Non-linear normal modes and non-parametric system identification of non-linear oscillators

The Karhunen-Loeve (K-L) decomposition procedure is applied to a system of coupled cantilever beams with non-linear grounding stiffnesses and a system of non-linearly coupled rods. The former system possesses localized non-linear normal modes (NNMs) for certain values of the coupling parameters and has been studied in the literature using various asymptotic techniques. In this work, the K-L method is used to locate the regions of such localized motions. The method yields orthogonal modes that best approximate the spatial behaviour of the beams. In order to apply this method simultaneous time series of the displacements at several points of the system are required. These measurements are obtained by a direct numerical integration of the governing partial differential equations, using the assumed modes method. A two-point correlation matrix is constructed using the measured time-series data, and its eigenvectors represent the dominant K-L modes of the system; the corresponding eigenvalues give an estimate of the participations (energies) of these modes in the dynamics. These participations are used to estimate the dimensionality of the system and to identify regions of localized motion in the coupling parameter space. The same approach is applied to a system of non-linearly coupled rods. Through the comparison of system response reconstructions of the responses using a simple K-L mode and a number of physical modes, it is shown that the K-L modes can be used to create lower-order models that can accurately capture the dynamics of the original system.