Stability Analysis of Central Difference Method for Dynamic Real-time Substructure Testing

Transcription

1 009 Aercan Control Conference Hyatt Regency Rverfront, St. Lous, MO, USA June 10-1, 009 FrB0. Stablty Analyss of Central Dfference Method for Dynac Real-te Substructure Testng B. Wu, L. Deng, Z. Wang, and X. Yang Abstract Ths paper studes the stablty of the central dfference ethod (CDM) for real-te substructure test consderng the ass of specen (.e., experental substructure). To obtan correct reacton nerta force, an explct acceleraton forulaton s assued for the CDM. The analytcal work shows that the stablty of the algorth decreases wth ncreasng specen ass f the experental substructure s a pure nerta specen. The algorth becoes unstable whatever the te ntegraton nterval,.e., uncondtonally unstable, when the ass of specen equal or greater than that of ts nuercal counterpart. For the case of dynac specen, the algorth s uncondtonally unstable when there s no dapng n the whole test structure; a dapng wll ake the algorth stable condtonally. The behavor of the CDM for vanshng te ntegraton nterval s verfed wth the zero-stablty analyss ethod for coupled ntegraton. Part of the analytcal results s valdated by an actual test. I. INTRODUCTION EAL-te substructure testng (RSTng) s a hybrd R sulaton technque, n whch a structure s splt nto two parts: the crtcal coplex part called experental substructure to be tested physcally and n real-te, whle the reander called nuercal substructure s sulated nuercally by the coputer [1]. As a core eleent of the analytcal part of RSTng, nuercal ntegraton plays a key role n a successful test. Many ntegraton ethods popular n PDT are explct only for dsplaceent. When an explct velocty s requred for dapng specen (experental substructure) n RSTs, soe extra forulaton has to be assued. Ths assupton very possbly does not confor to the orgnal velocty forulaton and ths ay change the nuercal behavor of the ntegraton algorth. For the central dfference ethod (CDM) n RSTng wth dapng specen, reference [] showed that the stablty decreases wth ncreasng dapng of the experental substructure, contrastng constant stablty lt of standard CDM. Slar proble ay exst for plct algorth pleented n a RST. Reference [3] found that Newark average acceleraton Ths work was supported by Grant and fro the Natonal Scence Foundaton of Chna, and Chna Mnstry of Educaton through Progra for New Century Excellent Talents n Unversty. B. Wu s wth the School of Cvl Engneerng, Harbn Insttute of Technology, HARBIN CHINA (e-al: bn.wu@ht.edu.cn ). L. Deng and Z. Wang are wth the School of Cvl Engneerng, Harbn Insttute of Technology, HARBIN CHINA. X. Yang s wth Chna Southwest Archtectural Desgn and Research Insttute, CHENGDU CHINA (e-al: yangxd_to@163.co ). ethod ay lose the uncondtonal stablty for dapng specen. For a dynac experental substructure, the loadng fashon should be explctly and properly specfed to obtan correct reacton of the substructures due to ts nerta as well as stffness and dapng. Ths ay change the nuercal behavor of ntegraton algorths whch norally do not nclude any explct expressons for acceleraton. Reference [4] showed that the Newark average acceleraton ethod becae condtonally stable when the loadng coands for the dynac substructure were sent off as a lnear rap functon of te. Although here have been successful applcatons of explct algorths to nerta specen and soe dscussons on stablty related to te delay [5] or couplng between nuercal and experental substructures [6], the pact of specen ass on the stablty of a specfc ntegraton ethod has not been explored. Ths paper wll focus on the stablty ssue of the CDM arsng n ts pleentaton to RSTng wth dynac experental substructure. II. FORMULATION OF THE CDM FOR RST CONSIDERIING SPECIMEN MASS In an RST, the acceleraton-dependent force (nerta force) and velocty-dependent force (dapng force) exhbted by the specen are ntroduced nto the easured reacton force together wth the statc restorng force when the specen s a dynac substructure. Therefore, the te-dscretzed equatons of oton of the nuercal substructure at the th te step n an RST can be expressed n a ore general and precse for as M a + C v + K d + R ( a, v, d ) = F (1) N N, N N, N N, E EC, EC, EC, and, when the CDM s eployed, the velocty and acceleraton are approxated by N, = ( N, + 1 N, 1)/Δt N, = ( N, + 1 N, + N, 1)/Δt v d d () a d d d (3) where M, C, and K are the ass, daper, and stffness atrces of the nuercal substructure respectvely; R s the reacton force; a, v, and d are acceleraton, velocty, and dsplaceent vectors; F s external exctaton on the /09/$ AACC 516

2 nuercal substructure; Δt s the ntegraton te nterval; subscrpts N, E, and C denote nuercal substructure, experental substructure, and couplng degree-of-freedo of nuercal and experental substructures, respectvely. Substtutng () and(3) nto (1) gves ( t ) ( t) d = [ M /Δ t + C / Δ ] 1 N, + 1 N N { F ( KN MN /Δ t ) dn, [ MN /Δt C / Δ ] d R ( a, v, d )} N N, 1 E EC, EC, EC, To obtan accurate dynac reacton force, the velocty and acceleraton on the couplng degree-of-freedo at the (+1)th step, whch denoted by v EC,+1 and a EC,+1 respectvely, have to be deterned and posed onto the specen together wth d EC,+1. Ths just can not be realzed through () and (3), as the dsplaceent at the (+)th step s not yet avalable. For an actuator controlled n a tradtonal dsplaceent ode, the acheveent of the explct velocty or acceleraton target s dependent on how the dsplaceent coand s ssued wth respect wth te. To ths end, we assue a constant acceleraton n the te nterval fro t to t +1, resulted n by a dsplaceent coand profle as a quadratc functon n te: ( ) 0.5 ( ) EC, + 1 EC, EC, EC, + 1 (4) d () t = d + ν t t + a t t (5) Note that, for sesc tests, the coands should nclude the ground oton to guarantee an absolute acceleraton nput to the dynac specen and hence obtan correctly reacton force due to nerta. Lettng d EC, =d NC, and v EC, =v NC,, and substtutng () nto the above equaton and lettng t=t +1 ental a d d d (6) EC, + 1 = ( NC, + 1 NC, + NC, 1)/Δt It s nterestng to note that a a (7) = EC, + 1 NC, accordng to (3). By coparng (6) wth (3), we see that the acceleraton at the couplng degree-of freedo of the experental substructure s obvously no longer consstent to those of nuercal substructure deterned by the standard CDM. Ths rases the ssue of possble change of nuercal behavor, especally the stablty, of the odfed CDM over the standard one, whch wll be dscussed next. III. STABILITY ANALYSIS We restran our dscusson wthn lnear systes and the nuercal substructure s of sngle-degree-of-freedo (SDOF). Two cases are consdered: one s wth a specen of pure ass as shown Fg. 1, and the other wth a SDOF dynac specen as shown Fg.. A. Pure Inerta Specen When the experental substructure s just an nerta specen, ts reacton force at the th step s easly obtaned as R = M a (8) E, E EC, where a EC, s deterned by (6). Substtutng (8) nto (4) yelds ( ) d = [ M /Δ t + C / Δ t ] N N { F [ KN ( MN ME)/Δ t ] d [( M N ME) / Δ t CN /(Δ t)] d 1 ME / Δ t d } Based on the above equaton, the dsplaceent responses of free vbraton between two adjacent te steps can be related n a recursve for as = + 1 where = [ d d d ] T (9) Y AY (10) a11 a1 a13 Y , A = wth a11 = [ (1 + γ ) γ ]/ (1 + ξn ), a1 = ( ξn + γ 1)/ (1 + ξn ), a13 = γ /(1 + ξn ), ( ) =Δ tω =Δ t KN MN + ME, ξn= CN/ ( MN ω ), γ =M E /M N. The atrx A s usually called aplfcaton atrx and ts egenvalues deterne the nuercal behavor of an ntegraton algorth. In partcular, the aplfcaton atrx defnes the stablty condton of an ntegraton algorth through ( ) 1 ρ A (11) where s the spectral radus of A, whch s defned as ρ (A) =ax( λ ), andλ are the egenvalues of A. The characterstc equaton of A can be obtaned as ( ) ( ) ( ) 3 1+ ξn λ + 1+ γ + γ λ + 1 ξ γ λ+ γ = 0 N (1) We defne a stablty lt as the axu of the s values such that 1 for any (0, s ). Lettng [] denote the stablty lt, one ay obtan the stablty lt of the odfed CDM for pure nerta specen as [7] 517

3 [ ] γ γ = (1 )/ (1 + ) (13) The above equaton ndcates that () the stablty lt only exsts when γ <1 (the non-exstence of stablty lt s called uncondtonally unstable n ths paper); () the stablty lt decreases wth ncreasng γ ; and () the stablty lt has no relatonshp wth dapng fro nuercal substructure. The nterestng behavor of uncondtonal nstablty can also be proved through the concept of zero-stablty of coupled ntegraton. The proof s provded n the appendx. The spectral radus ρ (A) aganst s plotted n Fg. 3a wth ξ N =0 and dfferent γ values. The stablty lts obtaned fro Fg. 3a are dentcal to those calculated usng (13). Fg. 3b shows the dagras of spectral radus ρ (A) aganst wth γ =0.5 and dfferent ξ N values. The ndependence of stablty lt upon ξ N s easly seen and s consstent wth the observaton fro(13). The sulated dsplaceent responses of free vbraton wth varous values are shown n Fg. 4 where M N =300kg, γ =0.8, ω=πs -1, ξ N =0. and ntal condton s d 0 =1c and v 0 =0. Here ω s fxed whle Δt changes to acheve dfferent values. The correspondng [] s 0.67 obtaned wth (13). When >0.67, the unstable response s observed n Fg. 4. It s also seen that the sulated result approaches exact one wth reducng slar to standard CDM. B. Dynac specen For the RST of a structure as shown Fg. subject to sesc exctaton, R E (t) s only related to the acceleraton a EC (not d EC or v EC ) at the couplng degree-of-freedo through followng equatons. RE, () t = [ CE ve, () t + KE de, ()] t (14) M a () t + C v () t + K d () t = M ( a + a )(15) E E, E E, E E, E EC, g, n whch a E, v E, and d E are the acceleraton, velocty, and dsplaceent of experental substructure relatve to the nuercal substructure; a g s the ground acceleraton; t [t -1, t ]. Free vbraton response of the experental substructure s obtaned usng Duhael s ntegral on (15). Accordngly, the analytcal soluton of reacton force s derved. Its substtuton together wth (6) nto (4) gves d = c d + c d + c d + c d + c v Δ t (16) N, N, N, 1 3 N, 4 E, 5 E, n whch c j s are constants related to structural paraeters and Δt; ther expressons can be found n [7]. Lettng Y + 1 = dn, + 1 dn, dn, 1 de, ve, Δt (17) one ay easly get the correspondng aplfcaton atrx A. It s dffcult to obtan the analytcal expresson of spectral radus of the atrx A due to atheatcal coplexty. T Therefore, the nuercal analyses were carred out to nvestgate the spectral characterstcs. Fg. 5a shows the results of undaped cases wth frequency rato γ ω equal to 1, where γ ω = ωe / ωn, ω E = KE/ ME, ω N = KN/ MN ; the horzontal coordnate s defned as =ω N Δt. It s seen that ρ (A) s always greater than unty, ndcatng unstable response, however sall the ass rato and are. Ths eans that the CMD s uncondtonal unstable for a dynac specen f there s no dapng assocated n the test structure. Ths contrasts the condtonal stablty of the case wth pure nerta specen and the ass rato lower than 1. Nonetheless the nstablty n the case of dynac specen s not that serous for sall ass rato and snce the spectral radus s very close to unty as shown n Fg. 5a, f only the testng duraton s not too long. It s also seen n Fg. 5a that the nstablty s proved wth reduced γ, as the spectral radus becoes closer to 1. The elnaton of uncondtonal nstablty proble can be acheved by addng a dapng to the structure. Ths s llustrated n Fg. 5b where dapng ratos of experental and nuercal substructures are both 5%, and γ ω =1. The dapng rato here are defned as ξ E =C E /(M E ω E ), and ξ N =C N /(M N ω N ). Fg. 6 shows the sulated and exact dsplaceent responses of the free vbratons wth ξ E =ξ N =0.05, γ =γ ω =1. The ntal condtons are: d N0 =1c, d E0 =-1c, v N0 =v E0 =0. Dfferent s are consdered n the sulaton. The stablty lt of ths case s 0.44 fro Fg. 5b. It s observed fro Fg. 6 that the response s unstable when = 0.45, whch verfes the result of spectral analyss. Although the stable responses of RST are attaned for saller s, the good agreeent wth the exact soluton s seen only for an as sall as IV. NUMERICAL SIMULATION OF RST WITH SHAKING TABLE In Sectons I and II, the dynacs of physcal loadng syste s not consdered n order to ephasze the nuercal behavors of the algorth tself. A shakng table s used n ths secton as a transfer syste of the RST wth dynac specen, and the stablty perforance wll be nvestgated through soe nuercal sulatons, n whch the lnear odel of the Rce unversty shakng table developed by [8] s adopted heren. All the paraeters of the shakng table are the sae as n [8] except the control gans specfed n ths paper. The exctaton s the El Centro (NS, 1940) earthquake record. A. Pure Inerta Specen The paraeters of the nuercal substructure and experent substructures for the nuercal sulatons are: M N =300kg, ω=πs -1, C N =0, and K E =C E =0. Δt =0.01s. The PID control gans are K P =39.4 5V/, K I =0, K D =8.5V s/, and the feed-forward and dfferental pressure control gans are K ff =1. V s/ and K dp = V /N, respectvely. The saplng frequency of the dgtal control s 1000 Hz. The nuercal sulaton results wth dfferent ass ratos and 518

4 the exact soluton of the dsplaceent responses are shown n Fg. 7. The exact soluton s calculated by usng LSIM coand n Matlab. It s seen that the response becoes unstable when γ = Ths s consstent wth the results of theoretc analyss n Secton III. It s also seen that the response approaches the exact soluton wth saller γ. B. Dynac Specen The paraeters of the nuercal substructure, as shown n Fg., are M N =500kg, ω N =πs -1, and ξ N =0.05. The paraeters of the experent substructure are dentcal to the nuercal substructure,.e., γ =γ ω =γ c =1, wth γ C = C E /C N. The PID control gans of the shake table controller are: K P =39.4 5V/, K I =0, K D =7.5V s/, and the feed-forward and dfferental pressure control gans are K ff =1.5 V s/ and K dp = 10-7 V /N, respectvely. Accordng the analyss of Secton III, the stablty lt s equal to 0.44 n ths case. The dsplaceent responses of the nuercal substructure wth dfferent values are shown n Fg. 8. The unstable response s clearly seen when =0.471>[]=0.44. The better result s obtaned as expected wth saller. V. VALIDATION TEST A valdaton test was carred out at the Mechancal and Structural Testng Center of the Harbn Insttute of Technology. The scheatc dagra of the whole test structure s shown n Fg. 1. The experental substructure was a pure ass ade of cast ron wth M E =116kg. A photograph of the experental substructure nstalled on the MTS servo-hydraulc actuator s shown n Fg. 9. The crcular frequency of the whole structure was kept πs -1 for all test cases. The dapng fro the nuercal substructure was assued zero. The ntegraton te nterval was 0.01and the saplng frequency was 104 Hz. Fgures 10-1 show the dsplaceent coands and responses of free vbraton wth dfferent ass ratos. The ntal condtons are d 0 =0, v 0 =3.14c/s. The stable result was obtaned wth γ =0.1 and the decayng response s observed n Fg. 10, probably due to the frcton force between the gudng coluns and the ron ass. Wth ths frcton force, the response reaned stable wth γ =1.01>1 as seen n Fg. 11. Further ncreasng γ by reducng M N resulted n an unstable tendency of response as seen n Fg. 1. The test was ternated before t went volently. Although the test results were not exactly the sae as predcted by the analytcal work n the prevous sectons, the nfluence of the specen ass on the stablty of RSTng wth the CDM has been confred. It s should be noted that, for pure dapng specen, the stablty s reduced by ncreasng dapng level of the specen []. The reason for the dfferent effects of dapng could be attrbuted to by the dfferent dapng echans of the experental substructure. The dapng n ths paper s coulob-typed frcton, whle a vscous dapng s assued n []. The dsplaceent responses tracked the coands very well n all these cases as shown n Fg The te delay of the dsplaceent responses to the correspondng coands was around s. The focus of ths paper was the effects of ass rato on the stablty, and all the test cases were subject to the sae te delay, therefore the ssues about te delay were not dscussed. Nevertheless, the effects of errors such as those due to te delay fro testng facltes and relevant prove technques are apparently portant and thus should be nvestgated further n future study. It s worth notng that Neld et al. [9] has carred out a slar nerta specen test usng shakng-table as the transfer syste and looked at the ass dvson fro control pont of vew. It was shown there that the dvson ethod of the eulated syste nto the substructure and the nuercal odel s hghly sgnfcant to the overall perforance of the syste. VI. CONCLUSION The CDM s odfed wth an explct acceleraton forulaton to obtan correct reacton force of dynac specen n a RST. The analytcal work, nuercal sulaton of the RST wth a shakng table and actual test have all shown that the stablty of the algorth decreases wth ncreasng ass rato of experental over nuercal substructures. VII. APPENDIX: ZERO-STABILITY OF COUPLED CDM In a hybrd sulaton of dynac syste, the soluton reles on the two ntegratons n the te doan: one s nuercal and the other s physcal. The nuercal ntegraton s coupled wth physcal ntegraton through the nterface between the nuercal and experental substructures. A necessary condton for stable soluton of the dscrete coupled syste s zero-stablty,.e., the syste s stable as the te ntegraton nterval approaches to zero [10]. The objectve of ths appendx s to confr the behavour of the CDM revealed n the thrd secton n ths paper, usng the ethod of zero-stablty analyss for coupled ntegraton. For the case of the pure nerta specen, the outputs of the nuercal and experental substructures at the th step can be expressed as yn, = an, 1 (A1) y = M u (A) E, E E, and the nputs are related to the outputs by u u = y (A3) = y (A4) N, E, E, N, Wth (1), and (A1)- (A4) we get 519

5 n whch y = G+ H y (A5) + 1 KN / MN CN / MN G = N, ME KN / MN MECN / M X N (A6) 0 1/ M N H = ME / MN 0 (A7) X = [ d v T ], and y = [ y y T ] N, N, N, N. E. As t s not dffcult to prove that X N, s a constant for vanshng Δt, G s apparently a constant vector wth (A6). Therefore the stablty of the syste output s deterned by ρ(h), the spectral radus of H. Ths s easly obtaned as ( ) = ME / MN = γ ρ H (A8) [4] V.T. Nguyen and U. E. Dorka, Applcaton of dgtal technque n a control syste for real-te sub-structure testng, 4th World Conference on Structural Control and Montorng, San Dego, USA, 006. [5] T. Horuch and T. Konno, A new ethod for copensatng actuator delay n real-te hybrd experents, Phl. Trans. R. Soc. Lond. A 359, pp , 001. [6] O.S. Burs, A. Gonzalez-Buelga, L. Vulcan, S.A. Neld, and D.J. Wagg, Novel couplng rosenbrock-based algorth for real-te dynac substructure testng, Earthquake Engneerng and Structural Dynacs, 007, 37, pp [7] X. Yang, Nuercal sulaton of real-te substructure testng wth shakng table, Dssertaton for the Master Degree Engneerng, Harbn Insttute of Technology, 007. [8] J.P. Conte, and T.L. Trobett, Lnear dynac odelng of a un-axal servo-hydraulc shakng table syste, Earthquake Engneerng and Structural Dynacs, 000, 9(9), pp. 1375~1404. [9] S. A. Neld, D. P. Stoten, D. Drury and D. J. Wagg. Control ssues relatng to real-te substructurng experents usng a shakng table, Earthquake Engneerng and Structural Dynacs, 005, 34, pp [10] R. Kubler and W. Schehlen, two ethod of sulator couplng. Matheatcal and coputer odelng of dynacal systes, vol.6 No, pp , 000. Therefore the output s zero-stable only when C N K N M E γ 1 (A9) K N Ths eans that any γ greater than unty wll lead to unstable response, whch coply wth what (13) ndcates. For the case of dynac specen, the atrces of G and H are obtaned as G K / M C / M XN, E, + 1 KE C X E (A10) 0 1/ M N H = 0 0 (A11) N N N N = For the physcal substructure wth contnuous oton, X E,+1 reans constant for vanshng Δt. Then G s agan a constant vector. ρ(h) s calculated as zero wth (A11), whch eans that zero-stablty s ensured no atter how great γ s. Ths atches the result as shown n Fg.5, where =1 for =0, ndcatng a stable response. Fg. 1. Coputaton scheatc of structure n RST wth pure nerta specen Fg.. Coputaton scheatc of structure n RST wth dynac specen γ =1.0 []=0.67 γ =0.8 []=1.16 γ =0.5 []=1.81 []=1.63 []= γ =0.1 γ =0 0.6 γ = K N C N M N (a) ξ N =0 K E CE ξ N =0. ξ N =0 M E REFERENCES [1] M.S. Wllas and A. Blakeborough, Laboratory testng of structures under dynac loads: an ntroductory revew, Phl. Trans. R. Soc. Lond. A 359, pp , 001. [] B.Wu, H. Bao, J. Ou, and S. Tan, Stablty and accuracy analyss of central dfference ethod for real-te substructure testng. Earthquake Engneerng and Structural Dynacs, 34, pp , 005. [3] B. Wu, Q. Wang, P.B. Shng, and J. Ou, Equvalent force control ethod for generalzed real-te substructure testng wth plct ntegraton. Earthquake Engneerng and Structural Dynacs, 36, pp , ξ N =0.8 ξ N =1.0 ξ N =0.5 []= (b) γ =0.5 Fg. 3. Spectral radus of CDM for RST wth pure nerta specen 50

6 Dsplaceent (c) =0.05 =0.65 =0.68 Dsplaceent() =0.068 =0.408 = Te (s) Fg. 4. Free vbraton responses wth pure nerta specen Fg. 8. Nuercal Sulaton result of RST usng shakng table (dynac specen) γ =0.01 γ =0.10 γ =1.00 γ = (a) ξ N=ξ E =0, γ ω=1 γ =0.01 γ =0.10 γ = γ = (b) ξ N=ξ E =0.05, γ ω=1 Fg. 5. Spectral radus of CDM for RST wth dynac specen Dsplaceent() Fg. 9. Photograph of test setup for RST Fg. 10. Test result wth γ =0.1 Coand Response Dsplaceent(c) =0.05 =0.43 =0.45 Dsplaceent() Coand Response Fg. 6. Free vbraton responses wth dynac specen Fg. 11. Test result wth γ =1.01 Dsplaceent(c) γ =0.10 γ =0.80 γ =1.003 Dsplaceent() Coand Response Fg. 7. Nuercal Sulaton result of RST usng shakng table (pure nerta specen) Fg. 1. Test result wth γ =1.3 51