Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory

The nonlinear flexural vibration properties of double layered viscoelastic nanoplates are investigated based on nonlocal continuum theory. The von Kaman strain-displacement relation is employed to model the geometrical nonlinearity. Based on the classical plate theory, the formulations are derived by the Hamilton's principle in conjunction with Eringen's nonlocal elasticity theory, and are further discretized by the Galerkin's method. The coordinate transformation is adopted to obtain the nonlinear governing equations of motion in the modal coordinate system. On the basis of these equations, the frequency responses of double layered nanoplates with simply supported and clamped boundary conditions are derived by the method of multiple scales. The influences of small scale and other structural parameters (e.g. the aspect ratio of the plate, van der Walls (vdW) interaction and the viscidity of the plate) on the nonlinear vibration characteristics are discussed. From the result, the vdW interaction has obvious effects on the nonlinear frequency corresponding to the second nonlinear normal mode (NNM). The nonexistence of the internal resonance is also induced from the vdW forces between the plates. The influence of the elastic matrix is also discussed. The hardening nonlinearity is observed for the primary resonance. Additionally, some interesting phenomena different from the linear vibraticin are observed. (C) 2014 Elsevier B.V. All rights reserved.