A MULTIPLE-SCALES ANALYSIS OF NONLINEAR, LOCALIZED MODES IN A CYCLIC PERIODIC SYSTEM

In this work the nonlinear localized modes of an n-degree-of-freedom ( DOF) nonlinear cyclic system are examined by the averaging method of m ultiple scales. The set of nonlinear algebraic equations describing th e localized modes is derived and is subsequently solved for systems wi th various numbers of DOF. It is shown that nonlinear localized modes exist only for small values of the ratio (k/mu), where k is the linear coupling stiffness and mu is the coefficient of the grounding stiffne ss nonlinearity. As (k/mu) increases the branches of localized modes b ecome nonlocalized and either bifurcate from ''extended'' antisymmetri c modes in inverse, ''multiple '' Hamiltonian pitchfork bifurcations ( for systems with even-DOF), or reach certain limiting values for large values of (k/mu) (for systems with odd-DOF). Motion confinement due t o nonlinear mode localization is demonstrated by examining the respons es of weakly coupled, perfectly periodic cyclic systems caused by exte rnal impulses. Finally, the implications of nonlinear mode localizatio n on the active or passive vibration isolation of such structures are discussed.