Dynamical behavior of vibro-impact machinery near a point of codimension two bifurcation

A dual component system with vibro-impact is considered. Local codimension two bifurcation of maps, involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously, is analyzed by using the center manifold and normal form method for maps. The period n single-impact motion and Poincaré map of the vibro-impact system are derived analytically. Stability and local bifurcation of single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with the codimension two bifurcation is obtained. Local behavior of the dual component system with vibro-impact, near the point of codimension two bifurcation, is analyzed. It is found that near the point of codimension two bifurcation, there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion commonly exists near the point of codimension two bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The period two fixed points are symmetrical about the corresponding period one point. The results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of Hopf bifurcation of period two points. Two actual examples, the impact-forming machinery and inertial shaker, are chosen to analyze further the phenomena of codimension two bifurcation of maps, and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro-impact systems near the points of codimension two bifurcations.