Solution of stochastic eigenvalue problem by improved stochastic inverse power method (I-SIPM)

Abstract

Eigenvalue analysis is an important problem in a variety of fields. In structural mechanics in the field of naval architecture and ocean engineering, eigenvalue problems commonly appear in the context of, e.g. vibrations and buckling. In eigenvalue analysis, the physical characteristics are often considered as deterministic, such as mass, geometries, stiffness in the structures. However, in many practical cases, they are not deterministic. Such uncertainties may cause serious problems because the influence of the uncertainties is in general unknown. To solve the stochastic eigenvalue problem, in this article, we have proposed two methods. First, the improved stochastic inverse power method (I-SIPM) based on response surface methodology is proposed. The method is different with previous stochastic inverse power method. The minimum eigenvalue and eigenvector of stochastic eigenvalue problems can be evaluated using the proposed method. Second, the stochastic Wielandt deflation method (SWDM) is proposed to evaluate ith (i > 1) eigenvalues and eigenvectors of stochastic eigenvalue problems. This is very important for solving natural mode and buckling mode analysis problem. Next, two example problems are investigated to show the validity of two new methods compared with a Monte-Carlo simulation, i.e. the vibration problem of a discrete 2-DOF system and the buckling problem of a continuous beam. Finally, the uncertainty estimation for the dynamic damper problem is discussed using proposed method. The probability of the natural frequency falling into the range to be avoided is shown when the dynamic damper has a stochastic mass and stiffness.