DAAG LOCALIZAION IN OUPU-ONLY SYSS: A LXIBILIY BASD APPROACH Dionisio Bernal an Burcu Gunes Department of Civil an nvironmental nineerin, 47 Snell nineerin Center, Northeastern University, Boston A 0115, U.S.A ABSRAC o assemble flexibility matrices from vibration sinals the input most be measure an that there must be, at least, one co-locate sensor actuator pair. echniques to localize amae that use chanes in flexibility have not, therefore, foun application in the important case of ambient excitation. he central topic of this paper is the iscussion of an approach that allows extension of flexibility base amae localization to cases where only output measurements are available. he funamental iea is that lack of eterministic information on the input can be partially compensate by knowlee of the structure of the mass matrix. In particular, when the inverse of the mass matrix in the coorinates efine by the sensors can be assume iaonal then matrices that iffer from the flexibility by a scalar multiplier can be assemble an these matrices can replace the flexibility with no loss of useful information. NONCLAUR U, D, D DLV nsi WSI K C Ψ Ψ m m n unamae flexibility matrix amae flexibility matrix chane in flexibility matrix amae locatin vector normalize stress inex weihte stress inex mass matrix stiffness matrix ampin matrix isplacement partition of the complex moes complex moes at sensor locations normalization constants number of output sensors number of moes INRODUCION When experimental ata is use to improve a mathematical moel the parameters that are caniates for upatin are typically selecte with uiance from a priori knowlee on what aspects of the moel are most uncertain. In the amae ientification problem, however, uncertainty or sensitivity can not be use as criteria to arrive at a set of free parameters. Since the use of a lare parameter space leas to ill-conitionin an non-uniqueness methos that can provie obective information about amae without the nee to refer to a etaile moel of the structure are of consierable practical importance. Amon the techniques that attempt to localize amae without reference to a moel, those that operate with chanes in moe shapes have receive sinificant attention [1,,3]. A ifficulty often face when usin moe shape chanes to locate amae, however, is the fact that the appropriate pairin of moes from the reference to the potentially amae-state is not always apparent. One way to et aroun this ifficulty is to use all the available moes to assemble flexibility matrices an then focus on the chane of these matrices [4,5]. he fact that the flexibility is ominate by the lower moes (which are typically the ones that can be ientifie experimentally) an that it can be assemble at whatever sensor coorinates are available are convenient features of the approach. An important limitation on the use of flexibility is the fact that these matrices can only be assemble from the ata when the input is measure [6,7]. Since in civil enineerin structures a full characterization of the input is often impractical, the flexibility-base amae localization has not been consiere as a viable option in builins or bries. his paper shows, however, that while it is true that flexibility matrices can
not be assemble from vibration ata for output only systems, matrices that iffer from the flexibility by a sinle scalar multiplier can be obtaine when some conitions prevail. Specifically, the paper shows that matrices that are proportional to the flexibility can be compute exclusively from the measure ata if the inverse of the mass matrix can be assume iaonal over the coorinates efine by the available sensors. he obective of this paper is to iscuss the issues associate with the computation of the flexibility proportional matrices an to illustrate how these matrices can be use in amae localization applications. he paper is oranize as follows. he first section presents a summary of a recently evelope technique for interroatin chanes in flexibility about amae localization. he technique, esinate as the Damae Locatin Vector Approach, (DLV) locates the amae by inspectin stress fiels create by vectors that are containe in the null space of the chane in flexibility [8]. he next section illustrates how moal orthoonality can be combine with knowlee of the structure of the inverse of the mass matrix to arrive at matrices that iffer from the flexibility by a sinle scalar multiplier. his evelopment is mae for a viscously ampe system but no limitation on the nature of the ampin (classical or not classical) is introuce. he theoretical part of the paper conclues with a iscussion on how to ensure that the missin scalar in the flexibility matrix, compute for the reference an the amae states, are essentially the same. A numerical example on a 4 DO system illustrates the techniques iscusse. H DLV CHNIQU he Damae Locatin Vector (DLV) approach provies a systematic way for interroatin chanes in flexibility matrices with respect to the localization of the amae. In this section the basic features of the technique are reviewe. A more etaile iscussion of the theoretical backroun, as well as iscussion on robustness an other issues may be foun in [8]. As shall be evient from the results in this section, a missin scalar multiplier in the flexibility matrices is immaterial in the DLV localization, provie the scalar can be mae the same in the reference an the amae states. he basic iea in the DLV approach is that the vectors that span the null-space of the chane in flexibility (from the unamae to the amae states) when treate as static loas on the system, lea to stress fiels that are zero over the amae elements. Depenin on the number an location of the sensors the intersection of the null stress reions ientifie by the DLVs may or may not exclusively contain amae elements. lements that are unamae but which cannot be theoretically iscriminate from the amae ones by chanes in flexibility (for a iven set of sensors) are esinate as inseparable. he steps of the DLV localization can be summarize as follows: 1. Compute the chane in flexibility as; D = U D (1). Obtain a sinular value ecomposition of D, namely; s1 D = U 0 0 s V where s are small sinular values. or ieal conitions s contains zeros an the DLV vectors are simply the columns of V associate with the null space. or the noisy conitions that prevail in practice, however, the values in s are never equal to zero an a cutoff has to be establishe to select the imension of the null space. he vectors in V that can be treate as DLVs for noisy conitions can be selecte as follows (see ref. [8] for the mathematical support) a) Compute the stresses in an unamae moel of the structure usin the columns in V as loas. b) Reuce the internal stresses in every element (for a iven loa vector) to a sinle characterizin stress, σ (strain enery per unit characterizin imension shoul be proportional to σ ). c) Desinate the reciprocal of the maximum value of the characterizin stress as c. Compute the svn inex for every vector in V as; where; q q () s c svn = (3) s c s q c q =max (s c ) for =1: m (4) he vectors for which svn 0.0 can be treate as DLVs. Once the set of DLV vectors has been ientifie the localization proper is carrie out as follows: 3. Compute, for each DLV vector, the normalize stress inex vector as;
nsi i σ i = (5) σ i max system has a full set of eienvectors (repeate eienvalues are permitte) the solutions to eq.10 can be oranize as; 4. Compute the vector of weihte stress inices, WSI, as; nlv nsi WSI = i = 1 svn nlv where svn i = max (svn i, 0. 015) an nlv is the number of DLV vectors. he potentially amae elements are those havin WSI <1. LXIBILIY ARICS O WIHIN A SCALAR ULIPLIR his section evelops expressions for the inverse of the mass an the stiffness (the flexibility) in terms of the eienvalues of the system an the corresponin eienvectors at the sensor coorinates. It is assume, for enerality, that the ampin is not classical so the moes are irreucibly complex. he expressions obtaine for -1 an K -1 are vali for a particular scalin of the moes which, as shall be shown, can only be enforce to within a missin scalar when the input is not measure. We bein by consierin the homoeneous equation for a time invariant finite-imensional linear system with viscous issipation, namely; i i (6) x& + Cx& + Kx = 0 (7) where, C an K R n x n are the mass, ampin an stiffness matrices. or a state vector efine usin isplacements an velocities the first orer form of eq.7 that preserves symmetry is; where, C =, 0 Assumin a solution eienvalue problem; y& = G y (8) K G = 0 λt 0, x y = (9a,b,c) x & y = φ e eq.8 leas to the φ λ = G φ (10) where Ψ = ΨΛ Ψ Λ, Λ = (11a,b) Ψ Λ Λ Φ Ψ = Φ = ΨΛ 1 1, an Λ = λ [ φ φ L ] λ O an the superscript stans for complex conuate. (1a,b) he symmetry of an G can be exploite to show that; Φ Φ rom where it follows that; G e Φ = (13) e G Φ = (14) = Φ = Φ e e Φ Φ (15) (16) quatin the inverse of the matrices an G in eq.9a an b with the results in eqs.15 an 16 one ets a number of equalities of which we list the ones that are relevant for our purposes: 1 = R ( ΨΛ Λ Ψ 1 ) (17) = R ( Ψ e Λ Ψ ) (18) K 1 = R ( Ψ Ψ ) (19) R ( Ψ Λ Ψ ) = 0 (0) R ( Ψ Ψ ) = 0 (1) rom eqs. 17 an 18 one conclues that; e Λ = e () an thus the previous expressions are reuce to; which, since the matrices are real, yiels real or complex conuate eienvalues λ. Assumin that the = R ( ΨΛ Ψ ) (4)
he iaonal matrix 1 K = R ( Ψ Ψ ) (5) R ( Ψ Λ Ψ ) = 0 (6) can be compute from the ata without knowlee of the system matrices in the eterministic input case. In the stochastic case, however, this is not possible an one must introuce some apriori knowlee to procee. Consier as an introuction the case where there is enouh information about the mass matrix to allow the computation of the partition of the inverse over the sensor coorinates. If there are m output sensors then one can use eqs.4 an 6 to set up m(m+1) equations to solve for the complex constants. Assumin all the moes are ientifie one can easily show that the number of sensors require to ientify the normalizin constants has to satisfy m 0. 5( 8n + 1 1) where n is the number of moes (orer is n). or example, if there are 10 moes in the system the normalizin constants can be obtaine from knowlee of the inverse of the mass if m 4. Note that if only a truncate moal space is available then the approach woul lea to some approximation in the constants because one woul be equatin a convere physical quantity (the left sie of eq.4) with a moally truncate approximation. In this paper we purse a less restrictive assumption than that of presumin complete knowlee of -1. In particular, we procee by assumin simply that -1 is iaonal. It is evient, of course, that what we re oin is sayin that we know the off-iaonal part of -1 to be zero. In this case we lose m equations an, because the equations are now all equal to zero we can compute the normalizin factors only up to r unetermine constants, where r is the nullity of the resultin coefficient matrix. It is possible to show that a necessary conition for the nullity to be one (the smallest it can be) is that the number of sensors satisfy m n. So, for n =10, for example, one nees at least 5 sensors. he comment mae previously with respect to equatin a physical quantity (in this case the zeros in the off iaonals of -1 ) to a truncate approximation hols without moification in this case also. he specifics use to arrive at the normalization constants in the case where -1 is assume iaonal are escribe next. Define; H R I m, m, + H i = Ψ Ψ Λ (7) q.4, therefore, can be written in the omain of real numbers as = n = 1 ( H H ) (8) R, R, I, I, where R an I are the real an the imainary components of. akin the upper (lower) trianular portion of -1 (without the main iaonal) an placin it in vector form one can write n 0 = ( Hˆ ˆ R, R, HI, I, = 1 ) (9) where the orer of the entries in the vectors Ĥ is arbitrary as lon as one is consistent in efinin Ĥ R, an Ĥ I,. We now efine the vector β as all the real components of followe by all the imainary components, namely [ R, 1 R, L I, 1 I, L ] β = (30) With the precein notation eq.9 can be written as; where; H β = 0 (31) = [ Ĥ 1,Ĥ,... Ĥ 1, Ĥ... ] (3) H R, R, I, I, It follows then that β (which contains the require normalizin constants orere as per eq.30) is in the null space of the matrix H. One can further restrict the subspace that contains β by takin avantae of the relationship in eq.6. Inee, followin the same approach use to pass from eq.9 to eq.31 one fins that eq.6 ives; S β = 0 (33) where the only ifference is that the assembly of S inclues the iaonal of the matrix in eq.6. Combinin eqs.31 an 33 one ets; H β = Y β = 0 S (34) So β is in the null space of Y. If all the moes are available, the nullity of Y is one an β can be compute to within a sinle scalar. he complex constants (to within a scalar) are then iven by eq.30. an the flexibility proportional matrix is evaluate with eq.5. In practice, of course, one
selom obtains all the moes an, as a result, the matrix Y in eq.34 proves to be full rank. An approximate solution can be obtaine by takin β as the sinular vector associate with the smallest sinular value of Y. COPAIBILIY O H SCALAR ULIPLIR he scalar that is missin in the flexibility matrices of the previous section is arbitrary an is not necessarily the same for the unamae an the amae states. In orer to preserve the null space when takin the ifference of the flexibility proportional matrices it is necessary to ensure that the missin constant is consistent. wo proceures have been examine thus far for ensurin compatibility. he first one is base on the iea that the mass matrix has not chane as a result of the amae. If this is true an the moal space is complete then the scalin factor can be auste so that the inverse of the mass in the two states (as iven by eq.8) are the same. Of course, in practice approximations are inevitable an one can not preten that it will be possible to make the two -1 expressions ientical. One can, however, efine a norm an aust the scalin to make this norm the same in the two states. When all the moes are not available the contribution of the available moes to -1 may iffer in the two states an one can not arue that the error in makin the missin scalar compatible erives exclusively from roun off an imprecision. hus far we have looke at two proceures for attainin approximate compatibility between the flexibility proportional matrices of the reference an the amae states. he first approach uses the trace of the -1 as the metric that shoul be equal in the two states an the secon, which oes not use information in -1, is outline next. Assume that the unamae an amae flexibility proportional matrices are U an D while the true moally truncate but, unetermine matrices, are U an D. One can then write; an L U L D = α (37) = η (38) Solvin for L in eq.37 an substitutin the result in eq.38 one ets; U D η = α (39) which shows that the esire η/α ratio is a real eienvalue of the matrix on the left sie. Since there are several real eienvalues, however, the solution oes not immeiately point out which is the correct result. An approach that helps in reucin ambiuity is to first normalize the matrices an in such a way that the η/α ratio is necessarily D U less than 1. his is the case, for example, if the flexibility proportional matrices are normalize to equal trace because the true value of the trace in the amae flexibility is larer (assumin that the effect of amae is more than the inevitable error in the computations). If this scalin is first introuce then one can iscar not only the η/α ratios that are complex but also those that are reater than 1. Unfortunately, there appear to be cases where the solution is still not unique after this is one. While research on the compatibility of the scalar continues, the limite numerical experience aine thus far suests that the simple approach base on the trace of -1 may be sufficiently accurate. NURICAL XAPL he basic steps of amae localization usin the DLV approach for unmeasure input are illustrate usin the simple system shown in fi.1. oal truncation an error in the ientification are not contemplate in this example. an = α (35) U U k 1 =000 1 3 k =1000 k 3 =3000 k 4 =4000 = η (36) D D where α an η are unetermine constants. Assume L is a vector in the effective null space of the moally truncate true chane in flexibility. Reconizin that u L DL one can write (replacin the with = for simplicity); c 1 =40 m 1 m m 3 m 4 c 1 =30 c 1 =0 c 1 =10 k 1 =1500 (after amae) (m 1 =0, m =4, m 3 =8, m 4 =3) iure 1 System consiere
Output sensors exist at coorinates 1, an 3 an amae is simulate as 5% reuction in the stiffness of sprin k 1. he first step is to extract arbitrarily scale complex moe shapes Ψ m at the sensor coorinates. his can be one usin any suitable stochastic ientification alorithm [9]. or the system in fi.1 the noise free results in the unamae state are; -0.04+0.058i -0.059-0.030i -0.047+0.033i -0.001-0.006i -0.067+0.167i -0.035-0.036i 0.00-0.033i 0.018+0.016i -0.084+0.04i 0.09+0.07i -0.01+0.019i 0.018+0.009i he secon step is to obtain the normalization constants from eq.34. his requires that one assemble H an S. he computations for the first moe are illustrate in the followin. Assemblin H 1 from eq.7 one ets; 0.017+0.00i 0.051+0.057i 0.06+0.071i H 1 = 0.051+0.057i 0.149+0.161i 0.180+0.199i 0.06+0.071i 0.180+0.199i 0.19+0.47i he upper trianular part of the matrix (without the main iaonal) contains, in this case, 3 numbers. he real an the imainary components of these numbers are use to form the vectors H 1an H 1, namely; Ĥ R,1 = ˆ R, ˆ I, 0.051 0.057 0.06 Ĥ I,1 = 0.071 0.180 0.199 where the orer selecte is arbitrary. Repeatin these steps for all the moes one obtains the vectors neee to form H (see eq.3). he result is; 0.051-0.07 0.073-0.034-0.057 0.381 0.399-0.041 0.06 0.071-0.047-0.018-0.071-0.99-0.9-0.040 0.180-0.01-0.031-0.044-0.199-0.34 0.154 0.174 where the columns of H corresponin to the first moe are hihlihte. he same approach is repeate to form S. One ets; 0.008-0.007i 0.01-0.00i 0.06-0.05i S 1 = 0.01-0.00i 0.060-0.059i 0.074-0.07i 0.06-0.05i 0.074-0.07i 0.09-0.087i Placin the results in vector form ives; 0.008-0.007 0.01-0.00 Ŝ 0.060-0.059 R,1 = Ŝ I,1 = 0.06-0.05 0.074-0.07 0.09-0.087 where we note that the main iaonal is also inclue since it is also zero accorin to eq.6. he matrix S is assemble by repeatin the process for all the moes, the results is; 0.008-0.039 0.040-0.000 0.007-0.05-0.00 0.001 0.01-0.035-0.030 0.00 0.00-0.009 0.00-0.00 0.060-0.07 0.00-0.011 0.059 0.00 0.007-0.000 0.06 0.07 0.017 0.00 0.05 0.008-0.001-0.001 0.074 0.0-0.011-0.009 0.07-0.001-0.004-0.003 0.09-0.017 0.006-0.006 0.087-0.000 0.00-0.004 Combinin H an S one obtains the matrix Y. rom inspection of the sinular values, s, it is evient that the nulity is one. he sinular values s an the nullspace β are shown below. 0.684-0.33 0.43 0.1191 0.191 0.0743 s = 0.117 β = 0.1491 0.073 0.3386 0.053-0.5149 0.009 0.5603 7.18-17 -0.3999 rom β one reaily ets the complex constants, from eq.30. he inverse of the mass an stiffness matrices can then be calculate from eqs.4 an 5. One ets; U = α 0.41 0.00 0.00 0.004 0.004 0.004 0.00 0.34 0.00 K U = α 0.004 0.01 0.01 0.00 0.00 0.5 0.004 0.01 0.017 he exact values for these matrices are; 0.050 0.000 0.000 0.050 0.050 0.050 e -1 = 0.000 0.04 0.000 K e -1 = 0.050 0.150 0.150 0.000 0.000 0.031 0.050 0.150 0.08 x10- from where it can be seen that the ifference is a simple scalar multiplier.
ollowin the same steps outline previously the system matrices extracte for the structure in the amae state are; D = α 0.41 0.00 0.00 0.005 0.005 0.005 0.00 0.34 0.00 K D = α 0.005 0.014 0.014 0.00 0.00 0.5 0.005 0.014 0.018 where the trace of -1 has been normalize to unity in both cases to ensure compatibilty of the missin scalar. Once the flexibility proportional matrices are calculate within a common multiplier, the SVD of the chane in these matrices yiels the sinular values an riht sie sinular vectors shown. 0.0041 0.58-0.81 0.1 s= 5.41x10-17 V = 0.58 0.31-0.76 3.44x10-18 0.58 0.50 0.64 Inspection of the sinular values shows that there are DLVs. Applyin the two vectors ientifie as DLVs as loas at the sensor coorinates one ets the axial force istribution N shown in fi. N N k 1 k k 3 k 4 m 1 m m 3 m 4 DLV1 c 1 c c 3 c 4 0.64 0.64 0 DLV 0 0.1 0.80 0.50 iure Axial force istribution 0.50 Both vectors, as expecte, locate the amae correctly at the first sprin. CONCLUSIONS his paper extens the DLV amae localization technique to the important case where the input can not be eterministically characterize. his is one by showin that flexibility proportional matrices can be extracte from the ata without knowlee of the input an that the unetermine missin scalar can be mae consistent for the reference an the amae states. he paper shows that computation of the flexibility proportional matrix is theoretically exact only when the moal basis is complete. xamination of the relationships involve suests, however, that oo approximations of the truncate flexibility shoul result when the truncate space provies a reasonable approximation for -1. he experience erive from numerical experiments has thus far supporte this expectation. RRNCS [1] West, W.., "Illustration of the use of moal assurance criterion to etect structural chanes in an orbiter test specimen." Proc. Air orce Conference on Aircraft Structural Interity, pp.1-6, 1984. [] ox, C. H. J., "he location of efects in structures: A comparison of the use of natural frequency an moe shape ata." Proc. of the 10th International oal Analysis Conference, pp.5-58, 199. [3] Salawu, O. S. an Williams, C., "Damae location usin vibration moe shapes." Proc. of the 1 th International oal Analysis Conference, pp. 933-939, 1994. [4] Doeblin, S. C., arrar, C. R., Prime,. B. an Schevitz, D. W., Damae ientification an health monitorin of structural an mechanical systems from chanes in their vibration characteristics: a literature review, Los Alamos, New exico, 1996. [5] Paney, A. K. an Biswas,., "Damae etection in structures usin chanes in flexibility." Journal of Soun an Vibration, Vol. 169, No.1, pp.3-17, 1994. [6] Bernal, D., "xtractin flexibility matrices from state-space realizations." COS 3 Conference, ari, Spain, Vol.1, pp.17-135, 000. [7] Alvin, K.. an Park, K. C., "Secon-orer structural ientification proceure via state-spacebase system ientification", AIAA Journal, Vol.3, No., pp.397-406, 1994. [8] Bernal, D., "Loa Vectors for Damae Localization", to appear in Journal of nineerin echanics, ASC, January 00. [9] Van Overschee, P., an oor B. L. R., Subspace Ientification for Linear Systems: heory, Implementation, Applications, Kluwer Acaemic Publishers, Boston, 1996.