Free vibrations of a two-layer nonlinear composite beam with different boundary conditions

The nonlinear free vibrations of a two-layer elastic composite beam are investigated. Different boundary conditions, both symmetric and not symmetric with respect to the beam midpoint, equal on both layers and different on each layers, are considered. The analysis is developed by means of the multiple time scale method, and at each order of the asymptotic development, we obtain different information. The first order terms provide the linear natural frequencies. The first, the second and the third natural frequencies are computed explicitly. The next order terms provide the nonlinearity coefficients measuring the nonlinear amplitude dependence of the natural frequencies, i.e. the curvature of the backbone curve. Both the linear frequencies and the nonlinear coefficients are found to be dependent on two dimensionless parameters only and, for boundary conditions different on each layer, also of the ratio between the axial stiffnesses of each layer.