Abstract
Two vibro-impact systems are considered. The period n single-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. Stability and local bifurcations of single-impact periodic motions are analyzed by using the Poincaré maps. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. It is found that near the point of codim 2 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 is commonly existent near the point of codim 2 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 results from simulation shows that there exists an interest torus doubling bifurcation occurring near the value of Hopf-flip bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transit to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems.
Highlights
Impact oscillators arise whenever the components of a vibrating system collide with rigid obstacles or with each other
The purpose of the present study is to focus attention on codim 2 bifurcation of single-impact periodic motion
The results from qualitative analysis and numerical simulation shows that the vibro-impact systems, near the points of codim 2 bifurcations, can exhibit richer and more complicated dynamical behavior
Summary
Impact oscillators arise whenever the components of a vibrating system collide with rigid obstacles or with each other. Such systems with repeated impacts exist in a wide variety of engineering applications, in mechanisms and machines with clearances or gaps. Researches into repeated impact dynamics have important significance in optimization design of machinery with rigid obstacles or clearances, noise suppression and reliability analyses, etc. The presence of the non-linearity and discontinuity complicates the dynamic analysis of repeated impact systems considerably, but it can be described theoretically and numerically by discontinuities in good agreement with reality. The results from qualitative analysis and numerical simulation shows that the vibro-impact systems, near the points of codim 2 bifurcations, can exhibit richer and more complicated dynamical behavior
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