Explicit solutions in the stochastic dynamics of structural systems

The paper presents a procedure to derive in explicit form the stationary response of a linear structure subjected to Gaussian white noise stochastic excitation. Namely, the analytical relationship between the second order statistical moments of the response and the structural parameters (element stiffness and modal damping ratio) is furnished. The method is based on the solution of complex eigenvalue problems, one for each variable structural parameter, possessing a number of eigenvalues different from zero much smaller than the problem dimension. If a single structural quantity is treated as a parameter then the exact explicit solution is found. When more parameters are present, the explicit solution is approximate and the introduction of cross terms is suggested to obtain more accurate predictions. The aforementioned explicit solution is exploited herein in the field of uncertain structures. The structural parameters are modeled as random variables and a Monte Carlo procedure is adopted to get the conditional, given the structural parameters, probability density function of the second order moments of the response. The efficiency and the accuracy of the proposed procedure are evidenced by numerical applications.