Evaluation of Time Delay Margin for Added Damping of SDOF Systems in Real-Time Dynamic Hybrid Testing (RTDHT) under Seismic Excitation

Transcription

1 Evaluatio of Tie Delay Margi for Added Dapig of SDOF Systes i Real-Tie Dyai Hybrid Testig (RTDHT) uder Seisi Exitatio Saffet Ayasu & Bai Oztur Faulty of Egieerig Nigde Uiversity, Nigde, 51100, TURKEY SUMMARY: Tie delay is ow to be a iportat issue i strutural otrol systes whih ai at better dyai perforae of strutures durig a earthquae. Tie delay, whih adversely affets the stability ad perforae of the atively otrolled strutures, builds up i the syste aily due to sesors, ad atuatio ad ouiatio delays. This paper presets a pratial, siple ad exat ethod to opute the delay argi of SDOF systes i Real-Tie Dyai Hybrid Testig (RTDHT) uder seisi exitatio. Firstly, the trasedetal harateristi equatio of delayed syste is overted ito a polyoial without the trasedetality suh that its real roots oiide with the iagiary roots of the harateristi equatio exatly. A siple riterio to deterie the delay-depedey of the syste stability is developed usig the polyoial without the trasedetality. Afterwards, a expressio i ters of syste paraeters suh as ass,, dapig, ad stiffess, is derived for oputig the delay argi. Moreover, the variatio patter of delay argi with respet to these paraeters will also be ivestigated theoretially i order to idetify ey paraeters for stability. Fially, the theoretial delay argi results are verified by usig the tie doai siulatio apabilities of MATLAB progra. The outoe of this study will help us to siulate ad atiipate the seisi behavior of SDOF systes i real-tie dyai hybrid testig (RTDHT) with tie delay better durig a earthquae. Keywords: Tie delay, SDOF systes i real-tie dyai hybrid testig, Delay-depedet Stability 1. INTRODUCTION The real-tie dyai hybrid testig (RTDHT), whih was developed reetly [1-4], provides a strutural syste oposed of a experietal substruture ad a uerial substruture. Two substrutures are oupled suh that the target fores, aeleratios or displaeets are applied to the experietal substruture, ad the easured reatios are provided to the uerial substruture i retur. Aordigly, seisi respose of the etire struture a be evaluated. I this study, a liear SDOF syste, havig ass, dapig ad sprig oeffiiets of, ad, respetively, is osidered. The SDOF syste, whih is show i Figure 1.1, is oposed of a experietal substruture ad a uerial substruture. The subsripts e ad will be used i order to defie the experietal ad the uerial substrutures, respetively. Figure 1.1. Sheati represetatio of the substrutured SDOF syste odel

2 Mass, dapig ad sprig oeffiiets are defied as: e + =, e Mass, dapig ad sprig ratios are give as: e e =, =, Mass, dapig ad sprig proportio fators are give as: + =, e + = (1.1) e = (1.) e e e ρ = =, ρ = =, ρ = = (1.3) Therefore, the experietal ad uerial parts of ass, dapig ad sprig oeffiiets are: = e = (1.4a) = e = (1.4b) = e = (1.4) There is a ievitable tie delay i the trasfer syste respose deterioratig the test stability, beause of the iheret dyais of a hydrauli servo syste or a atuator.the dyai equatio of the SDOF syste osiderig tie delay is defied as [5]: x ( t) + x ( t) + x ( t) + x ( t ) + x ( t ) + x ( t ) = f ( t) (1.5) e e e where displaeet, tie delay ad exterally applied fore are show as: x, f(t) ad, respetively. It is well ow that tie delays a degrade the perforae of otrol systes ad a eve ae losed-loop syste ustable. Therefore, i the desig of a otroller, tie delays ust be tae ito aout ad ethods should be developed to estiate the axiu aout of tie delay (delay argi) that the syste a tolerate without losig its stability. Suh owledge o the delay argi ould also be helpful i the otroller desig for ases uertaity i the etwor-idued delays is uavoidable. There are several ethods i the literature to opute delay argis of tie-delayed otiuous systes. The oo startig poit of the is the deteriatio of all the iagiary roots of the harateristi equatio. The existig proedures a be lassified ito the followig five distiguishable approahes: i) Shur-Coh (Herite atrix foratio) [6-9]; ii) Eliiatio of trasedetal ters i the harateristi equatio [10]; iii) Matrix peil, Kroeer su ethod [6-9]; iv) Kroeer ultipliatio ad eleetary trasforatio [11]; v) Reasius substitutio [1-14]. These ethods dead uerial proedures of differet oplexity ad they ay result i differet preisios i oputig iagiary roots. Aog these ethods, Reasius substitutio ow also as pseudo-delay tehique has bee suessfully applied to the stability aalysis of SDOF with load-rate idepedet ad depedet restorig fores ad the ritial tie delays that ause istability of the syste have bee aalytially deteried [15]. I this paper, we ipleet the diret ethod reported i [10] to estiate the delay argi of SDOF systes i Real-Tie Dyai Hybrid Testig (RTDHT) with tie delays. The proposed ethod is

3 a aalytially elegat proedure that first overts the trasedetal harateristi equatio ito a polyoial without the trasedetality by eliiatig the expoetial ter of the harateristi equatio. This proedure does ot use ay approxiatio or trasforatio to eliiate the trasedetality of the harateristi equatio. Therefore, it is exat ad the real positive roots of the ew polyoial oiide with the iagiary roots of the harateristi equatio exatly. The resultig polyoial without the trasedetality also eables us to easily deterie the delay-depedey of the syste stability ad the sesitivities of rossig roots (root tedey) with respet to the tie delay. This is a rearable feature of the proposed ethod. The, a easy-to-use forula is derived to opute delay argis i ters of test struture paraeters suh as ass, dapig ad stiffess. It ust be also stated that the proposed ethod has bee suessfully applied ito the atively otrolled liear SDOF systes uder seisi exitatio [16], stability aalysis of tie-delayed eletri power systes ad geerator exitatio otrol syste to opute the delay argi for stability [17-18]. The delay argis are oputed for a wide rage of syste paraeters ad the theoretial delay argi results are verified by usig tie-doai siulatio apabilities of MATLAB/Siuli [19].. STABILITY ANALYSIS.1. Proble Forulatio For stability aalysis of SDOF test struture show i Fig. 1.1, the harateristi equatio is required. I the absee of a exteral exitatio fore, the harateristi equatio ould be obtaied easily by perforig Laplae trasfor of Eq. 1.5 as follows ( ) e e e ( s, ) = P( s) + Q( s) e = s + s + + s + s + e = 0 (.1) Usig Eq. 1. ad 1.3, the harateristi equatio of Eq..1 ould be writte i ters of the ass, dapig, ad sprig proportio fators as: ( ) ( ) ( s, ) = P( s) + Q( s) e = p s + p s + p + q s + q s + q e = 0 ( s, ) = (1 ρ ) s + (1 ρ ) ξω s + (1 ρ ) ω + ρ s + ρ ξω s + ρ ω e = 0 where ω refers to udaped atural frequey, ω =, while ξ refers to dapig ratio, ξ =. The oeffiiets of the polyoials of P(s) ad Q(s) are as follows: ( ω ) (.) = 1 ρ; 1 = (1 ρ ) ξω; 0 = (1 ρ ) ω p p p = ρ; 1 = ρξω; 0 = ρω q q q (.3) The ai goal of the stability studies of delayed systes is to deterie oditios o the delay for ay give set of syste paraeters that will guaratee the stability of the syste. As with the delayfree syste (i.e. = 0 ), the stability of the tie-delayed SDOF syste depeds o the loatios of the roots of syste s harateristi equatio defied by Eq... It is obvious that the roots are futios of the tie delay,. As hages, loatios of soe of the roots ay hage. For the syste to be asyptotially stable, all the roots of the harateristi equatio of Eq.., say s = s 1,s,...,s ust lie i the left half of the oplex plae. That is ( ( i )) 0 for ax real s < s s or C (.4) i s i

4 where C represets the left half plae of the oplex plae. Depedig o syste paraeters, there are two differet possible types of asyptoti stability situatios due to the tie delay, [7, 10]: i) Delay-idepedet stability: The harateristi equatio Eq.. is said to be delayidepedet stable, if the stability oditio of Eq..4 holds for all positive ad fiite values of the delay, [0, ). ii) Delay-depedet stability: The harateristi equatio of Eq.. is said to be delaydepedet stable, if the oditio of Eq..4 holds for soe values of delays belogig i the delay iterval, [0, ), ad is violated for other values of delay. jω = 0 1 = * * 1 jω = * + < * < 1 = 0 σ jω Figure.1. Illustratio of eigevalue oveet with respet to tie delay I the delay-depedet ase, the roots of the harateristi equatios ove as the tie delay, ireases startig fro = 0. Fig..1 shows the oveet of the roots. Note that the delay free syste ( = 0 ) is assued to be stable. This is a realisti assuptio sie for the pratial values of syste paraeters, the SDOF test syste is stable whe the total delay is egleted. Observe that as the tie delay, is ireased, a pair of oplex eigevalue oves i the left half of the oplex plae. For a fiite value of > 0 they ross the iagiary axis ad pass to the right half plae. The tie delay value at whih the harateristi equatio has purely iagiary eigevalues is the upper boud o the delay size, ow as delay argi, for whih the syste will be stable for ay give delay less tha this boud, <. I order to haraterize the stability property of Eq.. opletely, we first eed to deterie whether the syste for ay give set of paraeters is delay-idepedet stable or ot, ad if ot, to opute the delay argi i ters of syste paraeters. The stability proble of iterest a be stated as follows: Give: A tie-delay liear syste or its harateristi equatio of Eq..1 or.. Deterie: If it is delay idepedet stable or ot; if ot (the tie-delay syste is delay depedet stable); fid the delay argi. I the followig setio, we preset a pratial approah that gives a riterio for evaluatig the delay depedey of stability ad a aalytial forula to opute the delay argi for the delay-depedet ase [10].

5 .. Solutio Method A eessary ad suffiiet oditio for the syste to be asyptotially stable is that all the roots of the harateristi equatio of Eq.. lie i the left half of the oplex plae. I the sigle delay ase, the proble is to fid values of for whih the harateristi equatio of Eq.. has roots (if ay) o the iagiary axis of the s-plae. Clearly, ( s, ) = 0 is a ipliit futio of s ad whih ay, or ay ot ross the iagiary axis. Assue for sipliity that ( s, 0) = 0 has all its roots i the left half-plae. That is, the delay free syste is stable. If for soe, ( s, ) = 0 has a root o the iagiary axis at s = jω, so does ( s, ) = 0, for the sae value of ad ω. Hee, looig for roots o the iagiary axis redues to fidig values of for whih ( s, ) = 0 ad ( s, ) = 0 have a oo root. That is, s P( s ) + Q( s ) e = 0 ad P( s ) + Q( s) e = 0 (.5) By eliiatig expoetial ters i Eq..5, we get the followig polyoial: P( s ) P( s) Q( s ) Q( s) = 0 (.6) If we replae s by jω i Eq..6, we have the followig polyoial i s ω [5, 10]: W ( ω ) = P( jω) P( jω) Q( jω) Q( jω) = 0 (.7) Substitutig P( s ) = ps + p1s + p0 ad Q( s) = qs + q1s + q0 polyoials give i Eq..3 ito Eq..6, we obtai a augeted harateristi equatio as W ( ω ) = ( p q ) ω + ( p q + q q p p ) ω + p q = 0 (.8) Please ote the augeted polyoial of Eq..8 is fiite i ω ; it is idepedet of. Moreover, the trasedetal harateristi equatio with sigle delay give i Eq.. is ow overted ito a polyoial without trasedetality give by Eq..8 ad its positive real roots oiide with the iagiary roots of Eq.. exatly. The roots of this polyoial ay easily be deteried by stadard ethods. Depedig o the roots of Eq..8, the followig situatio ay our: i) The polyoial of Eq..8 does ot have ay positive real roots, whih iplies that the harateristi equatio of Eq.. does ot have ay roots o the jω -axis. I that ase, the syste is stable for all 0, idiatig that the syste is delay-idepedet stable. ii) The polyoial of Eq..8 has at least oe positive real root, whih iplies that the harateristi equatio of Eq. has at least a pair oplex eigevalues o the jω -axis. I that ase, the syste is delay-depedet stable. It ould be easily show that the polyoial of Eq..8 has oly oe positive real root, say ω, whih idiates that the SDOF test syste is delay-depedet stable. Oe the value of ω have bee foud for a give set of paraeters, the orrespodig value of the delay argi a be obtaied as follows:

6 jω ( ) P ( ) ( ) j ω, = jω + Q jω e = 0 jω P( j ) ω e = os ( ω ) jsi( ) ω = Q( jω ) P ( jω ) P( j ) os ( ) Re ω ω ad si( ) I = ω Q ( j ) = ω Q( jω ) Fro Eq..10, we a deterie a aalytial forula for the delay argi as follows: (.9) * 1 = Ta ω -1 P( jω ) I Q ( j ω ) P( jω ) Re Q( jω ) (.10) * A aalytial forula for the agle ad for the upper boud o the delay size for the SDOF struture a easily be obtaied by substitutig P( s) ad Q( s ) polyoials give i Eq.. ad.3 ito Eq..10 as follows: 3 * 1 1 ( pq1 p1q ) ω + ( p1q0 p0q1) ω Ta 4 ω ( pq ) ω + ( p0q + pq0 p1q1 ) ω p0q0 = (.11) 3. THEORETICAL AND SIMULATION RESULTS I this setio, the delay argi of SDOF systes i Real-Tie Dyai Hybrid Testig (RTDHT), is oputed usig the expressio give by Eq..11. Theoretial delay argi results are also verified by usig Matlab/Siuli. The paraeters of the SDOF test struture is as follows [5]: ρ = 0.5; ρ = ρ = 0.5; ξ = 0.05; ω = 31.4 rad / s Substitutig give paraeters ito Eq..3,.8, ad.11 the rossig frequey ad the orrespodig delay argi are foud to be ω = rad/s ad = 1. 6 s. The theoretial delay argi result is verified by usig Matlab/Siuli. Fig ad 3. show the displaeet respose of the SDOF test struture subjeted to the exteral exitatio supposed as a siusoidal groud otio with a aplitude of 0.1 g ad frequey of 4 Hz for three differet tie delays = 1. 3 s, = 1. 7 s ad = 13 s. As a be see fro Fig. 3.1, the respose has sustaied osillatios idiatig the argial stability of the syste for = 1. 7 s. Reall that theoretial delay argi was foud to be = 1. 6 s. It is lear that the differee betwee the theoretial delay argi ad the oe obtaied by siulatio is just 0.1 s, whih is egligible. Fig also learly illustrates that the respose is stable for a tie delay saller tha the delay argi ( = 1. 3 s = 1. 7 s ). *

7 argially stable ( = 1.7 s) stable ( = 1.3 s) 0.5 Displaeet () Tie (s) Figure 3.1. The respose of the displaeet for = 1. 3 s ad = 1. 7 s : Stable ad argially stable ases Fig. 3. gives the ustable ase with growig osillatios idiatig a ustable oditio for a tie * delay larger tha the delay argi ( = 13 s = 1. 7 s ). These resposes idiate that the proposed ethod ould be effetively used to deterie the stability delay argi of SDOF strutures Displaeet () Tie (s) Figure 3.. The respose of the displaeet for = 13 s : Ustable ase Moreover, the effet of the dapig proportio, ρ o the delay argi is ivestigated for two differet udaped atural frequeies, ω = 31.4 rad / s ad ω = 15.7 rad / s. The delay argi results are preseted i Table 3.1.

8 Table 3.1. Delay Margi Results for Differet Values of Mass Proportio Fator Mass * Delay Margi (s) proportio fator ρ ω = 31.4 rad / s ω = 15.7 rad / s Fig. 3.3 shows the variatio of the delay argi with respet to the dapig proportio, ρ. It is lear that for a give udaped atural frequey, the delay argi ireases as ρ is varied i the rage of ρ = Moreover, a irease i the udaped atural frequey ω results i a derease i the delay argi value for a give ass proportio ρ w = 31.4 rad/s w = 15.7 rad/s Delay Margi (s) Mass Proportio Figure 3.3. The variatio of the delay argi with respet to the ass proportio for two differet values of udaped atural frequey 4. CONCLUSIONS I this paper, a aalytial ethod is preseted to opute the delay argi for SDOF systes i Real-Tie Dyai Hybrid Testig (RTDHT) uder seisi exitatio. To assess the effet of a tie delay o the stability of SDOF struture, a aalytial expressio is developed to opute the delay argi i ters of syste paraeters. The siulatio results are foud to be i good agreeet with the delay argi obtaied fro the aalytial expressio. The ability to predit the delay argi for SDOF strutures is of sigifiae. The delay argi iforatio is vital towards deteriig whe a syste beoes ustable, to set the requireets for the opesatio ethods to redue delay, ad thus its adverse ipat o the syste dyais.

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