Non-linear vibration and dynamic instability of internally-thickness-tapered composite plates under parametric excitation

Internally-tapered composite plates are formed by terminating or dropping-off some of the plies in the laminates at pre-determined locations, which is an important method for stiffness tailoring and weight saving in these structures. In the present work, the dynamic instability of internally-thickness-tapered laminated composite plates subjected to harmonic in-plane loading is studied based on non-linear vibration analysis. The non-linear von Karman strains associated with large deflections and curvatures are considered. The in-plane displacements are determined from the two in-plane force-equilibrium equations of motion of non-linear large deflection tapered plate. Consequently, the in-plane force-resultants can be obtained from the in-plane displacements and further applying the boundary conditions. Then the general Galerkin method is used for the moment-equilibrium equation of motion to satisfy spatial dependence in the partial differential equation of motion to produce a set of non-linear Mathieu-Hill equations. These equations are ordinary differential equations, with time-dependency. By applying the Bolotin's method to these equations, the dynamically-unstable regions, stable-, and unstable-solutions amplitudes of the steady-state vibrations are obtained. The non-linear dynamic stability characteristics of symmetric cross-ply laminates with different taper configurations are examined. A comprehensive parametric study is carried out to examine and compare the effects of the taper angles, magnitudes of both tensile and compressive in-plane loads, aspect ratios of the tapered plate including length-to-width and length-to-average-thickness ratios on the instability regions and the parametric resonance particularly the steady-state vibrations amplitude. For linear vibrations, the present results show good agreement with that available in the literature which were obtained based on linear analysis.