Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, June 7, 1999; final revision, Apr. 24, 2000. Associate Editor: A. A. Ferri.

All dynamic systems exhibit some degree of internal damping. Recent investigations have shown that a fractional derivative model provides a better representation of the internal damping of a material than an ordinary derivative model does. For a survey of fractional damping models and their applications to engineering systems, the readers are referred to Rossikhin and Shitikova 1 and the references therein. Traditionally, the Newton’s law is used to model such nonconservative systems, and when a Lagrangian, Hamiltonian, variational, or other energy-based approach is used, it is modified so that the resulting equations match those obtained using the Newtonian’s approach.

Several attempts...

1.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
1997
, “
Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids
,”
Appl. Mech. Rev.
,
50
, No.
1
, pp.
15
67
.
2.
Riewe
,
F.
,
1996
, “
Nonconservative Lagrangian and Hamiltonian Mechanics
,”
Phys. Rev. E
,
53
, No.
2
, pp.
1890
1899
.
3.
Riewe
,
F.
,
1997
, “
Mechanics With Fractional Derivative
,”
Phys. Rev. E
,
55
, No.
3
, pp.
3581
3592
.
4.
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York.
5.
Suarez
,
L. E.
, and
Shokooh
,
A.
,
1997
, “
An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives
,”
ASME J. Appl. Mech.
,
64
, pp.
629
635
.
6.
Miller, K. S., and Ross, B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York.
7.
Mainardi, F., 1997, “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Springer-Verlag, New York, pp. 291–348.
8.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, New York.
9.
Yuan, L., and Agrawal, O. P., 1998, “A Numerical Scheme for Dynamical Systems Containing Fractional Derivatives, Proceedings of the DETC98, ASME Design Engineering Technical Conferences, Paper No. DETC98/MECH 5857, Sept. 13–16, Atlanta, GA.
You do not currently have access to this content.