Fractional calculus and small angle motions for mechanical systems

Theoretical mechanics


Аuthors

Aleroeva H. T.

Moscow technical university of communications and informatics, MTUCI, 8a, Aviamotornaya str., Moscow, 111024, Russia

e-mail: binsabanur@gmail.com

Abstract

Many problems of mechanics and mathematical physics associated with perturbation of normal operators with discrete spectrum, reduced to consideration in Hilbert space of a compact operator , which called for a compact as a weak perturbation or as operator of a Keldysh type. In present paper we consider operators of Keldysh type, associated with boundary value problems for differential equations of second order with fractional derivatives in lower terms. Such problems simulate various physical processes. In particular, the oscillations of a string in viscous media, changes of deformable and strength characteristics of polymer concrete during the loading and etc. Since, considered boundary value problems simulate the oscillations of physical systems, then those problems shall to have a basic oscillational properties. In case when fractional derivatives have order less than 1, such oscillational properties are well-known. In given paper, this properties were proved for order , and there is shown, that mechanical systems, which are described by differential equations of second order with fractional derivative in lower term, are very sensitive to changes of fractional damping order. For example, if we consider a fractional damped van der Pol equation then periodic, quasi-periodic and chaotic motions existed, when the order of fractional damping is less than 1. When the order of fractional damping is , then there are chaotic motions only. This partly explains why oscillational properties (all eigenvalues are primary, and main tone has no nodes), obtained for fractional derivative order is less than 1, not available for fractional derivative order more than 1 but less than 2. In addition, in paper was shown, that operator, generated by the differential expression of second order with fractional derivatives in lower terms and boundary conditions of Sturm-Liouville type, is an Keldysh’s operator. From this, in particular, follows the completeness of a system of eigenfunctions and associated functions for this operator.

Keywords:

asphalt concrete, string oscillations in viscous media, fractional derivative, oscillational properties, operators of a Keldysh type

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