Abstract
Motivated by observations of low-frequency “snaking” motions of circular sawblades encountered during cutting operations, we study the large amplitude transverse vibrations of an imperfect, flexible, lightly damped, circular plate spinning near a critical speed resonance. Of particular interest are the effects of imperfection and the conditions under which low-frequency, amplitude and phase modulated traveling waves, that is snaking motions, can arise in this system. This is accomplished through a local bifurcation analysis of the first-order averaged equations governing the nonlinear interaction, under conditions of a 1 : 1 internal resonance, of a pair of backward and forward traveling waves. Experiments are performed at reduced ambient pressure on a rotating steel disk with imperfection induced through attached point masses, to confirm qualtitatively the analytical predictions. The experiments also indicate an unexplained, additional superposed slow motion on all solution branches.
Adaptive energy-based bilinear control of first-order 1-D hyperbolic PDEs: Application to a one-loop parabolic solar collector trough
