Geodesic
On one mathematical model in elastic stability theory
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 1, pp. 29-39

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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. 
@article{VUU_2019_29_1_a2,
     author = {A. S. Zapov},
     title = {On one mathematical model in elastic stability theory},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {29--39},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2019_29_1_a2/}
}
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A. S. Zapov. On one mathematical model in elastic stability theory. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 1, pp. 29-39. http://geodesic.mathdoc.fr/item/VUU_2019_29_1_a2/