Mechanical Systems and Signal Processing (1999) 13(3), 437}448Article No. mssp.1998.0192, available online at http://www.idealibrary.com on

08

ARTIFICIAL BOUNDARY CONDITIONS FORMODEL UPDATING AND DAMAGE DETECTION

J. H. GORDIS

Naval Postgraduate School, Department of Mechanical Engineering~Code ME/Go Monterey,CA 93943-5100, U.S.A.

(Accepted 18 October 1998)

The updating of "nite-element models commonly makes use of measured modal para-meters. The number of modal parameters measured in a typical modal test is small, while thenumber of model parameters to be adjusted can be large. In this paper, it is shown that thenatural frequencies for the structure under a variety of boundary conditions are availablefrom any square, spatially incomplete frequency response function data, without any actualphysical alteration of the boundary conditions, hence the term &arti"cial boundary condi-tions'. This approach can provide potentially a large number of additional and distinct modefrequencies from the same modal test. The simplest example of this approach is that thedriving-point antiresonance frequencies correspond to the natural frequencies obtainedfrom the structure with the driving-point degree of freedom constrained to ground. Thisresult is developed generally for multiple measured points, and is shown to be related toprevious results concerning omitted coordinate systems and spatially incomplete test data.The approach is applied to sensitivity-based model updating and damage detection.

( 1999 Academic Press

1. INTRODUCTION

The improvement, or &updating' of a "nite-element (FE) model is often a necessarystep in order for the model to be used with con"dence in the prediction of structureresponse. The inaccuracy in the prediction of dynamic response is gauged most often by theinaccuracy in the prediction of modal parameters. This inaccuracy of the FE model isreduced (the model is &improved') by the adjustment of selected physical and materialparameters which de"ne the model. These parameters can include dimensional properties ofstructural elements, moduli of elasticity, and densities, for example. The selected parametersare adjusted such that the predicted modal parameters calculated from the updated modelare brought into closer agreement with the corresponding parameters identi"ed in amodal test, i.e. the measured parameters. The selection of physical parameters tobe adjusted is based on a consideration of the accuracy of the parameter values, and froma need to compensate for structural dynamic behaviour not accounted for by the FEformulation.

A typical FE model may be de"ned by a large number (of the order of 102}103) ofphysical parameters. However, a typical modal test of the structure modelled yields a smallnumber (of the order of 101) of modal parameters to be used to guide the adjustment ofthe model parameters. This disparity in the number of known parameters (measuredmodal parameters) vs the number of parameters to be adjusted de"nes an underdeter-mined problem. A further and signi"cant di$culty is that those parameters which are trulyin error (if any) are unknown. The solution of this particular problem is known as errorlocalisation.

88}3270/99/030437#12 $30.00/0 ( 1999 Academic Press

438 J. H. GORDIS

There exists a recognised need to increase the size of the known parameter data-base. For example, Ref. [1] describes procedures referred to as &perturbed boundarycondition (PBC)' testing, where additional con"gurations of the structure are independentlytested. These con"gurations can include di!erent boundary conditions and addition ofmass at selected points on the structure. These procedures require physical modi"-cations be made to the structure, and an additional modal test for each additionalcon"guration.

This paper shows that a large number of additional and distinct mode frequencies can beeasily identi"ed from the same modal test performed to identify the standard system modefrequencies, without the need for any physical modi"cation of the structure. These addi-tional and distinct mode frequencies correspond exactly to those mode frequencies foundwhen combinations of measured coordinates are restrained to ground. Hence, these addi-tional frequencies are associated with di!erent boundary conditions for the structure. Itshould be noted that only frequency information is obtained.

The practicality of this result lies in the fact that these boundary conditions need not beactually applied, hence the term arti"cial boundary conditions' (ABC). As a simple exampleof this, consider the spectrum of antiresonances for any driving-point frequency responsefunction. The driving-point antiresonance frequencies correspond to the mode frequenciesof the structure with the driving-point degree of freedom (dof ) restrained ( &pinned') toground. The ABCs correspond to ideal constraints, i.e. &pins', and therefore can easily beimposed on an FE model. This scheme would yield a separate FE model for each ABCcon"guration, yet only one set of measured test data. Each FE model is identical except forthe boundary conditions, and each model can be used to generate independent sensitivitydata, for example.

The ABCs are implicitly de"ned in any set of spatially incomplete frequency responsefunction data, and are those boundary conditions which de"ne an omitted coordinate setsystem or OCS [2, 3]. This system is de"ned by the set of all unmeasured coordinates.For a test system, this set is of in"nite dimension. ABC con"guration frequenciesare available from any set of (spatially incomplete) test data due to the fact that a spatiallyincomplete frequency response function (FRF) matrix is identically equivalent to the FRFmatrix which is calculated from the exact dynamic reduction. In short, a measured FRFmatrix represents an in"nite-dimensional system dynamically reduced to the measuredcoordinates. This is a central fact in this work, and a complete development of this fact isfound in [2, 3].

The OCS has been shown to be central to the e!ective performance of advanced test-analysis models (TAMs) in model correlation, such as in cross-orthogonality calculations[4]. These background concepts will be brie#y developed in what follows.

The ABC mode frequencies not only provide a greater number of frequencies for thesystem, but also provide a means to reduce or eliminate ill-conditioning in the solution ofsensitivity equations. Since the system is being arti"cially restrained at various combina-tions of measured coordinates, the sensitivity matrix columns found from the arti"ciallyrestrained con"guration can be linearly independent of those columns calculated for thebaseline con"guration of the structure. It is also possible that a greater number of linearlyindependent columns can be found from the ABC con"guration exclusively, than from thebaseline con"guration. The use of ABC con"gurations can allow the discrimination oferrors for model parameters which have dependent or nearly dependent sensitivity matrixcolumns. These uses will be demonstrated with simple examples. The reader is referred toRefs. [2}4] for complete expositions of the background theory and its rami"cations inmodel reduction and identi"cation. The theory will be brie#y reviewed here to providea self-contained exposition.

439MODEL UPDATING AND DAMAGE DETECTION

2. THEORY

The equations which de"ne the ABC and OCS will be developed. First, fundamentalrelations in model reduction which de"ne the OCS are examined. The analogous relationsin the frequency domain are then developed which will allow the identi"cation of ABCcon"guration natural frequencies.

2.1. OCS AND MODEL REDUCTION

The equation of steady-state forced response for a linear structural dynamic system ata forcing frequency X (rad/s):

CCkaa

koa

kao

kooD!X2C

maa

moa

mao

mooDDG

xa

xoH"G

fafoH (1a)

where k and m are sti!ness and mass matrices, x and f vectors of generalised response andexcitation amplitudes, respectively. The subscript &a1 refers to the measured coordinates( &analysis coordinate set') and the subscript &o' refers to the coordinates not measured( &omitted coordinate set'). Equation (1a) can be written as

CZ

aaZ

oa

Zao

ZooD G

xa

x0H"G

fa

foH (1b)

where an impedance matrix is de"ned as Z"k!X2 m. Assuming no excitations act on theomitted coordinates, the following relationship between the omitted coordinate set and theanalysis coordinate set is found:

MxoN"[I!X2k~1

oom

oo]~1[!k~1

ookoa#X2k~1

oom

oa]Mx

aN. (2)

Equation (2) is the starting point for a class of physical coordinate model reductions [4].For example, setting X"0 yields the static or Guyan reduction. The derivation of the IRSmodel reduction transformation starting from equation (2) is shown in [4].

The origin of the omitted coordinate set system is seen from equation (2). The bracketedinverse term is replaced with its equivalent,

[I!X2k~1oo

moo

]~1"1

Det[I!X2k~1oo

moo

]Adj[I!X2k~1

oom

oo] (3)

where Det[ * ] indicates the determinant and Adj[ * ] indicates the adjoint matrix. Fromequation (3), it can be seen that the bracketed inverse term, and hence the exact relationshipbetween the analysis coordinate set and omitted coordinate set does not exist at thosefrequencies X which satisfy the equation

Det[I!X2k~1oo

moo

]"0. (4)

The frequencies which satisfy equation (4) are the eigenvalues of the system de"ned bykoo

and moo

, i.e. the omitted coordinate set system. This system is obtained by fullyconstraining to ground all coordinates in the analysis coordinate set. As shown in [4],equation (2) is the general starting point for the derivation of physical coordinate modelreduction transformations, and the existence of the inverse in equation (2) above, or theconvergence of the series used to replace this inverse, is dependent on the forcing frequencyX, or the placement of the omitted coordinate set system frequencies with respect to thesystem frequencies.

440 J. H. GORDIS

2.2. OCS AND FREQUENCY RESPONSE FUNCTION MATRICES

The process of instrumenting a structure with a "nite number of response transducersde"nes a reduced-order model, where the impedance of the reduced-order model is non-linearly dependent on the impedance of the full-order model [5]. Following [5], if the exactfull-order FRF model of a structure is considered, a FRF matrix of in"nite dimension(the superscript &x1 indicates an experimental quantity),

Hx"CH

aaH

oa

Hao

HooD (5)

where the number of coordinates in the omitted coordinate set is in"nite, then the FRFmatrix measured in a test is seen to be a matrix partition which has been extracted from thein"nite dimension matrix, i.e.

H1 x"Haa

(6)

where the overbar in equation (6) indicates a reduced model. That this measured matrixpartition represents a dynamically reduced model is seen as follows. From the partitionedidentity ZH"I,

Haa"(Z

aa!Z

aoZ~1

ooZ

oa)~1. (7)

The reduced-order impedance relation obtained using the exact dynamic reduction is [3]

MfaN"[Z

aa!Z

aoZ~1

ooZ

oa]Mx

aN. (8)

The common term in equations (7) and (8) makes clear that a spatially incomplete FRFmatrix represents a dynamically reduced model.

The presence of the OCS in a spatially incomplete FRF matrix is contained in thequantity Z~1

ooin equation (7). The formula for the matrix inverse given by equation (3)

applies here as well, and since every element in Z~1oo

is singular at the natural frequencies ofthe OCS, it is seen from equation (7) that elements of H~1

aawill be singular (or &large', for

a damped system) at the OCS natural frequencies.

2.3. ABC CONFIGURATION FREQUENCIES: EXAMPLES

The identi"cation of the ABC natural frequencies from measured FRF data usingequation (7) will be demonstrated with simple numerical examples. The "rst example, how-ever, will show that driving-point antiresonances correspond to the natural frequencies ofthe structure with the driving-point dof constrained to ground.

2.3.1. Example 1: Driving-Point Antiresonances Are ABC Frequencies

This is demonstrated "rst using basic principles. Consider the 2-dof system shown inFig. 1.A driving-point FRF is given by

Hii

(X)"p+r/1

(/ri)2

u2r!X2

(9)

where /riis a mass normalised mode shape element, u

ris the rth natural frequency, and X is

the forcing frequency. The frequency of the antiresonance of H11

(X) is given by

X2!/5*-3%4

"

R111

u22#R2

11u2

1R1

11#R2

11

(10)

Figure 1. A 2-dof system.

Figure 2. The 2-dof system with dof d1 restrained to ground.

Figure 3. H11

(X) vs X with X!/5*-3%4

"J2 (rad/s).

441MODEL UPDATING AND DAMAGE DETECTION

where the modal residue is given by Rrij"/r

i/rj. It is easily demonstrated that this frequency

is equal to the natural frequency of the system in Fig. 1, with the driving-point dofconstrained to ground, i.e. if m

1"m

2"1.0 and k

1and k

2"1.0 for the system shown in

Fig. 1, the frequency of the single antiresonance is X!/5*-3%4

"J2 rad/s, which equals the

single natural frequency of the ABC system (Fig. 2), which is u"J2 rad/s.

2.3.2. Example 2: Calculation of ABC frequencies for 2-dof system

The calculation of ABC frequencies is done using equation (7). A spatially incompleteFRF matrix is generated (either experimentally or analytically), and this FRF matrix(or submatrices thereof ) is inverted at each frequency. The elements of the resultingimpedance matrix are plotted vs frequency, and the singular frequencies are the ABC systemfrequencies. This process can be repeated for various submatrices of the original FRFmatrix, thereby generating multiple sets of independent ABC system frequencies. Example1 will be repeated, using the general approach.

The driving-point FRF H11

(X) for the system shown in Fig. 1 is plotted in Fig. 3. Thesystem has two modes at u

1"0.6180 rad/s and u

2"1.6180 rad/s. Using equation (10), the

antiresonance frequency is X!/5*-3%4

"J2 rad/s.The ABC system frequency is identi"ed by calculating H~1

aa(X), as per equation (7). The

analysis coordinate set contains the single-dof d1 (Fig. 1), and H~1aa

(X) is simply the scalarinverse of H

11(X). H~1

11(X) is plotted in Fig. 4 where the single singular frequency is evident,

and directly corresponds to the antiresonance frequency for this scalar case.

Figure 4. H11

(X) vs X with X!/5*-3%4

"J2 (rad/s).

Figure 5. A free}free beam with spatially incomplete transducer set.

Figure 6. Driving-point FRF, H11

(X) for free}free beam.

442 J. H. GORDIS

The singular frequency in H~111

(X) corresponds to the natural frequency of the ABCsystem obtained by constraining dof d1 (the single measured dof) to ground.

2.3.3. Example 3: Calculation of ABC frequencies for free}free beam

To demonstrate the calculation of ABC system frequencies consider the free}free beamshown in Fig. 5, modelled with 10 two-node beam elements. The beam properties are takenas: length"60 in, EI"5.5E5 lbf in2, o"0.283 lbf/in3, and cross-sectional area"0.75 in2.

Accelerometers are shown at dof 's d1, 5, 9, 13, 17, and 21, which are translationalcoordinates. The analysis coordinate set is therefore [1 5 9 13 17 21]. It is assumed thatexcitation has been applied at each dof in the analysis coordinate set, and hence the FRFmatrix is a 6]6 matrix. The impedance matrix H~1

aa(X) is calculated over a frequency range

of 0}800 Hz. The driving point FRF, H11

(X), is shown in Fig. 6.The ABC system frequencies are now identi"ed by calculating the impedance matrix

H~1aa

(X). If the 1, 1 element of this matrix is plotted (Fig. 7), the ABC system frequencies areevident.

Figure 7. ABC frequencies are the peaks of elements of H~1aa

(X).

Figure 8. The ABC con"guration for the measured coordinates.

Figure 9. ABC frequencies are the peaks of elements of H~1aa

(X).

443MODEL UPDATING AND DAMAGE DETECTION

These frequencies correspond exactly to the natural frequencies of the system obtainedwhen all measured coordinates are constrained to ground, as shown in Fig. 8.

As a check, the natural frequencies of the free}free beam, with six pin restraints at theanalysis coordinate set dof are calculated. These natural freuqencies are 346.5, 384.8, 482.4,610.0, and 735.0 Hz, which correspond to the peak frequencies in Fig. 7.

Additional sets of ABC con"guration frequencies are available from submatrices ofH

aa(X). For example, if analysis coordinate set dof 's 1 and 17 are taken, corresponding to

a more common two-shaker test, the 1, 1 element of H~1aa

(X) yields an additional set of ABCcon"guration frequencies, as shown in Fig. 9. The corresponding ABC con"guration systemis shown in Fig. 10.

Figure 10. The ABC con"guration for the a-set [1 17].

444 J. H. GORDIS

As a check, the natural frequencies for the system shown in Fig. 10 are calculated, and are,20.45, 63.57, 112.30, 215.05, 364.60, 543.96, 668.91, 846.93 Hz, which correspond to the peakfrequencies in Fig. 9.

2.4. ABC CONFIGURATION FREQUENCIES IN SENSITIVITY-BASED UPDATING

The ABC con"guration frequencies can be used in addition to, or instead of, the standardsystem mode frequencies in sensitivity-based model updating and damage detection. This isbecause the ABC frequencies correspond to the same structural system, but with di!erentboundary conditions. The governing equation for sensitivity-based updating is

MDuN"[¹]MDD<N (11)

where MDuN is a vector of natural frequency errors, MDD<N is the vector of changes to becalculated for speci"ed model parameters, or &design variable', and [¹] is the matrix of"rst-order sensitivities, where ¹

ij"Lu

i/LD<

j. Each ABC system de"nes additional rows of

equation (11). That is, each ABC system de"nes the following equation:

MDuiN"[¹ i]MDD<N (12)

where ¹kij"Luk

i/LD<

j, and uk

iis the ith natural frequency of the kth ABC con"guration

system. Quantities associated with the baseline (non-ABC) system will be denoted with thesuperscript &0'. By identifying the frequencies for a total of &K'ABC systems, the total systemof equations can be compiled:

GMDu0NMDu1N

FMDuKN H"

[¹0]

[¹1]

F[¹K]

MDD<N. (13)

Note that the degree of coupling between the ABC systems and the baseline system inequation (13) can be adjusted by deleting or retaining individual columns of the [¹K].For example, consider a system with one ABC con"guration. Partial coupling between thebaseline system and the ABC system is established if the design variables (D<'s) arepartitioned. In such a partitioning, some of the D<'s are associated exclusively with thebaseline system, MDD<0N, some are associated with both the baseline and the ABC system,MDD<0,1N, and some are associated exclusively with the ABC system, MDD<1N, i.e.

CMDu0NMDu1ND"C

[¹0,0]

[0]

[¹0,1]

[¹1,0]

[0]

[¹1,1]D GDD<0

DD<0,1

DD<1 H (14)

445MODEL UPDATING AND DAMAGE DETECTION

where MDD<N is so partitioned, and the partitions of the [¹k] are superscripted toemphasise this partial coupling of the equations for the two systems.

The use of ABC system sensitivities can eliminate or reduce the problem of poorlyconditioned or rank-de"cient [¹] matrices. For example, columns of [¹0] can be replacedwith columns of [¹k] in order to improve the conditioning. A related problem occurs, forexample, when trying to localise damage. Two closely spaced elements in a model will havenearly dependent columns of [¹0], preventing the discrimination of error or damagebetween the two elements. One column of [¹0] associated with one of the two elements canbe replaced with a column of [¹ k], and the associated baseline system natural frequencyreplaced with the ABC system frequency. These applications will be demonstrated asfollows.

2.4.1. Example 4: ABC sensitivity matrices can have improved conditioning

Consider again the beam of Example 2 (Fig. 5), in the context of structural damagedetection. The sensitivity matrix [¹] is generated for the undamaged structure model, inorder to solve for the changes in the element EI values representing damage, i.e.

MDu0N"[¹0]MDEIN. (15)

For the purpose of this example, sensitivities will be calculated for all 10 elements, and forthe "rst 10 modes, yielding a square [¹0] matrix of size 10]10. However, a calculation ofrank for this matrix indicates that it is rank de"cient,

Rank(¹0 )"5

and therefore will not provide a fully determined solution for the vector MDEIN. If excitationhas been applied at dof 's 1 and 17 (yielding a square FRF matrix), then it is possible toidentify the ABC system frequencies with these dof constrained to ground. Note that thischoice of ABC con"guration removes all symmetry from the system boundary conditions.The corresponding ABC system sensitivity matrix is generated, which is also of size 10]10.

MDu1N"[¹1]MDEIN (16)

and the calculated rank of [¹1] is

Rank(¹1)"10.

This [¹1] sensitivity matrix is full rank due to the asymmetric ABC con"guration.

2.4.2. Example 5: using ABC system sensitivities in damage detection

This example will demonstrate the use of a single ABC system (frequencies and sensitivitymatrix) in place of the standard system frequencies and sensitivities. The ABC system ofFigs 9 and 10 will again be used, corresponding to two-shaker excitation at dof 's 1 and 17.This ABC system yields rank"10 [¹1] matrix, as shown above.

A 10% reduction in EI is made at element d3 and a 15% reduction in EI is made atelement d4, yielding a simulated damaged structure. The natural frequencies of theundamaged (FE) and simulated damaged (Test) models are shown in Table 1.

If the baseline system is used, i.e. [¹0], the largest full rank submatrix of [¹0] is 5]10,and its condition number

Cond([¹0])"2.5e#02.

TABLE 1

FE and test frequencies (kHz) and error

Mode no. FE Test % error

1 0.0313 0.0307 1.92 0.0863 0.0841 2.53 0.1693 0.1674 1.14 0.2803 0.2763 1.45 0.4198 0.4144 1.36 0.5887 0.5818 1.27 0.7882 0.7759 1.68 1.0180 1.0049 1.39 1.2646 1.2471 1.4

10 1.6794 1.6571 1.3

Figure 11. Baseline system sensitivity solution for damage.

446 J. H. GORDIS

The solution for the &damage' is shown in Fig. 11, where the height of the bars correspondsto the magnitude of the element error. Given that the exact solution 0.1 in element d3, and0.15 in element d4, this solution provided by the baseline structure is poor.

If the ABC system sensitivities [¹1] and mode frequencies (FE and experimentallyidenti"ed) are used, one "nds that the condition number is actually double, i.e.

Cond([¹1])"5.1e#02

but the improved solution for the damage is shown in Fig. 12.

3. SUMMARY AND DISCUSSION

It has been shown that the natural frequencies for a structure under a variety of boundaryconditions are available from any square FRF matrix. The inversion of a spatially incom-plete FRF matrix at each frequency in the test bandwidth yields impedance spectra whichhave peaks at the frequencies corresponding to the natural frequencies of the structurerestrained to ground at all the measured coordinates. The practical bene"t of this result isthat no physical alterations need be made to the structure, and no additional testing isrequired, hence the term &arti"cial boundary conditions' (ABC). The ABC con"guration

Figure 12. ABC system sensitivity solution for damage.

447MODEL UPDATING AND DAMAGE DETECTION

frequencies provide a larger database for model updating. For each ABC con"guration, theFE model boundary conditions are altered in that &pins' are installed at the measuredcoordinates corresponding to the FRF matrix. This approach de"nes a single set ofmeasured FRFs and several FE models, di!ering only in the boundary conditions.

With respect to the identi"cation of the ABC frequencies (curve "tting), note that whilethe inverse of the measured FRF matrix de"nes an impedance matrix [equation (7)], nearthe ABC frequency the impedance spectra is dominated by the term Z~1

oo. This suggests the

use of standard single-degree-of-freedom modal parameter estimation to identify the un-damped ABC natural frequencies.

Given the baseline system frequencies, and additional ABC con"guration frequencies,structural sensitivities can be generated from the baseline FE model, and from the modelwith the boundary conditions applied corresponding to the spatially incomplete FRFmatrix. This expanded set of test data can be assembled into a linear system which can besolved in the context of either model updating or structural damage detection. It has beenshown that the ABC con"guration sensitivity matrix can be used in place of, or in additionto, the baseline con"guration sensitivities, and can reduce or eliminate conditioningproblems, such as those that arise when trying to discriminate damage in closely spacedelements.

A natural extension of this technique would be the addition, analytically, of lumpedmasses or springs to measured dof using frequency domain synthesis [6]. This approachprovides an exact modi"cation of the frequency response functions to provide modalinformation. The advantage of this approach is that modal vector information would beobtained for all measured dof.

ACKNOWLEDGEMENTS

This work is dedicated to Hannah Marie Gordis, in honour of her "rst year.

REFERENCES

1. S. LI, S. SHELLEY and D. BROWN 1995 Proceedings of the 13th International Modal AnalysisConference 1, 902}907. Perturbed boundary condition testing.

2. J. H. GORDIS 1992 Proceedings of the 34th AIAA/ASME/ASCE/AHS/ACS Structures, StructuralDynamics, and Materials Conference, 3050}3058. Spatial, frequency domain updating of linear,structural dynamic models.

448 J. H. GORDIS

3. J. H. GORDIS 1996 Modal Analysis 11, 83}95. Omitted coordinate systems and arti"cial constraintsin spatially incomplete identi"cation.

4. J. H. GORDIS 1994 Modal Analysis 9, 269}285. An analysis of the improved reduced system modelreduction procedure.

5. A. BERMAN 1984 Proceedings of the AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynam-ics, & Materials Conference 123}128. System identi"cation of structural dynamic models*theoret-ical and practical bounds.

6. J. H. GORDIS 1994 Shock and <ibration 1, 461}471. Structural synthesis in the frequency domain:a general formulation.

APPENDIX A: NOMENCLATURE

MD<N design variable vectorf load vectorH frequency response function matrixI identity matrixk sti!ness matrixm mass matrixR modal residueT sensitivity matrixx displacement vectorZ impedance matrix/ mass normalised mode shapeu natural frequency (rad/s)X forcing frequency (rad/s)

Subscripts/superscripts

a analysis seti dof indexj design variable indexo omitted setr mode indexx experimental quantity0 baseline system