Mathematical modeling of rotation of a free elastic solid with cavities containing fluids in a resistive medium

• Datum: 2026
• Hauptverfasser: Kononov, Yu., Кононов, Юрій
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2026
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/9958
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 531.36, 531.38, 517.93 Based on the well-known P. V. Kharlamov equations of motion of a system of coupled gyrostats and the S. L. Sobolev function of state, we develop a mathematical model of rotation in a resistive medium of a free elastic solid with two cavities completely filled with ideal incompressible fluid. The mathematical model of an elastic solid with an ideal fluid is presented in the form of a system of two elastically coupled solids filled with a fluid. The solids are connected with an elastic restoring Hooke hinge and subjected to the action of dissipative moments and moments supporting their rotation. In the case of two Lagrangian gyroscopes with arbitrary axisymmetric cavities containing fluids, we derive a transcendental characteristic equation and analyze it by taking into account the fundamental tone of the oscillations of fluid. It is proved that necessary conditions for the asymptotic stability can always be satisfied by increasing the elasticity coefficient of the hinge provided that the fundamental tone of the oscillations of fluids is greater than 1. The absence of internal resonance is demonstrated in the case where the first tones of the oscillations of fluids coincide. The analysis of necessary conditions for the asymptotic stability with respect to the angular velocity of uniform rotation turned out to be more complicated. These conditions impose restrictions not only on the fundamental tone of the oscillations of fluids but also on the attached equatorial moment of inertia of solids and the parameter of inertial coupling of the fluid. The investigation of the proposed quite simple mathematical model makes it possible to estimate the influence of elasticity and fluids on the stability of rotation of an elastic solid with fluid with an accuracy sufficient for the practical purposes.