Nedin V. Nonlinear oscillations of elastic structural elements with account of gyroscopic forces

The thesis is devoted to investigation of elastic rods oscillations during rotation with account of geometric nonlinearity, gyroscopic forces and axial external loads. The literature review on the research topic is done. The fundamental works of prominent scientists in the field of rod dynamics are considered, and the works of modern scientists who deal with the dynamics of rotating systems taking into account the actions of gyroscopic loads in complex motion are considered too. The mathematical modeling of the rotating elastic rods dynamics with account of geometric nonlinearity, linear inertia forces, centrifugal inertia forces, Coriolis inertia forces, gyroscopic moments, external axial and gravitational loads is done. The equations of oscillations are derived. The formulas for calculations of the critical rotational speeds and the natural oscillation frequencies are found via solving of the differential equations of oscillations. Using the programs that are developed on basis of these formulas were found and presented the results, which show the dependences of the critical rotational speeds on such parameters as: dimensions of the cross-section of rods, their length, values of axial compressing loads or longitudinal tensile load. Based on developed technique of computer modeling of oscillatory motion of rotating rods and with software, that is created using this technique, the research of the dynamics of transverse oscillations for different shafts and rotating rods have been done. It is shown how the gyroscopic forces have been influencing on the process of oscillatory motion by different rotating speeds. The diagrams that show the development of oscillation amplitudes in time were built. The diagrams that show the trajectories of moving for cross-sections of the rods, both in rotating coordinate system and stationary coordinate system were built, too. Using the specified software the bend forms of heavy rods with large length in process of oscillation were defined and presented for various moments of time. Such rods are modeling the work of drill strings. It is shown that the action of an axial force that is pointed to the lower end of the vertical heavy rod leads to the effect of twisting to a spiral of its lower part. This effect arises via action of gyroscopic moments, which begin to appear when the lower part of the rod starts to bend and this bending have been growing. It happens because the growth of bending leads to increase the angles of rod cross-sections’ turn, the velocity of change of which is part of the gyroscopic moment. In the process of rotating rods dynamics investigation under action of axial periodic forces for various objects, such ratios of rotational speeds and frequencies of axial loads changes have been found by which, even far before critical rotational speeds, arises the dynamic instability of system in case of direct and reverse precession. In course of thesis the techniques of dynamics research of investigated objects in specified statement were developed. Based on developed techniques, has been created the software using which the critical rotation speeds and natural oscillation frequencies are calculated and dynamic instability regions are found for objects with different parameters. Influence of gyroscopic effect on values of critical rotational speeds with different system parameters is studded. For various systems, the parameters at which they have dynamic instability and can quickly start intensive oscillations after loss of equilibrium are found.