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Vibro-impact dynamics near a strong resonance point

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Abstract

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a 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 two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T in, T on and T out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions.

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Authors and Affiliations

  1. School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Guanwei Luo & Yanlong Zhang

  2. Department of Engineering Mechanics, Southwest Jiaotong University, Chengdu, 610031, China

    Jianhua Xie

  3. School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Jiangang Zhang

Authors
  1. Guanwei Luo
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  2. Yanlong Zhang
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  3. Jianhua Xie
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  4. Jiangang Zhang
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Correspondence to Guanwei Luo.

Additional information

The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang.

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Luo, G., Zhang, Y., Xie, J. et al. Vibro-impact dynamics near a strong resonance point. Acta Mech Sin 23, 329–341 (2007). https://doi.org/10.1007/s10409-007-0072-7

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  • DOI: https://doi.org/10.1007/s10409-007-0072-7

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