Fractional Euler-Lagrange Equations Applied to Oscillatory Systems

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1 Mhemics 05, 3, 58-7; doi:0.3390/mh30058 Aricle OPEN ACCESS mhemics ISSN Frcionl Euler-grnge Euions Applied o Oscillory Sysems Sergio Adrini vid *, Crlos Alero Vlenim Jr. Universiy of São Pulo Pirssunung, Av. uue de Cxis Nore, , Pirssunung-SP, Brzil; E-Mil: crlos.vlenim@usp.r * Auhor o whom correspondence should e ddressed; E-Mil: sergiodvid@usp.r; Tel.: Acdemic Edior: Hri M. Srivsv Received: 4 Mrch 05 / Acceped: 5 April 05 / Pulished: 0 April 05 Asrc: In his pper, we pplied he Riemnn-iouville pproch nd he frcionl Euler-grnge euions in order o oin he frcionl nonliner dynmic euions involving wo clssicl physicl pplicions: Simple Pendulum nd he Spring-Mss-mper Sysem o oh ineger order clculus IOC nd frcionl order clculus FOC pproches. The numericl simulions were conduced nd he ime hisories nd pseudo-phse porris presened. Boh sysems, he one h lredy hd dmping ehvior Spring-Mss-mper nd he sysem h did no presen ny sor of dmping ehvior Simple Pendulum, showed signs indicing possile eer cpciy of enuion of heir respecive oscillion mpliudes. This implicion could men h if he selecion of he order of he derivive is convenienly mde, sysems h need greer inensiies of dmping or viring sorers my enefi from using frcionl order in dynmics nd possily in conrol of he foremenioned sysems. Therefer, we elieve h he resuls descried in his pper my offer greer insighs ino he complex ehvior of hese sysems, nd hus insige more reserch effors in his direcion. Keywords: Frcionl clculus; oscillory sysems; dynmic sysems; modeling; simulion.

2 Mhemics 05, Inroducion The heory of frcionl order clculus FOC des ck o he irh of he heory of differenil clculus, u is inheren complexiy delyed he pplicion of is ssocied conceps. In fc, frcionl clculus is nurl exension of clssicl mhemics. Perhps, his ckwrdness is due o he FOC s inheren complexiy nd o he curren lck of mening regrding is physicl nd geomeric inerpreion. The sic specs regrding he frcionl order clculus nd frcionl differenil euions cn e found in [ 5]. I is worh menioning h FOC cn coun on n ddiionl degree of freedom since he order of he derivives cn e rirry chnged o mch specific ehvior. This dvnge my enle he FOC o represen sysems wih high order dynmics nd complex nonliner phenomen, mking use of only few coefficiens. In fc, numerous mhemicins conriued o he hisory of frcionl clculus [6] wih every resercher using differen pproches nd soluions. A survey of useful formuls involving differen definiions ou FOC is provided y [7,8]. FOC is pplied in order o derive grngin mechnics of nonconservive sysems y [9]. Frcionl Hmilon nd frcionl Euler-grnge euions were considered in [0]. iner grngins in velociies were nlyzed using he frcionl clculus nd he Euler-grnge euions were derived y []. Here, he uhors invesiged wo exmples, he explici soluions of Euler-grnge euions were oined nd he recovery of he clssicl resuls ws discussed. In [], he uhors deduced h he frcionl Hmilonin sysems of Snilvsky from priculr les cion principle, sid o e cusl nd in his cse, he frcionl emedding ecomes coheren. In his pper, moived y he recen developmen of he frcionl Euler-grnge euions, we modeled wo clssicl physics pplicions nd we presened he numericl simulions resuls in order o provide possile comprison eween ineger nd frcionl dynmic ehvior. This pper is divided ino four secions. In Secion, we show he mehodology using frcionl Euler-grnge euions, he nlyicl modeling of wo physicl exmples nd condiions nd prmeers used in numericl simulions. In Secion 3, we presen some numericl simulions resuls oined nd in he finl secion, discussions nd conclusions re presened.. Mehodology.. Frcionl Euler-grnge Euions In his pper, we considered he Riemnn-iouville pproch nd n cion funcion ws used in he form [9,3,4]: S,, Where 0 β, 0 < γ <, 0 α. If ε indices he vriion of he funcion S, hen d S The Euion my e rewrien [4] s,, d

3 Mhemics 05, d S 3 Or lso, 0 ] [ ] [ d S 4 Thus, he Euler-grnge euions re wrien wih frcionl derivives, s he following 0 ] [ 5 For β = γ = nd ssuming h he grngin depends only on or on, he following is oined: 0 d d 6.. Applicions The foremenioned frcionl Euler-grnge euions were used s ool o invesige hrough modeling nd simulion he following pplicions: simple pendulum nd he spring-mss-dmper sysems.... The Simple Pendulum Modeling We considered simple pendulum whose clssicl nd known grngin my e wrien s: cos mgl ml ml 7 In his sysem, is he grngin, m is he mss of he pendulum, l represens he lengh of he wire nd, finlly, θ is he ngle. The pplicion of he Euler-grnge euion o his grngin lso provides he euion lso known s he moion euion 0 mglsen l m 8 Applying he grngin from E. 7 nd he E. 6 nd knowing h θ, i is possile o noice h mglsen ; ml ml ; ml d d 9 Thus, from he E. 6 one oins 0 ml ml mglsen l m 0 I cn e oserved s well h, for nlogy nd ssuming h he grngin depends only on, we cn wrie:

4 Mhemics 05, 3 6 ml ml mgl cos Thus, he frcionl Euler-grnge euion, in his cse, will e [ ] 0 Wih mglsen 3 ml ml mgl cos ml ml 4 Therewih, he frcionl euion for free sysems ecomes: [ ] sin 0 ml ml mgl 5 I is well known, for he IOC cse, h in sysems excied y exernl forces he righ side of E. 5 is no null. We cn pply he sme siuion for FOC. Thus, if we consider Q s n exernl force h my influence his sysem which in E. 5 is zero, we cn lso wrie: [ ml ml ] mgl sin Q 6 I is worh highlighing h if β = ineger order, i cn e esily verified h E. 5 is reduced o E. 0. Besides, in he E. 0, if α = ineger order, we oin ck, s expeced, he moion euion, E Spring-Mss-mper Sysem-Modeling Now, we considered dissipive sysem, regrding mss represened y m, spring siffness consn k nd dmper dmping consn c. The clssicl nd known grngin is given y Applying he Euler-grnge euion for non-conservive sysems Where So h, d d mx kx 7 Qi i 8 i Q i represens he dissipives forces, he following is oined: x k x ; m x x ; d d m x x 9

5 Mhemics 05, 3 6 Will provide he known moion euion d d x x Q i 0 m x cx kx Q where Q is he dissipive force. Using he grngin mx kx, in he euion 6 nd 8, so h x one cn oin: m x c x m x k x Q Applying he grngin h depends only on m The frcionl Euler-grnge euion will e: x, wrien s x k x 3 [ ] Q x 4 nd now so h, x k x x kx m x m x x 5 6 Therewih, he frcionl euion ecomes: kx ] [ m x c x 7 Or lso, [ x ] c x kx Q 8 Agin, if β = ineger order, i cn e esily verified h E. 6 is reduced o E. nd in he E., if α = ineger order we oin ck, s expeced, he known ineger moion euion E...3. Condiions nd Prmeers of he Simulions The previously menioned euions were numericlly simuled wih he inenion of sudying heir dynmic ehvior nd eslishing comprisons eween he resuls of he frcionl order sysems nd

6 Mhemics 05, 3 63 he ineger order sysems. These euions were simuled hrough Ml Simulink. The solver mehod/lgorihm of soluion used in ll cses ws ode3 Adms. Tle illusres he hree differen cses simuled, which involve he sence nd presence of exernl forces cing in he sysem. Tle. Simuled cses. Simple Pendulum Spring-Mss-mper Sysem Cse Exernl force Exernl force Cse A Q = 0 Q = 0 Cse B Q = A cosw Q = A cosw Cse C Q = A cos w l sin θ Q = Impulsive funcion On he oher hnd, Tle descries he pre-eslished vlues of prmeers nd coefficiens used in he simulions. Tle. Simulions prmeers for ll he hree cses. Simple Pendulum Spring-Mss-mper Sysem Mss m = kg Mss m = kg Accelerion of Accelerion of g = 9.8m/s² grviy grviy g = 9.8m/s² engh of he Siffness nd dmping l = m sring consns k = 5; c = 0. Coefficien u τ = Coefficien u τ = Coefficien α wih β = α τ = ; α = 0.4; α = 0.6; α = 0.9; α =.0; α =.; α =. Coefficien α wih β = α τ = ; α = 0.4; α = 0.6; α = 0.9; α =.0; α =.; α =. 3. Simulion Resuls 3.. Resuls Regrding he Simple Pendulum The resuls of he numericl simulions of he E. 6 regrding he simple pendulum re presened elow, wih differen vlues of α nd β α = β, ccording o Tle nd differen exernl exciions in crescen order of inensiy, from cse A o cse C foremenioned in Tle. The firs grph of ech cse compres he curves oined wih α smller or eul o.0 nd he second one compres he curves oined wih α igger or eul o.0.

7 Mhemics 05, 3 64 Cse A: Figure. Time hisory vlues of α. Cse B: Figure. Time hisory vlues of α. Figure 3. Time hisory vlues of α.

8 Mhemics 05, 3 65 Cse C: Figure 4. Time hisory vlues of α. Figure 5. Time hisory vlues of α. Figure 6. Time hisory vlues of α. Besides he grphs conining he simulions of he ngulr posiions for ll he hree cses of he simple pendulum, i is presened in Figures 7, 8 nd 9 he pseudo-phse porris for Cse B, wih differen vlues of α.

9 Mhemics 05, 3 66 Figure 7. Pseudo-phse porri Cse B for α = 0.9. Figure 8. Pseudo-phse porri Cse B for α =.0. Figure 9. Pseudo-phse porri Cse B for α =..

10 Mhemics 05, Resuls Regrding The Spring-Mss-mper Sysem The resuls of he numericl simulions of he E. 8 regrding he spring-mss-dmper sysem re presened elow, wih differen vlues of α nd β α = β, ccording o Tle nd differen exernl exciions in crescen order of inensiy, from cse A o cse C foremenioned in Tle. Once gin, he firs grph of ech cse compres he curves oined wih α smller or eul o.0 nd he second one compres he curves oined wih α igger or eul o.0. Cse A: Figure 0. Time hisory vlues of α. Figure. Time hisory vlues of α.

11 Mhemics 05, 3 68 Cse B: Figure. Time hisory vlues of α. Cse C: Figure 3. Time hisory vlues of α. Figure 4. Time hisory vlues of α.

12 Mhemics 05, 3 69 Figure 5. Time hisory vlues of α. Besides he grphs conining he simulions of he ngulr posiions for ll hree cses of he spring-mss-dmper sysem, i is presened in Figures 6, 7 nd 8 he pseudo-phse porris for Cse B, wih differen vlues of α. Figure 6. Pseudo-phse porri Cse B for α = 0.9. Figure 7. Pseudo-phse porri Cse B for α =.0.

13 Mhemics 05, iscussion nd Conclusions Figure 8. Pseudo-phse porri Cse B for α =.. The oined resuls poin o curious nd insiging specs of he effecs h rise from using frcionl orders in he differenil euions h represen he dynmics of he sudied sysems. I is worh noing in he firs cse simple pendulum h, in firs insnce, he prolem dels wih free oscillory sysem virory wihou he presence of dmping. When is moion euion is simuled, king ino ccoun he generl form of he dynmic h involves derivives of rirry order ineger or frcionl, he resuls ghered in Figures o 9 riefly show h: For α = ineger order, he sysem displys he expeced ehvior, h is, n oscillion whose mpliude is minined due o he sence of dmping. For vlues of α >, he sysem loses is oscillory chrcerisics nd is mpliudes seem o grow very fs nd indefiniely s shown in Figures, 4, nd 6. c However, for vlues of α <, Figures, 3 nd 5 imply h he sysem cuires dmping cpciy, which is indirecly proporionl o he vlue of α. For exmple, he smller he vlue of α, he igger he dmping cpciy. Thus, he sysem h is no dmped will disply chrcerisics h go from under dmping α = 0.9 o super dmping α = 0.4. On he oher hnd, oserving he second cse spring-mss-dmper sysem, i is possile o noice h: for α =, he dynmic moion euion of rirry order ineger or frcionl, represened y Figures 0 o 5, presens gin n expeced ehvior for ineger order sysems. Figures 0, nd 4 lso disply greer dmping cpciy in he sysem for vlues of α < s lredy hppened in he firs cse. I is noiced in his cse h Figures, 3 nd 5, which involve vlues of α <, no only seem o decrese even more drsiclly he mpliudes of he oscillions virions, u cn presen signs of nonliner phenomen, which cn e ojec of fuure invesigions. Anoher priculriy ghered from he resuls of he simulions is he presence, in some cses, of cerin insiliy in he ehvior of he sysem Figures. 7, 9 6 nd 8 nd even n indicive of possile occurrence of chos under some condiions. In fuure work, we would like invesige ou chos presence in hese frcionl dynmicl sysems using he rges ypunov Exponen E sed on he Wolf s lgorihm.

14 Mhemics 05, 3 7 In his pper, esides he mehod employed here is differen, we cn conclude h he decresing vlue of lph provides eer or greer enuion of he mpliudes of he oscillions. We elieve h he min resuls found in his work involve he oservion h when he vlue of lph decreses, he response of he sysem evolves from n under-dmped ehvior ino n over-dmped ehvior, i.e., here is n increse in he dmping cpciy of he sysems. These conclusions re new conriuion of his work. Th mens h if he choice of he order α of he derivive is convenienly mde, models h need igger inensiies of dmping or viring sorers my enefi from he use of he frcionl order in dynmics nd possily in conrol of he foremenioned sysems. I is expeced h he resuls descried in his pper my offer furher insighs ino he complex ehvior of hese sysems, nd hus insige more reserch effors in his direcion. Acknowledgmens This projec ws suppored y So Pulo Reserch Foundion FAPESP nd y fellowship of he en s Office for Reserch of he Universiy of So Pulo USP. Auhor Conriuions All uhors hve conriued eully. Conflics of Ineres The uhors declre no conflic of ineres. References. Oldhm, K.B.; Spnier, J. The Frcionl Clculus: Theory nd Applicions of iffereniion nd Inegrion o Arirry Order; Acdemic Press, Inc.,New York, NY, USA, Kils, A.A.; Srivsv, H.M.; Trujillo, J.J. Theory nd Applicions of Frcionl ifferenil Euions; Elsevier Science imied: Amserdm, he Neherlnds, 007; pp vid, S.A.; inres, J..; Pllone, E.M.J.A. Frcionl order clculus: hisoricl pologi, sic conceps nd some pplicions. Rev. Brs. Ensino Fís. 0, 33, Podluny, I. Frcionl ifferenil Euions; Acdemic Press: Wlhm, MA, USA, Blchndrn, K.; Trujillo, J.J. The nonlocl Cuchy prolem for nonliner frcionl inegrodifferenil euions in Bnch Spces. Nonliner Anl. Theory Mehods Appl. 00, 7, Mchdo, J.A.T.; Kirykov,V.; Minrdi, F. A Poser Aou he Old Hisory of Frcionl Clculus. Frc. Clc. Appl. Anl. 00, 3, Vlerio,.; Trujillo, J.J.; Rivero, M.; Mchdo, J.A.T.; Blenu,. Frcionl clculus: A survey of useful formuls. Eur. Phys. J. Specil Topics 03,, de Oliveir, E.C.; T.Mchdo, J.A. A Review of efiniions for Frcionl erivives nd Inegrl. Mh. Prol. Eng. 04, 6.

15 Mhemics 05, Ahmd-Rmi, E.N. A frcionl pproch o nonconservive grngin dynmicl sysems. FIZIKA A 005, 4, Trsov, E. T. Frcionl vriions for dynmicl sysems: Hmilon nd grnge pproches. J. Phys. A: Mh. Gen. 006, 39, Blenu,.; Avkr, T. grngins wih liner velociies wihin Riemnn-iouville frcionl derivives. Nuovo Cim. 004, B9, Cresson, J.; Inizn, P. Aou frcionl Hmilonin sysems. Phys. Scr. 009, T Agrwl, O.P. Frcionl vriionl clculus nd he rnsversliy condiions. J. Phys. A 006, 39, Muslih, S.M.; Blenu,. Frcionl Euler-grnge Euions of Moion in Frcionl Spce. J. Vi. Conrol 007, 3, y he uhors; licensee MPI, Bsel, Swizerlnd. This ricle is n open ccess ricle disriued under he erms nd condiions of he Creive Commons Ariuion license hp://creivecommons.org/licenses/y/4.0/.