A. Vakakis et al., A MULTIPLE-SCALES ANALYSIS OF NONLINEAR, LOCALIZED MODES IN A CYCLIC PERIODIC SYSTEM, Journal of applied mechanics, 60(2), 1993, pp. 388-397
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.