Савченко Н.В. Коливання та стійкість руху деяких неконсервативних механічних систем

Дисертація на здобуття наукового ступеня кандидата фізико-математичних наук зі спеціальності 01.02.01 – теоретична механіка. Слов’янськ, 2018 рік.Qualifying scientific work on the rights of manuscript. Dissertation for the degree of a Candidate of Physical and Mathematical Sciences (PhD) in the specialty 01.02.01 Theoretical mechanics (113 Applied mathematics). Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slavyansk, 2017.
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 dissertation consists of an introduction, five sections, a conclusion and a list of references to 118 scientific works by domestic and foreign authors, used as sources. The total volume of the dissertation is 144 pages. The work contains 12 figures and 1 table. The relevance of the chosen direction of research is confirmed by the conclusions in the first section of the dissertation, which gives an overview of the literature on the motion stability of nonlinear systems containing stable and neutral components; stability of movement with respect to a part of variables; the rotation stability of a heavy rigid body with a fixed point; the use of damping devices to stabilize the movement of mechanical systems and the study of the forces structure influence on the dynamics and movement stability of these systems. The general research methodology is described in the second section of the dissertation. Namely, it is indicated that the methods of analytical mechanics are used in the study of the equations of the pendulum systems motion and 6 the rotation of a heavy rigid body with a fixed point under the influence of the damping torque. When constructing estimates of the eigenvalues of the mechanical system, elements of the perturbation theory are used. Some results of the matrix theory are also used in the course of the study. A Lyapunov's direct method is used to solve the problem of stability of the equilibrium state of nonlinear mechanical systems. The reliability of the obtained results is proved by the results of numerical integration of differential equations of motion. At a comparative evaluation, the findings of the dissertation are confirmed by known results. 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. In the third section, we propose a method for constructing the Lyapunov function for the system of ordinary differential equations of order 2 m + l , whose linear part matrix has m pairs of purely imaginary 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 simple 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 stabilizing 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 7 absorber provides exponential stability of motion. Some aspects of the optimal configuration of the oscillations absorber are discussed. In the fourth section, 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. In the fifth section, 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 co efficient. 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 b o dy. 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.