fractional simple harmonic oscillator
TRANSCRIPT
FRACTIONAL SIMPLE HARMONIC OSCILLATOR
Akram A. Rousan1, Nabil Y. Ayoub
2 and Khetam Khasawinah
2
1 Department of Physical Sciences, Jordan University of Science and Technology,
P. O. Box 3030, Irbid-22110- JORDAN, e-mail: [email protected]
2 School of Applied Natural Sciences, German Jordanian University, Amman –
Jordan,
e-mail: [email protected]
3Physics Department, Yarmouk University, Irbid-Jordan, e-mail:
Abstract: A series solution of simple harmonic oscillator is presented using fractional calculus
of a general fractional differential equation with a damping term proportional to a fractional order
time derivative of the spatial coordinate. It is shown that the solution gives the undamped case
when =0 and the damped case when =1 (damping is proportional to the velocity). Damping
with different values of and the damping ratio are calculated, compared and discussed. An
explanation of the behavior of the oscillator under those conditions is proposed.
Mathematical Subject Classification: 26A33, 30B10, 33B15, 34C25, 34C27,
44A10
Keywords: Fractional differential equation; Simple harmonic oscillator; Damping;
Series solution; Intermediate stages.
1. Introduction
There is growing interest in fractional calculus, which enables
scientists to differentiate or integrate using fractional orders. Derivatives and
integrals to arbitrary orders are referred to as differintegrals. A full historical
survey of fractional calculus can be found in Oldham and Spanier [1] and Samko
et al [2]. Many useful books on fractional differential equations [3], applications
of fractional calculus [4] and fractional calculus [1] and [2] have been published.
Different authors use fractional calculus for different purposes.
Fractional equations give the most complete possible description of the system
memory (Kobelev and Romanov [5]), also the evolution of some physical
phenomena like multipoles and other problems in electromagnetism (Engheta [6]-
[15]) and gravity (Rousan et al [16]). Simple solutions using this technique are
presented by Kulish and Lage [17] and Kulish et al [18] on heat diffusion, and
Flores and Osler [19] on tautochrones. Attempts have been made to study
nonconservative systems using Lagrangian containing derivatives of fractional
order (Riewi [20] and [21] and Agrawal [22]) and potentials of arbitrary forces
(Rabei et al [23]).
On the problem of damping and viscoelasticity of dynamic systems
there has been a great deal of work . Yuan and Agrawal [24] presented a
numerical scheme to solve a fractionally damped single degree-of-freedom
spring-mass-damper forced system using Laguerre integral formula and
considering derivative of the order 0.5 in their example. A comparison of
numerical methods is found in Shokooh and Suarez [25]. A numerical method
was used by Koh and Kelly [26] in applying fractional derivatives of order 0.5 in
the formulation of stress-strain relationship of elastomers. Suarez and Shokooh
[27] proposed a closed form solution to the problem using an eigenvector
expansion method with fractional derivative of order 0.5 only. Laplace transform
was used by Bagley and Trovik [28] and [29] . Fourier transform was used by
Gaul et al [30] and [31] with fractional order of 0.5 only. Hartley and Lorenzo[32]
employed the Mittag-Leffler function to study the initialization problem for a
system of linear fractional order differential equations for discrete values of the
fractional order of derivation. Koeller [33] studied creep and relaxation functions
taking the order of derivation to vary from 0.05 to 0.35 to show that a continuous
transition from solid state to the fluid state occurs when the memory
parameter(order of derivation) varies from zero to one expressing the results in
terms of Mittag-Liffler function. Diethelm et al [34] recently offered a useful
discussion and comparison of different numerical solutions for fractional
equations.
In this work, the problem of harmonic oscillator is studied with the
damping term different from the usual one, which is proportional to the velocity,
but it is a fractional derivative of the spatial coordinate with time. A series
solution is presented and discussed, and it is not limited to the case of
semiderivative (fractional order of 0.5 ) but for any fractional order, with a
comparison between solutions for different fractional orders and the damping
ratios. Critical damping is also discussed in connection with the fractional order.
An attempt is made to explain the idea of fractional damping and how it is
different as a concept from the ordinary damping.
In future works the problem of forced oscillatory motion and of
diffusion processes will be explored employing fractional calculus following the
scheme discussed in the present work
2. The Fractional Simple Harmonic Oscillator
We start with writing the fractional equation of motion of the one-
dimensional simple harmonic oscillator in its general form as follows:
okxdt
xdb
dt
xdm
2
2
(1)
Where is non-integer and =0 corresponds to an undamped simple harmonic
oscillator and =1 corresponds to a damped simple harmonic oscillator.
Dividing equation (1) by m and applying using
22m
b, where
is the damping ratio and the power on was introduced for consistency of
dimensions [27], and m
k2 , and applying Laplace transform to the equation we
get:
02 22
2
2
Lxdt
xdL
dt
xdL
(2)
Consider the following initial conditions:
x(0)=x0 , since for =1 we have the initial condition 00
tdt
dx , then we
assume that for any 0
0
tdt
xd
, here it is clear that the condition
00
tdt
dx does not apply for any but only for =1, and we further assume
Cdt
xd
t
0
1
1
. And make use of the Laplace transform for derivatives of
fractional orders([1] p. 135), we get
0)()(2)( 2
1
12
0
2
sx
dt
xdsxssxsxs
ot
Rearranging the equation and solve it for x(s) one can write it as follows
222
2
0
2
2)(
ss
Csxsx (3)
Let x(s)=x1+x2, where:
222
0
12
ss
sxx and
222
2
22
2
ss
Cx
Rewriting x1 as follows
ssss
sxx
22
222
0
1
212
p
pp
p
o
ss
ss
x
)2()1(
21
22
2
0
2
2
1
2
212
2
0
0
)2
1(
)1(
pp
pp
p
ss
x
And by further expansion and rearranging terms, one can get:
0,
1)(2
)2(2
01!!
)2()!()1(
rprrp
rrprp
srp
rpxx
(4)
Similarly x2 can be written as follows:
0,
2)(2
)2(22
2!!
)2()!()1(2
rprrp
rrprp
srp
rpCx
(5)
and since x(s)=x1+x2 and using the following identity[31]:
)(
1 1
tL
s
One can get the displacement as a function of time:
1)(2
21
1)(2!!
)!()2()1()(
0,
0
)(2)(2
rrp
t
rrprp
xtrptx
rp
rrprrrprp
(6)
Here C can be taken according to the physical problem to be discussed. In this
paper we take it to be 1
0
x , to take care of dimensions. Expansion (6) is valid
for relatively short times t, however we may, using analytic continuation extend
the validity of an equation similar to expansion (6) for longer times t.
2-a: Simple Harmonic Oscillator (undamped)
In this case =0(no damping) and eq.(3) is reduced to this form
)1(
)(
2
2
0
ss
xsx
......)1(6
6
4
4
2
20
ssss
x
Taking the inverse Laplace transform we get
......)!4!2
1()(4422
0 tt
xtx
txtx cos)( 0
This is the solution of the simple harmonic oscillator with no damping.
Or putting =0 in equation (6) which makes x2=0 and hence r=0 then
0
0
22
)!2(
)1()( x
p
ttx
p
ppp
(7)
This is again
txtx cos)( 0
2-b: Damped Oscillator
When =1 the case is the usual damped oscillator with damping
proportional to the velocity, and the general fractional solution in equation (6)
becomes:
1)(2
21
1)(2!!
)!()2()1()(
0,
0
22
rrp
t
rrprp
xtrptx
rp
rprrprp
(8)
To prove that substitute for =1 in equation (2) and perform the Laplace
transform we get:
22
0)(
s
sxsx
02 2
2
2
Lx
dt
dxL
dt
xdL
0)()(2)( 2
00
2 sxxssxsxsxs
2202
2)(
ss
sxsx
2
1
2
1
102
1
20)(
2
)()(
sx
s
sxsx
Where )1( 222
1 , and the Inverse Laplace transform for this equation is
tttxtx 1
1
10 sincos)exp()(
This is the closed form solution of the damped oscillator having the damping
proportional to the velocity.
This result can be reached also by a series expansion of x(s) and the
Laplace inverse of the result will lead to the series result as in equation (7).
3. Results and Discussion
Semiderivative (fractional derivative of order 0.5) was adopted by
researchers as being capable of describing the frequency dependence of the
damping material quite satisfactorily ([27]-[29]). In this study we will not be
limited to semiderivative, but will take different values and the system will be
studied under such conditions. In the following we examine the effect of the
fractional order on the behavior of the oscillator having a certain damping ratio
employing the series solution derived in the previous section.
To see the effect of , consider a system where =10 rad/s, x0=0.001m,=0.1
v0=0 and takes the values of 0.00001,0.2,0.5,0.7 and 1,the value of 0.00001 is
taken to be close to zero. The results are displayed in fig. (1).
When =1, equation (1) becomes as follows:
okxdt
dxb
dt
xdm
2
,
and the system behaves as an “ordinary” (opposite to fractional) damped
oscillator. Whereas when =0 the damping term vanishes and the system is then
a free oscillator. For values of different from those limiting values the behavior
of the oscillator seems to be affected by ; as becomes greater than zero the
system seems to experience some damping and this damping increases as
increases until it reaches the upper limiting value ( =1). The chosen values for
the parameters of the oscillator make it an under damped oscillator. This can be
looked at as if the system was transformed from a free oscillator ( =0) to a
damped oscillator ( =1) in an evolution process (i.e. fractionally) that means that
the system is given (or allowed to have) a damping-like property, but the damping
ratio was not allowed to fully play its role until =1. So for values 10
the damping properties cannot be described by no means in terms of the properties
of an ordinary oscillator to be under damped, critically damped and over damped,
but an evolution process of damping character is being developed with until it is
equal to one then the material can be described as a damping material like oil in a
system consisting of a spring and a mass immersed in oil. The idea of evolution
was introduced by Engheta ([6]-[15]) in electric multipoles and electromagnetism
in general and was used by Rousan et al [16] in gravity.
If, on the other hand, is given a certain value and we change the
damping ratio we get the results that show the behavior of fractional oscillators
with different damping ratios, and how would that change with the fractional
order in view of the discussion above.
0.2 0.4 0.6 0.8 1 1.2
-0.001
-0.0005
0.0005
0.001
Figure (1): Oscillator with =10 rad/s, x0=0.001m,=0.1 v0=0 and
=0.00001,0.2,0.5,0.7,and 1(Damping increases as increases)
Figure (2) shows the results when =0.3 and =sqt(Pi),4,10,15,20 and 50 .
Those values of are purposely exaggerated in order to look for a value of at
which the curve will touch the zero axis revealing a critical damping behavior
[27], but non of the curves displays a critical damping behavior although all
values of the damping ratio are taken to be above the value the square root of Pi.
0.1 0.2 0.3 0.4 0.5 0.6
0.001
0.002
0.003
Figure (2) A system with =0.3 and =10, has the values of
sqt(Pi),4,10,15,20,50
(Frequency and damping increase as the damping ratio increases)
Close examination of the curves reveals that some of them touches the zero axis
( the time axis) once and then continues to oscillate above the equilibrium
position. This is clear in fig. (2) when =15 where the curve touches the zero axis
in the third period and when =50 it touches the zero axis in the second period.
This indicates that at high values of the curve will touch the zero axis at a
certain time and continue to oscillate but above the equilibrium position, a
behavior not found in ordinary damped oscillators. To make that clearer curves of
=0.1 and 50 are shown in fig. (3). Such a behavior was reported by Suarez and
Shokooh [27] but in the case of an applied external force.
0.2 0.4 0.6 0.8 1 1.2
-0.001
0.001
0.002
0.003
Figure (3) The same system as in fig. (2) but for =0.1 and 50
Figures (4) and (5) show the results of calculations for the cases in
which =0.5 and 0.7 where the same behavior is found and one can notice that
the curve of = 4.5 in fig. (4) where =0.5 the curve touches the zero axis
immediately after it reaches the maximum and instead of keeping approaching the
zero axis it rises again oscillating above the equilibrium position. The value of
at which such behavior is encountered can be thought as a value for critical
damping (c) at this value of , that is the fractional oscillator of order 0.5 has
critical damping at =4.5.When has values higher than c the whole curve
stays above the zero axis and oscillates above the equilibrium position.
0.1 0.2 0.3 0.4 0.5 0.6
-0.0005
0.0005
0.001
0.0015
0.002
Figure (4) A system with =0.5 and =10, has the values of
0.1,0.5,sqt(Pi),3,4.5,10 (Damping increases with damping ratio)
For a higher value of like 0.7, the results are shown in fig.(5) where
c=1.35 which is less than that when =0.5, which means that the value of c
approaches the ordinary critical value, which is approximately 0.2 in this example,
as the oscillator approaches the “ordinary” oscillator or being less “fractional” as
approaches the value of one.
According to the discussion above, It does not look obvious that the
fractional order =0.5 would represent the damping material best or any other
order. The results indicate that the damping character of the damping material is
not developed to give the known characteristics of damping materials until
becomes equal to one where the equation of the ordinary oscillator is used. That
view brings to mind that the damping material can be thought of as “pseudo-
damping” material in the case of fractional damped oscillator.
0.1 0.2 0.3 0.4 0.5 0.6
-0.0005
0.0005
0.001
Figure (5)) A system with =0.7 and =10, has the values of 0.1,0.5,1,1.35,sqt
(Pi), 3(Damping increases with damping ratio)
4. Conclusion
A series solution of fractional harmonic oscillator is introduced in this
work, which covers the solution of the equation of motion for all cases of the
unforced oscillator. Systems with different fractional orders (not limited to 0.5)
and with different damping ratios are then studied using the series solution offered
in this work. It is found that the oscillator does not act as an ordinary damped
oscillator with the three types of damping (under, critical and over damped) while
it is fractional but approaches the ordinary damped oscillator as the fractional
order approaches one.
The results indicate that there is no strong evidence of a fractional order, like the
order of 0.5 which represents best the damping materials as was suggested in the
literature ([27]-[29]). We suggest, instead, that the system goes through an
evolution process as the fractional order goes from zero (free) to one (damped)
letting it pass through intermediate stages where the system can have a damping
character and the material can be thought of as a pseudo-damping material. A sort
of critical damping in fractional oscillator can be thought of as the value of the
damping ratio at which the curve touches the zero axes after its first maximum
and oscillates above the equilibrium position.
References
[1] Oldham, K.B., and J. Spanier, The Fractional Calculus. Academic Press, New
York (1974)
[2]S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and
Derivatives, Theory and Applications. Gordon and Breach Science Publishers,
Langhorne, PA (1993)
[3]I. Podlubny, Fractional Differential Equations, Academic Press, San Diego,
(1999)
[4] Hilfer R., Applications of Fractional Calculus in Physics. World Scientific,
Singapore, (2000).
[5]V. Kobelev, and E. Romanov, Fractional Langevin Equation to Describe
Anomalous Diffusion, 4th
International Summer School/ Conference “Let’s Face
Chaos Through Nonlinear Dynamics”. 28 June-11- July (1999), Maribor, Slovenia
[6]N. Engheta, A Note on Fractional Calculus and the Image Method for
Dielectric Spheres.Journal of Electromagnetic Waves and Applications,
9(9),September (1995), 1179-1188.
[7]N. Engheta, Use of Fractional Integration to Propose Some Fractional
Solutions for the Scalar Helmholtz Equation. A chapter in Progress in
Electromagnetics Research (PIER) Monograph Series, 12, Jin A. Kong (ed.),
(1996), 107-132.
[8]N. Engheta, On Fractional Calculus and Fractional Multipoles in
Electromagnetism. IEEE Transactions on Antennas and Propagation, 44(4), April
(1996), 554-566.
[9]N. Engheta, Electrostatic Fractional Image Methods for Perfectly Conducting
Wedges and Cones. IEEE Transactions on Antennas and Propagation, 44(12),
December (1996), 1565-1574.
[10]N. Engheta, On the Role of Fractional Calculus in Electromagnetic Theory.
IEEE Antennas and Propagation Magazine, 39(4), August (1997), 35-46.
[11]N. Engheta, Fractional Curl Operator in Electromagnetics. Microwave and
Optical Technology Letters,17(2), February (1998), 86-91.
[12]N. Engheta, On Fractional Paradigm and Intermediate Zones in
Electromagnetism:
I. Planar Observation. Microwave and Optical Technology Letters, 22(4), August
(1999), 236-241.
[13]N. Engheta, Phase and Amplitude of the Fractional-Order Intermediate Wave.
Microwave and Optical Technology Letters, 21(5), June (1999), 338-343.
[14]N. Engheta, On Fractional Paradigm and Intermediate Zones in
Electromagnetism: II. Cylindrical and Spherical Observations. Microwave and
Optical Technology Letters, 23(2), October (1999), 100-103.
[15] N. Engheta, Fractional Paradigm in Electromagnetic Theory. A chapter in
Frontiers in (Electromagnetics), D. H. Werner and R. Mittra (eds.), IEEE Press,
chapter 12, (2000), 523-552.
[16] Akram A. Rousan, E.Malkawi, E. Rabei and H. Widyan.” Applications of
Fractional Calculus to Gravity”. Fractional Calculua & Applied Analysis. 5(2)
2002, 155-168
[17]V.V. Kulish and J.L. Lage, Fractional-Diffusion Solutions for Transient Local
Temperature and heat Flux. ASME J. Heat Transfer, 122(2), May (2000), 372-
376.
[18] V.V Kulish, J.L. Lage, P.L. Komarov, and P.E. Raad, A Fractional-Diffusion
Theory for Calculating Thermal Properties of thin Films from Surface Transient
Thermoreflectance Measurements. ASME J. Heat Transfer, 123 (2001), 1133-
1138.
[19] E. Flores and T. Osler, The Tautochrone under Arbitrary Potentials Using
Fractional Derivatives. Am. J.Phys. 67(1999), 718-722
[20] F. Riewe, Nonconservative Lagrangian and Hamiltonian Mechanics. Phys.
Rev. E 53(2)(1996), 1890-1899
[21] F. Riewe, Mechanics with Fractional Derivatives. Phys. Rev.E, 55(3)(1997),
3581-3592
[22] O.P.Agrawal, A new Lagrangian and a New Lagrange Equation of Motion
for Fractionally Damped systems. ASME J. appl. Mech. 68(2001) 339-341.
[23] E.M. Rabei, T. Alhalholy and A.A. Rousan, Potentials of Arbitrary Forces
with Fractional Derivatives. Int. J Modern Phys. 19(17-18)(2004), 3083-3092
[24] L. Yuan and Om.P. Agrawal, A Numerical Scheme for Dynamic Systems
Containing Fractional Derivatives. Proceedings of DETC’98, 1998 ASME Design
Engineering Technical Conferences, September 13-16, 1998, Atlanta, Georgia.
[25] A. Shokooh and L. Suarez, A Comparison of Numerical Methods Applied to
a Fractional Model of Damping Materials, J. Vibration and Control, 5 (3), (1999),
331-354.
[26] C. G. Koh and J.M. Kelly, Application of Fractional Derivatives to Seismic
analysis of Base-Isolated Models, Earthquake Engineering and Structural
Dynamics, 19, (1990), 229-241.
[27] L.E. Suares and A. Shokooh, An Eigenvector Expansion Method for the
Solution of Motion Containing Fractional Derivatives, J. Applied Mechanics, 64,
(1997), 629-635.
[28] R.L. Bagley and P.J. Trovic, Fractional Calculus-A Different Approach to the
Analysis of Viscoelasticaly Damped Structures, AIAA Journal, 21(5), (1983),
741-748.
[29] R.L. Bagley and P.J. Trovic, Fractional Calculus in the Transient Analysis of
Viscoelastically Damped Structures, AIAA Journal, 23(6), (1985), 918-925.
[30] L. Gaul, P. Klein and S. Kempfle, Impulse Response Function of an
Oscillator with Fractional Derivative in Damped Description, Mechanics research
Communications, Vol. 16(5), (1989), 297-305.
[31] L. Gaul, P. Klein and S. Kempfle, Damping Description Involving Fractional
operators, Mechanical Systems and Signal Processing, 5(2), (1991), 81-88.
[32] T.T. Hartley and C.F. Lorenzo, A Solution to the Fundamental Linear Order
Differential Equation, NASA TP- 1998-208693, December 1998.
[33] R.C. Koeler, Applications of Fractional Calculus to the Theory of
Viscoelasticity, J. Applied Mechanics, 51, (1984), 299-307.
[34] K. Diethelm, J.M. Ford, N.J. Ford and M. Weilbeer, Pitfalls in Fast
Numerical Solvers for Fractional Equations, J. Comput. Appl. Math. 186(2),
(2006), 482-503