DYNAMICAL BEHAVIOR OF THE INERTIAL SHAKER NEAR 1∶4 STRONG RESONANCE POINT

The local bifurcation of an inertial shaker, concerning one complex conjugate pair of eigenvalues of the Jacobian matrix of the mapping escaping the unit circle simultaneously, is investigated by using the center manifold theorem technique and normal form method of the mapping. A center manifold theorem technique is applied to reduce the Poincaré mapping to a two-dimensional one, and the normal form mapping associated with 1∶4 strong resonance is obtained. Thusly, the changing process of the bifurcation diagrams of the mapping near 1∶4 strong resonance point is discussed. The local dynamical behavior of an inertial shaker near 1∶4 strong resonance point is investigated by using qualitative analysis. The results from numerical simulation also illustrate that Neimark-Sacker bifurcation of periodic-impact motions and some complicated bifurcations, e.g., Ton and Tout types of tangent bifurcations of period-4 orbits, are found to exist in the inertial shaker near 1∶4 strong resonance point.