Revista Mexicana de Física

ISSN: 0035-001X
rmf@ciencias.unam.mx
Sociedad Mexicana de Física A.C.
México

Gómez-Aguilar, J.F.; Rosales-García, J.J.; Bernal-Alvarado, J.J.; Córdova-Fraga, T.; Guzmán-

Cabrera, R.
Fractional mechanical oscillators
Revista Mexicana de Física, vol. 58, núm. 4, agosto, 2012, pp. 348-352
Sociedad Mexicana de Física A.C.
Distrito Federal, México

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INVESTIGACIÓN Revista Mexicana de Fı́sica 58 (2012) 348–352 AGOSTO 2012

Fractional mechanical oscillators

J.F. Gómez-Aguilara,∗ , J.J. Rosales-Garcı́ab , J.J. Bernal-Alvaradoa ,
T. Córdova-Fragaa , and R. Guzmán-Cabrerab
a
Departamento de Fı́sica, División de Ciencias e Ingenierı́as Campus León, Universidad de Guanajuato,
Lomas del Bosque s/n, Lomas del Campestre, León Guanajuato, México,
e-mail: (jfga, bernal, teo)@fisica.ugto.mx
b
Departamento de Ingenierı́a Eléctrica, División de Ingenierı́as Campus Irapuato-Salamanca, Universidad de Guanajuato,
Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km, Comunidad de Palo Blanco, Salamanca Guanajuato México,
e-mail: (rosales, guzmanc)@ugto.mx
∗
Tel: +52 (477) 788-5100 ext. 8449.
Recibido el 15 de febrero de 2012; aceptado el 21 de mayo de 2012

In this contribution we propose a new fractional differential equation to describe the mechanical oscillations of a simple system. In particular,
we analyze the systems mass-spring and spring-damper. The order of the derivatives is 0 < γ ≤ 1. In order to be consistent with the physical
equation a new parameter σ is introduced. This parameter characterizes the existence of fractional structures in the system. A relation
between the fractional order time derivative γ and the new parameter σ is found. Due to this relation the solutions of the corresponding
fractional differential equations are given in terms of the Mittag-Leffler function depending only on the parameter γ. The classical cases are
recovered by taking the limit when γ = 1.

Keywords: Fractional calculus; mechanical oscillators; caputo derivative; fractional structures.

En esta contribución se propone una nueva ecuación diferencial fraccionaria que describe las oscilaciones mecánicas de un sistema simple.
En particular, se analizan los sistemas masa-resorte y resorte-amortiguador. El orden de las derivadas es 0 < γ ≤ 1. Para mantener la
consistencia con la ecuación fı́sica se introduce un nuevo parámetro σ. Este parámetro caracteriza la existencia de estructuras fraccionarias
en el sistema. Se muestra que existe una relación entre el orden de la derivada fraccionaria γ y el nuevo parámetro σ. Debido a esta relación
las soluciones de las correspondientes ecuaciones diferenciales fraccionarias estan dadas en terminos de la función de Mittag-Leffler, cuyas
soluciones dependen solo del orden fraccionario γ. Los casos clásicos son recuperados en el lı́mite cuando γ = 1.

Descriptores: Calculo fraccionario; oscilaciones mecanicas; derivada de caputo; estructuras fraccionarias.

PACS: 45.10.Hj; 46.40.Ff; 45.20.D-

1. Introduction
Although the application of Fractional Calculus (FC) has at- in comparison with the classical integer-order models, in
tracted interest of researches in recent decades, it has a long which such effects are in fact neglected.
history when the derivative of order 0.5 has been described by In a paper of Ryabov it is discussed the fractional os-
Leibniz in a letter to L’Hospital in 1695. A reviewing paper cillator equation involving fractional time derivatives of the
on applications and the formalism can be found in [1]. FC, in- Riemann-Liouville type [16]. Naber in [17], studied the
volving derivatives and integrals of non-integer order, is his- linearly damped oscillator equation, written as a fractional
torically the first generalization of the classical calculus [2-5]. derivative in the Caputo representation. The solution is
Many physical phenomena have “intrinsic” fractional order found analytically and a comparison with the ordinary lin-
description, hence, FC is necessary in order to explain them. early damped oscillator is made. In [18] was considered the
In many applications FC provides a more accurate model of fractional oscillator, being a generalization of the conven-
physical systems than ordinary calculus do. Since its success tional linear oscillator, in the framework of fractional calcu-
in the description of anomalous diffusion [6], non-integer lus. It is interpreted as an ensemble of ordinary harmonic
order calculus, both in one dimension and in multidimen- oscillators governed by a stochastic time arrow. Despite in-
sional space, has become an important tool in many areas troducing the fractional time derivatives the cases mentioned
of physics, mechanics, chemistry, engineering, finances and above seem to be justified, there is no clear understanding of
bioengineering [7-10]. Fundamental physical considerations the basic reason for fractional derivation in physics. There-
in favor of the use of models based on derivatives of non- fore, it is interesting to analyze a simple physical system and
integer order are given in [11-13]. Another large field which try to understand their fully behavior given by a fractional
requires the use of FC is the theory of fractals [14]. Frac- differential equation.
tional derivatives provide an excellent instrument for the de- The aim of this work is to give a simple alternative to
scription of memory and hereditary properties of various ma- construct fractional differential equations for physical sys-
terials and processes [15]. This is the main advantage of FC tems. In particular, we analyze the systems mass-spring and
 FRACTIONAL MECHANICAL OSCILLATORS 349

spring-damper in terms of the fractional derivative of the Ca- Zt

C γ 1 f (n) (η)
puto type. The analytical solutions are given in terms of the 0 Dt f (t) = dη, (4)
Γ(n − γ) (t − η)γ−n+1
Mittag-Leffler function depending on the parameter γ. 0

where n = 1, 2, . . . ∈ N and n−1 < γ ≤ n. We consider the

2. Fractional oscillator system case n = 1, i.e., in the integrand there is only a first deriva-
tive. In this case, 0 < γ ≤ 1, is the order of the fractional
We propose a simple alternative procedure for constructing derivative.
the fractional differential equation for the fractional oscilla-
The Caputo derivative operator satisfies the following re-
tor system. To do that, we replace the ordinary time derivative
lations
operator by the fractional one in the following way:
C γ C γ C γ
d dγ 0 Dt [f (t) + g(t)] = 0 Dt f (t) + 0 Dt g(t),
→ γ, 0<γ≤1 (1)
dt dt C γ
0 Dt c = 0, where c is constant. (5)
It can be seen that (1) is not quite right, from a physical point
of view, because the time derivative operator d/dt has dimen- For example, in the case f (t) = tk , where k is arbitrary
sion of inverse seconds s−1 , while the fractional time deriva- number and 0 < γ ≤ 1 we have the following expression for
tive operator dγ /dtγ has, s−γ . In order to be consistent with the fractional derivative operation,
the time dimensionality we introduce the new parameter σ in
the following way C γ k kΓ(k)
0 Dt t = tk−γ , (0 < γ ≤ 1) (6)
h 1 dγ i 1 Γ(k + 1 − γ)
= . 0<γ≤1 (2)
σ 1−γ dtγ s where Γ(k) and Γ(k + 1 − γ) are the Gamma functions. If
where γ is an arbitrary parameter which represents the order γ = 1 the expression (6) yields the ordinary derivative
of the derivative. In the case γ = 1 the expression (2) be-
comes an ordinary derivative operator d/dt. In this way (2) is C 1 k dtk
0 Dt t = = ktk−1 . (7)
dimensionally consistent if and only if the new parameter σ, dt
has dimension of time [σ] = s. Then, we have a simple proce- During the recent years the Mittag-Leffler function has
dure to construct fractional differential equations. It consists caused extensive interest among physicist due to its role
in the following; in an ordinary differential equation replace played in describing realistic physical systems with memory
the ordinary derivative by the following fractional derivative and delay. The Mittag-Leffler function is defined by the se-
operator ries expansion as
d 1 dγ ∞
→ 1−γ γ , 0 < γ ≤ 1. (3) X tm
dt σ dt Ea (t) = , (a > 0), (8)
m=0
Γ(am + 1)
The expression (3) is a time derivative in the usual sense,
because its dimension is s−1 . The parameter σ (auxiliary
where Γ(·) is the Gamma function. When a = 1, from (8)
parameter) represents the fractional time components in the
we have
system. This non-local time is called the cosmic time [19].
Another physical and geometrical interpretation of the frac- ∞
X X∞
tm tm
tional operators is given in [20]. E1 (t) = = = et . (9)
m=0
Γ(m + 1) m=0
m!
To analyze the dynamical behavior of a fractional sys-
tem it is necessary to use an appropriate definition of frac-
Therefore, the Mittag-Leffler function is a generalization of
tional derivative. In fact, the definition of the fractional
the exponential function.
order derivative is not unique and there exist several defi-
nitions, including: Grünwald-Letnikov, Riemann-Liouville, Now, we can write a fractional differential equation cor-
Weyl, Riesz and the Caputo representation. In the Caputo responding to the mechanical system, Fig. 1, in the following
case, the derivative of a constant is zero and we can define, way
properly, the initial conditions for the fractional differential
m d2γ x(t) β dγ x(t)
equations which can be handled by using an analogy with
2γ
+ 1−γ + kx(t) = 0,
the classical case (ordinary derivative). Caputo derivative im- σ 2(1−γ) dt σ dtγ
plies a memory effect by means of a convolution between the 0<γ≤1 (10)
integer order derivative and a power of time. For this reason,
in this paper we prefer to use the Caputo fractional derivative. where m is the mass, measured in Kg, β is the damped co-
The Caputo fractional derivative for a function of time, efficient, measured in N · s/m and k is the spring constant,
f (t), is defined as follows [5] measured in N/m [5].

Rev. Mex. Fis. 58 (2012) 348–352

350 J.F. GÓMEZ-AGUILAR, J.J. ROSALES-GARCÍA, J.J. BERNAL-ALVARADO, T. CÓRDOVA-FRAGA, AND R. GUZMÁN-CABRERA

Then in the case γ = 1 the solution of the Eq. (13) is a peri-

odic function given by

x(t) = x0 cosω0 t. (18)

Expression (18) is the well known solution for the case of

integer differential Eq. (11) with γ = 1.
Note that the parameter γ, which characterizes the frac-
tional order time derivative can be related to the σ parame-
ter, which characterizes the existence, in the system, of frac-
tional structures (components that show an intermediate be-
havior between a system conservative (spring) and dissipa-
tive (damper)). For example, for the system described by the
fractional equation (11), we can write the relation
F IGURE 1. Damped oscillator.
r
σ m
From Eq. (10) we obtain the particular cases: when γ = p m = σω0 , 0<σ≤ . (19)
k
k
1. β = 0

m d2γ x(t)
+ kx(t) = 0, 0 < γ ≤ 1 (11)
σ 2(1−γ) dt2γ
and

2. m = 0

β dγ x(t)
+ kx(t) = 0, 0 < γ ≤ 1, (12)
σ 1−γ dtγ
Equation (11) may be written as follows

d2γ x
+ ω 2 x(t) = 0, (13)
dt2γ
where
kσ 2(1−γ)
ω2 = = ω02 σ 2(1−γ) , (14)
m
is the angular frequency for different values of γ, and F IGURE 2. Mass-Spring system, γ = 1, γ = 0.75, γ = 0.5 and
γ = 0.25.
ω02 =k/m is the fundamental frequency of the system (i.e,
when γ = 1). The solution for the Eq. (13) with x(0) = x0
and ẋ(0) = 0 as the initial conditions, is given by
n o
x(t) = x0 E2γ − ω 2 t2γ , (15)

where
³ ´n
n o ∞
X − ω 2 t2γ
E2γ − ω 2 t2γ = , (16)
n=0
Γ(2γn + 1)

is the Mittag-Leffler function.

In the case γ = 1 from (14) we have ω 2 = ω02 = k/m
and (15) becomes hyperbolic cosine
½ ¾ Ãr ! Ãr !
k 2 k 2 k
E2 − t = ch − t = ch i t
m m m
F IGURE 3. Mass-Spring system, γ = 1, γ = 0.96, γ = 0.92 and
= ch(iω0 t) = cos ω0 t. (17) γ = 0.8.

Rev. Mex. Fis. 58 (2012) 348–352

 FRACTIONAL MECHANICAL OSCILLATORS 351

damping of fractional oscillator is intrinsic to the equation of

motion and not by introducing an additional force as in the
case of an ordinary damping harmonic oscillator. The frac-
tional oscillator should be considered as an ensemble average
of harmonic oscillators.
On the other hand, solution of the Eq. (12) is given by
n kσ 1−γ o
x̃(t) = x˜0 Eγ − tγ , (21)
β
where Eγ {} is the Mittag-Leffler function defined above.
For the case γ = 1, the expression (21) becomes
k
x̃(t) = x̃0 e− β t , (22)

which is the well-known solution for the integer differential

Eq. (12). In this case the relation between γ and σ is given
F IGURE 4. Damper-Spring system, γ = 1, γ = 0.75, γ = 0.5 and by
γ = 0.25. k k
γ = σ, 0<σ≤ . (23)
β β
The solution (21) of the fractional Eq. (12), taking into ac-
count the relation (23), may be written as follows
n o
x̃(t̃) = x̃0 Eγ − γ (1−γ) t̃γ , (24)

where t̃ = βk t is a dimensionless parameter. Figures 4 and 5,

show the solution of (24) for different values of γ.

3. Conclusion
In this work we have proposed a new fractional differential
equation of order 0 < γ ≤ 1 to describe the mechanical
oscillations of a simple system. In particular, we analyze
the systems mass-spring and spring-damper. In order to be
consistent with the physical equation the new parameter σ
F IGURE 5. Damper-Spring system, γ = 1, γ = 0.96, γ = 0.92 is introduced. The proposed equation gives a new universal
and γ = 0.8. behavior for the oscillating systems, Eqs. (20) and (24), for
equal value of the magnitude, δ = 1 − γ characterizing the
Then, the magnitude δ = 1 − γ characterizes the exis- existence of the fractional structures on the system. We also
tence of fractional structures in the system. It ispeasy to see found that there is a relation between γ and σ depending on
that, when γ = 1, equivalently σ = 1/ω0 = m/k, the the system studied, see the Eqs. (19) and (23). The analytical
value of δ is zero, which means that in the system there are solutions are given in terms of the Mittag-Leffler function.
no fractional structures. However, in the interval 0 < γ < 1, They depend on the parameter γ and preserve physical units
δ grows and tends to unity, fractional structures appear in the in the system parameters. The classical cases are recovered
mechanical system. by taking the limit when γ = 1.
Taking into account the expression (19), the solution (15) The general case of the Eq. (10) with respect to the pa-
of the Eq. (11) can be rewritten through γ by rameter γ and the classification of fractional systems depend-
n o
x(t̃) = x0 E2γ − γ 2(1−γ) t̃2γ , (20) ing on the magnitude δ will be made in a future paper.
We hope that this way of dealing with fractional differen-
where t̃ = tω0 is a dimensionless parameter. Plots for differ- tial equations can help us to understand better the behavior of
ent values of γ are shown in the Fig. 2 and 3. the fractional order systems.
As we can see from (20), the displacement of the frac-
tional oscillator is essentially described by the Mittag-Leffler Acknowledgments
function n o
E2γ − γ 2(1−γ) t̃2γ . This research was supported by CONACYT and PROMEP
Also it is proved that, if γ is less than 1 the displacement under the Grant: Fortalecimiento de CAs., 2011, UGTO-CA-
shows the behavior of a damped harmonic oscillator. The 27.

Rev. Mex. Fis. 58 (2012) 348–352

352 J.F. GÓMEZ-AGUILAR, J.J. ROSALES-GARCÍA, J.J. BERNAL-ALVARADO, T. CÓRDOVA-FRAGA, AND R. GUZMÁN-CABRERA

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