Nonlinear Dynamics Analysis Of Double Layered Nanoplate Structures

Since the success in discovering and fabrication of the carbon nanotubes, the nanoscience and nanotechnology develop rapidly and become a vigorous academic field. Extensive and profound researches have been performed. Due to the excellent chemical, electrical and mechanical performances, the nanostructures have been widely applied in mirco/nano electromechanical systems(MEMS/NEMS), biosensors,atomic force microscope and field emitters. Since the design and manufacture of such devices are heavily dependent on the insight of the mechanical properties of the nanostructures, it is significant to study the mechanical properties of nanostructures.In this paper, based on the nonlocal theory, the nonlinear free vibration and primary resonant characteristics of double layered nanoplates(DLNP) under the different boundary conditions are investigated systematically. The expressions of the nonlinear vibration frequencies and amplitude-frequency relationship are obtained. Moreover,the homoclinic and chaotic motions of the structure are investigated by the global perturbation technique.The nonlinear flexural vibration properties of the DLNP are studied. Considering the geometrical large deflection of the structure in the vibration, the nonlinear strain-displacement relation is employed to establish nonlinear dynamical equations.Under the simply supported and clamped boundary conditions with movable edges,the amplitude-frequency relationship of nonlinear primary resonance and the expression of nonlinear free vibration are obtained analytically by the method of multiple scales. The influences of the nonlocal effect and other structural parameters on the amplitude-frequency relationship and nonlinear vibration frequencies are analyzed and discussed. Meanwhile the nonexistence of the internal resonance of the DLNP is discovered.The nonlinear vibration properties of the DLNP embedded into elastic foundation are investigated. The frequency characteristics of the nonlinear free vibration of the DLNP with four different boundary conditions are compared. The elastic medium of the structure is taken into account. The influences of the elastic foundation’s stiffness coefficient on the amplitude-frequency relationships of the nonlinear primary resonance and the free vibration frequencies are discussed. It is observed that the nonlinear vibration properties of the structure are related to the fact that whether the edges are movable. Therefore, the influences of the aspect ratio of the DLNP on the first and second nonlinear vibration frequencies are discussed for the four kinds of boundary conditions, i.e. the simple supports with movable and immovable edges, and the clamped movable and immovable edges.The homoclinic and chaotic motions of the DLNP subjected to in-plane loads are systematically studied. The DLNP undergoes the synchronous buckling andasynchronous buckling when it is subjected to in-plane loads. Under these two kinds of buckling conditions, the homoclinic and chaotic motions are investigated by the generalized high dimensional Melnikov method. The criteria for transverse homoclinic orbits are established. Under the asynchronous buckling condition, the parameter space is divided into different regions for the different chaotic and hyperchaotic motions. The Lyapunov exponential spectrum and Lyapunov dimension are calculated in these different parameter regions. Moreover, the results obtained by the classical continuum theory and the nonlocal theory are compared with each other.The nonlocal parameter, i.e. small scale effect, on the homoclinic and chaotic motions are discussed under two kinds of buckling conditions.The homoclinic motions of the DLNP subjected to in-plane load and transverse harmonic excitation are investigated. The nonlinear dynamical equations are established by double mode Galerkin truncation method. It is observed that the rotary inertia of the structure can break the Hamiltonian symplectic symmetry of the unperturbed system. According to the literatures by Reddy and Amabili, the influence of rotary inertia could be neglected in the analysis of the nonlinear dynamics. The homoclinic phenomena are discussed by the generalized high dimensional Melnikov method for the different four buckling cases. The analytical criteria for homoclinic motion in different phase planes are presented. Finally, the influences of the small scale effect and the boundary conditions on the homoclinic motion are analyzed.The homoclinic phenomena and chaotic motions of double mode buckled DLNP subjected to parametrically excitation are investigated. Under the condition that the first and second modes of the structure undergo the synchronous buckling and asynchronous buckling, the generalized Melnikov method is employed to analyze the homoclinic and chaotic motions in the eight dimensional phase space. The results are compared with that of Lyapunov exponential spectrum and the results obtained by molecular dynamics in available open published literature. Furthermore, the influence of the small scale effect on the nonlinear dynamic properties such as the homoclinic motion of the structure is analyzed.