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.

As it is well known, the phenomena of impacts and dry frictions exist widely in a large number of engineering devices [

Various complex impact structures and models were designed and developed in the past decade; the study of these systems was extensively performed. Namely, since the 1960s, the theoretical and experimental analyses of an impact system under periodic excitation have been performed by Masri [

In order to overcome the difficulty of studying dynamical systems with an uncertain parameter, the orthogonal polynomial approximation (OPA) method has been utilized to study the stochastic dynamics of some kinds of smooth dynamical systems [

The remaining part of this paper is organized as follows: the equivalent high-deterministic system is derived by the OPA method in

As for stochastic vibro-impact system with nonlinear stiffness and damping under periodic impulse excitation and harmonic excitation, the non-smooth property is induced by the existence of a rigid barrier, as shown in

where

Particularly,

When the displacement is

When the constraint condition is

According to the OPA method, the responses of the system

where

The orthogonality of the polynomials is:

Based on the property of the Chebyshev orthogonal polynomials, the recurrent formula among them is:

As for the case without any constraints, when substituting

By

The expressions of

Substituting

In order to simplify

where

Since

By means of

the mean constraint condition

the mean constraint plane

and the mean jump condition

Substituting

and in case of

Naturally, in the deterministic case the parameters

Due to the deterministic property of _{0}. Correspondingly, the stochastic response of the equivalent system is denoted as EMR. Thus, the validity of the polynomial approximation for this kind of vibro-impact system with nonlinear stiffness and damping can be verified by comparing EMR_{0} with DMR. The differences between EMR and DMR can show the effect of the uncertain parameter on the responses.

In order to calculate the response, the initial condition of

Here, two different Poincaré sections (the phase plane and the constraint plane) are taken, while the value of _{1}A_{2}A_{3}A_{4}, C_{1}C_{2}C_{3}C_{4}, B_{1}B_{2}, D_{1}D_{2}). In the symbol _{0} also present very well the consistent base on the phase portraits in

In order to consider the effects of an uncertain parameter on period-doubling bifurcation, the following system parameters are chosen: _{0} are obtained as shown in _{0} from the periodic 6/2 motion turn to the approximate 3/1 motion. As for Poincaré section, the mark “▪” presents the corresponding Poincaré section, and the periodic properties also can be observed clearly from Poincaré section.

As for

As a conclusion, under the effect of the uncertain parameter with the decreasing frequency

In this section, to investigate the grazing bifurcation which is typical for the system with impact phenomenon, the same parameters are taken as in

When the frequency is

At the same time, in case of

In order to present the effect of an uncertain parameter on bifurcation, for different values of intensity

When it comes to the influence of the impulse signal on the bifurcation properties the following ideas are considered: the stronger impulse signals are chosen with

The phase portraits in

In this paper, the OPA method is applied in the vibro-impact system with an uncertain parameter under periodic impulse excitation and harmonic excitation, and the damping coefficient is considered as an uncertain parameter. In order to study the dynamical response, the ensemble mean response of an equivalent high-dimensional system is introduced and the impact conditions are also transformed by means of the mean value; then, the deterministic high-dimensional equivalent vibro-impact system is derived. Afterward, the constraint plane and the phase plane are chosen as the Poincaré section, respectively, with the response properties being consistent in the bifurcation diagrams. By combining the analysis of the phase portraits, it is evident that the approximation method is effective in this kind of vibro-impact system. Furthermore, it has been proved that besides period-doubling bifurcation, certain grazing bifurcation exists also in the stochastic nonlinear vibro-impact system. Under the influence of the uncertain parameter, the system responds not only with the characteristic of the smooth system which can make trajectories of the systems’ diffusion, but also with a special characteristic of a non-smooth system. At a critical point of bifurcation, the random factors with certain intensity make the dynamical behavior of the system change drastically. Furthermore, grazing bifurcation appears in advance with increasing random intensity. The existence of the stronger impulse force can induce the appearance of a diffusion phenomenon. An appropriate choice of impulse force can control the vibration and improve the response performance. Overall, the bifurcation analysis is helpful for further investigating of stochastic dynamics.

The authors are grateful for the support by the National Natural Science Foundation of China, the Bilateral Governmental Personnel Exchange Project between China and Slovenia for the Years 2021–2023, Slovenian Research Agency ARRS in Frame of Bilateral Project, the Fundamental Research Funds for the Central Universities, Joint University Education Project between China and East European.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12172266, 12272283), the Bilateral Governmental Personnel Exchange Project between China and Slovenia for the Years 2021–2023 (Grant No. 12), Slovenian Research Agency ARRS in Frame of Bilateral Project (Grant No. P2-0137), the Fundamental Research Funds for the Central Universities (Grant No. QTZX23004), Joint University Education Project between China and East European (Grant No. 2021122).

The authors confirm contribution to the paper as follows: Conceptualization, Methodology, Validation: Dongmei Huang; analysis and interpretation of results: Dongmei Huang, Dang Hong, Wei Li; Writing-Original Draft: Dongmei Huang, Dang Hong, Wei Li, Guidong Yang and Vesna Rajic. All authors reviewed the results and approved the final version of the manuscript.

Data will be made available on request.

The authors declare that they have no conflicts of interest to report regarding the present study.