Damage detection and localization in composite beam structures based on vibration analysis.
Khatir, Samir ; Belaidi, Idir ; Serra, Roger 等
1. Introduction
Composite materials are nowadays increasingly used as an
alternative to conventional materials, because of their high strength,
weight saving, specific rigidity, and mechanical flexibility especially
in the aerospace industry. The aim of vibration based damage detection
techniques is to determine the occurrence of structural damage, its
location and severity. The information produced by a damage assessment
process can play a vital role in the development of economical repair
and retrofit program. Rytter [1] proposed a classification in order to
allow a comparison between different techniques, which consists of four
levels. The first level is the detection, the second level is the
localization, the third level is the assessment, and finally the fourth
level, which is the consequence of damage, predicts the remaining life
and the actual safety of the structure in a certain state of damage. The
complete health state of a structure can be determined based on
presence, location, type and severity of damage (diagnostics) and
estimation of remaining useful life (prognostics) [2]. The concept of
dynamical invariants [3] in the SHM methodology is called Beacon-based
Exception Analysis for Multi-missions (BEAM).
Methods of identification of defects and their analysis in a
qualitative relation to the location of defect and its importance have
been studied in literature [4-5]. Noise does not affect stable low-order
dynamical models that can be created using POD based low-order model for
fault detection [6].
POD provides the most efficient way of capturing the dominant
components of an infinite-dimensional process with only (often
surprisingly) few modes. Various applications of POD to structural
dynamics were carried out in the literature [7-11]. The diagnostics of
various machines and mechanisms is an important problem to determine the
damaged structural elements. Solving of such problem, for example use
the method of resonance frequencies, the damaged structural element is
judged by the deviation of resonance frequency [12]. The location of
damage in the structure is more complicated for certain class of
structure, e.g. a beam-like structure, using vibration analysis. In this
approach, the beam structure is successively loaded with mass. A
harmonic force is used to excite the structure in the loaded zones. The
detection and localization of damage are indicated by the relationship
between vibration amplitudes of the additional mass and its location
[13]. Damage detection of a bridge structure based on computer
simulations of static displacement or strain data using POD [14] and
finite element analysis had shown a success detection. The damage
indicator based on mode shape data was introduced [15] to identify
damage in beam-like structures. A two-step procedure for damage
detection in structures from changes in curvature mode shapes was
proposed [16]. The damage identification and localization of some
complex mechanical system described in terms of reduced number of modes
using finite elements was reported [17], where an isolation procedure to
describe these parameters was followed. Mathematical simulations of
structural elements and dynamic behaviour due to loss of stiffness at
damaged area were presented [18].
The most existing damage detection study was based on modal
curvature and investigated the indicator value changes between the
intact and damaged states [19]. The processing of nonlinear features
associated with a damage event by quadratic time frequency distributions
for damage identification in a frame structure were studied [20].
A simple method for determining the stiffness matrix of structural
and mechanical systems using measured natural frequencies and
corresponding mode shapes was proposed [21]. The use of natural
frequency shifts for damage identification was proposed in several
research works, where the success of this parameter in the case of a
single damage location was proven [22]. The first four natural
frequencies of a simulated cantilever beam were used to locate a single
crack [23]. The identification of a single crack in an experimental
single story frame from shifts in the first three natural frequencies
was investigated [24] using a damage identification algorithm to locate
and identify the size of a single crack. The identification of a single
crack in a vibrating rod based on knowledge of the damage induced shifts
of a pair of natural frequencies was investigated [25]. A damage
identification methodology based on natural frequency changes to a
numerical model of a beam on an elastic foundation was studied [26].
Current electrodynamic vibrators and vibration rigs for monitoring
materials, structural elements and machine parts objects subjected to
vibrations and large accelerations were used to detect damage [27]. The
thrusting force and amplitude of oscillations in electrodynamic
vibrators are discussed along with broadening of the frequency range.
The results found in reference [28] identified single damage events
as stiffness loss, connection loosening and lump mass addition. Two
methods of damage assessment based on a relationship between modal
strain energy and measured modal properties tailored to single damage
cases [29] were used. Damage approach prediction in beam and plate
structures with initiated damage were presented [30]. The results
provided the basis for the development of diagnostics algorithms. The
location and severity of a single damaged element in a simulated planar
truss were determined by minimizing the square of the Residual Force
Vector (RFV) [31].
In the present work, a new damage identification method is applied
to a composite beam structure using genetic algorithm and particle swarm
optimization. By introducing the proper orthogonal decomposition with
radial basis function, a reduced model is built, the calculation of cost
function is minimized and more accurate results are obtained.
A finite element model of bi-dimensional monolithic beam reinforced
by a graphite-epoxy discretized into 10 elements is used to generate
vibration data. The damage resulted in reduction of stiffness with
levels of 5% and 25%, which are placed in different positions. A
comparative study between both results of (GA) and (PSO) using finite
element method indicates that PSO is better than GA. However, the
calculation of PSO takes a longer time, with small error between the
real and estimated damage. The PSO with POD is better than PSO with FEM
algorithm to detect and localize damage with higher accuracy and shorter
computational time. The effect of noise in this methodology is
considered in some cases by assigning noises to natural frequencies.
2. Theoretical background
2.1. Proper orthogonal decomposition (POD) with radial
basis functions (RBF)
POD with Radial Basis Functions (RBF) is used for the interpolation
of the data with previously reduced dimensionality by the POD. A group
of responses for a given system can be very effectively compressed using
the theory of a separate POD. This compression allows for a significant
reduction of the dimensionality. To make it clear, let us recall that
our goal is to define an approximation that should be used instead of FE
simulations. Therefore, we wish to find a function that depends on some
parameters collected in vector p such that:
f(p) = u. (1)
In Eq. (1), vector u collects the required output of the system and
represents the frequency vector of a damaged beam modelled using FEA.
However, since we already constructed a low order approximation of
these responses, they can be represented in the new truncated system by
the matrix of amplitudes. This practically means that RBF can be applied
in already reduced dimensionality, where responses are expressed as
amplitudes, and therefore the function we are looking for is in the
following form:
fa(p) = [bar.a]. (2)
The relationship previously defined between the responses expressed
in the reduced and full dimensionality holds for the functions f and fa.
Thus we can write:
f(p) = [bar.[PHI]]fa(p) = u. (3)
Applying the Radial Basis Functions (RBF) technique, the
approximation of fa is written as a linear combination of some basis
functions [g.sub.i], i.e.:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Or written in matrix form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Similar to other examples of Radial Basis Functions (RBF)
interpolation, after the basis functions are chosen, we need to solve
for the interpolation coefficients collected in matrix B.
Having in mind that the values of the function fa to be
approximated are collected in the matrix of amplitudes A in the reduced
space. This leads to the following equation:
B G = [bar.A]. (6)
Eq. (6) is solved for unknown matrix B, and then finally, by
combining Eqs. (5) with (3), we arrive to the required general formula
for the approximation of the system response for arbitrary parameter
combination, which is:
u [approximately equal to] [bar.[PHI]]B g(p). (7)
Eq. (7) was derived by performing the Radial Basis Functions (RBF)
interpolation of the system responses in the reduced space, which is
represented by amplitude matrix [bar.A] and further transformed by
pre-multiplying it by reduced POD basis by [32].
2.2. Genetic algorithm (GA)
The Genetic Algorithm (GA) is an evolutionary optimization method,
used efficiently for different kinds of optimization problems in last
decades [33]. In our study, 10 individuals, also called chromosomes,
represent the two damage parameters of position and severity, are
converted to binary encoding. The population evolves toward better
solution iteratively in a process inspired from the natural evolution,
where they are allowed to cross among themselves in order to obtain
favorable solutions. The fitness is the objective function value,
calculated in Eq. (10), as it will be explained latter under section 3.
The best feasible solutions have higher probability to be chosen as
parent of new individuals, where the properties of the parents are
combined by exchanging chromosomes parts, to produce two new designs.
Afterwards, the mutation is performed by randomly replacing the digits
of a randomly selected chromosome. These basic operators are repeated to
create the next generations until the maximum number of iterations is
reached [34].
2.3. Particle Swarm Optimization (PSO)
The Particle Swarm Optimization (PSO) is a method inspired by the
behavior of different kinds of flocks (birds, bees, fishes, etc.), which
is characterized by distinct social and psychological principles. These
principles lead the flock to adapt its physical movement towards food
seeking in a particular way, which ensures both the speed of the quest
and the avoidance of potential adversities, such as hostile predators.
This method has been given considerable attention in recent years among
the optimization research community.
It is pretty clear that PSO is a population based optimization
method built on the premise that social sharing of information among the
individuals can provide an evolutionary advantage. The fact that, as an
optimization method based on population data, PSO requires a relatively
small number of parameters, reduces the computational cost and
facilitates the implementation of the algorithm. Due to its simple
implementation, PSO can be used in both simple and large-scale
optimization problems. Therefore, PSO has been a rather attractive
optimization method in scientific circles.
The algorithm was first proposed by Kennedy and Eberhart. PSO has
been used widely in the recent years and has been modified in a variety
of versions that can handle the majority of optimization problems with
or without the presence of constraints.
According to the PSO method, a random population of candidate
solutions is considered to be a particle moving through the
multi-dimensional design space in search of the position of the global
minimum. The particles coexist and cooperate with each other to achieve
this position. Every particle can be characterized by its physical
position in the design space and its speed of movement. Furthermore,
each particle has the ability to remember the best position it has
passed so far or personal best (Pbest, Eq. (8)) and the best position
that any other particle of the swarm has passed so far or global best
(G_best, Eq. (9)).In every iteration the speed of the particle is
updated in a stochastic way. Finally, the old and new speed vectors are
used in order to update the position of the particle in an iterative
manner [35].
The update equations for the speed and the position of the
particles are in the following form:
{[v.sup.i] (t + 1)} = w {[v.sup.i] (t)} + [c.sub.1]
{[r.sub.1]}({[x.sup.Pb,j]} - {[x.sup.j] (t)}) + + [c.sub.2]
{[r.sub.2]}({ [x.sup.Gb] }-{[x.sup.j] (t)}). (8)
{[x.sup.j] (t +1 )} = {[x.sup.i] (t)} + [v.sup.i] (t +1). (9)
2.4. Description of test structure
We consider a clamped free beam of pure unidirectional composite of
AS4/3501-6 graphite-epoxy materials with symmetrical order of layers.
The finite element model is discretized in 10 elements as shown in Fig.
1. Each node of the finite element has three degrees of freedom, normal
displacement w along the z-axis, rotation y around the y-axis and
longitudinal displacement u along the x-axis. Since the beam is
macroscopically considered homogeneous, the shear correction coefficient
is the same as for isotropic beam, i.e. [K.sub.correction] = 5/6 [36].
The material properties and beam dimensions of AS4/3501-6 graphite
epoxy [37] are given in Table 1.
[FIGURE 1 OMITTED]
3. Objective function
A practical procedure implemented in a standalone software was
developed based on two different optimization algorithms: particle swarm
optimization and Genetic Algorithm. The detection and localization task
were maintained as an inverse identification problem, where the two
parameters of damage position and its level are calculated through the
fitness Eq. (10) where the [[omega].sup.r.sub.i] is the frequency of the
controlled beam, and [[omega].sup.c.sub.i] is the frequency calculated
using the proper orthogonal decomposition and finite element method:
Fitness = [[SIGMA].sup.n.sub.i] ([([[omega].sup.r.sub.i] -
[[omega].sup.c.sub.i]).sup.2] / [([[omega].sup.r.sub.i]).sup.2]) . (10)
3.1. Optimization parameters
In this study, we address the problem for damage detection and
localization in the beam by Genetic Algorithm (GA) and particle swarm
optimization (PSO) using FEM and new approach for damage detection and
localization using Proper Orthogonal Decomposition (POD).
The particle swarm optimization (PSO) and genetic algorithm (GA)
methods were used to minimize the fitness function. In PSO, coordinates
of the particles in a two dimensional space are the unknown parameters
for damage level and position, using 100 particles. In a second
approach, each of the 100 individuals contains two chromosomes
representing the required damage parameters. The maximum number of
iteration is set equal to 200.
After several applications, a crossover coefficient of 0.8 and
mutation of 0.1 were used in the GA parameters, while C1 = C2 = 2.0
considered in the PSO method.
3.2. POD-RBF Based damage detection technique
During the optimization process, we noticed many approaches that
are used to detect and locate damage in beam-like structures using
finite element method with optimization techniques. Generally, these
techniques require long computation time because each iterative
optimization method requires the calculation of location and level of
damage. However, our POD approach with RBF can converge in a very short
time, while it provides a solution with high accuracy. To build a
reduced model by POD-RBF method, 250 FEM results of the studied
structure were produced by varying the damage level from 0% to 50% using
a step of 2% in each of the 10 elements of the beam. The output
parameters considered for the damage identification process are the
first five natural frequencies.
[FIGURE 2 OMITTED]
The inverse problem is solved using PSO algorithm. The
Methodological approach to the damage detection and localization
problems is illustrated by a flow chart as shown in Fig. 2.
4. Results and discussion
The damage of beam structure was simulated by reducing the
stiffness of selected elements by varying amounts using FEM. The largest
damage (level 1) consists of 25% stiffness reduction at the center of
damage region followed by 5% reduction along length of the beam
structure. Three different damage locations were studied, namely, near
fixed end (D1), center of the plate (D2), and near free edge (D3), as
shown in Fig. 3. A modal analysis was performed to determine the natural
frequencies of the sane of beam structure and damage cases D1, D2, and
D3.
[FIGURE 3 OMITTED]
4.1. Damage detection and localization by genetic algorithm and
particle swarm optimization using FEM
GA and PSO were used to identify the parameters of the three
considered damage scenarios D1, D2 and D3, for damage located at the 2,
5 and 8 elements, respectively with a damage severity of 5 and 25%. The
error between real damage and estimated damage by GA and PSO is
calculated. A comparison of fitness evolution from three runs of both
algorithms is shown in Figs. 4, 5 and 6. The comparison of damage
location with GA and PSO are given in the Table 3.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
From Figs. 4 to 6 and Table 2, it can be seen that good results are
obtained using both PSO and GA algorithms, however PSO is more accurate
and faster than GA for the detection and localization of damage. It is
noticed that the errors in GA were considerably higher than the errors
in PSO. It should be noted that GA, manipulates different mechanisms
than PSO. In GA, chromosomes share information with each other so that
the whole population moves like one group towards an optimal area.
However, the PSO has one way information sharing mechanism, i.e. only
[x.sup.Gb], which gives out the information to others. The optimization
process takes a long time and sometimes the first operation doesn't
give the best results, even though several attempts have been made to
get the desired results.
4.2. Damage detection and localization by Particle Swarm
Optimization using POD
The proper orthogonal decomposition method (POD) with radial basis
function is a well-known model reduction method based on results of the
studied phenomenon called the snapshot method. To build a corresponding
model of our damaged beam, a snapshot represents a collection of 250
measurements u (see Eq. (1)) of different damage levels [1-25%] and
positions [1-10 element] were considered and the corresponding
frequencies are calculated using FEM.
PSO with POD and PSO with FEM were used to identify the parameters
of the three considered damage scenarios D1, D2 and D3 located at the 2,
5 and 8 elements, respectively, with a damage severities of 5 and 25%.
The error between real damage and estimated damage is calculated. A
comparison of fitness evolution for the three damage scenario of both
algorithms is shown in Figs. 7 to 9. The comparison with both approaches
is also given in the Table 3.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
From Figs. 7 to 9, it can be seen that the PSO with POD is faster
and more accurate than PSO with FEM algorithm to detect and localize
damage. The errors are listed in table 3 between real and estimated
results for both algorithms. It is noticed that the PSO with POD gives
good results with high accuracy and short computation time for locating
damage than PSO with FEM. The calculations were carried out using MATLAB
program in a PC with characterization Intel(R) core (TM) I3-2328 CPU2.20
GHz. Performing such a test with traditional approach, where system
responses are computed by FEM, to solve this current damage detection
problem, would require about one hour on an average computer. Moreover,
sometimes the desired results may not be obtained. However, using
previously calibrated POD-RBF procedure, the results are obtained in a
bit more than 50 s for the first iteration.
[FIGURE 9 OMITTED]
5. Effect of noise
In order to investigate the effect of noise on our damage detection
techniques, White Gaussian noise was added to previous results. To test
the accuracy of the method using PSO technique, the reference data of
the second damage scenario (D2) was manipulated to find natural
frequencies, when we consider noise levels of 5%, 10%, 25% and 50% as
shown in Table 4.
In Table 4, we compare damage positions and severities, after
introducing a perturbation level of 1%, 5%, 10% and 15%. For the cases
of 1%, 5%, and 10% noise, we note that there no significant difference
in the estimated damage levels. However, at a level of 15% noise, the
difference becomes notable.
6. Conclusion
An approach for detecting and locating damage in beam structures
based on model reduction has been investigated at a numerical level
using Genetic Algorithm (GA) and particle swarm optimization (PSO)
methods to determine damage severities and positions. The results of
finite element method (FEM) of the single damaged beam were used to
build the snapshot matrix, essential for building a lower order model by
the proper orthogonal decomposition. The frequencies of the controlled
beam were considered as references in our study.
In the first part of this paper, we run inverse computation using
FEM together with PSO and GA, applied to various damage scenarios. The
results, in the first part of this study, have shown that PSO using FEM
is favorable than GA in damage detection and localization. However, the
process takes a considerable amount of time, and requires several
iterations to get the desired results. In the second part of this study,
we used proper orthogonal decomposition POD with radial basis function
RBF to replace FEM in PSO optimization process. The results were found
in a very short computing time with high precision compared to FEM-PSO
technique. The efficiency of the approach was tested using data with
different noise levels.
http://dx.doi.org/10.5755/j01.mech.21.6.12526
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Received June 18, 2015
Accepted September 17, 2015
Samir Khatir *, Idir Belaidi *, Roger Serra **, Magd Abdel Wahab
***, Tawfiq Khatir ****
* Department Of mechanical engineering University Mhamed Bougara
Boumerdes, LEM Laboratory Research Team Modelling and Simulation in
Mechanical Engineering, 35000 Boumerdes, Algeria, E-mail:
Khatir_samir@hotmail.fr, idir.belaidi@gmail.com
** Laboratoire deMecanique etRheologie INSA Centre Val de Loire,
LMR, 3 Rue de la chocolaterie, 41000Blois, France, E-mail:
roger.serra@insa-cvl.fr
*** Applied Mechanics Laboratory Soete Faculty of Engineering and
Architecture Ghent University Technologiepark Zwijnaarde 903B-9052
Zwijnaarde, Belgium E-mail: Magd.AbdelWahab@UGent.be
**** Institute of science and technology University Centre Salhi
Ahmed, Naama 45000, Algeria, E-mail: khatir-usto@hotmail.fr
Table 1
Dimension and material of beam
Ply property Mean value
Length, m 0.75
Width, m 0.03
Thickness, m 0.005
Longitudinal modulus, GPa 141.96
Transverse Shear modulus, GPa 6
Density, kg x m-3 1600
Poisson's ratio v 0.42
[K.sub.Correction] 5/6
Table 2
Comparison between results for damage detection and
localization by PSO and GA using FEM
Damage Methods Damage Stiffness Error % Error %
element with element reduction Damage Stiffness
Stiffness element reduction
reduction (%)
D1-2-5 GA 2.011 4.997 0.011 0.002
PSO 1.999 4.998 0.001 0.002
D2-15-5 GA 4.993 4.957 0.007 0.043
PSO 5.004 4.999 0 0.001
D3-8-5 GA 7.998 5.000 0.002 0
PSO 7.999 5.000 0.001 0
D1-2-25 GA 2.023 24.995 0.023 0.005
PSO 1.999 24.998 0.001 0.002
D2-5-25 GA 4.995 24.957 0.005 0.043
PSO 5.000 24.999 0 0.001
D3-8-25 GA 7.991 24.9600 0.042 0.042
PSO 5.010 25.001 0.001 0.001
Table 3
Comparison between results for damage detection and
localization by PSO using FEM and POD
Damage Identification Damage Stiffness
element method element reduction
with Stiffness
reduction
(%)
D1-2-5% PSO-FEM 2.011 4.997
PSO-POD 2.000 5
D2-5-5% PSO-FEM 4.993 4.957
PSO-POD 4.9999 5.000
D3-8-5% PSO-FEM 7.998 5.000
PSO-POD 8.000 5.000
D1-2-25% PSO-FEM 2.023 24.995
PSO-POD 2.000 25.0000
D2-5-25% PSO-FEM 4.995 24.957
PSO-POD 5.000 24.999
D3-8-25% PSO-FEM 7.991 24.9600
PSO-POD 8.000 25.000
Damage Error % Error %
element Damage Stiffness
with Stiffness element reduction
reduction
(%)
D1-2-5% 0.011 0.002
0000 0000
D2-5-5% 0.007 0.043
0.0001 0000
D3-8-5% 0.002 0000
0.000 0000
D1-2-25% 0.023 0.005
0.000 0000
D2-5-25% 0.005 0.043
0000 0.0001
D3-8-25% 0.009 0.042
0000 0000
Table 4
Comparison between real damage and estimated
damage with noise
Damage Damage Stiffness Noise Damage Stiffness
scenario element reduction (%) element reduction
(%) with (%) with
noise noise
D2 6 15 1 5.99 14.987
D2 6 15 5 5.981 14.971
D2 6 15 10 5.974 14.968
D2 6 15 15 5.970 14.962
