טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBar Tamir
SubjectBifurcations of Self-Excitation Regimes in Van der Pol
Oscillator with Nonlinear Energy Sink
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Oleg Gendelman
Full Thesis textFull thesis text - English Version


Abstract

The following study investigates regimes of self-excitation in a classical self-excited oscillator, the Van Der Pol (VDP) oscillator, with an attached nonlinear energy sink (NES). This assembly presents various possible bifurcation regimes of the response, depending on the system parameters and the initial conditions (ICs). Initial equations are reduced by averaging to a 3 dimensional system. An assumption of a relatively small mass of the NES allows treating this averaged system as a singularly perturbed system and allows for separation of variables into two "slow" variables and one "super-slow" variable. Such approach, in turn, yields a complete analytical description of the possible response regimes. The system exhibits various response regimes; from almost unperturbed limit cycle oscillations (LCOs) to a complete elimination of the self-excitation, small-amplitude LCOs, chaotic-like responses (based on the multiple points in the Poincar'e' map) and excitation of quasiperiodic strongly modulated response (SMR). The latter can be excited by three distinct bifurcation mechanisms: The Canard explosion, a Shil'nikov bifurcation and a heteroclinic bifurcation. Some of the above oscillatory regimes can co-exist for the same values of system parameters. In this case, it is possible to establish the basins of attraction for the co-existing regimes. Direct numeric simulations demonstrate good coincidence with the analytic predictions.