Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Diagnosis performance of vibration-based structural identification
and health monitoring
Guido De Roeck1
Department of Civil Eng., Faculty of Engineering, University of Leuven, Kasteelpark Arenberg 40, 3001 Leuven, Belgium
email: [email protected]
1
ABSTRACT: Vibration-based Structural Identification has become an increasingly popular experimental technique, e.g. for
model calibration, control of structural behavior during construction, assessment of the efficiency of structural repair, … Main
breakthrough was the development of operational modal analysis (OMA), avoiding the use of artificial vibration sources.
Vibration-based Structural Health Monitoring is based on the principle that modal parameters of a structure are stiffness
dependent. Changes in natural frequencies, damping ratios, mode shapes or combinations can therefore be used as features to
detect and to identify damage. Challenges for the four levels of damage assessment will be treated. From the measurement side,
wireless sensors and optimal sensor placement strategies will be addressed.
KEY WORDS: System identification; Operational modal analysis; Elimination environmental influences; Damage assessment;
Wireless sensor networks; Optimal sensor placement.
1
INTRODUCTION
Vibration-based damage identification is based on the
principle that modal parameters of a structure are stiffness
dependent. Changes in natural frequencies, damping ratios,
and mode shapes can therefore be used as features to detect
and to identify damage. Compared to other approaches for
structural damage identification, vibration-based damage
identification has the advantages of (1) being nondestructive,
(2) being able to identify damage that is invisible at the
surface, (3) being ‘global’ because no a priori location of the
damage needs to be assumed as opposed to local methods
such as ultrasonic testing. A good overview of recent trends is
given in [18].
Output-only or operational modal analysis became very
popular as no artificial vibration source is needed which is
anyhow impossible in many cases. Powerful time domain
system identification algorithms like Stochastic Subspace
Iteration (SSI) have replaced the obsolete “peak-picking”
method [1], [2]. Although theoretically a white noise
(unmeasured) excitation is assumed, in practice these
algorithms provide accurate lower modes and good damping
estimates, even in case of (moderately) “colored” excitation.
Moreover, operational modal analysis can be used for a
permanent monitoring installation. In this case automated
system identification (i.e. automatic treatment of response
data with minimum or zero user intervention) is mandatory,
while it is highly advisable for periodic monitoring [6].
To cope with the quite distinct vibration levels during
operational and ambient loading, a large dynamic range of the
sensors is required and a 24 bits resolution for the A/D
conversion is advised. Main disadvantage is the dependence
on the temporal and spatial characteristics of the unmeasured
ambient loads, exciting a limited number of the lowest modes.
A possible to circumvent this drawback is to supplement the
present ambient excitation with a relatively small artificial
excitation. System identification algorithms to deal with this
hybrid stochastic-deterministic loading have been developed
successfully [3], [4].
Results can be used for calibration of numerical FE-models
inherently containing uncertainties especially related to
boundary conditions, joint stiffnesses, structural contribution
of non-bearing parts, material parameters, damping, …
Subsequently, properly calibrated FE-models can be used to
derive from response measurements the actual excitation like
moving loads on road or railway bridges and wind forces on
tall structures. Moreover, these models can be used to obtain
from the response measurements in a limited number of
sensors information in otherwise difficult to assess points, like
strains close to weldings or forces in bolts [5].
Other interesting applications relate to the follow-up of
critical phases during erection of constructions.
However, one of the main applications remains vibrationbased damage identification. In general four levels are
discriminated in damage assessment. Challenges for each of
these levels will be addressed.
The first level, damage detection (also called the “alarm
level”) is mainly situated in the context of permanent
monitoring. A damage related dynamic feature is permanently
registered and a statistically relevant deviation is considered
as a sign of possibly occurred damage. Probably this warning
will then be followed by a more thorough inspection.
Although a lot of potential damage features have been
proposed, natural frequencies and mode shapes are still the
most appealing. They are rather insensitive to the influence of
unknown excitation, easy to extract by applying state-of-theart system identification algorithms and many other features
(like strain energies) can just be derived from them.
Unfortunately, natural frequencies are not only sensitive to
damage, but also to changing environmental conditions such
as temperature variations, and moreover, their estimation from
vibration response data is also prone to experimental errors
[7],[8],[9]. So far the influence of the environment on mode
11
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
shapes and modal strains is less understood. Challenge is to
eliminate the environmental influences by prediction models
that rely just on estimated modal properties (i.e. output-only
models) or on the measured environment as input and the
estimated modal parameters as output (i.e. input-output
models). These models are trained by using data gathered in
undamaged condition.
Concerning levels two and three, localization and
quantification of damage, the most powerful methodology is
based on FE-model updating. It means that, by an
optimization process, differences between measured and
calculated modal parameters are minimized, taking
parameterized damage patterns as updating variables. A
probabilistic approach is necessary to account not only for the
remaining uncertainty after filtering the environmental
influence, but also the imperfections of the assumed numerical
model. So far, mostly quasi-linear behavior of the damaged
structure under small vibrations is assumed. Another
challenge for the future is updating of nonlinear models of the
damaged structure, which might be a step closer to level four.
To localize and to quantify damage can be an objective of a
permanent monitoring system interrogating a relatively low
number of sensors or the goal of a periodic inspection making
use of a more or less dense network of sensors. In the latter
case, the use of wireless sensors can greatly reduce the cost of
the measurement campaign.
Level four, prediction of the remaining life time, is still an
unresolved issue as changes in the dynamic behavior are
basically related to stiffness degradation and not directly to
strength reduction. Anyhow, in many cases there is at least
some relation between the two. In fatigue assessment,
vibration monitoring can deliver the true fatigue load. This
load can be introduced in a (calibrated) FE-model to predict
the remaining capacity by applying standard procedures for
fatigue analysis. This is of growing importance for bridges
because the dynamic loading due to traffic has increased
severely during the past decades.
A particular challenge for all four levels is the discovery of
small local damage. Not only natural frequencies and mode
shapes will hardly be affected but also its influence hidden in
the uncertainty blur. Therefore, the development of a
distributed strain sensor network able to cope with the very
low strain intensities during ambient excitation is a challenge
for future developments. Optical fiber sensors with Bragg
grating technology permanently attached to the structure could
be a good choice in this respect. Interrogation units are still
quite expensive but can be coupled and uncoupled when
adopting a periodic monitoring maintenance strategy. An
additional advantage of the measured strain field is that it can
be directly related to the stress field. Such a system will also
be able to measure quasi-static deformation as occurs in case
of shrinkage, creep, thermal expansion and very slowly
applied dynamic loads.
This paper will focus on some advances linked to the
previously mentioned challenges: the elimination of
environmental influences, the use of wireless sensors and the
design of an optimal sensor placement strategy.
12
2
FILTERING OF ENVIRONMENTAL INFLUENCES
It is well known that environmental changes like temperature
variations may have an important influence on features of the
dynamic characteristics such as natural frequencies [7],[8]. So
they may mask the influence of damage completely. A second
cause of inherent variance is due to estimation errors while
applying system identification algorithms to the measured
response data [9].
When monitored over a short period of time, in which the
rather slowly varying environmental influences remain
constant, a structure behaves like a linear time-invariant
system. However, over longer time spans, the dynamic
features change in a nonlinear way because of the nonlinear
relationships between environmental parameters (like
temperature) and stiffness of structural materials or boundary
conditions. Moreover, due to the large thermal inertia of most
structures, there can be a time delay between environmental
changes and adaptation of dynamic features. To connect
changes of dynamic features, e.g. natural frequencies, mode
shapes and/or modal strains, to damage, it is of paramount
importance to account for this environmental variability.
Therefore, global prediction models should be derived from
measurements over a long time span, preferentially covering
the complete range of possible environmental parameter
values.
A possible approach for constructing the global prediction
model is measuring the environmental factors that influence
the damage-sensitive features, and identifying a black-box
input model with these environmental factors as inputs and the
corresponding features as outputs. However, a major difficulty
with this input-output approach is to determine which
environmental (or even operational) influences should be
measured, and where the corresponding sensors should be
placed. This can be overcome by employing output-only
system identification methods, for which measurement of the
environmental parameters is not necessary.
2.1
Output-only prediction model [10],[11]
An output-only technique that has been applied for
eliminating environmental influences on features such as
natural frequencies is linear static principal component
analysis (PCA), estimating a static linear relationship between
the modal parameters and the unknown environmental
parameters. However, in practice, this relationship may be
strongly nonlinear. A very promising method for SHM in
changing environmental conditions is kernel principal
component analysis. This is a nonlinear version of PCA for
which the type of nonlinearity does not need to be defined
explicitly. Moreover, it is easy to implement and
computationally very robust and efficient. In [10], an
improved output-only technique for constructing a nonlinear
prediction model is developed and validated on real-life
monitoring data. It is based on Gaussian kernel PCA, where
the two parameters of the model are automatically determined.
The first parameter, which represents the standard deviation of
the Gaussian (or radial basis function) kernel is chosen in such
a way that the matrix of mapped output correlations is
maximally informative, as measured by Shannon’s
information entropy. The second parameter, which equals the
number of principal components in the mapped feature space,
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
is chosen in such a way that the retained principal components
amount for nearly all normal environmental and operational
variability.
In a first (training) step the available data are split in time
sequences that are short compared to the parameter variations,
and damage-sensitive features are estimated for each
individual sequence.
Then the second step consists in identifying a nonlinear
prediction model, where the time-varying features of the first
step are considered as outputs. The validity of this model can
be checked by applying it to new data of the still intact
structure.
After that, the structure can be monitored by continuously
repeating the first step. The features found in this way are
compared with the ones predicted by the model of the second
step. When the discrepancy becomes large, the model alone
insufficiently explains the evolution of the features, so the
structure may be damaged.
This methodology is illustrated by application to the data set
of the Z24, a bridge monitored for almost a year before
realistic damage patterns where introduced in a controlled
way. These tests were performed in the framework of the
Brite-EuRam project CT96 0277 SIMCES.
Figure 1. Z24 bridge.
The Z24 bridge was part of the road connection between the
villages of Koppigen and Utzenstorf, Switzerland,
overpassing the A1 highway between Bern and Zürich (Figure
1). It was a classical post-tensioned concrete two-cell boxgirder bridge with a main span of 30m and two side spans of
14m. The bridge, which dated from 1963, was demolished at
the end of 1998, because a new railway adjacent to the
highway required a new bridge with a larger side span. Before
complete demolition, the bridge was subjected to a short-term
progressive damage test, in which 17 different damage
scenarios were applied.
During the year before demolition, a long-term continuous
monitoring test took place to quantify the environmental
variability of the bridge dynamics. Every hour, 65536
acceleration samples from 16 sensors at different points and in
different directions were collected. The continuous monitoring
system was still running during the damage tests preceding the
demolition.
In [10] the Z24 data are used for to apply both linear and
kernel PCA. Selection of the modal parameters from the
stabilization diagram was done in an automated way. The first
four natural frequencies are taken as damage-sensitive
features. They correspond with a vertical bending mode
around 4 Hz, a lateral bending mode around 5 Hz, and two
modes combining vertical bending and torsion around 10 and
11 Hz.
A total of nt = 3000 data points, which amounts to about
50% of the number of data points available, are used for
training the model. The other data points are used for
monitoring. Figure 2 shows the result of the misfit ek when
applying linear PCA: as can be seen only part of the
environmental variation is covered by the training data.
Figure 2. Z24 bridge, misfit of the linear output-only model
constructed with 3000 data points. Blue: training data. Green:
monitoring data in undamaged condition. Red: monitoring
data in damaged condition.
For the application of the kernel PCA again a total of nt =
3000 data points are used for training the model. The
parameter in the radial basis function kernel is determined by
maximizing the information entropy. Next, the number of
retained principal components nu is determined based on the
criterion that nu should account for at least a certain fraction F
(e.g. 99%) of the normal variability.
Figure 3. Z24 bridge, misfit of the nonlinear output-only
model constructed with 3000 data points. Blue: training data.
Green: monitoring data in undamaged condition. Red:
monitoring data in damaged condition.
The evolution of the misfit is shown in Figure 3. In the
validation phase, the prediction error is not higher than during
the training phase when the bridge is still in undamaged
condition, except that there is a slight increase after 25 May,
when the undamaged bridge is subjected to environmental
conditions that are not covered by the training data. As soon
as progressive damage is applied to the structure, the
prediction error grows very significantly, so these unwanted
13
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
system changes can be clearly detected, either visually or with
a novelty detection algorithm.
It can be concluded that, when monitoring the misfit
between the model predictions and the observed natural
frequency data with linear PCA, it was difficult to
discriminate the damaged from the undamaged condition. On
the other hand, the nonlinear PCA approach allowed a clear
detection of the onset of damage.
2.2
Input-output prediction model [12]
As an alternative to eliminate the environmental influence, a
nonlinear prediction model, that takes the measured
temperatures as inputs and the estimated modal characteristics
(e.g. the natural frequencies) as outputs, can be constructed.
This influence is in general nonlinear and it affects different
modal characteristics in a different way, so nonlinear system
identification techniques are required. The nonlinear system
model is identified using modal characteristics that have been
obtained during a reference period in which the structure is
undamaged. During the subsequent monitoring period, the
values that are predicted by this model (using the measured
environmental variables such as temperatures as inputs) are
compared with the modal characteristics that are actually
observed. If this difference is small, then the variations in
measured modal characteristics can be well explained by
environmental variations. If on the other hand this difference
is large, then the observed variations in modal characteristics
are caused by other factors, possibly structural damage. It is
possible to account for uncertainty on both the predicted and
the measured modal characteristics. It is then possible to apply
robust decision rules for deciding whether or not the structure
may be damaged and should be inspected.
An efficient method for the identification of a nonlinear
input-output environmental model is proposed. The output of
the model is a sequence of vectors mk ∈ Rnm containing modal
characteristics that have been estimated from short-term
acceleration data. The input is a vector sequence of
corresponding environmental variables uk ∈ Rnu. Only the first
nt input-output samples (out of a total of ns samples) are used
as training data for identifying the model. After removal of the
DC input component, this model reads
mk = WTφ(uk) + b + ek
(1)
where uk ∈ Rnu is the vector with known inputs and ek an error
term that accounts for the misfit between the data and the
model. The term b is the DC component of the output. φ
represents a nonlinear mapping of (uk) onto a possibly very
high-dimensional or even infinite-dimensional mapped feature
space G:
φ : Rnu → G, uk → φ(uk)
(2)
In LS-SVM, the model (1) is identified in a least-squares
sense under the additional constraint that the regression
should be smooth.
Further, the uncertainties of the experimentally determined
modal characteristics can be quantified. The environmental
variables such as temperature are directly measured with high
14
accuracy, so their uncertainty is negligible with respect to that
of the modal characteristics. Next, propagating these
uncertainties through the SVM model delivers the probability
distribution of the normalized modal characteristics.
Once an environmental model is identified, the considered
structure can be monitored by regular experimental
determination of a modal characteristics vector mk and
comparing it with the corresponding environmental model
prediction ḿk An unwanted change causes the system to
behave differently as in the period during which the
environmental model was trained, so that the misfit ek grows.
In a stochastic analysis, the misfit is a random variable. Since
both mk and ḿk are normally distributed, the misfit, which is
the difference between both, is also normally distributed. A
probabilistic damage indicator can now be adopted. One
possibility is to use the Mahalanobis distance between the
misfit and zero as damage indicator.
Figure 4. Cross section of the Z24 showing the temperature
sensors.
The data of the Z24 bridge are used to investigate the
performance of LS-SVM in solving the monitoring problem.
The lowest two monitored natural frequencies, corresponding
to the first vertical and first lateral mode, are taken as outputs
and a subset of nine measured temperatures (Figure 4) as
inputs for the environmental model. A total of nt = 2664 data
points were used for training the model; in this way a large
portion of the temperature range is covered by the training
data. The other data points were used for monitoring. There
are approximately as many monitoring data in undamaged
condition as there are monitoring data in damaged condition.
The evolution of the misfit between the predictions of the
environmental model and the observed natural frequencies, as
measured by the Mahalanobis distance is shown in Figure 5.
In the validation phase, the prediction error is not significantly
higher than during the training phase when the bridge is still
in undamaged condition. As soon as progressive damage is
applied to the structure however, the prediction error grows
very significantly, so these unwanted system changes can be
clearly detected, either visually or with a novelty detection
algorithm.
The computational effort is largely reduced by changing the
objective function in the Support Vector Machines (SVM)
approach, so that a linear least squares problem needs to be
solved for identifying a nonlinear system model instead of a
quadratic programming one. The robustness issue is tackled
by quantifying the uncertainty on the modal characteristics
that are used as output data, and by propagating these through
the SVM model, so that the probability distribution of the
normalized modal characteristics is obtained.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Figure 5. Z24 bridge, Mahalanobis distance of the prediction
error from zero. Blue: training data. Green: monitoring data in
undamaged condition. Red: monitoring data in damaged
condition.
3
measurements, the modal parameters of the bridge structure
can be extracted. The ADC includes a high-performance,
linear-phase, digital and antialiasing filter; a high-pass
integrated filter; and digital control of the sample rate. The
signal conditioning circuit adapts common signals from the
ICP sensors to differential, filters the noise, and supplies the
necessary current for the ADCs.
The Zigbee protocol, defined in IEEE standard 802.15.4, is
used for the wireless synchronization module. This
synchronization module provides an initial reference to the
system, with less than 125-ns difference between all boxes.
The next pulses to the boxes are used to fix derivations in the
frequency clock.
For
communications
between
the
boxes
and
transmission/reception of data the 802.11 protocol, also
known as Wi-Fi, is exploited. It is possible to add as many
clients as there are boxes, and they are automatically assigned
an IP address. This feature provides a scalable architecture.
Measurements are stored temporarily on a 2-GB SD card.
This system has been tested successfully on a number of
bridges (Figure 7).
WIRELESS SENSOR NETWORKS
Because of the omission of cables, wireless networks (WNs)
have a lot of evident advantages, like the ease of system setup,
the largely reduced installation time, the possibility of local
data processing, data interpretation and anomaly detection.
Nevertheless, there is still a need for improvements of the
current WNs:
1) Reliability: packet collisions can occur in WNs because a
share transmission medium is used. Moreover, when the
distance between the nodes is too far, packets may not reach
the destination.
2) Scalability: to characterize the state of the structure, a lot
of raw data have to be sent over the air. This poses the
problem of the scalability for an increasing number of modes.
3) Synchronization: time-synchronization errors can cause
inaccuracy in the extracted mode shapes. Each sensor has its
own local clock, which is not initially synchronized with the
other sensor nodes. Synchronization errors affect the process
of obtaining mode shapes. There are two primary sources of
jitter: temporal jitter and spatial jitter. Temporal jitter takes
place inside a node, and spatial jitter occurs between different
nodes due to variation in node oscillator crystals and
imperfect time synchronization.
In a recent project a wireless measurement with high timesynchronization accuracy was developed [13]. Spatial jitter
was reduced to 125 ns, far below the 120 µs required for highprecision acquisition systems and much better than the 10-µs
current solutions, without adding complexity. Moreover, the
system is scalable to a large number of nodes to allow for
dense sensor coverage of real-world structures
The system has two components: a host laptop (server) and
boxes or nodes (clients). The architecture of a box is shown in
Figure 6. The acquisition module contains a signal
conditioning circuit, two low-noise ADCs with a sample rate
between 1 and 192 kHz and a 24-bit resolution, and a PIC32
microcontroller. The ADC has two differential inputs that
provide up to four channels in each box, allowing connection
of 4 piezoelectric ICP accelerometers. From the acceleration
Figure 6. Block diagram of a wireless box.
Figure 7. Wireless boxes in use for a bridge test.
4
OPTIMAL SENSOR PLACEMENT
In most of modal testing campaigns, there are more degreesof-freedom (DOFs) to be measured than sensors available.
The whole measurement grid then needs to be covered in
several phases by different measurement setups. A number of
reference locations is then selected and transducers at these
points are often kept fixed in all setups. The ideal location for
a reference sensor is a position where all modes have
15
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
relatively large modal amplitudes. However, there exists no
guidelines or general principles for evaluating whether a
chosen number of reference sensors leads to sufficiently
accurate mode shape estimates. In practice, even when
measuring large structures, often a very limited number of
reference sensors is allocated. The other transducers are
roved in successive setups, hence these transducers are often
termed `roving' sensors, so that by the end of the last setup,
vibration responses at all grid points will have been recorded.
So for any measurement campaign, the following questions
have to be answered: (i) the location of the reference sensors,
(ii) their number, (iii) the location of the roving sensors in the
different setups.
Answers can be given by applying Optimal sensor location
(OSL) approaches. Their practical applicability will be
illustrated on a large-scale operational modal test of a ﬁvespan steel truss railway bridge [15]. The measurement layout
was designed according to intuitive reasoning, based on the
modal results of a preliminary Finite Element analysis.
Afterwards, this test design is reviewed using OSL.
4.1
The Guadalquivir railway bridge
The Guadalquivir railway bridge in Andalusia, Spain, is a
five-span continuous twin steel truss with one track in each
direction (Figure 8). The spans have almost equal length:
50.48 + 50.94 + 50.94 + 50.94 + 50.61 (m). The abutments
are labeled as E-1 and E-2 and the four piers as P-1, P-2, P-3,
P-4 (Figure 9). The fixed bearings of the bridge are located on
top of pier P-2.
nodes, except a limited number of nodes in the upper chords
at the end portal frames.
Figure 10. Measurement layout: grid and position of reference
sensors.
The design of the measurement layout (Figure 10) was
based on the modal results of a preliminary Finite Element
analysis. To cover all measurement points, twenty-seven test
setups were needed. Four fixed reference sensors were placed
on the right main truss at bay numbers 9, 27, 43 and 63. These
reference sensors were positioned at the locations of
significant modal displacements of many lower modes. The
fifth fixed reference sensor was placed at node 63 on the left
main truss to provide comparison signals between the two
main trusses.
The remaining seven ‘roving’ sensors were distributed in a
way that, at each particular setup, there is at least one sensor
in each of the five spans. The last setup (27) was designed to
measure the vibration of the transversal portal frame and also
helped in distinguishing the global torsion modes. In this
setup, four sensors were installed on a few accessible upper
chord nodes at the end portal frames 0, 86 and 90. The
sampling frequency was 200 Hz and the measurement
duration was about fifteen minutes per setup. On average it
took about 30 minutes to complete one setup including the
time to relocate the roving sensors. Rail traffic was operating
normally, with up to 5 train passages during rush hours.
4.3
Figure 8. The Guadalquivir railway bridge.
Figure 9. Structural overview of the Guadalquivir bridge.
4.2
Test setup
The bridge was extensively tested by making use of twelve
wireless sensor units (GeoSIG GMS-18) which are connected
by a time-synchronous wiﬁ network. Each unit is equipped
with a triaxial accelerometer. Due to safety reasons, the
measurement could only be performed on the bottom chord
16
Modal analysis results
After the test operational modal analysis was performed using
the Matlab toolbox MACEC 3.2 [17]. Thirty-seven modes
have been identified within the frequency range from 2.78 Hz
to 14.21 Hz. The identiﬁed modal characteristics are given in
Table 3. The identiﬁed modes are all characterized by a low
damping ratio. Most of them have a mode shape that is almost
purely real, as evidenced by high MPC and low mean phase
deviation (MPD) values, thus indicating a high estimation
accuracy.
All modes with a natural frequency between 2.78 and 3.36
Hz are transverse modes. In the frequency range from 4.36 to
7.37 Hz, seven vertical modes are identiﬁed. The other six
modes in this frequency range are either transverse modes or
modes with coupled transverse and vertical displacements.
Mode 11 (5.16 Hz) and mode 12 (5.22 Hz) are closely spaced.
The quality of the higher transverse mode shapes 19 and 20 is
not as good as for the other four transverse modes of the same
frequency range. The ﬁrst two modes in the frequency range
from 8.46 to 14.21 Hz also belong to the transverse mode type
combined with vertical modal deformation. Almost all modes
from 23 onward are torsional modes of increasing frequency
(from 8.90 Hz to 14.21 Hz) and modal order, in both vertical
and transverse directions. Modes 31 and 32 can also be
considered as transverse/combination modes.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 1. Summary of identified modes (transverse ; vertical
 ; torsional/combined  ).
4.4
No.
Type
f
[Hz]
ξ
[%]
MPC
[-]
⁰
MP
[ ]
⁰
MPD
[ ]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37





































2.78
2.90
2.99
3.13
3.24
3.36
4.36
4.48
4.65
4.78
5.16
5.22
5.36
5.51
5.59
6.06
6.70
7.04
7.19
7.37
8.46
8.52
8.90
9.11
9.33
9.44
9.65
9.87
10.02
10.48
10.55
10.79
11.09
11.61
12.02
12.59
14.21
0.73
0.65
0.64
0.72
0.45
0.59
0.55
0.37
0.67
0.90
0.40
0.24
0.63
0.23
0.94
0.45
0.72
0.63
0.71
0.73
0.60
0.67
0.31
0.24
0.25
0.29
0.35
0.43
0.29
0.21
0.38
0.41
0.37
0.43
0.40
0.26
0.28
0.976
0.985
0.876
0.962
0.972
0.912
0.984
0.980
0.878
0.890
0.907
0.953
0.961
0.936
0.842
0.961
0.935
0.759
0.811
0.791
0.854
0.847
0.945
0.950
0.933
0.951
0.936
0.835
0.837
0.926
0.878
0.935
0.947
0.914
0.905
0.841
0.843
2.2
0.9
14.3
1.7
0.1
1.5
0.6
2.4
9.8
9.5
4.6
0.4
5
0.1
3.7
5.6
5.9
11
3.6
7.2
5.1
6.1
1.1
3
0.5
1
3.7
0.9
12.4
0.2
9.2
8
0.7
4.4
8
2.1
9.6
5.7
5.1
13
6.5
5.7
12.9
3.5
3.8
10.6
10.4
12.7
9.8
6.4
9.1
12.9
6.8
8.5
16.3
13.4
15.2
12.5
12.1
6.7
6.4
7.8
7.3
7.5
13.9
12.6
8.2
11.1
7.9
7.1
7.6
8.2
13
11.7
Comparison measured and calculated modes
In Figures 11 and 12 the vertical and the transverse modes are
compared. Some modes didn’t appear in the FE calculation.
The longitudinal displacements of the nodes at the lower
chords are relatively small compared to those in other
directions. Especially at the supports, the almost zero
longitudinal displacements indicate that they behave as
practically ﬁxed in the longitudinal direction. It seems that the
rollers allow free longitudinal displacement for static loading
and thermal expansion but behave as ﬁxed under small
amplitude vibration.
4.5
response and the modal coordinates of the finite element
model is
Review of the test design using optimal sensor
placement
After the field test, the original design of the test setups is
reviewed by using an optimal sensor location (OSL)
algorithm, which was recently developed [14]. Papadimitriou
and Lombaert have proposed to choose the sensor locations so
that the uncertainty of the identified modal coordinates, as
measured by their information entropy, is minimized. When
yk denotes the measured response, Φ the matrix with mode
shapes computed from the preliminary finite element model,
ξk the modal coordinates corresponding to this model and ek
the prediction error due to both modeling inaccuracies and
measurement errors, the relationship between the measured
Figure 11. The identified vertical modes and the
corresponding modes computed with the finite element model.
yk = L[Φ ξk + ek]
(3)
where L denotes the observation matrix. This matrix is
comprised of ones and zeros and maps, for a given
configuration, the measured DOFs to the DOFs of the finite
element model. Determining the optimal sensor configuration
is equivalent to choosing the observation matrix L in such a
way that the information entropy of the estimated modal
coordinates ξk is minimized. This is equivalent to maximizing
the determinant of the Fisher information matrix of ξk (as a
function of L), which reads
Q(L; Σ) = (L Φ)T (LΣLT)-1(L Φ)
(4)
where Σ denotes the covariance matrix of the prediction error
ek.
Since the objective function det Q(L; Σ) depends on the
covariance of the prediction error ek,, the same holds for the
optimal sensor configuration. The following model for the
prediction error covariance was proposed in [14]:
Σij =E[ek,iek,j ]= Σii Σ jj exp(-δij /λ)
(5)
17
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
where δij denotes the distance between DOFs i and j, and λ is a
scalar defining the degree of spatial correlation, called the
correlation length. A value of λ = 0 means there is no
correlation in the prediction error between any two
measurement DOFs.
Figure 12. The identified transverse modes and the
corresponding modes computed with the FE model.
An information entropy index (IEI) was also introduced as a
normalized value to compare the difference between any
configuration to the full configuration where all possible
measurement locations are equipped with sensors:
(6)
IEI is a function of three variables that are unknown in the
design stage. It depends on the number of sensors employed,
their configuration and the prediction error correlation matrix
among different channels. A large value of IEI means that the
information content is small in relation to the full
configuration. IEI should go down to unity as the number of
sensors deployed approaches the full mesh.
Two heuristic sequential sensor placement (SSP)
algorithms, the forward (FSSP) and the backward (BSSP),
18
were proposed for solving the optimal sensor configuration
problem as formulated in Equation (4).
FSSP starts without any sensor. The positions of sensors are
computed sequentially by placing one sensor at a time in the
structure at a position that results in the highest reduction in
information entropy. Specifically, at each iteration,
combinations with an additional sensor to the previous
configuration are considered, and the information entropy of
all new sensor configurations are evaluated. The one that
minimizes the information entropy is selected. On the
contrary, BSSP is accomplished in a reverse order, starting
with sensors placed at all measurable nodes on the structure
and removing successively one sensor at a time from the
position that results in the smallest increase in the information
entropy.
The actual test was very extensive with all truss nodal joints
measured. Now, we retrospect whether we can coarsen the
measurement grid and reduce the number of fixed reference
sensors to make more sensors available for roving, in this way
reducing the number of setups and so to speed up the test.
Since upper chord access is prohibited, therefore 2 × 91 =
182 measurable locations have to be considered. In the
transverse Y-direction it is expected that a sensor gives the
same response whether put on the left main truss node or the
right one of the same portal frame (section). Moreover, taken
into account the longitudinal plane of the bridge, measuring
on one truss plane is enough to characterize the dynamic
properties of the bridge. So, the problem of finding several
optimal locations for the fixed reference sensors is restricted
to one measurement line along the lower chord of either truss
plane, with 91 nodes from the section 0 to section 90. The
OSL problem is run considering seven FE vertical modes.
As the 91 measurable locations form a relatively dense grid,
it can already be anticipated that taking into account
correlation between sensors is important to maximize the
quality of the data. It is assumed that the prediction errors
between measurement channels are mutually correlated
depending on their relative distance. The correlation length is
chosen as = 0.001 (m), corresponding to the uncorrelated
scenario and = 2d = 5.66 (m), with d the typical length of
the truss bay and also the minimum distance between the two
consecutive nodes of the mesh, corresponding to the
correlated scenario.
The IEI for the correlated scenario is shown in Figure 13.
The change in IEI indices is signiﬁcant when the number of
sensors is less than 4. When there are more than ﬁve sensors,
they are distributed over all spans. If the number of sensors is
further increased, there is a tendency of a spread-out over the
measurement line. If we position 4 reference sensors,
according to the correlated scenario and BSSP, the
conﬁguration would be L = {9, 27, 44, 63} (Figure 14). This
setting has a fairly low IEI and is actually very close to the
choice L = {9, 27, 43, 63} based on intuition and experience
(see Figure 13 □).
Next, the method is employed again for one measurement
line but using all 19 FE mode shapes up to a frequency of 7.70
Hz (the ﬁrst 7 vertical modes, the ﬁrst 11 lateral ones and one
torsion mode) instead of the 7 vertical modes that were used
before. Question is now to choose the optimal position of the
triaxial sensors. It is assumed that there is no correlation
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
among the vertical, transverse and longitudinal directions. The
correlation length in any ﬁxed direction is taken the same as
before. Figure 15 suggests that at least 10 or 11 sensors have
to be used. In Figure 15 IEIs of the correlated one with 7
vertical modes using BSSP are superimposed. Figure 16
shows the arrangement of triaxial sensors for the correlated
scenario (from 6 to 20 sensors) when using 19 modes.
Figure 16. Optimal sensor location for the correlated scenario
when using 19 modes (× FSSP; ● BSSP).
Figure 13. IEI for the correlated scenario when using 7
vertical modes.
The previous analysis is not conclusive, however, on the
number of ﬁxed reference sensors that is actually needed. In
[16] the eﬀect of the number of reference sensors on the
identiﬁcation results is studied by considering three cases.
Case 1 is with only one reference sensor and its optimal
location is L = {43} (Figure 14, BSSP). Case 2 is with two
reference sensors at L = {27, 43}. Finally, case 3 is with three
reference sensors in conﬁguration L = {9, 27, 43}. Then we
compare the identification results with the previous case of
four reference sensors L = {9, 27, 43, 63} (Figure 10). From
this study [16] it can be concluded that, with optimal sensor
placement techniques, the number of references employed
(four in the present case) can be reduced to three or even to
two without a signiﬁcant loss of quality of the modal
estimates and without losing the ability to detect most modes
of interest. A single reference however does not result in
estimates of acceptable quality for most modes, even when
placed at an optimal position.
4.6
Roving strategy
The OSL methodology is used next to verify the strategy for
the positions of roving sensors in successive test setups. Two
roving strategies are often used in practice. One is to “rove in
modules" or RIM (distributed roving) and the other is “rove in
groups" or RIG (concentrated roving).
Figure 14. OSL for the correlated scenario when using 7
vertical modes (× FSSP; ● BSSP; □ actual test).
Figure 17. Roving configurations: in modules (top: RIM) or in
groups (bottom: RIG).
Figure15. IEI for the correlated scenario when using 19
modes.
The RIM case has been implemented in the actual test with
7 sensors in 13 setups to cover 91 bottom nodes in one
measurement line. In the RIM case the first setup would be L
= {0, 13, 26, 39, 52, 65, 78}. The next setup is built by
advancing every sensor in setup 1 by one truss bay to L = {1,
14, 27, 40, 53, 66, 79}. Following setups proceed successively
after one another. Finally, the last one - setup 13 - will have a
configuration of L = {12, 25, 38, 51, 64, 77, 90}.
In the RIG case, the first seven nodes L = {1, 2, 3, 4, 5, 6,
7} will be measured in the first setup. Then, the next seven
nodes are measured at configuration L = {7, 8, 9, 10, 11, 12,
13} in setup 2. In this roving strategy, the experiment finishes
at setup 13 where the last seven nodes are measured L = {84,
85, 86, 87, 88, 89, 90}.
The information entropy (IEI) values are computed directly
considering 19 FE mode shapes (7 vertical modes, 11 lateral
19
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
and one torsional mode up to a frequency of 7.70 Hz). In
general, the IEI values for setups in RIM case are signiﬁcantly
lower – consequently much richer information content – than
those calculated for the RIG case, except for setup 12 (Figure
18).
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[2]
[3]
[4]
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Figure 18. IEI values for the RIG and RIM strategies.
5
CONCLUSIONS
A crucial aspect for a correct diagnosis from vibration-based
structural health monitoring is the elimination of
environmental influences. Powerful output-only or inputoutput methods allow to reduce the remaining variance
significantly. The influence of environmental influences on
mode shapes and modal strains is less studied but could help
to discriminate structural changes, e.g. due to damage, from
environmental effects.
Wireless systems can reduce the measurement time for an
operational modal analysis drastically, certainly in the case of
large structures, and so contribute to an even more widespread
use of it. To keep the same performance as a wired system,
data synchronization, scalability and robustness are critical
issues.
A substantial improvement of the diagnosis capability to
discover also small local damages is possible if the rather
small dynamic strains during operation can be measured
accurately.
Optimal sensor placement methodologies, where the effect
of spatial correlation of sensors into the prediction error is
considered in order to avoid redundant information, provide
very valuable insights into important choices in the test
designing stage such as, an adequate measurement grid,
locations of reference sensors and a suitable roving scheme.
They are also very useful to decide upon the sensor positions
in case of a permanent monitoring system, usually based on a
limited number of sensors.
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[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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ACKNOWLEDGMENTS
This research is partially supported by the Spanish Ministry of
Science and Innovation (Sub-program INNPACTO 2010
Viadintegra research project).
20
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