Phase Shift Near Natural Frequencies of Non Linear Rods

We study the transient and steady state dynamics of a special class of motions of forced planar rods with exact geometric nonlinearity. The attractors of these motions are separated by a finite jump at a critical forcing frequency in an attractor diagram of the undistorted configuration generated by a quasi-static frequency sweep at fixed forcing amplitude. As the frequency of the forcing passes through this critical or jump frequency, the motion (trajectory) of the undistorted configuration changes basin of attraction. For forcing frequency slightly greater than the jump frequency, the response trajectories of the undistorted configuration pass near an unstable periodic attractor and undergo continuous phase shift while approaching a stable attractor. For forcing frequency slightly smaller than the jump frequency, the response trajectories of the undistorted configuration pass near the same unstable attractor and undergo no net phase angle when landing on the stable attractor that attracts them. The phase-shifting property reveals that the frequncy at which the jump occurs is indeed a natural frequncy of the nonlinear rod.