טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentPrivman Neta
SubjectDynamics of the 2D Oscillator with Internal Rotatory
Attachment: Effect of Geometric Nonlinearity
DepartmentDepartment of Design and Manufacturing Management
Supervisor Professor Yuli Starosvetsky


Abstract

Of late, the 1D and 2D response of dynamical systems comprising the primary structure with internal rotators has attracted substantial attention of various engineering communities. Many theoretical and experimental studies have shown that internal rotator can be used as a nonlinear energy sink, such that it can efficiently absorb energy from the impulsively loaded primary system in the broadband and irreversible fashion. Some very recent works have been majorly concerned with the 2D response of mechanical structure incorporating internal rotator. These works have shown that initial energy being supplied to the primary structure in a certain direction can be effectively redirected e.g. from horizontal to vertical vibrations. This mechanism is fully controlled by the motion of internal rotator. Contrary to the previous studies of 2D models, we investigate the effect of geometric nonlinearity on the 2D response of a similar type of mechanical model comprising the 2D oscillator with internal rotator. In fact, taking into account a geometrical nonlinearity enables a better approximation of the real mechanical models. It is worthwhile noting that in the previous studies, the nonlinearity of the local potential has been introduced in a somewhat artificial fashion. The mechanical model in the present study, includes the outer body mounted on the elastic support assembled from the four linear springs placed in horizontal and vertical directions. The mass of internal rotator is assumed to be significantly smaller than the mass of a primary structure.  In the present research we aimed at devising the analytical approach based on asymptotic analysis enabling to depict the stationary and non-stationary, 2D response of the outer element controlled by internal rotator. The analytical part of the study has involved the multi-scale procedure under assumption of small amplitude motion of the primary structure subject to the initial loading. Analytical model has been validated by numerical simulation of the original system under consideration.  It is worthwhile noting that analysis of the various system response regimes has revealed the new type of nonlinear normal modes which have not been reported before. In the present research we depict the complete bifurcation structure of stationary response regimes. In the second part of the study we focus on the special nonstationary system states manifested by the intense, energy flow from horizontal to vertical vibrations. Using the devised analytic procedure we determine the main system parameters that control the emergence and vanishing of these special system states. That is, the termination of intense beating states leading to the energy locking in a single direction of the system. Results of the asymptotic analysis are found to be in the fairly good correspondence with the numerical simulations of the real model.