In this work we propose the Step Matrix Multiplication based Path Integration method
(SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient
and accurate deterministic computation of the time-dependent response probability
density function by transforming the corresponding Chapman–Kolmogorov equation to
a matrix–vector multiplication using high-order numerical time-stepping and interpolation
methods. Additionally, the SMM-PI approach yields the computation of the joint probability
distribution for response and impact velocity, as well as the time between impacts
and other important characteristics. The method is applied to a nonlinear oscillator
with a pair of impact barriers, and to a linear oscillator with a single barrier,
providing relevant densities and analysing energy accumulation and absorption properties.
We validate the results with the help of stochastic Monte-Carlo simulations and show
the superior ability of the introduced formulation to compute accurate response statistics.
