Rigorous vibration analysis of double-layered orthotropic nanoplate system

A rigorous analytical symplectic method is introduced into the free vibration of a rectangular double-layered orthotropic nanoplate system. Eringen's nonlocal elasticity theory is taken to capture the small size effect. Meanwhile, it leads to a high-order differential governing equation in the Lagrangian system, which can only be analytically solved by the semi-inverse method with pre-determined trail functions in previous studies. To overcome this drawback, a Hamiltonian system is established by introducing a new total unknown vector consisting of the displacement amplitude, rotation angle, shear force and bending moment. The governing equation is reduced to a set of one-order ordinary differential equations so that they can be systematically solved by the method of separation of variables and the expansion of eigenfunctions. Analytical frequency equations are derived for the Levy-type edges and vibration modes are expressed in terms of the total unknown vectors. Comparison is presented to verify the accuracy of the symplectic method and comprehensive numerical examples are given also.