Non-stationary numerical covariance analysis of linear multi degree of freedom mechanical system subject to random inputs

Structural response of linear multi degree of freedom (MDoF) system subject to random Gaussian dynamic actions is defined by mean of vector and covariance matrix in state space. In case of non-stationary inputs, second-order spectral moments evaluation needs the solution of the so-called Lyapunov matrix differential equation. In this work a numerical scheme for its resolution is proposed, with reference to input processes modeled as linear filtered white noise with time-varying parameters, which is a common situation in amplitude and frequency variable loads. Numerical computational effort is minimized by taking into account symmetry characteristic of state space covariance matrix. As application of the proposed method a multi-storey building is analyzed to obtain reliability associated to maximum inter-storey exceeded over a given acceptable limit. It is assumed to be subject to seismic input described by a amplitude and frequency non-stationary process, by using a generalized non-stationary Kanai Tajimi seismic model. Structure is assumed as a plane shear frame MDoF system. Structural reliability evaluation is referred to "first time out-crossing" and different numerical benchmarks are considered.