Nc
(4)
is the number of cycles for failure that correspond to a specific harmonic stress level of ampli-
tude
obtained from available S-N curves [2],
[]EP
is the expected number of peaks per
unit time for the stress process,
()p
is the probability distribution of the stress cycles at
stress levels within
[ , ]d
, and
T
the period of observation. The parameters
c
and
are constants obtained from fatigue test experiments and depend on the material of the test
specimen.
The expected time of failure due to fatigue (fatigue lifetime)
life
T
corresponds to a critical
expected damage value
[] cr
E D D
which is often set equal to unity (
1
cr
D
). Using (3), the
fatigue lifetime is given by
cr
life
D
TD
(5)
where
D
is the expected damage rate given by
1
0
[ ] ( )D c E P p d
(6)
For Gaussian stochastic stress processes, the probability distribution of the stress cycles at
stress levels within
[ , ]d
is given by the Dirlik formula [5,10-11] as
22
2
() () ()
12
22
3
2
0
() ()
() 2
ZZZ
QR
E E Z
e e E Z e
QR
p
(7)
where
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
5
0
()
2
Z
(8)
max min
is the range of the stress process, and
1
E
,
2
E
,
3
E
,
R
and
Q
are specific
algebraic functions of the spectral moments
0
,
1
,
2
,
4
, given by
22
22 1 1
1 2 3 1 2
2
2
21
, , 1
11
m
xEE
E E E E E
R
(9)
2
232 21
2
1 2 1 1
1.25 ,1m
E E R xE
QR
E E E
(10)
1
2
1 2 1 2
1 2 1 2
04 0 2 0 4
,,
m
x
(11)
This is a semi-empirical probability density which is a mixture of one exponential and two
Rayleigh distributions. It has been derived by fitting the shape of a rain-flow range distribu-
tion via minimizing the normalized error between the rain-flow ranges and the above density
model. The spectral moments
4210 ,,,
constitute a base for the construction of the approx-
imate closed-form Dirlik formula for the probability density of the stress range. The Dirlik
formula constitutes an extension of the Rayleigh distribution to non-narrow band processes. It
is widely used for fatigue crack estimation under wide-band Gaussian stationary applied stress.
Extension to non-Gaussian stress processes requiring the skewness and kurtosis of the stress
process are available in the work by [12].
Using results from random vibration theory, the expected number of cycles
[]EP
per se-
cond for a stochastic process is given by the spectral moments of the process in the form
4
2
1
[] 2
EP
(12)
Substituting (12) into (6) and using the fact that
( ) ( )/ 2 (2 )/ 2p p p
, one obtains
the expected damage rate in the form
14
0
2
(4 ) (2 )D c p d
(13)
Using the Dirlik formula (7) and carrying out the integration analytically, the expected dam-
age rate simplifies to [6]
1 /2 /2
40 1 2 3
2
(8 ) (1 ) 2 1 ( | | )
2
a
D c DQ D R D
(14)
It is clear from the aforementioned formulation and equations (5) and (14) that the ex-
pected fatigue damage rate and, consequently, the fatigue accumulation during a time interval
T
at a point in the structure depends only on the spectral moments
i
,
1,0,2,4i
, of the
stress process
()t
. Using the definition of the spectral moments in (2), the spectral moments
and the fatigue predictions at a point of a structure eventually depend only on the power spec-
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
6
tral density
()S
of the stress process
()t
. The power spectral densities of the stress re-
sponse processes at a point can be calculated from measurements, provided that these meas-
urements are long enough to be considered stationary.
However, there is a limited number of points that can be instrumented in the structure. For
the points where measurements are not available, one has to make predictions of the stress
process and subsequently the power spectral density of the stress process, given the measure-
ments at other locations. This issue is addressed at the next section. Once these measurements
and predictions of the stresses are estimated at measured and unmeasured locations, the power
spectral densities and the corresponding damage accumulation due to fatigue are obtained,
using (5) and (14), everywhere in the structure. In this way, fatigue damage accumulation
maps for the entire structure are constructed from limited number of vibration measurements.
3 RESPONSE PREDICTIONS IN THE ENTIRE STRUCTURE USING
VIBRATION MEASUREMENTS
The objective of this section is to predict the stress response at all points (or DOFs) in a
structure using the measurements at a limited number of locations (or DOFs). This is achieved
using Kalman filter [7]. For convenience and without loss of generality, it is assumed in the
analysis that sensors that are placed in the structure measure the strains.
Consider the dynamic response of a linear structural system subjected to deterministic and
random excitations. Using a spatial discretization method, such as finite element analysis, the
equations of motion for a linear structure are given by the following set of
n
second order
differential equations
( ) ( ) ( ) ( ) ( )
uw
Mq t Cq t Kq t L u t L w t
(15)
where
1
() out
N
qt
is the displacement vector,
M
,
C
and
nn
K
are respectively the
mass, damping and stiffness matrices,
,1
() u in
N
ut
and
,1
() w in
N
wt
are the deterministic
and stochastic excitation vectors applied respectively at
,u in
N
and
,w in
N
DOFs, and
,u in
nN
u
L
and
,w in
nN
w
L
are matrices comprised of zeros and ones that maps the
,u in
N
and
,w in
N
excited DOFs to the
n
output DOFs. Throughout the analysis, it is assumed that the
system matrices
M
,
C
and
K
are symmetric. Let
()yt
be the vector that collects all meas-
urements at different locations of the structure at time
t
. These measurements are collected in
general from sensors such as accelerometers, strain gauges, etc.
Introducing the state vector
q
xq
(16)
the equation of motion can be written in the state space form
()
c c c
x A x B u t G w t
(17)
the measured output vector
yt
is given by the observation equation
y t H x Du t
(18)
where
c
A
,
c
B
and
c
G
are the state transition and observation matrices given by
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
7
11
0
c
I
AM K M C
(19)
11
00
and
cc
uw
BG
M L M L
(20)
respectively.
3.1 Kalman filter approach
Since measurements are available in digitized form, the presentation of the Kalman filter is
next given in discrete time. Using the sampling rate
1/ t
, the discrete-time state space model
corresponding to (17) and (18) is
1
11
k
k k k
x Ax Bu Gw
(21)
k
k k k
y Hx Du v
(22)
where
()
k
x x k t
and
()
k
y y k t
,
1, ,kN
, are the digitized state and output vectors,
and
c
At
Ae
(23)
is the state transition matrix for the discrete formulation. The random variables
k
w
and
k
v
represent the stochastic excitation and the measurement noise, respectively. They are assumed
to be independent, white and following normal probability distributions
( ) (0, )
k
p w N Q
and
( ) (0, )
k
p v N R
, where
Q
and
R
are the stochastic excitation and the measurement noise
covariances assumed to be constant, independent of time.
Kalman filter is used to estimate the state
ˆk
x
of the system described by (21) using the
measurements in the vector
k
y
in (22). Specifically, in the prediction step, an apriori state es-
timate
ˆk
x
of the state vector
k
x
of the system is estimated from equation [7]
11
ˆˆ
k k k
x Ax Bu
(24)
In the correction step, the measured value
k
y
is used to calculate an a posteriori state estimate
ˆk
x
, weighting the measured and estimated signals by the Kalman filter gain matrix
k
K
. This
is formulated by the equation
ˆ ˆ ˆ
[]
k k k k k k
x x K y Hx Du
(25)
where the Kalman gain factor is given by
1
[]
TT
k k k
K P H HP H R
(26)
and, for steady state response, the error covariance matrix
[ ( ) ]
T
k k k
P P E e e
, where
ˆ
kk
e x x
is the a priori error estimate, satisfies the discrete time Riccati equation:
1
()
T T T T T
P APA APH HPH R HPA GQG
(27)
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
8
Let
k
be a vector containing the digitized stresses at various locations of the structure.
Using structural mechanics theory, these stresses in the vector
k
are related to the state vec-
tor through a linear transformation
kk
x
, where
is the transformation matrix that asso-
ciates the state vector to the desired stresses in the entire structure. Consequently, an estimate
of the stresses
ˆk
at time
t k t
is related to the state vector estimate
ˆk
x
through the trans-
formation:
ˆˆ
kk
x
(28)
Herein, the response prediction vector
()t
is restricted to stresses at elements subjected to
uni-axial stress states required in lifetime fatigue estimation as described in Section 2.
Using the definition of the cross power spectral density and the Kalman filter equations (24)
and (25), the cross power spectral density
ˆ()S
of the stress response vector
ˆk
is given
with respect to the cross power spectral density
()
y
S
of the measurement vector
k
y
in the
form
1
ˆ( ) ( ) ( ) ( ) ( )
T T T T
xy
S S E j KS K E j
(29)
where
()Ej
is the matrix given by
( ) ( )
jt
E j Ie I KH A
(30)
and
I
is the identity matrix. Equation (29) relates the power spectral densities of the stresses
at various structural locations with the power spectral densities of the measured quantities in
k
y
available at the limited number of measured locations. This relation depends on the model
(e.g. a finite element model) used to represent the behavior of the structure and the assump-
tion that the excitation vector is broad-band so that the excitations can be modeled by zero-
mean stationary white noise processes with spectral density described by
[]
T
k l kl
E w w Q
,
where
kl
is the Kronecker delta.
It should be noted that in order to apply (27), an estimate of the zero-lag covariance
Q
of
the unknown input stochastic process has to be provided. This estimate is obtained using the
measurements
()yt
and the relation
()
yy yy
Q Q Q
between the covariance matrix
yy
Q
of the
measurement vector
()yt
and the covariance matrix
Q
of the excitation process. Using (22)
with
0D
, this relation is given by
T
yy xx
Q HQ H
(31)
where
xx
Q
is given by the discrete time Lyapunov equation in the form
0
TT
xx xx
A Q A Q GQG
(32)
The optimal values of the entries of the covariance matrix
Q
can be obtained by minimizing
the difference between the covariance matrix
()
yy yy
Q Q Q
predicted by the linear model giv-
en
Q
and the covariance matrix
1
1
ˆNT
yy
Q y y
N
obtained from the measurements in
y
,
1, ,N
. That is, the optimal value
opt
Q
is obtained by minimizing the objective function
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
9
22
ˆˆ
( ) || ( ) || / || ||
yy yy yy
J Q tr Q Q Q tr Q
(33)
with respect to the elements in
Q
. The optimal value
opt
Q
of
Q
is then substituted in (27) to
completely define the Riccati equation (27). The solution
P
of the Riccati equation is substi-
tuted in (26) in order to find
K
which is needed in (29).
3.2 Estimation of power spectral densities of stresses
The cross power spectral density matrix
()
y
S
of the sampled measurement vector
k
y
, in-
volved in (29), can be obtained using available signal processing techniques such as the
Welch technique [13]. Once
()
y
S
has been estimated from the measurements, equation (29)
is used to estimate the cross power spectral density
ˆ()S
of the stress response vector
ˆk
.
Alternatively, the PSD
ˆ()S
of the stress response vector
ˆk
is obtained by using equations
(24) and (25) for the Kalman filter to provide estimates
ˆk
x
of the system state vector which
are then used in equations (28) to estimate the stress vector
ˆk
. Finally, available signal pro-
cessing techniques such as the Welch technique are used to compute the PSD
ˆ()S
from the
sampled stress response vector
ˆk
. The length of the sampled time history should be suffi-
cient large in order for the estimates to be accurate.
The diagonal elements
ˆ
[ ( )]diag S
of the cross power spectral density matrix
ˆ()S
giv-
en in (29) contain the power spectral density estimates required for fatigue predictions using
equations (5) and (14).
4 APPLICATION
The applicability and effectiveness of the methodology is illustrated using simulated
“measurements” from a simple spring mass chain-like model of a structure. The stress state at
critical locations are uni-axial and the fatigue prediction methodology can be directly applied.
The measurements are assumed to be strain measurements. These measurements are simulat-
ed from the model of the structure using wide-band excitation resembling white noise.
Consider the class of
N
-DOF spring mass chain-like model fixed at the two ends as
shown in Figure 1. The model is used to represent a structure consisting of a series of bar and
body elements as shown in Figure 2. The structure consists of
N
bodies with the
i
-th body
having mass
i
m
. The
1i
and the
i
bodies are connected by elastic bar element which pro-
vide the stiffness
i
k
to the system. The number of bar (or spring) elements of the chain model
is
1N
. The material of the bar elements is considered to be steel. For steel bar elements, the
values of the fatigue constants in equation (4) are taken to be
88
4.06 10c
and
9.82
for
steel samples. The
i
-th bar element has length
i
L
, cross-sectional area
i
A
and modulus of
elasticity
i
E
. For simplicity, each bar element is represented by a spring element with stiff-
ness
/
i i i i
k E A L
as shown in Figure 1. Also, the nodal mass
i
m
in Figure 1 includes the ef-
fect of the
i
body mass and the lumped mass arising from the bar elements connected to node
i
. The
i
component
()
i
qt
of the vector
()qt
corresponds to the displacement of the node
i
of
the model. The system is subjected to an unmeasured excitation applied at node
e
.
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
10
Figure 1:
N
-DOF spring mass chain-like model.
Figure 2: Structure consisting of
N
masses and
1N
bar elements.
Property
Type
Property
Value
Units
Material
Steel
Modulus of
Elasticity,
E
2.1e11
2
/Nm
Radius
3.5e-3
m
Length
300e-3
m
Stiffness
16
kk
2.7e7
/Nm
Mass
1
m
21
Kg
Mass
1
m
15
Kg
Mass
1
m
12
Kg
Mass
1
m
15
Kg
Mass
1
m
21
Kg
Table 1: Geometric and material properties.
For demonstration purposes, the excitation vector
k
w
is assumed to be broadband and is
modeled by Gaussian stationary white noise vector process with constant spectral density ma-
trix. White noise provides a good approximation of the input whenever the correlation time of
the input is sufficiently small compared to the system time constants. The discrete state space
formulation of the equations of motion is used to simulated response time history data as well
as compute estimates of the covariance responses and the power spectral density of the re-
sponses using the white noise excitation applied at node
e
. In particular, the strain and stress
response time histories
k
and
k
, respectively, are simulated at all bar elements using the
state space formulation. The time discretization step used in simulated the sampled data is
3
0.5 10t
seconds. The simulated strain and stress response time histories are the refer-
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
11
ence stress response time histories that are considered to be the exact stress response time his-
tories for the excitations used. These response time histories and the corresponding power
spectral densities can be used with the fatigue prediction methodology to compute the lifetime
of the structure due to fatigue failure. Such predictions constitute the reference (exact) predic-
tions against which the predictions from the proposed Kalman filter approach should be com-
pared to for assessing the accuracy of the proposed methodology.
Let
o
be the set that contains the bar element numbers where the strains are measured. The
simulated sampled time history of the strain responses
k
associated with the nodes or DOFs
included in the set
o
are included in the measurement vector
()o
kk
y
. In practice, these
measurements are collected using appropriate sensors such as strain gauges. Let
p
be the set
that contains the bar element numbers where the stresses will be predicted. The set
p
is se-
lected to be
{1, , 1}pn
, i.e. it is assumed that stresses are predicted at all bar elements.
Results demonstrating the effectiveness of the proposed methodology are presented for a
five degree of freedom system (
5N
) shown in Figure 1. The nodal masses as well as the
geometric and material properties of the bars are listed in Table 1. The damping matrix
C
in
the equations of motion is chosen assuming that the system is classically damped. Rayleigh
damping is assumed, i.e.
C aM bK
. A single excitation is considered which is applied at
node
5e
. The strain response time histories
()o
kk
y
at the measured DOFs are used to
predict the stress response time histories at all bar elements identified in the set
p
using the
proposed Kalman filter approach.
Figure 3: Comparison between reference and estimated from Kalman filter stress response time histories.
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
12
Comparison between the reference (exact) stress time histories simulated by the model and
the estimated time histories from the Kalman filter are given in Figure 3 for all bar elements
in the set
p
, assuming that the measured strains are at bar elements
{2,4}o
. Similar com-
parisons for the power spectral densities of the stress time histories are shown in Figure 4. It
can be seen that the estimated stress time history and the power spectral densities of the stress
at the bar elements 2 and 4, where measurements are available, coincides with the correspond-
ing reference stress time histories and stress power spectral densities simulated by the model.
At the other bar elements, there is a discrepancy between the estimated and reference (exact)
stress time histories or power spectral densities. These differences are due to the estimation
error associated with the Kalman filter. However, it should be noted that the predictions from
the Kalman filter approach are quite good, especially for the high amplitudes around the reso-
nance peaks which mainly contribute to the fatigue process.
Figure 4: Comparison between reference and estimated from Kalman filter power spectral densities of the stress
responses.
Lifetime predictions due to fatigue are shown in Figure 5 for all six bar (spring) elements
of the structure as a function of the intensity
F
s
of the input white noise, where
F
s
is the
standard deviation of the input. The lifetime values in the Figure are obtained using the fa-
tigue prediction formula (5). For each bar element, there are two lifetime fatigue predictions.
The first prediction is based on the reference time histories simulated by the model and it is
used as the exact value against which to study the accuracy of the predictions from the pro-
posed methodology. The second fatigue-based lifetime estimate is the one predicted by the
methodology based on the use of Kalman filter approach to estimate the stress response time
histories at all bar elements.
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
13
Figure 5: Comparison of reference fatigue-based lifetime estimates and the estimates predicted by the Kalman
filter as a function of the excitation amplitude for the
5N
DOF model.
To study the effect of the number and location of sensors (measurement points) on the ac-
curacy of the predictions, Figures 6a and 6b presents predictions in the entire structure (all six
bar elements) using two sensors placed at bar elements
{2,4}o
and
{2,3}o
, respectively,
while Figures 6c and 6d present predictions using three and four sensors placed at bar ele-
ments
{1,2,4}o
and
{1,2,3,6}o
, respectively.
Figure 6: Comparison of reference fatigue-based lifetime estimates and the estimates predicted by the Kalman
filter for the
5N
DOF model.
Costas Papadimitriou, Claus-Peter Fritzen, Peter Kraemer, Evangelos Ntotsios
14
It can be seen that the estimates based on the Kalman filter predictions are quite close to
the reference fatigue values obtained from the actual simulations. Comparing the fatigue life-
time predictions in Figures 6a-d, it becomes clear that the accuracy of the predictions depend
on the number and location of sensors in the structure. One can improve the estimates consid-
erably using optimal sensor location methodologies [14].
5 CONCLUSIONS
A methodology for estimating damage due to fatigue on the entire body of a structure us-
ing spectral methods and output only vibration measurements at a limited number of locations
was presented. The fatigue predictions presented in this work are limited to structural mem-
bers subjected to a uni-axial stress state. These predictions can be extended using available
methods to structural members subjected to multi-axial stress state. Kalman filter is used for
predicting the power spectral densities of the stresses in the entire body of the structure using
the available measurements and a model of the structure. These predictions can be used to
construct fatigue accumulation and lifetime prediction maps consistent with measurements
provided by a monitoring system. The results from the proposed methodology are useful for
designing optimal maintenance strategies for metallic structures using information for the ac-
tual condition of a structure, collected from a sensor network placed at limited number of lo-
cations.
ACKNOWLEDGEMENTS
This research was funded by the Greek National Scholarship Foundation (IKY) within the
IKYDA program framework and by the Deutscher Akademischer Austausch Dienst (DAAD,
German Academic Exchange Service) within the PPP program. This research was also co-
funded 75% from the European Union (European Social Fund), 25% from the Greek Ministry
of Development (General Secretariat of Research and Technology) and from the private sec-
tor, in the context of measure 8.3 of the Operational Program Competitiveness (3rd Commu-
nity Support Framework Program) under grant 03-ΕΔ-524 (PENED 2003).
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