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.
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 24, 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.
Figure 5. Mutual benefits of
SHM and VSDDs
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.
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.
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.
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).
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 
and denominator 
polynomials as 
where w is
any frequency within the frequency range of interest in the specific problem.
The experimental TF 
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 
. The error formula was modified to a simpler form 
. Finally, by differentiating the norm of the simplified
error evaluated at known frequencies with respect to the unknown coefficients
of the polynomials and , a number of equations (equivalent to the number of the
unknown coefficients), are obtained. By solving these equations, the
coefficients of and are obtained.
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).
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.
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.
“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.
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).
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.
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.
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 semiactive hydraulic damper (SHD) and installed it in an
actual building (Kurata et al., 2000).
This was the first application of a semiactive
seismic building control system that continuously changes the device damping
coefficient. A forced vibration test was carried out by an exciter with a
maximum force of 100 kN to investigate the
building vibration characteristics and to determine the control system
performance. As a result, the primary resonance frequency and the damping ratio
of a building (without semiactive hydraulic
dampers) decreased as the exciting force increased due to the influence of
non-linear members such as plain concrete curtain walls. After the eight semiactive hydraulic dampers were installed in the
building, the control system performance was 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.
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Figure 6. ISB vibration absorber on I-35 bridge
(Patten et al., 1999)
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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).
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:
, 
(1)
where M, K, and Cd are the mass, stiffness and damping
matrices of the system, and C1, C2, 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
, 
(2)
where 
is the state
vector, 
is the system state matrix which is dependent on the
mass, damping, and stiffness matrices, 
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 
vector of measured
responses corrupted by sensor noise vector v.
Thus, the system can be represented by the transfer function
matrix H(jw).
Each element of H(jw)
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:
(3)
where B(jw)
and A(jw) are the numerator and denominator polynomials,
which may be expanded 
and 
, 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
, i=1, 2, …, nw (4)
is known at various discrete
frequency points. Therefore, the difference between the estimated theoretical
transfer function H(jw)
and the actual experimental one 
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 
vector
, which is to
be estimated, and of known controllable structural parameters, denoted by the
vector 
. The transfer function expression is thus modified to be:
(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
(6)
A conventional least-squares approach may be adopted to
solve this problem, forming a global square error
(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 , 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
(8)
where the symbol 
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
. 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:
(9)
which is, herein, denoted the least-squares
numerator (LSN) method.
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 
is large (i.e., often the case where H(jw)
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 
to the unknown
parameter vector ; then, the denominator of Eq. (6)
is estimated based on the vector 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
(10)
and the error is, thus, formed
as
(11)
and the squared error takes the
form:
(12)
Minimizing the sum of the square error in Eq. (12) will result in an updated estimate 
to the unknown
parameter vector . The iterations continue until the relative differences
between elements and the
corresponding elements of 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 , the use of multiple structural configurations, denoted by
the different values of known parameters , provided by VSDDs in a structure,
can generate more accurate estimates of than can be found with
a comparable amount of data in a conventional structure with a fixed . This is demonstrated for some examples in the following
section.
The least-squares identification with VSDDs
may be applied to various types of structures. In this study, both building and
bridge models are considered.
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, 
, and of the two floors, 
and 
, are used to generate a 
experimental transfer
function at nw distinct frequency values. (Note that xi
herein denotes the displacement of the ith
floor relative to the ground.) Let the unknown parameter vector be given by:
(13)
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A VSDD that can provide a number of distinct stiffness
levels is located in the first story of the structure. The VSDD is installed
in the lateral 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
 (14)
or to the damping coefficient of a variable damping
device
 (15)
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Figure 7. 2DOF shear building model
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Then, the theoretical transfer function can be written in
the polynomial form as:
(16)
Note that the denominator polynomial 
is the same for all
transfer functions from the same input. The numerator and denominator
polynomials can be expressed as:
A(s, q,
k)
= s4 + [q3+q4+q4q5]s3
+ [q1+q2+q2q5+q3q4+k]s2
+ [q1q4+q2q3+kq4]s
+ [q1q2+kq2]
B1(s, q, k)
= [q3]s3
+ [q1+q3q4+k]s2
+ [q1q4+q2q3+kq4]s
+ [q1q2+kq2] (17)
B2(s, q, k)
= [q3q4]s2
+ [q1q4+q2q3+kq4]s
+ [q1q2+kq2]
The parameter identification methods discussed in the
previous section can then be applied. The explicit reference to m1
has dropped out of the transfer function polynomials; it will be assumed here
that of all the parameters, only m1 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.
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Consider a bridge structure such as one shown in Fig.
10, which is a typical elevated highway bridge that consists of decks,
bearings, and piers. The behavior of the bridge deck and piers, with a
bearing between them, while complex, can be well approximated with the simple
2DOF model shown in Fig. 11c.
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 m1 is known.
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Figure 10. General view during construction of high
occupancy vehicle (HOV) lanes (ADOT, 2001)
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The theoretical polynomial transfer function matrix 
is defined similarly
to Eq. (16) where, here, 
is the transfer
function between the ground acceleration and the absolute acceleration of the
pier and 
is the transfer
function between the ground acceleration and that of the bridge deck. The
unknown parameter vector is defined similar to Eq. (13) but k1, is the stiffness of the
pier and k2 the stiffness of the
deck. The vector of known parameters 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.
Figure 11. 2DOF bridge model
Figure 12. Placement of VSDDs
in bridge
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.
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 for the six
degree-of-freedom model is, then, a set of unknown stiffness ki and damping ci coefficients as
follows:
(18)
Finally, the solution for the parameters was done using
the MATLAB® software and using the function fsolve( )
for the solution of the non-linear equations in the unknown parameters .
To demonstrate the benefits and the advantages of testing
a structure with VSDDs configured in multiple
settings, some numerical examples are considered.
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., k1 = 0.0, k2 = 0.1, ¼, k5 = 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 k = 0.0 ¾
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
nw 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.
Figure 14. Exact and noisy TF
magnitudes
Once the experimental transfer functions are generated, MATLAB® 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 , giving a number of equations that should equal zero. These
equations, which are cubic polynomials in for the parameterization in Eq. (13), are solved numerically using fsolve( )
function in the Optimization Toolbox™ with function tolerances and 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 ) 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 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.
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|
Relative % error in
estimate of k1
|
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Figure 15. Stiffness estimate
error levels with least-squares numerator method
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 ¾ much better than the least-squares numerator method.
Figure 17 shows the relative error in the estimates of the stories stiffness
for the conventional 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).
Relative % error in
estimate of k1
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|
Figure 17. Stiffness estimate
error levels for iterative method
with exact start for 2DOF model with VSDD in 1st story only
Figure 18. Damping estimate
error levels for the iterative method
with exact start for 2DOF model with VSDD in 1st 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 c1, and 36% and 40% for the estimation of the damping
coefficient in the second floor c2. 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 k1. Again, this indicates that using VSDDs
improved the stiffness estimates.
Figure 19. Stiffness estimate
error levels for the iterative method
with offset start for 2DOF model with VSDD in 1st 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.
Table 1. Estimate Means and
Coefficients-of-Variation for 2DOF Shear Building Model
