REMARKS ON MODELLING OF PASSIVE VISCOELASTIC DAMPERS

Transcription

1 REMARKS ON MODELLING OF PASSIVE VISCOELASTIC DAMPERS Roma Lewadows, Bartosz Chorążyczews Isttute of Structural Egeerg, Poza Uversty of Techology, ul. Potrowo 5., Pozań, Polad e-mal: Receved ; accepted Abstract. Ths paper cocetrate o revew of the results of realzed expermetal tests of vscoelastc dampers. The most popular ad well-ow models frequetly used to descrbg these dampers are here dscussed. Expermetal tests of desged by authors VE dampers ad the results of that tests are preseted. The geeralzed Maxwell model ad the Zeer model were used to represet the behavour of VE dampers. The tats of above models were determed by usg the proposed method of fttg metoed models to expermetal data. Keywords: vscoelastc (VE) dampers, rheologcal models, modellg of parameters, expermetal tests. Itroducto Gog beyod ew lmts desgg moder buldgs ad structures s carryg alog the ecessty of cludg ( absolutely ew dmeso) loads that were tlled ow eve gored. It s for example case of loads geerated by wds. The sgfcatos of them were appeared tall buldgs. Dyamc growth of moder agglomeratos ad shrg of free areas coerce locato of buldgs o the earthquae-threate terras. The evoluto of ew techologes ad ecessty of eepg varous maufacturg rgours s causg that bgger atteto s focused o dyamc loads whch have a fluece o producto les, maches ad worers. The passve vscoelastc dampers (VE dampers) are very ofte used all metoed cases. There are two ma types of these dampers. Frst type s vscoelastc lqud damper, whch ca have the shape of cylder flled wth slcoe gel (e.g. GERB dampers). The secod type s vscoelastc sold damper, whch st of e.g. three steel plates ad two layers of vscoelastc polymer (e.g. ISD Polymer or ). Applcato of passve dampers demads exact descrpto of behavour of these dampers. The smplest models, le the Maxwell model, the Kelv-Voght model [,,3] or the Burgers model are very ofte used to descrbg VE dampers. These models are able to represet correctly mechacal propertes of the dampers oly for dvdual frequeces. Because of that, ther applcato structure desgg s qute arrow. Ufortuately the mechacal propertes of the major umber of polymers are strogly frequecydepedet. As t was metoed, the smplest models allow descrbg the behavour of the VE dampers oly for selected frequecy of exctato. Stadard rheologcal models le the geeralzed Maxwell or geeralzed Vogt models are ofte used to descrbe propertes of may polymers. The model parameters are determed o a base of expermetal results. Stadard Mechacal Models ad ther practcal applcato descrbg the VE dampers were preseted by S.W. Par [4,5]. The vulerablty of these models s geeratg, certa cases, egatve (uphyscal) values of searched parameters (e.g. stffess) of the VE dampers. Authors have expereced smlar problems durg the detfcato of parameters of ther ow VE dampers. I some stuatos fdg of proper relaxato tmes for calculatos become troublesome. It should be cocered that relaxato tmes are geerally specfed a pror [4,6] ad ths s the ma source of problems. The ext group of models descrbg the behavour of VE dampers are the fractoal dervatve models. These models are able to reproduce the VE damper ature frequecy ad tme domas a effectve way. Usg fractoal calculus a umber of fractoal models ca be developed, e.g. fractoal Kelv-Voght model [7], fractoal Zeer model [8,9], fractoal Jeffreys model [] or fractoal Maxwell model. The Maxwell model ad the Zeer model were tested by authors. The problem to solve wth these models s about good ad effcet fttg them to expermetal data. Otherwse there are materals whch behavour exceeds beyod metoed models.

2 Ths wor cocetrate o revew of exstg models of vscoelastc dampers, ther advatages ad dsadvatages. It s a troducto to develop mathematcal model of the damper, whch parameters wll ot deped o frequecy of exctato. The am of the future vestgatos wll be developg of possble the most uversal model descrbg the behavour of VE damper a wde rage of frequeces (for stace from to 5 rad/s).. Exemplary expermetal results ad ther characterstc propertes The statc ad the dyamc tests are used to detfy propertes of the VE dampers. The dyamc tests are the most effectve way to catch the complex relatoshps betwee parameters of the damper ad the frequecy of exctato, ampltude of exctato ad temperature. I ths test, the VE damper s exposed to the harmoc varable exctato, e.g. dsplacemet. Geerally, the exctato chages harmocally wth tme [ - 5]. The metoed type of expermetal tests face wth some problems test system lmts form. They ca be performed oly for a relatvely arrow rage of frequeces. As wder rage of frequeces of exctatos you wat to overvew, the smaller ampltudes by test system ca be realze. The rage of avalable frequeces of exctato (whch test statos ca geerate) s dramatcally reduced for large ampltudes of dsplacemets. The propertes of vscoelastc materals are ofte preseted by the cocept of the complex parameter: X = X + X, (.) where: X s the real part of ths parameter (stffess), X s the magary part (dampg) ad = s the magary ut. Specfyg, X * ca be e.g. the complex stffess (f force-dsplacemet relato s aalyzed) or the complex shear modulus [4]: G * = G + G, (.) where: G s the storage modulus ad G s the loss modulus. The quotet of loss modulus ad storage modulus s called the loss factor η,.e.: η = G / G. (.3) I Fgs..., the chose, exemplary graphs of expermetal data, tae from lterature, are preseted. Preseted plots are showg how dfferet propertes of vscoelastc materal ca be. Observato of the loss factor plots (Fgs.,.6,.8 ad.) gves followg remars. The loss factor curve ca be qute tat (Fg..), ca grow wth the frequecy exctato (Fg..6) or ca decrease as t s show o Fgs.8 ad.. The forecastg of the behavour of VE damper beyod the measured frequeces s very dffcult. The storage ad loss modulus or stffess have geerally the same character ad all cases these quattes crease wth exctato frequecy. It ca be observe, that the major part of forgog expermetal data covers a relatvely arrow rage of frequeces. It gves us a questo about further course of dscussed parameters the frequecy doma. [MPa] 3,5 3,5,5,5,5,5 3 Frequecy [Hz] Shear storage modulus G Shear loss modulus G Fg. Shear storage ad loss modulus of VE sold damper wth vscoelastc materal maufactured by 3M (She, Soog, Chag, La, 995 [5]) Loss factor,6,,8,4,5,5 3 Frequecy [Hz] Fg. Loss factor η of VE sold damper (She, Soog, Chag, La, 995 [5]) Storage stffess [MN/m],5,5,5 4 6 Frequecy [Hz] Fg.3 Storage stffess K of VE sold damper (M, Km, Lee, 4 [])

3 3,6,5,5 Loss stffess [MN/m],5,5 Loss factor,4,3,, 4 6 Frequecy [Hz],9,5,5,5 3 3,5 Frequecy [Hz] Fg.5 Loss stffess K of VE sold damper (M, Km, Lee, 4 []) Fg.8 Loss factor η of VE sold damper at % stra (Aprle, Iaud, Kelly, 997 [6]), 44 Loss factor,8,4 Storage modulus [KPa] Frequecy [Hz],5,5,5 Frequecy [Hz] Fg.6 Loss factor η of VE sold damper (M, Km, Lee, 4 []) Fg.9 Shear storage modulus G of VE sold damper at 4 o C ad 5% stra (Hggs, Kasa, 998 []) Storage modulus [MPa] Storage stffess [N/m],,,5,5,5 3 3,5 Frequecy [Hz],,,, Frequecy [rad/s] Fg.7 Shear storage G modulus of VE sold damper at % stra (Aprle, Iaud, Kelly, 997 [6]) Fg. Storage stffess K of VE lqud damper (S.W. Par based o Mars tabular expermetal data, [4])

4 a) u(t) b) Loss stffess [N/m],, u(t) c q(t) u(t) c q(t) u(t),,,, Frequecy [rad/s] Fg. Loss stffess K of VE lqud damper (S.W. Par based o Mars tabular expermetal data, [4]) Loss factor,, Frequecy [rad/s] Fg. Loss factor η of VE lqud damper (data from [4]) 3. Aalyss of usage possbltes of rheologcal models to descrpto of dyamc behavour of VE dampers I ths secto the dyamc propertes of several rheologcal models wll be determed ad thers ablty to model qualtatvely the dyamc behavour of VE dampers wll be dered. 3.. The smple Kelv model The Kelv model, whch sts of sprg ad dashpot coected parallel (see Fg. 3.a) s govered by the equato ( q( t) +τ q& ( )) u ( t) = t, (3.) where t s tme, u (t) s the dampers force, q (t) s the dampers relatve dsplacemet, τ = c / s the relaxato tme. Moreover, ( o& ) = d ( o) / dt, c ad are the dampers stffess ad dampg factor, respectvely. Fg 3. The Kelv ad Maxwell rheologcal models If the damper s harmocally excted,.e. q( t) = q exp( λt), (3.) where = ad λ s the exctato frequecy the damper force the case of steady state vbrato ca be descrbed as u( t) = u exp( λt). (3.3) Moreover, t s easy to show that u ( + τ λ) q =, (3.4) what meas that the storage modulus K = s the tat fucto of λ, whle the loss modulus K = τ λ ad the loss factor η = K / K = τ λ are lear fuctos of λ. It s obvous that the smple Kelv model s ot able to model correctly the dyamc behavour of VE damper (see also Fgs ). The Kelv model s used [, 8, 9] to aalyze the dyamc behavour of buldgs wth the vscoelastc dampers. 3.. The smple Maxwell model The equato of Maxwell model, show Fg. 3.b, s gve by u ( t) + τ u& ( t) = τ q& ( t). (3.5) After troducg Eqs (3.) ad (3.3) to Eq (3.5) we obta what meas that λ u τ K =, + τ λ τ λ + = τ λ q, (3.6) + τ λ τ λ K =, + τ λ η =. (3.7) τ λ Fgures show how the storage ad loss modulus ad the loss factor vary wth the exctato frequecy for both the Kelv (dashed le) ad the Maxwell model (sold le). Accordg to the Maxwell model the storage modulus gradually growth wth λ ad the value of storage modulus s approxmately equal

5 for large values of λ τ λ. The fucto of loss modulus K ( ) has extremum, what ca be observed some expermets (see [7]). I ths model extremum s at λ e = /τ ad K ( λ e ) = /. Accordg to the Maxwell model the loss factor η (λ) always decreases wth λ what s stet wth some expermetal data (see Fgs..8 ad.). However, for λ, η (λ) what s ot observed expermetally. Moreover, K () = what s ot cofrmed some expermets (see Fg..3 ad.7). The Maxwell model s used [,, ] to aalyze the dyamc respose of buldgs. u( t) = u ( t), u t) = q( ), (3.8) = ( t u ( t) + τ u& ( t) = τ q& ( t), (3.9) where u (t) s the total damper force, u ( t ) s the force elastc sprg, u (t), ( =,,..., ) are the forces the Maxwell elemets, τ = c /. Moreover, s the stffess of elastc sprg ad c, are the stffess ad dampg factor of the -th Maxwell elemet, respectvely. The storage modulus λ e The loss factor λ e The exctato frequecy The exctato frequecy Fg 3. The storage modulus for the Kelv ad Maxwell model versus exctato frequecy Fg 3.4 The loss factor for the Kelv ad Maxwell model versus exctato frequecy u(t) The loss modulus λ e c c c c u(t) q(t) The exctato frequecy Fg 3.5 Scheme of the advaced rheologcal model Fg 3.3 The loos modulus for the Kelv ad Maxwell model versus exctato frequecy 3.3. Advaced rheologcal model of dampers I ths subsecto the model, show Fg. 3.5, whch elastc sprg coected parallel wth elemets of Maxwell type s dered. I a case = we have the well-ow stadard model. If we remove the elastc sprg from ths model (.e. whe = ) we obta the model used by Par [4] to descrbe behavour of the flud damper. I paper [] the metoed above model s used to aalyze the dyamc respose of buldg equpped wth dampers. The behavour of dered model ca be descrbed wth a help of the followg system of equatos: If damper s harmocally excted, the steady state respose of damper ca be descrbed by u( t) = u exp( λt), u ~ ( t) = u exp( λt), (3.) u ( t) = u ~ exp( λt), ( =,,..., ). (3.) Itroducg Eq (3.), (3.) ad (3.) to Eqs (3.8) ad (3.9) we obta u ~ = q, u ~ τ λ + u = τ λ q, (3.) + τ λ + = + λ = + q τ. (3.3)

6 It meas that the fuctos of storage ad loss modulus ad the fucto of loss factor are: = + K ( λ ) = +, (3.4) = + τ λ K ( λ ) =, (3.5) [ + ( τ j λ) ] τ λ = j=, j η( λ) = [ + ( τ j λ) ] + [ + ( τ jλ) ] j= = j=, j (3.6) May partcular models ca be obtaed f dfferet Maxwell elemets are tae to accout. These models have much more free parameters the smple models but the qualtatve propertes of geeralzed models are smlar to the smple Maxwell model or stadard model. Now, the propertes of stadard model wll be brefly dscussed. For =, from Eqs (3.4) (3.6) we obta results for the stadard model. I ths case the fucto K ( λ) s smlar as oe for the Maxwell model, but ow we have the tal stffess because K ( ) = what s agreemet wth expermetal results show Fgs..3 ad.7. The fucto of storage modulus s detcal as for the Maxwell model. However the fucto of loss factor η (λ) s dfferet, especally for small values of τ λ because η ( ) =. Fgures 3.6 show plots of fucto η (λ) for dfferet stffess rato of a = /. loss factor a=. a=.. a= odmesoal frequecy Fg. 3.6 The loss factor versus odmesoal frequecy stadard model 3.4. Fractoal Maxwell model of dampers Recetly, the ew rheologcal models based o fractoal calculus have bee proposed to descrbe VE dampers behavour [8,7,3]. Several fractoal rheologcal models were proposed [8,3]. Oe of them s the fractoal Maxwell model. The equato of the four parameters fractoal Maxwell model s gve by (see [8]) d u( t) β d q( t) u ( t) + τ = τ, (3.7) β dt dt The model parameters, τ,, β must be postve umbers, <, β ad β. Mars et al. [3] developed the fractoal Maxwell model whch parameters are complex umbers. Symbols le d u( t) dt deote the fractoal dervatves of order of u (t) wth respect to tme. Some valuable formato about fractoal calculus ca be foud [4]. Itroducg Eqs (3.) ad (3.3) to Eq (3.7) ad tag to accout that we obta where u d exp( λt) = ( λ ) exp( λt), (3.8) dt π π = + s, (3.9) ( ) ( K + K ) q = K ( + η) q =, (3.) βπ ( β ) π + β K = ( τ λ), (3.) π + + βπ ( β ) π s + s β K = ( τ λ), (3.) π + + η = βπ s + = βπ + β ( β ) π s. (3.3) ( β ) π I lterature also exst the fractoal Maxwell model wth three parameters. I ths case = β ad π + K = ( τ λ), π + + (3.4) π s K = ( τ λ), π + + (3.5)

7 π s η =. (3.6) π + The propertes of three parameters model, for dfferet values of parameter are show o Fgs O these fgures the odmesoal frequecy s defed as τ λ. storage modulus.e+5 =. 9.E+4 =.8 8.E+4 =.6 7.E+4 6.E+4 =.4 5.E+4 4.E+4 3.E+4.E+4.E+4.E odmesoal frequecy Fg. 3.7 Storage modulus of fractoal Maxwell model for dfferet values of parameter loss modulus 6.E+4 5.E+4 4.E+4 3.E+4.E+4.E+4 =. =.8 =.6 =.4.E odmesoal frequecy Fg. 3.8 Loss modulus of fractoal Maxwell model for dfferet values of parameter Now, the qualtatve agreemet of the fucto K ( λ) wth expermetal results s observed. The fucto of loss modulus could be very flat (for stat whe =. 4 ) what agree wth results show Fg.. or (for >. 4 ) ca well approxmate results show o Fgs..5 ad.. O the other had, the fucto of loss factor η (λ) does t agree wth the data show Fg..6 but t s qualtatve agreemet wth the values of expermetal loss factor preseted Fg... Moreover, η( ) = tg( βπ / ) what meas that ow the loss factor has a fte value for λ =. It s mportat dfferece comparso wth the loss factor fucto of the smple Maxwell model whch values go to fty f λ goes to zero. I cocluso, the tree parameter fractoal Maxwell model s qualtatve agreemet wth the expermetal data gve [4]. Fg. 3.9 Loss factor of fractoal Maxwell model for dfferet values of parameter Results of the fourth parameter fractoal Maxwell model (for =. 8 ad chose values of β parameter) are show o Fgs The results are qualtatvely smlar to the oes obtaed for the three parameter fractoal Maxwell model. Oe mportat excepto s that values of the loss modulus fucto ad the loss factor fucto ca be egatve (uphyscal) f the dfferece betwee values of ad β parameters are suffcetly large. storage modulus 9.E+4 8.E+4 7.E+4 6.E+4 5.E+4 4.E+4 3.E+4.E+4.E+4 β=.8 β=.7 β=.6.e odmesoal frequecy Fg. 3. Storage modulus of fractoal Maxwell model for dfferet values of β parameter ( =. 8 ) 4. The parameters detfcato of composte models of vscoelastc damper 4.. Descrpto of dampers ad expermetal test I the expermetal tests a VE dampers made of VHB 4959 materal (vscoelastc materal was maufactured by 3M) were used. The dampers were fabrcated from two layers of vscoelastc materal ad three steel plates. Cyclc test were performed a MTS 8 worstato. The samplg frequecy was 8 Hz ad the dampers were exposed to a harmoc varable dsplacemet.

8 4.E E+4 β=.8 loos modulus.e+4.e+4 β=.7 [N/m] 5.E+ β= E odmesoal frequecy Fg. 3. Loss modulus of the fractoal Maxwell model for dfferet values of β parameter ( =. 8 ) loss factor β=.8 β=.6 β= odmesoal frequecy Fg. 3. Loss factor of the fractoal Maxwell model for dfferet values of β parameter ( =. 8 ) Durg tests, two output fuctos u(t) (fucto of forces a tme doma) ad q(t) (fucto of dsplacemets a tme doma) were obtaed. The data were collected for varous ampltudes ad varous frequeces of exctato. The detfcato procedure sts of two steps. The frst step of detfcato procedure of the VE damper parameters (descrbed brefly below) was appled to each frequecy of exctato. For the gve frequecy of exctato the smple Maxwell model was used to represet the damper behavour. The measured dsplacemets of the damper were approxmated by usg the least-square method. The parameters c d (the dampg coeffcet of damper) ad d (the damper stffess) of the smple Maxwell model were determed wth a help of the least-square method. I detals ths procedure s descrbed [5]. The storage stffess, loss stffess ad the loss factor were calculated ad thers frequecydepedeces are show Fgs 4. ad 4.. The ext secto cotas the detaled descrpto of the secod step of detfcato procedure Frequecy [rad/s] Storage stffess K Loss stffess K Fg 4. The storage stffess ad the loss stffess from the expermets Loss factor,,8,6,4,, Frequecy [rad/s] Fg 4. The loss factor from the expermets 4.. Fttg models to expermetal data Havg, from expermet ad for dfferet exctato frequeces, the values of storage stffess, the loss stffess ad the loss factor the geeralzed Maxwell model ad the geeralzed Zeer model were used to descrbe the overall behavour of VE dampers. At the begg of the secod step of detfcato procedure the tal values of relaxato tmes are assumed. The fdg of proper relaxato tmes wll be descrbed later the ths secto. The least-square method was used to ft model parameters. I both cases, the followg fuctoal was used: N K ( λ ) K( λ = + ) J, (4.) = K e ( λ ) K e ( λ ) where: K (λ ) ad K (λ ) are the searched storage ad loss stffess of the model, K e (λ ) ad K e (λ ) are the storage ad the loss stffess calculated from expermetal data, N s the umber of frequeces, for whch expermetal data are ow. From statoary codtos of fuctoal (4.) followg system of equatos s determed:

9 A = g, (4.) where: A s the matrx of rj coeffcet, s the vector of searched model parameters j ad g s the vector of γ r free terms. The expressos for rj ad γ r are gve as follow: N λ τ rτ λτ rτ = rj +, = ( + λτ r )( + λτ ) K e ( λ ) K e ( λ ) (4.3) N λτ r λ τ r γ r = + + λ τ r K e ( λ ) K e ( λ ). (4.4) = The tats of the VE damper model are determed by solvg the system of equatos (4.). The value of fuctoal (4.) ca be also calculated. I ths procedure the approprate relaxato tmes τ ad umber of Maxwell elemets must be assumed a pror. If they are wrog, the system of equatos gves us egatve (uphyscal) values of stffess of Maxwell elemets. Now the mmum of fuctoal (4.) wth respect to the relaxato tmes must be determed. It s ot so smple, because sometmes fuctoal reaches ts mmum for egatve values of j. Authors used the Solver software (avalable Excel), to search the mmum of fuctoal (4.) wth respect to τ, but ths method does ot guaratee faultless results ad the mmzato procedure must be repeated startg from dfferet set of tal values of relaxato tmes. The computed tats ca be used to draw the hysteress plots, whch ca be compared wth smlar plots obtaed from expermetal data. The agreemet of geeralzed Maxwell model wth the expermetal data s show Fgs Coclusos The VE dampers, whch ca sgfcatly reduce vbrato of structures, are usually modelled wth a help of Kelv-Voght or Maxwell rheologcal models [, 8- ]. These models are smple but our ow expermetal results ad results prevously publshed show dfferet propertes of VE dampers. However, papers that descrbe results of expermetal testg of VE dampers are rather rare ad the expermetal data are gve for the restrcted rage of exctato frequeces. The propertes of vscoelastc materals used the expermetal test are demostratg sgfcat dffereces. For example, some tests show that the loss factor decrease wth the exctato frequecy whle results of others tests show a opposte relato betwee the loss factor ad the exctato frequecy. Force [N] , -,,9 Expermetal data Dsplcemet [m] Geeralzed Maxwell model Fg 4.4 Geeralzed Maxwell model ftted to expermetal data (VE damper excted by, mm dsplacemet at 5 Hz) 5 Force [N] Force [N] , -,,9 Dsplcemet [m] Expermetal data Geeralzed Maxwell model Fg 4.3 Geeralzed Maxwell model ftted to expermetal data (VE damper excted by, mm dsplacemet at Hz) It should be metoed that geeralzed models could well approxmate the behavour of VE damper oly some rage of frequeces, (see Fg. 4.6 for detals) , -,,9 Dsplcemet [m] Expermetal data Geeralzed Maxwell model Fg 4.5 Geeralzed Maxwell model ftted to expermetal data (VE damper excted by, mm dsplacemet at Hz)

10 [N/m] Frequecy [rad/s] expermetal K theoretcal K Fg 4.6 Comparso of expermetal storage stffess ad theoretcal storage stffess delvered by geeralzed Maxwell model (VE damper excted by, mm) Cosderg that, dfferet models ca be used to descrbe the VE dampers. The worst descrpto of the VE damper seems to be gve by the Kelv-Voght model. The geeralzed Maxwell model or the Zeer model s the better choce. The best ad most adequate qualtatve descrpto of behavour of VE dampers could be doe wth a help of the fractoal dervatve model. The reported expermetal results show that the storage stffess decrase wth the exctato frequecy for λ 5Hz. Ths fact eeds further vestgato. Acowledgemets The authors acowledge facal support receved from the Poza Uversty of Techology (Grat No. BW. -97/7) coecto wth ths wor Refereces. Sgh, M.P., Verma, N.P. ad Moresch, L.M., Sesmc aalyss ad desg wth Maxwell dampers, Joural of Egeerg Mechacs, 3, Vol. 9, p Lee, S.H., So, D.I., Km, J., M, K.W., Optmal desg of vscoelastc dampers usg egevalue assgmet, Earthquae Egeerg ad Structural Dyamcs, 4, Vol. 33, p Sophocleous A.A., Sesmc Cotrol of Regular ad Irregular Buldgs Usg Vscoelastc Passve Dampers, Joural of Structural Cotrol,, Vol. 8, p Par S.W., Aalytcal modelg of vscoelastc dampers for structural ad vbrato cotrol, Iteratoal Joural of solds ad Structures,, Vol.38, p Par S.W., Rheologcal Modelg of Vscoelastc Passve Dampers, Smart Structures ad Materals : Dampg ad Isolato, SPIE,, Vol. 433, p Syed Mustapha S.M.F.D., Phllps T.N., A dyamc olear regresso method for the determato of the dscrete relaxato spectrum, J.Phys. D: Appl. Phys., 3, 33, p Papoula K.D., Paosaltss V.P., Korovajchu I., Kurup N.V., Rheologcal represetato of fractoal dervatve models lear vscoelastcty, Rheologcal Acta, 7 ( press) 8. Prtz T., Aalyss of four-parameter fractoal dervatve model of real sold materals, Joural of Soud ad Vbrato, 996, 95(), p Ataacovc T.M., A modfed Zeer model of a vscoelastc body, Cotuum Mech. Thermody.,, 4, p Sog D.Y., Jag T.Q., Study o the ttutve equato wth fractoal dervatve for the vscoelastc fluds Modfed Jeffreys model ad ts applcato, Rheologca Acta, 998, Vol. 37, No. 5, p M K.W., Km J., Lee S.H., Vbrato tests of 5-storey steel frame wth vscoelastc dampers, Egeerg Structures, 4, 6, p Hggs Ch., Kasa K., Expermetal ad aalytcal smulato of wd respose for a full-scale VE-damped steel frame, Joural of Wd Egeerg ad Idustral Aerodyamcs, 998, 77&78, p Lewc D.E., Dehart D.W., Combg wood ad vscoelastc materal, Proceedgs of the World Coferece o Tmber Egeerg, Vacouver, Brtsh Columba, July, 4. Cho H., Km W.B., Lee S.J., A method of calculatg the o-lear sesmc respose of a buldg braced wth vscoelastc dampers, Earthquae Egeerg ad Structural Dyamcs, 3, Vol. 3, p She K.L., Soog T.T., Chag K.C., La M.L., Sesmc behavour of reforced cocrete frame wth added vscoelastc dampers, Egeerg Structures, 995, Vol. 7, No. 5, p Aprle A., Iaud J.A., Kelly J.M., Evolutoary Model of Vscoelastc Dampers for Structural Applcatos, Joural of Egeerg Mechacs, 997, p.55-56, 7. Mars N., Tme doma aalyss of geeralzed vscoelastc models, Sol Dyamcs ad Earthquae Egeerg, 995, Vol. 4, p , 8. Par, J.H., Km, J., M, K.W., Optmal desg of added vscoelastc dampers ad supportg braces, Earthquae Egeerg ad Structural Dyamcs, 4, Vol. 33, p Matsagar, V.A., Jagd, R.S., Vscoelastc damper coected to adjacet structures volvg sesmc solato, Joural of Cvl Egeerg ad Maagemet, 5, Vol., p Hatada, T., Kobor, T., Ishda, M., Nwa, N., Dyamc aalyss of structures wth Maxwell model, Earthquae Egeerg ad Structural Dyamcs,, Vol. 9, p Sgh, M.P., Moresch, L.M., Opmal placemet of dampers for passve respose cotrol, Earthquae Egeerg ad Structural Dyamcs,, Vol. 3, p Palmer, A., Rccardell, F., De Luca, A., Muscolo, G., State space formulato for lear vscoelastc dyamc systems wth memory, Joural of Egeerg Mechacs, 3, Vol. 9, p Mars N., Dargush G.F., Costatou M.C., Dyamc aalyss of vscoelastc-flud dampers, Joural of Egeerg Mechacs, 995, Vol., p Podluby I., Fractoal Dfferetal equatos, Academc Press, Lewadows R., Chorążyczews B., Waslewcz P., Evaluato of parameters of vscous ad vscoelastc dampers, Proceedgs of XXII Symposum Vbrato Physcal Systems (Eds. Cz. Cempel, J. Stefaa), Poza- Będlewo, May 6, Vol. XXII, p.3-8,