The Duffing oscillator under combined periodic and random excitations

•A couple relation for the deterministic and random parts is obtained.•A simplification for the couple relation is suggested.•The stationary responses of the deterministic and random parts are evaluated.•The accuracy of the proposed method is checked by numerical approaches.

The Duffing oscillator under combined periodic and random excitations is investigated by a simple technique. The system response is separated into the deterministic and random parts governed by two coupled differential equations. The couple relation is expressed through varying on time coefficients which are approximately replaced by their averaging values over one period. This simplification yields that the two coupled differential equations can be solved by averaging and equivalent linearization methods. The mean-square response of the system is compared with the numerical results obtained by the finite element and Monte Carlo simulation methods. The results obtained show the interaction between the periodic and random excitations on the system response.