Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations

• This work aims to deal with the stationary responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise.
• The original system is converted into an equivalent system relying on the viscoelastic force treatment and non-smooth transformation.
• The validity of the analytical method has been verified by utilizing Monte Carlo (MC) simulation results for a biquadratic Van der Pol VI oscillator with viscoelastic behavior in detail.
• The occurrence of stochastic P-bifurcation is explored by two angles.
• It shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.

This paper is mainly dealing with the stochastic responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise. By the aid of approximate conversion for the viscoelastic force, the original stochastic VI system is transformed into an equivalent stochastic system without viscoelastic term. Then, the equations of the converted system are simplified by non-smooth transformation, and the stochastic averaging method is employed to solve the above simplified system. A Van der Pol VI oscillator coupled with viscoelastic force is worked out in detail to illustrate the application of the mentioned method, and therewith the analytical solutions fit the numerical simulation results based on the original system. Therefore, the present analytical means of investigating this system is proved to be feasible. Additionally, the exploration of stochastic P-bifurcation by two different ways is also demonstrated in this paper through varying the value of the certain system parameters. Besides, it shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.