Nonlinear vibrations of a mechanical system with non-regular nonlinearities and uncertainties

•Concept of the Stochastic Multi-dimensional Harmonic Balance Method (Stochastic-MHBM).•Quasi-periodic stochastic dynamic response with uncertainties in linear/nonlinear elements.•Validation for a system with various types of nonlinearities: cubic, contact, friction. This paper presents the concept of the Stochastic Multi-dimensional Harmonic Balance Method (Stochastic-MHBM) in order to solve dynamical problems with non-regular non linearities in presence of uncertainties. To treat the nonlinearity in the stochastic and frequency domains, the Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) is proposed. The approach is demonstrated using nonlinear two-degree-of-freedom model with different types of nonlinearities (cubic nonlinearity, contact/no contact, friction). The quasi-periodic stochastic dynamic response is evaluated considering uncertainties in linear and nonlinear parts of the mechanical system. The results are compared with those obtained from the classical Monte Carlo Simulation (MCS). For various numerical tests, it is found that the results agreed very well whilst requiring significantly less computation.