A Combined Genetic and Eigensensitivity Algorithm for The

A combined genetic and eigensensitivity algorithm for thelocation of damage in structures

M.I. Friswell a, *, J.E.T. Pennyb, S.D. Garveyb

aDepartment of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, U.K.bDepartment of Mechanical and Electrical Engineering, Aston University, Birmingham B4 7ET, U.K.

Received 20 January 1997; accepted 28 April 1998

Abstract

Genetic algorithms have been the subject of considerable interest in recent years, since they appear to provide arobust search procedure for solving di�cult problems. Due to the way the genetic algorithm explores the region of

interest it avoids getting stuck at a particular local minimum and locates the global optimum. The genetic algorithmis slow in execution and is best applied to di�cult problems. This paper applies a genetic algorithm to the problemof damage detection using vibration data. The objective is to identify the position of one or more damage sites in astructure, and to estimate the extent of the damage at these sites. The genetic algorithm is used to optimize the

discrete damage location variables. For a given damage location site or sites, a standard eigensensitivity method isused to optimize the damage extent. This two-level approach incorporates the advantages of both the geneticalgorithm and the eigensensitivity methods. The method is demonstrated on a simulated beam example and an

experimental plate example. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Damage location; Genetic algorithm; Eigensensitivity

1. Introduction

Genetic algorithms are now frequently applied to

problems of maximising or minimizing a given objec-

tive function, often subject to some constraints.

Genetic algorithms have been applied to a wide range

of optimization problems in engineering which have

this form, such as the design of structures [1±4], pipe-

line optimization [5], ®nite element model updating [6±

8] and the problem of choosing the number and lo-

cation of actuators for the control of large ¯exible

space structures [9]. This paper approaches the pro-

blem of damage location by formulating an objective

function composed of measured and analytical vi-

bration data and applying a genetic algorithm to opti-

mize it. Other evolutionary strategies [10] and

algorithms such as simulated annealing [7, 11] have

been used for design optimization and model updating.

In recent years, there has been a considerable

demand for more accurate techniques to detect and

locate damage, particularly in large structures.

Damage will cause the sti�ness distribution in the

structure to change which may be detected by measur-

ing its dynamic response. Doebling et al. [12] gave an

overview of damage detection and location. Three dis-

tinct philosophies have been considered to locate the

damage in a structure using measured vibration data.

The ®rst group of methods uses techniques of ®nite el-

ement model updating and error localization to obtain

a corrected set of physical parameters that reproduce

the measured data. Mottershead and Friswell [13] gave

a survey of model updating methods, many of which

can also be used for damage location. The direct

model updating approach, where the whole sti�ness

matrix is updated, does not require parameterization

of the ®nite element model [14]. Otherwise, the model

Computers and Structures 69 (1998) 547±556

0045-7949/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(98 )00125-4

PERGAMON

* Corresponding author. E-Mail: M.I.Friswell@swansea.

ac.uk.

is parameterized and all the parametersupdated [15, 16]. Generally, the identi®cation will be

underdetermined and some form of regularizationshould be employed. Genetic algorithms have alsobeen used to identify damage in structures by optimiz-

ing the force residual [17] or modal residuals [18].A second group of methods, called forward

methods, assume a candidate set of possible damage

scenarios, which can include both damage mechanismand location. The change in dynamic response due tothe damage is predicted, usually in terms of the natural

frequencies. The predicted change in response of thestructure for all the damage scenarios is then comparedto the measured change and the closest damage case ischosen. Statistical methods are often used to identify

the most likely location of the damage. The di�erencebetween these methods and those from standard modelupdating is that only a subset of parameters are

assumed to be in error. Cawley and Adams [19±21]used this type of method and found that damage inspecimens fabricated from composite materials could

be detected. The method of Cawley and Adams hasthe advantage that damage location is identi®ed separ-ately from damage extent, although this only works

for a single damage site. Friswell et al. [22] proposed asimilar method but used a more re®ned statisticalanalysis. A similar approach is to directly ®nd the bestsubset of parameters that models the damage [23].

Neural networks are able to treat damage mechan-isms implicitly, so that it is not necessary to model thestructure in so much detail [24±30]. The method can

easily deal with non-linear damage mechanisms.Models are still required to provide the training casesfor the networks, and the algorithms should be robust

against systematic errors between the model used fortraining and the actual structure. For success, neuralnetworks require that the essential features in thedamaged structure are represented in the training data.

2. Overview of genetic algorithms

The striking feature of genetic algorithms is that

they are based on ideas from the science of geneticsand the process of natural selection. This cross fertili-zation from one ®eld of science to another has led tostimulating and fruitful applications in many ®elds and

particularly in computer science.The genetic algorithm works with an initial popu-

lation which may, for example, correspond to numeri-

cal values of a particular variable. The size of thispopulation may vary and is generally related to theproblem under consideration. The members of this

population are usually strings of zeros and ones i.e.binary strings. For example a small initial or ®rst gen-eration population may take the form:

1000010

1110000

1010101

1111001

1000001:

In practice, the population may be far larger than this

and the strings longer. The strings themselves may be

the encoded values of a variable or variables that we

are examining. This initial population is generated ran-

domly and we can use the terminology of genetics to

characterize it. Each string in the population corre-

sponds to a chromosome and each binary element of

the string to a gene. A new population must now

develop from this initial population and to do this we

implement the analogue of speci®c fundamental genetic

processes. These are:

. reproduction based on ®tness;

. crossover;

. mutation.

A set of chromosomes is selected at the reproduction

stage based on natural selection. Thus, members of the

population are chosen for reproduction on the basis of

their ®tness de®ned according to some speci®ed cri-

teria. The ®ttest are given a greater probability of

reproducing in proportion to the value of their ®tness.

The actual process of mating is implemented by

using the simple idea of crossover. This means that

two members of the population exchange genes. There

are many ways of implementing this crossover, for

example having a single crossover point or many cross-

over points. These crossover points are selected ran-

domly. A simple crossover is illustrated below for two

chromosomes selected according to ®tness. Here we

have randomly selected a crossover point after the

fourth digit.

1110 � 000;1010 � 101:After crossover this gives

1110 � 101;1010 � 000:Applying this procedure to our original population we

produce a new generation.

The ®nal process is mutation. Here, we randomly

change a particular gene in a particular chromosome.

Thus, 0 may be changed to a 1 or vice versa. The pro-

cess of mutation in a genetic algorithm occurs very

rarely and hence this probability of a change in a

string is kept very low. Mutation ensures genetic diver-

M. Friswell et al. / Computers and Structures 69 (1998) 547±556548

sity is maintained by generating new, random chromo-

somes.

The application of genetic algorithms to optimiz-

ation problems is relatively clear since the test of ®t-

ness can be based on the function we are trying to

maximize, thus the members of the population which

give the largest values of the objective function are

taken as the ®ttest. The chromosomes are viewed as a

concatenation of binary strings representing the values

individual variables in the optimization problem.

Continuous variables must be discretized, as described

later. Constraints are usually incorporated into the

objective function using a penalty function which

ensures that during the optimization process the sol-

ution will satisfy the constraints. This penalty must be

carefully chosen to ensure the constraints are satis®ed

and so the correct optimum is reached.

The reason why a genetic algorithm di�ers from a

simple direct search procedure is that it involves two

special features: crossover and mutation. Thus, starting

from an initial population the algorithm develops new

generations which rapidly explore the region of inter-

est. This is useful for di�cult optimization problems

and in particular for those where we wish to ®nd the

global maximum or minimum of function which has

many local maxima and minima. In this case, standard

optimization methods, such as the conjugate gradient

method, can locate only the local optimum. A genetic

algorithm, however, may locate the global optimum

although this is not guaranteed. This is due to the way

it explores the region of interest avoiding getting stuck

at a particular local minimum. Since we are dealing

with a stochastic process, the solution should be taken

as the best of a number of runs.

Genetic algorithms are a developing area of research

and many amendments could be made to the functions

which we have used to implement a genetic algorithm.

For example, the weighted roulette wheel selection can

be implemented in many di�erent ways, crossover can

be changed to multi-point crossover or other alterna-

tives. Elitism, where the best solution is always passed

on to the next generation, is a particular feature for

which good results have been reported. An alternative

coding for the variables of the problem such as Gray

coding have also been used by workers in the ®eld. It

will often be noticed that a genetic algorithm is slow in

execution, but it is usually applied to problems where

standard optimization techniques perform poorly; for

example those which have multiple optima and where

the global optimum is required. Problems which have

both real valued and integer variables pose particular

problems for standard algorithms. Since standard al-

gorithms often fail in these cases, the extra time taken

by the genetic algorithm is worthwhile. Having

described the basic principles of a genetic algorithm we

will now illustrate how it may be applied by consider-ing an engineering application.

3. Penalty functions for damage location

The location of damage in a structure using vi-bration data is very di�cult. We will make theassumption that damage causes a change in the sti�-ness of the structure. In many cases this is a gross

assumption and the distinction between strength andsti�ness should be made. For example, in a concretehighway bridge the concrete is held in compression by

steel tendons and the sti�ness of the bridge comes pre-dominantly from the concrete. The strength of thebridge comes from the steel tendons, since concrete has

little strength or sti�ness in tension. Thus, as a bridgeis damaged its dynamic response will change very littleuntil just before failure.

The simpli®ed problem to be addressed by thispaper is to identify the location and extent of a limitednumber of sti�ness reductions that best reproduce themeasured dynamics. The sti�ness reductions will be

modelled by a reduction in the Young's modulus for agiven element, and its location is speci®ed by the el-ement number. Localization of the damage is depen-

dent on a su�ciently re®ned mesh. A very ®ne meshcould be used with a ®tness penalty included to ensurethat adjacent elements are chosen. The examples that

follow only include beam and plate examples, althoughany structure with its associated ®nite element modelmay be used. A maximum of two damage sites will beallowed, although the extension to more sites is

straightforward. The measured data will be the naturalfrequencies and mode shapes of the structure.We next consider the objective function which is

based on the measured data. This will consist of threeterms: a term relating to the error in natural frequen-cies, Jo, a term relating to the error in mode shapes,

Jf, and a term to weight against two damage sites.These terms will be weighted to give a total error of

J �WoJo �WfJf �Wnsdns �1�where Wo, Wf, Wns are weighting factors and

dns � 0 if one site is damaged,1 if more than one site is damaged.

�The term added to weight against too many damagesites is very important. Suppose damage only occurs at

one site. Due to measurement noise and model inade-quacies, the modal model will not be reproducedexactly, even if the damage is correctly identi®ed.

Introducing a second damage site may make a minorimprovement in the correlation between test and analy-sis, although in reality one site is damaged. The ten-

M. Friswell et al. / Computers and Structures 69 (1998) 547±556 549

dency will always be to ®nd damage at every site.Thus, a penalty is introduced to weight against an

increased number of damage sites.

3.1. The frequency error

The obvious objective function for frequency errorsis just a weighted sum of squares of the di�erences in

the natural frequencies, provided the modes are pairedcorrectly. Any natural frequencies corresponding toanalytical and experimental modes that do not pair

with su�cient con®dence are simply omitted from theobjective function. Thus, one possibility for Jo is,

Jo �Xrj�1

Wojomj ÿ oaj

omj

!2

�2�

where omj and oaj are the jth measured and analyticalnatural frequency of the damaged structure, and r is

the number of measured modes. If required, the indi-vidual frequencies may be given di�erent weights,denoted by Woj, although often these weights are

taken as unity. The analytical natural frequencies, oaj,could be used in place of the measured frequencies inthe denominator in Eq. (2), although the di�erences inthese frequencies should be small and using the ana-

lytical frequencies makes the sensitivity calculationsmuch more di�cult.The problem with Eq. (2) is that the model for the

undamaged structure should correlate well with themeasured dynamics if the structure was undamaged. Ifperiodic measurements of a structure are undertaken,

so that measurements of the undamaged structureexist, then the model of the undamaged structure maybe updated to correspond closely to the undamagedstructure. If this updating is not performed, then the

algorithm to identify the location and extent of thedamage will try to compensate for both the originalmodelling error and the change due to the damage.

Obviously, the identi®cation of damage in this case isvery di�cult.A di�erent method to reduce the e�ect of modelling

errors is to consider changes in frequency from theundamaged to the damaged structure. Of course, thisassumes that the natural frequencies of the undamaged

structure were measured. In this case, Jo is given by

Jo �Xrj�1

Wojdomj ÿ doaj

omj

!2

�3�

where domj and doaj are the changes, due to damage,in the jth measured and analytical natural frequencies.Eq. (3) gives the penalty function that will be used in

subsequent sections.One possible disadvantage with Eqs. (2) and (3) is

that the range of the penalty function is not known.

Using the penalty function

Jo �Xrj�1

Wojo2

mj ÿ o2aj

o2mj � o2

aj

!2

�4�

in place of Eq. (2) overcomes this problem since the in-

dividual natural frequency error terms are between 0and 1. This may not be a great advantage, sinceEqs. (2) and (4) are approximately the same when the

di�erences between the measured and analytic frequen-cies are small. Calculating the sensitivities for Eq. (4) isalso di�cult. The problem of the relative weightingbetween the frequencies and mode shapes will be

addressed later. Expressions similar to Eq. (4), butusing frequency changes, may also be de®ned, andinstead of squaring the terms the absolute value may

be taken.

3.2. The mode shape error

The error in the mode shapes may be incorporatedinto a penalty function as

Jf �Xrj�1

Wfj�fmj ÿ faj�T�fmj ÿ faj� �5�

where fmj and faj are the jth measured and analyticalmode shapes. Only those degrees of freedom that aremeasured are picked out of the analytical mode shapes.To apply Eq. (5) correctly, both sets of mode shapes

should be normalized consistently, and typically massnormalization is used. Alternatively, the modes may benormalized using the modal scale factor (MSF) [31].

Using the MSF also makes sure that no modes are1808 out of phase. Of course the modes must also bereal if the ®nite element model does not include damp-

ing. The terms in Eq. (5) can also be weighted usingthe analytical mass or sti�ness matrix, reduced to themeasured degrees of freedom if necessary.An alternative method that does not require the

mode shape scaling, is to use the modal assurance cri-terion (MAC) [31]. Thus, the new penalty function is

Jf �Xrj�1

Wfj�1ÿMAC�fmj;faj�� 6�

where MAC(fmj, faj) is the MAC value for the jthpaired mode. This penalty function has the advantage

that each term is between 0 and 1, since the MACvalues are between 0 and 1. It is di�cult to use penaltyfunctions based on the MAC with eigensensitivity

methods, since the derivative of the penalty function isdi�cult and costly to obtain. The [1ÿMAC(fmj, faj )]terms may also be square rooted.


3.3. The relative weighting of frequencies and modeshapes

How much weight should be given to the errors inthe natural frequencies, compared to the errors in the

mode shapes? Although the mode shapes can providevaluable information, one has to be careful. The modeshapes are measured with less accuracy than the natu-

ral frequencies, and generally the mode shapes are lesssensitive to damage in the structure. As an example,consider a uniform cantilever beam with 15 elements.

Let the sti�ness of the fourth element from theclamped end be reduced to simulate damage. Figs. 1and 2 give the change in the natural frequency andmode shape (through the MAC value) for the ®rst ®ve

modes as the element sti�ness is reduced. Obviously,the sti�ness reduction has a relatively large e�ect onthe natural frequencies, as compared with the MAC

values. Because of the insensitivity of the modes to thedamage, only the natural frequency penalty functionwill be used in the following examples.

4. The application of the genetic algorithm

Let us ®rst consider the parameters to be optimized

by the genetic algorithm. One advantage of the geneticalgorithm is the ability to optimize a combination ofdiscrete and continuous variables. In this application

we have both discrete variables, the location of thedamage given by the element number, and continuousvariables, the extent of the damage given as a percen-

tage reduction in sti�ness.These continuous variables must be expressed in

terms of a ®xed number of binary digits. Thus, in rea-

lity, genetic algorithms optimize discrete variables, andcontinuous variables are approximated to the requiredaccuracy. For example, the damage extent variablesare in the range 0±100% and if 8 bits are allocated to

represent these variables, the maximum resolution is100/102310.1%. The location variables, or element

numbers, are already discrete variables. Zero will beincluded in this variable set to denote no damage.

Thus, if there are 15 elements, or possible damage lo-cations, in the ®nite element model then the location

variable will range from 0 to 15, requiring a 4 bitbinary number. If the model contains less elements,

then some numbers would have to be assigned a zero®tness. For example, if there were only 12 elements,

then the objective function would be zero for the vari-able values 13, 14 and 15.

The objective is to minimize the function given inEq. (1), whereas genetic algorithms try to ®nd the ®t-

test members of the population, or maximize an objec-tive function. This is easily recti®ed by subtracting J,de®ned in Eq. (1) from a suitable number. Negative ®t-

ness values will be taken to be zero. The choice of thisnumber is critical as it gives the range of ®tness values

for the algorithm, which is used to generate the prob-ability that a particular member of a generation will be

selected for the next generation. In general, the mostsatisfactory results are obtained when the population

has a good spread of ®tness values, from zero to the®ttest. The problems that occur are easily shown by an

example. Suppose the objective function we wish tominimize takes values between 0 and 1. If we subtract

these values from 100 we have ®tness values between99 and 100. The ®tness values are approximatelyequal, and all members of a generation will have ap-

proximately equal probability of survival to the nextgeneration. This is not desirable, as we want only the

®ttest to survive. If, instead, we subtract the value ofthe objective function from 1 we obtain ®tness values

between 0 and 1, and a more satisfactory performanceof the algorithm. The direct application of genetic al-

gorithms to damage detection was discussed by theauthors in an earlier paper [32].

Fig. 1. The e�ect of damage on the ®rst ®ve natural frequen-

cies of the cantilever beam (mode 1 w; 2 r; 3 q; 4 X; 5 +).

Fig. 2. The e�ect of damage on the MAC for the ®rst ®ve

modes of the cantilever beam (mode 1 w; 2 r; 3 q; 4 X; 5

+).


5. Combining eigensensitivity and genetic algorithms

Genetic algorithms may be used to directly minimize

the penalty function given in Eq. (1), by forcing thecontinuous variables to take on a ®nite number of dis-crete values. Genetic algorithms are best used to opti-

mize di�cult problems. In damage location and extentestimation most of the di�culty arises from trying tolocate the damage. Given one or two damage lo-

cations, then the optimum values for the extent of thedamage are easily calculated from sensitivity typemethods, based on a Taylor series expansion of thenatural frequencies and mode shapes in terms of the

unknown extent variables (see for example, Ref. [33]).In this way the length of the binary string representingeach member of the generation is considerably

reduced. In practice the penalty functions given byEq. (3) for frequency and Eq. (5) for mode shapes, aremost appropriate for the eigensensitivity method.

6. A simulated cantilever beam example

The location of damage using the combined eigen-sensitivity and genetic algorithm method was demon-

strated using a simulated steel, cantilever beam oflength 1 m and cross section 25�50 mm. Bending inthe plane of the thinner dimension only, was con-

sidered, that is in the most ¯exible of the two bendingplanes. The changes in the ®rst 5 natural frequencieswere used to locate the damage, and thus the penalty

function is given by Eq. (3).Four cases were tested: case 1 demonstrated the per-

formance of the method to locate a single damage site;in case 2 the damage was to an element near the free

end of the cantilever and hence di�cult to locate; incase 3 the damage was located at two sites; case 4demonstrated the performance of the method to sys-

tems with systematic errors, namely the addition ofextra mass into the ``measured'' system.In all cases the population had 10 members, 60% of

which were mated at every generation. Each binary bithad a 0.5% chance of mutation at every generation. Inall cases, the ®nite element model used to locate the

damage contained 15 elements, or 30 degrees of free-dom. These elements were numbered from the ®xedend. Only one run of the genetic algorithm was used

for each optimization. Mating was accomplished usinga single point crossover, with a randomly generatedcrossover point. The variables used for the genetic al-

gorithm optimization were the location of two damagesites. For a given set of damage sites the extent of thedamage was optimized using the eigensensitivity

approach. The genetic algorithm has been modi®ed sothat the most ®t member of a population survives intothe next generation, without mating or mutation. Thisis called elitism and ensures that the ``best'' solution is

maintained from generation to generation. The average®tness is also shown in the plots, although care mustbe exercised in interpreting these plots. The optimum

was always selected as the best member of a gener-ation, and the convergence of the average ®tnessshould not be taken as any sort of convergence to the

optimum. The increasing average ®tness values do indi-cate a reduction in the genetic diversity of the popu-lation.

6.1. Case 1

The simulated data was obtained from a ®nite el-

ement model with 15 elements. The model used tolocate the damage was the same as that used to simu-late the measurements. Damage was simulated as a30% reduction in the sti�ness of element 4. Table 1

shows the damaged and undamaged natural frequen-cies. The weights given to the terms in the objectivefunction were

Woj � 100; Wns � 0:25

and the objective function, Eq. (1), was subtractedfrom 1, giving ®tness values between 0 and 1.

Remember, negative values of ®tness are taken as zero.Fig. 3 shows the ®tness of the most ®t member of

the population, and also the average ®tness of the

population. The damage was correctly located at el-ement 4 at the sixth generation. The average ®tnessincreased with the generation number.

Table 1

The damaged and undamaged natural frequencies for cases 1 and 4

Natural frequencies (Hz)

Without added mass (case 1) With added mass (case 4)

Mode number Undamaged Damaged Undamaged Damaged

1 21.0 20.4 20.5 20.0

2 131.3 131.2 131.3 131.2

3 367.7 362.2 365.5 360.1

4 720.7 705.7 709.2 694.5

5 1191.7 1181.7 1176.0 1165.5


6.2. Case 2

Damage was simulated by a 30% reduction in sti�-ness at element 14. The cantilever beam model has

only 15 elements, and the damage was very close tothe free end of the cantilever beam. Changes in sti�-ness in this region have very little in¯uence on the

lower modes of the beam. Fig. 4 shows the results inthis case. The genetic algorithm correctly located thedamage, although the high average ®tness even at the

®rst generation shows that many estimated damage lo-cations ®tted the measured data reasonably well.

6.3. Case 3

Damage was simulated at two locations, by 30%

sti�ness reductions at elements 4 and 12, and theresults are shown in Fig. 5. The algorithm has less suc-cess here but still located the damage at elements 5and 12, which was very close to the actual damage lo-

cations. Note that the maximum ®tness expected inthis case was 0.75, since damage was located at 2 sites.

6.4. Case 4

This case demonstrated the e�ect of a systematic

error in the ``measurements''. Substantial discretization

errors in the ®nite element model are often found in

practice, although in our example these were relatively

small because the original ®nite element model was

su�ciently re®ned. The systematic error introduced

into the 15 element cantilever beam model was a mass

of 0.2 kg added at node 12. Table 1 shows the error in

the ®rst 5 natural frequencies of the damaged and

undamaged model, and shows the level of systematic

error introduced.

The evolution of the ®tness values is shown in Fig. 6.

The damage was correctly located in element 4, despite

the systematic error in the ``measured'' data. The per-

formance in many respects was better than that in case

1, without the systematic error. This is probably

because only a single run of the algorithm was per-

formed.

Fig. 3. Fitness values for the simulated beam problem, case

1(most ®t member ÐÐ; average ®tness - - - - -).








7. An experimental example

The second example consists of a 3 mm steel plate

which was clamped along one edge. The exposed plate

measured 305 mm � 357 mm and was clamped along

the shorter side. Fig. 7 shows the ®nite element mesh

used to model the undamaged plate. This mesh

resulted in a 48 degree of freedom model and produced

the estimated natural frequencies given in Table 2.

Table 2 also gives the natural frequencies of the plate

measured experimentally using impact excitation. The

12th mode was not measured because the acceler-

ometer was placed at a node of this mode. The analyti-

cal and experiment modes were easily paired in this

case using the displayed mode shapes. Clearly, the

®nite element mesh was very coarse and there were

considerable modelling and discretization errors pre-

sent. Any damage location method should be robust to

such errors, and, to increase the robustness, the di�er-

ence in natural frequencies between the undamaged

and damaged plate was used to locate the damage.

The plate was damaged by 4 saw cuts each of length

40 mm in the shape of a cross centred on element 4 of

the 48 degree of freedom ®nite element model, as

shown in Fig. 7.

The population had 10 members, 60% of which

were mated at every generation. Each binary bit had a

0.5% chance of mutation at every generation. The

weights given to the terms in the objective function

were

Woj � 1000; Wns � 0:25

and the objective function, Eq. (1), was subtracted

from 1, giving ®tness values between 0 and 1. Negative

values of ®tness were taken as zero. A maximum of 2

damage locations were assumed, and each location

variable was 4 bits long; zero was taken as no damage,

integers 1±12 as damage in the corresponding element

and integers 13, 14 and 15 were given a zero ®tness

value. Fig. 8 shows the ®tness of the best member of

Fig. 7. The experimental cantilever plate example, showing

the ®nite elements and the position of the saw cut.

Table 2

Analytical and experimental natural frequencies and frequency changes due to damage for the experimental cantilever plate

example

Mode number

Analytical undamaged natural

frequency (Hz) (48 degrees of

freedom model)

Experimental undamaged

natural frequency (Hz)

Experimental natural frequency

change due to damage (Hz)

1 19.1 20.0 0.4

2 55.7 56.7 1.0

3 119.7 124.6 3.8

4 192.8 198.1 4.7

5 214.6 212.6 3.0

6 341.6 353.9 10.9

7 367.6 380.8 3.0

8 405.6 427.5 12.0

9 544.7 530.1 9.0

10 602.3 639.9 17.5

11 663.6 690.7 24.5

12 665.8 Not measured Not measured

13 705.7 774.5 25.2

Fig. 8. Fitness values for the experimental cantilever plate

example(most ®t member ÐÐ; average ®tness - - - - -).


the population in each generation. The correct locationof the damage (element 4) was found in the 8th gener-

ation, showing the approach is robust to modellingand measurement errors.

8. Conclusions

A combined genetic algorithm and eigensensitivitymethod has been used to identify the location andmagnitude of damage from measured vibration data.

Essentially, the genetic algorithm is used to identifythe damage located and the eigensensitivity is used toidentify the damage extent. Damage at one and two

sites have been successfully located in the simulatedexample of a cantilever beam. Damage was also suc-cessfully location in an experimental cantilever plate.

The algorithm is robust to systematic errors in themeasured data, demonstrated in simulation by the ad-dition of a discrete mass and experimentally by using acrude model for the plate. One feature of the numeri-

cal example given is that genetic diversity was reducedas the generation number increased. Alternatively, thiscould be viewed as the convergence to a uniform ``op-

timum'' population, although it would not be guaran-teed that this population would be the globaloptimum. This may be a feature of the small number

of genes associated with each member of the popu-lation.

Acknowledgements

Dr Friswell gratefully acknowledges the support ofthe Engineering and Physical Sciences ResearchCouncil through the award of an advanced fellowship.

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