Nonlinear Dynamics of Cantilever FGM Cylindrical Shell under 1:2 Internal Resonance Relations

An analysis on the nonlinear dynamics of a cantilever functionally graded materials (FGM) cylindrical shell subjected to the transversal and static in-plane pre-applied excitation is presented in a thermal environment. Material properties are assumed to be temperature-dependent. Based on Reddy's first-order shell theory, the nonlinear governing equations of motion for the FGM cylindrical shell are derived using Hamilton's principle. Galerkin's method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. It is desirable to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the cantilever FGM cylindrical shell. The effects of external excitations on the internal resonance relationship and nonlinear dynamic response of FGM cylindrical shell are studied. The resonant case considered here is 1:2 internal resonance. A numerical method is used to find the nonlinear dynamic responses of the cantilever FGM cylindrical shell with different volume fraction index. It is found that the vibration amplitudes are not very sensitive to the value of the volume fraction index in the cases of 1:2 internal resonance. With the increasing of the volume fraction index, the value of w(1) and w(2) increased remarkably. There is a boundary value of the forcing amplitude, which divides the response into two parts for the case of different volume fraction index. One is the multiple periodic motion of the FGM cantilever cylindrical shell and the other is the quasi-period response. A larger value of volume fraction index leads to a larger region of quasi-period response.