Section
|
Mathematics
|
Title
|
On one mathematical model in elastic stability theory
|
Author(-s)
|
Zapov A.S.a
|
Affiliations
|
Yaroslavl State Universitya
|
Abstract
|
We consider a boundary-value problem for the nonlinear evolution partial differential equation, given in renormalized form. This problem appears in rotary system mechanics and describes the transverse vibrations of the rotating rotor of a constant cross-section from a viscoelastic material whose ends are pivotally fixed. The question of the stability of the zero equilibrium state is studied, the critical value of the rotor speed is found, above which continuous oscillations occur. Exact solutions of the boundary-value problem are found in the form of single-mode functions with respect to the spatial variable and functions periodic in time. The stability conditions for such solutions are derived, and in some cases an analysis of the stability conditions is given. The paper shows the absence of multimode time-periodic solutions. The basic and important (from an applied point of view) particular cases of this nonlinear boundary-value problem are analyzed. All the results of the analysis of a nonlinear boundary-value problem are analytical. Their conclusion is based on the qualitative theory of infinite-dimensional dynamical systems.
|
Keywords
|
nonlinear evolution equation, stability, oscillations of rotor systems, periodic solutions
|
UDC
|
517.957
|
MSC
|
35B10, 35B05
|
DOI
|
10.20537/vm190103
|
Received
|
24 September 2018
|
Language
|
Russian
|
Citation
|
Zapov A.S. On one mathematical model in elastic stability theory, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 29-39.
|
References
|
- Bolotin V.V. Nekonservativnye zadachi teorii uprugoi ustoichivosti (Nonconservative problems of the theory of elastic stability), Moscow: Nauka, 1961, 339 p.
- Pozdnyak E.V. Self-oscillation of rotors with many degrees of freedom, Izvestiya Akademii Nauk SSSR. Mekhanika Tverdogo Tela, 1977, no. 2, pp. 40-49 (in Russian).
- Kubyshkin E.P. Self-oscillatory solutions of a class of singularly perturbed boundary-value problems, Differ. Uravn., 1989, vol. 25, no. 4, pp. 674-685 (in Russian).
- Holmes P.J., Marsden J.E. Bifurcation of dynamical systems and nonlinear oscillations in engineering systems, Nonlinear Partial Differential Equations and Applications, Berlin: Springer, 1978, pp. 163-206. https://doi.org/10.1007/BFb0066411
- Holmes P.J. Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis, Journal of Sound and Vibration, 1977, vol. 53, no. 4, pp. 471-503. https://doi.org/10.1016/0022-460X(77)90521-1
- Holmes P.J., Marsden J.E. Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, 1978, vol. 14, no. 4, pp. 367-384. https://doi.org/10.1016/0005-1098(78)90036-5
- Paidoussis M.P., Issid N.T. Dynamic stability of pipes conveying fluid, Journal of Sound and Vibration, 1974, vol. 33, no. 3, pp. 267-294. https://doi.org/10.1016/S0022-460X(74)80002-7
- Kimball A.L. Internal friction theory of shaft whirling, General Electric Review, 1924, vol. 27, pp. 224-251.
- Kimball A.L. Internal friction as a cause of shaft whirling, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6, 1925, vol. 49, issue 292, pp. 724-727. https://doi.org/10.1080/14786442508634653
- Kulikov A.N. Attractors of a nonlinear boundary value problem arising in aeroelasticity, Differential Equations, 2001, vol. 37, issue 3, pp. 425-429. https://doi.org/10.1023/A:1019254818198
- Leonov G.A., Kuznetsov N.V., Kiseleva M.A., Solovyeva E.P., Zaretskiy A.M. Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor, Nonlinear Dynamics, 2014, vol. 77, no. 1-2, pp. 277-288. https://doi.org/10.1007/s11071-014-1292-6
- Kulikov A.N. Landau-Hopf scenario of passage to turbulence in some problems of elastic stability theory, Differential Equations, 2012, vol. 48, no. 9, pp. 1258-1271. https://doi.org/10.1134/S0012266112090066
- Kolesov Yu.S. Attractors of resonance wave type equations: Discontinuous oscillations, Mathematical Notes, 1994, vol. 56, no. 1, pp. 679-684. https://doi.org/10.1007/BF02110556
- Kulikov A.N. Some bifurcation problems of elastic stability theory and mathematical physics, Dr. Sci. (Phys.–Math.) Dissertation, Nizhny Novgorod, 2018, 299 p. (In Russian).
- Fischer A., Doikin A., Rulevskiy A. Research of dynamics of the rotor with three-film bearings, Procedia Engineering, 2016, vol. 150, pp. 635-640. https://doi.org/10.1016/j.proeng.2016.07.058
- Khulief Y.A., Al-Sulaiman F.A., Bashmal S. Vibration analysis of drillstrings with self-excited stick–slip oscillations, Journal of Sound and Vibration, 2007, vol. 299, no. 3, pp. 540-558. https://doi.org/10.1016/j.jsv.2006.06.065
- Segal I. Non-linear semi-groups, Annals of Mathematics, 1963, vol. 78, no. 2, pp. 339-364. https://doi.org/10.2307/1970347
- Banakh L.Y., Nikiforov A.N. Impact of aero-hydrodynamic forces on fast-rotating rotor systems, Izv. Ross. Akad. Nauk. Mekh. Tv. Tela, 2006, no. 5, pp. 42-51 (in Russian).
- Yakubov S.Y. Solvability of the Cauchy problem for abstract quasilinear second-order hyperbolic equations, and their applications, Tr. Mosk. Mat. Obs., 1970, vol. 23, pp. 37-60 (in Russian). http://mi.mathnet.ru/eng/mmo238
- Sobolevskiy P.E. Equations of parabolic type in a Banach space, Tr. Mosk. Mat. Obs., 1961, vol. 10, pp. 297-350 (in Russian). http://mi.mathnet.ru/eng/mmo123
- Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki (Equations of mathematical physics), Moscow: Nauka, 1966, 735 p.
|
Full text
|
|