Stochastic finite element analysis of a vibrating string

The response of a vibrating string subjected to spatial white noise excitation is analyzed by using the finite element (Galerkin) method. The discretization is achieved by using basis functions in the spatial and random spaces. The continuous time finite element equations are then integrated in time by using the central difference algorithm. It is shown that when the string is divided into N segments, the Galerkin approach leads to N(N?1) degrees of freedom which are governed by N subsets of equations. The subsets are similar in form. Furthermore, for a uniform string, they are governed by the same operator as that which governs the system of (N?1) degrees of freedom arising from the corresponding deterministic problem. Random initial conditions can also be treated by the same method, and the analysis of strings with initial conditions described by the standard Brownian process is carried out. Numerical results are given to illustrate the stochastic response.