Prediction and Control of Chaos in Nonlinear Rectangular Nano-plate on Elastic Foundation

In this study, nonlinear dynamics of non-classical Kirchhoff plate is analyzed for the first time and the region of chaotic behavior is predicted and controlled by designing the robust adaptive controller. Virtual displacement principle is employed to derive the governing equation of Kirchhoff micro-plate resting on nonlinear elastic foundation. In the governing equation, von Karman geometrical nonlinearity and couple stress theory is considered. Solving the eigen value governing equation for fully simply supported boundary conditions, results are validated. Next, Galerlkin method considering harmonic excitation of first mode is used to derive micro-plate forced vibration equation. Regardless of modal interaction, the chaos threshold is then investigated. Homoclinic orbits of unperturbed system is plotted and stable and unstable manifold transversely cut that is criteria to predict chaos according to Melnikov’s method is studied. Using maximum Lyapunov exponents numerical method, size-dependent chaos is also locally identified. Phase portrait, Poincare mapping and time response are plotted for different values of size ratios and the important effect of size on the chaotic behavior of micro-plats is presented. The results of this study show that chaos in the micro-plats is size dependent and using the non-classical theories to analyze the chaos of micro-plats is necessary. Subsequently, designing the robust adaptive fuzzy controllers, chaotic vibrations are completely eliminated from the system and the robust adaptive fuzzy controller is introduced as an effective method for controlling chaos in these systems. Results of this research can be used in the production process, optimization and control of nanoelectromechanical systems.