Non-Gaussian response distributions of non-linear MDOF-systems

A novel computational approach called “Local Statistical Linearization” (LSL) based on statistical equivalent linearization and Gaussian superposition has been introduced recently (4). The methodology allows to extend the concept of statistical equivalent linearization to proceed from estimates of the second moment properties of the stochastic response to estimates for the non-Gaussian probability distribution. Locally linearized non-linear systems do not have the properties of a linear system but approximate closely the non-linear characteristic of the system. The suggested approach employs the well developed equivalent linearization procedure or Gaussian closure to compute the non-Gaussian distribution of the response of a non-linear system. Gaussian closure is suggested for small systems with polynomial types of non-linearities. For MDOF-systems or hysteretic systems, statistical equivalent linearization is most instrumental. The LSL-procedures require non-zero mean computations of the local Gaussian densities. Since equivalent linearization is applicable for higher dimensions and FE-models, the suggested approach lends itself also to providing numerical solutions for higher dimensional cases. In this paper, the concept of LSL is outlined and the computations of probability density functions of non-linear MDOF-systems are discussed. Similar as in FE-analysis, the accuracy of the computed PDFs can be controlled by the level of discretization expressed by the number of of local Gaussian densities. The LSL-procedure has the potential to determine the tails of the distributions quite accurately, as shown in the numerical example, which makes this numerical approach an important tool for further reliability investigations. Moreover, the basic methodology is extended to increase the efficiency of the numerical procedure when dealing with non-linear MDOFsystems.