Semi-implicit integration algorithm for stochastic analysis of multi-degree-of-freedom structures

This paper presents a semi-implicit integration algorithm for random vibration problems that is appropriate for analyzing large structures, nonlinear hysteretic systems, and structural control problems. This semi-implicit approach results in a recursive expression for the mean and covariance response. A state-space representation of the equations of motion is adopted for deriving the algorithm. The solution of the state-space equations is first obtained, after which the expected value of the resulting equations is taken so as to obtain the first two moments. A stability condition for the method is also derived. Three numerical examples, a linear oscillator, a Duffing oscillator, and a multi-degree-of-freedom system with hysteretic supplemental damping devices, are provided to illustrate the effectiveness of the proposed method. Results compare well with Monte Carlo simulation, indicating that the semi-implicit integration algorithm is accurate and stable.