Pryhodko D. Oscillations of a cantilever rod with the supercritical static state with the base vertical vibrations

The dissertation (thesis) work is devoted to the laws of oscillations and stabilization of the rod with static supercritical state. Experimentally fixed effects of stabilization and resonance oscillations of a supercritical rod. Found parametric oscillation frequency range of the second and third eigenmodes and found the effect of simultaneous oscillations of the rod by two eigenmodes (the second and third). Experimentally derived frequecies of stabilization coincide well with theoretically calculated by the criterion of supercritical stabilizing rod, analytically constructed by Champneys and Fraser. Eigenmodes and its eigenfrequencies of supercritical rod oscillations for bend stiffness values, identified from the experiments, constructed numerically. Found values of eigenfrequencies indicate that the system under consideration has an internal resonance. A mathematical model of non-linear parametric resonance vibrations of a supercritical rod at their two-mode approximation, which is caused by an internal resonance of eigenfrequencies and results of experimental observations, as taking into account of the initial stress, and with its absence is derived. Shifts the resonance frequencies in the model with the initial stress are analytically obtained. A series of numerical experiments were condacted in order to identify and analyze the following two main classes of steady-state regimes: regular (which correspond to quasi-periodic oscillations) and chaotic (with positive largest Lyapunov exponents and continuous spectral densities). Chaotic steady-state regimes in the system with the initial stress are realized at less than about 10% frequency fluctuations. In addition, taking into account initial stress leads to a reduction in amplitude of the third eigenmode.