A NEW APPROACH TO OPERATIONAL MODAL ANALYSIS BASED ON MULTIVARIABLE TRANSMISSIBILITY FUNCTIONS

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1 A NEW APPROACH TO OPERATIONAL MODAL ANALYSIS BASED ON MULTIVARIABLE TRANSMISSIBILITY FUNCTIONS Wut Weijtjen 1, Gert de Sitter 2, Chrit Devriendt 3, Patrick Guillaume 4 ABSTRACT Tranmiibility baed peratinal mdal analyi (TOMA) i an alternative apprach t OMA uing pwer pectral denitie. It de n lnger aume the peratinal rce t behave a white nie and therer i unaected by harmnic r clring preent in the input pectrum. The irt verin TOMA allwed r identiicatin in the preence a ingle, pibly ditributed, dminant input rce. The current tate the art, plyreerence-toma (p-toma), can al be ued with multiple, independent, rce but require an increaed number lading cnditin. In thi article, a new apprach t p-toma will be intrduced. Thi new methd can unctin tarting rm tw lading cnditin. Furthermre, a new methd r detecting the number required reerence will be preented. Bth algrithm will be illutrated by mean a numerical experiment. Keywrd: Tranmiibility unctin, Harmnic, p-toma, Mdal Parameter Etimatin 1. Intrductin Operatinal Mdal Analyi (OMA) i a ield reearch with an increaingly large number applicatin and uccee. Hwever, the inability t meaure the input rce i nt nly the rean OMA wa develped, it can al becme the larget caue errr. Firt iteratin OMA technique, made peciic aumptin cnidering the nature the input rce. Fr intance [1 3], aume the input rce behave a white nie. While in a igniicant number applicatin thi aumptin i met, it i al ten vilated. The vilatin the white nie aumptin ten intrduce igniicant errr upn the etimated mdal parameter. Current lutin therere try t extend the white nie aumptin t allw r (varying) harmnic [4,] in the input pectrum. Other lutin prpe t remve the harmnic by angle ynchrnu averaging, [6], r t remve the rahmnic in the repne ceptrum, [7]. 1 ir. Wut Weijtjen, Vrije Univeriteit Bruel, wweijtje@vub.ac.be 2 ir. Gert de Sitter, Vrije Univeriteit Bruel, gdeitte@vub.ac.be 3 dr. ir. Chrit Devriendt, Vrije Univeriteit Bruel, cdevrien@vub.ac.be 4 pr. dr. ir. Patrick Guillaume, Vrije Univeriteit Bruel, paguilla@vub.ac.be

2 A dierent apprach i t ue Tranmiibility baed OMA (TOMA), [8, 9]. TOMA ue tranmiibility unctin btained ver dierent peratinal cnditin, a.k.a. lading cnditin, a a tarting pint. Tranmiibility unctin, unlike pwer pectral denitie, can becme independent the input pectrum and can therere be ued independent the input pectrum. Thank t thi prperty, tranmiibility unctin are al unaected by the niy behavir the input pectrum. Firt verin TOMA technique aumed nly a ingle dminant rce [8] and required tw dierent lading cnditin. Yet, when multiple, independent, rce excite the tructure, the methd i again dependent the input pectrum. Thi limitatin wa relved by the intrductin the plyreerence apprach [], but thi technique required at leat three lading cnditin. In thi article, an advanced way uing plyreerence-tranmiibility unctin r OMA (p-toma) will be hwn. Thi newet p-toma apprach nly require tw dierent lading cnditin and will therere be eaier t apply. Furthermre, a imple tl t identiy the number required reerence will be intrduced. 2. Tranmiibility baed peratinal mdal analyi (TOMA) 2.1. Prpertie the tranmiibility unctin Cnider a linear time invariant tructure with traner unctin H() C N N i a deined in eq. (1). F 1 () F 2 () X() = H() = H() F () (1). F N () Nw cnider that nly in a dicrete number (N ) input lcatin (K) a rce i applied: { } [ ] XL () HLK () = F X R () H RK () K () (2) In which X R () C Nr and X L () C N Nr are repectively the reerence and nn-reerence ubet a meaured requency dmain repne vectr X() C N. And H LK () C (N Nr) N, H RK () C Nr N are deined by X R () = H RK () F K () and X L () = H LK () F L (). Tranmiibility unctin, T () C (N Nr) Nr, were irt deined in [11] a llw: In [11] the llwing intereting prperty i hwn: X L () = T () X R () (3) T () = H LK ()H 1 RK () (4) In eence thi rmula tate that i the invere H RK () exit, when N r i chen equal t N, then the tranmiibility unctin becme lely dependent the ytem traner unctin H(). Thi mean T () i independent the pectral cntent the rce F K (). Thi imprtant prperty allw tranmiibility unctin t becme unaected by harmnic r clring in the rce pectrum r the niy rce pectrum itel. In [12] Eq.(3) wa extended t allw r multiple (N ) independent rce ditributed ver all (N i ) input lcatin. T l () = H L ()F l (H R ()F l ) 1 () in which the input rce are cnidered t behave a llw: F ditributed,l () = N i=1 F i,l µ i,l () = F l µ l () (6)

3 with N the number cnidered urce and ubcript l indicating a lading cnditin l. The rce are cnidered a a linear cmbinatin independent input urce, µ i,l (), which decribe the pectral cntent the individual rce. The ditributin ver the tructure i quantiied uing the (amplitude) ditributin vectr F i,l. All input urce can then be cmbined in the urce vectr, µ l (), and all ditributin vectr are cmbined in the ditributin matrix, F l. Eq. (6), imilar t Eq. (4), i independent µ l (), and therere unaected by the pectral cntent the excitatin rce. The dierence between the dicrete rce lcatin and the ditributed rce i illutrated in Fig.1. F m () F n () X 1 () X 2 ()... X N () F 2 µ 2 () F 1 µ 1 () X 1 () X 2 ()... X N () Figure 1 Illutratin the dierence between (let) dicrete input (i.e. Eq. (4)) and (right) the mre general cae ditributed input (i.e. Eq. ()). Bth cae have N = 2, and ne can eaily recgnize the (let) being a particular cae the ditributed input (right) In the cntinuatin thi article Eq. () will be ued, but recgnize that Eq. (4) i a particular cae Eq. () 2.2. A brie hitry (p-)toma TOMA wa irt decribed in [8] r the particular cae a ingle dminant excitatin urce, i.e. N = 1. It wa hwn that tw ingle reerence, tranmiibility unctin, N r = 1, btained under tw dierent lading cnditin, interect in the ytem ple, Fig.2.a. Yet, in practice the aumptin a ingle dminant excitatin urce i nt alway met and an expanin TOMA r multiple independent excitatin urce wa neceary. While TOMA till wrk even i the number urce exceed ne, the majr advantage TOMA i lt. It i illutrated in Fig.2.b that i N exceed the number chen reerence, N r, the tranmiibility unctin i again inluenced by the input pectrum. The lutin wuld be t ue plyreerence tranmiibility unctin, with N r = N, in rder t again btain determinitic tranmiibility unctin, Fig.2.c. Hwever, unlike ingle reerence tranmiibility unctin, plyreerence tranmiibility unctin d nt interect in the ytem ple. Thi implie that the baic idea behind TOMA wa nt applicable t plyreence tranmiibility unctin and a new algrithm wa neceary t expand TOMA t multiple reerence. Thi led t p-toma, which wa intrduced in []. The irt verin p-toma allwed r mdal parameter etimatin uing the plyreerence tranmiibility unctin. But it required at leat three lading cnditin t unctin. Thi limit the applicability the prped methd. A new lading cnditin i any igniicant change in the rce ditributin matrix, F l. Thi matrix change under dierent peratinal cnditin, e.g. dierent wind directin, etc. A t cnider at leat three dierent lading cnditin implie that ne ha t wait until thee three lading cnditin have ccurred and thi might igniicantly increae the meaurement time. 3. The recently develped advanced p-toma Thi ectin will give an inight in the general idea behind the mt recent algrithm r ply-reerence TOMA (p-toma), a practical implementatin i ummarized in Sectin. A mre extenive dicuin

4 Amplitude Amplitude Amplitude ω (a) N = N r = ω (b) N = 2, N r = ω (c) N = N r = 2 Figure 2 While tw ingle reerence tranmiibility unctin btained under tw dierent lading cnditin alway interect in the ytem ple, (a) and (b). The determinitic behavir i lt nce the number input urce exceed ne. On the ther ide, plyreerence tranmiibility unctin remain determinitic but d n lnger interect in the ytem ple. (c) Vertical dtted line indicate the lcatin the renance requencie. i prvided in [12]. The main prperty making TOMA pible i the act that in the ytem ple tranmiibility unctin becme independent the input ditributin, i.e. F l. Fr intance tw ingle-reerence tranmiibility unctin btained ver tw lading cnditin, will interect at the ytem ple. The plyreerence equivalent i readily und: lim T l () X R,l () = lim XL,l () (7) λ m λ m in which ubcript l indicating the l th lading cnditin. Making ue the prpertie the mdal mdel T l (λ m ) φ R,m = φ L,m (8) Thi equatin actually tate that in the ytem ple, λ m, the tranmiibility unctin alway relate the reerence part the aciated mde hape, φ R,m, t it nn-reerence part, φ L,m. Sme rearranging yield llwing exprein [T l (λ m ), I (N N r)] φ m = (9) With I (N N r) the quare (N N r ) dimeninal identity matrix. S at the ytem ple the rw [T l (λ m ), I (N N r)] are alway rthgnal t the aciated mde hape vectr φ m = { φ T R,m, φ T L,m }T. Mrever, Eq. (9) i independent the input, a the repne vectr i n lnger preent. Thi i nt the cae r the mre general exprein, utide the ytem ple: [T l (), I (N N r)] X l () = T l () X l () = () I the previu tep are repeated r N l dierent lading cnditin, then the ytem ple are und a thee value r which a vectr can be und rthgnal t all rw the N l btained intance T l (). Thi i illutrated in Fig.3. Nte that thee lutin are nly unique when the cnidered lading cnditin were uiciently dierent. I thi wuld nt be the cae ther vectr, beide the mde hape vectr, will exit that are al rthgnal t all cnidered rw vectr. Thi iue will al ccur i an inuicient number lading cnditin are cnidered. 4. Detecting the number required reerence The number required reerence can be determined uing lely the repne meaurement btained during a ingle lading cnditin, X() = X () + η(), with the true ytem repne, X (), and

5 uncrrelated N (, σ 2 )-ditributed meaurement nie, η(). The ubcript l i drpped r readability. The repne-autpwer can be calculated a: G XX () = 1 N X n () X H n () = 1 N ( ) ( ) H X,n () + η n () X,n () + η n () (11) which r a uiciently large number average, N, reduce t G XX () = G X X () + 1 N η n () η n H () (12) r which: G X X () = 1 N = 1 N X,n () X,n() H (13) H() F n () F n H ()H H () (14) = H() 1 N ( Fn () F H n ()) H H () (1) = H()G F F ()H H () (16) Due t the prperty rank(h()g F F ()H H ()) min{rank(h()), rank(g F F ())} = min{n i, N, N } (17) the rank G X X () will be N, the number uncrrelated urce. A N N i, i.e. ne can nt have mre rce acting n the tructure than the number input lcatin. Furthermre, N i aumed le than N, which i a neceary cnditin r TOMA. Hwever, t the preence nie the rank G XX () will nt be equal t N, a 1 N N η n() η H n () i rank N. Nnethele, the herein derived prperty can be ued t detect the number required reerence, N r = N, uing the ingular value, σ 1 () σ 2 ()... σ N (), G XX (). Thi i due t act that σ N () i linked t the true ytem repne while σ N+1() i linked t the nie level. In avrable cnditin the nie level huld be igniicantly maller than the ytem repne level and σ N () i a ew rder larger than σ N+1(). A igniicant dierence in tw cnecutive ingular value can therere be ued t identiy N, thi i illutrated in Sectin 6.. Hwever, whenever the nie level becme the ame magnitude the mallet repne level, thi prped methd will n lnger unctin. X1 X 2 T 2 T 1 φ m X1 X 2 T 2 T1 (a) λ m (b) = λ m Figure 3 Illutratin the prpertie T () with N = 2, making ue tw dierent lading cnditin in (a) X 1, X2 and cnequently T 1 and T 2 are till dependent the input ditributin, in (b) r = λ m thi dependency i lt and T 1 and T 2 will alway be rthgnal t the ame vectr.

6 . Implementatin The llwing ectin i a ummary the entire algrithm. Mre detail are prvided in [12]. 1. Meaure the ytem repne, X l (ω), uing a et N vibratinal enr, the ytem excited by unknwn input 2. Select the number reerence (N r ) r intance a uggeted in Sectin Select the reerence ignal X R,l (ω) and nn-reerence repne ignal X L,l (ω). 4. Ue a Let Matrix Fractin Mdel (LMFD) t decribe the tranmiibility unctin in a parametric way: T l (Ω) = D 1 l (Ω)N l (Ω) (18) with D l (Ω k ) = n D i= A i,l Ω i k N l (Ω k ) = n N i= B i,l Ω i k (19) and etimate the matrix ceicient [A i ], [B i ], r the uer-deined plynmial rder n D, n N, in the preerred dmain e.g. Laplace dmain (cntinuu-time), Ω k = jω k, r a a Z-dmain (dicrete-time), Ω k = e jω kt. Cntruct T l (Ω) r the data linked t ne lading cnditin. T l (Ω k ) = n max i= [ Ai,1, B i,1 ] Ωk (2) 6. Repeat the previu tep r N l lading cnditin, until a cmplete et lading cnditin i btained. 7. Cntruct the cmbined matrix Υ(Ω) and lve the plynmial eigenvalue prblem: T 1 (λ m ) A i,1, B i,1 Υ(λ m ) φ T 2 (λ m ) m = φ. n max A i,2, B i,2 m = i=. λi m φ m = (21) T Nl (λ m ) A i,nl, B i,nl 8. The ytem ple and mde hape are nw und within the et lutin the plynmial eigenvalue prblem, Eq.(21) 9. T dicriminate between phyical and nn-phyical ple within the et lutin ne can ue cmmnly ued criteria uch a r example: exceive damping rati ( ξ > %), untable ple (ξ ) and pure mathematical ple with ininite cmpnent. Other pibilitie are the cntructin a tabilizatin diagram and clutering algrithm. [13] 6. Numerical example 6.1. Prblem decriptin A numerical example i perrmed in the requency dmain, withut the preence leakage, uing a 4-DOF mechanical ytem a depicted in Fig.4. With renant requencie and damping rati hwn in Table1. The ytem i excited in all DOF by N = 2 ditributed input urce, with F l () = F 1,l µ 1 () + F 2,l µ 2 (). Bth urce are in act (band limited) white nie equence, with clring uperped n tp. The clring wa btained a the ytem repne a SDOF ytem t a (band limited) white nie input. The ple thee SDOF ytem are al prvided in table 1. Thee were imulated by varying the ditributin the tw rce, ee Table2. A mall amunt meaurement nie wa added t the imulated data.

7 Figure 4 The 4DOF mdel ued in the imulatin (m=1, c=., k=) Table 1 Renance requencie (Hz) and damping rati (%) the mdeled ytem. In additin the ple the input ignal are prvided (withut numbering) N re (Hz) ξ(%) Table 2 Lad ditributin vectr in the tw lading cnditin DOF Firt lading cnditin F1, (l = 1) F2, Secnd lading cnditin F1, (l = 2) F2,

8 6.2. Reult In Fig.(a) a clear gap i viible between the ecnd and third ingular value, crrectly indicating the need r tw reerence repne, i.e. N = 2. In thi the identiicatin the matrix ceicient Ai and Bi wa perrmed uing a leat quare apprach in which AnD wa cntrained t a (N Nr )dimeninal identity matrix. The etimatin cnider 6144 requency line and a dicrete time mdel, Ωk = e jωk T, wa ued in rder t btain a well cnditined bervatin matrice [14]. The quality the it, i.e. the reult tep 4 Sectin., i illutrated in Fig.(b). A cmparin a p-lscf etimatr Frequency(Hz) 2 Frequency(Hz) Frequency(Hz) T22 8 T21 Singular Value 2 T12 T Frequency (Hz) 12 Frequency(Hz) (a) (b) Figure (a) The ingular value GXX () hw a clear gap between the ecnd and third ingular value indicating that N = 2. (b) Illutratin the parametrically etimated (red) tranmiibility unctin n tp the nn-parametrically etimated (blue) tranmiibility unctin during ne lading cnditin wa ued n the btained GXX (), with N = 4, uing the ame amunt data. The crrepnding tabilizatin diagram i given in Fig.6(a). In Fig.6(b) the tabilizatin diagram r p-toma i pltted. In Stabilizatin Chart Stabilizatin Chart mdel rder mdel rder Frequency (Hz) (a) Crpwer baed p-lscf Frequency (Hz) (b) p-toma Figure 6 Stabilizatin diagram illutrate that the claic apprach (a) ha iue with the harmnic in the vicinity the irt mde. While the (b) p-toma apprach i unaected. thee igure a clear ditinctin i viible, while the ame amunt data i cnidered in bth algrithm p-toma clearly i le inluenced by the preence the harmnic. While in the cae the cr pwer nly the mde urthet away rm the harmnic reult in a clear table line. The btained reult r the prped p-toma apprach cincide nicely with the theretical value, Table 3.

9 Table 3 Reult p-toma N re (Hz) ξ(%) Cncluin In the intrductin thi article a hrt dicuin wa held abut the prpertie tranmiibility unctin. Then a mall recapitulatin n the tate the art in tranmiibility baed Operatinal Mdal Analyi (TOMA) wa given and it current limitatin were mentined. Thee are relved in the newly develped p-toma apprach, by uing ply-reerence tranmiibility unctin btained in tw lading cnditin. The newly prped methd tart rm etimating the matrix ceicient a mdel baed upn the input pectrum independent tranmiibility unctin. Multiple etimate the matrix ceicient are btained under dierent lading cnditin. When prperly cmbined the ytem ple and mde hape are btained a the lutin a plynmial eigenvalue prblem. A imulated pr cncept illutrate the applicability the prped technique and hwed it independence rm the pectral cntent the input. 8. Acknwledgement The inancial upprt the Intitute r the Prmtin Innvatin by Science and Technlgy in Flander (IWT) and the Reearch Cuncil (OZR) Vrije Univeriteit Bruel (VUB) are grateully acknwledged. Reerence [1] R. Brincker, L. Zhang, and P. Anderen. Mdal identiicatin utput-nly ytem uing requency dmain decmpitin. Smart Material & Structure, (3):441 44, June 21. [2] B. Peeter and G. de Reck. Reerence-baed tchatic ubpace identiicatin r utput-nly mdal analyi. Mechanical Sytem and Signal Prceing, 13(6):8 878, [3] E. Parl. Applicatin requency-dmain ytem identiicatin technique in the ield peratinal mdal analyi. PhD thei, Vrije Univeriteit Bruel, 23. [4] R. Pinteln, B. Peeter, and P. Guillaume. Cntinuu-time peratinal mdal analyi in the preence harmnic diturbance. Mechanical Sytem and Signal Prceing, 22():17 3, 28. [] R. Pinteln, B. Peeter, and P. Guillaume. Cntinuu-time peratinal mdal analyi in the preence harmnic diturbance-the multivariate cae. Mechanical Sytem and Signal Prceing, 24(1):9, January 2. [6] B. Peeter, B. Crneli, K. Janen, and H. Van der Auweraer. Remving diturbing harmnic in peratinal mdal analyi. In Prceeding Internatinal Operatinal Mdal Analyi Cnerence, Cpenhagen, Denmark, 27. [7] R.B. Randall, B. Peeter, J. Antni, and S. Manzat. New ceptral methd ignal pre-prceing r peratinal mdal analyi. In Prceeding ISMA 212, 212. [8] C. Devriendt and P. Guillaume. The ue tranmiibility meaurement in utput-nly mdal analyi. Mechanical Sytem and Signal Prceing, 21(7): , 27.

10 [9] W. Weijtjen, G. De Sitter, C. Devriendt, and P. Guillaume. Tranmiibility baed peratinal mdal analyi: n the ue the peud invere apprach. In Prceeding ISMA 212, Leuven, 212. [] C. Devriendt, G. De Sitter, and P. Guillaume. An peratinal mdal analyi apprach baed n parametrically identiied multivariable tranmiibilitie. Mechanical Sytem and Signal Prceing, 24():12 129, July 2. [11] N.M.M. Maia, J.M.M. Silva, and A.M.R. Ribeir. The tranmiibility cncept in multi-degree-reedm ytem. Mechanical Sytem and Signal Prceing, 1(1): , January 21. [12] W. Weijtjen, G. De Sitter, C. Devriendt, and P. Guillaume. Operatinal mdal analyi baed in multivariable tranmiibility unctin: Reviited. In Prceeding IMAC XXXI, 213. [13] G. De Sitter, C. Devriendt, and P. Guillaume. Tranmiibility-baed peratinal mdal analyi: Enhanced tabiliatin diagram. Shck and Vibratin, 19():8 97, 212. [14] P. Verbven. Frequency dmain ytem identiicatin r mdal analyi. PhD thei, Vrije Univeriteit Bruel, 22.