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Hindawi Publishing CorporationISRN Materials ScienceVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticlePredictability of Inverse Impact Force Location as

Affected by Measurement Noise

Abdelali El-Bakari,1 Abdellatif Khamlichi,2,3 Rachid Dkiouak,1

Ali Limam,4 and Eric Jacquelin5,6

Civil Engineering and Mechanics Laboratory, Abdelmalek Essaadi University, Tangier, Morocco Communications Systems and Detection Laboratory, Abdelmalek Essaadi University, Tetouan, Morocco

Department of Physics, Faculty of Sciences at Tetouan, P.O. Box , MHannech II, Tetouan, Morocco Civil and Environmental Engineering Laboratory, Institute of Applied Sciences at Lyon, Albert Einstein avenue, Villeurbanne Cedex, France

University of Lyon, Lyon, FranceIFSTTAR, LBMC, UMR-T, Universite Lyon , Villeurbanne, France

Correspondence should be addressed to Abdellati Khamlichi; [email protected]

Received June ; Accepted September

Academic Editors: J. L. C. Fonseca, . Matsumoto, and M. Saitou

Copyright Abdelali El-Bakari et al. Tis is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Te impact orce localization inverse problem is considered through a nonlinear optimization procedure. Te objective unction isderived in the particular case o elastic structures or which Maxwell-Betti theorem holds. Additional geometric constraints wereintroduced in order to stabilize optimum search. Te solution o the constrained non linear mathematical problem was perormedby means o two outstanding evolutionary algorithms that include Genetic Algorithm and Particle Swarm Optimization. Focuswas done on the robustness aspect o orce impact localization predictability when an additive white noise is assumed to perturbedstrain measurement. It was ound that the Genetic Algorithm ails to track the exact solution independently rom the noise level asan error wassystematicallypresent in thesolution. On theotherhand, theParticleSwarmOptimisation based algorithm perormedvery well even or noise levels as high as % o the measured strain signal.

1. Introduction

Identication o impact orce location or impact events

occurring on elastic structures can be perormed by variousmethods that were proposed in the literature []. oreview briey some o the important contributions in thisled, Martin and Doyle [] have described how to ndthe location o an impact orce using dynamic responsemeasurements. Tey proposed a solution procedureusing thespectral element method with a stochastic iterative search.Experimentally measured acceleration responses rom tworame structures were used to achieve orce localization byminimizing a tness unction. A Genetic Algorithm wasused to guess iteratively the minimum through monitoringthe actual error associated to a given sampling generation.Te process enabled to discriminate between good and

bad guesses and gave at convergence the correct impactlocation. An alternative technique which employs the arrival

time o each requency component o a pulse detected bymeans o wavelet transorm was proposed by Inoue et al.[]. But this approach suffers rom the lack o accuracyin measurement o small arrival times o signals. Yen andWu [, ] have used multiple strain responses along witha mutuality relationship based on Greens unctions andmeasured strains to achieve identication o orce locationon two-dimensional plate-like structures. Choi and Chang[] minimized the error between measured strain responsesin PZ sensors and numerically evaluated impact orcelocations. Shin [] proposed a technique or identiying theorce location usingmodal displacements and transientsignalmeasured by accelerometers.

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ISRN Materials Science

As mentioned by Doyle [], the implicit character oorce localization requires considering the inverse problemo orce identication as being rather associated to twodecoupled subproblems: localization o impact orce andreconstruction o the time orce signal. Te reason or thisis that the inverse problem o orce identication cannot be

handled by just adding the orce point coordinates as extraunknown parameters to the discrete vector o time orcesignal values. Tese positions intervene in act as implicitparameters, in contrast with the orce history values whichare explicit. Iterations are then required or solution o orcepositions unknowns in order to determine them at rst.reating all orce history values to be implicit parameters isnot a good choice since it would increase dramatically thenumber o unknowns and would consequently penalize thecomputational cost. When the orce localization is obtainedthrough the solution o a nonlinear mathematical program,the orce time signal reconstruction can then be perormedby means o a regularization deconvolution technique.

In this work, separation o the localization and recon-struction phases o the impact orce inverse problem isadopted. Elastic structures subjected to nonpunctual impactsor which the resulting orce eld can be assumed to beuniormaly distributed over a nite domain o the structureare considered. Te localization problem is solved by usingminimization procedures that are o evolutionary type. womethods are examined in the ollowing: Genetic Algorithm(GA) based strategy [] andthe Particle Swarm Optimization(PSO) approach [,]. Focus will be done on the particularrole related to noise affecting strain measurement as to theresulting perturbations that produce and which may impedethese algorithms to converge towards the exact problemsolution.

Te localization problem is derived straightorwardlyrom the reciprocity Maxwell-Betti theorem which is validor any elastic structure. However, as the direct expression othe tness unction ormed by this theorem is ill conditioned,because it admits a lot o trivial meaningless solutionswhich are associated to the xed boundary conditions, thetness unction is modied []. Te introduced modicationtransorms the nonlinear mathematical program to a uniquesolution problem and removes the trivial parasitic solutionswhich complicate optimization process convergence to thereal solution. Some extra constraints are also introducedin order to guide the exploration o the optimal solution;these are associated to the geometric pieces o evidence that

describe the domain containing the tracked unknowns.Te aimis to examine thepredictabilityo impactlocation

in this situation where noise is present in observation mea-surement. Tis means determining the amount o noise thatcan be tolerated and also selecting between the two proposedalgorithms the most suited one that can be used to conductsolution o orce localization problem with sufficientaccuracyand minimal error.

2. Materials and Methods

.. Direct Problem Formulation. Although the problem canbe stated or any elastic structural system, a simplied model

L

x0 + u

x0 u

x0

xi

F : An elastic beam with a uniorm rectangular cross-section and loadedwith a distributed uniorm pressure; our sensorspositions are indicated.

having the orm o a beam with a rectangular uniorm sectionis considered, Figure . Te beam is assumed to be simply

supported on both ends. It has length, width , and height .It is assumed to be made rom a homogeneous and isotropicelastic material with Youngs modulus and density. Teapplied orce modelling impact is assumed to result roma uniormly distributed pressure,, which is applied on arectangular patch as shown in Figure . Te pressurerectangleis assumed to be centred on0and o length2.

Te differential equation or transverse vibrations o abeam writes

2 (, )2 + (, )

+ 4 (, )

4 = (, ) , ()where

is the transverse displacement,

is the axial position,

is the time,is the cross-section area,is the density,isYoungs modulus,is the moment o inertia,is the viscousdamping coefficient, and(, )is the applied pressure over[0 , 0+ ].

Te considered boundary conditions are as ollows:

(0, )= (, )= 0,22(0, )=

22(, )= 0.

()

Using modal superposition and Duhamels integral, the tran-sient dynamic solution o the impacted beam in terms o axial

strainat location is given by the ollowing convolutionproblem: , ,0, =

00+

0

, , (, ) ,

()

where(, , )is the time response unction betweenpointand the sensor location.

ime discretisation o () under the assumption that thepressure (,) is uniorm yields the ollowing linear system:

= 0,, ()

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wheredesignates the vector containing the discrete valueso axial strain at point location indicated by index,is the

vector containing discrete time values o the applied uniormpressure, and(0, )is the oeplitz-like matrix.

Tis last has the ollowing orm:

= (1) 0 0 (2) (1) d ...

...... d 0 () ( 1) (1)

, ()

whereis the number o time steps.An explicit expression can be obtained or matrix. Let

us denote by the time step used in discretization and bythe order o modal truncation; the impulse response unctiongiving the deormation o the top ber or a section havingabscissais given by , 0,

= 42

=1

sin0 sin sin , , ,

()

with(, , ( )) = ( exp(( ))/1 2) sin(1 2( )) i ,and(, , ( )) = 0 i < , where = (22/2)/ and are, respectively, thecircular eigenrequency and damping ratio or a giveneigenmode... Localization Problem as Perturbed by the Presence of

Measurement Noise. Te problem o nding the impactlocation or the beam considered in the present study consistsin identiying the impact patch centre position0 andparameterdening the extent o the impacted zone. Using() to (), the responses measured by strain sensors placedat points having the abscissa and can be expressed,respectively, under the ollowing orm:

= 0,,= 0,. ()

Equation () can be used to prove the ollowing importantcommutativity property which does not contain the pressure

vector: 0, 0,= 0. ()

In this way, (0, ) appears to be the solution o the ollowingequation:

(, V) (, V)2 = 0, ()where is the Euclidian norm.

Tere are a lot o other trivial solutions or () or whichthe actor sin(/) sin(V/) appearing in the second

hand side o () vanishes or all values o [1,]. Teseare associated to the boundaries o the domain containing(0, ); that is,{ = 0; = ; V= 0; V= }. o get the exactsolution o impact location, the parasitic solutions should bewithdrawn rom ().

Denoting by the number o sensors used, the tnessunction to be minimized in order to nd the impact zoneparameters(0, )is proposed under the ollowing orm:0,

=Arg min(,V)

(, V)=

=1

=1 =

1

(, V) (, V) 2,

()

with= (, V)2 + (, V)2 dening the weightingcoefficients that are introduced in order to remove theparasitic solutions, as they vanish also or

{ = 0; = ;V

=0; V= }.In the presence o measurement noise, the perturbedtness unction takes the ollowing orm:

(, V)= =1

=1 =

1

(, V) (, V)2 ()

with

= 1 + ],= 1 +],

()

where is a random number belonging to the interval [1,1]and ]and ], respectively designate the noise level present inmeasurement delivered by sensor and sensor , respectively.

o stabilize the minimization procedure and obtain theunique physical solution, the unconstrained mathematicalprogram dened by () is constrained by adding the ollow-ing geometrical bounds conditions:

40 0

40 ,

40

2 .

()

o solve the mathematical program dened by () and ()(), the PSO algorithm and the GA are considered. Teirperormance will then be assessed as unction o the randomnoise level present in strain measurement.

.. PSO Algorithm. A PSO based method was proposedinitially by Eberhart and Kennedy [, ]. Tis approachhas gained since then considerable interest as being oneo the most promising optimization methods that is ableto provide high speed and high accuracy. PSO mimics thesocial behavior that a population o individuals adapts to itsenvironment by returningto promisingregions thatwerepre-

viously discovered []. Tis adaptation to the environment is

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a stochastic process that depends on boththe memory o eachindividual, called particle, and the knowledge gained by thepopulation, called swarm.

In the simplest numerical implementation o this method,each particle is characterised by our attributes: the position

vector in the search space, the velocity vector, the best

position achieved in its track, and the best position achievedby the swarm. Te process steps can be outlined as ollows.

Step . Generate the initial swarm involving given particlesplaced at random.

Step . Calculate the new velocity vector o each particle,based on its actual attributes.

Step . Calculate the new position o each particle rom thecurrent position and its new velocity vector.

Step . I the termination condition is satised, stop. Other-wise, go toStep .

o be more specic, the new velocity vector o the ith

particle at time + 1, denoted V+1 , is calculated accordingto the ollowing Shi and Eberhart [] ormula:

V+1 = V+ 11 + 22 . ()

In (),1 and2 are random numbers between and , is the best position o the ith particle in its track, andis the best position o the swarm. Tere are three problemdependent parameters that x perormance o thisalgorithm,namely, the inertia o the particle and the two trustparameters

1and

2.

Te new position o the ith particle at time, denoted+1 ,is then calculated as ollows:+1 = +V+1 , ()

where is the current position o theith particle at time.Teith particle actual position enables to determine the bestposition in itstrack . When considering all the particles, theglobal best position o the swarmis then obtained.

PSO algorithm works such that particles concentrate onthe best search position o the swarm. Tey cannot easilyescape rom the local optimal solution since the searchdirection vector V+1 calculated by () always includes the

direction vector to the best search position o the swarm. Tisshows the major eature o PSO algorithm as being a robustprocess o continuous enhancement or optimum search.

In the presence o constraints, a particle move should berestricted in order to remain in the easible solution space byexamining the given constraints. A modied PSO version wasintroduced or constrained problems in order to manage thissituation [].

.. Genetic Algorithm. Genetic Algorithm (GA) was rstlyintroduced by Holland []. It is a probabilistic optimizationmethod that is able to achieve global search by mimickingnatural biological evolution. GA operates on a population

o individuals called the set o potential solutions. Eachindividual is represented by an encoded string (chromosome)that contains the decision variables (genes). raditionally, GAuses binary strings as chromosome representation.

Te GA has an iterative procedure structure that com-prises generally the ollowing ve main steps.

Step . Creating an initial population (0).Step . Evaluation o the perormance o each individual orchromosome () o the population, by means o a tnessunction to be maximized.

Step . Selection o individuals or the reproduction o a newpopulation.

Step . Application o genetic operators: Crossover andMutation.

Step . Iteration o Stepstountil a termination criterion

is ullled.

In the localization problem considered in this work, thecandidate solution is the centre position0 and the extento the impact zone. Tese variables are then coded in achromosome using a binary coding scheme.

o start the algorithm, an initialpopulation o individuals(chromosomes) is dened. Te GA is congured, so that itcreates a xed number o initial individuals at random romthe whole easible solution space. An important parameter ininitialization is the population size. In general, the populationsize affects both the ultimate perormance and the efficiency

o GA and should be determined in a case by case study.

3. Results and Discussion

A pinned-pinned beam having the ollowing material and

geometric properties is considered: = 7.06 1010 Pa; =0.5m; = = 5 103 m; = 2660 kgm3;= 2%;0= 0.417 m; = 0.0417m. Te time interval considered hasthe duration= 1 s. Te beam is assumed to be subjectedto a hal sine pulse pressure having the shape and spectralcontent depicted in Figure . Te maximum pressure is takento be105 Pa.

Te rst ve modes were retained. Teir requencies aregiven by1= 46.72 Hz,2= 186.9Hz,3= 420.5Hz,4=747.5Hz, and5= 1168Hz. Te time step used was =3.4246104 s which satises largely Shannon condition withregards to5.

Four gauge strain sensors were used in this study. Teirlabeling and positions are indicated in able . Previousstudies conducted by means o PSO algorithm have shownthat, while considering the noise ree problem, localizationneeds at least three sensors to be achieved with adequateaccuracy. Here, a ourth sensor has been added in order toenhance the perormance o the localization inverse problemin the presence o measurement noise.

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0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

Time (s)

Pressure

(Pa

)

105

(a)

0 200 400 600 800 10000

0.5

1

1.5

2

Spectrum of the impact force signal

Psd

1012

(b)

F : Characteristics o the impulse impact pressure; (a) timehistory and (b) requency content.

: Positions o the gauge sensors considered or the measure-ment o axial strain at the upper beam ber.

Sensor Position

#1 5/24#2 /3#3 /2#4 2/3

Tree noise levels were considered in order to testrobustness o both PSO and GA based algorithms. Teycorrespond to the ollowing.

Case . Noise ree case where noise amplitude is set to thevalue]= 0% or the our sensors.Case . Noise level is xed at]= 2% or the our sensors.

Case . Noise level is xed at]= 5% or the our sensors.Te parameters used or GA were as ollows:

(i) stopping test:106;(ii) population size: ;

(iii) probability o intersection: ;(iv) probability o mutation: .;

(v) maximum number o generation: .

For GA algorithm the rst values o the unknown param-eters impact centre and extent were initialized with0= =1.25 102.

For PSO algorithm, the ollowing stability parameterswere used: = 0.4;1= 1.25;2= 0.5; size population: .

Figure gives evolution o impact location characteristicsas a unction o iterations in case o noiseless problem. Onecan see the difference existing between these two algorithms;PSO algorithm gives the exact solution o the impact location.On the opposite, GA has not converged to the right solution.

Figure gives evolution o impact location characteristicsas unction o iterations in case o noisy measurementconditions with a noise level ]= 2%. One can observe thatGA has not converged at all, as the calculated solution is tooar rom the exact solution. Te relative error reached %or the impact zone extent. Meanwhile, PSO based algorithmhas continued to give the exact solution with only a moderateerror, representing13% o the impact zone extent.

Figure gives the evolution o impact location character-istics as unction o iterations in case o noise level value givenby]= 5%. Unexpectedly, GA is better in this condition thanor noise level ]

= 2% as the relative error or impact zone

extent has decreased rom % to %. Moreover, one cansee that both GA and PSO algorithms ail to give the exactsolution as the error on the impact zone extent is too large,% and %, respectively.

ables and recall the obtained relative error as theunction o the algorithm used and noise level. One can seethat GA ails to predict the exact solution, with a behavior thatis notmonotonous,while PSO based algorithm perorms wellor small measurement noise levels not exceeding %. Teperormance decreases afer that when identiying the impactzone extent, while it remains sufficiently good or the impactzone centre as the relative error is smaller than %.

o emphasize the irregular behavior o GA, a comparison

has been made between the results o two runs o GA.Figure shows a comparison between the obtained evolu-tions. Tey are not the same because o the stochastic natureo GA. GA is ound thus to be inappropriate or the inverseorce location problem as it is too sensitive to noise level andits repeatability is too poor.

On the opposite, PSO algorithm runs always in the samemanner with perect repeatability.

4. Conclusion

Based on the separation approach that decouples orcelocation rom orce signal reconstruction in an inverse

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0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iteration number

Impactcentre

PSO

GA

(a)

PSOGA

0 20 40 60 80 1000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Iteration number

Impactextent

(b)

F : Noise ree case; evolution o impact zone characteristics.

(a) Impact centre and (b) impact zone extent.

: Relative error affecting impact zone extentas comparedwith the exact solution.

% % %

PSO 0 1.271 001 6.163 001GA 3.113 002 9.198 001 6.800 001

impact problem occurring on an elastic beam, robustnesso a particular localization procedure was analyzed. Tisuses a modied tness unction derived rom Maxwell-Betti theorem by applying some ltering coefficients that

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iteration number

Impactcentre

PSO

GA

(a)

0 20 40 60 80 1000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Iteration number

Impactextent

PSOGA

(b)

F : Noise level ]= 2%; evolution o impact zone character-istics. (a) Impact centre and (b) impact zone extent.

: Relative error affecting impact zone centre0as comparedwith the exact solution.

% % %

PSO 0 5.217 003 3.765 002GA 1.378 003 8.294 002 4.428 002

enable to remove parasitic solutions. Solution o the obtainedconstrained nonlinear mathematical program that providesthe impact zone location was perormed by GA and PSObased algorithms. Predictability o orce location was studied

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0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iteration number

Impactcentre

PSO

GA

(a)

0 20 40 60 80 1000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Iteration number

Impactextent

PSOGA

(b)

F : Noise level] = 5%; evolution o impact zone character-istics. (a) Impact centre and (b) impact zone extent.

as a unction o measurement noise level. It was ound thatPSO algorithm continues to achieve exact prediction o thesolution even in the presence o % o noise intensity, whileGA algorithm ails at that noise as the associated error wastoo large. Both o these algorithms ail to predict correctimpact extent when noise level reaches %, even i theycontinue to predict the position o the impact centre withrather acceptable accuracy.

o assess predictability in a large sense, system modelnoise should also be integrated in the uture in conjunctionwith measurement noise.

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iteration number

Impactcentre

GA 1

GA 2

(a)

0 20 40 60 80 1000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Iteration number

Impactextent

GA 1

GA 2

(b)

F : Noise level ]

= 5% or the our sensors; evolution o

impact centre and extent as unction o iterations (a) rst test GA and (b) second test GA .

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of Intelligent Material Systems and Structures, vol. , no. , pp., .

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