Fractional calculus and small angle motions for mechanical systems

Theoretical mechanics


Aleroeva H. T.

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



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.


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


  1. Keldysh M.V. Doklady AN SSSR, 1951, vol. 77, no. 1, pp. 11–14.

  2. Zveriaev E.M. Trudy MAI, 2014, no. 78:

  3. Zveriaev E.M. Olekhova L.V. Trudy MAI, 2015, no. 79:

  4. Nahushev A.M. Drobnoe ischislenie i ego primenenie (Fractional calculus and its application), Moscow, FIZMATLIT, 2003, 273 p.

  5. Aleroev T.S., Aleroeva H.T., Ning-Ming Nie, Yi-Fa Tang. Boundary Value Problems for Differential Equations of Fractional Order//Mem. Differential Equations Math. Phys. 49 (2010), pp. 19-82.

  6. Tseitlin A.I. Prikladnye metody resheniya kraevykh zadach stroitel’noi mekhaniki (Applied methods of solution for boundary value problems in structural mechanics), Moscow, Stroiizdat, 1984, 334 p.

  7. Ingman D., Suzdalnitsky J. Iteration method for equation of viscoelastic motion with fractional differential operator of damping, Comp. Methods Appl. Mech. Engrg. 190 (2001) pp. 5027–5036.

  8. Chen J.-H., Chen W.-C. Chaotic dynamics of the fractionally damped van der Pole equation, Chaos, Solitons and Fractals 35 (2008) pp. 188–198.

  9. Coffey W.T., Kalmykov Yu.P., Waldron J.T. The Langevin Equation. World Scientific Series in Contemporary Chemical Physics. Vol. 10, 2004. 704 pages.

  10. Yang, H.; Luo, G.; Karnchanaphanurach, P.; Louie, T. M.; Rech, I.; Cova, S.; Xun, L.; Xie, X. S. Science 2003, 302, pp. 262–266.

  11. Kekharsaeva E.R., Pirozhkov V.G. Trudy 6-i Vserossiiskoi nauchnoi konferentsii «Mekhanika kompozitsionnykh materialov i konstruktsii, slozhnykh i geterogennykh sred». Moscow, 16-18 November 2016, pp. 104 — 109.

  12. Aleroev T.S., Aleroeva H.T. Erratum to: «On the Eigenfunctions and Eigenvalues of a Class of Non-Selfadjoint Operators,» Lobachevskii Journal of Mathematics, 2016, Vol. 37, No. 6, p. 815.

  13. Aleroev T.S., Aleroeva H.T. On the Eigenfunctions and Eigenvalues of a Class of Non-Selfadjoint Operators, Lobachevskii Journal of Mathematics, 2016, Vol. 37, No. 3, pp. 227–230.

  14. Aleroev T.S., Kirane M., Tang Y.-F. Boundary-value problems for differential equations of fractional order, Journal of Mathematical Sciences, Vol. 10 (2013), No. 2, pp.158 — 175.

  15. Kato T. Teoriya vozmushchenii lineinykh operatorov (Perturbation theory of linear operators), Moscow, Mir, 1972, 739 p.

  16. Loginov B.V. Izvestiya AN UzSSR. Fiziko-matematicheskie nauki, 1963, no. 6, pp. 14-19.

  17. Larionov E.A., Zveriaev E.M., Aleroev T.S K teorii slabogo vozmushcheniya normal’nykh operatorov (The theory of weak perturbations of normal operators), Moscow, Keldysh Institute of Applied Mathematics, 2014, no. 14, 31 pages.

  18. Aleroev T.S. Kraevye zadachi dlya differentsial’nykh uravnenii s drobnymi proizvodnymi (Boundary value problems for differential equations with fractional derivatives), Doctor’s thesis, Moscow, MSU, 2000, 120 p.

  19. Gohberg I.C., Krejn M.G. Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil’bertovom prostranstve. (Introduction to the theory of linear nonselfadjoint operators in Hilbert space), Moscow, Nauka, 1965, 448 p.

Download — informational site MAI

Copyright © 2000-2021 by MAI