This paper considers the energy aspects of fractional-order elements defined by the equation: force is proportional to the fractional-order derivative of displacement, with order varying from zero to two. In contrast to the typically conservative assumption of classical physics that leads to the potential and kinetic energy expressions, a number of important nonconservative differences are exposed. Firstly, the considerations must be time-based rather than displacement or momentum based variables. Time based equations for energy behavior of fractional elements are presented and example applications are considered. The effect of fractional order on the energy input and energy return of these systems is shown. Importantly, it is shown that the history, or initialization, has a significant effect on energy response. Finally, compact expressions for the work or energy, are developed.

References

References
1.
Malti
,
R.
,
Cois
,
O.
,
Aoun
,
M.
,
Levron
,
F.
, and
Oustaloup
,
A.
,
2002
, “
Energy of Fractional Order Transfer Functions
,”
IFAC 15th Triennial World Congress
,
Barcelona, Spain
.
2.
Machado
,
J. T.
,
2013
, “
Fractional Generalization of Memristor and Higher Order Elements
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
, pp.
264
275
.10.1016/j.cnsns.2012.07.014
3.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
,
2013
, “
Energy Considerations in Fractional Systems
,”
Proceedings of ASME IDETC/CIE
,
Portland, Oregon
, August 4–7, Paper No. DETC2013-13247.
4.
Halliday
,
D.
, and
Resnick
,
R.
,
1974
,
Fundamentals of Physics
,
Wiley
,
New York
, p.
100
.
5.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
,
2008
, “
Initialization of Fractional Differential Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
2
), p.
021103
.10.1115/1.2833585
6.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
,
1999
, “
Initialization, Conceptualization, and Application in the Generalized Fractional Calculus
,” NASA Paper No. TP-1998-208415.
7.
Knopp
,
K.
,
1928
,
Theory and Application of Infinite Series
,
Hafner, New York
, p.
179
.
8.
Hartley
,
T. T.
,
Lorenzo
,
C. F.
,
Veillette
,
R. J.
, and
Adams
,
J. L.
,
2013
, “
On the Energy Stored in Fractional-Order Electrical Elements
” Proceedings of ASME IDETC/CIE,
Portland, Oregon
, August 4–7,
ASME
Paper No. DETC2013-13498.10.1115/DETC2013-13498
You do not currently have access to this content.