Study of Hopf curves in the time delayed active control of a 2DOF nonlinear dynamical system

Abstract

Within this paper, we are focusing on the time delay effects on the improved positive position feedback (PPF) controller for the oscillations of a 2DOF nonlinear dynamical system which represents a thin-walled pre-twisted rotating blade. Herein, we are improving the performance of the traditional positive position feedback controller (PPF) that suffers from the existence of two high amplitude peaks on both sides of its minimum amplitude point. The improvement involves coupling double nonlinear saturation controllers to the main system to notch down the high peaks. As an active control process, the time delay is inherent in the process which produces a locus of Hopf bifurcation points separating between the stable and unstable regions of operation. Variation of different parameters is studied to relate their effects with time delay on the system operation. To ensure the validity of our proposed work, a comparison between the approached multiple scales analytical solutions and the computed Rung-Kutta numerical solutions is done at the end of this paper.