Equivalent Linearization and Monte Carlo Simulation in Stochastic Dynamics

Equivalent Linearization and Monte Carlo Simulation are the most important techniques in analyzing large non-linear structural systems under random excitation. This paper reviews the development and the state-of-the-art of Equivalent Linearization and Monte Carlo Simulation in stochastic structural dynamics. General Remarks An extensive benchmark study [114] has revealed that most of the computational methods developed in stochastic structural dynamics are quite limited with respect to the dimension of the problem and consequently can not be applied to engineering problems of larger size. It appeared that only two approaches, namely Equivalent Linearization (EQL) and Monte Carlo Simulation (MCS), are suited to solve these problems. Other methods, such as numerical solutions based on the Fokker-Planck equation (FPE) by path integration (e.g., [137,94]), cell-mapping (e.g., [60,61]) and finite-elements (e.g., [15,129]), increase in complexity at least exponentially with the state space dimension . In fact, all numerical solutions of the FPE available in the literature belong to low dimensional problems with [141]. Both MCS and EQL are conceptional simple and easy to apply. There underlying concepts can be viewed as the poles between which most of the computational methods developed so far in stochastic dynamics can be settled: EQL, on the one hand, uses the simplest global approximation of the distribution of the response, while MCS is the most local approach without any a priori assumption. Whereas only biased estimates for the first two moments of the response are obtainable by the standard EQL procedure, MCS yields unbiased estimates for the probability density function of the nonlinear response. Moreover, the information that EQL provides is limited to the first two moments of the distribution of the response, while MCS techniques are aible to give additional information on the tails of the distribution and subsequently credible reliability information on the analyzed dynamical systems. 1. EQUIVALENT LINEARIZATION 1.