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# Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

Dongmei Huang1, Dang Hong2, Wei Li1,*, Guidong Yang1, Vesna Rajic3

1 School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China
2 Department of Mathematics, Changzhi University, Changzhi, 046011, China
3 Department for Statistics and Mathematics, Faculty of Economics, University of Belgrade, Belgrade, 11000, Serbia

* Corresponding Author: Wei Li. Email:

Computer Modeling in Engineering & Sciences 2024, 138(1), 509-524. https://doi.org/10.32604/cmes.2023.029215

## Abstract

In this paper, the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated. Firstly, by means of the orthogonal polynomial approximation (OPA) method, the nonlinear damping and stiffness are expanded into the linear combination of the state variable. The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of the mean value. Afterwards, the stochastic vibro-impact system can be turned into an equivalent high-dimensional deterministic non-smooth system. Two different Poincaré sections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response. Consequently, the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system. Furthermore, the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system. It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation. Increasing the random intensity may result in a diffusion-based trajectory and the impact with the constraint plane, which induces the topological behavior of the non-smooth system to change drastically. It is also found that grazing bifurcation appears in advance with increasing of the random intensity. The stronger impulse force can result in the appearance of the diffusion phenomenon.

## Keywords

APA Style
Huang, D., Hong, D., Li, W., Yang, G., Rajic, V. (2024). Bifurcation analysis of a nonlinear vibro-impact system with an uncertain parameter via OPA method. Computer Modeling in Engineering & Sciences, 138(1), 509-524. https://doi.org/10.32604/cmes.2023.029215
Vancouver Style
Huang D, Hong D, Li W, Yang G, Rajic V. Bifurcation analysis of a nonlinear vibro-impact system with an uncertain parameter via OPA method. Comput Model Eng Sci. 2024;138(1):509-524 https://doi.org/10.32604/cmes.2023.029215
IEEE Style
D. Huang, D. Hong, W. Li, G. Yang, and V. Rajic "Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method," Comput. Model. Eng. Sci., vol. 138, no. 1, pp. 509-524. 2024. https://doi.org/10.32604/cmes.2023.029215

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