probabilistic fault identification using vibration data and neural networks

Mechanical Systems and Signal Processing (2001) 15(6), 1109}1128doi:10.1006/mssp.2001.1386, available online at http://www.idealibrary.com on

PROBABILISTIC FAULT IDENTIFICATION USINGVIBRATION DATA AND NEURAL NETWORKS

TSHILIDZI MARWALA

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K.E-mail: [email protected]

(Received 3 November 1999, accepted 9 January 2001)

Bayesian formulated neural networks are implemented using hybrid Monte-Carlomethod for probabilistic fault identi"cation in structures. Each of the 20 nominally identicalcylindrical shells is arbitrarily divided into three substructures. Holes of 10}15 mm diameterare introduced in each of the substructures and vibration data are measured. Modalproperties and the coordinate modal assurance criterion (COMAC), with natural-frequency-vector taken as an additional mode, are utilised to train the modal-property-network andthe COMAC-network. Modal energies are calculated by determining the integrals of the realand imaginary components of the frequency response functions over bandwidths of 12% ofthe natural frequencies. The modal energies and the coordinate modal energy assurancecriterion (COMEAC) are used to train the modal-energy-network and the COMEAC-network. The average of the modal-property-network and the modal-energy-network aswell as the COMAC-network and the COMEAC-network form a modal-energy-modal-property-committee and COMEAC}COMAC-committee, respectively. Both committeesare observed to give lower mean square errors and standard deviations than their respectiveindividual methods. The modal-energy- and COMEAC-networks are found to give moreaccurate fault identi"cation results than the modal-property-network and the COMAC-network, respectively. For classi"cation (the presence or absence of faults) the modal-property-network is found to give the best results, followed by the COMEAC}COMAC-committee. The modal-energies and modal properties are observed to give betteridenti"cation of faults than the COMEAC and the COMAC data. The main advantage ofthe Bayesian formulation is that it gives identities of damage and their respective standarddeviations.

( 2001 Academic Press

1. INTRODUCTION

The identi"cation of faults in mechanical and aerospace structures at the manufacturingstage o!ers substantial economic bene"ts. Vibration methods [1] have been implementedwith varying degrees of success on identifying mechanical faults. These techniques can bebroadly classi"ed as being experimentally based or model-based.

Experimental methods use experimental data as a basis of fault identi"cation. Thesemethods use changes in vibration data with little or no assumptions about the analyticalbehaviour of the structure for fault identi"cation. The main shortcoming with thesemethods is that they are usually insensitive to faults of small magnitudes and experiencedi$culty on quantifying the severity of faults. The advantage of these methods is that theyare not computationally intensive. Model-based techniques modify a numerical model(such as "nite element model) to match measured vibration data as a basis for faultidenti"cation [2]. These methods are in principle capable of identifying the identity of faultsbut they rely on the accuracy of numerical models.

0888}3270/01/061109#20 $35.00/0 ( 2001 Academic Press

1110 T. MARWALA

The committee [3, 4] of neural networks, which employs frequency response functionsand modal properties simultaneously, has been found to give more reliable solutions thanthe two individual methods. The implementation of neural networks may be classi"ed asa non-deterministic optimisation problem. This is because when neural networks areimplemented, only the data are required instead of deterministic mathematical relations.The optimisation nature of neural networks causes a problem of not "nding a globaloptimum solution, especially if the number of parameters that indicate the identity of faultsis high. In order to avoid the high incidence of "nding local optimum solutions, it is oftendesirable to reduce the number of design variables. A method implemented to achieve thisobjective is the method of substructuring.

In this paper, the following issues are addressed:

(1) The committee approach is extended to probabilistic framework by using Bayesianformulation.

(2) The nature of the input data is investigated.(3) Experimentally validating this new framework by classifying and identifying faults in

20 cylindrical shells, which are arbitrarily divided into three substructures.

Faults are located within these three substructures. The parameters corresponding to eachsubstructure form a vector space, also known as the identity of fault and this information isde"ned as the substructure space. The information from the frequency response functionsand modal properties is transformed into substructure space using the weighted-average ofthe two independent neural networks. This approach performs fault identi"cation by usingchanges in vibration data resulting from the presence of faults despite the presence of otherchanges such as due to uncertainties of measured data because of variation in physicalproperties of a population of cylinders, uncertain measurement positions and changes insupport conditions. Bayesian approach is applied because it is easier to determine thecon"dence intervals of the identity of faults than the maximum likelihood approach [5]. Italso automatically penalises highly complex models and therefore is able to select anoptimal model without applying independent methods such as cross-validation as it is thecase for maximum likelihood approach.

2. THEORETICAL FORMULATION

In this section, substructuring of modal and frequency equations is introduced. Anyelastic structure may be expressed in terms of mass (M), damping (C) and sti!ness (K)matrices in time domain by

[M]MX(t)N#[C]MX (t)N#[K]MX(t)N"MF (t)N. (1)

If equation (1) is transformed into the frequency domain then the resulting equation is

M!u2[M]#iu[C]#[K]NMX(u)N"M f (u)N (2)

where u is the frequency while MXN the displacement and M f N the forcing function infrequency domain.

1111PROBABILISTIC FAULT IDENTIFICATION

2.1. FRF SUBSTRUCTURING

Suppose we are interested in locating fault as either in substructure 1, 2 or 3. Thenequation (2) may be partitioned into three super-elements as

A!u2

[M11

] [M12

] [M13

]

[M21

] [M22

] [M23

]

[M31

] [M32

] [M33

]

#ju

[C11

] [C12

] [C13

]

[C21

] [C22

] [C23

]

[C31

] [C32

] [C33

]

#

[K11

] [K12

] [K13

]

[K21

] [K22

] [K23

]

[K31

] [K32

] [K33

] B GX

1X

2X

3H"G

f1

f2f3H . (3)

Since displacements MXN and force M f N are not used directly, the force is assumed white,hence M f N has a unit force magnitude at all frequencies, and the displacement is replaced bythe frequency response functions. If any of the substructures has a fault, this would bere#ected by changes in frequency response functions of the three substructures. By compar-ing the relative changes of the frequency response functions as a result of faults, one may beable to deduce the presence and the location of faults.

The implicit relationship between physical properties of the structure, e.g. mass andsti!ness matrices, and the frequency response functions are used to identify faults instructures. Su$cient data of the modal energies extracted from frequency response func-tions x

1and their corresponding identities of fault y

1are obtained from experiment and

a functional mapping between the two is quanti"ed using neural networks by

y1"h(x

1). (4)

2.2. MODAL PROPERTY SUBSTRUCTURING

Equation (2) may be transformed into modal domain to form an eigenvalue equation forthe nth mode, which may be written in substructure domain as

A!u6 2n

[M11

] [M12

] [M13

]

[M21

] [M22

] [M23

]

[M31

] [M32

] [M33

]

#ju6n

[C11

] [C12

] [C13

]

[C21

] [C22

] [C23

]

[C31

] [C32

] [C33

]

#

[K11

] [K12

] [K13

]

[K21

] [K22

] [K23

]

[K31

] [K32

] [K33

] B G/1n1

/1n2

/1n3H"G

0

0

0H (5)

where u6nis the nth complex eigenvalue with its imaginary part corresponding to the natural

frequency unand M/1

nN is the nth complex mode shape vector.

If any of the substructures has a fault, this would be re#ected by changes in unand M/1

nN of

the three substructures. By comparing the relative changes of the modal properties of thesethree substructures due to faults, one may be able to deduce the presence and the location offaults. Similarly, a functional mapping between the identity of fault y

2and the modal

property x2

may be quanti"ed by

y2"f (x

2). (6)

1112 T. MARWALA

In this section, it is demonstrated how the method of substructuring can be used to reducethe order of the problem of fault identi"cation from several thousand substructures to three(three was chosen arbitrarily). Substructuring may be applied as a "rst step in faultdiagnostics by pointing to a larger area before localised methods such as acoustics methodsare applied.

3. NEURAL NETWORKS

Neural networks are viewed as parameterised graphs that make probabilistic assump-tions about the data. Learning algorithms are viewed as methods for "nding parametervalues that look probable in the light of the data. Learning processes occur by training thenetwork through either supervised or unsupervised learning. Unsupervised learning is usedwhen only the input data are available. Supervised learning is used when the input and theoutput are available and neural networks are used to approximate the functional mappingbetween the two.

There are several types of neural network architectures; namely multilayer perceptron(MLP) and radial basis function [5]. The MLP is chosen because it provides a complexnon-linear mapping between the input and the output.

A schematic illustration of the MLP is given in Fig. 1. This network architecture containshidden units and output units. The bias parameters in the "rst layer are shown as weightsfrom an extra input having a "xed value of x

0"1. The bias parameters in the second layer

are shown as weights from an extra hidden unit, with activation "xed at z0"1. The model

in Fig. 1 is able to take into account the intrinsic dimensionality of the data. Models of thisform can approximate any continuous function to an arbitrary accuracy if the number ofhidden units M is su$ciently large. By considering several layers expands the MLP.

Figure 1. Feed-forward network having two layers of adaptive weights.


The output units represent the identity of fault while the input units represent theparameters from the frequency response functions or modal properties. The non-linearmathematical relation that maps the input x to the output y is written as

yk(s)"

M+j/1

wkj

f Ad+i/1

wkj

xi#w

j0B#wk0

. (7)

Here wj0

and wk0

are bias parameters, d is the number of input units and M is the number ofhidden units. The sigmoid function f (f) implemented is

f (v)"1

1#e~v. (8)

In the maximum likelihood approach, the weights (wi) and biases (with subscripts 0 in

Fig. 1) in the hidden layers are varied until the error between the network prediction andthe output from the training data is minimised. Optimisation routine called scaled con-jugate gradient method [5] is utilised for training and backpropagation technique is used toevaluate the gradient of the error. The error between the network prediction and the outputfrom the training data is de"ned as

errori"

Ey!h (x)iE

Eh(x)iE

(9)

where y is the desired identity of damage and h(x)iis the output vector produced by the

network at the end of the ith iteration and EfE is the Euclidean norm of f. The analysisemployed in this section does not give the probability distribution of the weight vector andin the next section the MLP is reformulated using Bayesian approach.

3.1. PROBABILISTIC APPROACH

If the input vector x, is random (it has a mean and variance), because of the variationin physical properties of structures and variation of measurements, then the identity ofdamage y is probabilistic. This requires that the weight space be assigned a probabilitydistribution representing the relative degrees of belief in di!erent values for the weightvector. This implies that the mapping function between input vector and output vector hasa probability distribution. The weight space vector is initially assigned some prior distribu-tion. Once the data, in this case the processed frequency response functions or modalproperties and their respective identities of faults have been observed the weight vector canbe transformed into posterior distribution using Bayes' theorem. The posterior distributioncan then be used to evaluate the predictions of the trained network for data not used duringtraining [6, 7]. Bayes' theorem may be written as

p (w DD, x)"p(D Dw, x)p(w Dx)

p (D Dx). (10)

In equation (10) p (w Dx) is the prior probability distribution function of the weight spacein the absence of any data, w indicates the vector of adaptive weight and bias parametersand x is input data and D,(y1,2 , yN) is the output data. The quantity p (w DD, x) isthe posterior probability distribution after the data have been seen and p(D Dw, x) is thelikelihood function. The MLP network trained by supervised learning does not modelthe distribution of the input data.

1114 T. MARWALA

It may be shown that the posterior probability distribution of the weight given the datamay be derived from equation (10) to be [5]

pi(w DD)"

1

Zs

expA!E

i¹B"

1

Zs

expA!(b/2)+N

n/1Mh (x

n; w)!y

nN2!(a/2) +W

n/1w2n

¹ B(11)

where

Zs"+

i

expA!Ei

¹B (12)

with Zsa normalising constant, E

iis the error of the network given by the weight vector

from sample i, ¹ is the parameter called pseudotemperature and a as well b are theregularisation parameters. The probability in equation (11) is called Gibbs distribution andhas its origins in statistical mechanics. From equation (11) the following remarks may bemade:

(1) States of low error has a higher probability than states of high errors.(2) If parameter ¹ is reduced, the probability is concentrated on a smaller subset of

low-energy states.(3) The optimal weights correspond to the maximum probability.

The probability distribution of the weight vectors given the input data of equation (11) isobtained by sampling through the weight space using a procedure called hybridMonte-Carlo method [6, 7]. For each sample and for a given ¹ (we arbitrarily start withhigh value of ¹ so that the procedure will not be stuck in the region of local optimumdistribution).

(1) Choose the step size (*u) and the number of steps (¸) in trajectory.(2) From the initial weight vector (w

*/*5*!-), take ¸ steps each of size *u, in the weight

space in the direction that result with higher posterior probability [equation (11)]leading to vector w

/%8[this direction is obtained by determining the gradient of

p (w DD)].(3) If the error of the current sample is smaller than from the previous sample (*E(0)

then accept w/%8

. Otherwise, select a random number m of uniform distribution in therange [0, 1]. If m(exp(!*E/¹) then w

/%8is accepted, otherwise it is rejected.

Step 3 is called Metropolis criterion [8]. The procedure (1)}(3) is repeated until the requirednumber of samples is achieved. At each sample the value of ¹ is lowered at a rate slowerthan the exponential decay, until the rate at which samples are accepted stabilises. Thistechnique of lowering the value of ¹ is called simulated annealing [9] and it is implementedto avoid getting stuck in the region of local optimum distribution. Each of the weight-vectorfrom the accepted samples gives an output to the neural network. From the output of theseweight-vectors, the probability distribution (from the standard deviation and the mean ofthe output) of the output is obtained.

3.2. INPUT TO NEURAL NETWORK

The data to be used as the input the neural networks are the processed modal energiesand modal properties. The frequency response functions are transformed into modalenergies. These modal energies are de"ned as the integrals of the real and imaginarycomponents of the frequency response functions over frequency ranges that bind the natural


frequencies of the system (these ranges span over 12% of the natural frequencies). Two typesof processing are conducted to reduce the number of input data:

(1) The modal property and modal energy data with the 19 highest variance (these dataare chosen from data that have the lowest variance for the population of undamagedstructures).

(2) The coordinate modal assurance criterion (COMAC) and the coordinate modalenergy assurance criterion (COMEAC) data are used.

The modal properties that are repeatable for undamaged cylinders and have high variationfor damaged cylinders are used as inputs to the modal-property-network. The modalenergies that are repeatable for undamaged cylinders and have high variation for damagedcylinders are used as inputs to the modal-energy-network.

The COXAC is a criterion that measures the correlation between two sets of data of thesame dimension. The COXAC for coordinate i between the measured data X

mand the

median data for undamaged structures XMED

is

COXAC(i)"(+n

j/1DX

MED(i, j)X*

m(i, j) D)2

+nj/1

DXMED

(i, j) D2+nj/1

DXm(i, j) D2

. (13)

If X in equation(13) is substituted by mode shape vector then the COXAC is a familiarCOMAC [10]. When X

mand X

MEDare perfectly correlated then the COXAC for all degrees

of freedom is 1. Otherwise, when perfectly uncorrelated then the COXAC for all degrees offreedom of 0.

The natural frequencies and mode shapes are combined to form a single matrix by takingthe natural frequency vector as an additional mode. The correlation between the median ofthe modal property matrix from undamaged data and from each measured data, are used asinput data to the COMAC-network. The correlation between the median of the modalenergy matrix from undamaged data and from each measured data, are used as input datato the COMEAC-network.

To ensure that high order input values do not dominate the training, the input para-meters are normalised so that all the values of the input lie in the interval [0, 1] using

x/%8m

"

x0-$m

!min(x0-$m

)

max(x0-$m

)!min(x0-$m

)(14)

where xm

is a row of the input parameters.

3.3. COMMITTEE OF NEURAL NETWORKS

In this paper, a method illustrated schematically in Fig. 2 is applied using Bayesianformulated neural networks. This "gure illustrates how two neural networks are combinedto form a committee of neural networks. It has been shown before [3] that a committee ofnetworks which uses both frequency and modal domain data gives results that are morereliable than when the two networks are used individually. Detailed information on thecommittee method may be obtained from [3, 4].

4. EXPERIMENTAL EXAMPLE

In this section, an impulse hammer test is performed on each of the 20 steel seam-weldedcylindrical shells (1.75$0.02 mm thickness, 101.86$0.29 mm diameter and of height101.50$0.20 mm). These cylinders are rested on a &bubble wrap', to simulate a free}freeenvironment (see Fig. 3). The cylinders are excited using a modal hammer with sensitivity of

Figure 2. Illustration of committee of networks. Key: MP"modal property; >1"fault identity using data

from the frequency response functions; >2"fault identity using data from the modal property; c

1"weight given

to the frequency response function approach; c2"weight given to the MP-approach.

1116 T. MARWALA

4 pC/N, the mass of the head is 6.6 g, and a cut-o! frequency of 3.64 kHz. The response ismeasured using an accelerometer with a sensitivity of 2.6 pC/ms~2, which has a mass of19.8 g. Conventional signal processing procedures are applied to covert the time domainimpulse history and response data into frequency domain. The excitation and response datain the frequency domain are utilised to calculate the frequency response functions. From thefrequency response functions, modal energies are extracted.

Each cylinder is divided into three substructures, and holes of 10}15 mm in diameter aredrilled on each substructure (see Fig. 3). For an example for one cylinder the "rst type ofdamage is a zero-fault scenario and its identity is [0 0 0]. The second type of damage isa one-fault-scenario and if its magnitude is 11 mm is substructure 1 then its identity is[11 0 0]. The third type of damage is a two-fault scenario and if the fault magnitudes are 12and 13 mm in substructures 1 and 2, respectively, then the identity of this case is [12 13 0].The "nal type of damage is a three-fault scenario and if the magnitudes are 12 mm then theidentity of this case is [12 12 12]. For each damage case measurements are taken at least3 times by measuring the acceleration at a "xed position and roving the impulse position.One cylinder gives four damage scenarios and the total number of data collected is 264.

The structure is vibrated at 19 di!erent locations (see Fig. 3), nine on the upper ring of thecylinder and 10 on the lower ring of the cylinder. Some of the problems that are encounteredduring impulse testing include the di$culty to excite the structure at an exact the position(especially for an ensemble of structures) and that the direction of the hammer cannot beaccurately repeated. In Fig. 4, it may be observed that the repeatability of the measurementsof the frequency response functions is generally good at lower frequencies and as expectedbecome poor at higher frequencies. The presence of an accelerometer and the imperfectionof cylinders destroy the axis-symmetry of the structures. The incidence of repeated naturalfrequencies is destroyed.

Figure 3. Cylindrical shell of 1.75 mm thickness.

Figure 4. The frequency response functions from a population of 20 undamaged cylindrical shells.


Modal analysis is utilised to extract modal properties. The graph in Fig. 5 shows thedistribution of the "rst identi"ed mode shape for a population 20 undamaged cylindricalshells. This "gure shows that the identi"ed modes are repeatable. From the modal

Figure 5. Identi"ed mode shapes from a population of 20 undamaged cylindrical shells (413 Hz).

1118 T. MARWALA

properties of undamaged cylindrical shells (323 parameters for each fault case), 60 para-meters that have the least variance are chosen. From the chosen parameters and fordamaged cases, 19 parameters with the highest variances are chosen. The "rst modalproperty based network is trained using these 19 parameters. Modal properties are con-verted into the COMAC (note that this COMAC has a natural frequency vector as anadditional mode). The second modal property based network is trained using the COMACdata. Both networks have 19 input parameters, 11 hidden units and three output units(corresponding to three substructures).

Modal energies are calculated using the "rst seven natural frequencies. From the modalenergies of undamaged cylindrical shells (266 parameters, half from the imaginary part andthe other half from the real part of the frequency response functions from each fault case),60 parameters that have the least variance are chosen. From the chosen parameters and forboth undamaged and damaged cases 19 parameters with the highest variances are chosen.The "rst frequency response function-based network is trained using these 19 parameters.Modal energies are transformed into the COMEAC. The second frequency responsefunction-based network is trained using the COMEAC data. These networks have 19 inputparameters, 11 hidden units and three output units.

On training these networks, the coe$cient of prior a and the coe$cient b are initialisedto be 0.001 and 100, respectively [see equation (11)]. The number of samples ofweights retained when hybrid Monte-Carlo procedure is completed is 1000. The stepsize in each trajectory is 0.002. The number of steps in each trajectory is 200. From the264 measured data 80 are randomly chosen and used to generalise the fournetworks.

The modal-energy-network and the modal-property-network as well as the COMEAC-network and the COMAC-network are combined to form two committees (see Fig. 2).Comparisons between the trained networks are made.

TABLE 1

Exact fault identity and identi,ed fault for each substructure when mode shapes and naturalfrequencies are used as inputs to the neural network (bold shows the average fault given by 1000sample weights obtained using hybrid Monte-Carlo simulation; their respective standard

deviations are within parentheses)

Exact Identi"ed

Substr. 1 Substr. 2 Substr. 3 Substr. 1 Substr. 2 Substr. 3

1 0 0 0 0.0 (6.04) 0.6 (6.06) 1.2 (6.02)2 0 0 0 0.0 (6.04) 0.0 (6.09) 3.3 (6.08)3 11 0 11 12.1 (6.04) 0.0 (6.04) 12.6 (6.00)4 0 11 11 0.0 (6.09) 12.5 (6.11) 8.3 (6.05)5 0 13 11 0.0 (6.03) 12.5 (6.03) 12.4 (6.03)6 0 0 11 0.0 (6.03) 0.0 (6.07) 12.9 (6.07)7 13 0 0 11.8 (6.09) 0.0 (6.07) 0.0 (6.05)8 14 11 12 12.7 (6.09) 13.8 (6.10) 12.8 (6.02)9 13 13 13 12.8 (5.95) 12.1 (6.08) 12.3 (6.02)

10 12 11 14 12.2 (5.98) 12.9 (6.12) 11.9 (6.07)


5. RESULTS AND DISCUSSION

The modal-property-network requires 253 CPU min to train on a Pentium 200 MHzprocessor while the modal-energy-network requires 222 CPU min. The COMAC-networkrequires 153 CPU min while the COMEAC-network requires 281 CPU min.

In Table 1 the average identity and the standard deviation are shown. These results areobtained from 1000 sample of weights retained when training neural network with hybridMonte-Carlo simulation and modal properties as inputs. These fault cases were randomlychosen from 80 cases that were not used for training the network.

Table 2 shows the average identities of faults and their respective standard deviationswhen modal energies are used as inputs to the neural network.

Table 3 shows the average identities of faults and their respective standard deviationswhen the modal-energy-modal-property-committee is used.

The standard deviations in Tables 1}3 show that the committee approach gives thelowest standard deviations (highest probability of giving the correct solution), then themodal-energy-network and then the modal-property-network.

If 80 generalisation cases are used to classify whether there is fault or no fault presentgiven the input data, the results in Fig. 6 are obtained. This "gure shows that the total casesthat have faults but are classi"ed not to have faults ( fault/no fault case) is 1 for modal-property-network, 2 for both modal-energy-network and the modal-energy-modal-prop-erty-committee. The total cases that have no faults but are classi"ed to have faults(no fault/fault case) are 0 for modal-property-network as well as the modal-energy-modal-property-committee and 2 for modal-energy-network. The total scenarios that have nofaults and are classi"ed correctly (no fault/no fault case) are 36 for modal-property-networkand the modal-energy-modal-property-committee as well as 34 for modal-energy-network.The total cases that have faults and classi"ed correctly ( fault/fault case) are 43 formodal-property-network as well as 42 for both the modal-energy-modal-property-commit-tee and modal-energy-network. Figure 6 shows that for classifying purposes the modal-property gives the best results followed by the modal-energy-modal-property-committee.

TABLE 2

Exact fault identity and identi,ed fault for each substructure when modal energies are used asinputs to the neural network (bold shows the average fault given by 1000 sample weightsobtained using hybrid Monte-Carlo simulation; their respective standard deviations are within

parentheses)

Exact Identi"ed


1 0 0 0 0.0 (5.98) 5.4 (5.85) 0.0 (6.02)2 0 0 0 0.3 (5.77) 2.1 (5.92) 1.1 (6.02)3 11 0 11 12.4 (5.89) 0.07 (5.93) 13.0 (5.78)4 0 11 11 0.0 (5.90) 12.3 (5.95) 12.2 (5.87)5 0 13 11 0.1 (5.95) 12.3 (5.92) 12.2 (5.87)6 0 0 11 0.0 (5.96) 0.0 (5.97) 12.4 (5.88)7 13 0 0 11.8 (6.01) 0.1 (5.97) 0.0 (5.91)8 14 11 12 12.6 (6.02) 8.4 (6.06) 11.9 (5.90)9 13 13 13 11.6 (6.04) 12.1 (5.8) 11.7 (5.94)

10 12 11 14 12.4 (5.98) 11.3 (5.92) 8.6 (5.90)

TABLE 3

Exact fault identity and identi,ed fault for each substructure when modal properties and modalenergies are used simultaneously (bold shows the average fault, their respective standard


Exact Identi"ed


1 0 0 0 0.0 (5.50) 3.3 (5.34) 0.6 (5.39)2 0 0 0 0.2 (5.42) 1.2 (5.49) 2.1 (5.40)3 11 0 11 12.2 (5.41) 0.0 (5.41) 12.8 (5.27)4 0 11 11 0.0 (5.38) 12.4 (5.46) 10.5 (5.32)5 0 13 11 0.0 (5.46) 12.4 (5.44) 12.3 (5.39)6 0 0 11 0.0 (5.42) 0.0 (5.44) 12.6 (5.38)7 13 0 0 11.8 (5.47) 0.0 (5.44) 0.0 (5.35)8 14 11 12 12.6 (5.52) 10.9 (5.51) 12.3 (5.35)9 13 13 13 12.1 (5.53) 12.1 (5.43) 12.0 (5.34)

10 12 11 14 12.3 (5.36) 12.0 (5.44) 10.1 (5.37)

1120 T. MARWALA

The results showing the average prediction error (representing the absolute value of thedi!erence between the prediction and the true identity of faults) for 80 generalisation casesare shown in Fig. 7. This "gure shows that these three approaches are on average best ableto identify substructure 1 followed by substructure 3. It also shows that for identifyingsubstructure 1, the modal-property-network gives the best results followed by the commit-tee approach; for identifying substructure 2, the committee approach gives the best resultsfollowed by the modal-energy-network and for substructure 3 the modal-energy-networkgives the best results followed by the committee approach. If all substructures are con-sidered simultaneously, the committee approach is on average better than the two indi-vidual approaches.

Figure 6. The number of cases vs the FNF, NFF, NFNF and FF using modal property, modal energy andcommittee methods. Key: FNF"fault but no fault; NFF"no fault declares fault; NFNF"no fault declares nofault; FF"fault declares fault: , modal shape; , modal energy; h, committee.

Figure 7. The average prediction error of the absolute di!erence between the prediction and the true identity offaults for 80 fault cases not used for training: , modal properties; , modal energies; h, committee.


The results showing the standard deviation of the prediction error (representing theabsolute value of the di!erence between the prediction and the true identity of faults) for 80generalisation cases are shown in Fig. 8. This "gure shows that these three approaches givesimilar mean of standard deviations for all the substructures. It shows that the committeeapproach gives the best results (lower mean standard deviation) followed by the modal-energy-network.

Figure 8. The average standard deviation of the absolute di!erence between the prediction and the true identityof faults for 80 fault cases not used for training: , mode shapes; j, frequency energies; h, committee.

Figure 9. The % mean square error vs the weighting function given to the modal property approach (theremaining % is given to the FRF-approach). Key: - - COMEAC and COMAC, - modal properties and modalenergies directly.

1122 T. MARWALA

The graph showing mean square errors vs weighting functions on the modal-property-network is shown in Fig. 9 (with a solid line). This "gure demonstrates that the committeeapproach gives lower mean square errors than the individual methods. It also shows thatthe optimal committee is obtained by giving 45% weight to the modal-property-network.The graph showing the standard deviations of the absolute error between the predictedidentities of faults and the correct identity of faults vs weighting functions on the modal-property-network is shown in Fig. 10 (with a solid line). This "gure demonstrates that the

Figure 10. The % standard deviation of the error vs the weighting function given to the modal propertyapproach (the remaining % is given to the FRF-approach). Key: - - COMEAC and COMAC, - frequency responsefunctions and modal property.

TABLE 4

Exact fault identity and identi,ed fault for each substructure when the COMAC is used (boldshows the average fault, their respective standard deviations are within parentheses)

Exact Identi"ed


1 0 0 0 0.0 (6.07) 2.1 (5.95) 0.0 (6.03)2 0 0 0 0.0 (6.12) 0.0 (6.00) 0.0 (6.05)3 11 0 11 12.3 (5.94) 0.0 (6.05) 12.8 (5.86)4 0 11 11 0.0 (6.02) 13.6 (6.05) 12.4 (5.92)5 0 13 11 0.0 (5.95) 12.2 (6.04) 10.6 (5.95)6 0 0 11 0.0 (6.02) 0.0 (6.08) 11.4 (5.97)7 13 0 0 11.7 (5.99) 0.0 (6.06) 0.0 (5.93)8 14 11 12 12.6 (6.07) 9.6 (6.07) 12.1 (5.98)9 13 13 13 13.9 (6.06) 10.7 (6.01) 12.9 (5.96)

10 12 11 14 13.0 (6.03) 12.3 (6.03) 12.3 (6.06)


committee approach gives lower standard deviations than the individual methods. It alsoshows that the optimal committee is obtained by giving 45% weight to the modal-property-network. These graphs show that the modal-energy-network is better than the modal-property-network and the committee is better than the individual methods.

Table 4 shows the average identities of faults and their respective standard deviationsfrom 1000 sample of weights obtained using hybrid Monte-Carlo and selected parameters

TABLE 5

Exact fault identity and identi,ed fault for each substructure when the COMEAC are used(bold shows the average fault, their respective standard deviations are within parentheses)

Exact Identi"ed


1 0 0 0 0.7 (5.91) 0.0 (5.92) 2.4 (6.00)2 0 0 0 0.0 (5.98) 0.8 (5.92) 0.0 (5.87)3 11 0 11 12.2 (5.93) 0.0 (6.00) 12.7 (5.87)4 0 11 11 0.18 (5.92) 12.5 (6.01) 11.2 (6.01)5 0 13 11 0.0 (5.97) 12.9 (6.01) 6.23 (5.94)6 0 0 11 0.2 (5.87) 0.0 (6.00) 12.7 (6.05)7 13 0 0 11.8 (5.94) 0.0 (6.02) 0.0 (6.03)8 14 11 12 12.9 (5.87) 11.5 (6.06) 12.6 (5.96)9 13 13 13 8.5 (5.78) 11.8 (5.87) 11.4 (6.02)

10 12 11 14 13.2 (5.87) 11.2 (5.92) 12.8 (6.02)

TABLE 6

Exact fault identity and identi,ed fault for each substructure when the COMAC and theCOMEAC are used simultaneously (bold shows the average fault, their respective standard


Exact Identi"ed


1 0 0 0 0.4 (5.08) 1.0 (5.01) 1.3 (5.12)2 0 0 0 0.0 (5.18) 0.5 (5.05) 0.0 (5.23)3 11 0 11 12.2 (5.00) 0.0 (5.07) 12.7 (4.95)4 0 11 11 0.1 (5.02) 13.0 (5.03) 11.7 (5.07)5 0 13 11 0.0 (5.04) 12.6 (5.07) 8.2 (5.02)6 0 0 11 0.1 (4.99) 0.0 (5.08) 12.1 (5.08)7 13 0 0 11.8 (5.03) 0.0 (5.13) 0.0 (5.00)8 14 11 12 12.7 (5.13) 10.6 (5.11) 12.4 (5.00)9 13 13 13 10.9 (5.09) 11.3 (4.95) 12.1 (5.03)

10 12 11 14 13.1 (5.11) 11.7 (4.99) 12.5 (5.14)

1124 T. MARWALA

from the COMAC. These fault cases were chosen from 80 cases that were not used fortraining the networks.

Table 5 shows the average identities of faults and their respective standard deviations(1000 sample weights obtained using hybrid Monte-Carlo) and using the COMEAC asinputs.

Table 6 shows the average identities of faults and their respective standard deviationsfrom the COMEAC}COMAC-committee.

The standard deviations in Tables 4}6 show that the committee approach gives thelowest standard deviations (highest probability of giving the correct solution), then theCOMEAC-network and then the COMAC-network.

The 80 classi"cation results are shown in Fig. 11. This "gure shows that the totalfault/no-fault cases is 4 for COMAC-network, 0 for both COMEAC-network and the

Figure 11. The number of cases vs the FNF, NFF, NFNF and FF using COMAC, COMEAC and committeemethods. Key: FNF"fault but no fault; NFF"no fault declares fault; NFNF"no fault declares no fault;FF"fault declares fault: , COMAC; j, COMEAC; , committee.

Figure 12. The average prediction error of the absolute di!erence between the prediction and the true identity offaults for 80 fault cases not used for training: , COMAC , COMEAC; h, committee.


committee approach. The total no-fault/fault cases is 11 for the COMAC-network, 6 forCOMEAC-network and 2 using the COMEAC}COMAC-committee. The total no fault/no-fault cases is 25 for the COMAC-network, 30 for the COMEAC-network and 34 for theCOMEAC}COMAC-committee. The total number of fault/fault cases is 40 forthe COMAC-network and 44 for both the COMEAC-network as well as theCOMEAC}COMAC-committee. This "gure shows that overall, the performance of theCOMEAC}COMAC-committee is the best, followed by the COMEAC-network and thenthe COMAC-network.

When Figs 6 and 11 are compared it is found that for the fault/no-fault cases, theCOMEAC-network and the COMAC}COMEAC-network give the best results, followedby the modal-property-network. For the no-fault/fault cases, the modal-property-networkand the modal-energy-modal-property-committee give the best results followed by the

Figure 13. The average standard deviation of the absolute di!erence between the prediction and the trueidentity of faults for 80 fault cases not used for training: , COMAC; , COMEAC; h, committee.

1126 T. MARWALA

COMEAC}COMAC-committee and the modal-energy network. For the no falut/no-faultcases, the modal-property-network and modal-energy-modal-property-committee give thebest results followed by the COMEAC-COMAC-committee and the modal-energy-net-work. For the fault/fault cases, the COMEAC}COMAC-committee and the COMEAC-network give the best results followed by the modal-property-network. The overall classi-"cation shows that the modal-property-network gives the best results, followed by theCOMEAC}COMAC-committee and the modal-energy-modal-property-committee, fol-lowed by the modal-energy-network then the COMEAC-network and "nally the COMAC-network.

The results showing the average prediction error (representing the absolute value of thedi!erence between the prediction and the true identity of faults) for 80 generalisation casesare shown in Fig. 12. For substructures 1 and 2, the COMEAC-network is on average thebest followed by the COMEAC}COMAC-committee. For substructure 3, the COMAC-network is on average the best followed by the COMEAC}COMAC-committee. If allsubstructures are considered simultaneously, the COMEAC}COMAC-committee is onaverage better than the two individual approaches.

The results showing the standard deviation of the prediction error (representing theabsolute value of the di!erence between the prediction and the true identity of faults) for80 generalisation cases are shown in Fig. 13. This "gure shows that these three approachesgive similar mean of standard deviations for all the substructures. It shows that theCOMEAC}COMAC-committee gives the best results (lower mean standard deviation)followed by the COMEAC-network.

The graph showing mean square errors vs weighting functions on COMAC-network isshown in Fig. 9 (with dashed line). This "gure demonstrates that the COMEAC}COMAC-committee-network shows the least of errors than the individual methods. It also shows thatthe optimal committee is obtained by giving 40% weight to the COMAC-network. Thegraph showing standard deviations of the absolute error between the predicted identities offaults and the correct identity of faults vs weighting functions on the COMAC-networkis shown in Fig. 10 (with dashed lines). It also shows that the optimal committee is obtainedby giving 50% weight to the COMAC-network. These graphs demonstrate that theCOMEAC}COMAC-committee gives the best results followed by the COMEAC-network then the COMAC-network. The COMEAC-network performs better than the


COMAC-network for identifying faults. Using the modal-energies and modal propertiesshows better results than using the COMEAC and the COMAC as input data foridenti"cation of faults.

6. CONCLUSION

Four Bayesian formulated neural networks are trained using modal energies, modalproperties, the COMEAC and the COMAC data to perform probabilistic fault identi"ca-tion in a population of cylindrical shells. It is observed that using modal energies and modalproperties directly gives better results than using the COMEAC and COMAC. It isobserved that the Bayesian framework, which result with networks that give the identity offaults and the corresponding probability distribution (from the standard deviations), o!ersthe possibility of assessing the degree of con"dence the networks have on the given solution.The results show that the committee gives lower mean square errors and standard devi-ations than the individual networks. The modal-energy- and COMEAC-networks arefound to give more accurate identi"cation results than the modal-property-network and theCOMAC-network. For classi"cation of the presence or absence of faults the modal-property-network is found to give the best results, followed by the COMEAC}COMAC-committee and the modal-energy-modal-property-committee, then modal-energy-networkthen the COMEAC-network and "nally the COMAC-network. The committee is found tobe a reliable alternative to fault identi"cation if there is no prior knowledge as to whichmethod is better. It is also observed that the computational time required to implement thisprocedure is a!ordable.

REFERENCES

1. S. W. DOEBLING, C. R. FARRAR, M. B. PRIME and D. W. SHEVITZ 1996 ¸os Alamos National¸aboratory ¹echnical Report ¸A-13070-MS. Damage identi"cation and health monitoring ofstructural and mechanical systems from changes in their vibration characteristics: a literaturereview.

2. T. MARWALA and P. S. HEYNS 1998 American Institute of Aeronautics and Astronautics Journal 36,1494}1501. Multiple-criterion method for determining structural damage.

3. T. MARWALA and H. E. M. HUNT 1999 Mechanical Systems and Signal Processing 13, 475}490.Fault identi"cation using "nite element models and neural networks.

4. T. MARWALA 2000 American Society of Civil Engineers, Journal of Engineering Mechanics 126,43}50. On damage identi"cation using a committee of neural networks.

5. C. M. BISHOP 1995 Neural Networks for Pattern Recognition. Oxford: Oxford University Press.6. R. M. NEAL 1992 ;niversity of ¹oronto, ¹oronto, ¹echnical Report CRG-¹R-92-1 Bayesian

training of backpropagation networks by hybrid Monte Carlo method.7. S. DUANE, A. D. KENNEDY, B. J. PENDLETON and D. ROWETH 1987 Physics ¸etters 195, 216}222.

Hybrid Monte Carlo.8. N. METROPOLIS, A. W. ROSENBLUTH, M. N. RESENBLUTH, A. H. TELLER and E. TELLER 1953

Journal of Chemical Physics 21, 1087}1092. Equations of state calculations by fast computingmachines.

9. S. KIRKPATRICK, S. D. GELATT and M. P. VECCHI 1983 Science 220, 671}680. Optimization bysimulated annealing.

10. N. A. J. LIEVEN and D. J. EWINS 1988 Proceedings of the 6th International Modal AnalysisConference, 690}695. Spatial correlation of modeshapes: the coordinate modal assurance criterion(COMAC).

APPENDIX A: NOMENCLATURE

[C] damping matrixd number of input unitsD identity of fault matrixEi

error functions

1128 T. MARWALA

f (f) activation function of fM f N force vectorh(f) mapping function of f[K] sti!ness matrix[M] mass matrixM the number of hidden unitsN number of training datap(D Dw) the likelihood functionp(w Dx) weight probability distribution functionp(w DD) posterior probability distributionp(y Dx, w) distribution of the noise on ywj0

, wk0

bias parameters¹ pseudotemperature= number of weightsxn

nth input parameteryn

nth identity of damageMXN displacement vectorM/

nN, M/1

nN nth natural mode shape vector

un

natural frequency* complex conjugate

probabilistic fault identification using vibration data and neural networks

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