Steady-state response of systems with fractional dampers

Transcription

1 IOP Conferene Serie: Material Siene and Engineering PAPER OPEN ACCESS Steady-tate repone of ytem with frational damper o ite thi artile: R Lewandowi and A Lenowa 7 IOP Conf. Ser.: Mater. Si. Eng. 5 9 View the artile online for update and enhanement. Related ontent - Uing magnetorheologial fluid in an innovative hybrid biyle damper Y J Shiao and S Nguyen - Prinipal reonane repone of a tohati elati impat oillator under nonlinear delayed tate feedba * Huang Dong-Mei, Xu Wei, Xie Wen-Xian et al. - A miromahined anning grating interferometer for the out-of-plane vibration meaurement of MEMS B Kim, F L Degertein and R Kurfe hi ontent wa downloaded from IP addre on 4/5/8 at 3:

2 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 Steady-tate repone of ytem with frational damper R Lewandowi and A Lenowa Poznan Univerity of ehnology, Pozna, ul. Piotrowo 5, Poland roman.lewandowi@put.poznan.pl Abtrat. In thi paper, teady tate vibration of ytem with built-up vioelati damper are onidered. he damper are modeled uing the frational-derivative rheologial model. he Caputo type frational derivative definition i ued. In partiular, the teady tate vibration of ytem are analyed. he olution to the teady tate vibration i written uing real quantitie. he effet indued by hange of environmental temperature are alo onidered and, in thi ontext, the time-temperature uperpoition priniple i adopted. he reult of everal parametri tudie are alo deribed and diued in detail.. Introdution Modern truture are higher, lighter and more flexible, ontruted with the ue of material of higher trength, and optimally deigned. However, thee ytem are more ueptible to dynami loading and, in onequene, the amplitude of vibration of uh ytem are ometime too large; thi an mae it impoible to orretly utilize the truture or, in ome ae, it an detroy the truture. In uh ae, the truture vibration mut be redued. Many damper type are uefully developed to obtain a ignifiant redution of amplitude of exeive vibration []. Vioelati (VE) damper are a very promiing la of damper, to name jut one. o deribe the dynami behavior of uh damper, a number of rheologial model, haraterized by the frational derivative, an be ued. Steady tate vibration of a vibrating ytem with one degree of freedom and damping deribed by frational derivative were examined in paper []. Steady tate vibration aued by determiniti, harmonially hanging fore were onidered. In paper [3], the teady tate vibration of a linear and a non-linear ytem with one degree of freedom are analyzed. he damping of a ytem i deribed by mean of a frational derivative. In paper [4], non-linear teady tate vibration of arhe made by vioelati material are analyzed. he author are uing the reidue harmoni homotopy method. Paper [5] i onerned with the teady tate vibration of a two-member plane tru ytem. Vioelati propertie are deribed by the frational Kelvin-Voigt model. he effet of frational order and material modulu ratio on the ytem repone are tudied. An analyi of teady tate vibration of beam and frame with damper, haraterized by the frational derivative, i onduted in paper [6]. In the preent paper, teady tate vibration of ytem with and without VE damper are analyzed. o deribe the dynami behavior of damper, the rheologial model hown in Figure i ued. he equation deribing the model behavior ontain the frational derivative. he model preented in Figure i general beaue it ontain a number of impler model (both lai and frational) a peial ae, whih are often ued in the dynami analyi of ytem with the damper (ee [, 6, 7]). Content from thi wor may be ued under the term of the Creative Common Attribution 3. liene. Any further ditribution of thi wor mut maintain attribution to the author() and the title of the wor, journal itation and DOI. Publihed under liene by IOP Publihing Ltd

3 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9. Deription of vioelati damper Figure. Diagram of rheologial model of damper... Deription of pring-pot element Different type of rheologial model are ued for deribing the vibration of damper. Suh model onit of viou, elati and pring-pot element, onneted in different way. he mot often ued model are the viou, Kelvin and Maxwell model. A ombination of them provide more omplex model whih enable a more preie deription of damper. In model with a number of element, the number of parameter ignifiantly inreae. hi reult in muh more omplex equation of motion. o eliminate thee diadvantage, the frational model of damper are ued to deribe them. hi enable a redution of the number of damper parameter [7]. When frational model are ued, a preie deription of damper behavior with fewer parameter and wider frequeny of exitation range i poible. Suh model were propoed by Bagley and orvi in [8]; they were ued to deribe the dynami behavior of frame with damper and andwih beam, for example, in [9,]. In thi paper, the Caputo definition of frational derivative i ued: t dq( t ) / dt Dt q( = dt ( ), () Γ ( t t ) where Γ ( ) i a gamma funtion, < < i an order of frational derivative. Moreover, it i aumed that the lower limit of the integral in definition () wa moved to. hi derivative i ued to deribe the behavior of the pring-pot element a propoed by Sott-Blair in paper []. In Figure, the element are hown a the diamond. he behavior of the pring-pot element i haraterized by Eqn (): u( = Dt Δ q( () where u( i fore in the element, Δ q( i the differene of diplaement of the element end, and i the ontant of the model. he oeffiient ha an anomalou dimenion [ N / m]. he Sott- Blair element i one of whih the propertie are intermediate between thoe of elati and viou element. For =, the element behave lie a pring and for = it doe lie a viou damper... Steady tate olution for damper Let u onider a frational model of the damper hown in Figure. It i a general model, oniting of frational Kelvin and Maxwell element joined in parallel. In peial ae, the model an turn into one of many impler model of great pratial importane, e.g., Sott-Blair, Kelvin, Maxwell or Zener model, in either frational or laial verion (when = ). he fore in the analyzed model i a um of fore in both ontituent element: u ( = u ( + u(, (3) where index denote fore in the Kelvin element, and index i fore in the Maxwell element. he behavior of both element i deribed by the following equation: j u ( = ( + τ Dt )( q q ), u( + τ Dt u( = τ Dt ( q q j ), (4)

4 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 where τ = /, τ = /, ymbol and denote the tiffne of the Kelvin and Maxwell element, repetively, wherea q j and q denote diplaement of the end of the damper model. If the damper vibration are teady tate vibration, the following relationhip are valid: u( = u o λt + u in λt, u ( = u o λt + u in λt, (5) u ( = u o λt + u in λt, qi ( = qi o λt + qi in λt. (6) aing into aount that Dt oλt = λ o( λt + π / ), Dt in λt = λ in( λt + π / ), (7) and, after ubtituting relationhip (6) into Eqn (3) and (4), the following i obtained: u u ( j j = [ χ ( q q ) + χ ( q q )], u = [ χ ( q q ) + χ ( q q )], (8) ( j j = Θ ( q q ) + Θ ( q q ), u = Θ ( q q ) + Θ ( q q ), (9) ( j j ( j j where ymbol χ, χ, Θ and Θ are defined a: π π χ = + ( τ λ) o, χ = ( τ λ) in, () π ( τλ) ( τ λ) + o π ( τ λ) in Θ =, Θ =. () π π + ( τλ) o + ( τ λ) + ( τ λ) o + ( τλ) After ubtituting relationhip (8) and (9) into Eqn (3), the following i obtained: u = χ + Θ )( q q ) + ( χ + Θ )( q q ), () ( j j u = χ + Θ )( q q ) + ( χ + Θ )( q q ). (3) ( j j.3. Influene of temperature on damper parameter In order to determine the VE damper repone to hange of temperature, the time-temperature uperpoition priniple an be ued, a given by the following relationhip: K ( t, ) = K ( ~ t, ), (4) where K i alled the omplex modulu, t and are the referene time and referene temperature, repetively. he ymbol ~ denote the o-alled hift fator. he time-temperature uperpoition priniple an be applied to the frequeny domain and i ometime named a the frequenytemperature orrepondene priniple []: K( λ, ) = K( ~ λ, ), (5) where λ i the referene frequeny. In thi paper only the horizontal hift fator i taen into aount a the exiting literature implie that for vioelati material ued in damper the vertial hift fator eem to be equal to one (ee []). However, thi point need deeper theoretial and experimental tudy. he hift fator i alulated from ome empirial formula. he following William-Landel-Ferry formula i often ued: log ~ = CΔ /( C + Δ ), (6) where C and C are ontant and Δ =. Another formulae an alo be ued if they are more appropriate. 3

5 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 It i well nown (ee [7]) that the teady tate vibration of damper an be deribed uing the omplex modulu. he olution i in the following form: u λ ) = ( K ( λ) + ik ( λ))( q q ), (7) ( j where K '( λ) i the torage modulu, K "( λ) i the lo modulu and here i =. he torage and lo moduli an be written a follow (ee [7]): ( τ λ) [ ] [( τ λ) + o( π / ) ] + ( τ λ) o( π / ) + K '( λ) = χ + Θ =, (8) + ( τ λ) o( π / ) + ( τ λ) λ) ( τ in( π / ) K "( λ) = χ + Θ = ( τ λ) in( π / ) +. (9) + ( τ λ) o( π / ) + ( τ λ) Auming that parameter,, τ and τ are related to the referene temperature and the referene frequeny λ, Eqn (7) tae the form: ( τ λ ) [ ] [( τ λ ) + o( π / ) ] + ( τ λ ) o( π / ) + K '( λ, ) =. () + ( τ λ ) o( π / ) + ( τ λ ) When temperature i different from the referene value, the torage modulu an be deribed by: ~ τ λ [ ] [ τ λ π ] λ ~ ~ ( ~ ) ( ~ ) + o( / ) K '(, ) = + ( τ λ) o( π / ) + + ( ~ τ λ) o( π / ) + ( ~. () τ λ) aing into aount the frequeny-temperature orrepondene priniple, a expreed by Eqn (4), the above formula an be rewritten a follow: τ ~ λ [ ] [ τ ~ λ π ] λ ~ ~ λ ~ τ ~ ( ~ ( ~ λ π ( ~ ) ) + o( / ) K ( =, ) = + ( ) o( / ) + τ ~ λ) o( π / ) ( ~ τ ~. () + + λ) A omparion of Eqn () and () lead to the following reult: ~ ~ =, =, ~ τ ~ = τ, ~ τ ~ = τ (3) From Eqn (3), it an be etablihed that only the parameter and of the damper model will hange with hange of temperature aording to the relationhip: ~ =, ~ =, (4) where = ~ i the redued hift fator. 3. Steady tate vibration of ytem with damper he motion of the ytem with damper i deribed by the following equation [7]: M q ( + C q ( + K q( = P( + F( (5) In Eqn (5), the ymbol P( and F ( denote the vetor of exitation fore and the vetor of interating damper fore, repetively, M, C, K mean the matrix of ma, damping and tiffne, repetively. he vetor of diplaement of the ytem i denoted by q (. When the harmoni load deribed by Eqn (6) i applied: P ( = P o λt + P in λt, (6) the damper interating fore and teady tate vibration of the ytem an be deribed by the following equation: F ( = F o λt + F in λt, q ( = q o λt + q in λt. (7) 4

6 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 After introduing relationhip (6) and (7) into Eqn (5), the equation of amplitude i obtained: ( K λ M ) q + λcq = P + F, λ C q + ( K λ M ) q = P + F. (8) Moreover, the relationhip between the vetor F ( and fore in damper u i ( ( i =,,.., r ) are: r i= F ( = e u (, (9) i where e i i the alloation vetor deribing the poition of the i-th damper on the truture. Let u aume that the i-th damper i onneted with the point and j of whih the diplaement are q and q j, repetively (ee Figure ). he following relationhip an be written: i i q ( q ( = e q(, q q j = ei q, q q j = ei q. (3) j Figure. Diagram of a typial ytem with damper. aing into aount thee priniple, Eqn () and (3) an be rewritten a follow: u i ( i i + iθi ) i q ( i i + iθ i = χ e χ ) e q, (3) i u i ( i i i Θ i ) i q ( i i + iθi = χ e χ ) e q. (3) Index i mean that parameter with the index onern the i-th damper. If the Eqn (3), (3) and (7) are ubtituted into Eqn (9), the vetor F and F an be expreed a: () () () () F = ( K tν q + K tν ) q, F = ( K tν q + K tν q ), (33) where: K r r () () tν = [ i χi + i Θi ] Li, K tν = [ i χ i + iθ i ] L i, i= i= After ubtituting Eqn (33) in the amplitude equation (8), it tae the following form: () tν () tν i i iei L = e (34) ( K + K λ M ) q + ( λc + K ) q = P, (35) () t () tν ( λc + K ν ) q + ( K + K λ M ) q = P. (36) 4. Balane of energy in ytem with vioelati damper he balane of energy of the ytem in time t and t +, where = π / λ, an be formulated a follow: E ( + Et ( Ew ( =, E ( t + ) + Et ( t + ) Ew ( t + ) =. (37) he ymbol E (, E t ( and E w ( denote energy of the ytem, energy of the damper, and wor of the exitation fore in time t, repetively. he hange of energy during one period of the yle i: E ( t + ) E ( + Et ( t + ) Et ( Ew ( t + ) + Ew ( =. (38) 5

7 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 he energy of the ytem onit of the ineti, elati and diipated energie. he energy of the damper onit of the ineti and diipated energie. Beaue the ytem vibrate harmonially, hange of it ineti and elati energie are equal to zero. hen Eqn (38) an be rewritten in the form: Δ E + ΔEt = ΔEw, (39) where the ymbol Δ E, Δ Et and Δ Ew denote hange of the energy diipated by the ytem, diipated by the damper and wor of the external fore during one period of the yle, repetively. It an be aumed without loing the generality of onideration that t =. Change of both the elati and diipated energie of the i-th damper and vibrating harmonially with the period are alulated a follow: ΔEti = u( x ( dt, (4) where x( = q q j i the differene of diplaement of damper end (ee Figure ). After ubtituting relationhip (5) into (4) and integrating Eqn (4) with repet to time, the following i obtained: ΔE ti = π ( u x u x ) = π ( χ + Θ )( x + x ), (4) where x = q q j, x = q q j. he damper vibrate harmonially, it mean that the hange of elati energy i equal to zero in the damper and the relationhip (4) and (4) are the definition of hange of the energy diipated by the damper. If the viou damping fore are a load to the ytem, then a hange of the energy diipated by the ytem (without analyzing the damper) i given by Eqn (4): ΔE = q ( C q ( dt = πλ( q C q + q C q ), (4) t where C i the matrix of viou damping of the ytem. A hange of the wor of the exitation fore an be alulated from the formula: ΔE = P ( q ( dt = π ( P q P q ). (43) w emperature in VE damper an hange with environmental temperature and during the proe of energy diipation when diipated energy i hanged to the heat. he latter i o-alled the elfheating phenomenon. he inreae of temperature i important when truture are ubjeted to wind. More detailed analyi of effet of temperature on VE damper an be found in [-7]. 5. Reult of numerial analyi 5.. Example bai propertie of ytem with VE damper he ytem hown in Figure 3 wa analyzed. A fore P ( = P o λt wa applied upon it where P = [.,.,., 5. N]. It i aumed that: m = m = m3 = 44. g, m 4 =. g, = = 3 = 5. MN/m, 4 = 45. MN/m. he damping matrix wa determined baed on the equation: C = M + K where =. 34 and = In all example, damper parameter ued are imilar to parameter of VE material invetigated in the paper [8]. 6

8 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 Figure 3. Diagram of the analyed ytem. Sytem with and without vioelati damper, arranged a illutrated in Figure 3, were analyzed. he damper were haraterized by the following parameter: =. 7, =. MN /m, t = 5. MN/m, t =,3MN /m, t = 4.7 MN/m. he repone urve were plotted for the ytem without damper and with different rheologial damper model. he urve are hown in Figure 4. Moreover, the maximum amplitude of reonane vibration of a ytem with different type of damper are hown in able. t Figure 4. Repone urve for ytem with damper modeled by different rheologial model. able. Amplitude of vibration in reonane for ytem with damper of different type. Damper type Reonane frequeny [rad/] Firt reonane Amplitude of vibration [m] Seond reonane Reonane frequeny [rad/] Amplitude of vibration [m] no damper Spring pot Kelvin Maxwell Zener Figure 5 and 6 how the reult of alulation of the energy diipated by the damper. he ame reult are preented in able, where the maximum value of diipated energie and wor of external fore are given. Figure 5 how how the diipation energy hange with exitation frequeny for different type of damper. Figure 6 how hange of energy diipated by individual Zener damper v. exitation frequeny. he analogou graph are imilar a for other type of damper. In able 3, the maximum value of 7

9 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 energy diipated by the individual damper, modeled a different rheologial type are hown in both reonane region. Figure 5. Energy diipated by ytem with damper v. exitation frequeny. Figure 6. Energy diipated by firt and eond damper modeled by the Zener model. able. Maximum wor of external fore and maximum diipation energy. t reonane nd reonane Damper type Wor of external fore [J] Energy diipated by ytem [J] Energy diipated by damper [J] Wor of external fore [J] Energy diipated by ytem [J] Energy diipated by damper [J] no damper pring pot Kelvin Maxwell Zener

10 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 able 3. Maximum energy diipated by ingle damper in two reonane region. Damper type Energy diipated by damper [J] t reonane nd reonane t damper nd damper t damper nd damper Spring pot Kelvin Maxwell Zener Several remar an be formulated on the bai of alulation onduted by the author:. Amplitude of reonane vibration may be ignifiantly redued uing the mentioned damper.. he extent of redution of vibration depend on damper type. Damper modeled with the pringpot and Maxwell model redue the amplitude of vibration to a maller degree, ompared with damper modeled with the Kelvin and Zener model. 3. he reonane frequenie of vibration of the ytem with the pring pot and Maxwell model damper do not hange notieably, ompared with the ytem with no damper. In ontrat, frequenie of reonane vibration of ytem with the Kelvin and Zener type of damper may inreae ignifiantly. he differene depend on the proportion between the tiffne oeffiient of the ytem and the tiffne oeffiient of the damper. 4. In the ae diued in thi paper, the firt damper diipate energy mainly in the firt reonane region and the eond damper doe in the eond one, regardle of the damper model. 5.. Example temperature effet on repone of ytem with damper he ytem hown in Figure 3 wa analyzed. For the purpoe of alulation, the ame et of data a in Example i adopted. Damper are deribed with the frational Zener model. he following value of damper parameter were ued (valid for the referene temperature = o C ): =. 7, = N/m, = MN/m, 5 ref.84 N /m ref. N /m, C = 9. 3 and, =, = C =4.. Calulation were performed for the following range of temperature: min = 3 C, o max = 3 C. Figure 7 how hange of the hift fator value v. temperature. It an be notied that ~ dereae with inreaing temperature. Figure 8 illutrate the nature of hange of maximum vibration amplitude in the firt reonane region depending on damper temperature. he maximum amplitude dereae for the lower range of temperature and they tart to inreae after exeeding o = C. he hange of temperature aue ignifiant hange in the vibration amplitude. 6. Conluding remar he teady tate repone of ytem with vioelati damper are onidered in the paper. A et of frational rheologial model are onidered a model of damper. he Caputo frational derivative are ued to deribe vioelati damper. he impat of environmental temperature on the behavior of a ytem with vioelati damper i onidered uing the time-temperature uperpoition priniple. emperature wa found to have a ignifiant impat on maximum amplitude in the firt and eond reonane region. he unexpeted behavior of the ytem with damper wa oberved in the firt reonane region. o 9

11 IMS 7 IOP Publihing IOP Conf. Serie: Material Siene and Engineering (7) 9 doi:.88/ x/5//9 Figure 7. he hift fator ~ a a funtion of temperature of damper. Moreover, the equation whih deribe the balane of energy of the ytem with damper wa derived and ome remar onerning the diipated energy were formulated. In partiular, it wa hown that the energy diipated in the reonane region by the hoen damper provided valuable information about the damper effetivne and howed whih mode of vibration wa mainly damped. hi energy ould be a good indiator of the damper optimal poition. he typial repone urve of the ytem with damper are determined and ompared to ilutrate the propertie of ytem and poible redution of amplitude. Figure 8. Maximum reonane amplitude v. temperature of damper. Anowledgment he tudy wa partially upported by the National Siene Centre, Poland, a part of Projet No. DEC/3/9/B/S8/733, arried out in the year 4-7, and partially upported by the Poznan Univerity of ehnology a part of grant No. //DSPB/86. Referene [] Soong,. and Contantinou, M.C 994 Paive and Ative Strutural Vibration Control in Civil Engineering, CISM Leture Note (New Yor: Springer-Verlag) [] Huang C and Duan J.S 6 Steady-tate repone to periodi exitation in frational vibration ytem, J. Meh [3] Chen Y.M, Liu Q.X and Liu J.K 6 Steady tate repone analyi for frational dynami ytem baed on memory-free priniple and harmoni balaning, Int. J. Non-Lin. Meh [4] Leung A.Y., Yang H.X, Zhu P and Guo Z.J 3 Steady tate repone of frationally damped nonlinear vioelati arhe by reidue harmoni homotopy, Comp. Strut.

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