![](media/image1.jpeg){width="2.240972222222222in"
height="0.8215277777777777in"}

METRANS RESEARCH PROJECT

Project 01-10

PARAMETRIC FREQUENCY DOMAIN IDENTIFICATION\
USING VARIABLE STIFFNESS AND DAMPING DEVICES

FINAL REPORT

June 2003

Prof. Erik A. Johnson

Principal Investigator, Assistant Professor

Department of Civil and Environmental Engineering

University of Southern California

Mohamed I.S. Elmasry

Graduate Research Assistant

Department of Civil and Environmental Engineering

University of Southern California

PARAMETRIC FREQUENCY DOMAIN IDENTIFICATION\
USING VARIABLE STIFFNESS AND DAMPING DEVICES

Prof. Erik A. Johnson

Principal Investigator, Assistant Professor

Department of Civil and Environmental Engineering

University of Southern California

Mohamed I.S. Elmasry

Graduate Research Assistant

Department of Civil and Environmental Engineering

University of Southern California

June 2003

# ABSTRACT

Accurate diagnosis of structural health is a vital step in protecting
structures. Whether caused by acute events, such as earthquakes or other
natural disasters, or long-term degradation from environ­ment and human
use (and abuse), structural damage can threaten both human life and
economic loss. An autonomous monitoring system that has the capability
of predicting the location of damage would have a positive economic
impact, not to mention the potential for saving lives by giving quick
assess­ments of structural health and whether immediate evacuation of, or
re-routing, around a structure is necessary. The process of monitoring
structural health and identifying damage severity and location is
generally termed *struc­tural health monitoring* (SHM). The core of this
project is to investigate how variable stiffness and damping devices
(VSDD) can be most effectively used to identify local damage in bridge
and building structures. Using one or more VSDDs to modify the response,
simulated damage is detected, localized, and quantified. One fundamental
goal is to determine the best VSDD actions --- whether adding damping or
stiffness --- to precisely and robustly locate and identify bridge
damage. This study is based on simulation of bridge motion and other
structures due to ambient excitation sources. Several VSDD behaviors,
such as variable stiffness mode and variable damping mode, are studied
in the context of a frequency domain analysis. These analyses are
performed on simple models of bridge and building dynamics, *e.g.*, two
degree-of-freedom (2DOF) shear model systems and a six degree-of-freedom
(6DOF) model. Structural behavior is assumed locally linear before
damage and after damage. Generally, VSDDs were successful in improving
the damage identification in structures using the variable stiffness
mode but rather unsuccessful in the variable damping only mode. In
addition to demonstrating the potential of using variable stiffness and
damping devices to improve structural health monitoring, this study also
provides some insights into further avenues of future research to build
on the improvements studied herein.

**Keywords:** structural health monitoring, variable stiffness and
damping devices, parametric frequency-domain identification, damage
detection, structural control.

# TABLE OF CONTENTS

[**ABSTRACT ii**](#abstract)

[TABLE OF CONTENTS iv](#table-of-contents)

[TABLE OF FIGURES v](#table-of-figures)

[LIST OF TABLES vi](#list-of-tables)

[DISCLAIMER vii](#disclaimer)

[ACKNOWLEDGMENTS viii](#acknowledgments)

[1.0 INTRODUCTION 1](#introduction)

[1.1 SHM Benefits 1](#shm-benefits)

[1.2 Difficulties of Conventional SHM Approaches
1](#difficulties-of-conventional-shm-approaches)

[1.3 Use VSDDs to Improve SHM 3](#use-vsdds-to-improve-shm)

[1.4 Overview of this VSDD/SHM Research Project
4](#overview-of-this-vsddshm-research-project)

[2.0 LITERATURE REVIEW 6](#literature-review)

[2.1 Research on Structural Health Monitoring
6](#research-on-structural-health-monitoring)

[2.2 Modelling of Structures for SHM
7](#modelling-of-structures-for-shm)

[2.3 System Identification for SHM 7](#system-identification-for-shm)

[2.3.1 Frequency Domain Techniques 8](#frequency-domain-techniques)

[2.3.2 Time Domain Techniques 8](#time-domain-techniques)

[2.3.3 ERA and Subspace Identification Techniques
9](#era-and-subspace-identification-techniques)

[2.3.4 Structural Model Parameter Identification
9](#structural-model-parameter-identification)

[2.4 Variable Stiffness/Damping Devices
10](#variable-stiffnessdamping-devices)

[2.4.1 Passive, Active and Semiactive Devices
10](#passive-active-and-semiactive-devices)

[2.4.2 Types of Semiactive Devices 11](#types-of-semiactive-devices)

[2.4.3 Applications of Semiactive Control to Civil Structures
11](#applications-of-semiactive-control-to-civil-structures)

[2.4.4 Experimental and Full Scale Studies Using Semiactive Devices
12](#experimental-and-full-scale-studies-using-semiactive-devices)

[3.0 PROJECT OBJECTIVE AND SUMMARY 14](#project-objective-and-summary)

[4.0 PARAMETRIC FREQUENCY DOMAIN IDENTIFICATION WITH VSDDS
16](#parametric-frequency-domain-identification-with-vsdds)

[4.1 Least-Squares Numerator Method (LSN)
16](#least-squares-numerator-method-lsn)

[4.2 Iterative-Least Squares Numerator Method (ILSN)
18](#iterative-least-squares-numerator-method-ilsn)

[4.3 Illustrative Examples 18](#illustrative-examples)

[4.3.1 Two Degree-of-Freedom Shear Building Model
18](#two-degree-of-freedom-shear-building-model)

[4.3.2 Two Degree-of-Freedom Pier-Deck Bridge Model
20](#two-degree-of-freedom-pier-deck-bridge-model)

[4.3.3 Six Degree-of-Freedom Shear Building Model
21](#six-degree-of-freedom-shear-building-model)

[4.4 NUMERICAL EXAMPLES AND ANALYSIS OF RESULTS
22](#numerical-examples-and-analysis-of-results)

[4.4.1 Two Degree-of-Freedom Shear Building Model
22](#two-degree-of-freedom-shear-building-model-1)

[4.4.2 Two Degree-of-Freedom Pier-Deck Bridge Model
30](#two-degree-of-freedom-pier-deck-bridge-model-1)

[4.4.3 Six Degree-of-Freedom Shear Building Model
33](#six-degree-of-freedom-shear-building-model-1)

[4.5 Using VSDDs with Variable Damping Only
35](#using-vsdds-with-variable-damping-only)

[5.0 CONCLUSIONS AND RECOMMENDATIONS
37](#conclusions-and-recommendations)

[Appendix A: Computational Procedure for 6DOF Shear Model
39](#appendix-a-computational-procedure-for-6dof-shear-model)

[REFERENCES 42](#references)

# TABLE OF FIGURES

[Figure 1. Beam-column connection damage (Kiremedjian, 1999)
2](#__RefHeading___Toc44904747)

[Figure 2. Plastic hinging at top of column (Hall, 1995)
2](#__RefHeading___Toc44904748)

[Figure 3. Cracks through column flange and extending into web (Hall,
1995) 2](#__RefHeading___Toc44904749)

[Figure 4. SHM and variable stiffness/damping flow charts
3](#__RefHeading___Toc44904750)

[Figure 5. Mutual benefits of SHM and VSDDs
4](#__RefHeading___Toc44904751)

[Figure 6. ISB vibration absorber on I-35 bridge (Patten *et al*., 1999)
13](#__RefHeading___Toc44904752)

[Figure 7. 2DOF shear building model 19](#__RefHeading___Toc44904753)

[Figure 8. VSDD in 2^nd^ story of 2DOF model
20](#__RefHeading___Toc44904754)

[Figure 9. VSDDs in both stories of 2DOF model
20](#__RefHeading___Toc44904755)

[Figure 10. General view during construction of high occupancy vehicle
(HOV) lanes (ADOT, 2001) 20](#__RefHeading___Toc44904756)

[Figure 11. 2DOF bridge model 21](#__RefHeading___Toc44904757)

[Figure 12. Placement of VSDDs in bridge
21](#__RefHeading___Toc44904758)

[Figure 13. 6DOF model with VSDDs in first three stories
22](#__RefHeading___Toc44904759)

[Figure 14. Exact and noisy TF magnitudes
23](#__RefHeading___Toc44904760)

[Figure 15. Stiffness estimate error levels with least-squares numerator
method 24](#__RefHeading___Toc44904761)

[Figure 16. Denominator magnifies error at higher frequency with LSN
method. 24](#__RefHeading___Toc44904762)

[Figure 17. Stiffness estimate error levels for iterative method with
exact start for 2DOF model with VSDD in 1^st^ story only
25](#__RefHeading___Toc44904763)

[Figure 18. Damping estimate error levels for the iterative method with
exact start for 2DOF model with VSDD in 1^st^ story only
25](#__RefHeading___Toc44904764)

[Figure 19. Stiffness estimate error levels for the iterative method
with offset start for 2DOF model with VSDD in 1^st^ story only
26](#__RefHeading___Toc44904765)

[Figure 20. Stiffness error levels for the iterative method with exact
start for 2DOF model with VSDD in 2^nd^ story only
28](#__RefHeading___Toc44904766)

[Figure 21. Damping error levels for the iterative method with exact
start for 2DOF model with VSDD in 2^nd^ story only
28](#__RefHeading___Toc44904767)

[Figure 22. Stiffness error levels for the iterative method with exact
start for 2DOF model with VSDD in both 1^st^ and 2^nd^ stories
29](#__RefHeading___Toc44904768)

[Figure 23. Damping error levels for the iterative method with exact
start for 2DOF model with VSDD in both 1^st^ and 2^nd^ stories
29](#__RefHeading___Toc44904769)

[Figure 24. Exact and noisy TF magnitudes for 2DOF bridge model
31](#__RefHeading___Toc44904770)

[Figure 25. Stiffness estimate error levels for the iterative method
with exact start in 2DOF bridge model 32](#__RefHeading___Toc44904771)

[Figure 26. Damping estimates error levels for the iterative method with
accurate start in 2DOF bridge model 32](#__RefHeading___Toc44904772)

[Figure 27. Exact and noisy TF magnitudes for 6DOF shear building model
33](#__RefHeading___Toc44904773)

[Figure 28. Stiffness and damping estimates error levels for the
iterative method with exact starting guess in 6DOF model
34](#__RefHeading___Toc44904774)

[Figure 29. Stiffness estimates error levels for the iterative method
with exact starting guess in bridge model with multiple variations of
damping coefficients 36](#__RefHeading___Toc44904775)

[Figure 30. Damping estimates error levels for the iterative method with
exact starting guess in bridge model with multiple variations of damping
coefficients 36](#__RefHeading___Toc44904776)

# LIST OF TABLES

[Table 1. Estimate Means and Coefficients-of-Variation for 2DOF Shear
Building Model 27](#__RefHeading___Toc44904777)

[Table 2. Range of 3σ Stiffness and Damping Estimate Relative Errors for
2DOF Shear Model 30](#__RefHeading___Toc44904778)

[Table 3. Estimate Means and Coefficients-of-Variation for 6DOF Shear
Building Model 35](#__RefHeading___Toc44904779)

# DISCLAIMER

The contents of this report reflect the views of the authors, who are
responsible for the facts and the accuracy of the information presented
herein. This document is disseminated under the sponsorship of the
Department of Transportation, University Transportation Centers Program,
and California Department of Transportation in the interest of
information exchange. The U.S. Government and California Department of
Transportation assume no liability for the contents or use thereof. The
contents do not necessarily reflect the official views or policies of
the State of California or the Department of Transportation. This report
does not constitute a standard, specification, or regulation.

# ACKNOWLEDGMENTS

The authors gratefully acknowledge the partial support of this work by
Caltrans through the National Center for Metropolitan Transportation
Research (METRANS), and by the National Science Foundation under CAREER
grant 00-94030.

.

# 1.0 INTRODUCTION

Accurate diagnosis of structural health is a vital step in protecting
structures. Whether caused by acute events, such as earthquakes or other
natural disasters, or long- term degradation from environmental effects
and human use (and abuse), structural damage can threaten both danger to
human life and economic loss. The process of monitoring structural
health and identifying damage existence, severity and location is
generally termed *structural health monitoring* (SHM). Chang (1999)
defined structural health monitoring to be an "autonomous \[system\] for
the continuous monitoring, inspection, and damage detection of \[a
structure\] with minimum labor involvement."

Structural health monitoring has long been recognized as its own
particular category within system identification whose area is well
established. High fidelity modeling and accurate response estimation are
required for SHM for several reasons, foremost being the monitoring of
structural characteristics and response to predict the onset of failure
or the expected remaining life, and for the purposes of controlling the
motion of the structure.

By determining the model that best fits data taken from the structure,
certain structural characteristics may be identified. With
identification at different points in time --- periodic or shortly after
natural disasters --- changes in these characteristics may be monitored.
With damage models, changes in structural characteristics are used to
predict damage severity and location.

This report proposes using smart, controllable passive devices such as
Variable Stiffness and Damping devices (VSDDs) in structures to improve
SHM, and demonstrates the benefits over conventional passive structures.

## 1.1 SHM Benefits

Clear needs for structural health monitoring exist for both building and
bridge structures. For example, significant expenditures after the 1994
Northridge earthquake went to inspecting joints of steel buildings for
damage. These joints are often hidden within a structure, such as behind
walls or encased in concrete (for example, see Figs. 1--3), and require
removal of non-structural material, not to mention putting (at least
part of) the structure out of service during inspection. Similar
expenditures were required after the 1995 Hyogo-Ken Nanbu (Kobe)
earthquake as well (Mita, 1999). A recent report to the U.S. Congress by
the Federal Highway Administration (FHWA, 2002) indicates that
approximately 25% of the bridges in the U.S. are rated as deficient. It
is estimated that it will require an investment of 7 billion dollars per
year for the next two decades to rebuild or replace the bridge
infrastructure in the U.S. (Patten *et al.*, 1999). This all when many
public budgets are in the billions of dollars in the red (EBBT, 2002).
All of these facts make clear the essential need for effective
structural health monitoring systems.

## 1.2 Difficulties of Conventional SHM Approaches

One approach to SHM is based on global vibration methods. In applying
these methods, many global vibration SHM studies assume a class of
mathematical models that may represent the actual structure. When
considering damage detection, these studies have focused on identifying
changes in of modal parame­ters, such as natural frequencies, mode
shapes, and modal damping ratios, obtained from measured vibration
response. Recent state-of-the-art surveys of global vibration SHM
techniques applied to civil engineering applications are given by
Doebling *et al.* (1996, 1998).

![](media/image2.png){width="3.563888888888889in"
height="2.6444444444444444in"}

[]{#__RefHeading___Toc44904747 .anchor}Figure 1. Beam-column connection
damage (Kiremedjian, 1999)

![](media/image3.png){width="2.061111111111111in"
height="2.6444444444444444in"}

[]{#__RefHeading___Toc44904748 .anchor}Figure 2. Plastic hinging at top\
of column (Hall, 1995)

![](media/image4.png){width="3.1659722222222224in"
height="2.3965277777777776in"}

[]{#__RefHeading___Toc44904749 .anchor}Figure 3. Cracks through column
flange and extending into web (Hall, 1995)

Unfortunately, no global technique has been well established and
accepted as an overall successful approach (Sanayei *et al*., 1998).
Some explanatory reasons are that (*i*) models cannot perfectly predict
the full behavior of real structures, (*ii*) periodic environmental
effects such as thermally-induced variations may mask the effects of
damage on global vibration characteristics, (*iii*) measurement noise
can cause significant variation from one test to the next,
(*iv*) excitation is limited to ambient sources for most civil
structures, and (*v*) sensitivity of global vibration characteristics to
damage may be small. These can all lead to variations in the identified
model parameters characteristics that are not due to true changes in the
structure, raising uncertainty in damage estimates (Vanik *et al*.,
2000).

Using forced structural response strategies can overcome some of the
aforementioned problems. For example, forced response interrogation can
be timed to minimize periodic environmental effects from one test to the
next, and may provide greater excitation energy to decrease
signal-to-noise ratios in comparison with ambient excitation. For
structures with embedded active vibration control systems, the actuators
can be used to enhance damage detection by tuning the actuation signals
to directly increase closed-loop damage sensitivity of global vibration
characteristics (Ray and Tian, 1999).

In spite of the advantages of using forced response, large actuation
devices are not being used in a continuous manner for civil structures
(except a few isolated cases in Asia) due to the large power
requirements, concerns about stability and so forth, rendering them
impractical for damage mitigation or SHM of civil structures. Further,
building and bridge owners typically prefer their structures not to be
shaken deliberately, both for comfort of occupants or users of the
structures and to lessen any chance of risking further damage.

Given these limitations, one is restricted to analyzing response to
ambient excitation to perform SHM. Ambient excitation on bridge
structures takes a number of forms including wind, traffic, waves and
microtremors. The ambient excitation approach has several advantages
over approaches using forced vibration response. For example, for the
low amplitude excitations typically experienced during ambient
vibration, most structural systems are well characterized with linear
models. In addition, continuous ambient vibration tests can be performed
at a very low cost. However, while applications of ambient excitation
identification techniques are more acceptable than active control ones,
the sensitivity of the measured signals to noise is a pressing question.
Due to small structural response under such ambient excitations, the
signal-to-noise ratios are small enough to make SHM difficult and
results uncertain. Thus, solutions to these SHM difficulties must be
sought elsewhere.

## 1.3 Use VSDDs to Improve SHM

One approach that may help alleviate some of the SHM difficulties for
civil structures would be to use "smart" variable stiffness and damping
devices (VSDDs) --- controllable passive devices that have received
significant study for vibration mitigation (Spencer and Sain, 1997;
Symans and Constantinou, 1999) --- in a synergistic manner to provide
internal parametric changes to affect sensitivity to damage. VSDDs are
already being used to improve the performance of structures subjected to
natural hazards by mitigating vibration response. In addition to
providing near optimal structural control strategies for vibration
mitigation, these low-power and fail-safe devices can also provide
parametric changes to increase global vibration measurement sensitivity
for health monitoring. Further, the integration of smart damping and
health monitoring can exploit, in a synergistic manner, the common
aspects of both technologies as seen in the flowcharts in Fig. 4.

[]{#__RefHeading___Toc44904750 .anchor}Figure 4. SHM and variable
stiffness/damping flow charts

VSDDs can adjust the behavior of a structure by real-time modification
of stiffness and damping at discrete points within the structure. By
commanding different behavior for each VSDD in a structure, multiple
structural configurations can be tested, each of which can be designed
to increase the sensitivity to damage in different portions of the
structure (see Fig. 5). For example, consider a structure with four
stiffness devices with "on" and "off" settings; there are 2^4^, or 16,
distinct configurations, each of which can provide some information
about the structural characteristics. The SHM is provided, then, with
multiple signatures of the structure, each of which can provide
additional and, if done efficiently, mostly complementary information.
As an analogy to the previous discussion, consider taking a sequence of
photographs of the front of a house. If the camera is exactly in front
of the house for each picture, relatively little depth information can
be gathered. However, shifting the camera (or changing the lighting) can
give computer modeling software significantly new complementary
information.

![](media/image5.png){width="3.490972222222222in"
height="1.0666666666666667in"}

[]{#__RefHeading___Toc44904751 .anchor}Figure 5. Mutual benefits of SHM
and VSDDs

## 1.4 Overview of this VSDD/SHM Research Project

This report proposes using VSDDs in structures to improve SHM, and
demonstrates the benefits in contrast with conventional passive
struc­tures. Several structures are studied herein, with one or more
VSDDs installed. First 2DOF and then 6DOF shear structural models are
studied, each in several configurations: the 2DOF structure with a VSDD
in (*i*) the first story, (*ii*) the second story, and (*iii*) both
stories, and the 6DOF building with VSDDs in the first three stories.
Then, a 2DOF bridge structure model (Erkus *et al.*, 2002) is studied
with a VSDD attached in the bearing layer between the pier and the deck.
In each case, the VSDD is chosen to act as an ideal variable
stiffness/damping device, with one of several discrete stiffness/damping
values. Some examples from simulation are used to demonstrate the
benefits of the proposed method.

The focus herein is introducing a better approach for estimating the
structural dynamic parameters through the use of variable stiffness and
damping devices. Since controllable stiffness/damping devices are used
to give the parametric changes necessary for improved monitoring, the
bridge models must be *control-oriented* dynamic models --- *i.e.*,
low-order models that still capture most of the salient dynamic
characteristics of a real bridge, particularly in the locations of the
controllable devices and in the frequency ranges driven by the
excitation. These devices are located in the structures in the form of
lateral bracing or, in the bridge example, in the isolation layer
between the bridge deck and the pier supports. It is assumed that
transfer functions from ground acceleration to absolute horizontal
accelerations of the different degrees-of-freedom have been determined
experimentally through standard procedures with the VSDDs at various
stiffness or damping settings. Herein, these transfer functions are
modeled by the exact transfer functions plus the Fourier transform of a
Gaussian pulse process that would be typical of band-limited Gaussian
white sensor noise vector processes. A conventional least-squares
approach is adopted to estimate structural parameters. Because of the
VSDDs' useful properties, they were able to improve the sensitivity of
the identification. It is shown that using variable stiffness devices in
a structure subjected to repetitive ambient vibration simulation,
favorable results are obtained compared to other approaches.

While the VSDD approach can be used with other identification methods,
it is here investigated in the context of a parametric frequency domain
identification to determine structural parameters. Historically,
frequency-domain approaches have dominated modal testing literature for
many years. In the vast majority of modal testing, frequency response
functions are measured prior to using a modal identification algorithm
(Juang, 1994). Experienced modal testing personnel can deduce
considerable information by examining frequency response functions.

In this report, conventional parametric identification in the frequency
domain is described. The direct transfer function polynomial
identification is rather complex in the unknown parameters.
Consequently, one simplification discussed in the literature is
explained. However, it is shown that this simplification can often lead
to significant bias in parameter estimates due to unwanted magnification
of sensor noise effects. The numerical examples herein use a fairly
noisy signal to challenge the methods. An iterative method is proposed
that approximates the more direct method. These methods are then applied
to conventional structural identification, assuming no VSDD within the
structure, and then with VSDDs using several distinct stiffness settings
and (later) distinct damping settings.

It is shown that using the least squares approach on the modified
version of the error in transfer functions with known starting guesses,
both using VSDD and conventional identification techniques give
parameter estimates. The latter gives similar mean estimates but the
VSDDs approach gives rather smaller variation. For biased starting
guesses, the VSDDs approach gives more accurate estimate means than the
conventional approach though with slightly larger variation.

The choice of VSDD location(s), possibly key to efficient algorithms, is
also important. Though it seems beneficial to include these devices in
all stories, cost benefit analysis (of SHM as well as hazard vibration
mitigation) may indicate that using fewer devices is more economical.
Accordingly, in the 6DOF model studied herein, one VSDD is assumed in
each of the first three structural segments only. The results show again
the effectiveness of adding these devices in improving the mean
estimates of the structural parameters and their variations, even with a
quite noisy measurement signal.

In summary, this report documents a study that demonstrates the
potential of improving SHM by exploiting the controllable properties of
variable stiffness and damping devices. Further, the study raised
additional questions and insights that should be investigated in future
research. These conclusions and future directions comprise the last
chapter of the report.

# 2.0 LITERATURE REVIEW

The field of structural health monitoring has been quite rich with
research in the last few decades. The impact and benefits, both economic
and societal, of SHM have drawn the interest of many researchers. The
results of relevant work in the literature can be loosely broken down
into several related categories; some representative examples in each
category are briefly highlighted in this chapter. First, an overview of
SHM approaches is given, followed by a discussion of modeling issues.
Then several classes of system identification methods are described.
Finally, variable stiffness and damping devices are reviewed,
highlighting the various types and some applications in which they have
been studied.

## 2.1 Research on Structural Health Monitoring

Conventional research in SHM and damage detection for civil structures
can be roughly classified into local and global methods. Local SHM
methods detect changes in a structure in localized regions using, for
example, ultrasound, x-ray, piezoelectric devices and so forth.
Unfortunately, local SHM methods have limited range; *e.g.*,
piezoelectric devices have an effective range on the order of 30 cm
(Wang and Chang, 1999). For large civil structures, this would require a
staggering number of devices to monitor the entire structure.
Furthermore, other local methods usually require significant human
involvement or are limited to areas where damage might be expected to
occur. Consequently, service costs and errors in expected damage
locations limit the usefulness of such methods.

Conventional global vibration SHM methods (Doebling *et al.*, 1996,
1998) typically focus on identifying changes in modal parameters
(*e.g.*, natural frequencies, mode shapes, and modal damping ratios)
computed from measured vibration response. With identification at
different points in time --- periodic or shortly after natural disasters
--- changes in these characteristics may be monitored. However, these
approaches also have their difficulties. Global methods based on ambient
excitation are easy to implement as they require no additional
excitation source, but simply may not reveal certain defects since some
damage mechanisms are only strongly observable with narrowband
excitation that ambient sources alone cannot provide. Using known
excitation with force actuators can overcome this difficulty but, with a
few exceptions in Asia, such active devices are not used in civil
structures for other purposes so widespread permanent installation and
use is unlikely. As a result, no global technique has been well
established and accepted as an overall successful approach (Sanayei *et
al.*, 1998).

A benchmark study in structural health monitoring based on simulated
structural response data was developed by the joint IASC-ASCE task group
on structural health monitoring. This benchmark was created to
facilitate a comparison of various methods employed for the health
monitoring of structures. The focus of the problem is simulated
acceleration response data from an analytical model of an existing
physical structure. This problem was addressed and studied by various
researchers including Au *et al.* (2000), Bernal and Gunes (2000),
Caicedo *et al*. (2003), Corbin *et al.* (2000), Johnson *et al.*
(2003), Katafygiotis *et al*. (2000) and Luş and Betti (2000). This SHM
benchmark problem, both the definition and the application of a number
of SHM methods to solve the problem, will be published a special issue
of the Journal of Engineering Mechanics later in 2003.

SHM approaches, particularly ones based on global vibration, often
involve some level of structural modeling and some type of system
identification. The modeling, which is described in more detail in the
following section, whether discrete or continuous in nature, has the
difficulty that no model has the exact same dynamic properties as the
structure it is intended to represent. Fortunately, the small motion
levels typical of ambient vibration response can alleviate some of these
issues. In addition to various modeling approaches, there are numerous
system identification techniques, some of which are described in the
third section of this chapter.

## 2.2 Modelling of Structures for SHM

Picking the right model to represent a particular structure has been a
critical issue in the application of SHM. Capecchi and Vestroni (1999)
state that, from a theoretical point of view, it is convenient to
distinguish between continuous and discrete structures. Although all
structures are in fact continuous, structures in which concentrated
masses are dominant are considered discrete in dynamic analysis. In the
context of damage identification, structures are considered as discrete
if the damage cannot affect a portion smaller than the element. Thus
frames, for example, are seen as discrete structures whereas bridge
decks and pipelines are usually considered continuous. For continuous
structures, it is conceptually correct to pose the problem of localizing
of a crack, because its position affects the entire dynamics. For a
discrete structure the problem is different; for example, it is not
possible to precisely determine where, in an element, a crack exists
because the structure is modeled discretely, with all characteristics of
the element taken as a whole.

Despite extensive research in modeling, structural models cannot be
expected to perfectly predict the full behavior of the structure. For
example, the model may not account for effects such as thermally-induced
daily variations and amplitude dependence of modal parameters. Further,
the available measured information is restricted by limits on the amount
of instrumentation. Nonlinearities are also a major obstacle where the
model becomes very difficult to implement and each nonlinear system is a
special case of its own. Moreover, while finite element modeling is
convenient, it often produces a dynamic model that neither is of modest
order nor accurately captures the dynamics of the built structure. Some
reasons for the latter are mismodelling of structural elements,
differences between actual and modeled material properties and
dimensions, approximation in finite element derivations, and poor
convergence of the numerical model (Juang, 1994). For SHM, models of
extremely high order are of limited utility as, often, only a small
number of the lower modes of a structure can be identified with
confidence (Beck *et al.*, 2001). Additionally, a real structure often
has nonlinearities and higher frequency modes (or local vibration modes)
that may be missed in the analytical modeling, leading to what is
generally termed "model error." All of these difficulties make clear the
complexity of the modeling problem.

Using ambient vibration to excite structures may simplify the modeling
problem necessary for SHM. Basically, the model can be considered linear
in that case (see, *e.g.*, Beck *et al.*, 2001). This assumption is
based upon the facts that ambient excitations are small, and that civil
structures are highly rigid, making them behave linearly under small
external excitations. Linear behavior, as well as safety considerations,
have encouraged numerous researchers to adopt ambient excitation
approaches in SHM.

## 2.3 System Identification for SHM

As there is no unique and general mathematical definition for damage,
researchers tend to relate damage to changes in structural model
parameters (such as stiffness, damping, and masses) or sometimes modal
parameters (such as natural frequencies and mode shapes). This may be
done by assigning damage indices that are functions of the structural
parameters or by evaluating the damage as the reduction in one or more
of these parameters. Accordingly, identifying these parameters, whether
modal or model parameters, is crucial for SHM application.
Unfortunately, these parameters cannot be easily or directly
mathematically derived from structural response but rather require
extensive measurements and signal processing that are generally termed
*system identification* techniques. System identification is the process
of developing or improving a mathematical representation of a physical
system using experimental data (Juang, 1994). Thereafter, this model is
used to estimate the properties of the dynamic system such as stiffness,
damping, frequencies, etc., through computational techniques that use
the known input/output data.

System Identification (SI) techniques to study the actual states of
civil engineering structures have received considerable attention in
recent years, as extensive full-scale experimental studies are expensive
and often difficult to perform. Identification techniques are divided
into frequency domain and time domain methods. Some of the key
approaches among these techniques are discussed in the next two
sections, followed by brief summaries of ERA/subspace approaches and
some research in the literature focused specifically on structural model
parameter identification (both of which can be studied in the frequency
or time domains).

### 2.3.1 Frequency Domain Techniques 

Frequency-domain identification (parametric/non-parametric) techniques
in control engineering and system identification gained relevance with
stability and design methods based on frequency response measurements
(Juang, 1994). This approach began with the technique known as transfer
function (TF) analysis. Many frequency dependent methods, such as the
empirical transfer-function estimate (ETFE), bootstrap methods,
separable least squares methods, etc., are detailed by Ljung (1999).

One of the first methods that used the frequency domain data in the
identification of the transfer function (TF) of the system was
attributed to the efforts by Levy (1959). Levy's method, for a
single-input-single-output (SISO) system was based upon expressing the
TF in the form of frequency-dependent numerator $A(j\omega)$ and
denominator $B(j\omega)$ polynomials as
$H(j\omega) = B(j\omega\frac{)}{A}(j\omega)$ where *ω* is any frequency
within the frequency range of interest in the specific problem. The
experimental TF $\hat{H}(j\omega)$ is obtained from input/output
measured data for the studied system. Thereafter, Levy (1959) considered
the difference between the experimental and theoretical TFs as the error
$e(j\omega) = B(j\omega\frac{)}{A}(j\omega) - \hat{H}(j\omega)$. The
error formula was modified to a simpler form
$\hat{e}(j\omega) = B(j\omega) - A(j\omega)\hat{H}(j\omega)$. Finally,
by differentiating the norm of the simplified error evaluated at known
frequencies with respect to the unknown coefficients of the polynomials
$B(j\omega)$ and $A(j\omega)$, a number of equations (equivalent to the
number of the unknown coefficients), are obtained. By solving these
equations, the coefficients of $B(j\omega)$ and $A(j\omega)$ are
obtained.

### 2.3.2 Time Domain Techniques

Time domain methods as well became quite popular later in the last
century especially with the great advance in the computer systems
industry (Juang, 1994). Many time domain identification techniques are
explained thoroughly in Ljung (1999). Some examples of time domain
identification techniques are ARX, ARMA, ARMAX, ERA, recursive least
squares, etc. Also, Beck *et al.* (1994a) presented a methodology for
determination of modal characteristics of structures from its measured
ambient vibrations at several instrumented locations, an extension of
the MODE-ID algorithm that uses a Bayesian probability framework to
build a linear model based on a classical normal modes approach (Beck,
1978; Beck, 1990).

### 2.3.3 ERA and Subspace Identification Techniques

One example of non-parametric time-domain methods is given by Juang and
Pappa (1985), which proposed a method called the Eigensystem Realization
Algorithm (ERA) for modal identification from measured responses. This
method uses a singular-value decomposition to derive the basic
formulation for a minimum-order realization, which is an extended
version of the Ho-Kalman algorithm (Ho and Kalman, 1965). First, a block
Hankel matrix is obtained by arranging the pulse response data into the
blocks of the Hankel matrix. By examining the singular values of the
Hankel matrix, the order of the system is determined. A minimum-order
realization (**A**, **B**, and **C** state-space matrices) is
constructed using a shifted block Hankel matrix. By finding the
eigensolution of the realized state matrix, modal damping rates and
frequencies may be obtained. The method then evaluates coherence and
co-linearity accuracy parameters to separate system modes from noise
modes. Based on these accuracy parameters, the system model is
determined and the Hankel matrix based on identified state space
matrices is reconstructed and compared with the measurement data.

Some modifications were later considered to improve the ERA method.
Juang *et al.* (1988) introduced a modification to ERA algorithm, using
response data correlations (ERA/DC) rather than the pulse response
values in the formulation of the Hankel matrix. The ERA/DC modified
method was found to reduce measurement noise bias without model
over-specification. However, when over-specification is permitted and
singular value decomposition is used to obtain a minimum order
realization, both old and modified methods give equally good results for
the data used.

Other subspace techniques have also been were also introduced and
studied. Quek *et al*. (1999) introduced the Eigen-space Structural
Identification technique for tall buildings subjected to stationary
ambient excitations and based on the forward innovation model of the
Kalman filter sequence. The method used QR decomposition and Quotient
Singular Value Decomposition (QSVD) techniques, which are substituted
into a least-square formulation to obtain a non-unique solution. Luş *et
al*. (1999) presented an algorithm, based on the ERA and Observer/Kalman
filter Identification (OKID) approaches, that uses earthquake-induced
ground accelerations and structural vibrations as input/output data sets
for identification purposes.

### 2.3.4 Structural Model Parameter Identification

Having accurate and updated information about the condition of
structures, that may suffer hazardous shaking or loading, is crucial.
This would save many lives as well as a lot of money. Accordingly,
identifying the structural model parameters such as stiffness and
damping is done to predict the behavior of structures under expected
future loadings. Significant research effort, therefore, has been
directed to find methods to identify these parameters. Some methods,
such as that by Takewaki *et al.* (2000), have studied the
stiffness-damping simultaneous identification of building structures
using limited earthquake records with higher intensity level. Other
research has been based on probabilistic methods related to Bayesian
theory, such as Vanik *et al*. (2000) and Katafygiotis and Yuen (2001).

The focus of model parameter identification to achieve SHM is usually on
local loss of stiffness as a proxy for local damage (Capecchi and
Vestroni, 1999; Caicedo *et al.*, 2001; Elmasry and Johnson, 2002; Beck
*et al.*, 1994b, 2001). To achieve that, some research has sought to
identify relationships between change in the modal characteristics and
changes in structural properties such as mass, stiffness, and damping.
For example Bayesian methodologies have been used for identifying the
loss of structural stiffness, such as in Beck *et al.* (1994b, 2001).
Bayes' theorem is invoked to develop a probability density function
(PDF) for the model stiffness parameters conditioned on modal data and
the chosen class of models. The latter method estimates the natural
frequencies, damping ratios, and mode shape components of a linear model
with classical normal modes that best fits the measured data in a least
squares sense. Then the parameters of the structural model are
determined from the computed modal data. The critical assumption is that
the change in the structural model implies changes in the parts of the
real structure. As a first step, changes in the values of the modal
parameters are identified and then as a second step the corresponding
loss of stiffness within the structure is identified.

In other previous research work focusing on identifying these model
parameters, ambient vibration, such as ambient wind measurements, is
used for excitation. Béliveau and Chater (1984) outlined a procedure to
estimate parameters based on relatively simple ambient wind measurements
of story accelerations reduced to resonant frequencies, and
corresponding mode shapes. Also, several other research studies have
considered ambient vibration for identification of stiffness parameters
and, in turn, stiffness loss, such as Beck *et al*. (1994, 2001), and
Caicedo *et al*. (2001). Ray and Tian (1999) introduced a method
intended for smart structures embodying self-actuation and self-sensing
capabilities. The method enhances modal frequency sensitivity to damage
using feedback control.

## 2.4 Variable Stiffness/Damping Devices

"Smart" variable stiffness/damping devices (VSDDs), such as semiactive
dampers and controllable stiffness devices, are controllable passive
devices that potentially offer the reliability of passive devices, yet
maintain the versatility and adaptability of fully active systems (Dyke
*et al*., 1996). These devices have received significant study for
mitigating various types of natural hazards for many types of civil
structures (Spencer and Sain, 1997; Symans and Constantinou, 1999), and
are useful for improving SHM.

### 2.4.1 Passive, Active and Semiactive Devices

Control devices for civil structures can be divided into four classes
passive, active, semiactive and hybrid. Passive devices, generally, are
those that have fixed properties and require no energy to function. In
contrast, the controllable forces generated by active devices are
induced directly by energy (electrical or otherwise) put into the
device. Between passive and active are semiactive devices that are
passive devices with properties that are controllable by application of
a small amount of energy. Hybrid devices are combinations of the other
three classes. Each of these is discussed briefly in the following
paragraphs, with greater detail on semiactive devices.

Passive devices, such as visco-elastic dampers, viscous fluid dampers,
friction dampers, metallic dampers, tuned mass dampers, and tuned liquid
dampers can partially absorb structural vibration energy and reduce
response of the structure (Soong and Dargush, 1997). These passive
devices are relatively simple and easily replaced. However, the
effectiveness of passive devices is always limited due to the narrow
frequency ranges in which they tend to be effective, the dependence of
their force only on local information, and their inability to be
modified if goals (or design codes) change.

Active control devices, including active mass dampers and active tendon
systems, can reduce structural response more effectively than passive
devices because feedback and/or feed-forward control systems are used
(Housner *et al*., 1997). However, large power requirements during
strong earthquakes and other hazards hamper their implementation in
practice. Further, active devices have the ability to inject dynamic
energy into the structural system; if done improperly, this energy has
the potential to cause further damage to the structure. In particular,
this can occur when the assumptions used to design the control algorithm
are incorrect or do not have a proper characterization of the structural
dynamics.

In contrast, "smart" devices are controllable passive devices that
require small amounts of power to control certain passive behavior.
Moreover, these devices cannot add energy to the structural/mechanical
system; rather, they may only (temporarily) store and dissipate energy.
Furthermore, they offer highly reliable operation at a modest cost and
can be viewed as fail-safe in that they default to passive devices
should the control hardware malfunction (Dyke *et al*., 1996).

### 2.4.2 Types of Semiactive Devices

Different types of semiactive devices have been developed recently. One
type of is the semiactive damper, such as variable-orifice dampers,
controllable fluid dampers, and controllable friction devices.
Variable-orifice dampers use an electromechanical variable orifice to
alter the resistance to flow in a conventional hydraulic fluid.
Controllable fluid dampers are passive hydraulic dampers containing a
fluid, such as magnetorheological (MR) or electrorheological (ER) fluid,
with controllable yield stress (Spencer *et al*., 1997). Another type of
semiactive device is a semiactive stiffness device such those developed
by Kobori and Takahashi (1993), Patten *et al*. (1999) and Yang *et al*.
(1996). They are on-off hydraulic devices capable of providing mainly
variable damping and limited variable stiffness capability. Nagrarajaiah
and Ma (1996) introduced a variable stiffness device that consists of
four sets of spring elements and telescoping tube elements. Varying the
position of the springs with a servomotor produces the
continuously-variable stiffness.

Controllable fluid dampers use fluids with properties that can be
modified by some outside influence. MR or ER fluids change their
properties in the presence of a magnetic or electric field,
respectively. These fluids were originally developed in the 1940s
(Rabinow, 1948; Winslow, 1949), but few applications were foreseen at
that time. While ER fluids showed early promise for civil applications
(see, *e.g.*, Ehrgott and Masri, 1992), most of the attention of the
civil structural control community has shifted to using MR fluids due to
their insensitivity to impurities, relatively constant behavior over a
wide range of operating temperatures, and the low voltage required to
activate them (Spencer *et al.*, 1997). MR dampers typically consist of
a hydraulic cylinder containing micron-sized magnetically polarizable
particles suspended within a fluid. In the presence of magnetic field,
the particles polarize and form particle chains that resist fluid flow.
By varying the magnetic field, the mechanical behavior of an MR damper
can be modulated. Since MR fluids can be changed from a viscous fluid to
a yielding semisolid within milliseconds and the resulting damping force
can be considerably large with a low-power requirement, MR dampers are
applicable to large civil engineering structures.

### 2.4.3 Applications of Semiactive Control to Civil Structures

The idea of incorporating variable stiffness/damping devices in civil
structures is not new. These devices have been extensively researched
for base isolation of structures and other structural control
applications, particularly in the last decade. Some researchers have
investigated MR dampers for control of seismic response such as Dyke *et
al*. (1996), Spencer *et al*. (1997, 1998) and Yang *et al.* (2002). ER
dampers were studied for seismic response control by Ehrgott and Masri
(1992), Gavin *et al*. (1996a,b), Makris *et al*. (1996), and others.
Dyke *et al*. (1996) proposed a clipped-optimal force control algorithm
with acceleration feedback and obtained excellent results when this
algorithm was applied to control a seismically excited three story
scaled building model. Ribakov and Gluck (1999) investigated the
effectiveness of ER dampers in mitigating seismic response of frame
structures. They used an optimal linear passive control strategy to
determine the viscous constant of the ER damper and then use active
control strategy to determine control forces. Through numerical
simulation they found that ER dampers could reduce the peak displacement
response of a seven-story frame structure up to 65 per cent without
increases in base shear forces and accelerations. In Xu *et al*. (2000),
the force-displacement relationship of an MR damper or an ER damper,
based on a parallel-plate model, is first extended to include the
stiffness of chevron brace supporting the smart damper. An extensive
parameter study is performed in terms of the maximum yield shear stress
and the Newtonian viscosity of the fluid, the brace stiffness, and the
earthquake intensity. VSDDs are also studied for damping of stay cables
in suspended bridges. Johnson *et al*. (2003) investigated the potential
of improving the damping to these cables through the use of semiactive
damping devices. The response of the cables with a semiactive damper is
found to be reduced dramatically compared to the optimal passive linear
viscous damper for typical damper configurations, thus demonstrating the
efficacy of a semiactive damper for absorbing cable-vibratory damage.
Varadarajan and Nagrarajaiah (2000) introduce the use of semiactive
variable stiffness tuned mass damper to control the response of tall
buildings excited by wind. The results from the latter paper indicate
that using the semiactive variable stiffness tuned mass damper is able
to reduce the response similar to using an active tuned mass damper.

### 2.4.4 Experimental and Full Scale Studies Using Semiactive Devices

Since theoretical VSDD research has shown significant promise,
researchers have progressed to experimental work to apply semiactive
control in real world applications. Symans and Constantinou (1997)
describe shaking table tests of a multi-story scale-model building
structure subjected to seismic excitation and controlled by a semiactive
fluid damper control system. The semiactive dampers were installed in
the lateral bracing of the structure and the mechanical properties of
the dampers were modified according to control algorithms that utilized
the measured response of the structure.

Patten *et al*. (1999) reported the first successful full scale
demonstration of semiactive control technology, installing an
Intelligent Stiffener for Bridges (ISB) on an in-service bridge on
interstate I-35. The ISB consists of an otherwise generic stiffener,
retrofitted to a bridge, that is equipped with an adjustable hydraulic
link used to regulate the amount of stiffness (and damping) provided by
the stiffener as vehicles pass over the bridge. The ISB acts much like a
muscle, sometimes flexing, and other times remaining relaxed. A 12-volt
automobile battery energizes it. The performance of the installed ISB
system was assessed via experimental results (see Fig. 6). The results
indicate that the ISB system can add decades of service life to an
existing bridge.

+------------------------------------------------+---------------------+
| The Kajima Corporation has developed a         | ![](media/im        |
| semiactive hydraulic damper (SHD) and          | age6.png){width="1. |
| installed it in an actual building (Kurata *et | 7493055555555554in" |
| al*., 2000). This was the first application of | height="1.2         |
| a semiactive seismic building control system   | 173611111111111in"} |
| that continuously changes the device damping   |                     |
| coefficient. A forced vibration test was       | ![](media/im        |
| carried out by an exciter with a maximum force | age7.png){width="1. |
| of 100 kN to investigate the building          | 6736111111111112in" |
| vibration characteristics and to determine the | height="2.5         |
| control system performance. As a result, the   | 104166666666665in"} |
| primary resonance frequency and the damping    |                     |
| ratio of a building (without semiactive        | []{#__RefHe         |
| hydraulic dampers) decreased as the exciting   | ading___Toc44904752 |
| force increased due to the influence of        | .anchor}Figure 6.   |
| non-linear members such as plain concrete      | ISB vibration       |
| curtain walls. After the eight semiactive      | absorber on I-35    |
| hydraulic dampers were installed in the        | bridge (Patten *et  |
| building, the control system performance was   | al*., 1999)         |
| identified by a response control test for      |                     |
| steady-state vibration. The elements that      |                     |
| composed the semiactive damper system          |                     |
| demonstrated the specified performance and the |                     |
| whole system operated successfully,            |                     |
| considerably reducing the displacements at the |                     |
| roof of the structure.                         |                     |
|                                                |                     |
| To prove the scalability of MR fluid           |                     |
| technology to devices of size appropriate for  |                     |
| civil engineering applications, Yang *et al.*  |                     |
| (2002) study a MR fluid damper with a nominal  |                     |
| maximum damping force of 200 kN (20 tons). For |                     |
| design purposes, two quasi-static models, an   |                     |
| axisymmetric and a parallel-plate model, are   |                     |
| derived for the force-velocity relationship of |                     |
| the MR damper, and both models give results    |                     |
| that match closely match the experimental      |                     |
| data.                                          |                     |
+------------------------------------------------+---------------------+

# 3.0 PROJECT OBJECTIVE AND SUMMARY

The core of this research project has been to investigate whether and
how variable stiffness and damping devices can be effectively used to
identify local damage in bridge structures. A complete and conclusive
answer to this question is beyond the scope of a one-year project, but
it guides the direction of the ongoing research. This study is based on
simulation of bridge motion due to ambient excitation. Using one or more
variable stiffness and damping devices (VSDDs) to modify the response,
structural parameters are estimated. The critical assumption is that the
change in the structural model implies changes in the parts of the real
structure. The focus is usually on local loss of stiffness as a proxy
for local damage. The damage in this study is evaluated in terms of the
loss of stiffness within a structural model, found by comparing the
structural model parameters before and after damage occurrence given
some input/output data.

One fundamental goal is to determine the best VSDD actions --- whether
adding damping or stiffness --- to precisely and robustly locate and
identify bridge damage. Several VSDD behaviors, such as variable
stiffness mode and variable damping mode are considered. To reflect the
generality of the approach of using VSDDs in SHM, this study includes
bridge and building structures as well. Bridge and shear building model
behaviors are assumed locally linear before damage and after damage.
Some simple models of bridge dynamics are given herein to be used in
simulation. The system models necessary for this study are
*control-oriented* dynamic models --- *i.e.*, low-order models that
still capture the salient dynamic characteristics of a real bridge (or
frame structure), particularly in the locations of the controllable
devices and sensors, and across the frequency ranges driven by the
excitation.

After the development of control-oriented models, the next step involves
incorporating models of variable stiffness and damping devices into the
dynamic bridge model and/or shear building model systems. Ideal
stiffness devices with multiple discrete stiffness settings are used. A
least-squares identification method of the related transfer functions is
used to identify structural characteristics such as stiffness and
damping or their ratios to the corresponding masses. The responses are
simulated for the combined bridge/VSDD dynamic model and shear building
models. By using the various permutations of stiffness device settings,
multiple signatures are obtained that help better identify the
structural parameters. Then, controllable damping devices are tested
whether they can provide better information about structural model
properties.

The method used herein to identify the structure model parameters is
based on a least-squares error convergence technique. The error is
expressed as the difference between the measured transfer functions
(from the external excitation to the measured responses) and their
corresponding polynomial parametric forms. In order to obtain the
required structural parameters, the relationships between the structural
model parameters and the polynomial coefficients are obtained. The
problem is modeled as a single-input multi-output system (SIMO). One
simplification that is discussed in the literature (Levy, 1959) is used
to overcome the complexity of the direct transfer function polynomial
identification. However, this simplification, denoted the "Least Squares
Numerator Method," is hsown to lead to significant bias in parameter
estimates due to unwanted effects of sensor noise. Consequently, an
iterative method is proposed that approximates the more direct exact
method. The proposed method is called the "Iterative Least-Squares
Numerator Method." These methods are then applied to conventional
frequency domain structural identification, assuming no VSDD, and with
one or more VSDDs with several distinct stiffness settings.

It is shown that, using the parametric frequency-domain least-squares
approach with the proposed modified transfer function error and a known
starting guess, the VSDD approach gives mean parameter estimate similar
to that with a conventional structure but with rather smaller error
(*i.e.*, smaller variance).

# 4.0 PARAMETRIC FREQUENCY DOMAIN IDENTIFICATION WITH VSDDS

## 4.1 Least-Squares Numerator Method (LSN)

One method of identifying parameters of a dynamical system is by
representing transfer functions (TFs) in the frequency domain as ratios
of polynomials. The transfer functions generally are defined by the
ratio between the output and input signals. For example, consider a
linear structural model of the form:

$M\overset{¨}{x} + C_{d}\overset{˙}{x} + \text{Kx} = bƒ$,
$y = C_{1}x + C_{2}\overset{˙}{x} + dƒ + v$ (1)

where **M**, **K**, and **C**~d~ are the mass, stiffness and damping
matrices of the system, and **C**~1~, **C**~2~, and **d** are the output
influence matrices for the displacement, velocity and the external force
*f*. For simplicity of the method developed herein, the input force is
assumed to be a single scalar force. Similarly, one can write the model
in state-space form

$\overset{˙}{q} = \overset{\sim}{A}q + \overset{\sim}{B}ƒ$,
$y = \text{Cq} + Dƒ + v$ (2)

where $q = \begin{bmatrix}
x^{T} & {\overset{˙}{x}}^{T}
\end{bmatrix}^{T}$ is the state vector, $\overset{\sim}{A}$is the system
state matrix which is dependent on the mass, damping, and stiffness
matrices, $\overset{\sim}{B}$ is the input influence matrix, **C** is
the output influence matrix for the state vector **q,** and **D** is the
direct transmission matrix. In both equations, *f* is an excitation
force, and **y** is an $m \times 1$ vector of measured responses
corrupted by $m \times 1$ sensor noise vector **v**.

Thus, the system can be represented by the $m \times 1$ transfer
function matrix **H**(*jω*). Each element of **H**(*jω*) can be
expressed as the ratio of numerator and denominator polynomials at a
certain frequency with coefficients depending on matrices in Eqs. (1) or
(2). It is important to state that for many structural systems, the
denominator polynomial is the same for all transfer functions from the
same input. Therefore, identifying the denominator polynomial is crucial
in defining the system dynamics. The transfer function vector from a
single input to the outputs can consequently be written in polynomial
ratio form as:

$H(j\omega) = B(j\omega\frac{)}{A}(j\omega)$ (3)

where **B**(*jω*) and *A*(*jω*) are the numerator and denominator
polynomials, which may be expanded
$B_{r}(j\omega) = b_{n_{B} - 1}^{r}(j\omega)^{n_{B} - 1}b_{n_{B} - 2}^{r}(j\omega)^{n_{B} - 2} + \ldots + b_{0}^{r}$
and
$A(j\omega) = a_{n_{A}}(j\omega)^{n_{A}} + a_{n_{A} - 1}(j\omega)^{n_{A} - 1} + \ldots + a_{0}$,
where the *b*'s and *a*'s are real coefficients.

Assuming that the transfer function has been determined experimentally
through standard procedures from measured input and output data (Bendat
and Piersol, 2000), the experimental transfer function matrix, expressed
as

$\hat{H}({j\omega}_{i})$, *i*=1, 2, ..., *n~ω~* (4)

is known at various discrete frequency points. Therefore, the difference
between the estimated theoretical transfer function **H**(*jω*) and the
actual experimental one $\hat{H}(j\omega)$ represent the error equation
which is then used in the identification process of the parameters.

Parametric frequency-domain methods to match such theoretical and
measured transfer functions date back to the work of Levy (1959) who
parameterized a continuous-time TF by the coefficients of numerator and
denominator polynomials. One approach to this problem is to follow
Levy's procedure in determining the polynomial coefficients and then, as
a subsequent step, estimate structural parameters such as mass,
stiffness and damping coefficients. The latter could be through an
intermediate step of first identifying modal characteristics such as
modal frequency, damping ratio and mode shapes. In this study, however,
the parameterization method is chosen to be through the structural
parameters directly without calculating the coefficients of the
polynomials as an intermediate step.

For a structure with one or more variable stiffness and/or damping
devices, the properties of which are determined through a local control
system, some of the coefficients in the transfer function polynomials
may be adjusted through changing the VSDD control algorithms. Thus, it
is convenient to introduce notation to explicitly state that the
transfer function polynomials are functions of unknown structural
parameters, denoted by the $n \times 1$ vector $\theta$, which is to be
estimated, and of known controllable structural parameters, denoted by
the vector $\kappa$. The transfer function expression is thus modified
to be:

$H(j\omega) = B(j\omega,\theta,\kappa\frac{)}{A}(j\omega,\theta,\kappa)$
(5)

For a given structural model, the *A* and **B** polynomials are specific
known functions of their parameters. Substituting the measured TF in
place of the exact TF leaves a residual error **e** that may be defined
by

$e({j\omega}_{i},\theta,\kappa) = \frac{B({j\omega}_{i},\theta,\kappa) - A({j\omega}_{i},\theta,\kappa)\hat{H}({j\omega}_{i},\kappa)}{A({j\omega}_{i},\theta,\kappa)}$
(6)

A conventional least-squares approach may be adopted to solve this
problem, forming a global square error

$\Delta^{2}(\theta) = \sum_{i}^{}{e^{}({j\omega}_{i},\theta,\kappa) \cdot e({j\omega}_{i},\theta,\kappa)}$
(7)

where (·)^\*^ denotes complex conjugate transpose. The optimal choice of
the unknown parameters is found by minimizing the square error ---
*i.e*., take the derivatives of the square error Eq. (7) with respect to
the elements of unknown vector $\theta$, set them equal to zero, and
solve the resulting (generally nonlinear) equations. However, if there
are known controllable structural parameters in a structure with
multiple configurations --- which is the case when using VSDDs, for
example --- the square error equation can be augmented by using several
combinations of known controllable structural parameters

$\Delta^{2}(\theta) = \sum_{k}^{}{\sum_{i}^{}{e^{}(\text{jω}_{i},\theta,\kappa_{k}) \cdot e(\text{jω}_{i},\theta,\kappa_{k})}}$
(8)

where the symbol $\kappa_{k}$ denotes multiple distinct sets of
parametric changes to the structure. The error is, then, minimized
simultaneously for all configurations.

Because the residual error **e** in Eq. (6) is a ratio of polynomials,
the square error in Eq. (8) is an extremely complex function of the
unknown parameters $\theta$. One simplification, which has been
suggested and used in various studies in the literature, is to recognize
that the denominator of Eq. (6) is nonzero for systems with damping, so
minimizing the error in the numerator may prove sufficient (*e.g*.,
Levy, 1959). In other words, minimize the sum of the squares of:

$\overset{\sim}{e}({j\omega}_{i},\theta,\kappa_{k}) = B(\text{jω}_{i},\theta,\kappa_{k}) - A(\text{jω}_{i},\theta,\kappa)\hat{H}(\text{jω}_{i},\kappa_{k})$
(9)

which is, herein, denoted the *least-squares numerator* (LSN) method.

## 4.2 Iterative-Least Squares Numerator Method (ILSN)

It may be shown that the alternate error measure in the LSN method,
while simpler to solve, can be susceptible to strong bias from sensor
noise in frequency ranges where $A(\text{jω}_{i},\theta,\kappa)$is large
(*i.e.*, often the case where **H**(*jω*) is small). To avoid this bias,
and to avoid the difficulty in solving the least-squares problem for the
standard error measure **e**, an *iterative* method, described as
follows, is adopted here using an approximation to the denominator in
Eq. (6).

Assume that iteration *l* begins with a starting approximation
${\hat{\theta}}_{l - 1}$ to the unknown parameter vector $\theta$; then,
the denominator of Eq. (6) is estimated based on the vector
${\hat{\theta}}_{l - 1}$ of estimated parameters and is no longer a
function of these unknowns, but only in the frequency and the multiple
distinct sets of parametric changes to the structure

${\hat{A}}_{l}(\text{jω}_{i},\kappa_{\kappa}) \equiv \hat{A}(\text{jω}_{i},{\hat{\theta}}_{l - 1},\kappa_{\kappa})$
(10)

and the error is, thus, formed as

${\hat{e}}_{l}({j\omega}_{i},\theta,\kappa_{\kappa}) = \frac{B(\text{jω}_{i},\theta,\kappa_{\kappa}) - A(\text{jω}_{i},\theta,\kappa_{\kappa})\hat{H}(\text{jω}_{i},\kappa_{\kappa})}{{\hat{A}}_{l}(\text{jω}_{i},\kappa_{\kappa})}$
(11)

and the squared error takes the form:

$\Delta_{l^{2}}(\theta) = \sum_{k}^{}{\sum_{i}^{}{{\hat{e}}_{l^{}}(\text{jω}_{i},\theta,\kappa_{\kappa}) \cdot {\hat{e}}_{l}(\text{jω}_{i},\theta,\kappa_{\kappa})}}$
(12)

Minimizing the sum of the square error in Eq. (12) will result in an
updated estimate ${\hat{\theta}}_{l}$ to the unknown parameter vector
$\theta$. The iterations continue until the relative differences between
${\hat{\theta}}_{l - 1}$ elements and the corresponding elements of
${\hat{\theta}}_{l}$ are all below some threshold. (Absolute or relative
norms of the difference could also be used.) A maximum number of
iterations may also be set to stop the algorithm in the case that the
iterative method does not converge (though this termination criterion
was not required in this study as convergence always occurred within a
limited number of iterations).

Whichever method is used to estimate the unknown parameter vector
$\theta$, the use of multiple structural configurations, denoted by the
different values of known parameters $\kappa_{k}$, provided by VSDDs in
a structure, can generate more accurate estimates of $\theta$ than can
be found with a comparable amount of data in a conventional structure
with a fixed $\kappa$. This is demonstrated for some examples in the
following section.

## 4.3 Illustrative Examples

The least-squares identification with VSDDs may be applied to various
types of structures. In this study, both building and bridge models are
considered.

### 4.3.1 Two Degree-of-Freedom Shear Building Model

Consider the two degree-of-freedom (2DOF) shear building structure model
shown in Fig. 7. The structure is subject to ambient excitation from the
ground. Absolute accelerations measurements at the ground,
${\overset{¨}{x}}_{g}$, and of the two floors,
$({\overset{¨}{x}}_{1} + {\overset{¨}{x}}_{g})$ and
$({\overset{¨}{x}}_{2} + {\overset{¨}{x}}_{g})$, are used to generate a
$2 \times 1$ experimental transfer function at *n~ω~* distinct frequency
values. (Note that *x~i~* herein denotes the displacement of the *i*^th^
floor relative to the ground.) Let the unknown parameter vector be given
by:

![](media/image8.wmf){width="0.3909448818897638in"
height="0.4397637795275591in"} $\theta = \begin{bmatrix}
\frac{k_{1}}{m_{1}} & \frac{k_{2}}{m} & \frac{c_{1}}{m_{1}} & \frac{c_{2}}{m_{2}} & \frac{m_{2}}{m_{1}}
\end{bmatrix}^{T}$ (13)

+----------------------------------------+-----------------------------+
| A VSDD that can provide a number of    | []{                         |
| distinct stiffness levels is located   | #__RefHeading___Toc44904753 |
| in the first story of the structure.   | .anchor}Figure 7. 2DOF      |
| The VSDD is installed in the lateral   | shear building model        |
| bracing of the structure and the       |                             |
| mechanical properties of the dampers   |                             |
| are modified according to control      |                             |
| algorithms, which utilize the measured |                             |
| response of the structure. The device  |                             |
| is considered ideal and semiactive;    |                             |
| *i.e*., it can generate the desired    |                             |
| forces with no delay and with no       |                             |
| actuator dynamics (Ramallo *et al*.,   |                             |
| 2000). Therefore the known             |                             |
| controllable vector is related to the  |                             |
| stiffness of a variable stiffness      |                             |
| device                                 |                             |
|                                        |                             |
| $\kappa = \                            |                             |
| kappa = \frac{k_{\text{VSDD}}}{m_{1}}$ |                             |
| (14)                                   |                             |
|                                        |                             |
| or to the damping coefficient of a     |                             |
| variable damping device                |                             |
|                                        |                             |
| $\kappa = \                            |                             |
| kappa = \frac{c_{\text{VSDD}}}{m_{1}}$ |                             |
| (15)                                   |                             |
+----------------------------------------+-----------------------------+

Then, the theoretical transfer function can be written in the polynomial
form as:

$H(\text{jω},\kappa) = \begin{bmatrix}
\frac{B_{1}(\text{jω},\theta,\kappa)}{A(\text{jω},\theta,\kappa)} & \frac{B_{2}(\text{jω},\theta,\kappa)}{A(\text{jω},\theta,\kappa)}
\end{bmatrix}^{T}$ (16)

Note that the denominator polynomial $A(\text{jω},\theta,\kappa)$ is the
same for all transfer functions from the same input. The numerator and
denominator polynomials can be expressed as:

*    A*(*s*, **θ**, *κ*) = *s*^4^ +
\[*θ*~3~+*θ*~4~+*θ*~4~*θ*~5~\]*s*^3^ +
\[*θ*~1~+*θ*~2~+*θ*~2~*θ*~5~+*θ*~3~*θ*~4~+*κ*\]*s*^2^ +
\[*θ*~1~*θ*~4~+*θ*~2~*θ*~3~+*κθ*~4~\]s + \[*θ*~1~*θ*~2~+*κθ*~2~\]\
*   B*~1~(*s*, **θ**, *κ*) = \[*θ*~3~\]*s*^3^ +
\[*θ*~1~+*θ*~3~*θ*~4~+*κ*\]*s*^2^ +
\[*θ*~1~*θ*~4~+*θ*~2~*θ*~3~+*κθ*~4~\]*s* + \[*θ*~1~*θ*~2~+*κθ*~2~\]
(17)\
*   B*~2~(*s*, **θ**, *κ*) = \[*θ*~3~*θ*~4~\]*s*^2^ +
\[*θ*~1~*θ*~4~+*θ*~2~*θ*~3~+*κθ*~4~\]s + \[*θ*~1~*θ*~2~+*κθ*~2~\]

The parameter identification methods discussed in the previous section
can then be applied. The explicit reference to *m*~1~ has dropped out of
the transfer function polynomials; it will be assumed here that of all
the parameters, only *m*~1~ is known.

Different VSDD locations in the structure are studied in order to
investigate the best way of using these VSDDs to improve SHM through
better identification of the structural model parameters. Accordingly,
this example is also solved considering a VSDD that can provide a number
of distinct stiffness levels located in the second story of the
structure and in both stories as well, as shown in Figs. 8 and 9. This
will generally give an idea of how VSDD should be distributed in a
structure for good SHM. In all cases, the VSDD is chosen to act as an
ideal variable stiffness/damping device, with one of several discrete
stiffness/damping values. Some examples from simulation are used to
demonstrate the proposed method. This 2DOF model system identification
was solved for the variable stiffness VSDD and for a conventional
structure (no VSDD). In the simulations, each installed VSDD is assumed
to provide additional stiffness at five discrete levels: 0%, 10%, 20%,
30% and 40% of the stiffness of the story at which it is located.

[]{#__RefHeading___Toc44904754 .anchor}Figure 8. VSDD in 2^nd^ story of
2DOF model

[]{#__RefHeading___Toc44904755 .anchor}Figure 9. VSDDs in both stories
of 2DOF model

### 4.3.2 Two Degree-of-Freedom Pier-Deck Bridge Model

+--------------------------------+-------------------------------------+
| Consider a bridge structure    | ![](media/image9                    |
| such as one shown in Fig. 10,  | .jpeg){width="1.9909722222222221in" |
| which is a typical elevated    | height="2.9555555555555557in"}      |
| highway bridge that consists   |                                     |
| of decks, bearings, and piers. | []{#__RefHeading___Toc44904756      |
| The behavior of the bridge     | .anchor}Figure 10. General view     |
| deck and piers, with a bearing | during construction of high         |
| between them, while complex,   | occupancy vehicle (HOV) lanes       |
| can be well approximated with  | (ADOT, 2001)                        |
| the simple 2DOF model shown in |                                     |
| Fig. 11*c*. This 2DOF model    |                                     |
| may be used to represent a     |                                     |
| passive system with rubber     |                                     |
| bearings if the girder is      |                                     |
| continuous with one pier and   |                                     |
| one bearing, or for several    |                                     |
| piers and bearings with        |                                     |
| identical properties. Also,    |                                     |
| this model can be used for     |                                     |
| VSDD systems if the devices    |                                     |
| are attached as shown in Fig.  |                                     |
| 12 and commanded to provide    |                                     |
| identical force levels. It is  |                                     |
| assumed in this problem that   |                                     |
| the pier mass *m*~1~ is known. |                                     |
+--------------------------------+-------------------------------------+

The theoretical polynomial transfer function matrix
$H(\text{jω},\kappa)$ is defined similarly to Eq. (16) where, here,
$B_{1}(\text{jω},\theta,\kappa)/A(\text{jω},\theta,\kappa)$ is the
transfer function between the ground acceleration and the absolute
acceleration of the pier and
$B_{2}(\text{jω},\theta,\kappa)/A(\text{jω},\theta,\kappa)$ is the
transfer function between the ground acceleration and that of the bridge
deck. The unknown parameter vector $\theta$ is defined similar to Eq.
(13) but *k*~1~, is the stiffness of the pier and *k*~2~ the stiffness
of the deck. The vector of known parameters $\kappa$ denotes the
additional stiffness added through the VSDD connected between the pier
and the deck. In the simulations, the installed VSDD is assumed to
provide additional stiffness at five discrete levels: 0%, 10%, 20%, 30%
and 40% of the stiffness of the deck. A second, related example will use
five similar discrete levels of VSDD damping instead of stiffness.

[]{#__RefHeading___Toc44904757 .anchor}Figure 11. 2DOF bridge model

[]{#__RefHeading___Toc44904758 .anchor}Figure 12. Placement of VSDDs in
bridge

### 4.3.3 Six Degree-of-Freedom Shear Building Model

To demonstrate that these methods can be extended to more complex
problems, a six degree-of-freedom shear building model is also studied.
The shear building model is found to be a good representation for
general problems that may exist in multi-story (or multi degree of
freedom (MDOF) structures. This 6DOF model system identification was
solved for the variable stiffness VSDD and the conventional structure
(no VSDDs). In simulation, each installed VSDD is assumed to provide
additional stiffness at four discrete levels: 0%, 10%, 20%, and 30% of
the stiffness in the story at which it is located. The VSDD devices
included in the structure are also considered ideal; *i.e*., they can
generate the desired forces with no delay and with no actuator dynamics.
In this problem, the VSDDs are considered to be located in the first
three stories only, as shown in Fig. 13, to make best use of VSDDs in
the structure without escalating the expenditures too much.

It is important to note that applying the ILSN method for estimation of
parameters in case of higher degrees of freedom (such as the 6DOF model)
was very challenging in terms of RAM availability and computational
speed. Some details about the numerical procedure to solve these
difficulties are provided in Appendix A.

![](media/image11.wmf){width="0.3909448818897638in"
height="0.4397637795275591in"}

[]{#__RefHeading___Toc44904759 .anchor}Figure 13. 6DOF model with VSDDs
in first three stories

Here, masses are assumed a prior whereas the previous 2DOF model assumed
knowledge only of the first mass. The unknown parameter vector $\theta$
for the six degree-of-freedom model is, then, a set of unknown stiffness
*k~i~* and damping *c~i~* coefficients as follows:

$\theta = \lbrack\begin{matrix}
k_{1} & k_{2} & k_{3} & k_{4} & k_{5} & k_{6} & c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6}\rbrack
\end{matrix}^{T}$ (18)

Finally, the solution for the parameters was done using the
M[ATLAB]{.smallcaps}^®^ software and using the function fsolve( ) for
the solution of the non-linear equations in the unknown parameters
$\theta$.

## 4.4 NUMERICAL EXAMPLES AND ANALYSIS OF RESULTS 

To demonstrate the benefits and the advantages of testing a structure
with VSDDs configured in multiple settings, some numerical examples are
considered.

### 4.4.1 Two Degree-of-Freedom Shear Building Model

First, a numerical example of the 2DOF structure is studied. For
simplicity, the floor masses and story stiffnesses are taken to be unity
in the numerical model. The story damping coefficients are set to 0.05,
which results in 1.5% and 4.0% modal damping in the two modes,
respectively. The single VSDD is assumed to provide additional stiffness
in the story at which it is located with five discrete stiffness levels,
corresponding to an additional 0%, 10%, 20%, 30% and 40% stiffness;
*i.e*., *κ*~1~ = 0.0, *κ*~2 ~= 0.1, ..., *κ*~5~ = 0.4 . The primary
comparison reported here is the difference between:

-   the *VSDD* approach, using five experimental transfer function
    matrices, one per VSDD stiffness level, and

-   the *conventional* *structure* approach with *κ *= 0.0  to make for
    a fair comparison using the same amount of data, the conventional
    approach uses a square error based on five separate experimental
    transfer functions.

The experimental transfer functions are generated in MATLAB by using the
exact transfer functions plus the Fourier transform of a Gaussian pulse
process that would be typical of band-limited Gaussian white sensor
noise vector processes. The sensor noise used in generating the five
experimental *VSDD* transfer functions is the same as those used for the
*conventional structure* approach.

The number of evenly-spaced frequency points *n~ω~* is 51. Figure 14
shows a comparison, in both linear and log scales, of the exact transfer
function magnitudes that represents the true model with those of the
experimental transfer function corrupted by noise for zero VSDD
stiffness; this noise level provides a fairly challenging problem.

![](media/image12.wmf){width="3.459055118110236in"
height="2.572834645669291in"}

[]{#__RefHeading___Toc44904760 .anchor}Figure 14. Exact and noisy TF
magnitudes

Once the experimental transfer functions are generated,
M[ATLAB^®^]{.smallcaps} code employing the Symbolic Math Toolbox™ is
used to determine the total square error symbolically. The result is
then differentiated with respect to the unknown parameter vector
$\theta$, giving a number of equations that should equal zero. These
equations, which are cubic polynomials in $\theta$ for the
parameterization in Eq. (13), are solved numerically using fsolve( )
function in the Optimization Toolbox™ with function tolerances and
$\theta$ solution tolerances both set to 10^--5^. The iterative approach
uses a relative tolerance of 10^--4^ (which must be met for all elements
of the unknown parameter vector $\theta$) as a termination criterion.

###### Least-Squares Numerator Method Results

First, the expected bias in the least-squares numerator method is
verified by using the error definition in Eq. (9). The *VSDD* and
*conventional structure* approaches are used to determine the unknown
parameters. Each approach is performed 26 times, with 26 different seeds
for generating the random noise, to see the distribution of error levels
that may be expected with the least-squares numerator method. The
initial $\theta$ guess provided to the numerical equation solver is the
exact value. Figure 15, which shows the error in the stiffness estimates
for the 26 trials, demonstrates that both approaches give significant
error in the estimates as well as a wide systematic bias. This bias is
expected, as the denominator magnitude is quite large for higher
frequency points as shown in Fig. 16, causing the noise to significantly
skew the parameter estimates. Therefore, one can conclude that, as
expected, this simplified technique cannot be adopted for fair
comparison between the VSDD and *conventional structure* approaches
because the results are not accurate.

![](media/image14.png){width="3.35in" height="2.502083333333333in"}

[]{#__RefHeading___Toc44904761 .anchor}Figure 15. Stiffness estimate
error levels with least-squares numerator method

![](media/image15.png){width="4.02992125984252in"
height="2.8744094488188976in"}

[]{#__RefHeading___Toc44904762 .anchor}Figure 16. Denominator magnifies
error at higher frequency with LSN method.

###### Results with the Iterative Method

The iterative approach gives significantly better results. Using the
true parameter vector as an initial starting guess for the iterative
procedure, the algorithm converges, generally in 3--5 iterations, to
estimates that are fairly accurate  much better than the least-squares
numerator method. Figure 17 shows the relative error in the estimates of
the stories stiffness for the c*onventional structure* and *VSDD*
approaches. Here, 100 separate estimates were computed (each with a
different noise seed) to examine the variation due to sensor noise. The
graph also shows approximate one-, two- and three-sigma (standard
distribution) curves representing for the two approaches. The curves
were generated, assuming a Gaussian distribution, using the means and
covariance matrix of the 100 estimates. The variation in estimates of
first- and second-story stiffness using the *VSDD* approach are about
half and two-thirds, respectively, of those using the *conventional
structure* (without VSDDs).

![](media/image16.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904763 .anchor}Figure 17. Stiffness estimate
error levels for iterative method\
with exact start for 2DOF model with VSDD in 1^st^ story only

![](media/image17.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904764 .anchor}Figure 18. Damping estimate error
levels for the iterative method\
with exact start for 2DOF model with VSDD in 1^st^ story only

Damping estimates are, as expected, much less accurate for both
approaches. Figure 18 shows the error variation in damping coefficient
estimates using the ILSN method, and demonstrates that the *VSDD*
approach has again generated a smaller variation as sensor noise
changes. The relative error in this case is somewhat larger than that of
the stiffness estimates, equals 22% using *VSDDs* and 27% for the
conventional structure case in estimating the damping coefficient in the
first floor *c*~1~, and 36% and 40% for the estimation of the damping
coefficient in the second floor *c*~2~. This larger relative damping
estimate error, for both *VSDD* and *conventional structure* approaches,
compared to stiffness, is typical of most identification methods. Yet,
it is clear that the *VSDD* approach, while not making vast
improvements, shows some decrease in error compared to the
*conventional* approach.

An initial guess that is biased, offset from the correct parameter
vector by some amount, would be expected to give poorer estimates. This
is indeed the case. Using an initial parameter vector that is 20% higher
(in all components) than the exact values, 20 separate estimates were
computed. Figure 19 shows that both approaches, as anticipated, are much
less accurate in estimating the stories stiffness than with the exact
initial guess. Here, the *VSDD* approach gives estimate means that are
less biased than the *conventional structure* approach, though with
slightly larger variation in the estimate of *k*~1~. Again, this
indicates that using VSDDs improved the stiffness estimates.

![](media/image18.wmf){width="4.245138888888889in"
height="2.673611111111111in"}

[]{#__RefHeading___Toc44904765 .anchor}Figure 19. Stiffness estimate
error levels for the iterative method\
with offset start for 2DOF model with VSDD in 1^st^ story only

The computational effort to solve these problems symbolically was much
higher than anticipated. To compute one estimate with the correct
initial starting guess took approximately 15 minutes on a 600 MHz
Pentium III computer. This is based on using the symbolic Math toolbox
of MATLAB which is powered by MAPLE^®^. With an offset starting guess,
more iterations were required, both in the iterative approach outlined
above as well as within the numerical solver fsolve( ), requiring
slightly longer to converge. The estimate means and coefficients of
variation are shown for the two approaches in Table 1. The average
number of iterations required for the MATLAB solver to converge are
always less in case of *VSDD* approach than the *conventional structure*
approach as shown in Table 1.

[]{#__RefHeading___Toc44904777 .anchor}Table 1. Estimate Means and
Coefficients-of-Variation for 2DOF Shear Building Model

<table>
<colgroup>
<col style="width: 11%" />
<col style="width: 10%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
</colgroup>
<tbody>
<tr class="odd">
<td rowspan="3"><strong>Variables</strong></td>
<td rowspan="3"><strong>exact</strong></td>
<td colspan="4"><strong>mean</strong></td>
<td colspan="4"><strong>coeff. of variation[%]</strong></td>
</tr>
<tr class="even">
<td colspan="2"><em><strong>exact init. guess</strong></em></td>
<td colspan="2"><em><strong>offset init. guess</strong></em></td>
<td colspan="2"><em><strong>exact init. guess</strong></em></td>
<td colspan="2"><em><strong>offset init. guess</strong></em></td>
</tr>
<tr class="odd">
<td><strong>Conv. Struct.</strong></td>
<td><strong>VSDD</strong></td>
<td><strong>Conv. Struct.</strong></td>
<td><strong>VSDD</strong></td>
<td><strong>Conv. Struct.</strong></td>
<td><strong>VSDD</strong></td>
<td><strong>Conv. Struct.</strong></td>
<td><strong>VSDD</strong></td>
</tr>
<tr class="even">
<td><p><em>θ</em><sub>1</sub></p>
<p><em>θ</em><sub>2</sub></p>
<p><em>θ</em><sub>3</sub></p>
<p><em>θ</em><sub>4</sub></p>
<p><em>θ</em><sub>5</sub></p></td>
<td><p>1</p>
<p>1</p>
<p>0.05</p>
<p>0.05</p>
<p>1</p></td>
<td><p>0.9999</p>
<p>0.9972</p>
<p>0.0519</p>
<p>0.0450</p>
<p>0.9982</p></td>
<td><p>0.9999</p>
<p>0.9970</p>
<p>0.0531</p>
<p>0.0445</p>
<p>0.9975</p></td>
<td><p>1.0711</p>
<p>0.9337</p>
<p>0.0523</p>
<p>0.0477</p>
<p>1.0672</p></td>
<td><p>1.0241</p>
<p>0.9845</p>
<p>0.0521</p>
<p>0.0460</p>
<p>1.0183</p></td>
<td><p>0.0770</p>
<p>0.5310</p>
<p>7.4810</p>
<p>11.224</p>
<p>0.3780</p></td>
<td><p>0.0390</p>
<p>0.3660</p>
<p>5.0920</p>
<p>9.5930</p>
<p>0.2950</p></td>
<td><p>1.1990</p>
<p>1.2620</p>
<p>6.2690</p>
<p>9.2310</p>
<p>1.1220</p></td>
<td><p>2.4560</p>
<p>1.8750</p>
<p>7.1560</p>
<p>13.491</p>
<p>1.8370</p></td>
</tr>
<tr class="odd">
<td><p><em>k</em><sub>1</sub></p>
<p><em>k</em><sub>2</sub></p>
<p><em>c</em><sub>1</sub></p>
<p><em>c</em><sub>2</sub></p>
<p><em>m</em><sub>2</sub></p></td>
<td><p>1</p>
<p>1</p>
<p>0.05</p>
<p>0.05</p>
<p>1</p></td>
<td><p>0.9999</p>
<p>0.9955</p>
<p>0.0519</p>
<p>0.0449</p>
<p>0.9982</p></td>
<td><p>0.9999</p>
<p>0.9945</p>
<p>0.0531</p>
<p>0.0444</p>
<p>0.9975</p></td>
<td><p>1.0711</p>
<p>0.9964</p>
<p>0.0523</p>
<p>0.0509</p>
<p>1.0672</p></td>
<td><p>1.0241</p>
<p>1.0023</p>
<p>0.0521</p>
<p>0.0469</p>
<p>1.0183</p></td>
<td><p>0.0770</p>
<p>0.8930</p>
<p>7.4810</p>
<p>11.163</p>
<p>0.3780</p></td>
<td><p>0.0390</p>
<p>0.6570</p>
<p>5.0920</p>
<p>9.6220</p>
<p>0.2950</p></td>
<td><p>1.1990</p>
<p>1.0410</p>
<p>6.2960</p>
<p>9.1460</p>
<p>1.1220</p></td>
<td><p>2.4560</p>
<p>0.9100</p>
<p>7.1560</p>
<p>14.165</p>
<p>1.8370</p></td>
</tr>
<tr class="even">
<td>average no. of iterations</td>
<td>0</td>
<td>4.9800</td>
<td>4.0600</td>
<td>5.0500</td>
<td>4.8500</td>
<td colspan="4"></td>
</tr>
</tbody>
</table>

###### VSDD in Second or Both Stories

To understand the effect of VSDD location(s) on the estimation, this
2DOF numerical example is modified to consider a VSDD in the second
story. Trying to know the best location for VSDD within the structure,
this numerical example is extended to consider the existence of VSDD in
the second story, and in both stories. Again, comparison is made in each
case to the *conventional approach* with no VSDDs located in the
structure. The parametric frequency domain identification is applied and
the results with a VSDD in the second story or one in each of the two
stories are compared with those in *conventional structure* case.

From Fig. 20, it can be observed that the variation of the stiffness in
the second story was clearly reduced, giving better identification of
the second-story stiffness compared to a VSDD in the first story only.
However, the variation in estimation of the first-story stiffness *k*~1~
is larger with the second-story VSDD than with the *conventional*
approach. Figure 21 shows that the damping estimation is still poor, as
seen in the previous section. Yet, the VSDD still improves over the
*conventional structure* approach.

Finally, two VSDDs are considered, one located in each story. Several
simultaneous values of both VSDDs are considered. Each VSDD is
controlled to exert an extra stiffness of 0%, 20%, 30%, and 40% each in
its location. Five different profiles of simultaneous values of the
exerted extra stiffness by both VSDDs are considered in the
identification problem such that the values of additional stiffness are
(0%,0%), (0%,40%), (20%,30%), (30%,20%) and (40%,0%) of the stiffness of
each story, respectively. The results of the stiffness parameter
identification problem are compared, similarly to the previous cases, to
data from conventional structure simulations (so the amount of data is
the same in both cases).

![](media/image19.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904766 .anchor}Figure 20. Stiffness error levels
for the iterative method with\
exact start for 2DOF model with VSDD in 2^nd^ story only

![](media/image20.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904767 .anchor}Figure 21. Damping error levels
for the iterative method with\
exact start for 2DOF model with VSDD in 2^nd^ story only

![](media/image21.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904768 .anchor}Figure 22. Stiffness error levels
for the iterative method with exact\
start for 2DOF model with VSDD in both 1^st^ and 2^nd^ stories

The stiffness estimate results in Fig. 22 show that though the variation
of the stiffness identification of the second story is clearly decreased
and improved, the variation of the stiffness identification of the first
story is still more than the conventional approach case. However, some
improvement in the variation of the stiffness of the first story
compared to the case of a VSDD in the second story can be noticed. As a
result of that, it can be deduced that VSDDs may be most effective in
identifying the stiffness of the stories or levels where they are
located. However, multiple VSDD simultaneously may provide decreasing
returns in a 2DOF structure, which may not be a fact for more complex
structures. Comparing Figs. 18, 21 and 23, it is obvious that using a
VSDD in each story simultaneously results in better identification of
the damping coefficients of the first and second stories. The variation
of the damping coefficients is reduced significantly and the means are
more accurate.

![](media/image22.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904769 .anchor}Figure 23. Damping error levels
for the iterative method with exact\
start for 2DOF model with VSDD in both 1^st^ and 2^nd^ stories

[]{#__RefHeading___Toc44904778 .anchor}Table 2. Range of 3σ Stiffness
and Damping Estimate Relative Errors for 2DOF Shear Model

  ----------- ----------------- ------------ ------------ ------------ ------------
  **Initial   **Case**          ***k*~1~**   ***k*~2~**   ***c*~1~**   ***c*~2~**
  guess**                                                              

  *Exact*     Conventional      \[−0.24%,    \[−3.1%,     \[−20%,      \[−40%,
              structure         0.23%\]      2.2%\]       27%\]        20%\]

              VSDD in 1st story \[−0.13%,    \[−2.4%,     \[−10%,      \[−22%,
                                0.11%\]      1.4%\]       23%\]        5.0%\]

              VSDD in 2nd story \[−0.70%,    \[−2.0%,     \[−12%,      \[−20%,
                                0.70%\]      1.1%\]       16%\]        20%\]

              VSDDs in both     \[−0.50%,    \[−2.3%,     \[−5.0%,     \[−25%,
              stories           0.40%\]      1.5%\]       10%\]        10%\]

  *Biased*    Conventional      \[3.50%,     \[−3.5%,     \[−16%,      \[−26%,
              structure         12.00%\]     2.8%\]       25%\]        30%\]

              VSDD in 1st story \[−5.00%,    \[−2.5%,     \[−18%,      \[−46% ,
                                10.0%\]      3.0%\]       27%\]        34%\]
  ----------- ----------------- ------------ ------------ ------------ ------------

Table 2 show the ranges for three times standard deviation of stiffness
and damping estimate error. This is done for the different cases of the
2DOF shear building model. Note that all VSDD configurations in the 2DOF
structure model, the number of estimations was 100 times with exact
starting guesses and 26 times for with biased starting guesses.

### 4.4.2 Two Degree-of-Freedom Pier-Deck Bridge Model

The 2DOF bridge model shown in Fig. 11, provides an example that has
full-scale structural parameters. A VSDD is attached between the deck
and the pier as shown in Fig. 12 is assumed to provide an additional 0%,
10%, 20%, 30% and 40% stiffness; *i.e*., *κ*~1~ = 0.0, *κ*~2~ = 0.1,
\..., *κ*~5~ = 0.4. Numerical quantities for this model, drawn from
Erkus *et al.* (2002), are considered as an illustrative example
*k*~1~ = 15.791 MN/m, *k*~2~ = 7.685MN/m, *m*~1~ = 100 Mg (tons),
*m*~2~ = 500 Mg, *c*~1~ = 125.6 kN·s/m, and *c*~2~ = 196 kN·s/m. The
experimental transfer functions are simulated in MATLAB by using the
exact transfer functions plus the Fourier transform of a Gaussian pulse
process typical of band-limited Gaussian white sensor noise vector
processes. The noise induced in the experimental transfer function of
the 2DOF bridge model is shown in Fig. 24, in both linear and log
scales.

The iterative least-squares parametric frequency domain identification
is performed on this 2DOF bridge model, both with a VSDD in the
isolation layer between the deck and pier and without. The results show
error reductions in both stiffness and damping estimates (though with
the latter more modest than the former). The relative error in the
stiffness estimates, shown in Fig. 25, have some small bias  about 0.5%
in the estimate of the pier stiffness and about 2% in that of the
isolator  that exists both with and without the VSDD (though the bias
is to higher pier stiffness estimates with the VSDD, and lower without).
While the bias is similar, the VSDD approach shows notable reductions in
stiffness estimate variation, demonstrating that the VSDDs improve the
identification. Similar observations may be made regarding damping
estimates, as shown in Fig. 26. The VSDD approach slightly decreases the
bias in the pier damping coefficient estimate, and modestly decreases
the variation in both pier and isolator damping estimates.

  -----------------------------------------------------------------------
  ![](media/image23.wmf){width="5.333333333333333in"
  height="3.4270833333333335in"}

  ![](media/image24.wmf){width="5.333333333333333in"
  height="3.4270833333333335in"}
  -----------------------------------------------------------------------

[]{#__RefHeading___Toc44904770 .anchor}Figure 24. Exact and noisy TF
magnitudes for 2DOF bridge model

![](media/image25.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904771 .anchor}Figure 25. Stiffness estimate
error levels for the iterative\
method with exact start in 2DOF bridge model

![](media/image26.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904772 .anchor}Figure 26. Damping estimates
error levels for the iterative\
method with accurate start in 2DOF bridge model

### 4.4.3 Six Degree-of-Freedom Shear Building Model

The six degree-of-freedom structure has story and VSDD characteristics
similar to the 2DOF building structure discussed previously. The story
stiffnesses and floor masses are chosen, for simplicity, to be unity.
The damping coefficient in each story is 0.05. There are three VSDDs in
this structure, one in each of the first three stories, that can each
provide five discrete stiffness levels that are 0, 10, 20, 30 and 40
percent of the story stiffness. The transfer functions are generated
using the exact transfer function plus noise (shown in Fig. 27 for one
noise realization), and the estimation is performed 20 times (each with
a different random seed). In each estimation, the VSDD approach uses
four sets of VSDD stiffness combinations: (0%,0%,0%), (10%,20%,30%),
(20%,30%,10%), and (30%,10%,20%) in the first, second and third floors,
respectively. As before, for fair comparison, the conventional structure
approach uses four realizations of the transfer functions so that the
amount of data is the same as the VSDD approach.

![](media/image27.wmf){width="6.495833333333334in"
height="4.174305555555556in"}

[]{#__RefHeading___Toc44904773 .anchor}Figure 27. Exact and noisy TF
magnitudes for 6DOF shear building model

Figure 28 shows the improvement in the estimates of the stiffness and
damping coefficients when using VSDDs in the structural model. It is
quite evident that the variations are dramatically reduced  up to 3
times for stiffness estimates and up to 4 times in the damping
estimates. These results are similar to the improvements observed in
lower-order 2DOF shear building model and the 2DOF bridge model.
However, it can also be deduced that the expected improvements are more
contrasted for higher degree of freedom models (*i.e.*, more complex
models). The resulting estimates of the mean and the coefficient of
variations for the stiffness and damping coefficients are shown in Table
3.

  ----------------------------------------------------- -----------------------------------------------------
  ![](media/image28.wmf){width="2.9972222222222222in"   ![](media/image29.wmf){width="2.9972222222222222in"
  height="1.9597222222222221in"}                        height="1.9597222222222221in"}

  ![](media/image30.wmf){width="2.9972222222222222in"   ![](media/image31.wmf){width="2.9972222222222222in"
  height="1.9597222222222221in"}                        height="1.9597222222222221in"}

  ![](media/image32.wmf){width="2.9972222222222222in"   ![](media/image33.wmf){width="2.9972222222222222in"
  height="1.9597222222222221in"}                        height="1.9597222222222221in"}
  ----------------------------------------------------- -----------------------------------------------------

[]{#__RefHeading___Toc44904774 .anchor}Figure 28. Stiffness and damping
estimates error levels for the\
iterative method with exact starting guess in 6DOF model

[]{#__RefHeading___Toc44904779 .anchor}Table 3. Estimate Means and
Coefficients-of-Variation for 6DOF Shear Building Model

+-----------+--------+-----------+-----------+-----------+-----------+
| **Va      | **e    | **coeff.  |           |           |           |
| riables** | xact** | of        |           |           |           |
|           |        | variati   |           |           |           |
|           |        | on\[%\]** |           |           |           |
+-----------+--------+-----------+-----------+-----------+-----------+
|           |        | ***exact  |           | ***exact  |           |
|           |        | init.     |           | init.     |           |
|           |        | guess***  |           | guess***  |           |
+-----------+--------+-----------+-----------+-----------+-----------+
|           |        | **Conv.   | **VSDD**  | **Conv.   | **VSDD**  |
|           |        | Struct.** |           | Struct.** |           |
+-----------+--------+-----------+-----------+-----------+-----------+
| *k*~1~    | 1.0    | 1.0003    | 1.0001    | 0.18      | 0.066     |
|           |        |           |           |           |           |
| *k*~2~    | 1.0    | 0.9996    | 0.9998    | 0.097     | 0.034     |
|           |        |           |           |           |           |
| *k*~3~    | 1.0    | 1.0001    | 0.9999    | 0.163     | 0.068     |
|           |        |           |           |           |           |
| *k*~4~    | 1.0    | 1.0002    | 1.0003    | 0.219     | 0.074     |
|           |        |           |           |           |           |
| *k*~5~    | 1.0    | 0.9999    | 0.9999    | 0.172     | 0.062     |
|           |        |           |           |           |           |
| *k*~6~    | 1.0    | 0.9995    | 0.9998    | 0.135     | 0.042     |
+-----------+--------+-----------+-----------+-----------+-----------+
| *c*~1~    | 0.05   | 0.051095  | 0.05092   | 4.824     | 1.364     |
|           |        |           |           |           |           |
| *c*~2~    | 0.05   | 0.049145  | 0.049415  | 3.497     | 0.816     |
|           |        |           |           |           |           |
| *c*~3~    | 0.05   | 0.049555  | 0.049515  | 4.667     | 1.032     |
|           |        |           |           |           |           |
| *c*~4~    | 0.05   | 0.050210  | 0.050065  | 7.281     | 1.248     |
|           |        |           |           |           |           |
| *c*~5~    | 0.05   | 0.049230  | 0.049365  | 3.444     | 1.188     |
|           |        |           |           |           |           |
| *c*~6~    | 0.05   | 0.049560  | 0.049965  | 4.553     | 0.877     |
+-----------+--------+-----------+-----------+-----------+-----------+

## 4.5 Using VSDDs with Variable Damping Only

Whereas the previous sections investigated the effects of VSDDs
operating in a variable stiffness mode, it is possible that inducing
added damping will have an effect on the SHM as well. Using variable
damping would be of great interest since "smart" semiactive damping
devices have received extensive study for vibration mitigation purposes
(Soong and Spencer, 2002) and capitalizing on the synergies between
control and SHM would be a cost-effective solution. In this example, the
2DOF bridge model is studied with a variable damping device in the
isolation layer between the pier and deck. The damping levels of the
device are 0%, 10%, 20%, 30%, and 40% of the isolator damping
coefficient. As in the other numerical examples, the device is
considered ideal (no internal device dynamics), the transfer functions
are measured through standard means, and the least-squares parametric
frequency-domain identification technique is applied.

The results, shown in Figs. 29 and 30, indicate that the *VSDD*
approach, with the damping levels described above, did not differ
significantly from the *conventional structure* approach. The relative
error in stiffness estimates in Fig. 29 have similar bias in both
approaches and a slightly larger variation with the VSDD damping device.
Similar observations may be made about the damping estimates (Fig. 30).

One reason that the variable damping here did not provide any notable
improvement is the very small force levels generated by the damping
device. The damping forces in the isolation layer of this bridge model
are about one order of magnitude smaller than the stiffness forces. As a
result, changing the damping by this small amount has relatively little
effect. Using larger force levels in the variable damper may overcome
this difficulty and should be studied in the future.

![](media/image34.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904775 .anchor}Figure 29. Stiffness estimates
error levels for the iterative method with exact starting\
guess in bridge model with multiple variations of damping coefficients

![](media/image35.wmf){width="4.277083333333334in"
height="2.8069444444444445in"}

[]{#__RefHeading___Toc44904776 .anchor}Figure 30. Damping estimates
error levels for the iterative method with exact starting\
guess in bridge model with multiple variations of damping coefficients

# 5.0 CONCLUSIONS AND RECOMMENDATIONS

This report documents the initial study of the effectiveness of using
variable stiffness and damping devices to improve estimates of
structural parameters for structural health monitoring and damage
detection. VSDDs are controllable passive devices that have been shown
to have significant potential for mitigating structural response to
natural hazards. This study demonstrates that VSDDs also have strong
promise for use in SHM as well. Since VSDDs can be commanded to exert
various force time histories, the response of a structure may be altered
through the parametric changes effected by the VSDDs. The multiple
"snapshots" of structural characteristics provided by the VSDD approach
can provide additional information to make structural parameter
identification more accurate and efficient.

This study investigated VSDD/SHM by identifying structural parameters
--- mass, stiffness and damping coefficients --- based on measured
absolute acceleration transfer function data, using a parametric
frequency-domain least-squares identification method. For each numerical
example and configuration, the structural parameters were identified,
first with one or more VSDDs in the structure, and then with no VSDDs.
In all cases, simulated sensor noise is added to the exact transfer
function to replicate the noisy transfer functions that are typically
obtained through standard experimental techniques. In each VSDD
configuration, data is collected while the VSDDs are commanded to act in
one of several discrete stiffness or damping modes, with different noise
corrupting each subsequent data set. To make for a fair comparison, the
conventional structure approach is provided with the same amount of
data. The variation in identified structural parameters due to the
effects of random noise are studied by performing these identifications
a number of times, each with a different random seed to generate the
noise, giving a measure of both the mean and the variance of the
structural parameter estimates. While the VSDD approach is applicable to
a wide variety of structural identification methodologies, it is herein
studied in the context of a least-squares identification using the
frequency-domain transfer function representation of the input/output
dynamics of a structure. It is shown that one commonly-used
simplification of this identification method gives biased results, and
an alternate iterative approach is shown to give superior results. This
iterative least-squares identification, with and without VSDDs, are
applied to several numerical examples: a bridge pier/deck model and two
shear building models. The VSDD is commanded to act as a discrete
stiffness element or a discrete damping element, with stiffness or
damping coefficient based on a fraction of the corresponding coefficient
for the location where the VSDD is installed in the structure. When a
VSDD acts as a discrete multi-level damping element with coefficient a
fraction of that of the structure, little improvement is gained over the
conventional structure approach, probably due to the very low force
levels due to the VSDD damping. In contrast, the discrete stiffness
behavior is shown to improve estimates of the stiffness and damping
coefficients in the structure, particularly for the more complex
numerical example with several VSDDs.

While the results herein definitely show that there is the potential for
VSDDs to make a real contribution to structural health monitoring, a
number of open questions and future directions have arisen in the course
of this study. First, since the response to ambient excitation sources
is, generally, quite low, the force levels of the VSDDs in this study
were small --- in fact, orders of magnitude smaller than the forces the
structure is designed to withstand as well as, perhaps, two orders of
magnitude smaller than the peak forces capable with current VSDD
technology. Consequently, the VSDDs, especially in the discrete damping
mode, were less effective than would likely occur if they were commanded
to use larger (but still moderate compared to the structural capacity)
force levels. Future research should test if larger orce levels will
give further improvement in identifying structural parameters. Second,
VSDD behavior is commanded through a control system based on responses
in the structure; for example, to act as a stiffness element with a
particular stiffness level, the VSDD exerts a force proportional to the
relative displacement across it. Herein, these devices were considered
ideal and internal device dynamics were neglected. Future studies should
examine the effects of VSDD device dynamics on the SHM improvements.
Third, while it is convenient to consider (and explain) VSDDs in terms
of the added stiffness and damping they provide, these concepts are
inherently linear, time-invariant; and reliant only on local information
rather, VSDDs can be commanded to exert a range of forces that change
over time and cannot be characterized as either stiffness or damping or
combination of the two. The challenge lies in developing a methodology
for determining the best VSDD force time histories to improve the
structural parameter estimation in an even more robust and efficient
manner than studied herein.

# Appendix A: Computational Procedure for 6DOF Shear Model

While using the symbolic math capabilities of [Matlab]{.smallcaps} to
formulate the iterative least-squares frequency-domain identification is
convenient, it is not very efficient for larger problems, requiring the
development of a semi-numerical solution procedure. The core of this
alternate solution procedure lies in recognizing that the residual error
${\hat{e}}_{l}$ in Eq. (11) can be separated into a set of numerical
coefficients and a numeric representation of the symbolic unknowns. The
$r$^th^ element of the $m \times 1$ error ${\hat{e}}_{l}$ in Eq. (11)
can be modified as

${\hat{e}}_{\text{rl}}({j\omega}_{i},\theta,\kappa_{k}) = \frac{B_{r}({j\omega}_{i},\theta,\kappa_{k}) - A({j\omega}_{i},\theta,\kappa_{k}){\hat{H}}_{r}({j\omega}_{i},\kappa_{k})}{{\hat{A}}_{l}({j\omega}_{i},\kappa_{k})} = c_{\text{rl}}^{T}({j\omega}_{i},\kappa_{k})v_{r}(\theta)$
(A1)

where $c_{\text{rl}}({j\omega}_{i},\kappa_{k})$ is an
$n_{v_{r}} \times 1$ vector of numeric coefficients of the corresponding
elements of $v_{r}(\theta)$, which themselves are each of the form

$v_{\text{sr}}(\theta) = \prod_{i = 1}^{n}\theta_{i}^{p_{i}^{\text{rs}}}$,
$s = 1,\ldots,n_{v_{r}}$, $r = 1,\ldots,m$ (A2)

where the $p_{i}^{\text{rs}}$are non-negative integer exponents that can
be arranged in $n \times 1$ vectors $p^{\text{rs}}$ which can be
collected into set
$P_{r} = \left\{ p^{r1},\ldots,p^{\text{rn}_{v_{r}}} \right\}$. The
symbolic $v_{r}(\theta)$ is a function only of the unknown parameters
and is different for each element of the transfer function error. The
coefficient vector $c_{\text{rl}}({j\omega}_{i},\kappa_{k})$ depends on
the frequencies and the known parameters, and is different in each
iteration and for each element of the transfer function matrix. Since
the $v_{r}(\theta)$ can be defined by the numeric exponents of the
elements of the unknown $\theta$, they can be stored and manipulated
numerically just through the tensor $p_{i}^{\text{rs}}$.

Substituting Eq. (A1) into the square error Eq. (12), results in

$\Delta_{l}^{2}(\theta) = \sum_{r}^{}{\sum_{i}^{}{\sum_{k}^{}{v_{r}^{T}(\theta)c_{\text{rl}}^{\dagger}({j\omega}_{i},\kappa_{k})c_{\text{rl}}^{T}({j\omega}_{i},\kappa_{k})v_{r}(\theta)}}}$
(A3)

where $( \cdot )^{\dagger}$ denotes complex conjugation. The order of
summation and multiplication can be rearranged due to $v_{r}(\theta)$
being independent of $i$ and $k$, giving

$\Delta_{l}^{2}(\theta) = \sum_{r}^{}{v_{r}^{T}(\theta)\left\lbrack \sum_{i}^{}{\sum_{k}^{}{c_{\text{rl}}^{\dagger}({j\omega}_{i},\kappa_{k})c_{\text{rl}}^{T}({j\omega}_{i},\kappa_{k})}} \right\rbrack v_{r}(\theta)}$
(A4)

The quantity in brackets is purely numerical and can be computed quite
easily; let it be defined

$C_{\text{rl}} = \sum_{i}^{}{\sum_{k}^{}{c_{\text{rl}}^{\dagger}({j\omega}_{i},\kappa_{k})c_{\text{rl}}^{T}({j\omega}_{i},\kappa_{k})}}$,
$r = 1,\ldots,m$ (A5)

which incorporates all of the information for the data frequency points
${j\omega}_{i}$ and the different sets of known parameters $\kappa_{k}$.

The vectors $v_{r}(\theta)$ are not all the same. It is convenient to
define a single similar vector $v(\theta)$ that contains all of the
symbolic products of powers of all unknown $\theta$ contained in the
individual $v_{r}(\theta)$.

$v_{s}(\theta) = \prod_{i = 1}^{n}\theta_{i}^{p_{i}^{s}}$,
$s = 1,\ldots,n_{v}$ (A6)

where the $p_{i}^{s}$ are non-negative integer exponents that can be
arranged in $n_{v}$ $n \times 1$ vectors $p^{s}$ that can be collected
as set $P = \left\{ p^{1},\ldots,p^{n_{v}} \right\}$, which is the union
of the individual sets for each element of the transfer function,

$P = \underset{r = 1}{\overset{m}{}}P_{r}$ (A7)

The individual symbolic vectors $v_{r}(\theta)$ can be related to the
global symbolic vector by

$v_{r}(\theta) = T_{r}v(\theta)$, $r = 1,\ldots,m$ (A8)

where $T_{r}$ is a transformation matrix containing ones and zeros

$\left\lbrack T_{r} \right\rbrack_{\text{is}} = I(p^{\text{ri}},p^{s})$,
$r = 1,\ldots,m$, $i = 1,\ldots,n_{v_{r}}$, $s = 1,\ldots,n_{v}$ (A9)

where $I(x,y)$ is an indicator function that has value 1 when every
element of $x$ matches the corresponding element of $y$.

Substituting Eqs. (A5) and (A7) into the square error Eq. (A4) gives

$\Delta_{l}^{2}(\theta) = \sum_{r}^{}{v^{T}(\theta)T_{r}C_{\text{rl}}T_{r}v(\theta)}$
(A10)

which can be simplified since $v(\theta)$ is independent of transfer
function index $r$

$\Delta_{l}^{2}(\theta) = v^{T}(\theta)\left\lbrack \sum_{r}^{}{T_{r}^{T}C_{\text{rl}}T_{r}} \right\rbrack v(\theta) = v^{T}(\theta)C_{l}v(\theta)$
(A11)

where $C_{l}$is an $n_{v} \times n_{v}$ symmetric numeric matrix with
elements $c_{\text{rs}}^{l}$. Computing these elements can be done quite
efficiently; the slowest part is the process of forming the union of the
exponent sets and building the resulting transformation matrices
$T_{r}$.

Determining the least-square estimates of the unknowns then requires the
gradient of $\Delta_{l}^{2}(\theta)$

$\frac{\partial}{\partial\theta}\left\lbrack \Delta_{l}^{2}(\theta) \right\rbrack = 2\sum_{r = 1}^{n_{v}}{\sum_{s = 1}^{n_{v}}{c_{\text{rs}}^{l}v_{r}(\theta)\frac{\partial v_{s}(\theta)}{\partial\theta}}}$
(A12)

and the elements of its corresponding Jacobian

$\frac{\partial^{2}}{\partial\theta_{i}\partial\theta_{j}}\left\lbrack \Delta_{l}^{2}(\theta) \right\rbrack = 2\sum_{r = 1}^{n_{v}}{\sum_{s = 1}^{n_{v}}{c_{\text{rs}}^{l}\left\lbrack v_{r}\frac{\partial^{2}v_{s}}{\partial\theta_{i}\partial\theta_{j}} + \frac{\partial v_{r}}{\partial\theta_{i}}\frac{\partial v_{s}}{\partial\theta_{j}} + \frac{\partial v_{r}}{\partial\theta_{j}}\frac{\partial v_{s}}{\partial\theta_{i}} + \frac{\partial^{2}v_{r}}{\partial\theta_{i}\partial\theta_{j}}v_{s} \right\rbrack}}$
(A13)

The $j$^th^ element of the gradient can be simplified to a set of
products of powers of unknowns and the corresponding coefficients

$\frac{\partial}{\partial\theta_{j}}\left\lbrack \Delta_{l}^{2}(\theta) \right\rbrack = 2\sum_{r = 1}^{n_{v}}{\sum_{s = 1}^{n_{v}}{{\overset{\sim}{c}}_{\text{rs},j}^{l}{\overset{\sim}{v}}_{\text{rs},j}(\theta)}}$
(A14)

where

${\overset{\sim}{c}}_{\text{rs},j}^{l} = c_{\text{rs}}^{l}p_{j}^{s}$ and
${\overset{\sim}{v}}_{\text{rs},j}(\theta) = \prod_{i = 1}^{n}\theta_{i}^{p_{i}^{r} + p_{i}^{s} - \delta_{\text{ij}}} = \prod_{i = 1}^{n}\theta_{i}^{{\overset{\sim}{p}}_{i,j}^{\text{rs}}}$
(A15)

The elements of the Jacobian can be formed in a similar manner,
resulting in a set of coefficients and exponents.

Equation (A14), then, is a nonlinear equation that is the linear
combination of products of powers of the unknowns and can be solved
through standard numerical root solving algorithms --- this study used
the function fsolve( ) in [Matlab]{.smallcaps}'s Optimization Toolbox.
The coefficients ${\overset{\sim}{c}}_{\text{rs},j}^{l}$ and the
exponents ${\overset{\sim}{p}}_{i,j}^{\text{rs}}$can be computed in a
purely numeric procedure from the assumed structural model, and the root
solver is numerical. Thus, this alternate numerical solution avoids
symbolic computation entirely. The result is that the 2DOF
identification problems can be solved in a few seconds on a 2.4GHz
Pentium 4, whereas a mostly symbolic computation takes on the order of
several minutes on the same computer.

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