Eliminate the Nonlinear Oscillations of the Modified duffing Equation by using the Nonlinear Integrated Positive Position Feedback

In this article, the nonlinear integrated positive position feedback (NIPPF) control adds to a nonlinear dynamical system modeled as the well known Duffng oscillators. This control is proposed to mitigate system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow- ow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. Before and after (NIPPF) control the nonlinear systems' steady-state amplitude are examined, the comparison is made to validate the closeness between the numerical solution and the analytical perturbative one at time-history and frequency response curves.