Localized oscillations of a spatially periodic and articulated structure

Abstract

Localized oscillations are examined in flexural motions of a spatially periodic and articulated structure of uniform and rigid members connected with neighboring ones through a coupler giving nonlinear restoring moment. Formulation is made in terms of the Lagrangian with holonomic constraints for continuity of displacements at the junctions, and is also given in the form of constrained Hamiltonian system. Nonlinearity results from not only material response due to anharmonic potential in the restoring moment (linear plus hard, cubic spring) but also geometrically finite displacements, the latter of which induces longitudinal motions to couple, in turn, with the flexural (transverse) motions. In particular, the axial tension introduces nonlocal moment for rotation of each member in addition to the one due to nearest neighbors. Numerical calculations and asymptotic analysis show existence of time-periodic localized oscillations while the amplitude remains small. As it becomes larger, however, quasiperiodic oscillations emerge, and there also occur such cases that they are not decayed temporarily but are delocalized.