Abstract
Six different time-delayed controllers are introduced within this article to explore their efficiencies in suppressing the nonlinear oscillations of a parametrically excited system. The applied control techniques are the linear and nonlinear versions of the position, velocity, and acceleration of the considered system. The time-delay of the closed-loop control system is included in the proposed model. As the model under consideration is a nonlinear time-delayed dynamical system, the multiple scales homotopy method is utilized to derive two nonlinear algebraic equations that govern the vibration amplitude and the corresponding phase angle of the controlled system. Based on the obtained algebraic equations, the stability charts of the loop-delays are plotted. The influence of both the control gains and loop-delays on the steady-state vibration amplitude is examined. The obtained results illustrated that the loop-delays can play a dominant role in either improving the control efficiency or destabilizing the controlled system. Accordingly, two simple objective functions are introduced in order to design the optimum values of the control gains and loop-delays in such a way that improves the controllers' efficiency and increases the system robustness against instability. The efficiency of the proposed six controllers in mitigating the system vibrations is compared. It is found that the cubic-acceleration feedback controller is the most efficient in suppressing the system vibrations, while the cubic-velocity feedback controller is the best in bifurcation control when the loop-delay is neglected. However, the analytical and numerical investigations confirmed that the cubic-acceleration controller is the best either in vibration suppression or bifurcation control when the optimal time-delay is considered. It is worth mentioning that this may be the first article that has been dedicated to introducing an objective function to optimize the control gains and loop-delays of nonlinear time-delayed feedback controllers.