POMs analysis of randomly vibrating systems obtained from Karhunen–Loève expansion

The Karhunen–Loève (KL) decomposition establishes that a 2D random field can be expanded as a series involving a sequence of deterministic orthogonal functions with orthogonal random coefficients. The proper orthogonal decomposition (POD) method consists in detecting spatially coherent modes in the dynamics of a spatio-temporally varying system by diagonalizing the spatial correlation function given by an averaging operator. The KL expansion is applied here to the responses of randomly excited vibrating systems with a view to performing a POD in separated-variables (time and space) form.Discrete and continuous mechanical systems are considered in this study as well as stationary and transient (non-stationary) responses. An averaging operator involving time and ensemble averages is proposed to draw up the POD in separated-variables form from the associated KL expansion. The result obtained using this approach agrees with the classical POD in the case of deterministic or ergodic random signals. The associated proper orthogonal modes are interpreted in case of linear and nonlinear vibrating systems subjected to white noise excitation in terms of normal modes.