An improved path integration method for the stochastic soft-impact systems

This paper presents an improved path integration method for a soft-impact system under stochastic excitation, which focuses on the response of the system on the impact surface. The system involves complex impact processes, including contact, deformation, recovery, and disengagement. To address the technical challenges posed by the system discontinuity at the moment of impact, we establish a mapping relation between impact events to solve the system response. Considering that the non-smooth moment of such systems exists only at the moment of contact with the impact surface, we chose to select the impact surface as a Poincaré cross-section. Two independent mappings were established to describe the transition of the oscillator from leaving the obstacle to the next contact with the obstacle, and from contacting the obstacle to leaving the obstacle. These two consecutive mappings were integrated into the plane to form a unified mapping. This method was employed to investigate the response probability density function of the system for autonomous and non-autonomous systems, respectively. The effectiveness of the methodology was validated by the use of Monte Carlo simulations, in addition to the discovery of the stochastic P-bifurcation phenomenon.