Identification of Modal Parameters from Ambient Vibration Data by Modified Eigensystem Realization Algorithm *

Transcription

1 Journal of Aeronautcs, Astronautcs and Avaton, Seres A, Vol.42, No (2010) 79 Identfcaton of Modal Paraeters fro Abent Vbraton Data by Modfed Egensyste ealzaton Algorth * Dar-Yun Chang **, Chang-Sheng Ln, and Fang-He Su Deartent of Aeronautcs and Astronautcs, Natonal Cheng Kung Unversty No.1, Ta-Hsueh oad, Tanan 701, Tawan,.O.C. ABSTACT In the resent aer, we roose a odfcaton to the Egensyste ealzaton Algorth wth Data Correlaton (EA/DC) for odal-araeter dentfcaton of structural systes subjected to statonary whte-nose abent vbraton. By settng u a correlaton-functon atrx of statonary resonses and ntroducng an arorate factorzaton of the atrx, odal araeters of a syste could then be effectvely dentfed through the sngular-value decooston and egenvalue analyss. Through nuercal sulatons, effectveness and robustness of the roosed dentfcaton ethod s deonstrated. Keywords: Egensyste ealzaton Algorth wth Data Correlaton (EA/DC), Correlaton technque, Statonary abent vbraton I. INTODUCTION Exerental dentfcaton of odal araeters of a structure s usually carred out by easurng both ts nut and corresondng outut. Soe odal testng technques use free or ulse resonse of structures so that the nut (exctaton) need not be easured. However, there are stuatons where controlled exctaton or ntal exctaton ay not be avalable, such as the case of n-oeraton testng or n-flght easureent. Consequently, t s desrable to develo technques for odal-araeter dentfcaton drectly fro abent vbraton data,.e., wthout the need of nut easureent. Nuerous aers have been resented on syste dentfcaton, whch acqures ts estaton of ortant araeters fro the easured data. Ho and Kalan [1] roosed the ortant rncles of nal realzaton theory, whch uses a sequence of real atrces nown as Marov araeters (ulse resonse functons) to construct a state-sace reresentaton of a lnear syste. The error characterstcs due to nose, however, are yet to be estated. Zeger and McEwen [2] roosed a concet cobnng sngular-value decooston (SVD) and nal realzaton algorth. Aong follow-u develoents on SVD and nal realzaton algorth, Juang and Paa [3] roosed Egensyste ealzaton Algorth (EA), whch has been aled to dentfy odal araeters fro ulse resonse of a syste. Juang et al. also roosed a odfcaton of EA, generally nown as EA/DC (Egensyste ealzaton Algorth wth Data Correlaton) [4]. EA/DC uses data correlatons to reduce the nose effect n the rocess of odal-araeter dentfcaton. Ibrah et al. [5, 6] resented a te-doan odal-dentfcaton ethod, usually nown as ITD. ITD uses free-decay resonses of a structure to dentfy ts odal araeters. The ethod s effectve n dentfyng colex odes even wth closely saced odes [7]. Jaes et al. [8] develoed the so-called Natural Exctaton Technque (NExT), whch uses correlaton functons of easured resonse n cobnaton wth a te-doan araeter extracton schee under the assuton of whte-nose nut. Chang and Cheng [9] resented a correlaton technque for odal-araeter dentfcaton of a lnear, colex-ode syste subjected to statonary abent exctaton. Chang and Ln [10] further extended the correlaton technque for odal-araeter dentfcaton of a syste excted by non-statonary whte nose n the for of a roduct odel. In the resent aer, a odfcaton to EA/DC s resented for lnear systes excted by statonary * Manuscrt receved, Aug. 20, 2009, fnal revson, Dec. 11, 2009 ** To who corresondence should be addressed, E-al: dchang@al.ncu.edu.tw

2 80 Dar-Yun Chang Chang-Sheng Ln Fang-He Su whte-nose abent vbraton. By settng u a correlaton-functon atrx of the statonary abent resonses and ntroducng an arorate factorzaton of the atrx, odal araeters of a syste could then be dentfed usng the sngular-value decooston (SVD) and egenvalue analyss. Through nuercal sulaton, the alcablty and robustness s exaned of the roosed ethod for dentfyng odal araeters of a lnear syste fro easureents contanated wth nose. II. COELATION TECHNIQUE Jaes et al. [8] develoed the so-called Natural Exctaton Technque (NExT) usng the correlaton technque. It was shown that the cross-correlaton between two resonse sgnals of a lnear syste wth classcal noral odes and subjected to whte-nose nuts s of the sae for as free vbraton decay or ulse resonse. In cobnaton wth a te-doan araeter extracton schee, such as the ITD ethod, ths concet becoes a owerful tool for the dentfcaton analyss of structures under statonary abent vbraton. When a syste s excted by statonary whte nose, the cross-correlaton functon ( ) j statonary resonse sgnals shown to be [9] where shae, between two x t and j n r jr j r 1 rdr r r dr r x t can be A () ex( )sn( ) (1) r denotes the -th coonent of the r-th ode A a constant, and r the r-th odal ass. jr The result above shows that j τ n Eqn. (1) s a su of colex exonental functons (odal resonses), whch s of the sae atheatcal for as the free vbraton decay or the ulse resonse of the orgnal syste. Thus, the cross-correlaton functons evaluated of resonses data can be used as free vbraton decay or as ulse resonse n te-doan odal extracton schees so that easureent of whte-nose nuts can be avoded. It s rearable that the ter r Ajr n Eqn. (1) wll be dentfed as the ode-shae coonents. In order to elnate the A jr ter and retan the true ode-shae coonents r, all the easured channels are correlated aganst a coon reference channel, say x j. The dentfed coonents then all ossess the coon A jr coonent, whch can be noralzed out to obtan the desred ode shae r. III. MODIFIED EIGENSYSTEM EALIZATION ALGOITHM WITH DATA COELATION Juang et al. [4] roosed a odfcaton of EA called the Egensyste ealzaton Algorth wth Data Correlaton (EA/DC), whch uses data correlatons to reduce the nose effect for odal-araeter dentfcaton. It has been shown n the revous secton that the correlaton functons between two statonary resonse sgnals of a structure subjected to whte-nose nuts can be treated as free vbraton decay or as ulse resonse for further extracton of odal araeters. The correlaton functons can thus be used to extract odal araeters of a syste n the case of statonary abent vbraton. In the resent aer, a odfcaton to EA/DC s resented for lnear systes excted by statonary whte-nose abent vbratons. By settng u a correlaton-functon atrx of the statonary resonses and ntroducng a roer factorzaton of the atrx, odal araeters of a syste could then be dentfed through the sngular-value decooston (SVD) and egenvalue analyss. Consder an n-densonal, dscrete-te, lnear, te-nvarant dynac syste subjected to the exctaton of a zero-ean statonary whte nose u u, 1~. For fnte but suffcently long records, say of l te stes, we defne the correlaton-functon atrx T, whch s coosed of easured resonses fro every channel wth dfferent te-delayed sgnals, can be aroxately exressed as 1 T T T l Y Y (2) where yj, 1~, j l ~ Y sgnfes the easured resonses of the syste at the te-ste, and s the nuber of the channel of the easured resonse. Note that the correlaton-functon atrx T of the easured resonse of the syste could be obtaned fro a sngle sale functon of te through use of the ergodc roerty of statonary rando rocesses. Defne the data-correlaton atrx correlaton-functon atrx can be therefore wrtten as T T T T T T T T T to be fored usng the T, then n vew of Eqn.(2), (3) where s the nuber of the exanson channels n. Note that the correlaton-functon atrx T can be vewed as the ulse functons (also nown as the Marov araeters) n the conventonal EA/DC rocess and can be exressed as [11] 1 T CA G (4) where C s the n outut nfluence atrx for the

3 Identfcaton of Modal Paraeters fro Abent Vbraton Data by Modfed Egensyste ealzaton Algorth 81 state vector, A s the n n state atrx wth dynac 1 T characterstcs of the syste, and G 1 l X Y s n the so-called covarance atrx, where X 1 s the n l state-vectors atrx of the syste. Wth ths factorzaton, Eqn.(3) can be forulated as 1 2 CA G CA G CA G 1 1 CA G CA G CA G CA G CA G CA G C CA 1 2 A A G A G A G 1 CA P A To further reduce the effect of nose, and to ae the araeters estaton ore accurate, we construct a generalzed Hanel atrx U U as follows: 1 b1 1 2 b a1 a ab2 (5) a b where the ntegers a and b defne how any correlaton lags of atrx (6) are ncluded n the generalzed Hanel U analyss. To ae use of the roerty of consstency n the theory of syste dentfcaton, we tend to use ore easureent data n the analyss. Eqn.(6) sgnfes the use of exanded easureent sales through constructng a generalzed Hanel atrx U. Ths wll reduce the effect of nose. Substtutng Eqn.(5) nto Eqn.(6), the followng equaton can be derved: P P A U A A A a1 P A b1 a A b (7) dentfed. Based on syste realzaton theory, EA/DC constructs a dscrete state-sace odel of nal order usng the sngular-value decooston (SVD) analyss. To deterne the syste order, whch s defned as the nuber of structural odes nvolved n the resonses, U s SVD of the generalzed Hanel atrx erfored. The ran of the generalzed Hanel atrx U s the syste order, whch s just the nuber of the obvous non-zero n sngular values. Therefore, the state atrx A can then be obtaned by retanng the frst n sub-vectors as follows: 1 T n n n n 1 A S V U W S (8) where V n and W n are the atrces wth orthonoral roerty,.e., V T T n V n In W n W n, and S n s a dagonal atrx contanng the n largest sngular values. Denote the atrces of egenvalues and egenvectors of the n order state atrx A as Z dag z1, z2,, zn and 1, 2,, n, resectvely. The egenvalues contan the nforaton of odal frequences and dang ratos. Thus, once the state atrx A s obtaned fro easured data, the odal araeters of the structure syste of nterest can then be deterned by solvng the egenvalue roble assocated wth the state atrx A. The state atrx A thus can be exressed as A 1 Z (9) Substtutng Eqn.(9) nto Eqn.(4), the followng equaton can be derved: 1 T CZ G (10) where C C and 1 G G. Note that C contans the odal shaes of the syste. The egenvalues of the orgnal vbraton syste n the contnuous-te doan can be evaluated as follows [8]: ln z 1 t I 2 (11) where and I are the real and agery arts of, resectvely, and t s the salng nterval. The natural frequences and dang ratos of the syste could then be obtaned fro Eqn.(11): I 2 2 (12) where a and b are defned as generalzed observablty and controllablty atrces, resectvely. In order to dentfy syste odes effectvely, the su of and a should be greater than the nuber of odes to be I 2 2 (13)

4 82 Dar-Yun Chang Chang-Sheng Ln Fang-He Su It has been shown that a odfcaton to the Egensyste ealzaton Algorth wth Data Correlaton (EA/DC) s resented for odal-araeter dentfcaton fro statonary abent vbraton data. By settng u a correlaton-functon atrx of abent resonses of structure subjected to statonary whte nose and ntroducng an arorate factorzaton of the atrx, odal araeters of a syste could then be dentfed through SVD and the egenvalue analyss. IV. NUMEICAL SIMULATION AND DISCUSSION To deonstrate the feasblty and effectveness of the roosed ethod, we consder a lnear 6-dof chan odel wth vscous dang. A scheatc reresentaton of ths odel s shown n Fg.1. The ass atrx M, stffness atrx K, and the roortonal dang atrx C of the syste are gven as follows: M g K 600 N D 0.1M 0.001K N sec Consder that the abent vbraton nut can be odeled as statonary whte nose. The statonary whte nose s generated usng the sectru aroxaton ethod [12] as a zero-ean band-ass nose, whose 2 standard devaton s 0.02 N sec / rad wth a frequency range fro 0 to 50 Hz. The salng nterval s chosen as t 0.01 sec, and the salng erod s T Nt t sec. The statonary whte nose sulated serves as the exctaton force actng on the 6 th ass ont of the syste. The te sgnal of a sulated sale of the statonary whte nose and the ower sectru are shown n Fg.2 and Fg.3, resectvely. The dslaceent resonses of the syste were obtaned usng Newar s ethod. To exane the robustness of the roosed dentfcaton aroach under nosy condtons, 10% nose s added to the sulated dslaceent resonse data. For data wth suffcently long duraton, the correlaton-functon atrx T of the resonses of the Fgure 1 Scheatc lot of the 6-dof chan syste Exctaton (N) Tes (sec) Fgure 2 A sale functon of statonary whte nose n te doan Fgure 3 Power sectru of the sulated statonary whte nose

5 Identfcaton of Modal Paraeters fro Abent Vbraton Data by Modfed Egensyste ealzaton Algorth 83 syste could be estated fro a sngle sale functon of te (ergodcty assuton). Once the correlaton-functon atrx T was evaluated fro Eqn.(2), the data-correlaton atrx generalzed Hanel atrx and a U could be further constructed. The state atrx A can thus be estated through the SVD analyss and odal araeters of the syste could then be dentfed by solvng the egenvalue roble assocated wth the state atrx A. In theory, a contnuu structure has an nfnte nuber of degrees of freedo and an nfnte nuber of odes. In ractce, we do not now n advance how any odes are requred to descrbe the dynacal behavor of the observed structural syste. However, the nuber of ortant odes of the syste under consderaton could be roughly found by usng the sngular-value decooston analyss. The nuber of non-zero sngular values s the ran of the generalzed Hanel atrx U and s also the order of the syste. By exanng the dstrbuton of the sngular values assocated wth the resonses of the syste, we could deterne the order, as well as the nuber of odes to be dentfed, of the syste. The result s shown n Fg.4, fro whch the order of the syste odel was deterned to be 12. hgher due to the fact that the resonse of the syste generally has lower senstvty to these odal araeters than to the odal frequences. It s seen that good results of dentfcaton have been obtaned fro the sulated abent resonse data contanated wth 10% nose. The results of odal-araeter dentfcaton through the conventonal ITD [5,6] and the EA [3] are also suarzed n Table 1, Table 2, and Fg.5. It should be entoned that the conventonal ITD or the EA ethod uses only free-vbraton or ulse resonse of a structure to dentfy odal araeters. Therefore, n order to convert the orgnal forced resonses nto free-vbraton data, we need addtonal treatents, such as eloyng the correlaton technque. Furtherore, as to the correlaton technque, the choce of a reference channel for coutng correlaton functons s sgnfcant to the result of dentfcaton of odal araeters. The reference channel s usually chosen as the resonse channel whose Fourer sectru has rch frequency content around the odes of nterest [10]. If we choose a reference channel wth oor frequency content, the dentfcaton results wll also be oor. In contrast, n the resent ethod, we drectly coute the correlaton-functon atrx T, each row of whch s treated as the ulse resonse corresondng to each DOF. We could thus tae advantage of the coutaton of T to get rd of the addtonal treatent of convertng forced resonses nto the free-vbraton data and the selecton of reference channel. V. CONCLUSIONS Fgure 4 Sngular values assocated wth the resonses corresondng to statonary whte nose nut The odal frequences and dang ratos can be obtaned va Eqns.(11), (12) and (13) followng the egen-analyss of the state atrx A, and the ode shaes can be obtaned fro C as gven n Eqn.(10). The results of odal-araeter dentfcaton are suarzed n Table 1 and Table2, resectvely, fro whch we observe that n general the errors n natural frequences are less than 1 % and the errors n dang ratos are less than 10%. The dentfed ode shaes are also coared wth the exact ode shaes n Fg.5. The errors of dentfed dang ratos and ode shaes are soewhat A odfcaton to the Egensyste ealzaton Algorth wth Data Correlaton (EA/DC) s resented n ths aer for odal-araeter dentfcaton fro statonary whte-nose abent vbraton data. By settng u a correlaton-functon atrx T of statonary abent resonses and ntroducng an arorate factorzaton of the correlaton-functon atrx, odal araeters of a syste could then be dentfed va the sngular-value decooston and egenvalue analyss. Through nuercal sulatons, the effectveness and robustness of the roosed dentfcaton ethod s deonstrated n a oderately nosy condton. By tang advantage of the coutaton of correlaton-functon atrx T, we avod the addtonal treatent of convertng the orgnal forced-vbraton data nto the free-vbraton data and the selecton of reference channel for coutng correlaton. ACKNOWLEDGEMENT Ths research was suorted n art by Natonal Scence Councl of the eublc of Chna under the grant NSC E EFEENCES [1] Ho, B. L. and Kalan,. E., Effectve

6 84 Dar-Yun Chang Chang-Sheng Ln Fang-He Su Fgure 5 Coarson between the dentfed ode shaes and the exact ode shaes of the 6-dof syste subjected to statonary whte nose nut Constructon of Lnear State-Varable Models fro Inut/Outut Data, Proceedngs of the 3 rd Annual Allerton Conference on Crcut and Syste Theory, Montcello, IL Unversty of Illnos, 1965, [2] Zeger, H. P. and McEwen, A. J., Aroxate Lnear ealzatons of Gven Denson Va Ho s Algorth IEEE Transactons on Autoatc Control, Vol. AC-19, No. 2, Arl 1974, [3] Juang, J. N. and Paa,. S., An Egensyste ealzaton Algorth for Modal Paraeter Identfcaton and Model educton, Journal of Gudance, Control and Dynacs, Vol. 8, No. 5, Set.-Oct. 1985,

7 Identfcaton of Modal Paraeters fro Abent Vbraton Data by Modfed Egensyste ealzaton Algorth 85 Table 1 esults of natural frequency dentfcaton of the 6-dof. syste subjected to statonary whte nose nut Mode Natural Frequency (rad/sec) Exact EA/DC Error(%) EA Error(%) ITD Error(%) Table 2 esults of dang rato dentfcaton of the 6-dof. syste subjected to statonary whte nose nut Mode Dang ato (%) Exact EA/DC Error(%) EA Error(%) ITD Error(%) [4] Juang, J. N., Cooer, J. E. and Wrght J.., An Egensyste ealzaton Algorth usng Data Correlatons (EA/DC) for Modal Paraeter Identfcaton, Control-Theory and Advanced Technology, Vol. 4, No. 1,. 5-14, March, [5] Ibrah, S.. and Mulc, E. C., A Method for the Drect Identfcaton of Vbraton Paraeters fro Free esonse, Shoc and Vbraton Bulletn, Vol. 47, Part 4, Set. 1977, [6] Ibrah, S.. and Mulc, E. C., The Exerental Deternaton of Vbraton Paraeters fro Te esonses, Shoc and Vbraton Bulletn, Vol. 46, Part 5, Aug. 1976, [7] Paa,. S. and Ibrah, S.., A Paraetrc Study of Ibrah Te Doan Modal Analyss, Shoc and Vbraton Bulletn, Vol. 51, Part 3, 1981, [8] Jaes, G. H., Carne, T. G., and Lauffer, J. P., The Natural Exctaton Technque for Modal Paraeter Extracton fro Oeratng Wnd Turbnes, SAND , UC-261, Sanda Natonal Laboratores, [9] Chang, D.Y. and Cheng, M. S., Modal Paraeter Identfcaton fro Abent esonse, AIAA Journal, Vol. 37, 1999, [10] Chang, D. Y. and Ln, C. S., Identfcaton of Modal Paraeters fro Nonstatonary Abent Vbraton Data Usng Correlaton Technque, AIAA Journal, Vol. 46, No. 11, Nov. 2008, [11] Van Overschee, P. and De Moor, B., Subsace Algorths for the Stochastc Identfcaton Proble, Autoatca, Vol. 29, No. 3, 1993, [12] Shnozua, M. and Jan, C.-M., Dgtal Sulaton of ando Processes and ts Alcatons, Journal of Sound and Vbraton, Vol. 25, No. 1, 1972,