A Multiple-Scales Analysis of Nonlinear, Localized Modes in a Cyclic Periodic System

Abstract

In this work the nonlinear localized modes of an n-degree-of-freedom (DOF) nonlinear cyclic system are examined by the averaging method of multiple scales. The set of nonlinear algebraic equations describing the localized modes is derived and is subsequently solved for systems with various numbers of DOF. It is shown that nonlinear localized modes exist only for small values of the ratio (k/μ), where k is the linear coupling stiffness and μ is the coefficient of the grounding stiffness nonlinearity. As (k/μ) increases the branches of localized modes become nonlocalized and either bifurcate from “extended” antisymmetric modes in inverse, “multiple” Hamiltonian pitchfork bifurcations (for systems with even-DOF), or reach certain limiting values for large values of(k/μ) (for systems with odd-DOF). Motion confinement due to nonlinear mode localization is demonstrated by examining the responses of weakly coupled, perfectly periodic cyclic systems caused by external impulses. Finally, the implications of nonlinear mode localization on the active or passive vibration isolation of such structures are discussed.