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Analytical and experimental studies in modeling and monitoring of uncertain nonlinear systems using data-driven reduced‐order models

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Analytical and experimental studies in modeling and monitoring of uncertain nonlinear systems using data-driven reduced‐order models

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Content ANALYTICAL AND EXPERIMENTAL STUDIES IN MODELING AND
MONITORING OF UNCERTAIN NONLINEAR SYSTEMS USING
DATA-DRIVEN REDUCED-ORDER MODELS
by
Miguel R. Hernandez-Garcia
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CIVIL ENGINEERING)
July 2014
Copyright 2014 Miguel R. Hernandez-Garcia
Dedication
To the memory of my father, Jos e Miguel.
ii
Acknowledgements
First of all, I would like to thank my wife Tania Bibiana for her personal support and
patience at all times; and my parents, Elsa and Jose Miguel, and brother, Hugo Fernando,
who have given me their unequivocal support throughout, as always.
I would like to express my appreciation and gratitude to my advisor, Professor Sami
F. Masri, for his guidance, encouragement, and support throughout my doctoral studies
and research work. This thesis would not have been possible without his help, support
and patience. I'm also thankful to Dr. John Carey for his assistance and collaboration
during the experimental phase of this work. I also appreciate the advice and support of
Professors Roger Ghanem and Carter Wellford.
Amongst my fellow PhD students, I would like to thank Charles Devore, Fabian Rojas,
Leonardo Velderrain, Farzad Tasbihgoo, Armen Derkevorkian, Hae-Bum Yun, Reza Nay-
eri, Felipe Arrate, Mohammad Jahanshahi, Reza Jafarkhani for their friendship, support,
and making this time a memorable experience.
Finally, I acknowledge the partial nancial support by the National Science Founda-
tion (NSF), and the King Abdulaziz City for Science and Technology (KACST) through
contract number 32-710. I also also acknowledge the USC Viterbi School of Engineering
Fellowship for their generous support.
iii
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables vi
List of Figures viii
Abstract xix
Chapter 1 Introduction 1
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter 2 Development of a probabilistic data-based reduced-order rep-
resentation of nonlinear mechanical joints 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Identication of data-based reduce-order models . . . . . . . . . . . . . . . 6
2.3 Probabilistic representation of restoring force coecients . . . . . . . . . . . 8
2.4 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 3 A substructuring approach for detecting, locating, and quanti-
fying structural changes in chain-like systems 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Chain-like nonlinear system identication approach . . . . . . . . . . . . . . 31
3.3 LANL test-bed structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Detection of linear changes in a MDOF test-bed structure . . . . . . . . . . 34
3.5 Detection of nonlinear changes in a MDOF test-bed structure . . . . . . . . 56
3.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter 4 Experimental study of data-driven reduced-order modeling tech-
niques for detection of changes in structural systems 84
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Formulation of reduced-order modeling techniques . . . . . . . . . . . . . . 85
4.3 Detection of changes using reduced-order models of structural systems . . . 98
iv
4.4 Experimental results in six-story steel-frame laboratory structure . . . . . . 100
4.5 Experimental results in seven-story full-scale reinforced-concrete structure . 114
4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Chapter 5 Structural health monitoring and characterization of a re-congurable
structure based on a substructuring identication approach 130
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2 Re-congurable testbed structure . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3 Identication of reduced-order nonlinear models . . . . . . . . . . . . . . . . 142
5.4 Identication of a probabilistic reduced-order model of uncertain nonlinear
element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.5 Preliminary results on probabilistic model upscaling and stochastic dimen-
sion reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Chapter 6 Application of Statistical Monitoring Using Latent Variable
Techniques for Detection of Faults in Sensor Networks 185
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.2 Sensor failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.3 Statistical monitoring based on latent variables techniques . . . . . . . . . . 188
6.4 Sensor fault detection and identication . . . . . . . . . . . . . . . . . . . . 199
6.5 Application to case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Chapter 7 Conclusions 220
References 223
v
List of Tables
2.1 Chebyshev coecients C
ij
identied for one of the lap-type bolted joint
specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Power-series coecients a
ij
identied for one of the lap-type bolted joint
specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Summary of structural state conditions . . . . . . . . . . . . . . . . . . . . . 36
3.2 Summary of mean () and coecient of variation () of the identied restor-
ing force coecients for the LANL test-bed structure. . . . . . . . . . . . . 44
3.3 Summary of relative mean change (=
r
) and signal-to-noise ratio (=
r
)
in the identied restoring force coecients for the LANL test-bed structure.
Boldfaced table entries correspond to the detected structural changes. . . . 48
3.4 Global system identication. Comparison of =
r
in the identied mass-
normalized stinesses for LANL test-bed structure. Boldfaced table entries
correspond to the actual structural changes and false positives, respectively. 55
3.5 Global system identication. Comparison of =
r
in the identied mass-
normalized stinesses for LANL test-bed structure. Boldfaced and under-
lined table entries correspond to the actual structural changes and false
positives, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Comparison of numerical and experimental modal parameters . . . . . . . . 56
3.7 Relative changes in experimentally-identied natural frequencies . . . . . . 56
3.8 Summary of structural state conditions . . . . . . . . . . . . . . . . . . . . . 58
3.9 Masses and mass ratios estimated for the reference structural congurations. 59
3.10 Identied third-
oor Chebyshev coecients C
(3)
qr
for state#1 and state#14. 63
vi
3.11 Identied third-
oor power-series coecients a
(3)
qr
for state#1 and state#14. 64
3.12 Summary of mean () and coecient of variation () of the identied restor-
ing force coecients for the LANL test-bed structure. . . . . . . . . . . . . 67
3.13 Summary of =
r
and =
r
indices in the identied restoring force coef-
cients for the LANL test-bed structure. Boldfaced table entries correspond
to the detected changes based on established decision rule. . . . . . . . . . . 71
3.14 Summary of probability change quotient (PCQ) estimated for the restoring
force coecients identied for LANL test-bed structure. . . . . . . . . . . . 78
4.1 Summary of structural state conditions . . . . . . . . . . . . . . . . . . . . . 102
4.2 Summary of mean natural frequencies and mean damping ratios for the
rst six lateral modes in the x-direction and rst three lateral modes in
the y-direction of the structure identied from the reduced-order models
developed using SRIM, LSSID, and ChainID approaches. . . . . . . . . . . 114
4.3 Summary of structural state conditions . . . . . . . . . . . . . . . . . . . . . 120
4.4 Summary of mean natural frequencies for the rst two lateral modes of the
structure identied from the reduced-order models developed using SRIM,
LSSID, and ChainID approaches. . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1 Summary of structural state conditions . . . . . . . . . . . . . . . . . . . . . 147
5.2 Comparison of numerical and experimental modal parameters . . . . . . . . 154
5.3 Summary of detection ratio estimated for the a
(i)
10
and a
(i)
20
coecients.
Highlighted entries in the table correspond to the cases with signicant
changes in the coecients were detected using a detection threshold of

c
= 0:90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1 Typical sensor failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.2 Summary of fault detection rates of PCA, ICA, and MICA monitoring for
dierent types of sensor faults . . . . . . . . . . . . . . . . . . . . . . . . . . 213
vii
List of Figures
2.1 (a) Top and bottom elements used to assemble nine dierent inclined lap-
type bolted joint specimens to be tested in the big mass device (BMD).
(b) Overall view of the big mass device (BMD) experimental setup, and
detailed view of a test specimen in the BMD. Photos from Segalman et al.
[70] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Typical time-histories of relative displacement z, relative velocity _ z, and
restoring force G(z; _ z) obtained by processing experimental measurements
from swept sine vibration tests with maximum force levels of (a) 300 lb and
(b) 500 lb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 (a) Restoring force surface as a function of relative displacement and rela-
tive velocity of a typical inclined lap-type bolted joint specimen obtained
using the corresponding reduced-order representation built from experimen-
tal data. Notice the nonlinear nature of the microslip phenomena in the
bolted joint. (b) Comparison between the measured (solid line) and the
reconstructed restoring forces (dashed line) in a typical bolted joint speci-
men for each of the ve dierent force levels used during the dynamic tests.
Notice that the curves are virtually identical. . . . . . . . . . . . . . . . . . 19
2.4 (a) Relative contribution of eigenvalues
i
of the sample covariance matrix
C
a
of the identied restoring force coecients a
ij
. (b) Convergence of
loglikelihood logL

(p

) of polynomial chaos coecientsp

with respect to
the number of Monte Carlo samples used to estimate the probability density
function
^
f
^
(jp

). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Probability density functions of the dominant restoring force coecients
a
02
, a
20
, a
30
, and a
11
in the estimated using the probabilistic reduced-
order model of the inclined lap-type bolted joint. The pdf s were estimated
using 10000 Monte Carlo realizations of the random vector ^ a comprising
the restoring force coecients a
ij
. The red dots on each pdf correspond to
the actual a
ij
coecients obtained from experimental data. . . . . . . . . . 23
viii
2.6 (a) Mean and corresponding (b) error bounds of the restoring force surface
as a function of relative displacement and relative velocity of a typical in-
clined lap-type bolted joint specimen estimated from 10000 Monte Carlo
samples of random restoring force coecients generated with the proba-
bilistic reduced-order representation of the a
ij
built from experimental data. 23
2.7 Model of nonlinear 3DOF system with the lumped masses interconnected
using linear elements and anchored to a support interface at three dierent
locations using lap-type bolted joints. . . . . . . . . . . . . . . . . . . . . . 25
2.8 Average relative errors between the (a) mean and (b) standard deviation of
the displacements estimated using PC expansion with total degree s = 10
and PC expansions with smaller order s. . . . . . . . . . . . . . . . . . . . . 26
2.9 Time evolution of the (a) mean and (b) variance of the displacement and
velocity of critical component in complex sysmte under study. . . . . . . . . 27
2.10 Probability density functions of the two quantities of interest: (a) RMS
of the predicted displacement of the critical component (i.e., lumped mass
m
2
), and (b) RMS of the predicted velocity of the critical component, for
the structural congurations with one, two, and three lap-type bolted joints
in the structural system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 Typical structural topology for a nonlinear MDOF chain-like system. . . . . 35
3.2 LANL-4DOF test-bed structure schematic drawings. . . . . . . . . . . . . . 35
3.3 (a) Typical acceleration, velocity and displacement records obtained at 2
nd

oor. (b) Relative displacement and velocity time-histories computed be-
tween masses m
2
and m
1
are shown in the rst two rows of this gure.
The third row corresponds to a comparison between the measured restor-
ing force time-history for element G
(2)
and the reconstructed time-history
using the identied restoring force coecients (virtually identical curves). . 37
3.4 (a) Identied restoring force surface for elements G
(3)
, G
(2)
and G
(1)
in
the reference structural condition. (b) Identied change in the restoring
force surfaces in G
(3)
, G
(2)
and G
(1)
due to a 43.75% second-story stiness
reduction. Note that, for enhanced viewing, dierent amplitude scales are
used in the LHS and RHS columns of plots. . . . . . . . . . . . . . . . . . . 42
ix
3.5 Statistical representation of the identied restoring force coecients for the
reference structural condition. (a) Normal probability plots of the mass-
normalized stiness-like a
(2)
10
and mass-normalized damping-like a
(2)
01
coe-
cients. (b) Corresponding histograms, estimated probability density func-
tions by kernel density estimators (solid lines) and superposed Gaussian
distributions (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines
correspond to the pdf of the coecients in the baseline condition. The dot-
dashed and dashed lines show the coecients' pdf obtained in state#2 and
state#3, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7 Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines
correspond to the pdf of the coecients in the baseline condition. The dot-
dashed and dashed lines show the coecients' pdf obtained in state#4 and
state#5, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.8 Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines
correspond to the pdf of the coecients in the baseline condition. The dot-
dashed and dashed lines show the coecients' pdf obtained in state#6 and
state#7, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines
correspond to the pdf of the coecients in the baseline condition. The dot-
dashed and dashed lines show the coecients' pdf obtained in state#8 and
state#9, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.10 (a) Typical phase plots of the measured restoring forces G
(i)
and the rel-
ative displacements z
i
from State #14, which corresponds to the scenario
with a gap of 0.05 mm at 3
rd

oor. (b) Corresponding phase plots of the
reconstructed restoring forces
^
G
(i)
obtained by using the identied restor-
ing force coecients a
(i)
qr
. (c) Comparison between the measured restoring
force time-histories G
(i)
(solid lines), and the reconstructed time-histories
^
G
(i)
(dashed lines). Notice that the curves are virtually identical. Also, due
to the nature of the one-sided snubber, the restoring force associated with
third
oor is not-symmetric. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.11 Relative change in the restoring force surface in element G
(3)
with respect
to the reference structural condition due to the presence of a nonlinearity
(gap = 0.05 mm.) in the third-story of the LANL test-bed structure. . . . . 62
x
3.12 Mean () and coecient of variation () of the identied restoring force
coecients a
(i)
10
and a
(i)
30
for all the state congurations of LANL test-bed
structure. Top row for a
(i)
10
; lower row for a
(i)
30
. . . . . . . . . . . . . . . . . . 66
3.13 Stochastic representation of the identied restoring force coecients a
(3)
10
(for linear stiness) and a
(3)
30
(for cubic stiness) for the state#14. The
corresponding histograms, estimated probability density functions (solid
lines) and Gaussian distributions (dashed lines) are superimposed. . . . . . 68
3.14 Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) co-
ecients. The solid lines correspond to the pdf of the coecients from the
baseline condition in state#1. The dot-dashed and dashed lines show the
coecients' pdf obtained in state#10 and state#13, respectively, where a
nonlinear snubber was placed at the 3
rd

oor. Note that, for easy viewing,
the distributions were normalized. . . . . . . . . . . . . . . . . . . . . . . . 69
3.15 Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) co-
ecients. The solid lines correspond to the pdf of the coecients from
the baseline condition in state#1. The dot-dashed and dashed lines show
the coecients' pdf obtained in state#16 and state#17, respectively, where
a nonlinear snubber was placed at the 3
rd

oor, and an added mass was
placed at the 1
st

oor. Note that, for easy viewing, the distributions were
normalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.16 Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) co-
ecients. The solid lines correspond to the pdf of the coecients from the
baseline condition in state#18. The dot-dashed and dashed lines show the
coecients' pdf obtained in state#20 and state#21, respectively, where a
nonlinear snubber was placed at the 2
nd

oor, and a stiness reduction was
introduced on the 3
rd

oor. Note that, for easy viewing, the distributions
were normalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.17 Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) co-
ecients. The solid lines correspond to the pdf of the coecients from
the baseline condition in state#22. The dot-dashed and dashed lines show
the coecients' pdf obtained in state#23 and state#24, respectively. Note
that, for easy viewing, the distributions were normalized. . . . . . . . . . . 73
xi
3.18 Graphical representation of the indicesj=
r
j andj=
r
j computed for
the identied a
(i)
10
and a
(i)
30
coecients. The dashed horizontal line in the
RHS column of plots correspond the detection threshold used in the decision
rule (j=
r
j 2:0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.19 Comparison of overlapping areas between the probability density functions
of identied mass-normalized linear stiness-like (~ a
(i)
10
) and mass-normalized
cubic stiness-like (~ a
(i)
30
) coecients for state#14 and the corresponding
reference pdf s (state#1). The pdf of the coecients from state#1 and
state#14 are plotted in solid and dashed lines, respectively. The area of
intersection is used to obtain the probability change quotient (PCQ). . . . . 78
3.20 Summary of probability change quotient (PCQ) estimated for the a
(i)
10
and
a
(i)
30
coecients for each of the state conditions of LANL test-bed structure. 79
3.21 Evolution of the ratio between the rms of the nonlinear (f
(i)
NL
) and linear
(f
(i)
L
) components of the rst, second, and third-
oors restoring forces, with
the corresponding 1 error bars, for dierent third-
oor gap sizes. The
dashed lines depict the corresponding tted functions for the mean of the
kf
(i)
NL
k=kf
(i)
L
k ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.22 Evolution of the means of identied mass-normalized restoring force co-
ecients, with the corresponding 1 error bars, in the rst, second, and
third
oors for dierent third-
oor gap sizes. The dashed lines depict the
corresponding tted functions for the means of the a
(i)
10
and a
(i)
30
coecients. 81
4.1 Six-story steel frame testbed structure used for experimental study: (a)
Photo of the testbed structure at NCREE facilities; (b) Location of sensors
deployed in the testbed structure. . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Six-story steel frame testbed structure in one of the damaged structural
congurations. The 9.0-cm and 6.0-cm cuts made in the rst and second-

oor columns are shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3 Typical acceleration, velocity, and displacement time-histories at the top of
the structure in the x-axis direction are displayed in the top, middle, and
bottom plots, respectively. The measured velocities and displacements are
also compared to the velocities and displacements obtained by numerical
integration of the acceleration records. Note the close match between the
measured and numerically integrated displacements and velocities. . . . . . 103
xii
4.4 Comparison of the probability density distribution for the elements
~
k
11
and
~
k
22
in the diagonal of the mass-normalized stiness-like (
~
K) matrices of
the reduced-order models in the x- and y-direction of the test structure
estimated using SRIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Detection ratios and magnitude of relative changes
r
in the diagonal
elements of the stiness-like matrix
^
K of the (a) reduced-order models of
the test structure in the x-direction, and (b) and reduced-order models of
the test structure in the y-direction obtained using SRIM approach. The
horizontal line corresponds to the threshold = 0:90 used in this study to
determine if signicant changes in the parameters have been observed with
respect to the parameters from the reference condition. . . . . . . . . . . . . 108
4.6 Detection ratios and magnitude of relative changes
r
in the diagonal
elements of the stiness-like matrix
^
K of the (a) reduced-order models of
the test structure in the x-direction, and (b) and reduced-order models of
the test structure in the y-direction obtained using LSSID approach. The
horizontal line corresponds to the threshold = 0:90 used in this study to
determine if signicant changes in the parameters have been observed with
respect to the parameters from the reference condition. . . . . . . . . . . . . 109
4.7 Comparison of the probability density distribution for the stiness-like co-
ecientsa
(1)
10
anda
(2)
10
of the reduced-order models in the x- and y-direction
of the test structure estimated using ChainID approach. . . . . . . . . . . . 110
4.8 Detection ratios and magnitude of relative changes
r
in the diagonal
elements of the stiness-like coecientsa
(i)
10
of the interconnecting elements
in the (a) reduced-order models of the test structure in the x-direction,
and (b) and reduced-order models of the test structure in the y-direction
obtained using ChainID approach. The horizontal line corresponds to the
threshold = 0:90 used in this study to determine if signicant changes
in the parameters have been observed with respect to the parameters from
the reference condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.9 Mode shapes of the rst three lateral modes in the x- and y-direction esti-
mated from the reduced-order models built using the SRIM approach. . . . 115
4.10 Mode shapes of the rst three lateral modes in the x- and y-direction esti-
mated from the reduced-order models built using the LSSID approach. . . . 116
4.11 Mode shapes of the rst three lateral modes in the x- and y-direction esti-
mated from the reduced-order models built using the ChainID approach. . . 117
xiii
4.12 Detection ratios and magnitude of relative changes
r
in the (a) natural
frequencies and (b) damping ratios estimated for the test structure in the x-
direction obtained using SRIM approach. The horizontal line corresponds
to the threshold = 0:90 used in this study to determine if signicant
changes in the modal parameters have occurred with respect to the param-
eters from the reference condition. . . . . . . . . . . . . . . . . . . . . . . . 118
4.13 Detection ratios and magnitude of relative changes
r
in the (a) natural
frequencies and (b) damping ratios estimated for the test structure in the x-
direction obtained using LSSID approach. The horizontal line corresponds
to the threshold = 0:90 used in this study to determine if signicant
changes in the modal parameters have occurred with respect to the param-
eters from the reference condition. . . . . . . . . . . . . . . . . . . . . . . . 118
4.14 Detection ratios and magnitude of relative changes
r
in the (a) nat-
ural frequencies and (b) damping ratios estimated for the test structure
in the x-direction obtained using ChainID approach. The horizontal line
corresponds to the threshold = 0:90 used in this study to determine if
signicant changes in the modal parameters have occurred with respect to
the parameters from the reference condition. . . . . . . . . . . . . . . . . . . 119
4.15 View of the 7-story full-scale building slice structure . . . . . . . . . . . . . 119
4.16 First two lateral modes of reduced-order model obtained using (a) SRIM,
(b) LSSID, and (c) ChainID. . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.17 Probability density functions for the identied natural frequency (f) and
damping ratio () of the rst two lateral modes. . . . . . . . . . . . . . . . . 125
4.18 Mean () and coecient of variation () of the identied frequencies for the
rst two lateral modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.19 Mean () and coecient of variation () of the identied damping rations
for the rst two lateral modes . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.20 Change indicesj=
r
j andj=
r
j for the identied frequencies for the
rst two lateral modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1 Assembled modular re-congurable testbed structure. . . . . . . . . . . . . 132
5.2 Typical modular section (vertical view) of the test structure. . . . . . . . . 133
5.3 Modular section with nonlinear gap element (vertical view). . . . . . . . . . 133
5.4 Side view of mounted programmable nonlinear element . . . . . . . . . . . . 134
xiv
5.5 Modular section with nonlinear gap element (vertical view). . . . . . . . . . 135
5.6 Solid model of the custom-built electromagnetic exciter. . . . . . . . . . . . 136
5.7 Detail of the electromagnetic exciter showing the reaction load cell and the
linear support bearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.8 Electrical diagram for the operation of the electromagnetic exciter. . . . . . 139
5.9 Experimental measurements of normalized force in the neodymium magnet
generated for (a) forward current direction and (b) reverse current direction.140
5.10 Custom-built electromagnetic exciter installed on the structure . . . . . . . 142
5.11 Location of the accelerometers deployed on testbed structure. . . . . . . . . 147
5.12 Sample of (a) acceleration, velocity, and displacement time-histories, and
(b) relative displacement, relative velocity, and restoring force time-histories
from the linear system in the reference structural conguration (state#1). . 149
5.13 Identied restoring force coecients and corresponding estimated restoring
force surface for module 2 for structure in linear conguration (state#1) . . 151
5.14 Comparison of measured and estimated restoring forces for linear system
(state#1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.15 Probability density functions of the mass-normalized restoring force coef-
cients a
(2)
10
, a
(2)
01
, a
(2)
20
, and a
(2)
30
from the reduced-order representation of
structural module 2 built for the testbed structure in a conguration with
a linear dynamic response (state#1). . . . . . . . . . . . . . . . . . . . . . 153
5.16 Probability density function of the rst three natural frequencies of the
testbed structure in the linear reference conguration (state#1) estimated
using the mass-normalized stiness-like and damping-like coecients a
(i)
10
and a
(i)
01
from built reduced-order models. . . . . . . . . . . . . . . . . . . . 155
5.17 Estimated mode shapes for the rst three identied lateral modes of the
testbed structure in the linear reference conguration (state#1) estimated
using the mass-normalized stiness-like and damping-like coecients a
(i)
10
and a
(i)
01
from built reduced-order models . . . . . . . . . . . . . . . . . . . . 156
5.18 Sample of (a) acceleration, velocity, and displacement time-histories, and
(b) relative displacement, relative velocity, and restoring force time-histories
from the nonlinear system in state#7. . . . . . . . . . . . . . . . . . . . . . 157
xv
5.19 Identied restoring force coecients and corresponding estimated restoring
force surface for module 2 for testbed structure in nonlinear conguration
(state#7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.20 Comparison of measured and estimated restoring forces for nonlinear system
(state#7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.21 Comparison of the estimated probability density functions of the mass-
normalized restoring force coecientsa
(2)
10
,a
(2)
20
, anda
(2)
30
identied for struc-
tural module 2 in the corresponding linear and nonlinear congurations . . 160
5.22 Comparison of the estimated probability density functions of the mass-
normalized restoring force coecients a
(i)
10
and a
(i)
20
for all six structural
modules of the testbed structure in the linear and nonlinear congurations . 161
5.23 Comparison of the estimated probability density functions of the rst three
natural frequencies estimated for the testbed structure in the corresponding
linear and nonlinear congurations . . . . . . . . . . . . . . . . . . . . . . . 162
5.24 Probability density functions and corresponding 95%-condence bounds
for the a
(i)
10
and a
(i)
20
coecients identied for all structural modules of the
testbed structure in the reference conguration (state#1). The estimated
coecients from the testbed structure in state#5 conguration are shown
as red dots in each of the plots. . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.25 Probability density functions and corresponding 95%-condence bounds
for the a
(i)
10
and a
(i)
20
coecients identied for all structural modules of the
testbed structure in the reference conguration (state#1). The estimated
coecients from the testbed structure in state#7 conguration are shown
as red dots in each of the plots. . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.26 Estimated restoring force surface for structural module with variable-gap
element obtained for two dierent gap sizes from experimental measurements169
5.27 (a) Relative contribution of computed eigenvalues of the sample covariance
matrix of the restoring force coecientsa
qr
. (b) Loglikelihood logL

(p

) of
polynomial chaos coecientsp

as a function of the number of the monte
carlo samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.28 Probability density functions for coecients a
(2)
10
, a
(2)
20
, and a
(2)
30
obtained
using the polynomial chaos representation of the restoring force coecients
for the structural module with variable-gap size . . . . . . . . . . . . . . . . 171
xvi
5.29 Estimated mean restoring force surface for structural module with variable-
gap element and corresponding3 bounds obtained using the polynomial
chaos representation of the restoring force coecients. . . . . . . . . . . . . 172
5.30 Restoring force surfaces of variable-gap nonlinear element reconstructed
from realizations from the polynomial chaos representation of the of the
restoring force coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.31 Predicted mean acceleration, mean velocity, and mean displacement time-
histories of an uncertain nonlinear SDOF subjected to a deterministic band-
limited white-noise excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.32 Predicted2 bounds for dynamic response of the uncertain nonlinear
SDOF system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.33 Evolution of the mean value
!
and standard deviation
!
of the rst
random eigenfrequency ! for dierent number or Monte Carlo samples. . . 181
5.34 Comparison of probability density function f(!) (solid line) of rst eigen-
frequency obtained from the ne-scale model, and the probability density
functions f(!j^ p) obtained using a rst- (p = 1) and second-order (p = 2)
polynomial chaos expansions of the bending stiness (EI(p)) in the coarse-
scale model. (b) Comparison of probability density functions of bending
stiness (EI(p)) of the coarse-scale model obtained from the correspond-
ing rst- and second-order polynomial chaos representation. . . . . . . . . . 182
5.35 Probability density functions (pdf) obtained in the rst upscaling exam-
ple. (a) Comparison of probability density functionf(!) (solid line) of rst
eigenfrequency obtained from the ne-scale model, and the probability den-
sity functions f(!j^ p) obtained using a one- (n = 1) and two-dimensional
(n = 2) rst-order polynomial chaos expansions of the bending stiness
(EI) and mass per unit of length (A) in the coarse-scale model. . . . . . . 183
5.36 Probability density functions of (a) bending stiness (EI) and mass per unit
of length (A) of the coarse-scale model obtained from the corresponding
one-dimensional rst-order polynomial chaos representations. . . . . . . . . 183
5.37 Probability density functions of (a) bending stiness (EI) and mass per unit
of length (A) of the coarse-scale model obtained from the corresponding
two-dimensional rst-order polynomial chaos representations. . . . . . . . . 184
6.1 Model of the three-dimensional truss . . . . . . . . . . . . . . . . . . . . . . 203
6.2 Acceleration measurements in sensor 7 after simulating a bias, drift, scaling,
and hard-fault type of sensor faults . . . . . . . . . . . . . . . . . . . . . . . 204
xvii
6.3 Monitoring charts for the D-statistics and Q-statistic obtained under nor-
mal operating conditions of the sensor network for the (a) PCA model, (b)
ICA model, and (c) MICA model. The upper control limits correspond to
99% of condence level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.4 Monitoring charts for the D-statistics and Q-statistic obtained in the bias
sensor fault case for the (a) PCA model, (b) ICA model, and (c) MICA
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.5 Visualization of the original measured variables (i.e., sensors) and the cor-
responding representation of the measured data (i.e., sensor measurements)
in the space of the rst two retained latent variables (i.e., model space) in
the reduced-order model for the scenario with a bias fault in sensor 7. . . . 209
6.6 Visualization of the original measured variables (i.e., sensors) and the cor-
responding representation of the measured data (i.e., sensor measurements)
in the space of the rst two excluded latent variables for the scenario with
a bias fault in sensor 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.7 Variable contribution plots to theD- andQ-statistics obtained for an out-of-
control observation in the bias sensor fault case for the (a) PCA monitoring,
(b) ICA monitoring, and (c) MICA monitoring. . . . . . . . . . . . . . . . . 212
6.8 Location and direction of the deployed sensors on the Vincent Thomas
Bridge, San Pedro, CA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.9 Monitoring charts for the D-statistic and Q-statistic obtained in the bias
sensor fault case for the (a) PCA model, (b) ICA model, and (c) MICA
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.10 Comparison of acceleration measurements; from sensors 5, 10, 16, and 22;
obtained under normal operating and abnormal conditions. . . . . . . . . . 218
xviii
Abstract
Most of the available data-based methodologies developed for system identication and
health monitoring of complex nonlinear systems can be considered to be deterministic in
nature. These approaches use experimental measurements to characterize the complex
systems by means of nominal mathematical (e.g., parametric or non-parametric) models,
while neglecting the eects of aleatory and epistemic uncertainties that can be present in
real structures. The inherent stochastic nature of the systems components (i.e., random-
ness in structural, geometric and material properties); the variability in environmental and
operational conditions; and the uncertainties associated with the modeling, measurement
and data analysis process can lead to unreliable description and characterization of com-
plex nonlinear systems. Consequently, in order to develop robust and reliable models of
nonlinear systems, it is imperative to address the issues of quantication and propagation
of uncertainties.
This dissertation compiles analytical and experimental studies focused on implement-
ing, analyzing, and validating promising and robust data-driven methodologies to build
high-delity reduced-order models of uncertain complex nonlinear systems. These data-
based reduced-order methodologies were used in structural health monitoring applica-
tions, and in the modeling of critical structural components. Experimental datasets from
xix
dynamic tests performed in a small-scale lab structure at Los Alamos National Labora-
tory (LANL); a re-congurable test structure designed, built, and tested at University
of Southern California (USC); a scaled-down six-story steel-frame laboratory structure
at the National Center for Research in Earthquake Engineering (NCREE); and a seven-
story full-scale reinforced-concrete structure at the UCSD-NEES facilities, were used to
evaluate the eectiveness and reliability of the data-based reduced-order models for de-
tecting, locating and quantifying structural changes. In addition, sensor fault-detection
and identication techniques based on statistical monitoring using latent-variable tech-
niques, were implemented and evaluated for detecting and identifying faulty sensors using
measurements from an actual cable-supported bridge in the metropolitan Los Angeles
(CA) region. Finally, a general methodology for developing probabilistic reduced-order
models of critical structural components from experimental measurements was proposed.
This methodology was used to develop a probabilistic data-based reduced-order model
to characterize the mechanical behavior of a particular type of lap bolted joints with an
inclined interface directly from experimental data obtained from dynamic tests performed
at Sandia National Laboratories (SNL).
xx
Chapter 1
Introduction
1.1 Background and motivation
During the recent past, there has been an increasing interest in investigating nonlinear
dynamical systems for analyzing experimental measurements, developing analytical mod-
els, controlling, and identifying complex systems such as those appearing in the elds of
aerospace, civil and mechanical engineering. The extensive research done on simulation,
structural control and health monitoring have led to a signicant number of applications
and well-developed techniques for identication of complex dynamical systems. Proof
of these eorts is re
ected in the extremely broad technical literature available. Several
illustrative applications can be found in the proceedings of the International and Euro-
pean Workshops on Structural Health Monitoring [13, 14, 79], and the World Conference
on Structural Control and Monitoring [12, 37]. Excellent and careful reviews of existing
nonlinear identication techniques can be found in Adams and Allemang [2], Haber and
Keviczky [28], Worden and Tomlinson [85], Kerschen et al [42], and Farrar et al [20].
Nevertheless, most of the available identication techniques can be considered to be
deterministic due to the fact that measurements from a structural system are used to
1
characterize the nonlinearities by means of a nominal mathematical (e.g., parametric or
non-parametric) model, while neglecting the eects of aleatory and epistemic uncertain-
ties that can be present in real structures. The inherent stochastic nature of the systems'
components (i.e., randomness in structural, geometric and material properties); the vari-
ability in environmental and operational conditions; and the uncertainties associated with
the modeling, measurement and data analysis process can lead to unreliable description
and characterization of complex nonlinear systems. Consequently, in order to develop
robust and reliable models of nonlinear systems, it is imperative to address the issues of
quantication and propagation of uncertainties. Background information regarding the
approaches and challenges in the representation and propagation of uncertainties in struc-
tural systems is available in Ghanem and Spanos [24], Hemez [29], Soize [78], Ghanem et
al [25], Li and Chen [50], Masri et al [59], and Adhikari et al [3].
With the above discussion in mind, this analytical and experimental investigation
will focus on implementing, analyzing, and validating robust data-driven reduced-order
models of uncertain nonlinear systems that can be used in structural health monitoring
applications, and in the modeling of critical structural components.
1.2 Scope
Chapter 2 presents a general methodology for developing probabilistic reduced-order mod-
els of critical structural components from experimental measurements. This methodology
allows the estimation through analytical procedures of the uncertain bounds on the dy-
namic response of complex structural systems under dierent loading conditions. The
proposed methodology is used to develop a probabilistic data-based reduced-order model
2
to characterize the mechanical behavior of a particular type of lap bolted joints with an
inclined interface directly from experimental data obtained from dynamic tests performed
at Sandia National Laboratories (SNL).
Chapter 3 deals with the development and application of a robust data-driven reduced-
order models for detecting, locating and quantifying structural changes in uncertain chain-
like systems. Experimental data from a test-bed structure tested at Los Alamos National
Laboratory (LANL) are used to evaluate the eectiveness and reliability of the proposed
SHM methodology. The statistical and probabilistic description of the detected nonlin-
earities is additionally used to assess the model's detectability and reliability. Addition-
ally, a full-order nite element model of the test structure, as well as the results from
an experimental modal analysis were employed to validate the results obtained in this
change-detection study.
In Chapter 4, the results on an exploratory study focused on the detection and location
of structural changes in mid- and large-scale structures are presented. In this study,
experimental data from band-limited white-noise base excitation dynamic tests performed
in a scaled-down six-story steel-frame laboratory structure at the National Center for
Research in Earthquake Engineering (NCREE), and a seven-story full-scale reinforced-
concrete structure at the UCSD-NEES facilities, was used to assess the variability in a set
of features from reduced-order models developed using three dierent data-driven input-
output system identication approaches that have been successfully applied to analytical
and experimental structures
Chapter 5 describes a re-congurable test apparatus that was designed and built for
investigating complex dynamic systems with nonlinear structural components. Physical
3
experiments were performed to collect a statistically signicant ensemble of measurements
that were then used to investigate the development of reduced-order, low-complexity math-
ematical models for detecting and locating, in a probabilistic framework involving hypoth-
esis testing, linear and nonlinear structural changes at various locations within the testbed
structure.
In Chapter 6, three sensor fault-detection and identication techniques based on sta-
tistical monitoring, using latent-variable techniques, were implemented, evaluated, and
compared with respect to their capability to detect and identify faulty sensors using case
studies from an analytical three-dimensional truss and from an actual cable-supported
bridge in the metropolitan Los Angeles, California region.
4
Chapter 2
Development of a probabilistic data-based reduced-order
representation of nonlinear mechanical joints
2.1 Introduction
A formidable hurdle preventing the use of sophisticated computational tools, replicating
the behavior of existing physical structures, or simulating the response of complex non-
linear systems, is the lack of high-delity, physics-based, robust nonlinear computational
models that accurately characterize the behavior of such systems under arbitrary dynamic
environments. Related issues that are crucial to the development of useful computational
models (which, e.g., can be relied on to reduce the number of physical tests needed for certi-
cation of actual physical systems), on the basis of measurements obtained from nonlinear
physical models, are (1) the quantication of uncertainties that are inherently present in
the underlying physical prototype, (2) the determination of the corresponding uncertain-
ties in the identied reduced-order, reduced-complexity mathematical model, and (3) the
subsequent quantication of the propagation eects of these physical uncertainties in the
identied model, as well as the uncertainty bounds on the dynamic system response.
5
This paper presents a study of a general methodology for representing and propagating
the eects of uncertainties in complex nonlinear systems through the use of a reduced-
order, reduced-complexity, model-free representation, that allows the estimation through
analytical procedures of the uncertain system's response bounds when it is excited by a
dierent dynamic load than the one used to identify it. A nonparametric identication
approach based on the use of the restoring force is employed to obtain a probabilistic model
of the nonlinear system of interest. Subsequently, the probabilistic reduced-order model
is used in conjunction with polynomial chaos representations to predict the uncertainty
bounds on the nonlinear system response under transient dynamic loads.
2.2 Identication of data-based reduce-order models
Considering that the motion of a general deterministic nonlinear SDOF system subjected
to external excitation force can be mathematically represented by
m z(t) +G[x(t); _ x(t);q] =f(t) (2.1)
where m is the mass of the system, z(t), _ z(t), and z(t) are respectively the displacement,
velocity, and acceleration of the SDOF. The term G[x(t); _ x(t);q] corresponds to the non-
linear non-conservative restoring force in the system. The restoring force G is generally
assumed to be function of the system statesfz; _ zg and set of element-specic parameters
q. The directly applied forces in the system are represented by f(t).
6
A reduced-order model of the dynamic characteristics of the nonlinear non-conservative
element in the SDOF system is obtained by generating a truncated doubly-indexed se-
ries expansion, in a suitable basis, that approximates the real restoring force function
G[x(t); _ x(t);q] [53, 56, 57, 85, 87]. The approximating representation
^
G(z; _ z) for the
obtained restoring forces, in an orthogonal polynomial basis, is given by the following
expression:
G(z; _ z;q)
^
G(z; _ z) =
nz
X
i=0
n
_ z
X
j=0
C
ij
T
i
(z
0
)T
j
( _ z) (2.2)
where C
if
are the Chebyshev series coecients, T
k
() is the Chebyshev polynomial of
order k, and z
0
, _ z
0
are the normalized relative state variables. Subsequently, each of the
estimated restoring forces can be expressed as a power series of the form
^
G(z; _ z) =
nz
X
i=0
n
_ z
X
j=0
a
ij
z
i
_ z
j
(2.3)
where a
ij
are constant coecients, and z
i
, _ z
i
are the relative state variables. The appli-
cation of this non-parametric identication approach allows the capture of the dominant
features of the nonlinear elements into reduced-order, model-free representations [57].
7
2.3 Probabilistic representation of restoring force coecients
Mechanical systems are inherently stochastic due to unpredictable natural variability (ran-
domness) in the dynamical properties of the physical system. Additionally, the non-
parametric mathematical modeling used to represent the restoring force and, the mea-
surement errors will add to variability in the identied power-series coecients. In the
presence of all sources of variability associated to the non-parametric restoring force iden-
tication, the uncertain identied coecients a
ij
can be considered as random variables.
By relying on Karhunen-Loeve (KL) and polynomial chaos (PC) expansions, the ran-
dom restoring force coecients a
ij
can be characterized using a probabilistic reduced
representation of the form [7, 23, 26]
a ^ a =p
0
+
d
X
k=1
p

k
r
X
;jj=1
p

;k
H

()
k
(2.4)
where the random vector a with values in R
m
comprises all m random coecients in
the reduced-order representation of the restoring force (Eq. 2.3). The terms
k
and
k
correspond to thed most signicant eigenvalues and associated eigenvectors of the sample
covariance matrix
^
C
a
a =
1
n
n
X
k=1
a
(k)
(2.5)
C
a
=
1
n 1
n
X
k=1
(a
(k)
a)
(a
(k)
a) (2.6)
in whichfa
(k)
; 1 k ng correspond to a data set of n independent and identically-
distributed realizations or the random restoring force coecients.
8
The function H

() = h

1
(
1
) h

d
(
d
) in Eq. 2.4 corresponds to a multi-
dimensional Hermite polynomials with multi-index = (
1
;:::;
d
)2 R
d
and modulus
jj =
1
++
d
. Each termh

j
() is a one-dimensional normalized Hermite polynomial
of order
j
. The germ is a ddimensional vector of independent standard Gaussian
random variables. The termsp
0
2R
m
andp

2R
d
are vectors containing the unknown
coecients that dene the probabilistic representation of random vectora.
In order to determine the unknown parameters p = fp
0
;p

g in the reduced-order
representation of a (Eq. 2.4) from available data/observations, a two-step identication
procedure proposed by Arnst et al. [7] that relies on the maximum likelihood parameter
estimation, is followed.
In the rst step, the maximum likelihood estimate ^ p

of the parametersp

are obtained
by solving the optimization problem
^ p

= arg max
p

L

(p

) (2.7)
where the coecientsp

=fp

2R
d
; 1jjrg in the polynomial chaos representation
must satisfy
r
X
;jj=1
p

p
T

=I
d
(2.8)
9
in which I
d
is dd identity matrix. The function L

(p

), which corresponds to the
likelihood ofp

associated with the observationsf
(k)
j
; 1jmg, is dened as
L

(p

) =
n
Y
k=1
f
^
(
(k)
jp

) (2.9)
wheref
^
(jp

) denotes the probability density function of ^ for given set ofp

coecients.
The observations of
(k)
= (
(k)
1
;:::;
(k)
d
) can be readily obtained from the original data
a using the following relationship

(k)
j
=
(a
(k)
a)
T

j
p

j
(2.10)
In the next step, the coecients p
0
are estimated again by solving the optimization
problem
^ p
0
= arg max
p
0
L
0
(p
0
) (2.11)
where the likelihood ofp
0
is given by
L
0
(p
0
) =
n
Y
k=1
f
^ a
(a
(k)
jp
0
; ^ p

) (2.12)
and f
^ a
(jp
0
; ^ p

) corresponds to the probability density function of ^ a given the optimal
^ p

and a set ofp
0
coecients.
To implement the outlined procedure, it is necessary to estimate the likelihood func-
tions L

(p

) and L
0
(p
0
). These likelihood functions can be eciently computed using
Monte Carlo simulation and kernel density estimation [7, 18, 23]. To estimateL

(p

), rst
10
simulate a setf^
s
; 1sMg ofM realizations of ^ conditioned onp

by initially gener-
ating a setf
s
; 1sMg of independent samples of the germ and then computing the
realizations ^
s
=
P
r
;jj=1
p

H

(
s
). The probability density function
^
f
^
(jp

) can then
be estimated from the samplesf^
s
; 1sMg using the multivariate kernel density esti-
mation. Finally, the likelihood (Eq. 2.9) is calculated as
^
L

(p

) =
Q
n
k=1
^
f
^
(
(k)
jp

). This
procedure can be easily adapted to calculate the likelihood function L
0
(p
0
) (Eq. 2.12).
Finally, it should be noted that because the likelihood functions may have multiple local
maxima and gradients can be dicult to obtain, global-search gradient-free optimization
methods (e.g. simulated annealing, genetic algorithm) are better suited to solve the op-
timization problems dened in Eq. (2.7) and Eq. (2.11) than gradient-based approaches
[7, 23, 26]
2.4 Uncertainty propagation
Consider the discrete representation of a complex structure incorporating nonlinear non-
conservative structural components. It is assumed that the response of the system under
dynamic loading conditions can be described by an Ndimensional system of ordinary
dierential equations (ODE) of the form
M x(t) +C _ x(t) +Kx(t) +R[x(t); _ x(t);G] =F (t) (2.13)
where the vectorsx(t), _ x(t), and x(t) inR
N
are the displacement, velocity, and accelera-
tion of the system at a xed timet. The matricesM,C, andK inM
N
(R), characterize the
inertia, linearized damping, and linearized stiness forces. The vectorR[x(t); _ x(t);G] in
11
R
N
corresponds to the nonlinear forces in the system. The nonlinear forces inR(x; _ x;G)
are assumed to be function of the nonlinear non-conservative restoring forces G
k
(x; _ x;q)
of each of the nonlinear structural components in the system. The restoring forces G
k
are
generally function of the system statesfx; _ xg and set of element-specic parameters q.
The directly applied forces in the system are represented byF (t).
In this study, the nonlinear restoring forcesG
k
(x; _ x;q) will be dened using the prob-
abilistic data-based reduced-order representation
^
G
k
(x; _ x;
k
) developed from physical ex-
periments that characterize the dynamic behavior of the nonlinear non-conservative struc-
tural components. Therefore, the corresponding set of governing stochastic ODEs are then
denoted by
M x(t;) +C _ x(t;) +Kx(t;) +R[x(t;); _ x(t;);
^
G()] =F (t) (2.14)
where the elements of the vector
^
G[x; _ x;] are given by
^
G
k
(x; _ x;) =
mx
X
i=0
m
_ x
X
j=0
^ a
ij;k
()x
i
(t;) _ x
j
(t;) (2.15)
in which the restoring force coecients ^ a
k
() =f^ a
ij;k
; 0im
x
; 0jm
_ x
g are given
by Eq. (2.4).
The solution to the stochastic equation of motion of the 3DOF model (Eq. 2.14) can
be represented in the form of a multi-dimensional truncated polynomial chaos expansion
of the form [25]:
12
x(t;)
s
X
;jj=0
x

(t)H

(); _ x(t;)
s
X
;jj=0
_ x

(t)H

(); (2.16)
In the context of polynomial chaos expansions (PCE), the solution to Eq. (2.14)
is reduced to the computation of the polynomial chaos coecients x

(t) and _ x

(t) in
Eq. (2.16). In general, the algorithms for computing the PC coecients typically belong
to two categories: 1) intrusive, and 2) non-intrusive approaches. Non-intrusive approaches
rely on existing deterministic solvers and are used as a black box to compute the PC co-
ecients. Intrusive approaches, in the other hand, requires modications to the available
solvers [19]. In this study, a non-intrusive spectral projection (NISP) approach was used
solve for the PC coecients.
2.5 Application
In this section, the proposed methodology is used to develop a probabilistic data-based
reduced-order model to characterize the mechanical behavior of a particular type of lap
bolted joints with an inclined interface directly from experimental vibration data. These
type of inclined lap bolted joints are typically used in structural connections to attach
critical aerospace components or subsystems to surrounding support structures. It should
be noted that in this type of applications, the dynamic behavior of the bolted joints will
have important eects on the dynamic response of critical aerospace components because
practically all dynamic loads will be transmitted to the components through the support
elements. The dynamic experiments on these bolted joints were design and performed
13
at Sandia National Laboratories (SNL). Full details concerning this SNL test setup and
experiments are documented in Gregory et al. [27] and Segalman et al. [70].
2.5.1 Test specimens and experimental setup
A typical inclined lap-type bolted joint specimen consists of two stainless-steel components
that are joined with a bolt (Figure 2.1a). The top component is fabricated in one solid
piece, while the bottom component is a machined shell element. A hole in the mating
surface of the bottom component allows the bolt to pass freely and screw directly into a
tapped hole in the mating surface of the top component. The applied torque in the bolt
was specied so no relative slip would occur along the contact surface (macroslip) during
normal operating conditions. Therefore, the energy dissipation mechanism in the bolted
joint will be due to microslip phenomena occurring in the contact interface.
A total of three interchangeable top components and three bottom components were
manufactured. By combining all top and bottom pieces, nine nominally identical bolted
joint specimens were built. In order to characterize the mechanical behavior of the inclined
lap-type bolted joint, a series of dynamic experiments were performed on each of the
assembled test specimens using the big mass device (BMD). The BMD is an experimental
setup developed exclusively to investigate the microslip mechanism in mechanical joints,
and it can basically be modeled as single-degree-of-freedom (SDOF) system subjected to
base motions. An overall view of the BMD experimental setup is shown in Figure 2.1(b).
In this setup, the test specimen is used to connect a large inertial mass (200 lb), which
is hanged from a support frame by soft springs to balance the eects of the static load
in the joint, to the armature of an electrodynamic shaker. A detailed view of a bolted
14
(a) Inclined lap-type bolted joint specimens (b) Big mass device (BMD) setup
Figure 2.1: (a) Top and bottom elements used to assemble nine dierent inclined lap-type
bolted joint specimens to be tested in the big mass device (BMD). (b) Overall view of the
big mass device (BMD) experimental setup, and detailed view of a test specimen in the
BMD. Photos from Segalman et al. [70]
joint specimen attached to the large mass and the shaker in the BMD is shown in Figure
2.1(b). In this conguration, the test specimen essentially acts as a nonlinear spring in
the SDOF system. Two accelerometers installed in the top and bottom elements of the
test specimen were used to measure the accelerations in the axial direction of the mass
and the base, respectively. In addition, a load cell was placed between the test specimen
and the shaker to measure the applied force to the system. The test specimens were
then sequentially subjected to force-controlled swept-sine excitations with maximum force
levels of 100, 200, 300, 400, and 500 lb. Each bolted joint specimen was assembled and
tested four dierent times. Therefore, there were a total of 36 dynamic tests on the bolted
joints at ve dierent force levels.
15
2.5.2 Reduced-order model of inclined lap-type bolted joint
The thirty-six ensembles of measured force and acceleration time-histories obtained from
the dynamic experiments on the bolted joints were used to develop reduced-order repre-
sentation of the mechanical characteristics of the inclined lap-type bolted joint. In order to
obtained the nonlinear restoring forces in the joint, the relative displacements and relative
velocities across the bolted joint had to be estimated. Using the accelerations measured
above and below the joint specimen, the relative accelerations in the joint were calculated
initially. The relative accelerations were then integrated to obtained the relative displace-
ments and velocities. The linear component of the restoring force was estimated using a
least squares t of the inertial forces acting on the mass of the BMD in terms of the relative
displacements and velocities. The nonlinear restoring forces generated by the microslip
mechanism in the bolted joint can then be estimated by subtracting the linear restoring
forces from the response of the mass. Typical time-histories of the relative displacement
z, relative velocity _ z, and nonlinear restoring force G(z; _ z;q) obtained after initial pro-
cessing of the experimental measurements acquired from the swept sine vibration tests,
with maximum force levels of 300 lb and 500 lb, are displayed in Figure 2.2.
Once the nonlinear restoring force G(z; _ z;q), and the relative displacements z and
velocities _ z were obtained from all 36 dynamic tests, the time-domain non-parametric
identication technique (i.e., restoring force method) was applied to determine the corre-
sponding Chebyshev coecients C
ij
and power-series coecients a
ij
of the series expan-
sion, and build the associated reduced-order model for each of the joint specimens. It
should be noted that in this study, the reduced-order models for the nonlinear restoring
16
1 1.2 1.4 1.6 1.8 2
−2
−1
0
1
2
x 10
−4
t [s]
z [in]
1 1.2 1.4 1.6 1.8 2
−0.5
0
0.5
t [s]
˙ z [in/s]
1 1.2 1.4 1.6 1.8 2
−50
0
50
t [s]
G(z, ˙ z) [lb]
1 1.2 1.4 1.6 1.8 2
−2
−1
0
1
2
x 10
−4
t [s]
z [in]
1 1.2 1.4 1.6 1.8 2
−0.5
0
0.5
t [s]
˙ z [in/s]
1 1.2 1.4 1.6 1.8 2
−50
0
50
t [s]
G(z, ˙ z) [lb]
(a) Typical time-histories for 300-lb force level (b) Typical time-histories for 500-lb force level
Figure 2.2: Typical time-histories of relative displacementz, relative velocity _ z, and restor-
ing force G(z; _ z) obtained by processing experimental measurements from swept sine vi-
bration tests with maximum force levels of (a) 300 lb and (b) 500 lb.
force in the joint were built using 5
th
-order polynomial expansion. Therefore, the reduced-
order representation of the restoring force in the bolted joint specimens would be given
by
^
G(z; _ z) =
5
X
i=0
5
X
j=0
a
ij
z
i
_ z
j
(2.17)
Each of the 36 identied sets of a
ij
coecients denes a reduced-order model of the
joint. The Chebyshev coecients C
ij
and corresponding power-series coecients aij ob-
tained from one of 36 dynamic tests performed on the lap-type bolted joint are tabulated
in Table 2.1 and 2.2, respectively.
The presence of signicant high-order and cross-product terms in the restoring force
expansions can be used as an indicator of nonlinearities in the system. An analysis of
17
Table 2.1: Chebyshev coecients C
ij
identied for one of the lap-type bolted joint speci-
mens.
T
0
( _ z
0
) T
1
( _ z
0
) T
2
( _ z
0
) T
3
( _ z
0
) T
4
( _ z
0
) T
5
( _ z
0
)
T
0
(z
0
) 4.27 1.79 30.17 1.90 -1.06 -0.34
T
1
(z
0
) -6.92 7.76 -3.05 -0.18 0.65 0.10
T
2
(z
0
) -23.08 -2.44 0.32 -0.41 -1.10 -0.01
T
3
(z
0
) -9.29 -1.01 6.68 0.42 -0.43 0.03
T
4
(z
0
) 0.64 -0.01 1.31 0.09 -0.16 -0.03
T
5
(z
0
) 1.97 0.32 -0.66 0.02 -0.27 -0.05
Table 2.2: Power-series coecients a
ij
identied for one of the lap-type bolted joint spec-
imens.
_ z
0
_ z
1
_ z
2
_ z
3
_ z
4
_ z
5
z
0
2.7110
1
7.46 2.1410
2
1.0410
2
-3.1210
1
-1.3110
2
z
1
1.8610
5
4.9910
4
-7.2310
5
-1.0610
4
1.2610
5
-2.9610
5
z
2
-8.3310
8
-2.8510
7
-1.9410
8
-1.1410
9
2.9910
7
1.8910
9
z
3
-5.2610
12
-1.2910
11
8.9810
12
-2.8510
12
1.7010
13
1.8310
13
z
4
9.3110
14
4.1310
14
2.1510
16
7.6510
15
-4.6810
16
-2.7810
16
z
5
2.2710
19
1.8410
18
3.3210
19
3.4910
19
-2.7410
20
-1.7810
20
the relative contribution of the coecients in Table 2.1 to the restoring force showed the
signicance in the reduced-order model of the terms associated to C
02
, C
11
, C
20
, and
C
30
. Similar conclusions can be drawn by estimating the contribution of eacha
ij
z
i
_ z
j
term
in the power-series expansions, shown in Table 2.2, to the overall restoring force. The
signicance of these dominant terms was observed in all 36 reduced-order models.
A typical surface of the restoring forces
^
G(z; _ z) estimated using one of the reduced-
order models characterizing the microslip phenomena in a lap-type bolted joint, is dis-
played in Figure 2.3(a). It is important to note that in the proposed methodology for
developing data-based reduced-order models of structural components, no assumptions
have been made about the underlying dynamic characteristics of these particular lap-type
bolted joints; however, the intrinsic features, which are embedded in the experimental
measurements, can be inferred from the restoring force surfaces. Notice, for example, the
18
(a) Typical estimated restoring force surface (b) Measured and reconstructed restoring forces
Figure 2.3: (a) Restoring force surface as a function of relative displacement and rela-
tive velocity of a typical inclined lap-type bolted joint specimen obtained using the corre-
sponding reduced-order representation built from experimental data. Notice the nonlinear
nature of the microslip phenomena in the bolted joint. (b) Comparison between the mea-
sured (solid line) and the reconstructed restoring forces (dashed line) in a typical bolted
joint specimen for each of the ve dierent force levels used during the dynamic tests.
Notice that the curves are virtually identical.
non-symmetric nonlinear characteristics of the restoring forces associated to the microslip
mechanism in the bolted joints.
The three-dimensional plot in Figure 2.3(b) depicts both the experimental restoring
forces G(z; _ z;q) (solid line) and reconstructed restoring forces
^
G(z; _ z) (dashed line), as
a function of the measured relative displacements and velocities for the ve dierent
force levels used during the swept-sine vibration tests. Note that the dominant dynamic
features of the underlying physical phenomena in the inclined lap-type bolted joints have
been successfully captured by the associated reduced-order models.
19
2.5.3 Probabilistic reduced representation of random restoring force co-
ecients
In total, thirty-six reduced-order models were built from experimental data for a specic
inclined lap-type bolted joint. Each reduced-order model was dened by the thirty-six
coecients a
ij
in the reduced representation of the restoring force in the bolted joint
specimens given by Eq. (2.17). Therefore, thirty-six observations of the random vector
a, with values inR
36
, collecting the random restoring force coecients a
ij
were available.
Given the 36 observations of random vector a and based on the methodology presented
in section 2.3, a reduced probabilistic representation of a can now be constructed in the
form of
^ a() =p
0
+
d
X
k=1
p

k
r
X
;jj=1
p

;k
H

()
k
(2.18)
where
k
and
k
correspond to the d most signicant eigenvalues and associated eigen-
vectors of the sample covariance matrix
^
C
a
of random vector a, H

() is the multi-
dimensional normalized Hermite polynomial with multi-index and total degree of r,
and is a ddimensional vector of independent standard Gaussian random variables.
An analysis of the relative contribution of the eigenvalues of the sample covariance
matrix showed that the rst three eigenvalues were able to represent the 95% of the
observed variability approximately. A plot of the relative contribution of the rst 10
eigenvalues
i
of the matrix
^
C
a
is displayed in Figure 2.4(a). This suggest that a reduced
representation with d = 3 will be adequate for representing the variability in the random
restoring force coecients. It was also found that a reasonable approximation of was
20
achieve with a multi-dimensional polynomial chaos expansion of total degree r = 1. All
the results presented hereafter were obtained with d = 3 and r = 1.
The set of optimal parameters ^ p =f ^ p
0
; ^ p

g in the probabilistic reduced representation
of the restoring force coecientsa were identied by solving the associated optimization
problem
^ p

= arg max
p

^
L

(p

) ^ p
0
= arg max
p
0
^
L

(p
0
) (2.19)
where the approximations to the likelihood functions ofp

andp
0
are given by
^
L

(p

) =
36
Y
k=1
^
f
^
(
(k)
jp

)
^
L
0
(p
0
) =
36
Y
k=1
^
f
^ a
(a
(k)
jp
0
; ^ p

) (2.20)
in which the probability density functions
^
f
^
(jp

) and
^
f
^ a
(jp
0
; ^ p

) were obtained from
samples generated by Monte Carlo simulation and by using kernel density estimation.
The optimization problem was solve using a genetic algorithm optimization solver. Figure
2.4(b) displays the loglikelihood
^
L

(p

) for a specic value of p

obtained with dierent
number of Monte Carlo samples. It can be seen that reasonable convergence in the log-
likelihood function was achieved with approximately 10000 Monte Carlo realizations. All
results in this study shown hereafter were obtained with 10000 Monte Carlo samples.
The probabilistic representation of the restoring force coecients can now be used to
estimate the probability density function of all a
ij
coecients. Figure 2.5 shows the pdfs
for the four dominant restoring force coecients, a
02
, a
20
, a
30
, and a
11
, in the reduced-
order model of the inclined lap-type bolted joint. The pdfs were estimated from 10000
21
1 2 3 4 5 6 7 8 9 10
0.5
0.6
0.7
0.8
0.9
1
Index [-]
d
X
i=1
λ
i
. m
X
i=1
λ
i
[-]
0 5000 10000 15000
−190
−180
−170
−160
−150
−140
−130
Monte Carlo samples [-]
logL
δ
(p
δ
) [-]
(a) Eigenvalues of covariance matrixCa (b) Estimated loglikelihood ofp

coecients
Figure 2.4: (a) Relative contribution of eigenvalues
i
of the sample covariance matrixC
a
of the identied restoring force coecientsa
ij
. (b) Convergence of loglikelihood logL

(p

)
of polynomial chaos coecients p

with respect to the number of Monte Carlo samples
used to estimate the probability density function
^
f
^
(jp

).
Monte Carlo realizations of the random vector ^ a generated using the probabilistic rep-
resentation in Eq. (2.18). The restoring force coecient a
ij
that were directly identied
from experimental data (i.e., observations of random vector a) are shown as red dots on
each of the pdf s in Figure 2.5.
Using the probabilistic representation of the restoring force coecients built from
experimental data, the mean and error bounds on the restoring force surface for the
tested lap-type bolted joint can be estimated. The mean restoring force surface and
corresponding2 bounds are displayed in Figure 2.6. These estimates were obtained by
reconstructing the restoring forces over the plane dened by the relative displacementz and
relative velocity _ z from 10000 Monte Carlo samples of ^ a (i.e., realizations of the random
a
ij
restoring force coecients) generated using the identied probabilistic representation.
22
−20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
0.05
0.06
˜ a
02
[-]
pdf [-]
−15 −10 −5 0 5
0
0.05
0.1
0.15
˜ a
20
[-]
pdf [-]
−15 −10 −5 0 5
0
0.05
0.1
0.15
0.2
˜ a
30
[-]
pdf [-]
−2 0 2 4 6 8
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
˜ a
11
[-]
pdf [-]
Figure 2.5: Probability density functions of the dominant restoring force coecients a
02
,
a
20
, a
30
, and a
11
in the estimated using the probabilistic reduced-order model of the in-
clined lap-type bolted joint. The pdf s were estimated using 10000 Monte Carlo realizations
of the random vector ^ a comprising the restoring force coecients a
ij
. The red dots on
each pdf correspond to the actual a
ij
coecients obtained from experimental data.
(a) Estimated mean restoring force surface (b) Estimated2 bounds on restoring force surface
Figure 2.6: (a) Mean and corresponding (b) error bounds of the restoring force surface
as a function of relative displacement and relative velocity of a typical inclined lap-type
bolted joint specimen estimated from 10000 Monte Carlo samples of random restoring
force coecients generated with the probabilistic reduced-order representation of the a
ij
built from experimental data.
23
2.5.4 Dynamic response prediction of a complex system
To illustrate the application of the methodology under discussion, consider a complex
structural system in which inclined lap-type bolted joints are use to attach critical com-
ponents to a support structure. In order to predict the dynamic response of the system,
and assess the eects the microslip phenomena in the mechanical joints have on the over-
all response of the system, it is necessary to propagate the uncertainties on the dynamic
characteristics of the bolted joint observed during the experimental tests. To achieve this,
a high-delity probabilistic computational model of the complex structure is developed
by incorporating the identied probabilistic reduced-order model of the inclined lap-type
bolted joint into the deterministic model of the structure.
Let's assume that the complex system can be modeled as a 3DOF system as shown
in Figure 2.7. The three lumped masses are interconnected by means of arbitrary linear
elements and anchored to a support interface at three dierent locations using lap-type
bolted joints. The support interfaces are subjected to deterministic band-limited white-
noise-like motions. The objective is to quantify the uncertainty in the root-mean-square
(RMS) of the displacement and velocity of the critical component corresponding in this
case to the lumped mass m
2
in three dierent congurations with one, two, and three
bolted joints being used to support the structure.
The stochastic dimension of the dynamic problem would correspond to the total num-
ber of random variables required to characterize all input uncertainties in the joints. For
the rst case where just one bolted joints is use, only three i.i.d. Gaussian random vari-
ables = (
1
;
2
;
3
) are required in the probabilistic representation of the restoring force
coecients. Therefore, the solution to the stochastic equation of motion of the 3DOF
24
Figure 2.7: Model of nonlinear 3DOF system with the lumped masses interconnected using
linear elements and anchored to a support interface at three dierent locations using lap-
type bolted joints.
model can be represented in the form of a multi-dimensional truncated polynomial chaos
expansion of the form [25]:
x(t;)
s
X
;jj=0
x

(t)H

(); _ x(t;)
s
X
;jj=0
_ x

(t)H

(); (2.21)
A non-intrusive spectral projection (NISP) approach was used to estimate the PC co-
ecientsx

(t), _ x

(t). In this case, the projection integrals used to obtain the coecients
x

(t) are solved using numerical Gaussian quadrature rules with a full tensor product of
the univariate Gaussian quadrature integration.
x

(t) =
Z
x(t;)H

()p

()d

Nq
1
X
q
1
=1
Nq
2
X
q
2
=1
Nq
3
X
q
3
=1
x(t;
q
1
1
;
q
2
2
;
q
3
3
)H

(
q
1
1
;
q
2
2
;
q
3
3
)w
q
1
w
q
2
w
q
3
(2.22)
wheref
q
i
i
g andfw
q
i
g; q
i
= 1;:::;N
q
i
, denote, respectively, the set of Gaussian quadrature
nodes and corresponding weights associated with the probability density function p().
25
1 2 3 4 5
10
−15
10
−10
10
−5
PCE order [-]
ε
μ
[-]
1 2 3 4 5
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
PCE order [-]
ε
σ
[-]
(a) Relative error on estimated mean (b) Relative error on the estimated std. deviation
Figure 2.8: Average relative errors between the (a) mean and (b) standard deviation of the
displacements estimated using PC expansion with total degree s = 10 and PC expansions
with smaller order s.
Identical approach can be follow to obtain the coecients _ x

in the PC expansion of the
velocity _ x(t;).
Polynomial chaos expansions with order s = 10 for the displacement and velocity
responses of the system were initially obtained usingN
q
= 11 quadrature points along the
three dimensions of . By comparing the relative errors on the means "

and standard
deviation "

of the displacements and velocities estimated with PC expansions of smaller
order s and the responses obtained with s = 10, it was found that PC expansions with
total orders = 3 will be adequate to represent the stochastic solutionsx(t;) and _ x(t;).
Figure 2.8 shows the average relative errors "

and "

for dierent values of s.
Once the PC coecientsx

and _ x

were computed for PC expansions withs = 3, the
estimate of the mean and variance of the displacements x(t;) and velocities _ x(t;) can
be explicitly computed based on the identied PC coecients. The time evolution of the
expected valueE[x
2
] and the variance Var[x
2
] of the predicted displacement of the critical
component of the system under study are displayed in Figure 2.9(a). The time evolution
of the expected value E[ _ x
2
] and the variance Var[ _ x
2
] of the predicted velocity are shown
in Figure 2.9(b).
26
2.5 2.6 2.7 2.8 2.9 3
−2
−1
0
1
2
x 10
−4
t [s]
E[x
2
] [in]

2.5 2.6 2.7 2.8 2.9 3
0
0.5
1
1.5
x 10
−11
t [s]
Var[x
2
] [in
2
]

2.5 2.6 2.7 2.8 2.9 3
−0.2
−0.1
0
0.1
0.2
t [s]
E[ ˙ x
2
] [in/s]

2.5 2.6 2.7 2.8 2.9 3
0
0.5
1
1.5
2
x 10
−5
t [s]
Var[ ˙ x
2
] [in
2
/s
2
]

(a) Mean displacement and variance (b) Mean velocity and variance
Figure 2.9: Time evolution of the (a) mean and (b) variance of the displacement and
velocity of critical component in complex sysmte under study.
The number of random variables required to characterize all input uncertainties in
the cases with two and three bolted joints increase to d = 6 and d = 9, respectively. To
reduce the number of function evaluation when performing the multi-dimensional Gaussian
quadrature integration required to compute the PC coecients x

and _ x

with a full ,
Smolyak sparse grid of Gauss-Hermite nodes were used in the numerical integration. In
these cases, the order of in the PC expansions was s = 3 and the level of the sparse grid
quadrature was set to 4.
The probability density functions for the quantities of interest, the RMS of the dis-
placement and velocity of the critical component (i.e., lumped mass m
2
), for the three
cases where one, two, and three joints were used in the system are shown in Figure 2.10.
Note that as the number of bolted joints increases, the vibration levels in the critical
component are reduced.
27
0.465 0.47 0.475 0.48 0.485
0
50
100
150
200
250
300
x
rms
2
[10
−4
in]
pdf [10
4
in
−1
]

1 Joint
2 Joints
3 Joints
0.0505 0.051 0.0515 0.052 0.0525 0.053
0
1000
2000
3000
4000
5000
6000
˙ x
rms
2
[in/s]
pdf [s/in]

1 Joint
2 Joints
3 Joints
(a) RMS of displacement (b) RMS of velocity
Figure 2.10: Probability density functions of the two quantities of interest: (a) RMS of the
predicted displacement of the critical component (i.e., lumped mass m
2
), and (b) RMS
of the predicted velocity of the critical component, for the structural congurations with
one, two, and three lap-type bolted joints in the structural system
28
Chapter 3
A substructuring approach for detecting, locating, and
quantifying structural changes in chain-like systems
3.1 Introduction
3.1.1 Background and Motivation
Extensive research done during the recent past on structural control and health mon-
itoring of aerospace, civil, and mechanical structures have led to a signicant number
of well-developed system identication and damage detection techniques. Proof of that
is the extremely broad technical literature available on these emerging elds. Most of
these approaches have been developed by relying on linear dynamics-based models of the
structures. However, real structures can exhibit intrinsic nonlinear characteristics during
normal operation and, in many cases, the presence of damage in structural components
will additionally introduce nonlinear eects in the dynamic response of the system.
Among the types of damage that can cause a structure to have a nonlinear behavior
are: cracks that open and close under loading conditions (e.g., fatigue cracks, cracks
in brittle materials); rattling, impacting and sliding loose connections (e.g., loose bolts,
29
debond of glued connections); delamination and debonding in laminated materials (e.g.,
ber reinforced composite plates); and material nonlinearities associated with excesive
deformations (e.g., yielding of steel). These nonlinear phenomena are the most commonly
observed in real structures [1, 20]. Since damage identication could then be performed
by detecting, locating and quantifying the nonlinearities in structural systems [20, 21, 86],
the implementation of nonlinear system identication techniques can potentially improve
the capabilities and performance of structural health monitoring (SHM) strategies.
Excellent and careful reviews of existing nonlinear identication techniques can be
found in Adams and Allemang [2], Haber and Keviczky [28], Worden and Tomlinson
[85], Kerschen et al [42], Farrar et al [20], and Worden et al [86]. Several illustrative
applications can be found in the proceedings of the International and European Workshops
on Structural Health Monitoring [13, 14, 79], and the World Conference on Structural
Control and Monitoring [12, 37].
Nevertheless, most of the available identication techniques can be considered to be
deterministic due to the fact that measurements from a structural system are used to
characterize the nonlinearities by means of a nominal mathematical (e.g., parametric or
non-parametric) model, while neglecting the eects of aleatory and epistemic uncertain-
ties that can be present in real structures. The inherent stochastic nature of the systems'
components (i.e., randomness in structural, geometric and material properties); the vari-
ability in environmental and operational conditions; and the uncertainties associated with
the modeling, measurement and data analysis process can lead to unreliable detection of
nonlinearities and therefore, to unreliable damage detection. Consequently, robust and
30
reliable nonlinear identication and structural health monitoring methodologies should
address the issues of quantication and propagation of uncertainties.
Background information regarding the approaches and challenges in the representation
and propagation of uncertainties in structural systems is available in Ghanem and Spanos
[24], Hemez [29], Soize [78], Ghanem et al [25], Li and Chen [50], Masri et al [59], and
Adhikari et al [3].
3.2 Chain-like nonlinear system identication approach
Consider an MDOF chain-like system, consisting of a series of lumped masses m
i
inter-
connected byn arbitrary unknown nonlinear elementsG
(i)
, subjected to a base motionx
0
,
and/or directly applied forces F
i
. The nonlinear elements' restoring forces are assumed
to depend on the relative displacement and velocity across the terminals of each element,
in addition to a set of specic parametersp that characterize the various types of nonlin-
earities. The dierential equations of motion for the system under discussion, shown in
Fig. 3.1, can be written as [54, 61]
G
(n)
(z
n
; _ z
n
;p) =
F
n
m
n
x
n
G
(i)
(z
i
; _ z
i
;p) =
F
i
m
i
x
i
+
m
i+1
m
i
G
(i+1)
(z
i+1
; _ z
i+1
;p) (3.1)
for i =n 1;n 2;:::; 1
where G
(i)
(z
i
; _ z
i
;p) is the mass-normalized restoring force function of the nonlinear ele-
mentG
(i)
, x
i
is the absolute acceleration of the massm
i
, withz
i
and _ z
i
being the relative
31
displacements and velocities between two consecutive masses. These relative motion vari-
ables can be obtained from the absolute state variables of the masses and the moving
support.
From Eqn. 3.1, it is clearly seen that the restoring forces acting on all nonlinear ele-
ments in the chain-like system can be sequentially determined by starting the data pro-
cessing from the nth (tip) element of the chain. Within the context of this method, the
absolute accelerations x
i
are assumed to be available from measurements, as well as the
applied forces F
i
, the base excitation x
0
and the magnitude of the lumped masses m
i
.
After obtaining all the restoring force time histories, it is possible to generate a non-
parametric representation for each nonlinear element, in terms of a truncated doubly-
indexed series expansion in a suitable basis, that approximates the real restoring force
function [53, 56, 57, 87]. The approximating representation
^
G
(i)
(z
i
; _ z
i
) for the obtained
restoring forces, in an orthogonal polynomial basis, is given by the following expression:
G
(i)
(z
i
; _ z
i
;p)
^
G
(i)
(z
i
; _ z
i
) =
qmax
X
q=0
rmax
X
r=0
C
(i)
qr
T
q
(z
0
i
)T
r
( _ z
0
i
) (3.2)
where C
(i)
qr
are the Chebyshev series coecients, T
k
() is the Chebyshev polynomial of
order k, and z
0
i
, _ z
0
i
are the normalized relative state variables. Subsequently, each of the
estimated restoring forces can be expressed as a power series of the form
^
G
(i)
(z
i
; _ z
i
) =
qmax
X
q=0
rmax
X
r=0
a
(i)
qr
z
q
i
_ z
r
i
(3.3)
where a
(i)
qr
are constant coecients, and z
i
, _ z
i
are the relative state variables.
32
The application of this non-parametric identication approach allows the capture of the
dominant features of the nonlinear elements into reduced-order, model-free representations
[57]. Due to the fact that the non-parametric representation of the restoring force depends
on the relative (i.e., inter-story) state variables, the numerical implementation of the
chain-like system identication approach requires the availability of displacement and
velocity time-histories, which can be obtained by digital signal processing of the measured
accelerations. It should be emphasized that, since this proposed approach is entirely
data-driven, the number of degrees-of-freedom in the considered chain-like system will be
determined by the number of available sensors deployed on the structure.
In this identication approach, since no restrictions are imposed on the maximum or-
der of the series expansion, the accuracy of the restoring force t can be easily improved by
increasing the order of the Chebyshev polynomials. However, as is pointed out by Masri
et al [59], low-order ts are desired since the main goal of the identication process is to
construct reduced-order models of the elements in the system. By virtue of the orthgo-
nality property of the estimated Chebyshev coecients, the optimal model is obtained by
truncating the restoring force expansion to a suitable order, which is usually determined
through an analysis of the relative-contribution of the identied Chebyshev coecients
C
(i)
qr
and the normalized mean-square error [53, 59, 85].
3.3 LANL test-bed structure
Experimental data from a multiple degree-of-freedom test-bed structure, that has been
tested at the Los Alamos National Laboratory (LANL), were used to illustrate the ap-
plication of the methodology under discussion in detecting, locating and quantifying, and
33
dierentiating between linear and nonlinear structural changes in a test-bed structure.
Full details concerning this LANL test setup are documented in [21].
The laboratory three-story shear-building structure (see Fig. 3.2) consists of four alu-
minum plates (30:5 30:5 2:5 cm) connected by bolted joints to four aluminum columns
(17:7 2:5 0:6 cm) at each
oor. A movable assembly consisting of an additional sus-
pended column (15 2:5 2:5 cm) and an adjustable bumper, that can be attached at
each of the levels of the structure, is used to introduce an asymmetric impact nonlinearity
(a \snubber") in the system. The gap distance can be modied by adjusting the position
of the bumper to vary the level of the nonlinearity. The whole structure is mounted on
two rails to allow the system to slide only in one direction. An electro-dynamic shaker was
used to provide a band-limited random base excitation (20-150 Hz) to the test structure.
The deployed sensor network consists of four accelerometers and a force transducer.
The accelerometers were attached to each aluminum plate, along a vertical center line,
to measure the dynamic response of the 4DOF lab structure. The force transducer was
connected to the tip of the stinger to gauge the input force generated by the shaker. The
sensor's measurements were recorded at a sampling frequency of 322.58 Hz by a data
acquisition system.
3.4 Detection of linear changes in a MDOF test-bed struc-
ture
The structural changes in the system were physically simulated through pure variations in
either the mass or stiness of the reference structure. The mass of the system was modied
34
n
m
2
m
1
m
(1)
G
(2)
G
() n
G
2
F
n
F
1
F
0
x
x
Figure 3.1: Typical structural topol-
ogy for a nonlinear MDOF chain-like
system.

4
0.177
0.177
0.177
Accelerometer 4
0.174
A B
3
2
1
C D
3
2
1
Accelerometer 3
Accelerometer 2
Accelerometer 1
Column
Bumper
Shaker
2nd Floor
1st Floor
3rd Floor
Base
2nd Floor
1st Floor
3rd Floor
Base

0.305
Direction of Shaking
C D
B
A
0.305
Accelerometer 4
Force
Transducer

Figure 3: Basic dimensions of the three story frame structure.

The damage cases were simulated through the introduction of nonlinearities into the structure. A
bumper and a suspended column were used with different gaps in between them as shown in
Figure 4 (b). The gap between the bumper and the suspended column was varied (0.05, 0.10, 0.13,
0.15 and 0.20 mm) in order to introduce different degrees of nonlinearity (State#08 to 12).

The main goal of this study is to detect damage when the structure has undergone structural
changes caused by operational and environmental effects. For this purpose three state conditions
were set up (State#14 to 16).

The testing was repeated for different structural state conditions, and ten measurements were
recorded for each state.

Figure 3.2: LANL-4DOF test-bed structure
schematic drawings.
by attaching a 1.2 kg concentrated mass to the aluminum plates, while the changes in sti-
ness were introduced by reducing by half the cross-section thickness of selected columns.
This change in the cross-section corresponded to a 87.5% reduction in the column's sti-
ness. The nine structural state congurations considered in this study are summarized in
Table 3.1. A total of 90 acceleration data sets, involving all structural congurations, were
obtained from ten experimental tests performed under dierent external force realizations
(i.e., 10 data sets for each system congurations) in order to account for the variability in
the data. Although the test-bed structure had an adjustable nonlinear gap in the third
story, it was set to keep the system within the linear range during the dynamic tests
considered in this study.
35
Table 3.1: Summary of structural state conditions
State Condition Description
State#1 Reference condition -
State#2 19.1% base mass increment 1.2 kg. additional mass on the base
State#3 19.1% 1
st
-story mass increment 1.2 kg. additional mass on the 1
st
story
State#4 21.8% 1
st
-story stiness reduction 87.5% stiness reduction in column 1BD (1
st
story)
State#5 43.7% 1
st
-story stiness reduction 87.5% stiness reduction in columns 1AD and 1BD (1
st
story)
State#6 21.8% 2
nd
-story stiness reduction 87.5% stiness reduction in column 2BD (2
nd
story)
State#7 43.7% 2
nd
-story stiness reduction 87.5% stiness reduction in columns 2AD and 2BD (2
nd
story)
State#8 21.8% 3
rd
-story stiness reduction 87.5% stiness reduction in column 3BD (3
rd
story)
State#9 43.7% 3
rd
-story stiness reduction 87.5% stiness reduction in columns 3AD and 3BD (3
rd
story)
3.4.1 Sample data processing results
For the purposes of this study, the test-bed structure was considered as a 3DOF chain-like
system subjected to base motions, hence the simplied equations of motion (Eqn. 3.1)
could be rewritten as:
G
(3)
(z
3
; _ z
3
;p) = x
3
G
(2)
(z
2
; _ z
2
;p) = x
2
+
m
3
m
2
G
(3)
(z
3
; _ z
3
;p) (3.4)
G
(1)
(z
1
; _ z
1
;p) = x
1
+
m
2
m
1
G
(2)
(z
2
; _ z
2
;p)
In addition, only the acceleration time-histories recorded by the four accelerometers and
the
oor mass ratios m
i
=
m
i+1
m
i
were assumed to be available. Because the system's mass
is approximately uniformly distributed throughout the structure, the mass ratios m
i
were
considered equal to one. Since the displacement and velocity time-histories at measure-
ment stations are required to apply this identication approach, the acceleration records
were windowed, detrended, band-pass ltered and integrated. Typical top
oor vibration
records, including the measured accelerations and corresponding computed velocities and
36
13 13.5 14 14.5 15
−20
−10
0
10
20
¨x [m/s
2
]
13 13.5 14 14.5 15
−0.05
0
0.05
˙ x [m/s]
13 13.5 14 14.5 15
−1
0
1
x 10
−4
t [s]
x [m]
13 13.5 14 14.5 15
−1
0
1
x 10
−4
z
2
[m]
13 13.5 14 14.5 15
−0.05
0
0.05
˙ z
2
[m/s]
14 14.1 14.2 14.3 14.4 14.5
−10
−5
0
5
10
G
(2)
(z
2
, ˙ z
2
) [m/s
2
]
t [s]

Experimental Reconstructed
(a) Second-story vibration records (b) Response time-history records
Figure 3.3: (a) Typical acceleration, velocity and displacement records obtained at 2
nd

oor. (b) Relative displacement and velocity time-histories computed between masses
m
2
and m
1
are shown in the rst two rows of this gure. The third row corresponds
to a comparison between the measured restoring force time-history for element G
(2)
and
the reconstructed time-history using the identied restoring force coecients (virtually
identical curves).
displacements, are shown in Fig. 3.3(a). Note that the system displacements are on the
order of one tenth of a millimeter.
Due to the limited experimental data available, all vibration records were divided into
overlapped segments of 6.34 seconds, which include more than 100 fundamental periods of
the system, so as to obtain an enlarged collection of acceleration, velocity and displacement
time-histories. In total, 150 ensembles of time-history records were generated for each of
the structural congurations.
Once the relative displacements and velocities had been computed, the proposed time-
domain identication technique is then applied to the ensembles of vibration record seg-
ments to build the associated non-parametric reduced-order models for each element in
37
the 3DOF chain-like system, by determining the corresponding restoring force coecients.
The rst two plots in Fig. 3.3(b) illustrate sample time-histories of relative displacements
and velocities computed between the second and rst
oor of the 3DOF system in its
reference structural conguration. The third plot depicts the time-history records of the
measured and reconstructed mass-normalized restoring forces (basically identical curves)
in solid and dashed lines respectively, for the elementG
(2)
connecting the rst and second
stories. Figure 3.4(a) shows, as it is expected for linear elements, the estimated planar
restoring force surfaces over the normalized phase space for the reference condition (i.e.,
state#1). The reconstructed mass-normalized restoring forces
^
G
(i)
(z
i
; _ z
i
) were computed
using the estimated coecients and the corresponding sequence of Chebyshev polynomials
in the relative state variables.
It should be noted that, although the structure was kept within the linear range during
the dynamic tests, for the identication purposes of this study, the system was not as-
sumed linear. The restoring force identication was initially carried out using Chebyshev
polynomials of third-order in both normalized variables z
0
and _ z
0
. A relative-contribution
analysis of the identiedC
(i)
qr
indicated that the linear terms had the most signicant con-
tributions to the restoring forces while the eect of the nonlinear terms were negligible.
The non-parametric models for the elementsG
(i)
, in the reference and modied structural
congurations, were then reduced to their corresponding rst-order expansions by using
the orthogonality property of the identied Chebyshev coecients [53].
38
3.4.2 Change detection
As a consequence of structural changes, the dynamic characteristics and response time-
histories of any chain-like system are aected; hence, the estimated restoring force surface
of the interconnecting elements will exhibit variations with respect to the reference case. In
Fig. 3.4(b), the changes in the identied restoring force surface for the elements G
(3)
,G
(2)
and G
(1)
caused by 43.75% second-story stiness reduction (i.e., state#7) are displayed.
From Eqn. (3.2) and (3.3), it is observed that for a given order in the double-indexed
expansion, variations in the restoring force would have eects on the identied coecients,
since they characterize the governing dynamic features of the system. This makes the
restoring force coecients a suitable set of parameters for change detection applications
[57, 58, 87].
Despite the fact that the Chebyshev coecientsC
(i)
qr
have several useful properties (i.e.,
orthogonality and unbiasness with respect to model complexity) for the identication
and detection of changes in linear and nonlinear systems [87], their use in a stochastic
framework is inconvenient since they rely on normalized variables [25]. Consequently, the
equivalent de-normalized restoring force coecients a
(i)
qr
, corresponding to the dominant
terms in the expansion, were selected instead as the features to be used and analyzed in
this experimental study of change detection in uncertain chain-like systems.
Since the mass-normalized restoring force function for linear elements can be expressed
as:
G
(i)
(z
i
; _ z
i
) =
k
i
m
i
z
i
+
c
i
m
i
_ z
i
=

k
i
z
i
+ c
i
_ z
i
=a
(i)
10
z
i
+a
(i)
01
_ z
i
(3.5)
39
the changes in mass and stiness introduced in the LANL test-bed structure can be de-
tected and quantied through direct analysis of the a
(i)
10
and a
(i)
01
coecients, which here-
after are going to be also called the \mass-normalized stiness-like" and \mass-normalized
damping-like" terms, respectively. The global modal parameters of the system can be de-
termined through the eigen decomposition of the globalM
1
K andM
1
C matrices, which
are given by the following expressions:
M
1
K =
2
6
6
6
6
6
6
6
6
6
6
4
a
(1)
10
m
0
a
(1)
10
m
0
0 0
a
(1)
10
a
(1)
10
+a
(2)
10
m
1
a
(2)
10
m
1
0
0 a
(2)
10
a
(2)
10
+a
(3)
10
m
2
a
(3)
10
m
2
0 0 a
(3)
10
a
(3)
10
3
7
7
7
7
7
7
7
7
7
7
5
(3.6)
M
1
C =
2
6
6
6
6
6
6
6
6
6
6
4
a
(1)
01
m
0
a
(1)
01
m
0
0 0
a
(1)
01
a
(1)
01
+a
(2)
01
m
1
a
(2)
01
m
1
0
0 a
(2)
01
a
(2)
01
+a
(3)
01
m
2
a
(3)
01
m
2
0 0 a
(3)
01
a
(3)
01
3
7
7
7
7
7
7
7
7
7
7
5
(3.7)
Notice that these matrices were reconstructed by modeling the three-story structure as a
4DOF shear building [21].
40
By virtue of the normalization procedure embedded in the Chebyshev coecients, it
can be shown that the relative changes in the Chebyshev and de-normalized power-series
coecients are related by the equations:
C
10
C
r
10
=
a
10
(z
max
z
min
z
r
max
+z
r
min
) + a
10
(z
max
z
min
)
a
10
(z
r
max
z
r
min
)
(3.8)
C
01
C
r
01
=
a
01
( _ z
max
_ z
min
_ z
r
max
+ _ z
r
min
) + a
01
( _ z
max
_ z
min
)
a
01
( _ z
r
max
_ z
r
min
)
(3.9)
where ()
r
denotes the variables associated with the reference case. Clearly, straight in-
terpretation of variations in Chebyshev coecients as genuine changes in the structure
can be misleading because of their dependence on the extreme values of the relative state
variables.
It is worth pointing out that, although the identication results for all structural con-
ditions considered in this study indicated that the interstory restoring forces were entirely
characterize by the a
(i)
10
coecients, which is foreseen for shear building-type structures,
the coecients a
(i)
01
were also considered for the sake of completeness.
3.4.3 Implementation and statistical analysis
The results of implementing the chain-like system identication to build data-driven
reduced-order models of the LANL test-bed structure, and using the associated restoring
force coecients to detect structural changes in the system, are reported in this section.
The second-order statistics of coecients a
(i)
10
and a
(i)
01
, obtained from all data ensem-
bles and for each structural state condition, are summarized in Table 3.2. Looking at
41
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−15
−10
−5
0
5
10
15
z
′
3
˙ z
′
3
ˆ G
(3)
(z
3
, ˙ z
3
)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−3
−2
−1
0
1
2
3
z
′
3
˙ z
′
3
Δ
ˆ G
(3)
(z
3
, ˙ z
3
)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−15
−10
−5
0
5
10
15
z
′
2
˙ z
′
2
ˆ G
(2)
(z
2
, ˙ z
2
)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−3
−2
−1
0
1
2
3
z
′
2
˙ z
′
2
Δ
ˆ G
(2)
(z
2
, ˙ z
2
)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−15
−10
−5
0
5
10
15
z
′
1
˙ z
′
1
ˆ G
(1)
(z
1
, ˙ z
1
)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−3
−2
−1
0
1
2
3
z
′
1
˙ z
′
1
Δ
ˆ G
(1)
(z
1
, ˙ z
1
)
(a) Reference restoring force surfaces (b) Surfaces of change in restoring force
Figure 3.4: (a) Identied restoring force surface for elements G
(3)
, G
(2)
and G
(1)
in the
reference structural condition. (b) Identied change in the restoring force surfaces inG
(3)
,
G
(2)
and G
(1)
due to a 43.75% second-story stiness reduction. Note that, for enhanced
viewing, dierent amplitude scales are used in the LHS and RHS columns of plots.
42
the mean and coecient of variation of the restoring force coecients, it is noted
that the mass-normalized stiness-like coecients have a low variability, with coecients
of variation ranging from 2% to 6%, compared to the more scattered mass-normalized
damping-like coecients (i.e., coecients of variation between 50% and 250%). The dis-
persion of each coecient is evidently related to the level of importance (contribution)
in the characterization of the restoring forces. Clearly, from the change detection point
of view, the a
(i)
10
coecients are much more robust than the a
(i)
01
coecients. Since the
structure was tested under controlled laboratory conditions, the statistical variability in
the restoring force coecientsa
(i)
10
anda
(i)
01
observed in this experimental study is basically
due to modeling, measurement, and data processing errors.
In order to have a more appropriate description and characterization of the random-
ness in the restoring force coecients, their underlying probability distributions have to be
estimated. In an initial exploratory data analysis, normal probability plots indicated that
even though the mass-normalized coecients within the rst and third quartiles could be
reasonable assumed to have normal distributions, they deviated from Gaussianity in the
tails of the distributions. This non-Gaussianity in the distribution of the a
(i)
10
and a
(i)
01
co-
ecients can be attributable to inherent nonlinearities in the system's dynamic properties
as well as the model-order reduction performed in the identication procedure [57]. In
Fig. 3.5(a), the (representative example) probability plots of the second-
oor coecients
identied from the reference condition are shown. Finally, the stochastic representations
for the identied coecients a
(i)
10
and a
(i)
01
, for all structural conditions, were obtained by
kernel density estimation. Figure 3.5(b) displays the histograms of the mass-normalized
stiness-like and mass-normalized damping-like coecients for the second-
oor element in
43
Table 3.2: Summary of mean () and coecient of variation () of the identied restoring force coecients for the LANL test-bed
structure.
1
st

oor 2
nd

oor 3
rd

oor
a
(1)
10
=

k
1
a
(1)
01
= c
1
a
(2)
10
=

k
2
a
(2)
01
= c
2
a
(3)
10
=

k
3
a
(3)
01
= c
3
State (10
4
) (10
4
) (10
4
)
State#1 7.121 0.040 10.665 0.716 6.390 0.048 5.742 1.229 6.733 0.035 3.795 1.057
State#2 7.039 0.039 10.460 0.571 6.421 0.045 3.487 1.745 6.728 0.027 3.933 1.340
State#3 5.830 0.031 9.039 0.520 6.334 0.045 6.282 1.057 6.703 0.043 4.776 1.077
State#4 5.513 0.042 6.468 0.764 6.174 0.054 7.975 0.747 6.618 0.033 4.076 1.155
State#5 4.092 0.040 1.965 2.580 6.179 0.040 6.422 1.061 6.547 0.032 2.479 1.816
State#6 7.053 0.041 10.958 0.519 4.836 0.041 5.511 0.891 6.611 0.026 3.790 0.981
State#7 7.187 0.032 10.174 0.597 3.597 0.035 5.357 0.717 6.572 0.026 3.944 1.004
State#8 7.223 0.062 7.665 0.968 6.700 0.051 3.166 1.541 5.269 0.031 2.197 1.759
State#9 7.303 0.058 8.799 0.736 6.486 0.039 3.055 1.686 3.765 0.025 2.273 1.249
44
−5 0 5 10 15 20
0.003
0.05
0.25
0.50
0.75
0.95
0.997
a
(2)
01
Probability
5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2
x 10
4
0.003
0.05
0.25
0.50
0.75
0.95
0.997
a
(2)
10
Probability
−15 −10 −5 0 5 10 15 20 25 30
0
0.02
0.04
0.06
0.08
0.1
a
(2)
01
pdf
5 5.5 6 6.5 7 7.5
x 10
4
0
1
2
x 10
−4
a
(2)
10
pdf
(a) Normal probability plots (b) Histograms and probability density functions
Figure 3.5: Statistical representation of the identied restoring force coecients for
the reference structural condition. (a) Normal probability plots of the mass-normalized
stiness-like a
(2)
10
and mass-normalized damping-like a
(2)
01
coecients. (b) Corresponding
histograms, estimated probability density functions by kernel density estimators (solid
lines) and superposed Gaussian distributions (dashed lines).
the baseline structural condition, the estimated probability density functions (solid lines),
as well as the corresponding superposed Gaussian distributions (dashed lines).
To facilitate the visual analysis of the results for detecting structural changes in the
LANL test structure, the probability density functions of the reference-normalized coe-
cients ~ a
(i)
mn
= (a
(i)
mn

r
a
(i)
mn
)=
r
a
(i)
mn
, where
r
a
(i)
mn
indicates the mean value of the coecients
a
(i)
mn
from the reference condition, were estimated for all the structural conditions listed
in Table 3.1. Figures 3.6-3.9 display the pdfs of ~ a
(i)
10
and ~ a
(i)
01
for the baseline condition
in solid lines, while the probability functions from the modied structural congurations
are plotted with dot-dashed and dashed lines. First (top) rows correspond to third-story
coecients; middle rows to second-story, and bottom rows to rst-story.
Two normalized indices =
r
and =
r
were additionally employed to assess the
eectiveness and robustness of the chain-like system identication approach in detecting
45
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(1)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(2)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(3)
10
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(1)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(2)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(3)
01
pdf
(a) Normalized stiness-like coecients (~ a
(i)
10
) (b) Normalized damping-like coecients (~ a
(i)
01
)
Figure 3.6: Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines correspond to
the pdf of the coecients in the baseline condition. The dot-dashed and dashed lines show
the coecients' pdf obtained in state#2 and state#3, respectively.
and localizing structural changes. These dimensionless indices are found by dividing the
dierence in the mean values of the estimated restoring force coecients ( =
r
) by
the corresponding means (
r
) and standard deviations (
r
) from the reference case. The
dimensionless index =
r
, which corresponds to the relative change in the coecients,
can be used intuitively to assess the magnitude or level of \change" in the system, while a
measure of the statistical signicance of the detected changes can be gauged by the ratio
=
r
, which is also known as the signal-to-noise ratio (SNR). Relatively large values of
the latter index indicate that the existing mean dierences are not attributable just to
the normal variability within the coecients and therefore, they can be associated with
\genuine" changes in the system [58]. In this study, a rough threshold for detection of
j=
r
j 2:0 was established to determine, with at least a 95% of condence, when
46
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(1)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(2)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(3)
10
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(1)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(2)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(3)
01
pdf
(a) Normalized stiness-like coecients (~ a
(i)
10
) (b) Normalized damping-like coecients (~ a
(i)
01
)
Figure 3.7: Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines correspond to
the pdf of the coecients in the baseline condition. The dot-dashed and dashed lines show
the coecients' pdf obtained in state#4 and state#5, respectively.
signicant changes had occurred in the a
(i)
10
and a
(i)
01
coecients. The values of =
r
and =
r
for all eight scenarios with structural changes are summarized and presented
in Table 3.3. Table entries in boldface correspond to actual structural changes that were
detected correctly.
For the state#2 scenario, the tabulated results showed that the relative changes in
the mean values of all the restoring force coecients, including the 40% reduction in
the damping-like coecient for the second
oor, were not signicant since all =
r
were below the previously dened threshold. Therefore, by counting on the experimental
results, the presence of structural changes, in this case, is ruled out because the
uctuations
on the mean values can be associated with the randomness of the estimated restoring force
coecients. Notice that, in this scenario, an additional mass was placed at the baseplate
47
Table 3.3: Summary of relative mean change (=
r
) and signal-to-noise ratio (=
r
) in the identied restoring force coecients
for the LANL test-bed structure. Boldfaced table entries correspond to the detected structural changes.
1
st

oor 2
nd

oor 3
rd

oor
a
(1)
10
=

k
1
a
(1)
01
= c
1
a
(2)
10
=

k
2
a
(2)
01
= c
2
a
(3)
10
=

k
3
a
(3)
01
= c
3
State =r =r =r =r =r =r =r =r =r =r =r =r
State#2 -0.012 -0.285 -0.019 -0.027 0.005 0.100 -0.393 -0.320 -0.001 -0.023 0.036 0.034
State#3 -0.181 -4.480 -0.152 -0.213 -0.009 -0.179 0.094 0.076 -0.005 -0.130 0.259 0.245
State#4 -0.226 -5.579 -0.394 -0.549 -0.034 -0.695 0.389 0.317 -0.017 -0.491 0.074 0.070
State#5 -0.425 -10.506 -0.815 -1.138 -0.033 -0.678 0.118 0.096 -0.028 -0.796 -0.346 -0.328
State#6 -0.010 -0.235 0.027 0.038 -0.243 -5.019 -0.040 -0.033 -0.018 -0.523 -0.001 -0.001
State#7 0.009 0.229 -0.046 -0.064 -0.437 -9.019 -0.067 -0.055 -0.024 -0.689 0.039 0.037
State#8 0.014 0.352 -0.281 -0.393 0.049 1.002 -0.449 -0.365 -0.217 -6.257 -0.421 -0.398
State#9 0.025 0.629 -0.175 -0.244 0.015 0.312 -0.468 -0.381 -0.441 -12.688 -0.401 -0.379
48
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(1)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(2)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(3)
10
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(1)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(2)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(3)
01
pdf
(a) Normalized stiness-like coecients (~ a
(i)
10
) (b) Normalized damping-like coecients (~ a
(i)
01
)
Figure 3.8: Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines correspond to
the pdf of the coecients in the baseline condition. The dot-dashed and dashed lines show
the coecients' pdf obtained in state#6 and state#7, respectively.
of the structure. It is important to highlight that detecting structural changes located at
the system's base level, are beyond the capabilities of this methodology since this approach
relies on the relative motion between interconnected lumped-masses.
Similar to the previous scenario, in state#03 the structure underwent a change in
the mass (i.e., 19.1% mass increment), but in this case, the additional mass was placed
at the rst-
oor plate. The values obtained for =
r
showed a reduction of 18.1%
in the mean of the identied mass-normalized stiness-like coecient for the rst-
oor
element G
(1)
, while relatively low changes, around 1%, were observed in the second and
third-story elements G
(2)
and G
(3)
. In addition, the mean values of the mass-normalized
damping-like terms also changed by 15.2%, 9.4% and 25.9% in the rst, second and third

oor respectively. By examining the values of =
r
, it is clear that only the a
(1)
10
term
49
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(1)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(2)
10
pdf
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
5
10
15
20
25
˜a
(3)
10
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(1)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(2)
01
pdf
−8 −6 −4 −2 0 2 4 6 8
0
0.5
1
1.5
˜a
(3)
01
pdf
(a) Normalized stiness-like coecients (~ a
(i)
10
) (b) Normalized damping-like coecients (~ a
(i)
01
)
Figure 3.9: Comparison of probability density functions of identied mass-normalized
stiness-like and mass-normalized damping-like coecients. The solid lines correspond to
the pdf of the coecients in the baseline condition. The dot-dashed and dashed lines show
the coecients' pdf obtained in state#8 and state#9, respectively.
with the largest signal-to-noise ratio (=
r
=4:480) for this scenario had a signicant
reduction in the coecient's mean with respect to the reference case; while the variations
in the other coecients were negligible. Keeping in mind that changes in a
(i)
10
coecients
can be caused by modications in either or both the mass and stiness; hence, additional
assumptions and/or tests have to be made in order to infer further information about how
those physical parameters have been modied. This can be achieved by experimentally
estimating the mass in the sections of the chain-like system where the structural change
have been located. Detailed description of several approaches for mass estimation can be
found in [4, 54, 62, 84]. Assuming that only changes in mass could occur in this scenario,
the observed reduction in the a
(1)
10
coecient would correspond to a 22.1% increment in
the rst-
oor mass, which is fairly close to the actual 19.1% additional mass.
50
In Fig. 3.6(a), the estimated probability density functions of the reference-normalized
~ a
(i)
10
coecients obtained for state#2 (dot-dashed lines) and state#3 (dashed lines) are
compared with the baseline distributions from state#1 (solid lines). Clearly, the only sig-
nicant shift in the mass-normalized stiness-like coecients is observed in the structure's
rst-
oor. In the second and third
oor no dierence in the coecients' distributions were
observed after introducing the structural change. From inspection of Fig. 3.6(b), it is eas-
ily seen that the probability distributions of the mass-normalized damping-like coecients
did not dier appreciably among the dierent structural conditions.
In state#4 the stiness of the rst
oor was decreased by 21.88% by introducing a
87.5% stiness reduction in one of the four columns of the corresponding story. Similarly,
a 43.75%
oor stiness reduction was obtained for state#5 by decreasing the stiness
of two columns by a 87.5%. Since the structural changes in these scenarios consisted
of column-stiness reductions, it was expected that the estimated rst-
oor stiness-like
coecients would have the largest and most signicant variations among all other coe-
cients. By comparing the indices =
r
listed in Table 3.3, it is worth noting that the
detected relative changes in the mean of the identied coecienta
(1)
10
, for each of the above
mentioned scenarios, had in overall the highest levels of signicance, with corresponding
values of =
r
=5:579 and =
r
=10:506. As previously discussed, these results
mean that signicant and observable dierences exist between thea
(1)
10
coecients from the
baseline condition and each of the cases under discussion. Besides, the results also showed
that no false-positive changes in coecients a
(2)
10
and a
(3)
10
were detected. To appreciate
the capability of the presented methodology to assess the level of change in the structure,
51
a simple inspection of the =
r
indices for the relevant coecient is needed. The val-
ues of the corresponding detected mean changes in the rst-
oor stiness-like coecient
were =
r
=0:226 and =
r
=0:425 for the state#4 and state#5 respectively.
The magnitude of the identied mean reductions in a
(1)
10
correlate quite well with the real
story-stiness reductions in the experimental model. Notice that detected changes were
proportional to the magnitude of the structural modication introduced in the system.
Analysis of the probability density functions depicted in in Fig. 3.7 will lead to conclusions
similar to the already described ndings.
For the scenarios of state#6 through state#9, the normalized indices (Table 3.3) and
the density distributions of the a
(i)
10
coecients (Figs. 3.8-3.9), also showed the presence
of statistically signicant changes in the mass-normalized stiness-like coecients of the
second and third
oors. The magnitude and location of the detected changes closely
agree with the actual stiness modications introduced in system. Also note that no
false-positives in the a
(i)
10
coecients were observed for these scenarios. In contrast, the
mass-normalized damping-like coecients did not provide any useful information about
the condition of the structural system.
The results of this experimental study showed that the proposed detection approach
for chain-like systems was able, in a rigorous statistical framework and despite the in-
herent randomness in the dominant restoring force coecients, to condently detect the
presence of structural changes, accurately locate the structural section where the change
occurred (i.e., rst-
oor, second-
oor or third-
oor), and provide an accurate estimate of
the actual level of \change". It is important to point out that, for the established decision
rulej=
r
j 2:0, the minimum detectable (relative) change (MDC ) in the identied
52
restoring force coecients that can be reliably detected is given by MDC = 2:0
r
, where

r
indicates the coecient of variation of the coecients from the reference condition.
For the mass-normalized stiness-like terms a
(i)
10
, the corresponding minimum detectable
changes were 8%, 9% and 7% for the rst, second and third
oor respectively. Since the
magnitude of the structural changes was estimated quite accurately, it can be assumed
that, relative structural changes in the
oor stiness and mass of the LANL test-bed
structure below an approximate 9% could not be reliable detected by this proposed SHM
methodology.
3.4.4 Global modal analysis
A full-order nite element model of the test structure, as well as the results from the
experimental modal identication using the ERA algorithm [39] were employed to vali-
date the eectiveness of proposed change detection approach based on the chain system
identication methodology. The identication of the mass-normalized stinesses from
the experimental modal parameters (i.e., eigenvalues and mode shapes) was carried out
through the least-squares solution of the associated eigenvalue problem [11]. Similar to the
chain-like system identication, the variations in the mass-normalized stinesses can be
caused by either changes in the mass or the stiness of the structure. Therefore, a mass es-
timation would be necessary in order to dierentiate the type of structural change. Tables
3.4 and 3.5 summarize the relative changes (=
r
) and signal-to-noise ratios (=
r
)
of the mass-normalized stinesses

k
i
estimated from the analytical model (FEM), exper-
imental modal analysis (ERA), and the proposed algorithm (ChainID). Table entries in
53
boldface correspond to actual structural changes that were detected correctly while the
underlined ones were false positives.
From simple inspection of the tabulated =
r
values (Table 3.4), it can be observed
that both the ERA and ChainID algorithms were capable of estimating, with very good
level of accuracy, the magnitude of the dierent structural changes to which the LANL
test-bed system was subjected to. However, by comparing the =
r
indices in Table
3.5, the chain-like system identication methodology is shown to have better detection
robustness than the eigensystem realization algorithm. No false-positive changes were
detected by the approach proposed in this paper while the ERA had a false-positive rate
of 35.3% (6/17). Notice that this rate depends on the threshold for =
r
, and it clearly
can be increased or reduced by varying the levels of condence. Keep in mind that the
conditionj=
r
j 2:0 was assumed in order to detect structural changes beyond the
95% condence intervals of the reference distributions. The minimum detectable change
(MDC = 2:0
r
) in the mass-normalized stinesses identied with ERA was found to be
around 6%. However, because of the high rate of false-positives obtained with ERA, it
is not possible to assume that the estimated MDC would be a good indication for the
minimum structural change that can be detected by this approach.
Even though the use of modal properties to detect changes in the structure were out of
the scope of this study, it was interesting to check the consistency in the parameters ob-
tained by a global identication techniques such as ERA, and the presented methodology.
The numerical and experimental frequencies, as well as the damping ratios, for the refer-
ence condition are shown in Table 3.6. For all other scenarios, only the relative changes in
natural frequencies were summarized in Table 3.7. It can be seen that frequency changes,
54
Table 3.4: Global system identication. Comparison of =
r
in the identied mass-
normalized stinesses for LANL test-bed structure. Boldfaced table entries correspond to
the actual structural changes and false positives, respectively.
=r

k
1

k
2

k
3
State FEM ERA ChainID FEM ERA ChainID FEM ERA ChainID
State#2 - -0.098 -0.012 - 0.029 0.005 - -0.005 -0.001
State#3 -0.160 -0.102 -0.181 - -0.110 -0.009 - 0.037 -0.005
State#4 -0.218 -0.196 -0.226 - -0.006 -0.034 - 0.003 -0.017
State#5 -0.437 -0.390 -0.425 - -0.026 -0.033 - 0.008 -0.028
State#6 - -0.009 -0.01 -0.218 -0.240 -0.243 - -0.036 -0.018
State#7 - -0.032 0.009 -0.437 -0.456 -0.437 - -0.059 -0.024
State#8 - -0.007 0.014 - -0.008 0.049 -0.218 -0.213 -0.217
State#9 - 0.001 0.025 - -0.021 0.015 -0.437 -0.425 -0.441
Table 3.5: Global system identication. Comparison of =
r
in the identied mass-
normalized stinesses for LANL test-bed structure. Boldfaced and underlined table entries
correspond to the actual structural changes and false positives, respectively.
=r

k
1

k
2

k
3
State ERA ChainID ERA ChainID ERA ChainID
State#2 -3.260 -0.285 2.013 0.100 -0.496 -0.023
State#3 -3.388 -4.480 -7.610 -0.179 3.447 -0.130
State#4 -6.514 -5.579 -0.414 -0.695 0.293 -0.491
State#5 -12.968 -10.506 -1.815 -0.678 0.821 -0.796
State#6 -0.300 -0.235 -8.575 -5.019 -3.415 -0.523
State#7 -1.066 0.229 -17.029 -9.019 -5.542 -0.689
State#8 -0.236 0.352 -0.552 1.002 -9.807 -6.257
State#9 0.037 0.629 -1.480 0.312 -19.569 -12.688
for some specic scenarios and modes, were better estimated by either the ERA or the
ChainID method, but generally speaking, both approaches exhibited similar performance.
Furthermore, these results show some of the limitations that modal-based detection tech-
niques, especially those using natural frequencies, can experience in detecting, locating
and estimating the level of damage in real structures.
55
Table 3.6: Comparison of numerical and experimental modal parameters
f [Hz] [%]
Mode FEM ERA ChainID ERA ChainID
1 29.90 30.81 31.38 4.17 4.37
2 55.46 55.38 59.19 1.49 1.90
3 72.60 71.11 75.39 0.64 0.94
Table 3.7: Relative changes in experimentally-identied natural frequencies
=r
f
1
f
2
f
3
State FEM ERA ChainID FEM ERA ChainID FEM ERA ChainID
State#2 -0.036 -0.013 -0.000 -0.019 -0.021 -0.003 -0.005 -0.006 -0.007
State#3 -0.006 0.003 -0.027 -0.021 -0.016 -0.050 -0.031 -0.035 -0.026
State#4 -0.034 0.004 -0.043 -0.063 -0.063 -0.067 -0.023 -0.022 -0.039
State#5 -0.094 -0.021 -0.111 -0.133 -0.143 -0.156 -0.040 -0.041 -0.080
State#6 -0.065 -0.032 -0.079 0.001 0.006 -0.007 -0.052 -0.069 -0.061
State#7 -0.157 -0.081 -0.164 -0.013 0.014 -0.006 -0.108 -0.133 -0.102
State#8 -0.035 -0.021 -0.022 -0.062 -0.064 -0.059 -0.023 -0.022 -0.007
State#9 -0.095 -0.057 -0.094 -0.132 -0.135 -0.134 -0.039 -0.039 -0.027
3.5 Detection of nonlinear changes in a MDOF test-bed
structure
Three dierent baseline conditions, with the bumper assembly set sequentially at each
level of the structure, were established for this experimental study. Although the bumper
assembly was included in the baseline congurations, the gap space was set in order that
the reference response time-histories were within the linear dynamic range (i.e., no impacts
between suspended column and bumper).
With the bumper assembly located at the third
oor, the eects of ve dierent levels
of the gap-nonlinearity were introduced into the vibration response of the structure by
varying the spacing between the suspended column and the bumper. Although no specic
56
damage conditions were physically simulated in these scenarios, the bumper assembly was
actually used to add nonlinear characteristic eects, that can be observed in structures
with alternating stiness states caused by damage (i.e., cracks, loose connections), to the
system dynamics [10].
Similarly, nonlinear eects were imposed at the rst and second
oor by moving the
bumper assembly to the corresponding structural level. Finally, in addition to the gap-
nonlinearities, linear changes in the structure's stiness and mass were considered in order
to simulate the structural changes that can be expected in real structures due to opera-
tional and environmental variabilities [17, 21, 67, 77]. An additional 1.2 kg mass added at
specic locations and a 87.5% reduction in the stiness of selected columns were used to
introduce the linear structural changes in the system. The fteen structural congurations
considered in this study are summarized in Table 3.8. All structural congurations were
dynamically tested in order to generate the corresponding data sets of force and accelera-
tion time-histories. To account for the variability in the data, ten experimental tests were
performed under dierent external force realizations on each structural conguration (i.e.,
10 data sets for each system conguration).
3.5.1 Identication of the LANL test-bed structure
The acceleration time-histories recorded by the four accelerometers attached to the
oor
plates were windowed, detrended, ltered and integrated to numerically obtain the relative
displacements and velocities required to apply the identication approach under discus-
sion. In order to generate an enlarged collection of acceleration, velocity and displacement
time-histories, all vibration records were divided into overlapped segments of 6.34 seconds,
57
Table 3.8: Summary of structural state conditions
State Condition Description
State#1 Reference Bumper assembly in the 3
rd

oor
State#10 Nonlinear Gap = 0.20 mm
State#11 Nonlinear Gap = 0.15 mm
State#12 Nonlinear Gap = 0.13 mm
State#13 Nonlinear Gap = 0.10 mm
State#14 Nonlinear Gap = 0.05 mm
State#15 Linear/Nonlinear Gap = 0.10 mm and 19.1% mass increment on the base
State#16 Linear/Nonlinear Gap = 0.20 mm and 19.1% mass increment on the 1
st

oor
State#17 Linear/Nonlinear Gap = 0.10 mm and 19.1% mass increment on the 1
st

oor
State#18 Reference Bumper assembly moved to the 2
nd

oor
State#19 Linear 21.8% third-story stiness reduction
State#20 Nonlinear Gap = 0.15 mm
State#21 Linear/Nonlinear Gap = 0.15 mm and 21.8% third-story stiness reduction
State#22 Baseline Bumper assembly moved to the 1
st

oor
State#23 Nonlinear Gap = 0.15 mm
State#24 Linear/Nonlinear Gap = 0.15 mm and 21.8% third-story stiness reduction
which included more than 100 fundamental periods of the structure. In total, 150 ensem-
bles of time-history records were generated for each of the structural congurations listed
in Table 3.8.
For the purposes of this study, and based on the experimental setup previously de-
scribed, the LANL test-bed structure was treated as a 3DOF chain-like system subjected
to base motions; hence the dierential equations of motion, given in Eqn. 3.1, in the
absence of any directly-applied exciting forces could be rewritten as:
G
(3)
(z
3
; _ z
3
;p) = x
3
G
(2)
(z
2
; _ z
2
;p) = m
2
x
2
+G
(3)
(z
3
; _ z
3
;p) (3.10)
G
(1)
(z
1
; _ z
1
;p) = m
1
x
1
+G
(2)
(z
2
; _ z
2
;p)
58
Table 3.9: Masses and mass ratios estimated for the reference structural congurations.
1
st

oor 2
nd

oor 3
rd

oor
State m
1
[kg] m
1
m
2
[kg] m
2
m
3
[kg] m
3
State#1 6.565 1.000 6.565 1.016 6.675 -
State#18 6.565 1.038 6.819 0.941 6.422 -
State#22 6.819 0.962 6.565 0.978 6.422 -
where m
i
=
m
i+1
m
i
is the mass ratio between two consecutive
oors. Although the mass
ratios can, for practical reasons, be considered equal to one in structural change detection
applications even where the system's mass is not approximately uniformly distributed
throughout the structure, they were actually estimated based on the geometric and ma-
terial properties of the structure in the reference congurations to account for the small
mass variations caused by changing the bumper assembly location. The estimated masses
and corresponding mass ratios are shown in Table 3.9.
The structure's equations of motion given in Eqn. 3.10, in conjunction with the ab-
solute acceleration records and mass ratios were then employed to initially determine
the experimental mass-normalized restoring forces. Typical phase plots of the restoring
forces G
(i)
(z
i
; _ z
i
;p) and relative displacements z
i
, obtained from the structural congu-
ration with a gap of 0.05 mm in the third
oor (corresponding to state#14), are shown
in Fig. 3.10(a) for each of the interconnecting elements in the chain-like system. First
(top) row corresponds to third-story element, second (middle) row to second-story, and
third (bottom) row to rst-story. It can be clearly seen in the phase plots for the rst-
and second-
oor elements, as is expected for the linear elements of a shear-type frame
structure, constant linear trends in the experimental mass-normalized restoring forces.
In contrast, the restoring force phase plot for the third-
oor element showed the typical
59
bilinear hardening characteristics of a nonlinear gap. The discontinuity in the stinesses,
which was caused by the opening and closing of the gap, was observed to happen around
0.05 mm of relative displacement between the third and second
oor (which is the actual
gap size introduced in the physical system).
Once the restoring forces G(z
i
; _ z
i
;p), the relative displacements z
i
, and relative veloc-
ities _ z
i
were available for all structural congurations, the time-domain non-parametric
identication technique (i.e., restoring force method) was applied to determine the cor-
responding coecients of the series expansion and build the associated reduced-order
model-free representation for each element in the 3DOF chain-like system. It should be
noted that, for all structural congurations considered in this study, the nonparametric
reduced-order models for the elements G
(i)
in the system were built using third-order
Chebyshev polynomials in both normalized state variables z
0
and _ z
0
.
The reconstructed restoring forces
^
G
(i)
(z
i
; _ z
i
) were then computed using the estimated
C
(i)
qr
coecients and the corresponding sequence of Chebyshev polynomials. Figure 3.10(b)
depicts the phase plots of the reconstructed restoring forces obtained after processing the
experimental vibration records from the same data set shown in Fig. 3.10(a). Note that the
dominant dynamic features of the underlying physical phenomena in each of the elements
of the chain-like system have been successfully captured by the associated reduced-order
models. In Fig. 3.10(c), the time histories of the experimental and the reconstructed
restoring forces are displayed in solid and dashed lines, respectively. For this case, the
normalized mean square errors of the deviation between the records were below 7%.
60
−0.1 −0.05 0 0.05 0.1 0.15
−10
−5
0
5
10
z1 [mm]
G
(1)
[m/s
2
]
−0.1 −0.05 0 0.05 0.1 0.15
−10
−5
0
5
10
z2 [mm]
G
(2)
[m/s
2
]
−0.1 −0.05 0 0.05 0.1 0.15
−10
−5
0
5
10
z3 [mm]
G
(3)
[m/s
2
]
−0.1 −0.05 0 0.05 0.1 0.15
−10
−5
0
5
10
z1 [mm]
ˆ G
(1)
[m/s
2
]
−0.1 −0.05 0 0.05 0.1 0.15
−10
−5
0
5
10
z2 [mm]
ˆ G
(2)
[m/s
2
]
−0.1 −0.05 0 0.05 0.1 0.15
−10
−5
0
5
10
z3 [mm]
ˆ G
(3)
[m/s
2
]
3 3.1 3.2 3.3 3.4 3.5
−10
−5
0
5
10
t [s]
G
(1)
[m/s
2
]
3 3.1 3.2 3.3 3.4 3.5
−10
−5
0
5
10
t [s]
G
(2)
[m/s
2
]
3 3.1 3.2 3.3 3.4 3.5
−10
−5
0
5
10
t [s]
G
(3)
[m/s
2
]
(a) Experimental phase plots (b) Reconstructed phase plots (c) Response time-histories
Figure 3.10: (a) Typical phase plots of the measured restoring forces G
(i)
and the relative
displacements z
i
from State #14, which corresponds to the scenario with a gap of 0.05
mm at 3
rd

oor. (b) Corresponding phase plots of the reconstructed restoring forces
^
G
(i)
obtained by using the identied restoring force coecients a
(i)
qr
. (c) Comparison between
the measured restoring force time-histories G
(i)
(solid lines), and the reconstructed time-
histories
^
G
(i)
(dashed lines). Notice that the curves are virtually identical. Also, due
to the nature of the one-sided snubber, the restoring force associated with third
oor is
not-symmetric.
3.5.2 Change-sensitive features
As a direct consequence of changes in the dynamic characteristics of the chain-like system,
the restoring forces of the interconnecting elements will exhibit variations with respect to
the associated reference condition. As illustration, consider the state#14 where a 0.05
mm gap in the third-
oor was introduced in the structure. The interpolated surface
of the relative change in the experimental mass-normalized restoring force G
(3)
(z
3
; _ z
3
)
with respect to the reference structural condition (i.e., state#1), is shown in Fig. 3.11(a).
The corresponding approximating restoring force surface, reconstructed with a third-order
expansion, is displayed in Fig. 3.11(b). It is important to highlight that in the data
61
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−3
−2
−1
0
1
2
3
z
′
3
˙ z
′
3
ΔG
(3)
(z
3
, ˙ z
3
)
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−3
−2
−1
0
1
2
3
z
′
3
˙ z
′
3
Δ
ˆ G
(3)
(z
3
, ˙ z
3
)
(a) Actual restoring force surface (b) Reconstructed restoring force surface
Figure 3.11: Relative change in the restoring force surface in element G
(3)
with respect to
the reference structural condition due to the presence of a nonlinearity (gap = 0.05 mm.)
in the third-story of the LANL test-bed structure.
analysis procedure under discussion no assumptions have been made about the nature of
the structural changes; however, the intrinsic features of the changes, which are embedded
in the experimental measurements, can be inferred from the restoring force surfaces. In
Fig. 3.11, for example, notice the non-symmetric and nonlinear characteristics of the third-

oor restoring force changes generated by the asymmetric gap nonlinearity. Also note the
nonlinear dependence of the restoring force on the inter-story displacement and velocity.
It can be argued that the complex nonlinear dynamics involved in this case, are aecting
not only the stiness but also the damping of the third-story element in the chain-like
system.
Since the implemented non-parametric data-driven identication technique will adjust
the coecients in the double-indexed restoring force expansion in order to characterize
the new governing system dynamics, variations in the identied coecients will indicate
the existence of structural changes in the system. To illustrate this point, the resulting
third-story Chebyshev coecients C
(3)
qr
and corresponding power-series coecients a
(3)
qr
62
Table 3.10: Identied third-
oor Chebyshev coecients C
(3)
qr
for state#1 and state#14.
T
0
( _ z
0
) T
1
( _ z
0
) T
2
( _ z
0
) T
3
( _ z
0
)
T
0
(z
0
) 0.071 0.264 -0.031 0.009
T
1
(z
0
) 13.354 -0.029 -0.006 -0.008
T
2
(z
0
) -0.030 0.051 0.006 -0.021
T
3
(z
0
) 0.111 0.062 0.002 0.023
(a) Chebyshev coecients for state#1 (reference)
T
0
( _ z
0
) T
1
( _ z
0
) T
2
( _ z
0
) T
3
( _ z
0
)
T
0
(z
0
) 0.019 0.736 -0.132 -0.004
T
1
(z
0
) 11.698 0.837 0.367 -0.009
T
2
(z
0
) 0.506 0.151 -0.181 -0.109
T
3
(z
0
) 0.263 -0.056 -0.173 0.026
(b) Chebyshev coecients for state#14
from state#1 and state#14 are tabulated in Table 3.10 and Table 3.11, respectively. It
is clear that the restoring force coecients adapted to t the reduced-order model of the
third-story element to the new dynamic response. Furthermore, the presence of signicant
high-order and cross-product terms in the restoring force expansions can be used as an
indicator of nonlinearities in the system. An analysis of the contribution of the coecients
in Table 3.10(a) showed that coecients other than C
(3)
10
were initially negligible in the
reference structural conguration; on the other hand, the coecients for state#14 in Table
3.10(b) unveiled, in addition toC
(3)
10
andC
(3)
01
, the signicance to the reduced-order model
of terms corresponding toC
(3)
30
,C
(3)
20
,C
(3)
11
, andC
(3)
12
. Similar conclusions can be drawn by
estimating the contribution of each a
(3)
qr
x
q
_ x
r
term in the power-series expansions shown
in Table 3.11 to the overall restoring forces. It is clear, in this example, that signicant
nonlinear terms in the reduced-order models indicates the presence of the gap-nonlinearity
in the third-
oor section of the experimental test-bed structure. With regard to the
signicance of the terms in the power-series expansion, it was observed that the statistical
dispersion of eacha
(i)
qr
coecient is related to the level of importance (contribution) in the
characterization of the restoring forces.
63
Table 3.11: Identied third-
oor power-series coecients a
(3)
qr
for state#1 and state#14.
_ x
0
_ x
1
_ x
2
_ x
3
x
0
2.7010
2
1.583 -17.620 4.0610
2
x
1
6.5910
4
3.6310
3
-1.0110
5
-8.7010
6
x
2
-2.6610
6
8.9910
7
-5.2210
8
-1.5910
10
x
3
5.1110
10
-2.4710
11
7.5210
12
3.0510
14
(a) Power-series coecients for state#1 (reference)
_ x
0
_ x
1
_ x
2
_ x
3
x
0
-0.189 5.923 1.4810
2
3.2110
3
x
1
7.7810
4
1.9010
5
7.2010
6
-2.3810
7
x
2
1.1610
8
1.2010
9
-3.7210
10
-4.1710
11
x
3
1.0710
12
-3.8010
12
-4.0910
14
1.0810
15
(b) Power-series coecients for state#14
64
However, the use of Chebyshev coecients for detecting changes within a stochastic
framework is disadvantageous since they have embedded within them the normalization
of the associated state variables [25, 31]. In addition, due to the normalization procedure,
relative changes in the Chebyshev coecients can be erroneously interpreted as changes
in the structure [31]. Consequently, the restoring force coecients a
(i)
qr
were considered
as the suitable features to be used in applications for detecting linear/nonlinear changes
in the dynamics of structural systems. For the sake of convenience, only the equivalent
mass-normalized linear stiness-like (a
(i)
10
) and mass-normalized cubic stiness-like (a
(i)
30
)
terms were used as the change-sensitive features in this study.
3.5.3 Statistics of the identication results
The basic second-order statistics, the mean () and coecient of variation (), of the
identied restoring force coecientsa
(i)
10
anda
(i)
30
, obtained from all data ensembles in each
structural conguration of the LANL test structure, are summarized in Table 3.12 and
Fig. 3.12. It is observed that, in general, the mass-normalized stiness-like coecients have
a much lower variability than the cubic stiness-like terms. The coecient of variation
for a
(i)
10
ranges from 2% to 6%; while the coecient of variation for a
(i)
30
exhibited values
between 30% to 600%. However, as was mentioned previously, when the nonlinear eects
of the gap started to become signicant in the dynamics of the structure, noticeable
reductions in the dispersion of the a
(i)
30
coecients occurred. For example, the coecient
of variation for the third-
oor cubic stiness-like term decreased from an initial value of
1.597 in state#1 (reference condition) to 0.521 for the structural conguration with a
0.05 mm gap (state#14). Similar behavior was noted in the a
(1)
30
and a
(2)
30
in situations
65
#1 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24
0
0.2
0.4
0.6
0.8
1
State
μ
a
(i)
10
[× 10
5
]

1
st
story 2
nd
story 3
rd
story
#1 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24
0
0.5
1
State
μ
a
(i)
30
[× 10
12
]

1
st
story 2
nd
story 3
rd
story
#1 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24
0
0.02
0.04
0.06
0.08
State
δ
a
(i)
10

1
st
story 2
nd
story 3
rd
story
#1 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24
0
2
4
6
8
State
δ
a
(i)
30

1
st
story 2
nd
story 3
rd
story
(a) Means of a
(i)
10
and a
(i)
30
(b) Coecients of variation of a
(i)
10
and a
(i)
30
Figure 3.12: Mean () and coecient of variation () of the identied restoring force
coecients a
(i)
10
and a
(i)
30
for all the state congurations of LANL test-bed structure. Top
row for a
(i)
10
; lower row for a
(i)
30
.
with the bumper assembly located at the rst and second
oors, respectively. It can
be also noticed how the mean value of the mass-normalized cubic stiness-like coecients
increased as the gap size in the bumper assembly was reduced (i.e., level of the nonlinearity
was increased). Moreover, relatively slight variations in the mass-normalized stiness-
like coecients occurred as the severity of the nonlinearity increased. In the scenarios
where the stiness and mass of the structure were modied, notable reductions in the a
(i)
10
coecients were observed.
In order to have a probabilistic representation of the identied restoring force coef-
cients, the underlying probability density functions were estimated using the observed
data and kernel density estimators [72]. Figure 3.13 displays the histograms, the esti-
mated probability density functions (solid lines), and the corresponding Gaussian distri-
butions (dashed lines) of the mass-normalized stiness-likea
(3)
10
and mass-normalized cubic
stiness-like a
(3)
30
coecients for the third-
oor element in the state#14.
66
Table 3.12: Summary of mean () and coecient of variation () of the identied restoring force coecients for the LANL test-bed
structure.
1
st

oor 2
nd

oor 3
rd

oor
a
(1)
10
a
(1)
30
a
(2)
10
a
(2)
30
a
(3)
10
a
(3)
30
State (10
4
) (10
11
) (10
4
) (10
11
) (10
4
) (10
11
)
State#1 6.960 0.036 0.810 1.882 6.365 0.048 0.818 2.503 6.594 0.024 0.511 1.597
State#10 6.971 0.033 0.824 1.899 6.293 0.051 1.282 1.792 6.419 0.028 1.539 0.607
State#11 7.054 0.036 0.625 2.514 6.323 0.033 0.874 2.201 6.442 0.044 3.961 0.389
State#12 7.098 0.035 0.421 4.045 6.284 0.038 0.911 2.049 6.428 0.037 5.048 0.304
State#13 7.052 0.034 0.775 2.185 6.232 0.046 0.999 2.240 6.766 0.054 4.756 0.462
State#14 7.118 0.039 1.591 1.267 5.874 0.038 1.373 1.561 7.789 0.060 10.701 0.521
State#15 6.977 0.036 0.308 5.031 6.121 0.044 1.032 2.387 6.460 0.049 5.906 0.332
State#16 5.700 0.022 0.412 1.688 6.197 0.033 0.907 2.048 6.402 0.030 1.225 0.931
State#17 5.710 0.029 0.762 1.334 6.042 0.036 1.030 1.893 6.521 0.048 4.953 0.401
State#18 6.957 0.041 0.828 2.376 6.022 0.030 1.410 1.215 7.027 0.021 0.486 1.876
State#19 7.191 0.055 0.692 3.792 6.203 0.027 0.588 2.647 5.421 0.020 0.396 1.288
State#20 6.870 0.053 1.140 2.331 6.112 0.048 7.691 0.511 7.020 0.026 0.709 1.420
State#21 7.262 0.065 0.360 6.540 6.048 0.044 2.763 0.781 5.416 0.021 0.442 1.093
State#22 6.532 0.031 1.090 1.291 6.494 0.033 0.649 2.673 6.983 0.028 0.973 1.067
State#23 6.144 0.062 7.083 0.486 6.354 0.032 1.302 1.494 7.016 0.028 0.538 2.039
State#24 6.191 0.051 7.167 0.355 6.706 0.028 0.566 2.555 5.461 0.024 0.392 1.403
67
6 6.5 7 7.5 8 8.5 9 9.5
x 10
4
0
0.2
0.4
0.6
0.8
1
x 10
−4
a
(3)
10
pdf
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
x 10
12
0
0.2
0.4
0.6
0.8
1
x 10
−12
a
(3)
30
pdf
(a) Statistics for a
(3)
10
(b) Statistics for a
(3)
30
Figure 3.13: Stochastic representation of the identied restoring force coecients a
(3)
10
(for
linear stiness) and a
(3)
30
(for cubic stiness) for the state#14. The corresponding his-
tograms, estimated probability density functions (solid lines) and Gaussian distributions
(dashed lines) are superimposed.
To facilitate the graphical comparison and analysis of the stochastic characteristics
of the identied restoring force coecients, the probability density functions were esti-
mated for reference-normalized coecients ~ a
(i)
mn
rather than the actual identied coe-
cients. These new normalized coecients are mathematically dened as ~ a
(i)
mn
= (a
(i)
mn

r
a
(i)
mn
)=
r
a
(i)
mn
, where
r
a
(i)
mn
indicates the mean value of the coecients a
(i)
mn
from the corre-
sponding baseline conditions. The estimated density functions of ~ a
(i)
10
and ~ a
(i)
30
for illustra-
tive scenarios state#10 and state#13 are displayed in Fig. 3.14. In this gure, the refer-
ence pdf s are plotted in solid lines, whereas the pdf s from state conditions with structural
changes are shown in dot-dashed and dashed lines. First (top) row of graphs correspond
to coecients from third-story elements, middle row to second-story, and bottom row to
rst-story.
3.5.4 Detection and localization of nonlinearities
Two dimensionless indices =
r
and =
r
, based on the second-order statistics of
the coecients (Table 3.12 and Fig. 3.18), were employed to assess the eectiveness and
robustness of the approach under discussion for detecting and localizing nonlinearities in
68
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(1)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(2)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(3)
10
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(1)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(2)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(3)
30
pdf
(a) Mass-normalized linear stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.14: Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) coecients. The
solid lines correspond to the pdf of the coecients from the baseline condition in state#1.
The dot-dashed and dashed lines show the coecients' pdf obtained in state#10 and
state#13, respectively, where a nonlinear snubber was placed at the 3
rd

oor. Note that,
for easy viewing, the distributions were normalized.
the LANL test-bed structure. These indices were found by dividing the dierence in the
means of the identied restoring force coecients ( =
r
) by the corresponding
means (
r
) and standard deviations (
r
) from the corresponding reference cases. Since
variations in the restoring force coecients are related to changes in the dynamics of the
system, It was assumed that the index =
r
can be used to assess the magnitude or
level of \change" in the system. The statistical signicance of the relative change in the
means of the coecients can be gauged by the ratio =
r
. Relatively small values of
the latter index with respect to a predetermined threshold will indicate that the dierence
in the coecients' mean could be attributable just to normal variability within the a
(i)
qr
coecients. By contrast, large values of =
r
would therefore suggest the presence
69
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(1)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(2)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(3)
10
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(1)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(2)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(3)
30
pdf
(a) Mass-normalized linear stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.15: Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) coecients. The
solid lines correspond to the pdf of the coecients from the baseline condition in state#1.
The dot-dashed and dashed lines show the coecients' pdf obtained in state#16 and
state#17, respectively, where a nonlinear snubber was placed at the 3
rd

oor, and an
added mass was placed at the 1
st

oor. Note that, for easy viewing, the distributions were
normalized.
of \genuine" changes in the system [31, 58]. In this study, a threshold for detection of
j=
r
j 2:0 was established to determine, with an approximate 95% of condence,
when signicant changes had occurred in the a
(i)
10
and a
(i)
30
coecients. Notice that the
detectability depends on the dened threshold for =
r
in the decision rule. The values
of =
r
and =
r
for all thirteen scenarios with structural changes are summarized
in Table 3.13. Table entries in boldface correspond to signicant changes in the restoring
force coecients that were detected using the establishedj=
r
j 2:0 decision rule.
For the scenarios from state#10 through state#14, in which the severity of the non-
linearity was systematically increased by reducing the gap size in the bumper assembly
70
Table 3.13: Summary of =
r
and =
r
indices in the identied restoring force coecients for the LANL test-bed structure.
Boldfaced table entries correspond to the detected changes based on established decision rule.
1
st

oor 2
nd

oor 3
rd

oor
a
(1)
10
a
(1)
30
a
(2)
10
a
(2)
30
a
(3)
10
a
(3)
30
State =r =r =r =r =r =r =r =r =r =r =r =r
State#10 0.001 0.044 0.016 0.008 -0.011 -0.232 0.566 0.226 -0.026 -1.070 2.013 1.260
State#11 0.013 0.374 -0.228 -0.121 -0.006 -0.135 0.067 0.027 -0.023 -0.929 6.752 4.227
State#12 0.019 0.549 -0.480 -0.255 -0.012 -0.258 0.112 0.044 -0.025 -1.017 8.879 5.558
State#13 0.013 0.365 -0.043 -0.023 -0.020 -0.427 0.220 0.088 0.026 1.056 8.307 5.200
State#14 0.022 0.629 0.962 0.511 -0.077 -1.577 0.676 0.270 0.181 7.325 19.941 12.483
State#15 0.002 0.067 -0.619 -0.329 -0.038 -0.783 0.261 0.104 -0.020 -0.819 10.558 6.609
State#16 -0.181 -5.011 -0.491 -0.260 -0.026 -0.539 0.107 0.043 -0.029 -1.177 1.398 0.875
State#17 -0.179 -4.973 -0.060 -0.031 -0.050 -1.037 0.257 0.103 -0.011 -0.447 8.694 5.442
State#19 0.033 0.818 -0.163 -0.068 0.030 0.983 -0.582 -0.479 -0.228 -10.470 -0.185 -0.098
State#20 -0.012 -0.304 0.376 0.158 0.015 0.492 4.452 3.663 -0.001 -0.047 0.458 0.244
State#21 0.043 1.068 -0.565 -0.237 0.004 0.141 0.959 0.789 -0.229 -10.505 -0.091 -0.048
State#23 -0.059 -1.912 5.497 4.255 -0.021 -0.643 1.005 0.376 0.004 0.166 -0.447 -0.418
State#24 -0.052 -1.681 5.574 4.315 0.032 0.974 -0.128 -0.047 -0.218 -7.662 -0.597 -0.559
71
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(1)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(2)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(3)
10
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(1)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(2)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(3)
30
pdf
(a) Mass-normalized linear stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.16: Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) coecients. The
solid lines correspond to the pdf of the coecients from the baseline condition in state#18.
The dot-dashed and dashed lines show the coecients' pdf obtained in state#20 and
state#21, respectively, where a nonlinear snubber was placed at the 2
nd

oor, and a
stiness reduction was introduced on the 3
rd

oor. Note that, for easy viewing, the
distributions were normalized.
from 0.20 mm to 0.05 mm, the tabulated results showed statistically signicant changes in
the third-
oor mass-normalized cubic stiness-like coecients for the structural congura-
tions with \moderate" nonlinearities (gaps between 0.15 mm and 0.10 mm) and \severe"
nonlinearities (0.05 mm gap). For the case of the low-level nonlinearity (0.20 mm gap),
the value of =
r
for the a
(3)
30
coecient was below the established detection threshold;
hence, the observed changes could be interpreted as normal variations due to the inherent
randomness of the coecients rather than being caused by the presence of the nonlinear-
ity, It was also noted that, the indices =
r
and =
r
for thea
(3)
30
coecient increased
as the eects of the nonlinearity became more pronounced. On the other hand, the a
(3)
10
72
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(1)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(2)
10
pdf
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
˜a
(3)
10
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(1)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(2)
30
pdf
−15 −10 −5 0 5 10 15 20 25 30
0
0.5
1
˜a
(3)
30
pdf
(a) Mass-normalized linear stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.17: Comparison of probability density functions of identied mass-normalized
linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-like (~ a
(i)
30
) coecients. The
solid lines correspond to the pdf of the coecients from the baseline condition in state#22.
The dot-dashed and dashed lines show the coecients' pdf obtained in state#23 and
state#24, respectively. Note that, for easy viewing, the distributions were normalized.
#10 #11 #12 #13 #14 #15 #16 #17 #19 #20 #21 #23 #24
0
0.1
0.2
0.3
0.4
State
| Δμ/μ
r
| for a
(i)
10

1
st
story 2
nd
story 3
rd
story
#10 #11 #12 #13 #14 #15 #16 #17 #19 #20 #21 #23 #24
0
5
10
15
20
State
| Δμ/μ
r
| for a
(i)
30

1
st
story 2
nd
story 3
rd
story
#10 #11 #12 #13 #14 #15 #16 #17 #19 #20 #21 #23 #24
0
5
10
15
State
| Δμ/σ
r
| for a
(i)
10

1
st
story 2
nd
story 3
rd
story
#10 #11 #12 #13 #14 #15 #16 #17 #19 #20 #21 #23 #24
0
5
10
15
State
| Δμ/σ
r
| for a
(i)
30

1
st
story 2
nd
story 3
rd
story
(a) Normalized index =r (b) Normalized index =r
Figure 3.18: Graphical representation of the indicesj=
r
j andj=
r
j computed for
the identied a
(i)
10
and a
(i)
30
coecients. The dashed horizontal line in the RHS column of
plots correspond the detection threshold used in the decision rule (j=
r
j 2:0).
73
coecients, which characterize the linear behavior in the third-story linking element dy-
namics, were practically unaected when gaps larger than 0.10 mm were set between the
bumper and the suspended column. However, a 18.1% increment in the coecient's mean
was observed in the case of the severe nonlinearity. It is not surprising that the linear
dynamic features of the third
oor were accentuated because of the hardening-stiness
eects exerted by the 0.05 mm gap. An inspection of the indices estimated for the rst-
and second-
oor coecients indicated that the corresponding changes ina
(i)
10
anda
(i)
30
were
negligible for the scenarios under discussion.
For the rest of the state conditions (state#15 - state#17) within the rst group of
scenarios, in addition to nonlinearities in the third-
oor, changes in the mass of the struc-
ture were also introduced. In state#15, a signicant variation in the a
(3)
30
coecient was
observed; thus, only the presence of the nonlinearity could be experimentally determined.
The increment in the mass at the system's base level could not be detected since the pro-
posed methodology relies on the relative motion between interconnected lumped-masses.
However, notice that the additional mass when placed at the rst-
oor plate (state#16
and state#17) was easily detected by the mass-normalized stiness-like coecient. The
percentages of relative changes seen in the a
(1)
10
term correspond respectively to 22.1%
and 21.8% increments in the rst-story mass, which are fairly close to the actual 19.1%.
A more detailed discussion regarding the detection and quantication of changes in the
mass and stiness of the LANL test-bed structure can be found on Hernandez-Garcia et
al [31]. In regard to the nonlinearities in these structural congurations, it was observed
that the \moderate" nonlinearity, corresponding to the 0.10 mm gap (state#17), pro-
duced distinguishable dierences in the high-order coecient a
(3)
30
. On the contrary and
74
similarly to what was observed in state#10, the low-level nonlinearity (0.20 mm gap) was
undetectable.
The key ndings after analyzing the normalized indices =
r
and =
r
for the
second group of scenarios (state#19 - state#21) were analogous to those previously de-
scribed for state#10 through state#17. There were notorious changes with respect to the
corresponding reference case (state#18) in the third-story mass-normalized stiness-like
coecient for state#19 and state#21. The location and magnitude of the relative changes
in the a
(3)
10
coecients, did quite match the actual structural modications (i,e., 21.88%
reduction in the third-story stiness) made on the system. Even though the nonlinearities
in state#20 and state#21 were nominally identical (Table 3.8) and, signicant changes in
thea
(2)
30
coecients were expected in both scenarios, the presence of the 0.15 mm gap was
only detected for state#20.
For the last group of state conditions, and following the same line of analysis, note that
the nonlinearity in the rst-story linking element, as well as reduction in the third-story
stiness, were accurately detected and located. It is worth pointing out that the values
of =
r
and =
r
for the a
(1)
30
were practically the same for state#23 and state#24,
where a nonlinear snubber with a 0.15 mm gap was placed at the rst-
oor of the LANL
test structure. Similar observations were made in state#13 and state#17 (0.10 mm gap in
third story) for thea
(3)
30
coecients. It was expected that similar change patterns would be
observed in the a
(i)
30
coecients of the linking element where the snubber has been placed
at, as long as the location and severity of the nonlinearity were unaltered. Also notice that
this assumption is also valid for linear structural modications. For example, the added
mass at the rst-story (state#16 and state#17), and the third-story stiness reduction
75
(state#19 and state#21) caused practically identical changes in the a
(1)
10
and a
(3)
10
coe-
cients, respectively. Another important observation is that, due to the topology of the
LANL test structure (i.e., chain-like system) and the proposed substructuring method-
ology, any change in the dynamics of a linking element, will aect only the identied
restoring force coecients associated with the reduced-order model of that element.
So far, it has been shown that the basic statistics (i.e., mean and standard deviation)
of the identied coecients can be used in the detection of nonlinearities in the LANL
test-bed structure. Now, since probability density functions provide a complete charac-
terization of a random variable, the pdf s estimated for the a
(i)
10
and a
(i)
30
coecients were
used to probabilistically assess the detection capabilities of the proposed methodology.
Initial inspections of the estimated probability density functions for the reference-
normalized ~ a
(i)
10
and ~ a
(i)
30
coecients will lead to conclusions similar to the already afore-
mentioned ndings. For instance, is clear from Fig. 3.14 that the nonlinear snubber at
the third story of the LANL test structure caused evident changes in the stochastic char-
acteristics of the third-
oor restoring force coecients, even in the case of the 0.20 mm
gap. However, the changes in pdf of a
(3)
10
coecients were more related to variations in
their dispersion rather than actual shifts in the distributions. On the other hand, as the
level of the nonlinearity increased, the dierences in the pdf of the a
(3)
30
coecient were
more signicant. Also noticed that the pdf s of the rst- and second-story coecients were
virtually identical. By comparing the distributions plotted in Figs. 3.14-3.17, it could be
observed that, as mentioned before, the probability functions for the a
(i)
10
and a
(i)
30
coe-
cients estimated for linking elements that have undergone theoretically the same structural
changes shown to be basically similar.
76
In order to quantitatively assess the dierences in the probability density functions of
thea
(i)
10
anda
(i)
30
, the overlapping areas, between each estimated pdf and its corresponding
reference, were calculated. Figure 3.19 depicts the overlapping areas of the pdf curves esti-
mated for the coecients from state#1 (reference) and state#14. The area of intersection
gives a graphically probabilistic estimate of how similar two curves are [66]. Once the
overlapping area has been computed, and in accordance with Papadopolous and Garcia
[66], the probability change quotient (PCQ), which gives an indication of how dierent the
density functions are, were obtained for a
(i)
10
anda
(i)
30
coecients. Table 3.14 and Fig. 3.20
display the PCQ values computed for all test cases in this study.
Results indicated that the third-story cubic stiness-like coecients, in the cases with
0.15 mm to 0.05 mm gap snubbers, had denitely changed with respect to the baseline
state since PCQ values ranged from 0.82 to 0.95. On the other hand, the PCQs estimated
for a
(3)
30
in state#10 and state#16 (PCQ 0:4 and PCQ 0:32, respectively) indicated
that some changes occurred, but not at the level as the other state conditions in the
rst group of scenarios. It is also clear that distributions of a
(3)
10
in state#14, and a
(1)
10
in
state#16 and state#17, completely dierent from the reference pdf s. Additionally, note
that the probabilities of change in a
(3)
10
coecients were quite similar for cases with \low"
and \moderate" nonlinearities (PCQ 35%), while signicant changes were observed
for the \severe" nonlinearity. In overall, the values of the probability change quotient
correlated quite well with results from the analysis of the normalized indices =
r
and
=
r
. However, it should be mentioned, that deciding whether nonlinearities are present
or not in the system should be based on the normalized indices (i.e., =
r
and =
r
)
77
(a) Mass-normalized stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.19: Comparison of overlapping areas between the probability density functions of
identied mass-normalized linear stiness-like (~ a
(i)
10
) and mass-normalized cubic stiness-
like (~ a
(i)
30
) coecients for state#14 and the corresponding reference pdf s (state#1). The
pdf of the coecients from state#1 and state#14 are plotted in solid and dashed lines,
respectively. The area of intersection is used to obtain the probability change quotient
(PCQ).
Table 3.14: Summary of probability change quotient (PCQ) estimated for the restoring
force coecients identied for LANL test-bed structure.
PCQ
1
st

oor 2
nd

oor 3
rd

oor
State a
(1)
10
a
(1)
30
a
(2)
10
a
(2)
30
a
(3)
10
a
(3)
30
State#10 0.063 0.066 0.088 0.096 0.342 0.398
State#11 0.119 0.061 0.204 0.057 0.353 0.829
State#12 0.145 0.094 0.164 0.080 0.405 0.833
State#13 0.181 0.106 0.184 0.099 0.337 0.939
State#14 0.239 0.142 0.628 0.123 0.936 0.912
State#15 0.070 0.131 0.289 0.166 0.348 0.946
State#16 0.999 0.438 0.313 0.071 0.370 0.323
State#17 0.999 0.224 0.438 0.082 0.347 0.868
State#19 0.265 0.111 0.354 0.203 1.000 0.250
State#20 0.097 0.156 0.269 0.732 0.152 0.139
State#21 0.297 0.122 0.099 0.328 1.000 0.275
State#23 0.499 0.801 0.239 0.139 0.080 0.184
State#24 0.486 0.863 0.378 0.114 1.000 0.271
and the probability change quotient, since in some cases the solely used of each indicator
can be misleading.
3.5.5 Quantication of nonlinearities
In order to appreciate the eects that dierent gap-sizes in the snubber have on the nonlin-
ear dynamics of the LANL test structure, the nonlinear component of the restoring forces
78
#10 #11 #12 #13 #14 #15 #16 #17 #19 #20 #21 #23 #24
0
0.2
0.4
0.6
0.8
1
State
PCQ for a
(i)
10

1
st
story 2
nd
story 3
rd
story
#10 #11 #12 #13 #14 #15 #16 #17 #19 #20 #21 #23 #24
0
0.2
0.4
0.6
0.8
1
State
PCQ for a
(i)
30

1
st
story 2
nd
story 3
rd
story
(a) Mass-normalized stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.20: Summary of probability change quotient (PCQ) estimated for the a
(i)
10
and
a
(i)
30
coecients for each of the state conditions of LANL test-bed structure.
were compared against the contribution of linear forces (i.e., associated with damping and
stiness). The linear (f
(i)
L
) forces acting on each of the linking elements of the system were
obtained by reconstructing the restoring forces using only the equivalent linear damping-
like and stiness-like terms in the corresponding reduced-order model. Similarly, to nd
the nonlinear forces (f
(i)
NL
) all the high-order and cross-product terms in the model were
used in the reconstruction of the restoring forces. The ratio between the root-mean-square
of the nonlinear (kf
(i)
NL
k) and linear (kf
(i)
L
k) forces in each of the interconnecting elements
was computed for each of the data ensembles that were available in this study. The mean
and standard deviation of the ratioskf
(i)
NL
k=kf
(i)
L
k for the dierent gap sizes used in the
bumper assembly (i.e., snubber) to simulate dierent level of nonlinearities are displayed
in Fig. 3.21(a). Notice how the contribution of the nonlinear forces became more impor-
tant in the third-
oor restoring forces as the gap is reduced (i.e., level of nonlinearity is
increased). One of the assumptions in this study was that the severity of the nonlinearity
could be estimated by analyzing the magnitude of the changes in the dominant nonlinear
restoring force coecients. For comparison, the contribution of the nonlinear forces gen-
erated by the cubic stiness-like coecient a
(i)
30
alone are shown in Fig. 3.21(b). It is clear
79
that almost the 50% of the total nonlinear forces observed on the third story of the struc-
ture are associated to cubic stiness forces. Therefore, it is expected that the magnitude
of changes in the a
(i)
30
coecient, with respect to the reference coecients obtained when
the system was \completely" linear, would provide an indication of the severity of the
nonlinearity. In addition, as the level of the nonlinearity increased, the hardening-stiness
eects that the snubber introduced in the linear dynamics of the third story will be also
increased; hence, changes in the linear stiness-like coecient a
(i)
10
are expected too.
The evolution of the mean and standard deviation of the reference-normalized ~ a
(i)
10
and
~ a
(i)
30
coecients are displayed in Fig. 3.22. It is evident from Fig.3.22(b) that the magnitude
of relative changes in the a
(i)
30
coecient are denitely correlated, in a nonlinear manner,
with the level of the nonlinearity, which in this case is controlled by the size of the gap in
the snubber. Even though thea
(i)
10
coecients also showed a dependency on the severity of
the nonlinearity, it would not be a good indicator of the level of the nonlinearity since this
term would also capture changes in the mass and stiness of the structure. Furthermore, in
addition to the nonlinear detection capabilities of the restoring force coecients especially
the dominant nonlinear terms, it is possible to provide a quantication of the severity
of the nonlinearity by tracking the magnitude of the observed changes in the identied
nonlinear restoring force coecients.
3.6 Summary and conclusions
Experimental data from a test-bed structure, that has been tested at the Los Alamos
National Laboratory (LANL), were used to demonstrate the eectiveness, and evaluate
the robustness and reliability of a data-driven non-parametric identication technique for
80
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
Gap size [mm]
kf
NL1
k/kf
L1
k
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
Gap size [mm]
kf
NL2
k/kf
L2
k
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
Gap size [mm]
kf
NL3
k/kf
L3
k
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
Gap size [mm]
kf
NL1
k/kf
L1
k
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
Gap size [mm]
kf
NL2
k/kf
L2
k
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
Gap size [mm]
kf
NL3
k/kf
L3
k
(a) f
(i)
NL
estimated using all nonlinear a
(i)
qr coecients (b) f
(i)
NL
estimated using only a
(i)
30
coecients
Figure 3.21: Evolution of the ratio between the rms of the nonlinear (f
(i)
NL
) and linear (f
(i)
L
)
components of the rst, second, and third-
oors restoring forces, with the corresponding
1 error bars, for dierent third-
oor gap sizes. The dashed lines depict the corresponding
tted functions for the mean of thekf
(i)
NL
k=kf
(i)
L
k ratios.
0 0.05 0.1 0.15 0.2 0.25
−0.1
0
0.1
0.2
0.3
Gap size [mm]
˜a
(1)
10
0 0.05 0.1 0.15 0.2 0.25
−0.1
0
0.1
0.2
0.3
Gap size [mm]
˜a
(2)
10
0 0.05 0.1 0.15 0.2 0.25
−0.1
0
0.1
0.2
0.3
Gap size [mm]
˜a
(3)
10
0 0.05 0.1 0.15 0.2 0.25
0
10
20
30
Gap size [mm]
˜a
(1)
30
0 0.05 0.1 0.15 0.2 0.25
0
10
20
30
Gap size [mm]
˜a
(2)
30
0 0.05 0.1 0.15 0.2 0.25
0
10
20
30
Gap size [mm]
˜a
(3)
30
(a) Mass-normalized linear stiness-like coecients (b) Mass-normalized cubic stiness-like coecients
Figure 3.22: Evolution of the means of identied mass-normalized restoring force coef-
cients, with the corresponding 1 error bars, in the rst, second, and third
oors for
dierent third-
oor gap sizes. The dashed lines depict the corresponding tted functions
for the means of the a
(i)
10
and a
(i)
30
coecients.
81
detecting, locating, and quantifying nonlinearities in uncertain MDOF chain-like systems.
Although the applicability of the proposed methodology is restricted to a specic class
of structural systems, the chain-like topology encompasses many typical structures in the
aerospace, civil and mechanical engineering elds (e.g., tall buildings, transmission tow-
ers, oshore platforms, aircraft wings and windmill blades). The proposed identication
approach is used to develop data-based reduced-order models for each of the intercon-
necting elements, which can incorporate complex nonlinearities, in the MDOF chain-like
system. The estimated reduced-order models consist of a set of coecients (i.e., restoring
force coecients) of doubly-indexed series expansions of the restoring forces in the linking
elements. The identication of nonlinearities in the system was then accomplished by
analyzing and monitoring, in a stochastic framework, the coecients of the reduced-order
models.
The results of this experimental study clearly demonstrated that the dominant identi-
ed nonlinear restoring force coecients (i.e., high-order and cross-product terms) could
condently detect and locate the introduced gap-nonlinearities in the LANL test-bed struc-
ture despite the modeling, measurement and data processing uncertainties. It was also
noticed that the severity of the nonlinearity, which was controlled by varying the gap size
in the snubber, and the magnitude of the observed changes in the nonlinear coecients
were correlated; hence, the restoring force coecients could also be used to quantify the
nonlinearities. Additionally, it was shown that the linear restoring force coecients were
able to capture the variations in the operational and environmental conditions, which in
study were simulated as changes in the mass and stiness of the structure. Therefore, it
82
would be possible to distinguish between the structural changes caused by nonlinearities,
and the changes associated with environmental and operational conditions.
As long as the eects of operational and environmental conditions, intrinsic nonlin-
earities, and underlying damage mechanisms in a target structure are re
ected in the
monitored structural dynamic response, then the method under discussion may provide
a useful tool to accurately detect, locate (within each \decomposed" structural regions),
and quantify the \changes" in the structure as re
ected in the identied restoring force
surface.
83
Chapter 4
Experimental study of data-driven reduced-order modeling
techniques for detection of changes in structural systems
4.1 Introduction
During last years, extensive research on structural health monitoring have led to the devel-
opment of a signicant number of input-output system identication techniques. However,
most of these techniques are implemented in a deterministic manner, while neglecting the
unavoidable eects the variability in the environmental or operational conditions, as well
as the uncertainties in the modeling, measurement, and data analysis processes have on
the identied change-sensitive features (i.e., stiness-like parameters, modal parameters)
of real structures. The induced uncertainty in the identied parameters will aect the
robustness of structural health monitoring (SHM) methodologies for detecting and assess-
ing changes in the monitored structures. Robust and reliable SHM strategies that rely on
experimentally identied modal characteristics of structures to determine the presence of
structural changes should have embedded a statistical/stochastic framework that not only
84
quantify the uncertainty in the estimation of the structure's modal parameters but also
provide a measure of the uncertainty in the detection of structural changes.
In this exploratory study, experimental data from band-limited white-noise base exci-
tation dynamic tests performed in a scaled-down six-story steel-frame laboratory structure
at the National Center for Research in Earthquake Engineering (NCREE), and a seven-
story full-scale reinforced-concrete structure at the UCSD-NEES facilities, was used to
assess the variability in a set of features from reduced-order models developed using three
dierent data-driven input-output system identication approaches that have been suc-
cessfully applied to analytical and experimental structures [31, 32, 38, 54, 55, 74{76]: a sys-
tem realization algorithm using information matrices, a general time-domain least-squares
identication method, and a non-parametric chain- like system identication approach.
4.2 Formulation of reduced-order modeling techniques
4.2.1 System realization using information matrix (SRIM)
Consider that the input-output relationship in a high-dimensional complex structural sys-
tem can be represented, assuming zero-order hold sampling, by a reduced-order, linear,
time-invariant, discrete-time state-space model
x(k + 1) =Ax(k) +Bu(k)
y(k) =Cx(k) +Du(k)
(4.1)
wherek2Z is the time step index,x(k)2R
n
denotes the state variable vector,u(k)2R
r
is the input vector (i.e., directly applied forces and/or base excitations), and y(k)2R
m
85
corresponds to the output vector (i.e., sensor measurements). The system matrices A2
R
nn
,B2R
nr
,C2R
mn
, andD2R
mr
are unknown and will be determined from
the given input and output data u(k) andy(k). In practice, the inputu(k) and output
y(k) sequences are available only for k = 1; 2;:::;N, with N being the total number of
samples recorded.
After several algebraic manipulations, Equation 4.1 can be rewritten, for p time shifts
and s consecutive time steps, in a matrix form as
Y
p
(k) =O
p
X(k) +T
p
U
p
(k) (4.2)
where the matricesX(k)2R
ns
,Y
p
(k)2R
pms
, andU
p
(k)2R
prs
are given by
X(k) =

x(k) x(k + 1) x(k +s 1)

Y
p
(k) =
2
6
6
6
6
6
6
6
6
6
6
4
y(k) y(k + 1) y(k +s 1)
y(k + 1) y(k + 2) y(k +s)
.
.
.
.
.
.
.
.
.
.
.
.
y(k +p 1) y(k +p) y(k +p +s 2)
3
7
7
7
7
7
7
7
7
7
7
5
U
p
(k) =
2
6
6
6
6
6
6
6
6
6
6
4
u(k) u(k + 1) u(k +s 1)
u(k + 1) u(k + 2) u(k +s)
.
.
.
.
.
.
.
.
.
.
.
.
u(k +p 1) u(k +p) u(k +p +s 2)
3
7
7
7
7
7
7
7
7
7
7
5
(4.3)
86
The matrices O
p
2 R
pmn
and T
p
2 R
pmpr
are the observability matrix and the gen-
eralized Toeplitz matrix, respectively. Mathematically, these matrices are represented as
follows
O
p
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
C
CA
CA
2
.
.
.
CA
p1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
T
p
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
D
CB D
CAB CB D
.
.
.
.
.
.
.
.
.
.
.
.
CA
p2
B CA
p3
CA
p4
D
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(4.4)
Since the unknown system matricesA,B,C, andD are embedded in the observability
and generalized Toeplitz matrices, state-space system realizations can be obtained from
the matricesO
p
andT
p
, which can be estimated using the given input-output data. Juang
[38] showed that the matricesO
p
andT
p
can be calculated using the information matrix
R2 R
p(m+r)p(m+r)
, which consists of the auto- and cross-correlation matrices R
uu
2
R
prpr
,R
yy
2R
pmpm
, andR
yu
2R
pmpr
of the shifted input- and output-data matrices
U
p
(k) andY
p
(k), and is mathematically dened as
R =
2
6
6
4
R
yy
R
yu
R
T
yu
R
uu
3
7
7
5
=
1
s
2
6
6
4
Y
p
(k)
U
p
(k)
3
7
7
5

Y
T
p
(k) U
T
p
(k)

(4.5)
To determine the observability matrixO
p
, the following fundamental relationship was
established among the data correlations of the input-output matrices [38]
R
hh
=O
p
~
R
xx
O
T
p
(4.6)
87
where the quantitiesR
hh
2R
pmpm
and
~
R
xx
2R
nn
are dened as
R
hh
=R
yy
R
yu
R
1
uu
R
T
yu
(4.7a)
~
R
xx
=R
xx
R
xu
R
1
uu
R
T
xu
(4.7b)
The symmetric matrixR
hh
can therefore be obtained from the input-output data using the
information matrixR. However, notice thatR
hh
exists only if the input auto-correlation
matrixR
uu
is non-singular. To guaranteeR
uu
is invertible, the input block matrixU
p
(k)
has to be of full rank (i.e., rank(U
p
) =pr). From Equation (4.6), it is clear that in order
to estimate the observability matrix O
p
, the known matrix R
hh
has to be factored into
three matrices. By using the singular-value decomposition, the symmetric matrixR
hh
can
be decomposed as follows
R
hh
=V V
T
=

V
n
V
n
0

2
6
6
4

n
0
0
T

n
0
3
7
7
5
2
6
6
4
V
T
n
V
T
n
0
3
7
7
5
V
n

n
V
T
n
(4.8)
where the diagonal matrices
n
2R
nn
and
n
0
2R
n
0
n
0
contain then retained singular-
values and then
0
=pmn truncated (i.e., relatively small and negligible) singular-values,
respectively. The matricesV
n
2R
pmn
andV
n
0
2R
pmn
0
correspond to singular vectors
associated with the singular values in
n
and
n
0
. The integern will therefore determine
the order of the system.
88
From Equations (4.6) and (4.8), it is easy to show that
O
p
V
n
(4.9a)
~
R
xx

n
(4.9b)
Equation (4.9a) implies that the matrixV
n
is a representation of the observability matrix
O
p
and can consequently be used to estimate the system matricesA andC by
A =O
y
p
(1 : (p 1)m; :)O
p
(m + 1 :pm; :) (4.10a)
C =O
p
(1 :m; :) (4.10b)
whereO
y
p
(1 : (p 1)m; :) denotes the Moore-Penrose pseudoinverse of the matrix formed
with the rst (p 1)m rows ofO
p
, O
p
(m + 1 : pm; :) consists of the last (p 1)m rows
ofO
p
, andO
p
(1 :m; :) corresponds to the rst m rows of the observability matrixO
p
. In
order to solve for the state matrixA, the integers p and n should be chosen so that the
relationship pn
0
pmm +n is satised.
With matrices A andC already estimated, the unknown system matrices B andD
can now be obtained using the following key relationship involving the Toeplitz matrixT
p
V
T
n
0
T
p
=V
T
n
0
R
yu
R
1
uu
(4.11)
89
This equation, however, does not imply thatT
p
=R
yu
R
1
uu
[38]. Using the fact that the
Toeplitz matrix is partially known because it includes the matrices A and C, the term
V
T
n
0
T
p
can be partitioned as
=
2
6
6
4
D
B
3
7
7
5
(4.12)
where the matrices 2R
pn
0
r
and 2R
pn
0
(m+n)
are dened as
=
2
6
6
6
6
6
6
6
6
6
6
4
V
T
n
0
T
p
(:; 1 :r)
V
T
n
0
T
p
(:;r + 1 : 2r)
.
.
.
V
T
n
0
T
p
(:; (p 1)r + 1 :pr)
3
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
4
V
T
n
0
(:; 1 :m) V
T
n
0
(:;m + 1 :pm)V
n
(1 : (p 1)m; :)
V
T
n
0
(:;m + 1 : 2m) V
T
n
0
(:; 2m + 1 :pm)V
n
(1 : (p 2)m; :)
.
.
.
.
.
.
V
T
n
0
(:; (p 1)m + 1 :pm) 0
3
7
7
7
7
7
7
7
7
7
7
5
(4.13)
By denoting the known right-hand-side term in Equation (4.11) as =V
T
n
0
R
yu
R
1
uu
, it is
clear that is given by
=
2
6
6
6
6
6
6
6
6
6
6
4
(:; 1 :r)
(:;r + 1 : 2r)
.
.
.
(:; (p 1)r + 1 :pr)
3
7
7
7
7
7
7
7
7
7
7
5
(4.14)
90
and therefore, the matricesB andD can be computed by
2
6
6
4
D
B
3
7
7
5
=
y
(4.15)
where the rst m rows of
y
correspond to the matrixD, and the last n rows form the
matrixB.
The discrete-time state-space realization (i.e., A, B, C, D) of the dynamic system
obtained from the measured input-output data can be transformed into a continuous-time
representation using the following expressions
A
c
=
1
t
ln(A)
B
c
=A
c
(AI)
1
B
C
c
=C
D
c
=D
(4.16)
The modal parameters (e.g., mode shapes, natural frequencies, and damping ratios) of
the system can be estimated extracted from the continuous-time realization of the system.
The eigendecomposition or spectral decomposition of the matrixA
c
yields
A
c
=
1
(4.17)
91
where 2 C
nn
and 2 C
nn
are the complex eigenvector and complex eigenvalue
matrices, respectively. The complex eigenvalues
i
in matrix are mathematically related
to the natural frequencies !
i
and damping rations
i
of the system by
!
i
=
p
Re(
i
)
2
+ Im(
i
)
2

i
= Re(
i
)=!
i
(4.18)
The matrix of complex mode shapes in the coordinates of the sensors can be obtained
from the eigenvector matrix using the following transformation
=C
c

k
(4.19)
where k = 0 for displacement measurements, k = 1 for velocity measurements, and k = 2
for acceleration measurements.
In order to determine the parameters associated with the stiness and damping of the
reduced-order dynamic model from the identied state-space realization, an equivalence
transformation using the observability matrix can be carried out

A
c
=PA
c
P
1
=
2
6
6
4
C
c
C
c
A
c
3
7
7
5
A
c
2
6
6
4
C
c
C
c
A
c
3
7
7
5
1
(4.20)
where the matrix

A
c
2R
2m2m
will have the following form

A
c
=
2
6
6
4
0 I
M
1
K M
1
C
3
7
7
5
(4.21)
92
4.2.2 Least-squares system identication (LSSID)
Assume that a complex structural system subjected to the action of directly applied forces
and/or nonuniform support motions can be treated as a reduced-order discrete nonlinear
multi-degree-of-freedom (MDOF) system governed by the set of dierential equations:
M x(t) +f
L
(t) +f
N
(t) =g(t) (4.22)
with
f
L
(t) =C _ x(t) +Kx(t) +M
0
x
0
(t) +C
0
_ x
0
(t) +K
0
x
0
(t) (4.23)
wherex are the displacements of the n active degrees-of-freedom; x
0
are the prescribed
support motions of then
0
constrained degrees-of-freedom,g are the directly applied forces;
f
L
are the linear forces involvingx andx
0
;M,C,K are the matrices that characterize
the inertia, damping, and stiness forces associated with the active degrees-of-freedom
of the system; M
0
,C
0
,K
0
are the matrices that characterize the inertia, damping, and
stiness forces associated with support motions; andf
N
are the nonlinear nonconservative
forces involvingx as well asx
0
. The total number of degrees-of-freedom (i.e., n +n
0
) will
depend on the number of sensors distributed throughout the structure.
The rst step in the time-domain identication procedure involves determining the
equivalent linear reduced-order model of the structural system. Consider the linearized
version of the system under discussion and assume it is governed by
M x(t) +C _ x(t) +Kx(t) +M
0
x
0
(t) +C
0
_ x
0
(t) +K
0
x
0
(t) =g(t) (4.24)
93
For clarity of presentation, let the six matrices appearing in the previous equation be
denoted by
j
A; j = 1;:::; 6; respectively. Now assuming that the displacements, velocities,
and accelerations of the unconstrained and constrained DOFs, as well as the directly
applied excitations are measured at every time t
k
, k = 1; 2;:::;N; the linear model
(Eq. 4.24) can be recast into the generic format
R
i
=b
i
; i = 1;:::;n (4.25)
where the matrixR is assembled using the structural response measurements,
R =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
x
T
(t
1
) _ x
T
(t
1
) x
T
(t
1
) x
T
0
(t
1
) _ x
T
0
(t
1
) x
T
0
(t
1
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
x
T
(t
k
) _ x
T
(t
k
) x
T
(t
k
) x
T
0
(t
k
) _ x
T
0
(t
k
) x
T
0
(t
k
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
x
T
(t
N
) _ x
T
(t
N
) x
T
(t
N
) x
T
0
(t
N
) _ x
T
0
(t
N
) x
T
0
(t
N
)
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(4.26)
the vector
i
corresponds to the unknown elements of the ith row of the matrices
j
A,

i
=

1
A
i
:::
j
A
i
:::
6
A
i

T
(4.27)
and the vectorb
i
contains the measurements of the ith applied excitation
b
i
=

g
i
(t
1
) ::: g
i
(t
k
) ::: g
i
(t
N
)

T
(4.28)
94
It should be noted that depending on the assumptions made about the structural system,
the general representation given in Eq. (4.25) may be mathematically formulated in several
forms [55, 75, 76].
In general, the linear system of equation given in Eq. (4.25) is overdetermined (i.e., the
number of samplesN in the response measurements is much larger than the total number
of unknowns), and therefore, the corresponding least-squares solutions can be computed
by

i
=R
y
b
i
(4.29)
whereR
y
is the Moore-Penrose pseudoinverse of the measurements matrixR.
Once the linear system matrices have been determined, the nonlinear forces of the
reduced-order modelf
N
can now be calculated from Eq. (4.22)
f
N
(t) =g(t)M x(t)f
L
(t) (4.30)
Notice that the obtained vector f
N
will include not only the forces caused by the non-
linearities present in the complex structural system, but also measurement noise, and
residuals due to unmodeled linear dynamics that were not characterize by the identied
linear reduced-order model. However, since the equivalent linear reduced-order model has
a xed size (i.e., then active DOFs), as the nonlinearity eects become more important to
the dynamics of the system, it is expected that the termf
N
will be dominated by the con-
tribution of the nonlinear forces rather than the residual components. Therefore, to detect
95
the presence of nonlinearities and determine their location, the ratio between the root-
mean-square of the estimated nonlinear and linear forces for each active degree-of-freedom,
kf
N
i
krms
=
kf
L
i
krms
, is considered as the nonlinearity-sensitive feature within this approach.
Signicant changes in this ratio are assumed to be observed in the DOFs associated with
the sensors that are close to the actual nonlinearities.
4.2.3 Chain-like structures system identication (ChainID)
Consider an MDOF chain-like system, consisting of a series of lumped masses m
i
inter-
connected by n arbitrary unknown nonlinear elements G
(i)
, subjected to a base motion
x
0
, and/or directly applied forces F
i
. The nonlinear elements' restoring forces are as-
sumed to depend on the relative displacement and velocity across the terminals of each
element, in addition to a set of specic parameters p that characterize the various types
of nonlinearities.
The dierential equations of motion for the system under discussion, can be written
as [31, 32, 54]
G
(n)
(z
n
; _ z
n
;p
n
) =
F
n
m
n
x
n
G
(i)
(z
i
; _ z
i
;p
i
) =
F
i
m
i
x
i
+
m
i+1
m
i
G
(i+1)
(z
i+1
; _ z
i+1
;p
i+1
) (4.31)
for i =n 1;n 2;:::; 1
where G
(i)
(z
i
; _ z
i
;p
i
) is the mass-normalized restoring force function of the nonlinear ele-
mentG
(i)
, x
i
is the absolute acceleration of the massm
i
, withz
i
and _ z
i
being the relative
96
displacements and velocities between two consecutive masses. These relative motion vari-
ables can be obtained from the absolute state variables of the masses and the moving
support. Equation (5.1) can be rewritten in a more simplied form as
G
(i)
(z
i
; _ z
i
;p
i
) =
n
X
j=i
f
j
m
j

n
X
j=i
m
j
m
i
x
j
(4.32)
Within the context of this method, the absolute accelerations x
i
are assumed to be
available from measurements, as well as the applied forces F
i
, the base excitation x
0
and
the magnitude of the lumped masses m
i
.
After obtaining all the restoring force time histories, it is possible to generate a non-
parametric representation for each nonlinear element, in terms of a truncated doubly-
indexed series expansion in a suitable basis, that approximates the real restoring force
function [31, 32, 53, 57]. The approximating representation
^
G
(i)
(z
i
; _ z
i
) for the obtained
restoring forces, in an orthogonal polynomial basis, is given by the following expression:
G
(i)
(z
i
; _ z
i
;p)
^
G
(i)
(z
i
; _ z
i
) =
qmax
X
q=0
rmax
X
r=0
C
(i)
qr
T
q
(z
0
i
)T
r
( _ z
0
i
) (4.33)
where C
(i)
qr
are the Chebyshev series coecients, T
k
() is the Chebyshev polynomial of
order k, and z
0
i
, _ z
0
i
are the normalized relative state variables. Subsequently, each of the
estimated restoring forces can be expressed as a power series of the form
^
G
(i)
(z
i
; _ z
i
) =
qmax
X
q=0
rmax
X
r=0
a
(i)
qr
z
q
i
_ z
r
i
(4.34)
where a
(i)
qr
are constant coecients, and z
i
, _ z
i
are the relative state variables.
97
The application of this structural decomposition approach allows to independently
characterize the dominant dynamic features of each linking element in the chain-like sys-
tem into reduced-order models by adjusting the coecients in the restoring force expan-
sion. These restoring force coecients a
(i)
qr
have proven to be a suitable set of features in
detecting and localizing nonlinearities in structural systems [32]. The presence of nonlin-
earities in the system is then indicated by signicant high-order and cross-product terms
in the identied reduced-order models of the interconnecting elements. Additionally, the
structural sections where the nonlinearities are, will be determined by the nonlinear link-
ing elements location. It should be also emphasized that, since this proposed approach is
entirely data-driven, the number of degrees-of-freedom in the considered chain-like system
will be determined by the number of available sensors deployed on the structure.
4.3 Detection of changes using reduced-order models of struc-
tural systems
In this experimental study, two distinct sets of change-sensitive features estimated using
reduced-order models of structural system were employed in this experimental study: (1)
stiness-like and damping-like parameters of the identied reduced-order models, and (2)
the identied modal parameters (natural frequencies and damping ratios). It was assumed
that changes observed in these sets of features estimated from the reduced-order models
would be indicative of structural changes in the six-story test structure. The sensitivity
and robustness of the selected sets of features to structural changes in the presence of
uncertainty were used to evaluate the eectiveness of the developed reduced-order models
98
in detecting changes in structural systems. Structural changes can be either physically
introduced to simulate damage in the system, or caused by dierent damage mechanisms
developed during dynamic tests.
A classical hypothesis testing-based approach was used to determine if signicant
changes in the dierent sets of identied features have occurred with respect to the set
of features estimated from reduced-order models of the structural system in a reference
or baseline condition. A signicance level of = 0:05 was assumed in order to obtain
the bounds dening the two-tail condence region from the probability density function
estimated for each of the extracted features from the reference or baseline reduced-order
models. A threshold in the ratio of the total number of new observations of change-
sensitive features falling outside their condence bounds was used to assess the signicance
of any observed change in the parameters. In this study, this threshold has set to = 0:90.
Therefore, if more than the 90% of the new observations for a given parameter are out-
side the reference condence bounds, there is enough statistical evidence to conclude that
signicant changes were observed in that parameter. The ratio will be referred to as
detection ratio hereafter. In addition, the relative change (
r
) in the mean values of the
new estimated parameters with respect to the reference scenario was used to provide a
measure of the observed changes.
99
4.4 Experimental results in six-story steel-frame laboratory
structure
4.4.1 Test structure
The test structure used in this experimental study was a scaled-down six-story single-bay
steel frame structure (Figure 4.1a) mounted on a shaking table at the National Center for
Research in Earthquake Engineering (NCREE) at the National Taiwan University (NTU).
The test structure had a total height of 6.0 m with 1.0-m inter-story heights, and a plan
dimension of 1.0 m by 1.5 m. Steel plates (1:0 1:5 0:02 m), which were used as
oor
slabs, were welded to L-shape steel beams (0:05 0:05 0:005 m) and supported by four
rectangular cross-section steel columns (0:15 0:025 m) using bolted connections. The
total lumped mass on each
oor was 862 kg.
The sensor network deployed on the test structure consisted of 20 accelerometers, and
14 velocity and displacement transducers (Figure 4.1b). To measure the lateral dynamic
response of the structure, each
oor was instrumented with three accelerometers, two in
the x-direction (weak axis) and one in the y-direction (strong axis), one velocity transducer
in the x-direction, and one displacement sensor in the x-axis. The base accelerations were
recorded using two accelerometers, with each accelerometer oriented in the x- and y-axis.
Velocities and displacements of the shake table were measured only in the x-direction of
the structure.
A total of three structural states/congurations were considered in this study for de-
tecting structural changes using reduced-order models of the testbed structure. The rst
state conguration (state#1) corresponded to the baseline, or reference condition of the
100
Instrumentation
A Acceleration sensor (x-direction)
V Velocity sensor (x-direction)
D Displacement sensor (x-direction)
A Acceleration sensor (y-direction)
A
A
A
A
A
A
A
Y-axis view
V D A A
V D A A
V D A A
V D A A
V D A A
V D A A
V D A
X-axis view
(a) NCREE testbed structure (b) Sensor location
Figure 4.1: Six-story steel frame testbed structure used for experimental study: (a) Photo
of the testbed structure at NCREE facilities; (b) Location of sensors deployed in the
testbed structure.
test structure. In states 2 and 3, two columns at the rst and second
oors were cut
below the bolted
oor-column connection. For state 2, a 6.0-cm horizontal cut was made
in two of the rst-
oor columns. In state 3, the size of the cuts in the rst-
oor columns
was increased from 6.0 cm to 9.0 cm. Additionally, 6.0-cm cuts were made in two of the
second-
oor columns. Table 4.1 summarizes the three dierent state conditions of the test
structure used herein. Figure 4.2 displays a zoom-in image of the test structure in the
state#3 conguration. The 9.0-cm and 6.0-cm cuts made in the rst- and second-
oor
columns can be seen in these two images.
For each of the structural states, the testbed structure was subjected to a sequence of
base excitation tests by applying 120-second long band-limited white-noise base motions
in the two horizontal axes of the shake table simultaneously. During each dynamic test,
101
Table 4.1: Summary of structural state conditions
State Description
State#1 Baseline structural condition.
State#2 6.0-cm horizontal cut in two columns of rst
oor.
State#3 9.0-cm horizontal cut in two columns of rst
oor.
6.0-cm horizontal cut in two columns of second
oor.
Figure 4.2: Six-story steel frame testbed structure in one of the damaged structural con-
gurations. The 9.0-cm and 6.0-cm cuts made in the rst and second-
oor columns are
shown
the lateral response of the structure was measured in both the x-axis (weak axis) and
y-axis (strong axis). The acceleration, velocity, and displacement responses were acquired
at a sampling frequency of 200 Hz. In this study, however, only the acceleration mea-
surements in both x- and y-axis were assumed to be available. Additionally, by using
the rigid
oor assumption and basic kinematic relationships, the acceleration response of
the structure at each slab's geometric center were determined from the available accel-
eration measurements. The corresponding velocity and displacement time-histories were
then obtained through data processing and numerical integration. To assess the accuracy
102
10 11 12 13 14 15 16 17 18 19 20
−1.5
−1
−0.5
0
0.5
1
1.5
t [s]
¨ x
6
[ms
−2
]
10 11 12 13 14 15 16 17 18 19 20
−0.1
−0.05
0
0.05
0.1
t [s]
˙ x
6
[ms
−1
]

Integrated
Measured
10 11 12 13 14 15 16 17 18 19 20
−0.01
0
0.01
t [s]
x
6
[m]

Integrated
Measured
Figure 4.3: Typical acceleration, velocity, and displacement time-histories at the top of
the structure in the x-axis direction are displayed in the top, middle, and bottom plots,
respectively. The measured velocities and displacements are also compared to the ve-
locities and displacements obtained by numerical integration of the acceleration records.
Note the close match between the measured and numerically integrated displacements and
velocities.
of the implemented numerical integration approach, the integrated displacements and ve-
locities were compared with the response time histories measured in the x-axis. An error
analysis showed a close match between the estimated and measured displacement and ve-
locity time-histories. The top plot in Figure 4.3 displays a segment of typical acceleration
time-history estimated at the center of the top
oor from the accelerations measured at
both ends of the steel slab. The corresponding integrated and measured velocities and
displacements are shown in the middle and bottom plots.
103
4.4.2 Implementation and results
In this study, it was decided that only the lateral dynamic response at the geometric
center of each slab, in both the x- and y-direction, would be used to estimate the corre-
sponding reduced-order models of the test structure in each structural state/conguration.
Additionally, it was assumed that only the acceleration measurements were available.
Furthermore, all acceleration, velocity and displacement time-histories were divided
into overlapped segments of 50 seconds, generating 20 ensembles of input-output records
for each structural state. The vibration time-histories in each of these ensembles were
then employed to estimate 6DOF reduced-order models that approximately characterize
the dynamics of the test structure in each of the structural congurations (i.e., state 1, 2,
and 3) using each of the approaches described previously. Even though the total lumped
mass in each
oor was known beforehand, no a priori knowledge of the structure's mass
distribution was assumed in this study.
Twenty reference observations of the stiness-like parameters and modal parameters
were obtained from the reduced-order models built for the the structure in the baseline
condition (i.e., state#1) in both x- and y-directions. The probability density functions
and corresponding two-tail condence bounds for a signicance level of = 0:05 were
then computed using the reference observations of the stiness-like parameters and modal
parameters. For structural states#2 and #3, the presence of signicant changes in the
dierent set of parameters was determined by the number of new observations falling out-
side their condence bounds. If more than 90% (i.e., threshold in detection ratio = 0:9)
104
of the new observations from a given structural conguration were outside the parame-
ters' condence bounds, there is enough statistical evidence to conclude that signicant
changes were observed in that parameter.
4.4.2.1 Changes in stiness-like parameters
For the reduced-order models obtained using the SRIM and LSSID approaches specically,
the parameters used to investigate changes in the test structure were the diagonal elements
of the identied mass-normalized stiness-like matrices (
~
K =M
1
K) of the correspond-
ing reduced models. The use of the diagonal elements of identied stiness matrices
has been reported in dierent structural health monitoring studies [43, 51, 75, 76]. For
reduced-order models developed using the ChainID approach, on the other hand, the pa-
rameters used as indicators of structural changes in the test structure were the restoring
force coecients a
(i)
10
estimated for each inter-connecting link in the chain-like reduced-
order model. This coecients correspond to the mass-normalized stiness-like parameters
of each
oor in the test structure. It has been shown that the restoring force coecients in
the chain-like system identication approach are a suitable set of parameters for structural
change detection applications [31, 32].
Figure 4.4 displays the comparison of the probability density functions of the rst two
elements in the diagonal of the mass-normalized stiness-like matrices of the reduced-
order models of the test structure in the x- and y-direction. The pdf of the parameters
of the reduced-order models from the reference condition (state#1) is shown in a blue
solid line. The shaded region corresponds to the 95% condence region of the stiness-like
parameters. It is clear from Figure 4.4(a) that signicant reductions in the stiness-like
105
3400 3450 3500 3550 3600 3650
0
0.02
0.04
0.06
0.08
0.1
0.12
˜
k
11
[s
−2
]
pdf [s
2
]

3420 3440 3460 3480 3500 3520
0
0.05
0.1
0.15
˜
k
22
[s
−2
]
pdf [s
2
]

State 1 State 2 State 3
−5 −4 −3 −2 −1 0 1 2 3 4 5
x 10
5
0
0.2
0.4
0.6
0.8
1
1.2
x 10
−5
˜
k
11
[s
−2
]
pdf [s
2
]

−3 −2 −1 0 1 2 3
x 10
5
0
0.5
1
1.5
x 10
−5
˜
k
22
[s
−2
]
pdf [s
2
]

State 1 State 2 State 3
(a) Stiness-like parameters in x-direction (b) Stiness-like parameters in y-direction
Figure 4.4: Comparison of the probability density distribution for the elements
~
k
11
and
~
k
22
in the diagonal of the mass-normalized stiness-like (
~
K) matrices of the reduced-order
models in the x- and y-direction of the test structure estimated using SRIM.
parameters
^
k
11
^
k
22
in the x-direction estimated for the structure in the \damaged" con-
gurations. On the other hand, no changes in the parameters estimated for the structure
in the y-direction were observed since all
^
k
11
^
k
22
parameters from state#2 and state#3
were within the condence bounds of the parameters from the reference state#1.
All detection ratios and magnitude of relative changes
r
of the stiness-like pa-
rameters associated with the reduced-order models identied using SRIM for the test
structure in state#2 and state#3 scenarios are summarized in Figure 4.5. For state#2,
it is clear that signicant changes occurred in the
~
k
11
term of the reduced-order model in
the x-direction (Figure 4.5a) since all estimated parameters were outside the parameter's
condence bounds set for the reference condition. This change corresponded to an approx-
imately 2% reduction in the mean value of the stiness-like term
~
k
11
. Given the chain-like
topology of the structural system and based on a nite element idealization of the 6DOF
system, the rst term k
11
in the diagonal of the stiness matrix would correspond to
106
the sum of the inter-story stiness of the rst and second
oor. Therefore, a reduction
on the stiness-like term
~
k
11
was expected since the rst-story columns were cut in this
structural conguration. It should be noted then that the signicant changes observed on
the stiness-like parameter
~
k
11
of the reduced-order model did correlate with the physical
changes made in the test structure. On the other hand, the signicant changes detected
in the parameters
~
k
22
and
~
k
33
could not be associated to any actual structural change in
the test structure. Note that even thought the changes in
~
k
22
and
~
k
33
were statistically
signicant, the magnitudes (i.e., relative change
r
) were almost negligible. From the
detection ratios and magnitude of relative changes
r
shown in Figure 4.5(b), it can
be seen that although relatively large changes in the diagonal terms of the stiness-like
matrix obtained for the structure in the y-direction were observed, these changes were not
signicant. The maximum percentage of observations outside the condence bounds of
the stiness-like parameters was around a 30%.
Similar analysis can be done with the results obtained from the state#3 scenario.
The detected changes in
~
k
11
and
~
k
22
can be directly correlated to the physical changes
in the structure. Notice the increase in the magnitude of the relative change in the
stiness-like parameter
~
k
11
associated with the inter-story stiness of the rst and second

oor. In this case, an approximate 6% reduction in
~
k
11
was observed in average. In
addition, a statistically-signicant reduction of roughly a 2% in the term
~
k
22
was also
observed. It should be noted that the magnitude of the relative change in
~
k
22
for state#3,
it is approximately the same to the observed changes in
~
k
11
for state#2. In state#2 and
state#3, 6-cm cuts were made in two columns of the rst
oor and second
oor respectively.
Therefore, the magnitude of the relative change in the stiness-like parameters could
107
11 22 33 44 55 66
0
0.2
0.4
0.6
0.8
1
˜
k
ij
[-]
η [-]

11 22 33 44 55 66
−0.06
−0.04
−0.02
0
˜
k
ij
[-]
Δ
r
[-]
State 2 State 3
11 22 33 44 55 66
0
0.2
0.4
0.6
0.8
1
˜
k
ij
[-]
η [-]

11 22 33 44 55 66
−3
−2
−1
0
1
2
3
˜
k
ij
[-]
Δ
r
[-]
State 2 State 3
(a) Stiness-like parameters in x-direction (b) Stiness-like parameters in y-direction
Figure 4.5: Detection ratios and magnitude of relative changes
r
in the diagonal
elements of the stiness-like matrix
^
K of the (a) reduced-order models of the test structure
in the x-direction, and (b) and reduced-order models of the test structure in the y-direction
obtained using SRIM approach. The horizontal line corresponds to the threshold = 0:90
used in this study to determine if signicant changes in the parameters have been observed
with respect to the parameters from the reference condition.
be used as an indicator of the severity of the structural changes introduced in the test
structure. The signicant changes in
~
k
33
,
~
k
44
, and
~
k
55
could not be associated to physical
changes, and therefore, they can be considered as false-positives.
The detection ratios and magnitude of relative changes
r
of the stiness-like pa-
rameters associated with the reduced-order models identied using LSSID approach for
the test structure in state#2 and state#3 scenarios are summarized in Figure 4.6. From
these results, it can be seen that there is no clear correlation between the changes on
the diagonal elements of the stiness-like matrix of the identied reduced-order models
and the physical changes made to the test structure. From the reduced-order model ob-
tained in the x-direction of the structure, only a statistically-signicant reduction in the
parameter
~
k
11
of approximately 5% was observed for state#3 scenario. There was also an
108
11 22 33 44 55 66
0
0.2
0.4
0.6
0.8
1
˜
k
ij
[-]
η [-]

11 22 33 44 55 66
−0.05
0
0.05
˜
k
ij
[-]
Δ
r
[-]
State 2 State 3
11 22 33 44 55 66
0
0.2
0.4
0.6
0.8
1
˜
k
ij
[-]
η [-]

11 22 33 44 55 66
−4
−3
−2
−1
0
˜
k
ij
[-]
Δ
r
[-]
State 2 State 3
Figure 4.6: Detection ratios and magnitude of relative changes
r
in the diagonal
elements of the stiness-like matrix
^
K of the (a) reduced-order models of the test structure
in the x-direction, and (b) and reduced-order models of the test structure in the y-direction
obtained using LSSID approach. The horizontal line corresponds to the threshold = 0:90
used in this study to determine if signicant changes in the parameters have been observed
with respect to the parameters from the reference condition.
approximate 2% reduction in
~
k
11
estimated in state#2, but almost a 50% of the estimated
parameters were within the corresponding condence bounds. It can be seen also that
the magnitude of these changes in
~
k
11
in state#2 and state#3 increased proportionally
to the severity of the damage in the structure. The other elements in the diagonal of the
stiness-like matrix exhibited increments in their average magnitudes with respect to their
reference values. However, based on the detection rule set in this study, these changes
were not considered as signicant.
Figure 4.7 shows the comparison of the probability density functions of the stiness-like
parametersa
(1)
10
anda
(2)
10
for the rst and second inter-connecting elements of the idealized
chain-like reduced-order models of the test structure in the x- and y-direction. The pdf
of the parameters of the reduced-order models from the reference condition (state#1) is
shown in a blue solid line. The shaded region corresponds to the 95% condence region of
109
860 870 880 890 900 910 920 930 940 950 960
0
0.02
0.04
0.06
0.08
0.1
0.12
a
(1)
10
[s
−2
]
pdf [s
2
]

790 800 810 820 830 840
0
0.05
0.1
0.15
a
(2)
10
[s
−2
]
pdf [s
2
]

State 1 State 2 State 3
2000 2500 3000 3500 4000 4500 5000 5500 6000 6500
0
0.002
0.004
0.006
0.008
0.01
0.012
a
(1)
10
[s
−2
]
pdf [s
2
]

3100 3200 3300 3400 3500 3600 3700 3800 3900
0
0.005
0.01
0.015
0.02
0.025
0.03
a
(2)
10
[s
−2
]
pdf [s
2
]

State 1 State 2 State 3
Figure 4.7: Comparison of the probability density distribution for the stiness-like co-
ecients a
(1)
10
and a
(2)
10
of the reduced-order models in the x- and y-direction of the test
structure estimated using ChainID approach.
the stiness-like parameters. It is clear from Figure 4.7(a) that signicant changes in the
stiness-like coecientsa
(1)
10
anda
(2)
10
in the x-direction estimated for the test structure in
state#2 and state#3. Similarly, changes in thea
(1)
10
coecient estimated for the structure
in the y-direction were observed for state#2 and state#3. However, no changes were
observed in the a
(2)
10
coecient since the coecients estimated for state#2 and state#3
were within the condence bounds of the parameters from the reference state#1.
The detection ratios and magnitude of relative changes
r
estimated for the stiness-
like parameters a
(1)
10
and a
(2)
10
associated with the reduced-order models identied using
ChainID approach for the test structure in state#2 and state#3 scenarios are summarized
in Figure 4.8. For state#2, it is clear that signicant change occurred in thea
(1)
10
coecient
of the reduced-order model in the x-direction (Figure 4.8a). The magnitude of the relative
change in a
(1)
10
was in the order of 2.5%. For the stiness-like coecients a
(2)
ij
to a
(6)
ij
, no
signicant changes were detected because more than a 60% of the identied coecients in
110
state#2 were within their reference condence bounds. For state#3, it can be seen from
Figure 4.8(a) that only the observed changes in the a
(1)
10
and a
(2)
10
could be considered as
statistically signicant. These changes corresponded to reductions of approximately 5%
and 2% in thea
(1)
10
anda
(2)
10
, respectively. Notice that the detected changes in the stiness-
like coecientsa
(1)
10
anda
(2)
10
of the reduced-order model of the structure for the dynamics in
the x-direction can be directly correlated to the physical changes in the test structure made
in the columns of the rst and second
oors. In addition, the magnitude of the changes
in a
(1)
10
increased proportionally to the severity of the changes made to the test structure.
It should be noted also that no signicant changes with almost negligible magnitude were
observed in the a
(3)
10
, a
(4)
10
, and a
(5)
10
coecients. Even though a relatively large increment
of around 10% on the a
(6)
10
coecient was noticed for state#2 and state#3, these changes
were considered not signicant with respect to the coecients from the reference state#1.
4.4.2.2 Changes in modal parameters
Using the formulations in Section 4.2, the experimental modal parameters of the lateral
vibration modes of the test structure in the x- and y-direction were determined using the
20 input-output data ensembles from each of the structural state scenarios (Table 4.1).
The mean natural frequencies and mean damping ratios of the rst six lateral modes
in the weak axis of the structure (x-direction) and the rst three lateral modes in the
strong axis (y-direction) estimated from the reduced-order models built using the SRIM,
LSSID, and ChainID approaches for the reference structure, are summarized in Table 4.2.
The corresponding mode shapes for the rst three lateral modes in the x- and y-direction
are show in Figures 4.9 - 4.11. It should be noted that the natural frequencies of the
111
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
a
(i)
10
[-]
η [-]

1 2 3 4 5 6
−0.05
0
0.05
0.1
a
(i)
10
[-]
Δ
r
[-]
State 2 State 3
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
a
(i)
10
[-]
η [-]

1 2 3 4 5 6
−0.1
0
0.1
0.2
a
(i)
10
[-]
Δ
r
[-]
State 2 State 3
Figure 4.8: Detection ratios and magnitude of relative changes
r
in the diagonal
elements of the stiness-like coecients a
(i)
10
of the interconnecting elements in the (a)
reduced-order models of the test structure in the x-direction, and (b) and reduced-order
models of the test structure in the y-direction obtained using ChainID approach. The
horizontal line corresponds to the threshold = 0:90 used in this study to determine if
signicant changes in the parameters have been observed with respect to the parameters
from the reference condition.
lateral modes in the x- and y-direction estimated with SRIM and LSSID were practically
identical. The mean natural frequencies estimated from reduced-order models built in this
study did closely agree with the natural frequencies reported by Kim and Lynch [44]. The
mean values of the damping ratios obtained with the SRIM and the ones obtained with
LSSID models were very similar. However, the damping ratios of the lateral modes in the
x-direction were relatively smaller while the damping values of the modes in the y-direction
were larger with respect to the damping ratios reported by Kim and Lynch [44]. It should
be emphasized that while in this study the modal parameters reported correspond to the
average of the parameters identied from the 20 input-output data ensembles, the values
reported by Kim and Lynch [44] corresponded to just one input-output data ensemble.
112
As previously mentioned, change detection was based, in this study, on a classical
hypothesis testing-based approach. From the estimated probability density functions, and
assuming a signicance level of = 0:05, the bounds dening the two-tail condence region
of the modal frequencies and damping ratios identied in the reference structural condition
were obtained. New observations of the modal parameters, obtained from vibration data
ensembles from state#2 and state#3, were then compared against the lower and upper
condence bounds. The detection ratios and magnitudes of observed changes in the natural
frequencies and damping ratios are summarized in Figures 4.12 - 4.14.
From Figure 4.12, it is evident that signicant reductions on the natural frequencies for
all six lateral modes were observed in state#2 and state#3. The magnitude of the changes
in all estimated natural frequencies was proportional to the severity of the physical changes
in the test structure. Regarding the estimated damping rations for state#2, only the 15%
reduction in the damping of the sixth lateral mode could be considered as signicant change
in the damping when compared with the damping ratios from the reference condition. For
state#3, the damping ratios of the second, fourth, and sixth lateral modes had signicant
reductions from their reference values.
It can be seen in Figure 4.13 that only the changes in the frequencies of the second
and third lateral modes in the x-direction were detectable from the reduced-order models
generated for the structure in state#2. In addition, the frequencies of the rst four
modes did undergo signicant reductions in their mean values with respect to the reference
natural frequencies. Notice that an increment of approximately a 3% was observed in the
frequency of the sixth mode in both state#2 and state#3. However, since roughly the 50%
of the estimated frequencies were within the condence bounds set for !
6
in the reference
113
state#1. Signicant changes in the damping ratios
i
were only noticed in the second
mode in both state#2 and state#3.
An analysis of the detection ratios calculated for the natural frequencies and damping
ratios estimated from the models generated using ChainID approach showed that none of
observed changes in the modal parameters could be considered as statistically signicant,
therefore it was not possible to condently detect any change in the modal parameters
estimated using the ChainID approach.
Table 4.2: Summary of mean natural frequencies and mean damping ratios for the rst
six lateral modes in the x-direction and rst three lateral modes in the y-direction of the
structure identied from the reduced-order models developed using SRIM, LSSID, and
ChainID approaches.
SRIM LSSID ChainID
Mode ! [Hz] [%] ! (Hz) [%] ! (Hz) [%]
X-direction
1
st
Mode 1.123 1.21 1.123 0.99 1.136 0.82
2
nd
Mode 3.625 0.84 3.627 0.89 3.372 2.13
3
rd
Mode 6.327 0.77 6.334 0.78 5.201 2.98
4
th
Mode 9.202 0.77 9.218 0.80 6.860 4.09
5
th
Mode 12.087 0.50 12.010 0.47 8.257 4.16
6
th
Mode 14.335 0.34 13.294 0.45 9.220 3.28
Y-direction
1
st
Mode 2.247 1.62 2.226 1.63 2.125 0.84
2
nd
Mode 8.522 1.29 8.513 1.83 5.228 3.24
3
rd
Mode 15.273 2.09 15.825 1.45 8.664 4.22
4.5 Experimental results in seven-story full-scale reinforced-
concrete structure
4.5.1 Test structure and experimental setup
The test structure, which was designed and built to represent a slice of a 7-story full-scale
reinforced concrete wall building, consisted of a main rectangular wall (web wall) that
114
ω1 = 1.12 Hz ω2 = 3.63 Hz ω3 = 6.33 Hz
(a) Modeshapes in X-direction
ω 1 = 2.25 Hz ω 2 = 8.52 Hz ω 3 = 15.27 Hz
(a) Modeshapes in Y-direction
Figure 4.9: Mode shapes of the rst three lateral modes in the x- and y-direction estimated
from the reduced-order models built using the SRIM approach.
provided lateral force resistance and supported seven concrete slabs, a transverse wall
(
ange wall) providing transversal stability, an auxiliary post-tensioned wall for torsional
stability, and four gravity columns that carried axial loads. Slotted slab connections
that acted like pin-ended links were placed between the web and
ange walls in order to
minimize the transfer of out-of-plane shear and bending moment between the web and

ange walls while allowing the transfer of in-plane shear, moment and axial forces. Figure
4.15 shows a general view of the test structure and some of its components. A more
detailed description about the test structure is provided by Panagiotou and Restrepo
[63], Panagiotou et al. [64, 65];
115
ω1 = 1.12 Hz ω2 = 3.63 Hz ω3 = 6.33 Hz
(a) Modeshapes in X-direction
ω 1 = 2.23 Hz ω2 = 8.51 Hz ω3 = 15.83 Hz
(a) Modeshapes in Y-direction
Figure 4.10: Mode shapes of the rst three lateral modes in the x- and y-direction esti-
mated from the reduced-order models built using the LSSID approach.
A series of dynamic tests, including white noise and seismic base excitation tests, were
performed on the test structure using the UCSD-NEES shake table. The test structure
was progressively damaged by sequentially applying four historical earthquake records
of increasing intensity using the UCSD-NEES shake table. The low-intensity ground
motion (EQ1) and medium-intensity ground motion (EQ2) corresponded to the horizontal
components of the 1971 San Fernando earthquake (M
w
= 6:6) recorded at the Van Nuys
7-story hotel station. The third earthquake record of medium intensity (EQ3) and the
large-intensity ground motion (EQ4) were components of the 1994 Northridge earthquake
(M
w
= 6:7) recorded at the Oxnard Boulevard and Sylmar stations, respectively.
116
ω1 = 1.14 Hz ω2 = 3.37 Hz ω3 = 5.20 Hz
(a) Modeshapes in X-direction
ω1 = 1.98 Hz ω2 = 5.09 Hz ω3 = 8.54 Hz
(a) Modeshapes in Y-direction
Figure 4.11: Mode shapes of the rst three lateral modes in the x- and y-direction esti-
mated from the reduced-order models built using the ChainID approach.
In overall, six dierent structural states were considered in this study. State 1 cor-
responded to the undamaged/reference state of the test structure before being subjected
to any seismic excitation. States 2-4 were associated with the structural condition of the
building right after the rst (EQ1), second (EQ2), and third (EQ3) earthquake records
were applied. State 2 was characterized by limited yielding in the longitudinal reinforcing
steel of the web wall and widespread and visible cracking in the web wall up to the fourth

oor. In States 3 and 4, moderate yielding occurred in the web wall longitudinal reinforce-
ment. At this point, the bracing system between the slabs and the post-tensioned wall
was stiened before the last seismic base excitation test. This scenario corresponded to
State 5. Finally, the structural condition of the test structure after being subjected to the
117
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
ω
i
[-]
η [-]

1 2 3 4 5 6
−0.015
−0.01
−0.005
0
ω i [-]
Δr [-]
State 2 State 3
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
ζ i [-]
η [-]

1 2 3 4 5 6
−0.25
−0.2
−0.15
−0.1
−0.05
0
ζ
i
[-]
Δ
r
[-]
State 2 State 3
(a) Natural frequencies (b) Damping ratios
Figure 4.12: Detection ratios and magnitude of relative changes
r
in the (a) natural
frequencies and (b) damping ratios estimated for the test structure in the x-direction
obtained using SRIM approach. The horizontal line corresponds to the threshold =
0:90 used in this study to determine if signicant changes in the modal parameters have
occurred with respect to the parameters from the reference condition.
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
ω
i
[-]
η [-]

1 2 3 4 5 6
−0.01
0
0.01
0.02
0.03
0.04
ω i [-]
Δ
r
[-]
State 2 State 3
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
ζ i [-]
η [-]

1 2 3 4 5 6
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
ζ
i
[-]
Δ
r
[-]
State 2 State 3
(a) Natural frequencies (b) Damping ratios
Figure 4.13: Detection ratios and magnitude of relative changes
r
in the (a) natural
frequencies and (b) damping ratios estimated for the test structure in the x-direction
obtained using LSSID approach. The horizontal line corresponds to the threshold =
0:90 used in this study to determine if signicant changes in the modal parameters have
occurred with respect to the parameters from the reference condition.
118
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
ω
i
[-]
η [-]

1 2 3 4 5 6
−0.01
0
0.01
0.02
0.03
0.04
ω i [-]
Δ
r
[-]
State 2 State 3
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
ζ
i
[-]
η [-]

1 2 3 4 5 6
−0.4
−0.3
−0.2
−0.1
0
ζ
i
[-]
Δ
r
[-]
State 2 State 3
(a) Natural frequencies (b) Damping ratios
Figure 4.14: Detection ratios and magnitude of relative changes
r
in the (a) natural
frequencies and (b) damping ratios estimated for the test structure in the x-direction
obtained using ChainID approach. The horizontal line corresponds to the threshold =
0:90 used in this study to determine if signicant changes in the modal parameters have
occurred with respect to the parameters from the reference condition.
Flange Wall
Slabs
Web Wall
Gravity Columns
Slotted Slabs
Post-tensioned Wall
Braces
2.74 m 2.74 m
19.2 m
3.66 m
Braces
Post-tensioned Wall
Flange Wall
Web Wall
Gravity Column
Slotted Slab
4.87 m
8.13 m
3.66 m
Accelerometer
Plan View
Elevation
Figure 4.15: View of the 7-story full-scale building slice structure
fourth earthquake record (EQ4) was referred as State 6. In this structural state, localized
plasticity was developed at the base of the web wall, and a lap-splice failure with a large
119
Table 4.3: Summary of structural state conditions
State Description
State 1 Structure in reference condition
State 2 Structural condition after low-intensity ground motion EQ1 (San Fernando earthquake, 1971)
State 3 Structural condition after medium-intensity ground motion EQ2 (San Fernando earthquake, 1971)
State 4 Structural condition after medium-intensity ground motion EQ3 (Northridge earthquake, 1994)
State 5 Braces between slabs and post-tensioned wall were stiened
State 6 Structural condition after large-intensity ground motion EQ4 (Northridge earthquake, 1994)
split crack in the wall at the base of the second
oor also occurred. In addition to the
seismic tests, the test structure was subjected to band-limited white noise base excitation
tests for each of the structural states/conditions. The input records, in these white-noise
dynamic tests, were 8-minutes long with root-mean-square (RMS) amplitudes of 0.03 g
and 0.25 - 25 Hz bandwidth.
A simultaneous-sampling sensor network, including accelerometers, strain gages, po-
tentiometers, and displacement transducers, was deployed on the test structure. A total
of 106 channels were used to measure lateral, transverse, and vertical accelerations on the

oor slabs and the web wall at a sampling rate of 240 Hz.
4.5.2 Implementation and results
In this study, only the lateral acceleration time-histories recorded by the eight sensors
placed on the foundation and
oor slabs in a location right at the web wall and obtained
during the white-noise base excitation tests, were used to generate reduced-order models
of the test structure and monitor the structural changes caused by the dierent damage
mechanism developed in the test specimen during the seismic tests.
The raw acceleration measurements were initially processed and band-pass ltered
within the 0.25-25 Hz bandwidth. This frequency range corresponds to the modes of the
120
structure excited during the white-noise dynamic test. In order to obtain the velocity and
displacement time-histories that are required to implement two of the approaches under
discussion, the topology-free least-square identication algorithm and the substructuring
system identication approach for chain-like systems, the acceleration time-histories were
numerically integrated. Furthermore, all acceleration, velocity and displacement time-
histories were divided into overlapped segments of 60 seconds, which included more than
100 fundamental periods of the test structure. In total, 25 ensembles of vibration records
were generated for each structural state. The vibration records in each of these ensem-
bles were then used to generate the corresponding reduced-order models, through the
approaches considered in the study, and estimate their associated modal parameters (i.e.,
natural frequencies, damping ratios, and mode shapes). The mode shapes of the rst two
lateral modes estimated based on the reduced-order models of the test structure for one
of the 25 dierent realizations are shown in Figure 4.16.
The probability density functions (pdf ) of the estimated frequencies and damping ra-
tios for the rst two lateral modes of the reduced-order models generated for the reference
structural conguration (i.e., state 1) are shown in Figure 4.17. The mean values of the
natural frequencies for the rst mode estimated using the SRIM, LSSID, and ChainID were
1.71 Hz, 1.72 Hz, and 1.66 Hz, respectively. The estimates of the rst natural frequency
of the reduced-order models built using LSSID exhibited the lowest variability, with a
coecient of variation of 0.65%. The coecients of variation of the estimated frequencies
using SRIM and ChainID were 1.72% and 1.52%. Other published investigations have
been conducted on the identication of modal parameters of the 7-story full-scale struc-
ture tested on the UCSD-NEES shake table, although principally based on deterministic
121
(a) SRIM (b) LS (c) ChainID
Figure 4.16: First two lateral modes of reduced-order model obtained using (a) SRIM, (b)
LSSID, and (c) ChainID.
framework [60]. The mean values of the identied frequencies in this study are similar to
those obtained by Moaveni et al. [60] from the 0.03g RMS white noise base excitation test
data, which ranged between 1.66 Hz and 1.72 Hz.
By comparing the estimated pdf of the frequencies of the second lateral mode displayed
in Figure 4.17(b), it is clear that the model constructed using ChainID was not able to
represent accurately the dynamics of the experimental specimen. It should be noted that
this result was expected because the in corresponding reduced-order model a chain-like
topology was assumed; therefore, the identied second lateral mode corresponds to the
second mode of a shear-type structure. In this case, the mean and coecient of variation
of the estimated frequency were 4.02 Hz and 1.02%. On the other hand, the frequencies
of the second lateral mode identied using SRIM and LSSID, were consistent with the
expected frequencies of structures with a dominant bending beam-type dynamic behavior.
122
The mean values of the second mode frequencies obtained using SRIM and LSSID were
10.09 Hz and 9.54 Hz, respectively. The associated coecients of variation were 1.66%
and 1.40%. These estimated frequencies, however, are slightly lower than the frequencies
reported by Moaveni et al. [60] which were within the 10.7{11.88 Hz range.
The probability density functions of the damping ratios obtained for the reference
conguration are displayed in the bottom row of plots in Figure 4.17. The mean values
of the damping ratios estimated for the rst lateral mode were around 2.14%, 1.93%,
and 1.62% for the SRIM, LSSID, and ChainID approaches, respectively. Even though the
mean damping ratios were similar for the three approaches, the variability in the estimates
diered signicantly. The coecient of variation of the damping ratio for the rst mode
obtained from the model built using LSSID was 4.3%, while the coecients of variation
for the SRIM and ChainID approaches were around 22%. For the second mode, the means
of the identied damping ratios were signicantly dierent; for the SRIM case, the mean
damping ratio was 4.59%; for the LSSID, it was 6.88%; and for the Chain ID, the mean
was 2.09%. Regarding the variability in the damping estimates, it is clear from Figure
4.17(b) that the largest variability was observed from the ChainID model.
The basic second-order statistics, the mean () and standard deviation (), of the
identied natural frequencies and damping ratios for the rst two lateral modes in each
structural conguration/state of the test specimen, are summarized in Figures 4.18 and
4.19. It is clear, from Figures 4.18(a) and 4.18(b), that as the structural damage progresses
by sequentially subjecting the structure to earthquake base motions of increasing intensity,
the mean values of the identied natural frequencies for the damage states decreased with
respect to the reference case (i.e., state 1). It should also be noted that for state 5, due to
123
the stiening of the bracing system, the natural frequencies showed an slightly increment
from the values obtained in the previous structural state. Additionally, it can be observed
that the means of the natural frequencies identied for the rst lateral mode in each
structural conguration were similar despite of the approach used to build the reduced-
order models. For the second lateral mode, similar values for the mean frequencies were
observed in all structural states when the SRIM and LSSID approaches were employed. As
explained before, the signicant dierence in the identied frequencies for the second mode
obtained using the ChainID approach is attributed to the fact that chain-like topology is
assumed for the reduced-order model. The observed relative changes in the mean values
of the rst lateral mode frequencies for States 2{6, were comparable among the three
identication approaches. For example, the relative changes in the mean frequencies for
State 2 with respect to the frequencies in the reference State 1 were 7.1%, 9.5%, and
10.4% for SRIM, LSSID, and ChainID, respectively. For State 6, the relative changes
were, in the same order, 49.6%, 50.9%, and 49.2%. The coecients of variation of the
identied frequencies of the rst mode for all states ranged between 1.1% and 2.8% for
SRIM, between 0.7% and 1.5% for LSSID, and between 1.2% and 1.9% for ChainID.
For the second lateral mode, similar values for the mean frequencies were observed in
all structural states when the SRIM and LSSID approaches were employed. As explained
before, the signicant dierence observed in the frequencies for the second mode identied
using the ChainID approach is attributed to the fact that a chain-like topology for the
structure is assumed in the identication approach. However, the relative changes in the
mean frequencies, were comparable to the changes observed in the frequencies estimated
using SRIM and LSSID. The relative changes in the mean second-mode frequencies for
124
1.55 1.6 1.65 1.7 1.75 1.8 1.85
0
10
20
30
40
f [Hz]

0 0.01 0.02 0.03 0.04 0.05
0
200
400
600
ζ
SRIM LSSID ChainID
4 5 6 7 8 9 10 11
0
2
4
6
8
f [Hz]

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
0
20
40
60
80
ζ
SRIM LSSID ChainID
(a) First lateral mode (b) Second lateral mode
Figure 4.17: Probability density functions for the identied natural frequency (f) and
damping ratio () of the rst two lateral modes.
Table 4.4: Summary of mean natural frequencies for the rst two lateral modes of the
structure identied from the reduced-order models developed using SRIM, LSSID, and
ChainID approaches.
f
1
[Hz] f
2
[Hz]
State SRIM LS ChainID SRIM LS ChainID
State 1 1.71 1.72 1.66 10.09 9.54 4.02
State 2 1.59 1.55 1.48 9.26 8.84 3.58
State 3 1.42 1.27 1.23 8.32 7.50 2.95
State 4 1.24 1.13 1.11 7.50 6.47 2.68
State 5 1.23 1.22 1.19 7.21 6.80 2.88
State 6 0.86 0.84 0.84 4.75 4.29 2.05
State 2 were 8.2% and 7.4% for SRIM and LSSID, while a change of 11.0% was observed
in the frequency estimated using ChainID. Similarly, the changes in the frequencies for
State 6 with respect to State 1 were 52.9%, 55.0%, and 48.9%.
125
1 2 3 4 5 6
0
0.5
1
1.5
2
State
μ
f
[Hz]

1 2 3 4 5 6
0
0.01
0.02
0.03
State
δf
SRIM LSSID ChainID
1 2 3 4 5 6
0
2
4
6
8
10
State
μf [Hz]

1 2 3 4 5 6
0
0.02
0.04
0.06
State
δf
SRIM LSSID ChainID
(a) First lateral mode (b) Second lateral mode
Figure 4.18: Mean () and coecient of variation () of the identied frequencies for the
rst two lateral modes
1 2 3 4 5 6
0
0.01
0.02
0.03
0.04
State
μ
ζ

1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
State
δ
ζ
SRIM LSSID ChainID
1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
State
μ
ζ

1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
State
δ
ζ
SRIM LSSID ChainID
(a) First lateral mode (b) Second lateral mode
Figure 4.19: Mean () and coecient of variation () of the identied damping rations for
the rst two lateral modes
126
2 3 4 5 6
0
0.2
0.4
0.6
0.8
State
| Δμ/μ
r
|

2 3 4 5 6
0
20
40
60
80
State
| Δμ/σ
r
| SRIM LSSID ChainID
2 3 4 5 6
0
0.2
0.4
0.6
0.8
State
| Δμ/μ
r
|

2 3 4 5 6
0
10
20
30
40
50
State
| Δμ/σ
r
| SRIM LSSID ChainID
(a) First lateral mode (b) Second lateral mode
Figure 4.20: Change indicesj=
r
j andj=
r
j for the identied frequencies for the
rst two lateral modes
4.6 Summary and Conclusions
In this study, two dierent sets of features (i.e., modal parameters and stiness-like pa-
rameters) were used to evaluate the eectiveness of employing reduced-order models built
from experimental data utilizing three dierent system identication techniques, to detect
and locate physical changes in a mid- and a large-scale test structures was evaluated. One
of the important conclusions in this exploratory study is that structural health monitoring
methodologies required the implementation of statistical or stochastic framework in order
to handle and cope with the uncertainties associated not only with the environmental or
operational conditions, but also with the modeling, measurement, and data processing.
It is important to note that, the methodologies presented in this study have demon-
strated the capabilities for detecting and locating structural changes in a six-story steel-
frame and a seven-story full-scale reinforced-concrete structures, as long as the eects
127
of operational and environmental conditions, and underlying \damage" mechanisms are
re
ected in the structures' dynamic response.
In this experimental study, input-output data from band-limited white-noise base ex-
citation tests on a six-story single-bay steel frame structure was used to estimate reduced-
order models of the testbed structure in dierent structural conditions by implement-
ing three dierent system identication approaches: a system realization algorithm using
information matrices (SRIM), a general time-domain least-squares identication method
(LSSID), and a non-parametric chain-like system identication approach (ChainID). Vari-
ations in the estimated mass-normalized stiness-like parameters of the reduced-order
models were then used to detect the presence, and infer the location, of the actual struc-
tural changes made in the test structure. The results of this study showed that the statis-
tically signicant changes identied in the stiness-like parameters of the reduced-order
models built using the SRIM and ChainID could be correlated to the presence and loca-
tion of the actual physical changes made to the testbed structure, even in the presence of
modeling, measurement, and data processing errors using reduced-order representations.
However, in the case of the LSSID approach, the identied changes in the reduced-order
models could not be reliably correlated with the actual damage location in the structure.
The experimental results showed that the variability in the natural frequencies, related
solely to the input-output system identication approaches under consideration (assuming
no variations in the environmental and operational conditions during the duration of the
base motion tests), was on the average a 2%. It is also important to highlight that
even though the test structure had a dominant bending beam-type dynamic behavior,
the relative changes in the natural frequencies obtained by implementing the chain-like
128
system identication approach, which basically assumes that the structure has a shear-type
building behavior, were comparable to relative changes estimated using the topology-free
system identication approaches (i.e., SRIM and LSSID).
129
Chapter 5
Structural health monitoring and characterization of a
re-congurable structure based on a substructuring
identication approach
5.1 Introduction
In order to study and resolve some of the problems that hinder the development of reli-
able general-purpose computer simulation programs (which are capable of re
ecting pre-
cisely the dynamic behavior of distributed, nonlinear systems spanning the range from
large joint-dominated space structures, to intricate micro-electro-mechanical systems), a
comprehensive analytical and experimental study was conducted within this project to
investigate an important subset of the challenging issues discussed above.
By building a sophisticated, re-congurable test apparatus for investigating generic
types of complex nonlinear elements widely encountered in the aerospace, civil, and me-
chanical applications, this study utilized the test apparatus as a major element of a global
computer-controlled loop to automatically conduct a large number of physical experiments
130
to collect a statistically signicant ensemble of measurements from the stochastic test el-
ement whose randomized properties were automatically adjusted (for each test loop) by
using \smart" (adaptive) nonlinear components. The extensive statistical measurements
of the perturbed nonlinear system parameters as well as the corresponding physical system
response time history, were stored in an extensive database.
Subsequently, for each unique test, a corresponding mathematical model was identied
to mach the observed motions, and the uncertain parameters of the identied models were
then used to perform uncertainty propagation studies to establish probabilistic bounds on
the range of validity of response predictions based on the identied mathematical model(s).
Performing these experiments lead to a better understanding of the physics of the under-
lying phenomena and the associated uncertainties involving the physical parameters that
dene the nonlinear characteristic associated with the test article. By using a powerful
model-free approach to obtain computationally ecient reduced-order models, a general
framework was developed for the probabilistic representation and propagation of mea-
sured uncertainties in the stochastic nonlinear test articles, their related nonparametric
nonlinear model, and the corresponding probabilistic time-history response of the physical
system.
5.2 Re-congurable testbed structure
The re-congurable test structure is a modular multi-bay structure with removable el-
ements that was designed and built to perform studies on the detection, location, and
quantication of structural changes that physical structural elements can undergo due
to a variety of damage mechanisms. The fabricated laboratory structure consists of 6
131
Figure 5.1: Assembled modular re-congurable testbed structure.
modular sections each with the dimensions of 4 6 8 inches as shown in Figure 5.1.
As can be seen, each section has four supporting beam assemblies. The cross section of
each beam is 0.5 inch by 0.125 inch with a welded pad at each end. Each pad is bolted
to the adjacent section with four screws. At the interface between sections are one or
two steel plates depending on the desired mass distribution. This design allows to easily
modify the structure by removing or replacing any beam element without disassembling
the structure. Each beam assembly consists of a 0:5 0:125 7:75 inch steel rod with two
steel attachment pads. Furthermore, each pad is attached to the adjacent section with
the use of four screws. Figure 5.2 shows a typical module of the test structure.
5.2.1 Nonlinear gap element
To introduce nonlinear eects in the dynamics of the re-congurable testbed structure,
adjustable gap elements were added to specic modular sections in the structure. A rst
type of adjustable nonlinear gap element consisted of two hefty aluminum brackets with
an adjustment bolt between the two. A close up of the gap element is shown in Figure
132
Figure 5.2: Typical modular section (vertical view) of the test structure.
Figure 5.3: Modular section with nonlinear gap element (vertical view).
5.3. To adjust the gap, simply loosen the nut and adjust the length protrusion of the bolt.
Also shown in the close-up is the impact plate upon which the bolt head hits. This steel
plate can easily be replaced should it get worn from the impacts. Alternatively, the steel
impact plate can be replaced with rubber of various densities to vary the impact force
prole. Lastly, the brackets were designed with a large overlap and sucient distance
between the brackets to allow for a linear spring or other type of passage element to be
added should the need arise.
With the lessons learned from the manually adjustable gap nonlinear element, a com-
puter controlled programmable variable gap element was designed and built. The variable
gap element is designed around an RC-type servo drive. The servo motor drives a pair
133
Figure 5.4: Side view of mounted programmable nonlinear element
of lightweight delrin gears. Although the gears could be eliminated, they help isolate the
servo from any impact forces being generated. The driven gear is attached to the impact
shaft which is attached to a rotary impact plate. The impact shaft is supported by a
bearing/bearing support that is held rigidly connected to one of the section plates. The
rotary impact plate is a variable thickness disk. As the disk rotates, the distance between
the disk and the impact screw changes. The nominal gap distance is set by the position of
the impact screw and the gap variation is controlled by the rotary position of the rotary
impact plate. The impact screw is threaded into the impact screw support which is rigidly
attached to the opposing section plate. Impact is transmitted through the spacer to the
servo support with the goal of minimizing the force of the impact load on the servo. The
solid model of the variable gap element installed in one of the modular sections is shown
in Figure 5.4.
In operation, the servo is controlled with a PWM signal such that the duty cycle
determines the position of the servo shaft. For example, with the PWM signal having
a frequency of 500 Hz, a 10% duty cycle would produce a servo shaft position of -150
degrees, a duty cycle of 50% would produce a shaft position of 0 degrees, and duty cycle
134
Figure 5.5: Modular section with nonlinear gap element (vertical view).
of 90% would produce a shaft position of +150 degrees. Since the drive gear and driven
gear have the same pitch diameter, the rotation range of the servo shaft is equal to the
rotation of the rotary impact plate. The rotary impact plate was machined such that the
300 degree rotation corresponds to the gap variation of approximately 0.060 inches. This
value was selected based on studies performed with the xed gap element. Figure 5.5
shows a closeup view of the programmable gap element installed in structure.
5.2.2 Electromagnetic exciter
It was decided early in the research project that a free-standing force exciter capable of
generating a random force prole was needed. The exciter must be lightweight so as not to
adversely aect the dynamics of the structure and be able to run for long periods of time
without overheating. The force output must be broadband so as to excite the rst two
modes of the structure and be capable of fully-automatic operation for repetitive testing,
using the same random force prole. After searching for a commercially available force
exciter, it was determined that none existed that met the research project requirements.
The biggest drawbacks of the commercially available units was the weight. Therefore,
135
Figure 5.6: Solid model of the custom-built electromagnetic exciter.
designing and constructing a custom force exciter for the testbed was accomplished. Figure
5.6 shows the solid model of the custom-built electromagnetic exciter.
The exciter consists of ve independently wound coils on one ABS bobbin. Located
within the coils is a 1 inch diameter rare earth neodymium magnet attached to precision
bearing rod which is supported at both ends of the exciter with linear ball bushings. The
magnet is free to traverse the inside of the exciter with approximately 2 inches of travel.
With the use of the ball bushing to support the precision bearing rod, the friction is
negligible, allowing the magnet to
oat freely in the horizontal direction. The key to the
success of the exciter design is the ability to accurately control each of the coil currents
based on the controller demands. Each of the coils has 200 turns of 26 gauge magnet wire.
The coils are separated by a small air gap to provide convective cooling during long test
runs. With the magnet moving within the coil bobbin, a forced convection is generated.
Each coil is capable of 3 amps of continuous current and short duration current bursts
of 5 amps. The distance between the coils is optimized such that one of the neodymium
magnet faces is always within approximately 0.25 inches of the maximum
ux of one of the
coils providing a smoother generated force. With this design, the electromagnetic exciter
136
Figure 5.7: Detail of the electromagnetic exciter showing the reaction load cell and the
linear support bearing.
is able to generate broad-band random forces with a peak level of 6 lb with a frequency
response up to 25 Hz.
To measure the actual force being generated by the coils on the rare earth magnet,
the coil supports are attached to an aluminum plate, which is attached to the mounting
plate in the structure through a small linear bearing. The linear bearing decouples the
electromagnetic exciter from the mounting plate in the force excitation direction. The
small force gauge is then used to tie the two plates together. The force gauge serves two
purposes: It is used during the testing to help verify the actual force being generated and
second, which is more important, is used to calibrate the exciter. A close photo of the
small load cell is shown in Figure 5.7. Also shown in the Figure 5.7 is the linear bearing
located between the coil support plate and the structure attachment plate. The use of the
linear bearing forces all of the reaction load to pass through the small load cell.
The force imposed on the rare earth magnet is highly-dependent on the distance of
the magnet face to the location of the
ux density of the energized magnet. Therefore,
to generate a desired force, the position of the rare earth magnet relative to the coils,
137
and the current required to be supplied to the appropriate coils must be known. To
accomplish this, the position of the magnet is measured in real time with the use of an
optical encoder. The encoder is mounted on the base coil mounts and turned by a string
acting on the encoder shaft pulley. Using an encoder with a resolution of 2000 CPR in
quadrature mode and 1 inch shaft pulley, the position resolution of the rare earth magnet
is approximately 0.001 inches. To generate a desired force, the appropriate coil current
has to be turn on in the right direction and magnitude. The current direction is controlled
with the use of an H-bridge. The magnitude of the current is controlled with a PWM of the
H-bridge control circuit. For this design, ve separate H-bridges are needed. The optical
encoder is read with the use a Labjack U6 data acquisition unit which has quadrature
counter interface. The PWM drive signals are generated using separate Labjack U6 unit.
An electrical diagram showing the interconnection of the exciter, H-bridges and Labjack
units is provided in Figure 5.8.
The calibration for the exciter was fundamental to guarantee the performance require-
ments. The calibration procedure consists of measuring the force output of each of the
coils as a function of the position of the neodymium magnet. It is important to be precise
in positioning the magnet and to have a ne resolution requiring thousands of calibration
points for each coil. For the precision needed and the sheer number of data points needed,
it would not be possible to move the magnet manually; therefore, a calibration test appa-
ratus was designed and subsequently built. The custom-made calibrator was capable of
moving the precision shaft attached to the neodynium magnet in increments of less than
0.001 inches. The force gauge located between the coils attachment plate and the mount-
ing plate is then used to measure the force being generated by the coils on the rare earth
138

!"# $ %

"

!"&#'
!( )
!
" * +
, - +

. " Figure 5.8: Electrical diagram for the operation of the electromagnetic exciter.
magnet. The experimental measurements from the calibration procedure can be seen in
Figure 5.9. The calibration curves were generated in both forward and reverse current
directions. The calibration curves that were generated in the forward current direction,
that is in a positive current
ow in the coils, are shown in Figure 5.9(a), and those that
were generated in the reverse direction, that is a negative current in the coils are shown in
Figure 5.9(b). The calibration for all of the coils in both directions resulted in over 11000
data points. For implementation, the data was reduced by curve tting the data using
a seventh-order polynomial regression t, for each coil and for both current directions.
Comparing the calibration shown in the two gures gives some interesting insight into the
magnetic elds in the exciter. Looking at coil 1, which is the second coil from the left,
shows the force for the reverse current is negative of the forward current direction. The
same applies to the coils 2 and 3. This makes sense, as the magnetic eld is simply reversed
139
0.5 1 1.5 2
−0.4
−0.2
0
0.2
0.4
Displacement [in]
Forward Current
Normalized Force [−]

Coil 1
Coil 2
Coil 3
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Displacement [in]
Reverse Current
Normalized Force [−]

Coil 1
Coil 2
Coil 3
(a) Forward current direction (b) Reverse current direction
Figure 5.9: Experimental measurements of normalized force in the neodymium magnet
generated for (a) forward current direction and (b) reverse current direction.
when the current is reversed for a given magnet position. This indicates that there is very
little stiction in the exciter bearings. Comparing coil 1 force output to coil 3 force output
is also interesting. The output of the two coils are negative and are mirrors of each other,
for both the forward and reverse currents. This is due to the symmetry of the coils and
indicates that the force output of the coils are well balanced. By appropriate selection
of the appropriate coil, the current direction and the current magnitude, controlled using
the PWM, a very smooth excitation can be generated. With the calibration complete, the
exciter was installed on the structure.
The exciter is mounted on the end of the structure such that the neodymium magnet
motion is along the horizontal neutral axis of the structure. The mounted electromagnetic
exciter is shown in Figure 5.10. This will primarily excite the horizontal modes of the
structure. However, due to the non-symmetric mass distribution of the exciter and the
addition of the nonlinear gap and instrumentation, a torsion mode may be excited. The
system uses two computers, each running the LabVIEW software. One computer is dedi-
cated for the exciter control, with the second computer dedicated to the data acquisition.
Attached to the control computer are two Labjack U6 USB multifunction DAQ units. One
Labjack is used to control the ve H- bridges, one for each coil. The H-bridges control
140
signal consists of a current forward signal and a PWM signal to control the magnitude
of the current. The second Labjack is used to read the optical encoder. The Labjack
U6 has a built-in quadrature decoder to read the optical encoder. Hence, the control
computer knows the position of the neodymium magnet in real time and, with the use
of the LabVIEW software, can set the control of the current in each of the coils based
on a predened force prole. At the start of the test, the predened force prole is read
into a memory array after which the rst iteration begins. The force prole is typically
digitized with a t of 0.0005 second, which results in a force update rate of 2000 Hz. At
each iteration, the program reads the position of the neodymium magnet and determines
the appropriate coil to energize. The desired force is read from the array for the current
iteration and then sets the current direction control signal and the PWM signal based on
the calibration curves. Both the current direction control signal and the PWM signal are
sent to the appropriate H-bridge for the desired coil. The iterations are continued until
all of the values in the force array have been exhausted. The data acquisition computer,
which is also running LabVIEW, is equipped with a National Instruments NI SCSI data
acquisition hardware. In addition to recording the force cell output, all the structure's
accelerometer data is recorded. All the results and analysis presented in this report are
based on the data recording with the data acquisition computer.
141
Figure 5.10: Custom-built electromagnetic exciter installed on the structure
5.3 Identication of reduced-order nonlinear models
5.3.1 Overview of substructuring approach for chain-like systems (ChainID)
Consider an MDOF chain-like system, consisting of a series of lumped masses m
i
inter-
connected by n arbitrary unknown nonlinear elements G
(i)
, subjected to a base motion
x
0
, and/or directly applied forces F
i
. The nonlinear elements' restoring forces are as-
sumed to depend on the relative displacement and velocity across the terminals of each
element, in addition to a set of specic parameters p that characterize the various types
of nonlinearities.
142
The dierential equations of motion for the system under discussion, can be written
as [31, 32, 54]
G
(n)
(z
n
; _ z
n
;p
n
) =
F
n
m
n
x
n
G
(i)
(z
i
; _ z
i
;p
i
) =
F
i
m
i
x
i
+
m
i+1
m
i
G
(i+1)
(z
i+1
; _ z
i+1
;p
i+1
) (5.1)
for i =n 1;n 2;:::; 1
where G
(i)
(z
i
; _ z
i
;p
i
) is the mass-normalized restoring force function of the nonlinear ele-
mentG
(i)
, x
i
is the absolute acceleration of the massm
i
, withz
i
and _ z
i
being the relative
displacements and velocities between two consecutive masses. These relative motion vari-
ables can be obtained from the absolute state variables of the masses and the moving
support. Equation (5.1) can be rewritten in a more simplied form as
G
(i)
(z
i
; _ z
i
;p
i
) =
n
X
j=i
f
j
m
j

n
X
j=i
m
ij
x
j
(5.2)
where m
ij
= m
j
=m
i
is the ratio between the lumped masses m
j
and m
i
. Within the
context of this method, the absolute accelerations x
i
are assumed to be available from
measurements, as well as the applied forcesF
i
, the base excitationx
0
, and the magnitude
of the lumped masses m
i
.
After obtaining all the restoring force time histories, it is possible to generate a non-
parametric representation for each nonlinear element, in terms of a truncated doubly-
indexed series expansion in a suitable basis, that approximates the real restoring force
143
function [31, 32, 53, 57]. The approximating representation
^
G
(i)
(z
i
; _ z
i
) for the obtained
restoring forces, in an orthogonal polynomial basis, is given by the following expression:
G
(i)
(z
i
; _ z
i
;p)
^
G
(i)
(z
i
; _ z
i
) =
qmax
X
q=0
rmax
X
r=0
C
(i)
qr
T
q
(z
0
i
)T
r
( _ z
0
i
) (5.3)
where C
(i)
qr
are the Chebyshev series coecients, T
k
() is the Chebyshev polynomial of
order k, and z
0
i
, _ z
0
i
are the normalized relative state variables. Subsequently, each of the
estimated restoring forces can be expressed as a power series of the form
^
G
(i)
(z
i
; _ z
i
) =
qmax
X
q=0
rmax
X
r=0
a
(i)
qr
z
q
i
_ z
r
i
(5.4)
where a
(i)
qr
are constant coecients, and z
i
, _ z
i
are the relative state variables.
Since the mass-normalized restoring force function for linear elements can be expressed
as:
G
(i)
(z
i
; _ z
i
) =
k
i
m
i
z
i
+
c
i
m
i
_ z
i
=a
(i)
10
z
i
+a
(i)
01
_ z
i
(5.5)
the global modal parameters of the structure can be determined through the eigen-
decomposition of the matrixA, which is given as
A =
2
6
6
4
0 I
M
1
K M
1
C
3
7
7
5
=
1
(5.6)
where and are the complex eigenvector and complex eigenvalue matrices, respectively.
The matrices M
1
K and M
1
C can be obtain using the identied mass-normalized
144
stiness-like coecients a
(i)
10
and mass-normalized damping-like coecients a
(i)
01
, by the
following expressions,
M
1
K =
2
6
6
6
6
6
6
6
6
6
6
4
a
(1)
10
+a
(2)
10
a
(2)
10
0 0
a
(2)
10
a
(2)
10
+a
(3)
10
a
(3)
10
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 a
(6)
10
a
(6)
10
3
7
7
7
7
7
7
7
7
7
7
5
(5.7)
M
1
C =
2
6
6
6
6
6
6
6
6
6
6
4
a
(1)
01
+a
(2)
01
a
(2)
01
0 0
a
(2)
01
a
(2)
01
+a
(3)
01
a
(3)
01
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 a
(6)
01
a
(6)
01
3
7
7
7
7
7
7
7
7
7
7
5
(5.8)
The complex eigenvalues
i
in matrix are mathematically related to the natural
frequencies !
i
and damping rations
i
of the system by
!
i
=
p
Re(
i
)
2
+ Im(
i
)
2

i
= Re(
i
)=!
i
(5.9)
5.3.2 Data processing
Nine dierent structural states or congurations of the testbed structure were considered
for this study. The rst structural state (state#1) corresponded to the baseline or ref-
erence condition where the dynamic response of the test structure was within the linear
range. In the remaining states structural modication were made to the reference struc-
ture to introduce both linear and nonlinear changes in the system, and therefore, in the
145
dynamic response of the testbed structure. In states 2, one of the beam elements in the
fth structural module (module 5) was replaced by an element with a 40% reduction in
the cross section. Similar changes were introduced in state#3, where two of the four ele-
ments in the fth module were replaced by elements with 40% reduced cross-section. For
state#4, one of the bottom beam elements of module 5 was completely removed from the
structure. Another type of structural change made to the reference testbed structure was
the loosening of the bolted connections between structural modules. In state#5, for ex-
ample, the bolts in one of the connections between the modules 4 and 5 were loosened. In
order to introduce nonlinear structural changes into the testbed structure, two adjustable
gap elements. The level of the nonlinearity was varied by reducing or increasing the size of
the gap within the adjustable nonlinear element. The structural conguration in state#6
was set up by adding to the state#5 conguration two adjustable nonlinear elements to
modules 2 and 4. In both nonlinear elements, the gap was set to 0.1 mm. For state#7,
on the other hand, the introduced structural changes consisted only in the nonlinear gap
elements in modules 2 and 4, both with still the same 0.1-mm gap. For states 8 and 9,
just the nonlinear element in module 2 was used in these two structural congurations.
The gap size was set to 0.1 mm and 0.4 mm for state#8 and 9, respectively. Table 5.1
summarizes the dierent structural congurations of the testbed structure used herein.
For each of the structural congurations, the structure was subjected to a sequence of
forty (40) forced vibration tests under 30-second long band-limited white noise excitation.
The applied loads had a frequency bandwidth of 3-35 Hz, and a root-mean square value
(RMS) of 1 lb. A total of six variable-capacitance uniaxial accelerometers, with a frequency
bandwidth of DC-500 Hz, were installed to measure the lateral dynamic response in each
146
Table 5.1: Summary of structural state conditions
State Description
State#1 Reference condition
State#2 One (1) element with reduced cross-section in module 5
State#3 Two (2) elements with reduced cross-section in module 5
State#4 One (1) element removed from module 5
State#5 One (1) loose connection between module 4 and module 5
State#6 One (1) loose connection between module 4 and module 5
Two (2) nonlinear elements with a 0.1-mm gap in module 2 and module 4
State#7 Two (2) nonlinear elements with a 0.1-mm gap in module 2 and module 4
State#8 One (1) nonlinear element with a 0.1-mm gap in module 2
State#9 One (1) nonlinear element with a 0.4-mm gap in module 2
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 5 Sensor 6
Lateral vibration
Figure 5.11: Location of the accelerometers deployed on testbed structure.
of the six structural modules composing the testbed structure. The accelerometers were
attached to each of the carbon-steel plates, along the horizontal center line of the structure.
The acceleration measurements were acquired at a sampling frequency of 500 Hz. The
approximate location of the accelerometers, as well as the sensor and module numbering
used throughout this section of the report are displayed in Figure 5.11. In order to
obtain the displacement and velocity time-histories at each of the measurement stations,
the acceleration records were windowed, detrended, band-pass ltered and numerically
integrated. Typical time-histories of measured accelerations, and computed velocities and
displacements in module 2 from one of the tests performed on the testbed structure in the
reference conguration (state#1), are displayed on Figure 5.12(a).
It should be also emphasized that, since this proposed approach is entirely data-driven,
the number of degrees-of-freedom in the reduced-order model will be determined by the
147
number of available sensors deployed on the structure. In this case for example the testbed
structure was considered as a 6DOF chain-like system. In addition to the acceleration
time-histories recorded by the six accelerometers and the numerically integrated velocities
and displacements, the lumped-mass ratios m
ij
were also assumed to be available. Since
the system's mass is approximately uniformly distributed throughout the structure, the
mass ratios m
ij
were considered equal to one. For the forced dynamic tests performed
in the structure, where the load was applied at the tip of the structure, the simplied
equations of motion (Equation.5.2) can be rewritten as:
G
(i)
(z
i
; _ z
i
;p
i
) =f
6

6
X
j=i
x
j
i = 1;:::; 6
(5.10)
The structure's equations of motion given in Equation 5.10 were then employed to ini-
tially determine the experimental mass-normalized restoring forcesG
(i)
(z
i
; _ z
i
;p
i
). Typical
time-histories of the restoring force G
(i)
(z
i
; _ z
i
;p
i
), relative displacements z
i
, and relative
velocities _ z
i
for module 2 of the testbed structure (i.e., interconnecting elements in the
chain-like system), obtained from one of the dynamic tests, are shown in Figure 5.12(b).
It should be noted that the order of the relative motion observed during the performed
dynamic tests was in the order of tenths of a millimeter.
Once the mass-normalized restoring forces, and the relative displacements and ve-
locities had been computed, the proposed time-domain identication technique is then
applied to build the associated non-parametric reduced-order models for each element in
148
5 5.5 6 6.5 7 7.5 8
−10
−5
0
5
10
Time [s]
¨ x
2
[m/s
2
]
5 5.5 6 6.5 7 7.5 8
−0.1
−0.05
0
0.05
0.1
Time [s]
˙ x
2
[m/s]
5 5.5 6 6.5 7 7.5 8
−2
−1
0
1
2
x 10
−3
Time [s]
x
2
[m]
5 5.5 6 6.5 7 7.5 8
−1
−0.5
0
0.5
1
x 10
−3
Time [s]
z
2
[m]
5 5.5 6 6.5 7 7.5 8
−0.04
−0.02
0
0.02
0.04
Time [s]
˙ z
2
[m/s]
5 5.5 6 6.5 7 7.5 8
−20
−10
0
10
20
Time [s]
G
(2)
[m]
(a) (b)
Figure 5.12: Sample of (a) acceleration, velocity, and displacement time-histories, and (b)
relative displacement, relative velocity, and restoring force time-histories from the linear
system in the reference structural conguration (state#1).
the 6DOF chain-like system, by determining the corresponding mass-normalized restoring
force coecients a
(i)
qr
.
5.3.3 Identication of linear system
The testbed structure in the reference structural conguration (state#1) was used to
illustrate the results of implementing the proposed methodology for building reduced-
order models of a linear system. It should be noted that for the purposes of this study, the
system was not assumed linear. The restoring force identication approach was carried
out using Chebyshev polynomials of fth-order in both normalized variables z
0
and _ z
0
for
all structural modules in the testbed. A relative-contribution analysis of the identiedC
(i)
qr
for all six structural modules indicated that the linear term associated with the relative
149
displacements in the non-parametric representation (i.e., coecient C
(i)
10
), had the most
signicant contributions to the restoring force G
(i)
of each structural section, while the
eects of the nonlinear terms were negligible.
For the sake of brevity, only the identication results obtained for the structural module
2 will be presented and discussed in this section. The Chebyshev coecients C
(2)
qr
of the
non-parametric representation for the structural module 2 obtained from one of the forty
dierent tests are summarized graphically in Figure 5.13(a). It can be clearly seen that
the coecient C
(2)
10
was the most signicant term in the non-parametric representation of
the restoring force. A plot of the reconstructed mass-normalized restoring force surface
over the phase space obtained using the non-parametric representation of the restoring
forces (i.e., coecientsa
(2)
qr
and corresponding sequence of power-series in the relative state
variables z
2
and _ z
2
) in the structural module 2, is also displayed in Figure 5.13(b). In
addition to the reconstructed restoring force surface in Figure 5.13(b), the actual restoring
force measurements were also displayed in the plot as a point cloud. It can be clearly
seen also in 5.13(b), as is expected for linear elements, an almost planar restoring force
surface despite the fact that a fth-order expansion on both state variables (i.e., relative
displacement and relative velocity) was used to characterize the dynamics of the structural
modules.
To qualitatively assess how well the reduced-order model built using the identied
restoring force coecients was able characterized the dynamics of the structural module
2, the phase plots of the restoring force versus the relative displacement and relative ve-
locity, as well as the corresponding restoring force time-history are shown and compared
in Figure 5.14. As it can be seen in Figure 5.14(a), where the phase plot of the measured
150
0 1 2 3 4 5
0
1
2
3
4
5
r
q
C
(2)
qr

0
2
4
6
8
10
12
14
16
18
−5
0
5
−0.02
0
0.02
−30
−20
−10
0
10
20
30
z
2
[10
−4
m]
˙ z
2
[m/s]
G
(2)
(z
2
, ˙ z
2
) [m/s
2
]
(a) Restoring force coecients) (b) Restoring force surface
Figure 5.13: Identied restoring force coecients and corresponding estimated restoring
force surface for module 2 for structure in linear conguration (state#1)
restoring force G
(2)
(solid blue line) and relative displacements z
2
is compared to the re-
constructed restoring force (dashed red line), the estimated reduced-order model captured
the dominant linear characteristics in the dynamics of the structural module 2. Similarly,
the phase plots of the restoring forces and relative velocity are compared in Figure 5.14(b).
Additionally, a 3-second segment of the measured and reconstructed restoring force time-
histories are displayed in Figure 5.14(c). Notice that the built reduced-order model could
replicate the dominant dynamic features in the time response exhibit by the structural
module.
In total, forty reduced-order models were built using the non-parametric representation
of the restoring forces for each of the six structural modules in the testbed structure. In
other words, forty realizations of each of the mass-normalized restoring force coecients
a
(i)
qr
; i = 1;:::; 6, were obtained from the system identication procedure. In order to
describe and summarize the variability observed in the identied a
(i)
qr
coecients, the
probability density function (pdf) for each of the coecients was estimated from the
151
−8 −6 −4 −2 0 2 4 6 8
−30
−20
−10
0
10
20
30
z
2
[10
−4
m]
G
(2)
(z
2
, ˙ z
2
) [m/s
2
]
−0.05 0 0.05
−30
−20
−10
0
10
20
30
˙ z 2 [m/s]
G
(2)
(z 2, ˙ z 2) [m/s
2
]
(a) Phase plot of restoring force versus (b) Phase plot of restoring force versus
relative displacement relative velocity
5 5.5 6 6.5 7 7.5 8
−20
−15
−10
−5
0
5
10
15
20
Time [s]
G
(2)
[m/s
2
]

Measured
Reconstructed
(c) Restoring force time-history
Figure 5.14: Comparison of measured and estimated restoring forces for linear system
(state#1)
coecients' realizations using kernel density estimation. The estimated probability density
functions for a subset of the coecients identied for module 2 are shown in Figure 5.15.
The coecientsa
(2)
10
,a
(2)
20
, anda
(2)
30
, can be interpreted respectively as the mass-normalized
linear, quadratic, and cubic stiness-like terms in the reduced-order model, while the
a
(2)
01
coecient would correspond to the linear damping-like term. From the pdf of the
stiness-like coecients, it is clear that the linear stiness-like coecient a
(2)
10
has the
smallest variability among observations or realizations. The small variability observed in
the coecient a
(2)
10
could be explained by the fact that the largest contribution to the
restoring force G
(2)
came from the term in the restoring force expansion associated with
152
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
0
0.5
1
1.5
2
2.5
˜ a
(2)
10
[-]
pdf [-]
−60 −40 −20 0 20 40 60 80
0
0.005
0.01
0.015
0.02
0.025
0.03
˜ a
(2)
01
[-]
pdf [-]
−30 −20 −10 0 10 20 30 40 50
0
0.01
0.02
0.03
0.04
0.05
˜ a
(2)
20
[-]
pdf [-]
−20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
0.05
˜ a
(2)
30
[-]
pdf [-]
Figure 5.15: Probability density functions of the mass-normalized restoring force coe-
cientsa
(2)
10
,a
(2)
01
,a
(2)
20
, anda
(2)
30
from the reduced-order representation of structural module 2
built for the testbed structure in a conguration with a linear dynamic response (state#1).
the linear stiness, and therefore, the identication of the stiness-like coecient would
be more accurate.
In addition to the local representation of the dynamics of each module in the testbed
structure, the global characteristics (i.e., modal parameters) of the system can also be
estimated using the identied restoring force coecients, as previously explained. Due to
the random nature of the restoring force coecientsa
(i)
qr
, the estimated modal parameters
will be also random quantities. The probability density functions of the rst three nat-
ural frequencies estimated using the linear stiness-like and damping-like restoring force
coecients are shown in Figure 5.16. The mean values of the estimated frequencies for
the rst three lateral modes were !
1
= 5:77 Hz, !
2
= 15:65 Hz, and !
3
= 24:60 Hz. The
corresponding mode shapes are displayed in Figure 5.17. The estimated modal properties
were also compared to the parameters identied by implementing the system realization
153
using information matrix (SRIM) [38], which is a global identication technique, and the
modal parameters computed from the nite element model of the test structure. The
mean value of the frequencies, as well as the modal assurance criterion (MAC) value for
the mode shapes, are summarized in Table 5.2. The mean natural frequencies for the rst
two modes were comparable to the frequencies obtained from the FE analysis. However,
a slightly dierence was observed between the third mode natural frequency estimated
using SRIM and the frequencies from the FEM and ChainID approach. From the ob-
tained MAC values, it can be seen that the rst two mode shapes from the FEA, SRIM,
and ChainID were similar. The experimental mode shape for the third mode estimated
using SRIM and ChainID were also similar. However, a low MAC value was obtained for
the third mode shape when compared against the FEM mode shape. Even though the
damping was also identied experimentally, it was not reported since, as expected for a
very lightly-damped structure, the estimated damping was almost negligible. In average,
the estimated damping for the rst mode was around 0.5%.
Table 5.2: Comparison of numerical and experimental modal parameters
! [Hz] MAC [-]
Mode FEM SRIM ChainID FEM/SRIM FEM/ChainID ChainID/SRIM
1 5.36 5.60 5.77 0.997 0.994 0.999
2 15.82 17.42 15.65 0.826 0.881 0.964
3 25.84 29.58 24.60 0.676 0.582 0.887
5.3.4 Identication of nonlinear system
The identication results for the testbed structure congured with two nonlinear gap el-
ements installed respectively in module 2 and module 4 (state#7), are discussed in this
154
5.2 5.4 5.6 5.8 6 6.2
0
0.5
1
1.5
2
2.5
3
3.5
ω 1 [Hz]
pdf [Hz
−1
]
14.5 15 15.5 16 16.5
0
0.5
1
1.5
ω 2 [Hz]
pdf [Hz
−1
]
23 23.5 24 24.5 25 25.5 26 26.5
0
0.2
0.4
0.6
0.8
ω
3
[Hz]
pdf [Hz
−1
]
Figure 5.16: Probability density function of the rst three natural frequencies of the
testbed structure in the linear reference conguration (state#1) estimated using the mass-
normalized stiness-like and damping-like coecientsa
(i)
10
anda
(i)
01
from built reduced-order
models.
section to illustrate the capabilities of the proposed sub-structuring approach to charac-
terize the dynamics of the nonlinear structural modules. Figure 5.18(a) shows typical
acceleration, velocity, and displacement records obtained at second structural module
(Figure 5.11). The top and middle plots in Figure 5.18(b) shows the relative displacement
and velocity time-histories computed between the idealized rst and second degrees-of-
freedom. The bottom plot corresponds to the measured restoring force time-history for
the module 2 of the test structure.
Similarly to the identication of the testbed structure in the linear conguration,
Chebyshev polynomials of fth-order in both normalized variables z
0
and _ z
0
were also
used for the identication of all six modules in the structure. The contribution analysis
of the C
(i)
qr
coecients showed that the coecients C
(i)
10
for all six structural modules
had the largest contribution. In addition to the contribution of the linear C
(i)
10
coecient,
155
(a) First mode (b) Second mode
(c) Third mode
Figure 5.17: Estimated mode shapes for the rst three identied lateral modes of the
testbed structure in the linear reference conguration (state#1) estimated using the mass-
normalized stiness-like and damping-like coecientsa
(i)
10
anda
(i)
01
from built reduced-order
models
signicant contributions of high-order and cross-product terms in the representation of the
restoring forces were reveled in modules 2 and 4, while the contributions of these terms
for the remaining structural modules were negligible. The most signicant nonlinear and
cross-product terms in the reduced-order model of modules 2 and 4 corresponded to the
coecients C
(i)
20
, C
(i)
30
, C
(i)
21
, and C
(i)
22
. The presence of these signicant high-order and
cross-product terms can be used as an indicator of nonlinearities in the structural module.
To illustrate this point, the Chebyshev coecients C
(2)
qr
estimated from module 2 are
summarized in Figure 5.19(a). Moreover, a visual inspection of the reconstructed restoring
force surfaceG
(2)
generated using the non-parametric reduced-order representation of the
156
5 5.5 6 6.5 7 7.5 8
−10
−5
0
5
10
Time [s]
¨ x
2
[m/s
2
]
5 5.5 6 6.5 7 7.5 8
−0.1
−0.05
0
0.05
0.1
Time [s]
˙ x
2
[m/s]
5 5.5 6 6.5 7 7.5 8
−2
−1
0
1
2
x 10
−3
Time [s]
x
2
[m]
5 5.5 6 6.5 7 7.5 8
−5
0
5
x 10
−4
Time [s]
z
2
[m]
5 5.5 6 6.5 7 7.5 8
−0.04
−0.02
0
0.02
0.04
Time [s]
˙ z
2
[m/s]
5 5.5 6 6.5 7 7.5 8
−20
0
20
40
Time [s]
G
(2)
[m]
(a) Linear system (state#1) (b) Nonlinear system (state#7)
Figure 5.18: Sample of (a) acceleration, velocity, and displacement time-histories, and
(b) relative displacement, relative velocity, and restoring force time-histories from the
nonlinear system in state#7.
dynamic characteristics of module 2 can be used to veried that presence of nonlinear
eects in the structural module.
The plots in Figure 5.20 show the measured and reconstructed restoring force G
(2)
associated with the structural module 2 as it varies with the relative motion z
2
, the phase
plot of the restoring forceG
(2)
and relative velocity _ z
2
, as well as the restoring force time-
histories. The eects of the nonlinear element in the response of the structural module 2
are noticeable in both phase plots of the restoring force. In the phase plot of restoring force
and relative displacement (Figure 5.20(a)), the change in the restoring force slope, around
a relative displacement of z
2
= 0:1 mm, indicates a hardening eect in the introduced
nonlinearity. It should also be noted, that the embedded nonlinearity is not symmetric,
as the hardening is only observed in one side of the restoring force. Evidence of the
157
0 1 2 3 4 5
0
1
2
3
4
5
r
q
C
(2)
qr

−2
0
2
4
6
8
10
12
14
16
18
−6
−4
−2
0
2
4
−0.04
−0.02
0
0.02
−30
−20
−10
0
10
20
30
z
2
[10
−4
m]
˙ z
2
[m/s]
G
(2)
(z
2
, ˙ z
2
) [m/s
2
]
(a) Restoring force coecients) (b) Restoring force surface
Figure 5.19: Identied restoring force coecients and corresponding estimated restoring
force surface for module 2 for testbed structure in nonlinear conguration (state#7)
nonlinearity is also noticed in the plot of restoring force versus relative velocity (Figure
5.20(b)) as a pinching in the restoring force, which is typical signature of a impacts against
a rigid barrier where abrupt and rapid changes in the velocity take place. It should be
also noted, by comparing the measured and reconstructed restoring forces, that the non-
parametric reduced-order representation successfully captured the dominant features of
the underlying dynamics in the modules with the nonlinear-gap element attachment.
The estimated probability density functions for the random a
(2)
10
, a
(2)
20
, and a
(2)
30
coe-
cients identied for module 2 in the linear and nonlinear conguration, are shown in Figure
5.21. It is clear from the plots in Figure 5.21, that there were changes in the distribution
of the identied coecients. These changes could be correlated with the changes caused
by the dynamic eects introduced by the nonlinear gap element added to the structural
module 2. Similar observations were made when comparing the distributions of the identi-
eda
(4)
10
,a
(4)
20
, anda
(4)
30
coecients. It should be noted that in this structural conguration
(i.e., state#7), the testbed structure have also a nonlinear gap element aded to module 4.
158
−8 −6 −4 −2 0 2 4 6 8
−30
−20
−10
0
10
20
30
z
2
[10
−4
m]
G
(2)
(z
2
, ˙ z
2
) [m/s
2
]
−0.05 0 0.05
−30
−20
−10
0
10
20
30
˙ z 2 [m/s]
G
(2)
(z 2, ˙ z 2) [m/s
2
]
(a) Phase plot of restoring force versus (b) Phase plot of restoring force versus
relative displacement relative velocity
5 5.5 6 6.5 7 7.5 8
−15
−10
−5
0
5
10
15
Time [s]
G
(2)
[m/s
2
]

Measured
Reconstructed
(c) Restoring force time-history
Figure 5.20: Comparison of measured and estimated restoring forces for nonlinear system
(state#7)
The probability density functions estimated for the linear and quadratic stiness-like
coecients a
(i)
10
anda
(i)
20
of all six modules of the testbed structure in the linear (state#1)
and nonlinear (state#7) congurations are shown in Figure 5.22. It can be clearly seen
that the two nonlinear elements added to the testbed structure in the nonlinear congura-
tion (state#7) did cause noticeable changes in the distributions of the linear and quadratic
stiness-like coecients of the identied reduced-order models.for both structural mod-
ules 2 and 4, with respect to the estimated coecients from the linear structure (state#1).
On the other hand, the changes in the linear coecients a
i
10
for modules 1, 3, 5, and 6,
were almost negligible; and even though there were some changes in the correspondinga
(i)
20
159
1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
˜ a
(2)
10
[-]
pdf [-]

Linear (State 1)
Nonlinear (State 7)
−50 0 50 100 150 200
0
0.01
0.02
0.03
0.04
0.05
˜ a
(2)
20
[-]
pdf [-]

Linear (State 1)
Nonlinear (State 7)
−20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05
˜ a
(2)
30
[-]
pdf [-]

Linear (State 1)
Nonlinear (State 7)
Figure 5.21: Comparison of the estimated probability density functions of the mass-
normalized restoring force coecients a
(2)
10
, a
(2)
20
, and a
(2)
30
identied for structural module
2 in the corresponding linear and nonlinear congurations
coecients, those were not as pronounced or signicant as the changes in the a
2
20
anda
4
20
coecients. From this simple qualitative analysis, it could be concluded that the coe-
cients in the data-driven reduced-order representation will adjust in order to characterize
the new governing dynamics, variations in the identied a
(i)
qr
coecients could be consid-
ered as a suitable set of features to be used for detecting and locating linear/nonlinear
structural changes in the testbed structure, A quantitative assessment of the eectiveness
and robustness of approach under discussion for detecting and locating changes in the
testbed structure will be presented in the next section.
A similar analysis can be done to assess the eects of the added nonlinear elements in
the global parameters of the structure. From a simple inspection of the probability density
functions for the rst three natural frequencies estimated using the reduced-order repre-
sentation of testbed structure in the linear and nonlinear congurations shown in Figure
160
2 2.5 3 3.5
0
0.5
1
1.5
2
˜ a
(1)
10
[-]
pdf [-]

1.5 2 2.5 3 3.5
0
1
2
˜ a
(2)
10
[-]
pdf [-]
1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
1
2
˜ a
(3)
10
[-]
pdf [-]
1.4 1.6 1.8 2 2.2 2.4 2.6
0
1
2
3
˜ a
(4)
10
[-]
pdf [-]
1.4 1.5 1.6 1.7 1.8 1.9 2
0
2
4
6
˜ a
(5)
10
[-]
pdf [-]
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
0
2
4
6
˜ a
(6)
10
[-]
pdf [-]
Linear (State 1) Nonlinear (State 7)
−40 −30 −20 −10 0 10 20 30 40
0
0.02
0.04
0.06
˜ a
(1)
20
[-]
pdf [-]

−50 0 50 100 150 200
0
0.02
0.04
˜ a
(2)
20
[-]
pdf [-]
−80 −60 −40 −20 0 20 40 60
0
0.01
0.02
0.03
˜ a
(3)
20
[-]
pdf [-]
−50 0 50 100 150 200 250
0
0.01
0.02
0.03
˜ a
(4)
20
[-]
pdf [-]
−50 0 50
0
0.01
0.02
0.03
0.04
˜ a
(5)
20
[-]
pdf [-]
−50 −40 −30 −20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
˜ a
(6)
20
[-]
pdf [-]
Linear (State 1) Nonlinear (State 7)
(a) Linear stiness-like coecient a
(i)
10
(b) Quadratic stiness-like coecient a
(i)
20
Figure 5.22: Comparison of the estimated probability density functions of the mass-
normalized restoring force coecients a
(i)
10
and a
(i)
20
for all six structural modules of the
testbed structure in the linear and nonlinear congurations
161
5.2 5.4 5.6 5.8 6 6.2 6.4 6.6
0
0.5
1
1.5
2
2.5
3
3.5
ω 1 [Hz]
pdf [Hz
−1
]

Linear (State 1)
Nonlinear (State 7)
14.5 15 15.5 16 16.5 17
0
0.5
1
1.5
ω
2
[Hz]
pdf [Hz
−1
]

Linear (State 1)
Nonlinear (State 7)
23 23.5 24 24.5 25 25.5 26 26.5
0
0.2
0.4
0.6
0.8
ω
3
[Hz]
pdf [Hz
−1
]

Linear (State 1)
Nonlinear (State 7)
Figure 5.23: Comparison of the estimated probability density functions of the rst three
natural frequencies estimated for the testbed structure in the corresponding linear and
nonlinear congurations
5.23, it can be seen that indeed some changes were observed in the natural frequencies.
In the next section, a quantitative assessment of how signicant these changes are will be
presented.
5.3.5 Detection of changes due to linear and nonlinear modications
The sensitivity and robustness of the restoring force coecients a
(i)
qr
to structural changes
were evaluate using a classical hypothesis testing-based approach. Using the estimated
probability density functions, the bounds dening the two-tail condence region for a
signicance level of = 0:05 were obtained for the a
(i)
qr
coecients identied for the
structure in the reference structural conguration (state#1). New observations of the
a
(i)
qr
coecients obtained for the modied structure from states 2-9, were then compared
against the established lower and upper condence bounds. A threshold in the ratio of
the total number of new observations of thea
(i)
qr
coecients falling outside their condence
162
bounds, was used to assess the signicance of any observed change in the parameters. With
a small threshold, the In this study, this threshold has set to
c
= 0:90. Therefore, if more
than the 90% of the new observations for a given a
(i)
qr
coecient fall outside the reference
condence bounds, there is enough statistical evidence to conclude that signicant changes
were observed in that parameter. The ratio will be referred to as detection ratio hereafter.
The linear stiness-like a
(i)
10
and quadratic stiness-like a
(i)
20
coecients were used to
illustrate the proposed methodology for detection and location of structural changes, For
the sake of brevity, the results from only two of the nine scenarios will be discussed in
detailed. Those two scenarios are state#5,and state#7. In state#5, linear modication
to the testbed structure were made by loosening the connection between the structural
modules 4 and 5. In state#7, on the other hand, nonlinear changes were introduced to
the structure by adding nonlinear gap elements to the module 2 and 4, respectively.
In Figures 5.24 and 5.25, the displayed probability density function correspond to
the pdf of the coecients a
(i)
10
and a
(i)
20
estimated for all six structural modules of the
testbed structure from the reference conguration. The left-hand side plots correspond
to the linear stiness-like coecients a
(i)
10
, while the right-hand side of the same gure
correspond to the quadratic stiness-like coecients a
(i)
20
. The plots at the top of Figure
5.24 correspond to the coecients of the identied reduced-order representation of the
rst structural module (i.e., the module at the xed end of the structure). The bottom
row of plots correspond to the coecients of the structural module 6 (i.e., the module at
the free end of the structure). The shading region on each of the plots correspond to the
95%-condence region estimated for each of the a
(i)
10
and a
(i)
20
coecients. The red dots in
163
the plots correspond to new observations of the coecients estimated from the modied
structure.
From Figure 5.24, It can be clearly seen that all estimated linear stiness-like coef-
cients a
(4)
10
from the structural state#5 were outside the coecient's condence region,
while on the other hand, the none of the estimated quadratic stiness-like coecients a
(4)
20
fell outside the corresponding condence bounds established from the reference state#1.
Therefore, it can be asserted that signicant changes in the linear stiness-like coecients
estimated for structural module 4 in state#5 were detected with respect to the reference
condition, with a detection ratio of = 1:0; while no dierences at all were observed
( = 0:0) in the coecients a
(4)
20
from state#5 with respect to the reference coecients.
Regarding the coecients a
(5)
10
anda
(6)
10
, approximately the 73% and 60% of the identied
coecients from state#5 were outside the reference condence bounds. However, even
though the percentage of estimated coecients outside the bounds was relatively large,
with a detection threshold set to
c
= 0:90, these changes in the a
(5)
10
and a
(6)
10
coecients
could not be considered as signicant. In summary, these results indicate that a signicant
change in the linear dynamic properties of module 4 did occur in the testbed structure
with respect to the reference condition. It should be noted that these results partially
correlate with the actual structural changes in state#5. It was expected that a loosen
connection between modules 4 and 5 would reduced the lateral stiness of both structural
elements.
Similar interpretation can be done for the identication results from structural state#7.
It can be clearly seen in Figure 5.25 that a signicant number of identied a
(2)
20
and
a
(4)
20
coecients were outside the reference 95%-condence region. It can be also noticed
164
Table 5.3: Summary of detection ratio estimated for thea
(i)
10
anda
(i)
20
coecients. High-
lighted entries in the table correspond to the cases with signicant changes in the coe-
cients were detected using a detection threshold of
c
= 0:90.
Module 1 Module 2 Module 3 Module 4 Module 5 Module 6
State a
(1)
10
a
(1)
20
a
(2)
10
a
(2)
20
a
(3)
10
a
(3)
20
a
(4)
10
a
(4)
20
a
(5)
10
a
(5)
20
a
(6)
10
a
(6)
20
State#2 0.025 0.100 0.050 0.025 0.050 0.050 0.025 0.000 0.075 0.025 0.025 0.025
State#3 0.050 0.100 0.000 0.050 0.075 0.075 0.025 0.050 0.025 0.075 0.025 0.175
State#4 0.000 0.175 0.000 0.125 0.025 0.125 0.050 0.075 1.000 0.025 0.325 0.025
State#5 0.050 0.100 0.050 0.050 0.100 0.050 1.000 0.000 0.725 0.150 0.600 0.150
State#6 0.000 0.325 0.325 0.950 0.000 0.425 0.000 1.000 0.625 0.200 0.500 0.100
State#7 0.025 0.150 0.350 0.900 0.025 0.125 0.200 1.000 0.025 0.150 0.050 0.175
State#8 0.050 0.325 0.475 0.950 0.025 0.150 0.000 0.075 0.075 0.075 0.025 0.025
State#9 0.050 0.100 0.100 0.675 0.050 0.050 0.050 0.075 0.075 0.050 0.125 0.025
that most of the a
(i)
10
were within the reference condence bounds. Therefore, signicant
changes in the nonlinear dynamic properties of modules 2 and 4 were predicted to have
occurred. These results correlate with the actual physical modications made to the
testbed structure in state#7, where the nonlinear gap elements were added to modules 2
and 4.
The detection ratios of the linear stiness-like a
(i)
10
and quadratic stiness-like a
(i)
20
coecients for all structural states are summarized in Table 5.3.
5.4 Identication of a probabilistic reduced-order model of
uncertain nonlinear element
In this section, a general methodology for representing and propagating the eects of
uncertainties in complex nonlinear systems through the use of data-driven reduced-order
models, that allows the estimation through analytical procedures of the uncertain sys-
tem's response bounds. A nonparametric identication approach based on the use of the
165
2 2.5 3 3.5
0
0.5
1
1.5
2
˜ a
(1)
10
[-]
pdf [-]
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
0
1
2
˜ a
(2)
10
[-]
pdf [-]
1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
1.5
2
˜ a
(3)
10
[-]
pdf [-]
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
0
1
2
3
˜ a
(4)
10
[-]
pdf [-]
1.4 1.5 1.6 1.7 1.8 1.9 2
0
2
4
6
˜ a
(5)
10
[-]
pdf [-]
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
0
2
4
6
˜ a
(6)
10
[-]
pdf [-]
−20 −15 −10 −5 0 5 10 15 20 25
0
0.02
0.04
0.06
˜ a
(1)
20
[-]
pdf [-]
−30 −20 −10 0 10 20 30 40 50
0
0.02
0.04
˜ a
(2)
20
[-]
pdf [-]
−40 −30 −20 −10 0 10 20 30 40 50
0
0.01
0.02
0.03
˜ a
(3)
20
[-]
pdf [-]
−40 −30 −20 −10 0 10 20 30 40 50
0
0.01
0.02
0.03
˜ a
(4)
20
[-]
pdf [-]
−30 −20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
˜ a
(5)
20
[-]
pdf [-]
−30 −20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
˜ a
(6)
20
[-]
pdf [-]
Figure 5.24: Probability density functions and corresponding 95%-condence bounds for
the a
(i)
10
and a
(i)
20
coecients identied for all structural modules of the testbed structure
in the reference conguration (state#1). The estimated coecients from the testbed
structure in state#5 conguration are shown as red dots in each of the plots.
166
2 2.5 3 3.5
0
0.5
1
1.5
2
˜ a
(1)
10
[-]
pdf [-]
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
0
1
2
˜ a
(2)
10
[-]
pdf [-]
1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
1.5
2
˜ a
(3)
10
[-]
pdf [-]
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
0
1
2
3
˜ a
(4)
10
[-]
pdf [-]
1.5 1.6 1.7 1.8 1.9 2
0
2
4
6
˜ a
(5)
10
[-]
pdf [-]
1.1 1.2 1.3 1.4 1.5 1.6
0
2
4
6
˜ a
(6)
10
[-]
pdf [-]
−30 −20 −10 0 10 20 30
0
0.02
0.04
0.06
˜ a
(1)
20
[-]
pdf [-]
−40 −20 0 20 40 60 80 100 120 140
0
0.02
0.04
˜ a
(2)
20
[-]
pdf [-]
−60 −40 −20 0 20 40 60
0
0.01
0.02
0.03
˜ a
(3)
20
[-]
pdf [-]
−50 0 50 100 150 200
0
0.01
0.02
0.03
˜ a
(4)
20
[-]
pdf [-]
−40 −30 −20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
˜ a
(5)
20
[-]
pdf [-]
−30 −20 −10 0 10 20 30 40
0
0.01
0.02
0.03
0.04
˜ a
(6)
20
[-]
pdf [-]
Figure 5.25: Probability density functions and corresponding 95%-condence bounds for
the a
(i)
10
and a
(i)
20
coecients identied for all structural modules of the testbed structure
in the reference conguration (state#1). The estimated coecients from the testbed
structure in state#7 conguration are shown as red dots in each of the plots.
167
restoring force method is employed to obtain a stochastic model of the nonlinear sys-
tem of interest. Subsequently, the reduced-order stochastic model is used in conjunction
with polynomial chaos representations to predict the uncertainty bounds on the nonlinear
system response under transient dynamic loads.
Mechanical systems are inherently stochastic due to unpredictable natural variabil-
ity (randomness) in the dynamical properties of the physical system. Additionally, the
non-parametric mathematical modeling used to represent the restoring force and, the mea-
surement errors will add to variability in the identied power-series coecients. In the
presence of all sources of variability associated to the non-parametric restoring force iden-
tication, the uncertain identied coecients a
qr
can be considered as random variables.
5.4.1 Probabilistic representation of random restoring force coecients
The rst step in the procedure is the representation of all the model inputs (i.e., the
random coecients a
qr
in terms of a set of normal random variables. By adopting the
methodology proposed by Arnst et al. [7], polynomial chaos expansion (PCE) to charac-
terize the uncertainty in the data-driven reduced-order models, the randoma
qr
coecients
can be approximated by a truncated expansion in the form:
ap
0
+
d
X
j=1
p

j
r
X
;jj=1
p

j
H

()v
j
(5.11)
wherea is a vector containing all random restoring force coecients a
qr
. The parameters
p
0
andp

are the unknown parameters in the representation that must be estimated from
the available data,H

is a multi-dimensional normalized Hermite polynomial, and
j
and
v
j
are the eigenvalues and eigenvectors of the sample covariance matrix of the coecients
168
−5
0
5
−0.04
−0.02
0
0.02
0.04
−40
−20
0
20
40
z
2
[10
−4
m]
˙ z
2
[m/s]
G
(2)
[m/s
2
]
−5
0
5
−0.04
−0.02
0
0.02
0.04
−40
−20
0
20
40
z
2
[10
−4
m]
˙ z
2
[m/s]
G
(2)
[m/s
2
]
Figure 5.26: Estimated restoring force surface for structural module with variable-gap
element obtained for two dierent gap sizes from experimental measurements
a
qr
. A more detailed description of the methodology use to obtain the polynomial chaos
representation can be found in Arnst et al. [7]
To illustrate the proposed methodology to obtained a probabilistic representation of
identied reduced-order models of nonlinear elements, experimental measurements of the
response of the testbed structure were used. In this case, the adaptive nonlinear gap ele-
ment was installed in one of the modules of the testbed structure. This element, as was
previously described, can be used to introduce randomness in the gap size of the nonlin-
ear element. Forty dierent realization of the gap-size were generated. For each of the
realizations of the gap size, a dynamic test was performed on the testbed structure. The
proposed sub-structuring identication approach was then used to identify the restoring
force coecients a
qr
and build the corresponding reduced-order model of the nonlinear
element with the random gap size. Figure 5.26 shows the restoring force surfaces recon-
structed using the reduced-order representation for two dierent realizations of the size of
the gap in the nonlinear element.
169
1 2 3 4 5 6 7 8 9 10
0.5
0.6
0.7
0.8
0.9
1
Index [-]
d
X
i=1
λi
. m
X
i=1
λi [-]
0 5000 10000 15000
−400
−350
−300
−250
Monte Carlo samples [-]
logL
δ
(h
δ
) [-]
Figure 5.27: (a) Relative contribution of computed eigenvalues of the sample covariance
matrix of the restoring force coecients a
qr
. (b) Loglikelihood logL

(p

) of polynomial
chaos coecientsp

as a function of the number of the monte carlo samples.
Figure 5.27(a) shows the relative contribution of the eigenvalues of the covariance
matrix of the identied restoring force coecients a
qr
. This plot suggests that a reduced
representation of the stochastic dimension d = 5 is able to represent more than the 95%
of the variability in the identied restoring force coecients a
qr
. All results presented
hereafter were obtained with polynomial chaos expansion using of stochastic dimension
d = 5 and order r = 1.
The maximum likelihood methodology presented in Arnst et al. [7] was used to ob-
tained the optimal coecients p
0
and p

for the probabilistic characterization of the
uncertain restoring force coecients a
qr
in the reduced-order representation of the non-
linear element with random gag-size. This methodology requires an approximation of the
conditional probability density function f
a
(a
(j)
jp
0
;p

) of the random restoring force co-
ecientsa =fa
j
qr
g; j = 1;:::; 40. This approximation to the probability density function
were obtained by using kernel density estimation of samples generated by Monte Carlo
simulations. Figure 5.27(b) shows the variation of the loglikelihood logL

(p

) for a spe-
cic value ofp
d
with respect to the number of Monte Carlo samples. It is observed that
convergence of the loglikelihood is obtained for 10000 Monte Carlo realizations.
170
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
0
1
2
3
4
5
6
˜ a
(2)
10
[-]
pdf [-]
−2 0 2 4 6 8 10 12 14 16
0
0.05
0.1
0.15
0.2
0.25
˜ a
(2)
20
[-]
pdf [-]
−5 0 5 10 15 20
0
0.05
0.1
0.15
˜ a
(2)
30
[-]
pdf [-]
Figure 5.28: Probability density functions for coecientsa
(2)
10
,a
(2)
20
, anda
(2)
30
obtained using
the polynomial chaos representation of the restoring force coecients for the structural
module with variable-gap size
Figure 5.28 shows the probability density functions of three coecients a
10
, a
20
, and
a
30
in the reduced-order model of the nonlinear element with variable gap obtained using
the estimated polynomial chaos representation of the random restoring force coecients
a =fa
j
qr
g; j = 1;:::; 40. The identied coecients from experimental data are displayed
as red points over the plots of the probability density functions. It is possible now to
obtained bounds on the stochastic restoring force surface for the variable-gap nonlinear
element in the testbed structure using the probabilistic representation of the restoring force
coecients. The mean restoring force surface and corresponding3 bounds displayed
in Figure 5.29 were obtained using 15000 Monte Carlo realizations of the a
qr
restoring
force coecients using the estimated polynomial chaos representation. Two of the 15000
realizations of the stochastic restoring force surface are shown in Figure 5.30.
171
Figure 5.29: Estimated mean restoring force surface for structural module with variable-
gap element and corresponding3 bounds obtained using the polynomial chaos repre-
sentation of the restoring force coecients.
−5
0
5
−0.04
−0.02
0
0.02
0.04
−40
−20
0
20
40
z
2
[10
−4
m]
˙ z
2
[m/s]
G
(2)
[m/s
2
]
−5
0
5
−0.04
−0.02
0
0.02
0.04
−40
−20
0
20
40
z
2
[10
−4
m]
˙ z
2
[m/s]
G
(2)
[m/s
2
]
Figure 5.30: Restoring force surfaces of variable-gap nonlinear element reconstructed from
realizations from the polynomial chaos representation of the of the restoring force coe-
cients
5.4.2 Uncertainty Propagation
The responses of a dynamical system may be in
uenced by any randomness in model char-
acteristics (i.e., power-series coecients). Hence, any general functional representation of
uncertainty in the dynamical model outputs, should take into account uncertainties in all
identied restoring force coecients. For a general stochastic dynamical model with ran-
dom restoring force coecientsa =fa
(i)
qr
g are represented in terms of the set =f
j
g, the
172
response of the system can also be represented in terms of the same set, as the uncertainty
in the outputs is just due to the uncertainty of the restoring force coecients a
qr
.
Following this basic idea, the uncertainty in the response of a stochastic dynamical
system can be represented in the form of a multidimensional truncated PC expansion [25]:
x(t;)
u
X
;jj=0
x

(t)H

(); _ x(t;)
u
X
;jj=0
_ x

(t)H

(); (5.12)
where the summation limit depends on the highest order of the H

Hermite polynomials
(o), the dimension n = dim(), and it is given by:
u + 1 =
(n +o)!
n!o!
(5.13)
Using Equations (5.11) and (5.12), the restoring force equation, described in Equation
(5.4), for a stochastic SDOF system can be expressed as:
G(x; _ x;) =
qmax
X
q=0
rmax
X
r=0
0
@
p
0
qr
+
d
X
j=1
p

j
s
X
;jj=1
p

qrj
H

()v
j
1
A
0
@
u
X
;jj=0
x

(t)H

()
1
A
q
0
@
u
X
;jj=0
_ x

(t)H

()
1
A
r
(5.14)
With equations (5.14) in mind, the stochastic dierential equation of motion for a
SDOF is given mathematically by:
m
u
X
;jj=0
x

(t)H

() +G(x; _ x;) =F (t) (5.15)
173
In the context of polynomial chaos expansions (PCE), the solution to Equation 5.15 is
reduced to the computation of the polynomial chaos coecients x

(t), _ x

(t), and x

(t).
In general, the algorithms for computing the PC coecients typically belong to two cat-
egories: 1) intrusive, and 2) non-intrusive approaches. Non-intrusive approaches rely on
existing deterministic solvers and are used as a black box to compute the PC coecients.
Intrusive approaches, in the other hand, requires modications to the available solvers
[19]. However, in this report a Monte Carlo simulation approach and a standard explicit
4
th
-order Runge-Kutta integration scheme were adopted to propagate the uncertainty in
the reduced-order representation of the nonlinear element with a variable gap size and
determine the stochastic response of a SDOF subjected to a deterministic band-limited
white noise excitation. The time histories of mean displacement, mean velocity, and mean
acceleration predicted for an uncertain nonlinear SDOF with a nonlinear variable-gap el-
ement are displayed in Figure 5.31. The predicted2 bounds on the stochastic response
of the SDOF are shown in Figure 5.32.
5.5 Preliminary results on probabilistic model upscaling and
stochastic dimension reduction
This section includes the preliminary results obtained on the implementation of a proba-
bilistic model upscaling methodology proposed by Arnst and Ghanem [6] for characterizing
the information about an event or quantity of interest predicted from a ne-scale model,
by constructing a dierent model, at a (much) coarser scale, that provides statistically
equivalent predictions of the event or quantity of interest.
174
0 5 10 15 20 25 30
−10
−5
0
5
10
Time [s]
μ
¨ x
[m/s
2
]
0 5 10 15 20 25 30
−0.05
0
0.05
Time [s]
μ
˙ x
[m/s]
0 5 10 15 20 25 30
−4
−2
0
2
4
x 10
−4
Time [s]
μ
x
[m]
Figure 5.31: Predicted mean acceleration, mean velocity, and mean displacement time-
histories of an uncertain nonlinear SDOF subjected to a deterministic band-limited white-
noise excitation
5.5.1 Probabilistic model upscaling
In general, the objective of a probabilistic model upscaling is to achieve a perfect equality
between the probability density functions of the coarse-scale quantities of interest pre-
dicted by the ne-scale and the coarse-scale model. For practical purposes, however, this
condition has been relaxed. The redened objective is now to nd a random vector ^ that
minimizes a distance D between these probability density functions
^ = arg min

D

f(w);
~
f(w)

(5.16)
175
3.9 4 4.1 4.2 4.3 4.4
−10
0
10
Time [s]
¨ x [m/s
2
]

μ
¨ x μ
¨ x
± 2σ
¨ x
3.9 4 4.1 4.2 4.3 4.4
−0.05
0
0.05
Time [s]
˙ x [m/s]

μ
˙ x μ
˙ x
± 2σ
˙ x
3.9 4 4.1 4.2 4.3 4.4
−5
0
5
x 10
−4
Time [s]
x [m]

μ
x μ
x
± 2σ
x
Figure 5.32: Predicted2 bounds for dynamic response of the uncertain nonlinear SDOF
system
This optimization problem must be discretized in order to make it suitable for a nu-
merical solution. For this purpose, let be approximated by a truncated polynomial chaos
expansion of dimension n and order p
(p) =
p
X
;jj=0
p

H

(z) (5.17)
where then components or the random vectorz = (z
1
;:::;z
n
) are independent real Gaus-
sian random variables with zero mean and unit standard deviation, = (
1
;:::;
n
)2N
n
is a multi-index,jj =
1
++
n
, andH

(z) =h

1
(z
1
)h
n
(z
n
), in whichh

k
()
176
is the normalized Hermite polynomial of order
k
. The coecients of this expansion are
collected in the parameter setp =fp

j 0jjpg. The nite-dimensional optimization
problem thus obtained reads
^ p = arg min
p
D

f(w);
^
f(wjp)

(5.18)
where
~
f(wjp) denotes the probability density function of the quantity of interest ~ w((p))
predicted by the coarse-scale probabilistic model. A well-known distance-like measure
of the separation between two probability density functions is the relative entropy or
Kullback-Leibler divergence. This distance is dened mathematically as
D

f(w);
~
f(w)

=
Z
W
f(w) ln
f(w)
~
f(wjp)
dw (5.19)
In order to numerically compute the distance D

f(w);
~
f(wjp)

, the approach pro-
posed by Arnst and Ghanem [6], which is based on the nite element and Monte Carlo
simulation methods, was followed in this implementation. For a given set of polynomial
chaos coecientsp and a predetermined number of Monte Carlo samples m, the random
variablesw
h
(
s
) and ~ w
h
(
s
(p)) are initially obtained by computing the stochastic quan-
tity of interest with the ne-scale and coarse-scale nite element models associated to each
of the elements in the setsf
s
j 1 s mg andfz
s
j 1 s mg of independent and
identically distributed realizations of the ne-scale and coarse-scale basic random variables
andz, respectively. Notice that the realizations
s
(p) required to build the coarse-scale
nite element model, are obtained from the realizationsz
s
through Equation (5.17). Once
177
the quantity of interest has been computed with the ne-scale and coarse-scale probabilis-
tic models, the distance D can be approximated by
D
h;MC
(p) =
1
m
m
X
s=1
ln
f
h;MC
(w
h
(
s
))
~
f
h;MC
(w
h
(
s
)jp)
(5.20)
where f
h;MC
(w
h
(
s
)) and
~
f
h;MC
(w
h
(
s
)jp) are approximations of the probability den-
sity functions f(w) and
~
f(wjp) estimated from the samplesfw
h
(
s
)j 1 s mg and
f ~ w
h
(
s
(p))j 1smg, respectively. The probability density functions can be estimated
from the samples generated by Monte Carlo sampling using the kernel density estimation
method.
5.5.2 Fine-scale model
The ne-scale probabilistic model consists of a three-dimensional nite element model
made up of 5760 brick elements, 240 beam elements, and 48 rigid elements. The carbon-
steel plates were modeled using a linear, elastic, and isotropic material with an elastic
modulus E = 200 GPa, a mass density = 7850 kg/m
3
, and Poisson ratio = 0:3. The
carbon-steel beams, on the other hand, were modeled using a linear, elastic, and isotropic
random material with a stochastic elastic modulus. The random Young's modulus for
each of the beams was modeled by an independent log-normal random variable with a
mean of 200 GPa and standard deviation of 10 GPa. The ne-scale probabilistic model
of the testbed structure is then dened as a function of 24 basic random variables =
(
1
;:::;
24
).
178
5.5.3 Coarse-scale models
At the coarse scale, the structure was modeled as a cantilever Euler-Bernoulli beam of
lengthL = 1:22 m. In a rst coarse-scale probabilistic model, the bending stiness EI(p)
and linear density A(p) of the beam were assumed in the form
EI(p) = exp

p
X
=0
p

h

(z)
!
A(p) =

A
(5.21)
wherez is a standard Gaussian random variable, h

() is the normalized one-dimensional
Hermite polynomial of order , and the vectorp contains all the coecients of the poly-
nomial chaos representation. The mass density and cross-sectional area are assumed to
be = 7850 kg/m
3
and

A = 7:25 10
4
m
2
, respectively.
For the second coarse-scale probabilistic model, both the the bending stiness EI(p)
and linear density A(p) were assumed to be random variables with a polynomial chaos
representation given by
EI(p) = exp
0
@
p
X
;jj=0
p
EI

H

(z)
1
A
A(p) = exp
0
@
p
X
;jj=0
p
A

H

(z)
1
A
(5.22)
where, in this case, H

() is the multi-dimensional Hermite polynomial of the form
H

(z) =h

1
(z
1
)h
n
(z
n
), in which h

k
() is the normalized one-dimensional Her-
mite polynomial of order
k
. The n-dimensional random vectorz = (z
1
;:::;z
n
) consists
of independent Gaussian random variables with zero mean and unit standard deviation.
179
The parameter setp collects the coecients p
EI

and p
A

of the polynomial chaos expan-
sions. The exponential in the PC representations ensure positivity of the bending stiness
and linear density.
5.5.4 Stochastic upscaling and model reduction
The probabilistic model upscaling framework is implemented herein, to identify the coarse-
scale models, described in the previous section, that will predict the probabilistic infor-
mation of the rst eigenfrequency ! (i.e., quantity of interest w
h
= !) of the testbed
structure obtained from its corresponding ne-scale model.
The suitable set of coecients p for the rst and second coarse-scale probabilistic
models (Equations 5.21-5.22) are identied by minimizing the relative entropy or Kullback-
Liebler divergence,
^ p = arg min
p
D
h;MC
(p) (5.23)
where D
h;MC
(p) is the approximation to the relative entropy given in Equation (5.20).
Considering that the relative entropy D
h;MC
(p) may have multiple local minima and it
may be dicult to calculate gradients with respect to the parameters [6], the genetic
optimization method, which is a global-search gradient-free optimization method, was
used in this implementation.
All results to follow were obtained usingm = 3000 Monte Carlo samples of independent
realizations of used to simulate the Young's moduli of the beams in the ne-scale model of
the structure. Figure 5.33 shows the mean value
!
and standard deviation
!
of the rst
random eigenfrequency! computed from the ne-scale probabilistic model of the testbed
180
0 500 1000 1500 2000 2500 3000
5.32
5.34
5.36
5.38
5.4

Samples [-]
μ
ω
[Hz]
0 500 1000 1500 2000 2500 3000
0
0.01
0.02
0.03
0.04
0.05
0.06

Samples [-]
σ
ω
[Hz]
(a) Mean (b) Standard deviation
Figure 5.33: Evolution of the mean value
!
and standard deviation
!
of the rst random
eigenfrequency ! for dierent number or Monte Carlo samples.
structure as a function of the number of Monte Carlo realizations. It can be observed
that a reasonable convergence for the mean and standard deviation of the fundamental
frequency ! for 3000 Monte Carlo samples. The mean and standard deviation values of
the rst eigenfrequency converged to 5.35 Hz and 0.03 Hz, respectively.
For the rst coarse-scale model, the optimization problem (Equation 5.23) was solved
for rst-order (p = 1) and second-order (p = 2) polynomial chaos expansions of the
coarse-scale random bending stiness EI(p). The optimal values of the relative entropy
D
h;MC
(p) were 0.0059 and 0.0039 for the rst-order and second-order polynomial chaos
expansions, respectively. Notice that, based on the optimal values of relative entropy, the
polynomial expansion for the random bending stiness EI(p) can be truncated at order
p = 1, since no signicant reduction in the relative entropy resulted by increasing the order
of the expansion from p = 1 to p = 2. Figure 5.34(a) compares, the probability density
function f(!) of the rst eigenfrequency ! estimated using the ne-scale model, with the
probability density functions
~
f(!jp) predicted by the coarse-scale model for polynomial
chaos orders of p = 1 and p = 2. Notice that the rst coarse-scale model accurately
represents the uncertainty in the rst eigenfrequency. The estimated probability density
181
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7
0
2
4
6
8

ω [Hz]
pdf [Hz
−1
]

f (ω )
˜
f(ω |ˆ p), p = 1
˜
f(ω |ˆ p), p = 2
1700 1750 1800 1850 1900 1950
0
0.01
0.02

EI [N·m
2
]
pdf [N
−1
·m
−2
]

f (EI |ˆ p), p = 1
f (EI |ˆ p), p = 2
(a) First eigenfrequency (b) Bending stiness
Figure 5.34: Comparison of probability density function f(!) (solid line) of rst eigenfre-
quency obtained from the ne-scale model, and the probability density functions f(!j^ p)
obtained using a rst- (p = 1) and second-order (p = 2) polynomial chaos expansions of
the bending stiness (EI(p)) in the coarse-scale model. (b) Comparison of probability
density functions of bending stiness (EI(p)) of the coarse-scale model obtained from the
corresponding rst- and second-order polynomial chaos representation.
functions of the random bending stinessEI(p) obtained from the rst- and second-order
polynomial chaos representation using the corresponding optimal set of coecients ^ p are
shown in Figure 5.34(b). The mean and standard deviation of the estimated bending
stiness for the coarse-scale models were around 1848 Nm
2
and 23.6 Nm
2
, respectively.
For the second coarse-scale model, the optimization problem (Equation 5.23) was
solved, in this case, for one- and two-dimensional rst-order polynomial chaos expan-
sion of the bending stinessEI(p) and linear densityA(p), which are given by Equation
(5.22). The optimal values of the relative entropy were, this time,D
h;MC
(p) = 0:0054 and
D
h;MC
(p) = 0:0022 for the expansions of dimensionn = 1 andn = 2, respectively. Figure
5.35 compares the probability density functions of the fundamental frequency of the struc-
ture f(!) and
~
f(!jp) calculated with the ne-scale model and course-scale probabilistic
models, respectively. These results demonstrate that the second coarse-scale probabilistic
182
5 5.1 5.2 5.3 5.4 5.5 5.6
0
2
4
6
8
10
12

ω [Hz]
pdf [Hz
−1
]

f (ω )
˜
f(ω |ˆ p), n = 1
˜
f(ω |ˆ p), n = 2
Figure 5.35: Probability density functions (pdf) obtained in the rst upscaling example.
(a) Comparison of probability density function f(!) (solid line) of rst eigenfrequency
obtained from the ne-scale model, and the probability density functions f(!j^ p) obtained
using a one- (n = 1) and two-dimensional (n = 2) rst-order polynomial chaos expansions
of the bending stiness (EI) and mass per unit of length (A) in the coarse-scale model.
37.5 38 38.5
0
1
2
3
4
5

EI [N·m
2
]
pdf [N
−1
·m
−2
]
0.11 0.115 0.12 0.125
0
50
100
150
200
250
300
350

ρA [kg·m
−1
]
pdf [kg
−1
·m]
(a) Bending stiness (EI) (b) Mass per unit of length (A)
Figure 5.36: Probability density functions of (a) bending stiness (EI) and mass per unit
of length (A) of the coarse-scale model obtained from the corresponding one-dimensional
rst-order polynomial chaos representations.
model is also capable of characterizing the uncertainty in the rst structural eigenfre-
quency. The probability density functions of the bending stiness EI(p) and mass per
unit of lengthA(p), obtained from their corresponding optimal one- and two dimensional
polynomial chaos representations, are shown in Figures 5.36 and 5.37.
In summary, the two coarse-scale probabilistic models with stochastic dimensionsn = 1
andn = 2, were able to predict the uncertainty in the rst eigenfrequency of the ne-scale
183
50 100 150
0
0.01
0.02
0.03
0.04

EI [N·m
2
]
pdf [N
−1
·m
−2
]
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
2
4
6
8
10
12

ρA [kg·m
−1
]
pdf [kg
−1
·m ]
(a) Bending stiness (EI) (b) Mass per unit of length (A)
Figure 5.37: Probability density functions of (a) bending stiness (EI) and mass per unit
of length (A) of the coarse-scale model obtained from the corresponding two-dimensional
rst-order polynomial chaos representations.
model of the testbed structure. It should also be noted that the probabilistic upscaling
procedure did also achieve a reduction in the stochastic dimension since the dimension of
these coarse model are signicantly smaller than the ne-scale model, which had 24 basic
random variables (i.e. stochastic dimension N = 24).
184
Chapter 6
Application of Statistical Monitoring Using Latent Variable
Techniques for Detection of Faults in Sensor Networks
6.1 Introduction
During the recent past, extensive research in the eld of structural health monitoring has
produced a signicant number of well-developed algorithms that rely on accurate sensor
measurements for detecting and locating damage in a wide variety of civil, mechanical and
aerospace engineering applications. However, the performance of these structural health
monitoring algorithms can be aected by the presence of malfunctioning and/or faulty
sensors in the monitoring network. Therefore, robust sensor fault detection techniques
have to be an integral part in the development and implementation of structural health
monitoring schemes.
The approaches to sensor fault detection and identication, known also as sensor vali-
dation, can be categorized into three groups [36] : (1) signal-model based (i.e., correlation,
spectrum or wavelet analysis), (2) model-based (i.e., parameter estimation, state observers,
parity equations), and (3) data-driven (i.e., multivariate statistical monitoring).
185
Fault detection and identication approaches based on multivariate statistical monitor-
ing with latent variable techniques (e.g., principal component analysis), which have been
one of the most active research areas in the eld of process control [45, 52, 68], have proved
to be reliable and much easier to implement in complex systems, with densely-distributed
sensor networks, than model-based methods. These approaches have found wide ap-
plications in chemical, biological, and microelectronic industrial processes [16, 45, 47{
49, 52, 68]. However, this eld of research has not received much attention from the
structural health monitoring and smart structures community until recently. Details on
some recently proposed approaches for sensor fault detection and identication, and based
on latent variable methods and statistical monitoring, can be found in Friswell and Inman
[22], Kerschen et al. [41], Kullaa [46], and Shari et al. [71]. However, most of these studies
only considered one latent variable approach; the principal component analysis technique.
In this study three latent-variable methodologies, principal component analysis (PCA),
independent component analysis (ICA), and modied independent component analysis
(MICA), are implemented and evaluated with respect to their capability to detect and
identify faulty sensors, using synthetic as well as experimental measurements.
6.2 Sensor failure modes
A sensor fault can in general be dened as the occurrence of unexpected variations or ab-
normalities in a sensor output (i.e., measurement readout) when no atypical conditions or
changes in the monitored system are present [9, 36]. Sensor faults can be caused by several
factors, including defective components, manufacturing deciencies, weathering, normal
186
Table 6.1: Typical sensor failure modes
Type Mathematical representation
Bias y
f
(t) =y(t) + +(t)
Drift y
f
(t) =y(t) +(t) +(t)
Scaling y
f
(t) =
(t)y(t) +(t)
Hard fault y
f
(t) = +(t)
deterioration over time, excessive use, calibration errors, improper usage and handling,
and damaged wiring/connections.
In the related literature, sensor faults are usually categorized, based on the behavior
and characteristics of the sensor output, into four groups: bias, drift, scaling, and hard
fault. Table 6.1 summarizes the mathematical representation of the dierent four sensor
fault types. The variable y
f
(t) represents the faulty sensor readout, y(t) is the nominal
sensor measurement, and (t) corresponds to inherent noise. A bias fault is usually char-
acterized by a constant oset (i.e., ) in the sensor readings with respect to the reference
sensor signal. In the case of a time-varying oset factor (i.e., (t)), the sensor fault is
classied as a drift-type failure. Bias and drift faults can be also categorized as additive-
type sensor failures. A dierent type of failure occurs when the nominal sensor outputs
are scaled or multiplied by a factor (i.e.,
(t)), which can be constant or time-variant.
This multiplicative-type of sensor failure is known as scaling, gain failure, or precision
degradation. The last category, the hard-fault type, corresponds to cases where the sensor
readings are stuck at a particular constant value (i.e., ). A complete loss of signal can
also be represented by this type of fault by assuming that the output from the sensor is
zero (i.e., = 0). Balaban et al. [9] provided a representative list of typical faults and their
causes, in some of the most commonly used sensors in aerospace, civil, and mechanical
applications.
187
6.3 Statistical monitoring based on latent variables tech-
niques
Latent variable techniques are statistical approaches for developing reduced-order models
from observations/measurements on variables of multivariate phenomena. Consider a
data matrixX2R
mn
consisting ofm samples from a set ofn variablesx =fx
j
g
n
j=1
. In
general, the latent variable representation for X is typically given by set of equations of
the form
X =TV +E
T =XW
VW =I
r
(6.1)
where W 2 R
nr
and V 2 R
rn
dene the linear map from the n-dimensional original
variable space to an r-dimensional latent-variable space and the corresponding inverse
mapping, respectively. Here, the low-dimensional latent-variable space spanned by W
is the best approximating subspace of dimension r to the original space spanned by the
variables x. The matrix T 2 R
mr
containing the values of the set of latent variables
t =ft
j
g
r
j=1
, and the matrixW , are known as the latent-variable scores and latent-variable
loadings. The residual matrix E2 R
mn
contains the non-essential information lost in
the mapping transformation.
For multivariate statistical monitoring purposes, at least two complementary statis-
tics or indices are required when using data-driven latent-variable models [45]. The two
most commonly used indices are the D-statistic and Q-statistic. The D-statistic, also
188
known as the Hotelling's T
2
statistic, is a Mahalanobis-like measure of the variations of
the measured variables within the latent-variable subspace, whereas the Q-statistic, also
called the square prediction error (SPE), measures the discrepancy between the measured
variables and the latent-variable model. Depending on the latent-variable approach used,
these statistics will have a dierent mathematical representation.
There are dierent approaches that can be used to construct latent-variable represen-
tations of multidimensional data sets. These include the principal component analysis
(PCA), independent component analysis (ICA), and modied independent component
analysis (MICA). The PCA models the spaceX by nding a set of linearly uncorrelated
and orthogonal latent variables, the ICA, by obtaining statistically independent latent
variables, and the MICA by nding a set of independent and orthogonal latent variables.
6.3.1 Principal Component Analysis
Principal component analysis (PCA), one of the most widely used latent variable method-
ologies, is a multivariate statistical approach that relies on second-order statistics for per-
forming dimension reduction on high-dimensional datasets. Assume a multidimensional
data matrix X 2 R
mn
with an associated sample covariance matrix 2 R
nn
. The
matrix can be reduced into a canonical form as,
=E

X
T
X

=UU
T
(6.2)
189
where E [] is the expectation. The diagonal matrix = diag(
1
;:::;
n
)2 R
nn
con-
tains the ordered non-negative real eigenvalues (
1
> ::: >
n
) of the covariance ma-
trix. Since the matrix is symmetric positive denite, it is guaranteed that the matrix
U = [u
1
:::u
n
]2R
nn
is a full orthogonal set of real eigenvectors.
The best approximating subspace P = [u
1
:::u
r
]2R
nr
in terms of the variance of
the original data is then dened by the set of r dominant eigenfunctions of the covariance
matrix. The representation of the measured dataX on the principal component subspace
spanned byP is given by the linear mapping
T =XP (6.3)
In this case, the score matrixT2R
mr
consists of a set of uncorrelated latent variables
t =ft
j
g
r
j=1
ordered based on their corresponding variances
j
. The loss of information in
the low-dimensional projection can be estimated by re-mapping the PCA scores back to
the original space,
E =XTP
T
=X

I
n
PP
T

(6.4)
whereE2R
mn
denes the residual matrix, andI
n
2R
nn
is then-dimensional identity
matrix. Notice that the residual subspace spanned by I
n
PP
T
is orthogonal to the
principal components subspace spanned byP .
190
The corresponding monitoring statistics for a PCA model are dened as
T
2
=t
T

1
r
t =x
T
P
1
r
P
T
x
SPE =e
T
e =x
T

I
n
PP
T

x
(6.5)
where
r
is the diagonal matrix of the eigenvalues associated with the retainedr principal
components.
6.3.2 Independent Component Analysis
Independent component analysis (ICA) uses high-order statistics to decompose any given
dataset into a linear combination of statistically independent latent variables. Even though
several dierent algorithms have been proposed, one of the most well known ICA algo-
rithms is the fast xed-point ICA algorithm (FastICA) [34].
Consider that a set of n measured variablesx =fx
j
g
n
j=1
can be expressed as a linear
combination of a set of r statistical independent latent variables s =fs
j
g
r
j=1
with zero
mean (E[s
j
] = 0) and unit variance (E[s
2
j
] = 1), andrn. This relationship can then be
written as:
x =As +e (6.6)
whereA2R
nr
is the unknown mixing matrix, ande is a residual vector. The objective
of ICA consists therefore of estimating both the mixing matrix A and the independent
latent variabless, using only the observed variablesx. For this purpose, it is necessary to
191
nd a linear mappingW2R
rn
, so that the variables in the reconstructed set ^ s =f^ s
j
g
r
j=1
become as independent as possible. This linear transformation is then given as
s =Wx (6.7)
where WA =I
r
. Hereafter and without any loss of generality, it will be assumed that
r =n, unless otherwise it is specied.
The initial step in the implementation of ICA is to remove the cross-correlations among
the measured variables and obtain a set of mutually uncorrelated variablesz. This whiten-
ing or sphering can be achieved through classical PCA. Thus, the whitening transformation
can be expressed as
z =
1=2
U
T
x =Qx (6.8)
where Q2 R
nn
is the whitening matrix. The matrices and U are generated from
the eigendecomposition of the covariance matrix E[xx
T
] =UU
T
. From Eqs. (6.6) and
(6.8), the following relationship between the uncorrelated variablesz and the independent
latent variabless can be established
z =Qx =QAs =Bs (6.9)
192
whereB2R
nn
is an orthogonal matrix. Since s
j
are mutually independent and z
j
are
mutually uncorrelated, the following relationship is satised
E[zz
T
] =BE[ss
T
]B
T
=BB
T
=I
n
=B
T
B (6.10)
The independent components can now be estimated as follows
s =B
T
z =B
T
Qx (6.11)
By comparing Eqs. (6.7) and (6.11), the relationship between matricesW andB can be
expressed as
W =B
T
Q (6.12)
At this point, the ICA problem has been reduced to nding an orthogonal matrix B
that makes the elements in s statistically independent. The statistical independence of
each variable s
j
can be quantied using measurements of its non-Gaussianity, as kurtosis
or dierential entropy (negentropy) [34, 35]. An ecient and reliable approximation to
the negentropy is given as follows [34]:
J(y) (E [G(y)]E [G(v)])
2
(6.13)
193
where y is a variable with zero mean and unit variance, v is a Gaussian variable with the
same mean and variance asy, andG is an appropriate non-quadratic function. A suitable
general-purpose function for G is given by [33, 35]
G(u) =
1
a
log cosh(au) (6.14)
The fastICA algorithm, which is based on a xed-point iteration scheme, uses the approx-
imate form of negentropy and the Gram-Schmidt orthogonalization to nd each of the
columns b
j
of the matrix B, starting from an initial random guess, that maximize the
nongaussianity of b
T
j
z [34, 35]. Once the orthogonal matrix B has been obtained, the
independent components s and demixing matrix W can be calculated from Eqs. (6.11)
and (6.12), respectively.
Notice that at this point in the ICA algorithm, the n measured variablesx have just
been expressed as a linear combination of n statistically independent variables ^ s. In
order to achieve a dimension reduction with ICA, it is necessary to represent the observed
variables in terms of a small number of dominant components from all the estimated
independent components. However, ordering and determining how many independent
components should be extracted to establish a reduced-order ICA model are not trivial,
and there are no standard criteria [8, 15, 40, 47, 48]. In this study the Euclidean norm (L
2
)
is used to sort the rows of the matrix W and therefore, the corresponding independent
components. Once the independent components have been sorted, a low-dimensional
approximating subspace is obtained by selecting the r dominant rows of the matrix W .
194
From Eq. (6.6), the reconstruction error in the dimensionality-reduction projection can be
found as follows
e =xAs = (I
n
AW )x (6.15)
The implementation of the D-statistic, which is denoted in this case as I
2
, and the
Q-statistic for ICA monitoring are mathematically expressed as
I
2
=s
T
s =x
T
W
T
Wx
SPE =e
T
e =x
T

I
n
W
T
A
T

(I
n
AW )x
(6.16)
Notice that, since the independent components are assumed to have a unit variance, the
covariance matrixE[ss
T
] =I
r
, and therefore, the Mahalanobis distance is reduced to the
Euclidean distance in the latent-variable subspace.
6.3.3 Modied Independent Component Analysis
In order to overcome some drawbacks of the conventional ICA | random initialization
of the columns of matrix B can lead to dierent solutions, and no standard criterion
to sort the extracted independent components | Lee et al. [48] proposed a modied
independent component analysis (MICA) approach. The basic idea behind MICA is to
generate a consistent set of sortable independent components by rstly estimating a set
of few uncorrelated latent variables using PCA, and then transforming it into a set of
statistically independent latent variables by applying the fastICA algorithm.
195
Given a set of measured variables x = fx
j
g
n
j=1
, assume that there exists a linear
transformationW2R
rn
given by
y =Wx (6.17)
where the projected variables y
j
in the set y = fy
j
g
r
j=1
are statistically independent
and ordered by their variances, which are the same as the variances of the corresponding
principal components. Similar to traditional ICA, the rst step in MICA is to decorrelate
the observed variablesx through a whitening transformation using PCA. For convenience,
the whitening transformation (Eq. 6.8) is shown again below,
z =
1=2
t =
1=2
U
T
x =Qx (6.18)
where t =ft
j
g
n
j=1
is a set of uncorrelated variables, 2 R
nn
and U 2 R
nn
are the
eigenvalues and eigenvectors of the covariance matrix E[xx
T
], andzfz
j
g
n
j=1
corresponds
to the normalized PCA score vector with associated covariance matrix E[zz
T
] =I
n
.
One of the requirements established fory is that
E[yy
T
] =D (6.19)
196
where D = diag(
1
;:::;
r
)2 R
rr
consists of the rst r eigenvalues contained in the
matrix . In contrast to the original ICA algorithm, the modied ICA expresses the
independent components as a linear combination of the normalized PCA scores
y =C
T
z (6.20)
where the matrix C2R
nr
satises the condition C
T
C =D. A normalized version of
the previous equation is then dened by
y =D
1=2
y =D
1=2
C
T
z =

C
T
z (6.21)
where now,

C
T

C =I
r
and E[ y y
T
] =I
r
. Thus, the objective of MICA can be restated
as nding the matrix

C that transforms the normalized principal components z into a
set of statistically independent variables y. Although z is not independent, it has all
second-order statistical dependencies (i.e., mean and variance) removed, and therefore, it
can be used as an initial approximation to y [48]. The remaining high-order statistical
dependencies in y can then be reduced via fastICA. Consequently, the matrix

C can
be initialized as

C
T
= [I
r
j 0], where I
r
is the r-dimensional identity matrix and 0 is a
rectangular zero matrix of appropriate dimensions.
Once fastICA has found the optimal matrix

C that maximizes the negentropy of each
variable in y, the linear map W , from an original n-dimensional data space to a lower
197
r-dimensional latent space, and the corresponding inverse mapping A can be obtained
from
W =D
1=2

C
T

1=2
U
T
A =U
1=2

CD
1=2
(6.22)
where WA = I
r
. Similar to ICA, the error between the observed variables and the
predictions of the MICA model is given by
e =xAy = (I
n
AW )x (6.23)
The corresponding monitoring statistics for a MICA model are dened as
T
2
=y
T
D
1
y =x
T
W
T
D
1
Wx
SPE =e
T
e =x
T

I
n
W
T
A
T

(I
n
AW )x
(6.24)
6.3.4 Determination of Order of Dimensionality Reduction
A key issue in order-reduction of high-dimensional data sets is determining the dimension
of the low-dimensional approximating subspace (i.e., number of latent variables retained
in the statistical reduced-order model). The number of latent variables retained will de-
termine the quality and eectiveness of the reduced-order model. By choosing fewer latent
variables than required, the statistical model will not be able to characterize the dominant
underlying features in the data. On the other hand, by selecting more latent variables
than necessary, the model will be overparameterized and will include redundant informa-
tion and noise. Several dierent selection criteria have been developed and proposed to
198
determine the optimal number of latent variables. However, because the majority of these
criteria relied on monotonically increasing or decreasing indices, the decision to choose the
number of latent variables is still very subjective [69, 80].
Some of the most commonly used approaches in principal component analysis are
the Akaike information criterion (AIC), minimum description length (MDL), cumulative
percent variance (CPV), cross-validation, and variance of the reconstruction error (VRE).
A detailed description of these approaches can be found on Valle et al. [80].
6.4 Sensor fault detection and identication
The purpose of multivariate statistical monitoring using latent-variable techniques is, in
general, to continuously analyze and interpret high-dimensional, noisy, and correlated
sensor measurements in order to detect and identify abnormal sensor readings by using
statistical reduced-order models of the sensor-network measurements.
The reduced-order model, or latent-variable model, is built from historical or reference
measurements collected during fault-free and normal operating conditions of the sensors
in the network. When using latent-variable techniques, it is generally recommended to use
mean-centered and auto-scaled datasets. However, in cases where it is necessary to keep
the original importance of the measured variables, the data should not be scaled [16, 81].
Having established a latent-variable model, control limits for the corresponding mon-
itoring statistics (i.e., D-statistic and Q-statistic) during normal operating condition of
the sensor network have to be calculated. That is, obtain the upper condence limits that
determine whether the sensor measurements remain within a range of normal values and
only common-cause variations are present. In the specic case of PCA monitoring, the
199
condence limits for the Hotelling's T
2
and SPE statistics are generally computed with
approximate distributions by assuming that the latent variables follow a Gaussian distri-
bution [45, 68]. However, experimental data from an accelerometer-based sensor network
used in this study, showed that vibration measurements may not have such statistical
property. In addition, latent variables in ICA and MICA monitoring do not conform to
Gaussian distributions. Therefore, kernel density estimation is used to nd the condence
limits for the monitoring statistics [30, 48].
New sensor measurements are then continuously monitored and analyzed in order to
detect any type of atypical or abnormal operating situation associated with the eects of
sensor faults by statistically comparing the D-statistic and Q-statistic values of the new
observation against their established control limits for the normal operation of the sensor
network. In other words, sensor faults are detected if one of the monitoring statistics
(i.e., fault detection indices) is beyond its corresponding control limit. Once a fault is
detected, it is necessary to identify the sensor or group of sensors responsible for the
out-of-control condition. Contribution analysis methods examine the contributions of an
observed variable (i.e., sensor) to a monitoring statistic (i.e., fault detection index) in
a fault situation with the idea that the variables with large contribution are the likely
cause of the fault [5]. One of these methods is the partial decomposition contributions
(PDC), which decomposes a monitoring index as the summation of variable contributions
as follows
PDC
Index
j
=x
T
M
j

T
j
x (6.25)
200
where PDC
Index
j
is the contribution of the variable x
j
to a given monitoring statistic
Index, and
j
is the i-th column of the identify matrix. The notation of the D-statistics
and Q-statistics for PCA, ICA and MICA given in Eqs. (6.5), (6.16), and (6.24) can be
simplied in terms of a general index, Index(x), as
Index(x) =x
T
Mx (6.26)
Although contribution-based plots have been popularly used as simple fault identication
approaches that can be generated without prior fault knowledge [5, 48, 68], it has been
recently shown that some methods do not guarantee correct identication even for simple
sensor faults with large magnitudes [5].
Even though in this paper it was assumed that abnormal sensor readings were caused
solely by faulty sensors in the deployed network, it should be noted that abnormal readings
will also result from the presence of structural changes in the system being monitored,
or by variations in the environmental and/or operational conditions; therefore, the moni-
toring statistics may indicate an out-of-statistically-control situation. Implementation or
validation of methodologies to dierentiate the causes of out-of-control events are outside
the scope of this study, however, some general guidelines are presented below.
Assuming that failure of the sensors in a deployed network is an independent event and
that a malfunctioning sensor will not aect the other sensor readings, a possible approach
for distinguishing between out-of-control events in theD- andQ-statistics associated with
faulty sensors and structural changes is by reconstructing the \faulty" sensor readings
using the latent-variable models. More precisely, once a \faulty" sensor is identied or
201
isolated, the readings from that sensor are replaced by measurement predictions generated
using the measurements available from the remaining \healthy" sensors. If the monitoring
statistics return to an in-control situation, it can be inferred that the abnormal situation
was indeed caused by the identied faulty sensor. On the other hand, if the monitoring
statistics remain out-of-control, multiple sensors could have been involved in the abnor-
mal situation, and therefore, it could be attributed to structural changes in the system,
provided that the eects of these changes in the dynamic response of the structure were
captured by several of the deployed sensors. Now, in cases where sensor malfunctioning
or structural changes can be ruled out as the cause of out-of-control conditions, the pres-
ence of abnormal events can then be attributed to new normal operating conditions that
were not included or used to build the statistical latent-variable models. Sensor measure-
ments obtained from these new environmental or operational scenarios can then be used
to update the existing latent-variable models.
6.5 Application to case studies
To illustrate and compare the performance of statistical monitoring based on latent-
variable techniques for detecting and identifying faulty sensors, these methods were applied
to: (1) a virtual sensor network monitoring an analytical three-dimensional truss, and (2)
a real sensor network deployed on a cable-stayed bridge.
6.5.1 Analytical Three-dimensional Truss
The application of the proposed approaches will be initially illustrated using a numerical
model of a tridimensional 5-bay truss structure shown in Figure 6.1. This truss consisted of
202
Figure 6.1: Model of the three-dimensional truss
24 nodes and 97 members. One end of the truss is pinned to a vertical support plane. The
fundamental frequency of the truss is f
1
= 7:52 Hz. The rst fteen natural frequencies
were within 7{110 Hz. The structure was excited by four external forces applied directly,
in the vertical direction, at the bottom-chord nodes of the second and fth bay of the
truss. The applied forces were independent, band-limited, white-noise excitations with a
5{100 Hz bandwidth. The sensor network considered for this structure consisted of ten
accelerometers, with a sampling frequency of 250 Hz, placed at all the nodes in the top
chord of the truss. Five of the sensors measured accelerations in the vertical direction,
while the other ve did measure the lateral accelerations. A 5% RMS noise was added to
the accelerations obtained from the numerical simulation.
Two acceleration data sets were obtained from numerical simulations. One data set
was used as the reference measurements to build the latent-variable models of the sen-
sor network during normal operating condition (i.e., healthy state) and dene the corre-
sponding control limits for the fault detection indices (i.e., monitoring statistics). The
other dataset was used to generate dierent scenarios by simulating four sensor failure
modes: bias, drift, scaling, and hard fault. Both the training and testing data sets were
comprised of 15001 samples (i.e., 60 seconds duration). In each faulty scenario, the sensor
203
0 10 20 30 40 50 60
−1
−0.5
0
0.5
1
1.5
t
S7
0 10 20 30 40 50 60
−1
−0.5
0
0.5
1
1.5
2
t
S7
(a) Bias (b) Drift
0 10 20 30 40 50 60
−2
−1
0
1
2
t
S7
0 10 20 30 40 50 60
−1
−0.5
0
0.5
1
t
S7
(c) Scaling (d) Hard fault
Figure 6.2: Acceleration measurements in sensor 7 after simulating a bias, drift, scaling,
and hard-fault type of sensor faults
fault was introduced at 20 seconds (i.e., sample 5001) and continued until the end of the
time-history record. The four simulated sensor faults are shown in Figure 6.2.
After mean-centering and scaling into unit variance all the measured variables, the
PCA, ICA and MICA models are built using the reference/normal observations. Based on
the cumulative percent variance (CPV) criterion, six principal components were selected
to be retained in the model so that over 90% of the total variance of the normal operating
data was explained by the PCA model. Six latent variables were also retained in the ICA
and MICA models for fair comparison. For illustrative purposes, in this study, a second
D-statistic (T
2
e
for PCA and MICA, andI
2
e
for ICA), which is based on the excluded latent
variables (i.e., latent variables that were not used to build the statistical reduced-order
model) was considered just to show the eects sensor faults could have on the excluded
space. Even though this index can be used as an additional fault detection index that can
204
also compensate for the errors that could result when a non-optimal number of latent vari-
ables are retained in the reduced-order models [47, 73], the fault detection will be carried
out only with the standard D-statistic (i.e., T
2
and I
2
) and Q-statistic (i.e., SPE). The
99% condence limits of the D-statistics and Q-statistic for the PCA, ICA, and MICA
models were obtained using kernel density estimation techniques. The monitoring statis-
tics for each of the three models and the corresponding cumulative distribution functions
(cdf) during normal operation conditions of the sensor network are shown in Figure 6.3.
The horizontal dashed line indicates the corresponding 99% condence control limits.
The PCA, ICA, and MICA monitoring results for the bias fault case are shown in
Figure 6.4. From Figure 6.4a, it can be seen that for this sensor fault, the PCA-based
model was not able to detect the abnormal condition in itsT
2
monitoring chart. However,
the presence of the fault could be clearly detected in the monitoring charts forT
2
e
andSPE,
which are indices associated with the excluded and residual space, from the time t = 20
seconds up to the end of the timet = 60 seconds. The detection rates of theT
2
andSPE,
dened as the percentage of samples outside the 99% control limits, were 3.69% and 21.76%
respectively. In contrast to the PCA monitoring, the ICA and MICA monitoring charts in
Figures 6.4b and 6.4c showed that the bias fault could be detected by their corresponding
I
2
and T
2
statistics, and therefore, the eects of the introduced bias fault were captured
by the constructed ICA and MICA reduced-order models. It can also be noted that the
eects of the fault could be observed in the excluded latent variable space and detected
by the SPE statistic, and the I
2
e
and T
2
e
for the ICA and MICA respectively. The
detection rates of I
2
and SPE for the ICA-based approach were correspondingly 10.88%
and 12.81%, whereas the rates of the monitoring statistics T
2
and SPE for the MICA
205
0 1
0
10
20
30
cdf
T
2
0 10 20 30 40 50 60
t
0 1
0
10
20
30
cdf
T
2
e
0 10 20 30 40 50 60
t
0 1
0
2
4
6
8
cdf
SPE
0 10 20 30 40 50 60
t
0 1
0
10
20
30
cdf
I
2
0 10 20 30 40 50 60
t
0 1
0
5
10
15
20
25
cdf
I
2
e
0 10 20 30 40 50 60
t
0 1
0
10
20
30
40
cdf
SPE
0 10 20 30 40 50 60
t
(a) PCA monitoring (b) ICA monitoring
0 1
0
10
20
30
40
cdf
T
2
0 10 20 30 40 50 60
t
0 1
0
5
10
15
20
25
cdf
T
2
e
0 10 20 30 40 50 60
t
0 1
0
10
20
30
40
cdf
SPE
0 10 20 30 40 50 60
t
(c) MICA monitoring
Figure 6.3: Monitoring charts for the D-statistics and Q-statistic obtained under normal
operating conditions of the sensor network for the (a) PCA model, (b) ICA model, and
(c) MICA model. The upper control limits correspond to 99% of condence level.
206
model were 13.49% and 8.24%. Even though detection rates achieved with ICA monitoring
and MICA monitoring were below the the maximum detection rate obtained with the
PCA monitoring, the ICA- and MICA-based approaches exhibited better robustness in
detection performance since the fault could be detected in both the reduced-order model
space and the residual space.
In order to visualize the eects the bias fault in sensor 7 had in the retained and
excluded latent variables, the biplots of the latent variables 1 and 2 (i.e., rst two retained
latent variables in the reduced-order representation of the original data space), and latent
variables 7 and 8 (i.e., rst two excluded latent variables) were generated and are shown
in Figures 6.5 and 6.6, respectively. A biplot allows the visualization of each variable's
contribution to a specic set of latent variables, and how each observation is represented
in terms of those components. The axes in each biplot represent the latent variables,
and the measured variables (i.e., sensors) are therefore represented as vectors. The latent
variable scores are represented as points in the biplots. The scores plotted in the top row
of Figures 6.5 and 6.6 correspond to the nominal measurements (i.e., before introducing
the bias fault), while the bottom row of plots show a sample of latent variable scores
after the bias fault were introduced. In Figures 6.5a and 6.6a, the biplots for the PCA-
based approach are shown. Notice that a shift in the PCA scores along the direction of
sensor 7 is observed in the excluded latent variable space, while no signicant changes
occurred within the space spanned by the dominant latent variables that were retained in
the PCA model. Therefore, the fault could be easily detected by the monitoring statistics
T
2
e
and SPE, which are associated with the excluded latent variable and residual space,
respectively. This is exactly what was observed in the PCA monitoring charts (Figure
207
0 10 20 30 40 50 60
0
10
20
30
40
t
T
2
0 10 20 30 40 50 60
0
10
20
30
40
50
t
T
2
e
0 10 20 30 40 50 60
0
5
10
15
t
SPE
0 10 20 30 40 50 60
0
10
20
30
40
50
t
I
2
0 10 20 30 40 50 60
0
10
20
30
40
t
I
2
e
0 10 20 30 40 50 60
0
20
40
60
t
SPE
(a) PCA monitoring (b) ICA monitoring
0 10 20 30 40 50 60
0
10
20
30
40
t
T
2
0 10 20 30 40 50 60
0
10
20
30
40
50
t
T
2
e
0 10 20 30 40 50 60
0
20
40
60
t
SPE
(c) MICA monitoring
Figure 6.4: Monitoring charts for the D-statistics and Q-statistic obtained in the bias
sensor fault case for the (a) PCA model, (b) ICA model, and (c) MICA model.
208
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
PC 1
PC 2
Nominal
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
PC 1
PC 2
Bias fault
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5
0
0.5
1
1.5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
IC 1
IC 2
Nominal
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5
0
0.5
1
1.5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
IC 1
IC 2
Bias fault
−2 −1 0 1 2
−2
−1
0
1
2
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
MIC 1
MIC 2
Nominal
−2 −1 0 1 2
−2
−1
0
1
2
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
MIC 1
MIC 2
Bias fault
(a) PCA monitoring (b) ICA monitoring (c) MICA monitoring
Figure 6.5: Visualization of the original measured variables (i.e., sensors) and the corre-
sponding representation of the measured data (i.e., sensor measurements) in the space of
the rst two retained latent variables (i.e., model space) in the reduced-order model for
the scenario with a bias fault in sensor 7.
6.4a). Similar observations can be drawn for the ICA- and MICA-based approaches. It
can be seen in Figures 6.5b and 6.5c that, in contrast to the PCA model, the presence of
the bias fault could be observed in the space spanned by the latent variables retained in
the ICA and MICA models, as a shift in the corresponding scores along the direction of
sensor 7. It can also be seen, from Figures 6.6b and 6.6c, that the eects of the fault were
observable within the excluded latent variables too.
As was previously discussed, once a sensor fault is detected, the identication of
the faulty sensor or group of sensors responsible for the out-of-control condition can be
achieved through the use of the partial decomposition contributions (PDC). In this study,
the upper control limit for the contribution plots of the monitoring statistics for each vari-
able are calculated as the mean of the contributions plus three standard deviations. Even
though these limits are not considered to have a statistical signicance, they are very
209
−0.5 0 0.5
−0.5
0
0.5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
PC 7
PC 8
Nominal
−0.5 0 0.5
−0.5
0
0.5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
PC 7
PC 8
Bias fault
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
IC 7
IC 8
Nominal
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
IC 7
IC 8
Bias fault
−0.5 0 0.5
−0.5
0
0.5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
MIC 7
MIC 8
Nominal
−0.5 0 0.5
−0.5
0
0.5
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
MIC 7
MIC 8
Bias fault
(a) PCA monitoring (b) ICA monitoring (c) MICA monitoring
Figure 6.6: Visualization of the original measured variables (i.e., sensors) and the corre-
sponding representation of the measured data (i.e., sensor measurements) in the space of
the rst two excluded latent variables for the scenario with a bias fault in sensor 7.
helpful in detecting the contributions that are higher than the contributions of normal
operating condition data [48, 83].
Figure 6.7 displays the contributions (i.e., PDC) of the sensors to the monitoring
statistics in an out-of-control observation at sample 7501 (e.g., time t = 30 s) for each
of the latent variable approaches under discussion. From the contribution plot for the
PCA monitoring statistics, it is seen that sensor 7 exceeded its control limits and made
the largest contribution to the T
2
, T
2
e
, and SPE statistics. In the contribution plot for
T
2
, it can be also observed that sensor 2 had an important contribution to the abnormal
condition (i.e., out-of-control). From Figure 6.5a, it is clear that the vectors representing
sensors 2 and 7 are very close to each other, therefore, a contribution to the out-of-control
condition caused by the bias fault in sensor 7 can be mistakenly attributed to sensor 2. In
other words, the closest the sensors are in the latent variable space, the isolability of faults
occurring in any of those sensors will decrease. Similarly, the contributions of sensor 2
210
and 9 to theT
2
e
andSPE can be explained by looking at the representation of the sensors
in the excluded latent variable space in Figure 6.6a. Notice that sensors 2, 7 and 9 are in
the direction of the observed shift in the PCA scores. From the contribution plots for the
ICA and MICA monitoring statistics, which are shown in Figure 6.7b and Figure 6.7b,
it is observed that sensor 7 made the largest contribution to the out-of-control condition.
It is then clear that these contribution plots correctly indicated the major variable (i.e.,
sensor) aected by the bias fault.
The key ndings of this study after analyzing the fault detection and identication for
the drift, scaling, and hard-fault sensor failure modes were analogous to those previously
described for the bias fault case. The fault detection rates obtained using PCA, ICA, and
MICA monitoring for the sensor failures modes introduced to measurements in sensor 7,
are summarized in Table 6.2. In can be observed again, from Table 6.2, that the detection
rates achieved with ICA monitoring and MICA monitoring were below the maximum
detection rate obtained with the PCA monitoring. However, the ICA- and MICA-based
approaches exhibited better robustness in detection performance since the fault could be
detected in both the reduced-order model space and the residual space. In the related
literature on multivariate statistical monitoring using ICA and MICA, it has been shown
that the ICA- and MIC-based approaches detect faults more eciently than PCA, in
particular, when the measured/observed variables had non-Gaussian characteristics [47,
48]. The results reported by Hernandez-Garcia and Masri [30] also showed that the ICA
and MICA methods had better fault-detection capabilities than PCA when the monitored
system exhibited nonlinear characteristics. The slightly better detection rates observed
211
1 2 3 4 5 6 7 8 9 10
0
5
10
15
Sensor
PDC
T
2
1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
Sensor
PDC
T
2
e
1 2 3 4 5 6 7 8 9 10
0
1
2
3
Sensor
PDC
SPE
1 2 3 4 5 6 7 8 9 10
0
1
2
3
Sensor
PDC
I
2
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
Sensor
PDC
I
2
e
1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
Sensor
PDC
SPE
(a) PCA monitoring (b) ICA monitoring
1 2 3 4 5 6 7 8 9 10
0
5
10
15
Sensor
PDC
T
2
1 2 3 4 5 6 7 8 9 10
0
10
20
30
Sensor
PDC
T
2
e
1 2 3 4 5 6 7 8 9 10
0
10
20
30
Sensor
PDC
SPE
(c) MICA monitoring
Figure 6.7: Variable contribution plots to the D- and Q-statistics obtained for an out-
of-control observation in the bias sensor fault case for the (a) PCA monitoring, (b) ICA
monitoring, and (c) MICA monitoring.
212
Table 6.2: Summary of fault detection rates of PCA, ICA, and MICA monitoring for
dierent types of sensor faults
PCA monitoring ICA monitoring MICA monitoring
Sensor Fault T
2
SPE I
2
SPE T
2
SPE
Bias 3.69% 21.76% 10.88% 12.81% 13.49% 8.24%
Drift 6.41% 35.47% 20.55% 25.44% 26.34% 18.04%
Scaling 4.51% 7.41% 4.07% 8.29% 6.19% 7.38%
Hard Fault 1.60% 29.87% 16.17% 11.37% 17.66% 4.95%
for the PCA-based approach in this illustrative example could be attributable to the use
of the dynamic response of a linear system subjected to Gaussian white-noise excitations.
6.5.2 Cable-stayed Bridge
The Vincent Thomas bridge, completed in 1964, is located in San Pedro, California, and
connects Los Angeles with Terminal Island in the Port of Los Angeles. It is a a cable-
suspension bridge of approximately 1850 m and consisting of a main span of 457 m,
two suspended side spans of 154 m each, and two 10-span concrete approaches of 545 m
on both ends. In 1980 a strong-motion sensor network, consisting of 26 unidirectional
accelerometers and a digital recording system, was deployed on the bridge. The sensor
network is currently maintained by the California Strong Motion Instrumentation Program
(CSMIP) of the California Geological Survey (CGS). A layout of the location and direction
of all 26 sensors mounted on the bridge is shown in Figure 6.8. Since 2005, the Vincent
Thomas bridge (VTB) has been monitored with a web-based real-time monitoring system
developed at the University of Southern California [82, 88]. After an upgrade to the web-
based monitoring system, it was noticed that atypical measurements were being recorded
by several channels corresponding to sensors located on the west tower and main span of
the bridge. A signal-processing analysis of the acceleration measurements revealed that
213
consisting of a main span of 457m, two suspended side spans of 154m each, and two 10-span
cast-in-place concreteapproaches of545-m length on both ends. The roadway is 16-m wideand
accommodates four lanes of traffic. The bridge was completed in 1964 with 92000ton of
Portland cement, 13000ton of light weight concrete, 14100ton of steel and 1270ton of
suspension cables. The bridge was designed to withstand winds of up to 145kmph. A major
seismicretrofitwasperformedduringtheperiod1996–2000,includingavarietyofstrengthening
measures, and the incorporation of about 48 large-scale nonlinear passive viscous dampers.
2.2. VTB instrumentation
The VTB has been instrumented by the California Strong Motion Instrumentation Program
(CSMIP) of the California Geology Services (CGS), formerly known as the Division of Mines
andGeology(CDMG),formore than20years. Thestrong-motionrecordingsystemconsists of
26 accelerometers mounted on the bridge and an original analog recording system (later
converted to a digital recording system) located in the east anchor block. Figure 2 shows the
sensor locations for this system.
Significant motions have been recorded for the 1987 Whittier, 1994 Northridge, and several
other earthquakes. Analysis of these recordings has provided much information about the
dynamicresponseoflarge suspension bridges.The previous analogfilm recordingsystem, (used
until the mid-1990s) has proven to be very reliable, but the recorded data were limited in
dynamic range and difficult to convert to digital format appropriate for computer analysis.
Modern digital recording technology certainly can provide superior data quality and ease of
analysis. To demonstrate this, a temporary digital monitoring system with remote commu-
nications capability was installed in parallel with the existing analog recording system for the
VTB strong motion instrumentation between 3rd November and 5th December 1995. During
thisshorttimeperiod,alargeamountofambientvibrationdatawasrecorded.Thecapabilityof
remote real-time data monitoring was also demonstrated.
24
25
26
9
13
19
20
1
23
14
3
4
15 16
5
17 18
6
12
7
21
22
10
11
8
2
North
457.2 m
Figure2. Sensor locations and directions on the Vincent Thomas Bridge, San Pedro, CA.
COLLISION INCIDENT OF VTB 187
Copyright# 2007 John Wiley & Sons, Ltd. Struct. Control Health Monit. 2008; 15:183–206
DOI: 10.1002/stc
Figure 6.8: Location and direction of the deployed sensors on the Vincent Thomas Bridge,
San Pedro, CA.
the records from 12 out of 26 channels were abnormal. These channels corresponded to
sensors 1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18, and 23. Based on the fact that the sensors of
the aected channels clustered around the west tower and the main span, the cause of
the abnormal measurements was initially attributed to cabling issues rather than faulty
sensors. Even though this unusual situation was unrelated to the accelerometers, as CGS
later conrmed it after remotely testing each of the sensors, it provided an opportunity to
evaluate the use of statistical monitoring based on latent variable techniques for detection
of abnormal events and faults in sensor networks.
Because the hardware and software of web-based monitoring system had just been
upgraded when the abnormal event took place, there were no reference measurements
available that were recorded during normal operation with the new conguration of the
system. Therefore, one hour of acceleration measurements recorded once the sensor net-
work was serviced and the problem xed, were used as the reference data set to build
the latent variable models for statistical monitoring. The number of component used to
214
construct the PCA model was selected by the cumulative percent variance (CPV) crite-
rion. The rst ten principal components captured over the 90% of the total variance of
measurements during normal operating conditions. The number of ICA and MICA latent
variables was chosen also as ten, for fair comparison. The testing data set was generated
from a one-hour long set of measurements taken during the abnormal operating condition.
The horizontal dashed lines in Figure 6.9 represent the 99% control limits for the D- and
Q-statistic obtained during normal operation of the sensor network.
Figure 6.9 displays the PCA, ICA, and MICA monitoring charts for the rst eight-
minutes of acceleration measurements from the testing data set. As can be seen from these
monitoring charts, neither the PCA-, ICA-, nor the MICA-based monitoring approaches
were able to detect the abnormal condition in the sensing system, even though the mea-
surements, in almost half of the total number of channels in the sensor network, were
completely abnormal. Samples of normal and abnormal acceleration measurements from
sensors 5, 10, 16, and 22 are shown in Figure 6.10. From this gure, it is clear that the
channels involved in the abnormal condition experienced a complete loss of sensor data;
and, for the purposes of this study, this scenario can be considered as \complete failure"
of multiple sensors. Based on previous results, as well as results reported in related liter-
ature, it was initially assumed that the approaches implemented in this study would be
able to easily detect such scenario since the sensor fault detection performance depends
on the magnitude of the fault and quantity of faulty sensor; however, this was not the
case.
215
A further analysis of the implementation of the statistical monitoring approaches rev-
eled that these methodologies were not sensitive to scaling-type sensor faults with a multi-
plicative factor
(t)< 1:0 (see Table 6.1). Consider, without any loss of generality, a set of
noise-free measurements from sensors with multiplicative-type failure given byx
f
=
x,
where
contains the scaling factors, x is a vector of nominal sensor measurements, and
the operator represents the element-by-element multiplication. From Equation (6.5),
the corresponding PCA monitoring statistics for the faulty measurements will be given by
T
2
= (
x)
T
P
1
r
P
T
(
x)
SPE = (
x)
T

I
n
PP
T

(
x)
(6.27)
In the extreme case of complete failure of all sensors (i.e.,
= 0), T
2
and SPE will be
equal to zero and therefore, the sensor faults will not be detected. Identical conclusions
can be drawn for the ICA- and MICA monitoring methodologies.
6.6 Summary and conclusions
Three multivariate statistical approaches based on latent variable models developed from
historical data were examined. The PCA, ICA, and MICA approaches for monitoring a
sensor network are simple, straight-forward, and potentially useful techniques for struc-
tural health monitoring applications in complex systems equipped with densely-distributed
sensor networks because it is possible to build a reduced-order statistical model of the sen-
sor network just by using sensor measurements and by avoiding unnecessary assumptions
216
0 50 100 150 200 250 300 350 400 450 500
0
50
100
150
t
T
2
0 50 100 150 200 250 300 350 400 450 500
0
50
100
150
t
SPE
0 50 100 150 200 250 300 350 400 450 500
0
50
100
150
200
t
I
2
0 50 100 150 200 250 300 350 400 450 500
0
20
40
60
80
100
t
SPE
(a) PCA monitoring (b) ICA monitoring
0 50 100 150 200 250 300 350 400 450 500
0
50
100
150
200
250
t
T
2
0 50 100 150 200 250 300 350 400 450 500
0
20
40
60
80
t
SPE
(c) MICA monitoring
Figure 6.9: Monitoring charts for the D-statistic and Q-statistic obtained in the bias
sensor fault case for the (a) PCA model, (b) ICA model, and (c) MICA model.
and the construction of mathematical models. Additionally, these approaches are rela-
tively simpler and quicker to implement than model-based approaches, because of their
data-driven nature.
This study also highlights the promising capabilities, as well as the limitations, of sta-
tistical monitoring based on latent variable models for sensor fault detection and identi-
cation in sensor networks deployed on aerospace, civil and mechanical structures. Overall,
the PCA-, ICA-, and MICA-based monitoring approaches showed similar performance in
217
0 1 2 3 4 5 6 7 8 9 10
−0.02
−0.01
0
0.01
0.02
t
S5
0 1 2 3 4 5 6 7 8 9 10
−0.02
−0.01
0
0.01
0.02
t
S10
0 1 2 3 4 5 6 7 8 9 10
−0.02
0
0.02
t
S16
0 1 2 3 4 5 6 7 8 9 10
−0.05
0
0.05
t
S22
0 1 2 3 4 5 6 7 8 9 10
−0.02
−0.01
0
0.01
0.02
t
S5
0 1 2 3 4 5 6 7 8 9 10
−0.02
−0.01
0
0.01
0.02
t
S10
0 1 2 3 4 5 6 7 8 9 10
−0.03
−0.02
−0.01
0
0.01
0.02
t
S16
0 1 2 3 4 5 6 7 8 9 10
−0.05
0
0.05
t
S22
(a) Normal operating condition (b) Abnormal condition
Figure 6.10: Comparison of acceleration measurements; from sensors 5, 10, 16, and 22;
obtained under normal operating and abnormal conditions.
the detection of sensor faults. However, since ICA and MICA consider higher-order statis-
tics, the use of these techniques is especially suited for cases where the sensor measurements
would exhibit non-Gaussian characteristics, as in structures with inherent nonlinearities.
It was also observed that these latent variable-based approaches are more sensitive to
additive-type sensor faults (i.e., bias, drift), and multiplicative-type with scaling factors
larger than one. On the other hand, multiplicative-type faults with scaling factors smaller
than one, and sensor faults with a complete loss of signal could not be detected by any
of the implemented latent-variable monitoring approaches. Although contribution-based
methods are popularly used to perform fault identication, their results can be misleading
since they will basically select the sensors with large contributions as the likely faulty
sensors.
218
It is also necessary, in order to successfully integrate these methodologies in the struc-
tural health monitoring framework, to develop and implement methodologies for distin-
guishing among possible sources or causes (e.g., faulty sensor, structural changes, varia-
tions in operational or environmental conditions) of the abnormal sensor readings in the
deployed sensor network.
219
Chapter 7
Conclusions
This study shows that uncertain nonlinear dynamic systems can be analyzed by imple-
menting the Stochastic Restoring Force Method in conjunction with Polynomial Chaos
approaches. This straightforward implementation permits a robust characterization of
the model uncertainties in terms of stochastic power-series coecients. Using this rep-
resentation of the uncertain system and PCE approach for solving stochastic dierential
equations, it is possible to predict accurately the time evolution of dynamical systems in
the presence of stochastic uncertainty.
Analysis of the sample experimental results presented above has shown that a variety
of system identication approaches are capable of identifying, with reasonable accuracy,
the global structural dynamics characteristics (such as frequency values and corresponding
damping parameters) of the test article. The advantage of global identication approaches
such as the ones discussed above, is that no assumptions are made about the topology of
the structure being analyzed, and they do detect equivalent-linear changes in the under-
lying system.
220
However, if the topology of the structure resembles a chain-like system, then a powerful
sub-structuring approach can be used to decompose the structure into its constituent
elements, thus leading to a very ecient system identication approach that, not only
is more sensitive to local damage detection eects, but can also accurately distinguishes
structural changes associated with changes in the linear system properties, as opposed to
changes associated with inherent nonlinear phenomena within the specic substructure
of interest. Furthermore, the nonlinear nonparametric identication scheme used in this
study is ideally suited to dealing with complex nonlinear phenomena that are not amenable
to simple parametric representation.
As long as the eects of operational and environmental conditions, intrinsic nonlin-
earities, and underlying damage mechanisms in a target structure are re
ected in the
monitored structural dynamic response, then the nonparametric method under discussion
can provide a useful tool to accurately detect, locate (within each decomposed structural
region), and quantify the changes in the structure, as re
ected in the identied restoring
force surface.
In this experimental study it was also shown that, reduced-order models identied
using input-output data from experimental tests can be used to estimate physical changes
in the structures. Variations in the estimated mass-normalized stiness-like parameters of
the reduced-order models were then used to detect the presence, and infer the location,
of the actual structural changes made in the test structures. The results of this study
showed that the statistically signicant changes identied in the stiness-like parameters
221
of the reduced-order models could be correlated to the presence and location of the ac-
tual physical changes made to the testbed structures, even in the presence of modeling,
measurement, and data processing errors using reduced-order representations.
Finally, it was also shown that multivariate statistical approaches based on latent vari-
able models developed from historical data are simple, straight-forward, and potentially
useful techniques for detection and identication of faulty sensors in structural health
monitoring applications of complex systems equipped with densely-distributed sensor net-
works. These techniques allow the building of reduced-order statistical model of the sensor
network just by using sensor measurements and by avoiding unnecessary assumptions and
the construction of mathematical models.
222
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Abstract (if available)

Abstract Most of the available data‐based methodologies developed for system identification and health monitoring of complex nonlinear systems can be considered to be deterministic in nature. These approaches use experimental measurements to characterize the complex systems by means of nominal mathematical (e.g., parametric or non‐parametric) models, while neglecting the effects of aleatory and epistemic uncertainties that can be present in real structures. The inherent stochastic nature of the systems components (i.e., randomness in structural, geometric and material properties)

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Creator Hernandez-Garcia, Miguel Ricardo (author)

Core Title Analytical and experimental studies in modeling and monitoring of uncertain nonlinear systems using data-driven reduced‐order models

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 07/18/2014

Defense Date 05/08/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag nonlinear systems,OAI-PMH Harvest,reduced‐order models,Structural health monitoring,system identification,uncertainty quantification

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Masri, Sami F. (committee chair), Ghanem, Roger G. (committee member), Wellford, L. Carter (committee member)

Creator Email miguelrh@usc.edu,miguelricardohernandez@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-447047

Unique identifier UC11286766

Identifier etd-HernandezG-2719.pdf (filename),usctheses-c3-447047 (legacy record id)

Legacy Identifier etd-HernandezG-2719.pdf

Dmrecord 447047

Document Type Dissertation

Format application/pdf (imt)

Rights Hernandez-Garcia, Miguel Ricardo

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA