Probalistic vibration analysis of nearly periodic structures

The localization of modes of nearly periodic structures causes a concentration of energy in one part of the mode shape. It occurs for a disordered structure with weakly coupled subsystems. The forced excitation of localized modes may affect the maximum response amplitude of the system. Two nearly periodic structures are analyzed herein: a two span beam and a pair of coupled pendula. Results show that the sensitivity of the forced response to the degree of localization depends on a combination of the symmetry of the mode which is excited and the phase difference between the forces acting on each substructure. These results attempt to explain the range of contrasting conclusions of previous research on the effects of forced response on mistuned structures. Furthermore, a theoretical explanation of the results is given in terms of transfer admittance. A probabilistic analysis of the free and forced response of a nearly periodic structure is shown to be useful in the design of such structures which are sensitive to the degree of localization. The second moment method is used in the analysis with results verified by Monte Carlo simulation. The probabilities of localization and failure are calculated given the statistics of the system parameters, and the localization and failure tolerances respectively.