Elements of mathematical phenomenology and analogies of electrical and mechanical oscillators of the fractional type with finite number of degrees of freedom of oscillations: linear and nonlinear modes

Abstract

<p>Following the ideas about analogies, mathematical, qualitative, and structural, introduced by Mihailo Petrović in <italic>Elements of Mathematical Phenomenology</italic> (Serbian Royal Academy, Belgrade, 1911), in this paper we present our research results focused on analogies of fractional-type oscillation models between mechanical and electrical oscillators, with a finite number of degrees of freedom of oscillation. In addition to reviewing basic results, we investigate new constitutive relations and generalizations of the energy dissipation function, mechanical dissipative element of fractional type, and electrical resistor of dissipative fractional type. Those constitutive relations are expressed by means of the fractional order differential operator. By applying the Laplace transformation and the power series expansions, we determine and graphically present the approximate analytical solutions for eigen oscillations of the fractional type, as well as for forced oscillations, using a convolution integral. Tables with elements of mathematical phenomenology and analogies of oscillatory mechanical and electrical systems of fractional type are shown, as well as the principal fractional-type eigen-modes for a class of discrete mechanical or electrical oscillators, when these fractional-type modes are independent and there is no interaction between them. A number of theorems on the properties of independent modes of fractional type and the energy analysis of the discrete systems are also given.</p>