A STUDY OF RESPONSE SPECTRA OF ACTIVELY-CONTROLLED STRUCTURES DUE TO EARTHQUAKE

Transcription

1 13 th Wold Confeence on Eathquake Engineeing Vancouve, B.C., Canada August 1-6, 4 Pape No. 6 A STUDY OF RESPONSE SPECTRA OF ACTIVELY-CONTROLLED STRUCTURES DUE TO EARTHQUAKE Takashi MOCHIO 1 SUMMARY A newly esponse spectum is poposed in ode to evaluate simply the stuctual esponse of the activelycontolled stuctue subjected to eathquake. This pape adopts a linea single-degee-of-feedom stuctue with active mass dampe (AMD) system as an analytical model. Also, the eathquake wave is modeled as a stationay Gaussian andom pocess of which powe spectal density is equal to the Kanai-Tajimi spectum. The contol design is executed by using linea quadatic Gaussian contol stategy against the econstucted thee degee-of-feedom enlaged system to be coupled by soil, stuctue and AMD. Finally, the esponse amplification facto is given by the combination of the obtained statistical esponse values and the andom vibation theoy. Analytical esults ae compaed with numeical simulations, and both show a good ageement. As a esult, it seems that the validity of the poposed technique is confimed. INTRODUCTION Recently, the active contol technology is often applied to stuctues due to eathquake [1]. On the stuctual design fo such an actively-contolled stuctue, the detailed dynamical analysis is executed fom a initial design stage, since the design equies not only examination of stuctual integity but also estimation of contol chaacteistics against the stuctue. By the way, the dynamical seismic design of stuctues is often caied out by using esponse specta as functions of damping atio and natual peiod of a stuctue. Fo above-mentioned actively-contolled stuctue, howeve, the esponse spectum analysis is hadly executed, because the contol design has seveal paametes to be selected, and it consequently becomes moe difficult against applying the esponse specta with only two paametes such as damping atio and natual peiod to the design of actively-contolled stuctue with many design paametes. A newly esponse spectum, theefoe, is poposed in ode to evaluate simply the stuctual esponse of the actively-contolled stuctue subjected to eathquake. 1 Pofesso, Kinki Univesity, Wakayama, Japan. mochio@mec.waka.kindai.ac.jp

2 MATHEMATICAL MODEL TO DERIVE THE RESPONSE SPECTRA OF AN ACTIVELY- CONTROLLED STRUCTURE Fig. 1 shows the mathematical model to estimate the esponse specta of an actively contolled main stuctue due to eathquake. Although seveal kinds of active contol system have been developed fo the aseismic design, this pape adopts one of the simplest systems as shown in Fig.1, that is Active Mass Dampe (AMD) method, since the main pupose of this pape is not the development of a new active contol system but the poposal of a new esponse spectum fo stuctue with active contol system. f c m m d k y y d contolle sensos M z C K gound model w w&& Fig. 1 Mathematical model In Fig. 1, y, yd and z descibe absolute displacements of main stuctue, AMD and gound, espectively, and m,c and k mean mass, damping coefficient and stiffness to the stuctue, espectively, and also the paametes md and f show mass of AMD and contol foce. System equations fo the mathematical model shown in Fig. 1 ae given as follows: my && + c( y& z & ) + k( y z ) = f (1) d d my && = f () f = gu, (3) whee u is contol signal and g means tansfomed gain fom u to f. Suppose that the effect of soil/stuctue inteaction is negligible small, and then the gound acceleation z&& can be independently set up. The gound acceleation z&& may be fequently modeled as a esponse absolute acceleation though one-degee-of-feedom system subjected to Gaussian white noise w&&.

3 This model is well known as an atificial eathquake having so-called Kanai-Tajimi spectum [], and adopted in this pape. The govening equation, especially fo Fig. 1, is expessed by ( & ) ( ) Mz && C z& w K z w, (4) + + = and eplacing with v= z w in Fig. (4) yields whee && & &&, (5) v+ hωv+ Ω v = w C K hω =, Ω = M M. CONTROL RULE AND COVARIANCE OF STRUCTURAL RESPONSE By defining the elative displacements as follows, x = y z, xa = yd y (6) and coupling those vaiables with Eqs. (1) () (3) and (5), the following extended state equation elated to an actively-contolled stuctue can be obtained: X& = AX + Bu+ D w&&, (7) whee x 1 x& ω ξω Ω hω x a 1 X =, A =, x & a ω ξω v 1 v & Ω hω g m B = g 1 1+ m md m, D =, 1 c ξω = m k ω = m.

4 To achieve the active vibation contol, LQG algoithm is employed in this pape, and analysis is caied out unde the steady state conditions. Then a feedback gain vecto F is obtained as: [ ] 1 T FX, F B P , (8) u = = k k k k k k whee P is solution of the Riccati equation 1 T T Q PBB P + A P + PA=, (9) and Q, ae the weighting paametes in the following pefomance index: J = E + u T X QX, (1) whee E [ ] epesents the expected value elated to a andom vaiable. The esponse covaiance matix at steady state concening Eq. (7), V, σ κ κ κ κ κ V x xx& xxa xx& a xv xv& κxx & σ x& κxx & κ a xx && κ a xv & κxv && κx ax κxx σ T a x κ a xx κ a a xv κ & & a xv a& = E XX = κx& ax κx& ax& κx& ax σ a x& κ a x& av κx& av& κvx κvx κvx κ a vx σ a v κ & & vv& κvx & κvx && κvx & κ a vx && κ a vv & σv& satisfies the following Liapunov equation: ( ) ( ) T A -BF V+V A-BF +S=, (1) whee S is composed of the powe spectal intensity ζ fo a stationay Gaussian white noise wt && () of which the statistical chaacteistics may be expessed as follows, () =, () ( + ) = ( ) E w&& t E w&& t w&& t τ πζδ π. (13) Then S can be explained in the following fom: ( πζ ) T S = DD, (14) and consequently the covaiance of stuctual esponse can be obtained by solving Eq. (1). (11) DERIVATION OF RESPONSE SPECTRA Expession fo a esponse spectum concening the absolute esponse acceleation of stuctue ( pimay system ) is geneally classified into the following two pattens: 1) Ra ; maximum value of the absolute esponse acceleation of stuctue having natual peiod T and damping atio ξ R T ξ ; ) (, ) Ra maximum value of input acceleation (= esponse facto ),

5 whee the fome may be fequently used to know the absolute esponse acceleation of stuctue, the latte being convenient in ode to evaluate the esponse atio. The latte expession is, in this pape, adopted and also equal to the following tem, especially fo Fig. 1 : R y ξ = && && z max ) max. (15) Equation (15) is appoximately expessed by adopting the basic study [3] as follows : R && y theefoe, (, ) max = = && z max S.D. of absolute esponse acceleation of stuctue S.D. ; standad deviation S.D. of input acceleation R T ξ can be appoximately given by calculating y ( σ && z ) σ && and σ && z. ( σ && y ) (16) By Eqs. (1),(3) and (6), the following equation: g && y( = && x+ && z) = ξωx& ωx u (17) m is easily deived. Hence, substitution of Eq. (8) into Eq. (17) gives the equied squae of σ && y as gk1 gk gk3 && y = x + x& + xa σ ω σ ξω σ σ m m m gk4 gk5 gk6 + σ x& + σ a v + σv& m m m gk gk gk gk gk gk + ω ξω κ + ω κ + ω κ m m m m m m gk1 gk5 gk1 gk6 + ω κxv + ω κxv& m m m m xx && xxa xx& a gk gk3 gk gk4 + ξω κ xx & + a ξω κ xx && a m m m m gk gk5 gk gk6 + ξω κxv & + ξω κ m m m m xv && g kk g κ kkκ g kkκ m m m 3 4 xax& a 3 5 xav 3 6 xav& g g + kkκ + m m kkκ 4 5 x& av 4 6 x& av&

6 g kk 5 6κ vv + m Similaly, Eq. (5) yields ( ) &. (18) && && && &, (19) z = v+ w = hωv Ω v and then, the equied squae of σ && z can be also given by σ&& = 4h Ω σ + Ω σ + 4hΩ κ. () 4 3 z v& v vv& Finally, applying Eqs. (18) and () to Eq. (16) can appoximately poduce the (, ) R T ξ. EXAMINATION OF THE PROPOSED METHOD The esponse spectum fo an actively-contolled stuctue, in this pape, is deived fom Eq. (16). Hence, the validity of the poposed technique should be examined though the compaison between analytical esults and simulations. The following paametes, say ' efeence case ' fo the late convenience, ae pepaed fo the compaison: 1) paametes fo mass and contol md g =., =.98 m m qq qq 1 = 1, Q = ; ( qq= 1) ) paametes fo stuctual dynamics ω = π, ξ =.1. 3) paametes fo input Ω = 5 π, h =.6, ζ = 1 Table 1 shows the compaison between the analytical values following the definition of ight hand side in Eq. (16) and the numeical ones obtained by Monte Calo simulations accoding to the definition of left hand side in Eq. (16), as fo vaious combinations of paametes such as ξ, ω,qq (the emains ae the same as efeence case). The numeical values ae obtained though one hunded simulations by atificial eathquakes with 5-second peiod as the stationay andom pocess.

7 Table 1 Compaison of esults by analytical definition with those by numeical definition at Eq. (16) paametes qq ξ ω analytical value numeica l value.1 π π π π π π As seen in table 1, analytical values ae good ageements with numeical ones, and slightly exceed the numeical values in all cases, implying that the analytical estimation may be a kind of safety-sided design. Though this pape evaluates Eq. (16) unde the assumption of stationay andom pocess, the efeence [3] has examined it, but only fo non-active-contolled stuctue, with espect to some natual eathquakes as non-stationay andom pocess. Eventually, these esult in that the poposed method using Eq. (16) is consideed as a useful one in ode to obtain simply the esponse specta fo the actively-contolled stuctue subjected to eathquake. PARAMETRIC STUDY Diffeence of esponse specta with/without active contol R T, ξ with/without active contol ae given in Fig. and 3, espectively, with Response specta ( ) paamete ξ =.5,.1,.5 (the emains ae the same as efeence case). R ( ) T, ξ.4 Response facto ξ =.5 ξ =.1 ξ = Natual peiod T (sec) Fig. Response spectum R (, ) T ξ with active contol

8 R ξ =.5 ξ =.1 ξ =.5 Response facto Natual peiod T (sec) Fig. 3 Response spectum R (, ) T ξ without active contol Compaison of Fig. with Fig. 3 shows that the values of esponse facto with active contol become lowe than those without active contol in any fequency egion, and this means the usefulness of active contol technique to aseismic design at least elating to the paametes used in this pape. It is noted that esponse factos at highe fequency egion in Fig. tend to 1 which is geneal chaacteistics in esponse specta without active contol, since the contol ule adopted in this pape aims to educe the esponse level of x, x& by consideing moe weighting paametes at Eq. (1), which implies y z as x. Response specta with active contol Response specta with active contol can be simply obtained by the poposed method, concening the paametes of md m, g ( m ) and qq, as shown in Figs. 4,5,6, espectively. Damping atio ξ used in each figue is.1 as a geneal value. The emains of paametes ae the same as efeence case.

9 R Natual peiod T (sec) Fig. 4 Response spectum with active contol as function of md m R Response facto Response facto m d m g ( m ) Natual peiod T (sec) Fig. 5 Response spectum with active contol as function of g ( m )

10 R Response facto qq Natual peiod T (sec) Fig. 6 Response spectum with active contol as function of qq The paametes of md g m and qq ae a close elation to the pefomance of active contol. Incease m of each paamete, geneally speaking, implies the impovement of the pefomance as fo depessing the esponse level, especially at esonance egion, oppositely meaning that the esponse spectum with active contol moves towad to one without active contol as each paamete tends to zeo., ( ) By utilizing above mentioned esponse specta with active contol, one can easily evaluate not only the dynamic stuctual esponse of a stuctue but also the effects of educing the stuctual vibation though setting an active contol system, simultaneously. CONCLUSION The active contol technology is often applied to stuctues due to eathquake. On the stuctual design fo such an actively-contolled stuctue, the detailed dynamical analysis is executed fom a initial design stage, since the design equies not only examination of stuctual integity but also estimation of contol chaacteistics against the stuctue. On the othe hand, the dynamical seismic design of stuctues is also caied out fequently by using esponse specta as functions of damping atio and natual peiod of a stuctue. This pape descibes a newly esponse spectum in ode to evaluate simply the stuctual esponse of the actively-contolled stuctue subjected to eathquake, by coupling the concept of esponse specta with one of active contol.

11 This pape adopts a linea single-degee-of-feedom stuctue with active mass dampe system as an analytical model. Also, the eathquake wave is modeled as a stationay Gaussian andom pocess with the Kanai-Tajimi spectum. The contol design is executed by using linea quadatic Gaussian contol stategy. Analytical esults ae compaed with numeical simulations, and both show a good ageement. As a esult, it seems that the validity of the poposed technique is confimed. It is consideed that the newly developed technique is useful fo a simple estimation at initial seismic design stage fo an actively-contolled stuctue, though it is an appoximated method. REFERENCES 1. Chang JCH, Soong TT. " Stuctual contol using active tuned mass dampes. " Jounal of Engineeing Mechanics Division, ASCE; 16(6),198: Tajimi H. " A standad method of detemining the maximum esponse of a building stuctue duing an eathquake. " Poceedings of Second Wold Confeence on Eathquake Engineeing, Tokyo,Japan, Tajimi H. " Basic theoies on aseismic design of stuctues. " Repot of the Institute of Industial Science, 8(4), Univesity of Tokyo, 1959.