Chaotic Motion of a Nanoplate on a Nonlinear Foundation Under Linear and Nonlinear Displacement Time Delays

In this paper, the chaotic motion of a nanoplate on a nonlinear foundation under the combined effect of linear and nonlinear displacement time delays is investigated. In order to improve the stability of the system, a time delayed feedback proportional-derivative controller with a simple concept and an explicit tuning procedure is applied to the nanoplate system.

Based on the Melnikov method, the necessary analytic conditions for chaotic motion are established. Further, the effects of linear and nonlinear displacement feedback coefficients and their time delays on the necessary analytic conditions are discussed. Finally, the evolution of the dynamic behavior of the time delayed control system, including the bifurcation diagram, phase diagram, and Poincaré section diagram, is numerically simulated using the fourth order Langer–Kutta method.

The consistency between the analytical and numerical results verifies the correctness of the necessary conditions for the analysis. The results illustrate that time delay displacement feedback is a good method for chaotic motion suppression and provide a basis for further investigation of vibration in more complex nanostructured systems.

The increase in applied force leads to chaotic motion in the system, which is consistent with the results of other existing literature. With the implementation of linear and nonlinear displacement time delay control, the nonlinear dynamic characteristics of nanoplate systems change significantly. Specifically, four time delay parameters restrain the system from initial chaotic motion to periodic motion. By comparing the two time delay parameters under the same conditions, we find that the nonlinear displacement time delay parameters are more sensitive to the nonlinear vibration behavior of the system. The numerical simulation result of a critical chaotic state approximates to the theoretical prediction value by the Melnikov function method, which verifies the effectiveness for chaos control system.