Authors: Hedrih, Katica (Stevanović) 
Tenreiro Machado, J. A.
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Discrete fractional order system vibrations
Journal: International Journal of Non-Linear Mechanics
Volume: 73
First page: 2
Last page: 11
Issue Date: 1-Jul-2015
Rank: M21
ISSN: 0020-7462
DOI: 10.1016/j.ijnonlinmec.2014.11.009
Abstract: 
A theory of free vibrations of discrete fractional order (FO) systems with a finite number of degrees of freedom (dof) is developed. A FO system with a finite number of dof is defined by means of three matrices: mass inertia, system rigidity and FO elements. By adopting a matrix formulation, a mathematical description of FO discrete system free vibrations is determined in the form of coupled fractional order differential equations (FODE). The corresponding solutions in analytical form, for the special case of the matrix of FO properties elements, are determined and expressed as a polynomial series along time. For the eigen characteristic numbers, the system eigen main coordinates and the independent eigen FO modes are determined. A generalized function of visoelastic creep FO dissipation of energy and generalized forces of system with no ideal visoelastic creep FO dissipation of energy for generalized coordinates are formulated. Extended Lagrange FODE of second kind, for FO system dynamics, are also introduced. Two examples of FO chain systems are analyzed and the corresponding eigen characteristic numbers determined. It is shown that the oscillatory phenomena of a FO mechanical chain have analogies to electrical FO circuits. A FO electrical resistor is introduced and its constitutive voltage-current is formulated. Also a function of thermal energy FO dissipation of a FO electrical relation is discussed.
Keywords: Eigen fractional order mode | Fractional order element | Fractional order oscillator | Generalized function of fractional order dissipation of system energy | Matrix fractional order differential equation | Mechanical to electrical analogy
Publisher: Elsevier
Project: Dynamics of hybrid systems with complex structures. Mechanics of materials 

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