Journal of Mechanical Science and Technology 23 (2009) 1058~1064. Dynamic behaviors of nonlinear fractional-order differential oscillator

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1 Joural of Mechaical Sciece ad Techology 3 (9) 58~64 Joural of Mechaical Sciece ad Techology DOI.7/s Dyamic behaviors of oliear fracioal-order differeial oscillaor Wei ZHANG,*, Shao-kai LIAO ad Nobuyuki SHIMIZU 3 Deparme of Elecroic Egieerig, JiNa Uiversiy, Guagzhou, Chia Isiue of Mechaical Sciece ad Techology, Shezhe, Chia 3 Deparme of Mechaical ad Power Egieerig, Iwaki Meisei Uiversiy, Japa (Mauscrip Received December 4, 8; Revised March 6, 9; Acceped March 6, 9) Absrac The oliear dyamic behaviors of oscillaors described by fracioal-order differeial are preseed i his paper. The backgroud of he research is based upo wo egieerig pracices. Firs is ha he visco-elasic behaviors of some advaced polymeric maerials ca be accuraely modeled by he fracioal calculus cosiuive law. Secod is ha he ifluece of oliear visco-elasiciy described by he fracioal operaor cao be egleced i some cases such as he vibraio wih large displaceme or large srai ad hermo-visco-elasic coupled problems. The umerical scheme for solvig he oliear euaio of moio is developed. The resuls show ha because of he iroducio of oliear dampig modeled by he fracioal-order operaor, he bifurcaio ad chaos of he oscillaor appear i forced vibraio. Furhermore, he fracio value of he fracioal operaor evidely affecs he dyamic behavior of he oliear fracioal differeial oscillaor. Keywords: Nolieariy; Fracioal calculus; Bifurcaio; Chaos; Viscoelasiciy Iroducio The oliear behaviors of he mechaical sysems described by oliear fracioal calculus have begu o appear i lieraure i rece years due o he developme of fracioal calculus ad fracals. Addolfsso, Eelud, ad Larsso (5) sudied he models ad umerical procedures for oliear fracioal order viscoelasics fracioal differeiaio []. Meshaka, Sephae, ad Chrisia (5) demosraed how describig he viscoelasiciy by usig hermodyamic fucios is liked wih fracioal ype operaors [], while Nasuo ad Shimizu (5) preseed he experimeal evidece of oliear saic ad dyamic models of a fracioal derivaive viscoelasic This paper was preseed a he 4h Asia Coferece o Mulibody Dyamics(ACMD8), Jeju, Korea, Augus -3, 8. * Correspodig auhor. Tel.: , Fax.: address: zhagw@ju.edu.c KSME & Spriger 9 body [3]. These sudies remid us ha he oliear ifluece o boh he cosiuive relaioship of maerials ad mechaical behavior of srucures cao be igored i some cases of egieerig applicaios [6, 8]. I is herefore ecessary ha we expose, udersad, ad simulae he oliear characerisics of he dyamic sysems described by a fracioal-differeial operaor. Based o he sudies o viscoelasiciy of polymer maerials [3, 4], Zhag ad Shimizu proposed a oliear viscoelasic cosiuive model wih a fracioal-differeial operaor ( ) () = () () () σ ε D ε ε < < () where D ( ) is he Riema-Liouville fracioal derivaive operaor; σ() is he sress; ε() is he srai; ad,, ad are he model parameers. Thus, we ca obai he four-parameer oliear

2 W. Zhag e al. / Joural of Mechaical Sciece ad Techology 3 (9) 58~64 59 fracioal Kelvi-ype viscoelasic model. By adopig he oliear cosiuive law () o he oscillaor sysem, we ca obai a ype of a simple oliear dyamic sysem of a srucure described by a fracioal-differeial operaor Dx () D( c[ x ()] cx ()) kx () = f() () where x() deoes he o-dimesioal displaceme of he oscillaor, f() deoes he ouside exciaio, k describes he siffess of he oscillaor, ad c ad c refer o he liear ad oliear dampig, respecively, aced o he oscillaor. A special case of a Va der pol-like fracioal oscillaor ca be assumed as Dx ε x Dx x x f ( ) ( [ ( ) ]) ( ) ( ) ( ) = ( ) (3) where ε is a o-dimesioal posiive real umber. By referrig o (), (), ad (3), his paper preses he resuls of he oliear dyamic behaviors of he oscillaors described by fracioal-differeial euaios. The umerical schemes for solvig he oliear Es. () ad (3) are developed, which are based o he Zhag-Shimizu umerical algorihm of fracioal derivaive [5, 7]. The dyamic behaviors of he proposed oliear oscillaor sysem are compared wih he classical correspode oliear oscillaors. The resuls show ha because of he oliear dampig modeled by he fracioal-order operaor, he bifurcaio ad eve he chaos of he oscillaors appear i he forced vibraio. Furhermore, he fracio value of he fracioal operaor evidely affecs he dyamic behavior of he oliear fracioal oscillaor.. Numerical algorihm of fracioal derivaives The Riema-Liouville fracioal derivaive is defied as d f ( τ ) D f () = dτ Γ ( ) dx ( τ ) (4) < < where ( ) Γ deoes he Gamma fucio. By replacig f() wih x() i E. (4) ad assumig a leas he firs coiuous derivaive of x(), we ca obai [ ()] x x () x () D x() = { dτ} Γ ( ) ( τ) (5) < < From Zhag-Shimizu s umerical scheme [5, 7], we ca wrie [ ()] x ([ ] ) x( τ) ( τ) D x( ) = dτ Γ( ) ( τ) [ x() ] τ τ = dτ dτ Γ( ) ( ) τ (6) ( τ) = ( J J J) Γ( ) where J [ x() ] = ad J = τ dτ ( τ) If he rapezoidal iegraio mehod is adoped o calculae J = τ dτ, ( τ) we ca have τ J = dτ ( τ) x x i = ( ) ( ) ( ) x i x i = i For J, suppose ha E. (7) is (7) () τ = ητ ()( ) (8) τ where ητ () =, =, τ < <. If we furher ake = ( x x ) x x β β β (9) = α α () ( ) where α ad β are he parameers of he Newmarkype umerical iegraio. By subsiuig Es. (8) ad (9) for E. (), we ca derive ( ) ( ) x () τ = x α x αx τ () Likewise, by subsiuig Es. (9) ad () for J we ca obai

3 6 W. Zhag e al. / Joural of Mechaical Sciece ad Techology 3 (9) 58~64 x J = ( )( ) ( -α) α x ( )( ) 3 ( α) α ( )( )( 3) ( -α) α ( )( )( 3)( 4) () By subsiuig J, J, ad J for E. (6), we ca he fid ( x( ) ) D = Γ( ) ( α) x αx 3 4 [ x() ] x x ( ) ( x i i ) i= ( i ) x ( )( ) ( α) α x ( )( ) 3 ( α) α ( )( )( 3) ( )( )( )( ) (3) Thus, a he oliear umerical scheme of he fracioal-differeial Es. () ad (3) of oliear oscillaors ca be se up. Fig. illusraes he bifurcaio diagrams of he Duffig-like fracioal oscillaor wih differe orders of fracioal derivaives, which are solved by meas of our proposed umerical scheme. I ca be see ha he algorihm of he fracioal derivaive developed here shows he correcess ad effeciveess of he oliear umerical scheme. A deailed discussio o he Duffig-like fracioal oscillaor ca be foud i paper [8]. 3. Noliear dyamic behavior The oliear dyamic behavior of he oscillaor described i E. () is firs preseed here. Suppose he zero iiial codiio ad f()=fcos(π), ad se he model parameers a c =5, c =.5, k=(π), F=, α=.5, ad β=.5. If we chage oly he (a) = (b) =.75 Fig.. Bifurcaio diagram of he Duffig-like fracioal oscillaor. order of fracioal differeials, we ca obai he followig resuls as depiced i Fig.. The resuls show ha he orders of fracioal derivaives evidely affec he dyamic behavior of he oliear oscillaor. The icrease of he order of fracioal differeial has see he icrease of he degree of bifurcaio of he oscillaor. Isigh io his pheomeo exposes he maerial dampig characerizaio, wih he values of fracioal differeial orders for he dissipaio sysem modeled by fracioal calculus. This coclusio is more obvious whe compared wih he oscillaor described by he ieger differeial operaor if we keep he oher parameers uchaged, as show i Fig. 3. I ca be see from Fig. 3 ha he dyamic behavior of he oscillaor wih he ieger differeial operaor is more chaoic, ad is eergy dissipaio capabiliy is weaker ha he oe described here by he fracioal differeial operaor.

4 W. Zhag e al. / Joural of Mechaical Sciece ad Techology 3 (9) 58~64 6 (a) Order of fracioal differeial =.3 (b) Order of fracioal differeial =.5 (c) Order of fracioal differeial =.9 Fig.. Time respose, phase graph, ad power specrum of he oliear oscillaor. Fig. 3. Time respose, phase graph, ad power specrum of he oliear oscillaor whe he order of fracioal differeial =.. The sregh of ouside exciaio ca ceraily ifluece he oliear dyamic behavior of he oscillaor. Suppose ha c=5, c=, k=6π, =.5, α=.5, ad β=.5, ad le ampliude F chage from ui o 5. The umerical soluios ca be preseed as depiced i Fig. 4. As we prediced, he larger he ampliude of exciaio, he sroger he olieariy of he oscillaor is. Whe he sregh of he ouside exciaio reaches a poi, he chaos of he oscillaor occurs. We ca also assume he wi aracors of he oscillaor. Therefore, alhough his kid of oscillaor described i E. () is cosraied by he sroger dampig described by he fracioal calculus viscoelasic law, i ca sill evolve io srog ad ypical oliear dyamic behaviors, which are he same as he correspode oliear oscillaor wih ieger derivaive. If E. (3) of he Va der pol-like fracioal differeial oscillaor is reaed as a special case of E. (), he he oliear dyamic behaviors will appear as illusraed i Fig. 5, as we se ε=5, F=, adω=.5. The daa i Fig. 5 demosrae ha alog wih he chage of he orders of fracioal derivaive from a small o a large olieariy of his ype of oscillaor is he chage from srog o less complicaed. I oher words, he ifluece of he orders of fracioal deriva-

5 6 W. Zhag e al. / Joural of Mechaical Sciece ad Techology 3 (9) 58~64 (a) F= (b) F=5 (c) F= Fig. 4. Time respose, phase graph, ad power specrum of he oliear oscillaor whe he order of fracioal differeial =.5. =. Fig. 5. Ifluece of he orders of fracioal derivaive. =.3 =.75

6 W. Zhag e al. / Joural of Mechaical Sciece ad Techology 3 (9) 58~64 63 amic behavior of he oscillaor sysem as well i oher cases. Ackowledgeme This work is par of he projec suppored by he Naural Sciece Foudaio of Chia (No.8779: Shock Absorbig ad Eergy Dissipaio Mechaism of Advaced Polymer Dampig Maerials). Fig. 6. Bifurcaio diagram of he displaceme-fracioal derivaive orders. ive ideed srogly affecs he dyamic behavior of he sysem. I fac, i ca be idicaed ha he fracio values of he fracioal derivaives reflec he characerisics of he eergy dissipaio of such sysems as polymeric dampig maerials [8]. The bifurcaio diagram i Fig. 6 depics he dyamic behaviors of he fracioal differeial oscillaor alog wih he chages i he order of fracioal derivaive. The chaoic vibraio of hese kids of dyamic sysems ca be ecouered. Agai, he fracio values of he fracioal derivaives ifluece he dyamic saus of he sysems hey describe, ad hey ca surely reflec he characerisics of he eergy dissipaio of such sysems as polymeric dampig maerials. 4. Coclusios The oscillaor dyamic sysems associaed wih a ype of oliear viscoelasiciy modeled by a fracioal differeial operaor are proposed ad carefully sudied i his paper. The umerical scheme for he soluio of he oliear oscillaor is successfully developed. The mai resuls of he research i his paper ca be summarized as follows: The order of fracioal derivaive evidely affecs he dyamic behavior of he oliear oscillaor. The fracio values of he fracioal derivaives reflec he characerisics of he eergy dissipaio of such sysems as polymeric dampig maerials. The iroducio of he oliear dampig brigs abou o oly sroger eergy dissipaio i some cases, bu i may also icur he more complex dyamic Refereces [] K. Addolfsso, Mikael Eelud, S. Larsso, Models ad umerical procedures for oliear fracioal order viscoelasics fracioal differeiaio ad is applicaio, Germay: Bordeaux Uiversiy, ISBN: (5), [] Y. Meshaka, A. Sephae ad C. Chrisia, Describig viscoelasiciy usig hermodyamic fucios: lik wih fracioal ype operaors, Frace: FDA4, s IFAC workshop o fracioal differeiaio ad is applicaio 9-( 4). [3] H. Nasuo ad N. Shimizu, Noliear Saical ad Dyamical Models of Fracioal Derivaive Viscoelasic Body, Proc. 5 ASME Ieraioal Desig Egieerig Techical Cofereces ad Compuers ad Iformaio i Egieerig Coferece. Sepember 4-8, Log Beach, Califoria, USA(5). [4] H. Z. Su, Cosiuive law of viscoelasic maerials modeled by fracioal-order derivaives, Maserdegree hesis, JiNa Uiversiy, Chia (6, I Chiese). [5] Wei ZHANG ad Nobuyuki SHIMIZU, Numerical Algorihm for Dyamic Problems Ivolvig Fracioal Operaors, I. J. of JSME, Ser. C, 4, NO.3, (998) [6] Joh Guckeheimer ad Philip Holmes, Noliear oscillaios, dyamical sysems ad bifurcaios of vecor fields, Fifh Priig, New York: Spriger- Verlag(997). [7] W. Zhag, H. Xu ad N. Shimizu, Numerical aalysis formulaio of viscoelasic solids modeled by fracioal operaor, ACTA MECHANICA SINICA, 36 (5) Sep. (4) 67-6 (I Chiese). [8] S. K. Liao ad W. Zhag, Dyamics of fracioal Duffig oscillaor, Joural of Dyamics ad Corol, 6 () (8) -5 (I Chiese).

7 64 W. Zhag e al. / Joural of Mechaical Sciece ad Techology 3 (9) 58~64 Wei Zhag received his docorae degree i Mechaics a Xi a Jiaoog Uiversiy i Chia i 994. He we o he Uiversiy Of Sheffield i he UK as a academic visior from July 986 o July 987. He compleed his pos-docorae sudy i Iwaki Meisei Uiversiy i Japa hrough he suppor of he JSPS from May 997 o Jue 999. Dr. Zhag is currely a professor a he Deparme of Elecroic Egieerig of JiNa Uiversiy i Guagzhou, Chia. Nobuyuki Shimizu received his B.S. degree i Mechaical Egieerig a Yokohama Naioal Uiversiy ad his maser s ad docorae degrees i Mechaical Egieerig a Uiversiy of Tokyo i 968 ad 97, respecively. He we o he Califoria Isiue of Techology i Califoria, U.S.A. as a visiig associae from Ocober 976 o Sepember 977. Dr. Shimizu is currely a professor a he Deparme of Mechaical ad Power Egieerig of Iwaki Meisei Uiversiy i Japa. Shao-kai Liao received his B.E. degree i Civil Egieerig a JiNa Uiversiy i Guagzhou, Chia i 4 ad his maser s degree i Egieerig Mechaics a he same uiversiy i 6. Mr. Liao was oce a posgraduae sude of Professor Wei Zhag, ad he ow works i a isiue i Shezhe Ciy i Chia.