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Melnikov analysis of subharmonic motions for a class of bistable vibro-impact oscillators

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Abstract

In this paper, subharmonic motions of a bistable vibro-impact oscillator are studied by establishing the theoretical framework of the Melnikov analysis for an abstract non-smooth dynamical system. A ideal impacting map is employed to describe velocity changing during instantaneous collision with the bilateral rigid constraints. The unperturbed system without considering damping and external excitations is supposed to have a pair of homoclinic orbits connecting the origin to itself, and the inner and outer regions separated by the homoclinic orbits are assumed to be fully covered by periodic orbits, whose periods monotonically increase as they approach the homoclinic connections. Furthermore, periodic or homoclinic grazing of the unperturbed system can also occur by adjusting the position of the constraints. When a periodic perturbation is considered, the definitions of the Unilateral subharmonic orbits, the Bilateral subharmonic orbits and the Compound subharmonic orbits for this class of non-smooth systems are given by combining the impacting dynamics. The Melnikov functions for the first two types subharmonic orbits are also obtained and employed to detect the initial conditions for the existence of the corresponding subharmonic orbits. Finally, the bistable vibro-impact oscillator as an example is used to show the effectiveness of the developed Melnikov method for seeking subharmonic motions for this class of bistable vibro-impact oscillator. A numerical integration method is also introduced to overcome the difficulty of the Melnikov integration along the unperturbed periodic orbits.

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Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 12172376 and 11672326, the Fundamental Research Funds for the Central Universities through Grant No. 3122022PT09.

Funding

This study was founded by the National Natural Science Foundation of China through Grant Nos. 12172376 and 11672326 and the Fundamental Research Funds for the Central Universities through Grant No. 3122022PT09.

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Authors and Affiliations

  1. Research Institute of Science and Technology, Civil Aviation University of China, Tianjin, 300300, China

    Shuangbao Li

  2. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Ran Sun

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Li, S., Sun, R. Melnikov analysis of subharmonic motions for a class of bistable vibro-impact oscillators. Nonlinear Dyn 111, 1047–1069 (2023). https://doi.org/10.1007/s11071-022-07902-9

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