ON THE LINIARIZATION OF EXPERIMENTAL HYSTERETIC LOOPS

Transcription

1 ON THE LINIARIZATION OF EXPERIMENTAL HYSTERETIC LOOPS TUDOR SIRETEANU 1, MARIUS GIUCLEA 1, OVIDIU SOLOMON In this paper is presented a linearization ethod developed on the basis of the experiental hysteresis loop, by considering the ost general case, when there is no atheatical odel associated with the hysteretic behavior. The paraeters of linear odel are deterined for a predicted doinant freqency coponent in the response spectr of the echanical strctre eqipped by the stdied hysteretic device. The perforances of proposed ethod are illstrated on the basis of an experiental loop corresponding to a seisic protection device which is sed in base isolation of bildings. 1. INTRODUCTION The behavior of aterials, strctral eents or vibration isolators is described by hysteretic loops that are treated in a nified anner by a sing nonlinear differential eqation with no need to distingish different phases of the applied loading pattern. In practice, the Boc-Wen odel [1, ] is ostly sed within the following inverse prob approach: given a set of experiental inpt otpt data, how to adjst the Boc-Wen odel paraeters so that the otpt of the odel atches the experiental data. Usally, the experiental data are obtained by iposing cyclic relative otions between the onting ends on the testing rig of a sap aterial, strctral eent or vibration control device and by recording the evoltion of the developed force verss the iposed displaceent. Varios ethods where developed to identify the odel paraeters fro the experiental data of periodic vibration tests: analytical approaches, for exap [3] and different ethods based on genetic algoriths for ended versions of Boc-Wen odel, [4, 5, 6, 7]. One of the ost efficient techniqes for approxiating non-linear odels within the operating doain is the linearization ethod, both in deterinistic and stochastic systes. An iportant advantage of this approach is, nlike other ethods, it can readily be sed to deal with copx systes having any degrees of freedo and with copx types of excitations. There are any stdies abot 1 Institte of Solid Mechanics Roanian Acadey, Const. Mil, 15, RO Roanian-Aerican University, Expozitiei Bvd., 1B, RO Rev. Ro. Sci. Techn. Méc. Appl., Toe 55, Nº 1, P , Bcarest, 010

2 64 Tdor Siretean, Maris Gica, Ovidi Soloon statistical (or eqivant) linearization, [8, 9, 10] that have proved the efficiency of this approach. In the linearization techniqes literatre the linear odel is obtained by taking as reference the non-linear eqation which odels the hysteretic loop. This paper presents a linearization ethod that is developed only on the basis of experiental hysteretic loop, considering the general case, when there is no atheatical odel associated to the hysteretic behavior. The linear odel paraeters are deterined for a predicted doinant freqency coponent in the response spectr of the echanical strctre eqipped by the considered hysteretic device.. MATHEMATICAL MODEL In this paper is considered known the diensionss experiental hysteretic loop = sin t is the iposed cyclic in the coordinates and Φ where ( exp ) displaceent and Φ = Φ ( t) the force (Fig. 1). 1.0 f=1hz Φ() Φ () 0.5 Φ Fig. 1 Hysteretic loops: experiental ( ) and linear eqivant (- - -).

3 3 On the linearization of experiental hysteretic loops 65 The axi vale of hysteretic force is denoted by Φ and the dissipated energy per cyc associated to the experiental loop is given by, that is the area of srface enclosed by the hysteretic loop. In the n section is presented a linearization ethod and stdied its perforance sing an experiental loop of a seisic protection device Elastoeric Bearings (BIS) sed for base isolation of bildings [11]. In the proposed approach, the linear eqivant hysteretic force is given by: Φ =α+ z, (1) where α 0 and > 0 and the variab z satisfies the differential eqation (as in [9]): z& = az+ b+ & c, () a < (derived fro the stability condition) and [ a 1 ] = T, [ c 1 ] T with 0 a diensionss paraeter. = and b is N, the freqency response fnction corresponding to the inpt ( t) otpt Φ ( t ) is denoted by H ( ). and In order to deterine this fnction the iposed displaceent and velocity are written as: i ( t) = e t, (3) As ei t i = & t i e t. (4) Φ t = H, then by sing eqation (1), one can obtain: α i H z() t = e t, (5) H α z& i () t = i e t. (6) Therefore, the relation nber () becoes: H α H α i = a + bi+ c (7) and the following for of H( t ) is yielded H = α+ b + a a α c ab c i +, + a + a (8)

4 66 Tdor Siretean, Maris Gica, Ovidi Soloon 4 with the absolte vale and the phase having H = sin θ= ( α+ b) + a ( aα c ) + ( ab+ c) θ= tan + a ( ab c) ( b ) a( a c ) + 1, (9) α + + α, (10) ( ab + c) ( α+ b ) + a( aα c ) + ( ab+ c) N the following notations are considered: 1. (11) 1 =α+ b, = ( aα c), υ=. (1) Conseqently, the absolte vale of freqency response fnction, the phase and sin θ can be written: +υ H( ) = 1, + a (13) ( a) υ θ= tan 1 sin θ= + aυ, (14) ( υ a) ( + a )( +υ ) In this case, the energy dissipated per cyc by linear eqivant hysteretic force has the for: Fro the relations (15) and (16), since E > 0, one can derive:. (15) (16) ν sin θ( ) > 0 a <ν < 1. (17) a

5 5 On the linearization of experiental hysteretic loops 67 This ethod asse known a freqency, denoted, which is chosen by analyzing the freqency range of interest. The conditions for obtaining the linear odel paraeters are: sin θ is axi for the given freqency, that is dsinθ ( ) = 0 ; (18) d Φ freqency response fnction for eqals : H Φ = ; (19) energy dissipated per cyc by linear eqivant hysteretic force at eqals E d : The relation (18) iplies E = E. (0) d with the corresponding axi vale of sin θ( ) : = aυ, (1) sin sin θ = θ = ax υ a υ + a. () One can observe, as a < 0 (stability condition) then sing the eqation (1) is obtained υ< 0. An eqivant for of the relation (19) is 1 a +υ Φ =. (3) + By sing the relation (19), (0) and () one can derive ( a < 0 and υ< 0 ): υ a E υ+ a E = = d υ+ a π Φ υ+ π Φ. (4) d a Therefore, one has the paraeters α,, a, b, c and three relations (1), (3) and (4). As is readily seen fro (13), (16) and (3), the aplification factor H ( ) and dissipated energy E do not depend on the paraeters α and. Hence,

6 68 Tdor Siretean, Maris Gica, Ovidi Soloon 6 in the fraework of the proposed linearization ethod one can asse the particlar vales α= 0 and = 1. Conseqently, in this section is proposed the following: eqivant linearization algorith. 1. The vale of is chosen as fnction of the freqency range of interest.. Fro the relations (1) and (4) the vales of a and υ are obtained: a = µ + 1 µ 1, υ= 1 1 µ µ+. (5) π Φ where µ= > 1. 1 E d 3. The vales of b and c are deterined by the following relations: Φ µ+ 1 1 α αa = (sing (3)), = υ 1, b = and c =. µ 1 3. APPLICATION OF LINEARIZATION METHOD The efficiency of the proposed linearization ethod is illstrated for the hysteretic loop presented in figre 1, deterined by noralization of experiental data. The paraeters of the considered loop, involved in the identification of linear eqivant odel, are: E d = 1.187, = 0.9, Φ = The doinant freqency of the coponent of an oscillating syste eqipped by hysteretic device is assed to be -1 = πrad s (freqency f = 1Hz ). Then, by applying the previos eqivant linearization algorith one can find: a = 10.1, b = 1.67 and c = In figre are represented the experiental and the predicted hysteresis loops corresponding to the developed linear odel for three freqencies 0.5, 1 and 1.5 Hz. In order to show the qality of linear approxiation, in figre 3 are depicted the relative errors associated to the dissipated energy, E ( ) : ere E Ed =. (6) E On the graph is pointed ot the freqency range where the relative error is ss then 0%, which is the interval [0.7; 4] Hz. d

7 7 On the linearization of experiental hysteretic loops Experiental loop 1.0 f=0.5 Hz f=1.5 Hz f=1 Hz 0.5 Φ Fig. Hysteretic loops: experiental and linear eqivant for different freqencies. The efficiency of the proposed linearization algorith in ters of the approxiation of dissipated energy by the hysteretic device is assessed by the relative error was copted for a freqency range centered in. Usally, the linear viscos daping λ &, eqivant to the energy dissipated by a hysteretic device, is deterined fro: In this case, Ed λ=. (7) π =. (8) E E d

8 70 Tdor Siretean, Maris Gica, Ovidi Soloon 8 Figre 3 shows coparatively the relative errors of dissipated energy given by the proposed ethod and by the eqivant viscos daping..0 proposed linearization ethod viscos linearization ethod Relative error f [Hz] Fig. 3 Relative errors of dissipated energy. 4. CONCLUSIONS In this paper is presented an analytical ethod for the identification of a differential eqivant linear odel to approxiate experiental hysteretic loops. The linearization algorith reqires the deterination of only three paraeters which can be easily deterined fro the force-displaceent plot: aplitde of iposed cyclic displaceent, the axi force developed by the tested device and the dissipated energy per cyc, i.e. the loop area. Since the paraeters of the linear eqivant odel depend inherently on the cyc freqency, the linearization algorith is applied for a convenient choice of this freqency. The obtained reslts show that the dissipated energy per cyc, obtained for the proposed eqivant linear odel, is ss dependent on the cyc freqency than in the case of viscos eqivant odel. Moreover, the hysteretic crves portrayed by the linear eqivant odel are in good agreeent with the experiental loop arond the freqency sed for paraeters identification. If this freqency is a resonant

9 9 On the linearization of experiental hysteretic loops 71 freqency of an oscillating syste eqipped by the considered hysteretic device, then the proposed linearization ethod will provide a better description of syste response than the one predicted by viscos linearization. Acknowdgeent. The athors wish to express their gratitde to the Roanian Acadey for spporting this work throgh the Grant no. 36/ Theoretical and Experiental Analysis of the Dynaic Behavior of Degrading Strctres. Received March 6, 010 REFERENCES 1. BOUC, R., Forced vibration of echanical systes with hysteresis, Proceedings of the Forth Conference on Non-linear oscillation, Prage, Czechoslovakia, WEN, Y.K., Method for rando vibration of hysteretic systes, Jornal of the Engineering Mechanics Division, 10,, pp , SIRETEANU, T., GIUCLEA, M., MITU, A.M., An analytical approach for approxiation of experiental hysteretic loops by Boc-Wen odel, Proceedings of Roanian Acadey, Series A, 10, 1, GIUCLEA, M., SIRETEANU, T., MITU, A. M., Use of genetic algoriths for fitting the Boc- Wen odel to experiental hysteretic crves, Rev. Ro. Sci. Techn. Méc. Appl., 54, 1, pp. 3-10, SIRETEANU, T., GIUCLEA, M., MITU, A. M., Identification of an ended Boc-Wen odel with application to seisic protection throgh hysteretic devices, Coptational Mechanics, 45, 5, pp , SONG J., ARMEN Der KIUREGHIAN, Generalized Boc-Wen Model for Highly asyetric Hysteresis, Jornal of Engineering Mechanics, 13, 6, pp , KWOK, N.M., HA, Q.P., NGUYEN, M.T., LI, J., SAMALI, B., Boc Wen odel paraeter identification for a MR flid daper sing coptationally efficient GA, ISA Transactions, 46, pp , DOBSON, S., NOORI, M., HOU, Z., DIMENTBERG, M., BABER, T., Modelling and rando vibration analysis of SDOF systes with asyetric hysteresis, International Jornal of Non- Linear Mechanics, 3, 4, pp , HURTADO, J. E., BARBAT, A. H., Eqivant linearization of the Boc-Wen hysteretic odel, Engineering Strctres,, pp , ROBERTS, J. B., SPANOS, P. D., Rando vibration and statistical linearization, John Wiy and Sons, GUERREIRO, L, FERREIRA JP, COLATO GP, CASTELLANO G, BALDO P, Base isolation for seisic protection the new hospital in Lisbon, Proceedings of IABSE Syposi on Strctres and Extree Events, Septeber 14-16, Lisbon, Portgal, 005.