Stochastic dynamics analysis for unilateral vibro-impact systems under combined excitation

Vibro-impact system, as an important type of non-smooth system, exhibits intricately nonlinear characteristics. Inevitably, the vibro-impact system will encounter random excitations, but the conventional methods ar7e not eligible for simultaneous determination of its transient responses and reliabilities. Commonly, existing methods of applying non-smooth transformation tend to ignore the essential non-smooth characteristics of vibro-impact system. To this end, this paper proposes a unified framework based on direct probability integral method (DPIM) to simultaneously determine stochastic dynamic responses and reliabilities of unilateral vibro-impact systems under combined harmonic and random excitation without non-smooth transformation, and captures their complicated dynamical behaviors. Firstly, the impact velocity dependent coefficient of restitution is introduced to establish the motion equation of vibro-impact system. Secondly, the probability density integral equation (PDIE) for the unilateral vibro-impact system is derived from the perspective of probability conservation. Then, the PDIE and governing differential equation of the system is solved in a decoupled and efficient way. Moreover, the first-passage reliability is assessed by introducing extreme value mapping of the stochastic dynamic response. Numerical results of three typical examples using the proposed framework are compared with those using Monte Carlo simulation (MCS), quasi-MCS and from the reference, which highlights the advantages of DPIM in computing the stochastic responses and reliabilities of vibro-impact system under random excitations and random parameters. The stationary probability density functions exhibit periodic fluctuations under combined harmonic and stochastic excitation. Specially, the noise intensity and frequency of harmonic excitation pose the great influence on the reliabilities of systems.