On controlling the nonlinear vibrations of a Rectangular Thin Plate with Time delay feedback

In this paper, the time-delay feedback control is applied to reduce the dynamic vibration of a rectangular thin-plate. The motion of a rectangular thin -plate is formed of two-degree-of-freedom (2-DOF) with quadratic and cubic nonlinearities under external and parametric excitation forces. The analytical solution is obtained by the technique of multiple time scales method (MTSM). The effect of control is studied to show the safety regions in which the system amplitudes decreased at some values of time-delay. The system response is investigated numerically with and without time-delay feedback near-simultaneously internal and primary resonance using the Runge-Kutta fourth-order (RK-4) method (package ode45 in Matlab R2014a). The stability of the nonlinear vibrating system is studied by using the frequency response equations according to the Routh-–Hurwitz criterion. The effects of different parameters on the system behavior are investigated numerically. Also, numerical simulations have a good agreement with the analytical solution.  Finally, making a comparison with previously published work.