Fractional differential equations and kernels, and small vibrations of mechanical systems

Theoretical mechanics


Aleroeva H. T.1*, Aleroev T. S.2**

1. Moscow Technical University of Communications And Informatics, 8a, Aviamotornaya Str., Moscow, 111024, Russia
2. Moscow State University of Civil Engineering, MSUCE, 26, Yaroslavskoe shosse, Moscow, 129337, Russia



The article studies the Dirichlet boundary value problem for the equation of motion of an oscillator with viscoelastic damping in the case when the damping order is greater than one but less than two. Such problems simulate many physical processes, in particular, string vibrations in a viscous medium, changes in the deformation-strength characteristics of polymer concrete during loading, etc. Earlier, in the case, when the order of the fractional derivative is less than one, the authors established basic oscillation properties, using the methods of perturbation theory. They are as follows. All the frequencies are simple (i. e., the amplitude function of the natural oscillation of a given frequency is uniquely determined up to a constant factor); the natural oscillation with the lowest frequency  (the fundamental tone) does not have any nodes; the natural oscillations with frequency  (the jth overtone) has exactly j nodes; the nodes of two successive overtones alternate. It was also shown, that the operator, generated by the differential expression of a second order with fractional derivatives in lower terms and boundary conditions of Sturm-Liouville type, is a Keldysh’s operator. This makes it possible to establish the completeness of the systems of eigenfunctions and associated functions of boundary value problems induced by second-order differential expressions with fractional derivatives in the lower terms and Dirichlet boundary conditions.

In the case when the order of the fractional derivative is greater than one, the previously used technique based on perturbation theory for the study of oscillatory properties does not work. In this paper, we investigate the Green’s function of the corresponding problem, which allows us to establish the positivity of this Green’s function. From this fact, some oscillatory properties follow for the equation of motion of an oscillator with viscoelastic damping in the case when the order of the fractional derivative is greater than one. Since the application of the first eigenvalues is usually of the greatest interest in applied problems, the results obtained can be applied to evaluate the study of the natural oscillation with the lowest frequency, and also to establish that the fundamental tone has no nodes.


asphalt concrete, oscillating properties, fractional derivative, polymer concrete, Green function, fundamental tone


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