Savchenko N. Oscillations and stability of the motion of some non-conservative mechanical systems

The dissertation is devoted to the actual problems of modern theoretical mechanics which arise in the study of the motion stability of mechanical systems, and are described by nonlinear ordinary differential equations. The main subject of the applicant's research is non-conservative mechanical systems. The research area contains stability and stabilization problems for these mechanical systems. In the dissertation the scientific problem of constructive recording of Lyapunov functions for classes of non-conservative nonlinear mechanical systems in critical cases with application to the study of motion stability of stability problem of multibody system's dynamics is solved. We propose a method for constructing the Lyapunov function for the system of ordinary differential equations of order 2m + l, whose linear part matrix has m pairs of purely imaginary eigenvalues and l eigenvalues belonging to the open left complex half-plane, while the nonlinear part of the system has a special form. This approach seems more convenient than the well-known Kamenkov's principle of reduction. In the thesis two theorems were first formulated and proved, which allow us to establish constructively the asymptotic stability or instability of the solution of the system of the specified type. There has also been studied the problem of the equilibrium state stability of a double mathematical pendulum with a dynamic oscillator absorber. It is shown that adding the latter to the system makes the lower equilibrium state asymptotically stable. The dissertation has solved the problem of stabilization the equilibrium state of a pendulum oscillator by adding a dynamic absorber to it. It was found that in this case the addition of an absorber leads to uniform asymptotic stability with respect to a part of the variables. For a double physical pendulum, it is shown that attachment of the absorber provides exponential stability of motion. Some aspects of the optimal configuration of the oscillations absorber are discussed. There has been solved the problem of the motion stability of a linear mechanical system which is under the action of the forces structure (potential, gyroscopic, dissipative and circulating forces). In addition, self-value estimates for specific cases have been found, which allows us to estimate the attenuation rate of perturbed movements of the system. We have obtained necessary and sufficient conditions for the asymptotic stability of uniform rotations of the asymmetric gyroscope which is under the influence of the damping torque. These conditions impose restrictions on the distribution of masses in the body, the value of the rotation speed and the friction coefficient. An estimation of the damping torque influence on the gyroscope motion stability has been carried out. It was established that at rotation around the lower state of the relative equilibrium the motion becomes asymptotically stable. At rotation around the upper equilibrium state, the effect of the damping torque is twofold: the gyroscopically stabilized rotation of the body may lose the property of stability, but may become asymptotically stable. It is noteworthy that the last stabilization effect is possible only for a dynamically asymmetric body. The Lyapunov critical case is also investigated when the characteristic equation of the linear approximation system has a pair of purely imaginary roots. The new results obtained in the dissertation are basically of theoretical value. They are of interest to specialists in the field of theoretical mechanics, namely, they can be used to further develop the theory of motion stability of nonlinear mechanical systems.