טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentVictor Kislovsky
SubjectNonstationary Dynamics of the Damped and Parametrically
Driven Oscillatory Models: Analytical Study
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Starosvetsky Yuli


Abstract

Regimes of resonant energy transfer remain one of intensively studied topics of both applied physics and engineering sciences. These regimes are manifested by either weak or strong energy exchanges between the different parts of coupled oscillatory models. In fact these regimes are ubiquitous in a wide variety of physical and engineering problems, for instance: beating response in two-oscillator models, vibration absorption in structures, mitigation of seismic waves, chatter control of turning processes, energy transfer in oscillatory chains, auto-resonant control of coupled nonlinear waves, etc. However, in all the previous studies, less attention has been paid to the effect of essentially nonlinear coupling and parametric forcing on the dynamics of non-stationary regimes manifested by intense energy transfer in low and higher dimensional models. We initiated the current study with consideration of a nonstationary response emerging in the parametrically forced and nonlinearly coupled two-oscillator models, as well as a nonlinear Mathieu oscillator including the two-component parametric forcing. In the former model, we assumed some general coupling form, weak dissipation and parametric forcing applied on each oscillator. The main focus of the first part was the analytical description of nonstationary regimes exhibited by both models as well as the analysis of intrinsic mechanisms governing their formation and destruction. This work has been further extended to the parametrically driven oscillatory chains subjected to periodic boundary conditions and assuming essentially nonlinear coupling. In this part of the study, we report the existence of three special regimes namely a regime of nonlinear beats, as well as the standing and moving breathers. The first two regimes have been described analytically and analyzed for their stability and bifurcation structure. Our findings are confirmed by the bifurcation diagram constructed using the basic, numerical continuation method based on the Pseudo-Arclength predictor-corrector algorithm. The overall results of analytical study are in good agreement with the numerical ones.