Delayed control of an elastic beam

Abstract

Stabilization of longitudinal vibrations along an elastic beam with delayed feedback is analyzed. The beam is modeled as a series of equal masses connected by springs and dashpots. The degree of freedom (DoF) of the model is increased—first 1, then 3 and 9 DoF—with stability charts in terms of the gain and the time delay presented in each case for various levels of damping. With zero damping, for 9 DoF no stability region remains at all. Yet at the continuum case, when the partial delay differential equation is examined with zero damping, discrete stable intervals of gain are found at specific delay values. These intervals largely match the stability charts of the higher DoF models.