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In this paper, a time-delayed positive position feedback controller is proposed to reduce the horizontal vibration of a magnetically levitated body subjected to multi force excitations. This controller is coupled to the main system with 1:1 internal resonance. The method of multiple scales perturbation technique is employed to obtain an approximate solution which clarifies the nonlinear behavior for both amplitude and phase of the whole system. The effects of time delay magnitude are investigated to indicate the safe region of operation. Validation curves are involved to compare the approximate solution with the numerical simulation.

A nonlinear saturation controller (NSC) is applied in this work to reduce the oscillations of a rotating blade dynamical system running at unsteady rotating speed. The controller is coupled quadratically to the main system by designing its natural frequency to be one half of the main system natural frequency. This is done to setup an energy bridge between them to make use of saturation phenomenon. That phenomenon is advantageous when the excitation force increases; the whole energy in the main system is channeled to the controller. The two system modes of vibrations are found to be linearly coupled powerfully, so the controller is applied only to the first mode and, consequently, the second mode tracks it. The multiple scales perturbation technique (MSPT) is adopted to derive the steady state equations that describe the modulations of amplitudes and phases of the system before and after control. Then, a stability analysis is achieved via Lyapunov’s indirect method to determine the stable and unstable solutions depending on the real parts of the Jacobian matrix eigenvalues. Time history and different response curves of the controlled system are included for showing the controller effect. Eventually, validation curves and comparison with previously published work are included.

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