Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty

Abstract

An effective method is proposed in this paper for the symplectic eigenvalue problem of a periodic structure with uniform stochastic properties. Since structural parameters are uncertain, the symplectic transfer matrix of a typical substructure, which is described in the state space, also has stochastic properties. An effective spectral stochastic finite element method is adopted to ensure the symplectic orthogonality of the random symplectic matrix on the premise of certain precision. By means of the Rayleigh quotient method, the symplectic eigenvalue problem of random symplectic matrix is investigated. On the basis of this, the mean value and the standard deviation of the random eigenvalues are fully discussed. The comparison between the numerical results derived from the proposed method and the Monte-Carlo simulation indicates that the proposed method has high precision. This research provides a useful guidance for the dynamic analysis of periodic structures with stochastic properties.