Fractional mechanical oscillators
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 a 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.
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Digital revista |
| Language: | English |
| Published: |
Sociedad Mexicana de Física
2012
|
| Online Access: | http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0035-001X2012000400010 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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 a 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. |
|---|