Icasi2004

Proceedings of ICASI - 2004
International Conference on Advances in Structural Integrity
July 14-17, 2004, Indian Institute of Science, Bangalore, India

ICASI/XX-XXX

Time domain structural health monitoring with
magnetostrictive patches using five stage hierarchical neural
networks.
Ghosh D. P. (a) and Gopalakrishnan S. (b)

(a) Graduate student (b) Assoc. professor,
Dept. of Aerospace Engineering, Indian Institute of Science, Bangalore 560012

ABSTRACT

An integrated method for damage detection of composite laminates is presented in this paper using time
domain data obtained from magnetostrictive sensors and actuators and artificial neural networks (ANN)
identification with five stage hierarchical neural network (HNN). Magnetostrictive actuators are actuated
through an actuation coil, which vibrates the composite laminate. The presence of delamination, due to
induced magnetic field intensity, changes the stress response of the structure. This in turn changes the
magnetic flux intensity of the magnetostrictive sensor. The changes in the flux density are sensed through a
sensing coil as open circuit voltage. The ANN is applied to establish the mapping relationship between
structural damage status (location and severity) and sensor open circuit voltage. As ANN is prone to
overtraining and the dimension of input space is considerable high, a five-stage hierarchy of networks is
used for the identification procedure. The results of delamination damage detection for composite laminate
show that the method developed in this paper can be applied to structural damage detection and health
monitoring for various industrial structures. To demonstrate this approach, numerical simulations are
carried out on a composite cantilever beam to identify size and location of delamination using the sensor
data for a known actuation for a certain combination of sensor and actuator locations.

Keywords: Magnetostrictive, FEM, SHM, ANN, Hierarchical Neural Network, Inverse Problem.

1. INTRODUCTION may broadly refer to as functional materials.
With the availability of functional materials and
Composites have revolutionized structural the feasibility of embedding them into or
construction. They are extensively used in bonding them to composite structures, smart
aerospace, civil, mechanical and other industries. structural concepts are emerging to be attractive
Present day aerospace vehicles have composites for potential high performance structural
up to 60 % or more of the total material used. applications.1 A smart structure may be generally
More recently, materials, which can give rise to defined as one which has the ability to determine
mechanical response when subjected to non- its current state, decides in a rational manner on a
mechanical loads such as PZTs, Terfenol-D, set of actions that would change its state to a
SMAs, have become available. Such materials more desirable state and carries out these actions

in a controlled manner over a short period of important ways to ensure reliable operation and
time. With such features incorporated in a reduced maintenance cost for in-service
structure by embedding functional materials, it is structures. Magnetostrictive material such as
feasible to achieve technological advances such Terfenol-D, hitherto considered as only actuator
as vibration and noise reduction, high pointing material, was shown to be used for sensing
accuracy of antennae, damage detection, damage application in reference.7 In this work, the
mitigation etc.2, 3 authors proved this capacity experimentally by
passing a magnetic field on to an actuator
During the operation of a structure, damages may magnetostrictive patch and measured the voltage
develop, which will cause a change in the across the sensing patch to infer the presence of
strain/stress state of the structure and the damage.
vibration characteristics. By continuously In this paper we take this approach not only to
monitoring one or more of these response confirm its presence but also its location.
quantities, it is possible to assess the condition of Noncontact magnetostrictive strain sensor was
the structure for its structural integrity. Such a explored by Kleinke, D. K. et al.8 and the study
monitoring of the structure is called structural of magnetostrictive particulate actuator was done
health monitoring. Health monitoring application by Anjanappa, M. et al.9 Sensing of delamination
has been receiving great deal of attention all over in composite laminates using embedded
the world, due to possible significant impact on magnetostrictive material was studied by Krishna
safety and longevity of the structure. To Murty, A. V. et al.10 Authors [18] had developed
implement health-monitoring concept it is a new finite element formulation for inbuilt
necessary to have a number of sensors to magnetostrictive patches for performing
measure response parameters. These responses numerical simulations.
will then be post-processed to assess the The mathematical relationships between sensor
condition of structure. Mark Lin and Fu-Kuo open circuit voltage and structural damage status
Chang 4 built such a system when they developed (i.e., damage location and severity) are very
a built-in monitoring system for composite complex. It is not only strongly non-linear, but
structures using SMART layer containing a also often has no analytical solution. Deduction
network of actuators or sensors. from sensor output to practical damage status is
Change of structural dynamic performance mathematically classified as inverse problem,
caused by structural damage that is less than 1% and is very hard to compute precise solution
of the total structural size is unnoticeable. Yan using mathematical analysis. If one takes the
and Yam5 pointed out that when the crack length inherent law between sensor output and practical
in a composite plate equals 1% of the plate damage status as a black box, the mapping
length, the relative variation of structural natural relationship between these two state spaces can
frequency is only about 0.01 to 0.1%. This was be established using genetic algorithms (GAs) or
also shown by Nag et al.6 Therefore, using artificial neural networks (ANN). Thus, one need
vibration modal parameters, e.g., natural not know explicitly the inherent law in structural
frequencies, displacement or strain mode shapes, damage detection. Moslem and Nafaspour11 and
and modal damping are generally ineffective in Chou and Ghaboussi12 reported some researches
identifying small and incipient structural on structural damage detection using GAs, and
damage. It has been theoretically and they were successful in determining the severity
experimentally proved that local damage in a and locations of structural damage. However,
structure will cause the reduction of local GAs-based structural damage detection requires
structural stiffness, which leads to variation of repeatedly searching from numerous damage
dynamic performance of the whole structure. In parameters so as to find the optimal solution of
industry, using the time domain measured the objective function (measured data). The cost
structural vibration responses to identify and of computation limits the use of GA for damage
monitor structural damage is one of the detection applications.

ANN has particular advantage in establishing with different initial condition, learning rate,
accurate mapping relationships between sensor momentum rate, learning algorithm and learning
data and physical parameters of structural sequence.
damage. When classifications and identification
of structural damage needs to be carried out, the In this study, an integrated damage detection
required task is only to train the ANN in advance method is developed for composite laminate
using a set of known sensor data and damage through theoretical study and numerical
physical parameters of the structures that needs simulation. This method requires the generation
detection. Hung and Kao13 and Yun and Bahng14 of excitation and structural response
reported their researches on structural damage measurement using bonded magnetostrictive
detection using ANN, and their results showed patch actuator and sensors; which is used in
that ANN is a highly effective tool for hierarchical ANN framework for classification
identifying structural damage. and identification of structural damage.
By using the hierarchical scheme, a complicated
large-scale system is decomposed into a set of 2. FORWARD ANALYSIS
lower order subsystems and a coordination Application of magnetic field causes strain in the
process, and thus becomes tractable. Hierarchical magnetostrictive material (Terfenol-D) and the
structures can have more than two levels. stress, changes magnetic flux density of that
However, in practice, two-level structures are material.15
usually popular. Here five stages of hierarchy are The three-dimensional constitutive relationship
considered for structural health monitoring case. for magnetostrictive material is generally written
As the original sensor output (open circuit as
voltage) is high dimensional data hence the ANN
input space, it is quite impossible to train the {ε } = [ S {H } ]{σ } + [d ]T {H } (1)
network. To reduce the dimension of input space {B} = [d ]{σ } + [ µ {σ } ]{H } (2)
of ANN, different dimensional reduction
algorithms are used like ICA, PCA, peak value, The Equation- (1) is used as actuator and
peak location etc. But some times due to this Equation- (2) is used as sensor in the composite
dimensional reduction procedure, damage structure. Using these two equations, the time
signatures are lost in the lower dimension data, domain sensor response due to actuation in the
which leads to lose of the essential output actuator can be computed through finite element
uniqueness of ANN mapping. Here to deal with formulation [18]. In the forward analysis for a
this problem, output space is partitioned in given actuation history and a given damage
different overlapped subspace and different condition, sensor response history can be
experts are trained in these partitions. With in obtained for a particular structure. Where as, in
every expert, a number of validator networks are inverse problem, either actuation history or
trained. All of them have the same set of sample damage condition is unknown. Detail procedure
data in higher dimensional input space, but with for forward analysis can be obtained from [18].
different lower dimensional input space coming
from different dimensional reduction procedures.
Every ANN are trained and validated by a fixed 3. INVERSE ANALYSIS
set of sample set. Single validator network is Inverse problem can be classified in two
consisting of different ensembler network using category, force inverse problem and geometry
different partitioning of training set and inverse problem. In the force inverse problem,
validation set from the sample set. Similarly actuation history is unknown but the structure
single ensembler network is also classified with and its response is known. In geometry inverse
different multi layer perceptron (ANN). These problem applied force and response of structure
ANNs have different network architecture are known but structure is unknown. Damage
(hidden layers, hidden nodes), and are trained identification is the geometric inverse problem,
which can be solved by artificial neural network.

3.1 Artificial Neural Network the minimal training error. A small learning rate
Artificial neural networks (ANNs) can provide results in long learning times. A relatively large
non-linear parameterized mapping between a set learning rate results in faster learning but can
of inputs and a set of outputs with unknown also result in a chaotically learning behavior
function relationship. Thus ANNs are universal during training of the network. The function of
function approximators and are therefore the momentum term is to increase the size of the
attractive for automatically learning of the (non- learning steps when the direction in the weight
linear) functional relation between the input update is the same as the direction in the
variables and the output variables. A three-layer previous step. As with the learning rate, if the
network (Figure-1) with the sigmoid activation momentum term is too large the network will
functions can approximate any smooth mapping. display a chaotic learning behavior.
A typical supervised feed-forward multi layer If many training epochs are used, an ANN tends
neural network is called as a back propagation to overtrain the learning material (i.e. the
(BP) neural network. The structure of a BP accuracy on the training material is very high
neural network mainly includes the input layer whereas the accuracy on new instances is much
for receiving input data; the hidden layer for lower). In this work over training is avoided by
processing data; and the output layer to indicate dividing the training set into two groups, and
the identified results. In this study, ability of using one group of patterns to train the network
identifying structural damage status for an ANN while the other one is used for validating the
is acquired through training the neural network performance of the trained network. It was
using the known samples. Normally, many observed that the training error decreases along
training epochs are required before a set of with number of epoch while the validation error
weights is found that accurately fit the training decreases at first, bounces around, and then starts
material. increasing. The optimal learning is achieved at
The training of a BP neural network is a two-step the global minimum of validation error.
procedure. In the first step, the network If the number of hidden units in ANN is too
propagates input through each layer until an small, the modeling capacity of the network is
output is generated. The error between the output too low, and it is impossible for the learning rule
and the target output is then computed. In the to find an adequate model. If, on the other hand,
second step, the calculated error is transmitted the number of hidden units is too large, the
backwards from the output layer and the weights modeling capacity of the network is too
are adjusted to minimize the error. The training substantial, resulting in a strong inclination
process is terminated when the error is towards overlearning.
sufficiently small for all training samples. The Committee Machine
data set is separated into two parts, one for It is perhaps impossible to combine simplicity
training and the other for testing or validating the and accuracy in a single model of ANN. Single
network performance. The network parameters multi-layered perceptron (MLP) uses a black box
are determined, as is common practice, through approach to globally fit a single function into the
experimentation. This includes the number of data, thereby losing insight into the problem.
hidden nodes and the learning rates. Data This problem was studied [19] by partitioning
obtained from the magnetostrictive sensors the input and output space into a piecewise set of
described above, is used to train conventional subspaces, with each subspace having its own
back propagation networks to identify the expert.
delamination size and location of the composite Hierarchical Neural Network
laminate. With the common three-layer neural network
The accuracy of a trained network is measured architectures, networks lack internal structure; as
by calculating the mean square error (MSE) on a consequence, it is very difficult to discern
the training sample. The learning rate determines characteristics of the knowledge acquired by a
the size of the steps in the search space to find network in order to evaluate its reliability and

applicability. Alternative neural-network error gives more weightage of the neural
architecture is presented, based on a hierarchical network. In the execution phase, these
organization shown in Figure-3. By using the weightages are used to get weighted average of
hierarchical scheme, a complicated large-scale all neural networks output as the output of
system is decomposed into a set of lower order ensemble network. From the distribution of the
subsystems and a coordination process, and thus output of different neural networks and their
becomes tractable. A five-stage hierarchical corresponding weightage one measure of
neural network is designed by combining a overtraining can be computed. If the outputs of
multilayer perceptron first stage and mixture-of- different neural network are close (at least for
experts in the subsequent stages. The second those has more weightage) then networks are
stage mixture-of-experts, ensembler network, well trained otherwise it is overtrained. Every
learns to minimize the overtraining errors. The trained neural network is tested through the test
third stage mixture-of-experts, validation data set, which gives the testing errors as a
network, learns to minimize the validation errors. measure of generalization of the neural networks.
And the fourth stage mixture-of-experts, expert These testing errors with the weightages of the
networks, learns to minimize the error of network neural networks give the weighted average of
due to loss of information for input space testing error, as a measure of testing error of the
dimension reduction. And Finally fifth stage ensemble network. Next issue in the neural
committee machine choose the appropriate expert network is the generalization of the network
network from all expert networks. Each lower using the testing error of ensemble network.
level subsystem is solved independently for a Validation Network
fixed value of the coordination variable whose As, every neural network within the ensemble
value is adjusted by the upper level coordination network is trained with same training sample;
unit in an appropriate fashion so that the lower these neural networks need testing for
level subsystems resolve their problems. The generalization of the ensemble network. For
coordination is to provide a solution to the testing of network, sample data set is separated
overall system. Continuous exchange of into two groups, one for training and other for
information between the lower subsystems and testing (validation). After training of neural
the upper coordination unit will finally lead to a networks, with the training sample, the neural
better solution. The whole procedure is discussed networks simulate the testing sample and get the
as follows. testing error. These testing errors with their
Ensemble Network weightages give the measure of generalization of
Often artificial neural networks are prone to the ensemble network. However, every network
overtraining, where network trains the in the ensemble network is trained and tested
computational and experimental noises. And through the same set of training data set and
there is no direct rule to draw the line between testing data set respectively. To get a more
well training and overtraining for a set of training general input-output mapping from a set of
examples and network architecture. One of the sample data set, different division of training
indirect ways to get a measure of overtraining of sample and testing sample is essential. This can
the network is Ensemble Network. In ensemble be done through a systematic manner using
network, a number of neural networks are train validation network. Validation Network consists
with the same training samples but with different of some number of ensemble networks. These
initial condition, learning rate, training ensemble networks are trained and tested with
algorithm, network architecture and training same sample data set with different partition for
sequence (for sequential learning). In training training data set and testing data set. However,
phase, each network trains and generates training every validation network is train for a fixed set
error for the training samples. On the basis of of sample data set.
these training errors, weightages of the trained Dimensional Reduction of Input Space
neural network is determined, where less training

In time domain structural health monitoring, time and actuation combination. But as the input
histories of the sensor outputs (open circuit space subdivision is known a’ priory, committee
voltages) are the original input space for the machine for input space subdivision is not
input-output mapping, which is very high required.
dimensional. So, it is not possible to train the Conditional Expert
network taking full dimension of the original Always some dimension of output space is
input space (sensor output). To address this difficult to train than other dimension of output
issue, different dimension reduction procedures space. In SHM of composite laminate, depth
are available in the literature, which can reduce wise training is difficult than span wise training.
the dimension of the original input space keeping To solve this kind of problem some conditional
main features of the high dimensional data. But experts are trained. If the dimension of output
for structural health monitoring, suitable subset of some expert is less than the original
dimension reduction procedure is not available, dimension of output space, the value for the
which will reduce the dimension of the input missing dimension is required a ‘priory to
space preserving the signature of damage from execute the network. These experts are called
the high dimensional sensor output data. To Conditional Expert. Conditional experts are used
overcome this problem a number of reductions to get finer location for some dimensions in the
procedure is taken to increase the chance of output space, when locations for remaining
preserving the damage signature in the reduced dimensions are already known by other type of
input data sets. In the next section this issue is experts. For structural health monitoring problem
discussed in a systematic manner. in composite laminate, identification of layer
Expert wise location of the delamination is difficult
Expert consists of a number of validation using general type of experts. Ones the span wise
networks, which are train and tested through location is determined using general type of
same output of a sample data set but with experts, these conditional experts are trained
different lower dimensional input of the data set. taking samples from within that location.
These different input data sets come from Training and Testing of hierarchical network
different dimension reduction procedure of the In training phase, every multilayer perceptrons
original input set, which is high dimensional (MLP) are trained and tested with their
sensor output. So, every expert is a mapping corresponding training and testing samples.
from sensor output to the damage properties of These training will give their training and testing
the structure and performance of the expert error. Although network architectures and
depends on the performances of its validation learning algorithms are same, due to different
networks, which is trained through initial condition, learning rate, momentum rate
dimensionally reduced input sample set. and learning sequence the trained MLPs will be
Committee Machine different. Depending upon the training error
But some times, due to this reduction procedure every trained MLP is associated with their
the mapping looses the output uniqueness, which weightage. In the ensemble network these
is essential for the training of neural network. To weightages and their testing errors are considered
overcome this problem, output space is divided to calculate average testing error. These testing
with a set of overlapped subsets. Size of these errors are for every ensembler network. All
subsets are such that the output uniqueness ensemble networks within one validation
within a subset is preserved as well as sufficient network calculate are weighted average testing
number of sample data is available to train and error. This is the training and testing phase of the
test the network. Then for every subset one hierarchical network. Training and testing phase
expert is trained and tested taking sample data is limited within ANN and ensembler network.
from these subsets. Similar to the division of Execution of hierarchical network
output space, input space can be divided in In identification phase, execution samples (time
different subsets on the basis of sensor, actuator domain sensor output) are feed into every expert

to get their opinion and confidence. From this In this paper a numerical study on 12 layered
mutual information the location and size of the beam containing two patches, one acting as an
delamination is obtained with the level of actuator and the other as a sensor has been
confidence. Every expert pass on these execution presented. In order to evaluate the influence of
samples to their subordinates, validation delamination location and extent on structural
networks after assigned dimension reduction. dynamic characteristics, the situation with only
These validation networks also pass on the one delamination is considered in this study. In
sample to ensembler network. Similarly the finite element model, the delamination is
ensembler network gives these samples to their modeled keeping two elements in the same
subordinate MLP (ANN) network. These MLP location, and integrated bottom element from
execute these samples and give their opinion. As bottom layer to delamination layer and top
different MLP are trained differently, their element from delaminated layer to top layer. At
opinion will also be different. These opinion and delaminated zone, two nodes are created in the
their corresponding training weightage will same places, one is connected with top elements
create opinion and variance for the ensembler and other is connected with bottom elements.
network. After getting all the ensembler level
information, information on validation network Forward Analysis
will be created by fusing this ensembler level Numerical simulation is carried out by
information. In the execution phase of validation considering a unidirectional laminated composite
network, the opinion and variance of ensembler beam of total thickness 1.8 mm as shown in
network is used. Opinion of validation network Figure-1. Length and width of the beam is 500
is created by the opinion of ensembler network mm and 50 mm respectively. The beam is made
and their corresponding weighted average testing of 12 layers with thickness of each layer being
error. Similarly variance of validation network is 0.15 mm. Delamination is modeled as explained
created from variance of ensembler network with in the last section. Parametric studies are done by
their weighted average testing error. If this changing delamination size span wise of the
opinion is with in the range of inherited expert cantilever beam for each layer. Position of sensor
and variance is with in their accepted limit, this is fixed at 9th layer from bottom of the beam and
validation network is considered as active near the support of the beam, while the position
validation network. As the dimension reduction of actuator is fixed at 1st layer from bottom of
algorithm is different between different the beam and 425 mm apart from support. Size
validation networks, one weightage for of the actuator is 50 mm X 50 mm with 0.15 mm
dimension reduction is used to give more thickness and size of the sensor is 50 mm X 50
weightage for better algorithm. This operation mm with 0.3 mm thickness. Elastic modulus of
will be done for all experts. For execution in the composite is assumed 181 GPa and 10.3 GPa in
expert level, information are fused from parallel (E1) and perpendicular (E2) direction of
subordinate, validation networks. Active experts fiber. Poison ratio (ν), density (ρ) and shear
are determined depending upon the maximum modulus (G12) of composite are taken as 0.0, 1.6
number of active validation network within that gm/c.c. and 28 GPa respectively. Elastic
expert. Then weighted opinion and weighted modulus (Em), poison ratio (νm), shear modulus
variance among active validation network are (Gm) and density (ρm) of magnetostrictive
calculated, as the opinion and variance of the material are assumed 30 GPa, 0.0, 23 GPa and
hierarchical network (HNN). If more than one 9.25 gm/c.c. respectively. Magneto-mechanical
active expert are available, committee machine coupling coefficient is 15E-09 m/amp. Direct
calculate the opinion and variance of the HNN transient dynamic analysis has been done with
considering variance of all active experts. 500 time steps to calculate open circuit voltage
of the sensor in each time steps. Relative
4. NUMERICAL EXAMPLES permeability µr is the ratio of permeability of the
material and permeability of air is assumed as 10

for magnetostrictive material. Permeability at their time integrals are taken as the input space
vacuum or air is 400π nano-Henry/m. Number of of the four type of validation neural network.
coil turn in sensor (Ns) and actuator (N) is Every validation network consists of four
assumed 1000. Actuation current at actuator (I) is ensemble networks. These ensemble networks
taken as 1.0 Amp at three different frequencies. are trained and tested by same sample data set
but with different random partitioning between
Result and Discussion training and testing data sets. So every ensemble
Numerical results have been simulated for a network is trained and tested by a fixed set of
fixed position of sensor and actuator (x1 =25mm, training and testing data sets respectively. In
y1 =0.45mm, w1 =50mm, d1 =0.3mm, x2 order to identify the delamination length at each
=425mm, y2 =-0.825mm, w2 =50mm, d2 layer, one BP neural network with 15 inputs and
=0.15mm) combination, for different locations of 2 outputs are designed. One hidden layer of node
the delamination. Open circuit voltages in the strength 10 is taken as the net architecture.
sensor have been shown in 3-D plot. Figure-2 Every expert is trained by the sample data within
shows open circuit voltages in the sensor when their expertise location. These samples are for
the delamination is between 4th and 5th layer, the delamination in the corresponding location.
between 8th and 9th layer with input frequency Thus there are thirty experts for each actuation
5000Hz. frequency to predict the size and location of the
delamination. One numerical example is shown
Inverse Problem in the Table-1 for 200mm mid plane
As structural damage information is distributed delamination. Out of 30 experts, 8 experts are
in different vibration modes, and vibration active. Committee machine has fused the
modes with high frequencies are generally more information of these active experts and get the
sensitive to small damage, three different opinion and variance of HNN. Results give sub-
frequencies (50, 500, 5000Hz) are considered to mili-meter accuracy in span direction and 5
actuate the actuator. Thus input space is micrometer accuracy in depth wise direction. As
subdivided in three different subspaces. In this expert #8 gives the better result hence weightage
study, sinusoidal actuation current is considered is more. Table-2 shows the result for all
in the actuation coil. Three sin wave excitation validators of expert #8 and validator #1 gives the
with different frequencies are exerted on the better result. Table-3 showh the result for all
dynamic model of the composite laminate, and ensembler and ensembler #4 shows better result.
the vibration responses of 550 different cases are Table-4 shows the all multi layer perceptron
numerically simulated for each frequency. These (ANN) for expert #8, validator #1 and ensembler
550 cases include the intact laminate, laminates #4.
with delamination damage at different layers and A number of delamination identification is
of 50 different delamination sizes (10 mm to 500 performed using proposed Hierarchical neural
mm) at each layer. network and shown in the Figure-5. It is shown
For structural health monitoring Hierarchical that depth identification is difficult than span
Neural Network (HNN) is used. Training, testing wise identification.
and execution procedure of hierarchical neural
network is shown in Figure-4. Vibration 5. CONCLUSIONS
responses of higher dimension (500 time steps)
for a given delamination (open circuit voltage in The study demonstrates the use of vibration
magnetostrictive sensor), is preprocessed for response using magnetostrictive sensor and
dimension reduction in the input space of the actuator of an in-service structure for health
neural network. Different type of dimension information of the structure. The study also
reduction (PCA, ICA) can be used. Here for shows the feasibility of online damage detection
simplicity first fifteen optimum values and their and health monitoring using hierarchical ANN-
location of the sensor open circuit voltage and based identification. This study is successful in

classifying and identifying structural damage 10. Krishna Murty, A. V., Anjanappa, M.,
location and severity using the designed Wang, Z. and Chen, X. Sensing of
hierarchical neural network (HNN). The results delaminations in composite laminates
show that HNN is a powerful tool for using embedded magnetostrictive
establishing the mapping relationships between particle layers, Journal of Intelligent
open circuit voltages and the structural damage Material Systems Structures, Vol-10
status, and demonstrate the ability of HNN for October 1999,pp 825-835
structural damage detection. 11. K. Moslem and R. Nafaspour, Structural
damage detection by genetic algorithms.
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FIGURES AND TABLES

Figure 1: Laminated Beam with Actuator, Sensor and Delamination.

Figure 2: Actuation frequency 5000 Hz Figure-3

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