Implementaton of Complex Frequency Mappng to Low Order Frequency Doman algorthm for Operatonal Modal Analyss S. Chauhan, R. Martell, R.J. Allemang, D.L. Brown, Structural Dynamcs Research Laboratory Unversty of Cncnnat, Cncnnat, OH Emal: chauhas@emal.uc.edu, martelrf@emal.uc.edu, Randy.Allemang@uc.edu, Davd.L.Brown@uc.edu Nomenclature t, ω, z, s Tme, Frequency, Z, Laplace H(*) Frequency Response Functon Matrx N o Number of output degrees of freedom N Number of nput/reference degrees of freedom G FF (*) Input Force Power Spectrum G XX (*) Output Response Power Spectrum G + Postve power spectrum α, β Polynomal Coeffcents R, S Resdue Matrces λ Modal Frequency OMA Operatonal Modal Analyss RFP Ratonal Fracton Polynomal LSCF Least Squares Complex Frequency UMPA Unfed Matrx Polynomal Approach UMPA-LOFD UMPA Low Order Frequency Doman Abstract The use of complex frequency mappng n the polyreference least squares complex frequency (PolyMAX) algorthm demonstrated how t s possble to have better numercal characterstcs wth hgh order frequency doman based methods. The concept s appled successfully not only to other tradtonal, frequency doman expermental modal analyss methods but also n the feld of operatonal modal analyss. In ths paper, complex frequency mappng s used wth a low order frequency doman algorthm for the purpose of operatonal modal analyss. Its performance s evaluated and compared wth the low order frequency doman algorthm (UMPA- LOFD) whch does not use the complex frequency mappng. 1. Introducton The lac of frequency doman algorthms n Operatonal Modal Analyss (OMA) can be attrbuted to the nherent poor numercal characterstcs of frequency doman algorthms, specally n case of hgher order algorthms such as Ratonal Fracton Polynomal (RFP) algorthm [1]. Lmtng the frequency range and reducng the order of the model, normalzng the frequency range and usng orthogonal polynomals are some of the methods to reduce ths ll-condtonng problem. A major contrbuton towards mprovng the numercal characterstcs of the hgh order algorthms s the polyreference least squares complex frequency (PolyMAX) algorthm [2-4] that demonstrated the use of complex frequency mappng to enhance the numercal characterstcs. In [5], a low order mplementaton of
polyreference least squares complex frequency (PolyMAX) algorthm was presented that compared the tradtonal low order frequency doman algorthm to a complex Z mappng varaton. Recently a low order frequency doman algorthm (UMPA-LOFD) was developed for OMA usng Unfed Matrx Polynomal Approach (UMPA) [6]. Ths algorthm was shown to have good numercal characterstcs n comparson to hgh order frequency algorthm le RFP and results comparable to commonly used tme doman algorthms n OMA. Ths paper explores the possbltes of utlzng the complex Z mappng of generalzed frequency varable for the purpose of the UMPA-LOFD algorthm. The paper frst dscusses the UMPA-LOFD algorthm and the complex Z mappng. The UMPA-LOFD algorthm and ts complex Z mappng varaton are then appled to data collected over an analytcal system and a lghtly damped crcular plate. The obtaned results are compared to analyze the effectveness and performance of complex Z mappng n case of LOFD-OMA algorthm. 2. Theoretcal Bacground 2.1 UMPA LOFD Algorthm Ths secton dscusses the mathematcal formulaton of UMPA-LOFD algorthm [6]. Unfed Matrx Polynomal Approach or UMPA s essentally a general matrx polynomal model concept that recognzes that both tme and frequency doman models generate smlar matrx polynomal models [7, 8]. For a general multple nput, multple output case, the UMPA model n frequency doman s gven as m n ( j ) [ α ] [ H ( ω) ] = ( jω) [ β ] = 0 ω (1) = 0 where H s the frequency response functon matrx and α and β are matrx polynomal coeffcents. For order m = 2, Eq. 1 can be wrtten as 2 [ α ]( j ω ) [ α ]( jω ) + [ α ] [ H( ω )] = [ β ]( jω ) + [ ] 2 + 1 0 1 β0 (2) Ths basc equaton can be repeated for several frequences and the matrx polynomal coeffcents can be obtaned usng ether [α 2 ] or [α 0 ] normalzaton. Note that n Eqs. 3, 4; N o s the number of output degrees of freedom (output response locatons) and N s the number of nput degrees of freedom (nput force locatons). [α 2 ] Normalzaton [α 0 ] Normalzaton [ α ] [ α ] [ β ] [ β ] 0 ( jω ) [ H ( ω )] 1 ( jω ) [ H ( ω )] 0 ( jω ) [] I 1 ( jω ) [] I 4N0 N 2 ( jω ) [ H ( )] N0 N 0 1 0 1 = ω N0 4N [ α ] [ α ] [ β ] [ β ] 1 ( jω ) [ H ( ω )] 2 ( jω ) [ H ( ω )] 0 ( jω ) [] I 1 ( jω ) [] I 4N0 N 0 ( jω ) [ H ( )] N0 N 1 2 0 1 = ω N0 4N (3) (4) The coeffcents are then used to form a companon matrx and egenvalue decomposton can be appled to estmate the modal parameters.
The UMPA-LOFD algorthm for Operatonal Modal Analyss utlzes the above UMPA formulaton but nstead of worng wth frequency response functons, t wors wth power spectrums n the manner as descrbed below. The relatonshp between output response power spectra [G XX (ω)], nput force power spectra [G FF (ω)] and frequency response functon matrx [H] s gven by the followng relatonshp [4, 9] [ G ( )] [ H ( ω) ] [ G ( ω) ] [ H ( ω) ] H XX ω = (5) FF There are two mportant assumptons concernng the unmeasured exctaton n the OMA cases. 1) The most mportant and most obvous assumpton s that the power spectra of the nput force s assumed to be broadband and smooth. 2) The second and less obvous assumpton s that the exctaton s unformly dstrbuted from a spatal perspectve (N approachng N o, consderng the response s beng measured all over the structure). Ths means that the source of exctaton must be appled at a large number of degrees of freedom (as n wnd or wave exctaton) rather than at one or two degrees of freedom (as wth a shaer or a rotatng unbalance source). In all cases the exctaton s assumed to be broadband n ts frequency content. Thus, from Eq.5, t follows that the output response power spectra [G XX (ω)] s proportonal to the product [H(ω)][H(ω)] H and the order of output response power spectrum s twce that of frequency response functons. Ths also means that the power spectrum based UMPA model wll be twce the order of a frequency response functon based UMPA model [6]. Based on above dscusson [G XX (ω)] can be expressed n terms of frequency response functons as G XX ( ω ) [ G ( )] [ H ( ω) ][ I ][ H ( ω) ] H XX = = m ω (6) n = 0 H n [ β ]( jω) [ β ]( jω) 0 = 0 (7) m [ α ]( jω) [ ]( j ) α ω = 0 Snce (n < m), a partal fracton model can be formed for the output power spectrum. Ths partal fracton model for a partcular response locaton p and reference locaton q s gven by [6, 10] G pq ω R R S S = (8) ( ) N pq pq pq pq + + + = 1 jω λ jω λ jω λ ( λ ) jω ( ) Note that λ s the pole and R pq and S pq are the th mathematcal resdues. These resdues are dfferent from the resdue obtaned usng a frequency response functon based partal fracton model snce they do not contan modal scalng factor (as no force s measured). The roots of the above power spectrum based equaton wll be λ, λ, λ and λ for each model order 1 to N. To avod the numercal condtonng problems assocated wth the hgh order of power spectrum data, especally n frequency doman, an approach based on postve lag porton of the correlaton functons s used. In ths approach frst power spectrums are calculated from measured output tme responses and then respectve correlaton functons are obtaned by nverse Fourer transformaton. After ths step, the negatve lag porton of the correlaton functons s set out to zero and the postve power spectra [11] are obtaned by Fourer transformng ths resultant data. Ths s mathematcally represented as N R + pq R pq G ( ω pq ) = + = 1 jω λ jω λ (9) Note that the order of postve power spectrum s the same as that of the frequency response functons. Also postve power spectrums contan all the necessary system nformaton. Thus Eq. 2-4 can be formed based on [G XX (ω)] + data and second order UMPA model can be utlzed for the purpose of modal parameter estmaton. It should be noted that n case of OMA, N refers to those output degrees of freedom whch are consdered as references.
2.2 Complex Z Mappng The major motvaton behnd the development of the polyreference least squares complex frequency (PolyMAX) algorthm was to overcome the numercal problems nherent wth the hgh order frequency doman algorthms le Ratonal Fracton Polynomal (RFP) algorthm whch uses a transformaton from power polynomals to orthogonal polynomals to avod ths problem [5, 8]. The polyreference least squares complex frequency (PolyMAX) algorthm s essentally RFP algorthm wth complex Z mappng and wll subsequently be referred to n a generc sense as the RFP-Z algorthm. The RFP-Z algorthm replaced ths mathematcally cumbersome method by a trgonometrc mappng functon (complex Z mappng) [3, 8]. The generalzed frequency n case of UMPA-LOFD algorthm s just the normalzed power polynomal gven by s = j * (ω / ω max ) (10) Thus generalzed frequency varable s bounded by (-1,1) whch gves better numercal condtonng. The lower order of the algorthm also helps n resolvng the numercal ssues. The complex Z mappng on the other hand s gven by ( ω / ω ) j ω t j π max s = z = e = e (11) s m m j π m ( ω / ω max ) = z = e (12) Usng ths mappng the postve and negatve frequency ranges are mapped to the postve and negatve unt crcles n the complex plane respectvely. Ths yelds a real part of mappng functons whch are cosne terms and an magnary part whch are sne terms. Snce sne and cosne functons are mathematcally orthogonal, the numercal condtonng of ths mappng functon s good. 3. Case Studes In ths secton, two case studes are dscussed n vew of effect of complex Z mappng on both hgh and low order frequency doman Operatonal Modal Analyss algorthms. In addton to analyzng the modal parameter estmates, consstency dagrams are also studed to analyze the performance of the varous algorthms. 3.1 Analytcal 15 Degree of Freedom System A 15 degree of freedom system (Fgure 1) s excted at all 15 degrees of freedom by means of a whte random uncorrelated nput. The modal parameter estmaton process s then carred out usng RFP, RFP-Z, UMPA-LOFD and ts complex Z mappng varaton.
Fgure 1: Analytcal 15 Degree of Freedom System Table 1 shows the modal parameters obtaned by the four algorthms and the correspondng consstency dagrams are shown n Fgures 2-5. It s observed that all the algorthms gve good results though dampng s over determned for some of the modes. The modal parameter estmaton s subject to user experence and depends sgnfcantly on parameters such as selected frequency range, choce of reference responses and use of resduals to account for modes out of the frequency range of nterest, etc. It can be seen from Fgure 2 that the hgher order frequency doman algorthm, RFP, does not yeld a good consstency dagram. The consstency dagram deterorates severely as the order ncreases or a wde frequency range s chosen. Ths can be attrbuted to the llcondtoned matrces of the Van der Monde form [8]. The poor consstency dagram adds to the uncertanty of the obtaned modal parameters and leaves a lot on the user judgment. The applcaton of complex Z doman mappng mproves the consstency dagram sgnfcantly (Fgure 3) thus underlnng ts sgnfcance and contrbuton to the feld of parameter estmaton. The obtaned consstency dagrams are very clear and thus mae t easy for the correct modes to be pced. The numercal ssues assocated wth the RFP algorthm are not apparent n case of UMPA-LOFD algorthm [6]. Unle RFP whch nvolves power polynomals wth ncreasng powers of the frequency resultng n ll-condtoned matrces, the UMPA-LOFD algorthm beng a low order algorthm does not run nto these sorts of problems. The modal parameters estmated by UMPA-LOFD algorthm show good agreement wth the results obtaned usng other algorthms. Further the consstency dagram (Fgure 4) s much clear n comparson showng a remarably mproved performance n comparson to RFP. One mportant thng to note wth applcaton of UMPA-LOFD algorthm s that snce t s a low order algorthm t requres more spatal nformaton than that requred by RFP or RFP-Z. Fnally, complex Z mappng s appled to UMPA-LOFD to see f t results n any sgnfcant mprovement le n case of hgh order RFP algorthm. The obtaned consstency dagram as shown n Fgure 5 s very clear but does not provde any promnent mprovement over the UMPA-LOFD consstency dagram (Fgure 4). It s observed that unle UMPA-LOFD, the complex Z mappng verson s not able to pc modes outsde the selected frequency range of nterest. It s also observed that the complex Z mappng mplementaton of UMPA-LOFD cannot easly obtan the heavly damped modes n the range 68-74 Hz. Overall t s observed, that complex Z mappng when appled to UMPA-LOFD does not result n any sgnfcant advantage unle ts applcaton to the hgh order RFP algorthm. However no partcular judgment can be made as both UMPA-LOFD and ts complex Z mappng varaton are gvng equally good results.
True Modes UMPA-LOFD (Low Order, Frequency Doman) Table 1: Modal Parameter Estmates UMPA-LOFD wth Complex Z Mappng RFP (Hgh Order, Frequency Doman) RFP-z (Hgh Order, Complex Z Mappng) Damp Freq Damp Freq Damp Freq Damp Freq Damp Freq 1.0042 15.985 2.338 15.963 2.4028 15.9943 2.3531 15.9587 3.1379 15.976 1.9372 30.858 2.517 30.863 2.4530 30.8710 2.53 30.8519 2.8758 30.912 2.7347 43.6 3.043 43.680 2.9028 43.6425 2.9814 43.7026 2.9318 43.762 2.9122 46.444 3.431 46.437 3.3260 46.1123 3.4852 46.3877 3.3806 46.498 3.3375 53.317 3.932 53.209 3.5984 53.2767 3.5670 53.4123 3.0676 53.316 3.3454 53.391 3.296 53.306 3.3930 53.3452 3.3236 53.4890 3.5501 53.370 3.7145 59.413 4.430 59.116 3.9630 59.4063 4.1472 59.3597 3.7389 59.169 3.858 61.624 4.180 61.133 4.1202 61.2957 4.1735 61.4719 3.7502 61.240 4.2978 68.811 4.291 69.237 3.0884 68.9138 4.3943 68.9440 4.8688 68.612 4.5925 73.63 4.812 73.253 4.3108 72.5819 4.4734 73.2355 4.3604 73.220 2.6093 128.84 2.712 128.848 2.6453 128.8716 2.6415 128.9428 2.7789 128.965 2.4548 136.55 2.563 136.547 2.4747 136.5401 2.5578 136.5783 2.4850 136.684 2.3288 143.86 2.426 143.869 2.3172 143.8190 2.3947 143.8762 2.3207 143.919 2.221 150.83 2.314 150.799 2.1831 150.6178 2.2871 150.7683 2.2171 150.866 2.122 157.47 2.216 157.444 2.3106 157.0458 2.2041 157.4022 2.1305 157.408 Fgure 2: Consstency dagram for Ratonal Fracton Polynomal (RFP) algorthm (Analytcal system)
Fgure 3: Consstency dagram for Ratonal Fracton Polynomal (RFP-z) algorthm (Analytcal system) Fgure 4: Consstency dagram for Low Order Frequency Doman (UMPA-LOFD) algorthm (Analytcal system)
Fgure 5: Consstency dagram for UMPA-LOFD wth algorthm Complex Z mappng (Analytcal system) 3.2 Lghtly Damped Crcular Plate A crcular plate made of alumnum, as shown n Fgure 6, s excted randomly across the entre surface usng an mpact hammer. Response s taen at 30 locatons usng the accelerometers. Fgure 6: Expermental set up for the lghtly damped crcular plate The modal parameters estmated by varous algorthms, as lsted n Table 2, show good agreement wth each other and also wth the expermental modal analyss based modal parameters. The consstency dagrams for the varous algorthms for the same frequency range are shown n Fgures 7-10. It can be seen that the RFP-Z) gves the most clear consstency dagram. Applcaton of complex Z mappng on UMPA-LOFD algorthm also results n mprovng the consstency dagram but, as n the analytcal system, the effect s not as sgnfcant.
EMA based modal parameters UMPA-LOFD (Low Order, Frequency Doman) Table 2: Modal Parameter Estmates UMPA-LOFD (Low Order, Complex Z Mappng) RFP (Hgh Order, Frequency Doman) RFP-z (Hgh Order, Complex Z Mappng) Damp Freq Damp Freq Damp Freq Damp Freq Damp Freq 0.258 56.591 0.612 56.478 0.578 56.542 0.663 56.462 0.762 56.504 0.285 57.194 0.621 57.253 0.66 57.252 0.669 57.214 0.717 57.24 0.312 96.577 0.636 96.665 0.626 96.653 0.638 96.663 0.637 96.662 0.412 132.101 0.342 131.83 0.369 131.842 0.353 131.847 0.349 131.84 0.147 132.65 0.285 132.76 0.337 132.735 0.312 132.743 0.302 132.723 0.243 219.582 0.3 219.375 0.292 219.368 0.299 219.373 0.303 219.365 0.216 220.952 0.364 221.358 0.359 221.342 0.367 221.35 0.366 221.344 0.214 231.172 0.256 230.851 0.265 230.843 0.257 230.855 0.257 230.856 0.137 232.077 0.225 232.394 0.218 232.421 0.227 232.391 0.221 232.4 0.089 352.997 0.147 351.677 0.138 351.716 0.151 351.69 0.151 351.715 0.174 355.509 0.219 355.773 0.207 355.801 0.22 355.78 0.219 355.801 0.18 374.554 0.268 373.933 0.269 373.882 0.27 373.941 0.271 373.936 0.176 377.569 0.236 377.505 0.235 377.499 0.236 377.508 0.236 377.506 0.313 412.414 0.241 411.727 0.244 411.734 0.241 411.729 0.242 411.731 0.209 486.801 0.219 485.405 0.225 485.425 0.221 485.408 0.219 485.397 Fgure 7: Consstency dagram for Ratonal Fracton Polynomal (RFP) algorthm (Crcular plate)
Fgure 8: Consstency dagram for RFP-z algorthm (Crcular plate) Fgure 9: Consstency dagram for UMPA-LOFD algorthm (Crcular plate)
Fgure 10: Consstency dagram for UMPA-LOFD wth algorthm Complex Z mappng (Crcular plate) 4. Conclusons The concept of utlzng the complex Z mappng as n the polyreference least squares complex frequency (PolyMAX) algorthm, or genercally the RFP-Z algorthm, for the purpose of obtanng better numercal characterstcs for frequency doman algorthms s extended to UMPA-LOFD algorthm whch s low order frequency doman Operatonal Modal Analyss algorthm. It s shown that complex Z mappng sgnfcantly mproves the performance of the hgh order frequency doman algorthms le RFP. The effect n case of low order UMPA-LOFD s notceable but not very promnent. However, wth no apparent negatve ssues, applcaton of complex Z mappng to UMPA-LOFD can be consdered as another alternatve for the purpose of modal parameter estmaton usng output-only data. Further, ths study should also be appled to real lfe systems whch mght throw some more lght on the effectveness of complex Z mappng on UMPA-LOFD. Acnowledgements The author S. Chauhan s supported by a grant from Oho Department of Transportaton (ODOT) whch s gratefully acnowledged. The author s also thanful to Unversty of Cncnnat Infrastructure Insttute (UCII), Dr. A.J. Helmc, Dr. V.J. Hunt and Dr. J.A. Swanson for ther collaboraton and support. References [1] Rchardson, M., Forment, D.; Parameter estmaton from frequency response measurements usng ratonal fracton polynomals, Proceedngs of the 1st IMAC, Orlando (FL), USA, 1982. [2] Gullaume, P., Verboven, P., Vanlandut, S., H. Van Der Auweraer, Peeters, B.; A poly-reference mplementaton of the least-squares complex frequency-doman estmator, Proceedngs of the 21st IMAC, Kssmmee (FL), USA, 2003. [3] Peeters, B., H. Van Der Auweraer, Gullaume, P., Leurdan, J.; The PolyMAX frequency-doman method: a new standard for modal parameter estmaton, Shoc and Vbraton, 11 (2004) 395-409.
[4] Peeters, B., H. Van der Auweraer; POLYMAX: a revoluton n operatonal modal analyss, Proceedngs of the 1st Internatonal Operatonal Modal Analyss Conference, Copenhagen, Denmar, 2005. [5] Phllps, A.W., Allemang, R.J.; A low order mplementaton of the polyreference least squares complex frequency (LSCF) algorthm, Proceedngs of ISMA Internatonal Conference on Nose and Vbraton Engneerng, Katholee Unverstet Leuven, Belgum, 2004. [6] Chauhan, S., Martell, R., Allemang, R.J., Brown, D.L.; A low order frequency doman algorthm for operatonal modal analyss, Proceedngs of ISMA Internatonal Conference on Nose and Vbraton Engneerng, Katholee Unverstet Leuven, Belgum, 2006. [7] Allemang, R.J., Brown, D.L.,; A unfed matrx polynomal approach to modal dentfcaton, Journal of Sound and Vbraton, Volume 211, Number 3, pp. 301-322, Aprl 1998. [8] Allemang, R.J., Phllps, A.W.; The unfed matrx polynomal approach to understandng modal parameter estmaton: an update, Proceedngs of ISMA Internatonal Conference on Nose and Vbraton Engneerng, Katholee Unverstet Leuven, Belgum, 2004. [9] Bendat, J., Persol, A.; Random data: Analyss and measurement procedures, 2nd edton, Wley, New Yor, 1986. [10] Chauhan, S., Allemang, R.J., Brown, D.L.; Unfed Matrx Polynomal Approach for Operatonal Modal Analyss, Proceedngs of the 25th IMAC, Orlando (FL), USA, 2007. [11] Cauberghe, B.; Applcaton of frequency doman system dentfcaton for expermental and operatonal modal analyss, PhD Dssertaton, Department of Mechancal Engneerng, Vrje Unverstet Brussel, Belgum, 2004.