Stochastic Transient Variational Principle in Vibration Analysis

Abstract

The stochastic transient variational principle (STVP) is developed based on the second order perturbation techniques. This chapter discusses the dependence of virtual displacements on real displacements hence on the material properties and loading conditions of the structures. For stochastic structures involved with physical and/or geometrical stochastic parameters, the virtual displacements are regarded as stochastic quantities. Therefore, a general purposes version of STVP is developed for vibration analysis of linear continuum. Probabilistic distributions of random parameters are consistently incorporated. Stochasticity of all random quantities, as well as their variations, is taken into consideration. The second-order perturbation techniques are employed to expand all the random quantities involved in the energy functional. Accordingly, a set of deterministic recursive equations is obtained as the alternative expressions of the STVP. On the basis of the STVP, the stochastic finite element method (SFEM) is developed for vibration analysis so that the roundabout procedures for formulations of the SFEM are avoided.