J. F. Gómez*, J. J. Rosales**, J. J. Bernal*, M. Guía** ABSTRACT RESUMEN INTRODUCTION

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1 Universidad de Guanajuao Mahemaical modelling of he mass-spring-damper sysem - A fracional calculus approach Modelado maemáico del sisema masa-resore-amoriguador - Enfoque en cálculo fraccionario J. F. Gómez* J. J. Rosales** J. J. Bernal* M. Guía** ABSTRACT In his paper he fracional differenial equaion for he mass-spring-damper sysem in erms of he fracional ime derivaives of he Capuo ype is considered. In order o be consisen wih he physical equaion a new parameer is inroduced. This parameer characerizes he exisence of fracional componens in he sysem. A relaion beween he fracional order ime derivaive and he new parameer is found. Differen paricular cases are analyzed. RESUMEN En ese rabajo se presena la ecuación diferencial fraccionaria del sisema masa-resoreamoriguador en erminos de la derivada fraccionaria de ipo Capuo. Con la finalidad de ser consisenes con la ecuación física se inroduce un nuevo parámero. Ese parámero caraceriza la exisencia de componenes fraccionarias en el sisema. Se encuenra la relación enre el orden de la derivada fraccionaria y el nuevo parámero. Diferenes casos pariculares son analizados. Recibido: de febrero de 01 Acepado: 4 de abril de 01 Keywords: Fracional Calculus; mass-spring-damper sysem; Capuo derivaive; fracional componens. Palabras clave: Cálculo Fraccionario; sisema masa-resore-amoriguador; derivada de Capuo; componenes fraccionarias. INTRODUCTION Fracional calculus (FC) involving derivaives and inegrals of non-ineger order is he naural generalizaion of he classical calculus [1-4]. Many physical phenomena have inrinsic" fracional order descripion and so FC is necessary in order o explain hem. In many applicaions FC provides more accurae models of he physical sysems han ordinary calculus do. Since is success in descripion of anomalous diffusion [5] non-ineger order calculus boh in one and mulidimensional space has become an imporan ool in many areas of Physics Mechanics Chemisry Engineering Finances and Bioengineering [6-9]. Fundamenal physical consideraions in favor of he use of models based on derivaives of non-ineger order are given in [10-1]. Fracional derivaives provide an excellen insrumen for he descripion of memory and herediary properies of various maerials and processes. This is he main advanage of FC in comparison wih he classical ineger-order models -in which such effecs are in fac negleced. In [13] are discussed he fracional oscillaor equaion involving fracional ime derivaives of he Riemann-Liouville ype [14] considered he linearly damped oscillaor equaion wih he damping erm generalized o a Capuo fracional derivaive. A soluion is found analyically and a comparison wih he ordinary linearly damped oscillaor is made. Despie inroducing he fracional ime derivaives he cases menioned above seem o be jusified; here is no clear undersanding of he basic reason for fracional derivaion in physics. Therefore i is ineresing o analyze a simple physical sysem and ry o undersand heir fully behavior given by he fracional differenial equaion. *Deparameno de Ingeniería Física. División de Ciencias e Ingenierías Campus León. Universidad de Guanajuao. Lomas del Bosque s/n col. Lomas del Campesre C. P León Guanajuao México. Phone: +5 (477) fax: +5 (477) y +5 (477) ex jfga@fisica.ugo.mx. **Deparameno de Ingeniería Elécrica. División de Ingenierías Campus Irapuao-Salamanca. Universidad de Guanajuao. Carreera Salamanca-Valle de Saniago km Comunidad de Palo Blanco Salamanca Guanajuao México. Phone. (464) fax 311 ex Vol. N. 5 Julio-Agoso 01 5

2 Universidad de Guanajuao The presen work is ineresed in he sudy of a simple mechanical sysem consising of a mass a damped coefficien and a spring in he framework of he fracional derivaive. The posiion o follow: in order o change he ime derivaive operaor d/ by a fracional operaor d / ( represens he order of he derivaive) i is necessary o inroduce an addiional parameer σ which mus have dimension of seconds (in he case of he ime derivaive) o be consisen wih he dimension of he ordinary derivaive. The σ parameer characerizes he presence of fracional ime componens in he sysem. METHODS An oscillaing sysem in general is a mechanical sysem consising of hree kinds of elemens: a mass (m) measured in kg a damped coefficien measured in Ns/m and a spring consan k measured in N/m. The change wih respec o ime of he movemen x() is described by he second order homogeneous differenial equaion d x() dx () m + + kx ()=0. (1) In a real oscillaing sysem he damped coefficien is differen from zero and he free mechanical oscillaions become exinc due o he Joule effec. To compensae for he damping mechanical oscillaions a source v() should be include in he sysem. Therefore he differenial equaion ha governs he behavior of he sysem (mass-spring-damper) wih source has he form d x() dx () m + + kx ()= v ( ). () The erm kx() is very imporan because lack of i in equaions (1) and () imply ha i has no oscillaing sysem. For he equaions (1) and () i will be considered he following special cases: In he absence of he damped coefficien in he sysem i.e. = 0 i is had from (1) and () d x() m + kx ()=0 (3) and d x() m + kx ()= v ( ). (4) In he case when m = 0 i.e. in he absence of mass in he sysem hey are had from equaions (1) and () dx() + kx()=0 (5) and dx() + kx()= v( ). (6) The soluions of he equaions (1) and () are well known. I is imporan o noe ha fracional differenial equaions corresponding o oscillaing sysems have been sudied before replacing he ineger ime derivaive by fracional one on a purely mahemaical or heurisics basis [14]. The idea is o wrie he equaions (1) and () in erms of a fracional ime derivaive operaor. I is proposed o change he ordinary ime derivaive operaor by fracional one in he following way d d 0 < 1 (7) where is an arbirary parameer represening he order of he fracional ime derivaive operaor and in he case = 1 becomes ordinary ime derivaive operaor. Since he ime derivaive operaor has dimensions of inverse seconds s -1 hen from (7) i is seen ha he expression d 1 s (8) i is no a ime derivaive operaor in he usual sense because is dimensionaliy is s -. In order o be consisen wih ime dimensionaliy i is inroduced he parameer σ in he following way 1 d 1 1 σ (9) s such ha when = 1 he expression (9) becomes ordinary derivaive operaor d/. This dimensionaliy equaion is saisfied only if he parameer σ has dimension of seconds [σ] = s. Therefore i is possible o always change ordinary ime derivaive operaor by he fracional one in he following general form d 1 d n 1< n 1 σ (10) where n is ineger. These wo expressions represen ime derivaives as he dimensions are inverse seconds s -1. The parameer σ 1- can be called a fracional ime parameer represening he fracional ime componens in he sysem; is dimensionaliy is s 1-. This non-local ime is called in he lieraure he cosmic ime [15]. Anoher physical and geomerical inerpreaion of he fracional operaors is given in [16]. 6 Vol. N. 5 Julio-Agoso 01

3 Universidad de Guanajuao Hopefully ha fracional ime derivaive operaor defined in (10) will be very useful in he consrucion of fracional differenial equaions for he physical and engineering problem. On he oher hand Capuo's fracional derivaive of a funcion f ( ) is defined as an inegral ransform of ordinary derivaive o fracional one [4]: 1 f (n )(η ) C d η (11) 0 D f ( ) = Γ(n ) 0 ( η ) n +1 From he equaion (16) i is idenified ω = k σ (1 ) = ω0σ (1 ) m (0) as he fracional angular frequency where ω0 = k/m is he fundamenal frequency of he non-fracional sysem (i.e when = 1). where C indicaes Capuo derivaive evaluaed in 0 and order n = 1... N and n - 1 < n. I is considered he case n = 1 i.e. in he inegrand here is only firs derivaive. In his case 0 < 1 is he inerval of he fracional derivaive. On he oher hand from equaion (14) when k = 0 i is defined he fracional relaxaion ime as m τ = 1. (1) σ This fracional relaxaion ime arises in he experimenal sudies of he complex sysems [1]. When = 1 from his expression i is had he well known relaxaion ime = m/. For example in he case f ( ) = k where k is arbirary number and 0 < 1 i is had he following expression for he fracional derivaive operaion (which will be useful laer for solving he wave equaion wih fracional ime derivaive C0D d / ): Soluions of he equaions (14) and (15) may be obained using he Laplace ransform. Soluions of he homogeneous fracional differenial equaions (16) and (18) will be found in erms of he Miag-Leffler funcion [17]. (1) where Γ(k ) and Γ(k + 1- ) are he Gamma funcions. If = 1 he expression (1) yields o ordinary derivaive C 1k 0 D = k = k k 1. (13) Then using definiion (11) he homogeneous soluion of he equaion (16) is given by () where (3) RESULTS Now i is possible o wrie he ordinary differenial equaions (1-6) in he fracional form. Using he expression (10) he fracional differenial equaions corresponding o equaions (1) and () are given by is he Miag-Leffler funcion and x(0) = x0. Firs case: = 1 he expression (3) becomes hyperbolic cosine (14) and Then in his case i is had periodic funcion given by (15) from hese equaions can be deduced he paricular cases when = 0 m d x ( ) + kx ( ) = 0 0 < 1 (16) (1 ) σ d x ( ) σ1 d x ( ) 1 σ + kx ( ) = 0 + kx ( ) = v ( ) 0 < 1 0 < 1. ( ) (5) wih angular frequency. The expression (5) is he well known soluion of he ineger differenial equaion (3). Second case: = 1/ from () i is given (17) and m = 0 (4) (18) (19) (6) Noe ha he parameer -which characerizes he fracional order ime derivaive- can be relaed o he σ parameer -which characerizes he exisence in he sysem of fracional exciaions [18]-[19]. For example for he sysem described by he fracional equaion (16) i is possible o wrie he relaion Vol. N. 5 Julio-Agoso 01 7

4 Universidad de Guanajuao σ m = 0< σ. (7) m / k k Then he magniude δ = 1 - characerizes he exisence of fracional srucures in he sysem. I is easy o see ha when = 1 hen and herefore δ = 0 ha means ha in he sysem here are no fracal srucure. However in he case δ grows and ends o uniy because in he sysem are increasingly fracal exciaions. Taking accoun he expression (7) he soluion () of he equaion (16) can be rewrien hrough as follows. (8) forces as in he case of a damped harmonic oscillaor. A simple fracional oscillaor behaves like a damped harmonic oscillaor. Soluion of he equaion (18) is given by which in he case = 1 reduces o (9) k x 0()= x 0e (30) which is he well known soluion of he ineger differenial equaion (5). In his case he relaion beween and σ is given by = σk 0< σ k. (31) The soluion (9) of he fracional equaion (18) aking accoun he relaion (31) may be wrien as. (3) Figure 1. Mass-Spring Sysem = 1 = 096 = 09 = 08 = 075 = 050 and = 05. Figure 1 shows he soluion of (8) for differen values of. Respec o he figure 1 he displacemen of he fracional oscillaor is essenially described by he Miag-Leffler funcion for he considered iniial condiions. The funcion is I is showed by numerical calculaions ha he displacemen of he fracional oscillaor varies as a funcion of ime and how his ime variaion depends on he parameer. Also i is proved ha if is less han 1 he displacemen shows he behavior of a damped harmonic oscillaor. As a resul in consisen wih he case of simple harmonic oscillaor he oal energy of simple fracional oscillaor will no be a consan. The damping of fracional oscillaor is inrinsic o he equaion of moion and no inroduced by addiional Figure. Damper-Spring Sysem = 1 = 096 = 09 = 08 = 075 = 050 and = 05. Figure shows he soluion of (3) for differen values of. The equaions (17) and (19) are fracional linear non-homogeneous differenial equaions hen he general soluion is given by he sum of he homogeneous (16) and (18) soluions and paricular soluions. These soluions may be obained applying direc and inverse Laplace ransform. As resul he soluion for he fracional non homogenous equaion (17) is given by 8 Vol. N. 5 Julio-Agoso 01

5 Universidad de Guanajuao (33) Figure 3 shows he soluion of (36) for differen values of. From equaion (36) i is clear ha in he case = 1 i is had (37) where i has been used he expression (7). Taking accoun he expression (31) he soluion of he fracional non homogeneous equaion (19) has he form (34). I easy o see ha in he case = 1 he soluion (34) becomes (35) In his case hey are had damping vibraions due o he presence of he damped coefficien 0. The soluion (37) is performed in R < km and has wo consans of inegraions as i should be for second ordinary differenial equaions (1) (/m = χ is he coefficien of damping of vibraions). In his case he parameer and σ have he following relaion. (38) Then he soluion akes he form (36) (39) which represens he general soluion of he ordinary ineger equaion (6). Now i is possible o pass o he general soluions of equaions (14) and (15) for a mechanical sysem having all mechanical elemens; mass m damped coefficien and spring k. The soluion of he equaion (14) has he form (36) where (40) Because of condiion choose as an example i is possible o. (41) Finally he soluion (36) akes he following form. (4) For he case (14) has he form he soluion of he equaion (43). In he case when = 1 i is had x 1( ) = x 0e /m (1+ 1 4m / k ) (44) where x(0) = x 0 is he movemen in he spring in = 0. The soluion (44) corresponds o he equaion (1) and characerize he change of movemen x() on he spring and has aperiodic characer. Taking he following relaions (45) Figure 3. Soluion of (36) for differen values of = 1 = 096 = 09 = 08 = 075 = 050 and = 05. he soluion (43) akes he form Vol. N. 5 Julio-Agoso 01 9

6 Universidad de Guanajuao where (46). (47) If he condiion is fullfilled i is given he following region range of values i is possible o choose (48) hen he soluion (46) can be wrien as (49). (50) Figure 4 shows he soluion of (50) for differen values of. Figures 3 and 4 show he complee soluion of he sysem mass-spring-damper. Figure 3 and 4 show ha he energy is conserved in he oscillaor -is conservaive when = 1- while he fracional oscillaor (1/ < < 1) show a dissipaive naure. applicaions of fracional calculus have gained considerable populariy [0-1]. In spie of hese various applicaions here are some imporan challenges. For example physical inerpreaion for he fracional derivaive is no compleely clarified ye [15]. I has been presened a new fracional differenial equaion for he oscillaing sysems. The proposed equaion gives a new universal behavior for he oscillaing sysems characerizing he exisence of he fracal srucures on he sysem. I was also found ou ha here is a relaion beween and σ depending on he sysem in sudies. In all simulaions he compuaional ime is around 443 s using an Inel Core Duo 18 GHz 99 GB RAM. Wih he approach presened here i will be possible o have a beer sudy of he ransien effecs in he mechanical sysems. The discussion of he soluions (8) and (3) he general case of he equaion (14) wih respec o he parameer he classificaion of fracal sysems depending on he magniude of δ and a complee analysis of he soluions (33) and (34) for differen value of will be made in a fuure paper. I is hoped ha his way of dealing wih fracional differenial equaions may help o undersand he behavior of he fracional order sysems beer. ACKNOWLEDGMENTS The auhors acknowledge fruiful discussions wih V. I. Tkach. This work was suppored by CONACYT and PROMEP under he Gran: Foralecimieno de CAs 011 UGTO-CA-7. REFERENCES [1] Oldham K. B. and Spanier J. (1974). The Fracional Calculus. Academic Press. New York. Figure 4. Soluion of (50) for differen values of = 1 = 096 = 09 = 08 = 075 = 050 and = 05. CONCLUSSION Fracional calculus is a very useful ool in describing he evoluion of sysems wih memory which ypically are dissipaive and o complex sysems. In recen decades i has araced ineres of researches in several areas of science. Specially in he field of physics [] Miller K. S. and Ross B. (1993). An Inroducion o he Fracional Calculus and Fracional Differenial Equaions. Wiley. New York. [3] Samko S. G. Kilbas A. A. and Marichev O.I. (1993). Fracional Inegrals and Derivaives Theory and Applicaions. Gordon and Breach Science Publishers. Langhorne PA. [4] Podlubny I. (1999). Fracional Differenial Equaions. Academic Press. New York. [5] Rousan A. A. Ayoub N. Y. Alzoubi F. Y. Khaeeb H. Al-Qadi M. Hasan (Quasser) M. K. and Albiss B. A. (006). Fracional Calculus and Applied analysis 9(1): pp Vol. N. 5 Julio-Agoso 01

7 Universidad de Guanajuao [6] Agrawal O. P. Tenreiro-Machado J. A. and Sabaier I. (004). Fracional Derivaives and Their Applicaions. Nonlinear Dynamics 38. Springer-Verlag. Berlin. [7] Hilfer R. (000). Applicaions of Fracional Calculus in Physics. World Scienific. Singapore. [8] Wes B. J. Bologna M. and Grigolini P. (003). Physics of Fracional Operaors. Springer-Verlag. Berlin. [9] Magin R. L. (006). Fracional calculus in Bioengineering. Begell House Publisher. Rodding. [10] Capuo M. and Mainardi F. (1971). Pure and Applied Geophysics 91: p [11] Weserlund S. (1994). Causaliy Universiy of Kalmar. [1] Baleanu D. Günvenc Z. B. and Tenreiro Machado J. A. (010). New Trends in Nanoechnology and Fracional Calculus Applicaions. Springer. [13] Ryabov Ya. E. and Puzenko A. (00). Damped oscillaions in view of he fracional oscillaor equaion. Physical Review B. 66: p [14] Naber M. (010). Linear Fracionally Damped Oscillaor. Inernaional Journal of Differenial Equaions. Hindawi Publishing Corporaion. [15] Podlubny I. (00). Geomeric and physical inerpreaion of fracional inegraion and fracional differeniaion. Frac. Calc. App. Anal. 5(4): pp [16] Moshre-Torbai M. and Hammond J. K. (1998). Physical and geomerical inerpreaion of fracional operaors. J. Franklin Ins. 335B(6): pp [17] Seybold H. and Hilfer R. (008). Numerical algorihm for calculaing he generalized Miag-Leffler funcion. SIAM J. Numer. Anal 47 (1): pp [18] Gómez J. F. Rosales J. J. Bernal J. J. Tkach V. I. Guía M. Sosa M. and Córdova T. (011). RC Circui of Non-ineger Order. Symposium on Fracional Signals and Sysems. Insiue Polyechnic of Coimbra: pp [19] Rosales J. J. Gómez J. F. Bernal J. J. Tkach V. I. Guía M. Córdova T. and González A. (011). Fracional Elecric RLC Circui. Symposium on Fracional Signals and Sysems. Insiue Polyechnic of Coimbra: pp [0] Herrmann R. (011). Fracional Calculus. World Scienific Press. [1] Tarasov V. E. (011). Fracional Dynamics. Springer. HEP. Vol. N. 5 Julio-Agoso 01 11