Asymptotic analysis of nonlinear mode localization in a class of coupled continuous structures

Abstract

An asymptotic methodology is developed for studying the localized nonlinear normal modes of a class of conservative periodic continuous structures. Localized modes are synchronous free periodic oscillations which are spatially localized to only a limited number of components of the structure. Nonlinear mode localization is analytically studied by defining modal functions to relate the motion of an arbitrary particle of the structure to the motion of a reference point. Conservation of energy is imposed to construct a set of singular nonlinear partial differential equations governing the modal functions. This set is asymptotically solved using a perturbation technique. Criteria for existence of strongly or weakly localized modes are formulated, and are subsequently used for detecting nonlinear mode localization in the system. The orbital stability of the detected localized modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and using Floquet theory to analyze the resulting set of linear differential equations with periodic coefficients. Application of the general theory is given by studying the localized modes of a system of two weakly coupled, simply supported beams resting on nonlinear elastic foundations. The use of mode localization for the active or passive spatial confinement of impulsive responses of flexible structures is discussed.