Modeling and Optimal Active Control with Delay of the Dynamics of a Strongly Nonlinear Beam

We focus our attention on a hinged-hinged beam subjected to active control devices. The dynamical behavior of the nonlinear beam with hinged ends is described by the catastrophic single well or catastrophic two well $\phi^6$ potential. The control by sandwich and the one using piezoelectric absorber are investigated. The range of coupling or control parameters that can produce an effective control (reduction of amplitude, avoidance of unbounded motion and horseshoes chaos) are determined. Emphasis is laid on the effects of time-delay. Numerical simulation has been done to confirm and complement the analytical prediction.